CN111523275B - Margin and uncertainty quantitative structure reliability evaluation method based on fuzzy random parameters - Google Patents
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Abstract
The invention provides a method for evaluating reliability of a margin and uncertainty quantization structure based on fuzzy random parameters, which comprises the following steps: 1. determining a fuzzy random variable and a confidence coefficient gamma of a structure, and initializing an intercept set alpha =0; 2. constructing envelope distribution of each variable under the current intercept set; 3. sampling to obtain the front 4-order center distance of the envelope distribution according to the envelope distribution; 4. deducing to obtain the front 4-order center distance of the structural response boundary according to the structural response function and the front 4-order center distance of the envelope distribution; 5. fitting a structural response boundary distribution expression according to the 4-order center distance in front of the structural response boundary; 6. calculating the margin M under the alpha cut set according to the confidence coefficient and the structural response boundary distribution expression α And response uncertainty U α (ii) a 7. Computing confidence coefficient CF under alpha intercept α ＝M α /U α (ii) a Judging that alpha is less than 1, if so, increasing alpha = alpha + delta alpha, and turning to the step 2; otherwise, go to step 8; 8. obtaining the overall reliability evaluation result CF = min { CF }of the structure α },α∈[0,1]。
Description
Technical Field
The invention relates to a method for evaluating reliability of a Margin and Uncertainty Quantization (QMU) structure based on fuzzy random parameters, aims to evaluate the reliability of a structure containing inherent Uncertainty and cognition Uncertainty information, and belongs to the field of structural reliability analysis.
Background
In the first aspect, for a complex system or structure with insufficient statistical data and a small sample size, the traditional reliability solving method is difficult to calculate and characterize the reliability. After the united states stopped the nuclear weapons full system test in 1992, the military required a reliability assessment method that was independent of the test and based on cognitive uncertainty due to limitations in experimental data and uncertainty, so a margin and uncertainty Quantification (QMU) -based reliability assessment method arose, and a complete set of QMU methods was proposed in 2002 and defined as "decision support methods for complex technical decisions centered around performance thresholds and related margins of engineering systems under uncertain conditions" in response to the military requirements of the los armos national laboratory, the larns reformer national laboratory, and the sandia national laboratory. QMU reliability assessment was initially widely used in nuclear weapons storage reliability, risk management, and security certification. With the increasing emphasis on the efficiency and operability of statistical data insufficiency and small sub-sample problem processing, recent research is focused on problems such as structural risk assessment and complex system assessment.
In the QMU, a design point refers to a response of a device under a nominal working condition, also called a theoretical design output value, i.e. a generalized response; design operating range refers to the range resulting from inherent uncertainties or tolerances encompassing design points; a performance channel refers to an interval consisting of an upper and lower response limit, which correspond to a reliable boundary, i.e., generalized strength; the performance Margin M (Margin) refers to the distance between the lower limit of the design working range and the upper limit of the performance channel; uncertainty U (uncertaintiy) refers to the range of response fluctuations caused by cognitive Uncertainty; the Confidence coefficient CF = M/U (Confidence Factor) is an evaluation of the reliability of the performance. When CF > 1 corresponds to reliable or acceptable performance, CF ≦ 1 corresponds to failure or risk, and the size of the CF measures the degree of reliability/failure, which is a measure of confidence in system performance under limited data availability. The current QMU method takes into account various uncertainty theories. The method comprises a QMU (Proavailability QMU) method under a Probability framework, a QMU method based on evidence theory, a QMU method under a mixture of interval analysis and evidence theory, a QMU method based on a Probability box and the like.
The QMU theory evaluates the reliability of the object with unclear description due to incomplete information, and the CF is used as a reliability index to visually represent the reliability of the object containing the cognitive information, so that the QMU theory has certain advantages in processing the reliability evaluation problem under non-probability information. However, in the current research QMU theoretical system, a method considering both the inherent random distribution and the cognitive uncertainty based on the distribution parameters is lacked, and in the engineering practice, some statistical data of design parameters are completely mastered, and some data are incomplete, so that the situation including two types of information widely exists.
In the second aspect, the fuzzy random variable is used as an uncertainty variable with inherent and cognitive uncertainty characteristics, is different from a function family expression of single inherent uncertainty of a probability box and is also different from the evidence theory that probability distribution is carried out on each interval segment in a complicated manner, is a continuous function expression and has a probability distribution form, but the distribution parameter is a fuzzy number, and the fuzzy random variable integrates the characteristics of random distribution and incomplete information, so that the representation is simple. Fuzzy random variables are widely applied to reliability algorithms due to their unique properties. The Method comprises a Fuzzy First Order Reliability Method FFORM (Fuzzy First Order Reliability Method) based on Fuzzy random variables, a saddle point approximation Method based on a system time-varying Reliability problem, fuzzy Monte Carlo simulation based on a Fuzzy random variable theory, an interval finite element Method based on Fuzzy random variables, a Bayes Method based on Fuzzy random data and the like.
Based on the two aspects, the invention provides a method for evaluating the reliability of a margin (also called margin) and uncertainty quantization structure based on fuzzy random parameters.
Disclosure of Invention
Objects of the invention
The invention aims to apply fuzzy random variables to structural reliability assessment by adopting a margin and uncertainty Quantification (QMU) method, and aims to explore a novel QMU reliability assessment framework under inherent and cognitive mixed uncertainty.
Because parameters of the fuzzy random variable have different membership range ranges on each intercept set, the performance margin and uncertainty under each intercept set need to be solved. The invention adopts a quadratic fourth-order moment method based on envelope distribution to determine the boundary of structural performance output, wherein a Maximum Entropy Model (ME) is used for fitting a Fuzzy Probability Density Function (FPDF) of structural response.
(II) technical scheme
In order to achieve the purpose, the invention adopts the following technical scheme:
the invention discloses a method for evaluating reliability of a Margin and Uncertainty quantization structure based on fuzzy random parameters, which relates to a method for evaluating reliability of a Margin and Uncertainty Quantization (QMU) structure based on fuzzy random parameters, and comprises the following implementation steps of:
step (1), determining a fuzzy random variable and a confidence coefficient gamma of a structure, and initializing an intercept set alpha =0;
step (2), constructing envelope distribution of each fuzzy random variable under the current intercept set;
step (3), according to the envelope distribution of the fuzzy random variables, sampling and calculating the first 4-order center distance mu of the envelope distribution Xi ,i＝0,1,2,3,；4
Step (4), performing Taylor series expansion on the structural response function at the mean point of the nominal distribution for the first 2 times, and deducing the first 4-order center distance of the structural response boundary by combining the structural response function on the basis of the first 4-order center distance of the envelope distribution
And (5) fitting a coefficient a of the FPDF polynomial of the structural response boundary by taking the first 4-order center distance of the structural response boundary as a constraint on the basis of a maximum entropy method j J =0,1,2,3,4 to obtain the structural response FPDF boundary expression
Step (6), calculating the margin M under the alpha cut set according to the given confidence coefficient α ：M α Fuzzy Probability cumulative distribution function FPCDF (Fuzzy Probability distribution iv) of threshold and structural responsee Distribution Function) boundary is the difference of the values with the confidence coefficient of 0.5, and the FPCDF is the integral result of the FPDF expression in the step (5); calculating the uncertainty U of structural response under alpha truncation α ：U α Is the difference between the value of the fuzzy probability Cumulative Distribution Function FPCDF of the structural response under a given confidence coefficient and the value of the probability Cumulative Distribution Function CDF (current Distribution Function) of the nominal response of the structural response under a confidence coefficient of 0.5;
step (7) according to M α And U α Computing a confidence coefficient CF at the alpha cut α ＝M α /U α (ii) a Judging that alpha is less than 1, if so, increasing alpha = alpha + delta alpha, and returning to the step (2); otherwise, performing the step (8);
step (8), according to QMU definition, calculating the structure overall reliability evaluation result CF = min { CF } α },α∈[0,1]。
Wherein, the fuzzy random variable in step (1) refers to an uncertain variable whose basic distribution is random variable but whose distribution parameter is fuzzy number, taking fuzzy random normal distribution as an example: is provided with
And &>
Fuzzy mean and fuzzy standard deviation, respectively, of the fuzzy random variable, then the fuzzy random normal distribution can be expressed as->
Wherein, in step (1), "determining the fuzzy random variable and confidence γ of the structure, initializing the intercept α =0", the determination process is as follows:
under a given structural analysis object, determining fuzzy random variables and parameter distribution and fuzziness of the structural object according to expert experience; according to the engineering requirements, giving a confidence coefficient gamma epsilon [0,1].
In the step (2), in the step of constructing the envelope distribution of each fuzzy random variable under the current intercept set, the envelope distribution is an envelope line formed by the upper and lower boundaries of the fuzzy random variable FPCDF; the purpose of the envelope distribution is to comprehensively describe the boundary of the fuzzy random variable and to obtain the central moment of the boundary by a statistical method; the configuration of the envelope distribution is as follows:
is provided with
And &>
Respectively is a fuzzy mean value and a fuzzy standard deviation of the fuzzy random variable; all membership functions are assumed to be fuzzy trigonometric numbers; thus, the fuzzy average and the standard deviation can be expressed as ∑ or ∑ respectively>
And
wherein the upper scales L, M and U are the lower bound, middle bound and upper bound, respectively; at a given alpha cut set, the order moment of the upper (lower) bound of the fuzzy random variable FPCDF can be obtained by: in the mean interval->
And standard deviation interval
Upon acquisition, FPCDF upper bound +>
The sample point of (a) consists of two parts: in or on>
To the left side of
Sampling for a distribution in>
To the right side of>
For distribution sampling, the upper bound sampling point set isIs defined as->
Correspondingly, the FPCDF lower bound->
Is at>
To the left side of
Sampling for a distribution, at>
On the right side of (c)>
For distribution sampling, a lower bound set of sampling points is definedX α ＝{x 1 ,x 2 ，...,x n }; thus, the respective moment of the upper and lower bounds of the FPCDF can be based on ^ or ^>
AndX α rapidly obtaining the product by using a mathematical statistical method; in particular, for compliance with x to N (mu) M ,σ M ) The distribution of (a) is called a nominal distribution; FIG. 1 shows
And &>
Schematic diagram of the envelope distribution and nominal distribution of the fuzzy random variables of (1).
Wherein, the step (3) of "sampling and calculating the first 4-step center distance of the envelope distribution" refers to the step (2)
And (3) calculating the center distance, wherein the K-th order center distance calculation formula is as follows:
wherein, in step (3), "based on the envelope distribution of the fuzzy random variables," the first 4-step center distance μ of the envelope distribution is sampled and calculated Xi I =0,1,2,3,4", which works as follows:
μ X0 ＝1
x in the above equation when solving for the first 4 th order center distance of the upper bound of the envelope distribution i For the set of sampling points obtained from step (2)
The number of the sample points is N; x in the above equation when solving for the first 4 th order center distance of the lower bound of the envelope distribution i For the set of sampling points obtained from step (2)X α The number of the sample points is N.
Wherein, the step (4) of performing Taylor series expansion on the structural response function at the mean point of the nominal distribution to the first 2 times means:
assuming a structural response function of
Wherein +>
Is a fuzzy random variable; FCDF Upper (lower) bound ≦ for structural response under the alpha truncated set>
At a mean point of the nominal distribution->
Is paired and/or matched>
The taylor approximation expansion of the first two orders of magnitude is performed:
wherein, in the step (4), "taylor series expansion is performed on the structural response function at the mean point of the nominal distribution for the first 2 times, and the first 4-order center distance of the structural response boundary is obtained by combining the structural response function derivation based on the first 4-order center distance of the envelope distribution
The method comprises the following steps:
according to the first 4 moments of the FPCDF boundary obtained in the step (3)
And combining the above formula (the obtained previous 2-time Taylor series expansion formula) to obtain the first-order to fourth-order center distance of the structural response according to the following formula
And &>
Z when calculating the first to fourth center distances of the structural response upper bound Q Namely, it is
Z when calculating the first to fourth center distances of the structure response lower bound Q I.e. is>
Wherein, the 'maximum entropy method based on' in step (5) takes the first 4-order center distance of the structural response boundary as a constraint to fit the coefficient a of the FPDF polynomial of the structural response boundary j J =0,1,2,3,4, resulting in a structural response FPDF boundary expression
The method comprises the following steps:
to solve the structural response upper bound
Is/are>
For example, the maximum entropy model under the alpha cut can be expressed as follows (the same lower bound):
wherein c is a constant,
is->
Standardizing; solving the maximum entropy model by adopting a Lagrange multiplier method: />
λ i I =0,1,2,3,4 is the lagrangian undetermined coefficient, the approximate expression of the structural response function can be simplified as follows:
wherein, a 0 ＝1-λ 0 /c，a i ＝-λ 0 /c (i =1,2,3,4); on the other hand, alpha cuts the lower fourth moment
Having been derived therein, an approximate expression of the structural response function can be solved.
