Method for determining rigidity of pile top of horizontal load-bearing friction type pipe pile

Abstract

The invention discloses a method for determining the pile top rigidity of a horizontal load-bearing friction type pipe pile, which comprises the following steps of: adopting a theoretical method, regarding a soil body around the pile as a viscoelastic continuous medium, and deducing a pile top rigidity expression of the horizontal load-bearing friction type pipe pile by using an analytical method based on a proposed friction type pipe pile theoretical model under the horizontal load; the method has strict logicality; meanwhile, the influence of pile periphery soil mass resistance and pile core soil mass resistance on the pipe pile is considered, and the method is more practical; according to the method, related parameters are substituted for calculation, the image drawn according to the obtained result can accurately describe the condition of the horizontal pile top rigidity of the friction type pipe pile under the horizontal load effect, the method is efficient, simple and convenient, the calculation result is specific and clear, and good reference can be provided for actual application of the friction type pipe pile in the pile foundation field.

Description

Technical Field

[0001] The present invention relates to the technical field of pile foundations, and particularly to a method for determining the pile top stiffness of a horizontally loaded friction pipe pile. Background Art

[0002] In recent years, with the wide application of hollow pipe piles such as PHC pipe piles, PCC piles, and large-diameter pipe piles in engineering, the bearing characteristics of pipe piles have gradually become a research hotspot. Different from solid piles, pipe piles not only have soil around them but also contain a soil plug inside. Therefore, under the action of load, pipe piles exhibit unique bearing characteristics. Due to the actual needs in fields such as earthquake engineering, bridge engineering, and ocean engineering, many researchers have analyzed the pile-soil coupling response in soil. In the analysis of the response of pile foundations under horizontal loads, the horizontal pile top stiffness of the pile-soil system is a basic part, which requires establishing a real mathematical model to describe the interaction between the pile and the soil around it. Most of the current research focuses on end-bearing solid piles, and there are relatively few mathematical models that can describe the bearing characteristics of friction pipe piles under horizontal loads.

[0003] Existing models for describing the horizontal response characteristics of friction pipe piles are usually based on the boundary element integration method or the finite element method, and the solution or calculation process is relatively complex, and the main research object is solid piles. Therefore, the current research lacks a method that can accurately describe the horizontal pile top stiffness of friction pipe piles under horizontal loads. Summary of the Invention

[0004] In view of this, the purpose of the present invention is to propose a method for determining the pile top stiffness of a horizontally loaded friction pipe pile with a simple solution process and fast calculation.

[0005] To achieve the above technical purpose, the technical solution adopted by the present invention is as follows: A method for determining the pile top stiffness of a horizontally loaded friction pipe pile, comprising:

[0006] Obtaining the pipe pile mechanical parameters of the pipe pile to be analyzed and the soil mechanical parameters of the building foundation. The pipe pile to be analyzed is a cylindrical pipe pile, and the building foundation is homogeneous viscoelastic soil. The pipe pile mechanical parameters include pile body length information, outer pile radius information, inner pile radius information, pile body elastic modulus information, pile body moment of inertia information, and pile body cross-sectional area information. The soil mechanical parameters include soil layer thickness information, soil elastic modulus information, soil shear modulus information, and soil Poisson's ratio information in the building foundation;

[0007] And obtaining boundary condition information and continuous condition information of the horizontal displacement at the pile-soil interface;

[0008] Construct a theoretical model of a friction pile under horizontal loading. The theoretical model of the friction pile includes the control equation of the soil around the pile, the control equation of the soil in the pile core, the control equation of the pipe pile, and the control equation of the virtual soil column. The control equation of the virtual soil column is configured as the control equation of the virtual soil column below the pipe pile;

[0009] Solve the control equation of the soil around the pile to obtain the soil resistance around the pile, and solve the control equation of the soil in the pile core to obtain the soil resistance in the pile core;

[0010] Obtain the first initial horizontal displacement expression based on the soil resistance around the pile, the soil resistance in the pile core, and the control equation of the pipe pile, and obtain the second initial horizontal displacement expression based on the soil resistance around the pile, the soil resistance in the pile core, and the control equation of the virtual soil column. The first initial horizontal displacement expression is configured as the horizontal displacement expression of the virtual soil column below the pipe pile, and the second initial horizontal displacement expression is configured as the horizontal displacement expression of the virtual soil column below the pile tip. The first initial horizontal displacement expression contains a first undetermined coefficient, and the second initial horizontal displacement expression contains a second undetermined coefficient;

[0011] Solve the first undetermined coefficient and the second undetermined coefficient according to the boundary condition information and the continuous condition information of the horizontal displacement at the pile-soil interface;

[0012] Substitute the solved first undetermined coefficient back into the first initial horizontal displacement expression to obtain the first final horizontal displacement expression, and substitute the solved second undetermined coefficient back into the second initial horizontal displacement expression to obtain the second final horizontal displacement expression;

[0013] Obtain the pile top stiffness expression of the friction pile under horizontal loading according to the first final horizontal displacement expression and the second final horizontal displacement expression.

[0014] In some embodiments, the control equation of the soil around the pile is represented by formula (1), and formula (1) is as follows:

[0015]

[0016] In formula (1), r is the pipe pile radius information, which is the outer radius information of the pile, and η s is the compression coefficient of the soil, v s is the Poisson's ratio information of the soil, z is the normal depth, μ r is the radial displacement of the soil, μ θ is the circumferential displacement of the soil, is the partial derivative symbol;

[0017] The soil resistance around the pile is represented by formula (2), and formula (2) is as follows:

[0018]

[0019] In formula (2), f 1 (z) is the soil resistance around the pile, σ r1 is the soil stress around the pile, θ is the circumferential rotation angle, τ rθ1 is the soil shear force around the pile, A n1 is the first variable coefficient, r 1 is the outer radius information of the pile, S n1 is the first calculation variable, g n is the first eigenvalue of the eigenfunction, H is the soil layer thickness information, cos(g n z) is the eigenfunction, S n1 is expressed by formula (3), and formula (3) is as follows:

[0020]

[0021] In formula (3), G s is the soil shear modulus information, q n is the first calculation coefficient, is the first modified Bessel function of the second kind of the first order, K 1 (g n r 1 ) is the second modified Bessel function of the second kind of the first order, δ n1 is the second calculation variable, δ n1 is expressed by formula (4), and formula (4) is as follows:

[0022]

[0023] In formula (4), K 0 (q n r 1 ) is the first modified Bessel function of the second kind of the zero order, K 0 (g n r 1 ) is the second modified Bessel function of the second kind of the zero order.

