Slope Stability, Study Guides, Projects, Research of Design

Download Slope Stability and more Study Guides, Projects, Research Design in PDF only on Docsity! US Army Corps of Engineers® ENGINEERING AND DESIGN EM 1110-2-1902 31 Oct 2003 Slope Stability ENGINEER MANUAL AVAILABILITY Electronic copies of this and other U.S. Army Corps of Engineers (USACE) publications are available on the Internet at http://www.usace.army.mil/inet/usace-docs/. This site is the only repository for all official USACE engineer regulations, circulars, manuals, and other documents originating from HQUSACE. Publications are provided in portable document format (PDF). EM 1110-2-1902 31 Oct 03 ii Subject Paragraph Page Appendix C Stability Analysis Procedures – Theory and Limitations Appendix D Shear Strength Characterization Appendix E Chart Solutions for Embankment Slopes Appendix F Example Problems and Calculations Appendix G Procedures and Examples for Rapid Drawdown EM 1110-2-1902 31 Oct 03 1-1 Chapter 1 Introduction 1-1. Purpose and Scope This engineer manual (EM) provides guidance for analyzing the static stability of slopes of earth and rock-fill dams, slopes of other types of embankments, excavated slopes, and natural slopes in soil and soft rock. Methods for analysis of slope stability are described and are illustrated by examples in the appendixes. Criteria are presented for strength tests, analysis conditions, and factors of safety. The criteria in this EM are to be used with methods of stability analysis that satisfy all conditions of equilibrium. Methods that do not satisfy all conditions of equilibrium may involve significant inaccuracies and should be used only under the restricted conditions described herein. This manual is intended to guide design and construction engineers, rather than to specify rigid procedures to be followed in connection with a particular project. 1-2. Applicability This EM is applicable to all USACE elements and field operating activities having responsibility for analyzing stability of slopes. 1-3. References Appendix A contains a list of Government and non-Government references pertaining to this manual. Each reference is identified in the text by either the designated publication number or by author and date. 1-4. Notation and Glossary Symbols used in this manual are listed and defined in Appendix B. The notation in this manual corresponds whenever possible to that recommended by the American Society of Civil Engineers. 1-5. Basic Design Considerations a. General overview. Successful design requires consistency in the design process. What are considered to be appropriate values of factor of safety are inseparable from the procedures used to measure shear strengths and analyze stability. Where procedures for sampling, testing, or analysis are different from the procedures described in this manual, it is imperative to evaluate the effects of those differences. b. Site characterization. The stability of dams and slopes must be evaluated utilizing pertinent geologic information and information regarding in situ engineering properties of soil and rock materials. The geologic information and site characteristics that should be considered include: (1) Groundwater and seepage conditions. (2) Lithology, stratigraphy, and geologic details disclosed by borings and geologic interpretations. (3) Maximum past overburden at the site as deduced from geological evidence. (4) Structure, including bedding, folding, and faulting. (5) Alteration of materials by faulting. EM 1110-2-1902 31 Oct 03 1-2 (6) Joints and joint systems. (7) Weathering. (8) Cementation. (9) Slickensides. (10) Field evidence relating to slides, earthquake activity, movement along existing faults, and tension jointing. c. Material characterization. In evaluating engineering properties of soil and rock materials for use in design, consideration must be given to: (1) possible variation in natural deposits or borrow materials, (2) natural water contents of the materials, (3) climatic conditions, (4) possible variations in rate and methods of fill placement, and (5) variations in placement water contents and compacted densities that must be expected with normal control of fill construction. Other factors that must be considered in selecting values of design parameters, which can be evaluated only through exercise of engineering judgment, include: (1) the effect of differential settlements where embankments are located on compressible foundations or in narrow, deep valleys, and (2) stress-strain compatibility of zones of different materials within an embankment, or of the embankment and its foundation. The stability analyses presented in this manual assume that design strengths can be mobilized simultaneously in all materials along assumed sliding surfaces. d. Conventional analysis procedures (limit equilibrium). The conventional limit equilibrium methods of slope stability analysis used in geotechnical practice investigate the equilibrium of a soil mass tending to move downslope under the influence of gravity. A comparison is made between forces, moments, or stresses tending to cause instability of the mass, and those that resist instability. Two-dimensional (2-D) sections are analyzed and plane strain conditions are assumed. These methods assume that the shear strengths of the materials along the potential failure surface are governed by linear (Mohr-Coulomb) or nonlinear relationships between shear strength and the normal stress on the failure surface. (1) A free body of the soil mass bounded below by an assumed or known surface of sliding (potential slip surface), and above by the surface of the slope, is considered in these analyses. The requirements for static equilibrium of the soil mass are used to compute a factor of safety with respect to shear strength. The factor of safety is defined as the ratio of the available shear resistance (the capacity) to that required for equilibrium (the demand). Limit equilibrium analyses assume the factor of safety is the same along the entire slip surface. A value of factor of safety greater than 1.0 indicates that capacity exceeds demand and that the slope will be stable with respect to sliding along the assumed particular slip surface analyzed. A value of factor of safety less than 1.0 indicates that the slope will be unstable. (2) The most common methods for limit equilibrium analyses are methods of slices. In these methods, the soil mass above the assumed slip surface is divided into vertical slices for purposes of convenience in analysis. Several different methods of slices have been developed. These methods may result in different values of factor of safety because: (a) the various methods employ different assumptions to make the problem statically determinate, and (b) some of the methods do not satisfy all conditions of equilibrium. These issues are discussed in Appendix C. e. Special analysis procedures (finite element, three-dimensional (3-D), and probabilistic methods). (1) The finite element method can be used to compute stresses and displacements in earth structures. The method is particularly useful for soil-structure interaction problems, in which structural members interact with a soil mass. The stability of a slope cannot be determined directly from finite element analyses, but the EM 1110-2-1902 31 Oct 03 1-5 d. Establish the seepage and groundwater conditions in the cross section as measured or as predicted for the design load conditions. EM 1110-2-1901 describes methods to establishing seepage conditions through analysis and field measurements. e. Select loading conditions for analysis (see Chapter 2). f. Select trial slip surfaces and compute factors of safety using Spencer's method. In some cases it may be adequate to compute factors of safety using the Simplified Bishop Method or the force equilibrium method (including the Modified Swedish Method) with a constant side force (Appendix C). Appendix F provides example problems and calculations for the simplified Bishop and Modified Swedish Procedures. g. Repeat step f above until the “critical” slip surface has been located. The critical slip surface is the one that has the lowest factor of safety and which, therefore, represents the most likely failure mechanism. Steps f and g are automated in most slope stability computer programs, but several different starting points and search criteria should be used to ensure that the critical slip surface has been located accurately. h. Compare the computed factor of safety with experienced-based criteria (see Chapter 3). Return to any of the items above, and repeat the process through step h, until a satisfactory design has been achieved. When the analysis has been completed, the following steps (not part of this manual) complete the design process: i. The specifications should be written consistent with the design assumptions. j. The design assumptions should be verified during construction. This may require repeating steps b, c, d, f, g, and h and modifying the design if conditions are found that do not match the design assumptions. k. Following construction, the performance of the completed structure should be monitored. Actual piezometric surfaces based on pore water pressure measurements should be compared with those assumed during design (part d above) to determine if the embankment meets safe stability standards. 1-7. Unsatisfactory Slope Performance a. Shear failure. A shear failure involves sliding of a portion of an embankment, or an embankment and its foundation, relative to the adjacent mass. A shear failure is conventionally considered to occur along a discrete surface and is so assumed in stability analyses, although the shear movements may in fact occur across a zone of appreciable thickness. Failure surfaces are frequently approximately circular in shape. Where zoned embankments or thin foundation layers overlying bedrock are involved, or where weak strata exist within a deposit, the failure surface may consist of interconnected arcs and planes. b. Surface sloughing. A shear failure in which a surficial portion of the embankment moves downslope is termed a surface slough. Surface sloughing is considered a maintenance problem, because it usually does not affect the structural capability of the embankment. However, repair of surficial failures can entail considerable cost. If such failures are not repaired, they can become progressively larger, and may then represent a threat to embankment safety. c. Excessive deformation. Some cohesive soils require large strains to develop peak shear resistance. As a consequence, these soils may deform excessively when loaded. To avoid excessive deformations, particular attention should be given to the stress-strain response of cohesive embankment and foundation soils during design. When strains larger than 15 percent are required to mobilize peak strengths, deformations in EM 1110-2-1902 31 Oct 03 1-6 the embankment or foundation may be excessive. It may be necessary in such cases to use the shearing resistance mobilized at 10 or 15 percent strain, rather than peak strengths, or to limit placement water contents to the dry side of optimum to reduce the magnitudes of failure strains. However, if cohesive soils are compacted too dry, and they later become wetter while under load, excessive settlement may occur. Also, compaction of cohesive soils dry of optimum water content may result in brittle stress-strain behavior and cracking of the embankment. Cracks can have adverse effects on stability and seepage. When large strains are required to develop shear strengths, surface movement measurement points and piezometers should be installed to monitor movements and pore water pressures during construction, in case it becomes necessary to modify the cross section or the rate of fill placement. d. Liquefaction. The phenomenon of soil liquefaction, or significant reduction in soil strength and stiffness as a result of shear-induced increase in pore water pressure, is a major cause of earthquake damage to embankments and slopes. Most instances of liquefaction have been associated with saturated loose sandy or silty soils. Loose gravelly soil deposits are also vulnerable to liquefaction (e.g., Coulter and Migliaccio 1966; Chang 1978; Youd et al. 1984; and Harder 1988). Cohesive soils with more than 20 percent of particles finer than 0.005 mm, or with liquid limit (LL) of 34 or greater, or with the plasticity index (PI) of 14 or greater are generally considered not susceptible to liquefaction. The methodology to evaluate liquefaction susceptibility will be presented in an Engineer Circular, “Dynamic Analysis of Embankment Dams,” which is still in draft form. e. Piping. Erosion and piping can occur when hydraulic gradients at the downstream end of a hydraulic structure are large enough to move soil particles. Analyses to compute hydraulic gradients and procedures to control piping are contained in EM 1110-2-1901. f. Other types of slope movements. Several types of slope movements, including rockfalls, topples, lateral spreading, flows, and combinations of these, are not controlled by shear strength (Huang 1983). These types of mass movements are not discussed in this manual, but the possibility of their occurrence should not be ignored. EM 1110-2-1902 31 Oct 03 2-1 Chapter 2 Design Considerations 2-1. Introduction Evaluation of slope stability requires: a. Establishing the conditions, called “design conditions” or “loading conditions,” to which the slope may be subjected during its life, and b. Performing analyses of stability for each of these conditions. There are four design conditions that must be considered for dams: (1) during and at the end of construction, (2) steady state seepage, (3) sudden drawdown, and (4) earthquake loading. The first three conditions are static; the fourth involves dynamic loading. Details concerning the analysis of slope stability for the three static loading conditions are discussed in this chapter. Criteria regarding which static design conditions should be applied and values of factor of safety are discussed in Chapter 3. Procedures for analysis of earthquake loading conditions can be found in an Engineer Circular, “Dynamic Analysis of Embankment Dams,” which is still in draft form.. 2-2. Aspects Applicable to All Load Conditions a. General. Some aspects of slope stability computations are generally applicable, independent of the design condition analyzed. These are discussed in the following paragraphs. b. Shear strength. Correct evaluation of shear strength is essential for meaningful analysis of slope stability. Shear strengths used in slope stability analyses should be selected with due consideration of factors such as sample disturbance, variability in borrow materials, possible variations in compaction water content and density of fill materials, anisotropy, loading rate, creep effects, and possibly partial drainage. The responsibility for selecting design strengths lies with the designer, not with the laboratory. (1) Drained and undrained conditions. A prime consideration in characterizing shear strengths is determining whether the soil will be drained or undrained for each design condition. For drained conditions, analyses are performed using drained strengths related to effective stresses. For undrained conditions, analyses are performed using undrained strengths related to total stresses. Table 2-1 summarizes appropriate shear strengths for use in analyses of static loading conditions. (2) Laboratory strength tests. Laboratory strength tests can be used to evaluate the shear strengths of some types of soils. Laboratory strength tests and their interpretation are discussed in Appendix D. (3) Linear and nonlinear strength envelopes. Strength envelopes used to characterize the variation of shear strength with