Abaqus Example Problems Manual

ABAQUS Example Problems Manual

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ADAMS is a registered United States trademark of Mechanical Dynamics, Inc. ADAMS/Flex and ADAMS/View are trademarks of Mechanical Dynamics, Inc. CATIA is a registered trademark of Dassault Systémes. C-MOLD is a registered trademark of Advanced CAE Technology, Inc., doing business as C-MOLD. Compaq Alpha is registered in the U.S. Patent and Trademark Office. FE-SAFE is a trademark of Safe Technology, Ltd. Fujitsu, UXP, and VPP are registered trademarks of Fujitsu Limited. Hewlett-Packard, HP-GL, and HP-GL/2 are registered trademarks of Hewlett-Packard Co. Hitachi is a registered trademark of Hitachi, Ltd. IBM RS/6000 is a trademark of IBM. Intel is a registered trademark of the Intel Corporation. NEC is a trademark of the NEC Corporation. PostScript is a registered trademark of Adobe Systems, Inc. Silicon Graphics is a registered trademark of Silicon Graphics, Inc. SUN is a registered trademark of Sun Microsystems, Inc. TEX is a trademark of the American Mathematical Society.

UNIX and Motif are registered trademarks and X Window System is a trademark of The Open Group in the U.S. and other countries. Windows NT is a registered trademark of the Microsoft Corporation. ABAQUS/CAE incorporates portions of the ACIS software by SPATIAL TECHNOLOGY INC. ACIS is a registered trademark of SPATIAL TECHNOLOGY INC. This release of ABAQUS on Windows NT includes the diff program obtained from the Free Software Foundation. You may freely distribute the diff program and/or modify it under the terms of the GNU Library General Public License as published by the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA. This release of ABAQUS/CAE includes lp_solve, a simplex-based code for linear and integer programming problems by Michel Berkelaar of Eindhoven University of Technology, Eindhoven, the Netherlands. Python, copyright 1991-1995 by Stichting Mathematisch Centrum, Amsterdam, The Netherlands. All Rights Reserved. Permission to use, copy, modify, and distribute the Python software and its documentation for any purpose and without fee is hereby granted, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation, and that the names of Stichting Mathematisch Centrum or CWI or

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Corporation for National Research Initiatives or CNRI not be used in advertising or publicity pertaining to distribution of the software without specific, written prior permission. All other brand or product names are trademarks or registered trademarks of their respective companies or organizations.

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General conversion factors (to five significant digits) Quantity U.S. unit SI equivalent Length 1 in 0.025400 m 1 ft 0.30480 m 1 mile 1609.3 m 2 Area 1 in 0.64516 ´ 10-3 m2 1 ft2 0.092903 m 2 1 acre 4046.9 m2 Volume 1 in3 0.016387 ´ 10-3 m3 3 1 ft 0.028317 m 3 1 US gallon 3.7854 ´ 10-3 m3 Quantity Density Energy Force Mass Power Pressure, Stress

Conversion factors for stress analysis U.S. unit SI equivalent 1 slug/ft3 = 1 lbf s2/ft4 515.38 kg/m3 1 lbf s2/in4 10.687 ´ 106 kg/m3 1 ft lbf 1.3558 J (N m) 1 lbf 4.4482 N (kg m/s2) 2 1 slug = 1 lbf s /ft 14.594 kg (N s2/m) 175.13 kg 1 lbf s2/in 1 ft lbf/s 1.3558 W (N m/s) 2 1 psi (lbf/in ) 6894.8 Pa (N/m2)

Conversion factors for heat transfer analysis Quantity U.S. unit SI equivalent Conductivity 1 Btu/ft hr °F 1.7307 W/m °C 1 Btu/in hr °F 20.769 W/m °C Density 1 lbm/in3 27680. kg/m3 Energy 1 Btu 1055.1 J Heat flux density 1 Btu/in 2 hr 454.26 W/m2 Power 1 Btu/hr 0.29307 W Specific heat 1 Btu/lbm °F 4186.8 J/kg °C Temperature 1 °F 5/9 °C Temp °F 9/5 ´ Temp °C + 32° 9/5 ´ Temp °K - 459.67° Constant Absolute zero Acceleration of gravity Atmospheric pressure Stefan-Boltzmann constant

Important constants U.S. unit -459.67 °F 32.174 ft/s 2 14.694 psi 0.1714 ´ 10-8 Btu/hr ft2 °R4 where °R = °F + 459.67

SI unit -273.15 °C 9.8066 m/s2 0.10132 ´ 106 Pa 5.669 ´ 10-8 W/m2 °K4 where °K = °C + 273.15

Approximate properties of mild steel at room temperature Quantity U.S. unit SI unit Conductivity 28.9 Btu/ft hr °F 50 W/m °C 2.4 Btu/in hr °F Density 15.13 slug/ft3 (lbf s2/ft4) 7800 kg/m3 0.730 ´ 10-3 lbf s2/in4

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Elastic modulus Specific heat Yield stress

0.282 lbm/in 3 30 ´ 106 psi 0.11 Btu/lbm °F 30 ´ 103 psi

207 ´ 109 Pa 460 J/kg °C 207 ´ 106 Pa

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UNITED STATES Hibbitt, Karlsson & Sorensen, Inc. 1080 Main Street Pawtucket, RI 02860-4847 Tel: 401 727 4200 Fax: 401 727 4208 E-mail: [email protected], [email protected] http://www.abaqus.com Hibbitt, Karlsson & Sorensen (West), Inc. 39221 Paseo Padre Parkway, Suite F Fremont, CA 94538-1611 Tel: 510 794 5891 Fax: 510 794 1194 E-mail: [email protected] AC Engineering, Inc. 1440 Innovation Place West Lafayette, IN 47906-1000 Tel: 765 497 1373 Fax: 765 497 4444 E-mail: [email protected] ARGENTINA KB Engineering S. R. L. Florida 274, Of. 37 (1005) Buenos Aires, Argentina Tel: +54 11 4393 8444 Fax: +54 11 4326 2424 E-mail: [email protected]

AUSTRIA VOEST-ALPINE STAHL LINZ GmbH Department WFE Postfach 3 A-4031 Linz Tel: 0732 6585 9919 Fax: 0732 6980 4338 E-mail: [email protected] CHINA Advanced Finite Element Services Department of Engineering Mechanics Tsinghua University Beijing 100084, P. R. China Tel: 010 62783986

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This section lists various resources that are available for help with using ABAQUS, including technical and systems support, training seminars, and documentation.

Support HKS offers both technical (engineering) support and systems support for ABAQUS. Technical and systems support are provided through the nearest local support office. You can contact our offices by telephone, fax, electronic mail, or regular mail. Information on how to contact each office is listed in the front of each ABAQUS manual. Support information is also available by visiting the ABAQUS Home Page on the World Wide Web (details are given below). When contacting your local support office, please specify whether you would like technical support (you have encountered problems performing an ABAQUS analysis) or systems support (ABAQUS will not install correctly, licensing does not work correctly, or other hardware-related issues have arisen). We welcome any suggestions for improvements to the support program or documentation. We will ensure that any enhancement requests you make are considered for future releases. If you wish to file a complaint about the service or products provided by HKS, refer to the ABAQUS Home Page.

Technical support HKS technical support engineers can assist in clarifying ABAQUS features and checking errors by giving both general information on using ABAQUS and information on its application to specific analyses. If you have concerns about an analysis, we suggest that you contact us at an early stage, since it is usually easier to solve problems at the beginning of a project rather than trying to correct an analysis at the end. Please have the following information ready before calling the technical support hotline, and include it in any written contacts: · The version of ABAQUS that are you using. - The version numbers for ABAQUS/Standard and ABAQUS/Explicit are given at the top of the data (.dat) file. - The version numbers for ABAQUS/CAE and ABAQUS/Viewer can be found by selecting Help->On version from the main menu bar. - The version number for ABAQUS/CAT is given at the top of the input ( .inp) file as well as the data file. - The version numbers for ABAQUS/ADAMS and ABAQUS/C-MOLD are output to the screen. - The version number for ABAQUS/Safe is given under the ABAQUS logo in the main window. · The type of computer on which you are running ABAQUS.

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· The symptoms of any problems, including the exact error messages, if any. · Workarounds or tests that you have already tried. When calling for support about a specific problem, any available ABAQUS output files may be helpful in answering questions that the support engineer may ask you. The support engineer will try to diagnose your problem from the model description and a description of the difficulties you are having. Frequently, the support engineer will need model sketches, which can be faxed to HKS or sent in the mail. Plots of the final results or the results near the point that the analysis terminated may also be needed to understand what may have caused the problem. If the support engineer cannot diagnose your problem from this information, you may be asked to send the input data. The data can be sent by means of e-mail, tape, or disk. Please check the ABAQUS Home Page at www.abaqus.com for the media formats that are currently accepted. All support calls are logged into a database, which enables us to monitor the progress of a particular problem and to check that we are resolving support issues efficiently. If you would like to know the log number of your particular call for future reference, please ask the support engineer. If you are calling to discuss an existing support problem and you know the log number, please mention it so that we can consult the database to see what the latest action has been and, thus, avoid duplication of effort. In addition, please give the receptionist the support engineer's name (or include it at the top of any e-mail correspondence).

Systems support HKS systems support engineers can help you resolve issues related to the installation and running of ABAQUS, including licensing difficulties, that are not covered by technical support. You should install ABAQUS by carefully following the instructions in the ABAQUS Site Guide. If you encounter problems with the installation or licensing, first review the instructions in the ABAQUS Site Guide to ensure that they have been followed correctly. If this does not resolve the problems, look on the ABAQUS Home Page under Technical Support for information about known installation problems. If this does not address your situation, please contact your local support office. Send whatever information is available to define the problem: error messages from an aborted analysis or a detailed explanation of the problems encountered. Whenever possible, please send the output from the abaqus info=env and abaqus info=sys commands.

ABAQUS Web server For users connected to the Internet, many questions can be answered by visiting the ABAQUS Home Page on the World Wide Web at http://www.abaqus.com

The information available on the ABAQUS Home Page includes: · Frequently asked questions · ABAQUS systems information and machine requirements

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· Benchmark timing documents · Error status reports · ABAQUS documentation price list · Training seminar schedule · Newsletters

Anonymous ftp site For users connected to the Internet, HKS maintains useful documents on an anonymous ftp account on the computer ftp.abaqus.com. Simply ftp to ftp.abaqus.com. Login as user anonymous, and type your e-mail address as your password. Directions will come up automatically upon login.

Writing to technical support Address of HKS Headquarters: Hibbitt, Karlsson & Sorensen, Inc. 1080 Main Street Pawtucket, RI 02860-4847, USA Attention: Technical Support Addresses for other offices and representatives are listed in the front of each manual.

Support for academic institutions Under the terms of the Academic License Agreement we do not provide support to users at academic institutions unless the institution has also purchased technical support. Please see the ABAQUS Home Page, or contact us for more information.

Training All HKS offices offer regularly scheduled public training classes. The Introduction to ABAQUS/Standard and ABAQUS/Explicit seminar covers basic usage and nonlinear applications, such as large deformation, plasticity, contact, and dynamics. Workshops provide as much practical experience with ABAQUS as possible. The Introduction to ABAQUS/CAE seminar discusses modeling, managing simulations, and viewing results with ABAQUS/CAE. "Hands-on" workshops are complemented by lectures. Advanced seminars cover topics of interest to customers with experience using ABAQUS, such as engine analysis, metal forming, fracture mechanics, and heat transfer. We also provide training seminars at customer sites. On-site training seminars can be one or more days in duration, depending on customer requirements. The training topics can include a combination of material from our introductory and advanced seminars. Workshops allow customers to exercise ABAQUS on their own computers.

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For a schedule of seminars see the ABAQUS Home Page, or call HKS or your local HKS representative.

Documentation The following documentation and publications are available from HKS, unless otherwise specified, in printed form and through our online documentation server. For more information on accessing the online books, refer to the discussion of execution procedures in the user's manuals. In addition to the documentation listed below, HKS publishes two newsletters on a regular schedule: ABAQUS/News and ABAQUS/Answers. ABAQUS/News includes topical information about program releases, training seminars, etc. ABAQUS/Answers includes technical articles on particular topics related to ABAQUS usage. These newsletters are distributed at no cost to users who wish to subscribe. Please contact your local ABAQUS support office if you wish to be added to the mailing list for these publications. They are also archived in the Reference Shelf on the ABAQUS Home Page.

Training Manuals Getting Started with ABAQUS/Standard: This document is a self-paced tutorial designed to help new users become familiar with using ABAQUS/Standard for static and dynamic stress analysis simulations. It contains a number of fully worked examples that provide practical guidelines for performing structural analyses with ABAQUS. Getting Started with ABAQUS/Explicit: This document is a self-paced tutorial designed to help new users become familiar with using ABAQUS/Explicit. It begins with the basics of modeling in ABAQUS, so no prior knowledge of ABAQUS is required. A number of fully worked examples provide practical guidelines for performing explicit dynamic analyses, such as drop tests and metal forming simulations, with ABAQUS/Explicit. Lecture Notes: These notes are available on many topics to which ABAQUS is applied. They are used in the technical seminars that HKS presents to help users improve their understanding and usage of ABAQUS (see the "Training" section above for more information about these seminars). While not intended as stand-alone tutorial material, they are sufficiently comprehensive that they can usually be used in that mode. The list of available lecture notes is included in the Documentation Price List.

User's Manuals ABAQUS/Standard User's Manual: This volume contains a complete description of the elements, material models, procedures, input specifications, etc. It is the basic reference document for ABAQUS/Standard. ABAQUS/Explicit User's Manual: This volume contains a complete description of the elements, material models, procedures, input specifications, etc. It is the basic reference document for ABAQUS/Explicit.

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ABAQUS/CAE User's Manual: This reference document for ABAQUS/CAE includes three comprehensive tutorials as well as detailed descriptions of how to use ABAQUS/CAE for model generation, analysis, and results evaluation. ABAQUS/Viewer User's Manual: This basic reference document for ABAQUS/Viewer includes an introductory tutorial as well as a complete description of how to use ABAQUS/Viewer to display your model and results. ABAQUS/ADAMS User's Manual: This document describes how to install and how to use ABAQUS/ADAMS, an interface program that creates ABAQUS models of ADAMS components and converts the ABAQUS results into an ADAMS modal neutral file that can be used by the ADAMS/Flex program. It is the basic reference document for the ABAQUS/ADAMS program. ABAQUS/CAT User's Manual: This document describes how to install and how to use ABAQUS/CAT, an interface program that creates an ABAQUS input file from a CATIA model and postprocesses the analysis results in CATIA. It is the basic reference document for the ABAQUS/CAT program. ABAQUS/C-MOLD User's Manual: This document describes how to install and how to use ABAQUS/C-MOLD, an interface program that translates finite element mesh, material property, and initial stress data from a C-MOLD analysis to an ABAQUS input file. ABAQUS/Safe User's Manual: This document describes how to install and how to use ABAQUS/Safe, an interface program that calculates fatigue lives and fatigue strength reserve factors from finite element models. It is the basic reference document for the ABAQUS/Safe program. The theoretical background to fatigue analysis is contained in the Modern Metal Fatigue Analysis manual (available only in print). Using ABAQUS Online Documentation: This online manual contains instructions on using the ABAQUS online documentation server to read the manuals that are available online. ABAQUS Release Notes: This document contains brief descriptions of the new features available in the latest release of the ABAQUS product line. ABAQUS Site Guide: This document describes how to install ABAQUS and how to configure the installation for particular circumstances. Some of this information, of most relevance to users, is also provided in the user's manuals.

Examples Manuals ABAQUS Example Problems Manual: This volume contains more than 75 detailed examples designed to illustrate the approaches and decisions needed to perform meaningful linear and nonlinear analysis. Typical cases are large motion of an elastic-plastic pipe hitting a rigid wall; inelastic buckling collapse of a thin-walled elbow; explosive loading of an elastic, viscoplastic thin ring; consolidation under a footing; buckling of a composite shell with a hole; and deep drawing of a metal sheet. It is generally useful to look for relevant examples in this manual and to review them when embarking on a new class of problem.

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ABAQUS Benchmarks Manual: This volume (available online and, if requested, in print) contains over 200 benchmark problems and standard analyses used to evaluate the performance of ABAQUS; the tests are multiple element tests of simple geometries or simplified versions of real problems. The NAFEMS benchmark problems are included in this manual. ABAQUS Verification Manual: This online-only volume contains more than 5000 basic test cases, providing verification of each individual program feature (procedures, output options, MPCs, etc.) against exact calculations and other published results. It may be useful to run these problems when learning to use a new capability. In addition, the supplied input data files provide good starting points to check the behavior of elements, materials, etc.

Reference Manuals ABAQUS Keywords Manual: This volume contains a complete description of all the input options that are available in ABAQUS/Standard and ABAQUS/Explicit. ABAQUS Theory Manual: This volume (available online and, if requested, in print) contains detailed, precise discussions of all theoretical aspects of ABAQUS. It is written to be understood by users with an engineering background. ABAQUS Command Language Manual: This online manual provides a description of the ABAQUS Command Language and a command reference that lists the syntax of each command. The manual describes how commands can be used to create and analyze ABAQUS/CAE models, to view the results of the analysis, and to automate repetitive tasks. It also contains information on using the ABAQUS Command Language or C++ as an application programming interface (API). ABAQUS Input Files: This online manual contains all the input files that are included with the ABAQUS release and referred to in the ABAQUS Example Problems Manual, the ABAQUS Benchmarks Manual, and the ABAQUS Verification Manual. They are listed in the order in which they appear in the manuals, under the title of the problem that refers to them. The input file references in the manuals hyperlink directly to this book. Quality Assurance Plan: This document describes HKS's QA procedures. It is a controlled document, provided to customers who subscribe to either HKS's Nuclear QA Program or the Quality Monitoring Service.

Introduction This is the Example Problems Manual for ABAQUS. It contains many solved examples that illustrate the use of the program for common types of problems. Some of the problems are quite difficult and require combinations of the capabilities in the code. The problems have been chosen to serve two purposes: to verify the capabilities in ABAQUS by exercising the code on nontrivial cases and to provide guidance to users who must work on a class of problems with which they are relatively unfamiliar. In each worked example the discussion in the manual states why the example is included and leads the reader through the standard approach to an

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analysis: element and mesh selection, material model, and a discussion of the results. Input data files are provided for all of these cases. Many of these problems are worked with different element types, mesh densities, and other variations. This results in a relatively large number of input data files for some of the problems. Only a few of the input files are listed in the printed manual. The selection has been made to provide the most guidance to the user. All input files, both the ones that are listed in the printed manual and the ones that are referenced, are included with the ABAQUS release. The ABAQUS/Fetch utility is used to extract these input files from the compressed archive files provided with the ABAQUS release. For example, to fetch input file boltpipeflange_3d_cyclsym.inp, type abaqus fetch job=boltpipeflange_3d_cyclsym.inp

Parametric study script (.psf) and user subroutine ( .f) files can be fetched in the same manner. All files for a particular problem can be obtained by leaving off the file extension. The ABAQUS/Fetch execution procedure is explained in detail in ``Execution procedure for ABAQUS/Fetch,'' Section 3.2.9 of the ABAQUS/Standard User's Manual and the ABAQUS/Explicit User's Manual. It is sometimes useful to search the input files. The findkeyword utility is used to locate input files that contain user-specified input. This utility is defined in ``Execution procedure for querying the keyword/problem database,'' Section 3.2.8 of the ABAQUS/Standard User's Manual and the ABAQUS/Explicit User's Manual. In addition, all the input files included with the ABAQUS release can be accessed through the ABAQUS Input Files electronic book. This book is part of the ABAQUS online documentation collection and, as such, is fully searchable (with the exception of numeric strings and ABAQUS-specific terms). When reading the online version of the ABAQUS Benchmarks Manual, the ABAQUS Example Problems Manual, or the ABAQUS Verification Manual, the user can click on an input file name; the ABAQUS Input Files book will open to that file in a separate window. To reproduce the graphical representation of the solution reported in some of the examples, the output frequency used in the input files may need to be increased. For example, in ``Linear analysis of the Indian Point reactor feedwater line,'' Section 2.2.2, the figures that appear in the manual can be obtained only if the solution is written to the results file every increment; that is, if the input files are changed to read *NODE FILE, ..., FREQUENCY=1

instead of FREQUENCY=100 as appears now. In addition to the Example Problems Manual, there are two other manuals that contain worked problems. The ABAQUS Benchmarks Manual contains benchmark problems (including the NAFEMS suite of test problems) and standard analyses used to evaluate the performance of ABAQUS. The tests in this manual are multiple element tests of simple geometries or simplified versions of real problems. The ABAQUS Verification Manual contains a large number of examples that are intended as elementary verification of the basic modeling capabilities. The verification of ABAQUS consists of running the problems in the ABAQUS Example Problems Manual, the ABAQUS Benchmarks Manual, and the ABAQUS Verification Manual. Before a version

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of ABAQUS is released, it must run all verification, benchmark, and example problems correctly.

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Static Stress/Displacement Analyses

1. Static Stress/Displacement Analyses 1.1 Static and quasi-static stress analyses 1.1.1 Axisymmetric analysis of bolted pipe flange connections Product: ABAQUS/Standard A bolted pipe flange connection is a common and important part of many piping systems. Such connections are typically composed of hubs of pipes, pipe flanges with bolt holes, sets of bolts and nuts, and a gasket. These components interact with each other in the tightening process and when operation loads such as internal pressure and temperature are applied. Experimental and numerical studies on different types of interaction among these components are frequently reported. The studies include analysis of the bolt-up procedure that yields uniform bolt stress (Bibel and Ezell, 1992), contact analysis of screw threads (Fukuoka, 1992; Chaaban and Muzzo, 1991), and full stress analysis of the entire pipe joint assembly (Sawa et al., 1991). To establish an optimal design, a full stress analysis determines factors such as the contact stresses that govern the sealing performance, the relationship between bolt force and internal pressure, the effective gasket seating width, and the bending moment produced in the bolts. This example shows how to perform such a design analysis by using an economical axisymmetric model and how to assess the accuracy of the axisymmetric solution by comparing the results to those obtained from a simulation using a three-dimensional segment model. In addition, several three-dimensional models that use multiple levels of superelements are analyzed to demonstrate the use of superelements with a large number of retained degrees of freedom.

Geometry and model The bolted joint assembly being analyzed is depicted in Figure 1.1.1-1. The geometry and dimensions of the various parts are taken from Sawa et al. (1991), modified slightly to simplify the modeling. The inner wall radius of both the hub and the gasket is 25 mm. The outer wall radii of the pipe flange and the gasket are 82.5 mm and 52.5 mm, respectively. The thickness of the gasket is 2.5 mm. The pipe flange has eight bolt holes that are equally spaced in the pitch circle of radius 65 mm. The radius of the bolt hole is modified in this analysis to be the same as that of the bolt: 8 mm. The bolt head (bearing surface) is assumed to be circular, and its radius is 12 mm. The Young's modulus is 206 GPa and the Poisson's ratio is 0.3 for both the bolt and the pipe hub/flange. The gasket is modeled with either solid continuum or gasket elements. When continuum elements are used, the gasket's Young's modulus, E, equals 68.7 GPa and its Poisson's ratio, º, equals 0.3. When gasket elements are used, a linear gasket pressure/closure relationship is used with the effective "normal stiffness," Sn , equal to the material Young's modulus divided by the thickness so that Sn = 27.48 GPa/mm. Similarly a linear shear stress/shear motion relationship is used with an effective shear stiffness, St , equal to the material shear modulus divided by the thickness so that St = 10.57 GPa/mm. The membrane behavior is specified with a Young's modulus of 68.7 GPa and a Poisson's ratio of 0.3. Sticking contact conditions are assumed in all contact areas: between the bearing surface and the flange and between the gasket and the hub. Contact between the bolt shank and the bolt hole is

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Static Stress/Displacement Analyses

ignored. The finite element idealizations of the symmetric half of the pipe joint are shown in Figure 1.1.1-2 and Figure 1.1.1-3, corresponding to the axisymmetric and three-dimensional analyses, respectively. The mesh used for the axisymmetric analysis consists of a mesh for the pipe hub/flange and gasket and a separate mesh for the bolts. In Figure 1.1.1-2the top figure shows the mesh of the pipe hub and flange, with the bolt hole area shown in a lighter shade; and the bottom figure shows the overall mesh with the gasket and the bolt in place. For the axisymmetric model second-order elements with reduced integration, CAX8R, are used throughout the mesh of the pipe hub/flange. The gasket is modeled with either CAX8R solid continuum elements or GKAX6 gasket elements. Contact between the gasket and the pipe hub/flange is modeled with contact pairs between surfaces defined on the faces of elements in the contact region or between such element-based surfaces and node-based surfaces. In an axisymmetric analysis the bolts and the perforated flange must be modeled properly. The bolts are modeled as plane stress elements since they do not carry hoop stress. Second-order plane stress elements with reduced integration, CPS8R, are employed for this purpose. The contact surface definitions, which are associated with the faces of the elements, account for the plane stress condition automatically. To account for all eight bolts used in the joint, the combined cross-sectional areas of the shank and the head of the bolts must be calculated and redistributed to the bolt mesh appropriately using the area attributes for the solid elements. The contact area is adjusted automatically. Figure 1.1.1-4 illustrates the cross-sectional views of the bolt head and the shank. Each plane stress element represents a volume that extends out of the x-y plane. For example, element A represents a volume calculated as (HA ) ´ (AreaA ). Likewise, element B represents a volume calculated as (HB ) ´ (AreaB ). The sectional area in the x-z plane pertaining to a given element can be calculated as Area = 2

Z

X2 X1

1

1

[(R2 ¡ x2 ) 2 ]dx = [x(R2 ¡ x2 ) 2 + R2 arcsin (

x ¯¯X2 )] ; jRj X1

where R is the bolt head radius, Rbolthead , or the shank radius, Rshank (depending on the element location), and X1 and X2 are x-coordinates of the left and right side of the given element, respectively. If the sectional areas are divided by the respective element widths, WA and WB , we obtain representative element thicknesses. Multiplying each element thickness by eight (the number of bolts in the model) produces the thickness values that are found in the *SOLID SECTION options. Sectional areas that are associated with bolt head elements located on the model's contact surfaces are used to calculate the surface areas of the nodes used in defining the node-based surfaces of the model. Referring again to Figure 1.1.1-4, nodal contact areas for a single bolt are calculated as follows: A1 =

AC ; 4

A9 =

AF ; 4

1-18

Static Stress/Displacement Analyses

A2 =

AC ; 2

A4 =

AD ; 2

A3 = (AC + AD )=4;

A6 =

AE ; 2

A8 =

A5 = (AD + AE )=4;

AF ; 2

A7 = (AE + AF )=4;

where A1 through A9 are contact areas that are associated with contact nodes 1-9 and Ac through AF are sectional areas that are associated with bolt head elements C-F . Multiplying the above areas by eight (the number of bolts in the model) provides the nodal contact areas found under the *SURFACE INTERACTION options. A common way of handling the presence of the bolt holes in the pipe flange in axisymmetric analyses is to smear the material properties used in the bolt hole area of the mesh and to use inhomogeneous material properties that correspond to a weaker material in this region. General guidelines for determining the effective material properties for perforated flat plates are found in ASME Section VIII Div 2 Article 4-9. For the type of structure under study, which is not a flat plate, a common approach to determining the effective material properties is to calculate the elasticity moduli reduction factor, which is the ratio of the ligament area in the pitch circle to the annular area of the pitch circle. In this model the annular area of the pitch circle is given by AA = 6534.51 mm2, and the total area of the bolt holes is given by AH = 8¼ 82 = 1608.5 mm2. Hence, the reduction factor is simply 1 ¡ AH=AA = 0.754. The effective in-plane moduli of elasticity, E 10 and E 20 , are obtained by multiplying the respective moduli, E 1 and E 2, by this factor. We assume material isotropy in the r-z plane; thus, E 10 = E 20 = E 0 : The modulus in the hoop direction, E 30 , should be very small and is chosen such that E 0 =E 30 = 106. The in-plane shear modulus is then calculated based on the effective elasticity modulus: G012 = E 0 =2(1 + º ): The shear moduli in the hoop direction are also calculated similarly but with º set to zero (they are not used in an axisymmetric model). Hence, we have E 10 = E 20 = 155292 MPa, E 30 = 0.155292 MPa, G012 = 59728 MPa, and G013 = G023 = 0.07765 MPa. These elasticity moduli are specified using *ELASTIC, TYPE=ENGINEERING CONSTANTS for the bolt hole part of the mesh. The mesh for the three-dimensional analysis without superelements, shown in Figure 1.1.1-3, represents a 22.5° segment of the pipe joint and employs second-order brick elements with reduced integration, C3D20R, for the pipe hub/flange and bolts. The gasket is modeled with C3D20R elements or GK3D18 elements. The top figure shows the mesh of the pipe hub and flange, and the bottom figure shows both the gasket and bolt (in the lighter color). Contact is modeled by the interaction of contact surfaces defined by grouping specific faces of the elements in the contacting regions. For three-dimensional contact where both the master and slave surfaces are deformable, the SMALL SLIDING parameter must be used on the *CONTACT PAIR option to indicate that small relative sliding occurs between contacting surfaces. No special adjustments need be made for the material properties used in the three-dimensional model because all parts are modeled appropriately. Four different meshes that use superelements to model the flange are tested. A first-level superelement is created for the entire 22.5° segment of the flange shown in Figure 1.1.1-3, while the gasket and the bolt are meshed as before. The nodes on the flange in contact with the bolt cap form a node-based surface, while the nodes on the flange in contact with the gasket form another node-based surface.

1-19

Static Stress/Displacement Analyses

These node-based surfaces will form contact pairs with the master surfaces on the bolt cap and on the gasket, which are defined with *SURFACE as before. The retained degrees of freedom on the superelement include all three degrees of freedom for the nodes in these node-based surfaces as well as for the nodes on the 0° and 22.5° faces of the flange. Appropriate boundary conditions are specified at the superelement usage level. A second-level superelement of 45° is created by reflecting the first-level superelement with respect to the 22.5° plane. The nodes on the 22.5° face belonging to the reflected superelement are constrained in all three degrees of freedom to the corresponding nodes on the 22.5° face belonging to the original first-level superelement. The half-bolt and the gasket sector corresponding to the reflected superelement are also constructed by reflection. The retained degrees of freedom include all three degrees of freedom of all contact nodes sets and of the nodes on the 0° and 45° faces of the flange. MPC-type CYCLSYM is used to impose cyclic symmetric boundary conditions on these two faces. A third-level superelement of 90° is created by reflecting the original 45° second-level superelement with respect to the 45° plane and by connecting it to the original 45° superelement. The remaining part of the gasket and the bolts corresponding to the 45° - 90° sector of the model is created by reflection and appropriate constraints. In this case it is not necessary to retain any degrees of freedom on the 0° and 90° faces of the flange because this 90° superelement will not be connected to other superelements and appropriate boundary conditions can be specified at the superelement creation level. The final model is set up by mirroring the 90° mesh with respect to the symmetry plane of the gasket perpendicular to the y-axis. Thus, an otherwise large analysis (¼ 750,000 unknowns) when no superelements are used can be solved conveniently ( ¼ 80,000 unknowns) by using the third-level superelement twice. The sparse solver is used because it significantly reduces the run time for this model.

Loading and boundary conditions The only boundary conditions are symmetry boundary conditions. In the axisymmetric model uz = 0 is applied to the symmetry plane of the gasket and to the bottom of the bolts. In the three-dimensional model uy = 0 is applied to the symmetry plane of the gasket as well as to the bottom of the bolt. The µ =0° and µ =22.5° planes are also symmetry planes. On the µ =22.5° plane, symmetry boundary conditions are enforced by invoking suitable nodal transformations and applying boundary conditions to local directions in this symmetry plane. These transformations are implemented using the *TRANSFORM option. On both the symmetry planes, the symmetry boundary conditions uz = 0 are imposed everywhere except for the dependent nodes associated with the C BIQUAD MPC and nodes on one side of the contact surface. The second exception is made to avoid overconstraining problems, which arise if there is a boundary condition in the same direction as a Lagrange multiplier constraint associated with the *FRICTION, ROUGH option. In the models where superelements are used, the boundary conditions are specified depending on what superelement is used. For the first-level 22.5° superelement the boundary conditions and constraint equations were the same as for the three-dimensional model shown in Figure 1.1.1-3. For the 45° second-level superelement the symmetry boundary conditions are enforced on the µ =45° plane with the constraint equation uz + ux = 0. A transform could have been used as well. For the 90° third-level

1-20

Static Stress/Displacement Analyses

superelement the face µ =90° is constrained with the boundary condition ux = 0. A clamping force of 15 kN is applied to each bolt by using the *PRE-TENSION SECTION option. The pre-tension section is identified by means of the *SURFACE option. The pre-tension is then prescribed by applying a concentrated load to the pre-tension node. In the axisymmetric analysis the actual load applied is 120 kN since there are eight bolts. In the three-dimensional model with no superelements the actual load applied is 7.5 kN since only half of a bolt is modeled. In the models using superelements all half-bolts are loaded with a 7.5 kN force. For all of the models the pre-tension section is specified about half-way down the bolt shank. Sticking contact conditions are assumed in all surface interactions in all analyses and are simulated with the *FRICTION, ROUGH and *SURFACE BEHAVIOR, NO SEPARATION options.

Results and discussion All analyses are performed as small-displacement analyses. Figure 1.1.1-5 shows a top view of the normal stress distributions in the gasket at the interface between the gasket and the pipe hub/flange predicted by the axisymmetric (bottom) and three-dimensional (top) analyses when solid continuum elements are used to model the gasket. The figure shows that the compressive normal stress is highest at the outer edge of the gasket, decreases radially inward, and changes from compression to tension at a radius of about 35 mm, which is consistent with findings reported by Sawa et al. (1991). The close agreement in the overall solution between axisymmetric and three-dimensional analyses is quite apparent, indicating that, for such problems, axisymmetric analysis offers a simple yet reasonably accurate alternative to three-dimensional analysis. Figure 1.1.1-6 shows a top view of the normal stress distributions in the gasket at the interface between the gasket and the pipe hub/flange predicted by the axisymmetric (bottom) and three-dimensional (top) analyses when gasket elements are used to model the gasket. Close agreement in the overall solution between the axisymmetric and three-dimensional analyses is also seen in this case. The gasket starts carrying compressive load at a radius of about 40 mm, a difference of 5 mm with the previous result. This difference is the result of the gasket elements being unable to carry tensile loads in their thickness direction. This solution is physically more realistic since, in most cases, gaskets separate from their neighboring parts when subjected to tensile loading. Removing the *SURFACE BEHAVIOR, NO SEPARATION option from the gasket/flange contact surface definition in the input files that model the gasket with continuum elements yields good agreement with the results obtained in Figure 1.1.1-6 (since, in that case, the solid continuum elements in the gasket cannot carry tensile loading in the gasket thickness direction). The models in this example can be modified to study other factors, such as the effective seating width of the gasket or the sealing performance of the gasket under operating loads. The gasket elements offer the advantage of allowing very complex behavior to be defined in the gasket thickness direction. Gasket elements can also use any of the small-strain material models provided in ABAQUS including user-defined material models. Figure 1.1.1-7shows a comparison of the normal stress distributions in the gasket at the interface between the gasket and the pipe hub/flange predicted by the axisymmetric (bottom) and three-dimensional (top) analyses when isotropic material properties are prescribed for gasket elements. The results in Figure 1.1.1-7compare well with the results in Figure 1.1.1-5 from

1-21

Static Stress/Displacement Analyses

analyses in which solid and axisymmetric elements are used to simulate the gasket. Figure 1.1.1-8 shows the distribution of the normal stresses in the gasket at the interface in the plane z = 0. The results are plotted for the three-dimensional model containing only solid continuum elements and no superelements and for the four models containing the superelements described above.

Input files boltpipeflange_axi_solidgask.inp Axisymmetric analysis containing a gasket modeled with solid continuum elements. boltpipeflange_axi_node.inp Node definitions for boltpipeflange_axi_solidgask.inp and boltpipeflange_axi_gkax6.inp. boltpipeflange_axi_element.inp Element definitions for boltpipeflange_axi_solidgask.inp. boltpipeflange_3d_solidgask.inp Three-dimensional analysis containing a gasket modeled with solid continuum elements. boltpipeflange_axi_gkax6.inp Axisymmetric analysis containing a gasket modeled with gasket elements. boltpipeflange_3d_gk3d18.inp Three-dimensional analysis containing a gasket modeled with gasket elements. boltpipeflange_3d_super1.inp Three-dimensional analysis using the first-level superelement (22.5° model). boltpipeflange_3d_super2.inp Three-dimensional analysis using the second-level superelement (45° model). boltpipeflange_3d_super3_1.inp Three-dimensional analysis using the third-level superelement once (90° model). boltpipeflange_3d_super3_2.inp Three-dimensional analysis using the third-level superelement twice (90° mirrored model). boltpipeflange_3d_gen1.inp First-level superelement generation data referenced by boltpipeflange_3d_super1.inp and boltpipeflange_3d_gen2.inp. boltpipeflange_3d_gen2.inp Second-level superelement generation data referenced by boltpipeflange_3d_super2.inp and boltpipeflange_3d_gen3.inp. boltpipeflange_3d_gen3.inp

1-22

Static Stress/Displacement Analyses

Third-level superelement generation data referenced by boltpipeflange_3d_super3_1.inp and boltpipeflange_3d_super3_2.inp. boltpipeflange_3d_node.inp Nodal coordinates used in boltpipeflange_3d_super1.inp, boltpipeflange_3d_super2.inp, boltpipeflange_3d_super3_1.inp, boltpipeflange_3d_super3_2.inp, boltpipeflange_3d_cyclsym.inp, boltpipeflange_3d_gen1.inp, boltpipeflange_3d_gen2.inp, and boltpipeflange_3d_gen3.inp. boltpipeflange_3d_cyclsym.inp Same as file boltpipeflange_3d_super2.inp except that CYCLSYM type MPCs are used. boltpipeflange_3d_missnode.inp Same as file boltpipeflange_3d_gk3d18.inp except that the option to generate missing nodes is used for gasket elements. boltpipeflange_3d_isomat.inp Same as file boltpipeflange_3d_gk3d18.inp except that gasket elements are modeled as isotropic using the *MATERIAL option. boltpipeflange_3d_ortho.inp Same as file boltpipeflange_3d_gk3d18.inp except that gasket elements are modeled as orthotropic and the *ORIENTATION option is used. boltpipeflange_axi_isomat.inp Same as file boltpipeflange_axi_gkax6.inp except that gasket elements are modeled as isotropic using the *MATERIAL option. boltpipeflange_3d_usr_umat.inp User subroutine UMAT used in boltpipeflange_3d_usr_umat.inp. boltpipeflange_3d_usr_umat.f Same as file boltpipeflange_3d_gk3d18.inp except that gasket elements are modeled as isotropic with user subroutine UMAT. boltpipeflange_3d_solidnum.inp Same as file boltpipeflange_3d_gk3d18.inp except that solid element numbering is used for gasket elements.

References · Bibel, G. D., and R. M. Ezell, ``An Improved Flange Bolt-Up Procedure Using Experimentally Determined Elastic Interaction Coefficients,'' Journal of Pressure Vessel Technology, vol. 114, pp. 439-443, 1992. · Chaaban, A., and U. Muzzo, ``Finite Element Analysis of Residual Stresses in Threaded End

1-23

Static Stress/Displacement Analyses

Closures,'' Transactions of ASME, vol. 113, pp. 398-401, 1991. · Fukuoka, T., ``Finite Element Simulation of Tightening Process of Bolted Joint with a Tensioner,'' Journal of Pressure Vessel Technology, vol. 114, pp. 433-438, 1992. · Sawa, T., N. Higurashi, and H. Akagawa, ``A Stress Analysis of Pipe Flange Connections,'' Journal of Pressure Vessel Technology, vol. 113, pp. 497-503, 1991.

Figures Figure 1.1.1-1 Schematic of the bolted joint. All dimensions in mm.

Figure 1.1.1-2 Axisymmetric model of the bolted joint.

1-24

Static Stress/Displacement Analyses

Figure 1.1.1-3 22.5° segment three-dimensional model of the bolted joint.

1-25

Static Stress/Displacement Analyses

Figure 1.1.1-4 Cross-sectional views of the bolt head and the shank.

1-26

Static Stress/Displacement Analyses

Figure 1.1.1-5 Normal stress distribution in the gasket contact surface when solid elements are used to model the gasket: three-dimensional versus axisymmetric results.

1-27

Static Stress/Displacement Analyses

Figure 1.1.1-6 Normal stress distribution in the gasket contact surface when gasket elements are used with direct specification of the gasket behavior: three-dimensional versus axisymmetric results.

1-28

Static Stress/Displacement Analyses

Figure 1.1.1-7 Normal stress distribution in the gasket contact surface when gasket elements are used with isotropic material properties: three-dimensional versus axisymmetric results.

1-29

Static Stress/Displacement Analyses

Figure 1.1.1-8 Normal stress distribution in the gasket contact surface along the line z = 0 for the models with and without superelements.

Sample listings

1-30

Static Stress/Displacement Analyses

Listing 1.1.1-1 *HEADING BOLTED PIPE JOINT: AXISYMMETRIC MODEL *RESTART, WRITE, FREQUENCY=1 *WAVEFRONT MINIMIZATION *NODE, INPUT=boltpipeflange_axi_node.inp *ELEMENT, TYPE=CAX8R, INPUT=boltpipeflange_axi_element.inp *ELSET, ELSET=PID1, GENERATE 609,640 *ELSET, ELSET=PID2, GENERATE 42,48 50,56 58,64 66,168 193,216 477,484 577,608 641,704 ** ** Contact Between Gasket and Hub ** *ELSET, ELSET=PID3, GENERATE 485,492 *SURFACE,NAME=HUB_BOT 477,S3 478,S1 479,S3 480,S1 481,S3 482,S1 483,S3 484,S1 *SURFACE,NAME=GASKET PID3,S4 *CONTACT PAIR, INTERACTION=ROUGH, SMALL SLIDING, ADJUST=.1 HUB_BOT,GASKET *SURFACE INTERACTION,NAME=ROUGH *FRICTION,ROUGH *SURFACE BEHAVIOR, NO SEPARATION ** ** Contact Between Bolt and Hub **

1-31

Static Stress/Displacement Analyses

** Note: Areas associated with contact nodes are determined ** by first calculating out-of-plane surface areas represented ** by the contact faces of bottom-side bolthead elements, and ** then assigning ratios of these areas to the relative contact ** nodes which lie on the faces. ** *SURFACE,type=node,NAME=NBOLT1 5008, *SURFACE,type=node,NAME=NBOLT2 5007, *SURFACE,type=node,NAME=NBOLT3 5006, *SURFACE,type=node,NAME=NBOLT4 5005, *SURFACE,type=node,NAME=NBOLT5 5004, *SURFACE,type=node,NAME=NBOLT6 5003, *SURFACE,type=node,NAME=NBOLT7 5002, *SURFACE,type=node,NAME=NBOLT8 5001, *SURFACE,type=node,NAME=NBOLT9 5000, *SURFACE,type=node,NAME=NBOLT1B 5017, *SURFACE,type=node,NAME=NBOLT2B 5016, *SURFACE,type=node,NAME=NBOLT3B 5015, *SURFACE,type=node,NAME=NBOLT4B 5014, *SURFACE,type=node,NAME=NBOLT5B 5013, *SURFACE,type=node,NAME=NBOLT6B 5012, *SURFACE,type=node,NAME=NBOLT7B 5011, *SURFACE,type=node,NAME=NBOLT8B 5010, *SURFACE,type=node,NAME=NBOLT9B 5009, *ELSET, ELSET=PID36, GENERATE

1-32

Static Stress/Displacement Analyses

577,580 *ELSET, ELSET=PID36B, GENERATE 641,644 *SURFACE,NAME=HUB_BOLT PID36,S4 *SURFACE,NAME=HUBBOLTB PID36B,S4 *CONTACT PAIR, INTERACTION=BLT_HUB1, NBOLT1,HUB_BOLT NBOLT1B,HUBBOLTB *SURFACE INTERACTION,NAME=BLT_HUB1 12.899, *FRICTION,ROUGH *SURFACE BEHAVIOR, NO SEPARATION *CONTACT PAIR, INTERACTION=BLT_HUB2, NBOLT2,HUB_BOLT NBOLT2B,HUBBOLTB *SURFACE INTERACTION,NAME=BLT_HUB2 25.799, *FRICTION,ROUGH *SURFACE BEHAVIOR, NO SEPARATION *CONTACT PAIR, INTERACTION=BLT_HUB3, NBOLT3,HUB_BOLT NBOLT3B,HUBBOLTB *SURFACE INTERACTION,NAME=BLT_HUB3 36.012, *FRICTION,ROUGH *SURFACE BEHAVIOR, NO SEPARATION *CONTACT PAIR, INTERACTION=BLT_HUB4, NBOLT4,HUB_BOLT NBOLT4B,HUBBOLTB *SURFACE INTERACTION,NAME=BLT_HUB4 46.226, *FRICTION,ROUGH *SURFACE BEHAVIOR, NO SEPARATION *CONTACT PAIR, INTERACTION=BLT_HUB5, NBOLT5,HUB_BOLT NBOLT5B,HUBBOLTB *SURFACE INTERACTION,NAME=BLT_HUB5 52.378, *FRICTION,ROUGH *SURFACE BEHAVIOR, NO SEPARATION *CONTACT PAIR, INTERACTION=BLT_HUB6,

SMALL SLIDING, HCRIT=1.1

SMALL SLIDING, HCRIT=1.1

SMALL SLIDING, HCRIT=1.1

SMALL SLIDING, HCRIT=1.1

SMALL SLIDING, HCRIT=1.1

SMALL SLIDING, HCRIT=1.1

1-33

Static Stress/Displacement Analyses

NBOLT6,HUB_BOLT NBOLT6B,HUBBOLTB *SURFACE INTERACTION,NAME=BLT_HUB6 58.529, *FRICTION,ROUGH *SURFACE BEHAVIOR, NO SEPARATION *CONTACT PAIR, INTERACTION=BLT_HUB7, SMALL SLIDING, HCRIT=1.1 NBOLT7,HUB_BOLT NBOLT7B,HUBBOLTB *SURFACE INTERACTION,NAME=BLT_HUB7 63.107, *FRICTION,ROUGH *SURFACE BEHAVIOR, NO SEPARATION *CONTACT PAIR, INTERACTION=BLT_HUB8, SMALL SLIDING, HCRIT=1.1 NBOLT8,HUB_BOLT NBOLT8B,HUBBOLTB *SURFACE INTERACTION,NAME=BLT_HUB8 67.685, *FRICTION,ROUGH *SURFACE BEHAVIOR, NO SEPARATION *CONTACT PAIR, INTERACTION=BLT_HUB9, SMALL SLIDING, HCRIT=1.1 NBOLT9,HUB_BOLT NBOLT9B,HUBBOLTB *SURFACE INTERACTION,NAME=BLT_HUB9 33.842, *FRICTION,ROUGH *SURFACE BEHAVIOR, NO SEPARATION ** ** Mesh Refinement ** *MPC QUADR, 415, 406, 432, 458 QUADR, 441, 406, 432, 458 QUADR, 467, 458, 484, 510 QUADR, 493, 458, 484, 510 QUADR, 519, 510, 536, 562 QUADR, 545, 510, 536, 562 QUADR, 571, 562, 588, 614 QUADR, 597, 562, 588, 614 QUADR, 1966, 150, 166, 176 QUADR, 1953, 150, 166, 176 QUADR, 1950, 176, 192, 202 QUADR, 1937, 176, 192, 202

1-34

Static Stress/Displacement Analyses

QUADR, 1934, 202, 218, 228 QUADR, 1921, 202, 218, 228 QUADR, 1918, 228, 244, 254 QUADR, 1905, 228, 244, 254 *ELEMENT, TYPE=CPS8R, ELSET=PID7 496, 2016, 2018, 2032, 2030, 500, 2030, 2032, 2046, 2044, 504, 2044, 2046, 2060, 2058, 508, 2060, 5008, 5006, 2058, *ELEMENT, TYPE=CPS8R, ELSET=PID8 495, 2014, 2016, 2030, 2028, 499, 2028, 2030, 2044, 2042, 503, 2042, 2044, 2058, 2056, 507, 2058, 5006, 5004, 2056, *ELEMENT, TYPE=CPS8R, ELSET=PID9 494, 2012, 2014, 2028, 2026, 498, 2026, 2028, 2042, 2040, 502, 2040, 2042, 2056, 2054, 506, 2056, 5004, 5002, 2054, *ELEMENT, TYPE=CPS8R, ELSET=PID10 493, 2010, 2012, 2026, 2024, 497, 2024, 2026, 2040, 2038, 501, 2038, 2040, 2054, 2052, 505, 2054, 5002, 5000, 2052, *ELEMENT, TYPE=CPS8R, ELSET=PID11 509, 2010, 2024, 2088, 2076, 513, 2024, 2038, 2100, 2088, 517, 2038, 2052, 2112, 2100, 521, 2052, 5000, 2124, 2112, *ELEMENT, TYPE=CPS8R, ELSET=PID12 510, 2076, 2088, 2090, 2078, 514, 2088, 2100, 2102, 2090, 518, 2100, 2112, 2114, 2102, 522, 2112, 2124, 2126, 2114, *ELEMENT, TYPE=CPS8R, ELSET=PID13 511, 2078, 2090, 2092, 2080, 515, 2090, 2102, 2104, 2092, 519, 2102, 2114, 2116, 2104, 523, 2114, 2126, 2128, 2116, *ELEMENT, TYPE=CPS8R, ELSET=PID14 512, 2080, 2092, 2094, 2082, 516, 2092, 2104, 2106, 2094, 520, 2104, 2116, 2118, 2106,

2017, 2031, 2045, 2065,

2023, 2037, 2051, 5007,

2031, 2045, 2059, 2064,

2022 2036 2050 2059

2015, 2029, 2043, 2064,

2022, 2036, 2050, 5005,

2029, 2043, 2057, 2063,

2021 2035 2049 2057

2013, 2027, 2041, 2063,

2021, 2035, 2049, 5003,

2027, 2041, 2055, 2062,

2020 2034 2048 2055

2011, 2025, 2039, 2062,

2020, 2034, 2048, 5001,

2025, 2039, 2053, 2061,

2019 2033 2047 2053

2019, 2033, 2047, 2061,

2087, 2099, 2111, 2123,

2083, 2095, 2107, 2119,

2075 2087 2099 2111

2083, 2095, 2107, 2119,

2089, 2101, 2113, 2125,

2084, 2096, 2108, 2120,

2077 2089 2101 2113

2084, 2096, 2108, 2120,

2091, 2103, 2115, 2127,

2085, 2097, 2109, 2121,

2079 2091 2103 2115

2085, 2093, 2086, 2081 2097, 2105, 2098, 2093 2109, 2117, 2110, 2105

1-35

Static Stress/Displacement Analyses

524, *ELEMENT, 525, 529, 533, 537, *ELEMENT, 526, 530, 534, 538, *ELEMENT, 527, 531, 535, 539, *ELEMENT, 528, 532, 536, 540, *ELEMENT, 541, 545, 549, 553, 557, 561, 565, 569, 573, *ELEMENT, 542, 546, 550, 554, 558, 562, 566, 570, 574, *ELEMENT, 543,

2116, 2128, TYPE=CPS8R, 2082, 2094, 2094, 2106, 2106, 2118, 2118, 5009, TYPE=CPS8R, 2132, 2144, 2144, 2156, 2156, 2168, 2168, 5011, TYPE=CPS8R, 2134, 2146, 2146, 2158, 2158, 2170, 2170, 5013, TYPE=CPS8R, 2136, 2148, 2148, 2160, 2160, 2172, 2172, 5015, TYPE=CPS8R, 5000, 2192, 2192, 2206, 2206, 2220, 2220, 2234, 2234, 2248, 2248, 2262, 2262, 2276, 2276, 2290, 2290, 2304, TYPE=CPS8R, 2124, 2194, 2194, 2208, 2208, 2222, 2222, 2236, 2236, 2250, 2250, 2264, 2264, 2278, 2278, 2292, 2292, 2306, TYPE=CPS8R, 2126, 2196,

5009, 2118, ELSET=PID15 2144, 2132, 2156, 2144, 2168, 2156, 5011, 2168, ELSET=PID16 2146, 2134, 2158, 2146, 2170, 2158, 5013, 2170, ELSET=PID17 2148, 2136, 2160, 2148, 2172, 2160, 5015, 2172, ELSET=PID18 2150, 2138, 2162, 2150, 2174, 2162, 5017, 2174, ELSET=PID19 2194, 2124, 2208, 2194, 2222, 2208, 2236, 2222, 2250, 2236, 2264, 2250, 2278, 2264, 2292, 2278, 2306, 2292, ELSET=PID20 2196, 2126, 2210, 2196, 2224, 2210, 2238, 2224, 2252, 2238, 2266, 2252, 2280, 2266, 2294, 2280, 2308, 2294, ELSET=PID21 2198, 2128,

2121, 2129, 2122, 2117 2086, 2098, 2110, 2122,

2143, 2155, 2167, 5010,

2139, 2151, 2163, 2175,

2131 2143 2155 2167

2139, 2151, 2163, 2175,

2145, 2157, 2169, 5012,

2140, 2152, 2164, 2176,

2133 2145 2157 2169

2140, 2152, 2164, 2176,

2147, 2159, 2171, 5014,

2141, 2153, 2165, 2177,

2135 2147 2159 2171

2141, 2153, 2165, 2177,

2149, 2161, 2173, 5016,

2142, 2154, 2166, 2178,

2137 2149 2161 2173

2187, 2201, 2215, 2229, 2243, 2257, 2271, 2285, 2299,

2193, 2207, 2221, 2235, 2249, 2263, 2277, 2291, 2305,

2188, 2202, 2216, 2230, 2244, 2258, 2272, 2286, 2300,

2123 2193 2207 2221 2235 2249 2263 2277 2291

2188, 2202, 2216, 2230, 2244, 2258, 2272, 2286, 2300,

2195, 2209, 2223, 2237, 2251, 2265, 2279, 2293, 2307,

2189, 2203, 2217, 2231, 2245, 2259, 2273, 2287, 2301,

2125 2195 2209 2223 2237 2251 2265 2279 2293

2189, 2197, 2190, 2127

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Static Stress/Displacement Analyses

547, 2196, 2210, 2212, 2198, 2203, 2211, 2204, 2197 551, 2210, 2224, 2226, 2212, 2217, 2225, 2218, 2211 555, 2224, 2238, 2240, 2226, 2231, 2239, 2232, 2225 559, 2238, 2252, 2254, 2240, 2245, 2253, 2246, 2239 563, 2252, 2266, 2268, 2254, 2259, 2267, 2260, 2253 567, 2266, 2280, 2282, 2268, 2273, 2281, 2274, 2267 571, 2280, 2294, 2296, 2282, 2287, 2295, 2288, 2281 575, 2294, 2308, 2310, 2296, 2301, 2309, 2302, 2295 *ELEMENT, TYPE=CPS8R, ELSET=PID22 544, 2128, 2198, 2200, 5009, 2190, 2199, 2191, 2129 548, 2198, 2212, 2214, 2200, 2204, 2213, 2205, 2199 552, 2212, 2226, 2228, 2214, 2218, 2227, 2219, 2213 556, 2226, 2240, 2242, 2228, 2232, 2241, 2233, 2227 560, 2240, 2254, 2256, 2242, 2246, 2255, 2247, 2241 564, 2254, 2268, 2270, 2256, 2260, 2269, 2261, 2255 568, 2268, 2282, 2284, 2270, 2274, 2283, 2275, 2269 572, 2282, 2296, 2298, 2284, 2288, 2297, 2289, 2283 576, 2296, 2310, 2312, 2298, 2302, 2311, 2303, 2297 *ELSET,ELSET=PIN PID7,PID8,PID9,PID10,PID11,PID12,PID13,PID14,PID15,PID16,PID17, PID18,PID19,PID20,PID21,PID22 ** flange_elements *SOLID SECTION, ELSET=PID2, MATERIAL=MID2 1., ** ** Note: Thicknesses of plane stress bolthead elements are ** determined by first calculating the areas which the 3-D ** volumes represented by the 2-D elements project onto the ** 1-3 plane, and then dividing the areas by the respective ** element widths. ** ** bolthead_1 *SOLID SECTION, ELSET=PID7, MATERIAL=MID1 51.5976, ** bolthead_12 *SOLID SECTION, ELSET=PID18, MATERIAL=MID1 51.5976, ** bolthead_2 *SOLID SECTION, ELSET=PID8, MATERIAL=MID1 92.4522, ** bolthead_11 *SOLID SECTION, ELSET=PID17, MATERIAL=MID1 92.4522,

1-37

Static Stress/Displacement Analyses

** bolthead_3 *SOLID SECTION, ELSET=PID9, MATERIAL=MID1 117.058, ** bolthead_10 *SOLID SECTION, ELSET=PID16, MATERIAL=MID1 117.058, ** bolthead_4 *SOLID SECTION, ELSET=PID10, MATERIAL=MID1 135.37, ** bolthead_9 *SOLID SECTION, ELSET=PID15, MATERIAL=MID1 135.37, ** bolthead_5 *SOLID SECTION, ELSET=PID11, MATERIAL=MID1 164.887, ** bolthead_8 *SOLID SECTION, ELSET=PID14, MATERIAL=MID1 164.887, ** bolthead_6 *SOLID SECTION, ELSET=PID12, MATERIAL=MID1 188.382, ** bolthead_7 *SOLID SECTION, ELSET=PID13, MATERIAL=MID1 188.382, ** gasket_elements *SOLID SECTION, ELSET=PID3, MATERIAL=MID3 1., ** bolttrunk_1 *SOLID SECTION, ELSET=PID19, MATERIAL=MID1 78.6157, ** bolttrunk_4 *SOLID SECTION, ELSET=PID22, MATERIAL=MID1 78.6157, ** bolttrunk_2 *SOLID SECTION, ELSET=PID20, MATERIAL=MID1 122.446, ** bolttrunk_3 *SOLID SECTION, ELSET=PID21, MATERIAL=MID1 122.446, ** hole_elements *SOLID SECTION, ELSET=PID1, MATERIAL=MID4,ORIENT=RECT 1., **local orientation matching global system

1-38

Static Stress/Displacement Analyses

*ORIENTATION,NAME=RECT 1.0, 0.0, 0.0, 0.0, 1.0, 0.0 1, 0.0 ** bolt_material *MATERIAL, NAME=MID1 *ELASTIC, TYPE=ISOTROPIC 2.06E+5, 0.3 ** BOLTS ASSUME PLANE STRESS CONDITIONS. ** THERE ARE 8 BOLTS SO A=8* BOLT CROSS-SECTION AREA ** flange_material *MATERIAL, NAME=MID2 *ELASTIC, TYPE=ISOTROPIC 2.06E+5, 0.3 ** gasket_material *MATERIAL, NAME=MID3 *ELASTIC, TYPE=ISOTROPIC .687E+5, 0.3 ** hole_material *MATERIAL, NAME=MID4 *ELASTIC, TYPE=ENGINEERING CONSTANTS 155.29E3, 155.29E3, 155.29E-3, 0.3, 0.0, 0.0, 59.728E3, 77.65E-3, 77.65E-3, ** HOOP/INPLANE DIRECT MODULI RATIO = 1.E6 *NSET,NSET=GASKTEND,GENERATE 1993,2009 *NSET,NSET=BOLTEND,GENERATE 2304,2312 *NSET,NSET=BOLTMID 2308, *NSET,NSET=NOUT 2078,5008,2126,5017,2308 747,876,5100,2438,5117,2684,2796,9,5,1,34,26,18 *BOUNDARY GASKTEND,2 BOLTEND,2 ** ** Pre-tension section ** *NSET,NSET=NSECT 6001, *ELSET,ELSET=ESECT,GEN 557,560,1 *SURFACE,NAME=PSECT

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Static Stress/Displacement Analyses

ESECT,S2 *PRE-TENSION SECTION,SURFACE=PSECT,NODE=6001 ** *STEP, AMPLITUDE=RAMP, INC=20 *STATIC 1.,1. *CLOAD 6001,1,120000. *NODE PRINT,FREQ=1,NSET=NSECT U,RF *NODE PRINT, FREQUENCY=1,NSET=BOLTEND,TOTALS=YES U,RF *NODE PRINT, FREQUENCY=1,NSET=GASKTEND,TOTALS=YES U,RF *NODE PRINT, FREQUENCY=1,NSET=NOUT U,RF *NODE FILE, NSET=NOUT, FREQUENCY=1 U, RF *CONTACT PRINT,SLAVE=HUB_BOT, MASTER=GASKET *CONTACT FILE,SLAVE=HUB_BOT, MASTER=GASKET *CONTACT PRINT,MASTER=HUB_BOLT *CONTACT PRINT,MASTER=HUBBOLTB *PRINT, CONTACT=YES *END STEP

1.1.2 Elastic-plastic collapse of a thin-walled elbow under in-plane bending and internal pressure Product: ABAQUS/Standard Elbows are used in piping systems because they ovalize more readily than straight pipes and, thus, provide flexibility in response to thermal expansion and other loadings that impose significant displacements on the system. Ovalization is the bending of the pipe wall into an oval, noncircular configuration. The elbow is, thus, behaving as a shell rather than as a beam. Straight pipe runs do not ovalize easily, so they behave essentially as beams. Thus, even under pure bending, complex interaction occurs between an elbow and the adjacent straight pipe segments; the elbow causes some ovalization in the straight pipe runs, which in turn tend to stiffen the elbow. This interaction can create significant axial gradients of bending strain in the elbow, especially in cases where the elbow is very flexible. This example provides verification of shell and elbow element modeling of such effects, through an analysis of a test elbow for which experimental results have been reported by Sobel and Newman (1979). An analysis is also included with elements of type ELBOW31B (which includes

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Static Stress/Displacement Analyses

ovalization but neglects axial gradients of strain) for the elbow itself, and beam elements for the straight pipe segments. This provides a comparative solution in which the interaction between the elbow and the adjacent straight pipes is neglected. The analyses predict the response up to quite large rotations across the elbow, so as to investigate possible collapse of the pipe and, particularly, the effect of internal pressure on that collapse.

Geometry and model The elbow configuration used in the study is shown in Figure 1.1.2-1. It is a thin walled elbow with elbow factor ¸=

r2

Rt p = 0:167 1 ¡ º2

and radius ratio R=r =3.07, so the flexibility factor from Dodge and Moore (1972) is 10.3. (The flexibility factor for an elbow is the ratio of the bending flexibility of an elbow segment to a straight pipe of the same dimensions, for small displacements and elastic response.) This is an extremely flexible case because the pipe wall is so thin. To demonstrate convergence of the overall moment-rotation behavior with respect to meshing, the two shell element meshes shown in Figure 1.1.2-2 are analyzed. Since the loading concerns in-plane bending only, it is assumed that the response is symmetric about the midplane of the system so that, in the shell element model, only one-half of the system need be modeled. Element type S8R5 is used, since tests have shown this to be the most cost-effective shell element in ABAQUS (input files using element types S9R5, STRI65, and S8R for this example are included with the ABAQUS release). The elbow element meshes replace each axial division in the more coarse shell element model with one ELBOW32 or two ELBOW31 elements and use 4 or 6 Fourier modes to model the deformation around the pipe. Seven integration points are used through the pipe wall in all the analyses. This is usually adequate to provide accurate modeling of the progress of yielding through the section in such cases as these, where essentially monotonic straining is expected. The ends of the system are rigidly attached to stiff plates in the experiments. These boundary conditions are easily modeled for the ELBOW elements and for the fixed end in the shell element model. For the rotating end of the shell element model the shell nodes must be constrained to a beam node that represents the motion of the end plate. This is done using the *KINEMATIC COUPLING option as described below. The material is assumed to be isotropic and elastic-plastic, following the measured response of type 304 stainless steel at room temperature, as reported by Sobel and Newman (1979). Since all the analyses give results that are stiffer than the experimentally measured response, and the mesh convergence tests (results are discussed below) demonstrate that the meshes are convergent with respect to overall response of the system, it seems that this stress-strain model may overestimate the material's actual strength.

Loading The load on the pipe has two components: a "dead" load, consisting of internal pressure (with a closed

1-41

Static Stress/Displacement Analyses

end condition), and a "live" in-plane bending moment applied to the end of the system. The pressure is applied to the model in an initial step and then held constant in the second analysis step while the bending moment is increased. The pressure values range from 0.0 to 3.45 MPa (500 lb/in 2), which is the range of interest for design purposes. The equivalent end force associated with the closed-end condition is applied as a follower force because it rotates with the motion of the end plane.

Kinematic boundary conditions The fixed end of the system is assumed to be fully built-in. The loaded end is fixed into a very stiff plate. For the ELBOW element models this condition is represented the NODEFORM boundary condition applied at this node. In the shell element model this rigid plate is represented by a single node, and the shell nodes at the end of the pipe are attached to it by using the kinematic coupling constraint and specifying that all degrees of freedom at the shell nodes are constrained to the motion of the single node.

Results and discussion The moment-rotation responses predicted by the various analysis models and measured in the experiment, all taken at zero internal pressure, are compared in Figure 1.1.2-3. The figure shows that the two shell models give very similar results, overestimating the experimentally measured collapse moment by about 15%. The 6-mode ELBOW element models are somewhat stiffer than the shell models, and those with 4 Fourier modes are much too stiff. This clearly shows that, for this very flexible system, the ovalization of the elbow is too localized for even the 6-mode ELBOW representation to provide accurate results. Since we know that the shell models are convergent with respect to discretization, the most likely explanation for the excessive stiffness in comparison to the experimentally measured response is that the material model used in the analyses is too strong. Sobel and Newman (1979) point out that the stress-strain curve measured and used in this analysis, shown in Figure 1.1.2-1, has a 0.2% offset yield that is 20% higher than the Nuclear Systems Materials Handbook value for type 304 stainless steel at room temperature, which suggests the possibility that the billets taken for the stress-strain curve measurement may have been from stronger parts of the fabrication. If this is the case, it points out the likelihood that the elbow tested is rather nonuniform in strength properties in spite of the care taken in its manufacture. We are left with the conclusion that discrepancies of this magnitude cannot be eliminated in practical cases, and the design use of such analysis results must allow for them. Figure 1.1.2-4 compares the moment-rotation response for opening and closing moments, under 0 and 3.45 MPa (500 lb/in 2) internal pressure, and shows the strong influence of large-displacement effects. If large-displacement effects were not important, the opening and closing moments would produce the same response. However, even with a 1° relative rotation across the elbow assembly, the opening and closing moments differ by about 12%; with a 2° relative rotation, the difference is about 17%. Such magnitudes of relative rotation would not normally be considered large; in this case it is the coupling into ovalization that makes geometric nonlinearity significant. As the rotation increases, the cases with closing moment loading show collapse, while the opening moment curves do not. In both cases internal pressure shows a strong effect on the results, which is to be expected in such a thin-walled pipeline. The level of interaction between the straight pipe and the elbows is well illustrated by the

1-42

Static Stress/Displacement Analyses

strain distribution on the outside wall, shown in Figure 1.1.2-5. The strain contours are slightly discontinuous at the ends of the curved elbow section because the shell thickness changes at those sections. Figure 1.1.2-6 shows a summary of the results of this and the previous example. The plot shows the collapse value of the closing moment under in-plane bending as a function of internal pressure. The strong influence of pressure on collapse is apparent. In addition, the effect of analyzing the elbow by neglecting interaction between the straight and curved segments is shown: the "uniform bending" results are obtained by using elements of type ELBOW31B in the bend and beams (element type B31) for the straight segments. The importance of the straight/elbow interaction is apparent. In this case the simpler analysis neglecting the interaction is conservative (in that it gives consistently lower values for the collapse moment), but this conservatism cannot be taken for granted. The analysis of Sobel and Newman (1979) also neglects interaction and agrees quite well with the results obtained here. For comparison the small-displacement limit analysis results of Goodall (1978), as well as his large-displacement, elastic-plastic lower bound (Goodall, 1978a), are also shown in this figure. Again, the importance of large-displacement effects is apparent from that comparison. Detailed results obtained with the model that uses ELBOW31 elements are shown in the following figures. Figure 1.1.2-7 shows the variation of the Mises stress along the length of the piping system. The length is measured along the centerline of the pipe starting at the loaded end. The figure compares the stress distribution at the intrados (integration point 1) on the inner and outer surfaces of the elements (section points 1 and 7, respectively). Figure 1.1.2-8 shows the variation of the Mises stress around the circumference of two elements (451 and 751) that are located in the bend section of the model; the results are for the inner surface of the elements (section point 1). Figure 1.1.2-9 shows the ovalization of elements 451 and 751. A nonovalized, circular cross-section is included in the figure for comparison. From the figure it is seen that element 751, located at the center of the bend section, experiences the most severe ovalization. These three figures were produced with the elbow element postprocessing program FELBOWFOR (``Creation of a data file to facilitate the postprocessing of elbow element results: FELBOW,'' Section 11.1.6).

Shell-to-solid submodeling One particular case was analyzed using the shell-to-solid submodeling technique. The problem was created for verification purposes to check the interpolation scheme in the case of double curved surfaces. A solid submodel using C3D27R elements was created around the elbow part of the pipe, spanning an angle of 40°. The finer submodel mesh has three elements through the thickness, 10 elements around half of the circumference of the cylinder, and 10 elements along the length of the elbow. Both ends are driven from the global shell model made of S8R elements. The submodel results agree closely with the shell model. The *SECTION FILE option is used to output the total force and the total moment in a cross-section through the submodel.

Input files In all of the following input files, with the exception of elbowcollapse_elbow31b_b31.inp and elbowcollapse_s8r5_fine.inp, the step concerning the application of the pressure load is commented

1-43

Static Stress/Displacement Analyses

out. To include the effects of the internal pressure in any given job, uncomment the step definition in the appropriate input file. elbowcollapse_elbow31b_b31.inp ELBOW31B and B31 element model. elbowcollapse_elbow31_6four.inp ELBOW31 model with 6 Fourier modes. elbowcollapse_elbow32_6four.inp ELBOW32 model with 6 Fourier modes. elbowcollapse_s8r.inp S8R element model. elbowcollapse_s8r5.inp S8R5 element model. elbowcollapse_s8r5_fine.inp Finer S8R5 element model. elbowcollapse_s9r5.inp S9R5 element model. elbowcollapse_stri65.inp STRI65 element model. elbowcollapse_submod.inp Submodel using C3D27R elements.

References · Dodge, W. G., and S. E. Moore, "Stress Indices and Flexibility Factors for Moment Loadings on Elbows and Curved Pipes," Welding Research Council Bulletin, no. 179, 1972. · Goodall, I. W., "Lower Bound Limit Analysis of Curved Tubes Loaded by Combined Internal Pressure and In-Plane Bending Moment," Research Division Report RD/B/N4360, Central Electricity Generating Board, England, 1978. · Goodall, I. W., "Large Deformations in Plastically Deforming Curved Tubes Subjected to In-Plane Bending," Research Division Report RD/B/N4312, Central Electricity Generating Board, England, 1978a. · Sobel, L. H. and S. Z. Newman, "Elastic-Plastic In-Plane Bending and Buckling of an Elbow: Comparison of Experimental and Simplified Analysis Results," Westinghouse Advanced Reactors Division, Report WARD-HT-94000-2, 1979.

1-44

Static Stress/Displacement Analyses

Figures Figure 1.1.2-1 MLTF elbow: geometry and measured material response.

Figure 1.1.2-2 Models for elbow/pipe interaction study.

1-45

Static Stress/Displacement Analyses

Figure 1.1.2-3 Moment-rotation response: mesh convergence studies.

1-46

Static Stress/Displacement Analyses

Figure 1.1.2-4 Moment-rotation response: pressure dependence.

Figure 1.1.2-5 Strain distribution on the outside surface: closing moment case.

1-47

Static Stress/Displacement Analyses

Figure 1.1.2-6 In-plane bending of an elbow, elastic-plastic collapse moment results.

1-48

Static Stress/Displacement Analyses

Figure 1.1.2-7 Mises stress distribution along the length of the piping system.

Figure 1.1.2-8 Mises stress distribution around the circumference of elements 451 and 751.

1-49

Static Stress/Displacement Analyses

Figure 1.1.2-9 Ovalization of elements 451 and 751.

Sample listings

1-50

Static Stress/Displacement Analyses

Listing 1.1.2-1 *HEADING MLTF ELBOW: IN-PLANE BENDING. SYM. HALF S8R5 MODEL,LAMDA=.167 *NSET,NSET=N100 100, *NODE 100,0.,24.,-24. 101,0.,31.808,-24. 113,0.,16.192,-24. 900,0.,24.0,0. 901,0.,31.808,0. 913,0.,16.192,0. 2100,0.,0.,24. 2101,0.,0.,31.808 2113,0.,0.,16.192 4500,0.,-72.,24. 4501,0.,-72.,31.808 4513,0.,-72.,16.192 *NGEN,NSET=PYZ 100,900,100 101,901,100 113,913,100 2100,4500,100 2101,4501,100 2113,4513,100 *NGEN,LINE=C,NSET=PYZ 900,2100,100,0,0.,0.,0.,1. 901,2101,100,0,0.,0.,0.,1. 913,2113,100,0,0.,0.,0.,1. *NGEN,LINE=C,NSET=LOADEND 101,113,1,0,0.,24.,-24.,0.,0.,-1. *NGEN,LINE=C ,NSET=END2 901,913,1,0,0.,24., 0.0,0.,0.,-1. *NGEN,LINE=C,NSET=END3 2101,2113,1,0,0.,0.,24., 0.,1.,0. *NGEN,LINE=C ,NSET=FIXEDEND 4501,4513,1,0,0.,-72.,24., 0.,1.,0. *NFILL LOADEND,END2,8,100 END3,FIXEDEND,24,100 *NGEN,LINE=C

1-51

Static Stress/Displacement Analyses

1001,1013,1,1000,,,,0.,.130526192,-.9914448614 1101,1113,1,1100,,,,0.,.258819045,-.965925826 1201,1213,1,1200,,,,0.,.382683432,-.923879532 1301,1313,1,1300,,,,0.,.5,-.866025404 1401,1413,1,1400,,,,0.,.60876143,-.79335334 1501,1513,1,1500,,,,0.,.707106781,-.707106781 1601,1613,1,1600,,,,0.,.79335334,-.60876143 1701,1713,1,1700,,,,0.,.866025404,-.5 1801,1813,1,1800,,,,0.,.923879532,-.382683432 1901,1913,1,1900,,,,0.,.965925826,-.258819045 2001,2013,1,2000,,,,0.,.9914448614,-.130526192 *ELEMENT,TYPE=S8R5 101,101,301,303,103,201,302,203,102 1101,2101,2301,2303,2103,2201,2302,2203,2102 *ELGEN,ELSET=LEG 101,6,2,1,4,200,100 1101,6,2,1,12,200,100 *ELEMENT,TYPE=S8R5 501,901,1101,1103,903,1001,1102,1003,902 *ELGEN,ELSET=LBOW 501,6,2,1,6,200,100 *ELSET,ELSET=ALL LEG,LBOW *MATERIAL ,NAME=PIPE *ELASTIC 28.1E6,.2642 *PLASTIC 39440.,0. 50170.,.00473 54950.,.01264 58540.,.02836 61520.,.0491 76520.,.105 *SHELL SECTION,ELSET=LEG,MATERIAL=PIPE .37,7 *SHELL SECTION,ELSET=LBOW,MATERIAL=PIPE .41,7 *BOUNDARY PYZ,1 PYZ,5,6 FIXEDEND,1,6 *NSET,NSET=COUPLED_END,GENERATE 101,113,1

1-52

Static Stress/Displacement Analyses

*KINEMATIC COUPLING, REF NODE = 100 COUPLED_END, *NSET,NSET=NOUT N100,LOADEND,END2,END3,FIXEDEND *RESTART,WRITE,FREQUENCY=5 ** *STEP,INC=1,NLGEOM ** APPLY PRESSURE ** *STATIC ** *DLOAD,OP=NEW ** ALL,P,500. ** *CLOAD,OP=NEW,FOLLOWER ** 100,3,-4.7882E4 ** *EL PRINT,FREQUENCY=0 ** PE ** *NODE PRINT,NSET=N100 ** U ** RF ** *NODE FILE,NSET=NOUT ** U,RF ** *END STEP *STEP,INC=25,NLGEOM HOLD PRESSURE CONSTANT, APPLY BENDING ACTION *STATIC,RIKS .05, 1.,.01,.5,2.0,100,4,-.2 *BOUNDARY 100,4,4,-.2 *EL PRINT,FREQUENCY=0 PE, *NODE PRINT,NSET=N100 U, RF, *NODE FILE,NSET=NOUT U,RF *END STEP

1-53

Static Stress/Displacement Analyses

Listing 1.1.2-2 *HEADING MESH WITH 4 B31 + 6 ELBOW31B + 12 B31. *NSET,NSET=NPRT 100,4500 *NODE,NSET=LOADEND 100,0.,24.,-24. *NODE,NSET=END2 900,0.,24.0,0. *NODE,NSET=END3 2100,0.,0.,24. *NODE,NSET=FIXEDEND 4500,0.,-72.,24. *NGEN,NSET=PYZ 100,900,100 2100,4500,100 *NGEN,LINE=C,NSET=PYZ 900,2100,100,0,0.,0.,0.,1. *ELEMENT,TYPE=B31 101,100,300 1101,2100,2300 *ELGEN,ELSET=LEG1 101,4,200,100 *ELGEN,ELSET=LEG2 1101,12,200,100 *ELEMENT,TYPE=ELBOW31B 501,900,1100 *ELGEN,ELSET=LBOW 501,6,200,100 *ELSET,ELSET=ALL LEG1,LEG2,LBOW *MATERIAL ,NAME=PIPE *ELASTIC 28.1E6,.2642 *PLASTIC 39440.,0. 50170.,.00473 54950.,.01264 58540.,.02836 61520.,.0491 76520.,.105 *BEAM SECTION,SECTION=PIPE,ELSET=LEG1,

1-54

Static Stress/Displacement Analyses

MATERIAL=PIPE 8.01,.37 1.,0.,0. 18, *BEAM SECTION,SECTION=PIPE,ELSET=LEG2, MATERIAL=PIPE 8.01,.37 1.,0.,0. 18, *BEAM SECTION,SECTION=ELBOW,ELSET=LBOW, MATERIAL=PIPE 8.01,.41,24. 0.,24.,24. 7,18,6 *BOUNDARY PYZ,1 PYZ,5,6 FIXEDEND,1,6 *NSET,NSET=NOUT,GENERATE 100,4500,200 *NSET,NSET=NOUTRF 100,4500 *RESTART,WRITE,FREQUENCY=10 *STEP,INC=11,NLGEOM APPLY BENDING ACTION *STATIC,RIKS .05, 1.,.005,.5,2.0,100,4,-.2 *BOUNDARY 100,4,4,-.2 *MONITOR,NODE=100,DOF=4 *EL PRINT,FREQUENCY=0 PE, *NODE PRINT,NSET=NPRT U, RF, *NODE FILE,NSET=NOUT U, *NODE FILE,NSET=NOUTRF RF, *OUTPUT,FIELD *NODE OUTPUT,NSET=NOUT U, *OUTPUT,HISTORY

1-55

Static Stress/Displacement Analyses

*NODE OUTPUT,NSET=NOUT U, *OUTPUT,FIELD *NODE OUTPUT,NSET=NOUTRF RF, *OUTPUT,HISTORY *NODE OUTPUT,NSET=NOUTRF RF, *END STEP

1-56

Static Stress/Displacement Analyses

Listing 1.1.2-3 *HEADING MLTF ELBOW TEST, FULL MODEL USING ELBOW31, LAMDA=.167 FINE MESH WITH 8+12+24 ELEMENTS. SECTION INTEGRATION: 7,18,6 *NSET,NSET=NPRT 100,4500 *NODE,NSET=LOADEND 100,0.,24.,-24. *NODE,NSET=END2 900,0.,24.0,0. *NODE,NSET=END3 2100,0.,0.,24. *NODE,NSET=FIXEDEND 4500,0.,-72.,24. *NGEN,NSET=PYZ 100,900,100 2100,4500,100 *NGEN,LINE=C,NSET=PYZ 900,2100,100,0,0.,0.,0.,1. *ELEMENT,TYPE=ELBOW31 101,100,200 1101,2100,2200 *ELGEN,ELSET=LEG1 101,8,100,50 *ELGEN,ELSET=LEG2 1101,24,100,50 *ELEMENT,TYPE=ELBOW31 501,900,1000 *ELGEN,ELSET=LBOW 501,12,100,50 *ELSET,ELSET=ALL LEG1,LEG2,LBOW *MATERIAL ,NAME=PIPE *ELASTIC 28.1E6,.2642 *PLASTIC 39440.,0. 50170.,.00473 54950.,.01264 58540.,.02836 61520.,.0491

1-57

Static Stress/Displacement Analyses

76520.,.105 *BEAM SECTION,SECTION=ELBOW,ELSET=LEG1, MATERIAL=PIPE 8.01,.37 0.,100.,-12. 7,18,6 *BEAM SECTION,SECTION=ELBOW,ELSET=LEG2, MATERIAL=PIPE 8.01,.37 0.,-36.,100. 7,18,6 *BEAM SECTION,SECTION=ELBOW,ELSET=LBOW, MATERIAL=PIPE 8.01,.41,24. 0.,24.,24. 7,18,6 *BOUNDARY PYZ,1 PYZ,5,6 FIXEDEND,1,6 FIXEDEND,NODEFORM LOADEND,NODEFORM *NSET,NSET=NOUT,GENERATE 100,4500,100 *RESTART,WRITE,FREQUENCY=10 ** *STEP,INC=1,NLGEOM ** APPLY PRESSURE ** *STATIC ** *DLOAD,OP=NEW ** ALL,PI,500.,15.65,OPEN ** *CLOAD,OP=NEW,FOLLOWER ** 100,3,-9.5764E4 ** *CONTROLS,PARAMETERS=FIELD,FIELD=DISPLACEMENT ** 1.,1.,,1000. ** *CONTROLS,PARAMETERS=FIELD,FIELD=ROTATION ** 1.,1.,,6000. ** *EL PRINT,FREQUENCY=0 ** PE ** *NODE PRINT,NSET=N100 ** U ** RF ** *NODE FILE,NSET=NOUT ** U,RF

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Static Stress/Displacement Analyses

** *END STEP *STEP,INC=15,NLGEOM HOLD PRESSURE CONSTANT, APPLY BENDING ACTION *STATIC,RIKS .05, 1.,.001,.5,2.0,100,4,-.2 *BOUNDARY 100,4,4,-.2 *CONTROLS,PARAMETERS=FIELD,FIELD=DISPLACEMENT 1.,1.,,10000. *CONTROLS,PARAMETERS=FIELD,FIELD=ROTATION 1.,1.,,61000. *MONITOR,NODE=100,DOF=4 *EL PRINT,FREQUENCY=0 PE, *NODE PRINT,NSET=NPRT U, RF, *NODE FILE,NSET=NOUT U,RF *OUTPUT,FIELD *NODE OUTPUT,NSET=NOUT U,RF *OUTPUT,HISTORY *NODE OUTPUT,NSET=NOUT U,RF *END STEP

1.1.3 Parametric study of a linear elastic pipeline under in-plane bending Product: ABAQUS/Standard Elbows are used in piping systems because they ovalize more readily than straight pipes and, thus, provide flexibility in response to thermal expansion and other loadings that impose significant displacements on the system. Ovalization is the bending of the pipe wall into an oval--i.e., noncircular--configuration. The elbow is, thus, behaving as a shell rather than as a beam. This example demonstrates the ability of elbow elements (``Pipes and pipebends with deforming cross-sections: elbow elements,'' Section 15.5.1 of the ABAQUS/Standard User's Manual) to model the nonlinear response of initially circular pipes and pipebends accurately when the distortion of the cross-section by ovalization is significant. It also provides some guidelines on the importance of including a sufficient number of Fourier modes in the elbow elements to capture the ovalization accurately. In addition, this example illustrates the shortcomings of using "flexibility knockdown factors" with simple beam elements in an attempt to capture the effects of ovalization in an ad hoc manner for large-displacement analyses.

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Geometry and model The pipeline configuration used in the study is shown in Figure 1.1.3-1. It is a simple model with two straight pipe sections connected by a 90° elbow. The straight pipes are 25.4 cm (10.0 inches) in length, the radius of the curved section is 10.16 cm (4.0 inches), and the outer radius of the pipe section is 1.27 cm (0.5 inches). The wall thickness of the pipe is varied from 0.03175 cm to .2032 cm (0.0125 inches to 0.08 inches) in a parametric study, as discussed below. The pipe material is assumed to be isotropic linear elastic with a Young's modulus of 194 GPa ( 28:1 £ 106 psi) and a Poisson's ratio of 0.0. The straight portions of the pipeline are assumed to be long enough so that warping at the ends of the structure is negligible. Two loading conditions are analyzed. The first case is shown in Figure 1.1.3-1 with unit inward displacements imposed on both ends of the structure. This loading condition has the effect of closing the pipeline in on itself. In the second case the sense of the applied unit displacements is outward--opening the pipeline. Both cases are considered to be large-displacement/small-strain analyses. A parametric study comparing the results obtained with different element types (shells, elbows, and pipes) over a range of flexibility factors, k, is performed. As defined in Dodge and Moore (1972), the flexibility factor for an elbow is the ratio of the bending flexibility of the elbow segment to that of a straight pipe of the same dimensions, assuming small displacements and an elastic response. When the internal (gauge) pressure is zero, as is assumed in this study, k can be approximated as k=

1:66 ; ¸

where ¸=

Rt p ; r2 1 ¡ º 2

R is the bend radius of the curved section, r is the mean radius of the pipe, t is the wall thickness of the pipe, and º is Poisson's ratio. Changes in the flexibility factor are introduced by varying the wall thickness of the pipe. The pipeline is modeled with three different element types: S4 shell elements, ELBOW31 elbow elements, and PIPE31 pipe elements. The S4 shell element model consists of a relatively fine mesh of 40 elements about the circumference and 75 elements along the length. This mesh is deemed fine enough to capture the true response of the pipeline accurately, although no mesh convergence studies are performed. The pipe and elbow element meshes consist of 75 elements along the length. The results of the shell element model are taken as the reference solution. The reaction force at the tip of the pipeline is used to evaluate the effectiveness of the pipe and elbow elements. In addition, the ovalization values of the pipeline cross-section predicted by the elbow element models are compared. The elbow elements are tested with 0, 3, and 6 Fourier modes, respectively. In general, elbow element accuracy improves as more modes are used, although the computational cost increases accordingly. In

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addition to standard pipe elements, tests are performed on pipe elements with a special flexibility knockdown factor. Flexibility knockdown factors (Dodge and Moore, 1972) are corrections to the bending stiffness based upon linear semianalytical results. They are applied to simple beam elements in an attempt to capture the global effects of ovalization. The knockdown factor is implemented in the PIPE31 elements by scaling the true thickness by the flexibility factor; this is equivalent to scaling the moment of inertia of the pipe element by 1=k.

Results and discussion In the following the results obtained with the shell element model are taken as the reference solution. The tip reaction forces due to the inward prescribed displacements for the various analysis models are shown in Figure 1.1.3-2. The results are normalized with respect to those obtained with the shell model. The results obtained with ELBOW31 element with 6 Fourier modes show excellent agreement with the reference solution over the entire range of flexibility factors considered in this study. The remaining four models generally exhibit excessively stiff response for all values of k. The PIPE31 element model, which uses the flexibility knockdown factor, showed a relatively constant error of about 20% over the entire range of flexibility factors. The 0-mode ELBOW31 element model and the PIPE31 element model without the knockdown factor produce very similar results for all values of k. The normalized tip reaction forces due to the outward unit displacement for the various analysis models are shown in Figure 1.1.3-3. Again, the results obtained with the 6-mode ELBOW31 element compare well with the reference shell solution. The 0-mode and 3-mode ELBOW31 and the PIPE31 (without the flexibility knockdown factor) element models exhibit overly stiff response. The PIPE31 element model with the knockdown factor has a transition region near k=1.5, where the response changes from being too stiff to being too soft. Figure 1.1.3-4 and Figure 1.1.3-5 illustrate the effect of the number of included Fourier modes (0, 3, and 6) on the ability of the elbow elements to model the ovalization in the pipebend accurately in both load cases considered in this study. By definition, the 0-mode model cannot ovalize, which accounts for its stiff response. The 3-mode and the 6-mode models show significant ovalization in both loading cases. Figure 1.1.3-6 compares the ovalization of the 6-mode model in the opened and closed deformation states. It clearly illustrates that when the ends of the pipe are displaced inward (closing mode), the height of the pipe's cross-section gets smaller, thereby reducing the overall stiffness of the pipe; the reverse is true when the pipe ends are displaced outward: the height of the pipe's cross-section gets larger, thereby increasing the pipe stiffness. These three figures were produced with the aid of the elbow element postprocessing program FELBOWFOR (``Creation of a data file to facilitate the postprocessing of elbow element results: FELBOW,'' Section 11.1.6).

Parametric study The performance of the pipe and elbow elements investigated in this example is analyzed conveniently in a parametric study using the Python scripting capabilities of ABAQUS (``Scripting parametric studies,'' Section 25.1.1 of the ABAQUS/Standard User's Manual). We perform a parametric study in which eight analyses are executed automatically for each of the three different element types (S4, ELBOW31, and PIPE31) discussed above; these parametric studies correspond to wall thickness

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values ranging from 0.03175 cm to .2032 cm (0.0125 inches to 0.08 inches). The Python script file elbowtest.psf is used to perform the parametric study. The function customTable (shown below) is an example of advanced Python scripting (Lutz and Ascher, 1999), which is used in elbowtest.psf. customTable is designed to take an XYPLOT file from the parametric study and convert it into a new file of reaction forces versus flexibility factors ( k). Such advanced scripting is not routinely needed. It is needed in this case to overcome the limitation that a dependent variable such as k cannot be included as a column of data in an XYPLOT file. ############################################################### # def customTable(file1, file2): for line in file1.readlines(): print line nl = string.split(line,',') disp = float(nl[0]) bend_radius = float(nl[1]) wall_thick = float(nl[2]) outer_pipe_radius = float(nl[3]) poisson = float(nl[4]) rf = float(nl[6]) mean_rad = outer_pipe_radius - wall_thick/2.0 k = bend_radius*wall_thick/mean_rad**2 k = k/sqrt(1.e0 - poisson**2) k = 1.66e0/k outputstring = str(k) + ', ' + str(rf) + '\n' file2.write(outputstring) # #############################################################

Input files elbowtest_shell.inp S4 model. elbowtest_elbow0.inp ELBOW31 model with 0 Fourier modes. elbowtest_elbow3.inp ELBOW31 model with 3 Fourier modes. elbowtest_elbow6.inp

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ELBOW31 model with 6 Fourier modes. elbowtest_pipek.inp PIPE31 model with the flexibility knockdown factor. elbowtest_pipe.inp PIPE31 model without the flexibility knockdown factor. elbowtest.psf Python script file for the parametric study.

References · Dodge, W. G., and S. E. Moore, ``Stress Indices and Flexibility Factors for Moment Loadings on Elbows and Curved Pipes,'' Welding Research Council Bulletin, no. 179, 1972. · Lutz, M., and D. Ascher, Learning Python, O'Reilly, 1999.

Figures Figure 1.1.3-1 Pipeline geometry with inward prescribed tip displacements.

Figure 1.1.3-2 Normalized tip reaction force: closing displacement case.

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Figure 1.1.3-3 Normalized tip reaction force: opening displacement case.

Figure 1.1.3-4 Ovalization of the ELBOW31 cross-sections for 0, 3, and 6 Fourier modes: closing displacement case.

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Figure 1.1.3-5 Ovalization of the ELBOW31 cross-sections for 0, 3, and 6 Fourier modes: opening displacement case.

Figure 1.1.3-6 Ovalization of the ELBOW31 cross-sections for 6 Fourier modes: opening and closing displacement cases.

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Sample listings

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Static Stress/Displacement Analyses

Listing 1.1.3-1 *HEADING ELBOW31 elements with 6 ovalization modes *restart,write ************************ *** PARAMETER DEFINITION ************************ *PARAMETER # length a of the straight sections a = 10.0 # radius of the curved section (bend) bend_radius = 4.0 # outer radius of the pipe outer_pipe_radius = 0.5 # wall thickness of the pipe wall_thick = 0.08 # number of elements along the straight sections # of the pipe num_elem_s = 25 # number of elements around the bend num_elem_c = 25 # displacement at the end of the pipe disp = 1.0 # Young's modulus young = 28.1E6 # Poisson's ratio poisson = 0.0 # number of integration points through the # thickness nip_thru_thick = 5 # number of integration points around the pipe nip_around_pipe = 20 # number of ovalization modes numoval = 6 # # total number of elements along the length of # the pipe # num_elem = num_elem_c + 2*num_elem_s # ############################################### #

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# geometrical properties of the problem: # points along pipe c/l # x1 = a + bend_radius y1 = 0. x2 = a + bend_radius y2 = a x3 = a y3 = a + bend_radius x4 = 0. y4 = a + bend_radius # x1_plus_100 = x1 + 100.0 y4_plus_100 = y4 + 100.0 # # node control data # ninc = 1 n1 = 1 n1_plus_inc = n1 + ninc n2 = n1 + num_elem_s*ninc n3 = n2 + num_elem_c*ninc n4 = n3 + num_elem_s*ninc # ndummy = n4 + 1000 # # element control data # einc = 1 e1 = 1 e2 = e1 + (num_elem_s-1)*einc e2_plus_inc = e2 + einc e3 = e2_plus_inc + (num_elem_c-1)*einc e3_plus_inc = e3 + einc e4 = e3_plus_inc + (num_elem_s-1)*einc # # END OF PARAMETER SECTION # #******************* #** Node definitions #******************* *NODE ,,,0.

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*NODE ,, ,, ,, ,, *NGEN,NSET=straight_1 ,, **** *NGEN,line=c,nset=curved ,,,,,,0. **** *NGEN,NSET=straight_2 ,, *NSET,NSET=all_nodes straight_1,curved,straight_2 ** ********************** ** Material definition ********************** *MATERIAL,NAME=pipe *ELASTIC , ************************** ** Element data definition ************************** *ELEMENT,TYPE=elbow31 ,, *ELGEN,ELSET=all_elem ,,, *ELSET,ELSET=straight1,GENERATE ,, *ELSET,ELSET=curved,GENERATE ,, *ELSET,ELSET=straight2,GENERATE ,, ************************** ** Element definition ************************** *BEAM SECTION,SECTION=elbow,ELSET=straight1, MATERIAL=PIPE , , 0. , ,,

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*BEAM SECTION,SECTION=elbow,ELSET=straight2, MATERIAL=PIPE , , 0. , ,, *BEAM SECTION,SECTION=elbow,ELSET=curved, MATERIAL=PIPE , , , ,, ********************** ** Boundary conditions ********************** *BOUNDARY ,ysymm ,nowarp ,nooval ,3 ,xsymm ,nowarp ,nooval ,3 ********************** *STEP,nlgeom Apply displacements at the ends of the structure *STATIC 1.,1. *BOUNDARY ,1,1, ,2,2, *NODE PRINT,NSET=all_nodes, FREQ=9999 u, rf, *NODE FILE,NSET=all_nodes, FREQ=9999 u, rf, *el file,elset=curved,f=999 coord, *END STEP

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Listing 1.1.3-2 1.1.4 Indentation of an elastomeric foam specimen with a hemispherical punch Products: ABAQUS/Standard ABAQUS/Explicit ABAQUS/Design In this example we consider a cylindrical specimen of an elastomeric foam, indented by a rough, rigid, hemispherical punch. Examples of elastomeric foam materials are cellular polymers such as cushions, padding, and packaging materials. This problem illustrates a typical application of elastomeric foam materials when used in energy absorption devices. The same geometry as the crushable foam model of ``Indentation of a crushable foam specimen with a hemispherical punch, '' Section 1.1.7, is used but with a slightly different mesh. Design sensitivity analysis is carried out for a shape design parameter and a material design parameter to illustrate the usage of design sensitivity analysis for a problem involving contact.

Geometry and model The axisymmetric model (135 linear 4-node elements) analyzed is shown in Figure 1.1.4-1. The mesh refinement is biased toward the center of the foam specimen where the largest deformation is expected. The foam specimen has a radius of 600 mm and a thickness of 300 mm. The punch has a radius of 200 mm. The bottom nodes of the mesh are fixed, while the outer boundary is free to move. In ABAQUS/Standard a contact pair is defined between the punch, which is modeled by a rough spherical rigid surface, and a slave surface composed of the faces of the axisymmetric elements in the contact region. A point mass of 200 kg representing the weight of the punch is attached to the rigid body reference node. In ABAQUS/Explicit the punch is modeled as either an analytical rigid surface or a rigid surface defined with RAX2 elements. In ABAQUS/Explicit the friction coefficient between the punch and the foam is 0.8.

Material The elastomeric foam material is defined through the *HYPERFOAM option using experimental test data. The uniaxial compression and simple shear data whose stress-strain curves are shown in Figure 1.1.4-2 are defined with the *UNIAXIAL TEST DATA and *SIMPLE SHEAR TEST DATA options. Other available test data options are *BIAXIAL TEST DATA, *PLANAR TEST DATA and *VOLUMETRIC TEST DATA. The test data are defined in terms of nominal stress and nominal strain values. ABAQUS performs a nonlinear least-squares fit of the test data to determine the hyperfoam coefficients ¹i ; ®i ; and ¯i . Details of the formulation and usage of the hyperfoam model are given in ``Elastomeric foam behavior,'' Section 10.5.2 of the ABAQUS/Standard User's Manual and Section 9.3.2 of the ABAQUS/Explicit User's Manual; ``Hyperelastic material behavior,'' Section 4.6.1 of the ABAQUS Theory Manual; and ``Fitting of hyperelastic and hyperfoam constants,'' Section 4.6.2 of the ABAQUS Theory Manual. ``Fitting of elastomeric foam test data,'' Section 3.1.5 of the ABAQUS Benchmarks

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Manual, illustrates the fitting of elastomeric foam test data to derive the hyperfoam coefficients. For the material used in this example, ¯i is zero, since the effective Poisson's ratio, º, is zero, as specified through the POISSON parameter. The order of the series expansion is chosen to be N = 2 since this fits the test data with sufficient accuracy. N = 2 also provides a more stable model than the N = 3 case. The viscoelastic properties in ABAQUS/Standard are specified in terms of a relaxation curve (shown in Figure 1.1.4-3) of the normalized modulus M (t)=M0 , where M (t) is the shear or bulk modulus as a function of time and M0 is the instantaneous modulus, as determined from the hyperfoam model. This requires the use of the TIME=RELAXATION TEST DATA parameter of the *VISCOELASTIC option. The relaxation data are specified through the *SHEAR TEST DATA option but actually apply to both shear and bulk moduli when used in conjunction with the hyperfoam model. ABAQUS/Standard performs a nonlinear least-squares fit of the relaxation data to a Prony series to determine the coefficients, g Pi , and the relaxation periods, ¿i . A maximum order of NMAX=2 for fitting the Prony series is used. If creep data are available, the TIME=CREEP TEST DATA parameter is set to specify normalized creep compliance data. The ABAQUS/Explicit analysis is purely elastic. A rectangular material orientation is defined for the foam specimen, so stress and strain are reported in material axes that rotate with the element deformation. This is especially useful when looking at the stress and strain values in the region of the foam in contact with the punch in the direction normal to the punch (direction "22"). The rough surface of the punch is modeled by specifying a friction coefficient of 0.80 for the contact surface interaction through the *FRICTION option under the *SURFACE INTERACTION definition. Because of the unsymmetric nature of the friction material model, set UNSYMM=YES on the *STEP option in ABAQUS/Standard.

Procedure and loading definitions in ABAQUS/Standard The ABAQUS/Standard example is composed of two analyses. In the first case the punch is displaced statically to indent the foam, and the reaction force-displacement relation is measured for both the purely elastic and viscoelastic cases. In the second case the punch statically indents the foam through gravity loading and is then subjected to a dynamic impulsive loading. The dynamic response of the punch is sought as it interacts with the viscoelastic foam.

Case 1 During the first step the punch is displaced downward by a prescribed *BOUNDARY condition, indenting the foam specimen by a distance of 250 mm. The NLGEOM parameter is specified on the *STEP option, since the response involves large deformation. In the second step the punch is displaced back to its original position. Two analyses are performed--one using the *STATIC procedure for both steps and the other using the *VISCO procedure for both steps. During a *STATIC step the material behaves purely elastically, using the properties specified with the hyperfoam model. The *VISCO, *DYNAMIC, or *COUPLED TEMPERATURE-DISPLACEMENT procedure must be used to activate the viscoelastic behavior. In this case the punch is pushed down in a period of one second and then moved back up again in one second. The accuracy of the creep

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integration in the *VISCO procedure is controlled by the CETOL parameter and is typically calculated by dividing an acceptable stress error tolerance by a typical elastic modulus. In this problem, we estimate a stress error tolerance of about 0.005 MPa and use the initial elastic modulus, E 0 = 2 P ¹i = 0.34, to determine a CETOL of 0.01.

Case 2

This analysis is composed of three steps. The first step is a *VISCO step, where gravity loading is applied to the point mass of the punch. The gravity loading is ramped up in two seconds, and the step is run for a total of five seconds to allow the foam to relax fully. In the second step, which is a *DYNAMIC step, an impulsive load in the form of a half sine wave amplitude with a peak magnitude of 5000 N is applied to the punch over a period of one second. In the third step (also a *DYNAMIC step) the punch is allowed to move freely until the vibration is damped out by the viscoelastic foam. For a dynamic analysis with automatic time stepping, the value of the HAFTOL (half-step residual tolerance) parameter of the *DYNAMIC procedure controls the accuracy of the time integration. For systems that have significant energy dissipation, such as this heavily damped model, a relatively high value of HAFTOL can be chosen. We choose HAFTOL to be 100 times a typical average force that we estimate (and later confirm from the analysis results) to be on the order of 50 N. Thus, HAFTOL is 5000 N. For the second *DYNAMIC step we set INITIAL=NO to bypass calculation of initial accelerations at the beginning of the step, since there is no sudden change in load to create a discontinuity in the accelerations.

Procedure and loading definitions in ABAQUS/Explicit In ABAQUS/Explicit the punch is displaced downward by a prescribed boundary condition, indenting the foam by a depth of 250 mm. The punch is then lifted back to its original position. The whole analysis runs for 0.12 seconds. This analysis is comparable to the Case 1 ABAQUS/Standard analysis performed with the *STATIC procedure.

Design sensitivity analysis For the design sensitivity analysis (DSA) carried out with static steps in ABAQUS/Standard, the hyperfoam material properties are given using direct input of coefficients based on the test data given above. For N = 2, the coeficients are ¹1 = 0.16245, ¹2 = 3.59734E-05, ®1 = 8.89239, ®2 = -4.52156, and ¯i = 0.0. Due to the current limitations in the DSA capability, the viscoelastic properties are not included and the interaction between the rigid punch and the foam is assumed to be frictionless with small sliding (the small sliding assumption is necessary since finite sliding is not currently supported for DSA). The material parameter ¹1 is chosen as one of the design parameters. The other (shape) design parameter used for design sensitivity analysis, L, represents the thickness of the foam at the free end (see Figure 1.1.4-1). The z-coordinates of the nodes on the top surface are assumed to depend on L via the equation z = 300 + (L ¡ 300)r=600 . The r-coordinates are considered to be independent of L. To define this dependency in ABAQUS, the gradients of the coordinates with respect to L :

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dr =0 dL dz r = dL 600 are given under the *PARAMETER SHAPE VARIATION option.

Results and discussion This problem tests the hyperfoam material model in ABAQUS but does not provide independent verification of the model. The results for all analyses are discussed in the following paragraphs.

ABAQUS/Standard: Case 1 Deformation and contour plots for oriented S22 stress and E22 strain are shown for the viscoelastic foam in Figure 1.1.4-4. Even though the foam has been subjected to large strains, only moderate distortions occur because of the zero Poisson's ratio. The maximum (logarithmic) total strain is of the order of -1.8, which is equivalent to a stretch of ¸ = e-1.8= 0.17, or a nominal compressive strain of 83% indicating severe compression of the foam. In the viscoelastic case the stresses relax during loading and, consequently, lead to a softer response than in the purely elastic case, as shown in Figure 1.1.4-5. The force-displacement responses are shown in Figure 1.1.4-6. The purely elastic material is reversible, while the viscoelastic material shows hysteresis.

ABAQUS/Standard: Case 2 Various displaced configurations during the Case 2 analysis are shown in Figure 1.1.4-7. Displacement, velocity, and acceleration histories for the punch are shown in Figure 1.1.4-8. The displacement is shown to reach a steady value at the stress relaxation stage, followed by a severe drop due to the impulsive dynamic load. This is followed by a rebound and then finally by a rapid decay of the subsequent oscillations due to the strong damping provided by the viscoelasticity of the foam.

ABAQUS/Explicit Figure 1.1.4-9 shows a plot of the initial configurations. Figure 1.1.4-10 shows a contour plot of stress in the y-direction. Figure 1.1.4-11 shows a contour plot of logarithmic strain in the y-direction at 0.06 seconds when the maximum indentation is reached. The elements underneath the punch are seen to be subject to large strains. The history of the punch reaction force (reference node 1000) is shown in Figure 1.1.4-12. A plot of the punch displacement versus the punch reaction force is shown in Figure 1.1.4-13. Because of the purely elastic material behavior, there is no hysteresis and the punch reaction force in the unloading stage follows the same curve as during loading.

ABAQUS/Design Figure 1.1.4-14 shows the variation of the normalized sensitivity of CPRESS with respect to the design

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variables L and ¹1 on the contact surface. The sensitivities have been normalized by multiplying with the value of the design parameter and dividing by the maximum value of CPRESS. Figure 1.1.4-15 and Figure 1.1.4-16 show the contours of sensitivity of the displacement in the z-direction to the design paramaters L and ¹1 , respectively. Figure 1.1.4-17 and Figure 1.1.4-18 show the contours of sensitivity of S22 to the design parameters L and ¹1 , respectively. To provide an independent assessment of the results provided by ABAQUS, sensitivities were computed using the overall finite difference (OFD) technique. The central difference method with a perturbation size of 0.1% of the value of the design parameter was used to obtain the OFD results. Table 1.1.4-1 shows that the sensitivities computed using ABAQUS compare well with the overall finite difference results.

Input files indentfoamhemipunch_case1.inp Case 1 of the ABAQUS/Standard example using test data for both elastic and viscoelastic properties of the foam, which is statically deformed in two *VISCO steps. indentfoamhemipunch_case2.inp Case 2 in which the ABAQUS/Standard analysis is performed in three steps subjecting the punch to both static and dynamic loading. hyperfoam_anl.inp ABAQUS/Explicit analysis using an analytical rigid surface. hyperfoam.inp ABAQUS/Explicit model using a faceted rigid surface. indentfoamhemipunch_dsa.inp Design sensitivity analysis.

Table Table 1.1.4-1 Comparison of normalized sensitivities computed using ABAQUS and overall finite difference. Normalized ³ sensitivity ´ ¹1 dS22 S22 max d¹1 ¡ ¢ max L dS22 max S22 ³ dL´ max ¹1 du2 umax 1 2 ¡ d¹ ¢ max du2 L max u2 ³ dL max ´

¹1 CPRESS L CPRESS

dCPRESS

max

max

d¹1 ¡ dCPRESS ¢ max dL

max

ABAQUS 0.5024

OFD 0.5018

-0.1134 -0.0075

-0.1107 -0.0075

0.2918

0.2922

0.5578

0.5582

0.8012

0.8038

Figures

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Figure 1.1.4-1 Model for foam indentation by a spherical punch.

Figure 1.1.4-2 Elastomeric foam stress-strain curves.

Figure 1.1.4-3 Elastic modulus relaxation curve.

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Figure 1.1.4-4 Deformation and contour plots of viscoelastic foam, ABAQUS/Standard.

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Figure 1.1.4-5 Punch reaction force history: static and viscoelastic cases, ABAQUS/Standard.

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Figure 1.1.4-6 Punch reaction force versus displacement response: static and viscoelastic cases, (loading-unloading curves); ABAQUS/Standard.

Figure 1.1.4-7 Deformation plots for the visco and dynamic steps, ABAQUS/Standard.

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Figure 1.1.4-8 Displacement, velocity, and acceleration histories of the punch; ABAQUS/Standard.

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Figure 1.1.4-9 Initial (undeformed) configuration, ABAQUS/Explicit.

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Figure 1.1.4-10 Stress S22 contour plot at 0.06 s, ABAQUS/Explicit.

Figure 1.1.4-11 Logarithmic strain in the z-direction at 0.06 s, ABAQUS/Explicit.

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Figure 1.1.4-12 Punch reaction force, ABAQUS/Explicit.

Figure 1.1.4-13 Punch reaction force versus indentation, ABAQUS/Explicit.

Figure 1.1.4-14 Normalized sensitivities of contact pressure at the end of the analysis.

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Figure 1.1.4-15 Sensitivities at the end of the analysis for displacement in the z-direction with respect to L.

Figure 1.1.4-16 Sensitivities at the end of the analysis for displacement in the z-direction with respect to ¹1 .

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Figure 1.1.4-17 Sensitivities at the end of the analysis for stress S22 with respect to L.

Figure 1.1.4-18 Sensitivities at the end of the analysis for stress S22 with respect to ¹1 .

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Sample listings

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Listing 1.1.4-1 *HEADING STATIC INDENTATION OF A VISCOELASTIC, ELASTOMERIC FOAM WITH A HEMISPHERICAL PUNCH Measure the punch reaction force during the following 2 steps: Step 1: Displace punch 250 mm downwards. Step 2: Return punch to original position. Units: N, mm, sec *RESTART,WRITE,FREQUENCY=10 *NODE,NSET=ALLN 1,0.,300. 19,0.,0. 481,300.,300. 499,300.,0. 601,600.,300. 619,600.,0. *NODE,NSET=N9999 9999,0.,500. *NSET,NSET=N1 1, *NSET,NSET=N19 19, *NSET,NSET=N481 481, *NSET,NSET=N499 499, *NFILL,NSET=TOP,BIAS=.85 N1,N481,12,40 *NGEN,NSET=TOP 481,601,40 *NFILL,NSET=BOT,BIAS=.85 N19,N499,12,40 *NGEN,NSET=BOT 499,619,40 *NFILL,NSET=ALLN,BIAS=.9 TOP,BOT,9,2 *NSET,NSET=CENTER,GENERATE 3,19 *ELEMENT,TYPE=CAX4,ELSET=FOAM 1,3,43,41,1 *ELGEN,ELSET=FOAM

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Static Stress/Displacement Analyses

1,15,40,10,9,2,1 *ELSET,ELSET=CENT,GENERATE 1,141,10 61,69 *ELSET,ELSET=ETOP,GENERATE 1,101,10 *RIGID BODY,ANALYTICAL SURFACE=BSURF,REF NODE=9999 *SURFACE,TYPE=SEGMENTS,NAME=BSURF START,141.42,641.42 CIRCL,-1.,300.,0.,500. *SURFACE,NAME=ASURF ETOP,S3 *CONTACT PAIR,INTERACTION=ROUGH ASURF,BSURF *SURFACE INTERACTION,NAME=ROUGH *FRICTION 0.8, *SOLID SECTION,ELSET=FOAM,MATERIAL=FOAM, ORIENTATION=RECT *ORIENTATION,NAME=RECT,SYSTEM=RECTANGULAR 1.,0.,0., 0.,1.,0. 3,0. *MATERIAL,NAME=FOAM *HYPERFOAM,N=2,TEST DATA INPUT,POISSON=0.0, MODULI=INSTANTANEOUS *UNIAXIAL TEST DATA -.0217, -.05 -.0317, -.10 -.0367, -.15 -.0402, -.20 -.0433, -.25 -.0467, -.30 -.0504, -.35 -.0542, -.40 -.0604, -.45 -.0668, -.50 -.0759, -.55 -.0909, -.60 -.1083, -.65 -.1410, -.70 -.1933, -.75 -.2896, -.80 *SIMPLE SHEAR TEST DATA

1-88

Static Stress/Displacement Analyses

.0107, .08, .0030 .0373, .16, .0166 .0533, .24, .0366 .0853, .32, .0573 .1280, .40, .0817 .1653, .48, .1098 .2080, .56, .1394 .2560, .64, .1666 .2987, .72, .1904 *VISCOELASTIC,TIME=RELAXATION TEST DATA *SHEAR TEST DATA,SHRINF=0.5000 1.0000, 0.0001 0.9695, 0.001 0.9417, 0.002 0.8722, 0.005 0.7913, 0.010 0.7043, 0.020 0.6233, 0.050 0.5736, 0.100 0.5271, 0.200 0.5013, 0.500 0.5000, 1.000 *BOUNDARY BOT,1,2 CENTER,1 N9999,1 N9999,6 *STEP,NLGEOM,INC=200,AMPLITUDE=RAMP, UNSYMM=YES ** Step 1: Move down punch. *VISCO,CETOL=0.01 .0015,1.,,.05 *BOUNDARY 9999,2,,-250. *PRINT,CONTACT=YES *CONTACT CONTROLS,FRICTION ONSET=DELAY *CONTACT PRINT,SLAVE=ASURF *CONTACT FILE,SLAVE=ASURF,FREQUENCY=10 *ENERGY PRINT,FREQUENCY=5 *ENERGY PRINT,ELSET=FOAM,FREQ=5 *ENERGY PRINT,ELSET=CENT,FREQ=5 *EL PRINT,FREQUENCY=50,ELSET=CENT S, SINV,

1-89

Static Stress/Displacement Analyses

E, EE, CE, *NODE PRINT,FREQUENCY=25 U,RF *EL FILE,FREQUENCY=100,ELSET=CENT S,SINV,E,EE,CE *NODE FILE,NSET=N9999,FREQUENCY=10 U,RF ** ** ODB OUTPUT REQUESTS ** *OUTPUT, FIELD, FREQUENCY=4 *CONTACT OUTPUT, NSET=TOP, VARIABLE=PRESELECT *END STEP *STEP,NLGEOM,INC=200,AMPLITUDE=RAMP, UNSYMM=YES ** Step 2: Return punch to original position. *VISCO,CETOL=0.01 .0015,1.,,.05 *BOUNDARY,OP=MOD 9999,2,,0.0 *END STEP

1-90

Static Stress/Displacement Analyses

Listing 1.1.4-2 *HEADING DYNAMIC LOADING OF AN ELASTOMERIC, VISCOELASTIC FOAM BLOCK 3 steps: 1. Apply a gravity load on the point mass of the punch in a visco step and let the foam relax fully. 2. Apply a sinusoidal half-wave force on the punch and measure the displacement, velocity and acceleration response in a dynamic step. 3. After removing the force, continue measuring the dynamic response. ** UNITS: N, mm, sec The unit of mass is N-sec^2/mm. Observe that 1 kg = 1 N-sec^2/m = 0.001 N-sec^2/mm. Therefore, Force = Mass x Acceleration is consistently calculated. ** *RESTART,WRITE,FREQUENCY=10 *NODE,NSET=ALLN 1,0.,300. 19,0.,0. 481,300.,300. 499,300.,0. 601,600.,300. 619,600.,0. *NODE,NSET=N9999 9999,0.,500. *NSET,NSET=N1 1, *NSET,NSET=N19 19, *NSET,NSET=N481 481, *NSET,NSET=N499 499, *NFILL,NSET=TOP,BIAS=.85 N1,N481,12,40

1-91

Static Stress/Displacement Analyses

*NGEN,NSET=TOP 481,601,40 *NFILL,NSET=BOT,BIAS=.85 N19,N499,12,40 *NGEN,NSET=BOT 499,619,40 *NFILL,NSET=ALLN,BIAS=.9 TOP,BOT,9,2 *NSET,NSET=CENTER,GENERATE 3,19 *ELEMENT,TYPE=CAX4,ELSET=FOAM 1,3,43,41,1 *ELGEN,ELSET=FOAM 1,15,40,10,9,2,1 *ELSET,ELSET=CENT,GENERATE 1,141,10 61,69 *ELSET,ELSET=ETOP,GENERATE 1,111,10 *ELSET,ELSET=CORNER 1, *ELEMENT,TYPE=MASS,ELSET=PMASS 1001,9999 *MASS,ELSET=PMASS ** 200 kg = 200 N-sec^2/m = 0.2 N-sec^2/mm 0.2, *RIGID BODY,ANALYTICAL SURFACE=BSURF,REF NODE=9999 *SURFACE,TYPE=SEGMENTS,NAME=BSURF START,141.42,641.42 CIRCL,-1.,300.,0.,500. *SURFACE,NAME=ASURF ETOP,S3 *CONTACT PAIR,INTERACTION=ROUGH ASURF,BSURF *SURFACE INTERACTION,NAME=ROUGH *FRICTION 0.8, *SOLID SECTION,ELSET=FOAM,MATERIAL=FOAM, ORIENTATION=RECT *ORIENTATION,NAME=RECT,SYSTEM=RECTANGULA 1.,0.,0., 0.,1.,0. 3,0. *MATERIAL,NAME=FOAM

1-92

Static Stress/Displacement Analyses

*HYPERFOAM,N=2,TEST DATA INPUT,POISSON=0.0, MODULI=INSTANTANEOUS ** Stress: MPa = N/mm^2 *UNIAXIAL TEST DATA -.0217, -.05 -.0317, -.10 -.0367, -.15 -.0402, -.20 -.0433, -.25 -.0467, -.30 -.0504, -.35 -.0542, -.40 -.0604, -.45 -.0668, -.50 -.0759, -.55 -.0909, -.60 -.1083, -.65 -.1410, -.70 -.1933, -.75 -.2896, -.80 *SIMPLE SHEAR TEST DATA .0140, .08, .0046 .0334, .16, .0166 .0533, .24, .0366 .0853, .32, .0573 .1280, .40, .0817 .1653, .48, .1098 .2080, .56, .1394 .2560, .64, .1666 .2987, .72, .1904 *VISCOELASTIC,TIME=RELAXATION TEST DATA *SHEAR TEST DATA,SHRINF=0.5000 1.0000, 0.0001 0.9695, 0.001 0.9417, 0.002 0.8722, 0.005 0.7913, 0.010 0.7043, 0.020 0.6233, 0.050 0.5736, 0.100 0.5271, 0.200 0.5013, 0.500 0.5000, 1.000

1-93

Static Stress/Displacement Analyses

*DENSITY ** 10 kg/m^3 = 10 N-sec^2/m^4 = ** 1.E-11 N-sec^2/mm^4 1.E-11, *BOUNDARY BOT,1,2 CENTER,1 N9999,1 N9999,6 *AMPLITUDE,NAME=RAMP1,VALUE=RELATIVE ** Ramp to full load in 2 sec 0.,0., 2.,1., 10.,1. *AMPLITUDE,NAME=SINE,DEFINITION=PERIODIC, VALUE=ABSOLUTE ** Force amplitude = 5000 N of a ** half-sine wave for a 1 sec period 1,3.1416,0.,0. 0.,-5000. *STEP,NLGEOM,INC=100, UNSYMM=YES ** Step 1: Apply gravity force on the mass ** of the punch. 1 Apply gravity force *VISCO,CETOL=0.01 .01,5. *DLOAD,AMPLITUDE=RAMP1 ** g = 9810 mm/s^2 used in F = Mg PMASS,GRAV,9810.,0.,-1.,0. ** New output statements to generate ODB for the ** Visualizer tutorial *OUTPUT,FIELD,FREQUENCY=10 *CONTACT OUTPUT,SLAVE=ASURF,MASTER=BSURF, VARIABLE=PRESELECT *NODE OUTPUT U, *ELEMENT OUTPUT,ELSET=FOAM S,E *OUTPUT,HISTORY,FREQUENCY=1 *NODE OUTPUT,NSET=N9999 U2,V2,A2 *ELEMENT OUTPUT,ELSET=CORNER MISES,E22,S22 ** *PRINT,CONTACT=YES

1-94

Static Stress/Displacement Analyses

*ENERGY PRINT,FREQUENCY=5 *ENERGY PRINT,ELSET=FOAM,FREQ=5 *ENERGY PRINT,ELSET=CENT,FREQ=5 *CONTACT PRINT,SLAVE=ASURF *CONTACT FILE,SLAVE=ASURF,FREQUENCY=10 *EL PRINT,FREQUENCY=20,ELSET=CENT S, SINV, E, EE, CE, LOADS, *EL FILE,FREQUENCY=50,ELSET=CENT S,SINV,E,EE,CE *NODE PRINT,FREQUENCY=20 U, CF,RF *NODE FILE,NSET=N9999,FREQUENCY=10 U,CF,RF *END STEP *STEP,NLGEOM,INC=200, UNSYMM=YES ** Step 2: Apply dynamic load (half sine wave) ** to the punch 2 Apply dynamic load *DYNAMIC,HAFTOL=5000. .01,1. *DLOAD PMASS,GRAV,9810.,0.,-1.,0. *CLOAD,AMPLITUDE=SINE 9999,2 *NODE PRINT,FREQUENCY=20 U,V, A, CF,RF *NODE FILE,NSET=N9999,FREQUENCY=10 U,V,A,CF,RF *END STEP *STEP,NLGEOM,INC=300, UNSYMM=YES ** Step 3: Remove load and let punch/foam ** system vibrate freely 3 Remove load *DYNAMIC,HAFTOL=5000.,INITIAL=NO .01,10.

