A method for determining the horizontal displacement and internal force of adjacent pile foundations in loose areas caused by excavation

Abstract

The present invention provides a method for determining the horizontal displacement and internal force of adjacent pile foundations in a loosened area caused by excavation. Shield tunnel excavation disturbs the surrounding soil, causing a loosened area in the surrounding strata. This scheme assumes that the soil within the loosened 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 loosened area is established. Based on the Fourier series method, a method for determining the horizontal displacement and internal force of adjacent pile foundations in a loosened area caused by excavation is proposed. The method is used to calculate the horizontal displacement and internal force of pile foundations caused by shield construction adjacent to pile foundations. 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, and only the horizontal displacement of the pile end needs to be solved. w O 、 w L Two unknown quantities can be used to further obtain other unknown coefficients, and the calculation efficiency is high.

Description

Technical Field

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

[0002] Shield tunneling causes secondary stress and displacement fields in the soil. Furthermore, the disturbance of the in-situ soil during construction will destroy the integrity of the soil surrounding the tunnel, causing stratum loss and, in turn, creating a loosened zone in the stratum surrounding the tunnel. Existing theoretical analytical methods for calculating pile foundation deformation typically assume good integrity when calculating foundation reaction forces. This assumes the foundation is homogeneous, without considering the impact of loosening of the surrounding soil caused by tunnel excavation on the pile foundation stress. Within the loosened zone, the integrity of the soil is destroyed, and the assumption of a homogeneous foundation is inconsistent with the actual stratum conditions. Therefore, it is necessary to conduct research on the determination of the horizontal displacements and internal forces of pile foundations adjacent to the loosened zone caused by excavation, and to provide the corresponding analytical calculation formulas and steps for pile foundation displacements and internal forces. 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 situations where loose zones may occur in sandy strata, such as those surrounding newly built tunnels, the present invention combines a loose zone calculation model and method, assumes that both ends of the pile foundation are free ends, and relies on the two-parameter Vlasov foundation model to derive formulas for determining the horizontal displacement and internal force of adjacent pile foundations in a loose zone caused by excavation.

[0004] To achieve the above-mentioned object, the technical solution adopted by the present invention is: a method for determining the horizontal displacement and internal force of adjacent pile foundations in a loose area caused by excavation, wherein the analytical algorithm comprises the following steps:

[0005] Step 1: Determine whether the pile foundation edge enters the loose zone of the surrounding strata caused by tunnel excavation;

[0006] Step 2: Calculate the position coordinate range of the pile foundation within the loose area;

[0007] Step 3: 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 pile foundation is assumed to be an elastic foundation beam, satisfying the plane section assumption. The horizontal displacement control differential equation of the adjacent pile foundation caused by shield tunnel excavation is:

[0008]

[0009] 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 value away from the tunnel excavation side being positive; H(z) is a step function; t is the load transfer rate; k is the stiffness coefficient of the soil near the pile; B is the pile foundation diameter; U is the horizontal displacement of the stratum at the pile position caused by shield tunnel excavation;

[0010] Step 4: Determine the calculation formula for the horizontal displacement of pile foundation caused by shield tunnel excavation:

[0011]

[0012] 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;

[0013] Step 5: Determine the calculation formula for pile foundation bending moment and shear force caused by shield tunnel excavation

[0014]

[0015] Where, M is the pile foundation bending moment; Q is the pile foundation shear force;

[0016] 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:

[0017] w=w O α -1 β1+w L α -1 β2+α -1 β1

[0018] Where w is the column vector composed of the 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;

[0019] The undetermined coefficients wO and wL are determined by the following formula:

[0020]

[0021] Where S1~S6 are comprehensive coefficients.

[0022] Preferably, in step 1, the horizontal distance xT from the tunnel centerline 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 following quantitative relationship: This indicates that the edge of the pile foundation has entered the loose area.

[0023] 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].

