An analytical method for inverse reliability of tunnel face support pressure

Abstract

The present invention relates to an inverse reliability analysis method for tunnel face support pressure. A general expression for tunnel face support pressure that meets a target reliability index is adopted. By selecting numerical simulation results of seven test points, relevant parameters of the general expression for support pressure are determined, and then a standardized variable Y3 of support pressure can be calculated. Furthermore, an average value #imgabs1# of support pressure and a standard deviation #imgabs2# of support pressure can be determined through soil cohesion c and internal friction angle #imgabs0#. Furthermore, a design value x3 of support pressure can be calculated through the standardized variable Y3 of soil cohesion c and internal friction angle #imgabs3#, that is, the required support pressure is calculated. In this method, multiple iterations are not required, the error rate is lower, and the obtained design value x3 of support pressure is more accurate. Furthermore, there is no need to adopt continuous iterative adjustment of support pressure, so the process of obtaining the required support pressure is simpler, the amount of numerical calculations faced is smaller, and the efficiency is higher.

Description

Technical Field

[0001] The invention relates to the technical field of tunnel face support design, in particular to an inverse reliability analysis method for tunnel face support pressure. Background Art

[0002] During tunnel excavation, the tunnel face becomes an open surface. Under the action of initial geostress, the rock and soil on the tunnel face will undergo extrusion deformation toward the tunnel interior. For tunnels constructed in weak surrounding rock, the rock and soil have low strength parameters. Extrusion deformation of the tunnel face may continue after excavation and lead to tunnel face collapse. Tunnel face collapse threatens the lives of construction workers and the safety of machinery and equipment. It also causes excessive ground deformation, threatening the safety of nearby underground structures and above-ground buildings.

[0003] To ensure the stability of the tunnel face during tunnel construction in weak surrounding rock and avoid the adverse effects of face collapse, certain protective or reinforcement measures must be implemented during tunnel construction. One commonly used protective measure is to directly apply support pressure to the tunnel face. For a specific working condition, there is a limit support pressure at which the tunnel face is in a critically stable state. Generally, a larger support pressure can be applied to stabilize the face with a certain safety margin. Existing technology can determine the corresponding support pressure by setting a target safety factor.

[0004] However, in practical engineering, some parameters affecting tunnel face stability are uncertain, making the stability of the tunnel face a probabilistic problem. Therefore, when considering parameter uncertainty, it is necessary to determine a support pressure such that the tunnel face stability meets a given target reliability or failure probability under this support pressure.

[0005] Determining the corresponding tunnel face reliability or failure probability based on a given support pressure is a forward reliability analysis, which can be solved using conventional reliability analysis methods, including the first-order second-moment method and Monte Carlo method. Determining the corresponding support pressure based on a given reliability or failure probability is an inverse reliability analysis. This generally requires using different support pressures and conducting multiple forward reliability analyses to calculate the support pressure that meets the target reliability. This involves multiple steps, a complex process, and is difficult and time-consuming.

[0006] In summary, existing technologies for tunnel face support pressure reliability design generally involve two components: 1) forward reliability analysis, which involves calculating a reliability index given a given support pressure; and 2) iterative adjustment of the support pressure, which involves continuously adjusting the support pressure to ensure that the calculated reliability index approaches the target value. This process typically uses analytical methods to calculate the reliability index, requiring multiple iterations and making the accuracy of the results difficult to guarantee. Adjusting the support pressure also involves multiple iterations, each requiring extensive numerical calculations. This process is cumbersome, error-prone, and inefficient. Summary of the Invention

[0007] The purpose of the present invention is to: when the existing technology determines the corresponding support pressure according to a given reliability or failure probability, it is necessary to adopt different support pressures and perform multiple forward reliability analyses. First, the calculation of the reliability index generally adopts an analytical method, which requires multiple iterations and the accuracy of the calculation results is difficult to guarantee; second, the adjustment of the support pressure also involves multiple iterations, and each iteration requires a large amount of numerical calculations. The process is cumbersome, prone to errors, and inefficient. In response to the above problems, an inverse reliability analytical method for tunnel face support pressure is provided. By deducing a general expression for support pressure that meets any target reliability index, after numerical calculations are performed on several samples and the safety factor is obtained, the target support pressure can be quickly calculated.

[0008] In order to achieve the above object, the technical solution adopted by the present invention is:

[0009] A tunnel face support pressure inverse reliability analysis method includes the following steps:

[0010] The relevant parameters of the general expression of support pressure are determined by numerical simulation results of 7 test points. The general expression of support pressure is:

[0011]

[0012] Where Y3 is the standardized variable of support pressure, A′=b3, B′=a3, C′=a0+a1Y1+a2Y2+b1Y1 2 +b2Y2 2 a0, a1, b1, a2, b2, a3, b3 are the coefficients of the linear equation system composed of the limit state equations of the 7 test points, Y1 is the standardized variable of the soil cohesion c, and Y2 is the internal friction angle The standardized variable of support pressure is obtained by the relevant parameters of the general expression of support pressure, and the standardized variable Y3 of support pressure is obtained;

[0013] And through the soil cohesion c and internal friction angle Determine the average support pressure and support pressure standard deviation

[0014] Then according to the average support pressure and support pressure standard deviation The standardized variable Y3 of support pressure is converted to the required design value of support pressure x3. The conversion formula is:

[0015]

[0016] The stability of the tunnel face is affected by the cohesion c of the soil, the internal friction angle The influence of factors such as soil weight, tunnel shape and size, and support pressure. Existing studies generally use soil cohesion c and internal friction angle Considered as a random variable; soil density, tunnel shape and size as fixed parameters; support pressure as a deterministic variable. Based on this, the present invention uses the numerical simulation results of 7 test points to determine the relevant parameters of the universal expression of support pressure.

