Non-probabilistic credible reliability analysis method for composite material structure under truncated normal prior
By adopting truncated normal prior distribution and Bayesian theory in composite material structures, combining Hashin failure criterion and Taylor series expansion method, a non-probability credibility reliability analysis model is established, which solves the problem of multi-source uncertainty in composite material structures, and improves the accuracy of structural reliability analysis and engineering applicability.
Patent Information
- Application Number
- CN202510069418.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-01-16
- Publication Date
- 2025-05-06
AI Technical Summary
In the design and reliability analysis of composite material structures, traditional methods are difficult to effectively deal with multi-source uncertainty, resulting in uncertainty in structural mechanical responses and affecting the safety of the structure.
A truncated normal prior distribution is adopted, based on Bayesian theory and interval model, combined with Hashin failure criterion and Taylor series expansion method, a non-probability credibility and reliability analysis model is established to deal with the balance between conservatism and credibility of interval model.
By introducing new sample points to update interval parameters, a more reliable indicator is established that is more in line with the actual situation, which improves the accuracy of engineering applicability and structural reliability analysis.
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Abstract
Description
Technical Field
[0001] The present invention relates to the field of static strength analysis of aircraft composite materials considering uncertainty, and in particular to a non-probabilistic credible reliability analysis method for composite materials under truncated normal prior, which provides an important theoretical basis for the design and reliability analysis of aircraft composite materials structures. Background Art
[0002] Composite materials are multiphase solid materials obtained by artificially combining two or more substances with different physical and chemical properties. Compared with traditional metal materials, composite materials have the advantages of high stiffness, high strength, corrosion resistance, good vibration resistance, light weight, and strong designability. They are widely used in the aerospace field, such as aircraft, space stations and other structural systems. The use of composite materials is increasing. However, the anisotropy and discontinuity of composite materials make their structural failure modes more complicated, and the research on mechanical properties and failure modes faces many obstacles. At present, composite materials are gradually applied to large and complex structural systems, and they have long service time and harsh service environment. They are easily corroded by the environment, which leads to material aging. In addition, the coupling effect of load factors such as long-term effect of load, fatigue effect and mutation effect will inevitably cause damage accumulation and resistance attenuation, and then face the problems of difficulty in distinguishing and predicting failure modes. Therefore, it is very important to establish a refined analysis method for the mechanical properties of composite structures.
[0003] In traditional structural analysis, most designers regard structural parameters and external loads as deterministic quantities, and use deterministic models and deterministic methods to analyze and describe the performance of the structure. But in fact, in the design of engineering structures, especially in the analysis and design of composite materials, uncertainty is widespread. Specifically, in many cases, the load or service environment borne by the structure is uncertain; in addition, due to various errors and uncertainties that may exist in the construction, processing, assembly, measurement and modeling of the structure, the physical parameters, geometric parameters, boundaries and initial conditions of the structure and even the calculation model itself are also uncertain. The above uncertainties usually lead to uncertainty in the mechanical response of the aircraft structure, which in turn threatens the safety of the structure and flight safety. Therefore, in order to ensure safety, uncertainty factors must be taken into account and structural reliability must be considered.
[0004] Traditional probabilistic reliability analysis methods need to assume the probability density function of the uncertainty parameters in order to obtain the final reliability. However, in the aerospace field, the cost of a single test is relatively high, the number of samples that can be used is small, and there is a lack of real statistical information on the uncertain parameters. Summary of the invention
[0005] In order to solve the above technical problems, the present invention provides a non-probabilistic credible reliability analysis method for composite structures under truncated normal prior. In the case of small samples and poor information, the uncertainty of aircraft composite material structure material parameters is fully considered. Based on Bayesian theory, the truncated normal distribution is used as the prior distribution of the interval radius, and new sample points are introduced to update the interval radius. The connection between the non-probabilistic quantization set and the credible level is constructed. Combined with the Hashin failure criterion, the Taylor series expansion method is used to solve the critical failure equation of the system, and a non-probabilistic credible reliability analysis model containing interval uncertainty is established. The obtained results contain credibility indicators, which are more in line with the actual situation and have stronger engineering applicability.