[0024] In some embodiments, the control equation of the soil in the pile core is expressed by formula (5), and formula (5) is as follows:

[0025]

[0026] The soil resistance in the pile core is expressed by formula (6), and formula (6) is as follows:

[0027]

[0028] In formula (6), f 2 (z) is the soil resistance in the pile core, σr2 is the stress of the pile core soil mass, τ rθ2 is the shear force of the pile core soil mass, A n2 is the second variable coefficient, r 2 is the inner radius information of the pile, S n2 is the third calculation variable, S n2 is expressed by formula (7), and formula (7) is as follows:

[0029]

[0030] In formula (7), I 1 (q n r 2 ) is the first modified Bessel function of the first kind of order one, I 1 (g n r 2 ) is the second modified Bessel function of the first kind of order one, δ n2 is the fourth calculation variable, δ n2 is expressed by formula (8), and formula (8) is as follows:

[0031]

[0032] In formula (8), I 0 (q n r 2 ) is the first modified Bessel function of the first kind of order zero, I 0 (g n r 2 ) is the second modified Bessel function of the first kind of order zero.

[0033] In some embodiments, the pipe pile control equation is expressed by formula (9), and formula (9) is as follows:

[0034]

[0035] In formula (9), E p is the elastic modulus information of the pile body, I p is the moment of inertia information of the pile body,

[0036] u p (z) is the horizontal displacement of the pile body, and L is the pile body length information;

[0037] The virtual soil column control equation is expressed by formula (10), and formula (10) is as follows:

[0038]

[0039] In formula (10), G s is the complex Young's modulus of the virtual soil column below the pile tip, Ap is the cross-sectional area information of the pile shaft, A p = π(r 1 2 - r 2 2 ), u s (z) is the horizontal displacement of the virtual soil column below the pile tip, and H is the soil layer thickness.

[0040] In some embodiments, the first initial horizontal displacement expression is represented by formula (11), and formula (11) is as follows:

[0041]

[0042] In formula (11), a p , b p , c p , d p are the first undetermined coefficients, δ n3 is the fifth calculation variable, U pn is the sixth calculation variable, U pn = U pn1 + δ n3 U pn2 , U pn1 is the horizontal displacement of the pipe pile considering the soil around the pile, U pn2 is the horizontal displacement of the pipe pile considering the soil in the pile core, δ n3 is represented by formula (12), and formula (12) is as follows:

[0043]

[0044] The second initial horizontal displacement expression is represented by formula (13), and formula (13) is as follows:

[0045]

[0046] In formula (13), a s , b s are the second undetermined coefficients, U sn is the seventh calculation variable, U sn = U sn1 + δ n3 U sn2 , U sn1 is the horizontal displacement of the virtual soil column considering the soil around the pile, U sn2 is the horizontal displacement of the virtual soil pile considering the soil in the pile core,

[0047] In some embodiments, the first undetermined coefficient is represented by formula (14), and formula (14) is as follows:

[0048]

[0049] In formula (14), M is the first variable matrix, f n is the second variable matrix, N is the third variable matrix, M is represented by formula (15), f n is represented by formula (16), and N is represented by formula (17) and formula (18);

[0050] Formula (15) is as follows:

[0051]

[0052] Formula (16) is as follows:

[0053]

[0054] When the top of the pile is subjected to a unit displacement under the condition of rotational fixation, N is represented by formula (17), and formula (17) is as follows:

[0055] N = [1 0 0 0 0 0] T ;

[0056] When the top of the pile is subjected to a unit rotation under the condition of displacement fixation, N is represented by formula (18), and formula (18) is as follows:

[0057] N = [0 1 0 0 0 0] T ;

[0058] Substitute the solved first undetermined coefficient into the first initial horizontal displacement expression in reverse to obtain the calculation expression of the first variable coefficient, which is represented by formula (19), and formula (19) is as follows:

[0059]

[0060] In formula (19), X mn is the second calculation coefficient, T m is the third calculation coefficient, X mn is represented by formula (20), and T m is represented by formula (21);

[0061] Formula (20) is as follows:

[0062]

[0063] In formula (20), g m is the second eigenvalue of the eigenfunction, is the fourth calculation coefficient,

[0064] Formula (21) is as follows:

[0065]

[0066] After expanding formula (19), it is converted into formula (22) as shown below. Formula (22) is as follows:

[0067]

[0068]

[0069] In some embodiments, obtaining the pile top stiffness expression of a horizontally loaded friction pile based on the first final horizontal displacement expression and the second final horizontal displacement expression includes:

[0070] The pile top stiffness expression of a horizontally loaded friction pile is expressed by formula (23). Formula (23) is as follows:

[0071]

[0072] In formula (23), K hh is the first component of the pile top stiffness of a friction pile under horizontal load, K hr is the second component of the pile top stiffness of a friction pile under horizontal load, K rh is the third component of the pile top stiffness of a friction pile under horizontal load, K rr is the fourth component of the pile top stiffness of a friction pile under horizontal load;

[0073] K hh 、K hr 、K rh 、K rr are expressed by formula (24). Formula (24) is as follows:

[0074]

[0075] In formula (24), u p (0) is the horizontal displacement of the pile top of the pipe pile when z = 0, is the horizontal rotation angle of the pile top of the pipe pile when z = 0, Q p (0) is the horizontal stress of the pile top of the pipe pile when z = 0, M p (0) is the horizontal moment of the pile top of the pipe pile when z = 0,

[0076] Adopting the above technical solutions, compared with the prior art, the beneficial effects of the present invention are:

[0077] Based on the theory of elastodynamics, the present invention establishes the three-dimensional viscoelastic continuum control equations for the soil around the pile and the soil in the pile core, and constructs a complete theoretical system for the stiffness of the pile top of a horizontally loaded friction pipe pile through an analytical method. By introducing the coupled differential equations of radial, circumferential and normal displacements (Formulas 1, 5), and combining the series expansion method of Bessel functions and eigenfunctions (Formulas 2, 6), the dynamic attenuation characteristics of the soil resistance along the depth are accurately characterized. Using the virtual soil column segmented modeling strategy (Formulas 9, 10), the bending stiffness of the pile body is organically connected with the response of the pile tip soil to form the analytical solutions of the displacement field along the entire pile length (Formulas 11, 13). By constructing the stiffness matrix (Formula 23) and its component expressions (Formula 24), the differential relationships between the horizontal displacement, rotation angle and internal force at the pile top are transformed into quantifiable stiffness parameters, systematically revealing the mechanical mechanism of the pile-soil interaction under horizontal loads. Compared with the traditional method, the present invention fully considers the spatial coupling effect of the internal and external soil frictional resistances through the series solutions of the soil resistances around the pile and in the pile core (Formulas 2, 6) and the displacement correction terms (Formulas 11, 13), breaking through the limitations of a single soil model; using matrix equations (Formulas 14, 22) and the orthogonality of eigenfunctions to efficiently solve multiple boundary conditions simultaneously, and significantly improving the calculation efficiency through parameter reduction and series truncation strategies; based on the symmetry design of the stiffness matrix (Formula 23), directly correlating the displacement response at the pile top with the load conditions, providing a universal analytical framework for the stiffness evaluation under different constraint conditions. The present invention transforms the complex three-dimensional pile-soil interaction into an engineering-applicable stiffness calculation formula through rigorous mathematical derivation. The calculation results can be quickly obtained and visualized by substituting parameters, with both theoretical rigor and practical guiding value, providing a reliable theoretical tool for the design optimization and performance evaluation of friction pipe piles. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0079] Figure 1 is the flow chart for calculating the stiffness of the pile top of a horizontally loaded friction pipe pile;

[0080] Figure 2 is the theoretical model of a friction pipe pile in the soil layer;

[0081] Figure 3 is the schematic diagram of the change in the stiffness of the pile top of a horizontally loaded friction pipe pile under the first component of the stiffness of the pile top of a friction pipe pile under horizontal load;

[0082] Figure 4 It is a schematic diagram of the change in the stiffness of the pile top of a friction-type pipe pile under horizontal load, under the second component of the stiffness of the pile top of the friction-type pipe pile under horizontal load;

[0083] Figure 5 It is a schematic diagram of the change in the stiffness of the pile top of a friction-type pipe pile under horizontal load, under the fourth component of the stiffness of the pile top of the friction-type pipe pile under horizontal load. Specific implementation manners

[0084] The present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It should be specifically noted that the following embodiments are only used to illustrate the present invention, but do not limit the scope of the present invention. Similarly, the following embodiments are only partial embodiments of the present invention rather than all embodiments. All other embodiments obtained by those of ordinary skill in the art without creative efforts fall within the scope of protection of the present invention.

[0085] Please refer to Figures 1 to 5 , in order to achieve the above technical objectives, the technical solution adopted by the present invention is: a method for determining the stiffness of the pile top of a friction-type pipe pile under horizontal load, including:

[0086] S1. Obtain the pipe pile mechanical parameters of the pipe pile to be analyzed and the soil mechanical parameters of the building foundation. The pipe pile to be analyzed is a cylindrical pipe pile, and the building foundation is homogeneous viscoelastic soil. The pipe pile mechanical parameters include pile body length information, outer pile radius information, inner pile radius information, pile body elastic modulus information, pile body moment of inertia information, and pile body cross-sectional area information. The soil mechanical parameters include soil layer thickness information, soil elastic modulus information, soil shear modulus information, and soil Poisson's ratio information in the building foundation;

[0087] And obtain the boundary condition information and the continuous condition information of the horizontal displacement at the pile-soil interface;

[0088] S2. Construct a theoretical model of a friction-type pipe pile under horizontal load. The theoretical model of the friction-type pipe pile includes a control equation for the soil around the pile, a control equation for the soil in the pile core, a control equation for the pipe pile, and a control equation for the virtual soil column. The control equation for the virtual soil column is configured as the control equation for the virtual soil column below the pipe pile;

[0089] Solve the control equation for the soil around the pile to obtain the soil resistance around the pile, and solve the control equation for the soil in the pile core to obtain the soil resistance in the pile core;

[0090] S3. Obtain the first initial horizontal displacement expression based on the soil resistance around the pile, the soil resistance in the pile core, and the pipe pile control equation, and obtain the second initial horizontal displacement expression based on the soil resistance around the pile, the soil resistance in the pile core, and the virtual soil column control equation. The first initial horizontal displacement expression is configured as the horizontal displacement expression of the virtual soil column below the pipe pile, and the second initial horizontal displacement expression is configured as the horizontal displacement expression of the virtual soil column below the pile tip. The first initial horizontal displacement expression contains a first undetermined coefficient, and the second initial horizontal displacement expression contains a second undetermined coefficient;

[0091] S4. Solve the first undetermined coefficient and the second undetermined coefficient according to the boundary condition information and the continuous condition information of the horizontal displacement at the pile-soil interface;

[0092] S5. Substitute the solved first undetermined coefficient back into the first initial horizontal displacement expression in reverse to obtain the first final horizontal displacement expression, and substitute the solved second undetermined coefficient back into the second initial horizontal displacement expression in reverse to obtain the second final horizontal displacement expression;

[0093] Obtain the pile top stiffness expression of the horizontally loaded friction pipe pile according to the first final horizontal displacement expression and the second final horizontal displacement expression.

[0094] In this embodiment, the mechanical parameters of the soil and the pile body are shown in Table 1 and Table 2 below.

[0095] Table 1 Basic mechanical parameters of soil

[0096]

[0097] Table 2 Basic mechanical parameters of pipe pile

[0098]

[0099]

[0100] In the calculation, the total number of eigenvalues n of the series can be taken as n = 100 to satisfy the convergence.

