A semi-analytical method for response analysis of deep tunnel excavation

Abstract

The application relates to a semi-analytical method for analyzing the response of deep-buried tunnel excavation, which comprises the following steps: step 1) establishing a deep-buried circular tunnel excavation mechanical model, and solving the stress at the elastic-plastic boundary in the model by using a numerical method, wherein the stress comprises three components, namely radial stress, tangential stress and axial stress; step 2) dividing the plastic zone into n circular rings based on the finite difference method, and sequentially solving the stress components of each circular ring from the elastic-plastic boundary to the tunnel wall; step 3) solving the radius of each circular ring based on the stress components of the circular rings and the balance equation; step 4) decomposing the strain components, and solving the strain components of each circular ring based on the finite difference method; and step 5) solving the radial displacement based on the strain components. Compared with the prior art, the application has the advantages of improving the calculation efficiency of the three-dimensional semi-analytical method of finite difference.

Description

Technical Field

[0001] This invention relates to the field of tunnel excavation response analysis technology, and in particular to a semi-analytical method for deep-buried tunnel excavation response analysis. Background Technology

[0002] The mechanical response of surrounding rock after tunnel excavation is a complex problem. Simplified elastoplastic analysis of a circular tunnel model helps to deeply understand the convergence-constraint characteristics of the surrounding rock and facilitates rapid preliminary design. Compared to the complex modeling and analysis of commercial software, analytical and semi-analytical methods are widely used in parameter studies, constitutive model verification, and the understanding of nonlinear mechanics due to their simplicity in modeling and high computational speed.

[0003] Existing analytical methods employ simple two-dimensional strength criteria, neglecting the influence of intermediate principal stresses, such as the Mohr-Coulomb and Hoek-Brown criteria, to obtain rigorous analytical solutions. Plane strain problems using two-dimensional strength criteria only concern the in-plane rock response, ignoring the influence of axial stress. In the plastic zone, axial stress is generally the intermediate principal stress, which has a significant impact on the strength and stability of the surrounding rock, as verified by numerous studies. However, for three-dimensional Hoek-Brown criteria (such as the GZZ criterion), or even two-dimensional generalized Hoek-Brown criteria, the expressions are transcendental equations, unable to provide explicit analytical solutions. Numerical methods are needed to solve nonlinear equations and second-order differential equations, or to perform numerical integration calculations. Three-dimensional semi-analytical methods increase the number of equations and the complexity of expressions, both significantly increasing computational complexity and leading to issues with computational efficiency, accuracy, and numerical stability. Summary of the Invention

[0004] The purpose of this invention is to provide a semi-analytical method for deep tunnel excavation response analysis in order to improve the computational efficiency of the finite difference three-dimensional semi-analytical method.

[0005] The objective of this invention can be achieved through the following technical solutions:

[0006] A semi-analytical method for response analysis of deep-buried tunnel excavation, the method comprising:

[0007] Step 1) Establish a mechanical model for the excavation of a deep-buried circular tunnel. The model divides the surrounding rock outside the tunnel wall into a plastic zone and an elastic zone. The stress at the elastic-plastic boundary in the model is solved using numerical methods. The stress includes three components: radial, tangential and axial stress. The radial, tangential and axial stresses at the elastic-plastic boundary satisfy the stress consistency condition of the plastic zone and the Lamé solution of the stress in the elastic zone.

[0008] Step 2) Based on the finite difference method, the plastic zone is divided into n rings, and the stress components of each ring are solved sequentially from the elastic-plastic boundary to the cavity wall.

[0009] Step 3) Solve for the radius of each ring based on the stress components and equilibrium equations of each ring;

[0010] Step 4) Decompose the strain components and solve for the strain components of each annulus based on the finite difference method;

[0011] Step 5) Solve for radial displacement based on strain components.

[0012] Furthermore, the specific steps for solving the radial stress at the elastoplastic boundary in the model are as follows:

[0013] By combining the smooth GZZ criterion for stress invariants, the Lamé solution for the elastic zone stress under plastic zone boundary conditions, and the supplementary equation for axial stress, the radial stress at the elastoplastic boundary is obtained.

