简要的分析斜坡稳定性外文翻译

简要的分析斜坡稳定性外文翻译 INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J. Numer. Anal. Meth. Geomech., 23, 439}449 (1999) SHORT COMMUNICATIONS ANALYTICAL METHOD FOR ANALYSIS OF SLOPE STABILITY JINGGANG CAOs AND MUSHARRAF M. ZAMAN*t School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, OK 73019, U.S.A. SUMMARY An analytical method is presented for analysis of slope stability involving cohesive and non-cohesive soils.Earthquake effects are considered in an approximate manner in terms of seismic coe$cient-dependent forces. Two kinds of failure surfaces areconsidered in this study: a planar failure surface, and a circular failure surface. The proposed method can be viewed as an extension of the method of slices, but it provides a more accurate etreatment of the forces because they are represented in an integral form. The factor of safety is obtained by using the minimization technique rather than by a trial and error approach used commonly. The factors of safety obtained by the analytical method are found to be in good agreement with those determined by the local minimum factor-of-safety, Bishop's, and the method of slices. The proposed method is straightforward, easy to use, and less time-consuming in locating the most critical slip surface and calculating the minimum factor of safety for a given slope. Copyright ( 1999) John Wiley & Sons, Ltd. Key words: analytical method; slope stability; cohesive and non-cohesive soils; dynamic effect; planar failure surface; circular failure surface; minimization technique; factor-of-safety. INTRODUCTION One of the earliest analyses which is still used in many applications involving earth pressure was proposed by Coulomb in 1773. His solution approach for earth pressures against retaining walls used plane sliding surfaces, which was extended to analysis of slopes in 1820 by Francais. By about 1840, experience with cuttings and embankments for railways and canals in England and France began to show that many failure surfaces in clay were not plane, but signi"cantly curved. In 1916, curved failure surfaces were again reported from the failure of quay structures in Sweden. In analyzing these failures, cylindrical surfaces were used and the sliding soil mass was divided into a number of vertical slices. The procedure is still sometimes referred to as the Swedish method of slices. By mid-1950s further attention was given to the methods of analysis using circular and non-circular sliding surfaces . In recent years, numerical methods have also been used in the slope stability analysis with the unprecedented development of computer hardware and software. Optimization techniques were used by Nguyen,10 and Chen and Shao. While finite element analyses have great potential for modelling field conditions realistically, they usually require signi"cant e!ort and cost that may not be justi"ed in some cases. The practice of dividing a sliding mass into a number of slices is still in use, and it forms the basis of many modern analyses.1,9 However, most of these methods use the sums of the terms for all slices which make the calculations involved in slope stability analysis a repetitive and laborious process. Locating the slip surface having the lowest factor of safety is an important part of analyzing a slope stability problem. A number of computer techniques have been developed to automate as much of this process as possible. Most computer programs use systematic changes in the position of the center of the circle and the length of the radius to find the critical circle. Unless there are geological controls that constrain the slip surface to a noncircular shape, it can be assumed with a reasonable certainty that the slip surface is circular.9 Spencer (1969) found that