A Fast Algorithm for Wind-Induced Response of Long-Span Roof Structures Considering Quasi-Static Response of Higher-Order Modes

Abstract

The present invention discloses a fast algorithm for wind-induced response of long-span roof structures considering the quasi-static response of high-order modes, comprising the following steps: Step S11, establishing a finite element model of the long-span roof structure and determining the modal damping ratio of the long-span roof structure according to the design conditions to obtain the structural dynamic characteristics of the long-span roof structure; Step S12, determining the random wind load acting on the long-span roof structure through a wind tunnel test and calculating the Fourier transform function of the wind load corresponding to the truncation function of the random wind load; Step S13, calculating by using the modal acceleration method according to the structural dynamic characteristics and the Fourier transform function of the wind load to obtain the Fourier transform function of the structural displacement response; Step S14, performing power spectral density estimation by using the periodogram method according to the Fourier transform function of the structural displacement response to obtain the power spectral density matrix of the structural displacement response, and integrating the power spectral density matrix of the structural displacement response to obtain the root mean square of the displacement response of the long-span roof structure. In the present invention, the quasi-static response of high-order modes is taken into account, which has better convergence than the traditional method, and under the condition of the same accuracy, the calculation time and memory occupation of the present invention are relatively small.

Description

Technical Field

[0001] The present invention relates to the technical field of wind resistance analysis of long-span roof structures, and particularly to a fast algorithm for wind-induced response of long-span roof structures considering the quasi-static response of high-order modes. Background Art

[0002] Long-span roof structures are light in mass, large in flexibility, and dense in modal frequencies, and are typical wind-sensitive structures. Accurately calculating their wind-induced responses is the key to wind resistance design. Traditional calculation methods include the Complete Quadratic Combination (CQC) method, the Square Root of Sum of Squares (SRSS) method, the Pseudodynamic Excitation Method (PEM), and the Harmonic Excitation Method (HEM). The above methods are all carried out based on the mode superposition method. For large-scale structures, it is unnecessary and impossible to consider all their modes. Usually, the practice is to truncate the high-order modes and only retain the low-order modes of the structure for calculation. However, the high-order modes of long-span roof structures have an unnegligible contribution to the response, and modal truncation may cause a large difference between the calculation results and the actual response.

[0003] To address this problem, one type of method is to replace the free vibration modes with Ritz modes constructed based on the Proper Orthogonal Decomposition (POD) of wind loads (Ritz-POD method), and the latter has better response calculation accuracy under the same number of participating vibration mode orders. Another type of method is to continue to use the free vibration modes but compensate for the responses of high-order modes. The Modal Acceleration Method (MAM) is a conceptually clear modal truncation compensation method. By adding a correction term, it can take into account the quasi-static response of high-order modes and is very suitable for the dynamic response analysis of structures under low-frequency loads such as wind loads.

[0004] In existing research, the methods for calculating the wind-induced response of long-span roof structures using the MAM method are all carried out within the framework of the CQC method. However, for the response analysis of large-scale structures with a large number of degrees of freedom, the calculation and storage overheads of the CQC method are unacceptable in engineering analysis. Therefore, it is not feasible to calculate the wind-induced response of long-span roof structures according to the MAM method within the CQC framework. Summary of the Invention

[0005] Based on this, the object of the present invention is to provide a high-precision and fast algorithm for the wind-induced response of complex long-span roof structures considering quasi-static compensation. This algorithm has good convergence, and the calculation time and required memory are greatly reduced.

[0006] In a first aspect, the present invention provides a fast algorithm for the wind-induced response of long-span roof structures considering the quasi-static response of high-order modes, including the following steps:

[0007] Step S11: Establish a finite element model of the long-span roof structure, and determine the modal damping ratio of the long-span roof structure according to the design conditions to obtain the structural dynamic characteristics of the long-span roof structure, where the structural dynamic characteristics include vibration modes, natural vibration frequencies, stiffness matrix, and mass matrix;

[0008] Step S12: Determine the random wind load {p(t)} acting on the long-span roof structure through wind tunnel tests, and calculate the Fourier transform function {P(f)} of the wind load corresponding to the truncated function {p(t)} of the random wind load. The Fourier transform function of the wind load is {P(f)}:

[0009]

[0010] Step S13: Calculate using the modal acceleration method according to the structural dynamic characteristics and the Fourier transform function of the wind load to obtain the Fourier transform function of the structural displacement response;

[0011] Step S14: Perform power spectral density estimation using the periodogram method according to the Fourier transform function of the structural displacement response to obtain the power spectral density matrix of the structural displacement response, and integrate the power spectral density matrix of the structural displacement response to obtain the root mean square of the displacement response of the long-span roof structure.

