Theory of Plates and Shells (McGraw-Hill Classic Textbook Reissue Series) Second Edition

THEORY OF PLATES AND SHELLS S. TIMOSHENKO Professor Emeritus of Engineering Mechanics Stanford University S. WOINOWSKY-KRIEGER Professor of Engineering Mechanics Laval University

SECOND EDITION

MCGRAW-HILL

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THEORY OF PLATES AND SHELLS International Edition 1959 Exclusive rights by McGraw-Hill Book Co.— Singapore for manufacture and export. This book cannot be re-exported from the country to which it is consigned by McGraw-Hill. Copyright ©

1959 by McGraw-Hill, Inc.

All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 40 09 08 07 06 05 04 03 02 20 09 08 07 06 05 04 03 02 01 PMP BJE Library of Congress Catalog Card Number 58-59675

When ordering this title use ISBN 0-07-085820-9

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PREFACE

Since the publication of the first edition of this book, the application of the theory of plates and shells in practice has widened considerably, and some new methods have been introduced into the theory. To take these facts into consideration, we have had to make many changes and additions. The principal additions are (1) an article on deflection of plates due to transverse shear, (2) an article on stress concentrations around a circular hole in a bent plate, (3) a chapter on bending of plates resting on an elastic foundation, (4) a chapter on bending of anisotropic plates, and (5) a chapter reviewing certain special and approximate methods used in plate analysis. We have also expanded the chapter on large deflections of plates, adding several new cases of plates of variable thickness and some numerical tables facilitating plate analysis. In the part of the book dealing with the theory of shells, we limited ourselves to the addition of the stress-function method in the membrane theory of shells and some minor additions in the flexural theory of shells. The theory of shells has been developing rapidly in recent years, and several new books have appeared in this field. Since it was not feasible for us to discuss these new developments in detail, we have merely referred to the new bibliography, in which persons specially interested in this field will find the necessary information. S. Timoshenko S. Woinowsky-Krieger

NOTATION x, y, z Rectangular coordinates r, 0 Polar coordinates rx, ry Radii of curvature of the middle surface of a plate in xz and yz planes, respectively h Thickness of a plate or a shell q Intensity of a continuously distributed load p Pressure P Single load 7 Weight per unit volume (Tx, Radial stress in polar coordinates at, (re Tangential stress in polar coordinates r Shearing stress Txy, Txz, Tyz Shearing stress components in rectangular coordinates u, v, w Components of displacements e Unit elongation «*, «•/, fz Unit elongations in x, y, and z directions er Radial unit elongation in polar coordinates et, eo Tangential unit elongation in polar coordinates ttp, eo Unit elongations of a shell in meridional direction and in the direction of parallel circle, respectively yxy, Vxz, jyz Shearing strain components in rectangular coordinates 7r0 Shearing strain in polar coordinates E Modulus of elasticity in tension and compression G Modulus of elasticity in shear v Poisson's ratio V Strain energy D Flexural rigidity of a plate or shell Mx, My Bending moments per unit length of sections of a plate perpendicular to x and y axes, respectively Mxy Twisting moment per unit length of section of a plate perpendicular to x axis Mn, Mnt Bending and twisting moments per unit length of a section of a plate perpendicular to n direction Qx, Qy Shearing forces parallel to z axis per unit length of sections of a plate perpendicular to x and y axes, respectively Qn Shearing force parallel to z axis per unit length of section of a plate perpendicular to n direction JVx, Ny Normal forces per Unit length of sections of a plate perpendicular to x and y directions, respectively

Nxu

Shearing force in direction of y axis per unit length of section of a plate perpendicular to x axis Mr, Mt, MH Radial, tangential, and twisting moments when using polar coordinates Qr, Qt Radial and tangential shearing forces Nn Nt Normal forces per unit length in radial and tangential directions ri, r2 Radii of curvature of a shell in the form of a surface of revolution in meridional plane and in the normal plane perpendicular to meridian, respectively X 3 the calculations for a plate can be replaced by those for a strip without substantial error. T A B L E 8. NUMERICAL FACTORS a, /3, 7, 5, n FOR UNIFORMLY LOADED AND SIMPLY SUPPORTED RECTANGULAR

