J. Mt. Sc. (011) 8: 718 77 DOI: 10.1007/s1169-011-070- A Method Combnng Numercal Analyss and Lmt Equlbrum Theory to Determne Potental Slp Surfaces n Sol Slopes XIAO Shguo 1 *, YAN Lpng, CHENG Zhqang 1 Department of Geologcal Engneerng, Southwest Jaotong Unversty, Chengdu 61001, Chna Los Angeles Department of Water and Power, Los Angeles, USA Department of Appled Mechancs and Engneerng, Southwest Jaotong Unversty, Chengdu 61001, Chna *Correspondng author, e-mal: xaoshguo16@16.com Scence Press and Insttute of Mountan Hazards and Envronment, CAS and Sprnger-Verlag Berln Hedelberg 011 Abstract: Ths paper descrbes a precse method combnng numercal analyss and lmt equlbrum theory to determne potental slp surfaces n sol slopes. In ths method, the drecton of the crtcal slp surface at any pont n a slope s determned usng the Coulomb s strength prncple and the extremum prncple based on the rato of the shear strength to the shear stress at that pont. The rato, whch s consdered as an analyss ndex, can be computed once the stress feld of the sol slope s obtaned. The crtcal slp drecton at any pont n the slope must be the tangental drecton of a potental slp surface passng through the pont. Therefore, startng from a pont on the top of the slope surface or on the horzontal segment outsde the slope toe, the ncrement wth a small dstance nto the slope s used to choose another pont and the correspondng slp drecton at the pont s computed. Connectng all the ponts used n the computaton forms a potental slp surface extng at the startng pont. Then the factor of safety for any potental slp surface can be computed usng lmt equlbrum method lke Spencer method. After factors of safety for all the potental slp surfaces Receved: 1 October 010 Accepted: 1 Aprl 011 are obtaned, the mnmum one s the factor of safety for the slope and the correspondng potental slp surface s the crtcal slp surface of the slope. The proposed method does not need to pre-assume the shape of potental slp surfaces. Thus t s sutable for any shape of slp surfaces. Moreover the method s very smple to be appled. Examples are presented n ths paper to llustrate the feasblty of the proposed method programmed n ANSYS software by macro commands. Keywords: Sol slope; Stress feld; Potental slp surface; Slope stablty; Factor of safety; Numercal analyss; Lmt equlbrum method; ANSYS software Introducton In slope stablty analyses, numercal methods have the advantage that cannot be replaced by the lmt equlbrum method as they can smulate accurately the stress feld and the dsplacement feld n a slope. Before applcaton of the shear strength reducton technque, the numercal 718 转载
J. Mt. Sc. (011) 8: 718 77 smulaton of the slope stablty was focused on the analyss of characterstcs of stress feld and dsplacement feld n the slope (Dng et al. 1995; Duncan 1996; Zhang et al. 1998) and there were no good methods to determne reasonably defned potental slp surfaces and the factor of safety for the slope. The shear strength reducton technque provdes a practcal method to compute slope factor of safety usng numercal methods. However, there exst some unsolved ssues and dfferent opnons relatng to the condton to end computaton (the nstablty crteron) for the technque (Lu et al. 005; Luan et al. 00; Zhao et al. 005). Some authors compute the stress on a slp surface by nterpolatng the results obtaned from a fnte element method (Gam and Donald 1988; Zou et al. 1995), and then compute the factor of safety usng the force equlbrum equaton. Ths method avods some of the assumptons adopted n the lmt equlbrum method, but t s qute complcated and not easy for realzaton. On the bass of the prncple of optmalty, along wth the method of slces, a crtcal slp feld (CSF) n a slope s postulated recently (Zhu 001). The method can determne the mnmum factor of safety of a slope, but t s not easy to analyze complcated slope and t stll belongs to a knd of lmt equlbrum method. Some authors also propose to compute the slope factor of safety usng the fnte arc searchng method (Yn and Lv 005). Ths method needs to assume the slp surface to be a crcular arc shape and s not sutable for slp surfaces wth other shapes. In realty, the potental slp surfaces of sol slopes are not just the same as crcular arc surfaces. Assumng crcular slp surfaces s the obvous defect of the classcal lmt equlbrum method. For ths reason, a numercal method s stll chosen n ths paper. After the stress feld n the slope s obtaned usng the ANSYS software, the crtcal slp drecton of any pont n the slope can be defned by the strength prncple of the slope materal; then postons of the potental slp surfaces can be determned and factors of safety can be further computed usng lmt equlbrum method. By comparng the factors of safety, the slope stablty can be evaluated and no assumpton of the shape of slp surfaces s needed. The proposed method can reasonably well predct potental slp surfaces and t can be convenently realzed n computer programmng. 