A method for determining the rigidity of the top of a horizontally loaded friction-type pipe pile

Abstract

The application discloses a kind of horizontal load friction type pipe pile top rigidity determination method, adopt theoretical method, the soil around pile is regarded as viscoelastic continuous medium, and based on the proposed friction type pipe pile theoretical model under horizontal load, the horizontal load friction type pipe pile top rigidity expression is deduced using analytical method, with rigorous logic;The influence of the resistance of soil around pile and the resistance of pile core soil on pipe pile is considered simultaneously, which is more in line with reality;By substituting relevant parameters for calculation, the image drawn according to the obtained results can accurately describe the horizontal load friction type pipe pile horizontal top rigidity under horizontal load, the method is efficient and simple, the calculation result is specific and clear, and can provide a good reference for the practical application of friction type pipe pile in pile foundation field.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of pile foundation, in particular to a method for determining the rigidity of the top of a horizontal load-bearing friction-type pipe pile. BACKGROUND

[0002] In recent years, with the wide application of PHC pipe piles, PCC piles and large-diameter pipe piles in engineering, the bearing characteristics of pipe piles have gradually become a research hotspot. Unlike solid piles, pipe piles not only have soil around the pile, but also contain a core plug inside. Therefore, under the action of load, pipe piles exhibit unique bearing characteristics. Due to the practical needs of the fields of earthquake engineering, bridge engineering and marine engineering, many researchers have analyzed the pile-soil coupling response in soil. In the analysis of the response of pile foundation under horizontal load, the horizontal pile top stiffness of the pile-soil system is a basic part, which requires a real mathematical model to describe the interaction between the pile and its surrounding soil. Most of the current researches are aimed at end-bearing solid piles, and there are relatively few mathematical models that can describe the bearing characteristics of friction-type pipe piles under horizontal load.

[0003] The existing models for describing the horizontal response characteristics of friction-type pipe piles are usually based on the boundary element integral method or the finite element method, and the solving 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-type pipe piles under horizontal load. SUMMARY

[0004] Therefore, the purpose of the present application is to provide a method for determining the horizontal load-bearing friction-type pipe pile top stiffness, which has a simple solving process and fast calculation.

[0005] In order to achieve the above technical purpose, the technical solution adopted by the present application is as follows: a method for determining the horizontal load-bearing friction-type pipe pile top stiffness, 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 being a cylindrical pipe pile, the building foundation being a homogeneous viscoelastic soil, the pipe pile mechanical parameters including pile 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 including soil layer thickness information in the building foundation, soil body elastic modulus information, soil body shear modulus information and soil body Poisson's ratio information;

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

[0008] A theoretical model of a friction-type pipe pile under horizontal load is constructed, the theoretical model of the friction-type pipe pile including a soil body control equation of a soil body around the pile, a soil body control equation of a soil body in the pile core, a pipe pile control equation, and a virtual soil column control equation configured as a control equation of a virtual soil column below the pipe pile;

[0009] The soil body control equation of the soil body around the pile is solved to obtain a resistance of the soil body around the pile, and the soil body control equation of the soil body in the pile core is solved to obtain a resistance of the soil body in the pile core;

[0010] A first initial horizontal displacement expression is obtained according to the resistance of the soil body around the pile, the resistance of the soil body in the pile core, and the pipe pile control equation, and a second initial horizontal displacement expression is obtained according to the resistance of the soil body around the pile, the resistance of the soil body in the pile core, and the virtual soil column control equation, the first initial horizontal displacement expression being configured as a horizontal displacement expression of the virtual soil column below the pipe pile, the second initial horizontal displacement expression being configured as a horizontal displacement expression of a virtual soil column below the pile end, the first initial horizontal displacement expression containing a first undetermined coefficient, and the second initial horizontal displacement expression containing a second undetermined coefficient;

[0011] The first undetermined coefficient and the second undetermined coefficient are solved according to boundary condition information and horizontal displacement continuity condition information of a pile-soil interface;

[0012] The first undetermined coefficient is reversely substituted into the first initial horizontal displacement expression to obtain a first final horizontal displacement expression, and the second undetermined coefficient is reversely substituted into the second initial horizontal displacement expression to obtain a second final horizontal displacement expression;

[0013] A pile top stiffness expression of the friction-type pipe pile under horizontal load is obtained according to the first final horizontal displacement expression and the second final horizontal displacement expression.

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

[0015]

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

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

[0018]

[0019] In formula (2), f1(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 angle, τ rθ1 is the shear of the soil around the pile, A n1 is the first variable coefficient, r1 is the information of the outer radius of the pile, S n1 is the first calculation variable, g n is the first eigenvalue of the characteristic function, H is the information of the thickness of the soil layer, cos(g n z) is the characteristic function, S n1 is expressed by formula (3), and formula (3) is as follows:

[0020]

[0021] In formula (3), G s is the information of the shear modulus of the soil, q n is the first calculation coefficient, K1(g is the first second-type first-order modified Bessel function, K1(g n r1) is the second second-type first-order modified Bessel function, 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), K0(q n r1) is the first second-type zero-order modified Bessel function, K0(g n r1) is the second second-type zero-order modified Bessel function.

