Finite Element Analysis for Design Engineers

Finite Element Analysis for Design Engineers
Paul M. Kurowski
Table of Contents
Preface xi
Chapter 1 Introduction 1
1.1 What Is Finite Element Analysis? 1
1.2 What Is “FEA for Design Engineers?’ 2
1.3 Note on Hands-on Exercises and Illustrations in This Book 3
Chapter 2 From CAD Model to Finite Element Analysis Results . 5
2.1 Formulation of the Mathematical Model . 5
2.2 Selecting the Numerical Method to Solve the
Numerical Methods in Engineering Analysis . 7
Mathematical Model 7
2.2.2 Reasons for the Dominance of the Finite Element
Method 8
2.3 The Finite Element Model . 9
2.3.1 Meshing . 9
2.3.2 Formulationof Finite Element Equations . 10
2.3.3 Errors in FEA Results . 11
Chapter 3 Major Concepts of the Finite Element Model . 13
3.1 Formulation of a Finite Element 13
3.1.1 Shape Functions, Degrees of Freedom. and
Element Order . 13
3.1.2 Requirements to Be Satisfied by Shape Functions . 15
3.1.3 Artificial Constraints . 15
3.1.4 The Choices of Discretization . 17
3.2 Types of Finite Elements . 18
3.2.1 Element Dimensionality 19
3.2.2 Analysis Dimensionality . 20
3.2.4 Element Order and Element Type . 21
Element Modeling Capabilities . 22
3.2.3 Element Shape . 21
3.2.5
Chapter 4 Controlling Discretization Errors 25
4.1 Convergence Process . 25
4.1.2 Convergence Error 29
4.1.3 Solution Error 29
4.1.1 h-Convergence Process . 25
4.1.4 p-Convergence Process . 30
4.1.5 Variations in Convergence Processes . 32viii I Finite Element Analysis for Design Engineers
4.2 Problems with Convergence 34
4.2.1 Stress Singularity . 35
4.2.2 Displacement Singularity 38
4.3.1
4.3.2 L-Shaped Bracket 43
4.3.3 Spot-Welded Cantilever 45
4.3 Hands-on Exercises . 42
Tensile Strip with a Circular Hole . 42
Chapter 5 Finite Element Mesh 47
5.1 Meshing Techniques 47
5.1.1 Manual Meshing 47
5.1.2 Semi-Automatic Meshing . 48
5.2 Mesh Compatibility 50
5.1.3 Automatic Meshing . 48
5.2.1 Compatible Elements 50
5.2.2 Incompatible Elements 51
5.2.3 Forced Compatibility 52
5.3.2 Not Enough Elements to Represent Model Stiffhess 55
5.3.3 Incorrect Mapping to Geometry 56
5.3.4 Incorrect Conversion to Shell Model 57
5.4 Hands-on Exercises . 58
5.4.1 Hollow Cantilever Bracket 58
5.4.2 Beam in Bending . 59
5.3 Common Meshing Problems 52
5.3.1 Element Distortion 54
Chapter 6 Modeling Process 61
6.1.1 Definition of the Objective of Analysis 62
6.1.2 Selection of the Units of Measurement . 62
6.1.3 Geometry Creation 63
6.1.4 Defining Material Properties . 64
6.1.5 Defining Boundary Conditions . 64
6.2.2 Axial Symmetry 68
6.2.3 Realignment of Degrees of Freedom 69
6.1 Modeling Steps 61
6.2 Some Usehl Modeling Techniques . 65
6.2.1 Taking Advantage of Symmetry and Anti-Symmetry 65
6.3 Hands-on Exercises . 71
6.3.1 Cantilever Bracket with Symmetry Boundary
Conditions . 71
6.3.2 Cantilever Bracket with Anti-Symmetry
6.3.3 Shaft in Torsion . 72
Boundary Conditions 71
6.3.4 Plate in Bending 73Table ofcontents I ix
6.3.5 Ring in Bending 74
6.3.6 Pin-Supported Link . 75
Chapter 7 Types of Finite Element Analysis 77
7.1 Thermal Analysis . 77
7.1.1
7.1.3 Modeling Considerations in Thermal Analysis . 79
Heat Flow Induced by Prescribed Temperatures 77
7.1.2 Heat Flow Induced by Heat Load and Convection . 78
7.2 Nonlinear Analysis . 80
7.2.1 Nonlinear Material 81
7.2.2 Nonlinear Geometry 84
7.3 Modal Analysis 92
Differences Between Modal and Static Analyses . 93
7.3.3 Modal Analysis with Rigid Body Modes 94
7.3.4 Importance of Supports in Modal Analysis 94
7.3.5 Applications of Modal Analysis 95
7.2.3 Contact Stress Analysis . 89
7.3.1
7.3.2 Interpretation of Displacement and Stress Results
in Modal Analysis . 94
7.3.6 Pre-Stress Modal Analysis 97
7.3.7 Symmetry and Anti-Symmetry Boundary Conditions
in Modal Analysis . 97
7.3.8 Convergence of Modal Frequencies Results . 99
7.3.9 Meshing Considerations for Modal Analysis 99
Buckling Load Factor 100
7.4.2 Interpretation of Results of Buckling Analysis .100
Modal SuperpositionMethod 101
7.5.2 Time Response Analysis . 102
7.5.3 Frequency Response Analysis . 104
7.4 Buckling Analysis 99
7.4.1
7.5 Dynamic Analysis 101
7.5.1
7.6 Hands-on Exercises . 106
7.6.1 Crossing Pipes . 106
7.6.2 Radiator . 107
7.6.3 Cantilever Bracket, Elasto-Plastic Material 108
7.6.4 Slender Cantilever Beam in Bending 109
7.6.5 Round Membrane Under Pressure 110
7.6.6 Tuning Fork . 111
7.6.7 U-Shaped Support Bracket 112
