Femap Structural - Verification Guide

)(0$36WUXFWXUDO Verification Guide

Version 8.2

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Throughout this manual, you will see references to Windows. Windows refers to Microsoft® Windows NT, Windows 2000, Windows 95, Windows 98, Windows Me, or Windows XP. You will need one of these operating environments to run FEMAP for the PC. This manual assumes that you are familiar with the general use of the operating environment. If you are not, you can refer to the Windows User’s Guide for additional assistance. Similarly, throughout the manual all references to FEMAP, refer to the latest version of our software.

Table of Contents Proprietary and Restricted Rights Notice Overview Linear Statics Verification Using Theoretical Solutions Nodal Loads on a Cantilever Beam ....................................................................................4 Axial Distributed Load on a Linear Beam ..........................................................................6 Distributed Loads on a Cantilever Beam ............................................................................9 Moment Load on a Cantilever Beam ................................................................................12 Thermal Strain, Displacement, and Stress on Heated Beam ............................................15 Uniformly Distributed Load on Linear Beam ..................................................................18 Membrane Loads on a Plate .............................................................................................21 Thin Wall Cylinder in Pure Tension .................................................................................24 Thin Shell Beam Wall in Pure Bending ...........................................................................27 Strain Energy of a Truss ...................................................................................................30

Linear Statics Verification Using Standard NAFEMS Benchmarks Elliptic Membrane ............................................................................................................34 Cylindrical Shell Patch Test .............................................................................................39 Laminate Strip ..................................................................................................................42 Hemisphere-Point Loads ..................................................................................................44 Z–Section Cantilever ........................................................................................................47 Skew Plate Normal Pressure .............................................................................................49 Thick Plate Pressure .........................................................................................................53 Solid Cylinder/Taper/Sphere–Temperature ......................................................................58

Normal Modes/Eigenvalue Verification Using Theoretical Solutions Undamped Free Vibration - Single Degree of Freedom ...................................................65 Two Degrees of Freedom Undamped Free Vibration - Principle Modes .........................68 Three Degrees of Freedom Torsional System ..................................................................71 Two Degrees of Freedom Vehicle Suspension System ....................................................73 Cantilever Beam Undamped Free Vibrations ...................................................................76 Natural Frequency of a Cantilevered Mass ......................................................................78

Normal Modes/Eigenvalue Verification Using Standard NAFEMS Benchmarks Bar Element Test Cases ....................................................................................................82 Pin-ended Cross - In-plane Vibration ........................................................................83 Pin-ended Double Cross - In-plane Vibration ...........................................................86 Free Square Frame - In-plane Vibration ....................................................................89

72& Cantilever with Off-Center Point Masses ................................................................. 92 Deep Simply-Supported Beam .................................................................................. 95 Circular Ring - In-plane and Out-of-plane Vibration ................................................ 98 Cantilevered Beam .................................................................................................. 101 Plate Element Test Cases ................................................................................................ 104 Thin Square Cantilevered Plate -Symmetric Modes ............................................... 105 Thin Square Cantilevered Plate - Anti-symmetric Modes ...................................... 108 Free Thin Square Plate ............................................................................................ 111 Simply-Supported Thin Square Plate ...................................................................... 114 Simply-Supported Thin Annular Plate .................................................................... 117 Clamped Thin Rhombic Plate ................................................................................. 121 Cantilevered Thin Square Plate with Distorted Mesh ............................................. 124 Simply-Supported Thick Square Plate, Test A ....................................................... 129 Simply-Supported Thick Square Plate, Test B ........................................................ 133 Clamped Thick Rhombic Plate ............................................................................... 136 Simply-Supported Thick Annular Plate .................................................................. 140 Cantilevered Square Membrane .............................................................................. 144 Cantilevered Tapered Membrane ............................................................................ 148 Free Annular Membrane ......................................................................................... 152 Cantilevered Thin Square Plate ............................................................................... 156 Cantilevered Thin Square Plate #2 .......................................................................... 161 Axisymmetric Solid and Solid Element Test Cases ....................................................... 164 Free Cylinder - Axisymmetric Vibration ................................................................ 165 Thick Hollow Sphere - Uniform Radial Vibration .................................................. 168 Simply-Supported Annular Plate -Axisymmetric Vibration ................................... 171 Deep Simply-Supported Solid Beam ...................................................................... 174 Simply-Supported Solid Square Plate ..................................................................... 178 Simply-Supported Solid Annular Plate ................................................................... 182 Cantilevered Solid Beam ......................................................................................... 186

Verification Test Cases from the Societe Francaise des Mechaniciens Mechanical Structures - Linear Statics Analysis with Bar or Rod Elements ................. 191 Short Beam on Two Articulated Supports .............................................................. 192 Clamped Beams Linked by a Rigid Element .......................................................... 194 Transverse Bending of a Curved Pipe ..................................................................... 196 Plane Bending Load on a Thin Arc ......................................................................... 199 Nodal Load on an Articulated Rod Truss ................................................................ 201 Articulated Plane Truss ........................................................................................... 203 Beam on an Elastic Foundation ............................................................................... 206 Mechanical Structures - Linear Statics Analysis with Plate Elements ........................... 209 Plane Shear and Bending Load on a Plate ............................................................... 210 Infinite Plate with a Circular Hole .......................................................................... 212 Uniformly Distributed Load on a Circular Plate ..................................................... 215 Torque Loading on a Square Tube .......................................................................... 218 Cylindrical Shell with Internal Pressure .................................................................. 221

72&  Uniform Axial Load on a Thin Wall Cylinder ........................................................225 Hydrostatic Pressure on a Thin Wall Cylinder ........................................................229 Gravity Loading on a Thin Wall Cylinder ..............................................................232 Pinched Cylindrical Shell ........................................................................................236 Spherical Shell with a Hole .....................................................................................239 Uniformly Distributed Load on a Simply-Supported Rectangular Plate .................242 Uniformly Distributed Load on a Simply-Supported Rhomboid Plate ...................247 Shear Loading on a Plate .........................................................................................251 Mechanical Structures - Linear Statics Analysis with Solid Elements ...........................254 Solid Cylinder in Pure Tension ...............................................................................255 Internal Pressure on a Thick-Walled Spherical Container ......................................261 Internal Pressure on a Thick-Walled Infinite Cylinder ...........................................268 Prismatic Rod in Pure Bending ...............................................................................274 Thick Plate Clamped at Edges .................................................................................279 Mechanical Structures - Normal Modes/Eigenvalue Analysis .......................................284 Lumped Mass-Spring System ..................................................................................285 Short Beam on Simple Supports ..............................................................................288 Axial Loading on a Rod ..........................................................................................291 Cantilever Beam with a Variable Rectangular Section ...........................................294 Thin Circular Ring ...................................................................................................297 Thin Circular Ring Clamped at Two Points ............................................................300 Vibration Modes of a Thin Pipe Elbow ...................................................................303 Cantilever Beam with Eccentric Lumped Mass ......................................................307 Thin Square Plate (Clamped or Free) ......................................................................311 Simply-Supported Rectangular Plate ......................................................................314 Thin Ring Plate Clamped on a Hub .........................................................................317 Vane of a Compressor - Clamped-free Thin Shell ..................................................320 Bending of a Symmetric Truss ................................................................................323 Hovgaard’s Problem - Pipes with Flexible Elbows .................................................326 Rectangular Plates ...................................................................................................328 Stationary Thermal Tests - Steady State Heat Transfer Analysis ...................................330 Hollow Cylinder - Fixed Temperatures ...................................................................331 Hollow Cylinder - Convection ................................................................................334 Cylindrical Rod - Flux Density ...............................................................................337 Hollow Cylinder with Two Materials - Convection ................................................340 Wall - Convection ....................................................................................................344 Wall - Fixed Temperatures ......................................................................................347 L-Plate .....................................................................................................................350 Hollow Sphere - Fixed Temperatures, Convection .................................................353 Hollow Sphere with Two Materials -Convection ....................................................356 Thermo-mechanical Test - Linear Statics Analysis ........................................................360 Thermal Gradient on a Thin Pipe ............................................................................361 Index ...............................................................................................................................365

Overview This guide contains verification test cases for the FEMAP Structural solver. These test cases verify the function of the different FEMAP Structural analysis types using theoretical and benchmark solutions from well–known engineering test cases. Each test case contains test case data and information, such as element type and material properties, results, and references. The guide contains test cases for: •

Linear Statics verification using theoretical solutions

•

Linear Statics verification using standard NAFEMS benchmarks

•

Normal Modes/Eigenvalue verification using theoretical solutions

•

Normal Modes/Eigenvalue verification using standard NAFEMS benchmarks

•

Verification Test Cases from the Societe Francaise des Mechaniciens

Linear Statics Verification Using Theoretical Solutions The purpose of these linear statics test cases is to verify the function of the FEMAP Structural Statics Analysis software using theoretical solutions. The test cases are relatively simple in form and most of them have closed–form theoretical solutions. The theoretical solutions shown in these examples are from well–known engineering texts. For each test case, a specific reference is cited. All theoretical reference texts are listed at the end of this topic. The finite element method is very flexible in the types of physical problems represented. The verification tests provided are not exhaustive in exploring all possible problems, but represent common types of applications. This overview provides information on the following: •

understanding the test case format

•

understanding comparisons with theoretical solutions

•

references

Understanding the Test Case Format Each test case is structured with the following information: •

test case data and information - physical and material properties - finite element modeling (modeling procedure or hints) - units - solution type - element type - boundary conditions (loads, constraints)

•

results

•

references (text from which a closed–form or theoretical solution was taken)

Note: . The node numbers listed in each case refer to the node numbers in the neutral (.neu) files associated with this guide. If you remesh a model, or rebuild that model from scratch, your node numbering may differ.

 In addition to these example problems, test cases from NAFEMS (National Agency for Finite Element Methods and Standards, National Engineering Laboratory, Glasgow, U.K.) have been executed. Results for these test cases can be found in the next section, Linear Statics Analysis Verification Using NAFEMS Standard Benchmarks.

Understanding Comparisons with Theoretical Solutions While differences in finite element and theoretical results are, in most cases, negligible, some tests would require an infinite number of elements to achieve the exact solution. Elements are chosen to achieve reasonable engineering accuracy with reasonable computing times. Results reported here are results which you can compare to the referenced theoretical solution. Other results available from the analyses are not reported here. Results for both theoretical and finite element solutions are carried out with the same significant digits of accuracy. The closed–form theoretical solution may have restrictions, such as rigid connections, that do not exist in the real world. These limiting restrictions are not necessary for the finite element model, but are used for comparison purposes. Verification to real world problems is more difficult but should be done when possible. The actual results from the FEMAP Structural software may vary insignificantly from the results presented in this document. This variation is due to different methods of performing real numerical arithmetic on different systems. In addition, it is due to changes in element formulations which SDRC has made to improve results under certain circumstances. References The following references have been used in the Linear Statics Analysis verification problems presented: 1. Beer and Johnston, Mechanics of Materials, (New York: McGraw–Hill, Inc., 1992.) 2. Harris, C. O., Introduction to Stress Analysis, (1959.) 3. Roark, R. and Young, W., Formulas for Stress and Strain, 5th Edition, (New York: McGraw–Hill Book Company, 1975.) 4. Shigley, J. and Mitchel L., Mechanical Engineering Design, 4th Edition, (New York: McGraw–Hill Book Company, 1983.) 5. Timoshenko, S., Strength of Materials, Part I, Elementary Theory and Problems, (New YorK: Van Norstrand Reinhold Company, 1955.)

Nodal Loads on a Cantilever Beam The complete model and results for this test case are in file mstvl001.neu. Determine the deflection of a beam at the free end. Determine the stress at the end of the beam.

Test Case Data and Information Element Types bar

Units Inch

Model Geometry Length=480 in

Cross Sectional Properties •

Area = 30 x 30 in

•

Iy =Iz = 67500 in4

Material Properties •

E = 30 E+06 psi

Finite Element Modeling •

5 nodes

•

4 successive bar elements along the X axis

 Boundary Conditions Constraints Constrain the left end (node 1) of the beam in all six degrees.

Loads Set nodal force to 50,000 lb. in the negative Y direction.

Solution Type Statics

Results Beam End A1 Z Shear Force Stress (Node 1)

T2 Translation (Node 5)

Bench Value

5333.3

0.91022

FEMAP Structural

5333.3

0.913

0%

0.30%

Difference

Reference • Beer and Johnston, Mechanics of Materials, (New York: McGraw–Hill, Inc., 1992.) p. 716.

Axial Distributed Load on a Linear Beam The complete model and results for this test case are in file mstvl002.neu. Determine the stress, elongation, and constraint force due to an axial loading along a linear beam.

Test Case Data and Information Element Type bar

Units Inch

Model Geometry Length = 300 in

Cross Sectional Properties •

Area = 9 in2

•

square cross section (3 in x 3 in)

•

I = 6.75 in4

Material Properties E = 30E+6 psi

Finite Element Modeling •

31 nodes

 •

30 bar elements along the X axis, each 10 inches long.

Boundary Conditions Constraints Constrain one end of the beam (node 1) in all translations and rotations.

Loads Set the axial distributed load (force per unit length) to 1000lb/in for the 10–inch long element (element 30) furthest from the constrained end.

Solution Type Statics



Results Beam End A1 Axial Stress (Node 1)

T1 Constraint Force (Node 1)

T1 Translation (Node 2)

Bench value

1111.1

0.0111111

-10,000

FEMAP Structural

1111.1

0.0109258

-10,000

0

1.6%

0

Difference

Reference • Beer and Johnston, Mechanics of Materials, (New York: McGraw–Hill, Inc., 1992.) p. 76.

Distributed Loads on a Cantilever Beam The complete model and results for this test case are in file mstvl003.neu. Determine the deflection of a beam at the free end. Determine the stress at the midpoint of the beam and the reaction force at the restrained end.

Test Case Data and Information Element Type bar

Units Inch

Model Geometry •

Length = 480 in

Cross Sectional Properties •

Area = 900 in2

•

square cross section (30 in x 30 in)

•

Iy = Iz = 67500 in4

Material Properties E = 30 E+06 psi

Finite Element Modeling •

9 nodes

 •

8 successive bar elements along the X axis

Boundary Conditions Constraints Constrain the left end of the beam (node 1) in all translations and rotations.

Loads Define a distributed load on the elements of 250 lb/in in the negative Y direction.

Solution Type Statics

Results Beam End A1Z Bend Stress (node 1)

Total Translation (node 5)

Total Constraint Force (lb)

Bench Value

6,400.0

0.8190

120,000

FEMAP Structural

6,400.0

0.8225*

120,000

Difference

0.0%

0.43%

0

 * Includes shear deformation which is neglected in theoretical value. Reference • Beer and Johnston, Mechanics of Materials, (New York: McGraw–Hill, Inc., 1992.) p. 716.

Moment Load on a Cantilever Beam The complete model and results for this test case are in file mstvl004.neu. Determine the deflection of a beam at the free end. Determine the bending stress of the beam and the reaction force at the restrained end.

Test Case Data and Information Element Type bar

Units Inch

Model Geometry Length = 480 in

Cross Sectional Properties •

Area = 900 in2

•

square cross section (30 in x 30 in)

•

Iy = Iz = 67500 in4

Material Properties E = 30 E+06 psi

Finite Element Modeling •

9 nodes

 •

8 successive bar elements along the X axis.

Boundary Conditions Constraints Constrain the left end of the beam (node 1) in all translations and rotations.

Loads Set the Z–moment of the end node (node 5) to 2.5e+6 in–lb.

Solution Type Statics



Results Beam End A1 Z Bend Total Translation (in) Stress (psi) (node 5) (node 1)

Total Constraint Moment (lb.) (node 1)

Bench Value

555.6

0.1422

2.5E+06

FEMAP Structural

555.6

0.1422

2.5E+06

Difference

0

0

0

Reference • Beer and Johnston, Mechanics of Materials, (New York: McGraw–Hill Inc., 1992.) p. 716.

Thermal Strain, Displacement, and Stress on Heated Beam The complete model and results for this test case are in file mstvl007.neu. A beam originally 1 meter long and at -50° C is heated to 25° C. Determine the displacement and thermal strain on a cantilever beam. In case 1, fix the beam at the free end. In case 2, fix the beam at both ends. In both cases, determine the displacement, constraint forces, and stresses along the beam.

Test Case Data and Information Element Type bar

Units SI - meter

Model Geometry Length = 1 m

Cross Sectional Properties Area = 0.01 m2

Material Properties •

E = 2.068E+11 PA

•

Coeff. of thermal expansion = 1.2E-05 m/(m-C)

•

v = 0.3

Finite Element Modeling •

11 nodes

 •

10 bar elements on the X axis.

Boundary Conditions Constraints •

Case 1: Constrain the node on one end (node 1) of the beam in all translations and rotations.

•

Case 2: Constrain the nodes on both ends (nodes 1 and 11) of the beam in all translations and rotations.

Loads Set the temperature on all nodes to 25°C. Set the reference temperature to -50°C.

Solution Type Statics



Results Case: One Fixed End Total Translation (Node 11) (m)

Beam End A1 Axial Strain

Bench Value

9E-04

9E-04

FEMAP Structural

9E-04

9E-04

0

Difference

0

Case: Both Ends Fixed

Total Translation (m)

Total Constraint Force(N) (node 1)

Beam End A1 Axial Stress (Pa)

Bench Value

0

1.86+06

–1.86E+08

FEMAP Structural

0

1.86+06

–1.86E+08

Difference

0

0

0

Reference • Beer and Johnston, Mechanics of Materials, (New York: McGraw–Hill, Inc., 1992.) p. 65.

Uniformly Distributed Load on Linear Beam The complete model and results for this test case are in file mstvl008.neu. A beam 40 feet long is restrained and loaded with a distributed load of –833 lb. Determine the beam end torque stress and the deflection at the middle of the beam.

Test Case Data and Information Element Type bar

Units Inch

Model Geometry Length = 480 in

Cross Sectional Properties •

Rectangular cross section (1.17 in x 43.24 in)

•

Iz = 7892 in4

Material Properties •

E = 30E6 psi

Finite Element Modeling •

5 nodes

 •

4 successive bar elements that are each 10 feet long

Boundary Conditions Constraints Constrain nodes 2 and 4 in five degrees of freedom. Do not constrain rotation about Z.

Loads Define a distributed load (force per unit length) of -833 lb. (global negative Y direction) on the elements 1 and 4.

Solution Type Statics



Results Total Translation (in) (node 3)

Beam End A1 Z Bend Stress (psi) (node 3)

Bench Value

0.182

16,439

FEMAP Structural

0.182

16,439

Difference

0

0

Reference • Timoshenko, S., Strength of Materials, Part I, Elementary Theory and Problems, (New York: Van Norstrand Reinhold Company, 1955.) p. 98.

Membrane Loads on a Plate The complete model and results for this test case are in file mstvl009.neu. A circle is scribed on an unstressed aluminum plate. Forces acting in the plane of the plate cause normal stresses. Determine the change in the length of diameter AB and of diameter CD.

Test Case Data and Information Element Types plate

Units Inch

Model Geometry •

Length = 15 in

•

Diameter = 9 in

•

Thickness = 3/4 in

Material Properties •

E = 10 E+06 psi

•

Poisson’s ratio = 1/3

•

F(x)/l = 9,000 lb./in

•

F(z)/l = 15,000 lb./in

 Finite Element Modeling Create 1/4 of the model and apply symmetry boundary conditions. Then multiply the answer by 2 for correct results. Remember to account for the ratio of the circle diameter to plate length.

Boundary Conditions Constraints Constrain nodes along adjacent sides of the plate to allow only translation along the corresponding axis. •

Node 1: Fully constrain in all translations and rotation.

•

Nodes 2-6: Constrain in the Y and Z translations and the X and Z rotations.

•

Nodes 12, 13, 19, 25, 31: Constrain in the X and Y translations and the X and Z rotations.

Loads Set the elemental edge load to 9,000 lb./in in the X direction and 15,000 lb/in in the Z direction.

 Solution Type Statics

Results T1 Translation (in)

T3 Translation (in)

Bench Value

4.8E-03

14.4E-03

FEMAP Structural

4.8E-03

14.4E-03

Difference

0

0

Post Processing •

(T1 translation at node 7 - T1 translation at node 10) x2 = (.004-.0016) x2 = .0048

•

(T3 translation at node 7 - T3 translation at node 24) x2 = (.012-.0048) x2 = .0144

Reference • Beer and Johnston, Mechanics of Materials, (New York: McGraw–Hill, Inc., 1992.) p. 85.

Thin Wall Cylinder in Pure Tension The complete model and results for this test care are in file mstvl014.neu. Determine the stress and deflection of a thin wall cylinder with a uniform axial load.

Test Case Data and Information Element Type linear quadrilateral plate

Units Inch

Model Geometry •

R = 0.5 in

•

Thickness = 0.01 in

•

y = 1.0 in

Material Properties •

E = 10000 psi

•

v = 0.3

Finite Element Modeling •

25 nodes

•

Create 1/4 model of the cylinder with 16 linear quadrilateral plate elements and symmetry boundary conditions.

 Boundary Conditions Constraints •

Constrain node 1 in the X and Z translation and the Z rotation.

•

Constrain nodes 2-4 in the Z translation.

•

Constrain node 5 in the Y and Z translation and Z rotation.

•

Constrain nodes 6, 11, 16, and 21 in the X translation and Z rotation.

•

Constrain nodes 10, 15, 20, and 25 in the Y translation and Z rotation.

Loads •

Nodal forces of p/(pi)D = 3.1831 where p = 10 psi; Apply the following nodal forces:

•

Nodes 21, 25: .9757 pounds

•

Nodes 22, 23, 24: 1.9509 pounds

Solution Type Statics

Results Top Y Normal Stress T3 Translation (in) (psi)

T1 Translation (in)

1000.0

0.1

-0.015

FEMAP Structural 1000.0

0.1

-0.015

0

0

Bench Value

Difference

0

 Reference • Roark, R. and Young, W., Formulas for Stress and Strain, 6th Edition, (New York: McGraw–Hill Book Company, 1989.) p. 518, Case 1a.

Thin Shell Beam Wall in Pure Bending The complete model and results for this test case are in file mstvl015.neu. Determine the maximum stress, maximum deflection, and strain energy of a thin shell beam wall with a uniform bending load.

Test Case Data and Information Element Type linear quadrilateral plate

Units Inch

Model Geometry •

Length = 30 in

•

Width = 5 in

•

Thickness = 0.1 in

Material Properties •

E = 30E6 psi

•

v = 0.03

Finite Element Modeling •

14 nodes

 •

6 linear quadrilateral plate elements

Boundary Conditions Constraints Constrain the nodes at one end (nodes 7 and 14) in all translations and rotations.

Out–of–plane Loads Apply nodal forces (nodes 1 and 8) of p/w = 1.2 lbs/in. where p = 6.0 lb

Solution Type Statics



Results Plate Bottom Major Stress (psi) Node 7

Total Strain Energy (lb in)

4.320

21600

12.96

FEMAP Structural 4.242

20983

12.73

1.39%

2.16%

T3 Translation (in) Node 1 Bench Value

Difference

2.17%

Reference • Shigley, J. and Mitchel L., Mechanical Engineering Design, 4th Edition, (New York: McGraw–Hill, Inc., 1983.) pp. 134, 804.

Strain Energy of a Truss The complete model and results for this test case are in file mstvl016.neu. Determine the strain energy of a truss. The cross–sectional area of the diagonal members is twice the cross–sectional area of the horizontal and vertical members.

Test Case Data and Information Element Type rod

Units Inch

Model Geometry •

Length = 10 in

Cross Sectional Properties Cross sectional area (A) = 0.01 in2

Material Properties E = 30E6 psi

Finite Element Modeling •

4 nodes

•

5 rod elements

 Boundary Conditions Constraints •

Constrain node 1 in the X, Y, and Z translations and the X and Y rotations.

•

Constrain node 3 in the Y and Z translations and the X and Y rotations.

Loads •

Apply nodal force in Y direction on node 2; p = 300 lb

Solution Type Statics

Results Total Strain Energy (lb in) Bench Value

5.846

FEMAP Structural

5.846

Difference

0

Reference • Beer and Johnston, Mechanics of Materials, (New York: McGraw–Hill, Inc., 1992.) p. 588.

Linear Statics Verification Using Standard NAFEMS Benchmarks The purpose of these linear statics test cases is to verify the function of the FEMAP Structural Statics Analysis software using standard benchmarks published by NAFEMS (National Agency for Finite Element Methods and Standards, National Engineering Laboratory, Glasgow, U.K.). These standard benchmark tests were created by NAFEMS to stretch the limits of the finite elements in commercial software. All results obtained using the FEMAP Structural Statics Analysis software compare favorably with other commercial finite element analysis software. Results of these test cases using other commercial finite element analysis software programs are available from NAFEMS. A detailed discussion of the linear statics NAFEMS benchmarks can be found in the NAFEMS publication Background to Benchmarks, cited below. The results for all of these test cases illustrate the need for adequate mesh refinement for obtaining accurate stresses, especially when using linear elements. The linear triangular and linear tetrahedral elements are particularly poor performers for stress analysis and are not generally recommended.

Understanding the Test Case Format Each test case is structured with the following information: •

test case data and information - physical and material properties - finite element modeling (modeling procedure or hints) - units - finite element modeling information - boundary conditions (loads and constraints) - solution type

•

results

•

reference

Note: The node numbers listed in each case refer to the node numbers in the neutral (.neu) files associated with this guide. If you remesh a model, or rebuild that model from scratch, your node numbering may differ. References The following references have been used in these test cases:

 •

NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks, (Glasgow: NAFEMS, Rev. 3, 1990.)

•

Davies, G. A. O., Fenner, R. T., and Lewis, R. W., Background to Benchmarks, (Glasgow: NAFEMS, 1993).

Elliptic Membrane The complete model and results for this test case are in the following files: •

le101.neu (quadrilateral plane strain)

•

le102.neu (triangular plane strain)

•

le103.neu (quadrilateral plate)

This test is a linear elastic analysis of an elliptic membrane (shown below) using coarse and fine meshes of plane strain elements and plate elements. The plane strain elements use a plane stress element formulation. It provides the input data and results for NAFEMS Standard Benchmark Test LE1.

B

A

Y

X C

D

Ellipses:

2 x 2 Ellipse AC:  --- + y = 1 2

y 2 x 2 Ellipse BD:  ---------- +  ---------- = 1 2.75 3.25

Test Case Data and Information Physical and Material Properties •

Thickness = 0.1 m

•

Isotropic material

•

E = 210 x 103 MPa

 •

v = 0.3

Units SI

Finite Element Modeling •

plane strain (with plane stress element formulation) - linear and parabolic quadrilaterals coarse and fine mesh

•

plane strain (with plane stress element formulation) - linear and parabolic triangles coarse and fine mesh

•

plate - linear and parabolic quadrilaterals - coarse and fine mesh

 The fine mesh is created by approximately halving the coarse mesh.

