j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j STRESSES IN PLATES AND SHELLS STRESSES IN PLATES AND SHELLS A. C. Ugural, Ph.D. Profe."sol' and Chairmall Ml!l ' /wllic:al Eugim"'ring DeplIrtmem Faideigh Dickillson University McGraw-Hili Book Company New York St. Louis San Francisco Bogota Hamburg Johannes burg London Madrid Mexico Montreal N(' w Delhi Panama Paris Sito Paulo Singapoft: Sydney Tokyo Toronto This book was set in Times Roman. The editor was Frank J. Cerra; the production supervisor was Donna Piligra. Fairfield Graphics was printer and binder. STRESSES IN PLATES AND SHELLS Copyright 1981 by Inc. All rights reserved. Printed ill th!! Ullitt!d States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by means, electronic, mechanical, photocopying, recording, or otherwise, with om the prior written permission of the publisher. 1234567890FGFG8987654321 Library of Congress Cataloging in Publication Data Ugura), A C Stresses in plates and shells. Includes bibliographical references and index. I. Plates (Engineering) 2, Shells (Engineering) 3. Strains and stresses, l. Title. TA660.P6U39 624.1'776 80-13927 ISBN 0-07-065730-0 CONTENTS Preface List of Symbols 1 Elements of Plate Bending Theory 1.1 Introduction 1.2 General Behavior of Plates 1.3 Relations 1.4 Stresses and Stress Resultants 1.5 Variation of Stress within a Plate 1.6 The Governing Equation for Denection of Plates 1.7 Boundary Conditions 1.8 Methods for Solution of Plate Deflections 1.9 Strain Energy Methods 1.10 Mechanical Properties and Behavior of Materials Problems 2 Circular Plates 2.1 lot roduction 2.2 Basic Relations in Polar Coordinates 2.3 The AsixYlTImetricai Bending 2.4 Uniformly Loaded Circular Plates 2.5 Effect of Shear on the Plate Deflection 2.6 Circular Plates under a Concentrated Load at Its Center 2.7 Annular Plates with Simply Supported Outer Edges 2.8 Deflection and Stress by Superposition 2.9 The Ritz Method Applied to Circular Plates on Elastic Foundation 2.10 Asymmetrical Bending of Circular Plates 2.11 Deflection by the Reciprocity Theorem 2.12 Circular Plates of Variable Thickness under Nonuniform Load Problems ix xiii 1 I 2 4 6 10 12 14 16 20 22 25 27 27 27 31 32 36 37 40 41 42 44 46 47 55 v \'i rs 3 Rectangular Plates 3.1 Introduction 3.2 Navicr's Solution for Simply Supported Rectangular Plates 13 Simply Supported Rectangular Plates under Loadings 3.4 Levy's Solution for Rectangular Plates 3.5 Levy's Method Applied to Nonuniformly Loaded Rectangular 16 Rectangular Plates under Distributed Edge moments 3.7 Method of Superposition Applied to Bending of Rectangular Plates 3.8 The Strip Method 3.9 Simply Supported Continuous Rectangular Plates 3.10 Rectangular Plates Supported by Intermediate Columns 3.11 Rcclangular Plates Oil Elastic Foundation 3.12 The Ritl Method Applied to Bending of Rectangular Plates Problems 4 Plates of Various Geometrical Forms 4.1 Introduction 4.2 Method of Images 4.3 Equilateral Triangular Plate with Simply Supported Edges 4.4 Elliptical Plates 4.5 Stress Concentration around Holes in a Plate Problems 5 Plate Bending by Numerical Methods 5.1 Introduction 5.2 Finite Differences 5.3 Solution of the Finite Difference Equations 5A Plates with Curved Boundaries 5.5 The Polar Mesh 5.6 The Triangular Mesh 5.7 Properties of a Finite Element 5.8 Formulation of the Finite Elcmt:nt Method 5.9 Triangular Finite Element 5.10 Rectangular Finite Element Problems 6 Orthotropic Plates 6.1 Introduction 6.2 Basic Relationships 6.3 Determination of Rigidities 6.4 Rectangular Orthotropic Plates 6.5 Elliptic and Circular Orthotropic Plates 6.6 Multilayered Plates 6.7 The Finite Element Solution Problems 59 59 59 62 66 74 75 77 80 83 86 89 90 94 96 96 96 98 100 102 105 106 \06 106 III 119 122 123 126 128 129 133 137 140 140 141 142 144 147 148 150 152 7 Plates under Combined Lateral and Direct Loads 7.1 Introduction 7.2 Governing Equatio!l for the Deflection Surface 7.3 Compression of Plates. Buckling 7.4 Application of the Energy Method 7.5 The Finite Difference Solution 7.6 Plates with Small Initial Curvature 7.7 Bending to a Cylindrical Surface Problems 8 Large Deflections of Plates 8.1 Introduction 8.2 Plate Behavior When Dellcctions Are Large 8.3 Comparison of Small- and Large-Dellection Theories SA The Governing Equations for Large Deflections 8.5 Deflections by the Ritz Method 8.6 The Finite Element Solution Problems 9 Thermal Stresses III Plates 9.1 Introduction 9.2 Stress. Strain, and Displacement Relations 9.3 Stress Resultants 9.4 The Governing Differential Equations 9.5 Simply SupP0ricd Rectangular Plate Subject to an Arbitrary Temperature Distribution 9.6 Simply Supported Rectangular Plate with Temperature Distribution Varying over the Thickness 9.7 Analogy between Thermal and Isothermal Plate Problems 9.8 Axisymmctrically Heated Circular Plates Problems 10 Membrane Stresses 111 Shells 10.1 General Behavior and Common Theories of Shells 10.:; Load Resistance Action of a Shell 10.3 Geometry of Shells of Revolution 10.4 Symmetrically Loaded Shells of Revolution 10.5 Some Typical Cases of Shells of Revolution 10.6 Axially Symmetric Deformal ion 10.7 Asymmetrically Loaded Shells of Revolution 10.8 Shells of Revolution under Wind Loading 10.9 Cylindrical Shells of General Shape 10.10 Breakdown of Elastic Action in Shells Problems 11 Bending Stresses 111 Shells 11.1 Introduction J 1.2 Shell Stress Resultants CO:\TF0:TS \'jj J 53 153 153 156 159 163 167 16S 172 174 174 174 175 177 179 181 184 185 185 186 187 188 189 190 191 193 197 198 198 199 202 203 205 215 217 218 220 224 226 231 231 231 ,iii 11.3 11.4 11.5 11.6 11.7 11.9 11.10 11.11 11.12 12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 13 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 Force. Moment, and Displacement Relations Compound Stresses in Shells Strain Energy in the Bending and Stretching of Shells Axisymmetrically Loaded Circular Cylindrical Shells A Typical Case of the Loaded Cylindrical Shells Shells of Revolution under Axisymmctrical Loads Governing Equations for Axisymmetrical Displacements Compari:;on of Bending and Membrane Stresses The Finite Elcmcnt Representations of Shells of General Shape The Finite Element Solution ofAxisymmetrically Loaded Shells Prohlcms Applications to Pipes, Tanks, and Pressure Vessels T ntroduction Pipes Subjected to Edge Forces and Moments Reinforced Cylinders Cylindrical Tanks Thermal Stresses in Cylinders Thermal Stresses in Composite Cylinders Discontinuity Stresses in Pressure Vessels Cylindrical Vessel with Hemispherical Heads Cylindrical Vessel with Ellipsoidal Heads Cylindrical Vessel with Flat Heads Design Formulas for Pressure Vessels Problems Cylindrical Shells under General Loads Introduction Differential Equations of Equilibrium Kinematic Relationships The Governing Equations for Deflections A Typical Case of Asymmetrical Loading lnextensional Deformations Symmetrical Buckling under Uniform Axial Pressure Nonsymmetrical Buckling under Uniform Axial Compression Problems Appendixes A Fourier Series Expansions A.l Single Fourier Series A.2 Hair-Range Expansions A.3 Double Fourier Series B Solution of Simultaneous Linear Equations Rl Introduction B.2 The Gauss Reduction Method References Answers to Selected Problems Index 2.D 235 236 236 239 243 245 247 249 249 25:! 254 254 255 258 260 263 266 269 270 272 273 274 275 278 278 279 280 282 284 287 290 293 296 297 297 299 301 303 303 304 307 309 313 PREFACE The subject matter of this text is usually covered in one-semester senior and one-semester graduate level courses dealing with the analysis ,,{plates and shells. As sufficient material is provided for a full year of study. the book may stimulate the development of courses in aduam:ed statics and structural analysis. The cover-age presumes a knowledge of elementary mechanics of materials. The text is intended to serve a twofold purpose: to complement classroom lectures and to accommodate the needs of practicing engineers in the analysis of plate and shell structures. The material presented is applicable to aeronautical. astronautical, chemical. civil. mechanical, and ocean engineering: engineering mechanics; and science curricula. Emphasis is given computer oriented numerical techniques in the solution of problems resisting mzalytical approaches. The reader is helped to realize that a firm grasp of fundamentals is necessary to perform the critical interpretations, so important when computerbased solutions are employed. However. the stress placed upon numerical methods is not intended to deny the merit of ciassical analysis which is given a rather full treatment. The volume attempts to fill what the writer believes is a void in the world of texts. The book offers a simple, comprehensive. and methodical presentation of the principles of plate and shell theories and their applications to numerous struc-tural elements. including domes, pressure vessels, tanks. and pipes. Theories of failures are employed in predicting the behavior of plates and shells under combined loading. Above all. an eflort has been made to provide a visual inter-pretation of the basic equations and of the means by which loads are resisted in shells, plates, and beams. A balance is presented between the theory necessary to gain insight into the mechanics and the numerical solutions. both so useful in performing stress analysis in a more realistic setting. Throughout the text, the ix author has attcmpted to p r o v i d ~ the fundamcntals of theory and application necessary to prepare students for mon: advanced study and for professional practice. Development of the physical and mathematical aspects of the slIbject is deliberately pursued. The physical significance of the solutions, and practical applications, [lre given emphasis. With regard to