Large-span roof structure wind pressure coherence probability modeling method based on Vine Copula

Abstract

According to the Vine Copula-based large-span roof structure wind pressure coherence probability modeling method, a large-span roof wind pressure coherence function considering different wind direction angles is put forward, generalized extreme value distribution is adopted to estimate marginal probability distribution of coherence function parameters, probability dependency of the coherence function parameters is quantized based on Vine Copula, and the probability dependency of the coherence function parameters is quantized based on Vine Copula. The wind pressure coherence of the large-span roof structure is effectively simulated by considering the uncertainty of the wind field. The specific implementation process is as follows: A, preprocessing wind pressure measurement data of the large-span roof structure, and establishing a wind pressure data set; b, deducing a wind pressure coherence function model considering wind directions, and fitting coherence functions of wind pressures measured under different wind directions through a nonlinear least square method; c, estimating marginal probability distribution of coherence function parameters according to the parameter samples of the fitted coherence function; d, on the basis of Vine Copula, constructing a probability dependency relationship among different coherence function parameters; and E, generating a random parameter sample by adopting a conditional sampling method, and substituting parameters into the proposed coherence function to obtain a simulated wind pressure coherence function of the large-span roof structure.

Description

Technical Field

[0001] The invention relates to a method for modeling wind pressure coherence of a large-span roof structure, in particular to a method for modeling wind pressure coherence probability of a large-span roof structure based on consideration of wind direction influence and parameter dependency, and belongs to the field of structural wind engineering. Background Art

[0002] Large-span roof structures are widely used in large public buildings such as stadiums, high-speed railway stations, airport terminals and exhibition centers. They are usually characterized by light weight, high flexibility and low damping. They are prone to large deformation and vibration under wind loads and are typical wind-sensitive structures. The spatiotemporal correlation of wind pressure is a key factor in the wind-resistant design of large-span roof structures. The wind pressure coherence function, as a tool to quantify the statistical correlation of wind pressure, plays an important role in the evaluation of wind loads and the analysis of structural wind response. Therefore, the study of wind pressure coherence modeling for large-span roofs has important theoretical significance and practical value for the wind-resistant safety design of structures.

[0003] Existing wind coherence function models, such as the Davenport model and the Krenk model, are widely used to describe the spatial coherence of wind fields. However, they cannot fully consider the influence of wind direction angle and model parameter dependence on the roof wind pressure distribution law, which may lead to underestimation of the wind-induced response of large-span roof structures. In addition, the existing wind pressure coherence function model does not fully reflect the influence of turbulence characteristics and complex wind field environmental factors, and it is difficult to reflect the uncertainty of roof wind pressure distribution. It is difficult to fully describe the spatiotemporal variability of wind pressure coherence on large-span roofs using only traditional models. To this end, a wind pressure coherence function considering the influence of wind direction is proposed, and the marginal probability distribution of coherence function parameters is estimated by probability statistics method. Based on Vine Copula, the probability dependence of random parameters is quantified, and a wind pressure coherence function model of large-span roof structures considering parameter dependence is established. The uncertainty of the spatial distribution of roof wind pressure is effectively quantified, providing a more reliable wind load model for the wind-resistant design of large-span roof structures based on reliability. Summary of the invention

[0004] In view of the shortcomings of the prior art, the present invention proposes a probabilistic modeling method for wind pressure coherence of large-span roof structures based on Vine Copula, which effectively models the spatial coherence of wind pressure by analyzing the uncertainty and parameter dependence of the wind pressure coherence function. The specific contents include:

[0005] The probabilistic modeling method of wind pressure coherence of large-span roof structures based on Vine Copula includes the following steps:

[0006] A. Preprocess the wind pressure measurement data of the large-span roof structure and establish a sample data set of wind pressure coherence function;

[0007] B. The wind pressure coherence function model considering wind direction is derived, and the coherence function of wind pressure measured under different wind directions is fitted by nonlinear least square method;

[0008] C. Estimate the marginal probability distribution of the coherence function parameters based on the parameter samples of the fitted coherence function;

[0009] D. Construct the probability dependence between different coherence function parameters based on Vine Copula;

[0010] E. The conditional sampling method is used to generate random parameter samples, and the parameters are substituted into the proposed coherence function to obtain the simulated wind pressure coherence function of the large-span roof structure.