[0101] In this embodiment, by constructing a theoretical model of multi-region synergy, the analytical accuracy and calculation efficiency of the pile top stiffness of laterally loaded friction pipe piles are significantly improved. Specifically, in this embodiment, by simultaneously considering the mechanical responses of the soil around the pile, the soil in the pile core, and the virtual soil column at the pile tip, a complete theoretical framework including four types of control equations is established, breaking through the simplified assumptions of the traditional method for the interaction between the soil in the pile core and the soil at the pile tip, and effectively reflecting the co-deformation mechanism of the soil inside and outside the pipe pile. Based on the constitutive relationship of viscoelastic soil and the continuity condition of pile-soil displacement, the expressions of the resistances of the soil around the pile and the soil in the pile core are derived by the analytical method, and the displacement coordination equation is constructed in combination with the control equation of the virtual soil column, avoiding the empirical treatment of complex soil responses and ensuring the rigor of the theoretical model. Further, by introducing the boundary conditions and the displacement continuity conditions to solve the undetermined coefficients, seamless connection of the displacement fields of the virtual soil columns above and below the pile tip is achieved, solving the problem that the pile tip boundary effect in the traditional method is difficult to accurately characterize. The closed solution is constructed by the series expansion method, and the convergence requirement can be satisfied by a finite number of series, significantly reducing the calculation complexity and improving the calculation efficiency while ensuring the accuracy.

[0102] In this embodiment, by systematically integrating the pile body geometric parameters, material properties, and soil mechanics, a parametric stiffness analytical expression is established, which can directly relate the pile top stiffness to the design parameters, providing a theoretical basis for the evaluation of the horizontal bearing performance of pipe piles without relying on numerical iteration or experimental fitting.

[0103] In some embodiments, the control equation of the soil around the pile is expressed by formula (1), and formula (1) is as follows:

[0104]

[0105] In formula (1), r is the pipe pile radius information, that is, the outer radius information of the pile, η s is the compression coefficient of the soil, v s is the Poisson's ratio information of the soil, z is the normal depth, μ r is the radial displacement of the soil, μ θ is the circumferential displacement of the soil, is the partial derivative symbol;

[0106] The resistance of the soil around the pile is expressed by formula (2), and formula (2) is as follows:

[0107]

[0108] In formula (2), f 1 (z) is the resistance of the soil around the pile, σ r1 is the stress of the soil around the pile, θ is the circumferential rotation angle, τ rθ1 is the shear force of the soil around the pile, A n1 is the first variable coefficient, r 1is the pile outer radius information, S n1 is the first calculation variable, g n is the first eigenvalue of the eigenfunction H is the soil layer thickness information, cos(g n z) is the eigenfunction, S n1 is expressed by formula (3), and formula (3) is as follows:

[0109]

[0110] In formula (3), G s is the soil shear modulus information, q n is the first calculation coefficient K 1 (q n r 1 ) is the first modified Bessel function of the second kind of the first order, K 1 (g n r 1 ) is the second modified Bessel function of the second kind of the first order, δ n1 is the second calculation variable, δ n1 is expressed by formula (4), and formula (4) is as follows:

[0111]

[0112] In formula (4), K 0 (q n r 1 ) is the first modified Bessel function of the second kind of the zero order, K 0 (g n r 1 ) is the second modified Bessel function of the second kind of the zero order.

[0113] In this embodiment, based on the elastic dynamics theory, the control equation of the soil body is established. The control equation of the soil body around the pile is expressed by formula (1), and formula (1) is as follows:

[0114]

[0115] Among them, z is the normal depth, and the normal direction refers to the direction in which the pipe pile is vertically embedded in the soil layer.

[0116] Based on the control equation of the soil body around the pile, through derivation, the expression of the soil friction resistance of the soil body around the pile, that is, the soil resistance around the pile, is obtained and is expressed by formula (2) as:

[0117]

[0118] In this embodiment, by constructing the control equation of the soil around the pile and the expression of the resistance force based on elastodynamics, a refined description of the mechanical behavior of the pile-soil interaction is achieved, effectively improving the applicability and calculation efficiency of the theoretical model. By introducing the coupling effect of the radial displacement and the circumferential displacement, formula (1) comprehensively reflects the deformation coordination relationship of the soil in the three-dimensional space. Its differential form completely covers the mechanical equilibrium conditions in the radial, circumferential, and normal directions, and can accurately depict the dynamic response characteristics of the soil under complex stress states. Formula (2) correlates the soil resistance force with the stress components in integral form, and combines the series expansion method to express the resistance force as a linear combination of eigenfunctions, significantly simplifying the solution process of the resistance force distribution under complex boundary conditions. At the same time, through the correlation between the eigenvalue g n and the soil layer thickness H, an adaptive characterization of the soil layer characteristics is realized. Further, formulas (3) and (4) accurately describe the physical law of the radial decay of the soil stress field by introducing the second-kind modified Bessel function and its derivative relationship, and combine the compression coefficient η s for the non-linear correction of the Poisson's ratio v s , strengthening the characterization ability of the model for the compression characteristics of the soil. In addition, the construction of the parameters q n and δ n1 effectively coordinates the scale relationship between the eigenfunction and the Bessel function, ensuring that the form of the solution achieves a balance between mathematical rigor and engineering applicability. The overall derivation process reduces the three-dimensional partial differential equation to an analytically expressible series solution through strict mathematical transformations, avoiding both the accuracy loss that may be caused by traditional simplified assumptions and providing a convenient mathematical framework for subsequent parameter inversion through the orthogonality of the eigenfunctions, laying a theoretical foundation for the in-depth analysis of the pile-soil interaction mechanism.

[0119] In some embodiments, the control equation of the soil in the pile core is represented by formula (5), and formula (5) is as follows:

[0120]

[0121] The resistance force of the soil in the pile core is represented by formula (6), and formula (6) is as follows:

[0122]

[0123] In formula (6), f 2 (z) is the resistance force of the soil in the pile core, σ r2 is the stress of the soil in the pile core, τ rθ2 is the shear force of the soil in the pile core, A n2 is the coefficient of the second variable, r 2 is the information of the inner radius of the pile, S n2 is the third calculation variable, and S n2 is represented by formula (7), and formula (7) is as follows:

[0124]

[0125] In Equation (7), I 1 (q n r 2 ) is the first first-kind modified Bessel function of the first order, and I 1 (g n r 2 ) is the second first-kind modified Bessel function of the first order. δ n2 is the fourth calculation variable, and δ n2 is expressed by Equation (8), and Equation (8) is as follows:

[0126]

[0127] In Equation (8), I 0 (q n r 2 ) is the first first-kind modified Bessel function of the zero order, and I 0 (g n r 2 ) is the second first-kind modified Bessel function of the zero order.

