Question 7.6: For a particular structure, the allowable overtopping thresh......
For a particular structure, the allowable overtopping threshold Q_{0} is normally distributed with mean 10 and standard deviation 4 l/s/m-run, and the overtopping discharge under design storm conditions is normally distributed with mean 3 and standard deviation 3 l/s/m−run. Calculate the reliability index and the probability of failure, using the tabulated values of the cumulative distribution function Φ(z) provided in Appendix D.
For a linear reliability function G the probability of failure, P_{F}, may be calculated as P_{F} = Φ(−μ_{G}/σ_{G}) where
μ_{G} = μ_{R} – μ_{S} and \sigma ^{2}_{G} =\sigma ^{2}_{R} + \sigma ^{2}_{S}
So, from the information provided we have
μ_{G}=10-3=7 and \sigma _{G} ^{2}=16+9=25=>σ_{G} = 5 .
Therefore, the reliability index, β, = μ_{G}/σ_{G} = 7/5 = 1.4;
and the probability of failure is given by
P_{f} = 1 – Φ(β ) = 1 – 0.919 = 0.081.
The same methods can also be used to determine a threshold required to achieve a specified probability of failure when the loading is uncertain.