An analytical method for horizontal displacement of free-end pile foundations in a dual-parameter foundation model

Abstract

The present invention provides a method for analyzing the horizontal displacement of free-end pile foundations in a dual-parameter foundation model. The boundary conditions of pile foundations such as friction piles and isolation piles can be simplified to free-end pile foundations. Combined with the Fourier series method, a method for analyzing the horizontal displacement of free-end pile foundations in a dual-parameter foundation model is proposed. The method is used to calculate the horizontal displacement and internal force of pile foundations caused by shield construction adjacent to pile foundations. Compared with the numerical simulation method, the physical quantities required for calculation can be obtained by substituting the corresponding calculation parameters. The calculation process is simple. Compared with the theoretical method using the numerical method, the method only needs to solve the horizontal displacement of the pile end. w O 、 w L The related two-variable linear equation can be used to further obtain other unknown coefficients, which has high calculation efficiency and provides an effective theoretical method for the safety risk assessment of existing structures in shield construction adjacent to pile foundations.

Description

Technical Field

[0001] The present invention relates to the technical field of tunnel engineering design, and in particular to a method for analyzing horizontal displacement of free pile foundations at both ends of a dual-parameter foundation model. Background Art

[0002] In the context of large-scale urban rail transit construction, new shield tunnels inevitably involve construction adjacent to existing pile foundations. This can cause deformation in the pile foundations, impacting the normal operation and structural safety of the existing superstructure. Therefore, it is necessary to conduct a safety risk assessment of shield tunnel construction adjacent to pile foundations to provide a basis for optimizing design and construction plans. Accurately calculating the deformation of adjacent pile foundations caused by shield construction is key to safety risk assessments of surrounding buildings.

[0003] Methods for calculating the horizontal displacement of single piles caused by adjacent shield tunnel construction primarily include analytical methods and numerical simulations. The analytical method is primarily based on the elastic foundation beam model and the Winkler foundation model. This model simplifies the pile foundation into an elastic foundation beam, subject to the additional loads generated by tunnel excavation, and calculates the foundation reaction using the Winkler assumption. The Winkler foundation model has limitations in reflecting soil continuity and shear properties, resulting in significant deviations from actual results. Two-parameter foundation models, such as the Pasternak foundation model and the Vlazov foundation model, address this issue better.

[0004] Existing research on the calculation of pile foundation lateral displacements caused by shield tunneling under dual-parameter foundation models primarily relies on numerical methods such as the finite difference method, while analytical approaches are relatively limited. In an existing literature, Sun Yingjie, Shi Chenghua, Wang Zuxian, et al. present an analytical solution to the tunnel construction-induced response of adjacent pile foundations, taking into account the actual constraints at both ends. To reflect the actual constraints at both ends of the pile foundation and to account for the shear effect of the pile foundation, the pile foundation is simplified as a Timoshenko beam with generalized elastic constraints in a Vlasov foundation, and a simplified tunnel-soil-pile foundation calculation model is established. Based on the elastic foundation beam theory, a two-stage method is used to derive analytical solutions for the additional lateral deformation and internal forces of the pile body under arbitrary boundary constraints during adjacent tunnel construction. The analytical solution is compared with boundary element and finite element numerical solutions to verify the reliability of the analytical model. Finally, a simplified calculation method for the actual constraints at both ends of the pile foundation is proposed, and the influence of the pile top and pile end constraints on the lateral deformation and internal forces of the pile foundation is further analyzed. However, the calculation method obtained is based on a numerical method and no corresponding analytical calculation formula is obtained. Compared with the calculation method using the numerical method, the analytical method has higher calculation efficiency and more stable algorithm. Therefore, it is necessary to conduct research on the calculation method of the horizontal displacement of the pile foundation adjacent to the shield tunnel construction, and give the analytical calculation formula and calculation steps for the horizontal displacement of the free pile foundations at both ends caused by the shield tunnel construction in a dual-parameter foundation model. Summary of the Invention

[0005] The purpose of the present invention is to provide an analytical method for the horizontal displacement of free-end pile foundations at both ends of a dual-parameter foundation model. Considering that the displacement constraints of the pile ends of friction piles, isolation piles, etc. can be simplified to the case of free ends, according to the stratum displacement caused by shield tunnel excavation, based on the dual-parameter foundation model theory and combined with the basic physical and mechanical parameters of the pile foundation and soil, the calculation formulas for the horizontal displacement and internal force of existing single piles caused by adjacent shield construction are derived.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is: a method for analyzing the horizontal displacement of free pile foundations at both ends of a dual-parameter foundation model, the calculation method comprising the following steps:

