CN109829209B - Fuzzy reliability analysis method based on perturbation principle - Google Patents
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Abstract
The invention discloses a fuzzy reliability calculation method containing fuzzy random variables based on a perturbation principle, which is applied to the field of reliability. Aiming at the problem of nonlinear fuzzy random reliability of a structural function, the fuzzy random variable fuzzy characteristic parameter is expressed as the sum of a true value and a fuzzy perturbation trace of the structural fuzzy random variable by utilizing perturbation theory and fuzzy decomposition theory, is decomposed into a series of interval quantities under a horizontal truncated set, is equivalent to the random variable, and is calculated to obtain a mean value and a standard deviation. And simultaneously, carrying out equivalent selection on the perturbation trace. And decomposing the fuzzy probability density function and the fuzzy function of the structure into a series of interval quantities under a horizontal truncated set, and performing Taylor series expansion on the upper and lower limit values of the interval to obtain perturbation values of the joint probability density function and the structural function. And finally, according to the definition of the structural reliability, introducing a sigmoid function as a step function by using a direct integration method, and solving to obtain a fuzzy reliability interval. Compared with the traditional convex set method, the method provided by the invention has the advantages that the calculated reliability interval is narrower under the same confidence level, and the engineering application is more convenient.
Description
Technical Field
The invention belongs to the field of reliability, and particularly relates to a structural reliability calculation method containing fuzzy random variables.
Background
There is a large amount of uncertainty in the actual engineering problem, including stochastic uncertainty and cognitive uncertainty. For structural analysis problems containing fuzzy uncertainty, if conventional reliability theory and method are applied to describe the structural analysis problems, even if the conventional reliability analysis method is not properly used, the situation 0 that the calculation result is completely inconsistent with the actual situation can occur. Therefore, the fuzzy reliability problem of the structure research by applying the fuzzy reliability analysis theory and the method has a certain practical significance. Particularly for a structure containing fuzzy random variables, when the function and the probability density function are nonlinear, a convex set method commonly adopted in a fuzzy reliability analysis method is not suitable, and an obtained reliability interval is fuzzy.
Disclosure of Invention
In order to solve the technical problems, the invention provides a fuzzy reliability analysis method based on a perturbation principle, which performs perturbation analysis on a structural function and a probability density function on the basis of the perturbation principle, and obtains a structural fuzzy reliability value by using a direct integration method.
The invention adopts the technical scheme that: the fuzzy reliability analysis method based on the perturbation principle adopts the perturbation theory and the fuzzy decomposition theory to carry out perturbation quantification on fuzzy random variables, and converts the structural fuzzy reliability analysis problem into the structural non-probability reliability analysis problem. And carrying out perturbation analysis on the structural function and the probability density function, and obtaining the structural blurring reliability value by using a direct integration method.
Further, the method specifically comprises the following steps:
s1, adopting a perturbation theory to represent a fuzzy characteristic parameter of a structural fuzzy random variable as the sum of a true value and a fuzzy perturbation trace;
s2, decomposing fuzzy characteristic parameters of the structure fuzzy random variable into interval amounts under a series of horizontal truncated sets by utilizing a fuzzy decomposition theory, and equivalent to the random variable, and calculating to obtain a mean value and a standard deviation;
s3, equivalently selecting the perturbation micro-quantity according to the step S1 and the step S2;
s4, decomposing the fuzzy probability density function and the fuzzy function of the structure into a series of interval quantities under a horizontal intercept, and performing Taylor series expansion on the upper and lower limit values of the interval to obtain perturbation values of the joint probability density function and the structure function;
and S5, finally, according to definition of the structural reliability, introducing a sigmoid function as a step function by using a direct integration method, and solving to obtain a fuzzy reliability interval.
Further, the true value and the blur disturbance trace of the blur feature parameter in step S1 are calculated according to the following formula:
wherein,,to blur feature parameters, x= (X 1 ，X 2 ，…X n ) Is true.Is a blurry perturbation trace.
