## 3 Methods for the Structural Analysis of Diagrid Tall Buildings
\(\{M_{x}\}\) and \(\{M_{y}\}\) contain the out-of-plane moments acting in the X and Y directions, respectively, while \(\{\varphi_{x}\}\) and \(\{\varphi_{y}\}\) are the corresponding out-of-plane rotations. Finally, \(\{M_{z}\}\) and \(\{\varphi_{z}\}\) denote the torque moments and rotations acting around the vertical axis. Based on the expansion of the force and displacement vectors reported in Equation (13), the \(6N\times 6N\) stiffness matrix is partitioned accordingly, where each \(N\times N\) submatrix relates a force-moment vector to the generic displacement-rotation vector. The procedure for the calculation of the stiffness matrices is grounded in the displacements  
Figure 6: Scheme for the calculation of (**a**) shear rigidity; (**b**) bending rigidity, according to the modular method (MM). Used with permission from [PERSON] and [PERSON] [17]. method, and is similar to the scheme adopted by [PERSON] and [PERSON] in the MM [17]. The stiffness coefficients are obtained by applying unitary displacement-rotations at the floor levels, and evaluating the total reacting force-moments according to the compatibility, constitutive and equilibrium equations. The MBM is more general than the MM, since it does not only consider shear and bending rigidities (contained in the matrices \([K_{Fx,ox}]\), \([K_{Yy,yy}]\), \([K_{Mx,ox}]\) and \([K_{Mx,ox}]\)), but also the vertical and torsional ones, i.e., \([K_{Fx,0z}]\) and \([K_{Mx,yz}]\). Besides these 6 sub-matrices that lie on the diagonal of the stiffness matrix, the MBM also evaluates another 30 mixed submatrices, although only 15 of them need to be computed due to the symmetry properties of \([K]\). For regular-form diagrids, most of these out-of-diagonal matrices are zero; nevertheless, the evaluation of the matrices \([K_{Fx,ox}]\), \([K_{Yy,yy}]\), \([K_{Mx,ox}]\) and \([K_{Mx,xy}]\) is extremely important, as they contain the information regarding the coupling between shear and bending stiffnesses, and therefore concur with the correct definition of the lateral deflection. After the complete calculation of the stiffness coefficients of the 21 non-identical submatrices in Equation (13), the application of forces and moments at the floor levels leads to the evaluation of the corresponding displacements and rotations, through the inversion of Equation (13). Eventually, knowing the deformation of the structure, the compatibility and constitutive equations are applied once again, in order to find the axial forces in the diagonals. The MBM is applied to perform the structural analysis of the double-curvature Swiss Re Tower, and comparisons with FEM calculations show the consistency of the suggested method in the evaluation of both lateral and vertical displacements, as well as torsional rotations [18]. The MBM has not only been developed for the structural analysis of diagrids, but also to investigate the interaction between a diagrid tube and other resisting elements embedded within the building. To do so, the MBM has been built within the framework of the General Algorithm (GA), a matrix-based analytical methodology proposed in 1985 for the preliminary analysis of tall buildings [19]. The GA was first developed for the analysis of 3D civil buildings with moment resisting frames and closed-section shear walls. Later on, open-section shear walls were also taken into account [20], which observe [PERSON]'s theory of deformation and exhibit the warping effects typical of thin-walled structures [21]. In recent years, the GA has also permitted the study of the interaction between structures of different heights [22], secondary effects in tall buildings [23], and unconventionally-designed systems such as tapered and twisted towers [24]. It has also been applied to investigate the dynamic behavior of tall buildings [25], as well as real case studies in Northern Italy [26; 27]. The framework of GA takes into account only 3 DOFs per floor, namely the two horizontal displacements and one torsional rotation. To make the MBM suitable for insertion into the GA, [PERSON] et al.