## 4 Applications
### Example 2: Effectiveness of Rock Fall Protection in the Propagation Zone  
In this example, the slope in Figure 5 is characterised by a potential hazard due to the release of blocks of about 5 m\({}^{3}\) from highly active cliffs, for which a mean failure frequency of 1 event (1 block) every 20 years was determined. Assets at risk are the buildings of a hamlet and, farther downslope, a road. The hazard zoning along the slope profile, obtained by combining the rock fall mean frequency of failure and rock fall trajectory simulations results (energy and frequency of reach), is shown in Figure 5A. The percentage P\({}_{\text{r}}\) of trajectories reaching the location of the hamlet \(\mathrm{x_{H}}\) is assumed to be equal to 84%, the associated energy E to be around 1500 kJ and the height of rebound d of the blocks to be 2.0 m. Farther downslope, at the location of the road, these parameters change to \(\mathrm{P_{r}}\) = 20%, E = 150 kJ, d = 0.5 m.  
In order to protect the hamlet and the road, an earth embankment was built at location \(\mathrm{x_{P}}\), where the frequency of reach is 92%, the rock fall energy is around 1700 kJ and the height of rebound is about 3.6 m. This embankment was designed with an energy absorption capacity of Eopt = 2000 kJ and a height \(\mathrm{D_{opt}}\) = 4 m, so that for all trajectories E \(<\mathrm{E_{opt}}\) and for 95% of them d \(<\mathrm{D_{opt}}\), i.e., the embankment can intercept 95% of the trajectories crossing its location. For the safety of the assets in danger, it is considered that no hazard should affect the locations of the hamlet and the road, \(\mathrm{x_{H}}\) and \(\mathrm{x_{R}}\), respectively. With reference to the rock fall intensity-frequency diagram used, this means that a mitigation in this example can occur only if, immediately beyond \(\mathrm{x_{P}}\) and until the road, \(\mathrm{E_{m}}\approx 0\) and \(\mathrm{T_{m}}>300\) years (condition corresponding to the point at the low right corner of the diagram). Figure 5: Cont. Based the procedure in Section 3 for protection measures in the propagation zone, the performance capacity of the protection can be determined as follows, by analysing at first (a) the hazard without the embankment, then (b) the situation with embankment in perfect working order and, finally, (c) the one in which the structure is affected by certain factors degrading its operation. (a) From the information about the event introduced above, the hazard at each location \(\mathrm{x_{P}}\), \(\mathrm{x_{H}}\) and \(\mathrm{x_{R}}\) is given by:  
(a.1) location \(\mathrm{x_{P}}\):  
\[\mathrm{E=1700\ kJ}\qquad\mathrm{P_{r}=0.92}\quad\rightarrow\quad\mathrm{T=1/ (P_{f}\cdot P_{r})=1/(1/20\cdot 0.92)=22\ years}\]  
(a.2) location \(\mathrm{x_{H}}\):  
\[\mathrm{E=1500\ kJ}\qquad\mathrm{P_{r}=0.84}\quad\rightarrow\quad\mathrm{T=1/ (P_{f}\cdot P_{r})=1/(1/20\cdot 0.84)=24\ years}\]  
(a.3) location \(\mathrm{x_{R}}\):  
\[\mathrm{E=150\ kJ}\qquad\mathrm{P_{r}=0.20}\quad\rightarrow\quad\mathrm{T=1/ (P_{f}\cdot P_{r})=1/(1/20\cdot 0.20)=100\ years}\]  
(b) With an embankment in perfect working order, the likelihood of a block travelling beyond its location and the corresponding return period can be computed starting from the values of the frequency of reach (\(\mathrm{P_{r}}=92\%\)) in absence of measure and the percentage of trajectories intercepted by the measure (\(95\%\)), thanks to its height \(\mathrm{D_{opt}}\):  
\[\mathrm{P_{r,opt}=0.92\ (1-0.95)=0.046}\qquad\rightarrow\quad\mathrm{T_{opt }=1/(P_{f}\cdot P_{r,opt})=1/(1/20\cdot 0.046)=434\ years}\]  
In this case, therefore, even though \(4.6\%\) of the trajectories might still develop downslope beyond \(\mathrm{x_{P}}\), they constitute no actual hazard regardless of the energy characterising them, as \(\mathrm{T_{opt}>300\ years}\).