CN107229823A - A kind of probabilistic analysis method of wind effect extreme value - Google Patents
A kind of probabilistic analysis method of wind effect extreme value Download PDFInfo
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Abstract
The invention discloses a kind of probabilistic analysis method of wind effect extreme value, uncertainty of the non-gaussian wind effect extreme value at any quantile is calculated.In calculating process, mother stock cloth is estimated using HPM；The extreme value distribution is obtained using transfer process method；The empirical equation of normalization process extreme distribution parameters estimation；Gauss reasoning based on probability-distribution function and any quantile of extreme value, it is proposed that calculate probabilistic two kinds of analysis methods of any quantile of extreme value.The present invention is in the case where considering pneumatic randomness, and uncertainty of the estimation non-gaussian wind effect extreme value at any quantile is significant to structures under wind RELIABILITY DESIGN.
Description
Technical field
The invention belongs to structural wind resistance design field, specifically in the case where considering pneumatic randomness, two kinds non-
Probabilistic analysis method of the Gauss wind effect extreme value at any quantile.
Background technology
In structures under wind RELIABILITY DESIGN, blast and wind-excited responese are determining it is very heavy to timing away from interior extreme value
The work wanted.In in the past few decades, substantial amounts of research method be suggested to determine wind effect peak factor or
Extreme value (e.g., the Davenport 1964 of a certain quantile；Sadek and Simiu 2002；Chen and Huang 2009；Kwon
With Kareem 2011；Huang et al.2013；Yang et al.2013；Ding and Chen 2014；Huang et
al.2016b；Ma et al.2016).Wherein, based on Hermite multinomial models (Hermite polynomial model,
HPM transfer process method) due to that can carry out easy flexible adjustment to function shape according to skewness and kurtosis, and with compared with
The high degree of accuracy (e.g., Peng et al.2014, Huang et al.2016b), therefore be widely used.Recently,
Huang etc. (Huang et al. (2016a) and Luo and Huang (2016)) have studied the correlation structure of blast and its extreme value.
It is well known that due to numerous enchancement factors (such as pneumatic force environment, model choose, calibration error) presence, by
The blast extreme value and response extreme value that short-term sample (such as 10min or 1h) estimation is obtained have very big variability.Usually, exist
In the uncertainty study of extreme value estimation, these enchancement factors can be divided into two classes：Accidentalia (variable in randomness)
With perceptional factors (lacking understanding or data) (Kiureghian and Ditlevsen 2009).Minciarelli etc. have studied many
The uncertainty of perceptional factors (time interval, surface roughness and other errors in such as wind tunnel test) is planted to wind effect
The influence of extreme value.Nowadays, the overlength sample that the probabilistic research application wind tunnel test of more extreme values is surveyed.Yang and Tian profits
Various notebook data is used, the probabilistic model of non-gaussian coefficient of wind pres peak factor is modeled.Gavanski etc. uses Gumbel
Distribution fitting method, have studied the uncertain influence estimated blast extreme value of sampling duration and extreme value sampling frequency.Wu etc.
Shadow of the perceptional factors brought using overlength test data research finite sample data in each return period wind-excited responese extreme value
Ring.
Usually, on extreme value it is probabilistic research mainly for extreme value peak factor (i.e. Gumbel be distributed
57% quantile).However, the research of extreme value is also very necessary in structural wind resistance design at other quantiles.For example,
The coefficient of wind pres extreme value of 78% quantile is frequently used for the determination of designed wind load in design specification, to consider wind load or wind speed
Extreme value uncertainty (e.g., Cook 1990；Chen and Huang 2010).
The content of the invention
The technical problems to be solved by the invention are to provide a kind of probabilistic analysis method of wind effect extreme value, have
Effect calculates the uncertainty of any quantile of wind effect extreme value.
In order to solve the above technical problems, the technical solution adopted by the present invention is：
A kind of probabilistic analysis method of wind effect extreme value, comprises the following steps：
Step 1：Mother stock cloth is estimated using HPM
Blast and wind-excited responese often have certain non-Gaussian system, are estimated using the transfer process method based on HPM
The extreme value distribution of wind effect, i.e., estimate mother stock cloth using HPM.
Assuming that nongausian process Y (t) average, standard deviation, skewness and kurtosis are expressed as r1、r2、r3And r4, to it
There is X (t)=[Y (t)-r after normalization1]/r2.According to HPM, standard gaussian process U (t) is transformed into：
X=k [u+h3(u2-1)+h4(u3-3u)] (1)
In formula, k, h3And h4For the shape parameters of the HPM curves, pass through Newton-Raphson iterative non-thread
Property equation group is obtained (e.g., Ditlevsen et al.1996).HPM inverse function and the requirement of corresponding monotonicity (or have
Effect region) refer to pertinent literature (e.g., Choi and Sweetman 2010；Winterstein and MacKenzie 2013；
Huang et al.2016b).X (t) probability density function is expressed as：
In formula,For the probability density function of standard gaussian process.Further obtain Y's (t) by X (t) probability distribution
Probability distribution.
It is located at the process of effective coverage for skewness and kurtosis, HPM can be fitted its probability density function well；For
Degree of bias kurtosis is located at the process outside effective coverage, the degree of bias or kurtosis can be adjusted, fall it approximate in edges of regions
Estimate its extreme value.
Step 2：The extreme value distribution is obtained using transfer process method
After mother stock cloth is obtained, its extreme value (e.g., Sadek and Simiu 2002) is obtained using transfer process method.
The extreme value Cumulative Distribution Function (when away from T) of known standard gaussian process U (t) is：
In formula, λ0=ν0,uT is Gaussian process U (t) zero degree of transcendence；ν0,uFor process U (t) zero up-crossing rate, under
Formula is calculated：
In formula, f is frequency (Hz), SU(f) power spectral density function for being Gaussian process U (t), because extreme value surmounts to zero
Rate is not very sensitive (e.g., Sadek and Simiu 2002；Luo et al.2016), SU(f) using nongausian process Y's (t)
Power spectral density function SY(f) replace.After the extreme value distribution for obtaining Gaussian process U (t), it can be converted by equiprobability relation
Obtain nongausian process X (t) extreme value XpkWith corresponding probable value, be discussed in detail refer to pertinent literature (e.g., Sadek and
Simiu 2002；Huang et al.2015).As can be seen that on the basis of the only known degree of bias, kurtosis and zero degree of transcendence
On, the extreme value distribution of normalized nongausian process is tried to achieve using the conversion method based on Hermite multinomial models.
Research shows, nongausian process X (t) the extreme value distribution close to Gumbel distributions (e.g., Holmes and
Cochran 2003；Huang et al.2016b), it is expressed as：
In formula, δxAnd ψxThe location parameter and scale parameter of Gumbel distributions are represented respectively.Thus, former mistake is further obtained
Journey Y (t) the extreme value distribution expression formula, its distributed constant is：
δy=r1+r2δx (6)
ψy=r2ψx (7)
In formula, δyAnd ψyThe location parameter and scale parameter of process Y (t) the extreme value distribution are represented respectively.Note：If adopted
Extreme value is fitted with generalized extreme value distribution, form parameter keeps constant.
