A three-dimensional pile driving effect correction calculation method based on wave equation

Abstract

This invention discloses a three-dimensional pile driving effect correction calculation method based on the wave equation, comprising: determining the Poisson's ratio and Pochhammer reduction radius of the pile material; extending the original one-dimensional wave equation into a three-dimensional wave equation by introducing a lateral inertial effect, and spatially discretizing the equation using the central difference method to establish the difference expression of acceleration at each node of the pile; deriving the tridiagonal coefficient matrix satisfied by the acceleration vector based on the discretization scheme; during the time-step iterative calculation process, based on the displacement field of the current time step, sequentially calculating the spring resistance, displacement, and soil resistance around the pile for each pile element to obtain the acceleration vector under the one-dimensional wave model, and obtaining the three-dimensional corrected acceleration by solving the tridiagonal coefficient matrix, and updating the velocity and displacement for the next time step. This invention extends pile driving analysis from one-dimensional to three-dimensional, truly reflecting the propagation and attenuation characteristics of pile wave, and improving the calculation accuracy and numerical stability of pile foundation dynamic response.

Description

Technical Field

[0001] This invention belongs to the field of underground engineering technology, specifically a three-dimensional pile driving effect correction calculation method based on the wave equation. Background Technology

[0002] As a crucial foundation type in civil engineering, bridge engineering, and offshore wind power, the construction quality of pile foundations directly impacts the load-bearing capacity and safety of structures. During pile driving, the stress waves generated by hammering propagate along the pile and undergo complex dynamic interactions with the surrounding soil and the soil at the pile tip. Therefore, establishing a calculation method that can accurately describe the dynamic response of pile driving is of great significance for optimizing pile foundation construction parameters, predicting penetration depth, and evaluating bearing capacity.

[0003] Currently, dynamic analysis of pile foundations commonly employs dynamic analysis methods based on Smith's one-dimensional wave theory. This method discretizes the pile body into several mass elements and simulates the dynamic response of the pile and soil using springs and damping elements. It boasts advantages such as high computational efficiency and mature engineering applications, and has been widely used in pile driving process simulation and dynamic pile driving analysis programs. However, Smith's one-dimensional wave theory assumes that the pile stress wave propagates only along the axial direction, neglecting the influence of the lateral inertial effect of the pile cross-section on the wave propagation characteristics. When the pile diameter is large, the wall thickness is thick, or the stress wave frequency is high, this assumption will lead to certain errors in the calculation of wave velocity and dynamic response, thus affecting the accuracy of pile driving process prediction.

[0004] In recent years, three-dimensional wave theory based on the Pochhammer-Chree equations (a nonlinear partial differential equation describing the propagation of longitudinal deformation waves in an elastic rod) has been able to consider the influence of the pile's lateral inertia and cross-sectional deformation on stress wave propagation, thus providing a more accurate description of the three-dimensional dynamic response within the pile. However, existing three-dimensional wave analysis methods mostly employ numerical calculation methods such as finite element and finite difference methods, which involve large computational scales and low solution efficiency, making them difficult to directly apply to existing Smith dynamic analysis programs.

[0005] Therefore, while maintaining the basic computational framework of the original Smith dynamic analysis program, introducing three-dimensional wave theory into the existing program to realize dynamic analysis that considers the lateral inertial effect, while taking into account computational efficiency, computational stability and engineering applicability, has become a technical problem that urgently needs to be solved in this field. Summary of the Invention

[0006] To address the aforementioned problems in the existing technology, this invention provides a three-dimensional pile driving effect correction calculation method based on the wave equation, which extends the pile driving analysis from one-dimensional to three-dimensional, more realistically reflecting the propagation and attenuation characteristics of pile body wave, thereby improving the calculation accuracy and numerical stability of pile foundation dynamic response.

