Design Method of Composite Pile Foundation for Renovation and Expansion Based on Coordination of Stiffness of New and Old Foundations

Abstract

This invention discloses a design method for composite pile foundations for renovation and expansion based on the synergistic stiffness of new and old foundations. The method includes: dividing the overlap between the new and old foundations into multiple regions according to stiffness differences; establishing dynamic control equations for the cushion layer in each region; setting boundary conditions and continuity conditions for the cushion layer in each region; obtaining the general solution of the dynamic response of the cushion layer in each region; simultaneously solving the boundary conditions, continuity conditions, and the general solution of the cushion layer's dynamic response to obtain undetermined coefficients; substituting the determined undetermined coefficients back into the general solution expression; and then quantifying the impact of the stiffness difference between the new and old foundations through displacement and internal force expressions, thereby guiding the design of composite pile foundations for renovation and expansion. This invention introduces stiffness difference as a core factor into the dynamic model, deriving dynamic equations from the theory of vibrating elastic thin plates, fully considering the problems at both ends and the connection between the new and old foundations, and revealing the key role of pile-soil coordinated stiffness through zonal modeling, guiding foundation design and reducing differential deformation.

Description

Technical Field

[0001] This invention belongs to the field of computer-aided design technology and relates to a design method for composite pile foundations for renovation and expansion based on the synergy of stiffness between new and old foundations. Background Technology

[0002] With the rapid development of my country's economy and the continuous growth of traffic volume, many highways built in the early stages can no longer meet the ever-increasing traffic demands and urgently need to be upgraded and expanded. Among these upgrades, roadbed widening is one of the most common forms of highway reconstruction and expansion projects, involving widening existing roads to improve their traffic capacity. However, traditional roadbed widening methods often lead to problems such as large differential settlement between the old and new roadbeds and poor stability, resulting in pavement cracking, bridge approach slab settlement, and other defects. This not only seriously affects driving safety but also greatly reduces the service life and comfort of the road surface.

[0003] The pile-net composite foundation, by setting pile foundations beneath the widened roadbed, transfers the embankment load to the deep foundation, effectively reducing differential settlement between the old and new roadbeds and improving the overall stability of the roadbed. In recent years, this technology has been widely used in highway reconstruction and expansion projects in my country, achieving good results.

[0004] The successful application of pile-net composite foundation technology has provided both theoretical and practical support for highway reconstruction and expansion projects in my country, effectively improving the traffic capacity and service quality of highways and playing a significant role in ensuring national traffic safety and promoting healthy economic development. With the continuous improvement and maturation of related technologies, it is expected that more highway reconstruction and expansion projects will adopt this technology in the future, contributing even more to my country's transportation infrastructure construction.

[0005] Under dynamic loads such as vehicle loads, the dynamic response characteristics of new and old foundations with different stiffnesses differ significantly, potentially leading to fatigue damage or even cumulative failure in the connection area. However, existing design codes and methods often focus on static analysis, lacking mature and unified design criteria and quantitative analysis methods for such complex dynamic interactions. Therefore, traditional pile-net composite foundation design often treats the foundation as a homogeneous body, failing to consider the stiffness difference between new and old foundations. While this reduces design complexity, it can easily lead to excessive differential deformation, causing pavement cracking and posing a potential risk to the long-term safety of the project. Summary of the Invention

[0006] To address the aforementioned issues, this invention provides a design method for composite pile foundations for renovation and expansion projects based on the synergistic stiffness of new and old foundations. It incorporates stiffness differences as a core factor into the dynamic model, derives dynamic equations from the theory of vibrating elastic thin plates, fully considers the problems at both ends and connections of the new and old foundations, and reveals the crucial role of pile-soil coordinated stiffness through zonal modeling. This guides foundation design, reduces differential deformation, and solves the problems existing in the prior art.

[0007] The technical solution adopted in this invention is a design method for composite pile foundations for renovation and expansion based on the synergistic effect of stiffness between new and old foundations, comprising the following steps:

[0008] S1, the joint between the new foundation and the old foundation is divided into multiple regions according to the stiffness difference. The reinforced cushion layer is regarded as an elastic thin plate with bending stiffness. According to the thin plate vibration theory, the fourth-order non-homogeneous ordinary differential dynamic control equations of the cushion layer in each region are established respectively.

[0009] S2, set the boundary conditions and continuity conditions for the cushion layer in each region respectively;

[0010] S3, obtain the general solution of the dynamic response of the subbase in each region;

[0011] S4, combined with S2 and S3, yields the undetermined coefficients in the general solution. Substituting the determined undetermined coefficients back into the general solution expression, we obtain the displacement and internal force expressions for each region's cushion layer. By quantifying the impact of the stiffness difference between the old and new foundations through the displacement and internal force expressions, we can guide the design of the composite pile foundation for reconstruction and expansion.

[0012] Furthermore, S1 includes the following steps:

[0013] S11, Assuming the reinforced cushion layer is considered as a plane strain model, the vibration equations for the cushion layer plate in different regions are established:

[0014] (1)

[0015] (2)

[0016] (3)

[0017] (4)

[0018] In the formula, Indicates time, , , and The displacements are for the subbase plates in sections 1, 2, 3, and 4, respectively. , , and The bending stiffness of the cushion layer in sections 1, 2, 3 and 4 are respectively. Indicates the first Flexural stiffness of the subbase layer Indicates the first The composite elastic modulus of the section cushion layer, For the first Poisson's ratio of the subbase section Indicates the first Thickness of the subbase layer in the section Represents the Laplace operator. Indicates the first Viscous damping of the section cushion layer, For the first The density of the subbase layer, of which, The numbers l1, l2, l3, and l4 represent the first, second, third, and fourth sections, respectively. l1, l2, l3, and l4 represent different dividing points along the cross-sectional direction of the road, dividing the entire reinforced subbase into the first, second, third, and fourth sections. Represents the imaginary unit ; These represent the pile-soil coordinated dynamic impedance under the cushion layer in sections 1-4, respectively; x indicates the location.

