Method for determining horizontal displacement and internal force of adjacent pile foundations in loose area caused by excavation

Abstract

The invention provides a method for determining the horizontal displacement and internal force of adjacent pile foundations in a loose area caused by excavation, shield tunnel excavation disturbs surrounding soil to cause the loose area of surrounding strata, it is assumed that the soil in the loose area does not support the existing pile foundations, and a Vlasov two-parameter foundation model is combined to determine the horizontal displacement and internal force of the adjacent pile foundations in the loose area. A calculation model considering the horizontal displacement of the adjacent pile foundation caused by shield excavation in the loose area is established, and the method for determining the horizontal displacement and the internal force of the adjacent pile foundation in the loose area caused by excavation is provided on the basis of a Fourier series method and used for calculating the horizontal displacement and the internal force of the pile foundation caused by shield adjacent pile foundation construction. According to the method, the physical quantity needing to be calculated can be obtained by substituting the corresponding calculation parameters, the calculation process is simple and convenient, other undetermined coefficients can be further obtained only by solving the two unknown quantities of the horizontal displacement wO and wL of the pile end, and the calculation efficiency is high.

Description

Technical Field

[0001] The invention relates to the technical field of tunnel engineering design, and in particular to a method for determining horizontal displacement and internal force of adjacent pile foundations in a loose zone caused by excavation. Background Art

[0002] Shield tunnel excavation will cause the soil to generate secondary stress fields and displacement fields, and the disturbance of the in-situ soil caused by the construction will destroy the integrity of the soil in the strata around the tunnel, causing stratum loss, and then produce loose zones in the strata around the tunnel. The existing theoretical analytical methods for pile foundation deformation calculation usually assume that the foundation soil has good integrity when calculating the foundation reaction force, that is, the foundation is assumed to be homogeneous, and the influence of the loosening of the surrounding soil caused by tunnel excavation on the pile foundation force is not considered. Within the loose zone, the integrity of the soil is destroyed, and the assumption of a homogeneous foundation does not match the actual condition of the stratum. Therefore, it is necessary to conduct research on the determination of the horizontal displacement and internal force of the adjacent pile foundation in the loose zone caused by excavation, and provide the corresponding analytical calculation formulas and calculation steps for pile foundation displacement and internal force. Summary of the invention

[0003] The purpose of the present invention is to provide a method for determining the horizontal displacement and internal force of adjacent pile foundations in a loose zone caused by excavation. For the situation where loose zones may be generated in sandy strata and other strata around newly built tunnels, a loose zone calculation model and method are combined, assuming that both ends of the pile foundation are free ends, and relying on the dual-parameter Vlasov foundation model, a formula for determining the horizontal displacement and internal force of adjacent pile foundations in the loose zone caused by excavation is derived.

[0004] In order to achieve the above object, the technical solution adopted by the present invention is: a method for determining the horizontal displacement and internal force of the adjacent pile foundation of the loose zone caused by excavation, and the analytical algorithm includes the following steps: Step 1: Determine whether the pile foundation edge line enters the loose area of ​​the surrounding strata caused by tunnel excavation; Step 2, calculating the position coordinate range of the pile foundation within the loose area; Step 3: Fourier series method is used to solve the bending deformation of pile foundation under the influence of shield tunnel excavation. The dual-parameter foundation model adopts Vlasov foundation model. The pile foundation is assumed to be an elastic foundation beam, which satisfies the plane section assumption. The horizontal displacement control differential equation of the adjacent pile foundation caused by shield tunnel excavation is: Where: E is the elastic modulus of the adjacent pile foundation; I is the interface moment of inertia of the adjacent pile foundation; w is the horizontal displacement of the adjacent pile foundation, with the side away from the tunnel excavation as positive; H(z) is a step function; t is the load transfer rate; k is the stiffness coefficient of the soil on the pile side; B is the pile foundation diameter; U is the horizontal displacement of the stratum at the pile position caused by the shield tunnel excavation; Step 4: Determine the calculation formula for the horizontal displacement of pile foundation caused by shield tunnel excavation: Where, wn is the sine series term of the pile foundation horizontal displacement function; L is the calculated length of the pile foundation, that is, the pile length; wO and wL are the horizontal displacements of the pile top and pile bottom respectively; Step 5: 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 6: The expressions of pile foundation horizontal displacement w, pile foundation bending moment M, and pile foundation shear force Q in steps 3-5 contain the undetermined coefficients wn, wO, and wL, where the undetermined coefficient wn is determined by the following formula: w=w O α -1 β1+w L α -1 β2+α -1 β1 Wherein, w is the column vector composed of unknown coefficients w1, w2, …, wn; α is the comprehensive coefficient matrix composed of known calculation parameters; β1, β2, β3 are the comprehensive coefficient column vectors composed of known parameters; The undetermined coefficients wO and wL are determined by the following formula: In the formula, S1~S6 are comprehensive coefficients.

