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andanalyses.boundsthemethodof afailurecurveproblemof slidinganglescomputedconsistentstressverticalthe nonlinearCE Database subject headings: Slope stability; Soils; Limit analysis; Nonlinear analysis; Optimization; Failures.IntroductionThe determination of the slope stability is a very important issueto geotechnical engineers. Many researchers have attempted todevelop and elaborate the methods for slope stability evaluation.The proposed methods in the past for stability analysis may beclassified into the following four categories: 1! the limit equilib-rium including the traditional slices method, 2! the characteristicline method, 3! the limit analysis method including upper andlower bound approaches, and 4! the finite element or finite dif-ference numerical techniques. Among them, the slices method hasalmost dominated the geotechnical profession for estimating thestability of soil and rock slopes. This is due to the fact that theslices method is very simple, a lot of experiences have been ac-cumulated on the use of the method, and the method is the mostknown and widely accepted by practicing engineers. By dividingthe soil mass into slices and making certain assumptions regard-ing the interslice forces to satisfy the force/moment equilibriumconditions, the safety factor can be easily obtained. Since the firstslices method for slope stability problems appeared in 1927, ex-tensive research Janbu 1957; Morgenstern and Price 1965; Spen-cer 1967; Bishop 1995! was done on the development of theslices method. However, none of the slices methods can be con-sidered rigorous due to the arbitrary assumptions regarding theinterslice forces. Proof is given neither to that a statically admis-sible stress field is satisfied within the slices, nor to that the as-sumed failure mechanism is kinematically admissible. After hav-ing reviewed extensive literatures, Nash 1987! considered that Nequations of the Mohr-Coulomb MC! failure criteria and 3Nstatic equilibrium equations are available when a slope is dividedinto N slices with 6N22 unknowns. Thus, 2N22 assumptionsregarding the interslice forces are required to be made for theproblem to be solvable. Different assumptions lead to variousslices techniques. In recent years, many efforts have been made toavoid these assumptions. A limit analysis method, as an alterativemethod, for slope stability analyses has been used by many re-searchers Chen 1975; Sloan 1989; Michalowski 1994, 1995,2002; Donald and Chen 1997; Kim et al. 1999, 2002; Zhang1999!. By the principle of virtual work, a system of rigid slicesor wedges! is in an equilibrium state if the virtual work done byall external loads is equal to the internal energy dissipated for thevirtual displacements of the system consistent with the con-straints. The upper bound theorem established on the principle of1Lecturer, Dept. of Civil and Architectural Engineering,Changsha Railway College, The Central South Univ., Hunan 410075,China; Visiting Scholar, Dept. of Civil and Structural Engineering, TheHong Kong Polytechnic Univ., Hong Kong, China. E-mail:2Professor, Dept. of Civil and Structural Engineering, The Hong KongPolytechnic Univ., Hung Hom, Kowloon, Hong Kong, China. E-mail:.hkNote. Associate Editor: Victor N. Kaliakin. Discussion open until Au-gust 1, 2004. Separate discussions must be submitted for individual pa-pers. To extend the closing date by one month, a written request must befiled with the ASCE Managing Editor. The manuscript for this paper wassubmitted for review and possible publication on December 18, 2002;approved on September 23, 2003. This paper is part of the Journal ofEngineering Mechanics, Vol. 130, No. 3, March 1, 2004. ASCE, ISSN0733-9399/2004/3-267273/$18.00.JOURNAL OF ENGINEERING MECHANICS ASCE / MARCH 2004 / 267Slope Stability Analysis withXiao-Li Yang1Abstract: A linear failure criterion is widely used in slope stabilityhas the nature of nonlinearity. This paper computes rigorous upperwith a nonlinear yield criterion by employing the upper bound theoremlinear Mohr-Coulomb MC! failure criterion which circumscribesactual stability factor or limit load!. In this paper, an improvednonlinear failure criterion is proposed to estimate the stability factorthe generalized tangential technique, the curve of the nonlinearlinear MC failure criterion. The straight line is tangential to thelinear MC failure criterion is employed to formulate