Further, step S2 is to decompose the fuzzy characteristic parameters into interval variables x under a series of horizontal truncated sets λ And equivalent it to a random variable, e.g. uniformly distributed random variable, and then find the mean value thereofAnd standard deviation-> Represented in the horizontal intercept x λ The upper and lower limits of the interval below.
Further, the perturbation in step S3 measures the standard deviation of the equivalent random variableTaking out
Further, the perturbation expansion of the fuzzy probability density function and the fuzzy function of the structure described in step S4 is as follows:
further, the calculation formula of the fuzzy reliability of the structure described in step S5 is as follows:
P r representing structural reliability; p (P) rλ Representing the blur reliability under each horizontal intercept;representing a decomposition of the random variable in combination with the probability density function; />A decomposition formula representing a structural function; sigk (·) represents a step function.
The invention has the beneficial effects that: according to the structural fuzzy reliability analysis method based on the perturbation principle, perturbation quantification is carried out on fuzzy random variables by utilizing the perturbation theory and the fuzzy decomposition theory, and the structural fuzzy reliability analysis problem is converted into the structural non-probability reliability analysis problem. Firstly, the fuzzy characteristic parameters of the structure fuzzy random variable are expressed as the sum of true values and fuzzy perturbation trace amounts, the fuzzy characteristic parameters are decomposed into interval amounts under a series of horizontal truncated sets, the interval amounts are equivalent to random variables, and the mean value and the standard deviation are calculated. And simultaneously, carrying out equivalent selection on the perturbation trace. And decomposing the fuzzy probability density function and the fuzzy function of the structure into a series of interval quantities under a horizontal truncated set, and performing Taylor series expansion on the upper and lower limit values of the interval to obtain perturbation values of the joint probability density function and the structural function. And finally, according to the definition of the structural reliability, introducing a sigmoid function as a step function by using a direct integration method, and solving to obtain a fuzzy reliability interval. The reliability interval obtained by the method is in the interval range obtained by the convex set method, which proves that the method is feasible and effective, and compared with the convex set method, the method has the advantages that the interval is narrower and clearer under the same confidence level, and the engineering application is more convenient.
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FIG. 1 is a flow chart of an embodiment of the present invention;
FIG. 2 is a diagram of a simple beam according to an embodiment of the present invention;
FIG. 3 is a triangle membership function diagram of a fuzzy random variable provided by an embodiment of the present invention;
FIG. 4 is a triangle membership function diagram of a fuzzy random variable provided by an embodiment of the present invention;
fig. 5 is a reliability membership function diagram of a simply supported beam according to an embodiment of the present invention.
Detailed Description
The present invention will be further explained below with reference to the drawings in order to facilitate understanding of technical contents of the present invention to those skilled in the art.
According to the structural fuzzy reliability analysis method based on the perturbation principle, perturbation quantification is carried out on fuzzy random variables by utilizing the perturbation theory and the fuzzy decomposition theory, and the structural fuzzy reliability analysis problem is converted into the structural non-probability reliability analysis problem. First, the fuzzy characteristic parameters of the structural fuzzy random variable are expressed as the sum of its true and fuzzy perturbation micro-quantities and are decomposed into a series of region amounts under a horizontal intercept. And then decomposing the fuzzy probability density function and the fuzzy function of the structure into a series of interval quantities under a horizontal truncated set, and performing Taylor series expansion on the upper and lower limit values of the interval to obtain perturbation expansion of the joint probability density function and the structural function and interval values thereof. And according to the definition of the structural reliability, solving by using a direct integration method to obtain the interval reliability under different horizontal truncated sets.