[0012] Preferably, the step S13 includes:

[0013] Step S131: Construct the modal frequency response function matrix of the long-span roof structure, and the modal frequency response function matrix is [H(f)] = diag(H1, H2,..., H n ), and the i-th order frequency response function H i (f) is:

[0014]

[0015] where f i is the i-th order natural vibration frequency, and ξ i is the i-th order modal damping ratio;

[0016] Step S132: Construct the modal stiffness matrix of the long-span roof structure The modal stiffness matrix is:

[0017]

[0018] where [Ф t is the vibration mode matrix, [K] is the stiffness matrix, and the superscript T represents transpose;

[0019] Step S133: Calculate using the modal acceleration method based on the modal frequency response function matrix and the modal stiffness matrix to obtain the Fourier transform function {Y(f)} of the structural displacement response. The Fourier transform function {Y(f)} of the structural displacement response is as follows:

[0020]

[0021] Preferably, the step S14 includes:

[0022] Step S141: Perform power spectral density estimation using the periodogram method based on the Fourier transform function of the structural displacement response to obtain the power spectral density matrix [S yy (f)] of the structural displacement response. The power spectral density matrix [S yy (f)] of the structural displacement response is as follows:

[0023]

[0024] In the formula, T is the sampling time, and the superscript * represents the conjugate;

[0025] Step S142: Integrate the power spectral density matrix of the structural displacement response to obtain the root mean square of the displacement response of the long - span roof structure. The root mean square of the displacement response is as follows:

[0026]

[0027] In the formula, σ y,k is the root mean square of the displacement on the k - th degree of freedom of the structure, and S yy,k (f) is the auto - spectrum of the displacement response on the k - th degree of freedom, that is, the k - th element on the main diagonal of the power spectral density matrix of the structural displacement response.

[0028] On the second aspect, the present invention provides a fast algorithm for the wind - induced response of a long - span roof structure considering the quasi - static response of high - order modes, including the following steps:

[0029] Step S21: Establish a finite - element model of the long - span roof structure and determine the modal damping ratio of the long - span roof structure according to the design conditions to obtain the structural dynamic characteristics of the long - span roof structure. The structural dynamic characteristics include vibration modes, natural vibration frequencies, stiffness matrix, and mass matrix;

[0030] Step S22: Determine the random wind load {p(t)} acting on the long - span roof structure through a wind tunnel test and calculate the Fourier transform function {P(f)} of the wind load corresponding to the truncated function {p(t)}. The Fourier transform function {P(f)} of the wind load is as follows:

[0031]

[0032] Step S23: Calculate according to the structural dynamic characteristics and the Fourier transform function of wind load by using the simplified modal acceleration method to obtain the Fourier transform function of the structural displacement response in modal coordinates;

[0033] Step S24: Estimate the power spectral density by using the periodogram method according to the Fourier transform function of the structural displacement response to obtain the power spectral density matrix of the structural displacement response in modal coordinates, and integrate the power spectral density matrix of the modal response to obtain the root mean square of the displacement response of the long-span roof structure.

[0034] Preferably, the said Step S23 includes:

[0035] Step S231: Conduct proper orthogonal decomposition on the random wind load {p(t)}:

[0036]

[0037] where M is the order of the proper mode, a k (t) is the corresponding modal coordinate function, {G k} is the k-th proper mode;

[0038] Step S232: Calculate the m load compensation terms corresponding to the m proper modes, and the calculation method is:

[0039]

[0040] where is the k-th load compensation term, [I] is the identity matrix, [M] is the mass matrix, [Ф t is the mode shape matrix, and the superscript T represents transpose;

[0041] Step S233: Take the first load compensation term as the external load distribution form, solve the static equation to obtain the displacement response {y1} of the structure under , and orthogonalize the displacement response {y1} with the structural mass matrix to obtain the first quasi-static mode {ψ1}, where

[0042] The static equation is:

[0043]

[0044] where [K] is the stiffness matrix;

[0045] The quasi-static mode {ψ1}:

[0046]

[0047] where the superscript T represents transpose, and [M] is the mass matrix;

[0048] Step S234: Using the 2nd to m-th load compensation terms as the external load distribution form, solve the static equation to obtain the displacement response {y k} of the structure under the action, and use the Gram-Schmidt orthogonalization method to orthogonalize the displacement response {y k} with respect to mass with all the previous vectors to obtain a pure vector that does not contain the components of the previous vectors Then, orthogonalize the pure vector with the structure mass matrix to obtain the k-th quasi-static mode, where

[0049] The static equation is:

[0050]

[0051] In the formula, k ≥ 2, [K] is the stiffness matrix;

[0052] The pure vector is:

[0053]

[0054] In the formula, the superscript T represents the transpose, {ψ i} is the i-th quasi-static mode;

[0055] The k-th quasi-static mode {ψ k} = is:

[0056]

[0057] In the formula, the superscript T represents the transpose, [M] is the mass matrix;

[0058] Step S235: After obtaining m quasi-static modes by solving, denote the quasi-static mode matrix as matrix [ψ] = [{ψ1}, {ψ2}, {ψ3}... {ψ m}], calculate the modal stiffness matrix [K*] corresponding to the quasi-static modes and solve it to obtain the eigenvalues [p] and the eigenmatrix [A], and then perform an orthogonal transformation on the matrix [ψ] to obtain the transformed quasi-static mode matrix [Ф c , and the corresponding calculation formula is:

[0059] [K*] = [ψ] T [K][ψ];

[0060] [K * [A] = [A][p];

[0061] [Φ c = [ψ][A]; In the formula, the superscript T represents the transpose;

[0062] Step S236, calculate the Fourier transform function of the structural displacement response in modal coordinates. The specific method is as follows:

[0063] {Q t (f)} = [H(f)][Φ t T {P(f)};

[0064]

[0065]

[0066] In the formula, {Q t (f)} is the Fourier transform function of the truncated modal response, [H(f)] is the modal frequency response function matrix, [Ф t is the mode shape matrix, the superscript T represents the transpose, {Q c (f)} is the Fourier transform function of the compensated modal response, - is the modal stiffness matrix corresponding to the quasi-static mode, [I] is the identity matrix, [M] is the mass matrix, is the combination of {Q t (f)} and {Q c (f)}.

