Theories and applications of plate analysis classical numerical and engineering methods

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Theories and Applications of Plate Analysis Theories and Applications of Plate Analysis: Classical, Numerical and Engineering Methods R Szilard Copyright © 2004 John Wiley & Sons, Inc Theories and Applications of Plate Analysis Classical, Numerical and Engineering Methods Rudolph Szilard, Dr.-Ing., P.E Professor Emeritus of Structural Mechanics University of Hawaii, United States Retired Chairman, Department of Structural Mechanics University of Dortmund, Germany JOHN WILEY & SONS, INC This book is printed on acid-free paper Copyright  2004 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-750-4470, or on the web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, e-mail: permreq@wiley.com Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services please contact our Customer Care Department within the U.S at (800)-762-2974, outside the U.S at (317)-572-3993 or fax (317)-572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print, however, may not be available in electronic format For more information about Wiley products, visit our web site at www.wiley.com Library of Congress Cataloging-in-Publication Data Szilard, Rudolph, 1921Theories and applications of plate analysis : classical, numerical and engineering methods / by Rudolph Szilard p cm Includes bibliographical references and index ISBN 0-471-42989-9 (cloth) Plates (Engineering) I Title TA660.P6S94 2003 624.1 7765—dc21 2003043256 Printed in the United States of America 10 To the memory of my father, Dipl.-Ing Rudolph Seybold-Szilard, senior, who encouraged and inspired my career in structural mechanics Contents Preface xvii Symbols xxi I Introduction II Historical Background PART I Elastic Plate Theories and Their Governing Differential Equations 10 21 23 1.1 Classical Small-Deflection Theory of Thin Plates*1 23 1.2 Plate Equation in Cartesian Coordinate System* 26 1.3 Boundary Conditions of Kirchhoff’s Plate Theory* 35 1.4 Differential Equation of Circular Plates* 42 1.5 Refined Theories for Moderately Thick Plates 45 1.6 Three-Dimensional Elasticity Equations for Thick Plates 53 Membranes 57 1.7 Plate Theories and Analytical Solutions of Static, Linear-Elastic Plate Problems Asterisks (∗ ) indicate sections recommended for classroom use vii viii Contents 1.8 Summary* 60 Problems* 61 Exact and Series Solutions of Governing Differential Equations 62 2.1 Rigorous Solution of Plate Equation 62 2.2 Solutions by Double Trigonometric Series (Navier’s Approach)* 69 Solutions by Single Trigonometric Series ´ (Levy’s Method)* 75 2.4 Further Examples of Series Solutions 83 2.5 ´ Extensions of Navier’s and Levy’s Methods 92 2.6 Method of Images 97 2.7 Plate Strips 99 2.8 Rigorous Solution of Circular Plates Subjected to Rotationally Symmetric Loading* 110 Solutions of Membrane Problems 116 2.10 Series Solutions of Moderately Thick Plates 120 2.11 Summary* 126 Problems* 127 2.3 2.9 Further Plate Problems and Their Classical Solutions 129 3.1 Plates on Elastic Foundation* 129 3.2 Plates with Variable Flexural Rigidity 139 3.3 Simultaneous Bending and Stretching 147 3.4 Plates of Various Geometrical Forms 150 3.5 Various Types of Circular Plates 156 3.6 Circular Plate Loaded by an Eccentric Concentrated Force 161 Plates with Edge Moments 165 3.7 Contents 3.8 Solutions Obtained by Means of Superposition 168 3.9 Continuous Plates 173 Summary 179 Problems 180 3.10 Energy and Variational Methods for Solution of Lateral Deflections Introduction and Basic Concepts* 181 4.2 Ritz’s Method* 187 4.3 Galerkin’s Method and Its Variant by Vlasov* 196 4.4 Further Variational and Energy Procedures 212 4.5 Techniques to Improve Energy Solutions 226 4.6 Application of Energy