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

[0025]

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

[0027]

[0028] In formula (6), f2(z) is the resistance of the soil of the pile core, σ r2 is the stress of the soil of the pile core, τ rθ2 is the shear of the soil of the pile core, A n2 is the second variable coefficient, r2 is the information of the inner radius of the pile, S n2 is the third calculation variable, S n2By equation (7), equation (7) is as follows:

[0029]

[0030] In equation (7), I1(q n r2) is a first first-order modified Bessel function of the first kind, I1(g n r2) is a second first-order modified Bessel function of the first kind, δ n2 is a fourth calculation variable, and δ n2 By equation (8), equation (8) is as follows:

[0031]

[0032] In equation (8), I0(q n r2) is a first zero-order modified Bessel function of the first kind, and I0(g n r2) is a second zero-order modified Bessel function of the first kind.

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

[0034]

[0035] In equation (9), E p is pile body elastic modulus information, I p is pile body rotational inertia information,

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

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

[0038]

[0039] In equation (10), G s is a complex Young's modulus of a virtual soil column below the pile end, A p is pile body cross-sectional area information, A p = π(r1 2 -r2 2 ), and u s (z) is a horizontal displacement of the virtual soil column below the pile end, and H is soil layer thickness.

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

[0041]

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

[0043]

[0044] The second initial horizontal displacement expression is expressed by Equation (13) as follows:

[0045]

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

[0047] In some embodiments, the first undetermined coefficients are expressed by Equation (14) as follows:

[0048]

[0049] In Equation (14), M is a first variable matrix, f n is a second variable matrix, N is a third variable matrix, M is expressed by Equation (15), f n is expressed by Equation (16), and N is expressed by Equations (17) and (18).

[0050] Equation (15) is as follows:

[0051]

[0052] Equation (16) is as follows:

[0053]

[0054] When the pile top is subjected to a unit displacement with the rotation fixed, N is represented by Equation (17), which is as follows:

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

[0056] When the pile top is subjected to a unit rotation with the displacement fixed, N is represented by Equation (18), which is as follows:

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

[0058] The first variable coefficient is obtained by substituting the solved first undetermined coefficient into the first initial horizontal displacement expression, and the calculation expression of the first variable coefficient is represented by Equation (19), which is as follows:

[0059]

[0060] In Equation (19), X mn is a second calculation coefficient, T m is a third calculation coefficient, and X mn is represented by Equation (20), and T m is represented by Equation (21).

[0061] Equation (20) is as follows:

[0062]

[0063] In Equation (20), g m is a second eigenvalue of a characteristic function, is a fourth calculation coefficient,

[0064] Equation (21) is as follows:

[0065]

[0066] After Equation (19) is expanded, it is converted into Equation (22) as shown below:

[0067]

[0068] In some embodiments, the horizontal load-bearing friction type pipe pile top stiffness expression is obtained according to the first final horizontal displacement expression and the second final horizontal displacement expression, and the horizontal load-bearing friction type pipe pile top stiffness expression comprises:

[0069] The horizontal load-bearing friction type pipe pile top stiffness expression is expressed by formula (23) as follows:

[0070]

[0071] In formula (23), K hh is a first component of the horizontal load-bearing friction type pipe pile top stiffness, K hr is a second component of the horizontal load-bearing friction type pipe pile top stiffness, K rh is a third component of the horizontal load-bearing friction type pipe pile top stiffness, and K rr is a fourth component of the horizontal load-bearing friction type pipe pile top stiffness.

[0072] K hh , K hr , K rh , and K rr are expressed by formula (24) as follows:

[0073]

[0074] In formula (24), u p (0) is the pipe pile top horizontal displacement when z=0, is the pipe pile top horizontal rotation angle when z=0, Q p (0) is the pipe pile top horizontal stress when z=0, M p (0) is the pipe pile top horizontal bending moment when z=0,

[0075] By using the technical scheme, the present application has the beneficial effects that:

[0076] The present application is based on the theory of elasticity dynamics, establishes the three-dimensional viscoelastic continuum control equation of the soil around the pile and the soil in the pile core, and constructs a complete theoretical system of the rigidity of the top of the horizontal load friction type pipe pile by an analytical method. By introducing the coupling differential equations of the radial, hoop and normal displacements (formula 1, 5), combining the series expansion method of the Bessel function and the characteristic function (formula 2, 6), accurately representing the dynamic attenuation characteristics of the soil resistance along the depth, and using the segmented modeling strategy of the virtual soil column (formula 9, 10), the bending rigidity of the pile body and the response of the soil at the pile end are organically connected to form an analytical solution of the displacement field of the whole pile length (formula 11, 13). By constructing the rigidity matrix (formula 23) and its component expressions (formula 24), the differential relationship between the horizontal displacement, angle and internal force of the top of the pile is transformed into quantifiable calculation rigidity parameters, and the mechanical mechanism of the pile-soil cooperation under the action of horizontal load is systematically revealed. Compared with the traditional method, the present application fully considers the spatial coupling effect of the frictional resistance of the internal and external soil through the series solution of the soil resistance (formula 2, 6) and the displacement correction term (formula 11, 13), breaks through the limitation of the single soil model; the efficient simultaneous solution of the multi-boundary condition is realized by using the matrix equation (formula 14, 22) and the orthogonality of the characteristic function, and the calculation efficiency is significantly improved by the parameter dimension reduction and series truncation strategy; based on the symmetry design of the rigidity matrix (formula 23), the displacement response at the top of the pile is directly related to the load working condition, and a universal analytical framework is provided for the rigidity evaluation under different constraint conditions. The present application transforms the complex three-dimensional pile-soil interaction into the rigidity calculation formula which can be applied to engineering through rigorous mathematical derivation, the calculation results can be quickly obtained and visualized by parameter substitution, and the present application has both theoretical rigor and practical guiding value, and provides a reliable theoretical tool for the design optimization and performance evaluation of the friction type pipe pile. BRIEF DESCRIPTION OF DRAWINGS

[0077] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or the prior art description. Obviously, the drawings in the following description only some embodiments of the present application, and for those skilled in the art, other drawings can also be obtained without creative labor on the basis of these drawings.