normal stress can be linear or nonlinear, as shown in Figure 2-1. (a) Linear strength envelopes correspond to the Mohr-Coulomb failure criterion. For total stresses, this is expressed as: s = c + σ tan φ (2-1) EM 1110-2-1902 31 Oct 03 2-4 (4) Ductile and brittle stress-strain behavior. For soils with ductile stress-strain behavior (shear resistance does not decrease significantly as strain increases beyond the peak), the peak shear strength can be used in evaluating slope stability. Ductile stress-strain behavior is characteristic of most soft clays, loose sands, and clays compacted at water contents higher than optimum. For soils with brittle stress-strain behavior (shear resistance decreases significantly as strain increases beyond the peak), the peak shear resistance should not be used in evaluating slope stability, because of the possibility of progressive failure. A shear resistance lower than the peak, possibly as low as the residual shear strength, should be used, based on the judgment of the designer. Brittle stress-strain behavior is characteristic of stiff clays and shales, dense sands, and clays compacted at optimum water content or below. (5) Peak, fully softened, and residual shear strengths. Stiff-fissured clays and shales pose particularly difficult problems with regard to strength evaluation. Experience has shown that the peak strengths of these materials measured in laboratory tests should not be used in evaluating long-term slope stability. For slopes without previous slides, the “fully softened” strength should be used. This is the same as the drained strength of remolded, normally consolidated test specimens. For slopes with previous slides, the “residual” strength should be used. This is the strength reached at very large shear displacements, when clay particles along the shear plane have become aligned in a “slickensided” parallel orientation. Back analysis of slope failures is an effective means of determining residual strengths of stiff clays and shales. Residual shear strengths can be measured in repeated direct shear tests on undisturbed specimens with field slickensided shear surfaces appropriately aligned in the shear box, repeated direct shear tests on undisturbed or remolded specimens with precut shear planes, or Bromhead ring shear tests on remolded material. (6) Strength anisotropy. The shear strengths of soils may vary with orientation of the failure plane. An example is shown in Figure 2-2. In this case the undrained shear strength on horizontal planes (α = 0) was low because the clay shale deposit had closely spaced horizontal fissures. Shear planes that crossed the fissures, even at a small angle, are characterized by higher strength. (7) Strain compatibility. As noted in Appendix D, Section D-9, different soils reach their full strength at different values of strain. In a slope consisting of several soil types, it may be necessary to consider strain compatibility among the various soils. Where there is a disparity among strains at failure, the shear resistances should be selected using the same strain failure criterion for all of the soils. c. Pore water pressures. For effective stress analyses, pore water pressures must be known and their values must be specified. For total stress analyses using computer software, hand computations, or slope stability charts, pore water pressures are specified as zero although, in fact, the pore pressures are not zero. This is necessary because all computer software programs for slope stability analyses subtract pore pressure from the total normal stress at the base of the slice: normal stress on base of slice u= σ − (2-3) The quantity σ in this equation is the total normal stress, and u is pore water pressure. (1) For total stress analyses, the normal stress should be the total normal stress. To achieve this, the pore water pressure should be set to zero. Setting the pore water pressure to zero ensures that the total normal stress is used in the calculations, as is appropriate. (2) For effective stress analyses, appropriate values of pore water pressure should be used. In this case, using the actual pore pressure ensures that the effective normal stress (σ' = σ − u) on the base of the slice is calculated correctly. EM 1110-2-1902 31 Oct 03 2-5 Figure 2-2. Representation of shear strength parameters for anisotropic soil d. Unit weights. The methods of analysis described in this manual use total unit weights for both total stress analyses and effective stress analyses. This applies for soils regardless of whether they are above or below water. Use of buoyant unit weights is not recommended, because experience has shown that confusion often arises as to when buoyant unit weights can be used and when they cannot. When computations are performed with computer software, there is no computational advantage in the use of buoyant unit weights. Therefore, to avoid possible confusion and computational errors, total unit weights should be used for all soils in all conditions. Total unit weights are used for all formulations and examples presented in this manual. EM 1110-2-1902 31 Oct 03 2-6 e. External loads. All external loads imposed on the slope or ground surface should be represented in slope stability analyses, including loads imposed by water pressures, structures, surcharge loads, anchor forces, hawser forces, or other causes. Slope stability analyses must satisfy equilibrium in terms of total stresses and forces, regardless of whether total or effective stresses are used to specify the shear strength. f. Tensile stresses and vertical cracks. Use of Mohr-Coulomb failure envelopes with an intercept, c or c', implies that the soil has some tensile strength (Figure 2-3). Although a cohesion intercept is convenient for representing the best-fit linear failure envelope over a range of positive normal stresses, the implied tensile strength is usually not reasonable. Unless tension tests are actually performed, which is rarely done, the implied tensile strength should be neglected. In most cases actual tensile strengths are very small and contribute little to slope stability. (1) One exception, where the tensile strengths should be considered, is in back-analyses of slope failures to estimate the shear strength of natural deposits. In many cases, the existence of steep natural slopes can only be explained by tensile strength of the natural deposits. The near vertical slopes found in loess deposits are an example. It may be necessary to include significant tensile strength in back-analyses of such slopes to obtain realistic strength parameters. If strengths are back-calculated assuming no tensile strength, the shear strength parameters may be significantly overestimated. (2) Significant tensile strengths in uncemented soils can often be attributed to partially saturated conditions. Later saturation of the soil mass can lead to loss of strength and slope failure. Thus, it may be most appropriate to assume significant tensile strength in back-analyses and then ignore the tensile strength (cohesion) in subsequent forward analysis of the slope. Guidelines to estimate shear strength in partially saturated soils are given in Appendix D, Section D-11. (3) When a strength envelope with a significant cohesion intercept is used in slope stability computations, tensile stresses appear in the form of negative forces on the sides of slices and sometimes on the bases of slices. Such tensile stresses are almost always located along the upper portion of the shear surface, near the crest of the slope, and should be eliminated unless the soil possesses significant tensile strength because of cementing which will not diminish over time. The tensile stresses are easily eliminated by introducing a vertical crack of an appropriate depth (Figure 2-4). The soil upslope from the crack (to the right of the crack in Figure 2-4) is then ignored in the stability computations. This is accomplished in the analyses by terminating the slices near the crest of the slope with a slice having a vertical boundary, rather than the usual triangular shape, at the upper end of the shear surface. If the vertical crack is likely to become filled with water, an appropriate force resulting from water in the crack should be computed and applied to the boundary of the slice adjacent to the crack. (4) The depth of the crack should be selected to eliminate tensile stresses, but not compressive stresses. As the crack depth is gradually increased, the factor of safety will decrease at first (as tensile stresses are eliminated), and then increase (as compressive stresses are eliminated) (Figure 2-5). The appropriate depth for a crack is the one producing the minimum factor of safety, which corresponds to the depth where tensile, but not compressive, stresses are eliminated. (5) The depth of a vertical crack often can be estimated with suitable accuracy from the Rankine earth pressure theory for active earth pressures beneath a horizontal ground surface. The stresses in the tensile stress zone of the slope can be approximated by active Rankine earth pressures as shown in Figure 2-6. In the case where shear strengths are expressed using total stresses, the depth of tensile stress zone, zt, is given by: D D t 2cz tan 45 2 φ⎛ ⎞= ° +⎜ ⎟γ ⎝ ⎠ (2-4) EM 1110-2-1902 31 Oct 03 2-9 Figure 2-6. Horizontal stresses near the crest of the slope according to Rankine active earth pressure theory where c, φ, and F are cohesion, angle of internal friction, and factor of safety. In most practical problems, the factor of safety can be estimated with sufficient accuracy to estimate the developed shear strength parameters (cD and φD) and the appropriate depth of the tension crack. (6) For effective stress analyses the depth of the tension crack can also be estimated from Rankine active earth pressure theory. In this case effective stress shear strength parameters, c' and φ' are used, with appropriate pore water pressure conditions. 2-3. Analyses of Stability during Construction and at the End of Construction a. General. Computations of stability during construction and at the end of construction are performed using drained strengths in free-draining materials and undrained strengths in materials that drain slowly. Consolidation analyses can be used to determine what degree of drainage may develop during the EM 1110-2-1902 31 Oct 03 2-10 construction period. As a rough guideline, materials with values of permeability greater than 10-4 cm/sec usually will be fully drained throughout construction. Materials with values of permeability less than 10-7 cm/sec usually will be essentially undrained at the end of construction. In cases where appreciable but incomplete drainage is expected during construction, stability should be analyzed assuming fully drained and completely undrained conditions, and the less stable of these conditions should be used as the basis for design. For undrained conditions, pore pressures are governed by several factors, most importantly the degree of saturation of the soil, the density of the soil, and the loads imposed on it. It is conceivable that pore pressures for undrained conditions could be estimated using results of laboratory tests or various empirical rules, but in most cases pore pressures for undrained conditions cannot be estimated accurately. For this reason, undrained conditions are usually analyzed using total stress procedures rather than effective stress procedures. b. Shear strength properties. During construction and at end of construction, stability is analyzed using drained strengths expressed in terms of effective stresses for free-draining materials and undrained strengths expressed in terms of total stresses for materials that drain slowly. (1) Staged construction may be necessary for embankments built on soft clay foundations. Consolidated- undrained triaxial tests can be used to determine strengths for partial consolidation during staged construction (Appendix D, Section D-10.) (2) Strength test specimens should be representative of the soil in the field: for naturally occurring soils, undisturbed samples should be obtained and tested at their natural water contents; for compacted soils, strength test specimens should be compacted to the lowest density, at the highest water content permitted by the specifications, to measure the lowest undrained strength of the material that is consistent with the specifications. (3) The potential for errors in strengths caused by sampling disturbance should always be considered, particularly when using Q tests in low plasticity soils. Methods to account for disturbances are discussed in Appendix D, Section D-3. c. Pool levels. In most cases the critical pool level for end of construction stability of the upstream slope is the minimum pool level possible. In some cases, it may be appropriate to consider a higher pool for end-of-construction stability of the downstream slope. (Section 2-4). d. Pore water pressures. For free-draining materials with strengths expressed in terms of effective stresses, pore water pressures must be determined for analysis of stability during and at the end of construction. These pore water pressures are determined by the water levels within and adjacent to the slope. Pore pressures can be estimated using the following analytical techniques: (1) Hydrostatic pressure computations for conditions of no flow. (2) Steady-state seepage analysis techniques such as flow nets or finite element analyses for nonhydrostatic conditions. For low-permeability soils with strengths expressed in of total stresses, pore water pressures are set to zero for purposes of analysis, as explained in Section 2-2. 2-4. Analyses of Steady-State Seepage Conditions a. General. Long-term stability computations are performed for conditions that will exist a sufficient length of time after construction for steady-state seepage or hydrostatic conditions to develop. (Hydrostatic conditions are a special case of steady-state seepage, in which there is no flow.) Stability computations are EM 1110-2-1902 31 Oct 03 2-11 performed using shear strengths expressed in terms of effective stresses, with pore pressures appropriate for the long-term condition. b. Shear strength properties. By definition, all soils are fully drained in the long-term condition, regardless of their permeability. Long-term conditions are analyzed using drained strengths expressed in terms of effective stress parameters (c' and φ'). c. Pool levels. The maximum storage pool (usually the spillway crest elevation) is the maximum water level that can be maintained long enough to produce a steady-state seepage condition. Intermediate pool levels considered in stability analyses should range from none to the maximum storage pool level. Intermediate pool levels are assumed to exist over a period long enough to develop steady-state seepage. d. Surcharge pool. The surcharge pool is considered a temporary pool, higher than the storage pool, that adds a load to the driving force but often does not persist long enough to establish a steady seepage condition. The stability of the downstream slope should be analyzed at maximum surcharge pool. Analyses of this surcharge pool condition should be performed using drained strengths in the embankment, assuming the extreme possibility of steady-state seepage at the surcharge pool level. (1) In some cases it may also be appropriate to consider the surcharge pool condition for end of construction (as discussed in Section 2-3), in which case low-permeability materials in the embankment would be assigned undrained strengths. (2) For all analyses, the tailwater levels should be appropriate for the various pool levels. e. Pore water pressures. The pore pressures used in the analyses should represent the field conditions of water pressure and steady-state seepage in the long-term condition. Pore pressures for use in the analyses can be estimated from: (1) Field measurements of pore pressures in existing slopes. (2) Past experience and judgement. (3) Hydrostatic pressure computations for conditions of no flow. (4) Steady-state seepage analyses using such techniques as flow nets or finite element analyses. 