A method for calculating temporal displacement of a subjacent shield tunnel when a foundation pit excavation causes the shield tunnel to heave

Abstract

The application discloses a kind of calculation methods of temporal displacement of underlying shield tunnel upheaval when foundation pit excavation causes, comprising the following steps: (1) the related size of foundation pit and underlying shield tunnel and relative position relationship are clear, establish coordinate system, construct calculation analysis model;(2) considering the characteristics of soil consolidation deformation in tunnel-soil interaction process, equivalent to elastic half-space foundation model, equivalent to Euler-Bernoulli beam of shield tunnel is placed in elastic half-space inside, combined with Terzaghi-Rendulic three-dimensional consolidation theory, using the deformation coupling condition of tunnel and soil, the temporal solution of underlying shield tunnel upheaval displacement caused by foundation pit excavation is obtained.The application does not need to determine the ground reaction coefficient or soil shear layer parameters, avoids the problem that these parameters are complex and difficult to obtain accurately in two-stage method;Fully consider the characteristics of soil consolidation deformation, can more reasonably and accurately predict the upheaval displacement size of underlying shield tunnel caused by foundation pit excavation and its development process with time.

Description

Technical Field

[0001] This invention belongs to the field of underground engineering, and specifically relates to a method for calculating the temporal displacement of the underpass shield tunnel caused by the excavation of a foundation pit, taking into account the soil consolidation effect. Background Technology

[0002] With the rapid development of urban rail transit, the development of underground space adjacent to rail transit is also increasing. This inevitably leads to the excavation of foundation pits above shield tunnels. Ground excavation causes unloading of the soil above the shield tunnel, which generates additional stress and deformation, leading to cracking and misalignment, seriously affecting the structural safety of the shield tunnel and the operational safety of the subway. In response, many domestic and international scholars have conducted in-depth research on the deformation of adjacent existing tunnels caused by foundation pit excavation unloading, mainly focusing on three methods: finite element simulation, experimental research, and theoretical analysis. While finite element simulation can provide a relatively comprehensive analysis of the problem, it is labor-intensive and involves complex modeling. Experimental research often has the disadvantages of long cycles and high costs, and the accuracy of test results depends to some extent on the experience of the test personnel. Compared with finite element simulation and experimental research, theoretical analysis methods are simpler and faster, and are more suitable for preliminary design and guiding engineering construction.

[0003] Currently, the main theoretical method for analyzing tunnel uplift deformation is the two-stage stress-controlled method. The first stage assumes the tunnel does not exist and uses the Mindlin solution to calculate the additional stress at the tunnel location caused by the excavation. The second stage applies this additional stress as an external load to the tunnel and then simplifies the tunnel as an Euler-Bernoulli beam or Timoshenko beam resting on a Winkler foundation model or a multi-parameter foundation model to solve for the tunnel's deformation response. The two-stage analysis method is clear in its approach and the mechanical relationships are well-defined, and it has been applied to some extent in practical engineering. However, this method still has some shortcomings. For example, the soil stiffness parameters in the Winkler foundation model or multi-parameter foundation model are complex and difficult to obtain accurately. Furthermore, the characteristics of soil consolidation deformation are often ignored during the establishment of these foundation models, assuming that the soil stiffness characteristics are independent of time. As is well known, the essence of soil deformation under stress is the excess pore pressure generated in the soil under external loads, causing pore water to be expelled or drawn in, thereby leading to deformation of the soil skeleton, the so-called consolidation effect (referring to primary consolidation, ignoring secondary consolidation or creep of the soil). It is evident that the stiffness characteristics of foundation soil change at different times after loading, and foundation stiffness has a certain time-dependent effect. Most existing foundation models fail to reflect this characteristic of soil deformation, and the obtained tunnel heave deformations are constant values ​​independent of time. Therefore, to more fully reflect soil deformation behavior and obtain more reasonable underlying tunnel heave deformations, it is essential to consider the characteristics of soil consolidation deformation in foundation model analysis. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a method for calculating the temporal displacement of the underpass shield tunnel caused by the excavation of the foundation pit, which takes into account the characteristics of soil consolidation deformation and can more accurately and reasonably predict the magnitude of the underpass tunnel heave displacement and its development process over time.

