A reliability analysis method for rock slopes considering non-Gaussian correlated random variables

Abstract

The present invention provides a rock slope reliability analysis method that takes into account non-Gaussian correlated random variables, and belongs to the field of geological engineering reliability analysis. The method includes the following steps: analyzing the components of the anti-sliding force and sliding force of the rock slope and calculating them; constructing the functional function of the rock slope; determining the probability distribution information of the random variables; and calculating the reliability index of the rock slope. The core of the present invention lies in: 1. It fully considers the stress characteristics of the rock slope itself, and conducts reliability analysis based on this; 2. It directly places the design point iterative calculation of the first-order reliability method based on Copula theory in the physical space, giving the iterative process an intuitive and easy-to-understand physical meaning, while significantly enhancing the stability of the iterative process. This breakthrough technology not only accurately meets the actual needs of geological engineering, especially slope reliability analysis, but also provides a solid and reliable technical backing for rock slope engineering design.

Description

Technical Field

[0001] The present application relates to the field of geological engineering reliability analysis, and in particular to an efficient reliability analysis method for rock slopes taking into account non-Gaussian correlated random variables. Background Art

[0002] With the rapid advancement of engineering technology and the continued advancement of scientific research, the quantitative assessment of slope stability by treating geotechnical parameters as random variables and ingeniously integrating advanced probabilistic statistical theories and techniques has become a widely practiced and highly acclaimed scientific paradigm within the industry. Despite significant achievements in probabilistic slope stability assessment, researchers continue to face numerous challenges and difficulties in their long-term practical journey. These challenges stem primarily from the extreme complexity and multi-factor nature of slope stability assessment systems, forcing researchers to continuously explore new approaches and optimize existing methods, aiming to develop more efficient, accurate, and adaptable probabilistic assessment models and procedures to meet the increasingly complex needs of slope stability assessment.

[0003] In recent years, researchers have gradually realized that relying solely on Gaussian correlation structures to model the correlations between random variables often fails to fully and accurately reflect the complex correlations between geotechnical parameters in real-world scenarios, prompting the search for more sophisticated and adaptable modeling methods. Consequently, Copula theory has attracted considerable attention for its ability to accurately model complex multidimensional joint distributions between random variables. Its core principle lies in its ingenious decomposition of the multidimensional joint distribution into independent marginal distributions and a core Copula function. This revolutionary approach not only intuitively reveals the inherent connections between random variables but also precisely quantifies their correlation characteristics, providing a powerful tool for in-depth understanding and analysis of complex systems. Particularly noteworthy is the diversity of Copula function types, each with unique correlation structure characteristics, offering a wide range of options for detailed analysis of random variable correlations in diverse application scenarios. Furthermore, first-order reliability methods, with their exceptional efficiency and practicality, have emerged as a promising approach for addressing the challenges of reliability assessment for nonlinear performance functions. This method cleverly uses the first-order Taylor expansion of the performance function in an independent standard normal space near the design point to simplify complex nonlinear functions into intuitive and easy-to-solve linear expressions. This transformation not only greatly simplifies the calculation process and reduces the calculation cost, but also ensures the high accuracy and reliability of the evaluation results, becoming an indispensable and efficient tool for dealing with such problems.

[0004] However, given that the inherent characteristics of geotechnical parameters are often nonnormal and correlated, researchers face a key challenge when applying the first-order reliability method framework based on Copula theory: the need to map these complex, correlated, nonnormal geotechnical parameters into independent standard normal random variables. This transformation process is not only difficult to understand but also significantly exacerbates the nonlinearity of the performance function, thereby introducing potential instability into the iterative solution process for the design point. Specifically, this nonlinear enhancement directly leads to a sharp decline in the iterative convergence rate and sometimes even leads to convergence failure, manifested as periodic fluctuations or chaos within a specific solution set, significantly hindering computational efficiency and the reliability of the results. In some extreme cases, the iterations may even completely diverge and fail to converge to a valid solution, requiring researchers to adopt more sophisticated numerical processing strategies or algorithm optimization measures to ensure the stability of the analysis process and the accuracy of the results. These issues have significantly restricted the widespread application and in-depth development of the first-order reliability method based on Copula theory in the field of geotechnical engineering reliability analysis, and have led to a severe challenge in the balance between accuracy and practicality.

