CN106202707A - A kind of structural stress Strength Interference Model set analysis method for reliability based on Lycoperdon polymorphum Vitt confidence interval - Google Patents

Abstract

The invention discloses a kind of structural stress Strength Interference Model set analysis method for reliability based on Lycoperdon polymorphum Vitt confidence interval.Sample information for structural stress and intensive parameter is limited, statistical nature is difficult to determine and causes conventional probability thought cannot provide the problem that its average is effectively estimated, the present invention is based on grey correlation theory, first define topological relation and the grey distance measure of distance relation between finite sample, and then, by the grey generation of grey distance measure being obtained the estimated value of structural stress and strength mean value；By defining and solving gray density index, it is thus achieved that meet the stress under certain Lycoperdon polymorphum Vitt confidence level and the confidence interval of strength mean value；Finally, introduce the structuralreliability theory under non-Making by Probability Sets theoretical frame, derive the structural stress Strength Interference Model two dimension degree of reiability index with confidence level meaning, it is achieved from the reasonable mapping of the Analysis of structural reliability of sample with confidence evaluation.

Description

A kind of structural stress-Strength Interference Model set based on Lycoperdon polymorphum Vitt confidence interval is reliable Property analyze method

Technical field

The present invention relates to finite sample Estimating Confidence Interval and structure safety evaluation technical field, particularly to a kind of base In gray zone and structural stress-Strength Interference Model analysis method for reliability of meeting confidence degree requirement, for realizing work Lean information in journey, minority according under the conditions of parameter uncertainty rationally quantify and structural stress and intensity two-dimensional ensemble interfere mould Under type, the valid metric of security postures provides feasible treating method.

Background technology

The safety issue of structure, particularly structural strength problem are the cores that engineering design field is paid close attention to.Due to material The probabilistic effect of dispersibility, load significantly exists, and is limited by experimentation cost and cycle, often in Practical Project in addition Obtainable sample information is extremely limited, therefore, how to utilize small-scale sample data, completes the reasonable of parameter uncertainty Quantify and the accurate evaluation of safety of structure, there is significant engineering practical value.

If wanting to realize probabilistic quantization and the rational evaluation of safety of structure under finite sample, it is necessary first to setting up can The method for parameter estimation of letter.Traditional method for parameter estimation proposes according to probability mathematical statistics, is from large sample number According to statistical regularity consider a problem, and with being estimated value and evaluate at the probability that estimation interval occurs the confidence of estimation interval Degree.But owing to the test number (TN) of the most of structure problem of all restrictions is less, it is thus achieved that data sample amount less, it is impossible to according to Experience statistically determines its regularity of distribution before.Therefore, traditional probability mathematical statistics is applied to there will be bigger on the contrary Risk.

Additionally, Analysis of structural reliability is the most effective hands of structural safety Situation Assessment under currently processed condition of uncertainty Section, mainly includes probabilistic reliability method and the big class of Multidisciplinary systems method two.Probabilistic reliability theoretical developments is ripe, but sample This demand is big, and engineering adaptability is poor；Multidisciplinary systems based on sets theory analyzes method, when processing Small Sample Size There is natural advantage, but current method is not from test sample, does not also consider that the structure under confidence level requirement is reliable Property index computational problem.

To sum up, as based on how finite sample data, carry out the method for parameter estimation with confidence degree, and and then Realize meeting the structure Multidisciplinary systems analysis under confidence level requirement, be current academia and the focus of engineering circles extensive concern Problem.The present invention is directed to structural stress-Strength Interference Model, theoretical by introducing grey mathematics, and define the Lycoperdon polymorphum Vitt between sample Distance measure, it is achieved the quick acquisition of sample average confidence interval；Integrated structure non-Making by Probability Sets analysis method for reliability, finally Establish safety of structure under confidence level and reliability double standards rationally passes judgment on criterion, promotes structural uncertainty analysis The through engineering approaches process of method.

Summary of the invention

The technical problem to be solved in the present invention is: overcome the deficiencies in the prior art, it is provided that a kind of based on Lycoperdon polymorphum Vitt confidence interval Structural stress-Strength Interference Model set analysis method for reliability, take into full account Practical Project problem pilot scale sample sample size Restriction, based on grey mathematics theory, by definition sample between grey distance measure, build meet certain Lycoperdon polymorphum Vitt confidence The structural stress of degree requirement and strength mean value quantized interval；It is re-introduced into the structural reliability index theoretical based on non-Making by Probability Sets, That finally set up structure-oriented stress-intensity set Interference Model and that there is level of confidence Analysis of structural reliability method. Proposed method can realize the complete procedure checked by Small Sample Database to structural safety.