7.6.8 Unsupported and Hinge-Supported Beam 113
7.6.9 Hollow Cantilever Bracket, Modes Separation . 114
7.6.10 Beam in Compression . 115I Finite Element Analysis for Design Engineers
Chapter 8 Design Optimization . 117
8.2 Structural Optimization 117
Factors Defining an Optimization Process 119
8.2.2 Sensitivity Studies . 119
8.1 Topological Optimization 117
8.2.1
Chapter 9 Using Finite Element Analysis in the Design Process 123
9.1 DifferencesBetween CAD and FEA Geometries 123
9.1.1 Defeaturing 124
9.1.2 Idealization 125
9.1.3 Cleanup . 126
9.1.4 Common Meshing Problems . 128
9.1.5 Mesh Inadequacy 130
9.2 Integration of CAD and FEA Software . 131
9.2.1 StandaloneFEA Software . 131
9.2.2 Integrated FEA Software . 131
9.3 FEA Implementation 132
9.3.1 Positioning of CAD and FEA Activities . 132
9.3.2 Personnel Training 133
9.3.3 Software Selection Criteria . 136
9.3.4 Building Confidencein the FEA . 139
9.3.5 Expected Return on Investment 139
9.4 FEA Projects 140
9.4.1 Major Steps in an FEA Project 141
9.4.2 FEA Report . 144
9.4.3 Importance of Documentationand Backups .146
9.4.4 ContractingOut FEA Services 147
9.4.5 Common Errors in FEA Management 147
Chapter 10 Some Misconceptions and Frequently Asked
Questions . 151
10.1 FEAQuiz 151
10.2 Frequently Asked Questions About FEA 155
Chapter 11 Finite Element Analysis Resources . 169
Chapter 12 Glossary of Terms 171
Index 177
About the Author 187
INDEX
Index Terms Links
A
Analysis dimensionality 20
Anti-symmetry
boundary conditions, in modal
analysis 97
taking advantage of 65 165
Artificial constraints 15 99
Audit 146
Automeshing 21 48 90 128
137 152
Available design space 117
Axial symmetry 68
B
Backups, importance of 146
Base excitation 102 103
Basis functions 18
Beam elements 20 50t 125 128
130
Benchmarking 154
Bending 38
modeling 55
Bending stiffness 84 85f
Bending stresses 130f
Bilinear material 81Index Terms Links
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Boundary conditions 5 164
anti-symmetry 65
in modal analysis 97
assigning 64
defining 64
essential 64
natural 64
symmetry 65
in modal analysis 97
Boundary element method 8
Brick elements 21
Buckling analysis 99
interpretation of results 100
linear 137
Buckling load safety factor 100
C
CAD, confused with FEA 149
CAD geometry
in CAD-FEA process 132
converting to FEA geometry 63
difference from FEA geometry 123
requirements of 123
CAD models
in FEA 2
from, to FEA results 5
CAD software
integration with FEA software 131 136
requirements for 138
Chamfers 57Index Terms Links
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Cleanup 126
Compatibility
forced 52 53f
inter-element 157
internal 157
Compatible elements 50 159
Compressive stress 88 97
Computer-based simulations 3
Conservative load 86 87f
Constant strain 15 157
Constraints 6 119
artificial 15 99
Contact stress analysis 89 137
Continuous displacement field 51
Contracting out FEA services 147 148f
Convection, heat flow induced by 78
Convergence error 29 30f
Convergence
of modal frequencies results 99
problems with 34
Convergence analysis, objective of 161
Convergence processes 25
adaptive 34
h- 25
p- 30
types of 160
variations in 32
Curve, solution domain represented
as 6
Cyclic symmetry 69 70fIndex Terms Links
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D
Damping 92 93 94 104
coefficient of 102
Data of interest 7 25
Defeaturing 124 165
Deflection 5 154
Deformation
second-order 13
effect on stiffness 84 89
Degenerated elements 18 54 154 159
167
Degrees of freedom 10 13 27 29
nodal 156
realignment of 69
Design analysis 1
Design engineers 133
increasing responsibilities of 134t
Design optimization 117
Design process, FEA-driven 3f
Design space 117
Design variables 119
Dimensionality
of analysis 20
of elements 19
Direct integration method 101
Discretization
choices of 17
of continuum 155
Discretization errors 11 161Index Terms Links
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in buckling analysis 100
controlling 25
Discretization process 9
Displacement
in modal analysis 94
nodal 13
Displacement analysis 6
Displacement boundary conditions 5
Displacement components, linking 52
Displacement field 9
Displacement functions 21
second-order 13
Displacement singularity 38
Distortion, element 54
Documentation 149
importance of 146
Drafting 133
Dynamic analysis 101
and modal analysis 96
Dynamic effects 93
Dynamic stress 99
Dynamic time response analysis 102
E