Wherein said step (6) "calculates the margin M under the alpha-cut set according to the given confidence α 'and' calculation of structural response uncertainty U under alpha truncation α ", means that:
at a given level of truncation, M α For the point Y at which the structural response function upper boundary found in step (5) has a confidence of 0.5 function And a threshold value F TH The difference between them; uncertainty U α Is the point (U) at which the confidence of the upper boundary of the structural response function determined in step (5) is (1 + gamma)/2 function ) α And a point U at which the nominal response has a confidence of 0.5 0.5 The difference between the two; the nominal response refers to the output of a structural response function under the nominal distribution of fuzzy random variables; the formula for M and U can be expressed as:
M α ＝F TH -(Y function ) α ；U α ＝(U function ) α -U 0.5
in the case, for easy understanding, F TH ，(Y function ) α ，(U function ) α And U 0.5 Labeled Margin _ up, margin _ low, ucertainty _ up, and Uncertainty _ low, respectively.
Wherein said in step (7) "is according to M α And U α Computing a confidence coefficient CF at the alpha cut α ＝M α /U α ", the FQMU index under the present truncation is calculated according to the calculation result of step (6) as follows:
M α and U α The result calculated in step (6).
Wherein the "according to QMU definition, structure overall reliability evaluation result CF = min { CF ] is calculated in step (8) α },α∈[0,1]", means CF under all truncations α After the calculation, the overall reliability evaluation result of the structure is calculated according to the following formula:
CF＝min{CF α },α∈[0,1]
a schematic diagram and a flow chart of the QMU reliability evaluation based on fuzzy random variables are shown in fig. 2 and fig. 3, respectively.
(III) advantages and Effect of the invention
The structure reliability evaluation method of the invention combines the respective characteristics of QMU and fuzzy random variables, provides a new QMU index calculation framework, and has the advantages and effects that:
(1) The QMU reliability evaluation index under hybrid uncertainty is constructed, the reliability evaluation output under fuzzy random variable input can be well described, and the reliability evaluation under incomplete data can be guided;
(2) The method is convenient and quick to calculate, can represent the QMU measurement of the whole structure, can also represent the measurement under different cut sets, and provides reliability evaluation representation under different cognitive levels;
(3) The method not only considers the original definition of QMU, but also considers the characteristics of fuzzy number, and is a calculation frame for mixing uncertainty quantification and propagation;
(4) The reliability assessment method is scientific, has good manufacturability and has wide popularization and application values.
Description of the drawings (the sequence numbers, symbols, and code symbols in the drawings are as follows)
FIG. 1, fuzzy probability cumulative distribution function.
Fig. 2, QMU reliability evaluation indicators based on fuzzy random variables.
FIG. 3 is a flowchart of the reliability assessment method of the present invention.
Fig. 4, free beam diagram.
FIG. 5, x 1 And x 2 Fuzzy probability cumulative distribution function.
FIG. 6 is a diagram showing the reliability evaluation of the free beam QMU.
Figure 7 truss diagram.
FIG. 8, x 1 ，x 2 ，x 3 And x 4 Fuzzy probability cumulative distribution function.
Fig. 9 is a schematic diagram of truss QMU reliability evaluation.
The numbers, symbols and codes in the figures are explained as follows:
in FIG. 1, FPCDF is a fuzzy probability accumulation distribution function.
In FIG. 2, FPCDF is fuzzy probability cumulative distribution function, γ is confidence, P is confidence boundary, M is performance margin, Y function As structural response boundaries, F TH Is a threshold value.
In fig. 6, unrotaitaby _ up is an upper Uncertainty bound, unrotaitaby _ low is a lower Uncertainty bound, and Mrgin _ low is a lower performance margin bound.
In FIG. 7, A, B, C, D, E, F and G are truss fixed point designations. l is the concrete and rebar length.
In fig. 9, unrotaitaby _ up is an upper Uncertainty bound, unrotaitaby _ low is a lower Uncertainty bound, and Mrgin _ low is a lower performance margin bound.
The symbols and codes referred to in the present specification are as follows:
QMU (Qualification and Margin of Uncertainty) -Margin and Uncertainty quantification
CF = M/U (Confidence Factor) -Confidence coefficient
FFORM (Fuzzy First Order Reliability Method) -Fuzzy First Order Reliability Method based on Fuzzy random variables
ME (Maximum Encopy) -Maximum Entropy model
FPDF (Fuzzy Probability Density Function) — Fuzzy Probability Density Function
FPCDF (Fuzzy basic statistical Distribution Function) -Fuzzy Probability Cumulative Distribution Function
CDF (Current Distribution Function) -probability Cumulative Distribution Function
Detailed Description
The technical scheme of the invention is explained in detail by combining an example and a drawing.
Case 1:
the free support beam carrying the uniform load is shown in fig. 4. The length l, the section width b and the section height h are basic variables, i.e., l =4000mm, b =105mm, h =210mm,
is a fuzzy random variable whose distribution is normally distributed, i.e.
