A PNET bridge system reliability assessment method based on Copula

Abstract

The present invention belongs to the technical field of bridge evaluation, and specifically relates to a PNET bridge system reliability evaluation method based on Copula, comprising the steps of: determining the functional function of each component in the bridge system; calculating the failure probability and reliability index of a single bridge component; selecting the most suitable Copula function to determine the nonlinear correlation between the failure modes of each component of the bridge system; calculating the Kendall rank correlation coefficient based on the selected Copula function; and evaluating the reliability of the bridge system according to the calculated Kendall rank correlation coefficient and the PNET method. The present invention overcomes the drawbacks of the traditional PNET method in dealing with the reliability problem of the bridge system due to improper handling of the correlation problem of each failure mode, and uses the Copula function that effectively handles the nonlinear correlation of random variables to accurately analyze the correlation between the failure modes of each component in the bridge system, and integrates it into the traditional PNET method, thereby improving the evaluation accuracy of the reliability of the bridge system.

Description

Technical Field

[0001] The invention belongs to the technical field of bridge evaluation, and in particular relates to a Copula-based PNET bridge system reliability evaluation method. Background Art

[0002] Considering that there are a lot of random uncertainties involved in engineering structures, such as uncertainties in material properties, geometric dimensions, calculation modes, and loads. The use of deterministic analysis methods has certain limitations in both structural design and evaluation. Reliability theory based on probability theory is an important method for dealing with random problems and is widely used to solve performance evaluation problems in aerospace, machinery, and civil engineering. Reliability refers to the probability that a structure can complete its intended function within a specified time, and is usually measured by reliability indicators or failure probabilities. In recent years, the reliability evaluation of bridge components has made breakthrough progress in theoretical research and engineering practice, and relevant industry standards and specifications have been promulgated at home and abroad. However, the bridge structure is a complex structural system composed of multiple components connected in different ways. The reliability of a single component is difficult to evaluate the overall performance of the bridge system, and the bridge maintenance department is often more concerned about the reliability of the bridge system.

[0003] For the reliability assessment method of bridge system, point estimation method and interval estimation method with clear logical relationship are usually used. The point estimation method considers each failure mechanism as two situations: independent and completely correlated. The calculation process is simple, but the accuracy of the evaluation result is difficult to guarantee, and the calculation result of hyperstatic structure is often unreasonable. The interval estimation method analyzes and calculates the failure probability of bridge system, including wide limit method and narrow limit method. Among them, the wide limit method treats each failure mode in the system as two extreme situations: independent and completely correlated, ignoring the correlation between failure modes, and is difficult to apply to events with complex failure modes and high failure probability; the narrow limit method narrows the range of the evaluation interval by considering the linear correlation between failure modes in response to the drawbacks of the wide limit method in the assessment process. The PNET method can effectively consider the correlation between failure modes in response to the problems and defects of the point estimation method and the interval estimation method, and can obtain a more accurate calculation result compared with the point estimation method, the narrow limit method and the wide limit method.

[0004] However, when evaluating the reliability of a system using the traditional PNET method, a linear processing method is used to deal with the correlation between failure modes. For complex bridge structure systems, due to the homology of structural materials and loads, there is often a complex nonlinear correlation between failure modes. Summary of the invention

[0005] In order to solve the above problems and overcome the drawbacks of the traditional PNET method in dealing with the reliability of bridge systems due to improper handling of the correlation of various failure modes, the present invention provides a PNET bridge system reliability assessment method based on Copula, which uses the Copula function that effectively handles the nonlinear correlation of random variables to accurately analyze the correlation between the failure modes of various components in the bridge system and integrates it into the traditional PNET method, thereby improving the assessment accuracy of the reliability of the bridge system. The specific technical solution is as follows:

[0006] A Copula-based PNET bridge system reliability assessment method includes the following steps:

[0007] S1: Determine the functional functions of each component in the bridge system;

[0008] S2: Calculate the failure probability p of a single bridge component fi and reliability index β i ;

[0009] S3: Select the most suitable Copula function to determine the nonlinear correlation between the failure modes of each component of the bridge system;

[0010] S4: Calculate the Kendall rank correlation coefficient based on the selected Copula function;

[0011] S5: Evaluate the reliability of the bridge system based on the calculated Kendall rank correlation coefficient and PNET method.

