Slope Stability: The Bishop Method of Slices

The Bishop Method of Slices

The Bishop Method was introduced in 1955 by Alan Wilfred Bishop from the Imperial College in London. It is one of several Methods of Slices developed to assess the stability of slopes and derive the associated Factor of Safety (FoS).

The approach differs from the Ordinary Method of slices in terms of the assumptions made regarding the interslice forces. In particular, the Bishop method assumes that the shear interslice forces acting at the lateral sides of each slice i can be neglected, i.e., V_i-1 = V_i+1 (Figure 1). It is pointed out that every method of slices requires certain assumptions to be made since the problem is, a priori, indeterminate (there are more unknown parameters than available equations).

Calculations

The FoS for a typical circular failure surface will be derived. It is assumed that the failure surface lies within a single type of soil which behaves in accordance with the Mohr-Coulomb failure criterion:

Where τ is the shear strength of the soil, σ_n is the normal effective stress, c is the cohesion and φ the friction angle of the soil.

The shear force T acting at the base of each slice (Figure 1) depends on the cohesion, the friction angle, the effective normal force N’ and the FoS as:

Where L is the arc length of the slice’s base which is approximated to a line segment.

The maximum shear force (shear strength) of the slice develops once the FoS becomes equal to 1,00.

The equilibrium force in the vertical direction is taken into account:

Subsequently, the moment equilibrium around the center of rotation yields the following equation:

The moments that each slice generates about the center of rotation are depicted in Figure 2.

Given that the shear force T is equal to T_maximum/FoS, the FoS for the entire failure surface is:

Utilizing equation [4] and substituting the length of the slice’s base as a function of its width and inclination (L = B* sin(α)), equation [5] becomes:

In some cases, it is preferable to utilize the r_u coefficient to take into account the boundary water force (r_u = u/σ), where u is the pore water pressure and σ the normal stress at the point of interest. In that case, equation [6] is slightly simplified and is transformed as follows:

Due to the fact that the FoS appears in both sides of equations [6] and [7], the solution is implicit and requires an iterative procedure until convergence towards a single value is reached. Therefore, contrary to the Ordinary Method of slices, the Bishop Method does not favor hand calculations and requires a computational solution.

The Bishop method has been found to be adequately accurate providing minor variances from the actual FoS of slopes. Zhu (2008) suggests that its accuracy is similar to more complex methods developed in the following years (e.g., the Spencer method), while researchers have not fully understood the theoretical causes of this phenomenon. Spencer (1967) mentioned that the FoS is insensitive to the inclination of the interslice forces, hence the assumptions of the Bishop method do not highly impact the results.

Nevertheless, a significant limitation of the method is that it should be applied solely to circular surfaces in order to preserve its accuracy.

References

Craig, R.F. (2004). Craig's Soil Mechanics (7th ed.). CRC Press. pp. 354-356 doi.org/10.4324/9780203494103

Spencer, E. (1967). A method of analysis of the stability of embankments assuming parallel inter-slice forces, Géotechnique, 17, 11–26.

Samtani, N.C, Nowatzki, E.A. (2006). Soils and Foundations Reference Manual Volume 1. U.S. Department of Transportation, Federal Highway Administration, Washington, D.C. 20590.

Zhu D.Y. (2008). Investigations on the accuracy of the simplified Bishop method. Landslides and Engineered Slopes – Chen et al. (eds). Taylor & Francis Group, London, ISBN 978-0-415-41196-7