A method for calculating deformation of adjacent pile foundation considering space effect of foundation pit excavation

Abstract

This invention discloses a method for calculating the deformation of adjacent pile foundations considering the spatial effect of foundation pit excavation. The method comprises the following steps: First, determining the spatial distribution curve of the lateral deformation of the retaining wall; second, determining the free displacement field of the soil outside the pit induced by the spatial deformation of the foundation pit; and finally, determining the governing equations for the deformation of adjacent pile foundations using the Pasternak foundation and Euler-Bernoulli beam models. This invention takes into account the significant spatial effect of the surrounding soil deformation at different spatial locations of the pile foundation. When the pile foundation is near the center of the foundation pit, the surrounding soil deformation is often large. Therefore, considering the spatial shape of the soil deformation and the spatial distribution of the pile foundation can better reproduce the actual situation of foundation pit excavation and the response of the pile foundation.

Description

Technical Field

[0001] This invention relates to the field of underground structure design, specifically to a method for calculating the deformation of adjacent pile foundations that takes into account the spatial effect of foundation pit excavation. Background Technology

[0002] With the advancement of urbanization and the increasing scarcity of land resources, underground space development has been continuously driven, leading to a surge in the number of foundation pit projects, which are characterized by large scale, tight spacing, and depth. Foundation pit excavation inevitably disturbs the surrounding soil, disrupting its original stress balance. Lateral deformation of the soil causes movement and deformation of adjacent pile foundations, and may even result in damage or destruction. Numerous similar engineering cases exist both domestically and internationally. Therefore, accurately assessing the stress and deformation of adjacent pile foundations caused by foundation pit excavation is of paramount importance.

[0003] Existing methods for addressing the additional deformation and internal forces of adjacent pile foundations caused by foundation pit excavation mainly include numerical analysis, laboratory tests, and theoretical research. Numerical simulation can analyze the retaining structure and surrounding environment as a whole, effectively simulating the complex interactions between piles and soil. However, the accuracy of the response patterns of adjacent pile foundations obtained through numerical simulation depends on the rationality of soil parameter values, and the modeling is complex and labor-intensive. Model testing provides more intuitive results and can better reproduce the real soil stress field, but sample preparation is complex and costly. Theoretical methods have clear principles and shorter computation time, but existing theoretical studies are mostly based on two-dimensional plane assumptions. The deformation of the foundation pit and the distribution of soil pressure outside the pit caused by excavation have significant spatial effects. The horizontal deformation and passive earth pressure of the retaining structure are minimal at the corners of the pit, reaching their maximum in the middle. Existing theoretical studies do not consider the spatial distribution effects of the displacement and stress fields around the pit on adjacent pile foundations, thus resulting in significant discrepancies with actual conditions. Summary of the Invention

[0004] To address the shortcomings of the aforementioned background technology, this invention provides a method for calculating the deformation of adjacent pile foundations that considers the spatial effect of foundation pit excavation. This method can better reproduce the actual situation of foundation pit excavation and the response of the pile foundation.

[0005] This invention is achieved using the following technical solution:

[0006] A method for calculating the deformation of adjacent pile foundations considering the spatial effect of foundation pit excavation includes the following steps:

[0007] (1) Determine the spatial distribution curve of the lateral deformation of the retaining wall;

[0008] (2) Determine the free displacement field of the soil outside the pit induced by the spatial deformation of the pit;

[0009] (3) The deformation control equations of adjacent pile foundations were determined using Pasternak foundation and Euler-Bernoulli beam models.

[0010] In the above technical solution, further, in step (1), the spatial distribution of the lateral deformation of the enclosure wall is represented as follows:

[0011]

[0012] 0≤y≤L, 0≤z≤H+D

[0013] Where: T = 0.5L[0.069ln(H / L) + 1.03]; f max y is the maximum lateral deformation of the retaining wall; y is the horizontal position; z is the depth of the calculation point; H is the excavation depth of the foundation pit; D is the insertion depth of the retaining wall; L is the length of the retaining wall along the wall direction; H max The depth H where the maximum lateral deformation of the retaining wall is located. max =H.

