Sensitivity analysis applied to slope stabilization at failure V. Navarro a, * , A. Yustres a , M. Candel a , J. López a , E. Castillo b a University of Castilla-La Mancha, Avda. Camilo José Cela s/n, 13071 Ciudad Real, Spain b University of Cantabria, Avda. de los Castros s/n. 39005 Santander, Spain article info Article history: Received 13 November 2009 Accepted 3 March 2010 Available online 24 August 2010 Keywords: Landslides Slopes Failure Slope stabilization Design Clays abstract This article discusses how sensitivity analysis is a sound assessment tool for selecting the most efficient stabilization method of slopes at failure. A discretized form of the variational approach is used not only for performing sensitivity analysis but to locate the critical slip surface, i.e., the sensitivity analysis is car- ried out in the same way as it is done in optimization problems. This method supplies a robust formula- tion and methodology for obtaining the sensitivities of the safety factor with respect to both the soil parameters and the slope profile, stating the slope stabilization design as a relatively simple minimiza- tion problem. Two well known examples, as the Selset landslide and the Sudbury Hill slip are used to illustrate the application of the method and to highlight both its capabilities and limitations. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Today, engineers are not completely satisfied with the solutions to given problems and they also require knowledge of how these solutions depend on data. Thus, for instance, in minimization prob- lems, such as those associated with slope stability, it is not enough to know the optimal value of the objective function (the safety fac- tor) and the solution (slip line) where the minimum is attained. In this sense, approaches as the ‘‘safety maps” introduced by Baker and Leshchinsky [1] provides a valuable information to determine how and how much specific changes in the parameters of the sys- tem modify both the optimal objective function value and the opti- mal solution. In the field of slope stability, sensitivity analysis is generally conducted by means of a series of calculations in which each sig- nificant parameter is varied systematically over its maximum cred- ible range in order to determine its influence upon the safety factor [2]. If one is interested in characterizing the variation in safety when encounter minor modifications in the parameters, these incremental techniques define an approximation of the safety fac- tor gradient. In this case, for discrete problems (i.e., slices) the sen- sitivity may be calculated in a simpler and compact way by using the techniques that have been developed in the area of non-linear optimization [3]. When dealing with continuous problems as those linked to the variational approach of slope stability analysis, the formulation put forth by Castillo et al. [4] can be used. In any event, regardless of how the sensitivity analysis is done, when instability occurs, a sensitivity analysis allows to know which qualitative or quantitative actions are more appropriate to stabilize the given slope. Therefore, the sensitivity analysis is a use- ful tool able to provide a sound assessment for the selection of the slope stabilization method. Our main objective in this article is to analyze the use of this sensitivity analysis tool. 2. Conceptual basis of the method Both when limit equilibrium methods are used, and when the kinematic approach of limit analysis is applied, if the safety factor is defined as the ratio of the shear stress of the soil to the shear stress at failure, slope stability is generally evaluated as a ratio: F ¼ S D ¼ R b a Gx; yðxÞ; y 0 ðxÞ; y G ðxÞ; y 0 G ðxÞ; uðx; yÞ; p   dx R b a Q ðx; yðxÞ; y 0 ðxÞ; y G ðxÞ; y 0 G ðxÞ; uðx; yÞ; pÞ dx ð1Þ For the two-dimensional collapse mechanism defined in Fig. 1, a and b are the x-coordinates of the sliding line end points, y G (x) is the ground profile (ordinate at point x), y(x) is the ordinate of the sliding line at point x, and y 0 (x) is its first derivative. The u(x, y) function defines the distribution of the soil water pressure. Finally, vector p groups all the parameters together. In principle, it could be a vectorial field (if, for example, the spatial variation of the strength parameters is taken into account), though what usually happens is that it contains only a discrete number of parameters which are constant throughout the entire domain. G and Q are two functionals that define the actions over the system. When lim- it equilibrium methods are used, actions are identified with forces 0266-352X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2010.03.010 * Corresponding author. Address: Edificio Politécnico Avda. Camilo José Cela s/n, 13071 Ciudad Real, Spain. Tel.: +34 926295453; fax: +34 926295391. E-mail address: vicente.navarro@uclm.es (V. Navarro). Computers and Geotechnics 37 (2010) 837–845 Contents lists available at ScienceDirect Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo