Using Plastic Blocks in the Upper Bound Method to Study the Quasi-Static Stability of Earth Slopes

In geotechnical engineering, stability analyses are used to predict the maximum load, which can be supported by a geostructure without inducing the failure. Limit equilibrium, limit analysis, slip-line methods, and displacement finite element method are among the main methods used for performing stability analysis in geotechnical problems. Among the mentioned methods, limit analysis is based on plastic bounding theorems developed, and it assumes small deformations, a perfectly plastic material, and an associated flow rule. Despite the limitations arising from the assumption of a simple purely plastic material model, the ability of limit theorems to provide the bounds on the collapse load is one of their great advantages. This is an important advantage for complex practical problems where the failure load is difficult to estimate by other methods and the maximum error in the solution can be precisely bounded. So far, several studies have been conducted to investigate the stability of earth slopes under different loading conditions using the upper bound limit analysis approach, assuming a failure mechanism consisting of one or several rigid block(s). As the safety factor of the slope stability obtained by the upper bound method is greater than the actual one, the more appropriate failure mechanism led to the lower and closer to actual safety factor. The present study has used the failure mechanism consisted of plastic blocks of which the behavior is closer to reality than rigid blocks; hence, the results have been more appropriate. In this paper, using plastic blocks to determine the safety factor of the slope stability under earthquake loads has been made possible by applying some changes to the Sloan-Kleeman formulation, and a suitable method has been developed to determine the optimal dimensions of plastic blocks using mathematical techniques. Researchers believe that such parameters as the slope height/angle, the slope material strength characteristics and the horizontal acceleration coefficient of the earthquake force can be effectively used to evaluate the slope stability. They have varied the values of these parameters to determine the related safety factor. This paper, too, calculated the mentioned safety factor for various values of the mentioned parameters and compared its results with those of other researchers’ upper and lower boundary methods to evaluate the capability of the proposed method. Since the real solution lies between the lower and upper boundary solutions, comparisons are more valid in calculations where the latter are close to each other. Therefore, this paper not only compared its findings with the results of other studies conducted by the OptumG2 software prepared by the related researchers (a version of this software was provided to the authors), but also made more calculations and compared the results with those of the proposed method to allow for better evaluations. All the mentioned comparisons revealed that the accuracy of the proposed method was acceptable in applied problems (differing, generally, by less than 7% from the average results of the upper and lower boundaries). Increased slope angle and the horizontal acceleration coefficient of the earthquake force made the answers obtained from the proposed method closer to reality.