[0014] Furthermore, step (2) specifically includes:

[0015] A three-dimensional model is established using the image source method, constructing a pore with radius 'a' and an outer sphere with radius greater than 'a' and concentric with the pore: when the pore contracts, the outer sphere contracts inward; when the pore expands, the outer sphere expands outward.

[0016] The deformation of the retaining wall at (0, y0, z0) is f(0, y0, z0), where y0 is the coordinate along the length of the pit and z0 is the coordinate along the depth of the pit. Assuming a point P1(x1, y1, z1) outside the pit is a point on the outer sphere, after the pores shrink, the radial displacement of the outer sphere is S. r At this point, the following conditions are met:

[0017]

[0018] Where r is the distance between the center of the pore and P1(x1,y1,z1); since S r Since r is very small, the soil displacement components at P1(x1,y1,z1) are discarded after discarding higher-order infinitesimals.

[0019]

[0020] Similarly, the displacement components of each soil mass caused by pore expansion can be obtained.

[0021] The soil displacement caused by the normal stress after superimposing the displacement caused by pore shrinkage is:

[0022]

[0023] In the formula,

[0024] Next, we need to solve for the soil displacement caused by the shear stress. First, we need to find the magnitude of the shear stress, which can be obtained from the knowledge of elasticity:

[0025]

[0026] In the formula, γ zx With τ zx These are the surface shear strain and shear stress, respectively, and G is the soil shear modulus.

[0027] The formula for calculating the shear stress caused by pore contraction or expansion at (x,y,0) is as follows:

[0028]

[0029] When a horizontal concentrated force acts on the Earth's surface, the displacement produced by the force at any point can be obtained from the Cerruti problem displacement solution. However, the Cerruti problem displacement solution only provides the expression when the horizontal concentrated force acts at the origin of the coordinate system, while the shear stress does not only act at the origin. Therefore, a global coordinate system Oxyz and local coordinate systems O′x′y′z′ and O″x″y″z″ are established. From the Cerruti problem displacement solution, when the horizontal concentrated force F acts at the local coordinate system O′x′y′z′(0, 0, 0), the displacement components at P1′(x1′, y1′, z1′) are as follows:

[0030]

[0031] In the formula, E and ν are the elastic modulus and Poisson's ratio of the soil, respectively;

[0032] Apply the shear stress in the opposite direction to the surface, and let dF = τ. zx Given dxdy, it is easy to know that x1′=x1-x, y1′=y1-y, z1′=z1, and thus we can obtain the result when τ zx The soil displacement in the x-direction produced at P1(x1,y1,z1) acting on the global coordinate system Oxyz(x,y,0) is:

[0033]

[0034] In the formula,

[0035] Substitute τ zx Equation (7) is further simplified, and the shear stress τ on the entire surface is considered. zx Integrating, we get:

[0036]

[0037] S 1_x The shear stress τ on the entire surface of the earth zx The resulting soil displacement in the x-direction;

[0038] When τ zy When applied to the local coordinate system O″x″y″z″(0,0,0), the S in formula (6) can be applied. y′ The algorithm is readily known, and it is also easy to see that x1″=y1′, y1″=-x1′, z1″=z1′. Then, using formula (7), the shear stress τ on the entire surface can be obtained. zy The resulting soil displacement S in the x-direction at P1(x1,y1,z1) 2_x for

[0039]

[0040] Thus, the soil displacement in the x-direction at P1(x1,y1,z1) caused by the elimination of shear stress is obtained. Adding the displacement caused by shrinkage or expansion of pores and the displacement generated by the reverse applied shear stress, the calculation formula for the soil displacement in the x-direction at P1(x1,y1,z1) outside the pit caused by soil loss at (0,y0,z0) is as follows:

[0041] S _xi =S σ_x +S 1_x +S 2_x (10)

[0042] Considering that the excavation of the foundation pit causes the soil on both sides of the retaining wall to no longer satisfy the infinite half-space condition, the problem is transformed into an infinite half-space problem by mirroring. The original problem is transformed into solving the deformation at any point under the stratum loss of 2f(0,y0,z0). The retaining wall is divided into n infinitesimal elements, each with a size of dy0dz0. Using the principle of equivalent volume, the volume of each infinitesimal element is equivalent to the volume of a sphere.