1-95

Static Stress/Displacement Analyses

*DLOAD PMASS,GRAV,9810.,0.,-1.,0. *CLOAD,OP=NEW *END STEP

1-96

Static Stress/Displacement Analyses

Listing 1.1.4-3 *HEADING DYNAMIC LOADING OF AN ELASTOMERIC FOAM WITH A HEMISPHERICAL PUNCH (Analytical rigid surface definition) *RESTART,TIMEMARKS=YES,WRITE,NUM=6 *NODE,NSET=ALLN 1,0.,300. 19,0.,0. 481,300.,300. 499,300.,0. 601,600.,300. 619,600.,0. *NSET,NSET=N1 1, *NSET,NSET=N19 19, *NSET,NSET=N481 481, *NSET,NSET=N499 499, *NFILL,NSET=TOP N1,N481,12,40 *NGEN,NSET=TOP 481,601,40 *NFILL,NSET=BOT N19,N499,12,40 *NGEN,NSET=BOT 499,619,40 *NFILL,NSET=ALLN TOP,BOT,9,2 *NSET,NSET=CENTER,GENERATE 3,19 *ELEMENT,TYPE=CAX4R,ELSET=FOAM 1,3,43,41,1 *ELGEN,ELSET=FOAM 1,15,40,10,9,2,1 *ELSET,ELSET=CENT,GENERATE 1,141,10 61,69 *NODE,NSET=PUNCH 1000,0.,510.

1-97

Static Stress/Displacement Analyses

*NSET,NSET=NOUT 1000, *ELSET,ELSET=UPPER,GEN 1,141,10 *ELSET,ELSET=LOWER,GEN 9,149,10 *SOLID SECTION,ELSET=FOAM,MATERIAL=FOAM *MATERIAL,NAME=FOAM *HYPERFOAM,N=2,TEST DATA INPUT,POISSON=0.0 ** Stress: MPa = N/mm^2 *UNIAXIAL TEST DATA -.0217, -.05 -.0317, -.10 -.0367, -.15 -.0402, -.20 -.0433, -.25 -.0467, -.30 -.0504, -.35 -.0542, -.40 -.0604, -.45 -.0668, -.50 -.0759, -.55 -.0909, -.60 -.1083, -.65 -.1410, -.70 -.1933, -.75 -.2896, -.80 *SIMPLE SHEAR TEST DATA .0140, .08, .0046 .0334, .16, .0166 .0533, .24, .0366 .0853, .32, .0573 .1280, .40, .0817 .1653, .48, .1098 .2080, .56, .1394 .2560, .64, .1666 .2987, .72, .1904 *DENSITY ** 10 kg/m^3 = 10 N-sec^2/m^4 = **1.E-11 N-sec^2/mm^4 1.E-11, *BOUNDARY BOT,1,2

1-98

Static Stress/Displacement Analyses

CENTER,1 1,1 1000,1 1000,6 *AMPLITUDE,NAME=SMOOTH,DEFINITION=SMOOTH STEP, TIME=TOTAL TIME 0.0,0.0,0.06,-250.,0.12,0. *SURFACE,TYPE=ELEMENT,NAME=TARGET UPPER,S3 *SURFACE, NAME=IMPACTOR,TYPE=SEGMENTS START, 200.,510. CIRCL,0.,310., 0.,510. *RIGID BODY, REF NODE=1000, ANALYTICAL SURFACE =IMPACTOR *STEP *DYNAMIC,EXPLICIT ,0.12 *BOUNDARY,AMPLITUDE=SMOOTH 1000,2,2,1. *SURFACE INTERACTION,NAME=IMP_TARG *FRICTION 0.8, *CONTACT PAIR,INTERACTION=IMP_TARG IMPACTOR,TARGET *HISTORY OUTPUT,TIME=0.0008 *EL HISTORY,ELSET=UPPER S,LE,LEP,NE,NEP *NODE HISTORY,NSET=NOUT RF,U *ENERGY HISTORY ALLKE,ALLIE,ALLAE,ALLVD,ALLWK,ETOTAL, DT, *FILE OUTPUT,NUMBER INTERVAL=6, TIMEMARKS=YES *EL FILE,ELSET=FOAM S,LE,LEP,NE,NEP *NODE FILE U,RF *ENERGY FILE *END STEP

1-99

Static Stress/Displacement Analyses

Listing 1.1.4-4 *HEADING STATIC INDENTATION OF AN ELASTOMERIC FOAM WITH A HEMISPHERICAL PUNCH DSA Step: Displace punch 250 mm downwards. *PARAMETER L=0.0 MU1=0.162449 *DESIGN PARAMETER L,MU1 *RESTART,WRITE,FREQUENCY=10 *NODE,NSET=ALLN 1,0.,300. 19,0.,0. 481,300.,300. 499,300.,0. 601,600.,300. 619,600.,0. *NODE,NSET=N9999 9999,0.,500. *NSET,NSET=N1 1 *NSET,NSET=N19 19 *NSET,NSET=N481 481 *NSET,NSET=N499 499 *NFILL,NSET=TOP,BIAS=.85 N1,N481,12,40 *NGEN,NSET=TOP 481,601,40 *NFILL,NSET=BOT,BIAS=.85 N19,N499,12,40 *NGEN,NSET=BOT 499,619,40 *NFILL,NSET=ALLN,BIAS=.9 TOP,BOT,9,2 *NSET,NSET=CENTER,GENERATE 3,19 *PARAMETER SHAPE VARIATION, PARAMETER=L

1-100

Static Stress/Displacement Analyses

1, 0.000, 0.00000, 0.000 41, 0.000, 0.01463, 0.000 81, 0.000, 0.03185, 0.000 121, 0.000, 0.05210, 0.000 161, 0.000, 0.07592, 0.000 201, 0.000, 0.10396, 0.000 241, 0.000, 0.13693, 0.000 281, 0.000, 0.17573, 0.000 321, 0.000, 0.22137, 0.000 361, 0.000, 0.27507, 0.000 401, 0.000, 0.33823, 0.000 441, 0.000, 0.41257, 0.000 481, 0.000, 0.50000, 0.000 521, 0.000, 0.66667, 0.000 561, 0.000, 0.83333, 0.000 601, 0.000, 1.00000, 0.000 *ELEMENT,TYPE=CAX4,ELSET=FOAM 1,3,43,41,1 *ELGEN,ELSET=FOAM 1,15,40,10,9,2,1 *ELSET,ELSET=CENT,GENERATE 1,141,10 61,69 *ELSET,ELSET=ETOP,GENERATE 1,101,10 *RIGID BODY,ANALYTICAL SURFACE=BSURF,REF NODE=9999 *SURFACE,TYPE=SEGMENTS,NAME=BSURF START,141.42,641.42 CIRCL,-1.,300.,0.,500. *SURFACE,NAME=ASURF ETOP,S3 *CONTACT PAIR,INTERACTION=SMOOTH, SMALL SLIDING ASURF,BSURF *SURFACE INTERACTION,NAME=SMOOTH *SOLID SECTION,ELSET=FOAM,MATERIAL=FOAM, ORIENTATION=RECT *ORIENTATION,NAME=RECT,SYSTEM=RECTANGULAR 1.,0.,0., 0.,1.,0. 3,0. *MATERIAL,NAME=FOAM *HYPERFOAM,N=2 , 8.89239, 3.597339e-05, -4.52156, 0.0, 0.0 *BOUNDARY

1-101

Static Stress/Displacement Analyses

BOT,1,2 CENTER,1 N9999,1 N9999,6 *ELSET, ELSET=ER, GENERATE 1, 59, 1 *NSET, NSET=NU, GENERATE 41,601,40 *NSET, NSET=NR, GENERATE 59,619,40 *STEP,NLGEOM,INC=200,AMPLITUDE=RAMP, DSA ** Step 1: Move down punch. *STATIC .0015,1.,,.05 *BOUNDARY 9999,2,,-250. ** ** ODB OUTPUT REQUESTS ** *OUTPUT, FIELD, FREQUENCY=4 *NODE OUTPUT U, *ELEMENT OUTPUT S, *CONTACT OUTPUT, NSET=TOP, VARIABLE=PRESELECT *DESIGN RESPONSE, FREQUENCY=999 *ELEMENT RESPONSE S, *NODE RESPONSE U, *CONTACT RESPONSE CSTRESS, CDISP, *EL FILE, ELSET=ER, FREQUENCY=999 S, E, *NODE FILE, NSET=NU, FREQUENCY=999 U, *NODE FILE, NSET=NR, FREQUENCY=999 RF, *END STEP

1.1.5 Collapse of a concrete slab 1-102

Static Stress/Displacement Analyses

Products: ABAQUS/Standard ABAQUS/Explicit This problem examines the use of cracking models for the analysis of reinforced concrete structures. The geometry of the problem is defined in Figure 1.1.5-1. A square slab is supported in the transverse direction at its four corners and loaded by a point load at its center. The slab is reinforced in two directions at 75% of its depth. The reinforcement ratio (volume of steel/volume of concrete) is 8.5 ´ 10-3 in each direction. The slab was tested experimentally by McNeice (1967) and has been analyzed by a number of workers, including Hand et al. (1973), Lin and Scordelis (1975), Gilbert and Warner (1978), Hinton et al. (1981), and Crisfield (1982).

Geometric modeling Symmetry conditions allow us to model one-quarter of the slab. A 3 ´ 3 mesh of 8-node shell elements is used for the ABAQUS/Standard analysis. No mesh convergence studies have been performed, but the reasonable agreement between the analysis results and the experimental data suggests that the mesh is adequate to predict overall response parameters with usable accuracy. Three different meshes are used in ABAQUS/Explicit to assess the sensitivity of the results to mesh refinement: a coarse 6 ´ 6 mesh, a medium 12 ´ 12 mesh, and a fine 24 ´ 24 mesh of S4R elements. Nine integration points are used through the thickness of the concrete to ensure that the development of plasticity and failure are modeled adequately. The two-way reinforcement is modeled using the *REBAR option. Symmetry boundary conditions are applied on the two edges of the mesh, and the corner point is restrained in the transverse direction.

Material properties The material data are given in Table 1.1.5-1. The material properties of concrete are taken from Gilbert and Warner (1978). Some of these data are assumed values, because they are not available for the concrete used in the experiment. The assumed values are taken from typical concrete data. The compressive behavior of concrete in the cracking model in ABAQUS/Explicit is assumed to be linear elastic. This is a reasonable assumption for a case such as this problem, where the behavior of the structure is dominated by cracking resulting from tension in the slab under bending. The modeling of the concrete-reinforcement interaction and the energy release at cracking are of critical importance to the response of a structure such as this once the concrete starts to crack. These effects are modeled in an indirect way by adding "tension stiffening" to the plain concrete model. This approach is described in ``A cracking model for concrete and other brittle materials, '' Section 4.5.2 of the ABAQUS Theory Manual; ``Concrete,'' Section 11.5.1 of the ABAQUS/Standard User's Manual; and ``Cracking model for concrete,'' Section 10.4.1 of the ABAQUS/Explicit User's Manual. The simplest tension stiffening model defines a linear loss of strength beyond the cracking failure of the concrete. In this example three different values for the strain beyond failure at which all strength is lost (5 ´ 10-4, 1 ´ 10-3, and 2 ´ 10-3) are used to illustrate the effect of the tension stiffening parameters on the response. Since the response is dominated by bending, it is controlled by the material behavior normal to the crack planes. The material's shear behavior in the plane of the cracks is not important. Consequently, the choice of shear retention has no significant influence on the results. In ABAQUS/Explicit the shear

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Static Stress/Displacement Analyses

retention chosen is exhausted at the same value of the crack opening at which tension stiffening is exhausted. In ABAQUS/Standard full shear retention is used because it provides a more efficient numerical solution.

Solution control Since considerable nonlinearity is expected in the response, including the possibility of unstable regimes as the concrete cracks, the modified Riks method is used with automatic incrementation in the ABAQUS/Standard analysis. With the Riks method the load data and solution parameters serve only to give an estimate of the initial increment of load. In this case it seems reasonable to apply an initial load of 1112 N (250 lb) on the quarter-model for a total initial load on the structure of 4448 N (1000 lb). This can be accomplished by specifying a load of 22241 N (5000 lb) and an initial time increment of 0.05 out of a total time period of 1.0. The analysis is terminated when the central displacement reaches 25.4 mm (1 in). Since ABAQUS/Explicit is a dynamic analysis program and in this case we are interested in static solutions, the slab must be loaded slowly enough to eliminate any significant inertia effects. The slab is loaded in its center by applying a velocity that increases linearly from 0 to 2.0 in/second such that the center displaces a total of 1 inches in 1 second. This very slow loading rate ensures quasi-static solutions; however, it is computationally expensive. The CPU time required for this analysis can be reduced in one of two ways: the loading rate can be increased incrementally until it is judged that any further increase in loading rate would no longer result in a quasi-static solution, or mass scaling can be used (see ``Explicit dynamic analysis,'' Section 6.2.1 of the ABAQUS/Explicit User's Manual). These two approaches are equivalent. Mass scaling is used here to demonstrate the validity of such an approach when it is used in conjunction with the concrete model. Mass scaling is done by increasing the density of the concrete and the reinforcement by a factor of 100, thereby increasing the stable time increment for the analysis by a factor of 10 and reducing the computation time by the same amount while using the original slow loading rate. Figure 1.1.5-4 shows the load-deflection response of the slab for analyses using the 12 ´ 12 mesh with and without mass scaling. The mass scaling used does not affect the results significantly; therefore, all subsequent analyses are performed using mass scaling.

Results and discussion Results for each analysis are discussed in the following sections.

ABAQUS/Standard results The numerical and experimental results are compared in Figure 1.1.5-2 on the basis of load versus deflection at the center of the slab. The strong effect of the tension stiffening assumption is very clear in that plot. The analysis with tension stiffening, such that the tensile strength is lost at a strain of 10 -3 beyond failure, shows the best agreement with the experiment. This analysis provides useful information from a design viewpoint. The failure pattern in the concrete is illustrated in Figure 1.1.5-3, which shows the predicted crack pattern on the lower surface of the slab at a central deflection of 7.6 mm (0.3 in).

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Static Stress/Displacement Analyses

ABAQUS/Explicit results Figure 1.1.5-5 shows the load-deflection response of the slab for the three different mesh densities using a tension stiffening value of 2 ´ 10-3. Since the coarse mesh predicts a slightly higher limit load than the medium and fine meshes do and the limit loads for the medium and fine mesh analyses are very close, the tension stiffening study is performed using the medium mesh only. The numerical (12 ´ 12 mesh) results are compared to the experimental results in Figure 1.1.5-6 for the three different values of tension stiffening. It is clear that the less tension stiffening used, the softer the load-deflection response is. A value of tension stiffening somewhere between the highest and middle values appears to match the experimental results best. The lowest tension stiffening value causes more sudden cracking in the concrete and, as a result, the response tends to be more dynamic than that obtained with the higher tension stiffening values. Figure 1.1.5-7 shows the numerically predicted crack pattern on the lower surface of the slab for the medium mesh.

Input files ABAQUS/Standard input files collapseconcslab_s8r.inp S8R elements. collapseconcslab_s9r5.inp S9R5 elements. collapseconcslab_postoutput.inp *POST OUTPUT analysis. ABAQUS/Explicit input files mcneice_1.inp Coarse (6 ´ 6) mesh; tension stiffening = 2 ´ 10-3. mcneice_2.inp Medium (12 ´ 12) mesh; tension stiffening = 2 ´ 10-3. mcneice_3.inp Fine (24 ´ 24) mesh; tension stiffening = 2 ´ 10-3. mcneice_4.inp Medium (12 ´ 12) mesh; tension stiffening = 1 ´ 10-3. mcneice_5.inp Medium (12 ´ 12) mesh; tension stiffening = 5 ´ 10-4.

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Static Stress/Displacement Analyses

mcneice_6.inp Medium (12 ´ 12) mesh; tension stiffening = 2 ´ 10-3; no mass scaling.

References · Crisfield, M. A, "Variable Step-Length for Nonlinear Structural Analysis," Report 1049, Transport and Road Research Lab., Crowthorne, England, 1982. · Gilbert, R. I. and R. F. Warner, "Tension Stiffening in Reinforced Concrete Slabs," Journal of the Structural Division, American Society of Civil Engineers, vol. 104, ST12, pp. 1885-1900, 1978. · Hand, F. D., D. A. Pecknold, and W. C. Schnobrich, "Nonlinear Analysis of Reinforced Concrete Plates and Shells," Journal of the Structural Division, American Society of Civil Engineers, vol. 99, ST7, pp. 1491-1505, 1973. · Hinton, E., H. H. Abdel Rahman, and O. C. Zienkiewicz, "Computational Strategies for Reinforced Concrete Slab Systems," International Association of Bridge and Structural Engineering Colloquium on Advanced Mechanics of Reinforced Concrete, pp. 303-313, Delft, 1981. · Lin, C. S. and A. C. Scordelis, "Nonlinear Analysis of Reinforced Concrete Shells of General Form," Journal of the Structural Division, American Society of Civil Engineers, vol. 101, pp. 523-238, 1975. · McNeice, A. M., "Elastic-Plastic Bending of Plates and Slabs by the Finite Element Method, " Ph. D. Thesis, London University, 1967.

Table Table 1.1.5-1 Material properties for McNeice slab. Concrete properties: Properties are taken from Gilbert and Warner (1978) if available in that paper. Properties marked with a * are not available, and are assumed values. Young's modulus 28.6 GPa (4.15´106 lb/in2) Poisson's ratio 0.15 Uniaxial compression values: Yield stress 20.68 MPa (3000 lb/in 2)* Failure stress 37.92 MPa (5500 lb/in 2) Plastic strain at failure 1.5´10-3* Ratio of uniaxial tension to compression failure stress 8.36´10-2 Ratio of biaxial to uniaxial compression failure stress 1.16* Cracking failure stress 459.8 lb/in 2 (3.17 MPa) Density (before mass scaling) 2.246 ´ 10-4 lb s2/in4 (2400 kg/m3) "Tension stiffening" is assumed as a linear decrease of the stress to zero stress, at a strain of 5´10-4, at a strain of 10´10-4, or at a strain of 20´10-4. 1-106

Static Stress/Displacement Analyses

Steel (rebar) properties: Young's modulus Yield stress Density (before mass scaling)

200 GPa (29´106 lb/in2) 345 MPa (50´103 lb/in2) 7.3 ´ 10-4 lb s2/in4 (7800 kg/m3)

Figures Figure 1.1.5-1 McNeice slab.

Figure 1.1.5-2 Load-deflection response of McNeice slab, ABAQUS/Standard.

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Static Stress/Displacement Analyses

Figure 1.1.5-3 Crack pattern on lower surface of slab, ABAQUS/Standard.

Figure 1.1.5-4 Load-deflection response of McNeice slab, ABAQUS/Explicit; influence of mass scaling.

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Static Stress/Displacement Analyses

Figure 1.1.5-5 Load-deflection response of McNeice slab, ABAQUS/Explicit; influence of mesh refinement.

Figure 1.1.5-6 Load-deflection response of McNeice slab, ABAQUS/Explicit; influence of tension stiffening.

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Static Stress/Displacement Analyses

Figure 1.1.5-7 Crack pattern on lower surface of slab, ABAQUS/Explicit.

Sample listings

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Listing 1.1.5-1 *HEADING CORNER SUPPORTED TWO-WAY SLAB TESTED BY MCNEICE *NODE 1,0.,0. 7,18.,0. 61,0.,18. 67,18.,18. *NGEN,NSET=Y-SYM 1,7 *NGEN,NSET=X-SYM 1,61,10 *NGEN,NSET=LX2 61,67 *NGEN,NSET=LY2 7,67,10 *NSET,NSET=ONE 1, *NFILL X-SYM,LY2,6,1 *ELEMENT,TYPE=S8R,ELSET=SLAB 1,1,3,23,21,2,13,22,11 *ELGEN,ELSET=SLAB 1,3,2,1,3,20,3 *SHELL SECTION,ELSET=SLAB,MATERIAL=A1 1.75,9 *MATERIAL,NAME=A1 *ELASTIC 4.15E6,.15 *CONCRETE 3000.,0. 5500.,.0015 *FAILURE RATIOS 1.16 , .0836 *TENSION STIFFENING 1.,0. 0.,2.E-3 *REBAR,ELEMENT=SHELL,MATERIAL=SLABMT, GEOMETRY=ISOPARAMETRIC,NAME=YY SLAB,.014875,1.,-.435,4 *REBAR,ELEMENT=SHELL,MATERIAL=SLABMT,

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GEOMETRY=ISOPARAMETRIC,NAME=XX SLAB,.014875,1.,-.435,1 *MATERIAL,NAME=SLABMT *ELASTIC 29.E6, *PLASTIC 50.E3, *BOUNDARY Y-SYM,YSYMM X-SYM,XSYMM 67,3 *RESTART,WRITE,FREQUENCY=999 *STEP,INC=30 *STATIC,RIKS .05,1.,,,,1,3,-1. *CLOAD 1,3,-5000. *EL PRINT,FREQUENCY=10 S, SINV, E, PE, CRACK, *EL FILE,FREQUENCY=10 S, SINV, E, PE, CRACK, *OUTPUT,FIELD,FREQ=10 *ELEMENT OUTPUT S, SINV, E, PE, CRACK, *NODE FILE,NSET=ONE U, *OUTPUT,FIELD *NODE OUTPUT,NSET=ONE U, *END STEP

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Listing 1.1.5-2 *HEADING CORNER SUPPORTED TWO-WAY SLAB TESTED BY MCNEICE *NODE 1,0.,0. 7,18.,0. 61,0.,18. 67,18.,18. *NGEN,NSET=Y-SYM 1,7 *NGEN,NSET=X-SYM 1,61,10 *NGEN,NSET=LX2 61,67 *NGEN,NSET=LY2 7,67,10 *NSET,NSET=ONE 1, *NFILL X-SYM,LY2,6,1 *ELEMENT,TYPE=S4R,ELSET=SLAB 1,1,2,12,11 *ELGEN,ELSET=SLAB 1,6,1,1,6,10,6 *SHELL SECTION,ELSET=SLAB,MATERIAL=CONC 1.75,9 *REBAR,ELEMENT=SHELL,MATERIAL=SLABMT, GEOMETRY=ISOPARAMETRIC,NAME=YY SLAB,.014875,1.,-.435,4 *REBAR,ELEMENT=SHELL,MATERIAL=SLABMT, GEOMETRY=ISOPARAMETRIC,NAME=XX SLAB,.014875,1.,-.435,1 *MATERIAL,NAME=SLABMT *ELASTIC 29.E6, *PLASTIC 50.E3, *DENSITY 7.3e-2, *MATERIAL,NAME=CONC *ELASTIC

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4.15E6,.15 *DENSITY 2.246e-2, *BRITTLE CRACKING 459.8,0. 0.,2.e-3 *BRITTLE SHEAR 1.,0. 0.,2.e-3 *BOUNDARY Y-SYM,YSYMM X-SYM,XSYMM 67,3 *AMPLITUDE,NAME=RAMP,DEFINITION=SMOOTH STEP 0.,0.,1.,1. *RESTART,WRITE,NUM=1 *STEP *DYNAMIC,EXPLICIT ,1 *BOUNDARY,AMP=RAMP 1,3,3,-1 *MONITOR,NODE=1,DOF=3 *HISTORY OUTPUT, TIME= 5.00E-4 *ENERGY HISTORY ALLKE, ALLSE, ALLIE, ALLWK, ALLAE, ETOTAL *NODE HISTORY,NSET=ONE RF, U, *FILE OUTPUT,TIMEMARKS=YES,NUMBER=1 *NODEFILE RF, U, *ELFILE S, LE, CKE, CKLE, CKLS, CKEMAG, CKSTAT, CRACK, *END STEP

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1.1.6 Jointed rock slope stability Product: ABAQUS/Standard This example illustrates the use of the jointed material model in the context of geotechnical applications. We examine the stability of the excavation of part of a jointed rock mass, leaving a sloped embankment. This problem is chosen mainly as a verification case because it has been studied previously by Barton (1971) and Hoek (1970), who used limit equilibrium methods, and by Zienkiewicz and Pande (1977), who used a finite element model.

Geometry and model The plane strain model analyzed is shown in Figure 1.1.6-1together with the excavation geometry and material properties. The rock mass contains two sets of planes of weakness: one vertical set of joints and one set of inclined joints. We begin from a nonzero state of stress. In this problem this consists of a vertical stress that increases linearly with depth to equilibrate the weight of the rock and horizontal stresses caused by tectonic effects: such stress is quite commonly encountered in geotechnical engineering. The active "loading" consists of removal of material to represent the excavation. It is clear that, with a different initial stress state, the response of the system would be different. This illustrates the need of nonlinear analysis in geotechnical applications--the response of a system to external "loading" depends on the state of the system when that loading sequence begins (and, by extension, to the sequence of loading). We can no longer think of superposing load cases, as is done in a linear analysis. Practical geotechnical excavations involve a sequence of steps, in each of which some part of the material mass is removed. Liners or retaining walls can be inserted during this process. Thus, geotechnical problems require generality in creating and using a finite element model: the model itself, and not just its response, changes with time--parts of the original model disappear, while other components that were not originally present are added. This example is somewhat academic, in that we do not encounter this level of complexity. Instead, following the previous authors' use of the example, we assume that the entire excavation occurs simultaneously.

Solution controls The jointed material model includes a joint opening/closing capability. When a joint opens, the material is assumed to have no elastic stiffness with respect to direct strain across the joint system. Because of this, and also as a result of the fact that different combinations of joints may be yielding at any one time, the overall convergence of the solution is expected to be nonmonotonic. In such cases the use of *CONTROLS, ANALYSIS=DISCONTINUOUS is generally recommended to prevent premature termination of the equilibrium iteration process because the solution may appear to be diverging. As the end of the excavation process is approached, the automatic incrementation algorithm reduces the load increment significantly, indicating the onset of failure of the slope. In such analyses it is useful to specify a minimum time step to avoid unproductive iteration. For the nonassociated flow case UNSYMM=YES is used on the *STEP option. This is essential for

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obtaining an acceptable rate of convergence since nonassociated flow plasticity has a nonsymmetric stiffness matrix.

Results and discussion In this problem we examine the effect of joint cohesion on slope collapse through a sequence of solutions with different values of joint cohesion, with all other parameters kept fixed. Figure 1.1.6-2 shows the variation of horizontal displacements as cohesion is reduced at the crest of the slope (point A in Figure 1.1.6-1) and at a point one-third of the way up the slope (point B in Figure 1.1.6-1). This plot suggests that the slope collapses if the cohesion is less than 24 kPa for the case of associated flow or less than 26 kPa for the case of nonassociated flow. These compare well with the value calculated by Barton (26 kPa) using a planar failure assumption in his limit equilibrium calculations. Barton's calculations also include "tension cracking" (akin to joint opening with no tension strength) as we do in our calculation. Hoek calculates a cohesion value of 24 kPa for collapse of the slope. Although he also makes the planar failure assumption, he does not include tension cracking. This is, presumably, the reason why his calculated value is lower than Barton's. Zienkiewicz and Pande assume the joints have a tension strength of one-tenth of the cohesion and calculate the cohesion value necessary for collapse as 23 kPa for associated flow and 25 kPa for nondilatant flow. Figure 1.1.6-3 shows the deformed configuration after excavation for the nonassociated flow case and clearly illustrates the manner in which the collapse is expected to occur. Figure 1.1.6-4shows the magnitude of the frictional slip on each joint system for the nonassociated flow case. A few joints open near the crest of the slope.

Input files jointrockstabil_nonassoc_30pka.inp Nonassociated flow case problem; cohesion = 30 kPa. jointrockstabil_assoc_25kpa.inp Associated flow case; cohesion = 25 kPa.

References · Barton, N., "Progressive Failure of Excavated Rock Slopes," Stability of Rock Slopes, Proceedings of the 13th Symposium on Rock Mechanics, Illinois, pp. 139-170, 1971. · Hoek, E., "Estimating the Stability of Excavated Slopes in Open Cast Mines, " Trans. Inst. Min. and Metal., vol. 79, pp. 109-132, 1970. · Zienkiewicz, O. C. and G. N. Pande, "Time-Dependent Multilaminate Model of Rocks - A Numerical Study of Deformation and Failure of Rock Masses," International Journal for Numerical and Analytical Methods in Geomechanics, vol. 1, pp. 219-247, 1977.

Figures

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Figure 1.1.6-1 Jointed rock slope problem.

Figure 1.1.6-2 Horizontal displacements with varying cohesion.

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Figure 1.1.6-3 Deformed configuration (nonassociated flow).

Figure 1.1.6-4 Contours of frictional slip magnitudes (nonassociated flow).

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Sample listings

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Listing 1.1.6-1 *HEADING JOINTED ROCK SLOPE, 2 JOINTS, C=30, NONASSOC FLOW, .5 SH RET *NODE 1,0.,0. 11,100.,0. 23,272.9,0. 241,0.,74. 251,100.,74. 263,272.9,74. 731,140.4,144. 743,272.9,144. *NGEN,NSET=BLHS 1,241,40 *NGEN,NSET=BCEN 11,251,40 *NGEN,NSET=BRHS 23,263,40 *NGEN,NSET=TCEN 251,731,40 *NGEN,NSET=TRHS 263,743,40 *NFILL,BIAS=1.5,TWO STEP BLHS,BCEN,10 *NFILL,BIAS=.66666666,TWO STEP BCEN,BRHS,12 *NFILL,BIAS=.66666666,TWO STEP TCEN,TRHS,12 *NSET,NSET=SLHS,GENERATE 241,251 *NSET,NSET=SRHS,GENERATE 731,743 *NSET,NSET=BOT,GENERATE 1,23 *NSET,NSET=FILN 251,411,731 *ELEMENT,TYPE=CPE4 1,1,2,42,41 101,11,12,52,51 *ELGEN,ELSET=ALLE 1,6,40,1,10,1,10

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101,18,40,1,12,1,20 *SOLID SECTION,ELSET=ALLE,MATERIAL=ALLE *MATERIAL,NAME=ALLE *ELASTIC 2.8E7,.2 *JOINTED MATERIAL,JOINT DIRECTION=JOINT1 45.,22.5,30. *JOINTED MATERIAL,JOINT DIRECTION=JOINT2 45.,22.5,30. *JOINTED MATERIAL,SHEAR RETENTION .5, *ORIENTATION,NAME=JOINT1 1.,0.,0.,0.,1.,0. *ORIENTATION,NAME=JOINT2 .7934,-.6088,0.,.6088,.7934,0. *INITIAL CONDITIONS,TYPE=STRESS,GEOSTATIC ALLE,0.,144.,-3600.,0.,.333333 *RESTART,WRITE,FREQUENCY=5 *STEP, UNSYMM=YES *GEOSTATIC 1.,1. *DLOAD ALLE,BY,-25. *BOUNDARY BOT,2,2,0. BLHS,1,1,0. BRHS,1,1,0. TRHS,1,1,0. SLHS,1,2,0. TCEN,1,2,0. SRHS,1,2,0. *EL PRINT S,MISES,PRESS E, PE, PEQC, *NODE PRINT U,RF *OUTPUT,FIELD *NODE OUTPUT,NSET=FILN U, *OUTPUT,FIELD,FREQ=5 *ELEMENT OUTPUT

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PEQC, *OUTPUT,HISTORY *NODE OUTPUT,NSET=FILN U, *END STEP *STEP,INC=20 *STATIC .1,1.,.001,.1 *CONTROLS,ANALYSIS=DISCONTINUOUS *BOUNDARY,OP=NEW BOT,2,2,0. BLHS,1,1,0. BRHS,1,1,0. TRHS,1,1,0. *MONITOR,NODE=411,DOF=1 *END STEP

1.1.7 Indentation of a crushable foam specimen with a hemispherical punch Products: ABAQUS/Standard ABAQUS/Explicit In this example we consider a cylindrical specimen of crushable foam indented by a rough, rigid, hemispherical punch. The example illustrates a typical application of crushable foam materials when used as energy absorption devices. The effect of rate dependence of the foam is shown. Results are also presented for both the linear and porous elastic models in ABAQUS/Standard. This problem tests the crushable foam model but does not provide independent verification of it.

Geometry and model The axisymmetric model (135 CAX4 elements in ABAQUS/Standard, 135 CAX4R elements in ABAQUS/Explicit) analyzed is shown in Figure 1.1.7-1. The foam specimen has a radius of 600 mm and a thickness of 300 mm. The punch has a radius of 200 mm. The bottom nodes of the mesh are fixed, while the outer boundary is free to move.

Material The material's behavior is modeled using linear elasticity and piecewise linear strain hardening. Results from ABAQUS/Standard using porous elasticity and exponential strain hardening are also presented. The elastic material parameters are chosen so that the elastic stiffnesses of the linear model and of the porous elastic model are similar at the end of the first loading step, and the piecewise linear and exponential hardening curves are similar. The material parameters are E = 3.0 MPa (for the linear elastic model) ∙ = 0.09 (for the porous elastic model) º = 0.0 p0 = 0.2 MPa (This is the initial value of pc .)

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pt = 0.02 MPa ¾0 = 0.22 MPa ¸ = 1.7 K =1 ½ = 500 In addition, we use the following material properties for the rate-dependent case: D = 80 per sec p=1

Contact interaction The contact between the top exterior surface of the foam specimen and the rigid punch is modeled with the *CONTACT PAIR option. The specimen's surface is defined by means of the *SURFACE option. The spherical rigid punch is modeled as an analytical rigid surface with the *SURFACE option in conjunction with the *RIGID BODY option. An ABAQUS/Explicit model using RAX2 elements instead of an analytical rigid surface is also available. However, an analytical rigid surface provides a more accurate representation of curved geometries. Results for the analytical rigid surface case are presented here. The mechanical interaction between the contact surfaces is assumed to be rough frictional contact in ABAQUS/Standard. Therefore, the *FRICTION, ROUGH option--which enforces a no slip constraint between the two surfaces--is used as a suboption of the *SURFACE INTERACTION property option. In ABAQUS/Explicit the friction coefficient between the punch and the foam is 0.95. The maximum p shear traction due to friction is assumed to be ¾0 = 3, or 0.127 MPa.

Loading and controls The loading is applied by first moving the rigid surface punch into the foam specimen. This step is followed by a second unloading step in which the punch is returned to the original position. Very large deformations will take place, so the NLGEOM parameter is needed on the *STEP option in the ABAQUS/Standard analysis. For nonassociated flow cases UNSYMM=YES is used on the *STEP option. This is important to obtain an acceptable rate of convergence during the equilibrium iterations, since the nonassociated flow plasticity model used for the foam has a nonsymmetric stiffness matrix. In the rate-independent case the punch is moved down at a constant velocity within 0.6 seconds, while in the fast indentation problem the punch indents half of the thickness of the specimen over a period of 15 ms to demonstrate the plastic strain rate effect. In the latter case the velocity of the punch is ramped from zero to 20 m/s.

Results and discussion The overall load-deflection response of the foam specimen for all four ABAQUS/Standard models is shown in Figure 1.1.7-2. As expected, the rate-dependent cases predict higher punch forces. The deformed configuration of the mesh at the end of the first loading step (showing actual displacements) is shown in Figure 1.1.7-3. The magnitude of the plastic strain is given in the contour plot of Figure 1.1.7-4. The figure shows that the plastic strain magnitude in the vicinity of the punch approaches 100%.

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The results differ very little between the linear and porous elastic models. This might be expected since the problem involves loading well into the plastic range, so that elastic effects are not likely to be significant (as long as the elasticity is sufficiently stiff to be realistic). However, in a case like this, the linear elastic modeling is more efficient computationally. In addition, it is not possible to unload the specimen for the rate-dependent case with porous elasticity. The crushable foam model with linear elasticity is, therefore, recommended for most applications. Figure 1.1.7-5 shows a contour plot of the position of the yield surface at the end of the indentation in the rate-independent ABAQUS/Explicit case. Figure 1.1.7-6 shows the overall load-deflection response of the foam specimen for both the rate-independent and the rate-dependent cases. Again, the rate-dependent case predicts higher punch forces.

Input files ABAQUS/Standard input files foamindent_ratedep_porous.inp Rate-dependent case with porous elasticity, exponential hardening, and power law rate dependence. foamindent_piecerate_linear.inp Rate-dependent case with linear elasticity, tabular hardening, and power law rate dependence entered as a piecewise linear function. foamindent_rateindep_porous.inp Rate-independent case with porous elasticity and exponential hardening. foamindent_rateindep_linear.inp Rate-independent case with linear elasticity and tabular hardening. foamindent_postoutput.inp *POST OUTPUT analysis of foamindent_ratedep_porous.inp. ABAQUS/Explicit input files crushfoam_anl.inp Rate-independent case. crushfoam.inp Rate-independent case using a faceted surface representation. crushfoam_rate_anl.inp Rate-dependent case with power law rate dependence. crushfoam_rate.inp Rate-dependent case with power law rate dependence using a faceted surface representation.

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crushfoam_tabular_anl.inp Rate-dependent case with power law rate dependence entered as a piecewise linear function. crushfoam_tabular.inp Rate-dependent case with power law rate dependence entered as a piecewise linear function for a model using a faceted surface representation.

Figures Figure 1.1.7-1 Model for foam indentation by spherical punch.

Figure 1.1.7-2 Punch force versus penetration response, ABAQUS/Standard.

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Figure 1.1.7-3 Deformed configuration showing actual displacements, ABAQUS/Standard.

Figure 1.1.7-4 Contours of magnitude of plastic strain, ABAQUS/Standard.

Figure 1.1.7-5 Yield surface position contours, ABAQUS/Explicit.

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Figure 1.1.7-6 Punch force versus penetration response, ABAQUS/Explicit.

Sample listings

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Listing 1.1.7-1 *HEADING RATE DEPENDENT FOAM INDENTATION WITH ROUGH HEMISPHERICAL PUNCH *NODE,NSET=ALLN 1,0.,.3 17,0.,.1 19,0.,0. 481,.3,.3 497,.3,.1 499,.3,0. 601,.6,.3 617,.6,.1 619,.6,0. *NODE,NSET=N9999 9999,0.,.5 *NGEN,NSET=ALLN 1,17 17,19 481,497 497,499 601,617 617,619 *NSET,NSET=CL,GENERATE 1,19 *NSET,NSET=MID,GENERATE 481,499,1 *NSET,NSET=OUT,GENERATE 601,619,1 *NFILL,NSET=ALLN CL,MID,24,20 *NFILL,NSET=ALLN MID,OUT,6,20 *NSET,NSET=BOT,GENERATE 19,619,20 *ELEMENT,TYPE=CAX4,ELSET=FOAM 1,3,43,41,1 *ELGEN,ELSET=FOAM 1,15,40,10,9,2,1 *ELSET,ELSET=CENT,GENERATE 1,141,10 61,69

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*ELSET,ELSET=ETOP,GENERATE 1,111,10 *RIGID BODY,ANALYTICAL SURFACE=BSURF,REF NODE=9999 *SURFACE,TYPE=SEGMENTS,NAME=BSURF START,.14142,.64142 CIRCL,-.0001,.3,0.,.5 *SURFACE,NAME=ASURF ETOP,S3 *CONTACT PAIR,INTERACTION=ROUGH ASURF,BSURF *SURFACE INTERACTION,NAME=ROUGH *FRICTION,ROUGH *ELSET,ELSET=ELPRT CENT, *SOLID SECTION,ELSET=FOAM,MATERIAL=FOAM *MATERIAL,NAME=FOAM *POROUS ELASTIC .09,0.,.02E6 *FOAM .2E6,.02E6,.22E6,1.7,1. *RATE DEPENDENT 80.,1. *INITIAL CONDITIONS,TYPE=RATIO ALLN,1. *BOUNDARY BOT,1,2 N9999,1 N9999,6 *RESTART,WRITE,FREQUENCY=20 *STEP,INC=100,NLGEOM, UNSYMM=YES *STATIC 20.E-5,15.E-3,,1.E-3 *BOUNDARY 9999,2,,-.15 *MONITOR,NODE=9999,DOF=2 *PRINT,CONTACT=YES *ENERGY PRINT,FREQUENCY=5 *ENERGY FILE,FREQUENCY=5 *CONTACT PRINT,SLAVE=ASURF *CONTACT FILE,SLAVE=ASURF *EL PRINT,FREQUENCY=50,ELSET=CENT S, SINV,

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E, PE, *NODE PRINT,FREQUENCY=20 U,RF *NODE FILE,NSET=N9999 U,RF *END STEP

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Listing 1.1.7-2 *HEADING RATE DEPENDENT FOAM INDENTATION WITH ROUGH HEMISPHERICAL PUNCH *NODE,NSET=ALLN 1,0.,.3 17,0.,.1 19,0.,0. 481,.3,.3 497,.3,.1 499,.3,0. 601,.6,.3 617,.6,.1 619,.6,0. *NODE,NSET=N9999 9999,0.,.5 *NGEN,NSET=ALLN 1,17 17,19 481,497 497,499 601,617 617,619 *NSET,NSET=CL,GENERATE 1,19 *NSET,NSET=MID,GENERATE 481,499,1 *NSET,NSET=OUT,GENERATE 601,619,1 *NFILL,NSET=ALLN CL,MID,24,20 *NFILL,NSET=ALLN MID,OUT,6,20 *NSET,NSET=BOT,GENERATE 19,619,20 *ELEMENT,TYPE=CAX4,ELSET=FOAM 1,3,43,41,1 *ELGEN,ELSET=FOAM 1,15,40,10,9,2,1 *ELSET,ELSET=CENT,GENERATE 1,141,10 61,69

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*ELSET,ELSET=DBODY,GENERATE 1,111,10 *SURFACE,NAME=DSURF DBODY,S3 *RIGID BODY,ANALYTICAL SURFACE=RSURF,REF NODE=9999 *SURFACE,TYPE=SEGMENTS,NAME=RSURF START,.14142,.64142 CIRCL,-.0001,.3,0.,.5 *ELSET,ELSET=ELPRT CENT, *SOLID SECTION,ELSET=FOAM,MATERIAL=FOAM *MATERIAL,NAME=FOAM *ELASTIC 3.0E6 , 0.0 *FOAM,HARDENING=TABULAR 1.0,.02E6,.22E6,1.7,1. *FOAM HARDENING .20745E+05 , 0.000 .42916E+05 , 0.200 .75427E+05 , 0.400 .11738E+06 , 0.600 .16653E+06 , 0.800 .22000E+06 , 1.000 .24745E+06 , 1.100 .27494E+06 , 1.200 .30217E+06 , 1.300 .32890E+06 , 1.400 .35492E+06 , 1.500 .38006E+06 , 1.600 .40418E+06 , 1.700 .42720E+06 , 1.800 .44905E+06 , 1.900 .46969E+06 , 2.000 .50729E+06 , 2.200 .54008E+06 , 2.400 .56834E+06 , 2.600 .59247E+06 , 2.800 .61291E+06 , 3.000 .65083E+06 , 3.500 .67484E+06 , 4.000 .70810E+06 , 6.000 .71340E+06 , 11.000 *RATE DEPENDENT,TYPE=YIELD RATIO

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1.00, 0.0 126.0,10000.0 *CONTACT PAIR,INTERACTION=RDINT1 DSURF,RSURF *SURFACE INTERACTION,NAME=RDINT1 1., *FRICTION,ROUGH *BOUNDARY BOT,1,2 N9999,1 N9999,6 *RESTART,WRITE,FREQUENCY=20 *STEP,INC=100,NLGEOM, UNSYMM=YES *STATIC 20.E-5,15.E-3,,1.E-3 *BOUNDARY 9999,2,,-.15 *MONITOR,NODE=9999,DOF=2 *PRINT,CONTACT=YES *ENERGY PRINT,FREQUENCY=5 *CONTACT PRINT,SLAVE=DSURF,FREQUENCY=999 *CONTACT FILE,SLAVE=DSURF,FREQUENCY=999 *EL PRINT,FREQUENCY=50,ELSET=CENT S, SINV, E, PE, *NODE PRINT,FREQUENCY=20 U,RF *NODE FILE,NSET=N9999 U,RF *END STEP *STEP,INC=100,NLGEOM, UNSYMM=YES *STATIC 20.E-5,15.E-3,,1.E-3 *BOUNDARY 9999,2,,0.0 *END STEP

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Listing 1.1.7-3 *HEADING FOAM INDENTATION WITH ROUGH HEMISPHERICAL PUNCH *RESTART,TIMEMARKS=YES,WRITE,NUMBER=1 *NODE,NSET=ALLN 1,0.,0.3 19,0.,0. 481,0.3,0.3 499,0.3,0. 601,0.6,0.3 619,0.6,0. *NSET,NSET=N1 1, *NSET,NSET=N19 19, *NSET,NSET=N481 481, *NSET,NSET=N499 499, *NFILL,NSET=TOP N1,N481,12,40 *NGEN,NSET=TOP 481,601,40 *NFILL,NSET=BOT N19,N499,12,40 *NGEN,NSET=BOT 499,619,40 *NFILL,NSET=ALLN TOP,BOT,9,2 *NSET,NSET=CENTER,GENERATE 3,19 *ELEMENT,TYPE=CAX4R,ELSET=FOAM 1,3,43,41,1 *ELGEN,ELSET=FOAM 1,15,40,10,9,2,1 *ELSET,ELSET=CENT,GENERATE 1,141,10 61,69 *NODE,NSET=PUNCH 1000,0.,0.51 *ELSET,ELSET=UPPER,GEN 1,141,10

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*SOLID SECTION,ELSET=FOAM,MATERIAL=FOAM *MATERIAL,NAME=FOAM *ELASTIC 3.E6,0.0 *FOAM 1.0,0.02E6,0.22E6, *FOAM HARDENING 0.20745E5,0.0 0.42916E5,0.2 0.75427E5,0.4 0.11738E6,0.6 0.16653E6,0.8 0.22E6,1.0 0.24745E6,1.1 0.27494E6,1.2 0.30217E6,1.3 0.3289E6,1.4 0.35492E6,1.5 0.38006E6,1.6 0.40418E6,1.7 0.4272E6,1.8 0.44905E6,1.9 0.46969E6,2.0 0.50729E6,2.2 0.54008E6,2.4 0.56834E6,2.6 0.59247E6,2.8 0.61291E6,3.0 0.65083E6,3.5 0.67484E6,4.0 0.70810E6,6.0 0.71340E6,11.0 *DENSITY 500., *BOUNDARY BOT,1,2 CENTER,1 1,1 1000,1 1000,6 *AMPLITUDE,NAME=SMOOTH,TIME=TOTAL TIME, DEFINITION=SMOOTH STEP 0.0,0.0,0.6,1.

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*SURFACE,TYPE=ELEMENT,NAME=TARGET UPPER,S3 *SURFACE, NAME=IMPACTOR, TYPE=SEGMENTS START, 0.2,0.51 CIRCL, 0.,0.31, 0.,0.51 *RIGID BODY, REF NODE=1000, ANALYTICAL SURFACE =IMPACTOR *STEP *DYNAMIC,EXPLICIT ,0.6 *BOUNDARY,TYPE=DISPLACEMENT,AMPLITUDE=SMOOTH 1000,2,2,-0.2 *SURFACE INTERACTION,NAME=IMP_TO_FOAM *FRICTION,TAUMAX=0.127E6 0.95, *CONTACT PAIR,INTERACTION=IMP_TO_FOAM IMPACTOR,TARGET *FILE OUTPUT,TIMEMARKS=YES,NUMBER INTERVAL=1 *EL FILE LE,ERV *NODE FILE U, *END STEP

1.1.8 Notched beam under cyclic loading Product: ABAQUS/Standard This example illustrates the use of the nonlinear isotropic/kinematic hardening material model to simulate the response of a notched beam under cyclic loading. The model has two features to simulate plastic hardening in cyclic loading conditions: the center of the yield surface moves in stress space (kinematic hardening behavior), and the size of the yield surface evolves with inelastic deformation (isotropic hardening behavior). This combination of kinematic and isotropic hardening components is introduced to model the Bauschinger effect and other phenomena such as plastic shakedown, ratchetting, and relaxation of the mean stress. The component investigated in this example is a notched beam subjected to a cyclic 4-point bending load. The results are compared with the finite element results published by Benallal et al. (1988) and Doghri (1993). No experimental data are available.

Geometry and model The geometry and mesh are shown in Figure 1.1.8-1. Figure 1.1.8-2 shows the discretization in the vicinity of the notch, which is the region of interest in this analysis. Only one-half of the beam is modeled since the geometry and loading are symmetric with respect to the x= 0 plane. All dimensions

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are given in millimeters. The beam is 1 mm thick and is modeled with plane strain, second-order, reduced-integration elements (type CPE8R). The mesh is chosen to be similar to the mesh used by Doghri (1993). No mesh convergence studies have been performed.

Material The material properties reported by Doghri (1993) for a low-carbon ( AISI 1010), rolled steel are used in this example. A Young's modulus of E= 210 GPa and a Poisson's ratio of º = 0.3 define the elastic response of the material. The initial yield stress is ¾j0 = 200 MPa. The nonlinear evolution of the center of the yield surface is defined by the equation ®_ = C

1 pl pl (¾ ¡ ® )"¹_ ¡ ° ® "¹_ ; ¾0

where ® is the backstress, ¾ 0 is the size of the yield surface (size of the elastic range), "¹pl is the equivalent plastic strain, and C = 25.5 GPa and ° = 81 are the material parameters that define the initial hardening modulus and the rate at which the hardening modulus decreases with increasing p plastic strain, respectively. The quantity 2=3 C=° = 257 MPa defines the limiting value of the p ¹ = ® dev : ® dev ; further hardening is possible only through the change in the equivalent backstress ® size of the yield surface (isotropic hardening). The isotropic hardening behavior of this material is modeled with the exponential law pl

¾ 0 = ¾j0 + Q1 (1 ¡ e¡b "¹ ); where ¾ 0 is the size of the yield surface (size of the elastic range), Q1 = 2000 MPa is the maximum increase in the elastic range, and b = 0.26 defines the rate at which the maximum size is reached as plastic straining develops. The material used for this simulation is cold rolled. This work hardened state is represented by specifying an initial equivalent plastic strain "¹pl j0 = 0.43 (so that ¾ 0 = 411 MPa) and an initial backstress tensor 2

128 4 ®j0 = 0 0

0 ¡181 0

3 0 0 5 MPa; 53

using the *INITIAL CONDITIONS, TYPE=HARDENING option.

Loading and boundary conditions The beam is subjected to a 4-point bending load. Since only half of the beam is modeled, the model contains one concentrated load at a distance of 26 mm from the symmetry plane (see Figure 1.1.8-1). The pivot point is 42 mm from the symmetry plane. The simulation runs 3 1/2 cycles over 7 time units. In each cycle the load is ramped from zero to 675 N and back to zero. An amplitude curve is used to 1-137

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describe the loading and unloading. The increment size is restricted to a maximum of 0.125 to force ABAQUS to follow the prescribed loading/unloading pattern closely.

Results and discussion Figure 1.1.8-3 shows the final deformed shape of the beam after the 3 1/2 cycles of load; the final load on the beam is 675 N. The deformation is most severe near the root of the notch. The results reported in Figure 1.1.8-4 and Figure 1.1.8-5are measured in this area (element 166, integration point 3). Figure 1.1.8-4shows the time evolution of stress versus strain. Several important effects are predicted using this material model. First, the onset of yield occurs at a lower absolute stress level during the first unloading than during the first loading, which is the Bauschinger effect. Second, the stress-strain cycles tend to shift and stabilize so that the mean stress decreases from cycle to cycle, tending toward zero. This behavior is referred to as the relaxation of the mean stress and is most pronounced in uniaxial cyclic tests in which the strain is prescribed between unsymmetric strain values. Third, the yield surface shifts along the strain axis with cycling, whereas the shape of the stress-strain curve tends to remain similar from one cycle to the next. This behavior is known as ratchetting and is most pronounced in uniaxial cyclic tests in which the stress is prescribed between unsymmetric stress values. Finally, the hardening behavior during the first half-cycle is very flat relative to the hardening curves of the other cycles, which is typical of work hardened metals whose initial hardened state is a result of a large monotonic plastic deformation caused by a forming process such as rolling. The low hardening modulus is the result of the initial conditions on backstress, which places the center of the yield surface at a distance of q dev dev ® j0 : ®j0 = 228 MPa away from the origin of stress space. Since this distance is close to the

maximum possible distance (257 MPa), most of the hardening during the first cycle is isotropic.

These phenomena are modeled in this example primarily by the nonlinear evolution of the backstress, since the rate of isotropic hardening is very small. This behavior can be verified by conducting an analysis in which the elastic domain remains fixed throughout the analysis. Figure 1.1.8-5 shows the evolution of the direct components of the deviatoric part of the backstress tensor. The backstress components evolve most during the first cycle as the Bauschinger effect overcomes the initial hardening configuration. Only the deviatoric components of the backstress are shown so that the results obtained using ABAQUS can be compared to those reported by Doghri (1993). Since ABAQUS uses an extension of the Ziegler evolution law, a backstress tensor with nonzero pressure is produced, whereas the backstress tensor produced with the law used by Doghri (which is an extension of the linear Prager law) is deviatoric. Since the plasticity model considers only the deviatoric part of the backstress, this difference in law does not affect the other solution variables. The results shown in Figure 1.1.8-4 and Figure 1.1.8-5 agree well with the results reported by Doghri (1993).

Input files cyclicnotchedbeam.inp

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Input data. cyclicnotchedbeam_mesh.inp Element and node data.

References · Benallal, A., R. Billardon, and I. Doghri, "An Integration Algorithm and the Corresponding Consistent Tangent Operator for Fully Coupled Elastoplastic and Damage Equations, " Communications in Applied Numerical Methods, vol. 4, pp. 731-740, 1988. · Doghri, I., "Fully Implicit Integration and Consistent Tangent Modulus in Elasto-Plasticity, " International Journal for Numerical Methods in Engineering, vol. 36, pp. 3915-3932, 1993.

Figures Figure 1.1.8-1 Undeformed mesh (dimensions in mm).

Figure 1.1.8-2 Magnified view of the root of the notch.

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Figure 1.1.8-3 Deformed mesh at the conclusion of the simulation. Displacement magnification factor is 3.

Figure 1.1.8-4 Evolution of stress versus strain in the vicinity of the root of the notch.

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Figure 1.1.8-5 Evolution of the diagonal components of the deviatoric part of the backstress tensor.

Sample listings

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Listing 1.1.8-1 *HEADING NOTCHED BEAM UNDER CYCLIC LOADING *INCLUDE,INPUT=cyclicnotchedbeam_mesh.inp *SOLID SECTION,ELSET=ELALL,MATERIAL=MAT *MATERIAL,NAME=MAT *ELASTIC 2.1E5,.3 *PLASTIC,HARDENING=COMBINED,DATA TYPE=PARAMETERS 200.,25500.,81. *CYCLIC HARDENING,PARAMETERS 200.,2000.,.26 *INITIAL CONDITIONS,TYPE=HARDENING ELALL,.43,128.,-181.,53. *BOUNDARY PIVOT,2 XSYM,1 *AMPLITUDE,NAME=AMP 0.,1.,1.,0.,2.,1.,3.,0., 4.,1.,5.,0.,6.,1. *RESTART,WRITE,FREQ=1 *STEP first loading *STATIC .125,1.,,.125 *CLOAD, OP=NEW 2210,2,675. *NODE FILE,FREQ=8 U, *EL FILE,ELSET=REFINE,FREQ=8 S,E,PE,ALPHA *OUTPUT,FIELD,FREQ=8 *NODE OUTPUT U, *OUTPUT,HISTORY,FREQ=1 *ELEMENT OUTPUT,ELSET=REFINE S,E,PE,ALPHA *NODE PRINT,FREQ=0 *EL PRINT,FREQ=0 *END STEP *STEP,INC=100 additional unloadings and loadings

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*STATIC .125,6.,,.125 *CLOAD,OP=NEW,AMPLITUDE=AMP 2210,2,675. *END STEP

1.1.9 Hydrostatic fluid elements: modeling an airspring Products: ABAQUS/Standard ABAQUS/Explicit Airsprings are rubber or fabric actuators that support and contain a column of compressed air. They are used as pneumatic actuators and vibration isolators. Unlike conventional pneumatic cylinders, airsprings have no pistons, rods, or dynamic seals. This makes them better suited to handle off-center loading and shock. In addition, airsprings are considerably more flexible than other types of isolators: the airspring's inflation pressure can be changed to compensate for different loads or heights without compromising isolation efficiency. Dils (1992) provides a brief discussion of various practical uses of airsprings. In this section two examples of the analysis of a cord-reinforced rubber airspring are discussed. Static analyses are performed in ABAQUS/Standard, and quasi-static analyses are performed in ABAQUS/Explicit. The first example is a three-dimensional, half-symmetry model that uses finite-strain shell elements to model the rubber spring; three-dimensional, hydrostatic fluid elements to model the air-filled cavity; and rebars to model the multi-ply steel reinforcements in the rubber membrane. In addition, a three-dimensional, element-based rigid surface is used to define the contact between the airspring and the lateral metal bead. The second example is a two-dimensional, axisymmetric version of the first model that uses composite axisymmetric, finite-strain shell elements to model the cord-reinforced rubber spring; axisymmetric, hydrostatic fluid elements to model the air-filled cavity; and an axisymmetric, element-based rigid surface in the contact definition. The three-dimensional, 180° model uses a hyperelastic material model in conjunction with steel rebars to model the cord-reinforced rubber membrane, and the axisymmetric shell model uses a composite shell section consisting of a thin orthotropic elastic layer sandwiched between two hyperelastic layers. The orthotropic layer captures the mechanical properties of the rebar definition used in the 180° model. The orthotropic material constants have been obtained by performing simple tests on a typical element of the three-dimensional model. The three-dimensional shell model uses rebars with material properties that are initially identical to the properties of the composite shell section in the axisymmetric shell model. For comparison, ABAQUS/Standard input files for the two- and three-dimensional models that use finite-strain membrane elements along with rebars to model the cord-reinforced rubber spring, hydrostatic fluid elements to model the air-filled cavity, and an element-based rigid surface in the contact definition are also included. In all analyses the air inside the airspring cavity has been modeled as a compressible or "pneumatic" fluid satisfying the ideal gas law.

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Geometry and model The dimensions of the airspring have been inferred from the paper by Fursdon (1990). This airspring, shown in Figure 1.1.9-1, is fairly large and is used in secondary suspension systems on railway bogies. However, the shape of the airspring is typical of airsprings used in other applications. The airspring's cross-section is shown in Figure 1.1.9-2. The airspring is toroidal in shape, with an inner radius of 200 mm and an outer radius of 400 mm. The airspring has been idealized in the model as consisting of two circular, metal disks connected to each other via a rubber component. The lower disk has a radius of 200 mm, and the upper disk has a radius of 362.11 mm. The disks are initially coaxial and are 100 mm apart. The rubber component is doubly curved and toroidal in shape. The rubber is constrained in the radial direction by a circular bead 55 mm in radius that goes around the circumference of the upper disk. The rubber "hose" in the half-symmetry, three-dimensional model is modeled with 550 S4R finite-strain shell elements. The mesh in the upper hemisphere of the hose is more refined than that in the lower hemisphere, because the rubber membrane undergoes a reversal in curvature in the upper region as it contours the circular bead attached to the upper disk. The circular bead is modeled by an axisymmetric, element-based rigid surface. Contact with the rubber is enforced by defining a contact pair between this rigid surface and a surface defined on the (deformable) shell mesh in the contacting region. The metal disks are assumed to be rigid relative to the rubber component of the airspring. In the ABAQUS/Standard analysis both disks are modeled by means of boundary conditions and multi-point constraints. In the ABAQUS/Explicit analysis the lower metal disk is modeled by means of boundary conditions, while the upper disk is modeled as part of the rigid surface. The mesh of the rubber membrane and the contact master surface is shown in Figure 1.1.9-3. Three-dimensional F3D3 and F3D4 hydrostatic fluid elements are used to model the air-filled airspring cavity. The F3D4 elements are used to cover the portion of the boundary of the cavity that is associated with the rubber hose. These fluid elements share the same nodes as the S4R elements. To define the cavity completely and to ensure proper calculation of its volume (see ``Modeling fluid-filled cavities,'' Section 7.9.1 of the ABAQUS/Standard User's Manual and Section 7.4.1 of the ABAQUS/Explicit User's Manual), the F3D3 hydrostatic fluid elements are defined along the bottom and top rigid disk boundaries of the cavity, even though no displacement elements exist along those surfaces. All the hydrostatic fluid elements have been grouped into an element set named FLUID and share the cavity reference node 50000. The cavity reference node has a single degree of freedom representing the pressure inside the cavity. This node is specified on the property reference option for the hydrostatic fluid elements, *FLUID PROPERTY. Because of symmetry only half of the cavity boundary has been modeled. The cavity reference node 50000 has been placed on the model's symmetry plane, y = 0, to assure proper calculation of the cavity volume. Figure 1.1.9-4 shows the mesh of the airspring's cavity. To facilitate comparisons, the two-dimensional axisymmetric model uses the same cross-sectional mesh refinement as the 180° model. For the shell model the rubber component is modeled with 25 SAX1 shell elements. For the membrane model the SAX1 elements are replaced with either MAX1 elements or MGAX1 elements. The circular bead is modeled in ABAQUS/Standard with a segmented analytical rigid surface and in ABAQUS/Explicit with an element-based rigid surface constructed of RAX2 rigid elements. Contact with the hose is enforced by defining a contact pair between this rigid

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surface and a surface defined on the (deformable) shell mesh in the contacting region. The rigid metal disks are again modeled by boundary conditions and multi-point constraints in ABAQUS/Standard; in ABAQUS/Explicit the lower rigid metal disk is again modeled by boundary conditions, and the upper rigid metal disk is modeled as part of the rigid body. Two-dimensional FAX2 hydrostatic fluid elements are used to model the airspring cavity. The mesh of the rubber membrane and the contact master surface is shown in Figure 1.1.9-5, and the mesh of the cavity is shown in Figure 1.1.9-6.

Symmetry boundary conditions and initial shell curvature Symmetry has been exploited in the three-dimensional airspring model, and the plane y = 0 has been made a plane of symmetry. Since S4R shell elements are true curved shell elements, accurate definition of the initial curvature of the surface being modeled is required, especially on the plane of symmetry. If the user does not provide this information by specifying the normal to the surface at the shell nodes, ABAQUS will estimate the normal direction based on the coordinates of the surrounding nodes on the shell. Normals computed in this fashion will be inaccurate on the symmetry plane: they will have out-of-plane components, which will lead to convergence difficulties in ABAQUS/Standard and inaccurate results. To avoid these difficulties, direction cosines have been specified for all shell nodes in the model.

Material properties The walls of an airspring's rubber component are made from plies of symmetrically placed, positively and negatively oriented reinforcement cords. The walls of an actual component are made of several such layers. However, for the purposes of the three-dimensional example problem being considered, the airspring's wall is taken to be a rubber matrix with a single 6-mm-thick symmetric layer of positively and negatively oriented cords. The cords are modeled by uniformly spaced skew rebars in the shell elements. The rebars are assumed to be made of steel. The rubber is modeled as an incompressible Mooney-Rivlin (hyperelastic) material with C10 = 3.2 MPa and C01 = 0.8 MPa, and the steel is modeled as a linear elastic material with E = 210.0 GPa and º = 0.3. Skew rebar orientations in shell elements are defined by giving the angle between the local 1-axis and the rebars. The default local 1-direction is the projection of the global x-axis onto the shell surface (see ``Conventions,'' Section 1.2.2 of the ABAQUS/Standard User's Manual and the ABAQUS/Explicit User's Manual). It is for this reason, and to make the rebar definition uniform for all elements, that the axis of revolution of the airspring model has been chosen to be the global x-axis. Two rebar sets, PLSBAR and MNSBAR, have been defined with orientation angles of 18° and -18°, respectively. The cross-sectional area of the rebars is 1 mm 2, and they are spaced every 3.5 mm in the shell surface. The above rebar specification is simplified and somewhat unrealistic. A more realistic simulation would require different rebar definitions in each ring of elements in the airspring model. The reinforced plies used in the manufacture of the airspring are located in an initially cylindrical tube with uniform rebar angles. However, the transformation of these layers from a cylindrical geometry to a toroidal one gives the airspring a variable rebar angle and rebar spacing that is dependent on the radius from the axis of revolution of the torus and on the initial rebar angle (see Fursdon, 1990). In the axisymmetric shell model the airspring walls are modeled by a three-layer composite shell

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section. The two outer layers are each 2.5 mm thick and made up of the same Mooney-Rivlin material that is used in the 180° model. The middle "rebar" layer is 1 mm thick and is made up of an orthotropic elastic material that captures the mechanical behavior of the positively and negatively oriented rebar definition used in the three-dimensional airspring model. The plane stress orthotropic engineering constants are obtained by looking at the response of a typical element in the three-dimensional model (element 14) subjected to uniaxial extensions along the local 1- and 2-directions. Using a shell thickness of 1 mm, the in-plane states of stress and strain resulting from these two tests are "2 "1 ¾1 (MPa) ¾2 (MPa) Test -2 -2 1 1-direction 1.00 ´ 10 -8.75 ´ 10 2.48 ´ 10 -2.41 ´ 10-5 2-direction -1.05 ´ 10-3 1.00 ´ 10-2 -5.96 ´ 10-6 2.86 ´ 10-1 For a plane-stress orthotropic material the in-plane stress and strain components are related to each other as follows: µ

"1 "2

¶

=

µ

1=E1 ¡º21 =E2

¡º12 =E1 1=E2

¶µ

¾1 ¾2

¶

;

where E1 , E2 , º12 , and º21 are engineering constants. Solving for these constants using the above stress-strain relation and the results of the two uniaxial tests yields E1 = 2:48 £ 103 MPa E2 = 2:86 £ 101 MPa

º12 = 9:1 º21 = 0:1 The remaining required engineering constants--G12 , G13 , and G23 --play no role in the rebar layer definition. Consequently, they have been arbitrarily set to be equal to the shear modulus of the rubber, which is given by 2(C10 + C01 ). For the axisymmetric membrane model the bulk material is chosen to have the same material properties (Mooney-Rivlin hyperelastic) as those used in the 180° model and the axisymmetric shell model. The rebar parameters and material properties are chosen such that they capture the initial material properties of the sandwiched steel layer in the axisymmetric shell model. The principal material directions do not rotate in the axisymmetric shell model (they are the default element basis directions--the meridional and the hoop directions, respectively). However, they do rotate with finite strain in the axisymmetric membrane model as a result of the use of rebars. Initial stresses are applied to the rebars in the axisymmetric membrane model. In all analyses the air inside the airspring cavity has been modeled as an ideal gas with the following properties: its reference density is 1.774 kg/m 3 at a room temperature of 27° C and an ambient pressure of 101.36 kPa.

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Loading The airspring is first pressurized to 506.6 ´ 103 kPa (5 atms) while holding the upper disk fixed. This pressure is applied by prescribing degree of freedom 8 at the cavity reference node using the *BOUNDARY option. In this case the air volume is adjusted automatically to fill the cavity. In the next step the *BOUNDARY, OP=NEW option is used to remove the boundary condition on the pressure degree of freedom, thus sealing the cavity with the current air volume. In addition, during this step in the ABAQUS/Standard analysis the boundary condition on the vertical displacement degree of freedom of the rigid body reference node is removed, and in its place a downward load of 150 kN is applied. The next two steps in the ABAQUS/Standard axisymmetric model analysis are linear perturbation steps to test the axial stiffness of the airspring with the cavity pressure allowed to vary (closed cavity conditions) and with it fixed. The three-dimensional ABAQUS/Standard analysis contains three linear perturbation steps, all under variable cavity pressure (closed cavity) conditions: the first to test the axial stiffness of the airspring, the second to test its lateral stiffness, and the third to test the rotational stiffness for rocking motion in the symmetry plane. The ABAQUS/Explicit axisymmetric analysis concludes with a nonlinear step in which the airspring is subjected to a downward displacement of 75 mm. The ABAQUS/Standard axisymmetric analysis concludes with a nonlinear step in which the airspring is compressed by increasing the downward load to 240.0 kN. The three-dimensional analyses conclude with a nonlinear step in which the airspring is subjected to a lateral displacement of 20 mm.

Results and discussion Figure 1.1.9-7 and Figure 1.1.9-8 show displaced plots of the axisymmetric shell model at the end of the pressurization step, Step 1. It is of interest to compare the results from this model with those from the 180° model to validate the material model that was used for the rebar reinforcements in the axisymmetric model. A close look at the nodal displacements reveals that the deformation is practically identical for the two models. Moreover, the axial reaction force at the rigid body reference node is 156 kN for the axisymmetric model and 155 kN for the 180° model (after multiplication by a factor of 2). The cavity volume predicted by the axisymmetric model is 8.22 ´ 10-2 m3 versus 8.34 ´ 10-2 m3 for the 180° model (again, after multiplication by a factor of 2).

ABAQUS/Standard results Figure 1.1.9-9 shows a displaced plot of the axisymmetric model at the end of Step 2. The spring has undergone an upward displacement along its axis because the downward load being applied to it is slightly smaller than the axial reaction force at the end of the previous step. Once again, comparison between the results from both models reveals almost identical solutions. Linearized stiffnesses for the airspring are obtained from the linear perturbation steps. The stiffness is computed by dividing the relevant reaction force at the rigid body reference node by the appropriate displacement, which gives the airspring's axial stiffness under variable cavity pressure conditions as 826 kN/m for the axisymmetric model. From the results of Step 4 the axial stiffness under "fixed"

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cavity pressure conditions is 134 kN/m. The difference in axial stiffness between these two cases (a factor of 6) is the result of differences in cavity pressure experienced during axial compression. Under variable cavity pressure conditions, a fixed mass of fluid (air) is contained in a cavity whose volume is decreasing; thus, the cavity pressure increases. Under "fixed" cavity pressure conditions, the pressure is prescribed as a constant value for the step. For the 180° model the predicted stiffnesses are as follows: the axial stiffness is 821 kN/m, the lateral stiffness is 3.31 MN/m, and the rotational stiffness is 273 kN/m. Figure 1.1.9-10 shows a series of displaced plots associated with the compression of the axisymmetric airspring model during Step 5. Figure 1.1.9-11 shows the load-deflection curve corresponding to this deformation. The response of the airspring is slightly nonlinear; consequently, there is good agreement between the axial stiffness obtained with the linear perturbation analysis (Step 3) and that obtained from the slope of the load-displacement curve. Figure 1.1.9-12 shows a plot of cavity pressure versus the downward displacement of the rigid body in Step 5, which shows that the gauge pressure in the cavity increases by approximately 50% during this step. This pressure increase substantially affects the deformation of the airspring structure and cannot be specified as an externally applied load during the step since it is an unknown quantity. Figure 1.1.9-13 shows a plot of cavity volume versus the downward displacement of the rigid body in Step 5. The corresponding results for the axisymmetric membrane model are in good agreement with the above results. Figure 1.1.9-14 shows the displaced plot of the 180° model at the end of Step 6, in which a lateral displacement was applied to the airspring. Figure 1.1.9-15 shows the load-deflection curve corresponding to this deformation. Once again, good agreement is found between the lateral stiffness predicted from the linear perturbation analysis (Step 4) and that obtained from the slope of the load-displacement curve.

ABAQUS/Explicit results Figure 1.1.9-16 shows a series of displaced plots associated with the compression of the axisymmetric model during the second step. Figure 1.1.9-17 shows the load-deflection curve corresponding to this deformation. Although the displacement of the rigid body was applied over a short enough time period to cause significant inertial effects in the model, there is still good agreement between the slope of the load-displacement curve in this example and the slope of the load-displacement curve for the same analysis performed statically in ABAQUS/Standard. Figure 1.1.9-18 shows a plot of cavity pressure versus the downward displacement of the rigid body in Step 2, which shows that the gauge pressure in the cavity increases by approximately 50 percent during this step. This pressure increase substantially affects the deformation of the airspring structure and cannot be specified as an externally applied load during the step since it is an unknown quantity. Figure 1.1.9-19 shows a plot of cavity volume versus the downward displacement of the rigid body in Step 2. Figure 1.1.9-20 shows the displaced plot of the 180° model at the end of Step 2, in which a lateral displacement was applied to the airspring. Figure 1.1.9-21 shows the load-deflection curve corresponding to this deformation. Although there is a significant amount of noise that results from the contact conditions and the coarseness of the mesh, the load-deflection curve shows good agreement between the analysis performed quasi-statically in ABAQUS/Explicit and the same analysis performed statically in ABAQUS/Standard. The load versus displacement curve shown has been smoothed to

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eliminate some of the noise.

Input files hydrofluidairspring_3d_shell.inp Three-dimensional ABAQUS/Standard model using shell elements. hydrofluidairspring_axisymm.inp Axisymmetric ABAQUS/Standard model. airspring_s4r.inp Three-dimensional ABAQUS/Explicit model using shell elements. airspring_sax1.inp Axisymmetric ABAQUS/Explicit model. hydrofluidairspring_3d_mem.inp Three-dimensional ABAQUS/Standard model using membrane elements. hydrofluidairspring_max1.inp ABAQUS/Standard analysis using MAX1 elements with rebars. hydrofluidairspring_mgax1.inp ABAQUS/Standard analysis using MGAX1 elements with rebars.

References · Dils, M., ``Air Springs vs. Air Cylinders,'' Machine Design, May 7, 1992. · Fursdon, P. M. T., ``Modelling a Cord Reinforced Component with ABAQUS,'' 6th UK ABAQUS User Group Conference Proceedings, 1990.

Figures Figure 1.1.9-1 A cord reinforced airspring.

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Figure 1.1.9-2 The airspring model cross-section.

Figure 1.1.9-3 180° model: mesh of the rubber membrane and partial view of the axisymmetric contact master surface.

Figure 1.1.9-4 180° model: mesh of the airspring cavity.

Figure 1.1.9-5 Axisymmetric model: mesh of the rubber membrane and the contact master surface.

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Figure 1.1.9-6 Axisymmetric model: mesh of the airspring cavity.

Figure 1.1.9-7 Axisymmetric ABAQUS/Standard model: deformed configuration at the end of Step 1.

Figure 1.1.9-8 Axisymmetric ABAQUS/Explicit model: deformed configuration at the end of Step 1.

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Figure 1.1.9-9 Axisymmetric ABAQUS/Standard model: deformed configuration at the end of Step 2.

Figure 1.1.9-10 Axisymmetric ABAQUS/Standard model: progressive deformed configurations during Step 5.

Figure 1.1.9-11 Axisymmetric ABAQUS/Standard model: load-displacement curve for Step 5.

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Figure 1.1.9-12 Axisymmetric ABAQUS/Standard model: cavity pressure versus downward displacement in Step 5.

Figure 1.1.9-13 Axisymmetric ABAQUS/Standard model: cavity volume versus downward displacement in Step 5.

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Figure 1.1.9-14 180° ABAQUS/Standard model: deformed configuration at the end of Step 6.

Figure 1.1.9-15 180° ABAQUS/Standard model: load-displacement curve for Step 6.

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Figure 1.1.9-16 Axisymmetric ABAQUS/Explicit model: progressive deformed configurations during Step 2.

Figure 1.1.9-17 Axisymmetric ABAQUS/Explicit model: load-displacement curve for Step 2.

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Figure 1.1.9-18 Axisymmetric ABAQUS/Explicit model: cavity pressure versus downward displacement in Step 2.

Figure 1.1.9-19 Axisymmetric ABAQUS/Explicit model: cavity volume versus downward displacement in Step 2.

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Figure 1.1.9-20 180° ABAQUS/Explicit model: deformed configuration at the end of Step 2.

Figure 1.1.9-21 180° ABAQUS/Explicit model: load-displacement for Step 2.

Sample listings 1-157

Static Stress/Displacement Analyses

Listing 1.1.9-1 *HEADING AIRSPRING -- 180 DEG MODEL *RESTART,WRITE,OVERLAY *NODE, NSET = FPLANE 1,2.E-2, 0.,0. 4,2.E-2, 0.,20.E-2,0.,0.,1.0 6,0.,0.,20.E-2,0.,0.,1.0 26,0.,0.,40.E-2,0.,0.,-1.0 32,6.293E-2,0.,37.771E-2,-0.58431,0.,-0.81153 38,10.698E-2,0.,36.211E-2,0.,0.,-1.0 39,11.35E-2,0.,36.211E-2,0.,0., -1.0 40,12.E-2,0.,36.211E-2,0.,0.,-1.0 41,12.E-2,0.,0. *NGEN, LINE = C, NSET = FPLANE 6, 26, 2, , 0. , 0. , 30.E-2 , 0., 1., 0 26, 32, 1, , 0. , 0. , 30.E-2 32, 38, 1, , 10.698E-2, 0. , 43.211E-2 *NCOPY, OLD SET = FPLANE, CHANGE NUMBER = 100, SHIFT, MULTIPLE = 22 0., 0., 0. 0., 0., 0., 1., 0., 0., 8.18181818 *NODE, NSET = RIGID 10000, 12.E-2, 0. , 0. *NODE, NSET = CAVITY 50000, 9.E-2 , 0. , 0. ** ** RUBBER SPRING ** *ELEMENT, TYPE = S4R 2, 4, 6, 106, 104 26, 26, 27, 127, 126 *ELGEN, ELSET = SPRING1 2, 11, 2, 2, 22, 100, 100 *ELGEN, ELSET = SPRING2 26, 14, 1, 1, 22, 100, 100 *ELSET,ELSET=SPRING SPRING1,SPRING2 *SHELL SECTION, MATERIAL = RUBBER, ELSET = SPRING 6.0E-3, *HOURGLASS STIFFNESS ,,,100.

1-158

Static Stress/Displacement Analyses

*MATERIAL, NAME = RUBBER *HYPERELASTIC, POLYNOMIAL, N = 1 3.2E6, 0.8E6 *REBAR, ELEMENT = SHELL, MATERIAL = STEEL, GEOMETRY = SKEW, NAME = PLSREBAR SPRING, 1.E-6, 3.5E-3, 0.0, 18.0 *REBAR, ELEMENT = SHELL, MATERIAL = STEEL, GEOMETRY = SKEW, NAME = MNSREBAR SPRING, 1.E-6, 3.5E-3, 0.0, -18.0 *MATERIAL, NAME = STEEL *ELASTIC 2.1E11, 0.3 ** ** CONTACT DEFINITION ** *SURFACE, NAME = ASURF SPRING2,SNEG *RIGID BODY,ANALYTICAL SURFACE=RSURF, REF NODE=10000 *SURFACE,TYPE=REVOLUTION,NAME=RSURF, FILLET RADIUS=0.5E-2 0., 0., 0., 1., 0., 0. START, 47.E-2 , 5.198E-2 LINE, 41.711E-2, 5.198E-2 CIRCL, 36.211E-2, 10.698E-2, 41.711E-2, 10.698E-2 LINE, 36.211E-2, 11.9E-2 *CONTACT PAIR, INTERACTION = SMOOTH ASURF, RSURF *SURFACE INTERACTION, NAME = SMOOTH ** ** FLUID ELEMENTS ** *ELEMENT, TYPE = F3D3, ELSET = FLUID 6001, 1, 4, 104 6040, 40, 41, 140 *ELGEN, ELSET = FLUID 6001, 22, 100, 100 6040, 22, 100, 100 *ELEMENT, TYPE = F3D4, ELSET = FLUID 6002, 4, 6, 106, 104 6026, 26, 27, 127, 126

1-159

Static Stress/Displacement Analyses

*ELGEN, ELSET = FLUID 6002, 11, 2, 2, 22, 100, 100 6026, 14, 1, 1, 22, 100, 100 *PHYSICAL CONSTANTS, ABSOLUTE ZERO=-273.16 *FLUID PROPERTY, REF NODE=50000, TYPE=PNEUMATIC, ELSET=FLUID, AMBIENT=101.36E+3 *FLUID DENSITY, PRESSURE=0.0, TEMPERATURE=27.0 1.774, ** ** KINEMATIC CONSTRAINTS ** *NSET, NSET = N1M1, GENERATE 101, 2201, 100 *NSET, NSET = N4, GENERATE 4, 2204, 100 *NSET, NSET = N6, GENERATE 6, 2206, 100 *NSET, NSET = N40, GENERATE 40, 2240, 100 *NSET, NSET = N41, GENERATE 41, 2241, 100 *NSET, NSET = YPLN1, GENERATE 4, 26, 2 26, 40, 1 *NSET, NSET = YPLN2, GENERATE 2204, 2226, 2 2226, 2240, 1 *MPC TIE, N1M1, 1 TIE, N41, 10000 BEAM, N40, 10000 *BOUNDARY 1,1,3 N4,1,6 N6,2,3 YPLN1, 2 YPLN1, 4 YPLN1, 6 YPLN2, 2 YPLN2, 4 YPLN2, 6 RIGID,1,6

1-160

Static Stress/Displacement Analyses

** ** STEP 1 - INFLATION STEP (5 ATMOSPHERES) ** *STEP, NLGEOM, INC=20 *STATIC 0.02,1.0 *MONITOR, NODE=26, DOF=3 *PRINT, CONTACT = YES *BOUNDARY CAVITY,8,8, 5.066E5 *EL PRINT, FREQUENCY=99 *NODE PRINT, NSET=CAVITY, FREQUENCY=99 PCAV, CVOL *NODE PRINT, NSET=RIGID, FREQUENCY=99 U, RF, *NODE FILE, NSET=CAVITY PCAV, CVOL *OUTPUT,FIELD *NODE OUTPUT,NSET=CAVITY PCAV,CVOL *NODE FILE, NSET=RIGID U, RF, CF, *OUTPUT,FIELD *NODE OUTPUT,NSET=RIGID U, RF, CF, *END STEP ** ** STEP 2 - SEAL CAVITY, RELEASE GRIP ON ** RIGID SURFACE, AND LOAD RIGID SURFACE ** *STEP, NLGEOM, INC=100 *STATIC 1.0,1.0 *MONITOR, NODE=10000, DOF=1 *BOUNDARY, OP=NEW 1,1,3 N4,1,6 N6,2,3

1-161

Static Stress/Displacement Analyses

YPLN1, 2 YPLN1, 4 YPLN1, 6 YPLN2, 2 YPLN2, 4 YPLN2, 6 RIGID,2,6 *CLOAD RIGID,1,-75.E3 *END STEP ** ** STEP 3 - PERTURBATION STEP 1 ** UNIT AXIAL DISPLACEMENT ** *STEP, PERTURBATION *STATIC *BOUNDARY RIGID,1,1, 1.E-3 *NODE PRINT, NSET=CAVITY PCAV, CVOL *NODE PRINT, NSET=RIGID U, RF, *END STEP ** ** STEP 4 - PERTURBATION STEP 2 ** UNIT LATERAL DISPLACEMENT ** *STEP, PERTURBATION *STATIC *BOUNDARY RIGID,3,3, 1.E-3 *NODE PRINT, NSET=CAVITY PCAV, CVOL *NODE PRINT, NSET=RIGID U, RF, *END STEP ** ** STEP 5 - PERTURBATION STEP 3 ** UNIT ROTATION ** *STEP, PERTURBATION

1-162

Static Stress/Displacement Analyses

*STATIC *BOUNDARY RIGID,5,5, 1.E-3 *NODE PRINT, NSET=CAVITY PCAV, CVOL *NODE PRINT, NSET=RIGID U, RF, *END STEP ** ** STEP 6 - NONLINEAR LATERAL ** LOAD DEFLECTION CURVE ** *STEP, NLGEOM, INC=100 *STATIC 0.1, 1.0 *CONTROLS,PARAMETERS=FIELD ,1.0 *MONITOR, NODE=10000, DOF=3 *BOUNDARY RIGID,3,3, 2.E-2 *EL PRINT,REBAR S,E,RBANG,RBROT *END STEP

1-163

Static Stress/Displacement Analyses

Listing 1.1.9-2 *HEADING AIRSPRING -- AXISYMMETRIC MODEL *RESTART,WRITE *NODE 1, 0. , 2.E-2 4, 20.E-2 , 2.E-2 6, 20.E-2 , 0. 26, 40.E-2 , 0. 32, 37.771E-2, 6.293E-2 38, 36.211E-2, 10.698E-2 39, 36.211E-2, 11.35E-2 40, 36.211E-2, 12.E-2 500, 0. , 9.E-2 1000, 0. , 12.E-2 *NSET, NSET = RIGID 1000, *NSET, NSET = CAVITY 500, *NGEN, LINE = C 6, 26, 2, , 30.E-2 , 0. , 0., 0., 0., 1. 26, 32, 1, , 30.E-2 , 0. , 0. 32, 38, 1, , 43.211E-2, 10.698E-2, 0. ** ** RUBBER SPRING ** *ELEMENT, TYPE = SAX1 2, 4, 6 26, 26, 27 *ELGEN, ELSET = SPRING 2, 11, 2, 2 26, 14, 1, 1 *SHELL SECTION, COMPOSITE, ELSET = SPRING 2.5E-3, 3, RUBBER 1.0E-3, 3, STEEL 2.5E-3, 3, RUBBER *MATERIAL, NAME = RUBBER *HYPERELASTIC, POLYNOMIAL, N = 1 3.2E6, 0.8E6 *MATERIAL, NAME = STEEL *ELASTIC, TYPE=LAMINA 2.48E9, 2.86E7, 9.105, 8.0E6, 8.0E6, 8.0E6

1-164

Static Stress/Displacement Analyses

** ** CONTACT ELEMENTS ** *ELSET,ELSET=ECON,GENERATE 26,39,1 *RIGID BODY,ANALYTICAL SURFACE=BSURF,REF NODE=1000 *SURFACE,TYPE=SEGMENTS,FILLET RADIUS=0.5E-2, NAME=BSURF START, 41.711E-2, 5.198E-2 CIRCL, 36.211E-2, 10.698E-2, 41.711E-2, 10.698E-2 LINE, 36.211E-2, 12.E-2 *SURFACE,NAME=ASURF ECON,SPOS *CONTACT PAIR,INTERACTION=SMOOTH ASURF,BSURF *SURFACE INTERACTION,NAME=SMOOTH ** ** FLUID ELEMENTS ** *ELEMENT, TYPE = FAX2, ELSET = FLUID 201, 1, 4 202, 4, 6 226, 26, 27 299, 40, 1000 *ELGEN, ELSET = FLUID 202, 11, 2, 2 226, 14, 1, 1 *PHYSICAL CONSTANTS, ABSOLUTE ZERO=-273.16 *FLUID PROPERTY, REF NODE=500, TYPE=PNEUMATIC, ELSET=FLUID, AMBIENT=101.36E+3 *FLUID DENSITY, PRESSURE=0.0, TEMPERATURE=27.0 1.774, ** *MPC BEAM, 40, 1000 *BOUNDARY 1,1,3 4,1,6 6,1,1 RIGID,1,6 ** ** STEP 1 - INFLATION STEP (5 ATMOSPHERES)

1-165

Static Stress/Displacement Analyses

** *STEP, NLGEOM *STATIC 0.1,1.0 *MONITOR, NODE=26, DOF=1 *PRINT, CONTACT = YES *BOUNDARY CAVITY,8,8, 5.066E5 *CONTACT PRINT,SLAVE=ASURF *CONTACT FILE,SLAVE=ASURF *OUTPUT,FIELD *CONTACT OUTPUT,VARIABLE=PRESELECT,SLAVE=ASURF *EL PRINT, FREQUENCY=99 *NODE PRINT, NSET=CAVITY, FREQUENCY=99 PCAV, CVOL *NODE PRINT, NSET=RIGID, FREQUENCY=99 U, RF *NODE FILE, NSET=CAVITY PCAV, CVOL *OUTPUT,FIELD *NODE OUTPUT,NSET=CAVITY PCAV,CVOL *NODE FILE, NSET=RIGID U, RF CF, *OUTPUT,FIELD *NODE OUTPUT,NSET=RIGID U,RF CF, *END STEP ** ** STEP 2 - SEAL CAVITY, RELEASE GRIP ON ** RIGID SURFACE, ** AND LOAD RIGID SURFACE ** *STEP, NLGEOM *STATIC 1.0, 1.0 *MONITOR, NODE=1000, DOF=2 *BOUNDARY, OP=NEW 1,1,3 4,1,6 6,1,1

1-166

Static Stress/Displacement Analyses

RIGID,1 RIGID,6 *CLOAD RIGID,2,-150.E3 *END STEP ** ** STEP 3 - PERTURBATION STEP 1 ** UNIT AXIAL DISPLACEMENT WITH CAVITY ** PRESS FREE ** *STEP, PERTURBATION *STATIC *BOUNDARY RIGID, 2, 2, 1.E-3 *NODE PRINT, NSET=CAVITY PCAV, CVOL *NODE PRINT, NSET=RIGID U, RF *END STEP ** ** STEP 4 - PERTURBATION STEP 2 ** UNIT LOAD WITH CAVITY PRESS FIXED ** *STEP, PERTURBATION *STATIC *BOUNDARY CAVITY, 8, 8, 0 RIGID, 2, 2, 1.E-3 *NODE PRINT, NSET=CAVITY PCAV, CVOL *NODE PRINT, NSET=RIGID U, RF *END STEP ** ** STEP 5 - NONLINEAR LOAD DEFLECTION CURVE ** *STEP, NLGEOM, INC=25 *STATIC 0.1,1.0 *CLOAD RIGID,2,-240.E3 *END STEP

1-167

Static Stress/Displacement Analyses

Listing 1.1.9-3 *HEADING AIRSPRING -- 180 DEG MODEL *RESTART,WRITE,NUMBER INTERVAL=10 *NODE, NSET = FPLANE 1,2.E-2,0.,0. 4,2.E-2,0.,20.E-2,0.,0.,1.0 6,0.,0.,20.E-2,0.,0.,1.0 26,0.,0.,40.E-2,0.,0.,-1.0 32,6.293E-2,0.,37.771E-2,-0.58431,0.,-0.81153 38,10.698E-2,0.,36.211E-2,0.,0.,-1.0 39,11.35E-2,0.,36.211E-2,0.,0.,-1.0 40,12.E-2,0.,36.211E-2,0.,0.,-1.0 41,12.E-2,0.,0. *NGEN, LINE = C, NSET = FPLANE 6, 26, 2, , 0., 0., 30.E-2, 0., 1., 0. 26, 32, 1, , 0., 0., 30.E-2 32, 38, 1, , 10.698E-2, 0., 43.211E-2 *NCOPY,OLDSET=FPLANE,CHANGE NUMBER=100,SHIFT, MULTIPLE=22,NEWSET=NALL 0., 0., 0. 0., 0., 0., 1., 0., 0., 8.18181818 *NODE, NSET = RIGID 11000, 12.E-2, 0. , 0. *NODE, NSET = CAVITY 50000, 9.E-2 , 0. , 0. ** ** RUBBER SPRING ** *ELEMENT, TYPE = S4R 2, 4, 6, 106, 104 26, 26, 27, 127, 126 *ELGEN, ELSET = SPRING1 2, 11, 2, 2, 22, 100, 100 *ELGEN, ELSET = SPRING2 26, 14, 1, 1, 22, 100, 100 *ELSET,ELSET=SPRING SPRING1,SPRING2 *SHELL SECTION, MATERIAL = RUBBER, ELSET = SPRING 6.0E-3, *MATERIAL, NAME = RUBBER *HYPERELASTIC, POLYNOMIAL, N = 1

1-168

Static Stress/Displacement Analyses

3.2E6, 0.8E6 *DENSITY 1000., *REBAR,ELEMENT=SHELL,MATERIAL=STEEL,GEOMETRY=SKEW, NAME=PLSREBAR SPRING, 1.E-6, 3.5E-3, 0., 18.0 *REBAR,ELEMENT=SHELL,MATERIAL=STEEL,GEOMETRY=SKEW, NAME=MNSREBAR SPRING, 1.E-6, 3.5E-3, 0., -18.0 *MATERIAL, NAME = STEEL *ELASTIC 2.1E11, 0.3 *DENSITY 7.8E3, ** ** FLUID ELEMENTS ** *ELEMENT, TYPE = F3D3, ELSET = FLUID 6001, 1, 4, 104 6040, 40, 41, 140 *ELGEN, ELSET = FLUID 6001, 22, 100, 100 6040, 22, 100, 100 *ELEMENT, TYPE = F3D4, ELSET = FLUID 6002, 4, 6, 106, 104 6026, 26, 27, 127, 126 *ELGEN, ELSET = FLUID 6002, 11, 2, 2, 22, 100, 100 6026, 14, 1, 1, 22, 100, 100 *PHYSICAL CONSTANTS, ABSOLUTE ZERO=-273.16 *FLUID PROPERTY, REF NODE=50000, TYPE=PNEUMATIC, ELSET=FLUID, AMBIENT=101.36E+3 *FLUID DENSITY, PRESSURE=0.0, TEMPERATURE=27.0 1.774, ** ** RIGID BODY/SURFACE ** *NODE, NSET = RIGID 11000, 12.E-2, 0., 0. *NODE,NSET=RIGNODE 11001, 12.E-2, 0., 36.511E-2 11021, 6.5E-2, 0., 42.011E-2

1-169

Static Stress/Displacement Analyses

12000, 12.E-2, 0., 42.011E-2 *NGEN,NSET=RIGNODE,LINE=C 11001, 11021, 1, 12000 *NCOPY,OLDSET=RIGNODE,CHANGE NUMBER=100,SHIFT, MULTIPLE=22,NEWSET=NALL 0., 0., 0. 0., 0., 0., 1., 0., 0., 8.18181818 *ELEMENT,TYPE=R3D4,ELSET=RIGELEM 11001, 11001,11101,11102,11002 *ELGEN,ELSET=RIGELEM 11001, 22,100,100, 20,1,1 *ELEMENT,TYPE=MASS,ELSET=MASS_RIGID 11000, 11000 *MASS,ELSET=MASS_RIGID .5, ** ** KINEMATIC CONSTRAINTS ** *NSET, NSET = N1M1, GENERATE 101, 2201, 100 *NSET, NSET = N4, GENERATE 4, 2204, 100 *NSET, NSET = N6, GENERATE 6, 2206, 100 *NSET, NSET = N40, GENERATE 40, 2240, 100 *NSET, NSET = N41, GENERATE 41, 2241, 100 *NSET, NSET = YPLN1, GENERATE 4, 26, 2 26, 39, 1 *NSET, NSET = YPLN2, GENERATE 2204, 2226, 2 2226, 2239, 1 *BOUNDARY 1,1,6 N4,1,6 N6,2,3 YPLN1, 2 YPLN1, 4 YPLN1, 6 YPLN2, 2 YPLN2, 4

1-170

Static Stress/Displacement Analyses

YPLN2, 6 RIGID,1,6 ** ** STEP 1 - INFLATION STEP (5 ATMOSPHERES) ** *AMPLITUDE,NAME=RAMP,DEFINITION=SMOOTH STEP 0.,0., 1.E-2,1.0 *AMPLITUDE,NAME=RAMPSMOOTH,DEFINITION=SMOOTH STEP 0.,0., 2.E-2,1.0 *SURFACE,TYPE=ELEMENT, NAME = ASURF ASURF,SNEG *SURFACE,TYPE=ELEMENT, NAME = RSURF RIGELEM,SPOS *RIGID BODY,REFNODE=1,PINNSET=N1M1 *RIGID BODY,REFNODE=11000,ELSET=RIGELEM, TIENSET=RIG *STEP *DYNAMIC,EXPLICIT ,1.E-2 *BOUNDARY,AMPLITUDE=RAMP CAVITY,8,8, 5.066E5 ** ** CONTACT DEFINITION ** *NSET,NSET=RIG N40,N41 *ELSET,ELSET=ASURF,GEN 26,2126,100 27,2127,100 28,2128,100 29,2129,100 30,2130,100 31,2131,100 32,2132,100 33,2133,100 34,2134,100 35,2135,100 36,2136,100 37,2137,100 38,2138,100 *CONTACT PAIR, INTERACTION = SMOOTH ASURF, RSURF *SURFACE INTERACTION, NAME = SMOOTH

1-171

Static Stress/Displacement Analyses

** ** OUTPUT REQUESTS ** *FILE OUTPUT,NUMBER INTERVAL=2 *NODE FILE, NSET=CAVITY PCAV, CVOL *NODE FILE, NSET=RIGID U, RF, *HISTORY OUTPUT,TIME=1.E-5 *ENERGY HISTORY ALLKE,ALLIE,ALLAE,ALLSE,ETOTAL,DT,DTFB,ALLVD *ELSET,ELSET=TOP 39,2139,1039 *EL HISTORY,ELSET=TOP S,E *EL HISTORY,ELSET=TOP,REBAR=PLSREBAR S,E,RBANG,RBROT *NODE HISTORY,NSET=CAVITY PCAV,CVOL *NODE HISTORY,NSET=RIG U,RF *NODE HISTORY,NSET=RIGID U,RF *END STEP ** ** STEP 2 - SEAL CAVITY, LOAD FURTHER ** *STEP *DYNAMIC,EXPLICIT ,2.E-2 *BOUNDARY, OP=NEW,AMPLITUDE=RAMPSMOOTH 1,1,6 N4,1,6 N6,2,3 YPLN1, 2 YPLN1, 4 YPLN1, 6 YPLN2, 2 YPLN2, 4 YPLN2, 6 RIGID,1,2 RIGID,4,6

1-172

Static Stress/Displacement Analyses

RIGID,3,3, 2.E-2 *END STEP

1-173

Static Stress/Displacement Analyses

Listing 1.1.9-4 *HEADING AIRSPRING -- AXISYMMETRIC MODEL *NODE 1, 0. , 2.E-2 4, 20.E-2 , 2.E-2 6, 20.E-2 , 0. 26, 40.E-2 , 0. 32, 37.771E-2, 6.293E-2 38, 36.211E-2, 10.698E-2 39, 36.211E-2, 11.35E-2 40, 36.211E-2, 12.E-2 500, 0. , 9.E-2 1000, 0. , 12.E-2 *NSET, NSET = CAVITY 500, *NGEN, LINE = C 6, 26, 2, , 30.E-2 , 0. , 0., 0., 0., 1. 26, 32, 1, , 30.E-2 , 0. , 0. 32, 38, 1, , 43.211E-2, 10.698E-2, 0. ** ** RUBBER SPRING ** *ELEMENT, TYPE = SAX1 2, 4, 6 26, 26, 27 *ELGEN, ELSET = SPRING 2, 11, 2, 2 26, 14, 1, 1 *SHELL SECTION, COMPOSITE, ELSET = SPRING 2.5E-3, 3, RUBBER 1.0E-3, 3, STEEL 2.5E-3, 3, RUBBER *MATERIAL, NAME = RUBBER *HYPERELASTIC, POLYNOMIAL, N = 1 3.2E6, 0.8E6 *DENSITY 1000., *MATERIAL, NAME = STEEL *ELASTIC, TYPE=LAMINA 2.48E9, 2.86E7, 9.105, 8.0E6, 8.0E6, 8.0E6 *DENSITY

1-174

Static Stress/Displacement Analyses

7.8E3, ** ** RIGID BODY/SURFACE ** *NODE, NSET = RIGID 11000, 0. , 12.E-2 *NODE,NSET=RIGNODE 11001, 36.211E-2, 10.698E-2 11021, 41.711E-2, 5.198E-2 12000, 41.711E-2, 10.698E-2 *NGEN,NSET=RIGNODE,LINE=C 11001, 11021, 1, 12000 *ELEMENT,TYPE=RAX2,ELSET=RIGELEM 11001, 1000,40 11002, 40, 11002 11003, 11002, 11003 *ELGEN,ELSET=RIGELEM 11003, 19, 1, 1 *ELEMENT,TYPE=MASS,ELSET=MASS_RIGID 11000, 11000 *MASS,ELSET=MASS_RIGID .5, ** ** FLUID ELEMENTS ** *ELEMENT, TYPE = FAX2, ELSET = FLUID 201, 1, 4 202, 4, 6 226, 26, 27 299, 40, 1000 *ELGEN, ELSET = FLUID 202, 11, 2, 2 226, 14, 1, 1 *PHYSICAL CONSTANTS, ABSOLUTE ZERO=-273.16 *FLUID PROPERTY, REF NODE=500, TYPE=PNEUMATIC, ELSET=FLUID, AMBIENT=101.36E+3 *FLUID DENSITY, PRESSURE=0.0, TEMPERATURE=27.0 1.774, ** *BOUNDARY 1,1,3 4,1,6

1-175

Static Stress/Displacement Analyses

6,1,1 RIGID,1,6 *NSET,NSET=RIG 40,1000 *AMPLITUDE,NAME=RAMP,DEFINITION=SMOOTHSTEP 0.,0., .05,1.0 ** ** STEP 1 - INFLATION STEP (5 ATMOSPHERES) ** *SURFACE,TYPE=ELEMENT,NAME=ASURF,NOTHICK ECON,SPOS *SURFACE,TYPE=ELEMENT,NAME=BSURF RIGELEM,SNEG *RIGID BODY,REFNODE=11000,ELSET=RIGELEM *STEP *DYNAMIC,EXPLICIT ,.05 *RESTART,WRITE,NUMBER INTERVAL=1 *BOUNDARY,AMPLITUDE=RAMP CAVITY,8,8, 5.066E5 ** ** CONTACT DEFINITION ** *ELSET,ELSET=ECON,GENERATE 26,38,1 *CONTACT PAIR,INTERACTION=SMOOTH ASURF,BSURF *SURFACE INTERACTION,NAME=SMOOTH ** ** OUTPUT REQUESTS ** *ELSET,ELSET=EOUT 39, *FILE OUTPUT, NUMBER INTERVAL=1 *NODE FILE, NSET=CAVITY PCAV, CVOL *NODE FILE, NSET=RIGID U, RF CF, *HISTORY OUTPUT,TIME=1.E-4 *ENERGY HISTORY ALLSE,ALLKE,ALLAE,ALLIE,ETOTAL *NODE HISTORY,NSET=CAVITY

1-176

Static Stress/Displacement Analyses

PCAV,CVOL *NODE HISTORY, NSET=RIGID U, RF, CF *EL HISTORY,ELSET=EOUT S,E *DLOAD SPRING,VP,1E4 *END STEP ** ** STEP 2 - SEAL CAVITY (KEEP RIGID SURFACE FIXED) ** NONLINEAR LOAD DEFLECTION CURVE ** *STEP *DYNAMIC,EXPLICIT ,.05 *RESTART,WRITE,NUMBER INTERVAL=2 *BOUNDARY, OP=NEW,AMPLITUDE=RAMP 1,1,3 4,1,6 6,1,1 RIGID,1 RIGID,3,6 RIGID,2,2,-.075 *END STEP

1.1.10 Shell-to-solid submodeling of a pipe joint Product: ABAQUS/Standard Submodeling is the technique of analyzing a local part of a model with a refined mesh, based on interpolation of the solution from an initial, global model (usually with a coarser mesh) onto the nodes on the appropriate parts of the boundary of the submodel. This local refinement procedure provides a cost-effective approach to model enhancement. Shell-to-solid submodeling models a region with solid elements, when the global model is made up of shell elements. The purpose of this example is to demonstrate the shell-to-solid submodeling capability in ABAQUS.

Geometry and model The joint between a pipe and a plate is analyzed. A pipe of radius 10 mm and thickness 0.75 mm is attached to a plate that is 100 mm long, 50 mm wide, and 1 mm thick. The pipe-plate intersection has a fillet radius of 1 mm. Taking advantage of the symmetry of the problem, only one-half of the assembly is modeled. Both the pipe and the plate are assumed to be made up of aluminum with E =69 ´ 103 MPa and Poisson's ratio º = 0.3. The global model consists of S4R elements with the mesh layout as shown in Figure 1.1.10-1. Since a shell model is used, the fillet radius is not taken into consideration.

1-177

Static Stress/Displacement Analyses

In the submodel the joint and its vicinity are meshed using three-dimensional continuum elements (C3D20R) with four layers through the thickness (see Figure 1.1.10-2). The solid model extends 10 mm along the pipe length and has a radius of 25 mm in the plane of the plate. The submodel accurately models the fillet radius at the joint. Hence, the submodel capability makes it possible to calculate the stress concentration in the fillet. The problem could be expanded by adding a ring of welded material to simulate a welded joint (for this case the submodel would have to be meshed with new element layers representing the welded material at the joint). The example could also be expanded by including plastic material behavior in the submodel while using an elastic global model solution.

Loading The pipe in the global model is subjected to concentrated loads acting in the 1-direction applied at the nodes at the free end, representing a shear load on the pipe. The total value of all concentrated forces is equal to 10 N.

Kinematic boundary conditions In the global shell model the plate is clamped along all edges. In the solid submodel kinematic conditions are interpolated from the global model at two surfaces of the submodel, one lying within the pipe and the other within the plate. The default center zone size, equal to 10% of the maximum shell thickness, is used. Thus, only one layer of driven nodes lies within the center zone, and only these nodes have all three displacement components driven by the global solution. For the remaining driven nodes only the displacement components parallel to the global model midsurface are driven from the global model. Thus, a single row of nodes is transmitting the transverse shear forces from the shell solution to the solid model.

Results and discussion The loading and boundary conditions are such that the pipe is subjected to bending. The end of the pipe that is attached to the plate leads to deformation of the plate itself (see Figure 1.1.10-3). From a design viewpoint the area of interest is the pipe-plate joint where the pipe is bending the plate. Hence, this area is submodeled to gain better understanding of the deformation and stress state. Figure 1.1.10-4 shows the contours of the out-of-plane displacement component in the plate and shows the differences between the behavior in the models. As expected, the solid model is in good agreement with the displacement of the shell model around the joint. The stress concentration in the fillet radius is obtained with the solid model. The maximum Mises stress in this region is equal to 80.6 MPa, which is 51% more than the Mises stress in the shell model. Similarly, the maximum principal stress in the fillet region in the solid model is 54% higher than the corresponding stress in the shell model.

Input files shellsolidpipe_s4_global.inp S4 global model. shellsolidpipe_c3d20rsub_s4.inp

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C3D20R submodel, which uses the S4 global model. shellsolidpipe_s4r_global.inp S4R global model. shellsolidpipe_c3d20rsub_s4r.inp Key input data for the C3D20R submodel, which uses the S4R global model. shellsolidpipe_c3d20r_mesh.inp Remainder of the input data for the C3D20R submodel. shellsolidpipe_node.inp Node definitions for the S4R and S4 global models. shellsolidpipe_element.inp Element definitions for the S4R and S4 global models.

Figures Figure 1.1.10-1 Global shell model of pipe-plate structure.

Figure 1.1.10-2 Magnified solid submodel of the pipe-plate joint.

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Figure 1.1.10-3 Solid submodel overlaid on the shell model in the deformed state, using magnification factor of 20.

Figure 1.1.10-4 Comparison of out-of-plane displacement in plate for both models.

Sample listings

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Listing 1.1.10-1 *HEADING Global Shell Model ** **RESTART,WRITE *NODE,INPUT=shellsolidpipe_node.inp *ELEMENT, TYPE=S4R, INPUT=shellsolidpipe_element.inp ** ** plate ** *ELSET,ELSET=PLATE,GENERATE 1,96,1 193,288,1 ** pipe ** *ELSET,ELSET=PIPE,GENERATE 97,192,1 289,384,1 *SHELL SECTION,ELSET=PLATE,MATERIAL=MAT1 0.001, *SHELL SECTION,ELSET=PIPE,MATERIAL=MAT1 0.00075, ** ** Basic steel properties *MATERIAL,NAME=MAT1 *ELASTIC 69.0E9,0.3 ** ** built_in ** *NSET, NSET=BUILT_IN 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 222, 223, 224, 225, 226, 227, 228, 285, 286, 287, 288, 289, ** ** ysymm **

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*NSET, NSET=YSYMM 1, 2, 3, 4, 5, 6, 7, 8, 9, 118, 119, 120, 121, 122, 123, 124, 222, 229, 236, 243, 250, 257, 264, 271, 278, 330, 337, 344, 351, 358, 365, 372 ** ** load ** *NSET, NSET=LOAD 125, 133, 141, 149, 157, 165, 173, 181, 189, 197, 205, 213, 221, 379, 380, 381, 382, 383, 384, 385, 421, 422, 423, 424, 425, *NSET, NSET=force 133, 141, 149, 157, 165, 173, 181, 189, 197, 205, 213, 221, 380, 381, 382, 383, 384, 385, 421, 422, 423, 424, 425 *BOUNDARY BUILT_IN,ENCASTRE YSYMM,YSYMM ** ** step 1,Default ** *STEP Total load of 10.0 N in the 1-direction *STATIC *CLOAD force,1,0.416667 125,1,0.208333 379,1,0.208333 *NODE FILE U, *END STEP

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Listing 1.1.10-2 *HEADING Solid submodel for global model. ABAQUS job create ** **RESTART,WRITE,FREQ=999 *** **Node, element and node set definitions **are written into the file **shellsolidpipe_c3d20r_mesh.inp *** *INCLUDE,INPUT=shellsolidpipe_c3d20r_mesh.inp *NSET,NSET=CUT NPIPE,NPLATE *MATERIAL,NAME=ALUMINUM *ELASTIC 69.E9,0.3 *SOLID SECTION,ELSET=EALL,MATERIAL=ALUMINUM ** *SUBMODEL,SHELLTOSOLID,SHELLTHICKNESS=0.001 CUT, *BOUNDARY NYSYMM,YSYMM *STEP *STATIC *BOUNDARY,SUBMODEL,STEP=1 CUT, *NODE FILE,NSET=CUT U, *END STEP

1.1.11 Stress-free element reactivation Product: ABAQUS/Standard This example demonstrates element reactivation for problems where new elements are to be added in a stress-free state. Typical examples include the construction of a gravity dam, in which unstressed layers of material are added to a mesh that has already deformed under geostatic load, or a tunnel in which a concrete or steel support liner is installed. The *MODEL CHANGE, ADD option (``Element and contact pair removal and reactivation, '' Section 7.4.2 of the ABAQUS/Standard User's Manual)

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provides for this type of application directly because the strain in newly added elements corresponds to the deformation of the mesh since the reactivation. Verification of the *MODEL CHANGE capability is provided in ``Model change,'' Section 3.8 of the ABAQUS Verification Manual.

Problem description The example considers the installation of a concrete liner to support a circular tunnel. Practical geotechnical problems usually involve a complex sequence of construction steps. The construction details determine the appropriate analysis method to represent these steps accurately. Such details have been avoided here for the sake of simplifying the illustration. The tunnel is assumed to be excavated in clay, with a Young's modulus of 200 MPa and a Poisson's ratio of 0.2 (see Figure 1.1.11-1). The diameter of the tunnel is 8 m, and the tunnel is excavated 20 m below ground surface. The material surrounding the excavation is discretized with first-order 4-node plane strain elements (element type CPE4). The infinite extent of the soil is represented by a 30-m-wide mesh that extends from the surface to a depth of 50 m below the surface. The left-hand boundary represents a vertical symmetry axis. Far-field conditions on the bottom and right-hand-side boundaries are modeled by infinite elements (element type CINPE4). No mesh convergence studies have been performed to establish if these boundary conditions are placed far enough away from the excavation. An initial stress field due to gravitational and tectonic forces exists through the depth of the soil. It is assumed that this stress varies linearly with depth and that the ratio between the horizontal and vertical stress components is 0.5. The self weight of the clay is 20.0 kN/m 3. The excavation of the tunnel material is accomplished by applying the forces that are required to maintain equilibrium with the initial stress state in the surrounding material as loads on the perimeter of the tunnel. These loads are then reduced to zero to simulate the excavation. The three-dimensional effect of face advancement during excavation is taken into account by relaxing the forces gradually over several steps. The liner is installed after 40% relaxation of the loads. Further deformation continues to occur as the face of the excavation advances. This ongoing deformation loads the liner. In the first input file the 150-mm-thick liner is discretized with one layer of incompatible mode elements (element type CPE4I). These elements are recommended in regions where bending response must be modeled accurately. In the second input file beam elements are used to discretize the liner. The liner is attached rigidly to the tunnel. The concrete is assumed to have cured to a strength represented by the elastic properties shown in Figure 1.1.11-1 by the time the liner is loaded. The liner is not shown in this diagram. It is expected that an overburden load representing the weight of traffic and buildings exists after the liner is installed.

Analysis method The excavation and installation of the liner is modeled in four analysis steps. In the first step the initial stress state is applied and the liner elements are removed using the *MODEL CHANGE, REMOVE

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option. Concentrated loads that are in equilibrium with the initial stress field are applied on the perimeter of the tunnel. These forces were obtained from an independent analysis where the displacements on the tunnel perimeter were constrained. The reaction forces at the constrained nodes are the loads applied here. The second step begins the tunnel excavation by reducing the concentrated loads on the tunnel surface. The loads are reduced by 40% in this step before the liner is installed in the third step using the *MODEL CHANGE, ADD option. No deformation takes place in the soil or liner during the third step. In the fourth step the surface load is applied, and the excavation is completed by removing the remainder of the load on the tunnel perimeter. In problems involving geometric nonlinearities with finite deformation, it is important to recognize that element reactivation occurs in the configuration at the start of the reactivation step. If the NLGEOM parameter were used in this problem, the thickness of the liner, when modeled with the continuum elements, would have a value at reactivation that would be different from its original value. This result would happen because the outside nodes (the nodes on the tunnel/liner interface) displace with the mesh, whereas the inside nodes remain at their current locations since liner elements are inactive initially. This effect is not relevant in this problem because geometric nonlinearities are not included. However, it may be significant for problems involving finite deformation, and it may lead to convergence problems in cases where elements are severely distorted upon reactivation. This problem would not occur in the model with beam elements because they have only one node through the thickness. In the model where the liner is modeled with continuum elements, the problem can be eliminated if the inner nodes are allowed to follow the outer nodes prior to reactivation, which can be accomplished by applying displacement boundary conditions on the inner nodes. Alternatively, the liner can be overlaid with (elastic) elements of very low stiffness. These elements use the same nodes as the liner but are so compliant that their effect on the analysis is negligible when the liner is present. They remain active throughout the analysis and ensure that the inner nodes follow the outer nodes, thereby preserving the liner thickness.

Results and discussion Figure 1.1.11-2 shows the stress state at a material point in the liner. The figure clearly indicates that the liner remains unstressed until reactivated. Figure 1.1.11-3 compares the axial stress obtained from the CPE4I and beam elements at the top and bottom of the liner section. A cylindrical *ORIENTATION (``Orientations,'' Section 2.2.4 of the ABAQUS/Standard User's Manual) is used to orient the liner stresses in the continuum element model along the beam axis so that these stresses can be compared directly with the results of the beam element model. The small difference between the results can be attributed to the element type used in the discretization of the liner: the beam element model uses a plane stress condition, and the continuum element model uses a plane strain condition.

Input files modelchangedemo_continuum.inp *MODEL CHANGE with continuum elements. modelchangedemo_beam.inp

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*MODEL CHANGE with beam elements. modelchangedemo_node.inp Nodal coordinates for the soil. modelchangedemo_element.inp Element definitions for the soil.

Figures Figure 1.1.11-1 Geometry and finite element discretization.

Figure 1.1.11-2 Liner stress during analysis history.

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Figure 1.1.11-3 Axial stress along beam inside and outside.

Sample listings

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Listing 1.1.11-1 *HEADING STRESS FREE INCLUSION OF TUNNEL LINER (USING *MODEL CHANGE) Units : N, m *RESTART,WRITE ** *NODE,NSET=SOIL,INPUT=modelchangedemo_node.inp *NSET,NSET=TUNNEL,UNSORTED 100, 101, 102, 103, 104, 105, 106, 2105, 2104, 2103, 2102, 2101, 2100 *NODE 4000, 0.00, -16.00 4012, 0.00, -24.00 4100, 0.00, -16.15 4112, 0.00, -23.85 *NGEN,LINE=C 4000, 4012, 1, ,0.0, -20.0, 0.0, 0.0, 0.0, -1.0 4100, 4112, 1, ,0.0, -20.0, 0.0, 0.0, 0.0, -1.0 *NSET,NSET=LINER,GENERATE 4000, 4012 *NSET,NSET=RHS,GENERATE 8, 3108, 10 *NSET,NSET=BOT,GENERATE 3100, 3108 *NSET,NSET=XSYMM,GENERATE 100, 150, 10 200, 1000, 100 2100, 2150, 10 2200, 3200, 100 *NSET,NSET=XSYMM 4000, 4012, 4100, 4112 *NODE,NSET=NINF 209, 60.00, -20.00 309, 60.00, -17.77 409, 60.00, -14.98 509, 60.00, -11.50 609, 60.00, -10.10 709, 60.00, -8.347 809, 60.00, -6.158 909, 60.00, -3.420 1009, 60.00, 0.000

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2309, 60.00, -22.03 2409, 60.00, -25.02 2509, 60.00, -28.50 2609, 60.00, -29.90 2709, 60.00, -31.65 2809, 60.00, -33.84 2909, 60.00, -36.58 3009, 60.00, -40.00 3109, 60.00, -76.00 3200, 0.000, -76.00 3201, 2.245, -76.00 3202, 5.052, -76.00 3203, 8.560, -76.00 3204, 11.17, -76.00 3205, 14.44, -76.00 3206, 18.52, -76.00 3207, 23.62, -76.00 ** *ELEMENT,TYPE=CPE4,ELSET=SOIL, INPUT=modelchangedemo_element.inp *ELEMENT,ELSET=LINER,TYPE=CPE4I 4000, 4100, 4101, 4001, 4000 *ELGEN,ELSET=LINER 4000, 12 *ELSET,ELSET=SURFACE,GENERATE 900 , 907 , 1 *ELEMENT,TYPE=CINPE4,ELSET=ELINF 3100,3101,3100,3200,3201 3101,3102,3101,3201,3202 3102,3103,3102,3202,3203 3103,3104,3103,3203,3204 3104,3105,3104,3204,3205 3105,3106,3105,3205,3206 3106,3107,3106,3206,3207 3107,3108,3107,3207,3109 3108,3008,3108,3109,3009 2908,2908,3008,3009,2909 2808,2808,2908,2909,2809 2708,2708,2808,2809,2709 2608,2608,2708,2709,2609 2508,2508,2608,2609,2509 2408,2408,2508,2509,2409 2308,2308,2408,2409,2309

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2208,208,2308,2309,209 208,308,208,209,309 308,408,308,309,409 408,508,408,409,509 508,608,508,509,609 608,708,608,609,709 708,808,708,709,809 808,908,808,809,909 908,1008,908,909,1009 *ELSET,ELSET=EALL SOIL,ELINF ** *ORIENTATION,NAME=OR,SYSTEM=CYLINDRICAL 0.0, -20.0, 0.0, 0.0, -20.0, 10.0 3, 0.0 *SOLID SECTION,MATERIAL=CONCRETE,ELSET=LINER, ORIENTATION=OR *SOLID SECTION,MATERIAL=CLAY,ELSET=SOIL *SOLID SECTION,MATERIAL=CLAY,ELSET=ELINF *MATERIAL,NAME=CLAY *ELASTIC,TYPE=ISOTROPIC 0.2E9, 0.2 *MATERIAL,NAME=CONCRETE *ELASTIC,TYPE=ISOTROPIC 19.0E9 , 0.2 ** *MPC TIE, LINER, TUNNEL *INITIAL CONDITIONS,TYPE=STRESS,GEOSTATIC EALL, 0.0, 0.0, -1.52E6, -76.00, 0.5 *AMPLITUDE,NAME=RELAX,TIME=TOTAL TIME 0.0, 1.0, 1.0, 1.0, 2.0, 0.6, 3.0, 0.6, 4.0, 0.0 ** -------------------------------------*STEP step 1: add initial stress state & remove liner *STATIC *DLOAD SOIL, BY, -20.0E3 *MODEL CHANGE,REMOVE LINER, *CLOAD,AMPLITUDE=RELAX 100, 1, 5.4086E+04

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101, 1, 4.3918E+04 102, 1, 8.6901E+04 103, 1, 1.2732E+05 104, 1, 1.6185E+05 105, 1, 1.8949E+05 106, 1, 2.0701E+05 2100, 1, 8.2710E+04 2101, 1, 6.3287E+04 2102, 1, 1.2031E+05 2103, 1, 1.6549E+05 2104, 1, 1.9676E+05 2105, 1, 2.1052E+05 100, 2, 1.6652E+05 101, 2, 3.2459E+05 102, 2, 2.9838E+05 103, 2, 2.5160E+05 104, 2, 1.8487E+05 105, 2, 9.9587E+04 106, 2, -2142. 2100, 2, -2.4756E+05 2101, 2, -4.7534E+05 2102, 2, -4.1880E+05 2103, 2, -3.3399E+05 2104, 2, -2.2922E+05 2105, 2, -1.1476E+05 *BOUNDARY XSYMM, 1 *NODE PRINT,FREQUENCY=100,NSET=TUNNEL U, RF *EL PRINT,FREQUENCY=100,ELSET=LINER, POSITION=AVERAGED AT NODES S, *END STEP ** -------------------------------------*STEP,INC=100 step 2: relax tunnel stress 40 % *STATIC *END STEP ** -------------------------------------*STEP step 3: add liner stress free *STATIC *MODEL CHANGE,ADD

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LINER, *END STEP ** -------------------------------------*STEP,INC=100 step 4: relax tunnel stress to zero & apply surface load *STATIC *DLOAD SURFACE, P3, 50.0E3 *EL FILE,FREQUENCY=100,ELSET=LINER, POSITION=AVERAGED AT NODES S, *END STEP

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Listing 1.1.11-2 *HEADING STRESS FREE INCLUSION OF TUNNEL LINER (USING *MODEL CHANGE) Units : N, m *RESTART,WRITE ** *NODE,NSET=SOIL,INPUT=modelchangedemo_node.inp *NSET,NSET=TUN_IX,UNSORTED 101, 102, 103, 104, 105, 106, 2105, 2104, 2103, 2102, 2101 *NSET,NSET=TUN_IY,UNSORTED 100, 101, 102, 103, 104, 105, 106, 2105, 2104, 2103, 2102, 2101, 2100 *NSET,NSET=TUNNEL,GENERATE 100, 106 2100, 2105 *NODE 4000, 0.00, -16.0 4012, 0.00, -24.0 *NGEN,NSET=LINER,LINE=C 4000, 4012, 1, ,0.0, -20.0, 0.0, 0.0, 0.0, -1.0 *NSET,NSET=LIN_JX,GENERATE 4001, 4011 *NSET,NSET=LIN_JY,GENERATE 4000, 4012 *NSET,NSET=RHS,GENERATE 8, 3108, 10 *NSET,NSET=BOT,GENERATE 3100, 3108 *NSET,NSET=XSYMM,GENERATE 100, 150, 10 200, 1000, 100 2100, 2150, 10 2200, 3200, 100 *NSET,NSET=XSYMM 4000, 4012, 4100, 4112 *NODE,NSET=NINF 209, 60.00, -20.00 309, 60.00, -17.77 409, 60.00, -14.98 509, 60.00, -11.50

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609, 60.00, -10.10 709, 60.00, -8.347 809, 60.00, -6.158 909, 60.00, -3.420 1009, 60.00, 0.000 2309, 60.00, -22.03 2409, 60.00, -25.02 2509, 60.00, -28.50 2609, 60.00, -29.90 2709, 60.00, -31.65 2809, 60.00, -33.84 2909, 60.00, -36.58 3009, 60.00, -40.00 3109, 60.00, -76.00 3200, 0.000, -76.00 3201, 2.245, -76.00 3202, 5.052, -76.00 3203, 8.560, -76.00 3204, 11.17, -76.00 3205, 14.44, -76.00 3206, 18.52, -76.00 3207, 23.62, -76.00 ** *ELEMENT,TYPE=CPE4,ELSET=SOIL, INPUT=modelchangedemo_element.inp *ELEMENT,TYPE=B21 4000, 4000, 4001 *ELGEN,ELSET=LINER 4000, 12 *ELSET,ELSET=SURFACE,GENERATE 900 , 907 , 1 *ELEMENT,TYPE=CINPE4,ELSET=ELINF 3100,3101,3100,3200,3201 3101,3102,3101,3201,3202 3102,3103,3102,3202,3203 3103,3104,3103,3203,3204 3104,3105,3104,3204,3205 3105,3106,3105,3205,3206 3106,3107,3106,3206,3207 3107,3108,3107,3207,3109 3108,3008,3108,3109,3009 2908,2908,3008,3009,2909 2808,2808,2908,2909,2809

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2708,2708,2808,2809,2709 2608,2608,2708,2709,2609 2508,2508,2608,2609,2509 2408,2408,2508,2509,2409 2308,2308,2408,2409,2309 2208,208,2308,2309,209 208,308,208,209,309 308,408,308,309,409 408,508,408,409,509 508,608,508,509,609 608,708,608,609,709 708,808,708,709,809 808,908,808,809,909 908,1008,908,909,1009 *ELSET,ELSET=EALL SOIL,ELINF ** *BEAM SECTION,MATERIAL=CONCRETE,SECTION=RECT, ELSET=LINER 1.0 , 0.15 0.0 , 0.0 , -1.0 *SOLID SECTION,MATERIAL=CLAY,ELSET=SOIL *SOLID SECTION,MATERIAL=CLAY,ELSET=ELINF *MATERIAL,NAME=CLAY *ELASTIC,TYPE=ISOTROPIC 0.2E9, 0.2 *MATERIAL,NAME=CONCRETE *ELASTIC,TYPE=ISOTROPIC 19.0E9 , 0.2 ** *EQUATION 2, TUN_IX, 1, 1.0, LIN_JX, 1, -1.0 2, TUN_IY, 2, 1.0, LIN_JY, 2, -1.0 *INITIAL CONDITIONS,TYPE=STRESS,GEOSTATIC EALL, 0.0, 0.0, -1.52E6, -76.00, 0.5 *AMPLITUDE,NAME=RELAX,TIME=TOTAL TIME 0.0, 1.0, 1.0, 1.0, 2.0, 0.6, 3.0, 0.6, 4.0, 0.0 ** -------------------------------------*STEP step 1: add initial stress state

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*STATIC *DLOAD SOIL, BY, -20.0E3 *MODEL CHANGE,REMOVE LINER, *CLOAD,AMPLITUDE=RELAX 100, 1, 5.4086E+04 101, 1, 4.3918E+04 102, 1, 8.6901E+04 103, 1, 1.2732E+05 104, 1, 1.6185E+05 105, 1, 1.8949E+05 106, 1, 2.0701E+05 2100, 1, 8.2710E+04 2101, 1, 6.3287E+04 2102, 1, 1.2031E+05 2103, 1, 1.6549E+05 2104, 1, 1.9676E+05 2105, 1, 2.1052E+05 100, 2, 1.6652E+05 101, 2, 3.2459E+05 102, 2, 2.9838E+05 103, 2, 2.5160E+05 104, 2, 1.8487E+05 105, 2, 9.9587E+04 106, 2, -2142. 2100, 2, -2.4756E+05 2101, 2, -4.7534E+05 2102, 2, -4.1880E+05 2103, 2, -3.3399E+05 2104, 2, -2.2922E+05 2105, 2, -1.1476E+05 *BOUNDARY XSYMM, 1 XSYMM, 6 *NODE PRINT,FREQUENCY=100,NSET=TUNNEL U, RF *EL PRINT,FREQUENCY=100,ELSET=LINER, POSITION=AVERAGED AT NODES S, *END STEP ** -------------------------------------*STEP,INC=100

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step 2: relax tunnel stress 40 % *STATIC *END STEP ** -------------------------------------*STEP step 3: add liner stress free *STATIC *MODEL CHANGE,ADD LINER, *END STEP ** -------------------------------------*STEP,IN=100 step 4: relax tunnel stress to zero & apply surface load *STATIC *DLOAD SURFACE, P3, 50.0E3 *EL FILE,FREQUENCY=100,ELSET=LINER, POSITION=AVERAGED AT NODES S, *END STEP

1.1.12 Symmetric results transfer for a rubber bushing Products: ABAQUS/Standard ABAQUS/Design The *SYMMETRIC RESULTS TRANSFER option (``Transferring results from a symmetric mesh to a three-dimensional mesh,'' Section 7.7.2 of the ABAQUS/Standard User's Manual) allows the user to transfer the solution obtained from an axisymmetric analysis onto a three-dimensional model with the same geometry. It also allows the transfer of a symmetric three-dimensional solution to a full three-dimensional model. This capability can reduce the analysis cost of structures that undergo symmetric deformation, followed by nonsymmetric deformation later during the loading history. The use of the results transfer capability is illustrated in this example by considering the response of an axisymmetric bushing to loading imposed by a shaft that fits through the center of the bushing. Only results transfer from an axisymmetric to a three-dimensional model is considered here. The bushing consists of inner and outer steel tubes that are bonded to a central rubber cylinder ( Figure 1.1.12-1). It is assumed that the outer perimeter of the bushing is fully fixed. The loading conditions--which include axial, twisting, and bending loads--are typical of the loading conditions that would be applied to study the stiffness characteristics of the rubber bushing. Two loading sequences are considered in two separate analyses. The first analysis considers axial displacement of the shaft, followed by rotation of the rigid shaft about a transverse axis; the latter loading is called bending in these examples. The second analysis considers relative twisting between the shaft and the outer perimeter of the bushing, followed by the same bending load as in the first

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example. In both examples the first loading step is fully axisymmetric so that only a two-dimensional analysis is required. However, a full three-dimensional model is required to model the subsequent bending load. The *SYMMETRIC RESULTS TRANSFER option is used to transfer the axisymmetric solution to the full three-dimensional model. This solution becomes the base, or reference, state in the subsequent bending analysis. It is desirable to reduce the stress concentrations to increase the service life of the bushing. To this end the sensitivity of stresses in the axisymmetric model is studied with respect to two shape design parameters: the fillet radius, r, and the thickness, t, of the rubber bushing at the top and bottom ends where it is bonded to the inner steel tube.