[0007] The technical solution to achieve the above objectives is: A three-dimensional piling effect correction calculation method based on the wave equation includes: Step S1: Determine the Poisson's ratio of the pile material based on the pile parameters. Pochhammer reduced radius ; Step S2: The original one-dimensional wave equation is extended into a three-dimensional wave equation by introducing the transverse inertial effect, and the equation is spatially discretized using the central difference method to establish the difference expression of acceleration at each node of the pile. Step S3: Based on the discrete scheme of step S2, derive the tridiagonal coefficient matrix satisfied by the acceleration vector; Step S4: During the time-step iterative calculation process, based on the displacement field of the current time step, the spring resistance, displacement and soil resistance of each pile unit are calculated sequentially to obtain the acceleration vector under the one-dimensional wave model. The three-dimensional corrected acceleration is obtained by solving the tridiagonal coefficient matrix of step S3, and the velocity and displacement of the next time step are updated. Step S5: Monitor numerical stability. If the amplitude of high-frequency noise exceeds the allowable threshold, reduce the time step to suppress instability.

[0008] Preferably, in step S1, the Poisson's ratio of the pile material is determined according to the type of pile material. The Poisson's ratio for steel is 0.3, for concrete it is 0.2, and for timber it is selected within the range of 0.3-0.6 depending on the material orientation. Calculate the Pochhammer reduction radius based on the pile cross-section. ,in, For outer diameter , inner diameter Hollow piles: ; For a solid pile with diameter D: .

[0009] Preferably, step S2 includes: Establish a one-dimensional wave equation for the propagation of longitudinal stress waves in the pile: ; In the formula, The density of the pile material, For elastic modulus, This refers to the vertical displacement of the pile body. The time variable is the time of stress wave propagation in the pile. These are spatial coordinates along the length of the pile, used to locate the vibration state at different positions on the pile. The resistance force exerted by the soil on the pile body. This refers to the cross-sectional area of ​​the pile. By introducing a transverse inertia correction term based on Pochhammer-Chree theory into the one-dimensional wave equation, a three-dimensional wave equation is established: ; In the formula, the new item Used to describe the three-dimensional wave correction effect caused by the lateral inertia of the pile body; make The governing equations are then transformed into a solution form for nodal accelerations, namely: ; The pile body is divided into Smith discrete elements along the axial direction. Using the central difference scheme to discretize the second derivative in space at each discrete node, we obtain the discrete equations for the nodal accelerations: ; In the formula, For the first The acceleration of each discrete node in the pile body, i.e., the acceleration after three-dimensional wave correction to be determined. , and The first , , Vertical displacement of each node, , The first , Acceleration of discrete nodes in the pile body For the first The soil resistance around the pile in each pile unit For the first Cross-sectional area of ​​each pile unit is the length of the discrete element in the axial direction of the pile.

[0010] Preferably, step S3 includes: Let constant coefficient The discrete equations of step S2 are rearranged as follows: ; in, ; This indicates the first wave calculated according to Smith's one-dimensional wave theory. Individual node acceleration; By combining all discrete nodes, we can construct the acceleration matrix equation for all pile nodes: ; In the formula, It is a tridiagonal matrix with constant coefficients, and its main diagonal elements are: The left and right diagonal elements are , Let be the acceleration vector of the three-dimensional node to be determined. The acceleration vector calculated from the original one-dimensional model; Approximate boundary conditions are applied to the pile top and pile bottom boundaries. The corresponding boundary nodes satisfy and .

