外文翻譯--孔隙水壓力作用下土坡的極限分析

1、<p><b>　　附：英文翻譯</b></p><p>　　LIMIT ANALYSIS OF SOIL SLOPES SUBJECTED TO PORE-WATER PRESSURES</p><p>　　By J.Kim R.salgado, assoicite member, ASCE ,and H.S., member,ASCE </p>

2、;<p>　　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 neith

3、er 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

4、this paper, three nodded linear triangular finite elements are used to con</p><p>　　INTRODUCTION </p><p>　　Stability and deformation problem in geotechnical engineering are boundary-value proble

5、m; 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 soi

6、l, 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 plastic</p><p>　　Stab

7、ility 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-equilibriu

8、m 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

9、 divided into vertical slices. Global static-equilibrium</p><p>　　Limit analysis takes advantage of the upper-and lower-bound theorems of plasticity theory to bound the rigorous solution to a stability probl

10、em 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 vel

11、ocity field associated with an upper-bound solution is compatible with imposed displacements. In simple terms, under lower-bound loadings, </p><p>　　The effects of pore-water pressure have been considered in

12、 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

13、 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

14、of</p><p>　　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

15、accuracy of Bishop’s simplified method for slope stability analysis by comparing Bishop’s solution with lower-and upper-bound solution. The present study is an extension of previous research, where Bishop’s simplified li

16、mit-equilibrium solutions are compared with lower-and upper-bund solutions for simple</p><p>　　LIMIT ANALYSIS WITH PORE-WATER PRESSURE </p><p>　　Assumptions and Their implementation </p>

17、<p>　　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 possib

18、le states of stress in the form</p><p>　　F() = 0 (1)</p><p>　　Where F() = yield function; and = effective stress tensor.</p><p>　　Associated flow rule defines the p

19、lastic 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</p><p><b>　?。?）</b></p>&l

20、t;p>　　where = nonnegative plastic multiplier rate that is positive only when plastic deformations occur.</p><p>　　Eq. (2) is often referred to as the normality condition, which states that the direction o

21、f 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 slo

22、pes, where soils are not heavily constrained, are quite insensitive to whether the flow rule is associated or non-associated.</p><p>　　Principle of Virtual Work</p><p>　　Both the lower-and upper

23、 –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</p><p><b&g

24、t;　　（3）</b></p><p>　　Or （4）</p><p>　　Where = boundary loadings; = body forces not including seepage and buoyancy forces; = body forces including seepage and buoyancy forces

25、;= 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 .</p&

26、gt;<p>　　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 diss

27、ipation 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 pressu

28、res are considered as internal force, reducing the internal power dissipatio</p><p>　　Lower-bound Theorem</p><p>　　If the stress field within the soil mass is stable and statically admissible, t

29、hen 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 </p><p><b>　　(5)<

30、;/b></p><p>　　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 = veloc

31、ity fields.</p><p>　　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 s

32、train 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 an

33、alyzing various trial statically admissible stress fields.</p><p><b>　　中文對(duì)照翻譯：</b></p><p>　　孔隙水壓力作用下土坡的極限分析</p><p>　　摘要：極限平衡法一般用于土坡的穩(wěn)定性分析。然而，眾所周知的是，從極限分析法中獲得的解是不嚴(yán)密的，因?yàn)樗?/p>

34、不滿足靜態(tài)的允許條件，又不滿足動(dòng)態(tài)的允許條件。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(jiǎn)單但又嚴(yán)密的邊界。在這篇文章中，三點(diǎn)確定的三角形三邊有限元法被利用與構(gòu)造在下邊界分析中的靜態(tài)允許應(yīng)力場(chǎng)和上邊界分析中的速度場(chǎng)。通過(guò)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(wèn)題。在靜態(tài)和動(dòng)態(tài)的條件都滿足的基礎(chǔ)上，真實(shí)解應(yīng)該處在上下邊界所得的解之間。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

35、的有效應(yīng)力分析可以得出。作者對(duì)從不同地下水形式下簡(jiǎn)單土坡的極限分析所得的結(jié)果與極限平衡法中所得的結(jié)果作了比較。</p><p>　　概述：穩(wěn)定性和變形問(wèn)題在全球技術(shù)工程領(lǐng)域是一個(gè)邊界值問(wèn)題。微分方程必須用給定的邊界條件來(lái)解決。通過(guò)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。按照傳統(tǒng)的說(shuō)法，在土力學(xué)中，彈性理論是用來(lái)建立變形微分方程的，就象塑性理論是用來(lái)建立穩(wěn)定性問(wèn)題的微分方程一樣

