Calculation method of settlement deformation of pile-net composite foundation considering the effect of reinforcement

Abstract

The application discloses a kind of pile-net composite foundation settlement deformation calculation method considering the effect of reinforcement, including: the reinforced mat in pile-net composite foundation is regarded as elastic sheet, the mat is divided into inter-pile soil area and pile top area, respectively, the non-homogeneous partial differential dynamic control equation of considering the displacement of inter-pile soil and pile top area mat of reinforcement tensile force is established;Inter-pile soil and pile top area mat dynamic response general solution is obtained;The boundary condition and continuity condition of inter-pile soil and pile top area mat are set respectively;The undetermined coefficient in general solution is obtained by simultaneously solving boundary condition and continuity condition, dynamic response general solution, so as to obtain the definite analytical solution of inter-pile soil and pile top area mat dynamic response;Repeat iteration until convergence, finally take the maximum value of inter-pile soil area mat vertical displacement and the maximum value of pile top area mat vertical displacement as the settlement deformation of pile-net composite foundation.The application realizes the accurate quantitative evaluation of pile-net composite foundation settlement deformation under dynamic load.

Description

Technical Field

[0001] This invention belongs to the field of computer-aided design technology and relates to a method for calculating the settlement and deformation of pile-net composite foundations that takes into account reinforcement. Background Technology

[0002] Pile-net composite foundations are an efficient technology for treating soft soil foundations and have been widely used to control differential settlement in road and building engineering. Currently, the force-deformation characteristics of pile-net composite foundations have been extensively studied. However, most studies have been conducted under static embankment loads. Currently, with the increase in traffic volume, vehicle size, and speed, the impact of dynamic loads on pile-net composite foundations is becoming increasingly significant. In addition to static studies, some research has also investigated the dynamic response of pile-net composite foundations under high-speed train or highway vehicle loads. However, these studies have not revealed the characteristics of dynamic flexibility, pile-soil load sharing ratio, and vertical dynamic response of pile-net composite foundations, which is of great significance for the design of pile-net composite foundations. Summary of the Invention

[0003] To address the aforementioned issues, this invention provides a method for calculating the settlement deformation of pile-net composite foundations that considers reinforcement. Based on thin plate vibration theory, the cushion layer is divided into the soil region between piles and the pile top region according to the difference in support stiffness. The influence of reinforcement tension is also introduced, thereby achieving accurate quantitative assessment of the settlement deformation of pile-net composite foundations under dynamic loads.

[0004] The technical solution adopted in this invention is a method for calculating the settlement and deformation of a pile-net composite foundation considering reinforcement, comprising the following steps:

[0005] S1. The reinforced cushion layer in the pile-net composite foundation is regarded as an elastic thin plate. The cushion layer is divided into the soil area between piles and the pile top area. According to the thin plate vibration theory, non-homogeneous partial differential dynamic control equations for the displacement of the soil area between piles and the cushion layer in the pile top area considering the tensile force of the reinforcement are established respectively.

[0006] S2, obtain the general solution of the dynamic response of the soil between piles and the cushion layer in the pile top area;

[0007] S3 sets the boundary conditions and continuity conditions for the soil between piles and the cushion layer in the pile top area, respectively;

[0008] S4. Combine the boundary conditions and continuity conditions of the soil between piles and the cushion layer at the top of the piles, and the general solution of the dynamic response of the soil between piles and the cushion layer at the top of the piles to obtain the undetermined coefficients in the general solution. Substitute the determined undetermined coefficients back into the general solution expression to obtain the definite analytical solution of the dynamic response of the soil between piles and the cushion layer at the top of the piles.

[0009] S5. Substitute the assumed initial tensile force of the reinforcement into the analytical solution of the dynamic response to calculate the maximum vertical displacement of the soil between piles and the cushion layer in the pile top area. Use the maximum value as the settlement. Then update the reinforcement tensile force according to the constitutive relationship of the reinforcement. Repeat the iteration until convergence. Finally, use the maximum vertical displacement of the cushion layer in the soil between piles and the maximum vertical displacement of the cushion layer in the pile top area as the settlement deformation of the pile-net composite foundation.

[0010] Furthermore, S1 includes the following steps:

[0011] S11, under the given geometric shape and load conditions, the reinforced cushion layer, piles, and soil between piles are axisymmetric in a cylindrical coordinate system with the pile center as the origin. Therefore, the non-homogeneous partial differential dynamic control equations for the displacement of the soil between piles and the cushion layer in the pile top region are established as follows:

[0012] (1)

[0013] (2)

[0014] In the formula: , These represent the radial displacements of the pile top and the top of the soil cushion layer between the piles, respectively. For the tensile force of the reinforcing bars; The bending stiffness of the padding layer; For Laplace operator; r represents radial coordinate. For the complex damping of the cushion layer; Indicates the dynamic impedance of the pile foundation. Indicates the dynamic impedance of the soil surrounding the pile; Indicates the harmonic load when referring to a vertical embankment; The density of the cushion layer; The thickness of the padding layer; Indicates time; Represents the imaginary unit;

[0015] S12, due to harmonic load on the embankment Under the influence of the system, the entire system undergoes simple harmonic motion, and equations (1) to (2) simplify to:

[0016] (3)

[0017] (4)

[0018] In the formula, Represents the embankment load amplitude. , These are the vertical displacement amplitudes of the pile top and the top of the soil cushion layer between piles, respectively. This represents the excitation angular frequency.

[0019] Furthermore, S2 includes the following steps:

[0020] S21 decomposes the dynamic control equations for the displacement of the soil between piles and the cushion layer in the pile top region into homogeneous and non-homogeneous parts.

[0021] S22, Solve the homogeneous general solution using the modified Bessel function;

[0022] S23, combined with the particular solution, the complete displacement general solution of the soil between piles and the cushion layer in the pile top area is shown in equations (5) to (6):

[0023] (5)

[0024] (6)

[0025] (7)

[0026] (8)

[0027] In the formula: , , , , , , and All are undetermined coefficients; and These are the first and second type zero-order modified Bessel functions, respectively; , , and These are all intermediate parameters; It is the dynamic compliance coefficient of the cushion layer in the pile top area; This represents the dynamic compliance coefficient of the cushion layer in the area above the soil between piles; based on the correlation between cushion layer displacement and rotation, bending moment and shear force, the expressions for rotation, bending moment and shear force are obtained according to the general solution of displacement.

