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 |  موضوع: كتاب Mechanics of Laminated Composite Plates and Shells - Theory and Analysis السبت 24 أغسطس 2019, 12:52 am | |
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أخوانى فى الله
أحضرت لكم كتاب
Mechanics of Laminated Composite Plates and Shells - Theory and Analysis
Second Edition
J.N. Reddy
و المحتوى كما يلي :
Contents
Preface to the Second Edition xix
Preface to the First Edition . xxi
1 Equations of Anisotropic Elasticity, Virtual Work Principles,
and Variational Methods . 1
l.1 Fiber-Reinforced Composite Materials 1
l.2 Mathematical Preliminaries . 3
l.2.1 General Comments '" 3
1.2.2 Vectors and Tensors 3
l.3 Equations of Anisotropic Entropy 12
l.3.1 Introduction 12
1.3.2 Strain-Displacement Equations 13
l.3.3 Strain Compatibility Equations 18
1.3.4 Stress Measures . 18
l.3.5 Equations of Motion 19
1.3.6 Generalized Hooke's Law 22
l.3.7 Thermodynamic Principles 34
1.4 Virtual Work Principles . 38
1.4.1 Introduction 38
1.4.2 Virtual Displacements and Virtual Work 38
l.4.3 Variational Operator and Euler Equations . 40
l.4.4 Principle of Virtual Displacements . 44
1.5 Variational Methods . 58
l.5.1 Introduction 58
l.5.2 The Ritz Method . 58
1.5.3 Weighted-Residual Methods . 64
1.6 Summary . 71
Problems . 72
References for Additional Reading . 78
2 Introduction to Composite Materials 81
2.1 Basic Concepts and Terminology 81
2.l.1 Fibers and Matrix . '" , . 81
2.l.2 Laminae and Laminates . 83
2.2 Constitutive Equations of a Lamina . 85
2.2.1 Generalized Hooke's Law 85
2.2.2 Characteristics of a Unidirectional Lamina . 86x CONTENTS
2.3 Transformation of Stresses and Strains . 89
2.3.1 Coordinate Transformations . 89
2.3.2 Transformation of Stress Components . 90
2.3.3 Transformation of Strain Components . 93
2.3.4 Transformation of Material Coefficients 96
2.4 Plan Stress Constitutive Relations . 99
Problems 103
References for Additional Reading 106
3 Classical and First-Order Theories of Laminated
Composite Plates . 109
3.1 Introduction . 109
3.1.1 Preliminary Comments . 109
3.1.2 Classification of Structural Theories 109
3.2 An Overview of Laminated Plate Theories 110
3.3 The Classical Laminated Plate Theory 112
3.3.1 Assumptions . 112
3.3.2 Displacements and Strains . 113
3.3.3 Lamina Constitutive Relations . 117
3.3.4 Equations of Motion . 119
3.3.5 Laminate Constitutive Equations . 127
3.3.6 Equations of Motion in Terms of Displacements 129
3.4 The First-Order Laminated Plate Theory . 132
3.4.1 Displacements and Strains . 132
3.4.2 Equations of Motion . 134
3.4.3 Laminate Constitutive Equations . 137
3.4.4 Equations of Motion in Terms of Displacements 139
3.5 Laminate Stiffnesses for Selected Laminates . 142
3.5.1 General Discussion . 142
3.5.2 Single-Layer Plates . 144
3.5.3 Symmetric Laminates 148
3.5.4 Antisymmetric Laminates 152
3.5.5 Balanced and Quasi-Isotropic Laminates . 156
Problems 157
References for Additional Reading 161
4 One-Dimensional Analysis of Laminated Composite Plates . 165
4.1 Introduction . 165
4.2 Analysis of Laminated Beams Using CLPT . 167
4.2.1 Governing Equations . 167
4.2.2 Bending . 169
4.2.3 Buckling . 176
4.2.4 Vibration 182CONTENTS Xl
4.3 Analysis of Laminated Beams Using FSDT . 187
4.3.1 Governing Equations . 187
4.3.2 Bending . 188
4.3.3 Buckling . 192
4.3.4 Vibration 197
4.4 Cylindrical Bending Using CLPT . 200
4.4.1 Governing Equations . 200
4.4.2 Bending . 203
4.4.3 Buckling . 208
4.4.4 Vibration 209
4.5 Cylindrical Bending Using FSDT . 214
4.5.1 Governing Equations . 214
4.5.2 Bending . 215
4.5.3 Buckling . 216
4.5.4 Vibration 219
4.6 Vibration Suppression in Beams 222
4.6.1 Introduction . 222
4.6.2 Theoretical Formulation 222
4.6.3 Analytical Solution 227
4.6.4 Numerical Results . 230
4.7 Closing Remarks . 232
Problems 232
References for Additional Reading 242
5 Analysis of Specially Orthotropic Laminates Using CLPT 245
5.1 Introduction . 245
5.2 Bending of Simply Supported Rectangular Plates . 246
5.2.1 Governing Equations . 246
5.2.2 The Navier Solution . 247
5.3 Bending of Plates with Two Opposite Edges Simply Supported . 255
5.3.1 The Levy Solution Procedure 255
5.3.2 Analytical Solutions 257
5.3.3 Ritz Solution 262
5.4 Bending of Rectangular Plates with Various Boundary Conditions 265
5.4.1 Virtual Work Statements . 265
5.4.2 Clamped Plates 266
5.4.3 Approximation Functions for Other Boundary Conditions . 269
5.5 Buckling of Simply Supported Plates Under Compressive Loads . 271
5.5.1 Governing Equations . 271
5.5.2 The Navier Solution . 272
5.5.3 Biaxial Compression of a Square Laminate (k = 1) '" 273
5.5.4 Biaxial Loading of a Square Laminate 274