[0011] Preferably, step S4 includes: Step S41: Based on the displacement and velocity of each discrete node in the current time step, calculate the soil resistance and displacement of each element of the pile body according to the original Smith dynamic analysis program, establish the dynamic equilibrium relationship of the nodes, and obtain the node acceleration vector corresponding to the original one-dimensional wave equation. Step S42: Using the one-dimensional nodal acceleration vector from step S41 as the right-hand side of the three-dimensional correction equation, and combining it with the tridiagonal coefficient matrix from step S3, establish a set of linear equations for nodal acceleration considering lateral inertial effects. Step S43: The tridiagonal linear equation system is solved by the chasing method, and the three-dimensional corrected acceleration of each discrete node in the current time step is obtained by forward elimination and back substitution calculation. Step S44: Replace the one-dimensional nodal acceleration in the original Smith program with the three-dimensional corrected acceleration from step S43, and update the nodal velocities, displacements, and force states of each element according to the original program's time integral format to complete the calculation for the current time step. Step S45: Repeat steps S41 to S44 until all time steps are completed to obtain the dynamic response parameters of pile displacement, velocity, acceleration, stress wave propagation process, pile side resistance, pile end resistance and penetration.

[0012] Preferably, in step S43, the established tridiagonal linear equation system is solved using the chasing method, including: During the forward elimination process, let the lower diagonal elements, main diagonal elements, and upper diagonal elements of the tridiagonal matrix be respectively... , and The right-hand item is Then, the forward elimination process is recursively calculated as follows: ; ; In the formula, This is the upper diagonal advancement coefficient obtained after forward elimination, used to transform the original tridiagonal equation system into an upper triangular equation system, preparing for subsequent back-substitution calculations. The right-hand side advancement coefficient obtained after forward elimination is... To complete the back-substitution solution process, During the forward elimination process, the first Correction values ​​for the upper diagonal elements corresponding to each node. During the forward elimination process, the first Correction values ​​for the right-hand items corresponding to each node; After completing the forward elimination, back substitution calculations are performed starting from the last discrete node to solve for the three-dimensional corrected acceleration of each node in turn: ; ; In the formula, For the first Three-dimensional corrected acceleration of discrete nodes of the pile body The first result obtained after forward elimination The right-hand side correction values ​​of each node are directly used as the acceleration solution for the last node. During the forward elimination process, the first The right-hand side correction value corresponding to each node. During the forward elimination process, the first Correction values ​​for the upper diagonal elements corresponding to each node. This represents the total number of discrete nodes along the axial direction of the pile. The obtained three-dimensional corrected acceleration vector replaces the one-dimensional nodal acceleration in the original Smith dynamic analysis program, and the time step iteration program is returned to complete the update calculation of nodal velocity, nodal displacement and subsequent dynamic response parameters.

[0013] Preferably, in step S5, after the node dynamic response calculation is completed at each time step, it is monitored whether high-frequency numerical oscillations or non-physical oscillations occur at each discrete node. When the amplitude of high-frequency numerical oscillation exceeds the preset allowable threshold, it is determined that there is a trend of numerical instability in the calculation. The calculation is then re-executed after reducing the time step size. When the amplitude of high-frequency numerical oscillation does not exceed the preset allowable threshold, the current time step is maintained and subsequent time steps are iterated.

[0014] Compared with the prior art, the beneficial effects of the present invention are: This invention introduces a lateral inertia correction term based on the Pochhammer-Chree theory on the basis of the traditional Smith one-dimensional wave theory, and establishes a three-dimensional wave analysis method that considers the lateral inertia effect of the pile body. This method can more accurately describe the propagation process and dynamic response characteristics of stress waves in the pile body. This invention constructs a tridiagonal linear equation system for nodal acceleration and uses the catch-up method for rapid solution, achieving efficient coupling between the three-dimensional wave equation and the existing pile foundation dynamic analysis program. While ensuring calculation accuracy, it avoids the problems of large calculation volume and low solution efficiency caused by using complex three-dimensional finite element models, thus improving the calculation efficiency and numerical stability of the dynamic analysis program. This invention embeds a three-dimensional wave analysis module into the existing Smith dynamic analysis program. Only a three-dimensional correction calculation module needs to be added during the original time step iteration process. The three-dimensional dynamic analysis can be achieved without changing the overall calculation framework of the original program. It has good compatibility, scalability and engineering applicability. It can be applied to the dynamic response analysis of pile foundations under different pile types, different materials and different hammering conditions, and provides a more accurate and efficient calculation method for pile foundation construction process simulation and engineering design. Attached Figure Description