[0012] Preferably, the functional function in step S1 is specifically:

[0013] Z i (X) = g i (x1,x2,...,x k ),i=1,2,...,n;

[0014] Where: Z i (X) represents the functional function of the i-th component in the system, i = 1, 2, ..., n, n represents the number of components in the system; X represents a random variable group; x k Represents the function g of the i-th component i The kth random variable in .

[0015] Preferably, in step S2, according to the determined component function, the failure probability of a single bridge component is defined as shown in formula (2):

[0016]

[0017] Where: p firepresents the failure probability of the i-th component, f X (X) represents the joint probability density function of random variables, D f is the functional function Z i The domain of random variables when ≤0.

[0018] Preferably, the reliability index is calculated as follows:

[0019] According to the relationship between failure probability and reliability index, the reliability index β can be obtained: i The expression of is shown in formula (3):

[0020] β i =Φ -1 (1-p fi ); (3)

[0021] In the formula, β i represents the reliability index of the i-th component, Φ -1 Represents the inverse of the standard normal distribution function.

[0022] Preferably, the method for selecting the Copula function in step S3 is:

[0023] S31: Select an alternative Copula function;

[0024] S32: using the Monte Carlo method to sample the functional functions of n components m times respectively, generating a sample space of m×n, converting it into an empirical distribution sequence, and combining the samples of any two components in pairs;

[0025] S33: The parameter ρ in the candidate Copula function is fitted by the FitCopula function in the R language, and the AIC and BIC values ​​are calculated according to the Akaike information criterion and the Bayesian information criterion. The expressions are as follows:

[0026]

[0027]

[0028] Where: Represents the likelihood function value of the Copula function value of M samples; k represents the number of parameters in the Copula function, and M represents the number of samples;

[0029] S34: According to the minimum value principle of AIC and BIC, the most suitable Copula function is selected from the alternative Copula functions to describe the nonlinear correlation between the components in the multi-beam bridge structure system.

[0030] Preferably, the candidate Copula functions selected in step S31 specifically include Gaussian Copula function, Gumbel Copula function and Clayton Copula function.

[0031] Preferably, the calculation of the Kendall rank correlation coefficient based on the selected Copula function in step S4 specifically includes:

[0032] The relationship between the Gaussian Copula function, Gumbel Copula function, Clayton Copula function and the Kendall rank correlation coefficient is shown in equations (6), (7) and (8), respectively:

[0033] τ=(2arcsinρ) / π;(6)

[0034] τ=1-1 / α;(7)

[0035] τ=α / (2+α); (8)

[0036] In the formula, ρ and α are the parameters in the Copula function respectively; τ represents the Kendall rank correlation coefficient.

[0037] Preferably, the step S5 specifically includes the following steps:

[0038] S51: Calculate the failure mode Z of each n component of the bridge system i The failure probability P fi and the reliability index β i , the β of each failure mode i Arrange them in order from small to large to form n organizations;

[0039] S52: Find representative failure modes among n failure modes, using group screening; first determine the limit value of the correlation coefficient ρ0, the first group contains all n components, select the failure mode Z1 with the smallest reliability index as the representative component of the first group, and calculate the remaining n-1 failure modes Z through the selected Copula function i (i≠1) Correlation coefficient ρ with Z1 1i ; Correlation coefficient ρ 1i That is, it is equal to the Kendall rank correlation coefficient;

[0040] If 1i >ρ0, it is considered that the i-th component is highly correlated with the representative component 1 of the first group and can be replaced by the representative component 1; if ρ 1i <ρ0, it is considered that the i-th component and the representative component 1 are of low-level correlation and cannot replace each other;

[0041] These irreplaceable components are classified into the second group, and the component with the smallest reliability index is selected as the representative component of the second group. The above analysis process is repeated to find the next group of irreplaceable components until there is only one representative component.

