# A Novel Shape Finding Method for the Main Cable of
## 2 Shape Finding of the Main Cable with Eulerian Description  
### Governing Equations of Spatial Main Cable  
As shown in Figure 1a, a spatial main cable is described in a Eulerian coordinate system. The \(x\)-axis is the longitudinal direction along the bridge axis. The \(y\)-axis is in the vertical direction and the \(z\)-axis is in the lateral direction of the bridge. The tension forces of the cable at two ends are decomposed into three components, i.e., \(\left(H_{0},F_{y1},F_{z1}\right)\) and \(\left(H_{0},F_{y2},F_{z2}\right)\). The unit weight of the main cable is defined as \(q\). The projected length of the cable in the \(x\)-direction is \(L\). The total number of hangers is \(n\). The tension of \(i\)th (\(i=1,2,\,\ldots,\,n\)) inclined hanger is noted as \(T_{i}\), which consists of a vertical force of \(T_{yi}\) in the \(y\)-direction and a lateral component of \(T_{zi}\) in the \(z\) direction, as shown in Figure 1b. The connection position between the \(i\)th hanger and the main cable is at the coordinate of \(\left(x_{i},y_{i},z_{i}\right)\) or \(\left(x_{i},y_{i},-z_{i}\right)\). The other connection position at the bridge deck is defined as \(\left(x_{i},y_{di},z_{di}\right)\) or \(\left(x_{i},y_{di},-z_{di}\right)\). At the final state of the bridge, the lumped force transferred from each inclined hanger \(T_{i}\) can be determined from the dead load of the girder and self-weight of the hanger [25]. The distribution force due to the gravity of the main cable are also prescribed. To establish the equilibrium equations of a spatial main cable, four assumptions are employed in this study:  
1. The main cable of the suspension bridge is ideally flexible and unable to bear any bending moment. Only the axial tension is considered for the main cable. 2. The axial deformation of the main cable is tiny such that the area of the cross section for the main cable remains unchanged and the cable weight per unit length is a constant. 3. The linear elastic assumption for all materials is adopted. 4. All the loadings on the main cable are parallel to the \(y\)-\(o\)-\(z\) plane. The tension in the \(x\)-direction of the main cable is constant and defined as \(H_{0}\). The first three assumptions are commonly employed during the shape finding process at the preliminary design stage, e.g., [PERSON] et al. (2012); [PERSON] et al. (2016); [PERSON] et al. (2018). As reported by [PERSON] et al. (2018), the flexural stiffness of the main cable is insignificant and can be neglected during the construction process. Moreover, the main cable should work substantially below the yield stress such that the axial deformation is tiny and the variation of the cross-section area is negligible. For most modern suspension bridges, the hangers in are designed to incline only in the cross-sectional plane of the bridge. There is no or insignificant tension component of each danger along the longitudinal direction. Therefore, the tension of the main cable in the x-direction remains constant. The tension force along the axis of the main cable is assumed as \(\mathbf{F}\), which can be decomposed into the \(H_{0}\), \(F_{y}\) and \(F_{z}\) in the x-, y- and z-direction, respectively. The direction of \(\mathbf{F}\) is parallel to the cable element at the position of interest, i.e., \(d\mathbf{s}=(dx,\ dy,\ dz)\). The ratio between the \(F_{y}\), \(F_{z}\) and \(H_{0}\) can be expressed as:  
\[\frac{F_{y}}{H_{0}}=\frac{dy}{dx} \tag{1}\]  
\[\frac{F_{z}}{H_{0}}=\frac{dz}{dx} \tag{2}\]  
Taking the derivative of Equations (1) and (2) with the respect to \(x\) gives the form of  
\[\frac{dF_{y}}{dx}=H_{0}\frac{d^{2}y}{dx^{2}} \tag{3}\]  
\[\frac{dF_{z}}{dx}=H_{0}\frac{d^{2}z}{dx^{2}} \tag{4}\]  
Figure 1: The coordinate system and loading distribution of the spatial main cable: (**a**) Three-dimensional coordinate; (**b**) Two-dimensional coordinate of the cross section.