Limit analysis versus limit equilibrium for slope stability.pdf

Limit analysis versus limit equilibrium for slope stability.pdf

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Unformatted text preview: LIMIT ANALYSIS VERSUS LIMIT EQUILIBRIUM FOR SLOPE STABILITY By H. S. Yu,l Member, ASCE, R. Salgado,2 Associate Member, ASCE, S. W. Sloan,3 and J. M. Kim‘ ABSTRACT: This paper compares conventional limit-equilibrium results with rigorous upper and lower bound solutions for the stability of simple earth slopes. The bounding solutions presented in this paper are obtained by using two newly developed numerical procedures that are based on finite-element formulations of the bound theorems of limit analysis and linear programming techniques. Although limit-equilibrium analysis is used widely in practice for estimating the stability of slopes, its use may sometimes lead to significant errors as both kinematic and static admissibility are violated in the method. Because there are no exact solutions available against which the results of limit-equilibrium analysis can be checked, the present bounding solutions can be used to benchmark the results of limit-equilibrium analysis. In addition, the lower bound solutions obtained also can be applied directly in practice because they are inherently conservative. INTRODUCTION Background and Objectives Because of its practical importance, the analysis of slope stability has received wide attention in the literature. Limit- equilibrium analysis has been the most popular method for slope stability calculations. A major advantage of this approach is that complex soil profiles, seepage, and a variety of loading conditions can be easily dealt with. Many comparisons of limit-equilibrium methods [see, for example, Whitman and Bailey (1967), Fredlund and Krahn (1977), Duncan and Wright (1980), and Nash (1987)] indicate that techniques that satisfy all conditions of global equilibrium give similar results. Regardless of the different assumptions about the interslice forces, these methods (such as those of Janbu, Spencer, and Morgenstem and Price) give values of the safety factor that differ by no more than 5%. Even though it does not satisfy all conditions of global equilibrium, Bishop’s simplified method also gives very similar results. Partly because of this and partly because of its simplicity, the slice method of limit-equilibrium analysis proposed by Bishop (1955) has been used widely for predicting slope stability under both drained and undrained loading conditions. Because of the approximate and somewhat arbitrary nature of limit-equilibrium analysis, concern is often voiced about how accurate these types of solutions really are. The answer to this question is particularly important in cases where designs are based on slim margins of safety. The following are objectives of the present paper: (1) To present rigorous lower and upper bound solutions for the sta- bility of simple slopes in both homogeneous and inhomoge- neous soils; and (2) to check the accuracy of the method of Bishop (1955) by comparing its solutions against those derived from limit analysis. To overcome the difficulties of manually constructing both statically admissible stress fields and kine- matically admissible velocity fields, two newly developed nu- merical procedures are used to calculate both upper and lower bound solutions for the stability of simple earth slopes in both ’Sr. Lect, Dept. of Civ. Engrg., Univ. of Newcastle, NSW 2308, Aus- tralia. 2Assist. Prof, School of Civ. Engrg., Purdue Univ., IN 47907. 3Assoc. Prof, Dept. of Civ. Engrg., Univ. of Newcastle, NSW 2308, Australia. ‘Grad. Student, School of Civ. Engrg., Purdue Univ., IN. Note. Discussion open until June 1, 1998. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on January 11. 1996. This paper is part of the Jour- nal of Geotechnical and Geoenvironmental Engineering, Vol. 124, No. 1, January, 1998. ©ASCE, ISSN 1090-0241/98/0001—0001—0011/S4.00 + $.50 per page. Paper No. 12469. purely cohesive and cohesive-frictional soils. These numerical procedures are based on finite-element formulations of the bound theorems of limit analysis and the best lower and upper bound solutions are obtained, respectively, by optimizing stat- ically admissible stress fields and kinematically admissible ve- locity fields using linear programming techniques. Over the years, many limit-equilibrium computer codes have been developed to locate the most critical failure surface by using various search strategies. In this study, the limit-equi- librium results are obtained from the well-known slope stabil- ity computer code STABL with the option of Bishop’s sim- plified method (Kim et al. 1997). Limit Analysis Method The limit analysis method models the soil as a perfectly plastic material obeying an associated flow rule. With this ide- alization of the soil behavior, two plastic bounding theorems (lower and upper bounds) can be proved (Drucker et a1. 