variable in an analysis can be modeled as a

random variable as most properties and

parameters have some inherent variability and

f(c)

f( φ )

uncertainty. However, a few specific random

variables will usually dominate the analysis.

Including additional random variables may

φ

c

unnecessarily increase computational effort

without significantly improving results. When

in doubt, a few analyses with and without

certain random variables will quickly illustrate

βσlnFS

E[lnFS]

which are significant, as will the examination

of variance terms in a Taylor's series analysis.

f(lnFS)

For levee analysis, significant random

variables typically include material strengths,

soil permeability or permeability ratio, and

thickness of top stratum. Material properties

such as soil density may be significant, but

where strength and density both appear in an

ln(FS)

analysis, strength may dominate. An example

of a variable that can be represented

deterministically (nonrandom) is the density of

Figure B-1. The capacity-demand model

water.

the factor of safety, and the limit state is taken as the

A performance function and limit state are

condition ln *(FS) *= 0. The probability of failure is

identified.

then the shaded area corresponding to the condition ln

The expected value and standard deviation of

the performance function are calculated. In

obtained using standard statistical tables.

concept, this involves integrating the perfor-

mance function over the probability density

functions of the random variables. In prac-

tice, approximate values are obtained using

states can be defined using other measures, such as the

the expected value, standard deviation, and

exit gradient for seepage.

correlation coefficients of the random vari-

ables in the Taylor's series method or the

point estimate method.

reliability index is a *probability per structure*; it has no

The reliability index $ is calculated from the

time-frequency basis. Once a structure is constructed

or loaded as modeled, it either performs satisfactorily

expected and standard deviation of the perfor-

or not. Nevertheless, the $ value calculated for an

mance function. The reliability index is a

existing structure provides a rational comparative

measure of the distance between the expected

measure.

value of ln *(C/D) *or ln *(FS) *and the limit

state.

If a probability of failure value is desired, a

distribution is assumed and *Pr(f) *calculated.

As suggested by Figure B-1 for slope stability, a

reliability analysis includes the following steps:

Important variables considered to have

in the analysis and some significant uncertainty are

sufficient inherent uncertainty are taken as

taken as *random variables. *Instead of having precise

random variables and characterized by their

single values, random variables assume a range of

expected values, standard deviations, and

values in accordance with a *probability density*

correlation coefficients. In concept, every

B-3

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