## 2 Establishment of Basic Differential Equation
### Deformation Compatibility Equation  
There are seven unknowns in the balance differential Equation (9) of the three-tower self-anchored suspension bridge, such as \(\eta_{1},\eta_{2},\eta_{3},\eta_{4},\mathrm{H_{p1}},\mathrm{H_{p2}}\) and \(\mathrm{H_{p3}}\). Thus, three deformation compatibility equations are needed to solve the problem. In the deformation coordination equation of multi-tower self-anchored suspension bridge, the longitudinal displacement of the main tower top and the axial compression deformation of the stiffening girder should be considered. According to the deformation coordination conditions of three-tower self-anchored suspension bridge, three deformation coordination equations can be established. The horizontal projection shortening between the anchor points at both ends of the main cable is the axial compression of the stiffening beam, as shown in Equation (12). \[\sum_{i\ =\ 1}^{4}\left(\frac{H_{pi}}{E_{c}A_{c}}\int_{0}^{\mathrm{i_{1}}} \frac{dx}{\cos^{3}\varphi}\pm\alpha_{t}t\int_{0}^{\mathrm{i_{1}}}\frac{dx}{ \cos^{2}\varphi}-\int_{0}^{\mathrm{i_{1}}}\frac{dy_{i}}{dx}d\eta_{i}\ =\ \Delta\right. \tag{12}\]  
The relative displacement of the top of the left tower and the middle tower is the horizontal projection distance variation of the main cable of the left main span, as shown in Equation (13). \[\frac{H_{p2}}{E_{c}A_{c}}\int_{0}^{\mathrm{i_{2}}}\frac{dx}{\cos^{3}\varphi} \pm\alpha_{t}t\int_{0}^{\mathrm{i_{2}}}\frac{dx}{\cos^{2}\varphi}-\int_{0}^{ \mathrm{i_{2}}}\frac{dy_{2}}{dx}d\eta_{2}\ =\ \Delta_{2}-\Delta_{1} \tag{13}\]The relative displacement between the middle tower and the top of the right tower is the horizontal projection distance variation of the main cable of the right main span, as shown in Equation (14). \[\frac{\mathrm{H}_{\mathrm{p3}}}{\mathrm{E}_{\mathrm{c}}\mathrm{A}_{\mathrm{c}}} \int_{0}^{\mathrm{h}_{\mathrm{3}}}\frac{\mathrm{d}\mathrm{x}}{\cos^{3}\varphi} \pm\alpha_{\mathrm{t}}\mathrm{t}\int_{0}^{\mathrm{h}_{\mathrm{3}}}\frac{ \mathrm{d}\mathrm{x}}{\cos^{2}\varphi}-\int_{0}^{\mathrm{h}_{\mathrm{3}}}\frac{ \mathrm{d}\mathrm{y}_{\mathrm{3}}}{\mathrm{d}\mathrm{x}}\mathrm{d}\eta_{\mathrm{ 3}}\ =\ \Delta_{3}-\Delta_{2} \tag{14}\]  
where: \(\mathrm{E}_{\mathrm{c}}\mathrm{A}_{\mathrm{c}}\) = the axial stiffness of the main cable. \(\Phi\) = the horizontal inclination angle of the main cable, \(\cos\varphi\ =\ \frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\mathrm{s}}\). \(\alpha_{\mathrm{t}}\)t = the expansion coefficient of main cable temperature line; \(\mathrm{t}\) = the main cable temperature change; \(\Delta\) = the axial compression of the stiffening beam; \(\Delta_{1}\) = he horizontal displacement of the left tower top; \(\Delta_{2}\) = the horizontal displacement of the middle tower top; \(\Delta_{3}\) = the horizontal displacement of the right tower top. The left three items of Equations (12) to (14) respectively indicate the change of horizontal projection distance of the main cable caused by the stress, temperature change and deflection of the main cable. Compression of stiffening beam between anchorage points at both ends of main cable is negative, as shown in Equation (15). \[\Delta\ =\ -\mathrm{H}_{\mathrm{p1}}\sum\limits_{\mathrm{i}}^{4}\frac{\mathrm{ l}_{\mathrm{i}}}{\mathrm{E}_{\mathrm{i}}\mathrm{A}_{\mathrm{i}}}\pm\sum\limits_{ \mathrm{i}}^{4}\mathrm{l}_{\mathrm{i}}\alpha_{\mathrm{tbi}}\cdot\mathrm{t}_{ \mathrm{bi}} \tag{15}\]  
where: \(\mathrm{E}_{\mathrm{i}}\mathrm{A}_{\mathrm{i}}\) = the axial stiffness of each span stiffening beam. \(\alpha_{\mathrm{tbi}}\) = the thermal linear expansion coefficient of each span stiffening beam. \(\mathrm{t}_{\mathrm{bi}}\) = the temperature change of each span stiffening beam. Therefore, only the influence of the horizontal unbalance force of the main cables on both sides of each tower top on the horizontal displacement of the tower top was considered, and the bending effect caused by the vertical force of the tower top is ignored. The horizontal force balance equation of the main tower is shown in Equation (16). \[