[0011] Further, step A specifically includes:

[0012] A1. Obtain wind pressure measurement data of large-span roof structures under different wind directions;

[0013] A2. Calculate the net wind pressure coefficient C based on the wind pressure time history data measured on the inner and outer surfaces of the long-span roof p,net (t i ), the calculation formula is:

[0014]

[0015] Where: p out (t) and p in (t) are the wind pressure values ​​on the outer and inner surfaces of the roof, respectively; ρ is the density of the incoming air; V ∞ is the incoming wind speed at the reference point of the wind tunnel test;

[0016] A3. The net wind pressure coefficient data are processed using the Welch average periodogram method to estimate the wind pressure power spectrum density function at each measuring point, calculate the wind pressure coherence function between every two measuring points and construct a wind pressure coherence function data set. The coherence function calculation formula is:

[0017]

[0018] Where: f is the frequency; Coh ij (f) is the wind pressure coherence function between measuring points i and j; S ii (f) and S jj (f) are the auto-power spectra of wind pressure measuring points i and j respectively; S ij (f) is the cross power spectrum between wind pressure measuring points i and j; Co ij (f) and Qu ij (f) are the real and imaginary parts of the cross power spectrum, respectively, where Qu ij (f) is usually small and assumed to be zero.

[0019] Further, step B specifically includes:

[0020] B1. Calculate the downwind distance Δ between each two measuring points u , Across wind distance Δ v and vertical distance Δ w , the calculation formula is:

[0021]

[0022] Where: Δ x =|x i -x j |, Δ y =|y i -y j | are the distances between the measuring points i and j along the x-axis and y-axis in the horizontal plane xoy, respectively, z =|z i -z j | is the distance between measuring points i and j along the z-axis; (x i ,y i ,z i ) and (x j ,y j ,z j ) are the coordinates of measuring points i and j respectively; α is the angle between the horizontal wind direction and the positive direction of the y axis;

[0023] B2. Expand the Krenk coherence function model to three-dimensional form, which can be expressed as:

[0024]

[0025]

[0026] Where: is the correction frequency; L u is the downwind turbulence integral scale; C x , C y and C z are the attenuation coefficients along the x-axis, y-axis, and z-axis, respectively; is the average wind speed at measuring points i and j;

[0027] B3. Reduce the downwind distance Δ u , Across wind distance Δ v and vertical distance Δ w Substituting into the Krenk three-dimensional coherence function model, the wind pressure coherence function model considering wind direction can be obtained:

[0028]

[0029]

[0030] Where: A, B and C are the attenuation coefficients along the downwind direction, across the wind direction and vertical direction respectively;

[0031] B4. The wind pressure coherence function model considering wind direction is fitted by nonlinear least squares method to obtain the fitting parameters A, B, C and L of each coherence function sample. u .

[0032] Further, step C specifically includes:

[0033] C1. Set the determination coefficient R of the fitting curve 2 A threshold value is set, and fitting parameter samples of valid coherence function samples are screened according to the threshold value;

[0034] C2. Wind pressure coherence function parameters A, B, C and L for different wind directions u The generalized extreme value (GEV) distribution model is used to fit the parameter sample, and its cumulative distribution function is:

[0035]

[0036] Where: μ is the location parameter; σ is the scale parameter; k is the shape parameter. When k→0, the GEV distribution is a Gumbel distribution, that is, an extreme value type I distribution; when k<0, the GEV distribution is a Fréchet distribution with an unbounded upper tail, that is, an extreme value type II distribution; when k>0, the GEV distribution is a Weibull distribution with a finite upper tail, that is, an extreme value type III distribution;

[0037] C3. The Kolmogorov-Smirnov (KS) test is used to test the goodness of fit of the marginal probability distribution of the fitted model parameters.

