Construction existing pile foundation deformation analysis method considering pile tip elastic constraint

Abstract

The invention provides a construction existing pile foundation deformation analysis method considering pile tip elastic constraint, aiming at different constraint rigidities of pile tip displacement constraint under actual engineering conditions, a pile top and a pile bottom are regarded as a constraint system composed of elastic rotation constraint and elastic horizontal displacement constraint, and a Fourier series method is combined, so that the deformation of the pile tip displacement constraint is analyzed. The construction existing pile foundation deformation analysis method considering the pile tip elastic constraint 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 the calculation efficiency is high. Compared with a theoretical method by means of a numerical method, the method has the advantages that other undetermined coefficients can be further obtained only by solving the four unknown quantities of the horizontal displacement wO and wL of the pile end and the bending moment MO and ML of the pile end, the calculation efficiency is high, and an effective analytical calculation method is provided for safety risk evaluation of the 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 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 construction of urban rail transit, it is inevitable that new shield tunnels will be constructed adjacent to existing pile foundations, causing deformation of existing pile foundations and affecting the normal use and structural safety of the existing pile foundation superstructure. Therefore, it is necessary to conduct a safety risk assessment of the shield construction adjacent to the pile foundation to provide a basis for the optimization of the design and construction plan. Accurately calculating the deformation of the adjacent pile foundation caused by shield construction is the key to the safety risk assessment of surrounding buildings.

[0003] The deformation analysis methods of existing pile foundations under construction considering the elastic constraint of the pile end mainly include analytical methods and numerical simulation methods. The theoretical analytical method generally regards the pile foundation as an elastic foundation beam, which bears the stratum displacement load caused by tunnel excavation, and then calculates the displacement and internal force of the pile foundation under the influence of tunnel excavation. When dealing with the boundary conditions of the pile foundation, it is usually assumed that the pile end is a free end or a fixed end. However, in actual engineering, the structure or stratum that constrains the displacement of the pile end has different physical and mechanical parameters. Simply regarding the pile end as a free end or a fixed end cannot consider the influence of the actual displacement constraint stiffness of the pile end on the deformation and internal force of the pile foundation.

[0004] At present, there are related studies on the calculation method of pile foundation displacement considering the influence of the actual constraint at the pile end. 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 to 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 analytical solution of this paper is compared with the 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 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.

[0005] 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 deformation analysis method of existing pile foundation under construction considering the elastic constraint of the pile end, and give the corresponding analytical calculation formula 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 under construction taking into account the elastic constraint of the pile end. According to the stratum displacement caused by shield tunnel excavation and combined with the basic physical and mechanical parameters of the pile foundation and soil, the horizontal displacement and internal force calculation formula of the existing single pile caused by the adjacent shield construction is derived.

[0007] In order 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 under construction considering the elastic constraint of the pile end, 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. 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 pile foundation 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 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 pile foundation bending moment and shear force 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 、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 MO +A 4,n M L +A 5,n In the formula, 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 system of 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 system, that is, v=(v1v2 v3 v4) T .

[0008] 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

[0009] Preferably, for the Vlasov foundation model, there are:

[0010] 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.

[0011] Preferably, A1,n, A2,n, A3,n, A4,n, A5,n are determined by the following formula: Among them, φ 1,n ,φ 2,n ,φ 3,n ,φ 4,n The known functions are The coefficients of the sine series expansion are Un is the series coefficient of the sine series expansion of the known formation displacement function U(z), that is,

[0012] Preferably, the elements s11, s12, ..., s34, s44 of the coefficient matrix s of the system of equations are determined by the following formula: Among them, K su , K sd are the rotational restraint 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.

[0013] Preferably, the elements v1, v2, v3, and v4 of the right-hand side vector v of the system of equations are determined by the following formula:

[0014] The beneficial effects of the present invention are:

[0015] 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 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 adjacent pile foundation construction. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] 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.

[0017] 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.

[0018] 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.

[0019] 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.

[0020] 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 of the existing literature. DETAILED DESCRIPTION

[0021] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0022] 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, the present application is further described.

[0023] 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, the pile foundation is an elastic constraint, and Ksu=Ksd=1×109kN·m·rad-1, Ktu=Ktd=1×109kN·m-1.

[0024] 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:

[0025] 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.

[0026] 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

[0027] k is the stiffness coefficient of the soil around the pile, Pa / m. For the Vlasov foundation model, it is:

[0028] 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:

[0029] 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.

[0030] Step 2: Determine the calculation formula for the horizontal displacement of pile foundation caused by shield tunnel excavation:

[0031] Where wn is the sine series term of the pile foundation horizontal displacement function; L is the calculated length of the pile foundation, m; wO and wL are the horizontal displacements of the pile top and pile bottom, m; MO and ML are the bending moments of the pile top and pile bottom, N·m, respectively.

[0032] Step 3: Determine the calculation formula for pile foundation bending moment and shear force 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.

[0033] 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: 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 In the formula, 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: 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,

[0034] The unknown coefficients wO, wL, MO, and ML are 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 terms of the system of equations, that is, v=(v1v2 v3 v4)T.

[0035] Among them, the elements s11, s12, ..., s34, s44 of the coefficient matrix s of the equation system are determined by the following formula: Among them, K su , K sd are the rotational restraint stiffness of the pile top and pile bottom, N·m·rad -1 ; K tu , K td are the horizontal displacement restraint stiffness of the pile top and pile bottom, N·m -1 .

[0036] 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:

[0037] Substitute the known calculation parameters into the above formula to obtain the unknown coefficients wn, wO, wL, MO, ML, and then obtain the horizontal displacement, bending moment and shear force of the pile foundation 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 calculated 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 four unknown quantities wO, wL, MO, and ML to obtain other unknown parameters and then obtain the horizontal displacement and internal force of the pile foundation, and the calculation efficiency is significantly improved.

[0038] It should be noted that the parts not described in detail in the above embodiments are all prior art.

[0039] The above description is only 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 as above, it is not intended to limit the present invention. Any technician familiar with the profession can make some changes or modifications to equivalent embodiments of equivalent changes 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 within the protection scope of the present invention.

Claims

1. A method for analyzing deformation of existing pile foundations taking into account the elastic constraint of pile ends, 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. 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 pile foundation 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 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 pile foundation bending moment and shear force 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 、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 In the formula, 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 system of equations: w = sv In the formula, 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 system, that is, v=(v1v2 v3 v4) T .

2. The method for analyzing deformation of an existing pile foundation considering the elastic constraint of 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 along the horizontal direction, which is 3. The method for analyzing deformation of an existing pile foundation considering the elastic constraint of the pile end according to claim 2 is characterized by: For the Vlasov foundation model, we have:

4. The method for analyzing deformation of an existing pile foundation considering the elastic constraint of the pile end 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 deformation of an existing pile foundation considering the elastic constraint of 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 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, 6. The method for analyzing deformation of an existing pile foundation considering the elastic constraint of 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 restraint 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 the elastic constraint of 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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