A Numerical Simulation of Base Shear Forces and Moments Exerted by Waves on Large Diameter Pile

To solve the problem of the dynamic variation of wave force diameter of pile foundation for offshore wind turbines, wave force and moment of large diameter pile foundation can be calculated. In this paper, simulation technique is used to calculate the wave force andmoment of different large diameter pile foundation, and the base shear force andmoment of the interval 20-degree phase angle are obtained by the base line of the pile. Under the action of a certain load, the complete stress variation of the pile foundation is obtained. According to the basic principle of diffraction theory, the process curve of large diameter pile, and analysis of wave force, diffraction force changes in a certain period of time interval. The results show that the wave after the large diameter pile formed around the vortex, large diameter pile base shear, and moment dynamic change is nonlinear in a complete cycle, the diameter of the pile increases by 0.5m, and the wave force increases by about 5%, the results show that it provides certain reference value for the offshore pile foundation pile with large diameter primary site. Some significant results for practical application are discussed.


Introduction
The wave response of large diameter piles is one of the most important problems of offshore wind power pile foundation.Researchers have paid much attention on the problem of the large diameter pile surface waves.Ong et al. [1] provided a practical stochastic method based on assuming the waves to be a stationary narrowband random process and compared it with the Sumer and Freds data for 2D random waves plus current.Liu et al. [2] gave a review of the two-dimensional Reynolds-Averaged Navier-Stokes (RANS) equations with a Shear-Stress Transport (SST) - turbulence closure to establish a two-dimensional numerical model.The roles of local scour around submarine pipelines induced by the orbital fluid motion under surface water waves were discussed.The results that Si et al. [3] used the generalized KdV model (GKdV model for short) to calculate ISWs on a supposed rigid pile are of benefit to revealing the mechanism of destructive power exerted by large-amplitude ISWs on a pile.It is essential to make reliable assessments of the wave and current loads on the structure, as well as of the scour around the base of the foundation at the seabed.For example, by means of an analytical model subjected to vertical loading from the superstructures and lateral loading due to wind or wave actions, the interaction effects of vertical loads on the lateral responses of piles applied in such cases were further investigated numerically by Liang et al. [4].Studies on the solitary wave-induced loads on a submerged plate were very limited.Combining with the shift of polar coordinates, Liu et al. [5] examined the Bragg reflection of water waves by multiple submerged semicircular breakwaters.Experimental tests were carried out and the analytical results are in reasonable agreement with the experimental data.Li et al. [6] found that the extreme waves in deep waters are better reproduced than those in shallow waters, which is partly attributed to the limitations of the wave model used.In addition to the results and laboratory tests, various numerical models presented in this paper had also been developed to study the effect of wind data resolution on the simulation of long-term waves.Hayatdavoodi and Cengiz Ertekin [7,8] also conducted wave loads due to nonlinear waves of solitary and cnoidal type propagating over a submerged, horizontal, and thin plate to compare the results that they obtained with the available laboratory data and with linear Out-of-phase solutions of the problems.Accordingly, a new higher order boundary element method for the main diagonal elements was obtained.The proposed model can be used to simulate nonbreaking waves propagating on uneven bottoms or wave passing through surface-piercing structures, governing equations, and boundary conditions in the transformed plane which are first presented in [9][10][11].Lian et al. [12] based on the diffraction theory show that the analytical solution of the wave pressure and the wave force on the composite bucket foundation is accurately derived by assigning reasonable boundary conditions.
In this paper, different wave forces are calculated by using the numerical simulation method.The base shear forces and moments on changes of linear wave which goes through large diameter piles are obtained; wave force time history curves and stress changes can be defined as the pile foundation of offshore wind power to provide reference.

