## 2 Theoretical Problem
\begin{split}\frac{\partial}{\partial x}\bigg{[}A_{11}\left(\frac {\partial u}{\partial x}+\frac{1}{2}\left(\frac{\partial w}{\partial x}\right) ^{2}\right)-B_{11}\frac{\partial^{2}w}{\partial x^{2}}+N_{T}\bigg{]}\left( \frac{\partial w}{\partial x}\right)\\ +\frac{\partial^{2}w}{\partial x^{2}}\bigg{[}A_{11}\left(\frac{ \partial w}{\partial x}+\frac{1}{2}\left(\frac{\partial w}{\partial x}\right) ^{2}\right)-B_{11}\frac{\partial^{2}w}{\partial x^{2}}+N_{T}\bigg{]}\\ +\frac{\partial^{2}w}{\partial x^{2}}\bigg{[}B_{11}\left(\frac{ \partial u}{\partial x}+\frac{1}{2}\left(\frac{\partial w}{\partial x}\right) ^{2}\right)-D_{11}\frac{\partial^{2}w}{\partial x^{2}}+M_{T}\bigg{]}+F_{w}\\ =I_{0}\frac{\partial^{2}w}{\partial t^{2}}+\frac{\partial}{\partial x }\left(I_{1}\frac{\partial^{2}w}{\partial t^{2}}-I_{2}\frac{\partial^{3}w}{ \partial x\partial\partial t^{2}}\right)\end{split}  \tag{23}\]  
At the same time, the related boundary conditions at \(x=0\) and \(x=L\) are obtained as  
\[\left[A_{11}\left(\frac{\partial u}{\partial x}+\frac{1}{2}\left(\frac{ \partial w}{\partial x}\right)^{2}\right)-B_{11}\frac{\partial^{2}w}{\partial x ^{2}}+N_{T}\right]-\bar{N}=0\text{ or }\delta u=0 \tag{24}\]  
\[ \begin{split}\frac{\partial w}{\partial x}\bigg{[}A_{11}\left( \frac{\partial u}{\partial x}+\frac{1}{2}\left(\frac{\partial w}{\partial x} \right)^{2}\right)-B_{11}\frac{\partial^{2}w}{\partial x^{2}}+N_{T}\bigg{]}+ \frac{\partial}{\partial x}\bigg{[}B_{11}\left(\frac{\partial u}{\partial x} +\frac{1}{2}\left(\frac{\partial w}{\partial x}\right)^{2}\right)-D_{11}\frac{ \partial^{2}w}{\partial x^{2}}+M_{T}\bigg{]}\\ -\left[I_{1}\frac{\partial^{2}w}{\partial x^{2}}-I_{2}\frac{ \partial^{3}w}{\partial x\partial\partial t^{2}}\right]-\bar{\mathcal{V}}=0 \text{ or }\delta w=0\end{split}  \tag{25}\]\[\left[D_{11}\frac{\partial^{2}w}{\partial x^{2}}-B_{11}\left(\frac{\partial u}{ \partial x}+\frac{1}{2}\left(\frac{\partial w}{\partial x}\right)^{2}\right)-M_ {T}\right]-\tilde{M}=0\text{ or }\delta\left(\frac{\partial w}{\partial x}\right)=0 \tag{26}\]  
It is worth noticing that the differential governing equations of motion are coupled with respect to the displacement components \(u\) and \(w\). Hence, in order to obtain a solitary equation in terms of the lateral displacement, the contribution of rotational and axial inertia is neglected [36]. Moreover, the axial and lateral loads are omitted for a free vibration analysis. Therefore, Equations (22) and (23) take the following form  
\[\frac{\partial}{\partial x}\Bigg{[}A_{11}\left(\frac{\partial u}{\partial x}+ \frac{1}{2}\left(\frac{\partial w}{\partial x}\right)^{2}\right)-B_{11}\frac{ \partial^{2}w}{\partial x^{2}}+N_{T}\Bigg{]}=0 \tag{27}\]  
\[\frac{\partial}{\partial x}\left[A_{11}\left(\frac{\partial u}{\partial x}+ \frac{1}{2}\left(\frac{\partial w}{\partial x}\right)^{2}\right)-B_{11}\frac{ \partial^{2}w}{\partial x^{2}}+N_{T}\right]\left(\frac{\partial w}{\partial x }\right)+\frac{\partial^{2}w}{\partial x}\left[A_{11}\left(\frac{\partial u} {\partial x}+\frac{1}{2}\left(\frac{\partial w}{\partial x}\right)^{2}\right)- B_{11}\frac{\partial^{2}w}{\partial x^{2}}+N_{T}\right] \tag{28}\]  
Integrating Equation (27) with respect to the \(x\)-axis gives  
\[\frac{\partial u}{\partial x}=-\frac{1}{2}\left(\frac{\partial w}{\partial x} \right)^{2}+\frac{B_{11}}{A_{11}}\frac{\partial^{2}w}{\partial x^{2}}-\frac{ 1}{A_{11}}N_{T}+\frac{N_{0}}{A_{11}} \tag{29}\]  
Integrating both side of Equation (29) with respect to \(x\) yields  
\[u=\int_{0}^{x}-\frac{1}{2}\left(\frac{\partial w}{\partial x}\right)^{2}dx+ \frac{1}{A_{11}}(N_{0}-N_{T})x+\frac{B_{11}}{A_{11}}\frac{\partial w}{\partial x }+N_{1}(t) \tag{30}\]  
where \(N_{0}\) and \(N_{1}\) are two constants of integration with respect to \(x\), that must be determined by means of different boundary conditions. In this study, we selected three types of boundary conditions, namely simply supported (S-S), clamped-clamped (C-C) and clamped-Free (C-F) boundary conditions.