[0006] In order to achieve the above object, the present invention adopts the following technical scheme: A non-probabilistic credible reliability analysis method for composite structures under truncated normal prior is used for static strength and credible reliability analysis of aircraft composite structures under multi-source uncertainty, including the following steps: Step 1: Derive the stress-strain relationship of the composite laminate structure and use the non-probability interval method to quantify the uncertainty of material parameters in the composite structure under finite sample conditions; Step 2: Based on Bayesian theory, the interval parameter is considered to be an uncertainty parameter that obeys the truncated normal distribution. Sample points are introduced for updating, and the Markov chain Monte Carlo algorithm is used to construct the connection between the non-probability quantization set and the credibility. Step 3: Considering different failure modes, the Hashin failure criterion is used as the failure criterion to carry out static strength failure analysis of aircraft composite structures and construct the limit state equation; Step 4: Use the Taylor series expansion method to perform uncertainty propagation analysis and solve the Hashin coefficient response interval corresponding to the non-probability interval of the input uncertainty parameter; Step 5: Combined with the stress intensity interference theory, the response interval is compared with the limit state equation to establish a non-probabilistic Bayesian credible reliability analysis model.
[0007] Beneficial effects: The present invention establishes an interval uncertainty quantification and reliability analysis model with credibility for aircraft composite material structures under multi-source uncertainty. Aiming at the problem of inaccurate reliability assessment caused by insufficient sample information of composite material input variables, based on Bayesian theory and interval model, the interval radius is considered to be an uncertain parameter that obeys a truncated normal prior distribution, and is updated in combination with known limited sample data points. The relationship between the interval radius and the credibility level is obtained using the MCMC numerical method, and the Taylor series expansion method is used in combination with the Hashin criterion to calculate the credibility of the composite material structure under different failure modes. The simulation results show that the non-probabilistic Bayesian model proposed in the present invention that obeys a truncated normal distribution prior can better handle the balance between the conservatism and credibility of the interval model. With the introduction of new sample points, the reliability will continue to improve, providing a larger design space for the design of high-performance structures. BRIEF DESCRIPTION OF THE DRAWINGS
[0008] Figure 1 This is a flow chart of the credible reliability analysis of a composite material laminate structure with holes according to the present invention; Figure 2 It is a schematic diagram of the geometric parameters and ply angles of a typical composite material laminate with holes; Figure 3 It is the posterior distribution and posterior interval diagram of the longitudinal elastic modulus of the laminate structure under different prior distributions; Figure 4 is the posterior distribution and posterior interval diagram of the transverse elastic modulus of the laminate structure under different prior distributions; Figure 5 It is the posterior distribution and posterior interval diagram of shear elastic modulus of laminate structure under different prior distributions; Figure 6 is the posterior distribution and posterior interval diagram of Poisson's ratio of laminate structure under different prior distributions; Figure 7 is the posterior distribution and posterior interval diagram of shear strength of laminate structure under different prior distributions; Figure 8 is the posterior distribution and posterior interval diagram of transverse tensile strength of laminate structure under different prior distributions; Fig. 9 It is a graph showing the change of the updated value of the interval radius of the longitudinal elastic modulus of the laminate structure with the credibility under different prior distributions; Fig.10 It is a graph showing the variation of non-probabilistic reliability with credibility under different prior distributions of laminate structure. DETAILED DESCRIPTION
[0009] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all 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 protection scope of the present invention.
[0010] like Figure 1 As shown, the present invention proposes a non-probabilistic credible reliability analysis method for composite structures under truncated normal prior, comprising the following steps: Step (1) uses interval mathematical methods to obtain a non-probabilistic quantitative set of aircraft composite structures under multi-source uncertainties such as material parameters and external loads.
