600 600
gs (R, X, Y, w, t) =R r= R -( Y + X) (7-4)
wt2 w-t
where R is the yield strength, X and Y are the independent horizontal and vertical loads.
Another failure mode is the tip deflection exceeding the allowable displacement, Do
4L3 Y X
gD (E,X, Y, w,t) = D, D = D, +wjl j (7-5)
where E is the elastic modulus. The random variables are defined in Table 7-1.
Table 7-1. Random variables in the beam design problem
Random
X Y R E
variables
Normal Normal
Normal Normal
Di stributi on (40000,2000) (29E6, 1 .45E6)
(500, 100) lb (1000, 100) lb s i
The cross sectional area is minimized subject to two reliability constraints, which
require the safety indices for strength and deflection constraints to be larger than three
(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(76
based on probability of failure, or
minimize A = wt
such that
1 -P,, < 0
(7-7)
based on the probabilistic sufficiency factor. The reliability constraints are formulated in
the above forms, which are equivalently in terms of safety. The details of the beam
design are presented in Chapter 6.