Random vibration method for long-span bridge under spatially varying non-stationary earthquake

Abstract

The application discloses a kind of random vibration methods of long-span bridge under spatially varying non-stationary earthquake, technical points include: the motion equation of long-span bridge under spatially varying non-stationary earthquake is constructed by finite element method;Modal analysis is carried out to bridge superstructure;Recursive expression between state response vector and excitation vector is constructed;Each seismic input process is discretely characterized using spectral representation method, and it is mapped to the response of bridge support structure corresponding degree of freedom;The explicit relationship between the total response of superstructure in physical space and random input vector is constructed;Based on the explicit relationship between the total response and random input vector obtained, the time-frequency response statistics of bridge structure are evaluated.The application uses the above-mentioned random vibration method of long-span bridge under spatially varying non-stationary earthquake, overcomes the defects that the existing calculation method is low in efficiency and difficult to fully consider the spatial variability and complete non-stationarity of seismic motion, and is very convenient for computer programming implementation.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of bridge, and particularly relates to a random vibration method of long-span bridge under spatially varying non-stationary earthquake. BACKGROUND

[0002] For bridge structures in seismic active regions, ground motion is the most common potential natural disaster in their service process. Ground motion has inherent uncertainty, and a random process should be used to characterize its statistical characteristics. Accordingly, the bridge response induced by the ground motion should also be regarded as a random process, and a random vibration method should be used to probabilistically evaluate the bridge response, and the evaluation results can be further used for seismic design and time-varying reliability analysis of the bridge. When analyzing the random seismic response of long-span bridges, the complete non-stationarity and spatial variability of ground motion should be fully considered. The complete non-stationarity is reflected in the time domain and the frequency domain, which respectively correspond to the time-varying intensity and time-varying frequency components of the ground motion, and both kinds of non-stationarity have a significant influence on the seismic response of bridge structures. The spatial variability means that the ground motion borne by different supports of the bridge is not completely consistent, which is mainly caused by the traveling wave effect, partial coherence effect and local site effect of the ground motion.

[0003] At present, a variety of methods (such as the virtual excitation method, the evolutionary spectrum method and the Monte Carlo simulation) can be used for random vibration analysis of long-span bridges under non-stationary ground motion. However, these methods still have some problems in practical application, which are manifested in two aspects. First, the existing methods usually need to perform a large number of time-frequency integral operations or repetitive dynamic analysis. For example, the virtual excitation method needs to perform time history analysis of the bridge structure at a series of frequency points to obtain the response spectrum density in the complete frequency range, the evolutionary spectrum method needs to perform a large number of convolution integrals to evaluate the frequency response function and time-varying response statistics of the bridge structure, and the Monte Carlo simulation also needs to statistically analyze a large number of bridge response samples to obtain a convergent solution. The time-consuming calculation makes these methods face serious computing power bottlenecks when applied to large bridge structures. In addition, due to the inherent defects in applicability, the existing methods are difficult to consider the influence of the complete non-stationarity and spatial variability of ground motion on the random response of bridges. Specifically, some methods can fully consider the traveling wave effect, partial coherence effect and local site effect of ground motion, but are only applicable to time-domain non-stationary excitation determined by a uniform modulation function, and have difficulties in characterizing the frequency-domain non-stationarity of ground motion. In addition, some methods can consider the time-domain and frequency-domain non-stationarity of ground motion, but are only applicable to uniform ground motion, and cannot fully consider the influence of the spatial variability of ground motion on the random response of bridges.

[0004] Therefore, under the premise of fully considering the complete non-stationarity and spatial variability of ground motion, how to establish a more efficient and applicable random vibration method to realize the accurate evaluation of the statistical quantities of seismic response of long-span bridges becomes a problem to be solved by the technical personnel in the field. SUMMARY

[0005] The purpose of the present application is to provide a random vibration method for long-span bridges under spatially variable non-stationary earthquakes, so as to overcome the defects of the existing calculation methods that are low in efficiency and difficult to fully consider the spatial variability and complete non-stationarity of ground motion.