1 Analyss of Potental Slp Drecton 1.1 Basc assumptons (1) The geometrc shape and the nature of loadng for the slope satsfy the condtons of the plane stran state; thus the slope model can be smplfed as a plane stran problem. () Coulomb s strength prncple s approprate for the slope materal. 1. Defnton of factor of safety at a pont Accordng to the shear strength crteron descrbed n Rock and Sol Mechancs, whether the nstablty occurs at a pont n the slope depends on the shear stress and the shear strength at the pont. The factor of safety at the pont (F s ) s defned as the rato of the shear strength (S ) to the shear stress (τ ): F s =S /τ (1) For any pont n the slope, f F s >1, t means the pont s stable; f F s =1, the pont s n the crtcal state of nstablty. 1. Determnaton of potental slp drecton The magntude of shear stress n a pont s relevant to ts drecton. Accordng to the Coulomb s strength prncple, the magntude of shear strength n a pont s also relevant to ts drecton. Therefore, the factor of safety at a pont computed from Equaton (1) s relevant to ts drecton. In other word, the factor of safety F s of a pont s a functon of ts drecton. The stress state of a unt body for any pont n a slope nduced by self weght of sol or other appled loads on the slope s shown n Fgure 1. At ths pont, σ x, σ y, τ xy are the normal stress n the x- axs drecton, the normal stress n the y-axs drecton, the shear stress, respectvely; and they can be computed by nterpolaton of the stress feld for the neghborng elements obtaned from a numercal smulaton. The normal stress and shear stress on an oblque secton of the unt body n Fgure 1 are expressed by Equaton () and Equaton (), respectvely. The normal drecton of the oblque secton has an angle of α wth the 719
J. Mt. Sc. (011) 8: 718 77 horzontal drecton. coeffcent t on both sdes of Equaton (6) and settng to zero, an extremum equaton s derved as: X t X t X 0 (8) 1 = where X 1, X, X are coeffcents as expressed n Equaton (9): σ α = ( σ + σ ) + ( σ σ ) cos α τ sn α () x y x y + τ α = ( σ σ ) sn α τ cos α () x y Accordng to the Coulomb s strength prncple, the shear strength along the oblque secton s expressed as: S α c + σ α xy = tanφ (4) where c and φ are coheson and nternal frcton angle of the slope materal at the pont. Substtutng Equatons ()-(4) nto Equaton (1) leads to the followng expresson for the factor of safety at the pont n the slope: F s = S α τ α = = c + σ α tanφ ( σ σ ) sn α τ cos α (5) F s Fgure 1 Stress state of unt body n slope ( ) [ ] x By further arrangement, one can get: y = [( A1 A ) t + At + A1 + A ] ( at + at a) (6) where coeffcent t s equal to tanα, and A 1, A, A, a 1, a, a are the coeffcents expressed n Equaton (7). A = c 1 + a A = a tanφ A = a tanφ 1 tanφ a = ( σ + σ ) a a 1 = ( σ σ ) = τ xy x x y y xy xy (7) It s obvous that the factor of safety F s of a pont changes wth angle α n Equaton (5). The value of α ranges between 0 and 180. To be meanngful, F s s taken to be postve. The mnmum value of F s s the factor of safety at the pont and the correspondng angle α s the crtcal slp drecton. Performng dervatve operaton wth the X1 = ( A1 A ) a Aa X = A1 a X = + + Aa ( A1 A ) a (9) Thus, t can be obtaned by solvng Equaton (8): ( X ± X 4X1X ) ( X1) t = + (10) Wth t computed by Equaton (10), the mnmum value of F s s obtaned by Equaton (6). As there exst two solutons of t, there are two crtcal slp drectons. By the symmetry of the unt body, the two drectons are symmetrc about the vertcal axs and the absolute values of the two factor of safety along the two drectons must be equal. The two solutons of t are represented by t 1 and t and are computed by Equatons (11) and (1): ( X + X 4X1X ) ( X1) ( X X 4X1X ) ( X1) t = + (11) 1 t = + (1) The above-dscussed stress analyss s based on a unt body and the slope drecton s not consdered. The dfferent slope drecton wll nduce dfferent slp drecton. Practcally a slope always slps or has a slp tendency forward and down n the free surface. For a two-dmenson problem, there