## 3 Analytical Models of Lattice Structures Built with Polygon Hollow Shapes
### Analytical Models for Predicting the Behaviour of Selected Polygon Hollow Structures
#### 3.2.4 The Gibson-Ashby Model of a Circular Hollow Cell  
A circular unit cell illustrating the two mutually orthogonal planar in-plane directions of loading for a circular hollow cell is shown in Figure 8.  
Equation (8) represents stiffness in either of one of the two planar in-plane directions [18]:  
\[E_{y}^{*}=E_{x}^{*}=\frac{8.329}{(1-v^{2})}\bigg{(}\frac{t}{R}\bigg{)}^{3}E_{s} \tag{8}\]  
where symbol \(v\) represents the Poisson's ratio of the walls of the hollow cell. The circular hollow cell under planar in-plane compression loads deforms due to the contraction of  
Figure 8: A planar circular unit hollow cell showing two mutually orthogonal planar in-plane directions of loading. Figure 7: A planar unit square hollow cell showing two mutually orthogonal planar in-plane directions of loading. the circular unit cells in the direction of the applied load, as well as stretching in the perpendicular direction, the direction without an applied load [2; 8; 17; 18]. Therefore, the loading and perpendicular directions of the circular hollow structure experience a reduction and increase in radius, respectively, upon the application of the load. This, converts the circular cell into an ellipse [18]. From a comparison of Equations (5)-(8), it is evident that the circular hollow structure ranks first with regard to its transverse stiffness for values of the radius (\(R\)) of the cell equal to the lengths (\(L\)) of the other types of polygon hollow cells considered here. This is because, compared to polygon structures with corners, circular shapes transfer stress more uniformly. Stress in a circular hollow design tends to be distributed uniformly along its circumference, resulting in an efficient load transfer [2; 8]. Stress concentrations are often observed at the vertices of polygonal structures with non-circular unit cells, which in turn serve as weak points. Given that circular designs do not have corners, there is a reduced possibility of stress concentrations, leading to them being more resilient against localised deformation and deflection [8]. It is clear from the analytical models reviewed here that the order of the ranking of the four polygonal hollow cells goes from circular to hexagonal, triangular, and square cells, in decreasing order of transverse stiffness. Analytical models describing the load-bearing capacity of three-dimensional (3D) polygon hollow structures loaded in the cell walls' in-plane directions are lacking in the available literature. Presently available analytical models only describe the load-bearing capability of 3D polygon hollow structures for loads applied in the cell wall out-of-plane direction [2; 4; 7; 8; 16; 17; 18; 19; 23; 26]. This is due to the fact that all the identified studies [17; 27; 28; 29; 30; 31; 32; 33; 34; 35] investigating the crashworthiness of polygonal hollow structures are based on loading in the cell wall out-of-plane direction, as a result of its high capacity to absorb strain energy compared to loading in the cell wall in-plane directions. Therefore, there is a need to develop analytical models that describe the load-bearing capability of polygon hollow structures loaded along the cell wall in-plane direction. [PERSON] and [PERSON] [18] highlighted the fact that the cell wall in-plane loading of polygon structures discussed here could be simplified and represented by a rectangular plate or beam model subjected to either direct or buckling loads based on the \(t/L\) ratio that was used. They also highlighted the fact that the cell wall in-plane loading of polygon structures resulted in relatively smaller changes in the load-bearing capacity and stiffness, as the cell wall has insignificant in-plane anisotropy compared to the out-of-plane direction. Figure 9 shows a hexagonal structure subjected to a cell wall in-plane load [36]. Predicting the deformation behaviour of polygon structures subjected to cell wall in-plane loads accurately depends on the cell wall in-plane geometrical properties [17].