[0067] Preferably, in step S231, the k-th eigenmode {G k} is obtained by solving the eigenvalue problem of the covariance matrix [C pp corresponding to the random wind load {p(t)}, that is:

[0068] [C pp {G k} = λ k {G k};

[0069] In the formula, λ k is the k-th eigenvalue;

[0070] The modal coordinate function a k (t) corresponding to the k-th eigenmode {G k} is calculated as follows:

[0071]

[0072] In the formula, the superscript T represents the transpose.

[0073] Preferably, step S24 includes:

[0074] Step S241, according to the Fourier transform function of the structural displacement response, use the periodogram method to estimate the power spectral density, and the power spectral density matrix [S​qq , the power spectral density matrix [S of this modal response qq is:

[0075]

[0076] In the formula, T is the sampling time, and the superscript * represents conjugate;

[0077] Step S242, integrate the power spectral density matrix [S of the modal response qq to obtain the covariance matrix [C of the modal response qq :

[0078]

[0079] After calculating the covariance matrix [C of the modal response in step S243 qq , the root mean square value of the displacement response on the k-th degree of freedom can be expressed as:

[0080]

[0081] In the formula, c ij is the element in the i-th row and k-th column of the modal response covariance matrix, is the matrix The element in the k-th row and i-th column;

[0082] Among them, the matrix can be expressed as:

[0083]

[0084] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0085] First, the present invention takes into account the quasi-static response of high-order modes, has better convergence than traditional methods, and under the same accuracy conditions, the calculation time and memory occupation of the present invention are relatively small;

[0086] Second, compared with the modal acceleration method and the simplified modal acceleration method, the calculation accuracies of the two are comparable, and the latter requires less calculation time and memory occupation. Description of the Drawings

[0087] Figure 1 It is a schematic flow chart of a fast algorithm for wind-induced response of a long-span roof structure considering the quasi-static response of high-order modes in the first embodiment of the present invention;

[0088] Figure 2 It is a schematic flow chart of a fast algorithm for wind-induced response of a long-span roof structure considering the quasi-static response of high-order modes in the second embodiment of the present invention;

[0089] Figure 3 The architectural rendering of the engineering example in the present invention;

[0090] Figure 4 The test model in the present invention, the schematic diagram of the measuring point layout on the roof surface and the wind direction angle;

[0091] Figure 5 The natural vibration frequencies of the first 500 order free vibration modes of the structure calculated by the eigenvector method in the present invention;

[0092] Figure 6 The vertical pulsating displacement of the roof nodes calculated by the HEM method under different modal orders in the present invention and the corresponding R 2 fraction;

[0093] Figure 7 The calculation result of the HEM method for the south side cantilever nodes in the present invention;

[0094] Figure 8 The vertical pulsating displacement of the roof nodes calculated by the MAM-HEM method under different modal orders in the present invention and the corresponding R 2 fraction;

[0095] Figure 9 The root mean square of the base lift force calculated by different quasi-static modal orders in the present invention;

[0096] Figure 10 The comparison of the calculation results of the 30th order modal MAM-HEM and SSC-HEM in the present invention. Detailed implementation manners

[0097] The following description is used to disclose the present invention so that those skilled in the art can implement the present invention. The preferred embodiments in the following description are only examples, and those skilled in the art can think of other obvious variations.

[0098] Please refer to Figure 1 A fast algorithm for wind-induced response of long-span roof structures considering high-order modal quasi-static response provided in the first embodiment of the present invention includes the following steps:

[0099] Step S11, establish a finite element model of the long-span roof structure, and determine the modal damping ratio of the long-span roof structure according to the design conditions to obtain the structural dynamic characteristics of the long-span roof structure, where the structural dynamic characteristics include vibration modes, natural vibration frequencies, stiffness matrix and mass matrix;

[0100] Step S12, determine the random wind load {p(t)} acting on the long-span roof structure through wind tunnel tests, and calculate the Fourier transform function {P(f)} of the wind load corresponding to the truncated function {p(t)} of the random wind load. The Fourier transform function of the wind load is {P(f)}:

[0101]

[0102] Step S13: Calculate using the modal acceleration method according to the structural dynamic characteristics and the Fourier transform function of the wind load to obtain the Fourier transform function of the structural displacement response;

[0103] Step S14: Estimate the power spectral density using the periodogram method according to the Fourier transform function of the structural displacement response to obtain the power spectral density matrix of the structural displacement response, and integrate the power spectral density matrix of the structural displacement response to obtain the root mean square of the displacement response of the long-span roof structure.

[0104] Preferably, the said Step S13 includes:

[0105] Step S131: Construct the modal frequency response function matrix of the long-span roof structure, and this modal frequency response function matrix is [H(f)] = diag(H1, H2, …, H n ), and the i-th order frequency response function H i (f) is:

[0106]

[0107] where f i is the i-th order natural vibration frequency, and ξ i is the i-th order modal damping ratio;

[0108] Step S132: Construct the modal stiffness matrix of the long-span roof structure This modal stiffness matrix is:

[0109]

[0110] where [Ф t is the mode shape matrix, [K] is the stiffness matrix, and the superscript T represents the transpose;

[0111] Step S133: Calculate using the modal acceleration method according to the modal frequency response function matrix and the modal stiffness matrix to obtain the Fourier transform function {Y(f)} of the structural displacement response. This Fourier transform function {Y(f)} of the structural displacement response is:

[0112]

[0113] It should be noted that in the Fourier transform function {Y(f)} of the structural displacement response, the last two terms are correction terms, which are equivalent to considering the resonance response of the first n orders of the structure and the complete quasi-static response, compensating for the quasi-static response of the high-order mode response, and the calculation result is more accurate.

