BUCKLING ,, OF

CANTILEVER THIN PLATE

WITH PREE END SUBJECTED TO UNIFORM SHEAR

Jamea Obie Meng Yu 11 t

Theaia submitted to the Gi-aduate Faculty or the

Virginia Polyteo.bnio Institute

in candidacy tor the degree ot I'1A.STER OP SCIENCE

in

ENGINEERING MECHANICS

June, 1963

Blacksburg, Virginia

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TABLE OP COUTh"NTS

I. LISTS OF TABLES, FIGURES, AHD SYMBOLS

II. INTRODUCTION

III. ~dE I~1VESTIGATIOlif

1.

Page No. 3 s 8

2, SUb.f eot o:t Investigation Mathematical Prooedurea of Investigation

IV. lWMERI CAL RESULTS 25 v. DISCUSSION AND CONCLUSIONS 30

VI. APPENDIX 33 VII. ACKNOWLEDGEMENTS 36

VIII. BIBLI.OGRAPHY ~7

IX. VITA 39

.. 3 -

I. LISTS OF TABLES, FIGURES, Al1D SYMBOLS

Table No.

Piglll'e No.

1.

List of Tables

Title

Values of k._.L and Q(. 11 from n = l to n = 5 Function• a"" , s"" and z'"' for Arbitrary Value of n

!i\mctiOnS 01'1.ll I S ''"' and Z Mii to~ Arbitrary Values ot m and n

List of Figures

'l'itle

Diagl'am ot the riate

Curvea tor Maximum Stress at BllokUng State

Page No.

25

34

JS

Page Mo.

9

33

x . .. U· " a-:.

•J ... 1.J p. '

fl ( .. \ a• a.' •

u

u w p

( ~11\Q .. >, .. D

B

v

- 4 -

L1 st ot Symbols

Reotangular ooordinatea

Displaoements 1n the x; direction.a

Stress tensor 1n x, 17stam

Strain tensor in Xi, system

Boq foi-oes per unit volume in the x • direotions

Cubical dilatation

Parameters

Volume denaicy of strain enera

Strain enel'gJ of the whole elastic ayatem Work done by external forces

Potential energy ( P = U - W )

MaxSmum atrese at buckling state

PJ.ex.ural rigidit,' ot plate ( D = It~~~ v"> ) Modulus of elaaticit"J'

Poiaaon•s i-at1o

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II INTRODUCTION

As far back as 1757, the buckling problems o'£ stl"Uts

under different bounc:l.ary conditions were investigated by llll.er.coa) Lagrangeco•t) followed and made a more through

study to deter.mine the length 'Which a column must attain

to be bent by its own or applied weight. In 18451 E. ( o.\) Lamez-le found a more accurate differential equation

tor the buckling load of struts than Lagrange•a, and solved

this equation by the method or series. This modiiaioation

introduced into Lagttange•s result gives only a taotor 'Which

is negligible in moat praotical cases. In the year 1850. (o!)

G. Kirchhoff introdueed the ene~gy method for problems

ot elastic stability aa an extremum principle of meohanios

t.ilioh character:J.zea the conditions of equilibrium in an

elastic body. The ~irst one to apply the energy principle ( C>~) to the solution or buckling problems of plates ~ras Bryan,

Who applied the energy method to get a differential equa-

tion which governs the buckling load or the plate.

To consider the buckling problem as a boundary-Yalu•

p~oblem hQ.8 the advantage that the solution obtained is

accurate. But it is difficult to write do'Wll the govel"D.ing

equation and sometimes harder or even impossible to

ascertain and satiai'y the boundarry conditions. Then the

approximate method based on an energy criterion waa

- 6 -

1ntroduoed into the buckling problem. The main credit. appai-ently, belongs to Ra.yleighco") and Ritz;<o-i) the

former introduced the approximate method for the main

purpose ot finding the frequencies of vibrating systems,

the latter general.ized lla:y"leigh•e met~od into an extremum

problem which is widely used in math~tical physics.

Timoshenko developed Ritz method into a powerful tool to~

the treatment of buckling problems under various loading

and bound~~ conditions. This method is frequently used in

his book "IJ.'heo:ry ot Elastic Stability."

'lhe Ritz method leads to an approxi:mate value ot buckling load which is larger than the exact one. 'l'hie 11

due to the fact that more strain energy is needed to

maintain the assumed buckling configuration which deviates (06) from the true one. In 1935, Trefftz supplemented Ritz

method by developing a procedure tor the determination ot

a lower bound tor the buckling load. Thus tha degree ot

accuracy tor the buckling load obtained by the energy

method can be judged.

