# A Novel Shape Finding Method for the Main Cable of
## 3 Cases Study  
### Case 1: 2D Parallel Main Cables of an Earth-Anchored Suspension Bridge  
An earth-anchored suspension bridge with 2D parallel main cables is utilized, as shown in Figure 4. The main span is 1666 m, and the side span is 500 m with a sag of 172.64 m in the main span. The designed sag-span ratio is 9.65. A single box girder with the width of the 49.7 m is adopted. The height of the bridge tower is 265 m and the elevations of the two IP points at the tops of the pylons are 267.414 m. There are two parallel vertical cable planes, each of which consists of a main cable as well as 29 hangers in the side span  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Object** & **Coordinate System** & **Governing Equations** & **Axis Stiffness of Cable** \\ \hline entry 1 & Cable segment & Lagrangian (with respect to \(l_{0}\)) & Algebraic equations & Yes \\ entry 2 & Whole cable & Eulerian (with respect to \(x\)) & Differential equations & No \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison between the SCM and NFEM.  
Figure 3: Flowchart of the iteration scheme for solving the cable shape of segmental catenary method.  
and 103 hangers in the main span. The spacing between two parallel cables in z-direction is 41 m. The cable segment lengths in the \(x\)-direction are 35 m + 28 \(\times\) 16 m + 17 m in the side span and 17 m + 102 \(\times\) 16 m + 17 m in the main span. The cable self-weight per unit length \(q=54.088\) kN/m. The elastic modulus and the cross area of the main cable are \(E_{c}=1.96\)E11N/m\({}^{2}\) and \(A_{c}=0.706\) m\({}^{2}\). The origin of the Eulerian coordinate system is set at the midpoint of the bridge with zero elevation.  
The nodal force or the tension of each hanger \(T_{yi}\) in \(y\)-direction is estimated from the dead loads of the stiffening girder by introducing the boundary conditions of the girder. The displacements of the intersection point between the girder and pylon in \(y\)-direction are coupled. The intersection point between the girder and transition pier is coupled in \(y\)- and \(z\)-directions as well as the rotation degree of freedom with respect to the \(x\)-axis. The estimated hanger tension forces in y-direction are shown in Figure 5. The maximum value of 3781.2 kN occurs at the first hanger in the side span while the most hangers have the tension of 2372.5.2 kN. The sag is given as \(f=172.64\) m at the beginning, which indicates \(y_{\text{mid}}=267.414-172.640\) = 94.774 m.  
Two-layer iterations are implemented to perform the shape finding process for the main cable by following the scheme in Figure 2. The outer-layer and inner-layer iteration tolerances are set as \(\varepsilon_{1}=1.0\times 10^{-4}\) and \(\varepsilon_{2}=1.0\times 10^{-6}\). Figure 6 illustrates the iteration results, in which a convergent horizontal component of the axial force, i.e., \(H_{0}\) is achieved in five outer-layer iteration steps. Between each adjacent outer-layer iterations, less than four-step inner-layer iterations are required for solving the cable shape. The main cable shape from the initial input coordinates of to the final shape finding result that corresponds the final outer- and inner-layer iteration is plotted in Figure 7. The shape finding using the commonly used segment catenary method (SCM) is also conducted as a benchmark for comparison purpose. Both the final main cable shapes obtained from this study and the SCM are given in Figure 8a, showing a reasonable agreement with each other. The maximum relative difference at all hanger nodes is 0.0011%. Moreover, the parabolic curve of the main cable in the middle span calculated by the \(y=y_{IP}+4f\big{(}x^{2}-L^{2}/4\big{)}/L^{2}\) is also compared with the present solution, as shown in Figure 8b. The difference between these two methods is also insignificant with the maximum relative error of 0.0948%. This  
Figure 4: Layout of the suspension bridge using parallel main cables.  
Figure 5: Hanger tension forces in \(y\)-direction along the bridge span.