A soft rock tunnel support pressure reliability analytical design method and system

Abstract

This invention discloses an analytical design method and system for the reliability of support pressure in soft rock tunnels. The method involves shifting the sampling center to a sphere with the target reliability index as its radius in a standard normal space for central composite sampling, ensuring that the sampling points cover the reliability design points. Response surface functions of the ultimate support pressure are established in both the standard normal space and the original parameter space. The coordinates of the design points satisfying the target reliability index are directly and analytically calculated by simultaneously solving the equations of the standard normal space response surface function and the sphere or target reliability contour lines. The design points in the standard normal space are then transformed to the original parameter space. Finally, using the response surface function in the original parameter space, the failure threshold of the support pressure satisfying the target reliability index is calculated. This method fundamentally solves the technical problems of low computational efficiency, cumbersome process, and limited applicability of traditional reliability design methods.

Description

Technical Field

[0001] This invention relates to the field of tunnel engineering design, specifically to an analytical design method and system for the support pressure reliability of soft rock tunnels based on twin response surfaces. Background Technology

[0002] The geological strata surrounding soft rock tunnels have low material parameters and low shear strength. During construction, due to stress release, the excavation face may experience significant extrusion deformation, which could lead to face collapse and endanger the safety of construction personnel and equipment. To prevent face collapse during tunnel excavation, support pressure can be applied to the face.

[0003] For a long time, support pressure design has been deterministic, neglecting the uncertainty of rock mass material parameters. When reliability design is adopted, due to the complexity of the mechanical response of tunnel face construction, there is currently no explicit function, and numerical models need to be established. Traditional reliability analysis methods, such as the first-order second-moment method, are used, and multiple iterations are required to determine the support pressure that meets the reliability index. The process is extremely complex and computationally inefficient. Summary of the Invention

[0004] To address the aforementioned shortcomings, this invention provides an analytical design method and system for the support pressure reliability of soft rock tunnels, fundamentally solving the technical problems of low calculation efficiency, cumbersome process, and limited applicability of traditional reliability design methods.

[0005] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows: An analytical design method for the support pressure reliability of soft rock tunnels includes the following steps: S1: Offset the sampling center to a sphere with the target reliability index as the radius in the standard normal space and perform central composite sampling so that the sampling points cover the reliability design points; S2: Establish the response surface function of the ultimate support pressure in the standard normal space and the original parameter space, respectively; S3: Combine the standard normal spatial response surface function with the equation of the sphere or target reliability contour line, and directly calculate the coordinates of the design point that meets the target reliability index analytically. S4: Transform the design points in the standard normal space to the original parameter space; S5: Using the original parameter space response surface function, calculate the failure threshold of the support pressure applied to the tunnel face that meets the target reliability index.

[0006] In the above technical solution, in step S1, based on the sampling center, a sampling point matrix of the standard normal space is obtained using the center composite sampling method. The sampling point matrix includes at least the sampling center point and the sampling step distance offset along the positive and negative directions of the coordinate axes of each random variable. Axis points of multiple standard deviations.

[0007] In the above technical solution, in step S1, the target reliability index ranges from 2.0 to 4.2. It is generally determined based on engineering structural reliability design specifications, risk acceptance criteria, or cost-risk optimization analysis.

[0008] In the above technical solution, in step S1, the sampling step distance k ranges from 1 to 3.

[0009] In the above technical solution, in step S1, the sampling center coordinates are set to the negative quantile value of a certain coordinate of the radius target reliability index.

[0010] In the above technical solution, in step S1, the target reliability index On a sphere with radius , the coordinates of the sampling center are set to ( , , ).

[0011] In the above technical solution, in step S2, the sampling points are mapped to the original parameter space according to the actual distribution and correlation coefficient of the random variable to obtain the sampling points in the original parameter space; the sampling points in the two spaces are input into the numerical calculation model to obtain the ultimate support pressure response value; and twin response surface functions are fitted in the standard normal space and the original parameter space respectively.

[0012] Furthermore, the numerical calculation model is a numerical analysis model of tunnel face stability based on the Hoek-Brown failure criterion, and the ultimate support pressure is determined by the bisection method.

[0013] Furthermore, the response surface functions for the ultimate support pressure established in both the standard normal space and the original parameter space are quadratic polynomial response surfaces without cross terms.

[0014] In the above technical solution, in step S2, the standard normal spatial response surface function Twin response surface function in original parameter space as follows, and These are the design points for each space:

[0015]

[0016] in , There are 7 undetermined coefficients; each coefficient is calculated by using the coordinates of each sampling point in each space and the ultimate support pressure.