Geometry and model The bushing is 457.2 mm (18.0 in) long, with an outside diameter of 508.0 mm (20.0 in) and an inside diameter of 228.6 mm (9.0 in). The steel is elastic with Young's modulus = 206.0 GPa (3.0 ´ 107 psi) and Poisson's ratio = 0.3. The rubber is modeled as a fully incompressible hyperelastic material that at all strain levels is relatively soft compared to the steel. The nonlinear elastic behavior of the rubber is described by a strain energy function that is a second-order polynomial in the strain invariants. The first model is discretized with standard axisymmetric elements since the axial loading results in pure axisymmetric deformation; CAX4 elements are used for the steel components, and CAX4H elements are used for the rubber component. The second model is discretized with first-order, reduced-integration axisymmetric elements with twist (CGAX4R type elements) to accommodate the relative twisting between the shaft and outer perimeter of the bushing. Rigid elements (element type RAX2) are attached to the inside of the bushing in both models to represent the relatively stiff shaft. The use of these elements also simplifies the application of the loading conditions. The axisymmetric finite element mesh is shown in Figure 1.1.12-1. The corresponding three-dimensional models are generated using the *SYMMETRIC MODEL GENERATION, REVOLVE option (``Symmetric model generation,'' Section 7.7.1 of the ABAQUS/Standard User's Manual). The first model consists of a 180.0° revolution, and the second consists of a 360.0° revolution of the axisymmetric cross-section about the symmetry axis. The model generation capability converts the CAX4 and CGAX4R elements to C3D8 and C3D8R elements, respectively. It also converts the CAX4H and CGAX4RH elements to C3D8H and C3D8RH, respectively, and the RAX2 elements to R3D4 elements. Both models are discretized with eight elements along the circumference in a 180.0° segment. No mesh convergence studies were performed.

Design sensitivity analysis The objective is to modify the bushing geometry to lower the maximum axial stress. Sensitivity analysis is used to provide an approximate assessment of the change needed in the design parameters to achieve this objective. Only the axisymmetric model is considered. To carry out the design sensitivity analysis with respect to a shape design parameter, the gradients of the nodal coordinates with respect to the design parameter must be specified in the *PARAMETER SHAPE VARIATION option. One simple approach to obtaining these gradients is to perturb the shape design parameters r and t one at a time and to record the perturbed coordinates. The gradients are then found by

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numerically differencing the initial and perturbed nodal coordinates. The perturbation of the thickness is such that it causes the line of nodes connecting the thickness dimension to the fillet radius to rotate about the point of tangency of this line to the fillet radius.

Results and discussion In the first example an axial force of magnitude 10675.0 N (2400.0 lbs) is applied to the rigid body reference node in the first step, while the outer steel tube is fully fixed. This solution is transferred to a 180.0° segment of the three-dimensional model; the result forms the base state from which the subsequent bending loading is applied. Symmetry boundary conditions are applied on the surfaces of the three-dimensional model at µ = 0.0° and 180.0°. To ensure that the model is in equilibrium at the beginning of the three-dimensional analysis, a *STEP definition is included using boundary conditions and loading that are consistent with the state of the axisymmetric model. This step is completed in one loading increment. The deformed mesh is shown in Figure 1.1.12-2. A bending moment of 294.0 N m (2600.0 lbs in) is applied to the rigid body reference node in the second step. Figure 1.1.12-3 shows the resulting deformation. A 22% saving in analysis time is obtained in this example by taking advantage of symmetry conditions and the result transfer capability. In the second example a rotation of 8.6° (0.15 rad) with respect to the axis of symmetry is applied to the outer perimeter in the first step, while the shaft is held fixed. This solution is transferred to a full 360.0° three-dimensional model; the result forms the base state from which the subsequent bending loading is applied. An initial *STEP definition is again included, using the boundary conditions and loading that are consistent with the state of the axisymmetric model, to ensure that the three-dimensional model is in equilibrium. Figure 1.1.12-4 and Figure 1.1.12-5 show the deformed mesh that results from the twisting and bending loads, respectively. An 86% saving in analysis time is obtained in this example by transferring the axisymmetric analysis onto the three-dimensional model. The first example, the axial and bending problem, can also be solved using asymmetric-axisymmetric CAXAnn elements rather than by using the axisymmetric model generation and results transfer option (see ``Finite sliding between concentric cylinders--axisymmetric and CAXA models,'' Section 1.6.13 of the ABAQUS Verification Manual). Depending on the number of Fourier modes used, the CAXAnn elements should execute the analysis more quickly than the symmetric model generation followed by the results transfer option. However, the twisting and bending problem cannot be solved using CAXAnn elements; in this case, the axisymmetric model generation and the results transfer option are the most efficient method available. Figure 1.1.12-6 shows the contours of axial stress in the rubber part of the bushing at the end of the axisymmetric analysis. The maximum stress occurs near the top fillet close to the axis. Figure 1.1.12-7 and Figure 1.1.12-8 show the contours of the sensitivities of the axial stress for the shape design variables r and t, respectively. Table 1.1.12-1 shows the normalized sensitivities of the maximum axial stress, ¾max = 0.18 MPa (26.01 psi), with respect to the shape design variables. The normalization has been carried out by multiplying the sensitivities by a characteristic dimension (initial fillet radius r0 = 12.7 mm (0.5 in) and initial thickness t0 = 15.24 mm (0.6 in)) and dividing by the maximum stress. As can be inferred from this table, a change in the fillet radius influences the maximum stress to a larger extent than a change in the thickness of the rubber. Hence, it is desirable to change r to modify the stresses. To obtain approximately a 10% reduction in the maximum stress in the axial direction, the

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fillet radius is increased by ¢r =

¢¾max d¾max dr

:

Substituting for ¢¾max = 0:1¾max and d¾max =dr = 0.008 MPa/mm (28.75 psi/in) (see Figure 1.1.12-7) gives ¢r = 2.25 mm (0.09049 in). A reanalysis of the problem with the radius changed to r0 + ¢r = 14.99 mm (0.59049 in) yields a reduction of 8.8% in the maximum axial stress, which is slightly less than the goal of 10%. This is expected because of the nonlinearity of the problem; to achieve the 10% reduction, this process would have to be repeated, which is essentially an optimization problem.

Input files bushing_cax4_axi.inp Axisymmetric model with CAX4 elements. bushing_cax4_3d.inp Three-dimensional model created from CAX4 elements. bushing_cgax4r_axi.inp Axisymmetric model with CGAX4R elements. bushing_cgax4r_3d.inp Three-dimensional model created from CGAX4R elements. bushing_node.inp Node definitions. bushing_steel.inp Element definitions for the steel. bushing_rubber.inp Element definitions for the rubber. bushing_rigid.inp Element definitions for the rigid body. bushing_cax4_axi_dsa.inp Design sensitivity analysis for the axisymmetric model.

Table Table 1.1.12-1 Normalized sensitivities of the maximum stress.

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Parameter r t

r0 d¾max ¾max dr

-0.55 --

t0 d¾max ¾max dt

--0.11

Figures Figure 1.1.12-1 Axisymmetric cross-section.

Figure 1.1.12-2 Deformed mesh after axial loading.

Figure 1.1.12-3 Deformed mesh after axial loading followed by nonaxisymmetric loading.

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Figure 1.1.12-4 Deformed mesh after twist loading.

Figure 1.1.12-5 Deformed mesh after twist loading followed by nonaxisymmetric loading.

Figure 1.1.12-6 Variation of axial stress in the rubber after axial loading.

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Figure 1.1.12-7 Variation of the sensitivity of the axial stress with respect to an increase in the radius of the fillet, r.

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Figure 1.1.12-8 Variation of the sensitivity of the axial stress with respect to a decrease in the thickness of the rubber, t.

Sample listings

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Listing 1.1.12-1 *HEADING AXISYMMETRIC BUSHING *RESTART,WRITE *NODE,NSET=NALL,INPUT=bushing_node.inp *NODE 229,0.,0.,0. *ELEMENT, TYPE=CAX4,INPUT=bushing_steel.inp *ELEMENT, TYPE=CAX4H,ELSET=RUBBER,INPUT=bushing_rubber.inp *ELEMENT,TYPE=RAX2,ELSET=RIGID,INPUT=bushing_rigid.inp *RIGID BODY,ELSET=RIGID,REF NODE=229 *ELSET,ELSET=INNER,GENERATE 1,5,1 76,79,1 163,171,1 *ELSET,ELSET=OUTER,GENERATE 178,184,1 86,92,1 *NSET, NSET=OUT_ST 110, 111, 112, 113, 114, 115, 116, 222, 223, 224, 225, 226, 227, 228 *NSET,NSET=IN_ST 1, 2, 3, 4, 5, 6, 98, 100, 101, 209, 210, 211, 212, 213, 215, 216, 217, *SOLID SECTION, ELSET=INNER, MATERIAL=STEEL-MATERIAL 1., *SOLID SECTION, ELSET=OUTER, MATERIAL=STEEL-MATERIAL 1., *SOLID SECTION, ELSET=RUBBER, MATERIAL=RUBBER-MATERIAL 1., *MATERIAL, NAME=STEEL-MATERIAL *ELASTIC 3.E+7,0.3 *MATERIAL, NAME=RUBBER-MATERIAL *HYPERELASTIC, N=2 11.5796, 3.47492, 2.269385E-1, -1.77868E-1, 8.5253E-3, 0.0, 0.0 *STEP, NLGEOM *STATIC 1.0, 1.0 *BOUNDARY OUT_ST,1,2,0.0

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117,

99, 214,

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229,1,1 229,6,6 *CLOAD 229,2,2400.0 *NODE PRINT,FREQ=999 U, RF, *NODE FILE,FREQ=999 U, RF, *EL PRINT,FREQ=999 S, E, *EL FILE,FREQ=999 S, E, *END STEP

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Listing 1.1.12-2 *HEADING AXISYMMETRIC BUSHING *RESTART,WRITE,FREQ=999 *SYMMETRIC MODEL GENERATION,REVOLVE 0,0,0,0,0,1 1,0,0 180.0,8,1. *NSET,NSET=SYMM,GEN 1,228,1 1833,2060,1 *NSET,NSET=NOUT,GEN 917,1144,1 *ELSET,ELSET=EOUT,GEN 1555,1738,1 *SYMMETRIC RESULTS TRANSFER,STEP=1,INC=1 *TRANSFORM,NSET=NALL,TYPE=C 0.,0.,0.,0.,0.,10. *FILE FORMAT,ZERO INCREMENT *STEP,NLGEOM *STATIC 1.0, 1.0 *BOUNDARY OUT_ST,1,3 SYMM,2,2,0.0 229,1,2,0.0 229,4,6,0.0 *CLOAD 229,3,1200. *NODE PRINT,NSET=NOUT,FREQ=999 U, RF, *NODE FILE,NSET=NOUT,FREQ=999 U, RF, *EL PRINT,ELSET=EOUT,FREQ=999 S, E, *EL FILE,ELSET=EOUT,FREQ=999 S, E, *END STEP

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*STEP,NLGEOM,INC=20 *STATIC .1, 1.0 *BOUNDARY,OP=NEW OUT_ST,1,3 SYMM,2,2,0.0 229,2,2,0.0 *CLOAD 229,5,2600. *NODE PRINT,NSET=NOUT,FREQ=999 U, RF, *NODE FILE,NSET=NOUT,FREQ=999 U, RF, *EL PRINT,ELSET=EOUT,FREQ=999 S, E, *EL FILE,ELSET=EOUT,FREQ=999 S, E, *END STEP

1.1.13 Transient loading of a viscoelastic bushing Product: ABAQUS/Standard This example demonstrates the automatic incrementation capability provided for integration of time-dependent material models and the use of the viscoelastic material model in conjunction with large-strain hyperelasticity in a typical design application. The structure is a bushing, modeled as a hollow, viscoelastic cylinder. The bushing is glued to a rigid, fixed body on the outside and to a rigid shaft on the inside, to which the loading is applied. A static preload is applied to the shaft, which moves the inner shaft off center. This load is held for sufficient time for steady-state response to be obtained. Then a torque is applied instantaneously and held for a long enough period of time to reach steady-state response. We compute the bushing's transient response to these events.

Geometry and model The viscoelastic bushing has an inner radius of 12.7 mm (0.5 in) and an outer radius of 25.4 mm (1.0 in). We assume that the bushing is long enough for plane strain deformation to occur. The problem is modeled with first-order reduced-integration elements ( CPE4R). The mesh is regular, consisting of 6 elements radially, repeated 56 times to cover the 360° span in the hoop direction. The mesh is shown in Figure 1.1.13-1. No mesh convergence studies have been performed. The fixed outer body is modeled by fixing both displacement components at all the outside nodes. The

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nodes in the inner boundary of the bushing are connected, using the *KINEMATIC COUPLING option, to a node located in the center of the model. This node, thus, defines the inner shaft as a rigid body.

Material The material model is not defined from any particular physical material. The instantaneous behavior of the viscoelastic material is defined by hyperelastic properties. A polynomial model with N =1 (a Mooney-Rivlin model) is used for this, with the constants C10 = 27.56 MPa (4000 psi), C01 = 6.89 MPa (1000 psi), and D1 = 0.0029 MPa -1 (0.00002 psi -1). The viscous behavior is modeled by a time-dependent shear modulus, GR (t), and a time-dependent bulk modulus, KR (t), each of which is expanded in a Prony series in terms of the corresponding instantaneous modulus, ¶¶ t GR (t)=G0 = 1 ¡ 1 ¡ exp ¡ ¿i i=1 µ µ ¶¶ 2 X t P ¹ : ki 1 ¡ exp ¡ KR (t)=K0 = 1 ¡ ¿ i i=1 2 X

g¹iP

µ

µ

The relative moduli g¹iP and k¹iP and time constants ¿i are g¹iP k¹iP i ¿i ; sec 1 0.2 0.5 0.1 2 0.1 0.2 0.2 This model results in an initial instantaneous Young's modulus of 206.7 MPa (30000 psi) and Poisson's ratio of 0.45. It relaxes pressures faster than shear stresses.

Analysis The analysis is done in four steps. The first step is a preload of 222.4 kN (50000 lbs) applied in the x-direction to the node in the center of the model in 0.001 sec with a *STATIC procedure (``Static stress analysis,'' Section 6.2.2 of the ABAQUS/Standard User's Manual). The *STATIC procedure does not allow viscous material behavior, so this response is purely elastic. During the second step the load stays constant and the material is allowed to creep for 1 sec by using the *VISCO procedure (``Quasi-static analysis,'' Section 6.2.5 of the ABAQUS/Standard User's Manual). Since 1 sec is a long time compared with the material time constants, the solution at that time should be close to steady state. The CETOL parameter on the *VISCO option defines the accuracy of the automatic time incrementation during creep response. CETOL is an upper bound on the allowable error in the creep strain increment in each time increment. It is chosen as 5 ´ 10-4, which is small compared to the elastic strains. The third step is another *STATIC step. Here the loading is a torque of 1129.8 N-m (10000 lb-in) applied in 0.001 sec. The fourth step is another *VISCO step with a time period of 1 sec.

Results and discussion Figure 1.1.13-2 through Figure 1.1.13-5depict the deformed shape of the bushing at the end of each

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step. Each of the static loads produces finite amounts of deformation, which are considerably expanded during the holding periods. Figure 1.1.13-6shows the displacement of the center of the bushing in the x-direction and its rotation as functions of time.

Input file viscobushing.inp Input data for the analysis.

Figures Figure 1.1.13-1 Finite element model of viscoelastic bushing.

Figure 1.1.13-2 Deformed model after horizontal static loading.

Figure 1.1.13-3 Deformed model after first holding period.

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Figure 1.1.13-4 Deformed model after static moment loading.

Figure 1.1.13-5 Deformed model after second holding period.

Figure 1.1.13-6 Displacement and rotation of center of bushing.

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Sample listings

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Listing 1.1.13-1 *HEADING TRANSIENT LOADING OF A VISCOELASTIC BUSHING -CPE4R *NODE 1,0.5,0. 7,1.0,0. 999,0.,0. *NGEN,NSET=RADIAL 1,7 *NCOPY,CHANGE NUMBER=10,OLD SET=RADIAL, NEW SET=ALL,SHIFT,MULTIPLE=55 0.,0.,0.,0.,0.,1.,6.4285714 *NSET,NSET=INSIDE,GENERATE 1,551,10 *NSET,NSET=OUTSIDE,GENERATE 7,557,10 *ELEMENT,TYPE=CPE4R,ELSET=ONE 1, 1,2,12,11 *ELGEN,ELSET=ALL 1,6,1,1,55,10,10 *ELSET,ELSET=TWO 1,2 *ELSET,ELSET=RIM,GENERATE 1,21,10 2,22,10 401,421,10 402,422,10 *ELEMENT,TYPE=CPE4R 551,551,552,2,1 *ELGEN,ELSET=ALL 551,6,1,1 *KINEMATIC COUPLING, REF NODE=999 INSIDE,1,2 *BOUNDARY OUTSIDE,1,2 *MATERIAL,NAME=RUBBER *HYPERELASTIC,N=1,MODULI=INSTANTANEOUS 4000.,1000.,0.00002 *VISCOELASTIC,TIME=PRONY 0.2,0.5,.1

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0.1,0.2,.2 *SOLID SECTION,ELSET=ALL,MATERIAL=RUBBER 1., *NSET,NSET=CENTER 999, *STEP,NLGEOM *STATIC 0.001,0.001,0.0001 *CLOAD 999,1,50000. *EL PRINT,FREQUENCY=0 *NODE PRINT,NSET=CENTER *NODE FILE,FREQUENCY=150,NSET=RADIAL U,RF *EL FILE,FREQUENCY=150,ELSET=TWO S,E,CE,CEP *OUTPUT,VAR=PRESELECT,FIELD,FREQ=999 *OUTPUT,HISTORY,FREQ=1 *NODE OUTPUT,NSET=CENTER U, *OUTPUT,FIELD,Frequency=150 *NODE OUTPUT,NSET=RADIAL U,RF *OUTPUT,HISTORY,Frequency=150 *NODE OUTPUT,NSET=RADIAL U,RF *OUTPUT,FIELD,Frequency=150 *ELEMENT OUTPUT,ELSET=TWO S,E,CE,CEP *OUTPUT,HISTORY,Frequency=150 *ELEMENT OUTPUT,ELSET=TWO S,E,CE,CEP *END STEP *STEP,NLGEOM,INC=150 *VISCO,CETOL=5.E-4 0.04,1. *END STEP *STEP,NLGEOM *STATIC 0.001,0.001 *CLOAD 999,6,10000. *NODE FILE,FREQUENCY=150,NSET=RADIAL

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U,RF *EL FILE,FREQUENCY=150,ELSET=RIM S,E,CE,CEP *OUTPUT,FIELD,Frequency=150 *NODE OUTPUT,NSET=RADIAL U,RF *OUTPUT,HISTORY,FREQ=1 *NODE OUTPUT,NSET=CENTER U, *OUTPUT,HISTORY,Frequency=150 *NODE OUTPUT,NSET=RADIAL U,RF *OUTPUT,VAR=PRESELECT,FIELD,FREQ=999 *OUTPUT,FIELD,Frequency=150 *ELEMENT OUTPUT,ELSET=RIM S,E,CE,CEP *OUTPUT,HISTORY,Frequency=150 *ELEMENT OUTPUT,ELSET=RIM S,E,CE,CEP *END STEP *STEP,NLGEOM,INC=150 *VISCO,CETOL=5.E-4 0.04,1. *END STEP

1.1.14 Indentation of a thick plate Product: ABAQUS/Explicit This example illustrates the use of adaptive meshing in deep indentation problems with graded meshes.

Problem description A deep indentation problem is solved for both axisymmetric and three-dimensional geometries, as shown in Figure 1.1.14-1. Each model consists of a rigid punch and a deformable blank. The punch has a semicircular nose section and a radius of 100 mm. The blank is modeled as a von Mises elastic-plastic material with a Young's modulus of 3 ´ 106 MPa, an initial yield stress of 1.5 ´ 105 MPa, and a constant hardening slope of .45 ´ 105 MPa. Poisson's ratio is 0.3; the density is 1.0 ´ 10-5 kg/mm3. In both cases the punch is fully constrained except in the vertical direction. A deep indentation is made by moving the punch into the blank to a depth of 250 mm. The displacement of the punch is prescribed using the SMOOTH STEP parameter on the *AMPLITUDE option so that a quasi-static response is generated.

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Case 1: Axisymmetric model The blank is meshed with CAX4R elements and measures 300 ´ 300 mm. The punch is modeled as an analytical rigid surface using the *SURFACE, TYPE=SEGMENTS option in conjunction with the *RIGID BODY option. The bottom of the blank is constrained in the x- and z-directions, and symmetry boundary conditions are prescribed at r=0.

Case 2: Three-dimensional models Two models are analyzed. For one model the blank is meshed uniformly, while for the other a graded mesh is used. For both models the blank is meshed with C3D8R elements and measures 600 ´ 300 ´ 600 mm. The punch is modeled as an analytical rigid surface using the *SURFACE, TYPE=REVOLUTION option in conjunction with the *RIGID BODY option. The bottom of the blank is fully constrained.

Adaptive meshing A single adaptive mesh domain that incorporates the entire blank is used for each model. A Lagrangian boundary region type (the default) is used to define the constraints along the bottom of the plate for both models and along the axis of symmetry in two dimensions. A sliding boundary region (the default) is used to define the contact surface on the plate. To obtain a good mesh throughout the simulation, the number of mesh sweeps is increased to 3 using the MESH SWEEPS parameter on the *ADAPTIVE MESH option. For the graded three-dimensional model the SMOOTHING OBJECTIVE parameter is set to GRADED on the *ADAPTIVE MESH CONTROLS option to preserve the gradation of the mesh while adaptive meshing is performed.

Results and discussion Figure 1.1.14-2 to Figure 1.1.14-4show the initial configurations for the axisymmetric model, the three-dimensional uniform mesh model, and the three-dimensional graded mesh model. Although the punch is not shown in these figures, it is initially in contact with the plate. Figure 1.1.14-5 shows the final deformed mesh for the axisymmetric indentation. The meshing algorithm attempts to minimize element distortion both near and away from the contact surface with the punch. Figure 1.1.14-6 and Figure 1.1.14-7show the deformed mesh of the entire blank and a quarter-symmetry, cutaway view, respectively, for the three-dimensional model with an initially uniform mesh. Even under this depth of indentation, elements appear to be nicely shaped both on the surface and throughout the cross-section of the plate. Figure 1.1.14-8 and Figure 1.1.14-9show the deformed mesh of the entire plate and a quarter-symmetry, cutaway view, respectively, for the three-dimensional case with an initially graded mesh. Adaptive meshing with the graded smoothing objective preserves the mesh gradation throughout the indentation process while simultaneously minimizing element distortion. Preserving mesh gradation in adaptivity problems is a powerful capability that allows mesh refinement to be concentrated in the areas of highest strain gradients. A contour plot of equivalent plastic strain for the graded mesh case is shown in Figure 1.1.14-10.

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Input files ale_indent_axi.inp Case 1. ale_indent_sph.inp Case 2 with a uniform mesh. ale_indent_gradedsph.inp Case 2 with a graded mesh. ale_indent_sphelset.inp External file referenced by Case 2.

Figures Figure 1.1.14-1 Axisymmetric and three-dimensional model geometries.

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Figure 1.1.14-2 Initial configuration for the axisymmetric model.

Figure 1.1.14-3 Initial configuration for the three-dimensional model with a uniform mesh.

Figure 1.1.14-4 Initial configuration for the three-dimensional model with a graded mesh.

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Figure 1.1.14-5 Deformed configuration for the axisymmetric model.

Figure 1.1.14-6 Deformed configuration for the three-dimensional model with an initially uniform mesh.

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Figure 1.1.14-7 Quarter-symmetry, cutaway view of the deformed configuration for the three-dimensional model with an initially uniform mesh.

Figure 1.1.14-8 Deformed configuration for the three-dimensional model with an initially graded mesh.

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Figure 1.1.14-9 Quarter-symmetry, cutaway view of the deformed configuration for the three-dimensional model with an initially graded mesh.

Figure 1.1.14-10 Contours of equivalent plastic strain for the three-dimensional model with an initially graded mesh.

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Sample listings

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Listing 1.1.14-1 *HEADING ADAPTIVE MESHING EXAMPLE 3D SPHERICAL INDENTATION Units - N, mm, sec *RESTART,TIMEMARKS=YES,WRITE,NUM=10 *NODE,NSET=ALLN 1,0.,300. 10,0.,0. 401,600.,300. 410,600.,0. *NSET,NSET=N1 1, *NSET,NSET=N10 10, *NSET,NSET=N401 401, *NSET,NSET=N410 410, *NFILL,NSET=TOP2D N1,N401,40,10 *NFILL,NSET=BOT2D N10,N410,40,10 *NFILL,NSET=HEAD TOP2D,BOT2D,9,1 *NCOPY, SHIFT, OLD SET=HEAD, NEW SET=TAIL, CHANGE NUMBER=16400 0., 0., 600. 0., 0., -1., 0., 0., 1., 0. *NFILL, NSET=NALL HEAD, TAIL, 40, 410 *ELEMENT,TYPE=C3D8R 1,2,12,11,1,412,422,421,411 *ELGEN,ELSET=BLANK 1,40,10,1,9,1,40,40,410,360 *NCOPY, SHIFT, OLD SET=TOP2D, NEW SET=TOPLAST, CHANGE NUMBER=16400 0., 0., 600. 0., 0., -1., 0., 0., 1., 0. *NFILL, NSET=TOP TOP2D, TOPLAST, 40, 410 *NCOPY, SHIFT, OLD SET=BOT2D, NEW SET=BOTLAST,

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CHANGE NUMBER=16400 0., 0., 600. 0., 0., -1., 0., 0., 1., 0. *NFILL, NSET=BOT BOT2D, BOTLAST, 40, 410 *INCLUDE,INPUT=ale_indent_sphelset.inp *NODE,NSET=NOUT 100000,300.,410.,300. *ELEMENT,TYPE=MASS,ELSET=PMASS 100000,100000 *MASS,ELSET=PMASS 0.2, *SOLID SECTION,ELSET=BLANK,MATERIAL=MAT, CONTROLS=SECT *SECTION CONTROLS,HOURGLASS=STIFFNESS, KINEMATICS=ORTHOGONAL,NAME=SECT *MATERIAL,NAME=MAT *ELASTIC 3.0E6,0.3 *PLASTIC 0.15E5, 0.0 0.6E5, 1.0 *DENSITY 1.E-5, *BOUNDARY BOT,1,3, 100000,1, 100000,3, 100000,4,6, *AMPLITUDE,NAME=RAMPP,TIME=TOTAL TIME, DEFINITION=SMOOTH STEP 0.0,0.0,0.06,-250., *ELSET,ELSET=SMALL,GEN 6495,6505,1 6855,6865,1 7215,7225,1 *NSET,NSET=SMALL 7571,7581,7591,7981,7991,8001,8391,8401, 8411, *SURFACE,TYPE=ELEMENT,NAME=TARGET UPPER,S5 *SURFACE,TYPE=REVOLUTION,NAME=PUNCH 300., 400., 300.,300.,600., 300.

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START,100.,0. CIRCL,0.,-100.,0.,0. *RIGID BODY,REF NODE=100000, ANALYTICAL SURFACE =PUNCH *STEP *DYNAMIC,EXPLICIT ,0.06 *BOUNDARY,AMPLITUDE=RAMPP 100000,2,2,1. *SURFACE INTERACTION,NAME=IMP_TARG *CONTACT PAIR,INTERACTION=IMP_TARG PUNCH,TARGET *HISTORY OUTPUT,TIME=1.E-4 *EL HISTORY,ELSET=SMALL S,LE,LEP,NE,NEP,PEEQ *NODE HISTORY,NSET=SMALL RF,U *ENERGY HISTORY ALLKE,ALLIE,ALLAE,ALLVD,ALLWK,ETOTAL, DT, *FILE OUTPUT,NUMBER INTERVAL=6, TIMEMARKS=YES *EL FILE, ELSET=SMALL S,LE,LEP,NE,NEP *NODE FILE, NSET=NOUT U,RF *ENERGY FILE *ADAPTIVE MESH, ELSET=BLANK, MESH SWEEPS=3 *END STEP

1.1.15 Damage and failure of a laminated composite plate Product: ABAQUS/Standard This example demonstrates how the *USER DEFINED FIELD option and user subroutine USDFLD (``USDFLD,'' Section 23.2.36 of the ABAQUS/Standard User's Manual) can be applied to model nonlinear material behavior in a composite laminate. With this option it is possible to modify the standard linear elastic material behavior (for instance, to include the effects of damage) or to change the behavior of the nonlinear material models in ABAQUS. The material model in this example includes damage, resulting in nonlinear behavior. It also includes various modes of failure, resulting in abrupt loss of stress carrying capacity (Chang and Lessard, 1989). The analysis results are compared with experimental results.

Problem description and material behavior A composite plate with a hole in the center is subjected to in-plane compression. The plate is made of

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Static Stress/Displacement Analyses

24 plies of T300/976 graphite-epoxy in a [(-45/+45) 6 ]s layup. Each ply has a thickness of 0.1429 mm (0.005625 in); thus, the total plate thickness is 3.429 mm (0.135 in). The plate has a length of 101.6 mm (4.0 in) and a width of 25.4 mm (1.0 in), and the diameter of the hole is 6.35 mm (0.25 in). The plate is loaded in compression in the length direction. The thickness of the plate is sufficient that out-of-plane displacements of the plate can be ignored. The compressive load is measured, as well as the length change between two points, originally a distance of 25.4 mm (1.0 in) apart, above and below the hole. The plate geometry and loading are shown in Figure 1.1.15-1. The material behavior of each ply is described in detail by Chang and Lessard. The initial elastic ply properties are longitudinal modulus Ex =156512 MPa (22700 ksi), transverse modulus Ey =12962 MPa (1880 ksi), shear modulus Gxy =6964 MPa (1010 ksi), and Poisson's ratio ºxy =0.23. The material accumulates damage in shear, leading to a nonlinear stress-strain relation of the form 3 °xy = G¡1 xy ¾xy + ®¾xy ;

where Gxy is the (initial) ply shear modulus and the nonlinearity is characterized by the factor ®=2.44´10-8 MPa-3 (0.8´1-5 ksi-3). Failure modes in laminated composites are strongly dependent on geometry, loading direction, and ply orientation. Typically, one distinguishes in-plane failure modes and transverse failure modes (associated with interlaminar shear or peel stress). Since this composite is loaded in-plane, only in-plane failure modes need to be considered, which can be done for each ply individually. For a unidirectional ply as used here, five failure modes can be considered: matrix tensile cracking, matrix compression, fiber breakage, fiber matrix shearing, and fiber buckling. All the mechanisms, with the exception of fiber breakage, can cause compression failure in laminated composites. The failure strength in laminates also depends on the ply layup. The effective failure strength of the layup is at a maximum if neighboring plies are orthogonal to each other. The effective strength decreases as the angle between plies decreases and is at a minimum if plies have the same direction. (This is called a ply cluster.) Chang and Lessard have obtained some empirical formulas for the effective transverse tensile strength; however, in this model we ignore such effects. Instead, we use the following strength properties for the T300/976 laminate: transverse tensile strength Yt =102.4 MPa (14.86 ksi), ply shear strength Sc =106.9 MPa (15.5 ksi), matrix compressive strength Yc =253.0 MPa (36.7 ksi), and fiber buckling strength Xc =2707.6 MPa (392.7 ksi). The strength parameters can be combined into failure criteria for multiaxial loading. Four different failure modes are considered in the model analyzed here. · Matrix tensile cracking can result from a combination of transverse tensile stress, ¾y , and shear stress, ¾xy . The failure index, em , can be defined in terms of these stresses and the strength parameters, Yt and Sc . When the index exceeds 1.0, failure is assumed to occur. Without nonlinear material behavior, the failure index has the simple form, e2m

=

µ

¾y Yt

¶2

+

µ

¾xy Sc

¶2

:

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Static Stress/Displacement Analyses

With nonlinear shear behavior taken into consideration, the failure index takes the more complex form, e2m

=

µ

¾y Yt

¶2

2 4 =Gxy + 3®¾xy 2¾xy + : 2Sc2 =Gxy + 3®Sc4

· Matrix compressive failure results from a combination of transverse compressive stress and shear stress. The failure criterion has the same form as that for matrix tensile cracking: e2m

=

µ

¾y Yc

¶2

2 4 =Gxy + 3®¾xy 2¾xy + : 2Sc2 =Gxy + 3®Sc4

The same failure index is used since the previous two failure mechanisms cannot occur simultaneously at the same point. After the failure index exceeds 1.0, both the transverse stiffness and Poisson's ratio of the ply drop to zero. · Fiber-matrix shearing failure results from a combination of fiber compression and matrix shearing. The failure criterion has essentially the same form as the other two criteria: e2f s

=

µ

¾x Xc

¶2

+

2 4 =Gxy + 3®¾xy 2¾xy : 2Sc2 =Gxy + 3®Sc4

This mechanism can occur simultaneously with the other two criteria; hence, a separate failure index is used. Shear stresses are no longer supported after the failure index exceeds 1.0, but direct stresses in the fiber and transverse directions continue to be supported. · Fiber buckling failure occurs when the maximum compressive stress in the fiber direction ( ¡¾x ) exceeds the fiber buckling strength, Xc , independent of the other stress components: eb = ¡

¾x : Xc

It is obvious that, unless the shear stress vanishes exactly, fiber-matrix shearing failure occurs prior to fiber buckling. However, fiber buckling may follow subsequent to fiber shearing because only the shear stiffness degrades after fiber-matrix shearing failure. Fiber buckling in a layer is a catastrophic mode of failure. Hence, after this failure index exceeds 1.0, it is assumed that the material at this point can no longer support any loads. In this example the primary loading mode is shear. Therefore, failure of the plate occurs well before the fiber stresses can develop to a level where fiber buckling takes place, and this failure mode need not be taken into consideration. Chang and Lessard assume that after failure occurs, the stresses in the failed directions drop to zero immediately, which corresponds to brittle failure with no energy absorption. This kind of failure model usually leads to immediate, unstable failure of the composite. This assumption is not very realistic: in

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reality, the stress-carrying capacity degrades gradually with increasing strain after failure occurs. Hence, the behavior of the composite after onset of failure is not likely to be captured well by this model. Moreover, the instantaneous loss of stress-carrying capacity also makes the postfailure analysis results strongly dependent on the refinement of the finite element mesh and the finite element type used.

Material model implementation To simulate the shear nonlinearity and the failure modes (matrix failure in tension or compression and fiber-matrix shear failure), the elastic properties are made linearly dependent on three field variables. The first field variable represents the matrix failure index, the second represents the fiber-matrix shear failure index, and the third represents the shear nonlinearity (damage) prior to failure. The dependence of the elastic material properties on the field variables is shown in Table 1.1.15-1. To account for the nonlinearity, the nonlinear stress-strain relation must be expressed in a different form: the stress at the end of the increment must be given as a linear function of the strain. The most obvious way to do this is to linearize the nonlinear term, leading to the relation ³ ´ (i+1) (i) 2 (i+1) °xy ¾xy = G¡1 + ® ( ¾ ) ; xy xy where i represents the increment number. This relation can be written in inverted form as (i+1) ¾xy =

Gxy 1+

(i) ®Gxy (¾xy )2

(i+1) °xy ;

thus providing an algorithm to define the effective shear modulus. However, this algorithm is not very suitable because it is unstable at higher strain levels, which is readily demonstrated by stability analysis. Consider an increment where the strain does not change; (i+1) (i) e (i) = °xy = °xy : Let the stress at increment i have a small perturbation from ¾xy , the exact i.e., °xy (i)

(i)

e (i) + ¢¾xy . Similarly, at increment i+1, solution at that increment: ¾xy = ¾xy (i+1)

¾xy

(i+1)

e (i+1) = ¾xy + ¢¾xy

(i+1)

. For the algorithm to be stable, ¢¾xy

(i) ¢¾xy .

should not be larger than (i)

The perturbation in increment i+1 is calculated by substituting ¾xy in the effective shear e (i) modulus equation and linearizing it about ¾xy : (i+1) ¢¾xy

¡2®G2xy ¾xy °xy (i) = ¢¾xy ; 2 2 (1 + ®Gxy ¾xy )

e (i) : The perturbation in increment i+1 is larger than the perturbation in increment i if where ¾xy = ¾xy 2 2 2®G2xy ¾xy °xy > (1 + ®Gxy ¾xy ) ;

which, after elimination of °xy , reduces to the expression

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Static Stress/Displacement Analyses

3 ¡1 ®¾xy > Gxy ¾xy :

Hence, instability occurs when the "nonlinear" part of the shear strain is larger than the "linear" part of the shear strain. To obtain a more stable algorithm, we write the nonlinear stress-strain law in the form 3 3 = G¡1 °xy + ¯¾xy xy ¾xy + (® + ¯ )¾xy ;

where ¯ is an as yet unknown coefficient. In linearized form this leads to the update algorithm (1 +

(i) 3 (i) (i+1) ¯ (¾xy ) =°xy )°xy

=

³

G¡1 xy

+ (® +

(i) 2 ¯ )(¾xy )

´

(i+1) ¾xy ;

or, in inverted form, (i)

(i+1) ¾xy

=

(i)

1 + ¯ (¾xy )3 =°xy 1 + (® +

(i) ¯ )Gxy (¾xy )2

(i+1) Gxy °xy :

Following the same procedure as that for the original update algorithm, it is readily derived that a small (i) perturbation, ¢¾xy , in increment i reduces to zero in increment i+1 if ¯ = 2®. Hence, the optimal algorithm appears to be (i)

(i+1) ¾xy

=

(i)

1 + 2®(¾xy )3 =°xy 1+

(i) 3®Gxy (¾xy )2

(i+1) Gxy °xy :

Finally, this relation is written in terms of the damage parameter d: (i+1) (i+1) ¾xy = (1 ¡ d)Gxy °xy ;

where (i)

d=

(i)

(i)

3®Gxy (¾xy )2 ¡ 2®(¾xy )3 =°xy (i)

1 + 3®Gxy (¾xy )2

:

This relation is implemented in user subroutine USDFLD, and the value of the damage parameter is assigned directly to the third field variable used for definition of the elastic properties. The failure indices are calculated with the expressions discussed earlier, based on the stresses at the start of the increment:

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Static Stress/Displacement Analyses

e2m e2m

=

=

e2f s =

à à Ã

(i)

¾y Yt

(i) ¾y

Yc (i)

¾x Xc

!2 !2 !2

(i)

(i)

2(¾xy )2 =Gxy + 3®(¾xy )4 + 2Sc2 =Gxy + 3®Sc4 (i)

(i)

¾y(i) > 0;

if

¾y(i) < 0;

(i)

2(¾xy )2 =Gxy + 3®(¾xy )4 + 2Sc2 =Gxy + 3®Sc4 +

if

(i)

2(¾xy )2 =Gxy + 3®(¾xy )4 : 2Sc2 =Gxy + 3®Sc4

The values of the failure indices are not assigned directly to the field variables: instead, they are stored as solution-dependent state variables. Only if the value of a failure index exceeds 1.0 is the corresponding user-defined field variable set equal to 1.0. After the failure index has exceeded 1.0, the associated user-defined field variable continues to have the value 1.0 even though the stresses may reduce significantly, which ensures that the material does not "heal" after it has become damaged.

Finite element model The plate consists of 24 plies of T300/976 graphite-epoxy in a [(-45/+45) 6 ]s layup. Instead of modeling each ply individually, we combine all plies in the -45° direction and all plies in the +45° direction. Consequently, only two layers need to be modeled separately: 1. A layer in the -45° direction with a thickness of 1.715 mm (0.0675 in). 2. A layer in the +45° direction with a thickness of 1.715 mm (0.0675 in). The corresponding finite element model consists of two layers of CPS4 plane stress elements, with thicknesses and properties as previously discussed. The quarter-symmetry finite element model is shown in Figure 1.1.15-1. The implementation of nonlinear material behavior with user-defined field variables is explicit: the nonlinearity is based on the state at the start of the increment. Hence, the user must ensure that the time increments are sufficiently small, which is particularly important because the automatic time increment control in ABAQUS is ineffective with the explicit nonlinearity implemented in USDFLD. If automatic time incrementation is used, the maximum time increment can be controlled from within subroutine USDFLD with the variable PNEWDT. This capability is useful if there are other nonlinearities that require automatic time incrementation. In this example the only significant nonlinearity is the result of the material behavior. Hence, fixed time incrementation can be used effectively.

Results and discussion For this problem experimental load-displacement results were obtained by Chang and Lessard. The experimental results, together with the numerical results obtained by ABAQUS, are shown in Figure 1.1.15-2. The agreement between the experimental and numerical results is excellent up to the point where the load maximum is reached. After that, the numerical load-displacement curve drops off sharply, whereas the experimental data indicate that the load remains more or less constant. Chang and Lessard also show numerical results: their results agree with the results obtained by ABAQUS but do not extend to the region where the load drops off. The dominant failure mode in this plate is

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Static Stress/Displacement Analyses

fiber/matrix shear: failure occurs first at a load of approximately 12.15 kN (2700 lbs) and continues to grow in a stable manner until a load of approximately 13.5 kN (3000 lbs) is reached. Figure 1.1.15-3 shows the extent of the damage in the finite element model at the point of maximum load. In this figure an element is shaded if fiber/matrix shear failure has occurred at at least three integration points. These results also show excellent agreement with the results obtained by Chang and Lessard. As discussed earlier, the sharp load drop-off in the numerical results is the result of the lack of residual stress carrying capacity after the failure criterion is exceeded. Better agreement could be reached only if postfailure material data were available. Without postfailure data the results are very sensitive to the mesh and element type, which is clearly demonstrated by changing the element type from CPS4 (full integration) to CPS4R (reduced integration). The results are virtually identical up to the point where first failure occurs. After that point the damage in the CPS4R model spreads more rapidly than in the CPS4 model until a maximum load of about 12.6 kN (2800 lbs) is reached. The load then drops off rapidly. The problem is also analyzed with models consisting of S4R and S4 elements. The elements have a composite section with two layers, with each layer thickness equal to the thickness of the plane stress elements in the CPS4 and CPS4R models. The results that were obtained with the S4R and S4 element models are indistinguishable from those obtained with the CPS4R element model.

Input files damagefailcomplate_cps4.inp CPS4 elements. damagefailcomplate_cps4.f User subroutine USDFLD used in damagefailcomplate_cps4.inp. damagefailcomplate_node.inp Node definitions. damagefailcomplate_element.inp Element definitions. damagefailcomplate_cps4r.inp CPS4R elements. damagefailcomplate_cps4r.f User subroutine USDFLD used in damagefailcomplate_cps4r.inp. damagefailcomplate_s4.inp S4 elements. damagefailcomplate_s4.f User subroutine USDFLD used in damagefailcomplate_s4.inp. damagefailcomplate_s4r.inp

1-231

Static Stress/Displacement Analyses

S4R elements. damagefailcomplate_s4r.f User subroutine USDFLD used in damagefailcomplate_s4r.inp.

Reference · Chang, F-K., and L. B. Lessard, ``Damage Tolerance of Laminated Composites Containing an Open Hole and Subjected to Compressive Loadings: Part I--Analysis,'' Journal of Composite Materials, vol. 25, pp. 2-43, 1991.

Table Table 1.1.15-1 Dependence of the elastic material properties on the field variables. Material State Elastic Properties FV1 FV2 FV3 Ex Ey ºxy Gxy No failure 0 0 0 G Ex 0 xy Matrix failure 0 1 0 0 Ex Ey 0 Fiber/matrix shear 0 0 1 0 Ex Ey ºxy Shear damage 0 0 0 1 Matrix failure and fiber/matrix Ex 0 0 0 1 1 0 shear Matrix failure and shear Ex 0 0 0 1 0 1 damage Fiber/matrix shear and shear Ex Ey 0 0 0 1 1 damage Ex 0 All failure modes 0 0 1 1 1

Figures Figure 1.1.15-1 Plate geometry and loading.

1-232

Static Stress/Displacement Analyses

Figure 1.1.15-2 Experimental and numerical load displacement curves.

Figure 1.1.15-3 Distribution of material damage at maximum load.

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Sample listings

1-234

Static Stress/Displacement Analyses

Listing 1.1.15-1 *HEADING COMPOSITE PLATE WITH EXPLICIT DAMAGE AND CPS4 FAILURE MODEL ** UNITS: IN, LBS ** ------------------------** MODEL DEFINITION ** ------------------------*NODE,INPUT=damagefailcomplate_node.inp,NSET=NALL *ELEMENT,INPUT=damagefailcomplate_element.inp, TYPE=CPS4,ELSET=L1 *ELCOPY,ELEMENT SHIFT=1000,OLD SET=L1,NEW SET=L2, SHIFT NODES=0 *NSET,NSET=XSYMMTRY 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 190, 197, 204, 211, 218, 225, 232, 239, 246, 253, 260, 267, 1000 *NSET,NSET=YSYMMTRY 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 287, 294, 301, 308, 315, 322 *NSET,NSET=TOP 267, 268, 269, 270, 271, 272, 273, 408, 409, 410, 411, 412, 413 ** ** MATERIAL: NONLINEAR SHEAR WITH BUILT-IN ** EXPLICIT FAILURE ** ** FV1: MATRIX COMPRESSIVE/TENSILE FAILURE ** FV2: FIBER-MATRIX SHEAR FAILURE ** FV3: SHEAR NONLINEARITY (DAMAGE) PRIOR TO ** FAILURE ** TOTAL OF 2^3 = 8 STATES ** *MATERIAL,NAME=T300 *ELASTIC,TYPE=LAMINA,DEPENDENCIES=3 22.7E6,1.88E6,0.23,1.01E6,1.01E6,1.01E6,0.,0, 0,0 22.7E6,1.00E0,0.00,1.01E6,1.01E6,1.01E6,0.,1, 0,0 22.7E6,1.88E6,0.00,1.00E0,1.01E6,1.01E6,0.,0,

1-235

Static Stress/Displacement Analyses

1,0 22.7E6,1.00E0,0.00,1.00E0,1.01E6,1.01E6,0.,1, 1,0 22.7E6,1.88E6,0.23,1.00E0,1.01E6,1.01E6,0.,0, 0,1 22.7E6,1.00E0,0.00,1.00E0,1.01E6,1.01E6,0.,1, 0,1 22.7E6,1.88E6,0.00,1.00E0,1.01E6,1.01E6,0.,0, 1,1 22.7E6,1.00E0,0.00,1.00E0,1.01E6,1.01E6,0.,1, 1,1 *DEPVAR 3, *USER DEFINED FIELD ** ** LOCAL ORIENTATIONS: P45 AT +45, N45 AT -45 ** *ORIENTATION,NAME=P45 0.,1.,0.,-1.,0.,0. 3,+45.0 *ORIENTATION,NAME=N45 0.,1.,0.,-1.,0.,0. 3,-45.0 *SOLID SECTION,ELSET=L1,ORIENTATION=P45, MATERIAL=T300 0.06750, *SOLID SECTION,ELSET=L2,ORIENTATION=N45, MATERIAL=T300 0.06750, ** ** CONSTRAIN TOP FOR THE PURPOSES OF LOAD ** APPLICATION ** *EQUATION 2, TOP,2,1.0,1000,2,-1.0 ** ** CONSTRAIN OUTPUT NODE BETWEEN SURROUNDING NODES *MPC LINEAR,2000,197,204 *NSET,NSET=OUTPUT 1000,2000,204,197 *ELSET,ELSET=EOUT,GENERATE

1-236

Static Stress/Displacement Analyses

133,144 11133,11144 ** ------------------** ANALYSIS HISTORY ** ------------------*STEP,INC=200,NLGEOM *STATIC,DIRECT 0.05,1.0 *BOUNDARY XSYMMTRY,XSYMM YSYMMTRY,YSYMM 1000,2,,-0.027 *RESTART,WRITE,OVERLAY *NODE PRINT,NSET=OUTPUT U2,RF2 *NODE FILE,NSET=OUTPUT U,RF *EL FILE,ELSET=EOUT S,E SDV,FV NE,LE *EL PRINT,FREQUENCY=5,ELSET=EOUT S,E *END STEP

1.1.16 Analysis of an automotive boot seal Product: ABAQUS/Standard Boot seals are used to protect constant velocity joints and steering mechanisms in automobiles. These flexible components must accommodate the motions associated with angulation of the steering mechanism. Some regions of the boot seal are always in contact with an internal metal shaft, while other areas come into contact with the metal shaft during angulation. In addition, the boot seal may also come into contact with itself, both internally and externally. The contacting regions affect the performance and longevity of the boot seal. In this example the deformation of the boot seal, caused by a typical angular movement of the shaft, is studied. It provides a demonstration and verification of the finite-sliding capability in three-dimensional deformable-to-deformable contact in ABAQUS.

Geometry and model The boot seal with the internal shaft is shown in Figure 1.1.16-1. The boot seal and shaft are modeled as separate parts, each instanced once. Symmetry is utilized to model one-half of the boot seal. The corrugated shape of the boot seal tightly grips the steering shaft at one end, while the other end is

1-237

Static Stress/Displacement Analyses

fixed. The rubber seal is modeled with 1728 first-order, hybrid brick elements with two elements through the thickness. The seal has a nonuniform thickness varying from a minimum of 3.0 mm to a maximum of 4.75 mm at the fixed end. The rubber is modeled as a slightly compressible Mooney-Rivlin (hyperelastic) material with C10 = 0.752 MPa and D1 = 0.026 MPa -1. The internal shaft is considered to be rigid and is modeled as an analytical rigid surface; the radius of the shaft is 14 mm. The rigid body reference node is located precisely in the center of the constant velocity joint. Contact is specified between the rigid shaft and the regions on the inner surface of the seal that are likely to come into contact with the shaft. In Figure 1.1.16-2 the slave surface on the interior of the seal is shown. Contact is also specified between facing regions on the inner and outer surfaces of the seal. Figure 1.1.16-3 shows the surfaces on the interior of the seal that are likely to come into contact with each other, and Figure 1.1.16-4 shows the surfaces on the exterior that may come into contact with each other. The interactions between the appropriate pairs are specified by using the *CONTACT PAIR option. If a three-dimensional deformable master surface is defined from a large number of underlying elements and the equations would be ordered to account for any possible contact between the slave and master surface, the resulting wavefront would be very large. Hence, by default, ABAQUS/Standard employs an automated moving contact patch and equation reordering algorithm to reduce the wavefront and and solution time (see ``Common difficulties associated with contact modeling,'' Section 21.10.1 of the ABAQUS/Standard User's Manual). Alternatively, the user can specify a fixed, non-moving, contact patch size with the SLIDE DISTANCE parameter on the *CONTACT CONTROLS option. In such a case the maximum slide distance and patch location remain fixed throughout the analysis. This can be more efficient if the relative motion of the slave and master surfaces is limited to a few elements; although the wavefront might not be optimal, the avoidance of reordering may yield some cost savings. As an illustration, an additional input file has been included in which the SLIDE DISTANCE parameter is utilized. Due to the motion of the shaft, slave nodes will slide more in the longitudinal direction than in the circumferential direction. We can assume the maximum sliding distance of each slave node to be equal to the width of the master surface with which it comes into contact (see Figure 1.1.16-3 and Figure 1.1.16-4).

Loading The mounting of the boot seal and the angulation of the shaft are carried out in a two-step analysis. The inner radius at the neck of the boot seal is smaller than the radius of the shaft so as to provide a tight fit between the seal and the shaft. In the first step the initial interference fit is resolved, corresponding to the assembly process of mounting the boot seal onto the shaft. The automatic "shrink" fit method is utilized by including the SHRINK parameter on the *CONTACT INTERFERENCE option. The second step simulates the angulation of the inner shaft by specifying a finite rotation of 20° at the rigid body reference node of the shaft. Symmetry boundary conditions are specified for all nodes lying in the symmetry plane of the model. The wider end of the boot seal is constrained completely.

Results and discussion 1-238

Static Stress/Displacement Analyses

Figure 1.1.16-5 shows the deformed configuration of the model. The rotation of the shaft causes the stretching of one side and compression on the other side of the boot seal. The surfaces have come into contact on the compressed side. Figure 1.1.16-6 shows the contours of maximum principal stresses in the boot seal. Comparison of the analysis times when using fixed and automated contact patches shows that both analyses complete in approximately the same amount of time. This can be expected for this type of problem since the fixed contact patches are limited in size to a few elements. For the case with fixed contact patches the wavefront is somewhat larger, requiring more memory and solution time per iteration. However, this is offset by the time required to form new contact patches and to reorder the equations for the case with automatic contact patches.

Input files bootseal.inp Analysis with automatic contact patches. bootseal_fixed_cpatch.inp Analysis with fixed contact patches. bootseal_mesh.inp Node and element definitions.

Figures Figure 1.1.16-1 Undeformed model.

Figure 1.1.16-2 The non-shaded surface is the slave surface on the seal that may come into contact

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Static Stress/Displacement Analyses

with the shaft.

Figure 1.1.16-3 Surfaces on the interior of the seal that may come into contact with each other. The shaded surface is the master surface.

Figure 1.1.16-4 Surfaces on the exterior of the seal that may come into contact with each other. The shaded surface is the master surface.

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Static Stress/Displacement Analyses

Figure 1.1.16-5 Deformed configuration of the model.

Figure 1.1.16-6 Contours of maximum principal stress in the seal.

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Sample listings

1-242

Static Stress/Displacement Analyses

Listing 1.1.16-1 *HEADING Analysis of automotive boot seal. *RESTART,WRITE,FREQ=1 ** ** read nodes and elements from external file ** 1728 C3D8H elements ** 2849 nodes ** create element sets: OUT_SLAVE,OUT_MASTER, ** BOOT_IN,IN_SLAVE, ** and IN_MASTER ** create node sets: MGS,ZSYMM,and REFNODE ** *PART,NAME=SEAL *INCLUDE, INPUT=bootseal_mesh.inp *SOLID SECTION,ELSET=BOOT,MATERIAL=RUBBER *SURFACE, NAME=OUT_SLAV OUT_SLAVE,S4 *SURFACE, NAME=OUT_MAST OUT_MASTER,S4 *SURFACE, NAME=BOOT_IN BOOT_IN,S6 *SURFACE, NAME=IN_SLAVE IN_SLAVE,S6 *SURFACE, NAME=IN_MASTE IN_MASTER,S1 *END PART *PART,NAME=SHAFT *NODE,NSET=REFNODE 9999, 0.0000E+00,-3.3000E+01, 0.0000E+00 *RIGID BODY,ANALYTICAL SURFACE=SHAFT,REF NODE=9999 *SURFACE,TYPE=REVOLUTION,NAME=SHAFT START, 14.,150. LINE, 14.,0. *END PART *ASSEMBLY,NAME=BOOT-SEAL *INSTANCE,NAME=SEAL-1,PART=SEAL *END INSTANCE *INSTANCE,NAME=SHAFT-1,PART=SHAFT *END INSTANCE *NSET, NSET=NHIST

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Static Stress/Displacement Analyses

SHAFT-1.REFNODE, SEAL-1.1, SEAL-1.6001 *ELSET, ELSET=EHIST SEAL-1.6052, *END ASSEMBLY *MATERIAL,NAME=RUBBER *HYPERELASTIC,N=1 0.752,0., .026 *CONTACT PAIR,INTERACTION=OUT_SELF BOOT-SEAL.SEAL-1.OUT_SLAV, BOOT-SEAL.SEAL-1.OUT_MAST *SURFACE INTERACTION, NAME=OUT_SELF *CONTACT PAIR,INTERACTION=IN_SELF BOOT-SEAL.SEAL-1.IN_SLAVE, BOOT-SEAL.SEAL-1.IN_MASTE *SURFACE INTERACTION, NAME=IN_SELF *CONTACT PAIR,INTERACTION=SHAFT_C BOOT-SEAL.SEAL-1.BOOT_IN,BOOT-SEAL.SHAFT-1.SHAFT *SURFACE INTERACTION, NAME=SHAFT_C *BOUNDARY BOOT-SEAL.SEAL-1.MGS, 1, 3, 0.0 BOOT-SEAL.SEAL-1.ZSYMM, ZSYMM ** PERFORM INTERFERENCE FIT BETWEEN SHAFT AND BOOT *STEP,NLGEOM,INC=20 *STATIC .25,1. *CONTACT CONTROLS, SLAVE=BOOT-SEAL.SEAL-1.OUT_SLAV, MASTER=BOOT-SEAL.SEAL-1.OUT_MAST, SLIDEDISTANCE=15.0 *CONTACT CONTROLS, SLAVE=BOOT-SEAL.SEAL-1.IN_SLAVE, MASTER=BOOT-SEAL.SEAL-1.IN_MASTE, SLIDEDISTANCE=15.0 *BOUNDARY BOOT-SEAL.SHAFT-1.REFNODE, 1,6 *CONTACT INTERFERENCE, SHRINK BOOT-SEAL.SEAL-1.BOOT_IN,BOOT-SEAL.SHAFT-1.SHAFT *PRINT,CONTACT=YES *EL PRINT,FREQ=0 *NODE PRINT,NSET=BOOT-SEAL.SHAFT-1.REFNODE *NODE PRINT,FREQ=0 *NODE FILE, NSET=BOOT-SEAL.NHIST, FREQ=999 U, RF *ENERGY FILE, ELSET=BOOT-SEAL.SEAL-1.BOOT, FREQ=999 ELSE, *END STEP ** ROTATE SHAFT 20 DEGREES (.349 RADIANS) *STEP,NLGEOM,INC=20 *STATIC

1-244

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.1,1. *BOUNDARY, TYPE=VELOCITY BOOT-SEAL.SHAFT-1.REFNODE, 6,6, .349 *NODE FILE, NSET=BOOT-SEAL.NHIST, FREQ=1 U, RF *EL FILE, ELSET=BOOT-SEAL.EHIST, FREQ=1 S, *ENERGY FILE, ELSET=BOOT-SEAL.SEAL-1.BOOT, FREQ=1 ELSE, *END STEP

1.1.17 Pressure penetration analysis of an air duct kiss seal Product: ABAQUS/Standard Seals are common structural components that often require design analyses. ABAQUS can be used to perform nonlinear finite element analyses of seals and provide information needed to determine the seal performance. Information such as a load-deflection curve, seal deformation and stresses, and contact pressure distribution is readily obtained in these analyses. ABAQUS allows for pressure penetration effects between the seal and the contacting surfaces to be considered in these analyses, making routine analyses of seals more realistic and accurate. Analyses of clutch seals, threaded connectors, car door seals, and air duct kiss seals are some applications where pressure penetration effects are important. The surface-based pressure penetration capability is used to simulate pressure penetration between contacting surfaces. It is invoked by using the *PRESSURE PENETRATION option, which is described in ``Pressure penetration loading,'' Section 21.3.5 of the ABAQUS/Standard User's Manual. This capability is provided for simulating cases where a joint between two deforming bodies (for example, between two components threaded onto each other) or between a deforming body and a rigid surface (such as a soft gasket used in a joint) is exposed at one or multiple ends to a fluid or air pressure. This air pressure will penetrate into the joint and load the surfaces forming the joint until some area of the surfaces is reached where the contact pressure between the abutting surfaces exceeds the critical value specified on the *PRESSURE PENETRATION option, cutting off further penetration.

Geometry and model The major consideration in an air duct kiss seal design is to provide sealing while avoiding excessive closure force. A poorly designed air duct seal that minimizes the amount of effort to close the fan cowl door may fail to prevent leakage and reduce wind noise. The model used in this example is a simplified version of an air duct kiss seal. It illustrates how pressure penetration effects can be modeled using ABAQUS. The seal modeled is a rolled shape seal. An axisymmetric model of the seal is developed, as shown in Figure 1.1.17-1. The top horizontal rigid surface represents the air fan cowl door, and the bottom horizontal rigid surface represents the seal groove. The rolled seal is 2.54 mm (0.1 in) thick and 74.66

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mm (2.9 in) high; and its inner diameters at the top and bottom surfaces are 508.5 mm (20 in) and 528.3 mm (20.8 in), respectively. A folded metal clip is partially bonded to the top surface of the seal. The thickness of the metal clip is 0.48 mm (0.019 in). The material of the seal is taken to be an incompressible rubberlike material. To obtain the material constants, the Ogden form of the strain energy function with N = 4 is used to fit the uniaxial test data. The metal clip is made of steel, with a Young's modulus of 206.8 GPa (3.0 ´ 107 lb/in2) and a Poisson's ratio of 0.3. CAX4H elements are used to model the seal and the metal clip. The contact pair approach is used to model the contact between the top surface of the metal clip and the top rigid surface representing the fan cowl door, where the pressure penetration is likely to occur. The contact pair approach is also used to model the contact between the seal and the bottom rigid surface, the contact between the seal and the unbonded portion of the metal clip, and the self contact of the seal. The mechanical interaction between the contact surfaces is assumed to be frictional contact. Therefore, the *FRICTION option is used to specify friction coefficients. To increase computational efficiency, the SLIP TOLERANCE parameter on the *FRICTION option is used for the contact surfaces between the seal and the metal clip because the dimensions of these elements vary greatly. Fixed boundary conditions are applied initially to the reference node of the top rigid surface, 5001, and the reference node of the bottom rigid surface, 5002. The vertical edge at the bottom of the seal is constrained such that it cannot be moved in the x-direction. The bottom node of the vertical edge, 1, touches the bottom rigid surface and is held fixed in the y-direction. The top rigid surface is located initially 1.27 mm (0.05 in) above the top surface of the metal clip. The seal and the unbonded portion of the clip are loaded by air pressure on all of their inner surfaces and by contact pressure generated by closing the air fan cowl door. Two nonlinear static steps, all of which include large-displacement effects, are used to simulate these loading conditions. In the first step the top rigid surface moves 35.56 mm (1.4 in) downward in the y-direction, simulating the closing of the fan cowl door. In the second step the inner surface of the seal is subjected to a uniform air pressure load of 206.8 KPa (30.0 lb/in 2) since some gaps between the seal and the top rigid surface have been closed. The pressure penetration is simulated between the top surface of the metal clip ( PPRES), which includes 31 elements, and the top rigid surface ( CFACE). Air pressure penetration does not need to be modeled between the metal clip and the seal because they are well bonded. The *PRESSURE PENETRATION option is invoked to define the node exposed to the air pressure, the magnitude of the air pressure, and the critical contact pressure. The surface PPRES is exposed to the air pressure at node 597, with a pressure magnitude of 206.8 KPa (30.0 lb/in 2). A default value of zero for the critical contact pressure is used, indicating that the pressure penetration occurs only when contact at a slave node is lost.

Results and discussion The deformed configuration and the contours of the contact pressures on the seal at the end of Step 1 are shown in Figure 1.1.17-2 and Figure 1.1.17-3. A nonuniform contact pressure is observed along the surface of the seal. The contact pressure at the first five slave nodes is zero.

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The penetrating pressure loads are applied during Step 2. The air pressure is applied immediately to elements associated with the first five slave nodes since the contact pressure there is zero and the pressure penetration criterion is satisfied. The spread of the penetration is captured in Figure 1.1.17-4 through Figure 1.1.17-12, which show the deformed seal, the contact pressure profile, and the air pressure profile corresponding to load increments 2, 8, and 12 of Step 2. The pressures applied to the surface corresponding to these three increments are 3.23 KPa (0.469 lb/in 2), 23.37 KPa (3.39 lb/in 2), and 64.74 KPa (9.39 lb/in 2), respectively. Increased penetrating pressure loads applied in Step 2 further reduce the contact pressure, eventually causing complete air penetration through the seal. The seal was lifted off from the air fan cowl door except at the last slave node, 663, where the contact pressure is well-maintained due to imposed boundary conditions and the air pressures. The development of the weakening of the sealing is captured in Figure 1.1.17-13 through Figure 1.1.17-16, which show the deformed seal and the contact pressure profile corresponding to load increments 14 and at the end of Step 2. The pressures applied to the surface corresponding to these two increments are 85.5 KPa (12.4 lb/in 2) and 206.8 KPa (30.0 lb/in2), respectively. The behavior of the seal throughout the loading histories can be best described by plotting the air penetration distance as a function of the air pressure, as shown in Figure 1.1.17-17. It is clear that air penetration into the seal accelerates only when the pressure is on the order of 65.5 KPa (9.5 lb/in 2). The air completely penetrates through the seal when the pressure is 82.7 KPa (12.0 lb/in 2), which is approximately equal to 80% of the sea level atmospheric pressure.

Input files presspenairductseal.inp Pressure penetration simulation of an air duct kiss seal. presspenairductseal_node.inp Node definitions for the seal model. presspenairductseal_elem_metal.inp Element definitions for the metal part of the seal model. presspenairductseal_elem_rub.inp Element definitions for the rubber part of the seal model.

Figures Figure 1.1.17-1 Air duct kiss seal model.

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Figure 1.1.17-2 Deformed configuration of the seal at the end of Step 1.

Figure 1.1.17-3 Contact stress contours in the seal at the end of Step 1.

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Figure 1.1.17-4 Deformed configuration of the seal at Step 2, increment 2.

Figure 1.1.17-5 Contact stress contours in the seal at Step 2, increment 2.

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Figure 1.1.17-6 Air pressure contours in the seal at Step 2, increment 2.

Figure 1.1.17-7 Deformed configuration of the seal at Step 2, increment 8.

Figure 1.1.17-8 Contact stress contours in the seal at Step 2, increment 8.

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Figure 1.1.17-9 Air pressure contours in the seal at Step 2, increment 8.

Figure 1.1.17-10 Deformed configuration of the seal at Step 2, increment 12.

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Figure 1.1.17-11 Contact stress contours in the seal at Step 2, increment 12.

Figure 1.1.17-12 Air pressure contours in the seal at Step 2, increment 12.

Figure 1.1.17-13 Deformed configuration of the seal at Step 2, increment 14.

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Figure 1.1.17-14 Contact stress contours in the seal at Step 2, increment 14.

Figure 1.1.17-15 Deformed configuration of the seal at the end of Step 2.

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Figure 1.1.17-16 Contact stress contours in the seal at the end of Step 2.

Figure 1.1.17-17 Air penetration distance as a function of air pressure in the seal.

Sample listings

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Listing 1.1.17-1 *HEADING Surface-based Pressure Penetration Analysis of Air Duct Kiss Seal *NODE,NSET=NSEAL, INPUT=presspenairductseal_node.inp *ELEMENT,TYPE=CAX4H,ELSET=STEEL, INPUT=presspenairductseal_elem_metal.inp *ELEMENT,TYPE=CAX4H,ELSET=RUBFIB, INPUT=presspenairductseal_elem_rub.inp *ELSET,ELSET=FACE1,GENERATE 437,439,1 443,452,1 *ELSET,ELSET=FACE3,GENERATE 28,36,1 67,76,1 227,276,1 307,316,1 440,442,1 453,462,1 473,475,1 *ELSET,ELSET=FACE4,GENERATE 437,440,3 *ELSET,ELSET=PPRES 476,477,478,479,480,481,482,503,504, 505,506,507,508,509,510,511,512,513, 514,515,516,517,518,519,520,521,522, 533,534,535,536 *NSET,NSET=EDGE 1,2,3,4,5 *NODE,NSET=BCYL 5001,20.5,2.9,0. 5002,20.5,0.,0. *RIGID BODY,ANALYTICAL SURFACE=CFACE,REFNODE=5001 *SURFACE,NAME=CFACE,TYPE=SEGMENTS START,21.5,2.9, LINE,19.5,2.9 *SURFACE,NAME=PPRES PPRES,S3 *CONTACT PAIR,INTERACTION=PPEN PPRES,CFACE *SURFACE INTERACTION,NAME=PPEN

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*FRICTION .00001, *SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=EXPONENTIAL .0002,1000. *RIGID BODY,ANALYTICAL SURFACE=CFACE2, REF NODE=5002 *SURFACE,NAME=CFACE2,TYPE=SEGMENTS START,19.5,0., LINE,21.5,0. *ELSET,ELSET=ELBOT,GENERATE 1,9,1 *SURFACE,NAME=BOTTOM ELBOT, *CONTACT PAIR, INTERACTION=FLANGE BOTTOM,CFACE2 *SURFACE INTERACTION, NAME=FLANGE *FRICTION .1, *SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=EXPONENTIAL .0005,1000. *ELSET,ELSET=MA1,GENERATE 568,582,1 *ELSET,ELSET=SL1,GENERATE 77,122,1 *SURFACE,NAME=MASTER1 MA1, *SURFACE,NAME=SLAVE1 SL1, *CONTACT PAIR, INTERACTION=SELF1 SLAVE1,MASTER1 *SURFACE INTERACTION, NAME=SELF1 *FRICTION,SLIPTOLERANCE=.2 .1, *SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=EXPONENTIAL .005,1000. *ELSET,ELSET=MA2,GENERATE 307,315,1 *ELSET,ELSET=SL2,GENERATE 443,451,1 *SURFACE,NAME=MASTER2

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MA2, *SURFACE,NAME=SLAVE2 SL2, *CONTACT PAIR, INTERACTION=SELF2 SLAVE2,MASTER2 *SURFACE INTERACTION, NAME=SELF2 *FRICTION,SLIPTOLERANCE=.2 .1, *SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=EXPONENTIAL .0002,1000. *ELSET,ELSET=MA3,GENERATE 28,36,1 *ELSET,ELSET=SL3,GENERATE 227,270,1 *SURFACE,NAME=MASTER3 MA3, *SURFACE,NAME=SLAVE3 SL3, *CONTACT PAIR, INTERACTION=SELF3 SLAVE3,MASTER3 *SURFACE INTERACTION, NAME=SELF3 *FRICTION .1, *SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=EXPONENTIAL .0002,1000. *ELSET,ELSET=MA4,GENERATE 613,622,1 *ELSET,ELSET=SL4,GENERATE 550,562,1 *SURFACE,NAME=MASTER4 MA4, *SURFACE,NAME=SLAVE4 SL4, *CONTACT PAIR, INTERACTION=SELF4 SLAVE4,MASTER4 *SURFACE INTERACTION, NAME=SELF4 *FRICTION,SLIPTOLERANCE=.2 .1, *SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=EXPONENTIAL .001,1000.

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*ELSET,ELSET=MA5,GENERATE 603,610,1 *ELSET,ELSET=SL5,GENERATE 116,124,1 *SURFACE,NAME=MASTER5 MA5, *SURFACE,NAME=SLAVE5 SL5, *CONTACT PAIR, INTERACTION=SELF5 SLAVE5,MASTER5 *SURFACE INTERACTION, NAME=SELF5 *FRICTION .1, *SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=EXPONENTIAL .001,1000. *ELSET,ELSET=EALL STEEL,RUBFIB *SOLID SECTION,ELSET=RUBFIB,MATERIAL=RUBFIB *MATERIAL,NAME=RUBFIB *HYPERELASTIC,OGDEN,N=4,TEST DATA INPUT *UNIAXIAL TEST DATA -211.,-.4 -172.,-.35 -145.,-.30 -122.,-.25 -106.,-.20 -89.,-.15 -66.,-.1 -36.,-.05 0.,0. 132.,.05 231.,.1 297.,.15 363.,.2 594.,.25 924.,.3 1188.,.4 *SOLID SECTION,ELSET=STEEL,MATERIAL=STEEL *MATERIAL,NAME=STEEL *ELASTIC 3.0e+7,.3 *BOUNDARY

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EDGE,1,1 1,2,2 *RESTART,WRITE,FREQUENCY=2 *STEP,UNSYMM=YES,NLGEOM,INC=100 *STATIC .0125,1.,1.E-8,.05 *BOUNDARY,OP=MOD 5001,1,,0. 5001,2,,-1.4 5001,6,,0. 5002,1,,0. 5002,2,,0. 5002,6,,0. *EL PRINT,FREQUENCY=100 S, *NODE PRINT,FREQUENCY=100 U, *NODE PRINT,FREQUENCY=1,NSET=BCYL U1,U2,RF1,RF2 *EL FILE,FREQUENCY=0 *NODE FILE,FREQUENCY=100 U, *CONTACT PRINT,SLAVE=PPRES,MASTER=CFACE,FREQ=100 CSTRESS, *CONTACT FILE,SLAVE=PPRES,MASTER=CFACE,FREQ=100 CSTRESS, *OUTPUT,FIELD,FREQ=100 *CONTACT OUTPUT,SLAVE=PPRES,MASTER=CFACE CSTRESS, *CONTACT CONTROLS, FRICTION ONSET=DELAYED *END STEP *STEP,UNSYMM=YES,NLGEOM,INC=100 *STATIC .0125,1.,1.E-8,.05 *DLOAD,OP=NEW FACE1,P1,30. FACE3,P3,30. FACE4,P4,30. *PRESSURE PENETRATION,MASTER=CFACE,SLAVE=PPRES 597,,30.0 *EL PRINT,FREQUENCY=100 S, *NODE PRINT,FREQUENCY=100

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U, *NODE PRINT,FREQUENCY=1,NSET=BCYL U1,U2,RF1,RF2 *EL FILE,FREQUENCY=100 S, *NODE FILE,FREQUENCY=100 U, *CONTACT PRINT,SLAVE=PPRES,MASTER=CFACE,FREQ=2 CSTRESS,PPRESS *CONTACT FILE,SLAVE=PPRES,MASTER=CFACE,FREQ=2 CSTRESS,PPRESS *OUTPUT,FIELD,FREQ=2 *CONTACT OUTPUT,SLAVE=PPRES,MASTER=CFACE CSTRESS,PPRESS *END STEP

1.1.18 Self-contact in rubber/foam components: jounce bumper Product: ABAQUS/Standard The self-contact capability in ABAQUS is illustrated with two examples derived from the automotive component industry: this problem and the following one, which discusses a rubber gasket. These examples demonstrate the use of the single-surface contact capability available for two-dimensional large-sliding analysis. Components that deform and change their shape substantially can fold and have different parts of the surface come into contact with each other. In such cases it can be difficult to predict at the outset of the analysis where such contact may occur and, therefore, it can be difficult to define two independent surfaces to make up a contact pair. A jounce bumper, sometimes referred to as a "helper spring," is a highly compressible component that is used as part of the shock isolation system in a vehicle. It is typically located above the coil spring that connects the wheels to the frame. Microcellular material is used because of its high compressibility and low Poisson's ratio value at all but fully compressed configurations. The bumper is mounted in a mandrel with a diameter larger than the bumper's inner diameter ( Figure 1.1.18-1). The first step of the analysis solves this interference fit problem. The bumper initially sits against a fixed flat rigid surface on one end; on the other end, another flat rigid surface is used to model the compression of the bumper. The geometry of the bumper is such that it folds in three different locations. Three separate surfaces, one for each such location, are defined in this example. Each surface is allowed to contact itself. This modeling technique produces a more economical analysis because the scope of contact searches and the wavefront of the equation system are minimized.

Geometry and model

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The bumper is 76.0 mm (3.0 in) long and has an inside diameter of 20.0 mm (.8 in). The mandrel, which is modeled as a rigid surface, has a diameter of 22.0 mm (.9 in). The bumper is modeled with the hyperfoam material model. The compressible, nonlinear elastic behavior is described by a strain energy function. The irregular shape of the bumper makes use of an automatic triangular mesher convenient. The model is discretized with first-order triangular axisymmetric elements. In addition to the portions of the bumper's surface used to define self-contact, two additional regions are defined: one to model contact with the fixed surface and the other to model contact with the mandrel and the moving rigid surface. A small amount of friction (a Coulomb coefficient of 0.05) is applied to all of the surfaces.

Results and discussion The bumper analysis is a two-step process. In the first step the interference with the mandrel is resolved using the *CONTACT INTERFERENCE, SHRINK option (see ``Common difficulties associated with contact modeling,'' Section 21.10.1 of the ABAQUS/Standard User's Manual): the calculated initial penetration is allowed at the beginning of the step and scaled linearly to zero at the end of the step (Figure 1.1.18-2). In the second step the bottom surface compresses the bumper 42.0 mm (1.7 in) as a result of the application of displacement boundary conditions to the reference node of the surface (Figure 1.1.18-3). The high compressibility of the material is apparent, as well as the folding of the surface onto itself. Although a general knowledge of where the folding would occur was used in the definition of the self-contacting surfaces, it is not necessary to know exactly where the kinks in the surface will form. The energy absorbing capacity of the device is seen through the load versus displacement curve of the bottom surface (see Figure 1.1.18-4).

Input files selfcontact_bump.inp Jounce bumper model. selfcontact_bump_node.inp Node definitions for the bumper model. selfcontact_bump_element.inp Element definitions for the bumper model.

Figures Figure 1.1.18-1 Jounce bumper initial mesh.

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Figure 1.1.18-2 Bumper mesh after interference is resolved.

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Figure 1.1.18-3 Bumper mesh after crushing.

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Figure 1.1.18-4 Bumper load-displacement curve.

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Sample listings

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Listing 1.1.18-1 *HEADING SELF CONTACT: JOUNCE BUMPER ** *NODE, INPUT=selfcontact_bump_node.inp *NODE, NSET=DIES 1000, 50., 0. 1001, 50., 76.5 1002, 10., 80. ** *NSET, NSET=BOTDIE 1000, *NSET, NSET=TOPDIE 1001, *NSET, NSET=SHAFTDIE 1002, ** *ELEMENT, TYPE=CAX3, ELSET=ALLFEM, INPUT=selfcontact_bump_element.inp ** ** allfem ** *SOLID SECTION, ELSET=ALLFEM, MATERIAL=HYPRFOAM 1., ** ** hyprfoam ** *MATERIAL, NAME=HYPRFOAM ** *HYPERFOAM,N=1 3.0,11.5,.1 ******************************* ** ** Contact definition ** ******************************* *RIGID BODY,ANALYTICAL SURFACE=BOTDIE, REF NODE=1000 *SURFACE,NAME=BOTDIE, TYPE=SEGMENTS START, 0.00, 0.00 LINE, 50.00, 0.00 *RIGID BODY,ANALYTICAL SURFACE=TOPDIE,

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REF NODE=1001 *SURFACE,NAME=TOPDIE, TYPE=SEGMENTS START, 50.00, 76.50 LINE, 0.00, 76.50 *RIGID BODY,ANALYTICAL SURFACE=SHAFTDIE, REF NODE=1002 *SURFACE,NAME=SHAFTDIE, TYPE=SEGMENTS START, 11.00, 80.00 LINE, 11.00, -5.00 ** *ELSET, ELSET=BOTSKIN3, GEN 19, 61, 1 63, 91, 1 *ELSET, ELSET=BOTSKIN3 BOTSKIN3, 210 *ELSET, ELSET=BOTSKIN1, GEN 1321, 1328, 1 ** *SURFACE,NAME=BOTSKIN BOTSKIN3, S3 BOTSKIN1, S1 62, S2 62, S1 ** *ELSET, ELSET=TOPSKIN3, GEN 126, 140, 1 *ELSET, ELSET=TOPSKIN1 141, 1315, 1316, 1318, 1319 ** *SURFACE,NAME=TOPSKIN TOPSKIN3, S3 TOPSKIN1, S1 ** *ELSET, ELSET=SC1,GEN 1, 17, 1 ** *SURFACE,NAME=SC1 SC1, S3 ** *ELSET, ELSET=SC3,GEN 114, 124, 1 ** *SURFACE,NAME=SC3

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SC3, S3 ** *ELSET, ELSET=SC5,GEN 93, 104, 1 ** *SURFACE,NAME=SC5 SC5, S3 ** *CONTACT PAIR,INTERACTION=INT1 TOPSKIN,SHAFTDIE *CONTACT PAIR,INTERACTION=INT1 BOTSKIN,SHAFTDIE *CONTACT PAIR,INTERACTION=INT2 TOPSKIN,TOPDIE *CONTACT PAIR,INTERACTION=INT3 BOTSKIN,BOTDIE *CONTACT PAIR,INTERACTION=INT4 SC1,SC1 *CONTACT PAIR,INTERACTION=INT4 SC3,SC3 *CONTACT PAIR,INTERACTION=INT4 SC5,SC5 *SURFACE INTERACTION,NAME=INT1 *FRICTION .05, *SURFACE INTERACTION,NAME=INT2 *FRICTION .05, *SURFACE INTERACTION,NAME=INT3 *FRICTION .05, *SURFACE INTERACTION,NAME=INT4 *FRICTION .05, ** ******************** ** ** Boundary conditions ** ******************** *BOUNDARY BOTDIE, 1,6,0. TOPDIE, 1,6,0.

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SHAFTDIE, 1,6,0. **************************************** ** ** Step definition: Shrink fit ** **************************************** *STEP, INC=200, NLGEOM *STATIC 1.E-1, 1.0, *CONTACT INTERFERENCE,SHRINK TOPSKIN,SHAFTDIE BOTSKIN,SHAFTDIE *PRINT, CONTACT=YES *CONTACT FILE,FREQ=999 *EL PRINT, ELSET=ALLFEM, FREQ=999 S, E, *NODE PRINT, FREQ=999 *OUTPUT,FIELD,FREQUENCY=200 *NODE OUTPUT U,RF *ELEMENT OUTPUT,ELSET=ALLFEM S,E *CONTACT OUTPUT CSTRESS,CDISP *OUTPUT,HISTORY,FREQUENCY=1 *NODE OUTPUT,NSET=DIES U,RF *CONTACT CONTROLS, FRICTION ONSET=DELAYED *END STEP **************************************** ** ** Step definition ** **************************************** *STEP, INC=200, NLGEOM *STATIC 1.E-1, 1.0,1.e-5 , 0.1 *BOUNDARY,OP=MOD BOTDIE,2, ,42. *END STEP

1.1.19 Self-contact in rubber/foam components: rubber gasket 1-269

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Product: ABAQUS/Standard The self-contact capability in ABAQUS is illustrated with two examples derived from the automotive component industry: this problem and the preceding one, which discusses a jounce bumper. These examples demonstrate the use of the single-surface contact capability available for two-dimensional large-sliding analysis. Components that deform and change their shape substantially can fold and have different parts of the surface come into contact with each other. In such cases it can be difficult to predict at the outset of the analysis where such contact may occur and, therefore, it can be difficult to define two independent surfaces to make up a contact pair. This model is used to analyze an oil pan gasket, which enhances the sealing of the oil pan against the engine block. A primary objective of gasket designers is to reach or exceed a threshold value of contact pressure at the gasket bead/cover/engine block interfaces. Experience shows that, above such a threshold, oil will not leak. Another item of interest is the load-deflection curve obtained when compressing the gasket cross-section since it is indicative of the bolt load required to attain a certain gap between the oil pan and the engine block. Finally, the analysis provides details to ensure that stresses and strains are within acceptable bounds. The gasket is embedded in a plastic backbone. It has two planes of symmetry and a bead that, when compressed, provides the sealing effect (Figure 1.1.19-1). A flat rigid surface, parallel to one of the symmetry planes, pushes the gasket into the backbone. The geometry of the gasket is such that it folds in two different locations. In this model the entire free surface of the gasket and of the backbone is declared as a single surface allowed to contact itself. This modeling technique, although very simple, is more expensive because of the extensive contact searches required, as well as a larger wavefront of the equation system.

Geometry and model The gasket is modeled as a quarter of a plane strain section, initially in contact with a flat rigid surface. The clearance between the plastic backbone and the surface is 0.612 mm (.02 in). The height of the rubber bead in the gasket is 1.097 mm (.04 in). The backbone is modeled with a linear elastic material with a Young's modulus of 8000.0 GPa (1160 ksi) and a Poisson's ratio of 0.4. The gasket is modeled as a fully incompressible hyperelastic material, which is much softer than the backbone material at all strain levels. The nonlinear elastic behavior of the rubber is described by a strain energy function that is a first-order polynomial in the strain invariants. The model is discretized with lower-order quadrilaterals. Standard elements are used for the backbone, and hybrid elements are used for the rubber. The interface between the rubber and the backbone is assumed to be glued with no special treatment required. A single surface definition covers all of the free surface of the gasket and the backbone. Through the definition of contact pairs, this surface is allowed to contact both the rigid surface and itself. A small amount of friction (Coulomb coefficient of 0.05) is applied to the interface with the rigid surface, which is assumed to be lubricated. Sticking friction, through the option *FRICTION, ROUGH (``Interaction tangential to the surface,'' Section 21.3.4 of the ABAQUS/Standard User's Manual), is applied when the rubber contacts itself, denoting a clean surface.

Results and discussion 1-270

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The gasket analysis is a single-step procedure in which the rigid surface moves down almost all of the backbone clearance (0.61 mm or .02 in). The relative rigidity of the backbone forces the rubber gasket to fit inside the cavity provided by the backbone, folding in two regions ( Figure 1.1.19-2). Although the general vicinity of the location of the folds can be estimated from the initial configuration, their exact locations are difficult to predict.

Acknowledgements HKS would like to thank Mr. DeHerrera of Freudenberg-NOK General Partnership for providing these examples.

Input files selfcontact_gask.inp Gasket model. selfcontact_gask_node.inp Node definitions for the gasket model. selfcontact_gask_element1.inp Element definitions for the rubber part of the gasket model. selfcontact_gask_element2.inp Element definitions for the backbone part of the gasket model.

Figures Figure 1.1.19-1 Gasket initial mesh.

Figure 1.1.19-2 Gasket mesh after loading.

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Sample listings

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Listing 1.1.19-1 *HEADING SELF CONTACT: GASKET ** *NODE, INPUT=selfcontact_gask_node.inp, NSET=NSET1 *NODE, NSET=DIE 1001, 8.08006, 2.062 *NSET,NSET=NALL NSET1, DIE ** *NSET, NSET=BOTTOM 370, 404, 432, 460, 486, 508, 532, 561, 587, 609, 633, 657, 681, 694, 702, 703, 706, 707, 708, 709, 710, 711 ** *NSET, NSET=LEFT 126, 153, 186, 222, 256, 292, 330, 368, 369, 370 ** ** *ELEMENT, TYPE=CPE4H, ELSET=GASKET, INPUT=selfcontact_gask_element1.inp *ELEMENT, TYPE=CPE4, ELSET=BACKBONE, INPUT=selfcontact_gask_element2.inp ** *ELSET,ELSET=ALLFEM GASKET,BACKBONE ** ** gasket ** *SOLID SECTION, ELSET=GASKET, MATERIAL=ELSTMR01 1., ** ** backbone ** *SOLID SECTION, ELSET=BACKBONE, MATERIAL=PLASTIC 1., ** ** symm ** *BOUNDARY LEFT , 1,, 0. BOTTOM, 2,, 0.

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** ** plastic ** *MATERIAL, NAME=PLASTIC ** *ELASTIC, TYPE=ISO 8000., 0.4 ** ** Elastomer ** *MATERIAL, NAME=ELSTMR01 ** *HYPERELASTIC, N=1 0.35, 0.25 ***************************** ** ** Contact definition ** ***************************** *RIGID BODY,ANALYTICAL SURFACE=DIE,REF NODE=1001 *SURFACE,NAME=DIE, TYPE=SEGMENTS START, 8.080, 2.062 LINE, 0.000, 2.062 ** *ELSET, ELSET=SKIN4, GEN 1, 64, 1 626, 638, 1 ** *SURFACE,NAME=SKIN SKIN4, S4 65, S3 ** *CONTACT PAIR,INTERACTION=INT2 SKIN,DIE *CONTACT PAIR,INTERACTION=INT3 SKIN,SKIN *SURFACE INTERACTION,NAME=INT2 *FRICTION .05, *SURFACE INTERACTION,NAME=INT3 *FRICTION, ROUGH *CONTACT DAMPING, DEFINITION=DAMPING COEFFICIENT 1.0,0.02,

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** ******************** ** ** Boundary conditions ** ******************** *BOUNDARY, OP=NEW ** ** topdie ** DIE, 1,6, 0. **************************************** ** ** Step definition ** **************************************** *STEP, INC=200, NLGEOM *STATIC 0.05 , 1.0, , 0.05 *CONTACT CONTROLS,FRICTION ONSET=IMMEDIATE *BOUNDARY,OP=MOD DIE,2, ,-0.61 *PRINT, CONTACT=YES *CONTACT FILE,FREQ=999 *CONTACT PRINT,FREQ=999 *EL PRINT, ELSET=GASKET, FREQ=999 S, E, *NODE PRINT, FREQ=999 U, *OUTPUT,FIELD,FREQUENCY=200 *NODE OUTPUT,NSET=NALL U, *ELEMENT OUTPUT,ELSET=ALLFEM S, *CONTACT OUTPUT CSTRESS,CDISP *OUTPUT,HISTORY,FREQUENCY=1 *NODE OUTPUT,NSET=DIE U,RF *END STEP

1.1.20 Submodeling of a stacked sheet metal assembly 1-275

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Product: ABAQUS/Standard Sheet metal stampings stacked and fitted on top of each other and secured together via mechanical fasteners such as bolts or rivets are commonly used in the automotive industry. Examples include seat belt anchors and seating track assemblies. The submodeling capability in ABAQUS facilitates economical, yet detailed, prediction of the ultimate strength and integrity of such jointed assemblies. A global model analysis of an assembly is first performed to capture the overall deformation of the system. Subsequently, the displacement results of this global analysis are used to drive the boundaries of a submodeled region of critical concern. The submodeling methodology provides accurate modeling that is more economical than using a globally refined mesh in a single analysis. In a finite element analysis of such a structure, shell elements are commonly used to represent the sheet metal stampings. The nodes of each shell typically lie along the mid-plane of the shell thickness. The thickness of the shells is used in the structural calculations but is not taken into account in the contact calculations. Hence, a structure composed of a stack-up of several sheet stampings may have the nodes of each sheet all lying in the same spatial plane. This close proximity creates uncertainty in a submodel analysis since ABAQUS will not be able to determine the correct correspondence between the sheets in the submodel and the global model. Therefore, ABAQUS provides a capability that allows the user to specify particular elements of the global model that are used to drive a particular set of nodes in a submodel, which eliminates the uncertainty. This capability is demonstrated in this example problem.

Geometry and model The global model consists of five separate metal stampings meshed with S4R and S3R shell elements. An exploded view of the global finite element model is shown in Figure 1.1.20-1. The stampings are stacked one upon the other by collapsing the configuration in the 3-direction. All the shell elements are 0.5 mm thick, with all nodes positioned at the mid-surface of each shell. The separate meshes are connected together with BEAM-type MPCs between corresponding perimeter nodes on the large bolt holes through each layer. The nodes on the edges of the two small holes at the bottom of Layer 1 are constrained in all six degrees of freedom, representing the attachment point to ground. The translational degrees of freedom of the nodes around the perimeter of Layer 2 are also constrained, representing the far-field boundary condition in that plate. Several surface definitions are used to model the contact between the various adjacent layers. The contact definitions prevent unwanted penetration between shell element layers. The small-sliding contact formulation is employed. Most of the contact in this problem is between adjacent layers, but there is also direct contact between Layer 2 and Layer 4. To avoid overconstraints, it is important that no point on Layer 4 simultaneously contact Layer 3 and Layer 2; therefore, node-based surfaces are used for the slave surfaces. This precludes accurate calculation of contact stresses, but that is not important in this case since more accurate contact stresses are obtained in the submodel. All five stampings are made of steel and are modeled as an elastic-plastic material. The elastic modulus is 207,000 MPa, Poisson's ratio is 0.3, and the yield stress is 250 MPa. The *PLASTIC definition includes moderate strain hardening. The submodel stampings are truncated versions of the global model, located in the same physical position as the global model. In this case these are the regions of concern for high stresses and

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potential failure of the joint. The submodel is discretized with a finer mesh than the global model to provide a higher level of accuracy. Figure 1.1.20-2 shows an exploded view of the submodel. Because the stampings in the submodel contain the large bolt holes, the submodel contains BEAM-type MPCs in a manner analogous to that in the global model. The submodel has several surface definitions and contact pairs to avoid penetration of one stamping into another. The submodel contains no node-based surfaces, however. The contact is modeled as element-based surface-to-surface in each layer. The material definition and shell thicknesses in the submodel are the same as those in the global model.

Results and discussion The global model is loaded by enforcing prescribed boundary conditions on the protruding edge of Layer 3. This edge is displaced -5.0 mm in the 1-direction and -12.5 mm in the 3-direction. Figure 1.1.20-3 shows the deformed shape of the global model. The displacements at the nodes are saved to the results file for later use by the submodel analysis. The submodel driven nodes are loaded using the *BOUNDARY option with the SUBMODEL parameter. The perimeter nodes of each layer of the submodel that correspond to a "cut" out of the global geometry are driven by the interpolated nodal displacement results in the global results file. Each driven node set is in a separate shell layer. Therefore, the submodel contains multiple *SUBMODEL options, which designate the global model element sets to be searched for the responses that drive the submodel driven node sets. For example, the driven nodes in submodel Layer 1 (node set L1BC) are driven by the results for the global element set which contains the elements of (global) Layer 1 (element set LAYER1). Similar commands exist for Layers 2-4. Because submodel Layer 5 has no driven nodes, only four *SUBMODEL options are required. *SUBMODEL, L1BC, *SUBMODEL, L2BC, *SUBMODEL, L3BC, *SUBMODEL, L4BC ,

GLOBAL ELSET=LAYER1 GLOBAL ELSET=LAYER2 GLOBAL ELSET=LAYER3 GLOBAL ELSET=LAYER4

Figure 1.1.20-4 shows the deformed shape of the submodel. Figure 1.1.20-5 and Figure 1.1.20-6 show contour plots of the out-of-plane displacements in Layer 2 for the global model and submodel, respectively. In both cases the displacement patterns are similar; however, the maximum displacement predicted by the global model is about 7.8% larger than that predicted by the submodel.

Input files stackedassembly_s4r_global.inp S4R global model.

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stackedassembly_s4r_global_mesh.inp Key input data for the S4R global model. stackedassembly_s4r_sub.inp S4R submodel. stackedassembly_s4r_sub_mesh.inp Key input data for the S4R submodel.

Figures Figure 1.1.20-1 Exploded view of global model.

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Figure 1.1.20-2 Exploded view of submodel.

Figure 1.1.20-3 Deformed shape of global model.

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Figure 1.1.20-4 Deformed shape of submodel.

Figure 1.1.20-5 Out-of-plane displacement in Layer 2, global model.

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Figure 1.1.20-6 Out-of-plane displacement in Layer 2, submodel.

Sample listings

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Listing 1.1.20-1 *HEADING GLOBAL MODEL FOR SUBMODELING EXAMPLE STACKEDASSEMBLY_S4R_GLOBAL MULTIPLE *SUBMODEL OPTIONS *INCLUDE,INPUT=stackedassembly_s4r_global_mesh.inp *SHELLSECTION,ELSET=LAYER1,MATERIAL=STEEL 0.5, *SHELLSECTION,ELSET=LAYER2,MATERIAL=STEEL 0.5, *SHELLSECTION,ELSET=LAYER3,MATERIAL=STEEL 0.5, *SHELLSECTION,ELSET=LAYER4,MATERIAL=STEEL 0.5, *SHELLSECTION,ELSET=LAYER5,MATERIAL=STEEL 0.5, *MATERIAL,NAME=STEEL *DENSITY 7.8E-09, *ELASTIC 207000.,0.28 *PLASTIC 250.0,0.0 420.0,0.2 *SURFACE, NAME=BLAYER2 LA2SPOS,SPOS LA2SNEG,SNEG *SURFACE, NAME=TLAYER2 LA2SPOS,SNEG LA2SNEG,SPOS *SURFACE, NAME=TLAYER3 LAYER3,SNEG *SURFACE, NAME=TLAYER4 TLAYER4,SPOS *SURFACE, TYPE=NODE, NAME=CNODE1 CNODE1, *SURFACE, TYPE=NODE, NAME=CNODE3 CNODE3, *SURFACE, TYPE=NODE, NAME=CNODE4A CNODE4A, *SURFACE, TYPE=NODE, NAME=CNODE4B CNODE4B,

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*SURFACE, TYPE=NODE, NAME=CNODE5 CNODE5, *SURFACE INTERACTION,NAME=INTER1 *CONTACT PAIR,INTERACTION=INTER1, SMALL SLIDING CNODE1,BLAYER2 CNODE3,TLAYER2 CNODE4A,TLAYER3 CNODE4B,TLAYER2 CNODE5,TLAYER4 *BOUNDARY BC1,1,6 BC3,1,6 *RESTART,WRITE,OVERLAY *STEP,INC=999,NLGEOM *STATIC 0.025,1.0 *BOUNDARY BC2,3,3,-12.5 BC2,1,1,-5.0 *ELPRINT,FREQ=0 *NODEPRINT,FREQ=0 *NODEFILE,NSET=ALLNO,FREQ=100 U, *END STEP

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Listing 1.1.20-2 *HEADING SUBMODEL FOR SUBMODELING EXAMPLE STACKEDASSEMBLY_S4R_SUB MULTIPLE *SUBMODEL OPTIONS *INCLUDE,INPUT=stackedassembly_s4r_sub_mesh.inp *SHELLSECTION,ELSET=LAYER1,MATERIAL=STEEL 0.5, *SHELLSECTION,ELSET=LAYER2,MATERIAL=STEEL 0.5, *SHELLSECTION,ELSET=LAYER3,MATERIAL=STEEL 0.5, *SHELLSECTION,ELSET=LAYER4,MATERIAL=STEEL 0.5, *SHELLSECTION,ELSET=LAYER5,MATERIAL=STEEL 0.5, *MATERIAL,NAME=STEEL *ELASTIC 207000.0,0.3 *PLASTIC 250.0,0.0 420.0,0.2 *SURFACE, NAME=TLAYER1 OLAYER1,SNEG *SURFACE, NAME=TLAYER2 LAYER2,SNEG *SURFACE, NAME=BLAYER2 LAYER2,SPOS *SURFACE, NAME=TLAYER3 LAYER3,SNEG *SURFACE, NAME=BLAYER3 OLAYER3,SPOS *SURFACE, NAME=TLAYER4 OLAYER4,SNEG *SURFACE, NAME=BLAYER4 OLAYER4,SPOS *SURFACE, NAME=BLAYER5 LAYER5,SPOS *SURFACE INTERACTION,NAME=ROUGH *CONTACT PAIR,INTERACTION=ROUGH, SMALL SLIDING TLAYER1,BLAYER2

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BLAYER3,TLAYER2 BLAYER4,TLAYER3 BLAYER5,TLAYER4 BLAYER4,TLAYER2 *SUBMODEL,ABSOLUTE EXTERIOR TOLERANCE=2.75, GLOBAL ELSET=LAYER1 L1BC, *SUBMODEL,ABSOLUTE EXTERIOR TOLERANCE=2.75, GLOBAL ELSET=LAYER2 L2BC, *SUBMODEL,ABSOLUTE EXTERIOR TOLERANCE=2.75, GLOBAL ELSET=LAYER3 L3BC, *SUBMODEL,ABSOLUTE EXTERIOR TOLERANCE=2.75, GLOBAL ELSET=LAYER4 L4BC, *NSET,NSET=NLAYER2,ELSET=LAYER2 ** *STEP,INC=999,NLGEOM *STATIC 0.0025,1.0 *BOUNDARY,SUBMODEL,OP=NEW,STEP=1 L1BC,1,6, L2BC,1,6, L3BC,1,6, L4BC,1,6, *PRINT,CONTACT=YES *ELPRINT,FREQ=0 *ELFILE, FREQ=99999, ELSET=LAYER2 S, *NODEFILE, FREQ=99999, NSET=NLAYER2 U, *NODEPRINT,FREQ=0 *END STEP

1.2 Buckling and collapse analyses 1.2.1 Snap-through buckling analysis of circular arches Product: ABAQUS/Standard It is often necessary to study the postbuckling behavior of a structure whose response is unstable during part of its loading history. Two of the models in this example illustrate the use of the modified Riks method, which is provided to handle such cases. The method is based on moving with fixed

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increments along the static equilibrium path in a space defined by the displacements and a proportional loading parameter. The actual load value may increase or decrease as the solution progresses. The modified Riks method implemented in ABAQUS is described in ``Modified Riks algorithm,'' Section 2.3.2 of the ABAQUS Theory Manual. The other two models illustrate the use of viscous damping. One example applies viscous damping as a feature of surface contact, which allows for the definition of a "viscous" pressure that is proportional to the relative velocity between the surfaces. The implementation of this option in ABAQUS is described in ``Contact pressure definition,'' Section 5.2.1 of the ABAQUS Theory Manual. The other example applies volume proportional damping to the model. The implementation of this option is described in the automatic stabilization section of ``Solving nonlinear problems,'' Section 8.2.1 of the ABAQUS/Standard User's Manual. Three separate cases are considered here. The first is a clamped shallow arch subjected to a pressure load. Reference solutions for this case are given by Ramm (1981) and Sharafi and Popov (1971). The second case is the instability analysis of a clamped-hinged circular arch subjected to a point load. The exact analytical solution for this problem is given by DaDeppo and Schmidt (1975). The third case is a modification of the shallow arch problem in which the ends are pinned rather than clamped and the arch is depressed with a rigid punch.

Model and solution control The shallow circular arch is shown in Figure 1.2.1-1. Since the deformation is symmetric, one-half of the arch is modeled. Ten elements of type B21 (linear interpolation beams) are used. A uniform pressure is first applied to snap the arch through. The loading is then reversed so that the behavior is also found as the pressure is removed. The deep circular arch is shown in Figure 1.2.1-2. One end of the arch is clamped, and the other is hinged. A concentrated load is applied at the apex of the arch. The arch undergoes extremely large deflections but small strains. Because of the asymmetric boundary conditions, the arch will sway toward the hinged end and then collapse. The arch is almost inextensible for most of the response history. Sixty elements of type B31H are used. Hybrid elements are used because they are most suitable for problems such as this. The *CONTROLS option is used to set a very tight convergence tolerance because the problem contains more than one equilibrium path. If tight tolerances are not used, the response might follow a path that is different from the one shown. In the Riks procedure actual values of load magnitudes cannot be specified. Instead, they are computed as part of the solution, as the "load proportionality factor" multiplying the load magnitudes given on the loading data lines. User-prescribed load magnitudes serve only to define the direction and to estimate the magnitude of the initial increment of the load for a step. This initial load increment is the product of the ratio of the initial time increment to the time period and the load magnitudes given in the loading options. The user can terminate a Riks analysis by specifying either a maximum load proportionality factor or a maximum displacement at a node, or both. When a solution point is computed at which either of these limits is crossed, the analysis will stop. In any event, or if neither option is used, the analysis ends when the maximum number of increments for the step is exceeded.

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In snap-through studies such as these, the structure can carry increasing load after a complete snap. Therefore, the analysis is terminated conveniently by specifying a maximum load proportionality factor. For the clamped shallow arch the initial snap occurs at a pressure of about -1000 (force/length 2 units). Thus, -250 (force/length 2 units) seems to be a reasonable estimate for the first increment of load to be applied. Accordingly, an initial time increment of 0.05 is specified for a time period of 1.0 and a pressure load of -5000 (force/length 2 units). The solution will have been sufficiently developed at a pressure of about -2000 (force/length 2 units). Therefore, the analysis is terminated when the load proportionality factor exceeds 0.4. To illustrate the use of Riks in several steps, a second step is included in which the pressure is taken off the arch so that it will snap back toward its initial configuration. At any point in a Riks analysis, the actual load is given by P = P0 + ¸(Pref ¡ P0 ) , where P0 is the load at the end of the previous step, Pref is the load magnitude prescribed in the current step, and ¸ is the load proportionality factor. The arch is unloaded so that in the initial time increment, a pressure of approximately 0.15 P0 is removed. Using an initial time increment of 0.05 in a time period of 1.0, a load of Pref = ¡2P0 is prescribed for this restarted step. Furthermore, we want the analysis to end when all the load is removed and the arch has returned to its initial configuration. Therefore, a displacement threshold of 0.0 is set for the center of the arch. The analysis terminates when this limit is crossed. Because ABAQUS must pick up the load magnitude at the end of the initial Riks step to start the next step, any step following a Riks step can be done only as a restart job, using the *RESTART option with the END STEP parameter. For the deep clamped-hinged arch, the initial snap occurs at a load of about 900 (force units). The load magnitude specified is 100 (force units), and the maximum load proportionality factor is specified as 9.5. The shallow arch depressed with a rigid punch is shown in Figure 1.2.1-3. The analysis uses the same model of the arch as the first problem. However, the end is pinned rather than clamped, and load is applied through the displacement of the punch. The pinned boundary condition makes the problem more unstable than the clamped-end case. A preliminary analysis in which the arch is depressed with a prescribed displacement of the midpoint of the arch shows that the force will become negative during snap-through. Thus, if the arch is depressed with a rigid punch, the Riks method will not help convergence because, at the moment of snap-through, the arch separates from the punch, and the movement of the punch no longer controls the displacement of the arch. Therefore, damping is introduced to aid in convergence. Viscous damping with surface contact adds a pressure that is proportional to the relative velocity to slow down the separation of the arch from the punch. The viscous damping clearance is set to 10.0, and the fraction of the clearance interval is set to 0.9; the damping is constant for a clearance of up to 9.0. Since the arch is 4.0 units high, the distance traveled by the top of the arch from the initial position to the final snap-through position is 8.0 units. This distance is clearly larger than the clearance between the middle of the arch and the tip of the punch at any time during the analysis. Thus, the viscous damping is in effect for the whole period when the arch has separated from the punch. To choose the viscous damping coefficient, note that it is given as pressure per relative velocity. The

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relevant pressure is obtained by dividing the approximate peak force (10000.0) by the contact area (1.0). The relevant velocity is obtained by dividing the distance over which the top of the arch travels (8.0 from initial to snapped position, which can be rounded to 10.0) by the time (approximately 1.0, the total time of the step). A small percentage (0.1%) of this value is used for the viscous damping coefficient: ¹ = 0:001

(10000:0)(1:0) Ft = 1:0: = 0:001 Al (1:0)(10:)

With ¹ = 1.0, the analysis runs to completion. Another analysis was run with a smaller value of ¹ = 0.1, but the viscous damping was not sufficient to enable the analysis to pass the point of snap-through. Thus, a damping coefficient of 1.0 was determined to be an appropriate value. Alternatively, including the STABILIZE parameter on the *STATIC analysis procedure option applies volume proportional damping to the model. The default damping intensity is used in this case.

Results and discussion The results for the clamped shallow arch are shown in Figure 1.2.1-4, where the downward displacement of the top of the arch is plotted as a function of the pressure. The algorithm obtains this solution in 12 increments, with a maximum of three iterations in an increment. At the end of 12 increments the displacement of the top of the arch is about 7.5 length units. This represents a complete snap through, as the original rise of the arch was 4 length units. Figure 1.2.1-5 and Figure 1.2.1-6show a series of deformed configuration plots for this problem. Several other authors have examined this same case and have obtained essentially the same solution (see Ramm, 1981, and Sharafi and Popov, 1971). The results for the deep clamped-hinged arch are shown in Figure 1.2.1-7, where the displacement of the top of the arch is plotted as a function of the applied load. Figure 1.2.1-8 shows a series of deformed configuration plots for this problem. The arch collapses unstably at the peak load. Following this, the beam stiffens rapidly as the load increases. The ability of the Riks method to handle unstable response is well-illustrated by this example. The results of the preliminary analysis of the prescribed displacement of a pinned shallow arch are shown in Figure 1.2.1-9, with the displacement of the top of the arch plotted as a function of the reaction force at that point. This plot shows the negative force that develops during snap-through. A series of deformed configuration plots for the pinned shallow arch depressed with a punch and with viscous damping introduced is shown in Figure 1.2.1-10, with one plot showing the arch separated from the punch. Figure 1.2.1-11is a plot of the force between the punch and the top of the arch. The force is positive until snap-through, when the arch separates from the punch and a negative viscous force develops. Once the snap-through is complete, the force drops to zero as the punch continues to move down while separated from the arch. When the punch contacts the arch, a positive force develops again. Similar results are produced when the contact viscous damping is replaced by volume proportional damping. A sequence of configurations like Figure 1.2.1-10is obtained, in which separation of the arch

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from the punch occurs during snap-through. At the end of the analysis the amount of energy dissipated is similar to the amount dissipated with the viscous damping option.

Input files snapbuckling_shallow_step1.inp Initial analysis step for the shallow arch. snapbuckling_shallow_unload.inp Restart run to obtain the unloading response of the shallow arch. snapbuckling_deep.inp Deep arch. snapbuckling_shallow_midpoint.inp Shallow arch loaded by a fixed displacement of the midpoint. snapbuckling_shallow_punch.inp Shallow arch loaded by the displacement of a rigid punch. snapbuckling_b21h_deep.inp 60 elements of type B21H used for the deep clamped-hinged arch analysis. snapbuckling_b32h_deep.inp 30 elements of type B32H used for the deep clamped-hinged arch analysis. snapbuckling_restart1.inp Restart analysis of snapbuckling_shallow_step1.inp during the RIKS step. snapbuckling_restart2.inp Restart analysis of snapbuckling_restart1.inp during the RIKS step. This illustrates restarting an existing RIKS restart analysis. snapbuckling_shallow_stabilize.inp Same as snapbuckling_shallow_punch.inp with the surface contact viscous damping replaced by the volume proportional damping of *STATIC, STABILIZE.

References · DaDeppo, D. A., and R. Schmidt, "Instability of Clamped-Hinged Circular Arches Subjected to a Point Load," Transactions of the American Society of Mechanical Engineers , Journal of Applied Mechanics, pp. 894-896, Dec. 1975. · Ramm, E., "Strategies for Tracing the Nonlinear Response Near Limit Points," in Nonlinear Finite Element Analysis in Structural Mechanics , edited by W. Wunderlich, E. Stein and K. J. Bathe, Springer Verlag, Berlin, 1981.

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· Sharifi, P., and E. P. Popov, "Nonlinear Buckling Analysis of Sandwich Arches," Proc. ASCE, Journal of the Engineering Mechanics Division, vol. 97, pp. 1397-1412, 1971.

Figures Figure 1.2.1-1 Clamped shallow circular arch.

Figure 1.2.1-2 Deep clamped-hinged arch.

Figure 1.2.1-3 Pinned shallow arch with rigid punch.

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Figure 1.2.1-4 Load versus displacement curve for clamped shallow arch.

Figure 1.2.1-5 Deformed configuration plots for clamped shallow arch-Step 1.

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Figure 1.2.1-6 Deformed configuration plots for clamped shallow arch-Step 2.

Figure 1.2.1-7 Load versus displacement curves for deep clamped-hinged arch.

Figure 1.2.1-8 Deformed configuration plots for deep clamped-hinged arch.

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Figure 1.2.1-9 Force versus displacement curve for fixed displacement of pinned shallow arch.

Figure 1.2.1-10 Deformed configuration plots for pinned arch depressed with rigid punch.

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Figure 1.2.1-11 Force between the punch and the top of the pinned arch.

Sample listings

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Listing 1.2.1-1 *HEADING SHALLOW ARCH *NODE 1,0.,4.,0.,0.,1. 11,28.,0.,0.,.28,.96 *NGEN,LINE=C 1,11,1,0,0.,-96. *ELEMENT,TYPE=B21 1,1,2 *ELGEN,ELSET=ARCH 1,10 *SURFACE, NAME=SURFACE ARCH, SPOS *BEAM SECTION,SECTION=RECT,ELSET=ARCH,MATERIAL=A1 1.,2. *MATERIAL,NAME=A1 *ELASTIC 1.E7,.25 *BOUNDARY 1,1 1,6 11,1,6 *ELSET,ELSET=ONE 1, *NSET,NSET=ONE 1, *RESTART,WRITE,FREQUENCY=1 *STEP,NLGEOM,INC=50 LOADING *STATIC,RIKS .05,1.,0.,.2,.4 *DSLOAD SURFACE,P,5000. *PRINT,RESIDUAL=NO *EL PRINT,ELSET=ONE,SUMMARY=NO E, LOADS, *NODE PRINT,SUMMARY=NO U,RF *EL FILE,ELSET=ONE LOADS,

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S, E, *NODE FILE,NSET=ONE U, *MONITOR, NODE=1, DOF=2 *END STEP

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Listing 1.2.1-2 *HEADING UNLOADING OF SHALLOW ARCH *RESTART, READ, STEP=1, INC=12, WRITE, FREQUENCY=0, END STEP *STEP,NLGEOM,INC=25 *STATIC,RIKS .05,1.,,,,1,2,0. *DSLOAD SURFACE,P,-4685.6 *END STEP

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Listing 1.2.1-3 *HEADING DADEPPO'S ARCH *NODE 1, 95.37169508,-30.07057995 61,-95.37169508,-30.07057995 *NGEN,LINE=C 1,61,1,,0.0,0.0,0.0, 0.0,0.0,1.0 *NSET,NSET=APEX 31 , *ELEMENT,TYPE=B31H 1,1,2 *ELGEN,ELSET=EALL 1,60,1,1 *ELSET,ELSET=CENT 30,31 *BEAM SECTION,SECTION=RECT,ELSET=EALL, MATERIAL=MAT 1.0,2.289428295 *MATERIAL,NAME=MAT *ELASTIC 1.E6,.0 *BOUNDARY 1,1,6 61,1,3 *RESTART,WRITE,FREQUENCY=5 *STEP,NLGEOM,INC=150 *STATIC,RIKS 0.40,1.0,,,9.5 *CONTROLS, PARAMETERS=FIELD 5.0E-6, *CLOAD 31,2,-100.0 *EL PRINT,ELSET=CENT,FREQUENCY=5 SF, SE, *EL FILE,ELSET=CENT,FREQUENCY=10 SF, SE, *NODE PRINT,FREQUENCY=5 *NODE FILE,NSET=APEX U,

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CF, *NODE FILE RF, *END STEP

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Listing 1.2.1-4 *HEADING SHALLOW CLAMPED ARCH WITH DISPLACED MIDPOINT *NODE 1,0.,4.,0.,0.,1. 11,28.,0.,0.,.28,.96 *NGEN,LINE=C 1,11,1,0,0.,-96. *ELEMENT,TYPE=B21 1,1,2 *ELGEN,ELSET=ARCH 1,10 *BEAM SECTION,SECTION=RECT,ELSET=ARCH,MATERIAL=A1 1.,2. *MATERIAL,NAME=A1 *ELASTIC 1.E7,.25 *BOUNDARY 1,1 1,6 11,PINNED *NSET,NSET=NENDS 1,11 *RESTART,WRITE,FREQUENCY=2 *STEP,NLGEOM,INC=50 LOADING *STATIC .05,1., ,.05 *BOUNDARY,OP=MOD 1, 2,2, -9.0 *PRINT,RESIDUAL=NO *MONITOR,NODE=1,DOF=2 *EL PRINT,FREQUENCY=0 *NODE PRINT,SUMMARY=NO,FREQUENCY=2 U,RF *NODE FILE,NSET=NENDS U,RF *END STEP

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Listing 1.2.1-5 *HEADING SHALLOW CLAMPED ARCH LOADED WITH PUNCH, *NODE 1,0.,4.,0.,0.,1. 11,28.,0.,0.,.28,.96 100, 0., 9. *NGEN,LINE=C 1,11,1,0,0.,-96. *NSET,NSET=NENDS 1,11 *ELEMENT,TYPE=B21 1,1,2 *ELGEN,ELSET=ARCH 1,10 *BEAM SECTION,SECTION=RECT,ELSET=ARCH,MATERIAL=A1 1.,2. *MATERIAL,NAME=A1 *ELASTIC 1.E7,.25 *ELSET,ELSET=ET1 1, *ELSET,ELSET=ET2,GENERATE 2,4,1 *RIGID BODY,ANALYTICAL SURFACE=BSURF,REF NODE=100 *SURFACE,TYPE=SEGMENTS,NAME=BSURF START, 5.,9. CIRCL, 0.,4., 0.,9. *SURFACE,NAME=ASURF ET1,SPOS *CONTACT PAIR,INTERACTION=DAMP ASURF,BSURF *SURFACE INTERACTION,NAME=DAMP 1., *CONTACT DAMPING, DEFINITION=DAMPING COEFFICIENT 1.0,10.0,0.9 *SURFACE,NAME=CSURF ET2,SPOS *CONTACT PAIR,INTERACTION=DAMP CSURF,BSURF *BOUNDARY 1,1

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1,6 11,PINNED 100,1 100,6 *RESTART,WRITE,FREQUENCY=1 *STEP,NLGEOM,INC=100 LOADING *STATIC .05,1.,,.05 *BOUNDARY,OP=MOD 100,2,2,-9.0 *PRINT,RESIDUAL=NO *MONITOR, NODE=1, DOF=2 *CONTACT PRINT,SLAVE=ASURF,FREQUENCY=5 *CONTACT PRINT,SLAVE=CSURF,FREQUENCY=5 *CONTACT FILE,SLAVE=ASURF *CONTACT FILE,SLAVE=CSURF *EL PRINT,FREQUENCY=5 S,E *NODE PRINT,SUMMARY=NO,FREQUENCY=5 U,RF *END STEP

1.2.2 Laminated composite shells: buckling of a cylindrical panel with a circular hole Product: ABAQUS/Standard This example illustrates a type of analysis that is of interest in the aerospace industry. The objective is to determine the strength of a thin, laminated composite shell, typical of shells used to form the outer surfaces of aircraft fuselages and rocket motors. Such analyses are complicated by the fact that these shells typically include local discontinuities--stiffeners and cutouts--which can induce substantial stress concentrations that can delaminate the composite material. In the presence of buckling this delamination can propagate through the structure to cause failure. In this example we study only the geometrically nonlinear behavior of the shell: delamination or other section failures are not considered. Some estimate of the possibility of material failure could presumably be made from the stresses predicted in the analyses reported here, but no such assessment is included in this example. The example makes extensive use of material orientation as part of the *SHELL GENERAL SECTION option to define the multilayered, anisotropic, laminated section. The various orientation options for shells are discussed in ``Analysis of an anisotropic layered plate,'' Section 1.1.2 of the ABAQUS Benchmarks Manual. The *SHELL GENERAL SECTION option has two methods of defining laminated sections: defining the thickness, material, and orientation of each layer or defining the equivalent section properties

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directly. The last method is particularly useful if the laminate properties are obtained directly from experiments or a separate preprocessor. This example uses both methods with the *SHELL GENERAL SECTION option. Alternatively, the *SHELL SECTION option could be used to analyze the model; however, because the material behavior is linear, no difference in solution would be obtained and the computational costs would be greater.

