## 4 Applications
### Example 1: Effectiveness of Rock Fall Protection at the Source Area
(2) Proof of the mesh wire against puncturing of the nails, due to their action  
\[\mathrm{A}_{(2)}<\mathrm{R}_{\mathrm{opt},(2)}\] \[\mathrm{with:\;A}_{(2)}=30\;\mathrm{kN};\;\mathrm{R}_{\mathrm{ opt},(2)}=\tfrac{\mathrm{R}_{2}}{\mathrm{\gamma}_{1}}=\tfrac{80}{\mathrm{1.5}}\; \mathrm{kN}=53.3\;\mathrm{kN}\]  
and where \(\mathrm{A}_{(2)}\) is the design value of the pretension force F applied to the nails and \(\mathrm{R}_{\mathrm{opt},(2)}\) the resistance of the mesh against pressure in the nail direction (\(\mathrm{R}_{2}\) is determined from tests on real samples). (3) Proof of the nail against combined stresses of tension (due to the way nails operate) and shear (due to the rock blocks, which could slide off):  
\[\mathrm{A}_{(3)}<\mathrm{R}_{\mathrm{opt},(3)}\Rightarrow\;\sqrt{\left(\mathrm{ A}_{(2)}/\mathrm{T}_{\mathrm{r}}\right)^{2}+\left(\mathrm{A}_{(1)}/\mathrm{S}_{ \mathrm{r}}\right)^{2}}\leq 1.0\;\Rightarrow\;0.73<1.0\] \[\mathrm{with:\;T}_{\mathrm{r}}=\tfrac{\mathrm{f}_{y}\cdot\mathrm{ A}}{\mathrm{\gamma}_{3}}=\tfrac{500.616}{\mathrm{1.5}}\;\mathrm{N}=205.3\; \mathrm{kN};\]  
In this case, \(\mathrm{Tr}\) is the tensile resisting force of the nail and \(\mathrm{Sr}\) is the shear resisting force, computed as in proof 1. \(\mathrm{A}_{(1)}\) and \(\mathrm{A}_{(2)}\) are the same actions computed in proofs 1 and 2, respectively. (4) Proof of the mesh against shearing off in the nail direction, when a local rock block/compartment between two nails detaches and tends to slide down towards the lower nail, exerting an outward-directed force (Figure 4) on the mesh at its connection with that nail:  
\[\mathrm{A}_{(4)}<\mathrm{R}_{\mathrm{opt},(4)}\] Figure 3:0  
For this proof, \(\mathrm{A}_{(4)}\) is the design value of the maximum shear force on the net, due to the outward push of the rock block. This shear force is taken as equal and opposite to the stabilising force P of the mesh pressed against the slope, computed depending on the geometry of the equilibrium problem. \(\mathrm{R}_{\mathrm{opt},(4)}\) is the resistance of the mesh against shearing in the nail direction (again, determined from tests designed for this purpose). (5) Proof of the mesh against tensile stresses parallel to the direction of the slope, as a reaction to the sliding off of local blocks (between two nails) behind the mesh:  
Figure 3:1 Figure 3:2  
Here, \(\mathrm{A}_{(5)}\) is the design force on the net parallel to the direction of the slope, again computed depending on the geometry of the equilibrium problem, and \(\mathrm{R}_{\mathrm{opt},(5)}\) the resistance of the mesh against forces in the same direction (\(\mathrm{R}_{5}\) is also to be determined by means of specifically designed tests). Let it now be assumed that, after inspection, the current state of the protection is influenced by no particular factor of Group 0 (referring to Condition 0 analysis) and, on the other hand, by the factors: (i) corrosion of mesh, (ii) corrosion of the nails and (iii) damages to the mesh, belonging to Group N (Table 2), for which corresponding degrees of severity were defined as (i) moderate to high, (ii) moderate and (iii) low to very low, respectively. Let the corrosion issues in this problem be related to a not fully optimal engineering design of the measure (Condition 2), while the damages to the mesh attest to a lack of maintenance (Condition 1). In these conditions, and under the hypothesis of an appropriate calibration, the penalty coefficient \(\mathrm{m}\) for evaluating the reduced performance capacity of the system could be defined and assigned as:* Corrosion of mesh: low & r\({}_{21}\) = 0.93;
* Corrosion of nails: moderate & r\({}_{28}\) = 0.85;
* Damages to the mesh: low to very low & r\({}_{12}\) = 0.95. Since the main effect of corrosion is to reduce the section of the resisting elements, coefficients r\({}_{21}\) and r\({}_{28}\) influence basically all the proofs of safety for the protection system. In particular, r\({}_{21}\) affects proofs 2, 4 and 5, while r\({}_{28}\) affects proofs 1 and 3.