[0043]

[0044] Substituting formula (11) into formula (10) and eliminating 'a', we can obtain the soil displacement in the x-direction at P1(x1,y1,z1) outside the pit caused by the excavation:

[0045]

[0046] Furthermore, step (3) specifically includes:

[0047] Treating the pile foundation as an Euler-Bernoulli beam on the Pasternak foundation, the differential deformation governing equations are as follows:

[0048]

[0049] In the formula, E p I p and D e Let be the bending stiffness and equivalent width of the pile, respectively; w(z) be the additional lateral deformation of the pile foundation; and q(z) be the additional load at the pile foundation caused by the excavation of the foundation pit. The calculation formulas are as follows:

[0050]

[0051] In the formula, S x For lateral displacement of the soil; k, G s The two parameters for the Pasternak foundation are the foundation reaction modulus and the shear layer stiffness:

[0052]

[0053]

[0054] In the formula: η is the depth parameter, h is the pile embedment depth, and E s ν and ν are the elastic modulus and Poisson's ratio of the foundation soil, respectively. For multi-layer soil, the weighted average value is taken according to the soil layer thickness.

[0055]

[0056] In the formula, t is the shear layer thickness, taken as t = 11D. p D p The diameter of the pile foundation outside the pit;

[0057] According to the difference method, the pile is divided into n segments, each segment is l long, and there are a total of n+1 nodes. Formula (13) is rewritten as:

[0058]

[0059] Among them, w i Let q be the deformation of the i-th pile node; i Let be the load borne by the i-th pile node;

[0060] For a friction pile that is free at both ends, with the boundary conditions that the bending moment and shear force at both ends are zero, the following system of equations can be obtained:

[0061]

[0062] Each node i has a deformation governing equation. The n+1 deformation differential equations are written as a system of equations and then rewritten as matrix expressions, as follows:

[0063] (K1-K2+K3)w=P (20)

[0064]

[0065]

[0066]

[0067] P = D e [q0 q1 q2 … q n-2 q n-1 q n ] T (n+1) (twenty four)

[0068] For end-bearing piles with fixed ends but incompletely fixed tops, the boundary conditions are: rotation angles at both ends are 0, pile end displacement is 0, and pile top displacement is b.

[0069]

[0070] The matrix expression is obtained as follows:

[0071] (K1'-K2'+K3)w+J+N=P (26)

[0072] In the formula, K1', K2', J, and N are as follows:

[0073]

[0074]

[0075]

[0076]

[0077] The beneficial effects of this invention are as follows:

[0078] The method of this invention considers the influence of the spatial deformation of the retaining wall on the deformation of the soil outside the pit, thus more realistically reproducing the deformation of the soil outside the pit. At different spatial locations of the pile foundation, the deformation of the surrounding soil exhibits a significant spatial effect. When the pile foundation is near the middle of the pit, the deformation of the surrounding soil is often larger. Therefore, by considering the spatial shape of the soil deformation and the spatial distribution of the pile foundation, the actual situation of the pit excavation and the response of the pile foundation are better reproduced. Attached Figure Description

[0079] Figure 1 This is a schematic diagram of the image source method for solving the problem;

[0080] Figure 2 A schematic diagram of the Cerruti coordinate system;

[0081] Figure 3 This is a comparison curve of the horizontal displacement of the pile body calculated and measured by the method of the present invention. Detailed Implementation

[0082] The present invention provides a method for calculating the deformation of adjacent pile foundations considering the spatial effect of foundation pit excavation, comprising the following steps:

[0083] 1. Determine the spatial distribution curve of the lateral deformation of the retaining wall.