consideration of circular slip surfaces was as critical as logarithmic spiral slip surfaces for all practical purposes. Celestino and Duncan (1981), and Spencer (1981) found that, in analyses where the slip surface was allowed to take any shape, the critical slip surface found by the search was essentially circular. Chen (1970), Baker and Garber (1977), and Chen and Liu maintained that the critical slip surface is actually a log spiral. Chen and Liu12 developed semi-analytical solutions using variational calculus, for slope stability analysis with a logspiral failure surface in the coordinate system. Earthquake e!ects were approximated in terms of inertiaforces (vertical and horizontal) defined by the corresponding seismic coe$cients. Although this is one of the comprehensive and useful methods, use of /-coordinate system makes the solution procedure attainable but very complicated. Also, the solutions are obtained via numerical means at the end. Chen and Liu12 have listed many constraints, stemming from physical considerations that need to be taken into account when using their approach in analyzing a slope stability problem. The circular slip surfaces are employed for analysis of clayey slopes, within the framework of an analytical approach, in this study. The proposed method is more straightforward and simpler than that developed by Chen and Liu. Earthquake effects are included in the analysis in an approximate manner within the general framework of static loading. It is acknowledged that earthquake effects might be better modeled by including accumulated displacements in the analysis. The planar slip surfaces are employed for analysis of sandy slopes. A closed-form expression for the factor of safety is developed, which is diferent from that developed by Das. STABILITY ANALYSIS CONDITIONS AND SOIL STRENGTH There are two broad classes of soils. In coarse-grained cohesionless sands and gravels, the shear strength is directly proportional to the stress level: '' (1) ,,,,tanf /,f,where is the shear stress at failure, the effective normal stress at failure, /,and the effective angle of shearing resistance of soil. In fine-grained clays and silty clays, the strength depends on changes in pore water pressures or pore water volumes which take place during shearing. Under undrained ,uconditions, the shear strength cu is largely independent of pressure, that is=0. When ''drainage is permitted, however, both &cohesive' and &frictional' components are (,)c, observed. In this case the shear strength is given by (2) Consideration of the shear strengths of soils under drained and undrained conditions, and of the conditions that will control drainage in the field are important to include in analysis of slopes. Drained conditions are analyzed in terms of effective stresses, using ''values of determined from drained tests, or from undrained tests with pore (,)c, pressure measurement. Performing drained triaxial tests on clays is frequently impractical because the required testing time can be too long. Direct shear tests or CU tests with pore pressure measurement are often used because the testing time is relatively shorter. Stability analysis involves solution of a problem involving force and/or moment equilibrium.The equilibrium problem can be formulated in terms of (1) total unit weights and boundary water pressure; or (2) buoyant unit weights and seepage forces. The first alternative is a better choice, because it is more straightforward. Although it is possible, in principle, to use buoyant unit weights and seepage forces, that procedure is fraught with conceptual diffculties. PLANAR FAILURE SURFACE Failure surfaces in homogeneous or layered non-homogeneous sandy slopes are essentially planar. In some important