Gravity-induced stresses in finite slopes

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<ul><li><p>Pergamon </p><p>hit. J. Rock Mech. Min. Sci. &amp; Geomech. Ahstr. Vol. 31, No. 5, pp. 471~,83, 1994 Elsevier Science Ltd. Printed in Great Britain </p><p>0148-9062(93)E0032-J </p><p>Gravity-induced Stresses in Finite Slopes W. Z. SAVAGEf </p><p>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 90finite 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. </p><p>INTRODUCTION </p><p>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]. </p><p>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. </p><p>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. </p><p>GENERAL SOLUTION FOR GRAVITY- INDUCED STRESSES </p><p>Muskhelishvili [6, Chap. 19] gave a solution in terms of a single stress function analytic in the lower and upper </p><p>tU.S. Geological Survey, Box 25046, MS 966, Denver Federal Center, Denver, CO 80225, U.S.A. </p><p>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. </p><p>Muskhelishvili's [6, Chap. 19] expressions for the stresses caused by surface loading of a lower half-plane with a straight boundary are </p><p>~ + ~,. = 2[~(z) + ~(=)] {l) </p><p>and </p><p>a, . - a.,. + 2ia,:,. = 2[(~ - z )~ ' (z ) - ~(z ) - ~(z ) ] . (2) </p><p>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], </p><p>Taking complex conjugates and eliminating a, from equations (1) and (2) yields </p><p>a , . - ia,.,. = (z -- 5)~'(z) + clg(z) -- ~(5), (3) </p><p>which on the loaded portion of the straight boundary reduces to </p><p> +( t ) -~ ( t )=N- iT . (4) </p><p>Here, on the boundary, y=0, x=t , a~=N is the distributed normal load, and c r = T is the distributed </p><p>471 </p></li><li><p>472 SAVAGE: GRAVITY-INDUCED STRESSES IN FINITE SLOPES </p><p>X </p><p>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 </p><p>slope angle fl are also shown. </p><p>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 </p><p>@+ ( t ) - _ ( t ) = 0 . </p><p>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 </p><p>F+(t) -g( t )F (t)=f(t) </p><p>where g(t) and f(t) are given complex-valued functions. In particular, for the case represented by equation (4), </p><p>g(t) = 1, and the solution can be shown [6] to be given by the Cauchy integral </p><p>1 f + ~ N- iTdt. (5) q'(z)=~/~i _,. t - z </p><p>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. </p><p>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. </p><p>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 </p><p>a,. + a.,, = 2[q~(w) + q)(w)] (6) </p><p>and </p><p>f- ffw/ q o~, - a, + 2ia,, = 2 / ~ q~'(w) + ~(w) i . (7) </p><p>" L f (w) Equations (6) and (7) can be converted to expressions </p><p>analogous to equations (1) and (2) provided that q~(w) is analytically continued into the upper half-plane of the </p><p>u,v-coordinate sytem. This continuation i~ accom- plished by defining q~(w) in the upper half-plane as </p><p>.f(w) - , ., l:'(w) </p></li><li><p>SAVAGE: GRAVITY-INDUCED STRESSES IN FINITE SLOPES 473 </p><p>problem represented by equations (16) and (17) is given by the Cauchy integral, which, in this case, is written </p><p>1 f+~N-iTdt.