Transcript
Page 1: Gravity-induced stresses in finite slopes

P e r g a m o n

hit. J. Rock Mech. Min. Sci. & Geomech. Ahstr. Vol. 31, No. 5, pp. 471~,83, 1994 Elsevier Science Ltd. Printed in Great Britain

0148-9062(93)E0032-J

Gravity-induced Stresses in Finite Slopes W. Z. SAVAGEf

An exact solution for gravity-induced stresses in finite elastic slopes is presented. This solution, which is applied.for gravity -induced stresses in 15, 30, 45and 90°finite slopes, has application in pit-slope design, compares favorably with published finite element results for this problem and satisfies the conditions that shear and normal stresses vanish on the ground surface. The solution predicts that horizontal stresses are compressive along the top of the slopes (zero in the case of the 90 ° slope) and tensile away .from the bottom of the slopes, effects which are caused by downward movement and near-surface horizontal extension in front of the slope in response to grarity loading caused by the additional material associated with the finite slope.

INTRODUCTION

Previously, Savage et al. [1] and Savage and Swolfs [2] presented exact elastic solutions for the effect of topography on near-surface stresses caused by tectonic loading and gravity. These solutions, obtained by the Kolosov-Muskhelishvili method of complex potentials and conformal mapping, were applied to predict tectonic and gravitational stresses near isolated symmetric ridges and valleys and compared with hydraulic fracture measurements [3].

In this paper, an exact solution for gravity-induced stresses beneath a finite elastic slope of the type shown in Fig. 1 is presented. This solution, which has recently been applied to the case of a 90 ° slope [4], is based on a single analytic stress function and, unlike the solutions developed earlier [1, 2, 5], is more easily applied when the topography has sharp corners.

In what follows, this solution, an extension of a solution originally given by Muskhelishvili [6] for elastic half-spaces, is presented and then used to predict gravity-induced stresses beneath finite slopes of 15, 30, 45 and 90 ° . As the 90 ° case is a special case of the present, more general solution, it is repeated here for completeness. Also, comparisons are made with pre- viously published analytic [7, 8] and numerical [9, 10] models for gravity-induced stresses beneath finite slopes.

G E N E R A L S O L U T I O N F O R G R A V I T Y - I N D U C E D STRESSES

Muskhelishvili [6, Chap. 19] gave a solution in terms of a single stress function analytic in the lower and upper

tU.S. Geological Survey, Box 25046, MS 966, Denver Federal Center, Denver, CO 80225, U.S.A.

half-plane for stresses caused by distributed surface loading in an elastic half-space with a straight boundary. Muskhelishvili's [6] solution is herein extended to the case of an elastic half-space with an irregular boundary and then modified to give a general plane-strain solution for gravity-induced stresses under topographic features. Although this general solution is given in Savage [4], it is presented here for completeness.

Muskhelishvili's [6, Chap. 19] expressions for the stresses caused by surface loading of a lower half-plane with a straight boundary are

~ + ~,. = 2[~(z) + ~(=)] {l)

and

a , . - a.,. + 2ia,:,. = 2[ (~ - z ) ~ ' ( z ) - ~ ( z ) - ~ ( z ) ] . (2 )

Here • = x - iy, overbars indicate conjugate complex values, and q~(z) is a function analytic in both the lower and upper half-planes. The function q~(z) is defined in the upper half-plane by analytic continuation from the lower to upper half-plane through unloaded portions of the straight boundary. Note that the partial overbar in equation (2) has the meaning 7~(z) = q~(~); that is, 7~(z) is a function having the conjugate complex value of q~(z) at the point ~ [6, p. 288],

Taking complex conjugates and eliminating a, from equations (1) and (2) yields

a , . - ia,.,. = (z -- 5)~ ' (z) + clg(z) -- ~(5), (3)

which on the loaded portion of the straight boundary reduces to

• + ( t ) - ~ ( t ) = N - i T . (4)

Here, on the boundary, y = 0 , x = t , a ~ = N is the distributed normal load, and c r = T is the distributed

471

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472 SAVAGE: GRAVITY-INDUCED STRESSES IN FINITE SLOPES

X

II[I]ll] [lltlllllll]l I~l~l~ll~l~]'lllllll!l Fig. I. Finite slope of height b. The x, y-coordinate system and the

slope angle fl are also shown.

shear loading. Also, q~_ (t) represents the limiting values of ~(z) as z ~ t from the lower half-plane, and q~+(t) represents the limiting values of 4~(Y) as z ~ t from the upper half-plane. Finally, on the unloaded portions of the boundary

@+ ( t ) - • _ ( t ) = 0 .