[0005] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:

[0006] A method for calculating the temporal displacement of the underlying shield tunnel caused by foundation pit excavation includes the following steps:

[0007] (1) Determine the relevant dimensions and relative positional relationship between the foundation pit and the underlying shield tunnel, establish a coordinate system, and construct a calculation and analysis model;

[0008] (2) Considering the characteristics of soil consolidation deformation during the tunnel-soil interaction process, the foundation is equivalent to an elastic half-space foundation model, and the shield tunnel is equivalent to an Euler-Bernoulli beam placed inside the elastic half-space. Combining the Terzaghi-Rendulic three-dimensional consolidation theory, the deformation coupling condition between the tunnel and the soil is used to obtain the temporal solution of the heave displacement of the underlying shield tunnel caused by the excavation of the foundation pit.

[0009] The expression for the time-varying displacement of the underlying shield tunnel caused by the excavation of the foundation pit is:

[0010] {w} t =([I]+[δ]) t [D][K]) -1 {s} t

[0011] In the formula, {w} t Let be the column vector of the tunnel's uplift displacement at time t; [I] is the identity matrix; [δ] t To obtain the compliance matrix of the elastic half-space foundation at time t using the Terzaghi-Rendulic three-dimensional consolidation theory; {s} t To obtain the column vector of vertical displacement of the soil at the tunnel location caused by the excavation of the foundation pit at time t, using the Terzaghi-Rendulic three-dimensional consolidation theory; the expressions for the [K] and [D] matrices are as follows:

[0012]

[0013]

[0014] EI is the equivalent bending stiffness of the tunnel; h is the step size for dividing the tunnel nodes using the finite difference method; n+1 is the number of tunnel nodes; and D is the outer diameter of the tunnel.

[0015] According to the above scheme, in step (1), the calculation and analysis model construction process includes: establishing an xoy coordinate system with the pit center o as the origin, the pit length direction as the y-axis, and the width direction as the x-axis; drawing a perpendicular line from the pit center o to the tunnel axis, and then establishing an o′ξ coordinate system with the foot of the perpendicular o′ as the origin and the tunnel axis as the ξ-axis; the angle between the x-axis and the ξ-axis is θ, and the length of oo′ is d. The transformation formula between the two coordinate systems is:

[0016]

[0017] According to the above scheme, the compliance matrix [δ] of the elastic half-space foundation at time t is... t for:

[0018]

[0019] The elements in the compliance matrix are calculated using the following formula:

[0020] δ ij =w1(r ij ,0,z0)-w1(r ij ,0,H)-g1(r ij ,0,z0,t)

[0021] In the formula, w1(x,y,z) represents the final vertical displacement of the foundation calculated using the Mindlin solution; g1(x,y,z,t) represents the consolidation displacement of the foundation, calculated according to the Terzaghi-Rendulic three-dimensional consolidation theory; r ij =|j-ih(i,j=0,1,2,…,n); z0 is the vertical distance from the tunnel axis to the bottom of the pit; H is the unloading influence depth, taken as the ratio of the vertical additional stress of the foundation caused by excavation to the effective self-weight stress is 0.1.

[0022] According to the above scheme, the final vertical displacement w1(x,y,z) of the foundation is calculated using the Mindlin solution:

[0023]

[0024] In the formula, E is the resilient modulus of the foundation soil under effective stress conditions, and μ is the Poisson's ratio of the foundation soil under effective stress conditions. (x,y,z) are the coordinates of the calculation point, and (α,β,z0) are the coordinates of the load application point.

[0025] According to the above scheme, the foundation consolidation displacement g1(x,y,z,t) is calculated based on the Terzaghi-Rendulic three-dimensional consolidation theory:

[0026]

[0027]

[0028]

[0029] In the formula, C v The consolidation coefficient is expressed as: Where k is the soil permeability coefficient, γ w The specific gravity of water; This is the probability error function.