[0128] In this embodiment, based on the theory of elastodynamics, the control equation of the soil mass is established. The control equation of the soil mass in the pile core is expressed by Equation (5), and Equation (5) is as follows:

[0129]

[0130] Based on the control equation of the soil mass in the pile core, through derivation, the soil resistance of the pile core is obtained and expressed by Equation (6) as:

[0131]

[0132] In this embodiment, by establishing the elastodynamic control equation and the resistance expression of the pile core soil mass, the integrity and calculation accuracy of the pile-soil interaction analysis are significantly improved. Equation (5) reconstructs the differential balance relationship of the circumferential displacement, fully considering the coupling deformation characteristics of the pile core soil mass in the radial, circumferential, and normal directions. Its mathematical form completely characterizes the dynamic transfer mechanism of the internal stress field of the soil mass. Equation (6) uses a method combining integration and series expansion to express the pile core resistance as a linear combination of eigenfunctions. Through the eigenvalue g n , an efficient analysis of the resistance distribution along the depth is realized. At the same time, the first-kind modified Bessel functions I 1 and I 0 are used to accurately describe the radially increasing characteristics of the stress field of the pile core soil mass, forming a complement to the second-kind Bessel function of the soil around the pile, and completely covering the different mechanical responses of the soil inside and outside the pile. Equations (7) and (8) construct the parameter δ n2Coordinate the proportional relationship of different Bessel functions and combine with the soil shear modulus G s and Poisson's ratio v s physical correlation to ensure the convergence and engineering applicability of the analytical solution under the pile inner boundary conditions. The overall model reduces the three-dimensional control equation to a series solution that can be analytically expressed through strict mathematical derivation, which not only retains the physical essence of the soil dynamic response but also provides an efficient theoretical framework for the quantitative analysis of the pile core resistance distribution.

[0133] In some embodiments, the pipe pile control equation is represented by formula (9), and formula (9) is as follows:

[0134]

[0135] In formula (9), E p is the pile body elastic modulus information, I p is the pile body moment of inertia information, u p (z) is the horizontal displacement of the pile body, and L is the pile body length information;

[0136] The virtual soil column control equation is represented by formula (10), and formula (10) is as follows:

[0137]

[0138] In formula (10), G s is the complex Young's modulus of the virtual soil column below the pile tip, A p is the pile body cross-sectional area information, A p = π(r 1 2 - r 2 2 ), u s (z) is the horizontal displacement of the virtual soil column below the pile tip, and H is the soil layer thickness.

[0139] In this embodiment, this embodiment realizes the full-depth refined analysis of pile-soil interaction by constructing the first, virtual soil column control equation. Formula (9) dynamically couples the pile body bending stiffness with the resistances of the soil around the pile and the pile core, and accurately characterizes the balance relationship between the horizontal displacement u p (z) of the pile body and the soil resistances f 1 (z), f 2 (z) through a fourth-order differential equation. Its parameters E p I p integrate the inner and outer radius information of the pile through the pile body moment of inertia information I p to fully reflect the influence of the pipe pile cross-sectional characteristics on the deformation. Formula (10) establishes a second-order differential equation for the virtual soil column below the pile tip, and through the complex Young's modulus G sPhysical association with the pile cross-sectional area information A p Quantitatively relates the horizontal displacement u s (z) of the virtual soil column below the pile tip to the soil resistance within the depth range from L to H, forming an effective extension of the soil response below the pile tip. The two equations adopt a piecewise modeling strategy, focusing on the pile body deformation zone and the pile tip influence zone respectively, which not only maintains the continuity of the mechanical model at the pile-soil interface but also realizes the targeted description of the mechanical behavior in different regions through the difference in the order of differentiation. The overall model realizes the organic integration of geometric features and material properties through the parameters r 1 、r 2 and provides a unified theoretical framework for the collaborative deformation analysis of the pile-soil system under complex boundary conditions.

[0140] In some embodiments, the first initial horizontal displacement expression is represented by formula (11), and formula (11) is as follows:

[0141]

[0142] In formula (11), a p 、b p 、c p 、d p are the first undetermined coefficients, δ n3 is the fifth calculation variable, U pn is the sixth calculation variable, U pn = U pn1 + δ n3 U pn2 , U pn1 is the horizontal displacement of the pipe pile considering the soil around the pile, U pn2 is the horizontal displacement of the pipe pile considering the soil in the pile core, δ n3 is represented by formula (12), and formula (12) is as follows:

[0143]

[0144] The second initial horizontal displacement expression is represented by formula (13), and formula (13) is as follows:

[0145]

[0146] In formula (13), a s 、b s are the second undetermined coefficients, U sn is the seventh calculation variable, U sn = U sn1 + δ n3 U sn2 , U sn1 is the horizontal displacement of the virtual soil column considering the soil around the pile, U sn2 For the horizontal displacement of the virtual soil pile considering the soil in the pile core,

[0147] In this embodiment, the soil resistance around the pile obtained in the foregoing steps, i.e., formula (2), and the soil resistance in the pile core, i.e., formula (6), are substituted into the control equation of the pipe pile. After derivation and solution, the displacement of the pipe pile, i.e., the first initial horizontal displacement expression, is obtained and expressed as formula (11):

[0148]

[0149] The soil resistance around the pile obtained in the foregoing steps, i.e., formula (2), and the soil resistance in the pile core, i.e., formula (6), are substituted into the control equation of the virtual soil column. After derivation and solution, the displacement of the virtual soil column below the pile tip, i.e., the second initial horizontal displacement expression, is obtained and expressed as formula (13):

[0150]