Geometry and model The structure analyzed is shown in Figure 1.2.2-1and was originally studied experimentally by Knight and Starnes (1984). The test specimen is a cylindrical panel with a 355.6 mm (14 in) square platform and a 381 mm (15 in) radius of curvature, so that the panel covers a 55.6° arc of the cylinder. The panel contains a centrally located hole of 50.8 mm (2 in) diameter. The shell consists of 16 layers of unidirectional graphite fibers in an epoxy resin. Each layer is 0.142 mm (.0056 in) thick. The layers are arranged in the symmetric stacking sequence {§45/90/0/0/90/¨45} degrees repeated twice. The nominal orthotropic elastic material properties as defined by Stanley (1985) are E11 = 135 kN/mm2 (19.6 ´ 106 lb/in2), 2 E22 = 13 kN/mm (1.89 ´ 106 lb/in2), G12 = G13 = 6.4 kN/mm2 (.93 ´ 106 lb/in2), G23 = 4.3 kN/mm2 (0.63 ´ 106 lb/in2), º12 = 0.38, where the 1-direction is along the fibers, the 2-direction is transverse to the fibers in the surface of the lamina, and the 3-direction is normal to the lamina. The panel is fully clamped on the bottom edge, clamped except for axial motion on the top edge and simply supported along its vertical edges. Three analyses are considered. The first is a linear (prebuckling) analysis in which the panel is subjected to a uniform end shortening of 0.8 mm (.0316 in). The total axial force and the distribution of axial force along the midsection are used to compare the results with those obtained by Stanley (1985). The second analysis consists of an eigenvalue extraction of the first five buckling modes. The buckling loads and mode shapes are also compared with those presented by Stanley (1985). Finally, a nonlinear load-deflection analysis is done to predict the postbuckling behavior, using the modified Riks algorithm. For this analysis an initial imperfection is introduced. The imperfection is based on the fourth buckling mode extracted during the second analysis. These results are compared with those of Stanley (1985) and with the experimental measurements of Knight and Starnes (1984). The mesh used in ABAQUS is shown in Figure 1.2.2-2. The anisotropic material behavior precludes any symmetry assumptions, hence the entire panel is modeled. The same mesh is used with the 4-node shell element (type S4R5) and also with the 9-node shell element (type S9R5); the 9-node element mesh, thus, has about four times the number of degrees of freedom as the 4-node element mesh. The 6-node triangular shell element STRI65 is also used; it employs two triangles for each quadrilateral element of the second-order mesh. Mesh generation is facilitated by using the *NFILL and *NMAP options, as shown in the input data. In this model specification of the relative angle of orientation to define the material orientation within each layer, along with the *ELASTIC, TYPE=LAMINA option, makes the definition of the laminae properties straightforward. The shell elements used in this example use an approximation to thin shell theory, based on a

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numerical penalty applied to the transverse shear strain along the element edges. These elements are not universally applicable to the analysis of composites since transverse shear effects can be significant in such cases and these elements are not designed to model them accurately. Here, however, the geometry of the panel is that of a thin shell; and the symmetrical lay-up, along with the relatively large number of laminae, tends to diminish the importance of transverse shear deformation on the response.

Relation between stress resultants and generalized strains The shell section is most easily defined by giving the layer thickness, material, and orientation, in which case ABAQUS preintegrates to obtain the section stiffness properties. However, the user can choose to input the section stiffness properties directly instead, as follows. In ABAQUS a lamina is considered as an orthotropic sheet in plane stress. The principal material axes of the lamina (see Figure 1.2.2-3) are longitudinal, denoted by L; transverse to the fiber direction in the surface of the lamina, denoted by T ; and normal to the lamina surface, denoted by N: The constitutive relations for a general orthotropic material in the principal directions ( L; T; N ) are 8 ¾L > > > ¾ > > < T ¿LT ¾N > > > > > : ¿LN ¿T N

9 > > > > > =

2

C11 6 C12 6 6 0 =6 > 6 C14 > > 4 > > 0 ; 0

C12 C22 0 C24 0 0

0 0 C33 0 0 0

C14 C24 0 C44 0 0

0 0 0 0 C55 0

38 ²L 0 > > > 0 7> ² < T 7> 0 7 °LT 7 0 7> ²N > 5> > > 0 : °LN C66 °T N

9 > > > > > = > > > > > ;

:

In terms of the data required by the *ELASTIC, TYPE=ORTHO option in ABAQUS these are C11 = D1111 ;

C12 = D1122 ;

C14 = D1133 ;

C22 = D2222

C24 = D2233 ;

C33 = D1212 ;

C44 = D3333 ;

C55 = D1313

C66 = D2323 This matrix is symmetric and has nine independent constants. If we assume a state of plane stress, then ¾N is taken to be zero. This yields 8 ¾L > > > < ¾T ¿LT > > > : ¿LN ¿T N where

9 > > > =

2

Q11 6 Q12 6 =6 0 > 4 > 0 > ; 0

Q12 Q22 0 0 0

0 0 Q33 0 0

0 0 0 Q55 0

38 ²L 0 > > > 0 7 < ²T 7 0 7 °LT 5> > 0 > : °LN Q66 °T N

9 > > > = > > > ;

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;

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C14 C14 ; C44 C24 C14 = C12 ¡ ; C44 C24 C24 = C22 ¡ ; C44 = C33 ;

Q11 = C11 ¡ Q12 Q22 Q33

Q55 = C55 ; Q66 = C66 : The correspondence between these terms and the usual engineering constants that might be given for a simple orthotropic layer in a laminate is E1 ; 1 ¡ º12 º21 º12 E2 º21 E1 = = ; 1 ¡ º12 º21 1 ¡ º12 º21 E2 = ; 1 ¡ º12 º21 = G12 ;

Q11 = Q12 Q22 Q33

Q55 = G13 ; Q66 = G23 : The parameters used on the right-hand side of the above equation are those that must be provided as data on the *ELASTIC, TYPE=LAMINA option. If the (1; 2; N ) system denotes the standard shell basis directions that ABAQUS chooses by default, the local stiffness components must be rotated to this system to construct the lamina's contribution to the *SHELL GENERAL SECTION stiffness. Since Qij represent fourth-order tensors, in the case of a lamina they are oriented at an angle µ to the standard shell basis directions used in ABAQUS. Hence, the transformation is ¹ 11 = Q11 cos4 µ + 2(Q12 + 2Q33 ) sin2 µ cos2 µ + Q22 sin4 µ; Q ¹ 12 = (Q11 + Q22 ¡ 4Q33 ) sin2 µ cos2 µ + Q12 (sin4 µ + cos4 µ); Q ¹ 22 = Q11 sin4 µ + 2(Q12 + 2Q33 ) sin2 µ cos2 µ + Q22 cos4 µ; Q ¹ 13 = (Q11 ¡ Q12 ¡ 2Q33 ) sin µ cos3 µ + (Q12 ¡ Q22 + 2Q33 ) sin3 µ cos µ; Q ¹ 23 = (Q11 ¡ Q12 ¡ 2Q33 ) sin3 µ cos µ + (Q12 ¡ Q22 + 2Q33 ) sin µ cos3 µ; Q ¹ 33 = (Q11 + Q22 ¡ 2Q12 ¡ 2Q33 ) sin2 µ cos2 µ + Q33 (sin4 µ + cos4 µ); Q ¹ 55 = Q55 cos2 µ ¡ Q66 sin2 µ; Q ¹ 56 = Q55 sin µ cos µ ¡ Q66 sin µ cos µ; Q ¹ 66 = Q55 sin2 µ ¡ Q66 cos2 µ; Q

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¹ ij are the stiffness coefficients in the standard shell basis directions used by ABAQUS. where Q

ABAQUS assumes that a laminate is a stack of laminae arranged with the principal directions of each layer in different orientations. The various layers are assumed to be rigidly bonded together. The section force and moment resultants per unit length in the normal basis directions in a given layer can be defined on this basis as (N1 ; N2 ; N12 ) = (M1 ; M2 ; M12 ) = (V1 ; V2 ) =

Z Z Z

h=2

(¾1 ; ¾2 ; ¿12 ) dz; ¡h=2 h=2

(¾1 ; ¾2 ; ¿12 )z dz; ¡h=2 h=2

(¿13 ; ¿23 ) dz; ¡h=2

where h is the thickness of the layer. This leads to the relations 9 2 8 N1 > A11 > > > > > N A > > 6 2 > 12 > > > 6 > > > > N A 6 12 > > = 6 13 < M1 6B = 6 11 > M2 > 6 B12 > > > > 6 > > > > M12 > 6 B13 > > > > 4 0 > > ; : V1 > V2 0

A12 A22 A23 B12 B22 B23 0 0

A13 A23 A33 B13 B23 B33 0 0

B11 B12 B13 D11 D12 D13 0 0

B12 B22 B23 D12 D22 D23 0 0

B13 B23 B33 D13 D23 D33 0 0

0 0 0 0 0 0 E11 E12

9 38 ²1 > 0 > > > 0 7> > > ²2 > > > > 7> > > > 0 7> ° 12 > > = 7< 0 7 ∙1 ; 7 0 7> ∙2 > > > > 7> > > 0 7> ∙12 > > > > > 5> > E12 > ° ; : 13 > E22 °23

where the components of this section stiffness matrix are given by (Aij ; Bij ; Dij ) =

Eij =

Z Z

h=2 ¡h=2 h=2

¡h=2

2 ¹m Q ij (1; z; z ) dz; (i; j = 1; 2; 3)

¹m Q ®¯ ki kj dz; (i; j = 1; 2; and ®; ¯ = i + 4; j + 4):

¹ m depend on the material properties and fiber Here m indicates a particular layer. Thus, the Q ij orientation of the mth layer. The ki ; i = 1,2 parameters are the shear correction coefficients as defined by Whitney (1973). If there are n layers in the lay-up, we can rewrite the above equations as a summation of integrals over the n laminae. The material coefficients will then take the form

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Aij = Bij =

n X

¹m Q ij (hm ¡ hm¡1 );

m=1 n X

1 ¹ m (h2 ¡ h2m¡1 ); Q 2 m=1 ij m

n 1 X ¹m 3 Dij = Q (h ¡ h3m¡1 ); 3 m=1 ij m

Eij =

n X

m=1

¹m Q ®¯ (hm ¡ hm¡1 )ki kj ;

where the hm and hm¡1 in these equations indicate that the mth lamina is bounded by surfaces z = hm and z = hm¡1 : See Figure 1.2.2-4for the nomenclature. These equations define the coefficients required for the direct input of the section stiffness matrix method with the *SHELL GENERAL SECTION option. Only the [A], [B ], and [D ] submatrices are needed for that option. The three terms in [E ], if required, are defined using the *TRANSVERSE SHEAR STIFFNESS option. The section forces as defined above are in the normal shell basis directions. Applying these equations to the laminate defined for this example leads to the following overall section stiffness: 2

138:385 4 [A] = 44:0189 0

2

55:670 [D ] = 4 21:638 2:138

44:0189 138:385 0

21:638 58:521 2:138

3 0 5 kN/mm; 0 47:1831 3 2:138 2:138 5 kN-mm; 23:004

2

0 4 [B] = 0 0

0 0 0

∙

12:2387 [E ] = 0

3 0 05; 0 0 12:2387

¸

kN/mm,

or 2

790:239 4 [A] = 251:367 0

2

492:719 4 [D ] = 191:513 18:9245

251:367 790:239 0

3 0 5 £ 103 lb/in; 0 269:436

2

0 4 [B] = 0 0

191:513 517:951 18:9245

3 18:9245 18:9245 5 lb-in; 203:602

∙

49:573 [E ] = 0:002

Results and discussion

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0 0 0

3 0 05; 0

¸ 0:002 £ 103 lb/in : 52:967

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The total axial force necessary to compress the panel 0.803 mm (0.0316 in) is 100.2 kN (22529 lb) for the mesh of S9R5 elements, 99.5 kN (22359 lb) for the mesh of S4R5 elements, and 100.3 kN (22547 lb) for the mesh of STRI65 elements. These values match closely with the result of 100 kN (22480 lb) reported by Stanley (1985). Figure 1.2.2-5 shows the displaced configuration and a profile of axial force along the midsection of the panel (at z = L=2). It is interesting to note that the axial load is distributed almost evenly across the entire panel, with only a very localized area near the hole subjected to an amplified stress level. This suggests that adequate results for this linear analysis could also be obtained with a coarser mesh that has a bias toward the hole. The second stage of the analysis is the eigenvalue buckling prediction. To obtain the buckling predictions with ABAQUS, a *BUCKLE step is run. In this step nominal values of load are applied. The magnitude that is used is not of any significance, since eigenvalue buckling is a linear perturbation procedure: the stiffness matrix and the stress stiffening matrix are evaluated at the beginning of the step without any of this load applied. The *BUCKLE step calculates the eigenvalues that, multiplied with the applied load and added to any "base state" loading, are the predicted buckling loads. The eigenvectors associated with the eigenvalues are also obtained. This procedure is described in more detail in ``Eigenvalue buckling prediction,'' Section 6.2.3 of the ABAQUS/Standard User's Manual. The buckling predictions are summarized in Table 1.2.2-1and Figure 1.2.2-6. The buckling load predictions from ABAQUS are higher than those reported by Stanley. The eigenmode predictions given by the mesh using element types S4R5, S9R5, and STRI65 are all the same and agree well with those reported by Stanley. Stanley makes several important observations that remain valid for the ABAQUS results: (1) the eigenvalues are closely spaced; (2) nevertheless, the mode shapes vary significantly in character; (3) the first buckling mode bears the most similarity to the linear prebuckling solution; (4) there is no symmetry available that can be utilized for computational efficiency. Following the eigenvalue buckling analyses, nonlinear postbuckling analysis is carried out by imposing an imperfection based on the fourth buckling mode. The maximum initial perturbation is 10% of the thickness of the shell. The load versus normalized displacement plots for the S9R5 mesh, the S4R5 mesh, and the STRI65 mesh are compared with the experimental results and those given by Stanley in Figure 1.2.2-7. The overall response prediction is quite similar for the ABAQUS elements, although the general behavior predicted by Stanley is somewhat different. The ABAQUS results show a peak load slightly above the buckling load predicted by the eigenvalue extraction, while Stanley's results show a significantly lower peak load. In addition, the ABAQUS results show rather less loss of strength after the initial peak, followed quite soon by positive stiffness again. Neither the ABAQUS results nor Stanley's results agree closely with the experimentally observed dramatic loss of strength after peak load. Stanley ascribes this to material failure (presumably delamination), which is not modeled in his analyses or in these. Figure 1.2.2-8 shows the deformed configurations for the panel during its postbuckling response. The plots show the results for S4R5, but the pattern is similar for S9R5 and STRI65. The response is quite symmetric initially; but, as the critical load is approached, a nonsymmetric dimple develops and grows, presumably accounting for the panel's loss of strength. Later in the postbuckling response another wrinkle can be seen to be developing.

Input files 1-308

Static Stress/Displacement Analyses

laminpanel_s9r5_prebuckle.inp Prebuckling analysis for the 9-node (element type S9R5) mesh. laminpanel_s9r5_buckle.inp Corresponding eigenvalue buckling prediction. laminpanel_s9r5_postbuckle.inp Corresponding nonlinear postbuckling analysis. laminpanel_s9r5_buckle.inp Eigenvalue buckling prediction with direct input of shell section stiffness properties using the *SHELL GENERAL SECTION option. laminpanel_s9r5_postbuckle.inp Nonlinear postbuckling analysis with direct input of shell section stiffness properties using the *SHELL GENERAL SECTION option. laminpanel_s4r5_prebuckle.inp Prebuckling analysis using element type S4R5. laminpanel_s4r5_buckle.inp Eigenvalue buckling prediction using element type S4R5. laminpanel_s4r5_postbuckle.inp Nonlinear postbuckling analysis using element type S4R5. laminpanel_s4r5_node.inp Nodal coordinate data for the imperfection imposed for the postbuckling analysis using element type S4R5. laminpanel_s9r5_stri65_node.inp Nodal coordinate data for the imperfection imposed for the postbuckling analysis using element types S9R5 and STRI65. laminpanel_stri65_prebuckle.inp Prebuckling analysis using element type STRI65. laminpanel_stri65_buckle.inp Eigenvalue buckling prediction using element type STRI65. laminpanel_stri65_postbuckle.inp Nonlinear postbuckling analysis using element type STRI65. laminpanel_s4_prebuckle.inp Prebuckling analysis using element type S4.

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laminpanel_s4_buckle.inp Eigenvalue buckling prediction using element type S4. laminpanel_s4_postbuckle.inp Nonlinear postbuckling analysis using element type S4.

References · Knight, N. F., and J. H. Starnes Jr., "Postbuckling Behavior of Axially Compressed Graphite-Epoxy Cylindrical Panels with Circular Holes," presented at the 1984 ASME Joint Pressure Vessels and Piping/Applied Mechanics Conference, San Antonio, Texas, 1984. · Stanley, G.M., Continuum-Based Shell Elements, Ph.D. Dissertation, Department of Mechanical Engineering, Stanford University, 1985. · Whitney, J.M., "Shear Correction Factors for Orthotropic Laminates Under Static Loads," Journal of Applied Mechanics, Transactions of the ASME, vol. 40, pp. 302-304, 1973.

Table Table 1.2.2-1 Summary of buckling load predictions. Stanley 107.0 kN (24054 lb) Mode 1 S9R5 113.4 kN (25503 lb) S4R5 115.5 kN (25964 lb) S4 118.5 kN (26651 lb) STRI65 113.8 kN (25579 lb) Stanley 109.6 kN (24638 lb) Mode 2 S9R5 117.6 kN (26427 lb) S4R5 121.2 kN (27244 lb) S4 122.6 kN (27560 lb) STRI65 117.8 kN (26490 lb) Stanley 116.2 kN (26122 lb) Mode 3 S9R5 120.3 kN (27051 lb) S4R5 124.7 kN (28042 lb) S4 127.7 kN (28713 lb) STRI65 121.1 kN (27218 lb) Stanley 140.1 kN (31494 lb) Mode 4 S9R5 147.5 kN (33161 lb) S4R5 156.1 kN (35092 lb) S4 157.8 kN (35478 lb) STRI65 146.9 kN (33015 lb) Stanley 151.3 kN (34012 lb) Mode 5 S9R5 171.3 kN (38510 lb) S4R5 181.5 kN (40800 lb) S4 186.8 kN (41992 lb) STRI65 172.8 kN (38842 lb)

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Figures Figure 1.2.2-1 Geometry for cylindrical panel with hole.

Figure 1.2.2-2 Mesh for cylindrical panel with hole.

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Figure 1.2.2-3 Typical lamina.

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Figure 1.2.2-4 Typical laminate.

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Figure 1.2.2-5 Displaced shape and axial force distribution.

1-314

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Figure 1.2.2-6 Buckling modes, element types S4R5, S9R5, and STRI65.

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Figure 1.2.2-7 Load-displacement response.

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Figure 1.2.2-8 Postbuckling deformations: 10% h imperfection with S4R5.

1-317

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Sample listings

1-318

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Listing 1.2.2-1 *HEADING COMPOSITE CYLINDRICAL PANEL WITH CIRCULAR HOLES9R5 ** FROM STUDY IN THESIS OF G.M. STANLEY *RESTART,WRITE,FREQUENCY=1 *NODE 2001,15., 0., 0. 2017,15., 14.556, 0. 2033,15., 14.556, 14. 2049,15., 0. , 14. 2065,15., 0., 0. *NGEN,NSET=OUTSIDE 2001,2017 2017,2033 2033,2049 2049,2065 ** DEFINE CIRCLE *SYSTEM 15.,7.27802298,7.0 *NODE 9999, 0.,0. 1,0., -.70710678, -.70710678 17,0., .70710678,-.70710678 33,0., .70710678, .70710678 49,0., -.70710678, .70710678 65,0., -.70710678, -.70710678 *NGEN,LINE=C,NSET=HOLE 1,17,1,9999 17,33,1,9999 33,49,1,9999 49,65,1,9999 *SYSTEM 0.,0.,0. *NFILL,NSET=ALL HOLE,OUTSIDE,20,100 *NSET,NSET=XPARB,GENERATE 2001,2017 *NSET,NSET=XPART,GENERATE 2033,2049 *NSET,NSET=YPAR,GENERATE 2049,2065

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Static Stress/Displacement Analyses

2017,2033 *NSET,NSET=PROFILE,GENERATE 57,2057,100 25,2025,100 *NMAP,TYPE=CYLINDRICAL,NSET=ALL 0.,0.,0., 0.,0.,1. 15.,0.,0. 1.,3.8197307 ,1. *ELEMENT,TYPE=S9R5 1,1,201,203,3,101,202,103,2,102 *ELGEN,ELSET=ALL 1,31,2,1,10,200,100 *ELEMENT,TYPE=S9R5 32,63,263,201,1,163,264,101,64,164 *ELGEN,ELSET=ALL 32,10,200,100 *ELSET,ELSET=PROFILE,GENERATE 12, 912,100 13, 913,100 28, 928,100 29, 929,100 *SHELL GENERAL SECTION,ELSET=ALL,COMPOSITE, ORIENTATION=SECORI .0056,,LAMINA, 45. .0056,,LAMINA,-45. .0056,,LAMINA, 90. .0056,,LAMINA, 0. .0056,,LAMINA, 0. .0056,,LAMINA, 90. .0056,,LAMINA,-45. .0056,,LAMINA, 45. *** CENTER LINE .0056,,LAMINA, 45. .0056,,LAMINA,-45. .0056,,LAMINA, 90. .0056,,LAMINA, 0. .0056,,LAMINA, 0. .0056,,LAMINA, 90. .0056,,LAMINA,-45. .0056,,LAMINA, 45. *MATERIAL,NAME=LAMINA *ELASTIC,TYPE=LAMINA 19.6E6, 1.89E6, .38, .93E6, .93E6, .63E6

1-320

Static Stress/Displacement Analyses

*ORIENTATION,SYSTEM=CYLINDRICAL,NAME=SECORI 0.,0.,0., 0.,0., 1. 1, 0. *STEP *STATIC *BOUNDARY XPARB,1,6 XPART,1,2 XPART,4,6 YPAR,1,2 XPART,3,,-.0316 *EL PRINT,ELSET=PROFILE,POSITION=AVERAGED AT NODES SF, *EL FILE,ELSET=PROFILE,POSITION=AVERAGED AT NODES SF, *NODE PRINT,NSET=XPART,TOTALS=YES U, RF, *NODE PRINT,NSET=XPARB,TOTALS=YES U, RF, *NODE FILE,NSET=XPART U,RF *END STEP

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Listing 1.2.2-2 *HEADING COMPOSITE CYLINDRICAL PANEL WITH CIRCULAR HOLE-- S9R5 ** ** FROM STUDY IN THESIS OF G.M. STANLEY ** *RESTART,WRITE,FREQUENCY=1 *PREPRINT,ECHO=YES,MODEL=NO,HISTORY=NO *NODE 2001,15., 0., 0. 2017,15., 14.556, 0. 2033,15., 14.556, 14. 2049,15., 0. , 14. 2065,15., 0., 0. *NGEN,NSET=OUTSIDE 2001,2017 2017,2033 2033,2049 2049,2065 ** ** DEFINE CIRCLE ** *SYSTEM 15.,7.27802298,7.0 *NODE 9999, 0.,0. 1,0., -.70710678, -.70710678 17,0., .70710678,-.70710678 33,0., .70710678, .70710678 49,0., -.70710678, .70710678 65,0., -.70710678, -.70710678 *NGEN,LINE=C,NSET=HOLE 1,17,1,9999 17,33,1,9999 33,49,1,9999 49,65,1,9999 *SYSTEM 0.,0.,0. *NFILL,NSET=ALL HOLE,OUTSIDE,20,100 *NSET,NSET=XPARB,GENERATE 2001,2017

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*NSET,NSET=XPART,GENERATE 2033,2049 *NSET,NSET=YPAR,GENERATE 2049,2065 2017,2033 *NSET,NSET=PROFILE,GENERATE 57,2057,100 25,2025,100 *NMAP,TYPE=CYLINDRICAL,NSET=ALL 0.,0.,0., 0.,0.,1. 15.,0.,0. 1.,3.8197307 ,1. *EQUATION 2, 2033,3,1.,2041,3,-1. 2, 2034,3,1.,2041,3,-1. 2, 2035,3,1.,2041,3,-1. 2, 2036,3,1.,2041,3,-1. 2, 2037,3,1.,2041,3,-1. 2, 2038,3,1.,2041,3,-1. 2, 2039,3,1.,2041,3,-1. 2, 2040,3,1.,2041,3,-1. 2, 2042,3,1.,2041,3,-1. 2, 2043,3,1.,2041,3,-1. 2, 2044,3,1.,2041,3,-1. 2, 2045,3,1.,2041,3,-1. 2, 2046,3,1.,2041,3,-1. 2, 2047,3,1.,2041,3,-1. 2, 2048,3,1.,2041,3,-1.

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2, 2049,3,1.,2041,3,-1. *ELEMENT,TYPE=S9R5 1,1,201,203,3,101,202,103,2,102 *ELGEN,ELSET=ALL 1,31,2,1,10,200,100 *ELEMENT,TYPE=S9R5 32,63,263,201,1,163,264,101,64,164 *ELGEN,ELSET=ALL 32,10,200,100 *ELSET,ELSET=PROFILE,GENERATE 12, 912,100 13, 913,100 28, 928,100 29, 929,100 *SHELL GENERAL SECTION,ELSET=ALL 7.90239D5,2.51367D5,7.90239D5,-3.08578D-6,-7.94285D-5,2.69436D5,0.,0. 0., 4.92719D+02,0.,0.,0.,1.91513D+02,5.17951D+02,0. 0.,0., 1.89245D+01 , 1.89245D+01 ,2.03602D+02 *BOUNDARY XPARB,1,6 XPART,1,2 XPART,4,6 YPAR,1,2 *STEP *BUCKLE 5,,60,20 *CLOAD 2041,3,-1000. *EL PRINT,FREQUENCY=0 *NODE PRINT,FREQUENCY=0 *MODAL FILE *OUTPUT,HISTORY,FREQUENCY=1 *MODAL OUTPUT *END STEP

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Listing 1.2.2-3 *HEADING NONLINEAR POSTBUCKLING ANALYSIS--10% IMPERFECTIONWITH CONTRIBUTIONS FROM FIRST FOUR MODES COMPOSITE CYLINDRICAL PANEL WITH CIRCULAR HOLE ** ** FROM STUDY IN THESIS OF G.M. STANLEY ** *NODE,INPUT=PANEL9N.NOD *NSET,NSET=OUTSIDE,GENERATE 2001,2017 2017,2033 2033,2049 2049,2065 *NSET,NSET=HOLE,GENERATE 1,17,1 17,33,1 33,49,1 49,65,1 *NSET,NSET=XPARB,GENERATE 2001,2017 *NSET,NSET=XPART,GENERATE 2033,2049 *NSET,NSET=YPAR,GENERATE 2049,2065 2017,2033 *NSET,NSET=PROFILE,GENERATE 57,2057,100 25,2025,100 *EQUATION 2, 2033,3,1.,2041,3,-1. 2, 2034,3,1.,2041,3,-1. 2, 2035,3,1.,2041,3,-1. 2, 2036,3,1.,2041,3,-1. 2, 2037,3,1.,2041,3,-1. 2, 2038,3,1.,2041,3,-1.

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2, 2039,3,1.,2041,3,-1. 2, 2040,3,1.,2041,3,-1. 2, 2042,3,1.,2041,3,-1. 2, 2043,3,1.,2041,3,-1. 2, 2044,3,1.,2041,3,-1. 2, 2045,3,1.,2041,3,-1. 2, 2046,3,1.,2041,3,-1. 2, 2047,3,1.,2041,3,-1. 2, 2048,3,1.,2041,3,-1. 2, 2049,3,1.,2041,3,-1. *NSET,NSET=MASTER 2041, *ELEMENT,TYPE=S9R5 1,1,201,203,3,101,202,103,2,102 *ELGEN,ELSET=ALL 1,31,2,1,10,200,100 *ELEMENT,TYPE=S9R5 32,63,263,201,1,163,264,101,64,164 *ELGEN,ELSET=ALL 32,10,200,100 *ELSET,ELSET=PROFILE,GENERATE 12, 912,100 13, 913,100 28, 928,100 29, 929,100 *SHELL GENERAL SECTION,ELSET=ALL 7.90239D5,2.51367D5,7.90239D5,-3.08578D-6,-7.94285D-5,2.69436D5,0.,0. 0., 4.92719D+02,0.,0.,0.,1.91513D+02,5.17951D+02,0. 0.,0., 1.89245D+01 , 1.89245D+01 ,2.03602D+02 *BOUNDARY XPARB,1,6 XPART,1,2 XPART,4,6

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YPAR,1,2 *RESTART,WRITE,FREQUENCY=5 *STEP,NLGEOM,INC=14 *STATIC,RIKS 1.,1.,,,,2041,3,-0.08 *CLOAD 2041,3,-10000. *MONITOR,NODE=2041,DOF=3 *EL PRINT,ELSET=PROFILE,POSITION=AVERAGED AT NODES,FREQUENCY=0 SF, *NODE PRINT,NSET=XPART,TOTALS=YES,FREQUENCY=0 U, RF, *NODE PRINT,NSET=XPARB,TOTALS=YES,FREQUENCY=0 U, RF, *NODE FILE,FREQUENCY=1,NSET=MASTER U,CF *END STEP

1.2.3 Buckling of a column with spot welds Product: ABAQUS/Explicit This example illustrates the dynamic collapse of a steel column constructed by spot welding two channel sections. It is intended to illustrate the modeling of spot welds. ``Spot welds,'' Section 20.3.5 of the ABAQUS/Explicit User's Manual, discusses the spot weld modeling capabilities provided in ABAQUS/Explicit.

Problem description The pillar is composed of two columns of different cross-sections, one box-shaped and the other W-shaped,welded together with spot welds (Figure 1.2.3-1). The top end of the pillar is connected to a rigid body, which makes the deformation of the pillar easy to control by manipulating the rigid body reference node. The column with the box-shaped cross-section is defined to be the slave surface in contact with the column with the W-shaped cross-section. The box-shaped column is welded to the W-shaped column with five spot welds on either side of the box-shaped column. The columns are both composed of aluminum-killed steel, which is assumed to satisfy the Ramberg-Osgood relation between true stress and logarithmic strain: " = ¾=E + (¾=K )n ; where Young's modulus (E) is 206.8 GPa, the reference stress value (K) is 0.510 GPa, and the work-hardening exponent (n) is 4.76. The material is assumed to be linear elastic below a 0.5% offset yield stress of 170.0 MPa. (The 0.5% offset yield stress is defined from the Ramberg-Osgood fit by 1-327

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taking (" ¡ ¾=E ) to be 0.5% and solving for the stress.) Poisson's ratio is 0.3. The spot welds on the two sides of the box-shaped column are modeled with different yield forces and post-yield behavior to illustrate the two failure models. Spot welded nodes 5203, 15203, 25203, 35203, and 45203 are all located on the positive z-side of the box-shaped column, with node 5203 at the bottom end of the column and node 45203 at the top end of the column. The force to cause failure for the spot welds is 3000 N in pure tension and 1800 N in pure shear. Once the spot welds start to fail, the maximum force that they can bear is assumed to decay linearly with time over the course of 2.0 msec, which illustrates modeling of complete loss of strength over a given time period. These spot welds are shown in Figure 1.2.3-2. Spot welded nodes 5211, 15211, 25211, 35211, and 45211 are all located on the negative z-side of the box-shaped column, with node 5211 at the bottom end of the column and node 45211 at the top end of the column. The force to cause failure for these spot welds is 4000 N in pure tension and 2300 N in pure shear. The spot welds fail according to the damaged failure model, which assumes that the maximum forces that the spot welds can carry decay linearly with relative displacement between the welded node and the master surface. The welds are defined to fail completely once their total relative displacement reaches 0.3 mm, which illustrates modeling of loss of strength in the spot welds based on energy absorption.

Loading The bottom of the pillar is fully built-in. The reference node for the rigid body at the top of the pillar moves at a constant velocity of 25 m/sec in the y-direction, thus loading it in compression, together with a velocity of 2 m/sec in the z-direction that shears it slightly. At the same time the end of the pillar is rotated about the negative z-axis at 78.5 rad/sec and rotated about the negative x-axis at 7 rad/sec. This loading is applied by prescribing the velocities of the reference node of the rigid body that is attached to the top end of the compound pillar. The analysis is carried out over 10 milliseconds.

Results and discussion Figure 1.2.3-3 shows the deformed shape of the pillar after 5.0 msec. Figure 1.2.3-4shows the deformed shape of the pillar after 10.0 msec. Figure 1.2.3-5 and Figure 1.2.3-6show the status of the spot welds on the positive z-side of the column and the negative z-side of the column, respectively. In these figures a status of 1.0 means that the weld is fully intact, and 0.0 means that the weld has failed completely. Figure 1.2.3-7shows the load on spot weld node 25203 relative to the failure load. This relative value is called the bond load and is defined to be 1.0 when the spot weld starts to fail and 0.0 when the spot weld is broken. Figure 1.2.3-8 shows the time history of the total kinetic energy, the total work done on the model, the total energy dissipated by friction, the total internal energy, and the total energy balance.

Input files pillar.inp

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Input data for this analysis. pillar_rest.inp Used to test the restart capability with spot welds. pillar_ds.inp Analysis using the double-sided surface capability.

Figures Figure 1.2.3-1 Initial configuration of the compound pillar.

Figure 1.2.3-2 Initial configuration of the box-shaped column showing spot welds.

Figure 1.2.3-3 Deformed shape at 5.0 msec.

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Figure 1.2.3-4 Deformed shape at 10.0 msec.

Figure 1.2.3-5 Time histories of the status of all spot welds on positive z-side of column.

Figure 1.2.3-6 Time histories of the status of all spot welds on negative z-side of column.

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Figure 1.2.3-7 Time histories of the load on spot weld node 25203 relative to the failure load.

Figure 1.2.3-8 Time histories of the total kinetic energy, energy dissipated by friction, work done on the model, internal energy, and total energy.

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Sample listings

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Listing 1.2.3-1 *HEADING Buckling of car with spot welds ************************************************ ** *NODE,NSET=W0 101, 0., 0., 0.065 105, 0.02, 0., 0.065 109, 0.02, 0., 0.045 113, 0., 0., 0.045 117, 0., 0., 0.02 121, 0.02, 0., 0.02 125, 0.02, 0., 0. 129, 0., 0., 0. *NCOPY,CHANGE=50000,OLD=W0,NEW=W50,SHIFT 0.12,0.45,0. 0.,0.45,0., 0.,0.45,1., -20. *NGEN,LINE=P,NSET=WPILA 101,50101,1000,, 0.03,0.2,0.065 *NGEN,LINE=P,NSET=WPILB 105,50105,1000,, 0.05,0.2,0.065 *NGEN,LINE=P,NSET=WPILC 109,50109,1000,, 0.05,0.2,0.045 *NGEN,LINE=P,NSET=WPILD 113,50113,1000,, 0.03,0.2,0.045 *NGEN,LINE=P,NSET=WPILE 117,50117,1000,, 0.03,0.2,0.02 *NGEN,LINE=P,NSET=WPILF 121,50121,1000,, 0.05,0.2,0.02 *NGEN,LINE=P,NSET=WPILG 125,50125,1000,, 0.05,0.2,0.0 *NGEN,LINE=P,NSET=WPILH 129,50129,1000,, 0.03,0.2,0.0 *NFILL,NSET=WPIL WPILA,WPILB,4,1 WPILB,WPILC,4,1 WPILC,WPILD,4,1 WPILD,WPILE,4,1 WPILE,WPILF,4,1 WPILF,WPILG,4,1 WPILG,WPILH,4,1 ************************************************

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*ELEMENT,TYPE=S4R,ELSET=WPIL 101, 101,102,1102,1101 *ELGEN,ELSET=WPIL 101, 28,1,1, 50,1000,1000 ************************************************ ** box *NODE,NSET=B0 200, 0.003,0.,0.03725 201, 0.003,0.,0.042 205, 0.025,0.,0.042 209, 0.025,0.,0.023 213, 0.003,0.,0.023 214, 0.003,0.,0.02775 *NCOPY,CHANGE=50000,OLD=B0,NEW=B50,SHIFT 0.12,0.45,0. 0.,0.45,0., 0.,0.45,1., -20. *NGEN,LINE=P,NSET=BPILA 200,50200,1000,, 0.033,0.2,0.03725 *NGEN,LINE=P,NSET=BPILB 201,50201,1000,, 0.033,0.2,0.042 *NGEN,LINE=P,NSET=BPILC 205,50205,1000,, 0.055,0.2,0.042 *NGEN,LINE=P,NSET=BPILD 209,50209,1000,, 0.055,0.2,0.023 *NGEN,LINE=P,NSET=BPILE 213,50213,1000,, 0.033,0.2,0.023 *NGEN,LINE=P,NSET=BPILF 214,50214,1000,, 0.033,0.2,0.02775 *NFILL,NSET=BPIL BPILA,BPILB,1,1 BPILB,BPILC,4,1 BPILC,BPILD,4,1 BPILD,BPILE,4,1 BPILE,BPILF,1,1 ************************************************ *ELEMENT,TYPE=S4R,ELSET=BPIL 200, 200,201,1201,1200 *ELGEN,ELSET=BPIL 200, 14,1,1, 50,1000,1000 ************************************************ ** roof *NODE,NSET=ROOF 60000, 0.0125, 0., 0.0325

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*NCOPY,CHANGE=0,OLD=ROOF,NEW=ROOF,SHIFT 0.12,0.45,0. 0.,0.45,0., 0.,0.45,1., -20. *ELEMENT,ELSET=ROOF,TYPE=R3D4 60101, 49101,49102,50102,50101 60201, 49201,49202,50202,50201 *ELGEN,ELSET=ROOF 60101, 28,1,1 60201, 12,1,1 *ELEMENT,TYPE=MASS,ELSET=REF 60000, 60000 *MASS,ELSET=REF 1.E-3, ************************************************ ** *SHELL SECTION,ELSET=WPIL,MATERIAL=STEEL, SECTION INTEGRATION=GAUSS 0.002, *SHELL SECTION,ELSET=BPIL,MATERIAL=STEEL, SECTION INTEGRATION=GAUSS 0.003, ************************************************ ** *MATERIAL,NAME=STEEL *DENSITY 7850., *ELASTIC 206.8E9,0.3 *PLASTIC 170.0E6, 0.0000000E+00 180.0E6, 1.7205942E-03 190.0E6, 3.8296832E-03 200.0E6, 6.3897874E-03 210.0E6, 9.4694765E-03 220.0E6, 1.3143660E-02 230.0E6, 1.7493792E-02 240.0E6, 2.2608092E-02 250.0E6, 2.8581845E-02 260.0E6, 3.5517555E-02 270.0E6, 4.3525275E-02 280.0E6, 5.2722659E-02 290.0E6, 6.3235357E-02 300.0E6, 7.5197279E-02

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310.0E6, 8.8750519E-02 320.0E6, 0.1040458 330.0E6, 0.1212430 340.0E6, 0.1405106 350.0E6, 0.1620263 360.0E6, 0.1859779 370.0E6, 0.2125620 380.0E6, 0.2419857 390.0E6, 0.2744660 400.0E6, 0.3102303 410.0E6, 0.3495160 420.0E6, 0.3925720 430.0E6, 0.4396578 440.0E6, 0.4910434 450.0E6, 0.5470111 460.0E6, 0.6078544 470.0E6, 0.6738777 480.0E6, 0.7453985 490.0E6, 0.8227461 500.0E6, 0.9062610 510.0E6, 0.9962980 ************************************************ *NSET,NSET=BASE,GEN 101,129,1 200,214,1 *BOUNDARY BASE,ENCASTRE ************************************************ ** *SURFACE,TYPE=ELEMENT,NAME=BPIL BPIL,SPOS *SURFACE,TYPE=ELEMENT,NAME=WPIL WPIL,SPOS *RIGID BODY,ELSET=ROOF,REF=60000 *STEP *DYNAMIC,EXPLICIT ,10.E-3 *RESTART,WRITE,NUM=2, TIMEMARKS=NO ******************** *CONTACT PAIR BPIL, ******************** *NSET,NSET=WELDA,GEN

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5203,45203,10000 *NSET,NSET=WELDB,GEN 5211,45211,10000 *NSET,NSET=WELDS WELDA,WELDB *CONTACT PAIR,INTER=WELDS,WEIGHT=0. BPIL, WPIL *SURFACE INTERACTION,NAME=WELDS *FRICTION,TAUMAX=1.4E8 .3, *BOND WELDA, 3000., 1800., 0.5E-3, 2.E-3, 0., 0. WELDB, 4000., 2300., 0.5E-3, 0., 0.3E-3, 0.3E-3 ******************** *BOUNDARY,TYPE=VELOCITY 60000, 1,1, 0. 60000, 2,2, -25. 60000, 3,3, -2. 60000, 4,4, -7. 60000, 5,5, 0. 60000, 6,6, -78.5 ******************** *HISTORY OUTPUT,TIME=2.E-5 *NODE HISTORY,NSET=WELDS RF,U,BONDSTAT,BONDLOAD *NODE HISTORY,NSET=ROOF RF,U *ENERGY HISTORY ALLIE,ALLKE,ALLWK,ALLVD,ETOTAL,ALLFD,ALLSE, ALLAE,ALLCD,ALLPD,DT *FILE OUTPUT,TIMEMARKS=YES,NUM=1 *NODE FILE U, *ENERGY FILE *END STEP

1.2.4 Elastic-plastic K-frame structure Product: ABAQUS/Standard This example illustrates the use of the frame element FRAME2D. Frame elements (``Frame elements,'' Section 15.4.1 of the ABAQUS/Standard User's Manual) can be used to model elastic, elastic-plastic, and buckling strut responses of frame-like structures. The elastic response is defined by Euler-Bernoulli beam theory. The elastic-plastic response is modeled with nonlinear kinematic

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hardening plasticity concentrated at the element's ends, simulating the development of plastic hinges. The buckling strut response is a simplified, phenomenological representation of the highly nonlinear cross-section collapse and material yielding that takes place when slender members are loaded in compression; therefore, frame elements can be elastic, elastic-plastic, behave as struts (with or without buckling), or switch during the analysis to strut behavior followed by postbuckling behavior. Both the elastic-plastic and buckling strut responses are simplifications of highly nonlinear responses. They are designed to approximate these complex responses with a single finite element representing a structural member between connections. For parts of the model where higher solution resolution is required, such as stress prediction, the model should be refined with beam elements. The geometry in this example is a typical K-frame construction used in applications such as offshore structures (see Figure 1.2.4-1). A push-over analysis is performed to determine the maximum horizontal load that the structure can support before collapse results from the development of plastic hinges or buckling failure. During a push-over test, many structural members are loaded in compression. Slender members loaded in compression often fail due to geometric buckling, cross-section collapse, and/or material yielding. The buckling strut response, which models such compressive behavior, is added in separate simulations to investigate the effect of the compressive failure of critical members in the structure. Push-over analyses are either load or displacement control tests. A dead load is applied to the top of the structure representing the weight supported by the K-frame.

Geometry and model The structure consists of 19 members between structural connections. Hence, 19 frame elements are used: 17 elements with PIPE cross-sections of varying properties and 2 elements (the top platform) with I cross-sections. The plastic response of the elements is calculated from the yield stress of the material, using the plastic default values provided by ABAQUS. (The default values for the plastic response are based on experiments with slender steel members. For details on the default values, see ``Frame section behavior,'' Section 15.4.2 of the ABAQUS/Standard User's Manual.) The default plastic response includes mild hardening for axial forces and strong hardening for bending moments. The default hardening responses for a typical element in the model are shown in Figure 1.2.4-2 and Figure 1.2.4-3. A dead load of 444.8 kN (1.0 ´ 105 lb) is applied to the top of the K-frame, representing the part of the structure above the K-frame. Subsequently, the top platform is loaded or displaced horizontally. The load level or applied displacement is chosen to be large enough so that the entire structure fails by the formation of plastic hinges and, consequently, loses load carrying capacity. Three different models are investigated. A limit load is expected, since the goal of the analysis is to determine when the structure loses overall stiffness. Large- and small-displacement analyses are performed for all three models for comparison. (Frame elements assume that the strains are small. Large-displacement analyses using frame elements are valid for large overall rotations but small strains.) In the first model all elements use elastic-plastic material response. In the second model buckling is checked for all elements with PIPE cross-sections. The ISO equation is used as a criteria for buckling, and the default Marshall strut envelope is followed for the postbuckling behavior. The buckling strut envelope is calculated from the yield stress of the material and the default Marshall Strut

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theory. (For details on the default buckling strut envelope, see ``Frame section behavior,'' Section 15.4.2 of the ABAQUS/Standard User's Manual.) All frame members use the BUCKLING parameter on the *FRAME SECTION option to check the ISO criteria for the switching-to-strut algorithm. The displacement-control analyses are performed for both small and large displacements. In the third model the two elements that switched to strut behavior in the second model (elements 7 and 9) are replaced by frame elements with buckling strut response from the beginning of the analysis. To proceed beyond the unstable phase of the response, the Riks static solution procedure is used in the elastic-plastic problems. In large-displacement analysis with the switching algorithm and frame elements with buckling, the STABILIZE parameter is used on the *STATIC option to stabilize the results for the loads close to the limit load point. To decrease the number of solution iterations, the *CONTROLS option is used in some cases with the value of the ratio of the largest solution correction to the largest incremental solution set to 1.0, since displacement increments are very small for increments where switching occurs.

Results and discussion The structure is loaded or displaced to the point at which all load carrying capacity is lost. In the first model with elastic-plastic frame elements, the results for the linear and nonlinear geometries compare as expected. The limit load for the large-displacement analysis is reached at a load of 1141 kN (2.56 ´ 105 lb) as compared to a higher load of 1290 kN (2.91 ´ 105 lb) in the small-displacement analysis. The plastic hinge pattern is the same in both cases. The second model uses the switching algorithm. It shows that element 7 first violates the ISO equation (buckles) at a prescribed displacement equal to 1.85 ´ 10--2, before any elements form plastic hinges. The critical compressive force in this element is -303 kN (-68.12 ´ 103 lb) for the large-displacement analysis. Next, element 9 buckles after several elements develop plasticity. The frame elements with the switching algorithm predict the structural behavior in the most accurate way, checking the buckling criteria for all elements in the model and switching automatically to postbuckling behavior for highly compressed members (see the plastic and buckled frame elements in Figure 1.2.4-4). When the structure can no longer support horizontal loading, the patterns of plastic hinges for linear and nonlinear geometry are very similar. They only show small differences for loads close to the limit load. To compare the results of two different frame behaviors, the first and the third models are investigated (kframe_loadcntrl_nlgeom.inp and kframe_dispcntrl_buckle_nlgeom.inp). Load versus horizontal deflection curves for the large-displacement analyses are shown in Figure 1.2.4-5. The model with two elements using the buckling strut response becomes unstable as soon as the first element buckles. As the other elements deform and absorb the load no longer carried by the buckled element, the structure regains stiffness until the second element buckles. At this point the structure can no longer support additional horizontal loading because of the presence of buckled elements and the formation of plastic hinges. The limit load in the third model reaches only about 22% of the limit load in the model without buckling. The load displacement curves for the switching algorithm and for the example with elements 7 and 9 using the buckling strut response compare well.

Input files

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kframe_loadcntrl_nlgeom.inp Elastic-plastic analysis with load control; large-displacement analysis. kframe_loadcntrl.inp Elastic-plastic analysis with load control; small-displacement analysis. kframe_dispcntrl_switch_nlgeom.inp Elastic-plastic frame element with the switching algorithm and displacement control; large-displacement analysis. kframe_dispcntrl_switch.inp Elastic-plastic frame element with the switching algorithm and displacement control; small-displacement analysis. kframe_dispcntrl_buckle_nlgeom.inp Elastic-plastic and buckling strut response with load control; large-displacement analysis. kframe_dispcntrl_buckle.inp Elastic-plastic and buckling strut response with displacement control; small-displacement analysis.

Figures Figure 1.2.4-1 Two-dimensional K-frame structure.

Figure 1.2.4-2 Default hardening response for axial force in a typical element with PIPE cross-section

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(element 7 in the model).

Figure 1.2.4-3 Default hardening response for bending moments in a typical element with PIPE cross-section (element 7 in the model).

Figure 1.2.4-4 Results of analysis with switching algorithm: K-frame model with plastic and two buckled elements.

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Figure 1.2.4-5 Applied force versus horizontal displacement of the load point for the elastic-plastic model and the model including buckling strut response.

Sample listings

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Listing 1.2.4-1 *HEADING ELASTIC-PLASTIC K-FRAME: *RESTART,WRITE,FREQ=999 *NODE,NSET=ALL 1,-139.98,0.0 2, 3,139.98,0.0 4,-131.4996,72.0 5,0.0,72.0 6,131.4996,72.0 7,-110.9664,256.8 9, 110.9664,256.8 10,-104.3004,316.8 11,0.0,316.8 12,104.3004,316.8 *NSET,NSET=TEN 10 *ELEMENT,TYPE=FRAME2D,ELSET=BOTTOM 1,1,2 2,2,3 3,2,4 4,2,6 *ELEMENT,TYPE=FRAME2D,ELSET=MID 5,4,5 6,5,6 9,7,9 *ELEMENT,TYPE=FRAME2D,ELSET=PINNED1 7,5,7 8,5,9 *ELEMENT,TYPE=FRAME2D,ELSET=TOP 10,7,11 11,9,11 *ELEMENT,TYPE=FRAME2D,ELSET=W 12,10,11 13,11,12 *ELEMENT,TYPE=FRAME2D,ELSET=BOTTOM_C 14,1,4 15,3,6 *ELEMENT,TYPE=FRAME2D,ELSET=MID_COL 16,4,7 17,6,9

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18,7,10 19,9,12 *FRAME SECTION,SECTION=PIPE,ELSET=BOTTOM, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 4.3125,0.25 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=MID, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 3.125,0.156 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=PINNED1, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 3.125,0.156 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=TOP, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 4.3125,0.25 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=I,ELSET=W, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 7.01,14.02,14.52,14.52,0.71,0.71,0.44 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=BOTTOM_C, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 6.375,0.5 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=MID_COL, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 6.375,0.33 0.,0.,-1. 3.0E7,1.5E7 *BOUNDARY 1,1,2 3,1,2 *STEP,NLGEOM *STATIC *CLOAD

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10,2,-2.5E4 11,2,-5.0E4 12,2,-2.5E4 *EL PRINT,ELSET=PINNED1 ***** ** TO FIND ALL PLASTIC HINGES ** PRINT SEP FOR ALL ELEMENTS ***** SF, SEE, SEP, SALPHA, *FILE FORMAT,ZERO INCREMENT **NODE FILE,FREQ=100 *NODE FILE,FREQ=1 U, CF, *EL FILE,FREQ=100 SF, SEE, SEP, SALPHA, *ENERGY FILE,FREQ=100 *MONITOR,DOF=1,NODE=10 *END STEP *STEP,INC=78,NLGEOM ****** ** FOR THE LIMIT LOAD ANALYSIS INCREASE ** THE NUMBER OF INCREMENTS TO INC=100 ****** *STATIC,RIKS 0.001,1.0,,0.03 *controls,parameter=field ,1.0 *CLOAD 10,1, 2.75E5 *NODE FILE,FREQ=1,NSET=TEN U, CF, *END STEP

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Listing 1.2.4-2 *HEADING ELASTIC-PLASTIC K-FRAME WITH SWITCHING ALGORITHM *RESTART,WRITE,FREQ=999 *NODE,NSET=ALL 1,-139.98,0.0 2, 3,139.98,0.0 4,-131.4996,72.0 5,0.0,72.0 6,131.4996,72.0 7,-110.9664,256.8 9, 110.9664,256.8 10,-104.3004,316.8 11,0.0,316.8 12,104.3004,316.8 *ELEMENT,TYPE=FRAME2D,ELSET=BOTTOM 1,1,2 2,2,3 3,2,4 4,2,6 *ELEMENT,TYPE=FRAME2D,ELSET=MID 5,4,5 6,5,6 9,7,9 *ELEMENT,TYPE=FRAME2D,ELSET=PINNED1 7,5,7 8,5,9 *ELEMENT,TYPE=FRAME2D,ELSET=TOP 10,7,11 11,9,11 *ELEMENT,TYPE=FRAME2D,ELSET=W 12,10,11 13,11,12 *ELEMENT,TYPE=FRAME2D,ELSET=BOTTOM_C 14,1,4 15,3,6 *ELEMENT,TYPE=FRAME2D,ELSET=MID_COL 16,4,7 17,6,9 18,7,10 19,9,12

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*FRAME SECTION,SECTION=PIPE,ELSET=BOTTOM, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS,BUCKLING 4.3125,0.25 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=MID, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS,BUCKLING 3.125,0.156 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=PINNED1, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS,BUCKLING 3.125,0.156 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=TOP, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS,BUCKLING 4.3125,0.25 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=I,ELSET=W, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 7.01,14.02,14.52,14.52,0.71,0.71,0.44 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=BOTTOM_C, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS,BUCKLING 6.375,0.5 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=MID_COL, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS,BUCKLING 6.375,0.33 0.,0.,-1. 3.0E7,1.5E7 *ELSET,ELSET=OUTPUT 7,8,9 *NSET,NSET=TEN 10 *BOUNDARY 1,1,2 3,1,2 *STEP,INC=500,NLGEOM

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*STATIC,stabilize 0.01,1.0,,0.03 *controls,parameter=field ,1.0 *BOUNDARY ***** ** TO MAKE A PLOT AND COMPARE WITH THE LINEAR ** SOLUTION USE THE COMMENTED ** PRESCRIBED DISPLACEMENT **10,1,1,26. **10,2,2,10. **11,2,2,10. **12,2,2,10. ***** 10,1,1,5.72 10,2,2,2.2 11,2,2,2.2 12,2,2,2.2 *EL PRINT,FREQ=100,ELSET=OUTPUT SF, SEE, *EL PRINT,FREQ=100,ELSET=OUTPUT SEP, SALPHA, *FILE FORMAT,ZERO INCREMENT *NODE PRINT,NSET=TEN,FREQ=100 RF, *NODE FILE,NSET=TEN,FREQ=100 **** ** TO PLOT USE FREQ=1 **** U, RF, *EL FILE,FREQ=100,ELSET=OUTPUT SF, SEE, *EL FILE,FREQ=100 SEP, SALPHA, *ENERGY FILE,FREQ=50 *MONITOR,DOF=1,NODE=10 *END STEP

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Listing 1.2.4-3 *HEADING ELASTIC-PLASTIC, BUCKLING STRUT, DISPLACEMENT CONTROL, K-FRAME: *RESTART,WRITE,FREQ=999 *NODE,NSET=ALL 1,-139.98,0.0 2, 3,139.98,0.0 4,-131.4996,72.0 5,0.0,72.0 6,131.4996,72.0 7,-110.9664,256.8 9, 110.9664,256.8 10,-104.3004,316.8 11,0.0,316.8 12,104.3004,316.8 *NSET,NSET=TEN 10 *ELEMENT,TYPE=FRAME2D,ELSET=BOTTOM 1,1,2 2,2,3 3,2,4 4,2,6 *ELEMENT,TYPE=FRAME2D,ELSET=MID 5,4,5 6,5,6 *ELEMENT,TYPE=FRAME2D,ELSET=PINNED1 9,7,9 *ELEMENT,TYPE=FRAME2D,ELSET=PINNED2 7,5,7 *ELEMENT,TYPE=FRAME2D,ELSET=BRACER 8,5,9 *ELEMENT,TYPE=FRAME2D,ELSET=TOP 10,7,11 11,9,11 *ELEMENT,TYPE=FRAME2D,ELSET=W 12,10,11 13,11,12 *ELEMENT,TYPE=FRAME2D,ELSET=BOTTOM_C 14,1,4 15,3,6

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*ELEMENT,TYPE=FRAME2D,ELSET=MID_COL 16,4,7 17,6,9 18,7,10 19,9,12 *FRAME SECTION,SECTION=PIPE,ELSET=BOTTOM, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 4.3125,0.25 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=PINNED1, YIELDSTRESS=51.9E3,PINNED,BUCKLING 3.125,0.156 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=MID, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 3.125,0.156 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=PINNED2, YIELDSTRESS=51.9E3,PINNED,BUCKLING 3.125,0.156 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=BRACER, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 3.125,0.156 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=TOP, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 4.3125,0.25 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=I,ELSET=W, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 7.01,14.02,14.52,14.52,0.71,0.71,0.44 0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=BOTTOM_C, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 6.375,0.5

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0.,0.,-1. 3.0E7,1.5E7 *FRAME SECTION,SECTION=PIPE,ELSET=MID_COL, YIELDSTRESS=51.9E3,PLASTIC DEFAULTS 6.375,0.33 0.,0.,-1. 3.0E7,1.5E7 *ELSET,ELSET=OUTPUT 7,8,9 *BOUNDARY 1,1,2 3,1,2 *STEP,NLGEOM *STATIC,stabilize *CLOAD 10,2,-2.5E4 11,2,-5.0E4 12,2,-2.5E4 *EL PRINT,FREQ=50 SF, SEE, SEP, SALPHA, *FILE FORMAT,ZERO INCREMENT *NODE FILE,NSET=TEN,FREQ=50 U, RF, *EL FILE,FREQ=50,ELSET=OUTPUT SF, SEE, SEP, SALPHA, *ENERGY FILE,FREQ=50 *MONITOR,DOF=1,NODE=10 *END STEP *STEP,INC=200,NLGEOM *STATIC,stabilize 0.001,1.0,,0.02 *controls,parameter=field ,1.0 *BOUNDARY **** ** TO PLOT USE THE COMMENTED

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** PRESCRIBED DISPLACEMENT **10,1,1,26. **** 10,1,1,20.8 *NODE FILE,NSET=TEN,FREQ=50 **** ** TO PLOT USE FREQ=1 **** U, RF, *END STEP

1.2.5 Unstable static problem: reinforced plate under compressive loads Product: ABAQUS/Standard This example demonstrates the use of automatic techniques to stabilize unstable static problems. Geometrically nonlinear static problems can become unstable for a variety of reasons. Instability may occur in contact problems, either because of chattering or because contact intended to prevent rigid body motions is not established initially. Localized instabilities can also occur; they can be either geometrical, such as local buckling, or material, such as material softening. This problem models a reinforced plate structure subjected to in-plane compressive loading that produces localized buckling. Structures are usually designed for service loads properly augmented by safety factors. However, it is quite often of interest to explore their behavior under extreme accident loads. This example looks into a submodel of a naval construction structure. It is a rectangular plate reinforced with beams in its two principal directions ( Figure 1.2.5-1). The plate has symmetry boundary conditions along the longer edges and is pinned rigidly along the shorter sides. An in-plane load is applied to one of the pinned sides, compressing the plate. Gravity loads are also applied. The plate buckles under the load. The buckling is initially localized within each of the sections bounded by the reinforcements. At higher load levels the plate experiences global buckling in a row of sections closest to the applied load. Standard analysis procedures typically provide the load at which the structure starts to buckle. The user may be interested in knowing the structure's additional load carrying capacity. This information could translate, for instance, into knowing when the onset of global buckling takes place or how far into the structure damage propagates. In such situations more sophisticated analysis techniques are necessary. Arc length methods such as the Riks method available in ABAQUS are global load-control methods that are suitable for global buckling and postbuckling analyses; they do not function well when buckling is localized. Alternatives are to analyze the problem dynamically or to introduce damping. In the dynamic case the strain energy released locally from buckling is transformed into kinetic energy; in the damping case this strain energy is dissipated. To solve a quasi-static problem dynamically is typically an expensive proposition. In this example the automatic stabilization capability in ABAQUS, which applies volume proportional damping to the structure, is used.

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Geometry and model The model consists of a rectangular plate 10.8 m (425.0 in) long, 6.75 m (265.75 in) wide, and 5.0 mm (0.2 in) thick. This plate has several reinforcements in both the longitudinal and transverse directions (Figure 1.2.5-1). The plate represents part of a larger structure: the two longitudinal sides have symmetry boundary conditions, and the two transverse sides have pinned boundary conditions. In addition, springs at two major reinforcement intersections represent flexible connections to the rest of the structure. The mesh consists of S4 shell elements for both the plate and larger reinforcements and additional S3 shell and B31 beam elements for the remaining reinforcements. The entire structure is made of the same construction steel, with an initial flow stress of 235.0 MPa (34.0 ksi).

Results and discussion The analysis consists of two steps. In the first step a gravity load perpendicular to the plane of the plate is applied. In the second step a longitudinal compressive load of 6.46 ´ 106 N (1.45 ´ 106 lbf) is applied to one of the pinned sides of the plate. All the nodes on that side are forced to move equally by means of multi-point constraints. The analysis is quasi-static, but buckling is expected. The volume proportional damping stabilizing capability in ABAQUS is invoked with the *STATIC, STABILIZE option, with the default damping intensity. This option applies a damping coefficient such that the viscous dissipated energy extrapolated from the first increment to the total step is a small fraction (2.0 ´ 10-4) of the strain energy also extrapolated from the first increment to the total step. The algorithm works quite well in situations such as this problem, in which the first increment of a step is stable but instabilities develop later in the analysis. Initially local out-of-plane buckling develops throughout the plate in an almost checkerboard pattern inside each one of the sections delimited by the reinforcements (Figure 1.2.5-2). Later, global buckling develops along a front of sections closer to the applied load (Figure 1.2.5-3). The evolution of the displacements produced by the applied load is very smooth (Figure 1.2.5-4) and does not reflect the early local instabilities in the structure. However, when the global instability develops, the curve becomes almost flat, indicating the complete loss of load carrying capacity. An inspection of the model's energy content (Figure 1.2.5-5 and Figure 1.2.5-6) reveals that while the load is increasing, the amount of dissipated energy is negligible. As soon as the load flattens out, the strain energy also flattens out (indicating a more or less constant load carrying capacity), while the dissipated energy increases dramatically to absorb the work done by the applied loads.

Acknowledgements HKS would like to thank IRCN (France) for providing this example.

Input files unstablestatic_plate.inp Plate model. unstablestatic_plate_node.inp Node definitions for the plate model.

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unstablestatic_plate_elem.inp Element definitions for the plate model.

Figures Figure 1.2.5-1 Reinforced plate initial mesh.

Figure 1.2.5-2 Plate localized buckling.

Figure 1.2.5-3 Plate global buckling.

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Figure 1.2.5-4 Plate load-displacement curve.

Figure 1.2.5-5 Dissipated and strain energies as functions of load.

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Figure 1.2.5-6 Dissipated and strain energies as functions of displacement.

Sample listings

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Listing 1.2.5-1 *HEADING STABILIZED COMPRESSION OF REINFORCED PLATE *PREPRINT,ECHO=NO,HISTORY=NO,MODEL=YES ** ************************************************ ** FILE WITH NODE DEFINITIONS AND NSETS: ** COTE1 ** COTE2 ** COTE3 ** COTE4 ************************************************ ** *INCLUDE,INPUT=unstablestatic_plate_node.inp ** ************************************************ ** FILE WITH ELEMENT DEFINITIONS AND ELSETS: ** E0000001 144 B31 ** E0000002 144 B31 ** E0000003 6 S3 ** E0000004 8 S3 ** TOLE 1008 S4 ** E0000006 288 S4 ** E0000007 252 S4 ** E0000008 178 S4 ** E0000009 108 S4 ** E0000010 78 S4 ************************************************ ** *INCLUDE,INPUT=unstablestatic_plate_elem.inp ** *ELEMENT,TYPE=SPRING1,ELSET=EPONT 50000,340 50001,2881 *SPRING,ELSET=EPONT 3, 5.3E+08, *BEAM SECTION,MATERIAL=ACIER,SECTION=RECT, ELSET=E0000001 1.500000E-02, 1.500000E-02 0.000000E+00, 6.557236E-03,-9.999785E-01 *MATERIAL,NAME=ACIER

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*ELASTIC 2.100E+11, 3.000E-01 *PLASTIC 235.259E+06,0.0 471.9E+06,0.18837 *DENSITY 7.850E+03, *BEAM SECTION,MATERIAL=ACIER,SECTION=RECT, ELSET=E0000002 1.500000E-02, 1.500000E-02 0.000000E+00, 8.695324E-03,-9.999622E-01 *SHELL SECTION,ELSET=E0000003,MATERIAL=ACIER 7.000E-03, 3 *SHELL SECTION,ELSET=E0000004,MATERIAL=ACIER 1.000E-02, 3 *SHELL SECTION,ELSET=TOLE,MATERIAL=ACIER 5.000E-03, 3 *SHELL SECTION,ELSET=E0000006,MATERIAL=ACIER 6.000E-03, 3 *SHELL SECTION,ELSET=E0000007,MATERIAL=ACIER 7.000E-03, 3 *SHELL SECTION,ELSET=E0000008,MATERIAL=ACIER 1.000E-02, 3 *SHELL SECTION,ELSET=E0000009,MATERIAL=ACIER 1.000E-02, 3 *SHELL SECTION,ELSET=E0000010,MATERIAL=ACIER 1.000E-02, 3 *NSET,NSET=NM1 329, *ELSET,ELSET=HILOIRE E0000004,E0000009,E0000010 *ELSET,ELSET=BARROTS E0000003,E0000007,E0000008 *ELSET,ELSET=TOT TOLE,HILOIRE,BARROTS,E0000001,E0000002,E0000006 ** *MPC BEAM,COTE1,NM1 ** *BOUNDARY NM1,3,3 NM1,4,4 NM1,6,6

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COTE2,1,1 COTE2,3,3 COTE2,4,4 COTE2,6,6 8896,2,2 COTE3,2,2 COTE3,4,4 COTE3,6,6 COTE4,2,2 COTE4,4,4 COTE4,6,6 ** *RESTART,WRITE,F=10 ** ************************************************ ** *STEP,NLGEOM,INC=1 GRAVITY LOAD *STATIC 1.,1. *DLOAD TOT,GRAV,9.81,0,0,-1. *NODE FILE,NSET=NM1,FREQ=1 U,CF *EL FILE,ELSET=HILOIRE,FREQ=200 S,E,PE *ENERGY FILE *OUTPUT,FIELD *NODE OUTPUT U,VF *OUTPUT,FIELD,FREQ=200 *ELEMENT OUTPUT S,E,PE *OUTPUT,HISTORY *ENERGY OUTPUT,VARIABLE=ALL *NODE OUTPUT,NSET=NM1 U,CF *EL PRINT,FREQ=0 *NODE PRINT,FREQ=0 *END STEP ** *STEP,NLGEOM,INC=1000 BUCKLING LOAD

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*STATIC,STABILIZE 0.1,1.,, *CLOAD NM1,1,646.E+04 *END STEP

1.2.6 Buckling of an imperfection sensitive cylindrical shell Product: ABAQUS/Standard This example serves as a guide to performing a postbuckling analysis using ABAQUS for an imperfection sensitive structure. A structure is imperfection sensitive if small changes in an imperfection change the buckling load significantly. Qualitatively, this behavior is characteristic of structures with closely spaced eigenvalues. For such structures the first eigenmode may not characterize the deformation that leads to the lowest buckling load. A cylindrical shell is chosen as an example of an imperfection sensitive structure.

Geometry and model The cylinder being analyzed is depicted in Figure 1.2.6-1. The cylinder is simply supported at its ends and is loaded by a uniform, compressive axial load. A uniform internal pressure is also applied to the cylinder. The material in the cylinder is assumed to be linear elastic. The thickness of the cylinder is 1/500 of its radius, so the structure can be considered to be a thin shell. The finite element mesh uses the fully integrated S4 shell element. This element is based on a finite membrane strain formulation and is chosen to avoid hourglassing. A full-length model is used to account for both symmetric and antisymmetric buckling modes. A fine mesh, based on the results of a refinement study of the linear eigenvalue problem, is used. The convergence of the mesh density is based on the relative change of the eigenvalues as the mesh is refined. The mesh must have several elements along each spatial deformation wave; therefore, the level of mesh refinement depends on the modes with the highest wave number in the circumferential and axial directions.

Solution procedure The solution strategy is based on introducing a geometric imperfection in the cylinder. In this study the imperfections are linear combinations of the eigenvectors of the linear buckling problem. If details of imperfections caused in a manufacturing process are known, it is normally more useful to use this information as the imperfection. However, in many instances only the maximum magnitude of an imperfection is known. In such cases assuming the imperfections are linear combinations of the eigenmodes is a reasonable way to estimate the imperfect geometry (Arbocz, 1987). Determining the most critical imperfection shape that leads to the lowest collapse load of an axially compressed cylindrical shell is an open research issue. The procedure discussed in this example does not, therefore, claim to compute the lowest collapse load. Rather, this example discusses one approach that can be used to study the postbuckling response of an imperfection sensitive structure. The first stage in the simulation is a linear eigenvalue buckling analysis. To prevent rigid body motion,

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a single node is fixed in the axial direction. This constraint is in addition to the simply supported boundary conditions noted earlier and will not introduce an overconstraint into the problem since the axial load is equilibrated on opposing edges. The reaction force in the axial direction should be zero at this node. The second stage involves introducing the imperfection into the structure using the *IMPERFECTION option. A single mode or a combination of modes is used to construct the imperfection. To compare the results obtained with different imperfections, the imperfection size must be fixed. The measure of the imperfection size used in this problem is the out-of-roundness of the cylinder, which is computed as the radial distance from the axis of the cylinder to the perturbed node minus the radius of the perfect structure. The scale factor associated with each eigenmode used to seed the imperfection is computed with a FORTRAN program. The program reads the results file produced by the linear analysis and determines the scale factors so that the out-of-roundness of the cylinder is equal to a specified value. This value is taken as a fraction of the cylinder thickness. The final stage of the analysis simulates the postbuckling response of the cylinder for a given imperfection. The primary objective of the simulation is to determine the static buckling load. The modified Riks method is used to obtain a solution since the problem under consideration is unstable. The Riks method can also be used to trace the unstable and stable solution branches of a buckled structure. However, with imperfection sensitive structures the first buckling mode is usually catastrophic, so further continuation of the analysis is usually not undertaken. When using the *STATIC, RIKS option, the tolerance used for the force residual convergence criteria may need to be tightened to ensure that the solution algorithm does not retrace its original loading path once the limit point is reached. Simply restricting the maximum arc length allowed in an increment is normally not sufficient.

Parametric study There are two factors that significantly alter the buckling behavior: the shape of the imperfection and the size of the imperfection. A convenient way to investigate the effects of these factors on the buckling response is to use the parametric study capabilities of ABAQUS. A Python script file is used to perform the study. The script executes the linear analysis, runs the FORTRAN routine to create an input file with a specified imperfection size, and finally executes the postbuckling analysis. Before executing the script, copy the FORTRAN routine cylsh_maximp.f to your work directory using the ABAQUS fetch command, abaqus fetch job=cylsh_maximp.f

and compile it using the ABAQUS make command, abaqus make job=cylsh_maximp.f

Parametrized template input data are used to generate variations of the parametric study. The script allows the analyst to vary the eigenmodes used to construct the imperfection, out-of-roundness measure, cylindrical shell geometry (radius, length, thickness), mesh density, material properties (Young's modulus and Poisson's ratio), etc. The results presented in the following section, however, are based on an analysis performed with a single set of parameters.

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Results and discussion The results for both the linear eigenvalue buckling and postbuckling analyses are discussed below.

Linear eigenvalue buckling The Lanczos eigensolver is used to extract the linear buckling modes. This solver is chosen because of its superior accuracy and convergence rate relative to wavefront solvers for problems with closely spaced eigenvalues. Table 1.2.6-1 lists the first 19 eigenvalues of the cylindrical shell. The eigenvalues are closely spaced with a maximum percentage difference of 1.3%. The geometry, loading, and material properties of the cylindrical shell analyzed in this example are characterized by their axisymmetry. As a consequence of this axisymmetry the eigenmodes associated with the linear buckling problem will be either (1) axisymmetric modes associated with a single eigenvalue, including the possibility of eigenmodes that are axially symmetric but are twisted about the symmetry axis or (2) nonaxisymmetric modes associated with repeated eigenvalues (Wohlever, 1999). The nonaxisymmetric modes are characterized by sinusoidal variations (n-fold symmetry) about the circumference of the cylinder. For most practical engineering problems and as illustrated in Table 1.2.6-1, it is usually found that a majority of the buckling modes of the cylindrical shell are nonaxisymmetric. The two orthogonal eigenmodes associated with each repeated eigenvalue span a two-dimensional space, and as a result any linear combination of these eigenmodes is also an eigenmode; i.e., there is no preferred direction. Therefore, while the shapes of the orthogonal eigenmodes extracted by the eigensolver will always be the same and span the same two-dimensional space, the phase of the modes is not fixed and might vary from one analysis to another. The lack of preferred directions has consequences with regard to any imperfection study based upon a linear combination of nonaxisymmetric eigenmodes from two or more distinct eigenvalues. As the relative phases of eigenmodes change, the shape of the resulting imperfection and, therefore, the postbuckling response, also changes. To avoid this situation, postprocessing is performed after the linear buckling analysis on each of the nonaxisymmetric eigenmode pairs to fix the phase of the eigenmodes before the imperfection studies are performed. The basic procedure involves calculating a scaling factor for each of the eigenvectors corresponding to a repeated eigenvalue so that their linear combination generates a maximum displacement of 1.0 along the global X-axis. This procedure is completely arbitrary but ensures that the postbuckling response calculations are repeatable. For the sake of consistency the maximum radial displacement associated with a unique eigenmode is also scaled to 1.0. These factors are further scaled to satisfy the out-of-roundness criterion mentioned earlier.

Postbuckling response The modes used to seed the imperfection are taken from the first 19 eigenmodes obtained in the linear eigenvalue buckling analysis. Different combinations are considered: all modes, unique eigenmodes, and pairs of repeated eigenmodes. An imperfection size (i.e., out-of-roundness) of 0.5 times the shell thickness is used in all cases. The results indicate that the cylinder buckles at a much lower load than the value predicted by the linear analysis (i.e., the value predicted using only the lowest eigenmode of 1-362

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the system). An imperfection based on mode 1 (a unique eigenmode) results in a buckling load of about 90% of the predicted value. When the imperfection was seeded with a combination of all modes (1-19), a buckling load of 35% of the predicted value was obtained. Table 1.2.6-2 lists the buckling loads predicted by ABAQUS (as a fraction of linear eigenvalue buckling load) when different modes are used to seed the imperfection. The smallest predicted buckling load in this study occurs when using modes 12 and 13 to seed the imperfection, yet the results obtained when the imperfection is seeded using all 19 modes indicate that a larger buckling load can be sustained. One possible explanation for this is that the solution strategy used in this study (discussed earlier) involves using a fixed value for the out-of-roundness of the cylinder as a measure of the imperfection size. Thus, when multiple modes are used to seed the imperfection, the overall effect of any given mode is less than it would be if only that mode were used to seed the imperfection. The large number of closely spaced eigenvalues and innumerable combinations of eigenmodes clearly demonstrates the difficulty of determining the collapse load of structures such as the cylindrical shell. In practice, designing imperfection sensitive structures against catastrophic failure usually requires a combination of numerical and experimental results as well as practical building experience. The deformed configuration shown in Figure 1.2.6-2uses a displacement magnification factor of 5 and corresponds to using all the modes to seed the imperfection. Even though the cylinder appears to be very short, it can in fact be classified as a moderately long cylinder using the parameters presented in Chajes (1985). The cylinder exhibits thin wall wrinkling; the initial buckling shape can be characterized as dimples appearing on the side of the cylinder. The compression of the cylinder causes a radial expansion due to Poisson's effect; the radial constraint at the ends of the cylinder causes localized bending to occur at the ends. This would cause the shell to fold into an accordion shape. (Presumably this would be seen if self-contact was specified and the analysis was allowed to run further. This is not a trivial task, however, and modifications to the solution controls would probably be required. Such a simulation would be easier to perform with ABAQUS/Explicit.) This deformed configuration is in accordance with the perturbed reference geometry, shown in Figure 1.2.6-3. To visualize the imperfect geometry, an imperfection size of 5.0 times the shell thickness (i.e., 10 times the value actually used in the analysis) was used to generate the perturbed mesh shown in this figure. The deformed configuration in the postbuckling analysis depends on the shape of the imperfection introduced into the structure. Seeding the structure with different combinations of modes and imperfection sizes produces different deformed configurations and buckling loads. As the results vary with the size and shape of the imperfection introduced into the structure, there is no solution to which the results from ABAQUS can be compared. The load-displacement curve for the case when the first 19 modes are used to seed the imperfection is shown in Figure 1.2.6-4. The figure shows the variation of the applied load (normalized with respect to the linear eigenvalue buckling load) versus the axial displacement of an end node. The peak load that the cylinder can sustain is clearly visible.

Input files cylsh_buck.inp

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Linear eigenvalue buckling problem. cylsh_postbuck.inp Postbuckling problem. cylsh_maximp.f FORTRAN program to compute the scaling factors for the imperfection size. cylsh_script.psf Python script to generate the parametrized input files.

References · Arbocz, J., "Post-Buckling Behaviour of Structures: Numerical Techniques for More Complicated Structures," in Lecture Notes in Physics, Ed. H. Araki et al., Springer-Verlag, Berlin, 1987, pp. 84-142. · Chajes, A., "Stability and Collapse Analysis of Axially Compressed Cylindrical Shells," in Shell Structures: Stability and Strength , Ed. R. Narayanan, Elsevier, New York, 1985, pp. 1-17. · Wohlever, J. C., "Some Computational Aspects of a Group Theoretic Finite Element Approach to the Buckling and Postbuckling Analyses of Plates and Shells-of-Revolution ," in Computer Methods in Applied Mechanics and Engineering , vol. 170, pp. 373-406, 1999.

Tables Table 1.2.6-1 Eigenvalue estimates for the first 19 modes. Mode Eigenvalue number 1 11721 2, 3 11722 4, 5 11726 6, 7 11733 8, 9 11744 10, 11 11758 12, 13 11777 14, 15 11802 16, 17 11833 18, 19 11872

Table 1.2.6-2 Summary of predicted buckling loads. Mode used to Normalized buckling seed the load imperfection 1 0.902

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2, 3 4, 5 6, 7 8, 9 10, 11 12, 13 14, 15 16, 17 18, 19 All modes (1-19)

0.707 0.480 0.355 0.351 0.340 0.306 0.323 0.411 0.422 0.352

Figures Figure 1.2.6-1 Cylindrical shell with uniform axial loading.

Figure 1.2.6-2 Deformed configuration of the cylindrical shell (first 19 eigenmodes used to seed the imperfection; displacement magnification factor of 5.0).

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Figure 1.2.6-3 Perturbed geometry of the cylindrical shell (imperfection factor = 5 ´ thickness for illustration only; actual imperfection factor used = .5 ´ thickness).

Figure 1.2.6-4 Normalized applied load versus axial displacement at an end node (first 19 modes used to seed the imperfection).

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Sample listings

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Listing 1.2.6-1 import string import os # ######################################################################## # # THIS SCRIPT RUNS A SEQUENCE OF PARAMETERIZED INPUT FILES TO STUDY THE # POSTBUCKLING BEHAVIOR OF A LINEAR ELASTIC, AXIALLY LOADED, # CYLINDRICAL SHELL. # # PARAMETERS USED IN STUDY: # # SHELL THICKNESS: thickness (set by xthickne # LENGTH OF THE CYLINDER: length (set by xlength) # MEAN RADIUS OF THE CYLINDER: radius (set by xradius) # NUMBER OF NODES AROUND CIRCUMF: node_circum (set by nel_c) # NUMBER OF NODES ALONG LENGTH: node_length (set by nel_l) poisson (set by xpoisson # POISSON'S RATIO: # YOUNG'S MODULUS: young (set by xyoung) # APPLIED LOAD (BUCKLING ANALYSIS): tot_load (set by xload) # INTERNAL PRESSURE: int_press (set by press) # NUMBER OF BUCKLING MODES: num_modes (set by nmodes) # # ADDITIONAL PARAMETERS FOR POST-BUCKLING ANALYSIS # # APPLIED AXIAL LOAD (BASED ON # LOWEST BUCKLING MODE): eig1_load (set by xload1) # BUCKLING ANALYSIS RESULTS FILE NAME: buckle file (set by bklfname # *INCLUDE FILE WITH *IMPERFECTION DATA : imperf file (set by impfname # # ADDITIONAL VARIABLES # # IMPERFECTION SCALE FACTOR: RadialImperfFactor # PRECRIBED RADIAL IMPERFECTION: rad_imp # EIGENMODES TO SEED THE IMPERFECTION: eigmodes # SCALE FACTOR ASSOC. WITH EACH MODE: modefctr # -THIS IS A GUESS; THE TRUE FACTORS # WILL BE COMPUTED LATER. # ######################################################################## # # 1. DEFINE THE SHELL PARAMETERS AND MAX-OUT-OF ROUND IMPERFECTION FACTO

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xthickness = 0.01 xlength = 2.0 xradius = 5.0 nel_c = 240 nel_l = 21 xpoisson = 0.3 xyoung = 30.0e6 xload = 1.0 press = 1.0 nmodes = 19 radialImperfFactor = 0.5 # #

2. SPECIFY THE EIGENMODES WHICH WILL BE USED TO SEED THE IMPERFECTION (TAKE ALL MODES, 1 THROUGH NMODES)

eigmodes = [ ] for i in range(nmodes): eigmodes.append(i+1) #

3. MISC. DEFINITIONS

zero = 0 ######################################################################## # # PERFORM EIGENVALUE BUCKLING ANALYSIS # # CREATE THE STUDY buckle = ParStudy(par=('thickness','length','radius', 'node_circum','node_length','poisson','young', 'tot_load','int_press','num_modes')) #

DEFINE THE NAMES OF THE INPUT DECKS

names = ['cylsh_buck'] #

DEFINE THE PARAMETERS

buckle.define(DISCRETE,par='thickness') buckle.define(DISCRETE,par='length') buckle.define(DISCRETE,par='radius')

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buckle.define(DISCRETE,par='node_circum') buckle.define(DISCRETE,par='node_length') buckle.define(DISCRETE,par='poisson') buckle.define(DISCRETE,par='young') buckle.define(DISCRETE,par='tot_load') buckle.define(DISCRETE,par='num_modes') buckle.define(DISCRETE,par='int_press') #

SAMPLE THE PARAMETERS - INPUT THE APPROPRIATE VALUES

buckle.sample(VALUES,par='thickness',values=(xthickness)) buckle.sample(VALUES,par='length',values=(xlength)) buckle.sample(VALUES,par='radius',values=(xradius)) buckle.sample(VALUES,par='node_circum',values=(nel_c)) buckle.sample(VALUES,par='node_length',values=(nel_l)) buckle.sample(VALUES,par='poisson',values=(xpoisson)) buckle.sample(VALUES,par='young',values=(xyoung)) buckle.sample(VALUES,par='tot_load',values=(xload)) buckle.sample(VALUES,par='num_modes',values=(nmodes)) buckle.sample(VALUES,par='int_press',values=(press)) #

COMBINE THE SAMPLES INTO ANALYSES

buckle.combine(MESH,name='short') for temp in names: # #

GENERATE INPUT DECKS AND EXECUTION SCRIPT FOR VARIOUS TEMPLATES

buckle.generate(template=temp) #

EXECUTE RUNS SEQUENTIALLY

buckle.execute() #

GATHER RESULTS FOR FIRST BUCKLING MODE AND WRITE TO OUTPUT FILES

buckle.gather(results='mode',variable='MODAL',step=2) buckle.report(PRINT,par=('length','radius','thickness', 'poisson','young'),results=('mode.2'))

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# # # # # # #

3. STORE LOWEST EIGENVALUE IN EIG_VALS; COMPUTE RAD_IMP. STORE THE RESULTS (.FIL) FILE NAMES OF EACH ANALYSIS (IN BKLFNAME THE PARAMETER SETTINGS FOR EACH ANALYSIS (IN PARAMS), AND A DESCRIPTIVE STRING OF CHARACTERS FOR EACH ANALYSIS (IN XNAMES). STORE THE NUMBER OF NODES ALONG THE CIRCUMFERENCE (IN N CIRC) AND LENGTH (N_LGTH)

eig_vals=[ ] rad_imp =[ ] n_circ = [ ] n_lgth = [ ] bklfnames=[ ] xnames=[ ] params = [] i = -1 res = buckle.table.results for jname in buckle.job.keys(): i = i+1 des = buckle.job[jname].design thickness = des[0] r_imp = radialImperfFactor*thickness length = des[1] radius = des[2] ncrc = des[3] nlen = des[4] value = res[i][0] xname = buckle.job[jname].designName root = buckle.job[jname].root bklfname = root + '_' + xname bklfnames.append(bklfname) xnames.append(xname) param = des params.append(param) eig_vals.append(value) rad_imp.append(r_imp) n_circ.append(ncrc) n_lgth.append(nlen) # #

INTERMEDIATE STEP: RUN FORTRAN PROGRAM TO DETERMINE TRUE SCALE FOR IMPERFECTIONS

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# # # # # # # # # # # # # #

NOW CREATE THE INPUT FILE REQUIRED TO RUN THE FORTRAN PROGRAM; INCLUDES THE FOLLOWING: 1. NAME ASSIGNED TO THE OUTPUT FILE CREATED BY THE PROGRAM THIS FILE CONTAINS THE SCALE FACTOR FOR EACH EIGENMODE USED TO SEED THE IMPERFECTION AND WILL BE INCLUDED IN THE POST BUCKLING ANALYSIS FILE.) 2. RESULTS FILE NAME FOR EIGENVALUE BUCKLING ANALYSIS 3. NUMBER OF CIRCUMF. AND LONG. NODES 4. PRESCRIBED RADIAL IMPERFECTION (FROM RAD_IMP) 5. LIST OF EIGENMODES (FROM EIGMODES) FOLLOWED BY ZERO 6. LIST OF "GUESS" SCALE FACTORS (FROM MODEFCTR)

file = 'max_round_input.dat' impfnames = [ ] names = [ ] i = -1 for bklfname in bklfnames: i = i+1 impfname = bklfname + str('_imp') impfnames.append(impfname) names.append(bklfname) modefctr = [ ] for m in eigmodes: modefctr.append(rad_imp[i]) f1 = open(file,'w') f1.write(impfnames[i] + '\n') f1.write(names[i] + '\n') f1.write(str(n_circ[i]) + '\n') f1.write(str(n_lgth[i]) + '\n') f1.write(str(rad_imp[i]) + '\n') for imode in eigmodes: f1.write(str(imode) + '\n') f1.write(str(zero) + '\n') for ifctr in modefctr: f1.write(str(ifctr) + '\n') f1.close() #

RUN THE FORTRAN PROGRAM os.system('./cylsh_maximp.x')

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# # #

PERFORM THE POST-BUCKLING ANALYSIS CREATE THE STUDY

riks = ParStudy(par=('thickness','length','radius', 'node_circum','node_length','poisson','young', 'int press','eig1 load','buckle file','imperf file' #

DETERMINE THE APPROPRIATE LOAD LEVEL FOR THE ANALYSIS

xload1 = [ ] for l in eig_vals: xl = float(xload)*float(l) xload1.append(xl) #

DEFINE THE NAMES OF THE INPUT DECKS

names2 = ['cylsh_postbuck'] #

DEFINE THE PARAMETERS

riks.define(DISCRETE,par='thickness') riks.define(DISCRETE,par='length') riks.define(DISCRETE,par='radius') riks.define(DISCRETE,par='node_circum') riks.define(DISCRETE,par='node_length') riks.define(DISCRETE,par='poisson') riks.define(DISCRETE,par='young') riks.define(DISCRETE,par='eig1_load') riks.define(DISCRETE,par='buckle_file') riks.define(DISCRETE,par='imperf_file') riks.define(DISCRETE,par='int_press') #

SAMPLE THE PARAMETERS - INPUT THE APPROPRIATE VALUES

for i in range(len(xload1)): riks.sample(VALUES,par='thickness',values=(params[i][0])) riks.sample(VALUES,par='length',values=(params[i][1])) riks.sample(VALUES,par='radius',values=(params[i][2])) riks.sample(VALUES,par='node_circum',values=(params[i][3])) riks.sample(VALUES,par='node_length',values=(params[i][4])) riks.sample(VALUES,par='poisson',values=(params[i][5]))

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riks.sample(VALUES,par='young',values=(params[i][6])) riks.sample(VALUES,par='int_press',values=(params[i][8])) riks.sample(VALUES,par='eig1_load',values=(xload1[i])) riks.sample(VALUES,par='buckle_file',values=(bklfnames[i])) riks.sample(VALUES,par='imperf_file',values=(impfnames[i])) # # # # #

COMBINE THE SAMPLES INTO ANALYSES (APPEND XNAME TO THE JOBNAMES OTHERWISE THEY WILL NOT BE UNIQUE; ALSO PROVIDES A MEANS OF ASSOCIATING THE POSTBUCKLING ANALYSIS WITH ITS CORRESPONDING EIGENVALUE BUCKLING ANALYSIS.)

riks.combine(MESH,name=xnames[i]) for temp in names2: # #

GENERATE INPUT DECKS AND EXECUTION SCRIPT FOR VARIOUS TEMPLATES

riks.generate(template=temp) #

EXECUTE RUNS SEQUENTIALLY

riks.execute() riks.gather(results='lpf',variable='LPF',step=2,inc=LAST) #

GATHER RESULTS FOR HKS QA PURPOSES ONLY

riks.report(PRINT,par=('length','radius', 'thickness','poisson','young'), results=('lpf')) buck_res = buckle.table.results riks_res = riks.table.results resfile = open('cylsh_script.psr','w') for x in buck_res: resfile.write('%14.6g \n' % x[0]) for y in riks_res: resfile.write('%14.3g \n' % y[0])

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Listing 1.2.6-2 *heading Input file for the linear buckling analysis. *parameter # #geometric/load parameters (can be modified) # radius = 5.0 length = 2.0 thickness = 0.01 tot_load = 1.0 # #elastic material properties (can be modified) # young = 30e+06 poisson=0.3 # #number of buckling modes to be extractend (can be modified) # num_modes=20 # #internal pressure (can be modified) # int_press = 0.0 # #mesh parameters (can be modified) # node_circum = 240 node_length = 21 ## ##dependent parameters (do not modify) ## chn = node_circum*node_length-node_circum node_ang = -360.0/float(node_circum) node_tot = node_circum*node_length node_tmp = node_tot-node_circum+1 node_int = node_length-1 node_circum1 = node_circum+1 node_circum2 = node_circum+2 node_circum0 = node_circum-1 e1 = node_circum*2 p = -tot_load/float(node_circum)

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pn = tot_load/float(node_circum) # #end of parameter list # *node,system=c 1,,0.0,0.0 ,,,0.0 ,,0.0, ,,, *ngen,line=c,nset=bottom 1,,1,,0.0,0.0,0.0,0.0,0.0,1.0 *ncopy,new set=top,old set=bottom,shift,change number= 0.0,0.0, 0.0,0.0,0.0,0.0,0.0,1.0,0.0 *nfill bottom,top,, *element,type=s4 1,1,2,, ,,1,, *elgen,elset=cylinder 1,,1,1,,, ,1,,,,, *shell section, elset=cylinder, material=mat_1 , *material,name=mat_1 *elastic , *nset,nset=ends bottom,top *transform,type=c,nset=ends 0.0,0.0,0.0,0.0,0.0,1.0 *boundary ends,1,2 ends,4,4 ends,6,6 1,3 ** *step,nlgeom,inc=10 static preload for internal pressure *static 1.0,1.0 *monitor,node=, dof=3 *dload

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cylinder,p, *end step *step,nlgeom *buckle,eigensolver=lanczos , *cload top,3, bottom,3, *output,field *node output u, *node file,global=yes u, *end step

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Listing 1.2.6-3 *heading Input file for the postbuckling analysis. *parameter # # filenames # buckle_file = 'file1.dat' imperf_file = 'file2.dat' # # workaround to allow parametrization of a # filename read with *INCLUDE # line1 = '*include, input='+imperf_file # # geometric/load parameters # radius = 5.0 length = 2.0 thickness = 0.01 # # this is the pcritical for the 1st value from # the linear eigenvalue analysis # eig1_load = 1.18305e+4 # # elastic material properties # young = 30e+06 poisson=0.3 # # internal pressure # int_press = 0.0 # # mesh parameters # node_circum = 240 node_length = 21 ## ## dependent parameters (do not modify) ##

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chn = node_circum*node_length-node_circum node_ang = -360.0/float(node_circum) node_tot = node_circum*node_length node_tmp = node_tot-node_circum+1 node_int = node_length-1 node_circum1 = node_circum+1 node_circum2 = node_circum+2 node_circum0 = node_circum-1 e1 = node_circum*2 p = -eig1_load/float(node_circum) pn = eig1_load/float(node_circum) # # end of parameter list # *node,system=c 1,,0.0,0.0 ,,,0.0 ,,0.0, ,,, *ngen,line=c,nset=bottom 1,,1,,0.0,0.0,0.0,0.0,0.0,1.0 *ncopy,new set=top,old set=bottom,shift,change number= 0.0,0.0, 0.0,0.0,0.0,0.0,0.0,1.0,0.0 *nfill bottom,top,, **specify the imperfection as a function of modeshape amplitude *imperfection,file=,step=2 *element,type=s4 1,1,2,, ,,1,, *elgen,elset=cylinder 1,,1,1,,, ,1,,,,, *shell section, elset=cylinder, material=mat_1 , *material,name=mat_1 *elastic , *nset,nset=ends bottom,top *transform,type=c,nset=ends

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0.0,0.0,0.0,0.0,0.0,1.0 *boundary ends,1,2 ends,4,4 ends,6,6 1,3 ** *step,nlgeom,inc=10 static preload for internal pressure *static 1.0,1.0 *monitor,node=, dof=3 *dload cylinder,p, *end step ** *step,nlgeom,inc=60 postbuckling (riks) analysis *static,riks 0.05,1.0,,0.05,,,3,-0.1 *monitor,node=,dof=3 *controls,parameter=field,field=global 1.e-5, *cload top,3, bottom,3, *node file,freq=1,nset=top u,cf *output,field,variable=preselect,freq=10 *output,history,freq=1 *node output,nset=top u,cf *end step

1.3 Forming analyses 1.3.1 Upsetting of a cylindrical billet in ABAQUS/Standard: quasi-static analysis with rezoning Product: ABAQUS/Standard This example illustrates the use of the rezoning capabilities of ABAQUS/Standard in a metal forming application. The same problem is analyzed using the coupled temperature-displacement elements in

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``Upsetting of a cylindrical billet: coupled temperature-displacement and adiabatic analysis, '' Section 1.3.17. Coupled temperature-displacement elements are included in this example only for rezoning verification purposes; no heat generation occurs in these elements for this example. The same test case is done with ABAQUS/Explicit in ``Upsetting of a cylindrical billet in ABAQUS/Explicit,'' Section 1.3.2. When the strains become large in geometrically nonlinear analysis, the elements often become so severely distorted that they no longer provide a good discretization of the problem. When this occurs, it is necessary to "rezone": to map the solution onto a new mesh that is better designed to continue the analysis. The procedure is to monitor the distortion of the mesh--for example, by observing deformed configuration plots--and decide when the mesh needs to be rezoned. At that point a new mesh must be generated using the mesh generation options in ABAQUS or some external mesh generator. The results file output is useful in this context since the current geometry of the model can be extracted from the data in the results file. Once a new mesh is defined, the analysis is continued by beginning a new problem using the solution from the old mesh at the point of rezoning as initial conditions. This is done by including the *MAP SOLUTION option and specifying the step number and increment number at which the solution should be read from the previous analysis. ABAQUS interpolates the solution from the old mesh onto the new mesh to begin the new problem. This technique provides considerable generality. For example, the new mesh might be more dense in regions of high-strain gradients and have fewer elements in regions that are distorting rigidly--there is no restriction that the number of elements be the same or that element types agree between the old and new meshes. In a typical practical analysis of a manufacturing process, rezoning may have to be done several times because of the large shape changes associated with such a process. The interpolation technique used in rezoning is a two-step process. First, values of all solution variables are obtained at the nodes of the old mesh. This is done by extrapolation of the values from the integration points to the nodes of each element and averaging those values over all elements abutting each node. The second step is to locate each integration point in the new mesh with respect to the old mesh (this assumes all integration points in the new mesh lie within the bounds of the old mesh: warning messages are issued if this is not so, and new model solution variables at the integration point are set to zero). The variables are then interpolated from the nodes of the element in the old mesh to the location in the new mesh. All solution variables are interpolated automatically in this way so that the solution can proceed on the new mesh. Whenever a model is rezoned, it can be expected that there will be some discontinuity in the solution because of the change in the mesh. If the discontinuity is significant, it is an indication that the meshes are too coarse or that the rezoning should have been done at an earlier stage before too much distortion occurred.

Geometry and model The geometry is the standard test case of Lippmann (1979) and is defined in ``Upsetting of a cylindrical billet: coupled temperature-displacement and adiabatic analysis, '' Section 1.3.17. It is a circular billet, 30 mm long, with a radius of 10 mm, compressed between two flat, rigid dies that are defined to be perfectly rough.

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The mesh used to begin the analysis is shown in Figure 1.3.1-1. The finite element model is axisymmetric and includes the top half of the billet only since the middle surface of the billet is a plane of symmetry. Element type CAX4R is used: this is a 4-node quadrilateral with a single integration point and "hourglass control" to control spurious mechanisms caused by the fully reduced integration. The element is chosen here because it is relatively inexpensive for problems involving nonlinear constitutive behavior since the material calculations are only done at one point in each element. The contact between the top and lateral exterior surfaces of the billet and the rigid die is modeled with the *CONTACT PAIR option. The billet surface is defined by means of the *SURFACE option. The rigid die is modeled as an analytical rigid surface with the *SURFACE option in conjunction with the *RIGID BODY option. The mechanical interaction between the contact surfaces is assumed to be nonintermittent, rough frictional contact. Therefore, two suboptions are used with the *SURFACE INTERACTION property option: the *FRICTION, ROUGH suboption to enforce a no slip constraint between the two surfaces, and the *SURFACE BEHAVIOR, NO SEPARATION suboption to ensure that separation does not occur once contact has been established. No mesh convergence studies have been done, but the agreement with the results given in Lippmann (1979) suggests that the meshes used here are good enough to provide reasonable predictions of the overall force on the dies.

Material The material behavior is similar to that used in ``Upsetting of a cylindrical billet: coupled temperature-displacement and adiabatic analysis,'' Section 1.3.17, except that rate dependence of the yield stress is not included. Thermal properties are not needed in this case since the analysis is mechanical only (we assume the loading is applied so slowly that the response is isothermal).

Boundary conditions and loading Kinematic boundary conditions are symmetry on the axis (nodes at r =0, in node set AXIS, have ur =0 prescribed), symmetry about z =0 (all nodes at z =0, in node set MIDDLE, have uz =0 prescribed). The node on the top surface of the billet that lies on the symmetry axis is not part of the node set AXIS to avoid overconstraint: the radial motion of this node is already constrained by a no slip frictional constraint (see ``Common difficulties associated with contact modeling,'' Section 21.10.1 of the ABAQUS/Standard User's Manual). The uz -displacement of the rigid body reference node for the die is prescribed as having a constant velocity in the axial direction so that the total displacement of the die is -9 mm over the history of the upsetting. The first analysis is done in two steps so that the first step can be stopped at a die displacement corresponding to 44% upsetting. The second step carries the first analysis to 60% upsetting. The second analysis restarts from the first step of the first analysis with a new mesh. A FORTRAN routine is used to extract the coordinates of the nodes along the outer boundary of the original mesh at 44% upsetting. These coordinates are then used to define the outer boundary of the new mesh.

Results and discussion The results from the first mesh are illustrated in Figure 1.3.1-2. This figure shows the configuration

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and plastic strain magnitude that are predicted at 44% upsetting (73.3% of the total die displacement). The folding of the top outside surface of the billet onto the die is clearly visible, as well as the severe straining of the middle of the specimen. At this point the mesh is rezoned. The new mesh for the rezoned model is shown in Figure 1.3.1-3. It is based on placing nodes on straight lines between the outer surface of the billet and the axis of the billet. The final configuration and plastic strain magnitudes predicted with rezoning are shown in Figure 1.3.1-4. Figure 1.3.1-5 shows the predictions of the total upsetting force versus displacement of the die. The results shown on the plot include the results for the analysis that includes the rezoning and the data obtained when the original mesh is used for the entire analysis. The results show that the rezoning of the mesh does not have a significant effect, in this case, on the overall die force. The results compare well with the rate independent results obtained by Taylor (1981).

Input files rezonebillet_cax4r.inp Original CAX4R mesh. rezonebillet_cax4r_rezone.inp Rezoned CAX4R mesh; requires the external file generated by rezonebillet_fortran_cax4r.f. rezonebillet_fortran_cax4r.f FORTRAN routine used to access the results file of rezonebillet_cax4r.inp and generate a file containing the nodal coordinates of the outer boundary at 44% upsetting. rezonebillet_cax4i.inp Original CAX4I mesh. rezonebillet_cax4i_rezone.inp Rezoned CAX4I mesh. The FORTRAN routine given in file rezonebillet_fortran_cax4r.f is not used as part of this analysis sequence. rezonebillet_cgax4r.inp Original CGAX4R mesh. rezonebillet_cgax4r_rezone.inp Rezoned CGAX4R mesh; requires the external file generated by rezonebillet_fortran_cax4r.f. rezonebillet_cgax4t.inp Original CGAX4T mesh. rezonebillet_cgax4t_rezone.inp Rezoned CGAX4T mesh; requires the external file generated by rezonebillet_fortran_cgax4t.f. rezonebillet_fortran_cgax4t.f FORTRAN routine used to access the results file from file rezonebillet_cgax4t.inp and generate a

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file containing the nodal coordinates of the outer boundary at 44% upsetting. rezonebillet_deftorigid.inp Rigid die simulated by declaring deformable elements (SAX1) as rigid. The billet is meshed with CAX4R elements. rezonebillet_deftorigid_rezone.inp Rezoned CAX4R mesh. The rigid die is simulated by declaring deformable elements (SAX1) as rigid.

References · Lippmann, H., Metal Forming Plasticity, Springer-Verlag, Berlin, 1979. · Taylor, L. M., "A Finite Element Analysis for Large Deformation Metal Forming Problems Involving Contact and Friction," Ph.D. Thesis, U. of Texas at Austin, 1981.

Figures Figure 1.3.1-1 Axisymmetric upsetting example: initial mesh.

Figure 1.3.1-2 Deformed configuration and plastic strain at 44% upset.

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Figure 1.3.1-3 New mesh at 44% upset.

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Figure 1.3.1-4 Deformed configuration and plastic strain of original mesh at 60% upset.

Figure 1.3.1-5 Force-deflection response for cylinder upsetting. (Results from the rezoned mesh start at 73.6% of applied displacement.)

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Sample listings

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Listing 1.3.1-1 *HEADING AXISYMMETRIC UPSETTING PROBLEM REZONING *RESTART,WRITE,FREQUENCY=30 *NODE,NSET=RSNODE 9999,0.,.015 *NODE 1, 13,.01 1201,0.,.015 1213,.01,.015 *NGEN,NSET=MIDDLE 1,13 *NGEN,NSET=TOP 1201,1213 *NFILL MIDDLE,TOP,12,100 *NSET,NSET=AXIS,GENERATE 1,1201,100 *NSET,NSET=OUTER,GENERATE 13,1013,100 *NSET,NSET=NAXIS,GENERATE 1,1101,100 *ELEMENT,TYPE=CAX4R,ELSET=METAL 1,1,2,102,101 *ELGEN,ELSET=METAL 1,12,1,1,12,100,100 *ELSET,ELSET=ECON1,GENERATE 1101,1112,1 *ELSET,ELSET=ECON2,GENERATE 12,1112,100 *RIGID BODY,ANALYTICAL SURFACE=BSURF,REF NODE=9999 *SURFACE,TYPE=SEGMENTS,NAME=BSURF START,.020,.015 LINE,-.001,.015 *SURFACE,NAME=ASURF ECON1,S3 ECON2,S2 *CONTACT PAIR,INTERACTION=ROUGH ASURF,BSURF *SURFACE INTERACTION,NAME=ROUGH

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*FRICTION,ROUGH *SURFACE BEHAVIOR,NO SEPARATION *SOLID SECTION,ELSET=METAL,MATERIAL=EL *MATERIAL,NAME=EL *ELASTIC 200.E9,.3 *PLASTIC 7.E8,0.00 3.7E9,10.0 *BOUNDARY MIDDLE,2 NAXIS,1 *STEP,INC=200,AMPLITUDE=RAMP,NLGEOM 73.3 PERCENT OF DIE DISPLACEMENT *STATIC 0.015,1. *BOUNDARY 9999,1 9999,6 9999,2,,-.0066 *MONITOR,NODE=9999,DOF=2 *CONTACT PRINT,SLAVE=ASURF,FREQUENCY=40 *CONTACT FILE,SLAVE=ASURF,FREQUENCY=40 *EL PRINT, ELSET=METAL,FREQUENCY=40 S,MISES E, PEEQ, *NODE PRINT, FREQUENCY=10 *NODE FILE,NSET=RSNODE U,RF *NODE FILE,FREQUENCY=999 COORD, *OUTPUT, FIELD, OP=NEW, FREQUENCY=9999 *NODE OUTPUT U, *END STEP *STEP,INC=200,AMPLITUDE=RAMP,NLGEOM 100 PERCENT OF DIE DISPLACEMENT *STATIC 0.015,1. *BOUNDARY 9999,1 9999,6

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9999,2,,-.009 *MONITOR,NODE=9999,DOF=2 *EL PRINT, ELSET=METAL,FREQUENCY=40 S,MISES E, PEEQ, *NODE PRINT, FREQUENCY=10 *NODE FILE,NSET=RSNODE U,RF *END STEP

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Listing 1.3.1-2 *HEADING AXISYMMETRIC UPSETTING PROBLEM REZONED *RESTART,WRITE,FREQUENCY=30 *NODE,NSET=RSNODE 9999,0.,.0084 *NODE 1, 1001,0.,.0084 *NODE,NSET=OUTER,INPUT=BOUNDARY.OUT *NGEN,NSET=AXIS 1,1001,100 *NSET,NSET=NAXIS,GENERATE 1,901,100 *NSET,NSET=MIDDLE,GENERATE 1,13 *NSET,NSET=TOP,GENERATE 1001,1013 *NFILL AXIS,OUTER,12,1 *ELEMENT,TYPE=CAX4R,ELSET=METAL 1,1,2,102,101 *ELGEN,ELSET=METAL 1,12,1,1,10,100,100 *ELSET,ELSET=ECON1,GENERATE 901,912,1 *ELSET,ELSET=ECON2,GENERATE 12,912,100 *RIGID BODY,ANALYTICAL SURFACE=BSURF,REF NODE=9999 *SURFACE,TYPE=SEGMENTS,NAME=BSURF START,.020,.0084 LINE,-.001,.0084 *SURFACE,NAME=ASURF ECON1,S3 ECON2,S2 *CONTACT PAIR,INTERACTION=ROUGH ASURF,BSURF *SURFACE INTERACTION,NAME=ROUGH *SURFACE BEHAVIOR,NO SEPARATION *FRICTION,ROUGH *SOLID SECTION,ELSET=METAL,MATERIAL=EL

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*MATERIAL,NAME=EL *ELASTIC 200.E9,.3 *PLASTIC 7.E8,0.00 3.7E9,10.0 *BOUNDARY MIDDLE,2 NAXIS,1 *MAP SOLUTION,STEP=1,INC=16 *STEP,INC=200,AMPLITUDE=RAMP,NLGEOM *STATIC .03,1. *BOUNDARY 9999,1 9999,6 9999,2,,-.0024 *MONITOR,NODE=9999,DOF=2 *CONTACT PRINT,SLAVE=ASURF *CONTACT FILE,SLAVE=ASURF,FREQUENCY=40 *EL PRINT, ELSET=METAL,FREQUENCY=40 S,MISES E, PEEQ, *NODE PRINT, FREQUENCY=10 *NODE FILE,NSET=RSNODE U,RF *END STEP

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Listing 1.3.1-3 SUBROUTINE HKSMAIN C C C

PROGRAM

READSETS

INCLUDE 'aba_param.inc' C PARAMETER (MAXNODES = 500) DIMENSION ARRAY(513), JRRAY(NPRECD,513), NMEMS(MAXNODES) EQUIVALENCE (ARRAY(1), JRRAY(1,1)) C INTEGER LRUNIT(2,1) LOGICAL READNODES CHARACTER FNAME*80, ASET*8 C C C C C C C C C C C C C C

THIS PROGRAM WILL EXTRACT THE CURRENT COORDINATES OF THE NODES OF THE "OUTER" AT THE STEP/INC DEFINED BY THE PARAMETERS K STEP AND K INC. TH NUMBERS AND COORDINATES ARE WRITTEN TO THE OUTPUT FILE BOUNDARY.OUT IN SUITABLE FOR INPUT INTO AN ABAQUS INPUT FILE. FOR EXAMPLE rezonebillet_cax4r: PARAMETER (K_STEP = 1, K_INC = 16, FNAME = 'rezonebillet_cax4r') FOR EXAMPLE rezonebillet_cgax4t: PARAMETER (K_STEP = 1, K_INC = 17, FNAME = 'rezonebillet_cgax4t') THIS MAY BE USED TO EXTRACT COORDINATES FROM EXA rezonebillet cax4i U PARAMETER (K_STEP = 1, K_INC = 40, FNAME = 'rezonebillet_cax4i') LRUNIT(1,1)=8 LRUNIT(2,1)=2 LOUTF=0 NRU = 1

C CALL

INITPF (FNAME, NRU, LRUNIT, LOUTF)

C JOUT = 6 KEYPRV= 0 KSTEP = 0 KINC = 0 READNODES = .FALSE. NUMMEM= 0

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C JUNIT = LRUNIT(1,1) CALL DBRNU (JUNIT) C OPEN (UNIT=6,STATUS='UNKNOWN',FILE='BOUNDARY.OUT') C C C

READ RECORDS FROM RESULTS FILE, UP TO 100000 RECORDS: DO 50 IXX = 1, 99999 CALL DBFILE(0,ARRAY,JRCD) IF (JRCD .NE. 0 .AND. KEYPRV .EQ. 2001) THEN WRITE(0,*) 'END OF FILE' CLOSE (JUNIT) GOTO 100 ELSE IF (JRCD .NE. 0) THEN WRITE(0,*) 'ERROR READING FILE' CLOSE (JUNIT) GOTO 100 ENDIF

C KEY=JRRAY(1,2) C C C

RECORD 2000: INCREMENT START RECORD IF(KEY.EQ.2000) THEN KSTEP = JRRAY(1,8) KINC = JRRAY(1,9) END IF

C C C

RECORD 1931: NODE SET DEFINITION RECORD IF(KEY.EQ.1931) THEN

C 110

WRITE(ASET,110) array(3) FORMAT(a8) IF (ASET(1:5).EQ.'OUTER') THEN NUMMEM = JRRAY(1,1) - 3 IF (NUMMEM.GT.MAXNODES) THEN WRITE(0,*)'ERROR: TOO MANY NODES ON RECORD' WRITE(0,*)' INCREASE MAXNODES TO ',NUMMEM CLOSE (JUNIT) GOTO 100 END IF

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DO KMEM = 1,NUMMEM NMEMS(KMEM)=JRRAY(1,3+KMEM) END DO READNODES = .TRUE. END IF END IF C C C

RECORD 107: NODAL COORDINATES RECORD IF (KEY.EQ.107 .AND. READNODES) THEN IF (KSTEP.EQ.K_STEP .AND. KINC.EQ.K_INC) THEN KNODE = JRRAY(1,3) DO KMEM = 1,NUMMEM IF (KNODE.EQ.NMEMS(KMEM)) THEN WRITE(JOUT,70)KNODE,ARRAY(4),ARRAY(5) FORMAT(I5,2(',',D18.8)) END IF END DO END IF END IF

70

C KEYPRV = KEY C 50

CONTINUE

C 100 CONTINUE C STOP END

1.3.2 Upsetting of a cylindrical billet in ABAQUS/Explicit Product: ABAQUS/Explicit The example illustrates the forming of a small, circular billet of metal that is reduced in length by 60%. This is the standard test case that is defined in Lippmann et al. (1979), so some verification of the result is available by comparing the results with the numerical results presented in that reference. The same test case is done with ABAQUS/Standard in ``Upsetting of a cylindrical billet in ABAQUS/Standard: quasi-static analysis with rezoning,'' Section 1.3.1. The same problem is analyzed using the coupled temperature-displacement elements in ``Upsetting of a cylindrical billet: coupled temperature-displacement and adiabatic analysis,'' Section 1.3.17.

Problem description The specimen is a circular billet, 30 mm long with a radius of 10 mm, compressed between flat, rough,

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rigid dies. The finite element model is axisymmetric and includes the top half of the billet only, since the middle surface of the billet is a plane of symmetry. Axisymmetric continuum elements of types CAX4R and CAX6M are used to model the billet. The rigid die is modeled in several different ways, as described below. 1. The die is modeled as an analytical rigid surface using the *SURFACE, TYPE=SEGMENTS and the *RIGID BODY options. The rigid surface is associated with a rigid body by its specified reference node. 2. Axisymmetric rigid elements of type RAX2 are used to model the rigid die. Three input files that use progressively finer meshes to model the billet exist for this case. 3. The die is modeled with RAX2 elements, as in Case 2. However, the die is assigned a mass by specifying point masses at the nodes of the RAX2 elements. The reference node of the rigid die is repositioned at its center of mass by specifying POSITION=CENTER OF MASS on the *RIGID BODY option. 4. The die is modeled with RAX2 elements, as in Case 2. The rigid elements are assigned thickness and density values such that the mass of the die is the same as in Case 3. 5. The die is modeled with RAX2 elements, as in Case 2. The NODAL THICKNESS parameter is used on the *RIGID BODY option to specify the thickness of the die at its nodes. The same thickness value is prescribed as in Case 4. 6. Axisymmetric shell elements of type SAX1 are used to model the die, and they are included in the rigid body by referring to them on the *RIGID BODY option. The thickness and the material density of the SAX1 elements is the same as that of the rigid elements in Case 4. 7. The die is modeled with axisymmetric shell elements of type SAX1 and with axisymmetric rigid elements of type RAX2. The deformable elements are included in the rigid body by referring to them on the *RIGID BODY option. Both element types have the same thickness and density as in Case 4. A coefficient of friction of 1.0 is used between the rigid surface and the billet. This value is large enough to ensure a no-slip condition so that, when the billet comes in contact with the rigid surface, there is virtually no sliding between the two. The material model assumed for the billet is that given in Lippmann et al. (1979). Young's modulus is 200 GPa, Poisson's ratio is 0.3, and the density is 7833 kg/m 3. A rate-independent von Mises elastic-plastic material model is used, with a yield stress of 700 MPa and a hardening slope of 0.3 GPa. Kinematic boundary conditions are symmetry on the axis (all nodes at r = 0 have ur = 0 prescribed) and symmetry about z = 0 (all nodes at z = 0 have uz = 0 prescribed). The rigid body reference node for the rigid body is constrained to have no rotation or ur -displacement. In Case 1 and Case 2 the uz -displacement is prescribed using a velocity boundary condition whose value is ramped up to a velocity of 20 m/s and then held constant until the die has moved a total of 9 mm. In the remaining cases a concentrated force of magnitude 410 kN is applied in the z-direction at the reference node. The

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magnitude of the concentrated force is such as to ensure that the resulting displacement of the die at the end of the time period is the same as in Case 1 and Case 2. The total time of the analysis is 0.55 millisec and is slow enough to be considered quasi-static. For Case 1 several different analyses are performed to compare section control options and mesh refinement for the billet modeled with CAX4R elements. Table 1.3.2-1 lists the analysis options used. A coarse mesh (analysis COARSE_SS) and a fine (analysis FINE_SS) mesh are analyzed with the pure stiffness form of hourglass control (HOURGLASS=STIFFNESS). A coarse mesh (analysis COARSE_CS) is analyzed with the combined hourglass control. The default section controls, using the integral viscoelastic form of hourglass control (HOURGLASS=RELAX STIFFNESS), are tested on a coarse mesh (analysis COARSE). Since this is a quasi-static analysis, the viscous hourglass control option (HOURGLASS=VISCOUS) should not be used. All other cases use the default section controls. In addition, Case 1 of the above problem has been analyzed using CAX6M elements with both coarse and fine meshes.

Results and discussion Figure 1.3.2-1 through Figure 1.3.2-4 show results for Case 1, where the billet is modeled with CAX4R elements. The rigid die is modeled using an analytical rigid surface, and the pure stiffness hourglass control is used. Figure 1.3.2-1 shows the original and deformed shape at the end of the analysis for the coarse mesh. Figure 1.3.2-2 shows the original and deformed shape at the end of the analysis for the fine mesh. Figure 1.3.2-3 and Figure 1.3.2-4 show contours of equivalent plastic strain for both meshes. Corresponding coarse and fine mesh results for the billet modeled with CAX6M elements are shown in Figure 1.3.2-5 to Figure 1.3.2-8. The number of nodes used for the coarse and fine CAX6M meshes are the same as those used for the coarse and fine CAX4R meshes. However, the analysis using CAX4R elements performs approximately 55% faster than the analysis using CAX6M elements. The equivalent plastic strain distributions obtained from each analysis compare closely for both the coarse and fine meshes; the CAX6M elements predict a slightly higher peak value. In both analyses the folding of the top outside surface of the billet onto the die is clearly visible, as well as the severe straining of the middle of the specimen. Figure 1.3.2-9 is a plot of vertical displacement versus reaction force at the rigid surface reference node for Case 1 with the section control options identified in Table 1.3.2-1. Results for an analysis run with ABAQUS/Standard (labeled COARSE_STD) are included for comparison. The curves obtained using CAX4R and CAX6M elements are very close and agree well with independent results obtained by Taylor (1981). The results from analysis COARSE_SS are virtually the same as the results from analysis COARSE, but at a much reduced cost; therefore, such analysis options are recommended for this problem. The results for all the other cases (which use the default section controls but different rigid surface models) are the same as the results for Case 1 using the default section controls.

Input files upset_anl_ss.inp

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Coarse mesh (CAX4R elements) of Case 1 using STIFFNESS hourglass control. upset_fine_anl_ss.inp Fine mesh of Case 1 (CAX4R elements) usingSTIFFNESS hourglass control. upset_anl_cs.inp Coarse mesh of Case 1 (CAX4R elements) using COMBINED hourglass control. upset_anl.inp Coarse mesh of Case 1 (CAX4R elements) using the default section control options. upset_fine_anl.inp Fine mesh of Case 1 (CAX4R elements) using the default section control options. upset_case2.inp Coarse mesh of Case 2 (CAX4R elements). upset_fine_case2.inp Fine mesh of Case 2 (CAX4R elements). upset_vfine_case2.inp An even finer mesh of Case 2 (CAX4R elements) included to test the performance of the code. upset_case3.inp Case 3 using CAX4R elements. upset_case4.inp Case 4 using CAX4R elements. upset_case5.inp Case 5 using CAX4R elements. upset_case6.inp Case 6 using CAX4R elements. upset_case7.inp Case 7 using CAX4R elements. upset_anl_cax6m.inp Case 1 using the coarse mesh of CAX6M elements. upset_fine_anl_cax6m.inp Case 1 using the fine mesh of CAX6M elements.