[0028] Furthermore, S3 includes the following:

[0029] Continuity condition of pile-net composite foundation: Since the pile-net composite foundation is a continuous whole, on the surface of the pile body, i.e. At a point where the deflection, rotation, radial bending moment, and shear force of the cushion layer are continuous, then:

[0030] (9)

[0031] (10)

[0032] (11)

[0033] (12)

[0034] in, Indicates the radius of the pile; This indicates the vertical displacement amplitude of the cushion layer in the pile top region at the pile surface. This indicates the vertical displacement amplitude of the cushion layer in the area above the soil between piles at the pile surface. This indicates the rotation angle of the cushion layer at the pile top area on the pile surface; This indicates the rotation angle of the cushion layer above the soil between piles at the pile surface; This indicates the magnitude of the bending moment of the cushion layer at the pile top region on the pile surface; This indicates the bending moment amplitude of the cushion layer in the area above the pile body at the pile surface; This indicates the shear force amplitude of the cushion layer at the pile top region on the pile surface; This indicates the shear force amplitude of the cushion layer in the area above the pile body at the pile surface.

[0035] Boundary conditions for pile-net composite foundations:

[0036] exist At this point, the angle of the subbase is 0, then:

[0037] (13)

[0038] exist At this point, the cushion layers are connected and there is no torsional or shear deformation, then:

[0039] (14)

[0040] (15)

[0041] This indicates the rotation angle of the cushion layer in the pile top area at the pile center; This indicates the influence radius of the cushion layer in the soil zone between piles. The angle amplitude at the location; This indicates the influence radius of the cushion layer in the pile top area. The amplitude of dynamic shear force at the location.

[0042] Furthermore, S4 includes the following steps:

[0043] S41, Substitute the general solution expression containing undetermined coefficients obtained in S2 into the boundary and continuity conditions of S3 to construct a system of equations about the undetermined coefficients;

[0044] S42, Solve the system of equations to determine all undetermined coefficients;

[0045] S43, substitute the determined undetermined coefficients back into the general solution expression to obtain a definite analytical solution for the dynamic response.

[0046] Furthermore, S5 includes the following steps:

[0047] S51, Substitute the assumed initial tensile force of the reinforcement into equations (5) and (6) to solve the deflection function of the reinforced cushion layer, and calculate the maximum vertical displacement of the current soil between piles and the cushion layer in the pile top area respectively;

[0048] S52, the obtained maximum value is taken as the settlement. Substitute into equation (16):

[0049] (16)

[0050] and combined Calculate the updated tensile strength of the reinforcing bars. ,in For the tensile stiffness of the reinforcing material, Indicates the average strain of the reinforcing steel. and To fit the parameters; update the reinforcement tension. Substitute into equations (5) and (6) to calculate the maximum vertical displacement of the soil between piles and the cushion layer in the pile top area, respectively;

[0051] S53, select the convergence error, compare the maximum vertical displacement of the cushion layer in the soil area between piles obtained twice, compare the maximum vertical displacement of the cushion layer in the pile top area obtained twice, until the difference between the two is less than the convergence error, and finally take the maximum vertical displacement of the cushion layer in the soil area between piles and the maximum vertical displacement of the cushion layer in the pile top area as the settlement deformation of the pile-net composite foundation.

[0052] Furthermore, in S53, the convergence error is less than 0.001.

[0053] Furthermore, based on the vertical displacement amplitudes of the pile top and the top of the soil cushion layer between the piles obtained from S4, the dynamic stiffness of the pile-soil system is calculated using equations (17) to (18). Pile-soil load sharing ratio :

[0054] (17)

[0055] (18)

[0056] in, , These represent the vertical displacement amplitudes of the pile top and the top of the soil cushion layer between piles, respectively. Indicates the radius of the pile. Indicates the radius of influence;

[0057] By batch calculating displacements under different parameters, the variation law of dynamic stiffness and load sharing ratio can be obtained, thereby guiding the selection or optimization of the elastic modulus of the cushion layer, the thickness of the cushion layer, the pile spacing and the pile foundation modulus in engineering design.

[0058] The beneficial effects of this invention are:

[0059] Based on the vibration control equation of an elastic circular thin plate, and combined with the boundary and continuity conditions of the circular thin plate, this invention derives a dynamic and complete analytical solution for the displacement, rotation, bending moment, and shear force of a sand and gravel composite cushion layer on a pile-net composite foundation considering reinforcement. This solution can be used for the design of reinforced pile-net composite foundations.

[0060] This invention solves the problems of settlement deformation calculation and stiffness assessment of pile-net composite foundations under dynamic loads. First, considering the tensile force of the reinforcement and the effects of dynamic loads, the pile-net composite foundation is divided into the soil between piles and the area above the piles. Dynamic control equations for the composite foundation cushion layer are established for each of these different areas. Then, considering the boundary and continuity conditions of the cushion layer, an iterative method is used to solve for the settlement deformation of the pile-net composite foundation under dynamic loads. Batch calculations of displacements under different parameters are performed to obtain the variation law of dynamic stiffness and load sharing ratio, thereby guiding the selection or optimization of the cushion layer's elastic modulus, cushion layer thickness, pile spacing, and pile foundation modulus in engineering design. Attached Figure Description

[0061] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0062] Figure 1 This is a schematic diagram of the pile-net composite foundation distribution in an embodiment of the present invention.

[0063] Figure 2 This is the calculation model for the pile-net composite foundation in the embodiment of the present invention.

[0064] Figure 3 This is a schematic diagram of the iterative calculation process for the tensile force of reinforcing bars in an embodiment of the present invention.

[0065] Figure 4 This is a comparison between the embodiments of the present invention and the solutions of the prior art; where (a) is displacement, (b) is rotation angle, (c) is bending moment, and (d) is shear force.

