Using Probabilistic Sufficiency Factor to Estimate Additional Structural Weight to
Satisfy the Reliability Constraint
The following cantilever beam example (Figure 6-2) is taken from Wu et al. (2001)
to demonstrate the use of probabilistic sufficiency factor.
L=100"
Figure 6-2. Cantilever beam subject to vertical and lateral beading
There are two failure modes in the beam design problem. One failure mode is
yielding, which is most critical at the corner of the rectangular cross section at the fixed
end of the beam
600 600
gs (R, X, Y, w, t) =R r= R -( 2Y + X) (6-6)
wt2 w2
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
22-7
4L3 Y fX
gD (E, X, Y, w, t) = Do D = Do +I 67
Ew t t 2
where E is the elastic modulus. The random variables are defined in Table 6-1.
Table 6-1. Random variables in the beam design problem
Random variables X Y R E
Normal Normal
Normal Normal
Di stribution(40020) (961.5)
(500, 100) lb (1000, 100) lb 40,20 (96 .5)
\", '/"psi psi
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
Y
tX