[0114] Further, [K] in this step -1 {P(f)} is calculated by solving a system of linear equations, that is, by solving {Y s (f)} in the system of linear equations [K]{Y s (f)}={P(f)}. Since [K] is a sparse matrix, there are many fast solution algorithms, such as: Cholesky decomposition method, conjugate gradient method, generalized minimum residual method, etc. In the formula, {Y s (f)} represents the complete quasi-static response of the structure.

[0115] Preferably, step S14 includes:

[0116] Step S141, according to the Fourier transform function of the structural displacement response, use the periodogram method to estimate the power spectral density to obtain the power spectral density matrix [S yy (f)] of the structural displacement response. The power spectral density matrix [S yy (f)] of the structural displacement response is:

[0117]

[0118] In the formula, T is the sampling time, and the superscript * represents the conjugate;

[0119] Step S142, integrate the power spectral density matrix of the structural displacement response to obtain the root mean square of the displacement response of the long-span roof structure. The root mean square of the displacement response is:

[0120]

[0121] In the formula, σ y,k is the root mean square of the displacement on the k-th degree of freedom of the structure, and S yy,k (f) is the auto-spectrum of the displacement response of the k-th degree of freedom, that is, the k-th element on the main diagonal of the power spectral density matrix of the structural displacement response.

[0122] It should be further clarified that although the method in the first embodiment can effectively improve the response calculation accuracy, there are still the following problems in actual applications:

[0123] (1) In actual engineering, what we care about is the root mean square of the response in physical coordinates. For large structures, due to the large number of degrees of freedom, we usually do not directly calculate the power spectral density of the response in physical coordinates, but first calculate the power spectral density matrix of the modal response, obtain the covariance matrix of the modal response through integration, and then calculate the root mean square value of the response in physical coordinates, that is:

[0124]

[0125]

[0126]

[0127] wherein, [C qq and [S qq are the covariance matrix and the spectral density matrix of the modal response respectively, σ y,k is the root mean square value of the displacement of the k-th degree of freedom, and c ij is the element in the i-th row and k-th column of the modal response covariance matrix.

[0128] In the Fourier transform function {Y(f)} of the structural displacement response, due to the introduction of the quasi-static term, the calculation of the response term and the estimation of the power spectral density have to be carried out in the physical coordinates, which will increase the storage and calculation overheads, and is also not conducive to the subsequent calculation of the equivalent static wind load.

[0129] (2) For large-scale structures, the flexibility matrix [K] -1 is difficult to obtain, and the quasi-static response {Y s (f)} of the structure is obtained by solving a system of linear equations. In the wind tunnel test, to ensure the stability of the statistical results, it is necessary to ensure that there are enough sampling point numbers L, which means that the Fourier transform function {Y s (f)} of the structural displacement response needs to solve the system of linear equations at least L times repeatedly. Although there are many fast algorithms for solving such problems, the calculation overhead of solving the system of linear equations L times repeatedly is still very large.

[0130] The above two problems limit the application of the method in the first embodiment in actual large-scale projects. Therefore, it is necessary to simplify the Fourier transform function {Y(f)} of the structural displacement response, and the core of the simplification lies in the treatment of the quasi-static term.

[0131] Please refer to Figure 2 , a fast algorithm for wind-induced response of long-span roof structures considering the quasi-static response of high-order modes provided in the second embodiment of the present invention includes the following steps:

[0132] Step S21, establish a finite element model of the long-span roof structure, and determine the modal damping ratio of the long-span roof structure according to the design conditions to obtain the structural dynamic characteristics of the long-span roof structure, where the structural dynamic characteristics include vibration modes, natural vibration frequencies, stiffness matrix, and mass matrix;

[0133] Step S22, determine the random wind load {p(t)} acting on the long-span roof structure through a wind tunnel test, and calculate the Fourier transform function {P(f)} of the wind load corresponding to the truncated function {p(t)} of the random wind load. The Fourier transform function {P(f)} of the wind load is:

[0134]

[0135] Step S23: According to the structural dynamic characteristics and the Fourier transform function of the wind load, the simplified modal acceleration method is used for calculation to obtain the Fourier transform function of the structural displacement response in the modal coordinates.

[0136] Step S24: According to the Fourier transform function of the structural displacement response, the periodogram method is used for power spectral density estimation to obtain the power spectral density matrix of the structural displacement response in the modal coordinates, and the power spectral density matrix of the modal response is integrated to obtain the root mean square of the displacement response of the long-span roof structure.