Numerical Approach
2.1.Theoretical Background.On the ratio of the wavelength and the diameter of large diameter pile more than 0.2, the wave diffraction theory assumes that the fluid is homogeneous, incompressible, and powerful, nonviscous ideal fluid.The velocity potential of any point in the wave field is that Φ (, , , ) = Φ  (, , , ) + Φ  (, , , ) , where  is the position parallel to the horizontal direction of wave motion,  is the position perpendicular to the direction of wave motion,  is the depth from the bottom of the sea, Φ  (, , , ) is the velocity potential of undisturbed incident wave, Φ  (, , , ) is the velocity potential of scattered wave, and Φ(, , , ) is the total velocity potential of any point in the wave field where Φ(, , ) is total velocity potential after separation of time variable, Φ  (, , ) is the velocity potential of undisturbed incident wave after separation of time variable, and Φ  (, , ) is the velocity potential of scattered wave after separation of time variable.The random function of wave force acting on the large diameter pile is that where () is the total wave force random function,  is the wave density,  is the wave number,  is the water depth,  is the wave amplitude, and  1 () is Bessel function of the first kind. 1 () is Bessel function of the second kind, and () is the displacement wave relative to the static water.The wave force spectrum of playing a role in large diameter pile is that where  is the circular frequency,   () is the total wave force spectrum, and   () is the spectral density function.
is the wave density,  is the wave number,  is the water depth,  is the wave amplitude, and  1 () is Bessel function of the first kind. 1 () is Bessel function of the second kind.The governing equation of the surface wave after the pile foundation [2] can be written as  where  1 =  and  2 =  are the horizontal and vertical coordinates, respectively,   is the fluid velocity in the   -direction,    is the velocity of moving grid in the   -direction,  is the time,  is the pressure,  is the wave density,  is the kinematic viscosity of the fluid,   denotes the gravitational acceleration, and   is the mean strain rate tensor with   = (  /  +   /  )/2.The stress term in (6) reads       =   (  /  +   /  ) + 2  /3, where   is the turbulent eddy viscosity,  is the turbulence kinetic energy, and   is the Kronecker operator.

Numerical Model.
In the change process of studying wave force on the large diameter pile foundation of offshore wind power, the large diameter pile foundation structure is regarded as a dynamical system, and effect of wave and other loads is called system input, structural stress, deformation, and movement, known as the output of the system.The wave spectrum and the response spectrum of the large diameter pile foundation are connected through the transfer function which is a function of frequency about the ratio of stress amplitude and cyclic wave.Airy wave theory is derived from the waveform which is a sine curve, and the other wave theory is derived from the wave which is not pure sine wave.They can be reasonably used to generate the transfer function.
As shown in Figure 1 of wave force decomposition diagram, the wave force is decomposed into two parts of water power and water static; hydrodynamics is divided into coupling wave force and diffraction wave force, considering the influence of structure on the wave force; static wave force included the influence of gravity and stages of wave force change and finally comes down to wave positive response, namely, the wave force in  direction.
Taking the offshore wind pile foundation with a diameter of 4 m as an example, the mathematical model is established according to the specific parameters of Table 1.In Figure 2, 0001 point is free, 0000 point is fixed, the origin is the sea level, and the buried depth is 10m.Wave and current act on pile foundation are described in Figure 3.
In the dynamic response analysis, the base of the pile is the foundation of the modal analysis.The static and dynamic response of wave, coupled modes, and pure wave theory should pay attention to the influence of ground surface elevation and the soil microorganism.It is both the internal friction angle of each soil layer and undrained strength that directly influence the calculated value of dynamic response corresponding to each iteration step.In the bending moment diagram that is under pile foundation mud, it should adjust the mud position height of pile foundation according to the simulated curve of the graph; otherwise, it will affect variation tendency of the wave current coupling force in the static response.
The wave response curve of large diameter pile shows that the relationship between the depth and lateral displacement of pile foundation can be the approximate parabolic shape, when the displacement is 0 in the pile foundation depth of 3350 cm, the depth increases to 4000 cm; it is a curve of linear relation because of irregular wave force.From the relationship between the axial load and the depth of the pile, when the pile foundation depth is 0, the axial load reaches the maximum value about 3750 KN, the depth increases, and the axial load is smaller.The curve shows nexus between axial response of soil and pile foundation's depth; it presents a parabolic shape from the depth of the pile foundation 0 cm to 800 cm and is a constant value from 800 cm to 4000 cm approximately.

