A deformation analysis method for existing pile foundations considering elastic constraints at the pile end

Abstract

The present invention provides a method for analyzing the deformation of existing pile foundations under construction, taking into account the elastic constraint of the pile end. In view of the different stiffness of the pile end displacement constraint under actual engineering conditions, the pile top and pile bottom are regarded as a constraint system consisting of an elastic rotation constraint and an elastic horizontal displacement constraint. Combined with the Fourier series method, a method for analyzing the deformation of existing pile foundations under construction, taking into account the elastic constraint of the pile end, is proposed. The method is used to calculate the horizontal displacement and internal force of the pile foundation caused by the construction of the shield adjacent to the pile foundation. 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 and pile tip bending moment M O 、 M L The four unknown quantities can be used to further obtain other unknown coefficients, which has high calculation efficiency and provides an effective analytical calculation 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 deformation of an existing pile foundation under construction taking into account elastic constraints on pile ends. Background Art

[0002] In the context of large-scale urban rail transit construction, new shield tunnels inevitably involve construction adjacent to existing pile foundations, causing deformation and impacting the normal use and structural safety of the existing pile 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 assessment of surrounding buildings.

[0003] Existing methods for analyzing pile foundation deformation under construction that considers elastic constraints at the pile end primarily include analytical methods and numerical simulations. Theoretical analytical methods generally treat the pile foundation as an elastic foundation beam, subject to the ground displacement loads caused by tunnel excavation, and thus calculate the displacement and internal forces of the pile foundation under the influence of tunnel excavation. When addressing pile foundation boundary conditions, the pile end is typically assumed to be either free or fixed. However, in actual projects, the structures or strata constraining pile end displacement have different physical and mechanical parameters. Simply treating the pile end as either free or fixed fails to consider the impact of the actual displacement constraint stiffness on pile foundation deformation and internal forces.

[0004] Currently, relevant research has been conducted on methods for calculating pile foundation displacements that consider the influence of actual constraints at the pile ends. In an existing paper, Sun Yingjie, Shi Chenghua, Wang Zuxian, et al. published an analytical solution to the adjacent pile foundation response induced by tunnel construction, considering actual constraints at both ends. To reflect the actual constraints at both ends of the pile foundation and account for the shear effect of the pile foundation, the pile foundation was simplified as a Timoshenko beam with generalized elastic constraints in a Vlasov foundation, and a simplified tunnel-soil-pile foundation calculation model was established. Based on the elastic foundation beam theory, a two-stage method was used to derive an analytical solution for the additional lateral deformation and internal forces of the pile body under arbitrary boundary constraints during adjacent tunnel construction. The analytical solution was 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 was proposed, and the influence of the pile top and pile end constraints on the lateral deformation and internal forces of the pile foundation was further analyzed.

[0005] However, the calculation method obtained is based on the 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 deformation analysis method of existing pile foundations under construction considering the elastic constraint of the pile end, and to provide the corresponding analytical calculation formulas and calculation steps for pile foundation displacement and internal force. Summary of the Invention

[0006] The purpose of the present invention is to provide a method for analyzing the deformation of existing pile foundations during construction, taking into account the elastic constraint of the pile end. Based on the stratum displacement caused by shield tunnel excavation and combined with the basic physical and mechanical parameters of the pile foundation and soil, a calculation formula for the horizontal displacement and internal force of an existing single pile caused by adjacent shield construction is derived.

[0007] To achieve the above object, the technical solution adopted by the present invention is: a method for analyzing the deformation of an existing pile foundation considering the elastic constraint of the pile end, the calculation method comprising the following steps:

[0008] Step 1: Use the Fourier series method to solve the bending deformation of the pile foundation caused by shield tunnel excavation. The two-parameter foundation model adopts the Vlasov foundation model. The existing pile foundation is assumed to be an elastic foundation beam. The differential equation governing the horizontal displacement of the existing pile foundation caused by shield tunnel excavation is:

[0009]

[0010] 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;

[0011] Step 2: Determine the calculation formula for the horizontal displacement of pile foundation caused by shield tunnel excavation:

[0012]

[0013] 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 top and pile bottom respectively; M O 、M L are the bending moments at the pile top and pile bottom, respectively;

