Titel_SS06

where the lower bound corresponds to the case of full correlation and the upper bound to the
case of zero correlation.

For a parallel system the same considerations apply leading to the observation that the upper
bound corresponds to the situation where all failure modes are fully correlated and the lower
bound to the situation where all failure modes are uncorrelated, i.e.:

1 >

1

() min ()

n n

iFi i

i

PF P  PF



. ? (9.6)

Finally for mixed systems, i.e. systems consisting of both series and parallel systems it is
straightforward to reduce these into either a series system or a parallel system in a sequential
manner using Equations (9.5)-(9.6) to reduce the sub-systems (parallel or series) into one
component.

Example 9.1– Successive reduction of systems using the simple bounds

Consider the mixed system illustrated in Figure 9.3 where the components represent the
failure modes of a structural system.

1

2

3

4

56

Figure 9.3: Mixed system for systems reliability analysis.

For illustrational purposes it is assumed that the probabilities of the failure modes 1-6 are as
given below and furthermore that it is unknown to what extent the failure modes are
correlated:
2
PF() ( ) ( )11012 4PF PF

5
PF() () ()110 356 PF PF


The mixed system as illustrated in Figure 9.3 may be successively reduced to a series system
as shown in Figure 9.4. In this reduced system, the failure probabilities of the components
123  and 456  >?

>

need to be determined. Considering first the element 1 and

assuming no correlation use of Equation

 23

(9.2) gives 0) 11^5.

For the component

P(1 23)(1 1 0 ) (1 1^220 ^9

456  ?

210

first the probability of the failure event of the series sub-

system 5 must be considered, which by application of Equation
P(5 6)

6

1

(9.1) is determined to

 (1 1 1 0 )^52  ^5. Thereafter there is for 456    >? by application of

Equation (9.2), P(4 > 5 6) 110?  O   ^252102107.