Application of the subset simulation approach to spatially varying soils - Risk and Reliability in Geotechnical Engineering - page 557

of terms. For instance, when l ln x = l ln y = 1 m, the error is equal to 45% when M = 50 and it
decreases to 12% when M = 600.
A computer program was developed that allows one to compute the values of a random
field (for a given realization) at different points of the soil mass. These points (as will be seen
in Example 3) are located on the centers of gravity of the different elements of a determin-
istic mesh. The obtained values of the random field (for a given realization) will be used to
compute the system response (displacement at a given point of the structure, ultimate load
of a footing, safety factor, etc.).

15.5.2 example 2: Computation of the failure probability
by SS approach in the case of random variables
In order to illustrate the algorithm of SS methodology in a simple way, a numerical example
is provided herein. In this example, the SS approach was used to calculate the failure prob-
ability P f against bearing capacity failure of a strip footing of breadth B. The footing rests
on a ( c , φ) soil and it is subjected to a service vertical load P s . The soil cohesion c and the soil
angle of internal friction φ were considered as random variables. The following formula was
used for the computation of the ultimate bearing capacity:

γ B N N N

q = + +

u γ c q

2

(15.18)

in which

N = 2 ( − 1 )tan ϕ

q

γ

(15.19a)

π ϕ

π tan ϕ 2

Ne

q =

⋅ tan +

42

(15.19b)

N − 1

tanϕ

q

N =

(15.19c)

c

where N γ , N q , and N c are the bearing capacity factors due to soil weight, surcharge loading,
and cohesion, respectively. These coefficients are functions of the soil friction angle. On the
other hand, γ is the soil unit weight and q is the surcharge loading. The performance func-
tion used in the analysis is

P
P

u
s

G =−1 (15.20)

where P u is the ultimate footing load and P s is the footing applied load. As mentioned previ-
ously, only the soil cohesion and friction angle were considered as random variables. All the
other parameters were considered as deterministic. These parameters are given in Table 15.4 .
In this example, the intermediate failure probability p 0 of a given level j ( j = 1, 2, …,
m − 1) was arbitrarily chosen equal to 0.2. A small number of samples per level ( N s = 10
samples) were used to facilitate the illustration.
Table 15.5 presents (i) the values of c and φ of each sample for the successive levels, (ii) the
corresponding values of the performance function, and (iii) the values of the failure thresh-
olds C j for the different levels. Notice that only the first two levels and the last level for which