1 Introduction

With the vigorous development of the transportation and marine engineering, many temporary buried structures (wind-driven pile drilling platforms, oil drilling platforms, etc.) have been built in offshore areas. Compared with long-term buried offshore structures (wind turbine pile foundation, bridge foundation of cross-sea bridge, etc.), the stability and safety of shallow buried structures are much lower than those of permanently buried structures because of their engineering characteristics. However, during the drilling process, the temporary buried structures needs to provide a stable platform for equipment [1, 2]. Therefore, the stress analysis of the temporary buried structures under the action of waves can effectively provide theoretical support for the economic and safety design of temporary buried structures [3,4,5].

Previous studies mainly focused on the permanent buried structures at sea, while there was little research on pile foundation construction platforms under the action of waves [6, 7]. However, the impact of wave force on temporary buried structures is much greater than that of permanent facilities. Therefore, as an essential consideration in designing and constructing temporarily buried marine structures, wave force has become a focus of scholars' research. For the cylindrical pile foundation common in offshore engineering [8, 9]. The calculation method of wave force is selected according to the structural size (the ratio d/l of the diameter d of the cylinder to wavelength l). For small-scale structures with D/L < 0.2, the Morrison equation, which is semi-empirical and semi-theoretical, can be used, and the influence of structure size on the flow field can be ignored. For large-scale structures with D/L ≥ 0.2, the scattering effect of the system on incident waves and the free surface effect between the design and the fluid surface is apparent, so the wave force should be calculated by diffraction theory [10,11,12]. The boundary element model established by He and Geng [11] is used to numerically study the diffraction of three-dimensional objects with arbitrary shapes in regular waves. Using numerical simulation, Kang [13] studied the diffraction of a large diameter vertical cylinder under the action of periodic waves. Teng [14] introduced in detail the construction scheme design and construction lowering process of single-wall steel hanging box cofferdam in offshore engineering. When the above-mentioned scholars study the structural wave force by theoretical calculation, experimental research and numerical simulation, they all ignored the influence of pore flow on the structural wave force of high-porosity seabed [15,16,17]. Therefore, the numerical analysis of the impact of the seabed on wave force on the single pile and the results showed that the seabed attenuated wave propagation and increased wave force [13, 15, 18, 19]. However, the above research does not involve the influence of the dynamic process, structural form, the buried depth, wave parameters and other factors on structural wave force [20,21,22] (Fig. 1).

Fig. 1
figure 1

The large diameter offshore drilling platform

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Due to the scale effect and high cost of practical methods, the applicable conditions of theoretical research methods are limited. With the rapid improvement of computer efficiency, numerical research methods have been used widely. Therefore, this paper refers to Chen [13] and others to control the fluid movement based on the modified RANS equation, describes the influence of waves on the seabed, and establishes a three-dimensional coupling analysis model of the interaction between shallow consolidated structures under wave action, further studies the attenuation effect of the seabed on waves, the change of the flow field around the system, and the influence of wave parameters and the movement state of the structure on the wave force on the network, which provides a reference for the design and buried of the system.

2 The numerical model

In this paper, FlAC-3 D simulation software is used for modeling and analysis of the project. The simulation software is a program for three-dimensional explicit calculation based on finite difference method [23]. The model is designed by square network elements. The upper water layer is 20 m high, and the unit size is 0.5 m × 0.5 m. Figure 2 is a three-dimensional figure for numerical analysis of the stress model of shallow embedded offshore structures under wave action based on the drilling platform of a wind turbine pile foundation in Long Yuan. For six platforms with a weight of over 550 t and a span of 23 m, the drilling liners extend to the relatively hard rock seabed with a depth of 1.5 m. Starting from the velocity boundary on the left side, the wave enters the propagation zone with sufficient length along the positive direction of the x-axis. x, y and z represent Cartesian coordinates; dw stands for water depth; ds represents the thickness of the seabed; H0 represents the initial wave height; L stands for wavelength; D indicates the diameter of a single drilling liner; e means the relative buried depth, that is, the ratio of the structural depth dm extending into the rocky seabed to the seabed thickness; The length, width and height of the computational domain along the coordinate axis are Lx, Ly and Lz, so it is necessary to calculate the value of the computational field in advance to minimise the influence of boundary conditions on the numerical analysis results. According to the test results, the calculation domain satisfies the following requirements: Lx = 10 L, Ly = 100 D, Lz = 25 m. In the subsequent numerical analysis, the ratio of structure width to numerical flume width is always less than 0.2, so the influence of flume width on the calculation results can be ignored. Unless otherwise stated, the analysis parameters are shown in Table 1.

