Proc. Indian Acad. Sci. (Math. Sci.), Vol. 104, No. 2, May 1994, pp..425-433. �9 Printed in India.

Stresses in an elastic plate lying over a base due to strip-loading

RAJ K U M A R S H A R M A and N A T R A M G A R G Department of Mathematics, Maharshi Dayanand University, Rohtak 124001. India

MS received 27 January 1992; revised 25 October 1993

Abstract. The closed-form analytic expressions for the stresses at any point of an elastic plate coupling in different ways to a base as a result of a two-dimensional shear strip-loading are obtained. The contact between the horizontal layer and the base is either smooth-rigid or rough-rigid or welded. The variations of the shear stresses with the horizontal distance have been studied numerically. It is found that the effect of different boundary conditions on the stress field is significant and the stresses for an elastic layer lying over an elastic half- space differ considerably from those of an entire homogeneous elastic half-space.

Keywords. Elastic plate; rigid base; smooth base; shear stresses; shear surface loads.

1. Introduction

The solut ion of the p rob lem of the deformat ion of a horizontal ly layered elastic mater ia l under the act ion of surface loads is finding wide appl icat ions in engineering, geophysics and soil mechanics. The deformat ion of a muit i layered elastic half-space due to two-dimens ional and three-dimensional surface loadings has been studied by m a n y researchers, (Kuo [6], Singh [9], Pan [7], Chaudhur i and Bhowal [1], G a r g and Sha rma [2, 3] and others). Recently, G a r g and S h a r m a [4] discussed t h e deformat ion of an elastic layer coupl ing in different ways to a base due to a very long vertical strike-slip fault si tuated in the layer.

In the present paper , the c losed-form expressions for the stresses in an hor izontal plate of infinite lateral extent lying over a base due to s t r ip- loading are obtained. In geophysics, the elastic plate represents the crust of the earth. The interface between the plate and the base m a y be either welded or rigid. The rigid interface is ei ther smooth-r igid or rough-rigid. The deformat ion of the plate corresponding to each type of the interface is obtained. The deformat ion of an homogeneous entire half-space due to s t r ip- loading can be obta ined f rom our results as a par t icular case. Similarly, the deformat ion of an elastic layer lying over an elastic half-space can be recovered from our results. Finally, the var ia t ion of stresses is studied numerically.

2. Basic equations

Let (x, y, z) be the cartesian coord ina te system with z-axis vertical. Let (u, v, w) be the displacement components . We consider the ant iplane strain p rob l em in which

425

426 Raj Kumar Sharma and Nat Ram Garo

the displacement components are of the type

u=u(y,z), v = w - O

and u(y, z) satisfies the equilibrium equation

dZu d2u

OY 2 + ~ = 0

for zero body forces. The non-zero strains and stresses are

1 du 1 0u e12=2dyy ' e t 3 - 2 a z

c3u du q 2 = ~ , T13 =~Tz

/~ being the rigidity of the medium.

(1)

(2)

(3)

3. Formulation of the problem

We consider a horizontal elastic plate of thickness H and rigidity # lying over a base. The origin of the cartesian coordinate system (x, y, z) is taken at the upper boundary of the plate and the z-axis is drawn into the medium. The plate occupies the region 0 ~< z ~< H and the region z > H is the base over which the plate is lying (figure 1).

We assume that a shear-load R per unit area is acting over the strip lYl ~< a of the surface z = 0 in the positive x-direction. The boundary condition at the surface z = 0 is

O R lYl ~<a T13= lYl>a

(4)

The interface z = H between the plate and the base may be either smooth-rigid or rough-rigid or welded.

When the interface z = H is of the smooth-rigid type, the boundary condition at

� 9

�9 " H

RIGIOITY ELASTIC PLATE

BASE

Ip r

Figure I, Section of the model by the plane x = 0.

