## 2 Simplified Approaches to the Preliminary Design of Diagrid Tubes
After carrying out the structural analyses on the designed buildings, it is found that the stiffness-based methodology leads to very efficient structures as regards the top lateral deflection, which is very close to the target value. However, this approach usually leads to unsatisfactory results in terms of the inter-story drift of the upper modules, as well as in terms of member strength demand-to-capacity ratio (DCR). In fact, besides the case of \(\theta=79^{\circ}\), where only 0.3% of the diagonals fail the strength requirements, 26% and 23% of them exhibit DCR greater than 1, for \(\theta=64^{\circ}\) and \(69^{\circ}\), respectively. On the contrary, adopting the strength-based design, the fraction of elements with DCR greater than 1 is 0%, 0.5% and 0.3%, for \(\theta=64^{\circ}\), \(69^{\circ}\) and \(79^{\circ}\), respectively. However, with this approach, unsatisfactory results are obtained in terms of lateral deformability, especially in the case of \(69^{\circ}\) and \(79^{\circ}\)[13]. Therefore, stiffness-based approaches might lead to unsatisfactory strength results, while strength criteria might fail stiffness requirements. A compromise should then be found depending on the specific building characteristics. In both cases, large inter-story drifts are usually found at the upper modules. This issue has been thoroughly analyzed by [PERSON] et al. [11], and tackled by providing special internal systems, like SBSs. More details on this will be provided in Section 5. Further investigation regarding the suitability of stiffness- and strength-based criteria for the preliminary design is subsequently developed in a broader range of diagrid structures [14; 15]. In [14], [PERSON] investigates the effect of both approaches on 90-story tall diagrid tubes, with diagonal angles of \(60^{\circ}\), \(70^{\circ}\) and \(80^{\circ}\). The results are in line with the previous findings. For smaller diagonal angles, strength usually drives the design, while the stiffness-based approach leads to inadequate DCR values. For greater angles, the stiffness-based design prevails, while strength criteria lead to excessive lateral deflections. In the range of the optimal angle, both criteria concur in defining the member sizes. More recently, the effect of both slenderness and diagonal angle has been taken into account simultaneously for the preliminary design [15]. Diagrids with aspect ratios ranging from 2 to 8, and diagonal angles from \(50^{\circ}\) to \(80^{\circ}\), are considered. It is found that, for those with an aspect ratio from 2 to 4, the design is mainly governed by strength requirements, independently of the diagrid angle, and the \"premium for height\" is mostly linear with the increase of slenderness. Conversely, for aspect ratios greater than 6, the design is mainly driven by stiffness, and the weight increases more than linearly with the slenderness. Aspect ratios around 5 are found to be the threshold, where stiffness- and strength-based designs provide comparable solutions [15]. Based on these results, it is concluded that, due to the extreme rigidity of the diagrid tubular system, it is not always possible to know a priori whether stiffness- or strength-based criteria should be considered for the preliminary design. Both approaches are necessary and unavoidable, and none of them should be used without the other [13]. The geometrical diagrid parameters--e.g., building aspect ratio and diagonal angle--drive the prevalence of one over the other. In any case, simplified approaches for the preliminary design represent an effective way to quickly define and assess the structural characteristics and performance of the diagrid.