Fig. 2
figure 2

Schematic diagram of numerical calculation of long-span particles on the rocky seabed

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Table 1 Numerical analysis parameters
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2.1 The governing equations

2.1.1 The fluid motion control

In this paper, according to RANS equation [24], the fluid load caused by waves is controlled, and the convergence of the equation is analyzed by using k–í equation. The analysis formula is as follows.

$$\frac{\partial \langle {u}_{i}\rangle }{\partial {x}_{i}}=0$$
(1)
$$\frac{\partial \rho \langle {u}_{i}\rangle }{\partial t}+\frac{\partial \rho \langle {u}_{i}\rangle \langle {u}_{j}\rangle }{\partial {x}_{j}}=-\frac{\partial \langle p\rangle }{\partial {x}_{i}}+\frac{\partial }{\partial {x}_{j}}\left[\mu \left(\frac{\partial \langle {u}_{i}\rangle }{\partial {x}_{j}}+\frac{\partial \langle {u}_{j}\rangle }{\partial {x}_{i}}\right)\right]+\frac{\partial }{\partial {x}_{j}}\left(-\rho \langle {u}_{i}^{^{\prime}}{u}_{j}^{^{\prime}}\rangle \right)+\rho {g}_{i}-{F}_{d}{u}_{i}$$
(2)

where \({x}_{i}\) is Cartesian coordinate; \(\langle {u}_{i}\rangle \) is the overall average velocity component; this time \(\rho \mathrm{is fluid density};\langle p\rangle \) is fluid pressure; \(\mu \) is dynamic viscosity;\({g}_{i}\) is the acceleration of gravity; \(-{F}_{d}{u}_{i}\) is the resistance action term of porous media to water flow, that is, drag force. \(-\rho \langle {u}_{i}^{^{\prime}}{u}_{j}^{^{\prime}}\rangle \) is Reynolds stress term, and k–ε turbulence equation can be used to simulate turbulent flow at high Reynolds number.

Boussinesq [7] put forward the calculation of eddy viscosity hypothesis of water flow, so Reynolds stress term can be expressed as:

$$-\rho \langle {u}_{i}^{\mathrm{^{\prime}}}{u}_{j}^{\mathrm{^{\prime}}}\rangle ={\mu }_{\mathrm{t}}\left(\frac{\partial \langle {u}_{i}\rangle }{\partial {x}_{j}}+\frac{\partial \langle {u}_{j}\rangle }{\partial {x}_{i}}\right)-\frac{2}{3}\rho {\delta }_{ij}\kappa $$
(3)

where \({\mu }_{\mathrm{t}}\) is turbulent viscosity; \(\kappa \) is turbulence kinetic energy;\({\delta }_{ij}\) is the Cronech symbol.

Substituting formula (3) into formula (2), we can get

$$\frac{\partial \rho \langle {u}_{i}\rangle }{\partial t}+\frac{\partial \rho \langle {u}_{i}{u}_{j}\rangle }{\partial {x}_{j}}=-\frac{\partial }{\partial {x}_{i}}\left[\langle p\rangle +\frac{2}{3}\rho \kappa \right]+\frac{\partial }{\partial {x}_{j}}\left[{\mu }_{\mathrm{eff}}\left(\frac{\partial \langle {u}_{i}\rangle }{\partial {x}_{j}}+\frac{\partial \langle {u}_{j}\rangle }{\partial {x}_{i}}\right)\right]$$
(4)

where \({\mu }_{\mathrm{eff}}=\mu +{\mu }_{\mathrm{t}}\) is the total effective viscosity.

