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Analysis of rectangular plate resting on an elastic

half space using an energy approach

Anant R. Kukreti and Man-Gi Ko

School of Civil Engineering and Environmental Science, Uni versi ty of Ok lahoma,

Norman, OK 73019 USA

An analytical formulation using the principle of minimum potential energy is presented to predict the

jlexural hehuvior of a rectangular plate resting in smooth contuct with un elastic half spuce (soil

medium) and subjected to a uniformly distributed load. The procedure accounts for interaction between

the plate and the soil medium. Compatibility at the interface of the plate und soil medium is sutisfied

by integrating the Boussinesqs formula, which relates the contact stress and the soil surface defor-

mution. Analytical formulations for the following two approximations used to model the contact stress

distributions are presented: u power series expansion and use of Chebychev polynomials. In both the

formulations the integrations over the plate domain are unulytically derived by dividing the plate

surfuce urea into eight triangular zones and evaluating explicitly the integruls over euch zone and

summing the results. First, the boundary conditions at the free edges of the plate are satisfied by

expressing some of the selected generulized coordinutes uppearing

in the assumed function

in terms

of the other, and then the total energy of the system is minimized to evaluate the unknown independent

generalized coordinates. The process of selecting these generulized coordinates to satisfy the boundary

conditions is uutomuted. Results obtainedfor a square plate are compured with similar results reported

in the literature and with those obtuined from three-dimensional finite element analyses. Results of a

parametric study investiguting the effect of the relative stiflness of the plate with respect to the elastic

half space ure also presented.

Keywords: rectangular plate, soil medium, deflection function, contact stress, interaction, energy

principles

Introduction

The analysis of plate structures resting on a deformable

soil medium is of significance in the design of foun-

dation for various type of structures. The geometrical

shape of the plate generally used is circular or rectan-

gular. Most of the studies have been done for circular

plate problems because of its geometric simplicity. In

case of rectangular and square plate problems, even

though theoretical methods have been reported, there

are practical limitations to the use of these methods

because of the mathematical complexity of the prob-

lem. A numerical method, such as finite element method,

can be used to analyze such a problem. But this method

would be expensive, since three-dimensional elements

are required to model the system properly. As a com-

promise to these two problems, i.e., the mathematical

Address reprint requests to Prof. Kukreti at the School of Engi-

neering and Environmental Science, University of Oklahoma, Nor-

man, OK 73019, USA.

Received 15 January 1990; revised 6 January 1992; accepted 15 Jan-

uary 1992

complexity and computational efficiency, methods based

on energy principles have been suggested recently to

analyze the flexural behavior of a rectangular plate

resting on a deformable soil medium.

One of the earliest works to analyze the rectangular

plate-soil interaction problem was suggested by Gor-

bunov-Posadov,- who assumed a double power se-

ries expansion to approximate both the plate deflection

and the contact stress distribution functions and de-

termined the unknown coefficients by satisfying the

governing differential equation and the boundary con-

ditions at the free edges of the plate. This method was

subsequently extended by Gorbunov-Posadov and

Serebrjanyi4 to analyze large rectangular plates. Be-

cause of the mathematical complexities involved the

application of this method was found to be limited to

only certain geometric aspect ratios of the plate.

Another analytical technique, developed by Ze-

mochkin and Sinitsyn,5 assumes that the contact stress

at the plate-soil interface can be approximated by a

grid of small rectangular areas, each of which is sub-

jected to a unique uniform stress. The uniform stresses

acting over these small areas are then approximated

as equivalent concentrated forces, which in turn are

338 Appt. Math. Modelling, 1992, Vol. 16, July

0 1992 Butterworth-Heinemann


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Energy analysis of rectangular plate on elast ic half space: A . R. KuK ret i and M.-G. Ko

determined by satisfying the compatibility between the

plate and the soil surface displacements and making

use of the equilibrium equations of the plate. This work

was an extension of the work done by Zemochkin for

circular plates. This approximation fails to consider

the step contact stress gradient at the plate edges. Oda

has applied the point-matching technique for deter-

mining the contact stress distribution under symmet-

rically loaded rigid rectangular plates resting on a lin-

early deformable elastic half space. Conway et al.x and

Conway and Farnham have employed this method to

solve the two-dimensional genera1 loading problem.

BrownlO numerically integrated the governing differ-

ential equation to analyze a rectangular plate resting

on an elastic half space. Additional analytical and nu-

merical results for plates resting on elastic foundation

have been reported by Borowicka, Poulos and Davis,

Brothers et al.,13 Kondo et a1.,14 and others. A com-

prehensive exposition of some of these works is given

by Selvadurai. I5

Zaman and Faruqueh have applied an energy method

to analyze rectangular plates of varying flexural rig-

idities resting on an isotropic elastic half space. In this

study the deflected shape of the plate is assumed as a

polynomial in terms of the spatial coordinates, and then

the contact stress distribution at the interface of the

plate-half space is assumed as a direct function of the

assumed plate deflection instead of being found by

integrating the Boussinesqs formula. Also, in this pro-

cedure one does not have the luxury of controlling the

error in the solution by considering a larger number of

terms in the assumed deflection function.

Aleksandrov et al. investigated the plane contact

problem of indentation of an elastic plate into the sur-

face of a thick elastic layer, in the absence of friction

between them. The method is based on joining the local

solutions that are valid in the neighborhood of the edges

of the plate to the global solution that is valid at some

distance from them. Improved theory of plate flexure

is used to construct the global solution of the second

kind (classical theory of thin plates yields global so-

lution of the first kind). The local solutions are ob-

tained by examining the problem of interaction of the

elastic quarter and half planes.

Cheung and Zienkiewicz8 introduced the finite ele-

ment technique to analyze plates resting on elastic

foundations. Wang et a1.19

extended this procedure to

analyze rigid pavements by introducing artificial cuts

to limit the size of the slab. HuangzO has suggested the

use of an iterative scheme to make the coefficient ma-

trix banded and has shown that this reduces the com-

puter storage requirements. Two of the limitations of

the finite element so1ution8 are that they assume the

contact stress to be uniform around each nodal point

and they use rectangular plate-bending elements. The

first assumption leads to unrealistic contact stress dis-

tribution, and the second limits the analysis to rectan-

gular plates only. Svec and G1adwel12 introduced tri-

angular plate-bending elements with 33 degrees of

freedom in a bid to overcome these limitations. They

represented the contact stress distribution by a cubic

polynomial to lift the first limitation. Recently, Raja-

pakse and Selvadurai

22 discussed the applicability and

performance of certain Midlin type plate elements, based

on reduced integration techniques, to solve the flexural

interaction between an elastic plate and an elastic half

space.

