Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 162

Table 3.9 First stage of the Simplified Hasofer Lind method

Assumed distribution
(N or LN)

Capacity or
demand? (C or D?) B 1.0 1.5 1.28

8.1 8.7 8.4 N D

γ eq

18.3 18.0 18.1 N C γ bf

Tan δ 0.45 0.43 0.44 N C

FS 1.1 0.94 1.0

where Equation 3.13 is for a normal distribution of the factor of safety and Equation
3.25 is for a lognormal distribution of the factor of safety.

4. Stage two of the method is to calculate the “gradient” of the performance function.
This is simply the change in the factor of safety for a given change in the variable. The
value of each variable is changed by 10% (Filz and Navin 2006). Variables related to
capacity are increased by 10% from the final values calculated in step 3, and variables
related to demand are reduced by 10%. For example, γ eq at the end of step 3 has a value
of 8.4 kN/m 3 . γ eq is related to demand, and therefore is decreased by 10%, to 7.6 kN/m 3 ,
while the other variables remain at the values shown in Table 3.10. Similarly, the val-
ues of γ bf and tanδ, both related to capacity are increased (one at a time) by 10%. The
gradient is calculated as shown in Equations 3.26 and 3.27.

∆
∆

F
x

dg

=

(3.26)

dx

′ i

i

x − µ

x

′ =

i xi

(3.27)

i

σ

xi

5. Stage 3 is similar to stage one where trial values of the reliability index are used until
the calculated factor of safety is equal to one. However, in this stage the variables are
calculated using the gradients, the type of distribution of the variable, the standard
deviation of the variable, and the trial value of the reliability index. The first value of
the reliability index to use is the value found at the end of step 3 ( Table 3.9 ) . Then, the
values of the variables are calculated using the value of α calculated from Equation
3.28 through 3.30. If the factor of safety is greater than one, a higher value of the reli-
ability index should be used. A lower value of the reliability index should be used if the
calculated factor of safety is less than one. This iterative process is continued until a
factor of safety equal to one is calculated ( Table 3.11 ).

Table 3.10 Stages 2 and 3 of the Simplified Hasofer Lind
method

Gradient
= dg/dx ′ α

7.6 8.4 8.4 0.13 0.78 γ eq

18.1 18.1 19.9 0.03 0.14

γ bf

0.44 0.48 0.44 0.11 0.64

Tan δ

FS 1.1 1.1 1.1