[0011] Generally speaking, a single-layer plate can be regarded as an orthotropic material. According to the basic knowledge of macroscopic mechanics of composite materials, when the elastic mechanical parameters of the composite material are certain values, the stress-strain relationship of the composite single-layer plate can be written as follows: (1) Among them, "1" represents the longitudinal fiber direction, "2" represents the in-plane transverse direction, and "3" represents the thickness direction. The stress tensor and strain tensor are expressed using the Voigt method. , ,in, is the linear strain in the “1” direction, is the linear strain in the “2” direction, is the linear strain in the “3” direction, is the shear strain in the “2” and “3” planes, is the shear strain in the “1” and “3” planes, is the shear strain in the “1” and “2” planes, is the normal stress in the “1” direction, is the normal stress in the “2” direction, is the normal stress in the “3” direction, is the shear stress in the “2” and “3” planes, is the shear stress in the “1” and “3” planes, is the shear stress in the “1” and “2” planes. is the flexibility matrix, which is determined by the elastic parameters of the single-layer plate, where is the elastic modulus in the i direction, is the Poisson's ratio, is the shear modulus of the ij plane, where the ij plane represents the "2" "3", "1" "3", "1" "2" planes, and is measured by tensile or shear tests of composite materials; For plane stress problems , the stress-strain relationship becomes: (2) According to the symmetry of the stiffness matrix, Therefore, for the plane stress problem of a single-layer composite plate, there are four independent elastic parameters: , , , , is regarded as an uncertain quantity.
[0012] Due to various uncertainties in the manufacturing and external service processes, the mechanical properties of composite materials are dispersed. There are many ways to describe uncertainty based on the amount of sample data. Since the amount of sample data for aircraft is relatively small, it is appropriate to use non-probability interval mathematical methods to describe uncertainty. According to interval mathematical theory, uncertainty parameters can be expressed as interval variables: (3) in, is the interval vector of uncertain mechanical parameters. and are the corresponding lower and upper bounds respectively. is the i-th component of the interval vector. and are the lower and upper bounds respectively. m is the number of uncertain mechanical parameters. According to interval mathematics theory, equation (3) can be rewritten in the form of center value and radius: (4) (5) Among them, the nominal value of the interval is , the interval radius is Considering the elastic parameters of fiber reinforced composites , , , , strength parameters is the uncertainty parameter, that is: (6) Therefore, the non-probabilistic interval method can be used to quantify the uncertain mechanical parameters of composite laminate structures into interval variables.
[0013] Step (2) Based on Bayesian theory, new sample points are introduced to update the interval parameters and establish the connection between the non-probabilistic quantitative set and credibility.
[0014] First, we assume that the parameters of the non-probabilistic quantization model obey a certain prior distribution, and then use the known information to introduce new sample points to update the posterior distribution of the model parameters. Without loss of generality, the distribution parameter-interval radius in the interval quantization model is considered uncertain. According to Bayesian theory, the posterior probability density function of the interval parameter can be expressed as: (7) in, is the interval parameter, is a sample of size n, is the posterior probability density function, is the prior probability density, is the sampling density of the sample, is a normalization constant representing the marginal density of the sample.
[0015] We will now explore each component of Bayesian theory in more detail, starting with the prior density. It reflects the knowledge of the parameters before analyzing the data. There are usually three prior distribution choices: standard non-informative prior distribution; informative conjugate distribution; more general informative prior distribution. The conjugate prior distribution should not be specified just for computational convenience. If a conjugate prior distribution that provides sufficient representation of available information cannot be found before the experiment, using a non-informative prior distribution as the prior distribution of the distribution parameters helps to weaken the influence of subjectivity. Therefore, in order to reliably model the reliability of the structure, in the absence of prior information, the present invention will use a typical non-informative prior distribution-truncated normal distribution as the prior distribution of the interval radius.
[0016] The probability density function of the normal distribution is: (8) in, is the probability density function of the normal distribution, is the mean, is the standard deviation, and are constants. exp is an exponential function.
[0017] The probability density function of the truncated normal distribution is: (9) in, is the probability density function of the truncated normal distribution, For the upper and lower bounds, , is the distribution function of the normal distribution and is a constant.