[0137] Preferably, the step S23 includes:

[0138] Step S231: Perform proper orthogonal decomposition on the random wind load {p(t)}:

[0139]

[0140] where m is the order of the proper mode, a k (t) is the corresponding modal coordinate function, {G k} is the k-th proper mode;

[0141] Step S232: Calculate m load compensation terms corresponding to the m proper modes, and the calculation method is:

[0142]

[0143] where is the k-th load compensation term, [I] is the identity matrix, [M] is the mass matrix, [Ф t is the mode shape matrix, and the superscript T represents transpose;

[0144] Step S233: Taking the first load compensation term as the external load distribution form, solve the static equation to obtain the displacement response {y1} of the structure under the action of, and orthogonalize the displacement response {y1} with the structural mass matrix to obtain the first quasi-static mode {ψ1}, where

[0145] The static equation is:

[0146]

[0147] where [K] is the stiffness matrix;

[0148] Quasi-static mode {ψ1}:

[0149]

[0150] In the formula, the superscript T represents transpose, and [M] is the mass matrix;

[0151] Step S234: Take the 2nd to m-th load compensation terms as the external load distribution form, solve the static equation to obtain the displacement response {y k c} of the structure under the action of {p k}, and use the Gram - Schmidt orthogonalization method to orthogonalize the displacement response {y k} with respect to mass with all the previous vectors to obtain a pure vector that does not contain the components of the previous vectors Then, orthogonalize the pure vector with the structure mass matrix to obtain the k-th quasi - static mode, where,

[0152] The static equation is:

[0153]

[0154] In the formula, k≥2, [K] is the stiffness matrix;

[0155] The pure vector is:

[0156]

[0157] In the formula, the superscript T represents transpose, {ψ i} is the i-th quasi - static mode;

[0158] The k-th quasi - static mode {ψ k} = is:

[0159]

[0160] In the formula, the superscript T represents transpose, [M] is the mass matrix;

[0161] Step S235: After solving for m quasi - static modes, denote the quasi - static mode matrix as matrix [ψ] = [{ψ1}, {ψ2}, {ψ3}... {ψ m}], calculate the modal stiffness matrix [K*] corresponding to the quasi - static modes and solve it to obtain the eigenvalues [p] and the eigenmatrix [A], and then perform an orthogonal transformation on the matrix [ψ] to obtain the transformed quasi - static mode matrix [Ф c , and the corresponding calculation formula is:

[0162] [K*] = [ψ] T [K][ψ] ;

[0163] [K * [A] = [A][p] ;

[0164] [Φc = [ψ][A]; where the superscript T represents transpose;

[0165] Step S236, calculate the Fourier transform function of the structural displacement response in modal coordinates. The specific method is as follows:

[0166] {Q t (f)} = [H(f)][Φ t T {P(f)};

[0167]

[0168]

[0169] In the formula, {Q t (f)} is the Fourier transform function of the truncated modal response, [H(f)] is the modal frequency response function matrix, [Ф t is the mode shape matrix, the superscript T represents transpose, {Q c (f)} is the Fourier transform function of the compensated modal response, - is the modal stiffness matrix corresponding to the quasi-static mode, [I] is the identity matrix, [M] is the mass matrix, is the combination of {Q t (f)} and {Q c (f)}.

[0170] Preferably, in step S231, the k-th eigenmode {G k} is obtained by solving the eigenvalue problem of the covariance matrix [C pp corresponding to the stochastic wind load {p(t)}, that is:

[0171] [C pp {G k} = λ k {G k};

[0172] In the formula, λ k is the k-th eigenvalue;

[0173] The modal coordinate function a k corresponding to the k-th eigenmode {G k}(t) is calculated as follows:

[0174]

[0175] In the formula, the superscript T represents transpose.

[0176] Preferably, the said step S24 includes:

[0177] ​Step S241, according to the Fourier transform function of the structural displacement response, the periodogram method is used for power spectral density estimation, and the power spectral density matrix [S qq of the structural displacement response in the modal coordinates can be obtained. The power spectral density matrix [S qq of this modal response is:

[0178]

[0179] where T is the sampling time, and the superscript * represents conjugate;

[0180] Step S242, integrate the power spectral density matrix [S qq of the modal response to obtain the covariance matrix [C qq of the modal response:

[0181]

[0182] Step S243, after obtaining the covariance matrix [C qq of the modal response, the root mean square value of the displacement response on the k-th degree of freedom can be expressed as:

[0183]

[0184] where c ij is the element in the i-th row and k-th column of the covariance matrix of the modal response, is the element in the k-th row and i-th column of the matrix ;

[0185] where the matrix can be expressed as:

[0186]

[0187] Furthermore, it should be noted that in this embodiment, most of the calculations are restricted to the modal coordinates, which greatly saves the calculation time and storage overhead.

[0188] In summary, the present invention takes into account the quasi-static response of high-order modes, has better convergence than the traditional method (HEM), and under the same accuracy conditions, the calculation time and memory occupancy of the present invention are relatively small; and compared with the simplified modal acceleration method, the calculation accuracies of the modal acceleration method and the latter are equivalent, and the latter requires less calculation time and memory occupancy.

[0189] The following uses a specific engineering example to illustrate the method of the present invention in detail:

[0190] Please refer to Figure 3, The Shenzhen International Convention and Exhibition Center project is a super-large convention and exhibition complex, consisting of 16 standard exhibition halls, 2 multi-functional exhibition halls, 1 extra-large exhibition hall, 2 north-south entrance halls and a central corridor, etc. The total north-south span of the project reaches 1.7 km, and it is currently the single building with the largest floor area in the world. Its architectural renderings are as shown in Figure 3 . Taking the standard exhibition hall at the southwestern corner of the convention and exhibition center as the analysis object. The north-south length of the roof structure is about 250 m, there is an open passage with a width of 42 m in the middle, the east-west width is 210 m, the roof is wavy, the height at the highest position is about 26 m, and the maximum overhanging length is 12 m. Since the span and overhanging length of this roof structure are both large, with light mass and small damping ratio, it belongs to a wind-sensitive structure, and the actual wind load on the structure surface needs to be determined through wind tunnel tests.