From the aforementioned literature review it is ev-

ident that the analysis of a rectangular plate foundation

resting on an elastic half space is a complicated prob-

lem, since it involves two spatial variables, and the

boundary conditions have to be specified for all four

edges of the foundation, which involve second- and

third-order derivatives of the plate deflection function.

Gorbunov-Posadovs power series expansion tech-

nique for rectangular plates resting on an elastic half

space is mathematically reasonable, but because of the

complexity of the formulas obtained, their practical

usage is limited to a certain geometric aspect ratio of

the rectangular plates. Because of the nature of its

formulation, this method also cannot be automated and

involves large amounts of manual bookkeeping. The

technique developed by Zemochkin for circular plates

has been extended by Zemochkin and Sinitsyns to rec-

tangular plates by assuming uniform contact stress over

small plate zones. This approximation fails to consider

the step contact stress gradient at the plate edges. Za-

man and Faruqueslh variational method is easily ap-

plicable for rectangular plates with various geometric

aspect ratios. But in this approach they have assumed

that the contact stress is a direct function of the plate

deflection instead of finding this relationship by inte-

grating the Boussinesqs equation, which relates the

contact stress distribution function and the soil defor-

mation in the form of a double integration. Also, in

this procedure one does not have the luxury of con-

trolling the error in the solution by considering the

larger number of terms in the assumed deflection func-

tion. Various finite element applications have also been

suggested. But to obtain realistic results, each problem

has to be analyzed by using three-dimensional ele-

ments, which is computationally expensive. Also, it is

not possible to directly derive analytical expressions

describing the flexural behavior of the plate and also

the function describing the contact stress distribution

at the plate-soil interface.

In this paper an analytical formulation based on the

principle of minimum potential energy is presented to

predict the flexural behavior of a thin rectangular plate

resting in smooth contact with an elastic half space and

subjected to a uniformly distributed load. Compatibil-

ity between the plate and the elastic soil medium in-

terface is assured by integrating the Boussinesqs for-

mula, which relates the contact stress and the soil surface

deformation. Two analytical formulations are pre-

sented to mode1 the contact stress distribution: the

power series expansion and the Chebychev polyno-

mials. The results obtained from the analytical for-

mulations are compared with those reported by Gor-

bunov-Posadov2 and Zaman and Faruque16 and from

a three-dimensional finite element analysis using the

computer program package SAP IV.2

Appt. Math. Modelling, 1992, Vol. 16, July 339


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Energy analysis of rectangul ar plate on elastic half space: A . R. Kuk reti and M.-G. Ko

Analytical formulation using power series

approximation for the contact stress

Assumed contact stress and plate dejlection function

A thin rectangular plate resting on an elastic half

space is considered in this study. Such a plate of length

and width 2a and 2b, respectively, and thickness tp is

shown in Figure I. The plate is subjected to a uniformly

distributed load of intensity

p.

Since the problem is

symmetrical about the X- and y-axis, the contact stress

distribution, q(x, y),

at

the interface between the plate

and the half space is assumed as an even power series

function of the spatial coordinates, x and y, as follows:

q(x,y) = p i B;x*yQ

i,j=o

(1)

where n is an arbitrary integer and B; are the unknown

coefficients, called generalized coordinates.

The deformation function, w(x, y), of the soil me-

dium surface, which also represents the deflection

Elastic Ha&pace

Figure 1. Typical rectangular plate resting on an elastic half

space

function of the plate, is related to q(x, y) by the Bous-

sinesqs formula, which is given as follows24:

W X,Y) = ~

4(5,77)

~(, ~b [(5 - x)2 + (77 - Y)*l*

4

(2)

where u, is the Poissons ratio and G, is the shear

modulus of the soil medium, and 5 and 77denote a point

where contact stress is applied. The shear modulus,

G,, is related to Youngs modulus, E,, and u, of the

soil medium by the following relation:

G, = Es

2(1 + us)

Substituting (1) into (2) reduces w(x, y) to

Thus the problem reduces to the evaluation of the fol-

lowing integral:

a

b

I, =

t2;v2j

_a _b

I(5 -

4* +

(I -

Y)2112

4dv

(5)

where

i, j = 0,

1, 2, 3,

. . . , n.

To evaluate I,, the

rectangular plate is divided into eight triangular zones,

as shown

in Figure 2a. Taking

I$-x=pcosh

(W

7 - y = psinh

(6b)

where

p = [(5- x)2 + (77- Y)212

then

dtdq = pdpdh

(7)

(8)

and (5) becomes

zij = (I,), + (I,>, + * * * + (ZJ,

(9)

where

Al -l(A)

K-j), = j- / f(p, A) dpdA

0 0

(104

A2

rz(A)

UiJ2 = / j- f(p, A) dp dA

0 0

(lob)

U,), = j- /

.HP,

A) dp dA

m 0

(1Oc)

A4 ~0)

UiJ4 = j- j- f(p> N dpdA

7T 0

Cod)

340 Appl. Math. Modelling, 1992, Vol. 16, July


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Energy analysis of rectangular plate on elastic half space: A. R. Kuk reti and M.-G. Ko

Kjh = j- / S(P, A) dp dA

A* 0

(toe)

where f p, A) = x + p cos AP y + p sin Aj2j, A,,

AZ, . . , A8

and r,(A),

r2 h),

. . . ,

r,(A) are defined

in

Fig-

ure

2.

In view of

(100

(1Og)

where

i

0

i

=

s

s (i - s)

(11)

(12)

it is possible to develop explicit expressions for (I,),

(IOh)

to (I,),. These expressions are given in the appendix.

In view of (5), (9), (IO), and the equations presented

in the appendix, (4) reduces to

W X,Y) = (1 - u,)

&

$ 5

2 5

& jCiJJrX2i-.~y2.i-~

.~,-0J-o.,-Or-O

where

x [(h;)++I) + (h;)++( - I)z; + (h;)s+f+(- l)Zi

+ (/$;)++I(_ I)s+,Z;l + (&)+r+Z; + (&).\+l I(- l)Z?

+ (A;).++(- l)ZY + (h#++(- l)+z;]

(13)

2i 2j

Cij.sr =

( >( 1

t

(14)

s+f+ 1

Zg = In (u,~ + VFG&

Z;I=NT&I ,

=

G m

t - 1 z,

--

I 2

for I > 2

t

I

(13

hi=h;=a-x

hj=h:=a+x

h;=h;,=b-y

h;=h;,=b+y

h; b-y h; b+y h; b-v

*I=;=-

h,

U-X

4=7=-

h,

U-X

[d3 I = i

h

a x

(16)

h;

a-x

4

a+x

us=h;=-

l4i = h)5 = b-y

b

m-3

m=4

m-7

Figure 2.