As shown in fig. 1, a scheme flow chart of the present application is shown in the technical scheme of the present application: the structure blurring reliability analysis method based on the perturbation principle comprises the following steps:
s1, adopting a perturbation theory to represent a fuzzy characteristic parameter of a structural fuzzy random variable as the sum of a true value and a fuzzy perturbation trace;
s2, decomposing fuzzy characteristic parameters of the structure fuzzy random variable into interval amounts under a series of horizontal truncated sets by utilizing a fuzzy decomposition theory, and equivalent to the random variable, and calculating to obtain a mean value and a standard deviation;
s3, equivalently selecting the perturbation micro-quantity according to the step S1 and the step S2;
s4, decomposing the fuzzy probability density function and the fuzzy function of the structure into a series of interval quantities under a horizontal intercept, and performing Taylor series expansion on the upper and lower limit values of the interval to obtain perturbation values of the joint probability density function and the structure function;
and S5, finally, according to definition of the structural reliability, introducing a sigmoid function as a step function by using a direct integration method, and solving to obtain a fuzzy reliability interval.
Fuzzy random variable means that the characteristic parameter of the random variable is a fuzzy number, for example, the mean value or standard deviation of the random variable is a fuzzy number.
The step S1 specifically comprises the following steps:
for fuzzy characteristic parameters in fuzzy random variablesIt can be expressed as its true value x= (X 1 ，X 2 ，…X n ) Micro-scale of blurred perturbation>The sum of, i.e
The perturbation is expressed in trace amount as
The step S2 specifically comprises the following steps:
according to the fuzzy decomposition theorem, any fuzzy variable can be decomposed into a series of interval amounts under a horizontal truncated set, and the decomposition formula is as follows:
(2)
wherein,, represented in the horizontal intercept x λ The upper and lower limits of the interval of lower->
Interval variable x under horizontal intercept λ Equivalent to random variables, e.g. uniformly distributed random variables, and then averagingAnd standard deviation->
The step S3 specifically comprises the following steps:
quantification of fuzzy random variables is critical in selection of the shot quantity, and needs to be ensuredThe amount of uptake is much smaller than the amount of true value. Because the standard deviation can measure the degree of dispersion of a data set, the standard deviation can be used as a measure of uncertainty. When the shooting quantity of the fuzzy variable is selected, the shooting quantity is enabled to measure the standard deviation of the equivalent random variableTaking outThe true value X takes the mean of its equivalent random variables.
The step S4 specifically comprises the following steps:
in the structure reliability theory, the structure failure probability is P j ＝∫ g ( x ) ≤0 f (x) dx, wherein f (x) represents the joint probability density function and g (x) represents the structural function. The structural reliability calculation formula is P r ＝1-P j The structural reliability is:
wherein f (x) 1 ，x 2 ，…，x n) Combining probability density functions for random variables;
g(x 1 ，x 2 ，…，x n ) Is a structural function.
When the structural variable is a fuzzy random variable, the probability density function and the structural function are fuzzy, and can be decomposed into a series of interval quantities under a horizontal truncated set according to the fuzzy decomposition theorem, and the decomposition formula is as follows:
wherein,, respectively expressed in horizontal intercept f (x 1 ，x 2 ，…，x n ) The upper and lower limits of the interval under λ;
respectively expressed in horizontal intercept g (x 1 ，x 2 ，…，x n ) The upper and lower limits of the interval under λ.
When the probability density function and the structural function are nonlinear functions, the upper and lower limits of the interval under the horizontal intercept are subjected to Taylor series expansion at the equivalent mean value point, and the higher order terms above two times are omitted, so that the perturbation expansion of the joint probability density function and the structural function is obtained. As shown in formulas (5) and (6).
Wherein,,respectively->For x 1 ，x 2 ，…，x n The partial derivative of (2) is x 1 ＝X 1λ ，x 2 ＝X 2λ ，…，x n ＝X nλ A value at;
respectively->For x 1 ，x 2 ，…，x n The partial derivative of (2) is x 1 ＝X 1λ ，x 2 ＝X 2λ ，…，x n ＝X nλ A value at;
respectively->For x 1 ，x 2 ，…，x n The partial derivative of (2) is x 1 ＝X 1λ ，x 2 ＝X 2λ ，…，x n ＝X nλ A value at.