[0014] Furthermore, the specific steps for dividing the plastic region into n rings are as follows:

[0015] The surrounding rock in the plastic zone is divided into n rings of different thicknesses according to the equal radial stress difference. The outer circle of the i-th ring is called the (i-1)-th ring, the inner circle of the i-th ring is called the i-th ring, the outer circle of the 1-th ring is called the 0-th ring, the 0-th ring is the elastic-plastic boundary, and the radius of the 0-th ring is the radius of the elastic-plastic boundary. The inner circle of the n-th ring is called the n-th ring, the n-th ring is the tunnel wall, and the radius of the n-th ring is the tunnel excavation radius.

[0016] Furthermore, the equal radial stress difference is: (p s -p cr ) / n;

[0017] Where, p s p is the radial stress at the tunnel wall. cr denoted as radial stress at the boundary of the plastic zone, and n as the number of annulus rings.

[0018] Furthermore, the specific steps for solving the stress components of each ring are as follows:

[0019] Analyze the plastic zone consistency condition satisfied by the inner and outer stresses of the i-th ring, and use the mathematical approximation of difference instead of differential to obtain the linear expressions of the inner and outer stresses. Based on the linear expressions, the supplementary equation of out-of-plane stress, and the division of the rings, construct the linear relationship between the inner and outer stress components of the i-th ring. Given the outer stress components, solve the linear relationship between the stress components to obtain the stress components on the inner side of each ring.

[0020] Furthermore, when solving for the stress components of each ring, the solution is performed in the order from the elastoplastic boundary to the cavity wall, solving for the stress components of the first ring, the second ring, and so on until the nth ring. The stress on the outer side of the first ring is the stress at the elastoplastic boundary obtained in step 1).

[0021] Furthermore, the linear expressions for the inner and outer stresses are as follows:

[0022]

[0023] Where F represents the uniformity condition of the plastic region, σ r(i-1) σ θ(i-1) and σ z(i-1) Let σ represent the radial, tangential, and axial stresses on the outer side of the i-th circle, respectively. r(i) σ θ(i) and σ z(i) These represent the radial, tangential, and axial stresses on the inner side of the i-th circle, respectively.

[0024] Furthermore, the specific steps for solving the radius of each ring based on the stress components and equilibrium equations are as follows:

[0025] Analyze the difference scheme of the equilibrium equation satisfied by the stress components, and combine it with the stress components of each ring in step 2) to obtain the radius of the inner circle of each ring.

[0026] Furthermore, the strain components are decomposed, and the specific steps for solving the strain components of each annulus based on the finite difference method are as follows:

[0027] The strain components are divided into elastic strain and plastic strain. The elastic strain of each ring is obtained by Hooke's law. Then, the difference form after the separation of elastic and plastic strain is established. Combined with the difference scheme of the non-associated flow law, the plastic strain of each ring is solved.

[0028] Furthermore, the specific steps for solving the radial displacement based on strain components are as follows:

[0029] Write the tangential strain component in the strain components as a geometric equation in polar coordinates, and calculate the radial displacement of each point.

[0030] Compared with the prior art, the present invention has the following beneficial effects:

[0031] (1) Compared with the existing calculation methods that directly solve the stress consistency condition of the plastic zone using numerical means, the method of this invention adopts the finite difference method, which avoids a large number of iterative calculation processes and has the advantages of high calculation speed, high calculation accuracy and fast convergence. It can quickly calculate the stress, strain and displacement distribution after tunnel excavation and can provide a theoretical basis for the pre-design of deep buried tunnel support.