The mean and standard deviation std are fuzzy numbers, and the membership functions are shown in FIG. 5. The beam is made of 45-grade steel and has the strength of 850MPa. According to the basic theory of material mechanics, the maximum stress of a free support beam is:
the confidence coefficient is 0.9, the truncated increment delta alpha is 0.2, and the reliability of the method is evaluated by applying the method, and the process is as follows:
the invention relates to a margin and uncertainty quantification structure reliability assessment method based on fuzzy random parameters, which comprises the following implementation steps as shown in figure 3:
step (1), determining a fuzzy random variable of a structure, and initializing an intercept set alpha =0;
step (2), constructing envelope distribution of each fuzzy random variable under the current truncated set, wherein the envelope distribution is shown in figure 5 as an envelope distribution schematic diagram 1;
step (3), sampling and calculating the first 4-order center distance of the envelope distribution according to the envelope distribution of the fuzzy random variables, wherein the result is as follows;
TABLE 1X 1 FPDF upper bound front four-order center distance
| Cutting set | First moment of center distance | Second order center moment | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 199.9972 | 4.4593 | 0.0018 | 2.9920 |
| α＝0.2 | 204.0317 | 4.4909 | 0.0455 | 3.0230 |
| α＝0.4 | 208.0723 | 4.4682 | 0.0450 | 2.9850 |
| α＝0.6 | 212.1117 | 4.4810 | 0.0655 | 3.0006 |
| α＝0.8 | 216.1445 | 4.4659 | 0.1091 | 3.0012 |
| α＝1.0 | 220.1850 | 4.4611 | 0.1100 | 3.0220 |
TABLE 2X 1 FPDF lower bound front four-order center distance
| Cutting set | First moment of center distance | Second moment of center distance | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 199.9902 | 4.4813 | -0.0022 | 3.0215 |
| α＝0.2 | 195.9742 | 4.4804 | -0.0403 | 3.0102 |
| α＝0.4 | 191.9234 | 4.4758 | -0.0516 | 2.9899 |
| α＝0.6 | 187.9036 | 4.4817 | -0.0774 | 2.9944 |
| α＝0.8 | 183.8701 | 4.4729 | -0.0911 | 2.9944 |
| α＝1.0 | 179.8109 | 4.4697 | -0.1231 | 2.9997 |
TABLE 3X 2 FPDF upper bound front four-order center distance
| Cutting set | First moment of center distance | Second order center moment | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 105.0021 | 3.1746 | -0.0023 | 2.9932 |
| α＝0.2 | 108.0343 | 3.1593 | 0.0244 | 2.9970 |
| α＝0.4 | 111.0451 | 3.1602 | 0.0497 | 3.0177 |
| α＝0.6 | 114.0831 | 3.1659 | 0.0699 | 2.9930 |
| α＝0.8 | 117.0927 | 3.1642 | 0.1003 | 2.9982 |
| α＝1.0 | 120.1281 | 3.1569 | 0.1245 | 3.0362 |
TABLE 4X 2 FPDF lower bound front four-order center distance
| Cutting set | First moment of center distance | Second order center moment | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 105.0044 | 3.1669 | 0.0093 | 2.9986 |
| α＝0.2 | 101.9788 | 3.1584 | -0.0105 | 3.0039 |
| α＝0.4 | 98.9684 | 3.1561 | -0.0628 | 3.0090 |
| α＝0.6 | 95.9107 | 3.1458 | -0.0842 | 2.9982 |
| α＝0.8 | 92.9089 | 3.1542 | -0.0945 | 3.0155 |
| α＝1.0 | 89.8755 | 3.1546 | -0.1074 | 2.9959 |
Step (4), performing Taylor series expansion on the structural response function at the mean point of the nominal distribution for the first 2 times, and deducing the first 4-order center distance of the structural response boundary by combining the structural response function on the basis of the first 4-order center distance of the envelope distribution;
TABLE 5 structural response FPDF Top-bound fourth-order center distance
| Cutting set | First moment of center distance | Second moment of center distance | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 523.06 | 72.119 | 0.39257 | 2.915 |
| α＝0.2 | 518.29 | 69.894 | 0.38224 | 2.9443 |
| α＝0.4 | 513.99 | 67.313 | 0.36785 | 2.9536 |
| α＝0.6 | 509.83 | 65.523 | 0.35877 | 2.9655 |
| α＝0.8 | 505.96 | 63.275 | 0.35865 | 2.9835 |
| α＝1.0 | 502.19 | 61.211 | 0.3445 | 3.0098 |
TABLE 6 structural response FPDF lower bound front fourth order center distance
| Cutting set | First moment of center distance | Second order center moment | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 522.98 | 72.274 | 0.38368 | 2.9312 |
| α＝0.2 | 527.92 | 74.622 | 0.38514 | 2.9148 |
| α＝0.4 | 533.03 | 77.318 | 0.40879 | 2.8764 |
| α＝0.6 | 538.73 | 80.148 | 0.41401 | 2.8583 |
| α＝0.8 | 544.69 | 83.32 | 0.4342 | 2.8457 |
| α＝1.0 | 551.07 | 86.673 | 0.44234 | 2.8257 |
And (5) fitting a coefficient a of the FPDF polynomial of the structural response boundary by taking the first 4-order center distance of the structural response boundary as a constraint on the basis of a maximum entropy method j J =0,1,2,3,4, resulting in a structural response FPDF boundary expression
TABLE 7 structural response FPDF upper bound expression
| Cutting set | Coefficient of quartic term | Coefficient of cubic term | Coefficient of quadratic term | Coefficient of first order | Constant term |
| α＝0 | -0.40453 | 0.10742 | -0.027225 | -0.27951 | -0.95921 |
| α＝0.2 | -0.41888 | 0.099248 | -0.023438 | -0.26191 | -0.95279 |
| α＝0.4 | -0.42674 | 0.092331 | -0.021026 | -0.24606 | -0.94945 |
| α＝0.6 | -0.43356 | 0.087511 | -0.019143 | -0.23506 | -0.94646 |
| α＝0.8 | -0.43946 | 0.085566 | -0.017861 | -0.23107 | -0.94367 |
| α＝1.0 | -0.45162 | 0.077885* | -0.014723 | -0.21337 | -0.93821 |
TABLE 8 structural response FPDF lower bound expression
| Cutting set | Coefficient of quartic term | Coefficient of cubic term | Coefficient of quadratic term | Coefficient of first order | Constant term |
| α＝0 | -0.02469 | 0.10147 | -0.41366 | -0.26652 | -0.95519 |
| α＝0.2 | -0.026273 | 0.10418 | -0.40703 | -0.27208 | -0.95824 |
| α＝0.4 | -0.033161 | 0.12045 | -0.38308 | -0.30711 | -0.9687 |
| α＝0.6 | -0.035841 | 0.12584 | -0.37326 | -0.31817 | -0.97306 |
| α＝0.8 | -0.040592 | 0.13863 | -0.35926 | -0.34538 | -0.97893 |
| α＝1.0 | -0.044442 | 0.14683 | -0.34627 | -0.36186 | -0.9846 |
Step (6), calculating the margin M under the alpha cut set according to the given confidence coefficient α And structural response uncertainty U α (ii) a As shown in fig. 6 (α = 1) of the schematic diagram of QMU reliability evaluation based on fuzzy random variables, the calculation results at each truncation are shown in table 9.