PLATES

v = 0.3

6/a

WUUi qa* ( M »>»« = a— = /8ga2

CAOmax = /ffiga2

«2*)ma* = 75a

(Qy)m« = 7iga

(F*) m a x = 5ga

(FJ^x = 5iga

/? — nqa2

a

j3

/Si

7

71

5

5i

n

1.0 1.1 1.2 1.3 1.4

0.00406 0.00485 0.00564 0.00638 0.00705

0.0479 0.0554 0.0627 0.0694 0.0755

0.0479 0.0493 0.0501 0.0503 0.0502

0.338 0.360 0.380 0.397 0.411

0.338 0.347 0.353 0.357 0.361

0.420 0.440 0.455 0.468 0.478

0.420 0.440 0.453 0.464 0.471

0.065 0.070 0.074 0.079 0.083

1.5 1.6 1.7 1.8 1.9

0.00772 0.00830 0.00883 0.00931 0.00974

0.0812 0.0862 0.0908 0.0948 0.0985

0.0498 0.0492 0.0486 0.0479 0.0471

0.424 0.435 0.444 0.452 0.459

0.363 0.365 0.367 0.368 0.369

0.486 0.491 0.496 0.499 0.502

0.480 0.485 0.488 0.491 0.494

0.085 0.086 0.088 0.090 0.091

2.0 3.0 4.0 5.0 00

0.01013 0.01223 0.01282 0.01297 0.01302

0.1017 0.1189 0.1235 0.1246 0.1250

0.0464 0.0406 0.0384 0.0375 0.0375

0.465 0.493 0.498 0.500 0.500

0.370 0.372 0.372 0.372 0.372

0.503 0.505 0.502 0.501 0.500

0.496 0.498 0.500 0.500 0.500

0.092 0.093 0.094 0.095 0.095

Expression (e) can be used also for calculating shearing forces and reactions at the boundary. Forming the second derivatives of this expression, we find

Substituting this in Eqs. (106) and (107), we obtain

For the sides x = 0 and y = — b/2 we find

These shearing forces have their numerical maximum value at the middle of the sides, where

The numerical factors y and 71 are also given in Table 8. The reactive forces along the side x = 0 are given by the expression

The maximum numerical value of this pressure is at the middle of the side (y = 0), at which point we find

where 5 is a numerical factor depending on v and on the ratio 6/a, which can readily be obtained by summing up the rapidly converging series that occur in expression (q). Numerical values of 8 and of Si, which corresponds to the middle of the sides parallel to the x axis, are given in Table 8. The distribution of the pressures (q) along the sides of a square plate is shown in Fig. 63. The portion of the pressures produced by the

Ratio f 64

FIG.

twisting moments Mxy is also shown. These latter pressures are balanced by reactive forces concentrated at the corners of the plate. The magnitude of these forces is given by the expression

The forces are directed downward and prevent the corners of a plate from rising up during bending. The values of the coefficient n are given in the last column of Table 8.

The values of the factors a, 0, 0i, 8 as functions of the ratio b/a are represented by the curves in Fig. 64. In the presence of the forces R, which act downward and are by no means small, anchorage must be provided at the corners of the plate if the plate is not solidly joined with the supporting beams. In order to determine the moments arising at the corner let us consider the equilibrium of the element abc of the plate next to its corner (Fig. 65) and let us introduce, for the same purpose, new coordinates 1, 2 at an angle of 45° to the coordinates x, y in Fig. 59. We can then readily verify that the bending moments acting at the sides ab and cb of the element are Mx = —R/2 and M2 = -f-#/2, respectively, and that the corresponding twisting moments are zero. In fact, using Eq. (39), we obtain for the side ac, that is, for the element of the edge, given by a = — 45°, the bending moment Mn — Mi cos2 a + M2 sin2 a = 0 in accordance with the boundary conditions of a simply supported plate. The magnitude of the twisting moment applied at the same edge element is obtained in like manner by means of Eq. (40). Putting ot = —45° we have Mnt = ^ sin 2*(MX - M2) - ^

FIG.

