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thesis entitled

BUCKLING OF A RECTANGULAR PLATE ON AN
ELASTIC FOUNDATION, COMPRESSED IN TWO DIRECTIONS

presented by

Richard Charles Warren

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Engineering Mechanics

 

 

 

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BUCKLING OF A RECTANGULAR
PLATE ON AN ELASTIC FOUNDATION,

COMPRESSED IN TWO DIRECTIONS

BY

Richard Charles Warren

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Metallurgy, Mechanics & Material Science

1980

cf//w>dr?69

ABSTRACT
BUCKLING OF A RECTANGULAR

PLATE ON AN ELASTIC FOUNDATION,
COMPRESSED IN TWO DIRECTIONS

BY

Richard Charles Warren

The purpose of this research is to determine the first
linear buckling mode of a rectangular elastic plate, rest-
ing on a Wrinkler foundation under various edge loading
conditions. Two cases of boundary conditions are considered.
simply supported all around and clamped all around. The
solution for the simply supported case is found in closed
form, but a numerical approximation is employed for the
clamped case. The results are compared to solutions of
the circular plate by Wolkowisky and to Hetenyi's solutions

for beams on elastic foundations.

AC KNOWLEIXSEMEN TS

The author wishes to express his appreciation to the
following people:

To Dr. Robert Little, major advisor, for his construc-
tive criticism and for his ability to channel my energies.

To Dr. Altiero for taking time to discuss areas
related to this work.

To Dr. Mass for his suggestions and support.

To Dr. Wasserman for his encouragement and critical

review of this work.

ii

TABLE OF CONTENTS

LIST OF VARIABLES
LIST OF FIGURES . .

I. INTRODUCTION

II. MATHEMATICAL FORMULATION

III. SIMPLY SUPPORTED CASE

1. Uniform Loading

2. Non-Uniform Loading

IV. CLAMPED CASE

1. Uniform Loading

2. Non-Uniform Loading

V. CONCLUSIONS

BIBLIOGRAPHY . . .

iii

iv

Vi

l7
17
25
32
32
51
53
56

0|

2
= Eh , the flexural rigidity of the

12 (1 - v2)
plate
Young's modulus of plate material
Poisson's ratio of plate material
plate thickness
foundation stiffness

deflection in the x.y and z direction
respectively

Airy stress function
perturbation parameter

differential operator

eigenvalue

=l£_ 4:52-15.
D D D

=ll}. l£2_&
I) + (D) D

separation constants
dimensions of the plate
edge stresses

critical edge stress

= zgha2

DFZ
=az/Tg
real numbers

iv

Q

Q.

m

U)

p' enq

arguments of trigonometric functions

expansion coefficients of the eigenfunctions
X and Y

C or

1 C2

LIST OF FIGURES

Figure
3.1 Rectangular Plate on an Elastic Foundation
for Simply Supported Case . . . . _
3.2 Buckling of a Simply Supported Rectangular
Plate on a Wrinkler Foundation . .
3.3 Buckling of a Beam on a Wrinkler Foundation
4.1 Rectangular Plate on an Elastic Foundation
for Clamped Case . . . . . . . .
4.2 Buckling of a Clamped Rectangular Plate on
a Wrinkler Foundation. First Approximation
4.3 Buckling of a Clamped Rectangular Plate on
a Wrinkler Foundation. Second Approximation .
4.4 Buckling of a Clamped Rectangular Plate on
a Wrinkler Foundation. Third Approximation
4.5 Buckling of a Clamped Rectangular Plate on
a Wrinkler Foundation. Ninth Approximation
4.6 Buckling of a Beam on a Wrinkler Foundation.
Clamped Case . . . . . . . . . .
4.7 Convergence of the Eigenvalues for Each Value
of K for the Symmetric Case , , , , ,
4.8 Convergence of the Eigenvalues for Each Value

of K for the Antisymmetric Case

vi

Page

17

24

26

33

40

43

45

47

48

49

50

CHAPTER I

INTRODUCTION

The use of plates in nautical and aeronautical
construction spawned the need for buckling solutions
of plates of various shapes under different edge
loading conditions and has resulted in considerable
examination of these problems as evidenced in the
literature. The advent of solid propellent rocket motors.
the need for a better understanding of rock mechanics in
mining and drilling operations. and the increased demand
on highways and airfields have recently precipitated the
need for solutions to buckling of plates and shells on
various types of foundations. One such foundation model
is the Wrinkler foundation, composed of closely spaced
linearly independent springs whose restoring force is
linearly proportional to the deflection of the founda-
tion surface with opposite sense. The present research
provides the buckling solution for a rectangular plate,
resting on a Wrinkler foundation and clamped or simply
supported at its edges. The boundary conditions and

foundation model were chosen for the purpose of comparing

-2-

these results to Hetenyi's [1] solutions for beams on
elastic foundations. and Wolkowisky's [2] solutions for

,___- y“..-—u-r

the circular plate embedded in elastic springs.

Solutions to the buckling of plates on elastic founda-
tions by edge thrusts in the plane of the plate have be-
come numerous in the last fifteen years. with the solutions
generally following previous approaches where the elastic
foundations were not considered.

The mathematical solution for the non—linear boundary
value problem of the buckled plate was first given by
Friedrichs and Stoker [3]. A circular plate was considered,
under uniform radial pressure and simply supported edge
conditions. The non-linear von Karman plate equations [4]
were utilized and simplified to a system of two ordinary
non-linear differential equations where radial symmetry was
taken into account. Buckling occurs when the edge load
Pe reaches some critical load PE‘ The authors sought to
determine the stress state when the ratio Pe/PE became
greater than unity. A perturbation technique was used.
but this technique was manageable only for the first few
eigenvalues of Pe/PE and a power series method was needed
to obtain a higher range of values. The power series mefiiod
was useful, provided solutions for the first few eigenvalues
have been obtained but this method also became cumbersome
as the ratio increased further and the authors were eventu-

ally forced into an asymptotic analysis as the ratio became

larger than fifteen. Bodner [5] solved the clamped case of

-3-

the problem presented by Friedrichs and Stoker using their
scheme.

Kline and Hancock [6] found the buckling solution of
a circular plate on a Wrinkler type elastic foundation
for the clamped and simply supported cases. They used a
differential equation developed by Yi-Yuan Yu [7] for the
deflection of a circular plate on an elastic foundation
under the action of edge thrusts and lateral loads. This
equation is based on the classical small deflection theory
and hence, is a linear fourth order ordinary differential
equation. By assuming no lateral load. they obtained a
solution which gave the initial buckling load for any
given circular plate in the linear range.

Wolkowisky [2] extended the work of Friedrichs and
Stoker by placing the non-linear circular plate problem on
a Wrinkler type elastic foundation. This added a term
(the restoring force of the foundation) to the bending—
stretching equation of the non—linear von Karman plate ;
equations. Wolkowisky's work closely followed the method
used by Friedrichs and Stoker, Except that, instead of the ]
power series method. a numerical approach called the ‘
"shooting technique" was used after transforming the equa-
tions to a system of first order differential equations. ,i.