Subdivisions of the plate used to evaluate the integrals: (a) subdivided zones; (b) variables defining the geometry

of the subdivided zones

Appl. Math. Modelling, 1992, Vol. 16, July 341


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Energy analysis of rectangular plate on elastic half space: A. R. Kukreti and M.-G. K o

Equation (13), which describes the plate deflection

function, will be used to evaluate the strain energy of

the plate subsystem. In order to do this, partial deriv-

atives of this function up to the fourth order need to

be evaluated first. In its present form, i.e., equation

(13), it would be very cumbersome to evaluate these

derivatives. Thus (13) is expressed first in terms of

the normalized spatial coordinates, then the result-

ing expression is approximated by using Chebychev

polynomials, and finally the required derivatives are

evaluated. The normalized coordinates used are ex-

plained next.

Normulized contact stress and plate deflection

function

The spatial coordinates, x and y, are normalized to

coordinates X and Y, as follows:

X=X

a

(18)

Defining new hi as

/2,=/22=1-X

/2,=/2,=1+x

&=&=1-Y h7 = h 8 = 1 + Y (19)

and p = alb and x = bla = l/p, and substituting (18)

and these definitions into (1) and (4) give, respectively,

q X, Y) = p i i B,xziY2j

i=Oj=O

(20)

w(X, Y) = K i 2 B,H,(X, Y)

(21)

;=oj+o

where

B__ = a2ib2jB .

I J

(22)

K =

(1 - u.Jb

2rrG,

(23)

Hjj(X,

Y) =

2

2

x x C,X2--.~y2~r [P+{h;+ +I) + h+ +( _ ])f1

s=o r=o

+h

$+r+ I( _ 1)~; + h;;+f+l( _ 1)+1;} + x.\{h5+f+ 1;

+ h;++l(-])I: + ,$+,+I(_ I)17 + h$+f+l(- 1)+1f}]

(24)

Approximation using Chebychev polynomials to evuluate strain energy

The next step is to express

(24)

in a form that can be easily differentiated to evaluate the strain energy stored

in the plate system. This can be achieved by using Chebychev polynomials to approximate H,, i.e., by expressing

H,(X, Y) =

-Yh> ,)~~(y&)T,(y,)T,(X)T,.(Y)

where

ND

denotes an arbitrary number of subdivi-

sions; C denotes that the first term of the summation

should be divided by 2;

yk = cos (2k IT

2ND

.,,, cos(2t In

2ND

(26)

and T,(x) are Chebychev functions that can be con-

structed by using the following formulas:

T,+,(z) = 2zT,(z) - Tnpl(Z)

To(z) = 1

Substituting

(25)

into

(24)

gives

T,(z) = z

(27)

w(X,

Y) =

& $ $ _, $ Cijt&ktJ2;X2kY2+i+j

k-O-O-hJ-1

(28)

(25)

where

n n ND ND

r=Os=O k= /=I

(2%

k = 4k( _ 1bz

(i + k)

i + k(2k) (i - k)

(30)

and the symbol 4; is such that 6; = 1 if

i # 0

and

$o = l/2. Equation (28) can be abbreviated as

n

n

w X,

Y) =

K

x 2

A,X2;y2j

i=Oj=O

where

A, = i i DijrJL

r=Os=O

(31)

(32)

(33)



6/19

Energy analysis of rectangular plate on elastic half space: A . R. Kukreti and M-G. Ko

Equation (32) can

as follows:

[Al = [DI[Bl

also be expressed in a matrix form,

(34)

Thus the coefficients appearing in the deflection func-

tion, w(X, Y), and those appearing in the contact stress

function, &f, Y), are linearly related by the coeffi-

cients D,,,. But both the contact stress and the dis-

placement functions are related to the same unknown

coefficients or generalized coordinates, B,.

Finally, the plate displacement function, w(X, Y),

can be expressed as polynomial functions containing

the unknown coefficients, B;j, as follows:

W(X, Y) = Kc 2 x x Diir,,B,,X2iY7.i

(35)

i=Oj=Or=O 5-O

Application of plate bound ary condit ions

Along the edges of the plate the bending and the

twisting moments and the vertical shear force must

vanish. These boundary conditions for each free edge

were first expressed by Poisson, and later, KirchhoffZ6

proved that by using small deflection theory for thin

plates it is not possible to satisfy these three boundary

conditions simultaneously. Kirchhoff showed that the

twisting moment, M,,., in the xy-plane and the shearing

forces along x- and y-directions, Q.%, nd QV, respec-

tively, must be replaced by one boundary condition.

The argument for this is that the vertical edge forces,

V,

and

V,,

along x- and y-directions per unit length,

respectively, can be written as

(37)

where QX and Q? are the lateral shear forces and the

second terms &V,,,l~y and dM,,ldx represent the ad-

ditional shearing forces at the edges, produced by the

torsional moments, M,, = My.,. These additional shear-

ing forces are called Kirchhoffs supplementary forces.

Thus the boundary conditions at the four plate edges

are as follows (refer to

Figure I):

Zero moment condit ions

[M. l = =

-0,

1

= 0

*=0 384

[M,Jv=_ch=

-0,

1 =O

=h

38b)

Zero shear force condit ions

= -%[

5

d W

+ (2 - LI,)

1

=

axay2

0

x= fU

(39a)

PJv=tt, =

[Q,, + lickb

a3w a3w

= -DiJ

ay +

(2 - up)-

ax2ay , , =h =

0

Wb)

where

D,

= E,,t;1/[12( 1 - u,)] represents the flexural

rigidity of the plate. Substituting (28) and (31) into (38a)

and (39b) gives a set of

4(n +

1) or 4N linear simul-

taneous equations, which can be expressed in matrix

form as _

[WIIAI = 0 (40)

where

[Al =

A(l) -

40,

0)

A(2) 40, 1)

A(N)

A(N + I)

AtO,

n)

41, 1)

A(N*)

_

,Ah n)

(41)

and the elements of the matrix [WI are functions of i,

j, v,], p,

and x. Equation (40) represents 4N numbers

of linear simultaneous equations with N2 numbers of

unknown coefficients (or generalized coordinates). Us-

ing these relations, 4N number of generalized coor-

dinates can be eliminated by expressing them in terms

of the other (N - 4N) number of generalized coor-

dinates. To reduce the unknown coefficients from N

to (N - 4N), (40) is partitioned as follows:

(42)

However, the form of matrix [WI obtained for this

problem does not permit arbitrary partitioning. For

example, when N = 6, the form of matrix [W] obtained

is shown in

Figure 3.

