外文翻译--孔隙水压力作用下土坡的极限分析.docx

附：英文翻译LIMIT ANALYSIS OF SOIL SLOPES SUBJECTED TO PORE-WATER PRESSURESBy 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-equilibrium 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 triangular 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 solution 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. Results 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 for 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 differential 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 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, practical 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. 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 conditions 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-slice 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 rule 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. Limit 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 terms, 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 solution 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 that 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 pressures 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 studies 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 parabolic 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 mechanism, 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, =total 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 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 constructing 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 Bishops 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 without 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 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 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 Limit 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() = 0 (1)Where F() = yield function; and = effective 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 can be obtained though （2）where = nonnegative plastic multiplier rate that is positive only 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.Principle of Virtual Work Both the lower-and upper bound theorems are based on the principle of virtual work. The virtual work equation is applicable, given the assumption of small deformations before collapse, and can be expressed as either （3）Or （4）Where = boundary loadings; = body forces not including seepage and buoyancy forces; = body forces including seepage and buoyancy forces;= total stress tensor in equilibrium with and ; = effective stress tensor in equilibrium with and ; = Kronecker delta; p = pore-water pressure; and = strain rate tensor compatible with the velocity field . There is no need for , , and to be related to and in any particular way for (3) or (4) represent the rate of the external work, while the right-hand sides represent the rate of the internal power dissipation done by the assumed stress field and external loads on the assumed strain and velocity fields. The difference between (3) and (4) is the way to incorporate the effect of pore-water pressure: the pore-water pressures are considered as internal force, reducing the internal power dissipation, in (3), while they are considered external force in (4). By taking advantage of the normality condition , it can be easily shown that elastic stress and strain have no influence on the collapse load; that is, only plastic deformation occurs during plastic flow, and = .This makes limit analysis a simple method to solve stability problems, without loss of rigor, assuming rigid perfect plasticity.Lower-bound Theorem If the stress field within the soil mass is stable and statically admissible, then collapse does not occur; that is, the true collapse load is definitely greater than the applied load. This can be written in the form of the virtual work equation, using (3), as (5)Where = statically admissible stress field in equilibrium with the traction and body force not including the seepage and buoyancy force; = actual stress; = actual stain rate; and = velocity fields. In (5), the inequality is due to the principle of maximum plastic dissipation, according to which the actual strain rate field is always larger than the rate of work done on the actual strain rate field by a stress field not causing collapse. In (5), only the equilibrium condition and the stress boundary conditions not taken into account. The best lower bound to the true collapse load can be found by analyzing various trial statically admissible stress fields.中文对照翻译：孔隙水压力作用下土坡的极限分析摘要：极限平衡法一般用于土坡的稳定性分析。然而，众所周知的是，从极限分析法中获得的解是不严密的，因为它既不满足静态的允许条件，又不满足动态的允许条件。极限分析法充分利用了塑性体的上下边界原理，在求真实解中提供了一个相对简单但又严密的边界。在这篇文章中，三点确定的三角形三边有限元法被利用与构造在下边界分析中的静态允许应力场和上边界分析中的速度场。通过假设三角形顶点的线变量和元素变量，真实解应该是一个线形的约束问题。在静态和动态的条件都满足的基础上，真实解应该处在上下边界所得的解之间。在有限元公式中，要考虑包括了孔隙压力的影响，以便使饱和土坡的有效应力分析可以得出。作者对从不同地下水形式下简单土坡的极限分析所得的结果与极限平衡法中所得的结果作了比较。概述：稳定性和变形问题在全球技术工程领域是一个边界值问题。微分方程必须用给定的边界条件来解决。通过解决由平衡协调条件以及沙土的本构关系推出的微分方程，从而得到边界条件下的解。按照传统的说法，在土力学中，弹性理论是用来建立变形微分方程的，就象塑性理论是用来建立稳定性问题的微分方程一样 。为了获得这个解，荷载由小到大变化，直到足够大引起部分土体的滑坡。作为土体破坏的力学行为，完整的弹塑性分析以为是一个可能的方法。然而，这样一个弹塑性分析方法很少应用于实际问题当中，因为他的计算机太过复杂。站在实践的立场上，稳定性的最基本关注点应该是土体破坏条件。因此，真实解应该是通过关注即将发生的破坏条件的一个简单的方法中得来。自然土坡、填方土坡和挖方土坡的稳定性问题是土木工程领域碰到的最常见的问题。求解通常建立在滑移线方法上，极限平衡方法或极限分析法的基础上。由于它的简单，在实践中，极限平衡法是最被广泛使用的。极限平衡法大部分建立在分块理论的基础上，在这种理论中，假设有一个破坏的滑动面，而且在此之上的土体被划分为若干垂直土条，整个静态平衡条件下假设的失稳表面是被确定的，一个临界的滑动破裂面必须要找到，因为它的安全因数最小。在极限平衡法的发展过程中，要努力去做的是怎么降低通过内力假设的不确定性。但是，没有一种解的得来是建立在这样的极限分析法的基础上，甚至在严格的力学意义上讲，它都不算一个严密的解。在极限平衡法中，平衡方程并不是对土体的每一点都适用的。另外，在典型的假定滑动面方法中，流动法则是不满足的，同样，协调性条件和破坏前的本构关系也是不满足的。极限分析法充分利用了边界理论，得出了相对应用于上下边界的两个严密的解。极限分析法在以下两种意义上是严密的，一是土体在外加荷载作用下的平衡，下边界解所对应的应力场；二是与外加位移相协调，上边界所对应的速度场。就简单而言，下边界荷载作用下，滑动不会发生，但是如果下边界受到外加荷载的作用，则滑动可能立即发生。同样，在上边界作用外加荷载，滑坡也会立即发生。通过寻找下边界的最大可能解和上边界的最小可能解，真实解存在于他们之间的范围内。对于土坡稳定性问题，给定土体的性质或几何尺寸的基础上，知道土坡发生滑动的临界高度和发生部分滑坡的临界荷载才能得出解来。在过去，对于土坡稳定性的应用，大多数研究工作都集中在上边界法上，这是因为通过求解下边界适合静态允许应力场方程的解是一项很困难的任务。大多数先前的工作都基于总应力之上。对于有效应力的分析，考虑孔隙水压力的作用是很有必要的。在极限平衡法中，孔隙水压力是通过限定一个地下水表面和一个可能的流动网或者通过一个孔隙水压力比率模拟地下水条件推测出来的。相似的方法可以用于明确说明孔隙水压力作用下的极限分析。在大量的实践中，孔隙水压力的影响被看作集中考虑在土坡稳定性问题的上边界解上。Miller和Hamilton两人实验出了两种力学破坏类型：（1）刚体旋转；（2）刚体旋转和持续变形相结合。孔隙水压力被假定成流体静力学下的一个抛物线型的自由水表面，尽管他们的研究得出了正确的答案，但是，从物理学上解释他们的研究，在能量消散上是有争议的。在他们的理论中，孔隙水压力被看作是内力，在给定的滑坡机理下，它对降低内部能量消散是有影响的。孔隙水压力也可以看作是一种外力。在Michalowsk的研究中，假设刚体是沿着螺旋线破坏的。孔隙水压力被考虑用孔隙水压力比来表示：这里，u是孔隙水压力；是沙土的比重；z是土体表面以下的深度。它表示孔隙水压力在内部摩擦力等于零时的分析没有影响，这就证实了用总应力分析时。在另一项研究中，除了用不同形状的破裂面结合分块分析法时，Michalowski秉承了相同的方法。当这种努力结合