## ROTATIONAL SLOPE FAILURES

We will continue to use the limit equilibrium method, but instead of a planar slip surface of infi nite extent we will assume circular (Figure ) and noncircular (Figure ) slip surfaces of finite extent. We will assume the presence of a phreatic surface within the sliding mass.

Circular

and noncircular failure

surfaces and the forces

on a slice of soil

A free-body diagram of the postulated circular failure mechanism would show the weight of the soil within the sliding mass acting at the center of mass. If seepage is present, then the seepage forces, Js, which may vary along the fl ow path, are present. The forces resisting outward movement of slope are the shearing forces mobilized by the soil along the slip surface. We must now use statics to determine whether the disturbing forces and moments created by W and Js exceed the resisting forces and moments due to the shearing forces mobilized by the soil. However, we have several problems in determining the forces and moments. Here is a list.

- It is cumbersome, if not diffi cult, to determine the location of the center of mass, especially when we have layered soils and groundwater.

- The problem is statically indeterminate.

- We do not know how the mobilized shear strength, tm, of the soil varies along the slip surface.

- We do not know how the normal effective stress, srn, varies along the slip surface.

- We do not know how the seepage forces vary within the soil mass and along the failure surface.

- Even the weight of sliding mass is diffi cult to calculate because of soil layering (different unit weights of the soils) and complex geometry of some slopes.

One approach to solve our problem is to divide the sliding mass into an arbitrary number of slices and then sum the forces and moments of each slice. Of course, the larger the number of slices, the better the accuracy of our answer. Dividing the area inside the sliding mass into slices presents new problems. We now have to account for the internal forces or interfacial forces between the slices.

Let us consider an arbitrary slice, ABCD (Figure), and draw a free-body diagram of the forces acting on the slices, as illustrated in Figure. The forces have the following meaning:

- Wj is the total weight of a slice including any external load.

- Ej is the interslice lateral effective force.

- (Js)j is the seepage force on the slice.

- Nj is the normal force along the slip surface.

- Tj is the mobilized shear force along the slip surface.

- Xj is the interslice shear force.

- Uj is the force from the porewater pressure.

- zj is the location of the interslice lateral effective force.

- zw is the location of the porewater force.

- aj is the location of the normal effective force along the slip surface.

- bj is the width of the slice.

- lj is the length of slip surface along the slice