[0038] Further, step D specifically includes:

[0039] D1. Use the Spearman correlation coefficient to analyze the dependencies between model parameters, and determine the random parameters of the model for which the probability dependency relationship is to be established based on the dependencies between different parameters;

[0040] D2. Determine the Vine Copula structure and Copula function family for modeling parameter dependence, and select the Copula structure according to the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) so that the AIC function and the BIC function reach the minimum value. The calculation formulas of the AIC function and the BIC function are:

[0041] AIC=-2lnl(θ)+2kθ (9)

[0042] BIC=-2lnl(θ)+k θ ·lnN s (10)

[0043] Where: l(θ) is the likelihood function based on the observed data; k θ is the number of independently tuned parameters; N s is the number of data samples;

[0044] D3. Use the maximum likelihood estimation (MLE) to obtain the parameters of each pair of variables Copula function family and establish the probability dependence relationship between multiple parameters of the coherent function model. For each binary Copula function, the likelihood function is:

[0045]

[0046] Where: n is the number of data samples; c θ (U 1i ,U 2i ) is the binary Copula density function.

[0047] Further, step E specifically includes:

[0048] E1. Based on the marginal probability distribution of random parameters of coherent functions and the probability dependence of random parameters modeled by Vine Copula, the conditional sampling method is used to generate s-dimensional random parameter samples with probability dependence. First, random sample points (u1,u2,…,u s ), and then generate random parameter samples (x1, x2, …, x according to the Copula conditional marginal distribution function as follows: s ):

[0049]

[0050] Where: F is the conditional distribution function. The conditional distribution function of C-vine can be expressed as:

[0051] The conditional distribution function of D-vine can be expressed as:

[0052]

[0053] E2. Substitute the random parameter samples (x1, x2, …, x s ) is substituted into formula (6), that is, the wind pressure coherence function model considering the wind direction is obtained, and the wind pressure coherence function of the simulated large-span roof structure is obtained.

[0054] The advantages of the present invention are:

[0055] (1) The present invention proposes a wind pressure coherence function that takes wind direction into consideration. By considering the attenuation rate of the coherence function from the downwind direction, crosswind direction and vertical distance respectively, the present invention has stronger applicability and clearer physical meaning, and can comprehensively and effectively capture the changes in the coherence of roof wind pressure under different wind directions.

[0056] (2) When considering the random characteristics of the wind pressure coherence function, the present invention not only accurately characterizes the variability of each random parameter through the marginal probability distribution of the model parameters, but also effectively models the probabilistic dependence between different model parameters based on Vine Copula, which can comprehensively describe the uncertainty of wind loads on large-span roofs.

[0057] (3) The random simulation method of the coherence function of the wind pressure on a large-span roof proposed in the present invention can not only accurately reproduce the marginal probability distribution of each model parameter, but also effectively simulate the nonlinear dependency between the parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 is a flow chart of the present invention;

[0059] Figure 2 1 is a diagram showing the arrangement of wind pressure measurement points for a three-span cylindrical lattice shell wind tunnel test in Example 1 of the present invention;

[0060] Figure 3(a) to Figure 3(d) 3(a) is a distribution histogram of parameter A, FIG3(b) is a distribution histogram of parameter B, FIG3(c) is a distribution histogram of parameter C, and FIG3(d) is a distribution histogram of parameter L. u Distribution histogram of ;

[0061] Figure 4 This is a scatter plot of the random parameter matrix of the model at a wind speed of 60° in Example 1 of the present invention;

[0062] Figure 5 It is the optimal Copula structure diagram of Example 1 of the present invention;

[0063] Figure 6(a) to Figure 6(f) 6(a) is a comparison diagram of the experimental value and simulation value of the coherence function parameter of Example 1 of the present invention, wherein FIG. 6(a) is a comparison diagram of the parameter L u The marginal histogram of A and Figure 6(b) is the parameter L u The marginal histogram of B, Figure 6(c) is the parameter L u Figure 6(d) is the marginal histogram of parameters A and B, Figure 6(e) is the marginal histogram of parameters A and C, and Figure 6(f) is the marginal histogram of parameters B and C;

[0064] Figure 7 is a coherence function simulation sample diagram of embodiment 1 of the present invention;

[0065] Figure 8 is a comparison diagram of the coherence function test value and simulation value of the 50% and 95% quantiles of Example 1 of the present invention;

[0066] Fig. 9 is a partial wind pressure coefficient time history diagram of Example 2 of the present invention;

[0067] Fig.10 This is the average wind pressure distribution diagram at a wind direction of 60° in Example 2 of the present invention. DETAILED DESCRIPTION

[0068] The present invention is further described in detail below in conjunction with the accompanying drawings and two embodiments.