Linear Triangle B

B

Parabolic Triangle

Fine Mesh A

A C

D

C

D

C

D

B

B Coarse Mesh A

A C

D

Linear Quadrilateral

Parabolic Quadrilateral B

B Fine Mesh

A

A C

D

C

D

C

D

B

B Coarse Mesh

A

A C

D

Boundary Conditions Constraints •

Constrain the nodes along edge AB in the X translation.

•

Constrain the nodes along edge CD in the Y translation.

 Loads •

Uniform outward pressure on the elements on outer edge BD = 10MPa

•

Inner curved edge AC is unloaded

Solution Type Statics

Results Output - Plate Mid Y Normal Stress at point D

Node #

Element Type & Mesh

NAFEMS Bench Value (MPa)

FEMAP Structural Result (MPa)

Plane Strain Elements with a Plane Strain Formulation (le101): Node 4 Node 204 Node 104 Node 304

Node 4 Node 204 Node 104 Node 304

linear quad - coarse mesh linear quad - fine mesh parabolic quad - coarse mesh parabolic quad - fine mesh Plane Strain Elements with a Plane Strain Formulation (le102): linear triangle - coarse mesh linear triangle - fine mesh parabolic triangle - coarse mesh parabolic triangle – fine mesh

92.7 92.7 92.7 92.7

62.8 80.3 88.3 90.7

92.7 92.7 92.7 92.7

54.2 72.0 93.0 94.0

 Node 4 Node 204 Node 104 Node 304

Plate Elements (le 103): linear quad - coarse mesh linear quad - fine mesh parabolic quad - coarse mesh parabolic quad - fine mesh

92.7 92.7 92.7 92.7

66.4 82.3 88.6 91.7

References • NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks, (Glasgow: NAFEMS, Rev. 3, 1990.) Test No. LE1. •

Davies, G. A. O., Fenner, R. T., and Lewis, R. W., Background to Benchmarks, (Glasgow: NAFEMS, 1993).

Cylindrical Shell Patch Test The complete model and results for this test case are in the following files: •

le201a.neu (linear plate, case 1)

•

le201b.neu (parabolic plate, case 1)

•

le202a.neu (linear plate, case 2)

•

le202b.neu (parabolic plate, case 2)

This test is a linear elastic analysis of a cylindrical shell (shown below) using plate elements and two different loadings. It provides the input data and results for NAFEMS Standard Benchmark Test LE2.

Test Case Data and Information Physical and Material Properties •

Thickness = 0.01 m

•

Isotropic material

•

E = 210 x 103 MPa

•

v = 0.3

Units SI

Finite Element Modeling •

le201a and le202a: 9 nodes, 4 linear quadrilateral plates

•

le201b and le202b: 21 nodes, 4 parabolic quadrilateral plates Linear Quadrilaterals

A

B

Parabolic Quadrilaterals A

E

D

B

E C

D

C

 Boundary Conditions Constraints Fully constrain the nodes on edge AB in all translations and rotations. Constrain the nodes on edge AD and edge BC in the Z translation and X and Y rotations.

Case 1 Loading: •

Nodal moments along DC = 1.0 kNm/m: Node 3 = -125 Node 4 = -250 Node 9 = -125

Case 2 Loading: •

Nodal forces: Nodes 3, and 9 = 75,000N Node 4 = 150,000N

 •

Apply an elemental pressure on elements 1-4 = 600,000Pa

Solution Type Statics

Results Output - Plate Top Major Stress at point E (node 2)

Plate Element & Loading linear plate - case 1 (le201a) linear plate - case 2 (le202a) parabolic plate - case 1 (le201b) parabolic plate - case 2 (le202b)

NAFEMS Bench Value (MPa) 60.0 60.0 60.0 60.0

FEMAP Structural Result (MPa) 57.9 66.0 * 54.8 55.7 *

*Since the shapes of the plates are an approximation to a cylindrical surface, an edge load will not be in the correct direction. To get this result, the edge load must be input as nodal loads in the tangential direction. References • NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks, (Glasgow: NAFEMS, Rev. 3, 1990.) Test No. LE2. •

Davies, G. A. O., Fenner, R. T., and Lewis, R. W., Background to Benchmarks, (Glasgow: NAFEMS, 1993).

Laminate Strip The complete model and results for this test case are in the following file: •

r0031.neu

This test is a linear statics analysis of plate using plate elements with a laminate material. It provides the input data and results for NAFEMS Report R0031.

Test Case Data and Information Geometry 0° fiber direction Y

X 10

15

15

10 C

10N/mm 1 Z X

A

E

0.1 0.1 0.1

0° 90° 0°

0.4

90°

0.1 0.1 0.1

0° 90° 0°

B

Material Properties Laminate material: E = 1.0E5 MPa

ν 12 = 0.4

E2 = 5.0E3 MPa

G 12 = 3.0E3 MPa

ν 23 = 0.3

G 33 = 2.0E3 MPa

ν12 ν 21 ------- = ------E1 E2

D F E

 Units SI

Finite Element Modeling 8 x 40 4-noded shells (quarter model)

Boundary Conditions Constraints The one quarter model is: •

simply supported at A (Z=0)

•

reflective symmetry about X=25 and Y=5

Loads Line load of 10N/mm at C (X=25, Z=1).

Solution Type Statics

Results

Results Z deflection at E Bending stress at E Bending stress at F Interlaminar shear stress at D Shear stress at F

NAFEMS Bench Value (MPa) -1.06 683.9 -4.1 -

FEMAP Structural Result (MPa) -1.06 *668 601 **-4.1 -2.2

*Value extrapolated from FEMAP Structural results at F. (FEMAP Structural calculates stress at the center of the ply (F)). **Recovered from post-processing. Reference • NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks, (Glasgow: NAFEMS, Rev. 3, 1990.) Test No. R0031.

Hemisphere-Point Loads The complete model and results for this test care are in the following files: •

le301.neu (linear quadrilateral plate, coarse mesh)

•

le302.neu (linear quadrilateral plate, fine mesh)

•

le303.neu (parabolic quadrilateral plate, coarse mesh)

•

le304.neu (parabolic quadrilateral plate, fine mesh)

This test is a linear elastic analysis of hemisphere point loads (shown below) using coarse and fine meshes of plate elements. It provides the input data and results for NAFEMS Standard Benchmark Test LE3.

Test Case Data and Information Physical and Material Properties •

Thickness = 0.04 m

•

Isotropic material

•

E = 68.25 x 103 MPa

•

v = 0.3

Units SI

Finite Element Modeling plate - linear & parabolic quadrilaterals - coarse & fine mesh equally spaced nodes on AC, CE, EA   10 Point G at X = Y = Z =  -----1- Node 7  ---   2 3



Coarse Mesh

Fine Mesh E

E D

F

F

D

G G

A

A

C B

C B

Boundary Conditions Constraints •

Fully constrain point E in all translations and rotations.

•

Constrain the nodes along edge AE (symmetry about X–Z plane) in the Y translation, and X and Z rotations.

•

Constrain the nodes along edge CE (symmetry about Y–Z plane) in the X translation, and Y and Z rotations.

Loads •

Concentrated radial load outward at A = 2KN

 •

Concentrated radial load inward at C = 2KN

Solution Type Statics

Results

Test Case Number le301 le302 le303 le304

Plate Element & Mesh

linear quadrilateral plate - coarse mesh linear quadrilateral plate - fine mesh parabolic quadrilateral plate - coarse mesh parabolic quadrilateral plate - fine mesh

NAFEMS Bench Value(m) 0.185 0.185 0.185 0.185

FEMAP Structural Result at node 1 (point A) T1 Translation (m) 0.113 0.185 0.0861 0.171

References • NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks, (Glasgow: NAFEMS, Rev. 3, 1990.) Test No. LE3. •

Davies, G. A. O., Fenner, R. T., and Lewis, R. W., Background to Benchmarks, (Glasgow: NAFEMS, 1993).

Z–Section Cantilever The complete model and results for this test case are in the following files: •

le501.neu (linear quadrilateral plate)

•

le502.neu (parabolic quadrilateral plate)

This test is a linear elastic analysis of a Z–section cantilever (shown below) using plate elements. It provides the input data and results for NAFEMS Standard Benchmark Test LE5.

Test Case Data and Information Physical and Material Properties •

Thickness = 0.1 m

•

Isotropic material

•

E = 210 x 103 MPa

•

v = 0.3

Units SI

Finite Element Modeling •

Test 1: 36 nodes, 24 linear quadrilateral plate elements

•

Test 2: 95 nodes, 24 parabolic quadrilateral plate elements

Boundary Conditions Constraints •

Fully constrain the nodes on edges B1, B2, B3 in all translations and rotations.

 Loads •

Torque of 1.2MN applied at end C by two nodal forces (at nodes 9 and 27) of 0.6MN B1

B2 B3

C

Solution Type Statics

Results Output - Plate Top Von Mises Stress (σxx), point A, node 30 (compression)

Plate Element & Loading linear quad - point A/node 30 parabolic quad - point A/node 30

NAFEMS Bench Value (MPa) -108 -108

FEMAP Structural Result (MPa) -117.3 -109.2

References • NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks, (Glasgow: NAFEMS, Rev. 3, 1990.) Test No. LE5. •

Davies, G. A. O., Fenner, R. T., and Lewis, R. W., Background to Benchmarks, (Glasgow: NAFEMS, 1993).

Skew Plate Normal Pressure The complete model and results for this test case are in the following files: •

le601.neu (linear and parabolic quadrilateral)

•

le602.neu (linear and parabolic triangle)

This test is a linear elastic analysis of a plate (shown below) using plate elements. It provides the input data and results for NAFEMS Standard Benchmark Test LE6.

10m o 150

D C

o 30 E A

B

Test Case Data and Information Physical and Material Properties •

Thickness = 0.01m

•

Isotropic material

•

E = 210 x 103 MPa

•

v = 0.3

Units SI

 Finite Element Modeling •

plate - linear and parabolic quadrilaterals - coarse and fine mesh

•

plate - linear and parabolic triangles - coarse and fine mesh

Boundary Conditions Constraints (le601) •

Constrain nodes 1, 10, 35, and 44 in the X, Y, and Z translations.

•

Constrain nodes 4, 13, 38, 47 in the X and Z translations.

 •

Constrain all other nodes in the Z translation.

Constraints (le602) •

Fully constrain nodes 1, 10, 35, 44 in all directions and rotations.

•

Constrain all other nodes in the Z translation.

Loads •

Elemental pressure = -0.7KPa in the Z–direction

 Solution Type Statics

Results Output - Plate Bottom Major Stress on the bottom surface at the plate center.

Test Case Name le601 le601 le601 le601 le602 le602 le602 le602

Node # Node 9 Node 18 Node 43 Node 52 Node 9 Node 18 Node 43 Node 52

Plate Element & Mesh linear quad - coarse mesh linear quad - fine mesh parabolic quad - coarse mesh parabolic quad - fine mesh linear triangle - coarse mesh linear triangle - fine mesh parabolic triangle - coarse mesh parabolic triangle - fine mesh

NAFEMS Bench Value (MPa) 0.802 0.802 0.802 0.802 0.802 0.802 0.802 0.802

FEMAP Structural Result (MPa) 0.365 0.714 1.055 0.791 0.390 0.709 0.847 0.822

References • NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks, (Glasgow: NAFEMS, Rev. 3, 1990.) Test No. LE6. •

Davies, G. A. O., Fenner, R. T., and Lewis, R. W., Background to Benchmarks, (Glasgow: NAFEMS, 1993).

Thick Plate Pressure The complete model and results for this test case are in the following files: •

le1001.neu (linear and parabolic brick)

•

le1002.neu (linear and parabolic wedge)

•

le1003.neu (linear and parabolic tetrahedron)

This article provides the input data and results for NAFEMS Standard Benchmark Test LE10. This test is a linear elastic analysis of a thick (shown below) using coarse and fine meshes of solid elements. B’ B

B

A’ A

A

D’ C’ D

D

C

C

Ellipses: 2 x 2 Ellipse AD:  --- + y = 1 2

y 2 x 2 Ellipse BC:  ---------- +  ---------- = 1 2.75 3.25

Test Case Data and Information Physical and Material Properties •

Isotropic material

•

E=210x103 MPa

•

v = 0.3

 Units SI

Finite Element Modeling •

Solid brick

•

Solid wedge

•

Solid tetrahedron

Solid Brick Linear and parabolic, coarse and fine mesh.

Solid Wedge Linear and parabolic, coarse and fine mesh.



Solid Tetrahdron Linear and parabolic, fine mesh.

 Boundary Conditions Constraints •

Constrain the nodes on faces DCD’C’ and ABA’B’ in the X and Y translations.

•

Constrain the nodes on face BCB’C’ in the X and Y translation.

•

Constrain the nodes along the mid–plane in the Z translation.

Loads •

Uniform normal elemental pressure on the elements on the upper surface of the plate = 1MPa

•

Inner curved edge AD unloaded

Solution Type Statics



Results Output - Solid Y normal stress at point D3σyy

Test Case Name le1001 le1001 le1001 le1001 le1002 le1002 le1002 le1002 le1003 le1003

Node # N4 N204 N104 N304 N4 N204 N104 N304 N40 N171

Element Type & Mesh linear brick - coarse mesh linear brick - fine mesh parabolic brick - coarse mesh parabolic brick - fine mesh linear wedge - coarse mesh linear wedge - fine mesh parab wedge - coarse mesh parab wedge - fine mesh linear tetra - fine mesh parabolic tetra - fine mesh

NAFEMS Bench Value (MPa) -5.38 -5.38 -5.38 -5.38 -5.38 -5.38 -5.38 -5.38 -5.38 -5.38

FEMAP Structural Result (MPa) -6.31 -6.01 -5.73 -5.84 -3.52 -4.97 -5.53 -6.10 -2.41 -5.29

References • NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks, (Glasgow: NAFEMS, Rev. 3, 1990.) Test No. LE10. •

Davies, G. A. O., Fenner, R. T., and Lewis, R. W., Background to Benchmarks, (Glasgow: NAFEMS, 1993)

Solid Cylinder/Taper/Sphere–Temperature The complete model and results for this test case are in the following files: •

le1101a.neu (linear brick, coarse mesh)

•

le1101b.neu (linear brick, fine mesh)

•

le1102a.neu (parabolic brick, coarse mesh)

•

le1102b.neu (parabolic brick, fine mesh)

•

le1103a.neu (linear wedge, coarse mesh)

•

le1103b.neu (linear wedge, fine mesh)

•

le1104a.neu (parabolic wedge, coarse mesh)

•

le1104b.neu (parabolic wedge, fine mesh)

•

le1105a.neu (linear tetrahedron, coarse mesh)

•

le1105b.neu (linear tetrahedron, fine mesh)

•

le1106a.neu (parabolic tetrahedron, coarse mesh)

•

le1106b.neu (parabolic tetrahedron, fine mesh)

This test is a linear elastic analysis of a solid cylinder with a temperature gradient (shown below) using coarse and fine meshes of solid elements. It provides the input data and results for NAFEMS Standard Benchmark Test LE11.

Test Case Data and Information Physical and Material Properties •

Isotropic material

•

E = 210 x 103 MPa

•

v = 0.3

•

a = 2.3 x 10-4/oC

Units SI

 Finite Element Modeling •

Solid brick - linear (8–noded) and parabolic (20–noded) - coarse and fine mesh

•

Solid tetrahedron - linear (4–noded) and parabolic (10–noded) - coarse and fine mesh

•

Solid wedge - linear (6–nodes) and parabolic (15–noded) - coarse and fine mesh

Solid Brick Coarse and fine mesh:

Coarse and fine mesh:

Boundary Conditions Constraints •

Constrain the nodes on the XZ plane and on the opposite face in the Y translation.

•

Constrain the nodes on the YZ plane in the Z translation.

•

Constrain the nodes on the XY plane in the X translation.



Loads •

Nodal temperatures: linear temperature gradient in the radial and axial direction

2

1 --2 2

T°C = ( X + Y ) + Z

 Solution Type Statics

Results Output - Solid Y Normal Stress at point A. Note that the Y direction in the models corresponds to the Z direction in NAFEMS.

Case le1101a le1101b le1102a le1102b le1103a le1103b le1104a le1104b le1105a le1105b le1106a le1106b

Node # at Point A 30 71 67 159 33 74 71 187 8 8 8 8

Element Type & Mesh linear brick - coarse mesh linear brick - fine mesh parabolic brick - coarse mesh parabolic brick - fine mesh linear wedge - coarse mesh linear wedge - fine mesh parabolic wedge - coarse mesh parabolic wedge - fine mesh linear tetra - coarse mesh linear tetra - fine mesh parabolic tetra - coarse mesh parabolic tetra - fine mesh

NAFEMS Bench Value (MPa) -105 -105 -105 -105 -105 -105 -105 -105 -105 -105 -105 -105

FEMAP Structural Result (MPa) -95.7 -99.5 -93.9 -105.9 -9.49 -46.9 -88.5 -96.8 -31.4 -65.2 -89.6 -97.2

References • NAFEMS Finite Element Methods & Standards, The Standard NAFEMS Benchmarks, (Glasgow: NAFEMS, Rev. 3, 1990.) Test No. LE11. •

Davies, G. A. O., Fenner, R. T., and Lewis, R. W., Background to Benchmarks, (Glasgow: NAFEMS, 1993).



Normal Modes/Eigenvalue Verification Using Theoretical Solutions The purpose of these normal mode dynamics test cases is to verify the function of the FEMAP Structural Normal Modes/Eigenvalue Analysis software using theoretical solutions. The test cases are relatively simple in form and most of them have closed–form theoretical solutions. The theoretical solutions shown in these examples are from well known engineering texts. For each test case, a specific reference is cited. All theoretical reference texts are listed at the end of this topic. The finite element method is very flexible in the types of physical problems represented. The verification tests provided are not exhaustive in exploring all possible problems, but represent common types of applications. This overview provides information on the following: •

understanding the test case format

•

understanding comparisons with theoretical solutions

•

references

Understanding the Test Case Format Each test case is structured with the following information: •

test case data and information - physical and material properties - finite element modeling (modeling procedure or hints) - units - solution type - element type - boundary conditions (loads and constraints)

•

results

•

references (text from which a closed–form or theoretical solution was taken)

Note: The node numbers listed in each case refer to the node numbers in the neutral (.neu) files associated with this guide. If you remesh a model, or rebuild that model from scratch, your node numbering may differ.



Understanding Comparisons with Theoretical Solutions While differences in finite element and theoretical results are, in most cases, negligible, some tests would require an infinite number of elements to achieve the exact solution. Elements are chosen to achieve reasonable engineering accuracy with reasonable computing times. Results reported here are results which you can compare to the referenced theoretical solution. Other results available from the analyses are not reported here. Results for both theoretical and finite element solutions are carried out with the same significant digits of accuracy. The closed–form theoretical solution may have restrictions, such as rigid connections, that do not exist in the real world. These limiting restrictions are not necessary for the finite element model, but are used for comparison purposes. Verification to real world problems is more difficult but should be done when possible. The actual results from the FEMAP Structural software may vary insignificantly from the results presented in this document. This variation is due to different methods of performing real numerical arithmetic on different systems. In addition, it is due to changes in element formulations which SDRC has made to improve results under certain circumstances. References The following references have been used in the Normal Mode Dynamics Analysis verification problems presented: •

Blevins, R., Formulas For Natural Frequency and Mode Shape, 1st Edition, (New York: Van Norstrand Reinhold Company, 1979.)

•

Timoshenko and Young, Vibration Problems in Engineering, (New York: Van Norstrand Reinhold Company, 1955.)

•

Tse, F., Morse, I., and Hinkle, R., Mechanical Vibrations, Theory and Applications, (Boston: Allyn and Bacon, Inc., 1978.)

•

Tse, F., Morse, I., and Hinkle, R., Mechanical Vibrations, 2nd Edition, (Boston: Allyn and Bacon, Inc., 1978.)

Undamped Free Vibration - Single Degree of Freedom The complete model and results for this test case are in file mstvn002.neu. Determine the natural frequency of the system.

Test Case Data and Information Element Types •

rigid

•

mass

•

DOF springs

Units SI - meter

Model Geometry •

Length = 0.5 m

•

a = 0.3 m

Physical Properties •

mass = 20 Kg

•

k = 8 KN/m

Finite Element Modeling •

Create 5 rigid elements along the X axis. Each rigid should be 0.1m long.

•

Create a mass element on the end node.

 •

Create 3 DOF spring elements 0.2m from the mass element.

Boundary Conditions Constraints Constrain node 6 in all directions except the Z rotation. Constrain all other nodes in the X and Y translations and in the Z rotation.

Solution Type Normal Modes/Eigenvalue – Guyan method

Results Frequency (Hz) Bench Value FEMAP Structural Difference

1.90985 1.90986 0.0%

 Reference • Tse, F., Morse, I., and Hinkle, R., Mechanical Vibrations, Theory and Applications, (Boston: Allyn and Bacon, Inc., 1978.) p. 75.

Two Degrees of Freedom Undamped Free Vibration - Principle Modes The complete model and results for this test case are in file mstvn003.neu. Determine the natural frequencies of a dynamic system with two degrees of freedom.

Test Case Data and Information Element Types •

DOF springs

•

mass

Units SI- meter

Physical Properties •

mass = 1 kg

•

k = 1 N/m

Finite Element Modeling •

Create four nodes on the Y axis.

•

Create DOF three springs with stiffness of 1 N/m and with a stiffness reference coordinate system being uniaxial.

 •

Create mass elements with a mass of 1 kg.

Boundary Conditions Constraints •

Constraint Set 1: Constrain nodes 1 and 4 in all DOF. On the other nodes, constrain all DOF except the Y translation.

•

Constraint Set 2: On the inner nodes, constrain the Y translation. Use this set as the Master (ASET) DOF set.

Solution Type Normal Modes/Eigenvalue – Guyan method



Results Frequency of Mode 1 (Hz) Bench Value FEMAP Structural Difference

0.159155 0.159155 0.00%

Frequency of Mode 2 (Hz) 0.2756644 0.2756644 0.00%

Reference • Tse, F., Morse, I., and Hinkle, R., Mechanical Vibrations, 2nd Edition, (Boston: Allyn and Bacon, Inc., 1978.) pp. 145-149.

Three Degrees of Freedom Torsional System The complete model and results for this test case are in file mstvn004.neu. Determine the natural frequencies of a dynamic system with three degrees of freedom.

Test Case Data and Information Element Types •

DOF springs

•

mass

Units SI - meter

Physical Properties •

J = J1 = J2 = J3 = 0.1 (mass)

•

k = k1 = k2 = k3 = 1 N*m (stiffness)

Finite Element Modeling •

Create four nodes on the X axis.

•

Create three DOF springs with stiffness of 1 N*m and with a stiffness reference coordinate system being uniaxial.

•

Create three mass elements with a mass coordinate system = 1 and with mass inertia system of: 0.1, 0.0, 0.0, 0.0, 0.0, 0.0.

 Boundary Conditions Constraints •

Constraint Set 1: On one end node (node 1), constrain all DOF. On the other nodes, constrain all DOF except RX.

•

Constraint Set 2: On the other nodes (nodes 2-4), constrain the DOF in RX. Use this set as the Master (ASET) DOF set.

Solution Type Normal Modes/Eigenvalue – Guyan method

Results Frequency of Mode 1 (Hz) Bench Value 0.223986 FEMAP Structural 0.223986 Difference 0.00%

Frequency of Mode 2 (Hz) 0.627595 0.627595 0.00%

Frequency of Mode 3 (Hz) 0.906901 0.906901 0.00%

Reference • Tse, F., Morse, I., and Hinkle, R., Mechanical Vibrations, 2nd Edition, (Boston: Allyn and Bacon, Inc., 1978.) pp. 153–155

Two Degrees of Freedom Vehicle Suspension System The complete model and results for this test case are in file mstvn005.neu. Determine the natural frequencies of dynamic system with two degrees of freedom. Degrees of freedom are one translational and one rotational.

Test Case Data and Information Element Types 5 nodes, 4 elements: •

2 DOF springs

•

1 mass element

•

1 rigid element

Units SI - meter

Model Geometry •

Length1 = 1.6 m

•

Length2 = 2.0 m

•

r = 1.4 m (radius of gyration; J=m*r*r)

Physical Properties •

mass = 1800 kg

•

K1 = 42000 N/m

•

K2 = 48000 N/m

Finite Element Modeling •

Create five nodes in the X–Y plane with coordinates: N1 = (0, 0) N2 = (L2, 0) N3 = (-L1, 0)

 N4 = (L2, -1) N5 = (-L1, -1) •

Create a DOF spring with stiffness of k1 between nodes 3 and 5.

•

Create a DOF spring with stiffness of k2 between nodes 2 and 4.

•

Create a mass element with a mass coordinate system = 1 and with mass inertia system of: 0.0, 0.0, 3528, 0.0, 0.0, 0.0.

•

Create a three–noded rigid element using node 1 as the master node and nodes 2 and 3 as the slave nodes.

Boundary Conditions Constraints •

Constraint Set 1: Constrain nodes 1-3 in the X and Z translation and X and Y rotations. Constrain nodes 4-5 in the X, Y, and Z translations.

•

Constraint Set 2 (Master (ASET) DOF Set): Constrain nodes 1-3 in the Y translation and Z rotation.

 Solution Type Normal Modes/Eigenvalue – Guyan method

Results Frequency of Mode 1 (Hz) Bench Value FEMAP Structural Difference

1.086347 1.086347 0.00%

Frequency of Mode 2 (Hz) 1.495612 1.495612 0.00%

Reference • Tse, F., Morse, I., and Hinkle, R., Mechanical Vibrations, 2nd Edition, (Boston: Allyn and Bacon, Inc., 1978.) pp. 150-153.

Cantilever Beam Undamped Free Vibrations The complete model and results for this test case are in file mstvn006.neu. Determine the natural frequencies of a cantilever beam.

Test Case Data and Information Element Type bar

Units Inch

Model Geometry •

Length = 100 in

•

Height = 2 in

Physical and Material Properties •

w = 1 lb/in

•

J = .10

•

Poisson’s ratio = .3

Calculated Data •

A = h2 = 4 in2

•

I = h4/12 = 1.33333

•

G = E/2 x 1/1+nu = 11538461.54

•

m = w/g = 2.59067375E-3

•

Ip = Ixx + Iyy = 2.66666

Finite Element Modeling •

Create 11 nodes on X axis.

 •

Create 10 bars between the nodes.

Boundary Conditions Constraints •

Fully constrain one end node (node 1) in all directions and rotations.

