# A Novel Shape Finding Method for the Main Cable of
## 2 Shape Finding of the Main Cable with Eulerian Description
### Comparison with the Segmental Catenary Method  
The segmental catenary method is an exact solution for an element of the main cable deformed due to its self-weight. The lengths of each segment in \(x\)- \(y\)- and \(z\)-directions are solved as:  
\[l_{x}=\frac{F_{x}}{E_{c}A_{c}}l_{0}-\frac{F_{x}}{q}\left\{\sin\mathrm{h}^{-1} \left(\frac{F_{y}}{H_{0}}\right)-\sin\mathrm{h}^{-1}\left(\frac{F_{y}-ql_{0}}{ H_{0}}\right)\right\} \tag{34}\]  
\[l_{y}=\frac{F_{z}}{E_{c}A_{c}}l_{0}-\frac{F_{z}}{q}\left\{\sin\mathrm{h}^{-1} \left(\frac{F_{y}}{H_{0}}\right)-\sin\mathrm{h}^{-1}\left(\frac{F_{y}-ql_{0}}{ H_{0}}\right)\right\} \tag{35}\]  
\[l_{z}=\frac{F_{y}}{E_{c}A_{c}}l_{0}-\frac{F_{x}}{2E_{c}A_{c}}-\frac{H_{0}}{q} \left[\sqrt{1+\left(\frac{F_{y}}{H_{0}}\right)^{2}}-\sqrt{1+\left(\frac{F_{y} -ql_{0}}{H_{0}}\right)^{2}}\right] \tag{36}\]  
in which \(l_{x}\), \(l_{y}\) and \(l_{z}\) are the distances between the two nodes of a segment along the global \(x\)- \(y\)- and \(z\)-axes, \(F_{x}\), \(F_{y}\) and \(F_{z}\) are three components of the global nodal force, \(E_{c}\) is the elastic modulus of the main cable, \(A_{c}\) is the cross-sectional area, \(q\) is the self-weight of the main cable per unit length, \(l_{0}\) is the unstrained length of the cable segment. Figure 3 illustrates the iteration scheme of the segmental catenary method. For a two-pylon suspension bridge, a guess of the axial force \(F_{1}=\left(F_{x1},F_{y1},F_{z1}\right)\) at one end of the main cable is made first. The unstrained length \(l_{01}\) and \(l_{y1}\) and \(l_{z1}\) of first segment are obtained by solving the transcendental equations described in Equations (34) and (36). The axial force of next segment, which will be used to solve the \(l_{02}\) and \(l_{y2}\) and \(l_{z2}\) is then readily calculated by introducing the tensile force of the first hanger. \(F_{1}\) will be adjusted by comparing the coordinates of the IP at the other pylon in \(y\)- and \(z\)-directions, i.e., \(y_{IP}\) and \(z_{IP}\) with the calculated \(\hat{y}_{IP}\) and \(\hat{z}_{IP}\) as well as the \(y_{mid}\) with \(\hat{y}_{mid}\) to achieve a convergent solution.  
Figure 2: Flowchart of the iteration scheme for solving the cable shape.  
A comparison between the SCM and present NFEM is shown in Table 1. The SCM focuses on each cable segment to develop its analytical expression of the cable shape using algebraic equations (Equations (34) and (36)) in the Lagrangian coordinate system with respect the unstrained cable length \(l_{0}\). The present NFEM solves the differential equations of the whole cable in the Eulerian coordinate system with respect the bridge axis coordinate \(x\) using the nonlinear finite element approximation. The SCM requires the inputs of the axis stiffness of the cable, i.e., \(E_{c}A_{c}\) to determine the unstrained cable length while no information regarding the cable material or cross section is needed in the NFEM for solving the cable shape. It is noteworthy that the unstrained cable length, which is of great importance during the construction stage, can also be calculated using the NFEM by subtracting the elongation from the final length of the cable.