[0024] Preferably, the expression of the step function H(z) in step 3 is

[0025]

[0026] Preferably, in step 3, 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

[0027] Preferably, in step 3, for the Vlasov foundation model, there are:

[0028] Preferably, in step 3, for the Vlasov foundation model, the soil displacement field analytical formula proposed by Loganathan is:

[0029]

[0030] Where 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.

[0031] Preferably, the element αnm in step 6 is determined by the following formula:

[0032]

[0033] 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, where h0 is the constant term of the cosine series expansion of the step function H(z), that is,

[0034] Preferably, the elements β1, β2, and β3 in step 6 are determined by the following formula:

[0035]

[0036] 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,

[0037] Preferably, the comprehensive coefficients S1 to S6 in step 6 are determined by the following formula:

[0038] S1=γ1α -1 β1, S2 = γ1α -1 β2, S3 = -γ1α -1 β3

[0039] S4=γ2α -1 β1, S5 = γ2α -1 β2, S6 = -γ2α -1 β3

[0040] Among them, γ1 and γ2 are coefficient row vectors composed of known parameters, and their elements are determined by the following formula:

[0041]

[0042] The beneficial effects of the present invention are:

[0043] Shield tunnel excavation disturbs the surrounding soil, causing loose zones in the surrounding strata. This scheme assumes that the soil within the loose zone 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 zone 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 zone caused by excavation is proposed. This method is used to calculate the horizontal displacement and internal force of pile foundations caused by shield construction adjacent to pile foundations. 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, providing an effective analytical calculation method for the safety risk assessment of existing structures under shield construction adjacent to pile foundations. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] 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.

[0045] 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.

[0046] Figure 2 This is a simplified diagram of the Vlasov foundation model in which shield excavation in a loose area causes horizontal displacement of adjacent pile foundations in an embodiment of the present invention.

[0047] Figure 3 This is a simplified diagram for calculating the horizontal displacement of adjacent pile foundations caused by shield excavation in a loose area in an embodiment of the present invention.

[0048] Figure 4 3 is a comparison chart 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.

[0049] Figure 5 3 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.

[0050] Figure 6 3 is a comparison chart of the pile foundation shear force calculated in the embodiment of the present invention and the calculation results without considering the influence of the loose zone. DETAILED DESCRIPTION

[0051] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0052] The present invention discloses a method for determining the horizontal displacement and internal forces of adjacent pile foundations in loosened areas caused by excavation. To clarify the purpose and technical solution of this application, this application is further described in conjunction with Lin Xingtao's existing literature, "Soil Arching Effect of Shield Tunneling in Sandy Strata and Its Application Using a Method Combining Numerical Analysis with Theoretical Calculation."

[0053] 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 distance 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 loose zone calculation model and method are taken from the existing literature, namely, Lin Xingtao's "Soil Arching Effect of Shield Tunneling in Sandy Strata and Its Application Using a Method Combining Numerical Analysis and Theoretical Calculation", and the soil internal friction angle is taken as 10°.

[0054] The calculation process of the horizontal displacement and internal force of the pile foundation under the influence of shield construction is as follows:

[0055] Step 1: Determine whether the pile foundation edge enters the loose area of ​​the surrounding strata caused by tunnel excavation. Figure 1 For 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 following quantitative relationship: This indicates that the edge of the pile foundation has entered the loose area.

[0056] Step 2: Calculate the coordinate range of the pile foundation within the loosening zone. Let z be the coordinate axis along the axis of the adjacent pile foundation, and the top of the pile as the coordinate origin z = 0. Based on the selected loosening zone calculation model and the geometric relationship between the pile foundation and the loosening zone position, deduce the coordinate range of the pile foundation within the loosening zone. The obtained coordinate interval range is recorded as [zL, zR]. Figure 3 As shown in the figure, Among them, the height Hz of the loose area is calculated using the following formula:

[0057]

[0058] 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. In this embodiment, α=0.03.

[0059] Step 3: 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 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 the foundation support effect on the existing pile foundation, such as Figure 2 As shown in the figure, the soil on the side of the pile foundation facing away from the tunnel excavation is blocked by the pile foundation, and it is assumed that there is no loosening of the soil. The differential equation governing the horizontal displacement of the adjacent pile foundation caused by shield tunnel excavation is:

[0060]

[0061] 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 direction 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 near the pile; B is the pile foundation diameter; and U is the horizontal displacement of the stratum at the pile position caused by shield tunnel excavation.