5.5.5 Uniaxial Compression of a Rectangular Laminate (k = 0) . 274xii CONTENTS
5.6 Buckling of Rectangular Plates Under In-Plane Shear Load . 278
5.6.1 Governing Equation 278
5.6.2 Simply Supported Plates . 278
5.6.3 Clamped Plates 280
5.7 Vibration of Simply Supported Plates 282
5.7.1 Governing Equations . 282
5.7.2 Solution . 282
5.8 Buckling and Vibration of Plates with Two Parallel Edges
Simply Supported . 285
5.8.1 Introduction . 285
5.8.2 Buckling by Direct Integration . 287
5.8.3 Vibration by Direct Integration 288
5.8.4 Buckling and Vibration by the State-Space Approach . 288
5.9 Transient Analysis . 290
5.9.1 Preliminary Comments . 290
5.9.2 Spatial Variation of the Solution . 290
5.9.3 Time Integration . 292
5.10 Closure . 293
Problems 293
References for Additional Reading 296
6 Analytical Solutions of Rectangular Laminated Plates
Using CLPT . 297
6.1 Governing Equations in Terms of Displacements 297
6.2 Admissible Boundary Conditions for the Navier Solutions . 299
6.3 Navier Solutions of Antisymmetric Cross-Ply Laminates 301
6.3.1 Boundary Conditions 301
6.3.2 Solution . 304
6.3.3 Bending . 308
6.3.4 Determination of Stresses 309
6.3.5 Buckling . 317
6.3.6 Vibration 323
6.4 Navier Solutions of Antisymmetric Angle-Ply Laminates 326
6.4.1 Boundary Conditions 326
6.4.2 Solution . 328
6.4.3 Bending . 329
6.4.4 Determination of Stresses 330
6.4.5 Buckling . 335
6.4.6 Vibration 337
6.5 The Levy Solutions . 339
6.5.1 Introduction . 339
6.5.2 Solution Procedure . 342
6.5.3 Antisymmetric Cross-Ply Laminates 348
6.5.4 Antisymmetric Angle-Ply Laminates . 353CONTENTS xiii
6.6 Analysis of Midplane Symmetric Laminates . 356
6.6.1 Introduction . 356
6.6.2 Governing Equations . 356
6.6.3 Weak Forms . 357
6.6.4 The Ritz Solution 358
6.6.5 Simply Supported Plates . 358
6.6.6 Other Boundary Conditions 360
6.7 Transient Analysis . 361
6.7.1 Preliminary Comments . 361
6.7.2 Equations of Motion . 361
6.7.3 Numerical Time Integration 362
6.7.4 Numerical Results . 364
6.8 Summary 371
Problems 371
References for Additional Reading 375
7 Analytical Solutions of Rectangular Laminated Plates
Using FSDT . 377
7.1 Introduction . 377
7.2 Simply Supported Antisymmetric Cross-Ply Laminated Plates 379
7.2.1 Solution for the General Case 379
7.2.2 Bending . 381
7.2.3 Buckling . 388
7.2.4 Vibration 394
7.3 Simply Supported Antisymmetric Angle-Ply Laminated Plates 400
7.3.1 Boundary Conditions 400
7.3.2 The Navier Solution . 402
7.3.3 Bending . 404
7.3.4 Buckling . 405
7.3.5 Vibration 406
7.4 Antisymmetric Cross-Ply Laminates with Two Opposite
Edges Simply Supported .412
7.4.1 Introduction . 412
7.4.2 The Levy Type Solution . 413
7.4.3 Numerical Examples . 415
7.5 Antisymmetric Angle-Ply Laminates with Two Opposite
Edges Simply Supported .421
7.5.1 Introduction . 421
7.5.2 Governing Equations . 421
7.5.3 The Levy Solution . 423
7.5.4 Numerical Examples . 425
7.6 Transient Solutions . 430
7.7 Vibration Control of Laminated Plates . 437
7.7.1 Preliminary Comments . 437
7.7.2 Theoretical Formulation 438xiv CONTENTS
7.7.3 Velocity Feedback Control. . 438
7.7.4 Analytical Solution 439
7.7.5 Numerical Results and Discussion 441
7.8 Summary 442
Problems 444
References for Additional Reading . .445
8 Theory and Analysis of Laminated Shells 449
8.1 Introduction . 449
8.2 Governing Equations . 450
8.2.1 Geometric Properties of the Shell 450
8.2.2 Kinetics of the Shell . 4fj4
8.2.3 Kinematics of the Shell 455
8.2.4 Equations of Motion . 457
8.2.5 Laminate Constitutive Relations . 461
8.3 Theory of Doubly-Curved Shells 462
8.3.1 Equations of Motion . 462
8.3.2 Analytical Solution 463
8.4 Vibration and Buckling of Cross-Ply Laminated
Circular Cylindrical Shells . 473
8.4.1 Equations of Motion . 473
8.4.2 Analytical Solution Procedure 475
8.4.3 Boundary Conditions 479
8.4.4 Numerical Results . 480
Problems 483
References for Additional Reading . .483
9 Linear Finite Element Analysis of Composite Plates and Shells . .487
9.1 Introduction . 487
9.2 Finite Element Models of the Classical Plate Theory (CLPT) . 488
9.2.1 Weak Forms . 488
9.2.2 Spatial Approximations 490
9.2.3 Semidiscrete Finite Element Model . 499
9.2.4 Fully Discretized Finite Element Models 500
9.2.5 Quadrilateral Elements and Numerical Integration 503
9.2.6 Post-Computation of Stresses 510
9.2. 7 Numerical Results . 510
9.3 Finite Element Models of Shear Deformation Plate Theory (FSDT) . 515
9.3.1 Weak Forms . 515
9.3.2 Finite Element Model 516
9.3.3 Penalty Function Formulation and Shear Locking . 520
9.3.4 Post-Computation of Stresses 524
9.3.5 Bending Analysis 525
9.3.6 Vibration Analysis . 540
