# Nonlinear Vibration and Post-Buckling Behaviors of Metal and FGM Pipes Transporting Heavy Crude Oil
## 3 Nonlinear Governing Equations  
In order to apply [PERSON]'s principle, it is necessary to determine the total kinetic energy (\(T_{total}=T_{pipe}+T_{fluid}\)) and the strain energy of the FGM pipe transporting fluid (\(U\)). In this regard, based on Equations (1), (4), and (5), the kinetic energy of pipe and fluid as well as the strain energy of the pipe made of FGM are as follows:  
\[T_{pipe}=\frac{1}{2}m_{f}\int\limits_{0}^{L}\left(u_{,x}^{2}+w_{,t}^{2} \right)dx+\frac{1}{2}\int\limits_{0}^{L}\rho(r)\,l\varphi_{,t}^{2}dx \tag{6}\]  
\[T_{fluid}=\frac{1}{2}m_{f}\int\limits_{0}^{L}\left\{(\Gamma+u_{,t}+\Gamma u_{,x})\quad^{2}+(w_{,t}+\Gamma w_{,x})\quad^{2}\right\}dx \tag{7}\]  
\[U=\frac{1}{2}\int\limits_{0}^{L}\left\{\overline{EI}\varphi_{,x}^{2}+ \overline{EA}\bigg{(}u_{,x}+\frac{1}{2}w_{,x}^{2}\bigg{)}^{2}+K_{SF} \overline{GA}(w_{,x}-\varphi)^{2}+P_{0}\Big{(}2u_{,x}+w_{,x}^{2}\bigg{)} \right\}dx \tag{8}\]  
where \(I\) represents the moment of inertia associated with the pipe's cross-section, and \(m_{p}\) and \(m_{f}\) denote the pipe's and fluid's mass per unit length, respectively. \(\overline{EI}\), \(\overline{EA}\), and \(\overline{GA}\) are as follows:  
\[\overline{EI}=\int\limits_{0}^{2\pi}\int\limits_{r_{i}}^{r_{o}}E(r)r^{2}(sin \theta)^{2}rdrd\theta;\ \overline{EA}=\int\limits_{0}^{2\pi}\int\limits_{r_{i}}^{r_{o}}E(r)rdrd\theta; \ \overline{GA}=\int\limits_{0}^{2\pi}\int\limits_{r_{i}}^{r_{o}}G(r)rdrd\theta \tag{9}\]  
According to [PERSON]'s principle:  
\[\int\limits_{t_{1}}^{t_{2}}(\delta T_{total}-\delta U)dt=0 \tag{10}\]  
By substituting Equations (7) and (8) into Equation (10), applying integration by parts, and then equating the coefficients of \(\delta u\), \(\delta w\), and \(\delta\varphi\) to zero, the resulting partial differential equations of motion are obtained as follows:  
\[m_{f}\Big{(}u_{,tt}+2\Gamma u_{,xt}+\Gamma^{2}u_{,xx}\Big{)}+m_{p}u_{,tt}- \overline{EA}\bigg{(}u_{,x}+\frac{1}{2}w_{,x}^{2}\bigg{)}_{,x}-P_{0,x}=0 \tag{11}\]  
\[m_{f}\Big{(}w_{,tt}+2\Gamma w_{,xt}+\Gamma^{2}w_{,xx}\Big{)}+m_{p}w_{,tt}- \bigg{(}w_{,x}\bigg{[}\overline{EA}\bigg{(}u_{,x}+\frac{1}{2}w_{,x}^{2}\bigg{)} +P_{0}\bigg{]}\bigg{)}_{,x}-K_{SF}\overline{GA}(w_{,xx}-\varphi_{,x})=0 \tag{12}\]  
\[\overline{EI}\varphi_{,xx}+K_{SF}\overline{GA}(w_{,x}-\varphi)-\rho(r)I\varphi _{,tt}=0 \tag{13}\]  
In this regard, for the simply supported FGM pipes, the essential boundary conditions can be defined as follows:  
\[u=w=\varphi_{,x}=0\quad\quad\text{at $x=0$, $L$} \tag{14}\]By focusing solely on the transverse motion and assuming limited stretching, Equation (11) simplifies to:  
\[\left(u_{,x}+\frac{1}{2}w_{,x}^{2}\right)_{,x}=0 \tag{15}\]  
Therefore, using Equation (14) in the integration of Equation (15) results in the following expression for \(u\):  
\[T_{fluid}=\frac{1}{2}m_{f}\int\limits_{0}^{L}\left\{(\Gamma+u_{,t}+\Gamma u_{,x})\quad^{2}+(w_{,t}+\Gamma w_{,x})\quad^{2}\right\}dx \tag{7}\]0  
Substituting Equation (16) into Equation (12) yields the following governing equations of motion:  
\[T_{fluid}=\frac{1}{2}m_{f}\int\limits_{0}^{L}\left\{(\Gamma+u_{,t}+\Gamma u_{,x})\quad^{2}+(w_{,t}+\Gamma w_{,x})\quad^{2}\right\}dx \tag{7}\]1  
\[T_{fluid}=\frac{1}{2}m_{f}\int\limits_{0}^{L}\left\{(\Gamma+u_{,t}+\Gamma u_{,x})\quad^{2}+(w_{,t}+\Gamma w_{,x})\quad^{2}\right\}dx \tag{7}\]2  
By omitting the rotary inertia component \(\varphi_{,tt}\) and shear deformation term \((w_{,x}-\varphi)\) from Equation (18), and then integrating the simplified form with Equation (17), a reduced formulation can be obtained as follows:  
\[T_{fluid}=\frac{1}{2}m_{f}\int\limits_{0}^{L}\left\{(\Gamma+u_{,t}+\Gamma u_{,x})\quad^{2}+(w_{,t}+\Gamma w_{,x})\quad^{2}\right\}dx \tag{7}\]3  
It is worth noting that Equation (19) corresponds to the same form as that presented by [PERSON] and [PERSON] [26] in their investigation of Euler-Bernoulli pipes, highlighting consistency with their theoretical framework.