[0042] S53: Through the steps, the failure probability P of each representative component in the m groups can be obtained respectively fi (i=1,2,...,m), the bridge system reliability is calculated according to the following formula:

[0043]

[0044] Where: p f represents the failure probability of the bridge system.

[0045] The beneficial effects of the present invention are as follows: in the traditional bridge system reliability assessment methods, although the point estimation method and the interval estimation method have the characteristics of clear logical relationship and easy calculation, their calculation results tend to be conservative or dangerous because they only consider the correlation of each failure mode as two extreme cases of mutual independence and complete correlation. The PNET method can effectively consider the correlation between each failure mode in view of the problems and defects of the point estimation method and the interval estimation method, and can obtain a calculation result with higher accuracy compared with the point estimation method and the interval estimation method. However, the traditional PNET method cannot handle the complex nonlinear correlation between the failure modes in the bridge system, resulting in certain limitations in the assessment results. The method proposed in this patent avoids the drawbacks of the traditional PNET method in analyzing the reliability of the bridge system. The PNET method based on Copula can effectively consider the complex correlation problem of each failure mode in the bridge structure, and improve the accuracy of the PNET method in assessing the reliability of the bridge system. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following is a brief introduction to the drawings required for the specific embodiments or the description of the prior art. In all the drawings, similar elements or parts are generally identified by similar reference numerals. In the drawings, the elements or parts are not necessarily drawn according to the actual scale.

[0047] Figure 1 It is a schematic diagram of the process of the present invention. DETAILED DESCRIPTION

[0048] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0049] It should be understood that when used in this specification and the appended claims, the terms "include" and "comprises" indicate the presence of described features, integers, steps, operations, elements and / or components, but do not exclude the presence or addition of one or more other features, integers, steps, operations, elements, components and / or combinations thereof.

[0050] It should also be understood that the terms used in the present specification are only for the purpose of describing specific embodiments and are not intended to limit the present invention. As used in the present specification and the appended claims, unless the context clearly indicates otherwise, the singular forms "a", "an" and "the" are intended to include plural forms.

[0051] It should be further understood that the term "and / or" used in the present description and the appended claims refers to any and all possible combinations of one or more of the associated listed items, and includes these combinations.

[0052] This embodiment provides a PNET bridge system reliability assessment method based on Copula. Figure 1 As shown, the following steps are included:

[0053] S1: Determine the functional functions of each component in the bridge system; the functional functions are as follows: Z i (X) = g i (x1,x2,...,x k ),i=1,2,...,n;

[0054] Where: Z i (X) represents the functional function of the i-th component in the system, i = 1, 2, ..., n, n represents the number of components in the system; X represents a random variable group; x k Represents the function g of the i-th component i The kth random variable in .

[0055] S2: Calculate the failure probability p of a single bridge component fi and reliability index β i According to the determined component function, the failure probability of a single bridge component is defined as shown in formula (2):

[0056]

[0057] Where: p fi represents the failure probability of the i-th component, f X (X) represents the joint probability density function of random variables, D f is the functional function Z iThe domain of random variables when ≤0.

[0058] According to the relationship between failure probability and reliability index, the reliability index β can be obtained: i The expression of is shown in formula (3):

[0059] β i =Φ -1 (1-p fi ); (3)

[0060] In the formula, β i represents the reliability index of the i-th component, Φ -1 Represents the inverse of the standard normal distribution function.