the slope stabilityformulated in this way is minimized with respect to the locationslope stability problems a homogeneous soil slope with two slopethe proposed method. For the soil slope with two slope angles, thecomparison shows that the proposed method gives reasonable andslope with a tension crack, a statically admissible stress field is constructedfailure criterion. Lower bound solutions are obtained by satisfyingthe proposed method are equal to the lower bound solutions for theproposed approach. The influences of the strength parameters indiscussed in this paper.DOI: 10.1061/ASCE130:3267!Nonlinear Failure CriterionJian-Hua Yin2However, the strength envelope of almost all geomaterialson slope stability factors under the condition of plane strainof plasticity. A stability factor or a limit load! computed using aactual nonlinear failure criterion is an upper bound value of theusing a generalized tangential technique to approximate aslope on the basis of the upper bound theorem of plasticity. Usingcriterion is simplified as a set of straight lines according to theof the nonlinear failure criterion. The set of straight lines of theas a classical optimization problem. The objective functionbody center and the location of tangency point. Two typicaland a vertical cut slope with a tension crack! are analyzed usingresults are compared with published solutions by others. Thevalues of the stability factor of the slope. For the vertical cutfor the slope. The stress field does not violate the nonlinearequilibrium conditions. The upper bound solutions obtained fromcut slope. The agreement further supports the validation of thecriterion on the stability of slopes are also studied andvirtual work, which assumes a perfectly plastic soil model with anassociated flow rule, states that the internal energy dissipated byany kinematically admissible velocity field can be equated to thework done by external loads, and so enables a strict upper boundon the actual safety factor or stability factor to be deduced, wherethe equilibrium of forces is satisfied and no assumption regardingthe interslice forces is required.Until now, a linear MC failure criterion is commonly used inthe limit analysis of stability problems. The reason is probablythat a linear MC failure criterion can be expressed as circles. Thischaracteristic makes it possible to approximate the circles by afailure surface, which is a linear function of the stresses in thestress space for plane strain problems. Thus, based on the upperand lower bound theorems, formulations of the stability or bear-ing capacity problems are linear programming problems. Usingthe linear MC failure criterion, Lymser 1970!, Sloan 1989! andSloan and Kleeman 1995!, Kim et al. 1999, 2002! used the fi-nite element method and a linear programming method to solvethe stability or bearing capacity problems. In addition, the yieldsurfaces for both the von Mises and the linear MC failure criteriahave a linear form in the principal stress space for plane strainproblems. This property makes it possible to predict very pre-cisely failure mechanisms for geotechnical structures such asshallow foundations and slope stability problems. In general, theclassically assumed failure mechanisms are planar or circular forthe von Mises failure criterion and planar or logarithmic for thelinear MC failure criterion Chen 1975!.However, experiments have shown that the strength envelopeof geomaterials has the nature of nonlinearity Hoek 1983; Agaret al. 1985; Santarelli 1987!. When applying an upper boundtheorem to estimate the influences of a nonlinear failure criterionon bearing capacity or stability, the main problem, which manyresearchers have encountered, is how to calculate the rate of workdone by external forces and the rate of energy dissipation alongvelocity discontinuities. Suitable methods for solving this prob-lem can be mainly classified into two types. The first type ofmethod is using a variational calculus technique. Baker andFrydman 1983! applied the variational calculus technique to de-rive the governing equations for the bearing capacity of a stripfooting resting on the top horizontal surface of a slope. Zhang andChen 1987! converted the complex differential equations ob-tained using the variational calculus technique into an initial valueproblem and presented an effective numerical procedure, calledan inverse method, for solving a plane strain stability problemusing a general