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 |  موضوع: كتاب The Finite Element Method for Mechanics of Solids with ANSYS Applications الخميس 11 يوليو 2019, 1:17 pm | |
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أخوانى فى الله
أحضرت لكم كتاب
The Finite Element Method for Mechanics of Solids with ANSYS Applications
Ellis H. Dill
Haym Benaroya
Department of Mechanical and Aerospace Engineering
Rutgers University
و المحتوى كما يلي :
Contents
Preface xiii
Author xv
Chapter 1 Finite Element Concepts 1
1.1 Introduction .1
1.2 Direct Stiffness Method 2
1.2.1 Merging the Element Stiffness Matrices 3
1.2.2 Augmenting the Element Stiffness Matrix 5
1.2.3 Stiffness Matrix Is Banded 5
1.3 The Energy Method .5
1.4 Truss Example .7
1.5 Axially Loaded Rod Example . 13
1.5.1 Augmented Matrices for the Rod . 16
1.5.2 Merge of Element Matrices for the Rod . 17
1.6 Force Method . 18
1.7 Other Structural Components 21
1.7.1 Space Truss . 21
1.7.2 Beams and Frames . 21
1.7.2.1 General Beam Equations 24
1.7.3 Plates and Shells .26
1.7.4 Two- or Three-Dimensional Solids 26
1.8 Problems 26
References 28
Bibliography .28
Chapter 2 Linear Elasticity .29
2.1 Basic Equations .29
2.1.1 Geometry of Deformation 29
2.1.2 Balance of Momentum .30
2.1.3 Virtual Work 30
2.1.4 Constitutive Relations . 31
2.1.5 Boundary Conditions and Initial Conditions 33
2.1.6 Incompressible Materials . 33
2.1.7 Plane Strain 34
2.1.8 Plane Stress 34
2.1.9 Tensile Test . 35
2.1.10 Pure Shear 36
2.1.11 Pure Bending 36
2.1.12 Bending and Shearing 37vi Contents
2.1.13 Properties of Solutions .38
2.1.14 A Plane Stress Example with a Singularity in
Stress 40
2.2 Potential Energy 42
2.2.1 Proof of Minimum Potential Energy 44
2.3 Matrix Notation . 45
2.4 Axially Symmetric Deformations .48
2.4.1 Cylindrical Coordinates .48
2.4.2 Axial Symmetry .49
2.4.3 Plane Stress and Plane Strain .50
2.5 Problems 50
References 51
Bibliography . 52
Chapter 3 Finite Element Method for Linear Elasticity . 53
3.1 Finite Element Approximation 54
3.1.1 Potential Energy . 55
3.1.2 Finite Element Equations . 57
3.1.3 Basic Equations in Matrix Notation .58
3.1.4 Basic Equations Using Virtual Work .59
3.1.5 Underestimate of Displacements 60
3.1.6 Nondimensional Equations 61
3.1.7 Uniaxial Stress .63
3.2 General Equations for an Assembly of Elements 66
3.2.1 Generalized Variational Principle 68
3.2.2 Potential Energy .69
3.2.3 Hybrid Displacement Functional 69
3.2.4 Hybrid Stress and Complementary Energy 70
3.2.5 Mixed Methods of Analysis .72
3.3 Nearly Incompressible Materials . 75
3.3.1 Nearly Incompressible Plane Strain . 78
Bibliography .79
Chapter 4 The Triangle and the Tetrahedron 81
4.1 Linear Functions over a Triangular Region . 81
4.2 Triangular Element for Plane Stress and Plane Strain 84
4.3 Plane Quadrilateral from Four Triangles 88
4.3.1 Square Element Formed from Four Triangles 90
4.4 Plane Stress Example: Short Beam .93
4.4.1 Extrapolation of the Solution 96
4.5 Linear Strain Triangles 97
4.6 Four-Node Tetrahedron .98
4.7 Ten-Node Tetrahedron .99
4.8 Problems 99Contents vii
Chapter 5 The Quadrilateral and the Hexahedron 103
5.1 Four-Node Plane Rectangle . 103
5.1.1 Stress Calculations . 109
5.1.2 Plane Stress Example: Pure Bending . 110
5.1.3 Plane Strain Example: Bending with Shear . 112
5.1.4 Plane Stress Example: Short Beam 112
5.2 Improvements to Four-Node Quadrilateral . 115
5.2.1 Wilson–Taylor Quadrilateral 115
5.2.2 Enhanced Strain Formulation 118
5.2.3 Approximate Volumetric Strains . 122
5.2.4 Reduced Integration on the ? Term 125
5.2.5 Reduced Integration on the ? Term 126
5.2.6 Uniform Reduced Integration 127
5.2.7 Example Using Improved Elements . 130
5.3 Numerical Integration . 130
5.4 Coordinate Transformations 133
5.5 Isoparametric Quadrilateral 134
5.5.1 Wilson–Taylor Element 138
5.5.2 Three-Node Triangle as a Special Case of
Rectangle 138
5.6 Eight-Node Quadrilateral 139
5.6.1 Nodal Loads . 144
