## 2 Overview of Structural Analysis and Form-Finding of Elastic Timber Gridshells
### Numerical Modelling for Form-Finding  
A very different approach to consider residual stresses is to simulate the gridshell construction process. This approach is also necessary when the final shape of the structure depends on the internal strain forces.  
Modelling the forming process involves solving very important geometrical nonlinearities due to the large displacements and rotations that take place during the deformation. This problem has been addressed by the scientific community in recent years in the general field of active bending systems (e.g., [26; 27]) as well as in the specific field of elastic gridshells, mainly using the dynamic relaxation (DR) method and FEM.  
DR solvers with three degrees of freedom (DoF) per vertex have been widely used for the form-finding of single-layer elastic gridshells made of composite materials, with circular cross-sections and no slip at the joints [28; 29]. However, these models can not accurately represent the behaviour of rectangular cross-sections which present torsion and bending in the two axes of inertia. The 4 and 6 DoF methods overcome this limitation, but have been applied exclusively to single-layer gridshells, also without slip at the joints [30; 31; 32; 33; 34]. A 6 DoF method for the form-finding of multilayer timber gridshells was developed in [35]. It conducted form-finding by initially considering a single layer. Once the final shape was reached, the bending stiffness of the members was modified by attributing a new equivalent bending stiffness, corresponding to that of the considered multilayer system. The new stiffness was assigned according to the Gamma Method included in Eurocode 5 [36] for the analysis of mechanically connected composite beams, which assumes linear behaviour of the connections. This model did not consider the eccentricity of the connections. Subsequent work with a similar approach was carried out by [3].  
FEM simulations have also been used for the form-finding of single-layer gridshells with circular and rectangular cross-sections [1; 37; 38], including the eccentricity between the laths by means of small couplings that allow rotation around the vector perpendicular to the grid surface. Although these models do not include multilayer systems or semi-rigidity in the joints, the great advantage of this method is that it allows a complete mechanical description of the structure, making it possible the simultaneous use of different types of elements (Timoshenko beams, cables, couplings, springs, etc.) and assigning them the actual physical properties, including non-linear behaviour if necessary. In addition, the main FEM software packages have the possibility to activate and deactivate temporary elements at different stages to perform the simulation of the forming process. This has been done by two procedures: the imposition of a displacement field at certain grid nodes, or the use of temporary elastic contraction elements [39]. The latter has the advantage of not requiring knowledge of the intermediate path of the points during the curving process. In addition, the increasing interoperability of parametric design software and FEM allows virtually unlimited modelling capabilities.  
As gridshells are structures with high deformability and hyperstability, it is essential that the numerical models are properly calibrated, including the main elements that influence the stiffness of the structure, to correctly estimate the stresses and the global deformability. Unfortunately, very little research can be found that has validated the proposed models with full-scale tests. In single-layer elastic gridshells made of composite materials, some of the research mentioned above has compared structural analysis and form-finding simulations with the experimental results of full-scale prototype tests [37; 40]. However, in the case of timber, research in this direction is really scarce (e.g., [8]).