[0014] Step 3: Determine the calculation formula for the pile foundation bending moment and shear force caused by shield tunnel excavation:

[0015]

[0016] Where, M is the pile foundation bending moment; Q is the pile foundation shear force;

[0017] 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 、M O 、M L, where the unknown coefficient w n Determined by the following formula:

[0018] w n =A 1,n w O +A 2,n w L +A 3,n M O +A 4,n M L +A 5,n

[0019] Where A 1,n 、A 2,n 、A 3,n 、A 4,n 、A 5,n is the comprehensive coefficient;

[0020] Undetermined coefficient w O 、w L 、M O 、M L Determined by the following equations: w = sv

[0021] Where w is a vector of unknown coefficients, that is, w = (w O w L M O M L ) T ; s is the coefficient matrix of the equation system, that is v is the vector composed of the right-hand side of the equation, that is, v=(v1v2 v3 v4) T .

[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, A1,n, A2,n, A3,n, A4,n, and A5,n are determined by the following formula:

[0028]

[0029] Among them, φ 1,n 、φ 2,n 、φ 3,n 、φ 4,n The known functions The coefficients of the sine series expansion are U n is the series coefficient after the sine series expansion of the known formation displacement function U(z), that is,

[0030] Preferably, the elements s11, s12, ..., s34, s44 of the coefficient matrix s of the system of equations are determined by the following formula:

[0031]

[0032] Among them, K su , K sd are the rotational constraint stiffness of the pile top and pile bottom respectively; K tu , K td are the horizontal displacement constraint stiffness of the pile top and pile bottom, respectively.

[0033] Preferably, the elements v1, v2, v3, and v4 of the right-hand side vector v of the equation system are determined by the following formula:

[0034]

[0035] The beneficial effects of the present invention are:

[0036] In view of the different stiffness of pile end displacement constraints under actual engineering conditions, this scheme regards the pile top and pile bottom as a constraint system composed of elastic rotation constraints and elastic horizontal displacement constraints. Combined with the Fourier series method, a deformation analysis method for existing pile foundations under construction considering the elastic constraints of the pile ends is proposed. It is used to calculate the horizontal displacement and internal force of the pile foundation caused by the construction of the shield adjacent to the pile foundation. Compared with the numerical simulation method, the required physical quantities can be obtained by substituting the corresponding calculation parameters. The calculation process is simple. Compared with the theoretical method with the help of numerical methods, this method only needs to solve the four unknown quantities of pile end horizontal displacement wO, wL and pile end bending moments MO, ML to further obtain other unknown coefficients. The calculation efficiency is high, which provides an effective analytical calculation method for the safety risk assessment of existing structures under shield construction adjacent to pile foundations. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] 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.

[0038] 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.

[0039] 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.

[0040] 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.

[0041] 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

[0042] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0043] The present invention discloses a method for analyzing the deformation of existing pile foundations under construction considering the elastic constraints of the pile ends. In order to make the purpose of this application and the technical solution clearer, the existing literature, namely the analytical solution of the response of adjacent pile foundations 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.

[0044] 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, the pile foundation is an elastic constraint, and Ksu = Ksd = 1×109 kN·m·rad-1, Ktu = Ktd = 1×109 kN·m-1.

[0045] 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:

[0046]

[0047] 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.

[0048] 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

[0049] k is the stiffness coefficient of the soil around the pile, Pa / m. For the Vlasov foundation model, we have:

[0050] 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:

[0051]

[0052] 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.

[0053] Step 2: Determine the calculation formula for the horizontal displacement of pile foundation caused by shield tunnel excavation:

[0054]

[0055] Where wn is the sine series term of the pile foundation horizontal displacement function; L is the calculated length of the pile foundation, in m; wO and wL are the horizontal displacements of the pile top and bottom, in m; MO and ML are the bending moments at the pile top and bottom, in N·m, respectively.

[0056] Step 3: Determine the calculation formula for the pile foundation bending moment and shear force caused by shield tunnel excavation:

[0057]

[0058] Where M is the bending moment of the pile foundation, N·m; Q is the shear force of the pile foundation, N.

