Two-end free pile foundation horizontal displacement analysis method of two-parameter foundation model

Abstract

The invention provides a two-end free pile foundation horizontal displacement analysis method of a two-parameter foundation model, and provides a two-end free pile foundation horizontal displacement analysis method of the two-parameter foundation model by combining a Fourier series method for pile foundations with pile foundation boundary conditions such as friction piles and isolation piles capable of being simplified into two-end free pile foundations. The method is used for calculating pile foundation horizontal displacement and internal force caused by shield adjacent pile foundation construction, compared with a numerical simulation method, physical quantities needing to be calculated can be obtained by substituting corresponding calculation parameters, the calculation process is simple and convenient, and compared with a theoretical method with the help of a numerical method, the calculation efficiency is improved. According to the method, other undetermined coefficients can be further obtained only by solving the binary linear equation related to the horizontal displacement wO and wL of the pile end, the calculation efficiency is high, and an effective theoretical method is provided for safety risk evaluation of an existing structure in shield adjacent pile foundation construction.

Description

Technical Field

[0001] The 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 construction of urban rail transit, it is inevitable that new shield tunnels will be constructed adjacent to existing pile foundations, causing deformation of the pile foundations and affecting the normal use and structural safety of the existing superstructure. Therefore, it is necessary to conduct a safety risk assessment of the shield construction adjacent to the pile foundations to provide a basis for the optimization of the design and construction scheme. Accurately calculating the deformation of the adjacent pile foundations caused by shield construction is the key to the safety risk assessment of surrounding buildings.

[0003] The calculation methods for the horizontal displacement of a single pile caused by the adjacent construction of a shield tunnel mainly include analytical methods and numerical simulation methods. The analytical method calculation model is mainly based on the elastic foundation beam model and the Winkler foundation model, that is, the pile foundation is simplified to an elastic foundation beam, which is subject to the additional load generated by tunnel excavation, and the foundation reaction force is calculated based on the Winkler assumption. The Winkler foundation model has limitations in reflecting the continuity and shear characteristics of the soil, resulting in a large deviation between the calculation results and the actual results. The dual-parameter foundation models such as the Pasternak foundation model and the Vlazov foundation model have better solved this problem.

[0004] For the calculation of pile foundation horizontal displacement caused by shield tunnel excavation under dual-parameter foundation model, existing research mainly relies on numerical methods such as finite difference method, and there are few studies on analytical methods. In the existing literature, Sun Yingjie, Shi Chenghua, Wang Zuxian, et al. published the analytical solution of the response of adjacent pile foundation induced by tunnel construction considering the actual constraints at both ends: In order to reflect the actual constraint state at both ends of the pile foundation and consider the shear effect of the pile foundation, the pile foundation is simplified as a Timoshenko beam with generalized elastic constraints in the Vlasov foundation, and a simplified calculation model of tunnel-soil-pile foundation is established. Based on the elastic foundation beam theory, the two-stage method is used to derive the analytical solution of the lateral additional deformation and internal force of the pile body under arbitrary boundary constraints during the construction of the adjacent tunnel. The reliability of the analytical model is verified by comparing the analytical solution with the boundary element and finite element numerical solutions. Finally, a simplified calculation method for the actual constraints at both ends of the pile foundation is given, and the influence of the pile top and pile end constraints on the lateral deformation and internal force of the pile foundation is further analyzed. However, the calculation method obtained is based on the numerical method, and the corresponding analytical calculation formula is not obtained. Compared with the calculation method with the help of 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 the 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 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 situation 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 existing single pile horizontal displacement and internal force calculation formula caused by the adjacent construction of the shield 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: Step 1: Use the Fourier series method to solve the bending deformation of the pile foundation under the influence of 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, which satisfies the plane section assumption. The horizontal displacement control differential equation 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 side 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 on the side of the pile; U is the horizontal displacement of the stratum at the pile position caused by the 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: In the formula, 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 bending moment of the pile foundation; Q is the shear force of the pile foundation; 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 =α n w O +β n w L+γ n In the formula, α n , β n , γ n is the comprehensive coefficient; Undetermined coefficient w O 、w L Determined by the following formula: In the formula, S1~S6 are comprehensive coefficients.

[0007] Preferably, for the Vlasov foundation model, there are: 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, which is

[0008] Preferably, for the Vlasov foundation model, there are:

[0009] Preferably, for the Vlasov foundation model, the soil displacement field analytical formula proposed by Loganathan is adopted: Among them, ε0 is the stratum loss rate; x0 is the horizontal distance from the pile foundation axis to the tunnel axis; R is the tunnel excavation radius; H is the burial depth of the tunnel center.

[0010] Preferably, the comprehensive coefficients αn, βn, and γn are determined by the following formula: in, The known functions are The coefficients of the sine series expansion are U n is the series coefficient of the known formation displacement function U(z) after the sine series expansion, that is,

[0011] Preferably, the comprehensive coefficients S1 to S6 are determined by the following formula:

[0012] The beneficial effects of the present invention are:

[0013] This scheme can simplify the boundary conditions of pile foundations such as friction piles and isolation piles into pile foundations with free ends. Combined with the Fourier series method, an analytical method for the horizontal displacement of free pile foundations at both ends of a dual-parameter foundation model is proposed to calculate the horizontal displacement and internal force of pile foundations caused by the construction of shield adjacent 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 binary linear equation related to the horizontal displacement wO and wL of the pile end 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 adjacent to the shield construction pile foundation. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. 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 creative work.