[0018] The interval method assumes that all samples are equally likely to be distributed in the interval, and samples are drawn from a uniform distribution with a sampling density of: (10) Marginal density It only represents a normalization constant, which is: (11) Based on Bayesian theory, the posterior distribution probability density function is: (12) Given credibility , integrating the posterior distribution is: (13) in, For a given credibility, is the numerical solution of the exponential integral, is the updated value of the interval parameter under the given sample and credibility conditions. Since the exponential integral has no analytical solution, the Markov chain Monte Carlo method is needed to numerically solve the upper bound of the integral.
[0019] Markov chain Monte Carlo methods are a class of general computational methods used to generate samples from a posterior distribution. The basic goal of Markov chain Monte Carlo methods is to simulate values from the posterior distribution of a parameter vector. Then, based on these simulated values, possible parameter values or functions of parameter values are inferred. The specific implementation process is as follows: The first step is to assume is a q-dimensional real parameter vector. The first step is to generate a candidate point , defining the proposal density Used from generate , a common generation method is to generate the i-th component Add a mean zero normal deviation: (14) in, is the number of iteration steps, It is The candidate point vector of the step, is the standard deviation of the normal distribution, is an arbitrary constant.
[0020] The second step is to calculate the probability of the candidate value being accepted as the next simulated value of the sequence based on the candidate point, and define the acceptance probability r as: (15) Among them, data represents sample information, and represents the likelihood function density, and represents the proposal density, and the acceptance probability r represents the product of the ratio of the posterior density evaluated at the candidate parameter value to the current parameter value and the ratio of the proposal density of the current point and the candidate point.
[0021] The third step is to draw a random variable that follows a uniform distribution (0,1) , and Compared with r, if , then accept the candidate value and set ,if , then reject the candidate value and set .
[0022] Step (3) Use the Hashin failure criterion as the failure criterion to conduct static strength failure analysis of aircraft composite structures and construct the limit state equation.
[0023] There are various strength theories for fiber reinforced composite materials. The present invention adopts the Hashin failure criterion that distinguishes the failure modes to perform refined static strength analysis of composite material structures. For different failure modes, the Hashin failure criterion is described as follows: (16) in, They are respectively the stress along the fiber direction, the stress perpendicular to the fiber direction, and the shear stress. They are the Hashin failure coefficients of fiber tension, fiber compression, matrix tension, matrix compression, and matrix shear, respectively. If they are greater than 1, it is considered that the corresponding failure has occurred. They are fiber tensile strength, fiber compressive strength, matrix tensile strength, matrix compressive strength, and matrix shear strength, respectively, which can be measured through composite material tensile test or shear failure test and are regarded as uncertain quantities.
[0024] Step (4) uses the Taylor series expansion method to perform uncertainty propagation analysis and solve the response interval of the Hashin failure coefficient for different failure modes.
[0025] Assume the response function of the structure is: (17) In the present invention, the response function is the failure coefficient of each Hashin in formula (16). Given that the uncertainty level of the parameters is usually small and the response function is a monotonic function of the uncertainty parameters, in order to obtain the interval of the structural response function, the response function can be expanded by a first-order Taylor expansion, which uses the information of the nominal value point of the interval where the uncertainty parameters are located and / or the interval endpoints to evaluate the response limit of the response function in the entire interval: (18) in, represents the number of uncertain variables, are the upper and lower bounds of the interval of the response function, respectively. is the interval nominal value point response function value, It is the first-order derivative of the response function to each uncertainty variable, which can be obtained by combining composite material mechanics and finite element analysis method, and i is the index value.
[0026] Step (5) combines the stress intensity interference theory, compares the response interval with the limit state equation, and establishes a non-probabilistic Bayesian credible reliability analysis model.