In numerical analysis, the standard equation of k–ε turbulence is:

$$\frac{\partial \rho \kappa }{\partial t}+\frac{\partial \rho \langle {u}_{j}\rangle \kappa }{\partial {x}_{j}}=\frac{\partial }{\partial {x}_{j}}\left[\left(\mu +\frac{{\mu }_{t}}{{\sigma }_{K}}\right)\frac{\partial \kappa }{\partial {x}_{j}}\right]+\rho {P}_{K}-\rho \varepsilon $$
(5)
$$\frac{\partial \rho \varepsilon }{\partial t}+\frac{\partial \rho \langle {u}_{j}\rangle \varepsilon }{\partial {x}_{j}}=\frac{\partial }{\partial {x}_{j}}\left[\left(\mu +\frac{{\mu }_{t}}{{\sigma }_{\varepsilon }}\right)\frac{\partial \varepsilon }{\partial {x}_{j}}\right]+\frac{\varepsilon }{\kappa }\left({C}_{\varepsilon 1}\rho {P}_{K}-{C}_{\varepsilon 2}\rho \varepsilon \right)$$
(6)
$${\mu }_{\mathrm{t}}=\rho {C}_{\mu }\frac{{\kappa }^{2}}{\varepsilon }$$
(7)
$${P}_{K}=\frac{{\mu }_{\mathrm{t}}}{\rho }\left(\frac{\partial \langle {u}_{i}\rangle }{\partial {x}_{j}}+\frac{\partial \langle {u}_{j}\rangle }{\partial {x}_{i}}\right)\frac{\partial \langle {u}_{i}\rangle }{\partial {x}_{j}}$$
(8)

where κ is turbulence kinetic energy; ε is the dissipation rate of turbulence kinetic energy; Parameters\({C}_{\mu }\),\({\delta }_{k}\),\({\delta }_{\varepsilon }\),\({C}_{\varepsilon 1}\) and \({C}_{\varepsilon 2}\) satisfy [23,24,25]: \({C}_{\mu }\)= 0.09, \({\delta }_{k}\)=1.00, \({\delta }_{\varepsilon }\)=1.30, \({C}_{\varepsilon 1}\)=1.44 and \({C}_{\varepsilon 2}\)=1.92.

2.1.2 The movement control of drilling liner

Because of the rigidity of drilling liners, the movement of the drilling liners is usually regarded as a rigid body. According to the principle of kinematics, the movement of rigid body can be divided into translation and rotation. so the velocity VP at a particular position of the rigid body is related to the velocity VG and angular velocity of rigid body ω, which meets the following requirements [8].

$${V}_{\mathrm{P}}={V}_{\mathrm{G}}+\omega \times {\gamma }_{\mathrm{PG}},$$
(9)

where \({\gamma }_{\mathrm{PG}}\) is the distance vector from G ~ P of the centroid.

2.1.3 Interaction between shallow drilling liners and seabed strata under wave action

Under the action of waves, the drilling liner will move horizontally and vertically, and the wave pressure will further change the water pressure of the soil. The height of the liner is h, and the depth of the seabed is d. The circulating force along the x-direction under the action of waves makes the bottom of the wellbore move along the z-direction and x-direction in the buried stratum. According to Biot's consolidation theory [25], The effective stress variation of two-dimensional seabed soil satisfies the following equilibrium equation.

$$\frac{\partial {\sigma }_{\mathrm{x}}}{\partial \mathrm{x}}+\frac{\partial \tau }{\partial \mathrm{z}}=\frac{\partial \mathrm{p}}{\partial \mathrm{x}}$$
(10)
$$\frac{\partial {\sigma }_{x}}{\partial z}+\frac{\partial \tau }{\partial x}=\frac{\partial p}{\partial z}$$
(11)

where \({\sigma }_{\mathrm{x}}\)/Pa and \({\sigma }_{z}\)/Pa are effective stresses of soil in horizontal and vertical directions; \(\tau \)/Pa is shear stress;\(p\) is excess pore water pressure.