[0015] Figure 1 This is a simplified diagram for calculating the horizontal displacement of existing pile foundations caused by the construction of tunnel adjacent to pile foundations in an embodiment of the present invention.

[0016] 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.

[0017] Figure 3 It 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.

[0018] Figure 4 It 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

[0019] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0020] The present invention discloses a method for analyzing 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, Sun Yingjie, Shi Chenghua, Wang Zuxian, etc., which considers the actual constraints at both ends of the tunnel construction-induced adjacent pile foundation response, is combined with the data and calculation parameters and the calculation diagram as shown in the figure. Figure 1 As shown, the present application is further described.

[0021] After the tunnel is constructed near the existing pile foundation, the stratum loss rate ε0 is 2.5%, the soil elastic modulus Es is 24MPa, the Poisson's ratio υs is 0.5, the tunnel diameter D is 6m, the pile foundation elastic modulus E is 30GPa, the pile diameter B is 0.5m, the pile length L is 25m, the tunnel center burial depth H is 20m, and the horizontal distance x0 between the tunnel axis and the pile foundation axis is 4.5m. The calculation diagram is as follows: Figure 1 As shown, both ends of the pile foundation are free ends.

[0022] Step 1: For the pile foundation boundary conditions and load conditions described in this example, the pile foundation is regarded as an elastic foundation beam on the Vlasov foundation model. The control differential equation for the horizontal displacement of the existing pile foundation caused by tunnel excavation is:

[0023] 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 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.

[0024] 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

[0025] k is the stiffness coefficient of the soil around the pile, Pa / m. For the Vlasov foundation model, it is:

[0026] U is the horizontal displacement of the stratum at the pile position caused by shield tunnel excavation, and the analytical formula of soil displacement field proposed by Loganathan is taken as:

[0027] Among them, 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 burial depth of the tunnel center, m.

[0028] Step 2: Determine the calculation formula for the horizontal displacement of the free pile foundations at both ends caused by shield tunnel excavation: In the formula, 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.

[0029] 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 bending moment of the pile foundation, N·m; Q is the shear force of the pile foundation, N.

[0030] 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: w n =α n w O +β n w L +γ n In the formula, α n , β n , γ n is the comprehensive coefficient, which is determined by the following formula:

[0031] 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,

[0032] The undetermined coefficients wO and wL are determined by the following formula:

[0033] In the formula, S1~S6 are comprehensive coefficients, which are determined by the following formula:

[0034] Substitute the known calculation parameters into the above formula to obtain the unknown coefficients wn, wO, wL, and then obtain the pile foundation horizontal displacement w(z), bending moment M(z) and shear force Q(z) caused by the adjacent construction of the tunnel. The results are as follows: Figure 2 to Figure 4 As shown in the figure, in order to verify the correctness of the calculation method of the present invention, the calculation results of the analytical solution of the response of adjacent pile foundation induced by tunnel construction considering the actual constraints at both ends by Sun Yingjie, Shi Chenghua, Wang Zuxian, etc. are compared. It can be seen that the calculation results of this embodiment are basically consistent with the calculation results of the existing literature, which proves the correctness of the calculation method of the present invention. In the literature, the calculation of pile foundation displacement and internal force is carried out by means of finite difference method. Compared with the calculation method in the literature, the calculation method of the present invention only needs to solve the two-variable linear equation to obtain the unknown boundary conditions of the control differential equation, and the calculation efficiency is significantly improved.

[0035] It should be noted that the parts not described in detail in the above embodiments are all prior art.

[0036] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any form. Although the present invention has been disclosed as a preferred embodiment as above, it is not used to limit the present invention. Any technician familiar with this profession can make some changes or modify the technical contents disclosed above into equivalent embodiments without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent change and modification made to the above embodiments according to the technical essence of the present invention without departing from the content of the technical solution of the present invention should be covered within the protection scope of the present invention.

Claims

1. A method for analyzing horizontal displacement of free pile foundations at both ends of a dual-parameter foundation model, characterized in that: The calculation method comprises the following steps: Step 1: Use the Fourier series method to solve the bending deformation of the pile foundation under the influence of 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, which satisfies the plane section assumption. The horizontal displacement control differential equation 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 side 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 on the side of the pile; U is the horizontal displacement of the stratum at the pile position caused by the 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: In the formula, 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 bending moment of the pile foundation; Q is the shear force of the pile foundation; 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 In the formula, α n , β n , γ n is the comprehensive coefficient; Undetermined coefficient w O 、w L Determined by the following formula: In the formula, S1~S6 are comprehensive coefficients.

2. The horizontal displacement analysis method of free pile foundation 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 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 formula of soil displacement field proposed by Loganathan is adopted: Among them, ε0 is the stratum loss rate; x0 is the horizontal distance from the pile foundation axis to the tunnel axis; R is the tunnel excavation radius; H is the burial depth of the tunnel center.

5. The method for analyzing horizontal displacement of free pile foundations at both ends 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 are 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 pile foundations at both ends 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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