The energy method for solving buckling problems is

ve'l!y' effective. since in most cases, only the rirat term

retained in an asswned series for the deflection yields

accurate results. But the power of the energy method ia

mainl7 rooted in the fact that it can ba used as a gener-al

approach for the problems of stability.co~)( lo)

- 7 -

1he problem considered in this thesis was first dis• ( 11)

cussed 1n 1899 by L. lTandtl who obtained a governing

ditterential equation based on the equilibrium oonditiona ot a narrow rectangular cantilever beam. Due to the as.aumpt1one,

the result obtained by Pl'andtl is good only when the length

or the thiri cantilever ia much greater than the width. When

the length of the cantilever beoomes smaller and WD&Jler, then

its buokling behavior ia more and more like a plate rath~

than a cantilever beam. Baaed on this idea, the author baa

used the enGrgy approach to attack the same problem baaed on

thin plate theorr.

The assumption that the deflections and alopea of the

plate in the middle plane are small enough to be neglected,

is adopted. As this assumption is viola.ted, a prooedure

baaed on the en•l'87 method in thia general caae ia alao

discuaaed.

.. 8 -

III. THE INVESTIGATION

l. SUbject of Investigation

The buckling problems of thin elastic plates have been .fully developed tor different boundary conditions and

different types of loading. But the cantilever plate, 1,e. one edge fixed and the others free, offers some difficulties

in finding the critical load. The ti-ouble results from the

method of solving such problems. Tne two tools to investi-

gate both the stability and the buckling problems are the

differential equation and the energy method. In the differ-

ential equation method, we usually have to solve a boundSl")"-

value problem of fourth order in each coordinate x, y, the

solution of which contains a particular solution and a homo-

geneous solution. The form of the particu1ar solution

depends on the types of loading, but the homogeneous solu•

tion always appeal's as some combinations of trigonometric

and hyperbolic functions. Due to the properties of these

functions and their derivatives, it is i1npossible to make

the boundary conditions of both the tree and the .fixed edges

satisfied. The energy method is a general appz-oach to

stability problems l~~ich can be used as an efficient and

Gconomical tool to dotermine the critical load of' some

specific structures. In an approximate energy method, we have to assurr.a a buckling configuration instead of' solving

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a dif.ferential equation. T'nererore the depee ot accuraoy

of the result depends on the adequacy of the assumed

function, which is based on suggestions from other solutions.

experimental data and even intuition.

~ ... de.i,.:J 1~ ~I I~ aLl o-l"er~ ~ >c le o ~fet O • 1. I T

x.\ L

IC,

Pig.1 Diagram of the Plate

)CI

" t

In this thesis, the author intends to uae the energr

method to examine the critical. load of a cantilevered, thin•

rectangular plate o.f thiclmess b, lying in the x,x1 plane

and being fixed along the x, axis ( Fig.l ). !!he apace

occupied by the plate berore deformation is det'ined as

( h. b k b - t , o , · "i"" ) ' ic; ' ( t , L , 'f )

'!he applied stress a-;i. is assumed to be a "dead load"

and uniformly distributed over the surface xt = L, The so-called "dead load" means that the applied load remains

constant in both magnitude and direction during the

buckling process.

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S1noe this paper diacuaeea the buckling ot a thin

plate• all assumptions used in thin plate theOl'J will be

adopted in the subsquent diacussiona unless notod.

- ll -

2. Mathematica1 Procedures of Investigation

As long as the applied load is smaller than the

critical one, there will be no displacement in the x~

direction. But "1hen the load increases to a certain vAl,ue,

the plate wi11 be slightly twisted. The buckling load he~e

ia de.fined as one at which the plate starts twisting and

"Which keeps the plate at its first equilibrium oontiguration.

Let u~ (xi• a~ ) be the assumed displacement in the

x\ direction which satisfies the geometrical boundary

conditions at least, whore a~ are undetermined paramete:ra.

Then the strain energy and the work done by the external

forces can be formulated as U(a~) and W(a_) respectively.

The f'unction P(a") will be defined as

P(a_) = U(a~) - W{aft) ( 3.2.1 ) which is the potential energy. Since the buckling load

keeps the buckled plate in en equilibrium position, this

requires that the potential energy asnume a minimum value.

':Chua we have

= 0

OX'

~ p 'a ( w ( u - - ] 'T> Q • '() Q --- w ~t • ~1.) = ()

... " o-,1

The split in Eq.( 3.2.2b ) seems meaningless, but aotua1ly,

- 12 -

the term ~ ~rz does not contain a;t, if W is a linear and

homogeneous .function of ~1 • However, this is true within

the elastie limit. By carrying out the differentiation of

F.q.( 3.2.2b ), we get

( u - - ) .,, ( 'W ) w ~ ( u ·-w CS'"lt - r.t · :a ~ + --- · - - '5". ) = o v II. u,1 °i1 ~Q" W 11

( 3.2.3. ) since ~.1 is assumed to be constant duzaing deformation.

