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 |  موضوع: كتاب Finite Element Analysis of Solids and Structures الإثنين 15 يوليو 2024, 1:47 pm | |
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أخواني في الله
أحضرت لكم كتاب
Finite Element Analysis of Solids and Structures
Sudip S. Bhattacharjee
و المحتوى كما يلي :
Contents
Preface .xi
Author xv
Chapter 1 Introduction to Stress Analysis of Solids and Structures .1
Summary 1
1.1 Introduction – A Brief Summary of Key Historical
Developments 2
1.2 Role of Analysis/Simulation in Product Engineering 3
1.3 Static Equilibrium of Structures and Analysis of Forces –
Statically Determinate and Indeterminate Systems .5
1.4 Stiffness (Displacement) Method of Structural Analysis 7
1.5 Components of Stresses in a 3D Body 11
1.6 Variation of Stresses and Differential Equations of
Equilibrium 13
1.7 Stress Boundary Conditions 15
1.8 Practice Problems on Stress Field Visualization with
CAE Tools .16
Chapter 2 Strain–Displacement Relationship and Elasticity of Materials 19
Summary 19
2.1 Measurement of Deformation Intensity (Strain) .20
2.2 General Description of Deformation State in 3D Solids .22
2.3 Compatibility of Strain (Deformation Field) in a Body 23
2.4 Stress-Strain Relationships (Hooke’s Law) .23
2.5 Solution of Elasticity Problems Using Stress Distribution
Functions .28
2.6 Solution of Elasticity Problems Using Displacement
Variation Functions 30
2.7 Stiffness Method (Finite Element Method) of Structural
Analysis .32
2.8 Stiffness Properties of 1-D Truss Element Providing
Resistance to Axial Deformation Only 34
2.9 Higher Order Truss Element and Model Refinement 39
2.10 Model Preparation for Computer-Aided Analysis of
Structures with Finite Element Simulation Software 41
2.11 Practice Problems – Stress Analysis of a Cable with Finite
Element Analysis Software ABAQUS .43vi Contents
Chapter 3 Analysis of Solids Represented by 2D Stress Fields 47
Summary 47
3.1 Plane Strain – A Special Form of Elasticity Problem 48
3.2 Plane Stress – A Special Form of Elasticity Problem 50
3.3 Stress Functions for 2D Plane Strain and Plane Stress
Elasticity Problems 51
3.4 Example Use of Stress Function Approach to Solve
Elasticity Problems 54
3.5 Stiffness Method Analysis of Solids Represented by
2D Stress-Deformation Fields .60
3.6 Iso-Parametric Definition of 2D
Solid Finite Elements 63
3.7 Numerical Calculation of Stiffness Matrix for
Iso-Parametric 2D Solid Finite Elements 67
3.8 Higher Order Plane-Stress/Plane-Strain Elements 71
3.9 Constant Stress/Strain Triangular Element 73
3.10 Convergence of Finite Element Model Solutions 76
3.11 Selection of Element Types for 2D Stress Analysis
with FEA Software Packages 78
3.12 Description of Load Types in General Stress Analysis
Problems 82
3.13 Refined Finite Element Model Preparation with a
Pre-Processor (HyperMesh) 83
3.14 Practice Problems – Stress Analysis of a Plate Subjected
to In-Plane Stresses 84
Chapter 4 FEA Model Preparation and Quality Checks .85
Summary 85
4.1 Adaptation of Finite Element Mesh to Stress Field
Variations .85
4.2 Element Type Selection for a Given Stress Analysis
Problem 88
4.3 Initial Geometric Design of Structural Components .91
4.4 FEA Model Preparation for Case-Study: Plate with
Hole (by Using HyperMesh) .92
4.4.1 CAD Data Preparation for FEA Modeling .92
4.4.2 Finite Element Meshing of the CAD Geometry .94
4.4.3 Finite Element Mesh Quality Checks .95
4.4.4 Material and Part Property Assignment 98
4.4.5 Analysis Parameters – Boundary Constraints,
External Loads and Model Outputs 99
