Structural reliability analysis method based on evidence theory and multipoint linearization

Abstract

The application discloses a structure reliability analysis method based on evidence theory and multi-point linearization, which comprises the following steps: firstly, constructing an evidence variable identification framework and a basic credibility distribution function according to engineering experience or authoritative expert information; equivalently converting the evidence theory reliability analysis into a probability reliability analysis model; solving the most possible failure point and the reliability index through a first-order second-moment method; determining a linearization point through a multi-point linearization method; respectively performing a first-order Taylor expansion at the most possible failure point and the linearization point to obtain a hyperplane approximate true limit state function; and dividing a failure domain, a safety domain and an uncertain domain according to the position and distance of the center point of a joint focus element relative to the hyperplane. Only the extreme value analysis of the joint focus element in the uncertain domain is needed to calculate the credibility and the quasi-truth of the structure. The method is closer to the actual result than the first-order approximation result, and can be superior to the second-order approximation result, and expands the application of the evidence theory in the actual engineering reliability analysis problem.

Description

Technical Field

[0001] This invention relates to the field of structural or product reliability analysis technology, and in particular to the analysis of structural reliability sensitivity using evidence theory. Specifically, it relates to a structural reliability analysis method, apparatus, computer equipment, and storage medium based on evidence theory and multi-point linearization. Background Technology

[0002] In practical engineering problems, uncertainties such as material properties, structural geometric parameters, boundary conditions, initial conditions, and measurement errors are unavoidable. Most of these factors cause very small degrees of uncertainty; however, when multiple uncertainties are superimposed, the uncertainty is amplified, potentially leading to structural failure. Therefore, in product and structural design, it is essential to analyze all existing uncertainties, fully consider the role of different uncertainties, and then conduct uncertainty reliability analysis on locations prone to structural damage. Traditional structural reliability analysis requires certain information and data about the structure. However, in practical engineering applications, many uncertainties are simply treated as deterministic, leading to overly conservative reliability analysis results, material waste, and excessive structural weight. Evidence theory has a strong ability to handle cognitive uncertainty. It can estimate existing uncertainties using only existing information without additional assumptions, and can flexibly handle interval uncertainties from multiple sources by employing composition rules. Evidence theory uses two measurement standards—credibility and plausibility—to quantify the upper and lower limits of precise probability. Evidence theory can be equivalent to probability theory, interval analysis theory, and fuzzy theory in different situations.

[0003] The multi-point linearization method involves finding multiple tangent points (linearization points) on the limit state surface and performing a first-order Taylor expansion at these points to obtain multiple hyperplanes that approximate the original limit state function. This simplifies complex nonlinear problems into a series of linear problems, making the problem easier to handle and improving computational efficiency.

[0004] Therefore, applying the structural reliability analysis method based on evidence theory and multi-point linearization to structural reliability analysis is of great significance for the use of efficient evidence theory analysis methods in the field of structural or product reliability assessment in civil engineering, mechanical electronics, aerospace and other fields to conduct structural reliability and sensitivity analysis. Summary of the Invention

[0005] The purpose of this invention is to overcome the above-mentioned deficiencies in the prior art and provide a structural reliability analysis method, apparatus, computer equipment, and storage medium based on evidence theory and multi-point linearization.

[0006] The first objective of this invention is to disclose a structural reliability analysis method based on evidence theory and multi-point linearization, the structural reliability analysis method comprising the following steps:

[0007] S1. Specify the limit state function g(X) in the domain to be analyzed, which reflects the normal operating capability or critical safe operating state of the structure or product, where X = [X1, X2, ..., X]. i ,...,X n ], X i Let X be the i-th dimension of uncertain evidence variable, and n be the dimension of the problem. Based on engineering experience or information from authoritative experts, construct the uncertain evidence variable X. i The identification framework, single variable X i The basic credibility assignment function m i (X i ), Joint Coke A X and joint basic credibility assignment function Among them, the basic credibility assignment function m i (X i The value reflects the degree of confidence that the evidence proposition is true, and is a basic probability number between [0,1].

[0008] S2. Based on the equal probability transformation, establish the uncertain evidence variable X. i With uncertain evidence variable X i The corresponding standard normal distribution variable Y i Equivalent relationship between them: X i =(φ(Y) i )-m i (A i,1 )-…-m i (A i,j-1 ))×(U i,j -L i,j ) / m i (A i,j )+L i,j , where φ(Y) i Y is a standard normal distribution i The cumulative probability distribution function, A i,j The evidence variable X is uncertain. i The j-th focal element, m i (A i,j ) is A i,j The basic credibility assignment function, focal element A i,j =[L i,j U i,j ], j = 1, 2, ..., L i,j and U i,j Corresponding to focal element A i,j The lower and upper bounds of , jiaoyuan A i,jIt refers to a range, not a specific point.

[0009] S3. Obtain the most likely failure point Y using the first second moment method. * Reliability index β and most likely failure point Y * The unit vector α at that location Y Y * Transforming back to the original evidence space using an equal probability transformation yields the most probable failure point X in the original space. * ;

[0010] S4. Using the orthogonalization method, with α Y Construct an orthogonal matrix R, where the nth column of R is the unit vector α. Y The variable Y in the standard normal space is transformed into the variable Y' in the rotated standard normal space by the rotation transformation Y' = RY;

[0011] S5. In the rotated standard normal space, find linear points along the i = 1, 2, ..., n-1 directions, where each dimension can determine two linearized points Y′ on the limit state surface. i+ and Y′ i- Thus, 2(n-1) linearization points are found, and the linearization point Y′ in the rotated standard normal space is obtained. i+ and Y′ i- Transform to linearized point X' in the original evidence space i+ and X' i- ;

[0012] S6. In the evidence space, at the most likely point of failure X * and 2(n-1) linearized points X' i+ and X' i- Using the first-order Taylor expansion to approximate the true limit state function g(X), 2n-1 approximate hyperplanes are obtained:

[0013]

[0014]

[0015]

[0016] Among them, symbols This indicates finding the gradient of a function;

[0017] S7. Divide the absolute failure domain, absolute safety domain, and uncertainty domain of the focal element. The division rules are as follows: If the p-th joint focal element A Xp Center point C p With the hyperplane and The distance is greater than the focal radius, and Then the focal element belongs to the failure domain S1; if the p-th joint focal element A X Center point C p With the hyperplane and The distance is greater than the focal radius, and and Then the focal element belongs to the safe region S2; in all other cases, the focal element belongs to the uncertain region S3.