[0005] Preferably, in step 1, the horizontal distance xT from the central axis of the tunnel to the side edge of the pile foundation adjacent to the tunnel excavation, the width of the loose zone Dz and the tunnel excavation diameter D satisfy the quantitative relationship: This indicates that the edge of the pile foundation has entered the loose area.

[0006] Preferably, in step 2, z is set as the coordinate axis along the axis direction of the adjacent pile foundation, and the top of the pile is taken as the coordinate origin z=0. According to the selected loose zone range calculation model, combined with the geometric relationship between the pile foundation and the position of the loose zone, the position coordinate range of the pile foundation within the loose zone is derived, and the obtained coordinate interval range is recorded as [zL, zR].

[0007] Preferably, the expression of the step function H(z) in step 3 is:

[0008] Preferably, in step 3, 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, in step 3, for the Vlasov foundation model, there are:

[0010] Preferably, in step 3, for the Vlasov foundation model, the soil displacement field analytical formula proposed by Loganathan is: Among them, x0 is the horizontal distance from the pile foundation axis to the tunnel axis; R is the tunnel excavation radius; H is the buried depth of the tunnel center.

[0011] Preferably, the element αnm in step 6 is determined by the following formula: Among them, hn is the cosine term coefficient of the cosine series expansion of the step function H(z), that is, When n = m, take hn = h0, h0 is the constant term of the cosine series expansion of the step function H(z), that is

[0012] Preferably, the elements β1, β2, and β3 in step 6 are determined by the following formula: φ1,n and φ2,n are known functions respectively. The coefficients of the sine series expansion of the sine series term are, that is, Un is the series coefficient of the known formation displacement function U(z) after the sine series expansion, that is,

[0013] Preferably, the comprehensive coefficients S1 to S6 in step 6 are determined by the following formula: S1=γ1α -1 β1, S2 = γ1α -1 β2, S3 = -γ1α -1 β3 S4=γ2α -1 β1, S5 = γ2α -1 β2,S6=-γ2α -1 β3 Among them, γ1 and γ2 are coefficient row vectors composed of known parameters, and their elements are determined by the following formula:

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

[0015] Shield tunnel excavation disturbs the surrounding soil, causing loose areas in the surrounding strata. This scheme assumes that the soil in the loose area has no supporting effect on the existing pile foundation. Combined with the Vlasov two-parameter foundation model, a calculation model for the horizontal displacement of adjacent pile foundations caused by shield excavation in the loose area is established. Based on the Fourier series method, a method for determining the horizontal displacement and internal force of adjacent pile foundations in the loose area caused by excavation is proposed. It is used to calculate the horizontal displacement and internal force of pile foundations caused by shield adjacent pile foundation construction. Compared with the numerical simulation method, the required physical quantities can be obtained by substituting the corresponding calculation parameters. The calculation process is simple. Only the two unknown quantities wO and wL related to the pile end horizontal displacement need to be solved 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 of the working condition in which shield excavation in a loose area causes horizontal displacement of adjacent pile foundations in an embodiment of the present invention.

[0018] Figure 2 This is a simplified diagram of the Vlasov foundation model in which shield excavation in a loose zone causes horizontal displacement of adjacent pile foundations in an embodiment of the present invention.

