Download Structural reliability for structural engineers evaluating

S14
G. C. HART ET AL.
Failure occurs if the capacity (C) is less than the demand (D).
Define the failure event as
C<D
(3)
F ¼CD
(4)
and the ‘safety margin’, F, as
Therefore, the failure event can also be expressed as {F < 0}.
Because the capacity (C) and the demand (D) are jointly normal random variables, the safety
margin, F, is also a normal random variable.
The expected (or mean) value of the safety margin, F, is
D
¼C
F
(5)
s2F ¼ s2C þ s2D 2rCD sC sD
(6)
and the variance of the safety margin, F, is
The standard deviation of F, sF, is the square root of the variance of F.
The probability of failure, pF, for the case in which C and D are jointly normal is given by
pF ¼ P½C < D ¼ P½F < 0 ¼ ΦðbÞ
(7)
where P[. . .] is the probability of the event defined inside the square brackets, Φ(. . .) denotes the
standard normal cumulative distribution function (of a normal random variable with zero mean and
unit standard deviation) and
b¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F
D
Þ= s2C þ s2D 2rCD sC sD
¼ ðC
sF
(8)
is referred to in the literature as the reliability (or safety) index.
As shown in Figure 1, as the reliability index increases, there is less probability that a failure occurs.
Look at Equation (8) closely because the reliability index (b) takes a different form when C and D
are jointly log-normal random variables.
Probability of Failure, Pr(F 0) (%)
100
15.9 %
10
6.7 %
2.3 %
1
0.62 %
0.1
0.14 %
1
1.5
2
2.5
3
Reliability Index ( )
Figure 1. Probability of failure versus reliability index.
Copyright © 2012 John Wiley & Sons, Ltd.
Struct. Design Tall Spec. Build. 21, S12–S30 (2012)
DOI: 10.1002/tal