Semi-rigid Connections Handbook - Wai-Fah Chen
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semi-rigid connections handbook Edited by
Wai-Fah Chen Norimitsu Kishi Masato Komuro
Copyright © 2011 by J. Ross Publishing, Inc. ISBN: 978-1-932159-99-8 Printed and bound in the U.S.A. Printed on acid-free paper 10 9 8 7 6 5 4 3 2 1
Library of Congress Cataloging-in-Publication Data Semi-rigid connections handbook / edited by Wai-Fah Chen, Norimitsu Kishi, and Masato Komuro. p. cm. — (J. Ross Publishing civil & environmental engineering series) Includes bibliographical references and index. ISBN 978-1-932159-99-8 (hardcover : alk. paper) 1. Columns, Iron and steel--Specifications. 2. Steel framing (Building) 3. Deformations (Mechanics) 4. Structural dynamics. I. Chen, Wai-Fah, 1936- II. Kishi, Norimitsu, 1949- III. Komuro, Masato, 1969 TA492.C7S46 2010 624.1’772--dc22 2010035278 This publication contains information obtained from authentic and highly regarded sources. Reprinted material is used with permission, and sources are indicated. Reasonable effort has been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. All rights reserved. Neither this publication nor any part thereof may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher. The copyright owner’s consent does not extend to copying for general distribution for promotion, for creating new works, or for resale. Specific permission must be obtained from J. Ross Publishing for such purposes. Direct all inquiries to J. Ross Publishing, Inc., 5765 N. Andrews Way, Fort Lauderdale, FL 33309. Phone: (954) 727-9333 Fax: (561) 892-0700 Web: www.jrosspub.com
CIVIL AND ENVIRONMENTAL ENGINEERING SERIES Wai-Fah Chen, Editor-in-Chief
Semi-rigid Connections Handbook by Wai-Fah Chen, Norimitsu Kishi, and Masato Komuro Elastic Beam Calculations Handbook by Jih-Jiang Chyu Deepwater Foundations and Pipeline Geomechanics by William O. McCarron Sulfur Concrete for the Construction Industry: A Sustainable Development Approach by Abdel-Mohsen Onsy Mohamed and Maisa El Gamal
Contents
Preface.......................................................................................................................................vii About the Editors.......................................................................................................................ix
Section I. Specifications and Classifications
1 Classification and AISC Specification...................................................................................1 1.1 Classification of Connections......................................................................................................1 1.2 AISC Specification.......................................................................................................................22 References.....................................................................................................................................26
Section II. Effects of Semi-rigid Connections on Structural Members and Frames
2 Effects of Semi-rigid Connections on Structural Members and Frames............................27 2.1 Introduction.................................................................................................................................27 2.2 Effects of Semi-rigid Connections on Columns......................................................................32 2.3 Effects of Semi-rigid Connections on Beams..........................................................................40 2.4 Effects of Semi-rigid Connections on Frames.........................................................................45 2.5 Summary and Conclusions........................................................................................................67 References.....................................................................................................................................68
Section III. Steel Connection Database and Modeling
3 Types of PR Connections.....................................................................................................71 3.1 Single Web-angle Connections/Single Plate Connections.....................................................71 3.2 Double Web-angle Connections................................................................................................71 3.3 Top- and Seat-angle Connections with Double Web-angle...................................................73 3.4 Top- and Seat-angle Connections.............................................................................................73 3.5 Extended End-plate Connections/Flush End-plate Connections.........................................75 3.6 Header Plate Connections..........................................................................................................75 References.....................................................................................................................................77 4 Modeling of Connections.....................................................................................................79 4.1 General Remarks.........................................................................................................................79 4.2 Behavior Under Monotonic Loading........................................................................................80 4.3 Behavior Under Cyclic Loading.................................................................................................89 References.....................................................................................................................................92 5 Steel Connection Database...................................................................................................95 5.1 General Remarks.........................................................................................................................95 5.2 Parameter Definition for Connection Type...........................................................................100 5.3 Steel Connection Databank Program.....................................................................................129 v
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Semi-rigid Connections Handbook
5.4 Web site of Connections...........................................................................................................132 References...................................................................................................................................140
Section IV. Steel-concrete Composite Connections
6 Advanced Analysis of Steel and Composite Semi-rigid Frames........................................143 6.1 Structural Frames......................................................................................................................143 6.2 Advanced Analysis of Steel Frames.........................................................................................154 6.3 Advanced Analysis of Composite Frames..............................................................................211 Appendix 1 Elastic Stiffness Matrix, ke.........................................................................................252 Appendix 2 Geometric Stiffness Matrix, kg..................................................................................253 Appendix 3 Bowing Matrix, kb. .....................................................................................................254
Section V. Case Study
7 Case Studies for Second-order (Direct) Analysis of Semi-rigid Frames in Hong Kong......................................................................................................................255 7.1 Introduction...............................................................................................................................255 7.2 Second-order Integrated Design and Analysis......................................................................256 7.3 Modeling of Semi-rigid Jointed Members..............................................................................257 7.4 Examples.....................................................................................................................................259 7.5 Conclusions................................................................................................................................268 References...................................................................................................................................268 Appendix Experimental Data of SCDB A-1 Single Web Angle/Single Plate Connections..................................................................... A1-1 A-2 Double Web-Angle Connections........................................................................................ A2-1 A-3 Top- and Seat-angle with Double Web-angle Connections............................................. A3-1 A-4 Top- and Seat-angle Connections....................................................................................... A4-1 A-5 Extended End-plate Connections....................................................................................... A5-1 A-6 Flush End-plate Connections.............................................................................................. A6-1 A-7 Header End-plate Connections........................................................................................... A7-1 Index.........................................................................................................................................I-1
Preface
The steel framework is one of the most commonly used structural systems in modern construction. The analysis of such structural systems is governed by the assumptions adopted in the modeling of their structural elements, especially those concerning the behavior of beam-to-column connections. Conventional analysis and design of this framework are performed using the assumption of a fully rigid or ideally pinned connection. The assumption of a fully rigid connection implies that no relative rotation of the connection occurs and the end moment of the beam is completely transferred to the column. On the other hand, the pinned connection implies that no restraint for rotation of the connection exists and the connection moment is always zero. The popularity of these idealized models results from the fact that they are simple to use and easy to implement in the analysis and design of steel frameworks. However, as evident from experimental observations, all beam-to-column connections used in current practice possess some stiffness and fall between the two extreme cases of fully rigid and ideally pinned. In general, the connection that welds the beam directly to the flange of the column is considered fully rigid. However, the connection that fastens the beam to the column with some top and bottom angles and a web plate or with all angles, using either bolts or rivets, (with some angles and/or a plate, bolts or rivets,) displays a nonlinear behavior and lies somewhere between the fully rigid and perfectly pinned conditions, commonly called semi-rigid connections. The nonlinear semi-rigid connection behaviors can be seen clearly in many experimental results. On the other hand, single and double web-angles and header plate connections generally have much less connection stiffness and may be treated as pinned connections. The end-plate connection has connection stiffness close to that of a rigid condition, while the top- and seat-angle connection lies somewhere in between these two extremes and is usually treated as a semi-rigid connection. The recent development of the limit-state approach to design has focused particular attention on the limits of resistance and serviceability. To this end, we need accurate information regarding the behavior of structures throughout the entire range of loading up to ultimate load considering the characteristics of connection stiffness. Research on the topic of steel frames with semi-rigid connections (Partially Restrained (PR) construction or PR connection) has been conducted over the past 20 years. Despite significant research and development efforts, usage of PR principles has nevertheless been slow in coming to the profession caused in part by the lack of easy access to reliable test data on these connections and also due to the lack of software for practical implementation. With the publication of the 2005 AISC specifications as well as Eurocode 3, practical implementation of the use of PR connections in structural systems is now a real possibility. This handbook presents a simple and comprehensive introduction that will help design practitioners implement these new developments into engineering practice. Beginning with a discussion of the new specifications and classifications of these connections, we go on to show (on the basis of the collected connections database) practical mathematical models for computer implementation and provide case studies on these frames including composite construction. With the help of the user-friendly list of collected data in tabular vii
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Semi-rigid Connections Handbook
form with illustrative figures, information on semi-rigid connections is now available in a single publication and may ultimately result in its widespread usage among practitioners. The scope of the book is indicated by the table of contents with the following features: • Introduces the 2005 AISC specifications with a cross reference to the counterpart of the new Eurocode 3 on semi-rigid construction • Includes more than 700 semi-rigid connection test data in tabular form with figures • Provides connection models for analysis and design with case studies • Includes the recent development of steel-concrete composite connections with case studies We are grateful to the following contributors for their advice, help, and original work at various stages during the preparation of the book: S. L. Chan, Y. Goto, J. Y. R. Liew, and E. Lui. W. F. Chen N. Kishi M. Komuro
About the Editors
Dr. Wai-Fah Chen is a well-respected leader in the field of plasticity, structural stability, and structural steel design over the past half century. Having headed the engineering school and department at the University of Hawaii and Purdue University, Chen is a widely cited author for his contributions in the fields of mechanics, materials, and computing. He is the author or coauthor of 20 engineering books and the recipient of several national engineering awards, including the 1990 Shortridge Hardesty Award from the American Society of Civil Engineers and the 2003 Lifetime Achievement Award from the American Institute of Steel Construction. He is a member of the U.S. National Academy of Engineering and an honorary member of the American Society of Civil Engineers. Dr. Norimitsu Kishi received his Ph.D. in civil engineering from Hokkaido University-Japan in 1977. From March 1985 to March 1986, he was an overseas researcher at Purdue University under Professor W. F. Chen. He is currently professor and director of the Structural Mechanics Laboratory in the Department of Civil Engineering and Architecture at the Muroran Institute of Technology-Japan. Dr. Masato Komuro graduated with a degree in civil engineering in 1992 from the Muroran Institute of Technology-Japan, where he also obtained his Ph.D. in structural engineering in 2001. From May 2007 to April 2008, he was a visiting researcher in the Department of Civil & Environmental Engineering at the University of Hawaii-Manoa. He is currently assistant professor in the Department of Civil Engineering and Architecture at the Muroran Institute of Technology.
