Question 7.5: Derive an expression for the probability of failure of a roc......
Derive an expression for the probability of failure of a rock armour revetment. Use this to calculate the probability of failure for the specific conditions given below with the MVA method.
We take as the response function Van der Meer’s (1988) a,b,c formula for armour stability under deep water plunging waves, (see Section 9.4.3 for further details). For a given damage level, S_{d} , the formula provides an estimate of the required nominal median stone size D_{n50}. The failure function may be written as
G=R-S=a P^b S_d^{0.2} \Delta D_{n 50} \sqrt{\cot (\theta)}\left(\frac{g}{2 \pi}\right)^{-1 / 4}-H^{0.75} T_m^{0.5} N^{0.1}where H_{s} is the significant wave height; T_{m} is the mean wave period; Δ is the relative mass density; (ρ_{a} − ρ_{s})/ρ_{s},ρ_{s} is the density of seawater; ρ_{a} is the density of rock; θ is the angle of the face of the rubble mound; P is a permeability parameter; and N the number of waves, a = 6.2 and b = 0.18. We take a, b, Δ, H_{s} , T_{m} and D_{n50} as random variables. The first step is to calculate the partial derivatives. Performing this analytically gives:
\frac{\partial G}{\partial a} =R/a
\frac{\partial G}{\partial H_{s} } =-0.75S/H_{s}
\frac{\partial G}{\partial T_{m} } =-0.5S/T_{m}
\frac{\partial G}{\partial D_{n50} } =R/D_{n50}
\frac{\partial G}{\partial b} =R\ln \left(p\right)
\frac{\partial G}{\partial \Delta } =R/\Delta
where R and S have been used to denote the terms describing the strength and load in the reliability function. With a = N(6.2,0.62), b = N(0.18,0.02), Δ = N(1.59,0.13), H_{s}= N(3,0.3), D_{n50} = N(1.30,0.03), T_{m} = N(6,2) S_{d}= 8, cot(θ) = 2.0, P = 0.1 and the number of waves equal to 2000, the probability of failure may be estimated. Table 7.4 summarises the results of the MVA calculations.
\begin{aligned} & \mu_G=\mu_R-\mu_s=4.29 \\ & \sigma_{G}^2=\sum_{i=1}^6 \alpha_i^2=9.86 \end{aligned}so
\sigma _{G} = 3 1. 4
Therefore, the reliability index is
\beta = \frac{\mu _{G} }{\sigma _{G} }=1.37and the probability of failure is
Φ(−β) = 0.085.
In this relatively simple case we have calculated the probability of failure taking into account uncertainty in the parameter values of an empirical equation, construction materials and the random nature of waves. In passing, it is interesting to note that the result is sensitive to the rock size and density, not just the wave conditions. It may also be noted that the rock size of 1.3 m was originally determined using the standard deterministic method using the no damage value of S_{d} = 2. In estimating the failure probability at S_{d} = 8, corresponding to complete failure of the primary armour layer, it can be seen that it is quite low (e.g., about 5%).
Reliability methods can also be used to take into account uncertainties in threshold values and estimated loadings, as in the following example.
| Table 7.4 Application of MVA | ||||
| Variable | Mean | Std. dev | Partial derivative | α^{2 }_{i} |
| a | 6.2 | 0.62 | 2.63 | 2.64 |
| b | 0.18 | 0.02 | −37.38 | 0.56 |
| Δ | 1.59 | 0.13 | 10.2 | 1.76 |
| D_{n50} | 1.30 | 0.03 | 12.5 | 0.14 |
| T_{m} | 6.0 | 2.0 | −1.00 | 3.96 |
| H_{s} | 3.0 | 0.30 | −3.0 | 0.80 |