Large-span roof structure wind-induced response fast algorithm considering high-order modal quasi-static response

Abstract

The invention discloses a large-span roof structure wind-induced response fast algorithm considering high-order modal quasi-static response, and the algorithm comprises the following steps: S11, building a finite element model of a large-span roof structure, and determining the modal damping ratio of the large-span roof structure according to the design condition, so as to obtain the structural dynamic characteristics of the large-span roof structure; step S12, determining a random wind load acting on the large-span roof structure through a wind tunnel test, and calculating a wind load Fourier transform function of a truncation function corresponding to the random wind load; step S13, according to the dynamic characteristics of the structure and the wind load Fourier transform function, performing calculation by adopting a modal acceleration method, obtaining a Fourier transform function of the structure displacement response; and S14, according to the Fourier transform function of the structure displacement response, performing power spectral density estimation by adopting a periodogram method to obtain a power spectral density matrix of the structure displacement response, and performing integration on the power spectral density matrix of the structure displacement response to obtain a displacement response root-mean-square of the large-span roof structure. According to the method, quasi-static response of a high-order mode is considered, better convergence is achieved compared with a traditional method, and under the condition of the same precision, calculation time and memory occupation are relatively small.

Description

Technical Field

[0001] The invention relates to the technical field of wind-resistant analysis of large-span roof structures, and in particular to a fast algorithm for wind-induced response of large-span roof structures taking into account high-order modal quasi-static responses. Background Art

[0002] Large-span roof structures are typical wind-sensitive structures with light weight, high flexibility, and dense modal frequencies. Accurate calculation of their wind-induced response is the key to wind-resistant design. Traditional calculation methods include the complete quadratic combination (CQC) method, the root mean square sum (SRSS) method, the virtual excitation (PEM) method, and the harmonic excitation (HEM) method. The above methods are all based on the modal superposition method. For large structures, it is neither necessary nor possible to consider all their modes. The usual 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 large-span roof structures have a non-negligible contribution to the response. Modal truncation may cause a large difference between the calculation results and the actual response.

[0003] To address this problem, one method is to replace the free vibration mode with the Ritz mode (Ritz-POD method) based on the wind load POD decomposition. The latter has better response calculation accuracy under the same parametric modal order. Another method is to continue to use the free vibration mode, but compensate for the response of the higher-order mode. The modal acceleration (MAM) method is a modal truncation compensation method with a clear concept. By adding correction terms, it can take into account the quasi-static response of the higher-order mode, which is very suitable for the dynamic response analysis of structures under low-frequency loads such as wind loads.

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

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

[0006] In a first aspect, the present invention provides a fast algorithm for wind-induced response of a large-span roof structure considering high-order modal quasi-static response, comprising the following steps:

[0007] 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;

[0008] 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)}:

[0009]

[0010] 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;

[0011] 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.

[0012] Preferably, the step S13 comprises:

[0013] 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:

[0014]

[0015] In the formula, f i is the i-th order natural frequency, ξ i is the i-th order modal damping ratio;

[0016] Step S132: construct the modal stiffness matrix of the large-span roof structure The modal stiffness matrix for:

[0017]

[0018] In the formula, [Ф t ] is the vibration mode matrix, [K] is the stiffness matrix, and the superscript T indicates transposition;

[0019] 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:

[0020]

[0021] Preferably, the step S14 includes:

[0022] 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:

[0023]

[0024] Where T is the sampling time, and the superscript * indicates conjugation;

[0025] 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:

[0026]

[0027] 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.

[0028] In a second aspect, the present invention provides a fast algorithm for wind-induced response of a large-span roof structure considering high-order modal quasi-static response, comprising the following steps:

[0029] 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;

[0030] 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:

[0031]

[0032] 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 the modal coordinates;

[0033] Step S24, based on 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 in modal coordinates, and the power spectrum density matrix of the modal response is integrated to obtain the root mean square of the displacement response of the large-span roof structure.