Edges, mesh control applied to 49f
Eigenvalue 93
Eigenvector 92 93
Elastic stiffness 92 99
Elasticity, modulus of 81
Elasto-plastic model 36 81Index Terms Links
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Element level basis functions 18
Elements
2-D axisymmetric 20
2-D plane strain 20
2-D plane stress 20
beam 20 50t 125 128
130
brick 21
communication between 158
compatible 50 159
degenerated 18 54 128 154
159 167
dimensionality of 19
distortion of 54
first-order 13
difference from secondorder 157
h- 21 22f 23t 160
hexahedral 21
hybrid 157
incompatible 51 159
insufficient number of 55 167
improper use of 167
mapping of 18 19f
membrane 20 50t
modeling of 22
order of 13 18 21 128
130 131 153
p- 21 22f 23t 160
quadrilateral 21Index Terms Links
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second-order 13 14f
difference from first-order 157
shape of 21
shell 19 50t 125 128
165
incorrect conversion to 57
size of 17 49 128
solid 19 50t 128 151
tetrahedral 21 128 153
triangular 13 21
types of 21 22f 23t 50t
128
Engineering analysis problems
methods for solving 9f
Errors
discretization 11 161
estimation of 153
in FEA results 11
modeling 11 161
numerical 162
solution 12 162
Essential boundary conditions 64
Executive summary, in FEA reports 144 166
Exercises, hands-on
beam
in bending 59
in compression 115
cantilever bracket
elasto-plastic material 108
with anti-symmetry bound-ary conditions 71Index Terms Links
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with symmetry boundary
conditions 71
crossing pipes 106
hollow cantilever bracket 58
modes separation 114
introduction to 3
L-shaped bracket 43
pin-supported link 75
plate in bending 73
radiator 107
ring in bending 74
round membrane under pres-sure 110
shaft in torsion 72
slender cantilever beam in
bending 109
spot-welded cantilever 45
tensile strip with circular hole 42
tuning fork 111
unsupported and hingesupported beam 113
U-shaped support bracket 112
Extrusion 48
F
Faces, mesh control applied to 49f
FAQs about FEA 155
FE models 9
assignment of mass to 158
boundary conditions applied to 164
degrees of freedom in 156Index Terms Links
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major concepts of 13
techniques to reduce size of 165
FEA
alternatives to 168
basic steps of 155
from CAD models to results 5
compared to prototyping and
testing 139
concurrence with design pro-cess 2
confused with CAD 149
contracting out for services 147 148f
defining objective of 62
demonstrating usefulness of
to organization 139
for design engineers 2
equations for, formulation of 10 156
expectations of 148
expertise in 148
frequently asked questions
about 155
golden rule of 168
implementation of 132
incompetent analysis in 155
introduction to 1
level of detail in 154
limitations of 2
misconceptions about 151
objectives of 2
poorly defined 148
reasons for dominance of 8Index Terms Links
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resources for users 169
return on investment in 139
time constraints of 148
training in 135
types of 77 164
using in design process 123
FEA analysts 133
FEA geometry
difference from CAD geometry 123
in CAD-FEA process 132
requirements of 123
FEA projects
analyzing results in 144
checkpoints in 145f
common errors in management
of 147
evaluating results in 143
FE model in 143
from managerial point of view 140
initial questions for 141
major steps in 141
mathematical model in 142
modeling approach in, costeffective 142
reports in 144 166
scope of analysis in 142
FEA quiz 151
FEA results
correlating with experiment 153
errors in 11Index Terms Links
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high accuracy of 152
validating 139 140t
FEA software 135 152
analysis types available in 137
automesher in 137
and benchmarking 154
computing considerations for 139
cost considerations for 138 139
error reporting in 152
integration with CAD software 131 136
presentation tools in 137
proficiency in 148
requirements for 138
selection criteria 136
user control in 136
user support for 138 139
Fillets 6 36 38 57
Finite difference method 8
Finite element analysis (FEA), see
FEA
Finite element method (FEM)
defined 8
Finite elements (FEs) 9
formulation of 13
major assumptions of 155
types of 18 158
First-order elements 13 14f
difference from second-order 157
Fixed support 86f
Floating solids 126Index Terms Links
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Floating support 70 86f
