## 4 Numerical Modelling for Structural Analysis of TEL-Gridshells
### Geometry, Materials and General Aspects  
The geometry was modelled in Rhino and Grasshopper [45] reconstructed from the coordinates of the target points measured by photogrammetry, and not from the original design model. This takes into account construction imperfections and the small movements of the structure due to elastic recovery, minimising uncertainties arising from geometrical differences between the computational model and the tested prototype. The main family of latths was discretised into six segments in each span between nodes. Rotation around the weak axis of the cross-section was allowed at the end nodes of each of the 18 latths. The transverse latths were discretised into segments of 60 mm in length, corresponding to the screw spacing of the _Type 3_ joint. The cross-section of each member segment was oriented so that the weak axis coincided with the vector normal to the surface at the centre point of each segment. The points corresponding to each target (deformation control points) were implemented in the numerical model using small beam members that perpendicularly join the latth axis with the point corresponding to the centre of each target. In this way, a total correspondence was achieved between the coordinates of the targets in the numerical model and those obtained from the photogrammetry. This strategy also made it possible to detect the rotation by torsion of the members even if their ends do not move (Figure 9). After completion of the geometrical model in Rhino, it was exported to RFEM (Dlubal Software GmbH, Tiefenbach, Germany) [46], where all necessary structural information was added. The timber latths were modelled as beam elements (with axial, shear, bending and torsional stiffness), to which the actual cross-section and average elastic properties were assigned. The LVL supports were modelled as shell elements of 40 mm thickness. The shells were meshed so that one of their nodes coincided with the position of the LVDTs (control points A and B). The meshing of these elements was performed automatically in RFEM allowing triangular and quadrangular elements with a maximum side length of 0.5 m. No mesh refinement was applied. The reinforcement pieces were modelled as beams in their actual position, which were connected to the shell using small couplings. The general appearance of the developed geometrical and FEM models of the T \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \(d\) **(mm)** & \(\rho_{\text{mean,1}}\) & \(\rho_{\text{mean,2}}\) & \(\rho_{\text{mean}}\) & \(k_{iy}=k_{iz}=K_{\text{ser}}\) \\  & & (kg/mm\({}^{3}\)) & (kg/mm\({}^{3}\)) & (kg/mm\({}^{3}\)) & (N/mm) \\ \hline _Spring 1_ & 6 & 795 & 795 & 795 & 5848 \\ _Spring 2A_ & 8 & 795 & 795 & 795 & 7797 \\ _Spring 2B_ & 8 & 795 & 510 & 637 & 5589 \\ _Spring 3_ & 4 & 795 & 795 & 795 & 3898 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Stiffness of the springs. Figure 9: Movement of a deformation control point (k) due exclusively to torsion in the latth. the location of control points A, B, C, D and E, and the detail of a nodal joint can be seen in Figure 10.  
Linear and elastic material behaviour was assumed. The material properties of the beams are summarised in Table 4. For the shells, 2D orthotropic elastic material was taken. The average values of the mechanical properties of eucalyptus have been obtained from tests carried out by the authors (see Section 3.2), while those of pine LVL have been taken from data provided by the manufacturer (Metsa Wood, Espoo, Finland) [47]. The external loads were applied as point loads at the upper nodes of the corresponding joints. These were applied incrementally in eight steps until the experimental test load was reached. Geometrically non-linear large deformation analysis was performed. To better understand the behaviour of the structure, the influence of the LVL supports and the stiffness of the joints on the overall stiffness of the gridshell was analysed.