Method for analyzing torsional vibration of pile foundation in transverse isotropic unsaturated soil

Abstract

The invention discloses a torsional vibration analysis method for a pile foundation in transverse isotropic unsaturated soil. The method is different from the previous assumption that the pile-soil interface is in complete contact and has no relative slip, and the relative slip between the pile-soil contact surfaces is simulated by using the Kelvin model, so that the theoretical model better conforms to the practical engineering application. On the basis of a Biot three-phase porous medium framework and a fractional derivative theory, viscoelasticity and transverse isotropy characteristics of a soil framework and viscosity and inertia coupling effects among phases are considered at the same time, and a dynamic control equation described by fractional derivative viscoelasticity stress-strain constitutive description is deduced. And deducing an analytic solution of the torsional vibration response of the pile foundation in the frequency domain and a semi-analytic solution under half-sine pulse excitation in the time domain by utilizing a separation variable method, inverse Fourier transform and a convolution theorem. According to the method, the defects of longitudinal waves can be overcome, and theoretical guidance is provided for improving the accuracy and reliability of pile body integrity detection.

Description

Technical Field

[0001] The invention relates to the technical field of civil engineering, and in particular to a torsional vibration analysis method of a pile foundation in transversely isotropic unsaturated soil. Background Art

[0002] As an important form of foundation, pile foundation is widely used in the construction of high-rise buildings, bridges, docks and other large structures. In actual engineering, pile foundations are often subjected to various dynamic loads, such as earthquakes, wind vibrations and traffic loads, which cause torsional vibrations in the pile foundations. Most of the previous studies on the dynamic characteristics of piles and soils assumed that the pile-soil interface was in continuous contact, ignoring the relative slip phenomenon that may exist at the pile-soil interface in actual engineering. In addition, traditional pile foundation vibration analysis mostly uses saturated soil and linear elastic assumptions, ignoring the unsaturated characteristics and viscoelastic behavior of the soil. However, due to the influence of multiple factors such as the sedimentary environment, compaction process and pore structure during the formation of the soil, the soil in actual engineering is often in an unsaturated state with more complex mechanical properties, which is significantly different from saturated soil.

[0003] In recent years, with the continuous development of civil engineering technology, scholars have gradually realized the importance of the transversely isotropic characteristics and unsaturated viscoelastic behavior of soil in the vibration analysis of pile foundations. Some studies have attempted to consider the influence of relative slip between piles and soil on the torsional vibration of pile foundations, but these methods are usually oversimplified and fail to fully consider the actual characteristics of the soil. Therefore, a more accurate analysis method is urgently needed that can simultaneously consider multiple factors such as relative slip between piles and soils, transversely isotropy, unsaturation and viscoelasticity, so as to better predict and control the torsional vibration behavior of pile foundations under complex dynamic loads.

[0004] The present invention aims to solve the above technical problems and propose a new analysis method to provide a more reliable theoretical basis and technical support for pile foundation vibration analysis. Summary of the invention

[0005] The object of the present invention is to solve the above-mentioned problems and to provide a torsional vibration analysis method for pile foundations in transversely isotropic unsaturated soil.

[0006] To achieve the above object, the technical solution adopted by the present invention is:

[0007] According to one aspect of the present invention, there is provided a method for torsional vibration analysis of a pile foundation in transversely isotropic unsaturated soil, comprising the following steps:

[0008] S1: The following assumptions are introduced to establish a torsional vibration model of pile foundation in transversely isotropic unsaturated viscoelastic soil considering relative slip between pile and soil: the soil is assumed to be a transversely isotropic, viscoelastic, homogeneous three-phase unsaturated porous medium, and each phase is compressible; the upper surface of the soil is a free boundary, without normal or shear stress, and the surface is permeable to water and air; it is assumed that during the torsional vibration process, there is relative slip at the pile-soil interface, and the pile end is located in the bedrock; the vertical and radial displacements of the soil around the pile are not considered;

[0009] S2: The transversely isotropic and fractional derivative Kelvin viscoelastic models are introduced to construct the three-dimensional constitutive relationship of the soil around the pile, and the motion equation of the transversely isotropic unsaturated viscoelastic soil in the cylindrical coordinate system is obtained;

