Rapid calculation method for dynamic stiffness of complex layered foundation

Abstract

The invention discloses a rapid calculation method for the dynamic stiffness of a complex layered foundation, and the method comprises the steps: selecting the number M of discrete circles with minimum discrete wavelengths, and calculating the total number of the circles and the radiuses of the circles on an exposed foundation which fully covers a whole to-be-researched hydraulic structure according to the M; discretizing the clear foundation, and acting unit loads in three directions by taking the center of any circle as an observation point to obtain the displacement of the observation point in a frequency domain-wave number domain; calculating a flexibility coefficient of a frequency domain-space domain representing the relationship between the unit load and the observation point, obtaining a Green function between the two circles through the flexibility coefficient, and calculating the dynamic stiffness of the foundation under M based on the Green function; repeating the steps to obtain the dynamic stiffness under different values of M, and calculating undetermined parameters in a dynamic stiffness fitting formula according to the dynamic stiffness and the radius of the circle corresponding to two discretions; and selecting an empirical value of the M, calculating the radius of a circle bestrewing the whole exposed foundation, and calculating the infinite domain dynamic stiffness of the foundation after convergence according to the radius and a dynamic stiffness fitting formula.

Description

Technical Field

[0001] This invention relates to the field of foundation safety assessment technology, specifically to a rapid calculation method for the dynamic stiffness of complex layered foundations. Background Technology

[0002] In the seismic safety assessment of dams, the seismic response of the dam actually depends on the dynamic response of the dam-foundation-reservoir hydrodynamic interaction system. The solution for the dam's seismic response should be considered as an open wave system, not a closed vibration system. An open wave system requires consideration of the wave's dissipation into an infinite domain, i.e., considering the radiation damping of the infinite domain.

[0003] The dynamic stiffness of an infinite domain is mathematically rigorous, but existing techniques for solving the dynamic stiffness of an infinite domain mainly rely on the Green's function in the infinite domain. This leads to a huge computational burden in two ways: first, it requires a triple Fourier transform, which involves transforming the horizontal x and y axes in the wavenumber and spatial domains, as well as the time and frequency domains, resulting in a very large computational burden for triple infinite integrals; second, the Green's function has spatiotemporal coupling, affecting all points in space, thus the dynamic stiffness exhibits a full matrix.

[0004] Furthermore, since the foundation is formed by sedimentation and exhibits stratification along the depth direction, it is often assumed to be a horizontally layered foundation in numerical analysis. A classic algorithm for the infinite-domain dynamic stiffness of horizontally layered foundations is the calculation method given by Wolf in Chapter 7 of his book *Dynamic Soil Interaction*. This method can be summarized as follows: 1. Discretize the basic structure into multiple small circles, such as... Figure 1 As shown, for a foundation of 73.9m × 67.3m, it is discretized using small circles with a radius of 3.25m, resulting in a total of 150 small circles; 2. The Green's function is solved in the frequency domain-wavenumber domain to construct the dynamic stiffness in the frequency domain-wavenumber domain. For an exposed foundation, it is a 3×3 full load; 3. The inverse operation of the double Fourier integral is performed to obtain the dynamic stiffness matrix in the frequency domain-space domain. At this time, each small circle has 3 degrees of freedom, which is a 450×450 full matrix, requiring 405,000 infinite integrals to be calculated.

[0005] To achieve the required accuracy for calculating the dynamic stiffness in the infinite domain, the size of the small circle must be sufficient for wave analysis, such as taking one-tenth of the minimum wavelength. For example, to satisfy the analysis frequency of 20Hz, the shear wave velocity of the foundation material of 400m / s, and the wavelength of 20m, the diameter of the small circle could be 2m and the radius 1m. For large hydraulic structures, the length and width of the foundation can reach hundreds of meters, so the number of small circles would reach tens of thousands. The resulting stiffness matrix would be a full matrix, resulting in an enormous computational workload and very low computational efficiency.