Solution Type Normal Modes/Eigenvalue – SVI method

Results Mode 1&2 3&4 5 6&7 8 9 & 10

Bench Values (Hz) 6.9533571 43.575945 64.684410 122.01391 193.85388 238.75784

FEMAP Structural (Hz) 6.951037 43.54267 64.66795 121.8567 195.6024 238.6964

Difference -0.033% -0.076% -0.254% -0.128% 0.901% -0.026%

Reference • Blevins, R., Formulas For Natural Frequency and Mode Shape, 1st Edition, (New York: Van Norstrand Reinhold Company, 1979) pp. 108,193.

Natural Frequency of a Cantilevered Mass The complete model and results for this test case are in file mstvn007.neu. Determine the natural frequencies of a dynamic system consisting of a massless bar element and a mass element at the end.

Test Case Data and Information Element Types •

bar

•

mass

Units Inch

Model Geometry •

Length = 30 in

Physical and Material Properties •

Mass = 0.5 lbm

•

E = 30E6 psi

•

Density = 1.0E-06

•

I = 1.5 in 4

Finite Element Modeling •

Create 2 nodes on the X axis with coordinates (0,0,0) and (30,0,0).

•

Create a bar between nodes with shear area ratio=0.

 •

Create a mass on one node with mass of 0.5 lbm.

Boundary Conditions Constraints •

−Constraint Set 1: On the wall end (at node 1), constrain all DOF. On the mass end, constrain the DOF in Z, RX, and RY.

•

Constraint Set 2: On the mass end node, constrain the DOF in Z, Y, and RZ. Use this set as the Master (ASET) DOF set.

Solution Type Normal Modes/Eigenvalue – Guyan method

Results Natural Frequency (Hz) Bench Value FEMAP Structural Difference

15.9155 15.9154 0.00%



Reference • Tse, F., Morse, I., and Hinkle, R., Mechanical Vibrations, 2nd Edition, (Boston: Allyn and Bacon, Inc., 1978.) p. 72

Normal Modes/Eigenvalue Verification Using Standard NAFEMS Benchmarks The purpose of these normal mode dynamics test cases is to verify the function of the FEMAP Structural Normal Modes/Eigenvalue solver using standard benchmarks published by NAFEMS (National Agency for Finite Element Methods and Standards, National Engineering Laboratory, Glasgow, U.K.). These standard benchmark tests were created by NAFEMS to stretch the limits of the finite elements in commercial software. All results obtained using the FEMAP Structural software compare favorably with other commercial finite element analysis software. Results of these test cases using other commercial finite element analysis software programs are available from NAFEMS.

Understanding the Test Case Format Each test case is structured with the following information: •

test case data and information - units - material properties - finite element modeling information - boundary conditions (loads and constraints) - solution type

•

results

•

reference

Note: The node numbers listed in each case refer to the node numbers in the neutral (.neu) files associated with this guide. If you remesh a model, or rebuild that model from scratch, your node numbering may differ. Reference The following reference has been used in these test cases: •

NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.)

Bar Element Test Cases The normal mode dynamics test cases using the standard NAFEMS benchmarks include these bar element test cases: •

"Pin-ended Cross - In-plane Vibration"

•

"Pin-ended Double Cross - In-plane Vibration"

•

"Free Square Frame - In-plane Vibration"

•

"Cantilever with Off-Center Point Masses"

•

"Deep Simply-Supported Beam"

•

"Circular Ring - In-plane and Out-of-plane Vibration"

•

"Cantilevered Beam"

Pin-ended Cross - In-plane Vibration The complete model and results for this test case are in file nf001ac.neu. This test is a normal modes/eigenvalue analysis of a pin–ended cross (shown below) using bar elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 1. B

A

C

.125 m

D .125 m 5.0 m

Attributes of this test are: •

coupling between flexural and extensional behavior

•

repeated and close eigenvalues

Test Case Data and Information Units SI

Cross Sectional Properties Key–in section: •

Area = .015625 m2

Shear ratio:

 •

Y=0

•

Z=0

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m η = 0.29 ( Poissons ratio ) G = 8.01x10

10

Finite Element Modeling •

17 nodes

•

16 bar elements; four elements per arm

Boundary Conditions Constraints •

Constrain points A, B, C, D (nodes 2, 3, 4, 5) in all directions except for the Z rotation.

•

Constrain node point Z (node 1) in the Z translation and X rotation.

 •

Constrain all other nodes (6-17) in the Z translation and X and Y rotations.

Solution Type Normal Modes/Eigenvalue – SVI method

Results Mode # 1 2, 3 4 5 6, 7 8

Ref. Value (Hz) 11.336 17.709 17.709 45.345 57.390 57.390

Mesh linear linear linear linear linear linear

NAFEMS Target Value (Hz) 11.336 17.687 17.715 45.477 57.364 57.683

FEMAP Structural (Hz) 11.336 17.687 17.715 45.477 57.364 57.683

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 1.

Pin-ended Double Cross - In-plane Vibration The complete model and results for this test case are in file nf002ac.neu. This test is a normal modes/eigenvalue analysis of a pin–ended double cross (shown below) using bar elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 2.

.125 m

C B

D .125 m

A

E

H

F

5.0m

G

5.0 m

Attributes of this test are: •

coupling between flexural and extensional behavior

•

repeated and close eigenvalues

Test Case Data and Information Units SI

Cross Sectional Properties Key–in section: •

Area = .015625 m2

Shear ratio:

 •

Y=0

•

Z=0

Material Properties 9 N E = 200x10 ------2m kg ρ = 8000 ------3m

Finite Element Modeling •

33 nodes

•

32 bar elements; four elements per arm

Boundary Conditions Constraints •

Constrain points A, B, C, D, E, F, G, H (nodes 2-9) in all directions except for the Z rotation.

•

Constrain all other nodes 1, (10-33) in the Z translation and X and Y rotations.

 The following figure shows the boundary conditions.

Solution Type Normal Modes/Eigenvalue – SVI method

Results Mode # 1 2, 3 4,5, 6,7,8 9 10, 11 12,13, 14,15, 16

Ref. Value (Hz)

Mesh

NAFEMS Target Value (Hz)

FEMAP Structural Result (Hz)

11.336 17.709 17.709

linear linear linear

11.336 17.687 17.715

11.336 17.687 17.715

45.345 57.390 57.390

linear linear linear

45.477 57.364 57.683

45.477 57.364 57.683

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 2.

Free Square Frame - In-plane Vibration The complete model and results for this test are in file nf003ac.neu. This test is a normal modes/eigenvalue analysis of a free square frame (shown below) using bar elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 3. .125 m

.125 m 10.0m

10.0 m

Attributes of this test are: •

coupling between flexural and extensional behavior

•

rigid body modes (3 modes)

•

repeated and close eigenvalues

Test Case Data and Information Units SI

Cross Sectional Properties Shear ratio: •

Y = 1.0

•

Z = 1.0

 Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m

Finite Element Modeling •

16 nodes

•

16 bar elements; four elements per arm

Boundary Conditions Constraints •

Constraint Set 1: Constrain all nodes in the Z translation and X and Y rotations.

 •

Constraint Set 2 (Kinematic DOF): Constrain nodes 1 and 3 in the X and Y translation and the Z rotation.

Solution Type Normal Modes/Eigenvalue – SVI method

Results Mode # 4 5 6, 7 8 9 10, 11

Ref. Value (Hz) 3.261 5.668 11.136 12.849 24.570 28.695

Mesh linear linear linear linear linear linear

NAFEMS FEMAP Structural Target Value (Hz) (Hz) 3.262 5.665 11.145 12.833 24.664 28.813

3.259 5.662 11.127 12.793 24.611 28.700

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 3.

Cantilever with Off-Center Point Masses The complete model and results for this test is in file nf004a.neu. This test is a normal modes/eigenvalue analysis of a cantilever with off–center point masses (shown below) using bar elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 4. Attributes of this test are: •

coupling between torsional and flexural behavior

•

inertial axis non–coincident with flexibility axis

•

discrete mass, rigid links

•

close eigenvalues

Test Case Data and Information Units SI

Cross Sectional Properties Shear ratio: •

Y = 1.128

•

Z = 1.128

Material Properties 9 N E = 200x10 ------2m

g ρ = 8000k ------3m ν = 0.3

Finite Element Modeling •

8 nodes

•

9 elements

 five bar elements along cantilever two mass elements two rigid elements

Boundary Conditions Constraints •

Fully constrain point A (node 1) in all directions.

Solution Type Normal Modes/Eigenvalue – SVI method



Results

Mode # 1 2 3 4 5 6

FEMAP NAFEMS Target Structural Result Ref. Value (Hz) Value (Hz) (Hz) 1.723 1.727 7.413 9.972 18.155 26.957

1.723 1.727 7.413 9.972 18.160 26.972

1.722 1.726 7.410 9.947 18.051 26.712

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 4

Deep Simply-Supported Beam The complete model and results for this test are in file nf005ac.neu. This test is a normal mode dynamic analysis of a deep simply–supported beam. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 5. Attributes of this test are: •

shear deformation and rotary inertial (Timoshenko beam)

•

possibility of missing extensional modes when using iteration solution methods

•

repeated eigenvalues

Test Case Data and Information Units SI

Cross Sectional Properties Shear Ratio •

Y = 1.176923

•

Z = 1.176923

Material Properties 9 N E = 200x10 ------2m

g ρ = 8000k ------3m ν = 0.3

Finite Element Modeling •

6 nodes

 •

5 bar elements

Boundary Conditions Constraints •

Constrain the X, Y, Z translation an X rotation at point A (node 1)

•

Constrain the Y and Z translation at point B (node 10)

The boundary conditions are shown in the following diagram.

Solution Type Normal Modes/Eigenvalues – SVI method

Results Mode # 1, 2 3 4 5, 6 7 8, 9

Ref. Value (Hz) 42.649 77.542 125.00 148.31 233.10 284.55

NAFEMS Target Value (Hz) 42.568 77.841 125.51 145.46 241.24 267.01

FEMAP Structural Result (Hz) 42.710 77.841 125.52 150.76 241.24 301.08



Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 5.

Circular Ring - In-plane and Out-ofplane Vibration The complete model and results for this test are in file nf006ac.neu. This test is a normal modes/eigenvalue analysis of a circular ring using bar elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 6. Attributes of this test are: •

rigid body modes (six modes)

•

repeated eigenvalues

Test Case Data and Information Units SI

Cross Sectional Properties Shear ratio: •

Y = 1.128205

•

Z = 1.128205

Material Properties 9 N E = 200x10 ------2m

g ρ = 8000k ------3m ν = 0.3

Finite Element Modeling •

20 nodes

 •

20 bar elements

Boundary Conditions Constraints •

Constraint Set 1 (Kinematic DOF): Constrain nodes 10 and 11 in all directions and rotations.

Solution Type Normal Modes/Eigenvalue – SVI method



Results Mode # 7, 8 (out of plane) 9, 10 (in plane) 11, 12 (out of plane) 13, 14 (in plane) 15 (out of plane) 16 (in plane)

Ref. Value (Hz)

NAFEMS Target Value (Hz)

FEMAP Structural Result (Hz)

51.849

52.290

52.211

53.382

53.971

53.775

148.77

149.70

148.92

150.99

152.44

151.25

286.98

288.25

285.33

289.51

288.25

285.33

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 6.

Cantilevered Beam The complete model and results for this test case are in the following files: •

nf071a.neu (Test 1)

•

nf071b.neu (Test 2)

•

nf071c.neu (Test 3)

This test is a normal modes/eigenvalue analysis of a cantilevered beam. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 71. Attributes of this test are: •

ill–conditioned stiffness matrix

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

g ρ = 8000k ------3m

Finite Element Modeling Three tests - all use 8 bar elements and 9 nodes •

Test 1: a=b

•

Test 2: a = 10b

 •

Test3: a = 100b

Boundary Conditions Constraints •

Fully constrain point A (node 1) in all directions and rotations.

•

Constrain all other nodes in the Z translation and X and Y rotations.

Solution Type Normal Modes/Eigenvalue – SVI method Bar elements always use a consistent mass formulation.



Results

Mode #

Ref. Value (Hz)

1

1.010

2

6.327

3

17.716

4

34.717

5

57.390

6

85.730

Mesh

a=b a = 10b a = 100b a=b a = 10b a = 100b a=b a = 10b a = 100b a=b a = 10b a = 100b a=b a = 10b a = 100b a=b a = 10b a = 100b

FEMAP Structural Result (Hz) 1.0095 1.0095 1.0095 6.3223 6.3260 6.3289 17.693 17.791 17.819 34.675 34.854 35.061 57.422 60.595 64.751 86.135 101.673 104.654

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 71.

Plate Element Test Cases The normal mode dynamics test cases using the standard NAFEMS benchmarks include these plate element test cases: •

"Thin Square Cantilevered Plate -Symmetric Modes"

•

"Thin Square Cantilevered Plate - Anti-symmetric Modes"

•

"Free Thin Square Plate"

•

"Simply-Supported Thin Square Plate"

•

"Simply-Supported Thin Annular Plate"

•

"Clamped Thin Rhombic Plate"

•

"Cantilevered Thin Square Plate with Distorted Mesh"

•

"Simply-Supported Thick Square Plate, Test A"

•

"Clamped Thick Rhombic Plate"

•

"Simply-Supported Thick Square Plate, Test B"

•

"Simply-Supported Thick Annular Plate"

•

"Cantilevered Square Membrane"

•

"Cantilevered Tapered Membrane"

•

"Free Annular Membrane"

•

"Cantilevered Thin Square Plate"

•

"Cantilevered Thin Square Plate #2"

Thin Square Cantilevered Plate Symmetric Modes The complete model and results for this test case are in the following files: •

nf011alc.neu (linear quadrilateral plate, consistent mass)

•

nf011all.neu (linear quadrilateral plate, lumped mass)

•

nf011apc.neu (parabolic quadrilateral plate, consistent mass)

•

nf011apl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a thin, square, cantilevered plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 11a. Attributes of this test are: •

symmetric modes, symmetric boundary conditions along the cutting plane

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Test 1 and Test 2 (nf011alc and nf011all) •

45 nodes

•

32 linear quadrilateral plate elements - thickness = 0.05m

Test 2 and Test 3 (nf011apc and nf011apl) •

37 nodes

 •

8 parabolic quadrilateral plate elements - thickness = 0.05m

Mesh only half the plate (10m x 5m). Linear Quadrilateral Plates

Parabolic Quadrilateral Plates

Boundary Conditions •

Constraints (all tests)

•

Fully constrain nodes 1-5 in all translations and rotations.

•

Constrain nodes 6, 11, 16, 21, 26, 31, 36, 41 in the X and Y translations and X and Z rotations.

•

Constrain all other nodes in the X and Y translations and Z rotation.

Solution Type Normal Modes/Eigenvalue – SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass



Results

Mode #

Ref. Value (Hz)

1

0.421

2

2.582

3

3.306

4

6.555

5

7.381

6

11.402

Mesh linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

FEMAP Structural Result (lumped mass) (Hz) 0.415 0.414 2.507 2.444 3.117 3.081 5.984 6.018 7.241 6.954 10.387 10.493

FEMAP Structural Result (consistent mass) (Hz) 0.418 0.418 2.623 2.569 3.315 3.281 6.573 6.551 7.979 7.525 12.112 11.950

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 11a.

Thin Square Cantilevered Plate Anti-symmetric Modes The complete model and results for this test case are the in following files: •

nf011blc.neu (linear quadrilateral plate, consistent mass)

•

nf011bll.neu (linear quadrilateral plate, lumped mass)

•

nf011bpc.neu (parabolic quadrilateral plate, consistent mass)

•

nf011bpl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a thin, square, cantilevered plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 11b. Attributes of this test are: •

anti–symmetric modes

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf011blc.neu and nf011bll.neu) •

45 nodes, 32 linear quadrilateral plate elements - thickness = 0.05m

Tests 3 and 4 (nf011bpc.neu and nf011bpl.neu) •

37 nodes, 8 parabolic quadrilateral plate elements - thickness = 0.05m

 Mesh only half the plate (10m x 5m).

Boundary Conditions Constraints (all tests) •

Fully constrain nodes 1-5 in all directions.

•

Constrain nodes 6, 11, 16, 21, 26, 31, 36, 41 in the X, Y, Z translations and Z rotation.

•

Constrain all other nodes in the X and Y translations and Z rotation.

Solution Type Normal Modes/Eigenvalue – SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass



Results

Mode #

Ref. Value (Hz)

1

1.029

2

3.753

3

7.730

4

8.561

5

not available not available

6

Mesh linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

FEMAP FEMAP NAFEMS Structural Result Structural Result Target Value (lumped mass) (consistent mass) (Hz) (Hz) (Hz) 1.019 1.018 3.839 3.710 8.313 7.768 9.424 8.483 11.728 11.185 17.818 15.755

0.993 0.999 3.553 3.541 7.130 6.847 8.082 7.894 9.805 9.954 13.087 13.724

1.012 1.024 3.750 3.728 8.162 7.846 9.079 8.693 11.526 11.451 17.192 16.918

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 11b.

Free Thin Square Plate The complete model and results for this test case are in the following files: •

nf012lc.neu (linear quadrilateral plate, consistent mass)

•

nf012ll.neu (linear quadrilateral plate, lumped mass)

•

nf012pc.neu (parabolic quadrilateral plate, consistent mass)

•

nf012pl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a free thin square plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 12. Attributes of this test are: •

rigid body modes (three modes)

•

repeated eigenvalues

•

use of kinematic DOF for the rigid body mode calculation with the SVI eigensolver

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf012lc.neu and nf012ll.neu) •

81 nodes, 64 linear quadrilateral plate elements - thickness = 0.05m

Tests 3 and 4 (nf012pc.neu and nf012pl.neu) •

65 nodes, 16 parabolic quadrilateral plate elements - thickness = 0.05m

 Boundary Conditions Constraints •

Constraint Set 1: Constrain all the nodes in the X and Y translations and Z rotation.

•

Constraint Set 2 (Kinematic DOF): Constrain nodes 1 and 3 in all directions and rotations.

Solution Type Normal Modes/Eigenvalue – SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass



Results Mode #

Ref. Value (Hz)

4

1.622

5

2.360

6

2.922

7, 8

4.233

9

7.416

10

not available

Mesh linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

NAFEMS Target Value (Hz) 1.632 1.532 2.402 2.356 3.006 2.861 4.251 4.122 7.859 7.363 8.027 7.392

FEMAP Structural FEMAP Structural Result (lumped Result (consistent mass) (Hz) mass) (Hz) 1.570 1.567 2.246 2.183 2.815 2.750 3.912 3.879 6.902 6.586 6.903 6.586

1.615 1.619 2.394 2.364 2.990 2.930 4.218 4.186 7.751 7.494 7.884 7.494

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 12.

Simply-Supported Thin Square Plate The complete model and results for this test case are in the following files: •

nf013lc.neu (linear quadrilateral plate, consistent mass)

•

nf013ll.neu (linear quadrilateral plate, lumped mass)

•

nf013pc.neu (parabolic quadrilateral plate, consistent mass)

•

nf013pl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a simply–supported thin square plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 13. Attributes of this test are: •

well established

•

repeated eigenvalues

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf013lc.neu and nf013ll.neu) •

81 nodes, 64 linear quadrilateral plate elements - thickness = 0.05m

Tests 3 and 4 (nf013pc.neu and nf013pl.neu)

 •

65 nodes, 16 parabolic quadrilateral plate elements - thickness = 0.05m

Boundary Conditions Constraints •

Constrain all nodes in the X and Y translations and Z rotation.

•

Constrain the nodes along edges X = 0 and X = 10m in the Z translation and X rotation.

•

Constrain the nodes along edges Y = 0 and Y = 10m in the Z translation and Y rotation.

•

Fully constrain the DOF on the four corner nodes (9, 13, 41, 68).

Solution Type Normal Modes/Eigenvalue – SVI method Results were obtained two different ways:

 •

using lumped mass

•

using consistent mass

Results Mode #

Ref. Value (Hz)

1

2.377

2, 3

5.942

4

9.507

5, 6

11.884

7, 8

15.449

Mesh 4–noded 8–noded 4–noded 8–noded 4–noded 8–noded 4–noded 8–noded 4–noded 8–noded

FEMAP Structural Result (lumped mass) (Hz) 2.338 2.375 5.820 5.932 8.909 9.392 11.770 11.879 14.215 15.033

FEMAP Structural Result (consistent mass) (Hz) 2.399 2.383 6.206 6.034 9.873 9.831 13.375 12.590 16.786 16.734

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 13.

Simply-Supported Thin Annular Plate The complete model and results for this test case are in the following files: •

nf014lc.neu (linear quadrilateral plate, consistent mass)

•

nf014ll.neu (linear quadrilateral plate, lumped mass)

•

nf014pc.neu (parabolic quadrilateral plate, consistent mass)

•

nf014pl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a simply–supported thin annular plate meshed with shell elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 14. Attributes of this test are: •

curved boundary (skewed coordinate system)

•

repeated eigenvalues

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf014lc and nf014ll): •

192 nodes, 160 linear quadrilateral plate elements - thickness = 0.06m

Tests 3 and 4 (nf014pc and nf014pl)

 •

176 nodes, 48 parabolic quadrilateral plate elements - thickness = 0.06m

Boundary Conditions •

Constraint Set 1 (All Tests): Constrain all nodes in in the X and Y translation and Z rotation. Additionally constrain all nodes around the model’s circumference in the Z translation and X rotation.

•

Constraint Set 2 (Kinematic DOF): Tests 1 and 2: Constrain nodes 258 and 290 in the X and Y translations.

 Tests 3 and 4: Constrain nodes 21 and 133 in the X and Y translations.

Solution Type Normal Mode Dynamics - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass

Results Mode #

Ref. Value (Hz)

1

1.870

2, 3

5.137

4, 5

9.673

6

14.850

7, 8

15.573

9

18.382

Mesh linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

FEMAP Structural FEMAP Structural Result (lumped Result (consistent mass) (Hz) mass) (Hz) 1.859 1.840 5.175 5.111 9.686 9.672 14.188 13.946 15.326 15.547 17.594 17.380

1.877 1.873 5.249 5.151 9.983 9.713 15.412 14.924 16.176 15.708 19.088 18.521

Note: Reference value (Ref. Value) refers to the accepted solution to the problem.

 Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 14.

Clamped Thin Rhombic Plate The complete model and results for this test case are in the following files: •

nf015lc.neu (linear quadrilateral plate, consistent mass)

•

nf015ll.neu (linear quadrilateral plate, lumped mass)

•

nf015pl.neu (parabolic quadrilateral plate, lumped mass)

•

nf015pc.neu (parabolic quadrilateral plate, consistent mass)

This test is a normal modes/eigenvalue analysis of a clamped thin rhombic plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 15. Attributes of this test are: •

distorted elements

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf015lc.neu and nf015ll.neu): •

169 nodes, 144 linear quadrilateral plate elements - thickness = 0.05m

 Tests 3 and 4 (nf015pc.neu and nf015pl.neu): •

133 nodes, 36 parabolic quadrilateral plate elements - thickness = 0.05m

Boundary Conditions Constraints •

Completely constrain the nodes along all four edges of the part in all directions and rotations.

•

Constrain all other nodes in the X and Y translation and Z rotation.

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass



Results Mode #

Ref. Value (Hz)

1

7.938

2

12.835

3

17.941

4

19.133

5

24.009

6

27.922

Mesh linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

NAFEMS FEMAP Structural FEMAP Structural Target Value Result (lumped Result (consistent mass) (Hz) mass) (Hz) (Hz) 8.142 7.873 13.891 12.480 20.036 17.312 20.165 18.738 27.704 27.950 32.046 25.883

7.818 7.902 12.831 12.851 17.807 17.952 18.554 18.964 23.665 23.879 27.698 27.910

7.955 7.929 13.388 13.008 19.072 18.472 19.239 19.168 26.185 25.226 29.816 28.810

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 15.

Cantilevered Thin Square Plate with Distorted Mesh The complete model and results for this test case are in the following files: •

nf016a1.neu (16 parabolic quadrilateral plate, lumped mass)

•

nf016a2.neu (16 parabolic quadrilateral plate, consistent mass)

•

nf016b1.neu (16 parabolic quadrilateral plate, lumped mass)

•

nf016b2.neu (16 parabolic quadrilateral plate, consistent mass)

•

nf016c1.neu (4 parabolic quadrilateral plate, lumped mass)

•

nf016c2.neu (4 parabolic quadrilateral plate, consistent mass)

•

nf016d1.neu (4 parabolic quadrilateral plate, lumped mass)

•

nf016d2.neu (4 parabolic quadrilateral plate, consistent mass)

This test is a normal modes/eigenvalue analysis of a cantilevered thin square plate meshed with distorted plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 16. Attributes of this test are: •

distorted meshes

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling All tests - parabolic quadrilateral plate elements - thickness = 0.05m

 Four tests: •

Test 1 (nf016a1, nf016a2) - 65 nodes, 16 elements

•

Test 2 (nf016b1, nf016b2) - 65 nodes, 16 elements with specified nodes at the following XY coordinates:

X Coordinate 4.0 2.25 4.75 7.25 7.5 7.75 5.25 2.25 2.5

Y Coordinate 4.0 2.25 2.5 2.75 4.75 7.25 7.25 7.25 4.75



•

Test 3 (nf016c1, nf016c2) - 21 nodes, 4 elements

•

Test 4 (nf016d1, nf016d2) - 21 nodes, 4 elements with a specified node at X=4.0, Y=4.0.

Boundary Conditions

 Constraints (nf016a1 and nf016a2) •

Constrain the nodes along the model’s Y axis in the X, Y, and Z translations and in the Y and Z rotations.

•

Constrain all other nodes in the Z rotation only.

Constraints (nf016b1 and nf016b2) •

Fully constrain the nodes along the model’s Y axis in all directions.

•

Constrain all other nodes in the Z rotation only.

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass



Results

Mode #

Ref. Value (Hz)

1

0.421

2

1.029

3

2.582

4

3.306

5

3.753

6

6.555

Test 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

NAFEMS Target Value (Hz) 0.4174 0.4174 0.4144 0.4145 1.020 1.020 0.999 1.002 2.564 2.571 2.554 2.565 3.302 3.317 3.401 3.424 3.769 3.780 3.697 3.714 6.805 6.883 5.455 5.133

FEMAP FEMAP Structural Structural Result (consistent Result (lumped mass) (Hz) mass) (Hz) 0.4139 0.4135 0.4021 0.3999 0.9985 0.9967 0.9347 0.9202 2.444 2.445 2.132 2.112 3.082 3.072 2.707 2.697 3.540 3.535 3.136 3.077 6.018 5.994 5.458 5.459

0.4181 0.4182 0.4189 0.4192 1.024 1.024 1.021 1.025 2.569 2.566 2.708 2.698 3.281 3.280 3.449 3.430 3.728 3.731 3.913 3.881 6.551 6.552 7.108 6.858

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 16.