[0005] In summary, the field of geological engineering urgently needs an innovative reliability analysis method that combines robustness, efficiency, and practicality. This method can efficiently handle non-Gaussian random variables directly in physical space. The core of this method lies in directly placing the iterative calculation of the design point in physical space, giving the iterative process an intuitive and easy-to-understand physical meaning while significantly enhancing its stability. This groundbreaking technology not only precisely meets the practical needs of geological engineering, especially slope reliability analysis, but also provides solid and reliable technical support for engineering designers. It is also committed to playing a core role in the safety assessment, stable operation, and long-term maintenance of geological engineering projects. Summary of the Invention

[0006] In response to the deficiencies of the prior art, the present application provides an efficient reliability analysis method for rock slopes that considers non-Gaussian correlated random variables in physical space.

[0007] This application provides a rock slope reliability analysis method considering non-Gaussian correlated random variables using the following technical solutions:

[0008] A rock slope reliability analysis method considering non-Gaussian correlated random variables includes the following steps:

[0009] Step 1: Calculate the anti-sliding force R and sliding force Q of the rock slope;

[0010] According to the stress characteristics of rock slopes, the anti-sliding force R of rock slopes is composed of the friction force on the sliding surface, the weight of the slope W, the horizontal seismic force, the water pressure on the sliding surface, the water pressure on the tensile fracture surface, and the slope anchoring force. The sliding force Q of rock slopes is composed of the weight of the slope W, the horizontal seismic force, the water pressure on the tensile fracture surface, and the slope anchoring force. Their expressions are:

[0011]

[0012] Where c is the cohesion of rock, is the internal friction angle of rock, ψ p is the inclination of the sliding surface, H is the height of the rock slope, z is the area of ​​the tensile crack per unit width, r is the water filling depth coefficient of the tensile crack, α is the horizontal earthquake acceleration coefficient, γ w is the gravity of water, T is the anchoring force on the slope, and ε is the angle between the anchoring force and the normal of the sliding surface;

[0013] Step 2: Construct the functional function G of the rock slope according to the anti-sliding force and sliding force of the rock slope, and its expression is: G = RQ;

[0014] Step 3, determining the probability distribution information of the random variable according to the characteristics of the non-Gaussian correlated random variable;

[0015] Select the rock's cohesion c, the rock's internal friction angle The area of ​​the tensile crack per unit width z, the water-filled depth coefficient r of the tensile crack, and the horizontal earthquake acceleration coefficient α are taken as non-Gaussian correlated random variables and are denoted as random variables x n , n=1,2,3,4,5, then its probability distribution information includes:

[0016] 5 random variables x n The cumulative distribution function of n (x n );

[0017] 5 random variables x n The probability density function of n (x n );

[0018] Copula function C(F1(x1),F2(x2);θ) between random variables x1 and x2;

[0019] Conditional Copula function h(F1(x1), F2(x2); θ) between random variables x1 and x2;

[0020] The second-order conditional copula function d(F1(x1), F2(x2); θ) between the random variables x1 and x2;

[0021] Copula density function c(F1(x1), F2(x2); θ) between random variables x1 and x2;

[0022] Where θ is the Copula parameter;

[0023] Step 4: The reliability index β of the rock slope is efficiently calculated through the physical parameter space design point iteration technology.

[0024] Preferably, the calculation formula of the slope weight W is:

[0025]

[0026] Where, γ rock is the weight of the rock, ψ f is the inclination angle of the rock slope.

[0027] Preferably, the Copula parameter θ is obtained by inversely solving the correlation coefficient ρ, and the calculation formula is:

[0028]

[0029] Where μ1 is the mean value of rock cohesion c, μ2 is the internal friction angle of rock σ1 is the standard deviation of rock cohesion c, and σ2 is the internal friction angle of rock. The standard deviation of .

[0030] Preferably, the implementation process of step 4 is as follows:

[0031] Step 4.1, set the following parameters:

[0032] Set the maximum number of iterations N and the allowable error ε x , denote any iteration as the kth iteration, k = 1, 2, ..., N; denote the random variable x in the kth iteration n is x n (k) ;

[0033] Step 4.2, let the design point of the first iteration be x (1) , x (1) =[x1 (1) x2 (1) x3 (1) x4 (1) x5 (1) ] T =[μ1μ2μ3μ4μ5] T, where μ1 is the mean value of the rock cohesion c, and μ2 is the internal friction angle of the rock , μ3 is the mean value of the area z of the unit width of the tensile crack, μ4 is the mean value of the water-filled depth coefficient r of the tensile crack, μ5 is the mean value of the horizontal earthquake acceleration coefficient α, and the superscript “T” is the transpose of the vector and matrix;

