Verification Manual Verification Tests for the Strand7 Finite Element Analysis System

Strand7 Software

www.strand7.com

Verification Manual Verification tests for the Strand7 finite element analysis system

© Strand7 Pty Limited. All rights reserved

Release 2.4.6 November 2013

Edition 1

ii

© Copyright Strand7 Pty Limited. All rights reserved worldwide. This manual is protected by law.

No part of this manual may be copied or distributed, transmitted, stored in a retrieval system, or

translated into any human or computer language, in any form or by any means, electronic, mechanical,

magnetic, manual or otherwise, or disclosed to third parties.

Strand7 Pty Limited reserves the right to revise this publication without obligation of Strand7 Pty

Limited to notify any person or organisation of such revision.

This document is provided electronically in the PDF format. If you are a legitimate Strand7 user, you

may print one (1) copy of this manual for each copy of Strand7 you have purchased. No other

printing rights are extended to any third party.

Strand7 is a registered trademark of Strand7 Pty Limited.

ISBN 0-957-73451-4

Strand7 Pty Limited:

Suite 1, Level 5, 65 York Street

Sydney NSW 2000 Australia

info@strand7.com

www.strand7.com

iii

Preface

This manual documents a set of test problems used for the verification of the solvers in the Strand7

finite element analysis system. The tests come from numerous sources as detailed in the reference

section of each chapter. Each test gives the Strand7 result together with the target value published in

the reference. Where a percentage difference between the two is given, this percentage is calculated

using the following formula:

ValueTarget

ValueTarget Result Strand7100%

The data given in the Problem sketch of each test is intended to illustrate the problem in general terms

rather than to give all the data necessary to actually perform the test.

The results given in this edition of the manual were obtained using Strand7 Release 2.4.6 (released in

November 2013). Copies of each Strand7 model file can be found in the “Verification” folder of the

Strand7 installation.

The tests are divided into 11 chapters, each covering a specific solver category:

Chapter 1: Linear Static

Chapter 2: Linear Buckling

Chapter 3: Nonlinear Static

Chapter 4: Natural Frequency

Chapter 5: Harmonic Response

Chapter 6: Spectral Response

Chapter 7: Linear Transient Dynamic

Chapter 8: Nonlinear Transient Dynamic

Chapter 9: Quasi-Static Analysis

Chapter 10: Steady State Heat Transfer

Chapter 11: Transient Heat Transfer

Solver Scheme and Computer Setup

Results in this document were calculated using the Sparse Solver Scheme on a computer with the

following processor:

Intel(R) Core(TM) i5 CPU 750 @ 2.67GHz

iv

v

Contents

PREFACE ..................................................................................................................................... III Solver Scheme and Computer Setup ..................................................................................................................... iii

CHAPTER 1 ................................................................................................................................. 1

Linear Static ................................................................................................................................... 1

VLS1: Elliptic Membrane ......................................................................................................................................... 2 VLS2: Cylindrical Shell Patch Test ........................................................................................................................ 4 VLS3: Hemisphere under Point Loads ................................................................................................................. 5 VLS4: Z-Section Cantilever ..................................................................................................................................... 6 VLS5: Skew Plate under Normal Pressure ........................................................................................................... 7 VLS6: Thick Plate under Pressure .......................................................................................................................... 8 VLS7: Solid Cylinder / Taper / Sphere under Thermal Loading ................................................................... 9 VLS8: Circular Membrane - Edge Pressure ....................................................................................................... 10 VLS9: Circular Membrane - Point Load ............................................................................................................ 12 VLS10: Circular Membrane Parabolic Temperature ....................................................................................... 14 VLS11: Plate Patch Test ......................................................................................................................................... 16 VLS12: Solid Patch Test ......................................................................................................................................... 18 VLS13: Straight Cantilever Beam ........................................................................................................................ 19 VLS14: Twisted Beam ............................................................................................................................................ 21 VLS15: Curved Beam.............................................................................................................................................. 22 VLS16: Rectangular Plate Under Normal Pressure ........................................................................................ 23 VLS17: Scordelis-Lo Roof ..................................................................................................................................... 25 VLS18: Thick-Walled Cylinder ............................................................................................................................. 26 VLS19: Continuous Beam Under Linearly Distributed Load ...................................................................... 27 VLS20: Rigid Beam Supported by Wires .......................................................................................................... 28 VLS21: Built-in Beam Thermal Stress Problem ............................................................................................... 29 VLS22: Tapered Cantilever Beam ........................................................................................................................ 30 VLS23: Simply Supported Composite Beam .................................................................................................... 31 VLS24: Circular Clamped Plate under Normal Pressure ............................................................................... 33 VLS25: Frame With Pin Connections ................................................................................................................ 35 VLS26: Stretching of an Orthotropic Solid ....................................................................................................... 36 VLS27: Rectangular Plate on Elastic Foundation ............................................................................................ 37 VLS28: Beam on Elastic Foundation .................................................................................................................. 38 VLS29: Pipe Under Combined Bending and Torsion .................................................................................... 40 VLS30: Out-of-Plane Bending of a Curved Bar ............................................................................................... 41 VLS31: Laminated strip under three-point bending ....................................................................................... 42 VLS32: Wrapped thick cylinder under pressure and thermal loading ........................................................ 44 VLS33: Three-Layer sandwich shell under normal pressure loading .......................................................... 46