TABLE 9M α And U α The result of the calculation under each cut set
| Cutting set | Margin_up | Margin_low | Uncertainty_up | Uncertainty_low |
| α＝0 | 850 | 517.2 | 652.35 | 520 |
| α＝0.2 | 850 | 521.95 | 661.49 | 520 |
| α＝0.4 | 850 | 526.07 | 672.98 | 520 |
| α＝0.6 | 850 | 531.52 | 683.8 | 520 |
| α＝0.8 | 850 | 536.36 | 698.83 | 520 |
| α＝1.0 | 850 | 542.41 | 711.42 | 520 |
Step (7) according to M α And U α Computing a confidence coefficient CF at the alpha cut α ＝M α /U α . Judging alpha < 1, if yes, increasing alpha = alpha + deltaα, returning to the step (12); otherwise, performing the step (8); the results of FQMU calculations at each intercept are shown below;
TABLE 10 FQMU results
| Cutting set | FQMU |
| α＝0 | 2.514545 |
| α＝0.2 | 2.318538 |
| α＝0.4 | 2.117466 |
| α＝0.6 | 1.944322 |
| α＝0.8 | 1.753844 |
| α＝1.0 | 1.606885 |
Step (8), according to QMU definition, calculating the structure overall reliability evaluation result CF = min { CF } α },α∈[0,1]。
CF＝min{2.5145,2.3185,2.1175,1.9443,1.7538,1.6069}＝1.6069。
Case 2:
the roof truss is shown in fig. 7 with the top hanger bar and the compression bar reinforced with concrete and the bottom hanger bar and the tension bar made of steel.The evaluation is performed under the premise that the roof truss is assumed to bear the uniform load, and the uniform load q can be converted into the node load, wherein P = ql/4. The vertical displacement of the node can be obtained by mechanical analysis and can be expressed as a function of the basic variable, i.e.
Wherein A is c ，A s ，E c ，E s And l are the concrete cross-sectional area, the rebar cross-sectional area, the concrete elastic modulus, the rebar elastic modulus, and the concrete (rebar) length, respectively. The displacement threshold of the vertex C in the vertical direction is 3.1cm, according to the requirements of safety and reliability, and the limit state function can be constructed by using the condition. l =12m, q =2 × 10 4 N。
In this example, A c ，A s ，E c And E s Is a fuzzy random normal distribution. The mean μ and standard deviation σ of the variables are set as triangular blur numbers:
α varies from 0 to 1, and the blur number is evaluated at α:0.0, 0.2, 0.4, 0.6, 0.8 and 1.0.
The confidence coefficient is 0.95, the intercept increment delta alpha is 0.2, and the reliability of the method is evaluated by applying the method, wherein the process is as follows:
the invention relates to a margin and uncertainty quantification structure reliability assessment method based on fuzzy random parameters, which comprises the following implementation steps as shown in figure 3:
step (1), determining a fuzzy random variable of a structure, and initializing an intercept set alpha =0;
step (2), constructing envelope distribution of each fuzzy random variable under the current intercept set, as shown in figure 8 of a schematic diagram 1 of the envelope distribution;
step (3), sampling and calculating the first 4-order center distance of the envelope distribution according to the envelope distribution of the fuzzy random variables, wherein the result is as follows;
TABLE 11X 1 FPDF upper bound front four-order center distance
| Cutting set | First moment of center distance | Second order center moment | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 0.2500 | 0.1414 | -0.0014 | 2.9976 |
| α＝0.2 | 0.2528 | 0.1414 | 0.0221 | 3.0069 |
| α＝0.4 | 0.2564 | 0.1415 | 0.0491 | 3.0027 |
| α＝0.6 | 0.2595 | 0.1415 | 0.0731 | 3.0018 |
| α＝0.8 | 0.2623 | 0.1413 | 0.0940 | 3.0024 |
| α＝1.0 | 0.2658 | 0.1413 | 0.1173 | 3.0231 |
TABLE 12X 1 FPDF lower bound front four-order center distance
| Cutting set | First moment of center distance | Second order center moment | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 0.2500 | 0.1415 | 0.0002 | 2.9977 |
| α＝0.2 | 0.2472 | 0.1413 | -0.0191 | 2.9935 |
| α＝0.4 | 0.2437 | 0.1415 | -0.0476 | 2.9969 |
| α＝0.6 | 0.2406 | 0.1413 | -0.0715 | 3.0011 |
| α＝0.8 | 0.2375 | 0.1411 | -0.0953 | 3.0026 |
| α＝1.0 | 0.2345 | 0.1413 | -0.1203 | 3.0176 |
TABLE 13X 2 FPDF upper bound front four-order center distance
| Cutting set | First moment of center distance | Second moment of center distance | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 0.9998 | 0.2447 | -0.0003 | 3.0019 |
| α＝0.2 | 1.0034 | 0.2450 | 0.0196 | 2.9995 |
| α＝0.4 | 1.0073 | 0.2448 | 0.0403 | 3.0056 |
| α＝0.6 | 1.0105 | 0.2448 | 0.0559 | 2.9992 |
| α＝0.8 | 1.0144 | 0.2451 | 0.0790 | 2.9950 |
| α＝1.0 | 1.0181 | 0.2449 | 0.1027 | 3.0089 |
TABLE 14X 2 FPDF lower bound front four-order center distance
| Cutting set | First moment of center distance | Second order center moment | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 1.0000 | 0.2451 | 0.0036 | 3.0031 |
| α＝0.2 | 0.9964 | 0.2450 | -0.0201 | 2.9973 |
| α＝0.4 | 0.9928 | 0.2449 | -0.0399 | 2.9965 |
| α＝0.6 | 0.9891 | 0.2450 | -0.0597 | 3.0068 |
| α＝0.8 | 0.9856 | 0.2448 | -0.0782 | 3.0024 |
| α＝1.0 | 0.9820 | 0.2447 | -0.0949 | 3.0012 |
TABLE 15X 3 FPDF upper bound front four-order center distance