Equation (4) represents a special case of the Riemann-Hilbert boundary-value problem [11, 12]. This boundary-value problem requires the determination of a function F(z) analytic for all points of a region external to a contour on which z = t and

F + ( t ) - g ( t ) F ( t )=f ( t )

where g(t) and f ( t ) are given complex-valued functions. In particular, for the case represented by equation (4),

g(t) = 1, and the solution can be shown [6] to be given by the Cauchy integral

1 f + ~ N - iTdt. (5) q'(z)=~/~i _,. t - z

Thus, given the shear and normal loading on the straight boundary, stresses within the body can be obtained from equations (1) and (2), provided that the integral in equation (5) can be evaluated.

The above formulation is appropriate for a half-space with a straight boundary. For an irregular boundary, the x, y-coordinate system is replaced by a u, v-coordinate system, which is an orthogonal curvilinear system on the z = x + iy plane [1,2, 6]. The u, v-coordinate system is then mapped by the conformal transformation z =f (w) , to the w = u + iv plane where the u, v-coordinate system is orthogonal Cartesian and the solution is carried out in the u, v system.

Stresses in a half-space with a curvilinear boundary can be expressed in terms of two stress functions, q~(w) and 7J(w) [1, 2] as

a,. + a.,, = 2[q~(w) + q)(w)] (6)

and

f- ffw/ q o~, - a, + 2ia, , = 2 / ~ q~'(w) + ~ ( w ) i . (7)

" Lf (w) Equations (6) and (7) can be converted to expressions

analogous to equations (1) and (2) provided that q~(w) is analytically continued into the upper half-plane of the

u,v-coordinate sytem. This continuation i~ accom- plished by defining q~(w) in the upper half-plane as

.f(w) - , ., l:'(w) <~(w) = - , ~ ( w ) - . ~ < t , ( , , j - F7,,77 ~ ( ' " ) (8)

Rearranging equation (8) and taking complex conjugates yields

~ ( w ) = f ' (w) ~(w)~ ' (w) . (9) -f,(w~ [~(w) + • (.,)1 -./-~{#-i

Elimination of q'(w) from equation (7) gives the system

a~ + ~,. = 2[q~(w) + @(w)] (10)

and

2 a.,,- ax + 2iax,, - f ' ( w ) [ f (w) -]'(w)]q~ '(w )

- 2 f ' ( w ) [ ~ ( w l + q , ( w ) l (11) f ' (w)

in terms of a single stress function 4~(w) analytic in the lower and upper half-planes of the u, v-coordinate system.

Stress components in the u, v-coordinate system, in terms of two stress functions [6], are obtained from

a, + a,, = 2[~(w) + q~(w)] (12)

and

~,,+a,+2iau~=2 f(w) ~ ' ( w ) + 2 j ( )~V(w). (13) f ' (w) f ' (w)

For the single stress function ~(w), defined in equation (8), equation (13) becomes

a,,-~ru+2ia.,, L 7'~w) j

y'(w) _ - 2 ~ [ ~ ( w l + c b ( w ) l . (14)

Taking complex conjugates and eliminating ~r u from equations (12) and (14) leads to

a,,-- ia.,, = ~ ( w ) + ~ ( w ) -- , [ ~ ( ~ ) + ~ ( w ) ]

(15) + L- 7T77') J

On v = 0, where u = t, equation (15) reduces to

cP+(t)-q~ ( t ) = N - i T (16)

over the loaded portion of the boundary and

• + ( t ) - q) ( t ) = 0 (17)

elsewhere, provided that f _ ( t ) = f + ( t ) = f ( t ) and f ' _ ( t ) = f + ( t ) = f ' ( t ) . Here, a, .=N is the distributed normal load, a,,, = T is the distributed shear loading, • _(t) represents the limiting values of q~(w) as w ~ t from the lower half-plane, and q)+(t) represents the limiting values of q~(~) as w ~ t from the upper half- plane. As before, the solution to the boundary-value

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SAVAGE: GRAVITY-INDUCED STRESSES IN FINITE SLOPES 473

problem represented by equations (16) and (17) is given by the Cauchy integral, which, in this case, is written

1 f+~N-iTdt.-- (18) ¢ ' ( w ) = ~ _~ t w

Savage et al. [1] showed that gravity-induced stresses in an elastic half-space with an irregular boundary cause a distributed load on the straight boundary (v = 0, u = t), which is given by

1 Nw e -- iT,.e = - - p g Imf( t )

2(1 - /~ )

(1 - 2/~) ' ~ 4 ~ i k ~ p g l m f ( t ) j , . , (19)

where p is density, g is gravitational acceleration, #~ is Poissons' ratio and the imaginary part of the mapping function is indicated by Imf(w). The distributed load in equation (19) has been termed "fictitious" by Perloff et al. [5]. Thus, on the boundary u = t, v = 0,

O+ (t) -- ~_ (t) = N,<. - iT,,.,, (20)

the stress function for the gravitational loading problem is given by

1 .f +~ Nw<, - iT,,<, dt, (21) q~(w) = ~ i _~ t w

and the stress field is given by

1 a., + ay = 2[q~(w) + q~(w)] + - - (22a)

and

pg Imf(w) 2(1 - /~)

2 a y - G + 2iGy =f'(w----) [f(w) -T(w)]q~'(w)

.S (w) - zf--C~ [r~(w) + ~(w)]

(1 - 2 p ) -+ - - p g Imf(w). (22b)

(1 - # )

In the next section, this general solution is applied to the problem of determining gravity-induced stresses in finite slopes.