[0019] , , and Their sizes are not equal. This represents the uniformly distributed traffic dynamic load acting on the old road area; This represents the uniformly distributed dynamic traffic load acting on the middle lane of the widened area; This represents the uniformly distributed dynamic traffic load acting on the outer lane of the widened area. This represents the static and non-uniform loads generated by embankment filling and soil arching effect;

[0020] S12, since the upper load is considered as a simple harmonic load, the entire system undergoes simple harmonic steady-state vibration, so equations (1)-(4) are simplified to:

[0021] (5)

[0022] (6)

[0023] (7)

[0024] (8)

[0025] In the formula, and They are respectively and The magnitude of the load; Indicates the circular excitation frequency; and These are the vibration amplitudes of the cushion layer in sections 1, 2, 3, and 4, respectively. .

[0026] Furthermore, S2 includes the following:

[0027] When point B is a strongly connected point... The boundary conditions for the point are:

[0028] (9)

[0029] (10)

[0030] The boundary continuity condition of a point is:

[0031] (11)

[0032] (12)

[0033] (13)

[0034] (14)

[0035] The boundary continuity condition of a point is:

[0036] (15)

[0037] (16)

[0038] (17)

[0039] (18)

[0040] The boundary continuity condition of a point is:

[0041] (19)

[0042] (20)

[0043] (twenty one)

[0044] (twenty two)

[0045] The boundary conditions for the point are:

[0046] (twenty three)

[0047] (twenty four);

[0048] in, This indicates the angular complex amplitude of the first section of the cushion layer. This indicates the angular complex amplitude of the second section of the cushion layer. This indicates the angular complex amplitude of the third section of the cushion layer. This indicates the angular complex amplitude of the subbase in section 4;

[0049] This represents the complex amplitude of the transverse shear force in the first section of the cushion layer. This indicates the complex amplitude of the transverse shear force in the second section of the cushion layer. This represents the complex amplitude of the transverse shear force in the third section of the cushion layer. This indicates the complex amplitude of the transverse shear force in the fourth section of the cushion layer;

[0050] This represents the complex amplitude of the bending moment of the first section of the cushion layer. This represents the complex amplitude of the bending moment of the second section of the cushion layer. This represents the complex amplitude of the bending moment of the third section of the cushion layer. This represents the complex amplitude of the bending moment of the cushion layer in section 4.

[0051] Furthermore, the solution method for the general solution of the dynamic response of the cushion layer in each region of S3 is as follows:

[0052] Equations (5)-(8) are written in homogeneous form. Since equations (5)-(8) are fourth-order non-homogeneous ordinary differential equations, the solution consists of a general solution and a particular solution. Let the particular solution be in the form of... , By adding the general solution of the homogeneous equation to the particular solution of the nonhomogeneous equation, we obtain the general solution of the fourth-order nonhomogeneous ordinary differential equation corresponding to equations (5)-(8), and thus obtain the displacement expression of the cushion layer in each region; based on the correlation between the internal force and dynamic response displacement of the vibration theory of elastic thin plate, we obtain the expressions of the rotation angle, bending moment and shear force of the cushion layer in the corresponding region according to the displacement expression of the cushion layer in each region.

[0053] in, and These represent the particular solutions of the fourth-order nonhomogeneous ordinary differential equations for the vibration equations of the cushion layer in sections 1, 2, 3, and 4, respectively. and These represent the dynamic amplification factors of the cushion layers in sections 1, 2, 3, and 4, respectively. and These represent the complex amplitudes of the external dynamic loads on the cushion layer in sections 1, 2, 3, and 4, respectively. It is a constant.

[0054] Furthermore, the general solution of the fourth-order nonhomogeneous ordinary differential equation corresponding to equation (5) is:

[0055] (25)

[0056] The general solution of the fourth-order nonhomogeneous ordinary differential equation corresponding to equation (6) is:

[0057] (26)

[0058] The general solution of the fourth-order nonhomogeneous ordinary differential equation corresponding to equation (7) is:

[0059] (27)

[0060] The general solution of the fourth-order nonhomogeneous ordinary differential equation corresponding to equation (8) is:

[0061] (28)

[0062] and All of these represent the undetermined coefficients of the general solution of the vibration equation for the first section of the cushion layer. All of these represent the undetermined coefficients of the general solution of the vibration equation for the second section of the cushion layer. All of these represent the undetermined coefficients of the general solution of the vibration equation for the cushion layer in section 3. All of these represent the undetermined coefficients of the general solution of the vibration equation for the cushion layer in section 4;

[0063] and These represent the complex wave numbers of the cushion layers in sections 1, 2, 3, and 4, respectively.

[0064] Furthermore, S4 includes the following steps:

[0065] S41, Substitute the general solution expression containing undetermined coefficients obtained in S3 into the boundary and continuity conditions of S3 to construct a system of equations about the undetermined coefficients;

[0066] S42, Solve the system of equations to determine all undetermined coefficients;

[0067] S43, substitute the determined undetermined coefficients back into the general solution expression to obtain the definite analytical solution of the dynamic response, that is, the displacement expression of the cushion layer in each region.

[0068] Furthermore, in S4, based on the correlation between the internal force and dynamic response displacement of the elastic thin plate vibration theory, the internal force expression of the corresponding region cushion layer is obtained according to the displacement expression of the cushion layer in each region.