[0078] Figure 1 The flow chart for calculating the rigidity of the top of the horizontal load friction type pipe pile;

[0079] Figure 2 The theoretical model of the friction type pipe pile in the soil layer;

[0080] Figure 3 The schematic diagram of the rigidity change of the top of the horizontal load friction type pipe pile under the first component of the rigidity of the top of the friction type pipe pile under the action of the horizontal load;

[0081] Figure 4is a schematic diagram of the change of the pile top stiffness of the friction type pipe pile under the second component of the pile top stiffness under the action of the horizontal load;

[0082] Figure 5 is a schematic diagram of the change of the pile top stiffness of the friction type pipe pile under the fourth component of the pile top stiffness under the action of the horizontal load. DETAILED DESCRIPTION

[0083] The application will be described in further detail below with reference to the drawings and embodiments. It is particularly pointed out that the following embodiments are only used to illustrate the application, but do not limit the scope of the application. Similarly, the following embodiments are only part of the embodiments of the application, not all embodiments, and all other embodiments obtained by those skilled in the art without creative labor are within the scope of the application.

[0084] Please refer to Figures 1 to 5 In order to achieve the above technical purposes, the technical scheme adopted by the application is as follows: a method for determining the pile top stiffness of a friction type pipe pile under horizontal load, comprising:

[0085] S1, obtaining the pipe pile mechanical parameters of a pipe pile to be analyzed and the soil body mechanical parameters of a building foundation, the pipe pile to be analyzed being a cylindrical pipe pile, the building foundation being a homogeneous viscoelastic soil, the pipe pile mechanical parameters including 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 body mechanical parameters including soil layer thickness information in the building foundation, soil body elastic modulus information, soil body shear modulus information and soil body Poisson's ratio information;

[0086] and obtaining boundary condition information and continuous condition information of horizontal displacement of the pile-soil interface;

[0087] S2, constructing a theoretical model of a friction type pipe pile under horizontal load, the theoretical model of the friction type pipe pile including a pile surrounding soil body control equation, a pile core soil body control equation, a pipe pile control equation and a virtual soil column control equation, the virtual soil column control equation being configured as a control equation of a virtual soil column below the pipe pile;

[0088] solving the pile surrounding soil body control equation to obtain the resistance of the pile surrounding soil body, and solving the pile core soil body control equation to obtain the resistance of the pile core soil body;

[0089] S3, obtaining a first initial horizontal displacement expression according to the soil body resistance around the pile, the soil body resistance in the pile core, and the pipe pile control equation, and obtaining a second initial horizontal displacement expression according to the soil body resistance around the pile, the soil body resistance in the pile core, and the virtual soil column control equation, the first initial horizontal displacement expression being configured as a horizontal displacement expression of a virtual soil column below the pipe pile, the second initial horizontal displacement expression being configured as a horizontal displacement expression of a virtual soil column below the pile end, the first initial horizontal displacement expression containing a first undetermined coefficient, and the second initial horizontal displacement expression containing a second undetermined coefficient;

[0090] S4, solving the first undetermined coefficient and the second undetermined coefficient according to the boundary condition information and the continuous condition information of the horizontal displacement of the pile-soil interface;

[0091] S5, substituting the solved first undetermined coefficient into the first initial horizontal displacement expression in reverse to obtain a first final horizontal displacement expression, and substituting the solved second undetermined coefficient into the second initial horizontal displacement expression in reverse to obtain a second final horizontal displacement expression;

[0092] obtaining a pipe pile top stiffness expression of the horizontal load type according to the first final horizontal displacement expression and the second final horizontal displacement expression.

[0093] In the embodiment, the mechanical parameters of the soil body and the pile body are shown in the following Table 1 and Table 2.

[0094] Table 1 Basic mechanical parameters of soil body

[0095]

[0096] Table 2 Basic mechanical parameters of pipe pile

[0097]

[0098]

[0099] In the calculation, the total number of eigenvalues of the series n is taken as n = 100, which can meet the convergence.

[0100] The embodiment significantly improves the analytical precision and calculation efficiency of the horizontal load friction type pipe pile top stiffness by constructing a multi-region coordination theory model. Specifically, the embodiment establishes a complete theoretical framework including four types of control equations 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 end, breaking through the simplified assumption of the interaction between the soil in the pile core and the soil at the pile end in traditional methods, and effectively reflecting the collaborative deformation mechanism of the soil inside and outside the pipe pile. Based on the viscoelastic soil constitutive relation and the displacement continuity condition of the pile and soil, the pile-soil resistance expressions are derived by the analytical method, and the displacement coordination equation is constructed by combining the virtual soil column control equation, which avoids the empirical treatment of the complex soil response and ensures the rigor of the theoretical model. Further, by introducing the boundary conditions and displacement continuity conditions to solve the undetermined coefficients, the seamless connection of the displacement field of the virtual soil columns above and below the pile end is realized, solving the problem that the boundary effect at the pile end is difficult to accurately represent in traditional methods. The closed solution is constructed by the series expansion method, which can meet the convergence requirement by a limited number of series, significantly reducing the calculation complexity, improving the calculation efficiency while ensuring the accuracy.

