# An Improved Analytical Algorithm on Main Cable System of Suspension Bridge
## 2 Segmental Catenary Theory of Main Cable
### Basic Equations  
An elastic catenary cable element has been derived from the exact solution of the elastic catenary cable equation, deformed due to its self-weight [32; 33]. It can be formulated in three dimensional coordinates, but only two-dimensional formulation is described in this study.  
Consider a cable segment suspended between points \(i(x_{i},y_{i})\) and \(j(x_{j},y_{j})\), as shown in Figure 1. It is assumed that the cable:  
1. is perfectly flexible and can sustain only tensile forces;
2. is composed of a homogeneous material which is linearly elastic;
3. is subjected to a uniform distributed load \(q\) along the cable length; and,
4. the tensile stiffness of the cable is calculated using the cross-section before deformation.  
The relative distances between two nodes \((i,j)\) along the global \(x\), \(y\) axis, are denoted as \(l\) (\(l=x_{j}\)\(-\)\(x_{i}\)) and \(h\) (\(h=y_{j}-y_{i}\)), respectively, in Figure 1, which can be expressed as a function of the global nodal force \(H_{i}\) and \(V_{i}\) at the node \(i\) as:  
\[l=-\frac{H_{i}\cdot S_{0}}{EA}-\frac{H_{i}}{q}\left\{\ln\left(V_{i}+\sqrt{H_{i }^{2}+V_{i}^{2}}\right)-\ln\left(V_{i}-S_{0}\cdot q+\sqrt{H_{i}^{2}+\left(V_{i }-S_{0}\cdot q\right)^{2}}\right)\right\} \tag{1}\]  
\[h=\frac{q\cdot S_{0}^{2}-2V_{i}\cdot S_{0}}{2 EA}-\frac{1}{q}\left[\sqrt{H_{i} ^{2}+V_{i}^{2}}-\sqrt{H_{i}^{2}+\left(V_{i}-S_{0}\cdot q\right)^{2}}\right] \tag{2}\]  
Figure 1: An elastic catenary cable segment.  
The force equilibriums of the elastic catenary cable require that:  
\[\left\{\begin{array}{l}-H_{i}=H_{j}=H\\ V_{j}=-V_{i}+S_{0}q\\ T_{i}=\sqrt{H_{i}^{2}+V_{i}^{2}}\\ T_{j}=\sqrt{H_{j}^{2}+V_{j}^{2}}\end{array}\right. \tag{3}\]  
Equations (1) and (2) are defined as the basic equations for segmental catenary cable, showing the relation between the segmental forces and geometric parameters. Generally, the main cable is divided into several segments (number \(N\)), each segment establishes two basic equations, in total 2 times of \(N\) equations are obtained for the whole main cable system. In Equations (1)-(3), \(E\) is the elastic modulus; \(A\) is the cross sectional area, \(q\) is the self-weight of the unstressed main cable; \(l\) represents the span length of the cable segment, \(h\) represents the elevation difference of two ends, and \(S_{0}\) represents the unstressed length of cable segment; \(T_{i}\), \(T_{j}\) are the cable tension at the left (\(i\)) and right (\(j\)) ends of the cable segment, respectively; \(H_{i}\) and \(H_{j}\) are the horizontal component of cable tension at the left (\(i\)) and right (\(j\)) ends of the cable segment, respectively; and, \(V_{i}\) and \(V_{j}\) are the vertical component of cable tension of the left(\(i\)) and right (\(j\)) ends of the cable segment, respectively.  
From Equations (1) and (2), it can be found that for a cable segment with determined \(S_{0}\), \(H\), and \(V_{i}\), the length \(l\), and high difference \(h\) can be easily obtained; similarly, for a cable segment with determined \(S_{0}\), \(l\), and \(h\), the internal forces \(H\) and \(V\) can be easily solved. Thus, only three independent variables exist in these five variables (\(S_{0}\), \(H\), \(V_{i}\), \(l\) and \(h\)).