The soil skeleton is regarded as an ideal isotropic elastic medium, and the constitutive equation of the seabed unit soil is.

$${\sigma }_{\mathrm{x}}=2\mathrm{G}\left[\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\frac{\mu }{1-2\mu }{\varepsilon }_{\mathrm{v}}\right]$$
(12)
$${\sigma }_{x}=26\left[\frac{\partial w}{\partial z}+\frac{\mu }{1-2\mu }{\varepsilon }_{y}\right]$$
(13)
$${\tau }_{\mathrm{xx}}=2\mathrm{G}\left[\frac{\partial \mathrm{u}}{\partial \mathrm{z}}+\frac{\partial \mathrm{w}}{\partial \mathrm{N}}\right]={\tau }_{\mathrm{zx}}$$
(14)

where G/Pa is the shear modulus of soil;\(\mu \) is poisson′s ratio; \({\tau }_{\mathrm{zx}}\)/Pa is the shear stress perpendicular to the xx-axisand along the z direction;\({\tau }_{\mathrm{zx}}\) is the shear stress perpendicular to the z axis and along the x direction. Substituting formulas (9) to (11) into formulas (7) and (8) can be obtained:

$$G{\nabla }^{2}u+\frac{G}{1-2\mu }\frac{\partial {\varepsilon }_{v}}{\partial x}=\frac{\partial p}{\partial x}$$
(15)
$$G{\nabla }^{2}\mathrm{w}+\frac{G}{1-2\mu }\frac{\partial {\varepsilon }_{v}}{\partial z}=\frac{\partial p}{\partial z}$$
(16)

2.2 The boundary conditions

According to the method proposed by Dean [26] to evaluate the applicable range of wave conditions, a fifth-order Stokes wave boundary is applied at the entrance boundary (Xmin). The exit boundary (Xmax) [27] adopts the Sommerfeld radiation boundary. When the edges of the numerical flume (Ymin and Ymax) assume a symmetric limit, waves can slide freely along the edges of the flume without penetrating. The wall boundary is adopted at the bottom and the structural surface of the numerical water tank (Zmin) to ensure that the average velocity of the fluid is zero and can not penetrate. At the top of the calculation field (Zmax), the pressure boundary is used, and the initial height of the fluid is specified. The VOF method locates and tracks the change of the free liquid surface [28]. The liquid volume function satisfies.

$$\frac{\partial F}{\partial t}+\frac{\partial \left({u}_{i}F\right)}{\partial {x}_{i}}=0$$
(17)

The fluid volume function f represents the ratio of the fluid volume in the cell to the total fluid volume that can be accommodated. When F = 0, it means that the grid is full of gas, when F = 1, it means that the grid is full of fluid, and when 0 < F < 1, it means that there are both gas and liquid in the grid, and it contains free water. Referring to the actual engineering construction process [8], In this paper, only the vertical and uniform descending motion of the drilling liner is considered. That is, the translation speed of the specified structure, and the remaining five degrees of freedom are all 0. In addition, the semi-coupling between structural motion and fluid means that the influence of structural motion on the flow field should be considered. However, the influence of the flow field on the structure signal has not been felt.

As shown in Fig. 2, the red marked pipe body of offshore single pile foundation is a large diameter steel pipe foundation. For this type of structural system, the pile-soil interaction system can be simplified as shown in Fig. 2. In Fig. 2, the dotted line indicates the pile, and the seabed soil in the lower light-coloured area can be described by the partial differential equations composed of Eqs. (4), (12) and (13), and the appropriate boundary conditions are established. The solution of pile-soil interaction in the ocean environment can be obtained by solving the partial differential equations. The seabed surface meets the following boundary conditions.

$$\mathrm{z}=0,\left\{\begin{array}{c}{\sigma }_{x}=0\\ {\tau }_{\mathrm{xx}}=0\\ p={\mathrm{P}}_{\mathrm{z}}\end{array}\right.$$
(18)

\({P}_{z}\) is the wave pressure on the seabed, which is determined by the formula (3). The wave pressure on the seabed surface is determined by the formula (3).

$$\mathrm{z}=-\mathrm{d},\left\{\begin{array}{c}u=0\\ w=0\\ \frac{\partial \mathrm{p}}{\partial \mathrm{z}}=0\end{array}\right.$$
(19)