In their book. Beams, Plates and Shells on Elagtic
Foundations. Vlasov and Leont'ev [8] used series solutions
to solve buckling problems of rectangular plates compressed

by loads in one direction under various types of boundary

-4-

conditions. Datta [9] found the thermal buckling solutions
for triangular and elliptical plates on elastic foundations
by conformally transforming the boundary onto the unit
circle. He used the method of K. Munakata [1D]. who found
the buckling solution of the rectangular plate by con-
formal mapping of the rectangle into a unit circle. How-
ever. the resulting differential equation is considerably
more complicated than the original biharmonic equation.
and hence. this method seems of little value here.

The buckling solutions for plates on elastic founda- T

tions. uniformly compressed in one direction are treated

by: Datta [ll]. buckling of non-homogeneous rectangular
plates; Ariman [12]. buckling of thick plates. rectangular
and infinite: and Sabir [13]. a finite element solution ‘1
for a rectangular plate.

An infinitely long elastic plate. simply supported
at its long edges and compressed in the longitudinal
direction was investigated by Seide [14]. The plate rested
on but was not attached to a wrinkler type foundation.
The deflections were governed by two differential equations:
one for the region in contact and the other for the region
of separation. The boundary conditions took the form of
continuity of deflection. slope. moment. and shear in the
region of contact.

Jacquot [15] developed a method for the prediction of
buckling of plates under the influence of elastic constraints.

where each constraint was modeled in the form of a linear

 

v...il .I .II '

-5-

spring. A buckling equation from Timoshenko and Woinosky-
Krieger's book Theoryyof Plates and Shell; [16]. was used
with a product of Dirac delta functions added to the right
hand side to represent an. elastic constraint. Each elastic
constraint required a differential equation of this type.
This technique was illustrated by application to the buck-
ling of a square plate. simply supported at the edges.

'under uniform compression in two directions. with a single
elastic constraint.

The first portion of the present research determines
the first linear buckling mode of a rectangular elastic
plate under uniform (Oxx = Oyy = a constant along the edges)
compression on a wrinkler elastic foundation when the thick-
ness and foundation stiffness are allowed to vary. This
allows comparison to the circular case by Wolkowisky. This
is followed by the case for non-uniform loading (the loading

function is constant along each edge but ). The

0xx E Uyy
solution to the differential equation of the circular plate
admits the simply supported and the clamped boundary condi-
tions without difficulty. thus only the clamped case was
carried out in detail by Wolkowisky. This is not the case
for the rectangular plate so both cases are illustrated.
The mathematical formulation used in this problem is
based on the non-linear Foppl-von Karman plate equations.
which involve the plate dflection W and the Airy stress

function e as functions of the independent variables x

and y. These equations are written in operator form and

-6-

linearized using a perturbation technique. resulting in a
system of linear partial differential equations. To obtain
a buckling solution under a uniform load. it is necessary
to make a choice for the Airy stress function satisfying

= 0 = a constant on the boundary. For the non-

0
xx YY
uniform case w must be chosen so that Gxx and O

YY

are constants along the edge but can be independently

varied according to some specified parameter. It is found
that. when buckling initially occurs. no nodal lines are
present provided that the foundation stiffness is relatively
weak. However as the stiffness of the elastic foundation

is increased. more and more nodal lines may appear for the
first buckling mode.

A closed form solution for the buckling load of a
rectangular plate can be found for the simply supported
plate. but not for the clamped plate [17]. Some writers "
[18. 19] have given approximate solutions by the strain
energy technique: Sezawa [19] found a closed form solution
to the lateral vibration of a rectangular plate clamped at
its four edges by assuming the plate clamped at the mid-
points along the edges resulting in residual slopes along
some parts of the boundary.

Exact solutions for buckling of rectangular clamped
plates have been found by Taylor [17] and by Iyengar and
Narasimhan [20]. Taylor's method satisfied one set of

boundary conditions and the differential equation term by

term. and approximated the other set of boundary conditions.

-7-

while Iyengar and Narasimhan's approach satisfied both sets
of boundary conditions term by term and approximated the
differential equation. Both solutions involve the trunca-
tion of an infinite determinate. However. Taylor's method
requires a considerable amount of numerical work compared
to Iyengar and Narasimhan's method. and since both solutions
coincide. the method by Iyengar and Narasimhan is used.

The present research can be extended to finding the
buckling load beyond the first step of the perturbation
technique. However. the power series method used in the
solutions for circular plates has no analogue in the case
for the rectangular plate [21] and "an exact solution for
the rectangular plate valid for an unlimited range of the
[buckling load] presents seemingly insurmountable

difficulaties" [l4].

CHAPTER II

MATHEMATICAL FORMULATION

An elastic rectangular plate of constant thickness
is attached to an elastic foundation and is either simply
supported or clamped all around. The foundation exerts
a lateral force that is linearly proportional to the
deflection of the plate and the sense of which is opposite
to the deflection. If the behavior of the foundation in
any particular region is independent of the behavior of
the foundation in an adjacent region. the foundation is
called a Wrinkler foundation. The plate is edge loaded
in its plane and the following cases are considered:

1) uniform (the load is constant along each edge and
= o ). 2) non-uniform (the load is constant along

0
XX YY

each edge and Oxx # Oyy).

The differential equation governing the bending of an
elastic plate subjected to lateral loads can be expressed
in terms of the biharmonic operator and the lateral load
pz (XoY) :

DV4W = pz (XrY) o (2.01)

_9...

where W is the deflection of the middle surface.
Equation (2.01) is based on the assumption that no exter-
nal forces act parallel to the middle surface. The
external lateral load of the plate is carried by the
internal transverse shear s and by the internal bending

moments m. The moments and shear forces are related

by

amx 5m x
?;F +'—S§—'= SX (2.02)
am am
+--—§!-= 3
BY ax y
Equilibrium requires that
as as
__2S __X._
+ - -P .
ax By 2 (2 03)

Substitution of equations (2.02) into (2.03) yields

 

azmx 52m 52m
2 + 2 7-6295 + ___2y_ = -P (x.y) (2.04)
ax x Y ay 2

The bending moments are obtained by integration of the

normal stress components:

J«11/2 Ih/z
m = O zdz, m = O zdz
X -h/2 x Y -h/2
(2.05)
h/2 h/2
m = I G zdz and m = F o zdz
XY ‘h/Z XY YX d_h/2 XY
but since the stress tensor i s etric O = O and
S ’"m' xy yx
hence mxy = myx' The stresses are related to the strains

by the equations:

-10-

E

 

 

6X = l_v2 (€X+v€y)

o = E (e +v€) (2.06)
y l-v2 y x

O E

xy = 2(l-+v) exy

and the strains to the displacements by

 

 

2
ex = -2 Big) Ey = -2 gig; and exy = -2 ix? (2.07)
ax By Y
2 2
where 33?)! and -3: are the curvature change of the
BX 6y 2

. a W . .

deflected middle surface and - is the warping of

axay
the plate. Substitution of equations (2.07) into (2.06)
and (2.06) into (2.05) and equations (2.05) into (2.04)
yields equation (2.01). If edge loads are present.
equation (2.01) must be modified to handle bending and
stretching simultaneously. Since the foundation exerts
a lateral force whose sense is opposite to the plate dis—
placement. equation (1.01) is modified to include membrane
forces and the effect of the foundation and becomes