Following load 86 87f
Follow-up, to FEA reports 144 167
Force boundary conditions 5
Forced compatibility 52 53f
Frequencies, natural 94 95 97 101
Frequency 93
Frequency analysis 92 137
Frequency response analysis 104
Frequency sweep 104
G
Gap 51
Gauss points 162
Geometry
accuracy of representation 151
boundary conditions applied
to 164
defining, relative uncertainty
in 65
nonlinear 84
model 8 154
in FEA reports 144 166
nonlinear 84
surface 125
solid 125
wire-frame 125
see also CAD geometry, FEA
geometry
Geometry cleanup 126Index Terms Links
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Geometry creation 63 131
Glossary 171
H
h-Elements 21 22f 23t 160
Hands-on exercises, see under
Exercises
Heat flow
induced by heat load and con-vection 78
induced by prescribed temper-ature 77
Heat flux 78
Heat power 78
Hexahedral elements 21
Hybrid element 157
Hybrid meshing 50
I
Idealization 125
Incompatible elements 51 159
Inertial stiffness 92 99
Integration of CAD and FEA
software 131
Inter-element compatibility 15 157
Internal compatibility 15 157
Interpretation of results, in modal
analysis 94
L
Large deformation analysis 84
Legal liability 147Index Terms Links
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Linear analysis 81 137
compared to nonlinear analysis 86
Linking displacement components 52
Links 51
Load vector 11 80
Loads 6 10 64 154
164
buckling, safety factor 100
conservative 86 87f
defining, relative uncertainty
in 65
in FEA reports 144 166
following 86 87f
heat 78
non-conservative 86 87f
non-following 86 87f
orientation of 86
volume 64
Local cylindrical coordinate system 70
M
Machine design 135
Manual meshing 47
Mapping
element 18 19f
incorrect 56
Mass
assignment of, to FE model 158
units of 62t
Mass density, units of 62tIndex Terms Links
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Mass matrix 93
Master entity 52 53f
Materials, mechanics of 135
Materials properties 6
assigning 64
defining 64 143
relative uncertainty in 65
nonlinear 81
Mathematical models 1 10f 61
formulation of 5
selecting best 7
Measurement units, selecting 62
Membrane elements 20 50t
Membrane stiffness 84 85f
Mesh bias 49 128
Mesh compatibility 50
Mesh control 49f 131
Mesh refinement 25
non-uniform 32
Mesh size 151 153
Meshing techniques 47
Meshes
in FEA reports 144
inadequacy of 130
quality check tools for 130
Meshing 9 21 123 126
152
automatic 48
common problems with 52 128
considerations for modal analy-sis 99Index Terms Links
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in contact stress analysis 89
manual 47
semi-automatic 48
Minimum total potential energy 10 156
Modal analysis 92 137 154
applications of 95
comparison of results 98t
difference from static analysis 93
displacement results in 94
importance of supports in 94
interpretation of results 94
meshing considerations for 99
pre-stress 97
with rigid body modes 94
stress results 94
symmetry and anti-symmetry
boundary conditions in 97
Modal frequencies 95
convergence of results 99
Modal superposition method 96 99 101
in frequency response analysis 104
Model stiffness, insufficient elements to represent 55
Modeling, simplified 65 80f
Modeling errors 11 161
in buckling analysis 100
Modeling process 61
Modeling steps 61
Modeling techniques, useful 65
ModelsIndex Terms Links
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mathematical 1 10f 61
formulation of 5
selecting best 7
physical 1
see also FE models
Modes of vibration 92
Modified analysis 146 147
Modulus of elasticity 81
Multiple entities 126
N
Natural boundary conditions 64
Natural frequencies 92 94 95 97
101
Nodal degree of freedom 156
Nodal displacements 10 13 156
vector of 11 80
Nodal stresses 162
Non-conservative load 86 87f
Non-following load 86 87f
Nonlinear analysis 80 164
compared to linear analysis 86
Nonlinear buckling analysis 100
Nonlinear geometry 84
Nonlinear materials 81
Numerical error 161
Numerical methods
in engineering analysis 7
selecting 7
Nylon, in contact stress analysis 91Index Terms Links
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O
Optimization
design 117
factors defining process 119
goal of 119
structural 117
topological 117 118f
Orientation, of load 86
Overlap 51
P
p-Elements 21 22f 23t 160
Patch test 15 157
Personnel training 133 146 149
Physical models 1
Pin supports 70
Plane stress 2-D 13
Plasticity 36
Point supports 38
Poisson’s ratio 81
Polygons 57
