Calculation method for reliability of rockfill dam slope considering internal correlation of parameters

Abstract

The present invention discloses a method for calculating the reliability of the slope of a rockfill dam considering the internal correlation of parameters, which includes constructing a two-dimensional finite element model of the rockfill dam to be analyzed, searching for the slip channels of the slope of the rockfill dam, and then constructing the reliability function of the downstream slope of the rockfill dam to be analyzed; constructing a parameter multi-dimensional joint regular vine Copula function model; randomly extracting several groups of Duncan Zhang E-B model parameters from the parameter multi-dimensional joint regular vine Copula function model; calculating the slope stability reliability corresponding to each group of Duncan Zhang E-B model parameters by using the reliability function of the downstream slope of the rockfill dam, and counting the number of reliabilities less than zero as the number of failures; calculating the failure probability of the rockfill dam to be analyzed according to the total number of extracted Duncan Zhang E-B model parameters and the number of failures.

Description

Technical Field

[0001] The present invention relates to the technology of dam reliability analysis, and particularly to a calculation method for the reliability of rockfill dam slopes considering the internal correlation of parameters. Background Art

[0002] Rockfill dams, with their advantages of low cost, strong adaptability, and simple construction, have become one of the most promising types of dams. Especially with the deepening of national hydropower development, a large number of high rockfill dams have been built and put into operation, such as Changheba Dam (dam height 240m), Nuozhadu Dam (dam height 261.5m), Lianghekou Dam (dam height 293m), etc. However, due to complex occurrence environments, extremely high dam heights, and thick overburden layers, these high rockfill dams face huge challenges in safety assessment and risk control. Among them, the stability of the dam slope is one of the main reasons for the instability of rockfill dams. The instability of the slope will greatly affect the safety of hydraulic structures and the lives and property of people downstream. The reliability of dam slope stability has gradually become one of the common means for the safety assessment of dam structures. In China, the development of reliability analysis methods started relatively late, and currently, the research on the reliability analysis of slope stability is still in the stage of research and exploration. Many scholars have improved the theory of dam slope stability reliability based on parameter randomness, algorithm optimization, etc., making the theory of dam slope stability reliability develop in a more scientific and reasonable direction. However, traditional reliability analysis generally focuses on the uncertainty caused by the inherent spatial variability of material parameters, while ignoring the internal correlation between parameters. If the objectively existing internal correlation between random variables is not considered, the combination of random variables obtained by sampling is likely to deviate from the true nature of the rockfill material, resulting in the deviation of the dam slope stability reliability analysis results from the actual situation. Therefore, as an inherent property of the parameters themselves, the internal correlation between parameters needs to be considered in reliability analysis. Summary of the Invention

[0003] Aiming at the above deficiencies in the prior art, the calculation method for the reliability of rockfill dam slopes considering the internal correlation of parameters provided by the present invention solves the problem that the traditional reliability analysis results deviate from the actual situation and affect the accurate selection of slope protection strategies.

[0004] In order to achieve the above invention purpose, the technical solution adopted by the present invention is as follows:

[0005] Provide a calculation method for the reliability of rockfill dam slopes considering the internal correlation of parameters, which includes the following steps:

[0006] S1. Construct a two-dimensional finite element model of the rockfill dam to be analyzed, and search for the slip channels of the rockfill dam slope through the strength reduction search method;

[0007] S2. Construct the reliability function of the downstream dam slope of the rockfill dam to be analyzed according to the obtained slip channels;

[0008] S3. Obtain the Duncan-Chang E-B model parameters of several rockfill dams and the marginal distribution function of each model parameter, and calculate the Pearson correlation coefficient between each model parameter and any one of the remaining model parameters;

[0009] S4. Construct a parameter multi-dimensional joint regular vine Copula function model based on all the model parameters, their corresponding marginal distribution functions, and all the Pearson correlation coefficients;

[0010] S5. Randomly extract several groups of Duncan-Chang E-B model parameters from the parameter multi-dimensional joint regular vine Copula function model by the Monte-Carlo method;

[0011] S6. Calculate the reliability of each group of Duncan-Chang E-B model parameters using the reliability function of the downstream slope of the rockfill dam, and count the number of reliabilities less than zero as the number of failures;

[0012] S7. Calculate the failure probability of the rockfill dam to be analyzed based on the total number of extracted Duncan-Chang E-B model parameters and the number of failures.

[0013] Further, step S2 further includes:

[0014] S21. Calculate several groups of slope stability safety factors F by adjusting the friction coefficient, cohesion, normal stress, and / or shear stress of different units in the slip channel, where: s :

[0015]

[0016] where M is the total number of units on the slip channel; f k , c k are the friction coefficient and cohesion of the k-th unit on the slip channel respectively, and σ k , τ k are the normal stress and shear stress received by the k-th unit in the slip channel direction respectively; l k is the unit length of the k-th unit along the slip channel direction;

[0017] S22. Calculate the undetermined coefficients a, b i , c i using the response surface equation according to several groups of slope stability safety factors and their corresponding Duncan-Chang E-B model parameters. The expression of the response surface equation is:

[0018]

[0019] where x i is the i-th model parameter, 1 ≤ i ≤ 7; b i , c iThey are the undetermined coefficients corresponding to the i-th model parameter respectively; I is the total number of model parameters;

[0020] S23. Use the calculated undetermined coefficients a, b i , c i to update the response surface equation as the fitting formula:

[0021]

[0022] where K e is the tangent modulus coefficient; n is the tangent modulus exponent; K b is the bulk modulus coefficient; m is the bulk modulus exponent; R f is the failure ratio; are all non-linear strength indexes; K e , n, K b , m, R f , are the 7 model parameters included in the Duncan-Chang E-B model parameters;

[0023] S23. According to the fitting formula, construct the downstream dam slope reliability function Z = F s -1..

[0024] Furthermore, the calculation formula of the Pearson correlation coefficient in step S3 is:

[0025]

[0026] where x oi and x oj are the i-th and j-th model parameters of the o-th rockfill dam; and are the averages of the i-th and j-th model parameters of N rockfill dams respectively; N is the total number of rockfill dams; r ij is the Pearson correlation coefficient of the i-th and j-th model parameters; 1 ≤ i ≤ 7, 1 ≤ j ≤ 7, and i ≠ j.

[0027] Furthermore, step S4 further includes:

[0028] S41. By the maximum spanning tree method, generate the tree structure of each layer with the principle that the sum of the absolute values of the Pearson correlation coefficients in each layer of the tree structure is the largest;

[0029] S42. Determine the Copula function set;

[0030] S43. In the Copula function set, select the optimal Pair Copula function of the variables connected by each side line in the tree structure through the AIC criterion and the maximum likelihood estimation method, and obtain the optimal parameter estimation of the optimal Pair Copula function.

[0031] S44. Combining the tree structure and the optimal pair copula function of the edge, a parameter multi-dimensional joint regular vine copula function model is constructed.

[0032] Furthermore, the Copula function set includes Gaussian Copula, t Copula, Clayton Copula, Frank Copula, Gumbel Copula, Joe Copula and Indep Copula;

[0033] The tree structure of the regular vine Copula function model consists of six layers. The first layer of the tree structure has 7 nodes and 6 edges. With each additional layer, the number of nodes decreases by one and the number of edges decreases by one.

[0034] Furthermore, the rockfill dam includes a primary rockfill area and a secondary rockfill area, and the corresponding dam slope failure probability calculation is performed in both rockfill areas according to the method of steps S3 to S7;

[0035] The optimal pair copula function corresponding to the node edge and node edge of each layer of the tree structure in the main rockfill area is:

[0036] The node edges of the first layer tree structure Tree1 are m, n, K b ,n,n,R f , R f , K e , K e , and The optimal pair copula functions corresponding to these node edges are Joe, Gumbel180°, Student, Joe, Gumbel, and Gumbel;

[0037] The node edges of the second-level tree structure Tree2 are m, K b |n、K b , R f |n、K e ,n|R f , R f , and K e , The optimal Paiir Copula functions corresponding to these node edges are Independence, Gaussian, Independence, Independence, and Joe180°;

[0038] The node edges of the third-level tree structure Tree3 are m, R f|K b ,n, K b ,K e |R f ,n, n R f and R f e ,the optimal Pair Copula functions corresponding to these node edges are Independence, Clayton180°, Independence, and Gaussian respectively;

[0039] The node edges of the fourth - layer tree - like structure Tree4 are m, K e |R f ,K b ,n, K R f ,n and n K e ,R f ,the optimal Pair Copula functions corresponding to these node edges are Gaussian, Independence, and Independence respectively;

[0040] The node edges of the fifth - layer tree - like structure Tree5 are m, R f ,K b , n and K b , K e ,R f ,n, the optimal Pair Copula functions of the two node edges are both Independence;

[0041] The node edges of the sixth - layer tree - like structure Tree6 are m, K e ,R f ,K b ,n, and its optimal Pair Copula function is Independence;

[0042] The node edges of each layer of the tree - like structure in the secondary rock - fill area and the optimal Pair Copula functions corresponding to the node edges are:

[0043] The node edges of the first - layer tree - like structure Tree1 are R f ,n, m, n, n, K b 、 and The optimal Pair Copula functions corresponding to these node edges are Joe, Gumbel, Gumbel180°, Joe 180°, Joe, and Gumbel respectively;

[0044] The node edge of the second - layer tree structure Tree2 is R f , m|n, m, K b |n, n K e , and K b The optimal Pair Copula functions corresponding to these node edges are all Independence;

[0045] The node edge of the third - layer tree structure Tree3 is R f , K b |m, n, m, , n, b , and Ke, The optimal Pair Copula functions corresponding to these node edges are respectively Independence;

[0046] The node edge of the fourth - layer tree structure Tree4 is R f , m, n, m, K b , n and n, K b , and the optimal Pair Copula functions corresponding to these node edges are respectively Gaussian, Independence, and Independence;

[0047] The node edge of the fifth - layer tree structure Tree5 is R f , K b , m, n and m, K b , n, and the optimal Pair Copula functions of the two node edges are both Independence;

[0048] The node edge of the sixth - layer tree structure Tree6 is R f , K b , m, n, and its optimal Pair Copula function is Joe 180°.