2 2 2 2 2 2
cpaw.acpaW-2 EH2 iW l‘-w, (2.08)

ayZ ax2 ax2 ayZ BXBY axay h

‘2 4 _ a
hVW---

where m is the Airy stress function for plane stress
which is related to the average stress across the thickness

of the plate by the equations

2 2 2
C = M, 0' = M, and 0' 2. -_LE . (2. 09)
xx ayz yy 5X2 XY BXBY

The applied bending load. pz(x.y) in equation (2.01)

has been replaced by - E-W(x.y). the restoring force per

-11..

unit area of the elastic foundation. The constant. k. is

the foundation stiffness and h is the plate thickness.
3
Eh
12(l-v

the plate. where E is Young's modulus and v is Poisson's

The constant. D = is the flexural rigidity of

2)'
ratio. Equation (2.08) involves two unknown functions 0
and W. and so a second equation. called the compatibility
equation. which relates w and W is necessary. This
equation can be developed from the strain displacement
equations by using the assumption that the squares and
products of the slopes of the deflection of the middle
surface are of the same order of magnitude as the strains
there. A nonlinear relationship is obtained between the
strains and the variables U.V. and W. which are the
deflections of the plate in the x.y. and 2 directions.
respectively. A linear relationship is obtained between

the strains and the stresses such that

=iq lfifl2__1. -
Exx ax + 2(ax) - E(Oxx vay)
=5_V 1312:; -
Eyy by + 2(ay) 3(ny VOXX) (2°10)

‘XY 2 By ax ax By E xy

These expressions differ from the expressions for strain
in the linear theory by the quadratic terms in W. The
deflections U and V are eliminated through cross
differentiation of equation (2.10). and then equations
(2.09) are substituted for the stresses resulting in

2

1 4 _ 251.2 firm
EV cp— (axay) - 2 2. (2.11)

ax 5y

-12-

The solution to the rectangular plate with simply
supported boundary conditions (zero deflection and zero
normal moment along the edge) is found under uniform edge
loading and carried out in detail for the square plate.

The buckling loads are expressed as a function of the
foundation stiffness and plate thickness. The case for
non-uniform loading follows with particular examples of

Oxx = -Oyy and ny = 0. Oxx ¥ 0. The same loading
conditions and examples are solved for the clamped boundary
conditions (zero deflection and zero slope along the edge).

Equations (2.08) and (2.11) can be written in operator

form by defining

 

4 4 4
L1=V4=L+2-—g-—2-+-a-Z (2.12)
Bx Bx By By
and
2. 2 2 2 2 2
L=ML+flL-25W 5 (2.13)
2 2 2 2 BxBy BxBy '

2
BX BY By Bx
where L1 = L1(x.y) and L2 = L2(W(x.y)). Replacing
equations (2.08) and (2.11) with their equivalent form

in operator notation gives
(L +5)w = — L cp (2 14)
l D 2 '

_ _ 2
sz — E Llo . (2.15)

These equations can be linearized through a perturbation
technique if the functions W and m are expanded in
terms of some unspecified parameter 6. If expansions are

made having the form

-13-

_ - 2
W "" 5W1 + 8 W2 +000 (2016)

_ , 2
cp—cpo+-CDl+€C02 +000 I (2'17)

and these values for W and m are substituted into equa-
tion (2.14) and equation (2.15). this allows for a grouping

of terms of comparable effects in the buckling problem:

6 . Limo = O (2.18)

51 (L1 +%)wl _ 95 chpo (2.19)
L101 = 0 (2.20)

e2 : (L1 +%)w2 = %(chpl + 12ch) (2.21)
L2W1 = - i2; Llcp2 . (2.22)

For the solution to equation (2.19). an Airy stress
function $0. is needed that satisfies equation (2.18)
and e uations 2.09 such that O = 0 and O = O =

q ( ) xy xx w
some constant on the boundary. Therefore. mo must be

an even function in both x and y. and may be written

as
00 = -1(x2-+y2) (2.23)

where the negative sign has been chosen to indicate com-

pression. This gives

0' = O' = — .
xx YY 21 (2 24)
If $0 is substituted into equation (2.07). and it

is noted that L2 is an operator dependent upon Wl in

equation (2.19). L200 gives

L = -21Li/2W (2. 25)

2°”0

-14-

substituting this into (2.19) gives:

1.5. _ _ 1.1 1/2
(Ll-rD)Wl — D 21Ll ‘Wl (2.26)
or
(L +21 3 L1/2+1‘-)w = o (2 27)
l D l D l ' °
This can be factored into:
1/2 1/2 _
where
- lb - 12 2_l<.
cl- D (D) D (2.29)
and
_ lb Ml 2-h

Since the operators are commutative. this factorization is
unique.
The complete solution [26] to equation (2.28) can be

obtained by a linear combination of the solutions to the

equations
1/2 (1) _
(L1 +cl)Wl - 0 (2.31)
and
1/2 (2) _
(Ll +c2)Wl - O . (2.32)

Using separation of variables. the solution for Wl

takes the form. W1 = X(x)Y(y). yielding:

22:: r”. _
X 4"Y + cl — 0 (2.33)

and

2;”. X _
X 4-‘Y +-c2 - O (2.34)
or
x” + aix = o
2 (2.35)
Y” + a Y = 0
l
and
X” + agx = 0
(2.36)
r” + BZY = o
2
where
2 2 _
d1 + 61 - cl (2.37)
and
2 .2 _

The solution to W1 is then:
W1 = A1 Sln dlx Sln 81y + B1 Sln dlx cos Bly
+ Cl cos dlx Sln 81y + D1 cos alx cos 81y

+ A sin a x sin 82y + B

2 2 Sln 02x cos 52y

2

+ C cos a x sin 82y + D

2 2 cos dzx cos 82y . (2.39)

2
A fourth order partial differential equation in two
variables must be able to satisfy two boundary conditions
along each edge. For most values of 1. the only solution
is W1 5 0: the special values of 1 for which non-
trivial solutions exist are the eigenvalues and the corres-

ponding solutions Wl(x.y) are the eigenfunctions. The

-16..

homogeneous differential equation. in conjunction with the
homogeneous boundary conditions uniquely determines the
shape of the buckled plate together with a set of eigen-

values leaving the amplitude arbitrary.

-17-

CHAPTER III
SIMPLY SUPPORTED CASE

III.l Uniform Loading

Consider a uniform rectangular plate over a domain
defined by 0‘s x‘g a and 0 g_y g b. The boundaries of
the domain are straight lines x = 0.a and y = 0.b. and
the origin is chosen at one of the corners of the plate

as shown in Figure 3.1.