[0084] 2. Determine the free displacement field of the soil outside the pit induced by the spatial deformation of the foundation pit.

[0085] 3. Determine the deformation control equations of adjacent pile foundations using Pasternak foundation and Euler-Bernoulli beam models.

[0086] The specific implementation methods for each step are as follows:

[0087] 1. Spatial distribution of lateral deformation of the retaining wall

[0088]

[0089] 0≤y≤L, 0≤z≤H+D

[0090] Where: T = 0.5L[0.069ln(H / L) + 1.03]; f max y is the maximum lateral deformation of the retaining wall; y is the horizontal position; z is the depth of the calculation point; H is the excavation depth of the foundation pit; D is the insertion depth of the retaining wall; L is the length of the retaining wall along the wall direction; H max The depth H where the maximum lateral deformation of the retaining wall is located. max =H. f max and H max The design calculations or measured values ​​of the support structure can be used for analysis depending on the different stages of the project.

[0091] Solving the free displacement field of the soil outside the pit 2

[0092] Image source method ( Figure 1 A three-dimensional model is established with a pore radius of a. P1(x1,y1,z1) is located on the outer sphere. Due to pore contraction, P1 contracts inward and, after displacement, is located on the inner sphere. The radial displacement of the contraction is S. r (Similarly, when the pores expand, P1 expands outward and, after displacement, is located on the outer sphere, with the center of all spheres in the same position). At this time, the following conditions are met:

[0093]

[0094] The deformation of the retaining wall at (0, y0, z0) is f(0, y0, z0), where y0 is the coordinate along the length of the excavation pit and z0 is the coordinate along the depth of the excavation pit; the soil displacement S rSince the distance r between the calculation point P1(x1,y1,z1) and the shrinkage pore is very small, after discarding higher-order infinitesimals, the soil displacement components at a certain point P1(x1,y1,z1) outside the pit are (coordinate system as follows). Figure 1 (c)):

[0095]

[0096] Similarly, the displacement components of each soil mass caused by pore expansion can be obtained.

[0097] The soil displacement caused by normal stress is obtained by superimposing the displacement caused by pore shrinkage:

[0098]

[0099] In the formula,

[0100] Next, we will solve for the soil displacement caused by shear stress. First, we need to determine the magnitude of the shear stress, using knowledge from elasticity mechanics:

[0101]

[0102] In the formula, γ zx With τ zx Here, denoted as surface shear strain and shear stress, respectively, and G is the soil shear modulus.

[0103] The shear stress caused by pore contraction or expansion at (x,y,0) (e.g.) Figure 1 (c) shows that the positive direction of the shear stress in the figure is consistent with the convention in mechanics of materials (the actual direction is determined by the sign of the numerical value), and the calculation formula is as follows:

[0104]

[0105] When a concentrated horizontal force acts on the Earth's surface, the displacement produced by this force at any point can be obtained from the Cerruti problem's displacement solution. However, the Cerruti problem's displacement solution only provides an expression for the concentrated horizontal force acting at the origin, while shear stress does not only act at the origin. Therefore, a global coordinate system Oxyz and local coordinate systems O′x′y′z′ and O″x″y″z″ are established. Figure 2 From the displacement solution of the Cerruti problem, when a horizontal concentrated force F acts on the local coordinate system O′x′y′z′(0, 0, 0), the displacement components at P1′(x1′, y1′, z1′) are as follows:

[0106]

[0107] In the formula, E and ν are the elastic modulus and Poisson's ratio of the soil, respectively.

[0108] Apply the shear stress in the opposite direction to the surface, and let dF = τ. zx Given dxdy, it is easy to know that x1′=x1-x, y1′=y1-y, z1′=z1, and thus we can obtain the result when τ zx The soil displacement in the x-direction produced at P1(x1,y1,z1) acting on the global coordinate system Oxyz(x,y,0) is:

[0109]

[0110] In the formula,

[0111] Substitute τ zx Equation (7) is further simplified, and the shear stress τ on the entire surface is considered. zx Integrating, we get:

[0112]

[0113] S 1_x The shear stress τ on the entire surface of the earth zx The resulting soil displacement in the x-direction.