[0010] S3: Ignoring the mass exchange between phases, considering the effective stress principle and the geometric equation of the soil skeleton, the continuity equation for transversely isotropic unsaturated soil is derived;

[0011] S4: According to the assumptions in step S1, based on the continuum theory, the compressibility of the soil skeleton, pore water and pore gas is considered, and the governing equation of the unsaturated soil described by the fractional derivative is obtained;

[0012] S5: Establish the control equation of pile foundation torsional vibration based on Euler-Bernoulli bar; establish the relative slip model of pile-soil system;

[0013] S6: The dynamic control equations of the soil around the pile are derived by the variable separation method. Considering the boundary conditions of the pile-soil interface and the convolution theorem, the torsional dynamic response of the pile foundation in the frequency domain and time domain is derived.

[0014] Further, according to the assumption in step S1 of claim 1, the motion equation of the transversely isotropic unsaturated viscoelastic soil in step 2 is:

[0015] (1a)

[0016] (1b)

[0017] (1c)

[0018] (1d)

[0019] (1e) (1f)

[0020] In the formula, C 11 , C 12 , C 13 , C 33 , C 44and C 66 are the transversely isotropic material elastic constants, denote the stress tensor and displacement components of the soil around the pile, respectively. and are the model parameters of the fractional derivative viscoelastic constitutive model of soil around piles; , , and denote pore water pressure and pore gas pressure, respectively; for The Riemann-Liouville fractional derivative of order is expressed as:

[0021] (2)

[0022] In the formula, represents the gamma function.

[0023] Ignoring gravity, the motion equation of unsaturated elastic soil in the cylindrical coordinate system is:

[0024] (3a)

[0025] (3b)

[0026] (3c)

[0027] In the formula, , Represent the absolute mass density of soil skeleton, liquid phase and gas phase respectively; n is the porosity of soil, S r Indicates saturation.

[0028] Furthermore, according to the assumption in step S1 of claim 1 and the simultaneous equations (1)-(3), the dynamic motion equation of the transversely isotropic unsaturated viscoelastic soil considering only the annular displacement is derived as follows:

[0029] (4)

[0030] In the formula, , ; K s represents the bulk modulus of the solid skeleton,

[0031] Ignoring the mass exchange between phases and considering the effective stress principle, the continuity equation of transversely isotropic unsaturated soil is obtained as follows:

[0032] (5a)

[0033] (5b)

[0034] In the formula, , , , , , , , , , , m and k represent the van Genuchten parameters respectively.

[0035] Furthermore, according to the assumptions in step S1 and the simultaneous equations (4) and (5), the dynamic control equation of the transversely isotropic unsaturated viscoelastic soil can be obtained as follows:

[0036] (6a)

[0037] (6b)

[0038] (6c)

[0039] In the formula, , , ; , , , , , , .

[0040] Further, according to step S5, the torsional dynamic control equation of the pile foundation is:

[0041] (7)

[0042] In the formula, represents the torsion angle, , and denote the shear modulus, polar moment of inertia and mass density of the pile, respectively. .

[0043] Furthermore, in step S5, the relative sliding model of the pile and soil is:

[0044] The Kelvin model is used to simulate the relative slip between the transversely isotropic unsaturated soil layer and the end-bearing pile interface (r=r0), that is, it is simulated by linear springs and linear dampers. The shear stress of the soil at r=r0 is Equal to the friction resistance of the pile side :

[0045] (8)

[0046] According to the definition of the Kelvin model, at r=r0, the following relationship exists:

[0047] (9)

[0048] In the formula, It represents the relative sliding displacement between pile and soil; is the torsional angular amplitude of the pile body; is the circumferential displacement, and They represent the dynamic stiffness and dynamic damping coefficient of the pile-soil contact surface respectively. The damping coefficient is proportional to the frequency and satisfies , where D f and B f are constants respectively.