[0034] Step 4.3, calculate the performance function G(x (k) ), x (k) is the design point of the kth iteration, x (k) =[x1 (k) x2 (k) x3 (k) x4 (k) x5 (k) ] T ,G(x (k) ) by taking the design point x of the kth iteration (k) Substitute the function G of the rock slope and calculate the gradient vector of the kth iteration by the central difference method

[0035] Step 4.4, calculate the generalized lower triangular matrix A of the kth iteration (k) , its calculation expression is:

[0036]

[0037] Where A 21 (k) is the generalized lower triangular matrix A of the kth iteration (k) The element in the second row and first column of , whose calculation expression is:

[0038]

[0039] Where, d(F1(x1 (k) ),F2(x2 (k) ); θ) is the second-order conditional Copula function of the kth iteration, c(F1(x1 (k) ),F2(x2 (k) ), θ) is the Copula density function of the kth iteration, φ(·) is the probability density function of the standard normal distribution, and Φ -1 (·) is the inverse function of the cumulative distribution function of the standard normal distribution;

[0040] A 22 (k) is the generalized lower triangular matrix A of the kth iteration (k) The element in the second row and second column of , whose calculation expression is:

[0041]

[0042] Where, h(F1(x1 (k) ),F2(x2 (k) ); θ) is the conditional Copula function of the kth iteration;

[0043] Calculate the generalized standard deviation matrix B for the kth iteration (k) , its calculation expression is:

[0044]

[0045] Where, F n (x n (k) ) is the cumulative distribution function of the kth iteration, f n (x n (k) ) is the probability density function of the k-th iteration;

[0046] Calculate the generalized mean vector C of the kth iteration (k) , and its calculation formula is:

[0047]

[0048] Where C e (k) is an additional term of the Copula function, and its calculation expression is:

[0049]

[0050] Calculate the design point x for the k+1th iteration (k+1) , x (k+1) =[x1 (k+1) x2 (k+1) x3 (k+1) x4 (k+1) x5 (k+1) ] T , its calculation expression is:

[0051] x (k+1) =x (k) +λ (k) d (k)

[0052] Where λ (k) is the iteration step length, d (k) is the iteration direction, and its expression is:

[0053]

[0054] Step 4.5, if |||x is satisfied (k+1) -x (k) || / ||x(k+1) |||≤ε x , then let the design point x of the kth iteration be (k) is the design point x * , and record the design point x of the kth iteration * The corresponding random variable x n is x n * , x * =[x1 * x2 * x3 * x4 * x5 * ] T , let the generalized lower triangular matrix A of the kth iteration (k) is the generalized lower triangular matrix A at the design point * , let the generalized standard deviation matrix B of the kth iteration be (k) is the generalized standard deviation matrix B at the design point * , let the generalized mean vector C of the kth iteration (k) is the generalized mean vector C at the design point * , and go to step 4.7. On the contrary, if |||x (k+1) -x (k) || / ||x (k+1) |||>ε x , then go to step 4.6;

[0055] Step 4.6: If the number of iterations k is greater than or equal to the maximum number of iterations N, then let the design point x of the kth iteration be (k) is the design point x * , x * =[x1 * x2 * x3 * x4 * x5 * ] T , let the generalized lower triangular matrix A of the kth iteration (k) is the generalized lower triangular matrix A at the design point * , let the generalized standard deviation matrix B of the kth iteration be (k) is the generalized standard deviation matrix B at the design point * , let the generalized mean vector C of the kth iteration (k) is the generalized mean vector C at the design point * , and go to step 4.7. Otherwise, set the number of iterations k = k + 1 and return to step 4.3;

[0056] Step 4.7, calculate the reliability index β of the rock slope, and its calculation expression is:

[0057] β=[(x* -C * ) T ((B * ) T )- 1 ((A * ) T )- 1 (A * )- 1 (B * )- 1 (x * -C * )] 1 / 2

[0058] In summary, this application has the following beneficial technical effects:

[0059] The present invention aims to solve the difficult problems of iterative process divergence and unclear physical meaning encountered by the first-order reliability method based on Copula theory in the field of geological engineering in an independent standard normal space, and innovatively proposes a new method for rock slope reliability analysis that is directly implemented in the physical space, efficient, and targeted at non-Gaussian correlated random variables. Specifically, the present invention first analyzes the anti-sliding force and sliding force of the rock slope, then constructs the functional function of the rock slope, then determines the probability distribution information of the random variables, and finally calculates the reliability index of the rock slope. In the process of calculating the reliability index of the rock slope, the present invention abandons the iterative method of the traditional method in the independent standard normal space, and instead directly performs an iterative search for the design point in the physical space. This change not only gives the iterative process an intuitive physical background, which is easier for researchers and engineers to understand, but also greatly improves the stability and convergence of the iterative calculation, effectively avoiding the problem of iterative divergence. In summary, this invention not only provides researchers in the field of geological engineering with a set of efficient, accurate, and easy-to-understand rock slope reliability analysis tools, but also provides solid technical support for engineering designers in the safety assessment and design of geological engineering projects. It is expected to play a vital role in improving engineering safety performance and reducing risk assessment errors. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 It is a flow chart of the reliability analysis method of the present invention.

[0061] Figure 2 Schematic diagram of rock slope in an embodiment of the present invention.

[0062] Figure 3 is the cohesion c of the rock and the internal friction angle of the rock in the embodiment of the present invention Design point iteration cloud diagram.

[0063] Figure 4: is a design point iteration cloud diagram of the rock cohesion c and the tensile crack area per unit width z in the embodiment of the present invention.

[0064] Figure 5 3 is a design point iteration cloud diagram of the rock cohesion c and the water filling depth coefficient r of the tension fracture in the embodiment of the present invention.

[0065] Figure 6 1 is a design point iteration cloud diagram of the rock cohesion c and the horizontal seismic acceleration coefficient α in the embodiment of the present invention.

[0066] Figure 7 is the internal friction angle of the rock in the embodiment of the present invention And the design point iteration cloud diagram of the tensile crack area z per unit width.

[0067] Figure 8 is the internal friction angle of the rock in the present invention example Iterative cloud diagram of design points and water filling depth coefficient r of tension crack.

[0068] Figure 9 is the internal friction angle of the rock in the embodiment of the present invention and design point iteration cloud diagram of horizontal seismic acceleration coefficient α.

[0069] Figure 10 1 is a design point iteration cloud diagram of the tensile crack area z per unit width and the water filling depth coefficient r of the tensile crack in the embodiment of the present invention.

[0070] Figure 11 1 is a design point iteration cloud diagram of the tensile crack area z per unit width and the horizontal earthquake acceleration coefficient α in an embodiment of the present invention.

[0071] Figure 12 1 is a design point iteration cloud diagram of the water filling depth coefficient r of the tension fracture and the horizontal earthquake acceleration coefficient α in the embodiment of the present invention. DETAILED DESCRIPTION

[0072] The present invention will be described in detail below with reference to the accompanying drawings.

[0073] Figure 1 This is a flow chart of the reliability analysis method of the present invention. Figure 1 It can be seen that the rock slope reliability analysis method considering non-Gaussian correlated random variables of the present invention includes the following steps:

[0074] Step 1: Calculate the anti-sliding force R and sliding force Q of the rock slope;

[0075] According to the stress characteristics of rock slopes, the anti-sliding force R of rock slopes is composed of the friction force on the sliding surface, the weight of the slope W, the horizontal seismic force, the water pressure on the sliding surface, the water pressure on the tensile fracture surface, and the slope anchoring force. The sliding force Q of rock slopes is composed of the weight of the slope, the horizontal seismic force, the water pressure on the tensile fracture surface, and the slope anchoring force. Their expressions are:

[0076]

[0077] Where c is the cohesion of rock, is the internal friction angle of rock, ψ p is the inclination of the sliding surface, H is the height of the rock slope, z is the area of ​​the tensile crack per unit width, r is the water filling depth coefficient of the tensile crack, α is the horizontal earthquake acceleration coefficient, γ w is the gravity of water, T is the anchoring force on the slope, and ε is the angle between the anchoring force and the normal of the sliding surface;

[0078] In this embodiment, the calculation formula of the slope weight W is:

[0079]

[0080] Where, γ rock is the density of the rock, ψ f is the inclination angle of the rock slope.

[0081] The Copula parameter θ is obtained by inversely solving the correlation coefficient ρ, and the calculation formula is:

[0082]

[0083] Where μ1 is the mean value of rock cohesion c, μ2 is the internal friction angle of rock σ1 is the standard deviation of rock cohesion c, and σ2 is the internal friction angle of rock. The standard deviation of .

