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1957

The stability of rectangular plates with built-in supports at the The stability of rectangular plates with built-in supports at the

edges edges

John Bruce Miles

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THE STABILITY OF RECTANGULAR PLATES WITH

OOILT-IN SUPPORTS AT THE EDGES

BY

JOHN BRUCE MILES

A

THE3IS

5Wanitted to the faculty of the

SCHOOL OF MINES AND METALLURGY OF THE UNIVEIBITY OF MiffiClJRI

in partial fulfil~ent of the work required for the

Degree of

MASTER OF SCIENCE, MECHANICAL ENGINEERING MAJOR

Rolla, Missouri

1957

ii

A.Ct:HONLEDGEMENT

The author wishes to thank Professor Aaron J. Miles for suqg:est­

inq the topic of this thesis, and for giving invaluable assistance

towards the completion of the problem.

iii

TABLE OF CONTENTS

Page Acl::rl.owledg"elllent •••••••••••••••••••••••••••••••••• _ i

List of Illustrations •••••••••••• ·•••• • • • • • • • • • • • • i v

List of Plates .................................. v

Introduction •••••••••••••••••••••••••••••••••••• 1

Review of Literature •••••••••••••• :;. . ~ ~- ......... •.• 2

Notations. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 4

Discussion of Problent ••••••••••••••• 1. • • • • • • • • • • • 5

Conclusions •••••••••••••••• · -· •• . . . . . . . . . . . . . . . . . 25

Bibl.ie>g":"ra..phy •• •••••••••••••••••••••.• : • -··- ··-~ - · • • • • • • • • • 2.6

Vita ....•.••.•...••.••.••.••.•••.. •L•. ··~ •••• •-.... 27

LIST OF ILLUSTRATIONS

Fig. No.

1. Bketch of reci:angular p1a.te with

campressi ve edqe loading and bui1 t-in

Page

SU.PJ)O: rt S • • • • . • • • • • • • • • • • • • • • • • • • • • • • • • • • a • • • • • • • • • • . 6

2. Sketch showinq symmetrical b1;1ckling of plate.... • • 18

3. Sketch showi.nq unsymmetrical buckl inq of p1a te ••••• 18

iv

LIST OF PI.ATm3

Plate Noa Page 2

1. ~- v s- ¢ for symmetrical. bucklinq •••••••••••••••••••••• 23

2. if'- VS- ¢11. for unsyD~D.etri-eal. buckling •••••••••••••••• a ••• 24

1

INTRODUCTION

.<

The purpose of this inV'estigation is to derive an expres.sion

for the critical stress for a thin rectangular plate with ~ressive

edge loading. The load is applied on two -o}:pOsi te edges with the re-

ma.ining two edges built-in. (See Fig. 1) This is essentially a sta-

bility problem.

Rectangu.lar plate.s with compressive edge loading and various

edge restraints appear · in a number of engineering" structures. Bulk-

heads of ships could be so classified, as could certain components

of fabricated beams and colllJlUls. The new supersonic B-58 b<;mber

developed by Convair has a honeycomb wing structure dictated by heat

conduction requirements. This, too, involves rectangular plates with

compress! ve edqe loading.

It is hoped that this thesis will aid engineers in designing

rectangular plates with the particular edge restraint mentioned.

To further adTance this purpose, the author has shown the resultant

eQuations in graphical fonn. This greatly simplifies the final

equations, which are transcendental.

2

REVILW -OF LITERATURE

The outstanding contributor to the solution of the problem of (1)

buckl inq of rectangular plates is S. P. Timoshenk:o. He has developed

the basic differential .equations for buckled plates with various edqe

restraints, as well as solving these equations for a number of particu-

lar cases. In fact, Timoshenko solved the problem considered in this

thesis, using a different method than employed by the author.