[0049] Furthermore, in step S1, the boundary conditions of the pile and the soil are:

[0050] Unsaturated soil top and bottom boundary conditions:

[0051] (10a)

[0052] (10b)

[0053] Displacement boundary conditions at infinity for unsaturated soil:

[0054] (11)

[0055] The boundary conditions at the pile top and pile bottom are:

[0056] (12a)

[0057] (12b)

[0058] The shear stress of the soil at the pile-soil contact surface is equal to the friction resistance of the pile body, so:

[0059] (13)

[0060] Furthermore, step 6 includes the following specific steps:

[0061] S61: Solve the dynamic control equation of the soil around the pile. Specifically, substitute equations (6b) and (6c) into equation (6a) to obtain:

[0062] (14)

[0063] In the formula, ,

[0064] Using the separation of variables method, , substituting it into formula (14) we can get:

[0065] (15)

[0066] In the formula, , an index reflecting the degree of anisotropy of the material,

[0067] Formula (15) can be decomposed into:

[0068] (16a)

[0069] (16b)

[0070] In the formula, the constant and Need to meet:

[0071] (17)

[0072] The general solution of equation (16a) and equation (16b) is:

[0073] (18a)

[0074] (18b)

[0075] In the formula, and is an arbitrary constant whose value depends on the boundary conditions; and denote the modified Bessel functions of the first and second kinds, respectively,

[0076] Substituting equation (18) into equations (10) and (11), we can obtain:

[0077] (19a)

[0078] (19b)

[0079] Combining equations (18) and (19), we can obtain the circumferential displacement of the solid phase of the soil around the pile as:

[0080] (19)

[0081] Substituting equation (19) into equation (13), we can obtain the friction force of the soil around the pile on the pile:

[0082] (20)

[0083] S62: Solve the dynamic control equation of the pile and soil system around the pile, specifically:

[0084] Substituting formula (20) into formula (7), we can obtain:

[0085] (twenty one)

[0086] In the formula, is the pile shear wave velocity, ,

[0087] Solving equation (21), we can obtain the torsion angle of a single pile:

[0088] (twenty two)

[0089] In the formula, ,

[0090] Substituting formula (22) into formula (12), we can obtain:

[0091] (23a)

[0092] (23b)

[0093] Substituting equations (19), (20) and (22) into equations (8) and (9), we can obtain: (twenty four)

[0094] Multiply both sides of formula (24) , and integrating in the integral interval [0, H] yields:

[0095] (25)

[0096] In the formula,

[0097]

[0098] Substituting formula (25) into formula (22), we can obtain:

[0099] (26)

[0100] In the formula,

[0101]

[0102]

[0103] S63: The dimensionless torsional complex impedance of the pile top is:

[0104] (27)

[0105] Furthermore, formula (27) can be written as:

[0106] (28)

[0107] In the formula, and It represents dynamic stiffness and dynamic damping, which respectively reflect the ability of a single pile to resist deformation and the energy dissipation characteristics of the pile-soil system.

[0108] S64: According to formula (26), the expression of pile top velocity admittance can be obtained as:

[0109] (29)

[0110] In the formula,

[0111] S65: The unit impulse time domain response function of the pile can be obtained by calling the inverse Fourier transform:

[0112] (30)

[0113] In the formula, , ;

[0114] The torque excitation at the pile top can be simplified to a half-sine pulse excitation , where T is the pulse width; in addition, through the inverse Fourier transform and convolution theorem, the time domain response function of the pile top can be obtained as:

[0115] (31)

[0116] In the formula, , is the dimensionless pulse width.

[0117] Compared with the prior art, the present invention has the following beneficial effects:

[0118] It can be seen from the above technical scheme that the present invention considers the torsional vibration complex impedance calculation system of the pile foundation in the transversely isotropic unsaturated viscoelastic soil with relative slip between the pile and the soil, and the soil around the pile is considered to be a porous medium mixed with solid, liquid and gas, which is undoubtedly closer to the actual situation, and the invention can be completely degraded to the torsional vibration response of the pile foundation in saturated soil or single-phase elastic soil foundation. Therefore, the calculation method disclosed in the present invention has a wide range of applications, and the calculation results are more in line with actual engineering. It can provide a theoretical basis for the scientificity and reliability of the design and construction of pile foundations in engineering fields such as bridges, docks, subways, and highways. BRIEF DESCRIPTION OF THE DRAWINGS

[0119] Figure 1 It is a flow chart of the core steps corresponding to the method of the present invention;

[0120] Figure 2 is a schematic diagram of a fractional-order viscoelastic model corresponding to the method of the present invention;

[0121] Figure 3 It is a schematic diagram of a pile-soil model corresponding to the method described in the present invention. DETAILED DESCRIPTION

[0122] In order to make the technical means, creative features, objectives and effects achieved by the present invention easy to understand, the present invention is further explained below in conjunction with specific implementation methods.