[0015] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used in conjunction with embodiments of the invention to explain the invention and do not constitute a limitation thereof. In the drawings: Figure 1 This is a flowchart of a three-dimensional pile driving effect correction calculation method based on the wave equation according to the present invention; Figure 2 This is a detailed flowchart of a three-dimensional piling effect correction calculation method based on the wave equation of the present invention; Figure 3 This is a theoretical diagram of the three-dimensional wave equation model in this invention; Figure 4 This is the logical flowchart of step S4, the time-step iterative calculation in this invention. Figure 5 This is a schematic diagram comparing the final penetration calculation results obtained by the traditional one-dimensional wave model of small-diameter steel pipe and the three-dimensional analysis and calculation method of this invention in an embodiment of the present invention. Figure 6 This is a schematic diagram comparing the final penetration calculation results obtained by the traditional one-dimensional wave model of large-diameter steel pipe and the three-dimensional analysis and calculation method of this invention in an embodiment of the present invention. Detailed Implementation

[0016] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0017] like Figure 1 , 2 As shown, a three-dimensional piling effect correction calculation method based on the wave equation includes: Step S1: Determine the Poisson's ratio of the pile material based on the pile parameters. Pochhammer reduced radius .

[0018] In this embodiment, the Poisson's ratio of the pile material is determined based on the type of pile material. The Poisson's ratio for steel is 0.3, for concrete it is 0.2, and for timber it is selected within the range of 0.3-0.6 depending on the material orientation. Calculate the Pochhammer reduction radius based on the pile cross-section. ,in, For outer diameter , inner diameter Hollow piles: ; For a solid pile with diameter D: .

[0019] Its function is to provide accurate pile body fluctuation parameters for three-dimensional analysis and calculation methods, ensure reasonable correction of lateral inertial effects, and thus improve the calculation accuracy and reliability of pile foundation dynamic response analysis.

[0020] Step S2: The original one-dimensional wave equation is extended into a three-dimensional wave equation by introducing the transverse inertial effect, and the equation is spatially discretized using the central difference method to establish the difference expression of acceleration at each node of the pile.

[0021] In this embodiment, step S2 includes: Establish a one-dimensional wave equation for the propagation of longitudinal stress waves in the pile: ; In the formula, The density of the pile material, For elastic modulus, This refers to the vertical displacement of the pile body. The time variable is the time of stress wave propagation in the pile. These are spatial coordinates along the length of the pile, used to locate the vibration state at different positions on the pile. The resistance force exerted by the soil on the pile body. This refers to the cross-sectional area of ​​the pile. like Figure 3 As shown, considering the influence of the pile's lateral inertia on the propagation velocity and characteristics of stress waves, a lateral inertia correction term based on Pochhammer-Chree theory is introduced into the one-dimensional wave equation to establish a three-dimensional wave equation: ; In the formula, the new item Used to describe the three-dimensional wave correction effect caused by the lateral inertia of the pile body; Furthermore, to facilitate numerical solutions, the higher-order mixed partial derivatives are expressed as the second derivative of acceleration with respect to spatial coordinates, let... Transforming the governing equations into a solution for nodal accelerations, we have: This transforms the governing equations into a solution form for nodal accelerations; The pile body is divided into Smith discrete elements along the axial direction. Using the central difference scheme to discretize the second derivative in space at each discrete node, we obtain the discrete equations for the nodal accelerations: ; In the formula, For the first The acceleration of each discrete node in the pile body, i.e., the acceleration after three-dimensional wave correction to be determined. , and The first , , Vertical displacement of each node, , The first , Acceleration of discrete nodes in the pile body For the first The soil resistance around the pile in each pile unit For the first Cross-sectional area of ​​each pile unit is the length of the discrete element in the axial direction of the pile.

[0022] Step S3: Based on the discrete format of step S2, derive the tridiagonal coefficient matrix satisfied by the acceleration vector.