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Contributors S. L. Chan Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Hong Chen Offshore Technology Development Pte Ltd, Keppel Offshore & Marine, Singapore Yoshiaki Goto Dr. Eng. Professor, Department of Civil Engineering, Nagoya Institute of Technology, Japan Dennis Lam Department of Civil Engineering, University of Leeds, U.K. J. Y. Richard Liew Department of Civil Engineering, National University of Singapore, Singapore Y. P. Liu Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Eric M. Lui Meredith Professor, Department of Civil and Environmental Engineering, Syracuse University, NY Zhiling Zhang Graduate Student, Department of Civil and Environmental Engineering, Syracuse University, NY
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At J. Ross Publishing we are committed to providing today’s professional with practical, hands-on tools that enhance the learning experience and give readers an opportunity to apply what they have learned. That is why we offer free ancillary materials available for download on this book and all participating Web Added Value™ publications. These online resources may include interactive versions of material that appears in the book or supplemental templates, worksheets, models, plans, case studies, proposals, spreadsheets, and assessment tools, among other things. Whenever you see the WAV™ symbol in any of our publications, it means bonus materials accompany the book and are available from the Web Added Value™ Download Resource Center at www.jrosspub. com. Downloads for Semi-rigid Connections Handbook consist of updated connection data, the Steel Connection Data Bank (SCDB) program, references, and a link to the semi-rigid website at the Muroran Institute of Technology, Japan.
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1
Classification and AISC Specification
Yoshiaki Goto Dr. Eng. Professor, Department of Civil Engineering, Nagoya Institute of Technology, Japan
1.1 Classification of Connections...................................................................................................1 General • Eurocode 3 Classification System (CEN 2005) • AISC Classification System (AISC 2005) • Bjorhovde et al. Classification System • Nethercot et al. Classification System • Goto et al. Classification System • Comparison of Existing Classification Systems 1.2 AISC Specification.................................................................................................................. 22 Introduction • Connection Classification • Structural Analysis and Design for Frames with PR Connections References......................................................................................................................................... 26
1.1 Classification of Connections 1.1.1 General
A beam-to-column connection is generally subject to axial force, shear force, and bending moment for its in-plane behavior. However, the deformation of the connection caused by axial and shear forces are usually small when compared to the deformation caused by bending moment. Consequently, for practical purposes, only the effect of moment on the rotational deformation of connections shown in Figure 1.1 needs to be considered. The in-plane behavior of connections can be represented by the moment-rotation (M–θr ) curves illustrated in Figure 1.2. All types of actual beam-to-column connections possess some rotational stiffness that falls between the two extreme cases of fully rigid and ideally pinned. Thus, the modeling of connections as semi-rigid is more realistic. However, in engineering practice, some connections can be considered pinned if their stiffness is so small that the connections are incapable of transmitting any significant moment, thus permitting almost free rotation. Similarly, some connections can be considered rigid if their rigidity is so large that no significant slope discontinuity exists between the adjoining members. The assumption of ideally pinned or rigid connections considerably simplifies the design and analysis procedures of framed structures. For that reason, it is beneficial and practical to estimate whether the connec-
1
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Semi-rigid Connections Handbook
θr
Column Beam
M
FIGURE 1.1 Rotational deformation of a connection.
tions can be assumed rigid, semi-rigid, or pinned. The classification of connections should be made based on the behavior of the frames at serviceability and ultimate-limit states. Primary index properties that govern the moment-rotation characteristics of connections are stiffness, strength, and rotational capacity. These properties are important factors that are used for the classification. However, the requirement for strength and rotational capacity of connections is not always necessary. This is because the strength and rotational capacity of connections are determined during the design procedure, such that they exceed the calculated moment and rotation demand at the ultimate-limit state of frames. The classification systems of connections have been presented by Bjorhovde et al. (1990), Goto et al. (1998), Nethercot et al. (1998), Eurocode 3 (CEN, 2005), and ANSI/AISC360-06 (AISC, 2005). The following explains and discusses these existing classification systems.
1.1.2 Eurocode 3 Classification System (CEN 2005)
Eurocode 3 (2005) models connections as simple, semicontinuous, and continuous in structural analysis. In the simple connection model, a connection is assumed not to transmit bending moment and may be modeled as ideally pinned. In the semicontinuous connection model, the behavior of a connection has to be taken into account and modeling of the so-called semi-rigid
Classification and AISC Specification
3
θr
Moment, M
Column
T-stub
M
Beam
End Plate
Top and Seat Angle
Header Plate
Double Web Angle Single Web Angle Rotation, θr FIGURE 1.2 Moment-rotation curves of semi-rigid connections.
