Analysis of circular plate-elastic half-space interaction using an energy approach

Analysis of circular plateelastic half-space interaction using an energy approach M. M. Zaman,* A. R. Kukreti,t and A. Issa¢ School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, OK 73019, USA (Received November 1986; revised August 1987)

An analytical formulation is developed, based on an energy approach, to predict the flexural behavior of uniformly loaded thin flexible circular plates resting in smooth and continuous contact with an isotropic elastic halfspace. In this development, the deflected shape of the plate is aplsroximated by an even power series expansion in terms of the radial coordinate. Any number of terms in the series can be considered. The coefficients associated with the series are evaluated by making use of the principle of minimum potential energy. Analytical expressions are derived for the contact stress distribution, the plate deflection, and the plate radial moment. The results obtained from the proposed procedure compare very well with the existing solutions of similar problems.

Keywords: energy approach, flexibility, circular plates Introduction The flexural behavior of uniformly loaded circular plates, resting on an isotropic elastic half-space, has been investigated extensively, l-s'j°-~2 The problem has received significant attention due to wide application of circular plates as foundations of various structures. An excellent review of various investigations in this area is given by Selvadurai l° and Issa. 9 Seivadurai t° was the first to study this problem by an energy method. In this study, ~° the deflected shape of the plate was approximated in the form of a power series in terms of a radial coordinate r. Of the four assumed coefficients in the series, two were eliminated by invoking the Poisson-Kirchhoff boundary conditions applicable to the free edge of the plate, and the remaining two were evaluated by the minimization of the total potential energy of the system. Selvadurai ~° obtained an analytical expression for plate deflection and concluded that this method yields satisfactory results for a wide range of circular foundation plates of practical interest. * Assistant professor. t Associate professor. ~: Former graduate research assistant.

© 1988 Butterworth Publishers

Unfortunately, although the foundation deflection was predicted with good accuracy, some difficulties were reported regarding the prediction of flexural moments. It was observed that the flexural moments, when evaluated as derivatives of deflection, can be subjected to substantial error. To avoid this problem, Selvadurai ~° determined the flexural moments by a superposition method. This method is only applicable for obtaining flexural moments at the center of the plate. Although this method for analyzing flexural response of circular foundations made a significant contribution to the stateof-the-art, the aforementioned limitation of the method is undesirable, because in the actual design of foundation plates not only the central moments but also the variation of flexural moments in the radial and transverse directions are often required. The main objective of this paper is to present an analytical procedure, based on an energy approach, for analyzing the flexural behavior of circular foundations subjected to a distributed load of uniform intensity. Analytical expressions are obtained for plate deflections, flexural moments, and for contact stress distributions. The proposed method can be considered essentially as an extension of Selvadurai's I° method. However, it is more general because it can account for any desired number of terms from the power series

Appl. Math. Modelling, 1988, Vol. 12, J u n e

285

Circular plate-elastic half-space interaction: M. M. Zarnan, A. Ft. Kukreti, and A. Issa

l~ I-

iPT

2a

uating appropriate derivatives. The proposed formulation is discussed briefly in this section, and some related mathematical expressions are given in the Appendix.

J

-I

0

Isotropicelastic ~fspace

r

Plate deflection function The plate deflection function w(r) is approximated by the power series expansion

~

Figure I Uniformly loaded circular plate resting on an isotropic elastic half-space

expansion to represent the plate deflection; as such, the foundation deflections as well as the variation of flexural moments can be predicted with desired accuracy. Compared to other numerical techniques, such as the finite element method, the proposed method is computationally efficient and inexpensive.

n

[r\2i

w(r) = a E__oA,ta }

(I)

where n is an integer and the A~'s are the generalized coordinates or unknown coefficients. Invoking the Poisson-Kirchhoff boundary conditions ~°'~3 applicable to the free edge of the plate lets us express any two selected coefficients (A;) in terms of the remaining coefficients. In this study, the coefficients A, and A,_ ~ are rewritten in terms of the other coefficients by the following equations: F(3, n - I)A,,_ I + F(3, n)A,,