In this figure the darkened ele-

ments are nonzero elements, and the others left blank

will be equal to zero. From the figure it can be seen

that if the last 12 columns are chosen as the ones as-

sociated with the (N? - 4N) unknowns in vector [A],

then some of the rows, such as row numbers 17, 18.

23, and 24, will have all elements equal to zero, which

will cause singularity when solving (42) for

[A],

be-

cause [W] cannot be inverted. Therefore to avoid this

singularity problem, a systematic way to choose the

(N - 4N) columns related to the independent vari-

ables in vector [A] is necessary. The following algo-

rithm is suggested:

1. Rearrange the rows of [W] so that it is in row-

echelon form. In other words, after rearrangement,

if ai denotes the column number of the first nonzero

element in row

i,

then the following relation exists:

a, 5 a2 5 . . . 5 a,.

In this stage, vector {NR} is

used to record the rearrangement of the rows of the

matrix. If row number i of the original matrix ap-



7/19

Energy analysis of rectangular plate on elastic half space: A. R. Kukreti and M-G. Ko

I1

I I I I

r

12

I

Figure 3.

Form of matrix

[WI

for N = 6

o r n =

5)

2

3.

pears in row numberjof the rearranged matrix, then matrix

[WI,

which is of the order of (4N

x

N*),

make NR(j) = i.

can be expressed as follows:

The rearranged matrix has 4N rows and N* col-

umns. Next, choose 4N numbers b,, bZ . . . , bAN

from the set 1, 2, . . . ,

N2, which will correspond

to appear in the matrix [W]. The numbers

bj

are

chosen recursively as follows: (a)

6, = al ;

(b)

For i 2 2, define b i to be the least number in the

column associated with a nonzero entry in row

i ,

and which is also greater than all the previously

chosen

bi

(1 5 j 5

i -

1).

Now, sort the numbers

b,, bZ,

. . . ,

bdN o

that they

are arranged in increasing order, and store them in

the vector {NC}, the elements of which are NC(l),

NC(2), . . . ,

NC(4N). Finally, all the column num-

bers from 1 to N*, which do not appear in the set

b b

N ,

are stored in the vector NC(4N + I),

A&N + 2), . . . ,

NC(N*). Now the elements of

W i , ) =

W i , j )

if lsjs4N

W i , j )

if 4N + l%jlw

(43)

where

W i , j ) = W NR i ) , NC j ) ) for 1 5 i , 5 4N

W i , ) = W NR i ) , NC 4N + j ) )

for 1 5

i 5

4N

(44a)

lsjsN*-4N

(44b)

The rearranged form of matrix [W]

so

obtained is shown

in

F igu re 4

for N = 6. Substituting (35) into (40) re-

duces it to

wI[m~l = 0

(45)

where

WOWl), NC(l))

WWWl), NW))

WWR(l),

NCW*))

[WI =

wNwL

NC(l))

WWW),

NCW)

WWW, NCW*))

W NR 4N) , NC 2) )

W NR 4N ) , NC N* ) )

344

Appt. Math. Modelling, 1992, Vol. 16, July

(464


8/19

Energy analysis of rectangular plate on elastic half space: A. R. Kuk reti and M-G. Ko

[Dl =

[Bl =

DOW 1)>NC( 1))

DOGI),

NCG?))

D(NC(I), NC(N2))

DWCC3, NC( 1))

DWW , NCW

D(NC(2), NC(W))

D(N&), NC(I)) D(NC(N;), NC(2)) D(NC(N;, NC(W))

[ NwI)) 1

Equation (45) can be abbreviated as

[B,l = - [ wD,l

[ W&l[Bzl = 1

TI [ l

(49)

[WDl[Bl = 0

(47)

where [T] = - [

WD,] [ WDJ. Thus (49) expresses

where [ WDI = [ Wl[D], and then can be partitioned

the 4N dependent generalized coordinates in terms of

as follows:

the remaining (N - 4N) unknown independent gen-

eralized coordinates.

[B,l

Considering (491, the number of unknown coeffi-

[[W~,l[W~,ll ,B21 = 0

F

1

(48)

cients in (21) and (35) that define q(X, Y) and w(X, Y),

respectively, can be reduced as follows:

which can be opened and solved for [B,], giving

I. Equation (21) can be written as

r

I

NR(i) J

Figure 4. Form of matrix [WI after rearrangement for N = 6 (or n = 5)

Appl. Math. Modelling, 1992, Vol. 16,

July 345


9/19

Energy analysis of rectangul ar plate o n elastic half space: A . R. Kuk reti and M.-G. K o

4(X, Y)

- =

B(NC(l))ZQ(NC(l)) + B(NC(2))ZQ(NC(2)) + . . . + B(NC(4N))ZQ(NC(4N))

P

+ B(NC(4N + l))ZQ(NC(4N + I)) + . . . + B(NC(N))ZQ(NC(N))

(50)

where

ZQ(ZK) = X2Y2j

(51)

in which i and j are

But in view of (49),

relatedtoZKbytheformula:ZK=iN+ i+ lforOsisN- landOsjsN- 1.

(50) can be written as

y = [ZQ(NC(l))ZQ(NC(2)). . . ZQ(NC(4N))l

T(1,2) . . . T(1,4N)

T(2,2) . . . T(2,4N)

T(4N, 2). . . T(4N, 4N)

+ [ZQ(NC(4N + 1)) ZQ(NC(4N + 2)). . . ZQ(NC(N2))]

Np4N

X

= x G(f;)B(NC(4N + ZJ)

(52)

I,= I

where

GV;) = : ZQWCW)UV, 1;)

k l

+ ZQ(NC(4N + ZJ) (53)

in which 1; = 1 to (N2 - 4N).

2. Similarly, a new form for w(X, Y) can be developed

by using (49) and defining a new vector

{ZW} as

follows:

ZW(ZZ) = i i D,.yiiX2Y2

r=os=o

= $ D(ZK,ZZ)X2Y2 (54)

rk=I

where ZZvaries from 1 to 4N and r and s are related

to zk by the formula Ik = rN + s + 1. Then (35),

which defines w(X, Y), can also be expressed in

terms of the (N2 -

4N) number of independent

unknown generalized coordinates, as follows:

F =N"gNF(Z;)B(NC(4N + ;))

(55)

I,= I

where

F(Z;) = g

(ZW(NC(k))T(k,Z;)

k=I

+ ZW(NC(4N + Z;))) (56)

in which Zj = 1 to (N2 - 4N).