[0032] (2) This invention fully considers the influence of tunnel axial stress (intermediate principal stress) on the stress and displacement curve characteristics of the surrounding rock of the tunnel, and more realistically considers the three-dimensional nonlinear mechanical characteristics of deep high-stress unloaded rock mass, thereby improving the rationality and reliability of deep buried tunnel support design analysis. Attached Figure Description

[0033] Figure 1 This is a flowchart of the present invention;

[0034] Figure 2 This is a diagram of the surrounding rock analysis model for the tunnel of this invention;

[0035] Figure 3 This is a comparison chart of the stress calculation results of the method of the present invention with those of conventional direct solution methods and finite element methods;

[0036] Figure 4 This is a comparison chart of the radial displacement calculation results of the method of the present invention with those of conventional direct solution methods and finite element methods;

[0037] Figure 5 This is a comparison chart of the computation time of the method of the present invention and the conventional direct solution method;

[0038] Figure 6 This is a comparison chart showing the convergence of the tunnel wall displacement calculation results between the method of this invention and the conventional direct solution method. Detailed Implementation

[0039] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. These embodiments are based on the technical solution of the present invention and provide detailed implementation methods and specific operating procedures. However, the scope of protection of the present invention is not limited to the following embodiments.

[0040] The purpose of this invention is to provide a semi-analytical method for analyzing the excavation response of deep-buried tunnels considering the three-dimensional strength of the rock mass. This method can accurately and efficiently provide the stress-strain distribution in the plastic zone of the surrounding rock, providing theoretical support for the preliminary pre-design of tunnels. The method proposed in this invention includes the following steps:

[0041] Step 1) Based on the Lamé solution that the stress at the elastoplastic boundary should simultaneously satisfy the stress consistency condition in the plastic region and the stress in the elastic region, the stress at the elastoplastic boundary is solved numerically. The stress includes three components: radial, tangential and axial stress.

[0042] Step 2) Based on the finite difference method, the plastic region is divided into n rings, and the stress components of each ring are solved in turn.

[0043] Step 3) Based on the stress solutions and equilibrium equations for each ring, solve for the radius of each ring;

[0044] Step 4) Based on the elastoplastic strain separation and finite difference method, solve for the strain components of each annulus in sequence;

[0045] Step 5) Based on the strain solution and geometric equations, solve for the radial displacement.

[0046] The model in step 1) is as follows Figure 2 As shown.

[0047] The stress uniformity condition in the plastic zone in step 1) can be expressed as:

[0048] F=F(σ)=0 (1)

[0049] Where, σ=[σ r ,σ θ ,σ z ] represents the stress vector of the rock mass, σ r σ θ σ z Let F represent the radial, tangential, and axial stresses at a point in the surrounding rock, respectively. Different strength criteria result in different expressions for F. The smooth GZZ criterion, expressed as a stress invariant, can be represented as:

[0050]

[0051] Where, σ c m is the uniaxial compressive strength of the rock. b , s, and a are both empirical parameters reflecting the characteristics of the rock mass, and θ σ Let I1 and J2 be the first invariant of stress and the second invariant of stress deviator, respectively, and their expressions are as follows:

[0052]

[0053] Rock mass strength parameter m b The calculation methods for s and a are as follows:

[0054]

[0055] Wherein, GSI is the geological strength index, m i Here, denoted as , and D as , representing the indoor rock parameters.

[0056] The Lamé solution for the stress in the elastic region in step 1) can be specifically expressed as:

[0057]

[0058] Where p is the hydrostatic stress acting around the surrounding rock, q is the axial stress perpendicular to the excavation plane, and p s R0 is the support pressure exerted on the tunnel wall after excavation, and r is the radius of the calculation point.

[0059] The strain and radial displacement at each point in the elastic zone can be expressed as:

[0060]

[0061] Where, ε r and ε θ Let ν be the radial and tangential strain of the tunnel, ur be the radial displacement, G = E / 2(1+ν) be the shear modulus of the surrounding rock, E be the elastic modulus, and ν be Poisson's ratio.

[0062] When a plastic zone exists in the surrounding rock, the radial stress p at the boundary of the plastic zone is used. cr and radius R p Replacing the radial stress p at the tunnel wall respectively s Using the radius R0 as the boundary condition, the solution for the elastic region when the plastic region exists can be obtained.

[0063] At the boundary of the plastic region, r = R p From equation (5), we get:

[0064]

[0065] Step 1) requires a supplementary equation for the axial stress in the plastic zone, which has several forms. Considering the dilatational properties of the rock mass, the supplementary equation for the axial stress is taken as:

[0066]

[0067] Where ψ is the rock mass dilatation angle.

[0068] By using numerical methods and combining equations (2), (7), and (8), the three stress components at the boundary of the plastic zone can be obtained, especially the radial stress p. cr .

