result by the product of the associated probability

concentrations (e.g., *P*+-- = *P*1+ P2- P3- ) and summing

the terms. For example, two random variables result

Once the expected value and standard deviation of the

in four combinations of point estimates and four terms:

performance function have been determined using the

Taylor*s series or point estimate methods, the reli-

ability index can be calculated as previously described.

% *P*%& g X1% , *X*2&

If the reliability index is assumed to be the number of

standard deviations by which the expected value of a

% *P*& % g X1& , *X*2%

% *P*&& g X1& , *X*2&

normally distributed performance function (e.g.,

ln (*FS*)) exceeds zero, than the probability of failure

can be calculated as:

the point estimates and *2*N terms in the summation. To

obtain the expected value of the performance function,

the function *g(X*1,X2) is calculated *2*N times using all the

combinations and the exponent *M *is 1. To obtain the

where R(*-z*) is the cumulative distribution function of

standard deviation of the performance function, the

the standard normal distribution evaluated at *-z*, which

exponent *M *is taken as 2 and the squares of the

is widely tabulated and available as a built-in function

obtained results are weighted and summed to obtain

on modern microcomputer spreadsheet programs.

identity

2

Reliability indices for a number of components or a

and the standard deviation is the square root of the

number of modes of performance may be used to

variance.

estimate the overall reliability of an embankment.

There are two types of systems that bound the possible

(2) Correlated random variables. Correlation

cases, the series system and the parallel system.

between symmetrically distributed random variables is

treated by adjusting the probability concentrations

(*P * .... ). A detailed discussion is provided by

will perfrom unsatisfactorily if any one component

Rosenblueth (1975) and summarized by Harr (1987).

performs unsatisfactorily. If a system has *n *compo-

For certain geotechnical analyses involving lateral

nents in series, the probability of unsatisfactory

earth pressure, bearing capacity of shallow founda-

performance of the *ith *component is *p*i and its reli-

tions, and slope stability, often only two random

variables (*c *and N or tan N) need to considered as

or probability that all components will perform satis-

correlated. For two correlated random variables

factorily, is the product of the component reliabilities.

within a group of two or more, the product of their

concentrations is modified by adding a correlation

term:

D

' 1 & *p*1 1 & *p*2 1 & *p*3 ... 1 & *p*i ... 1 & *p*n

4

D

4

the system will only perform unsatisfactorily if all

components perform unsatisfactorily. Thus, the

reliability is unity minus the probability that all com-

function is evaluated for many possible values of the

ponents perform unsatisfactorily, or

random variables. A plot of the results will produce

an approximation of the probability distribution. Once

the probability distribution is determined in this

manner, the mean and standard deviation of the

distribution can be calculated.

B-10

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