[0017] In the above technical solution, in step S2, the random variable includes the uniaxial compressive strength of soft rock. Material constants , and geological strength parameter GSI.

[0018] In the above technical solution, in step S3, the standard normal spatial response surface function is tangent to the spherical or target reliability contour equation at the design point, and the gradients of the two are proportional at the tangency point. The coordinates of the design point are then solved. The solution is extended to the corresponding dimension according to the number of random variables. When the random variable is a resistance parameter positively correlated with the function, the design point is taken as the lower quantile value, i.e., the coordinates are negative.

[0019] In the above technical solution, in step S4, the design points of the standard normal space are transformed to the original parameter space based on the statistical characteristics of each random variable, including distribution type, mean, standard deviation, and correlation coefficient.

[0020] In the above technical solution, in step S4, for non-normal random variables, the Rosenblatt transformation or Nataf transformation is used to convert the design points in the U space into design points in the X space.

[0021] Therefore, the method of the present invention brings at least the following basic effects: 1) A center composite sampling strategy based on target reliability index is proposed to improve the local fitting accuracy of response surface.

[0022] 2) It is proposed to establish the ultimate support pressure response surface in the standard normal space (U space), and based on the fitted response surface, the target reliability contour lines and the Lagrange multiplier, the coordinates of the design point that meets the target reliability can be directly analyzed.

[0023] 3) It is proposed to establish a twin response surface of the standard normal space in the original parameter space (X space), map the design points of the standard normal space to the original parameter space, and use the response surface of the original parameter space to realize the direct calculation of the support pressure that meets the target reliability index.

[0024] Based on the above method, the present invention also provides a system. Specifically, it is as follows: An analytical design system for the support pressure reliability of soft rock tunnels based on twin response surfaces includes: The sampling module is configured to perform central composite sampling based on the target reliability, setting the sampling center in the standard normal space U to achieve the target reliability index. On a sphere with radius , for the resistance parameter positively correlated with the function, the sampling center coordinates are set to ( , , ), and obtain the sampling point matrix in space U; The spatial mapping module is configured to map the sampling points of the U space to the original parameter space X space according to the actual distribution and correlation coefficient of the random variables, so as to obtain the sampling points of the X space; The numerical calculation module is configured to input the sampling points of the U space and X space into the numerical calculation model respectively to obtain the ultimate support pressure response value; The twin response surface construction module is configured to fit a U-space response surface based on the U-space sampling points and their response values, and to fit an X-space response surface based on the X-space sampling points and their response values. The design point analysis module is configured to solve the design point coordinates in U-space using the Lagrange multiplier method based on the response surface coefficients in U-space. For resistance parameters that are positively correlated with the function, the design point coordinates are calculated directly. The spatial transformation module is configured to convert the design points in the U space into design points in the X space based on the distribution type, statistical parameters, and correlation coefficient matrix of the random variables. The support pressure calculation module is configured to substitute the design point coordinates in the X-space into the X-space response surface equation to calculate the reliability index that meets the target. The support pressure failure threshold.

[0025] In the above technical solution, the system further includes a parameter input module for inputting statistical parameters of random variables, including distribution type, mean, standard deviation and correlation coefficient matrix.

[0026] In the above technical solution, the system further includes: The verification module is configured to apply the calculated support pressure failure threshold to the numerical model of the tunnel face and verify the accuracy of the reliability index using numerical difference and improved first-order second-moment method.

[0027] In summary, this invention fundamentally solves the technical problems of low computational efficiency, cumbersome process, and limited applicability of traditional reliability design methods by using a central composite sampling strategy based on target reliability, a UX space twin response surface system, and analytical solutions for standard normal space design points. It provides an efficient, accurate, and practical analytical method and system for reliability design of support pressure in soft rock tunnels.

[0028] Compared with the prior art, the beneficial effects of this invention are reflected in the following four aspects: Existing technologies (such as traditional CCDs) employ center-based composite sampling, where the sampling center is fixed at the origin. Design points are often far from the sampling center, leading to poor local fitting accuracy of the response surface and excessive deviations in calculation results. This invention (RB-CCD) shifts the sampling center to the target reliability sphere, ensuring that sampling points cover or approach the design points. This improves the local fitting accuracy of the response surface and provides assurance for accurately determining the design points.