[0069] Example 1

[0070] like Figure 1 This embodiment relates to a probabilistic modeling method for wind pressure coherence of a large-span roof structure based on Vine Copula, comprising the following steps:

[0071] A. Preprocess the wind pressure measurement data of the large-span roof structure and establish a sample data set of wind pressure coherence function, including:

[0072] A1. A three-span cylindrical lattice shell wind tunnel test data, including roof wind pressure time history collected at 24 wind direction angles (0° to 345° with 15° intervals), the wind pressure measurement points are arranged as follows: Figure 2 As shown, the wind pressure data of the largest span No.1 roof is selected for method verification;

[0073] A2. Calculate the net wind pressure coefficient C based on the wind pressure time history data measured on the inner and outer surfaces of the long-span roof p,net (t i );

[0074] A3. The net wind pressure coefficient data are processed using the Welch average periodogram method to estimate the wind pressure power spectrum density function of each measuring point, calculate the wind pressure coherence function between every two measuring points and construct a wind pressure coherence function data set.

[0075] B. The wind pressure coherence function model considering wind direction is derived, and the coherence function of wind pressure measured under different wind directions is fitted by nonlinear least squares method, including:

[0076] B1. Calculate the downwind distance Δ between each two measuring points u , Across wind distance Δ v and vertical distance Δ w ;

[0077] B2. Reduce the downwind distance Δu , Across wind distance Δ v and vertical distance Δ w Substituting into Krenk's three-dimensional coherence function model, we get the wind pressure coherence function model considering wind direction;

[0078] B3. Fit the wind pressure coherence function model considering wind direction by nonlinear least squares method to obtain the fitting parameters A, B, C and L of each coherence function sample. u .

[0079] C. Estimate the marginal probability distribution of the coherence function parameters based on the parameter samples of the fitted coherence function, specifically including:

[0080] C1. Set the determination coefficient R of the fitting curve 2 The threshold value is 0.2, according to R 2 >0.2 to screen the fitting parameter samples of effective coherence function samples;

[0081] C2. Wind pressure coherence function parameters A, B, C and L for different wind directions u The generalized extreme value (GEV) distribution model is used to fit the parameter samples. The distribution fitting results of the coherence function parameters under the 60° wind direction are as follows: Figure 3(a) to Figure 3(d) As shown;

[0082] C3. The Kolmogorov-Smirnov (KS) test is used to test the goodness of fit of the marginal probability distribution of the fitted model parameters.

[0083] D. Construct the probability dependency between different coherence function parameters based on Vine Copula, including:

[0084] D1. Spearman correlation coefficient is used to analyze the dependency between model parameters. The random parameters of the model with probability dependency relationship to be established are determined according to the dependency between different parameters. The scatter plot of the random parameter matrix of the model under 60° wind direction is shown in the figure. Figure 4 As shown, ρ in each subgraph is the Spearman correlation coefficient;

[0085] D2. Determine the Vine Copula structure and Copula function family for modeling parameter dependence, and select the Copula structure based on the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) so that the AIC function and the BIC function reach the minimum value. The optimal Copula structure is Figure 5 The C-vine structure shown;

[0086] D3. Maximum likelihood estimation (MLE) is used to obtain the parameters of the Copula function family for each pair of variables, and the probability dependence relationship between multiple parameters of the coherent function model is established. The parameters of the optimal Copula structure are shown in Table 1.

[0087] Table 1 Optimal Copula structure and parameters

[0088]

[0089] E. Use conditional sampling method to generate random parameter samples, substitute the parameters into the proposed coherence function, and obtain the simulated wind pressure coherence function of the large-span roof structure. Under the working condition of 60° wind direction angle, the test value and simulation value of the coherence function parameter and the corresponding marginal distribution are as follows: Figure 6(a) to Figure 6(f) As shown, the coherence function simulation sample is as follows Figure 7 As shown in the figure, the comparison between the experimental and simulated values ​​of the coherence function at the 50% and 95% quantiles is shown in the figure. Figure 8 shown.