4.4.6 Definition of ABAQUS Analysis Step 101
4.4.7 Exporting the Model Input File for ABAQUS
Analysis 101Contents vii
4.5 Post-Processing of FEA Results 103
4.6 Practice Problems – Stress Analysis of Plate with Hole 103
Chapter 5 Stress Analysis of Axisymmetric and General 3D Solids 107
Summary 107
5.1 Axisymmetric – A Special Form of 3D Elasticity
Problems 107
5.1.1 Equations of Equilibrium in Polar Coordinates 109
5.1.2 Strain–Displacement Relationships 109
5.1.3 Stress–Strain Relationships 110
5.1.4 Compatibility Condition .111
5.2 Stress Analysis of Axisymmetric Example – Thick
Wall Cylinder .111
5.3 Finite Element Analysis of Axisymmetric Problems .115
5.4 Stress Analysis of Three-dimensional Bodies –
Concentrated Loads .117
5.4.1 Stresses in a Semi-Infinite Solid Subjected to
Concentrated Normal Force on the Boundary 117
5.4.2 Stresses in a Solid Beam Subject to
Concentrated Lateral Forces .119
5.5 Finite Elements for Stress Analysis of General 3D Solids 121
5.6 Three-Dimensional FEA Model Preparation and Element
Quality Checks 123
5.7 Practice Problems – Stress Analysis of Axisymmetric
and 3D Solids 124
Chapter 6 Deformation Analysis of Beams for Axial, Bending, Shear, and
Torsional Loads 127
Summary 127
6.1 Bending Stresses in a Beam 128
6.2 Stresses Due to Transverse Shear 129
6.3 Transverse Normal Stress in a Beam .132
6.4 Torsional Response of a Beam 133
6.5 Beam Response to Combined Load Effects 142
6.6 Elastic Bending Deflection of Beams 144
6.7 Stress Analysis of Curved Beams 146
6.8 Stiffness Properties of Prismatic Euler–Bernoulli
Beam Elements 148
6.9 Stiffness Properties of Beams Including Shear
Deformation .154
6.10 Analysis of Beams and Frames with FEA Software
Packages 158
6.11 Practice Problems: Load–Deflection Analysis of Beams 162viii Contents
Chapter 7 Analysis of 3D Thin-Wall Structures (Plates and Shells) 165
Summary 165
7.1 Bending Stresses and Strains in a Plate .166
7.2 Analytical Solutions for Plate Bending Deflections 169
7.3 In-plane Membrane Stress Resistance of a Shell 172
7.4 Bending Stiffness of Flat Plate Element 174
7.5 Flat Shell Element as a Combination of Plate Bending
and Membrane Elements .176
7.6 Shear Deformation in Plates 179
7.7 Curved Shell Elements 181
7.8 Shell Element Mesh Quality and Integration Rules 184
7.9 Analysis of Shells with FEA Software 186
7.10 Practice Problems: Load–Deflection Analysis of Shells .187
Chapter 8 Multi-Component Model Assembly .191
Summary 191
8.1 Element Compatibility and Convergence of Simulation
Results .191
8.2 Modeling of Kinematic Joints in
Structural Assemblies 194
8.3 Deformable Joint Elements for Part-to-Part Connections .198
8.4 Mesh-Independent Fasteners for Part-to-Part Connections .201
8.5 Simulation of Part-to-Part Interface Contacts 203
8.6 Thin-Layer Interface Elements 207
8.7 Modular Organization of Data in Multi-component
Model Assembly 209
8.8 Result Quality Checks .211
8.9 Practice Problems: Analysis with Multi-Component
Model Assemblies .213
Chapter 9 Interpretation of Stress Analysis Results for Strength and
Durability Assessment 217
Summary 217
9.1 Engineering Properties of Materials 218
9.2 Stress-Strain Results from Linear Elastic Finite Element
Analysis of Solids 219
9.3 Strength Assessment of Solids – Use of Material Failure
Theories .222
9.4 Post-Processing of Finite Element Stress Analysis Results 229
9.5 Stress Analysis for Durability (Fatigue Life) Assessment
of Structures .230Contents ix
9.6 Structural Safety Assessment and Quality of FEA
Stress Results .234
9.7 Stresses at Points of Discontinuity:
Stress Intensity Factor .235