[0006] To achieve the above purpose, the present application provides a random vibration method for long-span bridges under spatially variable non-stationary earthquakes, comprising the following steps:

[0007] S1, constructing the motion equation of the long-span bridge under the action of spatially variable non-stationary ground motion by the finite element method, and writing the motion equation as a block matrix form;

[0008] S2, performing modal analysis on the bridge superstructure, dividing the displacement response of the superstructure into dynamic response components and quasi-static components, and converting the dynamic response components to modal space;

[0009] S3, converting the modal equation corresponding to the dynamic response components of the superstructure to state space, and numerically discretizing the state equation at equidistant time points to construct the recursive expression between the state response vector and the excitation vector;

[0010] S4, decomposing the coherence function matrix of the seismic acceleration input vector by the Cholesky method, discretely representing each ground motion input process by the spectral representation method, and mapping it to the response of the corresponding degrees of freedom of the bridge support structure;

[0011] S5, calculating the mapping coefficient matrix between the dynamic response components of the superstructure in the modal space and the random input vector, and further constructing the explicit relationship between the total response of the superstructure in the physical space and the random input vector;

[0012] S6, calculating the cross-correlation matrix between each random input vector, and further evaluating the time-frequency response statistics of the bridge structure based on the explicit relationship between the total response and the random input vector obtained.

[0013] Preferably, S1 specifically comprises the following steps:

[0014] S101: constructing the motion equation of the long-span bridge under the action of spatially variable non-stationary ground motion by the finite element method, as follows:

[0015] ;

[0016] In the formula,t Time is represented; M, C, and K represent the mass, damping, and stiffness matrices of the bridge finite element model, respectively. The damping matrix is ​​constructed using the Rayleigh method: C = α M + β K, where α and β Z represents the Rayleigh damping coefficient; and These represent the displacement, velocity, and acceleration vectors of the bridge, respectively; P represents the external force vector of the bridge caused by the earthquake.

[0017] S102: Based on the degrees of freedom corresponding to the superstructure and substructure of the bridge, the equations of motion of the bridge are divided into blocks:

[0018] ;

[0019] In the formula, the subscript " a "and" b "These correspond to the degrees of freedom of the bridge superstructure and the supporting structure, respectively." ab "and" ba "Corresponds to the coupled degrees of freedom; P" b This represents the support force vector of the bridge model.

[0020] Preferably, S2 specifically includes the following steps:

[0021] S201: Perform modal analysis on the mass and stiffness matrices corresponding to the bridge superstructure to obtain the mass-normalized modal matrix Φ of the superstructure, and then calculate the modal stiffness and damping matrices of the superstructure:

[0022] ;

[0023] S202: The displacement response Z of the bridge superstructure a Divided into dynamic response components Z d and quasi-static component Z s and Z d Transform to modal space: Z d =ΦX d , where X d Z represents d The modal displacement vector.

[0024] Preferably, S3 specifically includes the following steps:

[0025] S301: Constructing the state response vector Where “T” represents the matrix transpose operator, Z represents d The modal velocity vector; X dThe corresponding modal dynamic equation is transformed into state space, and the parameter matrix A of the state equation is calculated based on the following formula , m 1=1, 2:

[0026] ;

[0027] In the formula, I represents a unit matrix;

[0028] S302: Numerically discretize the state equation at S equally spaced time points (t t 1, t 2, …, t S ) with a step size of Δt t , based on the assumption that the excitation vector changes linearly at each time step, a recursive expression between the state response vector and the excitation vector is constructed:

[0029] ;

[0030] In the formula, k = 1, 2, …, S , T, and are calculated according to the following formula:

[0031] ;

[0032] ;

[0033] .

[0034] Preferably, S4 specifically includes the following steps:

[0035] S401: For the seismic acceleration input process of the bridge N points, set the stationary spectral density function S j ( ω ) and the time-frequency modulation function A j ( ω , t ) of each process, j = 1, 2, …, N ;

[0036] S402: Set the coherence function matrix ρ ( ω ) of the seismic acceleration input vector;

[0037] S403: sequentially calculate the matrix Mequally spaced frequency points ω 1, ω 2, …, ω M ) are decomposed as ρ ( ω ) =B ρ ( )B T ( ),where B( ) is a lower triangular complex matrix, 1= 1, 2,…, , and p ” denotes the conjugate operator; M

[0038] S404: Decompose each seismic acceleration input process based on the spectral representation method, and construct N random input vectors: where △ ω denotes the frequency discretization interval, and represent two sequences of random variables defined as

[0039] ;

[0040] ;

