Application of the subset simulation approach to spatially varying soils - Risk and Reliability in Geotechnical Engineering - page 555

Finally, the failure probability P ( F ) is evaluated as follows:

m

∏

PP ( F ) = ( F ) PF F

j

( )

(15.17)

1 j − 1

j = 2

It should be emphasized that a normal PDF must be used herein as a target PDF P t .
However, a uniform PDF can be used as a proposal PDF P p .

15.5 eXaMPle aPPlICatIonS

The next sections are devoted to three application examples. The first example involves the
generation of a random field using K-L expansion. The second example involves the applica-
tion of SS methodology to a case where the uncertain parameters are modeled by random
variables. This example aims at computing the failure probability against bearing capacity
failure of a strip footing resting on a ( c , φ) soil and subjected to a service vertical load P s ,
where c and φ are considered as two random variables. Finally, the third example involves
the application of the SS approach to a case of a spatially varying soil. It aims at computing
the probability of failure against exceeding a tolerable vertical displacement of a strip foot-
ing resting on a spatially varying soil, where the soil Young's modulus was considered as a
random field.

15.5.1 example 1: generation of a random field by k-l expansion
This example aims at illustrating the practical implementation of some theoretical concepts
presented before for the generation of a random field. In this example, the soil Young's
modulus was considered as a 2D log-normal random field. The statistical parameters used
in the analysis are μ = 60 MPa, COV = 15%, and l ln x = l ln y = 1 m. A small number of K-L
terms ( M = 2) is considered for illustrative purposes. The half-widths in both the x- and y-
directions were taken as follows: a x = 7.5 m and a y = 3 m.
In order to generate a 2D random field, 1D random fields were first considered to calculate
the eigenvalues and eigenfunctions corresponding to the horizontal and vertical directions
according to Equations 15.3 through 15.7. They are shown in Table 15.1 . Notice that in this
table, the 1D eigenfunctions were computed based on the arbitrary values x = 0.5 m and
y = 3 m.
After the computation of the eigenvalues and eigenfunctions of the 1D random fields, the
product of all possible combinations was carried out according to Equations 15.8 and 15.9 .
The results of this process are shown in Table 15.2 .

Table 15.1 Eigenvalues and eigenfunctions of 1D random fields in the x and y directions, where
μ = 60 MPa, COV = 15%, l ln x = l ln y = 1 m, a x = 7.5 m, a y = 3 m, and M = 2
1D random field in the x-direction

1D random field in the y-direction

ω j x λ j x φ j x ( x = 05m . )

ω y λ y φ k y j k ( y = 3m )

1 0.18505 1.93378 0.34220 1 0.39740 1.72722 0.18802

2 0.37146 1.75749 0.06380 2 0.81861 1.19752 0.33382