nonlinear failure criterion. They gave numericalresults of stability factors of a simple infinite homogenous slopewithout surcharge. The second type of method is using a tangen-tial technique. Drescher and Christopoulos 1988! and Collinset al. 1988! proposed a simpler alternative tangent techniqueto evaluate the stability factors of an infinite and homogeneousslope without surcharge. They showed that upper bound limitanalysis solutions could be obtained by means of a series of linearfailure surfaces which are tangent to an exceed the actual nonlin-ear failure surface, together with utilizing the previously calcu-lated linear stability factors, NL, given by Chen 1975!. The ad-vantage of the tangential technique is that it avoids thecalculations of work and energy dissipation, but it has to utilizethe previously calculated linear stability factors, NL, given byChen 1975!.This paper develops an improved method using a generalizedtangential technique. This method employs the tangential line alinear MC failure criterion!, instead of the actual nonlinear failurecriterion, to formulate the work and energy dissipation. Using the268 / JOURNAL OF ENGINEERING MECHANICS ASCE / MARCH 2004generalized tangential technique to evaluate stability factors, itis not necessary to use the previously calculated linear stabilityfactors given by Chen 1975!. This paper extends the work of theslope stability analysis using a linear failure criterion by Chen1975! to that using a nonlinear failure criterion.Upper Bound Solutions with a Nonlinear FailureCriterionIn an upper bound limit analysis, solutions depend on the choicesof kinematically admissible velocity fields. To obtain better solu-tions lower upper bounds!, work has to be done for trial kine-matically admissible velocity fields, as many as possible. Rota-tional failure mechanisms have been considered when using anupper bound approach Chen 1975!. In the stability analysis of aslope, comparing with different translational failure mechanisms,Chen 1975! concluded that a rotational failure mechanism is themost efficient one and that the rotational failure mechanisms leadto lower critical heights or stability factors than those obtained byusing other translational failure mechanisms. The kinematical ad-missibility condition in the upper bound theorem requires that therotational failure surface for a perfect-plastic body collapse mustbe a log-spiral surface log-spiral line for plane strain problems!.Basic ideas in Chen 1975! on the rotational log-spiral surfacesare adopted in the method of the paper.A Generalized Tangential TechniqueA limit load computed from a linear failure surface, which alwayscircumscribes the actual nonlinear failure surface, will be anupper bound value on the actual limit load Chen 1975!. This isdue to the fact that the strength of the circumscribing the actualnonlinear failure surface is equal to or larger than that of theactual failure surface. In the present analysis, a tangential line toa nonlinear failure criterion at point M is used and shown in Fig.1. It can be seen that the strength of the tangential line equals orexceeds that of a nonlinear failure criterion at the same normalstress. Thus, the linear failure criterion represented by the tangen-tial line will give an upper bound on the actual load for the ma-terial, whose failure is governed by a nonlinear failure criterion.In fact, many researchers Lymser 1970; Sloan 1989; Sloan andKleeman 1995; Yu et al. 1998; Kim et al. 1999, 2002! haveadopted this approach in their limit analyses. The linear MC fail-ure criterion may be expressed as a circle, the equation of whichis expressed as (sx2sy)21(2txy)252c cos w2(sx1sy) sin w#2in the stress space for a plane strain condition. In the expression,c and w are the cohesion and friction angle for a linear MC failurecriterion. It shall be noted that c and w shall be the effectivecohesion and effective friction angle in the case of water pres-sures in the soil. In order to use the upper bound theorem ofplasticity to formulate the stability problem as a linear program-Fig. 1. Tangential line for a nonlinear failure criterionming problem, it is necessary to approximate the circle by anexterior polygon, which always circumscribes the linear MC fail-ure criterion. The upper bound solutions corresponding to theexterior polygon are more than or equal to the upper bound solu-tions corresponding to the linear MC failure criterion by a kine-matical