[0084] Figure 2 is a schematic diagram of a rock slope in an embodiment of the present invention. p =35°, rock slope height H = 60m, water density γ w =10kN / m 3 , slope anchoring force T = 2570 kN / m, angle ε between anchoring force and normal of sliding surface = 55°, rock density γ rock =26kN / m 3 , rock slope inclination ψ f =50°.

[0085] Step 2: Construct the functional function G of the rock slope according to the anti-sliding force and sliding force of the rock slope, and its expression is: G=RQ.

[0086] Step 3, determining the probability distribution information of the random variable according to the characteristics of the non-Gaussian correlated random variable;

[0087] Select the rock's cohesion c, the rock's internal friction angle The tensile crack area z per unit width, the water-filled depth coefficient r of the tensile crack, and the horizontal earthquake acceleration coefficient α are taken as non-Gaussian correlated random variables and are denoted as random variables x n , n=1,2,3,4,5, then its probability distribution information includes:

[0088] 5 random variables x n The cumulative distribution function of n (x n );

[0089] 5 random variables x n The probability density function of n (x n );

[0090] Copula function C(F1(x1),F2(x2);θ) between random variables x1 and x2;

[0091] Conditional Copula function h(F1(x1), F2(x2); θ) between random variables x1 and x2;

[0092] The second-order conditional copula function d(F1(x1), F2(x2); θ) between the random variables x1 and x2;

[0093] Copula density function c(F1(x1), F2(x2); θ) between random variables x1 and x2;

[0094] Where θ is the Copula parameter.

[0095] Step 4: The reliability index β of the rock slope is efficiently calculated through the physical parameter space design point iteration technique.

[0096] In this embodiment, the implementation process of step 4 is as follows:

[0097] Step 4.1, set the following parameters:

[0098] Set the maximum number of iterations N and the allowable error ε x , denote any iteration as the kth iteration, k = 1, 2, ..., N; denote the random variable x in the kth iteration n is x n (k) ;

[0099] Step 4.2, let the design point of the first iteration be x (1) , x (1) =[x1 (1) x2 (1) x3 (1) x4 (1) x5 (1) ] T =[μ1μ2μ3μ4μ5] T , where μ1 is the mean value of the rock cohesion c, and μ2 is the internal friction angle of the rock , μ3 is the mean value of the area z of the unit width of the tensile crack, μ4 is the mean value of the water-filled depth coefficient r of the tensile crack, μ5 is the mean value of the horizontal earthquake acceleration coefficient α, and the superscript “T” is the transpose of the vector and matrix;

[0100] Step 4.3, calculate the performance function G(x (k) ), x (k) is the design point of the kth iteration, x (k) =[x1 (k) x2 (k) x3 (k) x4 (k) x5 (k) ] T ,G(x (k) ) by taking the design point x of the kth iteration (k) Substitute the function G of the rock slope and calculate the gradient vector of the kth iteration by the central difference method

[0101] Step 4.4, calculate the generalized lower triangular matrix A of the kth iteration (k) , its calculation expression is:

[0102]

[0103] Where A 21 (k) is the generalized lower triangular matrix A of the kth iteration (k) The element in the second row and first column of , whose calculation expression is:

[0104]

[0105] Where, d(F1(x1 (k) ),F2(x2 (k) ); θ) is the second-order conditional Copula function of the kth iteration, c(F1(x1 (k) ),F2(x2 (k)), θ) is the Copula density function of the kth iteration, φ(·) is the probability density function of the standard normal distribution, and Φ -1 (·) is the inverse function of the cumulative distribution function of the standard normal distribution; A 22 (k) is the generalized lower triangular matrix A of the kth iteration (k) The element in the second row and second column of , whose calculation expression is:

[0106]

[0107] Where, h(F1(x1 (k) ),F2(x2 (k) ); θ) is the conditional Copula function of the kth iteration;

[0108] Calculate the generalized standard deviation matrix B for the kth iteration (k) , its calculation expression is:

[0109]

[0110] Where, F n (x n (k) ) is the cumulative distribution function of the kth iteration, f n (x n (k) ) is the probability density function of the k-th iteration;

[0111] Calculate the generalized mean vector C of the kth iteration (k) , its calculation expression is:

[0112]

[0113] Where C e (k) is an additional term of the Copula function, and its calculation expression is:

[0114]

[0115] Calculate the design point x for the k+1th iteration (k+1) , x (k+1) =[x1 (k+1) x2 (k+1) x3 (k+1) x4 (k+1) x5 (k+1) ] T , its calculation expression is:

[0116] x (k+1) =x (k) +λ (k) d (k)

[0117] Where λ (k) is the iteration step length, d (k) is the iteration direction, and its expression is:

[0118]

[0119] Step 4.5, if |||x is satisfied (k+1) -x (k) || / ||x (k+1) |||≤ε x , then let the design point x of the kth iteration be (k) is the design point x * , and record the design point x of the kth iteration * The corresponding random variable x n is x n * , x * =[x1 * x2 * x3 * x4 * x5 * ] T , let the generalized lower triangular matrix A of the kth iteration (k) is the generalized lower triangular matrix A at the design point * , let the generalized standard deviation matrix B of the kth iteration be (k) is the generalized standard deviation matrix B at the design point * , let the generalized mean vector C of the kth iteration (k) is the generalized mean vector C at the design point * , and go to step 4.7. On the contrary, if |||x (k+1) -x (k) || / ||x (k+1) |||>ε x , then go to step 4.6;

[0120] Step 4.6: If the number of iterations k is greater than or equal to the maximum number of iterations N, then let the design point x of the kth iteration be (k) is the design point x * , x * =[x1 * x2 * x3 * x4 * x5 * ] T , let the generalized lower triangular matrix A of the kth iteration (k) is the generalized lower triangular matrix A at the design point * , let the generalized standard deviation matrix B of the kth iteration be (k) is the generalized standard deviation matrix B at the design point* , let the generalized mean vector C of the kth iteration (k) is the generalized mean vector C at the design point * , and go to step 4.7. Otherwise, set the number of iterations k = k + 1 and return to step 4.3;

[0121] Step 4.7, calculate the reliability index β of the rock slope, and its calculation expression is:

[0122] β=[(x * -C * ) T ((B * ) T )- 1 ((A * ) T )- 1 (A * )- 1 (B * )- 1 (x * -C * )] 1 / 2 .

[0123] In this embodiment, the five random variables x n The statistical information of 5 random variables x is shown in Table 1. n The cumulative distribution function F n (x n ) and the probability density function f n (x n ) can be uniquely determined by statistical information.

[0124] Table 1 Five random variables x n Statistics

[0125]

[0126]

[0127] Among them, the cohesion c of the rock and the internal friction angle of the rock The calculation formula of the Copula function C(F1(x1),F2(x2);θ) between them is:

[0128]

[0129] The cohesion c of rock and the internal friction angle of rock The calculation formula of the conditional Copula function h(F1(x1),F2(x2);θ) is:

[0130]

[0131] The cohesion c of rock and the internal friction angle of rock The calculation formula of the second-order conditional Copula function d(F1(x1),F2(x2);θ) between is:

[0132]

[0133] The cohesion c of rock and the internal friction angle of rock The calculation formula of the Copula density function c(F1(x1),F2(x2);θ) between them is:

[0134]

[0135] Among them, the correlation coefficient ρ = -0.5, so the copula parameter θ is 0.1811.

[0136] In this embodiment, the iterative process of the design point is shown in Table 2.

[0137] The cohesion c of rock and the internal friction angle of rock The design point iteration cloud diagram is shown in Figure 3 The design point iteration cloud diagram of rock cohesion c and tensile crack area per unit width z is shown in Figure 4 The design point iteration cloud diagram of rock cohesion c and water filling depth coefficient r of tension crack is shown in Figure 5 The design point iteration cloud diagram of rock cohesion c and horizontal earthquake acceleration coefficient α is shown in Figure 6 , the internal friction angle of rock The design point iteration cloud diagram of the tensile crack area z per unit width is shown in Figure 7 , the internal friction angle of rock The design point iteration cloud diagram of the water filling depth coefficient r of the tensile crack is shown in Figure 8 , the internal friction angle of rock The design point iteration cloud diagram of the horizontal earthquake acceleration coefficient α is shown in Figure 9 The design point iteration cloud diagram of the tensile crack area z per unit width and the water filling depth coefficient r of the tensile crack is shown in Figure 10 The design point iteration cloud diagram of the unit width tensile crack area z and the horizontal earthquake acceleration coefficient α is shown in Figure 11 The design point iteration cloud diagram of the water filling depth coefficient r of the tension crack and the horizontal earthquake acceleration coefficient α is shown in Figure 12 , design point x * =[81.5684 30.7776 16.1050 0.82710.1215] T , the generalized lower triangular matrix A at the design point * The expression is:

[0138]

[0139] Generalized standard deviation matrix B at the design point * The expression is:

[0140]

[0141] The generalized mean vector C at the design point * :

[0142]

[0143] And the reliability index of rock slope is β = 2.7760.