[0034] Preferably, the step S23 includes:

[0035] Step S231, perform intrinsic orthogonal decomposition on the random wind load {p(t)}:

[0036]

[0037] Where M is the eigenmode order, a k (t) is the corresponding modal coordinate function, {G k} is the kth order eigenmode;

[0038] Step S232, calculate m load compensation items corresponding to m eigenmodes, the calculation method is:

[0039]

[0040] 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;

[0041] 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

[0042] The static force equation is:

[0043]

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

[0045] Quasi-static mode {ψ1}:

[0046]

[0047] Where, the superscript T represents transposition, [M] is the mass matrix;

[0048] 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

[0049] The static force equation is:

[0050]

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

[0052] Pure vector for:

[0053]

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

[0055] The kth quasi-static mode {ψ k = =

[0056]

[0057] Where, the superscript T represents transposition, [M] is the mass matrix;

[0058] 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:

[0059] [K*]=[ψ] T [K][ψ] ;

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

[0061] [Φ c ]=[ψ][A]; where the superscript T indicates transposition;

[0062] Step S236, calculating the Fourier transform function of the structural displacement response in modal coordinates, the specific method is:

[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 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.

[0067] Preferably, in step S231, the k-th order eigenmode {G k}By solving the covariance matrix [C pp ] is obtained by the eigenvalue problem, namely:

[0068] [C pp ]{G k}=λ k {G k};

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

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

[0071]

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

[0073] Preferably, the step S24 includes:

[0074] 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 [Sqq ], the power spectrum density matrix of the modal response [S qq ]for:

[0075]

[0076] Where T is the sampling time, and the superscript * indicates conjugation;

[0077] Step S242: power spectrum density matrix [S qq ] is integrated to obtain the covariance matrix of the modal response [C qq ]:

[0078]

[0079] Step S243, after obtaining the covariance matrix of the modal response [C qq ], the root mean square value of the displacement response on the kth degree of freedom can be expressed as:

[0080]

[0081] 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;

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

[0083]

[0084] Compared with the prior art, the present invention has the following beneficial effects:

[0085] First, the present invention takes the quasi-static response of the higher-order modes into account, and has better convergence than the traditional method. Moreover, under the same accuracy conditions, the calculation time and memory usage of the present invention are relatively small.

[0086] Second, compared with the simplified modal acceleration method, the modal acceleration method has comparable calculation accuracy, and the latter requires less calculation time and memory usage. BRIEF DESCRIPTION OF THE DRAWINGS

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

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

[0089] Figure 3 It is the architectural effect diagram of the engineering example in the present invention;

[0090] Figure 4 This is a test model of the present invention, a schematic diagram of the arrangement of measuring points on the roof surface and the wind direction angle;

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

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

[0093] Figure 7 This is the calculation result of the HEM method for the south cantilever node in the present invention;

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

[0095] Fig. 9 is the base lift root mean square calculated for different quasi-static modal orders in the present invention;

[0096] Fig.10 Comparison of the calculation results of 30-order mode MAM-HEM and SSC-HEM in the present invention. DETAILED DESCRIPTION

[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 described below are only examples, and those skilled in the art may think of other obvious variations.

[0098] See also Figure 1 In the first embodiment of the present invention, a fast algorithm for wind-induced response of a large-span roof structure considering high-order modal quasi-static response is provided, comprising the following steps:

[0099] 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;

[0100] 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)}:

[0101]

[0102] 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;

[0103] 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.

[0104] Preferably, the step S13 comprises:

[0105] 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:

[0106]

[0107] In the formula, f i is the i-th order natural frequency, ξ i is the i-th order modal damping ratio;

[0108] Step S132: construct the modal stiffness matrix of the large-span roof structure The modal stiffness matrix for:

[0109]

[0110] In the formula, [Ф t ] is the vibration mode matrix, [K] is the stiffness matrix, and the superscript T indicates transposition;

[0111] 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:

[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 resonant response and complete quasi-static response of the first n modes of the structure, compensating for the quasi-static response of the higher-order modes, and making the calculation result more accurate.