S O L U T I O N F O R G R A V I T Y - I N D U C E D S T R E S S E S IN FINITE ELASTIC SLOPES

The conformal transformation of a finite slope of height b and slope angle fl in x, y-coordinates into a

half-plane in rectangular u, v-coordinates is given by the Schwarz-Christoffel transformation [13],

dz d---~ = f ' ( w ) = L~-C-~j ,

or with

as

= [ w q- llUP

mp L w _ l _ ] ,

bp 2 r ~' ~" d~ z = --~ J.,,, [¢7-112.

which is integrated to give

bp m. b Fm.+l] z = f ( w ) =--m~--z 1 + - In Lmp - l J

b p12 - 1 - - ~ Pk cos(2kn/p)

7~ k = l

2b pill-- Q, sin(2kn /p ). (23) 7~ k = l

Here, z = x +iy , w = uOiv , p = n/fl, p = 2,4, 6, 8 , . . . ,

Pk = ln[mp 2 - 2mp cos(2k~z/p) + 1],

and

. ,F sin(2krt/p) -] Qk = tan- - - - - - Lm,,-cos(2l<.lp)]"

Note that this conformal transformation applies for even values of p, for example, for slope angles of 30 or 15 °. For odd values of p, for a slope angle of, say, 60 °, a different form of equation (23) must be obtained.

The conformal transformation given by equation (23) can be written in terms of real and imaginary parts (indicated by Re and Im) by using the polar coordinate system shown in Fig. 2. The result is

I-R2-] '/p F~ -- 1)0, + 02G j ,,< cosL ,, _1

b [(1 + Re mp) 2 + (Im mp)2 G + in Re

b p/2- I 2~ ~ R e P k c o s ( 2 k g / p )

k = l

b pl~ I + - Re Qk sin(2kn/p) (24a)

~ k = l

- 1 1 u

U,V

Fig. 2. Rectangular u, v-coordinate system and the polar coordinates R~, R 2, O~ and 02.

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474 SAVAGE: GRAVITY-INDUCED STRESSES 1N FINITE SLOPES

v = I m f ( w ) bp FR2I'".Dp-I)0,+0,] h _]__ . - - ~ R I L~,j sin[ P ._J "

[ ,[ Immp ] ,[ Imrn,, ] ] b x tan 1 + Ree mpJ + tan 1 - P,.e ;n,,JJ rt

p12 I ~9 p 2 [

× ~ I m e k c o s ( 2 k x / p ) + ~ ~ ImQksin(2kn/p) k=l k= l

(24b)

where

R , = ~ / ( u - 1) 2 + v 2, 01 = tan-I(v/(u - 1)),

Rem,= k ,j

= [R/]"'sin[°2- °,] Immp LR,J m p

Re P~ = lnFH + (Re m.} 2

R2= ~((u .q- l)2-Ft, 2,

02 = t a n l ( u / ( u + l))

'k = ln[[I + (Re mp) 2

- (Im mp) 2 - - 2 Re mp cos(2kr~/p)] 2

+ [2 Im mp Re m p - 2 Im m r cos(2kx/p)] 2]

, [ I m m p - s i n ( 2 k x / p ) ] Re Qk = tan [~ee ~ 7 ~ ~p~ j

- tan- ' [ I-~ee mp + sin ( 2kx /p ) l ~ - - - ~ o o s ~ J

Im Pk =

[ 2ImmpRemp-2ImmpCOS(2kzt /p) ] tan- ' l+(Re 2 mo) - (Im rnp)- - 2 lm mp cos(2krt/p)

and

Im Qk =

lnr(Re mp - cos(2krr /p )) 2 + (lm mp + sin(2krt /p ))21 L(Re mp - cos(2krc /p )) 2 + (Im mp - - ~ J "

This transformation is shown in Fig. 3 for a 4Y slope.

0od'i°° ' -1.00

-2.00 t

-3,00 t

-4.00 1

y/b -5.oo t

-6.00 t

;7oJ Fig. 3. Conforrr

-3,00 -200 -100 000 1.00 2 I I i i i i I I I "

x/b -2,00 2.00 3 .00 4.00

i /

hal mapping of the rectangular u, v-coordinate system in Fig. 2 into a 45 finite slope.