The technical solution used in the present invention is: a kind of structural stress-Strength Interference Model collection based on Lycoperdon polymorphum Vitt confidence interval Close analysis method for reliability, it is achieved step is as follows:

The first step: consider that structural stress S and intensity R have uncertainty, can carry out mathematical character by following formula:

$S\∈S^I=\[\underset{\‾}{S},\stackrel{\‾}{S}\]=\[S^c-S^r,S^c+S^r\]$

With

$R\∈R^I=\[\underset{\‾}{R},\stackrel{\‾}{R}\]=\[R^c-R^r,R^c+R^r\]$

Wherein, S^IAnd R^IRepresenting structural stress and the feasible interval of intensity respectively, S and R represents stress interval and intensity interval Lower bound,WithRepresent the interval upper bound with intensity interval of stress, S^cAnd R^cIt is the average of structural stress and intensity, S^rAnd R^rIt is Structural stress and the radius of intensity.Under normal circumstances, the average of structural stress and intensity meets R^c>S^c, additionally, the present invention ties Structure stress and intensity radius S^rAnd R^rValue be known quantity.

Second step: in order to realize structural stress S and the interval expression of intensity R in the first step, need to be by actual experimental condition Or numerical simulation means, the increasing sequence first obtaining description scheme stress S and intensity R finite sample is as follows:

S_data={ S_data(1),S_data(2),...,S_data(m)And R_data={ R_data(1),R_data(2),...,R_data(n)}

Wherein, m and n represents stress sample sequence and the sample size of intensity sample sequence respectively and meets: 3 ＜ m, n≤ 10.And then, based on grey correlation theory, define stress sample sequence S respectively_dataWith intensity sample sequence R_dataIn any sample Grey distance measure between this and observation sample is as follows:

$d_g(S_i,S_j)=\frac{\mathrm{\ξ}\left|\right|d(S_i,S_{data})|{|}_{\mathrm{\∞}}}{|S_i-S_j|+\mathrm{\ξ}\left|\right|d(S_i,S_{data})|{|}_{\mathrm{\∞}}},i,j=1,2,...,m$

With

$d_g(R_i,R_j)=\frac{\mathrm{\ξ}\left|\right|d(R_i,R_{data})|{|}_{\mathrm{\∞}}}{|R_i-R_j|+\mathrm{\ξ}\left|\right|d(R_i,R_{data})|{|}_{\mathrm{\∞}}},i,j=1,2,...,n$

Wherein, d_g(S_i,S_j) and d_g(R_i,R_j) represent respectively towards S_dataAnd R_dataGrey distance measure, | | d (S_i, S_data)||_∞With | | d (R_i,R_data)||_∞Represent stress and the intensity sample Infinite Norm corresponding to sample sequence respectively, | | for Signed magnitude arithmetic(al) accords with, and ξ is resolution ratio, and value is 0.5 here, i and j is counting index.

3rd step: according to the grey distance measure of second step definition, with S_dataAnd R_dataIn each sample as observation sample This, calculate the grey distance measure of itself and whole sample sequence respectively, and be averaging processing result, obtains towards completely should The averaged measure function of power and intensity sample sequenceWithAs follows:

$J_{S_i}=\frac{1}{m}\underset{j=1}{\overset{m}{\Σ}}{d}_g(S_i,S_j),i=1,2,...,m$

With

$J_{R_i}=\frac{1}{n}\underset{j=1}{\overset{n}{\Σ}}{d}_g(R_i,R_j),i=1,2,...,n$

Recycling method for normalizing, willWithIt is converted into weighting functionWithAnd then, theoretical in conjunction with grey mathematics In cumulative ash generate thought, calculate the Lycoperdon polymorphum Vitt estimated value of structural stress based on finite sample data and strength mean valueWithHere, weighting functionRepresent sample sequence S_dataMiddle sample S_iIn Lycoperdon polymorphum Vitt estimated valueRatio shared by, weighting functionRepresent sample sequence R_dataMiddle sample R_iIn Lycoperdon polymorphum Vitt estimated valueRatio shared by, then has:And

4th step: the Lycoperdon polymorphum Vitt estimated value that the 3rd step is calculatedWithAs new sample, it is added separately to primitive stress Sample sequence S_dataWith green strength sample sequence R_dataIn, then, sample sequence is updated to:

With

In order to probe into structural stress and strength mean value S^cAnd R^cWith Lycoperdon polymorphum Vitt estimated valueWithBetween mathematical, utilize The grey distance measure expression formula set up in two steps, sets up the following expression under gray density index θ:

With

Wherein,WithRepresent stress and the Lycoperdon polymorphum Vitt estimated value of strength mean value respectivelyWithSame sample Renewal sequenceWithBetween grey distance measure, assignment gray density index θ also solves above formula, respectively obtains current ash close Average S under degree level^cAnd R^cCan row bound, it may be assumed that

With

Wherein,S ^c (θ) and Represent the lower bound in the feasible interval of stress average under given gray density index θ and upper respectively Boundary,R ^c (θ) and Represent lower bound and the upper bound in the feasible interval of strength mean value under given gray density index θ respectively, here,WithIt is based respectively on Sample Refreshment sequenceWithTopological relation and distance between middle finite sample are closed System, reflects S^cAnd R^cWith Lycoperdon polymorphum Vitt estimated valueWithBetween matching degree；Gray density index θ meetsIts value is more Greatly, average S is shown^cAnd R^cOccur in Lycoperdon polymorphum Vitt estimated valueWithNeighbouring probability is the biggest.

5th step: the valued space of traversal gray density index θ, sets up the gray density letter of structural stress and strength mean value respectively Number f_θ(S^c) and f_θ(R^c), given level of confidence 1-α, utilize numerical integration method to calculate and meet Lycoperdon polymorphum Vitt confidence level requirement Stress and strength mean value S^cAnd R^cConfidence intervalWithHere, the value model of level of confidence 1-α Enclose and be: 0 ＜ 1-α≤1, in the present invention, level of confidence is entered as 0.975, i.e. α=0.025.

6th step: meet the average confidence interval that Lycoperdon polymorphum Vitt confidence level requires according to what the 5th step was tried to achieveWithIn conjunction with the structural stress known and intensity radius S^rAnd R^r, based on area ratio thought, for typical condition, it may be assumed that Structure two-dimensional stress-Strength Interference Model should meet:

$\underset{\‾}{S_{1-a}^c}-S^r\≤\underset{\‾}{R_{1-a}^c}-R^r\≤\stackrel{\‾}{S_{1-a}^c}+S^r\≤\stackrel{\‾}{R_{1-a}^c}+R^r$

Build and solve satisfied (1-α)²Under level of confidence, the set of structure two-dimensional stress-Strength Interference Model is reliable Property tolerance as follows:

$R_{s,{(1-\mathrm{\α})}^2}=1-\frac{{\[(\stackrel{\‾}{S_{1-a}^c}+S^r)-(\underset{\‾}{R_{1-a}^c}-R^r)\]}^2}{8S^r{R}^r}$

To sum up, the confidence degree that meets from sample can be realized and require the comprehensive of lower structure non-Making by Probability Sets reliability Evaluate.

Present invention advantage compared with prior art is:

The invention provides the new approaches processing under finite sample the Analysis of structural reliability containing confidence level requirement, make up and Perfect existing structure Reliability Analysis Theory and the limitation of method.First, the topology pass between stress and intensity sample is utilized System and range information, determined grey distance measure, and then established the stress under Lycoperdon polymorphum Vitt confidence level requirement and the amount of strength mean value Change interval；Again non-to confidence interval and structure Making by Probability Sets reliability theory is combined, it is achieved that from sample, comprise simultaneously The safety of structure of confidence level and reliability double standards is checked, and the minute design for structure provides the most theoretical Hold.

Accompanying drawing explanation

Fig. 1 is present invention structural stress based on Lycoperdon polymorphum Vitt confidence interval-Strength Interference Model fail-safe analysis flow chart；

Fig. 2 is the mathematical expression schematic diagram that the present invention is directed to structural stress-Strength Interference Model；

Fig. 3 is stress or strength mean value interval border schematic diagram under the different gray density indexs that the present invention proposes；

Fig. 4 is that under the Lycoperdon polymorphum Vitt confidence level that the present invention proposes, structural stress or average confidence interval calculate schematic diagram；

Fig. 5 is that the confidence level that meets that the present invention proposes requires that structural stress-Strength Interference Model set reliability calculating shows It is intended to.

Detailed description of the invention

Below in conjunction with the accompanying drawings and detailed description of the invention further illustrates the present invention.