application, often classical engineering examples are used to maintain simplicity and lucidity. The author has made a special effort to illustratc important principles and applications with numerical examples. A variety of problems is provided for solulion by the student. Thc International System of Units (SI) is used. The expression defining the small lateral deflection of the midplane of a thin plate is formulated two ways. The first utilizes the fundamental assumptions made in the elementary theory of beams. The second is based upon the diITeren-tial equations of equilibrium for the three-dimellsiollal stress. The former approach. which requires less mathematical rigor but more physical interpreta. tion, is regarded as more appealing to the engineer, and is equally used in the case of thin shells. Emphasized also are the energy aspecls of plate and shel! bending and buckling because of the importance of elJel'gy methods in the solu-tion of many real-life problems and in modern computational techniques. Be-cause of the introductory nature of this book, the classical approaches requiring extensive mathematical background are not treated. Recent publications dealing with shell theory include analytical presenta-tions generally valid for any shell under any kind ofloading. These fomlUlations usually necessitate the employment of tensor notation. vector analysis. and a system of curvilinear coordinates. The theory introduced in this text is a special case of the above. The equations governing shells arc developed only to the extent necessary for solving the mOre usual engineering problems. The finite element method is applied to treat plates of nonuniform thickness and arbitrary shape as well as to represent shells of arbitrary form. The volume may be divided into two parts. Chapters 1 to 9 contain the fundamental definitions and the analysis of plates. Chapters 10 to 13 deal with shells. The book is organized so that Chap. 1 must be studied first. The remain-ing chapters may be taken in any sequence except that Chaps. 10 and 11 should be read before Chaps. 12 and 13. Numerous alternatives are possible in making selections from the book for two single-semester courses. The chapters have been arranged in a sequence compatible with an orderly study of the analysis of plates and shells. This text oITers a wide range of fully worked out illustrative examples, ap-proximately 170 problem sets, many of which are drawn from engineering prac-tice, a multitude of formulas and tabulations of plate and shell theory solutions from which direct and practical design calculations can be made, analyses of plates and shells made of isotropic as well as composite materials under ordinary and high temperature loadings, numerical methods amenable to computer solu-tion. and applications of the formulas developed and of the theories of failures to increasingly important structural members. PRFf A(T xi Thanks arc due to thL' many studellts who ofkred constructive suggestions when drafts of this work were used as a text. Dr. S. K. Fellster read the entire manuscript and made many corrections for which the author is most grateful. He is indebted to Dr. B. Lefkowitz, who read Chaps. 7 and 8, and to Mrs. H. Stanek for her most skillful services in the preparation of tile manuscript. A. C. Uqural LIST OF SYMBOLS It area, constant a, b dimensions, outer and inner radii of annular plate D flexural rigidity [D = E('/12(1 - \,2)] [D] elasticity matrix E modulus of elasticity F resultant external loading on shell element G modulus of elasticity in shear g acceleration of gravity (::e 9,81 m/s2) " mesh width, numerical factor k modulus of elastic foundation, numerical factor. axial load factor for slender mem bers in compression [k] stiffness matrix of finite element [K] stiffness matrix of whole structure m, 11 integers, numerical factors M moment per unit distance, . moment-sum [M = (Mx + iH,.)I(1 + 1')] M* thermal moment resultant per unit distance M x' M)' bend ing moments per unit distance on x and)' planes ,\If xy twisting moment per unit distance on x plane M" M, radial and tangential moments per unit distance M,,, twisting moment per unit distance on radial plane M. meridional bending moment per unit distance on parallel plane M, meridional bending moment per unit distance on parallel p l a n ~ of conical shell M,,, twisting moment per unit distance on axial plane of cylindrical shell N normal force per unit distance N" critical compressive load per unit distance xiii xi" LlST Of ;(1./* thermal force resultant per unit distance N.p N \" normal forces per unit on x and J' planes N X.I' shearing force per unit distance on x plane and parallel to y axis N'J radial and tengcntial forces per unit distance N'I' meridional force per unit distance on parallel plane N,pf! shear force per unit distance on parallel pIa Ile and perpendicular to meridional plane. N Xi' shear force per unit distance on axial plane and parallel to y axis of cylindrical shell Ny normal force per unit distance on parallel plane of conical shell p intensity of distributed transverse load per unit area, pressure p* equivalent transverse load per unit area P concentrated force {Ql nodal force matrix of finite element Q." Q, shear force per unit distance on x and .v planes Q" Q" radial and tangential shear forces per unit distancc Q, shear force per unit distance on plane perpendicular to the axial plane of cylindrical shell Q" meridional shear force per unit distance on parallel plane R reactive forces r radius r, (i polar coordinates r" r,. radii of curvature o[ midsurface in xo and yz planes 1'1,"2 radii of curvature of midsurface in meridional and parallel planes, principal radii of curvature T surface forces per unit area, temperature thickness s distance measured along generator in conical shell fI, l", IV displacements in x, y, and z directions; axial, tangential, and radial displacements in shell midsurface U strain energy T\. eJTective shear force per unit distance on -' and y planes v" I" radial and tangential effective shear [orces per unit distance W work, weight x, y, z distances, rectangular coordinates (f. angle, coefficient of thermal expansion, numerical factor II angle, cylinder geometry parameter [P' = 3(1 _. ,,' l/a'('], numerical factor ;' shear strain, weight per unit volume or specific weight i'x)" j'y! i'.::x shear strains in the xy, y:, and zx planes LIST Of- SY\IBOLS X\-Ir!! shear strain in the 1'0 plane t5 deflect ion. finite difference o p ~ r a t o r . numerical factor, variational symbol (0; nodal displacement matrix of finite element f; norma] strain tx. ty, [;: normal strains in x. y, and z directions en l:" radial and tangential normal strains 1:", 10" normal strain of the parallel circle and of the merid ian Ii angie, angular nodal displacement K curvature v n (J ar ao Poisson's ratio potential energy normal stress normal stresses on the x, y, and z planes radial and tangential normal stresses meridional normal stress on parallel plane principal stresses compressive stress at critical load ultimate stress yield stress shear stress shear stresses on the x, y, and z planes and parallel to the y, z, and x directions c" and C3 are determined from the following conditions at the outer and inner edges w=o 111,=0 M,= 0 (r = a) (I' = b) (e) Upon substitution of the constants into Eqs. (d), the following expression for the plate deflection is obtained IV = 5?e!"b 1(1 __ [--2 + -"-__ ,b' 4D 1 (/' 2(1 + v) (I- --- b' a r' r 2b' 1 + v b rl +- In - + - In -- In --I (2.29) a2 a a2_b21 -v a a It is observed that if the radius of the hole becomes infinitesimally small. b' In (bla) vanishes. and Eg_ (2.29), by letting Ql = Pl2nb. reduces to Eq_ (2..25), as expected. 2.8 DEFLECTION AND STRESS BY SUPERPOSITION The integration procedures discussed in the foregoing sections for determining the elastic deflection and stress of loaded plates are generally applicable to other cases of plates_ It is noted. however, that the solutions to numerous problems with simple loadings are already available_ For complex configurations of loads therefore, the method of superposition may be used to good advantage to sim-plify the analysis_ The method is illustrated for the case of the annular plate shown in Fig_ 2.6a. The plate is simply supported along its outer edge and is subjected to a Po jUJIDHliiJJlli (aJ (b J (c) Figure 2.6 42 STRESsES 11'\ PLATES A:-":D SHELl.S Table 2.3 Variously loaded annular plates Uniform load: = "m," = k2(p(){//tl) Concentrated load: = k3(Pa2/EtI), a,nu = k4{Plt2) A. OtHer edge supported alb k , k, 1.5 0.414 0.976 Pn 2.0 0.664 1.44 I -JI' Fd . U""i 3.0 0.824 1.88 i---- u ---..-j b 4.0 0.830 2.08 5.0 0.813 2.19 B. Inner and outer 1.5 0.0062 0.273 t-'dges clamped 2.0 0.0329 0.71 -'lUJ I P1m.ti 3.0 0.110 1.54 4.0 0.179 2.23 %'i 5.0 0.234 2.80 . I .. ' c. Inner and oute-I" alb k, k , edges clamped 1.5 0.0064 0.22 2.0 0.0237 0.405 1 3.0 0.062 0.703 I 4.0 0.092 0.933 5.0 0.114 1.13 uniformly distributed load Po at its surface. Shown in Fig. 2.6b is a circular plate under a uniform load Po. Figure 2.6c is an annular plate carrying along its inner edge a shear force per unit circumferential length Po b/2 and a radial bending moment, defined by Eq. (2.18), 1'0(3 + v)(a' - b')j16. The solutions for each of the latter two cases are known from Sees. 2.4 and 2.7. Hence, the deflection and stress at allY point of the plate in Fig. 2.6a can be found by the of the results at that point for the cases indicated in Fig. 2.6b and 2.6c (Prob. 2.14). Employing similar procedures, annular plates with various load and edge conditions may be treated. Table 2.3 provides only the final results for several examples.7.' In all cases v. is taken as v = 0.3. Design calculations are often facilitated by this type of compilation. 