[0017] The inverse reliability analytical method for tunnel face support pressure of the present invention adopts a general expression of tunnel face support pressure that meets the target reliability index. By selecting the numerical simulation results of 7 test points, the relevant parameters of the general expression of support pressure are determined, and then the standardized variable Y3 of support pressure can be calculated. And the soil cohesion c and internal friction angle are used to calculate the inverse reliability analytical method for tunnel face support pressure. Ability to determine the average support pressure and support pressure standard deviation Then, the soil cohesion c, internal friction angle The standardized variable Y3 of the support pressure is used to calculate the design value of the support pressure x3, that is, the required support pressure. This method does not require multiple iterations, has a lower error rate, and the obtained design value of the support pressure x3 is more accurate. There is no need to adopt continuous iterative adjustment of the support pressure, and the process of obtaining the required support pressure is simpler, the amount of numerical calculations faced is smaller, and the efficiency is higher.

[0018] Preferably, according to the soil cohesion c and internal friction angle Determine the ultimate support pressure σ T , and take the average support pressure Take the standard deviation of support pressure Able to quickly and accurately obtain the average support pressure and support pressure standard deviation

[0019] Preferably, according to the soil cohesion c and internal friction angle The average value of the ultimate support pressure σ is determined by numerical calculation using the dichotomy principle. TThe ultimate support pressure σ can be obtained more quickly and accurately through the dichotomy principle. T .

[0020] Preferably, after obtaining the support pressure design value x3, the tunnel face is supported according to the obtained support pressure design value x3, and the designed tunnel face support structure has high reliability and lower cost.

[0021] Preferably, the steps for determining the relevant parameters of the general expression of support pressure by selecting the numerical simulation results of 7 test points are as follows:

[0022] S01. Select 7 test sites based on the central composite sampling strategy;

[0023] S02. Calculate the strength reduction factor of the 7 test points in turn to obtain the safety factor F s , thereby determining the performance function value Z of each test point:

[0024] Z=F s -1;

[0025] S03. Solve the linear equation system consisting of the limit state equations of the 7 test points and determine the 7 undetermined coefficients [a0, a1, b1, a2, b2, a3, b3] of the linear equation system;

[0026] S04, according to the set target reliability index β t And the coefficients a1 and a2 are solved, and the standardized variable Y1 of soil cohesion c and the internal friction angle are calculated. The standardized variable Y2, the formula for calculating Y1 and Y2 is:

[0027]

[0028] S05. Calculate the coefficients a′, B′, and C′ of the quadratic equation system for the standardized support pressure Y3 based on Y1, Y2, and the coefficients of the linear equation system;

[0029] The quadratic equation system for the standardized support pressure Y3 is:

[0030] A′Y3 2 +B′Y3+C′=0;

[0031] Where A′=b3, B′=a3, C′=a0+a1Y1+a2Y2+b1Y1 2 +b2Y2 2 ;

[0032] S06. Solve the roots of a quadratic equation. The formula for solving the roots of a quadratic equation is:

[0033]

[0034] By adopting the above method, the relevant parameters of the universal expression of support pressure can be accurately obtained, and the standardized variable Y3 of support pressure can be obtained quickly and accurately.

[0035] Preferably, in step S01, 7 test points are selected according to the test point table in Table 1;

[0036] Table 1 Test point table

[0037]

[0038] In Table 1, the standardized variables are 0, which represents the mean value; 1, which represents the mean value plus one standard deviation; and -1, which represents the mean value minus one standard deviation.

[0039] Preferably, the target reliability index β is derived based on the Lagrange multiplier t A general expression for tunnel face support pressure.

[0040] Further preferably, the derivation steps of the general expression of support pressure are as follows:

[0041] S1, let x i (i=1,2,3) represent soil cohesion c, internal friction angle and support pressure σ T Three variables, Y i Represent the standardized variable forms of these variables respectively:

[0042]

[0043] In formula (1), and is the mean and standard deviation of the variable;

[0044] S2. Define the tunnel face performance function in a three-dimensional parameter space consisting of three standardized variables:

[0045] Z=f(Y1,Y2,Y3); (2)

[0046] When the value of the function Z is greater than 0, it indicates that the tunnel face is in a stable state; when the value of the function Z is less than 0, it indicates that the tunnel face is in an unstable state; when the value of the function Z is equal to 0, it indicates that the tunnel face is in a limit state. The limit state expression is:

[0047] f(Y1,Y2,Y3)=0; (3)

