## 5 Numerical Modelling for Form-Finding of TEL-Gridshells
### Residual Stresses
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & & **Before Relaxation** & **After Relaxation** \\ \hline \multirow{2}{*}{Control points} & \(\left|\left|\mathrm{U}\right|_{\mathrm{C}}\mathrm{(mm)}\right)\) & 0.0 & 0.1 \\  & \(\left|\left|\mathrm{U}\right|_{\mathrm{D-E}}\mathrm{(mm)}\right)\) & 0.1 & 18.4 \\ \hline \multirow{4}{*}{Longitudinal grid} & N (kN) & \(+0.590/-0.849\) & \(+0.951/-1.984\) \\  & M\({}_{y}\) (kNm) & \(+0.015/-0.164\) & \(+0.017/-0.164\) \\  & M\({}_{z}\) (kNm) & \(+0.315/-0.315\) & \(+0.308/-0.308\) \\  & M\({}_{t}\) (kNm) & \(+0.033/-0.032\) & \(+0.032/-0.030\) \\ \hline \multirow{2}{*}{Transverse laths} & N (kN) & \(+0.918/-0.397\) & \(+2.472/-2.617\) \\  & M\({}_{y}\) (kNm) & \(+0.006/-0.044\) & \(+0.001/-0.033\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Model A (with semi-rigid connections). Displacements at control points C and D-E and maximum axial forces and moments due to the construction process, before relaxation (_Stage 4_) and after relaxation (_Stage 5_). The effect of the bracing system on the structural behaviour of gridshells has been studied by several authors. It is essential to provide in-plane shear stiffness, significantly reducing deformations under external loads [3,40] and therefore considerably increasing the critical buckling load. However, the influence of the bracing system on the final shape of the structure and on the distribution of residual stresses in the members is a subject that has been little addressed in the literature. Moreover, in the specific case of the TEL-gridshells, the analysis of the bracing system requires special attention since it is formed by strongly curved laths and, therefore, with very high bending stresses. Thus, in order to better understand the influence of the bracing system in this type of structure, the results of the full structure model (model B) are compared with those of the model without transverse laths (model C) resulting from the elimination of _Stages 3_ and \(4\). In both cases, slip-free connections are considered. Table 11 shows the displacements of models B and C at control points C and D-E, as well as the maximum values obtained for the stresses N, M\({}_{y}\), M\({}_{z}\) and M\({}_{t}\) in the longitudinal members of the grid. The results are shown before and after relaxation (_Stages 4_ and _5_). As is obvious, in _Stage 4_ before relaxation, the position of the nodes of both models is identical. In model C, without bracing system by removing the supports that hold the nodes in the target position, the whole structure deforms considerably. \begin{table}
\begin{tabular}{c c c c c} \hline  & _Type 1_ & _Type 2A_ & _Type 2B_ & _Type 3_ \\ \hline V\({}_{j}\) (kN) & 1.49 & 0.79 & 1.39 & 0.46 \\ \hline \end{tabular}
\end{table} 
Table 10: Model A (with semi-rigid connections). Maximum residual shear forces in the joints. Figure 15: Model A with semi-rigid connections: residual bending moment. In model C, the central point (control point C) descends 48.2 mm (L/135) and the curvatures of the laths in both axes are reduced, and consequently also the bending moments. The grid tends to return to the initial flat position, deforming more in the area where it is not bound, i.e., in the central strip of the structure. The quadrilaterals tend to close in the transverse direction of the grid, decreasing the length of their short diagonal. However, in the braced gridshell (model B), the overall geometry remains almost undeformed (except in the lower area of the transverse members, where there is a maximum displacement of 16.2 mm). The central point descends only 0.1 mm, and the bending moments before and after relaxation are practically the same. The final geometry of the structure after relaxation for both models is shown in Figure 16.  
The elements of the bracing system triangulate the quadrilaterals of the grid maintaining their shape. As the quadrilaterals tend to close, the bracing is compressed.