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The GLE Formulation and the Various Methods of Stability Analysis

The general limit equilibrium (GLE) formulation in SLOPE/W encompasses the key elements of the common limit equilibrium methods of stability analyses. It provides a useful framework for describing and understanding the other methods of analysis, and is especially helpful when the different methods are solving to different Factors of Safety, and you need to decide which one is the most appropriate. In general, all limit equilibrium methods make assumptions about which Factor of Safety equilibrium equations are going to be satisfied and how interslice shear and normal forces are handled.

The GLE formulation is considered to be a rigorous method, which satisfies two factor of safety equations; The factor of safety with respect to moment equilibrium (F_m), and another with respect to horizontal force equilibrium (F_ƒ). In addition, both interslice shear and normal forces are considered and the interslice shear forces are computed using the equation:

      X=Eλ ƒ(x)

Where, ƒ(x) is an interslice force function, λ is the percentage (in decimal form) of the function used, E is the interslice normal force and X is the interslice shear force. Figure 1 shows a typical half-sine interslice force function which is available for both the GLE and Morgenstern-Price (M-P) methods.

Figure 1: Interslice applied and specified functions

This function reveals that the interslice shear forces are going to be higher in the middle of the slip surface than they are near the entry and exit points. By contrast, a constant interslice force function, such as the one assumed with the Spencer Method, holds the relationship between interslice shear and normal forces constant across the entire slip surface. In Figure 1 the upper curve is the specified input function while the lower curve is the actual function used in the solution. The ratio between the two curves is called lambda (λ). During the solve process the lambda value is determined where the two Factor of Safety equations become equal (F_ƒ = F_m). The lambda value shown in Figure 1 is 0.43, which is apparent by looking at the value of the applied function where the specified function is 1.0. In the other rigorous methods (i.e., Spencer and M-P), the lambda value is determined as part of the iterative solve process and only the final lambda value is presented.

In the GLE method, a user input range of lambda values are used to compute F_m and F_ƒ, and these values are then stored. With these stored values, a plot such as the one shown in Figure 2 can be created that shows how F_m and F_ƒ vary with lambda. Such a plot makes it possible to understand the differences between the factors of safety from the various methods, and to understand the influence of the selected interslice force function.

Figure 2: A factor of safety versus lambda plot

Two of the primary assumptions of the Bishops Simplified method are that it ignores interslice shear forces altogether and that it satisfies only moment equilibrium (F_m). In GLE terminology, not considering shear forces means λ is zero. As a result, the Bishops Simplified factor of safety in Figure 2 falls on the moment curve where lambda is zero. Janbu’s Simplified method also ignores interslice shear forces and only satisfies force equilibrium (F_ƒ). The Janbu Simplified factor of safety consequently falls on the force curve where λ is zero. The GLE, Spencer and Morgenstern-Price (M-P) factors of safety are determined at the point where the two curves cross. At this point the factor of safety satisfies both moment and force equilibrium. Whether the crossover point is the Spencer or M-P factor of safety depends on the interslice force function selected when setting up the analysis. As mentioned earlier, Spencer only considers a constant X/E ratio for all slices while the M-P method can utilize any general appropriate function. Methods like the Corps of Engineers and Lowe-Karafiath factors of safety fall on the force curve, since they only satisfy force equilibrium. The position on the force curve depends on the procedure used to establish the inclinations of the interslice force resultant.

From Figure 2, you can see that for this particular solution, the Simplified Bishop and Janbu computed factors of safety are lower than the one determined using the more rigorous method. It can be dangerous to make the assumption that this will always be the case. Depending on the geometry, the soil properties and the shape of the critical slip surface, the resulting GLE factor of safety versus lambda plot could have a shape such as the one shown in Figure 3. In this case, the GLE method reveals that the less rigorous Bishop method is not offering a conservative solution and should therefore be used with caution.

Figure 3: A factor of safety versus lambda plot for a composite slip surface

The GLE method is very useful for understanding the reasons for differences between the various methods. It is not necessarily a method for routine analyses in practice, but it is an effective supplementary method useful for enhancing your confidence in the selection and use of the other more common methods. Please refer to the SLOPE/W Engineering Book for a more detailed discussion of the GLE formulation.

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