As it is shown in figure 1, the present invention proposes a kind of structural stress-Strength Interference Model set based on Lycoperdon polymorphum Vitt confidence interval Analysis method for reliability, comprises the following steps:

(1) as shown in Figure 2, it is considered to structural stress S and intensity R have uncertainty, mathematical character can be carried out by following formula:

$S\∈S^I=\[\underset{\‾}{S},\stackrel{\‾}{S}\]=\[S^c-S^r,S^c+S^r\]$

With

$R\∈R^I=\[\underset{\‾}{R},\stackrel{\‾}{R}\]=\[R^c-R^r,R^c+R^r\]$

Wherein, S^IAnd R^IRepresenting structural stress and the feasible interval of intensity respectively, S and R represents stress interval and intensity interval Lower bound,WithRepresent the interval upper bound with intensity interval of stress, S^cAnd R^cIt is the average of structural stress and intensity, S^rAnd R^rIt is Structural stress and the radius of intensity.Under normal circumstances, the average of structural stress and intensity meets R^c>S^c, additionally, the present invention ties Structure stress and intensity radius S^rAnd R^rValue be known quantity.

(2) in order to realize structural stress S and the interval expression of intensity R in the first step, need to be by actual experimental condition or number Value simulation means, the increasing sequence first obtaining description scheme stress S and intensity R finite sample is as follows:

S_data={ S_data(1),S_data(2),...,S_data(m)And R_data={ R_data(1),R_data(2),...,R_data(n)}

Wherein, m and n represents stress sample sequence and the sample size of intensity sample sequence respectively and meets: 3 ＜ m, n≤ 10.And then, based on grey correlation theory, define stress sample sequence S respectively_dataWith intensity sample sequence R_dataIn any sample Grey distance measure between this and observation sample is as follows:

$d_g(S_i,S_j)=\frac{\mathrm{\ξ}\left|\right|d(S_i,S_{data})|{|}_{\mathrm{\∞}}}{|S_i-S_j|+\mathrm{\ξ}\left|\right|d(S_i,S_{data})|{|}_{\mathrm{\∞}}},i,j=1,2,...,m$

With

$d_g(R_i,R_j)=\frac{\mathrm{\ξ}\left|\right|d(R_i,R_{data})|{|}_{\mathrm{\∞}}}{|R_i-R_j|+\mathrm{\ξ}\left|\right|d(R_i,R_{data})|{|}_{\mathrm{\∞}}},i,j=1,2,...,n$

Wherein, d_g(S_i,S_j) and d_g(R_i,R_j) represent respectively towards S_dataAnd R_dataGrey distance measure, | | d (S_i, S_data)||_∞With | | d (R_i,R_data)||_∞Representing stress and the intensity sample Infinite Norm corresponding to sample sequence respectively, it is concrete Expression formula is:

$\left|\right|d(S_i,S_{data})|{|}_{\mathrm{\∞}}=\underset{k}{max}\{|S_{data\left(k\right)}-S_i|,k=1,2,...,m\}$

With

$\left|\right|d(R_i,R_{data})|{|}_{\mathrm{\∞}}=\underset{k}{max}\{|R_{data\left(k\right)}-R_i|,k=1,2,...,n\}$

| | according with for signed magnitude arithmetic(al), ξ is resolution ratio, and value is 0.5 here, and i, j and k represent counting index.

(3) according to the grey distance measure of second step definition, with S_dataAnd R_dataIn each sample as observation sample, Calculate the grey distance measure of itself and whole sample sequence respectively, and result is averaging processing, obtain towards complete stress Averaged measure function with intensity sample sequenceWithAs follows:

$J_{S_i}=\frac{1}{m}\underset{j=1}{\overset{m}{\Σ}}{d}_g(S_i,S_j),i=1,2,...,m$

With

$J_{R_i}=\frac{1}{n}\underset{j=1}{\overset{n}{\Σ}}{d}_g(R_i,R_j),i=1,2,...,n$

Recycling method for normalizing, willWithIt is converted into weighting functionWithThat is:

With

And then, generate thought in conjunction with the cumulative ash in grey mathematics theory, calculate structure based on finite sample data The Lycoperdon polymorphum Vitt estimated value of stress and strength mean value:WithHere, weighting functionRepresent sample sequence Row S_dataMiddle sample S_iIn Lycoperdon polymorphum Vitt estimated valueRatio shared by, weighting functionRepresent sample sequence R_dataMiddle sample R_iAt ash Color estimated valueRatio shared by, then has:And

(4) the Lycoperdon polymorphum Vitt estimated value that the 3rd step is calculatedWithAs new sample, it is added separately to primitive stress sample Sequence S_dataWith green strength sample sequence R_dataIn, then, sample sequence is updated to:

With

In order to probe into structural stress and strength mean value S^cAnd R^cWith Lycoperdon polymorphum Vitt estimated valueWithBetween mathematical, utilize The grey distance measure expression formula set up in two steps, sets up the following expression under gray density index θ:

With

As it is shown on figure 3, due to grey distance measureWithThere is piecewise monotonic, i.e. when WithTime,WithMonotonic increase, whenWithTime,WithDull Successively decrease, whenWithTime,WithTake maximum 1.Therefore, once S in above-mentioned inequality^cAnd R^c? Knowing, gray density index θ can uniquely determine, otherwise, if θ is it is known that for the calculating of structural stress and strength mean value by equivalent conversion Boundary value problem is solved for piecewise function.I.e. whenWithTime, have:

$d_g\left(S^c,\hat{S}\right)=\frac{\mathrm{\ξ}\left|\right|d\left(S^c,S_{data}^{*}\right)|{|}_{\mathrm{\∞}}}{|S^c-\hat{S}|+\mathrm{\ξ}\left|\right|d\left(S^c,S_{data}^{*}\right)|{|}_{\mathrm{\∞}}}\≥\mathrm{\θ}\⇒S^c\≥\hat{S}-0.5\left(1-\mathrm{\θ}\right)\left|\right|d\left(S^c,S_{data}^{*}\right)|{|}_{\mathrm{\∞}}$

With

$d_g\left(R^c,\hat{R}\right)=\frac{\mathrm{\ξ}\left|\right|d\left(R^c,R_{data}^{*}\right)|{|}_{\mathrm{\∞}}}{|R^c-\hat{R}|+\mathrm{\ξ}\left|\right|d\left(R^c,R_{data}^{*}\right)|{|}_{\mathrm{\∞}}}\≥\mathrm{\θ}\⇒R^c\≥\hat{R}-0.5\left(1-\mathrm{\θ}\right)\left|\right|d\left(R^c,R_{data}^{*}\right)|{|}_{\mathrm{\∞}}$

Otherwise, whenWithTime, have:

$d_g\left(S^c,\hat{S}\right)=\frac{\mathrm{\ξ}\left|\right|d\left(S^c,S_{data}^{*}\right)|{|}_{\mathrm{\∞}}}{|\hat{S}-S^c|+\mathrm{\ξ}\left|\right|d\left(S^c,S_{data}^{*}\right)|{|}_{\mathrm{\∞}}}\≥\mathrm{\θ}\⇒S^c\≤\hat{S}+0.5\left(1-\mathrm{\θ}\right)\left|\right|d\left(S^c,S_{data}^{*}\right)|{|}_{\mathrm{\∞}}$

With

$d_g\left(R^c,\hat{R}\right)=\frac{\mathrm{\ξ}\left|\right|d\left(R^c,R_{data}^{*}\right)|{|}_{\mathrm{\∞}}}{|\hat{R}-R^c|+\mathrm{\ξ}\left|\right|d\left(R^c,R_{data}^{*}\right)|{|}_{\mathrm{\∞}}}\≥\mathrm{\θ}\⇒R^c\≤\hat{R}+0.5\left(1-\mathrm{\θ}\right)\left|\right|d\left(R^c,R_{data}^{*}\right)|{|}_{\mathrm{\∞}}$

By assignment gray density index θ and solve above formula, respectively obtain average S under current gray density level^cAnd R^cFeasible Border, it may be assumed that

With

Here,WithIt is based respectively on Sample Refreshment sequenceWithTopology between middle finite sample Relation and distance relation, reflect S^cAnd R^cWith Lycoperdon polymorphum Vitt estimated valueWithBetween matching degree；Gray density index θ meetsIts value is the biggest, shows average S^cAnd R^cOccur in Lycoperdon polymorphum Vitt estimated valueWithNeighbouring probability is the biggest.

(5) travel through the valued space of gray density index θ, set up the gray density function f of structural stress and strength mean value respectively_θ (S^c) and f_θ(R^c) and meet:

$f_{\mathrm{\θ}}\left(S^c\right)=\frac{\mathrm{\θ}}{{\∫}_{-\mathrm{\∞}}^{\hat{S}}\underset{\‾}{S^c\left(\mathrm{\θ}\right)}\·\mathrm{\θ}d\underset{\‾}{S^c\left(\mathrm{\θ}\right)}+{\∫}_{\hat{S}}^{+\mathrm{\∞}}\stackrel{\‾}{S^c\left(\mathrm{\θ}\right)}\·\mathrm{\θ}d\stackrel{\‾}{S^c\left(\mathrm{\θ}\right)}}$

With

$f_{\mathrm{\θ}}\left(R^c\right)=\frac{\mathrm{\θ}}{{\∫}_{-\mathrm{\∞}}^{\hat{R}}\underset{\‾}{R^c\left(\mathrm{\θ}\right)}\·\mathrm{\θ}d\underset{\‾}{R^c\left(\mathrm{\θ}\right)}+{\∫}_{\hat{R}}^{+\mathrm{\∞}}\stackrel{\‾}{R^c\left(\mathrm{\θ}\right)}\·\mathrm{\θ}d\stackrel{\‾}{R^c\left(\mathrm{\θ}\right)}}$