n- 2

Proposed analysis procedure Figure 1 shows a uniformly loaded circular plate resting on an isotropic elastic half-space. The uniform load, of intensity P0, is considered to be distributed over the entire surface of the plate. The circular plate, of thickness tp and radius a, is assumed to be thin and flexible. The elastic properties of the half-space medium are described by Young's modulus Es and Poisson's ratio vs. The interface between the plate and the half-space is assumed to be smooth and continuous. This assumption eliminates the frictional forces between the plate and the half-space, which means that only the vertical component of the contact stress is present. It also implies that the interface displacement can be uniquely represented as either the plate deflection or the half-space surface deformation. With these assumptions, the energy method can be employed to obtain analytical expressions for the plate deflections, the flexural moments, and the contact stress distribution at the plate-half-space interface. To accomplish this task, we assume the plate deflection is in the form of an even power series expansion in terms of the nondimensional radial coordinate r/a. The coefficients associated with the series are called the generalized coordinates. The assumed deflection function is then modified by invoking the appropriate PoissonKirchhoff boundary conditions, t~.~3 Using this modified deflection function, we evaluate the total potential energy functional as the sum of (i) the strain energy of the half-space medium, (ii) the strain energy of the circular foundation plate, and (iii) the work done by the external loading: The minimization of the total potential energy functional with respect to the generalized coordinates yields a set of simultaneous linear equations. Solving these equations yields values for the generalized coordinates. Finally, these values are substituted in the assumed deflection function, and the expressions for flexural moments are obtained by eval286

Appl. Math. Modelling, 1988, Vol. 12, June

= - ~ A ~ F ( 3 , i)

i=0

(2a)

F(4, n - 1)A,,_ i + F(4, n)A,

n-2 = - ~A~F(4,i)

i=0

(2b)

where F(3,i) = iz(i - I)

(3a)

F(4,i) = i(2i - i + vo)

(3b)

F(3,0) = F(4,0) = 0

(3c)

and ~,p denotes Poisson's ratio of the plate material. Using equations (2a) and (2b), we obtain the modified deflection function for the plate

n-2 w(r)= a{ i~0 a,.[ ( r ) 2' t

+

/r\2(,-k)] ]

)

jj,

(4)

in which A(k,i) is related to Ai, F(3,i), and F(4,i), as discussed in the Appendix.

Contact stress distribution Assuming that the interface between the plate and the half-space is smooth, Sneddon ~ showed that the distribution of the normal contact stress q(r) at the interface can be uniquely obtained from the classical theory of elasticity. Considering an isotropic elastic half-space subjected at its surface to the axisymmetrical vertical displacement w(r) for 0 -< r -< a, and using the integral transforms, one can show that q(r) is given by

t/

q(r) -

2(1

for0-
l d f th(t)dt ~,2) r dr ~ t 2 --- ~-

r

(5)

Circular plate-elastic half-space interaction: M. M. Zaman, A. R. Kukreti, and A. Issa where t

2 d f rw(r) dr h(t) = -~-~t J ~/t 2 _ r2

for0
0

(6)

Substituting e q u a t i o n (4) into e q u a t i o n s (5) and (6) and e v a l u a t i n g the required integrals give E~ 1 fn-2 F n q(P) = " r r ( 1 - v ~ ) l - p 2 ~ i ~ o a i L ~ = o ~ U ( i ) P Z U J

-11 ./

w h e r e p and g,,(i) are d i m e n s i o n l e s s quantities defined by p =

[ 2 ( i - u)]!!

'

+ kZ= O s(u,n -

'ff

Us = ~

rq(r) w(r) dr dO

(10)

Substituting the e x p r e s s i o n s for q(r) and w(r) f r o m equations (7) and (4), r e s p e c t i v e l y , and p e r f o r m i n g the required integrations g i v e

E~a 3

f n-2n-2

)

(ll)

where

[(2i)!!]2 [ ( 2 i - 1)!!] 2

k)A(k,i)[2(__n-_ k - u )

- 3],! [ 2 ( n - k ---~111

(n - 2) a n d u

I

~,,(i) = ~ , s(u,n - k)A(k,i)

= 0,I,2 .....

[2(n - k -

(8c)

(n - 2) a n d ( i + 1 ) - < u - < ( n t

[2(l - k) - 3] !!

k=o

[2(•- k)]![

[[2(n - k)] !!]2

-

n

~o(0) = 1

(8e)

xj for o d d j and j - > 0

(8g)

for e v e n j a n d j - > 0

(8h)

for j-< 0

2~r

(12)

a

2(1

j!!=2x4x6x---xj j!! = 1

(n - 2)

Up = -'~f o DfoP {[V2w(r)]2

(8f)

I x3 x5x--,

[2(U + n -- k) + 1]!!