[0027] Based on the interval stress-strength interference model, the reliability of the Hashin coefficient of different failure modes is obtained as follows: The response interval [ , ], and then compared with the critical value of the Hashin coefficient (equal to 1), the credibility can be obtained. The probability of structural failure for: ; (19) Credibility is The structural reliability is: (20) Example: In order to more fully understand the characteristics of the invention and its applicability to actual engineering, the present invention first establishes a credible reliability analysis model for a composite laminate structure with holes under static load, uses random sampling to obtain samples, calculates credible interval parameters based on Bayesian theory, uses Taylor series expansion method to perform uncertainty propagation analysis, and constructs the limit state equation according to the Hashin failure criterion, so as to calculate its credible reliability without failure under tensile load, and compares it with other literature methods.
[0028] like Figure 2 As shown, the geometric parameters of the laminate are as follows: the laminate is 100 mm long, 50 mm wide, and the ply angle is , there is a hole with a diameter of 10mm in the center of the plate. The mechanical properties of the laminate are as follows: longitudinal elastic modulus is 193GPa, transverse elastic modulus is 11.7GPa, Poisson's ratio is 0.35, shear modulus is 4.34GPa, longitudinal tensile strength is 2741MPa, longitudinal compressive strength is 1513MPa, transverse tensile strength is 70.4MPa, transverse compressive strength is 70.4MPa, and shear strength is 105MPa. The unit type is a two-dimensional shell unit, and a fixed support constraint is applied to one end of the laminate, and a displacement load is applied to the other end.
[0029] The center of the interval is considered fixed, and the interval radius is considered to be an uncertain parameter to be updated. Based on the non-probabilistic credible Bayesian reliability model update method, new sample points are introduced to update the interval radius. Different from other literatures that assume the prior distribution is uniform distribution and Pareto distribution, the method proposed in this invention adopts truncated normal distribution as the prior distribution of the parameter to be updated. Given a series of credibility, under different prior distributions, the posterior distribution and posterior interval of the updated elastic parameter interval radius are respectively as follows: Figure 3-Figure 8 As shown, it can be seen that, except for the Pareto prior distribution, the interval radius after the introduction of new samples is larger than the original interval radius and can envelop all sample points. Although the use of Pareto distribution in the prior distribution makes the update process easier to handle mathematically, it will cause the updated interval to no longer contain all sample points, and the accuracy will be reduced. Fig. 9 The radius of the interval of uncertainty material parameters after the three prior distributions are updated changes with the credibility level. As the credibility level increases, the updated interval radius of the three prior distributions will increase, which is consistent with actual engineering experience, that is, ensuring a high credibility level will lead to a broad estimate of the uncertainty and increase conservatism. In addition, the interval radius after the prior distribution is a truncated normal distribution is smaller than that of a uniform distribution, and can include all sample points, indicating that the method proposed in the present invention can reduce conservatism while ensuring the accuracy of the quantitative uncertainty. By changing the uncertainty parameters and combining the finite element analysis method, the first-order derivative value of the Taylor series expansion method is obtained, and then the uncertainty propagation analysis can be performed to obtain the response boundary of the Hashin failure coefficient, and the credible reliability is calculated based on the stress intensity interference theory. The results show that the main failure mode of the structure is matrix tensile failure. Fig.10 It is a graph showing the variation of non-probabilistic reliability of no matrix tensile failure under different prior distributions with the credible level. The reliability of the prior distribution using Pareto distribution is always equal to 1 and is insensitive to the credible level. As the credible level increases, the reliability of the prior distribution using truncated normal distribution and uniform distribution gradually decreases, which is consistent with actual engineering experience. Under the same credible level, the reliability of the prior distribution using truncated normal distribution is greater than that of the uniform distribution, indicating that the credible Bayesian reliability model proposed in the present invention can more accurately characterize the uncertainty of interval parameters and improve the reliability of the non-probabilistic reliability model.
[0030] The above are only specific steps of the present invention and do not constitute any limitation to the protection scope of the present invention; it can be extended to other reliability analysis fields of composite materials structures, and any technical solutions formed by equivalent transformation or equivalent replacement fall within the protection scope of the present invention.
[0031] The present invention does not elaborate on some of the well-known technologies that belong to those skilled in the art.