As shown in Figure 4, given level of confidence 1-α, utilize numerical integration method to calculate and meet Lycoperdon polymorphum Vitt confidence level requirement Stress and strength mean value S^cAnd R^cConfidence intervalWithAnd meet:

${\∫}_{\underset{\‾}{S_{1-a}^c}}^{\stackrel{\‾}{S_{1-a}^c}}{f}_{\mathrm{\θ}}\left(S^c\right)\·S^c{\mathrm{dS}}^c={\∫}_{\underset{\‾}{R_{1-a}^c}}^{\stackrel{\‾}{R_{1-a}^c}}{f}_{\mathrm{\θ}}\left(R^c\right)\·R^c{\mathrm{dR}}^c=1-\mathrm{\α}$

Here, the span of level of confidence 1-α is: 0 ＜ 1-α≤1, in the present invention, level of confidence is entered as 0.975, i.e. α=0.025.

(6) meet, according to what the 5th step was tried to achieve, the average confidence interval that Lycoperdon polymorphum Vitt confidence level requiresWithIn conjunction with the structural stress known and intensity radius S^rAnd R^r, based on area ratio thought, for allusion quotation as shown in Figure 5 Type operating mode, it may be assumed that structure two-dimensional stress-Strength Interference Model should meet:

$\underset{\‾}{S_{1-a}^c}-S^r\≤\underset{\‾}{R_{1-a}^c}-R^r\≤\stackrel{\‾}{S_{1-a}^c}+S^r\≤\stackrel{\‾}{R_{1-a}^c}+R^r$

Build and solve satisfied (1-α)²Under level of confidence, the set of structure two-dimensional stress-Strength Interference Model is reliable Property tolerance as follows:

$R_{s,{(1-\mathrm{\α})}^2}=1-\frac{{\[(\stackrel{\‾}{S_{1-a}^c}+S^r)-(\underset{\‾}{R_{1-a}^c}-R^r)\]}^2}{8S^r{R}^r}$

To sum up, the confidence degree that meets from sample can be realized and require the comprehensive of lower structure non-Making by Probability Sets reliability Evaluate.

Embodiment:

In order to understand the feature of this invention and the suitability actual to engineering thereof more fully, the present invention is directed to structural stress Analysis of structural reliability based on Lycoperdon polymorphum Vitt confidence interval is completed with the finite data information of intensity.Wherein, structural stress and intensity Radius known, it may be assumed that S^r=3, R^r=1.5.Based on passing experience, the data message characterizing structural strength is more, has 15 samples This point；Considering the complexity of engineering problem, the data message characterizing structural stress needs to obtain by individuality test, therefore data Measure less, have 5 sample points.After sorted, it is known that the incremental sample sequence of structural stress and intensity is:

S_data={ 44.3,46.9,48.1,50.0,51.4}

R_data=49.3,49.6,49.6,49.7,49.9,50.2,50.3,50.4,50.5,50.6,50.6,5 0.9, 51.0,51.2,51.4}

According to the proposed method, utilize the grey distance measure of definition, be calculated Lycoperdon polymorphum Vitt estimated value respectively:WithTheoretical according to grey mathematics, structural stress and strength mean value can be calculated respectively at confidence level Level is that the Lycoperdon polymorphum Vitt confidence interval under 0.975 is:

With

According to known radius information, can obtain Lycoperdon polymorphum Vitt confidence level further is 0.975 time structural stress and intensity Confidence interval is: S ∈ [45.36,51.30] and R ∈ [48.61,52.18], and then, this Stress-Strength Interference Model is meeting Structured set reliability under confidence level requires is:

$R_{s,{0.975}^2}=R_{s,0.95}=1-\frac{{\[(\stackrel{\‾}{S_{1-a}^c}+S^r)-(\underset{\‾}{R_{1-a}^c}-R^r)\]}^2}{8S^r{R}^r}=1-\frac{{(51.3-48.61)}^2}{8\×3\×1.5}=80.1\%$

Therefore, under conditions of confidence level meets 0.95, in the present embodiment, structural reliability lower limit reaches 80.1%.

