PSI - Issue 22

Mahdi Shadab Fara et al. / Procedia Structural Integrity 22 (2019) 345–352 Shadab Far and Huang / Structural Integrity Procedia 00 (2018) 000–000

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provides the minimum FS. The parameters x 1 to x n can be obtained using either the vertices of a non-circular surface or the center and radius of a circular surface Pandit et al., 2018. The assumption of a circular slip surface, although may simplify the calculation, is not necessarily capable of calculating the minimum FS because the real slide surfaces are mostly non-circular (Zolfaghari et al., 2005). In non-circular methods, however, the main challenge is that the cost function is often multi-modal and the domain is highly discontinuous. Hence, the gradient-based optimization methods developed to implement the non-circular method, such as the variation methods, the simplex method, and the conjugate gradient method usually face convergence problems and may easily be trapped in local minima (Cheng et al., 2007a). Hence, a shift toward the population-based optimization methods is taken to identify the critical failure surface (Sanaeirad and Kashani, 2016). Cheng et al., 2007b utilized six intelligent approaches including simulated an nealing, genetic algorithm, particle swarm optimization algorithm (PSO), single harmonic search algorithm (SHM), modified harmonic search algorithm (MHM), and Tabu search algorithm (TSA) to identify the critical surface. They then concluded that the SA method would provide a more accurate result, especially in cases where the cost function is discontinuous. However, the major problem with the SA methodology is a large number of iteration required to con verge. Combining SA and random walk process, Xiao Su, 2005 defined a dynamic boundary system and decreased the iteration number and subsequently facilitated the computing process. The methods mentioned above are primarily formulated as a deterministic problem, while the uncertainty involved in the problem is also a serious and challenging issue. Typically, estimating the critical slip surface based on FS may not necessarily comply with the expected risk of the project. The uncertainty involved in the geotechnical and hydraulic characteristics of the environment could make the actual answer to be quite di ff erent than the results obtained from deterministic analyses (El-Ramly et al., 2002; Hamedifar et al., 2014). Hence, probabilistic analysis of the slope stability problem turned as a new challenge. To the best of author’s knowledge, most of the available studies investigated the uncertainties in soil geotechnical properties and have not considered the probabilistic features of groundwater level in their formulation. However, the uncertainty involved in determining the groundwater level could severely a ff ect the probability of failure. In this study, the SA method is adopted with random-walk dynamic boundaries to determine the non-circular slip surface. Then, using the Monte Carlo sampling method, a probabilistic form of the SA method is developed to estimate the failure probability. Furthermore, the position of the groundwater level is defined as a random variable and incorporated into the established reliability problem. In other words, the groundwater table is not considered as a deterministic variable and it is assumed to change according to a probability distribution function with certain mean and standard deviation. This assumption helped us estimate the e ff ect of underground water table level on failure probability.

2. Modeling

2.1. Overview of the model

The model presented in this paper is applied to the Lodalen slope (El-Ramly et al., 2006;Hassan and Wol ff , 1999). The Lodalen slope is one of the benchmark cases for slope stability analysis. The geometric characteristics of the slope are presented in Fig. 1. Soil is defined by Mohr-Coulomb model. Cohesion, friction angle, and unit weight of soil are considered as 10 kPa, 27.1 ( ◦ ), and 19.1 (kN / m 3 ), respectively. The phreatic surface in this model is determined by the equation y = − 0 . 003 x 2 − 0 . 078 x + 22 . 765, which is marked by blue dot-line in Fig. 1.

2.2. Slope analysis using simulated annealing approach

In order to prevent inadmissible slip surface and to control the search domain of the SA algorithm, a dynamic boundary condition, initially introduced by Cheng et al., 2007b, was utilized in this work. As shown in Fig. 2a, to specify the dynamic boundary, the entire domain is divided into n sections and the values of x 1 , x 2 , ..., x n are determined. Next, y 2 min and y 2 max are found using the upper roof of slope and bedrock, respectively. Afterward, y 2 is randomly selected in the interval [ y 2 min , y 2 max ]. Further, the point G is calculated by intersecting the line connecting