[0006] Therefore, for large hydraulic structures, the foundation surface is wide and the number of discrete points is large, which increases the computational workload of infinite domain dynamic stiffness, resulting in long computation time and thus a long seismic safety evaluation cycle for hydraulic structures. Consequently, structural dynamic calculations based on infinite domain dynamic stiffness are rarely carried out in the seismic safety evaluation of hydraulic structures. Summary of the Invention

[0007] To address the aforementioned shortcomings in existing technologies, the present invention provides a rapid calculation method for the dynamic stiffness of complex layered foundations, which solves the problem that existing methods involve large computational loads when calculating the dynamic stiffness of foundations in an infinite domain, resulting in long seismic safety evaluation cycles for hydraulic structures.

[0008] To achieve the above-mentioned objectives, the technical solution adopted by this invention is as follows: A rapid method for calculating the dynamic stiffness of complex layered foundations is provided, comprising the following steps: S1. Select the number M of discrete circles with the minimum discrete wavelength, and calculate the total number of circles and the radius of the circles on the exposed foundation covering the entire hydraulic structure under study based on M. S2. Discretize the exposed foundation according to the total number of circles, and apply unit loads in three directions with the center of any circle as the observation point to obtain the displacement of the observation point in the frequency domain-wavenumber domain. S3. Based on the displacement and radius, calculate the frequency-space domain compliance coefficient representing the relationship between the unit load and the observation point, and obtain the Green's function between the two circles through the compliance coefficient. Calculate the dynamic stiffness of the foundation under M based on the Green's function. S4. Repeat steps S1 to S3 to obtain the dynamic stiffness under different values ​​of M. Then, based on the dynamic stiffness and the radius of the circle corresponding to the two discretizations, calculate the undetermined parameters in the dynamic stiffness fitting formula. S5. Select an empirical value for M, and calculate the radius of the circle covering the entire exposed foundation based on the empirical value. Then, calculate the dynamic stiffness of the foundation in the infinite domain after convergence based on the fitting formula of radius and dynamic stiffness.

[0009] Furthermore, the dynamic stiffness is A diagonal matrix with six directions: horizontal (x), horizontal (y), vertical (z), and rotational directions around the x, y, and z axes; the expression for each stiffness is as follows: in, Let be the stiffness in any direction of dynamic stiffness; a and b are undetermined parameters in any direction of dynamic stiffness. It is constructed using dynamic stiffness in six directions. The diagonal matrix forms the infinite domain dynamic stiffness of the foundation after convergence.

[0010] Furthermore, the expressions for calculating the total number of circles and the radius of the circles in step S1 are as follows: , Where n is the total number of circles; Let be the radius of the circle; The area of ​​the exposed foundation base; The shear wave velocity of the foundation; This represents the maximum value of the frequency band of interest for the hydraulic structure.

[0011] Furthermore, in step A2, when discretizing the exposed foundation, an overall coordinate system including Cartesian coordinates and cylindrical coordinates is constructed on the complex layered foundation of the hydraulic structure, and the exposed foundation is discretized on the overall coordinate system. The expression for calculating the dynamic stiffness of an exposed foundation under M based on the Green's function is as follows: in, The dynamic stiffness when the foundation is discretized into n circles; This is the transformation matrix relating the reference point coordinates to the observation point coordinates; This is the transformation matrix of the load from Cartesian coordinates to cylindrical coordinates; The compliance matrix for n circles; For transpose; Transformation matrix The expression is: in, , , and The transformation matrix between the observation point and the reference point is taken as the center of the 1st, 2nd, i, and nth circles, respectively; The coordinates of the reference point; Let the center of the i-th circle be the coordinates of the observation point. Transformation matrix The expression is: Where diag(·) is a diagonal matrix; , , and These are the transformation matrices from Cartesian coordinates to cylindrical coordinates for the loads acting on the 1st, 2nd, i, and nth circles, respectively. Let the angle between the line connecting the center of the i-th circle (the observation point) and the reference point and the x-axis be denoted by . Compatibility matrix of n circles The expression is: in, , and These are the Green's functions between the first circle and the first, second, and nth circles, respectively. , and These are the Green's functions between the second circle and the first, second, and nth circles, respectively. , and Let be the Green's functions between the nth circle and the 1st, 2nd, and nth circles, respectively.