[0061] S3: Select the most suitable Copula function to determine the nonlinear correlation between the failure modes of each component of the bridge system; the selection method of the Copula function is:

[0062] S31: Select an alternative Copula function;

[0063] S32: using the Monte Carlo method to sample the functional functions of n components m times respectively, generating a sample space of m×n, converting it into an empirical distribution sequence, and combining the samples of any two components in pairs;

[0064] S33: The parameter ρ in the candidate Copula function is fitted by the FitCopula function in the R language, and the AIC and BIC values ​​are calculated according to the Akaike information criterion and the Bayesian information criterion. The expressions are as follows:

[0065]

[0066]

[0067] Where: Represents the likelihood function value of the Copula function value of M samples; k represents the number of parameters in the Copula function, and M represents the number of samples;

[0068] S34: According to the minimum value principle of AIC and BIC, the most suitable Copula function is selected from the alternative Copula functions to describe the nonlinear correlation between the components in the multi-beam bridge structure system.

[0069] The candidate Copula functions include Gaussian Copula function, Gumbel Copula function and Clayton Copula function.

[0070] S4: Calculate the Kendall rank correlation coefficient based on the selected Copula function. The relationship between the Gaussian Copula function, the Gumbel Copula function, the Clayton Copula function and the Kendall rank correlation coefficient is shown in equations (6), (7) and (8), respectively:

[0071] τ=(2arcsinρ) / π;(6)

[0072] τ=1-1 / α;(7)

[0073] τ=α / (2+α); (8)

[0074] In the formula, ρ and α are the parameters in the Copula function respectively; τ represents the Kendall rank correlation coefficient.

[0075] S5: Evaluate the reliability of the bridge system based on the calculated Kendall rank correlation coefficient and PNET method. Specifically, it includes the following steps:

[0076] S51: Calculate the failure mode Z of each n component of the bridge system i The failure probability P fi and the reliability index β i , the β of each failure mode i Arrange them in order from small to large to form n organizations;

[0077] S52: Find representative failure modes among n failure modes, using group screening; first determine the limit value of the correlation coefficient ρ0, the first group contains all n components, select the failure mode Z1 with the smallest reliability index as the representative component of the first group, and calculate the remaining n-1 failure modes Z through the selected Copula function i (i≠1) Correlation coefficient ρ with Z1 1i ; Correlation coefficient ρ 1i That is, it is equal to the Kendall rank correlation coefficient;

[0078] If 1i >ρ0, it is considered that the i-th component is highly correlated with the representative component 1 of the first group and can be replaced by the representative component 1; if ρ 1i <ρ0, it is considered that the i-th component and the representative component 1 are of low-level correlation and cannot replace each other;

[0079] These irreplaceable components are classified into the second group, and the component with the smallest reliability index is selected as the representative component of the second group. The above analysis process is repeated to find the next group of irreplaceable components until there is only one representative component.

[0080] S53: Through the steps, the failure probability P of each representative component in the m groups can be obtained respectively fi (i=1,2,...,m), the bridge system reliability is calculated according to the following formula:

[0081]

[0082] Where: p f represents the failure probability of the bridge system.

[0083] In order to better understand the implementation process of the invention and its applicability to actual engineering, the failure probability of the bridge superstructure system composed of 7 beam components was calculated and analyzed. i as follows:

[0084] Z i =R i -M i1 -M i2 -M i3 ,i=1,2,...,7

[0085] In the formula, R i 、M i1 、M i2 and M i3 are all random variables, R i represents the resistance of the i-th main beam, M i1 、M i2 、M i3 They represent the load effects of the deadweight of the i-th main beam, the second-phase dead load and the vehicle load respectively. The values ​​of each random variable and their probability distribution are shown in Table 1.

[0086] Table 1 Statistical parameters of random variables

[0087]

[0088] Calculate the reliability index β of beams 1-7# i With the failure probability p fi As shown in Table 2 below.