Polynomial functions 13
Pre-stress, modal analysis 97
Prisms 21
Product design process, traditional 3f
Prototyping and testing, compared
to FEA 139
Q
Quadrilateral elements 21Index Terms Links
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R
Ramberg-Osgood model 81
Realignment of degrees of free-dom 69
Repetitive symmetry 69 70f
Resonance 92
Resonant frequencies 95
Restraints 10 64 154 168
defining, relative uncertainty
in 65
in FEA reports 144 166
Revolution 48
Rigid body modes, modal analysis
with 94
Rigid body motion 15 41 157
Rotation 13
S
S.I. system of measurements 62
Second-order deformation 13
Second-order displacement func-tions 13
Second-order elements 13 14f
difference from first-order 157
Semi-automatic meshing 48
Sensitivity studies 119
Shaker table test 104
Shape deformation, effect on stiffness 84 89
Shape functions 13 157
impact of choice of 15Index Terms Links
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linear 13
requirements to be satisfiedby 15
Sharp re-entrant corner 35 42t 80
Shell elements 19 50t 125 128
165
incorrect conversion to 57
Single-degree-of-freedom systems 101
Singularities
displacement 38
stress 35 83 130f
Slave entity 52 53f
Solid elements 19 50t 128 151
first-order tetrahedral 153
Solid geometry 125
Solution errors 12 29 30f 162
Static analysis, difference from
modal analysis 93
Steel, in contact stress analysis 91
Stiffness matrix 11 80 166
Strain 11 55
2-D plane 20
constant 15 157
distribution of 14
Strain gauge, placement of 96
Stress 11 32 55
2-D plane 13 15 20
averaged 162
compressive 97
concentrations of 153
distribution of 14Index Terms Links
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dynamic 99
in modal analysis 94
nodal 162
tensile 88 97
yield 81
Stress analysis 5 6 135 154
contact 89
Stress averaging 16 162
Stress convergence curve 35
Stress singularities 35 83 130f
Stress stiffening 97
Stress stiffness 99
Stress-strain 81
StressCheck® 4
Structural analysis 1
analogies to thermal analysis 77 78t
nonlinear behavior in 80
Structural optimization 117
supports
defining 64
fixed 86f
floating 70 86f
importance in modal analysis 94
pin 70
point 38
Surface, solution domain
represented as 6
Surface geometry 125
Swelling 88 89f
Symmetry, taking advantage of 65 165Index Terms Links
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Symmetry boundary conditions
in modal analysis 97
in thermal analysis 79
T
2-D analysis 13 166
2-D plane strain 20
2-D plane stress 13 15 20
3-D analysis 14f 20
Temperature 6 10 11 77
Tensile stress 88 97
Tetrahedral elements 21 128 153
Thermal analysis 1 77
modeling considerations in 79
steady-state 137
Time response analysis 102
Topological optimization 117 118f
Torsion 68f 88
Training, personnel 133
Translation 13 69
Triangular elements 13 21
U
Uncertainties, relative 65
Units of measurement, selecting 62
V
Vertex, mesh control applied to 49f
Vibration 135
modes of 92Index Terms Links
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Vibration analysis 101
Volume loads 64
Volume, solution domain
represented as 5
W
Weak spots 96
Wire-frame geometry 125
Y
Yield stress 81Index Terms Links
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Boundary conditions 5 164
anti-symmetry 65
in modal analysis 97
assigning 64
defining 64
essential 64
natural 64
symmetry 65
in modal analysis 97
Boundary element method 8
Brick elements 21
Buckling analysis 99
interpretation of results 100
linear 137
Buckling load safety factor 100
C
CAD, confused with FEA 149
CAD geometry
in CAD-FEA process 132
converting to FEA geometry 63
difference from FEA geometry 123
requirements of 123
CAD models
in FEA 2
from, to FEA results 5
CAD software
integration with FEA software 131 136
requirements for 138
Chamfers 57Index Terms Links
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Cleanup 126
Compatibility
forced 52 53f
inter-element 157
internal 157
Compatible elements 50 159
Compressive stress 88 97
Computer-based simulations 3
Conservative load 86 87f
Constant strain 15 157
Constraints 6 119
artificial 15 99
Contact stress analysis 89 137
Continuous displacement field 51
Contracting out FEA services 147 148f
Convection, heat flow induced by 78
Convergence error 29 30f
Convergence
of modal frequencies results 99
problems with 34
Convergence analysis, objective of 161
Convergence processes 25
adaptive 34
h- 25
p- 30
types of 160
variations in 32
Curve, solution domain represented