9.3.7 Transient Analysis . 542CONTENTS xv
9.4 Finite Element Analysis of Shells . 543
9.4.1 Weak Forms . 543
9.4.2 Finite Element Model 546
9.4.3 Numerical Results . 549
9.5 Summary 558
Problems 560
References for Additional Reading 560
10 Nonlinear Analysis of Composite Plates and Shells 567
10.1 Introduction 567
10.2 Classical Plate Theory 568
10.2.1 Governing Equations 568
10.2.2 Virtual Work Statement 569
10.2.3 Finite Element Model. 572
10.3 First-Order Shear Deformation Plate Theory 575
10.3.1 Governing Equations 575
10.3.2 Virtual Work Statements . 576
10.3.3 Finite Element Model . 578
10.4 Time Approximation and the Newton-Raphson Method 583
10.4.1 Time Approximations . 583
10.4.2 The Newton-Raphson Method 584
10.4.3 Tangent Stiffness Coefficients for CLPT . 586
10.4.4 Tangent Stiffness Coefficients for FSDT . 590
10.4.5 Membrane Locking . 594
10.5 Numerical Examples of Plates . 596
10.5.1 Preliminary Comments . 596
10.5.2 Isotropic and Orthotropic Plates 596
10.5.3 Laminated Composite Plates 601
10.5.4 Effect of Symmetry Boundary Conditions on Nonlinear
Response . 604
10.5.5 Nonlinear Response Under In-Plane Compressive Loads . 608
10.5.6 Nonlinear Response of Antisymmetric Cross-Ply Laminated
Plate Strips 608
10.5.7 Transient Analysis of Composite Plates . 612
10.6 Functionally Graded Plates 613
10.6.1 Background . 613
10.6.2 Theoretical Formulation 615
10.6.3 Thermomechanical Coupling 616
10.6.4 Numerical Results 617
10.7 Finite Element Models of Laminated Shell Theory . 621
10.7.1 Governing Equations 621
10.7.2 Finite Element Model. 622
10.7.3 Numerical Examples 625xvi CONTENTS
10.8 Continuum Shell Finite Element . 627
10.8.1 Introduction . 627
10.8.2 Incremental Equations of Motion 628
10.8.3 Continuum Finite Element Mode . 631
10.8.4 Shell Finite Element . 633
10.8.5 Numerical Examples . 638
10.8.6 Closure. . . . . . . . . . . . . . . . . . . . . . . . 644
10.9 Postbuckling Response and Progressive Failure of Composite
Panels in Compression . 645
10.9.1 Preliminary Comments . 645
10.9.2 Experimental Study . 645
10.9.3 Finite Element Models . 647
10.9.4 Failure Analysis 648
10.9.5 Results for Panel C4. . . . . . . . . . . 650
10.9.6 Results for Panel H4. . . . . . . . . . . 655
10.10 Closure . 658
Problems 658
References for Additional Reading 664
11 Third-Order Theory of Laminated Composite Plates and Shells 671
11.1 Introduction 671
11.2 A Third-Order Plate Theory 671
11.2.1 Displacement Field . 671
11.2.2 Strains and Stresses . 674
11.2.3 Equations of Motion 674
11.3 Higher-Order Laminate Stiffness Characteristics . 677
11.3.1 Single-Layer Plates . 678
11.3.2 Symmetric Laminates 680
11.3.3 Antisymmetric Laminates 681
11.4 The Navier Solutions. . . . . . . . . . . . . . . . . 682
11.4.1 Preliminary Comments . 682
11.4.2 Antisymmetric Cross-Ply Laminates . 684
11.4.3 Antisymmetric Angle-Ply Laminates 687
11.4.4 Numerical Results 689
11.5 Levy Solutions of Cross-Ply Laminates 699
11.5.1 Preliminary Comments . 699
11.5.2 Solution Procedure . 701
11.5.3 Numerical Results 704
11.6 Finite Element Model of Plates. . . . . . . 706
11.6.1 Introduction 706
11.6.2 Finite Element Model. . 707
11.6.3 Numerical Results 712
11.6.4 Closure . 714CONTENTS XVll
11.7 Equations of Motion of the Third-Order Theory of Doubly-Curved
Shells . 718
Problems 720
References for Additional Reading 721
12 Layerwise Theory and Variable Kinematic Models . 725
12.1 Introduction 725
12.1.1 Motivation 725
12.1.2 An Overview of Layerwise Theories . 726
12.2 Development of the Theory 730
12.2.1 Displacement Field . 730
12.2.2 Strains and Stresses . 733
12.2.3 Equations of Motion 734
12.2.4 Laminate Constitutive Equations 736
12.3 Finite Element Model . 738
12.3.1 Layerwise Model 738
12.3.2 Full Layerwise Model Versus 3-D Finite Element Model . 739
12.3.3 Considerations for Modeling Relatively Thin Laminates . 742
12.3.4 Bending of a Simply Supported (0/90/0) Laminate 746
12.3.5 Free Edge Stresses in a (45/-45)8 Laminate 753
12.4 Variable Kinematic Formulations 759
12.4.1 Introduction 759
12.4.2 Multiple Assumed Displacement Fields 762
12.4.3 Incorporation of Delamination Kinematics . 764
12.4.4 Finite Element Model . 766
12.4.5 Illustrative Examples . 769
12.5 Application to Adaptive Structures 780
12.5.1 Introduction 780
12.5.2 Governing Equations 783
12.5.3 Finite Element Model . 785
12.5.4 An Example 787
12.6 Layerwise Theory of Cylindrical Shells . 794
12.6.1 Introduction 794
12.6.2 Unstiffened Shells . 794
12.6.3 Stiffened Shells . 798
12.6.4 Postbuckling of Laminated Cylinders 806
12.7 Closure . 812
References for Additional Reading 816
Subject Index .