applications, planar slides may develop. This may happen in slope, where permeable soils such as sandy soil and gravel or some permeable soils with some cohesion yet whose shear strength is principally provided by friction exist. For cohesionless sandy soils, the planar failure surface may happen in slopes where strong planar discontinuities develop, for example in the soil beneath the ground surface in natural hillsides or in man-made cuttings. , ,, 图平面破坏 Figure 1 shows a typical planar failure slope. From an equilibrium consideration of the slide body ABC by a vertical resolution of forces, the vertical forces across the base of the slide body must equal to weight w. Earthquake effects may be approximated by including a horizontal acceleration kg which produces a horizontal force k= acting through the centroid of the body and neglecting vertical inertia.1 For a slice of unit thickness in the strike direction, the resolved forces of normal and tangential components N and ? can be written as (3) NWk,,(cossin),, (4) TWk,，(sincos),, where is the inclination of the failure surface and w is given by LWxxdxHxdx,,，,,,,,,(tantan)(tan),,0 (5) 2H,,,(cotcot),,2 where is the unit weight of soil, H the height of slope, is LHlH,,cot,cot,,,,, the inclination of the slope. Since the length of the slide surface AB is , the cH/sin, resisting force produced by cohesion is cH/sin a. The friction force produced by N is . The total resisting or anti-sliding force is thus given by Wk(cossin)tan,,,, (6) RWkcH,,，(cossin)tan/sin,,,, For stability, the downslope slide force ? must not exceed the resisting force R of the body. The factor of safety, Fs , in the slope can be defined in terms of effective force by ratio R/T, that is 1tan2,kc, (7) F,，tan,skHk，，,tan(sincos)sin(),,,,,, It can be observed from equation (7) that Fs is a function of a. Thus the minimum value of Fs can be found using Powell's minimization technique18 from equation (7). Das reported a similar expression for Fs with k=0, developed directly from equation (2) by assuming that , where is the average shear strength of the soil, and F,,,/,sfdf the average shear stress developed along the potential failure surface. ,d For cohesionless soils where c=0, the safety factor can be readily written from equation (7) as 1tan,k, (8) F,tan,sk，tan, It is obvious that the minimum value of Fs occurs when a=b, and the failure becomes independent of slope height. For such cases (c=0 and k=0), the factors of safety obtainedfrom the proposed method and from Das are identical. CIRCULAR FAILURE SURFACE Slides in medium-stif clays are often deep-seated, and failure takes place along curved surfaces which can be closely approximated in two dimensions by circular surfaces. Figure 2 shows a potential circular sliding surface AB in two dimensions with centre O and radius r. The first step in the analysis is to evaluate the sliding' or disturbing moment Ms about the centre of the circle O. This should include the self-weight w of the sliding mass, and other terms such as crest loadings from stockpiles or railways, and water pressures acting externally to the slope. Earthquake effects is approximated by including a horizontal acceleration kg which produces a horiazontal force kd=acting through the centroid of each slice and neglecting vertical inertia. When the soil above AB is just on the point of sliding, the average shearing resistance which is required along AB for limiting equilibrium is given by equation (2). The slide mass is divided into vertical slices, and a typical slice DEFG is shown. The self-weight of the slice is . The method assumes that the dWhdx,, resultant forces Xl and Xr on DE and FG, respectively, are equal and opposite, and parallel to the base of the slice EF. It is