Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 145

of failure” is used. To avoid the possibility that the true significance of an event may be
obscured by use of the term “probability of failure,” it is preferable to be more specific, and
to use descriptions such as “probability of retaining wall sliding,” or “probability of slope
instability,” and to describe the likely consequences of the event together with the probabil-
ity of its occurrence.

3.4.2 assumed distribution of the factor of safety
The Taylor Series and Point Estimate methods (PEMs) for estimating the probability of
failure require an assumption regarding the distribution of the factor of safety. Most often
it is assumed that the distribution is either normal or lognormal. It is usual practice and
usually conservative to assume that the factor of safety is normally distributed (Baecher and
Christian 2003, Filz and Navin 2006).
Table 3.3 shows probabilities that the factor of safety may be smaller than 1.0 based on a
normal distribution of the factor of safety. Table 3.4 shows the same information but assum-
ing lognormal distribution of the factor of safety. Table 3.5 indicates which assumption
results in a larger value of Pf; f ; where the cells in Table 3.5 are shaded, assuming a normal
distribution results in a greater value of Pf f ; where the table is not shaded, the normal Pf f is
smaller than that the lognormal Pf. f . Table 3.5 shows that it is conservative in most cases to
assume a normal distribution for the factor of safety. Only for low factors of safety and high
coefficients of variation will the lognormal distribution of the factor of safety be higher.

3.5 MethoDS oF eStIMatIng StanDarD DeVIatIonS

An essential element of the art of geotechnical engineering is the ability to estimate reason-
able values of parameters, using meager data plus experience, or using correlations with
results of in situ and index tests. In order to be able to estimate Pf, f , it is necessary first to esti-
mate the standard deviations of the parameters involved in computing the factor of safety.
Standard deviations can be estimated using the same types of judgment and experience used
to estimate average values of parameters.
Depending on the amount of data available, various methods can be used to estimate
standard deviations. Four methods that are applicable to various situations are described in
the following sections.

3.5.1 Computation from data
When sufficient data are available, the formula definition of σ can be used to calculate its
value:

∑ xx
n

n

( )

2

−

(3.12)

i

σ=

i 1

− 1

in which σ is the sample standard deviation, x i the i th value of the parameter ( x ), x the aver-
age value of the parameter x , and n the number of values of x , or the size of the sample.
Most scientific calculators, as well as spreadsheets, have routines for calculating standard
deviation using Equation 3.12 .