[0018] S8. Perform extreme value analysis on each joint focal element in the uncertainty domain S3 to determine the confidence level Bel(S3) and the similarity level Pl(S3) in the uncertainty domain S3.

[0019] S9. Calculate the reliability and similarity of the structure: The reliability Bel(G) and similarity Pl(G) of the structure are the sum of the reliability and similarity of the safe region and the uncertain region: Bel(G) = Bel(S2) + Bel(S3), Pl(G) = Pl(S2) + Pl(S3), where the reliability and similarity of the safe region S2 are...

[0020] Furthermore, the field to be analyzed includes structural or product reliability analysis and evaluation in fields such as civil engineering, mechanical and electronic engineering, or aerospace.

[0021] Further, in step S5, linear points are found along the i = 1, 2, ..., n-1 digits in the rotated standard normal space. To carry out this process, the initial value is set to i = 1, the rotated standard normal space vector is Y' = (0, 0, ..., 0), and the initial trial value is k. i =0.25, and k i+ =k i- =k i .

[0022] Furthermore, in step S5, it is assumed that Y' i+ and Y' i- The initial values ​​are respectively In the rotated standard normal space, two distinct linearization points are determined along the positive and negative directions of the i-th dimension, respectively.

[0023] Furthermore, in step S5, for Y' i+ and Y' i- Randomly select η1 and η2, and let η1 = β+1 and η2 = β-1, Y' i+ and Y' i-The last component is taken as η1 and η2 respectively, resulting in two interpolation points. The limit state function values ​​g1 and g2 of these two interpolation points are calculated in the original evidence space to prepare for the zero point on the limit state surface of the linear search.

[0024] Furthermore, in step S5, Calculate Y' i+ and Y' i- The corresponding interpolation points and The corresponding limit state function value g3 in the evidence space is obtained. The approximate zero point on the limit state surface is obtained by linear interpolation using two interpolation points.

[0025] Further, in step S5, η1 = η2, η2 = η3, g1 = g2, g2 = g3, and S5-4 and S5-5 are repeated until |g3| < ε, where |g3| is the absolute value of g3 and ε is a small positive number, thus obtaining the linearization point. and Through continuous iteration, the interpolation points gradually approach the limit state surface, thereby obtaining linearized points with higher accuracy.

[0026] Furthermore, in step S5, k is updated. i Values: Calculated separately in, Δ0∈0.001~0.1, k i Get the current k i ,k i+ ,k i- The minimum value in k. This method... i The value of k is chosen such that the original limit state function can be approximated with second-order accuracy in that direction at that linear point. i Being closer to the most likely failure point contributes more to structural failure.

[0027] Furthermore, in step S5, if i < n-1, i = i+1, repeat steps S5-2 to S5-6. Continue until the i = n-1th point, ensuring that a total of 2(n-1) linearization points are found, and the intersection of the failure domains determined by the 2(n-1) hyperplanes approximates the failure domain determined by the original nonlinear limit state function.

[0028] Furthermore, in step S5, k i+ =k i- =k i Repeat steps S5-2 to S5-5 along the i = 1, 2, ..., n-1 points respectively to obtain the last 2(n-1) linearization points. and The above steps yield the optimal k. i Under this value, find the final 2(n-1) linearization point.

[0029] Furthermore, in step S8, extreme value analysis is performed on each joint focal element within the uncertainty domain S3 to determine the confidence level Bel(S3) and the similarity level Pl(S3) within the uncertainty domain S3. The initial values ​​are Bel(S3) = 0 and Pl(S3) = 0. Subsequently, the confidence level and similarity level of each focal element are continuously accumulated.

[0030] Furthermore, in step S8, the maximum value of the focal element with the smallest range of values ​​is calculated when 2n-1 approximate hyperplanes are used as the limit state function. Evaluating the limit of a multi-segment approximate hyperplane under the worst-case scenario prepares for the calculation of reliability and similarity in the uncertainty domain.

[0031] Further, in step S8, the p-th joint focal element A in the uncertainty domain is calculated. Xp The maximum and minimum values ​​are determined. Extremum analysis is performed on the joint focal elements to identify the assumptions of each joint focal element in the uncertainty domain. and In each joint focal element A Xp The maxima on are g_max and g_max, respectively. i+ _max、g i- _max, and the minimum values ​​are g_min and g_min respectively. i+ _min、g i- _min, which together form the set of maxima G. max and the set of minimum values ​​G min The characteristics of the maxima set and the minima set determine the different methods for calculating reliability and similarity.

[0032] Furthermore, in step S8, it is assumed that G max R Describe set G max The maximum value of G min R Let set G represent min The maximum value of these two extreme values ​​determines the difference in the calculation methods for reliability and similarity. That is:

[0033]

[0034] Assume G max L Describe set G max The minimum value of G min L Describe set G min The minimum value, that is:

[0035]

[0036] Where max{} and min{} represent the maximum and minimum values, respectively;

[0037] Furthermore, in step S8, for the p-th joint focal element A within the uncertainty domain... Xp The reliability and similarity are calculated as follows:

[0038] (1) If And combined with coke element A Xp of Then Pl(S3) = Pl(S3) + m(A) Xp );like And combined with coke element A Xp of Then Bel(S3) = Bel(S3) + m(A) Xp ); for the case where the left side of the limit state surface is positive, and whether the reliability and similarity of the focal element in the uncertainty domain should be included in the total reliability and similarity based on the extreme value sign.

[0039] (2) If And combined with coke element A Xp of Then Bel(S3) = Bel(S3) + m(A) Xp );like And combined with coke element A Xp of Then Pl(S3) = Pl(S3) + m(A) Xp For the case where the left side of the limit state surface is negative, and based on the sign of the extreme value, whether the reliability and similarity of the focal element in the uncertainty domain should be included in the total reliability and similarity.