realized that these assumptions are necessary to keep the analytical solution of the slope stability problem addressed in this paper achievable and some of these assumptions would lead to restrictions in terms of applications (e.g.earth pressure on retaining walls). However, analytical solutions have a special usefulness in engineering practice, particularly in terms of obtaining approximate solutions. More rigorous methods, e.g. finite element technique, can then be used to pursue a detail solution. Bishop's rigorous method5 introduces a further numerical procedure to permit specialcation of interslice shear forces Xl and Xr . SinceXl and Xr are internal forces, must be zero for the whole section. Resolving ()XX,,lr prerpendicularly and parallel to EF, one gets (9) Thdxkhdx,，,,,,sincos (10) Nhdxkhdx,,,,,,coscsin xa,22 (11) ,arcsin,rab,,，r The force N can produce a maximum shearing resistance when failure occurs: (12) Rcdxhdxk,，,sec(cossin)tan,,,,, The equations of lines AC, CB, and ABY are given by y 22 (13) yxyhybrxa,,,,,,tan,,(),123 The sums of the disturbing and resisting moments for all slices can be written as lMrhkdx,，,,,(sincos)s,0 ll (14) ,,，，,，ryykdxryykdx()(sincos)()(sincos),,,,,,1323,,0L ,，rIkI(),sclMrchkdx,，,sec(cossin)tan,,,,,,,r,0 ll,，,,rcdxryykdxsec()(cossin)tan,,,,,23,,00 (15) l，,,ryykdx()(cossin)tan,,,,23,L 2,，,rcrIkItan()cs,,, 22 (16) LHlarbH,,，,,cot,(), laa, (17) ,arcsinarcsin,，rr LlIyydxyydx,,，,()sin()sin,,1323s,,0L (18) 2H1，，2,，,(cot)secabH,,,,23r，， LlIyydxyydx,,，,()cos()cos,,s1323,,0L 22tantanbrb,,2222，，,,，,,,，，2()()()rLarLa，，623rr (19) rLara,,,,,，,，,(tan)arcsin(tan)arcsinaHab,,,,,,22rr,,,, rla,1222，，,,，,，,,()arcsin()4()()bHrlablaHa，，26rr The safety factor for this case is usually expressed as the ratio of the maximum available resisting moment to the disturbing moment, that is crIkI,,,，,tan()McsrF,, (20) sMIkI()，,ssc When the slope inclination exceeds 543, all failures emerge at the toe of the slope, ch is called toewhi failure, as shown in Figure 2. However, when the slope heightH is relatively large compared with the undrained shear strength or when a hard stratum is 0under the top of the slope of clayey soil with, the slide emerges from the face of ,,3 the slope, which is called Face failure, as shown in Figure 3. For Face failure, the safety factor Fs is the same as ?oe failure1s using instead of H. ()Hh,0 For flatter slopes, failure is deep-seated and extends to the hard stratum forming the base of the clay layer, which is called Base failure, as shown in Figure 4.1,3 Following the same procedure as that for ?oe failure, one can get the safety factor for Base failure: ''crIkI,,,，,tan()cs (21) F,s''IkI，,，，sc ''where t is given by equation (17), andandare given by IIsc lll01'Iyyxdxyyxdxyyxdx,,，,，,sinsinsin，，，，，，s031323,,,ll000 (22) 3HHblH222,，,,,,，,，cot()()(2)(33)lllllabbHH,0112223rrr lll01,,,,，，，，，，I,y,ycosd，y,ycosd，y,ycosd,,,xxxc031323ll010 (23) Hl1r1,arba,,,,2220,，，,r,Hcot,b,Harcsin,arcsin，,,,,2r42r2r,,,, ,HHcot1,,,,222，，，，,,ratan,arcsin，4rl,ab，l,aH,a,,,,,22r6r,,,, 22其中，yyxyHybrxa,,,,,,,0,tan,,, (24) ，，123 1122 (25) ,,,，,，,,,,laHlaHlarbHcot,cot,，，0122 It can be observed from equations (21)~(25) that the factor of safety Fs for a given slope is a function of the parameters a and b. Thus, the minimum value of Fs can be found using the Powell's minimization technique. For a given single function f which depends on two independent variables, such as the problem under consideration here, minimization techniques are needed to find the value of these variables where f takes on a minimum value, and then to calculate the corresponding value of f. If one starts at a point P in an N-dimensional space, and proceed from there in some vector direction n, then any function of N variables f (P) can