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The limit equilibrium method consists in the study of the equilibrium between a rigid body, such as the slope, and of a slide surface of any shape (straight line, arc of a circle, logarithmic spiral). From such equilibrium are calculated shear stresses (τ) and compared to the available strength (τf) calculated according to Mohr-Coulomb's failure criterion. From this comparison we derive the first indication of stability as the Safety Factor:

 

FS=τf/τ

 

Among the various equilibrium methods some consider the total equilibrium of the rigid body (Culman), while others divide the body into slices to cater for its non homogeneity and consider the equilibrium of each of these (Fellenius, Bishop, Janbu, etc.).

 

Slice equilibrium methods are discussed below.

 

Fellenius (1927)

Method valid only for sliding surfaces of circular shape, intra interface forces (between slices) are neglected. Using this method cannot be taken into consideration the works of intervention.

 

Bishop (1955)

Method valid only for sliding surfaces of circular shape. No contribution of the forces acting on blocks is ignored using this method that was the first to describe the problems of conventional methods.

 

Janbu (1956)

Janbu has extended Bishop's method to Free Form surfaces. When free form (generic form) sliding surfaces are treated the arm of the forces changes (in case of circular surfaces it is constant and equal to the radius of the arc) and therefore it is more convenient to evaluate the moment equation at the angle of each block.

 

Morgenstern & Price (1965)

A relation is established between the components of the interslice forces of the type X = λ f(x)E, where λ is a scale factor and f(x), a function of the position of E and X, defining a relation between the variables of the force X and the force E inside the sliding mass. The function f(x) is arbitrarily chosen (constant, sine, half-sine, trapezoidal, etc.) and has little influence on the result, but it should be verified that the values ​​obtained for the unknowns are physically acceptable.

 

Abstract of article in Canadian Geotechnical journal (2002):

Slope-stability problems are usually analyzed using a variety of limit equilibrium methods of slices. When evaluating the stability conditions of soil slopes of simple configuration, circular potential slip surfaces are usually assumed and the Ordinary method (Fellenius 1936) and the simplified Bishop method (Bishop 1955) are commonly used, the latter being preferred due to its high precision. However, in many situations, the actual failure surfaces are found to deviated largely from circular shape or the potential slip surfaces are predefined by planes of weakness in rock slopes. In such cases, a number of methods of slices can be used to accommodate the non-circular shape of slip surfaces (Janbu 1954; Lowe and Karafiath 1960; Morgenstern and Price 1965; Spener 1967; U.S. Army Corps of Engineers 1967; and etc.), among which the Morgenstern-Price method (Morgenstern and Price 1965) is regarded as the most popular because it satisfies complete the equilibrium conditions and involves the least numerical difficulties. The basic assumption underlying the Morgenstern-Price method is that the ratio of normal to shear interslice forces across the sliding mass is represented by an interlace force function that is the product of a specified function f(x) and an unknown scaling factor l. According to the vertical and force equilibrium conditions for individual slices and the moment equilibrium condition for the whole sliding mass, two equilibrium equations are derived involving the two unknowns the factor of safety Fs and the scaling factor l, thereby rendering the problem determinate. Unfortunately, solving for Fs and l is very complex since the equilibrium equations are highly nonlinear and in rather complicated form. Some sophisticated iterative procedures (Morgenstern and Price 1967; Fredlund and Krahn 1977; Chen and Morgenstern 1983; Zhu et al. 2001) have been developed for such purposes. Although these procedures can give converged solutions to Fs and l, they are not easily accessible to general geotechnical designers who have to rely on commercial packages as a black box.

 

Spencer (1967)

The interface forces along the division surfaces of the individual slices are oriented parallel to one another and inclined with respect to the horizontal by an assigned angle.

 

Bell (1968)

The equilibrium is obtained by equating to zero the sum of the horizontal forces, the sum of the vertical forces and the sum of the moments with respect to the origin. Distribution functions of normal stresses are adopted.

 

Sarma (1973)

The method of Sarma is a simple, but accurate method for the analysis of slope stability, which allows to determine the horizontal seismic acceleration required so that the mass of soil, delimited by the sliding surface and by the topographic profile, reaches the limit equilibrium state (critical acceleration Kc) and, at the same time, allows to obtain the usual safety factor obtained as for the other most common geotechnical methods.

It is a method based on the principle of limit equilibrium of the slices, therefore, is considered the equilibrium of a potential sliding soil mass divided into n vertical slices of a thickness sufficiently small to be considered eligible the assumption that the normal stress Ni acts in the midpoint of the base of the slice.

 

Zeng Liang (2002)

Zeng and Liang carried out a series of parametric analyzes of a two-dimensional model developed by finite element code, which reproduces the case of drilled shafts (piles immersed in a moving soil).

The two-dimensional model reproduces a slice of soil of unit thickness and assumes that the phenomenon occurs in plane strain conditions in the direction parallel to the axis of the piles.

The model was used to investigate the influence on the formation of an arch effect of some parameters such as the distance between the piles, the diameter and the shape of the piles, and the mechanical properties of the soil. The authors identify the relation between the distance and the diameter of the piles (s/d), the dimensionless parameter determining the formation of the arch effect.

The problem appears to be statically indeterminate, with a degree of indeterminacy equal to (8n-4), but in spite of this it is possible to obtain a solution by reducing the number of unknowns and thus taking simplifying assumptions, in order to make determined the problem.

 

 

Numerical method of displacements

 

D.E.M. Discrete Element Method (1992)

With this method, the soil is modeled as a series of discrete elements, which we will call "slices", and takes into account the mutual compatibility between the slices. A slope in the present model is treated as comprised of slices that are connected by elastoplastic Winkler springs. One set of springs is in the normal direction to simulate the normal stiffness. The other set is in the shear direction to simulate the sliding resistance at the interface. The behavior of the normal and shear springs is elastoplastic. The normal springs do not yield in compression, but they yield in tension, with a small tensile capacity for cohesive soil (tension cutoff) and no tensile capacity for frictional soil.

 

 

 

 

 

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