Adaptive structures, 780
Admissible configurations, 38
Admissible displacements, 51
Admissible variations, 43, 53
Alternating symbol, 5
Analytical solution, 227, 257, 297,
377, 439, 463, 475
Angle-ply laminate, 142, 150
antisymmetric, 155, 326
Anisotropic body, 22
Anisotropic layer, 147, 680
Antisymmetric angle-ply laminates,
155, 326, 353, 400, 421, 687
Antisymmetric cross-ply laminate,
154, 301, 348, 379, 412
nonlinear response, 608
third-order theory, 684
Antisymmetric laminates,
144, 152-155, 301, 326, 681
Apparent moduli of an orthotropic
material, 103
Approximation functions, 59, 269-271
Asymmetric laminate, 144
Backward difference method, 363, 502
Balanced laminate, 156
Barlow points, 524
Basis vectors, 4
orthonormal, 4
BCIZ triangle, 495
Beam:
bending of, 169-176, 188-192
buckling of, 176-182, 192-197
Euler-Bernoulli theory of, 167,
168, 224
nonlinear bending of, 595
Reddy third-order theory of, 224
SUBJECT INDEX 821
Subject Index
Timoshenko theory of, 187, 188, 224
vibration of, 182-187, 197-200
Bending (static response):
of antisymmetric angle-ply plates
(CLPT), 329, 353
(FSDT), 404, 426
(TSDT),694
of antisymmetric cross-ply plates
(CLPT), 308, 345, 349
(FSDT), 381, 416
(TSDT),689
of beams, 169-176, 188-192
of doubly curved shells, 467
of plates (FEM), 500, 511, 525
of specially orthotropic plates, 246,
382
Betti's reciprocity theorem, 29
Bifurcation, 271
Body force, 7
Boundary conditions:
essential, 43, 59, 127
force, 43, 168
geometric, 43, 45, 59, 168
homogeneous, 43
natural, 43, 126, 127, 137, 735
of beams, 169
of cantilever (fixed-free) beams,
50, 175
of clamped (fixed-fixed) beams,
173, 175, 180, 185, 190, 196, 198
of free beams, 182, 184
of hinged-fixed beams, 182, 184
of simply supported beams, 172,
180, 184, 190, 196, 198
of simply supported plate strips,
205, 208822 MECHANICS OF LAMINATED COMPOSITE PLATES AND SHELLS
of simply supported plates, 246,
259, 271, 282, 290, 341, 439
88-1, 299, 300, 359, 379, 422,
465, 511, 597, 601, 625, 682
88-2, 301, 326, 400, 422, 511, 683
88-3, 597
Buckling
deflection, 176
loads of beams, 176
mode, 179, 180
of antisymmetric angle-ply plates,
(CLPT), 335, 354
(F8DT), 405, 428
of antisymmetric cross-ply plates,
(CLPT), 317, 347, 351
(F8DT), 388, 419
(T8DT),698
of beams, 68, 176-182, 192-197
of circular cylindrical shells, 473
of laminated plates (FEM), 500
of specially orthotropic plates, 271,
285, 393
under compressive loads, 271
under shear load, 278
CD-Continuity, 172, 699
C 1 -Continuity, 767
CO plate element, 519
C 1 plate element, 495
Cartesian coordinates, 4
Cauchy stress formula, 8, 18
Cauchy stress, 8, 18
Central difference method, 363, 502
Ceramic-metal, 617
Characteristic equation, 181, 182,
184, 265, 269-271
Characteristic polynomial,
see Characteristic equation
Classical plate theory (CLPT):
assumptions of, 113
boundary conditions, 126
cylindrical bending, 131, 200
displacement field, 114
equations of motion, 119-124,
246, 297, 568
finite element model of, 488
strains, 116, 117
Classical shell theory, 474
Closed-form solution, 166, 248
Codazzi conditions, 452
Coefficients:
of hygroscopic expansion, 36
of mutual influence, 104
of thermal expansion, 35
Collocation method, 65, 67
Composite material, 1
Compatibility equations, 18
Compliance coefficients, 27
Conditionally stable, 363
Configuration, 13
Conforming element:
rectangular element, 498
triangular element, 496
Conservation of energy, 34
Conservation of angular momentum,
20
Conservation of linear momentum, 19
Constant-average-acceleration
method, 363, 502
Constant strain triangle, 492
Constitutive equations, 12, 22
anisotropic material, 24
electroelastic, 37
hygrothermal elastic, 36, 99
hyperelastic, 23, 50
isothermal condition, 85
isotropic material, 31, 32
monoclinic material, 25, 26
of a lamina, 85, 118, 119
orthotropic material, 26-30
plane stress, 33, 99-101
thermoelastic, 35
transformed, 25
Continuum elements, 567, 631
Continuum shell finite element, 627
Contracted notation, 24
Convective heat transfer coefficient,
34
Coordinate system:
Cartesian, 5cylindrical, 6
material, 25
orthonormal Cartesian, 5
rectangular Cartesian, 5
transformation of, 89
Coupled ESL models, 780
Coupled layerwise models, 780
Cramer's rule, 308
Critical buckling load, 68, 176, 273
Critical time step, 363