[0007] Step 1: Use the Fourier series method to solve the bending deformation of the pile foundation caused by shield tunnel excavation. The dual-parameter foundation model adopts the Vlasov foundation model. The existing pile foundation is assumed to be an elastic foundation beam, satisfying the plane section assumption. The differential equation governing the horizontal displacement of the existing pile foundation caused by shield tunnel excavation is:

[0008]

[0009] Where: E is the elastic modulus of the existing pile foundation; I is the interface moment of inertia of the existing pile foundation; w is the horizontal displacement of the existing pile foundation, with the direction away from the tunnel excavation as positive; z is the coordinate along the axis of the existing pile foundation, with the top of the pile as the origin z = 0; t is the load transfer rate; B is the diameter of the pile foundation; k is the stiffness coefficient of the soil around the pile; U is the horizontal displacement of the stratum at the pile position caused by shield tunnel excavation;

[0010] Step 2: Determine the calculation formula for the horizontal displacement of the free pile foundations at both ends caused by shield tunnel excavation:

[0011]

[0012] Where w n is the sine series term of the pile foundation horizontal displacement function; L is the calculated length of the pile foundation; w O 、w L are the horizontal displacements of the pile base surface end and the terminal pile end, respectively;

[0013] Step 3: Determine the calculation formula for the bending moment and shear force of the free pile foundations at both ends caused by shield tunnel excavation:

[0014]

[0015] Where, M is the pile foundation bending moment; Q is the pile foundation shear force;

[0016] Step 4: The expressions of pile foundation horizontal displacement w, pile foundation bending moment M, and pile foundation shear force Q in steps 1-3 contain the unknown coefficient w. n 、w O、w L , where the unknown coefficient w n Determined by the following formula:

[0017] w n =α n w O +β n w L +γ n

[0018] Where, α n , β n , γ n is the comprehensive coefficient;

[0019] Undetermined coefficient w O 、w L Determined by the following formula:

[0020]

[0021] Where S1~S6 are comprehensive coefficients.

[0022] Preferably, for the Vlasov foundation model, there are: Among them, Es is the elastic modulus of soil; υs is the Poisson's ratio of soil; He is the thickness of the foundation elastic layer, which is 2.5 times the pile diameter D; h is the function of displacement along the horizontal direction, which is

[0023] Preferably, for the Vlasov foundation model, there are:

[0024] Preferably, for the Vlasov foundation model, the analytical expression of soil displacement field proposed by Loganathan is adopted:

[0025]

[0026] Where ε0 is the ground loss rate; x0 is the horizontal distance from the pile foundation axis to the tunnel axis; R is the tunnel excavation radius; and H is the burial depth of the tunnel center.

[0027] Preferably, the comprehensive coefficients αn, βn, and γn are determined by the following formula:

[0028]

[0029] in, The known functions The coefficients of the sine series expansion are U n is the series coefficient of the sine series expansion of the known formation displacement function U(z), that is,

[0030] Preferably, the comprehensive coefficients S1 to S6 are determined by the following formula:

[0031]

[0032] The beneficial effects of the present invention are:

[0033] This scheme can simplify the boundary conditions of pile foundations such as friction piles and isolation piles into pile foundations with free ends at both ends. Combined with the Fourier series method, an analytical method for the horizontal displacement of free-end pile foundations of a dual-parameter foundation model is proposed to calculate the horizontal displacement and internal force of pile foundations caused by shield construction adjacent to pile foundations. Compared with the numerical simulation method, the required physical quantities can be obtained by substituting the corresponding calculation parameters, and the calculation process is simple. Compared with the theoretical method with the help of numerical methods, this method only needs to solve the two-variable linear equation related to the pile end horizontal displacement wO and wL to further obtain other unknown coefficients. The calculation efficiency is high, which provides an effective theoretical method for the safety risk assessment of existing structures under shield construction adjacent to pile foundations. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0035] Figure 1 This is a simplified diagram for calculating the horizontal displacement of existing pile foundations caused by the construction of tunnels adjacent to pile foundations in an embodiment of the present invention.

[0036] Figure 2 The figure is a comparison chart of the pile foundation horizontal displacement calculated in the embodiment of the present invention and the calculation results in the existing literature.

[0037] Figure 3 The figure is a comparison chart of the pile foundation bending moment calculated in the embodiment of the present invention and the calculation results in the existing literature.