[0049] Furthermore, the tree structure multiplies the marginal distribution functions of the seven model parameters and the optimal pair copula functions corresponding to all node edges as the correlation of the regular vine copula function model parameters.

[0050] Furthermore, the method for selecting the marginal distribution function of the model parameters in step S3 includes:

[0051] S31, using KS test to determine if the model parameters meet the marginal distribution function;

[0052] S32, based on all marginal distribution functions that each model parameter conforms to, using the AIC criterion to calculate the AIC value corresponding to each marginal distribution function;

[0053] S33. Select the marginal distribution with the minimum AIC value as the optimal distribution of the model parameters.

[0054] Furthermore, for the main rockfill area, n1 and It obeys the truncated normal distribution, and the remaining model parameters obey the Weibull distribution; for the secondary rockfill area, n2, R f2 and The remaining model parameters all obey the Weibull distribution.

[0055] The beneficial effects of the present invention are: a large number of Duncan-Zhang EB model parameters of rockfill dams are counted, marginal distribution optimization is performed on each parameter, and the correlation between each pair of parameters is analyzed. The intrinsic correlation between the Duncan-Zhang EB model parameters is intuitively revealed through the tree structure using the regular vine Copula function. It can be found from the reliability calculation results of whether the intrinsic correlation of the model parameters is considered that the correlation between the model parameters has a certain influence on the reliability of the stability of the rockfill dam slope, and the reliability index considering the intrinsic correlation of the parameters is higher than the reliability index not considering the intrinsic correlation of the parameters. In this solution, when the intrinsic correlation of the parameters is considered, the model parameter pairs with higher correlation will appear in a combination with a higher probability when extracted by the Monte-Carlo method. The failure probability calculated in this way is consistent with the actual situation of the parameters, and the probability of the parameter group that does not conform to the actual correlation of the parameters is greatly reduced when the selection is performed.

[0056] The reliability analysis method for rockfill dam slope stability that considers the intrinsic correlation of parameters solves the problem that independent fluctuations of parameters may deviate from the intrinsic properties of the material, causing misjudgment of the risk of slope instability. The reliability index results are more reasonable and can reflect the true state of rockfill dam slope stability, so as to facilitate the selection of relatively reasonable maintenance strategies during dam maintenance and improve the safe and stable operation of the dam. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1Flow chart of the calculation method for the reliability of the rockfill dam slope considering the internal correlation of parameters.

[0058] Figure 2 Schematic diagram of the finite element model of the 0+103.00 section of the DL concrete face rockfill dam.

[0059] Figure 3 Schematic diagram of the plastic zone of the downstream slope of the DL dam.

[0060] Figure 4 Schematic diagram of the test results of the marginal distribution of parameter m1 in the main rockfill area.

[0061] Figure 5 Vine structure diagram of the material parameters of the main and secondary rockfill areas. (a) is the main rockfill area, and (b) is the secondary rockfill area.

[0062] Figure 6 Comparison diagram of the sampling results of some parameters of the rockfill material using the regular vine Copula function model and the Duncan-Chang E-B model parameters of 36 rockfill dams; (a) is the comparison diagram of the sampling results and the main rockfill area of 36 rockfill dams regarding parameter Comparison diagram; (b) is the comparison diagram of the sampling results and the main rockfill area of 36 rockfill dams regarding parameter K e ~K b Comparison diagram; (c) is the comparison diagram of the sampling results and the main rockfill area of 36 rockfill dams regarding parameter m~K b Comparison diagram; (d) is the comparison diagram of the sampling results and the main rockfill area of 36 rockfill dams regarding parameter m~n; (e) is the comparison diagram of the sampling results and the secondary rockfill area of 36 rockfill dams regarding parameter Comparison diagram; (f) is the comparison diagram of the sampling results and the secondary rockfill area of 36 rockfill dams regarding parameter K e ~K b Comparison diagram; (g) is the comparison diagram of the sampling results and the secondary rockfill area of 36 rockfill dams regarding parameter m~K b Comparison diagram; (h) is the comparison diagram of the sampling results and the secondary rockfill area of 36 rockfill dams regarding parameter m~n.

[0063] Figure 7 Comparison diagram of the joint sampling results of the model parameters of this scheme and the independent sampling results of the existing technology parameters; (a) is the comparison diagram of the sampling results of the main rockfill material parameters Comparison diagram; (b) is the comparison diagram of the sampling results of the main rockfill material parameter K b1 ~K e1 Comparison diagram; (c) is the comparison diagram of the sampling results of the main rockfill material parameter K b1 ~R f1 Comparison diagram; (d) is the comparison diagram of the sampling results of the secondary rockfill material parameters Comparison diagram; (e) is the comparison diagram of the sampling results of the secondary rockfill material parameter K b2 ~Ke2 Comparison diagram of sampling results; (f) is the comparison diagram of sampling results of the parameters m2 - n2 of the secondary rockfill; in (a) - (f), the left side is the joint sampling result of the model parameters of this solution, and the right side is the independent sampling result of the parameters of the existing technology. Detailed implementation manners

[0064] The following describes the detailed implementation manners of the present invention to facilitate those skilled in the art of this technology to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the detailed implementation manners. For those of ordinary skill in the art of this technology, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions created using the concept of the present invention are within the scope of protection.