as 6
Cyclic symmetry 69 70fIndex Terms Links
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D
Damping 92 93 94 104
coefficient of 102
Data of interest 7 25
Defeaturing 124 165
Deflection 5 154
Deformation
second-order 13
effect on stiffness 84 89
Degenerated elements 18 54 154 159
167
Degrees of freedom 10 13 27 29
nodal 156
realignment of 69
Design analysis 1
Design engineers 133
increasing responsibilities of 134t
Design optimization 117
Design process, FEA-driven 3f
Design space 117
Design variables 119
Dimensionality
of analysis 20
of elements 19
Direct integration method 101
Discretization
choices of 17
of continuum 155
Discretization errors 11 161Index Terms Links
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in buckling analysis 100
controlling 25
Discretization process 9
Displacement
in modal analysis 94
nodal 13
Displacement analysis 6
Displacement boundary conditions 5
Displacement components, linking 52
Displacement field 9
Displacement functions 21
second-order 13
Displacement singularity 38
Distortion, element 54
Documentation 149
importance of 146
Drafting 133
Dynamic analysis 101
and modal analysis 96
Dynamic effects 93
Dynamic stress 99
Dynamic time response analysis 102
E
Edges, mesh control applied to 49f
Eigenvalue 93
Eigenvector 92 93
Elastic stiffness 92 99
Elasticity, modulus of 81
Elasto-plastic model 36 81Index Terms Links
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Element level basis functions 18
Elements
2-D axisymmetric 20
2-D plane strain 20
2-D plane stress 20
beam 20 50t 125 128
130
brick 21
communication between 158
compatible 50 159
degenerated 18 54 128 154
159 167
dimensionality of 19
distortion of 54
first-order 13
difference from secondorder 157
h- 21 22f 23t 160
hexahedral 21
hybrid 157
incompatible 51 159
insufficient number of 55 167
improper use of 167
mapping of 18 19f
membrane 20 50t
modeling of 22
order of 13 18 21 128
130 131 153
p- 21 22f 23t 160
quadrilateral 21Index Terms Links
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second-order 13 14f
difference from first-order 157
shape of 21
shell 19 50t 125 128
165
incorrect conversion to 57
size of 17 49 128
solid 19 50t 128 151
tetrahedral 21 128 153
triangular 13 21
types of 21 22f 23t 50t
128
Engineering analysis problems
methods for solving 9f
Errors
discretization 11 161
estimation of 153
in FEA results 11
modeling 11 161
numerical 162
solution 12 162
Essential boundary conditions 64
Executive summary, in FEA reports 144 166
Exercises, hands-on
beam
in bending 59
in compression 115
cantilever bracket
elasto-plastic material 108
with anti-symmetry bound-ary conditions 71Index Terms Links
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with symmetry boundary
conditions 71
crossing pipes 106
hollow cantilever bracket 58
modes separation 114
introduction to 3
L-shaped bracket 43
pin-supported link 75
plate in bending 73
radiator 107
ring in bending 74
round membrane under pres-sure 110
shaft in torsion 72
slender cantilever beam in
bending 109
spot-welded cantilever 45
tensile strip with circular hole 42
tuning fork 111
unsupported and hingesupported beam 113
U-shaped support bracket 112
Extrusion 48
F
Faces, mesh control applied to 49f
FAQs about FEA 155
FE models 9
assignment of mass to 158
boundary conditions applied to 164
degrees of freedom in 156Index Terms Links
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major concepts of 13
techniques to reduce size of 165
FEA
alternatives to 168
basic steps of 155
from CAD models to results 5
compared to prototyping and
testing 139
concurrence with design pro-cess 2
confused with CAD 149
contracting out for services 147 148f
defining objective of 62
demonstrating usefulness of
to organization 139
for design engineers 2
equations for, formulation of 10 156
expectations of 148
expertise in 148
frequently asked questions
about 155
golden rule of 168
implementation of 132
incompetent analysis in 155
introduction to 1
level of detail in 154
limitations of 2
misconceptions about 151
objectives of 2
poorly defined 148
reasons for dominance of 8Index Terms Links
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resources for users 169
return on investment in 139
time constraints of 148
training in 135
types of 77 164
using in design process 123
FEA analysts 133
FEA geometry
difference from CAD geometry 123
in CAD-FEA process 132
requirements of 123
