# Size-Dependent Mechanical Behaviors of Defective FGM Nanobeam Subjected to Random Loading
## 2 Theory of FGM Timoshenko Nanobeam
### The Governing Equation of Timoshenko Beam Considering Size-Dependent Effect  
According to the Timoshenko beam theory (TBT), the kinematic relation can be written as:  
\[ \begin{array}{c}u(x,z)=u_{0}(x)+z\varphi(x)\\ w(x,z)=w_{0}(x)\end{array}  \tag{6}\]  
where \(u\) and \(w\) are axial and transverse displacement at any point in the beam, \(u_{0}\) and \(w_{0}\) are the axial and transverse displacement at neutral-plane, and \(\varphi\) is the rotation of the cross section. Thus, the strain of the Timoshenko beam, based on TBT, can be calculated as:  
\[ \begin{array}{c}\varepsilon_{xx}=\frac{\partial u_{0}}{\partial x}+z\frac{ \partial\varphi}{\partial x}=\varepsilon_{xx}^{0}+zk^{0}\\ \gamma_{xz}=\frac{\partial u_{0}}{\partial x}+\varphi\end{array}  \tag{7}\]  
where \(\varepsilon_{xx}^{0}\) and \(\gamma_{xz}\) are external and transverse shear strain, and \(k^{0}\) is the bending strain. Due to varying the material properties (Young's and shear moduli) in the FGM beam, the neutral plane is always not at the midplane. To find the neutral plane, the resultant axial force equals zero, which can be formulated as:  
\[\int_{A}\sigma_{xx}\mathrm{d}A=\int_{A}E(\widetilde{z})\varepsilon_{xx} \mathrm{d}A=0 \tag{8}\]  
According to the Equation (8), the position of neutral plane can be obtained as follows  
\[\widetilde{z}_{\mathrm{c}}=\frac{\int_{A}E(\widetilde{z})\widetilde{z} \mathrm{d}A}{\int_{A}E(\widetilde{z})\mathrm{d}A}=\frac{h(2E_{\mathrm{m}}+nE_{ \mathrm{c}})(n+1)}{2(E_{\mathrm{m}}+nE_{\mathrm{c}})(n+2)} \tag{9}\]  
where \(\widetilde{z}_{\mathrm{c}}\) is the distance of neutral plane from the bottom, and \(\widetilde{z}\) describes the \(z\)-directional position of the point in the coordinates of \(\widetilde{z}_{\mathrm{c}}\) as the origin. Furthermore, according to the principle of minimum potential energy, we can obtain the governing equation as follows  
\[ \begin{array}{c}\int_{0}^{l}\Bigl{[}\left(A_{xx}\frac{\partial u_{0}}{ \partial x}+B_{xx}\frac{\partial\varphi}{\partial x}-\mu\frac{\partial f}{ \partial x}\right)\frac{\partial u_{0}}{\partial x}\\ +\left(B_{xx}\frac{\partial u_{0}}{\partial x}+D_{xx}\frac{\partial\varphi}{ \partial x}-\mu\frac{\partial^{2}}{\partial x^{2}}\left[\frac{\partial}{ \partial x}\left(\overline{N}\frac{\partial u_{0}}{\partial x}\right)-q\right] \right)\frac{\partial\delta\varphi}{\partial x}\\ +\left(A_{xz}\left(\frac{\partial u_{0}}{\partial x}+\varphi\right)+\mu\frac{ \partial}{\partial x}\left[\frac{\partial}{\partial x}\left(\overline{N} \frac{\partial u_{0}}{\partial x}\right)-q\right]\right)\left(\frac{\partial \delta w_{0}}{\partial x}+\delta\varphi\right)\Bigr{]}\mathrm{d}x\\ -\int_{0}^{l}\Bigl{[}\overline{N}\left(\frac{\partial u_{0}}{\partial x} \frac{\partial w_{0}}{\partial x}\right)+q\delta w_{0}+f\delta u_{0}\Bigr{]} \mathrm{d}x+\left[N_{b}\delta u_{0}+\overline{\
abla}\delta w_{0}+\overline{ M}\delta\varphi\right]_{0}^{l}=0\end{array}  \tag{10}\]  
Then, equilibrium equations can be obtained:  
\[ \begin{array}{c}\frac{\partial N}{\partial x}+f=0\\ \frac{\partial Q}{\partial x}-\frac{\partial}{\partial x}\left(\overline{N} \frac{\partial w_{0}}{\partial x}\right)+q=0\\ \frac{\partial M}{\partial x}-Q=0\end{array}  \tag{11}\]  
Subject to the boundary conditions:  
\[ \begin{array}{c}u_{0}\text{ or }N\\ w\text{ or }V=Q-\overline{N}\frac{\partial w_{0}}{\partial x}\\ \varphi\text{ or }M\end{array}  \tag{12}\]  
where  
\[N=\int_{A}\sigma_{xx}\mathrm{d}A \tag{13}\]  
\[M=\int_{A}\widetilde{z}\sigma_{xx}\mathrm{d}A \tag{14}\]  
\[Q=\int_{A}k_{s}\sigma_{xz}\mathrm{d}A \tag{15}\]in which \(N\), \(M\), and \(Q\) stand for induced normal force, bending moment, and shear force, respectively; \(k_{s}\) is the shear correction factor.