2.9 THE RITZ METHOD APPLIED TO CIRCULAR PLATES ON ELASTIC FOUNDATION In the problems discussed thus far, support was provided at the plate edges, and the plate was assumed to undergo no deflection at these supports. We now consider the case of a plate supportedcontinllollsly along its bottom surface by a foundation, itself assumed to experience elastic deformation. The foundation reaction forces will be taken to be linearly proportional to the plate deflection at CIRCULAR Pl.ATES 43 any point. i.e., 'vI.:.. I-Jere l\' is the plate dc1kctioll and k is a constant, termed as the modulus the .!cJ/lI1dmiol1 or bedding COflstal1l or the foundation material, having the dimensions of force per unit surlacc area of plate per unit of deflection (e.g., Paim). The above assumption with respect to the nature of the support not only leads to equations amenable to solution, but approximates closely many real situations.' Examples of this type of plate include concrete slabs, bridge decks. floor structures, and airport runways. We shall apply the Ritz method (Sec. 1.9) to treat the bending of a circular plate of radius a resting freely on an elastic foundation and subjected to a center load P. In this case of axis ymmetrica I bending. the expression for strain energy given by Eq. (P2.1l) reduces to . f(d2W Id\l')2 2(1 -l')dWd'W.l U = nD I -,- +-- --'--'-, I'dI' 1 . 0 d,." I' dl' ,. dr dr (2.30) A solution can be assumed in the form of a series (a) in which en are to be determined from the condition that the potential energy n of the system in stable equilibrium is minimum. If we retain, for example, only the first two terms of Eq. (a) (b) and the strain energy, from Eq. (2.30), is then [J, = 4clDrra2(1 + v) (e) The strain energy owing to the de/ormation of the elastic foundation is determined as follows .21t .," U2 = J I !kw2r dr dll = trrk(coa2 + COC2a4 + !da6) (d) o '0 The work done by the load is given by W=P'(w),"o=Peo (e) The potential enerlo'Y, IT = U, + [J 2 - W, is thus n = 4clDrra2(1 + 1') + j: (e5a' + coc,a4 + - Peo Applying the minimizing condition, aIT/ilc, = 0, we find that Co = rr:a, [I + (1/3) + + V)/ka4j P (j) 44 STRESSES IN J>LATFS AND SHELLS Theil by substituting Egs. U) into (b), we obtain the maximum del1ection at the center (,. = 0): >I'm" = [I + + v)/";;;; J (2.31 ) An improved approximation results from retention of more terms of the series given by Eq. (a) 2,10 ASYMMETRICAL BENDING OF CIRCULAR PLATES In the foregoing sections, our concern was with the circular plates loaded axisymmetrically. We now turn to asymmetrical bending. For analysis of deflection and stress we must obtain appropriate solutions of the governing differential equation (2.5). Consider the case of a clamped circular plate of radius a and subjected to a linearly varying or hydrostatic loading represented by ,. I' = Po + 1'1 cos 0 a as shown in Fig. 2.7. The boundary conditions are w=o where W = Hlp + Wh. ow - =0 a,. (r = a) (a) (b) The particular solution corresponding to Po, referring to Sec. 2.4, is = l'or4/64D. For the linear portion of the loading, 1/ Plr5'cos 0 \ w =A--_-P a Introduction of the above into Eg. (2.5) yields A = 1/192D. We thus have 1'01'4 1'1'" cos 0 Wp = 64D + 192aD (c) f a o r L Figure 2.7 cmCULAR Pl.ATES 45 It is noted that the general method of obtaining the particular solution or Eq. (2.5), given in Prob. 2.20, follows a procedure identical with that described in Sec. 3.4 (Levy's solution for rectangular plates). The homogeneous solution IV" will be symmetrical in e; t h u s r ~ in Eq. (2.7) vanishes. Owing to the nature of p and wp' we take only the terms of series (2.8) containing the function /0 and /,. Tbe deflection w" and its derivatives (or slope, moment, and shear) must be finite at tbe center (r = 0). It follows that Bo = Do = B, = D, = 0 in expressions for f ~ andr,. Hence, (d) The conditions (b) combined with Eqs. (e) and (d) yield two equations in the four unknown constants Since the term in each pair of parentheses is independent of cos 0, a solution exists for all 0 provided that 4 poa 2 64D- + Ao + Coa = 0 which upon solution, leads to poa4 AD = - -----64D The deflection is theretore C,= 2p,a ----192D PIa' A, = '192D "' = Po (a2 _ r2)' +..J'.!.....!: (a2 _ ,,2)2 cos 0 64D I92Da The center displacement is (e) (2.32) (f) 46 STRESSES I ~ PLATES A0:D SHELLS We observe that. when the loading is uniform, 1', = 0 and Eq. (1.32) reduces to Eg. (2.14) as expected. The expressions for the bending and twisting moments are, h'mn Egs. (2.32) and (2.2) Po [ 2( )' 1', [1'3 ] M, = 16 a 1 + v - 1'-(3 + \')) - 48 -;; (5 + v)- ar(3 -+ \') cos iJ M = l', [(l2(1 + v) - 1"(1 + 'lv)] - '. [,,3 (5v + 1) - ar(3v + I)] cos 0 (2.33) II 16' - 48 a M,.!) = _ I ~ V p, ra( I - ~ ; : ) sin (I The case of the simply supported plate under hydrostatic loading can be treated in a similar way, 2.11 DEFLECTION BY THE RECIPROCITY THEOREM Presented in this section is a practical approach for computation of the cenler deflection of a circular plate with symmetrical edge conditions under asymmetri-calor nonuniform loading. The method utilizes the reciprocity theorem together with expressions for deflection ofaxisymmetricaUy bent plates. Consider, for example, the forces P, and P2 acting at the center and at r (any 0) of a circular plate with simply supported edge (Fig. 2.8). According to the reciprocity theorem,' due to E. Betti and Lord Rayleigh, we may write: (a) That is, the work done by P, owing to displacement w" due to 1'" is equal to the work done by P, owing to displacement IVI2 due to 1',. For the sake of simplicity let P, = 1, IV" = w" and IV" = "'(r). The deflection at the center IV, of a circular plate with a nonuniform loading p(r, IJ) but symmetric boundary conditions may therefore be determined through appli-cation of Eg. (a) as follows ,211: ,0 w, = I I p(r, (1)",(1')1' dr diJ (2.34) '0 '0 Clearly, ",(r) is the deflection at r due to a unit force at the center. In the cases of fixed and simply supported plates, Iv(r) is given by the expressions obtained by setting l' = 1 in Eqs. (2.22) and (2.25), respectively. , ........... - ... --) C'IRClJl.AR PLA1TS 47 To illustrate the application of the approach. reconsider the bending of the plate described in Sec. 2.10. Upon substituting 1'(1'. Ii) = Po + PI (ria) cos () and Eg. (2.22) into Eg. (2.34), setting I' = 1 and integrating. we have lV, = D- (" ((Po + p, ': cos 0)(2/'2 In I' + ,,' _,,2)1' d" dO = rr . 0 0 a " 64D The above is identical with the value given by Eg. (I) of the preceding section. 2.12 CIRCULAR PLATES OF VARIABLE THICKNESS UNDER NONUNIFORM LOAD In this section we discuss an approximate method for computing stresses and deflections in solid or anllllal circular plates of variable thickness, subjected to arbitrary lateral loading9 Except for the requirement ofaxisymmetry, no special restrictions are placed on the manner in which either the thickness or the lateral loading vary with the radial coordinate. Several applications immediately come to mind: turbine disks, clutches, and pistons of reciprocating machinery. Consider a circular plate (Fig. 2.9a), and the division of the plate into small (finite) ring segments, as in Fig. 2.9b. Note that radial lengths of the segments need not be equal, but that the thickness is taken as constant for each. For each element defined as in Fig. 2.9b. the development of Sec. 23 applies. In order to accommodate the substitution of a series of constant-thickness elements for the original structure of varying thickness, it is necessary to match slopes and moments at the boundary between adjacent segments. The boundary conditions are handled in the usual manner. As the method treats the plate as a collection of constant-thickness disks, it is unnecessary to determine an analytical expres-sion for thickness as a function of radius. Prior to illustrating the technique by means of a numerical example, the general calculation procedure is outlined. A knowledge of the derivation of the basic relationships [Sec. 2.12(b)) is not essen-tial in applying the method. --r , r I _ (a) (b) Figure 2.9 48 STRESSES IN PLATES AND SHELLS (11) Calculation Procedure The expressions developed in Sec. 2.12(b) may be so arranged as to facilitate the calculation process. Consider, with this end in mind, a plate subdivided into a number of annular elements with the applied lateral loading on each element denoted Q, the average load on that element (Fig. 2.9). We shan apply the notation i+v i-v ., p, = In p - s;r- (I - (r) 1- p' p2 ),' = ------- + '---- In p 4(1 + v) 2(1 - v) , I - p2 I + p' "Q = ---g;- + -g;;- In p (J. =t(1 -- fI') Et' D= 12(! _ 1'2) I+v I-v fi. = -- In p + --(I _ p') 4" 8" 1- p2 p2 2, = 4(1 + In p In addition, for each joint between adjacent elements, 3 n- tn+.! I '/ _. 3 tn (2.35) (2.36) The notation thus introduced is next applied in the determination of the follow-ing quantities. Bending moments The change in bending moment in proceeding from the inner edge to the outer edge of any element may be ascertained by rearranging Eqs. (2.44) as follows: Ll.M, = a(M" - M,,) + PJJ Ll.M, = - M,,) + PoQ (2.37) At the inner edge, the moments are either given or assumed. The outer moments acting on an element are then MrQ=Mrl+/lM, (2.38) M,o = M" + Ll.Mo Similarly, the moment increments corresponding to the interface between adja-cent elements arc Ll.M, = 0 (2.39) It now follows that the moments at the inner dege of the next element are found from (M"),,+l = (M,,), (Mo,)" + 1 = (Mo,)" + Ll.M, (2.40) Table 2.4 Given inner houndary conditions AIr" Clamped Solid plate Assumed values at inner boundary of plate (A and 8 are any arbitrary values B of 0) Particular HOJnogl!1l0US solution solution A-1;,,, = AI", M;r.