[0123] The present invention provides a torsional vibration analysis method for a pile foundation in a transversely isotropic unsaturated soil, and provides a torsional vibration analysis method for a pile foundation in a transversely isotropic unsaturated soil, comprising the following steps:

[0124] S1: The following assumptions are introduced to establish a torsional vibration model of pile foundation in transversely isotropic unsaturated viscoelastic soil considering relative slip between pile and soil: the soil is assumed to be a transversely isotropic, viscoelastic, homogeneous three-phase unsaturated porous medium, and each phase is compressible; the upper surface of the soil is a free boundary, without normal or shear stress, and the surface is permeable to water and air; it is assumed that during the torsional vibration process, there is relative slip at the pile-soil interface, and the pile end is located in the bedrock; the vertical and radial displacements of the soil around the pile are not considered;

[0125] S2: The transversely isotropic and fractional derivative Kelvin viscoelastic models are introduced to construct the three-dimensional constitutive relationship of the soil around the pile, and the motion equation of the transversely isotropic unsaturated viscoelastic soil in the cylindrical coordinate system is obtained;

[0126] S3: Ignoring the mass exchange between phases, considering the effective stress principle and the geometric equation of the soil skeleton, the continuity equation for transversely isotropic unsaturated soil is derived;

[0127] S4: According to the assumptions in step S1, based on the continuum theory, the compressibility of the soil skeleton, pore water and pore gas is considered, and the governing equation of the unsaturated soil described by the fractional derivative is obtained;

[0128] S5: Establish the control equation of pile foundation torsional vibration based on Euler-Bernoulli bar; establish the relative slip model of pile-soil system;

[0129] S6: The dynamic control equations of the soil around the pile are derived by the variable separation method. Considering the boundary conditions of the pile-soil interface and the convolution theorem, the torsional dynamic response of the pile foundation in the frequency domain and time domain is derived.

[0130] According to the assumption in step S1 of claim 1, the equation of motion of the transversely isotropic unsaturated viscoelastic soil in step 2 is:

[0131] (1a)

[0132] (1b)

[0133] (1c)

[0134] (1d)

[0135] (1e) (1f)

[0136] In the formula, C 11 , C 12 , C 13 , C 33 , C 44 and C 66 are the transversely isotropic material elastic constants, denote the stress tensor and displacement components of the soil around the pile, respectively. and are the model parameters of the fractional derivative viscoelastic constitutive model of soil around piles; , , and denote pore water pressure and pore gas pressure, respectively; for The Riemann-Liouville fractional derivative of order is expressed as:

[0137] (2)

[0138] In the formula, represents the gamma function.

[0139] Ignoring gravity, the motion equation of unsaturated elastic soil in the cylindrical coordinate system is:

[0140] (3a)

[0141] (3b)

[0142] (3c)

[0143] In the formula, , Represent the absolute mass density of soil skeleton, liquid phase and gas phase respectively; n is the porosity of soil, S r Indicates saturation.

[0144] Based on the assumptions in step S1 of claim 1 and the simultaneous equations (1)-(3), the dynamic motion equation of the transversely isotropic unsaturated viscoelastic soil considering only the circumferential displacement is derived as follows:

[0145] (4)

[0146] In the formula, , ; K s represents the bulk modulus of the solid skeleton.

[0147] Ignoring the mass exchange between phases and considering the effective stress principle, the continuity equation of transversely isotropic unsaturated soil is obtained as follows:

[0148] (5a)

[0149] (5b)

[0150] In the formula, , , , , , , , , , , m and k represent the van Genuchten parameters respectively.

[0151] Based on the assumptions in step S1 and the simultaneous equations (4) and (5), the dynamic governing equations for the transversely isotropic unsaturated viscoelastic soil are obtained as follows:

[0152] (6a)

[0153] (6b)

[0154] (6c)

[0155] In the formula, , , ; , , , , , , .