| Cutting set | First moment of center distance | Second order center moment | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 0.5999 | 0.2236 | -0.0014 | 3.0019 |
| α＝0.2 | 0.6040 | 0.2235 | 0.0248 | 3.0101 |
| α＝0.4 | 0.6077 | 0.2234 | 0.0482 | 3.0023 |
| α＝0.6 | 0.6113 | 0.2238 | 0.0716 | 2.9976 |
| α＝0.8 | 0.6151 | 0.2233 | 0.0968 | 3.0033 |
| α＝1.0 | 0.6190 | 0.2234 | 0.1174 | 3.0086 |
TABLE 16X 3 FPDF lower bound front four-order center distance
| Cutting set | First moment of center distance | Second order center moment | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 0.6001 | 0.2233 | 0.0002 | 3.0101 |
| α＝0.2 | 0.5962 | 0.2234 | -0.0237 | 2.9953 |
| α＝0.4 | 0.5926 | 0.2236 | -0.0452 | 2.9978 |
| α＝0.6 | 0.5884 | 0.2233 | -0.0693 | 2.9942 |
| α＝0.8 | 0.5850 | 0.2234 | -0.0990 | 3.0103 |
| α＝1.0 | 0.5810 | 0.2234 | -0.1158 | 3.0156 |
TABLE 17X 4 FPDF upper bound front four-order center distance
| Cutting set | First moment of center distance | Second order center moment | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 1.1003 | 0.2234 | 0.0064 | 2.9918 |
| α＝0.2 | 1.1039 | 0.2235 | 0.0243 | 3.0040 |
| α＝0.4 | 1.1075 | 0.2235 | 0.0481 | 2.9964 |
| α＝0.6 | 1.1110 | 0.2237 | 0.0710 | 3.0053 |
| α＝0.8 | 1.1152 | 0.2236 | 0.0952 | 3.0064 |
| α＝1.0 | 1.1189 | 0.2233 | 0.1218 | 3.0139 |
TABLE 18X 4 FPDF lower bound front four-order center distance
| Cutting set | First moment of center distance | Second order center moment | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 1.0999 | 0.2236 | 0.0006 | 2.9964 |
| α＝0.2 | 1.0960 | 0.2235 | -0.0247 | 3.0033 |
| α＝0.4 | 1.0924 | 0.2238 | -0.0524 | 3.0028 |
| α＝0.6 | 1.0884 | 0.2236 | -0.0738 | 2.9996 |
| α＝0.8 | 1.0848 | 0.2235 | -0.0944 | 3.0065 |
| α＝1.0 | 1.0809 | 0.2234 | -0.1194 | 3.0095 |
Step (4), performing Taylor series expansion on the structural response function at the mean value point of nominal distribution for the first 2 times, and deducing to obtain the first 4-order center distance of the structural response boundary by combining the structural response function on the basis of the first 4-order center distance of the envelope distribution;
table 19 structural response FPDF upper bound front four-order center distance
| Cutting set | First moment of center distance | Second moment of center distance | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 2.6128 | 0.16483 | 0.090345 | 2.988 |
| α＝0.2 | 2.6402 | 0.16593 | 0.10095 | 2.9889 |
| α＝0.4 | 2.6701 | 0.16677 | 0.11239 | 2.9893 |
| α＝0.6 | 2.6969 | 0.16775 | 0.12228 | 2.9886 |
| α＝0.8 | 2.726 | 0.16861 | 0.13401 | 2.9883 |
| α＝1.0 | 2.7564 | 0.16937 | 0.14595 | 2.992 |
Table 20 structural response FPDF lower bound front four-order center distance
| Cutting set | First moment of center distance | Second order center moment | Third order moment of center distance | Fourth order moment of center distance |
| α＝0 | 2.6129 | 0.1651 | 0.091039 | 2.9887 |
| α＝0.2 | 2.5856 | 0.16403 | 0.078831 | 2.9875 |
| α＝0.4 | 2.5572 | 0.16333 | 0.067401 | 2.9873 |
| α＝0.6 | 2.5283 | 0.16225 | 0.056372 | 2.9889 |
| α＝0.8 | 2.5019 | 0.16113 | 0.045482 | 2.9888 |
| α＝1.0 | 2.4746 | 0.16023 | 0.035446 | 2.9892 |
And (5) fitting a coefficient a of the FPDF polynomial of the structural response boundary by taking the first 4-order center distance of the structural response boundary as a constraint on the basis of a maximum entropy method j J =0,1,2,3,4, resulting in a structureResponsive FPDF boundary expressions
TABLE 21 structural response FPDF Upper bound expression
| Cutting set | Coefficient of quartic term | Coefficient of cubic term | Coefficient of quadratic term | Coefficient of first order | Constant term |
| α＝0 | -0.00154 | 0.015753 | -0.49293 | -0.046701 | -0.92212 |
| α＝0.2 | -0.0017546 | 0.017693 | -0.49219 | -0.052372 | -0.92241 |
| α＝0.4 | -0.0020442 | 0.019841 | -0.49112 | -0.058605 | -0.92284 |
| α＝0.6 | -0.0023732 | 0.022005 | -0.48989 | -0.064841 | -0.92334 |
| α＝0.8 | -0.0027115 | 0.023939 | -0.48858 | -0.070372 | -0.92389 |
| α＝1.0 | -0.0030148 | 0.02627 | -0.48769 | -0.077056 | -0.9242 |
TABLE 22 structural response FPDF lower bound expression
| Cutting set | Coefficient of quartic term | Coefficient of cubic term | Coefficient of quadratic term | Coefficient of first order | Constant term |
| α＝0 | -0.0015235 | 0.015863 | -0.49306 | -0.047035 | -0.92205 |
| α＝0.2 | -0.0013174 | 0.013668 | -0.49374 | -0.04059 | -0.9218 |
| α＝0.4 | -0.0011159 | 0.011626 | -0.49451 | -0.034577 | -0.92149 |
| α＝0.6 | -0.00087317 | 0.0096566 | -0.4956 | -0.028773 | -0.921 |
| α＝0.8 | -0.0007341 | 0.0077624 | -0.49614 | -0.023154 | -0.92078 |
| α＝1.0 | -0.00061792 | 0.0060301 | -0.49663 | -0.018003 | -0.92057 |
Step (6) calculating the margin M under the alpha cut set according to the given confidence coefficient α And structural response uncertainty U α (ii) a QMU reliability evaluation based on fuzzy random variables is schematically shown in fig. 2 as fig. 9 (α = 1), and the calculation results at each truncation are shown in table 23.
