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1、边坡稳定重力和渗透力易引起天然边坡、开挖形成的边坡、堤防边坡和土坝的不稳定性。最重要的边坡破坏的类型如图9.1所示。在旋滑中，破坏面部分的形状可能是圆弧或非圆弧线。总的来说，匀质土为圆弧滑动破坏，而非匀质土为非圆弧滑动破坏。平面滑动和复合滑动发生在那些强度差异明显的相邻地层的交界面处。平面滑动易发生在相邻地层处于边坡破坏面以下相对较浅深度的地方：破坏面多为平面，且与边坡大致平行。复合滑动通常发生在相邻地层处于深处的地段，破坏面由圆弧面和平面组成。 在实践中极限平衡法被用于边坡稳定分析当中。它假定破坏面是发生在沿着一个假想或已知破坏面的点上的。土的有效抗剪强度与保持极限平衡状态所要求的抗剪强度相
2、比，就可以得到沿着破坏面上的平均安全系数。问题以二维考虑，即假想为平面应变的情况。二维分析为三维（碟形）面解答提供了保守的结果。在这种分析方法中，应用总应力法，适用于完全饱和粘土在不条件排水下的情况。如建造完工的瞬间情况。这种分析中只考虑力矩平衡。此间，假定潜在破坏面为圆弧面。图9.2展示了一个试验性破坏面（圆心O，半径r，长度La）。潜在的不稳定性取决于破坏面以上土体的总重量(单位长度上的重量W）。为了达到平衡，必须沿着破坏面传递的抗剪强度表示如下：其中 F 是就抗剪强度而言的安全系数关于 O点力矩平衡： 因此 (9.1) 其它外力的力矩必须亦予以考虑。在张裂发展过程中，如图9.2所示，如果
3、裂隙中充满水，弧长La会变短，超孔隙水压力将垂直作用在裂隙上。有必要用一系列试验性破坏面来对边坡进行分析，从而确定最小的安全系数。 基于几何相似原理，泰勒9.9发表了稳定系数，用于在总应力方面对匀质土边坡进行分析。对于一个高度为H的边坡，沿着安全系数最小的破坏面上的稳定系数(Ns)为： (9.2)对于u =0的情况， Ns 的值可以从图9.3中得到。Ns值取决于边坡坡角和高度系数 D，其中DH 是到稳固地层的深度。吉布森和摩根斯特恩9.3发表了不排水强度cu(u =0)随深度线性变化的正常固结粘土边坡的稳定系数。在这种方法中，潜在破坏面再次被假定为以O为圆心，以r为半径的圆弧。试验性破坏面（A
4、C）以上的土体（ABCD），如图9.5所示，被垂直划分为一系列宽度为b的条块。每个条块的底边假定为直线。对于任何一个条块来说，其底边与水平线的夹角为，它的高，从中心线测量，为h。安全系数定义为有效抗剪强度(f)与保持边限平衡状态的抗剪强度(m)的比值，即： 每个条块的安全系数取相同值，表明条块之间必须互相支持，即条块间必须有力的作用。 作用于条块上的力（条块每个单元维上法向力）如下：1.条块总重量，W=b h（适当时用sat）2.作用于底边上总法向力，N（等于l）。总体上，这个力有两部分：有效法向力N（等于l ）和边界孔隙水压力U（等于ul），其中u是底边中心的孔隙水压力，而l是底边长度。3.