[0069] The beneficial effects of this invention are:

[0070] This invention divides the joint between the old and new foundations into different regions based on the stiffness difference, establishes the dynamic control equations and boundary and continuity conditions of the cushion layer for each region, and then solves for the deformation and internal forces of the cushion layer. The deformation and internal forces can reflect the influence of the stiffness difference between the old and new foundations, thereby guiding the design of the pile foundation composite foundation for reconstruction and expansion.

[0071] This invention introduces stiffness difference as a core factor into the dynamic model, derives the dynamic equation from the theory of vibrating elastic thin plates, fully considers the problems at both ends and the connection between the new and old foundations, analyzes the dynamic interaction between them, and reveals the key role of pile-soil coordinated stiffness through zonal modeling. This guides foundation design, reduces differential deformation, and thus more effectively controls the differential settlement between the new and old roadbeds, ensuring the overall quality and long-term durability of the widening project. Attached Figure Description

[0072] 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.

[0073] Figure 1 This is a schematic diagram of the roadbed widening of the pile-net composite foundation according to an embodiment of the present invention.

[0074] Figure 2 This is a calculation model for pile-net composite foundation under different loads in an embodiment of the present invention.

[0075] Figure 3 It represents the displacement of the pad layer under different excitation frequencies in the embodiments of the present invention; where (a) is the real part and (b) is the imaginary part.

[0076] Figure 4 is the rotation angle of the pad layer under different excitation frequencies in the embodiments of the present invention; where (c) is the real part and (d) is the imaginary part.

[0077] Figure 5 is the bending moment of the pad under different excitation frequencies in the embodiments of the present invention; where (e) is the real part and (f) is the imaginary part.

[0078] Figure 6 is the shear force of the pad layer under different excitation frequencies in the embodiments of the present invention; where (g) is the real part and (h) is the imaginary part.

[0079] Figure 7 It represents the displacement of the cushion layer under different stiffnesses of the old and new foundations in the embodiments of the present invention; where (a) is the real part and (b) is the imaginary part.

[0080] Figure 8 , is the rotation angle of the cushion layer under different stiffnesses of new and old foundations in the embodiments of the present invention; where (c) is the real part and (d) is the imaginary part.

[0081] Figure 9 is the bending moment of the cushion layer under different stiffnesses of new and old foundations in the embodiments of the present invention; where (e) is the real part and (f) is the imaginary part.

[0082] Figure 10 It is the shear force of the cushion layer under different stiffnesses of new and old foundations in the embodiments of the present invention; where (g) is the real part and (h) is the imaginary part.

[0083] Figure 11 It is the displacement of the foundation layer under different new and old foundation layer modulus ratios in embodiments of the present invention; where (a) is the real part and (b) is the imaginary part.

[0084] Figure 12 It is the rotation angle of the foundation layer under different modulus ratios of new and old foundation layers in embodiments of the present invention; where (c) is the real part and (d) is the imaginary part.

[0085] Figure 13 , is the bending moment of the underlying subbase layer under different modulus ratios of new and old subbase layers in embodiments of the present invention; where (e) is the real part and (f) is the imaginary part.

[0086] Figure 14 It is the shear force of the foundation layer under different new and old foundation layer modulus ratios in embodiments of the present invention; where (g) is the real part and (h) is the imaginary part.

[0087] Figure 15 It is the displacement of the lower cushion layer relative to the thickness of the new and old foundation cushion layers in different embodiments of the present invention; where (a) is the real part and (b) is the imaginary part.

[0088] Figure 16 It is the angle of the lower cushion layer with different thickness ratios of new and old foundation cushion layers in embodiments of the present invention; where (c) is the real part and (d) is the imaginary part.

[0089] Figure 17 It is the bending moment of the lower cushion layer with different thickness ratios of new and old foundation cushion layers in embodiments of the present invention; where (e) is the real part and (f) is the imaginary part.

[0090] Figure 18 It is the shear force of the lower cushion layer with different thickness ratios of new and old foundation cushion layers in embodiments of the present invention; where (g) is the real part and (h) is the imaginary part.

[0091] Figure 19It represents the displacement of the lower cushion layer under different upper load ratios in embodiments of the present invention; where (a) is the real part and (b) is the imaginary part.

[0092] Figure 20 It refers to the rotation angle of the subbase layer under different upper load ratios in embodiments of the present invention; where (c) is the real part and (d) is the imaginary part.

[0093] Figure 21 is the bending moment of the subbase layer under different upper load ratios in embodiments of the present invention; where (e) is the real part and (f) is the imaginary part.

[0094] Figure 22 It is the shear force of the subbase layer under different upper load ratios in the embodiments of the present invention; where (g) is the real part and (h) is the imaginary part.

[0095] Figure 23 It represents the displacement of the padding layer under different splicing methods in the embodiments of the present invention; where (a) is the real part and (b) is the imaginary part.

[0096] Figure 24 These are the corners of the padding layer under different splicing methods in the embodiments of the present invention; where (c) is the real part and (d) is the imaginary part.

[0097] Figure 25 is the bending moment of the pad under different splicing methods in the embodiments of the present invention; where (e) is the real part and (f) is the imaginary part.

[0098] Figure 26 It represents the shear force of the padding layer under different splicing methods in the embodiments of the present invention; where (g) is the real part and (h) is the imaginary part.

[0099] Figure 27 This invention relates to the effects of different splicing methods on the foundation cushion layer under static conditions in an embodiment of the invention; where (a) is displacement, (b) is rotation angle, (c) is bending moment, and (d) is shear force. Detailed Implementation

[0100] 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.

