PSI - Issue 22

Mahdi Shadab Fara et al. / Procedia Structural Integrity 22 (2019) 345–352

349

Shadab Far and Huang / Structural Integrity Procedia 00 (2018) 000–000

5

0.25

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Histogram of ϕ ( 0 ) Cases where LSF<0 Fitted distribution

Sample mean=10.02 Sample Std=2.185 Minimum=3.374 Maximum=16.51

Histogram of C (kPa) Cases where FS<1 Fitted distribution

Sample mean=27.1 Sample Std=1.708 Minimum=22 Maximum=32.22

0.2

0.15

0.1

Relative frequency

Relative frequency

0.05

0

2

4

6

8

10

12

14

16

18

22

24

26

28

30

32

34

Soil cohesion, C (kPa)

Soil friction, ϕ ( 0 )

(a) Cohesion, C (kPa)

(b) Friction angel of soil, ϕ ( 0 )

0.12

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Factor of safety (FS) Relative frequency Histogram of FS Cases where FS<1 Sample mean=1.062 Sample Std=0.09135 Minimum=0.5094 Maximum=1.372

Histogram of γ (kN/m 3 ) Cases where LSF<0 Fitted distribution

Sample mean=19.11 Sample Std=4 Minimum=7.215 Maximum=31.05

0.1

0.08

0.06

0.04

Relative frequency

0.02

0

5

10

15

20

25

30

35

Unit weight of soil, γ (kN/m 3 )

(c) Unit weight of soil, γ (kN/m 3 )

(d) Safety factor

Fig. 3. Histogram of random samples generated for (a) C , (b) φ , (c) γ , and (d) FS.

45

Failure probability (%) Final value

40

35

30

25

Failure probability (%)

20

15

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Sample number

Fig. 4. The convergence of Monte Carlo analysis.

A total of 2606 samples were obtained with SF¡1. Dividing it by the total number of samples, failure probability is calculated as P f = 2606 / 10000 = 0 . 2606 = 26 . 06%.

3.3. Failure probability versus water table position

So far, the calculations were performed for FS < 1. By defining the failure as FS < f d , where f d is a decision variable, the probability of failure can be calculated for di ff erent values of the safety factor. The results are presented