CN109829209A - Reliability analysis based on fuzzy method based on Perturbation Principle - Google Patents
Reliability analysis based on fuzzy method based on Perturbation Principle Download PDFInfo
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Abstract
The present invention discloses a kind of Calculation method of fuzzy reliability containing fuzzy random variable based on Perturbation Principle, is applied to reliability field.For the nonlinear Fuzzy-random Reliability problem of Structural functional equation, the present invention carries out perturbation quantization to fuzzy random variable using perturbation theory and fuzzy resolution theory, the fuzzy characteristics parameter of structural fuzzy stochastic variable is expressed as its true value and fuzzy perturbation is the sum of micro, and it is broken down into a series of amount of the section under horizontal cut sets, and it is equivalent to stochastic variable, mean value and standard deviation is calculated.Simultaneously to the micro equivalent selection of progress of perturbing.Secondly the f uzzy probabilistic density function of structure and Fuzzy function function decomposition are measured at a series of section under horizontal cut sets, and Taylor series expansion is carried out to its section upper lower limit value, obtain the perturbation value of joint probability density function and Structural functional equation.Sigmoid function is finally introduced as jump function, solution obtains Fuzzy Reliability section using immediate integration according to the definition of structural reliability.The method of the present invention is compared with traditional convex set method, and the reliability section being calculated is narrower under identical confidence level, engineer application of being more convenient for.
Description
Technical field
The invention belongs to reliability field, in particular to a kind of Calculating Method for Structural Reliability containing fuzzy random variable.
Background technique
There is a large amount of uncertainties, including stochastic uncertainty and cognition uncertainty in Practical Project problem.
For the structural analysis problem containing fuzzy uncertainty, it is if the reliability theory and method of application routine are described
It is very difficult, in addition irrelevantly using conventional reliability analysis method be likely to result in calculate result and actual conditions it is complete
Complete inconsistent feelings 0 occur.Therefore, have using the Fuzzy Reliability problem of reliability analysis based on fuzzy theory and technique study structure
There is certain realistic meaning.Especially for the structure containing fuzzy random variable, power function and probability density function are non-thread
When property, in usual reliability analysis based on fuzzy method frequently with convex set method it is just unsuitable, and obtained reliability section compared with
It is fuzzy.
Summary of the invention
In order to solve the above technical problems, the present invention proposes the reliability analysis based on fuzzy method based on Perturbation Principle, perturbing
On the basis of principle, perturbation analysis is carried out to Structural functional equation and probability density function, acquires structure using immediate integration
Fuzzy reliable angle value.
The technical solution adopted by the present invention are as follows: the reliability analysis based on fuzzy method based on Perturbation Principle, using perturbation theory
And fuzzy resolution theory has carried out perturbation quantization to fuzzy random variable, and structural fuzzy fail-safe analysis problem is converted into structure
Multidisciplinary systems problem analysis.Perturbation analysis is carried out to Structural functional equation and probability density function, utilizes immediate integration
Acquire Stucture Fuzzy Reliability value.
Further, specifically includes the following steps:
S1, the fuzzy characteristics parameter of structural fuzzy stochastic variable is expressed as by its true value and fuzzy perturbation using perturbation theory
It is the sum of micro；
S2, the fuzzy characteristics parameter decomposition of structural fuzzy stochastic variable is cut at a series of levels using fuzzy resolution theory
Section amount under collection, and it is equivalent to stochastic variable, mean value and standard deviation is calculated；
S3, according to step S1 and step S2, micro carry out equivalent selection to perturbing；
S4, by the f uzzy probabilistic density function of structure and Fuzzy function function decomposition at a series of section under horizontal cut sets
Amount, and Taylor series expansion is carried out to its section upper lower limit value, obtain taking the photograph for joint probability density function and Structural functional equation
Dynamic value；
S5, finally according to the definition of structural reliability, using immediate integration, introduce sigmoid function as step letter
Number, solution obtain Fuzzy Reliability section.