2-5. Analyses of Sudden Drawdown Stability a. General. Sudden drawdown stability computations are performed for conditions occurring when the water level adjacent to the slope is lowered rapidly. For analysis purposes, it is assumed that drawdown is very fast, and no drainage occurs in materials with low permeability; thus the term “sudden” drawdown. Materials with values of permeability greater than 10-4 cm/sec can be assumed to drain during drawdown, and drained strengths are used for these materials. Two procedures are presented in Appendix G for computing slope stability for sudden drawdown. (1) The first is the procedure recommended by Wright and Duncan (1987) and later modified by Duncan, Wright, and Wong (1990). This is the preferred procedure. (2) The second is the procedure originally presented in the 1970 version of the USACE slope stability manual (EM 1110-2-1902). This procedure is referred to as the USACE 1970 procedure and is described in further detail in Appendix G. Both procedures are believed to be somewhat conservative in that they utilize EM 1110-2-1902 31 Oct 03 3-2 Table 3-1 Minimum Required Factors of Safety: New Earth and Rock-Fill Dams Analysis Condition1 Required Minimum Factor of Safety Slope End-of-Construction (including staged construction)2 1.3 Upstream and Downstream Long-term (Steady seepage, maximum storage pool, spillway crest or top of gates) 1.5 Downstream Maximum surcharge pool3 1.4 Downstream Rapid drawdown 1.1-1.34,5 Upstream 1 For earthquake loading, see ER 1110-2-1806 for guidance. An Engineer Circular, “Dynamic Analysis of Embankment Dams,” is still in preparation. 2 For embankments over 50 feet high on soft foundations and for embankments that will be subjected to pool loading during construction, a higher minimum end-of-construction factor of safety may be appropriate. 3 Pool thrust from maximum surcharge level. Pore pressures are usually taken as those developed under steady-state seepage at maximum storage pool. However, for pervious foundations with no positive cutoff steady-state seepage may develop under maximum surcharge pool. 4 Factor of safety (FS) to be used with improved method of analysis described in Appendix G. 5 FS = 1.1 applies to drawdown from maximum surcharge pool; FS = 1.3 applies to drawdown from maximum storage pool. For dams used in pump storage schemes or similar applications where rapid drawdown is a routine operating condition, higher factors of safety, e.g., 1.4-1.5, are appropriate. If consequences of an upstream failure are great, such as blockage of the outlet works resulting in a potential catastrophic failure, higher factors of safety should be considered. (1) During construction of embankments, materials should be examined to ensure that they are consistent with the materials on which the design was based. Records of compaction, moisture, and density for fill materials should be compared with the compaction conditions on which the undrained shear strengths used in stability analyses were based. (2) Particular attention should be given to determining if field compaction moisture contents of cohesive materials are significantly higher or dry unit weights are significantly lower than values on which design strengths were based. If so, undrained (UU, Q) shear strengths may be lower than the values used for design, and end-of-construction stability should be reevaluated. Undisturbed samples of cohesive materials should be taken during construction and unconsolidated-undrained (UU, Q) tests should be performed to verify end-of- construction stability. d. Pore water pressure. Seepage analyses (flow nets or numerical analyses) should be performed to estimate pore water pressures for use in long-term stability computations. During operation of the reservoir, especially during initial filling and as each new record pool is experienced, an appropriate monitoring and evaluation program must be carried out. This is imperative to identify unexpected seepage conditions, abnormally high piezometric levels, and unexpected deformations or rates of deformations. As the reservoir is brought up and as higher pools are experienced, trends of piezometric levels versus reservoir stage can be used to project piezometric levels for maximum storage and maximum surcharge pool levels. This allows comparison of anticipated actual performance to the piezometric levels assumed during original design studies and analysis. These projections provide a firm basis to assess the stability of the downstream slope of the dam for future maximum loading conditions. If this process indicates that pore water pressures will be higher than those used in design stability analyses, additional analyses should be performed to verify long-term stability. e. Loads on slopes. Loads imposed on slopes, such as those resulting from structures, vehicles, stored materials, etc. should be accounted for in stability analyses. EM-1110-2-1902 31 Oct 03 3-3 3-2. New Embankment Dams a. Earth and rock-fill dams. Minimum required factors of safety for design of new earth and rock-fill dams are given in Table 3-1. Criteria and procedures for conducting each analysis condition are found in Chapter 2 and the appendices. The factors of safety in Table 3-1 are based on USACE practice, which includes established methodology with regard to subsurface investigations, drilling and sampling, laboratory testing, field testing, and data interpretation. b. Embankment cofferdams. Cofferdams are usually temporary structures, but may also be incorporated into a final earth dam cross section. For temporary structures, stability computations only must be performed when the consequences of failure are serious. For cofferdams that become part of the final cross section of a new embankment dam, stability computations should be performed in the same manner as for new embankment dams. 3-3. Existing Embankment Dams a. Need for reevaluation of stability. While the purpose of this manual is to provide guidance for correct use of analysis procedures, the use of slope stability analysis must be held in proper perspective. There is danger in relying too heavily on slope stability analyses for existing dams. Appropriate emphasis must be placed on the often difficult task of establishing the true nature of the behavior of the dam through field investigations and research into the historical design, construction records, and observed performance of the embankment. In many instances monitoring and evaluation of instrumentation are the keys to meaningful assessment of stability. Nevertheless, stability analyses do provide a useful tool for assessing the stability of existing dams. Stability analyses are essential for evaluating remedial measures that involve changes in dam cross sections. (1) New stability analysis may be necessary for existing dams, particularly for older structures that did not have full advantage of modern state-of-the-art design methods. Where stability is in question, stability should be reevaluated using analysis procedures such as Spencer’s method, which satisfy all conditions of equilibrium. (2) With the force equilibrium procedures used for design analyses of many older dams, the calculated factor of safety is affected by the assumed side force inclination. The calculated factor of safety from these procedures may be in error, too high or too low, depending upon the assumptions made. b. Analysis conditions. It is not necessary to analyze end-of-construction stability for existing dams unless the cross section is modified. Long-term stability under steady-state seepage conditions (maximum storage pool and maximum surcharge pool), and rapid drawdown should be evaluated if the analyses performed for design appear questionable. The potential for slides in the embankment or abutment slope that could block the outlet works should also be evaluated. Guidance for earthquake loading is provided in ER 1110-2-1806, and an Engineer Circular, “Dynamic Analysis of Embankment Dams,” is in draft form. c. Factors of safety. Acceptable values of factors of safety for existing dams may be less than those for design of new dams, considering the benefits of being able to observe the actual performance of the embankment over a period of time. In selecting appropriate factors of safety for existing dam slopes, the considerations discussed in Section 3-1 should be taken into account. The factor of safety required will have an effect on determining whether or not remediation of the dam slope is necessary. Reliability analysis techniques can be used to provide additional insight into appropriate factors of safety and the necessity for remediation. EM 1110-2-1902 31 Oct 03 3-4 3-4. Other Slopes a. Factors of safety. Factors of safety for slopes other than the slopes of dams should be selected consistent with the uncertainty involved in the parameters such as shear strength and pore water pressures that affect the calculated value of factor of safety and the consequences of failure. When the uncertainty and the consequences of failure are both small, it is acceptable to use small factors of safety, on the order of 1.3 or even smaller in some circumstances. When the uncertainties or the consequences of failure increase, larger factors of safety are necessary. Large uncertainties coupled with large consequences of failure represent an unacceptable condition, no matter what the calculated value of the factor of safety. The values of factor of safety listed in Table 3-1 provide guidance but are not prescribed for slopes other than the slopes of new embankment dams. Typical minimum acceptable values of factor of safety are about 1.3 for end of construction and multistage loading, 1.5 for normal long-term loading conditions, and 1.1 to 1.3 for rapid drawdown in cases where rapid drawdown represents an infrequent loading condition. In cases where rapid drawdown represents a frequent loading condition, as in pumped storage projects, the factor of safety should be higher. b. Levees. Design of levees is governed by EM 1110-2-1913. Stability analyses of levees and their foundations should be performed following the principles set forth in this manual. The factors of safety listed in Table 3-1 provide guidance for levee slope stability, but the values listed are not required. c. Other embankment slopes. The analysis procedures described in this manual are applicable to other types of embankments, including highway embankments, railway embankments, retention dikes, stockpiles, fill slopes of navigation channels, river banks in fill, breakwaters, jetties, and sea walls. (1) The factor of safety of an embankment slope generally decreases as the embankment is raised, the slopes become higher, and the load on the foundation increases. As a result, the end of construction usually represents the critical short-term (undrained) loading condition for embankments, unless the embankment is built in stages. For embankments built in stages, the end of any stage may represent the most critical short- term condition. With time following completion of the embankment, the factor of safety against undrained failure will increase because of the consolidation of foundation soils and dissipation of construction pore pressures in the embankment fill. (2) Water ponded against a submerged or partially submerged slope provides a stabilizing load on the slope. The possibility of low water events and rapid drawdown should be considered. d. Excavated slopes. The analysis procedures described in this manual are applicable to excavated slopes, including foundation excavations, excavated navigation and river channel slopes, and sea walls. (1) In principle, the stability of excavation slopes should be evaluated for both the end-of-construction and the long-term conditions. The long-term condition is usually critical. The stability of an excavated slope decreases with time after construction as pore water pressures increase and the soils within the slope swell and become weaker. As a result, the critical condition for stability of excavated slopes is normally the long-term condition, when increase in pore water pressure and swelling and weakening of soils is complete. If the materials in which the excavation is made are so highly permeable that these changes occur completely as construction proceeds, the end-of-construction and the long-term conditions are the same. These considerations lead to the conclusion that an excavation that would be stable in the long-term condition would also be stable at the end of construction. (2) In the case of soils with very low permeability and an excavation that will only be open temporarily, the long-term (fully drained) condition may never be established. In such cases, it may be possible to excavate a slope that would be stable temporarily but would not be stable in the long term. Design for such a EM 1110-2-1902 31 Oct 03 4-2 acceptable to simplify the problem by using fewer slices, by averaging unit weights of soil layers, and by simplifying the piezometric conditions. While verification of stability analysis results is still required, it is no longer required that results be verified using graphical hand calculations. Stability analysis results can be verified using any of the methods listed above. Examples of verifications of analyses performed using Spencer’s Method, the Simplified Bishop Method, and the Modified Swedish Method are shown in Figures 4-1, 4-2, 4-3, and 4-4. b. Verification using a second computer program. For difficult and complex problems, a practical method of verifying or confirming computer results may be by the use of a second computer program. It is desirable that the verification analyses be performed by different personnel, to minimize the likelihood of repeating data entry errors. c. Software versions. Under most Microsoft Windows™ operating systems, the file properties, including version, size, date of creation, and date of modification can be reviewed to ensure that the correct version of the computer program is being used. Also, the size of the computer program file on disk can be compared with the size of the original file to ensure that the software has not been modified since it was verified. In addition, printed output may show version information and modification dates. These types of information can be useful to establish that the version of the software being used is the correct and most recent version available. d. Essential requirements for appropriate use of computer programs. A thorough knowledge of the capabilities of the software and knowledge of the theory of limit equilibrium slope stability analysis methods will allow the user to determine if the software available is appropriate for the problem being analyzed. (1) To verify that data are input correctly, a cross section of the problem being analyzed should be drawn to scale and include all the required data. The input data should be checked against the drawing to ensure the data in the input file are correct. Examining graphical displays generated from input data is an effective method of checking data input. (2) The computed output should be checked to ensure that results are reasonable and consistent. Important items to check include the weights of slices, shear strength properties, and pore water pressures at the bottoms of slices. The user should be able to determine if the critical slip surface is going through the material it should. For automatic searches, the output should designate the