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The Bending and Stretching of Plates The bending and stretching of plates The bending and stretching of plates Second edition E. H. MANSFIELD Formerly Chief Scientific Officer Royal Aircraft Establishment Farnborough The right of the University of Cambridge to print and sell all manner of books was granted by Henry VIII in IS34. The University has printed and published continuously since IS84. CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE NEW YORK NEW ROCHELLE MELBOURNE SYDNEY CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521333047 © Cambridge University Press 1989 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1964 by Pergamon Press Second edition first published 1989 This digitally printed first paperback version 2005 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Mansfield, Eric Harold. 1923- The bending and stretching of plates/E.H. Mansfield. — 2nd ed. p. cm. Includes bibliographies and indexes. ISBN 0-521-33304-0 1. Elastic plates and shells. I. Title. QA935.M36 1989 88-29035 624.1'776-dc 19 CIP ISBN-13 978-0-521-33304-7 hardback ISBN-10 0-521-33304-0 hardback ISBN-13 978-0-521-01816-6 paperback ISBN-10 0-521-01816-1 paperback CONTENTS Preface page IX Principal notation X Part I. Small-deflexion theory Derivation of the basic equations 3 1.1 Stress-strain relations 4 1.2 Rotation of axes of reference 7 1.3 Equilibrium 9 1.4 Differential equation for the deflexion 10 1.5 Effect of forces in the plane of the plate 12 1.6 Thermal stress effects 16 1.7 General boundary conditions 19 1.8 Anisotropic plates 24 Rectangular plates 33 2.1 Plates with all edges simply supported - double Fourier series solution 33 2.2 Plates with two opposite edges simply supported - single Fourier series solution 37 2.3 Plates with two opposite edges clamped 43 2.4 General loading on plates with all edges clamped 46 Plates of various shapes 50 3.1 Circular plates 50 3.2 Uniformly loaded sector plate 58 3.3 Sector and wedge-shaped plates with general bound- ary conditions 59 3.4 Clamped elliptical plate 60 3.5 Simply supported equilateral triangular plate 61 3.6 Simply supported isosceles right-angled triangular plate 61 vi Contents 3.7 Clamped parallelogram plate 62 3.8 Singular behaviour at corners 67 3.9 Simply supported rhombic plate 69 3.10 Multi-layered plate with coupling between moments and planar strains 73 Plates whose boundaries are amenable to conformal transformation 76 4.1 Governing differential equation in complex coordinates 77 4.2 General solution for a clamped plate 85 4.3 General solution for a simply supported plate 87 4.4 Square plate with rounded corners 88 4.5 Anisotropic plates 91 Plates with variable rigidity 93 5.1 Flexure and torsion of a strip of variable rigidity 93 5.2 Torsion and flexure of strip with chordwise tempera- ture variation 97 5.3 Rectangular plate with exponential variation of rigidity 100 5.4 Rectangular plate with linear variation of rigidity 103 5.5 Circular plates 105 5.6 Circular plates with rotational symmetry 108 Approximate methods 111 6.1 The strain energy of an isotropic plate 111 6.2 Strain energy in multi-layered anisotropic plates 115 6.3 Principle of minimum total potential energy - Ritz method 116 6.4 The Galerkin method 120 6.5 A variational method 121 6.6 Variational methods for clamped plates 126 Part II. Large-deflexion theory General equations and some exact solutions 133 7.1 Governing differential equations for isotropic plates 133 7.2 Cylindrical deflexion of long strip 141 7.3 Uniformly loaded circular plate 144 7.4 Pure bending of strip with shallow double curvature 146 7.5 Flexure and torsion of flat strip of lenticular section 152 Contents vii 7.6 Elliptical plate with temperature gradient through the thickness 158 7.7 Elliptical plate subjected to certain normal loadings 161 7.8 Governing differential equations for anisotropic plates 162 8 Approximate methods in large-deflexion analysis 167 8.1 Perturbation method for normally loaded plates 167 8.2 Perturbation method in post-buckling problems 170 8.3 An energy method 175 8.4 Anisotropic plates 179 9 Asymptotic large-deflexion theories for very thin plates 183 9.1 Membrane theory 183 9.2 Tension field theory 188 9.3 Inextensional theory 204 9.4 The analogy between tension field theory and inextensional theory 219 Author index 225 Subject index 227 PREFACE In the first edition of this book, I attempted to present a concise and unified introduction to elastic plate theory. Wherever