Further, the true value of fuzzy characteristics parameter described in step S1 and fuzzy perturbation is micro is calculated according to the following formula:
Wherein,For fuzzy characteristics parameter, X=(X1, X2... Xn) it is its true value.It is micro for fuzzy perturbation.
Further, described in step S2 by fuzzy characteristics parameter decomposition at a series of interval variable x under horizontal cut setsλ,
And it is equivalent to stochastic variable, then such as equally distributed stochastic variable acquires its mean valueWith standard
Difference
Further, perturbation described in step S3 measures the standard deviation of its equivalent stochastic variableIt takes
Further, the perturbation expansion of the f uzzy probabilistic density function and Fuzzy function function of structure described in step S4
It is as follows:
Further, the calculating formula of the Fuzzy Reliability of structure described in step S5 is as follows:
Beneficial effects of the present invention: the structural fuzzy analysis method for reliability of the invention based on Perturbation Principle, using taking the photograph
Dynamic theoretical and fuzzy resolution theory has carried out perturbation quantization to fuzzy random variable, and structural fuzzy fail-safe analysis problem is converted
At structure Multidisciplinary systems problem analysis.Firstly, the fuzzy characteristics parameter of structural fuzzy stochastic variable is expressed as its true value
A series of section amount the sum of micro with fuzzy perturbation, and being broken down under horizontal cut sets, and it is equivalent to stochastic variable,
Mean value and standard deviation is calculated.Simultaneously to the micro equivalent selection of progress of perturbing.Secondly by the f uzzy probabilistic density function of structure
And Fuzzy function function decomposition is measured at a series of section under horizontal cut sets, and carries out Taylor series exhibition to its section upper lower limit value
It opens, obtains the perturbation value of joint probability density function and Structural functional equation.Finally according to the definition of structural reliability, using straight
Integration method is connect, introduces sigmoid function as jump function, solution obtains Fuzzy Reliability section.What the method for the present invention obtained
Reliability section illustrates that the present processes are feasible effective and of the invention in the interval range that convex set method obtains
Method is compared with convex set method, and section is narrower apparent under identical confidence level, engineer application of being more convenient for.
Detailed description of the invention
Fig. 1 is the solution of the present invention flow chart；
Fig. 2 is simply supported beam force diagram provided in an embodiment of the present invention；
Fig. 3 is fuzzy random variable provided in an embodiment of the present inventionTriangular membership figure；
Fig. 4 is fuzzy random variable provided in an embodiment of the present inventionTriangular membership figure；
Fig. 5 is the reliability membership function figure of simply supported beam provided in an embodiment of the present invention.
Specific embodiment
For convenient for those skilled in the art understand that technology contents of the invention, with reference to the accompanying drawing to the content of present invention into one
Step is illustrated.
The structural fuzzy analysis method for reliability based on Perturbation Principle of the application is managed using perturbation theory and fuzzy decompose
Perturbation quantization has been carried out by fuzzy random variable, structural fuzzy fail-safe analysis problem is converted into structure Multidisciplinary systems
Problem analysis.Firstly, the fuzzy characteristics parameter of structural fuzzy stochastic variable is expressed as its true value and fuzzy perturbation is the sum of micro,
And it is broken down into a series of amount of the section under horizontal cut sets.Then by the f uzzy probabilistic density function of structure and Fuzzy function letter
Number resolves into a series of amount of the section under horizontal cut sets, and carries out Taylor series expansion to its section upper lower limit value, is combined
The perturbation expansion and its interval value of probability density function and Structural functional equation.According to the definition of structural reliability, using straight
Integration method is connect to solve to obtain the interval reliability under different level cut set.