[0038] Figure 4 The figure is a comparison chart of the pile foundation shear force calculated in the embodiment of the present invention and the calculation results in the existing literature. DETAILED DESCRIPTION

[0039] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0040] The present invention discloses a method for analyzing the horizontal displacement of free pile foundations at both ends of a dual-parameter foundation model. In order to make the purpose of this application and the technical solution clearer, the existing literature, namely the analytical solution of the adjacent pile foundation response induced by tunnel construction considering the actual constraints at both ends, is combined with the data, calculation parameters and calculation diagram as shown in the following figure. Figure 1 As shown, this application is further described.

[0041] After the tunnel is constructed near the existing pile foundation, the stratum loss rate ε0 is 2.5%, the soil elastic modulus Es is 24 MPa, the Poisson's ratio υs is 0.5, the tunnel diameter D is 6 m, the pile foundation elastic modulus E is 30 GPa, the pile diameter B is 0.5 m, the pile length L is 25 m, the tunnel center burial depth H is 20 m, and the horizontal distance x0 between the tunnel axis and the pile foundation axis is 4.5 m. The calculation diagram is as follows: Figure 1 As shown, both ends of the pile foundation are free ends.

[0042] Step 1: For the pile foundation boundary conditions and load conditions described in this example, the pile foundation is considered as an elastic foundation beam on the Vlasov foundation model. The differential equation governing the horizontal displacement of the existing pile foundation caused by tunnel excavation is:

[0043]

[0044] Where: E is the elastic modulus of the existing pile foundation, Pa; I is the interface moment of inertia of the existing pile foundation, m4; w is the horizontal displacement of the existing pile foundation, m, with the direction away from the tunnel excavation side as positive; z is the coordinate along the axis of the existing pile foundation, m, with the pile top as the origin z = 0; B is the diameter of the pile foundation, m; t is the load transfer rate, N / m.

[0045] For the Vlasov foundation model, we have: Among them, Es is the elastic modulus of the soil; υs is the Poisson's ratio of the soil; He is the thickness of the foundation elastic layer, which is 2.5 times the pile diameter D; h is the function of the displacement along the horizontal direction, i.e., the x direction, which is

[0046] k is the stiffness coefficient of the soil around the pile, Pa / m. For the Vlasov foundation model, we have:

[0047] U is the horizontal displacement of the ground at the pile position caused by shield tunnel excavation, and the analytical formula for soil displacement field proposed by Loganathan is adopted:

[0048]

[0049] Where ε0 is the stratum loss rate; x0 is the horizontal distance from the pile foundation axis to the tunnel axis, m; R is the tunnel excavation radius, m; H is the tunnel center burial depth, m.

[0050] Step 2: Determine the calculation formula for the horizontal displacement of the free pile foundations at both ends caused by shield tunnel excavation:

[0051]

[0052] Where w n is the sine series term of the pile foundation horizontal displacement function; L is the calculated length of the pile foundation, m; w O 、w L are the horizontal displacements of pile top and pile bottom, m, respectively.

[0053] Step 3: Determine the calculation formula for the bending moment and shear force of the free pile foundations at both ends caused by shield tunnel excavation:

[0054]

[0055] Where M is the bending moment of the pile foundation, N·m; Q is the shear force of the pile foundation, N.

[0056] Step 4: The expressions of pile foundation horizontal displacement w, pile foundation bending moment M, and pile foundation shear force Q in steps 1-3 contain the undetermined coefficients wn, wO, and wL, where the undetermined coefficient wn is determined by the following formula:

[0057] w n =α n w O +β n w L +γ n

[0058] Where, α n , β n , γ n is the comprehensive coefficient, which is determined by the following formula:

[0059]

[0060] Among them, φ1,n and φ2,n are known functions respectively. The coefficients of the sine series expansion are Un is the series coefficient of the known formation displacement function U(z) after the sine series expansion, that is,

[0061] The undetermined coefficients wO and wL are determined by the following formula:

[0062]

[0063] Where S1 to S6 are comprehensive coefficients, which are determined by the following formula:

[0064]

[0065] Substituting the known calculation parameters into the above formula, we can obtain the unknown coefficients wn, wO, and wL, and then obtain the pile foundation horizontal displacement w(z), bending moment M(z), and shear force Q(z) caused by the adjacent tunnel construction. The results are as follows: Figures 2 to 4 As shown in the figure, to verify the correctness of the calculation method of the present invention, a comparison is made with the calculation results of the existing literature, namely, Sun Yingjie, Shi Chenghua, Wang Zuxian, etc., which considers the actual constraints at both ends and induced by tunnel construction in the adjacent pile foundation. It can be seen that the calculation results of this embodiment are basically consistent with the calculation results of the existing literature, proving the correctness of the calculation method of the present invention. In the literature, the calculation of pile foundation displacement and internal force uses the finite difference method. Compared with the calculation method in the literature, the calculation method of the present invention only requires solving a linear equation of two variables to obtain the unknown boundary conditions of the governing differential equation, which significantly improves the calculation efficiency.