Because the system is equilibrium. the strain energJ stO?'ed

in the system must be equal to the work done by the external

forces acting on the same s7atem , that is

or

w since -=- does not vanish at all. Eq. ( 3.2.3 ) will be "it

aatistied it and only if

And it follows tram Eq.( J.2.4 ) that

-a -~a" ( cs;" ) • o

Now the conclusion has been reaehed that the potential

- 13 -

energy to be a minimum is equivalent to saying that

and

a -- ( ~1) • 0 1) QI\

( 3.2.6a )

( 3.2.6b )

For an isoti-opic, homogeneous elastic body, we have

• _E-_ ( " 6 (Z. ) ~. ~ .. .. .J. 'J t+\J 1-1.V •3

~ '& I, t, ~. J • I t. ~.

( 3.2.8 )

where A = e,, , the cubical dilatation.

Here and in the sequel> summation convention is used.

That is, when a latin su.ffix is repeated in one term,

summation over the range ot l,2,3 with respect to that sutfix

is understood. And the partial differentiation is denoted

by a comma..

For a n~ow plate, that is, a plate whose thickness

is veey small compared with the other dimensions, the stresses

are assumed to be constant in the x" direction. In the

loading condition under consideration, we have

~' • 0 ( 3.2.9 ) In Eq.( 3.2.9 ) and all subsequent equations, the range or a

latin suffix being 1 1 2 1 3 is understood.

The statement, ~~ = o, can be verified in the

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f'ollo·wing ,.re:y. Since the faces of the plate at x~ I» =+-- t are

free of external loads•

°ii ( 'Ir. ' lt' ! ~ ) : 0

and these equations associated with the equilibrium equation

in the x~ direction,

cs.\ •• ,+ ~t·t .. ~.\t'\ s 0

der11and that

b Q"H '~ ( Jc• ' )(t ' 1 t ) : 0 ( 3.2.11 )

The stress cs;" ( ic,, 'IC 1 , ~ ~ ) 11 o and 1 ts derivative

~~ '-l ( ><,, x1 , ~ ~ ) = o means that the stress ~~ differs

f'rom zero veey slightly through the plate if the thickness

is small. Eas.( 3.2.9 ) have been obtained under the

assumption of a thin plate.

!Tom here on, a Greek suffix has a range of 1,2 and

ita surnmation is expanded over l and 2.

~pending Eqs.( 3.2.9 ) in terms of strains by Eq.(

3.2.7 ), we get

e ~°' = 0 ( 3.2.12a )

v e ~~ = - A ( 3. 2 .12b )

I • 'Z V

Substituting Eq.( 3.2.8 ) into F..qs.( 3.2.12 ), we have

Uoe. '~ = -u-4,oe ( 3.2.13a )

u~,<\ = v ( 3.2.13b ) -- u 1-v 0<,oe

By integrating Eq.( 3.2.13a ) with respect to x~· we have

- 15 -

the displacements as

Since u" is assumed to be a function of x, and x 1 only,

ue(. = -x"u_., .. + Coe

For a thin plate, the displaoements along the x, and

xt axia are comparatively very small, so we have the condi-

tions that u = 0 as x = O to evaluate the integration oc ~

constants, which results in Coi = o. Thus we find

uot = -x_. u" ,.,. ( 3.2.14 )

By differentiation ot :ais.( 3.2,14 ) with respect to x~·

( 3.2.15 ) Substituting the results ot Eq.( 3.2.lJb ) and Eq.( 3.2.15 )

into Eq.( 3.2.8 ), we got all strain components in terms of

u" and x 4 •

e.c~ = -x"u" '-'f ( 3.2.16a ) y ( 3.2.16b ) e~'\ =-xu

l - v .\ -\ "*'~

e -lot = 0 ( 3.2.160 )

T.he general expression of volume density of strain energy ia

I u = -;z tS"ij e~j

"" = --'---

were

+ _v_ ll'I. ) I- tv

= et II

- 16 -

t 1 t -4' e'lt "' e~\ "' 1.«i,1

( I .. ( 1-.,v)t ) ( 1.1.!,,, + ""~•!1) 1,.y1.

i (l-v)1. ( u.~''' 14~11'1 + 'Zw.!,,i>

Insei-ting the values of A1 and e.· e .. into the eXpression ~J •J

for volume density of strain energy, we get

IA.•~ (IL ) ~&l -~ )(t ( I 1 1 , .. v t(l-'1) ~,ot~ ~ 1 11

( 3.2.17 ) The total energy of the plate is obtained by integrating

Eq.( 3.2.17 ) over the whole deformed volume.

U • ( u.dv )v•

- _!:_ [ '('L ( 11' v v' • - u.i'" "~,11 ]dv

( 3.2.18 ) where v 1 denotes the deformed volume.

'1'he element of the deformed volume can with sufficient

aocuraoy be written as dx 1 d.x1.dxi, eince the displ.acementa

are small. And all terms in the bracket in Eq.( 3.2.18 ) are functions free of x~. Arter carrying out the integration

over x fl'om the 11mi t - ~ to + ~ , we get

u • ~ l ( ( ·~ ..... )t + HI- 11)( •!.11 - "~·· "~ •tt )) clA ( 3.2.19 )

- 17 -

mere dA = dx,dxt, the elemental area of the plate be.fore

deformation, and

1-- b" D = It Ct· .,i) , the i'lexui-al rigidity of the plate.