[0191] Please refer to Figure 4 . According to the design requirements, the upstream terrain of the building model is assumed to be Type A terrain, and the basic wind pressure is 0.75 kPa. Before the wind tunnel test, the required Type A terrain type was simulated in the test section with binary spires, baffles and roughness elements according to the regulations in the "Load Code for Building Structures" GB 50009-2012. The test incoming flow velocity is 10 m / s, the reference height is 0.25 m, and the wind pressure field of the building is measured at 36 wind direction angles at intervals of 10°. The pressure measurement model is made of rigid materials, with a scale ratio of 1 / 150. A total of 592 measuring points are arranged on the model, of which 192 are double-sided measuring points. The layout of the measuring points on the roof surface of the test model and the schematic diagram of the wind direction angle are as shown in Figure 4 . During the test, the model sampling frequency is 300 Hz, the single sampling duration is 30 s, and the number of sampling points is 9000. According to the similarity ratio conversion, it can be known that the prototype sampling frequency is 9.204 Hz, and the prototype sampling duration is 16.3 min.

[0192] Please refer to Figure 5 . The structure is modeled in the SAP2000 software, and the total number of degrees of freedom of the structure is 27126. The first 500 free vibration modes of the structure are calculated by the eigenvector method, and the natural vibration frequencies are as shown in Figure 5 . The natural vibration frequency corresponding to the 500th mode is 5.313 Hz, which has exceeded the maximum resolvable frequency in the test (4.602 Hz). Therefore, the results calculated by the calculation method in the first embodiment for the 500 free vibration modes can be considered to include all the resonance responses and background responses that can be calculated for the structure. The subsequent analysis will use this result as the standard value of the pulsating response result.

[0193] Please refer to Figure 6 (a) to Figure 8 (b), with R 2Taking the score as an indicator to evaluate the overall accuracy of the method, the wind-induced vibration calculation results of the harmonic excitation method (HEM) and the harmonic excitation method based on the modal acceleration method (MAM-HEM, i.e., the calculation method in the first embodiment of the present application) under different modal orders are compared to illustrate the importance of high-order modes and the effectiveness of the calculation method in the first embodiment in compensating for high-order modes.

[0194] It should be noted that in previous wind engineering practices, the modal order taken for wind-induced response calculation generally did not exceed 100 orders. Therefore, modes greater than 100 orders can be considered high-order modes.

[0195] Please refer to Figure 6 (a) and Figure 6 (b). As can be seen from Figure 6 (a), when the modal order is 160 or less at a wind direction angle of 0°, the overall accuracy of the calculation results is relatively low, and the pulsating displacements at many nodes are close to zero. As the modal number further increases, the accuracy of the calculation results gradually improves. When the 500th-order mode is taken, the calculation results are basically consistent with the standard values. Figure 6 (b) shows the R 2 score under all wind direction angles. It can be seen that there are also differences in the overall accuracy of the calculations under different wind direction angles. When the modal order is 160 or less, the R 2 score is generally low between wind direction angles of 0° to 150°. This interval is exactly the angle where the cantilever is directly facing the wind and needs to be focused on in wind resistance design. However, ignoring the influence of high-order modes makes it difficult to ensure the calculation accuracy at these wind direction angles.

[0196] Please refer to Figure 7 (a) and Figure 7 (b). To further illustrate the importance of high-order modes, Figure 7 (a) gives the calculation results of the vertical pulsating displacement of the node at the maximum south-side cantilever under all wind direction angles. It can be seen that when the modal order is 80 or less, the error of the pulsating displacement is very large, indicating that the first 80 modes do not well reflect the vibration situation at this position. The error is the largest at a wind direction angle of 50°. Therefore, Figure 7 (b) gives the mean square values of the modal responses of the top 20 larger modal responses of this response at a wind direction angle of 50° in descending order. It can be seen that although the modal orders of modes 117, 112, and 91 are very high, their mean square values of modal responses rank among the top 3. Among the modes contributing to the top 20, 7 modes have orders exceeding 100. The above results show that high-order modes have an unignorable contribution to the wind-induced response of this roof structure, and a suitable modal truncation compensation method should be adopted for pulsating displacements.

[0197] Please refer to Figure 8 (a) and Figure 8 (b) for comparison Figure 6, it can be seen that the use of the MAM-HEM method has significantly improved the calculation accuracy. The overall accuracy (R 2 = 0.9995) of the calculation results obtained by the MAM-HEM method with 30 modes at a wind direction angle of 0° is already better than that (R 2 = 0.9966) of the HEM method with 500 modes. At all wind direction angles, taking 30 modes and calculating according to the MAM-HEM method can ensure that the R 2 score is close to 1, which is already a very satisfactory accuracy. The comparison results show that the MAM-HEM method has better convergence than the HEM method. By compensating for the quasi-static response of high-order modes, relatively accurate results can be obtained using only a small number of modes.

[0198] Please refer to Figure 9 , which gives the comparison of the root mean square of the base lift calculated at different quasi-static modal orders at a wind direction angle of 0° (approximate solution, [K][Ф c {q c (t)} corresponding to the root mean square of the lift) and the accurate value ({p c (t)} corresponding to the root mean square of the lift) (the number of participating vibration modes is 80 orders). It can be seen from Figure 9 that as the order increases, the approximate solution calculated from the quasi-static modes rapidly approaches the accurate value. When the order is 30, the relative error between the approximate solution and the accurate value is less than 1%, which shows the rationality and efficiency of the approximation. The relative error between the root mean square of the quasi-static base lift calculated from the quasi-static modes and the exact solution reflects the accuracy of the approximation. Therefore, it can be used as a basis for controlling the number of quasi-static modes. In this application, 1% is selected as the threshold, and the generation of quasi-static modes is stopped when the relative error is less than 1%.