are two knds of confguratons, called left-up and rght-down slope wth sgn \ and rght-up and left-down slope wth sgn /. When angle α vares between 0 and 90 n the unt body shown n Fgure 1, t corresponds to left-up and rght-down slope and the crtcal slp drectons of the ponts on the slope are between 0 and 90. When angle α vares between 90 and 180 n the unt body, t corresponds to rght-up and left-down slope and the crtcal slp drectons of the ponts on the slope are between 90 and 180. It should be notced that the crtcal slp drecton on the horzontal segment before the slope toe s n the range of 90 <α<180 for the left-up and rghtdown slope, whereas t s n the range of 0 <α< 90 for the rght-up and left-down slope. In 70
J. Mt. Sc. (011) 8: 718 77 Equaton (10), takng the postve sgn on the rght hand sde corresponds to the left-up and rghtdown slope and takng the negatve sgn on the rght hand sde corresponds to the rght-up and left-down slope. In practcal stuatons, t s selfevdent that t has unque value when the slope drecton s defned. Therefore, the crtcal slp drecton of a pont n the slope s unque. Based on the coordnate system of the unt body shown n Fgure 1, substtutng t = t 1 nto Equaton (6) results n the mnmum factor of safety for the left-up and rght-down slope, whch s a postve value; substtutng t=t nto Equaton (6) results n the mnmum factor of safety for the rght-up and left-down slope, whch s a negatve value and the absolute value can be used for practcal purposes. For smplfcaton, only the left-up and rght-down slope model s appled by usng coordnate transformaton. Therefore, Equaton (11) s used drectly to determne the crtcal slp drecton n a pont and then the mnmum factor of safety at the pont can be computed by Equaton (6). For many avalable fnte element software tools, the basc nformaton obtaned from a fnte element stress analyss are σ x, σ y, and τ xy wthn each element. The known σ x, σ y, and τ xy at the Gauss numercal ntegraton ponts n each element are projected to the nodes and then averaged at each node. Wth the σ x, σ y, and τ xy known at the nodes, the stresses can be computed at any other ponts wthn the element. Snce the stress at a selected pont can be unquely worked out by some numercal smulaton methods, the crtcal slp drecton at the pont n a sol slope can be also unquely determned. After the crtcal slp drectons at the selected ponts n the slope have been defned, these ponts are connected wth a smooth curve along the crtcal slp drectons and ths curve s the potental slp surface for the two-dmenson problem. Because the potental slp surface s determned by computaton, there s no need to make any assumptons. There are countless potental slp surfaces for a slope. Accordng to the geometry and loadng condtons of a practcal slope, a slp surface always passes through the free surface or the horzontal segment outsde the slope toe. Therefore, computaton can started from a pont on the free surface or the horzontal segment outsde the slope toe. Frst, the crtcal slp drecton of the startng pont s determned, then the poston of another pont s obtaned by advancng a small dstance step along ths crtcal slp drecton nto the slope; and then the crtcal slp drecton of the obtaned pont s determned. Repeatng ths process untl reachng the top of slope can form a potental slp surface wth the ext pont at the startng pont for computaton. Choosng another poston of the startng pont for computaton (ext ponts of sldng surface), another potental slp surface can be obtaned by the same process. After further computaton of the factors of safety for all slp surfaces, the stablty of the slope can be evaluated quanttatvely. Computaton of Factor of Safety The above paragraphs descrbe a method to determne a potental slp surface n a sol slope and to compute the factor of safety of a pont n the slope. It should be mentoned that whether nstablty occurs n the slope s not determned by the factor of safety of a pont, but the factor of safety of a potental slp surface. In other words, when nstablty occurs at some ponts n the slope, the potental slp surface may not have the stablty falure; vce versa, when nstablty occurs on the potental slp surface, nstablty occurs surely at each pont on the potental slp surface. A slp surface conssts of a group of ponts, whose stablty s evaluated by the rato of the shear strength to the shear stress at the pont. To fully represent the stablty of the slp surface that s formed by connectng the ponts, the factor of safety of a potental slp surface F s can be computed usng the lmt equlbrum method after the potental slp surface s determned by the abovedescrbed method. Consderng the potental slp surfaces beng not crcular, Spencer method (Spencer 1967) whch assumes a potental sldng mass to be dvded nto a number of vertcal slces and parallel nter-slce forces can be properly used to calculate the factor of safety of the potental slp surface. In fact, the approach s combnaton of the numercal smulaton method and the lmt equlbrum method. So t can be called the composte method. After the factor of safety of each potental slp surface s obtaned, the mnmum 71