Simply-Supported Thick Square Plate, Test A The complete model and results for this test case are in the following files: •

nf021alc.neu (linear quadrilateral plate, consistent mass)

•

nf021all.neu (linear quadrilateral plate, lumped mass)

•

nf021apc.neu (parabolic quadrilateral plate, consistent mass)

•

nf021apl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a simply–supported thick square plate meshed with shell elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 21a. Attributes of this test are: •

well–established

•

repeated eigenvalues

•

effect of secondary restraints

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf021alc.neu and nf021all.neu)

 •

81 nodes, 64 linear quadrilateral plate elements - thickness = 1.0m

Tests 3 and 4 (nf021apc.neu and nf021apl.neu) •

65 nodes, 16 parabolic quadrilateral plate elements - thickness = 1.0m

Boundary Conditions Constraints •

Fully constrain the corner nodes in all directions and rotations.

•

Constrain the nodes along edges X=0 and X=10m in all directions, except the Y rotation.

•

Constrain the nodes along edges Y=0 and Y=10m in all directions, except the X rotation.

 •

Constrain all other nodes in the X and Y translation and Z rotation.

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass

Results

Mode #

Ref. Value (Hz)

1

45.897

2, 3

109.44

4

167.89

5, 6

204.51

Mesh linear parabolic linear parabolic linear parabolic linear parabolic

FEMAP FEMAP NAFEMSTar Structural Structural Result get Value Result (lumped (consistent mass) (Hz) mass) (Hz) (Hz) 46.659 45.936 115.84 110.41 177.53 170.38 233.40 212.81

45.50 46.165 108.70 110.32 160.63 167.30 204.75 204.59

46.35 45.830 114.12 109.38 174.29 169.75 227.05 208.20

 7, 8

256.50

9

336.62

10

336.62

linear parabolic linear parabolic linear parabolic

283.60 269.96 371.11 344.77 371.11 344.77

240.84 249.26 298.18 311.32 320.41 347.63

276.88 268.40 364.30 319.40 385.84 319.40

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 21a.

Simply-Supported Thick Square Plate, Test B The complete model and results for this test case are in the following files: •

nf021blc.neu (linear quadrilateral plate elements, consistent mass)

•

nf021bll.neu (linear quadrilateral plate elements, lumped mass)

•

nf021bpc.neu (parabolic quadrilateral plate elements, consistent mass)

•

nf021bpl.neu (parabolic quadrilateral plate elements, lumped mass)

This test is a normal modes/eigenvalue analysis of a simply–supported thick square plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 21b. Attributes of this test are: •

well–established

•

repeated eigenvalues

•

effect of secondary restraints

Test Case Data and Information Units SI

Material Properties 9 N E = 200X10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf021blc.neu and nf021bll.neu) •

81 nodes, 64 linear quadrilateral plate elements - thickness = 1.0m

 Tests 3 and 4 (nf021plc.neu and nf021pll.neu) •

65 nodes, 16 parabolic quadrilateral plate elements - thickness = 1.0m

Boundary Conditions Constraints •

Constrain the nodes along all edges in the X,Y, and Z translations and Z rotation.

•

Constrain all other nodes in the X and Y translation and Z rotation.

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

 •

using consistent mass

Results

Mode #

Ref. Value (Hz)

1

45.897

2, 3

109.44

4

167.89

5, 6

204.51

7, 8

256.50

9

336.62

10

336.62

Mesh linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

FEMAP FEMAP NAFEMS Structural Result Structural Result Target Value (lumped mass) (consistent mass) (Hz) (Hz) (Hz) 44.745 44.134 112.94 107.85 170.28 164.19 230.23 20.07 274.19 260.32 355.98 342.80 355.98 342.80

44.14 44.815 106.96 108.52 156.96 163.57 203.40 203.12 237.31 245.71 293.95 307.16 319.64 346.85

44.96 44.493 112.25 107.57 170.17 165.70 225.51 206.46 272.47 263.61 358.43 318.56 384.78 318.58

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 21b.

Clamped Thick Rhombic Plate The complete model and results for this test case are in the following files: •

nf022lc.neu (linear quadrilateral plate, consistent mass)

•

nf022ll.neu (linear quadrilateral plate, lumped mass)

•

nf022pc.neu (parabolic quadrilateral plate, consistent mass)

•

nf022pl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a thick clamped thick rhombic plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 22. Attributes of this test are: •

distorted elements

Test Case Data and Information Units SI

Material Properties 9 N E = 200X10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf022lc.neu and nf022ll.neu)

 •

121 nodes, 100 linear quadrilateral plate elements - thickness = 1.0m

Tests 3 and 4 (nf022pc.neu and nf022pl.neu) •

133 nodes, 36 parabolic quadrilateral plate elements - thickness = 1.0m

Boundary Conditions Constraints •

Fully constrain the nodes along all four edges in all directions and rotations.

 •

Constrain all other nodes in the X and Y translations and Z rotation.

Solution Type Normal Mode Dynamics - SVI Results were obtained two different ways: •

using lumped mass

•

using consistent mass

Results

Mode #

Ref. Value (Hz)

1

133.95

2

201.41

3

265.81

4

282.74

5

334.45

6

not available

Mesh linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

FEMAP FEMAP Structural NAFEMS Structural Result Target Value Result (lumped (consistent mass) (Hz) mass) (Hz) (Hz) 137.80 133.86 218.48 203.34 295.42 271.38 296.83 283.68 383.56 346.41 426.59 386.62

133.33 134.51 204.42 204.30 269.23 270.17 279.75 283.95 337.92 338.90 381.87 381.90

135.17 132.48 213.06 200.28 288.08 266.06 289.05 273.65 377.05 338.88 411.28 369.79



Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 22

Simply-Supported Thick Annular Plate The complete model and results for this test case are in the following files: •

nf023lc.neu (linear quadrilateral plate, consistent mass)

•

nf023ll.neu (linear quadrilateral plate, lumped mass)

•

nf023pc.neu (parabolic quadrilateral plate, consistent mass)

•

nf023pl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a simply–supported thick annular plate meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 23. Attributes of this test are: •

curved boundary (skewed coordinate system)

•

repeated eigenvalues

Test Case Data and Information Units SI

Material Properties 9 N E = 200X10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf023lc.neu and nf023ll.neu)

 •

192 nodes, 160 linear quadrilateral plate elements - thickness = 0.6m

Tests 3 and 4 (nf023pc.neu and nf023pl.neu) •

176 nodes, 48 parabolic quadrilateral plate elements - thickness = 0.6m

Boundary Conditions Constraints •

Constrain the nodes around the circumference in the X, Y, and Z translations and X and Z rotations.

 •

Constrain all other nodes in the X and Y translation and Z rotation.

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass

Results

Mode #

Ref. Value (Hz)

1

18.58

2, 3

48.92

4, 5

92.59

Mesh

linear parabolic linear parabolic linear parabolic

FEMAP FEMAP Structural NAFEMS Structural Result Result Target Value (consistent mass) (lumped mass) (Hz) (Hz) (Hz) 18.82 18.59 49.82 49.02 96.06 92.90

18.49 18.32 49.89 48.99 93.43 93.19

18.61 18.59 50.35 49.13 95.44 92.42

 6

140.15

7, 8

not available 166.36

9

linear parabolic linear parabolic linear parabolic

148.34 140.86 153.68 146.63 174.52 167.31

136.71 134.27 145.21 146.87 163.74 160.43

145.39 139.41 151.28 145.37 174.10 166.11

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 23.

Cantilevered Square Membrane The complete model and results for this test case are in the following files: •

nf031lc.neu (linear quadrilateral plate, consistent mass)

•

nf031ll.neu (linear quadrilateral plate, lumped mass)

•

nf031pc.neu (parabolic quadrilateral plate, consistent mass)

•

nf031pl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a cantilevered square membrane meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 31. Attributes of this test are well established.

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf031lc.neu and nf031ll.neu)

 •

81 nodes, 64 linear quadrilateral plate elements - thickness = 0.05m

Tests 3 and 4 (nf031pc.neu and nf031pl.neu) •

65 nodes, 16 parabolic quadrilateral plate elements - thickness = 0.05m

Boundary Conditions Constraints •

Fully constrain the nodes along the Y axis in all directions and rotations.

 •

Constrain all other nodes in the Z translation and X, Y, and Z rotations.

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass

Results

Mode #

Ref. Value (Hz)

1

52.404

2

125.69

3

140.78

4

222.54

5

241.41

6

255.74

Mesh

linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

NAFEMS Target Value (Hz) 52.905 52.635 126.11 125.87 143.20 141.47 228.85 224.59 247.90 243.26 260.61 256.76

FEMAP FEMAP Structural Structural Result Result (consistent mass) (lumped mass) (Hz) (Hz) 52.47 52.16 125.59 125.18 139.54 138.28 214.61 209.01 239.84 239.16 252.06 251.31

52.77 52.39 126.06 122.48 142.83 138.02 227.04 214.95 247.25 227.48 259.46 236.73



Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 31.

Cantilevered Tapered Membrane The complete model and results for this test case are in the following files: •

nf032lc.neu (linear quadrilateral plate, consistent mass)

•

nf032ll.neu (linear quadrilateral plate, lumped mass)

•

nf032pc.neu (parabolic quadrilateral plate, consistent mass)

•

nf032pl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a cantilevered tapered membrane meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 32. Attributes of this test are: •

shear behavior

•

irregular mesh

•

symmetry

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf032lc.neu and nf032ll.neu)

 •

153 nodes, 128 linear quadrilateral plate elements - thickness = 0.1m

Tests 3 and 4 (nf032pc.neu and nf032pl.neu) •

153 nodes, 32 parabolic quadrilateral plate elements - thickness = 0.1m

Boundary Conditions Constraints •

Fully constrain the nodes along the Y axis in all directions and rotations.

 •

Constrain all other nodes in the Z translation and the X, Y, and Z rotations.

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass

Results

Mode #

Ref. Value (Hz)

1

44.623

2

130.03

3

162.70

4

246.05

5

379.90

6

391.44

Mesh

linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

FEMAP FEMAP Structural NAFEMS Structural Result Result Target Value (consistent mass) (lumped mass) (Hz) (Hz) (Hz) 44.905 44.636 132.12 130.14 162.83 162.72 252.99 246.63 393.31 382.02 396.26 391.55

44.73 44.84 129.92 129.05 162.61 162.37 244.62 241.80 375.09 369.61 389.81 388.11

44.82 45.14 131.28 130.50 162.80 161.37 250.56 245.00 391.79 374.78 393.11 375.77



Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 32

Free Annular Membrane The complete model and results for this test case are in the following files: •

nf033lc.neu (linear quadrilateral plate, consistent mass)

•

nf033ll.neu (linear quadrilateral plate, lumped mass)

•

nf033pc.neu (parabolic quadrilateral plate, consistent mass)

•

nf033pl.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a free annular membrane meshed with plate elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 33. Attributes of this test are: •

repeated eigenvalues

•

rigid body modes (three modes)

•

kinematically incomplete suppressions

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf033lc.neu and nf033ll.neu)

 •

192 nodes, 160 linear quadrilateral plate elements - thickness = 0.06m

Tests 3 and 4 (nf033pc.neu and nf033pl.neu) •

176 nodes, 48 parabolic quadrilateral plate elements - thickness = 0.06m

Boundary Conditions Constraints •

Constraint Set 1 (DOF set): Tests 1 and 2: Constrain nodes 254 and 286 in the X and Y translations.

 Tests 3 and 4: Constrain nodes 7 and 19 in the X and Y translations.

•

Constraint Set 2: Constrain all other nodes in the Z translation and X, Y, and Z rotations.

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass



Results

Mode #

Ref. Value (Hz)

4, 5

129.24

6

226.17

7, 8

234.74

9, 10

264.66

11, 12

336.61

13, 14

376.79

Mesh

linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

FEMAP FEMAP Structural NAFEMS Structural Result Result Target Value (consistent mass) (lumped mass) (Hz) (Hz) (Hz) 129.51 126.48 225.46 224.27 234.92 232.95 272.13 264.81 340.34 335.70 391.98 378.60

127.71 126.66 224.52 222.82 229.67 230.12 263.86 262.45 328.44 329.09 368.15 368.48

128.70 126.15 225.22 218.17 234.94 225.14 270.83 257.67 339.93 311.38 389.38 361.52

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 33.

Cantilevered Thin Square Plate The complete model and results for this test case are in the following files: •

nf073ac.neu (Test 1 - parabolic quadrilateral plate, consistent mass)

•

nf073al.neu (Test 2 - parabolic quadrilateral plate, lumped mass)

•

nf073bc.neu (Test 3 - parabolic quadrilateral plate, consistent mass)

•

nf073bl.neu (Test 4 - parabolic quadrilateral plate, lumped mass)

•

nf073cc.neu (Test 5 - parabolic quadrilateral plate, consistent mass)

•

nf073cl.neu (Test 6 - parabolic quadrilateral plate, lumped mass)

•

nf073dc.neu (Test 7 - parabolic quadrilateral plate, consistent mass)

•

nf073dl.neu (Test 8 - parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a cantilevered thin square plate. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 73. Attributes of this test are: •

effect of master DOF selection on frequencies

Test Case Data and Information Units SI

 Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling 65 nodes, 16 parabolic quadrilateral plate elements - thickness = 0.05m

Boundary Conditions Constraints •

Constraint Set 1: Constrain the nodes along the Y axis in the X, Y, and Z translations and Y rotation.

•

Constraint Set (DOF set) 2: Create a constraint set to define a Master (ASET) DOF set (in Z direction) - four different placements:

 Tests 1 and 2:

Tests 3 and 4:

Tests 5 and 6:

 Tests 7 and 8:

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass

Results

Mode #

Ref. Value (Hz)

1

0.421

2

1.029

3

2.582

DOF Set

test 1 test 2 test 3 test 4 test 1 test 2 test 3 test 4 test 1 test 2 test 3 test 4

FEMAP FEMAP Structural NAFEMS Structural Result Result Target Value (consistent mass) (lumped mass) (Hz) (Hz) (Hz) 0.4174 0.4174 0.4175 0.4184 1.020 1.020 1.021 1.032 2.564 2.597 2.677 2.850

0.4139 0.4139 0.4140 0.4147 0.999 1.000 1.001 1.009 2.449 2.476 2.524 2.670

0.4182 0.4182 0.4183 0.4191 1.025 1.026 1.027 1.036 2.580 2.610 2.675 2.844

 4

3.306

5

3.753

6

6.555

test 1 test 2 test 3 test 4 test 1 test 2 test 3 test 4 test 1 test 2 test 3 test 4

3.302 3.345 3.365 3.571 3.769 3.888 4.035 5.466 6.805 7.517 7.495 -----

3.095 3.126 3.140 3.325 3.563 3.663 3.765 4.816 6.126 6.694 6.675 ------

3.314 3.352 3.362 3.555 3.781 3.891 4.023 5.414 6.798 7.498 7.479 ------

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 73.

Cantilevered Thin Square Plate #2 The complete model and results for this test case are in the following files: •

nf074c.neu (parabolic quadrilateral plate, consistent mass)

•

nf074l.neu (parabolic quadrilateral plate, lumped mass)

This test is a normal modes/eigenvalue analysis of a cantilevered thin square plate. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 74.

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling 65 nodes, 16 parabolic quadrilateral plate elements - thickness = 0.05m

 Boundary Conditions Constraints Constrain the nodes along the Y axis in the X, Y, and Z translations and the Y rotation.

Solution Type Normal Modes/Eigenvalues - SVI Results were obtained two different ways: •

using lumped mass

•

using consistent mass

Results

Mode # 1 2 3 4 5

Ref. Value (Hz) 0.471 1.029 2.582 3.306 3.753

FEMAP Structural Result (lumped mass) (Hz) 0.4139 0.999 2.444 3.082 3.540

FEMAP Structural Result (consistent mass) (Hz) 0.4181 1.024 2.569 3.281 3.728

Note: Reference value (Ref. Value) refers to the accepted solution to the problem.

 Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 74.

Axisymmetric Solid and Solid Element Test Cases The normal mode dynamics test cases using the standard NAFEMS benchmarks include these axisymmetric solid and solid element test cases: •

"Free Cylinder - Axisymmetric Vibration"

•

"Simply-Supported Annular Plate -Axisymmetric Vibration"

•

"Thick Hollow Sphere - Uniform Radial Vibration"

•

"Simply-Supported Solid Square Plate"

•

"Simply-Supported Solid Annular Plate"

•

"Deep Simply-Supported Solid Beam"

•

"Cantilevered Solid Beam"

Free Cylinder - Axisymmetric Vibration The complete model and results for this test case are in the following files: •

nf041lc.neu (linear axisymmetric solid quadrilateral, consistent mass)

•

nf041ll.neu (linear axisymmetric solid quadrilateral, lumped mass)

•

nf041pc.neu (parabolic axisymmetric solid quadrilateral, consistent mass)

•

nf041pl.neu (parabolic axisymmetric solid quadrilateral, lumped mass)

This test is a normal modes/eigenvalue analysis of a free cylinder meshed with axisymmetric elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 41. Attributes of this test are: •

rigid body modes (one mode)

•

coupling between axial, radial, and circumferential behavior

•

close eigenvalues

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf041lc.neu and nf041ll.neu):

 •

68 nodes, 48 linear axisymmetric quadrilateral solid elements

Tests 3 and 4 (nf041pc.neu and nf041pl.neu): •

43 nodes, 8 parabolic axisymmetric quadrilateral solid elements

Boundary Conditions Constraints •

Tests 1 and 2: Create a constraint set (Kinematic DOF set) to constrain nodes 1 and 68 in the X and Z translations.

•

Tests 3 and 4: Create a constraint set (Kinematic DOF set) to constrain nodes 1 and 51 in the X and Z translations.

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass



Results

Mode #

Ref. Value (Hz)

2

243.53

3

377.41

4

394.11

5

397.72

6

405.28

Mesh

linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

FEMAP Structural NAFEMS Result Target Value (lumped mass) (Hz) (Hz) 244.01 243.50 379.41 377.46 395.41 394.28 401.35 397.94 421.87 406.41

243.18 243.24 370.86 356.49 379.31 356.88 385.92 375.85 389.56 393.65

FEMAP Structural Result (consistent mass) (Hz) 243.96 243.50 378.15 377.46 394.42 394.30 398.00 397.97 406.85 406.44

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 41.

Thick Hollow Sphere - Uniform Radial Vibration The complete model and results for this test case are in the following files: •

nf042lc.neu (linear axisymmetric solid quadrilateral, consistent mass)

•

nf042ll.neu (linear axisymmetric solid quadrilateral, lumped mass)

•

nf042pc.neu (parabolic axisymmetric solid quadrilateral, consistent mass)

•

nf042pl.neu (parabolic axisymmetric solid quadrilateral, lumped mass)

This test is a normal modes/eigenvalue analysis of a thick, hollow sphere using axisymmetric solid elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 42. Attributes of this test are: •

curved boundary (skewed coordinate system)

•

constraint equations

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf042lc.neu and nf042ll.neu) •

22 nodes, 10 linear axisymmetric solid quadrilateral elements - α = 5°

Tests 3 and 4 (nf042pc.neu and nf042pl.neu)

 •

53 nodes, 10 parabolic axisymmetric solid quadrilateral elements

Boundary Conditions Constraints •

Constraint Set 1: Constrain all nodes in the Z translation.

•

Constraint Equations: Constrain all nodes at the same R’ are constrained to have same r’ displacement

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass



Results

Mode #

Ref. Value (Hz)

1

369.91

2

838.03

3

1451.2

4

2117.0

5

2795.8

Mesh

linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

FEMAP FEMAP Structural NAFEMS Structural Result Result Target Value (consistent mass) (lumped mass) (Hz) (Hz) (Hz) 370.64 370.01 841.20 838.08 1473.1 1453.0 2192.2 2131.7 2975.7 2852.8

369.91 369.49 831.80 832.72 1421.3 1433.7 2030.5 2072.9 2604.2 2706.3

370.08 369.83 839.49 837.77 1470.5 1450.85 2188.6 2117.3 2970.9 2799.5

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 42.

Simply-Supported Annular Plate Axisymmetric Vibration The complete model and results for this test case are in the following files: •

nf043lc.neu (linear axisymmetric solid quadrilateral, consistent mass)

•

nf043ll.neu (linear axisymmetric solid quadrilateral, lumped mass)

•

nf043pc.neu (parabolic axisymmetric solid quadrilateral, consistent mass)

•

nf043pl.neu (parabolic axisymmetric solid quadrilateral, lumped mass)

This test is a normal modes/eigenvalue analysis of a simply–supported annular plate meshed with axisymmetric elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 43. Attributes of this test are: •

well established

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf043lc.neu and nf043ll.neu):

 •

80 nodes, 60 linear axisymmetric solid quadrilateral elements

Tests 3 and 4 (nf043pc.neu and nf043pl.neu) •

28 nodes, 5 parabolic axisymmetric solid quadrilateral elements

Boundary Conditions Constraints Constrain point A (node 1) in the Z translation

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass



Results

Mode #

Ref. Value (Hz)

1

18.543

2

150.15

3

224.16

4

358.29

5

629.19

Mesh

linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

FEMAP Structural NAFEMS Result Target Value (lumped mass) (Hz) (Hz) 18.711 18.582 145.46 145.56 224.22 224.18 385.59 374.05 689.34 686.04

18.542 18.429 138.66 135.97 224.20 224.00 361.50 353.62 643.34 633.16

FEMAP Structural Result (consistent mass) (Hz) 18.570 18.582 140.24 140.56 224.20 224.18 371.48 374.05 673.79 686.05

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 43.

Deep Simply-Supported Solid Beam The complete model and results for this test case are in the following files: •

nf051lc.neu (linear solid brick, consistent mass)

•

nf051ll.neu (linear solid brick, lumped mass)

•

nf051pc.neu (parabolic solid brick, consistent mass)

•

nf051pl.neu (parabolic solid brick, lumped mass)

This test is a normal mode dynamic analysis of a deep, solid beam meshed with bricks. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 51. Attributes of this test are: •

skewed coordinate system

•

skewed restraints

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf051lc.neu, nf051ll.neu)

 •

88 nodes, 30 linear solid brick elements

Tests 3 and 4 (nf051pc.neu, nf051pl.neu) •

68 nodes, 5 parabolic solid brick elements

Boundary Conditions Constraints, Tests 1 and 2: •

Constrain node 7 in the X, Y, and Z translations.

•

Constrain node 8 in the X and Z translations.

•

Constrain node 87 in the Y and Z translations.

•

Constrain node 88 in the Z translation.

 •

Constrain all other nodes along the plane Y’ in the Y translation.

Constraints, Tests 3 and 4: •

Constrain node 10 in the X, Y, and Z translations

•

Constrain nodes 12 and 35 in the X and Z translations.

•

Constrain node 30 in the Y and Z translations.

•

Constrain node 71 in the Z translation.

•

Constrain all other nodes along the plane Y’ in the Y translation.

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass



Results

Mode #

Ref. Value (Hz)

1

38.200

2

85.210

3

152.23

4

245.53

5

297.05

Mesh

linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

NAFEMS Target Value (Hz) 42.881 38.821 93.817 88.451 170.67 159.34 286.12 259.20 318.86 307.92

FEMAP Structural Result (lumped mass) (Hz) 37.964 37.788 83.407 87.027 152.84 150.53 251.76 243.10 288.20 281.27

FEMAP Structural Result (consistent mass) (Hz) 38.282 38.269 83.977 87.659 157.63 157.49 265.02 259.00 298.43 306.02

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 51.

Simply-Supported Solid Square Plate The complete model and results for this test case are in the following files: •

nf052lc.neu (linear solid brick, consistent mass)

•

nf052ll.neu (linear solid brick, lumped mass)

•

nf052pc.neu (parabolic solid brick, consistent mass)

•

nf052pl.neu (parabolic solid brick, lumped mass)

This test is a normal modes/eigenvalue analysis of a simply–supported solid square plate meshed with bricks. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 52. Attributes of this test are: •

well established

•

rigid body modes (three modes)

•

kinematically incomplete suppressions

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Tests 1 and 2 (nf052lc.neu, nf052ll.neu)

 •

324 nodes, 192 linear solid brick elements

Tests 3 and 4 (nf052pc.neu, nf052pl.neu) •

155 nodes, 16 parabolic solid brick elements

Boundary Conditions Constraints •

Constraint Set 1: Constrain all the nodes along the four edges on the plane ZS = -0.5m in the Z translation.

 •

Constraint Set 2 (Kinematic DOF): Tests 1 and 2: Constrain nodes 36 and 264 in the X, Y, and Z translations. Tests 3 and 4: Constrain nodes 27 and 219 in the X and Y translation.

Solution Type Normal Modes/Eigenvalue - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass

Results

Mode #

Ref. Value (Hz)

4

45.897

5, 6

109.44

7

167.89

8

193.59

9, 10

206.19

Mesh

linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

FEMAP FEMAP Structural NAFEMS Structural Result Result Target Value (consistent mass) (lumped mass) (Hz) (Hz) (Hz) 51.654 44.762 132.73 110.52 194.37 169.08 197.18 193.93 210.55 206.64

44.115 44.502 106.73 107.94 156.48 161.44 193.58 193.16 200.14 185.60

45.318 44.796 113.96 110.54 173.30 169.11 196.77 193.92 209.56 206.65



Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 52.

Simply-Supported Solid Annular Plate The complete model and results for this test case are in the following files: •

nf053lc.neu (linear solid brick, consistent mass)

•

nf053ll.neu (linear solid brick, lumped mass)

•

nf053pc.neu (parabolic solid brick, consistent mass)

•

nf053pl.neu (parabolic solid brick, lumped mass)

This test is a normal modes/eigenvalue analysis of a solid annular plate using solid elements. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 53. Attributes of this test are: •

curved boundary (skewed coordinate system)

•

constraint equations

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling •

160 nodes, 60 linear solid bricks: α = 5°



•

68 nodes, 5 solid parabolic bricks α = 10°

Boundary Conditions Constraints, Tests 1 and 2: •

Constrain nodes 76-80 and 156-160 in the Y and Z translations.

•

Constrain all other nodes in the Y translation.

 •

Constraint equations: Constrain nodes at same R and Z are constrained to have same z displacement.

Constraints, Tests 3 and 4: •

Constrain nodes 11, 22, 33, 44, 66, 77, 88, and 99 in the Y and Z translations and X, Y, and Z rotations.

•

Constrain all other nodes in the Y translation and X, Y, and Z rotations.

•

Constraint equations: Constrain nodes at same R and Z are constrained to have same z displacement

Solution Type Normal Modes/Eigenvalues - SVI method Results were obtained two different ways: •

using lumped mass

•

using consistent mass



Results

Mode #

Ref. Value (Hz)

1

18.583

2

140.15

3

224.16

4

358.29

5

629.19

Mesh

linear parabolic linear parabolic linear parabolic linear parabolic linear parabolic

FEMAP FEMAP Structural NAFEMS Structural Result Result Target Value (consistent mass) (lumped mass) (Hz) (Hz) (Hz) 19.659 18.582 146.42 140.42 224.25 224.18 386.70 374.04 689.47 686.02

18.612 18.409 140.13 134.21 224.34 223.62 369.74 345.98 668.73 616.01

18.641 18.629 141.78 141.44 224.48 224.33 380.74 380.03 690.09 688.59

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 53.