[0114] Furthermore, [K] in this step -1 {P(f)} is calculated by solving a linear system of equations, that is, solving the linear system of equations [K]{Y s {Y in {Y(f)} = {P(f)} s (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, the step S14 comprises:

[0116] 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:

[0117]

[0118] Where T is the sampling time, and the superscript * indicates conjugation;

[0119] 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:

[0120]

[0121] 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.

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

[0123] (1) In practical engineering, we are concerned with the root mean square of the response in physical coordinates. For large structures, due to the large number of degrees of freedom, the power spectrum density of the response in physical coordinates is usually not calculated directly. Instead, the power spectrum density matrix of the modal response is calculated first, and the covariance matrix of the modal response is obtained by integration. Then, the root mean square value of the response in physical coordinates is calculated, that is:

[0124]

[0125]

[0126]

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

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

[0129] (2) For large structures, the flexibility matrix [K] -1 It is difficult to obtain the quasi-static response of the structure {Y s (f)} is obtained by solving the linear equations. In order to ensure the stability of the statistical results in the wind tunnel test, it is necessary to ensure that there are enough sampling points L, which means that the Fourier transform function of the structural displacement response {Y s (f)} At least L-order linear equations need to be repeatedly solved. Although there are many fast algorithms for solving this type of problem, the computational overhead of repeatedly solving L-order linear equations 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. The core of the simplification lies in the processing of quasi-static terms.

[0131] See also Figure 2 In the second embodiment of the present invention, a fast algorithm for wind-induced response of a large-span roof structure considering high-order modal quasi-static response is provided, comprising the following steps:

[0132] 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;

[0133] 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:

[0134]

[0135] 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 the modal coordinates;

[0136] Step S24, based on 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 in modal coordinates, and the power spectrum density matrix of the modal response is integrated to obtain the root mean square of the displacement response of the large-span roof structure.

[0137] Preferably, the step S23 includes:

[0138] Step S231, perform intrinsic orthogonal decomposition on the random wind load {p(t)}:

[0139]

[0140] Where m is the eigenmode order, a k (t) is the corresponding modal coordinate function, {G k} is the kth order eigenmode;

[0141] Step S232, calculate m load compensation items corresponding to m eigenmodes, the calculation method is:

[0142]

[0143] 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;

[0144] 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:

[0145] The static force equation is:

[0146]

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

[0148] Quasi-static mode {ψ1}:

[0149]

[0150] Where, the superscript T represents transposition, [M] is the mass matrix;

[0151] Step S234, using the 2nd to mth load compensation items as the external load distribution form, solving the static equation to obtain {p k c The displacement response of the structure under the action of {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

[0152] The static force equation is:

[0153]

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

[0155] Pure vector for:

[0156]

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

[0158] The kth quasi-static mode {ψ k = =

[0159]

[0160] Where, the superscript T represents transposition, [M] is the mass matrix;

[0161] 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:

[0162] [K*]=[ψ] T [K][ψ] ;

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

[0164] [Φc ]=[ψ][A]; where the superscript T indicates transposition;

[0165] Step S236, calculating the Fourier transform function of the structural displacement response in modal coordinates, the specific method is:

[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 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.

[0170] Preferably, in step S231, the k-th order eigenmode {G k}By solving the covariance matrix [C pp ] is obtained by the eigenvalue problem, namely:

[0171] [C pp ]{G k}=λ k {G k};

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

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

[0174]

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

[0176] Preferably, the step S24 includes:

[0177] 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:

[0178]

[0179] Where T is the sampling time, and the superscript * indicates conjugation;

[0180] Step S242: power spectrum density matrix [S qq ] is integrated to obtain the covariance matrix of the modal response [C qq ]:

[0181]

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

[0183]

[0184] 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;

[0185] Among them, the matrix It can be expressed as:

[0186]

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

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

[0189] The method of the present invention is described in detail below with a specific engineering example:

[0190] See also Figure 3The Shenzhen World Exhibition & Convention Center project is a super large exhibition complex, consisting of 16 standard exhibition halls, 2 multifunctional exhibition halls, 1 super large exhibition hall, 2 north and south entry halls and a central corridor. The total north-south span of the project is 1.7km, making it the largest single building in the world. The architectural renderings are as follows: Figure 3 As shown. The standard exhibition hall in the southwest corner of the exhibition center is taken as the analysis object. The roof structure is about 250m long from north to south, with an open passage 42m wide in the middle, and 210m wide from east to west. The roof is wavy, with a height of about 26m at the highest position and a maximum cantilever length of 12m. Since the roof structure has a large span and cantilever length, light weight, and a small damping ratio, it is a wind-sensitive structure, and a wind tunnel test is required to determine the actual wind load on the surface of the structure.