[0156] Step S5, the torsional dynamic control equation of the pile foundation is:

[0157] (7)

[0158] In the formula, represents the torsion angle, , and denote the shear modulus, polar moment of inertia and mass density of the pile, respectively. .

[0159] In step S5, the relative sliding model of pile-soil is:

[0160] The Kelvin model is used to simulate the relative slip between the transversely isotropic unsaturated soil layer and the end-bearing pile interface (r=r0), that is, it is simulated by linear springs and linear dampers. The shear stress of the soil at r=r0 is Equal to the friction resistance of the pile side :

[0161] (8)

[0162] According to the definition of the Kelvin model, at r=r0, the following relationship exists:

[0163] (9)

[0164] In the formula, It represents the relative sliding displacement between pile and soil; is the torsional angular amplitude of the pile body; is the circumferential displacement, and They represent the dynamic stiffness and dynamic damping coefficient of the pile-soil contact surface respectively. The damping coefficient is proportional to the frequency and satisfies , where D f and B f are constants respectively.

[0165] In step S1, the boundary conditions of the pile and soil are:

[0166] Unsaturated soil top and bottom boundary conditions:

[0167] (10a)

[0168] (10b)

[0169] Displacement boundary conditions at infinity for unsaturated soil:

[0170] (11)

[0171] The boundary conditions at the pile top and pile bottom are:

[0172] (12a)

[0173] (12b)

[0174] The shear stress of the soil at the pile-soil contact surface is equal to the friction resistance of the pile body, so:

[0175] (13)

[0176] Step 6 includes the following specific steps:

[0177] S61: Solve the dynamic control equation of the soil around the pile. Specifically, substitute equations (6b) and (6c) into equation (6a) to obtain:

[0178] (14)

[0179] In the formula, ,

[0180] Using the separation of variables method, , substituting it into formula (14) we can get:

[0181] (15)

[0182] In the formula, , an index reflecting the degree of anisotropy of the material,

[0183] Formula (15) can be decomposed into:

[0184] (16a)

[0185] (16b)

[0186] In the formula, the constant and Need to meet:

[0187] (17)

[0188] The general solution of equation (16a) and equation (16b) is:

[0189] (18a)

[0190] (18b)

[0191] In the formula, and is an arbitrary constant whose value depends on the boundary conditions; and denote the modified Bessel functions of the first and second kinds, respectively,

[0192] Substituting equation (18) into equations (10) and (11), we can obtain:

[0193] (19a)

[0194] (19b)

[0195] Combining equations (18) and (19), we can obtain the circumferential displacement of the solid phase of the soil around the pile as:

[0196] (19)

[0197] Substituting equation (19) into equation (13), we can obtain the friction force of the soil around the pile on the pile:

[0198] (20)

[0199] S62: Solve the dynamic control equation of the pile and soil system around the pile, specifically:

[0200] Substituting formula (20) into formula (7), we can obtain:

[0201] (twenty one)

[0202] In the formula, is the pile shear wave velocity, ,

[0203] Solving equation (21), we can obtain the torsion angle of a single pile:

[0204] (twenty two)

[0205] In the formula, ,

[0206] Substituting formula (22) into formula (12), we can obtain:

[0207] (23a)

[0208] (23b)

[0209] Substituting equations (19), (20) and (22) into equations (8) and (9), we can obtain: (twenty four)

[0210] Multiply both sides of formula (24) , and integrating in the integral interval [0, H] yields:

[0211] (25)

[0212] In the formula,

[0213]

[0214] Substituting formula (25) into formula (22), we can obtain:

[0215] (26)

[0216] In the formula,

[0217]

[0218]

[0219] S63: The dimensionless torsional complex impedance of the pile top is:

[0220] (27)

[0221] Furthermore, formula (27) can be written as:

[0222] (28)

[0223] In the formula, and It represents dynamic stiffness and dynamic damping, which respectively reflect the ability of a single pile to resist deformation and the energy dissipation characteristics of the pile-soil system.

