简要的分析斜坡稳定性外文翻译.doc

international journal for numerical and analytical methods in geomechanicsint. j. numer. anal. meth. geomech., 23, 439449 (1999)short communicationsanalytical method for analysis of slopestabilityjinggang caos and musharraf m. zaman*tschool of civil engineering and environmental science, university of oklahoma, norman, ok 73019, u.s.a.summaryan 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, bishops, 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.introductionone 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 signicantly 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 usingcircular 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 signicant e!ort and cost that may not be justied 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 strengththere are two broad classes of soils. in coarse-grained cohesionless sands and gravels, the shear strength is directly proportional to the stress level: （1）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 conditions, the shear strength cu is largely independent of pressure, that is=0. when drainage is permitted, however, both &cohesive and &frictional components are 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 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 surfacefailure 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.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） （4）where is the inclination of the failure surface and w is given by （5）where is the unit weight of soil, h the height of slope, is the inclination of the slope. since the length of the slide surface ab is , the 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 （6）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 （7）it can be observed from equation (7) that fs is a function of a. thus the minimum value of fs can be found using powells 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 the average shear stress developed along the potential failure surface.for cohesionless soils where c=0, the safety factor can be readily written from equation (7) as （8）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 surfaceslides 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 thecircle 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 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. bishops 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 prerpendicularly and parallel to ef, one gets （9） （10） （11）the force n can produce a maximum shearing resistance when failure occurs: （12）the equations of lines ac, cb, and aby are given by y （13）the sums of the disturbing and resisting moments for all slices can be written as (14) (15) （16） （17） （18） （19）the safety factor for this case is usually expressed as the ratio of the maximum available resisting moment to the disturbing moment, that is (20)when the slope inclination exceeds 543, all failures emerge at the toe of the slope, which is called toe 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 under the top of the slope of clayey soil with, the slide emerges from the face of 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.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: （21）where t is given by equation (17), andandare given by (22) （23）其中， （24） (25)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 powells 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 powells algorithm. the modiffed powells 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 powells 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.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 commentsthe 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 bishops (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 bishops 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 bishops 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 th