grm rrwBA(x) BA 0.5 2.5 0.5r (9.17)FORM analysis of this limit state function yields a probability of failure in B conditional on
failure in A equal to PF,B AO1.47 10-3.
In the case where failure first develops at location B the system to be analysed is shown in
Figure 9.11.
A
R R
10
B
W
Figure 9.11: Beam structure with yield hinge formed at location B.
The limit state function is in this case given as:
grmrrAB(x) AB 2 3 5 w (9.18)which is seen to be identical to the limit state equation given in Equation (9.17). FORM
analysis of the limit state equation given in Equation (9.18) thus yields the same result,
namelyPF,A BO1.47 10-3.
By consideration of the block diagram in Figure 9.8 and Equations (9.5)-(9.6) the simple
bounds for the failure probability of the beam can now be derived. First the parallel systems
defined as ABA and BAB are considered and for which there is:
1.41 10^5 P(AB A) 9.58 10^3 (9.19)
6.71 10^7 P(B A B) 1.47 10 ^3 (9.20)Finally for a systems reliability analysis at level 2 (or in this case mechanism level) the series
system of the two parallel systems, i.e. >ABA BAB ? >? is considered for which the
following simple bounds are established:
1.48 10^5 9.58 10
PF
^3 (9.21)
By consideration of the bounds given for the system reliability in Equation (9.16) and
Equation (9.21) it is seen that the lower bound on the reliability of the beam as determined at
level 1 is equal to the upper bound in the level 2 analysis. Accepting a more developed failure
in the beam before the beam is considered to be in a state of failure not unexpectedly reduces
the failure probability.