# Research on Predicting the Mechanical Characteristics of Deep-Sea Mining Transportation Pipelines
## 2 Deep-Sea Mining Transportation Pipeline Mechanics Analysis
### Dynamic Analysis of Deep-Sea Mining Transport Pipelines  
This study is based on the Euler-Bernoulli beam theory [39] to analyze the dynamics of deep-sea transport pipelines, considering the coupling of the fluid and pipeline. It is assumed that the deformation of the transport pipeline is a small displacement, and the internal fluid is an ideal incompressible fluid. By analyzing the forces on the pipeline element \(dz\), the motion control equations of the transport pipeline in a three-dimensional coordinate system are obtained. As shown in Figure 2, Figure 2 illustrates the force analysis on the pipeline wall, where \(T\) is the axial tension of the pipeline, \(F\) is the pressure of the internal fluid on the wall, \(q\) is the frictional resistance of the internal flow on the wall, and \(Q\) is the shear force of the pipeline. Figure 2 depicts the force analysis on the internal fluid of the pipeline. The force balance equation for the small section of the pipeline wall in Figure 2 is as follows, where Equation (6) represents the Z-direction balance equation, and Equation (7) represents the Y-direction balance equation:  
\[\frac{\partial T}{\partial z}+qS-F\frac{\partial y}{\partial z}-m_{r}g-\frac{ \partial}{\partial z}\bigg{(}Q\frac{\partial y}{\partial z}\bigg{)}+\frac{ \partial(A_{e}P_{e})}{\partial z}+\rho_{e}gA_{e}dz=0 \tag{6}\]  
\[-C\frac{\partial y}{\partial t}-m_{r}\frac{\partial^{2}y}{\partial t^{2}}+F+ qS\frac{\partial y}{\partial z}+\frac{\partial Q}{\partial z}+\frac{ \partial}{\partial z}\bigg{(}T\frac{\partial y}{\partial z}\bigg{)}+\frac{ \partial}{\partial z}\bigg{(}A_{e}P_{e}\frac{\partial y}{\partial z}\bigg{)}+ f=0 \tag{7}\]  
In the equations, \(S\) represents the circumference inside the conveying pipeline, \(m_{r}\) is the mass per unit length of the conveying pipeline, \(g\) is the acceleration due to gravity, \(A_{e}\) is the cross-sectional area outside the conveying pipeline, \(A_{e}P_{e}\) is the pressure acting on this outer cross-section, \(\rho_{e}\) is the density of the fluid outside the conveying pipeline, \(\rho_{e}gA_{e}dz\) represents the buoyancy acting on the outer cross-section, \(C\) is the damping  
Figure 2: Force analysis of micro-elements of deep-sea mining transportation pipelines: (**a**) force analysis on the pipeline wall; (**b**) fluid force analysis inside the pipeline. coefficient, shear force \(Q=-EI\partial^{3}y/\partial z^{3}\), where \(EI\) is the bending stiffness of the conveying pipeline, \(P_{e}\) is the static water pressure, and \(f\) is the external force acting on the Y-direction conveying pipeline. The force balance equation for the fluid element in Figure 2b is as follows, where Equation (8) represents the Z-direction balance equation, and Equation (9) represents the Y-direction balance equation:  
\[-\frac{\partial(A_{i}P_{i})}{\partial z}-qS-m_{f}g-m_{f}\frac{\partial V}{ \partial t}+F\frac{\partial y}{\partial z}=0 \tag{8}\]  
\[m_{f}\Big{(}\frac{\partial}{\partial t}+V\frac{\partial}{\partial z}\Big{)}^ {2}y-F-qS\frac{\partial y}{\partial z}-\frac{\partial}{\partial z}\Big{(}A_{ i}P_{i}\frac{\partial y}{\partial z}\Big{)}=0 \tag{9}\]  
In the equations, \(A_{i}\) represents the cross-sectional area inside the conveying pipeline, \(P_{i}\) is the internal fluid pressure, and \(m_{f}\) is the mass per unit length of the internal fluid in the conveying pipeline. Combining Equations (6) and (8), the following expression is derived:  
\[\frac{\partial(T-A_{i}p_{i}+A_{e}p_{e})}{\partial z}=\Big{(}m_{r}+m_{f}-\rho_ {e}A_{e}\Big{)}g+m_{f}\frac{\partial V}{\partial t} \tag{10}\]  
In the equation, \(V\) represents the fluid velocity inside the conveying pipeline.