Strain energy due to foundation plate bending (Up). T h e strain e n e r g y stored in the f o u n d a t i o n plate due to b e n d i n g is jt.~3

l)

(n - 2) and 1 = O,l

for u = 1,2 . . . . .

k=O

-

(8d)

[[2(n - k) - 1]!!] 2 for i = 1,2 . . . . .

[2(u + n - k)]!!

u) - 3]!!

~(n -- k ~ ~ ] ~

~.-t(i) = ~ s(l,k)A(k,i)

g.(0) = 0

[2(u + j ) + 1]!!

f o r i , j = 0,1,2 . . . . .

[[2(n - k)]!!] 2 [[2(n - k) - 1]!!] 2

x

[2(u + j ) ] [ !

+ ~ x(k,j')

i

k=O

f o r i = 1,2 . . . . .

,,=o

(8b)

[[2(n - k) - 1]!!] 2

f o r i = 1,2 . . . . .

x,(i,j3 = ~ s~.(i)

""

[[2(n - k)]!!] 2

x

j!!=

a

(8a)

~u(i) = s(u,i) [2(i - u) - 3]!!

x

2'tr

0 0

(7)

r a

Strain energy stored in the soil medium (U~). T h e strain e n e r g y stored in the soil m e d i u m is defined here as the w o r k d o n e by the c o n t a c t s t r e s s e s at the platesoil interface, and is e v a l u a t e d f r o m the e x p r e s s i o n

- /Yp)dw(r) r

d2w(r)]

~

drdO

~ f r

(13)

where d2w(r) 1 dw(r) V2w(r) = ~ + - ~ dr 2 r dr

(14)

and D o is the flexural rigidity o f the plate.~°'t3 In view o f e q u a t i o n (4), Up can be e x p r e s s e d as

Up = 7rDp

AiAjx2(i, j)

L i=0

(15)

(8i) where

x2( i,J)

Total potential energy functional T h e total potential e n e r g y functional U o f the p l a t e half-space s y s t e m is U=

U s + U p + Uv

8f2i2j 2 - (1 - % ) i j ( 2 j - 1)

+ fsI

+ ~ ;~(k,i) t=o

[- 2j2(n _ k)2 /L i + n - k - 1

(9)

w h e r e U~ = strain e n e r g y stored in the soil m e d i u m Up = strain e n e r g y due to the plate bending Uv = w o r k d o n e by the external load in the vertical direction

- (1 -- Vp)j(n - k ) ] 1

+ ~o,=%

2(2n

-

k -

/ -

1)

Appl. Math. Modelling, 1988, Vol. 12, June

287

Circular plate-elastic half-space interaction: M. M. Zaman, A. R. Kukreti, and A. Issa •[2(n - k)2(n - I) 2 (1 - Up)(n - k)(n - l)

-

(2n-21-

I)]} .1

for i,j = 0,1,2 . . . . .

(n - 2 )

(16)

Work done by the external load ( UD. The work done by the external (vertical) load of uniform intensity P0 is U~ = -

Pow(r) r dr dO

(17)

Substituting equation (4) into equation (17) and performing the integration, we get .,t - 2

(18)

Uv = -"rrPoa 3 ~_a Aix3(i)

w(p) and the contact stress distribution q(p) can be computed from equations (4) and (7), respectively, where p is the nondimensionai length (p = r/a). For convenience in presenting numerical results, the plate deflection is expressed in a nondimensional form ~(p) as follows: -fF'(P) - Poa( ! - ~) w(p)

(23)

Evaluation o f flexural moments The flexural moments M,(r) and Mo(r) are computed from the expressions

_ rd2.,(, -) dw(r)] M,.(r) = -D,.L~5--r2 + ," dr J n El dw(r) M~,(,') = - O o L ; d, +

(24)

d2w(r)-I

J

(25)

i=0

where

I

~

A(k,i)

.= n - k +

x3(i)= i + 1

1

for i = 0,1,2 . . . . .