[0066] Figure 5 This is the FEM model of the pile-net composite foundation in the embodiment of the present invention.

[0067] Figure 6 This is a comparison between the embodiment of the present invention and the solution of the FEM model; where (a) is displacement, (b) is bending moment, and (c) is shear force.

[0068] Figure 7 The displacement changes of the reinforced and unreinforced pads under different excitation frequencies in the embodiments of the present invention are shown in (a) and (b) are shown in (b).

[0069] Figure 8 The rotation angle changes of the reinforced and unreinforced pads under different excitation frequencies in the embodiments of the present invention; where (a) is the real part and (b) is the imaginary part.

[0070] Figure 9 The bending moment changes of reinforced and unreinforced pads under different excitation frequencies in embodiments of the present invention are shown in (a) for the real part and (b) for the imaginary part.

[0071] Figure 10 The shear force variation of reinforced and unreinforced pads under different excitation frequencies in embodiments of the present invention is shown in (a) as the real part and (b) as the imaginary part.

[0072] Figure 11 This refers to the change in the real part of the dynamic compliance when the elastic modulus of different pads is different in the embodiments of the present invention.

[0073] Figure 12 This refers to the change in the imaginary part of dynamic compliance when the elastic modulus of different pads is different in the embodiments of the present invention.

[0074] Figure 13 This refers to the variation in load sharing ratio when the elastic modulus of the cushion layer is different in the embodiments of the present invention.

[0075] Figure 14 This refers to the real part of the dynamic compliance variation when the pad thickness is different in the embodiments of the present invention.

[0076] Figure 15 This refers to the change in the imaginary part of dynamic compliance when the thickness of the padding layer is different in the embodiments of the present invention.

[0077] Figure 16 This refers to the variation in load sharing ratio when the thickness of the cushion layer is different in the embodiments of the present invention.

[0078] Figure 17 This refers to the change in the real part of the dynamic compliance when the pile spacing is different in the embodiments of the present invention.

[0079] Figure 18 This refers to the change in the imaginary part of the dynamic flexibility under different pile spacings in the embodiments of the present invention.

[0080] Figure 19 This refers to the variation in load sharing ratio under different pile spacings in embodiments of the present invention.

[0081] Figure 20This refers to the variation of the real part of the dynamic compliance when the pile foundation modulus is different in the embodiments of the present invention.

[0082] Figure 21 This refers to the variation of the imaginary part of the dynamic flexibility when the pile foundation modulus is different in the embodiments of the present invention.

[0083] Figure 22 This refers to the variation in load sharing ratio for different pile foundation moduli in embodiments of the present invention. Detailed Implementation

[0084] The technical solutions of the present invention will be clearly and completely described below with reference to the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0085] Technical concept of the present invention embodiments:

[0086] The reinforced cushion layer of the pile-net composite foundation is considered as an elastic thin plate and described using thin plate vibration theory. By solving the motion equations of the cushion layer and applying the boundary and continuity conditions of the pile-net composite foundation, the dynamic expressions for the displacement, rotation, bending moment, and shear force of the cushion layer under simple harmonic vibration loads are obtained. Furthermore, the dynamic flexibility of the reinforced cushion layer and the pile-soil load sharing ratio are introduced as evaluation indicators to understand the dynamic response under different cushion layer moduli, thicknesses, and pile spacings.

[0087] Figure 1 As shown, a plum blossom-shaped pile arrangement is used. This is the center-to-center distance between piles. Furthermore, the soil surrounding the piles is considered a homogeneous, single-phase continuous body overlying rigid bedrock. According to... Figure 2 The typical unit cell calculation model within the influence range of the single pile is analyzed. In this model, the reinforced cushion layer is considered as a perfectly elastic circular thin plate; the pile diameter is d, and the diameter of the single pile reinforcement range is... When the top of the slab is subjected to a vertical embankment harmonic load ( , Represents the embankment load amplitude. t represents the excitation circular frequency, and t represents time. Indicates the radius of the pile. This indicates the radius of the single pile reinforcement range, while Indicates the dynamic impedance of the pile foundation. This represents the dynamic impedance of the soil surrounding the pile. Furthermore, it is assumed that there is no relative sliding or separation between the pile and soil systems, satisfying perfect contact conditions.

[0088] An embodiment of the method for calculating the settlement and deformation of a pile-net composite foundation considering reinforcement includes the following steps:

[0089] S1, Establishment of the governing equations for the reinforced cushion layer;

[0090] Based on the theory of thin plate vibration, the dynamic control equations for the displacement of the composite cushion layer are established. Figure 2 Therefore, under the given geometric shape and load conditions, the reinforced cushion layer, piles, and soil between piles are axisymmetric in a cylindrical coordinate system with the pile center as the origin, thus ruling out [the possibility of [other possibilities]]. The influence of direction.

[0091] (1)

[0092] (2)

[0093] In the formula: , These represent the radial displacements of the pile top and the top of the soil cushion layer between the piles, respectively. For the tensile force of the reinforcing bar, , For the average strain of the reinforcing material, This refers to the tensile stiffness of the reinforcing steel. The flexural stiffness of the subbase (reinforced subbase). , The composite modulus of the subbase (reinforced subbase). Poisson's ratio for the subbase (reinforced subbase); This refers to the thickness of the subbase (reinforced subbase). For the Laplace operator, , This indicates that the fourth-order Laplace operator is the square of the Laplace operator; r represents the radial coordinate. For the complex damping of the subbase (reinforced subbase); This refers to the density of the subbase (reinforced subbase).

[0094] Due to harmonic loads on embankments Under the influence of , the entire system undergoes simple harmonic motion, then , , , Let be the vertical displacement amplitudes of the pile top and the top of the soil cushion between the piles, respectively. Therefore, equations (1) and (2) can be rewritten as:

[0095] (3)

[0096] (4)

[0097] For the sake of brevity, the following... This will be omitted.

[0098] S2, general solution of dynamic response of reinforced cushion layer;

[0099] Equation (3) is a non-homogeneous equation, and its complete solution consists of homogeneous solutions and particular solutions. Let:

[0100] (5)

[0101] , These are intermediate parameters.