vi

VLS34: Tapered Beam ............................................................................................................................................ 48 VLS35: Lateral Acceleration of a Tank of Fluid ............................................................................................... 53 VLS36: Rotation of a Tank of Fluid .................................................................................................................... 54 References ................................................................................................................................................................... 56

CHAPTER 2 ............................................................................................................................... 57

Linear Buckling .......................................................................................................................... 57

VLB1: Bar-Spring System ...................................................................................................................................... 58 VLB2: 3-Member Frame ........................................................................................................................................ 59 VLB3: Cantilever Beam........................................................................................................................................... 60 VLB4: Plate Column ................................................................................................................................................ 61 VLB5: Plate Column – Distorted Mesh .............................................................................................................. 63 VLB6: Simply Supported Rectangular Plate ..................................................................................................... 64 VLB7: Square Cross-Ply Laminate Plate ............................................................................................................ 65 VLB8: Cylindrical Shell ........................................................................................................................................... 67 References ................................................................................................................................................................... 68

CHAPTER 3 ............................................................................................................................... 69

Nonlinear Static .......................................................................................................................... 69

VNS1: Snap-Back of a Bar-Spring System ......................................................................................................... 70 VNS2: Straight Cantilever With End Moment ................................................................................................. 72 VNS3: Straight Cantilever With Axial End Point Load ................................................................................. 74 VNS4: Straight Cantilever With Lateral Point Load ....................................................................................... 76 VNS5: Limit Load GNL ........................................................................................................................................ 78 VNS6: Plane Strain Plasticity ................................................................................................................................. 81 VNS7: Plane Stress Plasticity ................................................................................................................................. 83 VNS8: Solid Plasticity ............................................................................................................................................. 85 VNS9: Pressurized Cylinder Plasticity ................................................................................................................. 87 VNS10: Two-Bar Assembly Plasticity ................................................................................................................. 91 VNS11: Rigid Punch Plasticity ............................................................................................................................. 98 VNS12: Axisymmetric Thick Cylinder .............................................................................................................. 101 VNS13: Nonlinear Equation Solution Test - Overlay Model ..................................................................... 103 VNS14: Square Plate Under Uniform Pressure .............................................................................................. 105 VNS15: Large Deflection Analysis of a Curved Cantilever.......................................................................... 107 VNS16: Toggle Mechanism ................................................................................................................................. 109 VNS17: Beam With Gap Lift-Off ...................................................................................................................... 110 VNS18: Large Deflection of a Uniformly Loaded Plate............................................................................... 111 VNS19: Large Deflection Eccentric Compression of a Slender Column ................................................. 112 VNS20: Large Deflection of Rectangular Plate with Line Load ................................................................ 113 VNS21: Hinged Cylindrical Shell ....................................................................................................................... 115 VNS22: Propped Cantilever With Gap Beam ................................................................................................. 117 VNS23: Belt Over a Pulley ................................................................................................................................... 119