[0040] A second objective of this invention is to disclose a structural reliability analysis device based on evidence theory and multi-point linearization, the reliability analysis device comprising:

[0041] The information construction module specifies the limit state function g(X) that reflects the normal operating capability or critical safe operating state of a structure or product in the domain to be analyzed, where X = [X1, X2, ..., X...]. i ,...,X n ], X i Let X be the i-th dimension of uncertain evidence variable, and n be the dimension of the problem. Based on engineering experience or information from authoritative experts, construct the uncertain evidence variable X. i The identification framework, single variable X i Basic credibility assignment function m i (X i ), Joint Coke A X and joint basic credibility assignment function

[0042] The equivalence relation module establishes the uncertain evidence variable X based on the equal probability transformation. i With uncertain evidence variable X i The corresponding standard normal distribution variable Y i Equivalent relationship between them: X i =(φ(Y) i )-m i (A i,1 )-…-m i (A i,j-1 ))×(U i,j -L i,j ) / m i (A i,j )+L i,j , where φ(Y) i Y is a standard normal distribution i The cumulative probability distribution function, A i,j The evidence variable X is uncertain. i The j-th focal element, m i (A i,j ) is A i,j The basic credibility assignment function, focal element A i,j =[L i,j U i,j ], j = 1, 2, ..., L i,j and U i,j Corresponding to focal element A i,j The lower and upper bounds;

[0043] The spatial transformation module uses the first-order second-moment method to find the most likely failure point Y. * Reliability index β and most likely failure point Y * The unit vector α at that location Y Y * Transforming back to the original evidence space using an equal probability transformation yields the most probable failure point X in the original space. * ;

[0044] The orthogonalization module employs an orthogonalization method, with α... Y Construct an orthogonal matrix R, where the nth column of R is the unit vector α. Y The variable Y in the standard normal space is transformed into the variable Y' in the rotated standard normal space by the rotation transformation Y' = RY;

[0045] The linear point determination module finds linear points along the i = 1, 2, ..., n-1 axes in the rotated standard normal space, where each dimension can determine two linearized points Y′ on the limit state surface. i+ and Y′ i- Thus, 2(n-1) linearization points are found, and the linearization point Y' in the rotated standard normal space is obtained. i+and Y' i- Transform to linearized point X' in the original evidence space i+ and X' i- ;

[0046] Taylor expansion module, in the evidence space, at the most likely point of failure X. * and 2(n-1) linearized points X' i+ and X' i- Using the first-order Taylor expansion to approximate the true limit state function g(X), 2n-1 approximate hyperplanes are obtained:

[0047]

[0048]

[0049]

[0050] Among them, symbols This indicates finding the gradient of a function;

[0051] The domain partitioning module divides the focal element into the absolute failure domain, the absolute safety domain, and the uncertainty domain. The partitioning rules are as follows: If the p-th joint focal element A... Xp Center point C p With the hyperplane and The distance is greater than the focal radius, and Then the focal element belongs to the failure domain S1; if the p-th joint focal element A X Center point C p With the hyperplane and The distance is greater than the focal radius, and and Then the focal element belongs to the safe region S2; in all other cases, the focal element belongs to the uncertain region S3.

[0052] The extreme value analysis module performs extreme value analysis on each joint focal element in the uncertainty domain S3 to determine the confidence level Bel(S3) and the similarity level Pl(S3) in the uncertainty domain S3.

[0053] The reliability calculation module calculates the reliability and similarity of the structure: the reliability Bel(G) and similarity Pl(G) of the structure are the sum of the reliability and similarity of the safe region and the uncertain region: Bel(G) = Bel(S2) + Bel(S3), Pl(G) = Pl(S2) + Pl(S3), where the reliability and similarity of the safe region S2 are...

[0054] A third objective of this invention is to provide a computer device, including a processor and a memory for storing a processor-executable program, wherein when the processor executes the program stored in the memory, it implements the aforementioned structural reliability analysis method based on evidence theory and multi-point linearization.

[0055] A fourth objective of this invention is a storage medium storing a program that, when executed by a processor, implements the aforementioned structural reliability analysis method based on evidence theory and multi-point linearization.

[0056] The present invention has the following advantages and effects compared with the prior art:

[0057] (1) The first-order approximate reliability method only performs a first-order Taylor expansion linearly approximating the original limit state function at the most likely failure point, which cannot accurately handle highly nonlinear failure surfaces and is prone to approximation errors. The second-order approximate reliability method provides higher accuracy through a second-order Taylor expansion at the most likely failure point, but its computational complexity and time consumption are relatively large. In contrast, the method of this invention selects multiple linearization points on the limit state function and performs a linear expansion at the linearization points, thus achieving a more comprehensive approximation of the original limit state function, resulting in more accurate results and convenient calculation. The computational accuracy may exceed that of the second-order approximate reliability method. It can be seen that the method of this invention may provide a better balance between accuracy and efficiency.

[0058] (2) In evidence theory, performing extreme value analysis on each joint focal element individually results in an exponential increase in computational cost as the number of evidence variables and intervals increase, leading to excessive costs. This invention divides the joint focal element into failure, safety, and uncertainty domains based on the position and distance of its center point relative to the hyperplane. Only the joint focal elements in the uncertainty domain require extreme value analysis, significantly reducing the number of joint focal elements requiring extreme value calculations and thus improving the efficiency of the entire analysis process.

[0059] (3) This invention proposes to determine suitable linear points on the limit state function based on the degree of change in the slope of the limit state function. Compared with the series of steps of traversing all linearized points on the limit state and then excluding hyperplanes with high correlation coefficients, the search process proposed in this invention is faster and more reasonable. Attached Figure Description

[0060] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this application, illustrate exemplary embodiments of the invention and, together with their description, serve to explain the invention and do not constitute an undue limitation thereof. In the drawings:

[0061] Figure 1 This is a flowchart of a structural reliability analysis method based on evidence theory and multi-point linearization disclosed in this invention;

[0062] Figure 2 A schematic diagram of the uncertainty domain range, which can be considered as a channel, is provided in the example of this invention;

[0063] Figure 3 This is a simplified diagram of the tubular cantilever beam in Example 2;

[0064] Figure 4 This is a structural block diagram of the structural reliability analysis device in Embodiment 3 of the present invention;

[0065] Figure 5 This is a structural block diagram of the computer device in Embodiment 4 of the present invention. Detailed Implementation

[0066] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0067] Example 1

[0068] This embodiment further illustrates the invention using a two-dimensional limit state function. The structural reliability analysis method based on evidence theory and multi-point linearization includes the following steps:

[0069] S1. Specify the limit state function g(x) = λ(x1-x2) of the structure to be analyzed. 2 -x1-x2-0.2, where x1 and x2 are evidence variables, both with an identification frame of [-2, 2]; the parameter λ reflects the nonlinearity of the limit state function. The focal element interval and BPA distribution information are shown in Table 1 below:

[0070] Table 1. BPA distribution table for evidence variables x1 and x2 taking 10 subintervals

[0071]

[0072]

[0073] S2. Based on the equal probability transformation, establish the uncertain evidence variable X. i With uncertain evidence variable X i The corresponding standard normal distribution variable Y i Equivalent relationship between them: X i =(φ(Y) i )-m i (A i,1 )-…-m i(A i,j-1 ))×(U i,j -L i,j ) / m i (A i,j )+L i,j , where φ(Y) i Y is a standard normal distribution i The cumulative probability distribution function, A i,j The evidence variable X is uncertain. i The j-th focal element, m i (A i,j ) is A i,j The basic credibility assignment function, focal element A i,j =[L i,j U i,j ], j = 1, 2, ..., L i,j and U i,j Corresponding to focal element A i,j The lower and upper bounds;

[0074] S3. Obtain the most likely failure point Y using the first second moment method. * Reliability index β and most likely failure point Y * The unit vector α at that location Y Y * Transforming back to the original evidence space using an equal probability transformation yields the most probable failure point X in the original space. * ;

[0075] S4. Using the orthogonalization method, with α Y Construct an orthogonal matrix R, where the nth column of R is the unit vector α. Y The variable Y in the standard normal space is transformed into the variable Y' in the rotated standard normal space by the rotation transformation Y' = RY;

[0076] S5. In the rotated standard normal space, find linear points along the i = 1, 2, ..., n-1 directions, where each dimension can determine two linearized points Y′ on the limit state surface. i+ and Y′ i- Thus, 2(n-1) linearization points are found, and the linearization point Y′ in the rotated standard normal space is obtained. i+ and Y′ i- Transform to linearized point X' in the original evidence space i+ and X' i- ;

[0077] S6. In the evidence space, at the most likely point of failure X * and 2(n-1) linearized points X' i+ and X' i-Using the first-order Taylor expansion to approximate the true limit state function g(X), 2n-1 approximate hyperplanes are obtained:

[0078]

[0079]

[0080]

[0081] Among them, symbols This indicates finding the gradient of a function;

[0082] S7. Divide the absolute failure domain, absolute safety domain, and uncertainty domain of the focal element. The division rules are as follows: If the p-th joint focal element A Xp Center point C p With the hyperplane and The distance is greater than the focal radius, and Then the focal element belongs to the failure domain S1; if the p-th joint focal element A X Center point C p With the hyperplane and The distance is greater than the focal radius, and and Then the focal element belongs to the safe region S2; in all other cases, the focal element belongs to the uncertain region S3.

[0083] S8. Perform extreme value analysis on each joint focal element in the uncertainty domain S3 to determine the confidence level Bel(S3) and the similarity level Pl(S3) in the uncertainty domain S3.

[0084] S9. Calculate the reliability and similarity of the structure: The reliability Bel(G) and similarity Pl(G) of the structure are the sum of the reliability and similarity of the safe region and the uncertain region: Bel(G) = Bel(S2) + Bel(S3), Pl(G) = Pl(S2) + Pl(S3), where the reliability and similarity of the safe region S2 are...

[0085] Table 2 shows the structural confidence intervals and relative errors calculated by the reliability analysis method disclosed in Example 1 and other methods (GA: Genetic Algorithm, FARM: First-order approximate reliability analysis method, SARM: Second-order approximate reliability analysis method). The confidence and similarity obtained by the genetic algorithm are used as the exact solution. When the threshold parameter λ changes from 0.03 to 2.1, it can be seen that as the nonlinearity of the limit state function gradually increases, the second-order approximate reliability analysis method currently performs the same as the reference solution, while the confidence and similarity results calculated by the first-order approximate reliability analysis method have increasingly larger errors compared to the reference solution. The method of this invention considers the influence of the nonlinearity of the limit state function, thus improving the calculation accuracy of reliability analysis. For example, when λ = 0.07, the confidence and similarity results of the first-order approximate reliability analysis method have errors of 4.69% and 2.48%, respectively; the confidence and similarity results of the method of this invention and the second-order approximate reliability analysis method have zero errors. When λ = 1.7, the reliability and similarity results of the first-order approximate reliability analysis method increase to 49.34% and 36.23%, respectively; the error of the second-order approximate reliability analysis method is 0; the reliability and similarity results of the method of this invention are 1.18% and 1.89%, respectively. It can be seen that the method of this invention has very small error and meets the accuracy requirements of structural interval reliability analysis.

[0086] Extremum analysis for each joint focal element would require 10 × 10 = 100 elements. However, this invention proposes an uncertainty domain, significantly reducing the computational ratio. When λ = 0.03, there are 31 joint focal elements within the uncertainty domain, resulting in a computational ratio of 0.31. The table shows that as the threshold parameter λ changes from 0.03 to 2.1, the computational ratio remains less than 1, meaning the number of focal elements requiring computation within the uncertainty domain is consistently less than the total number of focal elements. Therefore, proposing an uncertainty domain effectively reduces computation.

[0087] Table 2. Comparison of Relative Errors of Confidence Intervals for Various Methods in Example 1

[0088]

[0089] Example 2

[0090] Based on the above embodiments, this embodiment 2 further illustrates the invention using an engineering application structure. The structural reliability analysis method based on evidence theory and multi-point linearization includes the following steps:

[0091] S1. Specify the limit state function g(X) = S of the tubular cantilever beam to be analyzed. y -σ max Tubular cantilever beams are attached. Figure 3 S y Permissible yield strength, σ max Maximum stress at the lower surface of the fixed end. σmax The calculation formula is as follows:

[0092] Where σ x The calculation is as follows:

[0093] The first term in the formula is the stress caused by the axial force, and the second term is caused by the bending moment M. Where σ x A, M, l and τ zx These represent normal stress, area, bending moment, moment of inertia, and shear stress, respectively.

[0094]

[0095] The fixed parameters in the model are: L1 = 200 mm, L2 = 100 mm, θ1 = 30°, θ2 = 30°, P = 12000 N, T = 90 N·m. t, d, F1, and F2 are evidence variables, and their focal element intervals and BPA distribution information are shown in Table 3 below.