=O A = B iH;" = A B M;J,,"'" I'M:" = I'M;" = A M;,,=B = M;Q = M;,. CIRCULAR PLATES 49 When Egs. (2.37) to (2.40) are applied successively, beginning with the innermost element, the moments at any intermediate edge may be found in terms of the moment at the inner boundary of the disk. Boundary conditions The following steps are taken to satisfy the boundary con-ditions at the inner and outer edges of the disk: Step 1. Apply Egs. (2.37) to (2.40) to obtain a particular solution (denoted by a single prime). Begin the calculations with the appropriate inner boundary values specified in Table 2.4. Step 2. Repeat step 1 with Q = 0 to obtain a homogeneous solution (denoted by a double prime). Seep 3. Superimpose the values found in steps 1 and 2 to obtain the general solution: M!I= + (2.41 ) The constant k is calculated as indicated in Table 2.5 from the given bound-ary conditions at the outer edge of the disk. Table 2.5 Given outer boundary conditions M'b Clamped Formula for kin Eq. (2.41) k (M" _. M;,)/M;, k =< I'M;h) 50 STRESSES IN PLATES AND SHELLS To verilY the correclness of the final results, use thl? values found in the final step to perform the calculations indicated in slep 1. The results thus obtained should be the same as those already found at the conclusion of slep 3. Stresses On the basis of the moments now known. the bending stresses are calculated from: 6M, a , (' (a) The stresses thus found will show a stepped distribution throughout the disk owing to the nature of the analysis. The actual distribution of stress may be approximated adequately by drawing a smooth curve through the calculated points. Deflections The change in deflection, Ll.w, is for each element found by rearrang-ing Eq. (2.45): (2.42) where 111, and Mo are given by Eqs. (2.41). The total deflection at any point in the plate is thus computed by adding the increments given by Eq. (2.42). 150 kP3 (pressure) 102.25 kN (on 9kN diu. circle) (on I dia. circle) 36 kN (on 360mrn-dia. circle) (a) Figure 2.10 I"f) ::: 360 ----_._---r = 290 r = 230 /";; 180 ,." 130 r::: 90 (b) 1/=5 I" J 2 n=4 t" 16 t::: 20 n=2 t = 25 11=1 t" 30 OJ "-2.08 Shearing force (kN) (e) CIRCULAR J>l.ATES 5] The foregoing apply to statically determinate prohlems. \Vhel1 indeterminate siluations af\..' encountered. the above procedure remains applic-able but requires the superposition of a number of determinate cases. Example 2.2 Calculate the stress distribution and the detlections of the steel disk shown in Fig. 2.10a. The member is frce at the inner and outer edges. Divide the plate into five segments as in Fig. 2.lOb. Let E = 207 GPa and ,,= OJ. All dimensions are in millimeters. SOLUTION The uniform lateral loading applied to each segment is shown in Fig. 2.10c. The results of the complete calculation are presented in Tables 2.6 through 2.9. Table 2.6 lists the various coefficients calculated on the basis of Eqs. (2.35). By assuming for the free inner edge, = M" = O. = 100 (arbitrary), Eqs. (2.37) through (2.40) provide the particular solution (Table 2.7). The foregoing step is repeated for g = 0 to find the complementary solution (Table 2.8). The constant, k, is next obtained by applying the ex-pression for the outer edge (Table 2.5): k = At,,, - = 0 - = 236.995 25.125 The final bending moments M, and tv1. are then calculated using Eqs. (2.41). Following this, Eqs. (a) and (2.42) provide the final stresses and deflections (Table 2.9). Table 2.6 Plate coefficients Eh':ment number, II Symbol Units 2 3 4 5 r, m 0.090 0.130 0.180 0.230 0.290 ,. III 0.130 0.180 0.230 0.290 0.360 " I III 0.Q30 0.025 0.020 0.016 0.012 Q N -2080 94450 49970 37790 10720 0.26035 0.23920 0.19176 0.18550 0.17554 II, -0.05254 0.04699 -0.03615 -0.01411 -0.03215 II, -0.02354 .. 0.02024 -0.01456 --0.01365 -0.01259 0.421l0 -0.48800 -0.48800 -0.57813 )., -0.02575 --0.02924 -0.03271 -0.03280 -0.03271 0.22603 0.21324 0.18176 0.17549 0.16774 )'Q -0.00093 -0.00067 -0.00031 -0.00026 -0.00022 (r;/D)IO' m/N 3.305 10.948 34.912 108.40 395.97 52 IN PLATES AND SHELLS Table 2.7 Particular solution Element number. 11Symbol 2 3 4 5 0 135.318 -4314.148 135.318 -4313.148 5587.744 -6267.813 - 5954.571 100 88.241 1564.256 2520.007 122.91g -1821.611 -2824.638 -3396.600 3312.862 Table 2.8 Homogeneous solution Element number, n Symbol 2 .l 4 5 M;, 0 26.DJ5 30.833 29.830 27.639 M;. 26.035 30.833 29.830 25.125 M;; 100 46.094 15.657 lUJ17 13319 73.965 41.296 26.660 20.208 15.833 Table 2.9 Final moments, stresses, and deflectioll' Element number, n Symbol 2 3 4 5 M.; 0 6305.4g 2993.14 1481.89 282.49 Alro 6305.48 2993.14 1481.89 282.49 0 MOi 23799.5 11012.31 4516.44 2005.73 M,. 17652.26 7965.33 3493.72 1392.71 439.41 (a,,) x 10-' a 60.533 44.897 34.732 11.770 (a,.o) x 42.036 28.734 22228 6.621 0 (a,,) x 158.66) 105.718 67.747 26.522 (atl,,) x 10 (, 117.682 76.467 52.406 32.642 18.)09 (i1w) x 10' 1.778 2.999 3.182 3.769 (IV,,) x 104 1.778 4.078 6.548 9.730 13.499 (b) Basic Derivation The differential equation governing a circular plate of constant thickness, sub-jected to a constant shearing force Q = 2nrQ" from Eq. (2.9c) is Q ---2nrD (2.43) CIRCULAR Pl.ATES 53 fntegrating this expression twice. we obtain dl\' 1 1 Qr c, r - c, J - = - - -- (2 In r - 1) + ~ + ---d,. D 8" 2 /' (6 ) where C1 and ('2 are constants of integration which may vary from element to element. The derivative of Eg. (b) is -- = - - - (2 In I' + 1) + -'- --'i d'\\' 1 1 Q e Co J d,.2 D 8" I' ,." (c) Substituting Egs. (b) and (e) into Egs_ (2.9a) and (2.91, we have Q (I + v)c, "2 M, = - 8,,[2(1 + v) In I' + (1 -- v)] + --T-- - (1- v)7 Q (I + v)e, ('2 Mo= --[2(1 + v) In,.- (1- v)] +---- + (1 - v), 8" - 2 1'-(d) Solving for (', and c, at the inner edge 1', of the element in terms of bending moments M". M 0' , yields c, = Mo, + M" + Sf In r-l+v 2n' ('2 = [ M ~ = ~ ~ " -- ~ N (e) Next, Egs. (e) are substituted into Egs. (d) and the bending moments M" and Mo. at the outer edge of the same clement are found: 1 1 ,.2 M" = 2-- (MOi + M,,) - ')..y (M" - M,,) ~ r, Q ,. Q (1") +-(1 + v) In -'----(1 - v) 1--'-4n tv 8n r;; I 1 1'2 Moo = '2 (M" + M,,) + ' 2 ~ (Mo, - M,,) en Q ,. Q (?) +-(I+1')ln-'+--(I-v) I-..y 4n rv 8n ro Continuity across the joint between adjacent elements is satisfied by eguat-ing the radial moments M, and slopes dwjdr at each side of the interface. To accomplish this, d2wjd,.2 is eliminated between Egs. (2.9a) and (2.9b) with the result for adjacent elements 54 STRESSES IN PLATES J\i"iD SHELLS (M"" = (1 r')/)" II tr" (/' 0 /I (AJ" - I'M,;)" = (1- 1'2)D" " I!, )'/,,+1 (2.44) Because the quantities I' and dwidr must be equal ill both of the above expressions. It then follows that the second of Egs. (2.44) divided by the first leads to (g) Expressions (2.44) also indicate that in the event the edge of a disk is fixed, ( since dw ) - =0 dr (il) This relationship is applied in the construction of Table 2.4. The change in deflection w in proceeding from one edge to the other of the same element is found by first integrating Eq. (b) over the radial length of an element: .11'" dw Q .. ro Q ,r" C1 ,rIO ,ro 1 D I dr = - - I I' In r dr + -- I rill' + ..... I I'dI' + (2 I - dr WI dr 4n 8rr J "Yj "Y; r ( i) Performing the indicated operations, we obtain n c (,,2 ,,2) ,. 2 2 2 2- 1 Q - i i D tJ.w = --- [I" In r + r - I' In I' r.] + - .... - - c, In -8n I 0 0 (I I 2 2 - fo (j) Introducing c, and C2 from Eqs. (e), the above assumes the form DA MOi+Mrir;-r; l.\l-\! = , 2 2(1 + 1') (2.45) The boundary conditions at the inner and outer edges of the plate are related to the above expressions in the following manner. By direct substitution, the general solution of dw/dr in Eg. (2.43) may be shown to be expressed dw = (dW)' + k dr dr til' (2.46) ClRn'LAR PLATES 55 where the single prime denotes the particular solution, the double prime indi-cates the homoYl'Ill!oUS sollltion. and J.:. repr':-scllts a constant. The same relation-ship applies to the moments given by Eqs. (f). Referring to Table 2.4, note that the particular and homogeneous solutions may be selected so as to satisfy the conditions imposed at the inner edge of the disk regardless orthe value of k. The constant k can thus be selected to satisfy the prescribed condition at the outer edge only, as indicated in Table 2.5. PROBLEMS St'('s. 2.1 10 2.8 2.1 A pressure control system includes Ii thin steel disk which is to clost' an eit'ctrical circuit by deflecting 1 mm at the center when the pressure attains a value of 3 MPa. Calculate the required disk thickness if it has a radius of 0.030 m and is built-in at the edge. Let v = 0.3 and E = 200 GPa. 22 A cylindrical thick.walled vessel of 0.25 m radius and flat thin plate head is subjected to an internal pressure 7 MPa. Determine: (a) the thickness of the cylinder head if the allowable s.tress is limited to 90 MPa: (b) the maximum deflection of the cylinder head. Usc E = 200 GPa and v = 0.3. 2.3 An aluminum alloy (6061T6) flat simply supported disk valve of 0.2 m diameter and 10 mm thickness is subject to a water pressure of 0.5 MPa. What is the factor of safety, assuming failure to take place according to the maximum principal stress theory. The yield strength of the material is 241 MPa. 2.4 The flat head of a piston is considered (0 be a clamped circular plale of radius (I. The head is under a pressure ( ,.)" f} = fin ~ ~ where Po is constant. Derive the equation p a' [('.)' ('.)' I w = Si6D ~ - 3 ~ + 2 (P2.4) for Ihe resulting displacement. 2.5 10 2.7 For the circular plates loaded as shown in the Figs. A, B, and C of Table 2.2, verify the results provided for the maximum deflections. 2.8 A simply supported circular platt! i!> under II rotationally symmetric lateral load which increases from the center to the edge: Show that the expression (Pc.8) S6 STRt:SSES IN 1'1An:S AND SHELLS where represent lhe re.'iultillg displacement. 