Analysis on the Change Nephogram of the Wave through the Offshore Wind Power Pile Foundation. Numerical results
show that it generates stress change nephogram automatically every 1 S case and completes a diffraction process from contacting the pile foundation by the same kind of wave a whole diffraction process.It can be seen that Figure 4 presents the change nephogram of wave passing through the large diameter pile, with the instantaneous velocity of wave from 0.070 rad/s to 3.378 rad/s and the transition from the green zone to the blue region, finally it becomes two symmetrical red regions and then slightly takes blue, and the stress changes quickly, which is due to energy dissipation caused by the stress changes after wave goes through the large diameter pile.In Figures 4(h), 4(i), and 4(j), the stress nephogram includes colors of blue, green, black, and red; these three charts indicate that velocity potential of the scattered wave is the largest when wave passes the pile foundation, after the scattered wave is superposed on partial incident wave; the formation of new the surface wave field's stress changes greatly, whose value rises from 0.8185 to 1.180.Figure 4(o) shows that the incident wave is almost scattered after the wave goes through the large diameter pile, as a new wave field is formed, a full cycle wave is completed, and the next cycle wave starts again.
When touching pile, the wave forms a vortex due to drag force and inertia force; it began shaping from Figure 4(h) to the end of Figure 4(m).When the large diameter pile exists in the wave field, because of diffraction effect, the cylindrical wave of each direction can cause the outward divergence effect.Cylindrical scattered wave has a certain amplitude at the cylindrical surface, due to the continuous divergence of the wave front; even if the total wave energy remains unchanged, the amplitude of the cylindrical wave will also tend to zero.The changes of the stress and the wave spectrum are consistent, which shows the change of the wave force directly.

Wave Force Time Frequency Curve Analysis.
In Figure 5, it shows that the ratio of 4 m diameter and the wavelength is / = 0.4; it uses the spectrum analysis method and then obtains the wave response curve; the values of diffraction wave force in  direction are max = 148 KN ( = 1.45 Hz) and min = 70KN ( = 0.21 Hz).It can be seen that frequency ranges from 0.21 Hz to 1.45 Hz.In a full cycle, wave diffraction's angle is from −90 ∘ to −84 ∘ in the  plane, of which frequency ranges from 0.21 Hz to 1.42 Hz.The moment in the  direction is 0 to 2810 Pa, whose frequency's section varies from 0.21 Hz to 1.42 Hz, and diffraction angle changes from 90 ∘ to 104.8 ∘ in  plane; its frequency's section ranges from 0.21 Hz to 1.42 Hz.Frequency's selection depends on the required accuracy, but the unbefitting increase in the spectral range will also make the simulation accuracy decrease.In the iterative process, angle and frequency are identical to wave force's variation curve and phase change curve.Steady-state's moment is 3000 Pa, which is in accord with the wave theory.
In Figure 6, it shows that the ratio of 5 m diameter and the wavelength is / = 0.5; it uses the spectrum In Figure 7, it is shown that the ratio of 6 m diameter and the wavelength is / = 0.6; it uses the spectrum analysis method and then obtains the wave response curve of diffraction wave force in  direction of which the wave force values are max = 321 KN ( = 1.11Hz) and min = 145 KN ( = 0.21 Hz); frequency ranges from 0.21 Hz to 1.11 Hz, wave diffraction angle in  plane ranges from −90 ∘ to −79 ∘ , and frequency ranges from 0.21 Hz to 1.41 Hz.The wave moment in  direction ranges from 0 to 1.1 × 10 4 frequency changes from 0.21 Hz to 1.42 Hz, the angle of wave diffraction in the  plane ranges from 90 to 118 its frequency ranges from 0.21 Hz to 1.42 Hz, and the steady-state moment is 1.1 × 10 4 Pa.8 shows that 4 m diameter pile phase angle interval is 20 ∘ in wave dynamic response, the maximum base shear value is 87.58 KN, the minimum base shear value is 0.2 KN, and the static force value is 47.21 KN.Dynamic force is equal to the sum of the static force and inertia force.In the  direction, the wave force change is simulated, and the phase angle of a period is 20 ∘ , which is divided into 18 steps.A row or a column of data in a matrix of stiffness matrices among super cell file in the calculation process is slightly different; namely, there is no linear change, which leads to different base shear.

Figure 1 :
Figure 1: Schematic diagram of wave force decomposition.

Figure 2 :Figure 3 :
Figure 2: Distribution of pile foundation of offshore wind tower.

Figure 4 :
Figure 4: Wave after the pile foundation changes.

Figure 5 :
Figure 5: The wave force time history curve of 4 m pile with diameter.

Figure 6 :
Figure 6: The wave force time history curve of 5 m pile with diameter.

Figure 7 :
Figure 7: The wave force time history curve of 6 m pile with diameter.

Figures 9 ,
10, and 11  show that the diameters of 4 m, 4.5 m, 5 m, 5.5 m, 6 m, and 6.5 m pile foundation are in a cycle.With the increase of the diameter, the wave force and moment are nonlinearly increased.

Table 1 :
List of working parameter in offshore wind power pile structure system.