FEA projects
analyzing results in 144
checkpoints in 145f
common errors in management
of 147
evaluating results in 143
FE model in 143
from managerial point of view 140
initial questions for 141
major steps in 141
mathematical model in 142
modeling approach in, costeffective 142
reports in 144 166
scope of analysis in 142
FEA quiz 151
FEA results
correlating with experiment 153
errors in 11Index Terms Links
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high accuracy of 152
validating 139 140t
FEA software 135 152
analysis types available in 137
automesher in 137
and benchmarking 154
computing considerations for 139
cost considerations for 138 139
error reporting in 152
integration with CAD software 131 136
presentation tools in 137
proficiency in 148
requirements for 138
selection criteria 136
user control in 136
user support for 138 139
Fillets 6 36 38 57
Finite difference method 8
Finite element analysis (FEA), see
FEA
Finite element method (FEM)
defined 8
Finite elements (FEs) 9
formulation of 13
major assumptions of 155
types of 18 158
First-order elements 13 14f
difference from second-order 157
Fixed support 86f
Floating solids 126Index Terms Links
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Floating support 70 86f
Following load 86 87f
Follow-up, to FEA reports 144 167
Force boundary conditions 5
Forced compatibility 52 53f
Frequencies, natural 94 95 97 101
Frequency 93
Frequency analysis 92 137
Frequency response analysis 104
Frequency sweep 104
G
Gap 51
Gauss points 162
Geometry
accuracy of representation 151
boundary conditions applied
to 164
defining, relative uncertainty
in 65
nonlinear 84
model 8 154
in FEA reports 144 166
nonlinear 84
surface 125
solid 125
wire-frame 125
see also CAD geometry, FEA
geometry
Geometry cleanup 126Index Terms Links
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Geometry creation 63 131
Glossary 171
H
h-Elements 21 22f 23t 160
Hands-on exercises, see under
Exercises
Heat flow
induced by heat load and con-vection 78
induced by prescribed temper-ature 77
Heat flux 78
Heat power 78
Hexahedral elements 21
Hybrid element 157
Hybrid meshing 50
I
Idealization 125
Incompatible elements 51 159
Inertial stiffness 92 99
Integration of CAD and FEA
software 131
Inter-element compatibility 15 157
Internal compatibility 15 157
Interpretation of results, in modal
analysis 94
L
Large deformation analysis 84
Legal liability 147Index Terms Links
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Linear analysis 81 137
compared to nonlinear analysis 86
Linking displacement components 52
Links 51
Load vector 11 80
Loads 6 10 64 154
164
buckling, safety factor 100
conservative 86 87f
defining, relative uncertainty
in 65
in FEA reports 144 166
following 86 87f
heat 78
non-conservative 86 87f
non-following 86 87f
orientation of 86
volume 64
Local cylindrical coordinate system 70
M
Machine design 135
Manual meshing 47
Mapping
element 18 19f
incorrect 56
Mass
assignment of, to FE model 158
units of 62t
Mass density, units of 62tIndex Terms Links
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Mass matrix 93
Master entity 52 53f
Materials, mechanics of 135
Materials properties 6
assigning 64
defining 64 143
relative uncertainty in 65
nonlinear 81
Mathematical models 1 10f 61
formulation of 5
selecting best 7
Measurement units, selecting 62
Membrane elements 20 50t
Membrane stiffness 84 85f
Mesh bias 49 128
Mesh compatibility 50
Mesh control 49f 131
Mesh refinement 25
non-uniform 32
Mesh size 151 153
Meshing techniques 47
Meshes
in FEA reports 144
inadequacy of 130
quality check tools for 130
Meshing 9 21 123 126
152
automatic 48
common problems with 52 128
considerations for modal analy-sis 99Index Terms Links
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in contact stress analysis 89
manual 47
semi-automatic 48
Minimum total potential energy 10 156
Modal analysis 92 137 154
applications of 95
comparison of results 98t
difference from static analysis 93
displacement results in 94
importance of supports in 94
interpretation of results 94
meshing considerations for 99
pre-stress 97
with rigid body modes 94
stress results 94
symmetry and anti-symmetry
boundary conditions in 97
Modal frequencies 95
convergence of results 99
Modal superposition method 96 99 101
in frequency response analysis 104
Model stiffness, insufficient elements to represent 55
Modeling, simplified 65 80f