[0048] S3, given a support pressure Y3 = Y3 i, the three-dimensional parameter space Ω(Y1, Y2, Y3) will be reduced to the two-dimensional parameter space Ω(Y1, Y2), and the limit state hyperplane f(Y1, Y2, Y3=Y3 i )=0 is the projection of f(Y1,Y2,Y3)=0 on the plane perpendicular to the Y3 axis, and makes the stable state of the tunnel face meet the target reliability in the two-dimensional parameter space Ω(Y1,Y2);

[0049] S4, let β t Represents the target reliability index. In the two-dimensional parameter space, a circle with a radius of R = β can be defined at the origin. t The equation of the circle is as follows:

[0050] g(Y1,Y2)=Y1 2 +Y2 2 =β t 2 ; (4)

[0051] According to the definition of the reliability index, its size is equal to the minimum distance from the coordinate origin to the limit state plane in the standard parameter space. The reliability index of the tunnel face increases with the increase of support pressure. Depending on the size of the support pressure, there are three relationships between the position of the limit state plane and the circle:

[0052] 1) When the applied tunnel face support pressure σ T Less than the target value σ T * When the corresponding reliability index β is less than the target reliability index β t , limit state plane f(Y1,Y2,Y3=Y3 i )=0 and circle g(Y1,Y2)=β t 2 For the intersection relationship;

[0053] 2) When the applied tunnel face support pressure σ T Greater than the target value σ T * When the corresponding reliability index β is greater than the target reliability index β t , limit state plane f(Y1,Y2,Y3=Y3 i )=0 and circle g(Y1,Y2)=β t 2 For the relationship of separation;

[0054] 3) When the applied tunnel face support pressure σ T Equal to the target value σ T * When the corresponding reliability index β is equal to the target reliability index β t , limit state plane f(Y1,Y2,Y3=Y3i )=0 and circle g(Y1,Y2)=β t 2 For the relationship;

[0055] According to the Lagrange principle, the gradient at the tangent point of the curve is proportional, and there exists a Lagrange multiplier such that f(Y1, Y2, Y3=Y3 i )=0 gradient and g(Y1,Y2)=β t 2 Gradient satisfy:

[0056]

[0057] S5. Gradient proportionality means that the partial derivatives of the function with respect to all variables are proportional, so:

[0058]

[0059] S6. Tunnel face stability is a complex three-dimensional problem. Its limit state function has no explicit expression. It is approximated by a polynomial without cross terms:

[0060] f(Y1,Y2,Y3)=a0+a1Y1+a2Y2+a3Y3+b1Y1 2 +b2Y2 2 +b3Y3 2 =0; (7)

[0061] In formula (7), [a0, a1, b1, a2, b2, a3, b3] represents the coefficients of the equation that need to be determined;

[0062] S7, the limit state equation f(Y1,Y2,Y3 i ) is expressed as:

[0063] f(Y1,Y2,Y3)=f′(Y1,Y2)+f″(Y3)=0; (8)

[0064] In formula (8), f′(Y1,Y2)=a0+a1Y1+a2Y2+b1Y1 2 +b2Y2 2 , f″(Y3)=a3Y3+b3Y3 2 , for a given support pressure Y3=Y3 i , f″(Y3)=a3Y3+b3Y3 2 is a constant term;

[0065] S8. Combined with formula (4), f′(Y1, Y2) is sorted as:

[0066]

[0067] Then the partial derivative of the limit state function with respect to Y1 is:

[0068]

[0069] Then the function g(Y1,Y2)=Y1 2 +Y2 2 The partial derivative with respect to variable Y1 is:

[0070]

[0071] S9, Combining equations (6), (10) and (11), we can obtain:

[0072] a1=2λY1 (12)

[0073] S10. Arrange f′(Y1,Y2) as follows:

[0074]

[0075] Then the partial derivative of the limit state function with respect to Y2 is:

[0076]

[0077] Then the function g(Y1,Y2)=Y1 2 +Y2 2 The partial derivative with respect to the variable Y2 is:

[0078]

[0079] S11, Combining equations (6), (14) and (15), we can get:

[0080] a2=2λY2; (16)

[0081] S12, Combining formula (12) and (16), we can get:

[0082]

[0083] S13. Substituting formula (17) into formula (4), we can obtain the solutions of variables Y1 and Y2:

[0084]

[0085] Each random variable has both positive and negative solutions. Since cohesion and internal friction angle are both strength parameters of the soil and are positively correlated with the stability of the tunnel face, to ensure a certain degree of stability and reliability, the support pressure corresponding to the design point should be such that the sample with lower strength is located on the limit state plane. Therefore, the solutions of these two variables should be less than their average values. Since cohesion and internal friction angle of the soil are strength parameters, the intersection of the ellipse represented by the limit state plane and the target reliability should be the smaller value of the two parameters. When the mean value of the standardized variable is 0, the solutions of the two variables should both be negative, resulting in:

[0086]

[0087] Given the target reliability index β t When the specific value of is used, the variables Y1 and Y2 are fixed values, and the limit state equation is a quadratic equation about the standardized support pressure Y3:

[0088] A′Y3 2 +B′Y3+C′=0; (20)

[0089] Where A′=b3, B′=a3, C′=a0+a1Y1+a2Y2+b1Y1 2 +b2Y2 2 ;

[0090] S14. According to the root-finding formula of the quadratic equation, the general expression of the standardized support pressure can be obtained:

[0091]

[0092] By constructing a specific form of limit state function and constraint function and using the Lagrange multiplier principle, a universal expression for the support pressure that meets any target reliability index is derived.