[0101] The embodiment establishes a parameterized stiffness analytical expression by systematically integrating the pile geometry parameters, material properties, and soil mechanics, which can directly relate the pile top stiffness to the design parameters, providing a theoretical basis for the horizontal bearing performance evaluation of the pipe pile, and does not need to rely on numerical iteration or experimental fitting.

[0102] In some embodiments, the soil body control equation around the pile is represented by formula (1), which is as follows:

[0103]

[0104] In formula (1), r is the pipe pile radius information, i.e., 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 hoop displacement of the soil, is the partial derivative symbol;

[0105] The pile-soil resistance is represented by formula (2), which is as follows:

[0106]

[0107] In formula (2), f1(z) is the pile-soil resistance, σ r1 is the pile-soil stress, θ is the hoop angle, τ rθ1 is the pile-soil shear force, A n1 is the first variable coefficient, r1 is the outer radius information of the pile, S n1G is a first calculation variable, g n G is a first eigenvalue of a characteristic function, H is soil layer thickness information, cos(g n z) is a characteristic function, S n1 It is expressed by formula (3), and formula (3) is as follows:

[0108]

[0109] In formula (3), G s is soil shear modulus information, q n is a first calculation coefficient, K1(q n r1) is a first second-type first-order modified Bessel function, K1(g n r1) is a second second-type first-order modified Bessel function, δ n1 is a second calculation variable, δ n1 It is expressed by formula (4), and formula (4) is as follows:

[0110]

[0111] In formula (4), K0(q n r1) is a first second-type zero-order modified Bessel function, K0(g n r1) is a second second-type zero-order modified Bessel function.

[0112] In the embodiment, based on the elastic dynamics theory, the control equation of the soil body is established, and the soil body control equation of the soil around the pile is expressed by formula (1), and formula (1) is as follows:

[0113]

[0114] Wherein, z is the depth of the normal direction, and the normal direction refers to the direction in which the pile is vertically embedded into the soil layer.

[0115] Based on the soil body control equation of the soil around the pile, the soil body friction resistance expression of the soil around the pile is obtained through derivation, that is, the soil body resistance of the soil around the pile, which is expressed by formula (2):

[0116]

[0117] The present embodiment realizes a fine description of the mechanical behavior of pile-soil interaction by constructing pile-soil interaction control equations and resistance expressions based on elastic dynamics, effectively improving the applicability and calculation efficiency of the theoretical model. By introducing the coupling effect of radial displacement and hoop displacement, formula (1) comprehensively reflects the deformation coordination relationship of the soil in three-dimensional space, and its differential form completely covers the mechanical equilibrium conditions in the radial, hoop and normal directions, which can accurately describe the dynamic response characteristics of the soil under complex stress state. Formula (2) relates the soil resistance to the stress component through the integral form, and combines the series expansion method to express the resistance as a linear combination of characteristic functions, which significantly simplifies the solution process of the resistance distribution under complex boundary conditions, and realizes the adaptive characterization of the soil layering characteristics by the correlation of the eigenvalue g n and the thickness of the soil layer H. Further, formulas (3) and (4) accurately describe the physical law of the radial decay of the soil stress field by introducing the second type of modified Bessel function and its derivative relationship, and combining the nonlinear modification of the compression coefficient η s and the Poisson's ratio v s , which enhances the characterization ability of the model for the compression characteristics of the soil. In addition, the construction of parameters q n and δ n1 effectively coordinates the scale relationship between the characteristic function and the Bessel function, ensuring that the solution form balances between mathematical rigor and engineering applicability. The overall derivation process reduces the three-dimensional partial differential equation to an analytically expressed series solution through strict mathematical transformation, which not only avoids the precision loss caused by traditional simplifying assumptions, but also provides a convenient mathematical framework for subsequent parameter inversion through the orthogonality of the characteristic function, laying a theoretical foundation for the in-depth analysis of the pile-soil interaction mechanism.

[0118] In some embodiments, the pile-soil interaction control equation is represented by formula (5), which is as follows:

[0119]

[0120] The pile-soil interaction control equation is represented by formula (5), which is as follows:

[0121]

[0122] In formula (6), f2(z) is the pile-soil interaction control equation, σ r2 is the pile-soil interaction stress, τ rθ2 is the pile-soil interaction shear force, A n2 is the second variable coefficient, r2 is the pile radius information, S n2 is the third calculation variable, S n2 is represented by formula (7), which is as follows:

[0123]

[0124] In formula (7), I1(q n r2) is a first first-order modified Bessel function of the first kind, I1(g n r2) is a second first-order modified Bessel function of the first kind, δ n2 is a fourth calculation variable, δ n2 is expressed by formula (8), and formula (8) is as follows:

[0125]

[0126] In formula (8), I0(q n r2) is a first zero-order modified Bessel function of the first kind, I0(g n r2) is a second zero-order modified Bessel function of the first kind.