Claims
1. A non-probabilistic credible reliability analysis method for composite structures under truncated normal priors, used for static strength and credible reliability analysis of aircraft composite structures under multi-source uncertainty, characterized by: The steps include: Step 1: Derive the stress-strain relationship of the composite laminate structure and use the non-probability interval method to quantify the uncertainty of material parameters in the composite structure under finite sample conditions; Step 2: Based on Bayesian theory, the interval parameter is considered to be an uncertainty parameter that obeys the truncated normal distribution. Sample points are introduced for updating, and the Markov chain Monte Carlo algorithm is used to construct the connection between the non-probability quantization set and the credibility. Step 3: Considering different failure modes, the Hashin failure criterion is used as the failure criterion to carry out static strength failure analysis of aircraft composite structures and construct the limit state equation; Step 4: Use the Taylor series expansion method to perform uncertainty propagation analysis and solve the Hashin coefficient response interval corresponding to the non-probability interval of the input uncertainty parameter; Step 5: Combined with the stress intensity interference theory, the response interval is compared with the limit state equation to establish a non-probabilistic Bayesian credible reliability analysis model.
2. The non-probabilistic credible reliability analysis method for composite structures under truncated normal prior according to claim 1 is characterized in that: The first step comprises: The single-layer plate is regarded as an orthotropic material. According to the macroscopic mechanics of composite materials, when the elastic mechanical parameters of the composite material are determined, the stress-strain relationship of the composite single-layer plate is written as follows: （1） Among them, "1" represents the longitudinal fiber direction, "2" represents the in-plane transverse direction, and "3" represents the thickness direction. The stress tensor and strain tensor are expressed using the Voigt method. , ,in, is the linear strain in the "1" direction, is the linear strain in the "2" direction, is the linear strain in the "3" direction, is the shear strain in the "2" and "3" planes, is the shear strain in the "1" and "3" planes, is the shear strain in the "1" and "2" planes, is the normal stress in the "1" direction, is the normal stress in the "2" direction, is the normal stress in the "3" direction, is the shear stress in the "2" and "3" planes, is the shear stress in the "1" and "3" planes, is the shear stress in the "1" and "2" planes. is the flexibility matrix, which is determined by the elastic parameters of the single-layer plate, where is the elastic modulus in the i direction, is the Poisson's ratio, is the shear modulus of the ij plane, where the ij plane represents the "2" "3", "1" "3", and "1" "2" planes, and is measured by tensile or shear tests of composite materials; For plane stress problems , the stress-strain relationship becomes: （2） According to the symmetry of the stiffness matrix, ; For the plane stress problem of a single-layer composite plate, there are four independent elastic parameters , , , , regarded as an uncertain quantity; According to interval mathematics theory, the uncertainty parameter is expressed as an interval variable: （3） in, is the interval vector of uncertain mechanical parameters, and are the corresponding lower and upper bounds respectively; is the i-th component of the interval vector, and are the lower and upper bounds respectively, and m is the number of uncertain mechanical parameters. According to interval mathematics theory, formula (3) can be rewritten in the form of center value and radius as follows: （4） （5） Among them, the nominal value of the interval is , the interval radius is ; Considering the elastic parameters of fiber reinforced composites , , , , strength parameters is the uncertainty parameter, that is: （6） Therefore, the non-probabilistic interval method is used to quantify the uncertain mechanical parameters of the composite laminate structure into interval variables.
