Prob(s < 1) < P, (6-4)
where Pr is the required probability of failure. Birger' s probabilistic sufficiency factor Pyf
is the solution to
Prob(s < Pf) = P, (6-5)
It is the safety factor that is violated with the required probability Pr.
Figure 6-1 shows the probability density of the safety factor for a given design. The
area under the curve left to s=1 represents the probability that s<1, hence it is equal to
actual probability of failure. The shaded area in the figure represents the target
probability of failure, Pt. For this example, since it is the area left of the line s=0.8, Pyf=
0.8. The value of 0.8 indicates that the target probability will be achieved if we reduced
the response by 20 % or increased the capacity by 25% (1/0.8-1). For many problems this
provides sufficient information for a designer to estimate the additional structural weight.
For example, raising the safety factor from 0.8 to 1 of a stress-dominated linear problem
typically requires a weight increase of about 20% of the weight of the overstressed
components .
0.6 0.8 1.0 1.2 1.4 1.6 sy/g
Figure 6-1. Probability density of safety factor. The area under the curve left to s=1
measures the actual probability of failure, while the shaded area is equal to
the target probability of failure indicating that probabilistic sufficiency factor
= 0.8