From equation (24b) on the boundary where ~ .... 0 and u = t , v = l m l ' ( t ) = O when t > 1. When - l < t < l .

bp JR_21' P sinflp - 1 )n ] v = lm/ ( t ) = - f ~ R, " LR,J L p "I I 'mm'- -'-tan [ 'm" r l -

+ - t a n ' ] rt 1 + Re mpJ 1 7rt,lJJ bp,2 i

lm Pk cos(2kTt!P) ~ k = l

b p,'2 i

+ ~ ~ Im Q~ sin(2krt/p) k=l

where

Im Pk =

tan-

R. l + t

R, 1 - t

Re m, = [R l' " cos[_ ] LR,J LPJ

lm mp LRIJ

'F 2Imm'Remp-2Immvc°s(2kn/P) l Ll + ( Re m ~ Z [Im mp)2 ~ I-~mp c-~s( 2fflt /P ) J

and

Im Qk =

[-(Re mp - cos(2kn/p)) 2 + (Im mp + sin(2kn/p))21 n[(ffem,-cos(2krE/p))2+ (Im m v - - ~ J '

Finally, when t-%<-1, y = l m f ( t ) = - b , and when t = 1, x = y =0 . These relationships can be seen in Fig. 3.

Then, from equation (19), the distributed "fictitious" loading on v---0 in the u, v plane caused by gravity stresses associated with the finite slope in x ,y coordi- nates is given by

N, , , - iT , , ,=O

when t > / l . When - l < t <1 ,

f ' ( t ) _ e 2,~i, f ' ( t )

and

pgb[l + ( ! - 2 s t ) e 2i"'P]imf(t)" N.,,,-iT.,. 2(! -St)

Finally, when t < - 1 .

N.,,, - iT.,,,, = -- pgb.

Substituting the "fictitious" distributed loading into equation (21) gives the result

1 f+~N.,,-iT.,,,dt_ • ( w ) = ~ i .~ t w

_ pgb f -~ dt 2xi J , t - w

q- pg[l + (I - 2p)e -~*"~'] f +' Imf( t ) dt.

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SAVAGE: GRAVITY-INDUCED STRESSES 1N FINITE SLOPES 475

Evaluation of the integrals requires application of gener- alized function theory [14] and a result from the Riemann-Hilbert boundary-value problem [12].

The first integral

1 f - ' dt I i (w)=~ni _ ~ t - w

is written

1 f + ° ~ [ 1 - H ( t + l ) ] d t I, (w) = ~-~/ -= t - w

where H indicates the unit step function. The derivative of this integral with respect to w is

1 f + ~ [ l - H ( t + l ) ] d t I', (w) = ~n/~_ ~ (t - w) 2

[ ] - - d 1 1 [1 H(t + 1)] ~ . 2hi _~

Integration by parts yields,

I~ (w) - 2nil f+~H'(t+l_)dt_~ t - w

1 f + ~ 6 ( t + l ) d t I [w---~] 2hi _~ t -- w 2hi

where 6 represents the Dirac delta function. Integration with respect to w gives, aside from an arbitrary complex constant of integration, the first integral I~ (w) as,

1 f i _ _ d t _ 1 l n [ w + l ] . I t ( w ) = ~ _ ~ ¢ t - w 2hi

For the second integral, let

1 I l I m f ( t ) d t I2(w)=l . , , ~ , t - -w

and by partial integration,

1 f + l I m f ( t ) d t I X ( w ) = ~ 3-1 ( t - w ) 2

'f+ - Imf ( t )d 2hi _ j

_ 1 F!mf(1) Ira_f(- 1) 1 2hi L l - w l + w J

1 f+t Im f ' ( t )d t + ~ _, t - w •

For - l < t < l ,

Pb F l + t l'/P Imf ' ( t ) = ~ m-i--~/ sin(n/p),

also I m f ( l ) = 0 and Ira f ( - ! ) = - b . Hence,

- b pbEsin(~/p)] IX(w) = 2ni(w + 1 ) + ~ /

f+'[l+ tl';' dt × _,LT-LSj t -w"

To evaluate the integral

[sin~/p)]f+'[l + t l ' / p dt _, kT----~J t - w '

consider a functionf(z) which is analytic except for a cut along a portion of the real axis (x = t) extending from a to b. If the discontinuity across this cut is given by

f+ (t) - f _ (t) = 2i Imf+ (t),

then it can be shown from the theory of the Riemann-Hilbert boundary-value problem [12, Chap. 12] that the function f ( z ) is given by

1 I h I m f + ( t ) d t f ( z ) = ~ .], t----~, + E(z).

Here, the function f_ (t) represents the limiting values of f ( z ) as z ~ t from the lower half-plane, f+ (t) represents the limiting values of f (2) as z ~ t from the upper half-plane, and E(z) is an arbitrary entire function.

