[中英文翻译]边坡稳定汇编

土木建筑学院 土木0302班 学生邵明志 外文翻译 第2页共14页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。滑动边坡图瓦1边坡破坏类型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 度表小如下：其中F是就抗剪强度而言的安全系数.关于 O点力矩平衡:%吟因此图9.2 mu情况的分析(9.1)Wd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。FyH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（h =0的情况，Ns的值可以从图9.3中得到。尺值取决于边坡坡角B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（ 4u=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 。为圆心，以r为半径的圆弧。试验性破 坏面（AQ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a ,它的高，从中心线测量，为h。安全系数定义为有效抗剪强度（Pf）与保持边限平衡状态的抗剪 强度（Pm）的比值，即：F h %济南大学毕业设计用纸土木建筑学院土木0302班 学生邵明志 外文翻译 第4页共14页图9. 5条分法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Tsat）2 .作用于底边上总法向力，N （等于目）。总体上，这个力有两部分：有效法向力N（等 于-1 ）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 .底边上的剪力，T=r ml o4 .侧面上总法向力，E i和巳。5 .侧面上总剪力，Xi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。点的力矩，破坏弧AC上的剪力T的力矩总和，必须与土体 ABCDM量所 产生的力矩相等。对于任何条块，W的力臂为rsin a ,因此ETr=EWr sin a则，对于有效应力方面的分析：E W sins或者此 + ian式卬其中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零。解答包括了解出每 个条块垂直于底边的作用力，即：N=WCO S -ul因此，在有效应力方面的安全系数（公式 9.3）,由下式计算：也 + tan/(Wcosot 一 皿)Ursina(9.4)对于每个条块，Wcosx和Wsin a可以通过图表法确定。a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。如果小u=0,那么安全系 数为：LIV sin a(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:X -X2=0为了达到平衡，任何一个条块底边上的剪力为：7 = / k + N tan解答垂直方向上的力:,c7 N国 Ncosa + ulcosa + Tsina + -tansini H f ）很方便得到：l=b sec a从公式9.3,通过一些重新整理，(9.6)F = rN 仲 + W-邮砌sec atan a tan(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)济南大学毕业设计用纸土木建筑学院 土木0302班 学生邵明志 外文翻译 第15页共14页（适当时用Y sat）.对于任何条块,U% 丽因此公式9.7可写为：士丽占 般+呷一.树湫:不一一一F（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） 。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%0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Tu线性变化，因此可以表示为：F=m-T u（9.10）其中m和n是稳定系数。系数 m和n是0 ,小，c/ 丫及深度系数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，忽略端部效应。边坡与水平线成B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 何垂直条块侧面上白力是等值反向的，且破坏面上任意一点的应力状态是相同的.图9. 7平面层滑应用有效应力法，沿着破坏面上的土的抗剪强度为0 = c + (tr - u)tan 0安全系数为:b=1 一m)y +西九职上cosT = (I - m)y + zsinflcos/? h = /nzywcos2 flc =0和m=0 (即坡面与破坏面间的土接下来的特殊情况是需要引起注意的。如果 是不完全饱和的)，那么：F =步(9.11)tan fl如果c =0和m=1仰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. 一一 ,一 、 、 , 一一 、一 一、 一 全系数就是z的函数，如果z比规定值还小的话，B可能会超过小。应用总应力分析法，需使用抗剪强度参数cu和一，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，而入是尺寸效应系数。 入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 。这种分析包含了人和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 SlopesbyGravitational 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 failure 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 curve. In general, circular slips are associatedwith homogeneous soil conditions and non-circular slips with non-homogeneous conditions. Translational and compound slips occur where the form of the failure surface is influencedthe presence of an adjacent stratum of significantlydifferent strengt hTranslational 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 roughly parallel to the slope.Compound slips usually occur where the adjacent stratum is at greater depth the failure surface consisting of curved and plane section, sTyjm of Sge failureIn practice, limiting equilibrium methods are used in the analysis of slope stability. It is considered that failure is on the point of occurring along an assumed or a known failure surface. The 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 surface. The problem is considered in two dimensions, conditions of plane strain being assumed It has been shown that a two-dimensional analysis gives a conservative result for a failure on a three-dimensional(dish-shaped) surfac eThis analysis, in terms of total stress, covers the case of a fully saturated clay under undrained conditions, i.e. For the condition immediately after construction. Only moment equilibrium is considered in the analysis In 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(W per unit Length) above the failure surface. For equilibrium the shear strength which must be mobilized along the failure surface is expressed ast =册 F Fwhere F is the factor of safety with respect to shear strength Equating moments about OThereforeKg 91 The e.