2.9 Verify the result given by Eq. (2.8). [Him: Introduction of Eq. (2.7) into Eq. (2.6) to (N 0, 1. 2, ... ) (P2.9) (N 1,2, ... ) Solution of these equid imcnsionul equations can bc taken as: 1(' f,(r) = d" r" andf! (r) = ",/', wherein {I" and bIt are constants and the ):s arc the roots of auxiliary equation of Egs. (P2.9).] 2.10 Verify the result given by Eg. (2.13): (a) by integrating Eq. (2.12); (b) by expanding Eq. {2.lOa}, setting 1 = In r, and thereby transforming the resulting expression into an ordinary differential equa-tion with constant coefficients. 2.11 Show that Eq. (1.34) for the strain energy results in the following form in terms of the polar coordinates (I i"w 1 aWl'] +2(1 rdrdO for j'l() /,2 cO (P2J 1) 2.12 Calculate the maximunl deflection w in the annular plate loaded as shown in Fig. 2.5a by setting a = 2b, MJ = 2M2, and V= 0.3. 2.13 Determine the maximum displacement in tht! annular plate loaded as shown in Fig. 2.5b by setting a = 2h and \' = 0.3. 2.14 A pump diaphragm can be approximated as an annular plate under a uniformly distributed surface load Po and with outer edge simply supported (case A in Table 2.3). Compute, using the method of superposition, the maximum plate deflection for b = a/4 and v = 0.3. Compare the result with that given in the table. Set:s. 2.9 to 2.12 2.15 An aircraft window is approximated as a simply supported circular plate of radius a. The window is to a -uniform cabin pressure Po. Determine its maximum deflection, assuming that a diametrical section of the bent plate paraholic. Employ the Ritz method. Take v = 0.3. 2.16 Redo Prob. 2.15 for a simply supported circular plate that is loaded only by a concentrated center force P. 2.17 Determine maximum deflection of a structural steel circular plate with free end resting on a gravel-sand mixture foundation and submitted to a load P at its center. Use the Ritz method, taking the first three terms in Eq. (a) of Sec. 2.9. Let k = 200 MPa/m, Cl = 0.5 m. t = 40 mm, E = 200 GPa. and v = 0.3. CIR('UL,\H Pl._AlES 57 2.18:\ i,lmply 'iupported l.'irculal' plate loaded by u,:vlllmetrically distributed edgf described by AI, = I IU"COSIIO (t II) /_"0 I, 1 ... III this case, it is observed thaI \\'1' = 0 and thus \1' = \\'h reduces to II = I (A"r" + C"IJ"'2) cos nO "r/,\ ... Verify that the resuiting deflection is 11'= (P2.18) 2.19 Detennine the expression for the radial stress in the plate described in Prob. 2.18 by taking /1 = 0, 1. 2.20 The particular solution of Eq. (2.5), for an arbitrary loading p(r, 0) expanded in a Fourier series pit. U) 1',(1") + l: [P"(I") cos 110 + R.(t) sin 110] (a) J where 1 .' p.(,.) -/ rI>', 0) cos ,,0 dO n- .. r. (II O. 1. ... ) 1 ." R,,('") - I p(t, 0) sin "U dO (II 1,2 .... ) is expressed in the general form:6 I\'JI = Fo(r) + L [Fn(r) cos nO + Gn(r) sin nO (Pl.20) n""J Here /-'0(1'), F,,(r), !". When Eqs. (h) are inserted into Eq. (3.25). an expression ror the plate deflection is established. In the case or a square plate (a = h) the center deflection and the maxi-mum bending moment are found to be (Prab. 3.10) w = 0.0028 (x = i ' y = ) (i) Situations involving other combinations of boundary conditions, on the two opposite arbitrary sides of the plate, may be treated similarly as illustrated in the following example. Example 3.5 The unirorm load Po acts all a rectangular balcony reinforce-ment plate with opposite edges x = 0 and x = a simply supported, the third edge y = h free, and the edge y = 0 clamped (Fig. 3.6). Outline the derivation of the expression for the deflection surrace lV. SOLUTION For the situation described, the boundary conditions, Eqs. (1.25) to (1.27), are IV = 0 (l2w ---0 ox' -(x=O,x=a) (j) au.' (.1' = 0) (k) w=o - =0 0.1' (.1'=1 (/) __ -(l---.. .. // ///,"//' . x f: ,-iu.l_"-,, I "'\ Free t .1' Figure 3.6 PLATES 73 r--, -, --_t_ ...-I I I I , , t l' Figure 3.7 The particular and the homogeneous solutions are given by Egs. (3.20) and (3.16), respectively. both of which satisfy the conditions (j). Applications of Egs. (k) and (I) to w" + IVp leads to definite values of the constants A"" B,,,, em, Dm The deflection is then obtained by adding Ego. (3.16) and (3.20). Example 3.6 Derive an expression for the deflection surface of a very long and narrow rectangular floor-panel subjected to a uniform load of intensity Po (Fig. 3.7). Assume the edges x = a. x = (I, and y = 0 arc simply supported. SOLUTION The plate deflection can readily be obtained by superposing the solution for an infinite strip lVp given by Eg. (3.20) with a suitable solution w" of Eg. (1.18). It is observed from Eq. (3.14) that in order for the coefficients of Wh, .t;n. and its derivatives to vanish at y = ro, and C:n should be equated to zero. Hence. the homogeneous solution w" can be represented by the following expression IT:! nlnx w = '"" (8' + D' J')e-mff}'la sin ---"--Ir L If! m a (3.26) The above. of course. satisfies Eq. (1.18). The boundary conditions of sides x = 0 and x = a are fulfilled by Eqs. (120) and (126). It remains now to determine B;" and in such a manner as to satisfy the boundary conditions on side y = O. Substituting lV = lV11 + wo_ into w = 0 and a2,,\'/cy2 = 0 and setting y = O. we obtain two equations which after solution yield B;" = '-- 4Poa4/n' Dm' and D;' = mnB;'/2a. It then follows that the elastic surface is given by The caseS involving a clamped edge at y = 0 @1' a Fee edye at y = 0 may be treated in a like manner, applying Eqs. (1.25) and (1.27). respectively. 74 STRESSES 1:-"- PLATES AND SHILLS 3.5 LEVY'S METHOD APPLIED TO NONlJNIFORMLY LOADED RECTANGULAR PLATES Levy's approach is now applied to the treatment of bending problems of rectangu-lar plates under nonuniform loading which is a function of.\ only. Bounding the plate as shown in Fig. 3.4. and assuming that tbe edges s = 0 and x = (I are simply supported, the loading is expressed by the Fourier series: where ':.1:: mrrx p(x) = L 1'", Sill --. m.7.1,2.... a (3.28 ) 2 .4 and the bending moment at the fixed edge 5 poa'b" At r =8 2a4'+Sb4 (y = 0) (x = a) (a) (h) (e) IHOCTA:-':C;l'L ,\R PLATES Po.l In the' case 01 a sqllare plate (0 = h), Eqs. (a) to (e) red lice to \I' = O.OOJ72Pn((J.,.D IV! x = 0.OS02po ,,2 My = 0.03571'0 a' lv! { = 0.08931'oa' The dellect ion at the center of the plate is 33 percent greater and the hend ing. moment ;\1 r is 11 percent greater than the values obtained by the bending theory of plates (Example 3.4). 3.9 SIMPLY SUPPORTED CONTINOOUS RECTANGULAR PLATES When a uniform plate extends over a support and has more than one span along its length or width, it is termed continllous. In a continuo liS plate the several spans may be of varying length. Intermediate supports are provided in the form of beams or columns. Only a continuous plate with a rigid intermediate beam i,; treated in this section, i.c., the plate has zero delleetion along the axis of the supporting beam. We shall assume that the beam does not prevent rotation of the plate. Hence. the intermediate beam represents a simple support to the plate. Consider the two-span simply supported continuous plate, half of which is subjected to a uniform load of intensity Po (Fig. 3.1la). A convenient way of looking at the problem is to draw a free body diagram of each rectangular panel as shown in Fig. 3.111>. The distributed moment along the common edge may be represented by a Fourier series t b I ' (a) Figure .1.J I b ' ! (b) I c 2 (0) 84 STRESSES 11" PLATES AND SI'II:LLS \Vhcn the set of coefficients A1m is determined, the expressions ror thc boundary conditions can be handled with ease. The lateral deflection of each simply sup-ported replacement plate may be obtained by application of Levy's method. Referring to Figs. 3.5 and 3.11" we conclude from the symmetry in deflections that the general solution for plate I is given by Eq. (3.25) if y " replaced by x" x by y, and a by />. That is where (/ Similarly, for plate 2, setting Po = 0, the deflection expressed in terms of the coordinates X2 and y, and for a different set of constants is 00 W2 = L: [Em sinh A.mX2 + Fill cosh AmX2 m=t.3, .. The boundary conditions for plate I and plate 2 are represented as follows, respectively: w, =0 82wl (Xl = 0) --2 =0 OXl (e) a2}VI w w, = 0 -D--;- = L Mm sin }'mY (x, = a) ilx, .. and W2 = 0 82w2 (X2 = a) -=0 a2H.'2 ", IV2 = 0 -D-= L /1..1 m sin Am Y (X2 = 0) m= 1. 3 .... (d) We thus have eight equations (e) and (d) containing nine unknowns Am . ... , H." Mm. The required additional equation is obtained by expressing the condi-tion that the slopes must be the same for each panel at the middle support. This continuity requirement is expressed RECTANGl'LAR PL.'!'TI:S 85 Introducing Eqs. (3.44) and (3.45) into the above we have Application of the edge conditions (e) and (d) to Eqs. (3.44) and (3.45) leads to values for the constants as follows: 1 I coth }'m" ItV!m 4 = -- a --.-- _ .. -m ." '1' f) I sll1h I'ma .... 1./11 csch Ama [Mm - --... ~ - . --.. _-D 2}.m 21'0 'J + X41; (-1 + cosh Ima) and Mma ( 2) E = - - - ' ~ 1 "- coth ~ " In 2)'IJID' m I'm = 0 A1m , H = - .