## 2 Simplified Approaches to the Preliminary Design of Diagrid Tubes
In order to assess the relative contribution of bending versus shear deformation, [PERSON] et al. [10] introduce a non-dimensional parameter \(s\), given by the ratio of the bending to the shear contribution, i.e. :  
\[s=\frac{\chi^{*}H^{2}}{2\gamma^{*}H}. \tag{6}\]  
Combining Equations (5) and (6), and considering that the top lateral displacement is usually specified as a fraction of the total building height, i.e., \(u(H)=H/\alpha\) (\(\alpha\) usually being 500), one obtains the following relations between \(\gamma^{*}\), \(\chi^{*}\) and \(s\):  
\[\gamma^{*}=\frac{1}{(1+s)\alpha}, \tag{7}\]  
Figure 4.0  
Substituting Equations (7) and (8) into Equations (3) and (4), the member sizes can be obtained for the different values of the parameters. Adopting different \(s\) values leads to different preliminary sizing for the external diagonals. When \(s\) is extremely low, the shear deformation mode of the structure prevails over the bending mode, and this leads to excessive material usage in the flange members to limit the bending deflection. Conversely, when \(s\) is high, the bending deformation prevails, and the obtained cross-sectional areas are mainly governed by the web facades to limit the shear deformability. Therefore, an optimal value of \(s\) is shown to exist, \(s_{opt}\), which balances the need to limit both shear and bending deformability [10]. In this case, the member sizes at the higher stories are usually governed by the shear deformation, while the ones at the lower stories are mostly controlled by the bending deformation. The \(s_{opt}\) depends on the building aspect ratio--\((H/B)\)--and leads to the most efficient solutions that comply with the target maximum displacement while employing the minimum amount of material. For diagrid structures taller than 40 stories, with \(H/B\) greater than 5 and diagonal angles between \(60^{\circ}\) and \(70^{\circ}\), the empirical equation \(s_{opt}=H/B-3\) is proposed [10]. The other fundamental parameter that plays a key role in the preliminary design of diagrids is the diagonal inclination. Investigating a set of 20- to 60-story tall buildings, [PERSON] et al. show that, for diagrid structures with aspect ratios of about 7, the optimal angle is between \(65^{\circ}\) and \(75^{\circ}\), whereas for diagrids with aspect ratios of about 5 the optimal angle is lower, at around \(10^{\circ}\)[10]. This is due to the competition between shear and bending stiffnesses in governing the deformation mode, and their dependence on the diagonal angle. Shear rigidity is at a maximum when the inclination is about \(35^{\circ}\), while bending rigidity achieves its maximum value when the elements are vertical, i.e., \(\theta=90^{\circ}\). The optimal value for maximizing both shear and bending rigidity lies between these two. Since shear mechanism prevails in shorter buildings and bending prevails in taller ones, it is expected that, as the aspect ratio increases, the building behaves more like a bending beam, and the optimal angle thus increases [10]. This consideration has been strongly exploited in the analysis and design of diagrid systems, by considering various angle-based strategies and patterns for optimizing the diagrid performance. More details about this subject are reported in Section 4. The same stiffness-based approach reported in the previous paragraphs is also applied by [PERSON] to braced tubes in [12]. In this case, the shear force is carried by the external mega-diagonals, while the bending moment is carried by the perimeter vertical columns. Analyses based on 40- to 100-story tall braced tubes show that the optimal angle in this case is close to \(45^{\circ}\), and is less dependent on the building aspect ratio. This is due to the negligible involvement of external diagonals in carrying bending moment. For braced tubes with an aspect ratio greater than 6, [PERSON] suggests a different empirical equation for the optimal \(s\) value, i.e., \(s_{opt}=H/2B-1\)[12].