Slope Stability Analysis by the Limit Equilibrium Method: Fundamentals and Methods

Failures of natural and artificial slopes often pose formidable challenges in geotechnical engineering. This introductory chapter reviews types of slope movements and various limit equilibrium methods to determine the factor of safety for the sliding types of mass movements. The most popular and well known limit equilibrium methods—namely, normal, simplified Bishop, original Spencer, and Spencer—are discussed in the book. These methods are based on the method of slices and can be divided broadly into four categories, depending on the number of equilibrium equations to be satisfied: (1) methods that satisfy overall moment equilibrium, (2) methods that satisfy overall moment and overall force equilibrium, (3) methods that satisfy force equilibrium of each slice, and (4) methods that satisfy moment and force equilibrium of each slice.

This chapter describes three types of failure surface: circular, noncircular, and composite. It also explains the method of slices and the sliding-block analysis for determining the factor of safety. The two methods for analyzing the stability of slopes are total stress analysis and effective stress analysis. The major difference between these methods is that total stress analysis does not require knowledge of the pore pressure, whereas effective stress analysis does. Two examples illustrate the total stress analysis for circular and noncircular failure surfaces. The circular failure surface is analyzed with the well known Fellenius method, and the noncircular failure surface is analyzed with the sliding-block method. The drawback of the Fellenius method is discussed, and the normal method, which is based on the concept of submerged weight, is then introduced. Finally, the chapter discusses the minimum factors of safety required under various conditions.

Shear strength is one of the most important factors affecting the factor of safety. Subsurface investigations determine the nature and extent of materials that may have an effect on the stability of the slope. This chapter focuses on the following field tests for determining the shear strength of soils in the field: the standard penetration test, the Dutch cone test, the piezocone test, and the vane shear test. Typical laboratory tests that complement field tests include the direct shear test, the triaxial compression test, and the unconfined compression test. The chapter also presents the ranges of shear strength for various materials, including municipal wastes, and the correlations of effective friction angle or undrained shear strength with other index properties. Finally, it talks about the shear strength of granular soils with a curved strength envelope and the back-calculation of shear strength from failed slopes.

In the stability analysis of slopes, particularly those related to earth dams, estimating the location of the phreatic surface, or the line of seepage, is necessary. This chapter discusses various methods for determining the phreatic surface. If the slope has not been constructed and is complex in configuration, the phreatic surface can be obtained by drawing a flownet or using a finite-element computer program. The chapter provides details about flownets involving homogeneous cross-sections. It also discusses two methods for determining the point of exit depending on the angle of the filter drain, if the drainage system is provided within the dam, such as the use of porous shells, toe drains, or underdrains. Finally, the chapter briefly describes the application of pore pressure ratios.

When a slide takes place, determining the cause of the slide is necessary to take proper remedial measures to correct it. This chapter describes the field investigation that must be undertaken before planning and design of such remedial measures. The scope of a field investigation should include topography, geology, water, weather, and history of slope changes. After the field investigation, some preliminary planning is needed before the start of the permanent remedial works. Corrective methods can either decrease the driving forces or increase the resisting forces. The chapter describes various methods for correcting slides, including slope reduction or removal of weight, buttresses, surface drainage, subsurface drainage, retaining walls, anchor systems, bridging or tunneling, soil reinforcements, pile systems, vegetation and biotechnical stabilization, and hardening of soils. The selection of methods for stabilization also is discussed.

This chapter presents simplified methods for stability analysis of plane failure surfaces and includes a case of practical interest: the infinite slope in which a thin soil overburden is laid above a rock slope of considerable length. The factor of safety for triangular and trapezoidal fills on a sloping surface is determined in the chapter. Sliding-block analysis may be used when a layer of weak material occurs at the bottom of each block. A quadratic equation determines the factor of safety in the case of two sliding blocks; for three sliding blocks, a cubic equation is available. The active and passive earth pressure theory is used for the stability analysis of earth slopes.

Stability charts that apply only to cylindrical failure surfaces are the focus of this chapter. The chapter also includes charts for triangular and trapezoidal fills on rock or stiff slopes, triangular fills on soil slopes, and earth dams and embankments, and some new charts for the effective stress analysis of nonhomogeneous dams. The well known friction circle and logarithmic spiral methods are also presented.

The method of slices is a powerful tool for analyzing slopes of any configuration of soils and groundwater conditions. This chapter discusses four different limit equilibrium methods: normal, simplified Bishop, original Spencer, and Spencer. The important equations used in LEAME computer software are derived in the chapter, and some special techniques to solve these equations are also discussed, including Newton's method of tangent and the use of a relaxation factor. The use of spreadsheets to compute the factors of safety is also demonstrated in the chapter.

This chapter presents theoretical background for three-dimensional (3D) analysis and extends the method of slices to the method of columns. To simplify the analysis, the chapter covers only the normal, simplified Bishop, and original Spencer methods. The simplified Bishop method is recommended for circular failure surfaces and the original Spencer method for noncircular and composite failure surfaces. The purpose of 3D analysis is to include the end effects, so the same cross-section for 2D analysis is used for 3D analysis. Two types of failures are discussed, one with ellipsoidal ends and the other with planar ends. Major equations incorporated in LEAME are derived, and the use of spreadsheets to solve these equations is demonstrated. The equation for overall moment equilibrium used in all three methods is presented.

The safe design of a slope requires a factor of safety between 1 and 1.5. A more practical approach to safe design is the probabilistic method, in which each input parameter has a mean and coefficient of variation. This chapter presents some statistical concepts and equations, which are incorporated in LEAME, to determine the reliability of a design. Taylor's expansion applies to simple cases where the factor of safety can be expressed in a closed-form formula. For complex cases involving computer programming, the mean-value first order second moment method (MFOSM) applies. This simplified method is particularly useful in geotechnical engineering. The distribution function used most frequently in geotechnical engineering as a probabilistic model is called the normal or Gaussian distribution. Another distribution used widely in geotechnical engineering is the lognormal distribution.