Consider the function

mp(w)=F w +'l'~' F ~ q , o , , , , 0,,,, L~--~-~J = L~J

For - 1 < u < 1, 02= 0 and 0, = 4-n as v approaches zero from above and below. Thus, the limiting values of this function are defined as

FR211/P mp+ (t) = L~-~l_] [cos(n/p) - i sin(n/p)]

F R211/p m,_(t)=Lg, j [cos(n/p)+isin(n/p)]

and there is a discontinuity given by

_2 i l l+ t ] '/p me+ (t) -- mp_ (t) = L-i-L-~ j

x sin(n/p) = 2i Im mp+(t)

over a cut from - l < t < l along the real axis of the w-plane. Thus by Riemann-Hilbert boundary-value theory

w + ll' /" lf+'Immp+(t)dt 7-s5j =~ _, i -w

- s in (n /p) f+ ' [ l+t l" , at +E(w) n , LT2tA t -w

or

sin(n/p) f+'Fl +, l ' " dt Fw + ,l,,, ~- _, LT-SSj t-w-E(W)-L-C=-fj .

Henceforth, E(w), the arbitrary entire function will be taken as 1. Then,

- b ~i pb I',(w) - 2ni(w + 1) f ' (w) -~ 4hi

and integrating with respect to w it is found that

- b ! . pbw I2(w) = ~n/In(w + i) -- zt~f(w) 4- ~ .

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476 SAVAGE: GRAVITY-INDUCED STRESSES IN FINITE SLOPES

The stress function q~(w) is then

= ~ [1 - e 2in"P]i ln[w + 1]

ipgbp [ 1 + ( 1 - 2p )e 2,~,p ~ .

L J" ipg F l + (1 - 2# )e 2~/p . . q-L i - ; (25)

Substi tut ion of equat ion (25) and its derivative into equat ions (22a) and (22b), separat ion into real and imaginary parts , and addit ion and subtract ion lead to

R e f(w) -iT(w) 4 ~ ' ( w " ] +

÷Im,O,w)+ O .sinI2'b 0' ] #

+ pgy (26) 1-- /~

[f(w)-- y(w)-] , q a:.=.Ke[ 7 ' ~ ]~ (w)J +2 Red~(w)

-Im[~(w)+Cb(w)]sinI2(Op-O1)]+pgy (27)

and

Here,

. Ff(w)-y(w)-l, ,-1 a's=Jm~ f ' (~- j (w,j

°"] ,mI ,w) + o,w)l osp; 0')]

R F f (w) -~w)] . . . . eL ff tw,--0 when v = 0 a n d

R e F f ( w ) - f ( w ) ] . . . . pg Imf(w)[RjT" L .7'Tw5 f f two,- i----~ Lnd

x {[~R22~]tcostOz + (O2- O,)/P]

- cos[02 + (02 - 0, + 2n)/P]+½[1 - cos[(02 - O,)/p]

- ½[1 - 2U] [cos[(02 - 0, + 2n)/p] - cos[2n/p]]]}

when v < 0, and further

i F/(w) -y(w)q . . . . mL 7 ( - ~ - J q ' t w ) = 0

(28)

when c = 0 , and

i rf(w)-T(w)q , p g l m f ( w ) [ R , ] 'p m " - ~ /4~ (w

[ .f'u,'J J ) - [ - -T LR~.J

x { [ ~ l [ s i n [ O 2 + (O2-O,)/P]

- sin[02 + (02 - 0, + 2n)/p]] - ~[sin[(02 - 0, )/p]

+ ½[1 - 2/~][sin[(02 - 0, + 2n)/p] - sin[Zn/p]]]}

when v < 0.

Also,

Re 4~(w) -

Re[~(w) + q~(w)] -

pgb [ ~ - 2 ~ ] [[ l - c°s(2n /p

ppgb + I n R 2 sin(2n/p)] + [v - [1 - 2#]

8nil - ~ ]

x [u sin(2n/p) - v cos(2n/p)]] + Pg F 1 -21~] 4 L I - ~ j

x [Ref(w)sin(2n/p) - lm/(w)cos(2n/p)] pg I m f ( w )

411 --/~]

pgb [ ~ - 2 ~ ] ln R2 sin(2n /p

ppgb [I - 2~ ]u 4n L-(~- ~ j sin(Zn/p)

+ PgI~_2g~ ] Ref(w)sin(2n/p)

and

Im[7~(w) + q~(w)] - pgb I ~ - 2 ~ ]O2 sin(2n /P

ppgb [ 1 - 2 p ] v sin(2n/p) 4n k l - ~ J

Note the sign convent ion adopted in this paper; com- pressive stresses are negative.