Methods to Moderately Thick Plates 231 Summary* 234 Problems* 235 PART II Numerical Methods for Solution of Static, Linear-Elastic Plate Problems 181 4.1 4.7 ix Finite Difference Methods 237 247 5.1 Ordinary Finite Difference Methods* 247 5.2 Improved Finite Difference Methods 276 5.3 Finite Difference Analysis of Moderately Thick Plates 303 5.4 Advances in Finite Difference Methods 312 5.5 Summary and Conclusions* 314 Problems* 315 Gridwork and Framework Methods 6.1 Basic Concepts* 317 317 x Contents 6.2 Equivalent Cross-Sectional Properties 320 6.3 Gridwork Cells and Their Stiffness Matrices* 328 6.4 Computational Procedures for Gridworks 6.4.1 Procedures Using Commercially Available Programs* 6.4.2 Guidance for Gridwork Programming 336 337 343 Summary and Conclusions* 361 Problems* 362 6.5 Finite Element Method 364 7.1 Introduction and Brief History of the Method* 364 7.2 Engineering Approach to the Method* 370 7.3 Mathematical Formulation of Finite Element Method* 7.3.1 Consideration of Total System* 7.3.2 Formulation of Element Stiffness Matrices* 380 380 383 7.4 Requirements for Shape Functions* 389 7.5 Various Shape Functions and Corresponding Element Families 7.5.1 Polynomials and Their Element Families 7.5.2 Hermitian Elements 7.5.3 Other Element Families 392 393 399 403 7.6 Simple Plate Elements* 7.6.1 Rectangular Element with Four Corner Nodes* 7.6.2 Triangular Element with Three Corner Nodes* 406 7.7 Higher-Order Plate Elements 7.7.1 Rectangular Element with 16 DOF 7.7.2 Discrete Kirchhoff Triangular Element 418 418 423 7.8 Computation of Loads and Stress Resultants* 434 7.9 Moderately Thick Plate Elements 446 Thick-Plate Elements 453 7.10 406 411 Contents xi 7.11 Numerical Integration 458 7.12 Modeling Finite Element Analysis* 463 7.13 Programming Finite Element Analysis* 465 7.14 Commercial Finite Element Codes* 469 7.15 Summary and Conclusions* 472 Problems* 474 Classical Finite Strip Method 475 8.1 Introduction and Basic Concepts 475 8.2 Displacement Functions for Classical FSM 477 8.3 Formulation of the Method 481 8.4 Outline of Computational Procedures 489 8.5 Summary and Conclusions 494 Problems 495 Boundary Element Method 496 9.1 Introduction 496 9.2 Basic Concepts of Boundary Element Method 497 PART III Advanced Topics 10 Linear Considerations 505 507 10.1 Orthotropic Plates 507 10.2 Laminated and Sandwich Plates 10.2.1 Classical Laminated Plate Theory 10.2.2 Sandwich Plates 10.2.3 Moderately Thick Laminated Plates 530 531 534 539 10.3 Analysis of Skew Plates 546 10.4 Thermal Bending of Plates 561 1012 About the CD The conforming 16-DOF rectangular finite element† was employed in the pertinent FORTRAN programs It was also assumed that all nodal forces have the same time dependency The forcing function is expanded by using half-range Fourier series expansion containing only sine terms (see Sec A.1), as required in the harmonic analysis procedure Again, it is recommended that the user of WPP forced vibration programs first prints the corresponding User’s Guide, which gives detailed well-illustrated step-bystep instructions for their application As is the case with the static and free vibration parts of WPP, the forced vibration cannot be run directly from the CD-ROM Thus, it must first be installed on one of the hard disks of the user’s computer by creating a special folder for this purpose Like the other parts of WPP, the Dynamic Module2 (folder WPP df) contains the following files: User’s Guide, FORTRAN source codes, solved test problems, and WinPlatePrimer DF file To activate the harmonic analysis procedure, the user should click on the pertinent WPP df icon A first-time user of WPP DF should start with one of the solved test problems to get familiarized with the application of programs For this purpose, one should select the OPEN