-- ... coth I a III 2)'mD "III (3.47) (3.48) Having Eqs. (3.47) and (3.48) available we obtain. from Eq. (3.46). the moment coefficients M",. Equations (3.44) and (3.45) then give the dellection of the con-tinuous plate from which moments and stresses can also be computed. The foregoing approach may be extended to include the case of long rectan-gular plates with many supports. subjected to loading which is symmetric in y. In so doing, an equation similar to that of the three-moment equation of contin-uous beams is obtained.' It is noted that there are situations where the inter-mediate beams arc relatively flexible compared with the flexibility of the plate. The deflections and rotations of the plate arc not then taken as zero along the supports, but are functions of the hending and torsional stitTnesses of the sup-porting beams'" The design methods used in connect ion with continuous plates ut ilize the solutions derived in the foregoing sections and a number of approximations1i 86 STRESSES IN PLATES A ~ D SHELLS 3.10 RECTANGUl.AR PLATES SUPPORTED BY INTER MEDIA TE COLUMNS In this section we consider the bending of a thin continuous plate over many columns. To attain a simplified expression for the lateral deflection it is assumed that: the plate is suhjected to a uniform load Po. the column cross sections are so small that their reactions on the plate are regarded as point loads, the columns arc equally spaced in mutually perpendicular directions, and the dimensions of the plate are large as compared with the column spacing. The loregoing set of assumptions enables one to assume that the bending in all panels. away !i'om the boundary of the plate, is the same. We can therelore restrict our attention to the bending of one panel alone, and consider it as a unifomlly loaded rectangular plate (a x b) supported at the corners by the columns. The origin of coordinates is placecl at the center of a panel shown by the shaded area in Fig. 3.12a. Clearly, the maximum dellection occurs at the center of the panel. A solution can be obtained utilizing Levy's approach (Sec. 3.4). Accordingly, the deflection may be expressed as a combination of that associated with a strip with uniform load and fixed ends y = b/2 and that associated with a rectangu-lar plate. That is, P b* ( 4V2)2 IV = wp + w" = 3 ~ 4 D 1 - bi + Bo + L: Bm cosh_- + Em _. sinh w ( 1nny mny 111""'2,4. ... a a -...J 1--.2c '. . I> lal Figure 3.12 mny) mn.\: "--- COS ---a a (h) (3.49) RECTA!'..:Gl"L,\R PI.ATf'S 87 where Bu, Bm, and Em are constants of WI!' yet to he determined, The aboH" satisfies the boundary conditions for the rectangular panel along the x edges, thv -- = 0 ax and Eqs. (1.17) and (1.18). (a) For purposes of simplifying the analysis, the support forces are regarded as acting over short (infinitesimal) line segments, between x = a(2 - e and x = 1/2 + c or 2c (Fig. 3.12,,). The plate loading is transmitted to the columns by the vertical shear forces. From the conditions of symmetry of the bent panel we are led to conclude that the slope in the direction of the normal to the boundary and the vertical shear force vanish everywhere on the edges of the panel with the exception of the corner points. Thus, Qy=O (o J -30 10 0 30 20 Symmetric --6 -2 0 6 Symmdr:c . [ k J ~ " (" r 0 0 0 0 0 0 [k.d = -6 (l -8 6 \1, .h 30 . 15 0 -30 -15 () 60 -84 " -6 84 " " 84 -15 HI 0 15 0 -30 20 6 -, (l -6 2 (J 6 0 (l 0 0 0 (I 0 0 0 6 (l -6 ,) -, " -30 t5 0 30 IS " -60 30 () 60 " 6 " -84 -6 .. 6 -84 6 " :';4 --15 0 15 lO 0 - 30 10 0 30 20 6 2 0 -6 -R I) - 6 Il (, Il 0 \) I) \} 0 0 \} 0 0 0 0 -6 \} 6 \l - 6 -, .. (1" , [[') [ll) 10) 10]1 [I 0 0] IR) = (0) I') [0] (0) where ['I = 0 ; 0 [0] 10) ['I [OJ 00(/ [OJ [0) [OJ ['I :;; '" 136 STRESSFS [.....; A1\;D SHELLS -I I.il (a) Actual plate Figure 5.15 t--: I \ ,/ Sliding edges (b) Substitutt! plate te) Nodal force and dis plact'lll ents Example 5.8 Consider a square plate of sides II with two opposite edges x = and x = a simply supported and the remaining edges clamped (Fig. 5.15(/). Compute maximum value of 1\' if the plate is subjected to uni-formly distributed load of intensity Po. Take a = 2 m and \' = 0.3. SOLUTION Symmetry in deflection dictates that only one quarter-plate need be analyzed. provided that slic/illy-edge conditions (1.28) are introduced along the lines of symmetry. The substitute plate is shown ill Fig. 5.151>, For the sake of simplicity in calculations, we employ only one element per quarter-plate. A concentrated load Po(1 x 1)/4 = 1'0/4 is assigned to node I. The bound-ary constraints permit only a lateral displacement \VI at node 1 and a rotation Ox2 at node 2 (Fig. 5.15c). Nodal force and the displacement matrices are {QL = {Po/4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, O} {I}, = {Wi' 0, 0, 0, 0_'2,0,0,0,0,0,0, OJ The forcedisplacement relationship (5.26), together with the values of stiffness coefficients (Table 5.3), is then readily reduced to the form IPo!41 r' I 1 = lso7i-=o.9) [(60 + 60 + OJ x 30 + 0.35 x 84) (0 - 30 + -- 0,35 x 6) From the above, we obtain (0-30+0-0.35 x 6))111'11 (0 + 20 + 0 + 035 x 8) 10x,I Po H'J = H'max= 3 .1 The "exact" solution of thi' problem (see Example 3.7) is 0.3355J'oEi'. When the quarter-plate is divided into 4, 16, and 25 elements, the results art!, respectively: Wn1ax = O.3512Po/Et.) H'max = 0.3397Po/(3 Wn1llX = O.3378Po/Et3 It is apparent that the accuracy of the solution increases as the mesh is refined. PROBLEMS Secs. 5.1 to 5.3 S.1 Verify thaI the effective forces arc represented, tiS finite difference approximations at point o (Fig. 5.1b), in the form j) Vx = - ;;-J3 [W9 - 11'" - 2(3 - V){Wj -- 1\'3) + (2 - 1')(W, -- \\'(> - \\'7 + WIl)] ., (PS.I) Referring to Table 5.1. check the correctness orille reslllt for 5.2 Determine a finite difTercncc expression corresponding to \741\' = "ID CIt a nodal point () (Fig. 5.1h), for a rectanglliar mesh. Take.1 __ = "and L1y = k. 5.3 Determine the /inite difference cquivalent of M ... M,I' , and Vx at a nodal point 0 (Fig. 5.lh) for II rectangular mesh with .1.x = II and Ay == k. 5.4 Calculate the maximum stress al the nodal point 22 for the plate shown in Fig. 5.5. 5.5 Swept wing of an aircraft is approximated by a simply supported skew plate subjected to uniform loading Po (Fig. P5.5). Determine the deflection w at the nodal points 1 through 5. \, 'I , , a _:0. Free -_. II 3 Figure P5.5 Figure P5.6 5.6 Consider a uniformly loaded plate with two opposite sides supported. the third side clamped, and the fourth side free (Fig. P5.6). Determine: (tI) the displacement II at the nodttl 1 through 9; (IJ) the bending moments al nodal point 9. 138 STRESSES I ~ PLr\TE:S AND SHELLS T " J 1. a/3 L a/3 r " / " , .. , Figure PS.7 6 ~ , ___ =.t= J. i .... - ~ a ~ - ~ _ . a ___ -1-1 Figure PS.9 Figure PS.ll t I a -I t a II"" _. h, 4' II Ii, "" 11 6 - 5 , '. Figure PS.S Figure P5.1 0 Figure PS.12 , - ~ PLATE BY -"";t;MF:JUCAL METHODS J39 Sees. 5.4 (0 5.6 5.7 through 5.12 Each of the \ariously shaped plaks in thL' figurcs is simply supported at ;' , . . ' 2 13 I t ; ill = a I 4 t , .' I ! Fignre 6.2 Example 6.1 A square orthotropic plate is subjected to a uniform loading of intensity Po. Assume the plate edges are clamped and parallel with the principal directions of orthotropy. Determine the defiection, using the finite differencc approach by dividing the domain into equal nets with h = a/4. Take D, = Do, D,. = O.5Do, If = 1.248Do, and ''x = v, = 0.3. SOLUTION For this case, the governing cxpression for del1ectioll, Eq. (6.9), appears Po Do (e) Considerations of symmetry indicate that only one-quarter (shaded portion) of the plate need by analyzed (Fig. 6.2). The conditions that the slopes vanish at the edges are satisfied by numbering the nodes located outside the plate surface as shown in the figure (See Example 5.2). The values of ware zero on the boundary. Applying Fig. 6.1 at the nodes I, 2, 3, and 4, we obtain 118.718 -8.859 .. 6.859 2.429 -17.718 20.718 4.859 -6.859 - 13.718 4.859 19.718 -8.859 9.718]IWII 111 -13.718 J w2 i = E < ~ ~ ~ ) 1 , -17.718 ii' 11'3 ( Do \ 11 21.718 w,1 II The simultaneous solution of Eqs. (d) results in (d) The center deflection, W, = O.OO19po a4lDo, is about 24 percent more than the "cxact" value," O.OOI56poa4/Do. By decreasing the size of the mesh increment, the accuracy of the solution can be improved. ORTHOTROI'IC PLATES 147 6.5 ELLIPTIC AND CIRCULAH OHTHOTROPIC PLATES Consider an elliptic orthotropic plate with scmiaxes a and b, clamped at the edge and subjected to the uniformly distributed load Po (Fig. 4.3). Assume that the principal axes of the ellipse and the principal directions of the orthotropic mate-rial are parallel. The solution procedure follows a pattern similar to that described in Sec. 4.4. Thus, we Jet ( " IV = k 1 - a' (a) in which k is a constant to be determined. Substitution of the above into Eq. (6.9) leads to an expression which is satisfied when k _ Po a4b4 ,_ ... _ .. - 8 3b"Dx + 2a'b' H + 3a"D, (h) The expression describing the deflected surface of the plate is then W = - ..... 1 - - ---Po a4b" (. x' y')2 8 3h*Dx + 2a2b2H + 3a4D,. a' b' (6.17) This equation satisfies the boundary conditions for an elliptic plate with fixed edge, presented by Egs. (a) in Sec. 4.4 The maximum deflection occurs at the center of the plate and is given by (6.18) As anticipated, in the case of an isotropic plate, Egs. (6.17) and (6.18) reduce to Egs. (4.9) and (4.10), respectively. Expressions for the moments may then be obtained from Egs. (6.5). The result obtained above for an elliptic plate may readily be reduced to the case of a circular plate by setting b = a. For a built-in-edge orthotropic circular plate of radius a under uniform load, we have, from Eq. (6.17): where IV (a2 _ 1'2)2 MD, D, = t{3Dx + 2H + 3D,,) (6.19) (c) When Dx = D,. = H = D, Eq. (6.19) is identical with Eq. (2.14), the deflection formula for an isotropic circular plate. 148 S1RESSES I:\" Pl..dO] 1 ._ \'; . } I X (6.22) rifJ = ".