computers
& stmctwes
Vol.
45, No.
5/6, pp.
959-964,
1992
Printed
in Great
Britaiti.
004~7949B2
S5.00
+ 0.W
PerpamonpragLtd
SYSTEM RELIABILITY
USING TRIVARIATE
BIVARIATE INTEGRALS
K. ~MACHANDRAN
Department of Civil Engineering, Imperial College of science, Technology and
London SW7 2BU, U.K.
(Receiued 23 ~~re~~
1991)
AND
Medicine,
Abstract-Probabilistic
safety analysis of civil engineering structures (system reliability analysis) requires
the evaluation of multi-normal integrals and these multi-normal integrals can be estimated either from
upper and lower bounds on these probabilities or by first-order second-moment approximate methods or
by simulation methods. In this paper a new method based on conditional probabilities for the evaluation
of series and parallel structural systems reliabilities is presented. Examples are studied to compare the
accuracy of this new approach with other approximate methods.
INTRODUCI’ION
Reliability
theory
is now widely used in soil and
structural
engineering
to assess the probability
of
failure of earth slopes [l-3], propped cantilever [4,5],
offshore structures [6,7j and many other engineering
structures
[8-l
11. The technique
for evaluating
the
reliability
index
of a single
failure
mode
is well
established
and better statistical data are now being
used in these computations.
However, methods
for
calculating
the probability
of failure of a structural
system with more than three failure modes still lack
vigour,
and
occasionally
give unsafe
results.
The
calculation
of system reliability
usually requires the
evaluation
of a multi-normal
integral and the current
practice
is to use bounds
on the probability
as a
means
of
estimating
the
system
failure
prob-
ability [12-141 or to use an approximate
method for
the evaluation
of this multi-normal
integral
[ 151
or to
use improved
Monte Carlo methods [16]. Although
the available bounds [12, 13, 17] are reasonably
satis-
factory, there is a definite need for tinding an exact
or near-exact method for reliable estimation
of these
multi-normal
integrals.
In this paper, a near-exact
method
is presented
for the determination
of these
failure
probabilities
using
trivariate
and
bivariate
integrals.
PROBLEM DEACON
In reliability
analysis,
each failure mode is rep-
resented
by a continuous
function
(safety margin)
gi(x) of the statistical
variables x. Usually there are
a very large number
of possible failure modes in a
system; however, with a little experience, the number
of failure modes which have significant eikct on the
reliability of the system (predominant
modes) can be
substantially
reduced. The event g,(x) < 0 is referred
to as the ith failure event (mode) 4, and its prob-
ability of occurrence denoted by P(,F;,). It is assumed
that all P(Fj)
and the correlation
matrix p can be
obtained by suitable transformations
into an uncorre-
lated standard
normal
space n [17]. Then the event
1% > /%I,
i =
1, n is referred to as the event I$ and