Cantilevered Solid Beam The complete model and results for this test case are in the following files: •

nf072ac.neu (conventional numbering, consistent mass)

•

nf072al.neu (conventional numbering, lumped mass)

•

nf072bc.neu (unconventional numbering, consistent mass)

•

nf072bl.neu (unconventional numbering, lumped mass)

This test is a normal modes/eigenvalue analysis of a cantilevered solid beam. This document provides the input data and results for NAFEMS Selected Benchmarks for Natural Frequency Analysis, Test 72. Attributes of this test are: •

highly populated stiffness matrix

Test Case Data and Information Units SI

Material Properties 9 N E = 200x10 ------2m

kg ρ = 8000 ------3m ν = 0.3

Finite Element Modeling Two tests - both use solid parabolic brick elements

 •

Test 1: conventional node numbering

•

Test 2: unconventional node numbering

Boundary Conditions Constraints •

Constrain all nodes on the X=0 plane in the X, Y, and Z translations.

 •

Constrain all nodes on the Y=1m plane in the Y translation.

Solution Type Normal Modes/Eigenvalue – SVI Method

Results

Mode # 1 2 3 4 5 6

Mesh Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2 Test 1 Test 2

FEMAP FEMAP NAFEMS Structural Structural Target Value (lumped mass) (consistent mass) (Hz) (Hz) (Hz) 16.007 16.007 87.226 87.226 125.96 125.96 209.56 209.56 351.11 351.11 375.81 375.81

15.800 15.800 82.235 82.235 125.03 125.03 189.33 189.33 299.30 299.32 352.39 352.40

16.007 16.007 87.226 87.226 125.96 125.96 209.56 209.56 351.11 351.11 375.82 375.81

Note: Reference value (Ref. Value) refers to the accepted solution to the problem. Reference • NAFEMS Finite Element Methods & Standards, Abbassian, F., Dawswell, D. J., and Knowles, N. C., Selected Benchmarks for Natural Frequency Analysis (Glasgow: NAFEMS, Nov., 1987.) Test No. 72.

Verification Test Cases from the Societe Francaise des Mechaniciens The purpose of these test cases is to verify the function of the FEMAP Structural software using standard benchmarks published by SFM (Societe Francaise des Mecaniciens, Paris, France) in “Guide de validation des progiciels de calcul de structures.” Included here are: •

test cases on mechanical structures using linear statics analysis and normal modes/eigenvalue analysis

•

stationary thermal test cases using heat transfer analysis

•

a thermo–mechanical test case using linear statics analysis

Results published in “Guide de validation des progiciels de calcul de structures” are compared with those computed using the FEMAP Structural software.

Understanding the Test Case Format Each test case is structured with the following information: •

test case data and information - units - material properties - finite element modeling information - boundary conditions (loads and constraints) - solution type

•

results

•

reference

Note:

The node numbers listed in each case refer to the node numbers in the neutral (.neu) files associated with this guide. If you remesh a model, or rebuild that model from scratch, your node numbering may differ.

Reference The following reference has been used in these test cases:

 •

Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.)

Mechanical Structures - Linear Statics Analysis with Bar or Rod Elements The linear statics analysis test cases from the Societe Francaise des Mecaniciens include these bar and rod element test cases: •

"Short Beam on Two Articulated Supports"

•

"Clamped Beams Linked by a Rigid Element"

•

"Transverse Bending of a Curved Pipe"

•

"Plane Bending Load on a Thin Arc"

•

"Nodal Load on an Articulated Rod Truss"

•

"Articulated Plane Truss"

•

"Beam on an Elastic Foundation"

Short Beam on Two Articulated Supports The complete model and results for this test case are in file ssll02.neu. This test is a linear statics analysis of a short, straight beam with plane bending and shear loading. It provides the input data and results for benchmark test SSLL02/89 from “Guide de validation des progiciels de calcul de structures.” •

area = 31E-4m2

•

inertia = 2810E-8m4

•

Shear area ratio = 2.42

Test Case Data and Information Units SI

Material Properties E = 2E11 Pa ν = 0.3

Finite Element Modeling •

10 bar elements

•

11 nodes

The mesh is shown in the following figure:

 Boundary Conditions Constraints •

Constrain the nodes at both free ends of the beam (nodes 1 and 2) in all directions except for the Z rotation.

Loads •

On nodes 1-10, apply a load = 1E5 N/m in -Y direction

The boundary conditions are shown in the following figure:

Solution Type Statics

Results Total Translation at point B (Node 7) Bench Value FEMAP Structural Value Difference

-1.25926E-3 -1.25926E-3 0.00%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLL02/89.

Clamped Beams Linked by a Rigid Element The complete model and results for this test case are in file ssll05.neu. This test is a linear statics analysis of a straight, cantilever beam with plane bending and a rigid element. It provides the input data and results for benchmark test SSLL05/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties •

E = 2E11 Pa

•

I = (4/3)E-8m4

Finite Element Modeling •

20 bar elements

•

1 rigid element

•

26 nodes

The mesh is shown in the following figure:

Boundary Conditions Constraints •

Nodes 1 and 4: Fully constrained in all directions.

 Loads •

Node 3: Set nodal force = 1000 N in -Y direction

The boundary conditions are shown in the following figure:

Solution Type Statics

Results Node # Node 6 Node 3 Node 1 Node 1 Node 4 Node 4

Displacement Reaction Force Displacement Y (T2 Translation) Displacement Y (T2 Translation) Force Y (N) (T2 Constraint Force) Moment Rz (Nm) (R3 Constraint Moment) Force Y (N) (T2 Constraint Force) Rz moment (Nm) (R3 Constraint Moment)

Bench Value

FEMAP Structural

Difference

-0.125

-0.125

0.00%

-0.125

-0.125

0.00%

500

500

0.00%

500

500

0.00%

500

500

0.00%

500

500

0.00%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLL05/89.

Transverse Bending of a Curved Pipe The complete model and results for this test case are the following files: •

ssll07a.neu (linear beam)

•

ssll07b.neu (curved beam)

This test is a linear statics analysis (three–dimensional problem) of a curved pipe with transverse bending and bending–torque loading. It provides the input data and results for benchmark test SSLL07/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties E = 2E11 Pa ν = 0.3

Finite Element Modeling Test 1 (ssll07a) •

90 bar elements

•

91 nodes

Test 2 (ssll07b) •

90 curved beam elements

•

91 nodes

 The mesh for Test 1 is shown in the following figure:

Boundary Conditions Constraints •

Fully constrain node 91 in all translations and rotations.

Loads •

Create a nodal force at node 1 = 100 N in Z direction

The boundary conditions are shown in the following figure:

 Solution Type Statics

Results Node #

Displacement Moment

Point

Node 1 Node 1 θ=15°

Displacement Z (T3 Translation) Displacement Z (T3 Translation) Mt (Nm)* Mt (Nm)* Mf (Nm) Mf (Nm)

Bench Value 0.13462

74.1180 -96.5925

Test Number

FEMAP Structural

Difference

1

0.13465

0.02%

2

0.13464

0.01%

1 2 1 2

76.6709 75.8109 -96.3680 -95.2869

3.44% 1.02% 0.23% 1.35%

Mf = bending moment Mt = torsional moment *See “Post Processing” below

Post Processing Bar Element (ssll07a) List beam forces on element 167, second end •

Mf=Bar End BX2 Moment

•

Mt=Bar End BX1 Moment

Curved Beam Element (ssll07b) List beam forces on element 166, second end •

Mf=Bar End BX2 Moment

•

Mt=Bar End BX1 Moment

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLL07/89.

Plane Bending Load on a Thin Arc The complete model and results for this test case are in file ssll08.neu. This test is a linear statics analysis (plane problem) of a thin arc with plane bending. It provides the input data and results for benchmark test SSLL08/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties E = 2E11 Pa ν = 0.3

Finite Element Modeling •

11 nodes

•

10 bar elements

The mesh is shown in the following figure:

Boundary Conditions Constraints •

Node 2: Constrain the X, Y, and Z translations.

•

Node 1: Constrain the Y and Z translation only.

 •

Nodes 3-11: Constrain in the Z translation only.

Loads •

Force=100N in -Y direction

The boundary conditions are shown in the following figure:

Solution Type Statics

Results Node # Node 2 Node 1 Node 7 Node 1

Displacement Rz (rad) (R3 Rotation) Rz (rad) (R3 Rotation) Y (m) (T2 Translation) X (m) (T1 Translation)

Bench Value

FEMAP Structural

Difference

-3.0774E-2

-3.1097E-2

1.05%

3.0774E-2

3.1097E-2

1.05%

-1.9206E-2

-1.9342E-2

0.71%

5.3913E-2

5.3735E-2

0.33%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLL08/89.

Nodal Load on an Articulated Rod Truss The complete model and results for this test case are in file ssll11.neu. This test is a linear statics analysis of a plane truss with an articulated rod. It provides the input data and results for benchmark test SSLL11/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties •

E = 1.962E11 Pa

Finite Element Modeling •

4 nodes

•

4 rod elements

The mesh is shown in the following figure:

Boundary Conditions Constraints •

Nodes 3 and 17: Constrained in the X, Y, and Z translations only.

 •

Nodes 2 and 18: Constrained in the Z translation only.

Loads •

Node 2: Set Nodal force = 9.81E3 N in -Y direction

The boundary conditions are shown in the following figure:

Solution Type Statics

Results Node # Node 18 Node 18 Node 2 Node 2

Displacement X (m) (T1 Translation) Y (m) (T2 Translation) X (m) (T1 Translation) Y (m) (T2 Translation)

Bench Value

FEMAP Structural

Difference

0.26517E-3

0.26517E-3

0.00%

0.08839E-3

0.08839E-3

0.00%

3.47902E-3

3.47903E-3

~0.00%

-5.60084E-3

-5.6004E-3

~0.00%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLL11/89.

Articulated Plane Truss The complete model and results for this test case are in the following files: •

ssll14a.neu (4 bar elements)

•

ssll14b.neu (10 bar elements)

This test is a linear statics analysis of a straight cantilever beam with plane bending and tension–compression. It provides the input data and results for benchmark test SSLL14/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties •

E = 2.1E11 Pa

Finite Element Modeling Test 1 (ssll14a) •

4 bar elements

•

5 nodes

Test 2 (ssll14b) •

10 linear beam elements

•

11 nodes

The mesh for Test 1 is shown in the following figure:

 Boundary Conditions Test 1 (ssll14a) •

Constraints Nodes 1 and 4: Constrain in the X, Y, and Z translations. Nodes 2, 3, 8: Constrain in the Z translation only.

•

Loads Set forces and moments to the following numeric values: p = -3,000N/m (on element 4); F1 = -20,000N (on node 8); F2 = -10,000N (on node 2); M = -100,000Nm (on node 2)

Test 2 (ssll14b) •

Constraints Nodes 1 and 4: Constrain in the X, Y, and Z translations. Nodes 2, 3, 5-13: Constrain in the Z translation only.

•

Loads (ssll14b) Set forces and moments to the following numeric values: p = -3,000N/m (on elements 5-7); F1 = -20,000N (on node 8); F2 = -10,000N (on node 2); M = -100,000Nm (on node 2)

The boundary conditions are shown in the following figure:

Solution Type Statics



Results Node # 1

1 8

Displacement Reaction Force

Bench Value

V vertical (Y) 31500.0 reaction (N) (T2 Constraint Force) horizontal (x) reaction (N) 20239.4 (T1 Constraint Force) Y (m) (T2 Translation) -0.03072

Test Number

FEMAP Structural

Difference

1 2

33233.1 33233.1

5.50% 5.50%

1 2 1 2

20609.2 20609.3 -0.03106 -0.03161

1.82% 1.83% 1.10% 2.90%

Note: The software takes shear effect into account. Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLL14/89.

Beam on an Elastic Foundation The complete model and results for this test case are in file ssll16.neu. This test is a linear statics analysis (plane problem) of a straight beam with plane bending and an elastic support. It provides the input data and results for benchmark test SSLL16/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties •

E = 2.1E11 Pa

•

K = 8.4E5 N/m2

•

Each spring stiffness is set to: K*L/ (number of DOF spring elements).

Finite Element Modeling •

50 bar elements

•

49 DOF spring elements

•

51 nodes

The mesh is shown in the following figure:

 Boundary Conditions Constraints •

Nodes 1 and 51: Constrain in the X, Y, and Z translations.

•

Nodes 2-49: Constrain in the Z translation and X and Y rotations only.

Loads •

Set forces, moments, and distributed loads on element to the following numeric values: F = -10000 N (node 26) ; p = -5000 N/m (elements 1-50) ; M = 15000 Nm (node 51); M= -15000 Nm (node 1).

The distributed loads are shown below:

 The forces and moments are shown below:

Solution Type Statics

Results Node 51

26 26

Displacement Force, Moment rotation(rad) Rz (R3 rotation) reaction force (N) Y (T2 Constraint Force) disp. Y (m) (T2 Translation) M moment (Nm)* (Bar End BX3 Moment)

Bench Value

FEMAP Structural

Difference

-0.003045

-0.003041

0.36%

11674

11646

0.78%

-0.423326E-2 33840

-0.42270E-2 33286

0.41% 1.63%

*On element 26, second end Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLL16/89.

Mechanical Structures - Linear Statics Analysis with Plate Elements The linear statics analysis test cases from the Societe Francaise des Mecaniciens include these plate element test cases: •

"Plane Shear and Bending Load on a Plate"

•

"Infinite Plate with a Circular Hole"

•

"Uniformly Distributed Load on a Circular Plate"

•

"Torque Loading on a Square Tube"

•

"Cylindrical Shell with Internal Pressure"

•

"Uniform Axial Load on a Thin Wall Cylinder"

•

"Hydrostatic Pressure on a Thin Wall Cylinder"

•

"Gravity Loading on a Thin Wall Cylinder"

•

"Pinched Cylindrical Shell"

•

"Spherical Shell with a Hole"

•

"Uniformly Distributed Load on a Simply-Supported Rectangular Plate"

•

"Shear Loading on a Plate"

•

"Uniformly Distributed Load on a Simply-Supported Rhomboid Plate"

Plane Shear and Bending Load on a Plate The complete model and results for this test case are in file sslp01.neu. This test is a linear statics analysis (plane problem) of a plate with plane bending. It provides the input data and results for benchmark test SSLP01/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties E = 3E10 Pa ν = 0.25

Finite Element Modeling •

100 linear quadrilateral plate elements

•

126 nodes

The mesh is shown in the following figure:

Boundary Conditions Constraints •

Nodes 121-126: Fully constrain in all translations and rotations.

 Loads •

Set a shear force with parabolic distribution on width and constant distribution on thickness

•

Resultant force: p = 40 N.

The boundary conditions are shown in the following figure:

Solution Type Statics

Results Node # 3

Point Coordinates (L,y)

Centerline Displacement Y (mm) (T2 Translation)

Bench Value 0.3413

FEMAP Structural 0.3408

Difference 0.15%

The displacements are shown in the following figure:

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLP01/89.

Infinite Plate with a Circular Hole The complete model and results for this test case are in file sslp02.neu. This test is a linear statics analysis (plane problem) of a plate with tension–compression and a membrane effect. It provides the input data and results for benchmark test SSLP02/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties E = 3E10 Pa ν = 0.25

Finite Element Modeling Mapped meshing (with biasing) •

100 linear quadrilateral plate elements

•

121 nodes

The mesh is shown in the following figure:

 Boundary Conditions Constraints •

Nodes 1-11: Constrain in Y translation and X and Z rotations only.

•

Nodes 12-110: Constrain in Z translation only.

•

Nodes 111-121: Constrain in X translation and Y and Z rotations only.

Loads •

Tension force P = 2.5 N/mm**2 (in plane force of 2500 N/m)

The boundary conditions are shown in the following figure:

Solution Type Statics



Results Point Coordinates (a,0)

Node #

Stress

1

Bench Value

FEMAP Structural

Difference

7.5

7.52

0.26%

56

Plate Top Y Normal Stress (N/mm**2) 2.5 Plate Top Y Normal Stress

2.61

4.40%

111

Plate Top Y Normal Stress -2.5

-2.38

4.80%

σθ

 a, π ---  4  a, π ---  2

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLP02/89.

Uniformly Distributed Load on a Circular Plate The complete model and results for this test case are in the following files: •

ssls03a.neu (linear quadrilateral)

•

ssls03b.neu (linear triangle)

This test is a linear statics analysis (three–dimensional problem) of a circular plate fixed at the edge with transverse bending and a uniform load. It provides the input data and results for benchmark test SSLS03/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.1 ×10 Pa ν = 0.3

Finite Element Modeling Test 1 (ssl03a) - Free meshing: •

38 linear quadrilateral plate elements

•

50 nodes

 Test 2 (ssl03a) - Free meshing: •

53 linear triangular plate elements

•

38 nodes

Only 1/4 of the plate is meshed.

Boundary Conditions Constraints •

Constrain node 1 in all directions except for the Z translation.

•

Fully constrain nodes 2-3 and nodes 15-21 in all directions.

•

Constrain nodes 4-8 in the X translation and Y and Z rotations.

•

Constrain nodes 9-13 in the Y translation and X and Z rotations.

Note: Symmetric conditions are applied to the sides.

Loads •

Uniform elemental pressure p = -1000 Pa.

 Test 1 boundary conditions:

Solution Type Statics

Results Node # Node 1 Node 1

Point Center O Center O

T3 Translation (Displacement Z) w (m)

Bench Value -0.0065 -0.0065

Test Number 1 2

FEMAP Structural -0.0065 -0.0065

Difference 0.00% 0.00%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLS03/89.

Torque Loading on a Square Tube The complete model and results for this test case are in file ssls05.neu. This test is a linear statics analysis (three–dimensional problem) of a thin–walled tube loaded in torsion by pure shear at the free end. It provides the input data and results for benchmark test SSLS05/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.1 ×10 Pa ν = 0.3

Finite Element Modeling Mapped meshing •

160 linear quadrilateral plate elements

•

176 nodes

The mesh is shown in the following figure:

 Boundary Conditions Constraints •

Completely constrain nodes 1-5, 57-60, 112-115, and 167-169 in all translations and rotations.

Loads •

Torque equal to 10 Nm on the free end.

Note: This translates into an equivalent nodal force of ±12.5N. The boundary conditions are shown in the following figure:

Solution Type Statics

Results Node # 193 193 193 208 208 208

Displacement and Stress T2 Translation (m) R1 Rotation (rad) Plate Bottom Minor Stress (Pa) T2 Translation (m) R1 Rotation (rad) Plate Bottom Minor Stress (Pa)

Bench Value

FEMAP Structural

Difference

-0.617E-7 0.123E-4 -0.11E6

-0.617E-7 0.123E-4 -0.11E6

0.00% 0.00% 0.00%

-0.987E-7 0.197E-4 -0.11E6

-0.988E-7 0.197E-4 -0.11E6

0.10% 0.00% 0.00%

 Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLS05/89.

Cylindrical Shell with Internal Pressure The complete model and results for this test are in the following files: •

ssls06a.neu (linear quadrilateral, test 1)

•

ssls06b.neu (linear quadrilateral, test 2)

This test is a linear statics analysis of the thin cylinder loaded by internal pressure. It provides the input data and results for benchmark test SSLS06/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.1 ×10 Pa ν = 0.3

Finite Element Modeling Test 1 (ssls06a) - Mapped meshing •

100 linear quadrilateral plate elements

•

121 nodes

 Test 2 (ssls06b) - Mapped meshing •

400 linear quadrilateral plate elements

•

441 nodes

Boundary Conditions Constraints for Test 1 (ssls06a) •

Constrain node 1 in all directions except for the Y translation.

•

Constrain nodes 2-10 in the Z translation and X and Y rotations.

•

Constrain node 11 in all directions except for the X translation.

•

Constrain nodes 12, 23, 34, 45, 56, 67, 78, 89, 100, and 111 in the X translation and Y and Z rotations only.

•

Constrain nodes 22, 33, 44, 55, 66, 77, 88, 99, 110, 121 in the Y translation and X and Z rotations.

Constraints for Test 2 (ssls06b) •

Constrain node 1 in all directions except for the Y translation.

•

Constrain nodes 2-20 in the Z translation and X and Y rotations.

•

Constrain node 21 in all directions except for the X translation.

•

Constrain nodes 22, 43, 64, 85, 106, 127, 148, 169, 190, 211, 232, 253, 274, 295, 316, 337, 358, 379, 400, and 421 in the X translation and Y and Z rotations only.

•

Constrain nodes 42, 63, 84, 105, 126, 147, 168, 189, 210, 231, 252, 273, 294, 315, 336, 357, 378, 399, 420, 441 in the Y translation and X and Z rotations only.

Loads for Test 1 and Test 2 •

Internal pressure on the elements = 10000 Pa.

 The boundary conditions are shown in the following figure:

Solution Type Statics

Results Node #

Displacement and Stress

11

Bench Value 0.0

Test Number

FEMAP Structural

Difference

1

1.32

2

-0.139

1

4.98E5

0.40%

2

4.99E5

0.20%

σ11 ( Pa ) )

Plate Top Y Normal Stress 21 σ11 ( Pa ) )

Plate Top Y Normal Stress 111

5.00E5 σ22 ( Pa ) )

421

Plate Top X Normal Stress σ22(Pa) σ22 ( Pa ) )

Plate Top X Normal Stress

 121

2.38E-6

1

2.37E-6

0.42%

2

2.38E-6

0.00%

1

-1.42E-6

0.70%

2

-1.43E-6

0.00%

∆R ( m )

T1 Translation 441 ∆R ( m )

T1 Translation 121

-1.43E-6 ∆L ( m )

T3 Translation 441 ∆L ( m )

T3 Translation All results are averages. Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLS06/89.

Uniform Axial Load on a Thin Wall Cylinder The complete model and results for this test are in the following files: •

ssls07a.neu (parabolic quadrilateral plate, test 1)

•

ssls07b.neu (parabolic triangle plate, test 2)

This test is a linear static analysis of a thin cylinder loaded axially. It provides the input data and results for benchmark test SSLS07/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.1 ×10 Pa ν = 0.3

Finite Element Modeling Test 1 •

Meshed by revolving a meshed beam

•

200 parabolic quadrilateral plate elements

 •

661 nodes

Test 2 •

Meshed by free meshing on 1/8 of a cylinder

•

400 parabolic triangular plate elements

Boundary Conditions Constraints •

Constrain the nodes along one long edge in the Y translation and X and Z rotations.

•

Constrain the nodes along the other long edge in the X translation and the Y and Z rotations.

•

Constrain the nodes along the top short edge in the Z translation only.

•

Constrain node 1 in the Y and Z translations and the X and Z rotations.

 •

Constrain node 21 in the X and Z translations and Y and Z rotations.

Loads •

Uniform axial elemental pressures, q = 10000 N/m

The boundary conditions are shown in the following figure:

Solution Type Statics

Results Node #

Displacement and Stress

641

Bench Value

Test Number

FEMAP Structural

Difference

5.00E5

1

5.00E5

0.00%

5.00E5

2

5.00E5

0.00%

σ11 ( Pa )

Plate Top Y Normal Stress 641 σ11 ( Pa )

Plate Top Y Normal Stress

 641

0.0

1

0.0

0.0

2

0.0

-7.14E-7

1

-7.14E-7

0.0%

-7.14E-7

2

-7.14E-7

0.0%

9.52E-6

1

9.52E-6

0.0%

9.52E-6

2

9.52E-6

0.0%

σ22 ( Pa )

Plate Top X Normal Stress 641 σ22 ( Pa )

Plate Top X Normal Stress 641 ∆R ( m )

T1 Translation 641 ∆R ( m )

T1 Translation 641 ∆L ( m )

T3 Translation 641 ∆L ( m )

T3 Translation All results are averages. Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLS07/89.

Hydrostatic Pressure on a Thin Wall Cylinder The complete model and results for this test case are in file ssls08.neu. This test is a linear statics analysis of a thin cylinder loaded by hydrostatic pressure. It provides the input data and results for benchmark test SSLS08/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.1 ×10 Pa ν = 0.3

Finite Element Modeling •

200 parabolic quadrilateral plate elements

•

661 nodes

Cylinder is meshed by revolving a meshed beam. The mesh is shown in the following figure:

 Boundary Conditions Constraints •

Constrain the nodes on side A (from node 21 to node 661) in the X translation and Y and Z rotations.

•

Constrain the nodes on side B (from node 1 to node 641) in the Y translation,and X and Z rotation.

Loads •

Internal elemental pressures, p = p0*Z/L with p0=20000 Pa

The boundary conditions are shown in the following figure:

Solution Type Statics



Results Node Node 321

Point

Displacement and Stress

Any

Bench Value

FEMAP Structural

Difference

0.0

-0.0054E5

5.0E5

4.98E5

0.40%

2.38E-6

2.38E-6

0.00%

-2.86E-6

1.486E-6

0.00%

1.19E-6

1.19E-6

0.00%

σ11 ( Pa )

Plate Top Y Normal Stress Node 321

x=L/2 σ22 ( Pa )

Plate Top X Normal Stress Node 321

x=L/2 ∆R ( m )

T1 Translation Node 1

x=L ∆L ( m )

T3 Translation Node 321 ψ ( rad )

R2 Rotation ψ represents the rotation of a generator Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLS08/89.

Gravity Loading on a Thin Wall Cylinder The complete model and results for this test case are in file ssls09.neu. This test is a linear statics analysis of a thin cylinder loaded by its own weight. It provides the input data and results for benchmark test SSLS09/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.1 ×10 Pa ν = 0.3 11

γ = 7.85 ×10 Pa kg mass = 8002 ------3m

Finite Element Modeling •

65 linear quadrilateral plate elements (mapped meshing)

•

84 nodes

 The mesh is shown in the following figure:

Boundary Conditions Constraints •

Nodes 1, 5-16: Constrain in the Y translation and X and Z rotations.

•

Node 2: Constrain in all directions except for the X translation and Y rotation.

•

Nodes 3, 21-32: Constrain the X translation and Y and Z rotations.

•

Node 4: Constrain in the X and Z translations and the Y and Z rotations.

•

Nodes 33-36: Constrain in the Z translation only.