[0023] In this embodiment, since the introduction of the lateral inertia correction term means that the left and right sides of the discrete equation simultaneously contain the acceleration variables of each discrete node, a spatial coupling relationship is formed between the accelerations of each node. Therefore, the explicit solution method of calculating each node independently in the original one-dimensional model can no longer be used. Instead, an overall matrix equation needs to be established for synchronous solution. Thus, step S3 includes: Let constant coefficient The discrete equations of step S2 are rearranged as follows: ; in, ; This indicates the first wave calculated according to Smith's one-dimensional wave theory. Individual node acceleration; By combining all discrete nodes, we can construct the acceleration matrix equation for all pile nodes: ; In the formula, It is a tridiagonal matrix with constant coefficients, and its main diagonal elements are: The left and right diagonal elements are , Let be the acceleration vector of the three-dimensional node to be determined. The acceleration vector calculated from the original one-dimensional model; Approximate boundary conditions are applied to the pile top and pile bottom boundaries. The corresponding boundary nodes satisfy and .

[0024] In step S4, during the time-step iterative calculation, based on the displacement field of the current time step, the spring resistance, displacement, and soil resistance of each pile unit are calculated sequentially to obtain the acceleration vector under the one-dimensional wave model. The three-dimensional corrected acceleration is obtained by solving the tridiagonal coefficient matrix of step S3, and the velocity and displacement of the next time step are updated.

[0025] like Figure 4 As shown, step S4 includes: Step S41: Based on the displacement and velocity of each discrete node in the current time step, calculate the soil resistance and displacement of each element of the pile body according to the original Smith dynamic analysis program, establish the dynamic equilibrium relationship of the nodes, and obtain the node acceleration vector corresponding to the original one-dimensional wave equation.

[0026] Step S42: Using the one-dimensional nodal acceleration vector from step S41 as the right-hand side of the three-dimensional correction equation, and combining it with the tridiagonal coefficient matrix from step S3, establish a set of linear equations for nodal acceleration considering lateral inertial effects.

[0027] Step S43: The tridiagonal linear equation system is solved by the chasing method, and the three-dimensional corrected acceleration of each discrete node in the current time step is obtained by forward elimination and back substitution calculation. The tridiagonal linear equation system is solved using the chasing method, including: During the forward elimination process, let the lower diagonal elements, main diagonal elements, and upper diagonal elements of the tridiagonal matrix be respectively... , and The right-hand item is Then, the forward elimination process is recursively calculated as follows: ; ; In the formula, This is the upper diagonal advancement coefficient obtained after forward elimination, used to transform the original tridiagonal equation system into an upper triangular equation system, preparing for subsequent back-substitution calculations. The right-hand side advancement coefficient obtained after forward elimination is... To complete the back-substitution solution process, During the forward elimination process, the first Correction values ​​for the upper diagonal elements corresponding to each node. During the forward elimination process, the first Correction values ​​for the right-hand items corresponding to each node; After completing the forward elimination, back substitution calculations are performed starting from the last discrete node to solve for the three-dimensional corrected acceleration of each node in turn: ; ; In the formula, For the first Three-dimensional corrected acceleration of discrete nodes of the pile body The first result obtained after forward elimination The right-hand side correction values ​​of each node are directly used as the acceleration solution for the last node. During the forward elimination process, the first The right-hand side correction value corresponding to each node. During the forward elimination process, the first Correction values ​​for the upper diagonal elements corresponding to each node. This represents the total number of discrete nodes along the axial direction of the pile. The obtained three-dimensional corrected acceleration vector replaces the one-dimensional nodal acceleration in the original Smith dynamic analysis program, and the time step iteration program is returned to complete the update calculation of nodal velocity, nodal displacement and subsequent dynamic response parameters.

[0028] Step S44: Replace the one-dimensional node acceleration in the original Smith program with the three-dimensional corrected acceleration from step S43, and update the node velocity, displacement, and force state of each element according to the original program's time integration format to complete the calculation for the current time step.

