Reddy J.N. Theory and analysis of elastic plates and shells (Boca Raton, 2007). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаReddy J.N. Theory and analysis of elastic plates and shells. - Boca Raton: CRC Press, 2007. - 2nd ed. - 547 p.: ill. - Ref.: p.531-541. - Sub. ind.: p.543-547. - ISBN 978-0-8493-8415-8
 

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Оглавление / Contents
 
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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