In sum, the present invention proposes a kind of structural stress-Strength Interference Model set based on Lycoperdon polymorphum Vitt confidence interval Analysis method for reliability.The method utilizes limited structural stress and intensity sample data, closes by probing into the distance between data System and topological relation, in conjunction with grey correlation thought, construct sign stress and strength mean value excursion Lycoperdon polymorphum Vitt confidence interval.Enter And, quantized interval result is combined with structural stress-Strength Interference Model, finally achieves the structure meeting confidence level requirement Set reliability assessment.Method proposed by the invention can complete confidence level and reliability from limited sample data The rational evaluation of structural safety situation under double standards, has the distinctest engineering practical value.

Below it is only the concrete steps of the present invention, protection scope of the present invention is not constituted any limitation；Its expansible should For containing multi-source uncertainty large scale structure in limited experimentation and the set fail-safe analysis field under the conditions of meeting confidence level, all Use the technical scheme that equivalents or equivalence are replaced and formed, within the scope of all falling within rights protection of the present invention.

Non-elaborated part of the present invention belongs to the known technology of those skilled in the art.

Claims (7)

1. structural stress based on Lycoperdon polymorphum Vitt confidence interval-Strength Interference Model set analysis method for reliability, its feature exists As follows in realizing step:

The first step: consider that structural stress S and intensity R have uncertainty, can carry out mathematical character by following formula:

$S\∈S^I=\[\underset{\‾}{S},\stackrel{\‾}{S}\]=\[S^c-S^r,S^c+S^r\]$

With

$R\∈R^I=\[\underset{\‾}{R},\stackrel{\‾}{R}\]=\[R^c-R^r,R^c+R^r\]$

Wherein, S^IAnd R^IRepresent structural stress and the feasible interval of intensity respectively,SWithRRepresent under stress interval and intensity interval Boundary,WithRepresent the interval upper bound with intensity interval of stress, S^cAnd R^cIt is the average of structural stress and intensity, S^rAnd R^rIt it is structure Stress and the radius of intensity；

Second step: in order to realize structural stress S and the interval expression of intensity R in the first step, need to be by actual experimental condition or number Value simulation means, the increasing sequence first obtaining description scheme stress S and intensity R finite sample is as follows:

S_data={ S_data(1),S_data(2),...,S_data(m)And R_data={ R_data(1),R_data(2),...,R_data(n)}

Wherein, m and n represents stress sample sequence and the sample size of intensity sample sequence respectively, and then, manage based on grey correlation Opinion, defines stress sample sequence S respectively_dataWith intensity sample sequence R_dataLycoperdon polymorphum Vitt between middle arbitrary sample and observation sample away from As follows from estimating:

$d_g(S_i,S_j)=\frac{\mathrm{\ξ}\left|\right|d(S_i,S_{data})|{|}_{\mathrm{\∞}}}{|S_i-S_j|+\mathrm{\ξ}\left|\right|d(S_i,S_{data})|{|}_{\mathrm{\∞}}},i,j=1,2,\mathrm{...},m$

With

$d_g(R_i,R_j)=\frac{\mathrm{\ξ}\left|\right|d(R_i,R_{data})|{|}_{\mathrm{\∞}}}{|R_i-R_j|+\mathrm{\ξ}\left|\right|d(R_i,R_{data})|{|}_{\mathrm{\∞}}},i,j=1,2,\mathrm{...},n$

Wherein, d_g(S_i,S_j) and d_g(R_i,R_j) represent respectively towards S_dataAnd R_dataGrey distance measure, | | d (S_i,S_data)||_∞ With | | d (R_i,R_data)||_∞Represent stress and the intensity sample Infinite Norm corresponding to sample sequence respectively, | | transport for absolute value Operator, ξ is resolution ratio, i and j is counting index；

3rd step: according to the grey distance measure of second step definition, with S_dataAnd R_dataIn each sample as observation sample, Calculate the grey distance measure of itself and whole sample sequence respectively, and result is averaging processing, obtain towards complete stress Averaged measure function with intensity sample sequenceWithAs follows:

$J_{S_i}=\frac{1}{m}\underset{j=1}{\overset{m}{\Σ}}{d}_g(S_i,S_j),i=1,2,\mathrm{...},m$

With

$J_{R_i}=\frac{1}{n}\underset{j=1}{\overset{n}{\Σ}}{d}_g(R_i,R_j),i=1,2,\mathrm{...},n$

Recycling method for normalizing, willWithIt is converted into weighting functionWithAnd then, in conjunction with in grey mathematics theory Cumulative ash generates thought, calculates the Lycoperdon polymorphum Vitt estimated value of structural stress based on finite sample data and strength mean valueWith

4th step: the Lycoperdon polymorphum Vitt estimated value that the 3rd step is calculatedWithAs new sample, it is added separately to primitive stress sample Sequence S_dataWith green strength sample sequence R_dataIn, then, sample sequence is updated to:

With

In order to probe into structural stress and strength mean value S^cAnd R^cWith Lycoperdon polymorphum Vitt estimated valueWithBetween mathematical, utilize second step The grey distance measure expression formula of middle foundation, sets up the following expression under gray density index θ:

With

Wherein,WithRepresent stress and the Lycoperdon polymorphum Vitt estimated value of strength mean value respectivelyWithWith Sample Refreshment sequence RowWithBetween grey distance measure, assignment gray density index θ also solves above formula, respectively obtains current gray density level Lower average S^cAnd R^cCan row bound, it may be assumed that

With

Wherein,S ^c (θ) and Represent lower bound and the upper bound in the feasible interval of stress average under given gray density index θ respectively,R ^c (θ)WithRepresent lower bound and the upper bound in the feasible interval of strength mean value under given gray density index θ respectively；

5th step: the valued space of traversal gray density index θ, sets up the gray density function f of structural stress and strength mean value respectively_θ (S^c) and f_θ(R^c), given level of confidence 1-α, utilize numerical integration method to calculate the stress meeting Lycoperdon polymorphum Vitt confidence level requirement With strength mean value S^cAnd R^cConfidence intervalWith

6th step: meet the average confidence interval that Lycoperdon polymorphum Vitt confidence level requires according to what the 5th step was tried to achieveWithIn conjunction with the structural stress known and intensity radius S^rAnd R^r, based on area ratio thought, build also for typical condition Solve satisfied (1-α)²Under level of confidence, the set degree of reiability of structure two-dimensional stress-Strength Interference Model is as follows:

$R_{s,{(1-\mathrm{\α})}^2}=1-\frac{{\[(\stackrel{\‾}{S_{1-a}^c}+S^r)-(\underset{\‾}{R_{1-a}^c}-R^r)\]}^2}{8S^r{R}^r}$

To sum up, the confidence degree that meets from sample can be realized and require comprehensively commenting of lower structure non-Making by Probability Sets reliability Valency.

A kind of structural stress-Strength Interference Model set based on Lycoperdon polymorphum Vitt confidence interval the most according to claim 1 is reliable Property analyze method, it is characterised in that: in the described first step, the average of structural stress and intensity generally meets R^c>S^c, structural stress and Intensity radius S^rAnd R^rValue be known quantity.

A kind of structural stress-Strength Interference Model set based on Lycoperdon polymorphum Vitt confidence interval the most according to claim 1 is reliable Property analyze method, it is characterised in that: in described second step, sample size m and n of stress sample sequence and intensity sample sequence is full Foot: 3 < m, n≤10, resolution ratio ξ=0.5.

A kind of structural stress-Strength Interference Model set based on Lycoperdon polymorphum Vitt confidence interval the most according to claim 1 is reliable Property analyze method, it is characterised in that: weighting function in described 3rd stepRepresent sample sequence S_dataMiddle sample S_iEstimate in Lycoperdon polymorphum Vitt EvaluationRatio shared by, weighting functionRepresent sample sequence R_dataMiddle sample R_iIn Lycoperdon polymorphum Vitt estimated valueRatio shared by Example, then has:And

A kind of structural stress-Strength Interference Model set based on Lycoperdon polymorphum Vitt confidence interval the most according to claim 1 is reliable Property analyze method, it is characterised in that: in described 4th stepWithIt is based respectively on Sample Refreshment sequenceWithTopological relation between middle finite sample and distance relation, reflect S^cAnd R^cWith Lycoperdon polymorphum Vitt estimated valueWithBetween coupling journey Degree；Gray density index θ meetsIts value is the biggest, shows average S^cAnd R^cOccur in Lycoperdon polymorphum Vitt estimated valueWithNear Probability the biggest.

A kind of structural stress-Strength Interference Model set based on Lycoperdon polymorphum Vitt confidence interval the most according to claim 1 is reliable Property analyze method, it is characterised in that: in described 5th step, the span of level of confidence 1-α is: 0 < 1-α≤1, wherein, puts Confidence level is entered as 0.975, i.e. α=0.025.

A kind of structural stress-Strength Interference Model set based on Lycoperdon polymorphum Vitt confidence interval the most according to claim 1 is reliable Property analyze method, it is characterised in that: in described 6th step, typical condition structure two-dimensional stress-Strength Interference Model should meet:

$\underset{\&OverBar;}{S_{1-a}^c}-S^r\&le;\underset{\&OverBar;}{R_{1-a}^c}-R^r\&le;\stackrel{\&OverBar;}{S_{1-a}^c}+S^r\&le;\stackrel{\&OverBar;}{R_{1-a}^c}+R^r.$