Step 3：The empirical equation of normalization process extreme distribution parameters estimation
(first group is set up between two groups of variables：The degree of bias, kurtosis and zero degree of transcendence；Second group：Normalization process the extreme value distribution
Location parameter and scale parameter) direct relation formula, key step is as follows：
First, the excursion of the degree of bias, kurtosis and zero degree of transcendence is determined.According to (the Huang et such as Huang
The degree of bias kurtosis scatterplot distribution of al.2016a and b) research, typical building roofing blast and roof boarding climbing power is as shown in Figure 1.Figure
In, dotted line represents HPM effective coverage (Winterstein and MacKenzie (2013)).Solid line encloses region representation and is chosen
Degree of bias kurtosis excursion, it basically comprises all possible degree of bias kurtosis data group in practice.Based on two UF houses
The blast time-histories (Yang et al.2013) of roofing, it can be found that the excursion of zero up-crossing rate is 1.44~9.37.In engineering
In practice, the excursion of zero up-crossing rate is regarded as 1~10.Usually, determine extreme value when away from for 10min or 1hour.
Correspondingly, the excursion of zero degree of transcendence is 600~36000.
Secondly, for all valued combinations in above-mentioned first group of range of variables, tried to achieve pair based on HPM transfer process methods
The extreme value distribution (Gumbel) parameter value for the normalization nongausian process answered, is used as second group of variable.In specific zero degree of transcendence
The lower degree of bias kurtosis value for estimating to obtain is as shown in Figure 2 and Figure 3.Compared with scale parameter, perhaps location parameter surmounts to less zero
Number of times is more sensitive.
Finally, relation fitting is carried out to two groups of variables by multiple linear regression analysis, obtains following empirical equation：
In formula, δxAnd ψxCoefficient be given in Table 1.It can be seen that, the degree of bias, kurtosis and zero in known wind effect process
In the case of degree of transcendence, the empirical equation of gained can be used directly to estimate the extreme value distribution.
δ in the empirical equation of table 1xAnd ψxCoefficient
In order to verify the accuracy of the empirical equation, it is estimated using the relative error (RE) for being estimated parameter.With δx
Exemplified by, for the given degree of bias, kurtosis and zero degree of transcendence, its RE is expressed as：
In formula,For the parameter value tried to achieve by formula (8), δxHPM transfer processes are then directly based upon to try to achieve.For given change
Change the specific degree of bias kurtosis in region, in different λ0(excursion：Under 600-36000), δxRE maximums it is as shown in Figure 4.
Similarly, ψxRE maximums it is as shown in Figure 5.As can be seen that for δxAnd ψx, RE is substantially all below 3.5.In addition, in order to
The accuracy of empirical equation is further assessed, the RE of any quantile of extreme value can be defined using formula (10).Similarly, in different λ0
Under, the RE maximums of the quantile of extreme value 57% and 78% are as shown in Figure 6 and Figure 7.As can be seen that its RE is both less than 4%.Therefore,
The empirical equation is satisfactory to the estimated result for normalizing the extreme distribution parameters of process.
Step 4：The uncertainty estimation of extreme value
Yang and Tian etc. (e.g., Yang and Tian 2015) researchs show that blast crosses Cheng Qian's Fourth-order moment (such as average, mark
Accurate poor, skewness and kurtosis) there is significant variability or uncertainty.Correspondingly, the extreme value distribution can also have uncertainty.
Although the research such as Yang and Tian shows the uncertain influence very little to extreme value of zero degree of transcendence, it influences to need further
Discussion because change of Fig. 2 display locations parameter to zero degree of transcendence is more sensitive.
Preceding Fourth-order moment and zero degree of transcendence will be seen as stochastic variable to consider that it is uncertain.If R1,R2,R3And R4Point
Average, standard deviation, four stochastic variables of skewness and kurtosis, Λ Yong Lai not represented0Represent this stochastic variable of zero degree of transcendence.
First method：Analysis method (method 1) based on probability-distribution function
For Fourth-order moment before describing and the uncertainty of zero degree of transcendence, generally using probability distribution.Assuming that R3,R4And Λ0
Edge Cumulative Distribution Function be respectivelyWithIts correlation matrix Σ estimates from substantial amounts of sample
Meter is obtained or directly specified.So, non-gaussian variable R3,R4And Λ0Equiprobability relation and standard gaussian variable can be passed through respectively
Contact：
In formula, Φ is distributed for standard gaussian.Based on Jacobi conversion, R3,R4And Λ0Joint probability density function represent
For：
In formula,Ternary joint Gaussian probability-density function is represented, its correlation matrix is Σz, element leads in matrix
Cross element in Σ to try to achieve, such as Z1And Z2Coefficient correlation estimated by following formula：
In formula,For R3And R4Coefficient correlation；For corresponding gaussian variable Z1And Z2Coefficient correlation；E [] and
D [] represents the average and standard deviation of stochastic variable respectively.
Empirical equation establishes directly contacting between the degree of bias, kurtosis and zero degree of transcendence and position, scale parameter, by refined
Gram can be from R than conversion3,R4And Λ0Joint probability density function in obtain extreme value two parameter ΔxAnd ΨxJoint Distribution letter
Number.However, the presence of higher order term causes inverse function to solve the substantial amounts of iterative calculation of sufficiently complex and needs in empirical equation, expend
Plenty of time, reduce the efficiency of uncertainty analysis method.
In order to effectively obtain the joint probability function of extreme distribution parameters, by Monte Carlo simulation (MCS), specific side
Method is：First, converted and by Monte Carlo simulation based on Nataf, obtain one group of R3,R4And Λ0Analog sample, its maintain
Correlation structure (e.g., Huang et al.2016) between variable；Then, by empirical equation (8) and (9), it can obtain
Two parameter, Δs in the extreme value distributionxAnd ΨxSample；Then, the sample for two parameter in the extreme value distribution being obtained to simulation carries out probability
Density function is fitted, and is expressed asWithIts coefficient correlation is calculated from these samples and obtained.Finally, pass through
Following Jacobi conversion obtains ΔxAnd ΨxJoint probability density function
If Λ0Influence of the randomness to extreme value it is smaller, using the sample average λ of zero degree of transcendence0,mApproximately divided
Analysis.So as to which formula (12) can be reduced to：
Similarly, by MCS, ΔxAnd ΨxJoint probability density function can be by R3And R4Joint probability density function
Try to achieve.
