ETL 1110-2-547
30 Sep 95
result by the product of the associated probability
B-9. Determining the Probability of Failure
concentrations (e.g., P+-- = P1+ 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.
E Y M ' P% % g X1% , X2%
M
M
% P%& g X1% , X2&
If the reliability index is assumed to be the number of
standard deviations by which the expected value of a
M
M
% P& % g X1& , X2%
% P&& g X1& , X2&
normally distributed performance function (e.g.,
ln (FS)) exceeds zero, than the probability of failure
can be calculated as:
the point estimates and 2N terms in the summation. To
obtain the expected value of the performance function,
Pr f ' R &$ ' R &z
the function g(X1,X2) is calculated 2N 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.
E[Y2]. The variance can then be obtained from the
identity
B-10. Overall System Reliability
Var Y ' E Y 2 & E Y
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
a. Series system. In a series system, the system
(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 pi 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:
R ' R1 R2 R3 ... Ri ... Rn
D
' 1 & p1 1 & p2 1 & p3 ... 1 & pi ... 1 & pn
Pi% j& ' Pi&j% ' Pi& Pj% &
4
D
Pi% j% ' Pi&j& ' Pi% Pj% %
b. Simple parallel system. In a parallel system,
4
the system will only perform unsatisfactorily if all
components perform unsatisfactorily. Thus, the
c. Monte Carlo simulation. The performance
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
R ' 1 & p1 p2 p3 ... pi ... pn
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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