1952; Chen 1975). According to the upper bound theorem, if a set of external loads acts on a failure mechanism and the work done by the external loads in an increment of displacement equals the work done by the internal stresses, the external loads obtained are not lower than the true collapse loads. It is noted that the external loads are not necessarily in equilibrium with the in— ternal stresses and the mechanism of failure is not necessarily the actual failure mechanism. By examining different mecha- nisms, the best (least) upper bound value may be found. The lower bound theorem states if an equilibrium distribution of stress covering the whole body can be found that balances a set of external loads on the stress boundary and is nowhere above the failure criterion of the material, the external loads are not higher than the true collapse loads. It is noted that in the lower bound theorem, the strain and displacements are not considered and that the state of stress is not necessarily the actual state of stress at collapse. By examining different ad- missible states of stress, the best (highest) lower bound value may be found. The bound theorems of limit analysis are particularly useful if both upper and lower bound solutions can be calculated, because the true collapse load can then be bracketed from above and below. This feature is invaluable in cases for which an exact solution cannot be determined (such as slope stability problems), because it provides a built-in error check on the accuracy of the approximate collapse load. Limit-Equilibrium Method The limit-equilibrium method has been used to analyze slope stability problems in soil mechanics for many years by JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / JANUARY 1998 /1 assuming that soil at failure obeys the perfectly plastic Mohr- Coulomb criterion [e.g., Fellenius (1926) and Terzaghi (1943)]. However, over the last 30 years the analysis of slopes using the limit-equilibrium method has been significantly re- fined by using various methods of vertical slices [e.g., Bishop (1955), Janbu (1954), Morgenstcrn and Price (1965), and Spencer (1967)]. An excellent review of popular limit-equilib- rium techniques for predicting slope stability can be found in Nash (1987) or Graham (1984). Using a global equilibrium condition, the limit-equilibrium approach is purely static as it neglects altogether the plastic flow rule for the soil (i.e., constitutive relation). If the soil at failure is assumed to be a rigid perfectly plastic material obey- ing an associated flow rule, then collapse mechanisms selected by the limit-equilibrium method are usually kinematically in- admissible. In addition, the static admissibility of the stress field also is not satisfied, because some arbitrary assumptions are made to remove statical indeterminacy and, more impor- tantly, only a global equilibrium condition (rather than equi- librium conditions at every point in the soil) is satisfied. Mich- alowski (1989) showed that upper bound solutions based on kinematically admissible rigid-block velocity fields (associated with the linear Mohr-Coulomb criterion) also satisfy global force equilibrium equations. Hence an upper bound limit anal- ysis solution also may be regarded as a special limit-equilib- rium solution but not vice versa. However, by no means. can these two methods be regarded as equivalent (Collins 1974; Chen 1975). Based on the bound theorems of limit analysis, it can be concluded that, in general, the limit-equilibrium method is of an approximate and arbitrary nature and the re- sults obtained from this method are neither upper bounds nor lower bounds on the true collapse loads. Any attempt to val- idate the limit-equilibrium approach by comparing different limit-equilibrium solutions, without reference to a more rig- orous analysis, is considered to be inconclusive. PROBLEM OF STABILITY OF SIMPLE SLOPES The slope geometry analyzed in this paper is shown in Fig. 1. TWO types of analysis are considered: undrained stability analysis of purely cohesive slopes and drained stability anal- ysis of cohesive-frictional slopes. The purely cohesive soil un- der undrained loading conditions is modeled by a rigid per- fectly plastic Tresca yield criterion with an associated flow rule. The strength of the cohesive soil is determined by the undrained shear strength S“, which may increase linearly with depth as is the case in normally consolidated clays (Gibson and Morgenstcrn 1962; Hunter and Schuster 1968). Under drained loading conditions, a perfectly plastic Mohr—Coulomb model is used to describe the soil behavior. For this case, the strength parameters are the effective cohesion c’ and the ef- fective friction angle d)’. Both of these quantities are assumed to be constant throughout the slope. For simplicity, the effect of seepage (or pore pressures) on the stability of cohesive- frictional slopes has not been included in this study. The so- lutions obtained are therefore only relevant for fully drained loading conditions where the effect of pore pressures can