Work as ΔxAnd ΨxJoint probability density function try to achieve after, ΔyAnd ΨyAlso can further it try to achieve.Assuming that in R1=r1
And R2=r2Under conditions of, ΔyAnd ΨyConditional joint probability density function beCan according to conditional probability
：
In formula,Represent Δy,Ψy,R1And R2Joint probability density function,Represent R1
And R2Joint probability density function, can be tried to achieve according to formula (12).Become according to Jacobi and get following formula in return：
In formula, the order of Jacobian matrix can be pushed away by formula (6) and (7)：
Formula (15) is brought into following formula Δ to (17)yAnd ΨyJoint probability density function：
Obtain：
Process Y (t) is in any quantile q (0<q<1) extreme value YqIt is expressed as：
Yq=Δy-Ψyln(-lnq) (20)
When obtaining ΔyAnd ΨyJoint probability density function after, YqCumulative Distribution Function be expressed as：
Converted by Jacobi, YqProbability density function be then expressed as：
Second method：The analysis method (method 2) of Gauss reasoning based on any quantile of extreme value
Although method 1 has the satisfied degree of accuracy, when limited sample size, the method may is that infeasible.Typically
Ground, the extreme distribution parameters Δ of female processyAnd ΨyIt is considered as obeying independent same distribution.It can prove to obtain ΔyAnd ΨyIt is progressive
Tend to Gaussian Profile.Therefore, the Y in formula (20)qBy Gaussian distributed.The calculating side of its average and standard deviation is discussed below
Method.
In order to briefly introduce method 2, Λ will not be considered here0Uncertainty, it is, using λ0,mInstead of in formula (8)
λ0.In r3=E [R3] and r4=E [R4] when, δx(r3,r4,λ0,m) Taylor expansions be expressed as：
In formula, k=r3-E[R3] and l=r4-E[R4]；Rn is discrepance
N=2 and 1 are taken in formula (23), then ΔxAverage and variance approximate representation be：
In formula, Cov (R3,R4) it is R3And R4Covariance.Similarly, E [Ψ are obtainedx] and D [Ψx]。
Make η (r3,r4,λ0,m)=δx(r3,r4,λ0,m)ψx(r3,r4,λ0,m), in formula (24), with η0=η0(E[R3],E
[R4],λ0,m) replaceObtain its average.Therefore, ΔxAnd ΨxCovariance be：
Wherein
With
R1And R2It is related to female process, ΔxAnd ΨxIt is then related to the extreme value of the female process of normalization.Therefore, this two groups of parameters
The correlation seen generally very weak, R1And R2With ΔxAnd ΨxIt is separate.So as to ΔyAnd ΨyAverage directly by formula (6) and (7)
Try to achieve：
E[Δy]=E [R1]+E[R2]E[Δx] (27)
E[Ψy]=E [R2]E[Ψx] (28)
Deployed by Taylor, variance and covariance approximate representation are：
D[Δy]≈D[R1]+D[R2]E[Δx]2+D[Δx]E[R2]2 (29)
D[Ψy]≈D[R2]E[Ψx]2+D[Ψx]E[R2]2 (30)
Cov(Δy,Ψy)≈E[Δx]E[Ψx]D[R2]+E[Ψx]Cov(R1,R2)+E[R2]2Cov(Δx,Ψx) (31)
By the relational expression of formula (20), estimation obtains YqAverage and variance：
E[Yq]=E [Δsy]-ln(-lnq)E[Ψy] (32)
D[Yq]≈D[Δy]+ln2(-lnq)D[Ψy]-2ln(-lnq)Cov(Δy,Ψy) (33)
Compared with method 1, method 2 has easy and effective advantage, and it is applied to the situation of a small amount of sample.In addition,
In this method, it is thus only necessary to obtain average, the variance and covariance of the preceding Fourth-order moment of multivariate sample.These statistics can be from sample
Estimate to obtain in this, be either derived by or directly given from their probability density function.Moreover, calculating Yq
Average and variance the step of can be extended to further consider Λ0Probabilistic situation.However it is to be noted that：
If one or several wind effect samples are due to other factors (such as unexpected Changes in weather and the instrument not detected
Failure) and deviate overall a lot, the moments estimation in method 2 can be influenceed by serious, so as to cause extreme value uncertainty estimation
Inaccuracy.It is uncertain in extreme value on the contrary, method 1 is modeled to the full probability of these squares, it is to avoid drawbacks described above
There is stability in estimation.
Compared with prior art, the beneficial effects of the invention are as follows：The present invention is in the case where considering pneumatic randomness, estimation
Uncertainty of the non-gaussian wind effect extreme value at any quantile, it is significant to structures under wind RELIABILITY DESIGN.
Brief description of the drawings：
Fig. 1 is distributed scatter diagram for the degree of bias kurtosis of wind effect.
Fig. 2 is the location parameter for estimating to obtain under 5 kind of zero degree of transcendence.
Fig. 3 is the scale parameter for estimating to obtain under 5 kind of zero degree of transcendence.
Fig. 4 is different λ0Under relative error maximum (location parameter).
Fig. 5 is different λ0Under relative error maximum (scale parameter).
Fig. 6 is different λ0Under relative error maximum (57% quantile).
Fig. 7 is different λ0Under relative error maximum (78% quantile).
Fig. 8 is building model FL30.
Fig. 9 is roofing and point layout.
Figure 10 is R1And R2, R3And R4The coefficient correlation of this two groups of variables
Figure 11 is R1And R2, R3And R4The coefficient correlation of this two groups of variables
Figure 12 is R1And R2, R3And R4This two groups of variables coefficient correlation (Maximum
Value).
Figure 13 is R1And R2, R3And R4This two groups of variables coefficient correlation (Maximum).
Figure 14 is the average and amplitude of variation scatter diagram of zero degree of transcendence.
Figure 15 is the histogram of average.
Figure 16 is the histogram of variance.
Figure 17 is the histogram of the degree of bias.
Figure 18 is the histogram of kurtosis.
Figure 19 is joint probability density function
Figure 20 is joint probability density function
Figure 21 is the histogram of distributing position parameter.
Figure 22 is the histogram of distribution scale parameter.
Figure 23 is the joint probability density function of distributing position parameter and scale parameter.
Figure 24 is the joint probability density function of cloth location parameter and scale parameter.
Figure 25 is the marginal probability density function of distributing position parameter.
Figure 26 is the marginal probability density function of distribution scale parameter.
Figure 27 is the probability density function and its histogram of extreme value quantile 57%.
Figure 28 is the probability density function and its histogram of extreme value quantile 78%.
Figure 29 is the extreme value coefficient of variation of the coefficient of wind pres at 78% quantile.
Figure 30 is the frame models in house.
Figure 31 is the foundation B extreme value probability density function at quantile 57%.
Figure 32 is the foundation B extreme value probability density function at quantile 78%.
Embodiment
The present invention is further detailed explanation with reference to the accompanying drawings and detailed description, specific as follows：
Substantial amounts of blast sample will be used to assess the calculating performance of institute's extracting method, here, using this method rated wind pressure
The uncertainty of coefficient and charming appearance and behaviour structural response.Wind pressure data derives from Canada University of Western Ontario (University of
Western Ontario, UWO) boundary layer wind tunnel laboratory.Scaling factor is used for 1:50 test model (FL30), such as Fig. 8
Shown (no neighboring buildings；Under B class landforms).Roofing arranges 474 pressure taps altogether, and its distribution situation is as shown in Figure 9.It is used
Data are defined as under 120 ° of wind directions, sample frequency 400Hz, a length of 3h during sampling (on experimental model yardstick).If in actual chi
The wind speed that 10m highly locates in degree is 31.7m/s, then speed scaling factor is 1:5.Correspondingly, the sample frequency in physical size
It is respectively 40Hz and 30h with duration.30h overlength wind pressure data is divided into 180 sections of 10min data by the present invention.The model it is detailed
Thin information refers to document Peng et al. (2014).In order to preferably explain, wind pressure data will be multiplied by -1.
