# Dynamic Programming to Slope Stability Analysis

## Dynamic Programming to Slope Stability Analysis

The application of dynamic programming to slope stability analysis

Ha T.V. Pham and Delwyn G. Fredlund

• What made me most interested in this paper is that the dynamic programming method can be combined with a finite element stress analysis to provide a more complete solution for the analysis of slope stability because the technique overcomes the primarily difficulties associated with limit equilibrium methods. And a computer program named DYNPROG was developed determine the shape and location of the critical slip surface and the corresponding factor of safety.
• Definition of the factor of safety:
[pic]
where n is the number of discrete segments, (i is the shear stress actuated, (fi is the shear strength, and (Li is the length of the segment.
• The objective of this research program is to study the use of the dynamic programming method in solving practical slope stability problems. The critical slip surface is defined as the slip surface that yields the minimum value of an optimal function. It is assumed that the critical slip surface can be approximated by an assemblage of linear segments. Each linear segment connects two state points located in two successive stages.
• Theory of the dynamic programming method
The minimum of Fs can be found by using an auxiliary function G. The auxiliary function is also known as the return function:
[pic]
where Si are actuating forces acting on the ith segment of the slip surface, Ri are resisting forces acting on the ith segment of the slip surface, and n is the total number of discrete segments making up the slip surface.
The minimum value of the auxiliary function is Gm and is defined as:
[pic]
An optimal function, Hi(j), obtained at state point {j} located in stage [i] is introduced. The optimal function, Hi(j), is defined as the minimum of the return function, G, calculated from a state point for the initial stage to state point {j} located in stage [i].
Hi=1(k) = Hi(j) + Gi(j,k)
where Gi(j,k) is...