[0012] Furthermore, the expression for the Green's function is: in, Let be the Green's function between the i-th circle and the j-th circle; , and Let be the Green's functions in the r direction on the i-th circle after a unit load is applied in the x, y, and z directions of the j-th circle, respectively; , and Let be the Green's functions in the θ direction on the i-th circle when a unit load is applied in the x, y, and z directions respectively; , and Let be the Green's functions for applying a unit load in the x, y, and z directions on the j-th circle and in the z-direction on the i-th circle, respectively. Let be the radius of the circle; These are the first, second, and third order Bessel functions of the first kind, respectively; Wave number; Let be the distance between the i-th circle and the j-th circle; , and They are respectively frequency domain and spatial domain along Compliance coefficients in the r and z directions; The compliance coefficient for applying a z-axis load in the r-axis direction in the frequency-space domain; The angle between the line connecting the center of the i-th circle and the j-th circle as the observation point and the x-axis.

[0013] Furthermore, the formula for calculating the compliance coefficient in the frequency domain to spatial domain is: in, , and These represent the displacements of the observation point along the θ, r, and z directions in the frequency-wavenumber domain, respectively. , and These are the loads along the θ, r, and z directions in the frequency-wavenumber domain after a unit load is applied to the observation point; For frequency; , and They are respectively frequency domain and spatial domain along Compliance coefficients in the r and z directions; The compliance coefficient for applying a z-axis load in the r-axis direction in the frequency-space domain; The compliance coefficient for applying a r-direction load in the z-direction of the frequency-space domain.

[0014] Furthermore, the method for constructing the formula for calculating the frequency-space domain compliance coefficient includes: A global coordinate system, including Cartesian and cylindrical coordinate systems, is established on the complex layered foundation of the hydraulic structure to be studied. Establish a system of partial differential equations in the time and space domains in cylindrical coordinates: in, The density of the foundation; , and These are displacements in the r, θ, and z directions, respectively; , and Let be the normal stresses in the r, θ, and z directions, respectively. , Let r and θ be the shear stresses along the z-plane. , Let be the shear stress along the θ and z directions on the r-plane. , Let θ represent the shear stress along the r and z directions; r is the variable in the r direction. Taking a Fourier transform of the partial differential equations yields a second-order ordinary differential equation system in the frequency domain and wavenumber domain: , , , , , , in, and These are out-of-plane and in-plane fluctuations, respectively. It is a displacement vector. for The identity matrix; , , , , , and Both are coefficient matrices; H is the imaginary unit, and H is the conjugate transpose. and All are foundation Lamé constants; For frequency; Solving the system of second-order ordinary differential equations yields the compliance coefficients in the frequency and wavenumber domains. , , and : in, , and These represent the displacements of the observation point along the θ, r, and z directions in the frequency-wavenumber domain, respectively. , and These represent the loads along the θ, r, and z directions in the frequency-wavenumber domain after applying a unit load to the observation point.

[0015] Furthermore, when discretizing the open-ended foundation twice, the number M of the discrete minimum wavelength discrete circles takes the values ​​of 1 and 2 respectively; the empirical value is 20.

[0016] The beneficial effects of this invention are as follows: When calculating the dynamic stiffness of the foundation, this scheme first performs two discretizations to obtain the dynamic stiffness of the foundation of the hydraulic structure under study. Then, it fits the undetermined coefficients in the fitting formula between the dynamic stiffness and the radius of the small circle. Based on selected empirical values, it calculates the radius of the small circle at which stiffness converges. Combining this with the fitting formula, the dynamic stiffness value at which the foundation converges can be quickly calculated. By calculating the results of several discretizations with fewer circles, the refined results are extrapolated, significantly reducing the computational load of dynamic stiffness in the infinite domain and shortening the calculation cycle, thereby shortening the long seismic safety evaluation cycle of hydraulic structures. Attached Figure Description

[0017] Figure 1 This is a schematic diagram illustrating how the basic structure is discretized into multiple small circles in the background art.

[0018] Figure 2 A schematic diagram of an overall coordinate system including Cartesian and cylindrical coordinate systems constructed on a complex layered foundation.