Total potential energy functional

The total potential energy functional, UT, of the

plate-half space system is composed of the following

three contributions:

u, = UP -I- u, -I- u, (57)

where

U,

strain energy stored in the plate due to bending,

a

b

UP = %

1 J

[(w,, +

WYJ2

a1 -

qJ(w$ -

w,, W_,)l

x dy

a -b

&(b4wgx + aw&

+ 2a2b2wx,wyy) +

2(1 - u,)$w& - wxxwvr) dXdY

1



10/19

Energy analy sis of rectangular plat e on elasti c half space: A. R. Kuk reti and M.-G. Ko

Us

strain energy stored in the soil medium,

I I

U,=;ab

J-J

q X, Y)w X, Y) dXdY

59)

PIP1

U,

work done by the externally applied load.

I I

U, =

-pab

J-I

w X, Y) dX dY

-1-I

60)

Finally, in view of (52), (55), and (58)-(60) the total potential energy functional, UT, can be expressed as

I I

+ i Kab

J-J-(

'-4N N'-4N

c B ZW Z;)

c B Z,)G f;)

dXdY

- I - I

I,=

I,= I

I I

- pKab

II

N-4N

2 B Z,)F Z;) dX dY

61)

- I PI

I,=

M inimizati on of tot al potenti al energy functional

In order that (35) can adequately describe the plate

deflection and the contact stress acting at the plate-half

space interface, the independent generalized coordi-

nates, B(Zi), where Ii = 1 to (N* - 4N), should be

chosen such that the value of the total potential energy

functional is a relative minimum. According to the prin-

ciple of minimum potential energy, this can be achieved

by setting equal to zero the first derivative, with re-

spect to each of the independent generalized coordi-

nates, i.e.,

i3Ur

-=

aB Zi)

0

(62)

where 1; = 1 to N* - 4N. Substituting (61) into (62)

gives a set of linear simultaneous equations with

B,j

as

the unknowns, which can be written as

[XI[Bl =

IFI

(63)

where, after evaluating all the integrals appearing in

the expressions for the elements of the matrices [Xl

and [F], the following relationships are obtained:

+ + g 5 CRS i, Zi )Cli s j , r (? 6+m r ~ 1,c:~f2+m~) 3)

;=lj=]

+

y 5 5

CRS i , Zi )CRS j ,

Z ,,:\(T , F+T ,

i=,j=,

+ 4( 1 - 0,) 2 $ $ CRS i , ZJCZ?S , Zi )

64rslm

,+lJ-I

2 r + 1) - 1) 2 s + m) - 1)

+

:pKab 5 CRS i , ZJ

4

i= l

2 r + i ,) + 1) 2 s + j2) +

1)

continued



11/19

Energy analysis of rectangular plate o n elastic half space: A . R. Kuk reti and M.-G. K o

+ g T(k I-)

4

=

(2(r + ix) + 1)(2(S +j,) +

1)

+ :pKab 5 CRS(i, Zj)

4

i=

I

(2(r + id + 1)(2(s +j,) + 1)

+ f T(k,ZJ

4

k=l

(2(r + i2) +

1)(2(s +j,) + 1)

F(Zi) = 5 CRS(i I.)

4

i=

I

(2r + 1)(2s + 1)

where

CRS(i, Zi) = D(i, NC(4N + Z;)) + ff T(k,

Zi)D(i, NC(k))

%=I

(64

(65)

(66)

and r, s, 1, m, i], j,, iz, j,, i3, and j, are integers, which are obtained from the following relationships:

i=rN+s+ 1

j=IN+m+l NC(4N + ZJ =

i,N + j, + 1

NC(k) = &IV + j, + 1 NC(4N+Zj)=i,N+j,+ 1 (67)

Also in (63) and (64), both Ii and Zj vary from

i

to (IV*

- 4N). Thus the (IV2 - 4N) unknown independent

B(ZJ can be solved from

[Bl = [Xlr[Fl

(68)

Finally, substituting the results obtained from (68) into

(52) and (55), the contact stress distribution and the

deflection function of the plate can be evaluated.

Analytical formulation using Chebychev

polynomials approximation for contact stress

In this section the analytical formulation for a rectan-

gular plate resting on an elastic foundation when the

contact stress distribution function itself is approxi-

mated by Chebychev polynomials instead of a power

series is presented. Because the steps involved are

exactly similar to the ones presented in the previous

section, where the contact stress was approximated by

a power series, only the new resulting equations will

be presented in this section.

To approximate the contact stress distribution by

Chebychev polynomials, the spatial coordinates are

normalized by defining X and Y, as done before, by

(19). Using these normalized spatial coordinates, the

contact stress distribution between the plate and the

half space is assumed as

(69)

In view of (28) this equation can be written as

(70)

Similarly, the Boussinesqs formula (equation (2)), which

relates the contact stress to the plate deflection, can

be expressed by using the normalized spatial coordi-

nates, as follows:

(71)

Using the integration method presented in the previous

section, i.e., dividing the rectangular plate into eight

triangular zones (refer to Figure 2), integrating over

each triangular zone and then summing them up, re-

duces (71) to

w(X, Y) =

K i ~&&(X, Y)

i=Oj=O

where

K = (1 - u&b

87TG,

(72)

(73a)

348

Appl. Math. Modelling, 1992, Vol. 16, July


12/19

Energy analysis of rectangul ar plate on elastic half space: A. R. Kuk reti and M.-G. Ko

H..(X, y) = i i -$ 5 Cijk lstX*k~sy 2/~[p+(h;++z)

k=O/=O.~=O~=O

+ @++l(-l)]; + &++I + &+r +l(-l)SI; + ,~+I+(-l)~~)

+ Xq& ++lz; + /I;++( -

l)Z2 + h;++( - l)Z,7

+ &++l( _ I).TZf)]

zm = Amtan W dA

t

/

cos h

(73C)

0

(73b)

Cjklsr = ktj2y1

(y)(:,)/,s+ +, G3d)

Substituting (70) and (71) into (58)-(60), and then

using (62), gives an equation similar to (63) but with

and A has been defined earlier by (6a) and (6b).

the elements of matrix [X] defined by

7

X(Zi, Zj) = y 5 5 CRS(i, Zi)CRS(J , I) (2(r ~;5(~3~(~:(~~SI)t 1)

i=fj=]

7 7

+ 4( 1 - U,) Z$ $ $ CRS(i, ZJCRS( j, Zj)

64rslm

-IJ-I

(2(r + I) - 1)(2(s + m) - 1)

+ &Kab g CRS(i,ZJ 2 t,,t ,,

[

I

;=

I II = 0 I, = 0

(2(Z, +

r) +

1)(2(Z, + s) + I)

