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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Simulated annealing, FS Bishop = 1.074 Simulated annealing, FS Janbu = 1.011 Circular method, FS Bishop = 1.138 Circular method, FS Janbu = 1.023

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(a) Dynamic boundaries

(b) The results of SA and circular methods

Fig. 2. A presentation of (a) schematic dynamic boundaries and (b) the results of SA and circular methods.

surface and subsequently the FS value were calculated. The histogram of the generated samples is shown in Fig. 3. The red part denotes cases with FS less than 1. The mean, standard deviation, minimum, and maximum of samples are also shown in each figure. In total, 2136 less-than-zero samples were obtained. Dividing the less-than-zero samples by the total number of samples, the probability of slope failure was calculated as P f = 2136 / 10000 = 0 . 2136 = 21 . 36%. To ensure that the number of random samples used is su ffi cient and to assess the convergence trend of the Monte Carlo algorithm, the probability of failure was plotted against the random samples in a graph (Fig. 4). As can be seen, after using about 6000 random samples, the Monte Carlo algorithm converges with a negligible fluctuation. The mean FS obtained from Monte Carlo analysis is relatively less than the deterministic value. This means that the deterministic analysis may not necessarily show the most critical condition, and thus deterministic-based decision cannot meet the target risk. So far, the groundwater table is assumed to be constant. In this section, a method is proposed to define the ground water level as a random variable and implement it in the Monte Carlo analysis. For this purpose, two minimum and maximum boundaries for groundwater level should first be assumed so that the desired water table level is placed between them. Then, by defining the vertical distance between the minimum and desired water levels ( Y w ) and divid ing it by the vertical distance between the maximum and minimum water levels ( Y max ), normalized water table level, which is assumed to be between 0 and 1, is calculated as Y wn = Y w / Y max . The groundwater table can now be modeled as a random number between 0 and 1 with the desired probabilistic distribution function. In order to incorporate the groundwater level in the Monte Carlo analysis, the minimum and maximum water levels were considered as shown in Fig. 5a. It was also assumed that the Y wn has a normal distribution function with an average of 0.5 and a standard deviation of 0.17. Monte Carlo analysis was again utilized and 10,000 random samples were generated for each involved variables, including the groundwater table. The generated slip surfaces corresponding to each random samples are shown in Fig. 5b. The resulting histogram for Y wn and FS are shown in Fig. 6. The red parts represent the cases where FS < 1. 3.2. Modeling the underground water level as a random variable

Table 2. Probabilistic characteristics of the random variables involved in the problem. Parameter Probability distribution function Mean

Standard deviation

Min

Max

C (kPa)

Normal Normal Normal

10

2.21 1.72

16.63 32.26

3.37

φ ( ◦ )

27.1 19.1

21.94

γ (kN / m 3 )

4

31.1

7.1