## 4 Applications
### Example 2: Effectiveness of Rock Fall Protection in the Propagation Zone
Based on Equations (7), (8), (10) and (11), the reduced performance capacity of the embankment can then be evaluated as:  
\[\rm E_{eff}=E_{opt}\cdot\prod_{j}e_{0j}=E_{opt}=2000\ kJ\]  
\[\mathrm{E=1700\ kJ}\qquad\mathrm{P_{r}=0.92}\quad\rightarrow\quad\mathrm{T=1/ (P_{f}\cdot P_{r})=1/(1/20\cdot 0.92)=22\ years}\]0  
\[\mathrm{E=1700\ kJ}\qquad\mathrm{P_{r}=0.92}\quad\rightarrow\quad\mathrm{T=1/ (P_{f}\cdot P_{r})=1/(1/20\cdot 0.92)=22\ years}\]1  
\[\mathrm{E=1700\ kJ}\qquad\mathrm{P_{r}=0.92}\quad\rightarrow\quad\mathrm{T=1/ (P_{f}\cdot P_{r})=1/(1/20\cdot 0.92)=22\ years}\]2  
In spite of quite an important reduction of energy absorption capacity, the current conditions of the embankment still allow it to withstand the potential impacts of the event considered. On the other hand, after the reduction of the effective height, it results: \(\rm d>D_{red}\). It must be checked therefore, according to Equation (19), what increase in frequency of reach this decrease in effective height produces. Under the hypothesis that \(\rm D_{red}=3.2\) implies an increase in frequency of 33%, based, e.g., on the analysis of trajectory simulation results at \(x_{\rm P}\), the reduced value of the return period \(\rm T_{red}\) is given by:  
\[\mathrm{E=1700\ kJ}\qquad\mathrm{P_{r}=0.92}\quad\rightarrow\quad\mathrm{T=1/ (P_{f}\cdot P_{r})=1/(1/20\cdot 0.92)=22\ years}\]3By applying the same reduction at the locations \(x_{\rm H}\) and \(x_{\rm R}\), the hazard is equal to: (c.1) location \(x_{\rm P}\):  
\[\mathrm{E=1700\ kJ}\qquad\mathrm{P_{r}=0.92}\quad\rightarrow\quad\mathrm{T=1/ (P_{f}\cdot P_{r})=1/(1/20\cdot 0.92)=22\ years}\]4 (c.2) location \(x_{\rm H}\):  
\[\mathrm{E=1700\ kJ}\qquad\mathrm{P_{r}=0.92}\quad\rightarrow\quad\mathrm{T=1/ (P_{f}\cdot P_{r})=1/(1/20\cdot 0.92)=22\ years}\]5 (c.3) location \(x_{\rm R}\):  
\[\mathrm{E=1700\ kJ}\qquad\mathrm{P_{r}=0.92}\quad\rightarrow\quad\mathrm{T=1/ (P_{f}\cdot P_{r})=1/(1/20\cdot 0.92)=22\ years}\]6  
where \(\rm E^{*}\), \(\rm P_{\rm r}{}^{*}\) and \(\rm T^{*}\) are, like in case b), the values of frequency of reach and return period associated to the effect of the embankment. Again, it was put \(\rm E^{*}\approx 0\) at all locations, including \(x_{\rm H}\) and \(x_{\rm R}\). Despite the reduction in height and, consequently, in the return period value, Equation (19) is still satisfied, as \(\rm E<E_{\rm red}\) and \(\rm T_{\rm red}>T_{\rm m}=300\) years (at \(x_{\rm P}\) and beyond); therefore, the embankment would still be able to mitigate the existing threat (Figure 5C), as the obtained return period values demonstrate no relevance for hazard assessment and zoning (i.e., high return period value compared to the scale of life time of all elements potentially endangered: persons, buildings, roads/other engineering works and, up to some extent, environment). The intensity-frequency diagram in Figure 5C allows us to visualise how the hazard is reduced at the locations of interest in comparison to the absence of embankment; at the same time, it also shows how the hazard slightly increases (to still negligible levels, though) due to the reduced performance capacity of the measure, when compared to the situation of the embankment in optimal working order (gray arrows and hollow circles). Even though in presence of an embankment with reduced performance capacity it still results: \(\rm E<E_{\rm red}\) and \(\rm T_{\rm red}>T_{\rm m}=300\) years, the situation should be carefully dealt with, due to a considerable reduction of the optimal performance capacity (\(-14\%\) in terms of energy absorption capacity of the embankment and \(25\%\) reduction in the return period values). Measures to restore as much as possible the performance capacity of the embankment towards optimal should therefore be planned, such as, e.g., purge of the snow and blocks behind the structure, reinforcement of the damaged parts, etc.