## 4 Applications
### Example 2: Effectiveness of Rock Fall Protection in the Propagation Zone
By computing \(\mathrm{P_{r}}\) and \(\mathrm{T}\) in a similar way at the locations of the hamlet and the road, the hazard at all location of interest is given by, respectively:  
Figure 5: Example 2: hazard zoning obtained along the slope based on the presence of an earth embankment in the rock fall propagation zone. (**A**) Hazard zoning without embankment. (**B**) Hazard zoning with embankment in optimal working order. (**C**) Hazard zoning with embankment with reduced performance capacity. The mitigation effects at each location of interest are illustrated with reference to the Swiss rock fall intensity–frequency diagram. (b.1) location \(x_{\rm P}\):  
\[\rm E^{*}\approx 0\ kJ\qquad P_{\rm r,opt}=0.046\qquad\to\quad T_{\rm opt}=434\ years\] (b.2) location \(x_{\rm H}\):  
\[\rm E^{*}\approx 0\ kJ\qquad P_{\rm r}{}^{*}=0.84\ (1-0.95)=0.042\qquad\to\quad T^{*}=1/(1 /20\cdot 0.042)=476\ years\] (b.3) location \(x_{\rm R}\):  
\[\rm E^{*}\approx 0\ kJ\qquad P_{\rm r}{}^{*}=0.20\ (1-0.95)=0.010\qquad\to\quad T^{*}=1/(1 /20\cdot 0.010)=2000\ years\]  
In these computations, the energy \(\rm E^{*}\), \(\rm P_{\rm r}{}^{*}\) and \(\rm T^{*}\) at \(x_{\rm H}\) and \(x_{\rm R}\) are the values of energy, frequency of reach and return period obtained as a consequence of the effect of the embankment. The energy \(\rm E^{*}\) of the blocks travelling beyond the protection was considered negligible at each location, based on the considerations just made above. Figure 5B shows the new hazard zoning after mitigation at all locations beyond \(x_{\rm P}\) and, in particular, the effects of the embankment in optimal working order at \(x_{\rm P}\), \(x_{\rm H}\) and \(x_{\rm R}\), also in terms of hazard evolution on the intensity-frequency diagram. (c) Let it be assumed that the actual state of the embankment is affected by: (i) the presence of frequent snow, which can cumulate behind its upslope face (Group 0), (ii) a certain lack of the purge of blocks from previous events, (iii) some deformations from previous impacts and (iv) some slight sign of potential stability issues of its upslope face (all these three last factors belonging to Group N). Lack of purge and the presence of impact traces can be considered to be due to, e.g., maintenance problems (Condition 1), while the slight signs of instability to some issues in the original design (Condition 2). Let the qualitative evaluation of the influence of each factor and the values for the corresponding penalty coefficients, \(\rm e_{n}\) and \(\rm t_{n}\), be the following:  
- Frequent snow: moderate (Condition 0) \(\to\quad\)\(\rm t_{03}=0.85\);  
- Lack of purge: moderate to low (Condition 1) \(\to\quad\)\(\rm t_{15}=0.90\);  
- Deformations due to previous impacts: moderate (Condition 1) \(\rm e_{11}=0.88\);  
- Stability issues of upslope face: very low (Condition 2) \(\rm e_{23}=0.98\)\(\rm t_{23}=0.98\). The effects of snow cumulating behind the embankment and the presence of blocks at the same place are similar in that they mostly modify the effective height of the structure: thus, only a coefficient \(t\) was defined to model them. Previous impacts of blocks, on the other hand, can degrade the energy absorption capacity of the embankment; consequently, only a coefficient \(e\) was specified. Stability issues were instead considered to affect both parameters.