In order to compute the ori tical load by Eqs. ( 3.2 .• 6 ) ,

we have to find the expre3sion tor the work done by the

external i'orces.

BJ Clapeyron 1 s theocy C l4) , we have

(Ji'. u. dv + ( ~ u. ds = 2 ( udv Jv ' 4 Js " • Jv ( 3.2.20 )

,.mere u = ~ ~j e,j , the volume density of strain energy, and

F~ = components of body force pex- unit volume in the

x, direction.

The left hand side or Eq.( 3.2.20 ) 1s the work done by the

external i'orcea and can be expressed by

W = ( er:. eJ. dv ) v 'J ~J

The nonlinear t~rms ot displacement• must be included in the

expression tor the strain tensor, othel'Wise the function W(eij )

td.11 vanish identically. Therefore, we put I

e•i ="i"( And since

u .. ''3 + U· • J ...

~j= o, except ~1

- 18 -

By- tha assumption that the deflections and slopes in the x ,

x directions a.Pe Sl?lal.1 1 the third and fourth tel'm8 in the

parenthesis can be neglected. Then a simpler tol"Dl ot work done

by the external. forces is obtained,

w = ( CJ' ( u + u + u u )dv J 'V t'l I , 1 1 t 1 ~ ' I ~ , t

the function W can be formulated as tollOl'18:

Assume the applied load .. Oj'l = cr.1 at the free end is

uniformly distributed and remains constant during the

buolcling p~oceas. Therefore the integration can be carried

out over x_., since u-\ = u ~ ( x,, x 1 ) on1y.

By noticing that o;1 x" u~, 11 is an odd function ot x4,

and thus vanishes after integration,

w = b Gj11 JA u"" u~, 1 dA

SUbstituting Eq.( 3.2.22 ) and Eo.( 3.2.19 } into Eq.( 3.2.6a ) 7ields

er - t> fA [ (u-''-'" )t + 't ( l·Y) ( u.!,1t • "'°'"' '°'4•\'&) dA 11. - ~ b -------:-------__;-----1.--

J Lt"" "•, i d A. A

- 19 -

-Prom Eq.( 3.2.23 >. we notice that the load <rl'L wil.l

be completely determined if the f'unction u~(xi) ia known.

The disadvantage ot the energy method ia that we do not knov

the true expression of u~ .(.x~ ) which deaoribea the buokling

detmamation. But the benefit of this method is that we can

obtain an approximate result by an assumed .tunotion which

satisfies the geometl'ical boundB.J."1 conditions at least.

Let u 4 (x•) be expressed aa oO

u-4 ( x~) : L a"' t C 'IC,) Y..,. t 11.)

"'= 0

This expression does not satisfy all the bound&rJ' con<litiona

ot the plate, however, the geometrical boundal'Y' condition at

tlut fixed edge will be satisfied it the !"unctions t(.x, ) and

X,.(x1 ) are •uitabl7 chosen. )Jhen comparing buckling and

vibration phenomena. matlJ' similarities between them(•~)

l."ill !>e noticed. 'l'hus 1 t ia natu~al. to oh.OOl'f'I

x"' • C..()(i ( k' .. x1 ) - <..o<,~ ( K°,. ll1.)

~ o<.. ( .;i ... ( k .. ~1.) - 4i .. k (. k .. x..,,)) the n~l"mlll fUnctiona ot a cantilever beam Vibrating in its

( "') transverse direction. The values of k .. L and «n al'e to

be determined f'rom the following equations,

cos(k~L)oosh(k,.L} + l = O ( 3.2.26a }

( 3.2.26b )

.. 20 •

Since t~e boundary conditions oi' the vibrating beam

a:i:•o satisi'ied by x... the geometr1oa1 boundary condi tiona or

the cantilever plate are satisfied also. Thus the ohoica or

the function 1'(x1 } will be guided by the following conaidex--

ations:

(a) The product Of f (x, ) and X~ IllUSt not violate the

geometrical bound&l'J conditions.

(b) The closer the actual buckling configuration the

tunotion describes, the better the result that will be

obtained.