[0093] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are:

[0094] 1. The inverse reliability analytical method for tunnel face support pressure of the present invention adopts a general expression of tunnel face support pressure that meets the target reliability index. By selecting the numerical simulation results of 7 test points, the relevant parameters of the general expression of support pressure are determined, and then the standardized variable Y3 of support pressure can be calculated. And the soil cohesion c and internal friction angle are used to calculate the inverse reliability analytical method for tunnel face support pressure. Ability to determine the average support pressure and support pressure standard deviation Then, the soil cohesion c, internal friction angle The standardized variable Y3 of the support pressure is used to calculate the design value of the support pressure x3, that is, the required support pressure. This method does not require multiple iterations, has a lower error rate, and the obtained design value of the support pressure x3 is more accurate. There is no need to adopt continuous iterative adjustment of the support pressure, and the process of obtaining the required support pressure is simpler, the amount of numerical calculations faced is smaller, and the efficiency is higher.

[0095] 2. The inverse reliability analytical method for tunnel face support pressure described in the present invention constructs a specific form of limit state function and constraint function, and uses the Lagrange multiplier principle to derive a general expression for support pressure that meets any target reliability index. By selecting the numerical simulation results of test points, the relevant parameters of the support pressure expression are determined, thereby quickly calculating the required support pressure. BRIEF DESCRIPTION OF THE DRAWINGS

[0096] Figure 1 1 is a flow chart of the inverse reliability analysis method for tunnel face support pressure according to the present invention;

[0097] Figure 2 The position of the limit state plane and circle in the general expression of support pressure derived by this invention is β<β t Schematic diagram of;

[0098] Figure 3 The position of the limit state plane and circle in the general expression of support pressure derived by this invention is β>β t Schematic diagram of;

[0099] Figure 4 The position of the limit state plane and circle in the general expression of support pressure derived by this invention is β=β t Schematic diagram of;

[0100] Figure 5 The general expression for support pressure derived in this invention is Schematic diagram of;

[0101] Figure 6 Yes x3 1 =Verification result diagram of tunnel face failure probability under support pressure of 36.4kPa;

[0102] Figure 7 Yes x3 1 = 40.2kPa support pressure under tunnel face failure probability verification results. DETAILED DESCRIPTION

[0103] The present invention will be described in detail below with reference to the accompanying drawings.

[0104] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are only intended to illustrate the present invention and are not intended to limit the present invention. Unless otherwise specified, all conventional methods are used.

[0105] Example 1

[0106] This embodiment provides a method for analyzing the inverse reliability of tunnel face support pressure. Figure 1 , including the following steps:

[0107] The relevant parameters of the general expression of support pressure are determined by numerical simulation results of 7 test points. The general expression of support pressure is:

[0108]

[0109] Where Y3 is the standardized variable of support pressure, A′=b3, B′=a3, C′=a0+a1Y1+a2Y2+b1Y1 2 +b2Y2 2 a0, a1, b1, a2, b2, a3, b3 are the coefficients of the linear equation system composed of the limit state equations of the 7 test points, Y1 is the standardized variable of the soil cohesion c, and Y2 is the internal friction angle The standardized variable of support pressure is obtained by the relevant parameters of the general expression of support pressure, and the standardized variable Y3 of support pressure is obtained;

[0110] And through the soil cohesion c and internal friction angle Determine the average support pressure and support pressure standard deviation

[0111] Then according to the average support pressure and support pressure standard deviation The standardized variable Y3 of support pressure is converted to the required design value of support pressure x3. The conversion formula is:

[0112]

[0113] In this scheme, the target reliability index β is derived based on the Lagrange multiplier. t A general expression for tunnel face support pressure is derived. By constructing a specific form of limit state function and constraint function and using the Lagrange multiplier principle, a general expression for support pressure that meets any target reliability index is derived.

[0114] The stability of the tunnel face is affected by the cohesion c of the soil, the internal friction angle The influence of factors such as soil weight, tunnel shape and size, and support pressure. Existing studies generally use soil cohesion c and internal friction angle The soil weight, tunnel shape and size are considered as random variables; the support pressure is considered as a deterministic variable.

[0115] The derivation steps of the general expression of support pressure are as follows:

[0116] S1, let x i (i=1,2,3) represent soil cohesion c, internal friction angle and support pressure σ T Three variables, Y i Represent the standardized variable forms of these variables respectively:

[0117]

[0118] In formula (1), and is the mean value and standard deviation of the variable, that is, x1 represents the soil cohesion c, Y1 represents the standardized variable of soil cohesion c, represents the average value of soil cohesion c, represents the standard deviation of soil cohesion c, that is, x2 represents the internal friction angle Y2 represents the internal friction angle The standardized variable, represents the internal friction angle The average value of represents the internal friction angle The standard deviation of x3 represents the support pressure σ T , Y3 represents the support pressure σ T The standardized variable, represents the support pressure σ T The average value of represents the support pressure σ T The standard deviation of