[0096] Table 3. BPA distribution table for evidence variables t, d, F1, F2 taking four intervals.

[0097]

[0098] S2. Based on the equal probability transformation, establish the uncertain evidence variable X. i With uncertain evidence variable X i The corresponding standard normal distribution variable Y i Equivalent relationship between them: X i =(φ(Y) i )-m i (A i,1 )-…-m i (A i,j-1 ))×(U i,j -L i,j ) / m i (A i,j )+L i,j , where φ(Y) i Y is a standard normal distribution i The cumulative probability distribution function, A i,j The evidence variable X is uncertain. i The j-th focal element, m i (A i,j ) is A i,j The basic credibility assignment function, focal element A i,j =[L i,j U i,j ], j = 1, 2, ..., L i,j and U i,j Corresponding to focal element Ai,j The lower and upper bounds;

[0099] S3. Obtain the most likely failure point Y using the first second moment method. * Reliability index β and most likely failure point Y * The unit vector α at that location Y Y * Transforming back to the original evidence space using an equal probability transformation yields the most probable failure point X in the original space. * ;

[0100] S4. Using the orthogonalization method, with α Y Construct an orthogonal matrix R, where the nth column of R is the unit vector α. Y The variable Y in the standard normal space is transformed into the variable Y' in the rotated standard normal space by the rotation transformation Y' = RY;

[0101] S5. In the rotated standard normal space, find linear points along the i = 1, 2, ..., n-1 directions, where each dimension can determine two linearized points Y′ on the limit state surface. i+ and Y′ i- Thus, 2(n-1) linearization points are found, and the linearization point Y′ in the rotated standard normal space is obtained. i+ and Y′ i- Transform to linearized point X' in the original evidence space i+ and X' i- ;

[0102] S6. In the evidence space, at the most likely point of failure X * and 2(n-1) linearized points X' i+ and X' i- Using the first-order Taylor expansion to approximate the true limit state function g(X), 2n-1 approximate hyperplanes are obtained:

[0103]

[0104] Among them, symbols This indicates finding the gradient of a function;

[0105] S7. Divide the absolute failure domain, absolute safety domain, and uncertainty domain of the focal element. The division rules are as follows: If the p-th joint focal element A Xp Center point C p With the hyperplane and The distance is greater than the focal radius, and Then the focal element belongs to the failure domain S1; if the p-th joint focal element A X Center point Cp With the hyperplane and The distance is greater than the focal radius, and and Then the focal element belongs to the safe region S2; in all other cases, the focal element belongs to the uncertain region S3.

[0106] S8. Perform extreme value analysis on each joint focal element in the uncertainty domain S3 to determine the confidence level Bel(S3) and the similarity level Pl(S3) in the uncertainty domain S3.

[0107] S9. Calculate the reliability and similarity of the structure: The reliability Bel(G) and similarity Pl(G) of the structure are the sum of the reliability and similarity of the safe region and the uncertain region: Bel(G) = Bel(S2) + Bel(S3), Pl(G) = Pl(S2) + Pl(S3), where the reliability and similarity of the safe region S2 are...

[0108] Table 4. Comparison of Relative Errors of Confidence Intervals for Various Methods in Example 2

[0109]

[0110] Table 4 shows the structural confidence intervals and relative errors calculated by the reliability analysis method disclosed in Example 2, along with those calculated by other methods. The confidence level and similarity obtained from the genetic algorithm are used as the exact solution. The permissible yield strength S is changed... y The magnitude yields a series of limit state surfaces. Among them, the reliability and similarity results of the method of this invention and the first-order approximate reliability analysis method, along with the error of the reference solution, all meet the engineering requirements of less than 10%, when the allowable yield strength S... y For values ​​greater than 200, the error is within 2%. However, the results of the second-order approximate reliability analysis method change with the allowable yield strength S. y The increase of gradually deviates from the reference solution, especially in S. y When the value is 400, the calculation error of the confidence level reaches 53.19%.

[0111] Extremum analysis for each joint focal element requires a total of 4 4 =256, but this invention proposes a range of uncertainty domains, significantly reducing the calculation ratio. Table 4 shows that the calculation ratios are all less than 0.6, especially in S. y =400, the number of joint focal elements in the uncertainty domain is 91, and the calculation ratio is 0.36.

[0112] Example 3

[0113] like Figure 4As shown, this embodiment provides a structural reliability analysis device based on evidence theory and multi-point linearization. The device includes an information construction module 401, an equivalence relation module 402, a spatial transformation module 403, an orthogonalization module 404, a linear point determination module 405, a Taylor expansion module 406, a domain partitioning module 407, an extreme value analysis module 408, and a reliability calculation module 409. The specific functions of each module are as follows:

[0114] Information construction module 401 specifies the limit state function g(X) that reflects the normal working capability or critical safe working state of a structure or product in the domain to be analyzed, where X = [X1, X2, ..., X...]. i ,...,X n ], X i Let X be the i-th dimension of uncertain evidence variable, and n be the dimension of the problem. Based on engineering experience or information from authoritative experts, construct the uncertain evidence variable X. i The identification framework and basic confidence assignment function m i (X i ), Joint Coke A X and joint basic credibility assignment function

[0115] Equivalence module 402 establishes the uncertain evidence variable X based on the equal probability transformation. i With uncertain evidence variable X i The corresponding standard normal distribution variable Y i Equivalent relationship between them: X i =(φ(Y) i )-m i (A i,1 )-…-m i (A i,j-1 ))×(U i,j -L i,j ) / m i (A i,j )+L i,j , where φ(Y) i Y is a standard normal distribution i The cumulative probability distribution function, A i,j The evidence variable X is uncertain. i The j-th focal element, m i (A i,j ) is A i,j The basic credibility assignment function, focal element A i,j =[L i,j U i,j ], j = 1, 2, ..., L i,j and U i,j Corresponding to focal element A i,j The lower and upper bounds;

[0116] The space transformation module 403 uses the first-order second-moment method to find the most likely failure point Y. * Reliability index β and most likely failure point Y * The unit vector α at that location Y Y * Transforming back to the original evidence space using an equal probability transformation yields the most probable failure point X in the original space. * ;

[0117] Orthogonalization module 404 employs an orthogonalization method, with α... Y Construct an orthogonal matrix R, where the nth column of R is the unit vector α. Y The variable Y in the standard normal space is transformed into the variable Y' in the rotated standard normal space by the rotation transformation Y' = RY;