36、。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。作為土體破壞的力學(xué)行為，完整的彈塑性分析以為是一個(gè)可能的方法。然而，這樣一個(gè)彈塑性分析方法很少應(yīng)用于實(shí)際問(wèn)題當(dāng)中，因?yàn)樗挠?jì)算機(jī)太過(guò)復(fù)雜。站在實(shí)踐的立場(chǎng)上，穩(wěn)定性的最基本關(guān)注點(diǎn)應(yīng)該是土體破壞條件。因此，真實(shí)解應(yīng)該是通過(guò)關(guān)注即將發(fā)生的破壞條件的一個(gè)簡(jiǎn)單的方法中得來(lái)。</p><p>　　自然土坡、填方土坡和挖方土坡的穩(wěn)定性問(wèn)題是土木工程領(lǐng)域碰到的最常

37、見(jiàn)的問(wèn)題。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。由于它的簡(jiǎn)單，在實(shí)踐中，極限平衡法是最被廣泛使用的。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。在極限平衡法的發(fā)展過(guò)程中，要努力去做的是怎么降低通過(guò)內(nèi)力假設(shè)的不確定性。但是，沒(méi)有一種解的得來(lái)是建立

38、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。在極限平衡法中，平衡方程并不是對(duì)土體的每一點(diǎn)都適用的。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。</p><p>　　極限分析法充分利用了邊界理論，得出了相對(duì)應(yīng)用于上下邊界的兩個(gè)嚴(yán)密的解。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(chǎng)；二

39、是與外加位移相協(xié)調(diào)，上邊界所對(duì)應(yīng)的速度場(chǎng)。就簡(jiǎn)單而言，下邊界荷載作用下，滑動(dòng)不會(huì)發(fā)生，但是如果下邊界受到外加荷載的作用，則滑動(dòng)可能立即發(fā)生。同樣，在上邊界作用外加荷載，滑坡也會(huì)立即發(fā)生。通過(guò)尋找下邊界的最大可能解和上邊界的最小可能解，真實(shí)解存在于他們之間的范圍內(nèi)。對(duì)于土坡穩(wěn)定性問(wèn)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(lái)。在過(guò)去，對(duì)于土坡穩(wěn)定性的應(yīng)用，大多數(shù)研究工作都集中在上邊界

40、法上，這是因?yàn)橥ㄟ^(guò)求解下邊界適合靜態(tài)允許應(yīng)力場(chǎng)方程的解是一項(xiàng)很困難的任務(wù)。大多數(shù)先前的工作都基于總應(yīng)力之上。對(duì)于有效應(yīng)力的分析，考慮孔隙水壓力的作用是很有必要的。在極限平衡法中，孔隙水壓力是通過(guò)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(guò)一個(gè)孔隙水壓力比率模擬地下水條件推測(cè)出來(lái)的。相似的方法可以用于明確說(shuō)明孔隙水壓力作用下的極限分析。</p><p>　　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

41、問(wèn)題的上邊界解上。Miller和Hamilton兩人實(shí)驗(yàn)出了兩種力學(xué)破壞類型：（1）剛體旋轉(zhuǎn)；（2）剛體旋轉(zhuǎn)和持續(xù)變形相結(jié)合?？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(zhēng)議的。在他們的理論中，孔隙水壓力被看作是內(nèi)力，在給定的滑坡機(jī)理下，它對(duì)降低內(nèi)部能量消散是有影響的。孔隙水壓力也可以看作是一種外力。在Michalowsk的研究中，假設(shè)剛體是沿

42、著螺旋線破壞的。孔隙水壓力被考慮用孔隙水壓力比來(lái)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。它表示孔隙水壓力在內(nèi)部摩擦力等于零時(shí)的分析沒(méi)有影響，這就證實(shí)了用總應(yīng)力分析時(shí)。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。當(dāng)這種努力結(jié)合孔隙水壓力作用在土坡上邊界的影響中，作者沒(méi)有意識(shí)到就有效應(yīng)力而言有任何的下邊界極限分析要做。這可能因?yàn)樵诳紤]孔隙水壓力的情況下，構(gòu)造靜態(tài)允

43、許應(yīng)力場(chǎng)的難度增大。</p><p>　　這篇文章的目的有兩個(gè)：（1）就有效應(yīng)力而言，為土坡在孔隙水壓力作用下的極限分析提出了一個(gè)有限元的公式；（2）通過(guò)比較Bishop的上下邊界解來(lái)核實(shí)土坡穩(wěn)定性分析方法在被Bishop簡(jiǎn)化的極限平衡法所得的解與簡(jiǎn)單坡中不考慮孔隙水壓力作用，上下邊界所得的解相比較。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。Slon和Kleenman在考慮了上邊