And the interface between the liner and the seabed is impermeable to water. Assuming that the soil is not separated from the pile, the lateral displacement profile of the pile is linear. Under the action of linear waves, the time course of seabed soil displacement changes in sinusoidal curve, so does the time course of pile displacement. The left contact surface of pile-soil satisfies the following boundary conditions.

$$\left\{\begin{array}{c}u={\mathrm{y}}_{0}\cdot (1+z/l)cos(kx-\omega t)\\ \frac{\partial \mathrm{p}}{\partial \mathrm{x}}=0\end{array}\right.$$
(20)

where \({y}_{0}\)/m is the pile displacement at the mud surface; According to the equal transverse displacement on both sides of the pile, The boundary conditions of the pile-soil interface can be obtained.

$$\left\{\begin{array}{c}u={\mathrm{y}}_{0}\cdot (1+z/l)cos(kx-\omega t-kb)\\ \frac{\partial \mathrm{p}}{\partial \mathrm{x}}=0\end{array}\right.$$
(21)

where b(m) is the calculated width of the pile.

The left and right interfaces of seabed soil satisfy the following boundary conditions:

$$\left\{\begin{array}{c}{\sigma }_{\mathrm{x}}={\sigma }_{0}\\ \frac{\partial \mathrm{p}}{\partial \mathrm{x}}=0\end{array}\right.$$
(22)

where \({\sigma }_{0}\)/Pa is the confining pressure of soil. In the marine environment, the interaction system between the drilling casing and the earth can be expressed by the boundary conditions of partial differential equations composed of Eqs. (18), (19) and (20).

3 The model validation

In order to verify the accuracy of the model of the shallow-buried beam under wave action, and further study the stability of shallow-buried beam under different wave loads, the wave intensity around the interaction between wave and shallow-buried shaft and the stress of shallow-buried pile on rock seabed under wave action are analyzed.

3.1 Environmental wave intensity under the interaction of waves and shallow buried shaft

Mo et al. [8]. designed a series of flume tests to study wave and single pile interaction. The water tank is 309 m in length, 5 m in width and 7 m in depth. The single pile is fixed at the bottom of the horizontal flume at a distance of 111 m from the wave generator. A plurality of wavemeters (WG) is arranged at different positions around the structure. The data of three typical locations(as shown in the Fig. 3) around the structure were compared: front (WG1), side (WG2) and back (WG3). The wave parameters are water depth dw = 4.76 m, wave period T = 4 s, wave height H0 = 1.2 m, and single pile diameter D = 0.7 m.

Fig. 3
figure 3

Comparison of relative wave elevation time-history curves

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Figure 3 shows the time history curve of relative wave height. The wave height η is dimensionless by using ηmax and WGref of the experimental data of the reference wave gauge (WGref). Because the reference wave gauge (WGref) is far away from the structure, the influence of the form on the wavefront can be ignored. The graphic results show that the measured data are consistent with the numerical results, and some characteristic ripples are well simulated. For example, when the wave trough passes through the drilling casing, the experimental and numerical results of the wave height at the front end (WG1) show noticeable ripples. The results show that the numerical model can reasonably simulate the wave flow around the lining and provide reliable analysis results for the follow-up study of the wave forces on the structures under the wave action.

3.2 The force of shallow embedded shaft on the rocky seabed under wave action

To consider the influence of rock seabed characteristics on wave forces on structures, a three-dimensional coupling model of wave-seabed-structure interaction is established, in which the rigidity (n = 0) and elasticity (n = 0.5) of the seabed are considered, respectively. To verify the accuracy of wave force calculation and the necessity of considering seabed characteristics, the numerical results of wave force FX along the wave propagation direction of liners under different seabed conditions are compared with MacCamy’s theoretical results. Figure 4 shows FX time history curves under other seabed conditions. The comparison results show that the rigid seabed numerical analysis results agree with the MacCamy theoretical developments at the peak position. Still, there is a slight deviation at the valley position. This is mainly because the MacCamy theoretical formula does not consider the lateral force caused by vortex discharge behind the structure, which makes the amplitude of MacCamy’s hypothetical results at the valley position more significant than the numerical analysis results of the rigid seabed. In addition, the maximum value of FX in the elastic seabed is about 35% higher than that in the rigid seabed. It shows that the three-dimensional coupling model can accurately calculate the change of wave force on the structure under the condition of a rigid seabed, and the influence of seabed characteristics on the numerical value of wave force can not be ignored.