[0114] like Figure 2 As shown, when τ zy When applied to the local coordinate system O″x″y″z″(0,0,0), the S in formula (6) can be applied. y′ The algorithm is readily known, and it is also easy to see that x1″=y1′, y1″=-x1′, z1″=z1′. Then, by using the same calculation method as formula (7), the shear stress τ on the entire surface can be obtained. zy The resulting soil displacement S in the x-direction at P1(x1,y1,z1) 2_x for:

[0115]

[0116] Thus, the soil displacement in the x-direction at P1(x1,y1,z1) caused by the elimination of shear stress is obtained. Adding the displacements caused by shrinkage or expansion pores and the reverse applied shear stress, the calculation formula for the soil displacement in the x-direction at P1(x1,y1,z1) outside the pit caused by soil loss at (0,y0,z0) is as follows:

[0117] S _xi =S σ_x +S 1_x +S 2_x (40)

[0118] Considering that the excavation of the foundation pit causes the soil on both sides of the retaining wall to no longer satisfy the infinite half-space condition, a mirror treatment is used to transform it into an infinite half-space problem. The original problem is transformed into solving the deformation at any point under the stratum loss of 2f(0,y0,z0). The retaining wall is divided into n infinitesimal elements (n can be infinite), each element having a size of dy0dz0. Using the principle of equivalent volume, the volume of each element is equivalent to the volume of a sphere:

[0119]

[0120] Substituting formula (11) into formula (10) and eliminating 'a', we can obtain the soil displacement in the x-direction at P1(x1,y1,z1) outside the pit caused by the excavation:

[0121]

[0122] 3. Deformation control equations of pile foundations in the Pasternak foundation model

[0123] Treating the pile foundation as an Euler-Bernoulli beam on the Pasternak foundation, the differential deformation governing equations are as follows:

[0124]

[0125] In the formula, E p I p and D e Let be the bending stiffness and equivalent width of the pile, respectively; w(z) be the additional lateral deformation of the pile foundation; and q(z) be the additional load at the pile foundation caused by the excavation of the foundation pit, as follows:

[0126]

[0127] In the formula, S x The lateral displacement of the soil calculated above; k, G s The two parameters for the Pasternak foundation are the foundation reaction modulus and the shear layer stiffness:

[0128]

[0129]

[0130] In the formula: η is the depth parameter, h is the pile embedment depth, and E s ν and ν are the elastic modulus and Poisson's ratio of the foundation soil, respectively. For multi-layer soil, the weighted average value is taken according to the soil layer thickness.

[0131]

[0132] In the formula, t is the shear layer thickness, taken as t = 11D. p t = 11Dp D p The diameter of the pile foundation outside the pit.

[0133] According to the difference method, the pile is divided into n segments, each segment is l long, and there are a total of n+1 nodes. Formula (13) is rewritten as:

[0134]

[0135] w i Let q be the deformation of the i-th pile node; i Let be the load borne by the i-th pile node;

[0136] For a friction pile with both ends free, the boundary condition is that the bending moment and shear force at both ends are 0: as z→0, the bending moment M i =0, shear force Q i =0; z→pile length L p At that time, M n =0, Q n =0. The system of equations is as follows:

[0137]

[0138] Each node i has a deformation governing equation. The n+1 deformation differential equations are written as a system of equations and then rewritten as matrix expressions, as follows:

[0139] (K1-K2+K3)w=P (50)

[0140]

[0141]

[0142]

[0143] P = D e [q0 q1 q2 … q n-2 q n-1 q n ] T (n+1) (54)

[0144] For end-bearing piles with fixed ends but incompletely fixed tops, the boundary conditions are: when the rotation angles at both ends are 0, the pile end displacement is 0, and the pile top displacement is b: z→0, θ i =0, w i =b; z→L p When, θ i =0, w i =0.