[0019] Figure 3 This is a flowchart of a rapid calculation method for the dynamic stiffness of complex layered foundations. Detailed Implementation

[0020] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0021] refer to Figure 3 , Figure 3 A flowchart illustrating a rapid calculation method for the dynamic stiffness of complex layered foundations is shown, such as... Figure 3 As shown, the method S includes steps S1 to S5.

[0022] In step S1, the number M of discrete minimum wavelength discrete circles is selected, and the total number of circles and the radius of the circles on the exposed foundation covering the entire hydraulic structure under study are calculated based on M.

[0023] In implementation, the preferred expressions for calculating the total number of circles and the radius of the circles in step S1 of this scheme are as follows: , Where n is the total number of circles; Let be the radius of the circle; The area of ​​the exposed foundation base; The shear wave velocity of the foundation; This represents the maximum value of the frequency band of interest for the hydraulic structure.

[0024] In step S2, the exposed foundation is discretized according to the total number of circles, and a unit load in three directions is applied with the center of any circle as the observation point to obtain the displacement of the observation point in the frequency-wavenumber domain. When discretizing the exposed foundation in step S2, an overall coordinate system including Cartesian and cylindrical coordinate systems is constructed on the complex layered foundation of the hydraulic structure. For details, please refer to... Figure 2 The explicit foundation is discretized on the global coordinate system.

[0025] In step S3, based on the displacement and radius, the frequency-space domain compliance coefficient representing the relationship between the unit load and the observation point is calculated, and the Green's function between the two circles is obtained through the compliance coefficient. Based on the Green's function, the dynamic stiffness of the foundation under M is calculated.

[0026] In one embodiment of the present invention, the expression for calculating the dynamic stiffness of an exposed foundation under M based on the Green's function is as follows: in, The dynamic stiffness when the foundation is discretized into n circles; This is the transformation matrix relating the reference point coordinates to the observation point coordinates; This is the transformation matrix of the load from Cartesian coordinates to cylindrical coordinates; The compliance matrix for n circles; For transpose; Transformation matrix The expression is: in, , , and The transformation matrix between the observation point and the reference point is taken as the center of the 1st, 2nd, i, and nth circles, respectively; The coordinates of the reference point; Let the center of the i-th circle be the coordinates of the observation point. Transformation matrix The expression is: Where diag(·) is a diagonal matrix; , , and These are the transformation matrices from Cartesian coordinates to cylindrical coordinates for the loads acting on the 1st, 2nd, i, and nth circles, respectively. Let the angle between the line connecting the center of the i-th circle (the observation point) and the reference point and the x-axis be denoted by . Compatibility matrix of n circles The expression is: in, , and These are the Green's functions between the first circle and the first, second, and nth circles, respectively. , and These are the Green's functions between the second circle and the first, second, and nth circles, respectively. , and Let be the Green's functions between the nth circle and the 1st, 2nd, and nth circles, respectively.

[0027] The expression for the Green's function is: in, Let be the Green's function between the i-th circle and the j-th circle; , and Let be the Green's functions in the r direction on the i-th circle after a unit load is applied in the x, y, and z directions of the j-th circle, respectively; , and Let be the Green's functions in the θ direction on the i-th circle when a unit load is applied in the x, y, and z directions respectively; , and Let be the Green's functions for applying a unit load in the x, y, and z directions on the j-th circle and in the z-direction on the i-th circle, respectively. Let be the radius of the circle; These are the first, second, and third order Bessel functions of the first kind, respectively; Wave number; Let be the distance between the i-th circle and the j-th circle; , and They are respectively frequency domain and spatial domain along Compliance coefficients in the r and z directions; The compliance coefficient for applying a z-axis load in the r-axis direction in the frequency-space domain; The angle between the line connecting the center of the i-th circle and the j-th circle as the observation point and the x-axis.

[0028] In implementation, the preferred formula for calculating the compliance coefficient in the frequency domain to spatial domain is: in, , and These represent the displacements of the observation point along the θ, r, and z directions in the frequency-wavenumber domain, respectively. , and These are the loads along the θ, r, and z directions in the frequency-wavenumber domain after a unit load is applied to the observation point; For frequency; , and They are respectively frequency domain and spatial domain along Compliance coefficients in the r and z directions; The compliance coefficient for applying a z-axis load in the r-axis direction in the frequency-space domain; The compliance coefficient for applying a r-direction load in the z-direction of the frequency-space domain.