[0029] Existing technologies use iterative calculations to solve for design points, requiring multiple fittings of the response surface and calls to numerical models, resulting in low computational efficiency and convergence issues. This invention utilizes standard normal space response surface coefficients and directly calculates the design point coordinates analytically based on the Lagrange multiplier method, eliminating the need for iterative calculations, achieving high computational efficiency, and avoiding convergence problems.

[0030] Existing technologies, when establishing response surfaces in a single space, cannot account for non-normal parameters and correlations in the U-space response surface, and cannot utilize reliability contour relationships in the X-space response surface. This invention establishes a UX-space twin response surface. The U-space response surface is only used for analyzing design points, while the X-space response surface is directly used to calculate support pressure. This approach can solve problems related to non-normal and correlated random variables, and has a wide range of applications.

[0031] Existing technologies require repeated adjustments to support pressure, reliability calculations, and determinations of whether target values ​​are met, resulting in cumbersome design processes, high computational costs, and poor engineering practicality. This invention can directly calculate the support pressure required to meet the target reliability, requiring only seven calculations to fit the response surface. The process is simple, computationally efficient, and easy to promote and apply in engineering projects. Attached Figure Description

[0032] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings: Figure 1 This is a schematic diagram of the method flow according to an embodiment of the present invention; Figure 2 This is a two-dimensional schematic diagram of the central composite sampling based on reliability in this invention.

[0033] Figure 3 This is a schematic diagram illustrating the principle of analytical solution for reliability design points in the two-dimensional random variable U-space of this invention.

[0034] Figure 4 This invention provides a schematic diagram of the UX space design point conversion.

[0035] Figure 5 The numerical model diagram established in Embodiment 2 of the present invention.

[0036] Figure 6 This is a reliability verification curve for Embodiment 2 of the present invention. Detailed Implementation

[0037] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0038] Example 1 The flow chart of the analytical design method for the support pressure reliability of soft rock tunnels according to the present invention is as follows: Figure 1 As shown, it includes 5 steps: 1) Centered composite sampling based on target reliability For engineering problems without a displayed function, central composite sampling (CCD) is often used to select sampling points for numerical calculation. The results are then used to fit a quadratic response surface without cross terms. The sampling center for CCD is (0, 0, 0). The sampling matrix of the three-dimensional random variable in standard normal space (U space) is shown on the left side of Table 1. , and Let the standard deviations of the three random variables be denoted as . The sampling step size is typically 1 to 3.

[0039] This method of response surface fitting is a local fitting, and its fitting accuracy is highly dependent on the spacing between sampling points and the positional relationship between design points. When the spacing between sampling points is large, or when the design points are far from the sampling points, it is very easy to lead to inaccurate fitting results. In actual engineering, the design points for reliability design are often deviated from the sampling center, resulting in excessive deviations in the calculation results.

[0040] To address this issue, this invention proposes a Reliability Index-Based Centered Composite Sampling (RB-CCD). The method involves setting the sampling center of the centered composite sampling at a radius of [missing information]. The ball, among which The target reliability index. When all random variables are resistance parameters positively correlated with the function, the parameter value of the design point should be a certain lower quantile value, then the coordinates of the sampling center can be set to ( , , This ensures that the sampling points are as close as possible to or cover the design points, and the sampling matrix is ​​shown on the right side of Table 1. A schematic diagram of the sampling center and sampling matrix for two-dimensional sampling is shown below. Figure 2 As shown, this facilitates the visualization of the sampling center offset.

[0041] Table 1. Three-dimensional sampling matrices for central composite sampling and reliability-based central composite sampling.

[0042] The method of this invention is applicable in any dimension. In practice, three-dimensional sampling (three random variables) is typically used for engineering applications. The principles for determining the sampling dimension are shown in Table 2 below: Table 2. Principles for Determining Sampling Dimensions

[0043] 2) Establish a UX spatial twin response surface The conventional response surface methodology first selects a design parameter and then constructs a function response surface in either the standard normal space (U space) or the original parameter space (X space). This approach cannot account for variations in the design parameter, requires multiple fittings of the response surface, and is complex and computationally inefficient. Furthermore, response surfaces constructed in U space cannot account for non-normal or correlated parameters, while response surfaces constructed in X space cannot account for the functional relationship between the response surface and the target reliability contour lines, exhibiting numerous limitations.

[0044] Therefore, this invention proposes to establish twin response surfaces in U-space and X-space. Their expression is shown in the following formula, where... , There are 7 undetermined coefficients.

[0045] ; ; To solve for these coefficients, reliability-based central composite sampling is first performed in the U-space to obtain 7 sample points. Then, based on the actual distribution and correlation coefficient of the random variable, it is mapped to the X space to obtain sample points. The response values ​​are obtained by sequentially inputting them into numerical software for numerical calculation. .

