A method for evaluating deformation caused by excavation of a pile foundation with both ends fixed

Abstract

The present invention provides a method for evaluating deformation caused by excavation when the two ends of a pile foundation are fixed. For end-bearing piles, rock-embedded piles, and pile tops embedded in caps, the boundary conditions of the pile foundation can be simplified to pile foundations with both ends fixed. Combined with the Fourier series method, a method for evaluating deformation caused by excavation when the two ends of a pile foundation are fixed is proposed. The method is used to calculate the horizontal displacement and internal force of the pile foundation caused by shield construction 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 numerical methods, the method only needs to solve the shear force at the pile end. Q O 、 Q 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 evaluating deformation caused by excavation of a pile foundation with both ends fixed. Background Art

[0002] In the context of large-scale urban rail transit construction, the excavation of new shield tunnels inevitably involves the construction of adjacent existing pile foundations, causing deformation. Considering that the displacement constraints on pile foundations, such as end-bearing piles and rock-embedded piles, can be simplified to fixed constraints, existing research on methods for assessing excavation-induced deformation with fixed constraints on both ends primarily relies on numerical methods such as the finite difference method, while analytical methods are less studied. Compared with numerical methods, analytical methods are more efficient and have more stable algorithms. Therefore, it is necessary to conduct research on methods for assessing the deformation of pile foundations with fixed constraints on both ends caused by the excavation of new shield tunnels. Summary of the Invention

[0003] The purpose of the present invention is to provide a method for evaluating the deformation caused by excavation of a pile foundation with both ends fixed. Taking into account the situation where the displacement constraints at both ends of pile foundations such as end-bearing piles and rock-socketed piles can be simplified to fixed constraints, based on the ground displacement caused by shield tunnel excavation, the dual-parameter foundation model theory is combined with the basic physical and mechanical parameters of the pile foundation and soil, and the method for evaluating the deformation caused by excavation of a pile foundation with both ends fixed is derived.

[0004] To achieve the above-mentioned purpose, the technical solution adopted by the present invention is: a method for evaluating deformation caused by excavation of a pile foundation with both ends fixed, the calculation method comprising the following steps:

[0005] 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:

[0006]

[0007] 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;

[0008] Step 2: Determine the calculation formula for the horizontal displacement of the fixed pile foundations at both ends caused by shield tunnel excavation:

[0009]

[0010] Where w0 is the cosine series constant of the pile foundation horizontal displacement function; w n is the cosine series term of the pile foundation horizontal displacement function; L is the calculated length of the pile foundation; Q O , Q L are the shear forces at the pile top and pile bottom, respectively;

[0011] Step 3: Determine the calculation formula for the bending moment and shear force of the fixed pile foundations at both ends caused by shield tunnel excavation:

[0012]

[0013] Where, M is the pile foundation bending moment; Q is the pile foundation shear force;

[0014] 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 coefficients w0 and w n , Q O , Q L , where the unknown coefficients w0, w n Determined by the following formula:

[0015]

[0016] Where U0 is the constant term after the cosine series expansion of the known formation displacement function U(z), that is, α0, α n , β0, β n , γ n is the comprehensive coefficient;

[0017] Undetermined coefficient Q O , Q L Determined by the following formula:

[0018]

[0019] Where S1~S6 are comprehensive coefficients.

[0020] 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

[0021] Preferably, for the Vlasov foundation model, there are:

[0022] Preferably, for the Vlasov foundation model, the analytical expression of soil displacement field proposed by Loganathan is adopted:

[0023]

[0024] 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.

[0025] Preferably, the comprehensive coefficients α0, αn, β0, βn, and γn are determined by the following formula:

[0026]

[0027] in, The known functions The coefficient of the cosine series expansion is U n is the series coefficient after the cosine series expansion of the known formation displacement function U(z), that is,

[0028] Preferably, the comprehensive coefficients S1 to S6 are determined by the following formula:

[0029]

[0030] The beneficial effects of the present invention are:

[0031] This scheme can simplify the boundary conditions of pile foundations in cases such as end-bearing piles, rock-embedded piles, and pile tops embedded in caps to pile foundations fixed at both ends. Combined with the Fourier series method, a deformation assessment method caused by excavation with pile foundations fixed at both ends is proposed to calculate the horizontal displacement and internal force of the pile foundation caused by shield construction adjacent to the pile foundation. 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 shear force QO and QL to further obtain other unknown coefficients. The calculation efficiency is high, providing an effective theoretical method for the safety risk assessment of existing structures under shield construction adjacent to pile foundations. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] 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.