[0102] The homogeneous equation corresponding to equation (3) can be transformed into:

[0103] (6)

[0104] Based on the operator separation theory. ,in and They respectively satisfy the following equations:

[0105] (7)

[0106] (8)

[0107] Will Substituting into equations (7) and (8), we obtain the following two ordinary differential equations:

[0108] (9)

[0109] (10)

[0110] Equations (9) and (10) both satisfy the modified Bessel equation, and their general solution is:

[0111] (11)

[0112] (12)

[0113] , All of these are intermediate variables.

[0114] Combining (11) and (12), the homogeneous general solution of equation (3) is:

[0115] (13)

[0116] In the formula: , These are the first and second type zero-order modified Bessel functions, respectively; These are coefficients to be determined.

[0117] make Substituting into equation (3), we obtain the particular solution of equation (3):

[0118] (14)

[0119] In the formula , It is the dynamic flexibility coefficient of the cushion layer in the pile top area.

[0120] It is the displacement amplitude of the cushion layer in the pile top region under forced vibration under simple harmonic load. It is the total displacement amplitude of the cushion layer in the pile top region, which consists of two parts: the homogeneous solution response caused by the cushion layer's own vibration characteristics and the forced vibration response directly induced by the load. .

[0121] Combining equation (13), the displacement expression for the cushion layer in the pile top region is obtained as follows:

[0122] (15)

[0123] Similarly, the general solution of equation (4) can be obtained by letting:

[0124] (16)

[0125] (17)

[0126] This represents the reciprocal of the equivalent dynamic stiffness of the pile after considering the inertial effect of the reinforced cushion layer; and These are intermediate parameters.

[0127] The general solution of the corresponding homogeneous equation is:

[0128] (18)

[0129] In the formula: These are coefficients to be determined.

[0130] make Substituting into equation (4), we obtain the particular solution of equation (4):

[0131] (19)

[0132] In the formula , This indicates the dynamic compliance coefficient of the cushion layer in the area above the soil between piles; , These are the vertical displacement amplitudes of the pile top and the top of the soil cushion layer between piles, respectively. This indicates the displacement amplitude of the cushion layer above the soil between piles under forced vibration under simple harmonic load.

[0133] Combining equation (18), the general solution for the cushion layer in the top region of the soil between piles is obtained as follows:

[0134] (20)

[0135] S3 sets the boundary conditions;

[0136] exist At this point, the corner of the subbase is 0, that is:

[0137] (twenty one)

[0138] exist At this location, because the pile-net composite foundation is a continuous whole, therefore... The deflection, rotation, radial bending moment, and shear force of the cushion layer are continuous, that is:

[0139] (twenty two)

[0140] (twenty three)

[0141] (twenty four)

[0142] (25)

[0143] exist At this point, the cushion layers are connected without torsional or shear deformation, that is:

[0144] (26)

[0145] (27)

[0146] This indicates that the cushion layer in the pile top area is at the center of the pile ( The magnitude of the angle (angular displacement) at point ( ); This indicates that the cushion layer in the pile top area is on the pile body surface (pile radius). The vertical displacement amplitude at point (). This indicates the rotation angle of the cushion layer at the pile top area on the pile surface; This indicates that the cushion layer in the pile top area is on the pile body surface (pile radius). The magnitude of the bending moment at point (). This indicates that the cushion layer in the pile top area is on the pile body surface (pile radius). Shear force amplitude at () location; This indicates the influence radius of the cushion layer in the pile top area. The amplitude of dynamic shear force at the location.

[0147] This indicates that the cushion layer in the area above the soil between piles is on the pile surface (pile radius). The magnitude of the angle (angular displacement) at point ( ); This indicates that the cushion layer in the area above the soil between piles is on the pile surface (pile radius). The vertical displacement amplitude at point (). This indicates that the cushion layer in the area above the soil between piles is on the pile surface (pile radius). The magnitude of the bending moment at point (). This indicates that the cushion layer in the area above the soil between piles is on the pile surface (pile radius). Shear force amplitude at () location; This indicates the influence radius of the cushion layer in the soil zone between piles. The angle amplitude at the point; Equations (21), (26) to (27) above are boundary conditions; Equations (22) to (25) are continuity conditions.

[0148] S4, Solve for the coefficient matrix;

[0149] Based on the relationship between internal force and displacement in the theory of vibration of elastic thin plates:

[0150] (28)

[0151] In the formula ; ; ; .

[0152] The specific expressions for the dynamic control equations of the composite reinforced cushion layer are (15) and (20). According to equation (21), in At the time, at the corner .according to hour The trend, received By using equation (21) and the internal force expression (28), we can obtain the following: and When the subbase angle, bending moment, and shear force are as follows:

[0153] exist hour:

[0154] (29)

[0155] (30)

[0156] (31)

[0157] exist hour:

[0158] (32)

[0159] (33)

[0160] (34)

[0161] Based on the above, by combining equations (21) to (28) and equations (29) to (34), the undetermined coefficients can be obtained. The matrix equation is as follows:

[0162] (35)

[0163] In the formula:

[0164]

[0165]

[0166]

[0167]

[0168]

[0169]

[0170] Solving equation (35) yields the following results. The numerical solution is obtained by solving for all the undetermined coefficients and substituting them back into the general solution expression. This yields the definite solutions for the dynamic response of the thin plate in terms of displacement, rotation, bending moment, and shear force.

[0171] Furthermore, when the cushion layer is subjected to harmonic loads, the average vertical displacement is proportional to the load amplitude. To more comprehensively elucidate the dynamic characteristics related to the deformation of the cushion layer, the dynamic stiffness of the pile-soil relationship is introduced. Pile-soil load sharing ratio :

[0172] (36)

[0173] (37)

[0174] This indicates the magnitude of the dynamic load borne by the pile. This indicates the amplitude of the dynamic load borne by the soil between the piles; the dynamic stiffness of the pile-soil pair. The dynamic flexibility is the reciprocal of the dynamic flexibility; the pile-soil co-bearing law—the load borne by the pile and the soil between the piles is related to their own stiffness and cumulative deformation. The load sharing ratio between the pile and the soil between the piles is quantified by formula (37).

