## 4 Example Calculation and Numerical Analysis
Example 2: influence of different load speeds and river bed depth. Under the condition of different load speeds (\(v_{c}=0.2v_{s}\), \(0.5v_{s}\), \(1.2v_{s}\)), the river bed depth \(h_{w}=0.0\) m, 2.0 m, 4.0 m, 7.0 m, and the buried depth of the underwater tunnel is H = 1.0 m. From the river bed bedrock, \(h_{s}=15.0\) m, the displacement changes ofthe structural beam of the tunnel are analyzed. As shown in Figure 4, the viscoelastic beam of the underwater immersed tunnel adopts the standard solid model, and the calculated parameters: \(E_{p}^{(1)}=E_{p}^{(2)}=2.0\ \times\ 10^{9}\), \(\eta=3.0\ \times\ 10^{5}\) Pa, and the calculation parameters of saturated soil and fluid: \(\mu=2.0\times 10^{7}\) N/m\({}^{2}\), \(\phi=0.125\), \(\lambda=4.0\times 10^{7}\) N/m\({}^{2}\), \(\alpha=0.97\), \(M=2.44\times 10^{8}\) N/m\({}^{2}\), \(\rho_{s}=2.0\times 10^{3}\) kg/m\({}^{3}\), \(m=1890\) kg/m\({}^{3}\), \(v_{w}=1414\) m/s, \(b_{p}=1.0\times 10^{8}\) kg/m\({}^{3}\)s, \(\rho_{fw}=1.0\times 10^{3}\) kg/m\({}^{3}\). From Figure 4, the vertical deformation of the structural beam of the tunnel increases with the increase in riverbed water depth. However, when the water depth of the riverbed increases to a certain value, \(h_{w}=4.0\) m, the water depth of the riverbed has little impact on the vertical deformation of the beam. With load speed increasing, tunnel structure beam vertical vibration is enhanced, and when the riverbed water depth is larger, even at low speed, the tunnel structure beam vibration fluctuates. The greater the riverbed depth, the more significant the tunnel structure beam vibration, showing that the existence of riverbed water greatly reduces the ray wave speed of fluid-saturated soil bed foundation systems, therefore the speed of high water levels must be controlled. Figure 3: Comparison with reference results. (**a**) horizontal displacement, (**b**) vertical displacement. The black square refers to Ref. [26]. Figure 4: Vertical displacement change of the structural beam of the underwater immersed tunnel during different riverbed water depths. (**a**) \(v_{\rm c}=0.2\)\(v_{s}\), (**b**) \(v_{\rm c}=0.5\)\(v_{s}\), (**c**) \(v_{\rm c}=1.2\)\(v_{s}\). Figures 5 and 6 show the change in shear force and bending moment of the tunnel structure beam under different riverbed water depth \(h_{w}=0.0\) m, 2.0 m, 4.0 m, and 7.0 m. \(v_{c}v_{s}\) from Figures 5 and 6, the bending moment and shear force of the tunnel structure beam across the river increase with the increase in riverbed water depth. When the load moves at a low speed (\(<\)0.5), the bending moment and shear peak of the beam increase as the load speed increases, and when the load speed reaches a high speed, the bending moment and shear decrease as the load speed increases, and the distribution range is relatively narrow. With the increase in movement speed, the load is transferred more to the bed foundation to reduce the internal force of the beam. Figures 7-9 show the displacement and pore pressure changes of the observation point (A (0, 2.0 m)) in riverbed soil under different riverbed water \(h_{w}\) depths = 0.0 m, 2.0 m, 4.0 m, and 7.0 m.  
From Figures 7 and 8, the vertical, horizontal displacement, and hole pressure of the observation point increase as the water depth of the riverbed increases, and the existence of the riverbed water makes the vibration of the observation point more significant, indicating that when the load speed is small, such as \(v_{c}=0.2\;v_{s}\), the vertical and horizontal displacement of the temporal dynamic response of the observation point is basically symmetrically distributed on both sides of the moment of load arrival. With increasing load speed and displacement amplitude, the longer the vibration time of the observation point in the riverbed foundation, the more its symmetry about the moment of load arrival disappears. And the phenomenon of shock waves appears.