# A Novel Shape Finding Method for the Main Cable of
## 3 Cases Study
### Case 2: 3D Spatial Main Cables of an Earth-Anchored Suspension Bridge  
Three dimensional spatial main cables are applied to the same suspension bridge used in Case 1. The spacings between two cables at the position of the anchorage and the IP points in \(z\)-direction are set as 63.0 m and 3.0 m, respectively, as shown in Figure 10. The vertical (\(y\)-direction) force components of all hangers given in Figure 5 are identical to Case 1. Same tolerances are also employed in this case, say \(\varepsilon_{1}=1.0\times 10^{-4}\) and \(\varepsilon_{2}=1.0\times 10^{-6}\). The iteration process is shown in Figure 11. Similar to the Case 1, five outer-layer iterations are implemented to obtain a convergent \(H_{0}\). All inner-layer iterations are less than five steps to meet the tolerance requirement. The main cable shape from the initial input coordinates in \(y\)- and \(z\)-directions to the final target configurations is illustrated in Figure 12. The final shape of the 2D parallel main cables in Case 1 is also plotted in Figure 12a. As can be seen, the shapes in \(y\)-direction of 2D planar cable and 3D spatial cable match well with each other. This can be explained by comparing the governing equations for the cable shape described by Equations (9) and (11). The difference of the cable shapes in \(x\)-\(o\)-\(y\) plane for parallel and spatial cases in mainly attributed to the \(z^{\prime}\). For the long-span suspension bridge studied in this case, the slope in \(z\)-direction of the spatial cable is small as compared to the slope in \(y\)-direction, resulting in that the effect of the \({z^{\prime}}^{2}\) in Equations (9) and (10) is insignificant. This can also be demonstrated from the final main shape in Figure 12. The calculated maximum \({y^{\prime}}^{2}\) and \({z^{\prime}}^{2}\) using the central difference method are 0.2593 and 0.0033, respectively.  
Figure 8: Comparison of the cable shape for 2D parallel main cable: (**a**) NFEM vs. SCM; (**b**) NFEM vs. Parabolic theory.  
Figure 9: Static displacement of the final 2D parallel main cable achieved by the FEM.  
Figure 13 illustrates the solution of hanger tension components in \(z\)-direction along the bridge span. The positive tension indicates that the inclination of the hanger tilts inward. The maximum positive tension in \(z\)-direction occurs at the middle span with the value of 331.637 kN. The negative tensions for outward inclined hangers at two side spans reach the maximum for first or end hanger. This is because the vertical component of the tension for first or end hanger as shown in Figure 5 is large. Figure 14 compares the cable shape between this study and the results obtained by SCM. The maximum relative differences at all hanger nodes in \(y\)- and \(z\)-direction are 0.0022% and 0.7318%, respectively,  
Figure 11: Outer-layer and inner-layer iteration results for 3D spatial main cables in Case 2.  
Figure 12: The main cable shape from the initial coordinates to the target configuration for 3D spatial main cables: (**a**) \(y\)-direction; (**b**) \(z\)-direction.  
Figure 10: Top view of the suspension bridge using spatial main cables (A half of the bridge).  
suggesting a reasonable agreement with each other. To further validate the accuracy of the final cable shape, a finite element model of the spatial main cable is also established. The static displacement under the action of self-weight and hanger tension forces is illustrated in Figure 15. It can be found that the maximum displacements in \(y\)- and \(z\)-direction of the main cable are 0.008 m and 0.012 m, which are sufficiently small and acceptable for the engineering applications.  
Figure 14: Comparison of the cable shape obtained from the NFEM and SCM for 3D spatial main cables: (**a**) \(y\)-direction; (**b**) \(z\)-direction.  
Figure 13: Hanger tension components in z-direction along the bridge span.