[0175] By batch calculating displacements under different parameters, the variation law of dynamic stiffness and load sharing ratio can be obtained, thereby guiding the selection or optimization of the elastic modulus of the cushion layer, the thickness of the cushion layer, the pile spacing and the pile foundation modulus in engineering design.

[0176] S5, iterative calculation;

[0177] For solving the undetermined coefficients when the tensile force of the reinforcement is unknown, most engineering projects do not measure the deformation of the reinforcement beforehand. In such cases, a method based on existing literature 1 (Cao Xinwen, Qing Sanhui, Zhou Lixin. Experimental study on the reinforcement effect of geogrid in pile-mesh composite foundation [J]. Chinese Journal of Rock Mechanics and Engineering, 2006, 25(S1): 3162-3167.) can be used to study the average strain of the reinforcement under embankment load through field tests. With Settlement It can be approximated by the following relationship:

[0178] (38)

[0179] In the formula: and For fitting parameters, the parameters can be selected according to the table in Cao Xinwen et al., based on the settlement calculation location.

[0180] Based on the above research findings, the maximum deflection of the stiffened body was selected. As the settlement in equation (38) ,like Figure 3 As shown, the steps for iteratively calculating and solving for the undetermined coefficients are as follows:

[0181] (1) Assume the initial tensile force of the reinforcing bar. (It is recommended to use a smaller initial value), substitute the initial value into equation (15) and equation (20), solve for the deflection function of the reinforced cushion layer, and obtain the new value. and ;

[0182] (2) The obtained Value as settlement Substitute into equation (38) and combine with equation Find the updated tension Then re- Substituting equations (15) and (20), we solve for the deflection function of the reinforced cushion layer and obtain the results. and ; in step (2) , With step (1) , They are essentially the same, just before and after the update.

[0183] (3) Select the convergence error The two results and The results are compared, and the calculation stops when the difference between the two is less than the error; otherwise, the iterative calculation continues according to steps (2) and (3) until the error converges. The value is 0.001; after convergence, the maximum vertical displacement of the cushion layer in the soil area between piles and the maximum vertical displacement of the cushion layer in the pile top area are finally used as the settlement deformation of the pile-net composite foundation.

[0184] Note: The above method is based on the tensile stiffness of the reinforcing steel. The data was established based on test data for a 250 kN / m grid. When applying this data to other reinforcing materials, a stiffness correction factor needs to be considered. For important projects, it is recommended to verify and adjust the fitting parameters in conjunction with on-site monitoring data.

[0185] Calculation example:

[0186] The proposed solution was validated, and the dynamic characteristics of the pile-soil composite foundation under various pile-soil parameters were revealed through numerical example analysis. The excitation frequency was dimensionlessly reduced to... . Indicates the excitation angular frequency. It is a dimensionless quantity; unless otherwise specified, the parameter values ​​used can be found in Table 1.

[0187] Table 1 Material / Geometric Parameters of Pile-Network Composite Foundation

[0188]

[0189] First, by comparing and analyzing the solution with that in Reference 2 (Zhao Minghua, Liu Meng, Ma Binhui, et al. Calculation of pile-soil stress ratio and settlement of pile-net composite foundation based on elastic substrate theory [J]. Journal of Central South University (Natural Science Edition), 2016, 47(6): 2007-2014.), the correctness of the solution proposed in this embodiment is verified. By setting , The solution of this invention embodiment is degenerated into a static, unreinforced case, and the thickness of the cushion layer is changed. The solution is then compared with the solution obtained in Reference 2. Figure 4 As shown in (a)-(d), the solution proposed in this embodiment (the solution in this paper) is very consistent with the solution obtained in Reference 2, verifying the correctness of the solution obtained in this embodiment.

[0190] Secondly, the results obtained from the solutions of the embodiments of the present invention are compared with the finite element model (FEM) results of ABAQUS software. The FEM of the pile-net composite foundation is as follows: Figure 5As shown, in this finite element model, the cushion layer, piles, and soil are all linearly elastic and described by C3D8R elements. Model parameters are from Table 1, and the cushion layer is considered to have three thicknesses: 0.30m, 0.25m, and 0.20m. Boundary conditions are applied to the outer edges of the cushion layer and soil to restrict their lateral horizontal displacement (U). x U y ), while allowing vertical displacement in the vertical direction (U) z The soil and piles at the bottom of the model are fixed, in a vertical (U) orientation. z ) and level (U x U y There was no movement in the [direction]. The cushion layer, piles, and soil were in close contact, with no relative sliding or separation. Mesh density was controlled by locally subdividing each component into elements, with size controlled according to the number of elements. There were 40 elements at the outer edges of the cushion layer, piles, and soil; 20 elements along the centerline of the cushion layer surface; and 10 elements each along the centerlines of the pile and soil surfaces. A uniform load of 40 kN was applied to the top of the cushion layer. The seed constraint allowed only an increase in the number of elements. Mesh control properties employed a hexahedral element shape, utilizing a neutral axis algorithm to minimize mesh transformation, thereby reducing mesh deformation and mitigating mesh distortion. Experimental results are as follows: Figure 6 As shown in (a)-(c), it can be seen that the solution proposed in this embodiment of the invention (in this paper) is in very good agreement with the results of the finite element method (FEM). It should be noted that since the ABAQUS software does not have the function of extracting rotation angle results, therefore... Figure 6 The rotation angle was not compared.

[0191] Figures 7-10 The curves showing the variation of displacement, rotation, bending moment, and shear force of the reinforced and unreinforced subgrade under different excitation frequencies are presented. It can be observed that as the excitation frequency increases, the differences in the amplitudes of displacement, rotation, bending moment, and shear force gradually decrease. Therefore, this indicates that as the excitation frequency increases, the reinforcement gradually plays a role, and the displacement of the reinforced subgrade is smaller than that of the unreinforced subgrade. However, it is worth noting that... Figure 7 As can be seen from (a) and (b), the maximum displacement occurs at the edge representing the reinforced zone. This indicates that the strengthening effect is not significant and the role of the reinforcing material is not obvious, leading to significant deformation. Additionally, in... At this point, the continuity condition (22) is satisfied.

