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Failures Resulting from Static Loading 253
Interference—General
In the previous segments, we employed interference theory to estimate reliability when the
distributions are both normal and when they are both lognormal. Sometimes, however, it
turns out that the strength has, say, a Weibull distribution while the stress is distributed
lognormally. In fact, stresses are quite likely to have a lognormal distribution, because the
multiplication of variates that are normally distributed produces a result that approaches
lognormal. What all this means is that we must expect to encounter interference problems
involving mixed distributions and we need a general method to handle the problem.
It is quite likely that we will use interference theory for problems involving distri-
butions other than strength and stress. For this reason we employ the subscript 1 to
designate the strength distribution and the subscript 2 to designate the stress distribu-
tion. Figure 5–32 shows these two distributions aligned so that a single cursor x can be
used to identify points on both distributions. We can now write
Probability that

stress is less = dp(σ < x) = dR = F 2 (x) dF 1 (x)
than strength
By substituting 1 − R 2 for F 2 and −dR 1 for dF 1 , we have

dR =−[1 − R 2 (x)] dR 1 (x)
The reliability for all possible locations of the cursor is obtained by integrating x
from −∞ to ∞; but this corresponds to an integration from 1 to 0 on the reliability R 1 .
Therefore
0

R =− [1 − R 2 (x)] dR 1 (x)
1
which can be written
1

R = 1 − R 2 dR 1 (5–46)
0

Figure 5–32 f (S)
1
(a) PDF of the strength dF (x) = f (x) dx
1 1
distribution; (b) PDF of the
load-induced stress
distribution.

S
dx
(a)

x
Cursor
f ( )
2

F (x)
2
R (x)
2

(b)