-- (18) ' (w)=~ _~ t w </p><p>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 </p><p>1 Nw e -- iT,.e = - - p g Imf(t) </p><p>2(1 - /~) </p><p>(1 - 2/~) '~ 4 ~ik~pg lmf ( t ) j , . , (19) </p><p>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, </p><p>O+ (t) -- ~_ (t) = N,</p></li><li><p>474 SAVAGE: GRAVITY-INDUCED STRESSES 1N FINITE SLOPES </p><p>v=Imf(w) bp FR2I'".Dp-I)0,+0,] h _]__ . - -~RI L~,j sin[ P ._J " [ ,[ Immp ] ,[ Imrn,, ] ] b </p><p>x tan 1 + Ree mpJ + tan 1 - P,.e ;n,,JJ rt p12 I ~9 p2 [ </p><p> ~ Imekcos(2kx /p)+~ ~ ImQksin(2kn/p) k=l k=l </p><p>(24b) </p><p>where </p><p>R, = ~/ (u - 1) 2 + v 2, 01 = tan-I(v/(u - 1)), </p><p>Rem,= k ,j </p><p>= [R/]"'sin[2- ,] Immp LR,J m p </p><p>Re P~ = lnFH + (Re m.} 2 </p><p>R2= ~((u .q- l)2-Ft, 2, </p><p>02 = tan l (u / (u + l)) </p><p>'k = ln[[I + (Re mp) 2 </p><p>- (Im mp) 2 - - 2 Re mp cos(2kr~/p)] 2 </p><p>+ [2 Im mp Re mp- 2 Im m r cos(2kx/p)] 2] </p><p>, [ Immp-s in(2kx/p) ] Re Qk = tan [~ee ~ 7 ~ ~p~ j </p><p>- tan- ' [ I-~ee mp + sin ( 2kx /p ) l ~-- - ~oos~J </p><p>Im Pk = </p><p>[ 2ImmpRemp-2ImmpCOS(2kzt/p) ] tan-' l+(Re 2 mo) - (Im rnp)- - 2 lm mp cos(2krt/p) </p><p>and </p><p>Im Qk = </p><p>lnr(Re mp - cos(2krr /p )) 2 + (lm mp + sin(2krt /p ))21 L(Re mp - cos(2krc /p )) 2 + (Im mp - - ~ J " </p><p>This transformation is shown in Fig. 3 for a 4Y slope. </p><p>0od'i ' -1.00 </p><p>-2.00 t </p><p>-3,00 t </p><p>-4.00 1 </p><p>y/b -5.oo t </p><p>-6.00 t </p><p>;7oJ Fig. 3. Conforrr </p><p>-3,00 -200 -100 000 1.00 2 I I i i i i I I I " </p><p>x/b -2,00 2.00 3.00 4.00 i / </p><p>hal mapping of the rectangular u, v-coordinate system in Fig. 2 into a 45 finite slope. </p><p>From equation (24b) on the boundary where ~ .... 0 and u=t , v=lml ' ( t )=O when t &gt; 1. When - l</p></li><li><p>SAVAGE: GRAVITY-INDUCED STRESSES 1N FINITE SLOPES 475 </p><p>Evaluation of the integrals requires application of gener- alized function theory [14] and a result from the Riemann-Hilbert boundary-value problem [12]. </p><p>The first integral </p><p>1 f - ' dt I i (w)=~ni _~t -w </p><p>is written </p><p>1 f+~[1-H( t+ l ) ]d t I, (w) = ~-~/ -= t -w </p><p>where H indicates the unit step function. The derivative of this integral with respect to w is </p><p>1 f+~[ l -H( t+ l ) ]d t I', (w) = ~n/~_ ~ (t - w) 2 </p><p>[ ] - - d 1 1 [1 H(t + 1)] ~ . 