[0192] The peak value of the rotation angle of the reinforced cushion layer decreases with increasing excitation frequency, and reinforcement can significantly reduce the rotation angle of the cushion layer, such as... Figure 8As shown in (a) and (b), this indicates that due to different pile-soil stiffness, when the differential settlement between the pile and soil reaches a certain magnitude at the excitation frequency, the pile-soil system gradually tends towards steady-state vibration, and the difference in rotation angle decreases. Furthermore, in and At this point, the angle is 0, which satisfies the boundary conditions (21) and (26).

[0193] The bending moment of the subbase exhibits a trend of multiple peak points as the excitation frequency increases, such as... Figure 9 As shown in (a) and (b), with the increase of excitation frequency... The peak value gradually decreases within the range, when At this time, fluctuations occur because the soil around the pile is soft and has insufficient bearing capacity; and the wave crests show a decreasing trend, indicating that when the excitation frequency reaches... When this happens, the reinforcing material plays a role, effectively reducing the bending moment generated by the subbase. Furthermore, it was observed that the bending moment... The continuity condition (24) is satisfied at this point.

[0194] The shear force of the cushion layer increases with the increase of the excitation frequency. The peak shear force gradually decreases within the range. The fluctuation range is stable, such as Figure 10 As shown in (a) and (b). This is due to the different stiffness at the pile-soil interface. This difference leads to uneven settlement of the cushion layer, generating huge shear forces to resist this deformation, but at high frequencies, the system tends to a steady state, and the stiffness difference is no longer significant. Furthermore, in... At this time, the reinforcing materials play a role, bearing part of the shear force, thus significantly reducing the shear force of the reinforced cushion layer. Furthermore, as can be seen in the figure, the shear force at... Continuous, in and The value is zero at the point, which is consistent with the continuity condition of equation (25) and the boundary conditions of equations (21) and (27). The validity of the current solution is also confirmed by the above results and the compatibility between the system boundary conditions and continuity.

[0195] Figures 11-13The effect of the elastic modulus of the subgrade on pile-soil dynamic flexibility and load sharing ratio is shown. It can be seen that an increase in the elastic modulus leads to a decrease in dynamic flexibility. This indicates that as the subgrade modulus increases, its resistance to deformation increases, resulting in a decrease in elasticity and thus a decreasing trend in dynamic flexibility. Furthermore, an increase in the elastic modulus of the subgrade leads to a higher proportion of load borne by the soil at low frequencies. This is because the increased stiffness of the subgrade, due to the larger distance between piles, initially causes the soil to bear more load. As the modulus increases, the tensile force of the reinforcement gradually decreases, and the load gradually shifts towards the pile top. Therefore, the effect of increasing the subgrade modulus on the load sharing ratio shows a trend of first increasing and then decreasing.

[0196] Figures 14-16 The effect of cushion layer thickness on pile-soil dynamic flexibility and load sharing ratio is shown. It can be seen that increasing the cushion layer thickness leads to a decrease in flexibility, indicating an enhanced resistance to deformation under external forces. Furthermore, the figure shows that increasing the cushion layer thickness improves its resistance to deformation, thereby increasing the pile's load-bearing capacity. This phenomenon is attributed to stress concentration at the pile top, where the pile bears a greater load. At low frequencies and with a small cushion layer thickness, increasing the thickness increases the proportion of load borne by the soil between the piles, reduces stress concentration at the pile top, and significantly reduces the pile-soil load sharing ratio. Later, due to the presence of high-stiffness reinforcement materials, the "netting effect" generated by the reinforcement may transfer the load to the pile top. However, at this point, with increased thickness, the cushion layer stiffness may be too high, making it difficult for the pile to penetrate, thus exacerbating stress concentration at the pile top and increasing the pile-soil load sharing ratio. At high-frequency vibration, the pile-soil system is in steady-state vibration; increasing the cushion layer thickness enhances the soil arching effect, and the pile top load sharing ratio increases significantly.

[0197] Figures 17-19 This study explains the impact of pile spacing on pile-soil dynamic flexibility and load sharing ratio. Clearly, a larger pile spacing leads to increased dynamic flexibility. This means that within the same distribution area, a lower number of piles results in a larger bearing area for the cushion layer and soil, thus reducing stiffness. Consequently, the cushion layer is more prone to significant deformation. Furthermore, as pile spacing increases, the load sharing ratio decreases. This is because the increase in embankment load within the bearing range of a single pile exceeds the load transferred to the pile top by the soil arching effect, resulting in a smaller share ratio, and the reinforcement does not play a significant role. When the pile spacing increases to a certain distance, the load sharing ratio increases. This indicates that the reinforcement plays a role at this point, enhancing the transfer of load to the piles. This demonstrates that the reinforced cushion layer has a dual function: on the one hand, it enhances the load sharing of the piles through the "netting effect," and on the other hand, it weakens the effect of differential settlement by homogenizing stress.

[0198] Figures 20-22This illustrates the influence of pile modulus on pile-soil dynamic flexibility and load sharing ratio. The figure shows that flexibility decreases as pile modulus increases. This means that under the same conditions, a larger pile modulus results in a more rigid and less deformable pile. Furthermore, as pile modulus increases, load sharing ratio increases. This is because a larger pile modulus enhances pile stiffness, enabling it to bear the load more effectively, thus reducing the soil's share of the load and consequently increasing the load sharing ratio.