9.8 Practice Problems: Assessment of Structural Strength
and Durability 238
Chapter 10 Vibration Frequency Analysis of Structures with
FEA Model .241
Summary 241
10.1 Introduction – Dynamic Response of Structures .242
10.2 Vibration Frequency of a Single Degree of Freedom
Spring-Mass System 243
10.3 Forced Vibration Response and Resonance of Structures .245
10.4 Frequency Separation and Design Targets for Structures 249
10.5 Vibration Mode Shape and Frequency of
SDOF Structures 250
10.6 Vibration Frequencies of MDOF Systems 253
10.7 Calculation of System Mass Matrix for MDOF Systems 255
10.8 Numerical Calculation of Vibration Mode Shapes
and Frequencies of MDOF Systems 259
10.9 Vibration Frequency Analysis with ABAQUS .261
10.10 Practice Problems: Vibration Analysis of Structures .262
Chapter 11 Linear Dynamic Response Analysis of Structures .265
Summary 265
11.1 Linear Elastic Response of SDOF Systems to
Impulsive Loading .265
11.2 Response Spectrum of Linear Dynamic Systems 268
11.3 Time-Domain Analysis of Dynamic Structural
Response 270
11.4 Numerical Integration Parameters for Time-Domain
Analysis of Structures 271
11.5 Time Domain Analysis of MDOF Systems .273
11.6 Mode Superposition Method for Analysis of MDOF
Systems 276
11.7 Explicit Time-Domain Analysis of MDOF Systems .277
11.8 Linear Dynamic Response Analysis with ABAQUS .282
11.9 Practice Problems: Dynamic Response Analysis of
Structures 286x Contents
Chapter 12 Nonlinear Analysis of Structures 289
Summary 289
12.1 Simulation of Nonlinear Force-Deformation Response
of Structures .290
12.2 Nonlinear Material Models for Finite Element
Simulation of Structures 294
12.3 Simulation of Large Deformation Response –
Nonlinear Geometric Problems .301
12.4 Nonlinearity Arising from Changes to Inter-Body
Contacts .307
12.5 Nonlinear Dynamic Response Analysis of Structures .308
12.6 Material Failure Simulation in Nonlinear
Finite Element Analysis .310
12.7 Computational Methods for Structural Form
Simulation 311
12.8 Practice Problems: Nonlinear Response Analysis of
Structures .313
References .315
Index 319xi
Index
A
analysis of forces
statically determinate, 5
statically indeterminate, 5
aspect ratio, 96
average acceleration method, of dynamic
response analysis, 272
axisymmetric problems, 115
B
bending deflection of beams, 144
bending stiffness
of composite shell, 179
of plate, 174
bending stresses
in beams, 128
in dissimilar materials, 129
in plate, 166
bending theory, see Euler–Bernoulli
body force, 13, 33
boundary constraints, 9, 34, 99
buckling, thin-wall members, 4
bulk modulus of elasticity, 28
C
CAD (computer-aided-design) tools, 16
CAD data exchange formats, 91
CAE (computer aided engineering) tools, 16
central difference theorem of explicit
time-domain solution, 278
CFL law (Courant–Friedrichs–Lewy) of time
step limit in explicit dynamic
analysis, 279
cohesive joint element, 207
combined load effects on beam response, 142
compatibility condition
among strains, 23
plane strain, 50
plane stress, 51
in the polar coordinate, 111
completeness requirement, 191
concentrated forces/loads, 33, 45, 117, 119
connector element, 195, 198, 203
constant stress/strain, 73, 122
constitutive models, see stress–strain
relationship
contact force, 204
contact simulation, 204
contact stiffness, 204
convergence of finite element model solutions,
76, 191
crack-tip, 235
crash strength, 314
crashworthiness simulation, 258, 312
curvature of plate bending, 175
curved beams, 146
cumulative fatigue damage, 231
cut-plane, 212
cyclic load, 231
D
damping matrix, 274
damping ratio, 247