[0029] In step S4, steps S1 to S3 are repeated to obtain the dynamic stiffness under different values ​​of M. Then, based on the dynamic stiffness and the radius of the circle corresponding to the two discretizations, the undetermined parameters in the dynamic stiffness fitting formula are calculated. In this scheme, it is preferred that the number of discretization circles M with the minimum wavelength of discretization is 1 and 2 when the exposed foundation is discretized twice. The empirical value is 20.

[0030] In step S5, an empirical value for M is selected, and the radius of the circle covering the entire exposed foundation is calculated based on the empirical value. Then, the dynamic stiffness of the foundation in the infinite domain after convergence is calculated based on the fitting formula of radius and dynamic stiffness.

[0031] In this scheme, the dynamic stiffness is A diagonal matrix with six directions: horizontal (x), horizontal (y), vertical (z), and rotational directions around the x, y, and z axes; the expression for each stiffness is as follows: in, Let be the stiffness in any direction of dynamic stiffness; a and b are undetermined parameters in any direction of dynamic stiffness. It is constructed using dynamic stiffness in six directions. The diagonal matrix forms the infinite domain dynamic stiffness of the foundation after convergence.

[0032] In this invention, the method for constructing the formula for calculating the compliance coefficient in the frequency domain-spatial domain includes: A global coordinate system, including Cartesian and cylindrical coordinate systems, is established on the complex layered foundation of the hydraulic structure under study. This coordinate system can be referenced. Figure 2 In a Cartesian coordinate system, the positive z-axis points vertically downwards, while the horizontal x and y axes are determined by the right-hand rule. In a cylindrical coordinate system, the radial direction points outwards from the origin, and the circumferential direction is determined by the right-hand rule. A Cartesian coordinate system is a rectangular coordinate system. Figure 1 The xyz coordinate system in the text refers to a coordinate system where the coordinates are in cylindrical form. Figure 1 The r-θ-z coordinate system in the diagram.

[0033] Establish a system of partial differential equations in the time and space domains in cylindrical coordinates: in, The density of the foundation; , and These are displacements in the r, θ, and z directions, respectively; , and Let be the normal stresses in the r, θ, and z directions, respectively. , Let r and θ be the shear stresses along the z-plane. , Let be the shear stress along the θ and z directions on the r-plane. , Let θ represent the shear stress along the r and z directions; r is the variable in the r direction. Taking a Fourier transform of the partial differential equations yields a second-order ordinary differential equation system in the frequency domain and wavenumber domain: , , , , , , in, and These are out-of-plane and in-plane fluctuations, respectively. It is a displacement vector. for The identity matrix; , , , , , and Both are coefficient matrices; H is the imaginary unit, and H is the conjugate transpose. and All are foundation Lamé constants; For frequency; Solving the system of second-order ordinary differential equations yields the compliance coefficients in the frequency and wavenumber domains. , , and : in, , and These represent the displacements of the observation point along the θ, r, and z directions in the frequency-wavenumber domain, respectively. , and These represent the loads along the θ, r, and z directions in the frequency-wavenumber domain after applying a unit load to the observation point.

[0034] To illustrate how this approach can significantly reduce computational load and time, an example is provided below: Assume the foundation shear wave velocity of the hydraulic structure (specifically, a dam) is... The basis is The highest frequency of attention Since M is 20, the radius of the small circle at dynamic stiffness convergence is 1.25m according to the radius calculation formula. Using the traditional method, 1020 circles need to be arranged (calculated by the formula for the total number of circles). Using the calculation method given by Wolf in Chapter 7 of "Dynamic Soil-Interaction" mentioned in the background technology, 9,363,600 infinite integration calculations are required at each frequency point, i.e., 9 * 1020 * 1020 = 9,363,600. 9 is the 9 elements of the compliance matrix (in the Green's function expression). Such Green's functions need to be established between every two small circles, and a total of 1020 * 1020 functions need to be established. Each compliance matrix of 9 elements needs to be infinitely integrated, so the infinite integration at each frequency point is 9,363,600 times.