[0127] In the embodiment, based on the elastic dynamics theory, the control equation of the soil body of the pile core soil is established, and the control equation of the soil body of the pile core soil is expressed by formula (5), and formula (5) is as follows:

[0128]

[0129] Based on the soil body control equation of the pile core soil, the soil body resistance of the pile core soil is obtained through derivation, and is expressed by formula (6):

[0130]

[0131] The embodiment significantly improves the completeness and calculation accuracy of the pile-soil interaction analysis by establishing the elastic dynamics control equation and the resistance expression of the soil body of the pile core soil. Formula (5) fully considers the coupling deformation characteristics of the pile core soil in the radial direction, the hoop direction and the normal direction by balancing the differential item of the hoop displacement through reconstruction, and the mathematical form fully characterizes the dynamic transmission mechanism of the internal stress field of the soil body. Formula (6) expresses the pile core resistance as a linear combination of characteristic functions by using the combination method of integration and series expansion, realizes the efficient analysis of the distribution of the resistance along the depth, and accurately describes the radial increasing characteristics of the stress field of the pile core soil by using the first modified Bessel functions I1 and I0, which are complementary to the second Bessel functions of the surrounding soil, and completely cover the different mechanical responses of the soil bodies inside and outside the pile. Formulas (7) and (8) coordinate the proportional relationship of different Bessel functions by constructing parameters δ n n2 and δ s , combine the shear modulus G s ​the convergence and engineering applicability of the analytical solution under the boundary conditions in the pile. The overall model reduces the three-dimensional governing equation to an analytically expressed series solution through strict mathematical derivation, which not only preserves the physical nature of the dynamic response of the soil body but also provides an efficient theoretical framework for the quantitative analysis of the distribution of the pile core resistance.

[0132] In some embodiments, the pipe pile governing equation is represented by formula (9) as follows:

[0133]

[0134] 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 length information of the pile body.

[0135] The virtual soil column governing equation is represented by formula (10) as follows:

[0136]

[0137] 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 information of the pile body, A p = π(r1 2 -r2 2 ), and 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.

[0138] In the present embodiment, the first, virtual soil column governing equation is constructed to achieve fine analysis of the full depth of the pile-soil interaction. Formula (9) dynamically couples the bending stiffness of the pile body with the soil resistance around the pile and in the pile core, accurately characterizes the equilibrium relationship between the horizontal displacement u p (z) of the pile body and the soil resistance f1(z), f2(z) through a fourth-order differential equation, and the parameters E p I p The moment of inertia information I p of the pile body fuses the information of the inner and outer radii of the pile, and fully reflects the influence of the cross-sectional characteristics of the pipe pile on the deformation. Formula (10) establishes a second-order differential equation for the virtual soil column below the pile end, and through the physical correlation between the complex Young's modulus G s of the virtual soil column below the pile end and the cross-sectional area information A p of the pile body, the horizontal displacement u s(z) Quantitative connection with soil resistance in the depth range of L to H is established, forming an effective extension of the response of the soil below the pile tip. Two equations focus on the deformation zone of the pile body and the pile tip influence zone respectively through the piecewise modeling strategy, 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 of different regions through the difference in the order of differentiation. The overall model realizes the organic integration of geometric characteristics and material properties through parameters r1 and r2, providing a unified theoretical framework for the collaborative deformation analysis of the pile-soil system under complex boundary conditions.

[0139] In some embodiments, the first initial horizontal displacement expression is represented by equation (11) as follows:

[0140]

[0141] In equation (11), a p , b p , c p , d p are first undetermined coefficients, δ n3 is a fifth calculation variable, U pn is a 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 equation (12) as follows:

[0142]

[0143] The second initial horizontal displacement expression is represented by equation (13) as follows:

[0144]

[0145] In equation (13), a s , b s are second undetermined coefficients, U sn is a 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,

[0146] In this embodiment, the pile soil resistance obtained in the foregoing step, i.e., formula (2), the pile core soil resistance, i.e., formula (6), is substituted into the pipe pile control equation, and after derivation and solution, the displacement of the pipe pile, i.e., the first initial horizontal displacement expression, is obtained, which is expressed as formula (11):

[0147]

[0148] The pile soil resistance obtained in the foregoing step, i.e., formula (2), the pile core soil resistance, i.e., formula (6), is substituted into the virtual soil column control equation, and 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, which is expressed as formula (13):

[0149]

[0150] This embodiment realizes the fine analytical characterization of the horizontal displacement of the pile body and the soil column below the pile tip by constructing a segmented displacement expression. Formula (11) describes the overall bending deformation of the pile body as a cubic polynomial, reflects the high-order correction effect of the pile soil resistance on the displacement through the series term, and coordinates the contribution proportion of the pile soil through the parameter U pn The soil resistance coefficient A n1 , the Bessel function characteristic value g n , and the pile body stiffness E p I p are dynamically related, and the contribution proportion of the pile soil is coordinated through δ n3 , so as to ensure the mechanical continuity of the displacement solution at the pile-soil interface. Formula (13) adopts a linear function and a series term superposition form, and the quantitative relationship between the displacement of the virtual soil column below the pile tip and the soil resistance is established through the coupling of the complex Young's modulus G s of the virtual soil column and the pile body cross-sectional area information A p , and the parameter U sn inherits the pile-soil interaction characteristic value system, and realizes the seamless connection of the displacement field above and below the pile tip. The two expressions reserve the boundary condition freedom degrees through the undetermined coefficients a p , b p , c p , d p and c s , b s , and the convergence characteristics of the series term are considered, so as to balance the analytical accuracy and the calculation efficiency, form the displacement analytical framework covering the whole pile length, and provide a universal theoretical tool for the pile-soil collaborative deformation analysis under complex loads.