[0101] like Figure 1 The diagram shows a real-world example of widening a pile-mesh composite foundation. The soil surrounding the piles is considered a homogeneous, single-phase continuous body covering rigid bedrock. Figure 2 The calculation models of the embankment subbase under different load conditions are analyzed. In this model, the embankment subbase is considered as a perfectly elastic thin plate, with the top of the plate subjected to harmonic loads from the vertical embankment. ( , Represents the embankment load amplitude. (Represents the circular excitation frequency). Symbol , , , These represent the coordinated dynamic impedance of the pile and soil under different slabs. Furthermore, it is stipulated that there is no relative sliding or separation between the pile and soil systems, satisfying the perfect contact condition.

[0102] S1, Establishment of the governing equations for pile-net composite foundation;

[0103] like Figure 2 As shown, the composite structure composed of the sand and gravel cushion layer and the reinforcing material is a cushion plate. Due to the presence of the reinforcing material, it has a certain bending stiffness. The cushion plate can be treated as a thin plate for mechanical analysis (such as thin plate vibration theory). Based on the thin plate vibration theory, the dynamic control equation for the displacement of the composite cushion layer is established. From Figure 2 Therefore, under the given geometric shape and load conditions, the reinforced cushion layer can be considered as a plane strain model. Thus, the vibration equations for each cushion layer plate (corresponding to the horizontal plates q1-q4 in the figure) are established as follows:

[0104] (1)

[0105] (2)

[0106] (3)

[0107] (4)

[0108] In the formula, Indicates time, , , and The displacements are for the subbase plates in sections 1, 2, 3, and 4, respectively. , , and The bending stiffness of the cushion layer in sections 1, 2, 3 and 4 are respectively. Indicates the first Flexural stiffness of the subbase layer Indicates the first The composite elastic modulus of the section cushion layer, For the first Poisson's ratio of the subbase section Indicates the first Thickness of the subbase layer in the section Represents the Laplace operator. Indicates the first Viscous damping of the section cushion layer, For the first The density of the subbase layer, of which, The numbers q1, q2, q3, and q4 represent sections 1, 2, 3, and 4, respectively, corresponding to the horizontal directions q1-q4. l1, l2, l3, and l4 represent different boundary points along the cross-section of the road, dividing the entire reinforced subbase into sections 1, 2, 3, and 4. 1 represents the old road section, 0 ≤ x ≤ l1; 2 represents the middle lane of the widened section, l1 < x ≤ l2; 3 represents the outer lane of the widened section, l2 < x ≤ l3; and 4 represents the embankment section, l3 < x ≤ l4. The subbase and subbase slab are essentially the same.

[0109] , , and Their sizes are not equal. Indicates the effect on the old road area Uniformly distributed traffic dynamic loads; Indicates that the action applies to the middle lane of the widened area. Uniformly distributed traffic dynamic loads; Indicates that the action applies to the outer lane of the widened area. Uniformly distributed traffic dynamic loads; This represents the static and non-uniform loads generated by the embankment fill and the arching effect. Due to the self-weight of the embankment fill and the arching effect, the load transferred to the top of the subbase is no longer uniform, but varies with position x. The embankment self-weight load can be considered as the static load component. Represents the imaginary unit ; - These represent the pile-soil coordinated dynamic impedances under the cushion layer in sections 1-4, respectively.

[0110] Due to the upper load Treating it as a simple harmonic load, the entire system undergoes simple harmonic steady-state vibration. This represents the circular excitation frequency; therefore, in equations (1) to (4) , , , ; , , , Let be the vibration amplitudes of the cushion layers in sections 1, 2, 3, and 4, respectively. Therefore, equations (1)-(4) can be simplified to:

[0111] (5)

[0112] (6)

[0113] (7)

[0114] (8)

[0115] In the formula, , For the sake of brevity This will be omitted in subsequent formulas.

[0116] S2, Boundary and Continuity Conditions:

[0117] according to Figure 2 At the junction of the old and new foundations, i.e., point B (where there is a strong connection), the boundary continuity conditions at each point are as follows:

[0118] exist Point boundary conditions:

[0119] (9)

[0120] (10)

[0121] This indicates the angular complex amplitude of the first section of the cushion layer. This represents the complex amplitude of the transverse shear force in the first section of the cushion layer.

[0122] exist Boundary continuity condition of a point:

[0123] (11)

[0124] (12)

[0125] (13)

[0126] (14)

[0127] This indicates the angular complex amplitude of the second section of the cushion layer. This indicates the complex amplitude of the transverse shear force in the second section of the cushion layer. This represents the complex amplitude of the bending moment of the first section of the cushion layer. This represents the complex amplitude of the bending moment of the second section of the cushion layer.

[0128] exist Boundary continuity condition of a point:

[0129] (15)

[0130] (16)

[0131] (17)

[0132] (18)

[0133] This indicates the angular complex amplitude of the third section of the cushion layer. This represents the complex amplitude of the bending moment of the third section of the cushion layer. This represents the complex amplitude of the transverse shear force in the third section of the cushion layer.

[0134] exist Boundary continuity condition of a point:

[0135] (19)

[0136] (20)

[0137] (twenty one)

[0138] (twenty two)

[0139] This indicates the angular complex amplitude of the subbase in section 4. This represents the complex amplitude of the bending moment of the fourth section of the cushion layer. This represents the complex amplitude of the transverse shear force in the fourth section of the cushion layer.

[0140] exist Point boundary conditions:

[0141] (twenty three)

[0142] (twenty four)

[0143] Point O is located on the far left of the model, which is the left end point of the old road area subbase slab and also the origin of the coordinate system, corresponding to the starting position of the old road area load; Point A is located at the boundary between the old road area and the widening area, which is the right end point of the old road area subbase slab and also the left end point of the widening area load; Point B is located inside the widening area and is the boundary point of the load at the splice; Point C is located on the right side of the widening area, which is the right end point of the new road area load and also the left end point of the linear distributed load; Point D is located on the far right of the model, which is the right end point of the subbase slab and also the end point of the linear distributed load.