[0030] According to the above scheme, the vertical displacement column vector {s} of the soil at the tunnel location caused by the excavation of the foundation pit at time t is... t for:

[0031]

[0032] element s in vector i Calculate using the following formula:

[0033] s i (x,y,z0,t)=w2(x,y,z0)-w2(x,y,H)-g2(x,y,z0,t)

[0034] In the formula, w2(x,y,z) is the final vertical displacement of the soil at position z caused by the excavation of the foundation pit, calculated using the Mindlin solution; g2(x,y,z,t) is the consolidation displacement of the soil at position z caused by the excavation of the foundation pit, calculated according to the Terzaghi-Rendulic three-dimensional consolidation theory; (x,y) is the coordinate of tunnel node i in the xoy coordinate system, obtained by the transformation formula between the xoy coordinate system and the o′ξ coordinate system.

[0035] According to the above scheme, the final vertical displacement w2(x,y,z) of the soil at position z caused by the excavation of the foundation pit is calculated using the Mindlin solution:

[0036]

[0037] In the formula, L, B, and c are the length, width, and depth of the foundation pit, respectively; γ is the average unit weight of the soil within the excavation depth range of the foundation pit.

[0038] According to the above scheme, the consolidation displacement g2(x,y,z,t) of the soil at position z caused by the excavation of the foundation pit is calculated based on the Terzaghi-Rendulic three-dimensional consolidation theory:

[0039]

[0040]

[0041]

[0042] In the formula,

[0043] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0044] 1. The calculation method of this invention uses an elastic half-space foundation model for analysis. During the calculation process, it is not necessary to determine the foundation reaction coefficient or soil shear layer parameters, which avoids the problem that the values ​​of these parameters are complicated and difficult to obtain accurately in the two-stage method, and is more convenient for engineering applications.

[0045] 2. By equating the underlying shield tunnel with an Euler-Bernoulli beam in an elastic half-space foundation, the soil consolidation deformation characteristics are fully considered during the tunnel-soil interaction process, which more accurately reflects the characteristics of soil deformation changing over time. This yields a more realistic result of the underpass tunnel heave deformation and its time-varying behavior. Combined with the excavation sequence of the foundation pit, the magnitude of the underpass tunnel heave and its time-varying process can be predicted more accurately. Attached Figure Description

[0046] Figure 1 This is a plan view showing the relative positions of the foundation pit and the tunnel in an example of the present invention;

[0047] Figure 2 This is a longitudinal section view showing the relative positions of the foundation pit and the tunnel in an example of the present invention;

[0048] Figure 3 This is a diagram showing the relative positions of the foundation pit and tunnel, and the layout of monitoring points in an example of the present invention.

[0049] Figure 4 This is a comparison chart of the calculated and measured displacement results of the left tunnel in this invention example;

[0050] Figure 5 This is a comparison chart of the calculated and measured displacement results of the right tunnel in an example of the present invention. Detailed Implementation

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

[0052] The method for calculating the temporal displacement of the underpass shield tunnel caused by foundation pit excavation described in this invention considers the characteristics of soil consolidation deformation during the tunnel-soil interaction process. The foundation is equivalent to an elastic half-space foundation model, and the shield tunnel is equivalent to an Euler-Bernoulli beam placed inside the elastic half-space. Combining the Terzaghi-Rendulic three-dimensional consolidation theory, the deformation coupling condition between the tunnel and the soil is used to obtain the temporal solution of the underpass shield tunnel heave displacement caused by foundation pit excavation.