[0046] For the reliability design of tunnel face support pressure, the response value is the ultimate support pressure, i.e. Therefore, a twin response surface in the UX space can be established by sequentially performing central composite sampling based on reliability.

[0047] 3) Analysis of the design points of U-space This invention proposes an analytical solution for U-space reliability design points, the principle of which is explained in the two-dimensional random variable space as follows: Figure 3 As shown. The function of space U is... , and the target reliability contour line (radius is The circles are tangent to each other, and the point of tangency is the design point to be determined. The equation of a circle is: ; According to the Lagrange principle, the gradients of two functions at the point of tangency satisfy a proportional relationship, i.e. ,in Let be a Lagrange multiplier. For the partial derivative form, we have: ; ; function The partial derivatives for the two random variables are: ; ; Because it satisfies at the design point Function Can be converted to: ; ; The partial derivatives of the function with respect to the two random variables are: ; ; Solving the above equations simultaneously, we get: ; ; Equations and parameters of the same circle and The coordinates of the design point can be obtained as follows:

[0048]

[0049] Based on this principle, for the case of three random variables, the coordinates of the design point are:

[0050]

[0051]

[0052] When the random variable is a resistance parameter positively correlated with the function of resistance, its design point should be the lower quantile value, i.e.:

[0053]

[0054]

[0055] 4) Design points converted to X space Based on the distribution type of the random variable and the correlation coefficient matrix, the design points in space U can be determined. Design points converted to X space The transformation relationship diagram of the two-dimensional random variable is shown below. Figure 4 As shown.

[0056] 5) Calculate the support pressure failure threshold Design points of X space By substituting the coordinates into the response surface equation in the X space, the reliability index that satisfies the target can be calculated. Support pressure: .

[0057] Example 2 Selected diameter and burial depth are both The circular tunnel is excavated from soft rock strata and follows the HB failure criterion, which has three variables: uniaxial compressive strength. Material constants , and the geological intensity parameter GSI are normal random variables.

[0058] The mean values ​​of the random variables are respectively , and The standard deviations are respectively divided into , and .

[0059] The target reliability index is Seeking support and pressure.

[0060] 1) Centered composite sampling based on target reliability When the target reliability index is The sampling centers of the three random variables that are positively correlated with the function are ( , , ),Right now .

[0061] The sampling center is approximated as Set sampling interval The sampling matrices of the three random variables are shown in Table 3 below: Table 3 Sampling matrices for U-space and X-space

[0062] 2) Establish a UX spatial twin response surface The established numerical model is shown in Figure 5. The parameter values ​​of the X space of the seven sampling points are input in sequence. Using the bisection method, the collapse loads of the tunnel face are 24.92 kPa, 40.55 kPa, 5.23 kPa, 38.20 kPa, 6.02 kPa, 33.20 kPa and 5.08 kPa.

[0063] By using the U-space coordinates of 7 sampling points and the ultimate support pressure, the linear equation system was solved to obtain the coefficients of the U-space response surface. Therefore, the equation for the U-space response surface is determined as follows:

[0064] Similarly, by using the X-space coordinates of 7 sampling points and the ultimate support pressure, the system of linear equations is solved to obtain the coefficients of the X-space response surface. Therefore, the equation of the X-space response surface is determined as follows:

[0065] 3) Analysis of the design points of U-space Using the method proposed in this patent, the coordinates of the three random variables at the design point in space U are obtained as follows:

[0066]

[0067]

[0068] 4) Design points converted to X space Based on the statistical characteristics of the three random variables, including distribution type, mean, standard deviation, and correlation coefficient, the coordinates of the design points in U space can be transformed to X space as follows: , , .

[0069] 5) Calculate the support pressure failure threshold Substituting the coordinates of the design points in X-space into the X-space response surface equation, the failure threshold of the support pressure is calculated as follows:

[0070] This support pressure is what satisfies the target reliability index. The support pressure.

[0071] In order to verify the design support pressure, Applied to the tunnel face, using a conventional numerical difference method and an improved first-order second-moment method, the relationship between the iteration step and the reliability index is obtained as follows: Figure 6 As shown, the calculated reliability index is in high agreement with the target reliability index, verifying the accuracy of the method of the present invention.

[0072] It should be understood that those skilled in the art can make improvements or modifications based on the above description, and all such improvements and modifications should fall within the protection scope of the appended claims.