[0151] In some embodiments, the first undetermined coefficient is expressed by formula (14), and formula (14) is as follows:

[0152]

[0153] In formula (14), M is a first variable matrix, f n is a second variable matrix, N is a third variable matrix, M is expressed by formula (15), f n is expressed by formula (16), N is expressed by formula (17) and formula (18);

[0154] Formula (15) is as follows:

[0155]

[0156] Formula (16) is as follows:

[0157]

[0158] When the pile top is subjected to a unit displacement in the case of rotation fixation, N is expressed by formula (17), and formula (17) is as follows:

[0159] N = [1 0 0 0 0 0] T ;

[0160] When the pile top is subjected to a unit rotation in the case of displacement fixation, N is expressed by formula (18), and formula (18) is as follows:

[0161] N = [0 1 0 0 0 0] T ;

[0162] The first undetermined coefficient after solving is substituted into the first initial horizontal displacement expression in reverse to obtain a calculation expression of the first variable coefficient, which is expressed by formula (19), and formula (19) is as follows:

[0163]

[0164] In formula (19), X mn is a second calculation coefficient, T m is a third calculation coefficient, X mn is expressed by formula (20), T m is expressed by formula (21);

[0165] Formula (20) is as follows:

[0166]

[0167] In formula (20), g m is a second eigenvalue of a characteristic function, V n is a fourth calculation coefficient,

[0168] Formula (21) is as follows:

[0169]

[0170] Equation (19) can be expanded to Equation (22), which is as follows:

[0171]

[0172] In this embodiment, based on the first and second initial horizontal displacement expressions obtained from the aforementioned steps, and combined with the boundary conditions and the continuity condition of the pile-soil interface displacement, the undetermined constant 'a' is derived. p b p c p d p a s b s The calculation expression is, i.e., formula (14):

[0173]

[0174] It should be noted that the second undetermined coefficient is obtained through formula (12) in f n Li Jiang a n2 Transform into A n1 By substituting the relevant parameters, formula (22) can be used to calculate A. n1 Solve the problem; obtain the solution A. n1 Substituting the specific results into formula (14), the coefficient of the first variable can be solved; in the calculation, the total number of characteristic values ​​n of the series is taken as n=100 to satisfy the convergence.

[0175] A can be solved here. n1 Substitute into formula (11) to obtain the horizontal displacement of the virtual soil column below the pipe pile, and use formula (13) to obtain the horizontal displacement of the virtual soil column below the pile end.

[0176] This embodiment significantly improves the analytical efficiency and accuracy of the displacement field and resistance parameters of the pile-soil system by constructing a solution system that combines matrix equations and series expansion. Formula (14) uses the undetermined coefficient a p b p c p d p a s b s With soil resistance coefficient A n1 Through matrix M and vector f n The dynamic correlation between N and its matrix M is constructed using boundary conditions such as pile end displacement, rotation angle, and bending moment to ensure the physical rationality of the solution; the vector f n Through U pn U snThe continuity of the pile body and the soil column displacement is coordinated, and the difference of the N different pile top constraint conditions is combined to realize the unified solution of multiple working conditions. Formula (19) further combines the series term with the matrix inverse operation, eliminates the non-orthogonality of the series solution by constructing the cross term of the coefficient X mn and T m , forms a closed linear equation group (formula 22), and efficiently solves A n1 by using matrix operation. Formula (20) is segmented for m=n and m≠n, accurately captures the interaction effect of eigenvalues g m and g n , and combines the quantitative modification of the attenuation characteristics of the Bessel function with the parameter V n to ensure the convergence stability of the series solution. The overall method uses the cooperative strategy of matrix dimension reduction and series truncation (such as n=100) to greatly reduce the calculation complexity while ensuring the theoretical rigor, and provides an efficient analytical path for full-parameter coupling analysis of pile-soil systems.

[0177] In some embodiments, the horizontal load-bearing friction type pipe pile top stiffness expression obtained according to the first final horizontal displacement expression, the second final horizontal displacement expression includes:

[0178] The horizontal load-bearing friction type pipe pile top stiffness expression is expressed by formula (23), and formula (23) is as follows:

[0179]

[0180] In formula (23), K hh is the first component of the horizontal load-bearing friction type pipe pile top stiffness, K hr is the second component of the horizontal load-bearing friction type pipe pile top stiffness, K rh is the third component of the horizontal load-bearing friction type pipe pile top stiffness, and K rr is the fourth component of the horizontal load-bearing friction type pipe pile top stiffness.

