## 2 Governing Equations and Boundary Conditions
\begin{split} T&=\frac{1}{2}\int_{0}^{L}\Bigg{\{}\mu\Big{[}( i_{z})^{2}+2y_{c}i_{z}\dot{y}+(i_{y})^{2}-2z_{c}i_{y}\dot{y}\Big{]}\\ &\quad+I_{s}(\dot{y})^{2}+\rho A(\dot{u}_{x})^{2}-2\mu\big{(}z_{c} i_{x}\dot{a}_{z}+y_{c}i_{x}\dot{a}_{y}\big{)}\\ &\quad+2\rho I_{sy}\dot{\theta}_{z}\dot{y}+\rho I_{y}(\dot{\theta }_{z})^{2}+\rho I_{z}(\dot{\theta}_{y})^{2}+\rho\Gamma(\dot{y}^{\prime})^{2}+2 \rho\dot{y}^{\prime}\big{(}-\tilde{S}_{\omega}\dot{u}_{x}+J_{y\omega}\dot{ \theta}_{y}+J_{z\omega}\dot{\theta}_{z}\big{)}\Bigg{\}}dx\end{split}  \tag{10}\]  
The Lagrangian is given by \(L=T-U\), and the governing equations of motion and the boundary conditions are derived by means of [PERSON]'s principle, which is stated as  
\[\delta\int_{t_{1}}^{t_{2}}L\,dt=0 \tag{11}\]  
in which the boundary conditions are specified as  
\[\delta u_{x}=\delta u_{y}=\delta u_{z}=\delta\theta_{y}=\delta\theta_{z}=0\quad \text{at}\quad t=t_{1},t_{2} \tag{12}\]  
Further, we have  
\[ \begin{split}\delta\int_{t_{1}}^{t_{2}}L\,dt&=\delta \frac{1}{2}\int_{t_{1}}^{t_{2}}\int_{0}^{L}\Bigg{\{}\mu\Big{[}(\dot{u}_{z})^{2} +2y_{c}\dot{u}_{x}\dot{y}+(i_{y})^{2}-2z_{c}\dot{u}_{y}\dot{y}\Big{]}+I_{s}(\dot {y})^{2}+\rho A(\dot{u}_{x})^{2}\\ &\quad+2\rho I_{sy}\dot{\theta}_{z}\dot{y}-2\mu\big{(}z_{c}\dot{u }_{x}\dot{\theta}_{z}+y_{c}\dot{u}_{x}\dot{\theta}_{y}\big{)}+\rho I_{y}(\dot{ \theta}_{z})^{2}+\rho I_{z}(\dot{\theta}_{y})^{2}+\rho\Gamma(\dot{y}^{\prime}) ^{2}\\ &\quad+2\rho\dot{y}^{\prime}\big{(}-\tilde{S}_{\omega}\dot{u}_{x}+ J_{y\omega}\dot{\theta}_{y}+J_{z}\dot{\omega}\dot{\theta}_{z}\big{)}\\ &\quad-\bigg{[}EA(u_{x}^{\prime})-2 EA_{c}u_{x}^{\prime}\dot{ \theta}_{z}^{\prime}-2 EAy_{c}u_{x}^{\prime}\dot{\theta}_{y}^{\prime}+EI_{y}( \dot{\theta}_{z}^{\prime})^{2}+EI_{z}(\theta_{y}^{\prime})^{2}+2 EI_{zy}\theta _{y}^{\prime}\dot{\theta}_{z}^{\prime}\\ &\quad+k_{z}AG(u_{z}^{\prime}-\theta_{z})^{2}+k_{y}AG(u_{y}^{ \prime}-\theta_{y})^{2}-P\Big{(}(u_{z}^{\prime})^{2}+2y_{c}u_{z}^{\prime}\dot{ \theta}^{\prime}+(u_{y}^{\prime})^{2}-2z_{c}u_{y}^{\prime}\dot{y}^{\prime}\\ &\quad+\frac{I_{s}}{\mu}(\dot{\psi}^{\prime})^{2}\Big{)}+GJ(\dot{ \psi}^{\prime})^{2}+ET(\dot{\psi}^{\prime\prime})^{2}+2E\psi^{\prime\prime} \big{(}-\tilde{S}_{\omega}u_{x}^{\prime}+J_{y}\omega\dot{\theta}_{y}^{ \prime}+J_{z}\omega\dot{\theta}_{z}^{\prime}\big{)}\\ &\quad+2K_{y}\theta_{y}^{\prime}\dot{\psi}^{\prime}+2K_{z}\theta _{z}^{\prime}\dot{\psi}^{\prime}-(2u_{x}f_{x}+2u_{z}f_{z}+2u_{y}f_{y}+2m\dot {\psi}+2m_{y}\theta_{z}+2m_{z}\theta_{y})\Bigg{]}\Bigg{\}}dx\,dt\end{split}  \tag{13}\]  
A coordinate system is established with the shear center (SC) as the origin. The \(x\)-axis is aligned with the beam's longitudinal axis, while the yz-plane represents the cross-section of the beam. It is also designated as the principal sectoral pole. The principal sectoral zero is chosen such that  
\[\tilde{S}_{\omega}=J_{y\omega}=J_{z\omega}=0. \tag{14}\]  
Substituting Equations (6), (10), and (11) into Equation (13) and carrying out the usual steps yields the governing equations of motion and the boundary conditions. The boundary conditions at the ends where \(x=0\) or \(x=L\) are  
\[ \begin{split}&\big{[}-EA(u_{x}^{\prime})+EAz_{c}\theta_{z}^{ \prime}+EAy_{c}\theta_{y}^{\prime}\big{]}\delta u_{x}=0,\\ &\big{[}P(u_{y}^{\prime}-z_{c}\psi^{\prime})-k_{y}AG(u_{y}^{ \prime}-\theta_{y})\big{]}\delta u_{y}=0,\\ &\big{[}P(u_{z}^{\prime}+y_{c}\psi^{\prime})-k_{z}AG(u_{z}^{ \prime}-\theta_{z})\big{]}\delta u_{z}=0,\\ &\big{[}-EI_{y}\theta_{z}^{\prime}-EI_{zy}\theta_{y}^{\prime}-K_{z} \psi^{\prime}+EAz_{c}u_{x}^{\prime}\big{]}\delta\theta_{z}=0,\\ &\big{[}-EI_{z}\theta_{y}^{\prime}-EI_{zy}\theta_{z}^{\prime}-K_{y} \psi^{\prime}+EAy_{c}u_{x}^{\prime}\big{]}\delta\theta_{y}=0,\\ &\big{[}-EI\psi^{\prime\prime}\big{]}\delta\psi^{\prime}=0.\end{split}  \tag{15}\]Therefore, the following physical quantities are given by the following equations:  
\[