Environmental Engineering Reference
In-Depth Information
Getting more into the details of the previous chart ( Figure 11.13 ) , and the links with the
concepts previously introduced in this chapter, special attention should be devoted to the
following issues:
Step 1. Define the mathematical or numerical model that simulates the physical prob-
lem, as defined in Section 11.4.1. The selection of a mathematical or a more advanced
numerical model will depend on the complexity of the problem analyzed and the quan-
tity and quality of data available.
Step 2. Define the loading range that will make sense to estimate the fragility curve.
This range will delimit the x-axis of the fragility curve. Furthermore, the number
of loading cases analyzed in this range should be prechosen. The more the num-
ber of cases, the more accurate the results will be, but more time will be needed for
computations.
Step 3. As explained in Section 11.4.1, the engineer has to assess which variables will
be considered as subjected to none or very low uncertainty, and which variables have
necessarily to be treated as random.
Step 4. At least two different distributions should be defined for each variable: one
for natural uncertainty and one for epistemic uncertainty. Mean values, standard
deviations, and probability distribution should be estimated, based on available data.
Typical probability distributions are the uniform, normal, log-normal, triangular, and
beta distributions. These distributions are used partly because they fit the methods in
use and partly because we know how to solve the mathematics if we use them.
Step 5. Select the reliability method that will be used in the model to estimate failure
probability as explained in Section 11.4.1. Some examples of these methods are FOSM
Taylor's method, PEM, ASM Hasofer-Lind method, and Monte Carlo method. Monte
Carlo method will produce more accurate results, although computations will require
more time.
Step 6. For each loading case, compute the failure probability using the selected reli-
ability method and the probability distribution defined for the natural uncertainty.
The number of values of the random variables used to compute the failure probability
will depend on the reliability method chosen. For each group of variables sampled,
a full computation of the calculation/numerical model is required. When the failure
probability is represented versus the loading range, the fragility curve capturing natu-
ral uncertainty is obtained.
Step 7. First, groups of random variables are selected in the epistemic uncertainty dis-
tributions following the chosen reliability method. For each group of random variables
sampled, a fragility curve is estimated using the procedure explained in the previous
step but using these sampled values of random variables as new “averages” for the nat-
ural uncertainty distribution (which is assumed not to change). This family of fragility
curves will then capture both the epistemic and natural uncertainty of the structure.
Step 8. Check the outcomes and perform sensitivity analysis on any of the decisions
previously taken. This last one is a crucial step as the engineer should never get lost
in any mathematical approaches that may end up not representing sound engineering
judgment.
11.4.4 example of fragility analysis for stability
failure mode of an earth dam
The dam to be analyzed is a homogeneous earth fill dam. Its upstream slope is 23.5° and its
downstream slope is 28°, being the total height 16 m ( Figure 11.14 ) .
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