## 3 Governing Equations of CNTRC Beams  
The present research may shed light on the coupling effect of the thickness and shear deformation of CNTRC, so the unified beam theory was exploited in the beam's modeling. A unified higher-order shear deformation beam theory incorporating various beam theories was used to capture the shear deformation and rotation. Within the framework of higher-order beam theory, the deformations of the beam took place in the x- and z-directions. According to the shear deformation theory, the displacement field consisting of the axial displacement \(U_{1}\) and the transverse displacement \(U_{3}\), at any point, is as follows [3,6,13,16,24]:  
\[U_{1}(x,y,z,t)=u(x,t)-zw^{\prime}+H(z)\big{(}w^{\prime}-\theta(x,t)\big{)} \tag{13}\]  
\[U_{2}(\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{t})=0 \tag{14}\]  
\[U_{3}(\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{t})=\mathrm{w}(\mathrm{x}, \mathrm{t}) \tag{15}\]  
where the prime denotes the derivative with respect to variable \(x\); \(u(x,t)\) and \(w(x,t)\) are the axial and transverse displacements at the reference plane of the beam, respectively; \(\theta(x,t)\) is the bending rotation of the cross-section at any point of the reference plane; and \(\mathrm{t}\) is the time. The shape function \(H(z)\) describes the transverse shear stress distribution across the beam thickness. Some of these shape functions for different beam theories are:  
\[[PERSON]\ beam:\ H(z)=0\]  
\[[PERSON]\ beam:H(z)=z\]  
\[[PERSON]\ beam:\ H(z)=z^{3}\]  
\[[PERSON]\ beam:\ H(z)=ze^{-2(\frac{z}{h})^{2}}\]  
\[[PERSON]\ beam:\ H(z)=sin(\frac{\pi z}{2h})e^{0.03 cos(\frac{\pi z}{h})}\]  
\[[PERSON]\ beam:\ H(z)=tan(\frac{\pi z}{2h})e^{0.03 sec(\frac{\pi z}{h})}\]  
\[Trigonometric\ shear\ deformation\ theory:\ H(z)=\frac{h}{\pi}\sin\frac{\pi z}{h}\]  
\[U_{2}(\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{t})=0 \tag{14}\]0  
The expressions of the normal and shear strain components associated with the displacement field in Equations (13)-(15) are:  
\[U_{2}(\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{t})=0 \tag{14}\]1  
where the strain of the beam caused by the axial displacement \(\varepsilon_{1}\) is given by:  
\[U_{2}(\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{t})=0 \tag{14}\]2  
\[U_{2}(\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{t})=0 \tag{14}\]3  
and the strain caused by the curvature is:  
\[U_{2}(\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{t})=0 \tag{14}\]4  
such that the normal strain component (17) becomes:  
\[U_{2}(\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{t})=0 \tag{14}\]5  
where  
\[ \begin{array}{c}f(x,t)=\frac{\partial u}{\partial x}+\frac{1}{2}\Big{(} \frac{\partial w}{\partial x}\Big{)}^{2}-\frac{1}{8}\Big{(}\frac{\partial w}{ \partial x}\Big{)}^{4};\ g(x,t)=\frac{\partial^{2}w}{\partial x^{2}}\bigg{(} 1-\frac{3}{2}\Big{(}\frac{\partial w}{\partial x}\Big{)}^{2}\bigg{)};\\ h(x,t)=\frac{\partial^{2}w}{\partial x^{2}}-\frac{\partial\theta}{\partial x} \end{array}  \tag{21}\]  
The shear strain becomes:  
\[\gamma_{xz}=\frac{dH}{dz}\bigg{(}\frac{\partial w}{\partial x}-\theta\bigg{)} \tag{22}\]  
Now, based on the constitutive relation of the kth layer of the fiber-reinforced polymeric material, we obtain the stress:  
\[\sigma_{xx}^{(k)}=\overline{Q}_{11}^{(k)}\varepsilon_{xx},\ \sigma_{xz}^{(k)}= \overline{Q}_{55}^{(k)}\gamma_{xz}, \tag{23}\]  
where \(\overline{Q}_{ij}^{(k)}\) are transformed elastic constants in the \(z\)-direction defined by:  
\[\overline{Q}_{11}^{(k)}=Q_{11}^{(k)}cos^{4}\varphi+\frac{1}{2}\Big{(}Q_{12}^ {(k)}+2Q_{66}^{(k)}\Big{)}sin^{2}2\varphi+Q_{22}^{(k)}sin^{4}\varphi, \tag{24}\]  
\[\overline{Q}_{55}^{(k)}=Q_{44}^{(k)}sin^{2}\varphi+Q_{55}^{(k)}cos^{2}\varphi ;\ Q_{1}^{(k)}=\frac{E}{1-v_{21}v_{12}};\ Q_{12}^{(k)}=\frac{v_{12}E}{1-v_{21} v_{12}};\ Q_{22}^{(k)}=\frac{E}{1-v_{21}v_{12}} \tag{25}\]\[Q_{66}^{(k)}=G_{12},\ Q_{44}^{(k)}=G_{23},Q_{35}^{(k)}=G_{13} \tag{26}\]  
where \(\varphi\) is the direction of magnetic anisotropy; \(v_{12}\) and \(v_{21}\) are Poisson ratios; and \(G_{12}\), \(G_{13}\), \(G_{31}\), and \(G_{23}\) are the shear moduli.