[0199] Please refer to Figure 10 (a) and Figure 10 (b). Figure 10 (a) shows the calculation results of different methods when the number of participating vibration modes is 30 at a wind direction angle of 0°. It can be seen that the calculation results of the MAM-HEM method (R 2 = 0.9995) and the SSC-HEM method (R 2 = 0.9942) are basically the same and much better than that of the HEM method (R 2 = 0.8613). Figure 10 (b) gives the calculation results of the MAM-HEM method and the SSC-HEM method at all wind direction angles. It can be seen that using the SSC-HEM method can ensure that the R 2 is above 0.99, which once again proves the rationality and effectiveness of the simplification..

[0200] Table 1 presents the comparison of the computing time and memory overhead of the HEM method, the MAM-HEM method, and the SSC-HEM method. The comparison results of another project are also given in the table. However, due to space limitations, this application only discusses the computing efficiency. As can be seen from the table, in the Shenzhen Convention and Exhibition Center project, although satisfactory accuracy can be obtained by calculating 500-order modes according to the HEM method, the computing time and memory overhead are the largest. By adopting the MAM-HEM method in the first embodiment, the convergence is greatly improved by adding a static correction term to the response expression. Only 30-order participating modes are required, and the accuracy is basically the same as that of the HEM method. Although it is necessary to repeatedly solve the linear equations and store and estimate the spectrum of the response in the physical coordinates, the computing time and memory overhead are still less than those of the HEM method. By adopting the SSC-HEM method in the second embodiment, through the approximate treatment of the static correction term, while ensuring the accuracy is comparable to that of the MAM-HEM method, the computing time is reduced to 1 / 5 of the MAM-HEM method, and the memory overhead is only 1 / 8 of the HEM method. In the Nanhai Museum project with a larger problem scale, the advantage of simplification is further expanded. The computing time of the SSC-HEM method is only 1 / 12 of the MAM-HEM method, and the memory occupancy is only 1 / 6 of the MAM-HEM method. At the same time, it should be pointed out that if the existing MAM (modal acceleration)-CQC (complete quadratic combination) method is used to calculate the Shenzhen Convention and Exhibition Center project, only storing the residual flexibility matrix [F H with a size of 27126×27126 requires 5.48 GB of memory, which is obviously unacceptable.

[0201] Table 1 Comparison of computing time and memory overhead of different methods

[0202]

[0203] The above shows and describes the basic principles, main features, and advantages of the present invention. Those skilled in the art of this industry should understand that the present invention is not limited by the above embodiments. What is described in the above embodiments and the specification is only the principle of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and these changes and improvements all fall within the scope of the present invention claimed. The scope of protection required by the present invention is defined by the appended claims and their equivalents.

Claims

1. A fast algorithm for wind-induced response of large-span roof structures considering high-order modal quasi-static response, characterized in that: The following steps are involved: Step S11, establishing a finite element model of the long-span roof structure, and determining the modal damping ratio of the long-span roof structure according to the design situation, so as to obtain the structural dynamic characteristics of the long-span roof structure, wherein the structural dynamic characteristics include vibration mode, natural frequency, stiffness matrix and mass matrix; Step S12, determining the random wind load {p(t)} acting on the large-span roof structure through a wind tunnel test, and calculating the wind load Fourier transform function {P(f)} of the truncated function {p(t)} corresponding to the random wind load, the wind load Fourier transform function is {P(f)}: Step S13, according to the structural dynamic characteristics and the wind load Fourier transform function, the modal acceleration method is used to calculate to obtain the Fourier transform function of the structural displacement response, where: Step S131, constructing a modal frequency response function matrix of the large-span roof structure, the modal frequency response function matrix is ​​[H(f)] = diag(H1, H2, ..., H n ), the i-th order frequency response function H i (f) is: In the formula, f i is the i-th order natural frequency, ξ i is the i-th order modal damping ratio; Step S132: construct the modal stiffness matrix of the large-span roof structure The modal stiffness matrix for: In the formula, [Ф t ] is the vibration mode matrix, [K] is the stiffness matrix, and the superscript T indicates transposition; Step S133, according to the modal frequency response function matrix and the modal stiffness matrix, the modal acceleration method is used to calculate to obtain the Fourier transform function {Y(f)} of the structural displacement response, and the Fourier transform function {Y(f)} of the structural displacement response is: Step S14, according to the Fourier transform function of the structural displacement response, a periodogram method is used to perform power spectrum density estimation to obtain a power spectrum density matrix of the structural displacement response, and the power spectrum density matrix of the structural displacement response is integrated to obtain the root mean square of the displacement response of the large-span roof structure.

2. According to claim 1, a fast algorithm for wind-induced response of a large-span roof structure considering high-order modal quasi-static response is characterized in that: The step S14 comprises: Step S141, based on the Fourier transform function of the structural displacement response, the power spectrum density is estimated by using the periodogram method to obtain the power spectrum density matrix [S yy (f)], the power spectral density matrix of the structural displacement response [S yy (f)] is: Where T is the sampling time, and the superscript * indicates conjugation; Step S142, integrating the power spectrum density matrix of the structural displacement response to obtain the root mean square of the displacement response of the large-span roof structure, the root mean square of the displacement response is: In the formula, σ y,k is the root mean square of the displacement on the kth degree of freedom of the structure, S yy,k (f) is the displacement response autospectrum of the kth degree of freedom, that is, the kth element on the main diagonal of the power spectrum density matrix of the structural displacement response.