Fig. 4
figure 4

Time history curve of wave force FX on a single pile

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The numerical results show that the rock seabed can cause the attenuation of wave energy, but this does not mean that the wave force on the structure must definitely decrease. The wave forces in the designs will be affected by the wave conditions and the characteristics of rock seabed. Compared with rigid seabed, the rock characteristics of seabed will significantly affect the maximum FX. Simplifying the rock seabed into frozen impermeable solid and ignoring the rock characteristics of the seabed will underestimate the force of waves on the structure. Therefore, it is necessary to comprehensively consider the influence of wave parameters on seabed and structural characteristics, and deeply analyze the forces exerted by wave on structures.

4 Discussion

4.1 The influence of wave parameters

The stress of shallow-buried shaft under wave action is mainly affected by local geological and hydrological conditions, so without considering the influence of the seabed, the wave parameters are more critical to the horizontal wave force of liners [7, 10, 29], and the calculation method of horizontal wave force of various structures under wave action is introduced in detail in "Hydrographic Code for Ports and Waterways (JTS 145-2015) [30]. Because researchers pay more attention to the attenuation of waves on the rocky seabed, there is little research on wave forces on structures extending to the elastic seabed. Therefore, the correlation between incident wave conditions and wave forces acting on the cushion layer is studied and compared with strict seabed conditions. the effects of period T, wave steepness (H0/L) and relative water depth (dw/L) on wave forces on the cushion were studied.

Figures 5, 6 and 7 show FXmax curves under different wave periods T, wave steepness (H0/L) and relative water depth (dw/L). The results show that the FXmax value increases linearly with the increase of wave parameters, but the increased range is different; The maximum value of wave force FXmax is sensitive to wave height and water depth. This means that the liners extending into the elastic seabed are subjected to greater wave forces under the action of big waves and long periods. This is because the wave energy is significant, and the projection area of the transverse structure to the wave propagation direction is substantial; that is, the reference volume is large, which makes the force borne by the cushion large. In addition, the value of elastic seabed FXmax is greater than that of the rigid seabed. By comparing the analysis results, the importance of considering the seabed's strength in shallow-buried ocean engineering design is emphasised again.

Fig. 5
figure 5

Under different period T conditions Fxmax change curve

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Fig. 6
figure 6

Under different wave steepness (H0/L) Fxmax change curve

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Fig. 7
figure 7

Fxmax variation curves under different relative water depths (dw/L)

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4.2 The influence of structure characteristics

In engineering practice, the pile foundation's size will affect the structure's stress. In addition, during the initial structural design, various cross-section types, such as round, square or round end, will be compared, and the embedding depth will be set. At the same time, the pile foundation structure is lowered during construction. Therefore, it is beneficial for structural safety to study the influence of cross-section diameter d, shape, buried depth (e) and construction lowering speed (v) on the dynamic response of structures.

4.2.1 The influence of structure diameter D

To study the influence of single pile diameter on wave force on a single pile considering the porous characteristics of the seabed, the single pile diameter increased from 4 to 8 m under the same wave and seabed characteristics. Figure 8 is a time history curve of wave force FX on a single pile. As shown in the figure, with the wave propagation, the value of wave force FX oscillates with the wave period t, and the amplitude is stable when the wave is stable. In addition, the larger the diameter of a single pile, the larger the amplitude of wave force FX on a single pile. This is because under the action of waves, the larger the diameter of a single pile, the larger the lateral stress area of the structure. In engineering practice, the diameter of systems can be determined concerning the traditional calculation method under the condition of high permeability seabed.

