The Bending and Stretching of Plates PDF
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OTHER TITLES IN THE DIVISION ON SOLID AND STRUCTURAL MECHANICS Vol. 1. SAVIN- Stress Concentration around Holes Vol. 2. GOL'DENVEIZER- Theory of Elastic Thin Shells Vol. 3. NOWACKI- Thermoelasticity Vol. 4. COX- The Bucklingof Plates and Shells Vol. 5. MORLEY- Skew Plates and Structures O'THER DIVISIONS IN THE SERIES ON AERONAUTICS AND ASTRONAUTICS AERODYNAMICS PROPULSION SYSTEMS INCLUDING FUELS AVIONICS AVIATION AND SPACE MEDICINE FLIGHT TESTING ASTRONAUTICS MATERIALS SCIENCE AND ENGINEERING SYMPOSIA INTERNATIONAL SERIES OF MONOGRAPHS ON AERONAUTICS AND ASTRONAUTICS CHAIRMEN Th. von KARMAN H. L. DRYDEN Advisory Group for Aeronautical Deputy Administrator, Research and Development, National Aeronautics and North Atlantic Treaty Organization, Space Administration, 64 rue de Varenne, Paris VII, Washington 25, D.C., France U.S.A. HONORARY ADVISORY BOARD UNITED KINGDOM UNITED STATES GERMANY (cont.) A. M. Ballantyne G. Bock C. Kaplan A. D. Baxter H. Görtier J. Kaplan W. Cawood Ο. Lutz J. Keto J. S. Clarke A. W. Quick W. B. Klemperer Sir H. Roxbee Cox E. Kotcher Sir W. S. Farren AUSTRALIA Ε. H. Krause G. W. H. Gardner Col. N. L. Krisberg L. P. Coombes W. S. Hemp A. M. Kuethe S. G. Hooker J. P. Layton BELGIUM Ε. T. Jones I. Lees J. Ducarme W. P. Jones B. Lewis G. V. Lachmann P. A. Libby ITALY A. A. Lombard H. W. Liepmann G. Gabrielli B. P. Mullins J. R. Markham A. J. Murphy C. B. Millikan CANADA L. F. Nicholson W. F. Milliken, Jr. J. J. Green F. W. Page W. C. Nelson H. C. Luttman Sir A. G. Pugsley W. H. Pickering D. C. MacPhail H. B. Squire R. W. Porter D. L. Mordell L. H. Sterne L. E. Root A. D. Young G. S. Schairer SWEDEN F. R. Shanley Β. K. L. Lundberg UNITED STATES E. R. Sharp S. F. Singer POLAND H. J. Allen M. Alperin C. R. Soderberg F. Misztal R. L. Bisplinghoff J. Stack W. von Braun M. Stern FRANCE F. H. Clauser H. G. Stever L. Malavard M. U. Clauser G. P. Sutton M. Roy J. R. Dempsey R. J. Thompson W. S. Diehi L. A. Wood SPAIN C. S. Draper T. P. Wright Col. A. Pérez-Marin A. Ferri M. J. Zucrow JAPAN C. C. Furnas C. Gazley, Jr. I. Tani Ε. Haynes HOLLAND Ε. H. Heinemann H. J. van dee Maas RUSSIA Ν. J. Hoff C. Zwikker A. A. Ilyushin THE BENDING A ND STRETCHING OF PLATES by Ε. H. MANSFIELD, Sc.D. Senior Principal Scientific Officer Structures Department Royal Aircraft Establishment Farnborough PERGAMON PRESS OXFORD · LONDON · NEW YORK PARIS 1964 PERGAMON PRESS LTD. Headington Hill Hall, Oxford 4 and 5 Fitzroy Square, London W.l. PERGAMON PRESS INC. 122 East 55th Street, New York 22, N.Y. GAUTHIER-VILLARS ED. e 55 Quai des Grands-August ins, Paris, 6 PERGAMON PRESS G.m.b.H. Kaiserstrasse 75, Frankfurt am Main Distributed in the Western Hemisphere by THE MACMILLAN COMPANY · NEW YORK pursuant to a special arrangement with Pergamon Press Limited Copyright © 1964 PERGAMON PRESS LTD. Library of Congress Catalog Card Number 61-18328 MADE IN GREAT BRITAIN PREFACE I HAVE attempted in this monograph to present a concise, up-to-date and unified introduction to elastic plate theory. Wherever possible the approach has been to give a clear physical understanding of plate behav- iour. The style is thus more appropriate to engineers than mathemati- cians and although some of the topics have an aeronautical flavour, the monograph will also be of value to structural engineers in civil, mechan- ical and marine engineering, and to structural research workes and students. Small-deflexion theory is considered in Part I in six chapters entitled: Derivation of the basic equations — Rectangular plates — Plates of various shapes — Plates whose boundaries are amenable to conformai transfor- mation — Plates with variable rigidity — Approximate methods. Each chapter depends to some extent on the first, but is otherwise self-con- tained. There are three chapters in Part II on large-deflexion theory, entitled: General