(o) It we take x 1 as oonstant tempo~arily, then

u~ (xi) = u~ (x 1 ) which depends on the geometrical dimensions

and the loading conditions~ Howeve:i-, it is reasonable to

assume that it is a linee.r .tunetion of x, as the slope in

the x, direction is not large,

According to the above considerations, the function f(x,)

will be chosen as

t(z,) = l - ~· where

"' k -- ~x ' -'Z, I 1

Substituting Eq.( 3.2.25) and Eq.( 3.2.27 ) into Eq.(

3.2.,24 ) yields oO

u-\ (xi. ) = ~ a" ( 1 - ~· )X" (xt) ( 3.2.28 ) t\: I

- 21 -

Due to the aaaumed u~(x~) aa given in Eq.( 3.2.28 >. Eq.( 3.2.23 ) will be simplified to

(1..[ <•., .. )-t • t C 1-'r) < •~, ... ) 1 ) dA

(A it~,, "'~ •1. d A

since u~ = O " , II

After ditterentiating u~(x;) in Eq.( 3.2.28 ), we get

the following expressions: 00

\4<\1 I • - + ) (lll x .. b, oQ

"'°', 'l. : ( ~· - I ) ~ a.,k~ S,. 00

"'~ , l'l. = ~ L Q .. k .. s .. . 'll: I

oP

14~ 't'L = ( ~· • I ) L Gh k-.. 1 c .. l\-:: I where

c:. I 0 x .. J,. : - ~ ~

.. c,i ... ( k .. li.) + ~·°"~ ( k,. Xi~ - oc. II ( C:..0~( k .. x ... ) - "°~~ ( k' .. x,) J

c .. : ~ .. ::~ = (.Al~ ck~ x1. ') + <AJ\k < k .. ~"') • o< .. ( "'~ < k" .. 11") '° '7;" { < k .. ~i.) )

For the sake of compa.ctnesa in handling, the following

notations are introduced.

c .... $ c ... c .. 5 ... = s ... C.,,.

Attention should be paid to the tact that all expressions are

ayr.mietric Tr.Ti th resp Get to m a11d n except z.... By use of these notations, the following expressions ~~11 be shortened.

( )t u-\, 11

By substituting these expressions into Eq.( 3.2.29 ) and

carrying out the integration with respect to x, ovor the

range from - ~ to + ~ , the tinal expression for Oj2 is

obtained.

( 3.2.30 ) 'Where

oO

+ 2 La"'" k"'" S"'0" I : ) "'~ "'

l"or the sake of simplicity, the notations

- 23 -

0 f CIM\ dxt 0 "'" =

so = f s .. dxt ... 0 f z"'" dxt z." =

have been used 1n Eq.( 3.2.30 ). -Examining the expression ot tS'ii in Eq. ( 3.2.30 ) , w

see that a-tt ia a :tunction of n pai-ameters 'Which will be

determined by the critical equi11brium condition.

Substituting ~1 defined by Eq.( 3.2.30 ) into Eq.(

3.2.6b ) yields

~ (a-;:) D M~ - ~ ~ '?)QI\ It = tb~ M = 0

D Since ZbM ; O for non-tI>i vial solution or a,.,

'1>~ N ~ M 0 'O QI\ - lJ '?) Q... =

t.J 'lb -We reca1l that M = 0 Oj1 , hence

n=l,2,3, •••••••••••.••• m'

were m• is t.he number or parameters retained in the assumed

displacement u-\ {x~ ~ a..,). Since M and U are functions ot a~ which are in quadratic form, Eq.( 3.2.31 ) ,gf vea mt line&l',

homogeneous equatioria for a"'. T'ne neoess&ry· and aUf1'1oient

- 24 -

condition that we have a non-trivial solution tor a~ is that

the determinant ot a~ ia zero. That is

v = 0 ( 3.2.32 ) This is the chaPacteristie determinant tor buckling stress

( Gj-t>,t- 'Which is the only unknown quantity. The degree ot <Sit

in F.q.( 3.2.32 ) will be equal to the number of terms

retained in the aeries tor u~ (x,,· a"'}. Therefore, it ia

evident that it more terms are :-etained, more wottk is needed

to solv• this equation. A numerical nalculat1on will be

presented in the next seotion aa an illustration.

- 25 -

IV NUMERICAL RESULTS

For speeiric problems, certain d1ff1cu1ties are

encountered in rinding the de.t'inite integrals for the

funotions c.". s •• and z-~ that appear in Eq.( 3.2.30 ).

After a series of tedious calculations, two tables for these

integrals. were prepared and are pr~sented in the appendix

for application.

The values or k" L and ot,.. -vlhich make u~ (x~) in Eq. (

3.2.28 ) satisf7 the geometrical boundary conditions can be

determined by F.q.( 3.2.26 ).

Table l Values of k.L and «.a fl-oin n = l to n = 5

n l 2 3 4 k L l.875 4.694 1.sss 10.996 at~ -0.734 -1.108 -1.000 -1.000

As n becomes large, then very sensibly

or

kL=(2n-l)l ... ~

lim ( k,.. L - k 1\ • 1 L ) = 'It l\..+oO

By noticing the fact that

li.m ( eosh(k~L) - sinh(k.._L)) = lim ~L~~ ~~~

we have

s lJ.i..137

-1.000

I

e. t.a.

- 26 ..

cosh(k"'L) ~ sinh(k"L)

and

ex,. GOC. C le" L) ~ ~ ( \c11ol ) = - c,ir1. C k .. L) + "' ... "- < k.l.) -=:. - 1

for large k~L, since sin(k,.L) and cos(k.L) Va%7 bet~een zero

and absolute unity which is comparatively small.