1st, coefficient of wind pres
First, Fourth-order moment and the correlation of zero degree of transcendence before the coefficient of wind pres of each measuring point are studied；Secondly, zero is discussed
The uncertain influence to extreme value of degree of transcendence；Finally, the calculating performance of two kinds of analysis methods is assessed.
(1) Fourth-order moment and the correlation of zero degree of transcendence before
For each roofing measuring point, R1And R2, R3And R4The coefficient correlation of this two groups of variables is respectively such as Figure 10 and Figure 11 institutes
Show.As can be seen that the correlation between average and standard deviation is not very strong, the correlation between skewness and kurtosis is in many situations
0.8 is both greater than down.
R1And R3,R1And R4,R2And R3, and R2And R4The correlation of this four groups of variables can also be calculated and obtained.These coefficient correlations
Maximum it is as shown in figure 12.As can be seen that between average and the degree of bias (kurtosis) and standard deviation is related with the degree of bias (kurtosis)
Property is very weak, can be ignored, it was confirmed that ΔxAnd ΨxR can be considered separately from1And R2.It is explained as follows：Because ΔxAnd Ψx
Only R3And R4Function, and R1And R2It is substantially independent of R3And R4, so as to draw a conclusion：ΔxAnd ΨxIndependently of R1
And R2。
Similarly, Λ0And R3, and Λ0And R4The coefficient correlation of this two groups of variables can also be obtained by calculating.These are related
The maximum of coefficient is as shown in figure 13.As can be seen that zero degree of transcendence is substantially independent of skewness and kurtosis.
The uncertain influence to extreme value of (2) zero degree of transcendence
Before the uncertain influence to extreme value of zero degree of transcendence is inquired into, discuss that the variation of zero degree of transcendence is special first
Levy.Average, minimum value and the maximum of zero degree of transcendence are expressed as λ0,m,λ0,minAnd λ0,max, can be from 180 sections of wind pressure datas
Estimation is obtained.The amplitude of variation λ of zero degree of transcendence0,r=λ0,max-λ0,min.Figure 14 illustrates the zero of 474 pressure taps and surmounts secondary
Several averages and amplitude of variation scatter diagram, it is found that both has this linear approximate relationship.Calculate 180 sections of samples of all pressure taps
The coefficient of variation of this zero degree of transcendence, the scope for as a result showing its coefficient of variation is 1.3%~5.3%.It follows that zero surpassing
The variability of more number of times is relatively weak.
In order to study the uncertain influence to extreme value of zero degree of transcendence, respectively in λ0,m,λ0,minAnd λ0,maxUnder, calculate
The extreme value of a certain quantiles of X (t), can be expressed as xpk,m,xpk,minAnd xpk,max.In extreme value estimation, using 180 sections of samples
Skewness and kurtosis average calculated.As a result show, for all pressure taps, under quantile 57% and 78%,
(xpk,max-xpk,m)/xpk,m(xpk,m-xpk,min)/xpk,mThe greater be both less than 1.5%.It is concluded that：Zero surmounts secondary
Several uncertain influences to extreme value is little.Therefore, the uncertain factor of zero degree of transcendence will be ignored in subsequent analysis, is adopted
Analyzed with the average value of zero degree of transcendence in 180 segment datas.
The Performance Evaluation of (3) two kinds of analysis methods
The wind pressure data of measuring point A in Fig. 9 is by the calculating performance of the extracting method for illustrating.From 180 sections of wind pressure datas
It can calculate and obtain average, standard deviation, skewness and kurtosis.The histogram of average and standard deviation and the Gauss of its probability density function intend
Close curve difference as shown in Figure 15 and Figure 16.As can be seen that average and standard deviation substantially Gaussian distributed.According to center pole
It is worth theorem, with the increase of sample size, average and standard deviation can be close to Gaussian Profiles.The histogram of skewness and kurtosis is respectively such as
Shown in figure Figure 17 and Figure 18, due to both there is obvious non-Gaussian system, using HPM Fitted probability density functions, difference
As shown in figure Figure 17 and Figure 18.
In order to try to achieve the joint probability density function of average and standard deviation and skewness and kurtosis, its coefficient correlation should divide
Do not obtained by 180 sections of wind pressure data estimations not directly, its value is：WithDue to skewness and kurtosis
Probability density function is fitted using HPM, its corresponding coefficient correlation in formula (13)It can be obtained using formula is simplified
To (Blaise et al 2016；Luo and Huang 2016)：
In formula, ki,h3,iAnd h4,iFor Ri(i=3,4) model parameter.Obtained by calculatingCorrespondingly,
Joint probability density function can be tried to achieveWithIts difference is as illustrated in figures 19 and 20.
The extreme value distribution location parameter of normalization process and the histogram of scale parameter are as shown in figure 21 and figure.Pass through
MCS methods, can obtain the analog sample of the two parameters, and its probability density is as shown in figure 21 and figure.Can from figure
Go out, analog sample is close to initial data histogram.In addition, simulation obtain skewness and kurtosis coefficient correlation be 0.9932, with by
The coefficient 0.9955 that initial data is estimated is consistent.It was found that analog sample closely Gaussian Profile, it is explained as follows：ΔxAnd Ψx
For ΔyAnd ΨyLinear transformation, R1And R2Fixed value will be converged in probability as the variable in conversion.Therefore, ΔxAnd Ψx
Also will be close to Gaussian Profile.The coefficient correlation that the Gauss edge distribution and estimation obtained according to fitting is obtained, can be returned
One changes the joint probability density function of process the extreme value distribution location parameter and scale parameter, as shown in figure 23.
According to Figure 23 joint probability density function, the joint of original procedure Y (t) extreme value Gumbel distribution two parameters is general
Rate density function can be calculated by formula (19) and obtained, as shown in figure 24.The marginal probability density function of two parameters is then such as Figure 25
With shown in Figure 26.As can be seen that the analytic model that method 1 is derived by can be intended the histogram of two parameters well
Close.In addition, can also be derived by the Gauss analysis model of two parameters from method 2, as shown in Figure 25,26, fitting effect is same
Sample is fine.The probability density function its corresponding histogram point for the extreme value quantile 57% and 78% for estimating to obtain from formula (22)
Not as shown in Figure 27 and Figure 28.It can be found that its fitting effect is fine.In addition, from R1,R2,R3And R4Sample in, at two points
Extreme value average and standard deviation at site can be obtained by method 2, its corresponding probability density function as shown in Figure 27 and Figure 28,
It was found that they can well to calculate extreme value uncertainty.
At quantile 57% and 78%, the extreme value coefficient of variation of pressure tap A coefficient of wind pres is about 8%.It is all to survey
As shown in figure 29, it is both greater than 6.5% to the extreme value coefficient of variation of the coefficient of wind pres of pressure point at 78% quantile, and some reach
10% or even 20%.