44、界和下邊界的情況下，利用了三點(diǎn)構(gòu)成的三邊線性代數(shù)的方法將孔隙水壓力計(jì)算出來(lái)的。為了模仿應(yīng)力場(chǎng)和速度場(chǎng)，由三點(diǎn)組成的三邊線性元素就要被利用。公認(rèn)的力學(xué)理論包括剛體的轉(zhuǎn)動(dòng)和持續(xù)變形。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(lái)表達(dá)，那么，求解最佳上下邊界是在幾個(gè)簡(jiǎn)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。</p><p>　　孔隙水壓力作用

45、下的極限分析：</p><p>　　極限分析用了屈服準(zhǔn)則和應(yīng)力應(yīng)變關(guān)系：土體在流動(dòng)法則下假設(shè)成一個(gè)理想的塑性體。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：</p><p><b>　　(1)</b></p><p>　　這里，F(xiàn)是應(yīng)變函數(shù)，是有效應(yīng)力張量。</p><p>　　通過(guò)假設(shè)應(yīng)變函數(shù)F配合塑性體潛在的應(yīng)變

46、函數(shù)G，用伴隨的流動(dòng)法則來(lái)定義塑性體應(yīng)變率，塑性應(yīng)變率可以從中得出：</p><p><b>　　(2)</b></p><p>　　這里，是非負(fù)的塑性比值，也就是當(dāng)發(fā)生塑性變形時(shí)的正直。</p><p>　　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。理想的塑性體伴隨著流動(dòng)法則是一個(gè)合理的假設(shè)，如果考慮荷載下

47、伴隨非常大的位移。另外，理論研究表明，不管有沒(méi)有流動(dòng)法則的存在，當(dāng)土體沒(méi)有受到嚴(yán)重的受壓，土坡的坍塌荷載是很不敏感的。</p><p><b>　　內(nèi)部作用法則：</b></p><p>　　上邊界和下邊界原理都建立在內(nèi)部作用原則的基礎(chǔ)上，在發(fā)生坍塌前，給定了假設(shè)的小變形，內(nèi)部作用等式是可以運(yùn)用的，并且可以用以下的關(guān)系來(lái)表達(dá)：</p><p>

48、<b>　　(3)</b></p><p><b>　　或者</b></p><p><b>　　(4)</b></p><p>　　這里，是邊界荷載；是不包括滲透量和浮力的自重是包括滲透量和浮力的自重；是在和平衡下的總應(yīng)力張量；是在和平衡下的有效應(yīng)力張量；是Kronecker增量；P是孔隙水壓力；是

49、與速度場(chǎng)一致的應(yīng)變率張量。</p><p>　　不需要用任何特殊的方式通過(guò)（3）式和（4）式把,和聯(lián)系在一起。在（3）式和（4）式的左邊代表外部作用比率，而（3）式（4）式右邊代表外部能量消散比率，通過(guò)假設(shè)外加荷載的應(yīng)力場(chǎng)和假設(shè)應(yīng)變的速度場(chǎng)，（3）式和（4）式之間的區(qū)別在于組成有效孔隙水壓力的形式不同。（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。</p

50、><p>　　通過(guò)充分利用常態(tài)條件，顯而易見(jiàn)的是彈性應(yīng)力和應(yīng)變對(duì)坍塌荷載沒(méi)有影響，即：= .也就是說(shuō)，只有在塑性體流動(dòng)時(shí)，塑性變形才發(fā)生。假設(shè)剛體是理想的塑性體，用一個(gè)簡(jiǎn)單的極限分析去解決穩(wěn)定性問(wèn)題是沒(méi)有嚴(yán)格損失的。</p><p><b>　　下邊界理論：</b></p><p>　　如果土體內(nèi)的應(yīng)力場(chǎng)是穩(wěn)定的，靜態(tài)允許的，那么坍塌不會(huì)發(fā)生，也就

51、是說(shuō)，真實(shí)的坍塌荷載一定比計(jì)算荷載要大。這里可以用內(nèi)部作用等式的形式來(lái)表示，由（3）式得：</p><p><b>　　(5)</b></p><p>　　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。是真實(shí)的應(yīng)力，是真實(shí)的應(yīng)變率，是速度場(chǎng)。</p><p>　　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(chǎng)