Equat ions (26), (27) and (28) for gravity-induced stresses beneath finite slopes like that shown in Fig. 1 satisfy the condit ions that shear and normal stresses parallel and perpendicular to the ground surface vanish. N o r m a l and shear stresses on the ground surface are given by the t ransformat ion

I - a , . - a,.] "" + °" [ ~ ] c o s 2c, + at = 2 + a,.,. sin 2a,

a,. + a,. [-a,. - al-] a,, = ~ - L ~ j c o s 27 - a,, sin 2u,

and

O ' ) . m O " X .

o~,~ = 2 sin 2a + ¢,, cos 2:~,

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SAVAGE: GRAVITY-INDUCED STRESSES IN FINITE SLOPES 477

or, using equat ions (26), (27) and (28) for v = 0 and u = t, as

" = p-gb 2n [1 - cos(2n /p )]02

n I m f ( t ) cos(2n/p)l + ~

+ ~ ~ cos 2~

(29)

- ' F ' - 2 I 1-cos(2.u)lo= pgb i ~ L l - ~ j t

n I m f ( t ) cos(2nlp) + 2 [ n R e f ( t ) + ~ b

PU2 lnR2]sin(2n/p)}

1 [ 1 - 2 # ] I m f ( t ) 2 L 1 - ~ j - - T - c o s 2ct ( 3 0 )

pgb - 2n [02 sin(2n /p )]

1 - +~-b [~2~]Imf(t)[s in2~-sin(2n/p)] (31)

where a, and a~n are, respectively, the normal and shear stresses acting parallel to the gound surface and a n is the normal stress acting perpendicular to the ground surface. Here, on the g round surface to the left o f the toe o f the slope in Fig. 1, v = 0, u = t < - 1, y/b = - 1, and

= 0 . On the face of the slope, v = 0 and u = t , - 1 < t < 1 and a = fl = n /p (the slope angle shown in Fig. 1). Finally, u = t > 1 and v = 0, a = 0, x/b > 0 and y/b = 0 on the g round surface to the right o f the top corner o f the slope.

On the g round surface to the left o f the toe of the slope, where a = 0, equat ion (30) reduces to

". - ' f ' - + ['< pgb pg~-~ L 1-,u JL b

-in R2-2]sin(2nlp)] and equat ions (29) and (31) reduce to

an = a~.=O

and

O'Tn = if-D' = O.

As t ~ - ~ , x --, - ~ , and the horizontal stress becomes

a, 1 + - + In p pgb n p / 2 - I

- ~ l n [ 2 [ l - c o s ( 2 k n / p ) ] ] c o s ( 2 k n / p ) k = l

k sin(2kn/p) sin(2rGp) P k = l

As x ~ - ~ , for slopes of 15, 30, 45 and 90 ° (p = 12 ,6 ,4 ,2 ) , the normal ized horizontal stress has respective limiting values given by 0.118, 0.344, 0.623 and 1.0 times [(1 - 2#)/( i - ~)]. Note that to the left o f the toe of the slope in Fig. 1 the horizontal surface stress is zero when Poisson 's ratio, ~, is 1/2 and tensile when

is less than 1/2. This tension can be at t r ibuted to downward m o v e m e n t and near-surface horizontal exten- sion to the left o f the toe of the slope in response to the gravity loading caused by the slope.

Fo r the face of the slope, where v = 0, u = t and - 1 < t < 1, a~, a n and a~n are given by equat ions (29), (30) and (31) with 01 = - n , 02= 0 and a = f l = nip. Equat ions (29) and (31) reduce to zero and, thus, normal , a n, and shear stresses, a,n, perpendicular and parallel to the slope face in Fig. 1, vanish. Equat ion (30) gives the normal stress, a~, parallel to the slope surface a s

o- t

pgb i[l-2~']l-F'~Ref(t) P" inns]

L I - . I LL b 2

x s in (2n /p ) ]

lb [~---2l~l~ ]Im f( t )cos(2n /p ).

In the 90 ° case, surface stresses for - 1 < t < 1 reduce to a, n=axy=a n = a x = O and

a ~ _ % _ 1 [ l - 2 P l l m _ f ( t ) . (32) pgb pgb n L l - p ] b

To the right o f the top corner of the slope in Fig. 1, w h e r e u = t / > l a n d v = 0 , x / b > > - 0 a n d y / b = 0 , a = 0 ; equat ions (29), (30) and (31) reduce to

O" z O" x

pgb pgb

and

[ 1-2Pl[[nRef(w) 7~--~AL L b

- I n R 2 - 2 ] s i n ( 2 n / p ) ]

a, = a,.= 0

O'rn ~ O'~, ~ 0 .

For a 90 ° slope, the horizontal stress for x/b >~ 0 reduces to zero. As t ~ , x ~ and the horizontal stress becomes

pgb n p/2 1

- ~ ln[2[1 - cos(2kn/p)]]cos(2kn/p) k = l

2; p/2~l sin(2kn/p)}sin(2n/p)

which is zero when Poisson's ratio, p, is 1/2. As x ~ o o , for slopes o f 15, 30 and 45 ° (p = 12, 6 and 4), the horizontal stress has respective limiting values given by - 0 . 3 7 7 , - 0 . 7 8 0 and - 0 . 8 8 2 times [ ( 1 - 2 p ) / ( l - / ~ ) ] .