Respectively->For x 1 ，x 2 ，…，x n The partial derivative of (2) is x 1 ＝X 1λ ，x 2 ＝X 2λ ，…，x n ＝X nλ A value at;
the step S5 specifically comprises the following steps:
the present description solves for structural reliability using direct integration. One of the key problems in solving reliability with the direct integration method is regularization of the integration region, which can introduce a step function h (), converting equation (6) to equation (7). In view of the convenience of the subsequent derivation, a sigk (x) function is employed instead of the step function, as shown in formula (8), where sigk (x) =1/(1+exp (-kx)). In order to make the value of the sigk (x) function approach the step function as much as possible, the k value takes a larger value in the calculation, for example k=100.
Substituting the interval values of the joint probability density function and the structural function in the formulas (5 and 6) into the formula (8) to obtain the fuzzy reliability under each horizontal truncated set:
performing interval operation on the above method to obtain upper and lower boundaries of a reliability interval of the fuzzy reliability under the horizontal intercept:
wherein,,
thereby obtaining the interval of the fuzzy reliability under each horizontal interceptAnd finally, giving the fuzzy reliability (11) of the structure according to the fuzzy decomposition theorem.
The method of the present application is illustrated by the following specific examples:
as shown in fig. 2, the simple beam is provided with uniformly distributed load, and the length l, the section width b and the section height h of the simple beam are all basic variables, namely l=4000 mm, b=105 and h=210 mm.Is fuzzified random variable, obeys normal distributionMean>The membership function is a triangle distribution function as shown in fig. 3 for the blur amount. The material of the beam is 45 steel, the strength of which is +.>Is a fuzzy random variable subject to normal distribution +.> Is the amount of blurring, and its membership function is shown in fig. 4. And (5) obtaining the reliability of the structure.
Fuzzy random variables in this exampleAnd->Fuzzy vector field of structure>So that the blur vector field->Can be developed into the sum of true value and fuzzy perturbation trace, and is expressed as follows:
wherein: x= { X 1 ，X 2 }，
According to the fuzzy decomposition theorem, it is decomposed into interval amounts under the horizontal intercept:
wherein,,and calculating the mean value and standard deviation of the equivalent uniformly distributed random variables: />μ 2λ ＝550-30λ，/>Taking its perturbation +.>True value X 1λ ＝μ 1λ ，X 2λ ＝μ 2λ 。
The maximum stress of the simply supported beam is known as
According to the failure criterion of the simply supported beams, determining the structural function g (·) as follows:
irrespective of the influence of the variable dependence, the joint probability density function of the input variables can be written as:
according to the fuzzy decomposition theorem, decomposing the structural function and the probability density function into a series of region amounts under a horizontal truncated set, performing Taylor series expansion on the upper and lower limits of the region at the equivalent mean value point, and omitting the higher order items above two times to obtain the perturbation expansion of the structural function and the joint probability density function. As shown below.
Substituting the interval values of the structural function and the joint probability density function into a formula (9) by using a direct integration method to obtain the fuzzy reliability under each horizontal truncated set:
performing interval operation on the above method to obtain upper and lower boundaries of a reliability interval of the fuzzy reliability under the horizontal intercept:
wherein,,
substituting the above formula into formula (10) to obtain reliability interval value
Obtaining fuzzy reliability of structure by fuzzy decomposition theoremThe results are shown in Table 1, and the reliability membership functions are plotted according to the data in Table 1 as shown in FIG. 5. Meanwhile, the fuzzy reliability is obtained by adopting a convex set method for comparison.
