## 2 Overview of Structural Analysis and Form-Finding of Elastic Timber Gridshells
### Numerical Modelling for Structural Analysis from the Final Geometry  
For elastic timber gridshells used in building construction, a very practical approach is to model the geometry to be built by performing a non-linear geometric analysis at large deformations (usually solved with methods based on Newton-Rapson iteration scheme). The model must take into account the slip at the joints of the multilayer system. One possible strategy is to replace the slipping multilayer system with simple elements of equivalent stiffness. This approach was applied in the analysis of the Mannheim Multihalle gridshell. In this case, the multilayer structure was simplified into simple elements of reduced second moment of area with respect to the full composite member, assuming linear elastic behaviour of the joints [5]. A similar strategy was proposed for the analysis of ribbed shells [8; 14], which can be adequately modelled by applying the shear analogy method developed for layered timber beams [15]. This method transforms each multilayer timber rib into a model of two beams connected by inextensible bars with equivalent bending and shear stiffness, considering also a linear behaviour of the connections. In both cases, eccentricities between layers were not considered.  
Subsequent work included eccentricity between the layers of elastic gridshells through small rigid couplings. The use of couplings also makes it possible to incorporate the complete multilayer system without resorting to simplified elements of equivalent stiffness. One possible solution is to adjust the bending stiffness of the coupling so that it produces the same deformation as the expected slip at the joints [16]. This solution only assumes a linear behaviour of the joints. Numerical investigations carried out on the buckling analysis of elastic gridshells in composite materials [17] and timber [18] have shown the importance of considering eccentricities in the model of single-layer elastic gridshells, as they can significantly reduce the stiffness of the structure and consequently the critical buckling load. However, in multilayer gridshells, the effect of eccentricity is much less relevant [18].  
Numerous works in the field of simulation of timber composite elements e.g., [19; 20] have demonstrated the effectiveness of couplings (or link elements) and springs for modelling the behaviour of multilayer elements joined with mechanical fasteners. In these cases, the couplings are rigid and the slip at the joints is included from the stiffness of the springs. This approach has also been successfully used in the modelling of small-scale elastic timber gridshells [21] and allows easy consideration of linear and non-linear behaviour of the joints.  
Obviously, when directly modelling the final geometry, additional considerations on the residual stresses due to the forming process are necessary and depend on the individual case. For multilayer structures made with laths of very small thicknesses and large curvature radii, the stresses may be very small and could be neglected [8]. In structures built with wet timber that bend very slowly (several months), as in the case of the Weald & Downland Living Museum gridshell or the Savill Garden gridshell, the initial stressesdissipate to a large extent. Under these conditions, one approach could be to disregard the curving stresses, but to consider a reduction in the timber bending strength, similar to the standard procedure applied in the sizing of gluam arches [22]. However, in most cases it is necessary to take into account the forces resulting from the construction process, especially bending moments. These can be derived geometrically from a curvature analysis of the grid [8; 23] and considered as prestress loads [11] or strains loads [24; 25].