[0224] S64: According to formula (26), the expression of pile top velocity admittance can be obtained as:

[0225] (29)

[0226] In the formula,

[0227] S65: The unit impulse time domain response function of the pile can be obtained by calling the inverse Fourier transform:

[0228] (30)

[0229] In the formula, , ;

[0230] The torque excitation at the pile top can be simplified to a half-sine pulse excitation , where T is the pulse width; in addition, through the inverse Fourier transform and convolution theorem, the time domain response function of the pile top can be obtained as:

[0231] (31)

[0232] In the formula, , is the dimensionless pulse width.

[0233] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

[0234] In addition, it should be understood that although the present specification is described according to implementation modes, not every implementation mode contains only one independent technical solution. This description of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment may also be appropriately combined to form other implementation modes that can be understood by those skilled in the art.

Claims

1. A method for analyzing torsional vibration of pile foundations in transversely isotropic unsaturated soil, characterized in that: The following steps are involved: S1: The following assumptions are introduced to establish a torsional vibration model of pile foundation in transversely isotropic unsaturated viscoelastic soil considering relative slip between pile and soil: the soil is assumed to be a transversely isotropic, viscoelastic, homogeneous three-phase unsaturated porous medium, and each phase is compressible; the upper surface of the soil is a free boundary, without normal or shear stress, and the surface is permeable to water and air; it is assumed that during the torsional vibration process, there is relative slip at the pile-soil interface, and the pile end is located in the bedrock; the vertical and radial displacements of the soil around the pile are not considered; S2: The transversely isotropic and fractional derivative Kelvin viscoelastic models are introduced to construct the three-dimensional constitutive relationship of the soil around the pile, and the motion equation of the transversely isotropic unsaturated viscoelastic soil in the cylindrical coordinate system is obtained; S3: Ignoring the mass exchange between phases, considering the effective stress principle and the geometric equation of the soil skeleton, the continuity equation for transversely isotropic unsaturated soil is derived; S4: According to the assumptions in step S1, based on the continuum theory, the compressibility of the soil skeleton, pore water and pore gas is considered, and the governing equation of the unsaturated soil described by the fractional derivative is obtained; S5: Establish the control equation of pile foundation torsional vibration based on Euler-Bernoulli bar; establish the relative slip model of pile-soil system; S6: The dynamic control equations of the soil around the pile are derived by the variable separation method. Considering the boundary conditions of the pile-soil interface and the convolution theorem, the torsional dynamic response of the pile foundation in the frequency domain and time domain is derived.

2. The method for analyzing torsional vibration of pile foundation in transversely isotropic unsaturated soil according to claim 1, characterized in that: According to the assumption in step S1 of claim 1, the equation of motion of the transversely isotropic unsaturated viscoelastic soil in step 2 is: （1a） （1b） （1c） （1d） （1e） （1f） In the formula, C 11 , C 12 , C 13 , C 33 , C 44 and C 66 are the transversely isotropic material elastic constants, denote the stress tensor and displacement components of the soil around the pile, respectively. and are the model parameters of the fractional derivative viscoelastic constitutive model of soil around piles; , , and denote pore water pressure and pore gas pressure, respectively; for The Riemann-Liouville fractional derivative of order is expressed as: （2） In the formula, represents the gamma function; Ignoring gravity, the motion equation of unsaturated elastic soil in the cylindrical coordinate system is: （3a） （3b） （3c） In the formula, , Represent the absolute mass density of soil skeleton, liquid phase and gas phase respectively; n is the porosity of soil, S r Indicates saturation.

3. The method for analyzing torsional vibration of pile foundation in transversely isotropic unsaturated soil according to claim 2, characterized in that: According to the assumptions in step S1 of claim 1 and the simultaneous equations (1)-(3), the dynamic motion equation of the transversely isotropic unsaturated viscoelastic soil considering only the annular displacement is derived as follows: （4） In the formula, , ; K s represents the bulk modulus of the solid skeleton, Ignoring the mass exchange between phases and considering the effective stress principle, the continuity equation of transversely isotropic unsaturated soil is obtained as follows: （5a） （5b） In the formula, , , , , , , , , , , m and k represent the van Genuchten parameters respectively.

4. The method for analyzing torsional vibration of pile foundation in transversely isotropic unsaturated soil according to claim 3, characterized in that: According to the assumptions in step S1 and the simultaneous equations (4) and (5), the dynamic governing equations of the transversely isotropic unsaturated viscoelastic soil are obtained as follows: （6a） （6b） （6c） In the formula, , , ; , , , , , , .