be minimized along the line n by one-dimensional methods. Different methods will difer only by how, at each stage, they choose the next direction n. Powell "rst discovered a direction set method which producesN mutually conjugate directions.Unfortunately, a problem of linear dependence was observed in Powell's algorithm. The modiffed Powell's method avoids a buildup of linear dependence. The closed-form slope stability equation (21) allows the application of an optimization technique to locate the center of the sliding circle (a, b). The minimum factor of safety Fs min then obtained by substituting the values of these parameters into equations (22)~(25) and the results into equation (21), for a base failure problem (Figure 4). While using the Powell's method, the key is to specify some initial values of a and b. Well-assumed initial values of a and b can result in a quick convergence. If the values of a and b are given inappropriately, it may result in a delayed convergence and certain values would not produce a convergent solution. Generally, a should be assumed within$?, while b should be equal to or greater thanH (Figure 4). Similarly, equations (16)~(20) could be used to compute the Fs .min for toe failure (Figure 2) and face failure (Figure 3),except is used instead of H in the case of face failure. Hh,，，0 Besides the Powell method, other available minimization methods were also tried in this study such as downhill simplex method, conjugate gradient methods, and variable metric methods. These methods need more rigorous or closer initial values of a and b to the target values than the Powell method. A short computer program was developed using the Powell method to locate the center of the sliding circle (a, b) and to find the minimum value of Fs . This approach of slope stability analysis is straightforward and simple. RESULTS AND COMMENTS The validity of the analytical method presented in the preceding sections was evaluated using two well-established methods of slope stability analysis. The local minimum factor-of-safety (1993) method, with the state of the effective stresses in a slope determined by the finite element method with the Drucker-Prager non-linear stress-strain relationship, and Bishop's (1952) method were used to compare the overall factors of safety with respect to the slip surface determined by the proposed analytical method. Assuming k=0 for comparison with the results obtained from the local minimum factor-of-safety and Bishop's method, the results obtained from each of those three methods are listed in Table I. The cases are chosen from the toe failure in a hypothetical homogeneous dry soil slope having a unit weight of 18.5 kN/m3. Two slope configurations were analysed, one 1 : 1 slope and one 2 : 1 slope. Each slope height H was arbitrarily chosen as 8 m. To evaluate the sensitivity of strength parameters on slope stability, cohesion ranging from 5 to 30 kPa and friction angles ranging from 103 to 203 were used in the analyses (Table I). A number of critical combinations of c and were found to be unstable for the model slopes studied. The factors of safety obtained by the proposed method are in good agreement with those determined by the local minimum factor-of-safety and Bishop's methods, as shown in Table I. To examine the e!ect of dynamic forces, the analytical method is chosen to analyse a toe failure in a homogeneous clayey slope (Figure 2). The height of the slope H is 13.5 m; the slope inclination b is arctan 1/2; the unit weight of the soil c is 17.3 kN/m3; the friction angle is 17.3KN/m; and the cohesion c is 57.5 kPa. Using the conventional method of slices, Liu obtained the minimum safety factor Using the F,2.09smin proposed method, one can get the minimum value of