[0069] The radial stress p at the boundary of the plastic zone is obtained. cr It is also the critical stress determining the existence or absence of the plastic zone. When the support pressure is sufficient, p s >p cr At that time, the entire surrounding rock is in an elastic state, and no plastic zone is generated; only when the support pressure is insufficient, p s <p cr At that time, a plastic zone will appear near the cave wall.

[0070] In step 2), an approximate solution is given using the finite difference method, and the surrounding rock in the plastic zone is subjected to an equal radial stress difference Δσ. r =σ r(i) -σ r(i-1) =(p s -p cr Divide ) / n into n rings of different thicknesses.

[0071] If i = 0 at the boundary of the elastic-plastic region and i = n at the tunnel wall, then r (0) =R p r (n) =R0, where r is the inner side of the i-th annulus. (i) The outer side is r (i–1) .

[0072] In step 2), the stress component calculation is performed from the boundary of the elastoplastic zone toward the tunnel wall.

[0073] The stress calculation process for each ring is as follows: based on the known stress σ on the outer side of the i-th ring... (i-1) Solve for its inner stress σ (i) The stress σ on the outer side of the i-th ring (i-1) and inner stress σ (i) All should satisfy the plastic region consistency condition, i.e., F(σ) (i) )=F(σ (i-1) If ) = 0, then solve for F(σ). (i) The equivalent equation for ) = 0 is:

[0074] dF=F(σ (i) )-F(σ (i-1) )=0 (9)

[0075] For the yield stage of ideal elastoplastic and elastoplastic-brittle models, the consistency condition on the yield surface yields the following:

[0076]

[0077] By using the mathematical approximation of difference instead of differential, we obtain σ. (i) With σ (i-1) The linear expression:

[0078]

[0079] Among them, the calculation coefficients Let σ be the partial derivative of the strength criterion with respect to each principal stress. (i-1) =(σ r(i-1) ,σ θ(i-1) ,σ z(i-1) ( ) is known. Given the calculation coefficients for different strength criteria. Unlike the previous example, the calculated coefficients derived from the smooth GZZ strength criterion (2) are as follows:

[0080]

[0081] Where S is the stress deviator tensor;

[0082]

[0083] Based on equation (11), the supplementary equation for out-of-plane stress (8), and the rule for dividing the plastic zone into annular sections, σ can be obtained. (i) The linear relationship between the components:

[0084]

[0085] in:

[0086]

[0087] For a system of linear equations, its explicit expression can be easily written out directly and the solution obtained.

[0088] The stress at the boundary of the elasto-plastic region when boundary condition i = 0 is obtained from step 1). Based on the initial condition σ at the boundary of the elasto-plastic region... (0) All stress components can be calculated step by step.

[0089] In step 3), the solution for the radius of each ring is based on the stress and equilibrium equations of each ring obtained in step 2).

[0090] The stress components in the plastic zone should satisfy the equilibrium equation:

[0091]

[0092] Its difference scheme is expressed as:

[0093]

[0094] Rearranging equation (16) and combining it with the stress obtained in step 2), we can obtain the expression for the inner radius of the i-th ring:

[0095]

[0096] Among them, the boundary condition is r at the cave wall (n) =R p The radius of all locations can be calculated step by step from the cave wall into the depth of the surrounding rock.

[0097] Step 4) involves solving for the strain components. First, the strain component ε... (i) It is divided into two parts: elastic strain and plastic strain.

[0098]

[0099] The elastic strain can be obtained from Hooke's law, and the solution method for plastic strain is as follows.

[0100] The strain in the plastic region should satisfy the compatibility equation:

[0101]

[0102] The strain components can be divided into two parts: elastic strain and plastic strain. The difference form after separating the elastic and plastic strains is as follows:

[0103]

[0104] Consider the non-correlated flow rule:

[0105]

[0106] Where λ=(1+sinψ) / (1-sinψ) is the dilatation coefficient, which is related to the dilatation angle ψ. The difference scheme of equation (21) is:

[0107]

[0108] Combining equations (20) and (22), we can obtain and The expression is:

[0109]

[0110] Where, η (i) =1-r (i-1) / r (i) .