[0066] It should be noted that the parts not described in detail in the above embodiments are all prior art.

[0067] The above description is merely a preferred embodiment of the present invention and does not constitute any form of limitation to the present invention. Although the present invention has been disclosed as a preferred embodiment, it is not intended to limit the present invention. Any technician familiar with the present profession can make slight changes or modifications to equivalent embodiments using the technical contents disclosed above without departing from the scope of the technical solution of the present invention. However, any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solution of the present invention should be covered by the protection scope of the present invention.

Claims

1. A method for analyzing horizontal displacement of free-end pile foundations in a dual-parameter foundation model, characterized by: The calculation method comprises the following steps: Step 1: Use the Fourier series method to solve the bending deformation of the pile foundation caused by shield tunnel excavation. The dual-parameter foundation model adopts the Vlasov foundation model. The existing pile foundation is assumed to be an elastic foundation beam, satisfying the plane section assumption. The differential equation governing the horizontal displacement of the existing pile foundation caused by shield tunnel excavation is: Where: E is the elastic modulus of the existing pile foundation; I is the interface moment of inertia of the existing pile foundation; w is the horizontal displacement of the existing pile foundation, with the direction away from the tunnel excavation as positive; z is the coordinate along the axis of the existing pile foundation, with the top of the pile as the origin z = 0; t is the load transfer rate; B is the diameter of the pile foundation; k is the stiffness coefficient of the soil around the pile; U is the horizontal displacement of the stratum at the pile position caused by shield tunnel excavation; Step 2: Determine the calculation formula for the horizontal displacement of the free pile foundations at both ends caused by shield tunnel excavation: Where w n is the sine series term of the pile foundation horizontal displacement function; L is the calculated length of the pile foundation; w O 、w L are the horizontal displacements of the pile base surface end and the terminal pile end, respectively; Step 3: Determine the calculation formula for the bending moment and shear force of the free pile foundations at both ends caused by shield tunnel excavation: Where, M is the pile foundation bending moment; Q is the pile foundation shear force; Step 4: The expressions of pile foundation horizontal displacement w, pile foundation bending moment M, and pile foundation shear force Q in steps 1-3 contain the unknown coefficient w. n 、w O 、w L , where the unknown coefficient w n Determined by the following formula: w n =a n w O +b n w L +g n Where, α n , β n , γ n is the comprehensive coefficient; Undetermined coefficient w O 、w L Determined by the following formula: Where S1~S6 are comprehensive coefficients.

2. The method for analyzing horizontal displacement of free pile foundations at both ends of a dual-parameter foundation model according to claim 1 is characterized by: For the Vlasov foundation model, we have: Among them, E s is the elastic modulus of soil; s is the Poisson's ratio of soil; H e is the thickness of the foundation elastic layer, which is 2.5 times the pile diameter D; h is the function of displacement changing along the horizontal direction, which is 3. The method for analyzing horizontal displacement of free pile foundations at both ends of a dual-parameter foundation model according to claim 2 is characterized by: For the Vlasov foundation model, we have:

4. The method for analyzing horizontal displacement of free pile foundations at both ends of a dual-parameter foundation model according to claim 3 is characterized by: For the Vlasov foundation model, the analytical expression of soil displacement field proposed by Loganathan is adopted: Where ε0 is the ground loss rate; x0 is the horizontal distance from the pile foundation axis to the tunnel axis; R is the tunnel excavation radius; and H is the burial depth of the tunnel center.

5. The method for analyzing horizontal displacement of free-end pile foundations of a dual-parameter foundation model according to claim 4 is characterized in that: The comprehensive coefficient α n , β n , γ n Determined by the following formula: in, The known functions The coefficients of the sine series expansion are U n is the series coefficient of the sine series expansion of the known formation displacement function U(z), that is, 6. The method for analyzing horizontal displacement of free-end pile foundations of a dual-parameter foundation model according to claim 5 is characterized by: The comprehensive coefficients S1 to S6 are determined by the following formula:

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