References

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· H. Lippmann, editor, Metal Forming Plasticity, Springer-Verlag, Berlin, 1979. · Taylor, L. M., "A Finite Element Analysis for Large Deformation Metal Forming Problems Involving Contact and Friction," Ph. D. Dissertation, U. of Texas at Austin, 1981.

Table Table 1.3.2-1 Analysis options for Case 1 using CAX4R elements. Analysis Mesh Hourglass Label Type Control COARSE_SS coarse STIFFNESS FINE_SS fine STIFFNESS COARSE_CS coarse COMBINED COARSE coarse RELAX

Figures Figure 1.3.2-1 Undeformed and deformed shape for the coarse mesh (CAX4R) of Case 1 (using the STIFFNESS hourglass control).

Figure 1.3.2-2 Undeformed and deformed shape for the fine mesh (CAX4R) of Case 1 (using the STIFFNESS hourglass control).

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Figure 1.3.2-3 Contours of equivalent plastic strain for the coarse mesh ( CAX4R) of Case 1 (using the STIFFNESS hourglass control).

Figure 1.3.2-4 Contours of equivalent plastic strain for the fine mesh ( CAX4R) of Case 1 (using the STIFFNESS hourglass control).

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Figure 1.3.2-5 Undeformed and deformed shape for the coarse mesh (CAX6M) of Case 1.

Figure 1.3.2-6 Contours of equivalent plastic strain for the coarse mesh ( CAX6M) of Case 1.

Figure 1.3.2-7 Undeformed and deformed shape for the fine mesh (CAX6M) of Case 1.

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Figure 1.3.2-8 Contours of equivalent plastic strain for the fine mesh ( CAX6M) of Case 1.

Figure 1.3.2-9 Comparison of reaction force versus vertical displacement for the different analyses tested for Case 1.

Sample listings

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Listing 1.3.2-1 *HEADING AXISYMMETRIC UPSETTING PROBLEM -- COARSE MESH (WITH RIGID SURFACE) SECTION CONTROLS USED (HOURGLASS=STIFFNESS) *RESTART,WRITE,NUM=30 *NODE 1, 13,.01 1201,0.,.015 1213,.01,.015 *NGEN,NSET=MIDDLE 1,13 *NGEN,NSET=TOP 1201,1213 *NFILL MIDDLE,TOP,12,100 *NSET,NSET=AXIS,GEN 1,1201,100 *ELEMENT,TYPE=CAX4R,ELSET=BILLET 1,1,2,102,101 *ELGEN,ELSET=BILLET 1,12,1,1,12,100,100 *NODE, NSET=NRIGID 2003,0.01,.02 *SOLID SECTION,ELSET=BILLET,MATERIAL=METAL, CONTROL=B *SECTION CONTROLS, HOURGLASS=STIFFNESS, NAME=B *MATERIAL,NAME=METAL *ELASTIC 200.E9,.3 *PLASTIC 7.E8,0.00 3.7E9,10.0 *DENSITY 7833., *BOUNDARY MIDDLE,2 AXIS,1 2003,1 2003,3,6 *AMPLITUDE,NAME=RAMP

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0.,0., 2.E-4,1., 5.5E-4,1. *SURFACE,TYPE=ELEMENT,NAME=BILLET TOP,S3 SIDE,S2 *SURFACE,NAME=RIGID,TYPE=SEGMENTS START, 0.02,.015 LINE, 0.00,.015 *RIGID BODY, REF NODE=2003, ANALYTICAL SURFACE =RIGID *STEP *DYNAMIC,EXPLICIT ,5.5E-4 *BOUNDARY,TYPE=VELOCITY,AMPLITUDE=RAMP 2003,2,,-20. *ELSET,ELSET=TOP,GEN 1101,1112,1 *ELSET,ELSET=SIDE,GEN 12,1112,100 *SURFACE INTERACTION,NAME=RIG_BILL *FRICTION 1.0, *CONTACT PAIR,INTERACTION=RIG_BILL RIGID,BILLET *MONITOR,NODE=2003,DOF=2 *FILE OUTPUT,NUM=2,TIMEMARKS=YES *EL FILE PEEQ,MISES *HISTORY OUTPUT, TIME=4.E-7 *NODE HISTORY, NSET=NRIGID U2,RF2 *NODE FILE, NSET=NRIGID U,RF *END STEP

1.3.3 Superplastic forming of a rectangular box Product: ABAQUS/Standard In this example we consider the superplastic forming of a rectangular box. The example illustrates the use of rigid elements to create a smooth three-dimensional rigid surface. Superplastic metals exhibit high ductility and very low resistance to deformation and are, thus, suitable for forming processes that require very large deformations. Superplastic forming has a number of advantages over conventional forming methods. Forming is usually accomplished in one step rather than several, and intermediate annealing steps are usually unnecessary. This process allows the

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production of relatively complex, deep-shaped parts with quite uniform thickness. Moreover, tooling costs are lower since only a single die is usually required. Drawbacks associated with this method include the need for tight control of temperature and deformation rate. Very long forming times make this method impractical for high volume production of parts. A superplastic forming process usually consists of clamping a sheet against a die whose surface forms a cavity of the shape required. Gas pressure is then applied to the opposite surface of the sheet, forcing it to acquire the die shape.

Rigid surface The *SURFACE option allows the creation of a rigid faceted surface created from an arbitrary mesh of three-dimensional rigid elements (either triangular R3D3 or quadrilateral R3D4 elements). See ``Defining analytical rigid surfaces,'' Section 2.3.4 of the ABAQUS/Standard User's Manual, for a discussion of smoothing of master surfaces. ABAQUS automatically smoothes any discontinuous surface normal transitions between the surface facets.

Solution-dependent amplitude One of the main difficulties in superplastically forming a part is the control of the processing parameters. The temperature and the strain rates that the material experiences must remain within a certain range for superplasticity to be maintained. The former is relatively easy to achieve. The latter is more difficult because of the unknown distribution of strain rates in the part. The manufacturing process must be designed to be as rapid as possible without exceeding a maximum allowable strain rate at any material point. For this purpose ABAQUS has a feature that allows the loading (usually the gas pressure) to be controlled by means of a solution-dependent amplitude. The options invoked are *AMPLITUDE, DEFINITION=SOLUTION DEPENDENT and a target maximum *CREEP STRAIN RATE CONTROL. In the loading options the user specifies a reference value. The amplitude definition requires an initial, a minimum, and a maximum load multiplier. During a *VISCO procedure ABAQUS will then monitor the maximum creep strain rate and compare it with the target value. The load amplitude is adjusted based on this comparison. This controlling algorithm is simple and relatively crude. The purpose is not to follow the target values exactly but to obtain a practical loading schedule.

Geometry and model The example treated here corresponds to superplastic forming of a rectangular box whose final dimensions are 1524 mm (60 in) long by 1016 mm (40 in) wide by 508 mm (20 in) deep with a 50.8 mm (2 in) flange around it. All fillet radii are 101.6 mm (4 in). The box is formed by means of a uniform fluid pressure. A quarter of the blank is modeled using 704 membrane elements of type M3D4R. These are bilinear membrane elements with fully reduced integration and hourglass control. The initial dimensions of the blank are 1625.6 mm (64 in) by 1117.6 mm (44 in), and the thickness is 3.175 mm (0.125 in). The blank is clamped at all its edges. The flat initial configuration of the membrane model is entirely singular in the normal direction unless it is stressed in biaxial tension. This difficulty is prevented by applying a small biaxial initial stress of 6.89 kPa (1 lb/in 2) by means of the *INITIAL CONDITIONS,

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TYPE=STRESS option. The female die is modeled as a rigid body and is meshed with rigid R3D3 elements. The rigid surface is defined with the *SURFACE option by grouping together those faces of the 231 R3D3 elements used to model the die that face the contact direction. See Figure 1.3.3-1 for an illustration of the rigid element mesh. To avoid having points "fall off" the rigid surface during the course of the analysis, more than a quarter of the die has been modeled, as shown in Figure 1.3.3-2. It is always a good idea to extend the rigid surface far enough so that contacting nodes will not slide off the master surface. By default, ABAQUS generates a unique normal to the rigid surface at each node point, based on the average of the normals to the elements sharing each node. There are times, however, when the normal to the surface should be specified directly. This is discussed in ``Node definition,'' Section 2.1.1 of the ABAQUS/Standard User's Manual. In this example the flange around the box must be flat to ensure compatibility between the originally flat blank and the die. Therefore, an outer normal (0, 1, 0) has been specified at the 10 nodes that make up the inner edge of the flange. This is done by entering the direction cosines after the node coordinates. The labels of these 10 vertices are 9043, 9046, 9049, 9052, 9089, 9090, 9091, 9121, 9124, and 9127; and their definitions can be found in superplasticbox_node.inp.

Material The material in the blank is assumed to be elastic-viscoplastic, and the properties roughly represent the 2004 (Al-6Cu-0.4Zr)-based commercial superplastic aluminum alloy Supral 100 at 470°C. It has a Young's modulus of 71 GPa (10.3 ´ 106 lb/in2) and a Poisson's ratio of 0.34. The flow stress is assumed to depend on the plastic strain rate according to ¾ f = A("_pl )1=2 ; where A is 179.2 MPa (26. ´ 103 lb/in2) and the time is in seconds.

Loading and controls We perform two analyses to compare constant pressure loading and a pressure schedule automatically adjusted to achieve a maximum strain rate of 0.02/sec. In the constant load case the prestressed blank is subjected to a rapidly applied external pressure of 68.8 kPa (10 lb/in 2), which is then held constant for 3000 sec until the box has been formed. In the second case the prestressed blank is subjected to a rapidly applied external pressure of 1.38 kPa (0.2 lb/in 2). The pressure schedule is then chosen by ABAQUS. The initial application of the pressures is assumed to occur so quickly that it involves purely elastic response. This is achieved by using the *STATIC procedure. The creep response is developed in a second step using the *VISCO procedure. During the *VISCO step the parameter CETOL controls the time increment and, hence, the accuracy of the transient creep solution. ABAQUS compares the equivalent creep strain rate at the beginning and

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the end of an increment. The difference should be less than CETOL divided by the time increment. Otherwise, the increment is reattempted with a smaller time increment. The usual guideline for setting CETOL is to decide on an acceptable error in stress and convert it to an error in strain by dividing by the elastic modulus. For this problem we assume that moderate accuracy is required and choose CETOL as 0.5%. In general, larger values of CETOL allow ABAQUS to use larger time increments, resulting in a less accurate and less expensive analysis. In the automatic scheduling analysis the pressure is referred to an amplitude that allows for a maximum pressure of 1.38 MPa (200 lb/in 2) and a minimum pressure of 0.138 kPa (0.02 lb/in 2). The target creep strain rate is a constant entered using the *CREEP STRAIN RATE CONTROL option.

Results and discussion Figure 1.3.3-3 through Figure 1.3.3-5show a sequence of deformed configurations during the automatically controlled forming process. The stages of deformation are very similar in the constant load process. However, the time necessary to obtain the deformation is much shorter with automatic loading--the maximum allowable pressure is reached after 83.3 seconds. The initial stages of the deformation correspond to inflation of the blank because there is no contact except at the edges of the box. Contact then occurs at the box's bottom, with the bottom corners finally filling. Although there is some localized thinning at the bottom corners, with strains on the order of 100%, these strains are not too much larger than the 80% strains seen on the midsides. Figure 1.3.3-6 shows the equivalent plastic strain at the end of the process. The constant load case provides similar results. Figure 1.3.3-7 shows the evolution in time of the ratio between the maximum creep strain rate found in the model and the target creep strain rate. The load applied initially produces a low maximum creep strain rate at the beginning of the analysis. At the end the maximum creep strain rate falls substantially as the die cavity fills up. Although the curve appears very jagged, it indicates that the maximum peak strain rate is always relatively close to the target value. This is quite acceptable in practice. Figure 1.3.3-8 shows the pressure schedule that ABAQUS calculates for this problem. For most of the time, while the sheet does not contact the bottom of the die, the pressure is low. Once the die starts restraining the deformation, the pressure can be increased substantially without producing high strain rates.

Input files superplasticbox_constpress.inp Constant pressure main analysis. superplasticbox_autopress.inp Automatic pressurization main analysis. superplasticbox_node.inp Node definitions for the rigid elements. superplasticbox_element.inp

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Element definitions for the rigid R3D3 elements.

Figures Figure 1.3.3-1 Rigid surface for die.

Figure 1.3.3-2 Initial position of blank with respect to die.

Figure 1.3.3-3 Automatic loading: deformed configuration after 34 sec in Step 2.

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Figure 1.3.3-4 Automatic loading: deformed configuration after 63 sec in Step 2.

Figure 1.3.3-5 Automatic loading: deformed configuration after 83 sec in Step 2.

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Figure 1.3.3-6 Automatic loading: inelastic strain in the formed box.

Figure 1.3.3-7 History of ratio between maximum creep strain rate and target creep strain rate.

Figure 1.3.3-8 History of pressure amplitude.

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Sample listings

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Listing 1.3.3-1 *HEADING SUPERPLASTIC FORMING OF BOX WITH MEMBRANES - CONSTANT LOADING *RESTART, WRITE, FREQUENCY=30 ** ** PLATE DEFINITION ** *NODE 1, 4.0, 20.0, -4.0 23, 26.0, 20.0, -4.0 3201, 4.0, 20.0, -36.0 3223, 26.0, 20.0, -36.0 *NGEN, NSET=EDGE3 1,3201,100 *NGEN, NSET=EDGE5 23,3223,100 *NFILL, NSET=PLATE EDGE3,EDGE5,22,1 *NSET, NSET=EDGE1, GENERATE 1,23,1 *NSET, NSET=EDGE7, GENERATE 3201,3223,1 *ELEMENT, TYPE=M3D4, ELSET=PLATE1 1,1,2,102,101 *ELGEN, ELSET=PLATE1 1,21,1,1,31,100,100 *NSET,ELSET=PLATE1, NSET=NCONT *ELEMENT, TYPE=M3D4, ELSET=PLATE2 22,22,23,123,122 3101,3101,3102,3202,3201 *ELGEN, ELSET=PLATE2 22,32,100,100 3101,21,1,1 *ELSET,ELSET=PLATE PLATE1,PLATE2 *MEMBRANE SECTION, ELSET=PLATE, MATERIAL=SUPRAL .125, ** ** MATERIAL IS CLOSE TO SUPRAL100 AT 470C ** *MATERIAL, NAME=SUPRAL

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*ELASTIC 10.3E6, .34 *CREEP, LAW=TIME 1.48E-9, 2., 0. ** ** CONTACT surface ** *NODE, NSET=DIE 10000, 0.0, 0.0, 0.0 *RIGID BODY, ELSET=ERIGID, REFNODE=10000 *NODE, INPUT=superplasticbox_node.inp *ELEMENT,TYPE=R3D3,ELSET=ERIGID, INPUT=superplasticbox_element.inp *SURFACE,NAME=DIE ERIGID,SPOS *SURFACE,TYPE=NODE,NAME=SLAVES NCONT, *CONTACT PAIR, INTERACTION=DIE_NODE, smooth=0.2 SLAVES,DIE *SURFACE INTERACTION,NAME=DIE_NODE ** ** BOUNDARY AND INITIAL CONDITIONS ** *BOUNDARY EDGE1,3 EDGE7,2,3 EDGE3,1 EDGE5,1,2 10000,1,6 *INITIAL CONDITIONS, TYPE=STRESS PLATE, 1.0, 1.0 *NSET,NSET=NSELECT 101,121,1001,1101,1507,2509 1,113,1211,1308,1408,1508,1509,1705 ** ** STEP 1 ** *STEP, INC=50, NLGEOM, unsymm=yes *STATIC 1.E-4,1.0, *DLOAD PLATE,P,-10. *CONTACT PRINT, FREQUENCY=100

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*CONTACT FILE, FREQUENCY=100, NSET=NSELECT *EL PRINT, ELSET=PLATE , FREQUENCY=100 S, E CE, SINV, *PRINT, CONTACT=YES *NODE PRINT, NSET=EDGE3, FREQUENCY=100 U, *NODE FILE, NSET=EDGE3, FREQUENCY=100 U, *END STEP ** ** STEP 2 ** *STEP, INC=500, NLGEOM, unsymm=yes *VISCO, CETOL=0.005 0.005, 3000.0 *DLOAD PLATE,P,-10. *END STEP

1-414

Static Stress/Displacement Analyses

Listing 1.3.3-2 *HEADING SUPERPLASTIC FORMING OF BOX WITH MEMBRANES - AUTOMATIC LOADING *RESTART, WRITE, FREQUENCY=30 ** ** PLATE DEFINITION ** *NODE 1, 4.0, 20.0, -4.0 23, 26.0, 20.0, -4.0 3201, 4.0, 20.0, -36.0 3223, 26.0, 20.0, -36.0 *NGEN, NSET=EDGE3 1,3201,100 *NGEN, NSET=EDGE5 23,3223,100 *NFILL, NSET=PLATE EDGE3,EDGE5,22,1 *NSET, NSET=EDGE1, GENERATE 1,23,1 *NSET, NSET=EDGE7, GENERATE 3201,3223,1 *NSET, NSET=CENTER 1, *ELEMENT, TYPE=M3D4, ELSET=PLATE1 1,1,2,102,101 *ELGEN, ELSET=PLATE1 1,21,1,1,31,100,100 *NSET,ELSET=PLATE1, NSET=NCONT *ELEMENT, TYPE=M3D4, ELSET=PLATE2 22,22,23,123,122 3101,3101,3102,3202,3201 *ELGEN, ELSET=PLATE2 22,32,100,100 3101,21,1,1 *ELSET,ELSET=PLATE PLATE1,PLATE2 *MEMBRANE SECTION, ELSET=PLATE, MATERIAL=SUPRAL .125, ** ** MATERIAL IS CLOSE TO SUPRAL100 AT 470C

1-415

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** *MATERIAL, NAME=SUPRAL *ELASTIC 10.3E6, .34 *CREEP, LAW=TIME 1.48E-9, 2., 0. ** ** CONTACT surface ** *NODE, NSET=DIE 10000, 0.0, 0.0, 0.0 *RIGID BODY, ELSET=ERIGID, REFNODE=10000 *NODE, INPUT=superplasticbox_node.inp *ELEMENT,TYPE=R3D3,ELSET=ERIGID, INPUT=superplasticbox_element.inp *SURFACE,NAME=DIE ERIGID,SPOS *SURFACE,type=node,NAME=SLAVES NCONT, *CONTACT PAIR, INTERACTION=DIE_NODE, smooth=0.2 SLAVES,DIE *SURFACE INTERACTION,NAME=DIE_NODE ** ** BOUNDARY AND INITIAL CONDITIONS ** *BOUNDARY EDGE1,3 EDGE7,2,3 EDGE3,1 EDGE5,1,2 10000,1,6 *INITIAL CONDITIONS, TYPE=STRESS PLATE, 1.0, 1.0 *AMPLITUDE,DEFINITION=SOLUTION DEPENDENT,NAME=AUTO 1.,0.1,1000. *NSET,NSET=NSELECT 101,121,1001,1101,1507,2509 1,113,1211,1308,1408,1508,1509,1705 ** ** STEP 1 ** *STEP, INC=30, NLGEOM, unsymm=yes *STATIC

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2.E-3,1.0, *DLOAD PLATE,P,-0.2 *CONTACT PRINT, FREQUENCY=100 *CONTACT FILE, FREQUENCY=100, NSET=NSELECT *EL PRINT, ELSET=PLATE , FREQUENCY=100 S, E CE, SINV, *PRINT, CONTACT=YES *NODE PRINT, NSET=EDGE3, FREQUENCY=100 U, *NODE FILE, NSET=EDGE3, FREQUENCY=100 U, *END STEP ** ** STEP 2 ** *STEP, INC=500, NLGEOM, unsymm=yes *VISCO, CETOL=0.005 0.2, 2000.0 *DLOAD,AMPLITUDE=AUTO PLATE,P,-0.2 *CREEP STRAIN RATE CONTROL, ELSET=PLATE, AMPLITUDE=AUTO 0.02 , ** TO WRITE THE AUTOMATIC SOLUTION CONTROL ** VARIABLES AMPCU AND RATIO TO THE RESULTS ** FILE EVERY INCREMENT SUCH FILE HAS TO BE ** ACTIVATED *NODE FILE, NSET=CENTER, FREQUENCY=1 U, *END STEP

1.3.4 Stretching of a thin sheet with a hemispherical punch Products: ABAQUS/Standard ABAQUS/Explicit Stamping of sheet metals by means of rigid punches and dies is a standard manufacturing process. In most bulk forming processes the loads required for the forming operation are often the primary concern. However, in sheet forming the prediction of strain distributions and limit strains (which define the onset of local necking) are most important. Such analysis is complicated in that it requires consideration of large plastic strains during deformation, an accurate description of material response including strain hardening, the treatment of a moving boundary that separates the region in contact

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with the punch head from the unsupported one, and the inclusion of friction between the sheet and the punch head. The stretching of a thin circular sheet with a hemispherical punch is considered in this example.

Geometry and model The geometry of this problem is shown in Figure 1.3.4-1. The sheet being stretched has a clamping radius, r0 , of 59.18 mm. The radius of the punch, rp , is 50.8 mm; the die radius, rd , is 6.35 mm; and the initial thickness of the sheet, t0 , is 0.85 mm. Such a sheet has been tested experimentally by Ghosh and Hecker (1975) and has been analyzed by Wang and Budiansky (1978) using an axisymmetric membrane shell finite element formulation. The analysis is conducted statically in ABAQUS/Standard and dynamically in ABAQUS/Explicit such that inertial forces are relatively small. The initial configuration for the ABAQUS/Explicit analysis is shown in Figure 1.3.4-2. The sheet, the punch, and the die are modeled as separate parts, each instanced once. As an axisymmetric problem in ABAQUS/Standard the sheet is modeled using 50 elements of type SAX1 (or MAX1) or 25 elements of type SAX2 (or MAX2). The ABAQUS/Explicit model uses 50 elements of type SAX1. Mesh convergence studies (not reported here) have been done and indicate that these meshes give acceptably accurate results for most of the values of interest. To test the three-dimensional membrane and shell elements in ABAQUS/Standard, a 10° sector is modeled using 100 elements of type S4R, S4, or M3D4R or 25 elements of type M3D9R. All these meshes are reasonably fine; they are used to obtain good resolution of the moving contact between the sheet and the dies. In the ABAQUS/Standard shell models nine integration points are used through the thickness of the sheet to ensure the development of yielding and elastic-plastic bending response; in ABAQUS/Explicit five integration points are used through the thickness of the sheet. The rigid punch and die are modeled in ABAQUS/Standard as analytical rigid surfaces with the *SURFACE option in conjunction with the *RIGID BODY option. The top and bottom surfaces of the sheet are defined with the *SURFACE option. In ABAQUS/Explicit the punch and die are modeled as rigid bodies using the *RIGID BODY option; the surface of the punch and die are modeled either by analytical rigid surfaces or RAX2 elements. The rigid surfaces are offset from the blank by half of the thickness of the blank because the contact algorithm in ABAQUS/Explicit takes the shell thickness into account.

Material properties The material (aluminum-killed steel) is assumed to satisfy the Ramberg-Osgood relation between true stress and logarithmic strain: " = ¾=E + (¾=K )n ; where Young's modulus, E, is 206.8 GPa; the reference stress value, K, is 0.510 GPa; and the work-hardening exponent, n, is 4.76. The material is assumed to be linear elastic below a 0.5% offset yield stress of 170.0 MPa and the stress-strain curve beyond that value is defined in piecewise linear segments using the *PLASTIC option. (The 0.5% offset yield stress is defined from the Ramberg-Osgood fit by taking (" ¡ ¾=E ) to be 0.5% and solving for the stress.) Poisson's ratio is 1-418

Static Stress/Displacement Analyses

0.3. The membrane element models in ABAQUS/Standard are inherently unstable in a static analysis unless some prestress is present in the elements prior to the application of external loading. Therefore, an equibiaxial initial stress condition equal to 5% of the initial yield stress is prescribed for the membrane elements in ABAQUS/Standard.

Contact interactions The contact between the sheet and the rigid punch and the rigid die is modeled with the *CONTACT PAIR option. The mechanical interaction between the contact surfaces is assumed to be frictional contact, with a coefficient of friction of 0.275 in ABAQUS/Standard and 0.265 in ABAQUS/Explicit.

Loading The ABAQUS/Standard analysis is carried out in six steps; the ABAQUS/Explicit analysis is carried out in four steps. In each of the first four steps of the ABAQUS/Standard analysis either the die or the punch head is moved using the *BOUNDARY option. In ABAQUS/Explicit the velocity of the punch head is prescribed using the *BOUNDARY option; the magnitude of the velocity is specified with the *AMPLITUDE option. It is ramped up to 30 m/s at 1.24 milliseconds during the first step and then kept constant until time reaches 1.57 milliseconds at the end of the second step. It is then ramped down to zero at a time of 1.97 milliseconds at the end of the third step. In the first step of the ABAQUS/Standard analysis the die is moved so that it just touches the sheet. In the next three steps (the first three steps of the ABAQUS/Explicit analysis) the punch head is moved toward the sheet through total distances of 18.6 mm, 28.5 mm, and 34.5 mm, respectively. The purpose of these three steps is to compare the results with those provided experimentally by Ghosh and Hecker for these punch displacements. More typically the punch would be moved through its entire travel in one step. Two final steps are included in the ABAQUS/Standard analysis. In the first step the metal sheet is held in place and the contact pairs are removed from the model with the *MODEL CHANGE, TYPE=CONTACT PAIR, REMOVE option. In the second step the original boundary conditions for the metal sheet are reintroduced for springback analysis. However, this springback step is not included for the analyses using membrane elements, since these elements do not have any bending stiffness and residual bending stress is often a key determinant of springback. In the final step of the ABAQUS/Explicit analysis the punch head is moved away from the sheet for springback analysis. A viscous pressure load is applied to the surface of the shell during this step to damp out transient wave effects so that quasi-static equilibrium can be reached quickly. This effect happens within approximately 2 milliseconds from the start of unloading. The coefficient of viscous pressure is chosen to be 0.35 MPa sec/m, approximately 1% of the value of ½cd , where ½ is the material density of the sheet and cd is the dilatational wave speed. A value of viscous pressure of ½cd would absorb all the energy in a pressure wave. For typical structural problems choosing a small percentage of this value provides an effective way of minimizing ongoing dynamic effects. Static equilibrium is reached when residual stresses in the sheet are reasonably constant over time.

Results and discussion

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Static Stress/Displacement Analyses

Figure 1.3.4-2 shows the initial, undeformed profile of the blank, the die, and the punch. Figure 1.3.4-3 illustrates the deformed sheet and the punch and the die. Figure 1.3.4-4 shows a plot of the same system after the punch is lifted back, showing the springback of the sheet. Figure 1.3.4-5 and Figure 1.3.4-6 show the distribution of nominal values of radial and circumferential membrane strain in the sheet for 18.6 mm punch head displacement. Figure 1.3.4-7 and Figure 1.3.4-8 show the strain distributions at a punch head displacement of 28.5 mm, and Figure 1.3.4-9 and Figure 1.3.4-10 show the strain distributions at a punch head displacement of 34.5 mm. The strain distributions for the SAX1 models compare well with those obtained experimentally by Ghosh and Hecker (1975) and those obtained numerically by Wang and Budiansky (1978), who used a membrane shell finite element formulation. The important phenomenon of necking during stretching is reproduced at nearly the same location, although slightly different strain values are obtained. Draw beads are used to hold the edge of the sheet in the experiment, but in this analysis the sheet is simply clamped at its edge. Incorporation of the draw bead boundary conditions may further improve the correlation with the experimental data. A spike can be observed in the radial strain distribution toward the edge of the sheet in some of the ABAQUS/Standard shell models. This strain spike is the result of the sheet bending around the die. The spike is not present in the membrane element models since they possess no bending stiffness. The results obtained with the axisymmetric membrane models are compared with those obtained from the axisymmetric shell models and were found to be in good agreement. These analyses assume values of 0.265 or 0.275 for the coefficient of friction. Ghosh and Hecker do not give a value for their experiments, but Wang and Budiansky assume a value of 0.17. The coefficient of friction has a marked effect on the peak strain during necking and may be a factor contributing to the discrepancy of peak strain results during necking. The values used in these analyses have been chosen to provide good correlation with the experimental data. The distributions of the residual stresses on springback of the sheet in ABAQUS/Explicit are shown in Figure 1.3.4-11 and Figure 1.3.4-12.

Input files ABAQUS/Standard input files thinsheetstretching_m3d4r.inp Element type M3D4R. thinsheetstretching_m3d9r.inp Element type M3D9R. thinsheetstretching_max1.inp Element type MAX1. thinsheetstretching_max2.inp Element type MAX2.

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thinsheetstretching_s4.inp Element type S4. thinsheetstretching_s4r.inp Element type S4R. thinsheetstretching_sax1.inp Element type SAX1. thinsheetstretching_sax2.inp Element type SAX2. thinsheetstretching_restart.inp Restart of thinsheetstretching_sax2.inp. ABAQUS/Explicit input files hemipunch_anl.inp Model using analytical rigid surfaces to describe the rigid surface. hemipunch.inp Model using rigid elements to describe the rigid surface.

References · Ghosh, A. K., and S. S. Hecker, ``Failure in Thin Sheets Stretched Over Rigid Punches ,'' Metallurgical Transactions, vol. 6A, pp. 1065-1074, 1975. · Wang, N. M., and B. Budiansky, ``Analysis of Sheet Metal Stamping by a Finite Element Method,'' Journal of Applied Mechanics, vol. 45, pp. 73-82, 1978.

Figures Figure 1.3.4-1 Configuration and dimensions for hemispherical punch stretching.

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Figure 1.3.4-2 Initial ABAQUS/Explicit configuration.

Figure 1.3.4-3 Configuration for punch head displacement of 34.5 mm, ABAQUS/Explicit.

Figure 1.3.4-4 Final configuration after springback, ABAQUS/Explicit.

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Static Stress/Displacement Analyses

Figure 1.3.4-5 Strain distribution for punch head displacement of 18.6 mm, ABAQUS/Standard.

Figure 1.3.4-6 Strain distribution for punch head displacement of 18.6 mm, ABAQUS/Explicit.

Figure 1.3.4-7 Strain distribution for punch head displacement of 28.5 mm, ABAQUS/Standard.

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Figure 1.3.4-8 Strain distribution for punch head displacement of 28.5 mm, ABAQUS/Explicit.

Figure 1.3.4-9 Strain distribution for punch head displacement of 34.5 mm, ABAQUS/Standard.

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Static Stress/Displacement Analyses

Figure 1.3.4-10 Strain distribution for punch head displacement of 34.5 mm, ABAQUS/Explicit.

Figure 1.3.4-11 Residual stress on top surface after springback, ABAQUS/Explicit.

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Figure 1.3.4-12 Residual stress on bottom surface after springback, ABAQUS/Explicit.

Sample listings

1-426

Static Stress/Displacement Analyses

Listing 1.3.4-1 *HEADING WANG AND BUDIANSKY'S SPHERICAL PUNCH WITH SAX1 50 ELEMENTS, 9 LAYERS *RESTART,WRITE,FREQUENCY=250 *PREPRINT,ECHO=YES *PART,NAME=BLANK *NODE,NSET=MID 1,0.0,0.0 *NODE,NSET=REFD 401,50.59,0. *NFILL,BIAS=1.0,NSET=METND MID,REFD,40,10 *NODE,NSET=END 501,59.18,0. *NFILL,BIAS=1.0,NSET=METND REFD,END,10,10 *NSET,NSET=NODWR,GENERATE 1,501,10 *ELEMENT,TYPE=SAX1 1,1,11 *ELGEN,ELSET=METAL 1,50,10 *ELSET,ELSET=EDIE,GENERATE 42,49,1 *ELSET,ELSET=ECON,GENERATE 1,41,1 *SHELL SECTION,ELSET=METAL,MATERIAL=SAMP 0.85,9 *END PART *MATERIAL,NAME=SAMP *ELASTIC 206.8,0.3 *PLASTIC 0.1700000, 0.0000000E+00 0.1800000 , 1.7205942E-03 0.1900000 , 3.8296832E-03 0.2000000 , 6.3897874E-03 0.2100000, 9.4694765E-03 0.2200000, 1.3143660E-02 0.2300000, 1.7493792E-02 0.2400000, 2.2608092E-02

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0.2500000, 2.8581845E-02 0.2600000, 3.5517555E-02 0.2700000, 4.3525275E-02 0.2800000, 5.2722659E-02 0.2900000, 6.3235357E-02 0.3000000, 7.5197279E-02 0.3100000, 8.8750519E-02 0.3200000, 0.1040458 0.3300000, 0.1212430 0.3400000, 0.1405106 0.3500000, 0.1620263 0.3600000, 0.1859779 0.3700000, 0.2125620 0.3800000, 0.2419857 0.3900000, 0.2744660 0.4000000, 0.3102303 0.4100000, 0.3495160 0.4200000, 0.3925720 0.4300000, 0.4396578 0.4400000, 0.4910434 0.4500000, 0.5470111 0.4600000, 0.6078544 0.4700000, 0.6738777 0.4800000, 0.7453985 0.4900000, 0.8227461 0.5000000, 0.9062610 0.5100000 , 0.9962980 *PART,NAME=PUNCH *NODE,NSET=PUNCH 1000,0.,0. *RIGID BODY,ANALYTICAL SURFACE=BSURF,REF NODE=1000 *END PART *PART,NAME=DIE *NODE,NSET=DIE 2000,59.18,0.05 *RIGID BODY,ANALYTICAL SURFACE=DSURF,REF NODE=2000 *END PART *ASSEMBLY,NAME=FORM *INSTANCE,NAME=BLANK-1,PART=BLANK *SURFACE,NAME=ASURF ECON,SNEG *SURFACE,NAME=CSURF EDIE,SPOS

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*END INSTANCE *INSTANCE,NAME=PUNCH-1,PART=PUNCH *SURFACE,NAME=BSURF,TYPE=SEGMENTS START,0.0,0.0 CIRCL,50.8,-50.80,0.0,-50.80 *END INSTANCE *INSTANCE,NAME=DIE-1,PART=DIE *SURFACE,NAME=DSURF,TYPE=SEGMENTS START,61.00,0.05 LINE,59.18,0.05 CIRCL,52.83,6.4,59.18,6.4 LINE,52.83,8. *END INSTANCE *NSET,NSET=PUNK PUNCH-1.PUNCH,DIE-1.DIE *END ASSEMBLY *CONTACT PAIR,INTERACTION=ROUGH FORM.BLANK-1.ASURF,FORM.PUNCH-1.BSURF *CONTACT PAIR,INTERACTION=ROUGH FORM.BLANK-1.CSURF,FORM.DIE-1.DSURF *SURFACE INTERACTION,NAME=ROUGH *FRICTION 0.275, *BOUNDARY FORM.BLANK-1.1,1,1 FORM.BLANK-1.1,6,6 FORM.BLANK-1.501,1,1 FORM.BLANK-1.501,2,2 FORM.PUNCH-1.1000,6,6 FORM.PUNCH-1.1000,1,1 FORM.DIE-1.2000,1,1 FORM.DIE-1.2000,6,6 *STEP,INC=10,NLGEOM, UNSYMM=YES *STATIC 1.,1. *BOUNDARY FORM.DIE-1.2000,2,2,-0.05 *PRINT,RESIDUAL=NO,FREQUENCY=10 *EL PRINT,FREQUENCY=0 *NODE FILE,FREQUENCY=1000 U,RF COORD, *CONTACT FILE,SLAVE=FORM.BLANK-1.ASURF,FREQUENCY=1000

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*CONTACT FILE,SLAVE=FORM.BLANK-1.CSURF,FREQUENCY=1000 *END STEP *STEP,INC=2000,NLGEOM, UNSYMM=YES *STATIC 0.05,100.,1.E-5 *BOUNDARY FORM.PUNCH-1.1000,2,2,18.6 FORM.DIE-1.2000,2,2,-0.05 *MONITOR,NODE=FORM.PUNCH-1.1000,DOF=2 *NODE PRINT,NSET=FORM.PUNK,FREQUENCY=100 U,RF COORD, *END STEP *STEP,INC=2000,NLGEOM, UNSYMM=YES *STATIC 0.05,100.,1.E-5 *EL FILE,ELSET=FORM.BLANK-1.METAL,FREQUENCY=1000 5, S,E *BOUNDARY FORM.PUNCH-1.1000,2,2,28.5 FORM.DIE-1.2000,2,2,-0.05 *END STEP *STEP,INC=2000,NLGEOM, UNSYMM=YES *STATIC 0.05,100.,1.E-5 *BOUNDARY FORM.PUNCH-1.1000,2,2,34.5 FORM.DIE-1.2000,2,2,-0.05 *END STEP *STEP,INC=2000,NLGEOM, UNSYMM=YES *STATIC 100.,100. *BOUNDARY,FIXED FORM.BLANK-1.METND,1,2 FORM.BLANK-1.METND,6 *MODEL CHANGE,TYPE=CONTACT PAIR,REMOVE FORM.BLANK-1.ASURF,FORM.PUNCH-1.BSURF FORM.BLANK-1.CSURF,FORM.DIE-1.DSURF *EL FILE,ELSET=FORM.BLANK-1.METAL,FREQUENCY=1000 5, S, *END STEP

1-430

Static Stress/Displacement Analyses

*STEP,INC=2000,NLGEOM, UNSYMM=YES *STATIC 1.,100. *BOUNDARY,OP=NEW FORM.BLANK-1.1,1,1 FORM.BLANK-1.1,6,6 FORM.BLANK-1.501,1,1 FORM.BLANK-1.501,2,2 FORM.PUNCH-1.1000,6,6 FORM.PUNCH-1.1000,1,2 FORM.DIE-1.2000,1,2 FORM.DIE-1.2000,6,6 *MONITOR,NODE=FORM.BLANK-1.1,DOF=2 *EL FILE,ELSET=FORM.BLANK-1.METAL,FREQUENCY=1000 5, S, *END STEP

1-431

Static Stress/Displacement Analyses

Listing 1.3.4-2 *HEADING WANG AND BUDIANSKY'S SPHERICAL PUNCH WITH SAX1 ELEMENTS PUNCH AND DIE ARE ANALYTICAL RIGID SEGMENTS *PREPRINT,ECHO=YES *PART,NAME=BLANK *NODE 1,0.0,0.0 401,.05059,0. 501,.05918,0. *NGEN 1,401,10 401,501,10 *ELEMENT,TYPE=SAX1,ELSET=BLANK 1,1,11 *ELGEN,ELSET=BLANK 1,50,10 *ELSET,ELSET=CENTER,GEN 1,10,1 *SHELL SECTION,ELSET=BLANK,MATERIAL=SAMP 0.00085 *SURFACE, NAME=TOP BLANK,SPOS *SURFACE, NAME=BOTTOM BLANK,SNEG *END PART *MATERIAL,NAME=SAMP *DENSITY 7850. *ELASTIC 206.8E9,0.3 *PLASTIC 170.0E6, 0.0000000E+00 180.0E6, 1.7205942E-03 190.0E6, 3.8296832E-03 200.0E6, 6.3897874E-03 210.0E6, 9.4694765E-03 220.0E6, 1.3143660E-02 230.0E6, 1.7493792E-02 240.0E6, 2.2608092E-02 250.0E6, 2.8581845E-02

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Static Stress/Displacement Analyses

260.0E6, 3.5517555E-02 270.0E6, 4.3525275E-02 280.0E6, 5.2722659E-02 290.0E6, 6.3235357E-02 300.0E6, 7.5197279E-02 310.0E6, 8.8750519E-02 320.0E6, 0.1040458 330.0E6, 0.1212430 340.0E6, 0.1405106 350.0E6, 0.1620263 360.0E6, 0.1859779 370.0E6, 0.2125620 380.0E6, 0.2419857 390.0E6, 0.2744660 400.0E6, 0.3102303 410.0E6, 0.3495160 420.0E6, 0.3925720 430.0E6, 0.4396578 440.0E6, 0.4910434 450.0E6, 0.5470111 460.0E6, 0.6078544 470.0E6, 0.6738777 480.0E6, 0.7453985 490.0E6, 0.8227461 500.0E6, 0.9062610 510.0E6, 0.9962980 ** *PART,NAME=PUNCH *NODE,NSET=REF_NODE 1000,0.,.051225 *RIGID BODY, REF NODE=REF_NODE, ANALYTICAL SURFACE=PUNCH_BOT *SURFACE, NAME=PUNCH_BOT, TYPE=SEGMENTS START, .0508,.051225 CIRCL, 0.,0.000425, 0.,.051225 *END PART *PART,NAME=DIE *NODE,NSET=REF_NODE 2000, .05918,-.006775 *RIGID BODY, REF NODE=REF_NODE, ANALYTICAL SURFACE=DIE_TOP *SURFACE, NAME=DIE_TOP, TYPE=SEGMENTS START, .05283,-.030425

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LINE, .05283,-.006775 CIRCL, .05918,-0.000425, .05918,-.006775 LINE, .05930,-0.000425 *END PART ** *ASSEMBLY,NAME=ASSEMBLY-1 *INSTANCE,NAME=BLANK-1,PART=BLANK *NSET,NSET=NOUT 1, *ELSET,ELSET=EOUT 22,23,24,25 *END INSTANCE *INSTANCE,NAME=PUNCH-1,PART=PUNCH *NSET,NSET=PUNCH 1000, *END INSTANCE *INSTANCE,NAME=DIE-1,PART=DIE *END INSTANCE *END ASSEMBLY *BOUNDARY ASSEMBLY-1.BLANK-1.1,1,1 ASSEMBLY-1.BLANK-1.1,6,6 ASSEMBLY-1.BLANK-1.501,1,2 ASSEMBLY-1.BLANK-1.501,6,6 ASSEMBLY-1.PUNCH-1.1000,1,1 ASSEMBLY-1.PUNCH-1.1000,1,6 ASSEMBLY-1.DIE-1.2000,1,2 ASSEMBLY-1.DIE-1.2000,6,6 *AMPLITUDE,NAME=LOAD,TIME=TOTAL 0.,0.,1.24E-3,1.,1.57E-3,1.,1.97E-3,0., 3.97E-3,-.25 *RESTART,WRITE,NUM=2,TIMEMARKS=NO *STEP *DYNAMIC,EXPLICIT ,1.24E-3 *BOUNDARY,TYPE=VELOCITY,AMP=LOAD ASSEMBLY-1.PUNCH-1.1000,2,2,-30. *SURFACE INTERACTION,NAME=PUNCH_TOP *FRICTION 0.265, *CONTACT PAIR, INTERACTION=PUNCH_TOP, CPSET=PUNCH_DIE ASSEMBLY-1.PUNCH-1.PUNCH_BOT,ASSEMBLY-1.BLANK-1.TOP

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ASSEMBLY-1.DIE-1.DIE_TOP,ASSEMBLY-1.BLANK-1.BOTTOM *FILE OUTPUT,NUMBER INTERVAL = 1 *EL FILE S,LE *ENERGY FILE *HISTORY OUTPUT,TIME=0. *NODE HISTORY,NSET=ASSEMBLY-1.PUNCH-1.PUNCH U2,RF2 *NODE HISTORY,NSET=ASSEMBLY-1.BLANK-1.NOUT U,V *EL HISTORY,ELSET=ASSEMBLY-1.BLANK-1.EOUT STH MISES,S,LE *ENERGY HISTORY ALLKE,ALLSE,ALLWK,ALLPD,ALLIE,ALLVD,ETOTAL,ALLAE, ALLCD,ALLFD,DT *OUTPUT, FIELD, NUMBER INTERVAL=4, TIMEMARKS=NO *CONTACT OUTPUT, CPSET=PUNCH_DIE, VARIABLE=PRESELECT *END STEP *STEP *DYNAMIC,EXPLICIT ,.33E-3 *END STEP *STEP *DYNAMIC,EXPLICIT ,.40E-3 *END STEP *STEP ** ** Unloading step ** *DYNAMIC,EXPLICIT ,2.E-3 *DLOAD,OP=NEW ASSEMBLY-1.BLANK-1.BLANK,VP,0.35E6 *END STEP

1.3.5 Deep drawing of a cylindrical cup Product: ABAQUS/Standard Deep drawing of sheet metal is an important manufacturing technique. In the deep drawing process a "blank" of sheet metal is clamped by a blank holder against a die. A punch is then moved against the

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blank, which is drawn into the die. Unlike the operation described in the hemispherical punch stretching example (``Stretching of a thin sheet with a hemispherical punch, '' Section 1.3.4), the blank is not assumed to be fixed between the die and the blank holder; rather, the blank is drawn from between these two tools. The ratio of drawing versus stretching is controlled by the force on the blank holder and the friction conditions at the interface between the blank and the blank holder and the die. Higher force or friction at the blank/die/blank holder interface limits the slip at the interface and increases the radial stretching of the blank. In certain cases drawbeads, shown in Figure 1.3.5-1, are used to restrain the slip at this interface even further. To obtain a successful deep drawing process, it is essential to control the slip between the blank and its holder and die. If the slip is restrained too much, the material will undergo severe stretching, thus potentially causing necking and rupture. If the blank can slide too easily, the material will be drawn in completely and high compressive circumferential stresses will develop, causing wrinkling in the product. For simple shapes like the cylindrical cup here, a wide range of interface conditions will give satisfactory results. But for more complex, three-dimensional shapes, the interface conditions need to be controlled within a narrow range to obtain a good product. During the drawing process the response is determined primarily by the membrane behavior of the sheet. For axisymmetric problems in particular, the bending stiffness of the metal yields only a small correction to the pure membrane solution, as discussed by Wang and Tang (1988). In contrast, the interaction between the die, the blank, and the blank holder is critical. Thus, thickness changes in the sheet material must be modeled accurately in a finite element simulation, since they will have a significant influence on the contact and friction stresses at the interface. In these circumstances the most suitable elements in ABAQUS are the 4-node reduced-integration axisymmetric quadrilateral, CAX4R; the first-order axisymmetric shell element, SAX1; the first-order axisymmetric membrane element, MAX1; the first-order finite-strain quadrilateral shell element, S4R; and the fully integrated general-purpose finite-membrane-strain shell element, S4. Membrane effects and thickness changes are modeled properly with CAX4R. However, the bending stiffness of the element is low. The element does not exhibit "locking" due to incompressibility or parasitic shear. It is also very cost-effective. In the shells and membranes the thickness change is calculated from the assumption of incompressible deformation of the material. This simplifying assumption does not allow for the development of stress in the thickness direction of the shell, thus making it difficult to model the contact pressure between the blank and the die and the blank holder. This situation is resolved in the shell and membrane models by using the *SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=EXPONENTIAL option (``Interaction normal to the surface,'' Section 21.3.3 of the ABAQUS/Standard User's Manual) to impose the proper clamping pressure in the thickness direction of the shell or membrane between the blank and the die and the blank holder.

Geometry and model The geometry of the problem is shown in Figure 1.3.5-2. The circular blank being drawn has an initial radius of 100 mm and an initial thickness of 0.82 mm. The punch has a radius of 50 mm and is rounded off at the corner with a radius of 13 mm. The die has an internal radius of 51.25 mm and is rounded off at the corner with a radius of 5 mm. The blank holder has an internal radius of 56.25 mm.

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The blank is modeled using 40 elements of type CAX4R or 31 elements of either type SAX1, MAX1, S4R, or S4. An 11.25° wedge of the circular blank is used in the three-dimensional S4R and S4 models. These meshes are rather coarse for this analysis. However, since the primary interest in this problem is to study the membrane effects, the analysis will still give a fair indication of the stresses and strains occurring in the process. The contact between the blank and the rigid punch, the rigid die, and the rigid blank holder is modeled with the *CONTACT PAIR option. The top and bottom surfaces of the blank are defined by means of the *SURFACE option. The rigid punch, the die, and the blank holder are modeled as analytical rigid surfaces with the *RIGID BODY option in conjunction with the *SURFACE option. The mechanical interaction between the contact surfaces is assumed to be frictional contact. Therefore, the *FRICTION option is used in conjunction with the various *SURFACE INTERACTION property options to specify coefficients of friction. For the shell models the interaction between the blank and the blank holder is also assumed to be "softened" contact, as discussed previously. At the start of the analysis for the CAX4R model, the blank is positioned precisely on top of the die and the blank holder is precisely in touch with the top surface of the blank. The punch is positioned 0.18 mm above the top surface of the blank. The shell and membrane models begin with the same state except that the blank holder is positioned a fixed distance above the blank. This fixed distance is the distance at which the contact pressure is set to zero by means of the *SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=EXPONENTIAL option.

Material properties The material (aluminum-killed steel) is assumed to satisfy the Ramberg-Osgood relation between true stress and logarithmic strain: ² = (¾=K )1=n : The reference stress value, K, is 513 MPa; and the work-hardening exponent, n, is 0.223. The Young's modulus is 211 GPa, and the Poisson's ratio is 0.3. An initial yield stress of 91.3 MPa is obtained with these data. The stress-strain curve is defined in piecewise linear segments in the *PLASTIC option, up to a total (logarithmic) strain level of 107%. The coefficient of friction between the interface and the punch is taken to be 0.25; and that between the die and the blank holder is taken as 0.1, the latter value simulating a certain degree of lubrication between the surfaces. The stiffness method of sticking friction is used in these analyses. The numerics of this method make it necessary to choose an acceptable measure of relative elastic slip between mating surfaces when sticking should actually be occurring. The basis for the choice is as follows. Small values of elastic slip best simulate the actual behavior but also result in a slower convergence of the solution. Permission of large relative elastic displacements between the contacting surfaces can cause higher strains at the center of the blank. In these runs we let ABAQUS choose the allowable elastic slip, which is done by determining a characteristic interface element length over the entire mesh and multiplying by a small fraction to get an allowable elastic slip measure. This method typically

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gives a fairly small amount of elastic slip. Although the material in this process is fully isotropic, the *ORIENTATION option is used with the CAX4R elements to define a local orientation that is coincident initially with the global directions. The reason for using this option is to obtain the stress and strain output in more natural coordinates: if the *ORIENTATION option is used in a geometrically nonlinear analysis, stress and strain components are given in a corotational framework. Hence, in our case throughout the motion, S11 will be the stress in the r-z plane in the direction of the middle surface of the cup. S22 will be the stress in the thickness direction, S33 will be the hoop stress, and S12 will be the transverse shear stress, which makes interpreting the results considerably easier. This orientation definition is not necessary with the SAX1 or MAX1 elements since the output for shell and membrane elements is already given in the local shell system. For the SAX1 and MAX1 model, S11 is the stress in the meridional direction and S22 is the circumferential (hoop) stress. An orientation definition would normally be needed for the S4R and S4 models but can be avoided by defining the wedge in such a manner that the single integration point of each element lies along the global x-axis. Such a model definition, along with appropriate kinematic boundary conditions, keeps the local stress output definitions for the shells as S11 being the stress in the meridional plane and S22 the hoop stress. There should be no in-plane shear, S12, in this problem. A transformation is used in the S4R and S4 models to impose boundary constraints in a cylindrical system.

Loading The entire analysis is carried out in five steps. In the first step the blank holder is pushed onto the blank with a prescribed displacement of -17.5 ´ 10-6 mm. This value is chosen to obtain a reaction force that is approximately equal to the applied force. In the shell models this displacement corresponds to zero clearance across the interface, thus resulting in the application of a predetermined clamping pressure across the shell thickness via the *SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=EXPONENTIAL option. In the second step the boundary condition is removed and replaced by the applied force of 100 kN on the blank holder. This force is kept constant during Steps 2 and 3. This technique of simulating the clamping process is used to avoid potential problems with rigid body modes of the blank holder, since there is no firm contact between the blank holder, the blank, and the die at the start of the process. The two-step procedure creates contact before the blank holder is allowed to move freely. In the third step the punch is moved toward the blank through a total distance of 60 mm. This step models the actual drawing process. During this step the option *CONTROLS, ANALYSIS=DISCONTINUOUS is included since contact with friction tends to create a severely discontinuous nonlinearity and we wish to avoid premature cutbacks of the automatic time incrementation scheme. The last two steps are used to simulate springback. In the fourth step all the nodes in the model are fixed in their current positions and the contact pairs are removed from the model with the *MODEL CHANGE, TYPE=CONTACT PAIR, REMOVE option. This is the most reliable method for releasing contact conditions. In the fifth, and final, step the regular set of boundary conditions is reinstated and the springback is allowed to take place. This part of the analysis with the CAX4R elements is included

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to demonstrate the feasibility of the unloading procedure only and is not expected to produce realistic results, since the reduced-integration elements have a purely elastic bending behavior. The springback is modeled with more accuracy in the shell element models.

Results and discussion Figure 1.3.5-3 shows deformed shapes that are predicted at various stages of the drawing process for the CAX4R model. The profiles show that the metal initially bends and stretches and is then drawn in over the surface of the die. The distributions of radial and circumferential strain for all three models and thickness strain for the CAX4R model are shown in Figure 1.3.5-4. The thickness for the shell or membrane models can be monitored with output variable STH (current shell or membrane thickness). The thickness does not change very much: the change ranges from approximately -12% in the cylindrical part to approximately +16% at the edge of the formed cup. Relatively small thickness changes are usually desired in deep drawing processes and are achieved because the radial tensile strain and the circumferential compressive strain balance each other. The drawing force as a function of punch displacement is shown in Figure 1.3.5-5. The curves for the three models compare closely. The oscillations in the force are a result of the rather coarse mesh--each oscillation represents an element being drawn over the corner of the die. Compared to the shell models, the membrane model predicts a smaller punch force for a given punch displacement. Thus, toward the end of the analysis the results for punch force versus displacement for the MAX1 model are closer to those for the CAX4R model. The deformed shape after complete unloading is shown in Figure 1.3.5-6, superimposed on the deformed shape under complete loading. The analysis shows the lip of the cup springing back strongly after the blank holder is removed for the CAX4R model. No springback is evident in the shell models. As was noted before, this springback in the CAX4R model is not physically realistic: in the first-order reduced-integration elements an elastic "hourglass control" stiffness is associated with the "bending" mode, since this mode is identical to the "hourglass" mode exhibited by this element in continuum situations. In reality the bending of the element is an elastic-plastic process, so that the springback is likely to be much less. A better simulation of this aspect would be achieved by using several elements through the thickness of the blank, which would also increase the cost of the analysis. The springback results for the shell models do not exhibit this problem and are clearly more representative of the actual elastic-plastic process.

Input files deepdrawcup_cax4r.inp CAX4R model. deepdrawcup_cax4i.inp Model using the incompatible mode element, CAX4I, as an alternative to the CAX4R element. In contrast to the reduced-integration, linear isoparametric elements such as the CAX4R element, the incompatible mode elements have excellent bending properties even with one layer of elements through the thickness (see ``Geometrically nonlinear analysis of a cantilever beam,'' Section 2.1.2 of the ABAQUS Benchmarks Manual) and have no hourglassing problems. However, they are

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computationally more expensive. deepdrawcup_s4.inp S4 model. deepdrawcup_s4r.inp S4R model. deepdrawcup_sax1.inp SAX1 model. deepdrawcup_postoutput.inp *POST OUTPUT analysis of deepdrawcup_sax1.inp. deepdrawcup_max1.inp MAX1 model. deepdrawcup_mgax1.inp MGAX1 model.

Reference · Wang, N. M., and S. C. Tang, "Analysis of Bending Effects in Sheet Forming Operations," International Journal for Numerical Methods in Engineering, vol. 25, pp. 253-267, January 1988.

Figures Figure 1.3.5-1 A typical drawbead used to limit slip between the blank and die.

Figure 1.3.5-2 Geometry and mesh for the deep drawing problem.

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Figure 1.3.5-3 Deformed shapes at various stages of the analysis.

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Figure 1.3.5-4 Strain distribution at the end of the deep drawing step.

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Figure 1.3.5-5 Punch force versus punch displacement.

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Figure 1.3.5-6 Deformed shape after unloading.

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Sample listings

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Listing 1.3.5-1 *HEADING DEEP DRAWING OF CYLINDRICAL CUP WITH CAX4R *RESTART,WRITE,FREQUENCY=25 *NODE 101, 181,0.1 301,0.0,0.00082 381,0.1,0.00082 *NGEN,NSET=BOT 101,181,2 *NGEN,NSET=TOP 301,381,2 *NSET,NSET=WRKPC BOT,TOP *NODE,NSET=DIE 100,0.1,-0.05 *NODE,NSET=PUNCH 200,0.,.05 *NODE,NSET=HOLDER 300,0.1,0.05 *NSET,NSET=TOOLS PUNCH,DIE,HOLDER *NSET,NSET=CENTER 101,301 *ELEMENT,TYPE=CAX4R,ELSET=BLANK 201,101,103,303,301 *ELGEN,ELSET=BLANK 201,40,2,2 *ELSET,ELSET=ALL BLANK, *SOLID SECTION,MATERIAL=STEEL,ORIENTATION=LOCAL, ELSET=BLANK *ORIENTATION,NAME=LOCAL 1.,0.,0.,0.,1.,0. 0,0., *MATERIAL,NAME=STEEL *ELASTIC 2.1E11,0.3 *PLASTIC,HARDENING=ISOTROPIC 0.91294E+08, 0.00000E+00 0.10129E+09, 0.21052E-03

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0.11129E+09, 0.52686E-03 0.12129E+09, 0.97685E-03 0.13129E+09, 0.15923E-02 0.14129E+09, 0.24090E-02 0.15129E+09, 0.34674E-02 0.16129E+09, 0.48120E-02 0.17129E+09, 0.64921E-02 0.18129E+09, 0.85618E-02 0.19129E+09, 0.11080E-01 0.20129E+09, 0.14110E-01 0.21129E+09, 0.17723E-01 0.22129E+09, 0.21991E-01 0.23129E+09, 0.26994E-01 0.24129E+09, 0.32819E-01 0.25129E+09, 0.39556E-01 0.26129E+09, 0.47301E-01 0.27129E+09, 0.56159E-01 0.28129E+09, 0.66236E-01 0.29129E+09, 0.77648E-01 0.30129E+09, 0.90516E-01 0.31129E+09, 0.10497E+00 0.32129E+09, 0.12114E+00 0.33129E+09, 0.13916E+00 0.34129E+09, 0.15919E+00 0.35129E+09, 0.18138E+00 0.36129E+09, 0.20588E+00 0.37129E+09, 0.23287E+00 0.38129E+09, 0.26252E+00 0.39129E+09, 0.29502E+00 0.40129E+09, 0.33054E+00 0.41129E+09, 0.36929E+00 0.42129E+09, 0.41147E+00 0.43129E+09, 0.45729E+00 0.44129E+09, 0.50696E+00 0.45129E+09, 0.56073E+00 0.46129E+09, 0.61881E+00 0.47129E+09, 0.68145E+00 0.48129E+09, 0.74890E+00 0.49129E+09, 0.82142E+00 0.50129E+09, 0.89928E+00 0.51129E+09, 0.98274E+00 0.52129E+09, 0.10721E+01 *ELSET,ELSET=EDIE,GENERATE

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231,279,2 *ELSET,ELSET=EHOLDER,GENERATE 241,279,2 *ELSET,ELSET=EPUNCH,GENERATE 201,249,2 *RIGID BODY,ANALYTICAL SURFACE=BSURF,REF NODE=100 *SURFACE,TYPE=SEGMENTS,NAME=BSURF START,0.05125,-0.060 LINE,0.05125,-0.005 CIRCL,0.05625,0.0,0.05625,-0.005 LINE,0.1,0.0 *RIGID BODY,ANALYTICAL SURFACE=DSURF,REF NODE=300 *SURFACE,TYPE=SEGMENTS,NAME=DSURF START,0.1,0.00082 LINE,0.05630,0.00082 CIRCL,0.05625,.00087,.05630,.00087 LINE,0.05625,.06 *RIGID BODY,ANALYTICAL SURFACE=FSURF,REF NODE=200 *SURFACE,TYPE=SEGMENTS,FILLET RADIUS=.013, NAME=FSURF START,0.05,0.060 LINE,0.05,2.250782E-3 CIRCL,0.0,0.001,0.0,1.001 LINE,-0.001,0.001 *SURFACE,NAME=ASURF EDIE,S1 *SURFACE,NAME=CSURF EHOLDER,S3 *SURFACE,NAME=ESURF EPUNCH,S3 *CONTACT PAIR,INTERACTION=ROUGH1 ASURF,BSURF *CONTACT PAIR,INTERACTION=ROUGH2 CSURF,DSURF *CONTACT PAIR,INTERACTION=ROUGH3 ESURF,FSURF *SURFACE INTERACTION,NAME=ROUGH1 *FRICTION 0.1, *SURFACE INTERACTION,NAME=ROUGH2 *FRICTION 0.1, *SURFACE INTERACTION,NAME=ROUGH3

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*FRICTION 0.25, *STEP,INC=10,NLGEOM, UNSYMM=YES PUSH THE BLANKHOLDER DOWN BY A PRESCRIBED DISPLACEMENT *STATIC 1.,1. *BOUNDARY CENTER,1,1 DIE,1,1 DIE,2,2 DIE,6,6 PUNCH,1,1 PUNCH,2,2 PUNCH,6,6 HOLDER,1,1 HOLDER,2,2,-1.75E-8 HOLDER,6,6 *MONITOR,NODE=200,DOF=2 *CONTACT CONTROLS,FRICTION ONSET=DELAY *PRINT,CONTACT=YES *NODE PRINT,NSET=TOOLS,FREQUENCY=100 COORD,U,RF *EL PRINT,ELSET=ALL,FREQUENCY=500 S,E *NODE FILE,NSET=TOOLS,FREQUENCY=10 U,RF *CONTACT FILE,SLAVE=ASURF,FREQUENCY=10 *CONTACT FILE,SLAVE=CSURF,FREQUENCY=10 *CONTACT FILE,SLAVE=ESURF,FREQUENCY=10 *END STEP *STEP,INC=10,NLGEOM APPLY PRESCRIBED FORCE ON BLANKHOLDER AND RELEASE DISPLACEMENT *STATIC 1.,1. *BOUNDARY,OP=NEW CENTER,1,1 DIE,1,1 DIE,2,2 DIE,6,6 PUNCH,1,1 PUNCH,2,2

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PUNCH,6,6 HOLDER,1,1 HOLDER,6,6 *CLOAD HOLDER,2,-100000. *END STEP *STEP,INC=500,NLGEOM MOVE THE PUNCH DOWN *STATIC .01,1.,1.E-6 *CONTROLS,ANALYSIS=DISCONTINUOUS *BOUNDARY,OP=NEW CENTER,1,1 DIE,1,1 DIE,2,2 DIE,6,6 PUNCH,1,1 PUNCH,2,2,-.06 PUNCH,6,6 HOLDER,1,1 HOLDER,6,6 *CLOAD HOLDER,2,-100000. *END STEP *STEP,INC=100,NLGEOM FIX ALL NODES AND REMOVE THE CONTACT SURFACES *STATIC 1.,1.,1.,1. *BOUNDARY,FIXED WRKPC,1,2 *MODEL CHANGE,TYPE=CONTACT PAIR,REMOVE ASURF,BSURF CSURF,DSURF ESURF,FSURF *CLOAD,OP=NEW HOLDER,2,0. *END STEP *STEP,INC=50,NLGEOM, UNSYMM=NO REPLACE THE BOUNDARY CONDITIONS BY THE REGULAR SET *STATIC .1,1.,1.E-6 *BOUNDARY,OP=NEW 181,2

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CENTER,1,1 *END STEP

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Listing 1.3.5-2 *HEADING DEEP DRAWING OF CYLINDRICAL CUP WITH SAX1 *RESTART,WRITE,FREQUENCY=25 *NODE 101, 181,0.1 *NGEN,NSET=BOT 101,181,2 *NSET,NSET=WRKPC,GENERATE 121,181,2 *NODE,NSET=DIE 100,0.,-0.05 *NODE,NSET=PUNCH 200,0.,.05 *NODE,NSET=HOLDER 300,0.,0.05 *NSET,NSET=TOOLS PUNCH,DIE,HOLDER *NSET,NSET=CENTER 101, *ELEMENT,TYPE=SAX1,ELSET=BLANK 201,101,121 202,121,123 *ELGEN,ELSET=BLANK 202,30,2,2 *SHELL SECTION,MATERIAL=STEEL,ELSET=BLANK .00082,5 *MATERIAL,NAME=STEEL *ELASTIC 2.1E11,0.3 *PLASTIC,HARDENING=ISOTROPIC 0.91294E+08, 0.00000E+00 0.10129E+09, 0.21052E-03 0.11129E+09, 0.52686E-03 0.12129E+09, 0.97685E-03 0.13129E+09, 0.15923E-02 0.14129E+09, 0.24090E-02 0.15129E+09, 0.34674E-02 0.16129E+09, 0.48120E-02 0.17129E+09, 0.64921E-02 0.18129E+09, 0.85618E-02

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0.19129E+09, 0.11080E-01 0.20129E+09, 0.14110E-01 0.21129E+09, 0.17723E-01 0.22129E+09, 0.21991E-01 0.23129E+09, 0.26994E-01 0.24129E+09, 0.32819E-01 0.25129E+09, 0.39556E-01 0.26129E+09, 0.47301E-01 0.27129E+09, 0.56159E-01 0.28129E+09, 0.66236E-01 0.29129E+09, 0.77648E-01 0.30129E+09, 0.90516E-01 0.31129E+09, 0.10497E+00 0.32129E+09, 0.12114E+00 0.33129E+09, 0.13916E+00 0.34129E+09, 0.15919E+00 0.35129E+09, 0.18138E+00 0.36129E+09, 0.20588E+00 0.37129E+09, 0.23287E+00 0.38129E+09, 0.26252E+00 0.39129E+09, 0.29502E+00 0.40129E+09, 0.33054E+00 0.41129E+09, 0.36929E+00 0.42129E+09, 0.41147E+00 0.43129E+09, 0.45729E+00 0.44129E+09, 0.50696E+00 0.45129E+09, 0.56073E+00 0.46129E+09, 0.61881E+00 0.47129E+09, 0.68145E+00 0.48129E+09, 0.74890E+00 0.49129E+09, 0.82142E+00 0.50129E+09, 0.89928E+00 0.51129E+09, 0.98274E+00 0.52129E+09, 0.10721E+01 *RIGID BODY,ANALYTICAL SURFACE=HOLDER,REF NODE=300 *SURFACE,TYPE=SEGMENTS,NAME=HOLDER, FILLET RADIUS=0.001 START,0.12,1.75E-8 LINE,0.05625,1.75E-8 LINE,0.05625,.06 *RIGID BODY,ANALYTICAL SURFACE=DIE,REF NODE=100 *SURFACE,TYPE=SEGMENTS,NAME=DIE, FILLET RADIUS=0.00541

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START,0.05125,-0.060 LINE,0.05125,0.0 LINE,0.12,0.0 *RIGID BODY,ANALYTICAL SURFACE=PUNCH,REF NODE=200 *SURFACE,TYPE=SEGMENTS,NAME=PUNCH, FILLET RADIUS=.01341 START,0.05,0.060 LINE,0.05,2.250782E-3 CIRCL,0.0,.0001,0.0,1.0001 LINE,-0.001,0.0001 ** ** Contact with holder ** *ELSET,ELSET=HOLD_CON,GENERATE 222,260,2 *SURFACE,NAME=HLD_SURF HOLD_CON,SPOS *CONTACT PAIR,INTERACTION=FRIC1 HLD_SURF,HOLDER *SURFACE INTERACTION,NAME=FRIC1 *SURFACE BEHAVIOR, PRESSURE-OVERCLOSURE=EXPONENTIAL 1.75E-8,4.5E6 *FRICTION 0.0, ** ** Contact with die ** *ELSET,ELSET=DIE_CON,GENERATE 212,260,2 *SURFACE,NAME=DIE_SURF DIE_CON,SNEG *CONTACT PAIR,INTERACTION=FRIC2 DIE_SURF,DIE *SURFACE INTERACTION,NAME=FRIC2 *FRICTION 0.0, ** ** Contact with punch ** *ELSET,ELSET=PUN_CON2,GENERATE 202,230,2 *SURFACE,NAME=PCH_SURF

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PUN_CON2,SPOS *CONTACT PAIR,INTERACTION=FRIC3 PCH_SURF,PUNCH *SURFACE INTERACTION,NAME=FRIC3 *FRICTION 0.0, *SURFACE,TYPE=NODE,NAME=PCH_PNT 101, *CONTACT PAIR,INTERACTION=FRIC4 PCH_PNT,PUNCH *SURFACE INTERACTION,NAME=FRIC4 *FRICTION 0.0, ** ** *STEP,INC=10, UNSYMM=YES PUSH THE BLANKHOLDER DOWN BY A PRESCRIBED DISPLACEMENT *STATIC 1.,1. *BOUNDARY CENTER,1,1 CENTER,6,6 DIE,1,1 DIE,2,2 DIE,6,6 PUNCH,1,1 PUNCH,2,2 PUNCH,6,6 HOLDER,1,1 HOLDER,2,2,-1.75E-8 HOLDER,6,6 *MONITOR,NODE=200,DOF=2 *PRINT,CONTACT=YES *NODE PRINT,NSET=TOOLS,FREQUENCY=100 COORD,U,RF *EL PRINT,ELSET=BLANK,FREQUENCY=500 S,E STH, *CONTACT PRINT,FREQUENCY=500 *CONTACT FILE,FREQUENCY=500 *NODE FILE,NSET=TOOLS,FREQUENCY=10 U,RF

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Static Stress/Displacement Analyses

*END STEP *STEP,INC=10,NLGEOM, UNSYMM=YES APPLY PRESCRIBED FORCE ON BLANKHOLDER AND RELEASE DISPLACEMENT *STATIC 1.,1. *BOUNDARY,OP=NEW CENTER,1,1 CENTER,6,6 DIE,1,1 DIE,2,2 DIE,6,6 PUNCH,1,1 PUNCH,2,2 PUNCH,6,6 HOLDER,1,1 HOLDER,6,6 *CLOAD HOLDER,2,-100000. *CHANGE FRICTION,INTERACTION=FRIC2 *FRICTION 0.1, *CHANGE FRICTION,INTERACTION=FRIC1 *FRICTION 0.1, *CHANGE FRICTION,INTERACTION=FRIC3 *FRICTION 0.25, *END STEP *STEP,INC=500,NLGEOM,UNSYMM=YES MOVE THE PUNCH DOWN *STATIC .01,1.,1.E-6 *CONTROLS,ANALYSIS=DISCONTINUOUS *BOUNDARY,OP=NEW CENTER,1,1 CENTER,6,6 DIE,1,1 DIE,2,2 DIE,6,6 PUNCH,1,1 PUNCH,2,2,-.06 PUNCH,6,6

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Static Stress/Displacement Analyses

HOLDER,1,1 HOLDER,6,6 *CLOAD HOLDER,2,-100000. *END STEP *STEP,INC=1,NLGEOM, UNSYMM=YES FIX ALL NODES AND REMOVE THE CONTACT SURFACES *STATIC 1.,1.,1.,1. *BOUNDARY,FIXED WRKPC,1,6 *BOUNDARY,FIXED CENTER,1,6 *MODEL CHANGE,TYPE=CONTACT PAIR,REMOVE HLD_SURF,HOLDER PCH_PNT,PUNCH PCH_SURF,PUNCH DIE_SURF,DIE *CLOAD,OP=NEW HOLDER,2,0. *END STEP *STEP,INC=50,NLGEOM, UNSYMM=YES REPLACE THE BOUNDARY CONDITIONS BY THE REGULAR SET *STATIC .1,1.,1.E-6 *BOUNDARY,OP=NEW CENTER,1,1 CENTER,6,6 181,2,2 *END STEP

1.3.6 Extrusion of a cylindrical metal bar with frictional heat generation Products: ABAQUS/Standard ABAQUS/Explicit This analysis illustrates how extrusion problems can be simulated with ABAQUS. In this particular problem the radius of an aluminum cylindrical bar is reduced 33% by an extrusion process. The generation of heat due to plastic dissipation inside the bar and the frictional heat generation at the workpiece/die interface are considered.

Geometry and model The bar has an initial radius of 100 mm and is 300 mm long. Figure 1.3.6-1 shows half of the cross-section of the bar, modeled with first-order axisymmetric elements ( CAX4T elements in

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ABAQUS/Standard and CAX4RT elements in ABAQUS/Explicit). The die is assumed to be rigid. In ABAQUS/Standard the die is modeled with CAX4T elements, which are made into an isothermal rigid body with the *RIGID BODY, ISOTHERMAL option. The *SURFACE option is used to define the slave surface on the outside of the bar and the master surface on the inside of the die. To model a die that has no sharp corners and is smooth in the transition region, the SMOOTH parameter on the *CONTACT PAIR option is set to 0.48. In ABAQUS/Explicit the die is modeled with either an analytical rigid surface or discrete rigid elements (RAX2). The analytical rigid surface is defined using the *RIGID BODY option in conjunction with the *SURFACE option. The FILLET RADIUS parameter on the *SURFACE option is set to 0.075 to remove sharp corners in the transition region of the die. For simplicity we do not model any heat transfer in the die--we simply fix the temperature of the rigid body reference node and assume that no heat is transmitted between the bar and the die. Half the heat dissipated as a result of friction is assumed to be conducted into the workpiece; the other half is conducted into the die. 90% of the nonrecoverable work because of plasticity is assumed to heat the work material. More realistic analysis would include thermal modeling of the die. The ABAQUS/Explicit simulations are performed both with and without adaptive meshing.

Material model and interface behavior The material model is chosen to reflect the response of a typical commercial purity aluminum alloy. The material is assumed to harden isotropically. The dependence of the flow stress on the temperature is included, but strain rate dependence is ignored. Instead, representative material data at a strain rate of 0.1 sec-1 are selected to characterize the flow strength. The interface is assumed to have no conductive properties. Coulomb friction is assumed for the mechanical behavior, with a friction coefficient of 0.1. The *GAP HEAT GENERATION option is used to specify the fraction, fg , of total heat generated by frictional dissipation that is transferred to the two bodies in contact. Half of this heat is conducted into the workpiece, and the other half is conducted into the die.

Boundary conditions, loading, and solution control In the first step the bar is moved to a position where contact is established and slipping of the workpiece against the die begins. In the second step the bar is extruded through the die to realize the extrusion process. This is accomplished by prescribing displacements to the nodes at the top of the bar. In the third step the contact elements are removed in preparation for the cool down portion of the simulation. In ABAQUS/Standard this is performed in a single step: the bar is allowed to cool down using film conditions, and deformation is driven by thermal contraction during the fourth step. In ABAQUS/Explicit the cool down simulation is broken into two steps: the first introduces viscous pressure to damp out dynamic effects and, thus, allow the bar to reach static equilibrium quickly; the balance of the cool down simulation is performed in a fifth step. The relief of residual stresses through creep is not analyzed in this example. In ABAQUS/Explicit mass scaling is used to reduce the computational cost of the analysis; nondefault

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hourglass control is used to control the hourglassing in the model. The default integral viscoelastic approach to hourglass control generally works best for problems where sudden dynamic loading occurs; a stiffness-based hourglass control is recommended for problems where the response is quasi-static. A combination of stiffness and viscous hourglass control is used in this problem. For purposes of comparison a second problem is also analyzed, in which the first two steps of the previous analysis are repeated in a static analysis with the adiabatic heat generation capability. The adiabatic analysis neglects heat conduction in the bar. Frictional heat generation must also be ignored in this case. This problem is analyzed only in ABAQUS/Standard.

Results and discussion The following discussion centers around the results obtained with ABAQUS/Standard. The results of the ABAQUS/Explicit simulation are in close agreement with those obtained with ABAQUS/Standard. Figure 1.3.6-2 shows the deformed configuration after Step 2 of the analysis. Figure 1.3.6-3 and Figure 1.3.6-4 show contour plots of plastic strain and temperature at the end of Step 2 for the fully coupled analysis. The plastic deformation is most severe near the surface of the workpiece, where plastic strains exceed 100%. The peak temperature also occurs at the surface of the workpiece because of plastic deformation and frictional heating. The peak temperature occurs immediately after the radial reduction zone of the die. This is expected for two reasons. First, the material that is heated by dissipative processes in the reduction zone will cool by conduction as the material progresses through the postreduction zone. Second, frictional heating is largest in the reduction zone because of the larger values of shear stress in that zone. The peak surface temperature is approximately 106°C (i.e., ¢T ¼ 86°C). If we ignore the zone of extreme distortion at the end of the bar, the temperature increase on the surface is not as large for the adiabatic analysis (Figure 1.3.6-5) because of the absence of frictional heating. The surface temperatures in this analysis are approximately 80°C. As expected, the temperature field contours for the adiabatic heating analysis, Figure 1.3.6-5, are very similar to the contours of plastic strain, Figure 1.3.6-3, from the thermally coupled analysis. As noted earlier, excellent agreement is observed between the results obtained with ABAQUS/Explicit and ABAQUS/Standard. Figure 1.3.6-6 compares the effects of adaptive meshing on the element quality. The results obtained with adaptive meshing show significantly reduced mesh distortion. The material point in the bar that experiences the largest temperature rise during the course of the simulation is indicated (node 2029 in the model without adaptivity). Figure 1.3.6-7 compares the results obtained with ABAQUS/Explicit for the temperature history of this material point against the same results obtained with ABAQUS/Standard.

Input files ABAQUS/Standard input files metalbarextrusion_coupled_fric.inp Thermally coupled extrusion with frictional heat generation.

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metalbarextrusion_adiab.inp Extrusion with adiabatic heat generation and without frictional heat generation. metalbarextrusion_stabil.inp Thermally coupled extrusion with frictional heat generation and automatic stabilization. ABAQUS/Explicit input files metalbarextrusion_x_cax4rt.inp Thermally coupled extrusion with frictional heat generation and without adaptive meshing; die modeled with an analytical rigid surface; kinematic mechanical contact. metalbarextrusion_xad_cax4rt.inp Thermally coupled extrusion with frictional heat generation and adaptive meshing; die modeled with an analytical rigid surface; kinematic mechanical contact. metalbarextrusion_xd_cax4rt.inp Thermally coupled extrusion with frictional heat generation and without adaptive meshing; die modeled with RAX2 elements; kinematic mechanical contact. metalbarextrusion_xp_cax4rt.inp Thermally coupled extrusion with frictional heat generation and without adaptive meshing; die modeled with an analytical rigid surface; penalty mechanical contact.

Figures Figure 1.3.6-1 Mesh and geometry: axisymmetric extrusion, ABAQUS/Standard.

Figure 1.3.6-2 Deformed configuration, Step 2, ABAQUS/Standard.

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Figure 1.3.6-3 Plastic strain contours, Step 2, thermally coupled analysis (frictional heat generation), ABAQUS/Standard.

Figure 1.3.6-4 Temperature contours, Step 2, thermally coupled analysis (frictional heat generation),

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ABAQUS/Standard.

Figure 1.3.6-5 Temperature contours, Step 2, adiabatic heat generation (without heat generation due to friction), ABAQUS/Standard.

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Figure 1.3.6-6 Deformed shape of the workpiece: without adaptive remeshing, left; with adaptive remeshing, right; ABAQUS/Explicit.

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Figure 1.3.6-7 Temperature history of node 2029 (nonadaptive result), ABAQUS/Explicit.

Sample listings

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Static Stress/Displacement Analyses

Listing 1.3.6-1 *HEADING Extrusion *RESTART,WRITE,FREQUENCY=50 *NODE 1,0.,0. 61,0.,.3 2001,.1,0. 2061,.1,.3 *NODE, NSET=NREF 99999, 0.5 , 0.0 *NGEN,NSET=AXIS 1,61,1 *NGEN,NSET=OUTSIDE 2001,2061,1 *NFILL,NSET=ALL AXIS,OUTSIDE,20,100 *ELEMENT,TYPE=CAX4T,ELSET=WORK 1,1,201,203,3 *ELGEN,ELSET=WORK 1,30,2,1,10,200,100 *ELSET,ELSET=BOT,GENERATE 1,901,100 *ELSET,ELSET=SIDE,GENERATE 901,930,1 *ELSET,ELSET=TOP,GENERATE 30,930,100 ** *** Node & element definitions for die ** *NODE, NSET=CONTACT 10001, 0.250000000, -0.180000000 10002, 0.250000000, -0.114444000 10003, 0.250000000, -0.048888900 10004, 0.250000000, 0.016666700 10005, 0.250000000, 0.082222200 10006, 0.250000000, 0.147778000 10007, 0.250000000, 0.213333000 10008, 0.250000000, 0.278889000 10009, 0.250000000, 0.344444000 10010, 0.250000000, 0.410000000 10011, 0.188867000, -0.180000000

1-465

Static Stress/Displacement Analyses

10012, 0.189030000, -0.106122000 10013, 0.199098000, -0.041751700 10014, 0.199227000, 0.022784300 10015, 0.199356000, 0.087320200 10016, 0.199484000, 0.151856000 10017, 0.199613000, 0.216392000 10018, 0.199742000, 0.280928000 10019, 0.199871000, 0.345464000 10020, 0.200000000, 0.410000000 10021, 0.127733000, -0.180000000 10022, 0.127815000, -0.103061000 10023, 0.149548000, -0.045875800 10024, 0.149613000, 0.012106400 10025, 0.149677000, 0.070088700 10026, 0.149742000, 0.128071000 10027, 0.149806000, 0.186053000 10028, 0.149871000, 0.244035000 10029, 0.149935000, 0.302018000 10030, 0.149999000, 0.360000000 10031, 0.066600000, -0.180000000 10032, 0.066600000, -0.100000000 10033, 0.099999000, -0.050000000 10034, 0.099999000, 0.001428570 10035, 0.099999000, 0.052857100 10036, 0.099999000, 0.104286000 10037, 0.099999000, 0.155714000 10038, 0.099999000, 0.207143000 10039, 0.099999000, 0.258571000 10040, 0.099999000, 0.310000000 *ELEMENT, TYPE=CAX4T, ELSET=CONTACT 10001, 10001, 10002, 10012, 10011 10002, 10002, 10003, 10013, 10012 10003, 10003, 10004, 10014, 10013 10004, 10004, 10005, 10015, 10014 10005, 10005, 10006, 10016, 10015 10006, 10006, 10007, 10017, 10016 10007, 10007, 10008, 10018, 10017 10008, 10008, 10009, 10019, 10018 10009, 10009, 10010, 10020, 10019 10010, 10011, 10012, 10022, 10021 10011, 10012, 10013, 10023, 10022 10012, 10013, 10014, 10024, 10023 10013, 10014, 10015, 10025, 10024

1-466

Static Stress/Displacement Analyses

10014, 10015, 10016, 10026, 10025 10015, 10016, 10017, 10027, 10026 10016, 10017, 10018, 10028, 10027 10017, 10018, 10019, 10029, 10028 10018, 10019, 10020, 10030, 10029 10019, 10021, 10022, 10032, 10031 10020, 10022, 10023, 10033, 10032 10021, 10023, 10024, 10034, 10033 10022, 10024, 10025, 10035, 10034 10023, 10025, 10026, 10036, 10035 10024, 10026, 10027, 10037, 10036 10025, 10027, 10028, 10038, 10037 10026, 10028, 10029, 10039, 10038 10027, 10029, 10030, 10040, 10039 *SOLID SECTION, ELSET=CONTACT, MATERIAL=RIG *RIGID BODY, ELSET=CONTACT, ISOTHERMAL=YES, REF NODE=99999 *SOLID SECTION,ELSET=WORK,MATERIAL=METAL *MATERIAL,NAME=METAL *ELASTIC 6.9E10,.33 *PLASTIC ** STRAIN RATE APPX .1 ** 60.E6,0.0 ,20. 90.E6,.125 ,20. 113.E6,.25 ,20. 124.E6,.375 ,20. 133.E6,0.5 ,20. 165.E6,1.0 ,20. 166.E6,2.0 ,20. 60.E6,0. ,50. 80.E6,.125 ,50. 97.E6,.25 ,50. 110.E6,.375 ,50. 120.E6,0.5 ,50. 150.E6,1.0 ,50. 151.E6,2.0 ,50. 50.E6,0.0 ,100. 65.E6,.125 ,100, 81.5E6,.25 ,100. 91.E6,.375 ,100. 100.E6,0.5 ,100. 125.E6,1.0 ,100.

1-467

Static Stress/Displacement Analyses

126.E6,2.0 ,100. 45.E6,0.0 ,150. 63.E6,.125 ,150. 75.E6,.25 ,150. 89.E6,.5 ,150. 110.E6,1. ,150. 111.E6,2. ,150. *SPECIFIC HEAT 880., *DENSITY 2700., *CONDUCTIVITY 204.,0. 225.,300. *EXPANSION,ZERO=20.0 8.42E-5, *INELASTIC HEAT FRACTION .9, ** *** material properties are inconsequential *** for rigid elements *** *MATERIAL, NAME=RIG *ELASTIC 1.0E10,0.3 *SPECIFIC HEAT 880., *DENSITY 2700., *CONDUCTIVITY 204.,0. 225.,300. *EXPANSION,ZERO=20.0 8.42E-5, ** ** ** *NSET,NSET=TOP,GENERATE 61,2061,100 *NSET,NSET=ALL 1,2061,1 *INITIAL CONDITIONS,TYPE=TEMPERATURE ALL,20.

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Static Stress/Displacement Analyses

** *** surface definitions *** ** *ELSET, ELSET=INDIE, GEN 10019,10027,1 *ELSET, ELSET=BOTDIE, GEN 10001,10019,9 *SURFACE, NAME=RIGID, TYPE=ELEMENT INDIE, S3 BOTDIE,S4 *SURFACE, NAME=DEF1, TYPE=ELEMENT SIDE, S2 *SURFACE, NAME=DEF2, TYPE=ELEMENT BOT, S1 ** *** Interaction definitions ** *SURFACE INTERACTION, NAME=INTER *FRICTION 0.1 *GAP HEAT GENERATION 1.0, ** *** Contact pair definitions ** *CONTACT PAIR, INTERACTION=INTER, SMOOTH=0.48 DEF1, RIGID DEF2, RIGID ** *** elset for output purposes ** *ELSET,ELSET=EFILEOUT BOT,SIDE,TOP ** *** step 1 ** *STEP,INC=100,AMPLITUDE=RAMP,NLGEOM, UNSYMM=YES STABILIZE WORKPIECE INSIDE DIE *COUPLED TEMPERATURE-DISPLACEMENT,DELTMX=100. .1,1. *BOUNDARY NREF, 1, 2, 0.0 NREF, 6, 6, 0.0

1-469

Static Stress/Displacement Analyses

NREF,11,11,20.0 AXIS,1,1,0.0 TOP,2,2,-.000125 ALL,11,11,20.0 **2061,1,1,0.0 *PRINT,CONTACT=YES *NODE PRINT,FREQUENCY=10 U RF NT RFL *ENERGY PRINT,FREQUENCY=1 *END STEP ** *** step 2 ** *STEP,INC=800,AMPLITUDE=RAMP,NLGEOM, UNSYMM=YES EXTRUSION *COUPLED TEMPERATURE-DISPLACEMENT,DELTMX=100. .1,10. *BOUNDARY,OP=NEW NREF, 1, 2, 0.0 NREF, 6, 6, 0.0 NREF, 11,11,20.0 AXIS,1,1,0.0 TOP,2,2,-.25 **2061,1,1,0.0 *PRINT,CONTACT=YES *EL PRINT,ELSET=CONTACT,FREQUENCY=10 S E *EL FILE,FREQUENCY=999,ELSET=EFILEOUT S,E PE, *NSET,NSET=NFILEOUT AXIS,OUTSIDE,TOP *NODE FILE,FREQUENCY=999,NSET=NFILEOUT NT, *NODE PRINT,FREQUENCY=10 U RF NT RFL

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Static Stress/Displacement Analyses

*ENERGY PRINT,FREQUENCY=1 *END STEP ** *** step 3 ** *STEP,INC=200,AMPLITUDE=RAMP,NLGEOM, UNSYMM=YES REMOVE CONTACT PAIRS *COUPLED TEMPERATURE-DISPLACEMENT,DELTMX=100. .1,.1, *MODEL CHANGE,REMOVE, TYPE=CONTACT PAIR DEF1, RIGID DEF2, RIGID *PRINT,CONTACT=YES *NODE PRINT,FREQUENCY=10 U RF NT RFL *EL FILE,FREQUENCY=0 *NODE FILE,FREQUENCY=0 *ENERGY PRINT,FREQUENCY=1 *END STEP ** *** step 4 ** *STEP,INC=200,AMPLITUDE=RAMP,NLGEOM, UNSYMM=YES LET WORKPIECE COOL DOWN *COUPLED TEMPERATURE-DISPLACEMENT,DELTMX=100. 100.,10000., *FILM BOT,F1,20.,10. TOP,F3,20.,10. SIDE,F2,20.,10. *PRINT,CONTACT=YES *NODE PRINT,FREQUENCY=10 U RF NT RFL *ENERGY PRINT,FREQUENCY=1 *EL FILE,FREQUENCY=999,ELSET=EFILEOUT S,E PE,

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Static Stress/Displacement Analyses

*NODE FILE,FREQUENCY=999,NSET=NFILEOUT NT, *END STEP

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Static Stress/Displacement Analyses

Listing 1.3.6-2 *HEADING Extrusion - adiabatic analysis *RESTART,WRITE,FREQUENCY=50 *NODE 1,0.,0. 61,0.,.3 2001,.1,0. 2061,.1,.3 *NODE, NSET=NREF 99999, 0.5, 0.0 ** *NGEN,NSET=AXIS 1,61,1 *NGEN,NSET=OUTSIDE 2001,2061,1 *NFILL,NSET=ALL AXIS,OUTSIDE,20,100 *ELEMENT,TYPE=CAX4,ELSET=WORK 1,1,201,203,3 *ELGEN,ELSET=WORK 1,30,2,1,10,200,100 *ELSET,ELSET=BOT,GENERATE 1,901,100 *ELSET,ELSET=SIDE,GENERATE 901,930,1 *ELSET,ELSET=TOP,GENERATE 30,930,100 ** *** Node & element definitions for die ** *NODE, NSET=CONTACT 10001, 0.250000000, -0.180000000 10002, 0.250000000, -0.114444000 10003, 0.250000000, -0.048888900 10004, 0.250000000, 0.016666700 10005, 0.250000000, 0.082222200 10006, 0.250000000, 0.147778000 10007, 0.250000000, 0.213333000 10008, 0.250000000, 0.278889000 10009, 0.250000000, 0.344444000 10010, 0.250000000, 0.410000000

1-473

Static Stress/Displacement Analyses

10011, 0.188867000, -0.180000000 10012, 0.189030000, -0.106122000 10013, 0.199098000, -0.041751700 10014, 0.199227000, 0.022784300 10015, 0.199356000, 0.087320200 10016, 0.199484000, 0.151856000 10017, 0.199613000, 0.216392000 10018, 0.199742000, 0.280928000 10019, 0.199871000, 0.345464000 10020, 0.200000000, 0.410000000 10021, 0.127733000, -0.180000000 10022, 0.127815000, -0.103061000 10023, 0.149548000, -0.045875800 10024, 0.149613000, 0.012106400 10025, 0.149677000, 0.070088700 10026, 0.149742000, 0.128071000 10027, 0.149806000, 0.186053000 10028, 0.149871000, 0.244035000 10029, 0.149935000, 0.302018000 10030, 0.149999000, 0.360000000 10031, 0.066600000, -0.180000000 10032, 0.066600000, -0.100000000 10033, 0.099999000, -0.050000000 10034, 0.099999000, 0.001428570 10035, 0.099999000, 0.052857100 10036, 0.099999000, 0.104286000 10037, 0.099999000, 0.155714000 10038, 0.099999000, 0.207143000 10039, 0.099999000, 0.258571000 10040, 0.099999000, 0.310000000 *ELEMENT, TYPE=CAX4, ELSET=CONTACT 10001, 10001, 10002, 10012, 10011 10002, 10002, 10003, 10013, 10012 10003, 10003, 10004, 10014, 10013 10004, 10004, 10005, 10015, 10014 10005, 10005, 10006, 10016, 10015 10006, 10006, 10007, 10017, 10016 10007, 10007, 10008, 10018, 10017 10008, 10008, 10009, 10019, 10018 10009, 10009, 10010, 10020, 10019 10010, 10011, 10012, 10022, 10021 10011, 10012, 10013, 10023, 10022 10012, 10013, 10014, 10024, 10023

1-474

Static Stress/Displacement Analyses

10013, 10014, 10015, 10025, 10024 10014, 10015, 10016, 10026, 10025 10015, 10016, 10017, 10027, 10026 10016, 10017, 10018, 10028, 10027 10017, 10018, 10019, 10029, 10028 10018, 10019, 10020, 10030, 10029 10019, 10021, 10022, 10032, 10031 10020, 10022, 10023, 10033, 10032 10021, 10023, 10024, 10034, 10033 10022, 10024, 10025, 10035, 10034 10023, 10025, 10026, 10036, 10035 10024, 10026, 10027, 10037, 10036 10025, 10027, 10028, 10038, 10037 10026, 10028, 10029, 10039, 10038 10027, 10029, 10030, 10040, 10039 *SOLID SECTION, ELSET=CONTACT, MATERIAL=RIG *RIGID BODY, ELSET=CONTACT, REF NODE=99999 *SOLID SECTION,ELSET=WORK,MATERIAL=METAL *MATERIAL,NAME=METAL *ELASTIC 6.9E10,.33 *PLASTIC ** STRAIN RATE APPX .1 ** 60.E6,0.0 ,20. 90.E6,.125 ,20. 113.E6,.25 ,20. 124.E6,.375 ,20. 133.E6,0.5 ,20. 165.E6,1.0 ,20. 166.E6,2.0 ,20. 60.E6,0. ,50. 80.E6,.125 ,50. 97.E6,.25 ,50. 110.E6,.375 ,50. 120.E6,0.5 ,50. 150.E6,1.0 ,50. 151.E6,2.0 ,50. 50.E6,0.0 ,100. 65.E6,.125 ,100, 81.5E6,.25 ,100. 91.E6,.375 ,100. 100.E6,0.5 ,100. 125.E6,1.0 ,100.

1-475

Static Stress/Displacement Analyses

126.E6,2.0 ,100. 45.E6,0.0 ,150. 63.E6,.125 ,150. 75.E6,.25 ,150. 89.E6,.5 ,150. 110.E6,1. ,150. 111.E6,2. ,150. *SPECIFIC HEAT 880., *DENSITY 2700., *CONDUCTIVITY 204.,0. 225.,300. *EXPANSION,ZERO=20.0 8.42E-5, *INELASTIC HEAT FRACTION .9, *NSET,NSET=TOP,GENERATE 61,2061,100 *NSET,NSET=ALL 1,2061,1 ** *** material properties are inconsequential *** for rigid elements *** *MATERIAL, NAME=RIG *ELASTIC 1.0E10,0.3 *PLASTIC ** STRAIN RATE APPX .1 ** 60.E6,0.0 ,20. 90.E6,.125 ,20. 113.E6,.25 ,20. 124.E6,.375 ,20. 133.E6,0.5 ,20. 165.E6,1.0 ,20. 166.E6,2.0 ,20. 60.E6,0. ,50. 80.E6,.125 ,50. 97.E6,.25 ,50. 110.E6,.375 ,50. 120.E6,0.5 ,50.

1-476

Static Stress/Displacement Analyses

150.E6,1.0 ,50. 151.E6,2.0 ,50. 50.E6,0.0 ,100. 65.E6,.125 ,100, 81.5E6,.25 ,100. 91.E6,.375 ,100. 100.E6,0.5 ,100. 125.E6,1.0 ,100. 126.E6,2.0 ,100. 45.E6,0.0 ,150. 63.E6,.125 ,150. 75.E6,.25 ,150. 89.E6,.5 ,150. 110.E6,1. ,150. 111.E6,2. ,150. *SPECIFIC HEAT 880., *DENSITY 2700., *CONDUCTIVITY 204.,0. 225.,300. *EXPANSION,ZERO=20.0 8.42E-5, ** *** surface definitions *** ** *ELSET, ELSET=INDIE, GEN 10019,10027,1 *ELSET, ELSET=BOTDIE, GEN 10001,10019,9 *SURFACE, NAME=RIGID, TYPE=ELEMENT INDIE, S3 BOTDIE,S4 *SURFACE, NAME=DEF1, TYPE=ELEMENT SIDE, S2 *SURFACE, NAME=DEF2, TYPE=ELEMENT BOT, S1 ** *** Interaction definitions ** *SURFACE INTERACTION, NAME=INTER *FRICTION

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Static Stress/Displacement Analyses

0.1 *GAP HEAT GENERATION 1.0, ** *** Contact pair definitions ** *CONTACT PAIR, INTERACTION=INTER, SMOOTH=0.48 DEF1, RIGID DEF2, RIGID *INITIAL CONDITIONS,TYPE=TEMPERATURE ALL,20. *NSET, NSET=NALL CONTACT,ALL ** ** step 1 ** *STEP,INC=100,AMPLITUDE=RAMP,NLGEOM, UNSYMM=YES STABILIZE WORKPIECE INSIDE DIE *STATIC,ADIABATIC .1,1. *BOUNDARY NREF,1,2,0.0 NREF,6,6,0.0 AXIS,1,1,0.0 TOP,2,2,-.000125 *PRINT,CONTACT=YES *EL PRINT,ELSET=CONTACT,FREQUENCY=10 S E MISES PE *NODE PRINT,FREQUENCY=10 U RF *ENERGY PRINT,FREQUENCY=1 *OUTPUT, FIELD, FREQ=10 *ELEMENT OUTPUT, ELSET=WORK TEMP S E PE PEEQ *NODE OUTPUT, NSET=ALL

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Static Stress/Displacement Analyses

U *END STEP ** *** step 2 ** *STEP,INC=800,AMPLITUDE=RAMP,NLGEOM, UNSYMM=YES EXTRUSION *STATIC,ADIABATIC .1,10. *BOUNDARY,OP=NEW NREF,1,2,0.0 NREF,6,6,0.0 AXIS,1,1,0.0 TOP,2,2,-.25 *PRINT,CONTACT=YES *EL PRINT,ELSET=CONTACT,FREQUENCY=10 S,E *NODE PRINT,FREQUENCY=10 U RF *EL FILE,FREQUENCY=999 S E PE TEMP *OUTPUT, FIELD, FREQ=10 *ELEMENT OUTPUT, ELSET=WORK TEMP S E PE PEEQ *NODE OUTPUT, NSET=ALL U *ENERGY PRINT,FREQUENCY=1 *END STEP

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Listing 1.3.6-3 *HEADING EXTRUSION OF A BAR WITH FRICTIONAL HEAT GENERATION EXPLICIT [CAX4RT] *NODE 1,0.,0. 61,0.,.3 2001,.1,0. 2061,.1,.3 *NGEN,NSET=AXIS 1,61,1 *NGEN,NSET=OUTSIDE 2001,2061,1 *NFILL,NSET=ALL AXIS,OUTSIDE,20,100 *NSET, NSET=TEMP 2025, 2027, 2029, 2031 *ELEMENT, TYPE=CAX4RT, ELSET=WORK 1,1,201,203,3 *ELGEN,ELSET=WORK 1,30,2,1,10,200,100 *ELSET,ELSET=BOT,GENERATE 1,901,100 *ELSET,ELSET=SIDE,GENERATE 901,930,1 *ELSET,ELSET=TOP,GENERATE 30,930,100 *SOLID SECTION,ELSET=WORK,MATERIAL=METAL, CONTROLS=HGLASS *SECTION CONTROLS,NAME=HGLASS,HOURGLASS=COMBINED *MATERIAL,NAME=METAL *ELASTIC 6.9E10,.33 *PLASTIC ** STRAIN RATE APPX .1 ** 60.E6,0.0 ,20. 90.E6,.125 ,20. 113.E6,.25 ,20. 124.E6,.375 ,20. 133.E6,0.5 ,20. 165.E6,1.0 ,20. 166.E6,2.0 ,20.

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Static Stress/Displacement Analyses

60.E6,0. ,50. 80.E6,.125 ,50. 97.E6,.25 ,50. 110.E6,.375 ,50. 120.E6,0.5 ,50. 150.E6,1.0 ,50. 151.E6,2.0 ,50. 50.E6,0.0 ,100. 65.E6,.125 ,100, 81.5E6,.25 ,100. 91.E6,.375 ,100. 100.E6,0.5 ,100. 125.E6,1.0 ,100. 126.E6,2.0 ,100. 45.E6,0.0 ,150. 63.E6,.125 ,150. 75.E6,.25 ,150. 89.E6,.5 ,150. 110.E6,1. ,150. 111.E6,2. ,150. *SPECIFIC HEAT 880., *DENSITY 2700., *CONDUCTIVITY 204.,0. 225.,300. *EXPANSION,ZERO=20.0 8.42E-5, *INELASTIC HEAT FRACTION .9, *NODE, NSET=REFNODE 9999, 0.2, 0.0, 0.0 *ELEMENT, TYPE=HEATCAP, ELSET=CAP 99001,9999 *HEATCAP, ELSET=CAP 1., ** *NSET,NSET=TOP,GENERATE 61,2061,100 *NSET,NSET=ALL 1,2061,1 *INITIAL CONDITIONS,TYPE=TEMPERATURE

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ALL,20. *NSET,NSET=NFILEOUT AXIS,OUTSIDE,TOP *ELSET,ELSET=EFILEOUT BOT,SIDE,TOP *SURFACE, TYPE=S, NAME=DIE, FILLET=0.075 START,.25,-.18 LINE,.0866,-.18 LINE,.0666,-.18 LINE,.0666,-.17 LINE,.0666,-.15 LINE,.0666,-.1 LINE,.099999,-.05 LINE,.099999,0.0 LINE,.099999,.3 LINE,.099999,.31 LINE,.2,.41 *SURFACE,TYPE=ELEMENT, NAME=BAR BOT, S1 SIDE, S2 *RIGID BODY, REFNODE=9999, ISOTHERMAL=YES, ANALYTICAL SURFACE =DIE *STEP STABILIZE WORKPIECE INSIDE DIE *DYNAMIC TEMPERATURE-DISPLACEMENT, EXPLICIT ,1. *FIXED MASS SCALING, ELSET=WORK, FACTOR=1.E5 *CONTACT PAIR, INTERACTION=CONTACT BAR, DIE *SURFACE INTERACTION, NAME=CONTACT *FRICTION 0.1 , *GAP HEAT GENERATION 1.0,0.5 *BOUNDARY, TYPE=VELOCITY REFNODE,1,6,0.0 AXIS,1,1,0.0 TOP,2,2,-.000125 2061,1,1,0.0 *BOUNDARY ALL,11,11,20.0 REFNODE,11,11,20.0 *OUTPUT,FIELD,NUM=1,VAR=PRESELECT

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*OUTPUT,HISTORY,FREQ=50 *NODE OUTPUT, NSET=TEMP NT, *END STEP ** *STEP EXTRUSION *DYNAMIC TEMPERATURE-DISPLACEMENT, EXPLICIT ,10. *FIXED MASS SCALING, ELSET=WORK, FACTOR=1.E5 *BOUNDARY,OP=NEW, TYPE=VELOCITY REFNODE,1,6,0.0 AXIS,1,1,0.0 TOP,2,2,-.0249875 2061,1,1,0.0 *BOUNDARY,OP=NEW REFNODE,11,11,20.0 *FILE OUTPUT, NUM=1 *EL FILE,,ELSET=EFILEOUT S,E PE, *NODE FILE,NSET=NFILEOUT NT, *EL FILE,ELSET=EFILEOUT S,E PE, *NODE FILE,NSET=NFILEOUT NT, *OUTPUT, HISTORY, TIME INTERVAL=35.0 *NODE OUTPUT, NSET=TEMP NT, *OUTPUT, FIELD,NUM=4 *ELEMENT OUTPUT,ELSET=WORK S,PEEQ, TEMP, HFL *NODE OUTPUT NT, U *END STEP ** *STEP REMOVE CONTACT *DYNAMIC TEMPERATURE-DISPLACEMENT, EXPLICIT ,.1 *FIXED MASS SCALING, ELSET=WORK, FACTOR=1.E5

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*CONTACT PAIR, OP=DELETE BAR,DIE *BOUNDARY,OP=NEW, TYPE=VELOCITY REFNODE,1,6,0.0 AXIS,1,1,0.0 TOP,2,2,0.0 *END STEP ** *STEP LET WORKPIECE COOL DOWN--I (ADD VISCOUS PRESSURE) *DYNAMIC TEMPERATURE-DISPLACEMENT, EXPLICIT ,10. *FIXED MASS SCALING,ELSET=WORK,FACTOR=1.E6 *AMPLITUDE, NAME=RAMP,TIME=TOTAL TIME 0.0,0.0,11.1,0.0,10011.1,1.0 *FILM, AMP=RAMP BOT,F1,20.,10. TOP,F3,20.,10. SIDE,F2,20.,10. ** *DLOAD WORK,VP1,1.E8 WORK,VP2,1.E8 WORK,VP3,1.E8 WORK,VP4,1.E8 ** *FILE OUTPUT, NUM=2 *EL FILE,ELSET=EFILEOUT S,E PE, *NODE FILE,NSET=NFILEOUT NT, *OUTPUT, HISTORY, TIME INTERVAL=35.0 *NODE OUTPUT, NSET=TEMP NT, *OUTPUT, FIELD,NUM=2 *ELEMENT OUTPUT,ELSET=WORK S,PEEQ, TEMP, HFL *NODE OUTPUT NT, U *END STEP *STEP LET WORKPIECE COOL DOWN--II

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*DYNAMIC TEMPERATURE-DISPLACEMENT, EXPLICIT ,9990. *FIXED MASS SCALING,ELSET=WORK,FACTOR=4.E9 ** *DLOAD,OP=NEW ** *END STEP

1.3.7 Rolling of thick plates Product: ABAQUS/Explicit Hot rolling is a basic manufacturing technique used to transform preformed shapes into a form suitable for further processing. Rolling processes can be divided into different categories, depending on the complexity of metal flow and on the geometry of the rolled product. Finite element computations are used increasingly to analyze the elongation and spread of the material during rolling (Kobayashi, 1989). Although the forming process is often carried out at low roll speed, this example shows that a considerable amount of engineering information can be obtained by using the explicit dynamics procedure in ABAQUS/Explicit to model the process. The rolling process is first investigated using plane strain computations. These results are used to choose the modeling parameters associated with the more computationally expensive three-dimensional analysis. Since rolling is normally performed at relatively low speeds, it is natural to assume that static analysis is the proper modeling approach. Typical rolling speeds (surface speed of the roller) are on the order of 1 m/sec. At these speeds inertia effects are not significant, so the response--except for rate effects in the material behavior--is quasi-static. Representative rolling geometries generally require three-dimensional modeling, resulting in very large models, and include nonlinear material behavior and discontinuous effects--contact and friction. Because the problem size is large and the discontinuous effects dominate the solution, the explicit dynamics approach is often less expensive computationally and more reliable than an implicit quasi-static solution technique. The computer time involved in running a simulation using explicit time integration with a given mesh is directly proportional to the time period of the event. This is because numerical stability considerations restrict the time increment to µ

¢t ∙ min L

el

r

½ ¸ + 2¹

¶

;

where the minimum is taken over all elements in the mesh, Lel is a characteristic length associated with an element, ½ is the density of the material in the element, and ¸ and ¹ are the effective Lamé's constants for the material in the element. Since this condition effectively means that the time increment can be no larger than the time required to propagate a stress wave across an element, the computer time involved in running a quasi-static analysis can be very large. The cost of the simulation is directly proportional to the number of time increments required, n = T =¢t if ¢t remains constant, where T is

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the time period of the event being simulated. ( ¢t will not remain constant in general, since element distortion will change Lel and nonlinear material response will change the effective Lamé constants and density. But the assumption is acceptable for the purposes of this discussion.) Thus,

n = T max

Ã

1 Lel

s

¸ + 2¹ ½

!

:

To reduce n, we can speed up the simulation compared to the time of the actual process; that is, we can artificially reduce the time period of the event, T . This will introduce two possible errors. If the simulation speed is increased too much, the inertia forces will be larger and will change the predicted response (in an extreme case the problem will exhibit wave propagation response). The only way to avoid this error is to find a speed-up that is not too large. The other error is that some aspects of the problem other than inertia forces--for example, material behavior--may also be rate dependent. This implies that we cannot change the actual time period of the event being modeled. But we can see a simple equivalent--artificially increasing the material density, ½, by a factor f 2 reduces n to n=f , just as decreasing T to T =f . This concept, which is called "mass scaling," reduces the ratio of the event time to the time for wave propagation across an element while leaving the event time fixed, thus allowing treatment of rate-dependent material and other behaviors, while having exactly the same effect on inertia forces as speeding up the time of simulation. Mass scaling is attractive because it allows us to treat rate-dependent quasi-static problems efficiently. But we cannot take it too far or we allow the inertia forces to dominate and, thus, change the solution. This example illustrates the use of mass scaling and shows how far we can take it for a practical case.

Problem description A steel plate of an original square cross-section of 40 mm by 40 mm and a length of 92 mm is reduced to a 30 mm height by rolling through one roll stand. The radius of the rollers is 170 mm. The single roller in the model (taking advantage of symmetry) is assumed to be rigid and is modeled as an analytical rigid surface. The isotropic hardening yield curve of the steel is taken from Kopp and Dohmen (1990). Isotropic elasticity is assumed, with Young's modulus of 150 GPa and Poisson's ratio of 0.3. The strain hardening is described using 11 points on the yield stress versus plastic strain curve, with an initial yield stress of 168.2 MPa and a maximum yield stress of 448.45 MPa. No rate dependence or temperature dependence is taken into account. Coulomb friction is assumed between the roller and the plate, with a friction coefficient of 0.3. Friction plays an important role in this process, as it is the only mechanism by which the plate is pulled through the roll stand. If the friction coefficient is too low, the plate cannot be drawn through the roll stand. Initially, when a point on the surface of the plate has just made contact with the roller, the roller surface is moving faster than the point on the surface of the plate and there is a relative slip between the two surfaces. As the point on the plate is drawn into the process zone under the roller, it moves faster and, after a certain distance, sticks to the roller. As the point on the surface of the plate is pushed out of the process zone, it picks up speed and begins to move faster than the roller. This causes slip in the opposite direction before the point on the surface of the sheet finally loses contact with the roller.

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For plane strain computations a half-symmetry model with CPE4R elements is used. For the three-dimensional computations a one-quarter symmetry model with C3D8R elements is used. The roller is modeled with analytical rigid surfaces for both the two-dimensional and three-dimensional cases. For quasi-static rolling problems perfectly round analytical surfaces can provide a more accurate representation of the revolved roller geometry, improve computational efficiency, and reduce noise when compared to element-based rigid surfaces. The roller is rotated through 32° at a constant angular velocity of 1 revolution per second (6.28 rad/sec), which corresponds to a roller surface speed of 1.07 m/sec. The plate is given an initial velocity in the global x-direction. The initial velocity is chosen to match the x-component of velocity of the roller at the point of first contact. This choice of initial velocity results in a net acceleration of zero in the x-direction at the point of contact and minimizes the initial impact between the plate and the roller. This minimizes the initial transient disturbance. In each analysis performed in this example, the *FIXED MASS SCALING option is used to scale the masses of all the elements in the model by factors of either 110, 2758, or 68962. These scaling factors translate into effective roller surface speeds of 11.2 m/sec, 56.1 m/sec, and 280.5 m/sec. An alternative, but equivalent, means of mass scaling could be achieved by scaling the actual density (entered on the *DENSITY option) by the aforementioned factors. The element formulation for the two-dimensional (using CPE4R elements) and three-dimensional (using C3D8R elements) analyses uses the pure stiffness form of hourglass control (HOURGLASS=STIFFNESS). The element formulation is selected using the *SECTION CONTROLS option. In addition, the three-dimensional model (using C3D8R elements) uses the centroidal (KINEMATIC SPLIT=CENTROID) kinematic formulation. These options are economical yet provide the necessary level of accuracy for this class of problems. Other cases using more computationally intensive element formulations are included for comparison: analyses that use the default section control options and a mass scaling factor of 2758 and two- and three-dimensional analyses that use an element formulation intermediate in computational cost between the two previous formulations. For the sole purpose of testing the performances of the modified triangular and tetrahedral elements, the problem is also analyzed in two dimensions using CPE6M elements and in three dimensions using C3D10M elements.

Results and discussion Table 1.3.7-1 shows the effective rolling speeds and the relative CPU cost of the cases using the element formulations recommended for this problem. The relative costs are normalized with respect to the CPU time for the two-dimensional model (using CPE4R elements) with the intermediate mass scaling value. In addition, Table 1.3.7-2 compares the relative CPU cost and accuracy between the different element formulations of the solid elements using the intermediate mass scaling value.

Plane strain rolling (CPE4R elements) A plane strain calculation allows the user to resolve a number of modeling questions in two dimensions before attempting a more expensive three-dimensional calculation. In particular, an acceptable effective mass scaling factor for running the transient dynamics procedure can be

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determined. Figure 1.3.7-1 through Figure 1.3.7-3 show contours of equivalent plastic strain for the three mass scaling factors using the STIFFNESS hourglass control. Figure 1.3.7-4 through Figure 1.3.7-6 show contours of shear stress for the same cases. These results show that there is very little difference between the lowest and the intermediate mass scaling cases. All the results are in good agreement with the quasi-static analysis results obtained with ABAQUS. The results of the largest mass scaling case show pronounced dynamic effects. Table 1.3.7-1 shows the relative run time of the quasi-static calculation, and Table 1.3.7-2 compares the different element formulations at the same level of mass scaling. The intermediate mass scaling case gives essentially the same results as the quasi-static calculation, using about one-seventh of the CPU time. In addition to the savings provided by the mass scaling option, more computational savings are achieved using the chosen element formulation of STIFFNESS hourglass control; the results for this formulation compare well to the results for the computationally more expensive element formulations.

Three-dimensional rolling (C3D8R elements) We have ascertained with the two-dimensional calculations that mass scaling by a factor of 2758 gives results that are essentially the same as a quasi-static solution. Figure 1.3.7-7 shows the distribution of the equivalent plastic strain of the deformed sheet for the three-dimensional case using the CENTROID kinematic and STIFFNESS hourglass section control options. Figure 1.3.7-8 shows the distribution of the equivalent plastic strain of the deformed sheet for the three-dimensional case using the default section control options ( AVERAGE STRAIN kinematic and RELAX STIFFNESS hourglass). Table 1.3.7-1 compares this three-dimensional case with the plane strain and quasi-static cases, and Table 1.3.7-2 compares the three different three-dimensional element formulations included here with the two-dimensional cases at the same level of mass scaling. The accuracy for all three element formulations tested is very similar for this problem, but significant savings are realized in the three-dimensional analyses when using more economical element formulations.

Analyses using CPE6M and C3D10M elements The total number of nodes in the CPE6M model is identical to the number in the CPE4R model. The number of nodes in the C3D10M model is 3440 (compared to 3808 in the C3D8R model). The analyses using the CPE6M and C3D10M elements use a mass scaling factor of 2758. Figure 1.3.7-9 and Figure 1.3.7-10 show the distribution of the equivalent plastic strain of the plate for the two-dimensional and three-dimensional cases, respectively. The results are in reasonably good agreement with other element formulations. However, the CPU costs are higher since the modified triangular and tetrahedral elements use more than one integration point in each element and the stable time increment size is somewhat smaller than in analyses that use reduced integration elements with the same node count. For the mesh refinements used in this problem, the CPE6M model takes about twice the CPU time as the CPE4R model, while the C3D10M model takes about 5.75 times the CPU time as the C3D8R model.

Input files roll2d330_anl_ss.inp

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Two-dimensional case (using CPE4R elements) with a mass scaling factor of 2758 and the STIFFNESS hourglass control. roll3d330_rev_anl_css.inp Three-dimensional case (using C3D8R elements) with a mass scaling factor of 2758; an analytical rigid surface of TYPE=REVOLUTION; and the CENTROID kinematic and STIFFNESS hourglass section control options. roll2d66_anl_ss.inp Two-dimensional case (using CPE4R elements) with a mass scaling factor of 110 using the STIFFNESS hourglass control. roll2d330_anl_cs.inp Two-dimensional case (using CPE4R elements) with a mass scaling factor of 2758 using the COMBINED hourglass control. roll2d330_cs.inp Two-dimensional case (using CPE4R elements) with a mass scaling factor of 2758 using the COMBINED hourglass control and rigid elements. roll3d330_css.inp Three-dimensional case (using C3D8R elements) with a mass scaling factor of 2758, rigid elements, and the CENTROID kinematic and STIFFNESS hourglass section control options. roll3d330_ocs.inp Three-dimensional case (using C3D8R elements) with a mass scaling factor of 2758, rigid elements, and the ORTHOGONAL kinematic and COMBINED hourglass section control options. roll2d1650_anl_ss.inp Two-dimensional case (using CPE4R elements) with a mass scaling factor of 68962 using the STIFFNESS hourglass control. roll3d330_rev_anl_ocs.inp Three-dimensional model (using C3D8R elements) with a mass scaling factor of 2758; an analytical rigid surface of TYPE=REVOLUTION; and the ORTHOGONAL kinematic and COMBINED hourglass section control options. roll3d330_rev_anl.inp Three-dimensional model (using C3D8R elements) with a mass scaling factor of 2758, an analytical rigid surface of TYPE=REVOLUTION, and the default section control options. roll3d330_cyl_anl.inp Three-dimensional model (using C3D8R elements) with a mass scaling factor of 2758, an analytical rigid surface of TYPE=CYLINDER, and the default section control options. roll2d66.inp

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Two-dimensional model (using CPE4R elements) with a mass scaling factor of 110 and default section controls. roll2d330.inp Two-dimensional model (using CPE4R elements) with a mass scaling factor of 2758 and default section controls. roll2d1650.inp Two-dimensional model (using CPE4R elements) with a mass scaling factor of 68962 and default section controls. roll3d330.inp Three-dimensional model using rigid elements and default section controls. roll2d66_anl.inp Two-dimensional model (using CPE4R elements) with a mass scaling factor of 110 using analytical rigid surfaces and default section controls. roll2d330_anl.inp Two-dimensional model (using CPE4R elements) with a mass scaling factor of 2758 using analytical rigid surfaces and default section controls. roll2d1650_anl.inp Two-dimensional model (using CPE4R elements) with a mass scaling factor of 68962 using analytical rigid surfaces and default section controls. roll2d330_anl_cpe6m.inp Two-dimensional case (using CPE6M elements) with a mass scaling factor of 2758. roll3d330_anl_c3d10m.inp Three-dimensional case (using C3D10M elements) with a mass scaling factor of 2758. roll3d_medium.inp Additional mesh refinement case (using C3D8R elements) included for the sole purpose of testing the performance of the code.

References · Kobayashi, S., S. I. Oh, and T. Altan, Metal Forming and the Finite Element Method , Oxford University Press, 1989. · Kopp, R., and P. M. Dohmen, "Simulation und Planung von Walzprozessen mit Hilfe der Finite-Elemente-Methode (FEM)," Stahl U. Eisen, no. 7, pp. 131-136, 1990.

Tables

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Static Stress/Displacement Analyses

Table 1.3.7-1 Analysis cases and relative CPU costs. (The two-dimensional explicit analyses all use CPE4R elements and the STIFFNESS hourglass control. The three-dimensional explicit analysis uses C3D8R elements and the CENTROID kinematic, STIFFNESS hourglass section control options.) Analysis Type Mass Scaling Effective Roll Surface Relative CPU Factor Speed (m/sec) Time Explicit, plane strain 110.3 11.2 4.99 Explicit, plane strain 2758.5 56.1 1.00 Explicit, plane strain 68961.8 280.5 0.21 Implicit, plane strain quasi-static 6.90 Explicit, 3-D 2758.5 56.1 16.0

Table 1.3.7-2 Explicit section control options tested (mass scaling factor=2758.5). CPE4R and C3D8R elements are employed for the two-dimensional and three-dimensional cases, respectively. Spread values are reported for the half-model at node 24015. Analysis Type Section Controls Relative Kinematic Hourglass CPU Time Explicit, plane strain n/a STIFFNES 1.00 S Explicit, plane strain n/a RELAX 1.11 Explicit, plane strain n/a COMBINE 1.04 D 30.7 Explicit, 3-D AVERAGE RELAX STRAIN STIFFNES S Explicit, 3-D ORTHOGON COMBINE 21.2 AL D Explicit, 3-D CENTROID STIFFNES 16.0 S

Sprea d (mm) n/a n/a n/a 2.06

2.07 2.10

Figures Figure 1.3.7-1 Equivalent plastic strain for the plane strain case ( CPE4R) with STIFFNESS hourglass control (mass scaling factor=110.3).

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Static Stress/Displacement Analyses

Figure 1.3.7-2 Equivalent plastic strain for the plane strain case ( CPE4R) with STIFFNESS hourglass control (mass scaling factor=2758.5).

Figure 1.3.7-3 Equivalent plastic strain for the plane strain case ( CPE4R) with STIFFNESS hourglass control (mass scaling factor=68961.8).

Figure 1.3.7-4 Shear stress for the plane strain case (CPE4R) with STIFFNESS hourglass control (mass scaling factor=110.3).

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Static Stress/Displacement Analyses

Figure 1.3.7-5 Shear stress for the plane strain case (CPE4R) with STIFFNESS hourglass control (mass scaling factor=2758.5).

Figure 1.3.7-6 Shear stress for the plane strain case (CPE4R) with STIFFNESS hourglass control (mass scaling factor=68961.8).

Figure 1.3.7-7 Equivalent plastic strain for the three-dimensional case ( C3D8R) using the CENTROID kinematic and STIFFNESS hourglass section control options (mass scaling 1-493

Static Stress/Displacement Analyses

factor=2758.5).

Figure 1.3.7-8 Equivalent plastic strain for the three-dimensional case ( C3D8R) using the AVERAGE STRAIN kinematic and RELAX STIFFNESS hourglass section control options (mass scaling factor=2758.5).

Figure 1.3.7-9 Equivalent plastic strain for the plane strain case ( CPE6M) (mass scaling factor=2758.5).

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Static Stress/Displacement Analyses

Figure 1.3.7-10 Equivalent plastic strain for the three-dimensional case ( C3D10M) (mass scaling factor=2758.5).

