1、附：英文翻译 LIMIT ANALYSIS OF SOIL SLOPES SUBJECTED TO PORE-WATER PRESSURES By J.Kim R.salgado, assoicite member, ASCE ,and H.S., member,ASCE ABSTRACT: the limit-equilibrium method is commonly, used for slope stability analysis. However, it is well known that the solution obtained from the limit-equilibr
2、ium method is not rigorous, because neither static nor kinematic admissibility conditions are satisfied. Limit analysis takes advantage of the lower-and upper-bound theorem of plasticity to provide relatively simple but rigorous bounds on the true solution. In this paper, three nodded linear triangu
3、lar finite elements are used to construct both statically admissible stress fields for lower-bound analysis and kinematically admissible velocity fields for upper-bound analysis. By assuming linear variation of nodal and elemental variables the determination of the best lower-and upper-bound solutio
4、n maybe set up as a linear programming problem with constraints based on the satisfaction of static and kinematic admissibility. The effects of pro-water pressure are considered and incorporated into the finite-element formulations so that effective stress analysis of saturated slope may be done. Re
5、sults obtained from limit analysis of simple slopes with different ground-water patterns are compared with those obtained from the limit-equilibrium method. INTRODUCTION Stability and deformation problem in geotechnical engineering are boundary-value problem； differential equations must be solved fo
6、r given boundary conditions. Solutions are found by solving differential equations derived from condition of equilibrium, compatibility, and the constitutive relation of the soil, subjected to boundary condition. Traditionally, in soil mechanics, the theory of elasticity is used to set up the differ
7、ential equations for deformation problems, while the theory of plasticity is used for stability problems. To obtain solution for loadings ranging from small to sufficiently large to cause collapse of a portion of the soil mass, a complete elastoplastic analysis considering the mechanical behavior of
8、 the soil until failure may be thought of as a possible method. However, such an elastoplastic analysis is rarely used in practice due to the complexity of the computations. From a practical standpoint, the primary focus of a stability problem is on the failure condition of the soil mass. Thus, prac
9、tical solutions can be found in a simpler manner by focusing on conditions at impending collapse. Stability problem of natural slopes, or cut slopes are commonly encountered in civil engineering projects. Solutions may be based on the slip-line method, the limit-equilibrium method, or limit analysis
10、. The limit-equilibrium method has gained wide acceptance in practice due to its simplicity. Most limit-equilibrium method are based on the method of slices, in which a failure surface is assumed and the soil mass above the failure surface is divided into vertical slices. Global static-equilibrium c
11、onditions for assumed failure surface are examined, and a critical slip surface is searched, for which the factor of safety is minimized. In the development of the limit-equilibrium method, efforts have focused on how to reduce the indeterminacy of the problem mainly by making assumptions on inter-s
12、lice forces. However, no solution based on the limit-equilibrium method, not even the so called “rigorous” solutions can be regarded as rigorous in a strict mechanical sense. In limit-equilibrium, the equilibrium equations are not satisfied for every point in the soil mass. Additionally, the flow ru
13、le is not satisfied in typical assumed slip surface, nor are the compatibility condition and pre-failure constitutive relationship. Limit analysis takes advantage of the upper-and lower-bound theorems of plasticity theory to bound the rigorous solution to a stability problem from below and above. Li
14、mit analysis solutions are rigorous in the sense that the stress field associated with a lower-bound solution is in equilibrium with imposed loads at every point in the soil mass, while the velocity field associated with an upper-bound solution is compatible with imposed displacements. In simple ter
15、ms, under lower-bound loadings, collapse is not in progress, but it may be imminent if the lower bound coincides with the true solution lies can be narrowed down by finding the highest possible lower-bound solution and the lowest possible upper-bound solution. For slope stability analysis, the solut
16、ion is in terms of either a critical slope height or a collapse loading applied on some portion of the slope boundary, for given soil properties and/or given slope geometry. In the past, for slope stability applications, most research concentrated on the upper-bound method. This is due to the fact t