[0065] Reference Figure 1 , Figure 1 shows a flow chart of a calculation method for the reliability of the rockfill dam slope considering the internal correlation of parameters. As Figure 1 shown, this method S includes steps S1 to S7.

[0066] In step S1, a two-dimensional finite element model of the rockfill dam to be analyzed is constructed, and the slip channel of the rockfill dam slope is searched through the strength reduction search method;

[0067] To facilitate the understanding of the slip channel search, an example is used for illustration here. The maximum cross-section (0 + 103.00) of the DL concrete face rockfill dam is selected for two-dimensional finite element modeling. The simulation range extends 2 times the dam height in the upstream, downstream, and depth directions, and it is discretized into 1719 elements and 3364 nodes. Among them, there are 705 dam body elements and 1014 dam foundation elements. The constructed finite element model refers to Figure 2 .

[0068] The rockfill materials in the downstream rockfill area, secondary rockfill area, and main rockfill area are all mined from the lower dam site quarry, with the same lithology. At the same time, to coordinate the uneven deformation of the dam body, the downstream rockfill area and the secondary rockfill area adopt the same design control parameters as the main rockfill, and the gradation of the main rockfill and the downstream rockfill is the same. For simplicity in calculation, the upstream rockfill and the downstream rockfill are analyzed as the same material. Since the deformation of the concrete face rockfill dam mainly occurs in the rockfill area, only the main rockfill and the secondary rockfill are analyzed this time. The Duncan-Chang E - B model parameters of the materials in each partition of the dam body are selected as the dam design parameters, as shown in Table 1.

[0069] Table 1 Duncan-Chang E - B model parameters of the DL concrete face rockfill dam

[0070]

[0071] Search for the failure path of the dam slope of the DL panel rockfill dam by the strength reduction method. In the numerical model, gradually reduce the material strength parameters (cohesion and internal friction angle) until the calculation does not converge, that is, it is considered that the dam slope reaches the limit state. Use the corresponding stability discrimination criterion to determine the minimum stability safety factor F of the dam slope. s , and obtain the sliding failure surface of the dam slope, that is, the strip passing through the dam body composed of damaged elements.

[0072] The reduced shear strength parameters are calculated by the following formula:

[0073]

[0074] In the formula, c′, are the reduced values of the cohesion and internal friction angle of the dam body material respectively; F r is the reduction coefficient.

[0075] When the reduction coefficient increases to 2.26, the plastic zone of the dam slope is basically penetrated and the calculation no longer converges. The plastic zone of the dam slope is as shown in Figure 3 . The "Code for Design of Roller Compacted Earth and Rockfill Dams" SL274-2020 stipulates that for Class 1 earth and rockfill dams under normal operating conditions, the minimum safety factor [K] for the anti-sliding stability of the dam slope = 1.5. Therefore, the anti-sliding stability safety of the dam slope meets the design standard.

[0076] In step S2, according to the obtained slip channel, construct the reliability function of the downstream dam slope of the rockfill dam to be analyzed; in implementation, this solution preferably further includes step S2:

[0077] S21. By adjusting the friction coefficient, cohesion, normal stress and / or shear stress of different elements in the slip channel, calculate several groups of dam slope stability safety factors F s :

[0078]

[0079] Among them, M is the total number of elements on the slip channel; f k , c k are the friction coefficient and cohesion of the kth element on the slip channel respectively, σ k , τ k are the normal stress and shear stress received by the kth element in the slip channel direction respectively; l k is the element length of the kth element along the slip channel direction;

[0080] S22. According to several groups of dam slope stability safety factors and their corresponding Duncan-Chang E-B model parameters, use the response surface equation to calculate the undetermined coefficients a, b i , c i , and the expression of the response surface equation is:

[0081]

[0082] Among them, x i is the i-th model parameter, where 1 ≤ i ≤ 7; b i , c i are the undetermined coefficients corresponding to the i-th model parameter respectively; I is the total number of model parameters;

[0083] S23. Update the response surface equation as the fitting formula by using the calculated undetermined coefficients a, b i , c i :

[0084]

[0085] Among them, K e is the tangent modulus coefficient; n is the tangent modulus exponent; K b is the bulk modulus coefficient; m is the bulk modulus exponent; R f is the failure ratio; are all non-linear strength indexes; K e , n, K b , m, R f , are the 7 model parameters included in the Duncan-Chang E-B model parameters;

[0086] S23. According to the fitting formula, construct the downstream dam slope reliability function Z = F s -1.

[0087] In step S3, obtain the Duncan-Chang E-B model parameters of several rockfill dams and the marginal distribution function of each model parameter, and calculate the Pearson correlation coefficient between each model parameter and any one of the remaining model parameters.

[0088] In this solution, the Duncan-Chang E-B model parameters of 36 rockfill dams in China are statistically analyzed, and their mean values, standard deviations and coefficients of variation are calculated. Table 2 shows the statistical characteristics of these model parameters.