[0033] 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.

[0034] Figure 2The 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.

[0035] 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.

[0036] 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

[0037] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0038] The present invention discloses a method for evaluating deformation caused by excavation of a pile foundation with both ends fixed. 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.

[0039] 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 two ends of the pile foundation are fixed constraints.

[0040] 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:

[0041]

[0042] Where: E is the elastic modulus of the existing pile foundation, Pa; I is the interface moment of inertia of the existing pile foundation, m 4 ; 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 top of the pile as the origin z = 0; B is the diameter of the pile foundation, m; t is the load transfer rate, N / m.

[0043] 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

[0044] k is the stiffness coefficient of the soil around the pile, Pa / m. For the Vlasov foundation model, we have:

[0045] 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:

[0046]

[0047] 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.

[0048] Step 2: Determine the calculation formula for the horizontal displacement of the fixed pile foundations at both ends caused by shield tunnel excavation:

[0049]

[0050] Where w0 is the cosine series constant of the pile foundation horizontal displacement function; w n is the cosine series term of the pile foundation horizontal displacement function; L is the calculated length of the pile foundation, m; Q O , Q L are the shear forces at the pile top and pile bottom, N respectively.

[0051] Step 3: Determine the calculation formula for the bending moment and shear force of the fixed pile foundations at both ends caused by shield tunnel excavation:

[0052]

[0053] Where M is the bending moment of the pile foundation, N·m; Q is the shear force of the pile foundation, N.

[0054] 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 w0, wn, QO, and QL. The undetermined coefficients w0 and wn are determined by the following formula:

[0055]

[0056] Where U0 is the constant term after the cosine series expansion of the known formation displacement function U(z), that is, α0, αn, β0, βn, and γn are comprehensive coefficients, which are determined by the following formula:

[0057]

[0058] Among them, φ1,0, φ1,n, φ2,0, φ2,n are known functions respectively. The coefficient of the cosine series expansion is Un is the series coefficient of the cosine series expansion of the known formation displacement function U(z), that is,

[0059] The undetermined coefficients QO and QL are determined by the following formula:

[0060]

[0061] Where S1 to S6 are comprehensive coefficients, which are determined by the following formula:

[0062]

[0063] Substituting the known calculation parameters into the above formula, we can obtain the unknown coefficients w0, wn, QO, and QL, 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.

[0064] It should be noted that the parts not described in detail in the above embodiments are all prior art.

[0065] 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 evaluating deformation caused by excavation of a pile foundation with both ends fixed, 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 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 fixed pile foundations at both ends caused by shield tunnel excavation: Where w0 is the cosine series constant of the pile foundation horizontal displacement function; w n is the cosine series term of the pile foundation horizontal displacement function; L is the calculated length of the pile foundation; Q O , Q L are the shear forces at the pile top and pile bottom, respectively; Step 3: Determine the calculation formula for the bending moment and shear force of the fixed 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 coefficients w0 and w n , Q O , Q L , where the unknown coefficients w0, w n Determined by the following formula: Where U0 is the constant term after the cosine series expansion of the known formation displacement function U(z), that is, α0, α n , β0, β n , γ n is the comprehensive coefficient; Undetermined coefficient Q O , Q L Determined by the following formula: Where S1~S6 are comprehensive coefficients.

2. The method for evaluating deformation caused by excavation of a pile foundation with both ends fixed 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 evaluating deformation caused by excavation of a pile foundation with both ends fixed according to claim 2, characterized in that: For the Vlasov foundation model, we have:

4. The method for evaluating deformation caused by excavation of a pile foundation with both ends fixed 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 evaluating deformation caused by excavation of a pile foundation with both ends fixed according to claim 4, characterized in that: The comprehensive coefficients α0, α n , β0, β n , γ n Determined by the following formula: in, The known functions The coefficient of the cosine series expansion is U n is the series coefficient after the cosine series expansion of the known formation displacement function U(z), that is, 6. The method for evaluating deformation caused by excavation of a pile foundation with both ends fixed according to claim 5, characterized in that: The comprehensive coefficients S1 to S6 are determined by the following formula:

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