command from the File menu and click on a solved program Following this action, on the left-hand hand side of the screen appears the input table filled with corresponding node numbers, nodal coordinates, loading and boundary conditions, etc Simultaneously, the graphical representation of the plate problem in question is shown on the right-hand side of the screen The input data for the time dependency of the forcing function includes the number of Fourier terms and the time steps used in the corresponding expansion Scrolling down to the end in the Time Dependency column the user finds a command for PREVIEW FOURIER TRANSFORMATION AND TERMS By activating this command the result of this procedure is graphically illustrated at the bottom of the screen Next, the SAVE AND COMPUTE command in the menu bar should be activated By clicking on DISPLACEMENTS in the menu bar, we obtain at the left-hand side the results of the corresponding computation However, to obtain the pertinent graphical illustrations of the lateral displacements of the plate, one must click on the dotted line at the left-hand side of the screen located right below the menu bar This dotted line graphically represents the time interval of the forcing function As a default, the results of the last time step are presented graphically on the right-hand side of the screen, as shown in Fig A.4.7 By clicking at some other location on this dotted line, however, the results pertinent to the corresponding time step are presented The very same procedure should be used for obtaining the NODAL FORCES and STRESS AT THE CENTER OF ELEMENTS Figure A.4.7 illustrates the resulting screen picture of such a computation As in the case of static computation, one can select in the View file the desired way for a graphic presentation To obtain a hard copy of the numerical results, the user should consult the User’s Guide In general, the procedure for solution of the dynamic response of a new problem follows the above-described steps Consequently, it sufficient to deal here only with the apparent differences Again, the user should click on the NEW in the File menu The fundamental deviation in the input procedure involves the consideration of the time dependency of the forcing functions For this purpose, the user should fill in the table of Time Dependency, the activation command of which is under the menu bar at the left-hand side of the screen The required inputs are the number of time steps † See Sec 7.7.1 WinPlatePrimer Program System Figure A.4.7 1013 Screen picture of the results and the pertinent ordinates of the time-dependent forcing function and the number of Fourier terms to be used in the Fourier expansion Please note that the first time step must be zero Again, it must be remembered that all cells in the input tables must be filled in; otherwise the user cannot continue with the program If there is no input data, zero must be used After completion of the inputs, [enter] should be pressed to transfer these data to the computer Click also on PREVIEW OF FOURIER TRANSFORMATION to see the result of the already executed Fourier expansion of the forcing function in graphical form Then, activate SAVE AND COMPUTE to solve the dynamic response of the plate in question All other steps are identical to those already described above Index A Airy’s stress function, 65, 190, 616, 947, 964, 968, 970 Argyris, J H., 366 B Beam functions, 374, 378, 401, 480, 600, 866 Bending and stretching, 147–149, 511 solution of, 149–150 Bogner, F K., 399 Boundary conditions of curved boundaries, 40 finite difference methods, 253–260, 517 large deflections, 618 moderately thick plates, 46, 49 straight boundaries, 35–40 thermal bending, 564 thick plates, 56 Boundary element method, 496–504 discretization of boundary, in, 497, 501 fundamental