5.6.2 Plane Stress Example: Pure Bending . 145
5.6.3 Plane Stress Example: Bending with Shear . 145
5.6.4 Plane Stress Example: Short Beam 148
5.6.5 General Quadrilateral Element 148
5.7 Eight-Node Block 149
5.8 Twenty-Node Solid 152
5.9 Singularity Element . 152
5.10 Mixed U–P Elements . 154
5.10.1 Plane Strain 154
5.10.2 Alternative Formulation for Plane Strain . 158
5.10.3 3D Elements . 160
5.11 Problems 163
References 168
Bibliography . 169
Chapter 6 Errors and Convergence of Finite Element Solution 171
6.1 General Remarks . 171
6.2 Element Shape Limits 173
6.2.1 Aspect Ratio . 173
6.2.2 Parallel Deviation for a Quadrilateral 174
6.2.3 Large Corner Angle 175
6.2.4 Jacobian Ratio 175viii Contents
6.3 Patch Test . 176
6.3.1 Wilson–Taylor Quadrilateral 178
References 180
Chapter 7 Heat Conduction in Elastic Solids 181
7.1 Differential Equations and Virtual Work 181
7.2 Example Problem: One-Dimensional Transient Heat Flux . 185
7.3 Example: Hollow Cylinder 187
7.4 Problems 188
Chapter 8 Finite Element Method for Plasticity . 191
8.1 Theory of Plasticity . 191
8.1.1 Tensile Test . 194
8.1.2 Plane Stress 195
8.1.3 Summary of Plasticity 196
8.2 Finite Element Formulation for Plasticity . 197
8.2.1 Fundamental Solution 198
8.2.2 Iteration to Improve the Solution 199
8.3 Example: Short Beam 201
8.4 Problems 203
Bibliography .204
Chapter 9 Viscoelasticity 205
9.1 Theory of Linear Viscoelasticity .205
9.1.1 Recurrence Formula for History 210
9.1.2 Viscoelastic Example . 211
9.2 Finite Element Formulation for Viscoelasticity 215
9.2.1 Basic Step-by-Step Solution Method 216
9.2.2 Step-by-Step Calculation with Load Correction 217
9.2.3 Plane Strain Example . 218
9.3 Problems 219
Bibliography .220
Chapter 10 Dynamic Analyses . 221
10.1 Dynamical Equations 221
10.1.1 Lumped Mass . 221
10.1.2 Consistent Mass .222
10.2 Natural Frequencies 224
10.2.1 Lumped Mass .224
10.2.2 Consistent Mass .225
10.3 Mode Superposition Solution 225
10.4 Example: Axially Loaded Rod 227Contents ix
10.4.1 Exact Solution for Axially Loaded Rod .227
10.4.2 Finite Element Model .229
10.4.2.1 One-Element Model 229
10.4.2.2 Two-Element Model .230
10.4.3 Mode Superposition for Continuum Model of
the Rod . 232
10.5 Example: Short Beam 236
10.6 Dynamic Analysis with Damping 237
10.6.1 Viscoelastic Damping 238
10.6.2 Viscous Body Force 239
10.6.3 Analysis of Damped Motion by Mode
Superposition 240
10.7 Numerical Solution of Differential Equations . 241
10.7.1 Constant Average Acceleration 241
10.7.2 General Newmark Method .243
10.7.3 General Methods 244
10.7.3.1 Implicit Methods in General 244
10.7.3.2 Explicit Methods in General 244
10.7.4 Stability Analysis of Newmark’s Method 245
10.7.5 Convergence, Stability, and Error 246
10.7.6 Example: Numerical Integration for Axially
Loaded Rod 247
10.8 Example: Analysis of Short Beam .249
10.9 Problems 251
Bibliography . 253
Chapter 11 Linear Elastic Fracture Mechanics 255
11.1 Fracture Criterion 255
11.1.1 Analysis of Sheet 257
11.1.2 Fracture Modes .258
11.1.2.1 Mode I .258
11.1.2.2 Mode II .259
11.1.2.3 Mode III .259
11.2 Determination of K by Finite Element Analysis .260
11.2.1 Crack Opening Displacement Method .260
11.3 J-Integral for Plane Regions 263
11.4 Problems 267
References 268
Bibliography .268
Chapter 12 Plates and Shells .269
12.1 Geometry of Deformation .269
12.2 Equations of Equilibrium 270
12.3 Constitutive Relations for an Elastic Material . 271x Contents
12.4 Virtual Work 273
12.5 Finite Element Relations for Bending . 276
12.6 Classical Plate Theory .280
12.7 Plate Bending Example .282
12.8 Problems 287
References 288
Bibliography .289
Chapter 13 Large Deformations 291
13.1 Theory of Large Deformations 291
13.1.1 Virtual Work 292
13.1.2 Elastic Materials .293
13.1.3 Mooney–Rivlin Model of an Incompressible
Material 297
13.1.4 Generalized Mooney–Rivlin Model .298
13.1.5 Polynomial Formula . 301
13.1.6 Ogden’s Function 303
13.1.7 Blatz–Ko Model .304