Cross-ply laminate, 143, 150, 699
antisymmetric, 154, 301, 379
Cross product, 5
Curl operation, 6, 11
Cylindrical bending,
CLPT, 131, 200
FSDT, 141, 142, 214
FEM, 608
Cylindrical pressure vessel, 92
Cylindrical shell, 550, 794, 806
Cylindrical shell panel, 551, 557, 641
Deformation, 13
Deformation gradient tensor, 19
Delamination, 83, 764
Del operator, 5
Description of motion,
Eulerian, 13
Lagrangian, 13
material, 13
referential, 13
spatial, 13
Deviatoric, 32
Dielectric constants, 37
transformed, 102
Direct methods, 58
Direction cosines, 90
Discrete layer theory, 728
Displacement finite element model,
500
Dot product, 5
double, 10
Dilatation, 32
Divergence, 6, 11
Divergence theorem, 11
Double arrow notation, 20
SUBJECT INDEX 823
Double-dot product, 10
Doubly-curved shells, 462, 718
Doubly-curved shell panel, 550
Duhamel-Neumann law, 35
Dummy index, 5
Dyad,3
components of, 10
Effect of bending-stretching coupling:
on buckling load, 209, 335, 337,
394, 406
on deflection, 207, 314, 317, 331,
332, 353, 388, 404
on frequenc~ 324, 339, 399, 419
on stresses, 313, 314, 317, 331
Effect of bending-twisting coupling:
on deflection, 538
on frequency, 360
Effect of lamination angle:
on buckling load, 213, 338, 355,
409, 428, 699
on deflection, 213, 333, 353, 407,
426, 533, 539, 696
on frequency, 213, 339, 354, 411,
428, 698
Effect of length-to-height ratio:
on buckling load, 200, 201, 220,
395,396,409,410,699,716,717
on deflection, 195, 200, 217, 218,
385-387, 389, 392, 405, 407, 416,
417, 426, 427, 532, 533, 535, 636,
539, 690, 691, 694, 696, 706, 713
on frequency, 200, 201, 211, 223,
398-401, 410, 411, 429, 538, 540,
541, 697, 698, 716, 718
Effect of orthotropy:
on buckling load, 220, 277, 289,
290, 395, 321, 322, 336, 352, 355,
420, 699
on deflection, 218, 314, 318, 331,
333, 350, 406, 418, 426, 427, 538,
695
on frequency, 284, 285, 289, 290,
324, 340, 352, 354, 400, 420, 429,
697824 MECHANICS OF LAMINATED COMPOSITE PLATES AND SHELLS
on stresses, 314, 331, 406
Effect of plate aspect ratio:
on buckling load, 276, 277, 278,
321, 322, 336, 355
on deflection, 253, 313, 315, 318,
332, 392
on frequency, 285, 325, 340, 352,
354
on stresses, 253, 313, 315, 316
Effect of radius-to-thickness:
on deflection, 467, 468, 555, 557,
559
on stress, 555, 557, 559
Effect of rotary inertia:
on natural frequency, 285, 398, 399
Effect of shear deformation:
on buckling load, 395, 410, 421, 716
on deflection, 385-387, 405, 406,
437, 536, 538, 691, 695, 696, 705,
713
on frequency, 223, 398-400, 410,
420,697-700,716,717
on stresses, 385-387, 405, 406, 437,
537, 692, 693, 705, 707, 714, 715
on thermal deflection, 706
Effect of stacking sequence:
on buckling load, 186, 212
on deflection, 186, 212
on natural frequency, 186, 212
Eigenfunctions, 264, 269-271, 360
Eigenvalue problem, 67, 287, 323, 337
Eigenvalues, 68
Eigenvectors, 68see Eigenfunctions
Elastic, 22
Elastic compliances, 24, 27, 35
transformed, 97, 98
Elastic coefficients, 24
transformed, 101, 119
Electric displacement vector, 100
Electric potential, 101
Electroelasticity, 36
Electrostriction, 222
Engineering constants, 27-30, 86, 677
Engineering notation, 24
Enthalpy function, 37
Entropy density, 35
Epsilon-delta (1':-8) identity, 5
Equations of equilibrium, 19
cylindrical bending,
(CLPT),203
(FSDT), 215
elasticity, 19
Euler-Bernoulli beam theory,
46, 169
specially orthotropic plates, 246
Third-order beam theory, 224
Timoshenko beam theory, 224
Equations of motion of:
antisymmetric angle-ply plates,
(FSDT), 421, 422
antisymmetric cross-ply plates,
(CLPT),342
classical plate theory, 119-124, 297,
568
cylindrical bending,
(CLPT), 1:31
(FSDT), 141, 142
elasticity (3D), 19
Euler-Bernoulli beam theory,
46-49, 226
first-order plate theory, 134-142,
377, 378, 575
layerwise plate theory, 734
shells, 457-460, 463, 473, 620, 719
specially orthotropic plates, 246
symmetric laminates, 356, 357
Timoshenko beam theory, 57, 226
Third-order beam theory, 57, 226
Third-order plate theory, 674-676
Equivalent single-layer theory, 109
Error criterion, 585
Essential boundary condition,
see Boundary conditions
Euler-Bernoulli beam theory, 46,
167, 168, 224
Euler-Bernoulli hypotheses, 46
Euler-Lagrange equations, 44, 46, 49,
52, 55, 124, 136, 675, 735Eulerian description, 13
Exact solution, 165
