## 4 Search Results for Spine Biotensegrity Models
### Analysis Results Using Approach One  
Under approach one, a random choice of each stage's twist angle ranging from 10\({}^{\circ}\) to 60\({}^{\circ}\) has been selected for the form-finding analysis for the spine biotensegrity models. The symbol \(\alpha_{i}\) is defined by the twist angles for the \(i\)-th stage's triangular cells. Arrangement of the spine biotensegrity models with \(\alpha_{i}\geq 60^{\circ}\) is found to be not feasible due to the problem of closely spaced struts. Figure 11 shows the form-finding analysis with the application of the same twist angle 10\({}^{\circ}\) to 50\({}^{\circ}\) for all the triangular cells in the spine biotensegrity models (i.e., cases 10101010, 20202020, 30303030, 40404040, and 50505050). The case 10101010 denotes the model as \(\alpha_{1}\)=\(\alpha_{2}=\alpha_{3}=\alpha_{4}=10^{\circ}\). The cables in compression are slacked and should be removed. The slackened cables of SB1 are indicated by the red line (Figure 11). From the form-finding analysis, it is found that SB1 with \(\alpha_{1}=\alpha_{2}=\alpha_{3}=\alpha_{4}=50^{\circ}\) yields the least degree of slackness (9 slackened cables). Larger twist angles tend to reduce the number of slacked cables, but at \(\alpha_{1}=\alpha_{2}=\alpha_{3}=\alpha_{4}=20^{\circ}\), the number of slacked cables is maximum. For 10\({}^{\circ}\leq\alpha_{i}\leq 40^{\circ}\), slackened cables are found to occur throughout the model. For \(\alpha_{i}=50^{\circ}\), more slackened cables are found to occur at the first and third stages. The information about slackened cables at each trial case is an important input in the iterative analysis. Figure 11: Effect of same twist angles on numbers of slackened cables for SB1. Figure 12 shows the analysis with the application of various combinations of twist angles for triangular cells within the model (i.e., cases 10505050, 20505050, 30505050, and 40505050). It is found that cases where the \(\alpha_{1}\) for the triangular cell at stage 1 is less than 50\({}^{\circ}\) lead to a lesser number of slackened cables in comparison with other combinations of \(\alpha_{2}\), \(\alpha_{3}\), \(\alpha_{4}\) (Figure 12a). A similar effect of such combinations of twist angles is found in SB2 and SB3. When \(\alpha_{1}\) changes while the other \(\alpha_{i}\) are held at 50\({}^{\circ}\), the numbers of slackened cables ranged from three to nine for SB1, SB2, and SB3 are found (Figure 12b). In the model SB1 and SB3, the numbers of slackened cables are maximum at \(\alpha_{1}=20^{\circ}\). In the case of \(\alpha_{1}=10^{\circ}\) in SB1 and \(\alpha_{1}=40^{\circ}\) in SB3, the number of slackened cables is minimum. On the other hand, SB2 is  
4 to node 21). The controllable range of the nodal coordinates is less than 15 mm in the y direction. Few cases (i.e., cases A1 to H1) that are close to the self-equilibrated state are chosen to demonstrate the feasibility of the form-finding approach. Figure 14 shows the numbers and positions of slackened cables in SB1. It is seen that SB1 still preserves the spine anatomical parameters, especially the natural curvature of the spine. It is also important to note that the position of slackened cables changes even though the modifications of nodal coordinates are small. Table 1 presents the axial forces of the potential slackened cables of SB1. As can be seen in Table 1, although the slackened cables 40, 50, 59, and 68 may change into tension state easily, the remaining cables (particularly cable 51 and 65) remain slackened in most of the cases. In the case of B1, it is apparent that the case with the modification of node 5 instead of node 6 reduces the degree of slackness in most of the slackened cable (especially in cables 40, 42, 57, 65, and 67). Additionally, the modification of node 19 in case F1 clearly shows the effect of further reducing the degree of slackness in the slackened cables 51, 59, 65, and 66 compared to case E1. However, such modification in case F1 also leads to an increase in the degree of slackness in slackened cables 42 and 67.