[0119] S2. Define the tunnel face performance function in a three-dimensional parameter space consisting of three standardized variables:

[0120] Z=f(Y1,Y2,Y3); (2)

[0121] When the value of the function Z is greater than 0, it indicates that the tunnel face is in a stable state; when the value of the function Z is less than 0, it indicates that the tunnel face is in an unstable state; when the value of the function Z is equal to 0, it indicates that the tunnel face is in a limit state. The limit state expression is:

[0122] f(Y1,Y2,Y3)=0; (3)

[0123] S3, given a support pressure Y3 = Y3 i , the three-dimensional parameter space Ω(Y1, Y2, Y3) will be reduced to the two-dimensional parameter space Ω(Y1, Y2), and the limit state hyperplane f(Y1, Y2, Y3=Y3 i )=0 is the projection of f(Y1,Y2,Y3)=0 on the plane perpendicular to the Y3 axis, and makes the stable state of the tunnel face meet the target reliability in the two-dimensional parameter space Ω(Y1,Y2). In the two-dimensional parameter space Ω(Y1,Y2), the reliability index is a certain value, and its value depends on the limit state plane f(Y1,Y2,Y3=Y3 i )=0, which is determined by the size of the support pressure. In other words, by choosing the appropriate support pressure Y3=Y3 i , so that the stable state of the tunnel face meets the target reliability.

[0124] S4, let β t Represents the target reliability index. In the two-dimensional parameter space, a circle with a radius of R = β can be defined at the origin. t The equation of the circle is as follows:

[0125] g(Y1,Y2)=Y1 2 +Y2 2 =β t 2 ; (4)

[0126] According to the definition of the reliability index, its size is equal to the minimum distance from the coordinate origin to the limit state plane in the standard parameter space. The reliability index of the tunnel face increases with the increase of support pressure. Depending on the size of the support pressure, there are three relationships between the position of the limit state plane and the circle:

[0127] 1) If Figure 2 As shown in the figure, when the applied tunnel face support pressure σ T Less than the target value σ T * When the corresponding reliability index β is less than the target reliability index β t , limit state plane f(Y1,Y2,Y3=Y3 i )=0 and circle g(Y1,Y2)=β t 2 For the intersection relationship;

[0128] 2) If Figure 3 As shown in the figure, when the applied tunnel face support pressure σ T Greater than the target value σ T * When the corresponding reliability index β is greater than the target reliability index β t, limit state plane f(Y1,Y2,Y3=Y3 i )=0 and circle g(Y1,Y2)=β t 2 For the relationship of separation;

[0129] 3) If Figure 4 As shown in the figure, when the applied tunnel face support pressure σ T Equal to the target value σ T * When the corresponding reliability index β is equal to the target reliability index β t , limit state plane f(Y1,Y2,Y3=Y3 i )=0 and circle g(Y1,Y2)=β t 2 For the relationship;

[0130] According to the Lagrange principle, the gradient at the tangent point of the curve is proportional, that is, there is a Lagrange multiplier such that f(Y1, Y2, Y3=Y3 i )=0 gradient and g(Y1,Y2)=β t 2 Gradient satisfy Figure 5 As shown:

[0131]

[0132] S5. Gradient proportionality means that the partial derivatives of the function with respect to all variables are proportional, so:

[0133]

[0134] S6. Tunnel face stability is a complex three-dimensional problem. Its limit state function has no explicit expression. It is approximated by a polynomial without cross terms:

[0135] f(Y1,Y2,Y3)=a0+a1Y1+a2Y2+a3Y3+b1Y1 2 +b2Y2 2 +b3Y3 2 =0; (7)

[0136] In formula (7), [a0, a1, b1, a2, b2, a3, b3] represents the coefficients of the equation that need to be determined;

[0137] S7, the limit state equation f(Y1,Y2,Y3 i ) is expressed as:

[0138] f(Y1,Y2,Y3)=f′(Y1,Y2)+f″(Y3)=0; (8)

[0139] In formula (8), f′(Y1,Y2)=a0+a1Y1+a2Y2+b1Y1 2 +b2Y2 2 , f″(Y3)=a3Y3+b3Y3 2 , for a given support pressure Y3=Y3 i , f″(Y3)=a3Y3+b3Y3 2 is a constant term;

[0140] S8. Combined with formula (4), f′(Y1, Y2) is sorted as:

[0141]

[0142] Then the partial derivative of the limit state function with respect to Y1 is:

[0143]

[0144] Then the function g(Y1,Y2)=Y1 2 +Y2 2 The partial derivative with respect to variable Y1 is:

[0145]

[0146] S9, Combining equations (6), (10) and (11), we can obtain:

[0147] a1=2λY1 (12)

[0148] S10. Arrange f′(Y1,Y2) as follows:

[0149]

[0150] Then the partial derivative of the limit state function with respect to Y2 is:

[0151]

[0152] Then the function g(Y1,Y2)=Y1 2 +Y2 2 The partial derivative with respect to the variable Y2 is:

[0153]

[0154] S11, Combining equations (6), (14) and (15), we can get:

[0155] a2=2λY2; (16)

[0156] S12, Combining formula (12) and (16), we can get:

[0157]

[0158] S13. Substituting formula (17) into formula (4), we can obtain the solutions of variables Y1 and Y2:

[0159]

[0160] Each random variable has both positive and negative solutions. Since cohesion and internal friction angle are both strength parameters of the soil and are positively correlated with the stability of the tunnel face, to ensure a certain degree of stability and reliability, the support pressure corresponding to the design point should be such that the sample with lower strength is located on the limit state plane. Therefore, the solutions of these two variables should be less than their average values. Since cohesion and internal friction angle of the soil are strength parameters, the intersection of the ellipse represented by the limit state plane and the target reliability should be the smaller value of the two parameters. When the mean value of the standardized variable is 0, the solutions of the two variables should both be negative, resulting in:

[0161]

[0162] Given the target reliability index β t When the specific value of is used, the variables Y1 and Y2 are fixed values, and the limit state equation is a quadratic equation about the standardized support pressure Y3:

[0163] A′Y3 2 +B′Y3+C′=0; (20)

[0164] Where A′=b3, B′=a3, C′=a0+a1Y1+a2Y2+b1Y1 2 +b2Y2 2 ;

[0165] S14. According to the root-finding formula of the quadratic equation, the general expression of the standardized support pressure can be obtained:

[0166]

[0167] When the coefficients [a0, a1, b1, a2, b2, a3, b3] are known, they can be brought into the solution for Y3 and solved by The support pressure is converted to x3. Since the support pressure must be positive, the negative value can be discarded to obtain the final target support pressure.

[0168] For details, see Figure 1 , the steps to determine the relevant parameters of the general expression of support pressure and solve Y3 by selecting the numerical simulation results of 7 test points are as follows:

[0169] S01. Select 7 test sites based on the central composite sampling strategy;

[0170] In step S01, 7 test points are selected according to the test point table in Table 1;

[0171] Table 1 Test point table

[0172]

[0173] In Table 1, the standardized variables are 0 for the mean, 1 for the mean plus one standard deviation, and -1 for the mean minus one standard deviation.

[0174] S02. Calculate the strength reduction factor of the 7 test points in turn to obtain the safety factor F s , thereby determining the performance function value Z of each test point:

[0175] Z=F s -1;

[0176] S03. Solve the linear equation system consisting of the limit state equations of the 7 test points and determine the 7 undetermined coefficients [a0, a1, b1, a2, b2, a3, b3] of the linear equation system;

[0177] S04, according to the set target reliability index β t And the coefficients a1 and a2 are solved, and the standardized variable Y1 of soil cohesion c and the internal friction angle are calculated. The standardized variable Y2, the formula for calculating Y1 and Y2 is:

[0178]

[0179] S05. Calculate the coefficients A', B' and C' of the quadratic equation system for the standardized support pressure Y3 based on Y1, Y2 and the coefficients of the linear equation system;

[0180] The quadratic equation system for the standardized support pressure Y3 is:

[0181] A′Y3 2 +B′Y3+C′=0;

[0182] Where A′=b3, B′=a3, C′=a0+a1Y1+a2Y2+b1Y1 2 +b2Y2 2 ;

[0183] S06. Solve the roots of a quadratic equation. The formula for solving the roots of a quadratic equation is:

[0184]

[0185] By adopting the above method, the relevant parameters of the universal expression of support pressure can be accurately obtained, and the standardized variable Y3 of support pressure can be obtained quickly and accurately.

[0186] In addition to obtaining Y3, it is also necessary to obtain the average support pressure and support pressure standard deviation In this embodiment, the soil cohesion c and internal friction angle Determine the average support pressure and support pressure standard deviation Specifically, according to the soil cohesion c and internal friction angle The average value of the ultimate support pressure σ is determined by numerical calculation using the dichotomy principle. T The ultimate support pressure σ can be obtained more quickly and accurately through the dichotomy principle. T According to the soil cohesion c and internal friction angle Determine the ultimate support pressure σ T Later, take the average support pressure Take the standard deviation of support pressure Able to quickly and accurately obtain the average support pressure and support pressure standard deviation

[0187] Obtaining the average support pressure Support pressure standard deviation After the standardized variable Y3 of the support pressure is obtained, the conversion formula is used. The required support pressure design value x3 can be obtained. After obtaining the support pressure design value x3, the tunnel face is supported according to the obtained support pressure design value x3. The designed tunnel face support structure has high reliability and lower cost.

[0188] The inverse reliability analytical method for tunnel face support pressure described in this embodiment constructs a specific form of limit state function and constraint function, and uses the Lagrange multiplier principle to derive a general expression for support pressure that meets any target reliability index. The general expression for tunnel face support pressure that meets the target reliability index is adopted, and the relevant parameters of the general expression for support pressure are determined by selecting the numerical simulation results of 7 test points, thereby being able to calculate the standardized variable Y3 of support pressure, and the soil cohesion c and internal friction angle are used to calculate the support pressure. Ability to determine the average support pressure and support pressure standard deviation Then, the soil cohesion c, internal friction angle The standardized variable Y3 of the support pressure is used to calculate the design value of the support pressure x3, that is, the required support pressure. This method does not require multiple iterations, has a lower error rate, and the obtained design value of the support pressure x3 is more accurate. There is no need to adopt continuous iterative adjustment of the support pressure, and the process of obtaining the required support pressure is simpler, the amount of numerical calculations faced is smaller, and the efficiency is higher, thereby quickly and accurately calculating the required support pressure.