13.1.8 Logarithmic Strain Measure 306
13.1.9 Yeoh Model 307
13.1.10 Fitting Constitutive Relations to
Experimental Data .308
13.1.10.1 Volumetric Data .308
13.1.10.2 Tensile Test .308
13.1.10.3 Biaxial Test .309
13.2 Finite Elements for Large Displacements .309
13.2.1 Lagrangian Formulation . 311
13.2.2 Basic Step-by-Step Analysis 312
13.2.3 Iteration Procedure . 312
13.2.4 Updated Reference Configuration 313
13.2.5 Example I . 315
13.2.6 Example II 315
13.3 Structure of Tangent Modulus . 317
13.4 Stability and Buckling . 318
13.4.1 Beam–Column . 319
13.5 Snap-Through Buckling 319
13.5.1 Shallow Arch 323
13.6 Problems 324
References 326
Bibliography . 326
Chapter 14 Constraints and Contact . 327
14.1 Application of Constraints . 327
14.1.1 Lagrange Multipliers 327Contents xi
14.1.2 Perturbed Lagrangian Method . 329
14.1.3 Penalty Functions . 331
14.1.4 Augmented Lagrangian Method 332
14.2 Contact Problems . 333
14.2.1 Example: A Truss Contacts a Rigid Foundation 333
14.2.1.1 Load F
y > 0 Is Applied with Fx = 0 . 335
14.2.1.2 Loads Are Ramped Up Together:
Fx
= 27?, F
y = 12.8? 336
14.2.2 Lagrange Multiplier, No Friction Force . 337
14.2.2.1 Stick Condition . 338
14.2.2.2 Slip Condition . 338
14.2.3 Lagrange Multiplier, with Friction . 338
14.2.3.1 Stick Condition . 339
14.2.3.2 Slip Condition . 339
14.2.4 Penalty Method .340
14.2.4.1 Stick Condition . 341
14.2.4.2 Slip Condition . 341
14.3 Finite Element Analysis . 341
14.3.1 Example: Contact of a Cylinder with a Rigid
Plane . 342
14.3.2 Hertz Contact Problem . 343
14.4 Dynamic Impact 346
14.5 Problems 347
References 348
Bibliography .348
Chapter 15 ANSYS APDL Examples .349
15.1 ANSYS Instructions 349
15.1.1 ANSYS File Names 351
15.1.2 Graphic Window Controls 352
15.1.2.1 Graphics Window Logo 352
15.1.2.2 Display of Model 352
15.1.2.3 Display of Deformed and Undeformed
Shape White on White 352
15.1.2.4 Adjusting Graph Colors 352
15.1.2.5 Printing from Windows Version of
ANSYS . 353
15.1.2.6 Some Useful Notes . 353
15.2 ANSYS Elements SURF153, SURF154 353
15.3 Truss Example . 354
15.4 Beam Bending . 357
15.5 Beam with a Distributed Load 360
15.6 One Triangle 361
15.7 Plane Stress Example Using Triangles 364
15.8 Cantilever Beam Modeled as Plane Stress 366xii Contents
15.9 Plane Stress: Pure Bending .369
15.10 Plane Strain Bending Example 371
15.11 Plane Stress Example: Short Beam . 376
15.12 Sheet with a Hole . 379
15.12.1 Solution Procedure . 379
15.13 Plasticity Example . 381
15.14 Viscoelasticity Creep Test .387
15.15 Viscoelasticity Example 391
15.16 Mode Shapes and Frequencies of a Rod 394
15.17 Mode Shapes and Frequencies of a Short Beam .397
15.18 Transient Analysis of Short Beam . 398
15.19 Stress Intensity Factor by Crack Opening Displacement 400
15.20 Stress Intensity Factor by J-Integral 402
15.21 Stretching of a Nonlinear Elastic Sheet .405
15.22 Nonlinear Elasticity: Tensile Test 408
15.23 Column Buckling 412
15.24 Column Post-Buckling 415
15.25 Snap-Through 417
15.26 Plate Bending Example .420
15.27 Clamped Plate 423
15.28 Gravity Load on a Cylindrical Shell .425
15.29 Plate Buckling . 429
15.30 Heated Rectangular Rod . 432
15.31 Heated Cylindrical Rod . 434
15.32 Heated Disk . 438
15.33 Truss Contacting a Rigid Foundation 442
15.34 Compression of a Rubber Cylinder between Rigid Plates .446
15.35 Hertz Contact Problem 451
15.36 Elastic Rod Impacting a Rigid Wall 456
15.37 Curve Fit for Nonlinear Elasticity Using Blatz–Ko Model .460
15.38 Curve Fit for Nonlinear Elasticity Using Polynomial Model 464
Bibliography .469
Chapter 16 ANSYS Workbench 471
16.1 Two- and Three-Dimensional Geometry 471
16.2 Stress Analysis . 472
16.3 Short Beam Example . 473
16.3.1 Short Beam Geometry 473
16.3.2 Short Beam, Static Loading . 474
16.3.3 Short Beam, Transient Analysis . 476
16.4 Filleted Bar Example . 477
16.5 Sheet with a Hole .480
Bibliography .482
Index
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