Application of the subset simulation approach to spatially varying soils - Risk and Reliability in Geotechnical Engineering - page 564

To overcome this inconvenience, Santoso et al. (2011) proposed to modify the classical M-H
algorithm as follows:

a. A candidate realization ŝ is generated using the proposal PDF ( P p ). The candidate real-
ization ŝ is centered at the current realization s k .

b. Calculate the ratio r 1 = P t ( ŝ )/ P t ( s k ) and the ratio r 2 = P p ( s k | ŝ )/ P p ( ŝ | s k ), and then compute
the value r = r 1 r 2 .

c. If r ≥ 1, set s k +1 = ŝ ; otherwise, another candidate realization is generated. Candidate
realizations are generated randomly until the condition r ≥ 1 is satisfied.

d. Using the deterministic model, evaluate the value of the performance function G( s k +1 )
of the candidate realization that satisfies the condition r ≥ 1. If G ( s k +1 ) < C j (i.e., s k +1 is
located in the failure region F j ), one continues to retain the realization s k +1 obtained in
step c; otherwise, reject ŝ and set s k +1 = s k (i.e., the current realization s k is repeated).

These modifications reduce the repeated realizations and allow one to avoid the computa-
tion of the system response of the rejected realizations. This becomes of great importance
when the time cost for the computation of the system response is expensive (i.e., for the finite
element or finite difference models).

lISt oF SYMbolS

x horizontal coordinate

y vertical coordinate

μ mean value of a random variable or random field

σ standard deviation of a random variable or random field

μ ln mean value of an underlying normal random variable or random field

σ ln standard deviation of an underlying normal random variable or random field

l x horizontal autocorrelation length of a random field

l y vertical autocorrelation length of a random field

l lln x horizontal autocorrelation length of an underlying normal random field

l lln y vertical autocorrelation length of an underlying normal random field

M number of terms in K-L expansion

λ eigenvalue of the covariance function

ϕ eigenfunction of the covariance function

ξ vector of standard normal uncorrelated random variables

p 0 intermediate conditional failure probability

N s number of simulations in each level of subset simulation approach

b footing width

c soil cohesion

φ angle of internal friction of the soil

γ soil unit weight

N c , N q , bearing capacity factors

and N γ
q u ultimate bearing capacity

q s footing vertical pressure

m number of levels of subset simulation approach

P t target probability density function

P p proposal probability density function

P e probability of exceeding a tolerable settlement