[0059] 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, wL, MO, and ML, where the undetermined coefficient wn is determined by the following formula:

[0060] w n =A1,n w O +A 2,n w L +A 3,n M O +A 4,n M L +A 5,n

[0061] Where A 1,n 、A 2,n 、A 3,n 、A 4,n 、A 5,n is the comprehensive coefficient, which is determined by the following formula:

[0062]

[0063] in, The known functions The coefficients of the sine series expansion are U n is the series coefficient after the sine series expansion of the known formation displacement function U(z), that is,

[0064] The unknown coefficients wO, wL, MO, and ML are determined by the following equations: w = sv

[0065] Where w is a vector of unknown coefficients, that is, w = (w O w L M O M L ) T ; s is the coefficient matrix of the equation system, that is v is the vector composed of the right-hand side terms of the equation group, that is, v = (v1v2 v3 v4)T.

[0066] The elements s11, s12, ..., s34, and s44 of the coefficient matrix s of the system of equations are determined by the following formula:

[0067]

[0068] Among them, K su , K sd are the rotational constraint stiffness of the pile top and pile bottom, N·m·rad -1 ;K tu , K td are the horizontal displacement constraint stiffness of the pile top and pile bottom, N·m -1 .

[0069] The elements v1, v2, v3, and v4 of the vector v on the right side of the equation system are determined by the following formula:

[0070]

[0071] Substitute the known calculation parameters into the above formula to obtain the unknown coefficients wn, wO, wL, MO, and ML, and then obtain the horizontal displacement, bending moment, and shear force of the pile foundation 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 needs to solve the four unknown quantities wO, wL, MO, and ML to obtain the other unknown parameters and then calculate the pile foundation horizontal displacement and internal force, which significantly improves the calculation efficiency.

[0072] It should be noted that the parts not described in detail in the above embodiments are all prior art.

[0073] 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 deformation of existing pile foundations taking into account elastic constraints at the pile ends, characterized by: The calculation method includes 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 two-parameter foundation model adopts the Vlasov foundation model. The existing pile foundation is assumed to be an elastic foundation beam. 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 pile foundation 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 top and pile bottom respectively; M O 、M L are the bending moments at the pile top and pile bottom, respectively; Step 3: Determine the calculation formula for the pile foundation bending moment and shear force 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 、M O 、M L , where the unknown coefficient w n Determined by the following formula: w n =A 1,n w O +A 2,n w L +A 3,n M O +A 4,n M L +A 5,n Where A 1,n 、A 2,n 、A 3,n 、A 4,n 、A 5,n is the comprehensive coefficient; Undetermined coefficient w O 、w L 、M O 、M L Determined by the following equations: w = sv Where w is a vector of unknown coefficients, that is, w = (w O w L M O M L ) T ; s is the coefficient matrix of the equation system, that is v is the vector composed of the right-hand side of the equation, that is, v=(v1v2 v3 v4) T .

2. The method for analyzing deformation of an existing pile foundation considering elastic constraints at the pile end according to claim 1, characterized in that: 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 deformation of an existing pile foundation considering elastic constraints at the pile end according to claim 2, characterized in that: For the Vlasov foundation model, we have:

4. The method for analyzing deformation of an existing pile foundation considering elastic constraints at the pile end according to claim 3, characterized in that: 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 deformation of an existing pile foundation considering elastic constraints at the pile end according to claim 4, characterized in that: The A 1,n 、A 2,n 、A 3,n 、A 4,n 、A 5,n Determined by the following formula: Among them, φ 1,n 、φ 2,n 、φ 3,n 、φ 4,n The known functions The coefficients of the sine series expansion are U n is the series coefficient after the sine series expansion of the known formation displacement function U(z), that is, 6. The method for analyzing deformation of an existing pile foundation considering elastic constraints at the pile end according to claim 5, characterized in that: The coefficient matrix s of the system of equations has elements s 11 、s 12 ,…,s 34 、s 44 Determined by the following formula: Among them, K su , K sd are the rotational constraint stiffness of the pile top and pile bottom respectively; K tu , K td are the horizontal displacement constraint stiffness of the pile top and pile bottom, respectively.

7. The method for analyzing deformation of an existing pile foundation considering elastic constraints at the pile end according to claim 6, characterized in that: The elements v1, v2, v3, and v4 of the vector v on the right side of the equation system are determined by the following formula:

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