most critical slip surface, as well as what other slip surfaces were analyzed during the search. Checking this information thoroughly will allow the user to determine that the problem being analyzed was properly entered into the computer and the software is correctly analyzing the problem. e. Automatic search verification. Automatic searches can be performed for circular or noncircular slip surfaces. The automatic search procedures used in computer programs are designed to aid the user in locating the most critical slip surface corresponding to a minimum factor of safety. However, considerable judgment must be exercised to ensure that the most critical slip surface has actually been located. More than one local minimum may exist, and the user should use multiple searches to ensure that the global minimum factor of safety has been found. (1) Searches with circular slip surfaces. Various methods can used to locate the most critical circular slip surfaces in slopes. Regardless of the method used, the user should be aware of the assumptions and limitations in the search method. EM 1110-2-1902 31 Oct 03 4-3 Figure 4-1. Hand verification using force equilibrium procedure to check stability computations performed via Spencer’s Method – end-of-construction conditions (a) During an automatic search, the program should not permit the search to jump from one face of the slope to another. If the initial trial slip surface is for the left face of the slope, slip surfaces on the right face of the slope should be rejected. (b) In some cases, a slope may have several locally critical circles. The center of each such locally critical circle is surrounded by centers of circles that have higher values for the factor of safety. In such cases, when a search is performed, only one of the locally critical circles will be searched out, and the circle found may not be the one with the overall lowest factor of safety. To locate the overall critical EM 1110-2-1902 31 Oct 03 4-4 Figure 4-2. Verification of computations using a spreadsheet for the Simplified Bishop Method – upstream slope, low pool EM 1110-2-1902 31 Oct 03 4-7 (c) An alternative approach is to perform analyses for a suite of circles with selected center points, and to vary the radii or depths of the circles for each center point. The computed factors of safety can be examined to determine the location of the most critical circle and the corresponding minimum factor of safety. (2) Searches with noncircular slip surfaces. As with circular slip surfaces, various methods are used to search for critical noncircular slip surfaces. In all of these methods, the initial position of the slip surface is specified by the user and should correspond to the estimated position of the critical slip surface. (a) In most methods of limit equilibrium slope stability analysis, the equilibrium equations used to compute the factor of safety may yield unrealistic values for the stresses near the toes of slip surfaces that are inclined upward at angles much steeper than those that would be logical based on considerations of passive earth pressure. Trial slip surfaces may become excessively steep in an automatic search unless some restriction is placed on their orientation. (b) Because procedures for searching for critical noncircular slip surfaces have been developed more recently than those for circles, there is less experience with them. Thus, extra care and several trials may be required to select optimum values for the parameters that control the automatic search. The search parameters should be selected such that the search will result in an acceptably refined location for the most critical slip surface. The search parameters should be selected so that the final increments of distance used to shift the noncircular slip surface are no more than 10 to 25 percent of the thickness of the thinnest stratum through which the shear surface may pass. 4-3. Presentation of the Analysis and Results a. Basic requirements. The description of the slope stability analysis should be concise, accurate, and self-supporting. The results and conclusions should be described clearly and should be supported by data. b. Contents. It is recommended that the documentation of the stability analysis should include the items listed below. Some of the background information may be included by reference to other design documents. Essential content includes: (1) Introduction. (a) Scope. A brief description of the objectives of the analysis. (b) Description of the project and any major issues or concerns that influence the analysis. (c) References to engineering manuals, analysis procedures, and design guidance used in the analysis. (2) Regional geology. Refer to the appropriate design memorandum, if published. If there is no previously published document on the regional geology, include a description of the regional geology to the extent that the regional geology is pertinent to the stability analysis. (3) Site geology and subsurface explorations. Present detailed site geology including past and current exploration, drilling, and sampling activities. Present geologic maps and cross sections, in sufficient number and detail, to show clearly those features of the site that influence slope stability. EM 1110-2-1902 31 Oct 03 4-8 (4) Instrumentation and summary of data. Present and discuss any available instrumentation data for the site. Items of interest are piezometric data, subsurface movements observed with inclinometers, and surface movements. (5) Field and laboratory test results. (a) Show the location of samples on logs, plans, and cross sections. (b) Present a summary of each laboratory test for each material, using approved forms as presented in EM 1110-2-1906, for laboratory soils testing. (c) Show laboratory test reports for all materials. Examples are shown in Figures 4-5, 4-6, 4-7, 4-8, 4-9, and 4-10. (d) Discuss any problems with sampling or testing of materials. (e) Discuss the use of unique or special sampling or testing procedures. (6) Design shear strengths. Present the design shear strength envelopes, accompanied by the shear strength envelopes developed from the individual test data for each material in the embankment, foundation, or slope, for each load condition analyzed. An example is shown in Figure 4-11. (7) Material properties. Present the material properties for all the materials in the stability cross section, as shown in Figure 4-12. Explain how the assigned soil property values were obtained. In the case of an embankment, specify the location of the borrow area from which the embankment material is to be obtained. Discuss any factors regarding the borrow sites that would impact the material properties, especially the natural moisture content, and expected variations in the materials in the borrow area. (8) Groundwater and seepage conditions. Present the pore water pressure information used in the stability analysis. Show the piezometric line(s) or discrete pore pressure points in the cross section used in the analysis, as shown in Figure 4-12. If the piezometric data are derived from a seepage analysis, include a summary of the seepage analysis in the report. Include all information used to determine the piezometric data, such as water surface levels in piezometers, artesian conditions at the site, excess pore water pressures measured, reservoir and river levels, and drawdown levels for rapid drawdown analysis. (9) Stability analyses. (a) State the method used to perform the slope stability analysis, e.g., Spencer’s Method in a given computer program, Modified Swedish Method using hand calculations with the graphical (force polygon) method, or slope stability charts. Provide the required computer software verification information described in Section 4-1. (b) For each load condition, present a tabulation of material property values, show the cross section analyzed on one or more figures, and show the locations and the factors of safety for the critical and other significant slip surfaces, as shown in Figure 4-12. For circular slip surfaces, show the center point, including the coordinates, and the value of radius. (c) For the critical slip surface for each load condition, describe how the factor of safety results were verified and include details of the verification procedure, as discussed previously. EM 1110-2-1902 31 Oct 03 4-9 Figure 4-5. Triaxial compression test report for Q (unconsolidated-undrained) tests EM 1110-2-1902 31 Oct 03 4-12 Figure 4-8. Variation of τff with σ‘fc for R-bar test with isotropic consolidation EM 1110-2-1902 31 Oct 03 4-13 Figure 4-9. Triaxial compression test report for S (drained) tests – effective stress envelope EM 1110-2-1902 31 Oct 03 4-14 Figure 4-10. Direct shear test report – effective stress envelope EM 1110-2-1902 31 Oct 03 A-1 Appendix A References EM 1110-1-1804 EM 1110-1-1804. “Geotechnical Investigations,” U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. EM 1110-2-1601 EM 1110-2-1601. “Hydraulic Design of Flood Control Channels,” U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. EM 1110-2-1806 EM 1110-2-1806. “Earthquake Design and Evaluation for Civil Works Projects,” Washington, DC. EM 1110-2-1901 EM 1110-2-1901. “Seepage Analysis and Control for Dams,” U.S. Army Engineers Waterways Experiment Station, Vicksburg, MS. EM 1110-2-1902 EM 1110-2-1902. 1970. “Stability of Earth and Rock-Fill Dams,” U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. EM 1110-2-1906 EM 1110-2-1906. “Laboratory Testing of Soils,” U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. EM 1110-2-1912 EM 1110-2-1912. “Stability of Excavated and Natural Slopes in Soils and Clay Shales,” U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. EM 1110-2-1913 EM 1110-2-1913. “Design and Construction of Levees,” U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. EM 1110-2-2300 EM 1110-2-2300. “Earth and Rockfill Dams, General Design and Construction Considerations,” U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. ETL 1110-2-556 ETL 1110-2-556. “Risk Based Analysis in Geotechnical Engineering for Support of Planning Studies,” Washington, DC. American Society for Testing and Materials 1999 American Society for Testing and Materials. 1999. “D 2850: Standard Test Method for Unconsolidated- Undrained Triaxial Compression Test on Cohesive Soils,” West Conshohocken, PA. Bishop and Morgenstern 1960 Bishop, A. W., and Morgenstern, N. 1960. “Stability Coefficients for Earth Slopes,” Geotechnique, Vol 10, No. 4, pp 129-150. EM 1110-2-1901 31 Oct 03 A-2 Bishop 1955 Bishop, A. W. 1955. “The Use of the Slip Circle in the Stability Analysis of Slopes,” Geotechnique, Vol 5, No. 1, pp 7-17. Bjerrum 1973 Bjerrum, L. 1973. “Problems of Soil Mechanics and Construction on Soft Clays and Structurally Unstable Soils (Collapsible, Expansive and Others).” Proceedings of the Eighth International Conference on Soil Mechanics and Foundation Engineering, Moscow, Vol 3, pp 111-159. Bolton 1979 Bolton, M. 1979. A Guide to Soil Mechanics. A Halstead Press Book, John Wiley and Sons, New York, 439 pp. Budhu 2000 Budhu, M. 2000. Soil Mechanics and Foundations. John Wiley and Sons, 586 pp. Casagrande 1936 Casagrande, A. 1936. “Characteristics of Cohesionless Soils Affecting the Stability of Slopes and Earth Fills,” Originally published in Journal of the Boston Society of Civil Engineers, reprinted in Contributions to Soil Mechanics 1925-1940, Boston Society of Civil Engineers, pp 257-276. Celestino and Duncan 1981 Celestino, T. B., and Duncan, J. M. 1981. “Simplified Search for Non-Circular Slip Surfaces,” Proceedings, Tenth International Conference on Soil Mechanics and Foundation Engineering, International Society for Soil Mechanics and Foundation Engineering, Stockholm, A.A. Balkema, Rotterdam, Holland, Vol 3, pp 391-394. Chang 1978 Chang, K. T. 1978. “An Analysis of Damage of Slope Sliding by Earthquake on the Paiho Main Dam and its Earthquake Strengthening,” Tseng-hua Design Section, Department of Earthquake-Resistant Design and Flood Control Command of Miyna Reservoir, Peoples Republic of China. Ching and Fredlund 1983 Ching, R. K. H., and Fredlund, D. G. 1983. “Some Difficulties Associated with the Limit Equilibrium Method of Slices,” Canadian Geotechnical Journal, Vol 20, No. 4, pp 661- 672. Chirapunta and Duncan 1975 Chirapuntu, S., and Duncan, J. M. 1975. “The Role of Fill Strength in the Stability of Embankments on Soft Clay Foundations,” Geotechnical Engineering Research Report, Department of Civil Engineering, University of California, Berkeley. Coulter and Migliaccio 1966 Coulter, H.W., and Migliaccio, R. R. 1966. “Effects of the Earthquake of March 27, 1964 at Valdez, Alaska,” Geological Survey Professional Paper No. 542-C, U.S. Department of the Interior, Washington, DC. Duncan 1996 Duncan, J. M. 1996. “State of the Art: Limit Equilibrium and Finite-Element Analysis of Slopes,” Journal of Geotechnical Engineering, Vol 122, No. 7, July, pp 557-596. EM 1110-2-1902 31 Oct 03 A-3 Duncan and Buchignani 1975 Duncan, J. M., and Buchignani, A. L. 1975. “An Engineering Manual for Stability Studies,” Civil Engineering 270B, University of California, Berkeley, CA. Duncan, Buchignani, and DeWet 1987 Duncan, J. M., Buchignani, A. L., and DeWet, M. 1987. “An Engineering Manual for Slope Stability Studies,” Department of Civil Engineering, Geotechnical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA. Duncan, Horz, and Yang 1989 Duncan, J. M., Horz, R. C., and Yang, T. L. 1989. “Shear Strength Correlations for Geotechnical Engineering,” Virginia Tech, Department of Civil Engineering, August, 100 pp. Duncan, Navin, and Patterson 1999 Duncan, J. M, Navin, M., and Patterson, K. 1999. “Manual for Geotechnical Engineering Reliability Calculations,” Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA. Duncan and Wright 1980 Duncan, J. M., and Wright, S. G. 1980. “The Accuracy of Equilibrium Methods of Slope Stability Analysis,” Engineering Geology, Vol 16, No. 1/2, pp 5-17. Duncan, Wright, and Wong 1990 Duncan, J. M., Wright, S. G., and Wong, K. S. 1990. “Slope Stability During Rapid Drawdown,” H. Bolton Seed Symposium, Vol. 2, University of California at Berkeley, pp 253-272. Edris, Munger, and Brown 1992 Edris, E. V., Jr., Munger, D., and Brown, R. 1992. “User’s Guide: UTEXAS3 Slope Stability Package: Volume III Example Problems,” Instruction Report GL-87-1, U.S. Army Engineer Waterways Experi- ment Station, Vicksburg, MS. Edris and Wright 1992 Edris, E. V., Jr. and Wright, S. G. 1992. “User’s Guide: UTEXAS3 Slope Stability Package: Volume IV User’s Manual,” Instruction Report GL-87-1, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. Fellenius 1936 Fellenius, W. 1936. “Calculation of the Stability of Earth Dams,” Transactions, 2nd International Congress on Large Darns, International Commission on Large Dams, Washington, DC, pp 445-459. Fredlund 1989 Fredlund, D. G. 1989. “Negative Pore Water Pressures in Slope Stability,” Proceedings, Symposio Suramericano de Deslizamientos, Paipa, Columbia, pp 429-439. Fredlund 1995 Fredlund, D. G. 1995. “The Stability of Slopes with Negative Pore-Water Pressures.” Proceedings, Ian Boyd Donald Symposium on Modern Developments in Geomechanics, Monash University, Melbourne, Australia, pp 99-116. EM 1110-2-1901 31 Oct 03 A-6 Roscoe, Schofield, and Wroth 1958 Roscoe, K. H., Schofield, A. N., and Wroth, C. P. 1958. “On the Yielding of Soils,” Geotechnique, Institution of Civil Engineers, Great Britain, Vol 8, No. 1, pp 22-53. Scott 1963 Scott, R. F. 1963. Principles of Soil Mechanics, Addison-Wesley, Reading, MA. Seed 1979 Seed, H. B. 1979. “19th Rankine Lecture: Considerations in the Earthquake Resistant Design of Earth and Rockfill Dams,” Geotechnique, Vol 29, No. 3, pp 215-263. Skempton 1977 Skempton, A. W. 1977. “Slope Stability of Cuttings