3. The non-probabilistic credible reliability analysis method for composite structures under truncated normal prior according to claim 1 is characterized in that: The second step includes: First, it is assumed that the parameters of the non-probabilistic quantization model obey a certain prior distribution, and then new sample points are introduced using known information to update the posterior distribution of the model parameters; the distribution parameter-interval radius in the interval quantization model is regarded as uncertain; according to Bayesian theory, the posterior probability density function of the interval parameter is expressed as: （7） in, is the interval parameter, is a sample of size n, is the posterior probability density function, is the prior probability density, is the sampling density of the sample, is a normalization constant, which represents the marginal density of the sample; The typical non-informative prior distribution - truncated normal distribution is used as the prior distribution of the interval radius; The probability density function of the normal distribution is: （8） in, is the probability density function of the normal distribution, is the mean, is the standard deviation, and All are constants, exp is an exponential function; The probability density function of the truncated normal distribution is: （9） in, is the probability density function of the truncated normal distribution, For the upper and lower bounds, , is the distribution function of the normal distribution, is a constant; The interval method assumes that all samples are equally likely to be distributed in the interval, and samples are drawn from a uniform distribution with a sampling density of: （10） Marginal density represents a normalization constant, which is: （11） Based on Bayesian theory, the posterior distribution probability density function is: （12） Given credibility , integrating the posterior distribution is: （13） in, For a given credibility, is the numerical solution of the exponential integral, is the updated value of the interval parameter under given sample and confidence level conditions.
4. The non-probabilistic credible reliability analysis method for composite structures under truncated normal prior according to claim 3 is characterized in that: The Markov chain Monte Carlo method is used to numerically solve the upper bound of the integral, including: The first step is to assume is a q-dimensional real parameter vector. The first step is to generate a candidate point , defining the proposal density Used from generate , a common generation method is to generate the i-th component Add a mean zero normal deviation: （14） in, is the number of iteration steps, It is The candidate point vector of the step, is the standard deviation of the normal distribution, is an arbitrary constant, k is the index value, and i is the index value; The second step is to calculate the probability of the candidate value being accepted as the next simulated value of the sequence based on the candidate point, and define the acceptance probability r as: （15） Among them, data represents sample information, and represents the likelihood function density, and represents the proposal density, and the acceptance probability r represents the product of the ratio of the posterior density evaluated at the candidate parameter value to the current parameter value and the ratio of the proposal density of the current point and the candidate point; The third step is to draw a random variable that follows a uniform distribution (0,1) , and Compared with r, if , then accept the candidate value and set ,if , then reject the candidate value and set .
5. The non-probabilistic credible reliability analysis method for composite structures under truncated normal prior according to claim 1 is characterized in that: The third step comprises: The Hashin failure criterion that distinguishes failure modes is used to perform refined static strength analysis of composite materials. For different failure modes, the Hashin failure criterion is described as follows: （16） in, are the stress along the fiber direction, the stress perpendicular to the fiber direction, and the shear stress, They are the Hashin failure coefficients of fiber tension, fiber compression, matrix tension, matrix compression, and matrix shear, respectively. If they are greater than 1, it is considered that the corresponding failure has occurred; They are fiber tensile strength, fiber compressive strength, matrix tensile strength, matrix compressive strength, and matrix shear strength, respectively, which are measured through composite material tensile test or shear failure test and are regarded as uncertain quantities.
6. The non-probabilistic credible reliability analysis method for composite structures under truncated normal prior according to claim 5 is characterized in that: The fourth step comprises: Assume the response function of the structure is: （17） Response Function are the various Hashin failure coefficients in formula (16); the response function is subjected to a first-order Taylor expansion, which uses the information of the nominal value point of the interval where the uncertainty parameter is located and / or the interval endpoint to evaluate the response limit of the response function in the entire interval: （18） in, represents the number of uncertain variables, are the upper and lower bounds of the interval of the response function, respectively. is the interval nominal value point response function value, is the first-order derivative of the response function to each uncertainty variable, obtained by combining composite material mechanics and finite element analysis method, i is the index value, is the nominal value of the interval.
7. The non-probabilistic credible reliability analysis method for composite structures under truncated normal prior according to claim 1 is characterized in that: The fifth step comprises: Based on the interval stress-strength interference model, the reliability of the Hashin coefficient of different failure modes is obtained as follows: The response interval [ , ], and then compared with the critical value of the Hashin coefficient, the credibility is The probability of structural failure for: ； （19） The critical value of Hashin coefficient is equal to 1; Credibility is The structural reliability is: （20）。