Sample listings

1-495

Static Stress/Displacement Analyses

Listing 1.3.7-1 *HEADING Thick plate rolling: Plane Strain, ABAQUS/Explicit (Analytical rigid surfaces) SECTION CONTROLS USED (HOURGLASS=STIFFNESS) *RESTART,WRITE,NUM=10 ** *NODE ** Bar 1, 0., 0. 401, 0., 0.020 47, -00.092, 0. 447, -00.092, 0.020 ** *NGEN,NSET=BOTTOM 1,47,1 *NGEN,NSET=TOP 401,447,1 ** *NFILL,NSET=BAR BOTTOM,TOP,8,50 ** ***** Bar ** *ELEMENT,TYPE=CPE4R,ELSET=METAL 1,1,51,52,2 *ELGEN,ELSET=METAL 1,8,50,50,46,1,1 ** *ELSET,ELSET=TOP,GEN 351,396,1 *ELSET,ELSET=BACK,GEN 46,396,50 ** *SOLID SECTION,ELSET=METAL,MAT=C15,CONTROL=B 1., *SECTION CONTROLS, HOURGLASS=STIFFNESS, NAME=B ** *MATERIAL,NAME=C15 *ELASTIC 1.5E11,.3

1-496

Static Stress/Displacement Analyses

*PLASTIC 168.72E06,0 219.33E06,0.1 272.02E06,0.2 308.53E06,0.3 337.37E06,0.4 361.58E06,0.5 382.65E06,0.6 401.42E06,0.7 418.42E06,0.8 434.01E06,0.9 448.45E06,1.0 *DENSITY 7.85E3, ** ** Node for rigid surface *NODE 10000, 0.0409 , 0.185 *INITIAL CONDITIONS,TYPE=VELOCITY BAR,1,1.0367 ** ** *SURFACE, TYPE=SEGMENTS,NAME=ROLLER START, 0.040900, 0.015000 CIRCL, -.129100, 0.185000 , 0.0409 , 0.185 *SURFACE,TYPE=ELEMENT, NAME=SURF1 TOP,S2 *RIGID BODY, REF NODE=10000, ANALYTICAL SURFACE=ROLLER *STEP *DYNAMIC,EXPLICIT ,0.089286 *FIXED MASS SCALING,FACTOR=2758.5,ELSET=METAL ** ** Roller, Radius = .170 m ** *BOUNDARY BOTTOM,2,2 10000,1,2 ** *BOUNDARY,TYPE=VELOCITY 10000,6,6,6.2832 **

1-497

Static Stress/Displacement Analyses

*SURFACE INTERACTION,NAME=FRICT *FRICTION 0.3, *CONTACT PAIR,INTERACTION=FRICT SURF1,ROLLER ** ** *FILE OUTPUT,TIMEMARKS=YES,NUM=1 *EL FILE PEEQ,MISES,PE,LE *NODE FILE U, *ENERGY FILE *END STEP **

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Listing 1.3.7-2 *HEADING Thick plate rolling: 3-Dimensional, ABAQUS/Explicit SECTION CONTROLS USED (KINEMATIC=CENTROID, HOURGLASS=STIFFNESS) *RESTART,WRITE,NUM=4 *NODE ** Bar 1, 0., 0. 801, 0., 0.020 47, -00.092, 0. 847, -00.092, 0.020 24001, 0., 0. , -0.020 24801, 0., 0.020, -0.020 24047, -00.092, 0. , -0.020 24847, -00.092, 0.020, -0.020 *NGEN,NSET=BOT1 1,47,1 *NGEN,NSET=TOP1 801,847,1 *NGEN,NSET=BOT2 24001,24047,1 *NGEN,NSET=TOP2 24801,24847,1 *NFILL,NSET=ZSYMM BOT1,TOP1,8,100 *NFILL,NSET=SIDE BOT2,TOP2,8,100 *NFILL,NSET=BAR ZSYMM,SIDE,8,3000 *NSET,NSET=BOTTOM,GEN 1,47,1 3001,3047,1 6001,6047,1 9001,9047,1 12001,12047,1 15001,15047,1 18001,18047,1 21001,21047,1 24001,24047,1 **

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***** Bar ** *ELEMENT,TYPE=C3D8R,ELSET=METAL 1,2,1,3001,3002,102,101,3101,3102 *ELGEN,ELSET=METAL 1,8,100,100,46,1,1,8,3000,1000 ** *ELSET,ELSET=TOP,GEN 701,746,1 1701,1746,1 2701,2746,1 3701,3746,1 4701,4746,1 5701,5746,1 6701,6746,1 7701,7746,1 *ELSET,ELSET=BACK,GEN 46,746,100 1046,1746,100 2046,2746,100 3046,3746,100 4046,4746,100 5046,5746,100 6046,6746,100 7046,7746,100 *ELSET,ELSET=SIDE,GEN 7001,7046,1 7101,7146,1 7201,7246,1 7301,7346,1 7401,7446,1 7501,7546,1 7601,7646,1 7701,7746,1 ** *SOLID SECTION,ELSET=METAL,MAT=C15,CONTROL=C 1., *SECTION CONTROLS,KINEMATIC=CENTROID, HOURGLASS=STIFFNESS, NAME=C ** *MATERIAL,NAME=C15 *ELASTIC 1.5E11,.3

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*PLASTIC 168.72E06,0 219.33E06,0.1 272.02E06,0.2 308.53E06,0.3 337.37E06,0.4 361.58E06,0.5 382.65E06,0.6 401.42E06,0.7 418.42E06,0.8 434.01E06,0.9 448.45E06,1.0 *DENSITY 7.85E3, ** *NODE ** Reference node 30000, 0.0409 , 0.185 , -0.010 *INITIAL CONDITIONS,TYPE=VELOCITY BAR,1,1.0367 ** ***************** Step 1 *SURFACE, NAME=ROLLER, TYPE=REVOL 0.0409,0.185,-0.025,0.0409,0.185,0.005 START, 0.170,0.03 LINE, 0.170,0.0 *SURFACE,TYPE=ELEMENT, NAME=SURF1 TOP,S2 SIDE,S5 *RIGID BODY, REF NODE=30000, ANALYTICAL SURFACE=ROLLER *STEP *DYNAMIC,EXPLICIT ,0.089286 *FIXED MASS SCALING,FACTOR=2758.5,ELSET=METAL ** Roller, Radius = .170 m *BOUNDARY BOTTOM,2,2 ZSYMM,ZSYMM 30000,1,5 ** *BOUNDARY,TYPE=VELOCITY **30000,6,6,330.0

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30000,6,6,6.2832 ** *SURFACE INTERACTION,NAME=FRICT *FRICTION 0.3, *CONTACT PAIR,INTERACTION=FRICT SURF1,ROLLER ** *FILE OUTPUT,TIMEMARKS=YES,NUM=1 *EL FILE PEEQ,MISES,PE,LE *NODE FILE U, *ENERGY FILE *END STEP

1.3.8 Axisymmetric forming of a circular cup Products: ABAQUS/Standard ABAQUS/Explicit This example illustrates the hydroforming of a circular cup using an axisymmetric model. In this case a two-stage forming sequence is used, with annealing between the stages. Two analysis methods are used: in one the entire process is analyzed using ABAQUS/Explicit; in the other the forming sequences are analyzed with ABAQUS/Explicit, while the springback analyses are run in ABAQUS/Standard. Here, the import capability is used to transfer results between ABAQUS/Explicit and ABAQUS/Standard and vice versa.

Problem description The model consists of a deformable blank and three rigid dies. The blank has a radius of 150.0 mm, is 1.0 mm thick, and is modeled using axisymmetric shell elements, SAX1. The coefficient of friction between the blank and the dies is taken to be 0.1. Dies 1 and 2 are offset from the blank by half of the thickness of the blank, because the contact algorithm takes into account the shell thickness. To avoid pinching of the blank while die 3 is put into position for the second forming stage, the radial gap between dies 2 and 3 is set to be 20% bigger than the initial shell thickness. Figure 1.3.8-1 and Figure 1.3.8-2 show the initial geometry of the model. The three dies are modeled with either two-dimensional analytical rigid surfaces or RAX2 rigid elements. An analytical rigid surface can yield a more accurate representation of two-dimensional curved punch geometries and result in computational savings. Contact pressure can be viewed on the specimen surface, and the reaction force is available at the rigid body reference node. In addition, both the kinematic (default) and penalty contact formulations are tested. Results for the kinematic contact formulation using rigid elements are presented here. The blank is made of aluminum-killed steel, which is assumed to satisfy the Ramberg-Osgood relation between true stress and logarithmic strain,

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² = (¾=K )1=n ; with a reference stress value (K) of 513 MPa and work-hardening exponent (n) of 0.223. Isotropic elasticity is assumed, with Young's modulus of 211 GPa and Poisson's ratio of 0.3. With these data an initial yield stress of 91.3 MPa is obtained. The stress-strain behavior is defined by piecewise linear segments matching the Ramberg-Osgood curve up to a total (logarithmic) strain level of 107%, with Mises yield, isotropic hardening, and no rate dependence. The analysis that is performed entirely within ABAQUS/Explicit consists of six steps. In the first step contact is defined between the blank and dies 1 and 2. Both dies remain fixed while a distributed load of 10 MPa in the negative z-direction is ramped onto the blank. This load is then ramped off in the second step, allowing the blank to spring back to an equilibrium state. The third step is an annealing step. The annealing procedure in ABAQUS/Explicit sets all appropriate state variables to zero. These variables include stresses, strains (excluding the thinning strain for shells, membranes, and plane stress elements), plastic strains, and velocities. There is no time associated with an annealing step. The process occurs instantaneously. In the fourth step contact is defined between the blank and die 3 and contact is removed between the blank and die 1. Die 3 moves down vertically in preparation for the next pressure loading. In the fifth step another distributed load is applied to the blank in the positive z-direction, forcing the blank into die 3. This load is then ramped off in the sixth step to monitor the springback of the blank. To obtain a quasi-static response, an investigation was conducted to determine the optimum rate for applying the pressure loads and removing them. The optimum rate balances the computational time against the accuracy of the results; increasing the loading rate will reduce the computer time but lead to less accurate quasi-static results. The analysis that uses the import capability consists of four runs. The first run is identical to Step 1 of the ABAQUS/Explicit analysis described earlier. In the second run the ABAQUS/Explicit results for the first forming stage are imported into ABAQUS/Standard (using UPDATE=NO and STATE=YES on the *IMPORT option) for the first springback analysis. The third run imports the results of the first springback analysis into ABAQUS/Explicit for the subsequent annealing process and the second forming stage. By setting UPDATE=YES and STATE=NO on the *IMPORT option, this run begins with no initial stresses or strains, effectively simulating the annealing process. The final run imports the results of the second forming stage into ABAQUS/Standard for the second springback analysis.

Results and discussion Figure 1.3.8-3 to Figure 1.3.8-5show the results of the analysis conducted entirely within ABAQUS/Explicit using the rigid element approach and the kinematic contact formulation. Figure 1.3.8-3shows the deformed shape at the end of Step 2, after the elastic springback. Figure 1.3.8-4 shows the deformed shape at the end of the analysis, after the second elastic springback. Although it is not shown here, the amount of springback observed during the unloading steps is negligible. Figure 1.3.8-5 shows a contour plot of the shell thickness ( STH) at the end of the analysis. The thickness of

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the material at the center of the cup has been reduced by about 20%, while the thickness at the edges of the cup has been increased by about 10%. The results obtained using the import capability to perform the springback analyses in ABAQUS/Standard are nearly identical, as are those obtained using analytical rigid surfaces and/or penalty contact formulations.

Input files axiform.inp ABAQUS/Explicit analysis that uses rigid elements and kinematic contact. This file is also used for the first step of the analysis that uses the import capability. axiform_anl.inp Model using analytical rigid surfaces and kinematic contact. axiform_pen.inp Model using rigid elements and penalty contact. axiform_anl_pen.inp Model using analytical rigid surfaces and penalty contact. axiform_sprbk1.inp First springback analysis using the import capability. axiform_form2.inp Second forming analysis using the import capability. axiform_sprbk2.inp Second springback analysis using the import capability. axiform_restart.inp Restart of axiform.inp included for the purpose of testing the restart capability. axiform_rest_anl.inp Restart of axiform_anl.inp included for the purpose of testing the restart capability.

Figures Figure 1.3.8-1 Configuration at the beginning of stage 1.

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Figure 1.3.8-2 Configuration of dies in forming stage 2. (The dotted line shows the initial position of die 3.)

Figure 1.3.8-3 Deformed configuration after the first forming stage.

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Figure 1.3.8-4 Final configuration.

Figure 1.3.8-5 Contour plot of shell thickness.

Sample listings

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Listing 1.3.8-1 *HEADING SHEET METAL FORMING WITH ANNEALING DIE3 SLIDES DOWN FROM ABOVE TO AVOID INITIAL OVERCLOSURE *NODE,NSET=BLANK 1,0.,0.0005 41,.150,0.0005 *NGEN,NSET=BLANK 1,41,1 *ELEMENT,TYPE=SAX1,ELSET=BLANK 1, 1,2 *ELGEN,ELSET=BLANK 1, 40,1,1 *SHELL SECTION,ELSET=BLANK,MATERIAL=STEEL, SECTION INTEGRATION=GAUSS .001,5 *MATERIAL,NAME=STEEL *DENSITY 7800., *ELASTIC 2.1E11,0.3 *PLASTIC 0.91294E+08, 0.00000E+00 0.10129E+09, 0.21052E-03 0.11129E+09, 0.52686E-03 0.12129E+09, 0.97685E-03 0.13129E+09, 0.15923E-02 0.14129E+09, 0.24090E-02 0.15129E+09, 0.34674E-02 0.16129E+09, 0.48120E-02 0.17129E+09, 0.64921E-02 0.18129E+09, 0.85618E-02 0.19129E+09, 0.11080E-01 0.20129E+09, 0.14110E-01 0.21129E+09, 0.17723E-01 0.22129E+09, 0.21991E-01 0.23129E+09, 0.26994E-01 0.24129E+09, 0.32819E-01 0.25129E+09, 0.39556E-01 0.26129E+09, 0.47301E-01 0.27129E+09, 0.56159E-01

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0.28129E+09, 0.66236E-01 0.29129E+09, 0.77648E-01 0.30129E+09, 0.90516E-01 0.31129E+09, 0.10497E+00 0.32129E+09, 0.12114E+00 0.33129E+09, 0.13916E+00 0.34129E+09, 0.15919E+00 0.35129E+09, 0.18138E+00 0.36129E+09, 0.20588E+00 0.37129E+09, 0.23287E+00 0.38129E+09, 0.26252E+00 0.39129E+09, 0.29502E+00 0.40129E+09, 0.33054E+00 0.41129E+09, 0.36929E+00 0.42129E+09, 0.41147E+00 0.43129E+09, 0.45729E+00 0.44129E+09, 0.50696E+00 0.45129E+09, 0.56073E+00 0.46129E+09, 0.61881E+00 0.47129E+09, 0.68145E+00 0.48129E+09, 0.74890E+00 0.49129E+09, 0.82142E+00 0.50129E+09, 0.89928E+00 0.51129E+09, 0.98274E+00 0.52129E+09, 0.10721E+01 *NODE,NSET=DIE1 1001, 0.,-0.05 1002,.090,-0.05 1011,.100,-.040 199991, 0., -0.05 *NGEN,NSET=DIE1,LINE=C 1002,1011,1,,.090,-.040,0. *ELEMENT,TYPE=RAX2,ELSET=DIE1 1001, 1001,1002 *ELGEN,ELSET=DIE1 1001, 10,1,1 *NODE,NSET=DIE2 2001,.100,-.060 2002,.100,-.010 2011,.110, 0. 2012,.160, 0. 299991, .100, -.060 *NGEN,NSET=DIE2,LINE=C

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2002,2011,1,,.110,-.010,0.,0.,0.,-1. *ELEMENT,TYPE=RAX2,ELSET=DIE2 2001, 2001,2002 *ELGEN,ELSET=DIE2 2001, 11,1,1 *NODE,NSET=DIE3 ** raised by 0.05 from original shift outer ** surface inward by then lower by 0.005 ** (half the radius of curvature) ** further lower by 0.005 3001,.0 ,0.0044 3014,.050,-0.009 3015,.0888,-0.009 3024,.0988,0.001 3025,.0988,0.0405 399991,.0,0.0044 *NGEN,NSET=DIE3,LINE=C 3001,3014,1,, 0.0,-0.0956,0.,0.,0.,-1. 3015,3024,1,,.0888, 0.001, 0. *ELEMENT,TYPE=RAX2,ELSET=DIE3 3001, 3001,3002 *ELGEN,ELSET=DIE3 3001, 24,1,1 *BOUNDARY 1,XSYMM 199991,1,6 299991,1,6 399991,1,2 399991,6,6 *AMPLITUDE,NAME=R1,TIME=STEP TIME 0.,0., .8E-3,1.5E6, 1.7E-3,1.5E6, 3.E-3,1.E7, 3.5E-3,1.E7 *AMPLITUDE,NAME=R2,TIME=STEP TIME 0.,1.E7, 1.E-3,0. *AMPLITUDE,NAME=R3A,TIME=STEP TIME, DEFINITION=SMOOTH STEP 0.,0., 1.E-3, 1.0 *AMPLITUDE,NAME=R4,TIME=STEP TIME 0.,0., 1.E-3,.6E7 *AMPLITUDE,NAME=R5,TIME=STEP TIME 0.,.6E7, 1.E-3,0. *RESTART,WRITE,NUM=1 **

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** First downward pressure loading *SURFACE,TYPE=ELEMENT,NAME=TOP BLANK,SPOS *SURFACE,TYPE=ELEMENT,NAME=DIE1 DIE1,SPOS *SURFACE,TYPE=ELEMENT,NAME=DIE2 DIE2,SPOS *SURFACE,TYPE=ELEMENT,NAME=DIE3 DIE3,SNEG *SURFACE,TYPE=ELEMENT,NAME=BOTTOM BLANK,SNEG *RIGID BODY,ELSET=DIE3,REF NODE=399991 *RIGID BODY,ELSET=DIE1,REF NODE=199991 *RIGID BODY,ELSET=DIE2,REF NODE=299991 *STEP *DYNAMIC,EXPLICIT ,3.5E-3 *DLOAD,AMP=R1 BLANK,P,-1.0 *SURFACE INTERACTION,NAME=FRICT *FRICTION 0.1, *CONTACT PAIR,INTERACTION=FRICT DIE2,BOTTOM DIE1,BOTTOM *FILE OUTPUT,NUM=2, TIMEMARKS=YES *EL FILE STH, *OUTPUT,FIELD *ELEMENT OUTPUT STH, *NODE FILE U, *OUTPUT,FIELD *NODE OUTPUT U, *HISTORY OUTPUT,TIME=3.5E-6 *NSET,NSET=NOUT 1, *NODE HISTORY,NSET=NOUT U,V *ELSET,ELSET=EOUT 24,25,26

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*EL HISTORY,ELSET=EOUT STH, PEEQ,MISES,S,LE,PE *ENERGY HISTORY ALLKE,ALLSE,ALLWK,ALLPD,ALLAE,ALLCD,ALLFD,ALLIE, ALLVD,ETOTAL,DT *END STEP ** ** First springback *STEP *DYNAMIC,EXPLICIT ,1.E-3 *DLOAD,AMP=R2,OP=NEW BLANK,P,-1.0 *END STEP ** ** anneal *STEP *ANNEAL *END STEP ** ** gradually slide die 3 into position *STEP *DYNAMIC,EXPLICIT ,1.0E-3 *BOUNDARY, AMP=R3A 399991,2,2,-0.04 ***DLOAD,OP=NEW **BLANK, VP, 100.0 *CONTACT PAIR,INTERACTION=FRICT,OP=ADD DIE3,TOP *CONTACT PAIR,OP=DELETE DIE1,BOTTOM *END STEP ** ** Second upward pressure loading *STEP *DYNAMIC,EXPLICIT ,1.0E-3 *ELSET,ELSET=LOAD,GEN 1,26,1 *DLOAD,AMP=R4,OP=NEW LOAD,P,1.0

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*END STEP ** ** Second springback *STEP *DYNAMIC,EXPLICIT ,1.E-3 *DLOAD,AMP=R5,OP=NEW LOAD,P,1.0 *END STEP

1.3.9 Cup/trough forming Product: ABAQUS/Explicit This example illustrates the use of adaptive meshing in forging problems that include large amounts of shearing at the tool-blank interface; a cup and a trough are formed.

Problem description Three different geometric models are considered, as shown in Figure 1.3.9-1. Each model consists of a rigid punch, a rigid die, and a deformable blank. The outer top and bottom edges of the blank are cambered, which facilitates the flow of material against the tools. The punch and die have semicircular cross-sections; the punch has a radius of 68.4 mm, and the die has a radius of 67.9 mm. The blank is modeled as a von Mises elastic, perfectly plastic material with a Young's modulus of 4000 MPa and a yield stress of 5 MPa. The Poisson's ratio is 0.21; the density is 1.E-4 kg/mm 3. In each case the punch is moved 61 mm, while the die is fully constrained. The SMOOTH STEP parameter on the *AMPLITUDE option is used to ramp the punch velocity to a maximum, at which it remains constant. The SMOOTH STEP specification of the velocity promotes a quasi-static response to the loading.

Case 1: Axisymmetric model for cup forming The blank is meshed with CAX4R elements and measures 50 ´ 64.77 mm. The punch and the die are modeled as TYPE=SEGMENTS analytical rigid surfaces. Symmetry boundary conditions are prescribed at r=0. The finite element model is shown in Figure 1.3.9-2.

Case 2: Three-dimensional model for trough forming The blank is meshed with C3D8R elements and measures 50 ´ 64.7 ´ 64.7 mm. The punch and the die are modeled as TYPE=CYLINDER analytical rigid surfaces. Symmetry boundary conditions are applied at the x=0 and z=0 planes. The finite element model of the blank is shown in Figure 1.3.9-3.

Case 3: Three-dimensional model for cup forming The blank is meshed with C3D8R elements. A 90° wedge of the blank with a radius of 50 mm and a height of 64.7 mm is analyzed. The punch and the die are modeled as TYPE=REVOLUTION analytical rigid surfaces. Symmetry boundary conditions are applied at the x=0 and y=0 planes. The

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finite element model of the blank is shown in Figure 1.3.9-4.

Adaptive meshing A single adaptive mesh domain that incorporates the entire blank is used for each model. Symmetry planes are defined as Lagrangian boundary regions (the default), and contact surfaces are defined as sliding boundary regions (the default). Since this problem is quasi-static with relatively small amounts of deformation per increment, the default values for frequency, mesh sweeps, and other adaptive mesh parameters and controls are sufficient.

Results and discussion Figure 1.3.9-5 through Figure 1.3.9-7show the mesh configuration at the end of the forging simulation for Cases 1-3. In each case a quality mesh is maintained throughout the simulation. As the blank flattens out, geometric edges and corners that exist at the beginning of the analysis are broken and adaptive meshing is allowed across them. The eventual breaking of geometric edges and corners is essential for this type of problem to minimize element distortion and optimize element aspect ratios. For comparison purposes Figure 1.3.9-8 shows the deformed mesh for a pure Lagrangian simulation of Case 1 (the axisymmetric model). The mesh is clearly better when continuous adaptive meshing is used. Several diamond-shaped elements with extremely poor aspect ratios are formed in the pure Lagrangian simulation. Adaptive meshing improves the element quality significantly, especially along the top surface of the cup where solution gradients are highest. Figure 1.3.9-9 and Figure 1.3.9-10show contours of equivalent plastic strain at the completion of the forging for the adaptive meshing and pure Lagrangian analyses of Case 1, respectively. Overall plastic strains compare quite closely. Slight differences exist only along the upper surface, where the pure Lagrangian mesh becomes very distorted at the end of the simulation. The time histories of the vertical punch force for the adaptive and pure Lagrangian analyses agree closely for the duration of the forging, as shown in Figure 1.3.9-11.

Input files ale_cupforming_axi.inp Case 1. ale_cupforming_axinodes.inp External file referenced by Case 1. ale_cupforming_axielements.inp External file referenced by Case 1. ale_cupforming_cyl.inp Case 2. ale_cupforming_sph.inp Case 3. lag_cupforming_axi.inp

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Lagrangian solution of Case 1.

Figures Figure 1.3.9-1 Model geometries for each case.

Figure 1.3.9-2 Undeformed mesh for Case 1.

Figure 1.3.9-3 Undeformed mesh for Case 2.

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Figure 1.3.9-4 Undeformed mesh for Case 3.

Figure 1.3.9-5 Deformed mesh for Case 1.

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Figure 1.3.9-6 Deformed mesh for Case 2.

Figure 1.3.9-7 Deformed mesh for Case 3.

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Figure 1.3.9-8 Deformed mesh for Case 1 using a pure Lagrangian formulation.

Figure 1.3.9-9 Contours of equivalent plastic strain for Case 1 using adaptive meshing.

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Figure 1.3.9-10 Contours of equivalent plastic strain for Case 1 using a pure Lagrangian fomulation.

Figure 1.3.9-11 Comparison of time histories for the vertical punch force for Case 1.

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Sample listings

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Listing 1.3.9-1 *HEADING ADAPTIVE MESHING EXAMPLE BULK FORMING OF A CUP. Units - N, mm, sec *NODE, INPUT=ale_cupforming_axinodes.inp ** *ELEMENT, TYPE=CAX4R, ELSET=BLANK, INPUT=ale_cupforming_axielements.inp ** *SOLID SECTION, ELSET=BLANK, MATERIAL=AL1 ** *MATERIAL,NAME=AL1 *ELASTIC,TYPE=ISOTROPIC 4000,0.21 *PLASTIC,HARDENING=ISOTROPIC 5,0 5,0.22 *DENSITY 1.E-4, *ELSET, ELSET=BLANK_T, GEN 441, 450, 1 461, 470, 1 480, 480, 1 *ELSET, ELSET=BLANK_B, GEN 350, 440, 10 150, 240, 10 30, 40, 10 31, 39, 1 11, 20, 1 *NODE 900001,5,90,0 900003,0,-25,0 *NSET, NSET=XSYM, GEN 309,408, 11 67, 166, 11 1, 23, 11 551, 562, 11 *BOUNDARY 900003, 1, 6 900001, 1, 6 XSYM, XSYMM

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*NSET, NSET=REFN 900001, 900003 *AMPLITUDE,NAME=AMP,DEFINITION=SMOOTH STEP 0.0, 0.0, .5, 81.333, 1.,81.333 *RESTART, WRITE, NUMBER=30 *SURFACE,TYPE=ELEMENT,NAME=BLANK_B, REGION TYPE=SLIDING BLANK_B, *SURFACE, TYPE=SEGMENTS, NAME=DIE START, 0.0000000E+00, -0.1300000E+02 CIRCL, 0.7500000E+02 , 0.5200001E+02, 7.06897, 54.612727 *SURFACE,TYPE=ELEMENT,NAME=BLANK_T, REGION TYPE=SLIDING BLANK_T, *SURFACE, TYPE=SEGMENTS, NAME=PUNCH START, 80,144 LINE, 64,144 LINE, 64,134 CIRCL, 0,65.7,-4.44445,134 LINE, -1, 65.7 *RIGID BODY, REF NODE=900001, ANALYTICAL SURFACE =PUNCH *RIGID BODY, REF NODE=900003, ANALYTICAL SURFACE =DIE *STEP *DYNAMIC,EXPLICIT , 1. *BOUNDARY,TYPE=VELOCITY,AMPLITUDE=AMP 900001, 2,2, -1.00 *CONTACT PAIR, INTERACTION=TOP BLANK_T, PUNCH *SURFACE INTERACTION, NAME=TOP *FRICTION, TAUMAX=4. 0.1, *CONTACT PAIR, INTERACTION=BOTTOM BLANK_B, DIE *SURFACE INTERACTION, NAME=BOTTOM *FRICTION, TAUMAX=4. 0.1, *HISTORY OUTPUT, TIME INTERVAL=2.E-3 *NODE HISTORY, NSET=REFN U,RF *FILE OUTPUT,NUMBER INTERVAL=6, TIMEMARKS=YES *EL FILE, ELSET=BLANK_T MISES,PEEQ, *NODE FILE,NSET=REFN U,RF

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*ENERGY FILE *ADAPTIVE MESH, ELSET=BLANK *END STEP

1.3.10 Forging with sinusoidal dies Product: ABAQUS/Explicit This example illustrates the use of adaptive meshing in forging problems that incorporate geometrically complex dies and involve substantial material flow.

Problem description Three different geometric models are considered, as shown in Figure 1.3.10-1. Each model consists of a rigid die and a deformable blank. The cross-sectional shape of the die is sinusoidal with an amplitude and a period of 5 and 10 mm, respectively. The blank is steel and is modeled as a von Mises elastic-plastic material with a Young's modulus of 200 GPa, an initial yield stress of 100 MPa, and a constant hardening slope of 300 MPa. Poisson's ratio is 0.3; the density is 7800 kg/m 3. In all cases the die is moved downward vertically at a velocity of 2000 mm/sec and is constrained in all other degrees of freedom. The total die displacement is 7.6 mm for Cases 1 and 2 and 5.6 mm for Case 3. These displacements represent the maximum possible given the refinement and topology of the initial mesh (if the quality of the mesh is retained for the duration of the analysis). Although each analysis uses a sinusoidal die, the geometries and flow characteristics of the blank material are quite different for each problem.

Case 1: Axisymmetric model The blank is meshed with CAX4R elements and measures 20 ´ 10 mm. The dies are modeled as TYPE=SEGMENTS analytical rigid surfaces. The bottom of the blank is constrained in the z-direction, and symmetry boundary conditions are prescribed at r=0. The initial configuration of the blank and the die is shown in Figure 1.3.10-2.

Case 2: Three-dimensional model The blank is meshed with C3D8R elements and measures 20 ´ 10 ´ 10 mm.The dies are modeled as TYPE=CYLINDER analytical rigid surfaces. The bottom of the blank is constrained in the y-direction, and symmetry boundary conditions are applied at the x=0 and z=10 planes. The finite element model of the blank and the die is shown in Figure 1.3.10-3.

Case 3: Three-dimensional model The blank is meshed with C3D8R elements and measures 20 ´ 10 ´ 20 mm.The dies are modeled as TYPE=REVOLUTION analytical rigid surfaces. The bottom of the blank is constrained in the y-direction, and symmetry boundary conditions are applied at the x=0 and z=10 planes. The finite element model of the blank and the die is shown in Figure 1.3.10-4. The revolved die is displaced 1-522

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upward in the figure from its initial position for clarity.

Adaptive meshing A single adaptive mesh domain that incorporates the entire blank is used for each model. Symmetry planes are defined as Lagrangian boundary regions (the default), and contact surfaces are defined as sliding boundary regions (the default). Because the material flow for each of the geometries is substantial, the frequency and the intensity of adaptive meshing must be increased to provide an accurate solution. The value of the FREQUENCY parameter on the *ADAPTIVE MESH option is reduced from the default of 10 to 5 for all cases. The value of the MESH SWEEPS parameter is increased from the default of 1 to 3 for all cases.

Results and discussion Figure 1.3.10-5 and Figure 1.3.10-6show the deformed mesh and contours of equivalent plastic strain at the completion of the forming step for Case 1. Adaptive meshing maintains reasonable element shapes and aspect ratios. This type of forging problem cannot typically be solved using a pure Lagrangian formulation. Figure 1.3.10-7shows the deformed mesh for Case 2. A complex, doubly curved deformation pattern is formed on the free surface as the material spreads under the die. Element distortion appears to be reasonable. Figure 1.3.10-8and Figure 1.3.10-9 show the deformed mesh and contours of equivalent plastic strain for Case 3. Although the die is a revolved geometry, the three-dimensional nature of the blank gives rise to fairly complex strain patterns that are symmetric with respect to the planes of quarter symmetry.

Input files ale_sinusoid_forgingaxi.inp Case 1. ale_sinusoid_forgingaxisurf.inp External file referenced by Case 1. ale_sinusoid_forgingcyl.inp Case 2. ale_sinusoid_forgingrev.inp Case 3.

Figures Figure 1.3.10-1 Model geometries for each of the three cases.

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Figure 1.3.10-2 Initial configuration for Case 1.

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Figure 1.3.10-3 Initial configuration for Case 2.

Figure 1.3.10-4 Initial configuration for Case 3.

Figure 1.3.10-5 Deformed mesh for Case 1.

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Figure 1.3.10-6 Contours of equivalent plastic strain for Case 1.

Figure 1.3.10-7 Deformed mesh for Case 2.

Figure 1.3.10-8 Deformed mesh for Case 3.

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Figure 1.3.10-9 Contours of equivalent plastic strain for Case 3.

Sample listings

1-527

Static Stress/Displacement Analyses

Listing 1.3.10-1 *HEADING ADAPTIVE MESHING EXAMPLE 2D AXISYMMETRIC FORGING EXAMPLE Units - N, m, sec *RESTART, WRITE, NUMBER=10 *NODE 1, 0.00, 0.00 97, 0.02, 0.00 1165, 0.00, 0.01 1261, 0.02, 0.01 10000, 0.01, 0.02 *NGEN, NSET=BOT 1,97,1 *NGEN, NSET=TOP 1165,1261,1 *NFILL,NSET=NALL BOT, TOP, 12, 97 *NGEN, NSET=CENTER 1,1165,97 *ELEMENT, TYPE=CAX4R 1,1,2,99,98 *ELGEN, ELSET=METAL0 1,95,1,1,12,97,96, *ELEMENT, TYPE=CAX4R 96, 96,97,194,193 *ELGEN, ELSET=METAL1 96,12,97,96 *ELSET,ELSET=METAL METAL0,METAL1 *ELEMENT, TYPE=MASS, ELSET=PMASS 10000, 10000 *MASS, ELSET=PMASS 0.2, *ELSET, ELSET=UPPER, GEN 1057,1152,1 *ELSET, ELSET=SIDE, GEN 96,1152,96 *SOLID SECTION, ELSET=METAL, MATERIAL=STEEL *MATERIAL, NAME=STEEL *ELASTIC 200.E+9, .3

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*PLASTIC 1.E+8, 0.0 3.1E+9, 10.0 *DENSITY 7800.E+1, *BOUNDARY BOT, 2,2 CENTER, 1,1 10000,1,1 10000,6,6 *SURFACE, TYPE=SEGMENTS, NAME=RSURF, FILLET RADIUS=.001 *INCLUDE, INPUT=ale_sinusoid_forgingaxisurf.inp *SURFACE,TYPE=ELEMENT, NAME=TARGET, REGION TYPE=SLIDING UPPER, S3 SIDE, S2 *RIGID BODY, REF NODE=10000, ANALYTICAL SURFACE =RSURF *STEP *DYNAMIC, EXPLICIT ,.00038 *SURFACE INTERACTION, NAME=INTER *CONTACT PAIR, INTERACTION=INTER RSURF, TARGET *BOUNDARY, TYPE=VELOCITY 10000, 2, 2, -20. *HISTORY OUTPUT,TIME=0.0 *EL HISTORY,ELSET=UPPER S,LE,LEP,NE,NEP,PEEQ *ENERGY HISTORY ALLKE,ALLIE,ALLAE,ALLVD,ALLWK,ETOTAL, DT, *FILE OUTPUT,NUMBER INTERVAL=6, TIMEMARKS=YES *EL FILE S,LE,LEP,NE,NEP *NODE FILE U,RF *ENERGY FILE *ADAPTIVE MESH,ELSET=METAL,FREQUENCY=5, MESH SWEEPS=3 *END STEP

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1.3.11 Forging with multiple complex dies Product: ABAQUS/Explicit This example illustrates the use of adaptive meshing in forging problems that use multiple geometrically complex dies. The problem is based on a benchmark presented at the " FEM-Material Flow Simulation in the Forging Industry" workshop.

Problem description The benchmark problem is an axisymmetric forging, but in this example both axisymmetric and three-dimensional geometric models are considered. Each model is shown in Figure 1.3.11-1. Both models consist of two rigid dies and a deformable blank. The blank's maximum radial dimension is 895.2 mm, and its thickness is 211.4 mm. The outer edge of the blank is rounded to facilitate the flow of material through the dies. The blank is modeled as a von Mises elastic-plastic material with a Young's modulus of 200 GPa, an initial yield stress of 360 MPa, and a constant hardening slope of 30 MPa. The Poisson's ratio is 0.3; the density is 7340 kg/m 3. Both dies are fully constrained, with the exception of the top die, which is moved 183.4 mm downward at a constant velocity of 166.65 mm/s.

Case 1: Axisymmetric model The blank is meshed with CAX4R elements. A fine discretization is required in the radial direction because of the geometric complexity of the dies and the large amount of material flow that occurs in that direction. Symmetry boundary conditions are prescribed at r=0. The dies are modeled as TYPE=SEGMENTS analytical rigid surfaces. The initial configuration is shown in Figure 1.3.11-2.

Case 2: Three-dimensional model The blank is meshed with C3D8R elements. A 90° wedge of the blank is analyzed. The level of mesh refinement is the same as that used in the axisymmetric model. Symmetry boundary conditions are applied at the x=0 and z=0 planes. The dies are modeled as TYPE=REVOLUTION analytical rigid surfaces. The initial configuration of the blank only is shown in Figure 1.3.11-3. Although the tools are not shown in the figure, they are originally in contact with the blank.

Adaptive meshing A single adaptive mesh domain that incorporates the entire blank is used for each model. Symmetry planes are defined as Lagrangian boundary regions (the default), and contact surfaces are defined as sliding boundary regions (the default). Since this problem is quasi-static with relatively small amounts of deformation per increment, the defaults for frequency, mesh sweeps, and other adaptive mesh parameters and controls are sufficient.

Results and discussion Figure 1.3.11-4 and Figure 1.3.11-5show the deformed mesh for the axisymmetric case at an intermediate stage (t = 0.209 s) and in the final configuration ( t = 0.35 s), respectively. The elements

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remain well shaped throughout the entire simulation, with the exception of the elements at the extreme radius of the blank, which become very coarse as material flows radially during the last 5% of the top die's travel. Figure 1.3.11-6 shows contours of equivalent plastic strain at the completion of forming. Figure 1.3.11-7and Figure 1.3.11-8 show the deformed mesh for the three-dimensional case at t = 0.209 and t = 0.35, respectively. Although the axisymmetric and three-dimensional mesh smoothing algorithms are not identical, the elements in the three-dimensional model also remain well shaped until the end of the analysis, when the same behavior that is seen in the two-dimensional model occurs. Contours of equivalent plastic strain for the three-dimensional model (not shown) are virtually identical to those shown in Figure 1.3.11-6.

Input files ale_duckshape_forgingaxi.inp Case 1. ale_duckshape_forg_axind.inp External file referenced by the Case 1 analysis. ale_duckshape_forg_axiel.inp External file referenced by the Case 1 analysis. ale_duckshape_forg_axiset.inp External file referenced by the Case 1 analysis. ale_duckshape_forg_axirs.inp External file referenced by the Case 1 analysis. ale_duckshape_forgingrev.inp Case 2.

Reference · Industrieverband Deutscher Schmieden e.V.(IDS), "Forging of an Axisymmetric Disk," FEM-Material Flow Simulation in the Forging Industry, Hagen, Germany, October 1997.

Figures Figure 1.3.11-1 Axisymmetric and three-dimensional model geometries.

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Figure 1.3.11-2 Initial configuration for the axisymmetric model.

Figure 1.3.11-3 Initial configuration mesh for the three-dimensional model.

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Figure 1.3.11-4 The deformed mesh for the axisymmetric model at an intermediate stage.

Figure 1.3.11-5 The deformed mesh for the axisymmetric model at the end of forming.

Figure 1.3.11-6 Contours of equivalent plastic strain for the axisymmetric model at the end of forming.

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Figure 1.3.11-7 The deformed mesh for the three-dimensional model at an intermediate stage.

Figure 1.3.11-8 The deformed mesh for the three-dimensional model at the end of forming.

Sample listings 1-534

Static Stress/Displacement Analyses

Listing 1.3.11-1 *HEADING ADAPTIVE MESHING EXAMPLE FORGING WITH DUCK-SHAPED DIE (AXISYMMETRIC) Units - N, mm, sec *RESTART,WRITE,NUMBER INTERVAL=50 *NODE,INPUT=ale_duckshape_forg_axind.inp *ELEMENT, TYPE=CAX4R , ELSET=BLANK, INPUT=ale_duckshape_forg_axiel.inp *INCLUDE, INPUT=ale_duckshape_forg_axiset.inp *NSET,NSET=SIDE 1,83,164,245,326,407,488,569,650,731,812 *MATERIAL,NAME=BLANK *DENSITY 7340.e-9, *ELASTIC 2.E5, 0.3 *PLASTIC 360., 0. 390., 1. *SOLID SECTION,ELSET=BLANK,MATERIAL=BLANK *ELSET,ELSET=OUT,GEN 1,10,1 *NSET,NSET=REF 2000,2001 *SURFACE,TYPE=ELEMENT,NAME=BLANK, REGION TYPE=SLIDING BLANK, *SURFACE, NAME=TOP, TYPE=SEGMENTS *INCLUDE,INPUT=ale_duckshape_forg_axirs.inp *RIGID BODY,REF NODE=2000, ANALYTICAL SURFACE =BOT_1 *RIGID BODY,REF NODE=2001, ANALYTICAL SURFACE =TOP *STEP *DYNAMIC,EXPLICIT ,0.105 *BOUNDARY 2000,1 2000,3,6 2001,1,6 SIDE,1,1

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*BOUNDARY,TYPE=VELOCITY 2000,2,2,-166.652 *CONTACT PAIR,INTERACTION=SMOOTH BLANK,TOP BLANK,BOT_1 *SURFACE INTERACTION,NAME=SMOOTH *FILE OUTPUT,TIMEMARKS=YES,NUM=4 *EL FILE,ELSET=OUT PEEQ,MISES *NODE FILE,NSET=REF U, *ADAPTIVE MESH,ELSET=BLANK *END STEP

1.3.12 Flat rolling: transient and steady-state Product: ABAQUS/Explicit This example illustrates the use of adaptive meshing to simulate a rolling process using both transient and steady-state approaches, as shown in Figure 1.3.12-1. A transient flat rolling simulation is performed using three different methods: a "pure" Lagrangian approach, an adaptive meshing approach using a Lagrangian domain, and a mixed Eulerian-Lagrangian adaptive meshing approach in which material upstream from the roller is drawn from an Eulerian inflow boundary but the downstream end of the blank is handled in a Lagrangian manner. In addition, a steady-state flat rolling simulation is performed using an Eulerian adaptive mesh domain as a control volume and defining inflow and outflow Eulerian boundaries. Solutions using each approach are compared.

Problem description For each analysis case quarter symmetry is assumed; the model consists of a rigid roller and a deformable blank. The blank is meshed with C3D8R elements. The roller is modeled as an analytical rigid surface using the *SURFACE, TYPE=CYLINDER and *RIGID BODY options. The radius of the cylinder is 175 mm. Symmetry boundary conditions are prescribed on the right (z=0 plane) and bottom (y=0 plane) faces of the blank. Coulomb friction with a friction coefficient of 0.3 is assumed between the roller and the plate. All degrees of freedom are constrained on the roller except rotation about the z-axis, where a constant angular velocity of 6.28 rad/sec is defined. For each analysis case the blank is given an initial velocity of 0.3 m/s in the x-direction to initiate contact. The blank is steel and is modeled as a von Mises elastic-plastic material with isotropic hardening. The Young's modulus is 150 GPa, and the initial yield stress is 168.2 MPa. The Poisson's ratio is 0.3; the density is 7800 kg/m3. The *FIXED MASS SCALING option is used to scale the masses of all the blank elements by a factor of 2750 so that the analysis can be performed more economically. This scaling factor represents an approximate upper bound on the mass scaling possible for this problem, above which significant inertial effects would be generated.

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The *STEADY STATE DETECTION option is used to define the criteria for stopping the rolling analyses based on the achievement of a steady-state condition. The criteria used require the satisfaction of the steady-state detection norms of equivalent plastic strain, spread, force, and torque within the default tolerances. The exit plane for each norm is defined as the plane passing through the center of the roller with the normal to the plane coincident with the rolling direction. The SAMPLING parameter is set to PLANE BY PLANE for Case 1 through Case 3 for the steady-state detection norms to be evaluated as each plane of elements passes the exit plane. Case 4 requires that the SAMPLING parameter is set to UNIFORM since the initial mesh is roughly stationary due to the initial geometry and the inflow and outflow Eulerian boundaries. The finite element models used for each analysis case are shown in Figure 1.3.12-2. A description of each model and the adaptive meshing techniques used follows:

Case 1: Transient simulation--pure Lagrangian approach The blank is initially rectangular and measures 224 ´ 20 ´ 50 mm. No adaptive meshing is performed. The analysis is run until steady-state conditions are achieved.

Case 2: Transient simulation--Lagrangian adaptive mesh domain The finite element model for this case is identical to that used for Case 1, with the exception that a single adaptive mesh domain that incorporates the entire blank is defined to allow continuous adaptive meshing. Symmetry planes are defined as Lagrangian surfaces (the default), and the contact surface on the blank is defined as a sliding surface (the default). The analysis is run until steady-state conditions are achieved.

Case 3: Transient simulation--mixed Eulerian-Lagrangian approach This analysis is performed on a relatively short initial blank measuring 65 ´ 20 ´ 50 mm. Material is continuously drawn by the action of the roller on the blank through an inflow Eulerian boundary defined on the upstream end. The blank is meshed with the same number of elements as in Cases 1 and 2 so that similar aspect ratios are obtained as the blank lengthens and steady-state conditions are achieved. An adaptive mesh domain is defined that incorporates the entire blank. Because it contains at least one Eulerian surface, this domain is considered Eulerian for the purpose of setting parameter defaults. However, the analysis model has both Lagrangian and Eulerian aspects. The amount of material flow with respect to the mesh will be large at the inflow end and small at the downstream end of the domain. To account for the Lagrangian motion of the downstream end, the MESHING PREDICTOR option on the *ADAPTIVE MESH CONTROLS option is changed from the default of PREVIOUS to CURRENT for this problem. To mesh the inflow end accurately and to perform the analysis economically, the FREQUENCY parameter is set to 5 and the MESH SWEEPS parameter is set to 5. As in Case 2, symmetry planes are defined as Lagrangian boundary regions (the default), and the contact surface on the blank is defined as a sliding boundary region (the default). In addition, an Eulerian boundary region is defined on the upstream end using the *SURFACE, REGION

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TYPE=EULERIAN option. Adaptive mesh constraints are defined on the Eulerian surface using the *ADAPTIVE MESH CONSTRAINT option to hold the inflow surface mesh completely fixed while material is allowed to enter the domain normal to the surface. The *EQUATION option is used to ensure that the velocity normal to the inflow boundary is uniform across the surface. The velocity of nodes in the direction tangential to the inflow boundary surface is constrained.

Case 4: Steady-state simulation--Eulerian adaptive mesh domain This analysis employs a control volume approach in which material is drawn from an inflow Eulerian boundary and is pushed out through an outflow boundary by the action of the roller. The blank geometry for this analysis case is defined such that it approximates the shape corresponding to the steady-state solution: this geometry can be thought of as an "initial guess" to the solution. The blank initially measures 224 mm in length and 50 mm in width and has a variable thickness such that it conforms to the shape of the roller. The surface of the blank transverse to the rolling direction is not adjusted to account for the eventual spreading that will occur in the steady-state solution. Actually, any reasonable initial geometry will reach a steady state, but geometries that are closer to the steady-state geometry often allow a solution to be obtained in a shorter period of time. As in the previous two cases an adaptive mesh domain is defined on the blank, symmetry planes are defined as Lagrangian surfaces (the default), and the contact surface is defined as a sliding surface (the default). Inflow and outflow Eulerian surfaces are defined on the ends of the blank using the same techniques as in Case 3, except that for the outflow boundary adaptive mesh constraints are applied only normal to the boundary surface and no material constraints are applied tangential to the boundary surface. To improve the computational efficiency of the analysis, the frequency of adaptive meshing is increased to every fifth increment because the Eulerian domain undergoes very little overall deformation and the material flow speed is much less than the material wave speed. This frequency will cause the mesh at Eulerian boundaries to drift slightly. However, the amount of drift is extremely small and does not accumulate. There is no need to increase the mesh sweeps because this domain is relatively stationary and the default MESHING PREDICTOR setting for Eulerian domains is PREVIOUS. Very little mesh smoothing is required.

Results and discussion The final deformed configurations of the blank for each of the three transient cases are shown in Figure 1.3.12-3. The transient cases have reached a steady-state solution and have been terminated based on the criteria defined using the *STEADY STATE DETECTION option. Steady-state conditions are determined to have been reached when the reaction forces and moments on the roller have stabilized and the cross-sectional shape and distribution of equivalent plastic strain under the roller become constant over time. When using the *STEADY STATE DETECTION option, these conditions imply that the force, moment, spread, and equivalent plastic strain norms have stabilized such that the changes in the norms over three consecutive sampling intervals have fallen below the user-prescribed tolerances. See ``Steady-state detection,'' Section 7.7.1 of the ABAQUS/Explicit User's Manual, for a detailed discussion on the definition of the norms. Contours of equivalent plastic strain for each of the three transient cases are in good agreement and are shown in the final configuration of each blank in

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Figure 1.3.12-4. Figure 1.3.12-5 shows the initial and final mesh configurations at steady state. With the exception of Case 3 all analyses were terminated using the default steady-state norm tolerances. Case 3 required that the force and torque norm tolerances be increased from .005 to .01 due to the force and torque at the roller being rather noisy. To compare the results from the transient and steady-state approaches, the steady-state detection norms are summarized for each case in Table 1.3.12-1. The table shows a comparison of the values of the steady-state detection norms after the analyses have been terminated. The only significant difference is in the value of the spread norm for Case 4, which is higher than the others. The spread norm is defined as the largest of the second principle moments of inertia of the workpiece's cross-section. Since the spread norm is a cubic function of the lateral deformation of the workpiece, rather small differences in displacements between the test cases can lead to significant differences in the spread norms. Time history plots of the steady-state detection norms are also shown. Figure 1.3.12-9 and Figure 1.3.12-10 show time history plots of the steady-state force and torque norms, respectively, for all cases. The force and torque norms are essentially running averages of the force and moment on the roller and show good agreement for all four test cases. Figure 1.3.12-7 and Figure 1.3.12-8 show time history plots of the steady-state equivalent plastic strain and spread norms, respectively, for all cases. The equivalent plastic strains norms are in good agreement for all cases.

Input files lag_flatrolling.inp Case 1. ale_flatrolling_noeuler.inp Case 2. ale_flatrolling_inlet.inp Case 3. ale_flatrolling_inletoutlet.inp Case 4.

Table Table 1.3.12-1 Comparison of steady-state detection norms. Force norm Formulatio Spread norm Effective n plastic strain norm Case 1 1.349 E-7 .8037 -1.43 E6 Case 2 1.369 E-7 .8034 -1.43 E6 Case 3 1.365 E-7 .8018 -1.43 E6 Case 4 1.485 E-7 .8086 -1.40 E6

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Torque norm

3.59 E4 3.55 E4 3.61 E4 3.65 E4

Static Stress/Displacement Analyses

Figures Figure 1.3.12-1 Diagram illustrating the four analysis approaches used in this problem.

Figure 1.3.12-2 Initial configurations for each case.

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Figure 1.3.12-3 Deformed mesh for Cases 1-3.

1-541

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Figure 1.3.12-4 Contours of equivalent plastic strain for Cases 1-3.

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Figure 1.3.12-5 Deformed mesh for Case 4 (shown with initial mesh for comparison).

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Figure 1.3.12-6 Contours of equivalent plastic strain for Case 4.

Figure 1.3.12-7 Comparison of equivalent plastic strain norm versus time for all cases.

Figure 1.3.12-8 Comparison of spread norm versus time for all cases.

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Figure 1.3.12-9 Comparison of force norm versus time for all cases.

Figure 1.3.12-10 Comparison of torque norm versus time for all cases.

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Sample listings

1-546

Static Stress/Displacement Analyses

Listing 1.3.12-1 *HEADING ADAPTIVE MESHING EXAMPLE FLAT ROLLING - ADAPTIVE MESH, EULERIAN INLET NODES Units - N, m, second ** *RESTART,W,N=20 *NODE 1, -.0851, 0.00000, 0.00000 253, -.0851, 0.02000, 0.00000 42, -.0200, 0.00000, 0.00000 294, -.0200, 0.02000, 0.00000 2059,-.0851, 0.00000, 0.05000 2100,-.0200, 0.00000, 0.05000 2311,-.0851, 0.02000, 0.05000 2352,-.0200, 0.02000, 0.05000 *NGEN, NSET=BOT1 1,42,1 *NGEN, NSET=TOP1 253,294,1 *NFILL,NSET=FRONT BOT1, TOP1, 6, 42 *NGEN, NSET=BOT2 2059,2100,1 *NGEN, NSET=TOP2 2311,2352, 1 *NFILL,NSET=BACK BOT2,TOP2, 6, 42 *NFILL, NSET=BAR FRONT,BACK, 7, 294 *ELEMENT, TYPE=C3D8R 1, 1, 2, 295, 296, 338, *ELGEN, ELSET=BAR 1,41,1,1,6,42,41,7,294,246 *NSET,NSET=BOT,GEN 1,42,1 295,336,1 589,630,1 883,924,1 1177,1218,1

44, 337

1-547

43,

Static Stress/Displacement Analyses

1471,1512,1 1765,1806,1 2059,2100,1 *SOLID SECTION,ELSET=BAR,MAT=C15,CONTROLS=SECT 1., *SECTION CONTROLS, NAME=SECT,HOURGLASS=STIFFNESS ** *MATERIAL,NAME=C15 *ELASTIC 1.5E11,.3 *PLASTIC 168.72E06,0 219.33E06,0.1 272.02E06,0.2 308.53E06,0.3 337.37E06,0.4 361.58E06,0.5 382.65E06,0.6 401.42E06,0.7 418.42E06,0.8 434.01E06,0.9 448.45E06,1.0 *DENSITY 7.85E3, *********************************************** **** ROLL *NODE,NSET=REF 10000, 0.0409 , 0.185 *INITIAL CONDITIONS,TYPE=VELOCITY BAR,1,.30 *NSET,NSET=LEFT,GEN 1,294,1 *BOUNDARY LEFT,ZSYMM BOT,YSYMM *ELSET,ELSET=SIDE,GEN 1477,1722,1 *ELSET,ELSET=TOP,GEN 206,246,1 452,492,1 698,738,1 944,984,1 1190,1230,1

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1436,1476,1 1682,1722,1 *NSET,NSET=EULER,GEN 1,253,42 295,547,42 589,841,42 883,1135,42 1177,1429,42 1471,1723,42 1765,2017,42 2059,2311,42 *NSET,NSET=EULERINT,GEN 337,505,42 631,799,42 925,1093,42 1219,1387,42 1513,1681,42 1807,1975,42 *NSET, NSET=EULERSMALL1, GEN 295,547,42 589,841,42 883,1135,42 1177,1429,42 1471,1723,42 1765,2017,42 2059,2311,42 *NSET, NSET=EULERSMALL2, GEN 43,253,42 337,547,42 631,841,42 925,1135,42 1219,1429,42 1513,1723,42 1807,2017,42 2101,2311,42 *NSET,NSET=EQN,GEN 43,253,42 295,547,42 589,841,42 883,1135,42 1177,1429,42 1471,1723,42 1765,2017,42

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Static Stress/Displacement Analyses

2059,2311,42 *EQUATION 2, EQN,1,1.,1,1,-1. *ELSET, ELSET=EULER, GEN 1,1477,246 42,1518,246 83,1559,246 124,1600,246 165,1641,246 206,1682,246 *SURFACE, NAME=SURF1, REGION TYPE=SLIDING TOP,S5 SIDE,S2 *SURFACE, REGION TYPE=EULERIAN, NAME=EULER1 EULER,S6 *SURFACE,TYPE=CYLINDER,NAME=RIGID, FILLET RADIUS=.001 0.0409 , 0.185, 0.0, 0.05, 0.185,0.0 0.0409 , 0.185, -0.05 START,0.0,-0.175 CIRCL,-0.175,0.0,0.0,0.0 CIRCL,0.0,0.175,0.0,0.0 *RIGID BODY, REFNODE=10000, ANALYTICAL SURFACE = RIGID *STEP *DYNAMIC,EXPLICIT ,0.50 *STEADY STATE DETECTION,ELSET=BAR, SAMPLING=PLANE BY PLANE 1.0, 0., 0., .1, 0.0, 0.0 *STEADY STATE CRITERIA SSPEEQ, , .0409, 0., 0. SSSPRD, , .0409, 0., 0. SSTORQ, .01, .0409, 0., 0., 10000, 0., 0., 1. SSFORC, .01, .0409, 0., 0., 10000, 0., 1., 0. *BOUNDARY 10000,1,5 *BOUNDARY,TYPE=VELOCITY 10000,6,6,6.2832 *BOUNDARY,TYPE=VELOCITY,REGION TYPE=EULERIAN EULERSMALL1,3,3,0.0 EULERSMALL2,2,2,0.0

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*FIXED MASS SCALING, FACTOR=2750. *SURFACE INTERACTION,NAME=FRICT *FRICTION 0.3, *CONTACT PAIR,INTERACTION=FRICT SURF1,RIGID *FILE OUTPUT, NUMBER INTERVAL=10, TIMEMARKS=YES *EL FILE, ELSET=TOP MISES,PEEQ, *NODE FILE,NSET=REF U,RF *OUTPUT, FIELD,NUMBER INTERVAL=10 *ELEMENT OUTPUT MISES, PEEQ *NODE OUTPUT U, *OUTPUT,HISTORY,TIME INTERVAL=1.E-4 *NODE OUTPUT,NSET=REF RF2,RM3 *INCREMENTATION OUTPUT SSPEEQ,SSSPRD,SSTORQ,SSFORC *ADAPTIVE MESH, ELSET=BAR,FREQUENCY=5, MESH SWEEPS=5,CONTROLS=ALE *ADAPTIVE MESH CONSTRAINT EULER,1,3,0.0 *ADAPTIVE MESH CONTROLS,NAME=ALE,MESHING =CURRENT *ENDSTEP

1.3.13 Section rolling Product: ABAQUS/Explicit This example illustrates the use of adaptive meshing in a transient simulation of section rolling. Results are compared to a pure Lagrangian simulation.

Problem description This analysis shows a stage in the rolling of a symmetric I-section. Because of the cross-sectional shape of the I-section, two planes of symmetry exist and only a quarter of the section needs to be modeled. The quarter-symmetry model, shown in Figure 1.3.13-1, consists of two rigid rollers and a blank. Roller 1 has a radius of 747 mm, and roller 2 has a radius of 452 mm. The blank has a length of 775 mm, a web half-width of 176.5 mm, a web half-thickness of 24 mm, and a variable flange thickness. The finite element model is shown in Figure 1.3.13-2. The blank is meshed with C3D8R elements.

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Symmetry boundary conditions are applied on the y and z symmetry planes of the blank. The rollers are modeled as TYPE=REVOLUTION analytical rigid surfaces. Roller 1 has all degrees of freedom constrained except rotation about the z-axis, where a constant angular velocity of 5 rad/sec is specified. Roller 2 has all degrees of freedom constrained except rotation about the y-axis. An initial velocity of 5602.5 mm/sec in the negative x-direction is applied to the blank to initiate contact between the blank and the rollers. This velocity corresponds to the velocity of the rollers at the point of initial contact. The *VARIABLE MASS SCALING, TYPE=BELOW MIN option is used to scale the masses of all the blank elements so that a desired minimum stable time increment is achieved initially and the stable time increment does not fall below this minimum throughout the analysis. The loading rates and mass scaling definitions are such that a quasi-static solution is generated. The blank is steel and is modeled as a von Mises elastic-plastic material with a Young's modulus of 212 GPa, an initial yield stress of 80 MPa, and a constant hardening slope of 258 MPa. Poisson's ratio is 0.3; the density is 7833 kg/m 3. Coulomb friction with a friction coefficient of 0.3 is assumed between the rollers and the blank.

Adaptive meshing Adaptive meshing can improve the solution and mesh quality for section rolling problems that involve large deformations. A single adaptive mesh domain that incorporates the entire blank is defined. Symmetry planes are defined as Lagrangian boundary regions (the default), and the contact surface on the blank is defined as a sliding boundary region (the default). The default values are used for all adaptive mesh parameters and controls.

Results and discussion Figure 1.3.13-3 shows the deformed configuration of the blank when continuous adaptive meshing is used. For comparison purposes Figure 1.3.13-4 shows the deformed configuration for a pure Lagrangian simulation. The mesh at the flange-web interface is distorted in the Lagrangian simulation, but the mesh remains nicely proportioned in the adaptive mesh analysis. A close-up view of the deformed configuration of the blank is shown for each analysis in Figure 1.3.13-5 and Figure 1.3.13-6 to highlight the differences in mesh quality. Contours of equivalent plastic strain for each analysis are shown in Figure 1.3.13-5 and Figure 1.3.13-6. The plastic strain distributions are very similar. Figure 1.3.13-7 and Figure 1.3.13-8show time history plots for the y-component of reaction force and the reaction moment about the z-axis, respectively, for roller 1. The results for the adaptive mesh simulation compare closely to those for the pure Lagrangian simulation.

Input files ale_rolling_section.inp Analysis that uses adaptive meshing. ale_rolling_sectionnode.inp External file referenced by the adaptive mesh analysis.

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ale_rolling_sectionelem.inp External file referenced by the adaptive mesh analysis. ale_rolling_sectionnelset.inp External file referenced by the adaptive mesh analysis. ale_rolling_sectionsurf.inp External file referenced by the adaptive mesh analysis. lag_rolling_section.inp Lagrangian analysis.

Figures Figure 1.3.13-1 Geometry of the quarter-symmetry blank and the rollers.

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Figure 1.3.13-2 Quarter-symmetry finite element model.

Figure 1.3.13-3 Deformed blank for the adaptive mesh simulation.

Figure 1.3.13-4 Deformed blank for the pure Lagrangian simulation.

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Figure 1.3.13-5 Close-up of the deformed blank for the adaptive mesh simulation.

Figure 1.3.13-6 Close-up of the deformed blank for the pure Lagrangian simulation.

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Figure 1.3.13-7 Contours of equivalent plastic strain for the adaptive mesh simulation.

Figure 1.3.13-8 Contours of equivalent plastic strain for the pure Lagrangian simulation.

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Figure 1.3.13-9 Time history of the reaction force in the y-direction at the reference node of Roller 1.

Figure 1.3.13-10 Time history of the reaction moment about the z-axis at the reference node of Roller 1.

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Sample listings

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Listing 1.3.13-1 *HEADING ADAPTIVE MESHING EXAMPLE SECTION ROLLING Units - N, m, seconds *RESTART, WRITE,NUMBER=10 *SYSTEM -0.0949794,0.,0., -0.0949794,1.,0. -0.0949794,0.,1. *INCLUDE, INPUT=ale_rolling_sectionnode.inp *INCLUDE, INPUT=ale_rolling_sectionelem.inp *INCLUDE, INPUT=ale_rolling_sectionnelset.inp ** SECTION: METAL *SOLID SECTION, ELSET=METAL, MATERIAL=STEEL 1., *MATERIAL, NAME=STEEL *DENSITY 7833., *ELASTIC 2.12E+11, 0.281 *PLASTIC 8e+07, 0., 2.35e+08, 0.6 *ELEMENT, TYPE=ROTARYI, ELSET=ROTI 50000,10117 *ROTARY INERTIA, ELSET=ROTI 1.E-4,11.05,1.E-4 ** INITIAL CONDITION: VELOCITY *INITIAL CONDITIONS, TYPE=VELOCITY ROLNODES, 1, -4.187 ROLNODES, 2, 0. ROLNODES, 3, 0. ** STEP: STEP-1 ** *SURFACE, TYPE=REVOLUTION, NAME=VROLSURF -0.1800, 0., 0.6221, -0.1800, 100., 0.6221 START, 0.2, 0.2 LINE, 0.4354, 0.2 LINE, 0.45182, 0.035821 CIRCL, 0.45182,-0.035821, 0.093604, 6.9389E-18 LINE, 0.4354, -0.2 LINE, 0.2, -0.2

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*SURFACE, TYPE=REVOLUTION, NAME=HROLSURF -0.1800, 0.760, 0., -0.180, 0.760, 100. START, 0.547, 0.137 LINE, 0.71051, 0.12261 CIRCL, 0.747, 0.082765, 0.707, 0.082765 LINE, 0.747, -0.082765 CIRCL, 0.71051, -0.12261, 0.707, -0.082765 LINE, 0.547, -0.137 *RIGID BODY, REF NODE=10116, ANALYTICAL SURFACE =HROLSURF *RIGID BODY, REF NODE=10117, ANALYTICAL SURFACE =VROLSURF *INCLUDE, INPUT=ale_rolling_sectionsurf.inp *STEP *DYNAMIC, EXPLICIT , 0.227 *BOUNDARY, OP=NEW HROLREF, 1, 1 HROLREF, 2, 2 HROLREF, 3, 3 HROLREF, 4, 4 HROLREF, 5, 5 *BOUNDARY, OP=NEW, TYPE=VELOCITY HROLREF, 6, 6,-5. *BOUNDARY, OP=NEW VROLREF, 1, 1 VROLREF, 2, 2 VROLREF, 3, 3 VROLREF, 4, 4 VROLREF, 6, 6 *BOUNDARY, OP=NEW YSYM, YSYMM *BOUNDARY, OP=NEW ZSYM, ZSYMM *SURFACE INTERACTION, NAME=PRO-1 *FRICTION 0.3, ** INTERACTION: HOR_ROLL *CONTACT PAIR, INTERACTION=PRO-1 FLANGSURF, HROLSURF ** INTERACTION: VER_ROLL *CONTACT PAIR, INTERACTION=PRO-1 WEBSURF, VROLSURF

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*FILE OUTPUT, NUMBER INTERVAL=10, TIME MARKS=NO *NODE FILE, NSET=HROLREF RF, *HISTORY OUTPUT, TIME INTERVAL=0.0026 *ENERGY HISTORY ALLAE, ALLCD, ALLFD, ALLIE, ALLKE, ALLPD, ALLSE ALLVD, ALLWK, ETOTAL *NODE HISTORY, NSET=HROLREF RF2, RF3, RM2, RM3 *NODE HISTORY, NSET=VROLREF RF2, RF3, RM3, RM2 ** MASS SCALING: WHOLEMODEL 1 ** ROLLING *VARIABLE MASS SCALING,TYPE=BELOW MIN, FREQUENCY =50, ELSET=METAL,DT=2.E-5 *ADAPTIVE MESH,ELSET=METAL *END STEP

1.3.14 Ring rolling Product: ABAQUS/Explicit This example illustrates the use of adaptive meshing in a two-dimensional rolling simulation. Results are compared to those obtained using a pure Lagrangian approach.

Problem description Ring rolling is a specialized process typically used to manufacture parts with revolved geometries such as bearings. The three-dimensional rolling setup usually includes a freely mounted, idle roll; a continuously rotating driver roll; and guide rolls in the rolling plane. Transverse to the rolling plane, conical rolls are used to stabilize the ring and provide a forming surface in the out-of-plane direction. In this example a two-dimensional, plane stress idealization is used that ignores the effect of the conical rolls. A schematic diagram of the ring and the surrounding tools is shown in Figure 1.3.14-1. The driver roll has a diameter of 680 mm, and the idle and guide rolls have diameters of 102 mm. The ring has an initial inner diameter of 127.5 mm and a thickness of 178.5 mm. The idle and driver rolls are arranged vertically and are in contact with the inner and outer surfaces of the ring, respectively. The driver roll is rotated around its stationary axis, while the idle roll is moved vertically downward at a specified feed rate. For this simulation the x-y motion of the guide rolls is determined a priori and is prescribed so that the rolls remain in contact with the ring throughout the analysis but do not exert appreciable force on it. In practice the guide rolls are usually connected through linkage systems, and their motion is a function of both force and displacement. The ring is meshed with CPS4R elements, as shown in Figure 1.3.14-2. The ring is steel and is

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modeled as a von Mises elastic-plastic material with a Young's modulus of 150 GPa, an initial yield stress of 168.7 MPa, and a constant hardening slope of 884 MPa. The Poisson's ratio is 0.3; the density is 7800 kg/m3. The analysis is run so that the ring completes approximately 20 revolutions (16.5 seconds). The rigid rolls are modeled as TYPE=SEGMENTS analytical rigid surfaces. The driver roll is rotated at a constant angular velocity of 3.7888 rad/sec about the z-axis, while the idle roll has a constant feed rate of 4.9334 mm/sec and is free to rotate about the z-axis. All other degrees of freedom for the driver and idle rolls are constrained. A friction coefficient of 0.5 is defined at the blank-idle roll and blank-drive roll interfaces. Frictionless contact is used between the ring and guide rolls, and the rotation of the guide rolls is constrained since the actual guide rolls are free to rotate and exert negligible torque on the ring. To obtain an economical solution, the *FIXED MASS SCALING option is used to scale the masses of all elements in the ring by a factor of 2500. This scaling factor represents a reasonable upper limit on the mass scaling possible for this problem, above which significant inertial effects would be generated. Furthermore, since the two-dimensional model does not contain the conical rolls, the ring oscillates from side to side even under the action of the guide rolls. An artificial viscous pressure of 300 MPa sec/m is applied on the inner and outer surfaces of the ring to assist the guide rolls in preserving the circular shape of the ring. The pressure value was chosen by trial and error.

Adaptive meshing A single adaptive mesh domain that incorporates the ring is defined. Contact surfaces on the ring are defined as sliding boundary regions (the default). Because of the large number of increments required to simulate 20 revolutions, the deformation per increment is very small. Therefore, the frequency of adaptive meshing is changed from the default of 10 to every 50 increments. The cost of adaptive meshing at this frequency is negligible compared to the underlying analysis cost.

Results and discussion Figure 1.3.14-3 shows the deformed configuration of the ring after completing 20 revolutions with continuous adaptive meshing. High-quality element shapes and aspect ratios are maintained throughout the simulation. Figure 1.3.14-4 shows the deformed configuration of the ring when a pure Lagrangian simulation is performed. The pure Lagrangian mesh is distorted, especially at the inner radius where elements become skewed and very small in the radial direction. Figure 1.3.14-5 and Figure 1.3.14-6show time history plots for the y-component of reaction force on the idle roll and the reaction moment about the z-axis for the driver roll, respectively, for both the adaptive mesh and pure Lagrangian approaches. Although the final meshes are substantially different, the roll force and torque match reasonably well. For both the adaptive and pure Lagrangian solutions the plane stress idealization used here results in very localized through-thickness straining at the inner and outer radii of the ring. This specific type of localized straining is unique to plane stress modeling and does not occur in ring rolling processes. It is also not predicted by a three-dimensional finite element model. If adaptivity is used and refined meshing is desired to capture strong gradients at the inner and outer extremities, the initially uniform

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mesh can be replaced with a graded mesh. Although not shown here, a graded mesh concentrates element refinement in areas of strong gradients. The adaptive meshing technique preserves the initial grading when the SMOOTHING OBJECTIVE=GRADED parameter is used on the *ADAPTIVE MESH CONTROLS option.

Input files ale_ringroll_2d.inp Analysis that uses adaptive meshing. ale_ringroll_2dnode.inp External file referenced by the adaptive mesh analysis. ale_ringroll_2delem.inp External file referenced by the adaptive mesh analysis. guideamp.inp External file referenced by the adaptive mesh analysis. lag_ringroll_2d.inp Lagrangian analysis.

Figures Figure 1.3.14-1 Model geometry for the two-dimensional ring rolling analysis.

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Figure 1.3.14-2 Initial mesh configuration.

Figure 1.3.14-3 Deformed configuration after 20 revolutions using adaptive meshing.

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Figure 1.3.14-4 Deformed configuration after 20 revolutions using a pure Lagrangian approach.

Figure 1.3.14-5 Time history of the reaction force in the y-direction for the idle roll.

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Figure 1.3.14-6 Time history of the reaction moment about the z-axis for the driver roll.

Sample listings

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Listing 1.3.14-1 *HEADING ADAPTIVE MESHING EXAMPLE ROLLING OF A RING IN 2D (PLANE STRESS) WITH SMALL VISCOUS PRESSURE TO STABILIZE THE RING. Units - N, m, sec ** *NODE, INPUT=ale_ringroll_2dnode.inp *ELEMENT, TYPE=CPS4R, ELSET=RING, INPUT=ale_ringroll_2delem.inp *SOLID SECTION, ELSET=RING, MAT=C15, CONTROLS=SECT 0.119, *SECTION CONTROLS, NAME=SECT,HOURGLASS=STIFFNESS *MATERIAL,NAME=C15 *ELASTIC 1.5E11,.3 *PLASTIC 168.72E6,0.0 1053.00E6,1.0 *DENSITY 7.85E3, ***************** rigid bodies *** Driver roll *BOUNDARY, OP=NEW 1660, 1,, 0. 1660, 2,, 0. 1660, 3,, 0. ** *BOUNDARY, OP=NEW 1660, 4,, 0. 1660, 5,, 0. ** *** Idle roll *BOUNDARY, OP=NEW 1648, 1,, 0. 1648, 3,, 0. ** *BOUNDARY, OP=NEW 1648, 4,, 0. 1648, 5,, 0. *ELEMENT, TYPE=ROTARYI, ELSET=IDLEI 9000, 1648

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*ROTARY INERTIA, ELSET=IDLEI **0., 0., 0.3176634 0., 0., 0.3176634E4 *** *** Left guide roll *BOUNDARY, OP=NEW 1651, 3,, 0. 1651, 4,, 0. 1651, 6,, 0. *ELEMENT, TYPE=MASS, ELSET=MASSGUIDE 8001, 1651 *** Right guide roll *BOUNDARY, OP=NEW 1649, 3,, 0. 1649, 4,, 0. 1649, 6,, 0. ** *ELEMENT, TYPE=MASS, ELSET=MASSGUIDE 8002, 1649 *** *MASS, ELSET=MASSGUIDE **61.06563 61.06563E4, ************** *AMPLITUDE, NAME=IDLEVEL, DEFINITION=SMOOTH STEP 0., 1., 16.3, 1.0, 16.5, 0.0 *AMPLITUDE, NAME=DRVRVEL, DEFINITION=SMOOTH STEP 0., 0., 0.2, 1.0, 16.5, 1.0 *INCLUDE, INPUT=guideamp.inp *SURFACE,TYPE=ELEMENT, NAME=EXTERIOR, REGION TYPE=SLIDING EXTERIOR, *SURFACE, TYPE=SEGMENTS, NAME=LFTGUIDE START,-.39192431,.77060147,-.39192431,.87260147 CIRCL,-.49392431,.87260147,-.39192431,.87260147 CIRCL,-.39192431,.97460147,-.39192431,.87260147 CIRCL,-.28992431,.87260147,-.39192431,.87260147 CIRCL,-.39192431,.77060147,-.39192431,.87260147 *SURFACE,TYPE=ELEMENT, NAME=INTERIOR, REGION TYPE=SLIDING INTERIOR, *SURFACE, TYPE=SEGMENTS, NAME=RHTGUIDE START,.39192431,.77060147,.39192431,.87260147

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CIRCL,.28992431,.87260147,.39192431,.87260147 CIRCL,.39192431,.97460147,.39192431,.87260147 CIRCL,.49392431,.87260147,.39192431,.87260147 CIRCL,.39192431,.77060147,.39192431,.87260147 *SURFACE, TYPE=SEGMENTS, NAME=DRIVER START, 0., -.680 CIRCL,-0.680, 0., 0., 0. CIRCL, 0., 0.680, 0., 0. CIRCL, 0.680, 0., 0., 0. CIRCL, 0., -0.680, 0.,0. *SURFACE, TYPE=SEGMENTS, NAME=IDLE START, 0., 0.8585 CIRCL,-0.102, 0.9605, 0., 0.9605 CIRCL, 0., 1.0625, 0., 0.9605 CIRCL, 0.102, 0.9605, 0., 0.9605 CIRCL, 0., 0.8585, 0., 0.9605 *RIGID BODY, REF NODE=1648, ANALYTICAL SURFACE =IDLE *RIGID BODY, REF NODE=1649, ANALYTICAL SURFACE =RHTGUIDE *RIGID BODY, REF NODE=1651, ANALYTICAL SURFACE =LFTGUIDE *RIGID BODY, REF NODE=1660, ANALYTICAL SURFACE =DRIVER *STEP Ring Rolling process in Plane Strain. Reduce the thickness of the ring by 55.0% *DYNAMIC, EXPLICIT , 16.5, *FIXED MASS SCALING, ELSET=RING,FACTOR=2500. *********************** rigid surfaces: ********* driver roll *BOUNDARY, TYPE=VELOCITY, AMP=DRVRVEL 1660, 6, 6, -3.78884 ********* idle roll *BOUNDARY, TYPE=VELOCITY, AMP=IDLEVEL 1648, 2, 2, -4.933417E-3 ********* left guide roll *BOUNDARY,TYPE=DISPLACEMENT,AMP=X1651 1651,1,1,-1.0 *BOUNDARY,TYPE=DISPLACEMENT,AMP=Y1651 1651,2,2,1.0 ********* right guide roll

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*BOUNDARY,TYPE=DISPLACEMENT,AMP=X1649 1649,1,1,1.0 *BOUNDARY,TYPE=DISPLACEMENT,AMP=Y1649 1649,2,2,1.0 ******** define the contact pairs *ELSET, ELSET=INTERIOR, GEN 1, 1433, 8 *ELSET, ELSET=EXTERIOR, GEN 8, 1440, 8 *CONTACT PAIR, INTERACTION=FRIC EXTERIOR, DRIVER INTERIOR, IDLE *SURFACE INTERACTION, NAME=FRIC *FRICTION 0.5, *CONTACT PAIR EXTERIOR, LFTGUIDE EXTERIOR, RHTGUIDE *DLOAD INTERIOR, VP4, 3.0e8 EXTERIOR, VP2, 3.0e8 ****** *RESTART, WRITE, NUM=10 *NSET, NSET=QRTRPNTS 1, 406, 820, 1234, 9, 414, 828, 1242 *ELSET, ELSET=THRURING, GEN 1, 8, 1 *MONITOR, NODE=1660, DOF=6 *HISTORY OUTPUT, TIME INTERVAL=0.1 *NSET, NSET=REFNODES 1660, 1648, 1651, 1649 *NODE HISTORY, NSET=REFNODES U, UR3, RF, RM3 *NODE HISTORY, NSET=QRTRPNTS U, *EL HISTORY, ELSET=THRURING MISES, PEEQ, PRESS, ERV *ENERGY HISTORY *FILE OUTPUT, NUMBER INTER=5, TIME MARKS=YES *NODE FILE, NSET=REFNODES U, RF *NODE FILE, NSET=QRTRPNTS U,

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*EL FILE, ELSET=THRURING MISES, PEEQ, PRESS, ERV *ADAPTIVE MESH, ELSET=RING,FREQUENCY=50 *END STEP

1.3.15 Axisymmetric extrusion: transient and steady-state Product: ABAQUS/Explicit This example illustrates the use of adaptive meshing in simulations of extrusion processes with three axisymmetric analysis cases. First, a transient simulation is performed for a backward, flat-nosed die, extrusion geometry using adaptivity on a Lagrangian mesh domain. Second, a transient simulation is performed on the analogous forward, square die, extrusion geometry, also using adaptivity on a Lagrangian mesh domain. Finally, a steady-state simulation is performed for the forward extrusion geometry using adaptivity on an Eulerian mesh domain.

Problem description The model configurations for the three analysis cases are shown in Figure 1.3.15-1. Each of the models is axisymmetric and consists of one or more rigid tools and a deformable blank. The rigid tools are modeled as TYPE=SEGMENTS analytical rigid surfaces. All contact surfaces are assumed to be well-lubricated and, thus, are treated as frictionless. The blank is made of aluminum and is modeled as a von Mises elastic-plastic material with isotropic hardening. The Young's modulus is 38 GPa, and the initial yield stress is 27 MPa. The Poisson's ratio is 0.33; the density is 2672 kg/m 3.

Case 1: Transient analysis of a backward extrusion The model geometry consists of a rigid die, a rigid punch, and a blank. The blank is meshed with CAX4R elements and measures 28 ´ 89 mm. The blank is constrained along its base in the z-direction and at the axis of symmetry in the r-direction. Radial expansion is prevented by contact between the blank and the die. The punch and the die are fully constrained, with the exception of the prescribed vertical motion of the punch. The punch is moved downward 82 mm to form a tube with wall and endcap thicknesses of 7 mm each. The punch velocity is specified using the SMOOTH STEP parameter on the *AMPLITUDE option so that the response is essentially quasi-static. The deformation that occurs in extrusion problems, especially in those that involve flat-nosed die geometries, is extreme and requires adaptive meshing. Since adaptive meshing in ABAQUS/Explicit works with the same mesh topology throughout the step, the initial mesh must be chosen such that the mesh topology will be suitable for the duration of the simulation. A simple meshing technique has been developed for extrusion problems such as this. In two dimensions it uses a four-sided, mapped mesh domain that can be created with nearly all finite element mesh preprocessors. The vertices for the four-sided, mapped mesh are shown in Figure 1.3.15-1 and are denoted A, B, C, and D. Two vertices are located on either side of the extrusion opening, the third is in the corner of the dead material zone (the upper right corner of the blank), and the fourth vertex is located in the diagonally opposite corner. A 10 ´ 60 element mesh using this meshing technique is created for this analysis case and is shown in Figure 1.3.15-2. The mesh refinement is oriented such that the fine mesh along sides BC and DA will

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move up along the extruded walls as the punch is moved downward. An adaptive mesh domain is defined that incorporates the entire blank. Because of the extremely large distortions expected in the backward extrusion simulation, three mesh sweeps, instead of the default value of one, are specified using the MESH SWEEPS parameter on the *ADAPTIVE MESH option. The default adaptive meshing frequency of 10 is used. Alternatively, a higher frequency could be specified to perform one mesh sweep per adaptive mesh increment. However, this method would result in a higher computational cost because of the increased number of advection sweeps it would require. A substantial amount of initial mesh smoothing is performed by increasing the value of the INITIAL MESH SWEEPS parameter on the *ADAPTIVE MESH option to 100. The initially smoothed mesh is shown in Figure 1.3.15-2. Initial smoothing reduces the distortion of the mapped mesh by rounding out corners and easing sharp transitions before the analysis is performed; therefore, it allows the best mesh to be used throughout the analysis.

Case 2: Transient analysis of a forward extrusion The model geometry consists of a rigid die and a blank. The blank geometry and the mesh are identical to those described for Case 1, except that the mapped mesh is reversed with respect to the vertical plane so that the mesh lines are oriented toward the forward extrusion opening. The blank is constrained at the axis of symmetry in the r-direction. Radial expansion is prevented by contact between the blank and the die. The die is fully constrained. The blank is pushed up 19 mm by prescribing a constant velocity of 5 m/sec for the nodes along the bottom of the blank. As the blank is pushed up, material flows through the die opening to form a solid rod with a 7 mm radius. Adaptive meshing for Case 2 is defined in a similar manner as for Case 1. The undeformed mesh configurations, before and after initial mesh smoothing, are shown in Figure 1.3.15-3.

Case 3: Steady-state analysis of a forward extrusion The model geometry consists of a rigid die, identical to the die used for Case 2, and a blank. The blank geometry is defined such that it closely approximates the shape corresponding to the steady-state solution: this geometry can be thought of as an "initial guess" to the solution. As shown in Figure 1.3.15-4, the blank is discretized with a simple graded pattern that is most refined near the die fillet. No special mesh is required for the steady-state case since minimal mesh motion is expected during the simulation. The blank is constrained at the axis of symmetry in the r-direction. Radial expansion of the blank is prevented by contact between it and the die. An adaptive mesh domain is defined that incorporates the entire blank. Because the Eulerian domain undergoes very little overall deformation and the material flow speed is much less than the material wave speed, the frequency of adaptive meshing is changed to 5 from the default value of 1 to improve the computational efficiency of the analysis. The outflow boundary is assumed to be traction-free and is located far enough downstream to ensure that a steady-state solution can be obtained. This boundary is defined using the *SURFACE, REGION TYPE=EULERIAN option. A multi-point constraint is defined on the outflow boundary to keep the velocity normal to the boundary uniform. The inflow boundary is defined using the *BOUNDARY, REGION TYPE=EULERIAN option to prescribe a velocity of 5 m/sec in the vertical direction.

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Adaptive mesh constraints are defined on both the inflow and outflow boundaries to fix the mesh in the vertical direction using the *ADAPTIVE MESH CONSTRAINT option. This effectively creates a stationary control volume with respect to the inflow and outflow boundaries through which material can pass.

Results and discussion The results for each analysis case are described below.

Case 1 The use of the mapped meshing technique along with adaptive meshing allows the backward extrusion analysis to run to completion, creating the long tube with an endcap. Three plots of the deformed mesh at various times are shown in Figure 1.3.15-5. These plots clearly show how the quality of the mesh is preserved for the majority of the simulation. Despite the large amount of deformation involved, the mesh remains smooth and concentrated in the areas of high strain gradients. Extreme deformation and thinning at the punch fillet occurs near the end of the analysis. This thinning can be reduced by increasing the fillet radius of the punch. Corresponding contours of equivalent plastic strain are plotted in Figure 1.3.15-6. The plastic strains are highest along the inner surface of the tube.

Case 2 Adaptive meshing enables the transient forward extrusion simulation to proceed much further than would be possible using a pure Lagrangian approach. After pushing the billet 19 mm through the die, the analysis cannot be continued because the elements become too distorted. Since the billet material is essentially incompressible and the cross-sectional area of the die opening at the top is 1/16 of the original cross-sectional area of the billet, a rod measuring approximately 304 mm (three times the length of the original billet) is formed. Three plots of the deformed mesh at various times in the transient forward extrusion are shown in Figure 1.3.15-7. As in the backward extrusion case, the plots show that the quality of the mesh is preserved for a majority of the simulation. The last deformed shape has been truncated for clarity because the extruded column becomes very long and thin. Contours of equivalent plastic strain at similar times are shown in Figure 1.3.15-8. The plastic strain distribution developing in the vertical column does not reach a steady-state value, even at a height of 304 mm. The steady-state results reported in the discussion for Case 3 show that a steady-state solution based on the equivalent plastic strain distribution is not reached until much later. An absolute steady-state solution cannot be reached until the material on the upstream side of the dead material zone first passes along that zone and through the die opening. The dead material zone is roughly the shape of a triangle and is located in the upper right-hand corner of the die.

Case 3 The steady-state solution to the forward extrusion analysis is obtained at an extruded column height of 800 mm, which corresponds to pushing the billet 50 mm through the die. Thus, this analysis runs 2.5 times longer than Case 2.

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Contours of equivalent plastic strain in the middle and at the end of the simulation are shown in Figure 1.3.15-9. Time histories of the equivalent plastic strains on the outer edge of the extruded column at the outflow boundary and 27.5 mm below the outflow boundary are shown in Figure 1.3.15-10. The plastic strains at both locations converge to the same value by the end of the simulation, which indicates that the solution has reached a steady state. The final mesh configuration is shown in Figure 1.3.15-11. The mesh undergoes very little change from the beginning to the end of the analysis because of the accurate initial guess made for the steady-state domain shape and the ability of the adaptive meshing capability in ABAQUS/Explicit to retain the original mesh gradation. As a further check on the accuracy of the steady-state simulation and the conservation properties of adaptive meshing, a time history of the velocity at the outflow boundary is shown in Figure 1.3.15-12. The velocity reaches a steady value of approximately 80 m/s, which is consistent with the incompressible material assumption and the 1/16 ratio of the die opening to the billet size.

Input files ale_extrusion_back.inp Case 1. ale_extrusion_backnode.inp Node data for Case 1. ale_extrusion_backelem.inp Element data for Case 1. ale_extrusion_forward.inp Case 2. ale_extrusion_forwardnode.inp Node data for Case 2. ale_extrusion_forwardelem.inp Element data for Case 2. ale_extrusion_eulerian.inp Case 3. ale_extrusion_euleriannode.inp Node data for Case 3. ale_extrusion_eulerianelem.inp Element data for Case 3.

Figures

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Figure 1.3.15-1 Axisymmetric model geometries used in the extrusion analysis.

Figure 1.3.15-2 Undeformed configuration for Case 1, before and after initial smoothing.

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Figure 1.3.15-3 Undeformed configuration for Case 2, before and after initial smoothing.

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Figure 1.3.15-4 Undeformed configuration for Case 3.

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Figure 1.3.15-5 Deformed mesh at various times for Case 1.

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Figure 1.3.15-6 Contours of equivalent plastic strain at various times for Case 1.

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Figure 1.3.15-7 Deformed mesh at various times for Case 2.

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Figure 1.3.15-8 Contours of equivalent plastic strain at various times for Case 2.

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Figure 1.3.15-9 Contours of equivalent plastic strain at an intermediate stage and at the end of the analysis for Case 3.

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Figure 1.3.15-10 Time history of equivalent plastic strain along the outer edge of the extruded column for Case 3.

Figure 1.3.15-11 Final deformed mesh for Case 3.

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Static Stress/Displacement Analyses

Figure 1.3.15-12 Time history of material velocity at the outflow boundary for Case 3.

Sample listings

1-584

Static Stress/Displacement Analyses

Listing 1.3.15-1 *HEADING ADAPTIVE MESHING EXAMPLE BACKWARD EXTRUSION,MATERIAL: ALUMINIUM Units: N, m, second *RESTART,W,N=10 *NODE, INPUT=ale_extrusion_backnode.inp ** *ELEMENT, TYPE=CAX4R,ELSET=BLANK, INPUT=ale_extrusion_backelem.inp *NSET, NSET=BOT, GENERATE 979, 1001, 1 *NSET, NSET=RIGHT,GENERATE 911, 979, 1 *NSET, NSET=TOP, GENERATE 1, 92, 1 183, 183, 1 274, 274, 1 365, 365, 1 456, 456, 1 547, 547, 1 638, 638, 1 729, 729, 1 820, 820, 1 911, 911, 1 *NSET, NSET=LEFT 91, 182, 273, 364, 455, 546, 637, 728, 819, 910, 1001 ** *SOLID SECTION, ELSET=BLANK, MATERIAL=ALUMINIUM *MATERIAL, NAME=ALUMINIUM *ELASTIC 38E9,0.33 *PLASTIC 27E6,0 31E6,0.25 32.5E6,0.5 *DENSITY 2672, *BOUNDARY 9999,1,1

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Static Stress/Displacement Analyses

9999,6,6 LEFT,XSYMM BOT,YSYMM 9998,1,6 *NSET,NSET=REF 9999,9998 *ELSET, ELSET=OUT 1,2,3 *SURFACE,NAME=RIGHTS,TYPE=SEGMENTS START,0.028,-0.001 LINE,0.028,0.2 *SURFACE,TYPE=NODE,NAME=NSURF NCON, *SURFACE,NAME=TOPS,TYPE=SEGMENTS,FILLET =0.002 START,0.021,0.2 LINE,0.021,0.089 LINE,-0.0001,0.089 *RIGID BODY,REF NODE= 9998, ANALYTICAL SURFACE =RIGHTS *RIGID BODY,REF NODE= 9999, ANALYTICAL SURFACE =TOPS *STEP *DYNAMIC, EXPLICIT ,1.50337E-3 *BOUNDARY,TYPE=VELOCITY, AMP=STEP 9999,2,2,-60.0 *AMPLITUDE, DEFINITION=SMOOTH STEP, NAME=STEP 0.,0.,1.36667e-4,1.,1.36667e-3,1.,1.50337E-3, 0. *NSET,NSET=NCON TOP,RIGHT *CONTACT PAIR,INTERACTION=I1 NSURF,TOPS NSURF,RIGHTS *SURFACE INTERACTION,NAME=I1 *FILE OUTPUT, NUMBER=4, TIME MARKS=YES *NODE FILE, NSET=REF U,RF *EL FILE,ELSET=OUT MISES,PEEQ *ADAPTIVE MESH,ELSET=BLANK,FREQUENCY=10, MESH SWEEPS=3, INITIAL MESH SWEEPS=100 *ENDSTEP

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Static Stress/Displacement Analyses

1.3.16 Two-step forming simulation Product: ABAQUS/Explicit This example illustrates the use of adaptive meshing in simulations of a two-step, bulk metal forming process. The problem is based on a benchmark problem presented at the Metal Forming Process Simulation in Industry conference.

Problem description The model consists of two sets of rigid forming tools (one set for each forming step) and a deformable blank. The blank and forming die geometries used in the simulation are shown in Figure 1.3.16-1. The initial configurations of the blank and the tools for each step are shown in Figure 1.3.16-2 and Figure 1.3.16-4. All forming tools are modeled as discrete rigid bodies and meshed with R3D4 and R3D3 elements. The blank, which is meshed with C3D8R elements, is cylindrical and measures 14.5 ´ 21 mm. A half model is constructed, so symmetry boundary conditions are prescribed at the y=0 plane. The blank is made of a steel alloy that is assumed to satisfy the Ramberg-Osgood relation for true stress and logarithmic strain, ² = (¾=K )1=n ; with a reference stress value (K) of 763 MPa and a work-hardening exponent (n) of 0.245. Isotropic elasticity is assumed, with a Young's modulus of 211 GPa and a Poisson's ratio of 0.3. An initial yield stress of 200 MPa is obtained with these data. The stress-strain behavior is defined by piecewise linear segments matching the Ramberg-Osgood curve up to a total (logarithmic) strain level of 140%, with von Mises yield and isotropic hardening. The analysis is conducted in two steps. For the first step the rigid tools consist of a planar punch, a planar base, and a forming die. The initial configuration for this step is shown in Figure 1.3.16-2. The base, which is not shown, is placed at the opening of the forming die to prevent material from passing through the die. The motion of the tools is fully constrained, with the exception of the prescribed displacement in the z-direction for the punch, which is moved 12.69 mm toward the blank at a constant velocity of 30 m/sec consistent with a quasi-static response. The deformed configuration of the blank at the completion of the first step is shown in Figure 1.3.16-3. In the second step the original punch and die are removed from the model and replaced with a new punch and die, as shown in Figure 1.3.16-4. The removal of the tools is accomplished by deleting the contact pairs between them and the blank with the *CONTACT PAIR, OP=DELETE option. Although not shown in the figure, the base is retained; both it and the new die are fully constrained. The punch is moved 10.5 mm toward the blank at a constant velocity of 30 m/sec consistent with a quasi-static response. The deformed configuration of the blank at the completion of the second step is shown in Figure 1.3.16-5.

Adaptive meshing 1-587

Static Stress/Displacement Analyses

A single adaptive mesh domain that incorporates the entire blank is used for both steps. A Lagrangian boundary region type (the default) is used to define the constraints on the symmetry plane, and a sliding boundary region type (the default) is used to define all contact surfaces. The frequency of adaptive meshing is increased to 5 for this problem since material flows quickly near the end of the step.

Results and discussion Figure 1.3.16-6 shows the deformed mesh at the completion of forming for an analysis in which a pure Lagrangian mesh is used. Comparing Figure 1.3.16-5 and Figure 1.3.16-6, the resultant mesh for the simulation in which adaptive meshing is used is clearly better than that obtained with a pure Lagrangian mesh. In Figure 1.3.16-7 through Figure 1.3.16-9path plots of equivalent plastic strain in the blank are shown using the pure Lagrangian and adaptive mesh domains for locations in the y=0 symmetry plane at an elevation of z=10 mm. The paths are defined in the positive x-direction (from left to right in Figure 1.3.16-4 to Figure 1.3.16-6). As shown in Figure 1.3.16-7, the results are in good agreement at the end of the first step. At the end of the second step the path is discontinuous. Two paths are considered: one that spans the left-hand side and another that spans the right-hand side of the U-shaped cross-section along the symmetry plane. The left- and right-hand paths are shown in Figure 1.3.16-8and Figure 1.3.16-9, respectively. The solutions from the second step compare qualitatively. Small differences can be attributed to the increased mesh resolution and reduced mesh distortion for the adaptive mesh domain.

Input files ale_forging_steelpart.inp Analysis with adaptive meshing. ale_forging_steelpartnode1.inp External file referenced by the adaptive mesh analysis. ale_forging_steelpartnode2.inp External file referenced by the adaptive mesh analysis. ale_forging_steelpartnode3.inp External file referenced by the adaptive mesh analysis. ale_forging_steelpartnode4.inp External file referenced by the adaptive mesh analysis. ale_forging_steelpartelem1.inp External file referenced by the adaptive mesh analysis. ale_forging_steelpartelem2.inp

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External file referenced by the adaptive mesh analysis. ale_forging_steelpartelem3.inp External file referenced by the adaptive mesh analysis. ale_forging_steelpartelem4.inp External file referenced by the adaptive mesh analysis. ale_forging_steelpartelem5.inp External file referenced by the adaptive mesh analysis. ale_forging_steelpartsets.inp External file referenced by the adaptive mesh analysis. lag_forging_steelpart.inp Pure Lagrangian analysis.

Reference · Hermann, M. and A. Ruf, "Forming of a Steel Part," Metal Forming Process Simulation in Industry, Stuttgart, Germany, September 1994.

Figures Figure 1.3.16-1 Two-step forging process.

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Figure 1.3.16-2 Initial configuration for the first step.

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Figure 1.3.16-3 Deformed blank at the end of the first step.

Figure 1.3.16-4 Configuration at the beginning of the second step.

Figure 1.3.16-5 Deformed blank at the end of the second step for the adaptive mesh analysis.

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Static Stress/Displacement Analyses

Figure 1.3.16-6 Deformed blank at the end of the second step for the pure Lagrangian analysis.

Figure 1.3.16-7 Path plot of equivalent plastic strain at the end of the first step.

Figure 1.3.16-8 Path plot of equivalent plastic strain along the left side at the end of the second step.

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Static Stress/Displacement Analyses

Figure 1.3.16-9 Path plot of equivalent plastic strain along the right side at the end of the second step.

Sample listings

1-593

Static Stress/Displacement Analyses

Listing 1.3.16-1 *HEADING ADAPTIVE MESHING EXAMPLE TWO-STAGE FORGING OF A STEEL PART Units: N, mm, seconds *NODE,NSET=MOLD, INPUT=ale_forging_steelpartnode1.inp *NODE,NSET=DIE1, INPUT=ale_forging_steelpartnode2.inp *ELEMENT, TYPE=C3D8R, ELSET=MOLD, INPUT=ale_forging_steelpartelem1.inp *NODE,NSET=PUNCH1 2000001,-1.,-1.,29. 200000,-1.,-1.,29. 200001,44.,-1.,29. 200002,44.,20.,29. 200003,-1.,20.,29. *ELEMENT, TYPE=R3D4, ELSET=PUNCH1 200000,200000,200003,200002,200001 *NODE,NSET=BASE 3000001,-1.,-1.,.000001 300000,-1.,-1.,.000001 300001,44.,-1.,.000001 300002,44.,20.,.000001 300003,-1.,20.,.000001 *ELEMENT, TYPE=R3D4, ELSET=BASE 300000,300000,300001,300002,300003 *ELEMENT, TYPE=R3D4, ELSET=DIE1, INPUT=ale_forging_steelpartelem2.inp *NODE,NSET=PUNCH2, INPUT=ale_forging_steelpartnode3.inp *ELEMENT, TYPE=R3D4, ELSET=PUNCH2, INPUT=ale_forging_steelpartelem3.inp *ELEMENT, TYPE=R3D3, ELSET=PUNCH2, INPUT=ale_forging_steelpartelem4.inp *NODE,NSET=DIE2, INPUT=ale_forging_steelpartnode4.inp *ELEMENT, TYPE=R3D4, ELSET=DIE2, INPUT=ale_forging_steelpartelem5.inp *INCLUDE,INPUT=ale_forging_steelpartsets.inp *BOUNDARY 1000001,1,6

1-594

Static Stress/Displacement Analyses

2000001,1,2 2000001,4,6 3000001,1,6 4000001,1,6 5000001,1,6 YSYMM,YSYMM RIG1,YSYMM RIG2,YSYMM RIG3,YSYMM RIG4,YSYMM *SOLID SECTION,ELSET=MOLD,MATERIAL=C15, CONTROLS=SECT *SECTION CONTROLS,NAME=SECT,HOURGLASS=STIFFNESS, KINEMATICS=ORTHOGONAL ** *MATERIAL,NAME=C15 *ELASTIC 210000.,0.3 *PLASTIC 200.000, 0.000 246.934, 0.010 434.092, 0.100 514.439, 0.200 568.167, 0.300 609.658, 0.400 643.916, 0.500 673.331, 0.600 699.247, 0.700 722.501, 0.800 743.654, 0.900 763.100, 1.000 781.129, 1.100 797.960, 1.200 813.762, 1.300 828.672, 1.400 *DENSITY 7.85E-9, *RESTART,WRITE,NUM=2 *NSET,NSET=P1 2000001, *NSET,NSET=P2 4000001, *SURFACE,TYPE=ELEMENT,NAME=PUNCH2

1-595

Static Stress/Displacement Analyses

PUNCH2,SPOS *SURFACE,TYPE=ELEMENT,NAME=TOP,REGION TYPE=SLIDING TOP,S2 *SURFACE,TYPE=ELEMENT,NAME=DIE1 DIE1,SPOS *SURFACE,TYPE=ELEMENT,NAME=DIE2 DIE2,SPOS *SURFACE,TYPE=ELEMENT,NAME=SIDE, REGION TYPE=SLIDING SIDE2,S5 SIDE4,S4 *SURFACE,TYPE=ELEMENT,NAME=BOTTOM, REGION TYPE=SLIDING BOTTOM,S1 *SURFACE,TYPE=ELEMENT,NAME=BASE BASE,SPOS *SURFACE,TYPE=ELEMENT,NAME=PUNCH1 PUNCH1,SPOS *RIGID BODY,ELSET=DIE1,REF NODE=1000001 *RIGID BODY,ELSET=PUNCH1,REF NODE=2000001 *RIGID BODY,ELSET=BASE,REF NODE=3000001 *RIGID BODY,ELSET=PUNCH2,REF NODE=4000001 *RIGID BODY,ELSET=DIE2,REF NODE=5000001 *STEP *DYNAMIC,EXPLICIT ,.000423333 *BOUNDARY,TYPE=VELOCITY 2000001,3,3,-30000. *CONTACT PAIR,INTERACTION=FRIC BOTTOM,BASE BOTTOM,DIE1 SIDE,DIE1 TOP,PUNCH1 ** *SURFACE INTERACTION, NAME=FRIC *FRICTION 0.1, ** *HISTORY OUTPUT,TIMEINTERVAL=0.0000042 *NODE HISTORY,NSET=P1 U,RF *ADAPTIVE MESH,ELSET=MOLD,FREQUENCY=5, CONTROLS=TEST

1-596

Static Stress/Displacement Analyses

*ADAPTIVE MESH CONTROLS, NAME=TEST, TRANSITION FEATURE ANGLE=0.0 *OUTPUT,FIELD,number=2 *ELEMENT OUTPUT peeq,mises *node output u, *END STEP *STEP *DYNAMIC,EXPLICIT ,.00035 *BOUNDARY,TYPE=VELOCITY 4000001,3,3,-30000. *CONTACT PAIR,INTERACTION=FRIC,OP=DELETE BOTTOM,DIE1 SIDE,DIE1 TOP,PUNCH1 ** *CONTACT PAIR,INTERACTION=FRIC BOTTOM,DIE2 SIDE,DIE2 TOP,PUNCH2 ** *SURFACE INTERACTION, NAME=FRIC *FRICTION 0.1, ** *HISTORY OUTPUT,TIMEINTERVAL=0.0000035 *NODE HISTORY,NSET=P2 U,RF *EL HISTORY,ELSET=TOP PEEQ, EDT, *ENERGY HISTORY ALLKE,ALLIE,ALLAE,ALLVD,ALLWK,ETOTAL, DT, *ADAPTIVE MESH,ELSET=MOLD,FREQUENCY=5 *OUTPUT,FIELD,number=2 *ELEMENT OUTPUT peeq,mises *node output u, *END STEP

1-597

Static Stress/Displacement Analyses

1.3.17 Upsetting of a cylindrical billet: coupled temperature-displacement and adiabatic analysis Products: ABAQUS/Standard ABAQUS/Explicit This example illustrates coupled temperature-displacement analysis in a metal forming application. The case studied is an extension of the standard test case that is defined in Lippmann (1979); thus, some verification of the results is available by comparison with the numerical results presented in that reference. The example is that of a small, circular billet of metal that is reduced in length by 60%. Here the problem is analyzed as a viscoplastic case, including heating of the billet by plastic work. Such analysis is often important in manufacturing processes, especially when significant temperature rises degrade the material. The problem is also analyzed in ABAQUS/Standard using a porous metal material model. The same problem is used to illustrate mesh rezoning in ABAQUS/Standard in ``Upsetting of a cylindrical billet in ABAQUS/Standard: quasi-static analysis with rezoning,'' Section 1.3.1; the same test case is also used in ``Upsetting of a cylindrical billet in ABAQUS/Explicit,'' Section 1.3.2.

Geometry and model The specimen is shown in Figure 1.3.17-1: a circular billet, 30 mm long, with a radius of 10 mm, compressed between flat, rough, rigid dies. All surfaces of the billet are assumed to be fully insulated: this thermal boundary condition is chosen to maximize the temperature rise. The finite element model is axisymmetric and includes the top half of the billet only since the middle surface of the billet is a plane of symmetry. In ABAQUS/Standard elements of type CAX8RT, 8-node quadrilaterals with reduced integration that allow for fully coupled temperature-displacement analysis, are used. A regular mesh with six elements in each direction is used, as shown in Figure 1.3.17-1. In ABAQUS/Explicit the billet is modeled with CAX4RT elements in a 12 ´ 12 mesh. The contact between the top and the lateral exterior surfaces of the billet and the rigid die is modeled with the *CONTACT PAIR option. The billet surface is defined by means of the *SURFACE option. The rigid die is modeled as an analytical rigid surface or as an element-based rigid surface, using the *RIGID BODY option in conjunction with the *SURFACE option. The mechanical interaction between the contact surfaces is assumed to be nonintermittent, rough frictional contact in ABAQUS/Standard. Therefore, two options are used in conjunction with the *SURFACE INTERACTION property option: the *FRICTION, ROUGH option to enforce a no-slip constraint between the two surfaces and the *SURFACE BEHAVIOR, NO SEPARATION option to ensure that separation does not occur once contact has been established. In ABAQUS/Explicit the friction coefficient between the billet and the rigid die is 1.0. The problem is also solved in ABAQUS/Standard with the first-order fully coupled temperature-displacement CAX4T elements in a 12 ´ 12 mesh. Similarly, the problem is solved using CAX8RT elements and user subroutines UMAT and UMATHT to illustrate the use of these subroutines. No mesh convergence studies have been performed, but the comparison with results given in

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Lippmann (1979) suggests that these meshes provide accuracy similar to the best of those analyses. The ABAQUS/Explicit simulations are performed both with and without adaptive meshing.

Material The material definition is basically that given in Lippmann (1979), except that the metal is assumed to be rate dependent. The thermal properties are added, with values that correspond to a typical steel, as well as the data for the porous metal plasticity model. The material properties are then Young's modulus: 200 GPa Poisson's ratio: 0.3 Thermal expansion coefficient: 1.2´10-5 per °C Initial static yield stress: 700 MPa Work hardening rate: 300 MPa¡ ¢p "_pl = D (¾=¾ ± ) ¡ 1 ; D = 40/s, p = 5 Strain rate dependence: Specific heat: 586 J/(kg°C) Density: 7833 kg/m3 Conductivity: 52 J/(m-s-°C) q Porous material parameters: 1 = q2 = q3 = 1:0 Initial relative density: 0.95 (f0 = 0.05) Since the problem definition in ABAQUS/Standard assumes that the dies are completely rough, no tangential slipping is allowed wherever the metal contacts the die.

Boundary conditions and loading The kinematic boundary conditions are symmetry on the axis (nodes at r =0, in node set AXIS, have ur =0 prescribed), symmetry about z =0 (all nodes at z =0, in node set MIDDLE, have uz =0 prescribed). To avoid overconstraint, the node on the top surface of the billet that lies on the symmetry axis is not part of the node set AXIS: the radial motion of this node is already constrained by a no-slip frictional constraint (see ``Common difficulties associated with contact modeling,'' Section 21.10.1 of the ABAQUS/Standard User's Manual). The rigid body reference node for the rigid surface that defines the die is constrained to have no rotation or ur -displacement, and its uz -displacement is prescribed to move 9 mm down the axis at constant velocity. The reaction force at the rigid reference node corresponds to the total force applied by the die. The thermal boundary conditions are that all external surfaces are insulated (no heat flux allowed). This condition is chosen because it is the most extreme case: it must provide the largest temperature rises possible, since no heat can be removed from the specimen. DELTMX is the limit on the maximum temperature change allowed to occur in any increment and is one of the controls for the automatic time incrementation scheme in ABAQUS/Standard. It is set to 100°C, which is a large value and indicates that we are not restricting the time increments because of accuracy considerations in integrating the heat transfer equations. In fact, the automatic time incrementation scheme will choose fairly small increments because of the severe nonlinearity present in the problem and the resultant need for several iterations per increment even with a relatively large number of increments. The large value is used for DELTMX to obtain a reasonable solution at low cost.

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Static Stress/Displacement Analyses

In ABAQUS/Explicit the automatic time incrementation scheme is used to ensure numerical stability and to advance the solution in time. Mass scaling is used to reduce the computational cost of the analysis. The AMPLITUDE=RAMP parameter is included because the default amplitude variation for a transient, coupled temperature-displacement analysis is a step function, but here we want the die to move down at a constant velocity. Two versions of the analysis are run: a slow upsetting, where the upsetting occurs in 100 seconds, and a fast upsetting, where the event takes 0.1 second. Both versions are analyzed with the coupled temperature-displacement procedure. The fast upsetting is also run in ABAQUS/Standard as an adiabatic static stress analysis. The time period values are specified on the data line associated with the *COUPLED TEMPERATURE-DISPLACEMENT procedure, *DYNAMIC TEMPERATURE-DISPLACEMENT procedure, and the *STATIC procedure options. The adiabatic stress analysis is performed in the same time frame as the fast upsetting case. In all cases analyzed with ABAQUS/Standard an initial time increment of 1.5% of the time period is used; that is, 1.5 seconds in the slow case and 0.0015 second in the fast case. This value is chosen because it will result in a nominal axial strain of about 1% per increment, and experience suggests that such increment sizes are generally suitable for cases like this.

Results and discussion The results of the ABAQUS/Standard simulations are discussed first, beginning with the results for the viscoplastic fully dense material. The results of the slow upsetting are illustrated in Figure 1.3.17-2 to Figure 1.3.17-4. The results for the fast upsetting coupled temperature-displacement analysis are illustrated in Figure 1.3.17-5 to Figure 1.3.17-7; those for the adiabatic static stress analysis are shown in Figure 1.3.17-8 and Figure 1.3.17-9. Figure 1.3.17-2 and Figure 1.3.17-5 show the configuration that is predicted at 60% upsetting. The configuration for the adiabatic analysis is not shown since it is almost identical to the fast upsetting coupled case. Both the slow and the fast upsetting cases show the folding of the top outside surface of the billet onto the die, as well as the severe straining of the middle of the specimen. The second figure in each series (Figure 1.3.17-3 for the slow case, Figure 1.3.17-6 for the fast case, and Figure 1.3.17-8 for the adiabatic case) shows the equivalent plastic strain in the billet. Peak strains of around 180% occur in the center of the specimen. The third figure in each series (Figure 1.3.17-4 for the slow case, Figure 1.3.17-7 for the fast case, and Figure 1.3.17-9 for the adiabatic case) shows the temperature distributions, which are noticeably different between the slow and fast upsetting cases. In the slow case there is time for the heat to diffuse (the 60% upsetting takes place in 100 sec, on a specimen where a typical length is 10 mm), so the temperature distribution at 100 sec is quite uniform, varying only between 180°C and 185°C through the billet. In contrast, the fast upsetting occurs too quickly for the heat to diffuse. In this case the middle of the top surface of the specimen remains at 0°C at the end of the event, while the center of the specimen heats up to almost 600°C. There is no significant difference in temperatures between the fast coupled case and the adiabatic case. In the outer top section of the billet there are differences that are a result of the severe distortion of the elements in that region and the lack of dissipation of generated heat. The temperature in the rest of the billet compares well. This example illustrates the advantage of an adiabatic analysis, since a good representation of the results is obtained in about 60% of the computer time required for

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the fully coupled analysis. The results of the slow and fast upsetting of the billet modeled with the porous metal plasticity model are shown in Figure 1.3.17-10 to Figure 1.3.17-15. The deformed configuration is identical to that of Figure 1.3.17-2 and Figure 1.3.17-5. The extent of growth/closure of the voids in the specimen at the end of the analysis is shown in Figure 1.3.17-10 and Figure 1.3.17-13. The porous material is almost fully compacted near the center of the billet because of the compressive nature of the stress field in that region; on the other hand, the corner element is folded up and stretched out near the outer top portion of the billet, increasing the void volume fraction to almost 0.1 (or 10%) and indicating that tearing of the material is likely. The equivalent plastic strain is shown in Figure 1.3.17-11 (slow upsetting) and Figure 1.3.17-14 (fast upsetting) for the porous material; Figure 1.3.17-12 and Figure 1.3.17-15 show the temperature distribution for the slow and the fast upsetting of the porous metal. The porous metal needs less external work to achieve the same deformation compared to a fully dense metal. Consequently, there is less plastic work being dissipated as heat; hence, the temperature increase is not as much as that of fully dense metal. This effect is more pronounced in the fast upsetting problem, where the specimen heats up to only 510°C, compared to about 600°C for fully dense metal. Figure 1.3.17-16 to Figure 1.3.17-18 show predictions of total upsetting force versus displacement of the die. In Figure 1.3.17-16 the slow upsetting viscoplastic and porous plasticity results are compared with several elastic-plastic and rigid-plastic results that were collected by Lippmann (1979) and slow viscoplastic results obtained by Taylor (1981). There is general agreement between all the rate independent results, and these correspond to the slow viscoplastic results of the present example and of those found by Taylor (1981). In Figure 1.3.17-17 rate dependence of the yield stress is investigated. The fast viscoplastic and porous plasticity results show significantly higher force values throughout the event than the slow results. This effect can be estimated easily. A nominal strain rate of 6 sec is maintained throughout the event. With the viscoplastic model that is used, this effect increases the yield stress by 68%. This factor is very close to the load amplification factor that appears in Figure 1.3.17-17. Figure 1.3.17-18 shows that the force versus displacement prediction of the fast viscoplastic adiabatic analysis agrees well with the fully coupled results. Two cases using an element-based rigid surface to model the die are also considered in ABAQUS/Standard. To define the element-based rigid surface, the elements are assigned to rigid bodies using *RIGID BODY, ISOTHERMAL=YES. The results agree very well with the case when the analytical rigid surface is used. The automatic load incrementation results suggest that overall nominal strain increments of about 2% per increment were obtained, which is slightly better than what was anticipated in the initial time increment suggestion. These values are typical for problems of this class and are useful guidelines for estimating the computational effort required for such cases. The results obtained with ABAQUS/Explicit compare well with those obtained with ABAQUS/Standard, as illustrated in Figure 1.3.17-19, which compares the results obtained with ABAQUS/Explicit (without adaptive meshing) for the total upsetting force versus the displacement of the die against the same result obtained with ABAQUS/Standard. The agreement between the two solutions is excellent. Similar agreement is obtained with the results obtained from the ABAQUS/Explicit simulation using adaptive meshing. The mesh distortion is significantly reduced in

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this case, as illustrated in Figure 1.3.17-20.

Input files ABAQUS/Standard input files cylbillet_cax4t_slow_dense.inp Slow upsetting case with 144 CAX4T elements, using the fully dense material. cylbillet_cax4t_fast_dense.inp Fast upsetting case with 144 CAX4T elements, using the fully dense material. cylbillet_cax8rt_slow_dense.inp Slow upsetting case with CAX8RT elements, using the fully dense material. cylbillet_cax8rt_rb_s_dense.inp Slow upsetting case with CAX8RT elements, using the fully dense material and an element-based rigid surface for the die. cylbillet_cax8rt_fast_dense.inp Fast upsetting case with CAX8RT elements, using the fully dense material. cylbillet_cax8rt_slow_por.inp Slow upsetting case with CAX8RT elements, using the porous material. cylbillet_cax8rt_fast_por.inp Fast upsetting case with CAX8RT elements, using the porous material. cylbillet_cgax4t_slow_dense.inp Slow upsetting case with 144 CGAX4T elements, using the fully dense material. cylbillet_cgax4t_fast_dense.inp Fast upsetting case with 144 CGAX4T elements, using the fully dense material. cylbillet_cgax4t_rb_f_dense.inp Fast upsetting case with 144 CGAX4T elements, using the fully dense material and an element-based rigid surface for the die. cylbillet_cgax8rt_slow_dense.inp Slow upsetting case with CGAX8RT elements, using the fully dense material. cylbillet_cgax8rt_fast_dense.inp Fast upsetting case with CGAX8RT elements, using the fully dense material. cylbillet_c3d10m_adiab_dense.inp Adiabatic static analysis with fully dense material modeled with C3D10M elements.

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cylbillet_cax6m_adiab_dense.inp Adiabatic static analysis with fully dense material modeled with CAX6M elements. cylbillet_cax8r_adiab_dense.inp Adiabatic static analysis with fully dense material modeled with CAX8R elements. cylbillet_postoutput.inp *POST OUTPUT analysis, using the fully dense material. cylbillet_slow_usr_umat_umatht.inp Slow upsetting case with the material behavior defined in user subroutines UMAT and UMATHT. cylbillet_slow_usr_umat_umatht.f User subroutines UMAT and UMATHT used in cylbillet_slow_usr_umat_umatht.inp. ABAQUS/Explicit input files cylbillet_x_cax4rt_slow.inp Slow upsetting case with fully dense material modeled with CAX4RT elements and without adaptive meshing; kinematic mechanical contact. cylbillet_x_cax4rt_fast.inp Fast upsetting case with fully dense material modeled with CAX4RT elements and without adaptive meshing; kinematic mechanical contact. cylbillet_x_cax4rt_slow_adap.inp Slow upsetting case with fully dense material modeled with CAX4RT elements and with adaptive meshing; kinematic mechanical contact. cylbillet_x_cax4rt_fast_adap.inp Fast upsetting case with fully dense material modeled with CAX4RT elements and with adaptive meshing; kinematic mechanical contact. cylbillet_xp_cax4rt_fast.inp Fast upsetting case with fully dense material modeled with CAX4RT elements and without adaptive meshing; penalty mechanical contact.

References · Lippmann, H., Metal Forming Plasticity, Springer-Verlag, Berlin, 1979. · Taylor, L. M., "A Finite Element Analysis for Large Deformation Metal Forming Problems Involving Contact and Friction," Ph.D. Thesis, U. of Texas at Austin, 1981.

Figures

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Figure 1.3.17-1 Axisymmetric upsetting example: geometry and mesh (element type CAX8RT).

Figure 1.3.17-2 Deformed configuration at 60% upsetting: slow case, coupled temperature-displacement analysis, ABAQUS/Standard.

Figure 1.3.17-3 Plastic strain at 60% upsetting: slow case, coupled temperature-displacement analysis, ABAQUS/Standard.

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Figure 1.3.17-4 Temperature at 60% upsetting: slow case, coupled temperature-displacement analysis, ABAQUS/Standard.

Figure 1.3.17-5 Deformed configuration at 60% upsetting: fast case, coupled temperature-displacement analysis, ABAQUS/Standard.

Figure 1.3.17-6 Plastic strain at 60% upsetting: fast case, coupled temperature-displacement analysis, ABAQUS/Standard.

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Static Stress/Displacement Analyses

Figure 1.3.17-7 Temperature at 60% upsetting: fast case, coupled temperature-displacement analysis, ABAQUS/Standard.

Figure 1.3.17-8 Plastic strain at 60% upsetting: fast case, adiabatic stress analysis, ABAQUS/Standard.

Figure 1.3.17-9 Temperature at 60% upsetting: fast case, adiabatic stress analysis, ABAQUS/Standard.

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Static Stress/Displacement Analyses

Figure 1.3.17-10 Void volume fraction at 60% upsetting: porous material, slow coupled temperature-displacement analysis, ABAQUS/Standard.

Figure 1.3.17-11 Plastic strain at 60% upsetting: porous material, slow coupled temperature-displacement analysis, ABAQUS/Standard.

Figure 1.3.17-12 Temperature at 60% upsetting: porous material, slow coupled temperature-displacement analysis, ABAQUS/Standard.

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Static Stress/Displacement Analyses

Figure 1.3.17-13 Void volume fraction at 60% upsetting: porous material, fast coupled temperature-displacement analysis, ABAQUS/Standard.

Figure 1.3.17-14 Plastic strain at 60% upsetting: porous material, fast coupled temperature-displacement analysis, ABAQUS/Standard.

Figure 1.3.17-15 Temperature at 60% upsetting: porous material, fast coupled temperature-displacement analysis, ABAQUS/Standard.

Figure 1.3.17-16 Force-deflection response for slow cylinder upsetting, ABAQUS/Standard.

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Figure 1.3.17-17 Rate dependence of the force-deflection response, ABAQUS/Standard.

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Figure 1.3.17-18 Force-deflection response: adiabatic versus fully coupled analysis, ABAQUS/Standard.

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Static Stress/Displacement Analyses

Figure 1.3.17-19 Force-deflection response: ABAQUS/Explicit versus ABAQUS/Standard.

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Static Stress/Displacement Analyses

Figure 1.3.17-20 Deformed configuration of the at 60% upsetting: slow case; without adaptive meshing, left; with adaptive meshing, right (ABAQUS/Explicit).

