## Dynamic Programming in Slope Stability Computations

### June 29, 2010

• Truly unrestrained non-circular slip surface
• Can be used for weak layer detection in complex systems

A conventional slope stability analysis involving limit equilibrium methods of slices consists of the calculation of the factor of safety for a specified slip surface of predetermined shape. The determination of the location of the critical slip surface with the lowest factor of safety is part of the analysis. To render the inherently indeterminate analysis determinate, conventional limit equilibrium methods generally make use of assumptions regarding the relationship between the interslice forces. These assumptions become disadvantages to limit equilibrium methods, since the actual stresses acting along the slip surface are quite approximate and the location of the critical slip surface depends on the shape assumed by the analyst.

Limit Equilibrium methods (LEM) of slices have two major shortcomings:

• There is uncertainty regarding the shape of the critical slip surface
• There is a disregard for the actual total stresses within the soil mass

The assumptions related to the interslice force function in limit equilibrium methods are unnecessary when a finite element stress analysis is used to obtain the normal and shear stresses acting at the base of slices (Fredlund and Scoular 1999). A stress analysis provides normal and shear stresses through the use of the finite element numerical method with a "switch on" of the gravity forces. Subsequently, the equation for the factor of safety becomes linear. Assumptions regarding the uncertainty of the shape of the critical slip surface can be omitted when an appropriate optimization technique is introduced into the analysis. Optimization techniques have been developed by several researchers for over two decades and have provided a variety of approaches to determine the shape and location of the critical slip surface (Celestino and Duncan 1981; Nguyen 1985; Chen and Shao 1988; Greco 1996). Each approach has its own advantages and shortcomings. The main shortcoming associated with these approaches, however, is that the actual stresses within a slope are quite approximate. This disregard for a more accurate assessment of the stresses can lead to inaccuracies in the computation of the factor of safety and an inability to analyze more complex problems. 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. The disadvantage of the dynamic programming approach is that there are more variables to specify for the analysis, such as Poisson’s ratio and the elastic moduli of the soils involved.

The dynamic programming method for a slope stability analysis has not been widely used in engineering practice primarily because of the complexity of the formulation and the lack of verification of the computed results. Baker (1980) introduced an optimization procedure that utilized the algorithm of the dynamic programming method to determine the critical slip surface. In this approach, the associated factors of safety were calculated using the Spencer (1967) method of slices. Yamagami and Ueta (1988) enhanced Baker’s approach by combining the dynamic programming method with a finite element stress analysis to more accurately calculate the factor of safety (Fig. 1). The critical slip surface was assumed to be a chain of linear segments connecting two state points located in two successive stages. The resisting and the actuating forces used to calculate an auxiliary function were determined from stresses interpolated from Gaussian points within the domain of the problem. Yamagami and Ueta analyzed two example problems to illustrate the proposed procedure.

Zou et al. (1995) proposed an improved dynamic programming technique that used essentially the same method as that introduced by Yamagami and Ueta (1988). The modification made by Zou et al. was that the critical slip surface might contain a segment connecting two state points located in the same stage. The stability of a trial dam in Nong Ngu Hao, Bangkok, Thailand, was analyzed as part of the study of the proposed procedure.

Previous studies (Ha, 2003) have established the consistency of the method with against traditional limit equilibrium benchmark solutions as can be seen in the following figures. It should be noted that the algorithm implemented in SVSLOPE is consistent with the algorithm previously implemented in the SVDYNAMIC software (which has been discontinued). In the majority of the benchmarks in Ha Pham’s thesis the results of the Dynamic Programming algorithm are more credible than the results of the traditional grid and radius searching methodology.

Example of a Homogeneous Slope

Example of a Multilayered Slope

The true advantages of the method may be summarized as follows:

• True non-circular slip surface searches: the dynamic programming algorithm is primarily a searching method which is largely unrestrained. Therefore it is useful for identifying slip surfaces that may vary from the conventional circular shape.
• Weak layer identification: the dynamic programming methodology is useful in the scenario where the consultant must determine if a slip surface may follow a weak layer. The usefulness of dynamic programming in identifying if a slip surface will follow a weak layer was proven in the Molycorp – Questa Rock Pile Study. In the Molycorp project, the dynamic programming searching methodology was used to search for slip surfaces through two-dimensional complex geometry shown in the following figure.

Complex geometry for Molycorp waste rock pile

Point of Interest: the dynamic programming method must be considered a searching method primarily. The physics of the dynamic programming method are consistent with limit equilibrium formulations. Therefore added score the dynamic programming solution methodology is a limit equilibrium methodology.

### Advantages of the Dynamic Programming Algorithm

• The Location of the Critical Slip Surface is a part of the overall solution
• The Shape of the Critical Slip Surface is part of the overall solution
• Complex stress-strain models for soil behaviour (e.g., elasto-plastic) can be used in the finite element method for the computation of the stress state in the soil mass

### Case Studies

The following case studies have documented the use of the Dynamic Programming Method:

### Research

D.G. Fredlund, Gilson Gitirana Jr. - 2003
Analysis of Transient Embankment Stability Using the Dynamic Programming Method
56th Canadian Geotechnical Conference 4th Joint IAH-CNC/CGS Conference, September 28-October 1, 2003, Winnipeg, MB, Canada

H.T.V. Pham & D.G. Fredlund - 2003
The application of dynamic programming to slope stability analysis (PDF)

Arie Kreman, Yiannis Tsompanakis - 2010
Application of dynamic programming to evaluate the slope stability of a vertical extension to a balefill (PDF)
Waste Management and Research, 2010