[0035] Using the method proposed in this invention, M takes the values ​​1 and 2 in the two discretizations. The first time, 64 small circles are arranged, and the second time, 128 small circles are arranged. The dynamic stiffness is calculated for the 64 and 128 small circles, which requires a total of 9*(64*64+128*128)=184320 infinite integrations. Then, the dynamic stiffness is constructed using the dynamic stiffness corresponding to the two discretizations to obtain the dynamic stiffness fitting formula. The converged dynamic stiffness can then be obtained, and the obtained value is consistent with the dynamic stiffness obtained by 9,363,600 integrations.

[0036] The comparison shows that the present invention requires 184,320 infinite integral calculations at each frequency point to obtain the result of 9,363,600 integrals by the traditional method. The amount of calculation is about 1 / 50 of that of the traditional method. The traditional method takes 48 hours to complete the calculation, while the method proposed in this invention only takes less than 1 hour.

[0037] Therefore, this scheme reduces the amount of calculation by extrapolating the fitting formula of the refined division through several calculations of fewer circle divisions, thereby shortening the seismic safety evaluation cycle of the complex layered foundation of hydraulic structures.

Claims

1. A method for fast calculation of dynamic stiffness of complex layered ground, characterized by, The method comprises the steps of: S1, selecting the number M of discrete minimum wavelength discrete circles, and calculating the total number of circles and the radius of the circles on the entire exposed foundation of the water conservancy structure according to M; S2, discretizing the exposed foundation according to the total number of circles, and taking the center of any circle as an observation point to act on the unit load in three directions to obtain the displacement of the observation point in the frequency domain-wave number domain; S3, based on the displacement and the radius, calculating the frequency domain-space domain flexibility coefficient representing the relationship between the unit load and the observation point, and obtaining the Green function between two circles through the flexibility coefficient, and calculating the dynamic stiffness of the foundation under M based on the Green function; S4, repeating steps S1-S3 to obtain the dynamic stiffness under different values of M, and then calculating the undetermined parameters in the dynamic stiffness fitting formula according to the dynamic stiffness corresponding to the two times of discretization and the radius of the circle; S5, selecting an empirical value of M, and calculating the radius of the circle on the entire exposed foundation according to the empirical value, and calculating the dynamic stiffness of the foundation in the converged infinite domain according to the radius and the dynamic stiffness fitting formula.

2. The method for rapid calculation of dynamic stiffness of complex stratified ground according to claim 1, characterized in that, The dynamic stiffness is A diagonal matrix, 6 directions are horizontal x, horizontal y, vertical z, and rotation directions around x, y, and z axes. The expression of each stiffness of the dynamic stiffness is calculated as wherein, K is the dynamic stiffness in any direction; a and b are undetermined parameters in any direction of the dynamic stiffness. It is constructed using dynamic stiffness in six directions. The diagonal matrix forms the infinite domain dynamic stiffness of the foundation after convergence.

3. The method for rapid calculation of dynamic stiffness of complex stratified ground according to claim 1, characterized in that, The expression for calculating the total number of circles and the radius of the circle in step S1 is: ， where n is the total number of circles; is the radius of the circle; is the area of the footing; is the shear wave velocity of the foundation; is the maximum value of the frequency band of interest for the hydraulic structure.

4. The method for rapid calculation of dynamic stiffness of complex stratified ground according to claim 1, characterized in that, When discretizing the exposed foundation in step S2, a global coordinate system including a Cartesian coordinate system and a cylindrical coordinate system is constructed on the complex layered foundation of the water conservancy structure, and the exposed foundation is discretized in the global coordinate system; The expression for calculating the dynamic stiffness of the exposed foundation under M based on the Green function is: wherein, K is the dynamic stiffness corresponding to the foundation discretized into n circles; T is the transformation matrix related to the reference point coordinates and the observation point; P is the transformation matrix of the load from the Cartesian coordinate system to the cylindrical coordinate system; K is the flexibility matrix of the n circles; T is the transpose; conversion matrix The expression for the conversion matrix is: wherein, , , and are the transformation matrices of the observation points and the reference points with the centers of the 1st, 2nd, i-th and n-th circles as the observation points and the reference point, respectively; is the coordinate of the reference point; is the coordinate of the i-th circle center as the observation point. conversion matrix The expression for the conversion matrix is: where diag( ) is a diagonal matrix; 、 、 and are the transformation matrices of the 1st, 2nd, i-th and n-th circular load from the Cartesian coordinate system to the cylindrical coordinate system, respectively; is the angle between the line connecting the observation point and the reference point of the i-th circle and the x-axis. n-circles flexibility matrix The expression for the n-circles flexibility matrix is where, , and are the Green's functions between the 1st circle and the 1st, 2nd and n-th circles, respectively; , and are the Green's functions between the 2nd circle and the 1st, 2nd and n-th circles, respectively; , and are the Green's functions between the n-th circle and the 1st, 2nd and n-th circles, respectively.

