Practical risk assessment for embankments, dams, and slopes - Risk and Reliability in Geotechnical Engineering - page 443

11.4 MatheMatICal rootS anD nuMerICal eStIMatIon
oF FragIlItY CurVeS

11.4.1 Introduction
Uncertainties in geotechnical problems include parameter, system, and loading uncertain-
ties. As an example, uplift pressure under a concrete dam, probably the most important
factor influencing sliding stability (Ruggeri et al., 2004; Westberg, 2010), is a loading effect
subjected to considerable uncertainty.
To calculate the probabilities of the outcomes from a single node in an event tree, two
elements are needed:

• Mathematical or numerical model that simulates the physical problem
• Reliability method to be applied to the model

The selection of a mathematical or a more advanced numerical model will depend on
the complexity of the problem analyzed and the quantity and quality of data available. For
instance, a slope stability problem can be analyzed with a limit equilibrium model, such
as Taylor, Morgersten-Price, Janbu, Spencer, or other. But it can also be analyzed using a
numerical model implemented in a finite difference or finite element code.
Several reliability methods can be used. These methods include first-order second moment
(FOSM) Taylor's method, point estimate method (PEM), advanced second moment (ASM)
Hasofer-Lind method and Monte Carlo method.
These methods, except Monte Carlo, typically use linear approximations of the limit
state function g*(x 1 ,x 2 ,…,x n ). In addition, instead of considering the full probability density
function of the random variables, only the mean value and standard deviation are taken into
account. The result obtained using these techniques is the reliability index, β, defined as the
number of standard deviations, σ g* , between the expected value of the state function, E [g*],
and the value that represents the system failure, (g*) failure = g*(x 1 ,x 2 ,…,x n ) = 0 ( Equation
11.5 ) . This value gives a relative measure of the reliability of the system. Higher values of β
will mean higher margins of safety for the structure. On the other hand, the method does
not directly provide a value of the probability of failure.

E [] () g ∗ − g ∗ E [] g ∗ − 0 E [] g ∗

failure

β = = = (11.5)

σ σ σ

g ∗ g ∗ g ∗

As x 1 ,x 2 ,…,x n are random variables, the function g*(x 1 ,x 2 ,…,x n ) will be a random vari-
able as well, with a certain probability distribution, unknown in most cases. To derive a
failure probability value from β, it is necessary to make an additional hypothesis on the
type of probability distribution of g*(x 1 ,x 2 ,…,x n ). Using a probability distribution function
that is fully defined based on a mean and standard deviation of g*(x 1 ,x 2 ,…,x n ), for example,
normal or log-normal, the probability of failure can be derived.
The selection of the more adequate reliability method or combination of methods for a
geotechnical problem depends on factors such as the quantity and quality of data available,
probability distributions of the random variables, and order of magnitude of the searched
probability. For linear or quasi-linear problems, with normal or quasi-normal random vari-
ables, FOSM, PEM, and ASM methods provide a good choice for the engineer. For nonlin-
ear problems with non-Gaussian random variables and expected low probabilities of failure,
the Monte Carlo method will render more accurate results.