vii

VNS24: Elastoplastic Analysis of a Cantilever Bar ......................................................................................... 121 VNS25: Takeup Mechanism under Alternating Load ................................................................................... 122 VNS26: Cylindrical Hole in an Infinite Mohr-Coulomb Medium ............................................................ 124 VNS27: Strip Footing on a Mohr-Coulomb Material .................................................................................. 128 VNS28: Plastic Flow in a Punch ......................................................................................................................... 130 VNS29: Large Displacement and Large Strain Analysis of a Rubber Sheet ........................................... 132 VNS30: Stretching of a Square Membrane...................................................................................................... 133 VNS31: Circular Shell under Normal Pressure ............................................................................................... 136 VNS32: Equiaxial Tension of a Square Membrane ....................................................................................... 138 VNS33: Uniaxial Extension of a Rectangular Block ..................................................................................... 140 VNS34: Rubber Cylinder under Internal Pressure ........................................................................................ 142 VNS35: Rubber Cylinder Pressed Between Two Plates ............................................................................... 143 VNS36: Footing on Clay ...................................................................................................................................... 147 VNS37: Footing in Sand ...................................................................................................................................... 150 VNS38: Footing on Sand ..................................................................................................................................... 152 VNS39: Cam-Clay Soil under Triaxial Test ..................................................................................................... 154 VNS40: Partially Saturated Mohr-Coulomb Soil ........................................................................................... 156 VNS41: Strip Footing on a Mohr-Coulomb Soil .......................................................................................... 158 VNS42: Plastic Hinge - Square Section Beam ................................................................................................ 161 VNS43 Lee’s Frame ............................................................................................................................................... 163 References ................................................................................................................................................................. 166

CHAPTER 4 ............................................................................................................................. 169

Natural Frequency ................................................................................................................... 169

VNF1: Pin-Ended Cross - In-Plane Vibration ................................................................................................ 170 VNF2: Pin-Ended Double Cross - In-Plane Vibration ................................................................................. 171 VNF3: Free Square Frame - In-Plane Vibration ............................................................................................ 172 VNF4: Cantilever With Off Centre Point Masses .......................................................................................... 173 VNF5: Deep Simply Supported Beam .............................................................................................................. 174 VNF6: Free Circular Ring ................................................................................................................................... 175 VNF7: Thin Square Cantilevered Plate - Symmetric Modes ....................................................................... 176 VNF8: Thin Square Cantilevered Plate - Anti-Symmetric Modes ............................................................. 177 VNF9: Free Thin Square Plate ............................................................................................................................ 178 VNF10: Simply Supported Thin Square Plate ................................................................................................ 179 VNF11: Simply Supported Thin Annular Plate ............................................................................................. 181 VNF12: Clamped Thin Rhombic Plate ............................................................................................................ 182 VNF13: Cantilevered Thin Square Plate .......................................................................................................... 183 VNF14: Simply Supported Thick Square Plate – Part A ............................................................................. 186 VNF15: Simply Supported Thick Square Plate – Part B .............................................................................. 188 VNF16: Clamped Thick Rhombic Plate .......................................................................................................... 189 VNF17: Simply Supported Thick Annular Plate ............................................................................................ 190 VNF18: Cantilevered Square Membrane ......................................................................................................... 191 VNF19: Cantilevered Tapered Membrane ....................................................................................................... 192

viii

VNF20: Free Annular Membrane ...................................................................................................................... 193 VNF21: Free Cylinder Axisymmetric Vibration ............................................................................................. 194 VNF22: Thick Hollow Sphere - Uniform Radial Vibration ....................................................................... 195 VNF23: Deep Simply Supported 'Solid' Beam ............................................................................................... 196 VNF24: Simply Supported 'Solid' Square Plate .............................................................................................. 198 VNF25: Simply Supported Solid Annular Plate - Axisymmetric Vibration ............................................ 200 VNF26: Badly Conditioned Cantilever Beam ................................................................................................. 202 VNF27: Lateral Vibration of a Stretched Circular Membrane.................................................................... 203 VNF28: Lateral Vibration of a Stretched String ............................................................................................ 205 VNF29: Torsional Vibration of a Shaft With Three Disks .......................................................................... 206 VNF30: Cantilever with Balanced Off-Centre Point Masses ....................................................................... 208 VNF31: Natural Frequency of a Motor Generator ........................................................................................ 209 VNF32: Torsional Frequencies of a Drill Pipe ................................................................................................ 210 VNF33: Cantilever Beam on an Elastic Support ............................................................................................ 211 VNF34: Slosh Frequency of Water in a Rigid Container............................................................................. 213 VNF35: Laminated Spherical Shell .................................................................................................................... 214 References ................................................................................................................................................................. 215

CHAPTER 5 ............................................................................................................................. 217

