# Reinforced Drapery Meshes: A Design Method Accounting for Retaining Ropes Contribution
## 2 System Design
### Anchor Design  
The reinforced drapery mesh constitutes a mixed passive-active system to mitigate rockfall hazards. The anchors prevent instability of the cortical zone of the slope. Generally, the installed anchors are not prestressed, i.e., bolts, and thus, they are treated as elements that apply a resisting force only after the movement of the slope. Thus, by definition, bolts provide passive support; i.e., they are assumed to increase the resisting force of the slope along the sliding surface, as defined in Section 2. In the proposed approach, based on the limit equilibrium method on an infinite slope, the shear resistance contribution to slope stability due to the bolts is computed as a force along the sliding surface, \(C_{R,max}\) (Figure 6) [29; 30]. This approach is based on the assumption that the plastic work of the force acting on the bolts at the breaking surface (tensile and shear forces) is at its maximum; i.e., the maximum allowable yield tensile stress on the bolt is mobilized. The force on the bolts is caused by relative joint movements while also accounting for the dilatancy \(\delta\) of the sliding surface. The modulus and direction \(\iota\) of the resultant force \(R\) on the anchor vary as functions of the displacement of the cortical zone: referring to the plastic flow rule, the displacement is normal to the elliptical yield surface at \(R\), which can be found graphically by tracing a line orthogonal to the displacement and tangent to the elliptical yield surface (Figure 6). Thus, \(C_{R,max}\) is given by the sum of the frictional contribution, proportional to the component of \(R\) normal to the sliding surface, and the cohesive contribution, proportional to the component of \(R\) parallel to the sliding surface, according to:  
\[C_{R,max}=R\sin\left(\theta_{0}+\iota\right)+R\cos\left(\theta_{0}+\iota \right)\tan\phi \tag{1}\]  
where \(\theta_{0}\) is the inclination of the anchor with respect to the slope normal, and \(\phi\) is the friction angle of the sliding surface. The inclination \(\iota\) of the force and the inclination of the anchor are related according to:  
\[m=4\tan\iota=\cot\left(\theta_{0}+\delta\right). \tag{2}\]  
Following the Tresca criterion to evaluate normal-shear forces, the resultant force \(R\) can be computed as:  
\[R=N_{a,Rd}\sqrt{\frac{1+m^{2}/16}{1+m^{2}/4}} \tag{3}\]  
where \(N_{a,Rd}\) is the tensile yield resisting force of the anchor. Figure 5: Retaining effects of ropes (**a**) and failure if too slack (**b**). Once selected, the geometrical and mechanical parameters of the anchors and the driving and stabilizing forces along the sliding joint are computed, accounting for the self-weight of the rock among the anchors, the snow load, and other possible external loads. In the framework of the partial safety factor design approach [31], the design values of unfavorable \(F_{s,Ed}\) and favorable \(F_{s,Rd}\) actions can be computed, and the slope stability analysis can be performed by requiring:  
\[FS_{slope}=\frac{F_{s,Rd}+C_{R,max}}{F_{s,Ed}}\geq FS_{slope,min}\geq 1 \tag{4}\]  
where the minimum safety factor \(FS_{slope,min}\) is selected by the designer according to the reliability of the input parameters as well as the adopted method. Note that \(F_{s,Rd}\) and \(F_{s,Ed}\) are related to the sliding surface. For the adopted theory, the fulfillment of the inequality of Equation (4) also implies the structural assessment of the anchors themselves. Considering the limit condition and \(FS_{slope}=1\), the rate of work of the anchors can be evaluated as:  
\[\eta=\frac{F_{s,Ed}-F_{s,Rd}}{C_{R,max}}. \tag{5}\]  
Figure 6: Sketch of the force acting on the sliding surface considering global instability of the cortical band; \(E\) represents an external force. Thus, the tensile force acting on each anchor can be evaluated as \(\eta R\cos\iota\).