# Reinforced Drapery Meshes: A Design Method Accounting for Retaining Ropes Contribution
## 2 System Design
### Rope Design  
As explained in Section 2.2, the design of the mesh is performed considering individual portions that are laterally constrained. Consequently, the assessment of the ropes is also performed for individual portions (henceforth simply called rope), whose undeformed length \(L_{0}\), in the absence of a gravitational load, is computed as:  
\[L_{0}=l+\frac{8f_{0}^{2}}{3l} \tag{12}\]  
where \(l\) is the distance between anchors (Figure 4), and \(f_{0}\) is the initial deflection. Due to its gravitational load \(g_{r,Ed}\) (dead load), the rope installed between two anchors is subjected to a tensile force \(T_{r,g,Ed}\) that can be computed through an iterative procedure starting by assuming a deflection \(f_{g}\) equal to \(f_{0}\) and iteratively solving the following equations:\[T_{r,g,Ed}=\sqrt{\left(\frac{g_{r,Ed}l^{2}}{8f_{g}}\right)^{2}+\left(\frac{g_{r, Ed}l}{2}\right)^{2}} \tag{13}\]  
\[\Delta l_{g}=\frac{T_{r,g,Ed}L_{o}}{E_{r}A_{f}} \tag{14}\]  
\[f_{g}=\sqrt{\frac{3}{8}l(L_{0}+\Delta l_{g}-l)} \tag{15}\]  
until convergence, where \(\Delta l_{g}\) is elongation due to the gravity load, and \(E_{r}\) and \(A_{f}\) are the apparent elastic modulus and the nominal area of the rope, respectively.  
The main purpose of the rope is to transfer the loads from the mesh to the anchors, thus ensuring that working conditions are similar to those in laboratory tests by fixing the rope elongation \(\Delta l_{r}\), which should be smaller than a threshold \(\Delta l_{max}\). The latter is defined as:  
\[\Delta l_{max}=min\left[\frac{(T_{r,Rd}-T_{r,g,Ed})L_{0}}{E_{r}A_{f}},\frac{8 }{3}\frac{(f_{max}-f_{g})^{2}}{l}\right] \tag{16}\]  
where the first term of the right side is the maximum displacement at failure, and \(T_{r,Rd}\) is the maximum tensile resistance. The second term refers to the maximum deflection \(f_{max}\), defined by the designer. Based on the authors' experience, \(f_{max}=0.05l\).  
Thus, the maximum allowable tensile force is:  
\[T_{r,Ed,max}=\frac{E_{r}A_{f}\Delta l_{max}}{L_{0}} \tag{17}\]  
and the ultimate load \(q_{r,max}\) is:  
\[q_{r,max}=T_{r,Ed,max}\left[\frac{l^{4}}{64(f_{max}-f_{g})}+\frac{l^{2}}{4} \right]^{-1/2} \tag{18}\]  
where the maximum deflection \(f_{max}\) is:  
\[f_{max}=\sqrt{\frac{3}{8}l(L_{0}+\Delta l_{g}+\Delta l_{max}-l)}. \tag{19}\]  
The ropes are subjected to the load transferred from the mesh. In the hypothesis that the ropes provide an efficient constraint for the mesh, the proposed static scheme for the rope (subdivided into individual elements of length \(l\)) accounts for support at the anchorages, while, in addition to the fact that load configurations can vary from case to case, the worst scenario is considered. Consequently, assuming a horizontal/vertical rope pattern, a distributed load along its entire length is assumed, and two adjacent mesh panels are assumed to both load the rope. Combining the forces in the neighboring mesh panels results in a total force proportional to \(F_{m,Ed}\). Considering the proposed area of influence for each rope \(A_{r}\) as in Figure 4, the load \(q_{r,Ed}\) is equal to \(\frac{F_{m,Ed}}{l}\frac{A_{r}}{A_{m}}\). Adopting the same iterative procedure proposed in Equations (13)-(15), the tensile force acting on the rope \(T_{r,Ed}\) and its real deflection \(f_{r,Ed}\) can be computed. The safety of the rope must be assessed following the usual approaches for steel cables reported in Eurocode 3 [34]. The working rate is provided by the ratio between the acting loads \(q_{r,Ed}\) and \(q_{r,max}\).