Preface to the Second Edition .................................. xv
Preface ...................................................... xvii
About the Author .............................................. xix
1 Vectors, Tensors, and Equations of Elasticity ................ 1
1.1 Introduction ............................................ 1
1.2 Vectors, Tensors, and Matrices .......................... 2
1.2.1 Preliminary Comments ............................. 2
1.2.2 Components of Vectors and Tensors ................ 2
1.2.3 Summation Convention ............................. 3
1.2.4 The Del Operator ................................. 5
1.2.5 Matrices and Cramer's Rule ...................... 11
1.2.6 Transformations of Components ................... 14
1.3 Equations of Elasticity ................................ 18
1.3.1 Introduction .................................... 18
1.3.2 Kinematics ...................................... 18
1.3.3 Compatibility Equations ......................... 21
1.3.4 Stress Measures ................................. 23
1.3.5 Equations of Motion ............................. 25
1.3.6 Constitutive Equations .......................... 28
1.4 Transformation of Stresses, Strains, and Stiffnesses ... 32
1.4.1 Introduction .................................... 32
1.4.2 Transformation of Stress Components ............. 32
1.4.3 Transformation of Strain Components ............. 33
1.4.4 Transformation of Material Stiffnesses .......... 34
1.5 Summary ................................................ 35
Problems .................................................... 35
2 Energy Principles and Variational Methods ................... 39
2.1 Virtual Work ........................................... 39
2.1.1 Introduction .................................... 39
2.1.2 Virtual Displacements and Forces ................ 40
2.1.3 External and Internal Virtual Work .............. 42
2.1.4 The Variational Operator ........................ 46
2.1.5 Functionals ..................................... 47
2.1.1 Fundamental Lemma of Variational Calculus ....... 48
2.1.7 Euler-Lagrange Equations ........................ 49
2.2 Energy Principles ...................................... 51
2.2.1 Introduction .................................... 51
2.2.2 The Principle of Virtual Displacements .......... 52
2.2.3 Hamilton's Principle ............................ 55
2.2.4 The Principle of Minimum Total Potential
Energy .......................................... 58
2.3 Castigliano's Theorems ................................. 61
2.3.1 Theorem I ....................................... 61
2.3.2 Theorem II ...................................... 66
2.4 Variational Methods .................................... 68
2.4.1 Introduction .................................... 68
2.4.2 The Ritz Method ................................. 69
2.4.3 The Galerkin Method ............................. 83
2.5 Summary ................................................ 87
Problems .................................................... 87
3 Classical Theory of Plates .................................. 95
3.1 Introduction ........................................... 95
3.2 Assumptions of the Theory .............................. 96
3.3 Displacement Field and Strains ......................... 97
3.4 Equations of Motion ................................... 100
3.5 Boundary and Initial Conditions ....................... 105
3.6 Plate Stiffness Coefficients .......................... 110
3.7 Stiffness Coefficients of Orthotropic Plates .......... 115
3.8 Equations of Motion in Terms of Displacements ......... 118
3.9 Summary ............................................... 121
Problems ................................................... 121
4 Analysis of Plate Strips ................................... 125
4.1 Introduction .......................................... 125
4.2 Governing Equations ................................... 126
4.3 Bending Analysis ...................................... 126
4.3.1 General Solution ............................... 126
4.3.2 Simply Supported Plates ........................ 127
4.3.3 Clamped Plates ................................. 128
4.3.4 Plate Strips on Elastic Foundation ............. 129
4.4 Buckling under Inplane Compressive Load ............... 130
4.4.1 Introduction ................................... 130
4.4.2 Simply Supported Plate Strips .................. 132
4.4.3 Clamped Plate Strips ........................... 133
4.4.4 Other Boundary Conditions ...................... 134
4.5 Free Vibration ........................................ 135
4.5.1 General Formulation ............................ 135
4.5.2 Simply Supported Plate Strips .................. 138
4.5.3 Clamped Plate Strips ........................... 139