[0118] The linear point determination module 405 finds linear points along the i = 1, 2, ..., n-1 axes in the rotated standard normal space, where each dimension can determine two linearized points Y′ on the limit state surface. i+ and Y′ i- Thus, 2(n-1) linearization points are found, and the linearization point Y′ in the rotated standard normal space is obtained. i+ and Y′ i- Transform to linearized point X' in the original evidence space i+ and X' i- ;

[0119] Taylor expansion module 406, in the evidence space, at the most likely point of failure X * and 2(n-1) linearized points X' i+ and X' i- Using the first-order Taylor expansion to approximate the true limit state function g(X), 2n-1 approximate hyperplanes are obtained:

[0120]

[0121]

[0122]

[0123] Among them, symbols This indicates finding the gradient of a function;

[0124] Domain partitioning module 407 partitions the absolute failure domain, absolute safety domain, and uncertainty domain of the focal element. The partitioning rules are as follows: If the p-th joint focal element A Xp Center point C p With the hyperplane and The distance is greater than the focal radius, and Then the focal element belongs to the failure domain S1; if the p-th joint focal element A X Center point C p With the hyperplane and The distance is greater than the focal radius, and and Then the focal element belongs to the safe region S2; in all other cases, the focal element belongs to the uncertain region S3.

[0125] The extreme value analysis module 408 performs extreme value analysis on each joint focal element in the uncertainty domain S3 to determine the confidence level Bel(S3) and the similarity level Pl(S3) in the uncertainty domain S3.

[0126] The reliability calculation module 409 calculates the reliability and similarity of the structure: the reliability Bel(G) and similarity Pl(G) of the structure are the sum of the reliability and similarity of the safe region and the uncertain region: Bel(G) = Bel(S2) + Bel(S3), Pl(G) = Pl(S2) + Pl(S3), where the reliability and similarity of the safe region S2 are...

[0127] The specific implementation of each module in this embodiment can be found in Embodiment 1 above, and will not be repeated here. It should be noted that the device provided in this embodiment is only illustrated by the division of the above functional modules. In practical applications, the above functions can be assigned to different functional modules as needed, that is, the internal structure can be divided into different functional modules to complete all or part of the functions described above.

[0128] Example 4

[0129] This embodiment provides a computer device, which can be a computer, such as... Figure 5 As shown, the processor 502, memory, input device 503, display 504, and network interface 505 are connected via system bus 501. The processor provides computing and control capabilities. The memory includes a non-volatile storage medium 506 and internal memory 507. The non-volatile storage medium 506 stores the operating system, computer programs, and database. The internal memory 507 provides an environment for the operation of the operating system and computer programs in the non-volatile storage medium. When the processor 502 executes the computer programs stored in the memory, it implements the structural reliability analysis method based on evidence theory and multi-point linearization proposed in Embodiment 1, as follows:

[0130] S1. Specify the limit state function g(X) in the domain to be analyzed, which reflects the normal operating capability or critical safe operating state of the structure or product, where X = [X1, X2, ..., X]. i ,...,X n ], X i Let X be the i-th dimension of uncertain evidence variable, and n be the dimension of the problem. Based on engineering experience or information from authoritative experts, construct the uncertain evidence variable X. i The identification framework and basic confidence assignment function m i (X i ), Joint Coke A X and joint basic credibility assignment function Among them, the basic credibility assignment function m i (X i The value reflects the degree of confidence that the evidence proposition is true, and is a basic probability number between [0,1].

[0131] S2. Based on the equal probability transformation, establish the uncertain evidence variable X. i With uncertain evidence variable X i The corresponding standard normal distribution variable Y i Equivalent relationship between them: X i =(φ(Y) i )-m i (A i,1 )-…-m i (A i,j-1 ))×(U i,j -L i,j ) / m i (A i,j )+L i,j , where φ(Y) i Y is a standard normal distribution i The cumulative probability distribution function, A i,j The evidence variable X is uncertain. i The j-th focal element, m i (A i,j ) is A i,j The basic credibility assignment function, focal element A i,j =[L i,j U i,j ], j = 1, 2, ..., L i,j and U i,j Corresponding to focal element A i,j The lower and upper bounds of , jiaoyuan A i,j It is a range, not a specific point;

[0132] S3. Obtain the most likely failure point Y using the first second moment method. * Reliability index β and most likely failure point Y* The unit vector α at that location Y Y * Transforming back to the original evidence space using an equal probability transformation yields the most probable failure point X in the original space. * ;

[0133] S4. Using the orthogonalization method, with α Y Construct an orthogonal matrix R, where the nth column of R is the unit vector α. Y The variable Y in the standard normal space is transformed into the variable Y' in the rotated standard normal space by the rotation transformation Y' = RY;

[0134] S5. In the rotated standard normal space, find linear points along the i = 1, 2, ..., n-1 directions, where each dimension can determine two linearized points Y′ on the limit state surface. i+ and Y′ i- Thus, 2(n-1) linearization points are found, and the linearization point Y′ in the rotated standard normal space is obtained. i+ and Y′ i- Transform to linearized point X' in the original evidence space i+ and X' i- ;

[0135] S6. In the evidence space, at the most likely point of failure X * and 2(n-1) linearized points X' i+ and X' i- Using the first-order Taylor expansion to approximate the true limit state function g(X), 2n-1 approximate hyperplanes are obtained:

[0136]

[0137]

[0138]

[0139] Among them, symbols This indicates finding the gradient of a function;

[0140] S7. Divide the absolute failure domain, absolute safety domain, and uncertainty domain of the focal element. The division rules are as follows: If the p-th joint focal element A Xp Center point C p With the hyperplane and The distance is greater than the focal radius, and Then the focal element belongs to the failure domain S1; if the p-th joint focal element A X Center point C p With the hyperplane and The distance is greater than the focal radius, and and Then the focal element belongs to the safe region S2; in all other cases, the focal element belongs to the uncertain region S3.

[0141] S8. Perform extreme value analysis on each joint focal element in the uncertainty domain S3 to determine the confidence level Bel(S3) and the similarity level Pl(S3) in the uncertainty domain S3.