damping resistance, 247
deep beam, 146
deformation
axial, 10
bending, 88
distortional, 20
shear, 20
torsional, 134
degree of freedom
master, 195
multi-degree of freedom (MDOF) system, 9
single degree of freedom (SDOF) spring, 7
slave, 195
direction cosines, 137, 148, 181
displacement
control method of structural analysis, 293
field, 32, 34
method of structural analysis, 7
variation functions, 30
virtual, 32
distortional stress, 227
distributed loads, 45
ductile material, 3, 219
Duhamel integral, 266
durability, 230
dynamic amplification factor
damped system, 247
to impact force, 267
un-damped system, 246
dynamic equilibrium
MDOF, 273
SDOF, 247, 270
dynamic response
nonlinear, 308
of flexible spring-mass, 243, 271
of free rigid body mass, 242
to impact force, 266320 Index
dynamic response analysis
average acceleration method, 272
modal superposition method, 276
time domain analysis, 270
E
earthquake load, 249
eigenpair, 255
eigenvalue, 254
eigenvalue shift, 259
eigenvector, 254
elasticity property
matrix, 27, 294
tangential, 296
elastic modulus, 24, 219
element characteristic length, for explicit
dynamic analysis, 281
element compatibility, 191
element types
axisymmetric, 117
beam element, 154, 159
mass, 261
mixed, 192
shell elements, 186
3D elements, 122
truss elements, 34, 39
2D solids, 78, 88, 94
energy error, 102
engineering strain, 24, 218
engineering stress, 24, 218
equations of
dynamic equilibrium, 274
equilibrium in polar coordinates, 109
static equilibrium, 7, 11
strain compatibility, 23
stress compatibility, 28
stress equilibrium, 14
stress–strain relationship, 23
Euler–Bernoulli beam theory, 128, 146, 148,
151, 154, 160, 251
F
fatigue crack growth, 237
fatigue life, 230
FEA solvers, 16
finite elements
axisymmetric, 117
beam elements, 154, 160
plane-strain, 80
plane-stress, 78
shell, 176, 181, 186
3D solid, 122
truss, 34, 42
first moment of the beam section, 130
flexibility method of structural analysis, 6
flexural rigidity, 144, 169
force method pf structural analysis, 6
form-finding, 312
fracture modes, 236
fracture response simulation, 310
fracture toughness, 236
free-vibration response, 243
frequency response function, 284
frequency separation, 249
fundamental frequency of vibration, 252
G
gage length, 20
Gauss quadrature, 69
Green-Lagrange strains, 301
Guyan reduction, 259
H
hardening plasticity, 295
Hooke’s law, 23, 110, 168, 294
hourglass deformation, 71, 90, 184
householder-QR method, 260
high cycle fatigue, 230
hinge connection, 200
hydrostatic pressure, 28
hysteresis, 248
I
implicit time integration, 272
impulse, 266
in-compatible element deformation, 73, 80, 193
initial penetration, 206
interface contact, 203
interface element, 207
interpolation functions
beam deformation, 151
coordinate, 34
displacement, 36
general 3D solids, 122
3-node truss elements, 40
2D deformation fields, 60
2-nodetruss element, 35
intersection, 206
J
Jacobian
definition of, 66
determinant, 96
distorted 2D solid element, 66, 70
inverse of, 68
physical interpretation, 65
3D system, 122
triangular elements, 74Index 321
J-integral, 236
Johnson–Cook plasticity model, 295, 299, 309
joint elements, deformable, 198
K
kinematic constraint, 194
kinematic coupling, 196, 201
kinematic joints, 194
kinematic relations, 270
Kronecker delta, 254
L
Lagrangian element, 72, 80
Lagrangian method of nonlinear
structural analysis
total, 305
updated, 306
Lagrangian multiplier, 195, 204
Lame’s constants, 28
Lame’s equations, 13
Lanczos transformation method, 260
large deformation response, 301
linear acceleration method, of dynamic
response analysis, 272