Tim.oshenk:o lists three more or less similar methods of solving (2)

the problem under consideration. They are:

(1) The assumption can be made that the plate has same initial

curvature. Then the value of the load acting in the middle

plane of the plate which causes the lateral deflection to

become infinite is the critical load.

(2) Assume that the plate buckles slightly under the action of

forces acting in its middle plane, and then find the magni-

tude of the forces necessary to maintain the buckled condi-

tion. This method involves using the differential equation

of the deflection surface.

(3) The energy method requires obtaining ail expression for the

strain energy of a slightly buckled plate and an expression

(1) S. P. Timoshenko, Theory of Elastic Stability, McGraw-Hill Book Company, Inc., New York, N. Y., 1936, pp. 324-418.

(2) Ibid.

3

for the work done by the loads as they buckle the plate.

The condition of stability requires that the first varia-

tion of the difference o£ the two quantities mentioned be

zero.

The energy method will be used in this investigation, whereas

Timoshenko used the second mentioned method to solve the problem

unde.r consideration.

Other leading figures in the £ield of plate buckling have been (1) (2) (3)

G. H. Bryan, E. Chwalla, and A. Nadai, to mention a few. Their

work parallels Timoshenko' s in most respects.

(1) G. H. Bryan, Buckling of Compressed Plates, London Mathematical Society Proceedings, Volume XXLI, pp. 54.

( 2) E. Chwa.lla, Da.s Allgemeine Stabili tattsproblem d.er gedruckten, durch Randawinkel verstarken Platte, Ingenieur--Archiv.

(3) A. Nadai, Elastiche Platten, Julius Springer, Berlin, Gezma.ny.

SYMBOL

w

x andy

2h

a and b

i

J

D

y

c

4

NOTATIONS

SIGNIFICANCE

lateral deflection of the plate

coordinates of the middle plane of the plate

nota.tion for4 and~ , etc. dr.'& Jx t~y

modulus of elasticity

Poisson's Ratio

thickness of plate

dimensions of plate

positive integer

nonnal compressive stress

potential energy of plate s

e(RJ,) = flexural rigidity of the plate

/ll(l--,-) ifCJ,

4

l.!E .. vaJI:b· a function of y but not of x

constants of integration

5

DISCUSSION

Consider the rectangular plate loaded as shown in Fig. 1. The

plate has a length, a, a width, b, and thickness, 2h. A nonnal com-

pressive stress, q'" , acts along the sides x = 0, and x = a, while the

sides y ~ -b/2 and y = b/2 are built-in.

It is awarent th.c!:t some particular value of ,.- will cause the

plate to buckle slightly. This minimum value of q- , necessary to

produce buckling, is what is desired frOJil this investi-gation, and it

shall be called the critical stress.

As preTiously mentioned, the energy method will be used in sol vinq

this problem. The strain energy of a slightly buckled plate is (1) 1

The work done by the load is (2):

Q. ~ (2) 'W= r-r 2h..-: ufJxdy

D L~ 2. (1) SeeS. P. Ti.m.oshenko, Theory of Elastic Stability, pp. 307.

(2) Same as (1) except pp. 310.

6

Fig-. 1

X

7

The difference of equations 1 and 2 which v-rill be callfKI the

potential energy, J, is then:

The first variation of (3) when the critical load is reached

is zero. This statement might be validated by realizing that when

in a slightly buckled condition the elaBtic energy of the plate will

increase a certain amount due to an incremental increase in lateral

deflection of the plate. Further, the l'lOrk dene on t.he plate by the

load durinq this incremental increase in lateral deflection ~ctly

equals the increase in the plate' a elastic energy. Therefore, by

means of the Calculus of Variations:

8

It should be reca1led thatz

(a) or

(b)

or

Each of the six tezms of {4) will now be considered separately.

The first te.nn becom.~s:

{Q.r~ . (s) =. ~~ [w~x J"/x

0 -~ ./

Inteqratinq by parts:

(s)

dv = d(S'w)

v = tfw

9

Since tfw =0 when x = 0 or x = a the middle tezm in (5) drops

out.