safety factor from equation (20) as F,2.08 for k=0, which is very close to the value obtained from the slice method. smin When k"0)1, 0)15, or 0)2, one can getF,1.55,1.37, and 1)23, respectively,which smin shows the dynamic e!ect on the slope stability to be significant. CONCLUDING REMARKS An analytical method is presented for analysis of slope stability involving cohesive and noncohesive soils. Earthquake e!ects are considered in an approximate manner in terms of seismic coe$cient-dependent forces. Two kinds of failure surfaces are considered in this study: a planar failure surface, and a circular failure surface. Three failure conditions for circular failure surfaces namely toe failure, face failure, and base failure are considered for clayey slopes resting on a hard stratum. The proposed method can be viewed as an extension of the method of slices, but it provides a more accurate treatment of the forces because they are represented in an integral form. The factor of safety is obtained by using theminimization technique rather than by a trial and error approach used commonly. The factors of safety obtained from the proposed method are in good agreement with those determined by the local minimum factor-of-safety method (finite element method-based approach), the Bishop method, and the method of slices. A comparison of these methods shows that the proposed analytical approach is more straightforward, less time-consuming, and simple to use. The analytical solutions presented here may be found useful for (a) validating results obtained from other approaches, (b) providing initial estimates for slope stability, and (c) conducting parametric sensitivity analyses for various geometric and soil conditions. REFERENCES 1. D. Brunsden and D. B. Prior. Slope Instability, Wiley, New York, 1984. 2. B. F. Walker and R. Fell. Soil Slope Instability and Stabilization, Rotterdam, Sydney, 1987. 3. C. Y. Liu. Soil Mechanics, China Railway Press, Beijing, P. R. China, 1990. 448 SHORT COMMUNICATIONS Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 23, 439}449 (1999) 4. L. W. Abramson. Slope Stability and Stabilization Methods, Wiley, New York, 1996. 5. A. W. Bishop. &The use of the slip circle in the stability analysis of slopes', Geotechnique, 5, 7}17 (1955). 6. K. E. Petterson. &The early history of circular sliding surfaces', Geotechnique, 5, 275}296 (1956). 7. G. Lefebvre, J. M. Duncan and E. L. 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Principles of Geotechnical Engineering, PWS Publishing Company, Boston, 1994. 15. A. W. Skempton and H. Q. Golder. &Practical examples of the /"0 analysis of stability of clays', Proc. 2nd Int. Conf. SMFE, Rotterdam, Vol. 2, 1948, pp. 63}70. 16. L. Bjerrum, and T. C. Kenney. &E!ect of structure on the shear behavior of normally consolidated quick clays', Proc. Geotech. Conf., Oslo, Norway, vol. 2, 1967, pp. 19}27. 17. A. W. Skempton, &Long-term stability of clay slopes,' Geotechnique, 14, 77}102 (1964). 18. D. G. Liu, J. G. Fei, Y. J. Yu and G. Y. Li. FOR?RAN Programming, National Defense Industry Press, Beijing, P. R. China, 1988. 19. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes: ?he Art of Scienti,c Computing, Cambridge University Press, Cambridge, 1995. 20. M. G. Anderson and K. S. Richards. Slope Stability: Geotechnical Engineering and Geomorphology, Wiley, New York, 1987. 21. R. Baker. &Determination of critical slip surface in slope stability computations', Int. J. Numer. Anal. Meth. Geomech., 4, 333}359 (1980). 22. A. K. Chugh. &Variable factor of safety in slope stability analysis', Geotechnique, ?ondon, 36(1), 57}64 (1986). 23. B. M. Das. Principles of Soil Dynamics, PWS-Kent Publishing Company, Boston, 1993. 