[0189] Example 2

[0190] This embodiment provides a method for analyzing the inverse reliability of tunnel face support pressure. Based on Example 1, specific numerical descriptions are provided:

[0191] Consider a circular tunnel with a diameter and a depth of 10 m. The cohesion and friction angle of the soil are random variables with mean values μ and c =7kPa and The standard deviations are σ c =1.4kPa and Other parameters are fixed values, soil density is γ=18kN / m 3 , the elastic modulus is E=25MPa, and the Poisson's ratio is 0.3.

[0192] According to the method of the present invention, the reliability of the tunnel face support pressure is inversely analyzed to obtain the target reliability index β t The support pressure is determined as follows:

[0193] 1) According to the average value of soil cohesion μ c =7kPa and the average value of the internal friction angle The dichotomy principle is used to determine the ultimate support pressure σ using numerical calculations. T =23.3kPa. Take the average value of support pressure Support pressure standard deviation

[0194] 2) According to the central composite sampling strategy, the seven test points shown in Table 1 were selected.

[0195] 3) Calculate the strength reduction factor of the 7 test points in turn to obtain the safety factor F s , thereby determining the performance function value of the sample point, the results are shown in Table 2:

[0196] Table 2 Sample points and functional value table

[0197]

[0198]

[0199] 4) Solve the linear equation system consisting of the limit state equations of the 7 test points. Use the method of linear algebra to solve the homogeneous linear equation system. The method of linear algebra to solve the homogeneous linear equation system is the existing technology. The undetermined coefficients of the equation system are determined as follows:

[0200] [a0,a1,b1,a2,b2,a3,b3]=[0.33,0.055,0.005,0.115,0.005,0.355,0.025]; (23)

[0201] 5) Assuming the target reliability index β t =2, and the standardized variables Y1 and Y2 are calculated according to formula (19):

[0202]

[0203] 6) Calculate the coefficients S′, B′, and C′ of the quadratic equation based on Y1, Y2, and the coefficients of the equation.

[0204] A′=0.025, B′=0.355, C′=0.095; (25)

[0205] 7) Solve the roots of the quadratic equation according to formula (21).

[0206] Y3 1 =-0.273; Y3 2 =-13.927; (26)

[0207] 8) Convert the support pressure design value that meets the target reliability index.

[0208] x3 1 =40.2kPa; x3 2 =-277.9kPa; (27)

[0209] Since the support pressure must be positive, x3 is discarded. 2 =-277.9kPa, and the target reliability index β is obtained. t =2 support pressure is x3 1 =40.2kPa.

[0210] Example 3

[0211] Example 3, based on Example 2, selects different target reliability indicators to calculate the required support pressure. Since the coefficients of the limit state equation are known, the support pressure corresponding to all target reliability indicators can be quickly calculated using the general formula, as shown in Table 3. By discarding all negative values in Table 3, the support pressure corresponding to different target reliability indicators can be obtained.

[0212] Table 3 Support pressure required for different target reliability indicators

[0213]

[0214] In this embodiment, β in Table 3 is selected t =1.5 corresponding support pressure x3 1 =36.4kPa and β t =2.0 corresponding support pressure x3 1 =40.2kPa for accuracy verification. The verification method is direct Monte Carlo simulation, that is, taking enough samples and performing numerical calculations in sequence. Since Monte Carlo simulation requires the minimum number of samples to meet:

[0215] N min =100 / p f ; (28)

[0216] And β t =1.5 and β t =2.0 and the corresponding failure probabilities are p f = 0.07 and p f =0.023, the minimum number of samples required is N min =1429 and N min =4348. In this embodiment, N1=1500 and N2=4500 samples are randomly selected for numerical calculation and the stability of the tunnel face is judged. The calculation time is 3 days and 9 days respectively. The number of failure samples and the failure probability calculation results are as follows: Figure 6 and Figure 7 As shown. The failure probabilities calculated using the support pressure determined by the present invention are p f =0.072 and p f =0.026, which is consistent with the target reliability index β t =1.5 and β t =2.0 is very consistent with the corresponding failure probability, which verifies the calculation accuracy of the method of the present invention.

[0217] If support pressure is calculated based on the failure probability of direct Monte Carlo simulation, assuming that the required support pressure can be converged after 10 cycles, the conventional method would require calculations of 30 and 90 days, and even longer when the target failure probability is higher. The method of the present invention only requires a few hours of numerical simulation to determine the safety factor for seven samples. The derived general expression can then be used to calculate the required support pressure for any target failure probability, greatly improving computational efficiency. Verification results also demonstrate the high accuracy of this method and its wide application in the reliability design of support pressures considering soil parameter uncertainties.

[0218] The present invention uses the obtained general expression to calculate the tunnel face support pressure that meets the target reliability index. The support pressure design for any given target reliability index can be carried out by simply using the safety factor calculation results of 7 sample points, avoiding the errors caused by the iterative process of conventional reliability analysis methods. The method of the present invention also does not require continuous trial calculation of the support pressure and repeated forward reliability analysis processes, thereby greatly simplifying the calculation process and improving calculation efficiency.