Extensional stiffnesses, 128, 138
Failure analysis, 648
Failure criterion:
maximum stress, 648
Tsai-Wu, 649
Failure mode, 654
Fiber, 1, 81
Fick's second law, 35
Finite element method, 487, 567
Finite element model of:
layerwise theory, 738, 785
plates (CLPT), 488, 572
plates (FSDT), 516, 578
plates (TSDT), 706
shells, 543, 622, 633
variable kinematic formulation, 766
Finite strain, 15
First-order shear deformation theory
(FSDT):
boundary conditions, 137
displacement field, 132
equations of motion, 134-142, 575
finite element model of, 515
strains, 133, 134
First law of thermodynamics, 34
First Piola-Kirchhoff stress, 18
First-ply failure, 655
First variation, 40
Flexure stress formula, 20
Force boundary condition, 43
Force resultants, 122
Fourier's heat conduction law, 34
Fox-Goodwin scheme, 363
Free edge stresses, 753, 769, 779
Frequency, see Vibration
Full layerwise theory, 727
Functional, 41
extrema of, 42
linear, 41
quadratic, 41
Functionally graded plates, 613
Fundamental lemma, 42
SUBJECT INDEX 825
Galerkin's method, 65, 66, 279
363, 502
Gauss points, 508
Gauss quadrature, 506
Generalized Hooke's law, 22-33, 85
Generalized displacements, 133
Generally orthotropic layer, 146, 150,
680
Geometric boundary condition,
see Boundary conditions
Gibb's free energy function, 37
Global coordinates, 503
Global-local analysis, 759
Gradient operator, 6
Gradient theorem, 11
Green-Lagrange strain tensor,
14-16
Hamilton's principle, 53-57, 457,
707,719
Heat conduction equation, 34
Heat flux, 45
Helmholtz free-energy function, 35
Hermite interpolation, 495
Heterogeneous body, 22
Homogeneous, 22
Hooke's law,
see Generalized Hooke's law
Hygroscopic expansion coefficients,
36
Hygrothermal elasticity, 35
Hyperelastic, 22, 23, 50
Ideally elastic, 23
Ill-conditioned matrix, 348, 478
Index notation, 5
Infinitesimal strain tensor, 16
Initial conditions, 127, 137, 291, 441
In-plane inertia, 323
Integral relations, 10
Interlaminar stresses, 726
see Transverse stresses
Internal virtual work, 44
Internal work, 39, 44
Interpolation functions, 487
Invariant, 3
Isoparametric approximation, 504826 MECHANICS OF LAMINATED COMPOSITE PLATES AND SHELLS
Isotropic material, 2, 31, 32
Jacobian matrix, 506
Jacobian, 506
Jordan canonical form, 478
Kinematics, 12-16, 455
Kinetic energy, 53
Kinetics, 12, 454
Kirchhoff assumptions, 113
Kirchhoff free-edge condition, 127
Kronecker delta, 5
Lagrange interpolation, 491
Lagrange multiplier method, 521
Lagrangian description, 13
Lame coefficients, 452
Lame constants, 32
Lamina (ply), 2, 83
Laminate constitutive equations,
127-129, 137-139,461,736
Laminated beams, 167, 187
Laminated element, 567
Laminated plate theories:
classical (CLPT), 112-131
first order (FSDT), 132-142
third order, 112
Laminates:
antisymmetric, 144, 152-155,301,
326
asymmetric, 144
angle-ply, 150, 155, 326
balanced, 156
cross-ply, 150, 154, 301
generally orthotropic, 150
single-layer, 144-147
specially orthotropic, 149, 150, 245
symmetric, 145-151
Lamination scheme, 83
Laplace transform, 293
Layerwise theory:
displacement field of, 730
constitutive equations of, 736
equations of motion of, 734
finite element model of, 738, 785
of Reddy, 730
stiffnesses of. 736-738
strains of, 733
Least squares method, , 65 66
Levy's method, 255, 286, 475
Levy solutions:
antisymmetric angle-ply plates,
(CLPT),353
(FSDT), 423
antisymmetric cross-ply plates,
(CLPT),342
(FSDT), 413
(TSDT),699
specially orthotropic plates,
255-262, 286
Linear acceleration method, 363, 502
Linear functional, 41
Linearly independent set, 59
Local coordinates, 503
Locking:
membrane, 594
shear, 523
Macromechanical behavior 85
,
Magnetostriction, 222
Mass diffusitJvity tensor, 35
Mass diffusitivity, 35
Mass inertias, 122, 227, 458, 473
Master element, 504
Material coordinates, 13
Material compliance matrix, 97, 98
transformed, 97, 98
Material properties,
aluminum, 88
boron-epoxy, 88
glass-epoxy, 88
graphite-epoxy (AS), 88
graphite-epoxy (T), 88
graphite fabric-carbon, 30, 102
material I, 525, 625, 689
material 2, 320, 532, 625, 694
steel, 88
Material stiffnesses, 23-33
transformed, 96, 119
Material strengths, 649
Material symmetry, 25Matrix material, 1, 81
Maximum stress criterion, 648
Maxwell's relations, 36
Mean stress, 32
Membrane locking, spp Locking