W(X,Y)

 

 

 

 

a Ill III. III; MIA 'lll. I I

 

Figure 3.1 Rectangular Plate on an Elastic Foundation for

Simply Supported Case

-18-

In the case of simply supported edge conditions
all around. the requirement of no deflection and no normal

moment on the edge take the form

52w_
.__5 _ 0

W1 = O and along x = 0.a (3.01)
Bx
w_0 1:22- _
1 - and 2 - 0 along y — O.b . (3.02)
BY

Because equation (2.33) is composed of products of trigono-
metric functions. the boundary conditions of no deflection

or normal moment on the edge impose similar conditions on the
trigonometric coefficients. When the boundary conditions
Wl(0.y) = W1(x.0) = 0 are applied to equation (2.33). the

linear independence of the trigonometric functions gives

B1 = B2 = C1 = C2 = D1 = D2 = O

and equation (1.33) reduces to
W1(x.y) = A1 Sln dlx Sln 81y
+ A2 Sln 02x Sln 82y . (3.03)
When the boundary conditions Wl(a.y) = W2(x.b) = 0 are

used. equation (2.03) becomes

Wl(a.y) Al Sln dla Sln 81y

. . s
+ A2 Sln 02a Sln 12y (3.04)
Wl(x.b) = A1 Sln dlx Sin 81b
+ A2 Sln 02x Sln 82b . (3.05)

As indicated by equation (2.31). Q1 and 81 are

dependent on C1 and equation (1.33) denotes the dependency

-19-

of a and 8 on c Then d1.a2.cl and c can be

2 2 2° 2
eliminated by using equations (2.29). (2.30). (2.37) and
(2.38) so that equations (3.03) through (3.05) can be re-

written as

 

 

 

 

 

 

Wl(x.y) = A1 sin v ADE-312.- (.1511)Z_% x sin 81y
+ A2 sin @-8§+ ‘/(2l—311)2-% x sin 82y (3.06)
or
W1(a.y) = A1 sin \/-)‘Btl-B]2_- (L;)2-% a sin 81y
+A2 sin/%1"B§+ (BBQ)2_%asin BZY
= O (3.07)
Wl(x.b) — Al sin ‘%?-Bi- (%§)2-%'x sin 81b
+A2 sin %-B§+./(—%—l)2-%xsin 82b
= o .

For a general nontrivial solution to equations (3.07)
to exist. the arguments of the sine functions must be set

equal to some integer multiple of F. That is:

/1Db-5§- /(.XD_h)2-%a= nln- (3.08)

 

 

/X—g‘.-5§+(/(A§)2-%a= nzrr (3.10)

where nl.n .ml and m are integers not equal to zero.

2 2

-20-

Substituting (3.09) into (3.08) and (3.11) into
(3.10). and setting a = b for a square plate. and
choosing the integers nl.ml.n2.m2 equal to l for the

first buckling node. equation (3.08) can be written as

4
A-l/TT— AZ-K2=4 (3.12)
W2 4

 

and equation (3.10) becomes

4
A+-2§./-714- A2-K2=4 (3.13)
Tr

 

where the substitutions

2
—)‘h = —-’W (3.14)
D 2
2a
and
2
E -.E_

have been made. When K = O. the case in which the elastic
foundation is absent. equation (3.12) has no solution
implying that A1 = 0. For this case. equation (3.13) has
the solution A = 2 and A is arbitrary. The solution

2
to (3. 14) becomes

1 = -——' (3.16)

or

 

(3. 17)

which agrees with Timoshenko's solution [17] for the
.buckling of a simply supported rectangular plate.
As K increases. A1 = 0 remains valid. A increases

A2 remains arbitrary and the radical in equation (3.15)

-21-

approaches zero. At the point at which the radical in

equation (3.15) equals zero. the radical in equation (3.14)

also equals zero. Hence. A = 4. K 272 and A1 and

A2 are both arbitrary. For K > 272. equation (3.15)

has no solution and. hence. A2 = 0. Equation (3.14).
2

however. has the solution A = 4 at K = 2V and A

increases as K increases beyond 2W2; and Al is now
arbitrary. This process can be continued for different
values of n and m as K and A increase.

A more general approach can be taken by substituting

equation (3.06). with the substitutions of (3.08) through

(3.11) into the differential equation (2.20).

(Ll-+2gh L1/2+k)wl = 0 .
This gives:
nn" nn' mTI'
{((:)2+(mb)2]2- 235W (—§-)z+(-%—)21+§}A1 sin alxsinBly
n2)7r2 mTl'
+([(—) 2m+(—]5—)2]2-- gg—H—fi— +(-§—)2]
+ l<--]A sin a x sin 8 = O (3 18)
D 2 2 2y ‘
which implies that
n”1 2 2 2 21h “1" 2 m1" 2
(HT) +(Tm)]- —5-[(——a) +(b)]
ls -
+ D}A1 _ O (3.19)
and
1177' mn’ 1117' HIV
r__2._2 _2_22__21£ _2_2 _2_2
(1(3) +(b)] D[(a) +(bll
E _
+ D]A2 - O . (3.20)

-22-

Multiplying through by a4 and writing a general equation

for (3.19) or (3.20). depending on whether n and m or

1 1
I12 and m2 are used. gives
2 2 2 4
(n2-+3§'m2)2v4 - 23fl§—-(n2-+§§-m2)72 + £3- = 0 . (3.21)
b D b D

Using equations (3.12) and (3.13) and examining the case

a = b. for a square plate. yields

JR 2
(n2-tm2)2 - (nz-rm2)A + 5: = 0 (3.22)
v
This gives a series of quadratic curves in K. each of
2
which is tangent to the line K =‘%r A. To see this. one

can solve for K in (3.22) with the assumption that there
exists a line. K = rA. through the origin which is tangent
to

2 4

K = (n2-+m2)v A - (n 2 4

2”((2) 7r . (3.23)

If K = rA. the tangency requirements in the solution

of the quadratic for A in equation (3.23) dictate that the

discriminate vanishes. or r = g;. for which

with the points of tangency
A = 2(n2+m2) (3.25)
K = Tr2(n2+m2) (3.26)

To the left of these points of tangency A1 = 0 and A2

is arbitrary and to the right of these points Al is

arbitrary and A2 = 0.

Equation (3.22) is plotted in Figure 3.2.

-23-

The solid curve in Figure 2 is the line along which
buckling will occur. The numbers in parentheses represent
values for n and m: (n2.m2) to the left of the points
of tangency (equation (3.20)) and (nl.ml) to the right of
the points (equation (3.19)).

Letting (na.ma) represent the node numbers for one
curve and (nb.mb) the node numbers of the next consecutive
curve. and letting A = n: + m: and letting B = n: + mg,

then the intersection of two consecutive curves from two

equations of the form (3.22). occurs at the point

 

2 2 2 2 2 2
(na+ma) - (n'b+"lb) ___ A2 -B2
A-B

A = = A + B (3.27)
(n:+m:) - (n§+m.:)
and
K =V/A7r4(A-A) = F2 v/AB . (3.28)

For the intersection of the first two curves marked

(1,1) and (1.2). na = ma = l. and nb = mb = 2; A = 2

and B = 5 and therefore
A = 7 (3.29)

K =(/10 #2. (3.30)

For OIg K.g 10 F2, the buckling is symmetric in

two directions. For the next intersection. na = 1. ma = 2.
nb=mb=2 and A=13 and K=2W2/Ia.v/I5F2gK

g 2(/I6 r2. and the buckling is symmetric in one direction
and antisymmetric in the other direction. Continuing on.

for 2(/lO T2 < K < 4(/§_V2. buckling is antisymmetric in

two directions.

