## 2 Simplified Approaches to the Preliminary Design of Diagrid Tubes  
The first simplified stiffness-based approach to the preliminary design of diagrid systems was proposed by [PERSON] et al. in 2007 [10]. It is based on the evaluation of the shear and bending stiffness of the diagrid modules, aimed at limiting the lateral deflection of the structure. The building is treated as a vertical cantilever beam, fixed on the ground and subjected to lateral loads. Accordingly, the building undergoes horizontal displacements, which depend on the stiffness of the diagrid tubular structure. For the sake of the preliminary design, the contribution of the internal cores to the lateral stiffness of the building is neglected, as they are only designed to carry gravity loads. Figure 3: Fundamental diagrid geometrical features: (**a**) diagrid module and basic triangular element, used with permission from [PERSON] and [PERSON] [8]; (**b**) diagrid tubular configuration, used with permission from [PERSON] and [PERSON] [9]. The elementary diagrid module is depicted in Figure 4. The diagrid module covers a height \(h\) with two triangular elements. The diagonals have a length \(L_{d}\), and their inclination with respect to the horizontal plane is \(\theta\). Depending on the loading direction, each facade can act either as a web or a flange. \(V_{i}\) and \(M_{i}\) are the shear force and bending moment acting on the level of the \(i^{\text{th}}\) module. These are carried by the web and flange diagonals, respectively. Diagonals are assumed to be pinned at their end, thus carrying only axial force, and remain in the linear elastic regime. In this way, the cross-sectional areas of the web and flange members are the only factors that need to be obtained in order to accomplish the preliminary design. The shear stiffness \(K_{T,i}\) and bending stiffness \(K_{B,i}\) of the \(i^{\text{th}}\) diagrid module link the shear force \(V_{i}\) and bending moment \(M_{i}\) to the module displacement \(\Delta u_{i}\) and rotation \(\Delta\beta_{i}\), respectively. By applying compatibility, constitutive and equilibrium equations, \(K_{T,i}\) and \(K_{B,i}\) are obtained as follows:  
\[K_{T,i}=2N_{w}\bigg{(}\frac{A_{d,w;i}E}{L_{d}}\bigg{)}\cos^{2}\theta, \tag{1}\]  
\[K_{B,i}=N_{f}\bigg{(}\frac{B^{2}A_{d,f,i}E}{2L_{d}}\bigg{)}\sin^{2}\theta, \tag{2}\]  
where \(N_{w}\) and \(N_{f}\) are the total number of diagonals in the web and flange facade, respectively, \(A_{d,w;i}\) and \(A_{d,f,i}\) the cross-sectional area of the web and flange members, \(E\) the elastic modulus of the diagonals and \(B\) the web dimension. The displacement \(\Delta u_{i}\) and rotation \(\Delta\beta_{i}\) are equal to the product of the module height \(h\) and the shear and bending deformation, \(\gamma\) and \(\chi\), respectively. Specifying the desired values of shear and bending deformation, \(\gamma^{*}\) and \(\chi^{*}\), the member dimensions can be easily obtained as [10]:  
\[A_{d,w,i}=\frac{V_{i}L_{d}}{2N_{w}E\eta\gamma^{*}\cos^{2}\theta}, \tag{3}\]  
\[A_{d,f,i}=\frac{2M_{i}L_{d}}{N_{f}B^{2}E\eta\chi^{*}\sin^{2}\theta}. \tag{4}\]  
Since the horizontal load can act in either direction, the maximum value of the cross-sectional areas from Equations (3) and (4) should be assigned to each diagonal, which can act as either a web or flange member. The desired values of \(\gamma^{*}\) and \(\chi^{*}\) are specified based on the desired deformation mode of the building. Assuming that the building sway mechanism is equivalent to the deformation of a cantilever beam, the lateral deflection at the top of the building \(u(H)\) can be written as follows:  
\[u(H)=\gamma^{*}H+\frac{\chi^{*}H^{2}}{2}, \tag{5}\]  
Figure 4: Scheme of the elementary diagrid module for the definition of the stiffness-based approach to the diagrid preliminary design. Used with permission from [PERSON] et al. [10]. where \(\gamma^{*}H\) and \(\chi^{*}H^{2}/2\) are the shear and bending contribution, respectively.