be neglected. Recent work by Miller and Hamilton (1989) and Michalowski (1994) suggests that it is also possible to incor- porate the effect of pore pressures in limit analysis, but this extension will not be covered here. . The solutions for the simple slopes considered in this paper are relevant to excavations and man-made fills built on soil or rock (Taylor 1948; Gibson and Morgenstcrn 1962; Hunter and Schuster 1968; Duncan et al. 1987). Duncan et al. (1987) have showed that stability charts for simple slopes also can be used to obtain reasonably accurate answers for more complex prob— lems if irregular slopes are approximated by simple slopes and Firm Base FIG. 1. Geometry of Slmple Slopes and Parameters for De- ecrlblng Increasing Strength with Depth In Slope Stablllty Anal- yele carefully determined averaged values of unit weight, cohesion, and friction angles are used. FINITE-ELEMENT LIMIT ANALYSIS Because of the difficulties of constructing statically admis- sible stress fields manually, the application of limit analysis has in the past almost exclusively concentrated on the upper bound method (Chen 1975; Chen and Liu 1990). In fact, the authors are not aware of any rigorous lower bound solutions for the stability of slopes in cohesive-frictional soils. Although the upper bound solutions may be used as an estimate for the true collapse load, it is the lower bound solutions that are generally more useful in practice, because they are inherently conservative. Finite-Element Lower Bound Limit Analysis The use of finite elements and linear programming to com- pute rigorous lower bounds for soil mechanics problems ap- pears to have been first proposed by Lysmer (1970). Although Lysmer’s procedure was potentially very powerful, its useful- ness was limited initially by the slowness of the algorithms that were available for solving large linear programming prob- lems. In recent years, significant progress has been made in developing more efficient algorithms for solving large linear programming problems (Sloan 1988a). Detailed discussions of the recent developments in finite-element formulations of the lower bound theorem may be found in Anderheggen and Knopfel (1972), Bottero et a1. (1980), Sloan (1988b), and Yu and Sloan (1991a,b). In this study, the formulation of Sloan (1988b) has been used because it has proven to be very effi- cient and robust when applied to large practical problems. The lower bound formulation under conditions of plane strain uses the three types of elements shown in Fig. 2. The stress field for each of these elements is assumed to vary lin- early. The extension elements may be used to extend the so- lution over a semiinfinite domain and therefore provide a com- plete statically admissible stress field for infinite half-space problems. Because this paper is concerned mainly with the stability of finite depth slopes resting on a firm base, extension elements are needed only at the left and right boundaries of the problems (shown in Fig. 4). In fact, the extension elements shown in Fig. 2 can be used readily to extend the stress fields into a semiinfinite domain if a slope in an infinitely deep layer needs to be analyzed (i.e., the depth factor D = 00). Examples 2/ JOURNAL OF GEOTECHNICAL AND GEOENVIFIONMENTAL ENGINEERING /JANUARY 1998 (0,2. 0,2. r m) (ax3‘ay3’rxy3) (011,034,101) 3-noded triangular element Direction of Baension —e 4 Direction of Extension 4-noded rectangular extension element 3~noded triangular extension element X FIG. 2. Elements Used for Lower Bound Limit Analyels of how this can be done can be found in Yu and Sloan (1994). As will be shown later in this paper, the depth factor has a very small effect on the stability of slopes provided the value of depth factor D is greater than approximately 4 [in agree- ment with the slope stability charts presented by Taylor (1948)]. It may therefore be reasonable to assume that the stability solutions for a slope in a finite layer with a large D value (say D 2 4) would be very close to those of a slope in a layer of infinite depth. A lower bound solution is obtained by insisting that the stresses obey equilibrium and satisfy both the stress boundary conditions and the yield criterion. Each of these requirements imposes a separate set of constraints on the nodal stresses. In the lower bound finite-element analysis, statically admissible stress discontinuities are permitted at edges shared by adjacent triangles and also along borders between adjacent rectangular extension elements. The Mohr-Coulomb and Tresca yield functions can be shown to plot as a circle in the Zr,y versus (a, -- 0",) stress space, where 1;, is shear stress and 0’, and cry are normal stresses. To avoid nonlinear constraints occurring in the constraint matrix, a key idea behind the numerical lower bound technique is to use an internal linear approximation of the Mohr-Coulomb or the Tresca yield surface in the 27,,y ver- sus (0'; — 0),) stress space. In a typical lower bound analysis, a statically admissible