2hi _~ Integration by parts yields, </p><p>I~(w)- 2nil f+~H'(t+l_)dt_~ t - w 1 f +~6( t+ l )d t I [w---~] </p><p>2hi _~ t -- w 2hi </p><p>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, </p><p>1 f i _ _dt _ 1 ln [w+l ] . I t (w)=~ _~t -w 2hi </p><p>For the second integral, let </p><p>1 I l Imf ( t )d t I2(w)=l., , ~ , t --w </p><p>and by partial integration, </p><p>1 f+ l Imf ( t )dt IX (w)=~ 3-1 ( t -w) 2 </p><p>'f+ - Imf(t)d 2hi _ j _ 1 F!mf(1) Ira_f(- 1) 1 </p><p>2hi L l -w l+w J </p><p>1 f+t Imf ' ( t )dt +~ _, t -w </p><p>For - l</p></li><li><p>476 SAVAGE: GRAVITY-INDUCED STRESSES IN FINITE SLOPES </p><p>The stress function q~(w) is then </p><p>= ~ [1 - e 2in"P]i ln[w + 1] </p><p>ipgbp [ 1 + ( 1 - 2p )e 2,~,p ~ . L J" </p><p>ipg F l + (1- 2# )e 2~/p . . q-L i - ; (25) </p><p>Substitution of equation (25) and its derivative into equations (22a) and (22b), separation into real and imaginary parts, and addition and subtraction lead to </p><p>Re f(w) -iT(w) 4~' (w" ] + </p><p>Im,O,w)+ O .sinI2'b 0' ] # </p><p>+ pgy (26) 1--/~ </p><p>[f(w)-- y(w)-] , q a:.=.Ke[ 7 '~ ]~ (w)J +2 Red~(w) </p><p>-Im[~(w)+Cb(w)]sinI2(Op-O1)]+pgy (27) </p><p>and </p><p>Here, </p><p>. Ff(w)-y(w)-l, ,-1 a's=Jm~ f '(~- j (w,j </p><p>"] ,mI ,w) + o,w)l osp; 0')] </p><p>R Ff(w)-~w)] . . . . eL ff tw,--0 when v=0and </p><p>ReFf (w) - f (w) ] . . . . pg Imf(w)[RjT" L .7'Tw5 f f two,- i----~ Lnd </p><p>x {[~R22~]tcostOz + (O2- O,)/P] </p><p>- cos[02 + (02 - 0, + 2n)/P]+[1 - cos[(02 - O,)/p] </p><p>- [1 - 2U] [cos[(02 - 0, + 2n)/p] - cos[2n/p]]]} </p><p>when v &lt; 0, and further </p><p>i F/(w) -y(w)q . . . . mL 7( -~- Jq ' tw)=0 </p><p>(28) </p><p>when c = 0 , and </p><p>i rf(w)-T(w)q , pg lmf (w) [R , ] 'p m " -~ /4~ (w </p><p>[ .f'u,'J J ) - [--T LR~.J </p><p>x {[~ l [s in [O2 + (O2-O,)/P] </p><p>- sin[02 + (02 - 0, + 2n)/p]] - ~[sin[(02 - 0, )/p] </p><p>+ [1 - 2/~][sin[(02 - 0, + 2n)/p] - sin[Zn/p]]]} </p><p>when v &lt; 0. </p><p>Also, </p><p>Re 4~(w) - </p><p>Re[~(w) + q~(w)] - </p><p>pgb [ ~-2~ ] [[ l - cs(2n /p </p><p>ppgb +In R 2 sin(2n/p)] + [v - [1 - 2#] </p><p>8nil -~] </p><p>x [u sin(2n/p) - v cos(2n/p)]] + Pg F 1 -21~] 4L I -~ j </p><p>x [Ref(w)sin(2n/p) - lm/(w)cos(2n/p)] pg Imf (w) </p><p>411 --/~] </p><p>pgb [~-2~ ] ln R2 sin(2n /p </p><p>ppgb [I - 2~ ]u 4n L-(~- ~ j sin(Zn/p) </p><p>+ PgI~_2g~ ] Ref(w)sin(2n/p) </p><p>and </p><p>Im[7~(w) + q~(w)] - pgb I~-2~ ]O2 sin(2n /P </p><p>ppgb [1 - 2p] v sin(2n/p) 4n k l -~ J </p><p>Note the sign convention adopted in this paper; com- pressive stresses are negative. </p><p>Equations (26), (27) and (28) for gravity-induced stresses beneath finite slopes like that shown in Fig. 1 satisfy the conditions that shear and normal stresses parallel and perpendicular to the ground surface vanish. Normal and shear stresses on the ground surface are given by the transformation </p><p>I -a , . - a,.] "" + " [~]cos 2c, + at = 2 + a,.,. sin 2a, </p><p>a,. + a,. [-a,. - al-] a,, = ~ - L~ jcos 27 - a,, sin 2u, </p><p>and </p><p>O' ) . m O" X . </p><p>o~,~ = 2 sin 2a + ,, cos 2:~, </p></li><li><p>SAVAGE: GRAVITY-INDUCED STRESSES IN FINITE SLOPES 477 </p><p>or, using equations (26), (27) and (28) for v = 0 and u = t, as </p><p>"= p-gb 2n [1 - cos(2n /p )]02 </p><p>n Imf ( t ) cos(2n/p)l + ~ </p><p>+ ~ ~ cos 2~ </p><p>(29) </p><p>- ' F ' - 2 I 1-cos(2.u)lo= pgb i~ L l - ~ j t </p><p>n Imf ( t ) cos(2nlp) + 2[ n Ref ( t ) + ~ b </p><p>PU2 