Sample listings

1-612

Static Stress/Displacement Analyses

Listing 1.3.17-1 *HEADING - AXISYMMETRIC UPSETTING PROBLEM - SLOW CASE - COUPLED TEMPERATURE-DISPLACEMENT ANALYSIS - (SLOW UPSETTING) *RESTART,WRITE,FREQUENCY=30 *NODE,NSET=RSNODE 9999,0.,.015 *NODE 1, 13,.01 1201,0.,.015 1213,.01,.015 *NGEN,NSET=MIDDLE 1,13 *NGEN,NSET=TOP 1201,1213 *NSET,NSET=TOPBND,GENERATE 1201,1212,1 *NFILL MIDDLE,TOP,12,100 *NSET,NSET=AXIS,GENERATE 1,1101,100 *ELEMENT,TYPE=CAX8RT,ELSET=METAL 1,1,3,203,201,2,103,202,101 *ELGEN,ELSET=METAL 1,6,2,1,6,200,100 *ELSET,ELSET=ESID,GENERATE 6,506,100 *ELSET,ELSET=ETOP,GENERATE 501,506,1 *SURFACE,NAME=ASURF ESID,S2 ETOP,S3 *RIGID BODY,ANALYTICAL SURFACE=BSURF,REF NODE=9999 *SURFACE,TYPE=SEGMENTS,NAME=BSURF START,.020,.015 LINE,-.001,.015 *CONTACT PAIR,INTERACTION=SMOOTH ASURF,BSURF *SOLID SECTION,ELSET=METAL,MATERIAL=EL

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Static Stress/Displacement Analyses

*MATERIAL,NAME=EL *ELASTIC 200.E9,.3 *PLASTIC 7.E8,0.00 3.7E9,10.0 *RATE DEPENDENT 40.,5. *SPECIFIC HEAT 586., *DENSITY 7833., *CONDUCTIVITY 52., *EXPANSION 1.2E-5, *INELASTIC HEAT FRACTION 0.9, *SURFACE INTERACTION,NAME=SMOOTH *SURFACE BEHAVIOR,NO SEPARATION *FRICTION,ROUGH *BOUNDARY MIDDLE,2 AXIS,1 *STEP,INC=200,AMPLITUDE=RAMP,NLGEOM *COUPLED TEMPERATURE-DISPLACEMENT,DELTMX=100. 1.5,100.,5.E-8,5.0 *BOUNDARY RSNODE,1 RSNODE,6 RSNODE,2,,-.009 *PRINT,CONTACT=YES *CONTACT PRINT,SLAVE=ASURF,FREQUENCY=100 *CONTACT FILE,SLAVE=ASURF,FREQUENCY=200 *OUTPUT,FIELD,FREQ=200 *CONTACT OUTPUT,VARIABLE=PRESELECT,SLAVE=ASURF *CONTACT CONTROLS,FRICTION ONSET=DELAY *MONITOR,NODE=9999,DOF=2 *EL PRINT, ELSET=METAL,FREQUENCY=100 S,MISES E,PEEQ *NODE PRINT,FREQUENCY=25 U,RF,NT,RFL

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Static Stress/Displacement Analyses

*NODE FILE,NSET=RSNODE RF, *OUTPUT,FIELD *NODE OUTPUT,NSET=RSNODE RF, *NODE FILE,FREQUENCY=200 NT, *OUTPUT,FIELD,FREQ=200 *NODE OUTPUT NT, *END STEP

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Static Stress/Displacement Analyses

Listing 1.3.17-2 *HEADING - AXISYMMETRIC UPSETTING PROBLEM - WITH RIGID SURFACE - SECTION CONTROLS USED (HOURGLASS=STIFFNESS) - SLOW UPSETTING *NODE 1, 13,.01 1201,0.,.015 1213,.01,.015 *NGEN,NSET=MIDDLE 1,13 *NGEN,NSET=TOP 1201,1213 *NFILL MIDDLE,TOP,12,100 *NSET,NSET=AXIS,GEN 1,1201,100 *ELEMENT,TYPE=cax4rt,ELSET=BILLET 1,1,2,102,101 *ELGEN,ELSET=BILLET 1,12,1,1,12,100,100 *NODE, NSET=NRIGID 2003,0.01,.02 *SOLID SECTION,ELSET=BILLET, MATERIAL=METAL,CONTROL=B *SECTION CONTROLS, HOURGLASS=STIFFNESS, NAME=B *MATERIAL,NAME=METAL *ELASTIC 200.E9,.3 *PLASTIC 7.E8,0.00 3.7E9,10.0 *RATE DEPENDENT 40.,5. *SPECIFIC HEAT 586., *DENSITY 7833., *CONDUCTIVITY 52.,

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Static Stress/Displacement Analyses

*EXPANSION 1.2E-5, *INELASTIC HEAT FRACTION 0.9, *BOUNDARY MIDDLE,2 AXIS,1 2003,1 2003,3,6 *SURFACE,TYPE=ELEMENT,NAME=BILLET TOP,S3 SIDE,S2 *SURFACE,NAME=RIGID,TYPE=SEGMENTS START, 0.02,.015 LINE, 0.00,.015 *RIGID BODY, REF NODE=2003, ANALYTICAL SURFACE =RIGID *STEP *DYNAMIC TEMPERATURE-DISPLACEMENT, EXPLICIT ,100.0 *FIXED MASS SCALING, ELSET=BILLET,FACTOR=1.E+10 *BOUNDARY,TYPE=VELOCITY 2003,2,,-9.e-5 *ELSET,ELSET=TOP,GEN 1101,1112,1 *ELSET,ELSET=SIDE,GEN 12,1112,100 *SURFACE INTERACTION,NAME=RIG_BILL *FRICTION 1.0, *CONTACT PAIR,INTERACTION=RIG_BILL RIGID,BILLET *MONITOR,NODE=2003,DOF=2 *FILE OUTPUT,NUM=2 *EL FILE PEEQ,MISES *NODE FILE, NSET=NRIGID U,RF *NODE FILE NT, *OUTPUT,FIELD,VAR=PRESELECT,NUM=5 *OUTPUT,HISTORY *NODE OUTPUT,NSET=NRIGID

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Static Stress/Displacement Analyses

U2,RF2 *ENERGY OUTPUT ALLIE,ALLKE,ALLAE,ALLPD *END STEP

1.3.18 Unstable static problem: thermal forming of a metal sheet Product: ABAQUS/Standard This example demonstrates the use of automatic techniques to stabilize unstable static problems. Geometrically nonlinear static problems can become unstable for a variety of reasons. Instability may occur in contact problems, either because of chattering or because contact intended to prevent rigid body motions is not established initially. Localized instabilities can also occur; they can be either geometrical, such as local buckling, or material, such as material softening. This problem models the thermal forming of a metal sheet; the shape of the die may make it difficult to place the undeformed sheet exactly in initial contact, in which case the initial rigid body motion prevention algorithm is useful. Metal forming problems are characterized by relatively simply shaped parts being deformed by relatively complex-shaped dies. The initial placement of the workpiece on a die or the initial placement of a second die may not be a trivial geometrical exercise for an engineer modeling the forming process. ABAQUS accepts initial penetrations in contact pairs and instantaneously tries to resolve them; as long as the geometry allows for this to happen without excessive deformation, the misplacement of the workpiece usually does not cause problems. On the other hand, if the workpiece is initially placed away from the dies, serious problems may arise. Unless there are enough boundary conditions applied, singular finite element systems of equations result because one or more of the bodies has free rigid body motions. This typically arises when the deformation is applied through loads instead of boundary conditions. It is possible to eliminate this problem by modifying the model, which can be cumbersome for the analyst. Alternatively, the *CONTACT CONTROLS, APPROACH option allows initial placement of a body apart from others with smooth load-controlled motion until contact gets established. This example looks at the thermal forming of an aluminum sheet. The deformation is produced by applying pressure and gravity loads to push the sheet against a sculptured die. The deformation is initially elastic. Through heating, the yield stress of the material is lowered until permanent plastic deformations are produced. Subsequently, the assembly is cooled and the pressure loads are removed, leaving a formed part with some springback. Although the sheet is initially flat, the geometrical nature of the die makes it difficult to determine the exact location of the sheet when it is placed on the die. Therefore, an initial gap between the two bodies is modeled, as shown in Figure 1.3.18-1.

Geometry and model The model consists of a trapezoidal sheet 10.0 m (394.0 in) long, tapering from 2.0 m (78.75 in) to 3.0 m (118.0 in) wide, and 10.0 mm (0.4 in) thick. The die is a ruled surface controlled by two circles of radii 13.0 m (517.0 in) and 6.0 m (242.0 in) and dimensions slightly larger than the sheet. The sheet is initially placed over 0.2 m (7.9 in) apart from the die. The sheet has a longitudinal symmetry boundary condition, and one node prevents the remaining nodes from experiencing in-plane rigid body motion.

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Static Stress/Displacement Analyses

The die is fixed throughout the analysis. The sheet mesh consists of 640 S4R shell elements, while the die is represented by 640 R3D4 rigid elements. The material is an aluminum alloy with a flow stress of 1.0 ´ 108 Pa (14.5 ksi) at room temperature. A flow stress of 1.0 ´ 103 Pa (0.15 psi) at 400°C is also provided, essentially declaring that at the higher temperature the material will flow plastically at any stress. A Coulomb friction coefficient of 0.1 is used to model the interaction between the sheet and die.

Results and discussion The analysis consists of three steps. In the first step a gravity load and a pressure load of 1.0 ´ 105 Pa (14.5 psi) are applied, both pushing the sheet against the die. This step is aided by the automatic contact approach procedure to prevent unrestrained motion of the sheet. This procedure consists of the application of viscous pressure along the contact direction (normal to the die), opposing the relative motion between the sheet and the die. During the first increment of the step ABAQUS applies a very high amount of damping so that it can judge the magnitude of the external loads being applied, as well as determine the initial distances between the slave and master surfaces. Based on the results of this initial attempt a suitable damping coefficient is calculated and the increment is repeated, such that a smooth approach is produced during the initial part of the step. In this particular case five increments take place before the first contact point closes; from then on the sheet is pressed against the die. As soon as contact is established, the relative velocities between the sheet and the die decrease and become almost zero at the end of the step, which essentially eliminates the damping forces. In addition, ABAQUS ramps down the damping coefficient to zero from the middle of the step on. This guarantees that the viscous forces decrease to zero, thus avoiding any discontinuity in the forces at the start of the next step. The shape and relatively low curvatures of the die are such that the deformation at the end of the step is elastic (Figure 1.3.18-2). In the second step a two-hour heating (from room temperature to 360°C) and cooling (back to 50°C) cycle is applied to the loaded assembly. As a result of the decrease in flow stress permanent (plastic) deformation develops, as shown in Figure 1.3.18-3. Finally, in the third step the pressure load is removed and the springback of the deformed sheet is calculated, as depicted in Figure 1.3.18-4.

Acknowledgements HKS would like to thank British Aerospace Airbus, Ltd. for providing the basic data from which this example was derived.

Input file unstablestatic_forming.inp Thermal forming model.

Figures Figure 1.3.18-1 Initial placement of the sheet apart from the die.

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Static Stress/Displacement Analyses

Figure 1.3.18-2 Elastic deformation after gravity and pressure loading.

Figure 1.3.18-3 Permanent deformation produced by heating.

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Static Stress/Displacement Analyses

Figure 1.3.18-4 Springback.

Sample listings

1-621

Static Stress/Displacement Analyses

Listing 1.3.18-1 *HEADING THERMAL FORMING OF AN ALUMINUM SHEET S.I Units, Kg, M, s, N ** *RESTART, WRITE, FREQUENCY=20, OVERLAY ** ** Rigid body node definitions *NODE,NSET=RIGID 1, -1.800000, 0.000000, 0.000000 17, 1.800000, 0.000000, 0.000000 901, 0.000000, 0.000000, 13.000000 681, -1.300000, 10.300000, 0.000000 697, 1.300000, 10.300000, 0.000000 902, 0.000000, 10.300000, 6.000000 *NGEN,NSET=END1,LINE=C 1,17,1,901 *NGEN,NSET=END2,LINE=C 681,697,1,902 *NFILL,NSET=RIGID END1,END2,40,17 ** ** Plate node definitions *NODE,NSET=DEFORM 10001, -1.000000, 10.150000, 0.1725 10017, 1.000000, 10.150000, 0.1725 10681, -1.500000, 0.150000, 0.1725 10697, 1.500000, 0.150000, 0.1725 *NGEN,NSET=SHORT 10001,10017 *NGEN,NSET=LONG 10681,10697 *NFILL,NSET=DEFORM SHORT,LONG,40,17 ** ** Rigid body reference node *NODE,NSET=FIX 99999, 0., 0., 0. ** ** Plate element definitions *ELEMENT, TYPE=S4R, ELSET=SHEET 10001, 10001, 10002, 10019, 10018

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Static Stress/Displacement Analyses

*ELGEN,ELSET=SHEET 10001, 16, 1, 1, 40, 17, 16 *ELSET,ELSET=SHORT,GENERATE 10001,10016,1 ** ** Rigid surface element definitions *ELEMENT, TYPE=R3D4, ELSET=CONTACT 1, 1, 2, 19, 18 *ELGEN,ELSET=CONTACT 1, 16, 1, 1, 40, 17, 16 ** ** Plate 10mm in thickness *SHELL SECTION, ELSET=SHEET, MATERIAL=ALUMINUM 0.01, 5 ** *MATERIAL, NAME=ALUMINUM ** *ELASTIC 7.919714E10, 0.3, 292. *DENSITY 2.9e3, *EXPANSION 8.367e-6, ** *PLASTIC 1.e8,0,293. 1.e3,0,673. ** *RIGID BODY, ELSET=CONTACT, REF NODE=99999 ** *SURFACE, NAME=M1 CONTACT, SPOS *SURFACE, NAME=S1 SHEET, SPOS ** *CONTACT PAIR, INTERACTION=I1 S1, M1 *SURFACE INTERACTION, NAME=I1 *FRICTION, SLIP TOLERANCE=0.02 0.1, ** ** Nodes down the middle of the model on which ** boundary conditions are placed.

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Static Stress/Displacement Analyses

*NSET,NSET=MIDDLE 10009, 10026, 10043, 10060, 10077, 10094, 10111, 10128, 10145, 10162, 10179, 10196, 10213, 10230, 10247, 10264, 10281, 10298, 10315, 10332, 10349, 10366, 10383, 10400, 10417, 10434, 10451, 10468, 10485, 10502, 10519, 10536, 10553, 10570, 10587, 10604, 10621, 10638, 10655, 10672, 10689, ** ** *BOUNDARY 10009,2,2,0.0 MIDDLE,1,1,0.0 99999, 1,6,0.0 ** ** Temperature amplitude card for use ** during forming step ** *AMPLITUDE, NAME=TEMP_PROF, VALUE=ABSOLUTE 0,293., 3600,633. , 7200,323. ** ** Initial ambient temperature for the plate ** *INITIAL CONDITIONS, TYPE=TEMPERATURE DEFORM,293.,293.,293.,293.,293. ** ** --------------------------------------------** ** Step 1 - Apply gravity loading together with ** 1 Bar pressure loading on plate to force it ** into the die. *STEP, INC=10000, NLGEOM *STATIC 0.1, 1., 1.E-10 ** ** pressure + gravity ** *DLOAD, OP=NEW SHEET,GRAV,9.81, 0., 0., -1.

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Static Stress/Displacement Analyses

SHEET, P, 1.e5 ** *CONTACT CONTROLS,MASTER=M1, SLAVE=S1,APPROACH ** *NODE PRINT, FREQ=0 *EL PRINT, FREQ=0 *CONTACT PRINT, FREQ=999 *NODE FILE,NSET=SHORT,FREQ=999 U, *EL FILE, ELSET=SHORT, FREQ=999 S,E,PE *OUTPUT,FIELD,FREQ=20 *NODE OUTPUT U, *ELEMENT OUTPUT S,E,PEEQ *END STEP ** ** --------------------------------------------** ** Step 2 - Now perform the forming step. ** The temperature follows the profile on the ** *AMPLITUDE card *STEP, INC=10000, NLGEOM *STATIC 100.,7200.,1e-1,5.e2 ** *TEMPERATURE, OP=NEW, AMPLITUDE=TEMP_PROF DEFORM,1.,1.,1.,1.,1. ** *END STEP ** ** -------------------------------------------** ** Step 3 - Now release 1 Bar pressure ** (but maintain gravity load) *STEP, INC=10000, NLGEOM *STATIC 0.1, 1. ** *DLOAD, OP=NEW SHEET,GRAV,9.81, 0., 0., -1. **

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Static Stress/Displacement Analyses

*END STEP

1.4 Fracture mechanics 1.4.1 A plate with a part-through crack: elastic line spring modeling Product: ABAQUS/Standard The line spring elements in ABAQUS allow inexpensive evaluation of the effects of surface flaws in shell structures, with sufficient accuracy for use in design studies. The basic concept of these elements is that they introduce the local solution, dominated by the singularity at the crack tip, into a shell model of the uncracked geometry. The relative displacements and rotations across the cracked section, calculated in the line spring elements, are then used to determine the magnitude of the local strain field and hence the J -integral and stress intensity factor values, as functions of position along the crack front. This example illustrates the use of these elements and provides some verification of the results they provide by comparison with a published solution and also by making use of the shell-to-solid submodeling technique.

Problem description A large plate with a symmetric, centrally located, semi-elliptic, part-through crack is subjected to edge tension and bending. The objective is to estimate the Mode I stress intensity factor, KI , as a function of position along the crack front. Symmetry allows one quarter of the plate to be modeled, as shown in Figure 1.4.1-1. The 8-node shell element, S8R, and the corresponding 3-node (symmetry plane) line spring element LS3S are used in the model. A mesh using LS6 elements is also included. Only half-symmetry is used in this case. When LS6 elements are used, the shell elements on either side of an LS6 element must be numbered such that the normals to these shell elements point in approximately the same direction.

Geometry and model For each load case (tension and bending) two plate thicknesses are studied: a "thick" case, for which the plate thickness is 76.2 mm (3.0 in); and a "thin" case, for which the plate thickness is 19.05 mm (0.75 in). For both thicknesses the semi-elliptic crack has a maximum depth ( a0 in Figure 1.4.1-2) of 15.24 mm (0.6 in) and a half-length, c, of 76.2 mm (3.0 in). The plate is assumed to be square, with dimensions 609.6 ´ 609.6 mm (24 ´ 24 in). The material is assumed to be linear elastic, with Young's modulus 207 GPa (30 ´ 106 lb/in2) and Poisson's ratio 0.3. A quarter of the plate is modeled, with symmetry along the edges of the quarter-model at x =0 and y =0. On the edge containing the flaw (y =0), the symmetry boundary conditions are imposed only on the unflawed segment of the edge, since they are built into the symmetry plane of the line spring element being used (LS3S). The loading consists of a uniform edge tension (per unit length) of 52.44 kN/m (300 lb/in) or a uniform

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edge moment (per unit length) of 1335 N-m/m (300 lb-in/in).

Results and discussion The stress intensity factors for the thick and thin plates are compared with the detailed solutions of Raju and Newman (1979) and Newman and Raju (1979) in Figure 1.4.1-3 (tension load) and Figure 1.4.1-4 (bending load). These plots show that the present results agree reasonably well with those of Raju and Newman over the middle portion of the flaw (Á >30°), with better correlation being provided for the thick case, possibly because the crack is shallower in that geometry. The accuracy is probably adequate for basic assessment of the criticality of the flaw for design purposes. For values of Á less than about 30° (that is, at the ends of the flaw), the stress intensity values predicted by the line spring model lose accuracy. This accuracy loss arises from a combination of the relative coarseness of the mesh, (especially in this end region where the crack depth varies rapidly), as well as from theoretical considerations regarding the appropriateness of line spring modeling at the ends of the crack. These points are discussed in detail by Parks (1981) and Parks et al. (1981).

Shell-to-solid submodeling around the crack tip An input file for the case a0 =t= 0.2, which uses the shell-to-solid submodeling capability, is included. This C3D20R element mesh allows the user to study the local crack area using the energy domain integral formulation for the J -integral. The submodel uses a focused mesh with four rows of elements p around the crack tip. A 1= r singularity is utilized at the crack tip, the correct singularity for a linear elastic solution. Symmetry boundary conditions are imposed on two edges of the submodel mesh, while results from the global shell analysis are interpolated to two edges by using the submodeling technique. The global shell mesh gives satisfactory J -integral results; hence, we assume that the displacements at the submodel boundary are sufficiently accurate to drive the deformation in the submodel. No attempt has been made to study the effect of making the submodel region larger or smaller. The submodel is shown superimposed on the global shell model in Figure 1.4.1-5. The variations of the J -integral values along the crack in the submodeled analysis are compared to the line spring element analysis in Figure 1.4.1-3 (tension load) and Figure 1.4.1-4 (bending load). Excellent correlation is seen between the three solutions. A more refined mesh in the shell-to-solid submodel near the plate surface would be required to obtain J -integral values that more closely match the reference solution.

Input files crackplate_ls3s.inp LS3S elements. crackplate_surfaceflaw.f A small program used to create a data file containing the surface flaw depths. crackplate_ls6_nosym.inp LS6 elements without symmetry about y = 0.

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Static Stress/Displacement Analyses

crackplate_postoutput.inp *POST OUTPUT analysis. crackplate_submodel.inp Shell-to-solid submodel.

References · Newman, J. C., Jr., and I. S. Raju, "Analysis of Surface Cracks in Finite Plates Under Tension or Bending Loads," NASA Technical Paper 1578, National Aeronautics and Space Administration, December 1979. · Parks, D. M., "The Inelastic Line Spring: Estimates of Elastic-Plastic Fracture Mechanics Parameters for Surface-Cracked Plates and Shells," Journal of Pressure Vessel Technology, vol. 13, pp. 246-254, 1981. · Parks, D. M., R. R. Lockett, and J. R. Brockenbrough, "Stress Intensity Factors for Surface-Cracked Plates and Cylindrical Shells Using Line Spring Finite Elements," Advances in Aerospace Structures and Materials , Edited by S. S. Wang and W. J. Renton, ASME, AD-01, pp. 279-286, 1981. · Raju, I. S. and J. C. Newman Jr., "Stress Intensity Factors for a Wide Range of Semi-Elliptic Surface Cracks in Finite Thickness Plates," Journal of Engineering Fracture Mechanics, vol. 11, pp. 817-829, 1979.

Figures Figure 1.4.1-1 Quarter model of large plate with center surface crack.

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Figure 1.4.1-2 Schematic surface crack geometry for a semi-elliptical crack.

Figure 1.4.1-3 Stress intensity factor dependence on crack front position: tension loading.

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Static Stress/Displacement Analyses

Figure 1.4.1-4 Stress intensity factor dependence on crack front position: moment loading.

Figure 1.4.1-5 Solid submodel superimposed on shell global model.

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Static Stress/Displacement Analyses

Sample listings

1-631

Static Stress/Displacement Analyses

Listing 1.4.1-1 *HEADING QUARTER MODEL OF LARGE PLATE WITH CENTER SURFACE CRACK [S8R] *NODE 1,0.,0. 97,12.,0. 4801,0.,12. 4897,12.,12. *NGEN,NSET=ONY 1,4801,100 *NGEN,NSET=FREE 97,4897,100 *NGEN 1,97 101,197 201,297 401,497 601,697 1001,1097 1401,1497,2 2201,2297,2 3001,3097,2 3901,3997,2 4801,4897,2 *NSET,NSET=ONX 27,29,33,37,41,45,49,53,57 61,69,77,87,97 *ELEMENT,TYPE=LS3S,ELSET=LS 1,25,24,23 5,17,15,13 *ELGEN,ELSET=LS 1,4,-2,1 5,4,-4,1 *ELEMENT,TYPE=S8R,ELSET=SHELL 9,1,5,205,201,3,105,203,101 13,17,19,219,217,18,119,218,117 17,25,29,229,225,27,129,227,125 18,29,37,237,229,33,137,233,129 22,61,77,277,261,69,177,269,161 23,77,97,297,277,87,197,287,177 24,201,209,609,601,205,409,605,401

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Static Stress/Displacement Analyses

26,601,609,1409,1401,605,1009,1405,1001 28,217,221,621,617,219,421,619,417 31,617,621,1421,1417,619,1021,1419,1017 34,229,237,637,629,233,437,633,429 38,629,637,1437,1429,633,1037,1433,1029 42,261,277,677,661,269,477,669,461 43,661,677,1477,1461,669,1077,1469,1061 44,277,297,697,677,287,497,687,477 45,677,697,1497,1477,687,1097,1487,1077 46,1401,1417,3017,3001,1409,2217,3009,2201 48,1417,1421,3021,3017,1419,2221,3019,2217 50,1421,1429,3029,3021,1425,2229,3025,2221 58,1477,1497,3097,3077,1487,2297,3087,2277 47,3001,3017,4817,4801,3009,3917,4809,3901 49,3017,3021,4821,4817,3019,3921,4819,3917 51,3021,3029,4829,4821,3025,3929,4825,3921 59,3077,3097,4897,4877,3087,3997,4887,3977 *ELGEN,ELSET=SHELL 9,4,4,1 13,4,2,1 18,4,8,1 24,2,8,1 26,2,8,1 28,3,4,1 31,3,4,1 34,4,8,1 38,4,8,1 46,2,28,6 52,3,16,2 47,2,28,6 53,3,16,2 *ELSET,ELSET=PRINT 9,59,15,16,17 *MATERIAL,NAME=A1 *ELASTIC 30.E6,.3 *SHELL SECTION,ELSET=SHELL,MATERIAL=A1 3.0,3 *SHELL SECTION,ELSET=LS,MATERIAL=A1 3.0, *SURFACE FLAW,SIDE=POSITIVE,INPUT=CRACK.FLW ** DATA GENERATED FROM PROGRAM 7-1-1-2 *ELSET,ELSET=TOPL

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Static Stress/Displacement Analyses

47,49,51,53,55,57,59 *MPC QUADRATIC,203,201,205,209 QUADRATIC,207,201,205,209 QUADRATIC,211,209,213,217 QUADRATIC,215,209,213,217 QUADRATIC,218,217,219,221 QUADRATIC,220,217,219,221 QUADRATIC,222,221,223,225 QUADRATIC,224,221,223,225 QUADRATIC,1405,1401,1409,1417 QUADRATIC,1413,1401,1409,1417 QUADRATIC,1423,1421,1425,1429 QUADRATIC,1427,1421,1425,1429 QUADRATIC,1433,1429,1437,1445 QUADRATIC,1441,1429,1437,1445 QUADRATIC,1449,1445,1453,1461 QUADRATIC,1457,1445,1453,1461 *BOUNDARY ONY,1 ONY,5,6 ONX,2 ONX,4 ONX,6 1,3 *RESTART,WRITE,FREQUENCY=999 *STEP *STATIC 0.1,1.0 *CLOAD 4801,2,100. 4809,2,400. 4817,2,125. 4819,2,100. 4821,2,75. 4825,2,200. 4829,2,150. 4837,2,400. 4845,2,200. 4853,2,400. 4861,2,200. 4869,2,400. 4877,2,225.

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Static Stress/Displacement Analyses

4887,2,500. 4897,2,125. *EL PRINT,ELSET=LS JK, S, *EL PRINT,ELSET=PRINT COORD, S, E, *EL FILE,ELSET=LS JK, S, *EL FILE,ELSET=PRINT COORD, S, E, *NODE PRINT U, *NODE FILE U, *END STEP

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Static Stress/Displacement Analyses

Listing 1.4.1-2 CC--- PROGRAM TO GENERATE CRACK DEPTH DATA C PROGRAM CRACK C SUBROUTINE HKSMAIN C IMPLICIT REAL*8(A-H,O-Z) OPEN(UNIT=16,STATUS='NEW',ACCESS='SEQUENTIAL', 1 FORM='FORMATTED',FILE='CRACK.FLW') C=3. CC=C*C N=24 NNODE=N+1 X0=C/DBLE(N) X=0. DO 100 I=1,NNODE IF(I.GE.17) GO TO 1 IF((I/2)*2.EQ.I) GO TO 10 1 CONTINUE XX=X*X TMP=.2 Z=TMP*SQRT(CC-XX) WRITE(6,99) I,Z WRITE(16,99)I,Z 99 FORMAT(I5,', ',F10.7) 10 CONTINUE X=X+X0 100 CONTINUE REWIND 16 STOP END

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Static Stress/Displacement Analyses

Listing 1.4.1-3 *HEADING PLATE WITH A PART THROUGH CRACK: LS6 ELEMENTS *RESTART,WRITE *NODE 1,0.,0.,0.,0.,0.,1. 97,12.,0.,0.,0.,0.,1. 4801,0.,12.,0.,0.,0.,1. 4897,12.,12.,0.,0.,0.,1. *NGEN,NSET=ONY 1,4801,100 *NGEN,NSET=FREE 97,4897,100 *NGEN,NSET=TOP 1,97 101,197 201,297 401,497 601,697 1001,1097 1401,1497,2 2201,2297,2 3001,3097,2 3901,3997,2 4801,4897,2 *NSET,NSET=ONX 27,29,33,37,41,45,49,53,57 61,69,77,87,97 *NCOPY, OLD SET=TOP, NEW SET=BOTTOM, CHANGE NUMBER=10000, REFLECT=LINE 0.,0.,0.,12.,0.,0. *NSET,NSET=NALL TOP,BOTTOM *NSET,NSET=ONY,GENERATE 10001,14801,100 *NSET,NSET=ONX1 10027,10029,10033,10037,10041,10045,10049,10053, 10057,10061,10069,10077,10087,10097 *MPC TIE,ONX,ONX1 *ELEMENT,TYPE=S8R,ELSET=SHELLT 9,1,5,205,201,3,105,203,101

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Static Stress/Displacement Analyses

13,17,19,219,217,18,119,218,117 17,25,29,229,225,27,129,227,125 18,29,37,237,229,33,137,233,129 22,61,77,277,261,69,177,269,161 23,77,97,297,277,87,197,287,177 24,201,209,609,601,205,409,605,401 26,601,609,1409,1401,605,1009,1405,1001 28,217,221,621,617,219,421,619,417 31,617,621,1421,1417,619,1021,1419,1017 34,229,237,637,629,233,437,633,429 38,629,637,1437,1429,633,1037,1433,1029 42,261,277,677,661,269,477,669,461 43,661,677,1477,1461,669,1077,1469,1061 44,277,297,697,677,287,497,687,477 45,677,697,1497,1477,687,1097,1487,1077 46,1401,1417,3017,3001,1409,2217,3009,2201 48,1417,1421,3021,3017,1419,2221,3019,2217 50,1421,1429,3029,3021,1425,2229,3025,2221 58,1477,1497,3097,3077,1487,2297,3087,2277 47,3001,3017,4817,4801,3009,3917,4809,3901 49,3017,3021,4821,4817,3019,3921,4819,3917 51,3021,3029,4829,4821,3025,3929,4825,3921 59,3077,3097,4897,4877,3087,3997,4887,3977 *ELGEN,ELSET=SHELLT 9,4,4,1 13,4,2,1 18,4,8,1 24,2,8,1 26,2,8,1 28,3,4,1 31,3,4,1 34,4,8,1 38,4,8,1 46,2,28,6 52,3,16,2 47,2,28,6 53,3,16,2 *ELCOPY, ELEMENT SHIFT=10000, OLD SET=SHELLT, NEW SET=SHELLB,SHIFT NODES=10000 *ELEMENT,TYPE=LS6,ELSET=LS 1,25,24,23,10025,10024,10023 5,17,15,13,10017,10015,10013 *ELGEN,ELSET=LS

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Static Stress/Displacement Analyses

1,4,-2,1 5,4,-4,1 *ELSET,ELSET=SALL SHELLT,SHELLB,LS *ELSET,ELSET=SHELLS SHELLT,SHELLB *NORMAL SALL,NALL,0.,0.,1. *ELSET,ELSET=PRINT 9,59,15,16,17 *MATERIAL,NAME=A1 *ELASTIC 30.E6,.3 *SHELL SECTION,ELSET=SHELLS,MATERIAL=A1 .75,3 *SHELL SECTION,ELSET=LS,MATERIAL=A1 .75, *SURFACE FLAW,SIDE=POSITIVE 10001, 0.6000000 10003, 0.5979130 10005, 0.5916080 10007, 0.5809475 10009, 0.5656854 10011, 0.5454356 10013, 0.5196152 10015, 0.4873397 10017, 0.4472136 10018, 0.4235269 10019, 0.3968627 10020, 0.3665720 10021, 0.3316625 10022, 0.2904738 10023, 0.2397916 10024, 0.1713914 10025, 0.00 *ELSET,ELSET=TOPL 47,49,51,53,55,57,59 *ELSET,ELSET=BOTL 10047,10049,10051,10053,10055,10057,10059 *MPC QUADRATIC,203,201,205,209 QUADRATIC,207,201,205,209 QUADRATIC,211,209,213,217

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Static Stress/Displacement Analyses

QUADRATIC,215,209,213,217 QUADRATIC,218,217,219,221 QUADRATIC,220,217,219,221 QUADRATIC,222,221,223,225 QUADRATIC,224,221,223,225 QUADRATIC,1405,1401,1409,1417 QUADRATIC,1413,1401,1409,1417 QUADRATIC,1423,1421,1425,1429 QUADRATIC,1427,1421,1425,1429 QUADRATIC,1433,1429,1437,1445 QUADRATIC,1441,1429,1437,1445 QUADRATIC,1449,1445,1453,1461 QUADRATIC,1457,1445,1453,1461 QUADRATIC,10203,10201,10205,10209 QUADRATIC,10207,10201,10205,10209 QUADRATIC,10211,10209,10213,10217 QUADRATIC,10215,10209,10213,10217 QUADRATIC,10218,10217,10219,10221 QUADRATIC,10220,10217,10219,10221 QUADRATIC,10222,10221,10223,10225 QUADRATIC,10224,10221,10223,10225 QUADRATIC,11405,11401,11409,11417 QUADRATIC,11413,11401,11409,11417 QUADRATIC,11423,11421,11425,11429 QUADRATIC,11427,11421,11425,11429 QUADRATIC,11433,11429,11437,11445 QUADRATIC,11441,11429,11437,11445 QUADRATIC,11449,11445,11453,11461 QUADRATIC,11457,11445,11453,11461 *BOUNDARY ONY,1 ONY,5,6 4897,3 14897,3 10097,2 10097,4 *STEP *STATIC *CLOAD 10025,1,0., 10025,2,0. 10025,3,0. 10025,4,0.

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Static Stress/Displacement Analyses

10025,5,0. 10025,6,0. 4801,4,-100. 4809,4,-400. 4817,4,-125. 4819,4,-100. 4821,4,-75. 4825,4,-200. 4829,4,-150. 4837,4,-400. 4845,4,-200. 4853,4,-400. 4861,4,-200. 4869,4,-400. 4877,4,-225. 4887,4,-500. 4897,4,-125. 14801,4,100. 14809,4,400. 14817,4,125. 14819,4,100. 14821,4,75. 14825,4,200. 14829,4,150. 14837,4,400. 14845,4,200. 14853,4,400. 14861,4,200. 14869,4,400. 14877,4,225. 14887,4,500. 14897,4,125. *EL FILE JK, *END STEP

1.4.2 Conical crack in a half-space with and without submodeling Product: ABAQUS/Standard The purpose of this example is to verify that ABAQUS correctly evaluates contour integrals when the crack extension direction varies along the crack front. For the conical-shaped crack shown in Figure 1.4.2-1, the crack extension direction changes as the crack front is swept around the circle. The

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Static Stress/Displacement Analyses

problem is axisymmetric and can, therefore, also be modeled using axisymmetric elements. The contour integrals for the three-dimensional model are verified by comparing them to results using axisymmetric elements.

Full modeling vs. submodeling The full three-dimensional crack model has an RMS wavefront of 2800. The submodeling capability is used to obtain accurate results by running two, much smaller, models: first, a global model to get the displacement solution with moderate accuracy away from the crack tip; and then a submodel to obtain a more accurate solution, and, hence, more accurate J -integrals, along the crack front. These models have RMS wavefronts of less than 1700, allowing them to be run on a smaller computer than is required to run the full three-dimensional model.

Geometry and model The geometry analyzed is a conical crack in a half-space, as shown in Figure 1.4.2-1. The crack intersects the free surface at 45° and extends 15 units into the half-space. Pressure loading is applied on the region of the half-space surface circumscribed by the crack. The full three-dimensional and axisymmetric meshes are shown in Figure 1.4.2-2 and Figure 1.4.2-3, respectively. The full three-dimensional model represents one-quarter of the problem, using symmetry about the x-y and y-z planes, and is composed of 10 sectors parallel to the y-axis. In the region up to a distance of approximately 10 times the crack length away from the crack, reduced-integration elements (C3D20R and CAX8R) are used. Beyond this region infinite elements (CIN3D12R and CINAX5R) are used. The focused mesh surrounding the crack tip in a plane parallel to the y-axis consists of 8 rings of 16 elements. It encompasses half of the crack length and extends the same distance ahead of the crack tip. p To obtain the desired 1/ r strain singularity, all the nodes in each crack front node set are tied together using multi-point constraints; and on element edges radial to the crack front, the midside nodes are moved to the 1/4 point position. This improves the modeling of the strain field near the crack tip, which results in more accurate contour integral values. There are three regions of degenerate elements. At the crack tip collapsed elements are necessary to provide the desired singularity. The elements at the crack opening and the elements along the y-axis are collapsed to simplify the meshing. Figure 1.4.2-4 shows the displaced shape of the mesh near the crack for the three-dimensional case.

Submodel Submodeling is also used to solve the problem with smaller meshes. The global model represents the same problem as the full model, but with a coarser mesh. Only one ring of elements is used in the focused part of the global model mesh compared to eight rings in the full model. For the three-dimensional global model only five sectors of elements parallel to the 2-axis are used, as opposed to 10 sectors in the full model. The axisymmetric global model is shown in Figure 1.4.2-5. The submodels consist of only the focused region of the mesh around the crack tip and contain eight rings and, in the three-dimensional case, 10 sectors. Quarter-symmetry boundary conditions are applied to the three-dimensional submodel as well as to the global model. The axisymmetric and three-dimensional submodels are shown in Figure 1.4.2-6 and Figure 1.4.2-7, respectively. It is

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Static Stress/Displacement Analyses

assumed that the global model's coarse mesh is sufficiently accurate to drive the submodel: if the global model's displacement field far from the crack tip is accurate, the submodel can obtain accurate contour integral results. If two "driven" nodes on opposite faces of the crack share exactly the same location in the submodel, the submodeling capability is unable to assign the driven nodes to an element uniquely. The two nodes will behave as if they are tied together across the crack. In this example problem nodes 1033 and 65033 are defined to be approximately 0.01% of their typical element length away from their intended location along the crack. Moving the nodes this small amount does not affect the results but alleviates the assignment problem.

Results and discussion J -integral results are shown in Table 1.4.2-1. Neither the axisymmetric nor the three-dimensional global models used to drive submodels provide useful J values. The submodels driven by these global models, however, give J -integral results that differ from their related full models by only 2%. In the three-dimensional models the J -integrals on planes that include element corner nodes differ slightly from the J -integrals on planes that include only element midside nodes. The difference occurs because the strain singularity at the crack tip is reproduced on planes of nodes that include corner nodes, whereas on planes of nodes passing through midside nodes there is no singularity since we use 20-node elements. The use of 27-node elements adjacent to the crack line should eliminate this problem. In addition, the stress intensity factors and the T -stresses are calculated. The interaction integral method, in which the auxiliary plane strain crack-tip fields are employed, is used for their calculations. Since the crack front is very close to the symmetry axis, more refined meshes should be used to make the plane strain condition prevail locally around the crack front so that contour-independent results can be obtained. The calculated values of the stress intensity factors KI , KII , and KIII ; J -integral (estimated from both stress intensity factors and ABAQUS); and the T -stresses are shown in Table 1.4.2-2, Table 1.4.2-3, Table 1.4.2-4, Table 1.4.2-5, and Table 1.4.2-6, respectively. ABAQUS automatically outputs the J -integrals based on the stress intensity factors when the latter are evaluated. These J values are compared with the J values calculated directly by ABAQUS in Table 1.4.2-5, and good agreement is observed between them. The sign of KII is different in the three-dimensional model and in the axisymmetric model. This is not a problem since the sign of KII will depend on the order of the crack front node sets arranged for the contour integral computation.

Input files conicalcrack_3dglobal.inp Three-dimensional global model. conicalcrack_3dsubmodel.inp Three-dimensional submodel.

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Static Stress/Displacement Analyses

conicalcrack_full3d.inp Full three-dimensional model. conicalcrack_node.inp Node definitions for conicalcrack_full3d.inp. conicalcrack_element.inp Element definitions for conicalcrack_full3d.inp. conicalcrack_axiglobal.inp Axisymmetric global model. conicalcrack_axisubmodel.inp Axisymmetric submodel. conicalcrack_fullaxi.inp Full axisymmetric model. conicalcrack_3dsubmodel_rms.inp Three-dimensional submodel with refined meshes. conicalcrack_full3d_rms.inp Full three-dimensional model with refined meshes. conicalcrack_node_rms.inp Node definitions for conicalcrack_full3d_rms.inp. conicalcrack_element_rms.inp Element definitions for conicalcrack_full3d_rms.inp. conicalcrack_axisubmodel_rms.inp Axisymmetric submodel with refined meshes. conicalcrack_fullaxi_rms.inp Full axisymmetric model with refined meshes.

Tables Table 1.4.2-1 J -integral estimates (´ 10-7) for conical crack. Contour 1 is omitted from the average value calculations. Solution

Full

Crack Front Location Crack tip

1 5 1.360

Contour 2 3 6 7 1.331

1.336

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4 8 1.337

Average Value, Contours 2-8

Static Stress/Displacement Analyses

Axisymmetric

Corner nodes Full Three-dimensiona Midside l nodes

Submodel Axisymmetric

Crack tip

Corner nodes Submodel Three-dimensiona Midside l nodes

5 1.337 5 1.361 5 1.331 5 1.360 3 1.338 7 1.390 3 1.367 6 1.392 1 1.365 7 1.390 5 1.369 3

6 1.337 7 1.331 4 1.334 5 1.331 6 1.339 4 1.361 0 1.368 0 1.361 4 1.365 3 1.361 6 1.370 1

3 1.337 9 1.335 5 1.333 6 1.336 7 1.340 0 1.366 0 1.368 2 1.365 8 1.363 9 1.367 0 1.371 1

2 1.337 9 1.335 6 1.332 5 1.338 0 1.340 6 1.367 1 1.367 3 1.366 1 1.363 8 1.368 4 1.369 9

1.3364

1.3340

1.3379

1.3665

1.3646

1.3682

Table 1.4.2-2 Stress intensity factor KI estimates for conical crack using refined meshes. Contour 1 is omitted from the average value calculations. Solution Crack 1 Front

2

Contour 3

4

5

0.478 3 0.477 2 0.476 4 0.526 9 0.526 2 0.525 3

0.479 3 0.478 4 0.477 1 0.528 0 0.527 5 0.526 2

0.479 7 0.479 1 0.477 2 0.528 4 0.528 2 0.526 3

0.480 1 0.479 8 0.477 2 0.528 8 0.528 9 0.526 3

Location Full Axisymmetric

Crack tip

Corner Full nodes Three-dimensiona Midside l nodes Submodel Crack tip Axisymmetric Corner Submodel nodes Three-dimensiona Midside l nodes

0.487 7 0.472 4 0.492 8 0.537 3 0.521 0 0.543 5

Average Value, Contours 2-5 0.4794 0.4786 0.4770 0.5280 0.5277 0.5260

Table 1.4.2-3 Stress intensity factor KII estimates for conical crack using refined meshes. Contour 1

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Static Stress/Displacement Analyses

is omitted from the average value calculations. Solution Crack 1 2 Front

Contour 3

4

5

Location Full Axisymmetric

Crack tip

Corner Full nodes Three-dimensiona Midside l nodes Submodel Crack tip Axisymmetric Corner Submodel nodes Three-dimensiona Midside l nodes

-2.078 -2.039

Average Value, Contours 2-5 -2.040

2.015

2.041

-2.04 1 2.045

2.108

2.037

2.041

2.041

2.041

2.040

-2.090 -2.050

-2.052 -2.051

-2.051

2.057

2.057

2.056

2.053

2.053

2.052

2.027

2.053

-2.05 3 2.057

2.121

2.049

2.052

-2.041 -2.039 2.045

2.045

2.044

Table 1.4.2-4 Stress intensity factor KIII estimates for conical crack using refined meshes. Contour 1 is omitted from the average value calculations. Solution Crack 1 2 Front Location Corner Full nodes Three-dimensiona Midside l nodes Corner Submodel nodes Three-dimensiona Midside l nodes

0.000 0 0.015 0 0.000 0 0.016

0.000 0 0.014 0 0.000 0 0.016

Contour 3

4

5

0.000 0 0.014 0 0.000 0 0.015

0.000 0 0.013 0 0.000 0 0.015

0.000 0 0.013 0 0.000 0 0.014

Average Value, Contours 2-5 0.0000 0.0140 0.0000 0.015

Table 1.4.2-5 J -integral estimates (´ 10-7) for conical crack using refined meshes. JK denotes the J estimated from stress intensity factors; JA denotes the J estimated directly by ABAQUS. Contour Average Solution Crack 1 2 3 4 5 Value, Front (JK) (JK) (JK) (JK) (JK) Contours Location 1 2 3 4 5 2-5 (JA) (JA) (JA) (JA) (JA) Full Crack tip 1.382 1.330 1.334 1.333 1.332 1.332 1.377 1.331 1.335 1.336 1.336 1.334 Axisymmetric Corner 1.300 1.333 1.337 1.338 1.338 1.337

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Static Stress/Displacement Analyses

Full nodes Three-dimensiona Midside l nodes Submodel Crack tip Axisymmetric Corner Submodel nodes Three-dimensiona Midside l nodes

1.306 1.422 1.412 1.413 1.407 1.329 1.336 1.454 1.443

1.331 1.327 1.330 1.359 1.360 1.363 1.361 1.357 1.360

1.336 1.332 1.335 1.363 1.365 1.367 1.366 1.362 1.365

1.336 1.332 1.335 1.363 1.365 1.368 1.366 1.362 1.365

1.336 1.332 1.336 1.361 1.365 1.368 1.366 1.362 1.366

1.335 1.331 1.334 1.362 1.364 1.367 1.365 1.361 1.364

Table 1.4.2-6 T -stress estimates for conical crack using refined meshes. Contour 1-2 is omitted from the average value calculations. Solution Crack Front Location Full Crack tip Axisymmetric Corner Full nodes Three-dimensiona Midside l nodes Submodel Crack tip Axisymmetric Corner Submodel nodes Three-dimensiona Midside l nodes

1

Contour 3

2

-1.161 -0.981 -0.640 -0.971 -1.315 -0.973 -1.182 -0.983 -0.598 -0.974 -1.366 -0.976

-0.98 2 -0.97 6 -0.97 7 -0.98 5 -0.97 9 -0.98 0

Figures Figure 1.4.2-1 Conical crack in a half-space.

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Average Value, Contours 3-5 -0.981 -0.981 -0.981 4

5

-0.976 -0.976

-0.976

-0.978 -0.979

-0.978

-0.984 -0.984

-0.984

-0.979 -0.979

-0.979

-0.981 -0.982

-0.981

Static Stress/Displacement Analyses

Figure 1.4.2-2 Full three-dimensional mesh.

Figure 1.4.2-3 Full axisymmetric mesh.

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Static Stress/Displacement Analyses

Figure 1.4.2-4 Three-dimensional displaced shape.

Figure 1.4.2-5 Axisymmetric global model for use with submodeling.

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Static Stress/Displacement Analyses

Figure 1.4.2-6 Three-dimensional global model with submodel overlaid.

Figure 1.4.2-7 Axisymmetric global model with submodel overlaid.

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Static Stress/Displacement Analyses

Sample listings

1-651

Static Stress/Displacement Analyses

Listing 1.4.2-1 *HEADING FULL THREE-DIMENSIONAL CONICAL CRACK MODEL *WAVEFRONT MINIMIZATION,SUPPRESS ** INCLUDE NODE DATA *INCLUDE,INPUT=conicalcrack_node.inp ** INCLUDE ELEMENT DATA *INCLUDE,INPUT=conicalcrack_element.inp *MATERIAL,NAME=STEEL *ELASTIC 30.E6,0.3 *SOLID SECTION, MATERIAL=STEEL, ELSET=EALL *MPC TIE,TIP01,TIP02 TIE,TIP11,TIP12 TIE,TIP21,TIP22 TIE,TIP31,TIP32 TIE,TIP41,TIP42 TIE,TIP51,TIP52 TIE,TIP61,TIP62 TIE,TIP71,TIP72 TIE,TIP81,TIP82 TIE,TIP91,TIP92 TIE,TIP101,TIP102 TIE,TIP111,TIP112 TIE,TIP121,TIP122 TIE,TIP131,TIP132 TIE,TIP141,TIP142 TIE,TIP151,TIP152 TIE,TIP161,TIP162 TIE,TIP171,TIP172 TIE,TIP181,TIP182 TIE,TIP191,TIP192 TIE,TIP201,TIP202 *NSET,NSET=N571,GENERATE 57833,2057833,100000 *NSET,NSET=N572,GENERATE 57865,2057865,100000 *NSET,NSET=N65,GENERATE 65833,2065833,100000 *MPC

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Static Stress/Displacement Analyses

TIE,N572,N571 TIE,N65,N571 *NSET,NSET=N18,GENERATE 1833,2001833,100000 *NSET,NSET=N98,GENERATE 9833,2009833,100000 *MPC TIE,N18,N98 *MPC TIE,Y1,Y0 TIE,Y2,Y0 TIE,Y3,Y0 TIE,Y4,Y0 TIE,Y5,Y0 TIE,Y6,Y0 TIE,Y7,Y0 TIE,Y8,Y0 TIE,Y9,Y0 TIE,Y10,Y0 TIE,Y11,Y0 TIE,Y12,Y0 TIE,Y13,Y0 TIE,Y14,Y0 TIE,Y15,Y0 TIE,Y16,Y0 TIE,Y17,Y0 TIE,Y18,Y0 TIE,Y19,Y0 TIE,Y20,Y0 *BOUNDARY N0,3,3 N20,1,1 Y0,1,1 Y0,3,3 *STEP APPLY PRESSURE LOAD *STATIC 1.0,1.0 *DLOAD TOP9,P4,10. *CONTOUR INTEGRAL,CONTOURS=8,OUTPUT=BOTH T0,0.70710678,-0.70710678,0. T1,0.70492701,-0.70710678,0.055478959

1-653

Static Stress/Displacement Analyses

T2,0.69840112,-0.70710678,0.11061587 T3,0.68756936,-0.70710678,0.1650708 T4,0.67249851,-0.70710678,0.21850801 T5,0.65328148,-0.70710678,0.27059805 T6,0.63003676,-0.70710678,0.32101976 T7,0.60290764,-0.70710678,0.36946228 T8,0.5720614,-0.70710678,0.41562694 T9,0.53768821,-0.70710678,0.45922912 T10,0.5,-0.70710678,0.5 T11,0.45922912,-0.70710678,0.53768821 T12,0.41562694,-0.70710678,0.5720614 T13,0.36946228,-0.70710678,0.60290764 T14,0.32101976,-0.70710678,0.63003676 T15,0.27059805,-0.70710678,0.65328148 T16,0.21850801,-0.70710678,0.67249851 T17,0.1650708,-0.70710678,0.68756936 T18,0.11061587,-0.70710678,0.69840112 T19,0.055478959,-0.70710678,0.70492701 T20,0.,-0.70710678,0.70710678 *CONTOUR INTEGRAL,CONTOURS=8,OUTPUT=BOTH, TYPE=K FACTORS T0,0.70710678,-0.70710678,0. T1,0.70492701,-0.70710678,0.055478959 T2,0.69840112,-0.70710678,0.11061587 T3,0.68756936,-0.70710678,0.1650708 T4,0.67249851,-0.70710678,0.21850801 T5,0.65328148,-0.70710678,0.27059805 T6,0.63003676,-0.70710678,0.32101976 T7,0.60290764,-0.70710678,0.36946228 T8,0.5720614,-0.70710678,0.41562694 T9,0.53768821,-0.70710678,0.45922912 T10,0.5,-0.70710678,0.5 T11,0.45922912,-0.70710678,0.53768821 T12,0.41562694,-0.70710678,0.5720614 T13,0.36946228,-0.70710678,0.60290764 T14,0.32101976,-0.70710678,0.63003676 T15,0.27059805,-0.70710678,0.65328148 T16,0.21850801,-0.70710678,0.67249851 T17,0.1650708,-0.70710678,0.68756936 T18,0.11061587,-0.70710678,0.69840112 T19,0.055478959,-0.70710678,0.70492701 T20,0.,-0.70710678,0.70710678 *CONTOUR INTEGRAL,CONTOURS=8,OUTPUT=BOTH,

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Static Stress/Displacement Analyses

TYPE=T-STRESS T0,0.70710678,-0.70710678,0. T1,0.70492701,-0.70710678,0.055478959 T2,0.69840112,-0.70710678,0.11061587 T3,0.68756936,-0.70710678,0.1650708 T4,0.67249851,-0.70710678,0.21850801 T5,0.65328148,-0.70710678,0.27059805 T6,0.63003676,-0.70710678,0.32101976 T7,0.60290764,-0.70710678,0.36946228 T8,0.5720614,-0.70710678,0.41562694 T9,0.53768821,-0.70710678,0.45922912 T10,0.5,-0.70710678,0.5 T11,0.45922912,-0.70710678,0.53768821 T12,0.41562694,-0.70710678,0.5720614 T13,0.36946228,-0.70710678,0.60290764 T14,0.32101976,-0.70710678,0.63003676 T15,0.27059805,-0.70710678,0.65328148 T16,0.21850801,-0.70710678,0.67249851 T17,0.1650708,-0.70710678,0.68756936 T18,0.11061587,-0.70710678,0.69840112 T19,0.055478959,-0.70710678,0.70492701 T20,0.,-0.70710678,0.70710678 *EL PRINT S, *NODE PRINT U,RF *ENDSTEP

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Static Stress/Displacement Analyses

Listing 1.4.2-2 *HEADING AXISYMMETRIC CONICAL CRACK MODEL--GLOBAL WITH 1 RING *WAVEFRONT MINIMIZATION,SUPPRESS *SYSTEM 10.,0.,0. *NODE,SYSTEM=C 1833,0.,0.,0. 9833,0.,0.,0. 57833,0.,0.,0. 57865,0.,0.,0. 65833,0.,0.,0. 1001,15.,-45.,0. 65001,15.,-45.,0. 1033,8.,-45.,0. 65033,8.,-45.,0. 33033,25.,-45.,0. 41033,25.,-18.,0. 57033,10.,-18.,0. 9033,10.,-72.,0. 25033,25.,-72.,0. 41065,25.,0.,0. 57065,10.,0.,0. *SYSTEM 0.,0.,0. *NODE,SYSTEM=C 9865,0.,0.,0. 9065,15.,-90.,0. 25065,30.,-90.,0. 25865,170.,-90.,0. 25965,340.,-90.,0. 25833,170.,-60.,0. 25933,340.,-60.,0. 33833,170.,-45.,0. 33933,340.,-45.,0. 41833,170.,-30.,0. 41933,340.,-30.,0. 41865,170.,0.,0. 41965,340.,0.,0. ** **CRACK TIP REGION *NGEN,NSET=TIP

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1001,65001,1000 *NGEN,NSET=OUTER1 1033,9033,1000 *NGEN,NSET=OUTER2 9033,25033,1000 *NGEN,NSET=OUTER3 25033,33033,1000 *NGEN,NSET=OUTER4 33033,41033,1000 *NGEN,NSET=OUTER5 41033,57033,1000 *NGEN,NSET=OUTER6 57033,65033,1000 *NSET,NSET=OUTER OUTER1,OUTER2,OUTER3,OUTER4,OUTER5,OUTER6 *NFILL,NSET=JREGION,SINGULAR=1 TIP,OUTER,2,16 ** **SECTION 9 *NGEN,NSET=BOT9 9033,9065,1 *NGEN,NSET=TOP9 9833,9865,1 *NFILL,NSET=ALL9 BOT9,TOP9,16,50 ** **SECTION 25 *NGEN,NSET=BOT25 25033,25065,1 *NGEN,NSET=TOP25,LINE=C 25833,25865,1,9865 *NFILL,NSET=ALL25,BIAS=0.8 BOT25,TOP25,16,50 ** **SECTION 41 *NGEN,NSET=BOT41 41033,41065,1 *NGEN,NSET=TOP41,LINE=C 41833,41865,1,9865 *NFILL,NSET=ALL41,BIAS=0.8 BOT41,TOP41,16,50 ** **SECTION 925

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Static Stress/Displacement Analyses

*NFILL,NSET=ALL925 BOT9,BOT25,16,1000 ** **SECTION 2541 *NSET,NSET=BOT2533,GENERATE 25033,33033,1000 *NGEN,NSET=TOP2533,LINE=C 25833,33833,1000,9865 *NSET,NSET=BOT3341,GENERATE 33033,41033,1000 *NGEN,NSET=TOP3341,LINE=C 33833,41833,1000,9865 *NSET,NSET=BOT2541 BOT2533,BOT3341 *NSET,NSET=TOP2541 TOP2533,TOP3341 *NFILL,NSET=ALL2541,BIAS=0.8 BOT2541,TOP2541,16,50 ** **SECTION 4157 *NGEN,NSET=BOT57 57033,57065,1 *NFILL,NSET=ALL4157 BOT41,BOT57,16,1000 ** **SECTION 57 *NGEN,NSET=TOP57 57833,57865,1 *NFILL,NSET=ALL57,BIAS=1.0 BOT57,TOP57,16,50 ** **SECTION 5765 *NGEN,NSET=TOP5765 57833,65833,1000 *NFILL,NSET=ALL5765,BIAS=1.0 OUTER6,TOP5765,16,50 ** **SECTION 19 *NGEN,NSET=TOP19 1833,9833,1000 *NFILL,NSET=ALL19 OUTER1,TOP19,16,50 **

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Static Stress/Displacement Analyses

**INFINITE ELEMENT REGION *NGEN,NSET=INF25,LINE=C 25933,25965,8,9865 *NGEN,NSET=INF2533,LINE=C 25933,33933,4000,9865 *NGEN,NSET=INF3341,LINE=C 33933,41933,4000,9865 *NGEN,NSET=INF41,LINE=C 41933,41965,8,9865 *NSET,NSET=INF INF25,INF2533,INF3341,INF41 ** **CRACK TIP REGION ELEMENTS *ELEMENT,TYPE=CAX8R 1001,1001,1033,5033,5001,1017,3033,5017,3001 *ELGEN,ELSET=RING 1001,16,4000,4000 ** **ELEMENTS SECTION 9 *ELEMENT,TYPE=CAX8R 9033,9033,9233,9241,9041,9133,9237,9141,9037 *ELGEN,ELSET=SECT9 9033,4,8,8,4,200,200 *ELSET,ELSET=TOP9,GENERATE 9633,9657,8 ** **ELEMENTS SECTION 25 *ELEMENT,TYPE=CAX8R 25233,25233,25033,25041,25241,25133,25037, 25141,25237 *ELGEN,ELSET=SECT25 25233,4,8,8,4,200,200 ** **ELEMENTS SECTION 19 *ELEMENT,TYPE=CAX8R 1033,1033,1233,5233,5033,1133,3233,5133,3033 *ELGEN,ELSET=SECT19 1033,2,4000,4000,4,200,200 ** **ELEMENT SECTION 925 *ELEMENT,TYPE=CAX8R 13033,13033,9033,9041,13041,11033,9037, 11041,13037

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Static Stress/Displacement Analyses

*ELGEN,ELSET=SECT925 13033,4,8,8,4,4000,4000 ** **ELEMENT SECTION 2541 *ELEMENT,TYPE=CAX8R 29233,29233,29033,25033,25233,29133,27033, 25133,27233 *ELGEN,ELSET=SECT2541 29233,4,4000,4000,4,200,200 ** **ELEMENT SECTION 41 *ELEMENT, TYPE=CAX8R 41241,41241,41041,41033,41233,41141,41037, 41133,41237 *ELGEN,ELSET=SECT41 41241,4,8,8,4,200,200 ** **ELEMENT SECTION 4157 *ELEMENT,TYPE=CAX8R 41041,41041,45041,45033,41033,43041,45037, 43033,41037 *ELGEN,ELSET=SECT4157 41041,4,8,8,4,4000,4000 ** **ELEMENT SECTION 57 *ELEMENT, TYPE=CAX8R 57041,57041,57241,57233,57033,57141,57237, 57133,57037 *ELGEN,ELSET=SECT57 57041,4,8,8,4,200,200 ** **ELEMENT SECTION 5765 *ELEMENT,TYPE=CAX8R 57033,57033,57233,61233,61033,57133,59233, 61133,59033 *ELGEN,ELSET=SECT5765 57033,2,4000,4000,4,200,200 ** **INFINITE ELEMENTS *ELEMENT,TYPE=CINAX5R 41941,41841,41833,41933,41941,41837 *ELGEN,ELSET=INF41 41941,4,8,8

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Static Stress/Displacement Analyses

*ELEMENT,TYPE=CINAX5R 29933,29833,25833,25933,29933,27833 *ELGEN,ELSET=INF2541 29933,4,4000,4000 *ELEMENT,TYPE=CINAX5R 25933,25833,25841,25941,25933,25837 *ELGEN,ELSET=INF25 25933,4,8,8 *ELSET,ELSET=INF INF41,INF2541,INF25 ** *ELSET,ELSET=E1 RING,SECT9,SECT25,SECT41,SECT57,SECT19,SECT925, SECT2541,SECT4157,SECT5765,INF *MATERIAL,NAME=STEEL *ELASTIC 30.E6,0.3 *SOLID SECTION, MATERIAL=STEEL, ELSET=E1 ** **ADD BOUNDARY CONDITIONS *NSET,NSET=R9,GENERATE 9065,9865,50 *NSET,NSET=R925,GENERATE 9065,25065,1000 *NSET,NSET=R25,GENERATE 25065,25865,50 *NSET,NSET=YAXIS R9,R925,R25 *NSET,NSET=L57,GENERATE 57065,57865,50 *NSET,NSET=L4157,GENERATE 41065,57065,1000 *NSET,NSET=L41,GENERATE 41065,41865,50 *NSET,NSET=XAXIS TOP9,L57,L4157,L41 *BOUNDARY YAXIS,XSYMM ** **MPC'S TO TIE REDUNDANT NODES *NSET,NSET=TIP1,GENERATE 1001,64001,1000 *NSET,NSET=TIP2,GENERATE

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Static Stress/Displacement Analyses

2001,65001,1000 *MPC TIE,TIP1,TIP2 *MPC TIE,57865,57833 *MPC TIE,57833,65833 *MPC TIE,1833,9833 ** *STEP APPLY PRESSURE LOAD *STATIC 1.0,1.0 *DLOAD TOP9,P2,10. *CONTOUR INTEGRAL,CONTOURS=1,OUTPUT=BOTH TIP,0.707107,-0.707107 *CONTOUR INTEGRAL,CONTOURS=1,OUTPUT=BOTH, TYPE=K FACTORS TIP,0.707107,-0.707107 *CONTOUR INTEGRAL,CONTOURS=1,OUTPUT=BOTH, TYPE=T-STRESS TIP,0.707107,-0.707107 *EL PRINT S, *NODE PRINT U,RF *NODE FILE U, *OUTPUT,FIELD *NODE OUTPUT U, *ENDSTEP

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Static Stress/Displacement Analyses

Listing 1.4.2-3 *HEADING AXISYMMETRIC CONICAL CRACK MODEL SUBMODEL WITH 8 RINGS *WAVEFRONT MINIMIZATION,SUPPRESS *SYSTEM 10.,0.,0. *NODE,SYSTEM=C 1833,0.,0.,0. 9833,0.,0.,0. 57833,0.,0.,0. 57865,0.,0.,0. 65833,0.,0.,0. 1001,15.,-45.,0. 65001,15.,-45.,0. 1033,8.,-45.,0. 65033,8.,-45.,0. 33033,25.,-45.,0. 41033,25.,-18.,0. 57033,10.,-18.,0. 9033,10.,-72.,0. 25033,25.,-72.,0. 41065,25.,0.,0. 57065,10.,0.,0. *SYSTEM 0.,0.,0. *NODE,SYSTEM=C 9865,0.,0.,0. 9065,15.,-90.,0. 25065,30.,-90.,0. 25865,170.,-90.,0. 25965,340.,-90.,0. 25833,170.,-60.,0. 25933,340.,-60.,0. 33833,170.,-45.,0. 33933,340.,-45.,0. 41833,170.,-30.,0. 41933,340.,-30.,0. 41865,170.,0.,0. 41965,340.,0.,0. *NODE,SYSTEM=R 1033,15.656,-5.6579

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Static Stress/Displacement Analyses

65033,15.658,-5.6559 ** **CRACK TIP REGION *NGEN,NSET=TIP 1001,65001,2000 *NGEN,NSET=OUTER1 1033,9033,2000 *NGEN,NSET=OUTER2 9033,25033,2000 *NGEN,NSET=OUTER3 25033,33033,2000 *NGEN,NSET=OUTER4 33033,41033,2000 *NGEN,NSET=OUTER5 41033,57033,2000 *NGEN,NSET=OUTER6 57033,65033,2000 *NSET,NSET=OUTER OUTER1,OUTER2,OUTER3,OUTER4,OUTER5,OUTER6 *NFILL,NSET=JREGION,SINGULAR=1 TIP,OUTER,16,2 ** **CRACK TIP REGION ELEMENTS *ELEMENT,TYPE=CAX8R 1001,1001,1005,5005,5001,1003,3005,5003,3001 *ELGEN,ELSET=RINGS 1001,16,4000,4000,8,4,4 ** *MATERIAL,NAME=STEEL *ELASTIC 30.E6,0.3 *SOLID SECTION, MATERIAL=STEEL, ELSET=RINGS ** **MPC'S TO TIE REDUNDANT NODES *NSET,NSET=TIP1,GENERATE 1001,63001,2000 *NSET,NSET=TIP2,GENERATE 3001,65001,2000 *MPC TIE,TIP1,TIP2 *SUBMODEL OUTER, **

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Static Stress/Displacement Analyses

*STEP *STATIC 1.,1. *BOUNDARY,SUBMODEL,STEP=1 OUTER,1,2 *CONTOUR INTEGRAL,CONTOURS=8,OUTPUT=BOTH TIP,0.707107,-0.707107 *CONTOUR INTEGRAL,CONTOURS=8,OUTPUT=BOTH, TYPE=K FACTORS TIP,0.707107,-0.707107 *CONTOUR INTEGRAL,CONTOURS=8,OUTPUT=BOTH, TYPE=T-STRESS TIP,0.707107,-0.707107 *EL PRINT S, *NODE PRINT U,RF *ENDSTEP

1.4.3 Elastic-plastic line spring modeling of a finite length cylinder with a part-through axial flaw Product: ABAQUS/Standard The elastic-plastic line spring elements in ABAQUS are intended to provide inexpensive solutions for problems involving part-through surface cracks in shell structures loaded predominantly in Mode I by combined membrane and bending action in cases where it is important to include the effects of inelastic deformation. This example illustrates the use of these elements. The case considered is a long cylinder with an axial flaw in its inside surface, subjected to internal pressure. It is taken from the paper by Parks and White (1982). When the line spring element model reaches theoretical limitations, the shell-to-solid submodeling technique is utilized to provide accurate J -integral results. The energy domain integral is used to evaluate the J -integral for this case.

Geometry and model The cylinder has an inside radius of 254 mm (10 in), wall thickness of 25.4 mm (1 in), and is assumed to be very long. The mesh is shown in Figure 1.4.3-1. It is refined around the crack by using multi-point constraints (MPCs). There are 70 shell elements of type S8R in the symmetric quarter-model and eight symmetric line spring elements (type LS3S) along the crack. The mesh is taken from Parks and White, who suggest that this mesh is adequately convergent with respect to the fracture parameters (J -integral values) that are the primary objective of the analysis. No independent mesh studies have been done. The use of MPCs to refine a mesh of reduced integration shell elements (such as S8R) is generally satisfactory in relatively thick shells as in this case. However, it is not

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recommended for thin shells because it introduces constraints that "lock" the response in the finer mesh regions. In a thin shell case the finer mesh would have to be carried out well away from the region of high strain gradients. Three different flaws are studied. All have the semi-elliptic geometry shown in Figure 1.4.3-2, with, in all cases, c = 3a0 : The three flaws have a0 =t ratios of 0.25 (a shallow crack), 0.5, and 0.8 (a deep crack). In all cases the axial length of the cylinder is taken as 14 times the crack half-length, c: this is assumed to be sufficient to approximate the infinite length. An input data file for the case a0 =t =.5 without making the symmetry assumption about z =0 is also included. This mesh uses the LS6 line spring elements and serves to check the elastic-plastic capability of the LS6 elements. The results are the same as for the corresponding mesh using LS3S elements and symmetry about z =0. The formulation of the LS6 elements assumes that the plasticity is predominately due to Mode I deformation around the flaw and neglects the effect of the Mode II and Mode III deformation around the flaw. In the global mesh the displacement in the z-direction is constrained to be zero at the node at the end of the flaw where the flaw depth goes to zero. To duplicate this constraint in the mesh using LS6 elements, the two nodes at the end of the flaw (flaw depth = 0) are constrained to have the same displacements.

Material The cylinder is assumed to be made of an elastic-plastic metal, with a Young's modulus of 206.8 GPa (30 ´ 106 lb/in2), a Poisson's ratio of 0.3, an initial yield stress of 482.5 MPa (70000 lb/in 2, and constant work hardening to an ultimate stress of 689.4 MPa (10 5 lb/in2) at 10% plastic strain, with perfectly plastic behavior at higher strains.

Loading The loading consists of uniform internal pressure applied to all of the shell elements, with edge loads applied to the far end of the cylinder to provide the axial stress corresponding to a closed-end condition. Even though the flaw is on the inside surface of the cylinder, the pressure is not applied on the exposed crack face. Since pressure loads on the flaw surface of line spring elements are implemented using linear superposition in ABAQUS, there is no theoretical basis for applying these loads when nonlinearities are present. We assume that this is not a large effect in this problem. For consistency with the line spring element models, pressure loading of the crack face is not applied to the shell-to-solid submodel.

Results and discussion The line spring elements provide J -integral values directly. Figure 1.4.3-3 shows the J -integral values at the center of the crack as functions of applied pressure for the three flaws. In the input data the maximum time increment size has been limited so that adequately smooth graphs can be obtained. Figure 1.4.3-4 shows the variations of the J -integral values along the crack for the half-thickness crack ¹ ¹ y t, is used, where R (a0 =t =0.5), at several different pressure levels (a normalized pressure, p^ = pR=¾ is the mean radius of the cylinder). These results all agree closely with those reported by Parks and White (1982), where the authors state that these results are also confirmed by other work. In the region

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Static Stress/Displacement Analyses

Á 30° are shown in Figure 1.4.3-4.

Shell-to-solid submodeling around the crack tip An input file for the case a0 =t =0.25, which uses the shell-to-solid submodeling capability, is included. This C3D20R element mesh allows the user to study the local crack area using the energy domain integral formulation for the J -integral. The submodel uses a focused mesh with four rows of elements around the crack tip. A 1/r singularity is utilized at the crack tip, the correct singularity for a fully developed perfectly plastic solution. Symmetry boundary conditions are imposed on two edges of the submodel mesh, while results from the global shell analysis are interpolated to two surfaces via the submodeling technique. The global shell mesh gives satisfactory J -integral results; hence, we assume that the displacements at the submodel boundary are sufficiently accurate to drive the deformation in the submodel. No attempt has been made to study the effect of making the submodel region larger or smaller. The submodel is shown superimposed on the global shell model in Figure 1.4.3-5. In addition, an input file for the case a0 =t =0.25, which consists of a full three-dimensional C3D20R solid element model, is included for use as a reference solution. This model has the same general characteristics as the submodel mesh. See inelasticlinespring_c3d20r_ful.inp for further details about this mesh. One important difference exists in performing this analysis with shell elements as opposed to continuum elements. The pressure loading is applied to the midsurface of the shell elements as opposed to the continuum elements, where the pressure is accurately applied along the inside surface of the cylinder. For this analysis this discrepancy results in about 10% higher J -integral values for the line spring shell element analysis as compared to the full three-dimensional solid element model. Results from the submodeled analyses are compared to the LS3S line spring element analysis and full solid element mesh for variations of the J -integral values along the crack at the a normalized pressure ¹ is the mean radius of the cylinder. As seen in Figure 1.4.3-6, ¹ (¾y t) = 0.898, where R loading of pR= the line spring elements underestimate the J -integral values for Á 0 corresponds to the restrained mode addition technique. In the "Results and discussion" section below the solution obtained for the model without superelements (the "full model") is used as the reference solution. For the cyclic symmetry model without superelements, the eigenvalue extraction procedure was performed on the preloaded structure. The nonlinear static step has the centrifugal load applied to the blade. Fifty eigenvalues were requested using the Lanczos eigenvalue solver, which is the only eigensolver that can be used for *FREQUENCY analysis with the *CYCLIC SYMMETRY MODEL option. The *SELECT CYCLIC SYMMETRY MODES option is omitted; therefore, the eigenvalues are being extracted for all possible (three) cyclic symmetry modes. In the discussion that follows the

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solution obtained for the cyclic symmetry model is compared with the solution for the entire 360° model as the reference solution. A similar simulation was performed for the cyclic symmetry model with superelements but without the preload step. Twenty eigenvalues were extracted and compared with the reference solution obtained for the entire 360° model with superelements.

Results and discussion Results for the frequency analysis and the static analysis appear below.

Frequency analysis for models with superelements Frequencies corresponding to the 15 lowest eigenvalues have been extracted and are tabulated in Table 2.2.1-1 for each model. To study the effect of retaining dynamic modes during superelement generation, the superelement models were run after extracting 0, 5, and 20 dynamic modes during superelement generation. While the Guyan reduction technique (m =0) yields frequencies that are reasonable compared to those of the full model, the values obtained with 5 retained modes are much closer to full model predictions, especially for the higher eigenvalues. Increasing the number of retained modes to 20 does not yield a significant improvement in the results, consistent with the fact that in the Guyan reduction technique the choice of retained degrees of freedom affects accuracy, while for the restrained mode addition technique the modes corresponding to the lowest frequencies are by definition optimal. When superelements are used in an eigenfrequency analysis, it is to be expected that the lowest eigenfrequency in the superelement model is higher than the lowest eigenfrequency in the corresponding model without superelements. This is indeed the case for the single-level superelement analysis, but for the multi-level superelement analysis the lowest eigenfrequency is below the one for the full model. This occurs because the stress and load stiffness for the lowest level superelement (the blade) are generated with the root of the blade fixed, whereas in the full model the root of the blade will move radially due to the deformation of the hub under the applied centrifugal load. Hence, the superelement stiffness is somewhat inaccurate. Since the radial displacements at the blade root are small compared to the overall dimensions of the model (of order 10 -3), the resulting error should be small, as is observed from the results. Table 2.2.1-2 shows what happens if the NLGEOM parameter is omitted during the preloading steps. It is clear that the results are significantly different from the ones that take the effect of the preload on the stiffness into account. Note that in this case the lowest eigenfrequency in the superelement models is indeed above the lowest eigenfrequency in the model without superelements.

Static analysis for models with superelements A static analysis of the fan is carried out about the preloaded base state by applying a pressure load of 105 Pa normal to the blades of the fan. The axial displacement of the outer edge of the fan blade due to the pressure load is monitored at nodes along path AB, as shown in Figure 2.2.1-1. The results are shown in Figure 2.2.1-4; there is good agreement between the solutions for the superelement models and the full model. While superelements can be generated from models that exhibit nonlinear response, it must be noted

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Dynamic Stress/Displacement Analyses

that, once created, a superelement always exhibits linear response at the usage level. Hence, a preloaded superelement will produce a response equivalent to that of the response to a linear perturbation load on a preloaded full model. Consequently, the full model is analyzed by applying the centrifugal preload in a general step and the pressure load in a linear perturbation step. Since an analysis using superelements is equivalent to a perturbation step, the results obtained do not incorporate the preload deformation. Thus, if the total displacement of the structure is desired, the results of this perturbation step need to be added to the base state solution of the structure.

Static analysis and eigenvalue extraction for the cyclic symmetry model In the general static step that includes nonlinear geometry, the centrifugal load is applied to the datum sector. Only symmetric loads can be applied in general static steps with the *CYCLIC SYMMETRY MODEL option. The computed eigenvalues are identical with those obtained for the entire 360° model, as shown in the Table 2.2.1-1. The additional information obtained during the eigenvalue extraction is the cyclic symmetry mode number associated with each eigenvalue. In the case of 4 repetitive sectors, all the eigenvalues corresponding to cyclic symmetry mode 1 appear in pairs; the eigenvalues corresponding to modes 0 and 2 are single. The lowest first two eigenvalues correspond to cyclic symmetry mode 1, followed by the single eigenvalues corresponding to cyclic symmetry modes 2 and 0. For a comparison with the cyclic symmetry model option, the problem is also modeled with *MPC type CYCLSYM (see fansuperelem_mpc.inp). To verify the use of superelements with the cyclic symmetry model, it was determined that the results obtained with fansuperelem_cyclic.inp were identical to the results obtained with fansuperelem_1level_freq.inp.

Input files fan_cyclicsymmodel.inp Cyclic symmetry model with static and eigenvalue extraction steps. fansuperelem_1level_freq.inp Single-level superelement usage analysis with a frequency extraction step. fansuperelem_1level_static.inp Single-level superelement usage analysis with a static step. fansuperelem_multi_freq.inp Multi-level superelement usage analysis with a frequency extraction step. fansuperelem_multi_static.inp Multi-level superelement usage analysis with a static step. fansuperelem_freq.inp Frequency extraction without superelements. fansuperelem_static.inp

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Static analysis without superelements. fansuperelem_mpc.inp Single-level usage analysis demonstrating the use of cyclic symmetry MPCs. fansuperelem_gen1.inp Superelement generation for a single blade used in the multi-level superelement generation file fansuperelem_gen2.inp. fansuperelem_gen2.inp Multi-level superelement generation used in fansuperelem_multi_freq.inp and fansuperelem_multi_static.inp. fansuperelem_gen3.inp Single-level superelement generation used infansuperelem_1level_freq.inp, fansuperelem_1level_static.inp, and fansuperelem_mpc.inp. fansuperelem_cyclic.inp Single-level superelement with the cyclic symmetry model used in a frequency analysis.

Tables Table 2.2.1-1 Comparison of natural frequencies for single-level and multi-level superelements with the values for the model without superelements. With substructuring: 1 With substructuring: 2 Eigenvalue level levels no. cycles/sec m=0 m=5 m=20 m=0 m=5 m=20 1 6.9464 6.7893 6.7882 6.9191 6.7665 6.7654 2 6.9464 6.7893 6.7882 6.9191 6.7665 6.7654 3 8.0024 7.7148 7.7139 8.0082 7.7228 7.7219 4 8.2007 7.8817 7.8810 8.2079 7.8909 7.8903 5 11.343 11.021 11.010 11.308 10.986 10.976 6 11.343 11.021 11.010 11.308 10.986 10.976 7 12.513 11.916 11.897 12.291 11.760 11.741 8 14.683 14.354 14.301 14.671 14.303 14.252 9 17.862 14.432 14.432 17.745 14.470 14.470 10 18.921 14.776 14.772 18.913 14.814 14.810 11 21.150 14.776 14.772 21.010 14.814 14.810 12 21.150 15.990 15.952 21.010 16.001 15.963 13 28.449 17.773 17.696 28.417 17.652 17.575 14 28.986 19.029 19.012 29.001 19.030 19.013 15 28.986 21.234 21.077 29.001 21.082 20.928

Full model 6.7881 6.7881 7.7139 7.8810 11.009 11.009 11.895 14.303 14.432 14.771 14.771 15.948 17.696 19.001 21.075

Table 2.2.1-2 Comparison of natural frequencies for single-level and two-level superelements with

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the full model values without the use of the NLGEOM parameter. With Full Eigenvalue substructuring model no. cycles/sec 1 level 2 levels 1 4.4811 4.4811 4.4809 2 4.4811 4.4811 4.4809 3 4.5489 4.5489 4.5487 4 4.8916 4.8916 4.8914 5 9.5519 9.5519 9.5423 6 9.5519 9.5519 9.5423 7 9.7893 9.7894 9.7758 8 12.611 12.611 12.570 9 14.006 14.006 14.003 10 14.332 14.332 14.325 11 14.332 14.332 14.325 12 15.475 15.475 15.455 13 16.962 16.963 16.897 14 18.244 18.245 18.220 15 19.040 19.041 18.933

Figures Figure 2.2.1-1 Mesh used for the complete fan model.

Figure 2.2.1-2 Superelements generated.

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Figure 2.2.1-3 Datum sector for cyclic symmetry model.

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Figure 2.2.1-4 Displacements due to pressure loading along path AB.

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Sample listings

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Dynamic Stress/Displacement Analyses

Listing 2.2.1-1 *HEADING Superelement analysis of a fan Frequency analysis : One level superelement. Usage level: FAN=HUB+4xBLADES ** Z100: a superelement for a single blade; Z200: a superelement which contains the 1/4 hub and a superelement blade (Z100); Z300: a superelement which contains the 1/4 hub and a full blade. Requires superelement generation file fansuperelem_gen3.inp ** *RESTART,WRITE,FRE=1 ** *NODE 1, 3.50000, 3.00000, 4.00000 3, 3.46638, 3.18024, 4.00000 7, 3.18024, 3.46638, 4.00000 9, 3.00000, 3.50000, 4.00000 ** 31, 6.00000, 3.00000, 4.00000 33, 5.79830, 4.08144, 4.00000 37, 4.08144, 5.79830, 4.00000 39, 3.00000, 6.00000, 4.00000 ** 91, 6.00000, 3.00000, 0.00000 93, 5.79830, 4.08144, 0.00000 97, 4.08144, 5.79830, 0.00000 99, 3.00000, 6.00000, 0.00000 ** 100, 7.89643, 8.00249, 2.59810 9998, 3.00000, 3.00000, 4.00000 9999, 3.00000, 3.00000, 0.00000 *NGEN,LINE=C,NSET=NHUB 1,3,1,9998 3,7,1,9998 7,9,1,9998 *NGEN,LINE=C,NSET=RINGF 31,33,1,9998 33,37,1,9998

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37,39,1,9998 *NGEN,LINE=C,NSET=RINGB 91,93,1,9999 93,97,1,9999 97,99,1,9999 *NFILL,NSET=HUB NHUB,RINGF,3,10 RINGF,RINGB,6,10 ** *NSET,NSET=PART1,GENERATE 1,91,10 9,99,10 2,8,1 32,38,1 100,100,1 ** *NCOPY,CHANGE NUMBER=100,OLD SET=PART1,SHIFT 0.,0.,0. 3.,3.,-1.,3.,3.,1.,90. *NCOPY,CHANGE NUMBER=200,OLD SET=PART1,SHIFT 0.,0.,0. 3.,3.,-1.,3.,3.,1.,180. *NCOPY,CHANGE NUMBER=300,OLD SET=PART1,SHIFT 0.,0.,0. 3.,3.,-1.,3.,3.,1.,270. ** *NSET,NSET=PART1X,GENERATE 1,91,10 *NSET,NSET=PART1Y,GENERATE 9,99,10 *NSET,NSET=PART2X,GENERATE 109,199,10 *NSET,NSET=PART2Y,GENERATE 101,191,10 *NSET,NSET=PART3X,GENERATE 201,291,10 *NSET,NSET=PART3Y,GENERATE 209,299,10 *NSET,NSET=PART4X,GENERATE 309,399,10 *NSET,NSET=PART4Y,GENERATE 301,391,10 *MPC

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TIE,PART4X,PART1X TIE,PART2Y,PART1Y TIE,PART2X,PART3X TIE,PART4Y,PART3Y ** *NSET,NSET=NODEHUB,GENERATE 1, 9, 1 102, 108, 1 201, 209, 1 302, 308, 1 ** *ELEMENT,TYPE=Z300,ELSET=P1,FILE=FAN 901,1,2,3,4,5,6,7,8,9,11,19,21,29,31,32, 33,34,35,36,37,38,39,41,49,51,59,61,69,71,79,81, 89,91,99,100 *ELCOPY,ELEMENT SHIFT=1,OLD SET=P1, SHIFT NODES=100,NEW SET=P2 *ELCOPY,ELEMENT SHIFT=2,OLD SET=P1, SHIFT NODES=200,NEW SET=P3 *ELCOPY,ELEMENT SHIFT=3,OLD SET=P1, SHIFT NODES=300,NEW SET=P4 ** *SUPER PROPERTY,ELSET=P1 0.,0.,0. *SUPER PROPERTY,ELSET=P2 0.,0.,0. 3.,3.,0.,3.,3.,3.,90. *SUPER PROPERTY,ELSET=P3 0.,0.,0. 3.,3.,0.,3.,3.,3.,180. *SUPER PROPERTY,ELSET=P4 0.,0.,0. 3.,3.,0.,3.,3.,3.,270. ** *STEP Step 1: Eigenfrequency extraction *FREQUENCY 20, *BOUNDARY NODEHUB,ENCASTRE ** *EL PRINT,FREQ=0 *EL FILE,FREQ=0

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*NODE PRINT,FREQ=0 *MODAL FILE,FREQ=99 ** *END STEP

2.2.2 Linear analysis of the Indian Point reactor feedwater line Product: ABAQUS/Standard This example concerns the linear analysis of an actual pipeline from a nuclear reactor and is intended to illustrate some of the issues that must be addressed in performing seismic piping analysis. The pipeline is the Indian Point Boiler Feedwater Pipe fitted with modern supports, as shown in Figure 2.2.2-1. This pipeline was tested experimentally in EPRI's full-scale testing program. The model corresponds to Configuration 1 of the line in Phase III of the testing program. The experimental results are documented in EPRI Report NP-3108 Volume 1 (1983). We first verify that the geometric/kinematic model is adequate to simulate the dynamic response accurately. For this purpose we compare predictions of the natural frequencies of the system using a coarse model and a finer model, as well as two substructure models created from the coarse mesh. These analyses are intended to verify that the models used in subsequent runs provide accurate predictions of the lower frequencies of the pipeline. We then perform linear dynamic response analysis in the time domain for one of the "snap-back" loadings applied in the physical test ( EPRI NP-3108, 1983) and compare the results with the experimental measurements. The linear dynamic response analysis results are also compared with the results of direct integration analysis (integration of all variables in the entire model, as would be performed for a generally nonlinear problem). This is done primarily for cross-verification of the two analysis procedures. These snap-back response analyses correspond to a load of 31136 N (7000 lb) applied at node 25 in the z-direction, with the pipe filled with water. This load case is referred to as test S138R1SZ in EPRI NP-3108. We also compute the pipeline's response in the frequency domain to steady excitation at node 27 in the z-direction. Experimental data are also available for comparison with these results.

Geometry and model Geometrical and material properties are taken from EPRI NP-3108 (1983). The supports are assumed to be linear springs for the purpose of these linear analyses, although their actual response is probably nonlinear. The spring stiffness values are those recommended by Tang et al. (1985). The pipe is assumed to be completely restrained in the vertical direction at the wall penetration. In the experimental snap-back test used for the comparison (test S138R1SZ), the pipe is full of water. The DENSITY parameter on the *BEAM GENERAL SECTION option is, therefore, adjusted to account for the additional mass of the water by computing a composite (steel plus water) mass per unit length of pipe. The pipeline is modeled with element type B31. This is a shear flexible beam element that uses linear interpolation of displacement and rotation between two nodes, with transverse shear behavior modeled according to Timoshenko beam theory. The element uses a lumped mass matrix because this provides

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more accurate results in test cases. The coarse finite element model uses at least two beam elements along each straight run, with a finer division around the curved segments of the pipe to describe the curvature of the pipe with reasonable accuracy. Separate nodes are assigned for all spring supports, external loading locations, and all the points where experimental data have been recorded. The model is shown in Figure 2.2.2-2. This mesh has 74 beam elements. In typical piping systems the elbows play a dominant role in the response because of their flexibility. This could be incorporated in the model by using the ELBOW elements. However, ELBOW elements are intended for applications that involve nonlinear response within the elbows themselves and are an expensive option for linear response of the elbows, which is the case for this study. Therefore, instead of using elbow elements, we modify the geometrical properties of beam elements to model the elbows with correct flexibility. This is done by calculating the flexibility factor, k, for each elbow and modifying the moments of inertia of the beam cross-sections in these regions. The flexibility factor for an elbow is a function of two parameters. One is a geometric parameter, ¸, defined as ¸=

tR p ; r2 1 ¡ º 2

where t is the wall thickness of the curved pipe, R is the bend radius of the centerline of the curved pipe, r is the mean cross-sectional radius of the curved pipe, and º is Poisson's ratio. The other parameter is an internal pressure loading parameter, Ã. For thick sections (like the ones used in this pipe), Ã has negligible effect unless the pressures are very high and the water in this case is not pressurized. Consequently, the flexibility factor is a function of ¸ only. For the elbows in this pipeline ¸ = 0.786 for the 203 mm (8 in) section and ¸ = 0.912 for the 152 mm (6 in) section. The corresponding flexibility factors obtained from Dodge and Moore (1972) are 2.09 and 1.85. These are implemented in the model by modifying the moments of inertia of the beam cross-sections in the curved regions of the pipeline. ABAQUS provides two different options for introducing geometrical properties of a beam cross-section. One is the *BEAM GENERAL SECTION option, in which all geometric properties (area, moments of inertia) can be given without specifying the shape of the cross-section. The material data, including the density, are given on the same option. Alternatively, the geometrical properties of the cross-section can be given by using the *BEAM SECTION option. With this option the cross-section dimensions are given, and ABAQUS calculates the corresponding cross-sectional behavior by numerical integration, thus allowing for nonlinear material response in the section. When this option is used, the material properties--including density and damping coefficients--are introduced in the *MATERIAL option associated with the section. This approach is more expensive for systems in which the cross-sectional behavior is linear, since numerical integration over the section is required each time the stress must be computed. Thus, in this case we use the *BEAM GENERAL SECTION option. To verify that the mesh will provide results of adequate accuracy, the natural frequencies predicted with this model are compared with those obtained with another mesh that has twice as many elements

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in each pipe segment. Table 2.2.2-1 shows that these two meshes provide results within 2% for the first six modes and generally quite similar frequencies up to about 30 Hz. Based on this comparison the smaller model, with 74 beam elements, is used for the remaining studies (although the larger model would add little to the cost of the linear analyses, which for either case would be based on the same number of eigenmodes: only in direct integration would the cost increase proportionally with the model size).

Superelement models In ABAQUS the dynamic response of a superelement is defined by a combination of Guyan reduction and the inclusion of some natural modes of the fully restrained superelement. Guyan reduction consists of choosing additional physical degrees of freedom to retain in the dynamic model that are not needed to connect the superelement to the rest of the mesh. In this example we use only Guyan reduction since the model is small and it is easy to identify suitable degrees of freedom to retain. A critical modeling issue with this method is the choice of retained degrees of freedom: enough degrees of freedom must be retained so that the dynamic response of the substructure is modeled with sufficient accuracy. The retained degrees of freedom should be such as to distribute the mass evenly in each substructure so that the lower frequency response of each substructure is modeled accurately. Only frequencies up to 33 Hz are generally considered important in the seismic response of piping systems such as the one studied in this example, so the retained degrees of freedom must be chosen to provide accurate modeling of the response up to that frequency. In this case the pipeline naturally divides into three segments in terms of which kinematic directions participate in the dynamic response, because the response of a pipeline is generally dominated by transverse displacement. The lower part of the pipeline, between nodes 1 and 23, is, therefore, likely to respond predominantly in degrees of freedom 1 and 2; the middle part, between nodes 23 and 49, should respond in degrees of freedom 2 and 3; and the top part, above node 49, should respond in degrees of freedom 1 and 3. Comparative tests (not documented) have been run to verify these conjectures, and two superelement models have been retained for further analysis: one in which the entire pipeline is treated as a single superelement, and one in which it is split into three superelements. In the latter case all degrees of freedom must be retained at the interface nodes to join the superelements correctly. At other nodes only some translational degrees of freedom are retained, based on the arguments presented above. The choice of which degrees of freedom to retain can be investigated inexpensively in a case such as this by numerical experiments--extracting the modes of the reduced system for the particular set of retained degrees of freedom and comparing these modes with those of the complete model. The choices made in the superelement models used here are based somewhat on such tests, although insufficient tests have been run to ensure that they are close to the optimal choice for accuracy with a given number of retained variables. For linear analysis of a model as small as this one, achieving an optimal selection of retained degrees of freedom is not critical because computer run times are short: it becomes more critical when the reduced model is used in a nonlinear analysis or where the underlying model is so large that comparative eigenvalue tests cannot be performed easily. In such cases the inclusion of natural modes of the superelement is desirable. The superelement models are shown in Figure 2.2.2-2.

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Damping "Damping" plays an essential role in any practical dynamic analysis. In nonlinear analysis the "damping" is often modeled by introducing dissipation directly into the constitutive definition as viscosity or plasticity. In linear analysis equivalent linear damping is used to approximate dissipation mechanisms that are not modeled explicitly. Experimental estimates of equivalent linear damping, based on three different methods, are found in EPRI NP-3108 (see Table 7-6, Table 7-7, and Figure 7-15 of that report). For the load case and pipe configuration analyzed here, those results suggest that linear damping corresponding to 2.8% of critical damping in the lowest mode of the system matches the measured behavior of the structure, with the experimental results also showing that the percentage of critical damping changes from mode to mode. In spite of this all the numerical analyses reported here assume the same damping ratio for all modes included in the model, this choice being made for simplicity only. For linear dynamic analysis based on the eigenmodes, ABAQUS allows damping to be defined as a percentage of critical damping in each mode, as structural damping (proportional to nodal forces), or as Rayleigh damping (proportional to the mass and stiffness of the structure). Only the last option is possible when using direct integration, although other forms of damping can be added as discrete dashpots or in the constitutive models. In this case, results are obtained for linear dynamic analysis with modal and Rayleigh damping and for direct integration with Rayleigh damping.

Results and discussion Results are shown for four geometric models: the "coarse" (74 element) model, which has a total of 435 degrees of freedom; a finer (148 element) model, which has a total of 870 degrees of freedom; a model in which the pipeline is modeled as a single superelement (made from the coarse model), with 59 retained degrees of freedom; and a model in which the pipeline is modeled with three superelements (made from the coarse model), with 65 retained degrees of freedom. The first comparison of results is the natural frequencies of the system, as they are measured and as they are predicted by the various models. The first 24 modes are shown in Table 2.2.2-1. These modes span the frequency range from the lowest frequency (about 4.3 Hz) to about 43 Hz. In typical seismic analysis of systems such as this, the frequency range of practical importance is up to 33 Hz; on this basis these modes are more than sufficient. Only the first six modes of the actual system have been measured, so any comparison at higher frequencies is between the numerical calculations reported here and other similar computations. The results obtained with the four models correlate quite well between themselves, suggesting that the mesh and the choices of retained degrees of freedom in the superelement models are reasonable. It is particularly noteworthy that the results for the superelement models correspond extremely well with those provided by the original model, considering the large reduction in the number of degrees of freedom for the substructures. The results also correlate roughly with the analysis results obtained by EDS and reported in EPRI NP-3108: except for modes 3 and 4 the frequencies are within 10% of the EDS numbers. For the first three modes the ABAQUS results are lower than those reported by EDS. This suggests the possibility that the ABAQUS model may be too flexible. The SUPERPIPE values are

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significantly higher than any of the other data for most modes, and the ABAQUS and the EDS results diverge from the test results after the first four modes. The results of the time history analyses are summarized in Table 2.2.2-2to Table 2.2.2-5. These analyses are based on using all 24 modes of the coarse model. Typical predicted response plots are shown in Figure 2.2.2-3 to Figure 2.2.2-7. In many cases of regular, beam-type, one-dimensional structures, the first few modes will generally establish the dynamic behavior. Although the pipeline has an irregular shape, it is worth checking how much the higher modes influence the results. This is done in this case by comparing the results using the first six modes only with the results obtained with 24 modes. The highest discrepancy (20%) is found in the predicted accelerations at certain degrees of freedom. All other results show at most 5-10% differences (see Figure 2.2.2-3and Figure 2.2.2-4). This conclusion is also supported by the steady-state results. All the ABAQUS results are reasonably self-consistent, in the sense that Rayleigh and modal damping and modal dynamics and direct integration all predict essentially the same values. The choice of 2.8% damping seems reasonable, in that oscillations caused by the snap-back are damped out almost completely in 10 seconds, which corresponds to the measurements. Unfortunately there is poor correlation between predicted and measured support reactions and maximum recorded displacements. The test results and the corresponding computations are shown in Table 2.2.2-2and Table 2.2.2-3. All the models give essentially the same values. The initial reactions and displacements are computed for a snap-back load of 31136 N (7000 lb) applied at node 25 (node 417 in EPRI report NP-3108) in the z-direction. The maximum recorded displacements occur at node 27 (node 419 in EPRI report NP-3108) in the y- and z-directions. It is assumed that the supports are in the positions relative to the pipe exactly as shown in Figure 2.2.2-1. The scatter in the experimental measurements makes it difficult to assess the validity of the stiffness chosen for the spring supports. The maximum displacement predicted at node 27 in the z-direction is almost twice that measured. This again implies the possibility that, at least in the area near this node, the model is too flexible. The generally satisfactory agreement between the natural frequency predictions and poor agreement between the maximum displacements and reactions suggests that improved modeling of the supports may be necessary. In this context it is worthwhile noting that the experimental program recorded significantly different support parameters in different tests on the pipeline system. Table 2.2.2-4 shows the results for displacement and acceleration for node 27 (which has the largest displacement). All the computed results are higher than the experimental values. The largest discrepancies between the measurements and the analysis results are in the predictions of peak forces in the springs, summarized in Table 2.2.2-5. Results obtained with the various models differ by less than 10%: these differences are caused by the differences in the models, different types of damping, and--for the direct integration results--errors in the time integration (for the modal dynamic procedure the time integration is exact). The principal cause of the discrepancies between the measurements and the computed values is believed to be the assumption of linear response in the springs in the numerical models. In reality the spring supports are either rigid struts or mechanical snubbers (Configuration 2). Especially when snubbers are used, the supports perform as nonlinear elements and must be modeled as such to reflect the support behavior accurately. Interestingly, even with the assumption of linear support behavior, the character of the oscillation is well-predicted for many variables.

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The last group of numerical results are frequency domain calculations obtained using the *STEADY STATE DYNAMICS linear dynamic response option. The response corresponds to steady harmonic excitation at node 27 in the z-direction by a force with a peak amplitude of 31136 N (7000 lb). Such frequency domain results play a valuable role in earthquake analysis because they define the frequency ranges in which the structure's response is most amplified by the excitation. Although it is expected that the first few natural frequencies will be where the most amplification occurs, the results show clearly that some variables are strongly amplified by the fifth and sixth modes. This is observed both in the simulations and in the experimental measurements. Measured experimental results are available for the acceleration of node 33 (node 419 in EPRI NO-3108) in the z-direction and for the force in spring FW-R-21. The character of curves obtained with ABAQUS agrees well with the experimental results (see Figure 2.2.2-8and Figure 2.2.2-9), but the values differ significantly, as in the time domain results. The peak acceleration recorded is 2.0 m/s 2 (78.47 in/s 2), at the first natural frequency, while the analysis predicts 4.0 m/s 2 (157.5 in/s 2). Likewise, the peak force value recorded is 2.0 kN (450 lb), compared to 5.9 kN (1326 lb) predicted. The discrepancies are again attributed to incorrect estimates of the support stiffness or to nonlinearities in the supports.

Input files indianpoint_modaldyn_coarse.inp *MODAL DYNAMIC analysis with modal damping using the coarse model. indianpoint_modaldyn_3sub.inp *MODAL DYNAMIC analysis using the three substructure model. indianpoint_3sub_gen1.inp First superelement generation referenced by the analysis indianpoint_modaldyn_3sub.inp. indianpoint_3sub_gen2.inp Second superelement generation referenced by the analysis indianpoint_modaldyn_3sub.inp. indianpoint_3sub_gen3.inp Third superelement generation referenced by the analysis indianpoint_modaldyn_3sub.inp. indianpoint_sstate_sinedwell.inp *STEADY STATE DYNAMICS analysis corresponding to the sine dwell test performed experimentally using the coarse model. indianpoint_direct_beam_coarse.inp Direct integration analysis using the coarse model with the *BEAM SECTION option. indianpoint_sstate_modaldamp.inp *STEADY STATE DYNAMICS analysis with modal damping, covering a range of frequencies using the coarse model. indianpoint_modaldyn_1sub.inp

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*MODAL DYNAMIC analysis with one substructure. indianpoint_1sub_gen1.inp Superelement generation referenced by the analysis indianpoint_modaldyn_1sub.inp. indianpoint_direct_beamgensect.inp Direct integration using the coarse model with *BEAM GENERAL SECTION instead of *BEAM SECTION, which, thus, runs faster on the computer since numerical integration of the cross-section is avoided. indianpoint_modaldyn_elmatrix1.inp *MODAL DYNAMIC analysis that reads and uses the superelement matrix written to the results file in indianpoint_3sub_gen1.inp, indianpoint_3sub_gen2.inp, and indianpoint_3sub_gen3.inp. indianpoint_modaldyn_elmatrix2.inp Reads and uses the element matrix written to the results file in indianpoint_modaldyn_3sub.inp. indianpoint_modaldyn_elmatrix3.inp Reads and uses the superelement matrix written to the results file in indianpoint_1sub_gen1.inp. indianpoint_modaldyn_elmatrix4.inp Reads and uses the element matrix written to the results file in indianpoint_modaldyn_1sub.inp. indianpoint_modaldamp_rayleigh.inp *MODAL DAMPING analysis with modal Rayleigh damping using the coarse mesh with the *BEAM SECTION option. indianpoint_dyn_rayleigh_3sub.inp *DYNAMIC analysis with Rayleigh damping using the three substructure model. indianpoint_rayleigh_3sub_gen1.inp First superelement generation referenced by the analysis indianpoint_dyn_rayleigh_3sub.inp. indianpoint_rayleigh_3sub_gen2.inp Second superelement generation referenced by the analysis indianpoint_dyn_rayleigh_3sub.inp. indianpoint_rayleigh_3sub_gen3.inp Third superelement generation referenced by the analysis indianpoint_dyn_rayleigh_3sub.inp. indianpoint_modaldyn_unsorted.inp One substructure *MODAL DYNAMIC analysis with unsorted node sets and unsorted retained degrees of freedom. indianpoint_unsorted_gen1.inp Superelement generation with unsorted node sets and unsorted retained degrees of freedom referenced by the analysis indianpoint_modaldyn_unsorted.inp.

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indianpoint_lanczos.inp Same as indianpoint_modaldyn_coarse.inp, except that it uses the Lanczos solver and the eigenvectors are normalized with respect to the generalized mass. indianpoint_restart_normdisp.inp Restarts from indianpoint_lanczos.inp and continues the eigenvalue extraction with the eigenvectors normalized with respect to the maximum displacement. indianpoint_restart_bc.inp Restarts from indianpoint_lanczos.inp and continues the eigenvalue extraction with modified boundary conditions. indianpoint_overlapfreq.inp Contains two steps, which extract eigenvalues with overlapping frequency ranges.

References · Consolidated Edison Company of New York, Inc., EDS Nuclear, Inc., and Anco Engineers, Inc., Testing and Analysis of Feedwater Piping at Indian Point Unit 1, Volume 1: Damping and Frequency, EPRI NP-3108, vol. 1, July 1983. · Dodge, W. G., and S. E. Moore, "Stress Indices and Flexibility Factors for Moment Loadings in Elbows and Curved Pipe," WRC Bulletin, no. 179, December 1972. · Tang, Y. K., M. Gonin, and H. T. Tang, "Correlation Analysis of In-situ Piping Support Reactions," EPRI correspondence with HKS, May 1985.

Tables Table 2.2.2-1 Comparison of natural frequencies (Hz). Anco EDS SUPE ABAQUS R Mod (experiment PIPE coarse finer single e mesh mes super ) h 1 4.20 4.30 5.30 4.25 4.26 4.25 2 6.80 6.80 8.10 6.27 6.25 6.27 3 8.30 8.80 12.00 7.29 7.29 7.30 4 12.60 10.6 13.30 12.80 12.6 12.87 0 6 5 15.40 13.0 14.40 13.18 13.1 13.19 0 4 6 16.70 14.5 15.90 13.90 13.7 13.91 0 5 7 16.2 18.30 15.11 15.9 14.34

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three super s 4.25 6.27 7.30 12.86 13.20 13.92 14.39

Dynamic Stress/Displacement Analyses

0 8

19.40

16.30

9

20.20

16.89

10

22.20

17.43

11

18.02

12

19.58

13

23.43

14

23.99

15

24.27

16

24.80

17

26.82

18

29.53

19

30.61

20

30.95

21

31.52

22

33.50

23

39.09

24

39.86

8 16.0 7 16.8 1 17.8 2 19.0 7 20.1 0 21.4 5 22.1 3 23.5 8 24.1 5 26.8 4 30.1 8 30.6 0 32.5 8 33.1 1 35.0 8 39.6 5 43.2 5

16.24

16.31

16.43

16.43

17.17

17.20

18.10

18.10

20.05

20.01

23.98

24.00

24.47

24.47

24.97

24.96

25.34

25.28

27.63

27.56

30.31

30.55

31.08

31.06

31.43

31.43

32.00

31.98

33.76

33.77

39.75

39.97

42.98

42.97

Table 2.2.2-2 Comparison of initial support reactions. Snap-back Test No. S138R1SZ; 31136 N (7000 lb) at node 25, z-direction. NOD SUPPOR Anco TEST N (lb) E T 15 FW-R-11 -8000 (-1798.6 ) 22 FW-R-13 30000 (6744.6) 23 FW-R-14 -252 (-56.7) 35 FW-R-17 23625 (5311.4)

ABAQUS N (lb) -11712 (-2633) 29352 -3754 102

(6599) (-844) (-22.8)

2-881

Dynamic Stress/Displacement Analyses

35 39 39

FW-R-18 FW-R-20 FW-R-21

49 53 53 56 56

FW-R-23 FW-R-24 FW-R-25 FW-R-27 FW-R-28

10025 24000 -2450 0 8000 -4324 2000 432 156

(2553.8) (5395.7) (-5508.1 ) (1798.6) (-972.1) (449.6) (97.1) (35.1)

-18468 4212 25016

(-4152) ( 947) (5624)

24348 -4057 -816 -1801 -799

(5474) (-912) (-183) (-405) (-180)

Table 2.2.2-3 Comparison of maximum displacements. ABAQUS Anco Measured ABAQUS NODE NODE No. No. mm (in) mm (in) 27 419-Y -16.0 (-.630) -26.85 (-1.057) 27 419-Z 37.81 (1.49) 65.72 (2.587)

Table 2.2.2-4 Peak displacement and acceleration values at node 27. Variable Measured ABAQUS Modal, Modal, Direct (Anco) 2.8% modal Rayleigh integration damping damping uy (mm) -0.024/0.024 -0.029/0.029 -0.031/0.03 -0.031/0.031 1 uz (mm) -0.038/0.038 -0.058/0.066 -0.062/0.05 -0.063/0.068 9 -47.6/40.9 -42.1/50.8 -49.6/49.9 -83.8/91.0 u Äz (m/s2) The high acceleration amplitude reported for the ABAQUS direct integration analysis occurs only during the first few increments, after which it reduces to -31.6/48.6 m/s 2.

Table 2.2.2-5 Peak reaction forces at supports (in kN). Support Measured ABAQUS Modal, Modal, number (Anco) 2.8% modal Rayleigh damping damping FW-R-11 -16.44/19.22 -19.80/13.42 -19.90/14.26 FW-R-13 -15.10/29.91 -18.94/24.45 -19.46/23.61 FW-R-14 -7.22/12.00 -9.34/7.35 -10.23/10.00 FW-R-17 34.40/26.20 -7.50/10.59 -.17/9.25 FW-R-18 -14.30/14.40 -33.26/32.06 -33.58/31.61 FW-R-20 -25.60/26.90 -7.54/8.79 -7.98/8.50 FW-R-21 -24.50/23.80 -25.55/24.47 -26.38/25.30 FW-R-23 -15.30/16.00 -25.39/24.63 -26.06/25.36

2-882

Direct integration -21.82/15.76 -28.50/21.98 -12.54/9.03 -7.91/10.97 -33.46/32.63 -8.07/10.60 -27.78/25.26 -25.40/24.35

Dynamic Stress/Displacement Analyses

FW-R-24 FW-R-25 FW-R-27 FW-R-28

-9.61/7.30 -6.77/6.21 -3.76/3.04 -1.10/1.82

-7.17/6.87 -3.48/4.36 -4.12/4.00 -1.53/1.08

-7.69/7.20 -3.34/4.55 -3.78/3.80 -1.62/1.15

-8.23/8.64 -7.13/4.71 -4.29/4.43 -1.79/1.44

Figures Figure 2.2.2-1 Indian Point boiler feedwater line: modern supports, Configuration 1.

Figure 2.2.2-2 Basic mesh and superelement models.

2-883

Dynamic Stress/Displacement Analyses

Figure 2.2.2-3 z-displacement at node 27, modal analysis with 24 modes.

Figure 2.2.2-4 z-displacement at node 27, modal analysis with 6 modes.

2-884

Dynamic Stress/Displacement Analyses

Figure 2.2.2-5 z-displacement at node 27, direct integration analysis.

Figure 2.2.2-6 z-direction acceleration at node 27, modal analysis with 24 modes.

2-885

Dynamic Stress/Displacement Analyses

Figure 2.2.2-7 Force in spring support FW-R-11, modal analysis with 24 modes.

Figure 2.2.2-8 Comparison of z-direction acceleration at node 33 between experimental steady-state results (solid line) and ABAQUS (dashed line).

2-886

Dynamic Stress/Displacement Analyses

Figure 2.2.2-9 Comparison of force in spring support FW-R-21, between experimental steady-state results (solid line) and ABAQUS (dashed line).

2-887

Dynamic Stress/Displacement Analyses

Sample listings

2-888

Dynamic Stress/Displacement Analyses

Listing 2.2.2-1 *HEADING INDIAN POINT FEEDWATER LINE WITH SPRING SUPPORTS ** BEAM ELEMENTS WITH MODAL DYNAMICS, ** MODAL DAMPING *NODE 1, 0., 423., -234.96 3, 0., 423., -150.96 5, 0., 435., -138.96 6, 0., 474., -138.96 8, 0., 486., -126.96 10, 0., 486., -75.96 11, 0., 486., -51.96 12, 0., 486., -18.00 13, 0., 486., 9.00 15, 0., 486., 144.5 16, 0., 486., 159. 18, 8.484, 494.484, 171. 19, 8.484,494.484,171. 21, 16.93, 497.96, 171. 22, 19.8125, 497.96 , 171. 23, 29.125 ,497.96, 171. 25, 200.72, 497.96, 171.00 27, 260.72, 497.96, 171. 29, 272.72, 509.96, 171.00 31, 272.72, 569.964, 171.00 33, 280.44, 581.96, 180.19 35, 330.1 , 581.96, 239.3 36, 335.21, 581.96, 245.46 38, 342.91, 593.96, 254.65 39, 342.91 , 628. , 254.65 40, 342.91, 660., 254.65 42, 342.91, 706., 254.65 44, 340.22, 714.48, 256.91 46, 296.57, 771.47, 293.54 48, 282.36, 779.95, 289.80 49, 278.50, 779.95, 285.20 50,274.644, 779.95, 280.61 52, 266.93, 791.95, 271.42 53, 266.93, 801., 271.42 54, 266.93, 876.00, 271.42 56, 266.93, 990.96, 271.42

2-889

Dynamic Stress/Displacement Analyses

57, 266.93, 1000.27, 271.42 59, 278.88, 1012.27, 272.46 61, 335.26, 1012.27, 277.39 63, 343.40, 1012.27, 281.64 64, 366.97, 1012.27, 309.73 65, 369.52, 1012.27, 312.76 66, 379.16, 1012.27, 324.25 67, 388.8, 1012.27, 335.74 68, 389.11, 1012.27, 336.11 70, 396.83, 1024.27, 345.3 71, 396.83, 1027.27, 345.3 72, 396.83, 1033.27, 345.3 73, 396.83, 1040.95, 345.3 75, 389.93, 1049.95, 351.08 76, 380.74, 1049.95, 358.8 *NGEN 1,3 8,10 13,15 23,25 25,27 29,31 33,35 40,42 44,46 54,57 59,61 *NGEN,LINE=C 3,5,1,, 0., 435., -150.96 6,8,1,, 0., 474., -126.96 16,18,1,, 8.484, 494.48, 159.00 19, 21,1,, 16.932, 485.96, 171.00 27,29,1,, 260.724, 509.96, 171.00 31,33,1,, 280.44, 569.96, 180.19 36,38,1,, 335.21, 593.96, 245.46 42,44,1,, 333.71, 706.00, 262.37 46,48,1,, 288.85, 771.47, 284.35 50,52,1,, 274.64, 791.95, 280.61 57,59,1,, 278.88, 1000.27, 272.46 61,63,1,, 334.21, 1012.27, 289.34 68,70,1,, 389.11, 1024.27, 336.11 73,75,1,, 389.93, 1040.95, 351.08 **

2-890

Dynamic Stress/Displacement Analyses

** SPRING DEFINITIONS ** *NODE,NSET=SPRS 115, 24.91, 475.93, 144.5 122, 19.81, 497.96, 219.5 123, 29.13, 597.41, 160.55 135, 299.94, 555., 239.30 235, 330.10, 599.46, 239.30 139, 364.74, 628.00, 220.25 239, 359.58, 628.00, 291.83 149,278.50, 792.45, 285.20 153, 321.48, 801.00, 318.12 253, 314.43, 801.00, 212.09 156, 311.81, 990.96, 215.36 256,290.46, 1008.88, 299.46 *ELEMENT,TYPE=SPRINGA,ELSET=FWR11 1001,15,115 *ELEMENT,TYPE=SPRINGA,ELSET=FWR13 1002,22,122 *ELEMENT,TYPE=SPRINGA,ELSET=FWR14 1003,23,123 *ELEMENT,TYPE=SPRINGA,ELSET=FWR17 1004,35,135 *ELEMENT,TYPE=SPRINGA,ELSET=FWR18 1005,35,235 *ELEMENT,TYPE=SPRINGA,ELSET=FWR20 1006,39,139 *ELEMENT,TYPE=SPRINGA,ELSET=FWR21 1007,39,239 *ELEMENT,TYPE=SPRINGA,ELSET=FWR23 1008,49,149 *ELEMENT,TYPE=SPRINGA,ELSET=FWR25 1009,53,153 *ELEMENT,TYPE=SPRINGA,ELSET=FWR24 1010,53,253 *ELEMENT,TYPE=SPRINGA,ELSET=FWR27 1011,56,156 *ELEMENT,TYPE=SPRINGA,ELSET=FWR28 1012,56,256 *ELSET,ELSET=SPRINGS FWR11, FWR13, FWR14, FWR17, FWR18, FWR20, FWR21, FWR23, FWR24, FWR25, FWR27, FWR28 *SPRING ,ELSET=FWR11

2-891

Dynamic Stress/Displacement Analyses

17700. , *SPRING,ELSET=FWR13 119600., *SPRING ,ELSET=FWR14 403000., *SPRING ,ELSET=FWR17 97900., *SPRING,ELSET=FWR18 228000., *SPRING ,ELSET=FWR20 86300., *SPRING,ELSET=FWR21 86300., *SPRING,ELSET=FWR23 319000., *SPRING,ELSET=FWR24 56800., *SPRING,ELSET=FWR25 39100., *SPRING,ELSET=FWR27 55500., *SPRING,ELSET=FWR28 68000., ** ** PIPE DEFINITIONS ** *ELEMENT,TYPE=B31 1,1,2 14,13,14 20,19,20 28,25,26

2-892

Dynamic Stress/Displacement Analyses

53,49,50 *ELEMENT,TYPE=B31,ELSET=ONE 24,22,23 *ELGEN 1,12 14,5 20,3 24,3 28,24 53,27 *MPC BEAM,18,19 *ELSET,ELSET=D8 1,2,5,8,9,10,11,12,14,16,22 ,15 24,25,26,28,29,32,33,36,37,38,41,42,43,44,47,48 51,53,56,57,58,59,60,63,64,67,68,71 74, *ELSET,ELSET=D8E 3, 4, 6, 7,17,18,20,21,30,31,34,35,39,40 45,46,49,50,54,55,61,62,65,66,72,73 *ELSET,ELSET=BF57 69,70 *ELSET,ELSET=BWR 75, *ELSET,ELSET=D6 76,79 *ELSET,ELSET=D6E 77,78 *BEAM GENERAL SECTION,ELSET=D8 ,SECTION=GENERAL, DENSITY= .0010691 12.763 , 105.317 , , 105.317 , 210.635 1. , , -1. 27.9E6 , 10.73E6 *BEAM GENERAL SECTION,ELSET=D8E,SECTION=GENERAL, DENSITY=.0010691 12.763 , 50.439 , , 50.439 , 210.635 1. , , -1. 27.9E6 , 10.73E6 *BEAM GENERAL SECTION,ELSET=D6 ,SECTION=GENERAL, DENSITY=0.00102423 8.405 , 40.295 , , 40.295 , 80.589 1., 27.9E6 , 10.73E6

2-893

Dynamic Stress/Displacement Analyses

*BEAM GENERAL SECTION,ELSET=D6E,SECTION=GENERAL, DENSITY=0.00102423 8.405 , 21.828, ,21.828, 80.589 1., 27.9E6 , 10.73E6 *BEAM GENERAL SECTION,ELSET=BWR,SECTION=GENERAL, DENSITY=0.0013266 10.4806 , 67.143 , , 67.143 , 134.29 1. , , -1. 27.9E6 , 10.73E6 *BEAM GENERAL SECTION,ELSET=BF57,SECTION=GENERAL, DENSITY=0.002591 30.631 ,161.77 , , 161.77 , 323.53 1. , , -1. 27.9E6 , 10.73E6 *NSET,NSET=SMALL 1,25,27,45,55 *BOUNDARY 1,1,6 76,1,6 70,1 70,2 10,2 SPRS,1,3 *RESTART,WRITE,FREQUENCY=100 *STEP *FREQUENCY 24, *EL PRINT,ELSET=SPRINGS,FREQUENCY=1 S11,E11 *EL FILE,ELSET=SPRINGS ELEN, *EL PRINT,ELSET=ONE,FREQUENCY=1 SENER, ELSE, *EL FILE,ELSET=ONE ENER, ELEN, *NODE PRINT,FREQUENCY=0 *OUTPUT,FIELD *ELEMENT OUTPUT,ELSET=SPRINGS ELEN, *ELEMENT OUTPUT,ELSET=ONE

2-894

Dynamic Stress/Displacement Analyses

ENER, ELEN, *NODE OUTPUT *END STEP *STEP,PERTURBATION SNAP LOAD---APPLIED STATICALLY *STATIC *EL PRINT,ELSET=SPRINGS,FREQUENCY=1 S,E *CLOAD 25,3,7000. *NODE PRINT U, RF, *NODE FILE,NSET=SMALL,FREQUENCY=1 U,RF *OUTPUT,FIELD, FREQUENCY=1 *ELEMENT OUTPUT, ELSET=SPRINGS ELEN, ENER, S,E *ELEMENT OUTPUT,ELSET=ONE ELEN, ENER, S,E *NODE OUTPUT,NSET=SMALL U, RF, *OUTPUT,HISTORY,FREQUENCY=1 *NODE OUTPUT,NSET=SMALL U,RF *END STEP *STEP RELEASE LOAD *MODAL DYNAMIC,CONTINUE=YES 0.02,10.0 *MODAL DAMPING,MODAL=DIRECT 1,24,0.028 *PRINT,FREQUENCY=100 *NODE PRINT,NSET=SMALL,FREQUENCY=100 U, *NODE FILE,NSET=SMALL,FREQUENCY=100 U,V,A,RF

2-895

Dynamic Stress/Displacement Analyses

*EL FILE,ELSET=SPRINGS,FREQUENCY=100 S, *EL PRINT,ELSET=SPRINGS,FREQUENCY=100 S, *OUTPUT,FIELD,FREQUENCY=100 *NODE OUTPUT,NSET=SMALL U,V,A,RF *ELEMENT OUTPUT,ELSET=SPRINGS S, *OUTPUT,HISTORY,FREQUENCY=100 *NODE OUTPUT,NSET=SMALL U,V,A,RF *ELEMENT OUTPUT,ELSET=SPRINGS S, *END STEP

2-896

Dynamic Stress/Displacement Analyses

Listing 2.2.2-2 *HEADING INDIAN POINT FEEDWATER LINE WITH SPRING SUPPORTS ** BEAM ELEMENTS WITH STEADY STATE SINE DWELL *NODE 1, 0., 423., -234.96 3, 0., 423., -150.96 5, 0., 435., -138.96 6, 0., 474., -138.96 8, 0., 486., -126.96 10, 0., 486., -75.96 11, 0., 486., -51.96 12, 0., 486., -18.00 13, 0., 486., 9.00 15, 0., 486., 144.5 16, 0., 486., 159. 18, 8.484, 494.484, 171. 19, 8.484,494.484,171. 21, 16.93, 497.96, 171. 22, 19.8125, 497.96 , 171. 23, 29.125 ,497.96, 171. 25, 200.72, 497.96, 171.00 27, 260.72, 497.96, 171. 29, 272.72, 509.96, 171.00 31, 272.72, 569.964, 171.00 33, 280.44, 581.96, 180.19 35, 330.1 , 581.96, 239.3 36, 335.21, 581.96, 245.46 38, 342.91, 593.96, 254.65 39, 342.91 , 628. , 254.65 40, 342.91, 660., 254.65 42, 342.91, 706., 254.65 44, 340.22, 714.48, 256.91 46, 296.57, 771.47, 293.54 48, 282.36, 779.95, 289.80 49, 278.50, 779.95, 285.20 50,274.644, 779.95, 280.61 52, 266.93, 791.95, 271.42 53, 266.93, 801., 271.42 54, 266.93, 876.00, 271.42 56, 266.93, 990.96, 271.42 57, 266.93, 1000.27, 271.42

2-897

Dynamic Stress/Displacement Analyses

59, 278.88, 1012.27, 272.46 61, 335.26, 1012.27, 277.39 63, 343.40, 1012.27, 281.64 64, 366.97, 1012.27, 309.73 65, 369.52, 1012.27, 312.76 66, 379.16, 1012.27, 324.25 67, 388.8, 1012.27, 335.74 68, 389.11, 1012.27, 336.11 70, 396.83, 1024.27, 345.3 71, 396.83, 1027.27, 345.3 72, 396.83, 1033.27, 345.3 73, 396.83, 1040.95, 345.3 75, 389.93, 1049.95, 351.08 76, 380.74, 1049.95, 358.8 *NGEN 1,3 8,10 13,15 23,25 25,27 29,31 33,35 40,42 44,46 54,57 59,61 *NGEN,LINE=C 3,5,1,, 0., 435., -150.96 6,8,1,, 0., 474., -126.96 16,18,1,, 8.484, 494.48, 159.00 19, 21,1,, 16.932, 485.96, 171.00 27,29,1,, 260.724, 509.96, 171.00 31,33,1,, 280.44, 569.96, 180.19 36,38,1,, 335.21, 593.96, 245.46 42,44,1,, 333.71, 706.00, 262.37 46,48,1,, 288.85, 771.47, 284.35 50,52,1,, 274.64, 791.95, 280.61 57,59,1,, 278.88, 1000.27, 272.46 61,63,1,, 334.21, 1012.27, 289.34 68,70,1,, 389.11, 1024.27, 336.11 73,75,1,, 389.93, 1040.95, 351.08 ** ** SPRING DEFINITIONS

2-898

Dynamic Stress/Displacement Analyses

** *NODE,NSET=SPRS 115, 24.91, 475.93, 144.5 122, 19.81, 497.96, 219.5 123, 29.13, 597.41, 160.55 135, 299.94, 555., 239.30 235, 330.10, 599.46, 239.30 139, 364.74, 628.00, 220.25 239, 359.58, 628.00, 291.83 149,278.50, 792.45, 285.20 153, 321.48, 801.00, 318.12 253, 314.43, 801.00, 212.09 156, 311.81, 990.96, 215.36 256,290.46, 1008.88, 299.46 *NSET,NSET=NPDR 25,27,33,36,42,1,76,15,22,23,35,39,49,53,56 *ELEMENT,TYPE=SPRINGA,ELSET=FWR11 1001,15,115 *ELEMENT,TYPE=SPRINGA,ELSET=FWR13 1002,22,122 *ELEMENT,TYPE=SPRINGA,ELSET=FWR14 1003,23,123 *ELEMENT,TYPE=SPRINGA,ELSET=FWR17 1004,35,135 *ELEMENT,TYPE=SPRINGA,ELSET=FWR18 1005,35,235 *ELEMENT,TYPE=SPRINGA,ELSET=FWR20 1006,39,139 *ELEMENT,TYPE=SPRINGA,ELSET=FWR21 1007,39,239 *ELEMENT,TYPE=SPRINGA,ELSET=FWR23 1008,49,149 *ELEMENT,TYPE=SPRINGA,ELSET=FWR25 1009,53,153 *ELEMENT,TYPE=SPRINGA,ELSET=FWR24 1010,53,253 *ELEMENT,TYPE=SPRINGA,ELSET=FWR27 1011,56,156 *ELEMENT,TYPE=SPRINGA,ELSET=FWR28 1012,56,256 *ELSET,ELSET=SPRINGS FWR11, FWR13, FWR14, FWR17, FWR18, FWR20, FWR21, FWR23, FWR24, FWR25, FWR27, FWR28

2-899

Dynamic Stress/Displacement Analyses

*SPRING ,ELSET=FWR11 17700. , *SPRING,ELSET=FWR13 119600., *SPRING ,ELSET=FWR14 403000., *SPRING ,ELSET=FWR17 97900., *SPRING,ELSET=FWR18 228000., *SPRING ,ELSET=FWR20 86300., *SPRING,ELSET=FWR21 86300., *SPRING,ELSET=FWR23 319000., *SPRING,ELSET=FWR24 56800., *SPRING,ELSET=FWR25 39100., *SPRING,ELSET=FWR27 55500., *SPRING,ELSET=FWR28 68000., ** ** PIPE DEFINITIONS ** *ELEMENT,TYPE=B31 1,1,2 14,13,14 20,19,20

2-900

Dynamic Stress/Displacement Analyses

24,22,23 28,25,26 53,49,50 *ELGEN 1,12 14,5 20,3 24,3 28,24 53,27 *MPC BEAM,18,19 *ELSET,ELSET=D8 1,2,5,8,9,10,11,12,14,16,22 ,15 24,25,26,28,29,32,33,36,37,38,41,42,43,44,47,48 51,53,56,57,58,59,60,63,64,67,68,71 74, *ELSET,ELSET=D8E 3, 4, 6, 7,17,18,20,21,30,31,34,35,39,40 45,46,49,50,54,55,61,62,65,66,72,73 *ELSET,ELSET=BF57 69,70 *ELSET,ELSET=BWR 75, *ELSET,ELSET=D6 76,79 *ELSET,ELSET=D6E 77,78 *BEAM GENERAL SECTION,ELSET=D8 ,SECTION=GENERAL, DENSITY= .0010691 12.763 , 105.317 , , 105.317 , 210.635 1. , , -1. 27.9E6 , 10.73E6 *BEAM GENERAL SECTION,ELSET=D8E,SECTION=GENERAL, DENSITY=.0010691 12.763 , 50.439 , , 50.439 , 210.635 1. , , -1. 27.9E6 , 10.73E6 *BEAM GENERAL SECTION,ELSET=D6 ,SECTION=GENERAL, DENSITY=0.00102423 8.405 , 40.295 , , 40.295 , 80.589 1., 27.9E6 , 10.73E6

2-901

Dynamic Stress/Displacement Analyses

*BEAM GENERAL SECTION,ELSET=D6E,SECTION=GENERAL, DENSITY=0.00102423 8.405 , 21.828, ,21.828, 80.589 1., 27.9E6 , 10.73E6 *BEAM GENERAL SECTION,ELSET=BWR,SECTION=GENERAL, DENSITY=0.0013266 10.4806 , 67.143 , , 67.143 , 134.29 1. , , -1. 27.9E6 , 10.73E6 *BEAM GENERAL SECTION,ELSET=BF57,SECTION=GENERAL, DENSITY=0.002591 30.631 ,161.77 , , 161.77 , 323.53 1. , , -1. 27.9E6 , 10.73E6 *NSET,NSET=SMALL 25,27,33 *BOUNDARY 1,1,6 76,1,6 70,1 70,2 10,2 SPRS,1,3 *AMPLITUDE,NAME=AMP 0.0,1.0,2.0,18.46,2.4,26.54,2.8,36.16, 3.0,41.55,3.4,53.35,3.8,66.64,4.0,73.84, 4.1,77.58,4.2,81.44,4.3,85.34,4.4,89.35, 4.6,97.66,4.8,106.34,5.0,115.38,5.2,124.79, 5.4,134.58,5.6,144.73,5.8,155.26,6.0,166.15, 6.1,171.73,6.2,177.41,6.35,186.1,6.4,189.04, 6.6,201.04,6.8,213.41,7.0,226.15,7.2,239.26, 7.4,252.73,7.8,280.79,8.0,295.38 *STEP *FREQUENCY 24, *EL PRINT,ELSET=SPRINGS S11,E11 *NODE PRINT,FREQUENCY=0 *END STEP *STEP *STEADY STATE DYNAMICS,FREQUENCY SCALE=LINEAR 0.02,2.0,3,,

2-902

Dynamic Stress/Displacement Analyses

2.4,8.0,27,1.0 *CLOAD,AMPLITUDE=AMP 27,3,1.0 *MODAL DAMPING,MODAL=DIRECT 1,24,0.028 *EL PRINT,ELSET=SPRINGS,FREQUENCY=10 S11,E11 *NODE PRINT,NSET=SMALL,FREQUENCY=10 U, *NODE FILE,NSET=SMALL,FREQUENCY=10 U,V,A *MODAL PRINT,FREQUENCY=10 GU, GA, GPU, *EL FILE,ELSET=SPRINGS,FREQUENCY=10 S, *OUTPUT,FIELD,FREQUENCY=10 *NODE OUTPUT,NSET=SMALL U,V,A *ELEMENT OUTPUT,ELSET=SPRINGS S, *OUTPUT,HISTORY,FREQUENCY=10 *NODE OUTPUT,NSET=SMALL U,V,A *ELEMENT OUTPUT,ELSET=SPRINGS S, *END STEP

2.2.3 Response spectra of a three-dimensional frame building Product: ABAQUS/Standard The purpose of this example is to verify the different summation methods for natural modes in the *RESPONSE SPECTRUM procedure. To compare the five different methods that are available in ABAQUS, a three-dimensional model with closely spaced modes is examined.