[0090] Example 2

[0091] This embodiment relates to a method for estimating wind pressure distribution of a large-span roof structure using the large-span roof structure wind pressure coherence probability modeling method based on Vine Copula in Embodiment 1, comprising the following steps:

[0092] The first five steps of this embodiment are the same as steps A to E of embodiment 1. In addition, step F is added. The coherence function and Davenport wind power spectrum simulated in embodiment 1 are used to simulate the wind pressure coefficient time history of the No. 1 roof in embodiment 1 to estimate the wind pressure distribution of the roof in the area, which specifically includes:

[0093] F1. Using the wind pressure coherence function simulated in Example 1 and selecting the Davenport wind power spectrum, the wind pressure time history of No.1 roof is simulated by spectral representation, and the wind pressure coefficient time history of No.1 roof is calculated according to the wind speed at the reference point. Part of the wind pressure coefficient time history data is as follows Fig. 9 As shown;

[0094] F2. According to the simulated wind pressure coefficient time history, the average wind pressure coefficient C of No.1 roof is calculated using the 10-minute basic time interval. p,mean , where the average wind pressure distribution of No.1 roof at 60° wind direction is as follows Fig.10 shown.

[0095] The contents described in the implementation cases of this specification are merely an enumeration of the implementation forms of the inventive concept. The protection scope of the present invention should not be deemed to be limited to the specific forms described in the implementation cases. The protection scope of the present invention also extends to the equivalent technical means that can be conceived by those skilled in the art based on the inventive concept.

Claims

1. The probabilistic modeling method of wind pressure coherence of large-span roof structures based on Vine Copula includes the following steps: A. Preprocess the wind pressure measurement data of the large-span roof structure and establish a sample data set of wind pressure coherence function; B. The wind pressure coherence function model considering wind direction is derived, and the coherence function of wind pressure measured under different wind directions is fitted by nonlinear least square method; C. Estimate the marginal probability distribution of the coherence function parameters based on the parameter samples of the fitted coherence function; D. Construct the probability dependence between different coherence function parameters based on Vine Copula; E. The conditional sampling method is used to generate random parameter samples, and the parameters are substituted into the proposed coherence function to obtain the simulated wind pressure coherence function of the large-span roof structure.

2. The Vine Copula-based probabilistic modeling method for wind pressure coherence of large-span roof structures according to claim 1, characterized in that: Step A specifically includes: A1. Obtain wind pressure measurement data of large-span roof structures under different wind directions; A2. Calculate the net wind pressure coefficient C based on the wind pressure time history data measured on the inner and outer surfaces of the large-span roof p,net (t i ), the calculation formula is: Where: p out (t) and p in (t) are the wind pressure values ​​on the outer and inner surfaces of the roof, respectively; ρ is the density of the incoming air; V ∞ is the incoming wind speed at the reference point of the wind tunnel test; A3. The net wind pressure coefficient data are processed using the Welch average periodogram method to estimate the wind pressure power spectrum density function at each measuring point, calculate the wind pressure coherence function between every two measuring points and construct a wind pressure coherence function data set. The coherence function calculation formula is: Where: f is the frequency; Coh ij (f) is the wind pressure coherence function between measuring points i and j; S ii (f) and S jj (f) are the auto-power spectra of wind pressure measuring points i and j respectively; S ij (f) is the cross power spectrum between wind pressure measuring points i and j; Co ij (f) and Qu ij (f) are the real and imaginary parts of the cross power spectrum, respectively, where Qu ij (f) is usually small and assumed to be zero.