[0053] Example: An underground space utilization project in a certain city is located above a tunnel section, laid almost parallel to the tunnel's length. It is a multi-span, single-layer rectangular underground structure with a total length of 811.2m. The standard section width of the main foundation pit is 28.8m, and the pit depth ranges from 8.6m to 10.5m. To address the uncertainty regarding the uplift of the underlying strata and tunnel section caused by the excavation and unloading of the foundation pit, a section of the foundation pit was selected as a test section before construction. The dimensional parameters of the test section's foundation pit and the underlying shield tunnel were determined, and a computational analysis model was constructed based on their relative positional relationship, such as... Figures 1-2As shown. An xoy coordinate system is established with the center of the pit (o) as the origin, the length of the pit as the y-axis, and the width as the x-axis. A perpendicular line is drawn from the center of the pit (o) to the tunnel axis, and the foot of the perpendicular (o′) is used as the origin, with the tunnel axis as the ξ-axis, to establish an o′ξ coordinate system. The angle between the x-axis and the ξ-axis is θ, and the length of oo′ is d. The length of the test section pit is L = 20m, the width is B = 29.4m, the excavation depth is c = 9m, the vertical distance from the tunnel axis to the bottom of the pit is z0 = 9.15m, the outer diameter of the tunnel is D = 6m, the inner diameter is 5.4m, and the equivalent bending stiffness of the tunnel is EI = 1.76 × 10⁻⁶. 5 MN·m 2 Calculation assumptions: The tunnel maintains close contact with the surrounding soil, with no interface slippage; only the effect of elastic deformation is considered; the excavation of the foundation pit is completed instantaneously in one go; the permeability coefficient of the foundation is the same in all directions and remains unchanged during consolidation; the effect of precipitation is not considered. The relative positions of the tunnel and the foundation pit, and the locations of some monitoring points are shown below. Figure 3 As shown in Table 1, the soil parameters of the site area are as follows. Before the excavation of the test section foundation pit, the silty clay (soft plastic) soil layer below the bottom of the pit was reinforced by jet grouting to a depth of 3m below the bottom of the pit. The resilient modulus E and Poisson's ratio μ of the entire foundation soil under effective stress conditions were calculated based on the weighted average of each soil layer, and the resilient modulus E was taken as 4 times the compression modulus. Considering the soil reinforcement, the final foundation resilient modulus E was 96MPa and the Poisson's ratio μ was 0.27. In addition, the permeability coefficient of the silty clay (soft plastic) soil layer was calculated. Since the soil layer was reinforced by grouting, the permeability coefficient was reduced, so the permeability coefficient k was taken as 0.0015m / day in the calculation. The full-section excavation of the foundation pit in the test section took about 27 days, and at this time, most of the foundation pit bottom slab on the side near the left tunnel had been poured.

[0054] Table 1 Site Soil Parameters

[0055]

[0056] This invention utilizes the method of simulating an elastic half-space foundation model, and equating the shield tunnel with an Euler-Bernoulli beam placed within this elastic half-space. Combining this with the Terzaghi-Rendulic three-dimensional consolidation theory, and leveraging the deformation coupling condition between the tunnel and the soil, the time-varying solution for the heave displacement of the underlying shield tunnel caused by the excavation is obtained. When calculating the heave deformation of the left and right shield tunnels under this excavation, the required parameters are first determined based on the information in the case study. Then, the tunnel is divided into n+1 tunnel nodes with a step length h = 2.5m using the finite difference method. The heave deformation at 65 monitoring points calculated by this invention is compared with the measured data from the engineering site. The results are shown in […]. Figure 4 , Figure 5 .

[0057] from Figures 4-5 As can be seen, at the point when the full-section excavation of the foundation pit was completed (around day 27), the heave displacements of the left and right tunnels calculated by the method of this invention were basically consistent with the measured data. Moreover, for a period of time after the completion of the foundation pit excavation (around day 27 to day 35), the development trend of the heave displacement of the right tunnel calculated by the method of this invention was almost consistent with the measured data. The heave displacement of the left tunnel differed slightly from the measured data. This is because when the full-section excavation of the foundation pit was completed on day 27, most of the foundation pit bottom slab on the side closer to the left tunnel had already been poured, a point that was not considered in the calculation of the method of this invention. Overall, the theoretical calculation method proposed in this invention has a certain degree of accuracy and applicability.

[0058] This invention is not limited to the applications listed in the specification and embodiments. For those skilled in the art, various corresponding modifications and variations can be made according to this invention, and all such modifications and variations fall within the protection scope of the claims of this invention.