in Brown London Clay.” Proceedings, Ninth International Conference on Soil Mechanics and Foundation Engineering. Tokyo, Vol 3, pp 261-270. Spencer 1967 Spencer, E. 1967. “A Method of Analysis of the Stability of Embankments Assuming Parallel Inter-Slice Forces,” Geotechnique, Vol 17, No. 1, pp 11-26. Stark and Eid 1993 Stark, T. D., and Eid, H. T. 1993. “Modified Bromhead Ring Shear Apparatus,” Geotechnical Testing Journal, American Society for Testing and Materials, Vol 16, No. 1, Mar., pp 100-107. Stark and Eid 1994 Stark, T. D., and Eid, H. T. 1994. “Drained Residual Strength of Cohesive Soils,” Journal of Geotechnical Engineering, ASCE, Vol 120, No. 5, May, pp 856-871. Stark and Eid 1997 Stark, T. D., and Eid, H. T. 1997. “Slope Stability Analyses in Stiff Fissured Clays,” Journal of Geotechnical and Geoenvironmental Engineering,” ASCE, Vol 123, No. 4, Apr., pp 335-343. Taylor 1937 Taylor, D. W. 1937. “Stability of Earth Slopes,” Journal of the Boston Society of Civil Engineers, Vol 24, No. 3, July, pp 197-247, reprinted in Contributions to Soil Mechanics 1925-1940, Boston Society of Civil Engineers, 1940, pp 337-386. Whitman and Bailey 1967 Whitman, R. V., and Bailey, W. A. 1967. “Use of Computers for Slope Stability Analysis,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol 93, No. SM4, pp 475-498. Wright 1969 Wright, S. G. 1969. “A Study of Slope Stability and the Undrained Shear Strength of Clay Shales,” thesis presented to the University of California at Berkeley, California, in partial fulfillment of requirements for degree of Doctor of Philosophy. Wright 1982 Wright S. G. 1982. “Review of Limit Equilibrium Slope Analysis Procedures,” Technology Update Lecture, U. S. Bureau of Reclamation, Denver, CO, pp 1-36. EM 1110-2-1902 31 Oct 03 A-7 Wright 1991 Wright, S. G. 1991. “Limit Equilibrium Slope Stability Equations Used in the Computer Program UTEXAS3,” Geotechnical Engineering Software GS 91-2, Geotechnical Engineering Center, The University of Texas, Austin. Wright, Kulhawy, and Duncan 1973 Wright, S. G., Kulhawy, F. H., and Duncan, J. M. 1973. “Accuracy of Equilibrium Slope Stability Analysis,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol 99, No. 10, October, pp 783-791. Wright and Duncan 1987 Wright, S. G., and Duncan, J. M. 1987. “An Examination of Slope Stability Computation Procedures for Sudden Drawdown,” Miscellaneous Paper GL-87-25, U. S. Army Engineer Waterways Experiment Station, Vicksburg, MS. Youd et al. 1984 Youd, T. L., Harp, E. L., Keefer, D. K., and Wilson, R. C. 1984. “Liquefaction Generated by the 1983 Borah Peak, Idaho Earthquake,” Proceedings of Workshop XXVIII On the Borah Peak, Idaho Earthquake, Volume A, Open-File Report 85-290-A, U.S. Geological Survey, Menlo Park, CA, pp 625-634. EM 1110-2-1902 31 Oct 03 B-1 Appendix B Notation Dimensions: F indicates force, L indicates length A = cross-sectional area of slice (L2) b = width of slice (L) b = slope ratio = cot β (dimensionless) c = cohesion intercept for Mohr-Coulomb diagram plotted in terms of total normal stress, σ (F/L2) c' = cohesion intercept for Mohr-Coulomb diagram plotted in terms of total effective stress, σ' (F/L2) cR = cohesion intercept as determined from the R envelope (F/L2) cb = cohesion at the base of an embankment (F/L2) cavg = average cohesion over length of slip surface (F/L2) cD = ‘developed’ or ‘mobilized’ cohesion (F/L2) c'D = ‘developed’ or ‘mobilized’ cohesion (F/L2) CD = force because of the ‘developed’ or ‘mobilized’ cohesion on base of slice (F) C1 = term used to calculate side forces on a slice (F) C2 = term used to calculate side forces on a slice (F) C3 = term used to calculate side forces on a slice (F) C4 = term used to calculate side forces on a slice (F) d = depth factor = D/H (dimensionless) d = intercept value of failure envelope on ‘p-q’ diagram (F/L2) dh = horizontal moment arm (L) dV = vertical moment arm (L) d’ = intercept value of failure envelope on (σ1-σ3) vs. σ3 ‘modified’ Mohr-Coulomb diagram (F/L2) dR = intercept value of R failure envelope on ‘p-q’ diagram (F/L2) dcrack = depth of vertical ‘tension’ crack (L) dKc=1 = intercept value for τff vs. σ'fc shear strength envelope for isotropic consolidation (F/L2) D = depth from toe of slope to lowest point on the slip circle (L) e = void ratio (dimensionless) E = horizontal component of the interslice force (F) EA = active force acting on a wedge (F) EP = passive force acting on a wedge (F) F = factor of safety (defined with respect to shear strength) (dimensionless) EM 1110-2-1902 31 Oct 03 B-4 β = inclination from horizontal of the top of the slice (degrees) δ = inclination of the earth pressure force (degrees) ∆ℓ = length of bottom of slice (L) ∆u = change in pore water pressure, usually during shear (F/L2) ∆x = width of slice (F) ∆φ = change in friction angle (degrees) γ = total unit weight of soil (F/L3) γ’ = submerged unit weight of soil (F/L3) γm = moist unit weight of soil (F/L3) γw = unit weight of water (F/L3) γsat = saturated unit weight of soil (F/L3) λcφ = term to relate Pe with shear strength parameters (dimensionless) µq = surcharge correction factor (dimensionless) µw = submergence correction factor (dimensionless) µt = tension crack correction factor (dimensionless) µ'w = seepage correction factor (dimensionless) Ω = term used to calculate the inclination of the critical slip surface (dimensionless) φ = angle of internal friction for Mohr-Coulomb diagram plotted in terms of total normal stress, σ (degrees) φ' = angle of internal friction for Mohr-Coulomb diagram plotted in terms of effective normal stress, σ' (degrees) φD = ‘developed’ or ‘mobilized’ total stress angle of internal friction (degrees) φ'D = ‘developed’ or ‘mobilized’ effective stress angle of internal friction (degrees) φR = angle of internal friction as determined from the R envelope (degrees) φu = undrained friction angle (degrees) φsecant = secant value of friction angle (tan φsecant = τf/σf) (degrees) σ = total normal stress (F/L2) σ' = effective normal stress (F/L2) σ'v = effective vertical stress (F/L2) σ'vc = effective vertical stress for consolidation (F/L2) σ1f = major principal total stress at failure (F/L2) σ'1f = major principal effective stress at failure (F/L2) σ'1c = effective major principal stress for consolidation (F/L2) σ3f = minor principal total stress at failure (F/L2) σ'3f = minor principal effective stress at failure (F/L2) EM 1110-2-1902 31 Oct 03 B-5 σ'3c = minor principal effective stress for consolidation (F/L2) σ'3-critical = ‘critical’ effective confining pressure for critical state (F/L2) σ'c = effective normal stress on the slip surface at consolidation, before rapid drawdown (F/L2) σ'd = effective normal stress on the slip surface after drainage following rapid drawdown (F/L2) σ'fc = effective normal stress on the failure plane at consolidation (F/L2) σi = normal stress where R and S envelopes intersect (F/L2) σ'1/ σ'3 = principal stress ratio (dimensionless) (σ1-σ3) = principal stress difference (F/L2) (σ1-σ3)f = principal stress difference at failure (F/L2) τ = shear stress (F/L2) τc = shear stress on the slip surface at consolidation, before rapid drawdown (F/L2) τfc = shear stress on the failure plane at consolidation (F/L2) τff = shear stress on the failure plane at failure (F/L2) τff-Kc=1 = shear stress (strength) from envelope of τff vs. σ'fc for isotropic consolidation (F/L2) τff-Kc=Kf = shear stress (strength) from envelope of τff vs. σ'fc for maximum degree of anisotropic consolidation, Kc = Kf (F/L2) θ = inclination of the interslice force (degrees) Ψ = angle of inclination of failure envelope on ‘p-q’ diagram (degrees) Ψ’ = angle of inclination of failure envelope on (σ1-σ3) vs. σ3 ‘modified’ Mohr-Coulomb diagram (degrees) ΨR = angle of inclination of the R failure envelope on ‘p-q’ diagram (degrees) ΨKc=1 = slope angle for τff vs. σ'fc shear strength envelope for isotropic consolidation (degrees) EM 1110-2-1902 31 Oct 03 C-1 Appendix C Stability Analysis Procedures - Theory and Limitations C-1. Fundamentals of Slope Stability Analysis a. Conventional approach. Conventional slope stability analyses investigate the equilibrium of a mass of soil bounded below by an assumed potential slip surface and above by the surface of the slope. Forces and moments tending to cause instability of the mass are compared to those tending to resist instability. Most procedures assume a two-dimensional (2-D) cross section and plane strain conditions for analysis. Successive assumptions are made regarding the potential slip surface until the most critical surface (lowest factor of safety) is found. Figure C-1 shows a potential slide mass defined by a candidate slip surface. If the shear resistance of the soil along the slip surface exceeds that necessary to provide equilibrium, the mass is stable. If the shear resistance is insufficient, the mass is unstable. The stability or instability of the mass depends on its weight, the external forces acting on it (such as surcharges or accelerations caused by dynamic loads), the shear strengths and porewater pressures along the slip surface, and the strength of any internal reinforcement crossing potential slip surfaces. Figure C-1. Slope and potential slip surface b. The factor of safety. Conventional analysis procedures characterize the stability of a slope by calculating a factor of safety. The factor of safety is defined with respect to the shear strength of the soil as the ratio of the available shear strength (s) to the shear strength required for equilibrium (τ), that is: EM 1110-2-1902 31 Oct 03 C-4 Figure C-2. Typical slice and forces for method of slices Except for the weight of the slice, all of these forces are unknown and must be calculated in a way that satisfies static equilibrium. (1) For the current discussion, the shear force (S) on the bottom of the slice is not considered directly as an unknown in the equilibrium equations that are solved. Instead, the shear force is expressed in terms of other known and unknown quantities, as follows: S on the base of a slice is equal to the shear stress, τ, multiplied by the length of the base of the slice, ∆ , i.e., S = τ∆ (C-9) or, by introducing Equation C-5, which is based on the definition of the factor of safety, ( )u tan 'c 'S F F σ − ∆ φ∆= + (C-10) EM 1110-2-1902 31 Oct 03 C-5 Finally, noting that the normal force N is equal to the product of the normal stress (σ) and the length of the bottom of the slice (∆ ), i.e., N = σ ∆ , Equation C-9 can be written as: ( )N u tan 'c 'S F F − ∆ φ∆= + (C-11) (2) Equation C-11 relates the shear force, S, to the normal force on the bottom of the slice and the factor of safety. Thus, if the normal force and factor of safety can be calculated from the equations of static equilibrium, the shear force can be calculated (is known) from Equation C-11. Equation C-11 is derived from the Mohr-Coulomb equation and the definition of the factor of safety, independently of the conditions of static equilibrium. The forces and other unknowns that must be calculated from the equilibrium equations are summarized in Table C-1. As discussed above, the shear force, S, is not included in Table C-1, because it can be calculated from the unknowns listed and the Mohr-Coulomb equation (C-11), independently of static equilibrium equations. Table C-1 Unknowns and Equations for Limit Equilibrium Methods Unknowns Number of Unknowns for n Slices Factor of safety (F) 1 Normal forces on bottom of slices (N) N Interslice normal forces, E n – 1 Interslice shear forces, X n – 1 Location of normal forces on base of slice N Location of interslice normal forces n – 1 TOTAL NUMBER OF UNKNOWNS 5n – 2 Equations Number of Equations for n Slices Equilibrium of forces in the horizontal direction, ΣFx = 0 n Equilibrium of forces in the vertical direction, ΣFy = 0 n Equilibrium of moments n TOTAL NUMBER OF EQUILIBRIUM EQUATIONS 3n (3) In order to achieve a statically determinate solution, there must be a balance between the number of unknowns and the number of equilibrium equations. The number of equilibrium equations is shown in the lower part of Table C-1. The number of unknowns (5n – 2) exceeds the number of equilibrium equations (3n) if n is greater than one. Therefore, some assumptions must be made to achieve a statically determinate solution. (4) The various limit equilibrium methods use different assumptions to make the number of equations equal to the number of unknowns. They also differ with regard to which equilibrium equations are satisfied. For example, the Ordinary Method of Slices, the Simplified Bishop Method, and the U.S. Army Corps of Engineers’ Modified Swedish Methods do not satisfy all the conditions of static equilibrium. Methods such as the Morgenstern and Price’s and Spencer’s do satisfy all static equilibrium conditions. Methods that satisfy static equilibrium fully are referred to as “complete” equilibrium methods. Details of various limit equilibrium procedures and their differences are presented in Sections C-2 through C-7. Detailed comparison of limit equilibrium slope stability analysis methods have been reported by Whitman and Bailey (1967), Wright (1969), Duncan and Wright (1980) and Fredlund and Krahn (1977).1 e. Limitations of limit equilibrium methods. Complete equilibrium methods have generally been more accurate than those procedures which do not satisfy complete static equilibrium and are therefore preferable to 1 References information is presented in Appendix A. EM 1110-2-1902 31 Oct 03 C-6 “incomplete” methods. However, the “incomplete” methods are often sufficiently accurate and useful for many practical applications, including hand checks and preliminary analyses. In all of the procedures described in this manual, the factor of safety is applied to both cohesion and friction, as shown by Equation C-6. (1) The factor of safety is also assumed to be constant along the shear surface. Although the factor of safety may not in fact be the same at all points on the slip surface, the average value computed by assuming that F is constant provides a valid measure of stability for slopes in ductile (nonbrittle) soils. For slopes in brittle soils, the factor of safety computed assuming F is the same at all points on the slip surface may be higher than the actual factor of safety. (2) If the strength is fully mobilized at any point on the slip surface, the soil fails locally. If the soil has brittle stress-strain characteristics so that the strength drops once the peak strength is mobilized, the stress at that point of failure is reduced and stresses are transferred to adjacent points, which in turn may then fail. In extreme cases this may lead to progressive failure and collapse of the slope. If soils possess brittle stress- strain characteristics with relatively low residual shear strengths compared to the peak strengths, reduced strengths and/or higher factors of safety may be required for stability. Limitations of limit equilibrium procedures are summarized in Table C-2. Table C-2 Limitations of Limit-Equilibrium Methods 1. The factor of safety is assumed to be constant along the potential slip surface. 2. Load-deformation (stress-strain) characteristics are not explicitly accounted for. 3. The initial stress distribution within the slope is not explicitly accounted for. 4. Unreasonably large and or negative normal forces may be calculated along the base of slices under certain conditions (SectionC-l0.b and C-10.c). 