Stresses in an elastic plate 427

z = H is (Small and Booker [10])

zl3(z = H) = 0 (5)

whereas for the rough-rigid interface, the boundary condition is (Small and Booker [10])

u(z = H) = 0. (6)

When the interface z = H is welded, the continuity of the displacement and shear stress ~t3 implies (Singh [8])

u(z = n - ) = u(z = H + )

TI3(Z = H - - ) = T13(z ~-- H + ) . (7)

We shall find the deformation field at any point of the plate corresponding to each type of the boundary condition at the interface due to shear strip-loading.

4. Solution of the problem

For the solution of (2), we make use of Fourier transform. The Fourier transform of X(y, z) is defined as

X(k , z) = X(y , z )exp( iky )dy (8)

so that

l f~| X(y , z) = ~ - iky)dk. (9)

Taking the Fourier transform of (2), we get

(5 ,,0, Solution of (10) is

t~ = A e x p ( - [k[z) + Bexp([k[z) (11)

where A and B may be functions of k. Then

1 [Aexp( Iklz) + B e x p ( I k L z ) ] e x p ( - iky)dk. (12) U= 21t -,,o

The shear stresses are

i# I ~~ [Aexp(-- Iklz) + B e x p ( I k l z ) ] e x p ( - i ky )kdk (13)

~12= 2n .;-~o

zta = ~ [-- A e x p ( - Iklz) + B e x p ( I k l z ) ] e x p ( - iky) lk ldk (14) --00

428 Raj Kumar Sharma and Nat Ram Garg

Equations (4), (8) and (9) express the shear stress ~13 in the integral form at the upper boundary z = 0 of the plate. We find

-Rf sinka ~la = n -~o k exp( - iky)dk . (15)

Comparing (14) and (15), we obtain

2R ( sin ka ~ (16)

4.1 Smooth-rigid boundary at z = H

When the interface z = H is smooth-rigid, the boundary condition (5) and eq. (14) yield

A e x p ( - t k l H ) - Bexp(lkln) = 0. (17)

Solving (16)-(17), we have

A = 2R sin ka I 1 1 B = 2R sin ka #k lk~ 1 - e x p ( - E l k l H ) ' #klk~

exp(- 21kl--) ]. I1 - eXp(- 21kln)_i

(18)

The displacement and stresses for a smooth-rigid interface can be obtained directly from (12)--(14) and (18). Using the power series expansion for [1 - e x p ( - 21klH)]- 1, we obtain

u = - - sinka e x p ( - Iklz) + exp{-Ikl(2nH+z)} ~ - ~c n E I

+ ,=1 ~" exp{- Ik l (2nH - z)} ] exp(-klklikY)dk. (19)

Using the Appendix, the closed-form expressions for the stresses z12 and z13 are obtained as follows:

[ ~ (a + y)2 + z z r12 = l ~ zZ + ,flog(a+y)

= 1 [ ( a - y)~ + (2nil + z) 2

~a+Y)2+(2nH--z)2}] (20) + l o g _ y ) 2 + ( 2 n H z) 2 "

( ) 2az t an- 1 2a(2nH + z) + z ~---a 2 + , 1 y2 +(2nH + z) 2 - a 2

[ 2a(2nH - z) - t a n - l \ y 2 + ( 2 n H _ z ) 2 _ a 2 ) } ] . (21)

Stresses in an elastic plate 429

4.2 Rouoh-rigid interface

From (6) and (12), we get

A exp( - IklH) + Bexp([kiH) = O. (22)

From (16) and (22), the values of A and B are found to be

2Rsinka F 1 ] - 2 R s i n k a r exp(-2lklH) 1. A = - ~ LX+exp( -2 lk lH) ' B - /~klkl L l + e x p ( - 2 1 k l H ) l

(23)

The integral expression for the displacement, when the interface is rough-rigid, can be obtained from (12) and (23). We get

u = - - sinka exp ( - Iklz) + ( - l ) ' e x p { - l k l ( 2 n H + z ) } /[~t oo H = 1

exp(-- iky) dk. + _ ( - 1 ) ' e x p { - I k l ( E n H - z ) } (24)

k]kl

The closed-form expressions for stresses are found to be

- R .rlo(a + , ' + : z. ~ ( - , r ,r,og(a + ~)~- + (2.H + ,)~ ~2 =-~-~-~ L (a _ y)2 W z2 "t" = l ( (a -- y)z + ( 2 n H .-t- z) 2