[0144] S3, General solution for dynamic response of subbase:

[0145] Equation (5) can be written in homogeneous form as follows:

[0146] (25)

[0147] This indicates the bending stiffness of the padding layer.

[0148] The characteristic equation corresponding to equation (25) is:

[0149] (26)

[0150] These are characteristic roots introduced when solving the differential equation for the vibration of the cushion layer in section 1.

[0151] Then equation (26) can be rewritten as:

[0152] (27)

[0153] The general solution of the fourth-order homogeneous ordinary differential equation corresponding to equation (5) is:

[0154] (28)

[0155] , , and The coefficients to be determined in the general solution of the vibration equation of the first section cushion layer are represented by the coefficients to be determined by the boundary conditions of the system. This indicates the complex wave number of the first section of the cushion layer.

[0156] Meanwhile, equation (5) is a fourth-order nonhomogeneous ordinary differential equation, the form of which consists of a general solution and a particular solution. Let the form of the particular solution be... Substituting into equation (5), we get:

[0157] (29)

[0158] This represents a particular solution of the fourth-order nonhomogeneous ordinary differential equation representing the vibration equation of the first section of the cushion layer. This indicates the dynamic amplification factor of the first section of the subbase; It is the complex amplitude of the external dynamic load on the first section of the cushion layer.

[0159] The general solution of the fourth-order non-homogeneous ordinary differential equation corresponding to equation (5) is composed of the general solution of the homogeneous equation plus the particular solution of the non-homogeneous equation:

[0160] (30)

[0161] and They are essentially the same. It is a function representing the position x in the complete solution.

[0162] Therefore, based on the above general solution, the general solutions of formulas (6), (7), and (8) can be obtained similarly.

[0163] The homogeneous equations corresponding to formulas (6), (7), and (8) are:

[0164] (31)

[0165] (32)

[0166] (33)

[0167] The corresponding characteristic equation is:

[0168] (34)

[0169] (35)

[0170] (36)

[0171] , and These represent the characteristic roots introduced when solving the vibration differential equations for sections 2, 3, and 4 of the cushion layer, respectively.

[0172] make , , .

[0173] , and These represent the complex wave numbers of the cushion layers in sections 2, 3, and 4, respectively.

[0174] The general solution of the corresponding fourth-order homogeneous ordinary differential equation is:

[0175] (37)

[0176] (38)

[0177] (39)

[0178] Let the particular solutions of equations (6), (7), and (8) take the form of , , Substituting into equations (6), (7), and (8), we get:

[0179] (40)

[0180] (41)

[0181] (42)

[0182] (43)

[0183] Combining equations (40) to (43), we can obtain the general solutions of the fourth-order non-homogeneous ordinary differential equations corresponding to equations (6), (7), and (8), which consist of the general solution of the homogeneous equation plus the particular solution of the non-homogeneous equation:

[0184] (44)

[0185] (45)

[0186] (46)

[0187] , , , All of these represent the undetermined coefficients of the general solution of the vibration equation for the second section of the cushion layer. , , , All of these represent the undetermined coefficients of the general solution of the vibration equation for the cushion layer in section 3. , , , All of these represent the undetermined coefficients of the general solution of the vibration equation for the cushion layer in section 4, and their specific values ​​are determined by the boundary conditions of the system.

[0188] , and These represent the dynamic amplification factors of the cushion layer in sections 2, 3, and 4, respectively. and These represent the complex amplitudes of the external dynamic loads on the cushion layers of sections 2 and 3, respectively; constant term This is part of the special solution of the 4th section cushion layer.

[0189] In the vibration theory of elastic thin plates, the correlation between internal forces and displacements can be expressed as:

[0190] (47)

[0191] This represents the complex amplitude of the cross-sectional rotation of the cushion layer. The complex amplitude represents the lateral displacement (deflection) of the cushion layer. This represents the complex amplitude of the bending moment of the subbase. Indicates the flexural stiffness of the cushion layer. This indicates the complex amplitude of the transverse shear force of the subbase layer; and They are essentially the same. Represents a function with respect to position x.

[0192] In the formula, ; ; ; .

[0193] The specific expressions for the internal forces of each cushion layer are as follows:

[0194] exist hour:

[0195] (48)

[0196] (49)

[0197] (50)

[0198] exist hour:

[0199] (51)

[0200] (52)

[0201] (53)

[0202] exist hour:

[0203] (54)

[0204] (55)

[0205] (56)

[0206] exist hour:

[0207] (57)

[0208] (58)

[0209] (59)

[0210] , , and These represent the flexural stiffness of the cushion layer in sections 1, 2, 3, and 4, respectively.

[0211] S4, connecting the boundary continuity conditions (9)~(24), the displacement expressions (30), (44)~(46) of each cushion layer, and the internal force expressions (48)~(59), construct a system of equations about the undetermined coefficients (i.e., the following matrix of undetermined coefficients):

[0212] (60)

[0213] In the formula,

[0214]

[0215]

[0216]

[0217]

[0218]

[0219]

[0220]

[0221]

[0222]

[0223]

[0224]

[0225]

[0226]

[0227]

[0228]

[0229]

[0230]

[0231]

[0232]

[0233]

[0234]

[0235]

[0236] All other elements in equation (60) are 0.

[0237] The undetermined coefficients are determined by solving the matrix of undetermined coefficients. - These undetermined coefficients are integral constants in the displacement function of the cushion layer, determining the specific forms of displacement and internal forces. Once these constants are determined, substituting them back into the general solution expression can completely determine the displacement distribution (such as deflection and rotation) and internal force distribution (such as bending moment and shear force) of the cushion layer throughout the entire domain. The matrix equation ensures the continuity of physical quantities such as displacement, rotation, bending moment, and shear force at the junctions of different cushion layers (i.e., the boundary continuity condition). By solving this equation, it is possible to ensure that the mechanical behavior of the entire system is consistent at the boundary, avoiding unreasonable jumps or discontinuities, calculating the response of multi-layer cushion layers under load, and thus guiding the design of pile foundation composite foundations for renovation and expansion.