65

according to Eq. (r). Thus, the portion of the plate in the vicinity of the corner is bent to an anticlastic surface, the moments ±R/2 at the corner itself being of the same order of magnitude as the bending moments at the middle of the plate (see Table 8). The clamping effect of the corners of a simply supported plate is plainly illustrated by the distribution of the bending moments M1 and M2 of a square plate (Fig. 63). If the corners of the rectangular plate are not properly secured against lifting, the clamping becomes ineffective and the bending moments in the center portion of the plate increase accordingly. The values of (Mx) max and (M y ) max given in Table 8 must then be multiplied by some factor k > 1. The approximate expression1

may be used for that purpose. It should be noted that in the case of a polygonal plate with simply supported edges no single reactive forces arise at a corner point provided the angle between both adjacent sides of the plate is other than a right angle.2 Even in rectangular plates, however, no corner reactions are obtained if the transverse shear deformation is taken into account. In view of the strongly concentrated 1 Recommended by the German Code for Reinforced Concrete (1943) and basod on a simplified theory of thin plates due to H. Marcus; see his book " Die vereinfachte Berechnung biegsamer Platten," 2d ed., Berlin, 1925. 2 For a simple proof see, for example, H. Marcus, "Die Theorie e\astischer Gewebe," 2d ed., p. 46, Berlin, 1932.

reactive forces this shear deformation obviously is no longer negligible, and the customary thin-plate theory disregarding it completely must be replaced by a more exact theory. The latter, which will be discussed in Art. 39, actually leads to a distribution of reactive pressures which include no forces concentrated at the corners of the plate (see Fig. 81).

31. Simply Supported Rectangular Plates under Hydrostatic Pressure. Assume that a simply supported rectangular plate is loaded as shown in Fig. 66. Proceeding as in the case of a uniformly distributed load, we take the deflection of the plate in the form1

represents the deflection of a strip under the triangular load. This expression satisfies the differential equation

and the boundary conditions

The part Wi is taken in the form of a series

FIG.

66

where the functions Ym have the same form as in the preceding article, and m = 1, 2, 3, . . . . Substituting expressions (6) and (d) into Eq. (a), we obtain

where the constants Am and Bm are to be determined from the conditions

1

This problem was discussed by E. Estanave, op. cit. The numerical tables of deflections and moments were calculated by B. G. Galerkin, Bull. Polytech. InSt1 St. Petersburg, vols. 26 and 27, 1918.

From these conditions we find

In these equations we use, as before, the notation

Solving them, we find

The deflection of the plate along the x axis is

For a square plate a —

The deflection at the center of the plate is (uO—/i.iM> = 0.00203 2*£

(h)

which is one-half the deflection of a uniformly loaded plate (see page 116) as it should be. By equating the derivative of expression (g) to zero, we find that the maximum deflection is at the point x = 0.557a. This maximum deflection, which is 0.00206 q0a4/D, differs only very little from the deflection at the middle as given by formula (h). The point of maximum deflection approaches the center of the plate as the ratio b/a increases. For b/a = oo, as for a strip [see expression (b)], the maximum deflection is at the point x = 0.5193a. When b/a < 1, the point of maximum deflection moves away from the center of the plate as the ratio b/a decreases. The deflections at several points along the x axis (Fig. 66) are given in Table 9. It is seen that, as the ratio b/a increases, the deflections approach the values calculated for a strip. For b/a = 4 the differences in these values are about I^ per cent. We can always calculate the deflection of a plate for which b/a > 4 with satisfactory accuracy by using formula (6) for the deflection of a strip under triangular load. The bending moments Mx and My are found by substituting