TABLE 23M α And U α The calculation results under each cut set
| Cutting set | Margin_up | Margin_low | Uncertainty_up | Uncertainty_low |
| α＝0 | 3.5 | 2.6112 | 2.9441 | 2.61 |
| α＝0.2 | 3.5 | 2.6404 | 2.9759 | 2.61 |
| α＝0.4 | 3.5 | 2.6685 | 3.007 | 2.61 |
| α＝0.6 | 3.5 | 2.6936 | 3.0358 | 2.61 |
| α＝0.8 | 3.5 | 2.7226 | 3.0683 | 2.61 |
| α＝1.0 | 3.5 | 2.753 | 3.1002 | 2.61 |
Step (7) according to M α And U α Computing a confidence coefficient CF at the alpha cut α ＝M α /U α . Judging that alpha is less than 1, if so, increasing alpha = alpha + delta alpha, and returning to the step (12); otherwise, go to step (8). The results of FQMU calculations at each intercept are shown below;
TABLE 24 FQMU results
| Cutting set | FQMU |
| α＝0 | 2.660281 |
| α＝0.2 | 2.349276 |
| α＝0.4 | 2.094458 |
| α＝0.6 | 1.893847 |
| α＝0.8 | 1.696269 |
| α＝1.0 | 1.523868 |
Step (8),According to QMU definition, calculating the overall reliability evaluation result CF = min { CF } of the structure α },α∈[0,1]。
CF＝min{2.6603,2.3493,2.0945,1.8938,1.6963,1.5239}＝1.5239。
Claims (9)
1. A margin and uncertainty quantification structure reliability assessment method based on fuzzy random parameters is applied to roof trusses, top suspenders and compression bars are reinforced by concrete, and bottom suspenders and compression bars are made of steel; setting a roof truss to bear the uniformly distributed load, and carrying out P = ql/4 on the premise that the uniformly distributed load q is converted into a node load; the vertical displacement of the node is obtained by mechanical analysis and is expressed as a function of the basic variable, i.e.
Wherein A is c ，A s ，E c ，E s And l is the concrete cross-sectional area, the steel bar cross-sectional area, the concrete elastic modulus, the steel bar elastic modulus and the concrete length respectively; the displacement threshold value of the vertex C in the vertical direction is 3.1cm, and a limit state function is constructed by utilizing the condition; l =12m, q =2 × 10 4 N；
A c ，A s ，E c And E s Is a fuzzy random normal distribution; the mean μ and standard deviation σ of the variables are set as triangular blur numbers:
α varies from 0 to 1, and the blur number is evaluated at α:0.0, 0.2, 0.4, 0.6, 0.8 and 1.0; the confidence coefficient is 0.95, and the truncated increment delta alpha is 0.2; the method is characterized in that: the implementation steps are as follows:
step (1), determining a fuzzy random variable and a confidence coefficient gamma of a structure, and initializing an intercept set alpha =0;
step (2), constructing envelope distribution of each fuzzy random variable under the current intercept set;
step (3), according to the envelope distribution of the fuzzy random variables, sampling and calculating the first 4-order center distance mu of the envelope distribution Xi ,i＝0,1,2,3,4；
Step (4), performing Taylor series expansion on the structural response function at the mean point of the nominal distribution for the first 2 times, and deducing the first 4-order center distance of the structural response boundary by combining the structural response function on the basis of the first 4-order center distance of the envelope distribution
And (5) fitting out a coefficient a of an FPDF polynomial of the structural response boundary by taking the first 4-order center distance of the structural response boundary as a constraint based on a maximum entropy method j J =0,1,2,3,4, resulting in a structural response FPDF boundary expression
Step (6), calculating the margin M under the alpha cut set according to the given confidence coefficient α ：M α The difference between the threshold value and the value of the fuzzy probability cumulative distribution function FPCDF boundary of the structural response at the confidence coefficient of 0.5 is used, and the FPCDF is the integral result of the FPDF expression in the step (5); calculating the uncertainty U of structural response under alpha truncation α ：U α Is the difference between the value of the fuzzy probability cumulative distribution function FPCDF of the structural response under a given confidence coefficient and the value of the probability cumulative distribution function CDF of the nominal response of the structural response under the confidence coefficient of 0.5;
step (7) according to M α And U α Computing a confidence coefficient CF at the alpha cut α ＝M α /U α (ii) a Judging that alpha is less than 1, if so, increasing alpha = alpha + delta alpha, and returning to the step (2); otherwise, performing the step (8);
step (8), according to QMU definition, calculating the structure overall reliability evaluation result CF = min { CF } α },α∈[0,1]。
2. The method for evaluating reliability of the fuzzy random parameter based margin and uncertainty quantization structure according to claim 1, wherein:
the fuzzy random variable mentioned in step (1) refers to an uncertain variable whose basic distribution is random variable but whose distribution parameter is fuzzy number, taking fuzzy random normal distribution as an example: is provided with
And &>
Fuzzy mean and fuzzy standard deviation of the fuzzy random variable respectively, then the fuzzy random normal distribution can be expressed as->
In step (1), "determining the fuzzy random variables and confidence γ of the structure, initializing the truncated set α =0", the determination is performed as follows:
under a given structural analysis object, determining fuzzy random variables and parameter distribution and fuzziness of the structural object according to expert experience; according to engineering requirements, a confidence coefficient gamma epsilon [0,1] is given.
