Practical risk assessment for embankments, dams, and slopes - Risk and Reliability in Geotechnical Engineering - page 444

The trade-off between accuracy and time and cost of the analysis has to be carefully con-
sidered. For instance, the use of a finite element model with the Monte Carlo method will
only be justified when all the possible alternatives have proven to be insufficient to reach the
objectives of the analysis.
As long as the calculated probability corresponds to a situation where outcomes of previ-
ous nodes have already happened, the calculated probability is a conditional probability. If
the conditional probability of failure is calculated mapping all feasible load values, then a
fragility curve is obtained.
It is important to note that the event tree approach allows for a lot of flexibility when
considering the associated models to each node. Given a complex problem, the engineer can
(a) Decompose the process to a relatively high number of nodes, or
(b) Use an event tree with just a few nodes
In case (a), each node will represent a simpler process, which would be easier to simulate
by a mathematical or numerical model. In case (b), each node will represent a group of
interdependent processes, and the building of the associated simulation model will require
more effort.
The engineer has to evaluate the type of model that best suits his requirements, for exam-
ple, the choice between a 2D or a 3D model. In geotechnical problems where interfaces are
present, another decision is to choose between continuous models with previously defined
interfaces or resort to fracture mechanics models. If water is present, the interconnection
with hydromechanic phenomena has to be assessed. The flow regime is another issue that
has to be considered: will the flow be considered along the discontinuities or in the whole
domain? If numerical models are used, a decision has to be made on which one is the more
adequate constitutive model regarding the nature of the analyzed problem and the require-
ments of the study: elastic model, plasticity models, pore-elastic models, linear elastic frac-
ture models, or nonlinear fracture models.
In addition, the engineer has to assess which variables will be considered as deterministic
variables, subjected to none or very low uncertainty, and which variables may have values
not known with precision and therefore are subjected to uncertainty (either natural or epis-
temic). Once this separation between deterministic and random variables is done, random
variables should be characterized in a probabilistic manner. An estimation of mean values
and standard deviations and a hypothesis on their probability distribution has to be made
based on available data. Typical probability distributions are the uniform, normal, log-
normal, triangular, and beta distributions. An important decision regarding the selection of
probability distributions is related to the upper and lower truncation of unbounded distribu-
tions that theoretically extend in the [−∞, +∞] interval. In particular, imposing a lower limit
on resistance variables has a strong influence on the probability of failure.
In this context, FOSM techniques are suitable in the first stages of the analysis to identify
the key variables, which contribute to the variance of the state function.
Monte Carlo analysis is also a broadly used method to compute failure probability. This anal-
ysis is made generating n groups of values for the random variables, according to their proba-
bilistic distributions. The structural behavior is simulated for each group of values, obtaining a
number of situations nf f where the structure fails. Failure probability is computed dividing the
number of failures (n f ) by the number of groups generated (n). The higher the number of groups
of values, the more accurate the failure probability results. In order to characterize properly the
structures with a low failure probability, a high number of simulations are required.
The two main types of uncertainty already introduced in the chapter can be addressed
using a two-loop Monte Carlo analysis (Baraldi et al., 2008).