## 2 Establishment of Basic Differential Equation
### Equilibrium Differential Equation  
Considering the influence of the longitudinal stiffness of the main tower, the horizontal tension of the main cable under live load is not equal. Under the action of dead load, the micro-segment balance equation of the main cable of the left span is as follows:  
\[\mathrm{H_{gl}}\frac{\mathrm{d^{2}y_{1}}}{\mathrm{dx^{2}}}\ =\ \mathrm{g_{c}}+ \mathrm{f_{gl}} \tag{2}\]  
where: \(\mathrm{H_{gl}}\) = horizontal tension of main cable under dead load. \(\mathrm{y_{1}}\) = the vertical curve function of the main cable about \(\mathrm{x_{1}}\). \(\mathrm{g_{c}}\) = the dead weight of the main cable evenly  
Figure 1: Schematic diagram of three-tower self-anchored suspension bridge. distributed along the span; \(\rm f_{g1}=\) the main cable gets tension of each micro-segment of main cable when the cable force of the suspension cable is transmitted to the main cable. Assuming that the vertical curve of the stiffening beam under dead load is \(\rm z(x)\), and the axial force is \(\rm H_{g1}\), the equilibrium equation of the left span stiffening beam is Equation (3). \[\rm EI\frac{d^{4}z}{dx^{4}}+H_{g1}\;\frac{d^{2}z}{dx^{2}}\;=\;g_{b1}-f_{g1} \tag{3}\]  
where: \(\rm EI=\) the bending stiffness of stiffening beam; \(\rm g_{b1}=\) the dead load magnitude of stiffening beam. Substituting Equation (2) into Equation (3) to obtain the elastic Equation (4) of stiffened beam under dead load. \[\rm EI_{1}\frac{d^{4}z}{dx^{4}}+H_{g1}\;\left(\frac{d^{2}z}{dx^{2}}-\frac{d^{2} y_{1}}{dx^{2}}\right)\;=\;g_{b1}+g_{c} \tag{4}\]  
Under the action of live load \(\rm p_{1}(x)\), the vertical displacement of the stiffening beam and main cable on the left side span is \(\rm\eta_{1}(x)\), and the cable force of the suspension cable is \(\rm f_{1}\). The horizontal tension of the main cable is \(\rm H_{1}\;=\;H_{g1}+H_{p1}\), of which the horizontal tension of the main cable under live load is \(\rm H_{p1}\). Therefore, the micro-segment balance equation of the main cable under the combined action of dead load and live load is Equation (5). \[-H_{1}\;\left(\frac{d^{2}z}{dx^{2}}+\frac{d^{2}\eta_{1}}{dx^{2}}\right)\;=\;g _{c}+f_{1} \tag{5}\]  
The micro-segment equilibrium equation of stiffened beam under the combined action of dead load and live load is Equation (6). \[\rm EI_{1}\left(\frac{d^{4}z}{dx^{4}}+\frac{d^{4}\eta_{1}}{dx^{4}}\right)+H_{1 }\;\left(\frac{d^{2}z}{dx^{2}}+\frac{d^{2}\eta_{1}}{dx^{2}}\right)\;=\;g_{b1}+ p_{1}(x)-f_{1} \tag{6}\]  
Substituting Equation (5) into Equation (6) to obtain the elastic equation of stiffened beam under the combined action of dead load and live load, as shown in Equation (7). \[\rm EI_{1}\left(\frac{d^{4}z}{dx^{4}}+\frac{d^{4}\eta_{1}}{dx^{4}}\right)+H_{1 }\;\left(\frac{d^{2}z}{dx^{2}}-\frac{d^{2}y_{1}}{dx^{2}}\right)\;=\;g_{b1}+p_{ 1}(x)-g_{c} \tag{7}\]  
Subtract Equation (7) from Equation (4) to obtain the balance differential equation of the left span stiffened beam caused by live load, as shown in Equation (8). \[\rm EI_{1}\frac{d^{4}\eta_{1}}{dx^{4}}\;=\;p_{1}(x)+H_{p1}\;\frac{d^{2}y_{1}}{ dx^{2}}-H_{p1}\;\frac{d^{2}z}{dx^{2}} \tag{8}\]  
Using the same derivation method above, the equilibrium differential equations of other stiffened beams can be obtained. In addition, under normal circumstances, self-anchored suspension bridges usually adopt longitudinal floating system; that is, stiffening beams are not constrained in the longitudinal direction of the bridge. In the equilibrium state, \(\rm H_{p1}\;=\;H_{p4}\). Therefore, the balance differential equation of each span stiffening beam of three-tower self-anchored suspension bridge is shown in Equation (9).