Modeling errors 11 161
in buckling analysis 100
Modeling process 61
Modeling steps 61
Modeling techniques, useful 65
ModelsIndex Terms Links
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mathematical 1 10f 61
formulation of 5
selecting best 7
physical 1
see also FE models
Modes of vibration 92
Modified analysis 146 147
Modulus of elasticity 81
Multiple entities 126
N
Natural boundary conditions 64
Natural frequencies 92 94 95 97
101
Nodal degree of freedom 156
Nodal displacements 10 13 156
vector of 11 80
Nodal stresses 162
Non-conservative load 86 87f
Non-following load 86 87f
Nonlinear analysis 80 164
compared to linear analysis 86
Nonlinear buckling analysis 100
Nonlinear geometry 84
Nonlinear materials 81
Numerical error 161
Numerical methods
in engineering analysis 7
selecting 7
Nylon, in contact stress analysis 91Index Terms Links
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O
Optimization
design 117
factors defining process 119
goal of 119
structural 117
topological 117 118f
Orientation, of load 86
Overlap 51
P
p-Elements 21 22f 23t 160
Patch test 15 157
Personnel training 133 146 149
Physical models 1
Pin supports 70
Plane stress 2-D 13
Plasticity 36
Point supports 38
Poisson’s ratio 81
Polygons 57
Polynomial functions 13
Pre-stress, modal analysis 97
Prisms 21
Product design process, traditional 3f
Prototyping and testing, compared
to FEA 139
Q
Quadrilateral elements 21Index Terms Links
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R
Ramberg-Osgood model 81
Realignment of degrees of free-dom 69
Repetitive symmetry 69 70f
Resonance 92
Resonant frequencies 95
Restraints 10 64 154 168
defining, relative uncertainty
in 65
in FEA reports 144 166
Revolution 48
Rigid body modes, modal analysis
with 94
Rigid body motion 15 41 157
Rotation 13
S
S.I. system of measurements 62
Second-order deformation 13
Second-order displacement func-tions 13
Second-order elements 13 14f
difference from first-order 157
Semi-automatic meshing 48
Sensitivity studies 119
Shaker table test 104
Shape deformation, effect on stiffness 84 89
Shape functions 13 157
impact of choice of 15Index Terms Links
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linear 13
requirements to be satisfiedby 15
Sharp re-entrant corner 35 42t 80
Shell elements 19 50t 125 128
165
incorrect conversion to 57
Single-degree-of-freedom systems 101
Singularities
displacement 38
stress 35 83 130f
Slave entity 52 53f
Solid elements 19 50t 128 151
first-order tetrahedral 153
Solid geometry 125
Solution errors 12 29 30f 162
Static analysis, difference from
modal analysis 93
Steel, in contact stress analysis 91
Stiffness matrix 11 80 166
Strain 11 55
2-D plane 20
constant 15 157
distribution of 14
Strain gauge, placement of 96
Stress 11 32 55
2-D plane 13 15 20
averaged 162
compressive 97
concentrations of 153
distribution of 14Index Terms Links
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dynamic 99
in modal analysis 94
nodal 162
tensile 88 97
yield 81
Stress analysis 5 6 135 154
contact 89
Stress averaging 16 162
Stress convergence curve 35
Stress singularities 35 83 130f
Stress stiffening 97
Stress stiffness 99
Stress-strain 81
StressCheck® 4
Structural analysis 1
analogies to thermal analysis 77 78t
nonlinear behavior in 80
Structural optimization 117
supports
defining 64
fixed 86f
floating 70 86f
importance in modal analysis 94
pin 70
point 38
Surface, solution domain
represented as 6
Surface geometry 125
Swelling 88 89f
Symmetry, taking advantage of 65 165Index Terms Links
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Symmetry boundary conditions
in modal analysis 97
in thermal analysis 79
T
2-D analysis 13 166
2-D plane strain 20
2-D plane stress 13 15 20
3-D analysis 14f 20
Temperature 6 10 11 77
Tensile stress 88 97
Tetrahedral elements 21 128 153
Thermal analysis 1 77
modeling considerations in 79
steady-state 137
Time response analysis 102
Topological optimization 117 118f
Torsion 68f 88
Training, personnel 133
Translation 13 69
Triangular elements 13 21
U
Uncertainties, relative 65
Units of measurement, selecting 62
V
Vertex, mesh control applied to 49f
Vibration 135
modes of 92Index Terms Links
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Vibration analysis 101
Volume loads 64
Volume, solution domain
represented as 5
W
Weak spots 96
Wire-frame geometry 125
Y
Yield stress 81

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