connection is necessary in the structural analysis. In the continuous connection model, the behavior of a connection is assumed to have no effect on the analysis and may be modeled as so-called rigid. The above three types of connection models are classified based on the two independent criteria expressed in terms of rotational stiffness and strength, respectively. The scheme of modeling of connections is shown in Table 1.1, specifically for the case of nonlinear elastic-plastic structural analysis. This is because the types of connection models used in this analysis are determined based on both stiffness and strength of a connection in contrast to
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Semi-rigid Connections Handbook
TABLE 1.1 Types of EC3 connection models used in elastic-plastic structural analysis Type of connection model
Classification by stiffness1
Classification by strength2
Continuous
Rigid
Full strength
Semi-continuous
Rigid
Partial strength
Semi-rigid
Full strength Partial strength
Simple
Nominally pinned
Nominally pinned
Remarks: 1) See Table 1.2; 2) See Table 1.3
those used in elastic analysis or rigid-plastic analysis where either stiffness or strength is considered in the determination of connection modeling. As shown in Table 1.2, connections are classified as rigid, semi-rigid, or nominally pinned based on the initial rotational stiffness Rki of connections. The boundary between rigid and semirigid connections is determined, such that the load-carrying capacity of a semi-rigid portal frame is higher than 95 percent of the strength of the corresponding rigid frame. A frame model used to calculate the load-carrying capacity is a pin-based portal frame shown in Figure 1.3. The load carrying capacity is estimated by the Marchant-Rankin formula (Horne and Merchant, 1965). Since the effect of connection stiffness on the frame strength is dependent on whether the frame is braced or unbraced, the boundaries for the classification of connections are differently specified for these two types of frames. These boundaries are expressed in terms of a nondimensional initial connection stiffness parameter kb defined as kb = Rki /(EIb/Lb). Similar to the classification based on the rotational stiffness, connections are classified as full strength, partial strength or nominally pinned, depending on the ratio between the design mo-
TABLE 1.2 EC3 classification of connections by stiffness Connections Rigid Semi-rigid
kb = Rki /(EIb/Lb) Braced frames
8 < kb
Unbraced frames
25 < kb
Braced frames
0.5 ≤ kb ≤ 8
Unbraced frames Nominally pinned
25 < kb
Other requirement Kb/Kc ≥ 0.1 Kb/Kc < 0.1
0.5 ≤ kb ≤ 25 kb < 0.5
Remarks: Braced frames = those where the bracing system reduces the horizontal displacement by at least 80% Rki = initial rotational stiffness of connections Kb = the mean value of Ib /Lb for all the beams at the top of that story Kc = the mean value of Ic /Lc for all the columns in that story Ib = the second moment of inertia of a beam Ic = the second moment of inertia of a column Lb = the span of a beam Lc= the story height of a column
Classification and AISC Specification
5
Fc
Fc
EIb Rk
EIc
EIc
Lc
Lb FIGURE 1.3 Pin-based portal frame.
ment resistance of connections Mj,Rd and the design plastic moment resistance of the connected members such as beams Mb,pl,Rd and columns Mc,pl,Rd. The boundaries for this classification are different according to whether the connections are located at the top of columns or within the height of columns (Figure 1.4). The classification based on the connection strength is summarized in Table 1.3. Although the required rotational capacity of connections is not specified in EC3 classification, being different from the classifications by Bjorhovde et al. (1990), Goto et al. (1998) and Nethercot et al. (1998), the connections must be designed such that the deformation of connections imposed by the internal forces and moment is within the deformation capacity of the connections. An almost similar scheme is adopted by the AISC specification (AISC, 2005). EC3 considers the behavior of frames at the ultimate-limit state in the classification of connections. However, it does not explicitly take into account the behavior at the serviceability limit state. Furthermore, in the evaluation of the load-carrying capacity of frames, EC3 adopts an approximate formula (i.e., the Merchant-Rankine formula). In addition, a single-story portal frame model used in this evaluation may be too simple to reflect the effect of layout and member details of frames in practice.
a
Mj,Rd
b
Mj,Rd
FIGURE 1.4 Full-strength of connections: (a) top of column; (b) within column height.
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Semi-rigid Connections Handbook
TABLE 1.3 EC3 classification of connections by strength Connection Fullstrength
Partialstrength
Nominally pinned
Mj,Rd
Top of column
M j ,Rd M j ,Rd 1.0 ≤ or M c ,pl ,Rd M b ,pl ,Rd
Within column height
M j ,Rd M j ,Rd 1.0 ≤ or 2M c ,pl ,Rd M b ,pl ,Rd
Top of column
M j ,Rd M j ,Rd 0.25 < and < 1.0 M M b ,pl ,Rd c ,pl ,Rd
Within column height
M j ,Rd M j ,Rd 0.25 < and < 1.0 2M c ,pl ,Rd M b ,pl ,Rd
Top of column
M j , Rd M j ,Rd or ≤ 0.25 M M b ,pl ,Rd c ,pl ,Rd
Within column height
M j , Rd M j ,Rd or ≤ 0.25 2M c ,pl ,Rd M b ,pl ,Rd
Other requirements
1. Capable of transmitting internal forces without developing significant moment that might adversely affect the members or the structure as a whole 2. Capable of accepting the resulting rotation under the design load
Remarks: Mj,Rd = the design moment resistance of a connection Mb,pl,Rd = the design plastic moment resistance of the connected beam Mc,pl,Rd = the design plastic moment resistance of the connected column
A precise elastic-plastic finite-displacement analysis showed that the EC3 boundary between rigid and semi-rigid connections is on the whole restrictive in terms of the ultimate strength of frames (Goto and Miyashita, 1995).
1.1.3 AISC Classification System (AISC 2005)
AISC specification classifies connections as fully restrained (FR), partially restrained (PR), and simple connections, based on the secant stiffness Rks at service load and the strength Mn, respectively, of a connection, as summarized in Table 1.4. This classification is applied to both sway and nonsway frames. Simple and FR connections may be idealized as pinned and rigid, respectively, for the purpose of structural analysis. For structures with PR connections, the connection flexibility must be estimated and included in the structural analysis by the semi-rigid connection model. The boundary between the fully restrained and partially restrained connections expressed in terms of the secant connection stiffness parameter RksLb/EIb at the service load is determined such that the ratio of the connection moment obtained from semi-rigid analysis to that obtained from rigid analysis is 90 percent. Lb and EIb are the length and bending rigidity of the connected beam. The connection moment to determine the classification boundary is calculated by using a simple beam with semi-rigid connections at both ends, as shown in Figure 1.5 (Leon, 1994).
Classification and AISC Specification
7
TABLE 1.4 AISC classification of connections Connections
Stiffness
Strength
Ductility
RksLb /EIb ≤ 2
• Satisfy shear force demand at strength-limit state
• Satisfies rotation demand at the strengthlimit state or
• M ≤ 0.02 Mpb at 0.02 radians
• vu = 0.03 radians
• Satisfy combined shear force and moment demand at strength-limit state
• Satisfies rotation demand at the strengthlimit state or
• Satisfy combined shear force and moment demand at strength-limit state
• Satisfies rotation demand at the strengthlimit state or
Simple
2 < RksLb /EIb < 20 PR
FR
With partialstrength of connected beam
20 ≤ RksLb /EIb
• vu = 0.03 radians
• vu = 0.03 radians • No requirement
With fullstrength of connected beam
Remarks: Rks = secant stiffness of connections at service load Ib = second moment of inertia of beam Lb = span of beam E = Young’s modulus Mpb = plastic moment of beam vu = rotational capacity
As for the flexural strength Mn of connection, an explicit condition in the classification is imposed on the simple connections with no flexural strength. A connection with its moment of M ≤ 0.2Mpb (Mpb = fully plastic moment of the connected beam) at the rotation of 0.02 radians is considered to have no flexural strength for design analysis. The strengths of FR and PR connections must be adequate to resist the connection moment demand imposed by the design load. Therefore, it is possible in the AISC specification for an FR connection to have strength less than that of a connected beam. In this case, the semi-rigid action of the connection will appear if the plastification occurs in the connection. This implies that the rigid connection model is only
w Rk
Rk Lb
FIGURE 1.5 Simple beam with semi-rigid connections at both ends.
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Semi-rigid Connections Handbook
applicable before the plastication of the connection. It is also possible for a PR connection to have strength greater than that of the beam. Considering the above, the strength requirement for FR connections in the AISC is not as strict as that for rigid connections in EC3 where rigid connections must always have strength greater than the full strength of the connected beams or columns. The rotation capacity θu of connections is determined such that the rotation of connections at the strength-limit state of frames will not exceed θu. If the strength of a connection is substantially higher than that of the connected beam, the ductility of the structural system is controlled by the connected beam. However, if a connection has marginally higher or partial strength of the connected beam, the connection may experience some deformation before the structural system reaches its strength-limit state. In the above case, the ductility demand for FR, PR, and simple connections is calculated by a structural analysis in which the behavior of the connections is considered. This implies that the ductility required of the connection will depend on the particular application. In the absence of accurate analysis, the rotation capacity of 0.03 radians is considered adequate. The rotation capacity of connections defined by the AISC is explained in 1.2.2.4.