Harmonic Response ................................................................................................................ 217

VHR1: Deep Simply Supported Beam under Distributed Load................................................................ 218 VHR2: Simply Supported Thin Square Plate ................................................................................................. 219 VHR3: Simply Supported Thick Square Plate ................................................................................................ 221 VHR4: Spring Mass System ................................................................................................................................ 223 VHR5: Harmonic Response of a Simply Supported Beam ......................................................................... 225 References ................................................................................................................................................................. 227

CHAPTER 6 ............................................................................................................................. 229

Spectral Response .................................................................................................................... 229

VSR1: Seismic Response of a Simply Supported Beam ............................................................................... 230 VSR2: Earthquake Response of a Three Storey Building ............................................................................ 231 VSR3: Earthquake Response of a Simple Frame ............................................................................................ 233 VSR4: Antenna Subjected to Wind Load ......................................................................................................... 234 VSR5: Column Under Base Excitation ............................................................................................................. 236 VSR6: Rigid Slab Subject to a Base Acceleration........................................................................................... 237 References ................................................................................................................................................................. 238

CHAPTER 7 ............................................................................................................................. 239

Linear Transient Dynamic .................................................................................................... 239

VLT1: Deep Simply Supported Beam under Distributed Load ................................................................. 240 VLT2: Simply Supported Thin Square Plate ................................................................................................... 242 VLT3: Simply Supported Thick Square Plate ................................................................................................. 243

ix

VLT4: Transient Response of Spring to a Step Excitation .......................................................................... 244 VLT5: Response of a Cantilever Beam to an Impulse .................................................................................. 246 VLT6: Displacement Propagation along a Bar with Free Ends .................................................................. 247 References ................................................................................................................................................................. 248

CHAPTER 8 ............................................................................................................................. 249

Nonlinear Transient Dynamic ............................................................................................. 249

VNT1: Shallow Spherical Cap with a Concentrated Apex Load ................................................................ 250 VNT2: Weight Bouncing on an Elastic Platform ........................................................................................... 252 VNT3: Simply Supported Beam with Restrained Motion ........................................................................... 254 VNT4: Large Lateral Deflection of Unequal Stiffness Springs ................................................................... 256 VNT5: Large Rotation of a Swinging Pendulum .......................................................................................... 258 VNT6: Large Rotation of a Beam Pinned at One End ................................................................................ 259 VNT7: Uniaxial Load Secondary Creep ........................................................................................................... 261 VNT8: Uniaxial Load Primary Creep ............................................................................................................... 263 VNT9: Bi-axial Load Primary Creep – Stepped Load .................................................................................. 265 VNT10: Stress Relaxation Primary and Secondary Creep ........................................................................... 267 VNT11: Floating Block ........................................................................................................................................ 269 VNT12: Box Slides to Rest .................................................................................................................................. 271 References ................................................................................................................................................................. 273

CHAPTER 9 ............................................................................................................................. 275

Quasi-Static Analysis .............................................................................................................. 275

VQS1: Uniaxial Load Secondary Creep ........................................................................................................... 276 VQS2: Uniaxial Load Primary Creep ................................................................................................................ 278 VQS3: Bi-axial Load Primary Creep – Stepped Load ................................................................................... 280 VQS4: Stress Relaxation Primary and Secondary Creep .............................................................................. 282 VQS5: Creep and Shrinkage of Simply Supported Composite Beam under Distributed Load ......... 284 References ................................................................................................................................................................. 287

CHAPTER 10 .......................................................................................................................... 289

Steady State Heat Transfer ................................................................................................... 289

VSH1: 1D Heat Transfer with Radiation ........................................................................................................ 290 VSH2: 2D Heat Transfer with Convection ..................................................................................................... 291 VSH3: 2D Steady State Heat Conduction and Convection ........................................................................ 292 VSH4: Steady State Heat Transfer in a Solid Steel Billet ............................................................................. 294 VSH5: Steady State Heat Transfer through Building Corner ..................................................................... 295 References ................................................................................................................................................................. 298

CHAPTER 11 .......................................................................................................................... 299

Transient Heat Transfer ........................................................................................................ 299

VTH1: 1D Transient Heat Transfer .................................................................................................................. 300 VTH2: 2D Transient Heat Conduction and Convection ............................................................................. 302

x

VTH3: Transient Heat Conduction with Heat Generation ......................................................................... 303 VTH4: Axisymmetric Transient Heat Conduction and Convection ......................................................... 304 References ................................................................................................................................................................. 305