load-deformation response
monotonic, 293
snap-back, 293
load types, 82
low cycle fatigue, 230
M
mass matrix
calculation, 255
consistent, 256
lumped, 257
master–slave concept, 204
master surface, 204
material failure simulation, 310
material failure theories, 222
matrix equilibrium equations, 9
measurement of deformation intensity, 20
mechanics of materials, 2
membrane analogy, 138
membrane element, 183
membrane stress resistance of shell, 172
mesh-dependent localization of failure, 311
mesh-independent fasteners, 201
mesh quality checks, 95, 123, 184
mesh refinement
effect on predicted results, 76
h-adaptivity, 86
p-adaptivity, 86
selective, 90
Mindlin’s plate deformation theory, 179
Miner’s rule, 231
Modak’s T-method, for dynamic response
analysis, 272
modal damping, 274, 277
modal superposition method of dynamic response
analysis, 269, 276
model preparation
a-to-g key process steps, 102
axisymmetric problem, 116
for contact simulation, 206
multi-component, 209
plate with hole, 92
three-dimensional FEA, 123
modular organization of data, 209
Mohr circle of stress, 226
moment-curvature relationship of plates, 176
moment of inertia, 153
moment release, 194
momentum, 242
MPC (multi-point constraint), 195
multi-component model, 209
N
neutral axis, 10, 56, 128, 130, 147
Newmark integration, 272
Newton–Raphson method, 291
Newton’s second law, 242
nodes, 33
nonlinear analysis
contact problems, 307
dynamic response, 308
force-deformation response, 290
geometric problems, 301, 306
material models, 294
numerical damping, in dynamic response
analysis, 272
numerical integration
examples, 68
Gauss quadrature, 69
of higher order elements, 72
of mass matrix, 256
parameters for time-domain dynamic
response analysis, 271
reduced, 90
of stiffness function, 70, 122
P
parasitic shear, 89
Paris law, 237
part property, 98
part-to-part connections, 198, 201
part-to-part contact, 203
patch test, 76
penalty method, 204
penalty stiffness, 204322 Index
period of vibration, 244
perturbation of stress flow, 86
Piola–Kirchoff stresses, 305
plane strain, 48
plasticity models
associated, 298
non-associated, 298
plastic potential function, 298
plastic strain, 295
plate bending theory
Kirchoff’s, 166
Mindlin’s, 179
Poisson’s ratio, 25
polar coordinates, 107
polar moment of inertia, 141
post-processors, 16
pre-processors, 16, 83
principal strain, 225
principal stress, 223
principle of superposition, of stresses, 25
prismatic member, 148
product development, automotive, 4
R
radius of curvature, 147
Rayleigh damping coefficient, 274
Rayleigh’s quotient, 255
reference node, 196
resonance, 245
response spectrum, 268
result quality checks, 77, 103, 211, 234
rigid body, 196
rigid body motion, 9, 191
S
SDOF (single degree of freedom) system, 7
self-contact, 206
semi-infinite solid, 119
shape functions
beam deflection, 151
general 3D solids, 121
higher order elements, 71
iso-parametric definition, 64
2D deformation fields, 60
shear center, 143
shear correction factor, 146, 156, 158, 183
shear deformation
of cantilever beam, 59
in plates, 179
shear flow through beam, 143
shear locking of beam, 158
shear modulus, 27
shear stress distribution
in beam, 131
in plate, 168
simulation model limitations, 4
singular element, 236
size effect, 220
skeletal
beam, 10, 148
structural members, 10
skew, 96
slave node, 204
slender beam, 154, 158
S–N curve, 231
softening response of structures, 293
software
CAD, 16, 91–94, 102, 123, 186, 214
CAE tools, 16
FEA solvers, 16, 211, 212
finite element simulation, 78, 83
post-processor, 16
pre-processor, 16, 48, 83, 84, 90, 92, 93,
