## 5 Strategies for Tackling Excessive Inter-Story Drifts and Stability Phenomena in Diagrids  
In the previous Section, we have seen that the external diagonal layout can be properly modified to meet the necessary stiffness and strength requirements. Accordingly, the external mega-tracings can extend over multiple stories. As pointed out by [PERSON] et al. [11], this can give rise to two local structural issues, which need to be carefully addressed by the designer: (a) the instability of interior columns, and (b) excessive inter-story drifts. Both of them are mainly due to the lack of flexural  
Figure 12: Diagrid structures with vertical, convex and concave facades. Used with permission of Taylor & Francis Ltd. (www.tandfonline.com), from [PERSON] et al. [36]. resistance in the diagonals. This section investigates these local issues, as reported in the fundamental work of [PERSON] et al. [11]. The first local issue is shown in Figure 13a, where a 3-story diagrid module is represented. Elements A and B represent the external diagonal and the interior column, respectively. The column usually extends over the full height of the building and is subjected to high compression forces due to gravity loads, which might induce Eulerian buckling. The column's resistance to the lateral buckling mode relies on the external diagrid structure, which fully braces the interior column only at the panel points. Within the module height, the multi-story buckling mode is only prevented by the flexural resistance of the diagonals. If this is not enough, the multi-story sway mechanism takes place, which can occur at lower buckling loads than the 1-story mode. Based on a simplified analytical formulation, [PERSON] et al. [11] propose a simple equation in order to check whether the flexural resistance of the diagonals is sufficient to the purpose, specifically:  
\[n_{dg}I_{dg}>\left(k^{2}-1\right)\sin\theta n_{col}I_{col}, \tag{20}\]  
\(n_{dg}\) and \(I_{dg}\) being the number of diagonals along the perimeter and their inertia moment, respectively, \(k\) the number of intra-module stories, \(\theta\) the diagonal inclination, \(n_{col}\) and \(I_{col}\) the number and inertia moment of the gravity columns, respectively. If Equation (20) is not satisfied, the internal columns buckle in a multi-story sway mode (Figure 13a). In this case, either the columns are designed to sustain greater buckling loads or a secondary system is necessary. The second local issue is related to the excessive inter-story drift of intra-module floors. As in the case of column stability, the lateral displacements of these floors rely on the flexural resistance of the mega-diagonals. Based on the scheme reported in Figure 13b, a simple expression is obtained by [PERSON] et al. to assess the need for additional systems in the limitation of inter-story drifts [11]:  
\[\frac{\left(\sum_{i=1}^{k-1}F_{i}\right)\sin\theta L^{2}}{24 EI_{dg}n_{dg}}< \frac{500-\alpha}{500\alpha}, \tag{21}\]  
\(F_{i}\) being the horizontal force applied at the \(i^{\text{th}}\) intra-module floor, \(L\) the total span of the diagonal, \(E\) the elastic modulus, \(\alpha\) the limiting factor for the inter-story drift (usually \(\alpha=300\)), and \(k\), \(I_{dg}\) and \(n_{dg}\) with the same meaning reported above. If Equation (21) is not satisfied, either the inertia moment of the diagonals \(I_{dg}\) is increased or, again, an additional structural system is needed. Since the diagonals in diagrid systems are usually designed to carry only axial load, their flexural resistance is often not enough to prevent the multi-story sway mode of interior gravity columns, as well  
Figure 13: Local issues in the design of diagrid tall buildings: (a) stability of interior columns; (b) excessive inter-story drift of intra-module floors. Used with permission from [PERSON] et al. [11]. as the excessive inter-story drift of intra-module floors. For this reason, [PERSON] et al.