Consider n.ow the second tenn of (4).

Integrating by parts: Q..

u = [ w"t Jx 0 .

Q.

diJ = £ wlt>~y J~dy

10

Since S' wy = 0 for y = ~ h/2, the first tenn in (6) drops out.

Integrating by parts: Q.

u = J w,.,yJ-,

y = -b/2.

Next the third tezm of (4).

Inteq~le this by parts, lettinq

u = [ · Wyy-L . --~ 7

11

V= J W X

Usinq i~eqration by :parts on the second te.na of (7)

U = ~~cv)'y~a} * = J(clw)

v= Sw

The middle tenn in (7) drops out since Jw= 0 at x = 0 or x = a.

12

The first t .enn above drops out si.nc;:e W y = 0 when x = 0 or x = a

Iutegratinq again by parts . &

u = [ w.t•y o&c

The first te:DD. here drops out sinceJtu= 0 at y = -b/2 or y = +b/2.

13

The fifth te~ of (4) is:

Integrating by parts:

u = ~w o4 )0 G)'y ,

The first tenn dro,ps out since dr~y= 0 at y = ~ b/2

Integrating by parts:

u = .[~yyyJ~ 0

V= dw

The first teDil drops out since/w = 0 a.t y = ! b/2

The · last tenn of (4) is then:

The first tellll drops out sincedw = 0 at x = 0 or a

Colleotinq tezms of equatiollS ( 5) throuqh ( 9) :

14

€/)= o{I)_4111JIIIi-I .. R~x11 +- W~fY.Y +~r~~ J,clx J',._,

+ ojf.~~Hf~y)o/~~::= o

15

The Tariation in the defleetion, df4l , is an arbitrary quantity

. except at the boundaries of the plate where dw is zero. Consequently,

forcfJ to be zero for all variations, the inteqrands of { 4) must be

zero.

There£ore:

f2) ~If + ~ Wyy -= o7 . ~"=' ~

X•4

Equation (12) is the same as saying the Jllallent is zero when x = 0

or x = a, since the left hand side of (12) is the bending mau.ent around (1)

an axis parallel to the y axis.

Additional boundary conditions which (11) must satisfy are:

{!g) W = 0 when x z: 0 or x = a

~1/) "-' = 0 when y = -b/2 or y == b/2

~~ ~ = 0 when y = -b/2 or y = b/2

The asSUIDption is made that a solution to (11) is of the fonn

W = Y sin i.VI a

See Timoshenko, Theory of Elastio Stability, pp. 338.

(2)

(1)

(2) This asSUD~Ption was firat sugqested by M. Levy. See Sur 1' equi1ibre e1astique dlune plaque reata.Dgulaire, by M. Leu", Ccaiptes Rendu.s, · de 1'Acadetie des Sciences, Paris, Fra.noe, T01. 129, 1899, pp. 535-540.

In (16) Y is a function of · y only and i is a positive inteqer,

which physic:ally is the number of half sine lRl.Tes the plate buckles

into in the x di.:m.ension. Also let ~ • .i:.!r a..

(16) into (11)

- Rhr: 1l . Q(Y y = 0 ·-- oC $/# "'

0

Di Tiding . by sin o<. x gives:

{t7b)

• Then substituting

This is an ordina.zy differential equation with constant co-

efficients. The gener,al solution of sudh an equation is:

where

!7

Therefore:

(1) If the plate buc.kles syanetrica.lly with respect to the x-a.xis,

the odd functions in (19) must drop out. This means that c1 and c3

are zero.

ConTersely, if the plate buckles in one direction for positiTe (2)

values of y aad in the opposite direction for negative values of y,

the even tenn.s in Y then drop out. In this case c2 and C 4 are zero.