TABLE 9. NUMERICAL FACTOR a FOR DEFLECTIONS OF A SIMPLY SUPPORTED RECTANGULAR PLATE UNDER HYDROSTATIC PRESSURE q = qox/a

b>a w = <xqoaA/D, y = 0 b/a

x = 0.25a

x = 0.50a

x = 0.60a

x = 0.75a

1 1.1 1.2 1.3 1.4

0.00131 0.00158 0.00186 0.00212 0.00235

0.00203 0.00243 0.00282 0.00319 0.00353

0.00201 0.00242 0.00279 0.00315 0.00348

0.00162 0.00192 0.00221 0.00248 0.00273

1.5 1.6 1.7 1.8 1.9

0.00257 0.00277 0.00296 0.00313 0.00328

0.00386 0.00415 0.00441 0.00465 0.00487

0.00379 0.00407 0.00432 0.00455 0.00475

0.00296 0.00317 0.00335 0.00353 0.00368

2.0 3.0 4.0 5.0 oo

0.00342 0.00416 0.00437 0.00441 0.00443

0.00506 0.00612 0.00641 0.00648 0.00651

0.00494 0.00592 0.00622 0.00629 0.00632

0.00382 0.00456 0.00477 0.00483 0.00484

expression (e) for deflections in Eqs. (101). the expression for Mx becomes

Along the x axis (y = 0)

The first sum on the right-hand side of this expression represents the bending moment for a strip under the action of a triangular load and is equal to (qQ/6)(ax — x3/a). Using expressions (/) for the constants Am and Bm in the second sum, we obtain

The series thus obtained converges rapidly, and a sufficiently accurate value of Mx can be realized by taking only the first few terms. In this

TABLE 10. NUMERICAL FACTORS 0 AND /3I FOR BENDING MOMENTS OF SIMPLY SUPPORTED RECTANGULAR PLATES UNDER HYDROSTATIC PRESSURE q — qox/a

v = 0.3, b > a Mx = /Sa2^0,

My = Pia2qQ,

y = 0

y = 0

b/a X =

X =

X =

X =

X =

X =

X =

X =

0.25a

0.50a

0.60a

0.75a

0.25a

0.50a

0.60a

0.75a

1.0 1.1 1.2 1.3 1.4

0.0132 0.0156 0.0179 0.0200 0.0221

0.0239 0.0276 0.0313 0.0346 0.0376

0.0264 0.0302 0.0338 0.0371 0.0402

0.0259 0.0289 0.0318 0.0344 0.0367

0.0149 0.0155 0.0158 0.0160 0.0160

0.0239 0.0247 0.0250 0.0252 0.0253

0.0245 0.0251 0.0254 0.0255 0.0254

0.0207 0.0211 0.0213 0.0213 0.0212

1.5 1.6 1.7 1.8 1.9

0.0239 0.0256 0.0272 0.0286 0.0298

0.0406 0.0431 0.0454 0.0474 0.0492

0.0429 0.0454 0.0476 0.0496 0.0513

0.0388 0.0407 0.0424 0.0439 0.0452

0.0159 0.0158 0.0155 0.0153 0.0150

0.0249 0.0246 0.0243 0.0239 0.0235

0.0252 0.0249 0.0246 0.0242 0.0238

0.0210 0.0207 0.0205 0.0202 0.0199

2.0 3.0 4.0 5.0 oo

0.0309 0.0369 0.0385 0.0389 0.0391

0.0508 0.0594 0.0617 0.0623 0.0625

0.0529 0.0611 0.0632 0.0638 0.0640

0.0463 0.0525 0.0541 0.0546 0.0547

0.0148 0.0128 0.0120 0.0118 0.0117

0.0232 0.0202 0.0192 0.0187 0.0187

0.0234 0.0207 0.0196 0.0193 0.0192

0.0197 0.0176 0.0168 0.0166 0 0165

_J

way the bending moment at any point of the x axis can be represented by the equation ( M , ) H = Pqoa2 (k) where /3 is a numerical factor depending on the abscissa x of the point. In a similar manner we get (My)y=0

= P1Q0O,2

(I)

The numerical values of the factors /3 and 0i in formulas (Zc) and (J) are given in Table 10. It is seen that for b g: 4a the moments are very close to the values of the moments in a strip under a triangular load. Equations (106) and (107) are used to calculate shearing forces. From the first of these equations, by using expression (j), we obtain for points on the x axis

The general expressions for shearing forces Qx and Qy are

(m)

(n)

