## 3 Methods for the Structural Analysis of Diagrid Tall Buildings  
In the academic literature, the most common procedure for dealing with the structural analysis of diagrid systems is the Finite Element Method (FEM). However, simplified methodologies have also been developed for a quick evaluation of the diagrid's overall structural behavior. [PERSON] et al. [16] have proposed a hand-based method for the evaluation of the axial stress in the diagrid members. The method is based on the analysis of the internal forces arising in the basic triangular element due to gravity and vertical loads, taking also into account the effect of the horizontal and vertical curvatures of the diagrid facade. Although it does not allow the direct calculation of the displacements of the structure, this methodology has proven effective in the computation of the axial forces in the diagonals. It is applied to three real case studies, the Swiss Re Building (London),the Hearst Headquarters (New York) and West Tower (Guangzhou), and the axial stresses arising from hand-calculations show a very good correspondence with FEM results. Design considerations regarding the optimal diagonal inclination for the investigated cases are also provided. More recently, [PERSON] and [PERSON] have developed a simplified methodology, called the modular method (MM), for performing the structural analysis of diagrid tubes with an arbitrary polygonal shape [17]. So far, most of the research has been focused on rectangular diagrids, having vertical facades acting as webs or flanges (Figures 4 and 5); however, little attention has been paid to diagrids with polygonal shapes. The modular method relies on the modularization of the diagrid, and the calculation of the lateral stiffness of the diagrid modules in order to compute the total lateral deflection. The lateral displacement \(u_{i}\) of the \(i^{\text{th}}\) module can be obtained by superimposing the contribution of the shear displacement \(u_{V,i}\) and bending displacement \(u_{M,i}\). Based on the evaluation of the shear and bending rigidity of the \(i^{\text{th}}\) module, \(K_{V,i}\) and \(K_{M,i}\), the two contributions can be computed as follows:  
\[u_{V,i}=\frac{V_{1}}{K_{V,1}}+\frac{V_{2}}{K_{V,2}}+\ldots+\frac{V_{i}}{K_{V,i }}, \tag{9}\]  
\[u_{M,i}=\frac{M_{1}}{K_{M,1}}hi+\frac{M_{2}}{K_{M,2}}h(i-1)+\ldots+\frac{M_{i }}{K_{M,i}}h[i-(i-1)]. \tag{10}\]  
where \(V_{i}\) and \(M_{i}\) are the shear force and bending moment at the level of the \(i^{\text{th}}\) module, respectively, and \(h\) the height of the module. The key to the MM is the calculation of the shear and bending rigidities, \(K_{V,i}\) and \(K_{M,i}\), which is based on the usual assumptions for diagrid tubes: the diagonals are only subjected to axial stress and remain in the linear elastic regime; the building floors behave as rigid bodies without any internal deformation; the intra-module floors are neglected for the calculation of the modular rigidities. Shear rigidity is defined as the total shear force \(F_{V}\) required for unitary horizontal displacement of the module \(\Delta\
u\) (Figure 6a), and bending rigidity is defined as the bending moment \(M\) required for unitary floor rotation \(\Delta\beta\) (Figure 6b). Applying independently unitary floor displacements and rotations, and computing the total shear force and bending moment, leads to the direct evaluation of \(K_{V,i}\) and \(K_{M,i}\). The calculation of the shear force and bending moment is based on the geometrical compatibility equations, the constitutive relations of the diagonals, and finally the equilibrium equations at the level of the floor.