[0151] In this embodiment, by constructing a piecewise displacement expression, a refined analytical characterization of the horizontal displacement of the pile body and the soil column below the pile tip is achieved. Formula (11) describes the overall bending deformation of the pile body with a cubic polynomial, and combines series terms to reflect the high-order correction effect of the soil resistance around the pile and in the pile core on the displacement. Through the parameter U pn The soil resistance coefficient A n1 and the eigenvalue g of the Bessel function n are dynamically correlated with the pile body stiffness E p I p Using δ n3 coordinates the contribution ratios of the soil around the pile and in the pile core to ensure the mechanical continuity of the displacement solution at the pile-soil interface. Formula (13) adopts the superposition form of a linear function and series terms. Through the coupling of the complex Young's modulus G s of the virtual soil column below the pile tip and the pile body cross-sectional area information A p a quantitative relationship between the displacement of the virtual soil column and the soil resistance is established. Its parameter U sn inherits the eigenvalue system of pile-soil interaction to achieve seamless connection of the displacement fields above and below the pile tip. The two expressions retain the degrees of freedom of the boundary conditions through the undetermined coefficients a p b p c p d p and c s b s Combined with the convergence characteristics of the series terms, taking into account the analytical accuracy and calculation efficiency, a displacement analytical framework covering the entire pile length is formed, providing a general theoretical tool for the analysis of pile-soil co-deformation under complex loads.

[0152] In some embodiments, the first undetermined coefficient is represented by formula (14), and formula (14) is as follows:

[0153]

[0154] In formula (14), M is the first variable matrix, f n is the second variable matrix, N is the third variable matrix, M is represented by formula (15), f n is represented by formula (16), and N is represented by formula (17) and formula (18);

[0155] Formula (15) is as follows:

[0156]

[0157] Formula (16) is as follows:

[0158]

[0159] When the top of the pile is subjected to a unit displacement under the condition of rotational fixation, N is represented by formula (17), and formula (17) is as follows:

[0160] N = [1 0 0 0 0 0] T ;

[0161] When the top of the pile is subjected to a unit rotation under the condition of displacement fixation, N is represented by formula (18), and formula (18) is as follows:

[0162] N = [0 1 0 0 0 0] T ;

[0163] Substitute the solved first undetermined coefficient back into the first initial horizontal displacement expression to obtain the calculation expression of the first variable coefficient, which is represented by formula (19), and formula (19) is as follows:

[0164]

[0165] In formula (19), X mn is the second calculation coefficient, T m is the third calculation coefficient, X mn is represented by formula (20), and T m is represented by formula (21);

[0166] Formula (20) is as follows:

[0167]

[0168] In formula (20), g m is the second eigenvalue of the eigenfunction, Vn is the fourth calculation coefficient,

[0169] The formula (21) is as follows:

[0170]

[0171] After expanding the formula (19), it is converted into the formula (22) as shown below. The formula (22) is as follows:

[0172]

[0173] In this embodiment, based on the first initial horizontal displacement expression and the second initial horizontal displacement expression obtained according to the foregoing steps, combined with the boundary conditions and the continuous conditions of the pile-soil interface displacement, after derivation, the calculation expressions of the undetermined constants a p , b p , c p , d p , a s , b s are given, that is, the formula (14):

[0174]

[0175] It should be noted that the second undetermined coefficient is obtained by converting a n into A n2 in f n1 through the formula (12). By substituting relevant parameters, the formula (22) can solve for A n1 . Substituting the specific result of the solved A n1 into the formula (14), the first variable coefficient can be solved; in the calculation, taking the total number of eigenvalues n of the series as n = 100 can meet the convergence.

[0176] Here, A n1 can be solved and then substituted into the formula (11) to obtain the horizontal displacement of the virtual soil column below the pipe pile, and the horizontal displacement of the virtual soil column below the pile tip obtained by the formula (13).

[0177] In this embodiment, by constructing a solution system combining matrix equations and series expansions, the analytical efficiency and accuracy of the displacement field and resistance parameters of the pile-soil system are significantly improved. The formula (14) expresses the undetermined coefficients a p , b p , c p , d p , a s , b s and the soil resistance coefficient A n1 through the matrix M and the vector f n, N is dynamically associated, and its matrix M is constructed through boundary conditions such as pile tip displacement, rotation angle, and bending moment to ensure the physical rationality of the solution; vector f n Through U pn , U sn Coordinate the continuity of the displacement of the pile shaft and the soil column, and combine the differential representation of N for different pile top constraint conditions to achieve unified solution for multiple working conditions. Formula (19) further combines the series terms with the matrix inverse operation. By constructing the cross terms of coefficients X mn and T m , eliminate the non - orthogonality of the series solution, form a closed linear equation system (Formula 22), and efficiently solve A n1 . Formula (20) performs piece - wise processing for m = n and m ≠ n to accurately capture the interaction effects of eigenvalues g m and g n , and combine the quantitative correction of the attenuation characteristics of the Bessel function by parameter V n to ensure the convergence stability of the series solution. The overall method adopts a collaborative strategy of matrix dimensionality reduction and series truncation (such as n = 100), which significantly reduces the computational complexity while ensuring theoretical rigor, providing an efficient analytical path for the full - parameter coupling analysis of the pile - soil system.

[0178] In some embodiments, the pile top stiffness expression of the horizontally loaded friction pile is obtained according to the first final horizontal displacement expression and the second final horizontal displacement expression, including:

[0179] The pile top stiffness expression of the horizontally loaded friction pile is expressed by Formula (23), and Formula (23) is as follows:

[0180]

[0181] In Formula (23), K hh is the first component of the pile top stiffness of the friction pile under horizontal load, K hr is the second component of the pile top stiffness of the friction pile under horizontal load, K rh is the third component of the pile top stiffness of the friction pile under horizontal load, K rr is the fourth component of the pile top stiffness of the friction pile under horizontal load;

[0182] K hh , K hr , K rh , K rr are expressed by Formula (24), and Formula (24) is as follows:

[0183]

[0184] In Formula (24), u p$(0)$ is the horizontal displacement of the pile top of the pipe pile when $z = 0$. is the horizontal rotation angle of the pile top of the pipe pile when $z = 0$, $Q$ p $(0)$ is the horizontal stress of the pile top of the pipe pile when $z = 0$. $M$ p $(0)$ is the horizontal bending moment of the pile top of the pipe pile when $z = 0$.

