Probabilistic modeling method of wind pressure coherence for long-span roof structures based on Vine Copula

Abstract

A Vine Copula-based probabilistic modeling method for wind pressure coherence of large-span roof structures is proposed. A wind pressure coherence function for large-span roof structures considering different wind directions is proposed. The marginal probability distribution of the coherence function parameters is estimated using a generalized extreme value distribution. The probabilistic dependence of the coherence function parameters is quantified using Vine Copula, enabling effective simulation of wind pressure coherence for large-span roof structures considering wind field uncertainty. The specific implementation process is as follows: A. Preprocessing the measured wind pressure data for large-span roof structures to establish a wind pressure dataset; B. Derivation of a wind pressure coherence function model that considers wind direction and fitting the coherence function of the measured wind pressure under different wind directions using a nonlinear least squares method; C. Estimate the marginal probability distribution of the coherence function parameters based on the parameter samples of the fitted coherence function; D. Constructing the probabilistic dependence between the parameters of different coherence functions using Vine Copula; E. Using conditional sampling to generate random parameter samples, these parameters are substituted into the proposed coherence function to obtain the simulated wind pressure coherence function for large-span roof structures.

Description

Technical Field

[0001] The present invention relates to a wind pressure coherence modeling method for a large-span roof structure, in particular to a wind pressure coherence probability modeling method for 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] Long-span roof structures are widely used in large public buildings such as stadiums, high-speed rail stations, airport terminals, and convention centers. They are typically lightweight, flexible, and have low damping properties. They are susceptible to significant deformation and vibration under wind loads, making them typical wind-sensitive structures. The spatiotemporal correlation of wind pressure is a key factor in the wind-resistant design of long-span roof structures. The wind pressure coherence function, as a tool for quantifying the statistical correlation of wind pressure, plays an important role in wind load assessment and structural wind response analysis. Therefore, research on wind pressure coherence modeling for long-span roofs has important theoretical and practical implications 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 fail to fully consider the effects of wind direction angle and model parameter dependence on roof wind pressure distribution, potentially leading to underestimation of the wind-induced response of long-span roof structures. Furthermore, existing wind pressure coherence function models do not fully reflect the influence of turbulence characteristics and complex wind field environmental factors, making it difficult to reflect the uncertainty of roof wind pressure distribution. Traditional models alone are insufficient to fully describe the spatiotemporal variability of wind pressure coherence on long-span roofs. To address this issue, a wind pressure coherence function that considers the influence of wind direction is proposed. Probabilistic statistical methods are used to estimate the marginal probability distribution of the coherence function parameters. The probabilistic dependence of random parameters is quantified based on the Vine Copula. A parameter-dependent wind pressure coherence function model for long-span roof structures is established. This effectively quantifies the uncertainty of the spatial distribution of roof wind pressure, providing a more reliable wind load model for the reliability-based wind-resistant design of long-span roof structures. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this paper proposes a probabilistic modeling method for wind pressure coherence of large-span roof structures based on Vine Copula. By analyzing the uncertainty and parameter dependence of the wind pressure coherence function, this method effectively models the spatial coherence of wind pressure. The specific contents include:

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

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

[0007] B. Derive a wind pressure coherence function model considering wind direction, and fit the coherence function of wind pressure measured under different wind directions by nonlinear least squares 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. Constructing the probabilistic dependency 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 wind pressure coherence function of the simulated large-span roof structure.

[0011] Further, step A specifically includes:

[0012] A1. Obtain wind pressure measurement data for long-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. Use the Welch average periodogram method to process the net wind pressure coefficient data, 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 autopower spectra of wind pressure measuring points i and j respectively; S ij (f) is the cross power spectrum between wind pressure measurement points i and j; Co ij (f) and Qu ij (f) are the real and imaginary parts of the cross power spectrum, where Qu ij (f) is usually small and assumed to be 0.

[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. Extending the Krenk coherence function model to three-dimensional form, it 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, we can obtain the wind pressure coherence function model considering wind direction:

[0028]

[0029]

[0030] Where: A, B and C are the attenuation coefficients along the downwind, acrosswind and vertical directions respectively;

[0031] B4. 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 .

[0032] Further, step C specifically includes:

[0033] C1. Set the coefficient of determination 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. Use the Kolmogorov-Smirnov (KS) test to perform a goodness of fit test on 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. Based on the dependencies between different parameters, determine the random parameters of the model for which probabilistic dependencies are to be established.

[0040] D2. Determine the Vine Copula structure and family of Copula functions for modeling parameter dependencies. Select the Copula structure based on the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) to minimize the AIC and BIC functions. The AIC and BIC functions are calculated as follows:

[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 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, a conditional sampling method is used to generate s-dimensional random parameter samples with probability dependence. First, random sample points (u1, u2, ..., u) that obey uniform distribution are generated in the interval [0, 1]. s ), and then generate random parameter samples (x1, x2, ..., x according to the Copula conditional marginal distribution function. 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. Set the random parameter sample (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, crosswind and vertical distances respectively, it 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 wind pressure coherence function for large-span roofs proposed in this invention can not only accurately reproduce the marginal probability distribution of each model parameter, but also effectively simulate the nonlinear dependence between parameters. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0059] Figure 2 Layout diagram 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 The distribution histogram of ;

[0061] Figure 4 This is a scatter plot of the random parameter matrix of the model under a 60° wind direction according to Example 1 of the present invention;

