from which it is seen that the uncertainty associated with the strength of ductile parallel
systems tends to approach zero when the number of components is large and when the
individual components are more or less identical.
In case that the individual components of the parallel system are behaving brittle at failure it
can still be shown under certain conditions that the strength of the system is Normal
distributed with mean value:
RS nr 0 (1 F rR( ) ) 0 (9.11)and standard deviation:
22
RRRSnrF r 00 ()(1F r()) 0 (9.12)
where is chosen as the value maximising the function r 0 rFr(1 R( ) ). In this case is also
evident that the basic feature that the coefficient of variation:
0
0()(1 ())
(1 ( ) )
RR
RFr Fr
COV
nFr(^0) (9.13)
approaches zero for sufficiently large number of components.
Equations (9.11)-(9.12) are often used for the probabilistic modelling of the strength of
parallel wire cables. Due to the fact that the elastic elongation of cables under normal
conditions e.g. for suspension bridges and cable stayed bridges is already very large under
normal loading conditions and that the development of plasticity in individual wires develops
over a relatively short length of the wire (3-5 times the diameter) the failure of the individual
wires under extreme load conditions will always be brittle in nature.
For the reliability analysis of structural systems a number of methods have been developed,
see e.g. Thoft-Christensen and Murotzu (1986) where two principally different methods are
described, i.e. the -unzipping method and the fundamental mechanism method. An in depth
description of these two methods will not be made in the following but rather the different
approaches will be illustrated by a simple example. The reader is referred to the text of Thoft-
Christensen and Murotzu (1986) for further details.
Example 9.2– System reliability analysis
Consider the simple beam illustrated in Figure 9.7 with the point load W acting at mid span.
The beam is assumed to fail only in bending and is furthermore assumed to behave ductile at
failure.
It is assumed that the plastic moment of the beam R is Normal distributed with parameters
RR300, 30
WW100, 20
. The load W is also Normal distributed with parameters
.