[0181] K hh , K hr , K rh , and K rr are expressed by formula (24), and formula (24) is as follows:

[0182]

[0183] In formula (24), u p (0) is the horizontal displacement of the pipe pile top at z=0, is the horizontal rotation angle of the pipe pile top at z=0, and Q p (0) is the horizontal stress of the pipe pile top at z=0, Mp (0) is a horizontal bending moment of the pile top of the pipe pile when z=0,

[0184] In the embodiment, by constructing the pile top stiffness matrix expression, the systematic analysis characterization of the pile top stiffness characteristics of the friction type pipe pile is realized.Formula (23) comprehensively characterizes the coupling effect of the horizontal displacement and rotation of the pile top in the form of the stiffness matrix, K hh , K hr , K rh , K rr respectively correspond to the stiffness responses under different constraint conditions, and completely reflect the mechanical coupling characteristics of the pile top under the action of the horizontal load.Formula (24) directly relates the stiffness components to the third-order derivative (shear force) and the second-order derivative (bending moment) of the displacement field by the differential relationship between the pile top displacement u p (0), the rotation angle (0) and the internal force Q p (0), M p (0), and combines the pile body stiffness parameters E p I p to establish a quantitative relationship between the stiffness analytical solution and the material characteristics.This method converts the complex pile-soil interaction into an analytically expressed stiffness matrix through high-order differential operation of the displacement expression, avoids the calculation redundancy of the traditional trial method or numerical iteration, and supports the unified solution of the pile top under the horizontal force, bending moment and their combination working conditions through the orthogonality design of the stiffness components, thereby providing a stiffness evaluation tool that is rigorous in theory and practical in engineering for the pile foundation design.

[0185] Compared with the prior art, the technical scheme has the beneficial effects that:

[0186] ​The present application is based on the theory of elasticity dynamics, establishes the three-dimensional viscoelastic continuum control equation of the soil around the pile and the soil in the pile core, and constructs a complete theoretical system of the rigidity of the top of the horizontal load friction type pipe pile by an analytical method. By introducing the coupling differential equations of the radial, ring and normal displacements (formula 1, 5), combining the series expansion method of the Bessel function and the characteristic function (formula 2, 6), the dynamic attenuation characteristics of the soil resistance along the depth distribution are accurately characterized, and the virtual soil column segmentation modeling strategy (formula 9, 10) is used to organically connect the bending stiffness of the pile body and the response of the soil at the pile end, and the analytical solution of the displacement field of the whole pile length is formed (formula 11, 13). By constructing the stiffness matrix (formula 23) and its component expression (formula 24), the differential relationship between the horizontal displacement, angle and internal force of the pile top is transformed into the stiffness parameters that can be quantitatively calculated, and the mechanical mechanism of the pile-soil cooperation under the action of horizontal load is revealed. Compared with the traditional method, the present application considers the space coupling effect of the frictional resistance of the soil around and in the pile through the series solution of the soil resistance (formula 2, 6) and the displacement correction term (formula 11, 13), breaks through the limitation of the single soil model; the efficient simultaneous solution of the multi-boundary condition is realized by using the matrix equation (formula 14, 22) and the orthogonality of the characteristic function, and the calculation efficiency is significantly improved by the parameter dimension reduction and series truncation strategy; based on the symmetry design of the stiffness matrix (formula 23), the displacement response of the pile top and the load working condition are directly related, and an universal analytical framework is provided for the rigidity evaluation under different constraint conditions. The present application converts the complex three-dimensional pile-soil interaction into the rigidity calculation formula that can be applied to engineering by rigorous mathematical derivation, the calculation results can be quickly obtained and visualized by parameter substitution, and has both theoretical rigor and practical guiding value, and provides a reliable theoretical tool for the design optimization and performance evaluation of the friction type pipe pile.

[0187] The above only describes some embodiments of the present application, and does not limit the protection scope of the present application, and any equivalent device or equivalent process transformation using the content of the specification and drawings, or direct or indirect application in other related technical fields, are also included in the patent protection scope of the present application.

Claims

1. A method for determining the rigidity of the top of a horizontally loaded friction pile, characterized in that, The method comprises: obtaining pipe pile mechanical parameters of a pipe pile to be analyzed and soil body mechanical parameters of a building foundation, the pipe pile to be analyzed being a cylindrical pipe pile, the building foundation being homogeneous viscoelastic soil, the pipe pile mechanical parameters comprising pile body length information, pile outer radius information, pile inner radius information, pile body elastic modulus information, pile body rotational inertia information, and pile body cross-sectional area information, and the soil body mechanical parameters comprising soil layer thickness information, soil body elastic modulus information, soil body shear modulus information, and soil body Poisson's ratio information in the building foundation; and obtaining boundary condition information and pile-soil interface horizontal displacement continuity condition information; constructing a friction-type pipe pile theoretical model under horizontal load, the friction-type pipe pile theoretical model comprising a pile-surrounding-soil body control equation, a pile-core-soil body control equation, a pipe pile control equation, and a virtual soil column control equation, the virtual soil column control equation being configured as a control equation of a virtual soil column below the pipe pile; solving the pile-surrounding-soil body control equation to obtain pile-surrounding-soil body resistance, and solving the pile-core-soil body control equation to obtain pile-core-soil body resistance; obtaining a first initial horizontal displacement expression according to the pile-surrounding-soil body resistance, the pile-core-soil body resistance, and the pipe pile control equation, and obtaining a second initial horizontal displacement expression according to the pile-surrounding-soil body resistance, the pile-core-soil body resistance, and the virtual soil column control equation, the first initial horizontal displacement expression being configured as a horizontal displacement expression of the virtual soil column below the pipe pile, the second initial horizontal displacement expression being configured as a horizontal displacement expression of a virtual soil column below a pile end, the first initial horizontal displacement expression containing a first undetermined coefficient, and the second initial horizontal displacement expression containing a second undetermined coefficient; solving the first undetermined coefficient and the second undetermined coefficient according to the boundary condition information and the pile-soil interface horizontal displacement continuity condition information; reversely substituting the solved first undetermined coefficient into the first initial horizontal displacement expression to obtain a first final horizontal displacement expression, and reversely substituting the solved second undetermined coefficient into the second initial horizontal displacement expression to obtain a second final horizontal displacement expression; obtaining a horizontal load friction-type pipe pile top stiffness expression 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 pile according to claim 1, wherein, The pile-surrounding-soil body control equation is represented by formula (1), and the formula (1) is as follows: In formula (1), r is pipe pile radius information, that is, pile outer radius information, 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 hoop displacement of the soil, is the partial derivative symbol; The pile-surrounding-soil body resistance is represented by formula (2), and the formula (2) is as follows: In Equation (2), f1(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 angle, τ rθ1 is the shear of the soil around the pile, A n1 is the first variable coefficient, r1 is the radius information outside the pile, S n1 is the first calculation variable, g n is the first eigenvalue of the characteristic function, H is the thickness information of the soil layer, cos(g n z) is the characteristic function, S n1 is expressed by Equation (3) as follows: In formula (3), G s is the shear modulus information of the soil body, q n is a first calculation coefficient, K1(q n r1) is a first second-type first-order modified Bessel function, K1(g n r1) is a second second-type first-order modified Bessel function, δ n1 is a second calculation variable, δ n1 is expressed by formula (4) as follows: In Equation (4), K0(q n r1) is a first second-kind zero-order modified Bessel function, and K0(g n r1) is a second second-kind zero-order modified Bessel function.