[0238] Calculation example:

[0239] The proposed solution was validated, and the dynamic response characteristics of the pile-net composite foundation under spliced ​​subgrade loading were revealed through numerical example analysis. The excitation frequency was dimensionless. Unless otherwise specified, the parameter values ​​used can be found in Table 1 ( ).

[0240] Table 1 Material / Geometric Parameters

[0241]

[0242] Figures 3-6 The curves showing the variation of displacement, rotation angle, bending moment, and shear force of each cushion layer under different excitation frequencies are presented in the figure. It can be observed that as the excitation frequency increases, the amplitudes of displacement, rotation, bending moment, and shear force gradually increase. Therefore, this indicates that within a certain range, an increase in load leads to an increase in differential settlement. It is noteworthy that... Figure 3 As can be seen from (a) and (b), the maximum displacement occurs Nearby, this indicates significant differential settlement at the joint, forming a "reverse bending basin" phenomenon, and the overall displacement tends to decrease with increasing excitation frequency. Additionally, in , , At this location, the continuous displacement image corresponding to the continuity condition equations (11), (15), and (19) of the pile-net composite foundation is obtained.

[0243] The peak value of the rotation angle of the reinforced cushion increases with the increase of the excitation frequency, such as... Figure 4 As shown in (c) and (d), this indicates that due to different pile-soil stiffness, the unevenness of pile-soil settlement is significant within a certain range of excitation frequency, and the difference in rotation angle will increase. Furthermore, in At this point, the angle is 0, which conforms to the continuity boundary condition equation (9).

[0244] The bending moment of the subbase shows a trend of increasing peak value with increasing excitation frequency, such as... Figure 5 As shown in (e) and (f), and appears in Nearby. This is because the new road has a better load-bearing capacity than the old road, which suffers from insufficient load-bearing capacity; in The value is 0 at this point, which is consistent with the boundary conditions of equation (23). Furthermore, the bending moment is observed at... , , The continuity conditions (13), (17), and (21) are satisfied.

[0245] The peak shear force of the cushion layer gradually increases within a certain range as the excitation frequency increases, such as... Figure 6 As shown in (g) and (h), this is due to the different stiffness of the old and new road sections. This difference leads to uneven settlement of the subbase, generating enormous shear forces to resist this deformation. It can be seen that the shear force... , , Continuous, in and The value is zero at the given point, which is consistent with the boundary conditions of equations (10) and (24) and the continuity conditions of equations (14), (18) and (22). The validity of the current solution is also confirmed by the compatibility of the above results with the system boundary conditions and continuity.

[0246] Figures 7-10 This demonstrates the effect of the subgrade layer on the difference in stiffness between the old and new foundation layers. From Figure 7 Figures (a) and (b) show a positive correlation between differential settlement and stiffness difference. The figures indicate that settlement of the new foundation mainly occurs near the joint, while settlement of the old foundation is smaller. Differential settlement is more significant when the stiffness of the new foundation is less than that of the old foundation; however, once the new and old foundations reach the same stiffness, further increasing the stiffness of the new foundation to make it greater than that of the old foundation does not decrease differential settlement. Therefore, the greater the stiffness difference between the two foundations, the more significant the differential settlement between the new and old foundations.

[0247] The peak value of the rotation angle of the cushion layer first decreases and then increases with the increase of stiffness difference, such as... Figure 8 As shown in (c) and (d), this indicates that due to the different pile-soil stiffness of the old and new foundations, the uneven settlement of the pile and soil occurs. When the stiffness difference reaches a certain magnitude, the old and new foundations tend to be in the same state, and the difference in rotation angle will decrease. In addition, at the edge of the new foundation, as the stiffness of the new foundation increases, the rotation angle value also decreases, indicating that it is less prone to deflection.

[0248] The bending moment of the cushion layer shows a trend of decreasing peak value as the stiffness difference increases, such as... Figure 9 As shown in (e) and (f), as the difference in stiffness between the old and new foundations increases, the peak value within the old foundation gradually decreases, and fluctuations occur near the foundation junction. This is because the soil around the piles is soft soil with insufficient bearing capacity. This indicates that the side load of the new foundation will change the stress distribution of the old foundation, and the greater the difference in stiffness, the more obvious the gradient change of the bending moment.

[0249] As the difference in stiffness between the old and new foundations increases, the shear stress at the joint of the foundation layer will increase significantly, such as... Figure 10 As shown in (g) and (h), 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. At the same time, the bonding effect of the splice interface itself is weak, making it more prone to shear slip under external loads, further amplifying the increasing trend of shear force.

[0250] Figures 11-14 This demonstrates the influence of different modulus ratios of new and old foundation layers on the dynamic response of the foundation layer. From Figure 11 (a) and (b) in the text, and Figure 12 As can be seen from (c) and (d) in the figure, the change in modulus difference has little effect on differential settlement and rotation angle. Figure 13 Figures (e) and (f) show that the peak value of the bending moment increases with the increase of the modulus difference; this is because the soil around the pile is soft soil with insufficient bearing capacity; the larger the modulus of the new road subgrade, the greater the stiffness of the new road and the less prone it is to deformation. Therefore, the greater the stiffness difference between the new foundation and the old road, the more obvious the gradient change of the bending moment. At the same time, to reduce this differential deformation, shear force is generated inside the subgrade to resist this deformation, thus the peak value of the shear force increases with the increase of the modulus. Figure 14 As can be seen from (g) and (h) in the text.