二 0 anwhsis.Wd(9.1)The moments of any additional forces must be taken into account In 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 Ills with water . It is necessary to analyze the slope for a number of trial failure surfaces in order that the minimum factor of safety can be determine dBased on the principle of geometric similarity, Taylor9.9published stability coefficients for the analysis of homogeneous slopes in terms of total stres s For a slope of height H the stability coefficient (Ns) for the failure surface along which the factor of safety is a minimum is(9.2)For the case of u =0 values of Ns can be obtained from Fig.93The coefficient N depends on the slope angle 0 and the depth facwh ere DH is the depth to a firm stratum.Gibson and Morgenstern 9.3 published stability coefficients for slopes in normally consolidated clays in which the undrained strength u( u =0) varies linearly with depth.I- ig. 7、1 tie mel tiocE of slicesIn this method the potential failure surface in section, is again assumed to be a circular arc with centre O and radius r. The soil mass (ABCD) above a trial failure surface (AC) is divided by vertical planes into a series of slices of width b, as shown 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 is height, measured on the centre-1ine,is h. The factor of safety is defined as the ratio of the available shear strength( f)tor the shear strength( m) iwhich must be mobilized 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 are1 .The total weight of the slice, W=y b h (sat/where appropriate)2 .The total normal force on the base N (equal to . dn) general thisforce has two components, the effective normal force N(equal too- l ) and the boundarywater force U(equal to ul ), where u is the pore water pressure at the centre of the base and l is the length of the base3 .The shear force on the base T= iml.4 .The total normal forces on the sides, E and E2.5 .The shear forces on the sides Xi and X2.Any external forces must also be included in the analysisThe problem is statically indeterminate and in order to obtain a solution assumptions must be made regarding the interslice forces E and X the resulting solution for factor of safety is not exact.Considering moments about Q 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 ABCD For any slice the lever arm of W is rsin a , thereforeE Tr= E Wr sin aNow,T = r/=? I F=工亚血工.卬For an analysis in terms of effective stres sES + rftan 修 I 二lIVsinaOrcLb + lan(9.3)Sllsinxwhere La is the arc length AC . Equation 9.3 is exact but approximations are introduced in determining the forces N. For a given failure arc the value of F will depend on the way in which the forces N are estimatedIn this solution it is assumed that for each slice the resultant of the interslice forces is zero The solution involves resolving the forces on each slice normal to the ba ei.e.N=WCOS -ulHence the factor of safety in terms of effective stress (Equation 9.3) is given byIosinacrLf + tan W cos - ui)(9.4)for eachThe components WCOS andWsin a can be determined graphically济南大学毕业设计用纸slice. Alternatively , the value of a can be measured or calciAgtein, a series of trial failure surfaces must be chosen in order to obtain the minimum factor of safety. This solutionunderestimates the factor of safet ythe error, compared with more accurate methods of analysis is usually within the range 5-2%.For an analysis in terms of total stress the parameters Cand 而 are used and the value of u in Equation 9.4 is zero. If 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 resultant forces on the sides of theslices are horizontal, i.e.Xl-X2=0For equilibrium the shear force on the base of any slice is丁 =/ + 却tan 在)Resolving forces in the vertical direction:力VNc&s i + ulcos i + sin i + tan i1 疝 aFF一卜=修一机触，正005! COSI + FfIt is convenient to substitutel=b sec aFrom Equation 9.3, after some rearrangemen,t(9.6)琲加产:+吧吧The pore water pressure can be related to the totalpoint by means of the dimensionless pore pressure rat jo defined asu调(sat where appropriate). For any slice, u r -,W/bHence Equation 9.7 can be written:(9.7)fill pressure(9.8)F(9.9)at anyAs the factor of safety occurs on both sides of Equation 99a process of successive approximation must be used to obtain a solution but convergence is rap idDue 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 computer More complex slope geometry and different soil strata can be introduce dIn 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 design. Again, the factor of safety determined by this method is an underestimate but the error is unlikely to exceed % and 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 satisfied. Spencer showed that the accuracy of the Bishop simplified method, in which only moment equilibrium is satisfied, is due to the insensitivity of the moment equation to the slope of the interslice force sDimensionless 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 given slope angle and given soil properties the factor of safety varies linearly with u and can thus be expre ssed asF=m-n u(9.10)where, m and n are the stability coefficients. The coefficients, m and n are functions of p the dimensionless number c/丫 and the depth factor D.Using the Fellenius method of slices, determine the factor of safety, in terms of effective stress, of the slope shown in Fig.9.6 for the given failure surface The unit weight of the soil, both above and below the