[0144] Table 2 Iterative process of design points

[0145]

[0146]

[0147] As can be seen from the results of this example, the rock slope reliability analysis method of the present invention, which considers non-Gaussian correlated random variables, converges quickly and accurately to the correct design point on the limit state curve in only eight iterations. The entire process is intuitively executed in physical space, completely eliminating cumbersome variable conversion steps, making it perfectly suitable for processing complex reliability analysis scenarios involving non-Gaussian correlated random variables. Therefore, the core advantage of the present invention is that it gives the iterative process an intuitive physical meaning, greatly promoting the intuitive understanding and operational convenience of researchers and engineers. At the same time, by optimizing the algorithm structure, it significantly enhances the robustness and convergence efficiency of the iterative calculation, effectively circumvents the iterative divergence problem that may be encountered in traditional methods, and ensures the reliability and accuracy of the analysis results. For the prevention and mitigation of rock slope disasters in geological engineering, the application of the present invention not only builds a solid line of defense to prevent the occurrence of geological disasters, but also demonstrates extraordinary effectiveness in reducing potential disaster losses, contributing indispensable technical support to the harmonious and stable development of the social economy and the sustainable goals of environmental protection.

Claims

1. A rock slope reliability analysis method considering non-Gaussian correlated random variables, characterized in that: The following steps are involved: Step 1: Calculate the anti-sliding force R and sliding force Q of the rock slope; Step 2: Construct the functional function G of the rock slope according to the anti-sliding force and sliding force of the rock slope, and its expression is: G = RQ; Step 3, determining the probability distribution information of the random variable according to the characteristics of the non-Gaussian correlated random variable; Select the rock's cohesion c, the rock's internal friction angle The area of ​​the tensile crack per unit width z, the water-filled depth coefficient r of the tensile crack, and the horizontal earthquake acceleration coefficient α are taken as non-Gaussian correlated random variables and are denoted as random variables x n , n=1,2,3,4,5; determine the random variable x n Probability distribution information of ; Step 4, calculate the reliability index β of the rock slope by the physical parameter space design point iteration technique; Step 4.1, set the following parameters: Set the maximum number of iterations N, the allowable error εx, and denote any iteration as the kth iteration, k = 1, 2, ..., N; denote the random variable x in the kth iteration n is x n (k) ; Step 4.2, let the design point of the first iteration be x (1) , x (1) =[x1 (1) x2 (1) x3 (1) x4 (1) x5 (1) ] T =[μ1 μ2 μ3μ4 μ5] T , where μ1 is the mean value of the rock cohesion c, and μ2 is the internal friction angle of the rock , μ3 is the mean value of the area z of the unit width of the tensile crack, μ4 is the mean value of the water-filled depth coefficient r of the tensile crack, μ5 is the mean value of the horizontal earthquake acceleration coefficient α, and the superscript "T" is the transpose of the vector and matrix; Step 4.3, calculate the performance function G(x (k) ), x (k) is the design point of the kth iteration, x (k) =[x1 (k) x2 (k) x3 (k) x4 (k) x5 (k) ] T ,G(x (k) ) by taking the design point x of the kth iteration (k) Substitute the function G of the rock slope and calculate the gradient vector of the kth iteration by the central difference method Step 4.4, calculate the generalized lower triangular matrix A of the kth iteration (k) , the generalized standard deviation matrix Bk of the kth iteration, the generalized mean vector C of the kth iteration (k) and the design point x of the k+1th iteration (k+1) ; Step 4.5, if |||x is satisfied (k+1) -x (k) || / ||x (k+1) |||≤ε x , then let the design point x of the kth iteration be (k) is the design point x * , let the generalized lower triangular matrix A of the kth iteration (k) is the generalized lower triangular matrix A at the design point * , let the generalized standard deviation matrix B of the kth iteration be (k) is the generalized standard deviation matrix B at the design point * , let the generalized mean vector C of the kth iteration (k) is the generalized mean vector C at the design point * , and go to step 4.

7. On the contrary, if |||x (k+1) -x (k) || / ||x (k+1) |||>ε x , then go to step 4.6; Step 4.6: If the number of iterations k is greater than or equal to the maximum number of iterations N, then let the design point x of the kth iteration be (k) is the design point x * , let the generalized lower triangular matrix A of the kth iteration (k) is the generalized lower triangular matrix A at the design point * , let the generalized standard deviation matrix B of the kth iteration be (k) is the generalized standard deviation matrix B at the design point * , let the generalized mean vector C of the kth iteration (k) is the generalized mean vector C at the design point * , and go to step 4.

