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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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