[0041] In the formula, denotes the modulo operator; B jl ( ), l = 1, 2, …, N , denotes the element in the th row and the j th column of the matrix B( l ), and represents the complex phase of B jl ( ); represents an independent random variable uniformly distributed in the interval [0, 2π];

[0042] S405: For each discrete time point t k , k = 1, 2, …, S , construct the mapping coefficient vector between each seismic displacement, velocity and acceleration input process and its corresponding random input vector Θ j : , m ​​2=0, 1, 2, wherein, , p 2= 1, 2, …, 2 M , is expressed as:

[0043] ;

[0044] While , m 3= 0, 1, by integral operation on ;

[0045] S406: using the obtained random input vector Θ j Discrete representation is made to each seismic displacement, velocity and acceleration input process, and it is mapped to the response of the corresponding degree of freedom of the bridge support structure:

[0046] ;

[0047] In the formula, m 2 respectively correspond to the displacement, velocity and acceleration input of ground motion; E j represents the influence vector between Z b and the first j seismic input.

[0048] Preferably, S5 specifically comprises the following steps:

[0049] S501: calculating the dynamic displacement and velocity response components of the superstructure in the modal space at discrete time points t k The mapping coefficient matrix between the above and the random input vector Θ j and , wherein, and are respectively composed of the upper half row and lower half row elements of Γ jk , and Γ jk The initial value is 0 and is calculated according to the following recursive expression:

[0050] ;

[0051] S502: using the finite difference formula, calculating the dynamic acceleration response components of the superstructure in the modal space at discrete time points t k The mapping coefficient matrix between the above and the random input vector Θ j

[0052] ;

[0053] ​​S503: Introducing the mapping relationship between the quasi-static response component of the upper structure and the response of the support structure, and combining the mapping coefficient matrix between the dynamic response component and the random input vector obtained in S501 and S502, an explicit relationship between the total response of the upper structure in the physical space and the random input vector is constructed:

[0054] ;

[0055] wherein, represents the explicit relationship coefficient matrix between the total response and the random input vector.

[0056] Preferably, S6 specifically comprises the following steps:

[0057] S601: Calculate the cross-correlation matrix between each random input vector Θ j and Θ l based on the following formula:

[0058] ;

[0059] wherein, denotes the mathematical expectation operator, represents the random vector , ] T and , ] T the cross-correlation matrix between

[0060] ;

[0061] wherein, , , and all represent row vectors with a dimension of 1× N , which are calculated according to the following formula:

[0062] ;

[0063] ;

[0064] ;

[0065] ;

[0066] wherein, “⊗” represents the element-by-element multiplication operator between two matrices;

[0067] S602: Based on the explicit relationship between the total response and the random input vector obtained, the time-frequency response statistics of the bridge structure are evaluated, and the response The cross-correlation matrix between is expressed as:

[0068] ;

[0069] In addition, by expanding the coefficient matrix into , the response is further obtained at discrete time points t k and frequency points :

[0070] .

[0071] Therefore, the present application adopts the above-mentioned spatially varying non-stationary seismic random vibration method of long-span bridges, and the technical effects are as follows:

[0072] (1) The method can derive an explicit expression of the random seismic response of long-span bridges, which can be directly used for efficient evaluation of the time-frequency response statistics of the bridge, greatly reducing the calculation time of non-stationary random seismic analysis of large bridge structures.

[0073] (2) The method is suitable for any form of time-frequency modulation function and seismic motion model determined by the coherence function, and can fully consider the influences of traveling wave effect, partial coherence effect, local site effect and complete non-stationarity of seismic excitation on the random response of the bridge.

[0074] (3) The core step of the method is to calculate a series of coefficient matrices of the explicit expression of the bridge response by recursion, avoiding the large number of time-frequency integrals or repetitive dynamic analysis involved in traditional random vibration methods, which is very convenient for computer programming.