[0029] Step S45: Repeat steps S41 to S44 until all time steps are completed to obtain the dynamic response parameters of pile displacement, velocity, acceleration, stress wave propagation process, pile side resistance, pile end resistance and penetration.

[0030] Step S5: Monitor numerical stability. If the amplitude of high-frequency noise exceeds the allowable threshold, reduce the time step to suppress instability.

[0031] In the embodiment, after the node dynamic response calculation is completed at each time step, it is monitored whether each discrete node exhibits high-frequency numerical oscillation or non-physical oscillation phenomena. When the amplitude of high-frequency numerical oscillation exceeds the preset allowable threshold, it is determined that there is a trend of numerical instability in the calculation. The calculation is then re-executed after reducing the time step size. When the amplitude of high-frequency numerical oscillation does not exceed the preset allowable threshold, the current time step is maintained and subsequent time steps are iterated.

[0032] Example 1: This embodiment selects a single pile foundation as the analysis object to compare and study the influence of pile cross-sectional dimensions and lateral inertia effect on the dynamic penetration process. Two types of steel pipe piles with different cross-sectional dimensions are analyzed: the smaller steel pipe pile has an outer diameter of 3m and a wall thickness of 40mm; the larger steel pipe pile has an outer diameter of 6m and a wall thickness of 300mm. Both pile types are 80m long, and the pile material is steel with an elastic modulus of 2.1×10⁻⁶. 11 Pa, density is 7850 kg / m³ 3 The Poisson's ratio was set to 0.3, and the corresponding Pochhammer reduced radius was calculated based on different cross-sectional dimensions. The dynamic parameters for pile driving were set as follows: hammer drop distance of 2.037 m, hammer efficiency of 0.95, and a hydraulic hammer MENCK3500S was used for dynamic penetration analysis. A non-uniform soil layer model was established based on actual geological conditions, and parameters for pile side soil resistance and pile end resistance were set at different depths. The maximum pile side soil resistance was 50 kPa, the maximum pile bottom soil resistance was 2000 kPa, and the maximum elastic displacement was 2.54 mm. A pile-soil dynamic interaction calculation model was established.

[0033] To verify the improvement effect of the method of the present invention compared with the traditional one-dimensional wave model, dynamic penetration analysis of piles with different cross-sectional dimensions was carried out using the traditional one-dimensional wave model and the three-dimensional analysis and calculation method of the present invention, and the final penetration obtained by calculation was compared.

[0034] Figure 5 The curve shows a comparison of the final penetration calculation results obtained from the traditional one-dimensional wave model and the three-dimensional analysis and calculation method of this invention for small-diameter steel pipes. Figure 5 It can be seen that the penetration variation trends obtained by the two methods are basically consistent, and the calculation results are quite close. Because the cross-sectional dimensions of small-diameter piles are small, the lateral inertial effect of the pile body is relatively weak, and the influence of the three-dimensional effect on the propagation process of axial stress waves is limited; therefore, the calculation results of the two methods are quite close. As the burial depth increases, the lateral inertial effect gradually accumulates along the pile body during propagation. The three-dimensional analysis method will be smaller than the traditional one-dimensional model, but the overall difference is small.

[0035] Figure 6 The curves show a comparison of the final penetration depth calculation results obtained from the traditional one-dimensional wave model and the three-dimensional analysis and calculation method of this invention for large-diameter steel pipe piles. Figure 5 It is evident that the difference between the calculation results of the two methods increases significantly as the pile cross-sectional dimensions increase. This is primarily because large-diameter piles possess greater cross-sectional inertia, and the effects of lateral inertia and cross-sectional deformation on the axial stress wave propagation process are more pronounced. Traditional one-dimensional wave models cannot account for these three-dimensional effects, leading to deviations in the calculation results. The method of this invention, by introducing a lateral inertia correction term, effectively reflects the influence of cross-sectional dimension changes on the dynamic penetration response, thus improving the accuracy of the analysis of the large-diameter pile driving process.