[0199] This invention, based on the theory of vibrating thin plates and combined with the boundary conditions of thin plate continuity, derives the dynamic deflection equation and internal force expression of reinforced cushion layers. Through numerical examples, it systematically analyzes the dynamic response laws of cushion layer displacement, rotation angle, bending moment, and shear force. The study introduces dynamic compliance as an evaluation index, focusing on the influence of key design parameters such as elastic modulus, thickness, pile spacing, and pile foundation modulus on the mechanical behavior of the cushion layer. The main conclusions are as follows:

[0200] (1) The excitation frequency has a significant impact on the dynamic response of the subgrade. As the excitation frequency increases, the variation amplitudes of the subgrade's displacement, rotation angle, bending moment, and shear force decrease, gradually reaching a stable vibration state. In the low-frequency excitation range, the variation amplitudes of the subgrade's displacement, rotation angle, bending moment, and shear force increase with the increase of the excitation frequency; after the excitation frequency continues to increase and reaches the high-frequency range, the variation amplitudes of the subgrade's displacement, rotation angle, bending moment, and shear force decrease with the increase of the excitation frequency. This solves the problem of subgrade resonance or excessive deformation under dynamic loads in practical engineering. By analyzing the frequency response, the design can be optimized to avoid harmful vibration frequencies and ensure structural stability and durability.

[0201] (2) Pile spacing and pile modulus are key factors affecting the dynamic flexibility of the cushion layer. As the pile spacing increases, the dynamic flexibility of the reinforced cushion layer increases significantly, indicating that the increase in pile spacing weakens the overall stiffness of the cushion layer, resulting in a more significant dynamic response. However, the increase in pile modulus significantly reduces the dynamic flexibility and increases the overall stiffness. Therefore, in practical engineering, it is crucial to reasonably control the pile spacing and pile modulus to optimize the dynamic performance of the cushion layer.

[0202] (3) The composite modulus and thickness of the subbase have relatively small effects on the dynamic flexibility of the subbase, but there are still certain regularities. The larger the composite modulus, the smaller the dynamic flexibility of the subbase, indicating that the reinforcing material with a high modulus can enhance the overall stiffness of the subbase and suppress its dynamic response; similarly, the increase in subbase thickness will also reduce the dynamic flexibility, indicating that the increase in thickness helps to improve the deformation resistance of the subbase. However, compared with the pile spacing, the composite modulus and thickness have a relatively limited effect on the dynamic flexibility.

[0203] Some existing technologies treat the composite cushion layer as an elastic circular thin plate, and calculate the general solution of the composite cushion layer based on the physical and mechanical parameters of the thin plate. The obtained solution is an analytical boundary value; the reinforcement effect and the role of the reinforcement are not considered. There is a qualitative difference in the support conditions and dynamic constraint effect of the pile top and the pile inter-pile soil on the cushion layer. This invention divides the pile-net composite foundation into the pile inter-pile soil and the pile top range, and establishes the dynamic control equations of the composite foundation cushion layer in different ranges respectively, accurately capturing the core feature of the inherent non-uniformity of the pile-net composite foundation; the establishment of equations in different regions means that a partial differential equation needs to be solved in each region, and then these equations are coupled by applying continuity conditions (displacement, rotation angle, bending moment, shear force continuity) on the boundary. The problem of equation coupling and solution in multiple regions under complex boundary conditions is successfully handled, and the solution of the whole system is obtained. In formula (1), " The figure represents the influence of reinforcement tension on the dynamic response of the cushion layer. Reinforcement tension significantly alters the vibration modes and dynamic response of the cushion layer, adding the tensile effect of the reinforcement. Introducing reinforcement tension causes the control equations of the pile-net composite foundation to couple with the cushion layer displacement and reinforcement tension, making analytical solutions impossible. Decoupling methods are required, increasing the technical difficulty. Thin-plate vibration theory is used to treat the reinforced cushion layer of the composite foundation as an elastic plate, while the piles and soil act as spring dampers supporting the cushion layer. By separating variables and combining the boundary and continuity conditions of the pile-net composite foundation, the control equations of the cushion layer are solved, and a frequency domain analytical solution for the dynamic characteristics of the pile composite foundation is proposed. The pile-soil dynamic flexibility and pile-soil load sharing ratio are defined to evaluate the dynamic characteristics of the pile composite foundation. Analysis of the influence of cushion layer modulus, thickness, and pile spacing shows that the reinforced cushion layer can effectively share the load and reduce uneven settlement, but its dynamic response is significantly affected by the type and arrangement of reinforcement materials.

[0204] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention are included within the scope of protection of the present invention.

Claims

1. A method for calculating settlement and deformation of pile-net composite foundations considering reinforcement, characterized in that, Includes the following steps: S1. The reinforced cushion layer in the pile-net composite foundation is regarded as an elastic thin plate. The cushion layer is divided into the soil area between piles and the pile top area. According to the thin plate vibration theory, non-homogeneous partial differential dynamic control equations for the displacement of the soil area between piles and the cushion layer in the pile top area considering the tensile force of the reinforcement are established respectively. S2, obtain the general solution of the dynamic response of the soil between piles and the cushion layer in the pile top area; S3 sets the boundary conditions and continuity conditions for the soil between piles and the cushion layer in the pile top area, respectively; S4. Combine the boundary conditions and continuity conditions of the soil between piles and the cushion layer at the top of the piles, and the general solution of the dynamic response of the soil between piles and the cushion layer at the top of the piles to obtain the undetermined coefficients in the general solution. Substitute the determined undetermined coefficients back into the general solution expression to obtain the definite analytical solution of the dynamic response of the soil between piles and the cushion layer at the top of the piles. S5, substitute the assumed initial tensile force of the reinforcement into the analytical solution of the dynamic response, calculate the maximum vertical displacement of the soil between piles and the cushion layer in the pile top area, take the maximum value as the settlement, update the reinforcement tensile force according to the constitutive relationship of the reinforcement, repeat the iteration until convergence, and finally take the maximum vertical displacement of the cushion layer in the soil between piles and the maximum vertical displacement of the cushion layer in the pile top area as the settlement deformation of the pile-net composite foundation. S1 includes the following steps: S11, under the given geometric shape and load conditions, the reinforced cushion layer, piles, and soil between piles are axisymmetric in a cylindrical coordinate system with the pile center as the origin. Therefore, the non-homogeneous partial differential dynamic control equations for the displacement of the soil between piles and the cushion layer in the pile top region are established as follows: （1） （2） In the formula: , These represent the radial displacements of the pile top and the top of the soil cushion layer between the piles, respectively. For the tensile strength of the reinforcing bars; The bending stiffness of the padding layer; For Laplace operator; r represents radial coordinate. For the complex damping of the cushion layer; Indicates the dynamic impedance of the pile foundation. Indicates the dynamic impedance of the soil surrounding the pile; Indicates the harmonic load when referring to a vertical embankment; The density of the cushion layer; The thickness of the padding layer; Indicates time; Represents the imaginary unit; S12, due to harmonic load on the embankment Under the influence of the system, the entire system undergoes simple harmonic motion, and equations (1) to (2) simplify to: （3） （4） In the formula, Represents the embankment load amplitude. , These are the vertical displacement amplitudes of the pile top and the top of the soil cushion layer between piles, respectively. This represents the excitation angular frequency.