[0062] The expression of the step function H(z) in step 3 is:

[0063]

[0064] Where L is the pile length.

[0065] In step 3, 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

[0066] In step 3, for the Vlasov foundation model, there are:

[0067] In step 3, for the Vlasov foundation model, we have: Take the soil displacement field analytical formula proposed by Loganathan:

[0068]

[0069] Where 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.

[0070] Step 4: Determine the calculation formula for the horizontal displacement of pile foundation caused by shield tunnel excavation:

[0071]

[0072] Where wn is the sine series term of the pile foundation horizontal displacement function; wO and wL are the horizontal displacements of the pile top and pile bottom, respectively.

[0073] Step 5: Determine the calculation formula for pile foundation bending moment and shear force caused by shield tunnel excavation

[0074]

[0075] Where M is the bending moment of the pile foundation; Q is the shear force of the pile foundation.

[0076] 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:

[0077] w=w O α -1 β1+w L α -1 β2+α -1 β1

[0078] Where w is the column vector composed of the 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;

[0079] The undetermined coefficients wO and wL are determined by the following formula:

[0080]

[0081] Where S1~S6 are comprehensive coefficients.

[0082] The element αnm in step 6 is determined by the following formula:

[0083]

[0084] 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, where h0 is the constant term of the cosine series expansion of the step function H(z), that is,

[0085] In step 6, the elements β1, β2, and β3 are determined by the following formula:

[0086]

[0087] 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,

[0088] The comprehensive coefficients S1 to S6 in step 6 are determined by the following formula:

[0089] S1=γ1α -1 β1, S2 = γ1α -1 β2, S3 = -γ1α -1 β3

[0090] S4=γ2α -1 β1, S5 = γ2α -1 β2, S6 = -γ2α -1 β3

[0091] Among them, γ1 and γ2 are coefficient row vectors composed of known parameters, and their elements are determined by the following formula:

[0092]

[0093] Substituting the known calculation parameters into the above formula, the unknown coefficients wn, wO, and 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. The results are compared with the calculation results of zL=zR without considering the loose area, as shown in the following example: Figures 4 to 6 As shown in the figure, since the integrity of the soil in the loose zone is destroyed, the force acting 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.

[0094] It should be noted that the parts not described in detail in the above embodiments are all prior art.

[0095] 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 determining the horizontal displacement and internal force of adjacent pile foundations in a loosened area caused by excavation, characterized by: The parsing algorithm consists of the following steps: Step 1: Determine whether the pile foundation edge enters the loose zone of the surrounding strata caused by tunnel excavation; Step 2: Calculate the position coordinate range of the pile foundation within the loose area; Step 3: 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 pile foundation is assumed to be an elastic foundation beam, satisfying 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 value away from the tunnel excavation side being positive; H(z) is a step function; t is the load transfer rate; k is the stiffness coefficient of the soil near the pile; B is the pile foundation diameter; U is the horizontal displacement of the stratum at the pile position caused by shield tunnel excavation; Step 4: Determine the calculation formula for the horizontal displacement of pile foundation caused by shield tunnel excavation: Where w n is the sine series term of the pile foundation horizontal displacement function; L is the calculated length of the pile foundation, 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 pile foundation bending moment; Q is the pile foundation shear force; 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 a comprehensive coefficient matrix composed of known calculation parameters; β1, β2, β3 are comprehensive coefficient column vectors composed of known parameters; Undetermined coefficient w O 、w L Determined by the following formula: Where 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 With the loose zone width 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 axis 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 [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 changing 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: Where 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.

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: The element α in step 6 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. The method for determining the 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 The coefficients of the sine series expansion of the sine series term are, that is, U n is the series coefficient after the sine series expansion of the known formation displacement function U(z), that is, 10. The method for determining the 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:

Images