## 3 Differential Equilibrium Equations
### Hybrid Section
\begin{array}{l}b_{11}=-\frac{(\delta^{2}-\gamma^{2})ch(\gamma x)sin(\delta x )+2\gamma\delta sh(\gamma x)\cos(\delta x)}{2\gamma\delta}\\ b_{21}=\frac{(\delta^{2}-\gamma^{2})ch(\gamma x)\cos(\delta x)-2\gamma\delta sh (\gamma x)\sin(\delta x)}{2\gamma\delta}\\ b_{31}=\frac{(\delta^{2}-\gamma^{2})sh(\gamma x)\sin(\delta x)+2\gamma\delta ch (\gamma x)\cos(\delta x)}{2\gamma\delta}\\ b_{41}=-\frac{(\delta^{2}-\gamma^{2})sh(\gamma x)\cos(\delta x)-2\gamma\delta ch (\gamma x)sin(\delta x)}{2\gamma\delta}\\ b_{12}=\frac{(3\delta^{2}-\gamma^{2})\gamma sh(\gamma x)\sin(\delta x)+\left(3 \gamma^{2}-\delta^{2}\right)\delta ch(\gamma x)\cos(\delta x)}{2\gamma\delta ^{2}+\gamma^{2})}\\ b_{22}=-\frac{(3\delta^{2}-\gamma^{2})\gamma sh(\gamma x)\cos(\delta x)-\left(3 \gamma^{2}-\delta^{2}\right)\delta ch(\gamma x)\sin(\delta x)}{2\gamma\delta (\delta^{2}+\gamma^{2})}\\ b_{32}=-\frac{(3\delta^{2}-\gamma^{2})\gamma ch(\gamma x)\sin(\delta x)+\left(3 \gamma^{2}-\delta^{2}\right)\delta sh(\gamma x)\cos(\delta x)}{2\gamma\delta (\delta^{2}+\gamma^{2})}\\ b_{42}=\frac{(3\delta^{2}-\gamma^{2})\gamma ch(\gamma x)\cos(\delta x)-\left(3 \gamma^{2}-\delta^{2}\right)\delta sh(\gamma x)\sin(\delta x)}{2\gamma\delta (\delta^{2}+\gamma^{2})}\\ b_{13}=\frac{ch(\gamma x)\sin(\delta x)}{2\gamma\delta EI},\ b_{23}=-\frac{ch( \gamma x)\cos(\delta x)}{2\gamma\delta EI},\ b_{33}=-\frac{sh(\gamma x)\sin( \delta x)}{2\gamma\delta EI},\ b_{43}=\frac{sh(\gamma x)\cos(\delta x)}{2 \gamma\delta EI}\\ b_{14}=\frac{\delta ch(\gamma x)\cos(\delta x)-\gamma sh(\gamma x)\sin(\delta x )}{2\gamma\delta(\delta^{2}+\gamma^{2})EI}\\ b_{24}=\frac{\delta ch(\gamma x)\sin(\delta x)+\gamma sh(\gamma x)\cos(\delta x)}{2 \gamma\delta(\delta^{2}+\gamma^{2})EI}\\ b_{34}=-\frac{\delta sh(\gamma x)\cos(\delta x)-\gamma ch(\gamma x)\sin(\delta x )}{2\gamma\delta(\delta^{2}+\gamma^{2})EI}\\ b_{44}=-\frac{\delta sh(\gamma x)\sin(\delta x)+\gamma ch(\gamma x)\cos(\delta x )}{2\gamma\delta(\delta^{2}+\gamma^{2})EI}\\ \end{array} \]  
(2) When \(H_{q}-N=2\sqrt{EIk}\)\[K_{jc}= \begin{bmatrix}xsh(\eta x)&xch(\eta x)&sh(\eta x)&ch(\eta x)\\ sh(\eta x)+\eta xch(\eta x)&ch(\eta x)+\eta xsh(\eta x)&\eta ch(\eta x)&\eta sh(\eta x )\\ -EI\{2\eta ch(\eta x)+\eta^{2}xsh(\eta x)\}&-EI\{2\eta sh(\eta x)+\eta^{2}xch( \eta x)\}&-EI\eta^{2}sh(\eta x)&-EI\eta^{2}ch(\eta x)\\ -EI\eta^{2}\{3 sh(\eta x)+\eta xch(\eta x)\}&-EI\eta^{2}\{3 ch(\eta x)+\eta xsh( \eta x)\}&-EI\eta^{3}ch(\eta x)&-EI\eta^{3}sh(\eta x)\end{bmatrix} \]  
The inverse matrix is:  
\[K_{jc}^{-1}= \begin{bmatrix}-\dfrac{ch(\eta x)}{2}&\dfrac{sh(\eta x)}{2}&- \dfrac{ch(\eta x)}{2\eta EI}&\dfrac{sh(\eta x)}{2\eta^{2}EI}\\ \dfrac{sh(\eta x)}{2}&-\dfrac{ch(\eta x)}{2}&\dfrac{sh(\eta x)}{2\eta EI}&- \dfrac{ch(\eta x)}{2\eta^{2}EI}\\ -\dfrac{2 sh(\eta x)-\eta xch(\eta x)}{2}&\dfrac{3 ch(\eta x)-\eta xsh(\eta x)} {2\eta EI}&\dfrac{ch(\eta x)-\eta xsh(\eta x)}{2\eta^{3}EI}\\ \dfrac{2 ch(\eta x)-\eta xsh(\eta x)}{2}&-\dfrac{3 sh(\eta x)-\eta xch(\eta x)} {2\eta EI}&-\dfrac{sh(\eta x)-\eta xch(\eta x)}{2\eta^{3}EI}\end{bmatrix} \]  
(3) When \(H_{q}-N>2\sqrt{EIk}\)  
\[K_{jc}= \begin{bmatrix}sh(\eta x)&ch(\eta x)&sh(\varepsilon x)&ch(\varepsilon x )\\ \eta ch(\eta x)&\eta sh(\eta x)&\varepsilon ch(\varepsilon x)&\varepsilon sh( \varepsilon x)\\ -EI\eta^{2}sh(\eta x)&-EI\eta^{2}ch(\eta x)&-EI\varepsilon^{2}sh(\varepsilon x )&-EI\varepsilon^{2}ch(\varepsilon x)\\ -EI\eta^{3}ch(\eta x)&-EI\eta^{3}sh(\eta x)&-EI\varepsilon^{3}ch(\varepsilon x )&-EI\varepsilon^{3}sh(\varepsilon x)\end{bmatrix} \]  
The inverse matrix is  
\[K_{jc}^{-1}=