+ s Z(k, Zj) 2 5 t;l;t$,m

1

k=

I~=o,=o

(2(Z, + r) + 1)(2(Z, + s) + 1)

2

+ Kab 5 CRS(i, ZJ

I

5 5 t;,i t ,y

1

i=l IA= 0 I, = 0

(2(Zk + r) + ,)(2(Z, + s) + ,)

,),(2(z + s) + ,)

1

(74)

k= II_=0 I,=0

I

where

il,

.A,

k, _h, 4, .A

and r and s are related to

NC(4N + ZJ, NC(k) and NC(4N + Zj), respectively,

by the following relations:

NC(4N+ZJ=i,N+j,+ 1

(75a)

NC(k)=i,N+j,+ 1

(75b)

NC(4N + Zj) = i3N + j, + 1

(75c)

i=rN+s

(75d)

and CRS(i, .ZJ is the same as given by (66). The ele-

ments of matrix [Fl in (63) will remain the same and

are given by (64). Therefore, (68) can now be used with

appropriate values for the elements of matrix [X] to

solve for the unknown independent coefficients

B(i).

Finally, by substituting these coefficients into (70) and

(72), the contact stress distribution and the deflection

function of the plate, respectively, can be evaluated.

Numerical examples

In this section the analytical formulations presented in

the previous two sections are numerically investigated

for an isolated square plate (a/b = I) foundation resting

on an elastic half space and subjected to a uniformly

distributed load of intensity 4 = 1 psi. The results

obtained are compared with those obtained by a three-

dimensional finite element solution using the computer

program package SAP IV.23 Also, the flexural mo-



13/19

Energy analysis of rectangul ar plate on elastic half space: A. R. Kuk reti and M-G. Ko

ments obtained are compared with the results reported

by Gorbunov-Posadov2 and Zaman and Faruque.i6 In

this section, first, the nondimensional parameters used

to compare the results are explained. Then the nu-

merical results obtained are presented and compared.

Finally, the effect of the variation of the main nondi-

mensional parameter selected is presented.

Nondimensional quantities

Various nondimensional parameters, which repre-

sent the relative stiffness of the plate-half space, have

been developed by different researchers.4,5,27 These

parameters combine the different variables represent-

ing the properties of the plate-half space into one vari-

able, thus facilitating the case studies to be conducted

to predict the behavior of the plate foundation by vary-

ing only one variable instead of the seven parameters

identified in the previous subsections, which were u,

b, t,, Ep, u,, ES,

and u,, where

a

and

b

represent the

dimension of the plate, E is the modulus of elasticity

and u is Poissons ratio, and subscripts p and s rep-

resent the quantities pertaining to the plate and the soil

medium, respectively.

A relative stiffness parameter,

K,,

as defined by

Gorbunov-Posadov and Serebrjanyi,4 is used in this

study. This dimensionless parameter is defined as fol-

lows:

127r(l - u;) E, a * b

2

Kg= (1-u;) E, t, t,

00

(76)

The value of this parameter is a measure of the foun-

dation flexibility. A value of

K, = 0

represents a per-

fectly rigid plate, whereas K, = m represents theoret-

ically a perfectly flexible plate. But, as suggested by

Gorbunov-Posadov and SerebrjanyL4 for practical pur-

poses the plate can be treated as a rigid plate if K, 5

S/m,

where

a/b

represents the aspect ratio of the

plate. In this study, numerical examples for K, = 5

(2, 4000, 30, 30, 4.605, 0.2, 0),

K, =

10 (5, 3000, 30,

30, 5.4835, 0.2, 0), K, = 15 (2, 4000, 30, 30, 3.1935,

0.2, 0) and

K, =

20 (2, 4000, 30, 30,2.9015, 0.2,O) are

presented, where the seven elements in the parenthesis

represent the values E, (psi), Ep (psi), a (in.), b (in.),

t, (in.), up, and u,, respectively. In addition, the plate

deflection function, w(x, y), the contact stress distri-

bution function,

q(x, y),

and the flexural moments of

the plate, M, and My, along the X- and y-axis, respec-

tively, are nondimensionalized, as follows:

x&Y) =

ES

p4l -

uf)

w(x,Y)

4(x,

Y)

K&Y)

= -

P

MAX, Y)

Kc&Y) = ~

pa*

fi,kY) =

MJx, Y)

pa2

(77a)

Vb)

(77c)

G7d)

where all functions with an overbar denote nondimen-

sional variables.

Finite element model used

A three-dimensional eight-noded isoparametric ele-

ment with three transitional degrees of freedom per

node is used in this study. Due to the geometric sym-

metry of the problem, only one quarter part of the plate

is modeled by using the finite elements. The finite ele-

ment mesh selected is shown in Figure 5. As shown

in this figure, the effective zone of the soil medium

considered in the analysis is bounded by 5 times the

plate dimension in the z-direction (i.e., 5 x 2a) and

1.5 times the plate dimension in the x- and y-direction

(i.e., 1.5 x 2a), respectively. This boundary is con-

sistent with the values reported by Boussinesq and

Westergaad.28 The nodes on the planes of symmetry

and those on the planes defining the boundary of the

soil medium are constrained such that no translation

normal to these planes occurs. The finite element mesh

shown in Figure 5 contains 1512 elements, which in-

clude 72 three-dimensional plate elements. This finite

element model is analyzed by using the computer pro-

gram package SAP IV.

23The results obtained are used

to validate the two formulations presented in this pa-

per.

Results when the contact stress is approximated by

power series

A comparison of the nondimensional plate deflec-

tion obtained for

K, =

10 is shown in

Figure 6.

It is

observed that the deflection obtained by the method

presented in this paper does not change significantly

as the number of terms in the power series is increased

from 4 to 5. In fact, the two solutions obtained, as

shown in

Figure 6,

more or less overlap each other.

Also the present analysis predicted the displacement

at the plate center, which was 4.5% less compared to

the results obtained by the three-dimensional finite ele-

ment analysis. In

Figure 7

the nondimensional flexural

moments for a plate obtained with K, = 10 are com-

tt?

t

10a

1

(Not to scale)

Figure 5.

Finite element mesh used for the square plate prob-

lem



14/19


0.0

0.1 0.2. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Nomalized distance along x- and y-axis

Figure 6. Comparison of the plate deflection variation obtained

using the power series approximation and finite element method

(FEM) for Kg = 10

0.04

0.03

0.02

0.01

0.004 . . . . . .