4.6 Transient Analysis ..................................... 140
4.6.1 Preliminary Comments ........................... 140
4.6.2 The Navier Solution ............................ 140
4.6.3 The Ritz Solution .............................. 142
4.6.4 Transient Response ............................. 143
4.6.5 Laplace Transform Method ....................... 144
4.7 Summary ................................................ 146
Problems ................................................... 146
5 Analysis of Circular Plates ................................ 149
5.1 Introduction .......................................... 149
5.2 Governing Equations ................................... 149
5.2.1 Transformation of Equations from Rectangular
Coordinates to Polar Coordinates ............... 149
5.2.2 Derivation of Equations Using Hamilton's
Principle ...................................... 153
5.2.3 Plate Constitutive Equations ................... 158
5.3 Axisymmetric Bending .................................. 160
5.3.1 Governing Equations ............................ 160
5.3.2 Analytical Solutions ........................... 162
5.3.3 The Ritz Formulation ........................... 166
5.3.4 Simply Supported Circular Plate under
Distributed Load ............................... 167
5.3.5 Simply Supported Circular Plate under Central
Point Load ..................................... 171
5.3.6 Annular Plate with Simply Supported Outer
Edge ........................................... 174
5.3.7 Clamped Circular Plate under Distributed
Load ........................................... 178
5.3.8 Clamped Circular Plate under Central Point
Load ........................................... 179
5.3.9 Annular Plates with Clamped Outer Edges ........ 181
5.3.10 Circular Plates on Elastic Foundation .......... 185
5.3.11 Bending of Circular Plates under Thermal
Loads .......................................... 188
5.4 Asymmetrical Bending .................................. 189
5.4.1 General Solution ............................... 189
5.4.2 General Solution of Circular Plates under
Linearly Varying Asymmetric Loading ............ 190
5.4.3 Clamped Plate under Asymmetric Loading ......... 192
5.4.4 Simply Supported Plate under Asymmetric
Loading ........................................ 193
5.4.5 Circular Plates under Noncentral Point Load .... 194
5.4.6 The Ritz Solutions ............................. 196
5.5 Free Vibration ........................................ 200
5.5.1 Introduction ................................... 200
5.5.2 General Analytical Solution .................... 201
5.5.3 Clamped Circular Plates ........................ 202
5.5.4 Simply Supported Circular Plates ............... 204
5.5.5 The Ritz Solutions ............................. 204
5.6 Axisymmetric Buckling ................................. 207
5.6.1 Governing Equations ............................ 207
5.6.2 General Solution ............................... 208
5.6.3 Clamped Plates ................................. 209
5.6.4 Simply Supported Plates ........................ 209
5.6.5 Simply Supported Plates with Rotational
Restraint ...................................... 210
5.7 Summary ............................................... 211
Problems ................................................... 212
6 Bending of Simply Supported Rectangular Plates ............. 215
6.1 Introduction .......................................... 215
6.1.1 Governing Equations ............................ 215
6.1.2 Boundary Conditions ............................ 216
6.2 Navier Solutions ...................................... 217
6.2.1 Solution Procedure ............................. 217
6.2.2 Calculation of Bending Moments, Shear Forces,
and Stresses ................................... 221
6.2.3 Sinusoidally Loaded Plates ..................... 225
6.2.4 Plates with Distributed and Point Loads ........ 228
6.2.5 Plates with Thermal Loads ...................... 233
6.3 Levy's Solutions ...................................... 236
6.3.1 Solution Procedure ............................. 236
6.3.2 Analytical Solution ............................ 237
6.3.3 Plates under Distributed Transverse Loads ...... 242
6.3.4 Plates with Distributed Edge Moments ........... 246
6.3.5 An Alternate Form of the Lévy Solution ......... 249
6.3.6 The Ritz Solutions ............................. 254
6.4 Summary ............................................... 258
Problems ................................................... 259
7 Bending of Rectangular Plates with Various Boundary
Conditions ................................................. 263
7.1 Introduction .......................................... 263
7.2 Lévy Solutions ........................................ 263
7.2.1 Basic Equations ................................ 263