1.1.4 Bjorhovde et al. Classification System
A classification system proposed by Bjorhovde et al. (1990), as shown in Figure 1.6, is expressed in terms of nondimensional parameters of M/Mpb and θr/{Mpb(5d)/EIb}, where d and EIb are the depth and bending rigidity of the connected beam, respectively, and Mpb is the fully plastic moment capacity of the beam. As a reference beam length, 5d is assumed. Therefore, the initial stiffness of a connection is expressed in the form EIb/(ᾱd) where ᾱd is the equivalent length of the beam that gives the beam the same stiffness as the connection. If it is assumed that Lb is equal to 5d, the Bjorhovde’s classification parameter ᾱ is related to the EC3 (CEN, 2005) connection stiffness parameter kb as kb = 5/ᾱ. Based on the stiffness and strength of connections, connections are classified as rigid, semirigid, and flexible. Flexible connections may be modeled as ideally pinned in structural analysis. In addition, minimum rotational capacity inversely proportional to M/Mpb is specified for the respective connections. The Bjorhovde’s classification system is intended for the case where prior knowledge concerning the member and structural details is unavailable. M / Mpb
M=
EIb 2d
θr M=
1.0
EIb
10d
θr
Rigid 0.7
Ductility requirement
Semi-rigid 0.2 0
Flexible
θr 1.2
2.7
Mpb (5d) EIb
FIGURE 1.6 Classification (Bjorhovde et al., 1990).
Classification and AISC Specification
9
However, this classification system is considerably approximate in nature, because it does not consider the overall behavior of frames. In fact, a precise elastic-plastic geometrically nonlinear analysis showed that the behavior of rigid frames in some cases is not assured by the Bjorhovde’s condition of rigid connections where the moment capacity of connections is higher than the 70 percent of the fully plastic moment of the connected beam (Goto and Miyashita, 1995).
1.1.5 Nethercot et al. Classification System
In this classification system, a classification of connections is made based on their stiffness, strength, and rotational capacity. The categories used in the classification are fully-connected, partially-connected, pin-connected and non-structural connections. These categories are specified both for serviceability and for ultimate-limit states as illustrated in Figures 1.7 and 1.8, respectively. In the classification, parameter α defined as α = 8(EIc/Lc)/(EIb/Lb) is considered for classification. Nethercot’s classification of connections considers the stiffness of a column and a beam that a connection is connecting. In the category of fully-connected connections, the behavior of connections is assumed not to affect the behavior of framed structures and may be modeled as so-called rigid. In the category of partially-connected connections, the behavior of connections has to be taken into account and modeling of the so-called semi-rigid connection is necessary in structural analysis. Pin-connected connections are not assumed to transmit bending moment and may be modeled as ideally pinned. The non-structural connections are those that do not satisfy the requirement for strength and rotational capacity. The non-structural connections are not regarded as a structural element. In the classification at the ultimate-limit state, as shown in Figure 1.8, the stiffness boundary between the fully-connected connection and partially-connected connection is determined such that the ratio of the connection moment obtained from semi-rigid analysis to that obtained from rigid analysis is 95 percent. In addition, the fully-connected connections and partially-connected connections with the stiffness kb = Rki/(EIb/Lb) greater than 1/(0.53 − 1/α) are required to have a moment capacity at least equal to the connected beam moment capacity. In this case, a required M / Mpb
fully-connected zone
1.0 2/3
0.25 0
partially-connected zone non-structual zone pin-connected zone 1/6-1/α
1/3
(7α−2)/(8α)
(20+12α+α2)/(70α2−20α) FIGURE 1.7 Classification at serviceability limit state (Nethercot et al., 1998).
θr Mpb Lb EIb
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Semi-rigid Connections Handbook
M / Mpb
fully-connected zone
1.0
partially-connected zone
non-structual zone 0.25 0
pin-connected zone 0.53−1/α
0.75/α+0.37
0.9
(2+α)/(38α)
θr Mpb Lb EIb
FIGURE 1.8 Classification at ultimate-limit state (Nethercot et al., 1998).
rotational capacity is not specified for connections, considering that the rotation will be mainly concentrated at the beam end. The stiffness boundary between partially-connected and pin-connected connections at the ultimate-limit state is calculated under the condition that the partially-connected connection moment is 25 percent of rigid connection moment. To obtain the connection moment, a single span subframe under uniformly distributed beam load as illustrated in Figure 1.9 is used. This subframe model may be interpreted as a subassemblage of nonsway frames. Since partially-connected connections and pin-connected connections generally undergo some amount of rotation, the minimum required rotational capacity, which is inversely proportional to M/Mpb is specified for the connections with the initial stiffness kb = Rki/(EIb/Lb) less than 1/(0.53 − 1/α). In the classification at the serviceability limit state shown in Figure 1.7, the stiffness boundary between the fully-connected and partially-connected connections is determined as kb = (70α2 − 20α)/ (20 + 12α + α2) such that the ratio of the beam displacement obtained from rigid frame analysis to that obtained from semi-rigid frame analysis is 90 percent. In addition, the fully-connected connections and partially-connected connections with the stiffness kb = Rki/(EIb/Lb) greater than (2/3)/ (1/6 − 1/α) are required to have a moment capacity that exceeds at least 2/3 of the connected-beam moment capacity. The stiffness boundary between partially-connected and pin-connected connections is calculated as kb = 2α/(7α − 2) under the condition that the beam deflection ratio between partially-connected connections and pin-connected connections is 90 percent. This boundary is applied to the range where θrEIb/(MpbLb) ≥ (7α − 2)/(2α). In the range θrEIb/(MpbLb) ≤ (7α − 2)/(2α), the strength boundary between partially-connected and pin-connected connections is given at the location of 25 percent of the connected beam moment capacity. Similar to the classification at the ultimate-limit state, the minimum rotational capacity inversely proportional to M/Mpb is specified for the partially-connected and pin-connected connections with the initial stiffness kb = Rki/(EIb/Lb) less than 2/{3(1/6 − 1/α)}. Although the rotational capacity and strength of connections are specified for both serviceability and ultimate-limit states in the Nethercot et al. classification system, the required rotational capacity and strength at the ultimate-limit state normally dominates. So, there is little reason to
Classification and AISC Specification
11
EIc
EIc
Lc
EIb EIc
Rk
EIc
Lc
Lb FIGURE 1.9 Single-span subframe under uniformly distributed beam load.
specify these values in the classification at the serviceability limit states. Furthermore, it will be cumbersome to use different connection models in the frame analysis for design, depending on whether the limit state of concern is the ultimate or serviceability limit state.
1.1.6 Goto et al. Classification System
Goto et al. (1998) proposed a more detailed and precise classification system between rigid and semi-rigid connections considering the overall behavior of frames not only at the ultimate-limit state but also at the serviceability limit state. Herein, corrections are made to the mistakes in the original paper (Goto et al., 1998). In addition, a modification is made to improve the original classification scheme. To take into account the behavior of frames in the classification of connections, several subassemblages are used. These subassemblages approximately represent the behavior of the respective parts of the multistory multibay frames shown in Figure 1.10. The subassemblages are chosen by considering the deformation patterns of the respective parts of the sway and nonsway frames illustrated in Figure 1.11. The subassemblages so chosen are summarized in Figure 1.12 together with loading patterns. In this figure, the notations As ~ Fs and An ~ Fn are used to express the respective subassemblages that represent the parts of the frames in Figure 1.11.
12
Semi-rigid Connections Handbook
Lb
Lc
Lc
Lc
Ic
Ic
Ic
Lb
Ib
Ic
Ib
Lb
Ib
Ic
Ib
Ic
Ic Ib
Ic
Ib
Ic
Column Lc: length Ic: second moment of inertia
Ib
Ic Beam
Lb: length Ib: second moment of inertia
Ib
Ic
Ib
Ic
(Pinned base)
: Connection
Lc
Ic
(Fixed base)
FIGURE 1.10 Multistory and multibay frame. Cs
As
Ds
Bs Es
Fs
(Pinned base)
Bs
Ds
(Fixed base)
a
An
Cn
Bn
Dn
En
Bn
Dn
Fn
(Pinned base)
(Fixed base)
b FIGURE 1.11 Deformation pattern of multistory, multibay frames: (a) sway; (b) nonsway.