[0075] The technical solutions of the present application will be further described in detail below with the help of the accompanying drawings and examples. BRIEF DESCRIPTION OF DRAWINGS

[0076] Figure 1 is a flow chart of the spatially varying non-stationary seismic random vibration method of long-span bridges disclosed by the present application;

[0077] Figure 2 is a schematic diagram of a three-span high-pier continuous rigid frame bridge model used in the embodiment of the present application;

[0078] Figure 3 is a comparison chart of longitudinal and transverse displacement variances of the bridge mid-span node, wherein Figure 3 (a) is a comparison chart of longitudinal displacement variances of the bridge mid-span node, Figure 3 (b) is a comparison chart of transverse displacement variances of the bridge mid-span node;

[0079] Figure 4 This is a comparison diagram of the longitudinal and lateral displacement spectral densities of the mid-span node of the bridge in an embodiment of the present invention. Figure 4 (a) is a comparison diagram of the longitudinal displacement spectral density at the mid-span node of the bridge. Figure 4 (b) is a comparison diagram of the lateral displacement spectral density at the mid-span node of the bridge. Detailed Implementation

[0080] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0081] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0082] Example

[0083] by Figure 2 Taking a three-span high-pier continuous rigid frame bridge as an example, the calculation process and performance of the method proposed in this invention are further demonstrated. The span arrangement of the rigid frame bridge is 70 m + 120 m + 70 m, and the two piers are assumed to have a height of 76 m. Each main girder span and each pier has the same geometric and material parameters, as shown in Table 1. Except for the degrees of freedom around the Y-axis at support points 1 and 4, the remaining degrees of freedom of the bridge at the four support points are restricted. It is assumed that the seismic wave is incident horizontally on the four support points of the bridge, and the angle between the earthquake propagation direction and the main girder of the bridge is π / 6.

[0084] Table 1 Geometric and material parameters of main beams and piers

[0085]

[0086] like Figure 1 As shown, the specific steps in this embodiment are as follows:

[0087] S1: The equations of motion for a long-span bridge under spatially varying non-stationary ground motion are constructed using the finite element method, and expressed as a block matrix form. Specifically:

[0088] S101: The equations of motion for a three-span high-pier continuous rigid frame bridge under spatially varying non-stationary ground motion are constructed using the finite element method. Both the main girder and piers are modeled using spatial Euler beam elements, with each element having a length of 2 m. The equations of motion can be written as follows:

[0089] ;

[0090] In the formula, t Time is represented; M, C, and K represent the mass, damping, and stiffness matrices of the bridge finite element model, respectively. The damping matrix is ​​constructed using the Rayleigh method: C =α M+ β K, where α and β The values ​​represent the Rayleigh damping coefficients, taken as 0.03 Hz and 2.1 × 10⁻⁶ Hz, respectively. -3 s; Z, and These represent the displacement, velocity, and acceleration vectors of the bridge, respectively; P represents the external force vector of the bridge caused by seismic motion.

[0091] S102: Based on the degrees of freedom corresponding to the superstructure and substructure of the bridge, the equations of motion of the bridge are divided into blocks:

[0092] ;

[0093] In the formula, the subscript " a "and" b "These correspond to the degrees of freedom of the bridge superstructure and the supporting structure, respectively." ab "and" ba "Corresponds to the coupled degrees of freedom; P" b This represents the support force vector of the bridge model.

[0094] M a M represents the mass matrix of the superstructure in the finite element model of the bridge; b M represents the mass matrix of the supporting structure in the finite element model of the bridge; ab The mass matrix M represents the coupled structure of the bridge finite element model; ba The mass matrix represents the coupled structure of the bridge finite element model;

[0095] C a C represents the damping matrix of the superstructure in the finite element model of the bridge; b C represents the damping matrix of the supporting structure in the finite element model of the bridge; ab C represents the damping matrix of the coupled structure in the finite element model of the bridge; ba The damping matrix represents the coupled structure of the bridge finite element model;

[0096] K a K represents the stiffness matrix of the superstructure in the finite element model of the bridge. b K represents the stiffness matrix of the supporting structure in the finite element model of the bridge. ab K represents the stiffness matrix of the coupled structure in the finite element model of the bridge; ba The stiffness matrix representing the coupled structure of the bridge finite element model;

[0097] Z a Z represents the displacement vector of the superstructure in the finite element model of the bridge; b Z represents the displacement vector of the supporting structure in the finite element model of the bridge; abdenotes the displacement vector of the bridge finite element model coupling structure; ba denotes the displacement vector of the bridge finite element model coupling structure;

[0098] a denotes the velocity vector of the bridge finite element model superstructure; b denotes the velocity vector of the bridge finite element model support structure; ab denotes the velocity vector of the bridge finite element model coupling structure; ba denotes the velocity vector of the bridge finite element model coupling structure;

