NON-LINEAR BENDING OF A CIRCULAR PLATE UNDER NORMAL PRESSURE

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Xerox University Microfilms 3 0 0 N orth Z e e b R oad A nn A rb o r, M ichigan 48106

ld3907 ioIn «Bo5

13-2,1.9(09 s

Bromberg, Eleazer, 1916•' ITon-lin8ar bending of a circular plate under normal pressure. New York, 1950. [ij.],72 typewritten leaves, diagrs. 29cm. Thesis (Ph.D.) - New York Univer­ sity, Graduate School, 1950o Bibliography: p.71-72. C 57474 '

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THIS DISSERTATION HAS BEEN M ICRO FILM ED E XA C TLY AS RECEIVED.

LIBRAM OP IS P TORI UNITORSITY U M V lR S ITT HEIGHTS

NON-LINEAR BENDING OP A CIRCULAR PLATE UNDER NORMAL PRESSURE

By Eleazer Bromberg

A dissertation in the department of mathematics submitted in partial fulfillment of the require­ ments for the degree of Doctor of Philosophy at New York University*

A p r il

1950

PREFACE

The author wishes to express his thanks to Professor J. J. Stoker for suggesting this problem and for his guidance and advice;

to Professor

K. 0. Friedrichs for many helpful discussions and suggestions;

and to Dr. E. Isaacson for assistance

in some of the calculations.

7,0 114 3

CONTENTS

1

Introduction

Page

2

Mathematical

3

Perturbation Method

13

4

Power Series

25

Formulation

Method

1 4

5 Asymptotic Solution

38

Bibliography

71

ILLUSTRATIONS

Section 3 PERTURBATION METHOD Figure

1 Radial Stress,

Case I

Page 16

2 Circumferential Stress, Case I

17

3 Slope,

Case I

18

4 Bending Moment,

Case I

19

5 Stress,

Case II

21

6 Stress,

Case III

22

7 Slope,

Cases II, III

23

Section 4 POWER SERIES METHOD Figure

8 Radial Stress,

Case I

Page 28

9 Circumferential Stress, Case I

29

10 Slope,

Case I

30

11 Bending Moment,

Case I

31

12 Stress,

Case II

32

13 Slope,

Case II

33

14 Stress,

Case III

34

15 Slope,

Case III

35

Section 5 ASYMPTOTIC SOLUTION Figure 16 Slope, 17 Radial

Stress,

18 Slope, 19 Radial

Stress,

20 Slope, 21 Radial

Stress,

Case II

Page 54

Case II

58

Case III

61

Case III

62

Case I

68

Case I

69

1 - INTRODUCTION

The complete analysis of stress and strain dis­ tributions in a thin plate under a uniform normal load and supported in some manner at its edge is of j

interest for both practical and mathematical reasons. Practically, this information is useful in achieving safe and economical design of walls and floors, as for example, in high pressure tanks.

Mathematically,

the formulation of the problem as derived by v. KarmanC6]1 for the bending of thin plates with large deflections, leads to a pair of non-linear partial differential equations for which special methods of solution have had to be developed in many cases. In the present investigation the plate is assumed to be circular and to retain radial symmetry under loading.

This leads to a considerable simplifica­

tion, as the v. Karman equations become ordinary rather than partial differential equations. The methods employed to solve these equations essentially p is the pressure on the surface, h is the thiek^ Numbers in brackets refer to references which are listed in the bibliography at the end.

ness, and R is the radius of the plate*

When this

ratio is small, a perturbation method leads to a set of equations which can be solved without difficulty* For somewhat larger values of this parameter a power series expansion about the origin can be used. However, as the parameter becomes larger, the convergence of the power series gets steadily worse, requiring an increasing number of terms for even moderate accuracy and ultimately making this method impractical.

This development is related to the

presence of an "edge effect” or "boundary layer" phenomenon which is characterized mathematically by the fact that the coefficient of the highest order derivative approaches zero as the parameter gets larger so that in the limit, the term containing the highest order derivative vanishes.

A special asymp­

totic treatment, which resembles the Prandtl boundarylayer theory for the flow of a viscous fluid around an obstacle, is required in order to obtain solutions for large values of this parameter C&G . The normally loaded circular plate has been studied by a number of writers under various simpli­ fying assumptions [7,5>] .

