Titel_SS06

The reliability analysis of the block diagrams illustrated in Figure 9.2 is quite simple provided
the failure events of the different failure modes are uncorrelated. This special case will be
considered in the following.

For series systems the probability of failure is then simply given as:

1

11(1(

n

FS

i

PP PF



 .  i))

Fi

)

(9.1)

where is the probability of system survival, i.e. the probability that none of the failure

modes in the series system fail and is the probability of failure for failure mode i. It

should be noted that a series system of failure modes, which are uncorrelated would not
necessarily fail in the failure mode with the larger failure probability. Due to the fact that the
failure modes are uncorrelated there is a probability that failure will take place in any of the
failure modes.

PS

PF()i

For parallel systems the probability of system failure is given by:

1

()

n

F

i

PP



. (9.2)

When the failure modes are correlated the simple expressions for the failure probability are no
longer valid.

If the failure events of the parallel or series systems may be described by linear safety margins
in terms of Normal distributed basic variables the corresponding systems failure probabilities
may be calculated by use of the multivariate Normal probability distribution function i.e.
Equation (9.1) (series systems) becomes:

PPFS n ,11(, (9.3)

and Equation (9.2) (parallel systems) becomes:

PFn, (,) (9.4)

where is the vector of reliability indexes for the individual failure modes and  is the

correlation coefficient matrix. Equations



(9.3)-(9.4) forms the basis for first order and second
order reliability analysis of systems as described in e.g. Madsen et al. (1986) and implemented
in several commercially available software packages.

So-called simple bounds on the failure probability may be established on the basis of simple
considerations.

For a series system in which all failure modes are fully correlated it is realised that the failure
probability is equal to the failure probability of the failure mode with the largest failure
probability, i.e. in this case a system where the weakest link may be clearly identified. As the
correlation between the failure modes will be somewhere between zero and one, the simple
bounds on the failure probability for a series system may thus be given as:

1 >?

1

max ( ) 1 (1 ( ))

n n

i iF i

i

 PF P PF



 . (9.5)