Claims

1. A method for analytical design of support pressure reliability in soft rock tunnels, characterized in that... Includes the following steps: S1: Offset the sampling center to a sphere with the target reliability index as the radius in the standard normal space and perform central composite sampling so that the sampling points cover the reliability design points; S2: Establish the response surface function of the ultimate support pressure in the standard normal space and the original parameter space, respectively; S3: Combine the standard normal spatial response surface function with the equation of the sphere or target reliability contour line, and directly calculate the coordinates of the design point that meets the target reliability index analytically. S4: Transform the design points in the standard normal space to the original parameter space; S5: Using the original parameter space response surface function, calculate the failure threshold of the support pressure applied to the tunnel face that meets the target reliability index.

2. The analytical design method for the support pressure reliability of soft rock tunnels according to claim 1, characterized in that... In step S1, based on the sampling center, a sampling point matrix of the standard normal space is obtained using the center composite sampling method. The sampling point matrix includes at least the sampling center point and the sampling step distance offset along the positive and negative directions of the coordinate axes of each random variable. Axis points of multiple standard deviations.

3. The analytical design method for the support pressure reliability of soft rock tunnels according to claim 1, characterized in that... In step S1, the target reliability index ranges from 2.0 to 4.

2.

4. The analytical design method for the support pressure reliability of soft rock tunnels according to claim 1, characterized in that... In step S1, the target reliability index On a sphere with radius , the coordinates of the sampling center are set to ( , , ).

5. The analytical design method for the support pressure reliability of soft rock tunnels according to claim 1, characterized in that... In step S2, the sampling points are mapped to the original parameter space according to the actual distribution and correlation coefficient of the random variables to obtain the sampling points in the original parameter space; the sampling points in the two spaces are input into the numerical calculation model to obtain the ultimate support pressure response value; twin response surface functions are fitted in the standard normal space and the original parameter space respectively.

6. The analytical design method for the support pressure reliability of soft rock tunnels according to claim 1, characterized in that... The numerical calculation model is a numerical analysis model of tunnel face stability based on the Hoek-Brown failure criterion, and the ultimate support pressure is determined by the bisection method.

7. The analytical design method for the reliability of soft rock tunnel support pressure according to claim 1, characterized in that... The response surface function for the ultimate support pressure is established in both the standard normal space and the original parameter space. Both functions are quadratic polynomial response surfaces without cross terms.

8. The analytical design method for the support pressure reliability of soft rock tunnels according to claim 1, characterized in that... In step S2, the standard normal spatial response surface function Twin response surface function in original parameter space as follows, and These are the design points for each space: ； ； in , There are 7 undetermined coefficients; each coefficient is calculated by using the coordinates of each sampling point in each space and the ultimate support pressure.

9. The analytical design method for the support pressure reliability of soft rock tunnels according to claim 1, characterized in that... In step S3, the standard normal spatial response surface function is tangent to the spherical or target reliability contour equation at the design point, and the gradients of the two are proportional at the tangency point. The coordinates of the design point are then solved. The scope is expanded to the corresponding dimension according to the number of random variables. When the random variable is a resistance parameter positively correlated with the function, the design point is taken as the lower quantile value, i.e., the coordinates are negative.

10. An analytical design system for the reliability of soft rock tunnel support pressure based on twin response surfaces, characterized in that... include: The sampling module is configured to perform central composite sampling based on the target reliability, setting the sampling center in the standard normal space U to achieve the target reliability index. On a sphere with radius , for the resistance parameter positively correlated with the function, the sampling center coordinates are set to ( , , ), and obtain the sampling point matrix in space U; The spatial mapping module is configured to map the sampling points of the U space to the original parameter space X space according to the actual distribution and correlation coefficient of the random variables, so as to obtain the sampling points of the X space; The numerical calculation module is configured to input the sampling points of the U space and X space into the numerical calculation model respectively to obtain the ultimate support pressure response value; The twin response surface construction module is configured to fit a U-space response surface based on the U-space sampling points and their response values, and to fit an X-space response surface based on the X-space sampling points and their response values. The design point analysis module is configured to solve the design point coordinates in U-space using the Lagrange multiplier method based on the response surface coefficients in U-space. For resistance parameters that are positively correlated with the function, the design point coordinates are calculated directly. The spatial transformation module is configured to convert the design points in the U space into design points in the X space based on the distribution type, statistical parameters, and correlation coefficient matrix of the random variables. The support pressure calculation module is configured to substitute the design point coordinates in the X-space into the X-space response surface equation to calculate the reliability index that meets the target. The support pressure failure threshold.

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