equations and some exact solutions — Approximate methods in large-deflexion theory — Asymptotic large-deflexion theories for very thin plates. These asymptotic theories are membrane theory, tension field theory and inextensional theory. To restrict the monograph to its present size it has been necessary to omit virtually all numerical results and to focus attention on the methods of solution. Numerous illustrative examples are presented but the steps in the analyses are kept to a minimum. Attention is also drawn to the classes of plate problem for which relatively simple solutions are available. Certain topics are quite new while others have not previously appeared in book form. I am grateful to Professor W. S. Hemp for the invitation to write the monograph, and to my wife Ola for her encouragement and for typing the manuscript. Ε. H. MANSFIELD Farnborough [ix] PRINCIPAL NOTATION x, y, ζ Cartesian coordinates, χ, y in plane of plate r, θ, ζ cylindrical coordinates, r, 0 in plane of plate U, V, w displacements in x, y, ζ directions' € By, €γ direct and shear strains in plane ζ = const. χ χ C, (Tyt Ty direct and shear stresses in plane ζ = const. x X Ν Ν Ν direct and shear forces per unit length in plane of plate QX> Qy transverse shear forces per unit length Qr> Qe M, My, M x xy bending and twisting moments per unit length M, M, M r e r9 curvatures, etc. Ή>χ> Xy* Xy X E. G Young's modulus and shear modulus ν Poisson's ratio t plate thickness μ Ht D flexural rigidity, £/3/{12(l — v2)} k foundation modulus transverse edge support stiffness, or r/r Q x X rotational edge support stiffness, or complex potential function Ψ complex potential function, or — dw/dr q transverse loading per unit area coefficients in Fourier expansions for q <ln> Imn a, b typical plate dimensions Φ force function w w particular integrals lf p complementary functions Ρ point load t time, or tangent to boundary η normal to boundary s distance along boundary Ψ angle between tangent to boundary and *-axis [xi] PRINCIPAL NOTATION XÜ U strain energy Π potential energy Δ deflexion at reference point α angle between generator and x-axis M total moment about a generator a i) distance along a generator dx2 dy2 dxdy dxdy dy2 dx2 CHAPTER I 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 the 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', i.e. 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'. Possible sources of error arising from these assumptions are discussed later. [3] 4 THE BENDING AND STRETCHING OF PLATES 1.1 Stress-strain Relations Let us consider now the state of stress in a plate with an arbitrary small deflexion w(x, y) (See Fig. 1.1). The mid-plane is a neutral plane and accordingly ο FIG. 1.1 we shall focus attention on the state of strain, and hence the state of stress, in a plane at a distance ζ from the mid-plane. The slopes of the mid-plane dw dw are — and — so that the displacements u and ν in the x^-plane at a dx dy distance ζ from the mid-plane are given by dw \ and 1.1 dw V = —ζ—· dy J The strains in this *,>>-plane are therefore given by du dx d*w — z dv 1.2 du ^ dv ε dy dx DERIVATION OF THE BASIC EQUATIONS 5 Now by virtue of assumption (ii) of para. 1 a state of plane stress exists in the χ,^-plane so that the strains ε e, e are related to the stresses a % χ9 y xy x a, r by the relations y xy ε = — (σ -va) χ Ε χ y e = — {a—va) 1.3 y Ε y x 2(1 +v) Equations (1.2) and (1.3) may be combined to give Ez lldd2*ww dV σ„ = — Ez ld*w , d*w\ y + v 1.4 y = ~\ t \— ' dx* 1 — v2 \c Ez d*w 1 +v dxdy These stresses vary linearly through the thickness of the plate and are equivalent to moments per unit length acting on an element of the plate as shown in Fig. 1.2. Thus, ζσάζ χ λ , d*w\ n = — D\ \-v \dx2 dy*f M. ZOyUZ 1.5 id2w \dy* dx*J M = zTdz xy xy 2* 2 dW = —D(l —v) dxdy where the flexural rigidity D of the plate is defined by Efi D = 1.6 12(1-v2)
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