Loads •

Body load: Translational acceleration in the Z direction

 The boundary conditions are shown in the following figure:

Solution Type Statics

Results Node # Node 2

Point

Displacement and Stress

x=0

Bench Value

FEMAP Structural

3.14E5

3.02E5

0.0

-1578 to 1578

σ11 ( Pa )

Plate Top X Normal Stress Node 1

Any σ22 ( Pa )

Plate Top Y Normal Stress

Difference 3.82%

 Node 2

x=0

-4.49E-7

-4.39E-7

2.00%

2.99E-6

2.99E-6

0.00%

-1.12E-7

-1.12E-7

0.00

∆R ( m )

T1 Translation Node 1

x=L z∆ ( m )

T3 Translation Node 10

x-L ψ ( rad )

R2 Rotation Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLS09/89.

Pinched Cylindrical Shell The complete model and results for this test case are in the following files: •

ssls20a.neu (linear triangle plate)

•

ssls20b.neu (linear quadrilateral plate)

This test is a linear statics analysis of a cylindrical shell with nodal forces, F, pinching as shown. It provides the input data and results for benchmark test SSLS20/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 6

E = 10.5x10 Pa ν = 0.315

Finite Element Modeling Test 1 (ssls20a) - Free meshing •

296 linear triangle plate elements

•

173 nodes

 Test 2 (ssls20b) - Mapped meshing •

140 linear quadrilateral plate elements

•

165 nodes

Boundary Conditions Constraints •

Free conditions. To set free boundary conditions, use symmetry about XY, XZ and YZ planes.

•

Node 1: Fully constrain except for the X translation.

•

Node 2, 5-13: Constrain in the Y translation and the X and Z rotations.

•

Node 3: Fully constrain except for the Y translation.

•

Node 4, 27-35: Constrain in the X translation and the Y and Z rotations.

•

Nodes 14-26: Constrain the Z translation and the X and Y rotations.

Loads •

Nodal forces Fy = -25 N at point D

 The boundary conditions are shown in the following figure:

Solution Type Statics

Results Point D D

Displacement Displacement Y (Node 3) (T2 Translation) Displacement Y (Node 3) (T2 Translation)

Bench Value

Test Number

FEMAP Structural

Difference

-113.9E-3

1

-114.4E-3

0.44%

-113.9E-3

2

-113.3E-3

0.53%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLS20/89.

Spherical Shell with a Hole The complete model and results for this test case are in the following files: •

ssls21a.neu (Test 1, linear quadrilateral plate)

•

ssls21b.neu (Test 2, linear triangular plate)

•

ssls21c.neu (Test 3, parabolic quadrilateral plate)

This test is a linear statics analysis of a spherical shell with a hole with nodal forces. It provides the input data and results for benchmark test SSLS21/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 7

E = 6.285x10 Pa ν = 0.3

Finite Element Modeling Test 1 (ssls21a) •

100 linear quadrilateral plate elements

•

121 nodes

 Test 2 (ssls21b) •

200 linear triangular plate elements

•

121 nodes

Test 3 (ssls21c) •

100 parabolic quadrilateral plate elements

•

341 nodes

All tests are executed with mapped meshing.

Boundary Conditions Constraints •

Constrain nodes 1-11 in the X translation and Y and Z rotations.

•

Constrain nodes 111-121 in the Z translation and X and Y rotations.

•

Free condition

 s Note: To set free boundary conditions, use symmetry about XY and YZ planes.

Loads •

Nodal forces F = 2 Newtons Due to the symmetric boundary conditions, only half of the load is applied.

The boundary conditions are shown in the following figure:

Solution Type Statics

Results Point A(R,0,0)

T1 Translation u (m) node 111 node 111 node 421

Bench Value 94.0E-3 94.0E-3 94.0E-3

Test Number 1 2 3

FEMAP Structural 103.3E-3 103.7E-3 98.6E-3

Difference 9.91% 10.32% 4.89%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLS21/89.

Uniformly Distributed Load on a Simply-Supported Rectangular Plate The complete model and results for this test case are in the following files: •

ssls24a.neu (Test 1, coarse mesh)

•

ssls24b.neu (Test 2, fine mesh)

•

ssls24c.neu (Test 3, very fine mesh)

This test is a linear statics analysis of a plate with pressure loading and simple supports. It provides the input data and results for benchmark test SSLS24/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 7

E = 1.0x10 Pa ν = 0.3

Finite Element Modeling Test 1 (ssls24a): length/thickness=1 •

100 linear quadrilateral plate elements

 •

121 nodes

Test 2 (ssls24b): length/thickness=2 •

200 linear quadrilateral plate elements

•

231 nodes

Test 3 (ssls24c): length/thickness=5 •

500 linear quadrilateral plate elements

 •

561 nodes

Boundary Conditions Constraints •

Fully constrain node 1 in all translations and rotations.

•

Constrain the nodes on all edges in the Z translation only.

Loads •

Set pressure = 1 N/m**2 in the -Z direction

 The boundary conditions are shown in the following figure:

Solution Type Statics

Results Center Node 61z direction (T3 Translation) 116z direction (T3 Translation) 281z direction (T3 Translation)

Length/ Thickne ss

Parameter

1.0

Bench Value

Test

FEMAP Structural

Difference

0.00444

1

0.00453

2.03%

0.01110

2

0.01110

0.0%

0.1417

3

0.01402

1.06%

α

2.0 α

5.0 α

 61x component top surface (Plate Top X Normal Stress) 116x component top surface (Plate Top Y Normal Stress) 281x component top surface (Plate Top Y Normal Stress)

1.0

2874

1

2905

1.00%

6102

2

6065

0.61%

7476

3

7332

1.93%

β

2.0 β

5.0 α

2

βqb Max σ = σ b = ----------2 t 4

– αqb Max y = --------------3 Et

Where: q= distributed load b = dimension t = thickness E = elastic modules β values of reference from the “Guide de Validation” are incorrect. The correct values are extracted from “Formulas for Stress and Strain (Roark/Young)”. Note that the plate top surface corresponds to the side of the plate with negative global z coordinates. Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLS24/89.

Uniformly Distributed Load on a Simply-Supported Rhomboid Plate The complete model and results for this test case are in the following files: •

ssls25a.neu (Test 1)

•

ssls25b.neu (Test 2)

This test is a linear statics analysis (three–dimensional problem) of a plate with pressure and transverse bending. It provides the input data and results for a test similar to benchmark test SSLS25/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 6

E = 36.0x10 Pa ν = 0.3

Finite Element Modeling •

Length/thickness=2

•

linear quadrilateral plate elements

Test 1 (ssls25a) θ = 30°

 Test 2 (ssls25b) θ = 45°

Boundary Conditions Constraints •

Fully constrain node 231 in all translations and rotations.

•

Constrain the nodes along the edges of the mesh in the Z translation.

 Loads •

Elemental pressure = 1 N/m**2 in the -Z direction

Solution Type Statics

Results Test Case Test 1 ssls25a

Parameters

Z displacement α = 0.118 θ = 30°

ssls25a

-3.277x10E-3m Y stress

β = 0.570

Test 2 ssls25b

Bench Center location Value

-5.70x10E3N/m2 Z displacement

α = 0.108 θ = 45°

ssls25b

-3.0x10E-3m Y stress

β = 0.539

-5.39x10E3N/m2

FEMAP Structural

Difference

Z displacement (T3 Trans- 4.27% lation) at node 116 -3.137x10E-3m Y stress (Plate Top Y Nor- 1.07% mal Stress) at node 116 -5.761x10E3N/m2 Z displacement (T3 Trans- 3.53% lation) at node 116 -2.894x10E-3m Y stress (Plate Top Y Nor- 0.76% mal Stress) at node 116 -5.349x10E3N/m2



Max σ =βqb

2 4

αqb Max y = -----------3 Et

Where: q= distributed load b = dimension t = thickness E = elastic modules Values of reference from the “Guide de validation” are incorrect. The correct values are extracted from “Formulas for Stress and Strain (Roark/Young),” table 26, case number 14a. Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLS25/89.

Shear Loading on a Plate The complete model and results for this test case are in the following files: •

ssls27a.neu (Test 1)

•

ssls27b.neu (Test 2)

•

ssls27c.neu (Test 3)

This test is a linear statics analysis of a thin plate with torque and shear loading. It provides the input data and results for benchmark test SSLS27/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 7

E = 1.0x10 Pa ν = 0.25

Finite Element Modeling Test 1 (ssls27a) - Mindlin (element formulation) •

6 linear quadrilateral plate elements

•

14 nodes

Test 2 (ssls27b) - Kirchhoff (element formulation) •

6 linear quadrilateral plate elements

 •

14 nodes

Test 3 (ssls27c) - Mindlin (element formulation) •

48 linear quadrilateral plate elements

•

75 nodes

All tests are executed with mapped meshing.

Boundary Conditions Constraints •

Fully constrain the nodes on side AD in all translations and rotations.

Loads •

Create a nodal force Fz = -1N at point B.

•

Create a nodal force -Fz = 1N at point C.

The boundary conditions are shown in the following figure: D

A

C D

Solution Type Statics



Results at Location C Displacement Node (Total T3 Translation) 14 14 75

Bench Value 3.537E-2 3.537E-2 3.537E-2

Test Number 1 2 3

FEMAP Structural 5.335E-2 3.382E-2 3.750E-2

Difference 50.83% 4.38% 6.02%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLS27/89.

Mechanical Structures - Linear Statics Analysis with Solid Elements The linear statics analysis test cases from the Societe Francaise des Mecaniciens include these solid element test cases: •

"Solid Cylinder in Pure Tension"

•

"Internal Pressure on a Thick-Walled Spherical Container"

•

"Internal Pressure on a Thick-Walled Infinite Cylinder"

•

"Prismatic Rod in Pure Bending"

•

"Thick Plate Clamped at Edges"

Solid Cylinder in Pure Tension The complete model and results for this test case are in the following files: •

sslv01a.neu (parabolic tetrahedron, free meshing)

•

sslv01b.neu (linar brick, mapped meshing)

•

sslv01c.neu (linear quadrilateral axisymmetric solid, mapped meshing)

•

sslv01d.neu (linear triangular axisymmetric solid, free meshing)

This test is a linear statics analysis of a solid cylinder with tension–compression. It provides the input data and results for benchmark test SSLV01/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.0x10 Pa ν = 0.30

Finite Element Modeling Test 1 (sslv01a) - Free meshing •

155 parabolic tetrahedron elements

•

342 nodes

 Test 2 (sslv01b) - Mapped meshing •

192 linear brick elements

•

259 nodes

Test 3 (sslv01c) - Mapped meshing •

48 linear quadrilateral axisymmetric solid elements

•

65 nodes

Test 4 (sslv01d) - Free meshing •

28 linear triangular axisymmetric solid elements

•

24 nodes

 Boundary Conditions Constraints •

Uniaxial deformation of the cylinder section

Constraints (sslv01a) •

Nodes 1, 17-19: Constrain in the Y and Z translations.

•

Nodes 2, 14-16: Constrain in the X and Z translations.

•

Node 3: Constrain in the X, Y, and Z translations.

•

Nodes 4, 59-63: Constrain in the X and Y translations.

•

Nodes 5, 20-22, 33-45, 200-226: Constrain in the Y translation.

•

Nodes 6, 23-25, 46-58, 173-199: Constrain in the X translation.

•

Nodes 7-13, 64-72: Constrain in the Z translation.

Constraints (sslv01b) •

Constrain node 1, 10,19, and 28 in the Y and Z translation.

•

Constrain nodes 2-8, 11-17, 20-26, and 29-35 in the Z translation.

•

Constrain nodes 9, 18, 27, and 36 in the X and Z translation.

•

Constrain node 37 in the X, Y, and Z translations.

•

Constrain nodes 54, 63, 72, 81, 99, 108, 117, 126, 144, 153, 162, 171, 189, 198, 207, 216, 234, 243, 252, 261, 279, 288, 297, and 306 in the X translation.

•

Constrain nodse 82, 127, 172, 217, and 307 in the X and Y translation.

 •

Constrain nodes 46, 55, 64, 73, 91, 100, 109, 118, 136, 145, 154, 163, 181, 190, 199, 208, 226, 235, 244, 253, 271, 280, 289, and 298 in the Y translation.

Constraints (sslv01c) •

Constrain nodes 13, 26, 39, and 52 in the Z translation.

•

Constrain node 65 in the X and Z translations.

Constraints (sslv01d) •

Constrain node 1 in the X and Z translation

•

Constrain nodes 2, 5, 6, and 7 in the Z translation.

 Loads (all tests) •

Set uniformly distributed force -F/A on the free end in the Z direction

•

Elemental pressure, F/A = 100 MPa

Loads, Tests 1 and 2

Loads, Tests 3 and 4:

Solution Type Statics

Results Node # 6 279 1 4 4 307 53

Displacements T3 Translation T3 Translation T3 Translation T3 Translation T3 Translation T3 Translation T3 Translation

Bench Value 1.5E-3 1.5E-3 1.5E-3 1.5E-3 1.5E-3 1.5E-3 1.5E-3

Test # 1 2 3 4 1 2 3

FEMAP Structural 1.5E-3 1.5E-3 1.5E-3 1.5E-3 1.5E-3 1.5E-3 1.5E-3

Difference 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

 3 T3 Translation 1.5E-3 4 1.5E-3 0.00% 37 T3 Translation 1E-3 1 1E-3 0.00% 189 T3 Translation 1E-3 2 1E-3 0.00% 5 T3 Translation 1E-3 3 1E-3 0.00% 25 T3 Translation 1E-3 4 1E-3 0.00% 41 T3 Translation 0.5E-3 1 0.5E-3 0.00% 99 T3 Translation 0.5E-3 2 0.5E-3 0.00% 9 T3 Translation 0.5E-3 3 0.5E-3 0.00% 29 T3 Translation 0.5E-3 4 0.5E-3 0.00% 6 T2 Translation -0.15E-3 1 -0.15E-3 0.00% 279 T2 Translation -0.15E-3 2 -0.15E-3 0.00% 1 T1 Translation -0.15E-3 3 -0.15E-3 0.00% 4 T1 Translation -0.15E-3 4 -0.15E-3 0.00% 37 T1 Translation -0.15E-3 1 -0.15E-3 0.00% 189 T1 Translation -0.15E-3 2 -0.15E-3 0.00% 5 T2 Translation -0.15E-3 3 -0.15E-3 0.00% 25 T1 Translation -0.15E-3 4 -0.15E-3 0.00% 41 T1 Translation -0.15E-3 1 -0.15E-3 0.00% 99 T2 Translation -0.15E-3 2 -0.15E-3 0.00% 9 T1 Translation -0.15E-3 3 -0.15E-3 0.00% 29 T1 Translation -0.15E-3 4 -0.15E-3 0.00% Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLV01/89/89.

Internal Pressure on a Thick-Walled Spherical Container The complete model and results for this test case are in the following files: •

sslv03a.neu (Test 1, linear solids)

•

sslv03b.neu (Test 2, parabolic solids)

•

sslv03c.neu (Test 3, linear axisymmetric solids)

•

sslv03d.neu (Test 4, parabolic axisymmetric solids)

This test is a linear statics analysis of a thick sphere with internal pressure. It provides the input data and results for benchmark test SSLV03/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 5

E = 2.0x10 Pa ν = 0.30

Finite Element Modeling Test 1 (sslv03a) - Mapped meshing •

1600 linear brick elements

 •

1898 nodes

Test 2 (sslv03b) - Mapped meshing •

250 parabolic brick elements

•

1256 nodes

Test 3 (sslv03c) - Mapped meshing •

400 linear quadrilateral axisymmetric solid elements

 •

451 nodes

Test 4 (sslv03d) - Mapped meshing •

400 parabolic quadrilateral axisymmetric solid elements

•

1301 nodes

Boundary Conditions Constraints •

The equivalent of the center of the sphere being fixed is modeled via symmetric boundary conditions.

Constraints - Tests 1 and 2:

 Constraints - Tests 3 and 4:

Loads •

Uniform radial elemental pressure = 100 MPa

The boundary conditions are shown in the following figure: Pressure -Tests 1 and 2:

Pressure - Tests 3 and 4:

 Solution Type Statics

Results Results for Point R = 1m Point r=1 m

Node #

Displacement Stress

1

Bench Value

Test Number

FEMAP Structural

Difference

-100

1

-90.07

9.93%

-100 -100

2 3

-104.33 -95.50

4.33% 4.50%

-100

4

-94.81

5.19%

71.43

1

72.04

0.85%

71.43

2

73.70

3.18%

71.43

3

69.20

3.12%

71.43

4

69.50

2.70%

0.4E-3

1

0.40E-3

0.00%

σ Π ( MPa )

1 41 41

Solid Z Normal Stress Solid Z Normal Axisym C1 Radial Stress Axisym C1 Radial Stress

1 σ θ ( MPa )

Solid Y Normal Stress 1 σ θ ( MPa )

Solid Y Normal Stress 41 σ θ ( MPa )

Axisym C1 Azimuth Stress 41 σ θ ( MPa )

1

Axisym C1 Azimuth Stress u (m) T3 Translation

 1 41 41

u (m) T3 Translation u (m) T3 Translation u (m) T3 Translation

0.4E-3

2

0.40E-3

0.00%

0.4E-3

3

0.41E-3

2.50%

0.4E-3

4

0.40E-3

0.00%

Results for Point R = 2m

Point r=2 m

Node #

Displacement Stress

1826

Bench Value

Test Number

FEMAP Structural

Difference

0

1

-.041

N/A

0

2

-.649

N/A

0

3

-.233

N/A

0

4

-.430

N/A

21.43

1

21.18

1.16%

21.43

2

21.76

1.53%

21.43

3

21.39

0.19%

21.43

4

21.58

0.70%

σ Π ( MPa )

Solid Z Normal Stress 2221 σ Π ( MPa )

1 1

Solid Z Normal Stress Axisym C1 Radial Stress Axisym C1 Radial Stress

1826 σ θ ( MPa )

Solid Y Normal Stress 2221 σ θ ( MPa )

1 1

Solid Y Normal Stress Axisym C1 Radial Stress Axisym C1 Radial Stress

 1826 2221 1 1

u (m) T3 Translation u (m) T3 Translation u (m) T3 Translation u (m) T3 Translation

1.5E-4

1

1.50E-4

0.00%

1.5E-4

2

1.50E-4

0.00%

1.5E-4

3

1.53E-4

2.00%

1.5E-4

4

1.50E-4

0.00%

All results are averaged. Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLV03/89.

Internal Pressure on a Thick-Walled Infinite Cylinder The complete model and results for this test case are in the following files: •

sslv04a.neu (solid, linear brick)

•

sslv04b.neu (solid, parabolic brick)

•

sslv04c.neu (solid, axisymmetric quadrilateral)

•

sslv04d.neu (solid, axisymmetric parabolic)

This test is a linear statics analysis of a thick cylinder with internal pressure. It provides the input data and results for benchmark test SSLV04/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 5

E = 2.0x10 Pa ν = 0.30

Finite Element Modeling All tests are executed with mapped meshing.

Test 1 (sslv04a) - Mapped meshing •

400 solid (linear brick) elements

•

902 nodes

Test 2 (sslv04b) - Mapped meshing •

240 solid (parabolic brick) elements

•

1873 nodes

 FE Model - Tests 1 and 2:

Test 3 (sslv04c) - Mapped meshing •

600 axisymmetric (linear quadrilateral solid) elements

•

656 nodes

Test 4 (sslv04d) - Mapped meshing •

600 axisymmetric (parabolic quadrilateral solid) elements

•

1911 nodes

FE Model - Tests 3 and 4:

Boundary Conditions Constraints (sslv04a) •

Nodes 1-41, 452-492: Constrain in the X translation.

•

Nodes 411-451, 862-902: Constrain in the Z translation.

 Constraints (sslv04b) •

Nodes 1-61, 1038-1098, 2075-2135: Constrain in the X translation.

•

Nodes 977-1037, 2014-2074, 3051-3111: Constrain in the Z translation.

Constraints (sslv04c) •

Nodes 1-41: Constrain in the Z translation.

Constraints (sslv04d) •

Nodes 1-81: Constrain in the Z translation.

Loads (all tests) •

Unlimited cylinder

•

Internal elemental pressure p = 60 MPa

Boundary Conditions - Tests 1 and 2:

Boundary Conditions - Tests 3 and 4:

 Solution Type Statics

Results All results are averaged.

Results for R=0.1m Test Case

Point

sslv04a

r=0.1 m

Displacement Stress

Bench Value

Node #

FEMAP Structural

Difference

-60

411

-57.07

4.88%

-60

977

-60.97

1.62%

-60

616

-58.03

3.28%

-60

1831

-59.98

0.03%

100

411

99.69

0.31%

100

977

100.98

0.98%

100

616

100.77

0.77%

100

1831

99.98

0.02%

80

411

79.35

0.81%

σ r ( MPa )

sslv04b sslv04c sslv04d

Solid X Normal Stress Solid X Normal Stress Axisymm C1 Radial Stress Axisymm C1 Radial Stress

sslv04a σ θ ( MPa )

Solid Z Normal Stress sslv04b σ θ ( MPa )

sslv04c sslv04d

Solid Z Normal Stress Axisymm C1 Azimuth Stress Axisymm C1 Azimuth Stress

sslv04a τ max ( MPa )

Solid Max Shear Stress

 sslv04b

Solid Max Shear Stress u (m) T1 Translation T1 Translation T1 Translation T1 Translation

sslv04a sslv04b sslv04c sslv04d

80

977

80.97

1.21%

59E-6

411

59E-6

0.00%

59E-6 59E-6 59E-6

977 616 1831

59E-6 59E-6 59E-6

0.00% 0.00% 0.00%

Results for R=0.2m Test Case

Point

sslv04a

r=0.2m

Bench Value

Displacement Stress

Node #

FEMAP Structural

Difference

0

451

-.006

NA

0

1037

-.250

NA

0

656

-.253

NA

0

1911

.002

NA

40

451

39.70

0.75%

40

1037

40.25

0.62%

40

656

40.61

1.53%

40

1911

39.90

0.25%

20

451

20.10

0.50%

σ r ( MPa )

sslv04b sslv04c sslv04d

Solid X Normal Stress Solid X Normal Stress Axisymm C1 Radial Stress Axisymm C1 Radial Stress

sslv04a σ θ ( MPa )

sslv04b sslv04c sslv04d

Solid Z Normal Stress Solid Z Normal Stress Axisymm C1 Aximuth Stress Axisymm C1 Aximuth Stress

sslv04a τ max ( MPa )

Solid Max Shear Stress

 sslv04b sslv04a sslv04b sslv04c sslv04d

Solid Max Shear Stress u (m) T1 Translation T1 Translation T1 Translation T1 Translation

20

1037

20.25

1.25%

40E-6

451

40E-6

0.00%

40E-6 40E-6 40E-6

1037 656 1911

40E-6 39.9E-6 40E-6

0.00% 0.25% 0.00%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLV04/89.

Prismatic Rod in Pure Bending The complete model and results for this test case are in the following files: •

sslv08a.neu (Test 1, solid elements, linear tetrahedrons)

•

sslv08b.neu (Test 2, solid elements, parabolic tetrahedrons)

•

sslv08c.neu (Test 3, solid elements, linear bricks)

•

sslv08d.neu (Test 4 solid elements, parabolic bricks)

This test is a linear statics analysis of a solid rod with bending. It provides the input data and results for benchmark test SSLV08/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 5

E = 2.0x10 Pa ν = 0.30

Finite Element Modeling Test 1 (sslv08a) - Free meshing •

198 solid (linear tetrahedron) elements

•

76 nodes

Test 2 (sslv08b) - Free meshing •

198 solid (parabolic tetrahedron) elements

•

409 nodes

 FE Model - Tests 1 and 2:

Test 3 (sslv08c) - Mapped meshing •

48 solid (linear brick) elements

•

117 nodes

Test 4 (sslv08d) - Mapped meshing •

48 solid (parabolic brick) elements

•

381 nodes

FE Model - Tests 3 and 4:

Boundary Conditions Constraints (sslv08a) •

Nodes 29, 33: Constrain in the X and Z translations.

•

Nodes 30-32, 34, 39, 40: Constrain in the Z translation.

•

Node 57: Constrain in the X, Y, and Z translations.

 Constraints (sslv08b) •

Nodes 127, 131: Constrain in the X and Z translations.

•

Nodes 128-130, 132-146, 188-195: Constrain in the Z translation only.

•

Node 187: Constrain in the X, Y, and Z translations.

Constraints (sslv08c) •

Nodes 1-4, 6-9: Constrain in the Z translation.

•

Node 5: Constrain in the X, Y, and Z translations.

Constraints (sslv08d) •

Nodes 1-8, 10, 12, 14-21: Constrain in the Z translation.

•

Nodes 9, 13: Constrain in the X translation.

•

Nodes 11: Constrain in the X, Y, and Z translations.

Loads (all tests) •

Set moment Mx equal to (4/3)E+7 N.m

Boundary Conditions - Tests 1 and 2:

 Boundary Conditions - Tests 3 and 4:

Solution Type Statics

Results Test #

Node #

1

5

2

5

3

75

4

245

1 2 3 4 1 2 3 4 1 2 3 4

26 90 77 251 19 40 76 249 5 5 75 245

Displacement/ Stress Solid Z Normal Stress (Pa) Solid Z Normal Stress (Pa) Solid Z Normal Stress (Pa) Solid Z Normal Stress (Pa) T2 Translation T2 Translation T2 Translation T2 Translation T3 Translation T3 Translation T3 Translation T3 Translation T1 Translation T1 Translation T1 Translation T1 Translation

Bench Value

FEMAP Structural

Difference

-10E6

-4.268E6

57.00%

-10E6

10.03E6

0.30%

-10E6

10.07E6

0.70%

-10E6

10.01E6

0.10%

4E-4 4E-4 4E-4 4E-4 2E-4 2E-4 2E-4 2E-4 0.15E-4 0.15E-4 0.15E-4 0.15E-4

2.964E-4 4E-4 4E-4 4.044E-4 1.460E-4 2E-4 2E-4 2.010E-4 7.449E-6 0.1514E-4 0.1480E-4 0.1511E-4

26.00% 0.00% 0.00% 1.10% 27.00% 0.00% 0.00% 0.50% 50.34% 0.93% 1.33% 0.73%

 1 2 3 4

8 8 73 241

T1 Translation T1 Translation T1 Translation T1 Translation

-0.15E-4 -0.15E-4 -0.15E-4 -0.15E-4

-6.2620E-6 -0.1509E-4 -0.1480E-4 -0.1511E-4

58.20% 0.60% 1.33% 0.73%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLV08/89.