approach.In the past, the effects of pore water pressure were consideredin the calculation of upper bound solutions to slope stability prob-lems. In general, pore water pressures are obtained by numericalseepage analyses, a flow net method, or using a pore water pres-sure ratio. Miller and Hamilton 1989! considered pore waterpressures as internal forces, and used two kinds of failure mecha-nisms to analyze the slope stability. Michalowski 1995, 2002!and Kim et al. 1999, 2002! regarded the pore water pressures asexternal forces for clear physical meaning, and analyzed the ef-fects of pore water pressure on the slope stability. In the presentanalysis, the effects of pore water pressure are not considered, butit can be incorporated into the objective function so that the ef-fective stress analysis of saturated slopes can be done conve-niently, using the idea of Michalowski 1995, 2002!. Since nopore water pressures are considered in the following work, forsimplicity, total stresses and total strength parameters are usedinstead of effective ones.In the present study, a linear failure criterion represented in alinear MC diagram by a tangential line, which is always tangen-tial to the curve of a nonlinear failure criterion, is adopted forlimit analysis. In general, a nonlinear failure criterion can be ex-pressed as Zhang and Chen 1987; Drescher and Christopoulos1988!t5c011sn/st!1/m(1)where snand t5normal and shear stresses on the failure surface,respectively; and the parameter values of c0, st, and m are de-termined by tests. When m51, Eq. 1! reduces to the well-knownlinear MC failure criterion. If a stress state represented by a vec-tor from the origin is increased from zero, yielding will happenwhen the vector reaches the curve in the (sn,t) space. The tan-gential line to the curve at the location of tangency point M asshown in Fig. 1 is expressed ast5ct1sntan wt(2)where wt5tangential frictional angle; ct5intercept of the straightline on the t-axis; sn5normal stress; and t5shear stress. The ctand wtat point M are determined by the following two expres-sions:tan wt5dtdsn51mstc0S11sMstD12m!/m(3)ct5m21mc0Smsttan wtc0D1/12m!1sttan wt(4)In Eq. 3!, the stress sMis the value of normal stress at thetangency point M as shown in Fig. 1. In order to ensure that thetangential line always lies outside of the curve, and that thestrength corresponding to the tangential line is more than or equalto that of the corresponding nonlinear curve, the requirement ism.1 to be satisfied.The generalized tangential technique is that instead of usinga nonlinear failure criterion in Eq. 1!, a linear MC failure crite-rion in Eq. 2! with a tangential line to the nonlinear failure cri-terion is employed to calculate the rate of external work andenergy dissipation. The stability factor is obtained by minimizingthe objective function with respect to the location of tangencyJOURNALpoint and the location of sliding body center. In this way, weobtain an upper bound value of the stability factor of slopes.Rate of External Work and Energy DissipationBased on the upper bound theorem, equating the energy dissipa-tion rate along velocity discontinuities to the work rate of externalforce in any kinematically admissible velocity will lead to a sta-bility factor or a limit load! that is not less than or at most equalto the actual stability factor or limit load!. The best upper boundsolutions are obtained by an optimization method. In the presentstudy, the work rate of the external force is done by the self-weight of the soil mass, and the internal energy is dissipated alongvelocity discontinuities due to plastic shearing. Due to the usageof the tangential line a linear MC failure criterion!, a log-spiralfailure mechanism can be adopted in the upper bound analysis, asshown in Fig. 2. The region BAC rotates as a rigid body about thecenter of rotation O with the material below the logarithmic spiralsurface remaining at rest. The internal energy is only dissipatedalong the log-spiral sliding surface, while the external rate ofwork is done by the soil mass weight bounded by the boundaryline BAC and sliding surface.The work rate of external force self-weight! is formulated asWext5gEAVdA (5)where A5area of the cross section of the soil mass above thefailure surface; g5unit self-weight of the soil mass; and V5velocity jump vector along a log-spiral failure curve. The internalenergy dissipation rate along the velocity discontinuities is formu-l