(n - 2)

(19)

M,.(p) = ~

Determination o f the unknown coefficients According to the principle of minimum potential energy, the generalized coordinates should be chosen such that the value of the total potential energy functional U is a relative minimum. This is expressed in a mathematical form as 0U -- = 0 c)A;

for i = 0,1,2 . . . . .

(n - 2)

(20)

In view of equations (9), (11), (15), and (18), equation (20) can be expressed as

OA.

i=0

j=O

u

• [ Esa3 L1 -

v2xI(i,J) + 7rDpx2(i,j)]

n--2~

- 7rPoa ~ ~;~

(Ai)x3(i)

(21)

i=0

Noting that a ( A y a A , , = a,.,, and a(AAyOA, = A~Sj,, + Aft,.,, where 6u is the K r o n e c k e r delta, we can write equation (21) in the form

n-2 A f Esa 3 . = ~. i t 1_-~--~; D('(t'u) + X,(u.i)] dAu i=o OU

+ rrDpDo(i,u) + Xz(u,i)] I 3 - 1rPoa3X3(u) = 0

(22)

Equation (22) represents a set of linear simultaneous equations which can be solved to obtain the A,-'s. Once the values of the A;'s are obtained, the plate deflection

288

Substituting equation (4) into equations (24) and (25) and performing the required differentiations, we obtain the expressions for the flexural moments Mr(p) and Mo(p). For convenience in presenting numerical results, the flexural moments are expressed in nondimensional form as

Appl. Math. Modelling, 1988, Vol. 12, June

to-

Mo(p) = ~

Mr(p)

(26a)

Mo(p)

(26b)

I

The expressions for Mr(p) and Mo(p) are given in the Appendix. Further, for convenience, a relative rigidity factor R o is defined 9"~° as Rp - 6(I

3 6 \ E( ~J) v~)Kp = ~{EP']

(27)

From the numerical results presented in the following section, it can be clearly seen that equations (24) and (25) can accurately predict the flexural moments, provided a sufficient number of terms are retained in the expression of w(r) (equation (4)).

Numerical

results

To demonstrate the efficiency of the proposed formulation, plates with various relative rigidities (K~, see equation (27)) were analyzed. Typical results are presented in this section. Three values of K o are considered here: K 0 = 0.1 for a relatively flexible plate, Kp = I for a plate with intermediate rigidity, and K o = 10 for a plate with high rigidity. For most practical cases, the material used for the foundation plate is either concrete or steel, for which Poisson's ratio vr can be considered to be 0.15 to 0.30. Only these two limiting values are selected here for vo. The analyses were performed for values o f n varying from 3 to 20, where n is the number of terms considered in the series expansion of the plate deflection function w(r) given by equation (4). It was observed that the

Circular plate-elastic half-space interaction: M. M. Zaman, A. R. KukretL and A. Issa 1.5 -

- -

Brown

------

n>~12

.......

n=3

(1969)

-

1.5-

------

n>~12

i

. . . . . . . . .

0.2

i

. . . . . . .

0.4

i

. . . . . . . .

0.6

0.5

r . . . . . . . .

0.8

1.0

. . . . . . . . .

0.0

i

. . . . . . . . .

I

0.2

. . . . . . . .

/

Contact stress distribution Kp = 0.1 a n d Vp = 0 . 3 0

i

0.4

. . . . . . . . .

0.6

P

Figure 2

(1969)

1.0

0 . 5 ' . . . . . . . . .

Brown

....... n=3

1.o i Z .......

0.0

-

i

. . . . . . . .

0.8

i

1.0

P versus radial distance for

computing cost increases significantly with increasing n. As for the problems considered here, reasonably good results were obtained for n <- 20, so higher values were not tried. For n = 3, the results obtained from the present study identically matched those obtained by Selvadurai. *m In the following discussion, an "exact solution" is defined as the lowest number of terms in w(r) for which any additional terms do not significantly (less than 0. I%) improve the results.

Figure 3

Contact stress distribution K n = 1 a n d vp = 0 . 3 0

- -

Brown

------

n>-12

1.5-

. . . . . . .

=

0.5

. . . . . . . . .

i

. . . . . . . . .

0.2

0.0

i

.

.

.

.

.

.

i

0.4

. . . . . . . . .

0.6

i

. . . . . . . . .