[0019] Figure 3 This is a simplified calculation diagram of the horizontal displacement of adjacent pile foundations caused by shield excavation in a loose area in an embodiment of the present invention.

[0020] Figure 4 It is a comparison diagram of the horizontal displacement of the pile foundation calculated in the embodiment of the present invention and the calculation result without considering the influence of the loose zone.

[0021] Figure 5 It is a comparison diagram of the pile foundation bending moment calculated in the embodiment of the present invention and the calculation result without considering the influence of the loose zone.

[0022] Figure 6 It is a comparison diagram of the pile foundation shear force calculated in the embodiment of the present invention and the calculation result without considering the influence of the loose zone. DETAILED DESCRIPTION

[0023] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0024] The present invention discloses a method for determining the horizontal displacement and internal force of adjacent pile foundations in a loosened area caused by excavation. In order to make the purpose of this application and the technical solution clearer, this application is further explained in conjunction with the existing literature, namely, Lin Xingtao's method combining numerical analysis with theoretical calculation in the soil arch effect of shield tunneling in sandy soil and its application.

[0025] After the construction of the tunnel adjacent to the pile foundation, the stratum loss rate ε0 is 5%, the soil elastic modulus Es is 24 MPa, the Poisson's ratio υs is 0.5, the tunnel excavation diameter D is 8.6 m, the pile foundation elastic modulus E is 30 GPa, the pile diameter B is 1 m, the pile length L is 25 m, the tunnel center burial depth H is 20 m, the horizontal spacing x0 between the tunnel axis and the pile foundation axis is 5 m, and the two ends of the pile foundation are assumed to be free ends. The calculation model and method of the loose zone are taken from the existing literature, namely, Lin Xingtao's "Soil Arch Effect of Shield Tunneling in Sandy Strata and Its Application", which uses a method combining numerical analysis and theoretical calculation. The internal friction angle of the soil is 10°.

[0026] The calculation process of the horizontal displacement and internal force of the pile foundation under the influence of shield construction is as follows:

[0027] Step 1: Determine whether the pile foundation edge line enters the loose area of ​​the surrounding strata caused by tunnel excavation. Figure 1 In the working condition shown, the horizontal distance xT from the tunnel centerline to the side line of the pile foundation adjacent to the tunnel excavation, the width of the loose zone Dz and the tunnel excavation diameter D satisfy the quantitative relationship: This indicates that the edge of the pile foundation has entered the loose area.

[0028] Step 2: Calculate the position coordinate range of the pile foundation within the loose zone. Let z be the coordinate axis along the axis of the adjacent pile foundation, and take the top of the pile as the coordinate origin z=0. According to the selected loose zone range calculation model, combined with the geometric relationship between the pile foundation and the position of the loose zone, deduce the position coordinate range of the pile foundation within the loose zone, and record the obtained coordinate interval range as [zL,zR]. Figure 3 As shown in the figure, Among them, the loose zone height Hz is calculated using the following formula:

[0029] Wherein, ε0 is the formation loss rate; α is the soil expansion coefficient, for loose sand or medium dense sand, α<0.08, for dense sand, 0.1<α<0.24, and in this embodiment, α=0.03.

[0030] Step 3: Fourier series method is used to solve the bending deformation of pile foundation under the influence of shield tunnel excavation. The dual-parameter foundation model adopts Vlasov foundation model. The pile foundation is assumed to be an elastic foundation beam, which satisfies the plane section assumption. After the shield tunnel is excavated, a loose zone is generated in the surrounding strata. It is assumed that the soil on the side of the pile foundation adjacent to the tunnel excavation in the loose zone disappears, such as Figure 2 As shown in the figure, the soil on the side of the pile foundation away from the tunnel excavation is blocked by the pile foundation, and it is assumed that the soil is not loose. The control differential equation of the horizontal displacement of the adjacent pile foundation caused by shield tunnel excavation is: Where: E is the elastic modulus of the adjacent pile foundation; I is the interface moment of inertia of the adjacent pile foundation; w is the horizontal displacement of the adjacent pile foundation, with the side away from the tunnel excavation as positive; H(z) is a step function; t is the load transfer rate; k is the stiffness coefficient of the soil on the pile side; B is the pile foundation diameter; U is the horizontal displacement of the stratum at the pile position caused by shield tunnel excavation.