5. Iterative, trial and error, solutions may not converge in certain cases (Section C-10d). f. Shape of the slip surface. All of the limit equilibrium methods require that a potential slip surface be assumed in order to calculate the factor of safety. Calculations are repeated for a sufficient number of trial slip surfaces to ensure that the minimum factor of safety has been calculated. For computational simplicity the candidate slip surface is often assumed to be circular or composed of a few straight lines (Figure C-3). However, the slip surface will need to have a more complicated shape in complex stratigraphy. The assumed shape is dependent on the problem geometry and stratigraphy, material characteristics (especially anisotropy), and the capabilities of the analysis procedure used. Commonly assumed shapes are discussed below. (1) Circular. Observed failures in relatively homogeneous materials often occur along curved failure surfaces. A circular slip surface, like that shown in Figure C-3a, is often used because it is convenient to sum moments about the center of the circle, and because using a circle simplifies the calculations. A circular slip surface must be used in the Ordinary Method of Slices and Simplified Bishop Method. Circular slip surfaces are almost always useful for starting an analysis. Also, circular slip surfaces are generally sufficient for analyzing relatively homogeneous embankments or slopes and embankments on foundations with relatively thick soil layers. (2) Wedge. “Wedge” failure mechanisms are defined by three straight line segments defining an active wedge, central block, and passive wedge (Figure C-3b). This type of slip surface may be appropriate for slopes where the critical potential slip surface includes a relatively long linear segment through a weak material bounded by stronger material. A common example is a relatively strong levee embankment founded on weaker, stratified alluvial soils. Wedge methods, including methods for defining or calculating the inclination of the base of the wedges, are discussed in Section C-1g. EM 1110-2-1902 31 Oct 03 C-9 Figure C-4. Search with constant radius Figure C-5. Search with circles through a common point EM 1110-2-1902 31 Oct 03 C-10 Figure C-6. Search with circles tangent to a prescribed tangent line consistent with the computed factor of safety. This assumption for the inclination of the active and passive wedges is only appropriate where the top surfaces of the active and passive wedges are horizontal but provides reasonable results for gently inclined slopes. Common methods for searching for the inclination of the base of the wedges are shown in Figure C-7b. One technique, used where soil properties and inclinations of the base of each wedge vary in the zone of the active and passive wedges, is to assume that the bottoms of the wedges are inclined at α = θ ± φ'D/2. The value of θ is then varied until the maximum interslice force is found for the active wedge and minimum interslice force is found for the passive wedge. A second search technique, where the bases of the active and passive wedges are considered to be single planes, is to vary the value of α until a maximum interslice force is obtained for the entire group of active wedge segments and the minimum is found for the entire group of passive wedge segments. (3) General shapes. A number of techniques have been proposed and used to locate the most critical general-shaped slip surface. One of the most robust and useful procedures is the one developed by Celestino and Duncan (1981). The method is illustrated in Figure C-8. In this method, an initial slip surface is assumed and represented by a series of points that are connected by straight lines. The factor of safety is first calculated for the assumed slip surface. Next, all points except one are held fixed, and the “floating” point is shifted a small distance in two directions. The directions might be vertically up and down, horizontally left and right, or above and below the slip surface in some assumed direction. The factor of safety is calculated for the slip surface with each point shifted as described. This process is repeated for each point on the slip surface. As any one point is shifted, all other points are left at their original location. Once all points have been shifted in both directions and the factor of safety has been computed for each shift, a new location is estimated for the slip surface based on the computed factors of safety. The slip surface is then moved to the estimated location and the process of shifting points is repeated. This process is continued until no further reduction in factor of safety is noted and the distance that the shear surface is moved on successive approximations becomes minimal. EM 1110-2-1902 31 Oct 03 C-11 Figure C-7. Search schemes for wedges (4) Limitations and precautions. Any search scheme employed in computer programs is restricted to investigating a finite number of slip surfaces. In addition, most of these schemes are designed to locate one slip surface with a minimum factor of safety. The schemes may not be able to locate more than one local minimum. The results of automatic searches are dependent on the starting location for the search and any constraints that are imposed on how the slip surface is moved. Automatic searches are controlled largely by the data that the user inputs into the software. Regardless of the software used, a number of separate searches should be conducted to confirm that the lowest factor of safety has been calculated. EM 1110-2-1902 31 Oct 03 C-14 α = inclination of the bottom of the slice u = pore water pressure at the center of the base of the slice ∆ = length of the bottom of the slice As shown in Table C-3, there is only one unknown in the Ordinary Method of Slices (F), and only one equilibrium equation is used (the equation of equilibrium of the entire soil mass around the center of the circle). Table C-3 Unknowns and Equations for the Ordinary Method of Slices Procedure Unknowns Number of Unknowns for n Slices Factor of safety (F) 1 TOTAL NUMBER OF UNKNOWNS 1 Equations Number of Equations for n Slices Equilibrium of moments of the entire soil mass 1 TOTAL NUMBER OF EQUILIBRIUM EQUATIONS 1 (1) Two different equations have been used to compute the factor of safety by the OMS with effective stresses and pore water pressures. The first equation is shown above as Equation C-12. Equation C-12 is derived by first calculating an “effective” slice weight, W', by subtracting the uplift force due to pore water pressure from the weight, and then resolving forces in a direction perpendicular to the base of the slice (Figure C-9). The other OMS equation for effective stress analyses is written as: ( )c ' W cos u tan ' F Wsin ⎡ ∆ + α − ∆ φ ⎤⎣ ⎦= α ∑ ∑ (C-13) Equation C-13 is derived by first resolving the force because of the total slice weight (W) in a direction perpendicular to the base of the slice and then subtracting the force because of pore water pressures. Equation C-12 leads to more reasonable results when pore water pressures are used. Equation C-13 can lead to unrealistically low or negative stresses on the base of the slice because of pore water pressures and should not be used. (2) External water on a slope can be treated in either of two ways: The water may simply be represented as soil with c = 0 and φ = 0. In this case, the trial slip surface is assumed to extend through the water and exit at the surface of the water. Some of the slices will then include water and the shear strength for any slices whose base lies in water will be assigned as zero. The second way that water can be treated in an analysis is to treat the water as an external, hydrostatic load on the top of the slices. In this case, the trial slip surface will only pass through soil, and each end will exit at the ground or slope surface (Figure C-10). For the equations presented in this appendix as well as the examples in Appendixes F and G, the water is treated as an external load. Treating the water as another “soil” involves simply modifying the geometry and properties of the slices. (3) In the case where water loads act on the top of the slice, the expression for the factor of safety (Equation C-12) must be modified to the following: ( ){ }2 P c ' W cos Pcos u cos tan ' F M Wsin R ⎡ ⎤∆ + α + α − β − ∆ α φ⎣ ⎦= α − ∑ ∑∑ (C-14) EM 1110-2-1902 31 Oct 03 C-15 Figure C-10. Slice for Ordinary Method of Slices with external water loads where P = resultant water force acting perpendicular to the top of the slice β = inclination of the top of the slice MP = moment about the center of the circle produced by the water force acting on the top of the slice R = radius of the circle (Figure C-10). The moment, MP, is considered to be positive when it acts in the opposite direction to the moment produced by the weight of the sliding mass. b. Limitations. The principal limitation of the OMS comes from neglecting the forces on the sides of the slice. The method also does not satisfy equilibrium of forces in either the vertical or horizontal directions. Moment equilibrium is satisfied for the entire soil mass above the slip surface, but not for individual slices. EM 1110-2-1902 31 Oct 03 C-16 (1) Factors of safety calculated by the OMS may commonly differ as much as 20 percent from values calculated using rigorous methods (Whitman and Bailey 1967); in extreme cases (such as effective stress analysis with high pore water pressures), the differences may be even larger. The error is generally on the safe side (calculated factor of safety is too low), but the error may be so large as to yield uneconomical designs. Because of the tendency for errors to be on the “safe side,” the OMS is sometimes mistakenly thought always to produce conservative values for the factor of safety. This is not correct. When φ = 0, the OMS yields the same factor of safety as more rigorous procedures, which fully satisfy static equilibrium. Thus, the degree to which the OMS is conservative depends on the value of φ and whether the pore pressures are large or small. (2) Although Equation C-12 does not specifically include the radius of the circle, the equation is based on the assumption that the slip surface is circular. The OMS can only be used with circular slip surfaces. c. Recommendation for use. The OMS is included herein for reference purposes and completeness because numerous existing slopes have been designed using the method. As the method still finds occasional use in practice, occasions may arise where there is a need to review designs by others that were based on the method. Also, because the OMS is simple, it is useful where calculations must be done by hand using an electronic calculator. The method also may be used to overcome problems that may develop near the toe of steeply exiting shear surfaces as described in Section C-10.b. C-3. The Simplified Bishop Method a. Assumptions. The Simplified Bishop Method was developed by Bishop (1955). This procedure is based on the assumption that the interslice forces are horizontal, as shown in Figure C-11. A circular slip surface is also assumed in the Simplified Bishop Method. Forces are summed in the vertical direction. The resulting equilibrium equation is combined with the Mohr-Coulomb equation and the definition of the factor of safety to determine the forces on the base of the slice. Finally, moments are summed about the center of the circular slip surface to obtain the following expression for the factor of safety: ( ) P c ' x W Pcos u x sec tan ' m F M Wsin R α ⎡ ∆ + + β − ∆ α φ ⎤ ⎢ ⎥ ⎣ ⎦= α − ∑ ∑∑ (C-15) where ∆x is the width of the slice, and mα is defined by the following equation, sin tan 'm cos Fα α φ= α + (C-16) The terms W, c', φ', u, P, MP, and R are as defined earlier for the OMS. Factors of safety calculated from Equation C-15 satisfy equilibrium of forces in the vertical direction and overall equilibrium of moments about the center of a circle. The unknowns and equations in the Simplified Bishop Method are summarized in Table C-4. Because the value of the term mα depends on the factor of safety, the factor of safety appears on both sides of Equation C-15. Equation C-15 cannot be manipulated such that an explicit expression is obtained for the factor of safety. Thus, an iterative, trial and error procedure is used to solve for the factor of safety. EM 1110-2-1902 31 Oct 03 C-19 Figure C-12. Typical slice and forces for Modified Swedish Method Table C-5 Unknowns and Equations for Force Equilibrium Methods Unknowns Number of Unknowns for n Slices Factor of safety (F) 1 Normal forces on bottom of slices (N) n Resultant interslice forces, Z n – 1 TOTAL NUMBER OF UNKNOWNS 2n Equations Number of Equations for n Slices Equilibrium of forces in the horizontal direction, ΣFx = 0 n Equilibrium of forces in the vertical direction, ΣFy = 0 n TOTAL NUMBER OF EQUILIBRIUM EQUATIONS 2n EM 1110-2-1902 31 Oct 03 C-20 • When effective stresses are used to define the shear strengths, e.g., for analyses of steady-state seepage, a choice can be made between having the interslice forces (Z) represent either the total force or only the effective force. If the interslice forces are chosen to represent the effective force, the corresponding forces due to water pressures on the sides of the slice are calculated and included as additional forces in the analysis. In the equations presented in this appendix, the interslice forces for the Modified Swedish Method are represented as effective forces when effective stresses are used to characterize the shear strength. However, the equations and examples with effective interslice forces can easily be converted to represent interslice forces as total forces by setting the forces that represent water pressures on the sides of the slice to zero. • The original version of the Modified Swedish Method represented interslice forces as effective forces whenever effective stress analyses were performed (USACE 1970). In contrast, many computer programs represent the interslice forces as total forces. Fundamentally, representation of interslice forces as effective forces is sound and feasible for effective stress analyses because the pore water pressures are known (defined) when effective stress analyses are performed. However, there are a number of reasons why it is appropriate to represent interslice forces as total forces, particularly in computer software: (1) In complex stratigraphy, it is difficult to define and compute the resultant force from water pressures on the sides of each slice. (2) In many analyses, total stresses are used in some soil zones, and effective stresses are used in others; the shear strengths of freely draining soils are represented using effective stresses; while the shear strengths of less permeable soils are represented using undrained shear strengths and total stresses. Interslice water pressures can only be calculated when effective stresses are used for all materials. Thus, interslice forces must be represented as total forces in the cases where mixed drained and undrained shear strengths are used. (3) It makes almost no difference whether interslice forces are represented as effective or total forces when complete static equilibrium is satisfied, e.g., when Spencer’s Method is used to calculate the factor of safety. Thus, in Spencer’s Method total interslice forces are almost always used. The Modified Swedish Method is recommended for hand-checking calculations made with Spencer’s Method. Accordingly, when the Modified Swedish Method is used to check calculations made using Spencer’s Method, it is logical that the interslice forces should be total forces. • Regardless of whether the interslice forces represent total or effective forces, their inclination must be assumed. The inclination that is assumed is the inclination of either the total force or the effective force, depending on how the interslice forces are represented. The Corps of Engineers’ 1970 manual states that the side forces should be assumed to be parallel to the “average embankment slope”. The “average embankment slope” is usually taken to be the slope of a straight line drawn between the crest and toe of