0.8

i

1.0

P

Figure 4

Contact stress distribution

versus radial distance for

Kp = 10 a n d v n = 0 . 3 0

2.0-

. . . . . .

4 .

-~..~.

-----

n >.12

-......

n=3

1.8-

"',., ..\

Foundation p/ate deflection. The deflection function for the foundation plate is defined by equation (4). For Kp = 0.1 the nondimensional plate deflections are presented in Figure 5. As in the case of contact stress distribution, here also the series expansion in equation (4) converged for n = 12. Even for n = 3, the predicted deflection values were close to the exact solution. This

(1969)

1.0 ~

Contact stress distribution. For Kp = 0.1, I, and I0, the (normal) contact stress distribution (see equation (7)) at the foundation plate-half-space interface is shown in Figures 2-4, respectively. It can be seen that for n -> 12, the proposed analytical formulation and the work reported by Brown 2 give very close results, the difference being less than I% at the plate center (i.e., at p = 0). For lower values of n (n = 3), the quality of prediction becomes poor, especially for plates with low relative rigidity. This indicates that for relatively rigid plates (Kp -> 1) only a few terms in the series expansion of equation (4) suffice to adequately describe the distribution of contact stresses, whereas for relatively flexible plates (Kp -< 0.1), more terms (n -> 12) are needed. Figure 4 also shows that for very flexible plates (Kp -< 0.1), qo assumes a value of unity, whereas, for very rigid plates (Kp -> 10), q0 becomes 0.5. These results agree with the classical solution of the theory of elasticity.~a.~3

versus radial distance for

",\

1.6

',N. "N\

",,.

1.4

0.0

0.2

0.4

0.6

0.8

1.0

P * An error ~4 was detected in the expression o f A3 given by Selvadurai, m The term (1188/5) A~ should read (648/5) A.~.

Figure 5

Plate deflection Kp = 0.1 a n d Vp = 0 . 3 0

function

versus

radial distance

Appl. Math. Modelling, 1988, Vol. 12, June

for

289

Circular plate-elastic half-space interaction: M. 114.Zaman, A. R. KukretL and A. Issa - -

2.0--

V = 0.30 P V p = 0.15

(1969)

Brown

n;~15

0.10-

.....

n=3

0.08 1.8

o.o6 ~

W'o

~

0.04

~.~..-..

1.6 0.02 0.00 .... 0.0

1.4 .........

I ......... -1

-2

I ......... 0

I ......... 1

I

I

.....

0.2

I . . . . . .

6

Plate central

deflection

versus

"

"1 . . . . . . .

I

0.8

1.0

P Plate radial moment 10 a n d up = 0 . 3 0

Figure

n=

"

0.6

2

l o g 10 K p

Figure

I

0.4

rigidity

relative

for

9

v e r s u s r a d i a l d i s t a n c e f o r Kp =

12

Brown

0.10 -

(1969)

- -

V p = 0.30

....

V

P

= 0.15

n~>15 0.10

.....

n=3

0.09

Mo

0.06

0.05 -

M

0.04 0.02

""

0.00 ......... 0.0

I ........

0.2

I ........

0.4

I. . . .

0.6

I

.....

0.8

0.00-"

I 1.0

......... -2

P Plate radial moment 0.1 a n d vp = 0 . 3 0 Figure

7

I ......... -1

I ......... 0

I ......... 1

I

2

log 10 K p

v e r s u s r a d i a l d i s t a n c e f o r Kp = Figure

10

Place

central moment

versus

plate relative rigidity

f o r n = 15

0.10 -

- -

Brown

---

n>~15

.....

n=3

(1969)

parameter Kp is shown in Figure 6. Note that the plate deflection is not significantly affected by vp and that for an infinitely rigid plate (Kp ~ ~c) ii;o approaches 1.57 (i.e., 7r/2), but for very flexible plates (Kp---> 0) ~o approaches 2.00. These results agree perfectly with the classical theory of elasticity.~.~3

0.08 0.06 • 0.04 0.02 -

~

~

0.00 I .......

0.0

I

.....

0.2

I

I

0.4

........