(
)
(
)
V
V
P
Dn, Fn
column2
δs
beam2
}= loads at serviceability limit state
δs
beam1
column1
P = load at ultimate limit state
V H
Cn
column1
As, An Bs, Bn Cs, Cn Ds, Dn Es, En Fs, Fn Lc/2 Lc/2 Lc/2 Lc/2 Lc/2 Lc/2 Lc Lc/2 Lc Lc/2 --Lb/2 Lb/2 Lb/2 Lb/2 Lb/2 Lb/2 Lb/2 Lb/2 Lb/2 ----
Bn, En
δs
beam2
V
δs
Ds, Fs
column2
beam1
beam2
V
FIGURE 1.12 Subassemblages.
Note : As ~ Fs , An ~ Fn denote the subassemblages of multistory multibay frames shown in Fig. 1.11.
column1 column2 beam1 beam2
column2
beam1
H
P )
An
δs
beam1
V
(
column1
)
column1
Cs
δs
H
)
beam1
V
(
P
Bs, Es
column1
beam1
beam2
column1
(
P
As
column2
δs
H )
column1
δs
H
beam1
P
(
H
beam1
)
column1
)
P
(
P
(
P
Classification and AISC Specification 13
14
Semi-rigid Connections Handbook
It is assumed that the initial connection stiffness affects the behavior of frames at the serviceability limit state and the boundary value of the initial stiffness between rigid and semi-rigid connections is derived based on the elastic displacement δs. This elastic displacement for sway subassemblages is calculated under horizontal force H, while elastic displacement for nonsway subassemblages are calculated under vertical force V. At the serviceability limit state, the effect of vertical compressive force P on the displacement δs is small; therefore, the elastic displacement is calculated by the small displacement theory, ignoring the effect of P. The criterion used to classify the connections as rigid is determined based on the tolerance Δ defined by (Equation 1.1): Δ = (δs − δr)/δr
(1.1)
where δs is a displacement of a subassemblage with semi-rigid connections and δr is a displacement of the corresponding rigid subassemblage. The boundary of the initial stiffness Rkib between rigid and semi-rigid connections so determined in terms of nondimentionalized initial stiffness parameter k bb = Rkib /(EIb/Lb) are summarized in Table 1.5, classified according to the respective types of subassemblages. In this table, to compare with EC3 boundary between rigid and semi-rigid connections, numerical results are also shown for Δ = 0.05 where G = (EIb/Lb)/(EIc/Lc) = 1.4 is assumed similar to the EC3 classification. In Table 1.5, the mistakes in the original paper (Goto et al., 1998) are corrected. To consider the connection behavior at the ultimate-limit state, the nonlinear moment-rotation relations are assumed to be expressed by the three-parameter model (Kishi et al.,1993) shown in Equation 1.2: m = muθ (1 + θ n )1/ n
(1.2)
TABLE 1.5 Goto et al. (1998) classification for initial stiffness of connections
k bb = Rkib / (EIb / Lb )
k bb = R bki / (EIb / Lb ) G = (EIb / Lb ) / (EIc / Lc ) = 1.4 D = 0.05
As, Ds, Es
6 (1+G )∆
50.0
Bs
6 (1+G / 2)∆
70.6
Cs, Fs
6 (1+ 2G )∆
31.6
An, Dn
6 2 − (4G + 1)(G + 1)∆ G + 1
6.7
Bn
12 4 − (4G + 1)(G + 1)∆ G + 1
17.4
Cn
6 2 − (8G + 1)(2G + 1)∆ 2G + 1
10.0
Subassemblages Sway
Nonsway
Remarks: G = (EIb/Lb)/(EIc/Lc), Δ = (ds − dr)/dr
Classification and AISC Specification
15
where m = M/Mpb, mu = M u M pb , θ = θr θ 0 , θ0 = Mu/Rki. M, Mu, and Mpb are connection moment, ultimate-connection moment capacity, and fully plastic moment of connected beam, respectively. – = 1.0 according to the Equation 1.2 has different shapes as illustrated in Figure 1.13 for m u values of n. As can be seen from Equation 1.2, the connection curve is uniquely determined by three parameters. That is, ultimate-moment capacity Mn, initial stiffness Rki, and shape parameter n. The shape parameter n is determined such that the connection model given by Equation 1.2 best fits the experimental data. In the original classification system by Goto et al. (1998), empirical equations (Kishi and Chen, 1993) based on a statistical analysis of the connection databank by Chen and Kishi (1989) were used in order to estimate the values of n for the respective connection types. This is to reduce the independent parameters of Equation 1.2 to Rki and Mu in view of the simplification in the classification procedure. However, the empirical equations for the estimation of the shape parameter n are considerably approximate and may impair the accuracy of the classification system. Therefore, the boundary moment-rotation curves are herein determined in terms of Rki, Mu, and n, being different from the original classification system (Goto et al., 1998). The boundary values for the initial stiffness Rki between rigid and semi-rigid connections are determined, as shown in Table 1.5, from the behavior of subassemblages at the serviceability limit state. On the other hand, the boundary values for the ultimate-moment capacity Mu and shape parameter n are expressed in terms of interaction equations that are determined by the behavior of subassemblages at the ultimate-limit state. In order to classify the connections to be rigid from the ultimate behavior of frames, EC3 uses the following criterion, which only considers the ultimate strength of the frames: Fur − Fus ≤ 0.05 Fur
(1.3)
where Fur, Fus are ultimate strengths, respectively, of rigid and semi-rigid frames.
m
n= ∞
1.0
n=6
0.5
n=4 n=2
0
0.5
1.0
FIGURE 1.13 Three-parameter power model.
1.5
2.0
mu = 1
2.5
θ
16
Semi-rigid Connections Handbook
The criterion expressed by Equation 1.3, however, may be insufficient because the displacement of frames at the ultimate-limit state is not reflected. Therefore, the following classification criterion is used based on the tolerance Δu, which takes into account both strength and displacement at the ultimate-limit state: ( F − F ) 2 (d − d ) 2 ur us us ur (1.4) + ≤ ∆u Fur dur where (Fur, dur) are the coordinate values of the limit point of an equilibrium curve for a rigid subassemblage expressed in terms of force-displacement relation, while (Fus, dus) are the coordinate values nearest to (Fur, dur) of an equilibrium curve for the corresponding semi-rigid subassemblage, as shown in Figure 1.14. It should be noted that (Fus, dus) is not necessarily a limit point of the equilibrium curve for the semi-rigid subassemblage. As the tolerance Δu, the following value is used: ∆ u = (0.05)2 + (0.05)2 ≅ 0.07
(1.5)
Based on the classification criterion for the ultimate behavior of semi-rigid subassemblages given by Equation 1.4 and Equation 1.5, the boundary values between rigid and semi-rigid connections are determined for the ultimate-moment capacity Mu and shape parameter n expressed in terms of interaction equations. The ultimate behaviors of the subassemblages are calculated by geometrically and materially nonlinear analysis with monotonically increasing the vertical displacement at the loading point of compressive column load P (Figure 1.12). In this analysis, the horizontal load H and the vertical load V are ignored. The analysis method used for this calculation precisely considers the geometrical and material nonlinearities in the structural response.
Fs F , r Fur Fur
1.0
0.07
Rigid subassemblage a b c a Subassemblage with connections b classified as rigid
c Subassemblage with connections classified as semi-rigid
0
1.0
d ds , r dur dur
FIGURE 1.14 Equilibrium curves of subassemblages with connections classified as rigid.