the slope (Figure C-12). All side forces are assumed to have the same inclination. The assumption of side forces parallel to the average embankment slope has been shown to sometimes lead to unconservative results in many cases – the calculated factor of safety is too large - when compared to more rigorous procedures which satisfy both force and moment equilibrium such as Spencer’s Method or the Morgenstern and Price procedure. The degree of inaccuracy is greater when total interslice forces are used. It is probably more realistic and safer to assume that the interslice forces are inclined at one-half the average embankment slope when total forces are used. • To avoid possibly overestimating the factor of safety, some engineers in practice have assumed that the interslice forces are horizontal in the Modified Swedish Method. The assumption of horizontal interslice forces in procedures that only satisfy force equilibrium, and not moment equilibrium, is EM 1110-2-1902 31 Oct 03 C-21 sometimes referred to as the “Simplified Janbu” Method. This assumption, however, may significantly underestimate the value of the factor of safety. Accordingly, “correction” factors are sometimes applied to the value for the factor of safety calculated by the "Simplified Janbu” Method to account for the assumption of horizontal interslice forces (Janbu 1973). Some confusion exists in practice regarding whether the so-called “Simplified Janbu” Method should automatically include using the “correction” factors or not. Care should be exercised when reviewing results of slope stability calculations reported to have been made by the “Simplified Janbu” Method to determine whether a correction factor has been applied or not. • Lowe and Karafiath (1960) suggested assuming that the interslice forces are inclined at an angle that is the average of the inclinations of the slope (ground surface) and shear surface at each vertical interslice boundary. Unlike the other assumptions described above, with Lowe and Karafiath’s assumption the interslice force inclinations vary from slice to slice. This assumption appears to be better than any of the assumptions described earlier, especially when the side forces represent total, rather than effective, forces. Lowe and Karafiath’s assumption produces factors of safety that are usually within 10 percent of the values calculated by procedures which satisfy complete static equilibrium (Duncan and Wright 1980). (4) The force equilibrium equations for the Modified Swedish Method may be solved either graphically or numerically. Both the graphical and numerical solutions require an iterative, trial and error procedure to compute the factor of safety. A factor of safety is first assumed; force equilibrium is then checked. If force equilibrium is not satisfied, a new factor of safety is assumed and the process is repeated until force equilibrium is satisfied to an acceptable degree. The graphical and numerical procedures are each described separately in the sections that follow. b. Graphical solution procedure. A solution for the factor of safety by any force equilibrium procedure (including the Modified Swedish Method) is obtained by repeatedly assuming a value for the factor of safety and then constructing the force vector polygon for each slice until force equilibrium is satisfied for all slices. A typical slice and the forces acting on it for a case where there is no surface or pore water pressure is shown in Figure C-12. The forces consist of the slice weight (W), the forces on the left and right sides of the slice (Zi and Zi+1), and the normal and shear forces on the base of the slice (N and S). The interslice force, Zi, represents the force on the upslope side of the slice, while Zi+1 represents the force on the downslope side. Thus, Zi acts on the right side of the slice for a left facing slope and on the left side of the slice for a right- facing slope. The shear force on the bottom of the slice is expressed as: ( )1S c N tan F = ∆ + φ (C-17) or D DS c N tan= ∆ + φ (C-18) where Dc and Dφ are the developed shear strength parameters. In drawing the force polygons, the shear and normal forces are represented by a force resulting from cohesion, Dc ∆ , and a force, FD, representing the resultant force as a result of the normal force (N) and the frictional component of shear resistance ( DN tan φ ). These forces are illustrated for a slice in Figure C-13b. The force Dc ∆ acts parallel to the base of the slice, while the force FD acts at an angle Dφ from the normal to the base EM 1110-2-1902 31 Oct 03 C-24 Figure C-14. Forces for Modified Swedish Method with water ( )2 i i 1 tan 'sinC U U cos F+ φ α⎡ ⎤= − α +⎢ ⎥⎣ ⎦ (C-20b) ( ) ( )3 tan 'C P sin cos F φ⎡ ⎤= α − β − α − β⎢ ⎥⎣ ⎦ (C-20c) EM 1110-2-1902 31 Oct 03 C-25 ( )4C c ' u tan ' F ∆= − φ (C-20d) ( ) ( )tan 'sin n cos Fα φ α − θ = α − θ + (C-21) (1) The quantities iZ and i 1Z + represent the forces on the upslope and downslope sides of the slice, respectively, Ui and Ui+1 represent the water pressure forces on the upslope and downslope sides of the slice, and θ represents the inclination of the interslice forces. The remaining terms in Equation C-19 are the same as those defined earlier for the Ordinary Method of Slices and Simplified Bishop Methods. Equation C-19 is applied beginning with the first slice where iZ = 0 and proceeding slice-by-slice until the last slice is reached. Here it is assumed that calculations are performed proceeding from the top of the slope to the bottom of the slope, regardless of the direction that the slope faces. The calculated interslice force i 1Z + for the downslope side of the last slice (toe of the slip surface) should be zero if a correct value has been assumed for the factor of safety. If the force on the downslope side of the last slice is not zero, a new value is assumed for the factor of safety and the process is repeated until the force on the downslope side of the last slice is zero. Example calculations for the Modified Swedish Method using both the numerical solution and the graphical procedure are presented in Appendix F. (2) When the quantities, Ui and Ui+1, that represent water pressures on the sides of the slice are not zero, the interslice forces, Zi and Zi+1, represent forces in terms of effective stress. When total stresses are used, the quantities, Ui and Ui+1, are set to zero and the interslice forces then represent the total forces, including water pressures. The quantities, Ui and Ui+1, can also be set equal to zero for effective stress analyses and the side forces are then the total side forces. Total interslice forces are used in much of the computer software for slope stability analyses, but effective forces are recommended when the side forces are assumed to be parallel to the average embankment slope, as discussed in Section C-4a. d. Limitations. The principal limitation of the Modified Swedish Method is that calculated factors of safety are sensitive to the assumed interslice force inclination. Depending on the inclination assumed for the interslice forces, the factor of safety may be either underestimated or overestimated compared to the value calculated by more rigorous methods that fully satisfy static equilibrium. The sensitivity of the method appears to be due in large part to the fact that moment equilibrium is not satisfied. e. Recommendations for use. The force equilibrium procedure is the only method considered to this point that can be utilized for analyses with general shaped, noncircular slip surfaces. Although the force equilibrium method is not as accurate as Spencer’s Method (described next) for analyses of general-shaped noncircular slip surfaces, the force equilibrium method is much simpler and is therefore suitable for hand calculations, whereas Spencer’s Method is too lengthy for hand calculations. Accordingly, the force equilibrium method is recommended for use in hand calculations where noncircular slip surfaces are being analyzed. If the force equilibrium method is being used to check calculations that were performed using Spencer’s Method, the side force inclination used for the hand calculations should be the one calculated by Spencer’s Method (Section C-5). Spencer’s Method and the force equilibrium procedure should produce identical results when the same side force inclination is used in both method. The Modified Swedish Method is useful where existing slopes have been designed using the method and are being analyzed for new conditions, such as updated pore pressure information, or where alterations are to be made. Using the same method will allow meaningful comparison of results to those from previous analyses. For all new designs, preference should be given to the Simplified Bishop (circular slip surfaces) and Spencer (noncircular slip surfaces) Methods. EM 1110-2-1902 31 Oct 03 C-26 f. Verification procedures. As described above, either numerical or graphical procedures can be used in the Modified Swedish Method. Depending on which procedure was first used to compute the factor of safety (numerical or graphical), the other procedure can be used for verification. Thus, if the factor of safety was computed using the numerical procedure with Equation C-19, the force vector polygons can be drawn to confirm that force equilibrium has been satisfied. Likewise, if the graphical procedure was used to compute the factor of safety, the numerical solution (Equation C-19) can be used to compute the side forces and verify that equilibrium has been satisfied. C-5. Spencer’s Method a. Assumptions. Spencer’s Method assumes that the side forces are parallel, i.e., all side forces are inclined at the same angle. However, unlike the Modified Swedish Method, the side force inclination is not assumed, but instead is calculated as part of the equilibrium solution. Spencer’s Method also assumes that the normal forces on the bottom of the slice act at the center of the base – an assumption which has very little influence on the final solution. Spencer’s Method fully satisfies the requirements for both force and moment equilibrium. The unknowns and equations involved in the method are listed in Table C-6. Table C-6 Unknowns and Equations for Spencer’s Methods Unknowns Number of Unknowns for n Slices Factor of safety (F) 1 Inclination of interslice forces (θ) 1 Normal forces on bottom of slices (N) n Resultant interslice forces, Z n – 1 Location of interslice normal forces n – 1 TOTAL NUMBER OF UNKNOWNS 3n Equilibrium Equations Equations Number of Equations for n Slices Equilibrium of forces in the horizontal direction, ΣFx = 0 n Equilibrium of forces in the vertical direction, ΣFy = 0 n Equilibrium of moments n TOTAL NUMBER OF EQUILIBRIUM EQUATIONS 3n Although Spencer (1967) originally presented his method for circular slip surfaces, Wright (1969) showed that the method could readily be extended to analyses with noncircular slip surfaces. A solution by Spencer’s Method first involves an iterative, trial and error procedure in which values for the factor of safety (F) and side force inclination (θ) are assumed repeatedly until all conditions of force and moment equilibrium are satisfied for each slice. Then the values of N, Z, and yt are evaluated for each slice. b. Limitations. Spencer’s Method requires computer software to perform the calculations. Because moment and force equilibrium must be satisfied for every slice and the calculations are repeated for a number of assumed trial factors of safety and interslice force inclinations, complete and independent hand-checking of a solution using Spencer’s Method is impractical (Section C-5d). c. Recommendations for use. The use of Spencer’s Method for routine analysis and design has become practical as computer resources improve. The method has been implemented in several commercial computer programs and is used by several government agencies. Spencer’s Method should be used where a statically complete solution is desired. It should also be used as a check on final designs where the slope stability computations were performed by simpler methods. d. Verification procedures. Complete and independent hand-checking of a solution using Spencer’s Methods is impractical because of the complexity of the method and the lengthy calculations involved. Instead the force equilibrium procedure is recommended, using either the graphical or numerical solution methods. When checking Spencer’s Method using the force equilibrium procedure, the side force inclination EM 1110-2-1902 31 Oct 03 C-29 Figure C-16. Infinite slope For shear strengths expressed in terms of effective stresses and zero cohesion (c' = 0), the factor of safety is given by: ( )u tan 'sF σ − φ = = τ τ (C-24) where u is the pore water pressure at the depth of the shear plane. Letting ru = u/γz and substituting the expressions for σ and τ from Equations C-22 and C-23 into Equation C-24, gives: ( )2 ucos r tan 'sF cos sin β − φ = = τ β β (C-25) Equation C-25 can also be written as: ( )2 u tan 'F 1 r 1 tan tan φ ⎡ ⎤= − + β⎣ ⎦β (C-26) For the special case of no pore water pressure (u = 0; ru = 0) Equation C-26 reduces to: tan 'F tan φ= β (C-27) (1) The stability equation for an infinite slope can also be written for conditions involving seepage through the slope, as shown in Figure C-17. EM 1110-2-1902 31 Oct 03 C-30 Figure C-17. Infinite slope with parallel flow lines The factor of safety for an infinite slope with seepage can be expressed as follows (Bolton 1979): w s sat ' tan tan tan 'F tan γ − γ α β φ= γ β (C-28) where γ' =γsat – γw = submerged unit weight of soil γw = unit weight of water γsat = saturated unit weight of soil αs = angle between the flow lines and the embankment face (Figure C-17) β = inclination of the slope measured from the horizontal φ' = angle of internal friction expressed in terms of effective stresses The cohesion, c', is assumed to be zero because the infinite slope analysis is primarily applicable to cohesionless soils. (2) For the case where the direction of seepage is parallel to the slope (αs = 0), with the free surface of seepage at the ground surface, the factor of safety can be expressed as: sat ' tan 'F tan γ φ= γ β (C-29) Similarly, for the case of horizontal seepage (α = β) 2 w sat ' tan tan 'F tan γ − γ β φ= γ β (C-30) EM 1110-2-1902 31 Oct 03 C-31 c. Limitations. The equations for infinite slope factor of safety given by Equations C-24 through C-30 are applicable only to slopes in cohesionless materials. They apply to slopes in nonplastic silt, sand, gravel, and rock-fill where c' = 0. Charts for analysis of infinite slopes in materials with c' > 0 are given in Appendix E. d. Recommendations for use. The method is useful for evaluating the stability with respect to shallow sliding of slopes in cohesionless soils. C-8. Simple Approximations Simple approximations are sometimes useful for preliminary estimates of stability prior to more rigorous and complete calculations. Two such simplified approaches are discussed below. a. At-rest earth pressure method. The at-rest earth pressure method is used to estimate the potential for lateral spreading and horizontal sliding of an embankment, as shown in Figure C-18. (1) Assumptions. The method compares the at-rest earth pressure on a vertical plane through the embankment to the shear resistance along the base of the embankment to one side of the plane. The method is only partly a limit-equilibrium method, because the at-rest earth pressures are calculated independently of any equilibrium conditions and, then, compared to the limiting shear resistance. (2) Limitations. The method is not intended as a primary design method but only as a method to perform supplemental checks. It is applicable only to embankments. (3) Recommendations for use. Ensuring that an embankment has an adequate factor of safety by this analysis will