Thick Plate Clamped at Edges The complete model and results for this test case are in the following files: •

sslv09a10.neu (Test 1, parabolic brick, length/thickness =10)

•

sslv09a20.neu (Test 1, parabolic brick, length/thickness =20)

•

sslv09a50.neu (Test 1, parabolic brick, length/thickness =50)

•

sslv09a75.neu (Test 1, parabolic brick, length/thickness =75)

•

sslv09a100.neu (Test 1, parabolic brick, length/thickness =100)

•

sslv09b10.neu (Test 2, linear plate, length/thickness =10)

•

sslv09b20.neu (Test 2, linear plate, length/thickness =20)

•

sslv09b50.neu (Test 2, linear plate, length/thickness =50)

•

sslv09b75.neu (Test 2, linear plate, length/thickness =75)

•

sslv09b100.neu (Test 2, linear plate, length/thickness =100)

This test is a linear statics analysis of a square thick plate with pressure and transverse bending. It provides the input data and results for benchmark test SSLV09/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.1x10 Pa ν = 0.30

Finite Element Modeling Test 1 - Mapped meshing •

25 parabolic brick elements

•

228 nodes

 •

length/thickness =10, 20, 50, 75, 100

Test 2 - Mapped meshing •

25 linear quadrilateral plate elements

•

36 nodes

•

length/thickness =10, 20, 50, 75, 100

Test 2 is done using plate elements with the following thickness values: •

length/thickness =10, t=0.1

•

length/thickness =20, t=0.05

•

length/thickness =50, t=0.02

•

length/thickness =75, t=0.01333

•

length/thickness =100, t=0.01

 Boundary Conditions Constraints – Test 1 •

Fully constrain the nodes on edges AB, A’B’, AD, and A’D’ in all translations and rotations.

•

Constrain the nodes on edge BC and B’C’ in the X translation and Y and Z rotations.

•

Constrain the corner nodes at C and C’ in all translations and rotations except for the Z translation.

•

Constrain the nodes on edge DC and D’C’ in the Y translation and X and Z rotations.

Constraints – Test 2 •

Fully constrain the nodes on edges AB and AD in all translations and rotations.

•

Constrain the nodes on edge BC in the X translation and Y and Z rotations.

•

Constrain the corner nodes at C in all translations and rotations except for the Z translation.

•

Constrain the nodes on edge DC in the Y translation and X and Z rotations.

Loads •

Load case 1: Elemental pressure p = 1E6 Pascals in -Z direction

•

Load case 2: Point C Nodal force F = 2.5E5 N in -Z direction

Boundary conditions for Test 1:

 Boundary conditions for Test 2:

Solution Type Statics

Results Test Case 1 (T3 Translation at location C)

File Name

Length/ Node Thick- Load Case # ness

sslv09a10 sslv09a10 sslv09a20 sslv09a20 sslv09a50 sslv09a50 sslv09a75 sslv09a75 sslv09a100 sslv09a100

10 10 20 20 50 50 75 75 100 100

Pressure Force Pressure Force Pressure Force Pressure Force Pressure Force

242 242 242 242 242 242 242 242 242 242

Analytical -.6552E-4 -.29146E-3 -.52416E-3 -.23317E-2 -.81900E-2 -.36433E-1 -.27641E-1 -.12296 -.65520E-1 -.29146

Reference FEM

FEMAP Structural

-.76231E-4 -.42995E-3 -.53833E-3 -.25352E-2 -.80286E-2 -.35738E-1 -.26861E-1 -.11837 -.63389E-1 -.27794

-.735942E-4 -.426662E-3 -.523376E-3 -.242500E-2 -.778247E-2 -.346276E-1 -.259820E-1 -.114411 -.612191E-1 -.268120

Difference 12.32% 46.38% 0.15% 4.00% 4.98% 4.96% 6.00% 6.95% 6.56% 8.00%

 Test Case 2 (T3 Translation at location C) Length/ Node Thick- Load Case # ness

Part Name sslv09b10 sslv09b10 sslv09b20 sslv09b20 sslv09b50 sslv09b50 sslv09b75 sslv09b75 sslv09b10 0 sslv09b10 0

Analytical

Reference FEM

FEMAP Structural

Difference

10 10 20 20 50 50 75 75 100

Pressure Force Pressure Force Pressure Force Pressure Force Pressure

1 1 36 36 36 36 36 36 1

-.6552E-4 -.29146E-3 -.52416E-3 -.23317E-2 -.81900E-2 -.36433E-1 -.27641E-1 -.12296 -.65520E-1

-.78661E-4 -.41087E-3 -.55574E-3 -.25946E-2 -.83480E-2 -.37454E-1 -.28053E-1 -.12525 -.66390E-1

-.797294E-4 -.395973E-3 -.564973E-3 -.260199E-2 -.849953E-2 -.381471E-1 -.285676E-1 -.127845 -.676175E-1

21.69% 35.86% 8.69% 11.59% 3.78% 4.70% 3.35% 3.97% 3.20%

100

Force

1

-.29146

-.29579

-.302292

3.72%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SSLV09/89.

Mechanical Structures - Normal Modes/Eigenvalue Analysis The normal modes/eigevanlues test cases from the Societe Francaise des Mecaniciens include: •

"Lumped Mass-Spring System"

•

"Short Beam on Simple Supports"

•

"Axial Loading on a Rod"

•

"Thin Circular Ring"

•

"Cantilever Beam with a Variable Rectangular Section"

•

"Thin Circular Ring Clamped at Two Points"

•

"Vibration Modes of a Thin Pipe Elbow"

•

"Cantilever Beam with Eccentric Lumped Mass"

•

"Thin Square Plate (Clamped or Free)"

•

"Simply-Supported Rectangular Plate"

•

"Thin Ring Plate Clamped on a Hub"

•

"Vane of a Compressor - Clamped-free Thin Shell"

•

"Bending of a Symmetric Truss"

•

"Hovgaard’s Problem - Pipes with Flexible Elbows"

•

"Rectangular Plates"

Lumped Mass-Spring System The complete model and results for this test case are in file sdld02.neu. This test is a normal modes/eigenvalue analysis of an elastic link with lumped mass. It provides the input data and results for benchmark test SDLD02/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties Spring constant

Finite Element Modeling •

8 mass elements

•

9 DOF springs

•

8 nodes

The mesh is shown in the following figure:

Boundary Conditions Constraints •

Constrain all the nodes (1-8) in all translations and rotations except for the X translation.

 The boundary conditions are shown in the following figure:

Solution Type Normal Modes/Eigenvalue - SVI method

Results The mode shapes results are exact. The multiplication coefficient is 0.4642 for mode 1 and 0.4642 for mode 8.

Frequency Results: Bench Value (Hz)

Normal Mode 1 2 3 4 5 6 7 8

5.5274 10.8868 15.9155 20.4606 24.3840 27.5664 29.9113 31.3474

FEMAP Structural (Hz) 5.5274 10.8868 15.9155 20.4606 24.3840 27.5664 29.9113 31.3474

Difference 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

 Mode Shapes Results: Normal Mode 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8

Bench Value

Point P1 P2 P3 P4 P5 P6 P7 P8 P1 P2 P3 P4 P5 P6 P7 P8

0.1612 0.3030 0.4082 0.4642 0.4642 0.4082 0.3030 0.1612 0.1612 -0.3030 0.4082 -0.4642 0.4642 -0.4082 0.3030 -0.1612

FEMAP Structural 0.3473 0.6527 0.8794 1.0000 1.0000 0.8794 0.6527 0.3473 -0.3473 0.6527 -0.8794 1.0000 -1.0000 0.8794 -0.6527 0.3473

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLD02/89, p. 178.

Short Beam on Simple Supports The complete model and results for this test case are in the following files: •

sdll01a.neu

•

sdll01b.neu

This test is a modal analysis of a straight short beam with simple supports both inline and offset. It provides the input data and results for benchmark test SDLL01/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2x10 Pa ν = 0.3 kg ρ = 7800 ------3m

Finite Element Modeling Problem 1 (sdll01a) •

10 bar elements

•

11 nodes

 Problem 2 (sdll01b) •

10 bar elements

•

2 rigid elements (master node 4 to slave node 2; master node 3 to slave node 1)

Boundary Conditions Constraints •

Node 1: Constrain in all directions and rotations, except the Z rotation.

•

Node 2: Constrain in all directions and rotations, except for the X translation and Z rotation.

•

Constrain all other nodes in the Z translation and the X and Y rotations.

Loads •

no load case

The boundary conditions for both problems are shown in the following figure:

 Solution Type Normal Modes/Eigenvalue – SVI method

Results Problem 1: Frequency Results Bench Value (Hz)

Normal Mode Bending 1 Tension 1 Bending 2 Bending 3 Tension 2 Bending 4

431.555 1265.924 1498.295 2870.661 3797.773 4377.837

FEMAP Structural (Hz) 431.555 1267.226 1503.171 2904.096 3833.003 4493.912

Difference 0.03% 0.10% 0.33% 1.16% 0.93% 2.65%

Problem 2: Frequency Results Bench Value (Hz)

Mode number 1 2 3 4 5

392.8 902.2 1591.9 2629.2 3126.2

FEMAP Structural (Hz) 394.3 922.4 1641.0 2800.0 3291.2

Difference 0.38% 2.24% 3.08% 6.50% 5.28%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLL01/89.

Axial Loading on a Rod The complete model and results for this test case are in the following file: •

sdll05a.neu

•

sdll05b.neu

This test is a modal analysis of a simply–supported beam with stress stiffening. It provides the input data and results for benchmark test SDLL05/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2x10 Pa kg ρ = 7800 -------m3

Finite Element Modeling •

10 bar elements

•

11 nodes

The mesh is shown in the following figure:

 Boundary Conditions Problem 1 (sdll05a): •

Node 1: Leave the Z rotation free and constrain the node in all other translations and rotations.

•

Node 2 : Leave the X translation and Z rotation free and constrain in all other translations and rotations.

Problem 2 (sdll05b): •

Node 1: Leave the Z rotation free and constrain the node in all other translations and rotations.

•

Node 2: Leave the X translation and Z rotation free and constrain the node in all other translations and rotations.

•

Load Set 1 (node 2): Define a nodal force = to 1E5N in the -X direction. Ensure that Stress Stiffening is turned on in the analysis set.

Solution Type Normal Modes/Eigenvalue - SVI method



Results Frequency Results: Normal Mode sdll05a sdll05a sdll05b sdll05b

Mode 1 Mode 3 Mode 1 Mode 3

Bench Value (Hz) 28.702 114.807 22.434 109.080

FEMAP Structural (Hz) 28.672 114.351 22.399 108.61

Difference 0.10% 0.40% 0.16% 0.43%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLL05/89.

Cantilever Beam with a Variable Rectangular Section The complete model and results for this test case are in the following file: sdll09a.neu This test is a modal analysis of a straight cantilever beam with a variable section. It provides the input data and results for benchmark test SDLL09/89 from “Guide de validation des progiciels de calcul de structures.” b0

b0 β = -----b1 b1

Test Case Data and Information Units SI

Material Properties 11

E = 2x10 Pa kg ρ = 7800 -------m3

Finite Element Modeling •

10 beam elements (tapered)

•

11 nodes

 The mesh is shown in the following figure:

Boundary Conditions •

Constrain node 1 in all directions.

•

Constrain all other nodes in the Z translation and X and Y rotations only.

•

no load case

The boundary conditions are shown in the following figure:

Solution Type Normal Modes/Eigenvalue - SVI method



Results Frequency Results Normal Mode

β

4

1 2 3 4 5

Bench Value (Hz) 54.18 171.94 384.40 697.24 1112.28

FEMAP Structural (Hz) 54.13 171.36 381.70 688.89 1092.92

Difference 0.09% 0.34% 0.70% 1.20% 1.74%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLL09/89.

Thin Circular Ring The complete model and results for this test case are in file sdll11.neu. This test is a modal analysis of a thin curved beam. It provides the input data and results for benchmark test SDLL11/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 10

E = 7.2x10 Pa ν = 0.3 kg ρ = 2700 ------3m

Finite Element Modeling •

36 bar elements

•

36 nodes

The mesh is shown in the following figure:

 Boundary Conditions Constraints •

Unconstrained (free) conditions

•

Create 1 constraint set (Kinematic DOF set) to fully constrain the 3 nodes shown below (nodes 7, 21, 30).

Loads •

no load case

The boundary conditions are shown in the following figure:

Solution Type Normal Modes/Eigenvalue - SVI method

Results Frequency Results Bench Value (Hz)

Normal Mode Modes 1-6 Modes 7, 8 Modes 9, 10 Modes 11, 12 Modes 13, 14

0 318.36 511 900.46 1590

FEMAP Structural (Hz) 0 318.99 508 900.19 1569

Difference 0.00% 0.20% 0.59% 0.03% 1.32%

 Modes 15, 16 1726.55 1721.56 0.29% Modes 17, 18 2792.21 2774.91 0.62% Modes 19, 20 3184 3116 2.14% Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLL11/89.

Thin Circular Ring Clamped at Two Points The complete model and results for this test case are in file sdll12.neu. This test is a modal analysis of a thin curved beam. It provides the input data and results for benchmark test SDLL12/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 10

E = 7.2x10 Pa ν = 0.3 kg ρ = 2700 ------3m

Finite Element Modeling •

29 bar elements

•

29 nodes

The mesh is shown in the following figure:

 Boundary Conditions •

Points A and B (nodes 1 and 2): Fully constrained in all directions

•

All other nodes: Constrained the Z translation and X and Y rotations only.

•

no load case

The boundary conditions are shown in the following figure:

Solution Type Normal Modes/Eigenvalue - SVI method

Results Frequency Results Bench Value (Hz)

Normal Mode 1 2 3 4 5 6 7

235.3 575.3 1105.7 1405.6 1751.1 2557.0 2801.5

FEMAP Structural (Hz) 235.9 575.1 1102.7 1398.0 1740.8 2536.6 2723.0

Difference 0.25% 0.03% 0.27% 0.54% 0.59% 0.80% 2.80%

 Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLL12/89.

Vibration Modes of a Thin Pipe Elbow The complete model and results for this test case are in the following files: •

sdll014a.neu

•

sdll014b.neu

•

sdll014c.neu

This test is a modal analysis of a straight cantilever beam, and a thin curved beam. It provides the input data and results for benchmark test SDLL14/89 from “Guide de validation des progiciels de calcul de structures.” A

C

L

B

L D

Test Case Data and Information Units SI

 Material Properties 11

E = 2.1x10 Pa ν = 0.3 kg ρ = 7800 ------3m

Finite Element Modeling Problem 1 (sdll14a) where L=0 and Problem 2 (sdll14b) where L=0.6: •

18 bar elements

•

19 nodes

Problem 3 (sdll14c) where L=2: •

28 bar elements

•

29 nodes

The FE model is shown below:

 Boundary Conditions Problem 1 (sdll14a): •

Fully constrain points C and D (nodes 1 and 2) in all translations and rotations.

Problem 2 (sdll14b) and Problem 3 (sdll14c): •

Fully constrain points C and D (nodes 1 and 4) in all translations and rotations.

•

Constrain point B (node 2) in the X and Z translations.

•

Constrain point C (node 3) in the Y and Z translations.

Solution Type Normal Modes/Eigenvalue - SVI method



Results Problem 1 (sdll14a) Frequency Results:

0

Bench Value (Hz)

Normal Mode

L 1 2 3 4

44.23 119 125 227

FEMAP Structural (Hz) 44.11 119 126 225

Difference 0.27% 0.00% 0.80% 0.88%

Problem 2 (sdll14b) Frequency Results:

0.6

Bench Value (Hz)

Normal Mode

L 1 2 3 4

33.4 94 100 180

FEMAP Structural (Hz) 33.3 94 99 184

Difference 0.30% 0.00% 1.00% 2.22%

Problem 3 (sdll14c) Frequency Results:

2

Bench Value (Hz)

Normal Mode

L 1 2 3 4

17.9 24.8 25.3 27

FEMAP Structural (Hz) 17.7 24.4 24.9 26.67

Difference 1.12% 1.61% 1.58% 0.01%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLL14/89.

Cantilever Beam with Eccentric Lumped Mass The complete model and results for this test case are in the following files: •

sdll15a.neu

•

sdll15b.neu

This test is a modal analysis of a straight cantilever beam and a mass element. It provides the input data and results for benchmark test SDLL15/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.1x10 Pa kg ρ = 7800 ------3m

Finite Element Modeling Problem 1 (sdll15a) •

10 bar elements

•

1 mass element at point B

•

11 nodes A

B

 Problem 2 (sdll15b) •

10 bar elements

•

1 rigid element from point B to point C

•

1 mass element at point C

•

12 nodes

A

C

B

Boundary Conditions Constraints: •

Fully constrain point A (node 1) in all translations and rotations.

Solution Type Normal Modes/Eigenvalue - SVI



Results Frequency Results: Bench Value (Hz)

Normal Mode

yc 0

1,2 3,4 5,6 7 8 9,10 1 2 3 4 5 6 7 8

1

FEMAP Structural (Hz)

1.65 16.07 50.02 76.47 80.47 103.20 1.636 1.642 13.46 13.59 28.90 31.96 61.61 63.93

1.65 15.91 48.75 76.48 80.84 98.53 1.635 1.640 13.37 13.52 28.68 31.54 59.97 61.82

Difference 0.00% 1.00% 2.54% 0.01% 0.46% 4.53% 0.06% 0.12% 0.67% 0.52% 0.76% 1.31% 2.66% 3.30%

Mode Shapes Results: Normal Mode

yc 1

1 2 3 4

Modal Displacement wc/wb uc/vb uc/vb wc/wb

•

wc=T3 translation at point C

•

wb= T3 translation at point B

•

uc=T1 translation at point C

•

vb= T2 translation at point B

Bench Value 1.030 0.148 2.882 -0.922

FEMAP Structural 1.030 0.148 2.845 -0.956

Difference 0.00% 0.00% 1.28% 3.69%

 Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLL15/89.

Thin Square Plate (Clamped or Free) The complete model and results for this test case are in the following files: •

sdls01a.neu

•

sdls01b.neu

This test is a normal modes/eigenvalue analysis (three–dimensional problem) of a thin plate. It provides the input data and results for benchmark test SDLS01/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.1x10 Pa ν = 0.3 kg ρ = 7800 ------3m

Finite Element Modeling •

100 linear quadrilateral plate elements

•

121 nodes

The mesh is shown in the following figure: A

D

B

C

 Boundary Conditions •

Problem 1 (sdls01a): Constrain the nodes along side BC in all translations and rotations.

•

Problem 2 (sdls01b) : Free plate; Create a constraint set (Kinematic DOF set) to constrain the three nodes shown below (nodes 1, 11, and 111) in all translations and rotations.

Solution Type Normal Modes/Eigenvalue - SVI method

Results Problem 1 (sdls01a) Frequency Results: Normal Mode 1 2 3

Bench Value (Hz) 8.7266 21.3042 53.5542

FEMAP Structural (Hz) 8.6719 21.1474 53.9586

Difference 0.63% 0.74% 0.76%

 4 5 6

68.2984 77.7448 136.0471

68.4467 77.7814 135.783

0.21% 0.05% 0.19%

Problem 2 (sdls01b) Frequency Results: Normal Mode 7 8 9 10,11

Bench Value (Hz) 33.7119 49.4558 61.0513 87.5160

FEMAP Structural (Hz) 32.9104 47.4165 59.1873 83.0785

Difference 2.38% 4.12% 3.05% 5.07%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLS01/89.

Simply-Supported Rectangular Plate The complete model and results for this test case are in file sdls03.neu. This test is a normal modes/eigenvalue analysis (three–dimensional problem) of a thin plate. It provides the input data and results for benchmark test SDLS03/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.1x10 Pa ν = 0.3 kg ρ = 7800 ------3m

Finite Element Modeling •

150 linear quadrilateral plate elements

•

176 nodes

The mesh is shown in the following figure:

 Boundary Conditions •

Constrain the Z translation of the nodes on all sides of the plate.

•

Create a constraint set to define the Master (ASET) DOFs on nodes 47, 55, 119. Constrain these nodes in all directions except for the Z translation.

•

no load case

The boundary conditions are shown in the following figure:

Solution Type Normal Modes/Eigenvalue - SVI method

Results Frequency Results: Bench Value (Hz)

Normal Mode 4 5 6 7 8 9

35.63 68.51 109.62 123.32 142.51 197.32

FEMAP Structural (Hz) 35.21 67.21 108.96 121.13 138.30 187.94

Difference 1.18% 1.90% 0.60% 1.78% 2.95% 4.75%

 Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLS03/89.

Thin Ring Plate Clamped on a Hub The complete model and results for this test case are in file sdls04.neu. This test is a normal modes/eigenvalue analysis (three–dimensional problem) of an annular thin plate. It provides the input data and results for benchmark test SDLS04/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.1x10 Pa ν = 0.3 kg ρ = 7800 ------3m

Finite Element Modeling Mapped meshing •

400 linear quadrilateral plate elements

•

440 nodes

The mesh is shown in the following figure:

 Boundary Conditions Constraints •

Fully constrain all the nodes on the inner ring as shown below.

Loads •

no load case

The boundary conditions are shown in the following figure:

Solution Type Normal Modes/Eigenvalue – SVI

Results Frequency Results: Normal Mode 1 2, 3 4, 5 6, 7 8, 9 10, 11 12, 13 14, 15 16, 17 18

Bench Value (Hz) 79.26 81.09 89.63 112.79 not available not available not available not available not available 518.85

FEMAP Structural (Hz) 79.41 81.05 89.64 113.45 158.38 226.02 317.04 433.04 527.51 532.19

Difference 0.19% 0.05% 0.01% 0.58%

2.57%

 19, 20 528.61 561.91 6.30% 21, 22 559.09 576.90 3.18% 23 609.70 612.63 0.48% Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLS04/89.

Vane of a Compressor - Clampedfree Thin Shell The complete model and results for this test case are in the following files: •

sdls05a.neu (linear quadrilateral, coarse mesh)

•

slds05b.neu (linear quadrilateral, fine mesh)

This test is a normal modes/eigenvalue analysis (three–dimensional problem) of a cylindrical thin shell. It provides the input data and results for benchmark test SDLS05/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.0685x10 Pa ν = 0.3 kg ρ = 7857.2 ------3m

Finite Element Modeling - Coarse Mesh Mapped meshing •

100 linear quadrilateral plate elements

•

121 nodes

 The coarse mesh is shown in the following figure:

Finite Element Modeling - Fine Mesh Mapped Meshing •

225 linear quadrilateral plate elements

•

256 nodes

The fine mesh is shown in the following figure:

 Boundary Conditions Fully constrain the nodes on one side as shown in the following figure:

Solution Type Normal Modes/Eigenvalue - SVI method

Results Frequency Results: Bench Value (Hz)

Normal Mode 1 2 3 4 5 6

85.6 134.5 259.0 351.0 395.0 531.0

FEMAP Structural coarse mesh (Hz) 85.6 138.2 249.8 345.9 386.5 549.8

FEMAP Structural fine mesh (Hz) 85.7 138.3 248.0 343.7 386.0 537.7

Reference Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLS05/89.

Bending of a Symmetric Truss The complete model and results for this test case are in file sdlx01.neu. This test is a normal modes/eigenvalue analysis (plane problem) of a straight cantilever beam structure. It provides the input data and results for benchmark test SDLX01/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11

E = 2.1x10 Pa ν = 0.3 kg ρ = 7800 ------3m

Finite Element Modeling •

24 bar elements

•

24 nodes

The mesh is shown in the following figure:

 Boundary Conditions Constraints •

Fully constrain nodes 1 and 4 in all translations and rotations.

•

Constrain nodes 2-3 and 5-24 in the Z translation and X and Y rotations.

The boundary conditions are shown in the following figure:

Solution Type Normal Modes/Eigenvalue – SVI

Results Frequency Results: Bench Value (Hz)

Normal Mode 1 2 3 4 5 6 7 8 9

8.8 29.4 43.8 56.3 96.2 102.6 147.1 174.8 178.8

FEMAP Structural (Hz) 8.8 29.4 43.8 56.3 96.2 102.7 147.4 175.3 179.3

Difference 0.00% 0.00% 0.00% 0.00% 0.00% 0.10% 0.20% 0.29% 0.28%

 10 11 12 13

206.0 206.9 0.44% 266.4 268.1 0.64% 320.0 322.4 0.75% 335.0 338.7 1.10% Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLX01/89.

Hovgaard’s Problem - Pipes with Flexible Elbows The complete model and results for this test case are in file sdlx02.neu. This test is a normal modes/eigenvalue analysis (three–dimensional problem) of a straight, thin curved cantilever beam. It provides the input data and results for benchmark test SDLX02/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Material Properties 11 E = 1.658x· 10 Pa

ν = 0.3 kg ρ = 13404.106 ------3m

Units SI

Finite Element Modeling •

25 bar elements

•

26 nodes

The mesh is shown in the following figure:

 Boundary Conditions •

Fully constrain nodes 1 and 6 in all translations and rotations.

The boundary conditions are shown in the following figure:

Solution Type Normal Modes/Eigenvalue - SVI

Results Frequency Results: Bench Value (Hz)

Normal Mode 1 2 3 4 5 6 7 8 9

10.18 19.54 25.47 48.09 52.86 75.94 80.11 122.34 123.15

FEMAP Structural (Hz) 10.40 19.87 25.36 47.71 51.80 82.84 85.20 125.53 127.64

Difference 2.16% 1.69% 0.43% 0.79% 2.01% 9.09% 6.35% 2.61% 3.65%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLX02/89.

Rectangular Plates The complete model and results for this test case are in file sdlx03.neu. This test is a normal modes/eigenvalue analysis (three–dimensional problem) of a thin plate with rigid body modes. It provides the input data and results for benchmark test SDLX03/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11 E = 2.1x· 10 Pa

ν = 0.3 kg ρ = 7800 ------3m

Finite Element Modeling •

300 linear quadrilateral plate elements

•

320 nodes

The mesh is shown in the following figure:

 Boundary Conditions Constraints •

Constraint Set 1 (Kinematic DOF Set): Fully constrain nodes 2, 69, and 84 in all translations and rotations.

The boundary conditions are shown in the following figure:

Solution Type Normal Modes/Eigenvalue - SVI

Results Frequency Results: Bench Value (Hz)

Normal Mode 7 8 9 10 11 12

584 826 855 911 1113 1136

FEMAP Structural (Hz) 586 824 854 904 1072 1140

Difference 0.34% 0.24% 0.11% 0.76% 3.68% 0.35%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. SDLX03/89.

