## 3 Differential Equilibrium Equations
### Cable-Stayed Section
\), and the value of each element is as follows:  
\[b_{11} =-\frac{\big{(}\lambda^{2}-\beta^{2}\big{)}ch(\beta x)\sin(\lambda x )+2\beta Ash(\beta x)\cos(\lambda x)}{2\beta\lambda}\] \[b_{21} =\frac{\big{(}\lambda^{2}-\beta^{2}\big{)}ch(\beta x)\cos(\lambda x )-2\beta\lambda sh(\beta x)\sin(\lambda x)}{2\beta\lambda}\] \[b_{31} =\frac{\big{(}\lambda^{2}-\beta^{2}\big{)}sh(\beta x)\sin(\lambda x )+2\beta\lambda ch(\beta x)\cos(\lambda x)}{2\beta\lambda}\] \[b_{41} =-\frac{\big{(}\lambda^{2}-\beta^{2}\big{)}sh(\beta x)\cos( \lambda x)-2\beta\lambda ch(\beta x)\sin(\lambda x)}{2\beta\lambda}\] \[b_{12} =\frac{\big{(}3\lambda^{2}-\beta^{2}\big{)}\beta sh(\beta x)\sin (\lambda x)+\big{(}3\beta^{2}-\lambda^{2}\big{)}\lambda ch(\beta x)\cos( \lambda x)}{2\beta\lambda(\lambda^{2}+\beta^{2})}\] \[b_{22} =-\frac{\big{(}3\lambda^{2}-\beta^{2}\big{)}\beta sh(\beta x) \cos(\lambda x)-\big{(}3\beta^{2}-\lambda^{2}\big{)}\lambda ch(\beta x)\sin( \lambda x)}{2\beta\lambda(\lambda^{2}+\beta^{2})}\] \[b_{32} =-\frac{\big{(}3\lambda^{2}-\beta^{2}\big{)}\beta ch(\beta x)\sin (\lambda x)+\big{(}3\beta^{2}-\lambda^{2}\big{)}\lambda sh(\beta x)\cos( \lambda x)}{2\beta\lambda(\lambda^{2}+\beta^{2})}\] \[b_{42} =\frac{\big{(}3\lambda^{2}-\beta^{2}\big{)}\beta ch(\beta x)\cos (\lambda x)-\big{(}3\beta^{2}-\lambda^{2}\big{)}\lambda sh(\beta x)\sin( \lambda x)}{2\beta\lambda(\lambda^{2}+\beta^{2})}\] \[b_{13} =\frac{ch(\beta x)\sin(\lambda x)}{2\beta\lambda EI},b_{23}=- \frac{ch(\beta x)\cos(\lambda x)}{2\beta\lambda EI},b_{33}=-\frac{sh(\beta x) \sin(\lambda x)}{2\beta\lambda EI},b_{43}=\frac{sh(\beta x)\cos(\lambda x)}{2 \beta\lambda EI}\] \[b_{14} =\frac{\lambda ch(\beta x)\cos(\lambda x)-\beta sh(\beta x)\sin (\lambda x)}{2\beta\lambda(\lambda^{2}+\beta^{2})EI},\ b_{24}=\frac{\lambda ch (\beta x)\sin(\lambda x)+\beta sh(\beta x)\cos(\lambda x)}{2\beta\lambda( \lambda^{2}+\beta^{2})EI}\] \[b_{21} =\frac{\big{(}\lambda^{2}-\beta^{2}\big{)}ch(\beta x)\cos(\lambda x )-2\beta\lambda sh(\beta x)\sin(\lambda x)}{2\beta\lambda}\]0  
When \(N=2\sqrt{EIk}\)  
\[K_{xl}= \begin{bmatrix}xsin(\phi x)&xcos(\phi x)&sin(\phi x)&cos(\phi x)\\ sin(\phi x)+\phi xcos(\phi x)&cos(\phi x)-\phi xsin(\phi x)&\phi cos(\phi x)&- \phi sin(\phi x)\\ EI\phi\{\phi xsin(\phi x)-2 cos(\phi x)\}&EI\phi\{\phi xcos(\phi x)+2 sin(\phi x )\}&EI\phi^{2}sin(\phi x)&EI\phi^{2}cos(\phi x)\\ EI\phi^{2}\{3 sin(\phi x)+\phi xcos(\phi x)\}&EI\phi^{2}\{3 cos(\phi x)-\phi xsin( \phi x)\}&EI\phi^{2}cos(\phi x)&-EI\phi^{2}sin(\phi x)\end{bmatrix} \]  
and its inverse matrix is as follows:  
\[K_{xl}^{-1}= \begin{bmatrix}\frac{\phi cos(\phi x)}{2}&-\frac{sin(\phi x)}{2}&- \frac{cos(\phi x)}{2\phi EI}&\frac{sin(\phi x)}{2\phi^{2}EI}\\ -\frac{\phi sin(\phi x)}{2}&-\frac{cos(\phi x)}{2}&\frac{sin(\phi x)}{2\phi EI}& \frac{cos(\phi x)}{2\phi^{2}EI}\\ \frac{2 sin(\phi x)-\phi xcos(\phi x)}{2}&\frac{3 cos(\phi x)+\phi xsin(\phi x)}{2 \phi}&\frac{xcos(\phi x)}{2\phi EI}&-\frac{cos(\phi x)+\phi xsin(\phi x)}{2 \phi^{3}EI}\\ \frac{2 cos(\phi x)+\phi xsin(\phi x)}{2}&-\frac{3 sin(\phi x)-\phi xcos(\phi x)}{2 \phi}&-\frac{xsin(\phi x)}{2\phi EI}&\frac{sin(\phi x)-\phi xcos(\phi x)}{2 \phi^{3}EI}\end{bmatrix} \]  
When \(N>2\sqrt{EIk}\)\[K_{xl}= \begin{bmatrix}sin(\phi x)&cos(\phi x)&sin(\psi x)&cos(\psi x)\\ \phi cos(\phi x)&-\phi sin(\phi x)&\psi cos(\psi x)&-\psi sin(\psi x)\\ EI\phi^{2}sin(\phi x)&EI\phi^{2}cos(\phi x)&EI\psi^{2}sin(\psi x)&EI\psi^{2}cos( \psi x)\\ EI\phi^{3}cos(\phi x)&-EI\phi^{3}sin(\phi x)&EI\psi^{3}cos(\psi x)&-EI\psi^{3} sin(\psi x)\end{bmatrix} \]  
and its inverse matrix is as follows:  
\[K_{xl}^{-1}= \begin{bmatrix}-\psi^{2}sin(\phi x)&-\frac{\psi^{2}cos(\phi x)}{ \phi}&\frac{sin(\phi x)}{EI}&\frac{cos(\phi x)}{\phi EI}\\ -\psi^{2}cos(\phi x)&\frac{\psi^{2}sin(\phi x)}{\phi}&\frac{cos(\phi x)}{EI}& -\frac{sin(\phi x)}{\phi EI}\\ \phi^{2}sin(\psi x)&\frac{\phi^{2}cos(\psi x)}{\psi}&-\frac{sin(\psi x)}{EI}& -\frac{cos(\psi x)}{\psi EI}\\ \phi^{2}cos(\psi x)&-\frac{\phi^{2}sin(\psi x)}{\psi}&-\frac{cos(\psi x)}{EI} &\frac{sin(\psi x)}{\psi EI}\end{bmatrix} /(\phi^{2}-\psi^{2})\]