5、底边上的剪力，T=ml。4.侧面上总法向力, E1和E2。5.侧面上总剪力，X1 和X2任何的外力也必须包含在分析之中。这是一种静不定问题，为了得到解决，就必须对于条块间作用力E 和X作出假定：安全系数的最终解答是不准确的。考虑到围绕O点的力矩，破坏弧AC上的剪力T的力矩总和，必须与土体ABCD重量所产生的力矩相等。对于任何条块，W的力臂为rsin，因此Tr=Wr sin则， 对于有效应力方面的分析：或者 (9.3) 其中La是弧AC的长度。公式9.3是准确的，但是当确定力N时引入了近似。对于给定的破坏面，F的取值将决定于力N的计算方法。 在这种解法中，假定对于任何一个条块，条间的相互作用力为
6、零。解答包括了解出每个条块垂直于底边的作用力，即：N=WCOS-ul因此，在有效应力方面的安全系数（公式9.3），由下式计算： (9.4)对于每个条块，Wcos和Wsin可以通过图表法确定。的取值可以通过测量或计算得到。同样地，也必须选择一系列试验性的破坏面来获得最小的安全系数。这种解法所得的安全系数：与更精确的分析方法相比，其误差通常为5-2%。 应用总应力法分析时，使用参数Cu 和u，公式9.4中u取零。如果u=0，那么安全系数为： (9.5) 因为N没有出现在公式9.5中，故得到的安全系数F值是精确的。在这种解法中，假定条块侧面的力是水平的，即：Xl-X2=0为了达到平衡，任何一个条块底
7、边上的剪力为： 解答垂直方向上的力： (9.6)很方便得到： l=b sec从公式9.3，通过一些重新整理， (9.7)孔隙水压力通过孔压比，可以与任何点的与总“填充压力”相联系，定义为： (9.8)(适当时用sat)对于任何条块, 因此公式9.7可写为: (9.9) 因为安全系数出现在公式9.9的两边，必须使用一系列近似，才能获得解答，但收敛很快。基于计算的重复性，需要选择充分数量的试验性破坏面。条分法特别适合于计算机解答。可以引入更复杂的边坡几何学和不同的土层。 在大多数问题中，孔压力比的取值ru在整个破坏面上是不一致的，但一旦存在独立的高孔压区，通常在设计中采用平均值（单位面积上的荷重）
8、。同样的，这种方法确定的安全系数过低，但误差不超过7，多数情况下小于2。斯班瑟 9.8 提出了一种分析方法，在此法中，条块间的作用力是水平的，且满足力和力矩平衡。斯班瑟得到了只满足力矩平衡的毕肖普简化解，其精确度取决于边坡条块间作用力力矩平衡的不敏感性。 基于公式9.9的匀质土边坡的稳定系数，是由毕肖普和摩根斯特恩9.2发表的。由此可见，对于给定坡角和给定土性的边坡，安全系数随u 线性变化，因此可以表示为：F=m-u (9.10)其中m和n是稳定系数。系数 m 和 n 是,， c/及深度系数 D的函数。假定潜在破坏面与边坡面平行,所在深度与边坡长度相比很小。那么,边坡可以看作无限长,忽略端部效
9、应。边坡与水平线成角,破坏面深度为z如图9.7中所示。水位线在破坏面以上高度mz (0m1)处,与边坡平行。假定稳定渗流发生在与边坡平行的方向上。任何垂直条块侧面上的力是等值反向的,且破坏面上任意一点的应力状态是相同的. 应用有效应力法,沿着破坏面上的土的抗剪强度为: 安全系数为:,和表达为:接下来的特殊情况是需要引起注意的。如果 c=0 和 m=0 (即坡面与破坏面间的土是不完全饱和的),那么: (9.11)如果c=0 和m=1(即水位线与边坡面一致) ，那么： (9.12)应当注意的是，当c=0 时，安全系数是与深度无关的。如果c 大于零，那么安全系数就是z 的函数，如果z 比规定值还小的
10、话，可能会超过 。 应用总应力分析法，需使用抗剪强度参数cu 和u ，而u取值为零。摩根斯特恩和普莱斯9.4提出了一般分析法，此法满足所有的边界条件和平衡条件，破坏面可以是任何形状，圆弧，非圆弧或符合型。破坏面以上的土体被划分为一系列垂直的平面，问题通过假定每部分之间垂直边界上的作用力E 和X的关系 而转化为静定。这个假定的形式为X=f(x)E (9.13)其中f(x)是描述随土体而变化的比值X/E 的形式的任意函数，而是尺寸效应系数。的值是在解安全系数F时一同获得的。在每个垂直边界上能够确定作用力E 和X的值及作用点。对于任意的假定函数 f(x) ，有必要仔细地检查解答，以确定其在物理学上的
11、合理性（即破坏面以上土体中没有剪切破坏或张力）。函数f(x)的选择对于F的计算值的影响不能超过 5% ，通常假定f(x)=l。 这种分析包含了和F值相互作用的复杂过程，如摩根斯特恩和普莱斯9.5所描述的那样，计算机的运用是必不可少的。 贝尔9.1 提出了一种满足所有平衡情况，假定破坏面可能是任何形状的分析方法。土体被划分成一系列垂直的条块，通过沿着破坏面上的法向作用力的假想分配，转化为静定问题。 萨尔玛 9.6 基于条分法发展了一种方法，在此法中，产生极限平衡所要求的临界地震加速度是确定的。这种分析方法在分析中假定了条块间垂直作用力的分配。同样的，满足所有的平衡条件，破坏面可以是任何形状。静安
12、全系数是土的抗剪强度必须减小，以致于临界加速度为零时的系数。 计算机的使用对于贝尔法和萨尔玛法来说，是必不可少的。所有的解答必须要检查，以确保它们在物理学上是可以接受的。Stability of SlopesGravitational and seepage forces tend to cause instability in natural slopes, in slopes formed by excavation and in the slopes of embankments and earth dams. The most important types of slope fail
13、ure are illustrated in Fig.9.1.In rotational slips the shape of the failure surface in section may be a circular arc or a non-circular curveIn general，circular slips are associated with homogeneous soil conditions and non-circular slips with non-homogeneous conditionsTranslational and compound slips
14、 occur where the form of the failure surface is influenced by the presence of an adjacent stratum of significantlydifferent strengthTranslational slips tend to occur where the adjacent stratum is at a relatively shallow depth below the surface of the slope:the failure surface tends to be plane and r
15、oughly parallel to the slope.Compound slips usually occur where the adjacent stratum is at greater depth，the failure surface consisting of curved and plane sectionsIn practice, limiting equilibrium methods are used in the analysis of slope stability. It is considered that failure is on the point of
16、occurring along an assumed or a known failure surfaceThe shear strength required to maintain a condition of limiting equilibrium is compared with the available shear strength of the soil，giving the average factor of safety along the failure surfaceThe problem is considered in two dimensions，conditio
17、ns of plane strain being assumedIt has been shown that a two-dimensional analysis gives a conservative result for a failure on a three-dimensional(dish-shaped) surfaceThis analysis, in terms of total stress，covers the case of a fully saturated clay under undrained conditions, i.e. For the condition
18、immediately after constructionOnly moment equilibrium is considered in the analysisIn section, the potential failure surface is assumed to be a circular arc. A trial failure surface(centre O，radius r and length La)is shown in Fig.9.2. Potential instability is due to the total weight of the soil mass