[0089] Table 2 Reliability index and failure probability calculation results of each component

[0090] 1# beam 2# beam 3# beam 4# beam 5# beam 6# beam 7# beam <![CDATA[β i ]]> 3.0202 3.5873 3.5999 3.6232 3.6035 3.5909 3.0223 <![CDATA[p fi ]]> <![CDATA[1.26×10 3 ]]> <![CDATA[1.67×10 4 ]]> <![CDATA[1.59×10 4 ]]> <![CDATA[1.45×10 4 ]]> <![CDATA[1.57×10 4 ]]> <![CDATA[1.65×10 4 ]]> <![CDATA[1.25×10 3 ]]>

[0091] The Monte Carlo sampling method is used to analyze the performance function Z of each beam. iSampling was performed to obtain samples, and the sample size n = 1500. The sample data was processed using R language, and the AIC and BIC values ​​of several Gaussian Copula, Gumbel Copula and Clayton Copula were analyzed, as shown in Table 3. The Copula function that best describes the correlation of the failure modes between the two beams was determined, and then the Kendall rank correlation coefficient was calculated based on the Copula function parameters.

[0092] Table 3 AIC and BIC values ​​of failure modes corresponding to different Copula functions

[0093]

[0094]

[0095] According to the minimum AIC and BIC principles, it can be seen from Table 3 that the Gaussian Copula function is most suitable for describing the correlation between the two failure modes in this example. Then, the correlation coefficient is calculated based on the relationship between the kendall rank and the Gaussian Copula function in formula (6).

[0096] Table 4 Calculation results of nonlinear correlation coefficients between failure modes

[0097] 1# beam 2# beam 3# beam 4# beam 5# beam 6# beam 7# beam 1# beam 1 0.8407 0.8535 0.8521 0.8469 0.8473 0.8375 2# beam 1 0.8617 0.8615 0.8653 0.8622 0.8475 3# beam 1 0.8618 0.856 0.8617 0.8503 4# beam 1 0.8662 0.8604 0.8415 5# beam 1 0.8566 0.8461 6# beam 1 0.8473 7# beam 1

[0098] The reliability of the bridge system is evaluated based on the calculated Kendall rank correlation coefficient and PNET method:

[0099] (1) The reliability index β of each component i Arrange them in ascending order, first determine the limit value of the correlation coefficient ρ0 = 0.85, and group the screening results as shown in Table 5 below:

[0100] Table 5 Calculation results of representative components using the PNET method

[0101]

[0102]

[0103] (2) Then, based on the representative components of the four groups, namely, beams 1#, 7#, 2#, and 6#, the PNET method is used to calculate the reliability of the bridge system:

[0104] P f =P f1 +P f2 +P f6 +P f7 =2.85×10 -3 .

[0105] In summary, the present invention patent proposes a Copula-based PNET bridge system reliability assessment method, which can effectively solve the drawbacks of the traditional PNET method in dealing with bridge system reliability assessment problems. The traditional PNET method screens the representative mechanism in the system by comparing the linear correlation of each component, but cannot consider the nonlinear relationship between components due to the homology of the constituent materials and the acting loads of the bridge components, resulting in difficulty in obtaining objective and reasonable system failure probability calculation results. The present invention patent improves the traditional PNET method by introducing the Copula theory that can effectively handle the nonlinear correlation of random variables, which can more accurately screen the representative mechanism in the system, thereby improving the accuracy, objectivity and rationality of the bridge system failure probability calculation results.

[0106] Those of ordinary skill in the art will appreciate that the units of each example described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, computer software, or a combination of the two. In order to clearly illustrate the interchangeability of hardware and software, the composition of each example has been generally described in terms of function in the above description. Whether these functions are performed in hardware or software depends on the specific application and design constraints of the technical solution. Professional and technical personnel can use different methods to implement the described functions for each specific application, but such implementation should not be considered to be beyond the scope of the present invention.

[0107] In the embodiments provided in the present application, it should be understood that the division of units is merely a logical function division, and there may be other division methods in actual implementation, for example, multiple units may be combined into one unit, one unit may be split into multiple units, or some features may be ignored.

[0108] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein by equivalents. These modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present invention, and they should all be included in the scope of the claims and specification of the present invention.