0.6

I ......... 0.8

I

1.0

P

Figure

1 and

Plate radial moment vp = 0 . 3 0 8

v e r s u s r a d i a l d i s t a n c e f o r Kp =

indicates that only a few terms (n = 3) in the series expansion of equation (4) suffice to adequately describe the foundation plate deflection. For n = 12 and vp = 0.15 and 0.30, the variation of central plate deflection ii;o with the relative rigidity

290

Appl. Math. Modelling, 1988, Vol. 12, June

Plate radial moment. The radial moment of the foundation plate is defined by equation (All). For Kn = 0.1, I, and 10, and =,p = 0.3, the variation of radial moment is shown in Figures 7-9, respectively. It is observed that, as far as the radial moment variation is concerned, the series expansion in equation (4) is convergent for n = 15. Also, for n = 15, the predicted radial moment values compare favorably with those of Brown 2 for a wide range of relative rigidity (0.1 <-Kp <-- 10). For n = 3, on the other hand, the predicted flexural moment values show substantial error for both flexible and rigid plates. For two limiting values of ~'r (0.15 and 0.30) and different values of Kp, the variation of the radial moment at the plate center (Mo) is presented in Figure 10. As may be expected, ii;o ---> 0 for infinitely flexible

Circular plate-elastic ha~f-space interaction: M. 114.Zarnan, A. R. Kukreti, and A. Issa plates (Kp ~ 0). It is also seen that for very_flexible and very rigid (Kp >- 10) plates, Mo is affected very little (less than 0.1%) by a change in Kp. Further, for an infinitely rigid plate (Kp ~ oo), w0 0.0801 and 0.0734, for up = 0.15 and 0.30, respectively. These results show excellent agreement with those obtained from the classical theory of elasticity.ll'13

(K o <- 0. I)

formly distributed anti-symmetrical load. Dokl. Akad. Nauk. 9

10 II

Conclusions An analytical formulation was developed to predict the flexural behavior of uniformly loaded circular plates resting in smooth contact with an isotropic elastic halfspace. The formulation is based on an energy approach and is applicable for plates with a wide range of relative rigidity. The deflected shape of the foundation plate is approximated by an even power series expansion in terms of the radial coordinate. The coefficients associated with the series are evaluated by the principle of minimum potential energy. The formulation is quite general, because any number of terms can be included in the power series expansion. For flexible plates, more terms are required in the series to adequately describe its flexural behavior. Analytical expressions are derived for plate deflection and radial moment as well as for contact stress distribution. Parametric studies are performed to examine the effects of plate relative rigidity and Poisson's ratio of the plate material on its flexural behavior. For a relatively rigid plate, the deflection is mainly a rigid-body displacement; consequently, the first term in the series is predominant, and only a few terms are necessary to describe the plate behavior. Accurate prediction of flexural moments, on the other hand, requires more terms in the assumed plate deflection function, as compared with the prediction of plate deflections and contact stress distributions.

12 13 14 15

U.S.S.R. 1947, 4, 129-132 Issa, A. Analysis of cylindrical storage tanks and circular plate foundations resting on isotropic elastic halfspace using an energy method. M.S. thesis, University of Oklahoma, Norman, Okla., 1985 Selvadurai, A. P. S. The interaction between a uniformly loaded circular plate and an isotropic elastic half-space: a variational approach. J. Struct. Mech. 1979, 7, 231-246 Selvadurai, A. P. S. Elastic analysis of soil-foundation interaction. Developments in Geotechnical Engineering, Vol. 17. Elsevier, New York, 1979 Smith, 1. M. A finite element approach to elastic soil-structure interaction. Can. Geotech. J. 1970, 7, 95-105 Timoshenko, S. P. and Woinowsky-Krieger, S. Theory. of Plates and Shells, 2nd ed. McGraw-Hill, New York, 1959 Issa, A. and Zaman, M, M. A cylindrical tank-foundation-halfspace interaction using an energy approach. Comput. Meth. Appl. Mech. Engng. 1986, 56, 47-60. Zemochkin, B. N. Analysis of Circular Plates on Elastic Foundation. Mosk. Izd. Voenno. lnzh. Akad., Moscow, 1939

Appendix

Expressions for A(k,i) Equations (2a) and (2b) can be expressed in matrix form as follows:

[F]{AF} = {BF} where F(3, n - 1) [F] = LF(4, n I) A,,_ l

{AF}={A,,

References

2 3 4 5

6 7 8

(A2) (A3)