Page 8: Gravity-induced stresses in finite slopes

478 SAVAGE: ( IRAVITY-INDUCEI) STRESSES IN FINITE SLOP[S

2 0 0

..O

(9)

.5 I-

b

150

100

0 5 0

- 0 0 0

Z---Y----'" "7,, t

j " ' ' - \x

-0.50

Surface Sfress /.1,= 1/3

. . . . 15 ° S l o p e - - - 3 0 ° S l o p e - - 4 5 ° S l o p e

- 6 0 0 -4 .00 -2 .00 C' 00 2.00 4 O0 6 O0

x/b Fig. 4. Variation of the surface stress, ~r/'pgb, with x/b for a Poisson's

ratio of 1/3 and slope angles of 15, 30 and 45.

This compressive horizontal stress is in response to near- surface extension to the left of the toe of the slope in Fig. 1. A similar response, that is, near-surface tension in valleys and compression in ridges, is also found from the exact solution for gravity-induced stresses in these topographic features [1].

Finally, since ~r, the normal stress acting perpendicu- lar to the ground surface, and a~,, the shear stress acting parallel to the ground surface, are zero just to the left and just to the right of the corners at t = - 1 and t = 1, these stresses are taken to be zero for t = - I and t = 1. Except for the 90 ° case, the normal stress parallel to the ground surface a~, becomes infinite as t approaches - 1. The normal stress parallel to the ground surface, a~, is continuous at t = 1.

The variation with x/b of surface horizontal stress, a~/pgb, for # = 1/3 and slope angles of 15, 30 and 45-, is shown in Fig. 4. Taking, for example, the variation of horizontal surface stress for a slope of 4 5 (the first curve on the right in Fig. 4), it is seen that the surface horizontal stress to the left of the toe approaches a limiting tensile value of 0.315 pgb at x / b = - 6 . The surface-parallel stress becomes infinitely tensile at the toe (x/b = - 1 ) . This reflects the concentration of horizontal stress near the sharp lower corner at the toe of the slope. The stress parallel to the slope sur- face, where - 1 < x/b < O, varies from being infinitely tensile to compressive. The surface-parallel stress re- mains compressive from the crest of the slope onwards, that is for x/b ~ O, approaching a limiting compressive value of -0 .441 pgb at x/b = 6. As can be seen, this pattern of horizontal stress variation is repeated for slope angles of 30 and 15'. As has been previously noted, surface-parallel stresses for the 9 0 slope are given by [ ( 1 - 2 / , ) / ( 1 - # ) ] for x/b < 0 , by equation (32) for x/b = 0, and are zero for x/b ~ O.

Figure 5-8 show, respectively, contours of gravity- induced dimensionless normal, a~/pgb and a,/pgb, the shear stresses, ~r,,/pgb, greatest principal stresses ~r~/pgb, least principal stresses a2/pgb and maximum shear stresses z~/pgb in the vicinity of a finite slope with slope

angles 15, 30, 45 and 90 when Poisson's ratio is 13. Note the concentration of stress at the bottom of the toes of the slopes and the tensile region near the bottom and to the left of the bottom of the slopes. Also note that the effects of the slopes on gravity-induced normal and shear stresses dissipate with depth. Finally, these examples are, of course, only special cases. This solution applies for the usual range of isotropic Poisson ratios and to all finite slopes given by the mapping used here [equation (23)].

COMPARISON WITH PREVIOUS SOLUTIONS

Stresses calculated using equations (26), (27) and (28) compare favorably with published finite element sol- utions for gravity-induced stresses in finite slopes. In particular, the results shown in Figs 5-8 are similar to results from finite element solutions to this problem presented by Wang and Sun [9]. Figures 5-8 and Wang and Sun's figures for a~/pgb, ~2/pgb and ~m,x/pgb from the finite element solutions show stress concentrations at the toes of the slopes and similar stress distributions away from the slopes. Also, contours of Zmax/pgb for the 9 0 slope [Fig. 8(f)] compare well with contours of vm,x/pgb determined by the finite element method [10, Fig. 8]. However, contours of the least compressive principal stresses given by the present solution and Sitar and Clough's numerical solution [10, Fig. 8] are only qualitatively similar. Differences between Sitar and Clough's contours and those given here can probably be attributed to the proximity of the lower boundary of their finite element mesh to the 90 '~' finite slope.

Results obtained from equations (26), (27) and (28) and the above-quoted elastic and finite element solutions differ from results obtained by Silvestri and Tabib [7, 8] for gravity-induced stresses near finite slopes. This is particularly true near slope faces where, for example, the results of Silvestri and Tabib [7, 8] fail to show the concentrations of stress at the base of the slopes seen in the finite element solutions and predicted by the exact solution presented here. Also, their results for g~,/pgb do not compare well with results obtained with the present solution. However, Silvestri and Tabib 's [7, 8] results for the normal stresses a~/pgb and a,/pgb can be seen to approach those given by the present solution away from the slopes. As explained in Savage [4], differences be- tween results obtained from the present solution and results obtained from the solution of Silvestri and Tabib [7, 8] are a consequence of the approximate nature of their approach to this problem.