[0099] a denotes the acceleration vector of the bridge finite element model superstructure; b denotes the acceleration vector of the bridge finite element model support structure; ab denotes the acceleration vector of the bridge finite element model coupling structure; ba denotes the acceleration vector of the bridge finite element model coupling structure;

[0100] S2: modal analysis is performed on the bridge superstructure, the displacement response of the superstructure is divided into dynamic response component and quasi-static component, and the dynamic response component is converted to modal space. Specifically:

[0101] S201: modal analysis is performed on the mass and stiffness matrices corresponding to the bridge superstructure, the modal number is taken as 120, the mass normalized modal matrix Φ of the superstructure is obtained, and then the modal stiffness and damping matrices of the superstructure are calculated:

[0102]

[0103] S202: the displacement response Z of the bridge superstructure is divided into dynamic response component Z and quasi-static component Z and the Z is converted to modal space: Z = ΦX, where X denotes the modal displacement vector of Z. a d s d d d d d

[0104] S3: the modal equation corresponding to the dynamic response component of the superstructure is converted to state space, and the state equation is numerically discretized at equidistant time points to construct the recursive expression between the state response vector and the excitation vector. Specifically:​​​​​​​​

[0105] S301: Constructing the state response vector where "T" denotes the matrix transpose operator, denotes the modal velocity vector of Z d ; X d converts the corresponding modal dynamic equation into the state space, and the parameter matrix A of the state equation is calculated based on the following formula, , m 1=1, 2::

[0106] ;

[0107] In the formula, I represents the unit matrix.

[0108] S302: Numerically discretize the state equation at S equally spaced time points (t t 1, t 2, …, t S ) with a step size of Δ t . Among them, S takes the value of 5001, t 1 and t S are set to 0 and 50 s respectively, and Δ t takes 0.01 s. Based on the assumption that the excitation vector changes linearly at each time step, the recursive expression between the state response vector and the excitation vector is constructed as follows:

[0109] ;

[0110] In the formula, k = 1, 2, …, S , T, and are calculated according to the following formula:

[0111] ;

[0112] ;

[0113] .

[0114] S4: Decompose the coherence function matrix of the seismic acceleration input vector using the Cholesky method, discretely represent each ground motion input process using the spectral representation method, and map it to the response of the corresponding degrees of freedom of the bridge support structure. Specifically:

[0115] S401: For the bridge N= Seismic acceleration input process at 4 support points, setting the stationary spectral density function for each process. S j ( ω and time-frequency modulation function A j ( ω , t )( j = 1, 2, ..., N The following formula is used to determine the result:

[0116] ;

[0117] ;

[0118] Where i represents the imaginary unit; stationary autospectral parameters S 0j , ω gj , ξ gj , ω fj , ξ fj and modulation function parameters c j and ω 0j ( j =1, 2, …, N The values ​​for ) are listed in Table 2.

[0119] Table 2. Stationary autospectral and modulation function parameters of the seismic acceleration input process.

[0120]

[0121] S402: Define the coherence function matrix of the seismic acceleration input vector. ρ ( ω ), its elements ρ jl ( ω )( j , l = 1,2, …, N It is determined by the following formula:

[0122] ;

[0123] ;

[0124] In the formula, V app This represents the apparent wave velocity, which is taken as 1000 m / s in this embodiment; ( j, l =1, 2, 3, 4) represents the first, second, third, fourth, and fourth digits. j and l The mapping of the horizontal distance between bridge support points in the direction of seismic wave propagation; Re(•) and Im(•) represent the real and imaginary part operators; the other coherent model parameters are respectively set to: A 0 = 0.736 ε = 0.147, e = 5210 m, ω 0 = 6.85 rad / s r = 2.78.

[0125] S403: Using the Cholesky method sequentially in M Equally spaced frequency points ( ω 1, ω 2, …, ω M ) on ρ ( ω Decompose: ρ ( =B ( B T ( ), where B( ) represents a lower triangular complex-valued matrix. p 1 = 1, 2, ..., M “ " denotes the conjugate operator, M The value is 1201. ω 1 and ω M Set them to 0 and 60 rad / s respectively.