[0251] Figures 15-18This figure illustrates the effect of the thickness ratio of the new and old foundation layers on the foundation layer itself. As can be seen from the figure, changes in the thickness ratio have a relatively small impact on differential settlement, but a significant impact on rotation angle, bending moment, and shear force. This is because the greater the thickness of the foundation layer in the new foundation, the greater its stiffness and the less prone it is to deformation. Therefore, the greater the stiffness difference between the new foundation and the old road, the greater the internal force required within the foundation layer to resist this differential deformation. Consequently, the peak values ​​of bending moment and shear force increase with the increase of the modulus.

[0252] Figures 19-22 This illustrates the impact of different superstructure loads on the foundation cushion layer. It can be seen that the greater the difference in superstructure loads between the old and new foundations, the more pronounced the difference in their dynamic responses. As the load difference increases, the dynamic response increases; this is because a greater difference in superstructure loads means the old and new foundations bear different loads, resulting in different deformations. To resist these deformations, the spliced ​​cushion layer needs to generate larger internal forces, thus increasing the peak bending moment and shear force with increasing modulus.

[0253] Figures 23-26 This illustrates the impact of different splicing methods on the foundation cushion layer. From... Figure 23 (a) and (b) in the text, and Figure 24 As shown in (c) and (d), the strongly connected approach exhibits greater stability, while the unconnected approach results in greater displacement. This implies that the strongly connected approach can effectively reduce differential settlement at the joint. Furthermore, compared to the weakly connected approach, the strongly connected approach can effectively reduce the rotation angle of the old subbase and reduce the rotation angle difference at the joint. Figure 25 (e) and (f) in the middle, and Figure 26 Figures (g) and (h) show that the peak values ​​of shear force and bending moment coincide with the peak values ​​of displacement, indicating a correlation between them. Furthermore, the internal force response is significantly stronger when the connection is strong compared to when there is no connection. This suggests that when the old and new subbase layers are forcibly connected, the entire subbase layer generates substantial internal forces to resist deformation.

[0254] Figure 27 This illustrates the impact of different splicing methods on the foundation cushion layer under static high-fill conditions. From Figure 27 As shown in (a) and (b), the strong connection method exhibits greater stability, which is largely consistent with the dynamic situation. Furthermore, compared to the weak connection method, the strong connection effectively reduces the rotation angle of the old pad layer and reduces the rotation angle difference at the splice. Figure 27 The trends shown in (c) and (d) are also similar to those under dynamic conditions. This indicates that when the old and new subgrade layers are forcibly connected, a large internal force is generated within the entire subgrade layer to resist deformation. Therefore, it can be seen that connecting the old and new subgrade layers, whether under dynamic or static conditions, can effectively reduce differential settlement at the joint.

[0255] Existing technologies lack zonal modeling. Traditional pile-net composite foundation designs often treat the foundation as a homogeneous body, failing to consider the stiffness difference between the old and new foundation layers. This invention introduces stiffness difference as a core factor into the dynamic model. Based on the theory of vibrating elastic thin plates, the dynamic deflection and internal force expressions of the cushion layer are derived and solved. Furthermore, numerical examples are used to analyze the dynamic variations of displacement, rotation, bending moment, and shear force of the cushion layer. Simultaneously, the dynamic variations of displacement, rotation, bending moment, and shear force at the joint between the old and new cushion layers are systematically analyzed under different important design parameters such as elastic modulus, thickness, and pile-soil coordinated stiffness. This invention treats the boundary and continuity conditions of different regions according to stiffness differences between the old and new foundation layers. This increases the difficulty of model building, and the handling of boundary conditions involves complex mathematical techniques. The coupling effect at the joint is not easily understood intuitively. Stiffness difference affects dynamic response, but multiple parameters such as elastic modulus, thickness, and pile-soil coordinated stiffness are coupled, and their dynamic relationships are difficult to infer through simple experience.

[0256] This invention reveals the core role of pile-soil coordinated stiffness, a key parameter, in dynamic load transfer. For example, considering displacement changes... Figure 7 (a) and (b) in the text. / The ratio range is 0.75-1.25. If the ratio is too large or too small, the displacement change will be large (the differential settlement will be more significant). In actual design, the specific limit of differential settlement for the project should be combined with the method of the embodiment of the present invention to perform inversion analysis and determine an optimal stiffness ratio range for the project, so as to guide the design of the pile foundation composite foundation for renovation and expansion.