solution for, 496–497 Buckling of rectangular plates See also Cases 158–162 of Plate formulas on accompanying CD FS, inplane load, 917, 926 S, inplane load, 916, 931 SF, inplane load, 925, 927 Buckling, general See also Stability analysis analytical methods, in, 911–919 concepts of, 905–910 energy methods in, 919–928 numerical methods, in, 928–946 finite difference approach, with, 928–937 finite element approach, with, 938–947 C Categories of plates, 5–7 CD-ROM accompanying the book Plate formulas, 1003–1005 WPP program system, 1004–1012 Cheung, Y K., 17, 18, 366, 518 Chladni, E F F., 10, 17 Note: The following letters refer to the boundary conditions of the plates: C = clamped, F = free edge, S = simple supported edge Theories and Applications of Plate Analysis: Classical, Numerical and Engineering Methods R Szilard Copyright © 2004 John Wiley & Sons, Inc 1015 1016 Index Circular plates, dynamic analysis of, 808 See also Cases 138–144 of Plate formulas on accompanying CD Circular plates, static solution of See also Cases 1–64 of Plate formulas on accompanying CD C, concentrated force, 159 C, uniform load, 66, 113 S, eccentric force, 161 S, hydrostatic load, 157 S, linearly increasing load, 114 Code number technique, 347, 355, 416, 444 Colatz, L., 16, 282 Conjugate plate analogy, 121, 124, 303, 308–310, 450–452 Continuous plates, 173–179 engineering solutions for, 676–718 finite difference method for, 270 finite element method for, 558–559 flat slabs, 583–587, 718–727 influence surface of, 576 loading patterns for, 685 point supported, 578–583 stencils for, 683 yield-line analysis of, 764–769 Convergence in buckling analysis, 940 Convergence in dynamic analysis effect of mass representation, 865 Convergence, static finite elements membrane element, 666 orthotropic, 522 rectangular conforming, 423 rectangular nonconforming, 410, 433 requirements for, 590 tests for, 591, 593 triangular DKT, 428–429 funicular polygon method of, 295 gridwork method of, 335 ordinary finite difference method of, 262, 518 Corner forces, 36, 551, 760–761 Conservation of energy, 215, 221–223 Courant, R., 17, 18, 365 Curvatures constants, 391, 591 vector of, 373 D Damping of motion concept of, 883–885 viscous damping with, 885 Differential equations of plates in buckling, 912, 915 in Cartesian coordinate system, 31 in motion, 802 in oblique coordinate system, 548 in polar coordinate system, 42 in thermal bending, 564 laminated and sandwich, 533–535 moderately thick, 45, 47–49 orthotropic, 510–511 thick of, 53–57 with bending and stretching, 148 with large deflections, 616 with variable thickness, 139, 143 Discrete Kirchhoff Triangular (DKT) element, 423–426 Donnell’s approximation, 303 Duhamel’s integral, 795, 888 Dynamic analysis using classical methods circular frequency in, 790 Note: The following letters refer to the boundary conditions of the plates: C = clamped, F = free edge, S = simple supported edge Index differential equation for, 802–803 equation of dynamic equilibrium in, 789, 802 equation of motion in, 789 equivalent one DOF system with, 796–800 general concepts of, 787–789 initial condition of, 791 inpulse in, 794 membranes for, 810–814 natural frequency in, 790 Navier method, 791, 805 transient response, 793–796 vibration of spring-mass in, 788 Dynamic analysis using finite differences, 845–856 Dynamic analysis using finite elements forced vibration, 870–882 free vibration, 856–859 mass matrix, 860–866 Rayleigh quotient with, 859 Dynamic load factors, 792, 795, 797, 832, 834, 848, 870, 878, 888 Dynamic matrix, 860, 866 E Edge moments, 165–168 Effective width bending, in, 104–106 postbuckling, in, 970–973 Eigenfunctions column buckling of, 205–206, 479–480 vibrations of beam of, 202–203, 478–479 Eigenvalue problems, 846, 852, 886, 907, 922, 924, 930, 936, 939 1017 