It is as shown in Figure 1 the program flow chart of the application, the technical solution of the application are as follows: the structure mould based on Perturbation Principle
Paste analysis method for reliability, comprising:
S1, the fuzzy characteristics parameter of structural fuzzy stochastic variable is expressed as by its true value and fuzzy perturbation using perturbation theory
It is the sum of micro；
S2, the fuzzy characteristics parameter decomposition of structural fuzzy stochastic variable is cut at a series of levels using fuzzy resolution theory
Section amount under collection, and it is equivalent to stochastic variable, mean value and standard deviation is calculated；
S3, according to step S1 and step S2, micro carry out equivalent selection to perturbing；
S4, by the f uzzy probabilistic density function of structure and Fuzzy function function decomposition at a series of section under horizontal cut sets
Amount, and Taylor series expansion is carried out to its section upper lower limit value, obtain taking the photograph for joint probability density function and Structural functional equation
Dynamic value；
S5, finally according to the definition of structural reliability, using immediate integration, introduce sigmoid function as step letter
Number, solution obtain Fuzzy Reliability section.
Fuzzy random variable refers to that the characteristic parameter of stochastic variable is fuzzy number, as the mean value or standard deviation of stochastic variable are
Fuzzy number.
Step S1 specifically:
For the fuzzy characteristics parameter in fuzzy random variableIt can be expressed as its true value X=
(X1, X2... Xn) and fuzzy perturbation it is microThe sum of, i.e.,
The trace expression that then perturbs is
Step S2 specifically:
According to fuzzy resolution theorem, any one fuzzy variable is decomposed into a series of amount of the section under horizontal cut sets,
Then breakdown is as follows:
Wherein,It indicates in horizontal cut set xλUnder section bound,
By the interval variable x under horizontal cut setλIt is equivalent to stochastic variable, such as equally distributed stochastic variable, then acquired
Its mean valueWith standard deviation
Step S3 specifically:
The quantification of fuzzy random variable, key is in the selection of Perturbation, needs to guarantee that Perturbation is smaller than true value amount
It is more.Because standard deviation can measure the dispersion degree of a data set, standard deviation can be used as a kind of probabilistic
Estimate.When choosing the Perturbation of fuzzy variable, perturbation is made to measure the standard deviation of its equivalent stochastic variableIt takesTrue value X takes the mean value of its equivalent stochastic variable.
Step S4 specifically:
In Structural Reliability Theory, structural realism isWherein f (x) indicates joint probability
Density function, g (x) indicate Structural functional equation.Calculation of structure reliability degree formula is Pr=1-Pj, then structural reliability are as follows:
In formula, f (x1, x2..., xn) it is stochastic variable joint probability density function；
g(x1, x2..., xn) it is Structural functional equation.
When structure variable is fuzzy random variable, probability density function and Structural functional equation are all fuzzy, roots
According to fuzzy resolution theorem, a series of section amount being decomposed under horizontal cut sets, then breakdown is as follows:
Wherein,
It is illustrated respectively in horizontal cut set f (x1, x2..., xn)λUnder section bound；
It is illustrated respectively in horizontal cut set g (x1, x2..., xn)λUnder section bound.
When probability density function and Structural functional equation are nonlinear function, by the section bound under its horizontal cut set
Carry out Taylor series expansion at equivalent averages point, and omit high-order term more than two times, just obtain joint probability density function and
The perturbation expansion of Structural functional equation.Such as formula (5), (6) are shown.
Wherein,RespectivelyTo x1, x2..., xnPartial derivative exist
x1=X1λ, x2=X2λ..., xn=XnλThe value at place；
RespectivelyTo x1, x2..., xnPartial derivative exist
x1=X1λ, x2=X2λ..., xn=XnλThe value at place；
RespectivelyTo x1, x2..., xnPartial derivative exist
x1=X1λ, x2=X2λ..., xn=XnλThe value at place.
RespectivelyTo x1, x2..., xnPartial derivative exist
x1=X1λ, x2=X2λ..., xn=XnλThe value at place；
Step S5 specifically:
This explanation solves structural reliability using immediate integration.With immediate integration solve reliability critical issue it
One is exactly the regularization of integral domain, can introduce jump function h (), and formula (6) is converted to formula (7).In view of subsequent derivation side
Just it realizes, jump function is replaced using sigk (x) function, as shown in formula (8), wherein sigk (x)=1/ (1+exp (-
kx)).In order to make the value of sigk (x) function approach jump function as far as possible, k value takes biggish value, such as k=100 in calculating.