[0111] The initial condition i = 0 in equation (23) is at the boundary of the elastic-plastic region, where the plastic strain is 0. The plastic strain at all locations can be calculated step by step.

[0112] The radial displacement solution in step 5) can be written in polar coordinates as follows:

[0113]

[0114] The radial displacement of each point can be calculated using the following formula:

[0115] u r =ε θ r (25)

[0116] The following is a practical experimental analysis:

[0117] The in-situ mechanical and engineering parameters of the rock mass are set as follows: GSI0 = GSI r =50,m i =6, D=0, σ c =40MPa, E=2GPa, υ=0.35, ψ=10°, R0=8m, p=q=30MPa, p s =1MPa, n=1000.

[0118] The calculation results are as follows Figure 3 and Figure 4 The solid line data is shown. Figure 3 and Figure 4 This is a comparison chart showing the stress and radial displacement calculation results of the method proposed in this invention, the conventional direct solution method, and the finite element method.

[0119] To verify the effectiveness of the calculation method of this invention, the above calculation model was calculated using a conventional direct solution method, and the calculation results were compared with those of the method of this invention. The method of this invention differs from the conventional direct solution method in the handling of stress at each point in the plastic zone in step 2). According to the circular division rule of the plastic zone, i.e., division by equal radial stress difference, the radial stress σ of each ring in the plastic zone... r(i) =p cr +iΔσ r Given that the radii of each ring are unknown, the solution for the three-dimensional stress components of each ring is only σ. θ(i) and σ z(i) There are two unknowns. The conventional direct solution method is to eliminate σ using the axial stress supplementary equation. z(i) The stress consistency condition in the plastic zone is solved directly using numerical methods, and σ is obtained. θ(i) , and thus obtain σ z(i) The solution for all stress components is obtained step by step. The method of this invention uses a mathematical approximation of difference instead of differentiation, and gives σ... (i) The explicit expressions for each component avoid the numerical iteration process required to solve the consistency conditions of the plastic region, significantly improving the calculation speed and achieving higher calculation accuracy. Figure 3 and Figure 4 The comparison of the calculation results shows that the results of the calculation method of the present invention are consistent with those of the conventional direct solution method, with a negligible small deviation caused by the finite difference method, which verifies the effectiveness of the method of the present invention.

[0120] To further verify the effectiveness of the calculation method of this invention, finite element software was used to simulate tunnel excavation and calculate the stress and deformation of the surrounding rock after excavation. The tunnel excavation model was a thick-walled cylinder with an excavation radius of 8m and an outer diameter of 120m. The remaining rock mass parameters were consistent with the numerical calculations. Figure 3 and Figure 4 The comparison of the calculation results shows that the error of the calculation method of the present invention is very small compared with the finite element simulation results, which further verifies the effectiveness of the calculation method of the present invention.

[0121] To further verify the efficiency of the calculation method of the present invention, different plastic zone division numbers n were used for calculations of the two calculation methods, the present invention and the conventional direct solution, and the calculation time and calculation results of the two calculation methods were compared with the change of n. Figure 5 and Figure 6 The graphs show a comparison of the calculation time of the proposed method and the conventional direct solution method, as well as a comparison of the convergence of the calculation results of the tunnel wall displacement. Figure 5The comparison of calculation time results shows that, compared with the conventional direct solution method, the calculation method of the present invention can effectively improve the calculation speed and save calculation time, thus verifying the efficiency of the method of the present invention. Figure 6 The calculation results of the tunnel wall displacement show that, for the same number of plastic zone divisions, the calculation results of the method of the present invention are closer to the true value of the tunnel wall displacement than those of the conventional direct solution method, and have smaller errors; the number of divisions required to reach the allowable error is smaller, which verifies that the method of the present invention has better numerical convergence effect.