Due to the properties ot hyperbolic tunctions and Cl("'

as n becomes large. the following simplifications are

observed.

C"' ~ COS (k,.x1 ) - Sin(k11 X1) ( 4.1.2a ) ( 4.1.2b ) S"' ~ coa (k11 x 1 ) + sin (k1 x 1 ) - 2ainh (k11 x 1 )

0 0 0 Then the fUnotions c~ ... s."' and ZMll tor lal"ge values of k .. x1 can be easily obtained by the following approximate expres-

aions.

0 ~ ~i .. < r"' -k'. ) lit + uxck ..... ~") li ( 4.1.3a ) c ..... k-11\ - k' .. 1(111 ... k' ..

0 cai11tk'n.- k.) x .. _ Goe.(~"'+ I(,.'\ X1 s ...... = k"'- k11 ~ ... +kl'\

- - t [le M '"•< k.11 1 •••l«k,•, l • lc. "'H lr.l, l ...,i,. <Ir.•, l]

- _ t ( k'" t,i11. ( ~111 X1 ) ~"' ( ~ .. X,) - I(• wft( k11 l(1 ) '- 1~\.. ( k11 X~) )

- k!t, k;( le, ••· \l:.x,1 •Mk Or .x, 1 + I::,. .... < Ir.•, l """ < k:.x,l J

- t (k. c,i11o(k""~1 ) .... ,k(k'.x1 ) • ~ .. ~~<k' .. 1t"')~l .. ~(~x,))

.... 27 -

+ \c!: k.:' ( k .. '-'"' < k.'t~) t.K" ( k' .. t") .. k. "'°'-( k ... x") -.i .. ~ ( ~.x,))

- k! ~ t ~ [ le• t,i• ( k,. X. Hi.I.. ( Ii:'. i~H \:, ~ ( li:'.xtl loo.I. ( le;,\) ]

For"ic_, = k., the tunctiona CM•' s.A and z.~ oan be obtained

tram Ecf.s.( 4.1.3 ) 'bJ' using the limit process.

It will be reealled that Eqs.( 4.1.3 ) are only

appl1oable tor largo values ot k..,L.

Por simplicit7, only one tem in Eq.( 3.2.28 ) is

retained. Then the values o~ the .f'unctiona c:j~ , s:j~ and

Z: I ~ are obtained from Table A-l aa rollows: 01 L C11 0 = 0.981L

0 I L. Z 11 0 = 1.430L

From Eq.{ 3.2.30 ) for n = l, we have

l \ I~ \.\ t ' 0 I L t ( 1- ") 'l t 0 I L 1\1 : l'L Q, ~. C,, 0 + k Q, L c. 0 If, ":111

- 20 -

CojL OIL Substituting the values ot k,, " 0 1 8 11 0 •1L and z .. 0 into

and rearranging the expressions for N and M yield t

a1 ( 1 N = kL-l 1'4· 1.41 b:

M = 2.68la1

By differentiation, we get

~ N a. ( t ~ ( t ) • ·L~ -z.".'ltat h + 1e ... oe 1-v) L .,, Q, "

From Eq.( 3.2.31 } torn= 1, the equation tor deternrl.ning

the buckling stress becomes

By reru:Tsnging, we obtain

For convenience 1n comparison ~Ji.th other results, the

maximum stress ~mich occurs at the remotest fibre at the

fixed edge can be obtained from the follol-dng equation.

Substituting ( a-;'I.),., defined in Eq. ( 4.1.4 ) into Eq. ( 4.1.5 )

- 29 -

~b~ and recaJ.ling D = ,, .. yields

11. < 1-v•)

( ~ ) = ~. 4~.., E-b' ( 0. '"~ MW C.1' L I t I - y

L lilhere /\ = - • "

!. - o. ~4., 1' l•Y ,.._)

( 4.1.6 )

V= 0.3 for most stl"Uct~al steels. then Eq.( 4.1.6 ) becomes

(. G: ) _ ! . 4 9, (:-\~ '1 ( O ~ '1. I 1'Glc <.r - L I ' Q. 1= .j. 0.1.'11 "' )

( 4.1.7 )

- 30 -

V. DISCUSSION AND COMCLUSIONS

Por the sake of comparison, Eq.( 4.1.7 ) and the result

obtained originally by Prandtl Cli) are plotted aa curve• c, and ct respectively in Fig.2, p.33. Due to the lack ot experimental data, the author cannot give a definite valid

range ot A in his result, howeve~, aome qualitative conclu•

aiona can still be derived by reasoning.

The maximum stress at the buckling state is a function

ot hand L to first order in Ct, but to higher order in a,. The curve O, is divided into three parts by the intersection

points with c1 at ~.= 0.656 and ~t= 3.090. The left pa.rt ot 0, goea to intini ty as ~ goes to zei-o.

It shows that the buckling stress obtained trom Eq.( 4.1.4 ) ia much larger than that obtained trom Prandtl'a result.

