## Slope Stability Theory

Geology and Soil Mechanics, UW-Stout

### Slope Stability

An exposed ground surface that stands at an angle with the horizontal is called an unrestrained slope. The slope can be natural or artificial. If the ground surface is not horizontal, a component of gravity will tend to move the soil downward as in figure 1. If the component of gravity is large enough and the soils internal shear strength is small enough, a slope failure can occur. When an engineer checks a slope against potential failure, they determine and compare the shear stress developed along the most likely rupture surface with the shear strength of the soil. The stability analysis of a slope is not an easy task. Evaluation of variables such as the soil stratification and its in-place shear strength may prove difficult. Water seepage through the slope and the choice of a potential slip surface add to the complexity of the problem.

In fact, one should keep in mind the words of specialists:

"Slides may occur in almost every conceivable manner, slowly and suddenly, and with or without any apparent provocation. Usually, slides are due to excavation or to undercutting the foot of an existing slope. However, in some instances, they are caused by a gradual disintegration of the structure of the soil, starting at hair cracks which subdivide the soil into angular fragments. In others, they are caused by an increase of the porewater pressure in a few exceptionally permeable layers, or by a shock that liquefies the soil beneath the slope. Because of the extraordinary variety of factors and processes that may lead to slides, the conditions for the stability of slopes usually defy theoretical analysis."

[Source: Terzaghi, K., and Peck, R.B., 1967, Soil mechanics in engineering practice (2nd edition), New York, John Wiley & Sons, Inc., p. 729]

Figure 1

There are several different techniques used to help determine the safety factor of a slope. This sheet will introduce you to one of these techniques.

Stability Analysis Using A Circularly Cylindrical Failure Surface

One can model the slope failure as a 3-dimensional cylinder that rotates about a point in space called O. Lets take a brief look at the theory of such an approach. In essence (using figure 2), the weight W1 of the soil tends to cause the slip by producing a moment (or torque), Md , about point O.

Figure 2

Both the soil cohesion and the weight of the soil W2 produce another moment, MR , tending to resist failure. The moment causing the failure can be expressed as

.

The moment produced by the shear strength of the soil is

and is in the counter-clockwise direction. The variable cd is the cohesion of the soil measured in units of (force)/(area), the arc length is rq with the angle being in radians, the 1 represents one unit of distance into the page. For equilibrium to be maintained

for every point O and radius r selected. The most probable slip surface can be determined by varying the position in space of point O and the radius r to obtain the largest cd required to maintain the above inequality.

Example: Determine the necessary soil cohesion to prevent failure when W1= 10 kN, l1 =1.2m, W2=1.6kN, l2 =0.5m, r =7m, and q =120o.

.

Computer Program To Determine the Minimum Cohesion

I have developed an Excel Spreadsheet program (~9 Mbytes) that will calculate the minimum necessary cohesion (for a cohesive soil) for a slope to remain stable for a given slope failure surface. Please read the description before using this program. This program allows you to choose the shape of the surface (blue line in figure below) along with the cylinder center and radius (red circle in figure below). The program's output is the minimum soil cohesion needed to remain stable for this configuration. This file is rather big, so you'll need a good download speed and sufficient RAM.

For questions or comments regarding these pages contact Dr. Alan Scott / scotta@uwstout.edu / this page was last updated July 25, 2001