The interval value of joint probability density function and Structural functional equation in formula (5,6) is substituted into formula (8), is obtained
Fuzzy Reliability under each horizontal cut set:
Interval arithmetic is carried out to above formula, obtains reliability section bound of the Fuzzy Reliability under horizontal cut set:
Wherein,
To obtain section of the Fuzzy Reliability under each horizontal cut setIt is last to be divided according to fuzzy
Solution theorem provides the Fuzzy Reliability (11) of structure.
The present processes are illustrated below by way of specific example:
If the simply supported beam of uniform load is born, as shown in Fig. 2, its length l, cross-sectional width b, depth of section h, load?
For basic variable, i.e. l=4000mm, b=105, h=210mm.For fuzzy random variable, Normal DistributionThe mean value of distribution parameterFor fuzzy quantity, membership function is Triangle-Profile letter as shown in Figure 3
Number.The material of beam is 45 steel, intensityIt is fuzzy random variable, Normal DistributionIt is mould
Paste amount, membership function are as shown in Figure 4.Seek the reliability of structure.
Fuzzy random variable in this exampleWithConstitute the fuzzy vector field of structureTherefore fuzzy vector
?It is deployable for its true value and the sum of fuzzy perturbation is micro, be expressed as follows:
Wherein: X={ X1, X2,
According to fuzzy resolution theorem, it is broken down into the amount of the section under horizontal cut set:
Wherein, And calculate its equivalent uniform point
The mean value and standard deviation of cloth stochastic variable:μ2λ=550-30 λ,Take its PerturbationTrue value X1λ=μ1λ,
X2λ=μ2λ。
Known by the mechanics of materials, which is
According to the failure criteria of simply supported beam, Structural functional equation g () is determined are as follows:
The influence of correlation of variables is not considered, and the joint probability density function of input variable is writeable are as follows:
According to fuzzy resolution theorem, Structural functional equation and probability density function are resolved under a series of horizontal cut sets
Section amount, carries out Taylor series expansion for its section bound, and omit high-order term more than two times at equivalent averages point, just
To the perturbation expansion of Structural functional equation and joint probability density function.As follows.
Using immediate integration, the interval value of the above Structural functional equation and joint probability density function is substituted into formula
(9) in, the Fuzzy Reliability under each horizontal cut set is obtained:
Interval arithmetic is carried out to above formula, obtains reliability section bound of the Fuzzy Reliability under horizontal cut set:
Wherein,
Above formula is substituted into formula (10), reliability interval value is obtained
The Fuzzy Reliability of structure is obtained by fuzzy resolution theoremCalculated result column and table 1, and root
It is drawn shown in reliability membership function Fig. 5 according to 1 data of table.Meanwhile convex set method being used to acquire Fuzzy Reliability to compare.
Reliability value under 1 different level cut set of table
| Confidence level | Reliability section based on perturbation method | Reliability section based on convex set method |
| λ=0 | [0.4283,0.5290] | [0.0344,0.9137] |
| λ=0.1 | [0.4304,0.5209] | [0.0791,0.8634] |
| λ=0.2 | [0.4325,0.5128] | [0.1202,0.8166] |
| λ=0.3 | [0.4346,0.5048] | [0.1635,0.7674] |
| λ=0.4 | [0.4368,0.4969] | [0.2069,0.7182] |
| λ=0.5 | [0.4389,0.4890] | [0.2483,0.6710] |
| λ=0.6 | [0.4397,0.4802] | [0.2899,0.6240] |
| λ=0.7 | [0.4431,0.4730] | [0.3324,0.5756] |
| λ=0.8 | [0.4452,0.4652] | [0.3710,0.5313] |
| λ=0.9 | [0.4473,0.4573] | [0.4100,0.4865] |
| λ=1 | 0.4494 | 0.4454 |
According to table 1 and Fig. 5, fuzzy perturbation analysis approach is relative to traditional convex set method, the Fuzzy Reliability being calculated
Section is narrower, is more clear.When such as λ=0.1, Fuzzy Reliability section that fuzzy perturbation analysis approach is calculated be [0.4304,
0.5209] and Fuzzy Reliability section that convex set method is calculated is [0.0791,0.8634], it is seen that perturbation method acquires reliable
It is narrower than convex set method to spend interval range.