-24-

:oflumoCSOM Hodxcfiuz m :0 oumdm

“wasmcmuuou oouhomQSm xamfifim m mo mcflaxosm N.m ousmfim

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

EN m. 0. m 0
u M x 0
e \ .h<\ 0
%\\ \“Wx @\ \
(4. (xx \\ :nN
\\ \\ Ha
\\ \ Ox
\
\\ \ 0oz4
\\ \x .o...
\\\ \
\\\ W4
\\ O
\\ (ms
\\\
[p pr m rm to r .00.
.Ill 0. I. III I.
I. 1 1 1 E
or 9 / r. .1
\\ + . .j l. 0
r... 1 r E- -
* ‘4 (fie i4 ficld‘

-25-

The critical load when K = 0. occurs at:
2

Ocr = -2)( = - 91’2— (n2+m2) (3.31)
a h
or for a nonsquare plate
2 2
0 r = -21 = - m7.— (n2+m2 22.) (3.32)
C a h b

which corresponds to Timoshenko's solution [15].‘

These results compare favorably with Heteyni's [1]
results for a simply supported beam on an elastic foundation
as illustrated in Figure 3.3 where k is the foundation
stiffness. l the length and E1 is the flexural rigidity
of the beam. These values are plotted against Nor/Ne
which is the ratio of the critical buckling load to the
Euler load for a hinged end bar of length l and flexural

rigidity EI.

III.2 The Case of Nonuniform Loading

From equation (2.13):

(Ll+%)Wl = —L (3.33)

D 2ch
choose

m0 = —x(px2-+qy2) for the nonuniform case. (3.34)

where p and q are values to be chosen depending on the
desired loading conditions.

From equation (3.34). the edge stresses are

-25-

:oflumocsom Hoaxcfluz m :0 Emon m m0 mcflaxoom m.m ousmflm

 

0: ON 0. O. n O
Illl _. . p q . 4
- - x _.c
. x
- e x
- \x
\ x
s \
\s N I:
\ To...
\
\
x
x
x
x
\
\ ”I:
\
\
\\
\
\ oo-
\\
\
\
\
VI: of
\
4W?
.00.

 

G = ~21p (3.36)
and

c = O . (3.37)

From the right hand side of equation (3.33).

 

 

azwl 32wl
L (W )0 = -21(q + p ) . (3.38)
2 l 0 2 2
Ax By
Equation (3.33) can then be written as
(L +h)W - - —-2)‘h( a—i—W21+p:zwll (3 39)
1 D 1‘ D q ax2 yi '
or
L W + 2AQ( iii”. 3:2)w +‘E w = 0 (3 40)
1 1 D q 2 P 1 D 1 '
Bx By
This can be rewritten as
2. .
[(Ll/Z 21h q)__ 32 + (Ll/2 _2___1h we +51“, = O (3.41)
2 D 2 D l
Bx By

Again as in equations (2.25) and (2.26). if solutions are
sought in the form

L1/2

1 w = -Ew. (3.42)

then substitution into equation (3.41) yields:

 

[(- C+2-1>5—h q)-§—2-+ (- c+—Lh-p)-L2-+ §le = (3.43)
52X ayz
or
if: 32 k
(P 2+0 2+D)Wl = O. (3.44)
Bx Ay
where
p: -E+%flq (3.45)

Q= -E+%b-p (3.46)

-28-

when p = q. P = Q and equation (3.43) becomes. using
equation (3.42). a quadratic equation in c. The solution
is the superposition of the solutions to equation (3.42)
for the two different values of ‘3. This is the case for
uniform loading previously developed. When p ¥ q. the
solution to equation (3.44) involves only one value of ‘3.

Separation of variables in equation (3.44) leads to

Px”Y + QXY” + % XY = 0. (3.47)
or
PX” + aix = 0 (3.48)
or" + 5§Y = o. (3.49)
where
a: + B; =‘% . (3.50)

The solutions to equations (3.48) and (3.49) are
a a

X = A' sin —%,x + C’ cos -; x. P ¥ 0 (3.51)
/P V53
v
and
B3 83
Y = B’ sin 7:;y + D’ COS'j: y. Q # O (3.52)
x/Q \/Q

(When p = q. equation (3.41) reduces to equation (2.21).
For P or Q equal to zero. this reduction is not possible.)

From equation (3.51) and (3.52).

. O‘3 . B3 . “3 B3
W=Asan-;xs1n—y+BSin—xcos:y
v/’P \/Q v43 \/Q
a B3 0‘3 83
+ C cos -—'x sin -—-y + D cos - x cos 7: y. (3.53)

V’P Vi? Vi; \/Q

_29_

Applying the boundary conditions for the simply supported

case X(O) = Y(O) = O and X”(0) = Y”(0) = 0 yields
B=C=D=O. (3.54)

so that

W = A sin —; x sin 41'y . (3.55)

./P /6
Applying the boundary conditions for the other two edges:

X(a) = Y(b) = 0 and X”(a) = Y”(b) = 0 yield'

a3
A sin -'a = O (3.56)
)3
and
B3
A sin —-'b = O . (3.57)
x/Q

For a nontrivial solution to exist (for A ¥ 0). the
arguments of the trigonometric functions must be integer

multiples of W or

-§-a = nv (3.58)
P
and
B3
--'b = nmy (3.59)
v43

where m and n are integers not equal to zero.
Using equations (3.45). (3.46) and (3.50) in (3.58)

and (3.59) gives

 

1-62
'B—q-C

and

-30-

: _ (3.61)

    

D

Solving for 83 in (2.61) and substituting this into

(3.60) yields

 

= —- 8.62)

where c can be determined by substituting (3.55) into (3.42)

and using (3.58) and (3.59):

2 2
._ a B 2 2
c = —P3— + 3 " 7T2 L'i'fl—

'-— — ( ) . (3.63)
Q a2 b2

Substituting (3.63) into (3.62) and solving for 1 yields:

 

 

2 2
k 2 1:2
1= 2 2 + a 2 2 . (3.64)
2h(pm _an )FZ 2h(pm .an )
b2 a2 b2 2

1. equation (3.64) reduces to equation (3.21)

For p = q

for the case of uniform loading. Furthermore. for k = 0.

p = 0. q = l. and for k = 0. p = q 1. equation (3.64)
reduces to Timoshenko's solutions for the buckling of a
rectangular plate compressed in one direction and uniformly

compressed in two directions respectively.

=- O =-O
The Case For p q. xx yy

Let p = -q = l. a = b. then

2 2
A = ka + (n

+m2)2 DVZ
2 2

(3.65)
2h(m2-n2)v 2a h(m2-n2)

-31-

 

m # n and the first buckling node is then m = 2.
_ ka2 ZSDWZ
A I ___2 + _—2——
6h? 63 h
or
_ _ kaz ZSDWZ
Ocr — ‘2X _ - 2 - 2
3hr 3a h
For k = 0.
2
G = ‘8033 l 3
cr 2
a h

which is the value given by Brush and Almroth [21].

(3.67)

(3.68)

-32-

CHAPTER IV
CLAMPED CASE

IV.l Uniform Loading

The buckling solution for the clamped rectangular
plate is not as straightforward as the solution for the
simply supported case. A function satisfying the boundary
conditions and the differential equation has not been
found ([22]. [16]) or cannot be found ([23]). Various
approximate methods have been used. and the one found to
be the most useful (based on rapidity of convergence and
amount of numerical work) is the method by Iyengar and
Narasimhan.

The displacement function W(x.y) is expanded in a
double orthogonal series composed of hyperbolic and
trigonometric functions. The boundary conditions lead
to two transcendental equations which can be satisfied by
choosing the arguments of the functions. The satisfaction
of the differential equation involves an infinite determi-
nant which is truncated for an approximate solution.