[0219] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for analyzing the inverse reliability of tunnel face support pressure, characterized in that: The following steps are involved: The steps for determining the relevant parameters of the general expression of support pressure by selecting the numerical simulation results of 7 test points are as follows: S01. Select 7 test sites based on the central composite sampling strategy; S02. Calculate the strength reduction factor of the 7 test points in turn to obtain their safety factor , thereby determining the performance function value of each test point : ； S03. Solve the linear equation system consisting of the limit state equations of the 7 test points and determine the 7 unknown coefficients of the linear equation system. ; S04. According to the set target reliability index and the coefficients to be solved and , calculate soil cohesion Standardized variables and internal friction angle Standardized variables ,calculate and The formula is: ； S05, according to 、 and the coefficients of the linear equations, calculate the normalized support pressure The coefficients of the quadratic equation 、 and ; About Standardized Support Pressure The quadratic equation system is: ； Where, , , ; S06. Solve the roots of the quadratic equation to obtain the general expression of support pressure, which is: Where, is the standardized variable of support pressure; Through soil cohesion and internal friction angle Determine the average support pressure and support pressure standard deviation ; Then according to the average support pressure and support pressure standard deviation , standardized variables of support pressure Convert the required support pressure design value , the conversion formula is: 。 2. The inverse reliability analysis method for tunnel face support pressure according to claim 1 is characterized in that: According to soil cohesion and internal friction angle Determine the ultimate support pressure , and take the average support pressure , take the standard deviation of support pressure .

3. The inverse reliability analysis method for tunnel face support pressure according to claim 2 is characterized in that: According to soil cohesion and internal friction angle The average value of the ultimate support pressure is determined by numerical calculation using the dichotomy principle. .

4. The inverse reliability analysis method for tunnel face support pressure according to claim 1 is characterized in that: Obtaining the design value of support pressure After that, according to the support pressure design value Support the tunnel face.

5. The inverse reliability analysis method for tunnel face support pressure according to claim 1 is characterized in that: In step S01, 7 test points are selected according to the test point table in Table 1; Table 1 Test point table In Table 1, the standardized variables are 0, which represents the mean value; 1, which represents the mean value plus one standard deviation; and -1, which represents the mean value minus one standard deviation.

6. The inverse reliability analysis method for tunnel face support pressure according to any one of claims 1 to 5, characterized in that: Derivation of target reliability index based on Lagrange multipliers A general expression for tunnel face support pressure.

7. The inverse reliability analysis method for tunnel face support pressure according to claim 6 is characterized in that: The derivation steps of the general expression of support pressure are as follows: S1. Order ( ) represent soil cohesion , internal friction angle and support pressure Three variables, Represent the standardized variable forms of these variables respectively: In formula (1), and is the mean and standard deviation of the variable; S2. Define the tunnel face performance function in a three-dimensional parameter space consisting of three standardized variables: When the function When the value of is greater than 0, it means that the tunnel face is in a stable state; when the function When the value is less than 0, it means the tunnel face is unstable. When the value is equal to 0, it means that the tunnel face is in the limit state. The limit state expression is: S3. Given a support pressure , the three-dimensional parameter space It is reduced to a two-dimensional parameter space , the limit state hyperplane at this time yes In vertical The projection on the plane of the axis, and the stable state of the tunnel face satisfies the two-dimensional parameter space Reliability of target within S4. Let Represents the target reliability index. In the two-dimensional parameter space, a circle with a radius of centered at the origin can be defined. The equation of the circle is as follows: According to the Lagrange principle, the gradient at the tangent point of the curve is proportional, and there exists a Lagrange multiplier such that Gradient and Gradient satisfy: S5. Gradient proportionality means that the partial derivatives of the function with respect to all variables are proportional, so: S6. Tunnel face stability is a complex three-dimensional problem. Its limit state function has no explicit expression. It is approximated by a polynomial without cross terms: In formula (7), Indicates the coefficients of the equation that need to be determined; S7, the limit state equation Expressed as: In formula (8), , , for a given support pressure , is a constant term; S8, combined with formula (4), Arranged as: Then the limit state function is The partial derivative of is: Then the function For variables The partial derivative of is: S9, Combining equations (6), (10) and (11), we can obtain: S10, will Arranged as: Then the limit state function is The partial derivative of is: Then the function For variables The partial derivative of is: S11. Combining equations (6), (14) and (15), we can obtain: S12, Combining formula (12) and (16), we can get: S13. Substitute formula (17) into formula (4) to obtain the variable and Solution: Since the cohesion and internal friction angle of the soil are strength parameters, the intersection of the limit state plane and the ellipse represented by the target reliability should be the smaller value of the two parameters. When the mean value of the standardized variable is 0, the solutions of the two variables should both be negative, resulting in: Reliable indicators for a given target When the specific value of and To determine the value, the limit state equation is given by the normalized support pressure The quadratic equation of one variable is: Where, , , ; S14. According to the root-finding formula of the quadratic equation, the general expression of the standardized support pressure can be obtained: 。

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