3. The Vine Copula-based probabilistic modeling method for wind pressure coherence of large-span roof structures according to claim 1, characterized in that: Step B specifically includes: B1. Calculate the downwind distance Δ between each two measuring points u , Across wind distance Δ v and vertical distance Δ w , the calculation formula is: Where: Δ x =|x i -x j |, Δ y =|y i -y j | are the distances between the measuring points i and j along the x-axis and y-axis in the horizontal plane xoy, respectively, z =|z i -z j | is the distance between measuring points i and j along the z-axis; (x i ,y i ,z i ) and (x j ,y j ,z j ) are the coordinates of measuring points i and j respectively; α is the angle between the horizontal wind direction and the positive direction of the y axis; B2. Expand the Krenk coherence function model to three-dimensional form, which can be expressed as: Where: is the correction frequency; L u is the downwind turbulence integral scale; C x , C y and C z are the attenuation coefficients along the x-axis, y-axis, and z-axis, respectively; is the average wind speed at measuring points i and j; B3. Reduce the downwind distance Δ u , Across wind distance Δ v and vertical distance Δ w Substituting into the Krenk three-dimensional coherence function model, the wind pressure coherence function model considering wind direction can be obtained: Where: A, B and C are the attenuation coefficients along the downwind direction, across the wind direction and vertical direction respectively; B4. The wind pressure coherence function model considering wind direction is fitted by nonlinear least squares method to obtain the fitting parameters A, B, C and L of each coherence function sample. u .

4. The Vine Copula-based probabilistic modeling method for wind pressure coherence of a large-span roof structure according to claim 1, wherein step C specifically comprises: C1. Set the determination coefficient R of the fitting curve 2 A threshold value is set, and fitting parameter samples of valid coherence function samples are screened according to the threshold value; C2. Wind pressure coherence function parameters A, B, C and L for different wind directions u The generalized extreme value (GEV) distribution model is used to fit the parameter sample, and its cumulative distribution function is: Where: μ is the location parameter; σ is the scale parameter; k is the shape parameter. When k→0, the GEV distribution is a Gumbel distribution, that is, an extreme value type I distribution; when k<0, the GEV distribution is a Fréchet distribution with an unbounded upper tail, that is, an extreme value type II distribution; when k>0, the GEV distribution is a Weibull distribution with a finite upper tail, that is, an extreme value type III distribution; C3. The Kolmogorov-Smirnov (KS) test is used to test the goodness of fit of the marginal probability distribution of the fitted model parameters.

5. The Vine Copula-based probabilistic modeling method for wind pressure coherence of large-span roof structures according to claim 1, characterized in that: Step D specifically includes: D1. Use the Spearman correlation coefficient to analyze the dependencies between model parameters, and determine the random parameters of the model for which the probability dependency relationship is to be established based on the dependencies between different parameters; D2. Determine the Vine Copula structure and Copula function family for modeling parameter dependence, and select the Copula structure according to the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) so that the AIC function and the BIC function reach the minimum value. The calculation formulas of the AIC function and the BIC function are: AIC=-2lnl(θ)+2k θ (9) BIC=-2lnl(θ)+k θ ·lnN s (10) Where: l(θ) is the likelihood function based on the observed data; k θ is the number of independently tuned parameters; N s is the number of data samples; D3. Use the maximum likelihood estimation (MLE) to obtain the parameters of each pair of variables Copula function family and establish the probability dependence relationship between multiple parameters of the coherent function model. For each binary Copula function, the likelihood function is: Where: n is the number of data samples; c θ (U 1i ,U 2i ) is the binary Copula density function.

6. The Vine Copula-based probabilistic modeling method for wind pressure coherence of a large-span roof structure according to claim 1, characterized in that: Step E specifically includes: E1. Based on the marginal probability distribution of random parameters of coherent functions and the probability dependence of random parameters modeled by Vine Copula, the conditional sampling method is used to generate s-dimensional random parameter samples with probability dependence. First, random sample points (u1,u2,…,u s ), and then generate random parameter samples (x1, x2, …, x according to the Copula conditional marginal distribution function as follows: s ): Where: F is the conditional distribution function. The conditional distribution function of C-vine can be expressed as: The conditional distribution function of D-vine can be expressed as: E2. Substitute the random parameter samples (x1, x2, …, x s ) is substituted into formula (6), that is, the wind pressure coherence function model considering the wind direction is obtained, and the wind pressure coherence function of the simulated large-span roof structure is obtained.

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