+ log _ y)2 + (2nil - z) 2 J A (25)

2az l)~{tan_xC22a(2nH+z ) Z13 ---- : [ tan- 1 (y2_l.z2_a2) -t- .~1 (-- + ( 2 n H . 4 _ 2 ) 2 _ a 2 ]

- t a n - ' (y2 2a(2nH-z) ~ ~ l + (2nil -- z) 2 -- a 2 ] J J" (26 )

4.3 Welded interface

Let/~o be the rigidity of the half-space z > H. The displacement u in the region z >/H is of the type

1 ,4 o exp ( - Iklz)exp(- iky)dy (27) u 2n _|

in which the coefficient Ao is to be determined from the boundary conditions. Then

/'tO r ~ zla = - ~ j_ | Ao exp(-Iklz)exp(-/ky)lkldk. (28)

Equations (7), (12), (14), (27) and (28) yield the relations

A e x p ( - [klH) + Bexp(lkIH) = Aoexp(-- lklH) (29)

430 Raj Kumar Sharma and Nat Ram Gar#

/~ [ - A e x p ( - IkiH) + Bexp(IklH)] = - /%Aoexp(- - Ik lH) .

Solving (16), (29) and (30), we get

A~ = 1 - M e x p ( - 2lklH) ktkl

where

2 R [ 1 ] sinka A = -~- 1 - M e x p ( - 21kill) k-~

2 R [ Mexp(-EIklH) lsinka

(30)

(31)

S = #~go, M = (S - 1)/(S + 1) = (/~ - ,%)/(# + #o)- (32)

Using (31), we obtain the deformation field for the welded interface as follows:

u = - - e x p ( - Iktz) + M" e x p { ( - Ikl(2nH + z)} K~ .= l

+ , ~ t M"exp { - Ik l (2nH - z ) } ] k l r l ~ exp(-- iky)dk (33)

[ (a + y)2 + z z ~log (a + y):+(2nH+ z) 2 r12 = lOg (a--y)2 + z 2 ~- ~ M"

= 1 [ (a - y)Z + (2nH + z) 2

+ l o g ( a + y ) 2 + ( 2 n H - z ) 2 t ] (34) Ca -- y)2 + (2nil z) 2

~ 1 3 = _ ~ [ t a n _ 1 ( 2az / ~ M, {tan_ l ( y 2 2a(2nH+z)a2 ) \y2 + z 2 _ a 2 j + ,-1 + (2nil + z) 2 -

--tan-'( 2a(2nH -- z) ~ l y2 +(2nH_z)2_a2 l j . j . (35)

We observe that the deformation fields for the smooth-rigid and rough-rigid interfaces can also be obtained from the deformation field for the welded interface, respectively, on substituting M = 1 and M = - 1.

5. N u m e r i c a l r e s u l t s

We study numerically the variation of the shear stresses r and ~1 a with the horizontal distance y for fixed values of other quantities a, S and z. The width of the strip is assumed to be ff2H. It has been seen earlier that the values of S affect the shear stresses only when the interface between the elastic plate and the base is welded.

Figures 2 and 3 exhibit the variation of the stress T12. Two values of the ratio S of rigidities are taken, namely, S = 0-25 and S = 4.0. ~ = 0.25 implies that the rigidity of the elastic plate is less than that of the base and S = 4-0 means that the elastic plate is of high rigidity as compared to that of the base. Stresses at the points at

Stresses in an elastic plate 431

o

l _o.o,.\ - O . o s - ~

I -o.06.

- 0.(]7. ~ , , o 0.2. o.~. o.+. o. . . ~ 1.'=. +:,.

Y

Figure Z Variat ion of the dimensionless stress x~2/R with the horizontal distance y when a = 0.H and z = 0-25H.