Classification and AISC Specification
17
Furthermore, initial geometrical imperfections as well as residual stresses are taken into account, following the ECCS recommendation (ECCS, 1984). The stress-strain relation used for the respective subassemblages and initial imperfections are shown in Figure 1.15. Extensive numerical analyses are carried out on the twelve types of subassemblages (Figure 1.12) with various member sizes to identify the boundary values for the connection moment – (= M /M ) and shape parameter n, following the criterion given by Equation 1.4 and capacity m u u pb Equation 1.5. In Equation 1.4, Fur and Fus are the ultimate values of P for the respective subassemblages, while dur and dus correspond to the ultimate-vertical displacement at the loading point
− σr = 0.005 L
− σr
+σr
+σr
σr = 0.3 fy L
− σr
L
a b
− σr
+σr +σr
σ Est = 0.02 E
fy
E
E
εy
εst = 10 εy
15 %
−fy c FIGURE 1.15 Imperfections and material properties of subassemblages: (a) initial geometric imperfections; (b) residual stress; (c) stress-strain relation.
ε
2 fy
0.0340
0.0426
0.0329
En
Fn
0.0296
Cn
Dn
0.0340
0.0050
Fs
An
0.0035
Es
Bn
0.0242
−0.2143
Ds
0.0036
−0.0077
0.0101
0.0068
0.0234
0.0101
−0.0015
0.0522
0.0054
−0.0170
0.0263
−0.2933
−0.0321
Bs
Cs
−0.0026
−0.0020
As
c2
c1
Subassemblage
0.1224
0.0327
−0.3594
−1.3965
−0.1281
−0.3594
0.0097
−0.3322
−0.0208
0.0688
0.3465
0.0413
c3
0.4415
0.7482
1.3748
3.4374
1.0543
1.3748
0.9887
1.4543
0.9793
1.0498
1.3991
0.9931
c4
−0.0299
−0.0893
−0.1771
−0.0161
−0.2008
−0.1771
−0.0207
−0.1051
0.0112
−0.0352
−0.2221
−0.0231
c5
0.0703
−0.1739
−1.2436
−2.4585
−1.1667
−1.2436
−0.0196
−0.9124
−0.0470
0.0649
−1.0559
−0.1252
c6
0.0974
0.1159
0.2559
−0.1592
0.3115
0.2559
0.0111
0.1507
0.0112
−0.0242
0.2122
0.0385
c7
−0.0016
0.0018
−0.0002
0.0030
−0.0030
−0.0002
0.0002
−0.0125
−0.0032
0.0081
−0.0041
0.0000
c8
0.2883
0.3105
0.8971
1.5685
0.7797
0.8971
0.0472
0.7557
0.0216
−0.0126
0.5701
0.0554
c9
TABLE 1.6 Coefficients c1 ~ c10 for interaction equations between boundary moment capacity mub and boundary shape parameter nb
−0.0203
−0.0106
−0.0097
−0.0001
−0.0064
−0.0097
−0.0005
0.0754
−0.0095
0.0095
0.0517
−0.0036
c10
18 Semi-rigid Connections Handbook
Classification and AISC Specification
19
– b = Mb M of P. Based on the numerical data obtained for the boundary moment capacity m u pb u b and boundary shape parameter n , the interaction equations between these two quantities for the respective subassemblages are empirically derived through curve-fitting techniques. All the empirical interaction equations are assumed here to have the following form: n b = c1 + c2 G + c3 λ + c5Gλ + c6 λ mub + c7 Gmub + c8 G 2 + c9 λ 2 + c10 (mub )2 (1.6) m where c1 ~ c10 are curve-fitting coefficients determined by the least square method, according to the types of subassemblages. G and λ that denote relative stiffness and normalized column slenderness ratio, respectively, are expressed by:
G=
(I (I
b c
Lb ) Lc )
L λ = c πr
,
(1.7a)
fy
(1.7b)
E
where r is the radius of gyration of member cross section; E is Young’s modulus; fy is yield stress of member material. G and λ are introduced to consider the layout and member details of the subassemblages. The values of the curve-fitting coefficients c1 ~ c10 applicable to the following parameter range are summarized in Table 1.6:
0.7 ≤ G ≤ 4.2, 0.2 ≤ λ ≤ 1.0, 0.8 ≤ mu ≤ 1.2
(1.8a-c)
The required rotational capacity θu is defined as the maximum rotation θ max of the connections that is experienced up to the maximum load carrying capacity of the semi-rigid frames (Goto and Miyashita, 1995). The connections classified as rigid generally exhibit smaller rotation than do the connections that have the boundary moment-rotation characteristics between rigid and semi-rigid connections. This implies that the required rotational capacity calculated based on the boundary moment-rotation characteristics between rigid and semi-rigid connections will provide an upper bound that will be safe for the design of connections classified as rigid. Therefore, the required rotational capacities θu for the respective subassemblages are calculated by the nonlinear numerical analyses, assuming that the connection behavior is expressed by the boundary moment-rotation curve between rigid and semi-rigid connection as given by: m = mubθ /(1 + θ n )1/ n n
n
(1.9)
where θ = θr θ 0 and θ0 = M /R . Based on these numerical results, the required rotational capacities θu = θu θ 0 for the respective subassemblages classified as rigid are approximated by the empirical formulas as shown: b u
b ki
θu = d1 + d 2 G + d3 λ + d5Gλ + d6 λ mub + d 7 Gmub + d8 G 2 + d9 λ 2 + d10 (mub )2 (1.10) m where d1 ~ d10 are the curve-fitting coefficients shown in Table 1.7. These coefficients are obtained by the least square method for the range given by Equation 1.8. The validity of the proposed classification system is precisely examined for four types of sway and nonsway frames (Goto et al., 1998).
0.0455
−0.0819
0.0674
−0.0067
−0.0746
−0.0022
−0.0208
−0.0093
−0.0601
−0.0160
−0.0057
−0.1933
Bs
Cs
Ds
Es
Fs
An
Bn
Cn
Dn
En
Fn
0.0147
−0.0058
−0.0489
−0.1395
−0.0359
−0.0489
0.0004
0.0602
0.0062
−0.0280
0.0024
−0.0054
As
d2
d1
Subassemblage
−0.0243
0.0292
0.0883
1.8213
0.1597
0.1193
−0.0345
−0.6142
−0.0204
0.1349
−0.2690
−0.0021
d3
TABLE 1.7 Coefficients d1 ~ d10 for formulas to predict θu
0.6966
1.0029
0.9737
1.7711
1.0099
0.9820
1.0047
1.1700
0.9981
0.9662
1.1447
1.0006
d4
−0.0513
−0.0139
−0.0785
−0.3256
−0.0473
−0.0830
0.0219
0.1798
0.0119
−0.0358
0.0539
0.0070
d5
1.5172
−0.0052
−0.0104
−5.7450
−0.0854
−0.0559
0.0131
0.0040
0.0302
−0.1781
−0.0769
0.0150
d6
0.0828
0.0003
0.0531
0.3240
0.0116
0.0549
−0.0037
−0.0437
−0.0038
0.0348
−0.0034
−0.0009
d7
0.0067
0.0023
0.0153
0.0397
0.0099
0.0156
−0.0007
−0.0171
−0.0017
0.0039
−0.0098
−0.0008
d8
−0.0959
0.0146
0.1319
−0.5660
0.0389
0.1246
−0.0278
−0.0588
−0.0466
0.0860
−0.0001
−0.0294
d9
0.0714
−0.0002
0.0095
−0.0753
−0.0069
0.0076
0.0033
0.0725
0.0065
−0.0336
0.0485
0.0035
d10
20 Semi-rigid Connections Handbook
Classification and AISC Specification
21
1.1.7 Comparison of Existing Classification Systems 1.1.7.1 General
Classification of connections is normally made in terms of the index properties such as stiffness, strength, and rotation capacity. Among these index properties, AISC classification (AISC, 2005) adopts only stiffness except for the classification of simple connections where strength requirement is considered in addition to the stiffness. Eurocode 3 (CEN, 2005) adopts both stiffness and strength. The other classification systems (Bjorhovde et al., 1990; Nethercot et al., 1998; Goto et al., 1998) utilize all three indexes. Herein, the respective classification systems are compared with one another in terms of stiffness, strength, and rotation capacity.