lnR2]sin(2n/p)} </p><p>1[1- 2#] Imf ( t ) 2 L 1 - ~ j - -T - cos 2ct (30) </p><p>pgb - 2n [02 sin(2n /p )] </p><p>1 - +~-b [~2~]Imf(t)[sin2~-sin(2n/p)] (31) </p><p>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 ground surface to the left of the toe of the slope in Fig. 1, v = 0, u = t &lt; - 1, y/b = - 1, and </p><p>=0. On the face of the slope, v=0 and u=t , - 1 &lt; t &lt; 1 and a = fl = n/p (the slope angle shown in Fig. 1). Finally, u = t &gt; 1 and v = 0, a = 0, x/b &gt; 0 and y/b = 0 on the ground surface to the right of the top corner of the slope. </p><p>On the ground surface to the left of the toe of the slope, where a = 0, equation (30) reduces to </p><p>". - ' f ' - + ['&lt; pgb pg~-~ L 1-,u JL b </p><p>-in R2-2]sin(2nlp)] and equations (29) and (31) reduce to </p><p>an = a~.=O </p><p>and </p><p>O'Tn = if-D' = O. </p><p>As t ~ - ~ , x --, - ~ , and the horizontal stress becomes </p><p>a, 1 + - + In p pgb n p/2 - I </p><p>- ~ ln[2[ l -cos(2kn/p) ] ]cos(2kn/p) k=l </p><p>k sin(2kn/p) sin(2rGp) P k=l </p><p>As x~-~, for slopes of 15, 30, 45 and 90 (p = 12,6,4,2), the normalized 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 of the toe of the slope in Fig. 1 the horizontal surface stress is zero when Poisson's ratio, ~, is 1/2 and tensile when </p><p>is less than 1/2. This tension can be attributed to downward movement and near-surface horizontal exten- sion to the left of the toe of the slope in response to the gravity loading caused by the slope. </p><p>For the face of the slope, where v = 0, u = t and - 1 &lt; t &lt; 1, a~, a n and a~n are given by equations (29), (30) and (31) with 01 =-n , 02= 0 and a =f l = nip. Equations (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. Equation (30) gives the normal stress, a~, parallel to the slope surface as </p><p>o- t </p><p>pgb i[l-2~']l-F'~Ref(t) P" inns] </p><p>L I - . I LL b 2 </p><p>x sin(2n/p)] </p><p>lb [~---2l~l~ ]Im f( t )cos(2n /p ). </p><p>In the 90 case, surface stresses for - 1 &lt; t &lt; 1 reduce to a, n=axy=a n=ax=O and </p><p>a~_ % _1 [ l -2P l lm_ f ( t ) . (32) pgb pgb n L l -p ] b </p><p>To the right of the top corner of the slope in Fig. 1, whereu=t /&gt; l andv=0, x /b&gt;&gt;-0andy/b=0, a=0; equations (29), (30) and (31) reduce to </p><p>O" z O" x </p><p>pgb pgb </p><p>and </p><p>[ 1-2Pl[[nRef(w) 7~--~AL L b - In R2- 2 ]s in (2n /p) ] </p><p>a, = a,.= 0 </p><p>O'rn ~ O'~, ~ 0 . </p><p>For a 90 slope, the horizontal stress for x/b &gt;~ 0 reduces to zero. As t~, x ~ and the horizontal stress becomes </p><p>pgb n p/2 1 </p><p>- ~ ln[2[1 - cos(2kn/p)]]cos(2kn/p) k=l </p><p>2; p/2~l sin(2kn/p)}sin(2n/p) </p><p>which is zero when Poisson's ratio, p, is 1/2. As x~oo, for slopes of 15, 30 and 45 (p = 12, 6 and 4), the horizontal stress has respective limiting values given by -0 .377, -0 .780 and -0 .882 times [ (1 -2p) / ( l - /~) ] . </p></li><li><p>478 SAVAGE: ( IRAVITY-INDUCEI) STRESSES IN FINITE SLOP[S </p><p>200 </p><p>....</p></li></ul>