## 4 Applications
### Example 1: Effectiveness of Rock Fall Protection at the Source Area  
The slope represented in Figure 3 is affected by a rock fall scenario characterised by a maximum block size of around 4 m\({}^{3}\), falling on average every 100 years, according to data and information collected about the unstable cliffs at its top. Rock fall simulations of this particular event provide values for energy, height of rebound and runout distances which, combined with the mean temporal frequency of detachment estimated, provide the hazard zoning illustrated (Figure 3). This hazard threatens a mountain road at the toe of the rocky cliff (x\({}_{\text{R}}\)), located in the high hazard zone, and a village, more downslope (x\({}_{\text{V}}\)), situated in the low hazard zone. A protection measure constituted by a system of mesh wire and nails has been installed for a few decades at the source area to prevent blocks from detaching and reaching these elements. The characteristics of the system in terms of geometry are also shown in Figure 3. Under the condition of optimal performance, this measure is expected to provide the mitigation shown in the same figure (hatched zones). Figure 3: Example 1: hazard zoning obtained along the slope without and with the effects of a protection system installed at the rock fall source area. The mitigation effects all over the slope are illustrated with reference to the Swiss rock fall intensity–frequency diagram. The assessment of the performance capacity of the system can be conducted according to the procedure described in Section 3 for protection measures at the source area, by recalling the safety proofs necessary for this type of system to operate correctly and by evaluating the current conditions of the protection (to define the corresponding penalty coefficients). The proofs of safety i for the protection can be presented assuming a mode of operation similar to rock fall mesh wire and nails-based systems currently available e.g., Geobrugg 2020 [22]. With reference to Figure 4, under the hypothesis of a Mohr-Coulomb failure criterion, the proofs for the protection system installed are, in optimal working conditions:  
(1) Proof of the nail against the sliding-off of superficial rock blocks/compartments within the unstable rock layer:  
\[\text{A}_{(1)}<\text{R}_{\text{opt},(1)}\] \[\text{with : }\text{A}_{(1)}=85.5\,\text{kN};\text{ R}_{\text{opt},(1)}=\tfrac{\left(\text{f}_{\text{f}}/\sqrt{3}\right)\text{A}}{ \gamma_{1}}=\tfrac{\left(500/\sqrt{3}\right)616}{1.5}\,\text{N}=118.7\,\text{ kN}\]  
In the relationships above, \(\text{A}_{(1)}\) is the design shear force the nail has to bear and transmit to the stable part of the cliff, generated by the action of the block potentially  
Figure 4: Scheme of the equilibrium problem for the proofs of safety of the protection system assumed. Top: equilibrium problem for proofs of safety 1, 2 and 3; Bottom: equilibrium problem for proofs of safety 4 and 5. Destabilising and stabilising actions in the general equilibrium problem, as well as the forces in the nails, are marked in black; specific resulting actions \(\text{A}_{(i)}\) and corresponding resistances \(\text{R}_{(i)}\) to be considered for the proofs of safety are marked in red and green, respectively. sliding along the failure surface. This force is determined by the geometry of the problem. \(\mathrm{R}_{\mathrm{opt},(1)}\) is, on the other hand, the resistance of the nail to shear stress, function of the shear stress at failure \(\mathrm{\tau}_{y}=\mathrm{f}_{y}/\surd 3\) and the cross section A of the nail.