## 2 Simplified Approaches to the Preliminary Design of Diagrid Tubes
It has to be noted that, in the same paper, the same analysis has been applied to diagrids with a broader range of heights than previously analyzed, i.e., from 40 to 100 stories. As a result, the author proposes a new empirical equation for the \(s_{opt}\) for diagrid structures with aspect ratios greater than 6, i.e., \(s_{opt}=H/B-2\). In the cases investigated by [PERSON] [10; 12], it is found that the stiffness requirements drive the preliminary design, and the strength criteria are usually fulfilled. Only a few members in the leeward facade of the building are found to fail when the maximum allowable displacement is increased, i.e., \(\alpha<500\). However, thanks to the high rigidity of the diagonalized facades, which make the diagrid structure highly efficient, strength requirements may be of paramount importance, and in specific cases, they might even govern the design criteria, as suggested by [PERSON] et al. in [13]. In this paper, a simplified strength-based methodology for the preliminary design of diagrid tubes is provided. Figure 5 shows the adopted scheme for the development of the strength-based approach. Both gravity and lateral loads are applied to the building. Assuming that the internal core occupies 25% of the floor area, the diagrid carries 37.5% of the gravity load at the level of the m\({}^{\text{th}}\) module, \(Q_{m}\) (Figure 5a). This vertical loading condition generates a uniform compression state in all the \(n_{k}\) diagonals of the module, \(N_{m,k,Q}=0.375Q_{m}sin\theta/2n_{k}\). Lateral loads generate the bending moment \(M_{m}\) and shear force \(V_{m}\) at the module level. The former induces a uniform compression state in the diagonals of the leeward flange, a uniform tension state in the windward flange and a linear distribution of tension-compression axial forces in the webs, depending on the distance \(d_{i}\) of the \(i^{\text{th}}\) diagonal from the center of the floor (Figure 5b). This leads to the expression of the axial force \(N_{m,k,M}=\pm\,M_{m}d_{k}sin\theta/2\Sigma d_{i}\). Conversely, the shear force induces only tension-compression stresses in the web diagonals, therefore \(N_{m,k,V}=\pm\,V_{m}cos\alpha_{k}cos\theta/2\Sigma cos\alpha_{i}\), \(\alpha\) being the direction of the horizontal force with respect to the orientation of the diagrid module (Figure 5c). Considering all the loading conditions, one obtains the total axial force in the generic diagonal, as \(N_{m,k}=N_{m,k,Q}+N_{m,k,M}+N_{m,k,V}\). This value is finally used to define the member size, based on the tensile strength and the buckling compressive resistance of the diagonal. In the same paper, the authors also propose an analytical formulation, based on the [PERSON] and Timoshenko beam theories, to obtain an alternative optimal \(s\) value for the stiffness-based approach, i.e., \(s_{opt}=0.19H^{2}/tan\theta L^{2}\). The strength- and stiffness-based approaches are simultaneously applied in the preliminary design of a rectangular, 100-story tall diagrid tube, considering three different diagonal angles (\(64^{\circ}\), \(69^{\circ}\) and \(79^{\circ}\)), under both gravity and wind loads. The results show that, on the broad side of the building, strength requirements always prevail at the upper modules, whereas stiffness criteria drive the design of the lower modules. Conversely, on the shorter side, strength prevails over stiffness for the entire  
Figure 4.1 Scheme of the elementary diagrid module for the definition of the strength-based approach to the diagrid preliminary design, under: (**a**) gravity loads; (**b**) overturning moment; (**c**) shear force. Used with permission from [PERSON] et al. [13]. height of the building with \(\theta=64^{\circ}\), and stiffness prevails for \(\theta=79^{\circ}\), while in the case of \(69^{\circ}\) (which is close to the optimal angle inclination) the stiffness- and strength-based approaches provide almost the same result [13].