INDEX......................................................................................................................................... 307

307

Index

A

Actuator ...................................................... 109

Ambient temperature ... 288, 289, 300, 301, 302

Analytical solution

beam on elastic foundation ....................... 38

buckling of bar spring system ................... 58

built-in beam thermal stress problem ........ 29

circular plate under normal pressure ......... 33

continuous beam under LDL ................... 27

equiaxial tension of a rubber membrane . 138

footing on clay ....................................... 147

frame with pin connections ...................... 35

harmonic response.................................. 223

lateral vibration of stretched string ......... 205

rigid beam supported by wires.................. 28

stretching of an orthotropic solid.............. 36

vibration of a stretched membrane.......... 203

Arc-length method .............. 136, 158, 161, 163

B

Beam attribute

distributed load ................................ 27, 240

offset ........................................................ 40

pipe radius ............................................... 41

rotational end-release................ 35, 170, 171

support ............................................ 38, 211

taper ......................................................... 48

Beam element .............................................. 283

1D heat .......................................... 288, 298

beam . 27, 29, 35, 38, 48, 59, 60, 72, 74, 76,

107, 109, 110, 112, 117, 119, 121, 138,

142, 161, 163, 170, 171, 172, 173, 175,

202, 208, 210, 211, 218, 225, 230, 233,

236, 237, 244, 246, 247, 254, 260

cable ......................................................... 28

contact ................................................... 271

normal gap ..................... 110, 117, 119, 252

pipe .................................... 40, 41, 209, 210

spring/damper . 70, 109, 123, 205, 206, 223,

231, 234, 244, 252, 256

takeup gap ............................................. 123

thick beam ............................. 174, 218, 240

truss ...... 28, 29, 30, 31, 58, 70, 78, 91, 109,

121, 138, 142, 205, 258

zero gap ......................... 117, 144, 252, 254

Beam model

antenna .................................................. 234

bar-spring system ............................... 58, 70

belt through a pulley .............................. 119

cantilever .............................. 60, 72, 74, 246

curved ................................................ 107

with distributed support ..................... 211

with off-centre masses......................... 208

cantilever column ................................... 236

continuous beam ...................................... 27

eccentric compression ............................. 112

free circular ring ..................................... 175

free square frame .................................... 172

heat transfer ................................... 288, 298

Lee’s Frame ............................................ 163

on elastic foundation ................................ 38

on lift-off supports ................................. 110

pin connection ......................................... 35

pinned at one end ................................... 260

plane frame ................ 35, 59, 170, 171, 233

rigid beam supported by wires ................. 28

simply supported ............................ 218, 225

with rubber stopper ............................ 254

spring-mass system ................................. 223

stretched string ....................................... 205

stringer .................................................... 30

strut, shallow and deep ............................. 78

swinging pendulum ................................ 258

three disk system .................................... 206

three storey building .............................. 231

308 Strand7 Verification Manual

unequal stiffness springs ......................... 256

BEF (beam on elastic foundation) .......... 38, 211

Boundary nonlinearity . 110, 117, 119, 252, 254

Brick attribute

convection coefficient ..................... 300, 301

face load

normal pressure .................................. 150

heat source ............................................. 301

surface pressure .............................. 8, 26, 36

Brick element .. 8, 9, 18, 26, 36, 42, 54, 85, 140,

144, 150, 196, 198, 200

3D heat .......... 289, 290, 292, 293, 300, 301

Brick model

‘solid’ beam ............................................ 196

annular plate ........................................... 200

cylindrical structure .................................... 9

heat transfer.................................... 300, 301

square plate ............................................ 198

steel billet ....................................... 292, 293

thick elliptical plate ..................................... 8

thick-walled cylinder ................................. 26

unit cube .................................................. 85

Buoyancy ..................................................... 269

C

Cantilever

beam buckling .......................................... 60

column ................................................... 236

curved beam ........................................... 107

plate column ............................................. 61

square membrane ........................... 191, 192

square plate ............................ 176, 177, 183