Claims

1. A Copula-based PNET bridge system reliability assessment method, characterized by: The following steps are involved: S1: Determine the functional functions of each component in the bridge system; S2: Calculate the failure probability of the i-th bridge component Reliability Index ; , Indicates the number of components in the system; S3: Select the most suitable Copula function to determine the nonlinear correlation between the failure modes of each component of the bridge system; S4: Calculate the Kendall rank correlation coefficient based on the selected Copula function; S5: Evaluate the reliability of the bridge system based on the calculated Kendall rank correlation coefficient and PNET method; The step S5 specifically comprises the following steps: S51: Computational Bridge Systems Failure modes of each component The failure probability and reliability index , the failure modes of Arrange them in order from small to large to form institutions; S52: Find the representative failure mode among the failure modes, and use the group screening method; first determine the limit value of the correlation coefficient , Group 1 contains all Component, select the failure mode with the smallest reliability index As the representative component of Group 1, the remaining Failure Mode and The correlation coefficient , ; Correlation coefficient That is, it is equal to the Kendall rank correlation coefficient; like , then it is considered that The component is highly related to the representative component 1 of group 1 and can be replaced by the representative component 1; if , then it is considered that The component and representative component 1 are low-level related and cannot replace each other; These irreplaceable components are classified into the second group, and the component with the smallest reliability index is selected as the representative component of the second group. The above analysis process is repeated to find the next group of irreplaceable components until there is only one representative component. S53: Through the steps, we can get The failure probability of each representative component in the group , the bridge system reliability is calculated according to the following formula: ； Where: represents the failure probability of the bridge system.

2. A Copula-based PNET bridge system reliability assessment method according to claim 1, characterized in that: The functional function in step S1 is specifically: ； Where: Indicates the system The functional functions of the components, , Indicates the number of components in the system; represents a group of random variables; Indicates Component Function Middle A random variable.

3. A Copula-based PNET bridge system reliability assessment method according to claim 2, characterized in that: In step S2, according to the determined component function, the failure probability definition of a single bridge component is obtained as shown in formula (2): ；（2） Where: Indicates The failure probability of each component is represents the joint probability density function of random variables, For functional functions The domain of the random variable when .

4. A Copula-based PNET bridge system reliability assessment method according to claim 3, characterized in that: The reliability index is calculated as follows: According to the relationship between failure probability and reliability index, the reliability index can be obtained: The expression of is shown in formula (3): ；（3） In the formula, Indicates The reliability index of each component is Represents the inverse of the standard normal distribution function.

5. A Copula-based PNET bridge system reliability assessment method according to claim 1, characterized in that: The method for selecting the Copula function in step S3 is: S31: Select an alternative Copula function; S32: Monte Carlo method is used to sample the functional functions of n components m times respectively to generate The sample space of is converted into an empirical distribution sequence, and the samples of any two components are combined in pairs; S33: Fitting parameters in alternative Copula functions using the FitCopula function in R , and calculate the AIC and BIC values ​​according to the Akaike information criterion and the Bayesian information criterion. The expressions are as follows: ；（4） ；（5） Where: Represents the likelihood function value of the Copula function value of M samples; k Represents the number of parameters in the Copula function, and M represents the number of samples; S34: According to the minimum value principle of AIC and BIC, the most suitable Copula function is selected from the alternative Copula functions to describe the nonlinear correlation between the components in the multi-beam bridge structure system.

6. A Copula-based PNET bridge system reliability assessment method according to claim 5, characterized in that: The candidate Copula functions selected in step S31 specifically include Gaussian Copula function, Gumbel Copula function and Clayton Copula function.

7. A Copula-based PNET bridge system reliability assessment method according to claim 6, characterized in that: Calculating the Kendall rank correlation coefficient based on the selected Copula function in step S4 specifically includes: The relationship between the Gaussian Copula function, Gumbel Copula function, Clayton Copula function and the Kendall rank correlation coefficient is shown in equations (6), (7) and (8), respectively: ；（6） ；（7） ；（8） In the formula, , They are the parameters in the Copula function respectively; represents the Kendall rank correlation coefficient.

Images