0.0 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Normalized distance along x- and y-axis

Figure 7. Comparison of the moment variation obtained by

using the power series approximation and that reported in the

literature for Kg = 10

pared with the existing solutions. The results obtained

for the moment at the plate center by the method de-

veloped in this study is 29.7% higher than the value

reported by Gorbunov-Posadov2 and 12.2% higher than

the value reported by Zaman and Faruque.lh But the

boundary moments predicted by the method developed

in this study converge to zero, as they should, but those

reported by Zaman and Faruque did not do so. So the

method developed in this study predicts more correct

boundary values than those reported by Zaman and

Faruque.

Figure 8

shows the contact stress distribution

obtained by the power series approximation when

n = 4 and II = 5. In this figure the results are also

compared with those obtained by the three-dimen-

sional finite element analysis.

Results when the contact stress is approximated by

Chebychev polynomial

The variation of the plate deflection along the X- or

y-axis obtained by this approximation for the relative

stiffness,

K, = 5,

10, 15, and 20, respectively, as shown

in

Figures 9a-9d,

respectively. In these figures the

plate deflection obtained by the three-dimensional fi-

nite element analysis is also shown. It should be noted

in these figures that the variation of the plate deflection

obtained by the Chebychev polynomial approximation

for n = 4,5,6 more or less overlap each other, showing

that the deflection results predicted do not change sig-

nificantly as n is increased beyond 4. In Figure 9, com-

paring the results obtained for the plate deflection by

using the Chebychev polynomial approximation and

the finite element analysis, it is concluded that the two

results agree reasonably well, giving a maximum dif-

ference of 5.1% for K, = 5, 4.5% for K, = 10, 4.5%

for

K, =

15 and 4.25% for

K, =

20 between the two

solutions. The variance of the flexural moment for

K, = IO

along with X- or y-axis obtained by using the

Chebychev polynomial approximation is compared with

the results reported by Zaman and Faruque16 and Gor-

bunov-Posadov2 in

Figure IO.

As shown in this figure,

the results obtained by the Chebychev approximation

for

n =

4,5,6 more or less overlap each other, showing

that flexural moments predicted do not change signif-

icantly as

n

is increased beyond 4. It can be seen from

Figure 10

that moment results obtained at the plate

center by using the Chebychev polynomial approxi-

mation are 28.1% higher than the value reported by

Gorbunov-Posadov2 and 10.8% higher than the value

reported by Zaman and Faruque. But the moment

variation obtained by using the Chebychev polyno-

mials converges to a zero value at the boundary, whereas

0.0

F.E.M.

0.0 0.1 0.2 0.3 0.4 0.5 0.8 0.7 0.6 0.9 1 C


Figure 8. Contact stress distribution obtained by using the power

series approximation and FEM for Kg = 10



15/19

Energy analysis of rectangular plate on elastic half space: A. R. Kukreti and M.-G. Ko

o.~~.

1.4

11,

1.0

0.8

0.0

0.4

0.2

Normalized distance along x-

and y-axis

2.4r

2.2 C

FEM

2.0 . / _

I.0

7 / 7

n-4

1.6

n-5 n-6

1.4,

(5

1.0

0.0

0.6,

0.4

0.2,

0.0 .

. ,

0.0 0.1

0.2

0.3 0.4

0.9 0.9 0.7 0.9

0.9


and y-axis

o.oC

.

0.0 0.1

0.2 0.3 0.4 0.5 0:q 0.7

0.1 0.9 1.0


and y-axis

2.4

2.2 d

FE.hi

2.0 I

I

1.8

1 7 7

1.0,

s-4

n-5

r-6

1.4

1.2

1.0

0.8,

0.6

0.4

0.2

0.01 .

. . . .

0.0 0.1

0.2. 0.3 0.4 03 0.9 0.7

0.8 0.0 1


and y-axis

Figure 9. Plate deflection obtained by using the Chebyshev polynomial and FEM for KS = (a), 5, (b) 10, (c) 15, and (d) 20

0.041

\\ w

I

\\ \

0.05

0.02

0.01

0.004 . .

.

0.0 0.1 0.2

0.3 0.4 0.5 0.9 0.7 0.0

0.9


Figure 10. Comparison of the moment variation obtained by

using the Chebyshev polynomial and that reported in the liter-

ature for Kg = 10

352 Appl. Math.

Modelling, 1992, Vol. 16,

July

it did not do so in the results reported by Zaman and

Faruque. In Figures

l la-lld

the contact stress dis-

tribution obtained by using the Chebychev approxi-

mation for

K, = 5,

10, 15, and 20, respectively, is

presented along with the results obtained by the three-

dimensional finite element analysis. From these figures

it can be seen that for each K, value the variation of

the contact stress predicted near the center of the plate

by using the Chebychev polynomials does not change

significantly as n is increased, whereas the variation

of contact stress gradient near the boundary increases

rapidly as n is increased, which implies that as the

number of polynomial terms is increased, better pre-

diction for the contact stress distribution can be ex-

pected.

Effect of relative stiffness parameter on the results

predicted

Figure 12 shows the variation of the plate deflection,

obtained by using the Chebychev polynomial approx-

imation for K, = 5, 10, 15, and 20, respectively. In


16/19

Energy analysis of rectangul ar plate o n elastic half space: A . R. Kuk reti and M.-G. Ko

5.0

a

4.5

3.5

5.0.

2.5'

2.0'

1.s'

1.0

0.3

0.04

. . . . . . .

a.0 0.1

0.2 ?I 0.4 0.5

0.8 0.7 .8 0.9


and y-axis

5.0

TC

I

4.5

4.0

cl

cfl

3.0

c,

0

2.5

s

2.0'

0

5 '='

Q)

.: 1.0'

B-l,

I 0.5,

ii 0.01

0.0 .1

0.2 0.3 0.4 0.5 0.8 0.7 0.6 0.0 .0


and y-axis

5.0

b

4.5'

n

3.0'

23'

2.0'

1.5

F.E.M.

1.0

0.5

a.04

. . . . . . . .

J

0.0

0.1 0.2

OJ 0.4 0.5

0.6 0.7

0.8 0.9 .0


and y-axis

5.0

d

o.oi

. . . . . . . .-.-

0.0

0.1 0.2 0.3 0.4 0.s 0.6 0.7 0.1 0.9 1

Normalized distance along

X-

and y-axis

Figure 11. Comparison of the contact stress distribution obtained by using the Chebychev polynomial at FEM for Kg = (a) 5, (6) 10,

(c) 15, and (d) 20

0.d .

