## 4 Optimization of Diagrid Performance Based on the Geometrical Features
Figure 8: Different diagonal angle patterns diagrids: (**a**) varying-angle with steeper diagonals at the base; (**b**) uniform-angle; (**c**) varying-angle with steeper angle at the top, used with permission from [PERSON] [12]; (**d**) varying-angle with straight diagonals; (**e**) varying-angle with curved diagonals, used with permission from [PERSON] and [PERSON] [29]. In the solutions provided by [PERSON] with variable angles, the diagonals do not remain straight across their length over the full height of the building, because of their changing direction at the interface of two diagrid modules with different angles. To overcome this, [PERSON] et al. [31] propose a different strategy for the generation of varying-angle diagrid tubes. As shown in Figure 8d, a graphic approach is suggested for generating a varying-angle pattern with straight diagonals that extend over the full height of the building. This pattern is governed by two fundamental parameters, the top angle \(\theta_{1}\) and the bottom angle \(\theta_{2}\). The stiffness- and strength-based design criteria are applied to a set of 30- to 75-storey tall, varying-angle diagrids with straight diagonals, with aspect ratios ranging from 3.6 to 9. Several \(\theta_{1}\)-\(\theta_{2}\) combinations are considered to investigate the optimal solutions under gravity and wind loads. Based on the results, the following empirical formulas are suggested for the optimal values of \(\theta_{1}\) and \(\theta_{2}\), depending on the building's aspect ratio \(H\)/B:  
\[\theta_{1,opt}=\left\{ \begin{array}{ll}\theta_{2,opt},&H/B\leq 3.5\\ \frac{1}{\left(1+\ln\frac{H/B}{35}\right)^{\frac{H/B}{2}}}\left(\theta_{2,opt} -\arcsin\frac{1}{\sqrt{3}}\right)+\arcsin\frac{1}{\sqrt{3}},&H/B>3.5\end{array}  \right., \tag{16}\]  
\[\theta_{2,opt}=\arctan\frac{H/B}{1+0.475\sqrt{\frac{H/B}{4.75}}}. \tag{17}\]  
As \(H\)/B increases, the optimal bottom angle \(\theta_{2,opt}\) increases, while the optimal top angle \(\theta_{1,opt}\) decreases. A critical value of the aspect ratio, \((H\)/B\()_{crit}\), is found, which defines the interface between the efficiency of uniform- versus varying-angle diagrids, meaning that below \((H\)/B\()_{crit}\), uniform-angle diagrids are more efficient, while above this value varying-angle structures provide the most economical solutions. In this paper, \((H\)/B\()_{crit}\) is found to be 4.5-5; smaller than the value of 7 previously suggested by [PERSON] [12; 30]. This is mainly due to the different definition of the diagonal pattern. For aspect ratios less than \((H\)/B\()_{crit}\), the bottom angle rather than the top angle drives the design. Conversely, for greater aspect ratios, the top angle becomes one of the determining factors [31]. In a following paper, [PERSON] and [PERSON] [29] propose an additional diagrid configuration, where the varying-angle solution is obtained with curved diagonals (Figure 8e). In the same paper, they also consider seismic loads in the evaluation of the optimal diagrid pattern. It is found that, for varying-angle straight diagonals, the optimal bottom angle \(\theta_{2,opt}\) is not affected by the load type, thus Equation (17) holds also for seismic loads. However, the optimal top angle \(\theta_{1,opt}\) is always close to the lower limit for seismic loads, i.e., \(\theta_{1,opt}=\arcsin1/\sqrt{3}\), thus correcting Equation (16). In the case of diagrids with curved diagonals, they propose the following equations for the optimal angles, which are valid for both wind and seismic loads:  
\[\theta_{1,opt}=0.8\left(\frac{H/B}{8}\right)^{\frac{1}{8}}\theta_{2,opt}, \tag{18}\]  
\[\theta_{2,opt}=\arctan(H/B), \tag{19}\]  
\(H\)/B being in the range 3.6-9. With these values, the optimal top angle \(\theta_{1,opt}\) lies in the range 50-70\({}^{\circ}\), greater than the top angles in diagrids with straight diagonals (35\({}^{\circ}\)-45\({}^{\circ}\)). Thus, the lesser difference between \(\theta_{1,opt}\) and \(\theta_{2,opt}\) in this case results in a small curvature of the diagonals [29].