Elastic modulus secant, 647 tangent, 647 Elasticity matrix, 372, 386, 388, 437, 483, 602 Energy methods for moderately thick plates, 231–233 Energy methods in bending basic concepts of, 181–186 Galerkin method, 196–200, 206–208 minimum of potential, 187, 384 Ritz method, 187–196, 381 shape functions for, 191, 202, 205 virtual work, with, 182, 383 Vlasov approach, 200–206 Energy See also Strain energy total potential, 189, 381 Engineering methods See also Yield-line analysis bridge decks, for, 727–730 degree of fixity, 733–739 elastic web analogy, 676–689 flat slabs for, 718–727 moment distribution, 700–709 Pieper and Martins approach, 710–718 simplified slope defection, 689–699 skew plates for, 730–733 Equations See also Differential equations equilibrium, 26, 335, 346, 377, 383 variational, 199 Equivalent stress, stain, 646 stress distribution of, 658 Error distribution technique, 226–231 Note: The following letters refer to the boundary conditions of the plates: C = clamped, F = free edge, S = simple supported edge 1018 Index F Finite difference method, ordinary boundary conditions for, 253–260, 517 convergence of, 264, 518 large deflections for, 624–626 load representation, in, 258–262 orthotropic plates for, 516, 526–528 plate equation, in, 251–252 sandwich plates for, 542–545 skew plates for, 549–554 static analysis with, 238, 247–276 Finite difference methods, improved funicular polygon method, 285–292, 296–297 boundary conditions of, 293–294, convergence of, 295 higher approximations, with, 277, 300–302 multilocal method, 278–285, 297 boundary conditions of, 284–289 successive approximation, 292–295, 298–300 Finite element method element stiffness matrices, 409, 420–423 assembly of, 377 formulation of, 372–377 orthotropic, 519–523, 528 large deflections for, 626–629 engineering approach for, 370–380 history of, 366 introduction to, 364 mathematical formulation of, 380–383 alternative, 385 minimum of potential, 384 virtual work, 383 nonlinear procedures, 626–661 Finite strip method, classical computational procedures for, 488–493 concepts of, 475–477 formulation of, 481–488 stiffness matrices for, 485–486 Flat slabs, plates, 583–587 analytical solution of, 579–589 approximate analysis of, 718–728, 778–780 Flexural rigidity, 31 orthotropic, 511–515 reduced, 657, 660 transformed, 533 Fourier, J., 986 Fourier series, 985–997 coefficients for, 987–988, 991, 994–996 double, 994–997 fast, 996 single, 986–994 Frequency circular, 790 natural, 790 G Galerkin’s method in buckling of plates, 923, 927 large-deflection analysis, 617–619, 622–623 static analysis, 196–200, 206–208 vibration of plates, 818–823 Generalized coordinates, 385 Geometrical forms, various elliptical, C, uniform load, 151 sector-shape, FS, uniform load, 153 triangular, S, uniform load, 152 Note: The following letters refer to the boundary conditions of the plates: C = clamped, F = free edge, S = simple supported edge Index Geometric nonlinearity, procedures of, combined , 643–645 incremental (Euler), 638–641 iterative (Newton-Raphson), 641–643 Germain, Sophie, 12 Gridwork method in dynamic analysis convergence of, 865 mass matrices, 864–865 Gridwork method in static analysis concepts of, 317–320 cross-sectional properties for Hrennikoff’ s model, 321 other models, 329 Salonen’s model, 321–323 load representation in, 340 programming, 343–348 sandwich plates for, 537 skew plates for, 555–558 solutions with, 348–360 stiffness matrices for, Hrennikoff’s cell, 333 nonlinear, 634–637 Salonen’s cell, 334 H Harmonic analysis procedure, 871–873, 881 even, odd functions, 988–990 motion, 804, Hermitian elements, 399, 404 methods, 278, 285 polynomials, 201–402 shape functions, 401–402 Hill equation, 949 Homogeneous equation, 63, 71, 77, 82, 86, 90 Hrennikoff, A., 16, 18, 320, 366 1019 I Influence