straight beam ................................ 19, 72, 74

tapered beam ............................................ 30

tapered membrane .................................. 192

Z-section beam ........................................... 6

Composites ........................................ 42, 44, 46

Concrete creep ............................................. 283

Concrete shrinkage ...................................... 283

Conduction .. 290, 292, 293, 298, 300, 301, 302

Contact analysis ................... 117, 123, 252, 254

with friction ........................................... 119

Convection .. 289, 290, 292, 293, 300, 301, 302

Coupling

membrane and bending, between ............. 65

torsional and flexural behaviour .............. 173

Creep ... 261, 263, 265, 267, 275, 277, 279, 281

primary .......................... 263, 265, 277, 279

primary and secondary.................... 267, 281

secondary ....................................... 261, 275

stress relaxation .............................. 267, 281

Curved beam ................................................. 22

D

Diagram

bending moment ................................ 27, 38

shear force .......................................... 27, 38

Distorted mesh .................... See Mesh distortion

Duncan-Chang soil model ........... 147, 150, 152

E

Earthquake .................................. 231, 233, 236

Eccentric compression ................................. 112

Elastoplastic analysis .... 121, 124, 128, 130, 158

Element load, non-consistent ........................... 8

F

Fluid element ................................... 53, 54, 213

Footing

circular, on clay ...................................... 147

in sand ................................................... 150

on sand .................................................. 152

strip, on Mohr-Coulomb material ........... 128

strip, on Mohr-Coulomb soil .................. 158

Forced convection ............................... 292, 293

Free

annular membrane .................................. 193

cylinder .................................................. 194

thin square plate ..................................... 178

Freedom

multiple freedom cases .... 16, 23, 65, 81, 83,

85, 87

Friction ............................................... 119, 271

G

Gap element

normal ........................... 110, 117, 119, 252

takeup .................................................... 123

zero ................................ 117, 144, 252, 254

Geometric nonlinearity

finite strain ...... 78, 132, 133, 136, 138, 140,

142, 144

large deflection ...... 72, 74, 76, 78, 107, 109,

111, 112, 113, 115, 132, 133, 136, 138,

142, 144, 250, 256, 258, 260, 271

large rotation.................................. 258, 260

post-buckling ........................... 74, 112, 163

snap-back ................................................. 70

H

Heat generation ........................................... 301

Hinged cylindrical shell ............................... 115

I

Ill-conditioned matrix .................................. 202

Impulse load ................................................ 246

Incompressible material ................................. 26

Insulated (boundary) ................................... 289

Insulator ...................................................... 300

Interlaminar shear stress .......................... 42, 46

Isotropic hardening ............. 81, 83, 85, 87, 121

K

Kinematic hardening ............................. 91, 121

L

Laminates ............................ 42, 44, 46, 65, 214

Lee’s Frame ................................................. 163

Lift-off ........................................................ 110

Linear viscoelasticity .................................... 283

Link

master/slave ........................ 35, 91, 103, 123

rigid ................................................. 28, 173

sector-symmetry ......... 26, 33, 195, 200, 203

Loading

base excitation ........ 230, 231, 233, 236, 237

biaxial ............................................ 265, 279

end moment ............................................. 72

impulse .................................................. 246

linearly distributed ................................... 27

normal pressure ............ 7, 8, 10, 23, 33, 105

self-weight................................................ 25

step load......... 240, 242, 243, 244, 247, 250

thermal .................................... 9, 14, 29, 91

torque ........................................................ 6

uniaxial .......... 261, 263, 267, 275, 277, 281

uniformly distributed ............................. 240

wind ...................................................... 234

Locking

due to incompressibility ........................... 26

membrane .................................................. 5

M

Mass matrix

consistent ...... 170, 171, 172, 175, 176, 177,

178, 179, 181, 182, 183, 186, 188, 189,

190, 191, 192, 193, 194, 214

lumped .. 170, 171, 172, 175, 176, 177, 178,

179, 181, 182, 183, 186, 188, 189, 190,

191, 192, 193, 194

Master-Slave Links ...................................... 283

Material

Duncan-Chang model ............ 147, 150, 152

fluid ........................................... 53, 54, 213

incompressible ......................................... 26

isotropic hardening .......... 81, 83, 85, 87, 98

kinematic hardening ................................. 91

laminate ......................... 42, 44, 46, 65, 214

modified cam-clay .................................. 154