. . . . . . . . I

0.0 0.1

0.2 0.3 0.4 0.5 0.6 0. 7 0.8 0.0 1. 0


0.0 0.1 0.2 0.3 0. 4 0. 5 0.6 0.7 0.1 0.9 1


Figure 12. Influence of the relative rigidity on the nondimen-

Figure 13.

sional plate deflection

Influence of the relative rigidity on the nondimen-

sional plate flexural moment



17/19



Figure 14. Influence of the relative rigidity on the nondimen-

sional contact stress

this figure it can be seen that as the plate becomes

stiffer, in other words, as the relative stiffness of plate

K , decreases, the deflection at the plate center de-

creases. Figure 13 shows the variation of the flexural

moment of the plate alongx- ory-axis obtained by using

the Chebychev polynomial approximation for

K, = 5,

10, 15, and 20. In this figure it can be seen that as the

plate becomes stiffer the flexural moment of the plate

around the plate center increases. F&JUYP 4 shows the

variation of the contact stress at the plate-soil interface

obtained by using the Chebychev polynomial approx-

imation for

K, = 5,

10, 15, and 20. In this figure it can

be seen that as the plate becomes more flexible, in

other

words, as the relative stiffness of the plate

K,

increases, the contact stress around the center of the

plate also increases.

Conclusions

An analytical procedure is presented to predict the

flexural behavior of a rectangular plate resting on an

elastic half space. The procedure accounts for the in-

teraction between the plate foundation and the soil

medium, and in this paper, only the case in which the

whole plate is subjected to a uniformly distributed load

is presented. The method can be extended to include

other types of loadings in the expression for Ul given

in (60). The analytical formulation is based on the prin-

ciple of minimum potential energy. The contact stress

distribution function is approximated in the following

two ways: by a power series expansion and by Che-

bychev polynomials. In both the approximations the

basic variables appearing in each term are the spatial

coordinates, x and y, describing the geometry of the

middle plane of the plate, and the coefficients asso-

ciated with each term are called the generalized co-

ordinates. By satisfying the force-related boundary

conditions (or natural boundary conditions) at the free

edges of the plate, some of the generalized coordinates

are expressed in terms of the other coefficients. The

remaining unknown coefficients are called the inde-

pendent generalized coordinates. The minimization of

the total potential energy functional with respect to the

independent generalized coordinates results in a set of

linear simultaneous equations, which are solved to de-

termine the values of the unknown generalized coor-

dinates. These values are then substituted back into

the plate deflection function and the contact stress dis-

tribution function, and finally, the flexural moments in

the plate are computed.

Thus in this paper an analytical method has been

presented to analyze the rectangular plate-soil foun-

dation interaction problem, in which the natural bound-

ary conditions at the plate edges are satisfied exactly.

Satisfaction of these boundary conditions leads to a

more correct prediction of the contact stress distri-

bution between the rectangular plate and soil foun-

dation, which is largest near the plate edges. The en-

ergy method presented by Zaman and Faruquej fails

to do this, and so do all the finite element applications.

If the problem is solved using finite element method,

there are other limitations also. As pointed out by

Cheung and Zienkiewicz and Svec and Gladwell,*

use of plate-bending elements requires the contact stress

variation around each nodal point to be approximated,

which may not be realistic. If a similar type of finite

element is used to discretize the plate and soil medium,

as done in this paper by using three-dimensional finite

elements for both, then this assumption for the contact

stress variation is not needed. But in such a case the

bending moment and shear forces created in the plate

(or foundation slab), which are needed for design, can-

not be determined directly. If the nodal displacement

values obtained from the finite element analysis are

used,

the second derivatives (or curvature) and third

derivatives have to be first evaluated by using some

finite difference formulas and then substituted into the

standard expressions relating bending moments and

shears to these derivatives, using the thin bending plate

theory. This approach not only is cumbersome, but

also may give more error in the results and may require

a very fine finite element mesh to reduce this error to

acceptable limits. In the method presented in the pa-

per, once the unknown coefficients (B,) in the plate

deflection function, w(x, y), have been computed, ex-

plicit analytical expressions can be written for varia-

tion of bending moments (M,, M,,, and Mxy) and shear

forces

(V,

and

V,)

for the complete plate domain. The

method presented in this paper is easy to automate by

using the expressions presented and does not require

extensive preparation of input data, as is typically needed

for a finite element analysis. The method presented

would require the geometric dimensions of the plate

(a, 6, f,,), distributed load intensity (p), properties of

the plate and soil foundation material (E,,

EP, u,,

and

u,), and the number of terms to be considered in the

approximate function to be the input data. No boun-

daries of the soil medium are needed to be considered,

as would be required for a finite element analysis.

In comparing the numerical results obtained with



18/19

Energy analysis of rectangular plate on elastic half space: A. R. Kuk reti and M.-G. Ko

those obtained by finite element method we conclude

that the Chebychev polynomial approximation predicts

better results, especially for the contact stress distri-

bution, than the simple power series approximation.

Both methods predicted more conservative plate mo-

ments than those obtained by the methods suggested

by Gorbunov-Posadov and Zaman and Faruque. I6 Also,

both methods modelled the boundary conditions ac-

curately. Convergent solutions for plate deflections were

obtained by both methods. However, one has to be

careful that for a particular choice of n the matrix [X]

does not become ill conditioned, which could be pos-

sible if a large value of n is needed to obtain a con-

vergent solution. As expected, it was found that as the

plate becomes more flexible relative to the soil me-

dium, then the contact stress around the center of the

plate increases, indicating that the soil-structure inter-

action will be of importance.

References

I

2

3

4

5

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Appendix

After evaluating the integrations in (lOa)-(10h) the

expressions obtained for (Iij),,,, where m =

8 are, respectively, as follOws:

1,2, . . .

)

2i 2j

@)* = 2 2

C,,n*-sy*j~r(h;)s+r+l(_

s=or=o

2; 2j

( ), = c c CijsrX*i-sy2~r(h;)s+r+ ( -

.,=Ot=O

(Al)

,*I:

W)

,.TI;

(A3)

2 i

* . i

(Q, = 2

x CiJstX*;-SY*j~t(h~)s+r+

( - ,).s+l;

.Y=Ol=O

(A4)

2;

2.i

(I,), = C x CijslXzi-SYZi~r(h;)\+r+ 1-I

(As)

.y=O,=O

2i

*j

(& =

c. 2 C~.,tX*;-sy*j~ h~)J+r+ -

1)z:

.r=Of=O

646)



19/19

Energy analysis of rectangu lar plate on elastic half space: A . R. Kuk reti and M.-G. K o

2 i

2j

(I,), = c. 2 C,,X*-sy2j- h;)s+t+ - l)Zi

where

s=Ot=O

u,

b47)

c= o &

I

du for i= 1,2,...,8

b49)

2 i

2 j

analysis of rectangular plate resting on an elastic.pdf

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