[0142] S9. Calculate the reliability and similarity of the structure: The reliability Bel(G) and similarity Pl(G) of the structure are the sum of the reliability and similarity of the safe region and the uncertain region: Bel(G) = Bel(S2) + Bel(S3), Pl(G) = Pl(S2) + Pl(S3), where the reliability and similarity of the safe region S2 are...

[0143] Example 5

[0144] This embodiment provides a storage medium, which is a computer-readable storage medium, storing a computer program. When the computer program is executed by a processor, it implements the structural reliability analysis method based on evidence theory and multi-point linearization of Embodiment 1 above, as follows:

[0145] S1. Specify the limit state function g(X) in the domain to be analyzed, which reflects the normal operating capability or critical safe operating state of the structure or product, where X = [X1, X2, ..., X]. i ,…,X n ], X i Let X be the i-th dimension of uncertain evidence variable, and n be the dimension of the problem. Based on engineering experience or information from authoritative experts, construct the uncertain evidence variable X. i The identification framework and basic confidence assignment function m i (X i ), Joint Coke A X and joint basic credibility assignment function Among them, the basic credibility assignment function m i (X i The value reflects the degree of confidence that the evidence proposition is true, and is a basic probability number between [0,1].

[0146] S2. Based on the equal probability transformation, establish the uncertain evidence variable X. i With uncertain evidence variable X i The corresponding standard normal distribution variable Y i Equivalent relationship between them: X i =(φ(Y)i )-m i (A i,1 )-…-m i (A i,j-1 ))×(U i,j -L i,j ) / m i (A i,j )+L i,j , where φ(Y) i Y is a standard normal distribution i The cumulative probability distribution function, A i,j The evidence variable X is uncertain. i The j-th focal element, m i (A i,j ) is A i,j The basic credibility assignment function, focal element A i,j =[L i,j U i,j ], j = 1, 2, ..., L i,j and U i,j Corresponding to focal element A i,j The lower and upper bounds of , jiaoyuan A i,j It is a range, not a specific point;

[0147] S3. Obtain the most likely failure point Y using the first second moment method. * Reliability index β and most likely failure point Y * The unit vector α at that location Y Y * Transforming back to the original evidence space using an equal probability transformation yields the most probable failure point X in the original space. * ;

[0148] S4. Using the orthogonalization method, with α Y Construct an orthogonal matrix R, where the nth column of R is the unit vector α. Y The variable Y in the standard normal space is transformed into the variable Y' in the rotated standard normal space by the rotation transformation Y' = RY;

[0149] S5. In the rotated standard normal space, find linear points along the i = 1, 2, ..., n-1 directions, where each dimension can determine two linearized points Y′ on the limit state surface. i+ and Y′ i- Thus, 2(n-1) linearization points are found, and the linearization point Y′ in the rotated standard normal space is obtained. i+ and Y′ i- Transform to linearized point X' in the original evidence space i+ and X' i- ;

[0150] S6. In the evidence space, at the most likely point of failure X * and 2(n-1) linearized points X' i+ and X' i- By using the first-order Taylor expansion to approximate the true limit state function g(X), 2n-1 approximate hyperplanes are obtained;

[0151] S7. Divide the absolute failure domain, absolute safety domain, and uncertainty domain of the focal element. The division rules are as follows: If the p-th joint focal element A Xp Center point C p With the hyperplane and The distance is greater than the focal radius, and Then the focal element belongs to the failure domain S1; if the p-th joint focal element A X Center point C p With the hyperplane and The distance is greater than the focal radius, and and Then the focal element belongs to the safe region S2; in all other cases, the focal element belongs to the uncertain region S3.

[0152] S8. Perform extreme value analysis on each joint focal element in the uncertainty domain S3 to determine the confidence level Bel(S3) and the similarity level Pl(S3) in the uncertainty domain S3.

[0153] S9. Calculate the credibility and similarity of the structure: The credibility Bel(G) and similarity Pl(G) of the structure are the sum of the credibility and similarity of the safe region and the uncertain region: Bel(G) = Bel(S2) + Bel(S3), Pl(G) = Pl(S2) + Pl(S3).

[0154] The storage medium described in this embodiment can be a disk, optical disk, computer memory, random access memory (RAM), USB flash drive, portable hard drive, etc.

[0155] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A structural reliability analysis method based on evidence theory and multi-point linearization, characterized in that, The structural reliability analysis method includes the following steps: S1. Specify the limit state function of the tubular cantilever beam to be analyzed. ,in, Indicates the permissible yield strength. This indicates the maximum stress at the lower surface of the fixed end. , Let n be the i-th dimension of uncertain evidence variable, and n be the dimension of the problem. Based on engineering experience or information from authoritative experts, uncertain evidence variables are constructed. Recognition framework, single variable Basic credibility assignment function United Coking Coal and joint basic credibility assignment function ; The calculation formula is as follows: , The calculation formula is as follows: In the formula , , , and These represent normal stress, area, bending moment, moment of inertia, and shear stress, respectively. S2. Based on the equal probability transformation, establish uncertain evidence variables. With uncertain evidence variables Corresponding standard normal distribution variable Equivalent relationship between them: ,in, It is a standard normal distribution The cumulative probability distribution function, Uncertain evidence variables The Each jiao yuan, yes The basic credibility assignment function, focal element , , and Corresponding to focal lengths The lower and upper bounds; S3. Obtain the most likely failure point using the first-order second-moment method. Reliability indicators and the most likely failure point unit vector at ,Will Transforming back to the original evidence space using an equal probability transformation yields the most probable failure point in the original space. ; S4. Use the orthogonalization method to... Construct an orthogonal matrix orthogonal matrix The nth column is a unit vector Through rotation transformation Variables in standard normal space Variables transformed to rotated standard normal space ; S5. In the rotated standard normal space, find linear points along the i=1, 2, ..., n-1 directions respectively, where each dimension can determine two linearized points on the limit state surface. and to find Linearization points, rotating the linearization points in the standard normal space. and Transform to linearized points in the original evidence space and ; S6. In the evidence space, at the point of most likely failure. and linearization points and The true limit state function is approximated using a first-order Taylor expansion. ,get Approximate hyperplanes: ， ， ， Among them, the symbol " " indicates finding the gradient of the function; S7. Divide the absolute failure domain, absolute safety domain, and uncertainty domain of the focal element according to the following rules: If the first... A joint coke element center point With the hyperplane , and The distance is greater than the focal radius, and , , Then the focal element belongs to the failure domain. If the first A joint coke element center point With the hyperplane , and The distance is greater than the focal radius, and , and Then the focal element belongs to the security domain. In all other cases, the focal element belongs to the uncertain region. ; S8. For the uncertain region Extremum analysis is performed on each joint focal element within the domain to determine the uncertainty region. Credibility within Similarity ; S9. Calculating the reliability and fidelity of a structure: Reliability of the structure Similarity The confidence level and similarity are summed by the confidence level and similarity in the secure region and the uncertain region: , Among them, security domain Credibility and Realism .

