Practical risk assessment for embankments, dams, and slopes - Risk and Reliability in Geotechnical Engineering - page 445

It has to be taken into account that Monte Carlo techniques require a large number of
simulations and can make its practical application unfeasible, in particular, if combined
with complex numerical finite-element models. The concept of limit state surface can be
used to avoid this problem. The limit state or response surface, which in the general case is
a hyper-surface*, sets the boundary between the safe and failure domains. This limit state
surface can be approximated using the numerical model to calculate a few points on it.
Then, a statistical adjustment can be performed to obtain an estimation of the surface posi-
tion and shape from the calculated points. Once the limit state surface is approximated, it is
no longer necessary to use the numerical model to calculate the probability of failure, since
it is enough to generate samples from the probability distributions of the random variables
in a Monte Carlo fashion, verifying how many of them lay in the failure domain.

11.4.2 Conditional probability of failure versus FS
The end nodes of an event tree typically present two possible outcomes: failure/no failure.
When the system reaches the final node condition, all previous outcomes from all the nodes
along the analyzed branch have already happened. At this point, it is interesting to evaluate
the obtained probability of failure against the FS.
In normal practice, factors of safety are calculated and compared with reference values
to assess the safety condition of a structure, thus considering the structure as “safe” if
the calculated FS is higher than the reference value or “unsafe” otherwise. As it has been
pointed out elsewhere (Hoek, 2007; Smith, 2003) if uncertainties are incorporated in the
analysis, the FS considered as a mathematical construct, becomes another random variable,
so the probability of failure is the probability P(FS < 1). According to this definition, it may
be expected that a higher FS will imply a lower probability of failure, but it has been shown
(Silva et al., 2008; Smith, 2002) that higher factors of safety do not always correspond with
lower probabilities of failure due to the uncertainties involved in the analysis.
Ching (2009) has demonstrated with his theorem of equivalence that under some cir-
cumstances, equivalence between FS and probability of failure can exist. If Z equals the
uncertain variables, θ the design variables, and D is the allowable design region in the θ
space, the limit state function is denoted as g[Z,θ]. The nominal limit state function g n (θ) is
a positive function of θ. As an example, g n (θ) can be defined as g[Z,θ] with Z fixed at certain
chosen nominal values, that is, their mean values. The system is in failure when g[Z,θ] < 1.
According to this, the FS is defined in Equation 11.6.

g

n (θ

(11.6)

≥ 1

FS

The probability of failure can be defined as in Equation 11.7.

∫

∗

PgZ ([ ,] θ < 1 θ ) = p ZI gZ ( θ )([,] ⋅ θ < 1 ) dZ ≤ P f (11.7)

where P f ∗ is the desirable probability of failure and I(g[Z,θ] < 1) is a function that equals 1
when g[Z,θ] < 1 and 0 otherwise. Let the normalized limit state function be defined as the
limit state function divided by a nominal limit state function. If the probability distribu-
tion of the normalized limit state function is invariant over the design region, then pairs of

* An n-dimensional surface in a space of dimension n + 1, which represents the set of solutions to a single equation.