[0126] S404: Based on the spectral representation method, the various seismic acceleration input processes are decomposed and constructed. N A random input vector: ( j = 1, 2, ..., N ). Among them, △ ω This represents the frequency discrete interval, with a value of 0.05 rad / s. and ( j = 1, 2, …, N ; p 1 = 1, 2, …, M () represents a sequence of two random variables, defined as

[0127] ;

[0128] ;

[0129] In the formula, Indicates the modulo operator; B jl ( )( j , l = 1, 2, …, N ; p 1 = 1, 2, …, M ) represents matrix B( ) No. j Line and number l The element corresponding to the column; represent B jl ( The complex phase of ) ( l = 1, 2, …, N ; p 1 = 1, 2, …, M ) represents an independent random variable that is uniformly distributed on the interval [0, 2π].

[0130] S405: For each discrete time point t k ( k = 1, 2, …, S Construct each seismic displacement, velocity, and acceleration input process and its corresponding random input vector Θ. j ( j = 1, 2, ..., N The mapping coefficient vector between: , m 2 = 0, 1, 2, where, , p 2 = 1, 2, …, 2 M , can be represented as:

[0131] ;

[0132] and ( m 3 = 0, 1) can be obtained by... The integral operation is obtained, specifically expressed as:

[0133] ;

[0134] ;

[0135] In the formula, The calculation can be performed using the following formula:

[0136] ;

[0137] S406: Discretize each seismic displacement, velocity and acceleration input process using the obtained random input vector Θ j j = 1, 2,..., N ) and map them to the responses of the corresponding degrees of freedom of the bridge support structure:

[0138] ;

[0139] wherein, m 2= 0, 1, 2 respectively correspond to the displacement, velocity and acceleration input of the ground motion; E j j = 1, 2,..., N ) represents the influence vector between the Z b th seismic input and the first j th seismic input.

[0140] S5: Calculate the mapping coefficient matrix between the dynamic response components of the superstructure in the modal space and the random input vector, and further construct the explicit relationship between the total response of the superstructure in the physical space and the random input vector. Specifically:

[0141] S501: Based on the dynamic response component-excitation vector expression obtained in S3 and the excitation vector-random input vector expression obtained in S4, calculate the mapping coefficient matrix between the dynamic displacement and velocity response components of the superstructure in the modal space and the random input vector Θ t k k = 1, 2,..., S ) at discrete time points j j = 1, 2,..., N ) and Θ and . Among them, and are composed of the upper half and lower half elements of Γ jk , and the latter is initialized to 0 and can be calculated according to the following recursive expression:

[0142] ;

[0143] S502: Using the finite difference formula, calculate the dynamic acceleration response components of the superstructure in the modal space at discrete time points t k k = 1, 2,..., S ​​​​​) on the random input vector Θ j ( j = 1, 2, …, N The mapping coefficient matrix between )

[0144] ;

[0145] S503: Introducing the mapping relationship between the quasi-static response components of the superstructure and the response of the supporting structure, and combining the mapping coefficient matrix between the dynamic response components and the random input vector obtained in S501 and S502, an explicit relationship between the total response of the superstructure and the random input vector in physical space is constructed:

[0146] ;

[0147] in, The coefficient matrix represents the explicit relationship between the total response and the random input vector.

[0148] S6: Calculate the cross-correlation matrix between each random input vector, and then evaluate the time-frequency response statistics of the bridge structure based on the obtained explicit response vector-random input vector relationship. Specifically:

[0149] S601: Calculate each random input vector Θ based on the following formula. j and Θ l ( j , l = 1, 2, ..., N The cross-correlation matrix between them:

[0150] ;

[0151] In the formula, Represents the mathematical expectation operator. Represents a random vector [ , ] T and[ , ] T The cross-correlation matrix between them can be expressed as:

[0152] ;

[0153] In the formula, C jp C lp S jp and S lp ( j = 1, 2, …, N ; p = 1, 2, …, M) represent two row vectors with dimension 1 x N , which can be calculated as follows:

[0154] ;

[0155] ;

[0156] ;

[0157] ;

[0158] In the formula, "⊗" represents the element-wise multiplication operator between two matrices.

[0159] S602: Evaluate the time-frequency response statistics of the bridge structure based on the explicit relationship between the obtained total response and the random input vector. For example, the cross-correlation matrix between the responses and can be represented as:

[0160] ;

[0161] In addition, by expanding the coefficient matrix into , the bilateral spectral density of the response ( m 2= 0, 1, 2) at discrete time points t k ( k = 1, 2, …, S ) and frequency points ( p 1= 1, 2, …, M ) can be further obtained:

[0162] .