[0031] The expression of the step function H(z) in step 3 is: Where L is the pile length.

[0032] In step 3, 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

[0033] In step 3, for the Vlasov foundation model, there are:

[0034] In step 3, for the Vlasov foundation model, we have: Take the soil displacement field analytical formula proposed by Loganathan: Among them, x0 is the horizontal distance from the pile foundation axis to the tunnel axis; R is the tunnel excavation radius; H is the buried depth of the tunnel center.

[0035] Step 4: Determine the calculation formula for the horizontal displacement of pile foundation caused by shield tunnel excavation: Where wn is the sinusoidal series term of the pile foundation horizontal displacement function; wO and wL are the horizontal displacements of the pile top and pile bottom, respectively.

[0036] Step 5: 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.

[0037] Step 6: The expressions of pile foundation horizontal displacement w, pile foundation bending moment M, and pile foundation shear force Q in steps 3-5 contain the undetermined coefficients wn, wO, and wL, where the undetermined coefficient wn is determined by the following formula: w=w O α -1 β1+w L α -1 β2+α -1 β1 Wherein, w is the column vector composed of unknown coefficients w1, w2, …, wn; α is the comprehensive coefficient matrix composed of known calculation parameters; β1, β2, β3 are the comprehensive coefficient column vectors composed of known parameters;

[0038] The undetermined coefficients wO and wL are determined by the following formula: In the formula, S1~S6 are comprehensive coefficients.

[0039] The element αnm in step 6 is determined by the following formula: Among them, hn is the cosine term coefficient of the cosine series expansion of the step function H(z), that is, When n = m, take hn = h0, h0 is the constant term of the cosine series expansion of the step function H(z), that is

[0040] In step 6, the elements β1, β2, and β3 are determined by the following formula: φ1,n and φ2,n are known functions respectively. The coefficients of the sine series expansion of the sine series term are, that is, Un is the series coefficient of the known formation displacement function U(z) after the sine series expansion, that is,

[0041] The comprehensive coefficients S1 to S6 in step 6 are determined by the following formula: S1=γ1α -1 β1, S2 = γ1α -1 β2, S3 = -γ1α -1 β3 S4=γ2α -1 β1, S5 = γ2α -1 β2,S6=-γ2α -1 β3 Among them, γ1 and γ2 are coefficient row vectors composed of known parameters, and their elements are determined by the following formula:

[0042] Substituting the known calculation parameters into the above formula, the unknown coefficients wn, wO, wL are obtained, and then the horizontal displacement, bending moment and shear force of the pile foundation caused by the adjacent construction of the tunnel are obtained, and compared with the calculation results of zL=zR without considering the loose area, as shown in the figure. Figures 4 to 6 As shown in the figure, since the integrity of the soil in the loose zone is destroyed, the force on the pile foundation in the loose zone is reduced, and the horizontal displacement, bending moment and shear force of the pile foundation are all reduced compared with the case where the influence of the loose zone is not considered.

[0043] It should be noted that the parts not described in detail in the above embodiments are all prior art.

