In circular failure surfaces, when the soil is under conditions that allow for partial or full drainage (drained conditions), its shear strength is controlled by both its cohesion and its frictional components. The frictional strength depends on the effective stress acting on a soil element which is not constant along the entire surface. For this reason, the method of slices was introduced by W. Fellenius. The soil mass above the assumed failure surface is divided into a specified number of vertical columns with equal widths. Therefore, the bottom part of each slice is part of the failure surface. An example illustration of the division of a slope in 9 slices, for a pre-defined failure surface, of equal width is depicted in Figure 1.
Each slice can then be considered as a distinguished body in which force and moment equilibrium must be held. Interaction between the slices results in internal forces that all of the developed methods of slices take into consideration. A slice is also subjected to its weight, the reaction applied by the stable part of the slope at its base, as well as any additional forces that can be applied by external factors (ex., seismic loading). For each slice, the base is assumed to be an inclined straight line. The height of each column is measured along its centerline.
The forces acting on a typical slice (without considering any external loads) are depicted in Figure 2.
The collinearity of the interslice forces is not a minor assumption since it violates Newton’s 3rd law. The reaction forces between the slices are not assumed to be opposite. According to Whitman and Bailey (1967), the Ordinary Method could result in errors in the estimated FoS as high as 60%.
where a is the inclination of each slice, R is the radius of the circular failure surface, and d is the level arm of the weight force of each slice to the center of the selected circular surface.
The Ordinary Method is the only one according to which the FoS calculated via a linear equation, therefore, it enables simple and quick calculations.
As mentioned above, the calculations needed to perform the Ordinary Method of slices are rather simple. Here, a step-by-step guide on how to to estimate the FoS based on the Ordinary Method on a selected circular surface will be presented.
It should not be neglected that the Method of Slices is a 2-dimensional method hence, a representative cross-section of the slope (or embankment) of interest must be developed. The cross-section should be drawn on a suitable scale which must be the same for X and Y axes.
A circular surface must be selected to perform the analysis. To fully define the surface, the center and radius of the selected surface are needed.
Divide the surface into several slices (8-15 usually) as depicted in Figure 1. The widths of the slices do not necessarily have to be the same, therefore, the division is made so that modeling difficulties can be addressed. For example, the existence of more than one soil layer would affect the division. The base of each slice should be lying on top of a single layer so that shear strength parameters are uniform along the base of each slice. Moreover, the top boundaries of the slices should be located at geometric breaks of the slope to facilitate the measurement of geometric features. An example is illustrated in Figure 4. The base of each slice is located within one soil layer, while the top boundaries are selected to enable simpler shapes which are required for the calculation of the weight of each slice.
where: hw is the distance between the centroid point (midpoint of the base) of the slice and the water table and αw is the slope of the water table.
Each slice is approximated as a rectangular parallelepiped and its area is calculated as:
where Hi and Di are the average height and width of slice i.
Then, the unit weight of the soil γ (dry or saturated) is used to calculate the total weight of the slice (kN/m, along the direction perpendicular to the cross-section):
It is assumed that the shear strength of the soil layers is estimated based on the Mohr-Coulomb failure criterion, therefore:
where τ is the shear strength, σn is the normal effective stress, c is the cohesion, and φ the friction angle of the soil.
Firstly, the frictional resisting forces that act at the base of each slice are calculated. The normal effective force is derived via the following equation:
where a is the slope angle of the base of the slice assessed, u is the pore water pressure, and L the arc length of the base which can be approximated as a line segment.
The corresponding frictional force is:
The impact of the pore water pressure is negative in terms of slope stability since it decreases the normal effective force and, consequently, the frictional resisting component.
In addition, the cohesive resisting force is a function of the material’s cohesion and the length of the slice’s base L:
Consequently, the total resisting force for slice i is:
The above equation demonstrates why a slice must be located within one soil layer. Otherwise, we would have to make an assumption on which friction angle and cohesion to utilize.
The tangential driving force is parallel to the base of the slope and, in cases where no external or dynamic loads act on the slice, it is equal to the tangent component of the weight W (see Figure 3).
where a is the inclination of the slice’s base.
The Factor of Safety is derived by adding the resisting forces and dividing them with the sum of the driving forces (Equation 2):
Despite the fact that the ratio of resisting to driving forces is different for each slice, the FoS is considered to be equal for all of them since the slope is an entity that has been artificially divided into a number of sections to enable the aforementioned procedure.
Over the years, more sophisticated methods that made less significant assumptions were developed. The Ordinary Method was found to be more conservative providing lower FoS than other methods. However, its principles have been the driving factor for a multitude of extensive research efforts aiming at the optimization of the results of the Limit Equilibrium Method.
Fredlund, D.G. Krahn, J. (1977). Comparison of slope stability methods of analysis. Canadian Geotechnical Journal. 14(3): 429-439. https://doi.org/10.1139/t77-045
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.
Whitman R.V., Bailey W.A., (1967). Use of computer for slope stability analysis. ASCE Journal of the Soil Mechanics and Foundation Division, 93(SM4). doi.org/10.1061/JSFEAQ.0001 003
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