## 10 New Evolutions of Grid Structures: Hexgrids and Beyond
The gravity loads are equally distributed between the central core and the perimetral hexagrid, regardless of the specific hexagonal pattern, whereas 50-80% of the lateral loads are absorbed by the hexagrids, with differences depending on the arrangement of the hexagon module [64]. Hexagrids are also investigated under dynamic loading conditions, such as earthquakes, as well as under arbitrary member removals, to assess their resistance against progressive failure. In [65], [PERSON] and [PERSON] show that the shear mode deformation in hexagrids is usually greater than that occurring in diagrids, and hexagrids exhibit greater ductility under dynamic loadings. In a following paper, 28- and 48-story tall buildings with diagrid and hexagrid solutions are investigated in terms of resistance against progressive collapse, upon removal of corner elements [66]. The outcomes of non-linear static and dynamic analyses show that, although the specific geometrical configurations play an important role, hexagrids seem to be less vulnerable to progressive failure than diagrids, as they show greater potential for force redistribution. Hexagrids are not the only further development of grid tubular systems in tall buildings. [PERSON] et al. [67] investigate the efficiency of hexagrids compared to another grid system, the pentagrid. The latter is based on the arrangement of various pentagons on the surface of the building, where all the elements are designed to share a similar amount of stress. From the outcomes of the structural analysis, the authors find that the pentagrid is more structurally efficient than the hexagrid, although the cost of constructability of the pentagrid might be superior [67]. Other grid evolutions also include octagrids and Voronoi-like grid systems. Voronoi tessellation has been exploited in recent works as a new solution for grid tubular systems [68; 69; 70]. [PERSON] and [PERSON] focus their attention on the evaluation of the mechanical characteristics of irregular Voronoi-like patterns for tall buildings [69]. Starting from a regular hexagrid solution, irregularity in the pattern is applied through random parametric generation, to realize more  
Figure 18: (a) Analogy between the hexagrid tube and an orthotropic solid membrane; (b) hexagonal module and unit cell. Used with permission from [PERSON] et al. [63]. irregular building models (Figure 19a). The effect of the varying-density pattern in the irregular Voronoi-like grid is also taken into account by the researchers (Figure 19b). Static and dynamic analyses are carried out on square 351-m tall buildings. The outcomes reveal that cell irregularities do not affect the lateral stiffness significantly, and that the gradual rarefication of the pattern is a suitable strategy to optimize the lateral response [69]. The mathematical and numerical framework for the stiffness homogenization procedure of Voronoi-like grid tubes is thoroughly presented in [70], where the authors define the concept of the testing volume element (TVE), which replaces the RVE used in regular hexagrid structures. Based on numerical analyses, the polynomial expressions of the correction factors for the mechanical properties of the homogenized tube, when dealing with irregular Voronoi-like grids, are proposed. FE calculations for various 351-m buildings are also carried out to investigate the efficiency of different Voronoi-like patterns [70].