water table is 20 kN /m and the relevant shear strength parameters are c =10 kN/mnd=29.The factor of safety is given by Equation 9.4.The soil mass is divided into slices l.5 m wide.The weight (W) of each slice is given byW=T bh=20X51 油=30h kN/mThe height h for each slice is set off below the centre of the base and the normal and tangential components hcos a and hsin a respectively are determined graphicaas shown in Fig.9.6.ThenWcosa =30h cos aW sin a =30h sin aThe pore water pressureat the centre of the base of each slice is taken to b碰zw, where zwyis 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/ze, where ze is the vertical distance below the point of intersection of the water table and the equipotential through the centre of the slice base The error involved is on the safe sideThe arc length (La) is calculated as 14.35 mm The results are given in Table 9.1E Wcosa =30 x 17.50=525kNmEW sin a =30X8.45=254kmE (wcos -M)=525 132=393kN/m*。工“ + tan 力E(Wc口sot u/) EWsinot(tO x 14 35) + (0 554 x 393)2541435 + 218 ，小昨9看Table 9.!Sluxh COS Qh sin jiuino1叫IkN m2向卜ikNym)1075-0 155-91-559121 80-0 10b5U17 732 700 4016215525 432Sr do1*11 605J 45b75J7 11 7029 16MO2-35心1 9522 071 902250255O80550-95024501750H451435It 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 ignored. The slope is inclined at angle0 to thehorizontal and the depth of the failure plane is z. as shown in section in Fig.97The water table is taken to be parallel to the slope at a height of mz (0m1)above the failure plane. Steady seepage is assumed to be taking place in a direction parallel to the sloperhe forces on the sides of any vertical slice are equal and opposite and the stress conditions are the same at every point on the failure plane.J-ifL 胃 FLinc crrivlaliniLil lipIn terms of effective stress, the shear strength of the soil along the failure plane isT/=c + (r u)tanand the factor of safety isF = %The expressions for o-, r and ii are:仃=(1 - tn)y + 用%J 2 8s2#T = (1 -帅 + IM、)工品?ssfu = mzyw cos7The following special cases are of interest If c =0 and m=0 (i.e. the sobetween the surface and the failure plane is not fully saturated),thenAtan 由tan /?(9.11)If clan fi=0 and m=1(i.e. the water table coincides with the surface of the s opeen：(9.12)m may exIt should be noted that when c=0 the factor of safety is independent ofthe depth z. If c is greater than zetbe factor of safety is a function of z, and provided z is less than a critical valueFor a total stress analysis the shear strength parameters ond 加 are used with a zero value of u.Morgenstern and Price9.4developed a general analysis in which all boundary and equilibrium conditions are satisfied and in which the failure surface may be any shap e circular, non-circular or compound. The soil mass above the failure plane is divided into sections by a number of vertical planes and the problem is rendered statically determinate by assuming a relationship between the forces E and X on the vertical boundaries between each section This assumption is of the formX=X f(x)E(9.13)where f(x)is an arbitrary function describing the pattern in which the ratio X/E varies across the soil mass and 入 is a scale factoralue of 入 is obtained as part of theioo along with the factor of safety F . The values of the forces E and X and the point of application of E can be determined at each vertical boundary For any assumed function f(x) it is necessary to examine the solution in detail to ensure that it is physically reasonable (i.e. no shear failure or tension must be implied within the soil mass above the failure surface). The choice of the function f(x) does not appear to influence the computed value of F by more than about 5% and f(x)=l is a common assumption.The analysis involves a complex process of iteration for the values of , described by 入 and F Morgenstern and Price9.5, and the use of a computer is essential.Bell 9.1 proposed a method of analysis in which all the conditions of equilibrium are satisfied and the assumed failure surface may be of any shape The soil mass is divided into a number of vertical slices and statical determinacy is obtained by means of an assumed distribution of normal stress along the failure surfaceSarma 9.6 developed a method, based on the method of slices, in which the critical earthquake acceleration required to produce a condition of limiting equilibrium is determined. An assumed distribution of vertical interslice forces is used in the analysis Again , all the conditions of equilibrium are satisfied and the assumedfailure surface may be of any shape. The static factor of safety is the factor by which the shear strength of the soil must be reduced such that the critical acceleration is zeroThe use of a computer is also essential for the Bell and Sarma methods and all solutions must be checked to ensure that they are physically acceptab le 来源： (岩土英语)