From equation (1.20)

(L -+2Afl Ll/2-+%)Wl = O . (4.01)

l D l

-33..

W(X.y)

IIIIIIA'IIiIIIIIIII

 

 

 

 

 

Figure 4.1 Rectangular Plate on an Elastic Foundation
for Clamped Case

The symmetric form of buckling can be described by expanding

the function W1 in the following form. with the origin
of the coordinates chosen as in Figure 4.1.

23 Z) Amnmen
m=l n=l

ll

W1

cos amx/a ch dmx/a cos BnY/b ch BnY/b

cos Bn " ch Bn ) '

 

 

 

)(

K“ I
DEA \ -'
m n mn cos am ch am

(4.02)

The boundary conditions for the clamped case take the form
Bwl

W1 = O. —:: = O at X = ea (4.03)

-35-

5W

_ 1
W1 - O. BY

 

= O at y = ib . (4.04)

Wl = 0 is automatically satisfied at x = :a. y = 1b.

The conditions on %¥- and 23- can be met at the boundaries

provided the following equations are satisfied

tan am + tanh am = O (4.05)
tan 8n + tanh 8n = O . (4.06)

From this
am = Bm: 01 = 2.3650. a2 = 5.4978.. . (4.07)

Substituting (4.02) into (4.01) yields

 

4 4 2
E Z Amn[ ((1:4-35 d:+-k—IaD—) XmYn + 2532— QiliXIIY”
m n b b
_ 2M1. a2 (CLZX’IY +13 CLZX Y/I) 1 _ O (4 08)
D m m n b2 n m n . ' .
where
cos omx/a ch omx/a
x” = + —_ (4.09)
m cos a ch 0
m m
and
cos Buy/b ch any/b
Y” = + ’ (4010)

 

n COS 5n ch 6n

The functions considered here form complete sets in
II
their intervals1 so that Xm can be expressed as an
expansion of Xm' and likewise Y; can be expressed as

an expansion of Yn:

X” = Z) d x (4.11)
p:

 

1 These functions satisfy the normal modes of vibration of
a beam and hence satisfy a self-adjoint differential
equation.

Y” = Z) e Y . (4.12)
q:

Because of orthogonality. the coefficients d can be

mp'enq
determined by multiplying (4.11) through Xp and (4.12)

through by Yq and integrating from -a to a. This

 

 

yields:
a a 2
j_a XmXpdx = amp j_a Xpdx (4.13)
or r.a
X”de
J_a mp
= (4.14)
mp a 2
I X dx
-a
where
a 2
f x dx = 2a (4.15)
-a P
a -2 tanh a
J“ 3.1;an . “-5 + i 1'
-a p m ch 0 cos a
m m
p=m (4.16)
“a 8d a
X”X dx = a tanh a - tanh
J-a m p 4 4( m m op up)
a -a

p¥m . (4.17)

The same procedure is carried out for Yn and Y; and it

is found that

d = e p d = .5499, d = -04356' d

nm nm 11 12 = -'0805'

21

d .8181. (4.18)

22

For a square plate. a = b and equation (4.08) becomes

-3 7..

232A [(d4+d4+l<- a4)X Y +201 Ci. X”Y”
m n mn m n D m n m n m n

2
ZAha 2 ll 2 ll _
—D (ameYn + anmYn) ] — 0 (4. 19)

or

ZZA [(d4+d4+-]5-a4)XY +2a2d Zd X 23d Y
m n mn m n D m n m p mp p nq q
q
2
-mmzy 3d X 1&2
D m n p mp p n q

BM

1 =
xm E danq) J 0 . (4. 20)

Using only the first term of the series as a first approxi-

mation. (m = n = 1) gives

 

 

 

4 2
4 2 ka 41ha 2
All[201(l-+dll)-+ D - D oldll]Xl(x)Yl(y)
= O (4.21)
or for All # 0.
2 4(1-rd2 ) + 533-- 34233 82d - 0 (4 22)
0‘1 11 D D 1 11 ’ '
or solving for A:
2
6.63D ka
X = + ’ (4023)
aZh 12.3h
then
G _ -21 _ -l3.25D -ka2 (4 24)
cr aZh 6.15h .

For k = 0 this compares with Timoshenko‘s solution of

 

1131§2,. For k ¥ 0. let
a h 2
A = Zth (4.25)
Dr
K = 827% . (4.26)

then equation (4.24) can be written as

-38-

2

A - 1.647(10' )K - 1.343 = o . (4.27)

This is plotted as the "symmetric" curve in Figure 4.2.
For antisymmetric buckling. sines and hyperbolic sines

are used instead of cosines and hyperbolic cosines. This

 

 

 

time let
sin a x/a sh a x/a sin 8 y/b
Wl = Z Z XmYn = Z Z Amn( sinma - shma ) ( sinnB
m n m n m m n
sh Bny/b) )
-‘-——————' (4.28
sh an
The boundary conditions give
cot am - coth am = O (4.29)
cot 8n - coth 8n = O (4.30)
which implies that
on = an. a1 = 3.9266. a2 = 7.0686... . (4.31)

Substituting (4.28) into (4.1) gives a similar equation as

to (4.08), where

Sin amx/a sh amx/a

x” = . + -—-———-- (4.32)
m Sln am sh am

 

Sln any/b sh any/b

Yn = sin a +-_j§;7;—-' (4°33)
n n

 

Using (4.11) and (4.12). the equivalent equations for (4.16)

and (4.17) are
a

 

 

j x”x dx = a[’2 COth GD+- 1 4- l ],p = m (4.34)
m p a . 2 2
~a Sln a sh a
P p p
a 8a2a
F X”X dx = -—E—-[a coth a -a coth o ]p i m (4.35)
"-a m p 4 4 m m p p

a-oc
p m

-39-

Again. using the same procedure for Yn and Ya. yields

d22

.8585. (4.36)

Using only the first term of the series leads to

4 2
4 2 ka _ 4Xha 2 _
231(1-+d11) + D D alldll — o (4.37)
or
1 _ 16.9D + ka2 (4 38)
' 2 46.02h ° '
a h
This can be written as
A - 4.403(10‘3)K - 3.261 = o . (4.39)

This equation is plotted as the "antisymmetric" curve in
Figure 4.2.
For the second order approximation, m = l. n = 1.2.