[0145]

[0146] The matrix expression is obtained as follows:

[0147] (K1'-K2'+K3)w+J+N=P (56)

[0148] In the formula, K1', K2', J, and N are as follows, and the rest are the same as before, so they will not be repeated.

[0149]

[0150]

[0151]

[0152]

[0153] When b = 0, it is completely fixed.

[0154] Example

[0155] The foundation pit retaining structure adopts a combination of bored piles and internal bracing. The excavation depth H of the foundation pit is 9m, and the length L is 120m; the insertion depth of the retaining piles D is 9m; the maximum deformation of the retaining wall is H. max The maximum horizontal displacement of the retaining wall is 9m; max Approximately 32mm. The distance x1 from the edge of the pit to the pile foundation outside the pit is 2.4m, and the location y1 is 40m; the pile length L p The length is 18m, and the pile diameter is D. p The length is 0.8m, and the elastic modulus E of the pile body is... p The Pa is 28 GPa; the pile body is located in a silty clay layer, the Poisson's ratio υ is 0.35, and the elastic modulus E is 0.35. s It is 16 MPa.

[0156] The deformation of adjacent pile foundations is calculated based on the method of this invention and compared with measured data. Figure 3 It can be seen that the calculation method of the present invention can be well matched with the measured data, indicating that the method of the present invention can better reproduce the actual situation of foundation pit excavation and the response of pile foundation.

Claims

1. A method for calculating the deformation of adjacent pile foundations considering the spatial effect of foundation pit excavation, characterized in that, Includes the following steps: (1) Determine the spatial distribution curve of the lateral deformation of the retaining wall; (2) Determine the free displacement field of the soil outside the pit induced by the spatial deformation of the pit; (3) The deformation control equations of adjacent pile foundations were determined using the Pasternak foundation and Euler-Bernoulli beam models; In step (1), the spatial distribution of lateral deformation of the retaining wall is represented as follows: In the formula: ;f max This represents the maximum lateral deformation of the retaining wall; The horizontal position is z; the calculated point depth is z; the excavation depth of the foundation pit is H; the insertion depth of the retaining wall is D; the length of the retaining wall along the wall direction is L; H max The depth at which the maximum lateral deformation of the retaining wall is located. ; The specific steps (3) are as follows: Treating the pile foundation as an Euler-Bernoulli beam on the Pasternak foundation, the differential deformation governing equations are as follows: （1） In the formula, E p I p and D e These are the bending stiffness and equivalent width of the pile, respectively. Additional lateral deformation for pile foundations, The formula for calculating the additional load at the pile foundation caused by the excavation of the foundation pit is as follows: （2） In the formula, This represents the lateral displacement of the soil. , The two parameters for the Pasternak foundation are the foundation reaction modulus and the shear layer stiffness: （3） （4） In the formula: Here, h is the depth parameter, and E is the pile embedment depth. s and These are the elastic modulus and Poisson's ratio of the foundation soil, respectively. For multi-layered soil, the weighted average value is taken according to the soil layer thickness. （5） In the formula, t is the shear layer thickness, taken as t=11D. p D p The diameter of the pile foundation outside the pit; According to the difference method, the pile is divided into n segments, each segment is l long, and there are a total of n+1 nodes. Formula (1) is rewritten as: （6） in, The deformation of the i-th pile node; Let be the load borne by the i-th pile node; For a friction pile that is free at both ends, with the boundary conditions that the bending moment and shear force at both ends are zero, the following system of equations can be obtained: （7） Each node i has a deformation governing equation. The n+1 deformation differential equations are written as a system of equations and then rewritten as matrix expressions, as follows: （8） （9） （10） （11） （12） For end-bearing piles with fixed ends but incompletely fixed tops, the boundary conditions are: rotation angles at both ends are 0, pile end displacement is 0, and pile top displacement is... : （13） The matrix expression is obtained as follows: （14） In the formula, , , , as follows: （15） （16） （17） （18）。

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