[0163] To demonstrate the results, the longitudinal and transverse displacements of the bridge mid-span nodes (denoted as u x and u y ) are taken as the response quantities of interest, and the time-varying variances of u x and u y obtained by the method of the present application are plotted in Figure 3 , Figure 4 and the power spectral densities of u x and u y at t = 10 s are given. As a comparison,Figure 3 and Figure 4 The results obtained by the Monte Carlo simulation method through statistical analysis of 10000 groups of response samples are also given. It can be seen that the results obtained by the two methods are in good agreement, indicating that the method has good calculation accuracy. In addition, the computer time for calculating the response statistics of the method and the Monte Carlo simulation method is 290.6 s and 8.3 h respectively, indicating that the method has a significant advantage in calculation efficiency.

[0164] Therefore, the application adopts the above-mentioned random vibration method of long-span bridges under spatially varying non-stationary earthquakes, which overcomes the defects of the existing calculation method that the efficiency is low and it is difficult to fully consider the spatial variability and complete non-stationarity of ground motion.

[0165] Finally, it should be noted that: the above examples are only used to illustrate the technical solutions of the present application and not to limit it, although the present application has been described in detail with reference to the preferred embodiments, those skilled in the art should understand: the technical solutions of the present application can still be modified or replaced by equivalents, and these modifications or equivalent replacements cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present application.

Claims

1. A method for random vibration of long span bridges under spatially varying non-stationary earthquake, characterized in that, The method comprises the following steps: S1, constructing a motion equation of a long-span bridge under spatially varying non-stationary seismic action by a finite element method, and writing the motion equation in a block matrix form; S2, performing modal analysis on the superstructure of the bridge, dividing the displacement response of the superstructure into a dynamic response component and a quasi-static component, and converting the dynamic response component to a modal space; S3, converting the modal equation corresponding to the dynamic response component of the superstructure to a state space, and performing numerical discretization on the state equation at equidistant time points to construct a recursive expression between a state response vector and an excitation vector; S4, decomposing a coherence function matrix of a seismic acceleration input vector by a Cholesky method, discretely representing each seismic motion input process by a spectral representation method, and mapping the random input vector to the response of the corresponding degree of freedom of the bridge support structure, specifically comprising the following steps: S401: For the bridge N seismic acceleration input process of each support point, set the stationary spectral density function of each process S j ( ω ) and time-frequency modulation function A j ( ω , t ), j = 1, 2,..., N ; S402: Set the coherence function matrix of the seismic acceleration input vector ρ ω )​ S403: Using the Cholesky method sequentially in M Equally spaced frequency points ( ω 1, ω 2, …, ω M ) on ρ ( ω Decompose: ρ ( ) = B ( B T ( ), where B( ) represents a lower triangular complex-valued matrix. p 1 = 1, 2, ..., M " " denotes the conjugate operator; S404: decompose each seismic acceleration input process based on spectral representation method, construct N a random input vector: wherein, △ ω represents the frequency discrete interval, and represent two random variable sequences, defined as ； ； where denotes the modulo operator; B jl l = 1, 2, …, N denotes the element of matrix B ) corresponding to the j row and the l column, represents the complex phase of B jl represents an independent random variable uniformly distributed over the interval [0, 2π].​​​​ S405: For each discrete time point t k , k = 1, 2, …, S , construct the mapping coefficient vector between each seismic displacement, velocity and acceleration input process and its corresponding random input vector Θ j : , m 2 = 0,1, 2, wherein, , p 2 = 1, 2, …, 2 M , is expressed as: ； And , m 3 = 0, 1, obtained by integrating the operation of ; S406: Utilize the obtained random input vector Θ j Each seismic displacement, velocity, and acceleration input process is discretely represented and mapped to the response of the corresponding degrees of freedom of the bridge support structure: ； wherein m 2 corresponds to the displacement, velocity and acceleration input of the ground motion, respectively; E j denotes the Z b impact vector between the first j ground motion input. S5, calculating a mapping coefficient matrix between the dynamic response component of the superstructure in the modal space and the random input vector, and further constructing an explicit relationship between the total response of the superstructure in the physical space and the random input vector, specifically comprising the following steps: S501: Calculate the dynamic displacement and velocity response components of the superstructure in the modal space at discrete time points t k The mapping coefficient matrix between the upper and random input vector Θ j and where, and are composed of the upper and lower half elements of Γ jk respectively, while Γ jk is initialized to 0 and computed according to the following recursive expression:​ ； S502: Calculate the dynamic acceleration response components of the superstructure in the modal space at discrete time points using the finite difference formula t k The mapping coefficient matrix between the upper and the random input vector Θ j : ； S503: introducing a mapping relationship between the quasi-static response component of the superstructure and the response of the support structure, and combining the mapping coefficient matrix between the dynamic response component and the random input vector obtained in S501 and S502 to construct an explicit relationship between the total response of the superstructure in the physical space and the random input vector: ； wherein, denotes the explicit relation coefficient matrix between the total response and the random input vector; S6, calculating the cross-correlation matrix between each random input vector, and further evaluating the time-frequency response statistics of the bridge structure based on the explicit relationship between the total response and the random input vector.