[0036] In summary, this embodiment verifies, through comparative analysis of the traditional one-dimensional wave model and the three-dimensional analysis and calculation method of the present invention, that the method of the present invention can effectively consider the lateral inertial effect of the pile body, improve the problem of insufficient description of the dynamic response of the pile body in the dynamic penetration analysis of large-diameter and long piles by the traditional one-dimensional model, and at the same time maintain high computational efficiency and numerical stability, and has good engineering application value.

[0037] Finally, it should be noted that the above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A three-dimensional piling effect correction calculation method based on the wave equation, characterized in that, include: Step S1: Determine the Poisson's ratio of the pile material based on the pile parameters. Pochhammer reduced radius ; Step S2: The original one-dimensional wave equation is extended into a three-dimensional wave equation by introducing the transverse inertial effect, and the equation is spatially discretized using the central difference method to establish the difference expression of acceleration at each node of the pile. Step S3: Based on the discrete scheme of step S2, derive the tridiagonal coefficient matrix satisfied by the acceleration vector; Step S4: During the time-step iterative calculation process, based on the displacement field of the current time step, the spring resistance, displacement and soil resistance of each pile unit are calculated sequentially to obtain the acceleration vector under the one-dimensional wave model. The three-dimensional corrected acceleration is obtained by solving the tridiagonal coefficient matrix of step S3, and the velocity and displacement of the next time step are updated. Step S5: Monitor numerical stability. If the amplitude of high-frequency noise exceeds the allowable threshold, reduce the time step to suppress instability.

2. The method for calculating the correction of three-dimensional piling effect based on the wave equation according to claim 1, characterized in that, In step S1, the Poisson's ratio of the pile material is determined according to the type of pile material. The Poisson's ratio for steel is 0.3, for concrete it is 0.2, and for timber it is selected within the range of 0.3-0.6 depending on the material orientation. Calculate the Pochhammer reduction radius based on the pile cross-section. ,in, For outer diameter , inner diameter Hollow piles: ； For a solid pile with diameter D: 。 3. The method for correcting the three-dimensional piling effect based on the wave equation according to claim 2, characterized in that, Step S2 includes: Establish a one-dimensional wave equation for the propagation of longitudinal stress waves in the pile: ； In the formula, The density of the pile material, For elastic modulus, This refers to the vertical displacement of the pile body. The time variable is the time of stress wave propagation in the pile. These are spatial coordinates along the length of the pile, used to locate the vibration state at different positions on the pile. The resistance force exerted by the soil on the pile body. This refers to the cross-sectional area of ​​the pile. By introducing a transverse inertia correction term based on Pochhammer-Chree theory into the one-dimensional wave equation, a three-dimensional wave equation is established: ； In the formula, the new item Used to describe the three-dimensional wave correction effect caused by the lateral inertia of the pile body; make The governing equations are then transformed into a solution form for nodal accelerations, namely: ； The pile body is divided into Smith discrete elements along the axial direction. Using the central difference scheme to discretize the second derivative in space at each discrete node, we obtain the discrete equations for the nodal accelerations: ； In the formula, For the first The acceleration of each discrete node in the pile body, i.e., the acceleration after three-dimensional wave correction to be determined. , and The first , , Vertical displacement of each node, , The first , Acceleration of discrete nodes in the pile body For the first The soil resistance around the pile in each pile unit For the first Cross-sectional area of ​​each pile unit is the length of the discrete element in the axial direction of the pile.