orthotropic ............................. 36, 42, 44, 46

perfect plasticity ..................... 101, 103, 105

plasticity .... 81, 83, 85, 87, 91, 98, 121, 124,

128, 130, 156, 158

310 Strand7 Verification Manual

rubber .... 132, 133, 136, 138, 140, 142, 144

user-defined plate ..................................... 37

Material nonlinearity

hyperelasticity 132, 133, 136, 138, 140, 142,

144

plasticity .... 81, 83, 85, 87, 91, 98, 101, 103,

105, 121, 124, 128, 130, 156, 158, 161

soil ......................................... 147, 150, 152

Material reference system ............................... 36

Maximum deflection ............ 230, 231, 236, 237

Maximum response force ............................. 233

Membrane model

annular membrane .................................. 193

circular ..................................................... 14

elliptic ........................................................ 2

lateral vibration ...................................... 203

ring .................................................... 10, 12

square, cantilevered ........................ 191, 192

stretched circular .................................... 203

tapered, cantilevered ............................... 192

Mesh distortion ............................... 31, 63, 183

aspect ratio ............................................... 23

skew ..................................... 7, 19, 182, 189

taper ............................................. 19, 22, 30

warping .................................................... 21

Modal damping ........................... 240, 242, 243

Modulus of subgrade reaction .................. 37, 38

Modulus vs Time Table ............................... 283

N

Newmark method ..................................... 247

Node attribute

initial temperature .......... 298, 300, 301, 302

prescribed displacement ... 16, 18, 70, 78, 81,

83, 85, 98, 121, 128, 130, 132, 133, 138,

140, 144

prescribed temperature .. 288, 289, 290, 292,

293, 300

with time table .................................... 298

restraint in UCS 10, 12, 181, 190, 195, 196,

200

rotational mass ............... 206, 208, 209, 210

temperature ................................ 29, 91, 109

translational mass .. 173, 208, 209, 223, 231,

233, 234, 244, 258

translational stiffness ........................... 58, 70

Nonlinear buckling ................................ 74, 112

O

Off-centre point mass .................................. 173

Orthotropic material .................... 36, 42, 44, 46

Over-consolidation ...................................... 154

P

Patch test

bending .................................................... 16

brick ......................................................... 18

cylindrical shell ........................................... 4

membrane ................................................ 16

Peak value ............ 218, 219, 240, 242, 243, 244

Pin connection ............................................... 35

Pipe model

drill pipe ................................................. 210

motor-generator ..................................... 209

out-of-plane bending of a curved bar ........ 41

pipe under combined bending and torsion 40

Plastic hinge ................................................ 161

Plate attribute

convection coefficient ............. 300, 301, 302

edge load

normal shear ............................. 19, 21, 22

pressure . 2, 4, 10, 19, 26, 61, 63, 87, 101,

133, 142, 147, 150

shear ......................................... 19, 21, 22

face pressure ....... 4, 7, 23, 33, 105, 242, 243

face support .............................................. 37

heat source ............................................. 301

in-situ stress............................................ 154

In-situ Stress .................................. 156, 158

pre-stress .............................. 64, 65, 67, 124

Plate element

2D heat .......... 288, 289, 290, 298, 300, 301

3D membrane ... 2, 10, 12, 14, 16, 113, 132,

133, 136, 203

axisymmetric ..... 54, 87, 101, 124, 147, 152,

154, 194, 195

axisymmetric heat ................................... 302

plane strain .... 26, 81, 87, 98, 124, 128, 130,

144, 150, 156, 158, 213

plane stress .... 2, 10, 12, 14, 16, 53, 83, 103,

132, 191, 192, 193, 261, 263, 265, 267,

271, 275, 277, 279, 281

plate/shell .... 2, 4, 5, 6, 7, 10, 12, 14, 16, 19,

21, 22, 23, 25, 33, 37, 42, 44, 46, 61, 63,

64, 65, 67, 105, 111, 115, 176, 177, 178,

179, 181, 182, 183, 186, 188, 189, 190,

203, 214, 219, 221, 237, 242, 243, 250,

269

shear panel ......................................... 30, 31

user-defined ............................................. 37

Plate model

annular plate .................. 181, 190, 191, 192

cantilever plate ............................... 176, 177

circular plate ............................................. 33

column buckling ....................................... 61

curved beam ............................................. 22

cylinder ............................................ 87, 194

footing in sand ....................................... 150

footing on clay ....................................... 147

footing on sand ...................................... 152

heat transfer .. 288, 289, 290, 298, 300, 301,

302

hollow sphere ......................................... 195

on elastic foundation ................................ 37

overlay model ......................................... 103

rectangular membrane ............................ 113

rectangular plate ....................................... 23