[0185] In this embodiment, by constructing the expression of the pile top stiffness matrix, a systematic analytical characterization of the pile top stiffness characteristics of friction pile is realized. Formula (23) comprehensively characterizes the coupling effect of the horizontal displacement and rotation of the pile top in the form of a stiffness matrix, $K$ hh 、$K$ hr 、$K$ rh 、$K$ rr respectively correspond to the stiffness responses under different constraint conditions, and fully reflect the mechanical coupling characteristics of the pile top under the action of horizontal load. Formula (24) passes through the pile top displacement $u$ p $(0)$, rotation angle $(0)$ and internal force $Q$ p $(0)$, $M$ p $(0)$ differential relationship, directly correlates the stiffness components to the third derivative (shear force) and second derivative (bending moment) of the displacement field, combined with the pile body stiffness parameters $E$ p $I$ p , establishes a quantitative relationship between the analytical solution of stiffness and material properties. This method transforms the complex pile-soil interaction into an analytically expressible stiffness matrix through high-order differential operations of the displacement expression, avoids the computational redundancy of traditional trial methods or numerical iterations, and at the same time supports the unified solution of the pile top under horizontal force, bending moment and their combined working conditions through the orthogonality design of the stiffness components, providing a theoretically rigorous and engineering practical stiffness evaluation tool for pile foundation design.

[0186] Adopting the above technical solution, compared with the prior art, the beneficial effects of the present invention are as follows:

[0187] Based on the theory of elastodynamics, this invention establishes the three-dimensional viscoelastic continuum control equations for the soil around the pile and the soil in the pile core, and constructs a complete theoretical system for the stiffness of the pile top of a horizontally loaded friction pipe pile through the analytical method. By introducing the coupled differential equations of radial, circumferential and normal displacements (Formulas 1, 5), and combining the series expansion method of Bessel functions and eigenfunctions (Formulas 2, 6), the dynamic attenuation characteristics of the soil resistance along the depth are accurately characterized. Using the virtual soil column segmented modeling strategy (Formulas 9, 10), the bending stiffness of the pile shaft and the soil response at the pile tip are organically connected to form the analytical solutions of the displacement field along the entire pile length (Formulas 11, 13). By constructing the stiffness matrix (Formula 23) and its component expressions (Formula 24), the differential relationships between the horizontal displacement, rotation angle and internal force at the pile top are transformed into quantifiable stiffness parameters, systematically revealing the mechanical mechanism of pile-soil interaction under horizontal loads. Compared with traditional methods, this invention fully considers the spatial coupling effect of the internal and external soil frictional resistances through the series solutions of the soil resistances around and in the pile core (Formulas 2, 6) and the displacement correction terms (Formulas 11, 13), breaking through the limitations of a single soil model; uses matrix equations (Formulas 14, 22) and the orthogonality of eigenfunctions to efficiently solve the simultaneous equations of multiple boundary conditions, and significantly improves the calculation efficiency through the parameter reduction and series truncation strategies; based on the symmetry design of the stiffness matrix (Formula 23), directly correlates the pile top displacement response with the load conditions, providing a general analytical framework for the stiffness evaluation under different constraint conditions. This invention transforms the complex three-dimensional pile-soil interaction into an engineering-applicable stiffness calculation formula through rigorous mathematical derivation. The calculation results can be quickly obtained and visualized by substituting parameters, combining theoretical rigor and practical guiding value, providing a reliable theoretical tool for the design optimization and performance evaluation of friction pipe piles.

[0188] The above are only some embodiments of the present invention, and thus do not limit the protection scope of the present invention. Any equivalent device or equivalent process transformation made using the content of the specification and drawings of the present invention, or directly or indirectly applied in other related technical fields, shall be similarly included in the patent protection scope of the present invention.

Claims

1. A method for determining the top stiffness of a horizontally loaded friction type pipe pile, characterized in that: The method comprises: Obtaining the mechanical parameters of the pipe pile to be analyzed and the soil mechanical parameters of the building foundation, wherein the pipe pile to be analyzed is a cylindrical pipe pile, and the building foundation is a homogeneous viscoelastic soil, the mechanical parameters of the pipe pile include pile body length information, pile outer radius information, pile inner radius information, pile body elastic modulus information, pile body moment of inertia information, and pile body cross-sectional area information, and the soil mechanical parameters include soil layer thickness information, soil elastic modulus information, soil shear modulus information, and soil Poisson's ratio information in the building foundation; and, obtaining boundary condition information and continuity condition information of horizontal displacement of pile-soil interface; Constructing a theoretical model of friction type pipe piles under horizontal load, the theoretical model of friction type pipe piles includes a control equation of soil around the pile, a control equation of soil in the pile core, a control equation of the pipe piles and a control equation of a virtual soil column, wherein the control equation of the virtual soil column is configured as a control equation of a virtual soil column under the pipe piles; Solving the soil control equation around the pile to obtain the soil resistance around the pile, and solving the soil control equation of the pile core to obtain the soil resistance of the pile core; A first initial horizontal displacement expression is obtained according to the pile surrounding soil resistance, pile core soil resistance, and pipe pile control equation, and a second initial horizontal displacement expression is obtained according to the pile surrounding soil resistance, pile core soil resistance, and virtual soil column control equation, the first initial horizontal displacement expression is configured as a horizontal displacement expression of a virtual soil column below the pipe pile, the second initial horizontal displacement expression is configured as a horizontal displacement expression of a virtual soil column below the pile end, the first initial horizontal displacement expression contains a first undetermined coefficient, and the second initial horizontal displacement expression contains a second undetermined coefficient; Solving the first undetermined coefficient and the second undetermined coefficient according to the boundary condition information and the continuity condition information of the pile-soil interface horizontal displacement; Substituting the solved first undetermined coefficient in reverse into the first initial horizontal displacement expression to obtain a first final horizontal displacement expression, and substituting the solved second undetermined coefficient in reverse into the second initial horizontal displacement expression to obtain a second final horizontal displacement expression; The expression for the top stiffness of the horizontally loaded friction type pipe pile is obtained according to the first final horizontal displacement expression and the second final horizontal displacement expression.