1.1.7.2 Classification of Connection Stiffness
The connection stiffness is the most important index in the classification systems. The effect of connection stiffness on the overall behavior of frames is governed by the layout and details of members such as beams and columns. In addition, this effect is different according to whether the frames are sway or nonsway. Normally, the classification based on sway frames tends to yield a more restrictive criterion. Eurocode 3 (CEN, 2005) specifies different classification criteria for initial connection stiffness according to whether the horizontal displacement is restrained or not. However, the model used to derive a classification criterion is a pin-based portal frame (Figure 1.3). Furthermore, nondimensional connection stiffness parameters are simplified to include only the bending rigidity of the connected beam. This implies that the rigidity of the connected column is ignored. AISC specification (AISC, 2005) uses the secant connection stiffness at the service load for classification. A simple beam model with semi-rigid connections at both ends (Figure 1.5) is used to derive the criteria for classification. As a result, this classification system ignores the sway behavior of frames as well as the rigidity of columns. Bjorhovde’s classification criterion (Bjorhovde et al., 1990) that uses the initial stiffness of connections is intended for the case where prior knowledge concerning the member and structural details is unavailable. Therefore, this classification system ignores the overall behavior of frames. Nethercot’s classification system (Nethercot et al., 1998) adopts the initial stiffness of connections as a stiffness index. This classification takes into account the bending rigidities of the connected beam and column. The classification criteria are derived both for serviceability limit state and for ultimatelimit state. These criteria are based on a single-span nonsway subframe under uniformly distributed load applied on the beam, as illustrated in Figure 1.9. This subframe model may be interpreted as a subassemblage of nonsway frames. Although Nethercot considers the bending rigidities of the connected beam and column in the classification, the submodel used to derive the classification are too simple to reflect the actual behavior of frames, being similar to the AISC and EC3 classification. Furthermore, it will be cumbersome to use different connection models in the frame analysis for design, depending on whether the limit state of concern is the ultimate or serviceability limit state. A more detailed classification criterion for the boundary between rigid and semi-rigid connections is proposed by Goto et.al. (1998) in terms of initial connection stiffness. To take into account the behavior of frames in the classification of connections, several subassemblages are used to derive the classification criterion such that the subassemblages will appropriately represent the behavior of the respective parts of the multistory multibay frames with sway or nonsway displacement. As a result, this criterion includes the effect of the bending rigidities of the connected beams and columns.
1.1.7.3 Connection Strength and Rotation Capacity
In AISC specification, the required strength and rotation capacity of connections depend on a particular application. That is, the strength of connections is designed to resist the moment
22
Semi-rigid Connections Handbook
demand under the design load, while the rotational capacity must be larger than the rotational demand of connections at the strength-limit state of frames. Almost similar to AISC specification, connections are designed in EC3 to have a rotational capacity that meets their rotational deformation demand under the design load. The rotational demand has to be calculated by frame analysis except for continuous connections (CEN, 2005) and FR connections (AISC, 2005) that have the strength larger than that of the connected beams. Specifically, in the case of semicontinuous connections (CEN, 2005) and PR connections (AISC, 2005), as well as in the case of FR connections with the partial strength of a connected beam (AISC, 2005), the connection behavior has to be appropriately reflected in the frame analysis. As can be seen from the connection modeling explained above, it will be more appropriate to classify the FR connections with the partial strength of a connected beam (AISC, 2005) as PR connections.
1.2 AISC Specification 1.2.1 Introduction
In the 2005 ANSI/AISC Specification (AISC, 2005), connections are largely classified into two types: Simple and moment connections. A simple connection transmits a negligible moment across the connection. In the analysis of the structure, simple connections may be assumed to allow unrestrained relative rotation between the framing elements being connected. These connections may be modeled as pinned. A moment connection transmits moment across the connection. Two types of moment connections—fully restrained (FR) moment connection and partially restrained (PR) moment connections—are permitted. An FR connection transfers moment with a negligible rotation between the connected members. In the analysis of the structure, the FR connection may be generally assumed to allow no relative rotation and be modeled as rigid. A PR connection transfers moment but the rotation between connected members is not negligible. In the analysis of the structure, the PR connection is modeled as semi-rigid and the momentrotation response characteristics of the PR connection shall be included in the connection model. For simple and FR connections, the connection proportions are established after the final analysis of the structural design is completed. In contrast, the design of PR connections is inherently iterative because the values of connection proportions must be assumed in order to establish the force-deformation characteristics of the connections that are needed to carry out the structural analysis. Herein, the design of semi-rigid connections is explained and discussed based on the commentary of the 2005 ANSI/AISC specification.
1.2.2 Connection Classification 1.2.2.1 General
A connection is a medium through which forces and moments are transmitted from one member to another. A beam-to-column connection is generally subject to axial force, shear force, bending moment, and torsion. Regarding its in-plane behavior, the effect of torsion can be excluded. Furthermore, the effects of axial and shear forces are usually small compared to that of bending moment. Consequently, for practical purposes, only the effect of moment on the rotational deformation needs to be considered regarding the deformation of connections. As depicted in Figure 1.1, the connection rotates by an amount θr when a moment M is applied. The angle θr corresponds to the relative rotation between the beam and the column at the connection. Therefore, in the AISC specification, the in-plane behavior of semi-rigid connection is represented by the M − θr curve, referred to as moment-rotation curve. The moment-rotation curve of a connection is assumed to represent the behavior of a region consisting of the column and beam along with the connecting elements. The primary index properties that govern the moment-rotation charac-
Classification and AISC Specification
23
teristics of connections adopted in the AISC specification are stiffness, strength, and ductility that will be explained in the next three sections.
1.2.2.2 Connection Stiffness
Most connections exhibit nonlinear behavior at low moment-rotation levels. Therefore, the initial stiffness of a connection Rki inadequately characterizes the connection behavior at service load levels. Furthermore, many connection types do not exhibit a reliable initial stiffness, or they exist only for a small moment-rotation range. The secant stiffness Rks at service loads shown in Figure 1.16 is taken as an index property of connection stiffness for the connection classification. Rks is defined as Ms/θs where Ms and θs are the moment and rotation, respectively, at service loads.
1.2.2.3 Connection Strength
The strength of a connection is the maximum moment Mn in the moment-rotation relation shown in Figure 1.16. The strength of a connection can be determined based on an ultimate-limit-state model of the connection, or from a physical test. If the moment-rotation response does not exhibit a peak load, then the strength can be taken as the moment at a rotation of 0.02 radians.
1.2.2.4 Connection Ductility
The connection ductility (the rotation capacity) θu can be defined as the value of the connection rotation at the point where the resisting strength of connection has dropped to 0.8Mn as shown in Figure 1.16. If the connection does not exhibit a 0.2Mn drop of the resisting moment beyond 0.03
M(θr )
Moment, M
Rki
Rks
Mn
0.20 Mn
Ms
θs
θn
Rotation, θr
θu
FIGURE 1.16 Definition of stiffness, strength, and ductility characteristics of the moment-rotation response for partially restrained connections.
24
Semi-rigid Connections Handbook
radians, the connection ductility θu is taken as 0.03 radians. The latter criterion is intended to apply to connections where there is no significant loss in strength until large rotations occur.