[0257] 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 design method for a reconstruction pile composite foundation based on the stiffness coordination of new and old foundations, characterized in that, It comprises the following steps: S1, the new foundation and the old foundation lap joint place is divided into several areas according to the stiffness difference, the reinforced cushion is regarded as the elastic thin plate with bending stiffness, and the four-order non-homogeneous ordinary differential dynamic control equation of each area cushion is established according to the thin plate vibration theory; S2, the boundary condition and the continuity condition of each area cushion are respectively set; S3, the general solution of the dynamic response of each area cushion is obtained; S4, the undetermined coefficients in the general solution are obtained by combining S2 and S3, the determined undetermined coefficients are substituted back into the general solution expression, and then the displacement expression and the internal force expression of each area cushion are obtained, the influence of the stiffness difference between the new and old foundations is quantified through the displacement and internal force expressions, so as to guide the design of the pile foundation composite foundation of the reconstruction and expansion; The S1 comprises the following steps: S11, the reinforced cushion is assumed to be a plane strain model, and the vibration equation of the cushion plate in different areas is established: （1） （2） （3） （4） wherein, denotes time, , , and are the displacements of the 1st, 2nd, 3rd and 4th segment mat, respectively, , , and are the bending stiffness of the 1st, 2nd, 3rd and 4th segment mat, respectively; denotes the bending stiffness of the segment mat, denotes the complex elastic modulus of the segment mat, is the Poisson's ratio of the segment mat, denotes the thickness of the segment mat, denotes the Laplacian operator, denotes the viscous damping of the segment mat, is the density of the segment mat, wherein, , respectively, denote the 1st, 2nd, 3rd and 4th segment, and l1, l2, l3 and l4 represent different demarcation points along the cross-sectional direction of the road, dividing the whole reinforced mat into the 1st, 2nd, 3rd and 4th segment; denotes the imaginary unit ; - denote the pile-soil coordinated dynamic impedance under the 1st-4th segment mat, respectively; x denotes the position; , , and are not equal in size, represents the uniform traffic dynamic load acting on the old road area; represents the uniform traffic dynamic load acting on the middle lane of the widened area; represents the uniform traffic dynamic load acting on the outer lane of the widened area; represents the static and non-uniform load generated by embankment fill and soil arching effect; S12, since the upper load is regarded as a simple harmonic load, the whole system does simple harmonic steady vibration, so that formula (1)-(4) are simplified as: （5） （6） （7） （8） wherein , , and are the amplitudes of the load; , , and respectively; denotes the circular excitation frequency; , , and are the amplitudes of the vibration of the 1st, 2nd, 3rd and 4th section mat respectively; , ; The solving method of the general solution of the dynamic response of each area cushion in the S3 is: The formula (5)-(8) is written into homogeneous equation form, and since the formula (5)-(8) is a fourth-order non-homogeneous ordinary differential equation, the solution form is composed of a general solution and a particular solution, and the form of the particular solution is , , , The general solution of the homogeneous equation is added to the particular solution of the non-homogeneous equation to obtain the general solution of the fourth-order non-homogeneous ordinary differential equation corresponding to the formula (5)-(8), so as to obtain the displacement expression of each area cushion layer; based on the correlation between the internal force and the dynamic response displacement of the elastic thin plate, the expressions of the rotation angle, the bending moment and the shear force of the corresponding area cushion layer are obtained according to the displacement expression of each area cushion layer; where, , , and represent the particular solution of the fourth order non-homogeneous ordinary differential equation of the 1st, 2nd, 3rd and 4th segment mat foundation vibration equation respectively; , , and represent the dynamic magnification factor of the 1st, 2nd, 3rd and 4th segment mat foundation respectively; , and represent the complex amplitude of the external dynamic load of the 1st, 2nd and 3rd segment mat foundation respectively; is a constant.

2. The design method of the reconstruction pile composite foundation based on the stiffness coordination of the old and new foundations according to claim 1, characterized in that, The S2 comprises the following contents: The B point position is in strong connection, The boundary condition of the point is: （9） （10） The boundary continuity condition of a point is: （11） （12） （13） （14） The boundary continuity condition of a point is: （15） （16） （17） （18） The boundary continuity condition of a point is: （19） （20） （21） （22） The boundary conditions for the points are: （23） （24）； wherein, represents the corner complex amplitude of the 1st section pad, represents the corner complex amplitude of the 2nd section pad, represents the corner complex amplitude of the 3rd section pad, represents the corner complex amplitude of the 4th section pad; represents the transverse shear complex amplitude of the 1st section pad, represents the transverse shear complex amplitude of the 2nd section pad, represents the transverse shear complex amplitude of the 3rd section pad, represents the transverse shear complex amplitude of the 4th section pad; M1represents the bending moment complex amplitude of the first section mat layer, M2represents the bending moment complex amplitude of the second section mat layer, M3represents the bending moment complex amplitude of the third section mat layer, M4represents the bending moment complex amplitude of the fourth section mat layer; The O point is the starting position of the load in the old road area, and is also the coordinate origin; the A point is located at the boundary between the old road area and the widening area; the B point is located inside the widening area, and is the boundary point of the splicing load; the C point is the right end point of the load in the new road area; and the D point is the right end point of the cushion plate.

3. The design method of the reconstruction pile composite foundation based on the stiffness coordination of the old and new foundations according to claim 1, characterized in that, The general solution of the four-order non-homogeneous ordinary differential equation corresponding to formula (5) is: （25） The general solution of the four-order non-homogeneous ordinary differential equation corresponding to formula (6) is: （26） The general solution of the four-order non-homogeneous ordinary differential equation corresponding to formula (7) is: （27） The general solution of the four-order non-homogeneous ordinary differential equation corresponding to formula (8) is: （28） , , and all represent undetermined coefficients of the general solution of the vibration equation of the first section pad, , , , all represent undetermined coefficients of the general solution of the vibration equation of the second section pad, , , , all represent undetermined coefficients of the general solution of the vibration equation of the third section pad, , , , all represent undetermined coefficients of the general solution of the vibration equation of the fourth section pad; , , and denote the complex wave numbers of the 1st, 2nd, 3rd and 4th segment mat layers, respectively.

4. The design method of the reconstruction pile composite foundation based on the stiffness coordination of the old and new foundations according to claim 1, characterized in that, The S4 comprises the following steps: S41, the general solution expression containing the undetermined coefficients obtained by S3 is substituted into the boundary and continuity conditions of S3, and an equation group about the undetermined coefficients is constructed; S42, the equation group is solved to determine all the undetermined coefficients; S43, the determined undetermined coefficients are substituted back into the general solution expression, so that the determined analytical solution of the dynamic response, i.e., the displacement expression of each area cushion, is obtained.

5. The design method of the reconstruction pile composite foundation based on the stiffness coordination of the old and new foundations according to claim 4, characterized in that, In the S4, based on the correlation between the internal force and the dynamic response displacement of the elastic thin plate vibration theory, the internal force expression of the corresponding area cushion is obtained according to the displacement expression of each area cushion.

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