【正文】 dwater patterns are pared with those obtained from the limitequilibrium method.INTRODUCTION Stability and deformation problem in geotechnical engineering are boundaryvalue problem。 and (2) a bination of rigid rotation and continuous deformation. Porewater 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 porewater pressures were considered as internal forces and had the effect of reducing internal energy dissipation for a given collapse mechanism, has been disputed. Porewater pressures may also be regarded as external force. In a study by Michalowski, rigid body rotation along a logspiral failure surface was assumed, and porewater pressure was calculated using the porewater pressure ratio ru=u/ǐz, where u=porewater pressure, ǐ=total unit weight of soil, and z=depth of the point below the soil surface. It was showed that the porewater 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 Φ=0. In another study, Michalowski followed the same approach, except for the use of failure surface with different shapes to incorporate the effect of porewater pressure on upperbound analysis of slopes, the writers are not aware of any lowerbound 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 porewater pressures. The objectives of this paper are (1) present a finiteelement formulation in terms of effective stresses for limit analysis of soil slopes subjected to porewater pressures。 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 nonassociated.Principle of Virtual Work Both the lowerand 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 including seepage and buoyancy forces。 = effective stress tensor in equilibrium with and 。 p = porewater pressure。 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.Lowerbound Theorem If the stress field within the soil mass is stable and statically admissible, then collapse does not occur。 = actual stress。 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.中文對(duì)照翻譯：孔隙水壓力作用下土坡的極限分析摘要：極限平衡法一般用于土坡的穩(wěn)定性分析。極限分析法充分利用了塑性體的上下邊界原理，在求真實(shí)解中提供了一個(gè)相對(duì)簡單但又嚴(yán)密的邊界。通過假設(shè)三角形頂點(diǎn)的線變量和元素變量，真實(shí)解應(yīng)該是一個(gè)線形的約束問題。在有限元公式中，要考慮包括了孔隙壓力的影響，以便使飽和土坡的有效應(yīng)力分析可以得出。概述：穩(wěn)定性和變形問題在全球技術(shù)工程領(lǐng)域是一個(gè)邊界值問題。通過解決由平衡協(xié)調(diào)條件以及沙土的本構(gòu)關(guān)系推出的微分方程，從而得到邊界條件下的解。為了獲得這個(gè)解，荷載由小到大變化，直到足夠大引起部分土體的滑坡。然而，這樣一個(gè)彈塑性分析方法很少應(yīng)用于實(shí)際問題當(dāng)中，因?yàn)樗挠?jì)算機(jī)太過復(fù)雜。因此，真實(shí)解應(yīng)該是通過關(guān)注即將發(fā)生的破壞條件的一個(gè)簡單的方法中得來。求解通常建立在滑移線方法上，極限平衡方法或極限分析法的基礎(chǔ)上。極限平衡法大部分建立在分塊理論的基礎(chǔ)上，在這種理論中，假設(shè)有一個(gè)破壞的滑動(dòng)面，而且在此之上的土體被劃分為若干垂直土條，整個(gè)靜態(tài)平衡條件下假設(shè)的失穩(wěn)表面是被確定的，一個(gè)臨界的滑動(dòng)破裂面必須要找到，因?yàn)樗陌踩驍?shù)最小。但是，沒有一種解的得來是建立在這樣的極限分析法的基礎(chǔ)上，甚至在嚴(yán)格的力學(xué)意義上講，它都不算一個(gè)嚴(yán)密的解。另外，在典型的假定滑動(dòng)面方法中，流動(dòng)法則是不滿足的，同樣，協(xié)調(diào)性條件和破壞前的本構(gòu)關(guān)系也是不滿足的。極限分析法在以下兩種意義上是嚴(yán)密的，一是土體在外加荷載作用下的平衡，下邊界解所對(duì)應(yīng)的應(yīng)力場；二是與外加位移相協(xié)調(diào)，上邊界所對(duì)應(yīng)的速度場。同樣，在上邊界作用外加荷載，滑坡也會(huì)立即發(fā)生。對(duì)于土坡穩(wěn)定性問題，給定土體的性質(zhì)或幾何尺寸的基礎(chǔ)上，知道土坡發(fā)生滑動(dòng)的臨界高度和發(fā)生部分滑坡的臨界荷載才能得出解來。大多數(shù)先前的工作都基于總應(yīng)力之上。在極限平衡法中，孔隙水壓力是通過限定一個(gè)地下水表面和一個(gè)可能的流動(dòng)網(wǎng)或者通過一個(gè)孔隙水壓力比率模擬地下水條件推測(cè)出來的。在大量的實(shí)踐中，孔隙水壓力的影響被看作集中考慮在土坡穩(wěn)定性問題的上邊界解上?？紫端畨毫Ρ患俣ǔ闪黧w靜力學(xué)下的一個(gè)拋物線型的自由水表面，盡管他們的研究得出了正確的答案，但是，從物理學(xué)上解釋他們的研究，在能量消散上是有爭議的?？紫端畨毫σ部梢钥醋魇且环N外力?？紫端畨毫Ρ豢紤]用孔隙水壓力比來表示：這里，u是孔隙水壓力；是沙土的比重；z是土體表面以下的深度。在另一項(xiàng)研究中，除了用不同形狀的破裂面結(jié)合分塊分析法時(shí)，Michalowski秉承了相同的方法。這可能因?yàn)樵诳紤]孔隙水壓力的情況下，構(gòu)造靜態(tài)允許應(yīng)力場的難度增大。在這篇文章中，在平面應(yīng)變條件下，上下邊界的極限分析是要考慮孔隙水壓力影響的。為了模仿應(yīng)力場和速度場，由三點(diǎn)組成的三邊線性元素就要被利用。就分點(diǎn)的應(yīng)力和孔隙水壓力或者速度而言，用平衡方程、協(xié)調(diào)條件、流動(dòng)法則、屈服準(zhǔn)則、邊界條件的線性代數(shù)等式來表達(dá)，那么，求解最佳上下邊界是在幾個(gè)簡單土坡的構(gòu)造和地下水形式下被考慮的，解是以諾莫圖的形式給出。在這個(gè)理想的塑性體假設(shè)表明了可能的應(yīng)力狀態(tài)形式： (1)這里，F(xiàn)是應(yīng)變函數(shù)，是有效應(yīng)力張量。等式（2）通常被認(rèn)為是常態(tài)條件，就表面當(dāng)量而言，塑性應(yīng)變率的方向關(guān)系是垂直的。另外，理論研究表明，不管有沒有流動(dòng)法則的存在，當(dāng)土體沒有受到嚴(yán)重的受壓，土坡的坍塌荷載是很不敏感的。不需要用任何特殊的方式通過（3）式和（4）式把,和聯(lián)系在一起。（3）式中，孔隙水壓力被認(rèn)為是內(nèi)力，它減少了內(nèi)部的能量消散，而在（4）式中，孔隙水壓力被認(rèn)為是外力。假設(shè)剛體是理想的塑性體，用一個(gè)簡單的極限分析去解決穩(wěn)定性問題是沒有嚴(yán)格損失的。這里可以用內(nèi)部作用等式的形式來表示，由（3）式得： (5)這里是不包括滲透量和浮力的自重與摩擦力平衡時(shí)的靜態(tài)允許應(yīng)力。在（5）式中，不等式的成立是由于最大塑性消散原則，根據(jù)在真實(shí)應(yīng)變率場下的真實(shí)坍塌應(yīng)力場得出的內(nèi)部作用比率總是比在真實(shí)應(yīng)變率場下而引起坍塌的應(yīng)力場得出的比率要大而得出。35