Membrane strains, 117
Mesh generation, 488
Metric, 450
Micromechanics, 85
Mindlin plate theory,
see First-order plate theory
Minimum total potential energy, 50
Mixed finite element model, 521
Moisture concentration, 35
Moment resultants, 122
Monoclinic material, 25, 85
Multiple model analysis,
see Global-local analysis
Multiple model methods, 109, 759,
762
Multistep methods, 759
Natural boundary condition, 43, 126
127, 137, 735
Natural coordinates, 494, 504
Navier's method, 247
Navier's solutions:
antisymmetric angle-ply plates,
(CLPT),326
(FSDT),402
(TSDT),687
antisymmetric cross-ply plates,
(CLPT), 301
(FSDT),379
(TSDT),684
beam, 228
cylindrical shell, 801
doubly curved shells, 465
specially orthotropic plates,
247, 272
Newmark's integration schemes, 362,
502, 583
Newton's second law, 7, 19, 44, 53
Newton-Raphson iteration scheme,
584
SUBJECT INDEX 827
modified, 585
Nonconforming element:
rectangular, 497
triangular, 496
Nonion form, 9
Nonlinear analysis of:
bending of plates, 596
buckling of plates, 608, 645
transient response, 612
shell, 625, 638
Normal derivative, 12
Normal stress, 7, 31
Normalized coordinates, 504
Numerical integration, 506
Numerical time integration,
see Time approximation schemes
Orthotropic lamina, 100
Orthotropic material, 26, 85
Orthotropic piezoelectric lamina, 118
Partial layerwise theory, 727
Particular solution, 59
Particulate composites, 81
Penalty function method, 520
Penalty parameters, 521
Period of vibration, 363
Permutation symbol, 5
Petrov-Galerkin method, 65
Physical components, 4
Piezoelectric effect, 36
Piezoelectric moduli, 37, 100
transformed, 102, 119, 438
Piezoelectric resultants, 129, 569
Plane of material symmetry, 25
Plane strain, 165
Plane stress reduced stiffnesses,
33, 100, 677
Plane stress, 33, 165
Plates, 131
classical theory of, 112-131
first-order theory of, 132 -142
equivalent single-layer, 110
specially orthotropic, 145, 149, 245
third-order theory of, 671-677
Ply, 97828 MECHANICS OF LAMINATED COMPOSITE PLATES AND SHELLS
Poisson effect, 119
Polarization charge, 36
Polarization vector, 36
Polyads, 10
Postbuckling response, 645, 806
Potential energy, 53
Primary variables, 43, 126, 137,
227, 490, 516, 546, 676, 735
Principle:
of conservation of energy, 34
of minimum total potential energy,
44, 50-53
of superposition, 27
of virtual displacements, 44, 45, 120
134, 457, 631, 707, 734, 795
thermodynamics, 34-37
Progressive failure, 645
Pure extension, 17
Pure shear, 17
Pyroelectric constants, 37
Pyroelectric effect, 36
Quadratic functional, 41
Quasi-isotropic laminate, 156
Reciprocal relations, 28
Rectangular Cartesian, 4
Reddy's layerwise theory, 730
Reddy's third-order beam theory, 224
Reddy's third-order plate theory,
671-677
Reduced integration, 523
Referential description, 13
Residual, 584
Resultants:
force, 122
higher-order, 677
moment, 122
piezoelectric, 129
thermal, 128, 146, 147
Riks-Wempner method, 585
Ritz approximation, 62, 279, 280
see Ritz method
Ritz method, 58-62
Rotatory inertia,
see Rotary inertia
Rotary inertia, 125
Sanders shell theory, 449
Scalars, 3
Scalar product, 5
Second law of thermodynamics, 34
Second-order plate theory, 111
Second Piola-Kirchhoff stress 19
,
Secondary variables, 43, 126, 137,
227, 490, 516, 546, 676, 735
Self-starting scheme, 364
Semidiscrete finite element model
499, 547
Separable solution, 361
Sequential methods, 759
Serendipity elements, 495
Series solution, 166
,
Set of admissible configurations, 38
Shear correction coefficient, 57, 135
Shear correction factors, 455
Shear coupling, 168
Shear-extensional coupling, 26
Shear locking, see Locking
Shear stress, 7, 31
Shell, 449
Simplified third order theory, 57
Single subscript notation, 24, 85
Spanning set, 59
Spatial description, 13
Specific heat, 34
Specially orthotropic laminate, 245
Specially orthotropic layer,
145, 150, 151, 679, 681
Specially orthotropic plates,
245, 382
Specially orthotropic solution, 335
Spherical shell panel, 639, 641, 644
Stability, see Buckling
Stability, numerical, 502
Stable equilibrium, 176
Stacking sequence, 83
see lamination scheme
State-space approach, 260, 288,
345, 414, 425, 477, 703
Static condensation, 308Stiffnesses:
bending, 128
bending-extensional, 128
extensional, 128, 138
of antisymmetric angle-ply plates,