[0089] Table 2 Statistical characteristics of Duncan-Chang E-B model parameters of 36 rockfill dams

[0090]

[0091] During implementation, the preferred method for selecting the marginal distribution function of the model parameters in step S3 of this solution includes:

[0092] S31. Use the K-S test to determine the marginal distribution function that the model parameters conform to;

[0093] Most geotechnical material parameters obey Weibull, truncated normal, lognormal, extreme value, and t-distributions, which are often used to accurately describe soil uncertainty. This approach uses these five marginal distributions to analyze parameter distribution types. First, the KS test is performed to determine which marginal distribution type the parameters conform to.

[0094] The Kolmogorov-Smirnov test is a test method that compares a frequency distribution F(x) with a preset distribution G(x) using the cumulative distribution function. Since the test does not require the distribution of the data in advance, the KS test is a non-parametric test method. The empirical distribution function of the sample is calculated as follows:

[0095]

[0096] Where n is the sample size to be tested; I(x i ≤x) is the indicator function, when x i When ≤x, the value is 1, otherwise the value is 0.

[0097] The five distributions mentioned above are used as the preset distribution G(x). Table 3 shows the cumulative distribution and probability density function of these five distributions.

[0098] Table 3 Cumulative probability functions and marginal distribution functions of five marginal distributions

[0099]

[0100]

[0101] The KS statistic is used to evaluate the degree of difference between the empirical distribution function of sample data and the theoretical distribution function. The calculation method is:

[0102] D n =max|F(x)-G(x)|

[0103] Based on the sample size n = 36 and the significance level α = 0.05, the KS critical value D is calculated by looking up the table. a =0.226. If D n <D a , accept the original hypothesis and the sample follows the preset distribution. Otherwise, the sample does not conform to the preset distribution.

[0104] The marginal distribution test results of each parameter in the primary and secondary rockfill areas are shown in Tables 4 and 5. It can be seen from Tables 4 and 5 that most parameters are consistent with the five preset distributions. Taking m1 as an example, the statistical value D of the five preset distributions is n 0.090, 0.116, 0.077, 0.080 and 0.067 respectively, Figure 4Shows the results of fitting five distributions.

[0105] Table 4 Test results of the marginal distributions of Duncan-Chang E-B parameters in the main rockfill area

[0106]

[0107]

[0108] Table 5 Test results of the marginal distributions of Duncan-Chang E-B parameters in the secondary rockfill area

[0109]

[0110]

[0111] S32. According to all the marginal distribution functions that each model parameter conforms to, calculate the AIC value corresponding to each marginal distribution function using the AIC criterion;

[0112] The AIC criterion is the Akaike information criterion, which is based on the concept of entropy to evaluate the goodness of fit of the model to the data and is a standard for measuring the goodness of fit of a statistical model. Therefore, this paper uses the AIC criterion to select the optimal marginal distribution function. Traverse all marginal distribution types, and the marginal distribution with the smallest calculated AIC value is the optimal marginal distribution. The calculation formula for the AIC value is:

[0113] AIC = 2K - 2ln(L)

[0114] In the formula, K is the number of parameters of the marginal distribution type; L is the likelihood function, and here the probability density function of the marginal distribution type is taken.

[0115] S33. Select the marginal distribution with the smallest AIC value as the optimal distribution of the model parameters. Specifically: for the main rockfill area, n1 and follow the truncated normal distribution, and the remaining model parameters all follow the Weibull distribution; for the secondary rockfill area, n2, R f2 and follow the truncated normal distribution, and the remaining model parameters all follow the Weibull distribution.

[0116] During implementation, the calculation formula for the Pearson correlation coefficient in step S3 of this solution is:

[0117]

[0118] where, x oi and x oj are the i-th and j-th model parameters of the o-th rockfill dam; and are the average values of the i-th and j-th model parameters of N rockfill dams respectively; N is the total number of rockfill dams; r ij is the Pearson correlation coefficient of the i-th and j-th model parameters; 1 ≤ i ≤ 7, 1 ≤ j ≤ 7, and i ≠ j.

[0119] The value range of the correlation coefficient r is [-1, 1]. 1 and -1 indicate that the variables completely follow linear correlation, and 0 represents no correlation between variables. The specific correlation level division is shown in Table 6.

[0120] Table 6 Correlation Level

[0121]

[0122] The Pearson coefficients of each pair of parameters are shown in Tables 7 and 8. Among them, the non-linear strength indexes and in the main and secondary rockfill areas have Pearson coefficients of 0.857 and 0.870 respectively, showing significant linear correlation. In addition, there is a high correlation between K e1 and R f1 , K e1 and K e1 and as well as between n1 and K b1 . Through correlation analysis, the correlation relationships between parameters are revealed, laying a solid foundation for the subsequent establishment of a multi-dimensional joint vine copula function.

[0123] Table 7 Pearson Coefficient Table of Duncan-Chang E-B Model Parameters in the Main Rockfill Area

[0124]

[0125] Table 8 Pearson Coefficient Table of Duncan-Chang E-B Model Parameters in the Secondary Rockfill Area

[0126]

[0127]

[0128] In step S4, according to all the model parameters, their corresponding marginal distribution functions, and all the Pearson correlation coefficients, a parameter multi-dimensional joint regular vine Copula function model is constructed;

[0129] In an embodiment of the present invention, step S4 further includes:

[0130] S41. By the maximum spanning tree method, a tree structure of each layer is generated with the principle that the sum of the absolute values of the correlation coefficients in each layer of the tree structure is the largest;

[0131] S42. Determine the Copula function set; preferably, the Copula function set in this solution includes Gaussian Copula, tCopula, Clayton Copula, Frank Copula, Gumbel Copula, Joe Copula, and Indep Copula.