[0122] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A semi-analytical method for analyzing the response of deep-buried tunnels to excavation, characterized in that, The method comprises: Step 1) establishing a deep-buried circular tunnel excavation mechanics model, wherein the surrounding rock outside the hole wall is divided into a plastic zone and an elastic zone, a numerical method is used to solve the stress at the elastic-plastic boundary in the model, the stress comprises three components, namely radial stress, tangential stress and axial stress, the radial stress, the tangential stress and the axial stress at the elastic-plastic boundary satisfy the plastic zone stress consistency condition and the elastic zone stress Lame solution; Step 2) based on the finite difference method, the plastic zone is divided into n annuli, and the stress components of each annulus are sequentially solved from the elastic-plastic boundary to the hole wall; Step 3) based on the stress components of each annulus and the balance equation, the radius of each annulus is solved; Step 4) the strain components are decomposed, and the strain components of each annulus are solved based on the finite difference method; Step 5) the radial displacement is solved based on the strain components; The specific steps of dividing the plastic zone into n annuli are as follows: The plastic zone surrounding rock is divided into n annuli with different thicknesses according to the equal radial stress difference, the outer circle of the i-th annulus is the (i-1)-th circle, the inner circle of the i-th annulus is the i-th circle, the outer circle of the first annulus is the 0-th circle, the 0-th circle is the elastic-plastic boundary, the radius of the 0-th circle is the radius of the elastic-plastic boundary, the inner circle of the n-th annulus is the n-th circle, the n-th circle is the hole wall, and the radius of the n-th circle is the tunnel excavation radius; The radial stress difference is: ; wherein, is the radial stress at the tunnel wall, is the radial stress at the plastic zone boundary, n is the number of annuli; The radial stress difference is: ; wherein, is the radial stress at the tunnel wall, is the radial stress at the plastic zone boundary, n is the number of annuli; The specific steps of solving the stress components of each annulus are as follows: The plastic zone consistency condition satisfied by the inner stress and the outer stress of the i-th annulus is analyzed, the mathematical approximation of difference instead of differential is adopted to obtain the linear expression of the inner stress and the outer stress, the linear relationship between the inner stress and the outer stress of the i-th annulus is established based on the linear expression, the out-of-plane stress supplement equation and the division of the annulus, the linear relationship between the stress components is solved under the condition that the outer stress component is known, and the inner stress component of each annulus is obtained; The linear expression of the inner stress and the outer stress is as follows: wherein denotes a plastic zone consistency condition, , and denote the outer radial, tangential and axial stresses of the ith circle, , and denote the inner radial, tangential and axial stresses of the ith circle.

2. The semi-analytical method for analyzing the response of a deep tunnel excavation according to claim 1, wherein, The specific steps of solving the radial stress at the elastic-plastic boundary in the model are as follows: The radial stress at the elastic-plastic boundary is solved by simultaneously solving the smooth GZZ criterion expressed by the stress invariant, the Lame solution of the elastic zone stress under the plastic zone boundary condition and the axial stress supplement equation.

3. The semi-analytical method for analyzing the response of a deep tunnel excavation according to claim 1, wherein, When solving the stress components of each annulus, the inner stress components of the first annulus, the second annulus and the n-th annulus are sequentially solved from the elastic-plastic boundary to the hole wall, and the outer stress of the first annulus is the stress at the elastic-plastic boundary obtained in step 1).

4. The semi-analytical method for analyzing the response of a deep tunnel excavation according to claim 1, wherein, The specific steps of solving the radius of each annulus based on the stress components of each annulus and the balance equation are as follows: The difference format of the balance equation satisfied by the stress components is analyzed, and the radius of the inner circle of each annulus is obtained by combining the stress components of each annulus in step 2).

5. The semi-analytical method for response analysis of deep tunnel excavation according to claim 1, wherein, The specific steps of solving the strain components of each annulus by decomposing the strain components based on the finite difference method are as follows: The strain components are divided into elastic strain and plastic strain, the elastic strain of each annulus is obtained through Hooke's law, then the difference form after the separation of the elastic-plastic strain is established, and the plastic strain of each annulus is solved by combining the difference format of the non-associated flow rule.

6. The semi-analytical method for response analysis of deep tunnel excavation according to claim 1, wherein, The specific steps of solving the radial displacement based on the strain components are as follows: The tangential strain component in the strain component is written as a geometric equation form in the polar coordinate system, and the radial displacement of each point is obtained.

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