3. A fast algorithm for wind-induced response of large-span roof structures considering high-order modal quasi-static response, characterized in that: The following steps are involved: Step S21, establishing a finite element model of the long-span roof structure, and determining the modal damping ratio of the long-span roof structure according to the design situation, so as to obtain the structural dynamic characteristics of the long-span roof structure, wherein the structural dynamic characteristics include vibration mode, natural frequency, stiffness matrix and mass matrix; Step S22, determine the random wind load {p(t)} acting on the large-span roof structure through a wind tunnel test, and calculate the wind load Fourier transform function {P(f)} of the truncated function {p(t)} corresponding to the random wind load, and the wind load Fourier transform function {P(f)} is: Step S23, according to the structural dynamic characteristics and the wind load Fourier transform function, a simplified modal acceleration method is used to calculate to obtain the Fourier transform function of the structural displacement response in modal coordinates, where: Perform proper orthogonal decomposition on the random wind load {p(t)}: Where m is the eigenmode order, a is k (t) is the corresponding modal coordinate function, {G k } is the kth order eigenmode; The kth eigenmode {G k }By solving the covariance matrix [C pp ] is obtained by the eigenvalue problem, namely: [C pp ]{G k }=λ k {G k }; In the formula, λ k is the kth eigenvalue; The kth eigenmode {G k } corresponding to the modal coordinate function a k (t) is calculated as follows: In the formula, the superscript T represents transposition; Step S24, according to the Fourier transform function of the structural displacement response, the power spectrum density is estimated by using the periodogram method to obtain the power spectrum density matrix of the structural displacement response under the modal coordinates, and the power spectrum density matrix of the modal response is integrated to obtain the displacement response root mean square of the large-span roof structure, where: Step S241, based on the Fourier transform function of the structural displacement response, the power spectrum density is estimated by using the periodogram method to obtain the power spectrum density matrix [S qq ], the power spectrum density matrix of the modal response [S qq ]for: Where T is the sampling time, and the superscript * indicates conjugation; Step S242: power spectrum density matrix [S qq ] is integrated to obtain the covariance matrix of the modal response [C qq ]: Step S243, after the covariance matrix of the modal response is calculated [C qq ], the root mean square value of the displacement response on the kth degree of freedom can be expressed as: In the formula, c ij is the i-th row and k-th column element of the modal response covariance matrix, For the matrix The element in row k and column i; Among them, the matrix It can be expressed as: In the formula, [Ф t ] is the vibration mode matrix, [Ф c ] is the quasi-static modal matrix.

4. The fast algorithm for wind-induced response of a large-span roof structure considering high-order modal quasi-static response according to claim 3 is characterized in that: The step S23 comprises: Step S232, calculate m load compensation items corresponding to m eigenmodes, the calculation method is: In the formula, is the kth load compensation term, [I] is the unit matrix, [M] is the mass matrix, [Ф t ] is the vibration mode matrix, and the superscript T indicates transposition; Step S233, using the first load compensation term as the external load distribution form, solve the static equation to obtain The displacement response {y1} of the structure under the action is orthogonalized with the structural mass matrix to obtain the first quasi-static mode {ψ1}, where The static force equation is: Where [K] is the stiffness matrix; Quasi-static mode {ψ1}: Where, the superscript T represents transposition, [M] is the mass matrix; Step S234, using the 2nd to mth load compensation items as the external load distribution form, solve the static equation to obtain The displacement response of the structure under the action {y k }, and use the Gram-Schmidt orthogonalization method to transform the displacement response {y k Orthogonalize all previous vectors with respect to mass to obtain a pure vector that does not contain any previous vector components Then the pure vector Orthogonal processing is performed with the structural mass matrix to obtain the kth quasi-static mode, where The static force equation is: Where k ≥ 2, [K] is the stiffness matrix; Pure vector for: In the formula, the superscript T represents transposition, {ψ i } is the i-th quasi-static mode; The kth quasi-static mode {ψ k = = Where, the superscript T represents transposition, [M] is the mass matrix; Step S235: after solving and obtaining m quasi-static modes, the quasi-static mode matrix is ​​recorded as matrix [ψ] = [{ψ1}, {ψ2}, {ψ3}... {ψ m }], calculate the modal stiffness matrix [K*] corresponding to the quasi-static mode and solve it to obtain the eigenvalue [p] and eigenmatrix [A], and then perform an orthogonal transformation on the matrix [ψ] to obtain the transformed quasi-static modal matrix [Ф c ], the corresponding calculation formula is: [K*]=[ψ] T [K][ψ]; [K * ][A]=[A][p]; [F c ]=[ψ][A]; In the formula, the superscript T represents transposition; Step S236, calculating the Fourier transform function of the structural displacement response in modal coordinates, the specific method is: {Q t (f)}=[H(f)][Φ t ] T {P(f)}; In the formula, {Q t (f)} is the Fourier transform function of the modal response, [H(f)] is the modal frequency response function matrix, [Ф t ] is the vibration matrix, the superscript T represents the transpose, {Q c (f)} is the Fourier transform function of the compensated modal response, is the modal stiffness matrix corresponding to the quasi-static mode, [I] is the unit matrix, [M] is the mass matrix, For {Q t (f)} and {Q c (f)} combination.

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