3. The method for evaluating reliability of the fuzzy random parameter based margin and uncertainty quantization structure according to claim 1, wherein:
in the step (2), in the step of constructing the envelope distribution of each fuzzy random variable under the current truncation, the envelope distribution is an envelope line formed by the upper and lower boundaries of the fuzzy random variable FPCDF; the purpose of the envelope distribution is to comprehensively describe the boundary of the fuzzy random variable and can use a statistical method to obtain the central moment of the boundary; the configuration of the envelope distribution is as follows:
is provided with
And &>
Respectively is a fuzzy mean value and a fuzzy standard deviation of the fuzzy random variable; all membership functions are assumed to be fuzzy trigonometric numbers; thus, the fuzzy average and the standard deviation can be expressed as ∑ or ∑ respectively>
And &>
Wherein the superscripts L, M and U are the lower bound, the middle bound and the upper bound, respectively; at a given alpha cut set, the order moment of the upper (lower) bound of the fuzzy random variable FPCDF can be obtained by: in the mean interval->
And standard deviation interval->
After obtaining, the FPCDF upper bound
The sample point of (a) consists of two parts: in or on>
To the left side of>
Sampling for a distribution, at>
To the right side of>
The upper bound set of sample points is defined as @, sampled for a distribution>
Correspondingly, FPCDF lower bound +>
Is at>
To the left side of>
Sampling for a distribution, at>
To the right side of
For distribution sampling, a lower bound set of sampling points is definedX α ＝{x 1 ,x 2 ，...,x n }; thus, the respective moments of the upper and lower bounds of the FPCDF can be based on ^ or ^ 4>
AndX α rapidly obtaining the product by using a mathematical statistical method; in particular, for compliance with x to N (mu) M ,σ M ) The distribution of (a) is referred to as the nominal distribution.
4. The method for evaluating reliability of the fuzzy random parameter based margin and uncertainty quantization structure according to claim 1, wherein:
the "sampling and calculating the first 4 th-order center distance of the envelope distribution" described in step (3) refers to the step (2)
(X α ) And (3) calculating the center distance, wherein the K-th order center distance calculation formula is as follows:
"sampling and calculating the first 4-order center distance μ of the envelope distribution based on the envelope distribution of the fuzzy random variables" described in step (3) Xi I =0,1,2,3,4", which works as follows:
μ X0 ＝1
x in the above equation when solving for the first 4 th order center distance of the upper bound of the envelope distribution i For the set of sampling points obtained from step (2)
The number of the sample points is N; x in the above equation when solving for the first 4 th order center distance of the lower bound of the envelope distribution i For the set of sampling points obtained from step (2)X α The number of the sample points is N.
5. The method for evaluating reliability of the fuzzy random parameter based margin and uncertainty quantization structure according to claim 1, wherein:
the step (4) of "expanding the structural response function to the first 2 times at the mean point of the nominal distribution" means:
setting the structural response function as
Wherein->
Is a fuzzy random variable; FCDF Upper (lower) bound ≦ for structural response under the alpha truncated set>
At a mean point of the nominal distribution->
Is paired and/or matched>
The taylor approximation expansion of the first two orders of magnitude is performed:
in the step (4), "taylor series expansion is performed on the structural response function at the mean point of the nominal distribution for the first 2 times, and the first 4-order center distance of the structural response boundary is obtained by combining the structural response function derivation based on the first 4-order center distance of the envelope distribution
The method comprises the following steps:
according to the first 4-order moment mu of the FPCDF boundary obtained in the step (3) Xij J =0,1,2,3,4, and the first 2 taylor series expansions obtained by combining the above formula are used to obtain the first to fourth center distances of the structural response according to the following formula
And &>
Z when calculating the first to fourth center distances of the structural response upper bound Q Namely, it is
Z when calculating the first to fourth center distances of the structure response lower bound Q I.e. is>
6. The method for evaluating reliability of the fuzzy random parameter based margin and uncertainty quantization structure according to claim 1, wherein:
the 'maximum entropy method based on' in step (5) is used for fitting the coefficient a of the FPDF polynomial of the structural response boundary by taking the first 4-order center distance of the structural response boundary as a constraint j J =0,1,2,3,4, resulting in a structural response FPDF boundary expression
The method comprises the following steps:
to solve the structural response upper bound
Is/are>
For example, the maximum entropy model under the alpha cut can be expressed as follows (the same holds for the lower bounds):
wherein c is a constant,
is->
Standardizing; solving the maximum entropy model by adopting a Lagrange multiplier method: />
λ i I =0,1,2,3,4 is the lagrange undetermined coefficient, the approximate expression of the structural response function is simplified as follows:
wherein, a 0 ＝1-λ 0 c，a i ＝-λ 0 c (i =1,2,3,4); on the other hand, alpha cuts the lower fourth moment
Derived therein, the approximate expression of the structural response function can be solved。
7. The method for evaluating reliability of the fuzzy random parameter based margin and uncertainty quantization structure according to claim 1, wherein:
in step (6), "computing the margin M at α -cut set according to given confidence α 'and' calculation of structural response uncertainty U under alpha truncation α ", means:
at a given level of truncation, M α For the point Y at which the structural response function upper boundary found in step (5) has a confidence of 0.5 function And a threshold value F TH The difference between them; uncertainty U α Is the point (U) at which the upper boundary of the structural response function determined in step (5) has a confidence level of (1 + gamma) 2 function ) α And the point U at which the nominal response has a confidence of 0.5 0.5 The difference between the two; the nominal response refers to the output of a structural response function under the nominal distribution of fuzzy random variables; the formula for M and U can be expressed as:
M α ＝F TH -(Y function ) α ；U α ＝(U function ) α -U 0.5
in the case, for easy understanding, F TH ，(Y function ) α ，(U function ) α And U 0.5 Labeled Margin _ up, margin _ low, ucertainty _ up, and Uncertainty _ low, respectively.
8. The method for evaluating reliability of the fuzzy random parameter based margin and uncertainty quantization structure according to claim 1, wherein:
said "according to M" in step (7) α And U α Computing a confidence coefficient CF at the alpha cut α ＝M α U α ", the FQMU index under the present truncation is calculated according to the calculation result of step (6) as follows:
M α and U α The result calculated in step (6).
9. The method for evaluating reliability of the fuzzy random parameter based margin and uncertainty quantization structure according to claim 1, wherein:
"according to QMU definition, calculate the overall reliability assessment result CF = min { CF } in step (8) α },α∈[0,1]", means CF under all truncations α After the calculation, the overall reliability evaluation result of the structure is calculated according to the following formula:
CF＝min{CF α },α∈[0,1]。
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| 尼早 ; 邱志平 ; .分布参数为模糊变量的结构体系模糊可靠性分析方法及应用.航空学报.(第05期),全文. * |
| 谢朝阳 ; 李贵杰 ; 彭忠明 ; 黄洪钟 ; .基于证据理论和代理模型的QMU分析.电子科技大学学报.(第01期),全文. * |
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