24. S. L. Huang and K. Yamasaki. &Slope failure analysis using local minimum factor-of-safety approach', J. Geotech. Engng. ASCE, 119(12), 1974}1987 (1993). 25. S. L. Kramer. Geotechnical Earthquake Engineering, Prentice Hall, Englewood Cli!s, NJ, 1996. 26. D. Leshchinsky and C. Huang. &Generalized three dimensional slope stability analysis', J. Geotech. Engng. ASCE, 118(11), 1748}1764 (1992). 27. K. S. Li and W. White. &Rapid evaluation of the critical surface in slope stability problems', Int. J. Numer. Anal. Meth. Geomech., 11(5), 449}473 (1987). 28. D. W. Taylor. Fundamentals of Soil Mechanics, Wiley, Toronto, 1948. 29. U. S. Federal Highway Administration, Advanced ?echnology for Soil Slope Stability, U.S. Dept. of Transportation, Washington, DC, 1994. 30. Spencer (1969). 31. Celestino and Duncan (1981). 32. Spencer (1981). 33. Chen (1970). 34. Baker and Garber (1977). 35. Bishop (1952). 简要的分析斜坡稳定性的方法 JINGGANG CAOs 和 MUSHARRAF M. ZAMAN 诺曼底的俄克拉荷马大学土木环境工程学院 摘要 本文给出了解析法对边坡的稳定性分析,包括粘性和混凝土支撑。 地震被认为是用和振动相似的方式产生的地震从属效应。这篇论文涉及到了两种破坏面:一个平面的破坏面,一个圆形的破坏面，这个合适的方法可以被视为切割方法的延伸，但是它提供了更加精确的计算力的方法，因为他采用的是积分的方法。安全的方法是利用最小化的技术,而不是一般的由一个反复的试验方法。 安全的因素所获得的分析方法是符合最初最低基本安全因素的方法—切割法。推荐的方法是基于最危险滑动面的直接的，最简单的去用并且最快的计算，和计算该斜坡的最小安全系数。 关键词:解析方法;岩质边坡稳定性;有粘性和无粘性土;动力学因素;平坦的破坏面;圆形破坏面;估算最小值的方法;影响安全性的因素 介绍 最早的用在分析土应力的方法被认为是库伦在1773年提出来的。他的解决挡土墙土应力的方法用的是滑动面，在1820年法国这个被延伸用来分析边坡。直到1840年，英国和法国的铁路和隧道的钻凿和路堤经验表明了许多泥土中的破坏面不是平的，而是没有规律的弯曲的。1916年，不规则的破坏面在码头结构破坏中出现在瑞典。分析了这些破坏面之后，圆柱体截面被采用，并且滑移土体被分成了一定数量的条形体。这个解决程序有时候也被称为“瑞典条分法”。到十九世纪五十年代中期，人们的注意力转移到了用圆形和非圆形滑动面的分析上了。近些年来，随着电脑的硬件和软件史无前例的发展，数值分析法已经被用在了边坡稳定性分析上。最好的方法是Nguyen,and Chen and Shao用的，当有限元分析有模拟真实的土质情况的时候，他们一直需要巨大的人力和物理，这些可能是没有结果的。 这个方法的滑动到数片仍然在被使用,它形成了许多现代分析基础,然而,大多数的这些方法的使用条款所有切片使计算边坡稳定性分析中所涉及的重复性和艰苦的过程。 定位的滑动面具有最低的安全系数的分析,是一个边坡稳定问重要的一部 分。大量的计算机技术已经发展到自动化许多这样的过程。大多数的计算机程序在中心的位置,利用半径的长度使用系统的变化找到临界圆。 除非有地质控制去约防止动面称为一个圆形状,它可以被认为是某一个合理的圆形边坡。 承担合理的滑动面是circular.9斯潘塞(1969)发现考虑的圆形滑移面和对数螺旋滑动面是同样临界的使用目的。Celestino和邓肯(1981年)、斯潘塞(1981年)的研究发现,在分析滑动面形状可以发生任何变形的地方滑动面被证明基本上都是圆形的。陈(1970),贝克和Garber(1977年)、陈、Liu12坚持滑动面实际上是一个切削螺旋型的。为解决边坡稳定性分析，陈和Liu12发表的解析在坐标系里是利用,,变分微积分,和对数螺旋线破裂面的来分析的。地应力是几乎按照地震系数定义,, 的惯性力来估计的。虽然利用坐标系来解决是一个综合的测验和有用的方,, 法,但是这种方法是十分复杂的。 同时,采用数值方法最后也能解决问题。陈和刘列出了对边坡稳定性分析时需要考虑的很多,出于物理因素的限制。 在此研究中圆形滑动面是用于粘土质斜坡分析框架内的一个。 分析方法。所提出的方法比陈和刘提出的方法更直接、更简单地震效应也包含在相似的总体框架相对静载的方法中。地震效应可以在位移模拟的分析方法中被更好的模拟是公认的。平缓的滑动面用来分析砂性的斜坡。一个安全系数的解析表达式发展并且被应用了，这是不同于Das所提出的分析方法的。 稳定性分析条件和土壤应力 有两种级别的土壤。在无粘性土和砂石中，剪切力是和应力成正比的: '' (1) ,,,,tanf //,f,,是破坏时的剪切力，是破坏时的正应力，是土壤的摩擦角。 在粉质粘土和细粘土中，应力取决于孔隙水压力或者是水在剪切过程中占 cu得体积。在没有排水措施的前提下，剪切力很大程度上是和压力无关的，也就 ,''u是说=0.当有排水设施时，不管是密实的或者是有摩擦的，他们的系数都是(,)c,遵循上述规律的。 这种情况下，剪切强度如下式: 考虑到剪力强度在排水和不排水条件下的不同，所以排水情况在边坡分析中是非常重要的。排水条件是依据应力值确定的，用的是排水和不排水条件下的孔隙压 ''力测试来确定系数的。对粘土采用三周压缩试验的排水方法通常是不合适的，(,)c, 因为所需要的测试时间可太长了。经常采用直剪试验或CU测试孔隙水压力是因为测试时间是相对较短的。 稳定性分析包括解决涉及力和力矩平衡的问题。(1)利用容重和水压力界限可以来解决平衡问题，或者通过公式(2)利用浮容重和渗流压力。第一个方案是比较好的选择，是因为他更加直接，他的步骤只是存在概念上的不同。 二维破坏面 均匀的或者是不均匀的砂性边坡的破裂面是二维的。在一些重要的二维滑坡的应用是可以应用的。这种方法可以用在可渗透性的土壤，比如砂性土和砾石，或者是有内聚力的砂性土，这种土的剪切力是由摩擦力提供的。对于无粘性的砂性土，边坡的二维破坏面可能发生在较大的二维间断点发育的地方，比如在自然或者是人工的山体土壤的下面是自然的土质中。 , ,, 图平面破坏 图1显示一典型的平面失败的斜坡。把滑动体ABC上的平衡力竖向分解，作用在滑动体上的垂直力一定是平衡与滑动体的自重W的。振动力是接近包括同一水平线上的重力加速度，它产生了一个水平方向的作用在滑动体重心上的力KW，并且不考虑竖直方向的惯性。对于作用力方向的一个单位层厚度，已知常力及其分量N、T可以按下式: (3) NWk,,(cossin),, (4) TWk,，(sincos),, 其中为破坏面的倾斜角，W按照下式计算: LWxxdxHxdx,,，,,,,,,(tantan)(tan),,0 (5) 2H,,,(cotcot),,2 式中，是土壤的容重，H是边坡的高， ，是边坡的LHlH,,cot,cot,,,,, 倾斜。 滑动面长AB为，摩擦阻力为，N产生的摩擦力是H/sin,cH/sin, ，总的抗滑力按下式给出: Wk(cossin)tan,,,, (6) RWkcH,,，(cossin)tan/sin,,,, 边坡的下滑力T是一定不会超过坡体本身的抗滑力R的。安全因素可以由RFs和T的比例来确定，也就是: 1tan2,kc, (7) F,，tan,skHk，，,tan(sincos)sin(),,,,,, 可以由公式(7)得出:是的。所以得最小值可以通过公式(7)的最FF,ss 小值来确定。Das利用K=0提出了一个相似的的表达式,是从公式(2)中直接假Fs 定而得出的，其中，是土壤的平均剪应力，是潜在破裂面的平均剪F,,,/,,sfdfd应力。 对于无粘性土壤，C=0，安全系数可以很容易的有公式(7)得出: 1tan,k, (8) F,tan,sk，tan, 很明显的，当时，有最小的安全系数，并且这时破坏是独立于边坡高度的。,,, 对于c=0和k=0的情况，安全系数通过其他合适的方法和Das提出的方法是完全相同的。 