stress field is sought that maximizes either a collapse load over some part of the boundary or the magnitude of body forces acting within a region. Both the collapse load and soil unit weight can be used to define an objective function for linear programming calculations. In the simple slope problem con- sidered in this paper, it is convenient to find a static stress field that maximizes the unit weight. As a result, we will treat the unit weight as the unknown and optimize it directly. A typical lower bound mesh, together with the applied boundary con- ditions, is shown in Fig. 4(a). Note that the rectangular exten- sion elements are arranged to ensure that the computed stress field is statically admissible throughout the slope, so that the solution is a rigorous lower bound solution. Finite-Element Upper Bound Limit Analysis The first formulation of the upper bound theorem, which used constant-strain triangular finite elements and linear pro- gramming, appears to have been developed by Anderheggen and Knopfel (1972) who analyzed plate problems. This for- mulation was later generalized by Bottero et a1. (1981) and Sloan (1989) to include velocity discontinuities in plane strain limit analysis. When these constant-strain finite-element for- mulations are used, the grid must be arranged so that four triangles form a quadrilateral, with the central node lying at the intersection of the diagonals. If this pattern is not used, the elements cannot provide a sufficient number of degrees of freedom to satisfy the incompressibility condition that accom- panies undrained failure (Nagtegaal et al. 1974). To overcome this limitation, Yu et al. (1992) developed a six-noded qua- dratic element for upper bound limit analysis. This formulation can be used to model an incompressible velocity field without resorting to special grid arrangements and also is more effi- cient than an equivalent three—noded formulation with the same number of nodes. However, it does suffer from the same shortcomings as the formulation of Bottero et a1. (1980) and Sloan (1989), in that the direction of shearing for each velocity discontinuity must be specified a priori. Very recently, a new upper bound formulation that permits a large number of dis- continuities in the velocity field has been derived by Sloan and Kleeman (1995). This method is again based on the three— noded triangle, but a velocity discontinuity may occur at any edge that is shared by a pair of adjacent triangles, and the sign of shearing is chosen automatically during the optimization process to give the least-dissipated energy rate. In this study, the upper bound formulation developed by Sloan and Kleeman (1995) is used to calculate upper bound solutions for slope stability. A typical constant-strain triangular element used in the upper bound analysis is shown in Fig. 3. (”2aV2) (“3, v3) (“I‘vl) (111,112,...,A,,) FIG. 3. Trlangular Element Used for Upper Bound LImlt Anal- yele a,,=0 1=0 ,xsaaaaxmfi ,xaaaaaaaan ma” &%%%%%%X%M Z%§%%%%%%%%%%%%XE % XX% XXX QV XXXXXXQXD IXXXXXX XXXXXXX u=v= v=0 0 % XX XXXXXX X%X%%X XXXXXX %X%XXX XXXXXX XX X 0 (b) FIG. 4. Typlcal FInlte Element Meshes Used In leIt Analyels JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING /JANUAFIY 1998 / 3 Within each element, the velocities are assumed to vary line- arly. Each node has two velocity components and each element has p plastic multiplier rates in, X2, X,, where p is the number of planes used to linearize the yield criterion. An upper bound solution is obtained by requiring that the velocity fields obey an associated plastic flow rule and satisfy the velocity boundary conditions. Each of these requirements imposes a separate set of constraints on the nodal velocities. To remove the stress terms from the flow rule equations, and thus provide a linear relationship between the unknown nodal velocities and plastic multiplier rates, an external linear ap- proximation to the yield surface in the stress space of 213,, against (a, — 0,) is employed to ensure that the solution ob- tained is truly an upper bound. In a typical upper bound analysis, a kinematically admis- sible velocity field is sought that minimizes the amount of dissipated power. The dissipated power can be expressed in terms of the unknown plastic multiplier rates and the discon- tinuity parameters used to define an objective function [see Sloan and Kleeman (1995)]. For the slope stability problem considered in this paper, we will minimize the unit weight directly and this can be achieved by equating the power ex- pended by the external loads to the internal power dissipation [see Sloan (1995) for details]. A typical upper bound mesh is shown in Fig. 4(b). This is very similar to the equivalent lower bound grid of Fig. 4(a) except that the extension elements are no longer necessary because a rigid velocity boundary condi- tion used in the upper bound limit analysis will ensure that the solution obtained is a rigorous upper...
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