19、(W per unit Length) above the failure surfaceFor equilibrium the shear strength which must be mobilized along the failure surface is expressed aswhere F is the factor of safety with respect to shear strengthEquating moments about O： Therefore (9.1) The moments of any additional forces must be taken
20、into accountIn the event of a tension crack developing ，as shown in Fig.9.2，the arc length La is shortened and a hydrostatic force will act normal to the crack if the crack fills with waterIt is necessary to analyze the slope for a number of trial failure surfaces in order that the minimum factor of
21、 safety can be determined Based on the principle of geometric similarity，Taylor9.9published stability coefficients for the analysis of homogeneous slopes in terms of total stressFor a slope of height H the stability coefficient (Ns) for the failure surface along which the factor of safety is a minim
22、um is (9.2)For the case ofu =0，values of Ns can be obtained from Fig.9.3.The coefficient Ns depends on the slope angleand the depth factor D，where DH is the depth to a firm stratumGibson and Morgenstern 9.3 published stability coefficients for slopes in normally consolidated clays in which the undra
23、ined strength cu(u =0) varies linearly with depthIn this method the potential failure surface，in section，is again assumed to be a circular arc with centre O and radius rThe soil mass (ABCD) above a trial failure surface (AC) is divided by vertical planes into a series of slices of width b, as shown
24、in Fig.9.5.The base of each slice is assumed to be a straight lineFor any slice the inclination of the base to the horizontal isand the height, measured on the centre-1ine,is h. The factor of safety is defined as the ratio of the available shear strength(f)to the shear strength(m) which must be mobi
25、lized to maintain a condition of limiting equilibrium, i.e. The factor of safety is taken to be the same for each slice，implying that there must be mutual support between slices，i.e. forces must act between the slicesThe forces (per unit dimension normal to the section) acting on a slice are：1.The t
26、otal weight of the slice，W=b h (sat where appropriate)2.The total normal force on the base，N (equal to l)In general thisforce has two components，the effective normal force N(equal tol ) and the boundary water force U(equal to ul )，where u is the pore water pressure at the centre of the base and l is
27、 the length of the base3.The shear force on the base，T=ml.4.The total normal forces on the sides, E1 and E2.5.The shear forces on the sides，X1 and X2.Any external forces must also be included in the analysis The problem is statically indeterminate and in order to obtain a solution assumptions must b
28、e made regarding the interslice forces E and X：the resulting solution for factor of safety is not exact Considering moments about O，the sum of the moments of the shear forces T on the failure arc AC must equal the moment of the weight of the soil mass ABCDFor any slice the lever arm of W is rsin,the
29、reforeTr=Wr sinNow, For an analysis in terms of effective stress，Or (9.3)where La is the arc length ACEquation 9.3 is exact but approximations are introduced in determining the forces NFor a given failure arc the value of F will depend on the way in which the forces N are estimated In this solution
30、it is assumed that for each slice the resultant of the interslice forces is zeroThe solution involves resolving the forces on each slice normal to the base，i.e.N=WCOS-ulHence the factor of safety in terms of effective stress (Equation 9.3) is given by (9.4)The components WCOSand Wsincan be determine
31、d graphically for each sliceAlternatively，the value of can be measured or calculatedAgain，a series of trial failure surfaces must be chosen in order to obtain the minimum factor of safetyThis solution underestimates the factor of safety：the error，compared with more accurate methods of analysis，is us