In the middle part

than that in curve c1 • Since the buckling stress obtained

by the energy method is a1wa:ys greater than the true value,

curve C, gives a more satisfactory result than c1 which 1a

obtained by Prandtl. The values of ~, a.Tid )\1 may shift more

or leas as we retain mo1·e terms in the series u'\ (x~) • But 1

anyhow, we can aay that the energy method based on thin plate

theory gives satisi'aotory result for )... between and around

o.656 and 3.090.

In the right part, 0 1 deviates from c1 rui-ther and

- 31 -

further as ). becomes larger and large?'. Then the curve c. gives a less satisfactory approximation. But aa ~ becomes

greater than a certain value. the max:1mum st?'esaea defined

by both Eq.( 4.1.7 ) and Prandtl•11 result will be beyond the

elastic limits f'or most structural materials. Thus curves C1

and Ot do not apply in the case that ~ ia larger than a oer-

tain va1ue 'Which varies accol'ding to the material and the

dimensions ot the plate under consideration.

As~ becomes large, at.( 4.1.7 ) doea not hold, even it

( ~o.x >c.r remains in the elastic range, since the detlec-

tiona and alopea in the x, and Xt direotions are no longer

small enough to be neglected. '!hen the potential. energy will

be .formulated in tel'!tUI ot u ~ ?'ather than u ~ only. In this

general case, we ma;y assume the displacements in the form

(l) x a. " ..

In these aeries, X~ are certain admissible tunctiona ot x~ 1ihioh satiety the geometrical boundary conditions; a<.~l are

~

unknown parameters, independent of .x., which will be deter-~

mined by the cl'itical equilibrium condition.

Since the displacements u. are assumed, the potential • enel"gJ for the whole elastic body attar integration will be

a function ot a1~). Thus ~

p = P( fi~l ) •

- 32 -

where 1 = i. 2. .3

n = i. 2, 31 •·~~·••••• n The unknown parameters J~) corresponding to the diaplaoementa

~

in Fq,. ( s.1 .• 1 ) 111hioh will produce equilibrium oan be found by minim! zing the eXpreasion for P (a<t ) ,

'1> p ( Q(~) ) __ _...;•:_ = 0 '?) al~'

"' Prom Bq.( .5.1.2 ), we obtain a system ot Jn lineal' homogeneous equations, \ihere n ia the number ot tenna re•

tained in each ot the seriea defined by Eq. ( s.1.1 ) • In

order to guai-antee that the system has a non-trivial aolu-

tion, the determinant ot the ooetficienta must vanish. The

ooettioienta are tunctions ot geometrical dimenaiona,

elaatio properties and the extemal load Wioh ia the only

unlmom to be tound. i'heretore, the lowest value ot the

e.xtel'nal load, tor which the ch8l'aoter1stio determinant

vaniahea, and Wioh corresponds the critical value of the

load, will be determined.

I

4.,o j I l I f--QU~1 I Ol-.l o~ LLleVt- <., I

hl I J - ( ~1A, ) :: o. i;4 't 1..A.e1 t:-b'L - 4- O."l b1 ~ I - 't C..'(. -,.....

I I -·· !:-GUAllOU . i 01- C:.UQVb C1: r

~.o ~

"' I hL . -·· ( 6M~x ) r..~ :: 1

11 \

t. 421 t-bt

! '

. ----------. ·-· ..

I ... ' ' ' .

k~ I

.. . . ~ /fc.'Z.. OBfli1\1t-D BY r1m1or1_ 01 )

' ' ---·-I'-... .. - - - -·· .... - t- --· . -

I I

-"" 4 lo::;? ..._...

~ '2.0

J. ....J r ..s:: <.'()

tj

~

1, c; o.eoi:.;

I I I I i I

' ~0•:.")0 ·--· .. ..

__ ... ____

0 O.b.ob IO 1.41~ 1.0. ~.o S.o f:. 0 i,'.J

------·------~-

- 34 -

·--- . --~-J.-~_IJCTIOij~ c~Y\ So,.,!\. z:r\ -- -- ... -. -.

~~ Sl\l ( 1. ~ x.,.) I ·at...1 I - o1.,_1. o< '1 - --6. ~

-' !-0~(~1( X-i) oC 0(. I - o<1.

"'T - --a.-l<'I\ ~ . 'Z

~ n, SI ~\.I ( 1. \(.,. ~1_) l+-<.,.~ I"" ol...1.