[0191] See also Figure 4 According to the design requirements, the topography upstream of the building model is assumed to be Class A topography, and the basic wind pressure is 0.75kPa. Before the wind tunnel test, the required Class A topography type was simulated in the test section using binary spires, baffles and rough elements in accordance with the provisions of the national "Code for Loads on Building Structures" GB 50009-2012. The test incoming wind speed is 10m / s, the reference height is 0.25m, and the wind pressure field of the building is measured at 36 wind directions 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 test model, the arrangement of measuring points on the roof surface and the wind direction angle are shown as follows Figure 4 As shown in the figure, the sampling frequency of the model during the test is 300Hz, the single sampling duration is 30s, and the number of sampling points is 9000. According to the similarity ratio conversion, it can be seen that the prototype sampling frequency is 9.204Hz and the prototype sampling duration is 16.3min.

[0192] See also Figure 5 The structure was modeled in SAP2000 software, and the total degree of freedom of the structure was 27126. The first 500 free vibration modes of the structure were calculated by the eigenvector method, and the natural frequencies were as follows: Figure 5 As shown, the natural frequency corresponding to the 500th mode is 5.313 Hz, which has exceeded the maximum resolvable frequency of the test (4.602 Hz). Therefore, the result of the 500th free vibration mode calculated by the calculation method in the first embodiment can be considered to include all the resonant responses and background responses that can be calculated for the structure. Subsequent analysis will use this result as the standard value of the pulsation response result.

[0193] See also Figure 6 (a) to Figure 8 (b) R 2The score is used as an indicator to judge the overall accuracy of the evaluation method. The wind vibration calculation results of the harmonic excitation (HEM) method 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) method 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 high-order modal compensation.

[0194] It should be noted that in previous wind engineering practices, the modal order used in wind-induced response calculations generally does not exceed 100, so modes greater than 100 can be considered high-order modes.

[0195] See also Figure 6 (a) and Figure 6 (b) Figure 6 (a) It can be seen that when the modal order is 160 or below at a wind direction angle of 0°, the overall accuracy of the calculation results is low, and the pulsating displacements at many nodes are close to zero. As the number of modes increases, the accuracy of the calculation results gradually improves. When the 500th mode is taken, the calculation results are basically consistent with the standard value. Figure 6 (b) shows the R under all wind angles 2 It can be seen that the overall accuracy of the calculation under different wind directions is also different. When the modal order is 160 or less, the R 2 The scores are generally low. This range happens to be the angle where the cantilever directly faces the wind, which requires special attention in wind-resistant design. However, ignoring the influence of high-order modes makes it difficult to guarantee the calculation accuracy under these wind direction angles.

[0196] See also Figure 7 (a) and Figure 7 (b), to further illustrate the importance of higher-order modes, Figure 7 (a) shows the calculation results of the vertical pulsating displacement of the node at the maximum cantilever on the south side under all wind directions. It can be seen that when the modal order is 80 or below, the error of the pulsating displacement is very large, which indicates that the first 80 modes do not reflect the vibration situation at this location well. The error is the largest at a wind direction of 50°, so Figure 7 (b) The RMS values ​​of the top 20 modal responses at a wind direction angle of 50° are given in descending order. It can be seen that although the 117th, 112th, and 91st order modes have high orders, their modal response RMS values ​​are ranked in the top 3. Among the top 20 contributing modes, the orders of 7 modes are over 100. The above results show that high-order modes have a non-negligible contribution to the wind-induced response of the roof structure, and a suitable modal truncation compensation method should be used for the pulsating displacement.