[0062] Figure 5 This 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 and simulated values ​​of the coherence function parameters of Example 1 of the present invention, wherein FIG6(a) is a comparison diagram of the parameter L u With the marginal histogram of A, Figure 6(b) is the parameter L u With the marginal histogram of B, Figure 6(c) is the parameter L u and C, 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 Example 1 of the present invention;

[0065] Figure 8 1 is a comparison chart of the experimental and simulated values ​​of the coherence function at the 50% and 95% quantiles of Example 1 of the present invention;

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

[0067] Figure 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 will be further described in detail below with reference to 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 large-span roof structures and establish a sample data set of wind pressure coherence functions, including:

[0072] A1. Wind tunnel test data for a three-span cylindrical lattice shell, including roof wind pressure time histories collected at 24 wind direction angles (0° to 345° in 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 are 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. Use the Welch average periodogram method to process the net wind pressure coefficient data, estimate the wind pressure power spectral density function at each measuring point, calculate the wind pressure coherence function between every two measuring points, and construct a wind pressure coherence function dataset.

[0075] B. Derive a wind pressure coherence function model that takes wind direction into account, and use the nonlinear least squares method to fit the coherence function of wind pressure measured under different wind directions, specifically 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 coefficient of determination R of the fitting curve 2 The threshold 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. Use the Kolmogorov-Smirnov (KS) test to perform a goodness of fit test on the marginal probability distribution of the fitted model parameters.

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

[0084] D1. Use the Spearman correlation coefficient to analyze the dependencies between model parameters. According to the dependencies between different parameters, the random parameters of the model with the probability dependency relationship to be established are determined. The scatter plot of the random parameter matrix of the model under the 60° wind direction is as follows: 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 probabilistic dependence relationship between multiple parameters of the coherence 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 the conditional sampling method to generate random parameter samples, substitute the parameters into the proposed coherence function, and obtain the wind pressure coherence function of the simulated large-span roof structure. Under the working condition of 60° wind direction angle, the experimental value and simulation value of the coherence function parameters 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 Vine Copula-based probabilistic modeling method for wind pressure coherence of a large-span roof structure in Example 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 roof No. 1 in embodiment 1 and estimate the wind pressure distribution on the roof in the area. Specifically, the following steps are performed:

[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 the No.1 roof is simulated using the spectral representation method. The wind pressure coefficient time history of the No.1 roof is calculated according to the wind speed at the reference point. Some wind pressure coefficient time history data are as follows: Figure 9 As shown;

[0094] F2. Based on 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 Figure 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 scope of protection of the present invention should not be regarded as limited to the specific forms described in the implementation cases. The scope of protection of the present invention also extends to equivalent technical means that can be conceived by those skilled in the art based on the inventive concept.

Claims

1. A probabilistic modeling method for wind pressure coherence of large-span roof structures based on Vine Copula includes the following steps: A. Preprocess the wind pressure measurement data of large-span roof structures and establish a sample data set of wind pressure coherence functions; B. Derive a wind pressure coherence function model considering wind direction, and fit the coherence function of wind pressure measured under different wind directions by nonlinear least squares method; C. Estimating the marginal probability distribution of the coherence function parameters based on the parameter samples of the fitted coherence function; specifically including: C1. Set the coefficient of determination 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 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. Use the Kolmogorov-Smirnov test to test the goodness of fit of the marginal probability distribution of the fitted model parameters; D. Constructing the probabilistic dependency 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 wind pressure coherence function of the simulated 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 is characterized in that: Step A specifically includes: A1. Obtain wind pressure measurement data for long-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 long-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. Use the Welch average periodogram method to process the net wind pressure coefficient data, 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 autopower spectra of wind pressure measuring points i and j respectively; S ij (f) is the cross power spectrum between wind pressure measurement points i and j; Co ij (f) and Qu ij (f) are the real and imaginary parts of the cross power spectrum, where Qu ij (f) is usually small and assumed to be 0.

3. The Vine Copula-based probabilistic modeling method for wind pressure coherence of large-span roof structures according to claim 1 is characterized in that: Step B specifically includes: B1. Calculate the downwind distance between each two measuring points △ u , Across-wind distance △ v and vertical distance △ w , the calculation formula is: Where: x =|x i -x j |、△ 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. Extending the Krenk coherence function model to three-dimensional form, it 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. Change the downwind distance to △ u , Across-wind distance △ v and vertical distance △ w Substituting into the Krenk three-dimensional coherence function model, we can obtain the wind pressure coherence function model considering wind direction: Where: A, B and C are the attenuation coefficients along the downwind, acrosswind and vertical directions respectively; B4. 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 .

4. 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. Based on the dependencies between different parameters, determine the random parameters of the model for which probabilistic dependencies are to be established. D2. Determine the Vine Copula structure and family of Copula functions for modeling parameter dependencies. Based on the Akaike information criterion (AIC) and the Bayesian information criterion (BIC), select the Copula structure so that the AIC and BIC functions reach their minimum values. The AIC and BIC functions are calculated as follows: Where: 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 maximum likelihood estimation (MLE) to obtain the parameters of each pair of copula function families and establish the probabilistic 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.

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 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, a conditional sampling method is used to generate s-dimensional random parameter samples with probability dependence. First, random sample points (u1, u2, ..., u1) that obey uniform distribution are generated in the interval [0, 1]. s ), and then generate random parameter samples (x1, x2, ..., x according to the Copula conditional marginal distribution function. 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. Set the random parameter sample (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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