2. The method of random vibration of long span bridges under spatially varying non-stationary earthquake according to claim 1, wherein, S1 specifically comprises the following steps: S101: constructing a motion equation of a long-span bridge under spatially varying non-stationary seismic action by a finite element method, in the following form: ； wherein t denotes time; M, C and K represent the mass, damping and stiffness matrices of the bridge finite element model, respectively, the damping matrix being constructed using the Rayleigh method: C = α M + β K, wherein α and β denote the Rayleigh damping coefficients; Z, and represent the displacement, velocity and acceleration vectors of the bridge, respectively; P represents the external force vector on the bridge due to the ground motion. S102: dividing the bridge motion equation according to the respective degrees of freedom of the superstructure and the support structure of the bridge: ； where the subscripts a ” and b ” correspond to the degrees of freedom of the superstructure and the support structure, respectively, ab ” and ba ” correspond to the coupled degrees of freedom; P b denotes the support force vector of the bridge model.

3. The method of random vibration of long span bridges under spatially varying non-stationary earthquake according to claim 1, wherein, S2 specifically comprises the following steps: S201: performing modal analysis on the mass and stiffness matrices corresponding to the superstructure of the bridge to obtain a mass-normalized modal matrix Φ of the superstructure, and further calculating the modal stiffness and damping matrices of the superstructure: ； S202: Convert the displacement response Z a of the bridge superstructure into dynamic response component Z d and quasi-static component Z s , and convert Z d to modal space: Z d = ΦX d , where X d is the modal displacement vector of Z d .

4. The method of random vibration of long span bridges under spatially varying non-stationary earthquake according to claim 1, wherein, S3 specifically comprises the following steps: S301: Constructing the state response vector where "T" denotes the matrix transpose operator, denotes Z d the modal velocity vector; X d corresponding modal dynamic equation into the state space, the parameter matrix A of the state equation is calculated based on the following formula, , m 1=1,2: ； In the formula, I represents a unit matrix; S302: The state equations are... S Equally spaced time points ( t 1, t 2, …, t S Numerical discretization is performed on the [aspect name] with a step size of Δ. t Based on the assumption that the excitation vector changes linearly at each time step, a recursive expression is constructed between the state response vector and the excitation vector: ； wherein k = 1, 2, …, S , T, and are calculated according to the following formula: ； ； 。 5. The method of random vibration of long span bridges under spatially varying non-stationary earthquake according to claim 1, wherein, S6 specifically comprises the following steps: S601: Calculate each random input vector Θ based on the following formula j and the cross-correlation matrix between Θ l and Θ wherein denotes the mathematical expectation operator, denotes the random vector , ] T and , ] T the cross-correlation matrix between and is denoted by ； wherein , , and each represent a row vector of dimension 1 x N and are calculated according to the following equations: ； ； ； ； In the formula, "⊗" represents an element-by-element multiplication operator between two matrices; S602: evaluate the time-frequency response statistics of the bridge structure based on the explicit relationship between the acquired total response and the random input vector, response The cross-correlation matrix between and is expressed as: The cross-correlation matrix between and is expressed as: ； Furthermore, by expanding the coefficient matrix as , one further obtains the response at discrete time points t k and frequency points ​ 。

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