Limit Equilibrium Methods

The advantage of Limit Equilibrium Methods is that they can be translated into analytical equations. The more basic methods can relatively simply be worked out by hand for one or just a few defined failure surfaces. However, by using modem software, thousands of potential failure surfaces can be verified in order to find the governing mechanism with the lowest factor of safety. 

For the selection of a calculation method and/or type of slip surface, the following guidelines could be considered  

In the method of slices the factor of safety is usually expressed as the ratio of the available soil strength (stabilising forces) over the driving forces or moments, where summation is made over all slices within the failure surface  

In the method of slices basically circular failure surfaces are considered. However some slice methods have also been developed to analyse non-circular failure surfaces and arbitrarily defined multi-planar failure surfaces. In Table 8.12 an overview is given of different slice methods, the type of failure surface considered and the equilibrium equations that taken into account. 

Limit Equilibrium Methods (LEM s) that verify the stability by means of soil blocks within the failure surface require more complex calculations of all forces acting in between the blocks. These methods have also been implemented in software for a quick analysis. 

Does not require knowledge of in situ initial pore pressure distribution 

Requires only limited laboratory test program to determine characteristics of pore water pressure generation. This assumes availability of effetive stress-strain relation 

No tmiversaUy accepted effective stress-strain relation is at present available 

Source Chaney, R.C., Methods of predicting the deformation of the seabed due to cydic loading. Proceedings of the Symposium on Seabed Mechanics. International Union of Theoretical and Applied Mechanics, University of Newcastle, Callaghan, NSW, Australia, 159-167,1984. Republished with kind permission from Springer Science + Business Media. 

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Submarine slopes are commonly fairly gentle, uniform, and homogeneous over considerable horizontal distances (see Figure 11.8). Thus in many cases infinite slope limiting equilibrium methods are applicable. Assume an element of an infinite slope as shown in Figure 11.10. 

Limit equilibrium method is still the most widely used method to analyze slopes stability. It includes Fellenius s method (or Sweden method). Simplified Bishop s Method (SBM), Janbu s method, Sarma s method, Morgenstern Price s method and Spencer s method, et al. Among these methods, Fellenius s method and SBM are the most popular method in engineering field. 

There are several methods available for analysis of the stability of slopes in soils (Morgenstern, 1995). Most of these may be classed as limit equilibrium methods, in which the... 

To calculate the landslide stability, a special method, named transmitting coefficient method (TCM) is widely used but only in China. It is also a limit equilibrium method, and is appropriate for different sliding surface shapes. The TCM is illustrated in Figure 4. The specific equations are as follows ... 

Although slope mass ratings for rock slopes and a new system for soil slopes are used in this paper, different evaluation systems (e.g., factors of safety using the limit equilibrium method and numerical simulation, and the probability of failure from probabilistic analysis) may also be used. 

Keywords slope stability, open-pit mine, limit equilibrium method, underground mining... 

Figure 4. Calculated slip surface of the slope from limit equilibrium method.

Thus, it should be recognized that AASf/TO LRFD (AASHTO 2012) includes overall or global stability under the service Umit states design due to complexities involved in defining load and resistance factors when soil acts to exert both load and resistance simultaneously. Slope stability analysis for LRFD is performed in the same manner as for WSD, except that resistance factor, as stated above, is taken as the inverse of the FS determined based on the limit equilibrium methods. 

As shown in Figure 11.5, limit equilibrium methods involve discretization of the slide mass into a number of slices to take into account the inherent irregularities associated with the geometry of the... 

A detailed mathematical formulation of various limit equilibrium equations can be found in Krahn (2004). Because the number of unknowns in these equations is larger than the number of equations available, additional assumptions need to be introduced. Such assumptions are typically made in respect to the magnitude and orientation of the interslice forces. Major differences among various limit equilibrium methods are associated with the specific equilibrium equations used and the assumptions made with respect to the interslice forces. Table 11.2 outlines the equations used by various limit equilibrium methods of slope stability analysis together with the assumptions related to interslice forces associated with each method. 

The limit equilibrium methods forming the framework of slope stability/instability analysis generally accept the Mohr-Coulomb failure criterion, which can be expressed in terms of effective or total stresses. The Mohr-Coulomb shear strength in terms of effective stresses is as follows ... 

Limit Equilibrium Methods which are based on the solution of one or more equilibrium equations (moment, vertical, horizontal) for an assumed failure surface ... 

Different Limit Equilibrium Methods exist. The most frequently used is the method of slices in which the soil within the failure surface is subdivided into slices with a certain width. The equilibrium of these shces is written in terms of moment, vertical and horizontal stability (Figure 8.21). A slope stability analysis should, as a... 

In those events that a combined failure of soil and structural elements can be expected, due consideration must be given to the soil-structure interaction in terms of their relative stiffness. These cases include, for instance, failure surfaces intersecting structural members such as piles, flexible walls or armouring (e.g. geotextiles). Such an approach is, however, not always feasible in Limit Equilibrium Methods. The Finite Element Method is a more appropriate tool for the analysis of soil-structure interaction. 

While limit equilibrium methods fail to give any information regarding developed tensile and shear forces along nails and they cannot evaluate local stability of structure, numerically derived pseudostatic approaches have been developed to investigate behavior of soU-nailed walls under... 

The limit equilibrium method determines the forces required for equilibrium, taking into account the geometry of the slope, the pore pressure u, and the loads. When layers of reinforcement are introduced, it is possible to define the envelope of the maximum strength available in each layer, which depends on the Mction between the reinforcement and the soil, the characteristics of the reinforcement, and the compatibility of deformations with the functionality of the project. The diagram of the available forces must overlap that of the required forces, as shown in Fig. 15.4. 

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