[0132] S43. In the Copula function set, select the optimal Pair Copula function for the variables connected by each side line in the tree structure through the AIC criterion and the maximum likelihood estimation method, and obtain the optimal parameter estimation of the optimal Pair Copula function; taking Gaussian_copula as an example, the code for fitting the pair copula function between variables is shown in Table 9.

[0133] Table 9

[0134]

[0135]

[0136] S44. Combine the tree structure and the optimal Pair Copula functions of the side lines to construct a parameter multi-dimensional joint regular vine Copula function model. Among them, the tree structure multiplies the marginal distribution functions of the 7 model parameters and the optimal Pair Copula functions corresponding to all node side lines as the correlation of the regular vine Copula function model parameters.

[0137] The tree structure of the regular vine Copula function model includes six layers. The first layer of the tree structure has 7 nodes and 6 side lines; each time an additional layer is added, the number of nodes decreases by one and the number of side lines decreases by one.

[0138] In this solution, the rockfill dam includes a main rockfill area and a secondary rockfill area. The failure probabilities of the corresponding dam slopes of the two rockfill areas are calculated in the manner of steps S3 to S7; the schematic diagrams of the regular vine Copula function models of the main rockfill area and the secondary rockfill area in this solution can be referred to Figure 5 . The optimal Copula function type and parameter estimation results of the Duncan-Chang E-B model parameters in the main rockfill area and the optimal Copula function type and parameter estimation results of the Duncan-Chang E-B model parameters in the secondary rockfill area can be referred to Table 10 and Table 11 respectively.

[0139] Table 10 Optimal Copula Function Type and Parameter Estimation Results of Duncan-Chang E-B Model Parameters in the Main Rockfill Area

[0140]

[0141]

[0142] Table 1 Optimal Copula function type and parameter estimation results of Duncan-Chang E-B model parameters in the first rockfill area

[0143]

[0144] The established multi-dimensional joint probability distribution Vine Copula function model is simulated for joint sampling 1000 times, and the Pearson correlation coefficients between parameters are calculated, as shown in Tables 12 and 13. Through the joint sampling of parameters, it shows that the correlation of the sampled data set is close to that of the sample ( Figure 6 The sample in this refers to the Duncan-Chang E-B model parameters of the 36 selected dams), and the parameter pairs with relatively large original correlations, such as the main rockfill area The Pearson correlation coefficients and Kendall rank correlation coefficients between the sample set and the sampling set are 0.857 and 0.899, 0.682 and 0.712 respectively, and the correlations are all good; the parameter pairs with relatively weak original correlations, such as m and R in the secondary rockfill area f The Pearson correlation coefficients and Kendall rank correlation coefficients between the sample set and the sampling set are 0.019 and 0.081, 0.060 and 0.114 respectively, and there is no obvious correlation.

[0145] The sampling results of some parameter pairs are plotted into a graph, and the results are shown in Figure 6 . The joint sampling distribution results of the parameters can well cover the original sample distribution, and the Vine Copula function model of the multi-dimensional joint probability distribution of Duncan-Chang E-B model parameters of the concrete face rockfill dam can reasonably describe the internal correlation law between parameters and the joint probability distribution of parameters.

[0146] Table 12 Correlation coefficient table of joint sampling results of Duncan-Chang E-B model parameters in the main rockfill area

[0147]

[0148] Table 13 Correlation coefficient table of joint sampling results of Duncan-Chang E-B model parameters in the secondary rockfill area

[0149]

[0150] In step S5, several groups of Duncan-Chang E-B model parameters are randomly extracted from the parameter multi-dimensional joint regular vine Copula function model by the Monte-Carlo method;

[0151] In step S6, the reliability of each group of Duncan-Chang E-B model parameters is calculated by using the reliability function of the downstream slope of the rockfill dam, and the number of reliabilities less than zero is counted as the number of failures;

[0152] In step S7, the failure probability of the rockfill dam to be analyzed is calculated according to the total number and failure number of the extracted Duncan-Chang E-B model parameters.

[0153] Taking μ - 3σ and μ + 3σ as the upper and lower limits, the random variables are truncated, and 1×10 9 times of sampling are carried out by the Monte-Carlo method to calculate the failure probability p f of the dam slope stability and the reliability index β. When sampling, two situations are considered: there is an internal correlation between parameters and there is no internal correlation between parameters.

[0154] When not considering the internal correlation of parameters, each parameter follows its original marginal distribution. The marginal distribution models of each parameter are established and sampling simulation is carried out. It can be calculated through the surrogate model of the dam slope stability function that the failure probability p f is 2.51×10 -6 , and the reliability index β is 4.56. When considering the internal correlation of parameters, sampling simulation is carried out through the established multi-dimensional joint distribution model of the Duncan-Chang E-B model parameters of the rockfill material. It can be calculated through the surrogate model of the dam slope stability function that the failure probability p f is 5.42×10 -7 , and the reliability index β is 4.87. Both situations meet the standard of the minimum target reliability index β t of 4.2 for the ultimate limit state of the bearing capacity of the Grade-I structural members in the persistent design situation specified in GB50199-2013 "Unified Standard for Reliability Design of Water Conservancy and Hydropower Engineering Structures". The dam slope safety risk level is relatively low and the operation is reliable.