Stationary Thermal Tests - Steady State Heat Transfer Analysis The stationary thermal test cases for steady-state heat transfer analysis from the Societe Francaise des Mecaniciens include: •

"Hollow Cylinder - Fixed Temperatures"

•

"Hollow Cylinder - Convection"

•

"Cylindrical Rod - Flux Density"

•

"Hollow Cylinder with Two Materials - Convection"

•

"Wall - Fixed Temperatures"

•

"Wall - Convection"

•

"Hollow Sphere - Fixed Temperatures, Convection"

•

"L-Plate"

•

"Hollow Sphere with Two Materials -Convection"

Hollow Cylinder - Fixed Temperatures The complete model and results for this test case are in file htpla01.neu. This test is a steady–state heat transfer analysis of a 2D axisymmetric cylinder with fixed temperatures. It provides the input data and results for benchmark test TPLA01/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties W λ = 1 ----- °C m

Finite Element Modeling Two tests: •

Test 1 - 5 linear quadrilateral axisymmetric solid elements

•

Test 2 - 5 parabolic quadrilateral axisymmetric solid elements

The meshes are shown in the following figure:

Boundary Conditions •

One temperature set:

 Internal temperature Ti = 100°C

External temperature Te = 20°C

Solution Type Steady–State Heat Transfer



Results Temperature Results (0 degrees Celsius): Radius(m) 0.30 0.31 0.32 0.33 0.34 0.35

Bench Value 100.00 82.98 66.51 50.54 35.04 20.00

FEMAP Structural 5 linear quads. 100.00 82.98 66.51 50.54 35.04 20.00

FEMAP Structural 5 parabolic quads. 100.00 82.98 66.51 50.54 35.04 20.00

Total Heat Flux Results (W/m**2): Radius (m) 0.30 0.31 0.32 0.33 0.34 0.35

Bench Value 1729.91 1674.11 1621.79 1572.64 1526.39 1482.78

FEMAP FEMAP Structural Structural 5 linear quads. 5 parabolic quads. 1701.69 1674.68 1622.32 1573.13 1526.84 1504.39

1701.70 1674.69 1622.32 1573.13 1526.83 1504.38

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. TPLA01/89.

Hollow Cylinder - Convection The complete model and results for this test case are in file htpla03.neu. This test is a steady–state heat transfer analysis of a 2D axisymmetric cylinder with convection. It provides the input data and results for benchmark test TPLA03/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties W λ = 40 ----- °C m

Finite Element Modeling Three tests: •

Test 1 - 10 linear axisymmetric quadrilateral solid elements

•

Test 2 - 2 linear axisymmetric quadrilateral solid elements

•

Test 3 - 2 parabolic axisymmetric quadrilateral solid elements

The meshes are shown in the following figure:

 Boundary Conditions Elemental Convection •

Convection on internal surface (nodes 3, 14, 16): W hi = 150.0 ------2- °C m Ti = 500°C

•

Convection on external surface (nodes 12, 15, 17): W he = 142.0 ------2- °C m Ti = 20°C

Solution Type Steady–State Heat Transfer

Results Temperature and Element Total Heat Flux Ti (°C)

Bench Value 272.27

FEMAP Structural 10 linear quads. 272.35

FEMAP Structural 2 linear quads. 272.17

FEMAP Structural 2 parabolic quads. 272.35

 Te (°C)

205.05 34160.00

204.51 33637.10

204.66 31746.69

204.51 31792.7

26276.90

26508.40

27824.15

27853.8

W ϕi  ------2-   m

W ϕe  ------2-   m ϕ --- = ϕ2πR L

So: ϕ W --- = 34173.82 ⋅ 2 ⋅ π ⋅ 0.300 = 64416.13 ----L m

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. TPLA03/89.

Cylindrical Rod - Flux Density The complete model and results for this test case are in file htpla05.neu. This test is a steady–state heat transfer analysis of a 2D axisymmetric rod with fixed temperatures and flux density. It provides the input data and results for benchmark test TPLA05/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties W λ = 33.33 ----- °C m

Finite Element Modeling •

20 linear quadrilateral axisymmetric solid elements

•

42 nodes

The mesh is shown in the following figure:

 Boundary Conditions Nodal Temperatures •

z = 0 (nodes 1 and 3): Set temperature to 0°C

•

z = 1 (nodes 2 and 4): Set temperature to 500°C

Elemental Heat Flux •

Cylindrical surface (elements 1-20): W Set flux ϕ to – 200 -------m2

The boundary conditions are shown in the following figure:

Solution Type Steady–State Heat Transfer



Results Temperature Results (degrees C): Node # Node 3 Node 41 Node 39 Node 37 Node 35 Node 33 Node 31 Node 29 Node 27 Node 25 Node 4

z (m) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Bench Value 0.00 -4.00 4.00 24.00 56.00 100.00 156.00 224.00 304.00 396.00 500.00

FEMAP Structural 0.00 -4.02 3.98 23.97 55.97 99.97 155.97 223.97 303.98 395.98 500.00

Difference 0.00% 0.50% 0.50% 0.13% 0.05% 0.03% 0.02% ~0.00% ~0.00% 0.01% 0.00%

Results are post–processed on the internal surface. Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. TPLA05/89.

Hollow Cylinder with Two Materials - Convection The complete model and results for this test case are in file htpla08.neu. This test is a steady–state heat transfer analysis of a 2D axisymmetric cylinder with two materials and convection. It provides the input data and results for benchmark test TPLA08/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties •

Material 1: W λ 1 = 40.0 ----- °C m

•

Material 2: W λ 2 = 20.0 ----- °C m

Finite Element Modeling •

7 linear quadrilateral axisymmetric solid elements

•

16 nodes

 The mesh is shown in the following figure.

Boundary Conditions Elemental Convection •

Convection on internal surface: W hi = 150.0 ------2- °C m Ti = 70°C

•

Convection on external surface: W hi = 200.0 ------2- °C m Ti = ( – 15° )C



Solution Type Steady–State Heat Transfer

Results Node # Node 9 Node 14 Node 16 Node 9

Temperature/ Element X Heat Flux Ti (°C) Tm (°C) Te (°C)

Bench Value

FEMAP Structural

Difference

25.42 17.69 12.11 6687.44

25.42 17.69 12.11 6577.88

0.00% 0.00% 0.00% 1.64%

5732.09

5733.33

0.02%

5422.25

5496.59

1.37%

W ϕi  ------2- m

Node 14 W ϕm  ------2- m

Node 16 W ϕe  ------2- m

ϕ --- = ϕ2πR L

 So: ϕ W --- = 5733.33 ⋅ 2 ⋅ π ⋅ 0.35 = 12608.25 ----L m

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. TPLA08/89.

Wall - Convection The complete model and results for this test case are in file htpl03.neu. This test is a steady–state heat transfer analysis of a 1D wall with fixed convection. It provides the input data and results for benchmark test TPLL03/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties W λ = 1.0 ----- °C m

Finite Element Modeling •

1 linear quadrilateral plate element

•

4 nodes

The plate element thickness is set to 1m. The mesh is shown in the following figure:

 Boundary Conditions Elemental Convection •

Convection on internal surface: W hA = 20.0 ------2- °C m TA = – 20.0°C

•

Convection on external surface: W hB = 10.0 ------2- °C m TB = 500°C

•

Convection coefficient is defined as energy / (length*time*temperature) in the current system of units.

The boundary conditions are shown in the following figure:

A

Solution Type Steady–State Heat Transfer

B



Results Temperature Results (Degrees Celsius): Node # Node 1 (Temp) Node 4 (Temp) Node 1 (Flux)

Temperature Flux

Bench Value

FEMAP Structural

Difference

TA (°C)

21.71

21.71

0.00%

TB (°C)

416.58

416.57

∼0.00%

ϕ (W/m**2)

834.2

834.3

0.01%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. TPLL03/89.

Wall - Fixed Temperatures The complete model and results for this test case are in file htpl01.neu. This test is a steady–state heat transfer analysis of a 1D wall with fixed temperatures. It provides the input data and results for benchmark test TPLL01/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information The mesh is shown in the following figure:

Units SI

Material Properties W λ = 0.75 ----- °C m

Finite Element Modeling •

5 beam (line 2) elements

•

6 nodes

 Boundary Conditions Nodal Temperatures •

Internal temperature Ti = 100°C ( node 1 )

•

External temperature Te = 20°C ( node 6 )

The boundary conditions are shown in the following figure:

Solution Type Steady–State Heat Transfer

Results Temperature Results (Degrees Celsius): Length: x (m)

Node # Node 1 Node 2 Node 3 Node 4 Node 5 Node 6

0.00 0.01 0.02 0.03 0.04 0.05

Bench Value 100.0 84.0 68.0 52.0 36.0 20.0

FEMAP Structural 100.0 84.0 68.0 52.0 36.0 20.0

Difference 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

 The flux calculated with the software is exact: Ω ϕ = 1200 -----2µ

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. TPLL01/89.

L-Plate The complete model and results for this test case are in the following files: •

htpp01a.neu (linear quadrilateral)

•

htpp01b.neu (parabolic quadrilateral)

This test is a steady–state heat transfer analysis of a 2D L–plate with fixed temperatures. It provides the input data and results for benchmark test TPLP01/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties W λ = 1.0 ----- °C m

Finite Element Modeling Two tests: •

21 nodes, 12 linear quadrilateral plate elements

•

53 nodes, 12 parabolic quadrilateral plate elements

The mesh is shown in the following figure:

 Boundary Conditions Nodal Temperatures •

AF side: Set temperature to 10°C

•

DE side: Set temperature to 0°C

The boundary conditions are shown in the following figure: F

E

C

A

D

B

Solution Type Steady–State Heat Transfer

Results Temperature Results (Degrees Celsius):

Node 8 9 10

FEMAP Bench Values Structural linear quads. 7.869 5.495 2.816

7.861 5.502 2.845

% Difference 1.10 0.13 1.03

FEMAP Structural parabolic quads. 7.883 5.519 2.834

% Difference 0.18 0.43 0.64

 19 18 20 17 6 16 21 15 14 5

8.018 8.026 0.10 8.015 0.04 5.680 5.669 0.19 5.666 0.25 2.881 2.959 2.71 2.877 0.14 8.514 8.505 0.11 8.519 0.06 6.667 6.667 0.00 6.667 0.00 2.972 2.990 0.61 2.963 0.30 9.001 9.015 0.16 9.108 1.20 8.640 8.661 0.24 8.669 0.34 9.316 9.294 0.24 9.283 0.35 9.009 8.996 0.14 8.961 0.53 Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. TPLP01/89.

Hollow Sphere - Fixed Temperatures, Convection The complete model and results for this test case are in file htpv02.neu. This test is a steady–state heat transfer analysis of a 3D sphere with fixed temperatures and convection. It provides the input data and results for benchmark test TPLV02/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties W λ = 1.0 ----- °C m

Finite Element Modeling •

500 solid (brick and wedge) elements

•

666 nodes

The test is executed on 1/8 of a mapped meshed sphere. The mesh is shown in the following figure:

 Boundary Conditions Elemental Convection •

Convection on internal surface: W hi = 30 ------2- °C m Ti = 100°C(elements 401-500)

Nodal Temperature •

Set external surface temperature Te to 20°C(nodes 1-111)

The boundary conditions are shown in the following figure:

Solution Type Steady–State Heat Transfer



Results Temperature results (Degrees C): Radius r (m) 0.3 0.31 0.32 0.33 0.34 0.35

Bench Value

Node # 566 455 344 233 122 11

65.00 54.84 45.31 36.36 27.94 20.00

FEMAP Structural 64.87 54.74 45.24 36.32 27.92 20.00

Difference 0.20% 0.18% 0.15% 0.11% 0.07% 0.00%

Element X Heat Flux results (W/m**2): Radius r (m) 0.3 0.31 0.32 0.33 0.34 0.35

Node # 566 455 344 233 122 11

Bench Value 1050.00 983.35 922.85 867.47 817.47 771.43

FEMAP Structural 1019.34 987.57 926.90 871.65 821.21 797.11

Difference 2.92% 0.43% 0.43% 0.48% 0.45% 3.32%

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. TPLV02/89.

Hollow Sphere with Two Materials Convection The complete model and results for this test case are in the following files: •

htpv04a.neu (linear brick)

•

htpv04b.neu (parabolic tetrahedron)

•

htpv04c.neu (axisymmetric solid)

This test is a steady–state heat transfer analysis of a 3D sphere with two materials and convection. It provides the input data and results for benchmark test TPLV04/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties •

Material 1: W λ 1 = 40.0 ----- °C m

•

Material 2: W λ 2 = 20.0 ----- °C m

Finite Element Modeling Three tests:

 •

Test 1 - 888 nodes, 700 solid (brick and wedge) elements

•

Test 2 - 3818 nodes, 2192 solid parabolic tetrahedron elements

•

Test 3 - 23 nodes, 4 axisymmetric solid parabolic quadrilateral elements

The test is executed on 1/8 of a mapped meshed sphere.

 Boundary Conditions Elemental Convection •

Convection on internal surface: W hi = 150.0 ------2- °C m Ti = 70°C

•

Convection on external surface: W he = 200.0 ------2- °C m Te = ( – 9° )C

The boundary conditions are shown in the following figure:

Solution Type Steady–State Heat Transfer



Results Temperature Results (Degrees Celsius):

Bench Value

Temperature

Ti (C°) Tm (C°) Te (C°)

25.06 17.84 13.16

FEMAP Structural linear brick (htpv04a) N1 25.03 N556 17.84 N778 13.18

FEMAP Structural parabolic tetrahedron (htpv04b) N19 25.06 N9 17.84 N5 13.15

FEMAP Structural axisymmetric solid (htpv04c) N2 25.01 N6 17.75 N5 13.17

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. TPLV04/89.

Thermo-mechanical Test - Linear Statics Analysis The stationary thermal-mechanical test cases for linear statics analysis from the Societe Francaise des Mecaniciens include: •

"Thermal Gradient on a Thin Pipe"

Thermal Gradient on a Thin Pipe The complete model and results for this test case are in file hsla01.neu. This test is a thermo–mechanical linear statics analysis of a thin pipe with thermal gradient and plane strain. It provides the input data and results for benchmark test HSLA01/89 from “Guide de validation des progiciels de calcul de structures.”

Test Case Data and Information Units SI

Material Properties 11 E = 1x· 10 Pa

ν = 0.3 –5

10 Coefficient of expansion: α = 1x ----------C°

Finite Element Modeling •

500 axisymmetric (linear quadrilateral solid) elements

•

561 nodes

The mesh is shown in the following figure:

 Boundary Conditions Constraints •

Constrain nodes 1-11 in the X and Z translations.

Nodal Temperature •

Radial temperature ( 1 – ( r – Ri ) ) T = Ti ⋅ -------------------------------- with Ti=100°C ( Re – Ri )

The boundary conditions are shown in the following figure:

Solution Type Statics



Results Point

Bench Value

Stress

r = Ri

FEMAP Structural

Difference

0

-0.85E6

-74.07E6

-74.20E6

0.18%

-3.95E6

-3.89E6

1.52%

1.306E6

1.40E6

1.22%

0

-0.65E6

68.78E6

68.53E6

σ r ( Pa ) σ θ ( Pa )

r=(Re+Ri)/2 σ r ( Pa ) σ θ ( Pa )

r=Re σ r ( Pa ) σ θ ( Pa )

Post Processing Value

Definition = the axisymmetric C1 radial stress at node 265

σr

= the axisymmetric C4 Azimuth stress at node 265 σθ

=the axisymmetric C1 radial stress at node 270 σr

=the axisymmetric C1 Azimuth stress at node 270 σθ

= the axisymmetric C1 radial stress at node 275 σr

0.36%

 Value

Definition = the axisymmetric C2 Azimuth stress at node 275

σθ

Reference • Societe Francaise des Mecaniciens, Guide de validation des progiciels de calcul de structures, (Paris, Afnor Technique,1990.) Test No. HSLA01/89.



Index A Annular membrane 152 Annular plate 117, 140, 171, 182 Anti-symmetric modes 108 Articulated plane truss 203 Articulated rod truss 201 Articulated supports 192 Axial distributed load 6 Axial loading 291 Axisymmetric solid elements 165, 168, 255, 268, 331, 334, 337, 340 Axisymmetric vibration 165, 171 B Bar elements 76, 78, 83, 92, 95, 98, 101, 192, 194, 196, 199, 203, 206, 288, 291, 297, 300, 303, 307, 323, 326 Beam 4, 6, 9, 12, 15, 18, 95, 101, 174, 186, 192, 194, 206, 288, 294, 307 Beam elements 294, 347 Bending 27, 196, 199, 274, 323 Bending load 210 C Cantilever 92 Cantilever beam 4, 9, 12, 76, 101, 294, 307 Cantilever mass 78 Cantilevered plate 105 Cantilevered solid beam 186 Cantilevered square membrane 144 Cantilevered tapered membrane 148 Cantilevered thin square plate 124, 156, 161 Circular hole 212 Circular plate 215 Circular ring 98 Clamped beams 194

Clamped thick rhombic plate 136 Clamped thin rhombic plate 121 Clamped-free thin shell 320 Compressor 320 Convection 334, 340, 344, 353, 356 Curved beam elements 196 Curved pipe 196 Cylindrical rod 337 Cylindrical shell 39, 42, 221 D Deep simply-supported beam 95 Deep simply-supported solid beam 174 Displacement 15 Distorted mesh 124 Distributed loads 9, 18 E Elastic foundation 206 Elliptic membrane 34 F Fixed temperatures 331, 347, 353 Flux density 337 Free annular membrane 152 Free cylinder 165 G Gravity loading 232 H Heated beam 15 Hemisphere point loads 44 Hollow cylinder 331, 334, 340 Hollow sphere 353, 356 Hovgaard’s Problem 326 hsla01.neu 361 htpl01.neu 347 htpl03.neu 344 htpla01.neu 331 htpla03.neu 334 htpla05.neu 337 htpla08.neu 340 htpp01a.neu 350

 htpp01b.neu 350 htpv02.neu 353 htpv04a.neu 356 htpv04b.neu 356 htpv04c.neu 356 Hydrostatic pressure 229 I Infinite plate 212 In-plane vibrations 83, 98 Internal pressure 221, 261, 268 K Kirchhoff formulation 251 L le1001.neu 53 le1002.neu 53 le1003.neu 53 le101.neu 34 le102.neu 34 le103.neu 34 le1101a.neu 58 le1101b.neu 58 le1102a.neu 58 le1102b.neu 58 le1103a.neu 58 le1103b.neu 58 le1104a.neu 58 le1104b.neu 58 le1105a.neu 58 le1105b.neu 58 le1106a.neu 58 le1106b.neu 58 le201a.neu 39, 42 le201b.neu 39, 42 le202a.neu 39, 42 le202b.neu 39, 42 le301.neu 44 le302.neu 44 le303.neu 44 le304.neu 44

le501.neu 47 le502.neu 47 le601.neu 49 le602.neu 49 Linear beam 6, 18 Linear Statics 4, 6, 9, 12, 15, 18, 21, 24, 27, 30, 34, 39, 42, 44, 49, 53, 58, 192, 194, 196, 199, 201, 203, 210, 212, 215, 218, 221, 225, 229, 232, 236, 239, 242, 247, 251, 261, 268, 274, 361 L-Plate 350 Lumped mass 285, 307 M Mass elements 65, 68, 71, 78, 285, 307 Membrane 21 Membrane loads 21 Mindlin formulation 251 Moment load 12 mstv1001.neu 4 mstv1002.neu 6 mstv1003.neu 9 mstv1004.neu 12 mstv1007.neu 15 mstv1008.neu 18 mstv1009.neu 21 mstv1014.neu 24 mstv1015.neu 27 mstv1016.neu 30 mstvn002.neu 65 mstvn003.neu 68 mstvn004.neu 71 mstvn005.neu 73 mstvn006.neu 76 mstvn007.neu 78 N Natural frequency 78 ne014ll.neu 117 nf001ac.neu 83 nf002ac.neu 86

 nf003ac.neu 89 nf004a.neu 92 nf005ac.neu 95 nf006ac.neu 98 nf011alc.neu 105 nf011all.neu 105 nf011apc.neu 105 nf011apl.neu 105 nf011blc.neu 108 nf011bll.neu 108 nf011bpc.neu 108 nf011bpl.neu 108 nf0121c.neu 111 nf012ll.neu 111 nf012pc.neu 111 nf012pl.neu 111 nf013lc.neu 114 nf013ll.neu 114 nf013pc.neu 114 nf013pl.neu 114 nf014lc.neu 117 nf014pc.neu 117 nf014pl.neu 117 nf015lc.neu 121 nf015ll.neu 121 nf015pc.neu 121 nf015pl.neu 121 nf021alc.neu 129 nf021all.neu 129 nf021apc.neu 129 nf021apl.neu 129 nf021blc.neu 133 nf021bll.neu 133 nf021bpc.neu 133 nf021bpl.neu 133 nf0221c.neu 136 nf022ll.neu 136 nf022pc.neu 136 nf022pl.neu 136

nf023lc.neu 140 nf023ll.neu 140 nf023pc.neu 140 nf023pl.neu 140 nf031ll.neu 144 nf031llc.neu 144 nf031pc.neu 144 nf031pl.neu 144 nf032lc.neu 148 nf032ll.neu 148 nf032pc.neu 148 nf032pl.neu 148 nf033lc.neu 152 nf033ll.neu 152 nf033pc.neu 152 nf033pl.neu 152 nf041lc.neu 165 nf041ll.neu 165 nf041pc.neu 165 nf041pl.neu 165 nf042lc.neu 168 nf042ll.neu 168 nf042pc.neu 168 nf042pl.neu 168 nf043lc.neu 171 nf043ll.neu 171 nf043pc.neu 171 nf043pl.neu 171 nf051lc.neu 174 nf051ll.neu 174 nf051pc.neu 174 nf051pl.neu 174 nf052lc.neu 178 nf052ll.neu 178 nf052pc.neu 178 nf052pl.neu 178 nf053lc.neu 182 nf053ll.neu 182 nf053pc.neu 182

 nf053pl.neu 182 nf071a.neu 101 nf071b.neu 101 nf071c.neu 101 nf072ac.neu 186 nf072al.neu 186 nf072bc.neu 186 nf072bl.neu 186 nf073ac.neu 156 nf073al.neu 156 nf073bc.neu 156 nf073bl.neu 156 nf073cc.neu 156 nf073cl.neu 156 nf073dc.neu 156 nf073dl.neu 156 nf074c.neu 161 nf074l.neu 161 Nodal loads 4, 201 Normal Modes/Eigenvalue 65, 68, 71, 76, 78, 83, 92, 95, 98, 101, 105, 108, 114, 117, 121, 124, 129, 133, 136, 140, 144, 148, 152, 156, 161, 165, 168, 171, 174, 178, 182, 186, 285, 291, 294, 297, 300, 303, 307, 311, 314, 317, 320, 323, 326, 328 O Off-center point masses 92 Out-of-plane vibration 98 P Patch test 39, 42 Pinched cylindrical shell 236 Pin-ended cross 83 Pipes 326 Plane bending 199, 210 Plane strain elements 34 Plane truss 203 Plate elements 34, 39, 42, 44, 49, 105, 108, 114, 117, 121, 124, 129, 133, 136,

140, 148, 152, 156, 161, 210, 212, 215, 218, 221, 225, 229, 232, 236, 239, 242, 247, 251, 279, 311, 314, 317, 320, 328, 344, 350 plate elements 144 Pressure 53, 221, 229, 268 Prismatic rod 274 Pure bending 27, 274 Pure tension 24, 255 R Rectangular plates 328 Rhombic plate 121, 136 Rhomboid plate 247 Rigid elements 65, 194 Rod elements 201 S sdld02.neu 285 sdll014a.neu 303 sdll014b.neu 303 sdll014c.neu 303 sdll01a.neu 288 sdll01b.neu 288 sdll05a.neu 291 sdll05b.neu 291 sdll09a.neu 294 sdll11.neu 297 sdll12.neu 300 sdll15a.neu 307 sdll15b.neu 307 sdls01a.neu 311 sdls01b.neu 311 sdls03.neu 314 sdls04.neu 317 sdls05a.neu 320 sdls05b.neu 320 sdlx01.neu 323 sdlx02.neu 326 sdlx03.neu 328 Shear loading 251

 Short beam 192, 288 Simply-supported annular plate 117, 171 Simply-supported rectangular plate 242, 314 Simply-supported rhomboid plate 247 Simply-supported solid annular plate 182 Simply-supported solid square plate 178 Simply-supported thick annular plate 140 Simply-supported thick square plate 133 Simply-supported thin square plate 114 Single DOF 65 Skew plate normal pressure 49 Solid cylinder 58, 255 Solid elements 53, 58, 174, 178, 182, 186, 255, 268, 274, 279, 353, 356, 361 Solid sphere 58 Solid square plate 178 Solid taper 58 Spherical shell 239 Spring elements 65, 68, 71, 206, 285 Square tube 218 ssll02.neu 192 ssll05.neu 194 ssll07a.neu 196 ssll07b.neu 196 ssll08.neu 199 ssll11.neu 201 ssll14a.neu 203 ssll14b.neu 203 ssll16.neu 206 sslp01.neu 210 sslp02.neu 212 ssls03a.neu 215 ssls03b.neu 215 ssls05.neu 218 ssls06a.neu 221 ssls06b.neu 221

ssls07a.neu 225 ssls07b.neu 225 ssls08.neu 229 ssls09.neu 232 ssls20a.neu 236 ssls20b.neu 236 ssls21a.neu 239 ssls21b.neu 239 ssls21c.neu 239 ssls24a.neu 242 ssls24b.neu 242 ssls24c.neu 242 ssls25a.neu 247 ssls25b.neu 247 ssls27a.neu 251 ssls27b.neu 251 ssls27c.neu 251 sslv01a.neu 255 sslv01b.neu 255 sslv01c.neu 255 sslv01d.neu 255 sslv03a.neu 261 sslv03b.neu 261 sslv03c.neu 261 sslv03d.neu 261 sslv04a.neu 268 sslv04b.neu 268 sslv04c.neu 268 sslv04d.neu 268 sslv08a.neu 274 sslv08b.neu 274 sslv08c.neu 274 sslv08d.neu 274 sslv09a10.neu 279 sslv09a100.neu 279 sslv09a20.neu 279 sslv09a50.neu 279 sslv09a75.neu 279 sslv09b10.neu 279

 sslv09b100.neu 279 sslv09b20.neu 279 sslv09b50.neu 279 sslv09b75.neu 279 Steady-State Heat Transfer 331, 334, 337, 340, 344, 347, 350, 353, 356 Strain energy 30 Stress 15 Symmetric modes 105 Symmetric truss 323 T Tapered beam elements 294 Tapered membrane 148 Temperatures 58, 331, 347, 353 Tension 24 Thermal gradient 361 Thermal strain 15 Thick annular plate 140 Thick hollow sphere 168 Thick plate 279 Thick plate pressure 53 Thick square plate 129, 133 Thick-walled infinite cylinder 268 Thick-walled spherical container 261 Thin arc 199 Thin circular ring 297, 300 Thin pipe 361 Thin pipe elbow 303 Thin ring plate 317 Thin shell 320 Thin shell beam wall 27 Thin square cantilevered plate 105, 108 Thin square plate 124, 156, 161, 311 Thin wall cylinder 24, 225, 229, 232 Three DOF 71 Torque loading 218 Torsional system 71 Transverse bending 196 Truss 30

Two DOF 68 U Undamped free vibration 65, 68 Undamped free vibrations 76 Uniform axial load 225 Uniform radial vibration 168 Uniformly distributed load 215, 242, 247 V Vibrations 65, 68, 76, 83, 98, 165, 168, 171, 303 W Wall 344, 347