2nd, wind-excited responese
Based on building model FL30, design framework structural model.According to Steel Structural Design related specifications (GB50017-
2003；CECS 102-2002), topology layout is carried out, and the material and section of beam, post and purlin etc. are determined.
FEM model is set up in Sap2000, as shown in figure 30.Post bottom uses hold-down support, and beam and side column use rigid joint, beam with
Center pillar then uses hinged joint.Beam and column uses steel Q345, and purlin then uses steel Q235.By 30h blast, (wind speed is
31.7m/s) input in the FEM model, obtain 180 segment structures response time course data.
Due to post bottom moment of flexure (base bending moments, BBM) in structure design it is extremely important, using foundation B
Post bottom moment of flexure carry out analysis calculating.As coefficient of wind pres, the uncertainty of zero degree of transcendence, which will be ignored, to be disregarded.In a point position
At point 57% and 78%, it can be calculated by method 1 and 2 and obtain extreme value probability density function, respectively as shown in Figure 31 and Figure 32.Can be with
It was found that, two methods can be estimated extreme value response uncertainty very well.Under two quantiles, the variation lines of its extreme value
Number is all about 7%.
Note：ΔyAnd ΨyGauss Joint Distribution
The extreme value of wind effect is Ypk.Assuming that：YpkProbability density function beHereObey
Gumbel is distributed；θ=[Δy,Ψy]；N represents the number of wind effect extreme value, and its value is sufficiently large.The log-likelihood letter of parameter
Number is：
The maximal possibility estimation of parameter can be obtained in following constraint：
In formula,In Ypk,i, i=1, the precondition of the independent same distribution of 2 ..., n obediences
Under,It is present and converges on θ0。
In θ0Place, the single order Taylor of formula (36) is expanded into：
In formula, For as θ=θ0When value；Rn' makes for discrepance
It can be seen from weak law of large number,B (Chung 1974) is converged on probability.
So as to：
NoteWithAccording to center extreme value theorem, as n → ∞,
In formula,Represent that there is zero-mean,The Gaussian Profile of variance, it can thus be concluded that：
That is, θ=[Δy,Ψy] obey two variable Gaussian Profiles.
Claims (3)
1. a kind of probabilistic analysis method of wind effect extreme value, it is characterised in that comprise the following steps：
Step 1：Estimate mother stock cloth using based on Hermite multinomial models HPM, i.e., the transfer process method estimation based on HPM
The extreme value distribution of wind effect；
Assuming that nongausian process Y (t) average, standard deviation, skewness and kurtosis are expressed as r1、r2、r3And r4, it is normalized
After have X (t)=[Y (t)-r1]/r2, standard gaussian process U (t) is transformed into according to HPM：
X=k [u+h3(u2-1)+h4(u3-3u)] (1)
In formula, k, h3And h4For the shape parameters of the HPM curves；X (t) probability density function is expressed as：
In formula,For the probability density function of standard gaussian process, Y (t) probability distribution is obtained by X (t) probability distribution；
Step 2：After mother stock cloth is obtained, its extreme value is obtained using transfer process method；
The extreme value Cumulative Distribution Function of known standard gaussian process U (t) is：
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<mo>-</mo>
<mo>-</mo>
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<mo>(</mo>
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<mo>)</mo>
</mrow>
</mrow>
In formula, λ0=ν0,uT is Gaussian process U (t) zero degree of transcendence；ν0,uFor process U (t) zero up-crossing rate, by following formula meter
Calculate：
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<mi>d</mi>
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</mrow>
</msqrt>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, f is frequency, SU(f) power spectral density function for being Gaussian process U (t)；
After the extreme value distribution for obtaining Gaussian process U (t), the pole for obtaining nongausian process X (t) is converted by equiprobability relation
Value XpkWith corresponding probable value；
Nongausian process X (t) the extreme value distribution is distributed close to Gumbel, is expressed as：
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<mi>X</mi>
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<mo>=</mo>
<mi>exp</mi>
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<mi>exp</mi>
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<mi>p</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>-</mo>
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<mi>&delta;</mi>
<mi>x</mi>
</msub>
</mrow>
<msub>
<mi>&psi;</mi>
<mi>x</mi>
</msub>
</mfrac>
<mo>)</mo>
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<mo>&rsqb;</mo>
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<mo>-</mo>
<mo>-</mo>
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</mrow>
</mrow>
In formula, δxAnd ψxThe location parameter and scale parameter of Gumbel distributions are represented respectively, obtain the extreme value distribution of former process Y (t)
Expression formula, its distributed constant is：
δy=r1+r2δx (6)
ψy=r2ψx (7)
In formula, δyAnd ψyThe location parameter and scale parameter of process Y (t) the extreme value distribution are represented respectively；
Step 3：The empirical equation of normalization process extreme distribution parameters estimation
The direct relation formula set up between two groups of variables, first group：The degree of bias, kurtosis and zero degree of transcendence, second group：Normalization process
The location parameter and scale parameter of the extreme value distribution：
First, the excursion of the degree of bias, kurtosis and zero degree of transcendence is determined；
Secondly, for all valued combinations in first group of range of variables, corresponding normalizing is tried to achieve based on HPM transfer process methods
Change the extreme distribution parameters value of nongausian process, be used as second group of variable；
Finally, relation fitting is carried out to two groups of variables by multiple linear regression analysis, obtains following empirical equation：
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1
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<mi>b</mi>
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<mi>b</mi>
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<mn>2</mn>
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</msub>
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<mi>ln&lambda;</mi>
<mn>0</mn>
</msub>
<mo>+</mo>
<msub>