CONCLUSIONS

An exact solution for gravity-induced stresses beneath finite slopes has been presented. This solution is a special case of a general solution based on a single analytic stress function for gravity-induced stresses near topo- graphic features. This general solution, which is an

Page 9: Gravity-induced stresses in finite slopes

SAVAGE: G R A V I T Y - I N D U C E D STRESSES IN FINITE SLOPES 479

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* & I

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. D

e-i

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Od

0 t ~

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¢:::

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8

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x

R M M S 3 f , ~

Page 10: Gravity-induced stresses in finite slopes

480 SAVAGE: GRAVITY-INDUCED STRESSES IN FINITE SLOPES

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e.,

O

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3

,.6 mb b7

Page 11: Gravity-induced stresses in finite slopes

SAVAGE: GRAVITY-INDUCED STRESSES IN FINITE SLOPES 481

0

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o d I

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7 t~

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r ~

x

?

Page 12: Gravity-induced stresses in finite slopes

482 SAVAGE: GRAVITY-INDUCED STRESSES IN FINITE SLOPES

\

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Page 13: Gravity-induced stresses in finite slopes

SAVAGE: GRAVITY-INDUCED STRESSES 1N FINITE SLOPES 483

extension o f a so lu t ion or iginal ly given by Muskhe l - ishvili [6] for elastic half-spaces, uses con fo rma l m a p p i n g to t r ans fo rm a t o p o g r a p h i c feature descr ibed in a curvi- l inear coo rd ina t e system into a rec tangu la r Car tes ian coo rd ina t e system. The stress funct ion for the gravi ty- induced stresses is then der ived in the rec tangu la r coor- d ina te system by eva lua t ion o f a Cauchy integral and by analy t ic con t inua t ion . Grav i t y stresses near the topo- graphic feature, that is, in the curvi l inear coo rd ina t e system, are then ob ta ined by the usual rules o f tensor t r ans fo rma t ion .

The so lu t ion presented in this pape r for gravi ty- induced stresses benea th finite s lopes compares favor- ab ly with publ i shed finite e lement results for this p rob lem, but it does not compa re favorably , par t i cu la r ly near the slopes, with the so lu t ion p r o p o s e d by Silvestri and T a b i b [7, 8]. The present so lu t ion satisfies the con- d i t ions tha t shear and no rma l stresses vanish on the g round surface and predic ts the existence o f compress ive hor i zon ta l stresses a long the top o f the s lopes (zero hor i zon ta l stresses in the case o f the 90 ° slope) and tensile ho r i zon ta l stresses in f ront o f the s lopes- -ef fec ts caused by d o w n w a r d movemen t and hor izon ta l exten- sion o f the surface in f ront o f the slope in response to gravi ty load ing by the add i t iona l mate r ia l associa ted with the finite slope.

Accepted for publication II November 1993.

REFERENCES

I. Savage W. Z., Swolfs H. S. and Powers P. S. Gravitational stresses in long symmetric ridges and valleys. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 22, 291 302 (1985).

2. Savage W. Z. and Swolfs H. S. Tectonic and gravitational stress in long symmetric ridges and valleys. J. Geophys. Res. 91, 3677-3685 (1986).

3. Swolfs H. S., Savage W. Z. and Ellis W. L. An evaluation of topographically induced stresses at Yucca Mountain, Nevada. In U.S. Geological Survey Bulletin 1790 (Edited by Carr M. D. and Yount J. C,), pp. 95 101 (1988).

4. Savage W. Z. Gravity-induced stresses near a vertical cliff. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 30, 325-330 (1993).

5. Perloff W. H., Baladi G. Y. and Harr M. E. Stress distributions within and under long elastic embankments. Highway Res. Rec. 181, 12-40 (1967).

6. Muskhelishvili N. I. Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoof, Leiden, The Netherlands (1953).

7. Silvestri V. and Tabib C. Exact determination of gravity stresses in finite elastic slopes, Pt. I. Theoretical considerations. Can. Geotech. J. 20, 47-54 (1983a).

8. Silvestri V. and Tabib C. Exact determination of gravity stresses in finite elastic slopes, Pt. II. Applications. Can. Geotech. J. 20, 55~0 (1983b).

9. Wang F. D. and Sun M. C. A systematic analysis of pit slope structures by the stiffness matrix method. U.S. Bureau of Mines Report of Investigations 7343 (1970).

10. Sitar N. and Clough G. W. Seismic response of steep slopes in cemented soils. ASCE J. Geotech. Engng 109, 210-227 (1983).

1 I. Gakhov F. D. Boundary Value Problems. Dover Publications, Mineola, N.Y. (1966).

12. Wyld H. W. Mathematical Methods for Physics. Benjamin, Read- ing, Mass. (1976).

13. Churchill R. V. Complex Variables. McGraw-Hill, New York (1948).

14. Lighthill M. J. Introduction to Fourier Analysis and Generalized Functions. Cambridge Univeristy Press, Cambridge, U.K. (1964).