Assume syDB.etrica1 buckling, in which case ( 19) beccmes:

Substitutinq ·(20) into boundary conditions (14) and (15) respectiTely

qiyes:

(2 i) e.t Cos1 1_6 . f- c; .c.s; ¥ .:::: t:>

(?Z; ~ n1, s;,/ ;/ .. - t:; ~ shl ~ .. o . 3

(1) See Fig. 2.

(2) See Fig. 3.

18

Pig. 2

SYMMETRICAL BucHJ.IN•

Fig. 3

UNSYMMIITRICIIL BucKLIN•

19

Equations (21) and (22) could be satisfied by puttinq c2 and c4

equal to zero. However, this would yield a trivial solution. A use-

ful solution is foll.nd by puttinq the coefficient detenrrlnant of c2

and c4 equal to zero.

=0

This gives:

Dividing by: ~osh~yos¥) @1) nt2 4M r ~ vt~h'f-!/):;:~ 0

substituting:

. r = '(~:;-it Q,-1 ~,2= hrb a

20

Equation (23J then is the final equation for symmetrical buckling.

This shows the relationship between the dimensions of the plate, the

elastic properties of the material, and the critical stress.

In a s~lar manner the equation showinq these relationships

can be written for unsymmetrical buckling. For this configuration:

Substituting (24) into boundary conditions (14) and (15) re-

~ively gives:

4~ ~4d/J -r ~ ~~~s~l ::r o #. Ill

Also a useful solution of these equations must be obtained by

setting the coefficient dete~inant of cl and c3 equal to zero.

S/A/ ~ S",~~ z,.. z

:::0

~e~.s-~~" ~~lllf~ r -:z

21

Divid:Lnq by

Substituting ;[ =.,; 2 ;;-Ill am/

Equation ( 28) 1 then, is the unsy.DIIIletrical buckliDg .countezparl

of equation (23).

It is obvious that equations ( 23) and ( 28) are not practicable

by an engineer in design problems. To alleviate this situation,

graphs have been plotted showing the relatioRShip between ~ and ¢2. for both modes of buaklillg. These graphs are shown in Plates 1 and 2.

Points necessary for plottinq these graphs were obtained by trial and

error solutions of equations (23) and (28}.

The use of these graphs will be shown by means of an illustrative

problem. Consider a steel p1ate with E = 30 x 106 psi, and .A{= .25.

Let a = 18 inches, b = 10 iac::hes, al\d 2h = .1 inches.

AsSlUile first i = 1,. i.e. the plate buckles with one half-sine

wave in the x-direotion. Further 1 assume syJIIIletrica.l bucklinq.

rp2 .. i_q/;J = (1)(3.14}/tJ:. /.74

Q.. /B ;A • (to l.a.rqe to be read from graph)

22

Try i = 2

£= 8.83 (fran graph}

Try i = 3

¢2.= 3( /.~ e:. $.22

d::= 8.30

It is apparent when obtaining these values of ~ from the qraph

that i = 3 gives the lowest value of;e • Consequently the plate will

buckle into 3 half-sine waves in tlte x-direction.

r .. et.3o = V2:fl! ~ a 6 3

v-= (fJ.'3't0.:= (8.31;1 30•/0 (.1) 2 h b2 '!) (!oJI /2{/--:ls;)

'V• /8,400 ps/·- a.Bswer.

:-.r - -:- -· ~ ·-- -:-.

'"}" f

. - - --- o- ·---.-_:~--.-- -~------~-. ------ ---- ----. - ·-- -··-----:--- . . . .;,... . ~ .- .

, .. ' I ' ..

i --

. t.z. -- - · ----- ---~---+-~-~-

~~J~_.___---+-----+-----+---- ~ ~ V& ¢~--· · · f r ; .

Bv -----------1~--~--------~--~--------~--------4----------------

• · ;

' t

• !

. ~ .