(i) x)' 2(1 + v,) 'X)' Substituting strains defined by Eqs. (6.21) into the above, integrating over each layer, and summing the results. we obtain the stress resultants: I Mx I ." My' =2:: I I hi I ' '" .. , Xl" I (J x I'i) .. t7}";- Z dz irxy I (6.23) Stresses defined by Eqs. (1.8), for the ith layer, are (6.24) The general method of deriving the governing equation for multilayered plates follows a pattern identical with that described in Chap. 1. It can be shown that23 the differential equation (1.17) now becomes V4W= JL D, where the D, is the transformed flexural rigidity of laminated plates. (6.25) Layered plates of a symmetric structure about the midplane are of practical significance. For a plate of 211 + 1 symmetrical isotropic layers (Fig. 6.3) the transformed flexural rigidity is given by D, = [ (tf - t!, ,) + 1 (6.26) 3 j;;:.1 1 l-/ 1 '11+ 1 If boundary conditions, transverse load p, and the isotropic material proper-ties of each layer are known. Eq. (6.25) may be solved for w(x, y). The stress components in the ith layer may then be computed from Eq. (6.24). We observe that, upon introduction of transformed rigidity, solution of a multilayered plate problem reduces to that of a corresponding homogeneous plate. All analytical and numerical techniques are tll\ls equally applicable to homogeneous and laminated plates. 150 STRESSE'S I:'; PI ATFS A?\I) SHELLS 6.7 THE FI!'iITE ELEMENT SOLVTlON Tn Sees. 6.4 and 6.S. solutions of orthotropic plate problems were limited to simple cases in which there was uniformity of structural geometry and loading. In this section, the finite clement approach of Chap. 5 is applied for computation of displacement and stress in an orthotropic plate of arhilrClr.V shape and thickness, subjected to lIolllllli/arm loads. For plates made of any nonisotropic material it is necessary to rederive the elasticity matrix [DJ. When the principal directions of orthotropy are parallel to tbe directions of the x and y coordinates, the stress-generalized" strain" rela-tionship is given by Egs. (6.2). Written in the matrix form, they are as follows: E' E' I ax x [ - "x l'}. [ \' . 1'" hi = z ... k, , [ --l'x \'y [ " .. ,\. 0 0 0 B x 0 t:y (6.27a) G I x,' , or, succinctly J a\ = c[D*jl,.l ! k - . t 'J.:: (6.27b) The elasticity matrix. from Eg. (S.21), is theretore E' .. J 0 (6.28) 1 - ".1' o o G The principal dircctions of orthotropy usually do lIot coillcide with the x and y directions, however. Let us consider a plate in which x' and .v' represent the principal directions of the material (Fig. 6.4). The stress and generalized Figure 6.4 ORTHOTlWPIC 1'1.ATf:S 151 .. strain:' in the directions or these coord inates. arc related by .. -1 - "x'v)'. "xE; .. 1 - "x, \'r' E;. 1 - vxi'y (6.29a) o o or (6.29b) Equations for transformation of the strain components l:x, l?y, Yxy at any point of the plate, referring to Egs. (P1.2), are written in the following matrix form I cx' I .: ,:", .' = li'x'Y' I [COS' " sin' " sin (j' cos IX ] I Cx , sin'IX cos2 ex -sin IX cos (J. i C,,' (6.30a) -2 sin ex cos tI. 2 sin r:x cos r:x cos2 a - sin2 a I Concisely, {I;'} = [T](e} (6,30b) where [1') is called the strain transformation matrix. Similarly, the transforma-tion relating stress components in x. y, z to those in x', y', z' is written as {a} = [T]i{u'} (6.31 ) Upon introducing Egs, (6.29) together with Egs, (6.30) into the above, we obtain {u} = o[TYfD*'][T]{s} = z[D*]{"J (6.32) in which [D*] = [TFID*'][T] We thus have, from Eqs. (5.21) and (6.33), the expression 3 [DJ = !1 [TJ'[ D*'][T] L (6.33) (6.34) for the elasticity matrix of the orthotropic plate in which the principal directions of orthotropy are not oriented along the x and .v axes, With Eq, (6.28) or (6.34). explicit expressions'B of stiffness matrices for orthotropic plate elements may be evaluated as outlined in the preceding chap-ter. In the case of a rectallgular, orthotropic plate element, we obtain The coefficients [k,] to [k.] and [R] are listed in Table 5.3. For any particular 152 STRESSES t:-.: PLATES A1\1) SHELLS orthotropic; material, the appropriate values of the rigidit ics D" Dy, Dx)'. and Gx)' (Table 6.1) arc specified. The process of arriving at solutions for the orthotropic plates is identical to that described in Sees. 5.8 to 5.10. PROBLEMS Sees. 6.1 to 6.3 6.1 A plate is reinforced by single equidistant stiffeners (Table 6.1). Compute the rigidities. The plate and stiffeners ace made of steel with E = 200 GPa, v = OJ, s = 200 mm, t = 20 mm, and 1= 12 X 10'-1 m4. 6.2 Determine the rigidities of an ortholCopic steel bridge deck which may be approximated as a steel plate reinforced by II set of equidistant steel ribs (Table 6.1). Assume the following properties: t = Ir:;; 10 mm, 1., = 30 mm, S = 100 rum, v = 0.3, and E = 210 GPa. Torsional rigidity of one rib is:' C = JG = - ()G. 6.3 Show that Eq. {1.34} for the strain energy appears in the following form in the case of orthotropic plates: (P6.3) Sees. 6.4 to 6.7 6.4 A rectangular building floor slab made of a reinforced concrete material is subjected to a concentrated center load P .(Fig. 3.3). Determine expressions for: (a) the deflection surface; (b) the bending moment Mx. The plate edges can be assumed simply supported. Take b = 2a, m = 11 = I, I.>: = Ir_,/2, 1.,,. "'" lC)'/2, t = 0.2 m. = 200 OPa, . = 0.15, and Ec = 21.4 GPa. 6.5 Determine the value of the largest deneclion in the plate described in Proh. 6.4, if a = b. Retain Ihe first two terms of the series solution. 6.6 A simply supported square plate of sides a is subjected tn a uniform loud Po {Fig. 3.1). The plate is constructed from the material described in Prob. 6.1. What should be the value of Po for an allowable deflection \\'",ax = 1 mm. Retain only the first term of the series solution. 6.7 Determine. by taking n = III = 1, the center deflection of a simply supported square plate uni-formly loaded by po. Assume the plate is constructed of the material described in Prob. 6.2. 6.8 an expression for the deflection of an orthotropic damped rectangular plate under a uniform load Pu (Fig. 3.13). lise the Ritz method by retaining the first term of the series solution. Find the maximum deflection jf {/ = h. 6.9 A steel clamped manhole cover, subjected to uniform lond Po, consists of a Hat plate reinforced by equidistant steel stiffeners and is elliptical in form (Fig. 4.3). The material properties are given in Prob. 6.1 and {/ = 2h = 4 m. Compute the maximum displacement w, assuming that the principal x and y axes of the ellipse and the material coincide. 6.10 Redo Prob. 6.9 for a circular plate, Q = b. 6.11 Derive expressions for the bending moments of an orlhotropic elliptical plate with built-in edge. 6.12 A 5-mm thick large plate is fabricated of an orthotropic material having the properties: E:,. = = 2.2G = 13.6 GPa The angle between the principal directions of the material (x', y') and the reference axes (.\, y) is rJ. = 300 (Fig. 6.4). Determine the elasticity matrix of the plate. CHAP fER SEVEN PLATES UNDER COMBINED LATERAL AND DIRECT LOADS 7.1 INTRODUCTION The classical stress analysis relations of the small deformation theory of plates resulting from a lateral loading have been developed in the preceding chapters. Attention will now be directed to situations in which lateral and ill-plane or direct force systems act at a plate section. The latter forces are also referred to as the membrane jorces. These forces may be applied directly at the plate edges, or they may arise as a result of temperature changes (Chap. 9). To begin, the governing differential equations are modified to include the simultaneous action of the combined loading. This is followed by consideration of buckling stresses caused by in-plane compression, pure shear, and biaxial compression, upon application of equilibrium, energy, and finite difference methods, respectively. The problem of plates with small initial curvature under the action of combined forces is next discussed. The chapter concludes with consideration of a plate bent into a simple surface of practical importance. 7.2 GOVERNING EQUATION FOR THE DEFLECTION SURFACE The midplane is strained subsequent to combined loading, and assumption (2) of Sec. 1.2 is no longer valid. However, w is still regarded as small so that the remain-ing suppositions of Sec. 1.2 hold, and yet large enough so that the products of 153 154 STRESSES IN PLATES A:-.JD SHELLS (11) (/' ) Figure 7.1 the in-plane forces or their derivatives and the derivatives of ware of the same order of magnitude as the derivatives of the shear forces (Qx and Q),). Thus, as before, the stress resultants are given by Eqs. (.1.10) and (1.16). Consider a plate element of sides dx and dy under the action of direct forces Nx, N,., and N" = Ny., which are functions of x and y only. Assume the body forces to be negligible. The top and front views of such an clement are shown in Figs. 7.1a and b, respectively. The other resultants due to a lateral load, which also act on the element, are shown in Fig. 1.5. Referring to Fig. 7.1, from the equilibrium of N x dy forces, we obtain (Nx + dX)dY cos {3' - Nx dy cos {3 ox in which {3' = {3 + (o{3/h) dx. Writing cos {3 = (1 - sin2 {3)"2 = 1 - t sin2 {3 + ... = 1 ... [12 2 (a) and noting that for {3 small, {32/2 1 and cos {3 '>0 1, and that likewise, cos {3' '>0 1, Eg. (a) reduces to (INx/iJx) dx dy. The sum of the x components of N" dx is treated in a similar way. The condition L: Fx = 0 then leads to 0i'l" + oN" = 0 (7.1) ox oy Furthermore, the condition L: F). = 0 results in + of!, = 0 ax oy (7.2) To describe equilibrium in the z direction, it is necessary to consider the z components of the in-plane forces acting at each edge of the element. The z component of the normal forces acting on the x edges is equal to (I> ) PLATFS COMJlI:\'EJ) LATERAL AND DIRJ:CT LOADS ISS Ina,much '" /i and fi' are sm,dl. sin /i '" /i " hl'ex and sin /1' " Ii', and hellce , cjJ alV ,,211' II "II +- dx = cc- +,-- dx [!X (IX ('Xl Neglecting higher-order terms, Eg_ (b) is therefore -N, -+ (Nx + + dX) ex ('X ex ox-The z components of the shear forces N" on the x edges of the element are determined as follows. The slope of the deflection surface in the y direction on the x edges eguals ""'jay and awlc')' -+ (,,2wlex ?y) dx_ The: directed component of the shear forces is then (12W ONxOH> N"'----:;-- dx dv -+ ---'---- dx d)' . - rx oy . (Ix cy An expression identical to the above is found for the z projection of shear forces Nyx acting on the y edges: ;y2\-\> eN }>x i1W I N ---d,dy+-------,xdy )'x ax oy - fly ax For the forces in Figs_ 7.1 and 1.5, from I F, = 0, we thus have iJQx oQ) r2w tJ2w (32\-1.' -- + - -+ p + N --- + N .