95, 98
spot-welds, 197, 233
steady state dynamics, 285
stiffness matrix
of beam element, 153
of finite element, 33
of flat shell element, 180
fully integrated, 70
of general 3D solids, 122
membrane resistance, 177
positive definite, 9, 40
reduced integration, 71
spring elements, 7
system assembly, 8, 34, 39
tangent, 292, 305
of 2-node truss element, 37
stiffness method of structural analysis, 7, 32, 60
stiffness property
beam shear deformation, 156
calculation of finite elements, 32
Euler–Bernoulli beam element, 148
flat shell element, 177
joint element, 200
spring, 7
truss element, 34
strain
bi-directional, 21
compatibility, 23
components, 22
energy, 9, 34, 219
energy release rate, 237
failure, 3
Green-Lagrange, 301
hardening parameter, 296
measurement of, 20
normal, 20
plastic, 295
rosette, 21
shear, 20Index 323
thermal, 83
virtual, 32
strain–displacement relationship
axisymmetric problem, 115
beam element, 152
in differential element, 22
general 3D solid, 121
Green-Lagrange, 301
iso-parametric 2D solid, 67
nonlinear, 302
plane-strain, 48, 61
plane-stress, 50
plate bending, 167
in polar coordinates, 110
shell element, 181
two-node truss element, 36
triangular element, 76
stress
at the crack tip, 235
beam section, 10
boundary conditions, 15, 50
components in 3D solids, 11
concentration factor, 86, 234
equilibrium equations, 14, 49
initial, 33
intensity factor, 235
normal, 12, 129
Piola–Kirchoff, 305
shear, 12
singularity, 78, 88
tangential, 114
thermal, 83
torsional, 134
transverse shear, 129
vector, 13
stress function
in polar coordinates, 109
prismatic bar, 29
thin cantilever beam, 54
torsional problem, 136
for 2D stress field, 52, 84
stress flow
in a plate with hole, 86
torsion, 139
stress–strain relationship
axisymmetric, 110, 116
for bending deformation of beam, 152
for curved shell elements, 182
incremental, 299
nonlinear, 295
plane-strain, 49, 63
plane-stress, 51, 63
for plate bending, 168
strain-rate effect, 309
truss element, 36
structural adhesive, 233
structural analysis, step-by-step load control, 293
structural form simulation, 311
structural instability, 40
strum sequence check, 261
stamping of metal blank, 311
stress wave propagation, 280
subspace iteration method, 260
surface force, 33
T
tangent modulus, 297
Taylor’s series
for strain definition, 13
for time domain response, 272
thin-layer joint, 207
thin shell, 179
time domain analysis
error in predicted response, 273, 279
explicit method, 277
implicit method, 272
of MDOF system, 273
of SDOF system, 271
time step limit, 273, 276, 279
Timoshenko beam theory, 146, 156
torsional response of beam, 133
torsional rigidity, 135
torsional stiffness of member, 141
transformation matrix
of beam element stiffness matrix, 148
of global and local DOF of truss
element, 37
transverse normal stress in beam, 132
true stress, 24, 219
true strain, 24, 219
U
ultimate strength analysis of beam, 291
V
variable amplitude stress cycle, 232
vector of
applied loads, 11, 33
nodal displacements, 11, 33
vector iteration method, 260
velocity of stress wave, 280, 282
vibration frequency
MDOF, 253
numerical calculation, 259
SDOF, 244
vibration mode shape
MDOF, 259
numerical calculation, 259
SDOF, 250
simply supported beam, 250
simply supported plate, 252324 Index
virtual
cut-plane, 212
displacement, 32
work, 32
von Mises stress, 228, 297
W
warpage, 96
warpage-induced deformation, 135
Wilson θ method of dynamic response
analysis, 272
Winker formula, 147
Y
yield function, 297
yield strength, 3, 219
Young’s modulus, 24
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