7. Otherwise, set the number of iterations k = k + 1 and return to step 4.3; Step 4.7, calculate the reliability index β of the rock slope, and its calculation expression is: β=[(x * -C * ) T ((B * ) T ) -1 ((A * ) T ) -1 (A * ) -1 (B * ) -1 (x * -C * )] 1 / 2 。 2. A rock slope reliability analysis method considering non-Gaussian correlated random variables according to claim 1, characterized in that: The calculation of the anti-sliding force R and the sliding force Q of the rock slope described in step 1 is as follows: According to the force characteristics of the rock slope, the anti-sliding force R of the rock slope is composed of the friction force on the sliding surface, the slope weight W, the horizontal seismic force, the water pressure on the sliding surface, the water pressure on the tensile fracture surface, and the slope anchoring force. The sliding force Q of the rock slope is composed of the slope weight W, the horizontal seismic force, the water pressure on the tensile fracture surface, and the slope anchoring force. Their expressions are: Where c is the cohesion of rock, is the internal friction angle of rock, ψ p is the inclination of the sliding surface, H is the height of the rock slope, z is the area of ​​the tensile crack per unit width, r is the water filling depth coefficient of the tensile crack, α is the horizontal earthquake acceleration coefficient, γ w is the gravity of water, T is the anchoring force on the slope, and ε is the angle between the anchoring force and the normal of the sliding surface; The calculation formula of the slope weight W is: Where, γ rock is the density of the rock, ψ f is the inclination angle of the rock slope.

3. The rock slope reliability analysis method considering non-Gaussian correlation random variables according to claim 1 is characterized in that: The random variable x in step 3 n The probability distribution information includes: 5 random variables x n The cumulative distribution function of n (x n ); 5 random variables x n The probability density function of n (x n ); Copula function C(F1(x1), F2(x2); θ) between random variables x1 and x2; Conditional Copu1a function h(F1(x1), F2(x2); θ) between random variables x1 and x2; The second-order conditional copula function d(F1(x1), F2(x2); θ) between the random variables x1 and x2; Copula density function c(F1(x1), F2(x2); θ) between random variables x1 and x2; Among them, θ is the Copula parameter, which is obtained by inversely solving the correlation coefficient ρ. The calculation formula is: Where μ1 is the mean value of rock cohesion c, μ2 is the internal friction angle of rock σ1 is the standard deviation of rock cohesion c, and σ2 is the internal friction angle of rock. The standard deviation of .

4. A rock slope reliability analysis method considering non-Gaussian correlated random variables according to claim 3, characterized in that: The specific implementation process of step 4.4 is as follows: Step 4.4, calculate the generalized lower triangular matrix A of the kth iteration (k) , its calculation expression is: Where A 21 (k) is the generalized lower triangular matrix A of the kth iteration (k) The element in the second row and first column of , whose calculation expression is: Where, d(F1(x1 (k) ), F2(x2 (k) ); θ) is the second-order conditional Copula function of the kth iteration, c(F1(x1 (k) ), F2(x2 (k) ), θ) is the Copula density function of the kth iteration, φ(·) is the probability density function of the standard normal distribution, and Φ -1 (·) is the inverse function of the cumulative distribution function of the standard normal distribution; A 22 (k) is the generalized lower triangular matrix A of the kth iteration (k) The element in the second row and second column of , whose calculation expression is: Where, h(F1(x1 (k) ), F2(x2 (k) ); θ) is the conditional Copula function of the kth iteration; Calculate the generalized standard deviation matrix B for the kth iteration (k) , its calculation expression is: Where, F n (x n (k) ) is the cumulative distribution function of the kth iteration, f n (x n (k) ) is the probability density function of the k-th iteration; Calculate the generalized mean vector C of the kth iteration (k) , and its calculation formula is: Where C e (k) is an additional term of the Copula function, and its calculation expression is: Calculate the design point x for the k+1th iteration (k+1) , x (k+1) =[x1 (k+1) x2 (k+1) x3 (k+1) x4 (k+1) x5 (k+1) ] T , its calculation expression is: x (k+1) =x (k) +λ (k) d (k) Where λ (k) is the iteration step length, d (k) is the iteration direction, and its expression is:

Images