【正文】 idered in both lowerand upperbound limit analysis under planestrain conditions. Porewater pressures are accounted for by making modifications to the numerical algorithm for lowerand upperbound calculations using linear threenoded triangles developed by Sloan and Sloan and Kleeman. To model the stress field criterion, flow of linear equations in terms of nodal stresses and porewater pressures, or velocities, the problem of finding optimum lower and upperbound solutions can be set up as a linear programming problem. Lower and upperbound 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 POREWATER PRESSURE Assumptions and Their implementation Limit analysis uses an idealized yield criterion and stressstrain 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( 39。ij? = 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 pij? can be obtained though 39。A B A B Bi i i i ij ijS V VT v dS X v dV dV????? ? ? （ 4） Where AiT = boundary loadings。 39。 and Bij? = strain rate tensor patible with the velocity field Biv . There is no need for Aij? , AiT , and AijX? to be related to Bij? and Biv in any particular way for (3) or (4) represent the rate of the external work, while the righthand 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 porewater pressure: the porewater 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。 ij? = actual stress。極限分析法充分利用了塑性體的上下邊界原理，在求真實解中提供了一個相對簡單但又嚴密的邊界。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。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。然而，這樣一個彈塑性分析方法很少應(yīng)用于實際問題當(dāng)中，因為他的計算機太過復(fù)雜。 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。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，甚至在嚴格的力學(xué)意義上講，它都不算一個嚴密的解。極限分析法在以下兩種意義上是嚴密的，一是土體在外加荷載作用下的平衡，下邊界解所對應(yīng)的應(yīng)力場；二是與外加位移相協(xié)調(diào)，上邊界所對應(yīng)的速度場。對于土坡穩(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)生滑動的臨界高度和發(fā)生部分滑坡的臨界荷載才能得出解來。在極限平衡法中，孔隙水壓力是通過限定一個地下水表面和一個可能的流動網(wǎng)或者通過一個孔隙水壓力比率模擬地下水條件推測出來的?？紫端畨毫Ρ患俣ǔ闪黧w靜力學(xué)下的一個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。孔隙水壓力被考慮用孔隙水壓力比來表示： /ur u z?? 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。這可能因為在考慮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。為了模仿應(yīng)力場和速度場，由三點組成的三邊線性元素就要被利用。在這個理想的塑性體假設(shè)表明了可能的應(yīng)力狀態(tài)形式： 39。39。另外，理論研究表明，不管有沒有流動法則的存在，當(dāng)土體沒有受到嚴重的受壓，土坡的坍塌荷載是很不敏感的。Aij? 是在 AiT和 AijX 平衡下的有效應(yīng)力張量；ij?是 Kronecker 增量； P 是孔隙水壓力；Bij? 是與速度場 Biv 一致的應(yīng)變率張量。 通過充分利用常態(tài)條件，顯而易見的是彈性應(yīng)力和應(yīng)變對坍塌荷載沒有影響，即： ij? = Pij? .也就是說，只有在塑性體流動時，塑性變形才發(fā)生。()L L L Li i i i ij ij ij ij ijS V V VT v d S X v d V d V p d V? ? ? ? ??? ? ? ?? ? ? ? ()ij ij ijVVD d V d V? ? ????? (5) 這里 Lij? 是不包括滲透量和浮力的自重與摩擦力平衡時的靜態(tài)允許應(yīng)力。