Fig. 8
figure 8

The time history curve of FX under different pile foundation diameter D

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4.2.2 The influence of buried depth s of structures

According to the above analysis, the dynamic characteristics of the structure are influenced by the characteristics of elastic seabed, mainly due to the seepage inside the seabed. Then, the depth of the shallow-buried structures extends into the seabed is relatively shallow, so the elastic seabed has different influences on the stress of the shallow-buried structures according to the buried depth of liners. Figure 9 is the variation curve of FX and FXmax under the condition of relative burial depth of different structures. The results show that, with the increase of buried depth S, the change period of FX is the same, and FXmax increases linearly with the increase of buried depth. This is because the stress calculation length of shallow buried structures increases linearly. The calculated stress length of a single pile with s = 1 is always more meaningful than that of a single pile with s = 0, and the seabed thickness is ds.. In engineering practice, for the elastic seabed, the influence of seabed characteristics and the influence of lining buried depth on structural stress should be considered in the consolidation design.

Fig. 9
figure 9

FX and FXmax variation curves under different buried depth s

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4.3 The displacement analysis of structures in soil layer

According to the functional condition design of the offshore drilling platform of Long Yuan Group, the specific parameters are shown in Table 1. According to the model analysis, FX = 610 KN, calculated area: − 400 m ≤ x ≤ 400; − 100 mm ≤ z ≤ 0. Figure 5 shows the wave pressure of a boring steel cofferdam under different deflections. It can be seen from the figure that when the deflection of the steel cofferdam exceeds 10 mm and the soil conditions are unchanged, the earth pressure on the pile will decrease, which is consistent with the description in the code for pile foundation. Figure 10a and b shows the effective normal stress distribution of the soil around the steel cofferdam under different deflections. It can be seen from these figures that the deflection of the steel cofferdam has little effect on the normal stress of the underlying soil. Figure 10c and d represents the relative horizontal and vertical displacement distribution of soil in the free field. It can be seen from the figure that the horizontal and vertical displacement of the seabed soil gradually decrease along the depth direction, and become zero at a certain depth. The vertical displacement attenuates rapidly, and the influence of horizontal displacement on the bearing capacity of steel cofferdam can not be ignored. Based on the horizontal displacement distribution diagram and Wink foundation beam model, the response of the offshore pile foundation can be calculated accurately.

Fig. 10
figure 10

The analysis of calculation results

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5 Conclusion

The finite element numerical simulation method is used to analyze the wave forces of offshore shallow buried structures under various working conditions. The effects of the rigid and elastic seabed on buried structures, buried depth and structures, and structural characteristics on structural forces are studied, as well as the stability of shallow buried systems under different wave forces. The following conclusions and suggestions are obtained: three-dimensional coupled numerical model of waves on buried offshore structures in shallow water can better reflect the attenuation effect of waves, and can be further used to study the forces of waves on structures. During the process of design and construction of drilling pipe, the influence of seabed elasticity on structural wave forces should not be ignored. Due to the influence of wave energy and the volume of the structure, the structure extends into the deep-water elastic seabed. Under the action of big waves, high waves and long periods, the system is subjected to relatively large wave force. In structural design and construction, it is necessary to choose reasonable wave parameters.

Under the same wave conditions, the larger the diameter of the structure is, the larger the amplitude of FX is. Due to the influence of reference volume and surface curvature, the cross-sectional shape of the system will affect the stress of the structure. FXmax increases linearly with the increase of buried depth, and FX increases periodically in the process of lowering the structure at a uniform speed. The amplitude of FX is stable when the system is at rest on the seabed surface. The faster the descent speed, the faster the wave reaches stability, but the amplitude is the same. The numerical examples show that, under the action of waves, the horizontal displacement and vertical displacement of seabed soil gradually decrease along the depth direction, and the vertical displacement decreases rapidly. The influence of horizontal displacement on the bearing capacity of offshore pile foundations can not be ignored. The response of "passive pile" in the marine environment can be accurately calculated by combining the plane displacement distribution diagram obtained from the analysis of the Wink foundation beam model.

Through the related calculation and numerical analysis in this paper, the stability of shallow-buried offshore drilling platform under the geological conditions of rock seabed in South China Sea is analyzed and discussed. Therefore, it is necessary to calculate the stability of shallow water drilling platforms under different seabed conditions in the later research.