Practical reliability analysis and design by Monte Carlo Simulation in spreadsheet - Risk and Reliability in Geotechnical Engineering - page 302

different conditional probabilities. According to the Theorem of Total Probability (e.g., Ang
and Tang, 2007), the failure probability P ( F ) is, therefore, expressed as

m

∑ ΩΩ

PF () = PF (| )( ) P (7. 6)

i i

i = 0

where P ( F |Ω i ) is the conditional failure probability given sampling in Ω i and P (Ω i ) is the
probability of the event Ω i . P ( F |Ω i ) is estimated as the ratio of the failure sample number in
Ω i over the total sample number in Ω i . P (Ω i ) is calculated as

P ()

()
(

Ω
Ω

=−
=− =…−
=

1 p

0 0

i i + 1

P p p , i 1 , m 1

(7.7 )

i 0 0

P Ω ) p

m

m 0

Note that Ω i , i = 0, 1, 2, …, m , are mutually exclusive and collectively exhaustive events,
that is, iP0 k ∩ Ω j ) = 0 for k ≠ j and ∑

Ω 1 When P ( F |Ω i ), P (Ω i ), and P ( F ) are
obtained, the conditional probability P (Ω i | F ) is calculated using the Bayes' Theorem:

m

P () = .

i = 0 i

PF PF

(| )( ) ΩΩ P

F

i i

(7. 8)

(| ) Ω =

i

P ()

Then, the conditional probability P ( B , D | F ) of a specific combination of B and D is given
using the Theorem of Total Probability as

m

∑ (

∩ ΩΩ

PBDF (, |) = PBDF ,| ) PF (| ) (7. 9)

i i

i = 0

where P (Ω i | F ) is estimated from Equation 7.8 ; and P ( B , D | F ∩ Ω i ) is the conditional failure
probability of a combination of B and D given sampling in Ω i , and it is expressed as the ratio
of the number ( n

Ω , D ) of failure samples in Ω i with a combination of B and D over the total
failure sample number ( n

i B

| ∩Ω Ω Ω
Using Equations 7.6 through 7.9 , P ( F ) and P ( B , D | F ) are calculated using simulation
samples from Subset Simulation. Subsequently, P ( F | B , D ) is obtained in accordance with
Equation 7.4. Since the failure is defined as FS uls < 1 or FS sls < 1 for ULS and SLS require-
ments, respectively, two sets of conditional probabilities P ( F | B , D ) (i.e., the respective fail-
ure probabilities p f ULS and p f SLS of ULS and SLS failures for given B and D combinations)
are calculated for the drilled shaft design. Finally, the feasible design values of B and D
are determined by comparing the P ( F | B , D ) with the target failure probability p T . This sec-
tion only provides the conceptual framework. More details of the approach are illustrated
through a drilled shaft design example in Section 7.6.

Ω ) in Ω i , that is, PBDF

(, ) n / n .

i , BD

i i

7.4 ProbabIlIStIC FaIlure analYSIS uSIng SubSet
S I M u l atI o in

MCS can be treated as a “black box” that takes samples of the uncertain parameters as input
and returns failure probability or other reliability analysis results (e.g., complementary cumu-
lative distribution function (CCDF) and PDF) as output. It does not provide information on