Thus, H. Hencky [5] ne­

glected the effect of bending and treated the plate

as a membrane*

Solutions of the v* Karman equations

for two different sets of boundary conditions were obtained for a limited range of parameter values by S.

Way 0°G by means of a power series method* The edge effect was treated by boundary layer

methods by K. 0. Friedrichs and J. J. Stoker [3,4]

in

the analysis of the buckling of a circular plate under edge thrust, and subsequently by E. Bromberg and J* J. Stoker [ 0

in dealing with curved elastic sheets#

The asymptotic method was applied in a somewhat dif­ ferent way by W. Z4 Chien L£] to the case of a clamped and pinned circular plate under normal load as an ex­ tension of the Hencky problem* The purpose of the present investigation is to obtain complete mathematical solutions for the whole range of variation of the parameter from zero to infinity for several different boundary conditions. The outstanding problem was to clarify and extend the techniques of the asymptotic method, a3 will be shown.

4

2 - MATHEMATICAL FORMULATION

In deriving the linear equations for the stress­ es in a thin plate, the following assumptions are made:

the thickness of the plate, h, is small compared

to the other dimensions;

a normal to the middle sur­

face in the unstressed state remains a straight line normal to the middle surface in the stressed state; the normal stress on a surface parallel to the middle surface is neglected and the stresses in the middle surface are averages of the stresses over the thickness of the plate;

and the normal deflection of the middle

surface, w, is very small compared to the thickness h* Instead of the last assumption, v« Karm&n con­ sidered the deflection to be large and therefore that the squares of the slopes of the middle surface are of the same order of magnitude as the strains in the middle surface, yielding the following non-linear re­ lations between the strains £ u, v, w of the middle surface*

and the displacements

These expressions differ by the quadratic terms in ox

y

oy

from the expressions of the linear theory*

With these assumptions, v* Karman derived the following non-linear equations for the bending of a thin plate under normal pressure p:

.

2

(8.3)

XL v"i =(#4 j - Pit* I 'dxdy/ dx dy

where ^

is Poisson's ratio, S is the modulus of ©las-

ticity,

f =

, and is the Airy stress function

which is related to the stresses in the middle sur­ face by the relations

(2.4)

o-^ _

,

oiy -

^

,

y = 1

We shall confine ourselves in this study to a circular plate of radius R, and we assume that it has radial symmetry in the stressed state*

All de­

pendent variables are therefore taken to be functions of

r =1/xSyl

and (2*2) and (2.3) reduce to ordi­

nary differential equations;

namely,

6

(2.5)

/ « v 4w = i - + ° h

(2.6)

r \dr d r*-

c|r d r * J

' V 4§ = - - i - ^ ^ .

E.

The operator v

T

z

r d r dr*

can be expressed now as

(2 .7 )

r

dr v d r J

The radial and circumferential stresses O^, and 0^ are related to the Airy stress function by the ex­ pressions: r

j_ m The relations in (5.31) are not all independent. It can be shown by direct substitution of the power series (5.9) for

and

in the interior equations

(5.5) and (5.6), and of the solutions (5.24) and (5.25) for

and TCh into the boundary layer equa­

tions (5.13) and (5.14) that the same relations exist among the

and

as among the

and

•

In

both cases the solution of the nth set of equations leaves undetermined two of the or

9*0

*

, or

d^,

namely,

■^lae boundary conditions at

z * 0 and at y = 0 (z = 1) together with (5.31) then suffice to fix these constants.

This can best be

seen by returning to the problem at hand (Case II) and applying the condition at y = 0 (5.27) to the first order solution, from which it follows that

52

(5.32)

U ’= ~ IfcOL*-^ _ So? e

Cpi = 2.6G2 f l.447z i-.fe ^ x ^ .Z -S e z ^ + .lie x ^ ^ .O S O r5

(5.43)

+ .02.1 z.to+■ .009z.7 + .004-z6 + .002z* + . . -

uz = - 6 cp0(x < p j"




3 J- 1 W fl?

SV » - g j , *■»

60

_

Ua =

U;, + Uj. - (uio 4- u2lC Vlx

U, =

—

- 'h

II

- u ,vc,

x

(5.45)

£

U.=

U0 + c'^U, 4 0 ^ ^ . .

1 - u lz c ' X 1)

-t-. .

(5.46)

Figures 18 and. \9 contain graphs of u and cp for c*0.005 as determined by the power series method in section 4 and by the asymptotic approximation functions u 4 and