Claims

1. A method for calculating the temporal displacement of the underside shield tunnel caused by foundation pit excavation, characterized in that, Includes the following steps: (1) Clarify the relevant dimensions and relative positional relationship between the foundation pit and the underlying shield tunnel, establish a coordinate system, and construct a calculation and analysis model; In step (1), the calculation and analysis model construction process includes: establishing an xoy coordinate system with the center o of the foundation pit as the origin, the length direction of the foundation pit as the y-axis, and the width direction as the x-axis; drawing a perpendicular line from the center o of the foundation pit to the tunnel axis, and then using the foot of the perpendicular... Establish a system with the tunnel axis as the ξ-axis and the origin as the ξ-axis. ξ-coordinate system; the angle between the x-axis and the ξ-axis is θ, o If the length of the coordinate system is d, then the transformation formula between the two coordinate systems is: (2) Considering the characteristics of soil consolidation deformation during the tunnel-soil interaction process, the foundation is equivalent to an elastic half-space foundation model, and the shield tunnel is equivalent to an Euler-Bernoulli beam placed inside the elastic half-space. Combining the Terzaghi-Rendulic three-dimensional consolidation theory, the deformation coupling condition between the tunnel and the soil is used to obtain the temporal solution of the heave displacement of the underlying shield tunnel caused by the excavation of the foundation pit. The expression for the time-varying displacement of the underlying shield tunnel caused by the excavation of the foundation pit is: In the formula, Let be the column vector of the tunnel's uplift displacement at time t; It is the identity matrix; The compliance matrix of the elastic half-space foundation at time t is obtained by combining the Terzaghi-Rendulic three-dimensional consolidation theory; This is the column vector of vertical displacement of the soil at the tunnel location at time t caused by the excavation of the foundation pit, obtained by combining the Terzaghi-Rendulic three-dimensional consolidation theory; , The expressions for the matrices are as follows: Equivalent bending stiffness of the tunnel; The step size for dividing tunnel nodes using the finite difference method; The number of tunnel nodes; The outer diameter of the tunnel; The compliance matrix of the elastic half-space foundation at time t for: The elements in the compliance matrix are calculated using the following formula: In the formula, The final vertical displacement of the foundation calculated using the Mindlin solution; The foundation consolidation displacement is calculated based on the Terzaghi-Rendulic three-dimensional consolidation theory. ( ); This is the vertical distance from the tunnel axis to the bottom of the pit; To determine the depth of unloading influence, the ratio of the vertical additional stress on the foundation caused by excavation to the effective self-weight stress is taken as 0.

1. Final vertical displacement of the foundation The Mindlin solution was used for calculation: In the formula, The resilient modulus of the foundation soil under effective stress conditions. Poisson's ratio under effective stress conditions of the foundation soil; , , To calculate the coordinates of the point, These are the coordinates of the point where the load applies.

2. The method for calculating the temporal displacement of the underside shield tunnel caused by excavation of the foundation pit according to claim 1, characterized in that, The foundation consolidation displacement Calculations based on the Terzaghi-Rendulic three-dimensional consolidation theory: In the formula, The consolidation coefficient is expressed as: ,in The soil permeability coefficient, The specific gravity of water; , ; , where is the probability error function.

3. The method for calculating the temporal displacement of the underside shield tunnel caused by foundation pit excavation according to claim 1, characterized in that, The vertical displacement column vector of the soil at the tunnel location caused by the excavation of the foundation pit at time t for: elements in a vector Calculate using the following formula: In the formula, For the foundation pit excavation caused by the Mindlin solution calculation The final vertical displacement of the soil at the location; Caused by foundation pit excavation The consolidation displacement of the soil at the location was calculated based on the Terzaghi-Rendulic three-dimensional consolidation theory. For tunnel nodes The coordinates in the xoy coordinate system are determined by the xoy coordinate system and... The transformation formula between the ξ coordinate system is obtained.

4. The method for calculating the temporal displacement of the underside shield tunnel caused by foundation pit excavation according to claim 3, characterized in that, The excavation of the foundation pit caused Final vertical displacement of the soil at the location The Mindlin solution was used for calculation: In the formula, These are the length, width, and depth of the foundation pit, respectively. This represents the average unit weight of the soil within the excavation depth range of the foundation pit; , .

5. The method for calculating the temporal displacement of the underside shield tunnel caused by excavation of the foundation pit according to claim 3, characterized in that, The excavation of the foundation pit caused Consolidation displacement of soil at location Calculations based on the Terzaghi-Rendulic three-dimensional consolidation theory: In the formula, , .

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