Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 167

Table 3.13 Taylor Series reliability analysis of cantilever retaining wall
With all variables assigned their most likely
values

F = 1.40

Variable Values Factors of safety Δ F

Equivalent fluid unit
weight, γ eq
Most likely value plus σ

8.13 kN/m 3

F + = 1.224

6.01 kN/m 3

Most likely value minus σ F − = 1.634

− 0.410

Tangent of δ
Most likely value plus σ 0.55

F + = 1.539

0.45 0.279

Most likely value minus σ F − = 1.260

Backfill unit weight, γ bf
Most likely value plus σ

19.42 kN/m 3 F + = 1.434

18.28 kN/m 3 0.069

Most likely value minus σ F − = 1.365

deviation from its best estimate value. F 1 − is the factor of safety calculated with the
value of the first parameter decreased by one standard deviation.

In calculating F 1 + and F 1 − , the values of all of the other variables are kept at their most
likely values.

The values of Δ F 2 , and Δ F 3 , are calculated by varying the values of the second and
third variables (footing/sand friction angle, and backfill unit weight) by plus and minus
one standard deviation from their most likely values. The results of these calculations
are shown in Table 3.13 .

3. Substituting the values of Δ F into Equation 3.31, the value of the standard deviation of
the factor of safety (σ F ) is found to be 0.250, and the COV of the factor of safety ( V F ),
calculated using Equation 3.30 , is found to be 18%.

4. With both F M LV and V F known, the probability of failure and the reliability of the fac-
tor of safety can be determined. First, the reliability index is calculated using Equation
3.33 , assuming a normal distribution of the factor of safety.

F − 11400 . − 1

MLV

β = = = 160 . (3.33)

normal

σ 0 250 .

FS

The probability of failure can be calculated using the NORMSDIST function in Excel
based on the reliability index using Equation 3.34 .

P f ( =− 1 NORMSDIST ( β ) =− 1 NORMSDIST (. ) 160550 = . %

(3.34)

normal ) normal

The same analysis was used to compute the probability of failure by sliding on the clay,
and by bearing capacity failure in the clay. The standard deviation of the undrained shear
strength of the clay was estimated to be 24 kN/m 2 . The calculated probability of failure for
sliding in the clay was found to be 2.81%, and the calculated probability of failure for bear-
ing capacity failure was found to be 1.47%.
The results of the Taylor Series method assuming a normal distribution for the factor of
safety are compared to the results of the Monte Carlo method in Table 3.14. The results
show that the probabilities of failure computed using the Taylor Series method, assuming
a normal distribution for the factor of safety, differ from the values computed using the
Monte Carlo method. For the mode of failure involving sliding on the sand layer at the base