<mi>b</mi>
<mn>13</mn>
</msub>
<msubsup>
<mi>r</mi>
<mn>3</mn>
<mn>2</mn>
</msubsup>
<msup>
<mi>ln</mi>
<mn>2</mn>
</msup>
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<mi>r</mi>
<mn>4</mn>
</msub>
<msub>
<mi>ln&lambda;</mi>
<mn>0</mn>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
Step 4：Using the analysis method based on probability-distribution function, or Gauss reasoning based on any quantile of extreme value point
Uncertainty estimation of the analysis method to extreme value；If R1,R2,R3And R4It is respectively intended to represent average, standard deviation, skewness and kurtosis four
Individual stochastic variable, Λ0Represent this stochastic variable of zero degree of transcendence；
Method one：Analysis method based on probability-distribution function
Assuming that R3,R4And Λ0Edge Cumulative Distribution Function be respectivelyWithIts correlation matrix Σ
Estimation is obtained or directly specified from substantial amounts of sample；So, non-gaussian variable R3,R4And Λ0Respectively by equiprobability relation with
Standard gaussian variable is contacted：
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<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
<mo>=</mo>
<msup>
<mi>&Phi;</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>&lsqb;</mo>
<msub>
<mi>F</mi>
<msub>
<mi>R</mi>
<mn>3</mn>
</msub>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>r</mi>
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</msub>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>;</mo>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
<mo>=</mo>
<msup>
<mi>&Phi;</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>&lsqb;</mo>
<msub>
<mi>F</mi>
<msub>
<mi>R</mi>
<mn>4</mn>
</msub>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>r</mi>
<mn>4</mn>
</msub>
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</mrow>
<mo>&rsqb;</mo>
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<mi>z</mi>
<mn>3</mn>
</msub>
<mo>=</mo>
<msup>
<mi>&Phi;</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>&lsqb;</mo>
<msub>
<mi>F</mi>
<msub>
<mi>&Lambda;</mi>
<mn>0</mn>
</msub>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>&lambda;</mi>
<mn>0</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, Φ is distributed for standard gaussian；
Based on Jacobi conversion, R3,R4And Λ0Joint probability density function be expressed as：
In formula,Ternary joint Gaussian probability-density function is represented, its correlation matrix is Σz, element passes through Σ in matrix
Middle element is tried to achieve；
Method two：The analysis method of Gauss reasoning based on any quantile of extreme value
Do not consider Λ0Uncertainty, use λ0,mInstead of the λ in formula (8)0；In r3=E [R3] and r4=E [R4] when, δx(r3,r4,
λ0,m) Taylor expansions be expressed as：
In formula, k=r3-E[R3] and l=r4-E[R4]；Rn is discrepance；
N=2 and 1 are taken in formula (12), then ΔxAverage and variance approximate representation be：
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</mfrac>
<mfrac>
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<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<msubsup>
<mi>&delta;</mi>
<mi>x</mi>
<mn>0</mn>
</msubsup>
</mrow>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>r</mi>
<mn>3</mn>
<mn>2</mn>
</msubsup>
</mrow>
</mfrac>
<mi>D</mi>
<mo>&lsqb;</mo>
<msub>
<mi>R</mi>
<mn>3</mn>
</msub>
<mo>&rsqb;</mo>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mfrac>
<mrow>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<msubsup>
<mi>&delta;</mi>
<mi>x</mi>
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</msubsup>
</mrow>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>r</mi>
<mn>4</mn>
<mn>2</mn>
</msubsup>
</mrow>
</mfrac>
<mi>D</mi>
<mo>&lsqb;</mo>
<msub>
<mi>R</mi>
<mn>4</mn>
</msub>
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<mfrac>
<mrow>
<msup>
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<mn>2</mn>
</msup>
<msubsup>
<mi>&delta;</mi>
<mi>x</mi>
<mn>0</mn>
</msubsup>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>r</mi>
<mn>3</mn>
</msub>
<mo>&part;</mo>
<msub>
<mi>r</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
<mi>C</mi>
<mi>o</mi>
<mi>v</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>R</mi>
<mn>3</mn>
</msub>
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<msub>
<mi>R</mi>
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</msub>
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</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>13</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>D</mi>
<mo>&lsqb;</mo>
<msub>
<mi>&Delta;</mi>
<mi>x</mi>
</msub>
<mo>&rsqb;</mo>
<mo>&ap;</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<mfrac>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>&delta;</mi>
<mi>x</mi>
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</msubsup>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>r</mi>
<mn>3</mn>
</msub>
</mrow>
</mfrac>
<mo>&rsqb;</mo>
</mrow>
<mn>2</mn>
</msup>
<mi>D</mi>
<mo>&lsqb;</mo>
<msub>
<mi>R</mi>
<mn>3</mn>
</msub>
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<msup>
<mrow>
<mo>&lsqb;</mo>
<mfrac>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>&delta;</mi>
<mi>x</mi>
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</msubsup>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>r</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
<mo>&rsqb;</mo>
</mrow>
<mn>2</mn>
</msup>
<mi>D</mi>
<mo>&lsqb;</mo>
<msub>
<mi>R</mi>
<mn>4</mn>
</msub>
<mo>&rsqb;</mo>
<mo>+</mo>
<mn>2</mn>
<mfrac>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>&delta;</mi>
<mi>x</mi>
<mn>0</mn>
</msubsup>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>r</mi>
<mn>3</mn>
</msub>
</mrow>
</mfrac>
<mfrac>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>&delta;</mi>
<mi>x</mi>
<mn>0</mn>
</msubsup>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>r</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
<mi>C</mi>
<mi>o</mi>
<mi>v</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>R</mi>
<mn>3</mn>
</msub>
<mo>,</mo>
<msub>
<mi>R</mi>
<mn>4</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>14</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, Cov (R3,R4) it is R3And R4Covariance；Similarly, E [Ψ are obtainedx] and D [Ψx]；
Make η (r3,r4,λ0,m)=δx(r3,r4,λ0,m)ψx(r3,r4,λ0,m), in formula (13), with η0=η0(E[R3],E[R4],
λ0,m) replaceObtain its average；ΔxAnd ΨxCovariance be：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<mi>C</mi>
<mi>o</mi>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>&Delta;</mi>
<mi>x</mi>
</msub>
<mo>,</mo>
<msub>
<mi>&Psi;</mi>
<mi>x</mi>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>&ap;</mo>
<msup>
<mi>&eta;</mi>
<mn>0</mn>