2. The structural reliability analysis method based on evidence theory and multi-point linearization according to claim 1, characterized in that, The structural reliability analysis method can also be applied to the reliability analysis and evaluation of structures or products in fields such as civil engineering, mechanical and electronic engineering, or aerospace.

3. The structural reliability analysis method based on evidence theory and multi-point linearization according to claim 1, characterized in that, In step S5, linear points are found along the i=1, 2, ..., n-1 axes in the rotated standard normal space, where each dimension can determine two linearized points on the limit state surface. and to find The process of linearizing a point is as follows: S5-1, Set initial value Rotation of the standard normal space vector , ,make ; S5-2, Assumptions and The initial values ​​are respectively , ; S5-3, respectively for and Random selection , ,make , , and Take the last component separately and Two interpolation points are obtained, and the limit state function values ​​of these two interpolation points are calculated in the original evidence space. ; S5-4, ,calculate and The corresponding interpolation points and The corresponding limit state function value in the evidence space ; S5-5, , , , Repeat S5-4 and S5-5 until... , for Absolute value For small positive numbers, the linearization point is obtained. and ; S5-6, Update Values: Calculated separately ,in, , , Pick The minimum value in; S5-7, if , Repeat steps S5-2 to S5-6; S5-8 Repeat steps S5-2 to S5-5 along the i=1, 2, ..., n-1 lines respectively to obtain the final result. linearization points and .

4. The structural reliability analysis method based on evidence theory and multi-point linearization according to claim 1, characterized in that, In step S8, the uncertainty region is... Extremum analysis is performed on each joint focal element within the domain to determine the uncertainty region. Credibility within Similarity The process is as follows: S8-1、 、 ； S8-2, Calculate the focal element with the smallest range of values. The maximum value of the approximate hyperplane when it is the limit state function ; S8-3, Calculating the first [unclear] in the uncertainty region A joint coking element The maximum and minimum values, assuming , and In each joint focal element The maxima on are respectively , , The minimum values ​​are respectively , , They each constitute a set of maxima. and the set of minimum values ; S8-4, Assumptions Represents a set The maximum value, Represents a set The maximum value, that is: ， Assumption Represents a set The minimum value, Represents a set The minimum value, that is: ， in and Indicates the maximum and minimum values; S8-5, For the first in the uncertain domain A joint coke element : (1) If And together with Jiao Yuan of ,but ;like And together with Jiao Yuan of ,but ; (2) If And together with Jiao Yuan of ,but ;like And together with Jiao Yuan of ,but .

5. A structural reliability analysis device based on the structural reliability analysis method based on evidence theory and multi-point linearization as described in any one of claims 1 to 4, characterized in that, The reliability analysis device includes: The information construction module specifies the limit state function of a tubular cantilever beam to be analyzed. ,in, Indicates the permissible yield strength. This indicates the maximum stress at the lower surface of the fixed end. , Let n be the i-th dimension of uncertain evidence variable, and n be the dimension of the problem. Based on engineering experience or information from authoritative experts, uncertain evidence variables are constructed. Recognition framework, single variable Basic credibility assignment function United Coking Coal and joint basic credibility assignment function ; The calculation formula is as follows: , The calculation formula is as follows: In the formula , , , and These represent normal stress, area, bending moment, moment of inertia, and shear stress, respectively. The equivalence relation module establishes uncertain evidence variables based on equal probability transformations. With uncertain evidence variables Corresponding standard normal distribution variable Equivalent relationship between them: ,in, It is a standard normal distribution The cumulative probability distribution function, Uncertain evidence variables The Each jiao yuan, yes The basic credibility assignment function, focal element , , and Corresponding to focal lengths The lower and upper bounds; The space transformation module uses the first-order second-moment method to find the most likely failure point. Reliability indicators and the most likely failure point unit vector at ,Will Transforming back to the original evidence space using an equal probability transformation yields the most probable failure point in the original space. ; The orthogonalization module employs an orthogonalization method to... Construct an orthogonal matrix orthogonal matrix The nth column is a unit vector Through rotation transformation Variables in standard normal space Variables transformed to rotated standard normal space ; The linear point determination module finds linear points along the i=1, 2, ..., n-1 axes in the rotated standard normal space, where each dimension can determine two linearized points on the limit state surface. and to find Linearization points, rotating the linearization points in the standard normal space. and Transform to linearized points in the original evidence space and ; Taylor expansion module, in the evidence space, at the point of most likely failure. and linearization points and The true limit state function is approximated using a first-order Taylor expansion. ,get Approximate hyperplanes: ， ， ， Among them, the symbol " " indicates finding the gradient of the function; The domain partitioning module divides the focal element into the absolute failure domain, the absolute safety domain, and the uncertainty domain. The partitioning rules are as follows: If the first... A joint coke element center point With the hyperplane , and The distance is greater than the focal radius, and , , Then the focal element belongs to the failure domain. If the first A joint coke element center point With the hyperplane , and The distance is greater than the focal radius, and , and Then the focal element belongs to the security domain. In all other cases, the focal element belongs to the uncertain region. ; The extreme value analysis module is used for the uncertainty domain. Extremum analysis is performed on each joint focal element within the domain to determine the uncertainty region. Credibility within Similarity ; The reliability calculation module calculates the reliability and similarity of a structure: the reliability of the structure. Similarity The confidence level and similarity are summed by the confidence level and similarity in the secure region and the uncertain region: , Among them, security domain Credibility and Realism .

6. A computer device comprising a processor and a memory for storing a processor-executable program, characterized in that, When the processor executes the program stored in the memory, it implements the structural reliability analysis method according to any one of claims 1-4.

7. A storage medium storing a program, characterized in that, When the program is executed by the processor, it implements the structural reliability analysis method according to any one of claims 1-4.

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