[0044] 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 determining the horizontal displacement and internal force of adjacent pile foundations in a loose area caused by excavation, characterized in that: The parsing algorithm comprises the following steps: Step 1: Determine whether the pile foundation edge line enters the loose area of ​​the surrounding strata caused by tunnel excavation; Step 2, calculating the position coordinate range of the pile foundation within the loose area; Step 3: Fourier series method is used to solve the bending deformation of pile foundation under the influence of shield tunnel excavation. The dual-parameter foundation model adopts Vlasov foundation model. The pile foundation is assumed to be an elastic foundation beam, which satisfies the plane section assumption. The horizontal displacement control differential equation of the adjacent pile foundation caused by shield tunnel excavation is: Where: E is the elastic modulus of the adjacent pile foundation; I is the interface moment of inertia of the adjacent pile foundation; w is the horizontal displacement of the adjacent pile foundation, with the side away from the tunnel excavation as positive; H(z) is a step function; t is the load transfer rate; k is the stiffness coefficient of the soil on the pile side; B is the pile foundation diameter; U is the horizontal displacement of the stratum at the pile position caused by the shield tunnel excavation; Step 4: 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, that is, the pile length; w O 、w L are the horizontal displacements of the pile top and pile bottom, respectively; Step 5: 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 6: The expressions of pile foundation horizontal displacement w, pile foundation bending moment M, and pile foundation shear force Q in steps 3-5 contain the unknown coefficient w n 、w O 、w L , where the unknown coefficient w n Determined by the following formula: w=w O a -1 β1+w L a -1 β2+α -1 b1 Where w is the unknown coefficient w1, w2, ..., w n α is the comprehensive coefficient matrix composed of known calculation parameters; β1, β2, β3 are the comprehensive coefficient column vectors composed of known parameters; Undetermined coefficient w O 、w L Determined by the following formula: In the formula, S1~S6 are comprehensive coefficients.

2. The method for determining the horizontal displacement and internal force of adjacent pile foundations in a loosened area caused by excavation according to claim 1, characterized in that: The horizontal distance x from the tunnel centerline to the side line of the pile foundation adjacent to the tunnel excavation in step 1 T The width of the loose area is D z The quantitative relationship between the tunnel excavation diameter D satisfies: This indicates that the edge of the pile foundation has entered the loose area.

3. The method for determining the horizontal displacement and internal force of adjacent pile foundations in a loosened area caused by excavation according to claim 2, characterized in that: In step 2, z is set as the coordinate axis along the direction of the adjacent pile foundation axis, and the top of the pile is taken as the coordinate origin z=0. According to the selected loose zone range calculation model, combined with the geometric relationship between the pile foundation and the position of the loose zone, the position coordinate range of the pile foundation within the loose zone is derived, and the obtained coordinate interval range is recorded as [z L ,z R ].

4. The method for determining the horizontal displacement and internal force of adjacent pile foundations in a loosened area caused by excavation according to claim 3, characterized in that: The expression of the step function H(z) in step 3 is:

5. The method for determining the horizontal displacement and internal force of adjacent pile foundations in a loosened area caused by excavation according to claim 4, characterized in that: In step 3, for the Vlasov foundation model, there are: 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 6. The method for determining the horizontal displacement and internal force of adjacent pile foundations in a loosened area caused by excavation according to claim 5, characterized in that: In step 3, for the Vlasov foundation model, there are:

7. The method for determining the horizontal displacement and internal force of adjacent pile foundations in a loosened area caused by excavation according to claim 6, characterized in that: In step 3, for the Vlasov foundation model, we have: Take the soil displacement field analytical formula proposed by Loganathan: Among them, x0 is the horizontal distance from the pile foundation axis to the tunnel axis; R is the tunnel excavation radius; H is the buried depth of the tunnel center.

8. The method for determining the horizontal displacement and internal force of adjacent pile foundations in a loosened area caused by excavation according to claim 7, characterized in that: In step 6, the element α nm Determined by the following formula: Among them, h n is the cosine term coefficient of the cosine series expansion of the step function H(z), that is When n = m, take h n =h0, h0 is the constant term of the cosine series expansion of the step function H(z), that is, 9. A method for determining horizontal displacement and internal force of adjacent pile foundations in a loosened area caused by excavation according to claim 8, characterized in that: In step 6, the elements β1, β2, and β3 are determined by the following formula: The known functions are The coefficients of the sine series expansion of the sine series term are, that is, U n is the series coefficient of the sine series expansion of the known formation displacement function U(z), that is, 10. A method for determining horizontal displacement and internal force of adjacent pile foundations in a loosened area caused by excavation according to claim 9, characterized in that: The comprehensive coefficients S1 to S6 in step 6 are determined by the following formula: S1=γ1α -1 β1, S2=γ1α -1 β2, S3=-γ1α -1 b3 S4=γ2α -1 β1,S5=γ2α -1 β2,S6=-γ2α -1 β3 Among them, γ1 and γ2 are coefficient row vectors composed of known parameters, and their elements are determined by the following formula:

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