[0197] See also Figure 8 (a) and Figure 8 (b), contrast Figure 6It can be seen that the use of the MAM-HEM method has significantly improved the calculation accuracy. The overall accuracy of the results calculated by the MAM-HEM method for the 30th order mode at a wind direction angle of 0° (R 2 =0.9995) is better than the calculation result of the 500th modal HEM method (R 2 =0.9966). Under all wind directions, the 30th order mode can be calculated by MAM-HEM method to ensure R 2 The score is close to 1, which is 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 higher-order modes, more accurate results can be obtained using only a small number of modes.

[0198] See also Fig. 9 , gives the base lift root mean square calculated under different quasi-static mode orders at a wind direction angle of 0° (approximate solution, [K][Ф c ]{q c (t)}corresponding to the root mean square of lift) and the exact value ({p c (t)}corresponding to the root mean square of lift) (parametric vibration mode order is 80th order). Fig. 9 It can be seen that with the increase of the order, the approximate solution obtained by the quasi-static mode calculation quickly 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 quasi-static base lift root mean square calculated by the quasi-static mode and the exact solution reflects the accuracy of the approximation, so 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 quasi-static mode generation is stopped when the relative error is less than 1%.

[0199] See also Fig.10 (a) and Fig.10 (b) Fig.10 (a) shows the calculation results of different methods when the parametric vibration mode order is 30 under 0° wind direction angle. It can be seen that the MAM-HEM method (R 2 =0.9995) and SSC-HEM method (R 2 =0.9942) are basically consistent and much better than the HEM method (R 2 =0.8613). Fig.10 (b) shows the calculation results of MAM-HEM and SSC-HEM under all wind directions. It can be seen that the SSC-HEM method can ensure R 2 Above 0.99, this once again proves the rationality and effectiveness of the simplification.

[0200] Table 1 gives the calculation time and memory overhead comparison of the HEM method, MAM-HEM method and SSC-HEM method. The table also gives the comparison results of another project, but due to the limited space, this application only discusses the calculation efficiency. As can be seen from the table, in the Shenzhen Convention and Exhibition Center project, taking the 500th order mode and calculating by the HEM method can obtain satisfactory accuracy, but the calculation time and memory overhead are the largest. The MAM-HEM method in the first embodiment is adopted, and the convergence is greatly improved by adding the static correction term in the response expression. Only the 30th order parametric vibration modal accuracy is basically consistent with 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 calculation time and memory overhead are still less than the HEM method. The SSC-HEM method in the second embodiment is adopted. By approximating the static correction term, while ensuring that the accuracy is equivalent to that of the MAM-HEM method, the calculation 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, which is a larger problem, the advantage of simplification is further expanded. The calculation time of the SSC-HEM method is only 1 / 12 of that of the MAM-HEM method, and the memory usage is only 1 / 6 of that of the MAM-HEM method. It should also 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 the residual flexibility matrix [F H ] would require 5.48GB of memory, which is clearly unacceptable.

[0201] Table 1 Comparison of computation 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 should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions only describe the principles of the present invention. The present invention may be subject to various changes and improvements without departing from the spirit and scope of the present invention. These changes and improvements fall within the scope of the present invention. The scope of protection claimed by the present invention is defined by the attached 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; 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 S13 comprises: 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:

3. According to claim 2, 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.

4. 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 the modal coordinates; Step S24, based on 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 in modal coordinates, and the power spectrum density matrix of the modal response is integrated to obtain the root mean square of the displacement response of the large-span roof structure.

5. The fast algorithm for wind-induced response of a large-span roof structure considering high-order modal quasi-static response according to claim 4 is characterized in that: The step S23 comprises: Step S231, perform intrinsic 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; 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.

6. A fast algorithm for wind-induced response of a large-span roof structure considering high-order modal quasi-static response according to claim 5, characterized in that: In step S231, the k-th order 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.

7. A fast algorithm for wind-induced response of a large-span roof structure considering high-order modal quasi-static response according to claim 6, characterized in that: The step S24 comprises: 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:

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