[0155] From the reliability calculation results with or without considering the internal correlation of parameters, it can be found that the correlation between parameters has a certain influence on the reliability of the dam slope stability of the concrete face rockfill dam. The reliability index considering the internal correlation of parameters is higher than that without considering the internal correlation of parameters. Because when considering the internal correlation of parameters, the parameters with higher correlation will appear in the combination with a larger probability during the joint distribution sampling. Taking Figure 7 the sampling results of the parameters in the main rockfill area as an example, the joint distribution of the parameters is mainly concentrated in the middle shaded part, while the probability of the parameters appearing in the blank parts in the upper left corner and the lower right corner is relatively low, which reflects the joint distribution characteristics of the parameters; while in the independent sampling, the parameters follow their own marginal distribution, and the probability of appearing in the central part is the largest, and the probability decreases outward from the central part, ignoring the coupling characteristics of the parameters, and the sampling results of the parameters are distorted. The joint distribution sampling results consider the relationship between parameters, conform to the actual situation of the parameters, and greatly reduce the probability of the parameter groups that do not conform to the actual correlation of the parameters appearing during sampling.

[0156] This slope stability reliability analysis method for face-cement rockfill dams, which considers intrinsic parameter correlations, addresses the problem that independent parameter fluctuations can deviate from material properties, leading to misjudgment of slope instability risk. The resulting reliability index is more reasonable and reflects the true state of slope stability for face-cement rockfill dams. This slope stability reliability analysis method, which considers intrinsic parameter correlations, has both theoretical and practical application value for assessing the slope stability risk of high-pressure face-cement rockfill dams.

Claims

1. A calculation method for the reliability of the rockfill dam slope considering the internal correlation of parameters, characterized in that Including the steps: S1. Construct a two-dimensional finite element model of the rockfill dam to be analyzed, and search for the slip channels of the dam slope of the rockfill dam through the strength reduction search method; S2. According to the obtained slip channels, construct the reliability function of the downstream dam slope of the rockfill dam to be analyzed; S3. Obtain the Duncan Zhang E-B model parameters of several rockfill dams and the marginal distribution functions of each model parameter, and calculate the Pearson correlation coefficient between each model parameter and any one of the remaining model parameters; S4. According to all the model parameters, their corresponding marginal distribution functions, and all the Pearson correlation coefficients, construct a parameter multi-dimensional joint regular vine Copula function model; S5. Randomly extract several groups of Duncan Zhang E-B model parameters from the parameter multi-dimensional joint regular vine Copula function model by the Monte-Carlo method; S6. Calculate the reliability of each group of Duncan Zhang E-B model parameters using the reliability function of the downstream dam slope of the rockfill dam, and count the number of reliabilities less than zero as the number of failures; S7. Calculate the failure probability of the rockfill dam to be analyzed according to the total number of extracted Duncan Zhang E-B model parameters and the number of failures.

2. The reliability calculation method of the rockfill dam slope considering the internal correlation of parameters according to claim 1, characterized in that Step S2 further includes: S21. By adjusting the friction coefficient, cohesion, normal stress, and / or shear stress of different units in the slip channel, several groups of dam slope stability safety factors F are calculated s : Among them, M is the total number of units on the slip channel; f k , c k are respectively the friction coefficient and cohesion of the k-th unit on the slip channel, and σ k , τ k are respectively the normal stress and shear stress received by the k-th unit in the slip channel direction; l k is the unit length of the k-th unit along the slip channel direction; S22. According to several groups of dam slope stability safety factors and their corresponding Duncan-Chang E-B model parameters, using the response surface equation, calculate the undetermined coefficients a, b i , c i , and the expression of the response surface equation is: where x i is the i-th model parameter, 1 ≤ i ≤ 7; b i , c i are the undetermined coefficients corresponding to the i-th model parameter respectively; I is the total number of model parameters; S23. Use the calculated undetermined coefficients a, b i , c i to update the response surface equation as the fitting formula: Among them, K e is the tangent modulus coefficient; n is the tangent modulus exponent; K b is the bulk modulus coefficient; m is the bulk modulus exponent; R f is the failure ratio; are all non-linear strength indexes; K e , n, K b , m, R f , are the 7 model parameters included in the Duncan-Chang E-B model parameters; S23. According to the fitting formula, the reliability performance function of the downstream dam slope Z = F s -1 is constructed.

3. The reliability calculation method of the rockfill dam slope considering the internal correlation of parameters according to claim 1, characterized in that, The calculation formula for the Pearson correlation coefficient in step S3 is: where, x oi and x oj are the i-th and j-th model parameters of the o-th rockfill dam; and are the averages of the i-th and j-th model parameters of N rockfill dams, respectively; N is the total number of rockfill dams; r ij is the Pearson correlation coefficient of the i-th and j-th model parameters; 1 ≤ i ≤ 7, 1 ≤ j ≤ 7, and i ≠ j.