2. The method for determining the top stiffness of a horizontally loaded friction type pipe pile according to claim 1 is characterized in that: The control equation of the soil mass around the pile is expressed by formula (1), which is as follows: In formula (1), r is the pile radius information, which is the pile outer radius information, and η s is the soil compression coefficient, v s is the soil Poisson's ratio information, z is the normal depth, μ r is the radial displacement of the soil, μ θ is the circumferential displacement of the soil, To find the sign of partial derivative; The soil resistance around the pile is expressed by formula (2), which is as follows: In formula (2), f1(z) is the soil resistance around the pile, σ r1 is the soil stress around the pile, θ is the annular rotation angle, τ rθ1 is the soil shear force around the pile, A n1 is the first variable coefficient, r1 is the pile outer radius information, S n1 is the first calculated variable, g n is the first eigenvalue of the characteristic function, H is the soil thickness information, cos(g n z) is the characteristic function, S n1 It is expressed by formula (3), which is as follows: In formula (3), G s is the soil shear modulus information, q n is the first calculation coefficient, K1(q n r1) is the first second-order modified Bessel function, K1(g n r1) is the second first-order modified Bessel function of the second kind, δ n1 is the second calculated variable, δ n1 It is expressed by formula (4), which is as follows: In formula (4), K0(q n r1) is the first second-kind zero-order modified Bessel function, K0(g n r1) is the second modified Bessel function of the second kind, zero order.

3. The method for determining the top stiffness of a horizontally loaded friction type pipe pile according to claim 2, characterized in that: The pile core soil control equation is expressed by formula (5), which is as follows: The pile core soil resistance is expressed by formula (6), which is as follows: In formula (6), f2(z) is the pile core soil resistance, σ r2 is the soil stress in the pile core, τ rθ2 is the shear force of the pile core soil, A n2 is the second variable coefficient, r2 is the pile inner radius information, S n2 is the third calculated variable, S n2 It is expressed by formula (7), which is as follows: In formula (7), I1(q n r2) is the first first-order modified Bessel function of the first kind, I1(g n r2) is the second first-order modified Bessel function of the first kind, δ n2 is the fourth calculated variable, δ n2 It is expressed by formula (8), which is as follows: In formula (8), I0(q n r2) the first zero-order modified Bessel function of the first kind, I0(g n r2) is the second first kind zero-order modified Bessel function.

4. The method for determining the top stiffness of a horizontally loaded friction type pipe pile according to claim 3 is characterized in that: The pile control equation is expressed by formula (9), which is as follows: In formula (9), E p is the elastic modulus information of the pile body, I p is the moment of inertia information of the pile body, u p (z) is the horizontal displacement of the pile body, and L is the length of the pile body; The virtual soil column control equation is expressed by formula (10), which is as follows: In formula (10), G s is the complex Young's modulus of the virtual soil column below the pile end, A p is the cross-sectional area of ​​the pile body, A p =π(r1 2 -r2 2 ),u s (z) is the horizontal displacement of the virtual soil column below the pile end, and H is the thickness of the soil layer.

5. The method for determining the top stiffness of a horizontally loaded friction type pipe pile according to claim 4, characterized in that: The first initial horizontal displacement expression is expressed by formula (11), and the formula (11) is as follows: In formula (11), a p 、b p 、c p d p is the first undetermined coefficient, δ n3 is the fifth calculated variable, U pn is the sixth calculated variable, U pn =U pn1 +δ n3 U pn2 , U pn1 In order to consider the horizontal displacement of the pile when the soil around the pile is taken into account, U pn2 In order to consider the horizontal displacement of the pile when the pile core soil is δ n3 It is expressed by formula (12), which is as follows: The second initial horizontal displacement expression is expressed by formula (13), and the formula (13) is as follows: In formula (13), a s 、b s is the second undetermined coefficient, U sn is the seventh calculation variable, U sn =U sn1 +δ n3 U sn2 , U sn1 In order to consider the horizontal displacement of the virtual soil column when considering the soil around the pile, U sn2 In order to consider the horizontal displacement of the virtual soil pile when the pile core soil is 6. The method for determining the top stiffness of a horizontally loaded friction type pipe pile according to claim 5, characterized in that: The first undetermined coefficient is expressed by formula (14), and the formula (14) is as follows: In formula (14), M is the first variable matrix, f n is the second variable matrix, N is the third variable matrix, M is expressed by formula (15), f n It is expressed by formula (16), and N is expressed by formula (17) and formula (18); The formula (15) is as follows: The formula (16) is as follows: When the pile top is subjected to unit displacement while being rotationally fixed, N is expressed by formula (17), which is as follows: N＝[1 0 0 0 0 0] T ； When the pile top is subjected to unit rotation with the displacement fixed, N is expressed by formula (18), which is as follows: N＝[0 1 0 0 0 0] T ； Substitute the solved first undetermined coefficient into the first initial horizontal displacement expression in reverse order to obtain the calculation expression of the first variable coefficient, which is expressed by formula (19). Formula (19) is as follows: In formula (19), X mn is the second calculation coefficient, T m is the third calculation coefficient, X mn According to formula (20), T m It is expressed by formula (21); The formula (20) is as follows: In formula (20), g m is the second eigenvalue of the characteristic function, V n is the fourth calculation coefficient, The formula (21) is as follows: Formula (19) is expanded and converted into formula (22), which is as follows:

7. The method for determining the top stiffness of a horizontally loaded friction type pipe pile according to claim 6, characterized in that: According to the first final horizontal displacement expression and the second final horizontal displacement expression, the pile top stiffness expression of the horizontally loaded friction type pipe pile is obtained, which includes: The expression of the pile top stiffness of the horizontally loaded friction type pipe pile is expressed by formula (23), and the formula (23) is as follows: In formula (23), K hh K is the first component of the stiffness of the friction pile top under horizontal load, hr is the second component of the stiffness of the friction pile top under horizontal load, K rh K is the third component of the stiffness of the friction pile top under horizontal load, rr is the fourth component of the stiffness of the top of the friction pipe pile under horizontal load; K hh , K hr , K rh , K rr It is expressed by formula (24), which is as follows: In formula (24), u p (0) is the horizontal displacement of the pile top when z = 0, is the horizontal rotation angle of the pile top when z=0, Q p (0) is the horizontal stress at the top of the pile when z = 0, M p (0) is the horizontal bending moment at the top of the pile when z = 0,

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