7.2.2 Plates with Edges x = 0, α Clamped (CCSS) ...... 266
7.2.3 Plates with Edge x = 0 Clamped and Edge x = α
Simply Supported (CSSS) ........................ 273
7.2.4 Plates with Edge x = 0 Clamped and Edge x = α
Free (CFSS) .................................... 275
7.2.5 Plates with Edge x = 0 Simply Supported and
Edge x = α Free (SFSS) ......................... 277
7.2.6 Solution by the Method of Superposition ........ 280
7.3 Approximate Solutions by the Ritz Method .............. 283
7.3.1 Analysis of the Lévy Plates .................... 283
7.3.2 Formulation for General Plates ................. 287
7.3.3 Clamped Plates (CCCC) .......................... 291
7.4 Summary ............................................... 293
Problems ................................................... 293
8 General Buckling of Rectangular Plates ..................... 299
8.1 Buckling of Simply Supported Plates under
Compressive Loads ..................................... 299
8.1.1 Governing Equations ............................ 299
8.1.2 The Navier Solution ............................ 300
8.1.3 Biaxial Compression of a Plate ................. 301
8.1.4 Biaxial Loading of a Plate ..................... 302
8.1.5 Uniaxial Compression of a Rectangular Plate .... 303
8.2 Buckling of Plates Simply Supported along Two
Opposite Sides and Compressed in the Direction
Perpendicular to Those Sides .......................... 309
8.2.1 The Lévy Solution .............................. 309
8.2.2 Buckling of SSSF Plates ........................ 310
8.2.3 Buckling of SSCF Plates ........................ 312
8.2.4 Buckling of SSCC Plates ........................ 315
8.3 Buckling of Rectangular Plates Using the Ritz
Method ................................................ 317
8.3.1 Analysis of the Lévy Plates .................... 317
8.3.2 General Formulation ............................ 319
8.3.3 Buckling of a Simply Supported Plate under
Combined Bending and Compression ............... 321
8.3.4 Buckling of a Simply Supported Plate under
Inplane Shear .................................. 324
8.3.5 Buckling of Clamped Plates under Inplane
Shear .......................................... 326
8.4 Summary ............................................... 328
Problems ................................................... 328
9 Dynamic Analysis of Rectangular Plates ..................... 331
9.1 Introduction .......................................... 331
9.1.1 Governing Equations ............................ 331
9.1.2 Natural Vibration .............................. 331
9.1.3 Transient Analysis ............................. 332
9.2 Natural Vibration of Simply Supported Plates .......... 332
9.3 Natural Vibration of Plates with Two Parallel Sides
Simply Supported ...................................... 334
9.3.1 The Lévy Solution .............................. 334
9.3.2 Analytical Solution ............................ 335
9.3.3 Vibration of SSSF Plates ....................... 336
9.3.4 Vibration of SSCF Plates ....................... 338
9.3.5 Vibration of SSCC Plates ....................... 340
9.3.6 Vibration of SSCS Plates ....................... 341
9.3.7 Vibration of SSFF Plates ....................... 342
9.3.8 The Ritz Solutions ............................. 345
9.4 Natural Vibration of Plates with General Boundary
Conditions ............................................ 346
9.4.1 The Ritz Solution .............................. 346
9.4.2 Simply Supported Plates (SSSS) ................. 347
9.4.3 Clamped Plates (CCCC) .......................... 348
9.4.4 CCCS Plates .................................... 350
9.4.5 CSCS Plates .................................... 351
9.4.6 CFCF, CCFF, and CFFF Plates .................... 351
9.5 Transient Analysis .................................... 354
9.5.1 Spatial Variation of the Solution .............. 354
9.5.2 Time Integration ............................... 355
9.6 Summary ............................................... 356
Problems ................................................... 356
10 Shear Deformation Plate Theories ........................... 359
10.1 First-Order Shear Deformation Plate Theory ............ 359
10.1.1 Preliminary Comments ........................... 359
10.1.2 Kinematics ..................................... 359
10.1.3 Equations of Motion ............................ 361
10.1.4 Plate Constitutive Equations ................... 365
10.1.5 Equations of Motion in Terms of
Displacements .................................. 365
10.2 The Navier Solutions of FSDT .......................... 366
10.2.1 General Solution ............................... 366
10.2.2 Bending Analysis ............................... 368
10.2.3 Buckling Analysis .............................. 372