assist in limiting deformations where two or more materials with significantly different stress- strain behavior are present. A common example application is to zoned gravel or rock-fill dams with clay cores. b. Bearing capacity methods. Bearing capacity methods are useful for estimating the potential for weak, saturated, clay foundations to support embankments (Figure C-19). (1) Assumptions. These methods compare the ultimate bearing capacity of the foundation beneath an embankment to the total vertical stress imposed by the embankment. The vertical stress is calculated by multiplying the full height of the embankment by the total unit weight of the fill material. The bearing capacity of the foundation is calculated from the classical bearing capacity equations for a strip footing resting on the surface of the ground. For a saturated clay and undrained loading (φ = 0), the ultimate bearing capacity is computed as: ultq 5.14c= (C-31) Although more sophisticated approximations can be made, bearing capacity analyses should not be considered to be a substitute for detailed slope stability analyses. (2) Limitations. The bearing capacity methods are limited to homogeneous foundations where simple bearing capacity equations are applicable. These methods are also used primarily for evaluating short-term, undrained stability of embankments resting on soft, saturated clay foundations. These methods are intended only for preliminary analyses and for use as an approximate check of more rigorous and thorough analyses. EM 1110-2-1902 31 Oct 03 C-34 identify computational problems. Calculated values of normal forces (N) and interslice forces (Z) should be examined to be sure that their values are reasonable. Because most soils are not able to sustain significant tensile stresses, tensile forces should not exist on the sides or bottom of slices. Also the line of thrust (the locus of points describing the location of the interslice forces) should be within the sliding mass. Several specific computational problems are discussed below. b. Very large forces or tensile forces due to slip surface geometry. As shown in Figure C-20, the resultant force on the slip surface (FD) can become parallel or nearly parallel to the interslice force (Z) if the slip surface exits too steeply at the toe. When this condition occurs, very large, infinite, or negative, values may be calculated for these forces (Ching and Fredlund 1983). If FD and Z are parallel, division by zero occurs in the equilibrium equations, and the forces become infinite. If FD and Z are close to parallel, division by a very small number occurs in the equilibrium equations, and the values of FD and Z can be very large, either positive or negative. Factors of safety computed for such conditions are not meaningful. The condition of large positive or negative forces near the toe of the slope is usually caused by the slip surface exiting upward too steeply. The problem can be avoided by adjusting the inclination of the slip surface to conform more closely with the most critical slip surface that would be expected based on passive earth pressure theories. In the case where the ground surface and earth pressure (interslice) force are both horizontal, the inclination of the critical slip surface (shear plane) for passive earth pressure conditions is given by: 45 2 ′φ⎛ ⎞α = − ° −⎜ ⎟ ⎝ ⎠ (C-32) The negative sign arises from the sign convention used for the inclination of the shear surface in the slope stability equations. In the case of an inclined earth pressure (interslice) force, the inclination of the critical slip surface can be calculated from the following equation presented by Jumikis (1962): α = −Ω + φ (C-33) where tan tan (tan cot ) (1 tan cot ) arctan 1 tan (tan cot ) ⎡ ⎤φ + φ φ + φ + δ φ Ω = ⎢ ⎥ + δ φ + φ⎢ ⎥⎣ ⎦ (C-34) where δ is the inclination of the earth pressure force, which corresponds to θ in Figure C-20. The sign convention for α in Equation C-33 is such that α is negative for slip surfaces inclined upward at the toe of the slope. The existence of very large positive or negative values for the forces near the toe of the slope can lead to unreasonably large or unreasonably small values for the factor of safety. Depending on the procedure of slope stability analysis being used, the problem can be avoided in one of the following ways: (1) The slip surface can be flattened near the toe as described above: This is probably the best approach, but the use of noncircular slip surfaces is required. (2) The side force inclination can be changed: Of the procedures described in this manual, the Modified Swedish Method is the only one that allows the inclination of the side forces to be changed. It is also possible to change the assumed inclination for the side force with using the Morgenstern and Price method (Morgenstern and Price 1965). Changing the side force inclination to obtain a suitable solution with the Morgenstern and Price procedure can be time-consuming. EM 1110-2-1902 31 Oct 03 C-35 Figure C-20. Slice with parallel (co-linear) resultant force, FD, and interslice force, Z, leading to infinite values of these forces (3) The Ordinary Method of Slices can be used for the analysis. The problem described above does not occur with the OMS, because the OMS neglects side forces. However, the OMS is not accurate for effective stress analyses when pore pressures are high, and its use is undesirable for that reason. (4) The shear strength in the zone where the slip surface exits can be estimated assuming a simple passive earth pressure state of stress. The shear strength is then assigned to this zone as a cohesion with φ = 0. For cohesionless soil (c = 0), horizontal ground surface, and a horizontal earth pressure force, the shear strength spassive can be calculated from: 2 passive v 1s tan 45 1 cos 2 2 ′⎡ ⎤φ⎛ ⎞ ′= ° + − σ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ ′φ (C-35) where σ'v is the effective vertical stress. EM 1110-2-1902 31 Oct 03 C-36 In this case, the shear strength increases linearly with depth because the effective vertical stress also increases linearly with depth. Approach (4) is the only one that can be used to eliminate large positive or negative forces at the toe of the slope when the Simplified Bishop Method is used. Regardless of the procedure used to calculate the factor of safety, the details of the solution should be examined to determine if very large positive or negative forces are calculated for slices near the toe of the slope. If such conditions are found, one of the measures described above should be used to correct the problem. c. Tensile forces from cohesion. When soils at the crest of the slope have cohesion, the calculated values for the normal forces (N) and side forces (Z) in this area are often negative. Negative forces are consistent with what would be calculated by classical earth pressure theories for the active condition. The negative stresses result from the tensile strength that is implicit for any soil having a Mohr-Coulomb failure envelope with a cohesion intercept. This type of shear strength envelope implies that the soil has tensile strength, as shown in Figure C-21. Because few soils have tensile strength that can be relied on for slope stability, tensile stresses should be eliminated before an analysis is considered acceptable. Tensile stresses can be eliminated from an analysis by introducing a vertical tension crack near the upper end of the slip surface. The slip surface is terminated at the point where it reaches the bottom of crack elevation, as shown in Figure C-22. The appropriate crack depth can be determined in either of the following ways: (1) A range of crack depths can be assumed and the factor of safety calculated for each depth. The crack depth producing the minimum factor of safety is used for final analyses. The depth yielding the minimum factor of safety will correspond closely to the depth where tensile stresses are eliminated, but positive (driving) stresses are not. (2) The crack depth can be estimated as the depth over which the active Rankine earth pressures are negative. For total stresses and homogeneous soil the depth is given by: D crack D 2cd tan 45 2 = φ⎛ ⎞γ ° −⎜ ⎟ ⎝ ⎠ (C-36) where cD and φD = developed shear strength parameters cD = c/F tan φD = tan φ/F Similar expressions can be developed for the depth of tension for effective stresses and/or nonhomogeneous soil profiles. In some cases the depth of crack computed using Equation C-36 will be greater than the height of the slope. This is likely to be the case for low embankments of well-compacted clay. For embankments on weak foundations, where the crack depth computed using Equation C-36 is greater than the height of the embankment, the crack depth used in the stability analyses should be equal to the height of the embankment; the crack should not extend into the weak foundation. EM 1110-2-1902 31 Oct 03 C-39 zero, this may cause errors. As a result of roundoff and computer word length, the calculated point of intersection can be a considerable distance from the actual point of intersection. C-11. Selection of Method Some methods of slope stability analysis (e.g., Spencer’s) are more rigorous and should be favored for detailed evaluation of final designs. Some methods (e.g., Spencer’s, Modified Swedish, and the Wedge) can be used to analyze noncircular slip surfaces. Some methods (e.g., the Ordinary Method of Slices, the Simplified Bishop, the Modified Swedish, and the Wedge) can be used without the aid of a computer and are therefore convenient for independently checking results obtained using computer programs. Also, when these latter methods are implemented in software, they execute extremely fast and are useful where very large numbers of trial slip surfaces are to be analyzed. The various methods covered in this appendix are summarized in Table C-7. This table can be helpful in selecting a suitable method for analysis. Table C-7 Comparison of Features of Limit Equilibrium Methods Feature Ordinary Method of Slices Simplified Bishop Spencer Modified Swedish Wedge Infinite Slope Accuracy X X X Plane slip surfaces parallel to slope face X Circular slip surfaces X X X X Wedge failure mechanism X X X Non-circular slip surfaces – any shape X X Suitable for hand calculations X X X X X C-12. Use of the Finite Element Method a. General. The finite element method (FEM) can be used to compute displacements and stresses caused by applied loads. However, it does not provide a value for the overall factor of safety without additional processing of the computed stresses. The principal uses of the finite element method for design are as follows: (1) Finite element analyses can provide estimates of displacements and construction pore water pressures. These may be useful for field control of construction, or when there is concern for damage to adjacent structures. If the displacements and pore water pressures measured in the field differ greatly from those computed, the reason for the difference should be investigated. (2) Finite element analyses provide displacement pattern which may show potential and possibly complex failure mechanisms. The validity of the factor of safety obtained from limit equilibrium analyses depends on locating the most critical potential slip surfaces. In complex conditions, it is often difficult to anticipate failure modes, particularly if reinforcement or structural members such as geotextiles, concrete retaining walls, or sheet piles are included. Once a potential failure mechanism is recognized, the factor of safety against a shear failure developing by that mode can be computed using conventional limit equilibrium procedures. (3) Finite element analyses provide estimates of mobilized stresses and forces. The finite element method may be particularly useful in judging what strengths should be used when materials have very dissimilar stress-strain and strength properties, i.e., where strain compatibility is an issue. The FEM can help EM 1110-2-1902 31 Oct 03 C-40 identify local regions where “overstress” may occur and cause cracking in brittle and strain softening materials. Also, the FEM is helpful in identifying how reinforcement will respond in embankments. Finite element analyses may be useful in areas where new types of reinforcement are being used or reinforcement is being used in ways different from the ways for which experience exists. An important input to the stability analyses for reinforced slopes is the force in the reinforcement. The FEM can provide useful guidance for establishing the force that will be used. b. Use of finite element analyses to compute factors of safety. If desired, factors of safety equivalent to those computed using limit equilibrium analyses can be computed from results of finite element analyses. The procedure for using the FEM to compute factors of safety are as follows: (1) Perform an analysis using the FEM to determine the stresses for the slope. (2) Select a trial slip surface. (3) Subdivide the slip surface into segments. (4) Compute the normal stresses and shear stresses along an assumed slip surface. This requires interpolation of values of stress from the values calculated at Gauss points in the finite element mesh to obtain values at selected points on the slip surface. If an effective stress analysis is being performed, subtract pore pressures to determine the effective normal stresses on the slip surface. The pore pressures are determined from the same finite element analysis if a coupled analysis was performed to compute stresses and deformations. The pore pressures are determined from a separate steady seepage analysis if an uncoupled analysis was performed to compute stresses and deformations. (5) Use the normal stress and the shear strength parameters, c and φ, or c' and φ', to compute the available shear strength at points along the shear surface. Use total normal stresses and total stress shear strength parameters for total stress analysis and effective normal stresses and effective stress shear strength parameters for effective stress analyses. (6) Compute an overall factor of safety using the following equation: i i s F ∆ = τ ∆ ∑ ∑ (C-37) where si = available shear strength computed in step (4) τi = shear stress computed in step (3) ∆ = length of each individual segment into which the slip surface has been subdivided The summations in Equation C-37 are performed over all the segments into which the slip surface has been subdivided. (a) Studies have shown that factors of safety determined using the procedure described are, for practical purposes, equal to factors of safety determined using accurate limit equilibrium methods. EM 1110-2-1902 31 Oct 03 C-41 (b) “Local” (point-by-point) factors of safety can also be calculated using the stresses and shear strength properties at selected points in a slope. Some of the local factors of safety will be lower than the overall minimum factor of safety computed from Equation C-37 or limit equilibrium analyses. Local factors of safety of one or less do not necessarily indicate that a slope is unstable. Stresses will be redistributed from points of local failure to other points where the local factor of safety is greater than 1. As long as the overall factor of safety is greater than 1, the slope will be stable. c. Advantages and disadvantages. Where estimates of movements as well as factor of safety are required to achieve design objectives, the effort required to perform finite element analyses can be justified. However, finite element analyses require considerably more time and effort, beyond that required for limit equilibrium analyses and additional data related to stress-strain behavior of materials. Therefore, the use of finite element analyses is not justified for the sole purpose of calculating factors of safety.