J. Mt. Sc. (011) 8: 718 77 value s the factor of safety of the slope and the correspondng potental slp surface s the crtcal slp surface. Usng the above-descrbed method, the factor of safety of each pont on the potental slp surface can be computed at the same tme as the factor of safety of the potental slp surface s obtaned. It reveals quanttatvely the fact that the factors of safety are dfferent at dfferent ponts on the same potental slp surface, whch represents the status of stablty at each pont on the potental slp surface and s helpful to determne the gradual Fgure Programmng flow chart of the proposed method n ths paper 7
J. Mt. Sc. (011) 8: 718 77 falure process n the slope. Ths can not be represented by the shear strength reducton method. The proposed method n ths paper s benefcal for the analyss of relatve stablty at dfferent segments along the potental slp surface n the slope and thus t enables the comprehenson of the mechansm of the gradual falure process n the slope and stablzaton measures can be taken n segments for the complete slope treatment. The programmng flow chart of the proposed method n ths paper s shown n Fgure. Analyses of Practcal Examples To llustrate the use of the proposed method, two sol slopes are chosen as examples for analyss. 1) Example 1 A classcal sol slope (Hassots et al. 1997; Hull and Puolos 1999) wth ts dmensons n m s shown n Fgure. It has a heght of 1.7 m, a slope angle of 0. The slope sol has a unt weght of 19.6 kn/m, an nternal frcton angle of 10, a coheson of.94 kpa, an elastcs modulus of 50 MPa, and a Posson rato of 0.5. The load consdered s only the self weght of sol slope. condtons, the left and rght sdes of model were mposed wth horzontal dsplacement constrant only and the bottom of model was mposed wth both horzontal and vertcal dsplacement constrants. After the stress feld of the slope was obtaned from the numercal analyss, the potental slp surfaces were determned and the factors of safety were computed usng the proposed method that was realzed by the macro commands programmed n ANSYS software. Some of the potental slp surfaces and ther factors of safety computed usng Spencer method are shown n Fgure 4. Surface C s the crtcal slp surface and ts factor of safety s 1.056. For the same slope, slope stablty was analyzed usng shear strength reducton method (SSRM), smplfed Bshop method (SBM), and smplfed Janbu method (SJM); and the resulted factors of safety are 1.100, 1.110, 1.01, respectvely. The crtcal slp surfaces obtaned from the three methods are shown n Fgure 5. Fgure 4 Example 1 Results of the proposed method n Fgure Fnte element model of slope n Example 1 A fnte-element model was developed usng the ANSYS software to analyze the stress feld of the above sol slope. ANSYS software s a numercal smulaton software wth fnte element method developed by ANSYS Incorporaton, USA. It can carry out many mechancal computatons ncludng geomechanc problems. In the paper, the slope materal was smulated wth the elastoplastc consttutve relatonshp based on a nonassocatve plastc flow rule and the Mohr- Coulomb s falure crteron. The fnte-element dscretzaton usng 6-noded trangular elements s also shown n Fgure. For the boundary By the comparson of Fgure 4 and Fgure 5, t can be notced that there are some dfferences n the predcted crtcal slp surfaces between the proposed method and the other three methods. The shape of the crtcal slp surface predcted by the proposed method s not crcular arc, whch s concdent wth the actual stuaton n projects. The dstrbuton of factor of safety of a pont on the crtcal slp surface along the dstance from the slope toe s shown n Fgure 6. It can be seen that the stablty at these ponts on the crtcal slp surface s not the same, whch means they do not reach the lmt equlbrum state at the same tme and the occurrence of falure s not smultaneous at all the ponts on the crtcal slp surface but wth a sequental gradual process. For the dscussed example, the safety factors of the upper part of slope are larger than that of the mddle and lower 7