10.2.4 Natural Vibration .............................. 374
10.3 The Third-Order Plate Theory .......................... 376
10.3.1 General Comments ............................... 376
10.3.2 Displacement Field ............................. 376
10.3.3 Strains and Stresses ........................... 378
10.3.4 Equations of Motion ............................ 379
10.4 The Navier Solutions of TSDT .......................... 383
10.4.1 Preliminary Comments ........................... 383
10.4.2 General Solution ............................... 383
10.4.3 Bending Analysis ............................... 385
10.4.4 Buckling Analysis .............................. 387
10.4.5 Natural Vibration .............................. 389
10.5 Relationships between Solutions of Classical and
Shear Deformation Theories ............................ 390
10.5.1 Introduction ................................... 390
10.5.2 Circular Plates ................................ 391
10.5.3 Polygonal Plates ............................... 394
10.6 Summary ............................................... 399
Problems ................................................... 399
11 Theory and Analysis of Shells .............................. 403
11.1 Introduction .......................................... 403
11.1.1 Preliminary Comments ........................... 403
11.1.2 Classification of Shell Surfaces ............... 405
11.2 Governing Equations ................................... 407
11.2.1 Geometric Properties of the Shell .............. 407
11.2.2 General Strain-Displacement Relations .......... 413
11.2.3 Stress Resultants .............................. 414
11.2.4 Displacement Field and Strains ................. 415
11.2.5 Equations of Motion of a General Shell ......... 419
11.2.6 Equations of Motion of Thin Shells ............. 424
11.2.7 Constitutive Equations of a General Shell ...... 425
11.2.8 Constitutive Equations of Thin Shells .......... 428
11.3 Analytical Solutions of Thin Cylindrical Shells ....... 430
11.3.1 Introduction ................................... 430
11.3.2 Membrane Theory ................................ 431
11.3.3 Flexural Theory for Axisymmetric Loads ......... 436
11.4 Analytical Solutions of Shells with Double
Curvature ............................................. 441
11.4.1 Introduction and Geometry ...................... 441
11.4.2 Equations of Equilibrium ....................... 442
11.4.3 Membrane Stresses in Symmetrically Loaded
Shells ......................................... 444
11.4.4 Membrane Stresses in Unsymmetrically Loaded
Shells ......................................... 451
11.4.5 Bending Stresses in Spherical Shells ........... 458
11.5 Vibration and Buckling of Circular Cylinders .......... 464
11.5.1 Equations of Motion ............................ 464
11.5.2 Governing Equations in Terms of
Displacements .................................. 465
11.5.3 The Lévy Solution .............................. 466
11.5.4 Boundary Conditions ............................ 471
11.5.5 Numerical Results .............................. 472
11.6 Summary ............................................... 473
Problems ................................................... 473
12 Finite Element Analysis of Plates .......................... 479
12.1 Introduction .......................................... 479
12.2 Finite Element Models of CPT .......................... 480
12.2.1 Introduction ................................... 480
12.2.2 General Formulation ............................ 481
12.2.3 Plate-Bending Elements ......................... 483
12.2.4 Fully Discretized Finite Element Models ........ 486
12.2.5 Numerical Results .............................. 488
12.3 Finite Element Models of FSDT ......................... 491
12.3.1 Virtual Work Statements ........................ 491
12.3.2 Lagrange Interpolation Functions ............... 492
12.3.3 Finite Element Model ........................... 496
12.3.4 Numerical Results .............................. 498
12.4 Nonlinear Finite Element Models ....................... 504
12.4.1 Introduction ................................... 504
12.4.2 Classical Plate Theory ......................... 504
12.4.3 First-Order Shear Deformation Plate Theory ..... 508
12.4.4 The Newton-Raphson Iterative Method ............ 512
12.4.5 Tangent Stiffness Coefficients ................. 513
12.4.6 Membrane Locking ............................... 519
12.4.7 Numerical Examples ............................. 520
12.5 Summary ............................................... 524
Problems ................................................... 526
References ................................................. 531
Subject Index ................................................. 543
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