-L r

-+1 \ \ ~ I ..o.:,. I

j -o,,,. \ \ -~-0.02S- wELDED

l-~176 �9 0 0&- SMOOTH RIGID

- 0.o a,s , o o . ' , . ' ~ , . 1:2. 1 :6 , ' i . ' i , . ' i-,H

Y J

F i g u r e 3. V a r i a t i o n o f the d imens ion l e s s s t ress •;z/R wi th the h o r i z o n t a l d i s t a n c e y w h e n

a = 0 . H a n d z = 0-75H.

depths t l /4 and 3H/4 are calculated. It is observed that the stress ~12 for the welded contact lies between the corresponding values of z12 for the rough-rigid and smooth-rigid contacts, for different values of S.

Figures 4 and 5 show the variation of the stress zt3 with the distance y. In these figures, curves corresponding to different kinds of coupling of the elastic plate to a base have been drawn.

432 Raj Kumar Sharma and Nat Ram Garg

0.06~

0.021 f

- 0.02

-0.1

-OAt

; o, , i / I - 0 2 2 ~

-o.2a;] " ~ ' 4 . " ~ ' l . " ~ . . " l"s. 2 . y ;-

Fipee 4. Variation of dimensionless stress xx 3/R with y when a : 0.H, z : H/4 and S = 2.

O.OS~ ~ s M O O T H - R I G I D

t

-o.o,- , ~ J / / I s-2.o r , y / i .o.

-~176 / _o.o,1 ~ j ' -

0 0.tH 0,$H 1.2H I,SH 2H 2,tH ;I.SH Y

Figure 5. Variation ofxl3/R with y when a=O.H, z=0-75H and S=2.

Acknowledgement

One of the authors (NRG) is thankful to the University Grants Commission, New Delhi, for providing financial support. The authors are also grateful to the referee for providing valuable suggestions for the improvement of the paper.

Stresses in an elastic plate 433

Append ix (~ > 0)

(1) e x p ( - l k l ~ ) s i n k a e x p ( - i k y ) d k = - l l o g F ( y + a ) 2 + ~ 2 _~ Ikl 2 k(y a)2+~2

( ) (2) e x p ( - Ikl ~) exp( - i ky )dk = t an - l 2a~ _ ~ y2 + ~2 _ a 2

In integral (2), it is assumed that y2 + ~2/> a 2. If, however, y2 + ~2 < a 2, the quant i ty 7r is to be a d d e d on the r ight s ide [5] .

R e f e r e n c e s

[ I ] Chaudhuri P K and Bhowal S, Two-dimensional static response of a transversely isotropic multilayered non-homogeneous half-space to surface-loads, Geophys. Res. Bull. 2ml (1989) 77-87

[2] Garg N R and Sharma R K, Generation of displacements and stresses in a multilayered half-space due to strip-loading, Bull. Indian Soc. Earthquake Technol. 28 (1991) 1-26

[3] Garg N R and Sharma R K, Displacement and stresses at any point of a transversely isotropic multilayer~l half-space due to strip-loadings, Indian J. Pure Appl. Math. 22 (1991) 859-877

[4] Garg N R and Sharma R K, Deformation of an elastic layer coupling in different ways to a base due to a very long vertical strike-slip dislocation, Proc. Indian Acad. Sei. (Earth Planet. Sci.) I01 (1992) 255-268

[5] Gradshteyn I S and Ryzhik I M, Tables of Integrals, Series and Products (New York: Academic Press) (1980)

[6-] Kuo J T, Static response of a multilayercd medium under inclined surface loads, J. Geophys. Res. 74 (1969) 3195-3207

[7-] Pan E, Static response of a transversely isotropic and layered half-space to general surface loads, Phys. Earth Planet. Inter. 54 (1989) 353-363

[8-] Singh S J, Static deformation of a multilayered half-space by internal sources, J. Geophys. Res. 75 (1970) 3257-3263

[9-] Singh S J, Static deformation of a transversely isotropic multilayered half-space by surface loads, Phys. Earth Planet. Inter. 42 (1986) 263-73

[ 10"] Small J C and Booker J R, Finite layer analysis of layered elastic materfals using a flexibility approach. Part-I-strip-loading, Int. J. Num. Methods En#. 20 (1984) 1025-1037