. ' :·

--~---~--- -- --·- --- - -- -- - ------·-- --------+-------1---

------+------- -- -· ·-·- ·-··-------------+---

8

.. >-.·

: .

----.---~-o------=-----1t-----· .

. _· E-2:

.2

' 2. ¢ ... .

--- .. :.. -- - -... ----

.S 1 .. --·--- .. . _____ _

.

:L -·------·--+--

• I

•

! ~ . ,

·· r·i '!" ~ . !

; ... 1 l -.1 L~-

t·;.

•. !

.i • ··· ·

:

·:

: 1 I <I• f

' . . . . ) ... - ,~ I

:

..

I ! . . :.

i i ~:

i-

:

; .

; I

, '

i ;

1

... !

.• ! •·

.. , ~ ; .: !

25

OOBCLUBIONS

The same problem as the ~ple problem previously shown was (1)

worked by the author, using the tables prepared by Tim.oshenko from

the results of his investigation of the same loadinq conditions as

considered in this thesi·s. Timoshenk:o's tables sh~,_ a critical

stress of 18,500 psi as cCllll;Pa.red to 18,400 psi. This obviously is

quite a close correlatlon.

Referring again to the example problem worked out, it will be

noted that the plate buckled. into three half-sine waves in the x-

direction. As a qeaeral rule-, lon.q plates, large a/b ratio, tend to

buckle into more half-sine waves than short plates.

A plate, loaded as in this investigation and left free to buckle,

will buckle symmetrically rather than unSJDIIIletrically. UnsyDIBletrical

buckle occurs only when restraints prevent symmetrical bucklinqs, and

then at a higher stress than required for syJ~~Detrical buckling. This

occurrence is reasonable, since intuition would tell us that an un-

Symmetrically buckled plate would be stiffer than a syDIIletrically

buckled one. This also explains why£ is greater for a given ¢2

on graph 2 as compared to qrapb. 1.

(1} S. P. Timoshenko - Strength of Materials, Part II, D. VanNostrand Company, Inc., PriJtCJeton, N. J., pp. 197.

BffiLIOORAPHY

Bryan, G. H. - BuCkling of Compressed Plates. London Mathematical Society Proceedinqs, Volume XXLI, pp. 54.

Chwalla, E. - Das Allgemeine Stabili tattsproblem. der gedruckten, durch Randwinke1 verstarken Platte-. lnqertiear - Archi v.

Le-vy, M. -A pape-r on plate problems. Acad. Sci. Paris, Volume 129, pp. 535, 1899.

Miles, A. J. - Strength and Stability o£ Rectangular Plates on Elastic Beams. Thesis. University of Michigan, Ann Arbor, Michigan, 1935.

Nadai, A. - Elastiche Platten, Julius S:pringe:r, Berlin, Gezmany.

Timoshenko, S. P. - Theory of Elastic Stability. McGraw-Hill Book Company, Inc. · New York, New Yorl:.,. 1.·936, pp. 307, 310, 324-418.

Timoshenko, S. P. - Theory of E1asticity. MoGraw-Hill Book Company, Inc. New York, New York. 1934.

Timoshenko, S. P. - Strength of Materials, Part II. D. VanNostrand Company, Inc. Princeton, New Jersey, pp. 197.

27

VITA

The author was boD\ Febnta.ry 2, 193a," in st. Louis, Missouri.

His parents are Dr. and Mrs. Aaron J. Miles.

He att8nded the public schools of Rolla, Missouri, qradnatinq

from Rolla Hiqh School in 1951. In June or 1951, he entered the

Missouri SchOQl of Mines and Metallurgy, a.nd was graduated May,

1955 with a B. S. Degree in Mechauica.l Eltgineeri.nq. The last two

years he has S))ellt tea.chillg' in the Mechanics Depari:JaeJrt at Missouri

School of Mines, and workinq towards &D. M. s. :o.,qree in Mechanical

Enqineerinq, which degree he hopes to COJII)lete in May, 1957.