4. The calculation method for the reliability of the rockfill dam slope considering the internal correlation of parameters according to claim 1, characterized in that Step S4 further includes: S41. Through the maximum spanning tree method, generate the tree structure of each layer based on the principle that the sum of the absolute values of the Pearson correlation coefficients in each layer of the tree structure is the largest; S42. Determine the Copula function set; S43. In the Copula function set, select the optimal Pair Copula function of the variables connected by each side line in the tree structure through the AIC criterion and the maximum likelihood estimation method, and obtain the optimal parameter estimation of the optimal Pair Copula function; S44. Combine the tree structure and the optimal Pair Copula functions of the side lines to construct a parameter multi-dimensional joint regular vine Copula function model.

5. The calculation method of the reliability of the rockfill dam slope considering the internal correlation of parameters according to claim 4, characterized in that The Copula function set includes Gaussian Copula, t Copula, Clayton Copula, Frank Copula, Gumbel Copula, Joe Copula, and Indep Copula; The tree structure of the regular vine Copula function model includes six layers. The first layer of the tree structure has 7 nodes and 6 side lines; for each additional layer, the number of nodes decreases by one and the number of side lines decreases by one.

6. The calculation method for the reliability of the rockfill dam slope considering the internal correlation of parameters according to claim 5, characterized in that The rockfill dam includes a main rockfill area and a secondary rockfill area, and the failure probabilities of the corresponding dam slopes of both rockfill areas are calculated in the manner of steps S3 to S7; The node side lines and the optimal Pair Copula functions corresponding to the node side lines of each layer of the tree structure in the main rockfill area are: The node edges of the first - layer tree structure Tree1 are m, n, K b , n, n, R f 、R f , K e 、K e , and The optimal Pair Copula functions corresponding to these node edges are Joe, Gumbel180°, Student, Joe, Gumbel, and Gumbel respectively; The node edges of the second - layer tree - like structure Tree2 are m, K b |n, K b , R f |n, K e , n|R f , R f , and K e , The optimal Pair Copula functions corresponding to these node edges are Independence, Gaussian, Independence, Independence, and Joe180° respectively; The node edges of the third - layer tree - like structure Tree3 are m, R f |K b , n, K b , K e |R f , n, n, R f and R f , K e , and the optimal Pair Copula functions corresponding to these node edges are Independence, Clayton180°, Independence, and Gaussian respectively; The node edges of the fourth - layer tree structure Tree4 are m, K e |R f , K b , n, K b , R f , n and n, K e , R f , and the optimal Pair Copula functions corresponding to these node edges are Gaussian, Independence, and Independence respectively; The node edge of the fifth-layer tree structure Tree5 is m, R f , K b , n and K b , K e , R f , n, and the optimal Pair Copula function for the two node edges is Independence; The node edge line of the sixth-layer tree structure Tree6 is m, K e , R f , K b , n, and its optimal Pair Copula function is Independence; The node side lines and the optimal Pair Copula functions corresponding to the node side lines of each layer of the tree structure in the secondary rockfill area are: The node edges of the first - layer tree - like structure Tree1 are R f , n, m, n, n, K b and K e , K b , and The optimal Pair Copula functions corresponding to these node edges are Joe, Gumbel, Gumbel180°, Joe 180°, Joe, and Gumbel respectively; The node edges of the second - layer tree structure Tree2 are R f , m|n, m, K b |n, n, K e , and K b , The optimal Pair Copula functions corresponding to these node edges are all Independence; The node edges of the third - layer tree structure Tree3 are R f , K b |m, n, m, n, n, K b and K e The optimal Pair Copula functions corresponding to these node edges are Independence; The node edges of the fourth - layer tree structure Tree4 are R f , m, n, m K b , n and n, K b , and the optimal Pair Copula functions corresponding to these node edges are Gaussian, Independence, and Independence respectively; The node edges of the fifth - layer tree structure Tree5 are R f , K b , m, n and m, K b , n, the optimal Pair Copula functions of the two - node edges are both Independence; The node edge lines of the sixth-layer tree structure Tree6 are R f , K b , m, n, and its optimal Pair Copula function is Joe 180°.

7. The calculation method for the reliability of the rockfill dam slope considering the internal relationship of parameters according to claim 6, characterized in that The tree structure multiplies the marginal distribution functions of the 7 model parameters and the optimal Pair Copula functions corresponding to all node side lines as the correlation of the parameters of the regular vine Copula function model.

8. The reliability calculation method of the rockfill dam slope considering the internal correlation of parameters according to claim 6, characterized in that The method for selecting the marginal distribution function of the model parameters in step S3 includes: S31. Use the K-S test to determine the marginal distribution function that the model parameters conform to; S32. According to all the marginal distribution functions that each model parameter conforms to, calculate the AIC value corresponding to each marginal distribution function using the AIC criterion; S33. Select the marginal distribution with the smallest AIC value as the optimal distribution of the model parameters.

9. The calculation method of the reliability of the rockfill dam slope considering the internal relationship of parameters according to claim 8, characterized in that, For the main rockfill zone, n1 and follow a truncated normal distribution, and the remaining model parameters all follow a Weibull distribution; for the secondary rockfill zone, n2, R f2 and follow a truncated normal distribution, and the remaining model parameters all follow a Weibull distribution.

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