3. The method for determining the top stiffness of a horizontally loaded friction pile according to claim 2, wherein, The pile-core-soil body control equation is represented by formula (5), and the formula (5) is as follows: The pile-core-soil body resistance is represented by formula (6), and the formula (6) is as follows: In Equation (6), f2(z) is the resistance of the pile core soil, σ r2 is the stress of the pile core soil, τ rθ2 is the shear of the pile core soil, A n2 is the second variable coefficient, r2 is the information of the inner radius of the pile, S n2 is the third calculation variable, S n2 is expressed by Equation (7) as follows: In equation (7), Ii(q n r2) is a first first-kind first-order modified Bessel function, Ii(q n r2) is a second first-kind first-order modified Bessel function, δ n2 is a fourth calculated variable, δ n2 is expressed by equation (8) as follows: In equation (8), I0(q n r2) is the first first-kind zero-order modified Bessel function, I0(q 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 pile according to claim 3, wherein, The pipe pile control equation is represented by formula (9), and the formula (9) is as follows: 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. The virtual soil column control equation is represented by formula (10), and the formula (10) is as follows: In equation (10), G s is the complex Young's modulus of the virtual soil column below the pile tip, A p is the cross-sectional area information 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 tip, and H is the thickness of the soil layer.

5. The method for determining the top stiffness of a horizontally loaded friction pile according to claim 4, wherein, The first initial horizontal displacement expression is represented by formula (11), and the formula (11) is as follows: In Equation (11), a p , b p , c p , and d p are the first undetermined coefficients, δ n3 is a fifth calculation variable, U pn is a sixth calculation variable, U pn = U pn1 + δ n3 U pn2 , and U pn1 is the horizontal displacement of the pipe pile when the soil around the pile is considered, U pn2 is the horizontal displacement of the pipe pile when the soil in the pile core is considered, δ n3 is expressed by Equation (12) as follows: The second initial horizontal displacement expression is represented by formula (13), and the formula (13) is as follows: In formula (13), a s , b s are the second undetermined coefficients, U sn is a 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, 6. The method for determining the top stiffness of a horizontally loaded friction pile according to claim 5, wherein, The first undetermined coefficient is expressed by formula (14), which is as follows: In Equation (14), M is a first variable matrix, f n is a second variable matrix, N is a third variable matrix, M is expressed by Equation (15), f n is expressed by Equation (16), N is expressed by Equation (17) and Equation (18); The formula (15) is as follows: The formula (16) is as follows: When the pile top is subjected to a unit displacement in the case of rotation fixation, 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 a unit rotation in the case of displacement fixation, N is expressed by formula (18), which is as follows: N＝[0 1 0 0 0 0] T ； The first undetermined coefficient after being solved is reversely substituted into the first initial horizontal displacement expression to obtain a calculation expression of a first variable coefficient, which is expressed by formula (19), which is as follows: In Equation (19), X mn is a second calculation coefficient, T m is a third calculation coefficient, X mn is expressed by Equation (20), T m is expressed by Equation (21); The formula (20) is as follows: In Equation (20), g m is a second eigenvalue of the characteristic function, V n is a fourth calculation coefficient, The formula (21) is as follows: After formula (19) is expanded and converted, formula (22) is shown, which is as follows:

7. The method for determining the top stiffness of a horizontally loaded friction pile according to claim 6, wherein, According to the first final horizontal displacement expression, a second final horizontal displacement expression, a horizontal load friction type pipe pile top stiffness expression is obtained, which includes: The horizontal load friction type pipe pile top stiffness expression is expressed by formula (23), which is as follows: In formula (23), K hh is the first component of the pile top stiffness of the friction type pipe pile under the action of horizontal load, K hr is the second component of the pile top stiffness of the friction type pipe pile under the action of horizontal load, K rh is the third component of the pile top stiffness of the friction type pipe pile under the action of horizontal load, K rr is the fourth component of the pile top stiffness of the friction type pipe pile under the action of horizontal load. K hh , K hr , K rh , K rr By equation (24), which is as follows: In Equation (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,

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