The application is directed to the structural nonlinear reliability analysis based on fuzzy problem containing fuzzy random variable, utilizes perturbation theory
And fuzzy resolution theory quantifies fuzzy random variable, structural fuzzy integrity problem is converted into the non-probability of structure can
By property problem, the ambiguity of fuzzy random variable is set by Structural functional equation and probability density function to be transmitted to structure reliable
Then degree solves reliability using immediate integration.In immediate integration solution procedure, Structural functional equation is carried out first
Integral domain regularization；Then first order perturbation analysis has been carried out for Structural functional equation and probability density function, using straight
Integration method is connect to solve to obtain structural reliability.
The application method is compared with convex set method, and section is narrower under identical confidence level, apparent, answers convenient for engineering
With.
Those of ordinary skill in the art will understand that the embodiments described herein, which is to help reader, understands this hair
Bright principle, it should be understood that protection scope of the present invention is not limited to such specific embodiments and embodiments.For ability
For the technical staff in domain, the invention may be variously modified and varied.All within the spirits and principles of the present invention, made
Any modification, equivalent substitution, improvement and etc. should be included within scope of the presently claimed invention.
Claims (7)
1. the reliability analysis based on fuzzy method based on Perturbation Principle, which is characterized in that use perturbation theory and fuzzy resolution theory
Perturbation quantization has been carried out to fuzzy random variable, structural fuzzy fail-safe analysis problem is converted into structure Multidisciplinary systems point
Analysis problem.Then immediate integration is utilized, solution obtains Fuzzy Reliability section.
2. the reliability analysis based on fuzzy method according to claim 1 based on Perturbation Principle, which is characterized in that specifically include
Following steps:
S1, the fuzzy characteristics parameter of structural fuzzy stochastic variable is expressed as by its true value using perturbation theory and fuzzy perturbation is micro
The sum of；
S2, using fuzzy resolution theory by the fuzzy characteristics parameter decomposition of structural fuzzy stochastic variable under a series of horizontal cut sets
Section amount, and be equivalent to stochastic variable, mean value and standard deviation be calculated；
S3, according to step S1 and step S2, micro carry out equivalent selection to perturbing；
S4, the f uzzy probabilistic density function of structure and Fuzzy function function decomposition are measured at a series of section under horizontal cut sets,
And Taylor series expansion is carried out to its section upper lower limit value, obtain the perturbation of joint probability density function and Structural functional equation
Value；
S5, finally according to the definition of structural reliability, using immediate integration, introduce sigmoid function as jump function, ask
Solution obtains Fuzzy Reliability section.
3. the reliability analysis based on fuzzy method according to claim 2 based on Perturbation Principle, which is characterized in that step S1 institute
It states the true value of fuzzy characteristics parameter and fuzzy perturbation is micro is calculated according to the following formula:
Wherein,For fuzzy characteristics parameter, X=(X1, X2... Xn) it is its true value.For
Fuzzy perturbation is micro.
4. the reliability analysis based on fuzzy method according to claim 3 based on Perturbation Principle, which is characterized in that step S2 institute
It states fuzzy characteristics parameter decomposition into a series of interval variable x under horizontal cut setsλ, and it is equivalent to stochastic variable, such as
Then the stochastic variable of even distribution acquires its mean valueWith standard deviation
5. the reliability analysis based on fuzzy method according to claim 4 based on Perturbation Principle, which is characterized in that step S3 institute
The perturbation stated measures the standard deviation of its equivalent stochastic variableIt takes
6. the reliability analysis based on fuzzy method according to claim 5 based on Perturbation Principle, which is characterized in that the step
The f uzzy probabilistic density function of structure described in S4 and the perturbation expansion of Fuzzy function function are as follows:
7. the reliability analysis based on fuzzy method according to claim 6 based on Perturbation Principle, which is characterized in that the step
The calculating formula of the Fuzzy Reliability of structure described in S5 is as follows:
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