Using equation (4.20) for the symmetric case.

r 4 2 ka4 4
All-([201 (1 + all) +TJX1Y1 + 2aldlld12XlY2
‘ 2A%§3 O‘imdlllel"‘dlleyz)}
+ A12{2ai“§(dl1d21x1Y1'*dlld22X1Y2)'*(ai'*ag‘*h%i’xlyz
—-34%§3[a§d21x1Y1-+(aidll-+a§d22)xly J}
= o (4.40)
The linear independence of the functions XlYl and Xle

implies that

ka4_41ha2 azd ]
D D 1 11
2 2 2

I Zlha 2 a)
r F" -_ |, =

4 2
r ,

-40-

cOHumEonummm umufim .COMuMpczom Hoaxcfluz
m :0 mumHm Hmasmcmuomu pmmEmao m mo mafiaxonm N.¢ musmflm

ON 0. O. m

BIQEE>m

 

E’fi‘fid

 

 

400

 

-41-

and
r 4 21ha2 2 2 2
{A1lizaldl1d12" D OL1d12]+A12E‘23‘132d11‘5'22+111
4 ka4 21ha2 2 2 ~.
+ C12 +T-T(aldll +a2d22) ijle = O (4.42)

To satisfy these equations for all values of x and y,

the coefficients of X Y1 and X Y must vanish, or

 

 

 

 

1 1 2
P 4 .1,-
4 2 ka 2 2 7
20‘1(l'*‘311)+ D 2C3‘10‘2‘111‘121 A11
_ 43ha2 62 d _ 2111a2 azd
D 11 11 D 2 21
4
4 _4 4 ka 2 2
20‘1‘311‘312 a1'*a2'* D "*2a1a2d11d22 A12
2 2
_ tha 2 _ zgha 2 2
_ D O‘1‘5'12 D ‘0‘1‘111"‘O‘2dzz)_‘L 3
= o (4.43)

For the nontrivial solution to exist, the coefficients of

the determinant must vanish. or

4 2 4
ka _ 41ha 2 4 . 4 ka
D D C11‘311) [0‘1 + 0‘2 + D
2
2 2 2Xha 2 2
1a2d11d22 D (0‘1‘311‘H‘2‘322)1

4 2
(2al(l-+dll)-+

 

+ 2c

2
(2a2a2d d ._2Xha 2

4
1 2 11 21 D O‘2‘121)

(zaldlldlZ

 

= O (4.44)

This may be written

2 4 2
h k
660.28(Aj?—)2 g ) 42?
ka4
+ 1178.65 ———-+ 89.93.63 = o

D (4.45)

ka4)2

+ ( D

which can be expressed as

-42-

2 5 4

A - (5.49+.0208K2)A + 6.219 (10' )K

2

+ .0733K + 5.547 = 0 . (4.46)

Equation (4.46) is plotted as the symmetric curve in
Figure 4.3.
For the antisymmetric case. the same procedure is

carried out using equations (4.28) instead of (4.02):

4936.28 (431—5313)2 - (250213 + 154.88 113:) l-l‘gfi
+ (15%:2 + 4462.87 15-15—33 + 2746189.o4 = o (4.47)
or
A2 - (10.27+.0064K2)A + 8.32(1o'6)1<4 + .037K2

+ 22.84 = O . (4.48)

Equation (4.48) is plotted as the antisymmetric curve in
Figure 4.3.
For the third order approximation, m = 1.2, n = 1.2.

and equation (4.20) becomes

All([28i(l-+di)-+%-a4]XlYi-+Zai(di1X1Yl-+dlld12XlY2
+ dlld12X2Y1)-2‘%? azai(2dlleYl-+d12XlY2-+d12X2Y1)i
+ A12{(ai-+og-+%'a4)le2-+2aia§(dlld21XlYl
+ d12‘122X1Y2 +‘i’12‘321x2Y1) ' 2 ADA 3‘2 (0%anle
+ dgdlelYl + agdzleyz) }
+ A21[ (0‘: + mi +15” a4)X2Y1 + 20‘2“: (d11d21X1Y1
+ dzldlZXle +dlld22X2Yl) - 2 ADE a2[9.§ (621le1
+ d22x2Y1’ +aid11X2YlH = o (4.49)

-43-

:oflumsfixoummm pcoowm .cofiumpczom Hmdxcwuz
m :0 mumHm unasmcmuowu UoQEmHo m we mcflaxosm

m.¢ whsmwm

 

O.N 0p_ fi- 0. O
\ . 3
\\4
100
2.- 366 m
4
9.2
av 10h
9.508
.00.
&

 

-44-

which leads to the determinant

 

20.211: (1 +621) +1314 2aia§dlld21 26136281le1
‘ 4 2? azaidll ‘ 2 2? aZOL2221 ' 2'2? 3222 21
2aidlld12 ai-Fog-Ffi'a4 2a2a1d21d12
‘ 2 2? 32 idlz + 2aiagd11d22
- 2 %?'a2(aidll-+o§d22)
Zaidlldlz 2a1a2d12d21 a§-+ai-+%'a4
- 2 %? azoid12 + Zaiagdlldzz
‘ 223$“: 22
+ aidll)
= 0 (4.50)
The symmetric case may be written
A3 - (2.526 x 10'2K2-+9.646)A2 + (1.433 x 10’4K4
+ 1.83 x 10'1K2-+2.831 x 101)A - 2.351 x 10‘7K6
-4 4 1 2 1

- 5.349 x 10 K - 3.248 x 10’ K - 2.293 x 10 = 0,
(4.51)

which is plotted as the symmetric curve in Figure 4.4, and
the antisymmetric case may be written

A3 - (8.315 x 10’3K2-+1.729 x 101)A2 + (2.046 x 10‘5K4

+ 1.017 x lO-lK2-+9.488 X 101)A - 1.572 X 10-8K6

- 1.287 x 10’4K4 - 3.041 x 10’1K2 - 1.601 x 102 = 0.

(4.52)
which is plotted as the antisymmetric curve in Figure 4.4.

 

-45-

coflumexoummm phase .COHuMUCSOM Hoaxcfiuz
o co madam Hmaflmcmuomu @mQEMHo 0 mo mafiaxosm v.v musmflm

ON 9. O. m o

 

Tom
:40 A. .05

4. .A
.00;

 

-46-

The convergence for values of A near K = 0 is good
for lower order approximations. Results for larger values
of K or A requires the evaluation of higher order
approximations. A ninth order determinate is therefore
evaluated and the results are plotted in Figure 4.5. This
is followed by a plot of Hetenyi's solution for buckling
of a clamped beam on an elastic foundation in Figure 4.6.

The convergence for several orders of A vs. K for
the symmetric and the antisymmetric case is plotted in

Figures 4.7 and 4.8.

-47-

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-51-

The Case of Non-Uniform Loading IV.2

The case for non—uniform loading leads to equation

(3.40):
1 1 D ‘1 3x2 P ay2 1 D 1 '

Using equations (4.02) through (4.13), equation (4.21) is

replaced by

W

4 4 k a4 2 2
23 Z Amn[ (dm+ an+ D )Xm Yn + Zamocn 23 deXS 2 d
m n s r nr r

_ 1g 2 2 2
2 D a (amqyn E dmsxs + Oanxm 2:2 dnrYr) ]
= O. (4. 54)

The first term approximation (n = m = 1) yields

k 4 .Lh 2 2d
ll)+D a -2 D ac11d11(p+q)]XlYl

[A141(20L (l+d2
= 0 (4.55)
or

4 2 (k 4 1h 2 2d _
201(l-+dll) + D a - 2 ID a 01 dll(p-+q) — 0 (4.56)

which, for the symmetrflzcase becomes

 

 

 

 

2
81.5 + g-a4 - 6.15 xha (p-+q) = O (4.57)
Solving for 1:
2
13.25D ka
1 = . (4.58)
2 . +
a (p+q)h (6 15) (p q)h
which implies
2
G = -21 = _.__Z§;§2__ ka (4.58)

cr - 3.08h(p-+q) °

azh (p + q)

-52-

For p = 0,
O = _ 10.74F2D
cr a2h
which compares with
0 = 10.07W2D
cr azh

from Timoshenko and Gere [l7].

