Page 410 - Offshore Electrical Engineering Manual

Fault Trees 397

This may be proved for n events (see Reference 1) as follows:
P (a ⋅ b … n) = P (a) P (b) … P (n) for AND gates

P (a + b … n) = P (a) + P (b) +… P (n) for OR gates
Note that this is only true for independent events
Evaluation Using Event Failure Rates
For any number of events with constant failure rates input to an OR gate, it can be proved
(see Reference 1) that the output has a constant failure rate which is the sum of the failure
rates of the inputs. For any number of events with constant failure rates input to an AND
gate, it can be proved (see Reference 1) that the output failure rate after a given time t will
be a function of t. If each of the events is identical, as would be the case with the failure
rates for a number of generators in a system where each is capable of maintaining the full
system load, then without maintenance the output failure rate would tend to approach the
single unit failure rate after a certain number of hours (see Fig. 8.1.6). In a real situation

FIGURE 8.1.6
Graph of system failure rate against unit numbers, without maintenance.