surfaces, 571–578 continuous plate of, 576–577 loads for generation of, 573–574 singularity method in, 572 Initial imperfection in buckling, 973–975 Interaction curve in buckling, 910, 942 Internal forces and stresses, 30–34 acting on arbitrary planes, 33 Iterative approach large deflection for, 621 Richardson’s extrapolation for, 265 Runge’s principle in, 292 Stodolla-Vianello technique, 848, 855–856 J Johansen, W K., 16, 765, 750 K Kantorovich’s method, 214–215, 223–225 von K´arm´an, von, Th., 617, 947, 968, 970 Kirchhoff, G R., 14 publications of, 18, 35 supplementary forces of, 14, 37 L Laplacian operator in Cartesian coordinate system, 31 oblique coordinate system, 547 polar coordinate system, 44 Large-deflection analysis, 614–623 analytical solutions of, 617–623 differential equations of, 616–617 numerical buckling in, 947 Note: The following letters refer to the boundary conditions of the plates: C = clamped, F = free edge, S = simple supported edge 1020 Index Large-deflection analysis (continued ) numerical postbuckling in, 968 numerical solutions of, 624–644 Loads on supporting beams, 714 M Mapping, 412–414, 462 isoparemetric, 594–596 skew plates of, 549 Marcus, H., 676, 689 Mass matrix, explicit expressions for, 862, 864–865 convergence characteristics of, 865 Material nonlinearity procedures incremental (Euler), 648–649 initial strain approach, 651–653 initial stress approach, 653 iterative (Newton-Raphson, 649–651 Mathieu equation, 949, 952 Membrane forces due to large deflection, 616 Membranes, solution of, 116–117 circular, 118–119 dynamic analysis, in, 810–814 rectangular, 117–118 Method of images, 97–99 Mindlin, R D., 15, 18 Moderately thick plate, theories of circular plates for, 51 higher order, 49–51 Mindlin, 47–49 Reissner, 49–51 Speare and Kemp simplification, 47 Moderately thick plates, solutions of buckling of, 963–965 conjugate plate analogy with, 124–125, 308–310, 450–452 Donnell’s approximation for, 310–312 energy methods with, 231–233 finite difference method with, 308–310 finite strip method for, 606–609 free vibration for, 839–831 L´evy’s approach with, 121–124 Navier method with, 120–121 Wang’s formula for, 841 Moment diagrams approximate, 686, 713, 720, 731 reduction of, 582, 586, 684 Moments, internal Kirchhoff plates in, 31, 44 moderately thick plates in, 46–48 orthotropic plates in, 509 sandwich plates of, 536 thermal bending in, 567, Moment-sum, 34, 550, 930 Morley’s formula, 825, 880 N ´ 15, 16 N´adai, A., Navier, L., 13 lecture notes of, 18 Newmark method, 873 Nonlinearity combined, 656–661 geometric, 614–644 material, 645–655 Nonlinear solutions, incremental-iterative techniques, 657–661 Numbering of joint-points, 339 Numerical integration, 456–462 O Orthogonality of functions, 201, 480 Orthotropic plates, 507–529 Note: The following letters refer to the boundary conditions of the plates: C = clamped, F = free edge, S = simple supported edge Index analysis procedures with affine transformation, 766 buckling, 954–955 finite element method, 519–523, 528 finite difference method, 515–517, 526–528 gridwork method, 516 L´evy’s method, 524 Navier’s method, 523 differential equation of, 510 flexural rigidities of, 511–515 theory of, 508 P Patch test, 591 Periodic functions, 987 even or odd harmonic, 989 Plate strips, 99–109 CS, uniform load, 106 S, concentrated force, 109 S, uniform load, 63, 107 Plate theories, static elastic, 26–26 moderately thick, 45–51 thick, 53–57 Plates on elastic foundation, 129 L´evy solution, 132–135 S,concentrated force, 136–137 Navier solution, 130–132, 136 Poisson ratio, conversion of, 999 Polynomials for rectangular finite elements, 396–406 triangular finite elements, 394–396 Postbuckling, 966–977 Potential energy See also Energy minimum of, 187, 384, 908 nonlinear, 627 postbuckling, in, 970 