possible, the approach was to give a clear physical picture of plate behaviour. The presentation was thus geared more towards engineers than towards mathematicians, particularly to structural engineers in aeronautical, civil and mechanical engineering and to structural research workers. These comments apply equally to this second edition. The main difference here is that I have included thermal stress effects, the behaviour of multi-layered composite plates and much additional material on plates in the large- deflexion regime. The objective throughout is to derive 'continuum' or analytical solutions rather than solutions based on numerical techniques such as finite elements which give little direct information on the significance of the structural design parameters; indeed, such solutions can become simply number-crunching exercises that mask the true physical behaviour. E.H. Mansfield PRINCIPAL NOTATION a, b typical plate dimensions a,b,d defined by (1.96) A,B,D defined by (1.93) D flexural rigidity Et3/{12(1 - v2)} E, G Young modulus and shear modulus E defined by (1.89) k foundation modulus Lx, L2, L3 differential operators defined after (1.98) Mx,My,Mxy) ,,.,.. M M M ( ^en^mS anc* twisting moments per unit length T M" " ' (Mx,My,Mxy) Ma total moment about a generator n normal to boundary Nx,Ny,Nxy direct and shear forces per unit length in plane of plate T N (Nx9Ny,Nxy) P point load transverse shear forces per unit length q transverse loading per unit area ^n^mn coefficients in Fourier expansions for q r, 9, z cylindrical coordinates, r, 6 in plane of plate s distance along boundary t plate thickness t time, or tangent to boundary T temperature, torque 17 strain energy M, v9 w displacements in x, y, z directions wl9wp particular integrals w2, wc complementary functions x, y, z Cartesian coordinates, x, y in plane of plate a coefficient of thermal expansion, angle between generator and x-axis Principal notation xi £joe)>>£xy direct and shear strains in plane z = const. ST,KT thermal strain and curvature, see (1.56) and (1.57) T € (£x,£y,Sxy) rj distance along a generator 2 2 KX, Ky, Kxy curvatures, — (d w/dx \ etc. K (Kx,Ky92Kxy) v Poisson ratio II potential energy Q transverse edge support stiffness, or rjr^ ax>ay>?xy direct and shear stresses in plane z = const. cp complex potential function, or - dw/dr Q> force function X rotational edge support stiffness, or complex potential function \jj angle between tangent to boundary and x-axis dx2 ' dy2 2 2 2 2 2 2 fd g 2 8 f d g | d fd g dx2 dy2 dxdy dxdy dy2 dx2 + gW2f)} I SMALL-DEFLEXION THEORY 1 Derivation of the basic equations All structures are three-dimensional, and the exact analysis of stresses in them presents formidable difficulties. However, such precision is seldom needed, nor indeed justified, for the magnitude and distribution of the applied loading and the strength and stiffness of the structural material are not known accurately. For this reason it is adequate to analyse certain structures as if they are one- or two-dimensional. Thus the engineer's theory of beams is one-dimensional: the distribution of direct and shearing stresses across any section is assumed to depend only on the moment and shear at that section. By the same token, a plate, which is characterized by the fact that its thickness is small compared with its other linear dimensions, may be analysed in a two-dimensional manner. The simplest and most widely used plate theory is the classical small-deflexion theory which we will now consider. The classical small-deflexion theory of plates, developed by Lagrange (1811), is based on the following assumptions: (i) points which lie on a normal to the mid-plane of the undeflected plate lie on a normal to the mid-plane of the deflected plate; (ii) the stresses normal to the mid-plane of the plate, arising from the applied loading, are negligible in comparison with the stresses in the plane of the plate; (iii) the slope of the deflected plate in any direction is small so that its square may be neglected in comparison with unity; (iv) the mid-plane of the plate is a 'neutral plane', that is, any mid-plane stresses arising from the deflexion of the plate into a non- developable surface may be ignored. These assumptions have their counterparts in the engineer's theory of beams; assumption (i), for example, corresponds to the dual assumptions in beam theory that 'plane sections remain plane' and 'deflexions due to shear may be neglected'.
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