Geometry and model A four-story steel-frame building is analyzed. All columns in the building have the same geometric properties. However, as shown in Figure 2.2.3-1, the properties of the beams in Frames 1 and 2 are different, as compared to those in Frames 3 and 4, to move the center of mass of the structure away from its geometric center. Eigenvalue extraction performed on the model shows that many of the 30 modes that cover the frequency range up to 40 Hz are closely spaced. An acceleration spectrum based on the El Centro earthquake record is applied in the x-y plane. The FORTRAN program given in

2-903

Dynamic Stress/Displacement Analyses

frameresponsespect_acc.f is used to generate the spectrum. The frequency range is chosen between 0.1 Hz and 40 Hz, and the number of points at which the spectrum is calculated is set at 501. Only one spectrum curve is requested for 2% damping.

Results and discussion As described in ``Linear analysis of a rod under dynamic loading,'' Section 1.4.9 of the ABAQUS Benchmarks Manual, for structures with well-separated modes the TENP and the CQC methods reduce to the SRSS method, while the NRL and the ABS methods give similar results. Hence, for such structures, two summation rules would suffice, with ABS providing the more conservative results. However, when structures with closely spaced modes are analyzed, all five summation rules can yield very different results. This is even more apparent in three-dimensional problems. In the present example, the plane of the earthquake motion lies along the x-axis, so we expect that the structural response will be dominated by Frames 1 and 3 and will result in a significant base shear in the x-direction. All five methods are compared against a modal time history response using the same El Centro acceleration record in Table 2.2.3-1, where the base shear forces are summed up in the plane of each frame Si , where i is the frame number. This comparison shows that the best approximation is generated by the CQC method. The other methods overestimate the shear in the y-direction, and some of them underestimate the base shear in the x-direction. The CQC method is generally recommended for asymmetrical three-dimensional problems with closely spaced structural modes. This method takes into account the sign of the mode shapes through cross-modal correlation factors and can correctly predict the response in directions perpendicular to the direction of excitation.

Input files frameresponsespect_freq.inp *FREQUENCY analysis. frameresponsespect_rs.inp *RESPONSE SPECTRUM analysis. frameresponsespect_modal.inp *MODAL DYNAMIC analysis. frameresponsespect_acc.f FORTRAN program that will produce the acceleration spectrum needed to run frameresponsespect_rs.inp.

Table Table 2.2.3-1 Comparison of base shear forces for different summation methods. Method S1 (kip) S2 (kip) S3 (kip) S4 (kip) Time history -25.5 14.0 -37.0 -22.8 ABS 52.5 52.5 69.6 69.6

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Dynamic Stress/Displacement Analyses

SRSS TENP NRL CQC

20.9 33.1 29.3 26.6

20.9 33.1 29.3 14.6

26.8 38.8 37.2 31.6

26.8 38.8 37.2 22.1

Figure Figure 2.2.3-1 Three-dimensional frame system.

Sample listings

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Dynamic Stress/Displacement Analyses

Listing 2.2.3-1 *HEADING 3-D BUILDING SUBJECTED TO EARTHQUAKE RESP. SPECTRUM *RESTART,WRITE,FREQUENCY=99 *NODE,NSET=BOT 1, 5,200.,0.,0.,0. 9,350.,0.,0.,0. 13,550.,0.,0.,0. 17,550.,200.,0. 21,550.,350.,0. 25,550.,550.,0. 29,350.,550.,0. 33,200.,550.,0. 37,0.,550.,0. 41,0.,350.,0. 45,0.,200.,0. *NCOPY,OLD SET=BOT,NEW SET=TOP,SHIFT, CHANGE NUMBER=16000 0.,0.,400. 0.,0.,0.,0.,0.,10.,0. *NFILL BOT,TOP,16,1000 *NGEN 4021,4025,1 *NGEN,NSET=B1 4001,4005,1 4005,4009,1 4009,4013,1 *NGEN,NSET=B3 4013,4017,1 4017,4021,1 4021,4024,1 *NCOPY,OLD SET=B1,NEW SET=B2,REFLECT=POINT, CHANGE NUMBER=24 275.,275,100. *NCOPY,OLD SET=B3,NEW SET=B4,REFLECT=POINT, CHANGE NUMBER=24 275.,275,100. *NSET,NSET=BOT1 B1,B2,B3,B4

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Dynamic Stress/Displacement Analyses

*NCOPY,OLD SET=BOT1,NEW SET=TOP1,SHIFT, CHANGE NUMBER=12000 0.,0.,300. 0.,0.,0.,0.,0.,10.,0. *NFILL BOT1,TOP1,3,4000 ** ** NODE USED TO DEFINE THE SECTION ORIENTATION ** OF ELSET C1 *NODE 50001,1000,0 ** **CREATE ALL COLUMNS *ELEMENT,TYPE=B32,ELSET=C1 1,1,1001,2001,50001 *ELGEN,ELSET=C1 1,8,2000,1,12,4,8 **CREATE BEAMS IN FRAME 1 *ELEMENT,TYPE=B32,ELSET=B1 101,4001,4002,4003 *ELGEN,ELSET=B1 101,4,4000,1,6,2,4 **CREATE BEAMS IN FRAME 2 *ELEMENT,TYPE=B32,ELSET=B2 125,4013,4014,4015 *ELGEN,ELSET=B2 125,4,4000,1,6,2,4 **CREATE BEAMS IN FRAME 3 *ELEMENT,TYPE=B32,ELSET=B3 149,4025,4026,4027 *ELGEN,ELSET=B3 149,4,4000,1,6,2,4 **CREATE BEAMS IN FRAME 4 *ELEMENT,TYPE=B32,ELSET=B4 173,4037,4038,4039 *ELGEN,ELSET=B4 173,4,4000,1,5,2,4 *ELEMENT,TYPE=B32,ELSET=B4 193,4047,4048,4001 194,8047,8048,8001 195,12047,12048,12001 196,16047,16048,16001 *ELSET,ELSET=THREE

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Dynamic Stress/Displacement Analyses

1,9,104 *MATERIAL,NAME=STEEL *ELASTIC 30.E6, *DENSITY 0.000728, *BEAM SECTION,SECTION=BOX,MATERIAL=STEEL,ELSET=C1 14.,14.,1.5,1.5,1.5,1.5 *BEAM SECTION,SECTION=BOX,MATERIAL=STEEL,ELSET=B1 8.0,10.,1.0,1.0,1.0,1.0 0.,1.,0. *BEAM SECTION,SECTION=BOX,MATERIAL=STEEL,ELSET=B2 8.0,10.,1.0,1.0,1.0,1.0 -1.,0.,0. *BEAM SECTION,SECTION=BOX,MATERIAL=STEEL,ELSET=B3 12.,14.,1.2,1.2,1.2,1.2 0.,-1.,0. *BEAM SECTION,SECTION=BOX,MATERIAL=STEEL,ELSET=B4 12.,14.,1.2,1.2,1.2,1.2 1.,0.,0. *BOUNDARY BOT,1,6 *STEP *FREQUENCY 30, *NODE PRINT,NSET=TOP U, *EL PRINT,FREQUENCY=0 *EL FILE,ELSET=THREE SF, *OUTPUT,FIELD *ELEMENT OUTPUT,ELSET=THREE SF, *NODE FILE,NSET=TOP U, *OUTPUT,FIELD *NODE OUTPUT,NSET=TOP U, *MODAL FILE *OUTPUT,HISTORY,FREQUENCY=1 *MODAL OUTPUT *END STEP

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Dynamic Stress/Displacement Analyses

Listing 2.2.3-2 *HEADING RESPONSE SPECTRUM FOR 3-D BUILDING *RESTART,READ,STEP=1,INC=1,WRITE,FREQUENCY=0 *ELSET,ELSET=SMALL 1,9,104,112,116,148,152,160,172,176 *SPECTRUM,TYPE=ACCELERATION,INPUT=SPECTRUM.ACC, NAME=SPEC *STEP *RESPONSE SPECTRUM,SUM=ABS,COMP=ALGEBRAIC SPEC,1.,0.,0.,1. *MODAL DAMPING,MODAL=DIRECT 1,30, 0.02 *EL PRINT,ELSET=SMALL SF, *NODE PRINT,NSET=TOP U, *NODE PRINT RF, *NODE FILE,NSET=TOP U, *NODE FILE RF, *EL FILE,ELSET=SMALL SF, *END STEP *STEP *RESPONSE SPECTRUM,SUM=SRSS,COMP=ALGEBRAIC SPEC,1.,0.,0.,1. *MODAL DAMPING,MODAL=DIRECT 1,30, 0.02 *END STEP *STEP *RESPONSE SPECTRUM,SUM=SRSS,COMP=SRSS SPEC,1.,0.,0.,1. *MODAL DAMPING,MODAL=DIRECT 1,30, 0.02 *END STEP *STEP *RESPONSE SPECTRUM,SUM=CQC,COMP=ALGEBRAIC SPEC,1.,0.,0.,1.

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Dynamic Stress/Displacement Analyses

*MODAL DAMPING,MODAL=DIRECT 1,30, 0.02 *END STEP *STEP *RESPONSE SPECTRUM,SUM=NRL,COMP=ALGEBRAIC SPEC,1.,0.,0.,1. *MODAL DAMPING,MODAL=DIRECT 1,30, 0.02 *END STEP *STEP *RESPONSE SPECTRUM,SUM=NRL,COMP=SRSS SPEC,1.,0.,0.,1. *MODAL DAMPING,MODAL=DIRECT 1,30, 0.02 *END STEP *STEP *RESPONSE SPECTRUM,SUM=TENP,COMP=SRSS SPEC,1.,0.,0.,1. *MODAL DAMPING,MODAL=DIRECT 1,30, 0.02 *END STEP *STEP *RESPONSE SPECTRUM,SUM=TENP,COMP=ALGEBRAIC SPEC,1.,0.,0.,1. *MODAL DAMPING,MODAL=DIRECT 1,30, 0.02 *END STEP **************************** **multidirectional spectra** **************************** *STEP *RESPONSE SPECTRUM,SUM=SRSS,COMP=ALGEBRAIC SPEC,1.,0.,0.,1. SPEC,0.,1.,0.,1. *MODAL DAMPING,MODAL=DIRECT 1,30, 0.02 *END STEP *STEP *RESPONSE SPECTRUM,SUM=SRSS,COMP=SRSS SPEC,1.,0.,0.,1. SPEC,0.,1.,0.,1. *MODAL DAMPING,MODAL=DIRECT 1,30, 0.02

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Dynamic Stress/Displacement Analyses

*END STEP *STEP *RESPONSE SPECTRUM,SUM=TENP,COMP=SRSS SPEC,1.,0.,0.,1. SPEC,0.,1.,0.,1. *MODAL DAMPING,MODAL=DIRECT 1,30, 0.02 *END STEP *STEP *RESPONSE SPECTRUM,SUM=CQC,COMP=SRSS SPEC,1.,0.,0.,1. SPEC,0.,1.,0.,1. SPEC,0.,0.,1.,1. *MODAL DAMPING,MODAL=DIRECT 1,30, 0.02 *END STEP

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Dynamic Stress/Displacement Analyses

Listing 2.2.3-3 *HEADING 3-D BUILDING SUBJECTED TO EARTHQUAKE RECORD *RESTART,READ,STEP=1,INC=1,WRITE,FREQUENCY=0 *AMPLITUDE,VALUE=ABSOLUTE,TIME=STEP TIME, INPUT=QUAKE.AMP,NAME=EQ *ELSET,ELSET=ALL 104,112,116,120,124,128,136, 148,152,160,172,176,184,196 *STEP *MODAL DYNAMIC 0.01,10. *MODAL DAMPING,MODAL=DIRECT 1,30,0.02 *BASE MOTION,AMPLITUDE=EQ,DOF=1,SCALE=386.09 *NODE PRINT,NSET=TOP,FREQUENCY=100 U, *EL PRINT,ELSET=ALL,FREQUENCY=100 SF, *NODE PRINT,FREQUENCY=50 RF, ** IN ORDER TO GET RESULTS FOR TABLE 3.1.16-1, ** THE NODE FILE SHOULD BE WRITTEN WITH A ** FREQUENCY=2 **NODE FILE,FREQUENCY=2 *NODE FILE,FREQUENCY=100 RF, *END STEP

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Tire Analyses

3. Tire Analyses 3.1 Tire analyses 3.1.1 Symmetric results transfer for a static tire analysis Product: ABAQUS/Standard This example illustrates the use of the *SYMMETRIC RESULTS TRANSFER option as well as the *SYMMETRIC MODEL GENERATION option to model the static interaction between a tire and a flat rigid surface. The *SYMMETRIC MODEL GENERATION option (``Symmetric model generation,'' Section 7.7.1 of the ABAQUS/Standard User's Manual) can be used to create a three-dimensional model by revolving an axisymmetric model about its axis of revolution or by combining two parts of a symmetric model, where one part is the original model and the other part is the original model reflected through a line or a plane. Both model generating techniques are demonstrated in this example. The *SYMMETRIC RESULTS TRANSFER option (``Transferring results from a symmetric mesh to a three-dimensional mesh,'' Section 7.7.2 of the ABAQUS/Standard User's Manual) allows the user to transfer the solution obtained from an axisymmetric analysis onto a three-dimensional model with the same geometry. It also allows the transfer of a symmetric three-dimensional solution to a full three-dimensional model. Both these results transfer features are demonstrated in this example. The results transfer capability can significantly reduce the analysis cost of structures that undergo symmetric deformation, followed by nonsymmetric deformation later during the loading history. The purpose of this example is to obtain the footprint solution of a 175 SR14 tire subjected to an inflation pressure and a concentrated load on the axle, which represents the weight of the vehicle. The footprint solution is used as a starting point in ``Steady-state rolling analysis of a tire,'' Section 3.1.2, where the free rolling state of the tire rolling at 10 km/h is determined, and in ``Subspace-based steady-state dynamic tire analysis,'' Section 3.1.3, where a frequency response analysis is performed.

Problem description The different components of the tire are shown in Figure 3.1.1-1. The tread and sidewalls are made of rubber, and the belts and carcass are constructed from fiber reinforced rubber composites. The rubber is modeled as an incompressible hyperelastic material, and the fiber reinforcement is modeled as a linear elastic material. A small amount of skew symmetry is present in the geometry of the tire due to the placement and §20.0° orientation of the reinforcing belts. Two simulations are performed in this example. The first simulation exploits the symmetry in the tire model and utilizes the results transfer capability; the second simulation does not use the results transfer capability. Comparisons between the two methodologies are made. The first simulation is broken down into three separate analyses. In the first analysis the inflation of the tire by a uniform internal pressure is modeled. Due to the anisotropic nature of the tire construction, the inflation loading gives rise to a circumferential component of deformation. The resulting stress field is fully three-dimensional, but the problem remains axisymmetric in the sense that the solution 3-913

Tire Analyses

does not vary as a function of position along the circumference. ABAQUS provides axisymmetric elements with twist (CGAX) for such situations. These elements are used to model the inflation loading. Only half the tire cross-section is needed for the inflation analysis due to a reflection symmetry through the vertical line that passes through the tire axle (see Figure 3.1.1-2). We refer to this model as the axisymmetric model. The second part of the simulation entails the computation of the footprint solution, which represents the static deformed shape of the pressurized tire due to a vertical dead load (modeling the weight of a vehicle). A three-dimensional model is needed for this analysis. The finite element mesh for this model is obtained by revolving the axisymmetric cross-section about the axis of revolution. A nonuniform discretization along the circumference is used as shown in Figure 3.1.1-3. In addition, the axisymmetric solution is transferred to the new mesh where it serves as the initial or base state in the footprint calculations. As with the axisymmetric model, only half of the cross-section is needed in this simulation, but skew-symmetric boundary conditions must be applied along the mid-plane of the cross-section to account for antisymmetric stresses that result from the inflation loading and the concentrated load on the axle. We refer to this model as the partial three-dimensional model. In the last part of this analysis the footprint solution from the partial three-dimensional model is transferred to a full three-dimensional model and brought into equilibrium. This full three-dimensional model is used in the steady-state transport example that follows. The model is created by combining two parts of the partial three-dimensional model, where one part is the mesh used in the second analysis and the other part is the partial model reflected through a line. We refer to this model as the full three-dimensional model. A second simulation is performed in which the same loading steps are repeated, except that the full three-dimensional model is used for the entire analysis. Besides being used to validate the results transfer solution, this second simulation allows us to demonstrate the computational advantage afforded by the ABAQUS results transfer capability in problems with rotational and/or reflection symmetries.

Model definition In the first simulation the inflation step is performed on the axisymmetric model and the results are stored in the results files ( .res, .mdl, .stt, and .prt). The axisymmetric model is discretized with CGAX4H and CGAX3H elements. The belts and carcass are modeled by defining rebar in the continuum elements, and the road is defined as an analytical rigid surface. The axisymmetric results are read into the subsequent footprint analysis, and the partial three-dimensional model is generated by ABAQUS by revolving the axisymmetric model cross-section about the rotational symmetry axis. The *SYMMETRIC MODEL GENERATION, REVOLVE option is used for this purpose. The road is defined in the partial three-dimensional model. The results of the footprint analysis are read into the final equilibrium analysis, and the full three-dimensional model is generated by reflecting the partial three-dimensional model through a vertical line using the *SYMMETRIC MODEL GENERATION, REFLECT=LINE option. The line used in the reflection is the vertical line in the symmetry plane of the tire, which passes through the axis of rotation. The REFLECT=LINE parameter is used, as opposed to the REFLECT=PLANE parameter, to take into account the skew symmetry of the tire. The analytical rigid surface as defined in the partial three-dimensional model is transferred to the full

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Tire Analyses

model without change. The three-dimensional finite element mesh of the full model is shown in Figure 3.1.1-4. In the second simulation a datacheck analysis is performed to write the axisymmetric model information to the results files. The full tire cross-section is meshed in this model. No analysis step is performed. The axisymmetric model information is read in a subsequent run, and a full three-dimensional model is generated by ABAQUS by revolving the cross-section about the rotational symmetry axis. The *SYMMETRIC MODEL GENERATION, REVOLVE option is again used for this purpose. The road is defined in the full model. The three-dimensional finite element mesh of the full model is identical to the one generated in the first analysis. However, the inflation load and concentrated load on the axle are applied to the full model without making use of the results transfer capability. During the model generation from the axisymmetric to the three-dimensional meshes in both analyses, the axisymmetric CGAX4H and CGAX3H elements are converted into C3D8H and C3D6H elements, respectively. The footprint calculations in both analyses are performed with a friction coefficient of zero in anticipation of eventually performing a steady-state rolling analysis of the tire using the *STEADY STATE TRANSPORT option, as explained in ``Steady-state rolling analysis of a tire,'' Section 3.1.2. Since the results from the static analyses performed in this example are used in a subsequent time-domain dynamic example, there are a few features in the input files that would not ordinarily be included for purely static analyses. It is instructive to point out and to discuss these features briefly. The TRANSPORT parameter is included with the *SYMMETRIC MODEL GENERATION option to define streamlines in the model, which are needed by ABAQUS to perform streamline calculations during the *STEADY STATE TRANSPORT analysis in the next example problem. The TRANSPORT parameter is not required for any other analysis type except for *STEADY STATE TRANSPORT. The hyperelastic material, which models the rubber, has a *VISCOELASTIC, TIME=PRONY option included. This enables us to model viscoelasticity in the steady-state transport example that follows. As a consequence of defining a time-domain viscoelastic material property, the *HYPERELASTIC option includes the LONG TERM parameter to indicate that the elastic properties defined in the associated data lines define the long-term behavior of the rubber. In addition, all *STATIC steps include the LONG TERM parameter to ensure that the static solutions are based upon the long-term elastic moduli.

Loading As discussed in the previous sections, the loading on the tire is applied over several steps. In the first simulation the inflation of the tire to a pressure of 200.0 kPa is modeled using the axisymmetric tire model (tiretransfer_axi_half.inp) with a *STATIC analysis procedure. The results from this axisymmetric analysis are then transferred to the partial three-dimensional model (tiretransfer_symmetric.inp) in which the footprint solution is computed in two sequential *STATIC steps. The first of these static steps establishes the initial contact between the road and the tire by

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Tire Analyses

prescribing a vertical displacement of 0.02 m on the rigid body reference node. Since this is a static analysis, it is recommended that contact be established with a prescribed displacement, as opposed to a prescribed load, to avoid potential convergence difficulties that might arise due to unbalanced forces. The prescribed boundary condition is removed in the second static step, and a vertical load of N = 1.65 kN is applied to the rigid body reference node. The 1.65 kN load in the partial three-dimensional model represents a 3.3 kN load in the full three-dimensional model. The transfer of the results from the axisymmetric model to the partial three-dimensional model is accomplished by using the *SYMMETRIC RESULTS TRANSFER option. Once the static footprint solution for the partial three-dimensional model has been established, the *SYMMETRIC RESULTS TRANSFER option is used once again to transfer the solution to the full three-dimensional model ( tiretransfer_full.inp), where the footprint solution is brought into equilibrium in a single *STATIC increment. The results transfer sequence is illustrated in Figure 3.1.1-5. It is important to note that boundary conditions and loads are not transferred with the *SYMMETRIC RESULTS TRANSFER option; they must be carefully redefined in the new analysis to match the loads and boundary conditions from the transferred solution. Due to numerical and modeling issues the element formulation for the two-dimensional and three-dimensional elements are not identical. As a result, there may be slight differences between the equilibrium solutions generated by the two- and three-dimensional models. In addition, small numerical differences may occur between the symmetric and full three-dimensional solutions because of the presence of symmetry boundary conditions in the symmetric model that are not used in the full model. Therefore, it is advised that in a results transfer simulation an initial step be performed where equilibrium is established between the transferred solution and loads that match the state of the model from which the results are transferred. Since the transferred solution is applied in full at time t = 0, the external loads must also be applied in full at the beginning of the initial step. There is no benefit in reducing the magnitude of the loads to overcome convergence problems. To ensure that ABAQUS does not waste computational time by attempting smaller time increments if equilibrium cannot be attained, it is recommended that the initial step should consist of a *STATIC, DIRECT procedure with the initial time increment set to the total step time. In the second simulation identical inflation and the footprint steps are repeated. The only difference is that the entire analysis is performed on the full three-dimensional model (tiretransfer_full_footprint.inp). The full three-dimensional model is generated using the restart information from a datacheck analysis on an axisymmetric model of the full tire cross-section (tiretransfer_axi_full.inp).

Solution controls Since the three-dimensional tire model has a small loaded area and, thus, rather localized forces, the default averaged flux values for the convergence criteria produce very tight tolerances and cause more iteration than is necessary for an accurate solution. To decrease the computational time required for the analysis, the *CONTROLS option can be used to override the default values for average forces and moments. The default controls are used in this example.

Results and discussion

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Tire Analyses

The results from the two simulations are essentially identical. The peak Mises stresses and displacement magnitudes in the two models agree within 0.3% and 0.2%, respectively. The final deformed shape of the tire is shown in Figure 3.1.1-6. The computational cost of each simulation is shown in Table 3.1.1-1. The simulation performed on the full three-dimensional model took 2.5 times longer than the results transfer simulation --clearly demonstrating the computational advantage that can be attained by exploiting the symmetry in the model using the *SYMMETRIC RESULTS TRANSFER option.

Input files tiretransfer_axi_half.inp Axisymmetric model, inflation analysis (simulation 1). tiretransfer_symmetric.inp Partial three-dimensional model, footprint analysis (simulation 1). tiretransfer_full.inp Full three-dimensional model, final equilibrium analysis (simulation 1). tiretransfer_axi_full.inp Axisymmetric model, datacheck analysis (simulation 2). tiretransfer_full_footprint.inp Full three-dimensional model, complete analysis (simulation 2). tiretransfer_node.inp Nodal coordinates for both axisymmetric models.

Table Table 3.1.1-1 Comparison of normalized CPU times to perform the footprint analysis (normalized with respect to the total "No results transfer" analysis). No results Use results transfer transfer and symmetry conditions Inflation 0.002(a)+0.039(b) 0.36(e) (c) (d) Footprint 0.29 +0.061 0.64(e) Total 0.39 1.0 (a) axisymmetric model (b) equilibrium step in partial three-dimensional model (c) footprint analysis in partial three-dimensional model (d) equlibrium step in full three-dimensional model

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Tire Analyses

(e) full three-dimensional model

Figures Figure 3.1.1-1 Tire cross-section.

Figure 3.1.1-2 Axisymmetric tire mesh.

Figure 3.1.1-3 Partial three-dimensional tire mesh.

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Tire Analyses

Figure 3.1.1-4 Full three-dimensional tire mesh.

Figure 3.1.1-5 Results transfer analysis sequence.

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Tire Analyses

Figure 3.1.1-6 Deformed three-dimensional tire (Deformations scaled by a factor of 2).

Sample listings

3-920

Tire Analyses

Listing 3.1.1-1 *HEADING SYMMETRIC RESULTS TRANSFER FOR TIRE MODEL TIRETRANSFER_AXI_HALF AXISYMMETRIC HALF TIRE MODEL STEP 1: INFLATE TIRE TO 200 KPa UNITS: KG, M *RESTART,WRITE,FREQ=100 *NODE,NSET=NTIRE,INP=tiretransfer_node.inp *ELEMENT,TYPE=CGAX4H,ELSET=TREAD 1, 50, 55, 54, 49 2, 45, 50, 49, 44 3, 40, 45, 44, 39 5, 35, 40, 39, 34 7, 31, 35, 34, 30 *ELEMENT,TYPE=CGAX3H,ELSET=TREAD 4, 27, 31, 30 *ELEMENT,TYPE=CGAX4H,ELSET=SIDE 15, 27, 30, 28, 25 16, 24, 27, 25, 22 17, 21, 24, 22, 19 18, 18, 21, 19, 16 19, 15, 18, 16, 13 20, 12, 15, 13, 10 21, 30, 34, 32, 28 29, 9, 12, 10, 7 30, 6, 9, 7, 4 31, 3, 6, 4, 1 *ELEMENT,TYPE=CGAX4H,ELSET=BELT 35, 49, 54, 51, 46 36, 44, 49, 46, 41 37, 39, 44, 41, 36 38, 34, 39, 36, 32 *REBAR,ELEMENT=CONTINUUM,MATERIAL=BELT, GEOMETRY=ISO,NAME=BELT1 BELT, 0.2118683E-6, 1.16E-3, 70.0, 0.50, *REBAR,ELEMENT=CONTINUUM,MATERIAL=BELT, GEOMETRY=ISO,NAME=BELT2 BELT, 0.2118683E-6, 1.16E-3, 110.0, 0.83, *REBAR,ELEMENT=CONTINUUM,MATERIAL=CARCASS, GEOMETRY=ISO,NAME=CARCASS BELT, 0.4208352E-6, 1.00E-3, 0.0, 0.0, 3

3-921

3

3

Tire Analyses

SIDE, 0.4208352E-6, 1.00E-3, 0.0, 0.0, 3 *SOLID SECTION,ELSET=TREAD,MATERIAL=RUBBER *SOLID SECTION,ELSET=SIDE,MATERIAL=RUBBER *SOLID SECTION,ELSET=BELT,MATERIAL=RUBBER *MATERIAL,NAME=RUBBER *HYPERELASTIC,N=1,MODULI=LONG TERM 1.0e6, *VISCOELASTIC,TIME=PRONY 0.3, 0.0, 0.1 *DENSITY 1100., *MATERIAL,NAME=BELT *ELASTIC,TYPE=ISO 172.2E+09, 0.3 *DENSITY 5900., *MATERIAL,NAME=CARCASS *ELASTIC,TYPE=ISO 9.87E+9, 0.3 *DENSITY 1500., *NSET,NSET=RIM 6, 3, 1 *NSET,NSET=SYM 51, 54, 55 *ELSET,ELSET=SOLID,GENERATE 1, 76,1 *SURFACE,NAME=INSIDE BELT, S3 SIDE, S3 *********************************** *STEP,INC=100,NLGEOM=YES 1: INFLATION *STATIC, LONG TERM 0.25, 1.0 *BOUNDARY RIM, 1, 2 RIM, 5, SYM, 2, SYM, 5, *DSLOAD INSIDE, P, 200.E3 *EL FILE,FREQUENCY=50,POSITION=NODES

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Tire Analyses

S, E *NODE FILE,FREQUENCY=50 U *EL PRINT,FREQUENCY=50,REBAR,ELSET=SOLID S *NODE PRINT,FREQUENCY=100,TOTAL=YES U, RF *OUTPUT,FIELD,VARIABLE=PRESELECT,FREQ=50 *OUTPUT,HISTORY,FREQ=1 *END STEP

3-923

Tire Analyses

Listing 3.1.1-2 *HEADING SYMMETRIC RESULTS TRANSFER FOR TIRE MODEL 3D HALF TIRE MODEL STEP 0: TRANSFER TIRE INFLATION RESULTS FROM tiretransfer_axi_half AND GENERATE MODEL USING *SYMMETRIC MODEL GENERATION STEP 1: BRING TRANSFERRED AXISYMMETRIC RESULTS TO EQUILIBRIUM STEP 2: FOOTPRINT ANALYSIS (DISPLACEMENT CONTROL) STEP 3: FOOTPRINT ANALYSIS (LOAD CONTROL) UNITS: KG, M *RESTART,WRITE,FREQ=100 *NODE,NSET=ROAD 9999, 0.0, 0.0, -0.02 *SYMMETRIC MODEL GENERATION,REVOLVE,ELEMENT=200, NODE=200,TRANSPORT 0.0, 0.0, 0.0, 0.0, 1.0, 0.0 0.0, 0.0, 1.0 90.0, 3 70.0, 3 15.0, 7 10.0, 4 15.0, 7 70.0, 3 90.0, 3 *SYMMETRIC RESULTS TRANSFER, STEP=1, INC=4 *ELSET,ELSET=FOOT,GEN 1001, 4801, 200 1002, 4802, 200 1003, 4803, 200 1004, 4804, 200 1005, 4805, 200 1007, 4807, 200 *SURFACE,TYPE=CYLINDER,NAME=SROAD 0., 0.,-0.31657, 1., 0.,-0.31657 0., 1.,-0.31657 START, -0.3, 0. LINE, 0.3, 0. *RIGID BODY,REF NODE=9999,ANALYTICAL SURFACE=SROAD *SURFACE,NAME=STREAD FOOT, S3

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Tire Analyses

*CONTACT PAIR,INTERACTION=SRIGID STREAD, SROAD *SURFACE INTERACTION,NAME=SRIGID *FRICTION 0.0 *ELSET,ELSET=SECT,GENERATE 2800, 3200, 1 *NSET,NSET=SECT,GENERATE 2800, 3400, 1 *NSET,NSET=FOOT,ELSET=FOOT *NSET,NSET=NOUTP,GENERATE 1055, 5055, 200 *NSET,NSET=SYM1 51,54,55,3051,3054,3055 ** ** NODE SETS ASYMA,ASYMB,ASYMC, and ASYMD ** USED FOR ANTI_SYMMETRY BC's ** *NSET,NSET=ASYMA,GENERATE,UNSORTED 255, 2855, 200 254, 2854, 200 251, 2851, 200 *NSET,NSET=ASYMB,GENERATE,UNSORTED 5855, 3255, -200 5854, 3254, -200 5851, 3251, -200 *NSET,NSET=ASYMC,GENERATE,UNSORTED 255, 1055, 200 254, 2854, 200 251, 2851, 200 *NSET,NSET=ASYMD,GENERATE,UNSORTED 5855, 5055, -200 5854, 3254, -200 5851, 3251, -200 *EQUATION 2 ASYMA, 1, 1.0, ASYMB, 1, 1.0 *EQUATION 2 ASYMA, 2, 1.0, ASYMB, 2, 1.0 *EQUATION 2 ASYMC, 3, 1.0, ASYMD, 3, -1.0

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Tire Analyses

*FILE FORMAT,ZERO INCREMENT ******************************************** *STEP,INC=100,NLGEOM=YES 1: BRING TRANSFERRED RESULTS TO EQUILIBRIUM *STATIC, LONG TERM 1.0, 1.0 *BOUNDARY,OP=NEW RIM, 1, 3 ROAD, 1, 6 SYM1, 1, 2 *DSLOAD,OP=NEW INSIDE, P, 200.E3 *NODE PRINT,NSET=ROAD,FREQ=100 U, RF, *EL PRINT,FREQ=0 *NODE FILE,NSET=ROAD U, RF *OUTPUT,FIELD,VARIABLE=PRESELECT *END STEP ****************************************** *STEP,INC=100,NLGEOM=YES 2: FOOTPRINT (Displacement controlled) *STATIC, LONG TERM 0.2, 1.0 *BOUNDARY,OP=NEW RIM, 1, 3 ROAD, 1, 2 ROAD, 4, 6 SYM1, 1, 2 ROAD, 3, , 0.02 *NODE PRINT,NSET=ROAD,FREQ=100 U, RF, *EL PRINT,FREQ=0 *NODE FILE,NSET=ROAD U, RF *NODE FILE,NSET=FOOT,FREQ=100 *OUTPUT,FIELD,VARIABLE=PRESELECT,FREQ=5 *OUTPUT,HISTORY,FREQ=1 *NODE OUTPUT, NSET=ROAD U, RF *END STEP

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Tire Analyses

**************************************** *STEP,INC=100,NLGEOM=YES 3: FOOTPRINT (Load controlled) *STATIC, LONG TERM 1.0, 1.0 *BOUNDARY,OP=NEW RIM, 1, 3 ROAD, 1, 2 ROAD, 4, 6 SYM1, 1, 2 *CLOAD, OP=NEW ROAD, 3, 1650. *END STEP

3.1.2 Steady-state rolling analysis of a tire Product: ABAQUS/Standard This example illustrates the use of the *STEADY STATE TRANSPORT option in ABAQUS (``Steady-state transport analysis,'' Section 6.4.1 of the ABAQUS/Standard User's Manual) to model the steady-state dynamic interaction between a rolling tire and a flat rigid surface. A steady-state transport analysis uses a moving reference frame in which rigid body rotation is described in an Eulerian manner and the deformation is described in a Lagrangian manner. This kinematic description converts the steady moving contact problem into a pure spatially dependent simulation. Thus, the mesh need be refined only in the contact region--the steady motion transports the material through the mesh. Frictional effects, inertia effects, and history effects in the material can all be accounted for in a *STEADY STATE TRANSPORT analysis. The purpose of this analysis is to obtain free rolling equilibrium solutions of a 175 SR14 tire traveling at a ground velocity of 10.0 km/h (2.7778 m/s) at different slip angles. The slip angle is the angle between the direction of travel and the plane normal to the axle of the tire. Straight line rolling occurs at a 0.0° slip angle. An equilibrium solution for the rolling tire problem that has zero torque, T , applied around the axle is referred to as a free rolling solution. An equilibrium solution with a nonzero torque is referred to as either a traction or a braking solution depending upon the sense of T . Braking occurs when the angular velocity of the tire is small enough such that some or all of the contact points between the tire and the road are slipping and the resultant torque on the tire acts in an opposite sense from the angular velocity of the free rolling solution. Similarly, traction occurs when the angular velocity of the tire is large enough such that some or all of the contact points between the tire and the road are slipping and the resultant torque on the tire acts in the same sense as the angular velocity of the free rolling solution. Full braking (traction) occurs when all of the contact points between the tire and the road are slipping. A wheel in free rolling, traction, or braking will spin at different angular velocities, !, for the same ground velocity, v0 : Usually the combination of ! and v0 that results in free rolling is not known in

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advance. Since the steady-state transport analysis capability requires that both the rotational spinning velocity, !, and the traveling ground velocity, v0 , be prescribed, the free rolling solution must be found in an indirect manner. One such indirect approach is illustrated in this example. A finite element analysis of this problem, together with experimental results, have been published by Koishi et al. (1997).

Problem description and model definition A description of the tire and finite element model has been given in ``Symmetric results transfer for a static tire analysis,'' Section 3.1.1. To take into account the effect of the skew symmetry of the actual tire in the dynamic analysis, the steady-state rolling analysis will be performed on the full three-dimensional model, also referred to as the full model. Inertia effects are ignored since the rolling speed is low (v0 = 10 km/h). As stated earlier, the *STEADY STATE TRANSPORT capability in ABAQUS uses a mixed Eulerian/Lagrangian approach in which, to an observer in the moving reference frame, the material appears to flow through a stationary mesh. The paths that the material points follow through the mesh are referred to as streamlines and must be computed before a steady-state transport analysis can be performed. As discussed in ``Symmetric results transfer for a static tire analysis,'' Section 3.1.1, the streamlines needed for the steady-state transport analyses in this example were computed using the *SYMMETRIC MODEL GENERATION, REVOLVE, TRANSPORT option. This option generated the three-dimensional mesh by revolving the two-dimensional tire cross-section about the symmetry axis so that the streamlines followed the mesh lines. The incompressible hyperelastic material used to model the rubber in this example includes a time-domain viscoelastic component, which is enabled by the *VISCOELASTIC, TIME=PRONY option. A simple 1-term Prony series model is used. For an incompressible material a 1-term Prony series in ABAQUS is defined by providing a single value for the shear relaxation modulus ratio, g¹1P , and its associated relaxation time, ¿1 . In this example g¹1P = 0:3 and ¿1 = 0:1. The viscoelastic--i.e., material history--effects are included in a *STEADY STATE TRANSPORT step unless the LONG TERM parameter is used. See ``Time domain viscoelasticity,'' Section 10.6.1 of the ABAQUS/Standard User's Manual, for a more detailed discussion on modeling time-domain viscoelasticity in ABAQUS.

Loading As discussed in ``Symmetric results transfer for a static tire analysis,'' Section 3.1.1, it is recommended that the footprint analyses be obtained with a friction coefficient of zero (so that no frictional forces are transmitted across the contact surface). The frictional stresses for a rolling tire are very different from the frictional stresses in a stationary tire, even if the tire is rolling at very low speed; therefore, discontinuities may arise in the solution between the last *STATIC analysis and the first *STEADY STATE TRANSPORT analysis. Furthermore, varying the friction coefficient from zero at the beginning of the steady-state transport step to its final value at the end of the steady-state transport step ensures that the changes in frictional forces reduce with smaller load increments. This is important if ABAQUS must take a smaller load increment to overcome convergence difficulties while trying to obtain the steady-state rolling solution. 3-928

Tire Analyses

Once the static footprint solution for the tire has been computed, the steady-state rolling contact problem can be solved using the *STEADY STATE TRANSPORT option. The objective of the first simulation in this example is to obtain the straight-line, steady-state rolling solutions, including full braking and full traction, at different spinning velocities. We also compute the straight-line, free rolling solution. In the second simulation, free rolling solutions at different slip angles are computed. In the first and second simulations, material history effects are ignored by including the LONG TERM parameter on the *STEADY STATE TRANSPORT steps. The third simulation repeats a portion of the straight-line, steady-state rolling analysis from the first simulation; however, material history effects are included by omitting the LONG TERM parameter. A steady ground velocity of 10.0 km/h is maintained for all three simulations. In simulation 1 (rollingtire_brake_trac.inp) the full traction solution is obtained in the first *STEADY STATE TRANSPORT step by setting the friction coefficient, ¹, to its final value of 1.0 using the *CHANGE FRICTION option and applying the translational ground velocity together with a spinning angular velocity that will result in full braking. The *TRANSPORT VELOCITY and *MOTION options are used for this purpose. An estimate of the angular velocity corresponding to full braking is obtained as follows. A free rolling tire generally travels farther in one revolution than determined by its center height, H, but less than determined by the free tire radius. In this example the free radius is 316.2 mm and the vertical deflection is approximately 20.0 mm, so H = 294.2 mm. Using the free radius and the effective height, it is estimated that free rolling occurs at an angular velocity between ! = 8.78 rad/s and ! = 9.44 rad/s. Smaller angular velocities would result in braking, and larger angular velocities would result in traction. We use an angular velocity ! = 8.0 rad/s to ensure that the solution in the first steady-state transport step is a full braking solution (all contact points are slipping, so the magnitude of the total frictional force across the contact surface is ¹N ). In the second steady-state transport analysis step of the full model, the angular velocity is increased gradually to ! = 10.0 rad/s while the ground velocity is held constant. The solution at each load increment is a steady-state solution to the loads acting on the structure at that instant so that a series of steady-state solutions between full braking and full traction is obtained. In the second simulation (rollingtire_slipangles.inp) the free rolling solutions at different slip angles are computed. The slip angle, µ, is the angle between the direction of travel and the plane normal to the axle of the tire. In the first step the straight-line free rolling solution from the first simulation is brought into equilibrium. This step is followed by a *STEADY STATE TRANSPORT step where the slip angle is gradually increased from µ =0.0° at the beginning of the step to µ =3.0° at the end of the step, so a series of steady-state solutions at different slip angles are obtained. This is accomplished by prescribing a traveling velocity vector with components vx = v0 cos µ and vy = v0 sin µ on the *MOTION option, where µ =0.0° in the first steady-state transport step and µ =3.0° at the end of the second steady-state transport step. The final simulation in this example (rollingtire_materialhistory.inp) includes a series of steady-state solutions between full braking and full traction in which the material history effects are included.

Results and discussion Figure 3.1.2-1 and Figure 3.1.2-2 show the reaction force parallel to the ground (referred to as rolling 3-929

Tire Analyses

resistance) and the torque, T , on the tire axle at different angular spinning velocities. The figures show that free rolling, T = 0.0, occurs at an angular velocity of approximately 9.0 rad/s. Full braking occurs at spinning velocities smaller than 8.2 rad/s, and full traction occurs at velocities larger than 9.7 rad/s. At these spinning velocities all contact points are slipping, and the rolling resistance reaches the limiting value ¹N: Figure 3.1.2-3 and Figure 3.1.2-4 show shear stress along the centerline of the tire surface in the free rolling and full traction states, respectively. The distance along the centerline is measured as an angle with respect to a plane parallel to the ground passing through the tire axle. The dashed line is the maximum or limiting shear stress, ¹p, that can be transmitted across the surface, where p is the contact pressure. The figures show that all contact points are slipping during full traction. During free rolling all points stick. A better approximation to the angular velocity that corresponds to free rolling can be made by using the results generated by rollingtire_brake_trac.inp to refine the search about an angular velocity of 9.0 rad/s. The file rollingtire_trac_res.inp restarts the previous analysis from Step 3, Increment 11 (corresponding to an angular velocity of 9.006 rad/s) and performs a refined search up to 9.04 rad/s. Figure 3.1.2-5 shows the torque, T , on the tire axle computed in the refined search, which leads to a more precise value for the free rolling angular velocity of approximately 9.025 rad/s. This result is used for the model where the free rolling solutions at different slip angles are computed. Figure 3.1.2-6 shows the transverse force (force along the tire axle) measured at different slip angles. The figure compares the steady-state transport analysis prediction with the result obtained from a pure Lagrangian analysis. The Lagrangian solution is obtained by performing an explicit transient analysis using ABAQUS/Explicit. With this analysis technique a prescribed constant traveling velocity is applied to the tire, which is free to roll along the rigid surface. Since more than one revolution is necessary to obtain a steady-state configuration, fine meshing is required along the full circumference; hence, the Lagrangian solution is much more costly than the steady-state solutions shown in this example. The figure shows good agreement between the results obtained from the two analysis techniques. Figure 3.1.2-7 compares the free rolling solutions with and without material history effects included. The solid lines in the diagram represent the rolling resistance (force parallel to the ground along the traveling direction); and the broken lines, the torque (normalized with respect to the free radius) on the axle. The figure shows that free rolling, marked with bullet points, occurs at a higher angular velocity when history effects are included. It also shows that the rolling resistance increases when history effects are included. The influence of material history effects on a steady-state rolling solution is discussed in detail in ``Steady-state spinning of a disk in contact with a foundation, '' Section 1.5.2 of the ABAQUS Benchmarks Manual.

Acknowledgments HKS gratefully acknowledges Hankook Tire and Yokohama Rubber Company for their cooperation in developing the steady-state transport capability used in this example. HKS thanks Dr. Koishi of Yokohama Rubber Company for supplying the geometry and material properties used in this example.

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Input files rollingtire_brake_trac.inp Three-dimensional full model for the full braking and traction analyses. rollingtire_trac_res.inp Three-dimensional full model for the refined braking and traction analyses. rollingtire_slipangles.inp Three-dimensional full model for the slip angle analysis. rollingtire_materialhistory.inp Three-dimensional full model with material history effects.

Reference · Koishi, M., K. Kabe, and M. Shiratori, "Tire Cornering Simulation using Explicit Finite Element Analysis Code," 16th annual conference of the Tire Society at the University of Akron, 1997.

Figures Figure 3.1.2-1 Rolling resistance at different angular velocities.

Figure 3.1.2-2 Torque at different angular velocities.

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Figure 3.1.2-3 Shear stress along tire center (free rolling).

Figure 3.1.2-4 Shear stress along tire center (full traction).

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Figure 3.1.2-5 Torque at different angular velocities (refined search).

Figure 3.1.2-6 Transverse force as a function of slip angle.

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Figure 3.1.2-7 Rolling resistance and normalized torque as a function of angular velocity ( R=0.3162 m).

Sample listings

3-934

Tire Analyses

Listing 3.1.2-1 *HEADING STEADY-STATE ROLLING ANALYSIS OF A TIRE: ROLLINGTIRE_BRAKE_TRAC 3D FULL TIRE MODEL STEP 0: TRANSFER TIRE INFLATION FOOTPRINT RESULTS FROM TIRETRANSFER_FULL STEP 1: FULL BRAKING ANAYLSIS STEP 2: FULL TRACTION ANALYSIS UNITS: KG, M *RESTART,READ,STEP=1,INC=1 *FILE FORMAT,ZERO INCREMENT ****************************************** *STEP,INC=300,NLGEOM=YES,UNSYMM=YES 1: STRAIGHT LINE ROLLING (Full braking) *STEADY STATE TRANSPORT, LONG TERM 0.5, 1.0 *CHANGE FRICTION,INTERACTION=SRIGID *FRICTION,SLIP=0.01 1.0 *TRANSPORT VELOCITY NTIRE, 8.0 *MOTION,TYPE=VELOCITY,TRANSLATION NTIRE, 1, , 2.7778 *CONTACT PRINT,FREQ=100,NSET=NOUTP *CONTACT FILE,NSET=NOUTP,FREQ=100 *NODE FILE,NSET=NOUTP,FREQ=100 V, COORD *NODE FILE,NSET=ROAD U, RF *OUTPUT,HISTORY,FREQ=100,OP=ADD *NODE OUTPUT,NSET=NOUTP V, COORD *CONTACT OUTPUT,NSET=NOUTP CSTRESS, *END STEP ******************************************* *STEP,INC=300,NLGEOM=YES,UNSYMM=YES 2: STRAIGHT LINE ROLLING (Full traction) *STEADY STATE TRANSPORT, LONG TERM 0.1, 1.0, , 0.1 *RESTART,WRITE,FREQ=11

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*TRANSPORT VELOCITY NTIRE, 10.00 *END STEP

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Listing 3.1.2-2 *HEADING STEADY-STATE ROLLING ANALYSIS OF A TIRE: STEP 0: RESTART FROM rollingtire_trac_res STEP 1: GET EQUILIBRIUM STEP 2: FREE ROLLING AT DIFFERENT SLIP ANGLES UNITS: KG, M *RESTART,READ,STEP=5,INC=2,END STEP *FILE FORMAT,ZERO INCREMENT ****************************************** *STEP,INC=300,NLGEOM,UNSYMM=YES 1: STRAIGHT LINE FREE ROLLING *STEADY STATE TRANSPORT, LONG TERM 1.0, 1.0, *TRANSPORT VELOCITY NTIRE, 9.023 *END STEP ****************************************** *STEP,INC=300,NLGEOM,UNSYMM=YES 2: SLIP (3 degrees) *STEADY STATE TRANSPORT, LONG TERM 0.1, 1.0, , 0.1 *MOTION,TYPE=VELOCITY,TRANSLATION NTIRE, 1, , 2.774 NTIRE, 2, , 0.14538 *END STEP

3.1.3 Subspace-based steady-state dynamic tire analysis Product: ABAQUS/Standard This example illustrates the use of the *STEADY STATE DYNAMICS, SUBSPACE PROJECTION option to model the frequency response of a tire about a static footprint solution. The *STEADY STATE DYNAMICS, SUBSPACE PROJECTION option (``Subspace-based steady-state dynamic analysis,'' Section 6.3.7 of the ABAQUS/Standard User's Manual) is an analysis procedure that can be used to calculate the steady-state dynamic response of a system subjected to harmonic excitation. It does so by the direct solution of the steady-state dynamic equations projected onto a reduced-dimensional subspace spanned by a set of eigenmodes of the undamped system. If the dimension of the subspace is small compared to the dimension of the original problem--i.e., if a relatively small number of eigenmodes are used, the subspace method can offer a very cost-effective alternative to a direct-solution steady-state analysis. The purpose of this analysis is to obtain the frequency response of a 175 SR14 tire subjected to a 3-937

Tire Analyses

harmonic load excitation about the footprint solution discussed in ``Symmetric results transfer for a static tire analysis,'' Section 3.1.1). The *SYMMETRIC RESULTS TRANSFER and *SYMMETRIC MODEL GENERATION options are used to generate the footprint solution, which serves as the base state in the steady-state dynamics calculations.

Problem description A description of the tire being modeled has been given in ``Symmetric results transfer for a static tire analysis,'' Section 3.1.1. In this example we exploit the symmetry in the tire model and utilize the results transfer capability in ABAQUS to compute the footprint solution for the full three-dimensional model in a manner identical to that discussed in ``Symmetric results transfer for a static tire analysis,'' Section 3.1.1. Once the footprint solution has been computed, several steady-state dynamic steps are performed. Both the *STEADY STATE DYNAMICS, DIRECT and the *STEADY STATE DYNAMICS, SUBSPACE PROJECTION options are used. Besides being used to validate the subspace projection results, the direct steady-state procedure allows us to demonstrate the computational advantage afforded by the subspace projection capability in ABAQUS.

Model definition The model used in this analysis is essentially identical to that used in the first simulation discussed in ``Symmetric results transfer for a static tire analysis,'' Section 3.1.1, with CGAX4H and CGAX3H elements used in the axisymmetric model and rebar in the continuum elements for the belts and carcass. However, since no *STEADY STATE TRANSPORT steps are performed in this example, the TRANSPORT parameter is not needed during the symmetric model generation phase. In addition, instead of using a nonuniform discretization about the circumference, the uniform discretization shown in Figure 3.1.3-1 is used. The analysis procedures available with the *STEADY STATE DYNAMICS option are all frequency-domain procedures. In contrast, the *STEADY STATE TRANSPORT option discussed in ``Steady-state rolling analysis of a tire,'' Section 3.1.2, is a time-domain procedure. The incompressible hyperelastic material used to model the rubber in this example includes a frequency-domain viscoelastic model, which is activated by the *VISCOELASTIC, FREQUENCY=TABULAR option. This is different from the time-domain viscoelastic model (*VISCOELASTIC, TIME=PRONY) that was used in the steady-state transport example. The FREQUENCY=TABULAR option requires the user to provide tabular values of ! Format of input file. C LOUTF -- Format of output file: C 1 --> ABAQUS results file ASCII format. C 2 --> ABAQUS results file binary format. C JUNIT -- Unit number of file to be opened. C JRCD -- Error check return code: C .EQ. 0 --> No errors. C .NE. 0 --> Errors detected. C KEY -- Current record key identifier. C C==================================================================== C C The use of ABA_PARAM.INC eliminates the need to have different C versions of the code for single and double precision. C ABA_PARAM.INC defines an appropriate IMPLICIT REAL statement and C sets the value of NPRECD to 1 or 2, depending on whether the C machine uses single or double precision. C C==================================================================== C INCLUDE 'aba_param.inc' DIMENSION ARRAY(513), JRRAY(NPRECD,513) EQUIVALENCE (ARRAY(1), JRRAY(1,1)) C C==================================================================== C Set the dimensions of LRUNIT to be the maximum number of results C files to be joined. C C==================================================================== PARAMETER (MXUNIT=21) INTEGER LRUNIT(2,MXUNIT),LUNIT(10) CHARACTER FNAME*80 DATA LUNIT/1,5,6,7,9,11,12,13,20,28/ C C==================================================================== C Input the number of files to be joined and then the unit number and C format of each of the files. C C==================================================================== 5 CONTINUE

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WRITE(6,10) MXUNIT 10 FORMAT(1X,'Enter the number of files to be joined (MAX:',I3,'):') READ(5,'(I3)') NRU IF (NRU .GT. MXUNIT) GOTO 5 C C DO 40 INRU = 1, NRU 15 CONTINUE WRITE(6,20) INRU 20 FORMAT(1X,'Enter the unit number of input file #',I3,':') READ(5,*) LRUNIT(1,INRU) DO 41 K1=1,9 IF (LRUNIT(1,INRU) .EQ. LUNIT(K1)) THEN WRITE(6,*) 'ERROR! Unit number cannot be ',LUNIT(K1) GOTO 15 ENDIF 41 CONTINUE 42 CONTINUE WRITE(6,30) INRU 30 FORMAT(1X,'Enter the format of input file #',I3, 1 ' (1-ASCII, 2-binary):') READ(5,*) LRUNIT(2,INRU) IF (LRUNIT(2,INRU).NE. 1 .AND. LRUNIT(2,INRU) .NE. 2) THEN WRITE(6,*) 'ERROR! This number must be 1 or 2' GOTO 42 ENDIF 40 CONTINUE C C==================================================================== C Set LOUTF equal to the format of the output file. If this program C is to be used only to convert the file format from one type to C another, set NRU=1 (to read only one file) and specify a value of C LOUTF which is opposite to the value specified for LRUNIT(2,1). C C==================================================================== 45 CONTINUE WRITE(6,50) 50 FORMAT(1X,'Enter the format of the output file ', 1 '(1-ASCII, 2-binary):') READ(5,*) LOUTF IF (LOUTF .NE. 1 .AND. LOUTF .NE. 2) THEN WRITE(6,*) 'ERROR! This number must be 1 or 2' GOTO 45

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Postprocessing of ABAQUS Results Files

ENDIF C WRITE(6,60) 60 FORMAT(1X,'Enter the name of the input file(s) (w/o extension):') READ(5,'(A)') FNAME C CALL

INITPF (FNAME, NRU, LRUNIT, LOUTF)

C KEYPRV = 0 C C==================================================================== C Loop through NRU input files... C C==================================================================== DO 100 INRU = 1, NRU JUNIT = LRUNIT(1,INRU) CALL DBRNU (JUNIT) I2001 = 0 C==================================================================== C ...and cover a maximum of 10 million records in each file. C C==================================================================== DO 80 IXX2 = 1, 100 DO 80 IXX = 1, 99999 CALL DBFILE(0,ARRAY,JRCD) C WRITE(6,*) 'KEY/RECORD LENGTH = ', JRRAY(1,2),JRRAY(1,1) IF (JRCD .NE. 0 .AND. KEYPRV .EQ. 2001) THEN WRITE(6,*) 'END OF FILE #', INRU CLOSE (JUNIT) GOTO 100 ELSE IF (JRCD .NE. 0) THEN WRITE(6,*) 'ERROR READING FILE #', INRU CLOSE (JUNIT) GOTO 110 ENDIF C C==================================================================== C Initialize the flag to write a record to the file: C LWRITE=0 -- write disabled C LWRITE=1 -- write enabled C C==================================================================== LWRITE=1

11-1189

Postprocessing of ABAQUS Results Files

C C==================================================================== C For files other than the first, skip the 1900-series header records C except for the superelement path (1910; for superelement analyses), C output request (1911), heading (1922), and modal (1980; for natural C frequency extraction) records. In a merged file, the heading C record serves as a file delimiter. C C==================================================================== IF (INRU.GT.1) THEN IF (JRRAY(1,2).GE.1900 .AND. JRRAY(1,2).LE.1909) LWRITE=0 IF (JRRAY(1,2).GE.1912 .AND. JRRAY(1,2).LT.1922) LWRITE=0 C C==================================================================== C Skip the first 2001 record (this indicates the end of the header C records). C C==================================================================== IF (JRRAY(1,2) .EQ. 2001 .AND. I2001 .EQ. 0) THEN I2001 = 1 LWRITE = 0 ENDIF ENDIF C C==================================================================== C If this is the first input file, or if the write flag has not been C disabled for records in subsequent files, then write the data to C the output file. We are interested in retrieving the header C records (relevant 1900-series records), the increment start and C end records (2000 and 2001), the element header record, (1) and C the stress and strain records (11 and 21). C C==================================================================== IF (INRU .EQ. 1 .OR. LWRITE .EQ. 1) THEN KEY=JRRAY(1,2) IF((KEY.EQ.1900).OR.(KEY.EQ.1901).OR.(KEY.EQ.1902).OR. 1 (KEY.EQ.1910).OR.(KEY.EQ.1911).OR.(KEY.EQ.1921).OR. 2 (KEY.EQ.1922).OR.(KEY.EQ.1980).OR.(KEY.EQ.2000).OR. 3 (KEY.EQ.2001).OR.(KEY.EQ.1).OR.(KEY.EQ.11).OR. 4 (KEY.EQ.21)) THEN CALL DBFILW(1,ARRAY,JRCD) IF (JRCD .NE. 0) THEN WRITE(6,*) 'ERROR WRITING FILE'

11-1190

Postprocessing of ABAQUS Results Files

CLOSE (JUNIT) GOTO 110 ENDIF ENDIF ENDIF KEYPRV = JRRAY(1,2) 80 CONTINUE 100 CONTINUE 110 CONTINUE C RETURN END

11.1.3 Calculation of principal stresses and strains and their directions: FPRIN Product: ABAQUS/Standard This example illustrates the use of a FORTRAN program to read stress and strain records from an ABAQUS results file and to calculate principal stress and strain values and their directions.

General description This program shows how to retrieve integration point and nodal averaged stress and strain components from an ABAQUS results file and then compute principal values and directions using the ABAQUS subroutine SPRIND. Usage of this subroutine is documented in the program listing provided below, and further details about the interface to this subroutine are discussed in ``UMAT,'' Section 23.2.29 of the ABAQUS/Standard User's Manual. The results file created by the FJOIN program in ``Joining data from multiple results files and converting file format: FJOIN,'' Section 11.1.2, is used here to verify that the records that have been put together are retrievable. The previously generated results file was named fjoinxxx.fin. To use it as an input file for postprocessing program FPRIN, the file extension must be changed. This program will assume that the results file has the default .fil extension, which corresponds to FORTRAN unit 8.

Programming details The user should first review the general discussion on programming concepts and ABAQUS FORTRAN interfaces in ``User postprocessing of ABAQUS results files: overview,'' Section 11.1.1, and the detailed discussion of postprocessing given in Chapter 5, "File Output Format," of the ABAQUS/Standard User's Manual. When running program FPRIN (this program is named fprin.f on the ABAQUS release media), the user will be prompted for the file name that initializes FNAME. Other variables, such as LOUTF, NRU, LRUNIT(1,NRU), and LRUNIT(2,NRU), are initialized inside the program. INITPF and DBNRU are then called to complete the neccesary initializations and file connections. Data processing starts with a double DO-loop over all the records to be read, one-by-one, via a call to DBFILE. Each record

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is identified by its record key, which is stored in the second entry of the record. When records 1922 and 2000 are processed by program FPRIN, the heading and the current step and increment numbers are written out so as to provide a way to recognize the beginning of data in each analysis. Record type 1 is then examined to determine the output location of stress and strain, the number of direct and shear stress and strain components, and either the element number or the node number for which the records are written. The stress and strain records ( 11 and 21, respectively) will be filtered out for processing by the ABAQUS subroutine SPRIND. When a stress or strain record is passed into SPRIND, principal stresses or strains and the corresponding principal directions are calculated and returned in an unsorted order.

Program compilation and linking Before program execution, the FORTRAN program has to be compiled and linked. Both operations, as well as the inclusion of the aba_param.inc file, are performed by a single execution of the ABAQUS/Make procedure: abaqus make job=fprin

This may have to be repeated until all FORTRAN errors are corrected. After successful compilation, the program's object code is automatically linked with the ABAQUS object codes stored in the shared program library and interface library in order to build the executable program. Refer to Chapter 3, "Environment file," of the ABAQUS Site Guide to see which compile and link commands are used for a particular computer.

Program execution Before the program is executed, a results file must have been created. In this example the results file fjoinxxx.fin created by the FJOIN program discussed in ``Joining data from multiple results files and converting file format: FJOIN,'' Section 11.1.2, is used. This file must be renamed to fjoinxxx.fil since FORTRAN unit 8 (which is associated with the .fil file extension) is used in the program to read the file. When the program is executed using the command abaqus fprin, the prompt Enter the name of the input file (w/o .fil):

will appear. Enter fjoinxxx to define FNAME. The program processes the data and produces a file named pvalue.dat, which contains information about principal stresses and strains and their directions.

Results and discussion The computed principal stress and strain values and their directions are tabulated below. Analysis Principal Stress Strain Dir-1 Dir-2 Dir-3 Componen File ´ 105 ´ 10-3 t fjoin002.i 1 10.714 25.0 1.0 0.0 0.0 2 10.714 25.0 0.0 1.0 0.0 np 3 0.0 0.0 0.0 0.0 1.0 fjoin003.i 1 -2.8846 -12.5 0.707 -0.707 0.0

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np fjoin004.i np

2 3 1 2 3

2.8846 0.0 0.0 7.5 0.0

12.5 0.0 -7.5 25.0 0.0

0.707 0.0 1.0 0.0 0.0

Input file fprin.f Postprocessing program.

Sample listings

11-1193

0.707 0.0 0.0 1.0 0.0

0.0 1.0 0.0 0.0 1.0

Postprocessing of ABAQUS Results Files

Listing 11.1.3-1 SUBROUTINE HKSMAIN C==================================================================== C This program must be compiled and linked with the command: C abaqus make job=fprin C Run the program using the command: C abaqus fprin C==================================================================== C C Purpose: C C This program computes the principal stresses and strains and their C directions from stress and strain values stored in an ABAQUS C results file (.fil). C C Input File names: `FNAME.fil', where FNAME is the root file name of C the input file. C C Output File name: pvalue.dat C C==================================================================== C C Variables used by this program and ABAQUS subroutine SPRIND : C C NDI -- Number of direct components in stress/strain tensor. C NSHR -- Number of shear components in stress/strain tensor. C NDIP1 -- NDI + 1 C ARRAY -- Real array containing values read from results file C (.fil). Equivalenced to JRRAY. C JRRAY -- Integer array containing values read from results file C (.fil). Equivalenced to ARRAY. C FNAME -- Root file name of input file (w/o .fil extension). C NRU -- Number of results files (.fil) to be read. C LRUNIT -- Array containing unit number and format of results files: C LRUNIT(1,*) --> Unit number of input file. C LRUNIT(2,*) --> Format of input file. C LOUTF -- Format of output file: C 0 --> Standard ASCII format. C 1 --> ABAQUS results file ASCII format. C 2 --> ABAQUS results file binary format. C JUNIT -- Unit number of file to be opened. C JRCD -- Error check return code.

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C .EQ. 0 --> No errors. C .NE. 0 --> Errors detected. C KEY -- Current record key identifier. C JELNUM -- Current element number. C INTPN -- Integration point number. C LSTR -- Indicates type of principal value (stress/strain) and C ordering used: C For calculation of principal value (stress/strain): C 1 --> stress. C 2 --> strain. C For calculation of directions: C 1 --> stress. C 2 --> strain. C S -- Array containing stress tensor. C PS -- Array containing principal stresses. C ANPS -- Array containing directions of principal stresses. C E -- Array containing strain tensor. C PE -- Array containing principal strains. C ANPE -- Array containing directions of principal strains. C C==================================================================== C C The use of ABA_PARAM.INC eliminates the need to have different C versions of the code for single and double precision. C ABA_PARAM.INC defines an appropriate IMPLICIT REAL statement C and sets the value of NPRECD to 1 or 2, depending on whether C the machine uses single or double precision. C C==================================================================== C INCLUDE 'aba_param.inc' DIMENSION ARRAY(513), JRRAY(NPRECD,513), LRUNIT(2,1) EQUIVALENCE (ARRAY(1), JRRAY(1,1)) C C==================================================================== DIMENSION S(6), E(6), PS(3), PE(3), ANPS(3,3), ANPE(3,3) CHARACTER FNAME*80 C C==================================================================== C Get the name of the results file. C C==================================================================== WRITE(6,*) 'Enter the name of the input file (w/o .fil):'

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READ(5,'(A)') FNAME C C==================================================================== C Open the output file. C C==================================================================== OPEN(UNIT=9,FILE='pvalue.dat',STATUS='NEW') C NRU = 1 LOUTF = 0 LRUNIT(1,1) = 8 LRUNIT(2,1) = 2 C CALL INITPF(FNAME,NRU,LRUNIT,LOUTF) C JUNIT = 8 C CALL DBRNU(JUNIT) C C==================================================================== C Read records from the results (.fil) file and process the data. C Cover a maximum of 10 million records in the file. C C==================================================================== DO 1000 K100 = 1, 100 DO 1000 K1 = 1, 99999 CALL DBFILE(0,ARRAY,JRCD) IF (JRCD .NE. 0) GO TO 1001 KEY = JRRAY(1,2) C C==================================================================== C Get the heading (title) record. C C==================================================================== IF (KEY .EQ. 1922) THEN WRITE(9,1100) (ARRAY(IXX),IXX=3,12) 1100 FORMAT(1X,10A8) C C==================================================================== C Get the current step and increment number. C C==================================================================== ELSE IF (KEY .EQ. 2000) THEN

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1200

WRITE(9,1200) JRRAY(1,8), JRRAY(1,9) FORMAT(1X,'** STEP ',I2,' INCREMENT ',I3)

C C==================================================================== C Get the element and integration point numbers, JELNUM and INTPN, C and the location of INTPN (0--at int.pt., 1--at centroid, C 4--nodal average) and the number of direct and shear components C in the analysis. C C==================================================================== ELSE IF (KEY .EQ. 1) THEN JELNUM = JRRAY(1,3) INTPN = JRRAY(1,4) LOCATE = JRRAY(1,6) NDI = JRRAY(1,8) NSHR = JRRAY(1,9) NDIP1 = NDI + 1 IF(LOCATE.LE.1) THEN WRITE(9,1201) JELNUM, INTPN ,NDI,NSHR 1201 FORMAT(2X,'ELEMENT NUMBER = ',I8,5X, 1 'INT. PT. NUMBER = ',I2,5X, 2 'NDI/HSHR = ',2I2) ELSEIF(LOCATE.EQ.4) THEN WRITE(9,1191) JELNUM, NDI,NSHR 1191 FORMAT(2X,'NODE NUMBER = ',I8,5X, 1 'NDI/HSHR = ',2I2) END IF C C==================================================================== C Get the stress tensor. C C==================================================================== ELSE IF (KEY .EQ. 11) THEN WRITE(9,1202) 1202 FORMAT(3X,'STRESSES:') C DO 10 IXX = 1, NDI S(IXX) = ARRAY(IXX+2) 10 CONTINUE WRITE(9,1203) (S(IZZ), IZZ = 1, NDI) 1203 FORMAT(4X,'S11 = ',E12.5,' S22 = ',E12.5,' S33 = ',E12.5) DO 20 IYY = NDI + 1, NSHR + NDI S(IYY) = ARRAY(IYY+2)

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20 1204

CONTINUE WRITE(9,1204) (S(IZZ), IZZ = NDI + 1, NSHR + NDI) FORMAT(4X,'S12 = ',E12.5,' S13 = ',E12.5,' S23 = ',E12.5)

C C C==================================================================== C Calculate the principal stresses and corresponding principal C directions in unsorted order. C==================================================================== LSTR = 1 CALL SPRIND(S,PS,ANPS,LSTR,NDI,NSHR) WRITE(9,1205) PS(1), ANPS(1,1), ANPS(1,2), ANPS(1,3) 1205 FORMAT(4X,'PS1 = ',E12.5,/, 1 5X,'PD11 =',F8.3,2X,'PD12 =',F8.3,2X,'PD13 =',F8.3) WRITE(9,1206) PS(2), ANPS(2,1), ANPS(2,2), ANPS(2,3) 1206 FORMAT(4X,'PS2 = ',E12.5,/, 1 5X,'PD21 =',F8.3,2X,'PD22 =',F8.3,2X,'PD23 =',F8.3) WRITE(9,1207) PS(3), ANPS(3,1), ANPS(3,2), ANPS(3,3) 1207 FORMAT(4X,'PS3 = ',E12.5,/, 1 5X,'PD31 =',F8.3,2X,'PD32 =',F8.3,2X,'PD33 =',F8.3) C C C==================================================================== C Get the strain tensor. C C==================================================================== ELSE IF (KEY .EQ. 21) THEN WRITE(9,2202) 2202 FORMAT(3X,'STRAINS:') C DO 30 IXX = 1, NDI E(IXX) = ARRAY(IXX+2) 30 CONTINUE WRITE(9,2203) (E(IZZ), IZZ = 1, NDI) 2203 FORMAT(4X,'E11 = ',E12.5,' E22 = ',E12.5,' E33 = ',E12.5) DO 40 IYY = NDI + 1, NSHR + NDI E(IYY) = ARRAY(IYY+2) 40 CONTINUE WRITE(9,2204) (E(IZZ), IZZ = NDI + 1, NSHR + NDI) 2204 FORMAT(4X,'E12 = ',E12.5,' E13 = ',E12.5,' E23 = ',E12.5) C C C====================================================================

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C Calculate the principal strains and corresponding principal C directions in unsorted order. C==================================================================== LSTR = 2 CALL SPRIND(E,PE,ANPE,LSTR,NDI,NSHR) WRITE(9,2205) PE(1), ANPE(1,1), ANPE(1,2), ANPE(1,3) 2205 FORMAT(4X,'PE1 = ',E12.5,/, 1 5X,'PD11 =',F8.3,2X,'PD12 =',F8.3,2X,'PD13 =',F8.3) WRITE(9,2206) PE(2), ANPE(2,1), ANPE(2,2), ANPE(2,3) 2206 FORMAT(4X,'PE2 = ',E12.5,/, 1 5X,'PD21 =',F8.3,2X,'PD22 =',F8.3,2X,'PD23 =',F8.3) WRITE(9,2207) PE(3), ANPE(3,1), ANPE(3,2), ANPE(3,3) 2207 FORMAT(4X,'PE3 = ',E12.5,/, 1 5X,'PD31 =',F8.3,2X,'PD32 =',F8.3,2X,'PD33 =',F8.3) C END IF C 1000 CONTINUE 1001 CONTINUE C CLOSE (UNIT=9) C RETURN END

11.1.4 Creation of a perturbed mesh from original coordinate data and eigenvectors: FPERT Product: ABAQUS/Standard This example illustrates the use of a FORTRAN program to create a perturbed mesh by superimposing a small imperfection in the form of the weighted sum of several buckling modes on the initial geometry. The program retrieves the original nodal coordinates and the desired eigenvectors from an ABAQUS results file, then calculates new nodal coordinates for the perturbed mesh.

General description Collapse studies of a structure's postbuckling load-displacement (Riks) behavior are often conducted to verify that the critical buckling load and mode predicted by an eigenvalue buckling analysis are accurate. They are also done to investigate the effect of an initial geometric imperfection on the load-displacement response. A typical assumption is that an imperfection made up of a combination of the eigenmodes associated with the lowest eigenvalues will be the most critical. One method of PM introducing an imperfection of this type into the model is by adding i=1 ®i ui to the original mesh coordinates. In this case ui is the ith eigenmode, ®i is a scaling factor of the ith eigenmode, and M is

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the total number of eigenmodes extracted in the buckling analysis. Since the eigenvector is typically normalized to a maximum absolute value of one, ®i is usually some fraction of a geometric parameter, such as the shell thickness. The postprocessing program described below can be used to introduce an imperfection of this type into a model. The perturbation procedure is illustrated in ``Buckling of a cylindrical shell under uniform axial pressure,'' Section 1.2.3 of the ABAQUS Benchmarks Manual. An eigenvalue buckling analysis, fpert001, is run first. This analysis creates the results file, fpert001.fil, which contains the original nodal coordinates and the eigenvectors for the buckling modes. This results file is then used to generate a perturbed mesh for the postbuckling load-displacement analysis. The postprocessing program perturbs the original mesh using the relation 0

X =X+

M X

®i ui ;

i=1

where X0 is the vector containing the new global coordinates; X is the vector of original coordinates; M is the number of buckling modes; and ®i is the imperfection factor for the ith eigenvector, ui . The new coordinates are written to the file fpert002.015, which is read by the load-displacement analysis fpert002.

Programming details The general discussion on programming concepts and ABAQUS FORTRAN interfaces in ``User postprocessing of ABAQUS results files: overview,'' Section 11.1.1, should be reviewed before running or modifying this program. Review of the results file format in Chapter 5, "File Output Format," of the ABAQUS/Standard User's Manual is also recommended. The FPERT program (this program is named fpert.f on the ABAQUS release media) makes some assumptions concerning the type of results file it will be reading. Variables NRU, LRUNIT(1,NRU), and LRUNIT(2,NRU) are initialized within the program to 1, 8, and 2. These values indicate that one file will be read, the FORTRAN unit used will be 8, and the file type will be binary. See ``Accessing the results file information,'' Section 5.1.3 of the ABAQUS/Standard User's Manual, for more information on opening and initializing postprocessing files. Once the file specification parameters are set, the INITPF and DBNRU subroutines are called to open and ready the file, whose name is stored in FNAME, for reading. The file to which the perturbed coordinates are to be written can be directly opened using a FORTRAN OPEN statement. The ABAQUS file utilities are not necessary since the file is a plain text file. The records with the original nodal coordinates are read using the DBFILE routine and stored in the local array COORDS(3,8000). The first index of the COORDS array indicates the x-, y-, and z-coordinate of the node. The second index indicates the node number. The second dimension should be increased if there are more than 8000 nodes in a model. Components of the eigenvector are stored in the local array DISP(6,8000). This array holds up to 6 displacement terms for each node. The second dimension should be increased if there are more than 8000 nodes in a model. Subroutine NODEGEN, a subroutine local to this postprocessing program, is 11-1200

Postprocessing of ABAQUS Results Files

then called to compute the new nodal coordinates. Once all the requested mode shapes are computed, the new nodal coordinates are written to the plain text file opened earlier.

Program compilation and linking The ABAQUS/Make procedure is designed to compile and link this type of postprocessing program. It will also make the aba_param.inc file available during compilation. The ABAQUS/Make command to compile and link the FPERT program is as follows: abaqus make job=fpert

This command will have to be repeated if FORTRAN errors are discovered during the compilation or link. The commands used by the ABAQUS/Make procedure can be changed if necessary. The ABAQUS Site Guide lists the typical compile and link commands for each computer type.

Program execution Before the program is executed, an eigenvalue buckling job must have been run with ABAQUS. In this example the input file fpert001.inp is used to generate the results file fpert001.fil. When the FPERT program is executed using the command abaqus fpert, the first prompt will be Enter the name of the results file (w/o .fil):

Enter fpert001 to define FNAME. The second prompt will be Enter the mode shape(s) to be used in calculating the perturbed mesh (zero when finished):

Enter 1 followed by 0, since this is the only eigenvector available in the results file for this example. At the third prompt, Enter the imperfection factor to be introduced into the geometry for this eigenmode:

enter 0.25. This sets ® = 0.25, the shell thickness for this model. The program then processes the data and writes the nodal coordinates for the new mesh to fpert002.015.

Analysis description For a full discussion of the analysis, refer to ``Buckling of a cylindrical shell under uniform axial pressure,'' Section 1.2.3 of the ABAQUS Benchmarks Manual. The input file fpert001.inp (same file as bucklecylshell_s9r5_n3.inp) contains a 2 ´ 20 mesh of S9R5 elements and data lines for a buckling analysis. The input file fpert002.inp contains data lines for a Riks analysis using a perturbed mesh. The source code for the FPERT program is in fpert.f.

Results and discussion Plots produced by these analyses are shown in Figure 11.1.4-1 and Figure 11.1.4-2. Figure 11.1.4-1 is obtained from the eigenvalue buckling analysis and shows the original (cylindrical) mesh and the critical buckling mode. Figure 11.1.4-2 is generated when the load level has reached a local maximum (increment 8) in the Riks analysis using the perturbed mesh.

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Postprocessing of ABAQUS Results Files

Input files fpert001.inp Eigenvalue buckling analysis. fpert002.inp Riks analysis using a perturbed mesh. fpert.f Postprocessing program.

Figures Figure 11.1.4-1 Undeformed shape and eigenvalue buckling mode.

Figure 11.1.4-2 Deformed shape at first peak load in Riks analysis.

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Postprocessing of ABAQUS Results Files

Sample listings

11-1203

Postprocessing of ABAQUS Results Files

Listing 11.1.4-1 SUBROUTINE HKSMAIN C==================================================================== C This program must be compiled and linked with the command: C abaqus make job=fpert C Run the program using the command: C abaqus fpert C==================================================================== C C PURPOSE: C This program computes a perturbed mesh based on a user-specified C perturbation factor. The original coordinate data and C eigenvectors are read from an ABAQUS results (.fil) file. C C PROMPTS: C 1. `Enter the name of the results file (w/o .fil):' C 2. `Enter the mode shape(s) to be used in calculating the C perturbed mesh (zero when finished):' C 3. `Enter the imperfection factor to be introduced into the C geometry for this eigenmode:' C C==================================================================== C C INPUT FILE -- `FNAME'.fil C C OUTPUT FILE -- fpert002.015 C C==================================================================== C C The use of ABA_PARAM.INC eliminates the need to have different C versions of the code for single and double precision. C ABA_PARAM.INC defines an appropriate IMPLICIT REAL statement and C sets the value of NPRECD to 1 or 2, depending on whether the C machine uses single or double precision. C C==================================================================== C ARRAY = Described in Section 7.0.0 of the Verification manual C JRRAY = Described in Section 7.0.0 of the Verification manual C LRUNIT = Described in Section 7.0.0 of the Verification manual C DISP = Contains the eigenvector for a particular eigenmode C COORD = Original coordinate data C INODE = Original node label

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C IDOF = DOF for the element C JEIGNO = Array of mode shapes used for calculating the perturbed C mesh C FNAME = Name of the results file C NODEMAX = Number of nodes in the model C IELMAX = Number of elements in the model C==================================================================== INCLUDE 'aba_param.inc' DIMENSION ARRAY(513), JRRAY(NPRECD,513), LRUNIT(2,1) EQUIVALENCE (ARRAY(1), JRRAY(1,1)) C=================================================================== C ITOTAL must be greater than or equal to the number of nodes in the C model C=================================================================== PARAMETER (ITOTAL = 8000) C C==================================================================== C DIMENSION DISP(6,ITOTAL), COORD(3,ITOTAL) DIMENSION INODE(ITOTAL), IDOF(30), JEIGNO(10) CHARACTER FNAME*80,OUTFILE*(*) PARAMETER (OUTFILE = 'fpert002.015') C C==================================================================== C Define flags and counters. C C==================================================================== ICYCLE = 0 I1901 = 0 I101 = 0 I = 1 K = 1 C C==================================================================== C Define file access variables. C C==================================================================== NRU = 1 LRUNIT(1,NRU) = 8 LRUNIT(2,NRU) = 2 LOUTF = 0 C C====================================================================

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Postprocessing of ABAQUS Results Files

C Open output file. C C==================================================================== OPEN(UNIT=15,FILE=OUTFILE,STATUS='UNKNOWN',IOSTAT = J) IF (J .NE. 0) THEN WRITE(*,900) OUTFILE GOTO 950 ENDIF C C==================================================================== C Get the name of the results (.fil) file. C C==================================================================== WRITE(*,2000) WRITE(6,*) ' Enter the name of the results file (w/o .fil):' READ(5,'(A)', IOSTAT = J ) FNAME IF (J .NE. 0) GOTO 950 C C==================================================================== C Access ABAQUS libraries to set up input file. C C==================================================================== CALL INITPF (FNAME, NRU, LRUNIT, LOUTF) C JUNIT = LRUNIT(1,NRU) C CALL DBRNU (JUNIT) C C==================================================================== C Read a record from the input file. C On the first pass through the file obtain the number of nodes for C a diagnostic check. C==================================================================== CALL DBFILE (0, ARRAY, JRCD) DO WHILE (JRCD .EQ. 0) IF (JRRAY(1,2) .EQ. 1980) IEIGNO = JRRAY(1,3) IF (JRRAY(1,2) .EQ. 1921 ) THEN NODEMAX = JRRAY(1,8) IELMAX = JRRAY(1,7) ICYCLE = ICYCLE +1 ENDIF CALL DBFILE (0, ARRAY, JRCD) ENDDO

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Postprocessing of ABAQUS Results Files

C CALL DBFILE (2, ARRAY, JRCD) C=================================================================== C User is given a choice of eigenmodes for calculating the perturbed C mesh. C C=================================================================== WRITE(*,2010) NODEMAX, IELMAX WRITE(*,2015) IEIGNO 5 READ(5,*,ERR = 950) JEIGNO(I) IF (JEIGNO(I) .EQ. 0) GOTO 10 I=I+1 GOTO 5 C 10 CONTINUE CALL DBFILE (0, ARRAY, JRCD) C DO WHILE (JRCD .EQ. 0) C C==================================================================== C If this is the first pass through the file and the current record C is the nodal coordinate record, then read the original nodal C coordinates and the node numbers. Make sure that the third C coordinate exists before saving it. C C==================================================================== IF (JRRAY(1,2) .EQ. 1901 .AND. ICYCLE .LE. 1) THEN I1901 = I1901 + 1 INODE(I1901) = JRRAY(1,3) COORD(1,I1901) = ARRAY(4) COORD(2,I1901) = ARRAY(5) COORD(3,I1901) = 0.0D0 IF (JRRAY(1,1) .GE. 6) COORD(3,I1901) = ARRAY(6) C C==================================================================== C If this is the first pass through the file and the current record C is the active degree of freedom record, save the active d.o.f. C If the d.o.f. is active in the model, IDOF(XX) equals the C position of d.o.f. XX in the output arrays. If the d.o.f. is not C active, IDOF(XX) is zero for d.o.f. XX (i.e., for planar models C IDOF(1) = 1, IDOF(2) = 2, IDOF(3) = 0, IDOF(4) = 0, IDOF(5) = 0, C IDOF(6) = 3, etc.). ITRANS equals the number of translational C d.o.f.'s in the model.

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C C==================================================================== ELSE IF (JRRAY(1,2) .EQ. 1902) THEN DO 15 IXX = 1, JRRAY(1,1)-2 IDOF(IXX) = JRRAY(1,IXX+2) 15 CONTINUE ITRANS = 3 IF (IDOF(3) .EQ. 0) ITRANS = 2 C C==================================================================== C If the current record is the modal record, save the current C eigenvalue number. C C==================================================================== C ELSE IF (JRRAY(1,2) .EQ. 1980) THEN IEIGNO = JRRAY(1,3) DO J = 1, I-1 IF (JEIGNO(J) .EQ. IEIGNO) K = J ENDDO C C==================================================================== C If the current record is the displacement record and the current C eigenvalue was requested, read the displacement data. The data C will be in the coordinate system specified in the C `*NODE FILE,GLOBAL=' option. If nodal transformations were C performed and GLOBAL=NO was used, the displacements will be in C the local system. If nodal transformations were used and C GLOBAL=YES, the results will be in the global system. In all C other cases the results will be in the global system. Also, C make sure that degrees of freedom are active in the model before C saving them in the appropriate array location. C C==================================================================== C ELSE IF (JRRAY(1,2) .EQ. 101 .AND. IEIGNO .EQ. JEIGNO(K)) THEN I101 = I101 + 1 DISP(1,I101) = ARRAY(4) DISP(2,I101) = ARRAY(5) IF (IDOF(3) .NE. 0) DISP(3,I101) = ARRAY(IDOF(3)+3) IF (IDOF(4) .NE. 0) DISP(4,I101) = ARRAY(IDOF(4)+3) IF (IDOF(5) .NE. 0) DISP(5,I101) = ARRAY(IDOF(5)+3) IF (IDOF(6) .NE. 0) DISP(6,I101) = ARRAY(IDOF(6)+3)

11-1208

Postprocessing of ABAQUS Results Files

16 1

IF (INODE(I101) .EQ. 0) INODE(I101) = JRRAY(1,3) IF( I101 .EQ. NODEMAX ) THEN WRITE(6,16) JEIGNO(K) FORMAT(/,2X,'Nodal coordinate data being computed for', ' eigenvalue . . .',I5)

C C==================================================================== C FACTOR should be entered as a perturbation factor in terms of a C percentage value multiplied by a geometric parameter C (e.g., shell thickness) C==================================================================== C WRITE(6,2020) READ(5,*,IOSTAT = J) FACTOR IF (J .NE. 0) GOTO 950 ICYCLE = ICYCLE + 1 I101 = 0 C C==================================================================== C Compute the perturbed mesh. This section assumes that nodal C displacements are in the GLOBAL coordinate system. If they are C not, the correct transformations should be applied prior to C perturbing the mesh. The user should supply this coding. Also, C only the translational degrees of freedom should be used for the C perturbation (X = Xo + u). C==================================================================== C==================================================================== C Subroutine NODEGEN is the actual mesh generator. C==================================================================== CALL NODEGEN(COORD,DISP,I1901,FACTOR) ENDIF ENDIF C CALL DBFILE(0, ARRAY, JRCD) ENDDO C C==================================================================== C Next line is added for diagnostics C C==================================================================== IF (NODEMAX .NE. I1901) THEN WRITE(*,2025) NODEMAX,I1901 GOTO 950

11-1209

Postprocessing of ABAQUS Results Files

ENDIF C

30

IF (ICYCLE .LE. 1) THEN WRITE(*,30) FORMAT(2X,'. . . NO EIGENVECTORS WERE FOUND . . .',/ 1 2X,'The input file for the buckling analysis must contain:', 2 /,2X,'--> *NODE FILE U Unit number of input file. C lrunit(2,*) --> Format of input file. C loutf -- Format of output file: C 0 --> Standard ASCII format. C 1 --> ABAQUS results file ASCII format. C 2 --> ABAQUS results file binary format. C junit -- Unit number of file to be opened.

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C jrcd -- Error check return code. C .EQ. 0 --> No errors. C .NE. 0 --> Errors detected. C key -- Current record key identifier. C jelnum -- Current element number. C intpn -- Integration point number. C C================================================================== C C The use of ABA_PARAM.INC eliminates the need to have different C versions of the code for single and double precision. C ABA_PARAM.INC defines an appropriate IMPLICIT REAL statement C and sets the value of NPRECD to 1 or 2, depending on whether C the machine uses single or double precision. ABA_PARAM.INC is C referenced from the \site (for NT) or /site (for Unix) C ABAQUS release subdirectory when ABAQUS/Make is executed. C C================================================================== C include 'aba_param.inc' parameter (melem=10000,mnode=10000,mkey=3000,mintpt=360) dimension array(513),jrray(nprecd,513),lrunit(2,1) equivalence (array(1),jrray(1,1)) C C================================================================== C character fname*80 C C SOME DIMENSION STATEMENTS MAY BE NEEDED IF YOU ARE DOING ADDITIONAL C CALCULATIONS ON THE DATA C dimension var(513,melem),coords(3,mintpt) dimension iconn(3,melem),loc_el(melem) dimension naxipt(melem),iglb_nod(mnode),loc_nod(mnode) dimension islct(mkey),nintpt(melem),nodel(melem),iglb_el(melem) dimension xyz(3,mnode),xpos(mnode),tang1(3),tang2(3) dimension rot(3,3),xc(3),xpr(3,mintpt),scord(3,mintpt),xnorm(3) dimension isctchk(melem),icrdchk(melem),ivarchk(melem) C C================================================================== C C GET THE NAME OF THE RESULTS FILE.

11-1221

Postprocessing of ABAQUS Results Files

C C================================================================== C isect1 = 0 intpt1 = 0 icomp = 1 C write(6,*) 'Enter the name of the input file (w/o .fil):' read(5,'(A)') fname C 100 write(6,*) 'Enter the postprocessing option:' write(6,*) ' 1 - variation along the riser ' write(6,*) ' 2 - variation around the circumference' write(6,*) ' of the elbow' write(6,*) ' 3 - ovalization of elbow cross-section' read(5,*) ipost if (ipost .ne. 1 .and. ipost .ne. 2 .and. & ipost .ne. 3) then write(6,*)'Invalid entry - try again' go to 100 endif C write(6,*)'Enter the first element number' read(5,*) jel_1 C if (ipost .ne. 1) then jel_2 = jel_1 else write(6,*)'Enter the last element number' read(5,*) jel_2 endif if (jel_2 .lt. jel_1) then write(6,*)'last elem number must be greater than first' return endif jdiff = jel_2 - jel_1 + 1 if (jdiff .gt. melem) then write(6,*)'Too many elements - max is',melem return endif C if (ipost .lt. 3) then

11-1222

Postprocessing of ABAQUS Results Files

write(6,*)'Enter the section point number' read(5,*) isect1 endif C if (ipost .eq. 1) then write(6,*)'Enter the integration point number' read(5,*) intpt1 endif C if (ipost .lt. 3) then 998 999

write(6,999) format('Enter the key for the variable you wish to process:')

C write(6,500) 500 format(2x,'TEMP ...... 2',/, & 2x,'COORD...... 8',/, & 2x,'S.......... 11',/, & 2x,'SINV....... 12',/, & 2x,'SF......... 13',/, & 2x,'E.......... 21',/, & 2x,'PE......... 22',/, & 2x,'CE......... 23',/, & 2x,'IE......... 24',/, & 2x,'EE......... 25',/, & 2x,'THE........ 88',/, & 2x,'LE......... 89',/, & 2x,'NE......... 90',/, & 2x,'SP.........401',/, & 2x,'EP.........403',/, & 2x,'NEP........404',/, & 2x,'LEP........405',/, & 2x,'EEP........408',/, & 2x,'IEP........409',/, & 2x,'THEP.......410',/, & 2x,'PEP........411',/, & 2x,'CEP........412') C read(5,*) key C if (key .ne. 2 .and. key .ne. 8 .and. & key .ne. 11 .and. key .ne. 12 .and. & key .ne. 13 .and. key .ne. 21 .and.

11-1223

Postprocessing of ABAQUS Results Files

& & & & & & & &

key key key key key key key key

.ne. .ne. .ne. .ne. .ne. .ne. .ne. .ne.

22 .and. key .ne. 23 .and. 24 .and. key .ne. 25 .and. 88 .and. key .ne. 89 .and. 90 .and. key .ne. 401 .and. 403 .and. key .ne. 404 .and. 405 .and. key .ne. 408 .and. 409 .and. key .ne. 410 .and. 411 .and. key .ne. 412) then

C write(6,*)'Invalid key - try again' go to 998 C endif C C C C

Integration point coordinates available only at middle surface Overwrite the section point and set to the default if (key .eq. 8) isect1 = 0

C if (key .ge. 8) then write(6,*)'Enter the attribute number' read(5,*) icomp endif C if (key .eq. 13) then write(6,*)'Section force data written only once per element' if (ipost .eq. 1) then write(6,*)'Will use default integration & section point' write(6,*)' ' intpt1 = 1 isect1 = 0 elseif (ipost .gt. 1) then write(6,*)'Only use with option 1: Processing terminated' return endif endif C endif C write(6,*)' Enter the step number' read(5,*) istep1 write(6,*)' Enter the increment number' read(5,*) iinc1

11-1224

Postprocessing of ABAQUS Results Files

C C SELECT THE RECORDS TO BE PROCESSED C C IF THE ISLCT ARRAY IS SET TO 1 THE DATA WILL BE PROCESSED C IF THE ISLCT ARRAY IS SET TO 0 THE DATA WILL NOT BE PROCESSED C do ii = 1, mkey islct(ii) = 0 end do C C ELEMENT DEFINITION islct(1900)=0 C NODE DEFINITION islct(1901)=0 C INCREMENT START islct(2000)=0 C ELEMENT HEADER islct(1)=0 C COORD islct(8)=0 C if (ipost .lt. 3) islct(key) = 1 keyprv=0 C nels = 0 numnp = 0 C itimchk = 0 ielchk = 0 do i = 1, melem loc_el(i) = 0 isctchk(i) = 0 icrdchk(i) = 0 ivarchk(i) = 0 do j = 1, 513 var(j,i) = 0.0 end do end do C C================================================================== C C OPEN THE OUTPUT FILE. C

11-1225

Postprocessing of ABAQUS Results Files

C================================================================== open(unit=9,file='output.dat',status='unknown') C nru = 1 loutf = 0 lrunit(1,1) = 8 lrunit(2,1) = 2 C call initpf(fname,nru,lrunit,loutf) C junit = 8 C call dbrnu(junit) C C REWIND FILE C call dbfile(2,array,jrcd) C C C================================================================== C C Read records from the results (.fil) file and process the data. C Cover a maximum of 10 million records in the file. C C================================================================== C do 1000 k100 = 1, 100 do 1000 k1 = 1, 99999 C C READ RECORD FROM .FIL FILE AND PROCESS DATA. C call dbfile(0, array, jrcd) if (jrcd .ne. 0 .and. keyprv .eq. 2001) then write(6,*)'End of file' close(junit) go to 1001 else if (jrcd .ne. 0) then write(6,*)'Error reading input file' return endif C C LENGTH AND KEY RECORD NUMBER ARE ALWAYS NEEDED

11-1226

Postprocessing of ABAQUS Results Files

C lenf= jrray(1,1) key = jrray(1,2) C C SPECIFIC RECORD NUMBERS FOLLOW... C C================================================================== C C ELEMENT DEFINITIONS C C================================================================== if ( key .eq. 1900 ) then jelnum=jrray(1,3) if (jelnum .gt. melem) then write(6,*)'Maximum global elem number too large' write(6,*)'Increase melem' return endif ctype= array(4) if (jelnum .ge. jel_1 .and. & jelnum .le. jel_2) then ielchk = 1 nels = nels + 1 jel = nels iglb_el(jel) = jelnum loc_el(jelnum) = jel nodel(jel) = lenf - 4 naxipt(jel) = 1 if (ctype .ne. 'ELBOW31' .and. & ctype .ne. 'ELBOW31B' .and. & ctype .ne. 'ELBOW31C' .and. & ctype .ne. 'ELBOW32') then write(6,*)'Invalid element type encountered' return endif if (ctype .eq. 'ELBOW32')naxipt(jel) = 2 do kk=5,lenf ii=kk-4 iconn(ii,jel)=jrray(1,kk) end do endif keyprv = key goto 9099

11-1227

Postprocessing of ABAQUS Results Files

C C================================================================== C C NODE DEFINITIONS C C================================================================== else if ( key .eq. 1901 ) then jnod = jrray(1,3) if (jnod .gt. mnode) then write(6,*)'Maximum global node number exceeded' write(6,*)'Increase mnode' return endif numnp = numnp + 1 iglb_nod(numnp) = jnod loc_nod(jnod) = numnp xyz(1,numnp) = array(4) xyz(2,numnp) = array(5) xyz(3,numnp) = array(6) keyprv = key goto 9099 C C================================================================== C C CURRENT STEP AND INCREMENT NUMBER. C C================================================================== else if ( key .eq. 2000 ) then ttime=array(3) stime=array(4) sfreq=array(12) istep=jrray(1,8) iinc =jrray(1,9) C C SET THE FLAG ITIME IF THIS IS THE REQUESTED STEP/INC C if (istep .eq. istep1 .and. & iinc .eq. iinc1) then itime = 1 itimchk = 1 else itime = 0

11-1228

Postprocessing of ABAQUS Results Files

endif c keyprv = key goto 9099 C C================================================================== C C ELEMENT VARIABLES AT INTEGRATION POINTS WITHIN THE ELEMENT C C================================================================== C C ELEMENT DATA C else if ( key .eq. 1 ) then C C C

PROCESS THIS RECORD IF THIS IS THE CORRECT STEP/TIME if (itime .eq. 1) then jelnum = jrray(1,3) jel = loc_el(jelnum) ielem = 0

C C C

PROCESS THIS ELEMENT IF IT IS IN THE LIST OF DESIRED ELEMS if (jel .gt. 0) then ielem = 1 intpn = jrray(1,4) if (intpn .gt. mintpt) then write(6,*)'Max number of int points exceeded' write(6,*)'Increase mintpt' return endif isect = jrray(1,5) ilocn = jrray(1,6) if (ilocn .ne. 0) then write(6,*)'Element data must be at the int points' return endif crbar = array(7) ndi = jrray(1,8) nshr = jrray(1,9) ndir = jrray(1,10)

11-1229

Postprocessing of ABAQUS Results Files

&

nsfc = jrray(1,11) if (isect .eq. isect1)then isctflg = 1 isctchk(jel) = 1 else isctflg = 0 endif if ( (ipost .eq.1 .and. intpn .eq. intpt1) .or. ipost .gt. 1) then iptflg = 1 else iptflg = 0 endif endif endif keyprv = key goto 9099

C C================================================================== C C COORDINATES C C================================================================== else if ( key .eq. 8 .and. ipost .gt. 1) then C C PROCESS IF THIS IS THE CORRECT STEP/INC, ELEMENT, SECTION C POINT, AND INTEGRATION POINT C icheck = itime*ielem*iptflg if (icheck .eq. 1 ) then c if (ipost .eq. 3) nintpt(jel) = intpn icrdchk(jel) = 1 if (ipost .gt. 1) nintpt(jel) = intpn do kk=3,lenf ii=kk-2 coords(ii,intpn)=array(kk) end do C if (islct(key) .eq. 1 .and. ipost .lt. 3) then ivarchk(jel) = 1 jvar = intpn nintpt(jel) = intpn do kk=3,lenf

11-1230

Postprocessing of ABAQUS Results Files

ii=kk-2 var(ii,jvar)=array(kk) end do if (icomp .gt. ii) then write(6,*)'Invalid component' return endif endif C endif keyprv = key goto 9099 C C================================================================== C C VARIABLE RECORD C C================================================================== C PROCESS IF THIS IS THE DESIRED VARIABLE else if (islct(key) .eq. 1 .and. ipost .lt. 3) then C C C C

CONTINUE PROCESSING IF THIS IS THE CORRECT STEP/INC, ELEMENT, SECTION POINT, AND INTEGRATION POINT icheck = itime*ielem*isctflg*iptflg if (icheck .eq. 1 ) then ivarchk(jel) = 1 if (ipost .eq. 1) then jvar = jel else jvar = intpn endif nintpt(jel) = intpn do kk=3,lenf ii=kk-2 var(ii,jvar)=array(kk) end do if (icomp .gt. ii) then write(6,*)'Invalid component' return endif endif

11-1231

Postprocessing of ABAQUS Results Files

keyprv = key goto 9099 C C================================================================== C C ANYTHING ELSE C C================================================================== else c write (6,9000) key 9000 format(I5,' *** NO CODING FOR THIS RECORD ***') keyprv = key 9099 end if C C================================================================== C C END LOOPS C C================================================================== 1000 continue 1001 continue C C================================================================== C POSTPROCESS DATA HERE C================================================================== do ii = 1, nels nintpt(ii) = nintpt(ii)/naxipt(ii) end do C C================================================================== C VERIFY THAT APPROPRIATE DATA WERE WRITTEN TO RESULTS FILE C================================================================== c iquit = 0 if (ielchk .eq. 0) then write(6,*)'Desired element not found' return endif C if (itimchk .eq. 0) then write(6,*)'Desired step/increment not found' return endif C

11-1232

Postprocessing of ABAQUS Results Files

do iel = 1, nels if (isctchk(iel) .eq. 0 .and. ipost .lt. 3) then write(6,*)'Desired element and/or section point not found' return endif if (icrdchk(iel) .eq. 0 .and. ipost .gt. 1) then write(6,*)'Integration point coords not found' return endif if (ivarchk(iel) .eq. 0 .and. ipost .lt. 3) then write(6,*)'Desired element variable not found' return endif end do C C**************************************** C C PLOT VARIABLE ALONG ELBOW LENGTH C C**************************************** C if (ipost .eq. 1) then C nn_old = iconn(nodel(1),1) do iel = 1, nels nn_new = iconn(1,iel) if (iel .gt. 1 .and. nn_new .ne. nn_old) then write(6,*)'Error in element connectivity' return endif nnum_b = iconn(1,iel) nnum_b = loc_nod(nnum_b) nnum_e = iconn(nodel(iel),iel) nnum_e = loc_nod(nnum_e) if (iel .eq. 1) xpos(nnum_b) = 0.0 xlength = 0.d+0 do jj = 1, 3 delx = xyz(jj,nnum_e) - xyz(jj,nnum_b) xlength = xlength + delx*delx end do xlength = sqrt(xlength) xpos(nnum_e) = xpos(nnum_b) + xlength x_mid = 0.5*(xpos(nnum_e) + xpos(nnum_b))

11-1233

Postprocessing of ABAQUS Results Files

C C C

EXTRACT VARIABLE INFORMATION AND WRITE TO FILE

2000

data = var(icomp,iel) write(9,2000)x_mid,data format(5x,e17.9,5x,e17.9)

C nn_old=iconn(nodel(iel),iel) C end do return C C C**************************************** C C PLOT VARIABLE AROUND CIRCUMFERENCE OF ELBOW C C**************************************** C else if (ipost .eq. 2) then C xpos(1) = 0.0 C locel = loc_el(jel_1) iaxial = 1 C if (naxipt(locel) .eq. 2) then 3000 write(6,*)'Enter the axial station' read(5,*)iaxial if (iaxial .ne. 1 .and. iaxial .ne. 2) then write(6,*)'Try again - station must be 1 or 2' go to 3000 endif endif C do ipt = 1 , nintpt(locel) jpt = ipt + (iaxial - 1)*nintpt(locel) if (ipt .gt. 1) then xlength = 0.0 do jj = 1, 3 delx = coords(jj,jpt) - coords(jj,jpt-1) xlength = xlength + delx*delx end do

11-1234

Postprocessing of ABAQUS Results Files

xlength = sqrt(xlength) xpos(ipt) = xpos(ipt-1) + xlength endif C C C

EXTRACT VARIABLE INFORMATION AND WRITE TO FILE data = var(icomp,jpt) write(9,2000)xpos(ipt),data

C end do ipt = 1 jpt = ipt + (iaxial - 1)*nintpt(locel) data = var(icomp,jpt) xfinal = xpos(nintpt(locel)) + (xpos(2)- xpos(1)) write(9,2000)xfinal,data return C C**************************************** C C OVALIZATION PLOT C C**************************************** C else if (ipost .eq. 3) then C locel = loc_el(jel_1) iaxial = 1 C if (naxipt(locel) .eq. 2) then 3500 write(6,*)'Enter the axial station' read(5,*)iaxial if (iaxial .ne. 1 .and. iaxial .ne. 2) then write(6,*)'Try again - station must be 1 or 2' go to 3500 endif endif C C AVERAGE THE COORDS OF THE INTEGRATION POINTS TO GET THE CENTER C do i = 1, 3 xc(i) = 0.0

11-1235

Postprocessing of ABAQUS Results Files

end do C xnint = real(nintpt(locel)) C do ipt = 1 , nintpt(locel) jpt = ipt + (iaxial - 1)*nintpt(locel) C do i = 1, 3 xc(i) = xc(i) + coords(i,jpt)/xnint end do C end do C C C

SHIFT THE COORDINATES SO THAT CENTER OF SECTION IS AT ORIGIN do ipt = 1 , nintpt(locel) jpt = ipt + (iaxial - 1)*nintpt(locel) do i = 1, 3 scord(i,jpt) = coords(i,jpt) - xc(i) end do

C end do C C C C

DETERMINE TANGENT VECTOR 1 (FROM CENTER OF CROSS-SECTION TO INT PT 1) jpt = 1 + (iaxial - 1)*nintpt(locel) xmag = 0.0 do i = 1, 3 tang1(i) = coords(i,jpt) - xc(i) xmag = xmag + tang1(i)*tang1(i) end do xmag = sqrt(xmag) do i = 1, 3 tang1(i) = tang1(i)/xmag end do

C C C C

GUESS TANGENT VECTOR 2 (FROM CENTER OF CROSS-SECTION TO INT PT 2) jpt = 2 + (iaxial - 1)*nintpt(locel) xmag = 0.0 do i = 1, 3

11-1236

Postprocessing of ABAQUS Results Files

tang2(i) = coords(i,jpt) - xc(i) xmag = xmag + tang2(i)*tang2(i) end do xmag = sqrt(xmag) do i = 1, 3 tang2(i) = tang2(i)/xmag end do C C C

DETERMINE THE NORMAL VECTOR, TANG1 X TANG2 xnorm(1) = tang1(2)*tang2(3) - tang1(3)*tang2(2) xnorm(2) = - tang1(1)*tang2(3) + tang1(3)*tang2(1) tang1(1)*tang2(2) - tang1(2)*tang2(1) xnorm(3) = xmag = 0.0 do i = 1, 3 xmag = xmag + xnorm(i)*xnorm(i) end do xmag = sqrt(xmag) do i = 1, 3 xnorm(i) = xnorm(i)/xmag end do

C C C

REDEFINE TANGENT VECTOR 2 tang2(1) = - tang1(2)*xnorm(3) + tang1(3)*xnorm(2) tang2(2) = tang1(1)*xnorm(3) - tang1(3)*xnorm(1) tang2(3) = - tang1(1)*xnorm(2) + tang1(2)*xnorm(1)

C C C

DETERMINE THE ROTATION MATRIX do i = 1, 3 rot(1,i) = tang1(i) rot(2,i) = tang2(i) rot(3,i) = xnorm(i) end do

C C C

TRANSFORM COORDINATES INTO LOCAL SYSTEM do ipt = 1 , nintpt(locel) jpt = ipt + (iaxial - 1)*nintpt(locel) do i = 1, 3 xpr(i,ipt) = 0.0

11-1237

Postprocessing of ABAQUS Results Files

do j = 1, 3 xpr(i,ipt) = xpr(i,ipt) + rot(i,j)*scord(j,jpt) end do end do end do C C C

OUTPUT COORDS TO FILE

4000

do ipt = 1, nintpt(locel) write(9,4000)(xpr(i,ipt),i=1,2) format(5x,e17.9,5x,e17.9,5x,e17.9) end do

C ipt = 1 write(9,4000)(xpr(i,ipt),i=1,2) endif C close (unit=9) C return end

11-1238

Product Index ABAQUS/Standard Section 1.1.1

Axisymmetric analysis of bolted pipe flange connections

Section 1.1.2

Elastic-plastic collapse of a thin-walled elbow under in-plane bending and internal pressure

Section 1.1.3

Parametric study of a linear elastic pipeline under in-plane bending

Section 1.1.4

Indentation of an elastomeric foam specimen with a hemispherical punch

Section 1.1.5

Collapse of a concrete slab

Section 1.1.6

Jointed rock slope stability

Section 1.1.7

Indentation of a crushable foam specimen with a hemispherical punch

Section 1.1.8

Notched beam under cyclic loading

Section 1.1.9

Hydrostatic fluid elements: modeling an airspring

Section 1.1.10

Shell-to-solid submodeling of a pipe joint

Section 1.1.11

Stress-free element reactivation

Section 1.1.12

Symmetric results transfer for a rubber bushing

Section 1.1.13

Transient loading of a viscoelastic bushing

Section 1.1.15

Damage and failure of a laminated composite plate

Section 1.1.16

Analysis of an automotive boot seal

Section 1.1.17

Pressure penetration analysis of an air duct kiss seal

Section 1.1.18

Self-contact in rubber/foam components: jounce bumper

Section 1.1.19

Self-contact in rubber/foam components: rubber gasket

Section 1.1.20

Submodeling of a stacked sheet metal assembly

Section 1.2.1

Snap-through buckling analysis of circular arches

Section 1.2.2

Laminated composite shells: buckling of a cylindrical panel with a circular hole

Section 1.2.4

Elastic-plastic K-frame structure

Section 1.2.5

Unstable static problem: reinforced plate under compressive loads

Section 1.2.6

Buckling of an imperfection sensitive cylindrical shell

Section 1.3.1

Upsetting of a cylindrical billet in ABAQUS/Standard: quasi-static analysis with rezoning

Section 1.3.3

Superplastic forming of a rectangular box

Section 1.3.4

Stretching of a thin sheet with a hemispherical punch

Section 1.3.5

Deep drawing of a cylindrical cup

Section 1.3.6

Extrusion of a cylindrical metal bar with frictional heat generation

Section 1.3.8

Axisymmetric forming of a circular cup

Section 1.3.17

Upsetting of a cylindrical billet: coupled temperature-displacement and adiabatic analysis

Section 1.3.18

Unstable static problem: thermal forming of a metal sheet

0-1239

Section 1.4.1

A plate with a part-through crack: elastic line spring modeling

Section 1.4.2

Conical crack in a half-space with and without submodeling

Section 1.4.3

Elastic-plastic line spring modeling of a finite length cylinder with a part-through axial flaw

Section 1.4.4

Crack growth in a three-point bend specimen

Section 1.5.1

Springback of two-dimensional draw bending

Section 1.5.2

Deep drawing of a square box

Section 2.1.1

Nonlinear dynamic analysis of a structure with local inelastic collapse

Section 2.1.2

Detroit Edison pipe whip experiment

Section 2.1.5

Pressurized fuel tank with variable shell thickness

Section 2.1.6

Modeling of an automobile suspension

Section 2.1.12

Rigid multi-body mechanism

Section 2.2.1

Analysis of a rotating fan using superelements and cyclic symmetry model

Section 2.2.2

Linear analysis of the Indian Point reactor feedwater line

Section 2.2.3

Response spectra of a three-dimensional frame building

Section 3.1.1

Symmetric results transfer for a static tire analysis

Section 3.1.2

Steady-state rolling analysis of a tire

Section 3.1.3

Subspace-based steady-state dynamic tire analysis

Section 4.1.1

Thermally coupled analysis of a disc brake

Section 4.1.2

Exhaust manifold assemblage

Section 4.1.3

Coolant manifold cover gasketed joint

Section 4.1.4

Radiation analysis of a plane finned surface

Section 5.1.1

Eigenvalue analysis of a piezoelectric transducer

Section 5.2.1

Thermal-electrical modeling of an automotive fuse

Section 6.1.1

Hydrogen diffusion in a vessel wall section

Section 6.1.2

Diffusion toward an elastic crack tip

Section 7.1.1

Coupled acoustic-structural analysis of a car

Section 7.1.2

Fully and sequentially coupled structural acoustics of a muffler

Section 8.1.1

Plane strain consolidation

Section 8.1.2

Calculation of phreatic surface in an earth dam

Section 8.1.3

Axisymmetric simulation of an oil well

Section 8.1.4

Analysis of a pipeline buried in soil

Section 9.1.1

Jack-up foundation analyses

Section 9.1.2

Riser dynamics

Section 10.1.1

The cylinder whip problem

Section 11.1.2

Joining data from multiple results files and converting file format: FJOIN

Section 11.1.3

Calculation of principal stresses and strains and their directions: FPRIN

Section 11.1.4

Creation of a perturbed mesh from original coordinate data and eigenvectors: FPERT

0-1240

Section 11.1.5

Output radiation viewfactors and facet areas: FRAD

Section 11.1.6

Creation of a data file to facilitate the postprocessing of elbow element results: FELBOW

ABAQUS/Explicit Section 1.1.4

Indentation of an elastomeric foam specimen with a hemispherical punch

Section 1.1.5

Collapse of a concrete slab

Section 1.1.7

Indentation of a crushable foam specimen with a hemispherical punch

Section 1.1.9

Hydrostatic fluid elements: modeling an airspring

Section 1.1.14

Indentation of a thick plate

Section 1.2.3

Buckling of a column with spot welds

Section 1.3.2

Upsetting of a cylindrical billet in ABAQUS/Explicit

Section 1.3.4

Stretching of a thin sheet with a hemispherical punch

Section 1.3.6

Extrusion of a cylindrical metal bar with frictional heat generation

Section 1.3.7

Rolling of thick plates

Section 1.3.8

Axisymmetric forming of a circular cup

Section 1.3.9

Cup/trough forming

Section 1.3.10

Forging with sinusoidal dies

Section 1.3.11

Forging with multiple complex dies

Section 1.3.12

Flat rolling: transient and steady-state

Section 1.3.13

Section rolling

Section 1.3.14

Ring rolling

Section 1.3.15

Axisymmetric extrusion: transient and steady-state

Section 1.3.16

Two-step forming simulation

Section 1.3.17

Upsetting of a cylindrical billet: coupled temperature-displacement and adiabatic analysis

Section 1.5.1

Springback of two-dimensional draw bending

Section 1.5.2

Deep drawing of a square box

Section 2.1.3

Plate impact simulation

Section 2.1.4

Tennis racket and ball

Section 2.1.7

Explosive pipe closure

Section 2.1.8

Knee bolster impact with double-sided surface contact

Section 2.1.9

Cask drop with foam impact limiter

Section 2.1.10

Oblique impact of a copper rod

Section 2.1.11

Water sloshing in a baffled tank

Section 2.1.12

Rigid multi-body mechanism

Section 4.1.1

Thermally coupled analysis of a disc brake

ABAQUS/Design 0-1241

Section 1.1.4

Indentation of an elastomeric foam specimen with a hemispherical punch

Section 1.1.12

Symmetric results transfer for a rubber bushing

ABAQUS/Aqua Section 9.1.1

Jack-up foundation analyses

Section 9.1.2

Riser dynamics

ABAQUS/USA Section 10.1.1

The cylinder whip problem

0-1242