Question 7.4: The crest level of embankment over a reach is described by a......
The crest level of embankment over a reach is described by a normal distribution with mean 5 and standard deviation 0.5. This is often written as N(5,0.5). Monthly maximum water levels along the reach obey N(3,1). What is the probability of flooding?
Flooding occurs when water level > crest level. So the failure function can be written as
G = CL – WL .
The variables are normal and independent, so (from Equation 7.33)
\sigma ^{2}_{G}= \sigma ^{2}_{R}+ \sigma ^{2}_{S} (7.33)
\mu _{G} = 5 – 3 = 2,\sigma ^{2}_{G} = 0.5^{2} + 1^{2} = 1.25\Rightarrow \beta \frac{2}{\sqrt{1.25} } \left(by Equation 7.32\right)\beta = \frac{\mu_{G}}{\sigma _{G}} (7.32)
From Equation 7.34,
P_{F}=\Phi \left\{\frac{0-\mu _{G}}{\sigma _{G} } \right\}= \Phi \left\{\frac{\mu _{G}}{\sigma _{G} } \right\} =1- \Phi \left\{\frac{\mu _{G}}{\sigma _{G} } \right\} =\Phi \left(-β\right) (7.34)
P_{F}=\Phi \left\{\frac{0-2}{\sqrt{1.25} } \right\}=\Phi \left(-1.79\right) =0.037\approx 4%Thus, the probability of failure is approximately 4% per month.