rhombic plate ................................. 182, 189

simply supported rectangular plate ........... 64

skew plate .................................................. 7

spherical shell ......................................... 214

square ...................................... 37, 105, 111

square plate ............ 178, 179, 183, 186, 188

square, simply supported ................ 219, 221

straight cantilever beam ............................ 19

thick cylinder.................................... 26, 101

twisted beam ............................................ 21

Post-buckling .............................................. 163

Power spectral density (PSD) ...................... 234

Prescribed nodal deflection .... See Node attribute

Prescribed temperature288, 289, 290, 292, 293,

300, 301

Propagation ................................................. 247

R

Radial vibration ........................................... 195

Radiation .................................................... 288

Rayleigh damping ....................... 240, 242, 243

Repeated eigenvalues ........... 170, 171, 172, 175

Response spectrum

acceleration .................... 231, 233, 236, 237

displacement .......................................... 230

Rigid link .............................................. 28, 173

Rigid Links ................................................. 283

Rigid punch .................................................. 98

Rigid-body mode 172, 175, 178, 193, 194, 198

Rubber model

Mooney-Rivlin ....... 132, 133, 140, 142, 144

Neo-Hookean ........................................ 140

Ogden ............................................ 136, 138

S

Scordelis-Lo roof ........................................... 25

Secondary restraint .............................. 186, 188

Sector symmetry .......................................... 136

Seismic ................................ 230, 231, 233, 237

Shallow spherical cap ................................... 250

Shear panel model

simply supported composite beam ............ 31

tapered cantilever beam ............................ 30

Shell model

cylindrical ............................... 4, 25, 67, 115

hemispherical ............................................. 5

Scordelis-Lo roof ..................................... 25

spherical cap ........................................... 250

Slab model .................................................. 237

312 Strand7 Verification Manual

Sliding ......................................................... 271

Slosh ............................................................ 213

Soil

Mohr-Coulomb .............................. 156, 158

Spring elements ........................................... 283

Spring-mass system ...................................... 223

Standard deviation value .............................. 234

Stiffening effect .................................... 203, 205

Stress

beam bending stress............................ 40, 41

beam torsional shear stress .................. 40, 41

bending .............................................. 42, 46

Stretched membrane .................................... 203

Stretched string ............................................ 205

Subgrade reaction modulus ........ See Modulus of

subgrade reaction

Support

lift-off ..................................................... 110

Support on element

beam ................................................ 38, 211

plate face .................................................. 37

Symmetric modes ........................................ 176

T

Table

factor vs. position ................................... 269

Tapered beam ................................................ 48

Temperature distribution ................................. 9

parabolic .................................................. 14

Test

constitutive model ................ 81, 83, 85, 101

eigenvalue solver.... 170, 171, 172, 175, 178,

179, 193, 194, 198

ill-conditioning ....................................... 202

nonlinear equation solution .................... 103

structure orientation, effect of ................. 196

Thermal expansion ..................... 9, 91, 109, 203

Thermal load ................................................. 44

Thick beam ...................... See Timoshenko beam

Thick plate ... 186, 188, 189, 190, 191, 192, 221

Time table ................................................... 298

Timoshenko beam ....................... 174, 218, 240

Toggle mechanism ....................................... 109

Transverse shear deformation

beam ...................................... 174, 218, 240

plate 37, 186, 188, 189, 190, 191, 192, 221,

243

Twisted beam ................................................ 21

U

User-defined Coordinate System ..... 10, 12, 181,

190, 195, 196, 200

V

Vibration mode

anti-symmetric ....................................... 177

axisymmetric .................................. 194, 200

lateral ............................................. 203, 205

radial ...................................................... 195

symmetric............................................... 176

W

Wilson method ......................................... 247

Y

Yield criterion

Mohr-Coulomb ...... 124, 128, 130, 156, 158

von Mises . 81, 83, 85, 87, 98, 101, 103, 105

Z

Z-section cantilever .......................................... 6

Making finite element analysis easier.

Strand7 Pty Limited Suite 1, Level 5, 65 York Street Sydney NSW 2000 Australia Tel +61 2 9264 2977 Fax +61 2 9264 2066 info@strand7.com www.strand7.com

www.strand7.com