</msup>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mfrac>
<mrow>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<msup>
<mi>&eta;</mi>
<mn>0</mn>
</msup>
</mrow>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>r</mi>
<mn>3</mn>
<mn>2</mn>
</msubsup>
</mrow>
</mfrac>
<mi>D</mi>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mi>R</mi>
<mn>3</mn>
</msub>
<mo>&rsqb;</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mfrac>
<mrow>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<msup>
<mi>&eta;</mi>
<mn>0</mn>
</msup>
</mrow>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>r</mi>
<mn>4</mn>
<mn>2</mn>
</msubsup>
</mrow>
</mfrac>
<mi>D</mi>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mi>R</mi>
<mn>4</mn>
</msub>
<mo>&rsqb;</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mrow>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<msup>
<mi>&eta;</mi>
<mn>0</mn>
</msup>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>r</mi>
<mn>3</mn>
</msub>
<mo>&part;</mo>
<msub>
<mi>r</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
<mi>C</mi>
<mi>o</mi>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>R</mi>
<mn>3</mn>
</msub>
<mo>,</mo>
<msub>
<mi>R</mi>
<mn>4</mn>
</msub>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<mi>E</mi>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mi>&Delta;</mi>
<mi>x</mi>
</msub>
<mo>&rsqb;</mo>
</mrow>
<mi>E</mi>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mi>&Psi;</mi>
<mi>x</mi>
</msub>
<mo>&rsqb;</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>15</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein
<mrow>
<mfrac>
<mrow>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<msup>
<mi>&eta;</mi>
<mn>0</mn>
</msup>
</mrow>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>r</mi>
<mn>4</mn>
<mn>2</mn>
</msubsup>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<msubsup>
<mi>&delta;</mi>
<mi>x</mi>
<mn>0</mn>
</msubsup>
</mrow>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>r</mi>
<mn>4</mn>
<mn>2</mn>
</msubsup>
</mrow>
</mfrac>
<msubsup>
<mi>&psi;</mi>
<mi>x</mi>
<mn>0</mn>
</msubsup>
<mo>+</mo>
<mn>2</mn>
<mfrac>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>&delta;</mi>
<mi>x</mi>
<mn>0</mn>
</msubsup>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>r</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
<mfrac>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>&psi;</mi>
<mi>x</mi>
<mn>0</mn>
</msubsup>
</mrow>
<mrow>
<mo>&part;</mo>
<msub>
<mi>r</mi>
<mn>4</mn>
</msub>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<msubsup>
<mi>&psi;</mi>
<mi>x</mi>
<mn>0</mn>
</msubsup>
</mrow>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>r</mi>
<mn>4</mn>
<mn>2</mn>
</msubsup>
</mrow>
</mfrac>
<msubsup>
<mi>&delta;</mi>
<mi>x</mi>
<mn>0</mn>
</msubsup>
</mrow>
With
R1And R2It is related to female process, ΔxAnd ΨxIt is then related to the extreme value of the female process of normalization；
ΔyAnd ΨyAverage directly tried to achieve by formula (6) and (7)：
E[Δy]=E [R1]+E[R2]E[Δx] (16)
E[Ψy]=E [R2]E[Ψx] (17)
Deployed by Taylor, variance and covariance approximate representation are：
D[Δy]≈D[R1]+D[R2]E[Δx]2+D[Δx]E[R2]2 (18)
D[Ψy]≈D[R2]E[Ψx]2+D[Ψx]E[R2]2 (19)
Cov(Δy,Ψy)≈E[Δx]E[Ψx]D[R2]+E[Ψx]Cov(R1,R2)+E[R2]2Cov(Δx,Ψx) (20)
Pass through formula Yq=Δy-ΨyLn (- lnq) relational expression, estimation obtains YqAverage and variance：
E[Yq]=E [Δsy]-ln(-lnq)E[Ψy] (21)
D[Yq]≈D[Δy]+ln2(-lnq)D[Ψy]-2ln(-lnq)Cov(Δy,Ψy)。 (22)
2. a kind of probabilistic analysis method of wind effect extreme value as claimed in claim 1, it is characterised in that the step
Rapid 4 method one also includes：The joint probability function of extreme distribution parameters is taken by Monte Carlo simulation；
First, converted and by Monte Carlo simulation based on Nataf, obtain one group of R3,R4And Λ0Analog sample, its maintain become
Correlation structure between amount, obtains two parameter, Δs in the extreme value distributionxAnd ΨxSample；
Secondly, the sample for two parameter in the extreme value distribution being obtained to simulation carries out probability density function fitting, is expressed as
WithIts coefficient correlation is calculated from these samples and obtained；
Finally, Δ is obtained by Jacobi conversionxAnd ΨxJoint probability density function
If Λ0Influence of the randomness to extreme value it is smaller, using the sample average λ of zero degree of transcendence0,mCarry out approximate analysis, formula
(11) it is reduced to：
Equally, by Monte Carlo simulation, ΔxAnd ΨxJoint probability density function by R3And R4Joint probability density function
Try to achieve；
Work as ΔxAnd ΨxJoint probability density function try to achieve after, try to achieve ΔyAnd Ψy, when obtaining ΔyAnd ΨyJoint probability it is close
Spend after function, YqCumulative Distribution Function be expressed as：
<mrow>
<msub>
<mi>F</mi>
<msub>
<mi>Y</mi>
<mi>q</mi>
</msub>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mi>q</mi>
</msub>
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</mrow>
<mo>=</mo>
<munder>
<mrow>
<mo>&Integral;</mo>
<mo>&Integral;</mo>
</mrow>
<mrow>
<msub>
<mi>&delta;</mi>
<mi>y</mi>
</msub>
<mo>-</mo>
<msub>
<mi>&psi;</mi>
<mi>y</mi>
</msub>
<mi>l</mi>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mo>-</mo>
<mi>ln</mi>
<mi>q</mi>
<mo>)</mo>
</mrow>
<mo>&le;</mo>
<msub>
<mi>y</mi>
<mi>q</mi>
</msub>
</mrow>
</munder>
<msub>
<mi>f</mi>
<mrow>
<msub>
<mi>&Delta;</mi>
<mi>y</mi>
</msub>
<msub>
<mi>&Psi;</mi>
<mi>y</mi>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>&delta;</mi>
<mi>y</mi>
</msub>
<mo>,</mo>
<msub>
<mi>&psi;</mi>
<mi>y</mi>
</msub>
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</mrow>
<msub>
<mi>d&delta;</mi>
<mi>y</mi>
</msub>
<msub>
<mi>d&psi;</mi>
<mi>y</mi>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>24</mn>
<mo>)</mo>
</mrow>
</mrow>
Converted by Jacobi, YqProbability density function be then expressed as：
<mrow>
<msub>
<mi>f</mi>
<msub>
<mi>Y</mi>
<mi>q</mi>
</msub>
</msub>
<mrow>
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<msub>
<mi>y</mi>
<mi>q</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msubsup>
<mo>&Integral;</mo>
<mrow>
<mo>-</mo>
<mi>&infin;</mi>
</mrow>
<mi>&infin;</mi>
</msubsup>
<msub>
<mi>f</mi>
<mrow>
<msub>
<mi>&Delta;</mi>
<mi>y</mi>
</msub>
<msub>
<mi>&Psi;</mi>
<mi>y</mi>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mi>q</mi>
</msub>
<mo>+</mo>
<msub>
<mi>&psi;</mi>
<mi>y</mi>
</msub>
<mi>l</mi>
<mi>n</mi>
<mo>(</mo>
<mrow>
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<mi>ln</mi>
<mi> </mi>
<mi>q</mi>
</mrow>
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<mo>,</mo>
<msub>
<mi>&psi;</mi>
<mi>y</mi>
</msub>