stiffened plates of, 958 Principal stresses, 34 1021 Program system on CD-ROM forced vibration, 1011–1013 free vibration, 1010–1011 static analysis, 1007–1010 Punching shear, 583, 747 R Rayleigh’s method quotient, with, 817, 825, 859, 921, 960 stability analysis, in, 920, 926, 958 vibration analysis, in, 816 Rectangular plates, free vibrations of See also Cases 146–154 of Plate formulas on accompanying CD C, 821–823, 867 S, 805–807, 809, 828–830, 868–870 Rectangular plates, static solutions of See also Cases 74–105 of Plate formulas on accompanying CD C, concentrated force, 229, 415, 430, 442 C, thermal load, 569 C, uniformly loaded, 194, 206, 223 S, concentrated force, 89, 348, 430 S, hydrostatic load, 74 S, parabolic load, 84 S, partially loaded, 90, 756 S, prismatic load, 269, 296 S, small area loaded, 88 S, triangular load, 192 S, uniformly loaded, 72, 79, 355 SC, sinusoidal load, 81 SC, uniformly loaded, 208, 218, 221, 271, 298 SF, concentrated force, 491 Note: The following letters refer to the boundary conditions of the plates: C = clamped, F = free edge, S = simple supported edge 1022 Index Rectangular plates, static solutions of (continued ) SF, line load, 310 thermal load, 569 Reinforced concrete models of, 662–667 orthotropy of, 508 rigidity of, 511 Reissner, E., 15, 18 Resonance, 792, 834 Richardson’s extrapolation formula, 265 Rigid body motions, 390 Ritz method in dynamic analysis, 817–818, 821 large-deflection analysis, 620 postbuckling, 970 stability analysis, 922 static analysis, 187–195, 381 Robinson’s test, 593 Rotation of coordinate systems, 32, 330, 345, 382 S Shape functions for finite elements computed, 605 conforming, 390, 399, 403 finite plate strips, for, 478–481 moderately thick plates, for, 446–450, 456, 607 nonconforming, 390, 396, 407 polynomials, 392–405 requirements of, 389–392 triangular elements, for, 394–396, 424, 443 Shear correction factor, 48 Shear modulus, orthotropic, 509 Shear stresses, forces inplane, 27–28 moderately thick plates of, 46–47, 49–50 orthotropic plates of, 510 sandwich plates of, 536 thermal, 564 transverse, 46–47, 49 Sign conventions, 7, 347, 387 Singularity, 65, 572 Skew plates, 546–559 analytical solution of, 547–548 numerical solution, approximate methods for, 730–736 numerical solutions of, 549–559 Solution methods, classical exact, 63, 66–67 generalization of Navier’s approach, 92 homogeneous part of, 63 L´evy’s approach, 75–83, 85–87, 89–92 Navier’s approach, 69–75, 83–85, 87–89 particular part of, 65, 76 Spline finite strip method, 598–605 element stiffness matrix of, 603 formulation of, 600–604 load vector, for, 604 spline function, for, 598–600 Stress-strain curve, bilinear, 657 Stability analysis, analytical approaches basic concepts of, 905–910 bifurcation, in, 906 circular plates of, 914 dynamic approach, for, 909 energy methods for, 907–909, 919–928 equilibrium method for, 907, 911–919 snap-through in, 906 thermal buckling in, 961 under combined loads, 909–910 Stability analysis, numerical methods dynamic buckling in, 946–952 finite difference method with, 928–938 Note: The following letters refer to the boundary conditions of the plates: C = clamped, F = free edge, S = simple supported edge Index finite element method with, 938–943 gridwork method with, 943–946 Stiffened plates, buckling See also Cases 163–170 of Plate formulas on accompanying CD othotropic plates of, 954, 959–960 three or less stiffeners with, 955, 957 Stiffness matrices, explicit finite strips of, 485–486 gridwork of, 333–334 membrane finite element of, 669 nonlinear for, 628–630 orthotropic finite element of, 520 rectangular finite elements of, 409, 420–422 spline strip of, 603 triangular finite element, 427 Stodola-Vianello iteration, 827, 848, 855, 897, 930, 940 Strain energy, 182 bending, in, 184, 187, 188 