J. Mt. Sc. (011) 8: 718 77 (a) SSRM (b) Smplfed Bshop method (c) Smplfed Janbu method Fgure 5 Results of the classc methods n Example part of slope, whch ndcates that slope falure mode that begns at the mddle and upper part of slope s a type of towng shear-sldng rupture. The slope factors of safety computed by dfferent methods are shown n Table 1. Except that the slope safety factor computed by SJM s relatvely small, the other results are smlar. The Fgure 6 Dstrbuton of factor of safety of pont on the crtcal slp surface n Example 1 slope factor of safety computed by the proposed method s very close to that computed by the frcton crcle method, whch s a good verfcaton for the proposed method. For the above example, the nfluence of the step sze Ls on the slope factor of safety Fs s shown n Fgure 7. As the step sze ncreases, the slope factor of safety gradually decreases n an approxmately exponental fashon. The slope factor of safety s of hgh precson at smaller step sze. The slope factor of safety s 1.06 when the step sze s 0.1 m, whereas t s 1.04 when the step sze s.4 m. The slope factor of safety vares among varous step szes. In order to mantan enough precson for a practcal slope stablty problem, we suggest the step sze should not be more than 0.5 m when usng the proposed method. ) Example A sol slope wth ts szes n m s shown n Fgure 8. The slope materal parameters are shown n Table. Only loadng due to self weght of the sol s consdered n the analyss. 74
J. Mt. Sc. (011) 8: 718 77 Table 1 Comparson of slope factors of safety computed by dfferent methods n Example 1 Calculaton method Safety factor Frcton crcle method (Hassots et al. 1997) Taylor Method (Hassots et al. 1999) Lmt analyss upper bound method (Nan et al. 005) Smplfed Bshop method Smplfed Janbu method Shear strength reducton method Proposed method n ths paper 1.080 1.110 1.107 1.110 1.01 1.100 1.056 Table Materal parameters of the slope n Example Stratum Unt weght (kn/m ) Coheson (kpa) Angle of nternal frcton (degrees) Elastc modulus (MPa) Posson s rato Sol 1 18 7. 17.9 50 0. Sol 18.5 14.1 16.7 40 0.5 Sol 19 16.1 14.0 100 0.4 Fgure 9 Example Results of the proposed method n Fgure 7 Influence of the step sze Ls on the slope factor of safety Fs n Example 1 Fgure 8 Fnte element model of slope n Example Usng the proposed method, the predcted potental slp surfaces and ther slope factors of safety are shown n Fgure 9. Surface D s the crtcal slp surface and ts factor of safety s 1.14. For the same slope, slope stablty was analyzed usng SSRM, SBM, and SJM; and the resultng factor of safety s 1.090, 1.087, 1.011, respectvely. The crtcal slp surfaces obtaned from the three methods are shown n Fgure 10. The slope factor of safety computed by the proposed method s only 4.% bgger than that computed by the smplfed Bshop method, whch s a good verfcaton for the proposed method. Comparng Fgure 9 wth Fgure 10, t s obvous that there are some dfferences on the predcted crtcal slp surfaces between the proposed method and the other three methods. The shape of the crtcal slp surface predcted by the proposed method s not crcular arc too. The dstrbuton of factor of safety of a pont on the crtcal slde surface s shown n Fgure 11. It also ndcates that the slope falure s a gradual process. For the dscussed example, the factors of safety of a pont n the behnd part of slope are larger than those n the forehead, whch means the slope falure ntates at the forehead of slope and the falure mode belongs to the type of towng shearsldng rupture. Accordng to the analyss results of the two examples descrbed above, t s obvous that the shape of the slp surface determned usng the proposed method s not crcular arc, whch s dfferent from those determned usng some lmt equlbrum methods. Moreover the poston of the 75