4 4 0

I 0(. °" l .. oL._t kv. t.0~4 t 1.. \(.,_ )('\) - - a: '7.. 'l

I 7. K ;,1\l ()(1\)(1) ~l\J~ ( l(p,Xi' -z. o(. 0 •\I.a(.) I\

~"-~I~ lt,.X.,,) '-0~\.4 t k',.X~' 1 ~ot..,. 'I. I - oi<. 0

I 1.. .. (I + ~7.) K (.O~ (K',r,) 71~~ ( \(,_ X"') ' .. o<.. "lo(

W\

I I • ""-1. \( r.c.o~( ~ ... X1.) w.=\.I t I(.. 't'l) 0 - 'lo<

' k':. { \(W\'X'L) I o<. 'l. - o<

lJOTE-1 T~~ ~UIJC.ilOtJ '" f:'QUA.L 10 nu- ~UM o~ nH- P~ODUtT:i or- Tl.H-Tt:-~~c, 11-l '.TUf-- ~t~':»i '-01.U~l.I A~D T\.lC- lOefl.~t,PO~Pl~(:r- iE:--l?:M., HJ i\.lt:- c.0L.u1-.t1~ uNot-~ 11-Hi.i i:..uijc. itol.l. i:.oe H~'\.APLE-:

0 ' ' - o(1. ' 0( . ) ( ~ - (-- '>I~ ( '?f:'°X ) - - C.OC',( tic::~)+ - --- -- -· !Ill\ \(" '1, n 1. '? n. i.

c i

- 35 -

- 36 -

VII. ACKNOWLEDGEMENTS

The author wishes to thank Dr. R. Ghiourel, hia major

prof'eaaor, for all the guidance, auggestiona and encourage-

ment throughout his work.

He alao wishes to e.xpreaa his gratitude to Dr. D.

Fl-ederick for hie encouragement and kind help.

l.

2.

3.

4.

- 37 -

VIII., BIBLIOGRAPHY

I. Todhunter & K. Pearson, n A Ristory of the Theor:y; of Elasticity and of the Strength of Materials ', Dover Publications, 1960, Vol.l, P•39.

S8me ea l, Vol.l, p.62.

A. E. H. Love, " A Treatise on the Mathematical Theoey of Elasticity, 11 t~th Ed., Dover Publication, p.409.

Same as l. Vol.II, Part II1 p,40. 5. G. H. Beynn, 11 On the Stability of a Plane Plate under

Thrusts in Its Olin Plane with Apl:licat1on on the Buckling of the Sides ot a Ship , P~oo. London Math. Soc. 1891, p.59.

6. w. Ritz, " Uber eine Neue Methode zui- Loaung Oewlaaer Va.riationsprobleme der Mathemat1aohen Phyaik ", Zaitaohrift tur Beine und Angewandte Mathsmatik, 1909, p.l.

7. J. w. S. Ray].e1gh1 " The 'lbeoey or Sound 11 1 Dover Publications, Part I, p.109.

8. E. Tref'ttz, " Die Bestilmaung der Knicklaat Gedruckter, Rechtecld.ger Platten "• Zeit~ohrift tur Angewandte Mathematik und Mechanik• 19351 Vol.15t p.339.

9. c. E. Pearson, " Theol'etical Elasticity 0 , Harivard UniveJ-aity Presa; p.2o5.

10. A. E. Green& J. E. Adkins, " Large Elastic De:tormations & Non-linear Continum Mechanics "• Oxtord Presa p.273.

ll. L. Prandtl, 11 Ki!}persoheinungen "• Dissertation-Munich, 1 99.

12. D. Frederick, 11 Lecture Notes on Theory ot Platea ", 1961. Chapter II.

1.3. I. s. Sokolnikott, " Mathematical Theoey ot Elaaticity "• ¥cgraw-Hill, 1956, p.384.

24. 1.$.

.. 38 -

Same as 13, p.86.

P. Bleioh, " Buckling Str~ of Metal Structures "• Mograw-Bill, 1952. p.64.

s. Timoahenko & D. H. Young• " Vibration Problem.a in BDgineering '*, )rd Bd. 1 p.341.

s. Timoe.henko & Jamea M. Gere, " Theo17 ot Jllutic Stabil1t7 "• 2nd Ed. p.261.

The vita has been removed from the scanned document

BUCKLING OF CANTILEVER THIN PLATE WITH FREE END SUBJECTED TO UNIFORM SHEAR

by

James Chie Meng Yu

Abstract

This thesis is concel'lled with the buckling problem ot a oantilever thin plate with its tree end subjected to uni-

form shear. 'lhe same problem waa originally solved by

Prandtl in 1899, based on the equilibrium condition ot a

deep beam. The author baa uaed the energy- method baaed on

the thin plate theory to attack the problem.

After the displacement is aaaumed, the potential

energy can be tol'mulated. From the condition that the

potential energy asaumea a minimum value in an equilibriwn

configuration, results a system ot n linear homogineous

algebraic equations ot n parameters which are introduced in

the assumed displacement. For a non-trivial solution, the

determinant ot the coefficients must vanish. This gives a

characteristic eq~ation rrom which the buckling load is

determined. The author has obtained a curve tor maximum

stress at buckling state, l!lhich shows that the result 1a

better than that obtained b'f Prandtl in certain cases.

The energy method has been generalized to a t~ee

dimensional problem to consider the displacementa in all

directions.