(probability of failure less than 0.00135). The reliability-based design optimization
problem, with the width w and thickness t of the beam as design variables that are
deterministic, can be formulated as
minimize A = wt
such that
p 0.00135 <0(68
based on probability of failure, or
minimize A = wt
such that
3 < 0(6-9)
based on safety index, where Pis the safety index, or
minimize A = wt
such that
(6-10)
1- P, <0
based on the probabilistic sufficiency factor. The reliability constraints are formulated in
the above three forms, which are equivalently in terms of safety. The details of the beam
design are given later in the paper.
In order to demonstrate the utility of the P4, for estimating the required weight for
correcting a safety deficiency, it is useful to see how the stresses and the displacements
depend on the weight (or cross sectional area) for this problem. If we have a given design
with dimensions wo and to and a P,f of Pero, which is smaller than one, we can make the
structure safer by scaling both w and t uniformly by a constant c
w = cw,, t = ct, (6-11)
It is easy to check from (6-6) and (6-7) that the stress and the displacement will
then change by a factor of c3, and the area by a factor of c Since the P,f is inversely