4. The method for correcting the three-dimensional piling effect based on the wave equation according to claim 3, characterized in that, Step S3 includes: Let constant coefficient The discrete equations of step S2 are rearranged as follows: ； in, ； This indicates the first wave calculated according to Smith's one-dimensional wave theory. Individual node acceleration; By combining all discrete nodes, we can construct the acceleration matrix equation for all pile nodes: ； In the formula, It is a tridiagonal matrix with constant coefficients, and its main diagonal elements are: The left and right diagonal elements are , Let be the acceleration vector of the three-dimensional node to be determined. The acceleration vector calculated from the original one-dimensional model; Approximate boundary conditions are applied to the pile top and pile bottom boundaries. The corresponding boundary nodes satisfy and .

5. The method for calculating the correction of three-dimensional piling effect based on the wave equation according to claim 4, characterized in that, Step S4 includes: Step S41: Based on the displacement and velocity of each discrete node in the current time step, calculate the soil resistance and displacement of each element of the pile body according to the original Smith dynamic analysis program, establish the dynamic equilibrium relationship of the nodes, and obtain the node acceleration vector corresponding to the original one-dimensional wave equation. Step S42: Using the one-dimensional nodal acceleration vector from step S41 as the right-hand side of the three-dimensional correction equation, and combining it with the tridiagonal coefficient matrix from step S3, establish a set of linear equations for nodal acceleration considering lateral inertial effects. Step S43: The tridiagonal linear equation system is solved by the chasing method, and the three-dimensional corrected acceleration of each discrete node in the current time step is obtained by forward elimination and back substitution calculation. Step S44: Replace the one-dimensional nodal acceleration in the original Smith program with the three-dimensional corrected acceleration from step S43, and update the nodal velocities, displacements, and force states of each element according to the original program's time integral format to complete the calculation for the current time step. Step S45: Repeat steps S41 to S44 until all time steps are completed to obtain the dynamic response parameters of pile displacement, velocity, acceleration, stress wave propagation process, pile side resistance, pile end resistance and penetration.

6. The method for calculating the correction of three-dimensional piling effect based on the wave equation according to claim 5, characterized in that, In step S43, the established tridiagonal linear equation system is solved using the chasing method, including: During the forward elimination process, let the lower diagonal elements, main diagonal elements, and upper diagonal elements of the tridiagonal matrix be respectively... , and The right-hand item is Then, the forward elimination process is recursively calculated as follows: ； ； In the formula, This is the upper diagonal advancement coefficient obtained after forward elimination, used to transform the original tridiagonal equation system into an upper triangular equation system, preparing for subsequent back-substitution calculations. The right-hand side advancement coefficient obtained after forward elimination is... To complete the back-substitution solution process, During the forward elimination process, the first Correction values ​​for the upper diagonal elements corresponding to each node. During the forward elimination process, the first Correction values ​​for the right-hand items corresponding to each node; After completing the forward elimination, back substitution calculations are performed starting from the last discrete node to solve for the three-dimensional corrected acceleration of each node in turn: ； ； In the formula, For the first Three-dimensional corrected acceleration of discrete nodes of the pile body The first result obtained after forward elimination The right-hand side correction values ​​of each node are directly used as the acceleration solution for the last node. During the forward elimination process, the first The right-hand side correction value corresponding to each node. During the forward elimination process, the first Correction values ​​for the upper diagonal elements corresponding to each node. This represents the total number of discrete nodes along the axial direction of the pile. The obtained three-dimensional corrected acceleration vector replaces the one-dimensional nodal acceleration in the original Smith dynamic analysis program, and the time step iteration program is returned to complete the update calculation of nodal velocity, nodal displacement and subsequent dynamic response parameters.

7. The method for correcting the three-dimensional piling effect based on the wave equation according to claim 1, characterized in that, In step S5, after the node dynamic response calculation is completed at each time step, it is monitored whether high-frequency numerical oscillations or non-physical oscillations occur at each discrete node. When the amplitude of high-frequency numerical oscillation exceeds the preset allowable threshold, it is determined that there is a trend of numerical instability in the calculation. The calculation is then re-executed after reducing the time step size. When the amplitude of high-frequency numerical oscillation does not exceed the preset allowable threshold, the current time step is maintained and subsequent time steps are iterated.

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