The magnitude of the vertical reactions Vx and Vv along the boundary is obtained by combining the shearing forces with the derivatives of the twisting moments. Along the sides x = 0 and x — a these reactions can be represented in the form (o)

TABLE 11. NUMERICAL FACTORS 5 AND 5I FOR REACTIONS OF SIMPLY SUPPORTED RECTANGULAR PLATES UNDER HYDROSTATIC PRESSURE q — qox/a

v = 0.3, b > a Reactions 8qoa b

,

x =0

y

_ ~ "U

y~ 0.256

Reactions 5igo6

x =a

y

_n "U

y= 0.256

y = ±6/2

x= 0.25a

x= 0.50a

x= 0.60a

X = 0.75a

1.0 1.1 1.2 1.3 1.4

0.126 0.136 0.144 0.150 0.155

0.098 0.107 0.114 0.121 0.126

0.294 0.304 0.312 0.318 0.323

0.256 0.267 0.276 0.284 0.292

0.115 0.110 0.105 0.100 0.095

0.210 0.199 0.189 0.178 0.169

0.234 0.221 0.208 0.196 0.185

0.239 0.224 0.209 0.196 0.184

1.5 1.6 1.7 1.8 1.9

0.159 0.162 0.164 0.166 0.167

0.132 0.136 0.140 0.143 0.146

0.327 0.330 0.332 0.333 0.334

0.297 0.302 0.306 0.310 0.313

0.090 0.086 0.082 0.078 0.074

0.160 0.151 0.144 0.136 0.130

0.175 0.166 0.157 0.149 0.142

0.174 0.164 0.155 0.147 0.140

2.0 3.0 4.0 5.0 oo

0.168 0.169 0.168 0.167 0.167

0.149 0.163 0.167 0.167 0.167

0.335 0.336 0.334 0.334 0.333

0.316 0.331 0.334 0.335 0.333

0.071 0.048 0.036 0.029

0.124 0.083 0.063 0.050

0.135 0.091 0.068 0.055

0.134 0.089 0.067 0.054

and al^ng the sides y = ±6/2 in the form

in which 5 and 5i are numerical factors depending on the ratio b/a and on the coordinates of the points taken on the boundary. Several values of these factors are given in Table 11. The magnitude of concentrated forces that must be applied to prevent the corners of the plate rising up during bending can be found from the values of the twisting moments Mxy at the corners. Since the load is not symmetrical, the reactions Ri at x = 0 and y = ±6/2 are different from the reactions R2 at x = a and y = ±6/2. These reactions can be represented in the following form: Ri = niqoab

R2 = n2qoab

(q)

The values of the numerical factors n\ and n2 are given in Table 12. TABLE 12. NUMERICAL FACTORS nx AND n2 IN EQS. (ry) FOR REACTIVE FORCES Rl AND R2 AT THE CORNERS OF SlMPLY SUPPORTED RECTANGULAR PLATES UNDER HYDROSTATIC PRESSURE q = qox/a

v = 0.3, b > a b/a

m nj

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

3.0

4.0

5.0

0.026 0.026 0.026 0.026 0.025 0.024 0.023 0.022 0.021 0.021 0.020 0.014 0.010 0.008 0.039 0.038 0.037 0.036 0.035 0.033 0.032 0.030 0.029 0.028 0.026 0.018 0.014 0.011

Since a uniform load ^o is obtained by superposing the two triangular loads q = qox/a and qo(a — x)/a, it can be concluded that for corresponding values of b/a the sum n\ + n2 of the factors given in Table 12 multiplied by b/a must equal the corresponding value of n, the last column in Table 8. If the relative dimensions of the plate are such that a in Fig. 66 is greater than 6, then more rapidly converging series will be obtained by representing Wi and W2 by the following expressions: (r) (s) The first of these expressions is the deflection of a narrow strip parallel to the y axis, supported at y = ± 6/2 and carrying a uniformly distributed

TABLE 13. NUMERICAL FACTORS a FOR DEFLECTIONS OF SIMPLY SUPPORTED RECTANGULAR PLATES UNDER HYDROSTATIC PRESSURE q = gox/a

b