# Multi-Directional Shape Change Analysis of Biotensegrity Model Mimicking Human Spine Curvature
## 4 Numerical Examples
### Deformed Modes  
Figure 15 shows the deformed configurations at the final state for the selected cases in the uni-directional mode (series Xn, Xp, Yn and Yp). There are three apices in the spine biotensegriy models, namely, AP1, AP2 and AP3, which play a vital role as joints that permit bending. Generally, all of the spine biotensegriy models approach the target coordinates either by bending or by a combination of axial and bending deformations in cases of the uni-directional mode (i.e., series Xn, Xp, Yn and Yp). In these series, bending deformation primarily around AP2 was observed for a displacement of 200 mm (Figure 15a-c) and around all Aps (AP1 to AP3) for displacements of 400 mm and 600 mm (Figure 15d,e). However, due to the natural curvature in the YZ plane view, AP1 allows for greater bending compared to AP2 in most cases of series Yp (Figure 15f,g). Axial deformation accompanied by bending deformation was observed in most of the cases with a displacement of 600 mm, where the models bend about AP1 and AP3 while the segment between the two apices stretches (Figure 15g,h). Additionally, the deformation of spine biotensegriy models through expansion and contraction were observed (Figure 15d). The shape change analysis case Xp400 of SB2 can be found in Video S1: Shape Change of Biotensegriy Model Mimicking Human Spine in the Supplementary Materials section. Figure 16 shows the final state for series Zn and Zp in cases of the uni-directional mode. Spine biotensegriy models in series Zn achieved the targets by means of bending or axial deformation or a combination of both. Conversely, spine biotensegriy models in series Zp exhibit solely axial deformation. In series Zn, SB1 and SB2 demonstrate bending deformation with obvious bending about AP2 (Figure 16a,b). The bending deformation is mainly towards the positive y-axis, with the thoracic kyphosis (refer to the initial configuration in Figure 3 and the deformed configuration at the final state in Figure 15b) falling on the opposite side. On the other hand, SB3 demonstrates bending deformation about two axes. Specifically, the segments between AP1 and AP2 demonstrate bending deformation about the y-axis, whereas bending about the x-axis was observed in the upper segments (Figure 16c,d). In series Zp, all of the spine biotensegriy models approach the target through the lengthening of cables resulting in axial deformation. It is noted that the spine biotensegriy models were stacked up with upper cell sunk into the lower cell, forming sag depth at all of the apices (Figure 3). At the initial state, the lower triangular surface of the upper cell (denoted as l.t. in Figure 16e-g) is located below the upper triangular surface (denoted as u.t. in Figure 16e-g). The height of l.t increased and gradually exceeded the height of u.t during the shape change analysis. Such height increment was demonstrated  
Figure 14: Convergence history for 28 shape change analysis cases in the tri-directional mode for series XY400 for SB2. firstly at AP3 (Figure 16e), which is then followed by AP2 and later AP1 (Figure 16f,g), with the increase in target displacement magnitudes. Deformations in terms of contraction and expansion were also observed in both series Zn and Zp. Figure 15: Deformed configuration (**a**–**h**) at the final step for cases with the uni-directional mode, series Xn, Xp, Yn and Yp. Figure 16: Deformed configuration (a–g) at the final step for cases with the uni-directional mode, series Zn and Zp. Figure 17 shows the final state for cases of the bi-directional mode. In cases with target displacement magnitude of 200 mm, all spine biotensegrity models approach the target coordinates by means of bending deformation, either about one axis (Figure 17a) or two axes (Figure 17b,c). For cases with target displacement magnitude of 400 mm, the spine biotensegrity models generally demonstrate bending deformation with slight axial deformation (Figure 17d,e). Displacements of AP1 and AP3, which resulted in bending deformations with double curvature, were observed.