J. Mt. Sc. (011) 8: 718 77 (a) SSRM (b) Smplfed Bshop method (c) Smplfed Janbu method Fgure 10 Results of the classc methods n Example should be the connecton curve of ponts along ther crtcal slp drectons. Normally, ths curve s not crcular arc. Therefore, the assumpton about the crcular arc n the classcal methods s not accurate. 4 Conclusons Fgure 11 Dstrbuton of factor of safety of pont on the crtcal slde surface n Example crtcal slp surface obtaned usng the proposed method s dfferent from those obtaned usng the classc methods. The slope factor of safety computed from the proposed method s slghtly larger than those computed from the classcal methods. Based on the analyss of loadng condton n the slope, the crtcal slp surface After the stress feld of a sol slope s obtaned from numercal smulatons, the rato of shear strength to shear stress at any pont n the slope can be used as an analyss ndex and the crtcal sldng falure drecton at that pont can be determned by utlzng the Coulomb s strength prncple ncorporated wth the extremum prncple. The crtcal sldng falure drecton s n the tangental drecton of the potental slp surface passng through that pont. Startng from a pont on the slope surface or on the horzontal segment outsde the slope toe, the reverse computaton wth 76
J. Mt. Sc. (011) 8: 718 77 a small dstance ncrement step s performed to advance nto another pont n the slope. Ths process s repeated step by step untl a potental slp surface s formed. Choosng another startng pont can result n a dfferent potental slp surface wth the same process. Comparng the factors of safety computed usng all potental slp surfaces, the slope stablty can be evaluated quanttatvely. Snce there s no need to make any assumptons on the shape of potental slp surfaces, the proposed method s sutable for any shapes of potental slp surfaces. Furthermore, the varaton of the factor of safety of pont along the crtcal slp surface can be obtaned. Therefore, the characterstcs of the gradual falure of the slope can be assessed. The feasblty of the proposed method has been valdated by the two practcal examples dscussed n the paper. It s necessary to state that although the proposed method for determnng a potental slp surface s elucdated for sol slopes n ths paper, t s also applcable to rock slopes as long as the stress feld of rock slopes s obtaned accurately. The analyss n ths paper s only based on two-dmensonal slopes. For three-dmensonal slopes, further study s needed for the applcaton of the proposed method. References Dng X, Xu P, Xa X (1995) Rheologc deformaton and stablty analyss on rock mass durng unloadng excavaton process for hgh slope n lock area of the Three Gorges Project. Journal of Yangtze Rver Scentfc Research Insttute,1(4): 7-4. (In Chnese) Duncan JM (1996) State of the art: lmt equlbrum and fnte element analyss of slopes. Journal of Geotechncal Engneerng, 1(7): 477-596. Gam SK, Donald IB (1988) Determnaton of crtcal slp surfaces for slopes va stress-stran calculatons. In: 5th Australa New Zealand Conference Geomechancs 461-464, Sydney. Hassots S, Chmaeau JL, Gunarante M (1997) Desgn method for stablzaton of slopes wth ples. Journal of Geotechncal and Geoenvronmental Engneerng 1(4): 14-. Hassots S, Chmaeau JL, Gunarante M (1999) Desgn method for stablzaton of slopes wth ples (closure). Journal of Geotechncal and Geoenvronmental Engneerng 15(10): 91-914. Hull TS, Puolos H G (1999) Desgn method for stablzaton of slopes wth Ples (dscusson). Journal of Geotechncal and Geoenvronmental Engneerng 15(10): 911-91. Lu J, Luan M, Zhao S, Yuan F, Wang J (005) Dscusson on crtera for evaluatng stablty of slope n elastoplastc FEM based on shear strength reducton technque. Rock and Sol Mechancs 6(8): 145-148. (In Chnese) Luan M, Wu Y, Nan T (00) A crteron for evaluatng slope stablty based on development of plastc zone by shear strength reducton FEM. Journal of Dsaster Preventon and Mtgaton Engneerng (): 1-8. (In Chnese) Nan T, Luan M, Yang Q (005) Stablty analyss of slopes wth stablzng ples and ther smplfed desgn. Chnese Journal of Rock Mechancs and Engneerng 4(19): 47-4. (In Chnese) Spencer E (1967) A method of analyss of embankments assumng parallel nter-slce forces. Geotechnque 17: 11-6. Yn Z, Lv Q (005) Fnte element analyss of sol slope based on crcular slp surface assumpton. Rock and Sol Mechancs 6(10): 155-159. (In Chnese) Zhang Q, Zhu W, Chen W (1998) Analyss of elasto-plastc damage for hgh jonted slope of the Three Gorges Project shp lock durng unloadng due to excavaton. Journal of Hydraulc Engneerng (8): 19-. (In Chnese) Zhao S, Zheng Y, Zhang Y (005) Study on slope falure crteron n strength reducton fnte element method. Rock and Sol Mechancs, 6(): -6. (In Chnese) Zhu D (001) A method for locatng crtcal slp surfaces n slope stablty analyss. Canadan Geotechncal Journal 8(): 8-7. Zou JZ, Wllams DJ, Xong WL (1995) Search for crtcal slp surface based on fnte element method. Canadan Geotechncal Journal (1): -46. 77