5. The method for rapid calculation of dynamic stiffness of complex stratified ground according to claim 4, characterized in that, The expression for calculating the dynamic stiffness of the exposed foundation under M based on the Green function is: where, Gijis the Green's function between the ith circle and the jth circle; , and are the Green's functions in the r direction on the ith circle after a unit load is applied in the x, y, z direction on the jth circle, respectively; , and are the Green's functions in the 0 direction on the ith circle after a unit load is applied in the x, y, z direction on the jth circle, respectively; , and are the Green's functions in the z direction on the ith circle after a unit load is applied in the x, y, z direction on the jth circle, respectively; Riis the radius of the ith circle; J1, J2, and J3are the first, second, and third order Bessel functions of the first kind, respectively; k is the wave number; rijis the distance between the ith circle and the jth circle; , and are the flexibility coefficients in the frequency-space domain along the , r, and z directions, respectively; is the flexibility coefficient in the frequency-space domain r direction after a z direction load is applied; is the angle between the line connecting the centers of the ith circle and the jth circle as the observation point and the x axis.

6. The method for rapid calculation of dynamic stiffness of complex stratified ground according to claim 1 or 5, characterized in that, The calculation formula of the frequency domain-space domain flexibility coefficient is: where, , and are the displacements of the observation point in the frequency-wavenumber domain along the directions of θ, r and z, respectively; , and are the loads of the observation point in the frequency-wavenumber domain along the directions of θ, r and z, respectively, when a unit load is applied to the observation point; is the frequency; , and are the flexibility coefficients in the frequency-space domain along the directions of r and z, respectively; is the flexibility coefficient in the frequency-space domain of r direction when z direction load is applied; is the flexibility coefficient in the frequency-space domain of z direction when r direction load is applied.

7. The method for rapid calculation of dynamic stiffness of complex stratified ground according to claim 6, characterized in that, The construction method of the calculation formula of the frequency domain-space domain flexibility coefficient comprises: A global coordinate system including a Cartesian coordinate system and a cylindrical coordinate system is established on the complex layered foundation of the water conservancy structure to be studied; The partial differential equation group in the time domain-space domain is established in the cylindrical coordinate system: where is the density of the foundation; , and are the displacements in the r, θ, and z directions, respectively; , and are the normal stresses in the r, θ, and z directions, respectively, , are the shear stresses in the r and θ directions in the z plane, , are the shear stresses in the θ and z directions in the r plane, , are the shear stresses in the r and z directions in the θ plane; r is the r-direction variable; The Fourier transform is performed on the partial differential equation group to obtain the second-order ordinary differential equation group in the frequency domain-wave number domain: , , , , , , wherein and are out-of-plane and in-plane fluctuations, respectively; is a displacement vector, is is an identity matrix; , , , , , and are coefficient matrices; is an imaginary unit and H is a conjugate transpose, and are Lame constants of the foundation; is a frequency; Solving the second order ordinary differential equations yields the flexibility coefficients in the frequency domain and wave number domain , , and : where, , and are the displacements along the θ, r and z directions in the frequency-wavenumber domain for the observation point; , and are the loads along the θ, r and z directions in the frequency-wavenumber domain for the observation point under a unit load.

8. The method for rapid calculation of dynamic stiffness of complex stratified ground according to any one of claims 1 to 5 and 7, characterized in that, When discretizing the exposed foundation twice, the number M of discrete minimum wavelength discrete circles is 1 and 2 respectively; the empirical value is 20.

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