圆形破裂面 在中性粘土中的滑移面经常是较深的，并且破坏面为弯曲的表面，近似相当于两倍的二维的破坏面。图形二在二维极坐标系中给出了一个潜在的滑动面AB。分析的第一步是确定滑移和扰 动有关圆中心的。这个应该包括滑体的自重W和其他因素，例如:铁路的最值荷Ms 载和作用在边坡表面的水压力。地震效应包括一个水平的加速度KG，他提供了一个水平力KdW作用在每个片层的中心，并且忽略垂直方向上的惯性。当AB上的土壤刚刚好是处在滑动体上，沿着AB用来保持平衡的平均剪切强度是由公式(2)确定的。滑移体被分成竖直的条形体，DEFG就是一个典型的代表。条形体的自重是 。这种方法假设DE和FG上的合力和，分别是相等和相反的，并dWhdx,,XXlr 且是平行于片曾EF底部的。为了解决边坡稳定性问题，这些假设是不可缺少的，并且这些假设会成为软件计算中的限制条件(挡土墙的土压力)。然而在工程实践中特别是获得近似解的时候有一种特殊的解析模型。更严格的方法,有限元技术,可以用来追求细节的解决方案。Bishop的更加严谨的方法介绍了一种更加熟知的方法去得到剪力和。因为和是内力，所以在整个个体中一定是零。 XXXX()XX,,lrrllr (9) Thdxkhdx,，,,,,sincos (10) Nhdxkhdx,,,,,,coscsin xa,22 (11) ,arcsin,rab,,，r 当破坏发生时，力N可以产生一个最大的剪切阻力: (12) Rcdxhdxk,，,sec(cossin)tan,,,,, yyy123 AC，CB和AB的方程式中给出的，和，分别又下式计算: 22 (13) yxyhybrxa,,,,,,tan,,(),123 所有分层的抗力距总和可以写成如下形式: lMrhkdx,，,,,(sincos)s,0 ll (14) ,,，，,，ryykdxryykdx()(sincos)()(sincos),,,,,,1323,,0L ,，rIkI(),sclMrchkdx,，,sec(cossin)tan,,,,,,,r,0 ll,，,,rcdxryykdxsec()(cossin)tan,,,,,23,,00 (15) l，,,ryykdx()(cossin)tan,,,,23,L 2,，,rcrIkItan()cs,,, 其中， 22 (16) LHlarbH,,，,,cot,(), laa, (17) ,arcsinarcsin,，rr LlIyydxyydx,,，,()sin()sin,,1323s,,0L (18) 2H1，，2,，,(cot)secabH,,,,23r，， LlIyydxyydx,,，,()cos()cos,,s1323,,0L 22tantanbrb,,2222，，,,，,,,，，2()()()rLarLa，，623rr (19) rLara,,,,,，,，,(tan)arcsin(tan)arcsinaHab,,,,,,22rr,,,, rla,1222，，,,，,，,,()arcsin()4()()bHrlablaHa，，26rr 这个分析的安全系数通常表示为最大限度的阻力矩的扰动力矩，也就是， crIkI,,,，,tan()McsrF,, (20) sMIkI()，,ssc 当边坡的倾角超过54度，所有的破坏会发生在边坡的底角部分，这叫做底 角破坏，就如公式2中列出的一样。但是，当边坡高度H可以和不排水抗剪强度或 0者是一个边坡底下的坚硬的土层的，滑动体会发生在边坡的前面，叫做Face ,,3 failure，如公式3中所示。对于Face failure，安全系数和Toe破坏中的用Fs 代替H后的结构是相同的。 ()Hh,0 对于平面边坡，破坏较深的，并且延伸到粘性土的底部坚硬层，这个叫做Base 破坏，如公式4所示。按照和Toe破坏相同的步骤，可以得到Base破坏的安全系数: ''crIkI,,,，,tan()cs (21) F,s''IkI，,，，sc ''由公式(17)给出，和由下式给出: ,IIsc lll01'Iyyxdxyyxdxyyxdx,,，,，,sinsinsin，，，，，，s031323,,,0ll00 (22) 3HHblH222,，,,,,，,，cot()()(2)(33),lllllabbHH0112223rrr lll01,,,,，，，，，，I,y,ycosd，y,ycosd，y,ycosd,,,xxxc031323ll010 (23) Hl1r1,arba,,,,2220,，，,r,Hcot,b,Harcsin,arcsin，,,,,2r42r2r,,,, ,HHcot1,,,,222，，，，,,ratan,arcsin，4rl,ab，l,aH,a,,,,,22r6r,,,, 22其中， (24) yyxyHybrxa,,,,,,,0,tan,,,，，123 1122 (25) ,,,，,，,,,,laHlaHlarbHcot,cot,，，0122 从式(21)到式(25)中可以看到对于一个给定的边坡，安全系数是FS系数a和b的函数。因此，的最小值可以用Powell的求最小值方法来求得。对FS 于一个给定的函数f,取决于两个独立变量,如考虑以下问题,极小化方法需要找出这些变量的确切含义，f取得最小值的时候,然后去计算相应的f值。如果从N空间中的一点P开始，从这里沿着一些n矢量方向，通过一个单面的方法一些函数N,变量f(P)可以被最小化。在每一阶段，如何选择下一个方向n决定着采用了不同的方法。Powell首先发现了一种方法的方向，这种方法中的N符合数学中的共轭方向。不幸的是Powell的算法中存在着一个线性相关的问题。改进的Powell方法避免了线性相关性的增加。 不相关的坡面稳定性公式(21)包含找出滑动圆(a，b)的最优方法的应用。F的最小值可以通过向公式(22)--(25)、公式(21)的结果和底部破坏Frgure4，，S 中代入这些参数来获得。用Powell的方法的关键是找出a，b的原始值。选定恰当 的a，b初始值可以产生一个快速的收敛。 如果该数值a、b是不恰当的,这可能导致延误收敛性和某一个值可能产生一种不收敛的计算。一般情况下，应该假定a为a，b应该对于或大于H.(图4)。类似的，公式(16)-(20)应该用于toe破坏,L 中计算的最小值(图2)和face破坏(图3)，除了在face破坏中的代FHh,，，S0替H。 除了Powell的方法，其他可用的最小值求法也用在了这个方法中，如下坡单一法，共轭梯度法，可变标准法。这些方法比Powell需要更加严格更加接近目标值的a，b原始值。一个简短的计算机程序用的是Powell的方法去定位滑体圆的中心和发现的最小值。这种边坡稳定行分析的方法是直接简单的。 FS 结果和评论 这种分析方法的有效性提出在前面的部分中被评为使用两种方法对边坡稳定性分析为主。安全系数局部最小值的方法，利用边坡中的应力的有限单元法和德鲁克-普拉格的应力应变非线性关系，Bishop方法利用合适的分析方法确定的滑移面来比较全部的安全系数。假设K=0,去比较安全因素的局部最小值和Bishop法所得出的结果，这三种方法中的每一种所得出的结果列在了表1中。 为了查明动力的效应，分析方法的选择是为了分析一个匀质边坡的尖角破坏(图 12)。边坡的高H是13.5m，边坡的倾斜角是，土壤的容重是，摩17.3/kNmarctan2 擦角是7度，内聚力是57.5kpa。用切割的常规方法，刘得出的安全系数最小值 ，用适当的方法，可以从公式(20)中得到安全系数的最小值F,2.09smin F,2.08，其中K=0，这个安全系数是非常接近切割法得到的安全系数的。当smin K=0.1,0.15,0.2时，可以得到和1.23，这些显示的边坡上的动力效F,1.55,1.37smin 应都是重要的。 结束语 论文呈现了一种分析边坡包括粘性和非粘性土壤的稳定性的方法。地震效应被近似的当做一种依据地震独立系数的应力。这个论文当中包含了两种破坏面:一种是平坦的二维破坏面;和一种圆形的弯曲破坏面。圆形弯曲破坏下三种破坏情况分别叫做:尖角破坏、面破坏、和基础破坏，考虑的是在在坚硬底层上的粘性土边坡。 所提倡的方法是切割法的延伸方法，它提供了更加准确的解决各种力的方法，以为他们是以积分的形式表现出来的。安全系数是通过利用求公式的最小值来获得的，而不是通常所用的反复试验来接近结果的方法。 安全系数的局部最小值方法(有限单元法基础上的模拟法)决定了获得安全系数最小值的方法，如Bishop法和条分切割法。通过这些方法的比较可以知道，选择正确的分析方法是更加直接、快捷和容易操作的。这里提出的分析方法可以应用于(a)验证其他方法得出的结果;(b)对边坡稳定性提出最初的估算;(c)为各种几何体和土壤情况指出危险性分析的参数。 参考文献 1、D. 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