32、ually within the range 5-2%. For an analysis in terms of total stress the parameters Cu andu are used and the value of u in Equation 9.4 is zeroIf u=0 ,the factor of safety is given by (9.5)As N does not appear in Equation 9.5 an exact value of F is obtainedIn this solution it is assumed that the re
33、sultant forces on the sides of theslices are horizontal，i.e.Xl-X2=0For equilibrium the shear force on the base of any slice is Resolving forces in the vertical direction: (9.6)It is convenient to substitute l=b secFrom Equation 9.3，after some rearrangement， (9.7) The pore water pressure can be relat
34、ed to the total fill pressure at anypoint by means of the dimensionless pore pressure ratio，defined as (9.8)(sat where appropriate)For any slice, Hence Equation 9.7 can be written: (9.9) As the factor of safety occurs on both sides of Equation 9.9,a process of successive approximation must be used t
35、o obtain a solution but convergence is rapid Due to the repetitive nature of the calculations and the need to select an adequate number of trial failure surfaces，the method of slices is particularly suitable for solution by computerMore complex slope geometry and different soil strata can be introdu
36、ced In most problems the value of the pore pressure ratio ru is not constant over the whole failure surface but，unless there are isolated regions of high pore pressure，an average value(weighted on an area basis) is normally used in designAgain，the factor of safety determined by this method is an und
37、erestimate but the error is unlikely to exceed 7and in most cases is less than 2 Spencer 9.8 proposed a method of analysis in which the resultant Interslice forces are parallel and in which both force and moment equilibrium are satisfiedSpencer showed that the accuracy of the Bishop simplified metho
38、d，in which only moment equilibrium is satisfied, is due to the insensitivity of the moment equation to the slope of the interslice forces Dimensionless stability coefficients for homogeneous slopes，based on Equation 9.9，have been published by Bishop and Morgenstern 9.2.It can be shown that for a giv
39、en slope angle and given soil properties the factor of safety varies linearly with u and can thus be expressed asF=m-nu (9.10)where，m and n are the stability coefficientsThe coefficients，m and n arefunctions of,，the dimensionless number c/and the depth factor D.Using the Fellenius method of slices，d
40、etermine the factor of safety，in terms of effective stress，of the slope shown in Fig.9.6 for the given failure surfaceThe unit weight of the soil，both above and below the water table，is 20 kNm 3 and the relevant shear strength parameters are c=10 kN/m2 and=29.The factor of safety is given by Equatio
41、n 9.4.The soil mass is divided into slices l.5 m wide. The weight (W) of each slice is given by W=bh=201.5h=30h kNmThe height h for each slice is set off below the centre of the base and thenormal and tangential components hcosand hsinrespectively are determined graphically，as shown in Fig.9.6.ThenW
42、cos=30h cosW sin=30h sinThe pore water pressure at the centre of the base of each slice is taken to bewzw，where zw is the vertical distance of the centre point below the water table (as shown in figure)This procedure slightly overestimates the pore water pressure which strictly should be) wze，where
43、ze is the vertical distance below the point of intersection of the water table and the equipotential through the centre of the slice baseThe error involved is on the safe sideThe arc length (La) is calculated as 14.35 mmThe results are given inTable 9.1Wcos=3017.50=525kNmW sin=308.45=254kNm(wcos -ul
44、)=525132=393kNmIt is assumed that the potential failure surface is parallel to the surface of the slope and is at a depth that is small compared with the length of the slope. The slope can then be considered as being of infinite length，with end effects being ignoredThe slope is inclined at angle to the horizontal and the depth of the failure plane is zas shown in section in Fig.9.7.The water table is taken to be parallel to the slope at a height of mz (0m1)above the failure planeSteady seepage is assumed to be taking