-- + 1 N --,,--l};', CV \ ox2 } 8y2 .... xYax a.v + (ONx + + (o..l'jXl' + = 0 (c) ex 0)' ox Ox 0)' c7y It is observed from Egs. (7.1) and (7.2) that the terms within the parentheses in the above expression vanish. As the direct forces do not result in any moment along the edges of the element, Egs. (b) and (e) of Sec. 1.5, and hence Egs. (1.l6), are unchanged. Introduction of Eqs. (1.16) into Eg. (e) yields 02W ) + 2N -:--- (7 3) XJ' 8x iJy . Expressions (7.1), (7.2), and (7.3) are the governing differential equations for a thin plate, subjected to combined lateral and direct forces. It is observed that Eg. (1.17) is now replaced by Eg. (7.3) to determine the deflection surface of the plate. Either Navier's or Levy's method may be applied to obtain a solution. Example 7.1 A rectangular plate with simply supported edges is subject to the action of combined uniform lateral load Po and uniform tension N (Fig. 7.2). Derive the eguation of the deflection surface. 156 STRESSES 1:-': PtA rES A:,\:D SHELLS !-> -, ----(/------,.1 ---- _ 'I .. - .. -J'- .\" ! ::: 1 .. ,\, j ... --I .. , : I t " Figure 7.2 SOLUTION In this particular case, Nx = N = constant and N,. = N x.\" = 0, and hence Eqs. (7.1) and (7.2) are identically satisfied. The lateral load 1'0 can be represented by (Sec. 3.3): 16po ':Y) mI. I1lnx . mr)! I' = -i" I I -_. Sill -._. S1l1 ._", TC III /I 11111 a b (111, ,,= I, 3, ... ) Inserting the above in Eq. (7.3), we have D4w .., c4w a4w N alw 16po 00 00 1 . n11tx . mt)' a,:;;r + - 3x2 + t3y4 - 15 = TC2D 1m; SIn SIn b (7.4) The conditions at the simply supported edges, expressed by Eqs. (a) of Sec. 3.2, are satisfied by assuming a deflection of the form given by Eq. (3.lh). When this is introduced into Eq. (7.4), we obtain a = _._ __ 1 m" r (111' n')' N( m )' n6Dmn - +- +""-a' b' D na (m, 11 = 1. 3, ... ) The deflection is thus W = 16po f f . .."Lfl ("",x/a) sin (:,,,Y/I>2.1. n6D I'll /I + N(-"-'-)' ,,' b' D na (7.5) Upon comparison of Eqs. (3.6) and (7.5), we are led to conclude that the presence of a tensile (compressive) force decreases (increases) the plate deflection. 7.3 COMPRESSION OF PLATES. BUCKLING When a plate is compressed in its midplane, it becomes unstable and begins to huckle at a certain critical value of the in-plane force. Buckling of plates is qualitatively similar to column buckling.' However, a buckling analysis of the fonner case is not performed as readily as for the latter. Plate-buckling solutions using Eq. (7.3) usually involve considerable difficulty and subtlety,24 and the conditions that result in the lowest eigenvalue, or the actual buck lillY load, are not PLATES Ui'' cjJ --t ......... -ax OJ' (e) (d) : \ LARGE D1'.FLEClIO\;S Of PLATES 179 Insel'ling Egs. (d) into Eq. (,,) results in ?"rj) ,(04 ,'.' 'I( ,;'". )' + - -+: Zy4 = E , , ('-\1' (8.2 ) and introduction of Egs. (e) into Eq. (7.3) leads to + :2 ...'._._. + ?4W = L -+ + _.., .JY!..'! . nt1L\: . nnr Pm" = -I 1 M* SIn _. SIn _ ...... dx dy (9.17) ab . 0 . o II h Substitution of Eg. (9.16a), (9.17), and (9.16b) into Eq. (9.14) leads to a = _1.... Pm" , (9.18) "'" (1 - ,')rr2D (m/a)' + (n/b)' The det1ection w corresponding to the thermal loading M*(x. y) has thus been determined. 9,6 SIMPLY SUPPORTED RECTANGULAR PLATE WITH TEMPERATURE DISTRIBUTION VARYING OVER THE THICKNESS The solution of a simply supported plate subjected to nonunirorm heating such that the temperature varies through the thickness only, T(z), can readily be obtained from the results of Sec. 9.5. In this case, the thermal loading At* is STRESSES IN PLATES J9J cOllstant aod Eg. (9.17), after illlegratioo. leads to 4M* p"", =-[1 - (-1)"][1- (-1)"] 1[211111 . (a) Substitution of Egs. (a) and (9.18) into Eq. (9.16h) yields the following expression for dellection: 16M' . sin (mnx/a) sin (nnJ:/b) l\' = ... __ ._-- )',. --_. __ ..... - ... - ...... -(I - 1')Dn4 ";;;";;' 1Il,,[(m/aV + (Il/by] (111,11= 1,3, ... ) (9.19) The bending moments and stresses in the plate may now be calculated from Eqs. (9.6) and (9.8), As already noted in Example 3.2, while the expression for deflection (9.19) converges very rapidly, the relationship for moments does not. An alternate solution26 of Egs. (9.14) and (9.15), more suitable to the computa-tion of moments, may be obtained by the usc of simple series for It' and M* rather than the double series as before (Sec. 3.4). 9.7 ANALOGY BETWEEN THERMAL AND ISOTHERMAL PLATE PROBLEMS We now demonstrate that an analogy exists between the thermal and isothermal plate-bend ing problems, serving as a basis of a convenient procedure to deter-mine the deflection. The analogy is complete ollly for the determination qf dellectioll. The thermal stresses are ascertained by adding -(67)11 - I') to the stress components u., and (J" of the isothermal solution. Plates with clamped edges The problem of the bending of built-in plates as a result of nonuniform thermal load requires the solution of Eq. (9.10) together with the specified boundary conditions given in Table 9.1. Note that the boun-dary conditions for a clamped edge do not involve expl,icitly the temperature, Thus, it is observed from a comparison of Eqs. (9.9) and (9.10), that the solution sought is identical with that for the same shaped clamped plate subject to the equivalent transverse load p*. The thermal problem is therefore reduced to an isothermal one, and the res nits and techniques of the latter case are valid for the problem under consideration. Table 9.2 provides a list of some examples. Table 9.2 Plates with clamped edges Geometry Loading (p*) Solution Rectangular Uniform Sees. 3.7 and 3.12 Circular Uniform Sec. 2.4 (solid) Radial Sec. 9.8 AnnuJar Radial Sec.9.S Uniform Table 2.3 192 snU':SSFS r:-.: PLATES A:--iD SHELL!;) In the case or a platC' or tlrhirrarr .')I/(IIJ(, undergoing thermal variation f/Jroltyh tire (hie/dies:) only, we have V2;\J* = 0 and fl::: = o. According to the analogy. \I' = 0, and the corresponding stresses, from Eqs. are N.\T Txr=-. t (0 ) The first and the second terms represent the plane- and the hendino-stress com-ponents, respectively. Plates with simply supported or free edges An analogy also exists between heated and unheated plates with other than clamped supports. In this case a modification of the edge conditions is required inasmuch as they contain the temperature. At a simply supported edge of the analogous isothermal plate, IV = 0 as beforc, but a bending moment M*/(l - v) must be assumed to apply. In a like manner, at a free edge of the analogous unheated plate, a force equal to (Nv[*lcx)/(l - ,,) must be applied. It is thus observed that a thermal solution can always be determined by superposition various isothel'mal solutio1ls. Consider. for example. the bending caused by a nonuniform temperature distribution of a simply supported plate. The deflection of the plate is determined by adding the deflection of an unheated, simply supported plate subject to the surface load 1'*. to the deflection of an unheated plate carrying no transverse load but subject to the moment M*j(l - v) acting at its edges. The foregoing analogy is also useful in tbe experimental analysis of elastic heated plates. This is because it may be easier to test a plate at constant temper-ature with given transverse and edge loadings and then to impose upon it arbitrary temperature distributions.'9 Example 9.1 An aircraft window, which can be represented approximately as a simply supported circular plate, is subjected to unifoffil temperature T, and uniform temperature T, at the lower and the upper sLirfaces, respectively (Fig. 9.la). Determine the deflection and bending stress if the plate is free of p*::: 0 fht F A il/* I-v + CE===AJ M* (a) (b) (e) Figure 9.1 SrRf-.SSES E": PLATES t93 stress at rye. Assume that kmperature through the! hicklll'sS \'aries linearly and lhal T, > . SOLliTlOX The plate of Fig, 9.la is replaced by the plates shown in Figs, 9Jb and 9.1 c. The temperature difference between the faces is T) - T, and that between either face and the midsurlace is !J. T = !(T) + T,), Since for the present case M* is a function of z only, Egs. (9.11) give an equivalent loading p* = 0, The thermal stress resultant is, from Eq. (9,7), 11'1* = xE (' f.HT) + Ii) + HT, - '[i) ';2- !, dz = (T) -- 7;) (9.20) '-1:2 l If 1 For the plate of Fig. 9. I h, the bending stress at the faces is "E(tses at the faces of the plate, upon introduction of Eqs, (9.20) and (921b) into (9.8), are as follows: a, = (T, = 2 (=:--,;'j (T, - T,J (e) The resultant stress in the original plate is obtained by addition of the stresses given by Egs, (b) and (e): This is the result expected, Equation (92Ia) leads to the relationship rxa2 "'",,, =2'1 (7; - 12) for the maximum deflection of the original plate, 9.8 AXISYMMETRICALLY HEATED CIRCULAR PJ"ATES (922) Consider the bending of an axisymmetrically heated circular plate having simply supported or clamped edge conditions and subjected to temperatures varying with the /' and z coordinates, T(/', z), such that the equivalent transverse load 194 STRESSES IN PLATES AND SHELLS 1'* = p*(r). The expressions for moments, Eqs. for the situation described hecome M,.= + _ M* d .. - .. ' C, = -- ..... ------ .. -----...... ------- ......... ------ .. ------.----- ---- (al _ bI)l _ 4.111[>2 In2 (a/h) _ b2) tn (a/b)] + - {/2 - 211z In (a/h)J c.l stress resultant at ,. = u. Given a temperature distribution T(r, z), the deflection and moment in a solid or annular plate with simpl