}

~, A,FC3, i: i=O

{BF} = Borowicka, H. Influence of rigidity of a circular foundation slab on the distribution of pressure over the contact surface. Proceedings of the First International Conference on Soil Mechanics Found. Engineering, vol. 2, 1936, 144-149 Brown, P. T. Numerical analyses of uniformly loaded circular rafts on deep elastic foundations. Geotechnique 1969, 19, 301306 Chakravorty, A. K. and Ghosh, A. Finite difference solution for circular plates on elastic foundations. Int. J. Num. Meth. Engng. 1975, 9, 73-84 Cheung, Y. K. and Zienkiewicz, O. C. Plates and tanks on elastic foundations--an application of finite element method. Int. J. Solids Struct. 1965, 1,451-461 Faruque, M. O. and Zaman, M. M. Approximate analyses of uniformly loaded circular plates on isotropic elastic halfspace. Proceedings of the Ninth Congress of the National Academy of Engineering of Mexico, Leon, Mexico, 271-276 Habel, A. Die auf dem elastisch--isotropen halbraum auf ruhende zentral symmetrisch belastete elastische kreisplatte. Bauingenieur 1937, 18, 188-193 Holmberg, A. Cirkulara plattor med jamnt fordelad last pa elastiskt underlag. Betong 1946, 107 Ishkova, A. G. Exact solution of the problem of a circular plate in bending on the elastic halfspace under the action of a uni-

F(3, n)] F(4, n)

n-2 -

I

(AI)

(A4)

n - 3

- ~ AIF(4, i) i=0

Equation (AI) can be used to solve for

{AF} =

[F]-'

{AF},

giving

{BF}

(A5)

For convenience in writing, let the inverse of [F] be denoted by [f(i,j)]; i.e., [ F ] - ' = [f(i,j)]. Now, in view of equations (A3) and (A4), equation (A5) can be written as n -- 3

aih(k,i)

(A6)

i=0

where 2

A(k,i) = - ~ f(2 - k, l ) F ( I + 2,i)

(A7)

/=1

in which k = 0 and 1. In view of equation (3c), ,~(k,0) = 0

for k = 0 and 1

Appl. Math. Modelling, 1988, Vol. 12, June

(A8)

291

Circular plate-elastic half-space interaction: M. M. Zaman, A. R. Kukreti, and A. Issa

Evaluation offlexural moments

~ 2(n - k) (2n - 2k - 1) + 2~ a2 k=o

In o r d e r to obtain the e x p r e s s i o n s for the flexural m o m e n t s , we need the following derivatives: d---r

i=o

=

,=o

F2i ,.

i 2(n

.La p-'-' + *=o E

]

- k________)~(k,i)p_,,,,_,,_, a J

d12

c P \ i = o Lj=O

J

f 2 i ,.

V(i.j)B(j) i=0

"l

~Ev(i,j)B(i)~

"lap''-'

(A9)

d Zw =

v /,,-2 f .

+~[E

+

"=

k=0

i 2(n - k)(2n - 2k - I) + ~ a-" k=o

a

Similarly. in view of e q u a t i o n s (A9) and (AI0), the circumferential m o m e n t M.(p) can be e x p r e s s e d as foi-

l

lows:

"'~(k,i)P2('''-'-

[~tp/,,-'-f M } M.(p)= - D o 1 ( E ] E v(i,j)B(J)

where

\

i=0 kj=O

+

V(i,j) = E~a3,[xl(i,j) + xl(j,i)] 1 -

"

a

v~

+ lrDp[xz(i.j)+ x2(j,i)] B(i) = ~rPoa3x3(i)

(AI0) (Al 1)

Substituting equations (A9) and (Al0) into e q u a t i o n (24), one can obtain the required e x p r e s s i o n for the radial m o m e n t MA MAp) =

,) ~=o

,,-z i -Dp[S'~S'v(i,f)B(j)~

L,--%b% •

J

p_,-_

.h(k.i)pZ',,-" '~) J/

/ Mf M

"l

~o( E ~ E v(i,J)B(J)l \~=ow=o ~2i(2i-l) /' - 7 1 . aP-

-

~ 2(n-k)(2n-2k-l) + k"" a2 = 0 • h(k,i)

(AI3)

292 Appl. Math. Modelling, 1988, Vol. 12, June