equivalence of, 324–328 membranes for, 184, 189–190, 620 sandwich plates of, 535 Strains, 29–30, 187, 373, 508 large deflections of, 616 spline strip method, in, 602 thermal, 563 vector for, 382, 388, 454, 482, 666 Stress matrices, 436, 438, 439, 483, 521, 669 orthotropic, 509 resultants, 372, 378, 436–445, 509 sandwich plates, of, 536 1023 spline strips of, 603 thermal, 563 vector, 384, 364 Strutt diagram, 950 St¨ussi, F., 16, 285 Szilard, R., 65, 214, 226, 657 T Taylor series, 263, 277 Thermal bending, 561–570 governing equations of, 562–566 methods of solution of, 566–570 thermal loads, for, 564 thermal stretching, 566 Thermal buckling, 961–963 Thermal expansion coefficient, 562 Thick plate elements, 453–457 Timoshenko, S P., 15, 18, 19, 570, 657 Transformation matrices See also Rotation of coordinate systems matrices, 330–345, 376, 382 rotation matrices, 32, 376 Triangular plates, analysis of See also Cases 108–110 of Plate formulas on accompanying CD exact, 63 finite difference method with, 274–275 gridwork method with, 351–355 U Ultimate load technique See also Yield-line method Unit motions of nodal points, 374–375, 403 Units, SI conversion factors, for, 1002 prefixes of, 1001 Use of plates in engineering 3–6 Note: The following letters refer to the boundary conditions of the plates: C = clamped, F = free edge, S = simple supported edge 1024 Index V Variable thickness circular plates of, 140–146 rectangular plates of, 139–140 Variational methods, 185–186 equation of plate bending in, 199 stability analysis in, 908 Vehicle-bridge interaction, 890–892 Vibration, forced, analytical solutions for, 830–834 irregular surface, due to, 835, 892–894 moving loads, due to, 835–836 Navier method 831–833 pulsating loads, due to, 837 Vibration, forced, numerical methods See also Dynamic Module2, 1011–1013 finite element method, with, 870–878 harmonic analysis, 871–873, 880–882 Newmark’s method, 873–875 numerical integration, 873–882 Wilson’s method, 875–877 Vibration, free, numerical methods See also Dynamic Module1, 845–848, 1010–1011 finite difference, 845–856 finite element, 856–882 large amplitude, 895–899 Vibrations of plates, analytical procedures energy methods, 815–823 forced, 830–838 free, 789, 815–823 moderately thick plates of, 839–841 Morley’s formula, with, 825 Stodola-Vianello iteration, 827 using static deflections, 824–830 Vibrations of membranes circular, 812–814 rectangular, 810–811 Vlasov’s method in dynamic analysis, 818–821 large-deflection analysis, 619–621 static analysis, 200–206, 208–211 W Wang, C M., 126, 312, 841, 964 Wilson’s method, 875–877 Work external forces of, 182, 215 internal forces of, 182, 215 virtual, 182, 383 Y Yield-line analysis See also Cases 120–136 of Plate formulas on accompanying CD assumptions, 742–747 concentrated forces, in, 770–780 continuous slabs, for, 764–769 combined loadings, for, 763 deflections, estimating, in, 748–750 equilibrium method, in, 758–763 nodal forces in, 760–761 superposition theorem in, 747, 763 work methods in, 751–757 Z Zienkiewicz, O C., 17, 366, 414, 518, 653 Note: The following letters refer to the boundary conditions of the plates: C = clamped, F = free edge, S = simple supported edge John Wiley & Sons, Inc End-User License Agreement READ THIS You should carefully read these terms and conditions before opening the software packet(s) included with this book “Book” This is a license agreement “Agreement” between you and John Wiley & Sons, Inc “JWS” By opening the accompanying software packet(s), you acknowledge that you have read and accept the following terms and conditions If you not agree and not want to be bound by such terms and conditions, promptly return the Book and the unopened software packet(s) to the place you obtained them 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