Table 1 reliability values under different horizontal truncations
According to the table 1 and fig. 5, compared with the traditional convex set method, the fuzzy perturbation analysis method has the advantages that the calculated fuzzy reliability interval is narrower and clearer. When λ=0.1, the fuzzy reliability interval calculated by the fuzzy perturbation analysis method is [0.4304,0.5209] and the fuzzy reliability interval calculated by the convex set method is [0.0791,0.8634], and the reliability interval range obtained by the visible perturbation method is narrower than that obtained by the convex set method.
Aiming at the structural nonlinear fuzzy reliability analysis problem containing fuzzy random variables, the fuzzy random variables are quantized by utilizing perturbation theory and fuzzy decomposition theory, the structural fuzzy reliability problem is converted into structural non-probability reliability problem, the fuzziness of the fuzzy random variables is transferred to structural reliability through a structural function and a probability density function, and then the reliability is solved by utilizing a direct integration method. In the direct integration method solving process, firstly, regularization of an integration region is carried out on a structural function; and then performing first order perturbation analysis on the structural function and the probability density function, and solving by using a direct integration method to obtain the structural reliability.
Compared with the convex set method, the method has narrower interval and clearer interval under the same confidence level, and is convenient for engineering application.
Those of ordinary skill in the art will recognize that the embodiments described herein are for the purpose of aiding the reader in understanding the principles of the present invention and should be understood that the scope of the invention is not limited to such specific statements and embodiments. Various modifications and variations of the present invention will be apparent to those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the scope of the claims of the present invention.
Claims (3)
1. The fuzzy reliability analysis method based on the perturbation principle is characterized in that perturbation quantification is carried out on fuzzy random variables by adopting the perturbation theory and the fuzzy decomposition theory, and the structural fuzzy reliability analysis problem is converted into the structural non-probability reliability analysis problem; then, solving by using a direct integration method to obtain a fuzzy reliability interval;
the method specifically comprises the following steps:
s1, adopting a perturbation theory to represent a fuzzy characteristic parameter of a structural fuzzy random variable as the sum of a true value and a fuzzy perturbation trace;
s2, decomposing fuzzy characteristic parameters of the structure fuzzy random variable into interval amounts under a series of horizontal truncated sets by utilizing a fuzzy decomposition theory, and equivalent to the random variable, and calculating to obtain a mean value and a standard deviation;
s3, equivalently selecting the perturbation micro-quantity according to the step S1 and the step S2;
s4, decomposing the fuzzy probability density function and the fuzzy function of the structure into a series of interval quantities under a horizontal intercept, and performing Taylor series expansion on the upper and lower limit values of the interval to obtain perturbation values of the joint probability density function and the structure function;
s5, finally, according to definition of the structural reliability, introducing a sigmoid function as a step function by using a direct integration method, and solving to obtain a fuzzy reliability interval;
and (2) calculating the true value and the fuzzy perturbation trace of the fuzzy characteristic parameters in the step (S1) according to the following formula:
wherein,,to blur feature parameters, x= (X 1 ，X 2 ，…X n ) True value thereof; />Is a fuzzy perturbation trace;
step S2 of decomposing the fuzzy characteristic parameters into interval variables x under a series of level truncations λ And equivalent it to be uniformRandom variable of distribution, and then find its average valueAnd standard deviation-> Representing the upper and lower limits of the interval under the horizontal intercept x lambda;
step S3, measuring standard deviation of equivalent random variables of the perturbation micro-quantityGet->
2. The method according to claim 1, wherein the fuzzy probability density function and the perturbation expansion of the fuzzy function of the structure in the step S4 are as follows:
3. the method for analyzing the blur reliability based on the perturbation principle according to claim 2, wherein the structural blur reliability in step S5 is calculated as follows:
P r representing structural reliability; p (P) rλ Representing the blur reliability under each horizontal intercept;
representing a decomposition of the random variable in combination with the probability density function;
a decomposition formula representing a structural function; sigk (·) represents a step function.
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