682
of antisymmetric cross-ply plates,
682
of asymmetric laminates, 144
of balanced laminate, 156
of quasi-isotropic laminate, 156
of single isotropic layer, 145, 678
of single-layer plates, 144-147, 678
of symmetric laminates, 680
laminate, 142-157
layerwise theory, 736, 737
Strain-displacement relations, 13-16
Strain:
Green-Lagrange, 14-16, 629
infinitesimal, 16
hygrothermal, 36
moisture, 35
transformation of, 93, 94
thermal, 35, 36
Strain compatibility, 18
Strain energy, 3, 40, 50
complementary, 40
Strain energy density, 23, 33, 50
Strain gages, 87
Strain rate tensor, 34
Stress,
Cauchy, 8, 18
deviatoric, 32
dyadic, 8
mean, 32
measures, 18
first Piola-Kirchhoff, 18
second Piola-Kirchhoff, 19, 629
single subscript notation, 24, 91
tensor, 8
transformation of, 90, 91
vector, 7
Stress computation:
of antisymmetric angle-ply plates,
(CLPT),330
(FSDT),403
(TSDT),688
SUBJECT INDEX 829
of antisymmetric cross-ply plates,
(CLPT),309
(FSDT),381
(TSDT), 686
of beams, 169-172
of plates (FEM), 510, 524
of specially orthotropic plates, 250,
383
Subparametric formulation, 504
Summation convention, 5, 15
Superparametric formulation, 504
Surface metrics, 450
Symmetric laminate, 143, 148-151,
680
Tangent stiffness matrix, 584
Tensor product, 509
Tensor, 3, 7-10
first-order, 10
Green-Lagrange strain, 14, 15
mass diffusivity, 35
proollct, SOg
second-order, 10
third-order, 10
transformation of, 10
transpose of, 9
unit, 10
zeroth-order, 10
Thermal coefficients of expansion, 35
transformed, 99, 101, 119
Thermal conductivity tensor, 34
Thermodynamics, 12, 34-37
Third-order beam theory, 55-57, 224
Third-order plate theory, 671-677
bending of, 689, 712
buckling of, 698, 712
displacement field of, 671-673
equations of motion of, 674
finite element model of, 706
stiffnesses of, 676-682
Levy solution, 699
strains of, 674
vibration of, 696, 712830 MECHANICS OF LAMINATED COMPOSITE PLATES AND SHELLS
Three-point bending, 172
Time approximation schemes,
362-364
Timoshenko beam theory, 57, 187,
188, 224
Total Lagrangian formulation, 568,
627
Total potential energy, 44, 50-53,
266, 279, 522
Transformation of:
material coefficients, 25, 96, 97
strains, 93, 94
stresses, 90, 91
tensor components, 10
Transformation matrix, 26, 636
Transient analysis, 290, 361, 430,
612
Transverse force resultants, 122, 135
Transverse stresses from:
constitutive relations, 190, 403
686
equilibrium equations, 170-172,
250, 310, 382, 384, 403, 686, 688
Tsai-Wu criterion, 649
Uncoupled ESL models, 780
Undetermined parameters, 58
Uniaxial compression, 274
Unstable equilibrium, 176
Unsymmetric laminate, 145
Updated Lagrangian formulation,
568, 627
Variables,
primary, 43, 126, 137,
227, 490, 516, 546, 676, 735
secondary, 43, 126, 137,
227, 490, 516, 546, 676, 735
Variable kinematic formulation, 759
Variational operator, 40--42
properties of, 41
Variational methods, 58
collocation, 65, 67, 69
Galerkin, 65, 66, 68, 279
least squares, 65, 66, 69
Ritz, 58-62, 68, 262, 279, 280, 358
weighted-residual, 64-68
Vector product, 5
Vectors, 3
basis, 4
cross product of, 5
Vector space, 3
Velocity feedback control, 226, 438
Vibration, natural:
of antisymmetric angle-ply plates,
(CLPT), 337, 354
(FSDT), 406, 428
of antisymmetric cross-ply plates,
(CLPT), 323, 346, 351
(FSDT), 394, 419
of beams, 182-187, 197-200
of circular cylindrical shells,
473
of doubly curved shells, 468
of plates (FEM), 501, 515, 540
of specially orthotropic plates,
282, 285, 397
Vibration suppression,
of doubly curved shells, 469
of laminated beams, 222
of laminated plates, 437
Virtual complementary strain energy,
40
Virtual displacements, 38, 44, 45
principle of, 44, 45, 120, 134, 674
Virtual forces, 40
Virtual strain energy, 40, 120, 134,
457, 674
Virtual work, 38, 45, 54, 120, 134,
266, 675
Virtual work principles, 38-46, 120
Viscous dissipation, 34
Voit-Kelvin notation, 24
von Karman nonlinearity, 567, 620,
794
von Karman strains, 117, 620
Weak forms for:
laminated plates (CLPT), 488
laminated plates (FSDT), 515
laminated plates (TSDT), 707midplane symmetric plates, 357
specially orthotropic plate, 266
shells, 543
Weight functions, 64
Weingarten-Gauss relations, 451
Whiskers, 1, 81
Work:
external, 45
internal, 39, 45
virtual, 38, 45, 54
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