## 2 Calculation Method of Main Cable Curve: Segmented Catenary Method
### Piecewise Catenary  
The loading form of the main cable of suspension bridge under the weight of the hanger, a cable clamp, an anchor head, and the main girder is shown in Figure 2.  
In the \(i\)-section cable, the load on the cable is equivalent to a uniformly distributed load, as shown in Figure 3.  
Figure 1: Force diagram of the main cable under distributed load [27]. Figure 2: Scheme of the piecewise catenary of the main cable segment of a suspension bridge [27]. According to Figure 3, the force can be formulated as  
\[qds=q_{y}dx. \tag{4}\]  
Substitute Equation (4) into Equation (3), we have  
\[\frac{d^{2}y}{dx^{2}}=-\frac{q}{H}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}. \tag{5}\]  
Assume \(\frac{dy}{dx}=z\), and it can be written as  
\[x=-\frac{H}{q}\ln\Bigl{(}z+\sqrt{1+z^{2}}\Bigr{)}+t=-\frac{H}{q}sh^{-1}z+t, \tag{6}\]  
\[z=\frac{dy}{dx}=sh\Bigl{(}-\frac{q}{H}x+\frac{q}{H}t\Bigr{)}, \tag{7}\]  
where \(t\) represents the integral constants related to boundary conditions. Suppose that the differential equation is in catenary form:  
\[y=-\frac{H}{q}ch\Bigl{(}-\frac{q}{H}x+\frac{q}{H}t\Bigr{)}+m. \tag{8}\]  
As shown in Figure 3, the segmented cable is fixed at two points A(0, 0) and B(\(l,c\)), and when the coordinates of points A and B are substituted into Equation (8), we can formulate  
\[y=-\frac{H}{q}ch\Bigl{(}\frac{qt}{H}\Bigr{)}+m=0, \tag{9}\]  
Figure 3.0  
Then, we can obtain  
Figure 3.1  
Assume \(\beta=\frac{ql}{2H}\), and we derive  
Figure 3.2  
Furthermore, it can be written as  
Figure 3.3  
Figure 3.4 Scheme of cable force of \(i\)-section [27]. Figure 3.5  
Assume \(\alpha=sh^{-1}\Big{(}\frac{c\beta}{Ish\beta}\Big{)}+\beta\); then, we have  
Figure 3.6  
Finally, the analytical solution of catenary equation is obtained as  
Figure 3.7  
where \(\alpha=sh^{-1}\Big{(}\frac{c\beta}{Ish\beta}\Big{)}+\beta\) and \(\beta=\frac{ql}{2H}\). Based on the continuous boundary requirements, the shape of the main cable can be calculated as follows and the main cable can be divided into \(n\) sections. The equation of the main cable of \(i\)-section can be written as  
Figure 3.8  
where \(\alpha_{i}=sh^{-1}\Big{(}\frac{c_{i}\beta_{i}}{Ish\beta_{i}}\Big{)}+\beta_{i}\) and \(\beta_{i}=\frac{ql_{i}}{2H}\). As for the deformation compatibility condition,  
Figure 3.9  
Any point on the main cable should pass through a given point, and the given point of the main cable could be regarded as a special point. When passing through this point in the middle of the span, each node of the main cable should meet the boundary force balance condition as  
Figure 1:0  
Hence, the iterative calculation formulas can be established according to Equations (16)-(19). The detailed iteration format is as follows [28]:  
As \(q\), \(l\), and \(f\) (the rise of the main span) are known, we then assume  
Figure 1:1  
Figure 1:2  
Then,  
Figure 1:3  
Setting \(i=1\), then,  
Figure 1:4  
Figure 1:5  
Figure 1:6Combining with Equation (19), the next segments \(\alpha_{2}\) and \(\beta_{2}\) are established, and then, \(c_{2}\), \(c_{3}\),... \(c_{n}\) are obtained in turn. Using deformation compatibility conditions,  
Figure 1:7  
it is determined whether Equation (26) is true. If not true, the value of \(p_{i}\) is adjusted to reiterate from Equation (22) again. If true, it continues to calculate whether to pass the control point. If it does not pass, \(H_{0}\) is changed and reiterated from Equation (22). Until it passes, the calculation is over. The iterative process can be regarded by solving nonlinear equations:  
Figure 1:8