17、hat the construction of proper statically admissible stress fields for finding lower-bound solutions is a difficult task. Most previous work was based on total stresses. For effective stress analysis, it is necessary to calculate pore-water pressures. In the limit-equilibrium method, pore-water pres
18、sures are estimated from ground-water conditions simulated by defining a phreatic surface, and possibly a flow net, or by a pore-water pressure ratio. Similar methods can be used to specify pore-water pressure for limit analysis. The effects of pore-water pressure have been considered in some studie
19、s focusing on calculation of upper-bound solutions to the slope stability problem. Miller and Hamilton examined two types of failure mechanism: (1) rigid body rotation； and (2) a combination of rigid rotation and continuous deformation. Pore-water pressure was assumed to be hydrostatic beneath a par
20、abolic free water surface. Although their calculations led to correct answers, the physical interpretation of their calculation of energy dissipation, where the pore-water pressures were considered as internal forces and had the effect of reducing internal energy dissipation for a given collapse mec
21、hanism, has been disputed. Pore-water pressures may also be regarded as external force. In a study by Michalowski, rigid body rotation along a log-spiral failure surface was assumed, and pore-water pressure was calculated using the pore-water pressure ratio ru=u/z, where u=pore-water pressure, =tota
22、l unit weight of soil, and z=depth of the point below the soil surface. It was showed that the pore-water pressure has no influence on the analysis when the internal friction angle is equal to zero, which validates the use of total stress analysis with =0. In another study, Michalowski followed the
23、same approach, except for the use of failure surface with different shapes to incorporate the effect of pore-water pressure on upper-bound analysis of slopes, the writers are not aware of any lower-bound limit analysis done in term of effective stresses. This is probably due to the increased in cons
24、tructing statically admissible stress fields accounting also for the pore-water pressures. The objectives of this paper are (1) present a finite-element formulation in terms of effective stresses for limit analysis of soil slopes subjected to pore-water pressures； and (2) to check the accuracy of Bi
25、shops simplified method for slope stability analysis by comparing Bishops solution with lower-and upper-bound solution. The present study is an extension of previous research, where Bishops simplified limit-equilibrium solutions are compared with lower-and upper-bund solutions for simple slopes with
26、out considering the effect of pore-water pressure. In the present paper, the effect of pore-water pressure is considered in both lower-and upper-bound limit analysis under plane-strain conditions. Pore-water pressures are accounted for by making modifications to the numerical algorithm for lower-and
27、 upper-bound calculations using linear three-noded triangles developed by Sloan and Sloan and Kleeman. To model the stress field criterion, flow of linear equations in terms of nodal stresses and pore-water pressures, or velocities, the problem of finding optimum lower- and upper-bound solutions can
28、 be set up as a linear programming problem. Lower- and upper-bound collapse loadings are calculated for several simple slope configurations and groundwater patterns, and the solutions are presented in the form of chart. LIMIT ANALYSIS WITH PORE-WATER PRESSURE Assumptions and Their implementation Lim
29、it analysis uses an idealized yield criterion and stress-strain relation: soil is assumed to follow perfect plasticity with an associated flow rule. The assumption of perfect plasticity expresses the possible states of stress in the form F( ij) = 0 (1) Where F( ij) = yield function； and ij = effecti
30、ve stress tensor. Associated flow rule defines the plastic strain rate by assuming the yield function F to coincide with the plastic potential function G, from which the plastic strain rate pij can be obtained though pij ij ijGF （ 2） where = nonnegative plastic multiplier rate that is positive only
31、when plastic deformations occur. Eq. (2) is often referred to as the normality condition, which states that the direction of plastic strain rate is perpendicular to the yield surface. Perfect plasticity with an associated with very large displacements are of concern. In addition, theoretical studies show that the collapse loads for earth slopes, where soils are not heavily constrained, are quite insensitive to whether the flow rule is associated or non-associated.