-53-

CHAPTER V

CONCLUSIONS

The non-linear FSppl-von Karman plate equations are
written in operator form and linearized using a perturbation
technique. A choice of the Airy stress function 0 leads
to a linear homogeneous partial differential equation with
homogeneous boundary conditions. The problem becomes one
of determining the parameter 1 for which non-trivial
solutions exist.

The buckling solution to the rectangular plate on an
elastic foundation shows a similarity to the solution for
the circular plate on an elastic foundation. In both
cases the elastic foundation causes the plates to assume
a shape dependent on the stiffness of the foundation.

Also. if the foundation is absent or if the foundation
stiffness is very weak, the plates buckle into the first
mode with no nodal lines. For stiffer foundations. the
first buckling mode can assume shapes with many nodes.

An interesting phenomenon of the circular plate in
the linearized case was an even distribution of the "ridges"
and "valleys" for a given foundation stiffness. As the

edge load was increased. the solution of the resulting

-54-

non-linear boundary value problem yielded a boundary layer
effect or a migration of the ridges and valleys to the
edge of the plate. It would be of interest to solve the
non-linear boundary value problem of the rectangular

plate to detect the presence of a boundary layer effect.

The perturbation technique used in the present research
provides a means of going beyond the linearized case,
although this presents difficulty owing the nature of a two
dimensional operator. Solutions in the circular case were
easier to handle because the axisymmetric loading led to
an ordinary differential equation. The additional sequence
of differential equations beyond the first linearized
equation may provide an advantage over or be an aid to other
methods of solution in post buckling behavior as in the
case treated by Friedrichs and Stoker. This. of course,
can only be made apparent by further investigation.

A critical load vs. foundation stiffness plot was made
for both the simply supported and the clamped cases and
compared to plots of beams under the same boundary conditions.
This became an aid in determining the progression of buck-
ling from symmetric to antisymmetric and back to symmetric
buckling as the foundation stiffness increased.

A closed form solution for the buckling of a plate
clamped on four sides does not exist, consequently a
numerical method was used. The technique chosen was
singled out for its lack of numerical computation and for

its rapidity of convergence. When K = 0, good results

-55-

are obtained using only a second order determinant. However,
for large values of K it becomes necessary to evaluate

determinants of much higher order.

-55-

BIBLIOGRAPHY

[1] Hetenyi, M., Beams on Elastic Foundations, The Uni—
versity of Michigan Press. Ann Arbor. Michigan
(1946).

j/ [2] Wolkowisky, J.H.. Buckling of the Circular Plate
/’ Embedded in Elastic Springs. An Application to
Geophysics. Communications on Pure and Applied
Mathematigg. Vol. XXI, 639-667 (1969).

[3] Friedrichs, K.0. and Stoker, J.J., The Non-linear
Boundary Value Problem of the Buckled Plate.
American Journal of Mathematics. 63 (1941).

[4] Karman. Th.v.. Festigkeist problem im Maschinenbau.
Enz. d. math. Wiss.. Bd. IV (1910). 438-452.

[5] Bodner, S.R., Post Buckling Behavior of Clamped Cir-
cular Plate, Quarterly Journal of Applied Mathe-
matics, 12, 397-401 (1955).

l r. .f I v7 -’

/.;I’f.-

‘{4
C2

, [6] Kline. L.V. and Hancock, J.0., Buckling of Circular 2

 

(3 Plate on Elastic Foundation. Transactions of mwpvu.3132(
the ASME, 323-324 (1965). _ 46...;
fil-‘it

[7] Yu, Yi-Yuan, Axisymmetrical Bending of Circular Plates
Under Simultaneous Action of Lateral Load, Force
in Middle Plane. and Elastic Foundation, Journal
of Applied Mechanigs.‘;4, Transaction§_9f the
ASME, 79. 141-143 (1957).

[8] Vlasov. V.Z. and Leont'ev, N.N., Beams Plates and
//’ Shell§_on Elastic Foundations, Israel Program
for Scientific Translations, Jerusalem (1966).

[9] Datta. 8.. Thermal Buckling of Some Heated Plates
Placed on Elastic Foundation. Defense Science
Journal. 26, 3, 119-122 (1976).

[10] Munakata, K., On the Vibration and Elastic Stability
of a Rectangular Plate Clamped at its Four Edges,
Journal of Math. and Physics. 31. 1, 69-70 (1952).

'1 [ll] Datta. 5.. Buckling of a Non-homogeneous Rectangular éf'vfl
' Plate on Elastic Foundation, Journal of Structural £324
Engineering, 4, No. 3, 100-105 1976). 49g?

(" 5 a5
Sh». p.215. (my ,5: ‘

t”.

C-J

-57-

[12] Ariman, T., Buckling of Thick Plates on Elastic
Foundation, Bautechnik (Berlin), 46, No. 2,

[13] Sabir, A.B., The Application of the Finite Element
Method to the Buckling of Rectangular Plates and
Plates on Elastic Foundation, Stavebnicky Casopis
21, No. 10, 689-711 (1973).

-[14] Seide, P., Compressive Buckling of a Long Simply
Supported Plate on an Elastic Foundation, Journal
of the Aeronautical Sciences, (1958).

[15] Jacqout, R.G., Elastic Stability of Plates Subject to
Elastic Constraints, Journal of the Engineering
Mechanics Division, Proceedings of the ASCE,

—_-_

101, No. EMA, (Technical Notes), 501-505 (1975).

[16] Timoshenko, S.P. and Woinosky-Krieger, 5., Theory of
Plates and Shells, 2nd. Edition, McGraw-Hill Book
Co. Inc., New York, New York, 1959.

[17] Taylor, Von G.I., The Buckling Load for a Rectangular
Plate with Four Clamped Edges. Zxchr. f. angew.
Math. und Mech., Band 13, Heft 2, 147 (1933).

 

[18] Timoshenko. S.P. and Gere, J.M., Theory of Elastic
Stability, McGraw-Hill Book Co., New York (1961),
p. 387.

[19] Sezawa, K., On the Lateral Vibration of a Rectangular
Plate Clamped at Four Edges, Aeronautical Research
Iggtitute of the Tokyo Imperial University, Report
No. 70, 61-71 (1931).

[20] Iyengar, K.T., Sundara Raja and Narasimhan, K.D.,
Buckling of Rectangular Plates with Clamped or
Simply Supported Edges, Publication De L‘Institut
Mathematiqpe. Nouvelle serie, tome 5 (19), 1965,
31-40.

[21] Friedrichs, K.0. and Stoker, J.J., Buckling of the
Circular Plate Beyond the Critical Thrust, Journal
of Applied Mgghanics, 7-14 (1942).

[22] Szilard, R., Theory and Analysis of Plates, Prentice
Hall Inc., Englewood Cliffs, New Jersey, 1974.

[23] Brush, D.O. and Almroth, B.0., Buckling of Bars,
Plates, and Shells, McGraw-Hill Book Co. Inc.,
New York, New York, 1975.

[241

[25]

[26]

-53-

Love, A.E.H., A Treatise on the Mathematical Theory
of Elasticity, Dover Publications, New York
(1944).

Courant and Hilbert, Methods of Mathematical Physics,
Vol. 1, Interscience Publishers Inc., New York
(1966).

Meirovitch, L. Analytical Methods in Vibrations, The
Macmillan Company, Collier-Macmillan Limited,
London (1967).

 

 

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