5. The method for analyzing torsional vibration of pile foundation in transversely isotropic unsaturated soil according to claim 1, characterized in that: According to step S5, the torsional dynamic control equation of the pile foundation is: （7） In the formula, represents the torsion angle, , and denote the shear modulus, polar moment of inertia and mass density of the pile, respectively. .

6. A method for analyzing torsional vibration of pile foundation in transversely isotropic unsaturated soil according to claim 1, characterized in that: In step S5, the relative sliding model of pile-soil is: The Kelvin model is used to simulate the relative slip between the transversely isotropic unsaturated soil layer and the end-bearing pile interface (r=r0), that is, it is simulated by linear springs and linear dampers. The shear stress of the soil at r=r0 is Equal to the friction resistance of the pile side : （8） According to the definition of the Kelvin model, at r=r0, the following relationship exists: （9） In the formula, It represents the relative sliding displacement between pile and soil; is the torsional angular amplitude of the pile body; is the circumferential displacement, and They represent the dynamic stiffness and dynamic damping coefficient of the pile-soil contact surface respectively. The damping coefficient is proportional to the frequency and satisfies , where D f and B f are constants respectively.

7. A method for analyzing torsional vibration of pile foundation in transversely isotropic unsaturated soil according to claim 1, characterized in that: In step S1, the boundary conditions of the pile and the soil are: Unsaturated soil top and bottom boundary conditions: （10a） （10b） Displacement boundary conditions at infinity for unsaturated soil: （11） The boundary conditions at the pile top and pile bottom are: （12a） （12b） The shear stress of the soil at the pile-soil contact surface is equal to the friction resistance of the pile body, so: （13）。 8. The method for analyzing torsional vibration of pile foundation in transversely isotropic unsaturated soil according to claim 1, characterized in that: The step 6 comprises the following specific steps: S61: Solve the dynamic control equation of the soil around the pile. Specifically, substitute equations (6b) and (6c) into equation (6a) to obtain: （14） In the formula, , Using the separation of variables method, , substituting it into formula (14) we can get: （15） In the formula, , an index reflecting the degree of anisotropy of the material, Formula (15) can be decomposed into: （16a） （16b） In the formula, the constant and Need to meet: （17） The general solution of equation (16a) and equation (16b) is: （18a） （18b） In the formula, and is an arbitrary constant whose value depends on the boundary conditions; and denote the modified Bessel functions of the first and second kind, respectively, Substituting equation (18) into equations (10) and (11), we can obtain: （19a） （19b） Combining equations (18) and (19), we can obtain the circumferential displacement of the solid phase of the soil around the pile as: （19） Substituting equation (19) into equation (13), we can obtain the friction force of the soil around the pile on the pile: （20） S62: Solve the dynamic control equation of the pile and soil system around the pile, specifically: Substituting formula (20) into formula (7), we can obtain: （21） In the formula, is the pile shear wave velocity, , Solving equation (21), we can obtain the torsion angle of a single pile: （22） In the formula, , Substituting formula (22) into formula (12), we can obtain: （23a） （23b） Substituting equations (19), (20) and (22) into equations (8) and (9), we can obtain: （24） Multiply both sides of formula (24) , and integrating in the integral interval [0, H] yields: （25） In the formula, Substituting formula (25) into formula (22), we can obtain: （26） In the formula, S63: The dimensionless torsional complex impedance of the pile top is: （27） Furthermore, formula (27) can be written as: （28） In the formula, and It represents dynamic stiffness and dynamic damping, which respectively reflect the ability of a single pile to resist deformation and the energy dissipation characteristics of the pile-soil system. S64: According to formula (26), the expression of pile top velocity admittance can be obtained as: （29） In the formula, S65: The unit impulse time domain response function of the pile can be obtained by calling the inverse Fourier transform: （30） In the formula, , ; The torque excitation at the pile top can be simplified to a half-sine pulse excitation , where T is the pulse width; In addition, through the inverse Fourier transform and convolution theorem, the time domain response function of the pile top can be obtained as: （31） In the formula, , is the dimensionless pulse width.

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