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</mrow>
<msub>
<mi>d&psi;</mi>
<mi>y</mi>
</msub>
<mo>.</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>25</mn>
<mo>)</mo>
</mrow>
</mrow>
3. a kind of probabilistic analysis method of wind effect extreme value as claimed in claim 2, it is characterised in that try to achieve Δy
And ΨyConcretely comprise the following steps：
Assuming that in R1=r1And R2=r2Under conditions of, ΔyAnd ΨyConditional joint probability density function be
Obtained according to conditional probability：
<mrow>
<msub>
<mi>f</mi>
<mrow>
<msub>
<mi>&Delta;</mi>
<mi>y</mi>
</msub>
<msub>
<mi>&Psi;</mi>
<mi>y</mi>
</msub>
<msub>
<mi>R</mi>
<mn>1</mn>
</msub>
<msub>
<mi>R</mi>
<mn>2</mn>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>&delta;</mi>
<mi>y</mi>
</msub>
<mo>,</mo>
<msub>
<mi>&psi;</mi>
<mi>y</mi>
</msub>
<mo>,</mo>
<msub>
<mi>r</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>f</mi>
<mrow>
<msub>
<mi>&Delta;</mi>
<mi>y</mi>
</msub>
<msub>
<mi>&Psi;</mi>
<mi>y</mi>
</msub>
<mo>|</mo>
<msub>
<mi>R</mi>
<mn>1</mn>
</msub>
<mo>=</mo>
<msub>
<mi>r</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>R</mi>
<mn>2</mn>
</msub>
<mo>=</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>&delta;</mi>
<mi>y</mi>
</msub>
<mo>,</mo>
<msub>
<mi>&psi;</mi>
<mi>y</mi>
</msub>
<mo>)</mo>
</mrow>
<msub>
<mi>f</mi>
<mrow>
<msub>
<mi>R</mi>
<mn>1</mn>
</msub>
<msub>
<mi>R</mi>
<mn>2</mn>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>r</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>26</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula,Represent Δy,Ψy,R1And R2Joint probability density function,Represent R1And R2's
Joint probability density function；Become according to Jacobi and get following formula in return：
<mrow>
<msub>
<mi>f</mi>
<mrow>
<msub>
<mi>&Delta;</mi>
<mi>y</mi>
</msub>
<msub>
<mi>&Psi;</mi>
<mi>y</mi>
</msub>
<mo>|</mo>
<msub>
<mi>R</mi>
<mn>1</mn>
</msub>
<mo>=</mo>
<msub>
<mi>r</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>R</mi>
<mn>2</mn>
</msub>
<mo>=</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>&delta;</mi>
<mi>y</mi>
</msub>
<mo>,</mo>
<msub>
<mi>&psi;</mi>
<mi>y</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>f</mi>
<mrow>
<msub>
<mi>&Delta;</mi>
<mi>x</mi>
</msub>
<msub>
<mi>&Psi;</mi>
<mi>x</mi>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>&delta;</mi>
<mi>x</mi>
</msub>
<mo>,</mo>
<msub>
<mi>&psi;</mi>
<mi>x</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>|</mo>
<mi>J</mi>
<mo>|</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>27</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, the order of Jacobian matrix is pushed away by formula (6) and (7)：
<mrow>
<mo>|</mo>
<mi>J</mi>
<mo>|</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<msubsup>
<mi>r</mi>
<mn>2</mn>
<mn>2</mn>
</msubsup>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>28</mn>
<mo>)</mo>
</mrow>
</mrow>
Formula (26) is brought into following formula Δ to (28)yAnd ΨyJoint probability density function：
<mrow>
<msub>
<mi>f</mi>
<mrow>
<msub>
<mi>&Delta;</mi>
<mi>y</mi>
</msub>
<msub>
<mi>&Psi;</mi>
<mi>y</mi>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>&delta;</mi>
<mi>y</mi>
</msub>
<mo>,</mo>
<msub>
<mi>&psi;</mi>
<mi>y</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msubsup>
<mo>&Integral;</mo>
<mrow>
<mo>-</mo>
<mi>&infin;</mi>
</mrow>
<mi>&infin;</mi>
</msubsup>
<msubsup>
<mo>&Integral;</mo>
<mrow>
<mo>-</mo>
<mi>&infin;</mi>
</mrow>
<mi>&infin;</mi>
</msubsup>
<msub>
<mi>f</mi>
<mrow>
<msub>
<mi>&Delta;</mi>
<mi>y</mi>
</msub>
<msub>
<mi>&Psi;</mi>
<mi>y</mi>
</msub>
<msub>
<mi>R</mi>
<mn>1</mn>
</msub>
<msub>
<mi>R</mi>
<mn>2</mn>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>&delta;</mi>
<mi>y</mi>
</msub>
<mo>,</mo>
<msub>
<mi>&psi;</mi>
<mi>y</mi>
</msub>
<mo>,</mo>
<msub>
<mi>r</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
<msub>
<mi>dr</mi>
<mn>1</mn>
</msub>
<msub>
<mi>dr</mi>
<mn>2</mn>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>29</mn>
<mo>)</mo>
</mrow>
</mrow>
Obtain：
<mrow>
<msub>
<mi>f</mi>
<mrow>
<msub>
<mi>&Delta;</mi>
<mi>y</mi>
</msub>
<msub>
<mi>&Psi;</mi>
<mi>y</mi>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>&delta;</mi>
<mi>y</mi>
</msub>
<mo>,</mo>
<msub>
<mi>&psi;</mi>
<mi>y</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msubsup>
<mo>&Integral;</mo>
<mrow>
<mo>-</mo>
<mi>&infin;</mi>
</mrow>
<mi>&infin;</mi>
</msubsup>
<msubsup>
<mo>&Integral;</mo>
<mrow>
<mo>-</mo>
<mi>&infin;</mi>
</mrow>
<mi>&infin;</mi>
</msubsup>
<msub>
<mi>f</mi>
<mrow>
<msub>
<mi>R</mi>
<mn>1</mn>
</msub>
<msub>
<mi>R</mi>
<mn>2</mn>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>r</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>r</mi>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
<msub>
<mi>f</mi>
<mrow>
<msub>
<mi>&Delta;</mi>
<mi>x</mi>
</msub>
<msub>
<mi>&Psi;</mi>
<mi>x</mi>
</msub>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<msub>
<mi>&delta;</mi>
<mi>y</mi>
</msub>
<mo>-</mo>
<msub>
<mi>r</mi>
<mn>1</mn>
</msub>
</mrow>
<msub>
<mi>r</mi>
<mn>2</mn>
</msub>
</mfrac>
<mo>,</mo>
<mfrac>
<msub>
<mi>&psi;</mi>
<mi>y</mi>
</msub>
<msub>
<mi>r</mi>
<mn>2</mn>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<mfrac>
<mn>1</mn>
<msubsup>
<mi>r</mi>
<mn>2</mn>
<mn>2</mn>
</msubsup>
</mfrac>
<msub>
<mi>dr</mi>
<mn>1</mn>
</msub>
<msub>
<mi>dr</mi>
<mn>2</mn>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>30</mn>
<mo>)</mo>
</mrow>
</mrow>
Process Y (t) is in any quantile q (0<q<1) extreme value YqIt is expressed as：
Yq=Δy-Ψyln(-lnq)。
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| CN110779680B (en) * | 2019-10-18 | 2021-03-30 | 华南理工大学 | Method for detecting extreme value wind pressure of building envelope structure |
| CN112199826A (en) * | 2020-09-23 | 2021-01-08 | 中国电建集团华东勘测设计研究院有限公司 | Broadband non-Gaussian random extreme value analysis and prediction method for monitoring data |
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| CN113673094A (en) * | 2021-08-05 | 2021-11-19 | 中国建筑科学研究院有限公司 | Building dead weight load evaluation method based on reliable indexes |
| CN113673094B (en) * | 2021-08-05 | 2023-12-15 | 中国建筑科学研究院有限公司 | Building dead weight load assessment method based on reliable index |
| CN114239353A (en) * | 2021-12-14 | 2022-03-25 | 国网江苏省电力有限公司经济技术研究院 | Mounting optimization arrangement method for newly-increased base stations of shared towers based on extreme value response surface |
| CN114239353B (en) * | 2021-12-14 | 2023-08-04 | 国网江苏省电力有限公司经济技术研究院 | Extremum response surface-based shared pole tower newly-added base station mounting optimization arrangement method |
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