1.2.2.5 Classification Scheme
The AISC classification scheme for simple, FR, and PR connections is summarized in Table 1.4. The classification boundaries are expressed in terms of the nondimentionalized secant connection stiffness RksLb/EIb at service load where Lb and EIb are the length and the bending rigidity, respectively, of the connected beam. This classification is applied to both sway and nonsway frames, being different from the Eurocode3 (CEN 2005) where different classification boundaries are proposed according to whether the frame is braced or unbraced. As for the connection-moment strength Mn, an explicit condition in the classification is imposed only on the simple connections. The connection moment of M ≤ 0.2Mpb at 0.02 radians is considered to have no flexural strength for design. The strengths of FR and PR connections must be adequate to resist the connection moment and shear force demands imposed by the design load. Therefore, it is possible in the AISC specification for an FR connection to have strength less than that of a connected beam. Furthermore, it is also possible for a PR connection to have strength greater than the strength of the beam. The strength requirement for FR connections is not as strict as that for rigid connections in EC3 where rigid connections must always have strength greater than the full strength of the connected beams or columns. However, if the FR connections have marginally higher or partial strength of the connected beam, it is necessary to calculate the rotational demand of the connections under the factored load by using the semi-rigid modeling of connections. Therefore, in view of the simplification of design analysis procedure, it will be of little use to classify connections with marginally higher or partial strength of the connected beam as FR connections. This connection type should be classified as PR. The rotation capacity θu of connections is required to exceed their demands calculated at the strength-limit state of frames. In the absence of an accurate analysis, a rotation capacity θu of 0.03 radians is considered adequate. The above requirement must also be applied to the FR connections with marginally higher or partial strength of the connected beam. However, for the FR connections with strength substantially higher than that of the connected beam, there is no requirement in their rotation capacities. This is because the deformation is controlled by the plastification of the connected beam, and the FR connections exhibit an elastic behavior at the strength-limit state. Figure 1.17 schematically shows the examples of moment-rotation responses of simple, PR, and FR connections classified according to the AISC scheme.
1.2.3 Structural Analysis and Design for Frames with PR Connections
If the connections in a steel framework are classified as either simple or FR, it is allowed to perform the design analysis for the frame assuming that the connections are either ideally pinned or fully rigid. The assumption of fully rigid connection implies that no relative rotation of connection occurs, while the assumption of pinned connection implies that no restraint for rotation of connection exists and the connection moment is always zero. The assumption of the simple or the FR connection simplifies considerably the analysis and design procedures. The proportions of these connections can be established after the final analysis for the structural design is completed. Simple connections are generally designed under the shear force and must meet the required rotation without introducing moment strength and stiffness that may significantly alter the mode of response. On the other hand, FR connections will be designed for the combined effect of forces resulting from moment and shear and must have sufficient stiffness to transmit moment and maintain the original angle between connected members.
Classification and AISC Specification
FR
Rks =
Mn
Moment, M
Mpb
25
20EIb Lb
M(θr )
θu
θs
Mn
θs
PR
θu θs
Mn 0.03
θu
Rks =
Simple
2EIb Lb
Rotation, θr (radians) FIGURE 1.17 Classification of moment-rotation response of fully restrained (FR), partially restrained (PR), and simple connections.
When PR connections are used, the relevant moment-rotation relation must be included in the analysis of the structure to determine member forces, connection forces, displacements, and frame stability. Therefore, PR construction first requires that the connection moment-rotation characteristics must be known and second, requires that these characteristics be incorporated into the analysis and member design. In contrast to the design of simple and FR connections, it should be noted that the design of PR connections is inherently iterative because the connection proportions must be assumed in order to establish their moment-rotation characteristics that are used in the structural analysis. The PR connections must transfer the moment similar to the FR connections, whereas the PR connections must have the rotational capacities that meet the rotation demands similar to the simple connections. Therefore, the connections have to be designed under the combined effect of moment and shear such that their rotational capacity satisfies the rotational demand at the strength-limit state. Typical moment-rotation curves for many PR connections are available from existing connection databases. When utilizing the databases, care must be taken not to extrapolate data to sizes or conditions beyond those used to develop the database since other failure modes may control. For the connections out of the range of the databases, it may be possible to derive the momentrotation characteristics from tests, component models, or finite element studies. Details of connection databases and connection models are explained in Chapters 4 and 5. Usually, design of PR construction requires separate analyses for the serviceability and strengthlimit states. For the serviceability limit state, an analysis using linear springs with a stiffness given by Rks is sufficient if the resistance demanded of the connection is well below the strength. When
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Semi-rigid Connections Handbook
subjected to strength load combinations at the ultimate-limit states, a more careful procedure is needed so that the connection characteristics assumed in the analysis are consistent with those of the actual connection response. The response will become nonlinear as the applied moment approaches the connection strength. In particular, the effect of the connection nonlinearity on second-order moment and stability checks need to be considered. The design requirements for simple, PR, and FR connections are shown in Table 1.4.
References AISC, Specification for Structural Steel Buildings, ANSI/AISC 360-05, American Institute of Steel Construction, Chicago, IL, 2005. Bijorhovde, R., Colson, A., and Brozzeti, J., Classification system for beam-to-column connections, Journal of Structural Engineering, ASCE, 116(11), 3059-3076, 1990. CEN, Eurocode 3, Design of Steel Structures, Part 1-1: General rules and rules for buildings, EN 1993-1-1, Part 1-8: Design of joints, EN 1993-1-8, European Committee for Standardization, Brussels, Belgium, 2005. Chen, W. F. and Kishi, N., Semi-rigid steel beam-to-column connections: Data base and modeling, Journal of Structural Engineering, ASCE, 115(7), 105-119, 1989. European Commission for Constructional Steelwork (ECCS), Ultimate-limit state calculation of sway frames with rigid joints, Systems Publication No.33, Technical Working Group 8.2, Brussels, Belgium, 1984. Goto, Y. and Miyashita, S., Classification system for rigid and semi-rigid connections, Journal of Structural Engineering, ASCE, 124(7), 750-757, 1998. Goto, Y. and Miyashita, S., Validity of classification systems of semi-rigid connections, Engineering Structures, 17(8), 544-553, 1995. Horne, M. R. and Merchant, W., The stability of frames, Pergamon Press, 1965. Kishi, N., Chen, W. F., Goto, Y., and Matsuoka, K. G., Design aid of semi-rigid connections for frame analysis, AISC Engineering Journal, 30(3), 90-107, 1993. Leon, R. T., Composite semi-rigid construction, AISC Engineering Journal, 31(2), 54-67, 1998. Nethercot, D. A., Ahmed, T. Q., and Li, B., Unified classification system for beam-to-column connections, Journal of Constructional Steel Research, 45(1), 39-65, 1998.
2
Effects of Semi-rigid Connections on Structural Members and Frames
Eric M. Lui Meredith Professor, Department of Civil and Environmental Engineering, Syracuse University, NY Zhiling Zhang Graduate Student, Department of Civil and Environmental Engineering, Syracuse University, NY
2.1 Introduction............................................................................................................................. 27 Connection Behavior 2.2 Effects of Semi-rigid Connections on Columns................................................................. 32 Column Effective Length Factor • K Factor for Columns in Semi-rigid Frames 2.3 Effects of Semi-rigid Connections on Beams...................................................................... 40 Lateral Torsional Buckling Load for Compact Beams • Beams with Semi-rigid Connections 2.4 Effects of Semi-rigid Connections on Frames..................................................................... 45 Analysis of Semi-rigid Frames • Design of Semi-rigid Frames • Drift of Semi-rigid Frames 2.5 Summary and Conclusions.................................................................................................... 67
2.1 Introduction A connection is a medium through which forces and moments are transmitted from one member to another. In a conventional analysis and design of steel frameworks, the frames are often conveniently analyzed and designed under the simplifications that the connections behave either as ideally pinned or fully rigid. The use of the ideally pinned condition implies that no moment will be transmitted between the beam and the column. As far as rotation is concerned, the beam and column that are jointed together by a pin will behave independently. On the other extreme, the use of the fully rigid condition implies that no relative rotation will occur between the adjoining members. The angle between the beam and the column remains virtually unchanged as the frame deforms under the applied loads. Although the use of these idealized joint behaviors drastically simplifies the analysis and design procedures, the predicted response of the frame may not be realistic as most connections used in actual practice transmit some moment and experience some deformation. 27
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Semi-rigid Connections Handbook
According to the American Institute of Steel Construction (AISC, 2005), a connection is considered semi-rigid or partially restrained (PR) if its stiffness Rk falls in the range 2EIb/Lb
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