2. The method for calculating settlement and deformation of a pile-net composite foundation considering reinforcement as described in claim 1, characterized in that, S2 includes the following steps: S21 decomposes the dynamic control equations for the displacement of the soil between piles and the cushion layer in the pile top region into homogeneous and non-homogeneous parts. S22, Solve the homogeneous general solution using the modified Bessel function; S23, combined with the particular solution, the complete displacement general solution of the soil between piles and the cushion layer in the pile top area is shown in equations (5) to (6): （5） （6） （7） （8） In the formula: , , , , , , and All are undetermined coefficients; and These are the first and second type zero-order modified Bessel functions, respectively; , , and These are all intermediate parameters; It is the dynamic compliance coefficient of the cushion layer in the pile top area; This represents the dynamic compliance coefficient of the cushion layer in the area above the soil between piles; based on the correlation between cushion layer displacement and rotation, bending moment and shear force, the expressions for rotation, bending moment and shear force are obtained according to the general solution of displacement.

3. The method for calculating settlement and deformation of a pile-net composite foundation considering reinforcement as described in claim 1, characterized in that, S3 includes the following: Continuity condition of pile-net composite foundation: Since the pile-net composite foundation is a continuous whole, on the surface of the pile body, i.e. At a point where the deflection, rotation, radial bending moment, and shear force of the cushion layer are continuous, then: （9） （10） （11） （12） in, Indicates the radius of the pile; This indicates the vertical displacement amplitude of the cushion layer in the pile top region at the pile surface. This indicates the vertical displacement amplitude of the cushion layer in the area above the soil between piles at the pile surface. This indicates the rotation angle of the cushion layer at the pile top area on the pile surface; This indicates the rotation angle of the cushion layer above the soil between piles at the pile surface; This indicates the magnitude of the bending moment of the cushion layer at the pile top region on the pile surface; This indicates the bending moment amplitude of the cushion layer in the area above the pile body at the pile surface; This indicates the shear force amplitude of the cushion layer at the pile top region on the pile surface; This indicates the shear force amplitude of the cushion layer in the area above the pile body at the pile surface; Boundary conditions for pile-net composite foundations: exist At this point, the angle of the subbase is 0, then: （13） exist At this point, the cushion layers are connected and there is no torsional or shear deformation, then: （14） （15） This indicates the rotation angle of the cushion layer in the pile top area at the pile center; This indicates the influence radius of the cushion layer in the soil zone between piles. The angle amplitude at the location; This indicates the influence radius of the cushion layer in the pile top area. The amplitude of dynamic shear force at the location.

4. The method for calculating settlement and deformation of a pile-net composite foundation considering reinforcement as described in claim 1, characterized in that, S4 includes the following steps: S41, Substitute the general solution expression containing undetermined coefficients obtained in S2 into the boundary and continuity conditions of S3 to construct a system of equations about the undetermined coefficients; S42, Solve the system of equations to determine all undetermined coefficients; S43, substitute the determined undetermined coefficients back into the general solution expression to obtain a definite analytical solution for the dynamic response.

5. The method for calculating settlement and deformation of a pile-net composite foundation considering reinforcement as described in claim 1, characterized in that, S5 includes the following steps: S51, Substitute the assumed initial tensile force of the reinforcement into equations (5) and (6) to solve the deflection function of the reinforced cushion layer, and calculate the maximum vertical displacement of the current soil between piles and the cushion layer in the pile top area respectively; S52, the obtained maximum value is taken as the settlement. Substitute into equation (16): （16） and combined Calculate the updated tensile strength of the reinforcing bars. ,in For the tensile stiffness of the reinforcing material, Indicates the average strain of the reinforcing steel. and To fit the parameters; update the reinforcement tension. Substitute into equations (5) and (6) to calculate the maximum vertical displacement of the soil between piles and the cushion layer in the pile top area, respectively; S53, select the convergence error, compare the maximum vertical displacement of the cushion layer in the soil area between piles obtained twice, compare the maximum vertical displacement of the cushion layer in the pile top area obtained twice, until the difference between the two is less than the convergence error, and finally take the maximum vertical displacement of the cushion layer in the soil area between piles and the maximum vertical displacement of the cushion layer in the pile top area as the settlement deformation of the pile-net composite foundation.

6. The method for calculating settlement and deformation of a pile-net composite foundation considering reinforcement as described in claim 1, characterized in that, In S53, the convergence error is less than 0.

001.

7. The method for calculating settlement and deformation of a pile-net composite foundation considering reinforcement as described in claim 2, characterized in that, Based on the vertical displacement amplitudes of the pile top and the top of the soil cushion between the piles obtained from S4, the dynamic stiffness of the pile-soil system is calculated using equations (17) and (18). Pile-soil load sharing ratio : （17） （18） in, , These represent the vertical displacement amplitudes of the pile top and the top of the soil cushion layer between piles, respectively. Indicates the radius of the pile. Indicates the radius of influence; By batch calculating displacements under different parameters, the variation law of dynamic stiffness and load sharing ratio can be obtained, thereby guiding the selection or optimization of the elastic modulus of the cushion layer, the thickness of the cushion layer, the pile spacing and the pile foundation modulus in engineering design.

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