A method for simulation and stochastic dynamic analysis of random mainshock and aftershock sequences

Abstract

The present invention provides a method for simulating a random mainshock and aftershock sequence and a random dynamic analysis method, and belongs to the field of random dynamic analysis of earthquakes. The key contents include establishing the correlation between mainshock and aftershock parameters based on the Copula function, generating an earthquake sequence based on a random physical model, and performing random response analysis based on the DPIM method. The present invention can solve the problem that the existing random earthquake motion model ignores the influence of aftershocks on earthquake motion, and can reasonably establish a random relationship between mainshock and aftershock parameters from a probability perspective, and can overcome the limitations of traditional structural dynamic analysis methods, such as large amount of calculation and low efficiency, thereby greatly shortening the operation time and improving the calculation accuracy.

Description

Technical Field

[0001] The present application belongs to the field of random dynamic analysis of earthquakes, and relates to a simulation and random dynamic analysis method of a random mainshock and aftershock sequence. Technical Background

[0002] Most earthquakes occur in the form of earthquake sequences. Sometimes aftershocks are larger than the main shock and can cause serious damage to buildings. Therefore, when conducting structural analysis of random earthquakes, it is necessary to simulate the random mainshock-aftershock sequence and use appropriate methods for analysis.

[0003] Early random seismic models assumed that ground motion was singular, thus ignoring aftershocks or not distinguishing parameters between main shocks and aftershocks. Such models include white noise models, Kanai-Tajimi spectrum models, evolving power spectrum methods, and spectrum representation methods. Currently, measured earthquake sequences are often used in the analysis of main-aftershock structures. However, the number of measured earthquake sequences is limited and they do not belong to the same probability space, which is not convenient for random dynamics calculations. In fact, the main shock and aftershocks in the same earthquake event have a certain correlation, but in the current ground motion model based on deterministic relationships, the proportional relationship between the main shock and aftershock intensity is determined based on experience, ignoring the randomness of aftershock parameters, which should not be specified in fact. On the other hand, it is also assumed that the main shock is a stable process. In fact, due to the change of amplitude in the time domain and the uneven energy distribution, the main shock is not stable.

[0004] Traditional methods used in structural dynamic analysis, such as the Monte Carlo simulation (MCS) method, believe that the larger the number of samples, the higher the calculation accuracy. However, the amount of calculation for tens of thousands of finite element simulations of complex structures is huge and the efficiency is low. The probability density evolution method (PDEM) can handle the random vibration of structures with random parameters under random excitation, which is more efficient than MCS, but it requires solving the generalized density evolution equation (GDEE), thereby treating the random process of the static system as a dynamic situation, which adds extra work. Summary of the invention

[0005] In order to solve the problems existing in the above-mentioned prior art, the present invention provides a simulation method of a random mainshock and aftershock sequence and a random dynamic analysis method, which determines the intensity of aftershocks from a probability perspective, can reasonably give the correlation between the mainshock and aftershocks, and uses the DPIM method to perform random response analysis, thereby shortening the calculation time and improving the calculation accuracy.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is:

[0007] A method for simulating and analyzing random main shock and aftershock sequences and random dynamics comprises the following steps:

[0008] The first step is to simulate the random main shock and aftershock sequence.

[0009] S1: Obtain a large amount of measured sequential earthquake acceleration time history data according to the type of engineering site, and screen to obtain m groups of measured data;

[0010] S2: Invert m groups of measured data to obtain the corresponding number of groups of "source-path-site" model parameters;

[0011] S3: Perform probability statistics on the inverted model parameters to obtain the optimal distribution and its distribution parameters;

[0012] S4: Generate a set of representative points of multi-dimensional random variables of artificial main shock and aftershock earthquake motions, and generate N groups of random main shock and aftershock sequence groups;

[0013] S5: Establish correlation between artificial main shock and aftershock parameters;

[0014] Step 2: Calculate random dynamics

[0015] S6: According to the structural characteristics of the physical model of the analysis object, constraints are imposed to establish a finite element model of the engineering structure;

[0016] S7: Take each earthquake sequence as an external load and input it into the finite element model to perform a random dynamic calculation. Perform N calculations in total to traverse all samples, that is, perform N finite element simulations;

[0017] S8: From the random dynamic calculation results of step S7, N groups of time histories of one or more physical quantities of interest are extracted according to the research content, such as stress, displacement, safety factor, etc.;

[0018] The third step is to conduct random response analysis

[0019] S9: using direct probability integration method to perform probability calculation on the physical quantity in step S8 to obtain its probability distribution at each time (PDF / CDF);

[0020] S10: Calculate the dynamic reliability of the engineering structure according to the limit state or threshold.

[0021] Preferably, the specific method for obtaining and screening the measured data according to the type of the engineering site in step S1 is:

[0022] According to the Chinese seismic code, the site types of sequential earthquakes are divided into four categories: I, II, III, and IV. The influence of site types on structural response and damage state is studied. The criteria for screening a large amount of measured data are:

[0023] (1) The mainshock-aftershock combination is the “mainshock + the largest aftershock” in an earthquake event;

[0024] (2) The selected main shock and aftershock combination should be data recorded at the same station;

[0025] (3) To avoid soil-structure coupling, the station should be located in a free field or on the ground floor of a small building;

[0026] (4) Earthquakes that are unlikely to have an impact on the structure should not be selected, and the magnitude of the main aftershock should not be less than 5;

[0027] Finally, m groups of measured sequential earthquake acceleration time history data were screened and obtained.

[0028] Preferably, the specific method for inverting m groups of measured data in step S2 is:

[0029] Using the "source-path-site" model, the source, propagation path, and local site are modeled based on the random Fourier spectrum and random function model, and the key physical factors affecting the randomness of ground motion are abstracted as random variables. g ,ω g , a, b, c, d are the most basic physical parameter vectors of the main and aftershocks: A0 is the source amplitude coefficient, reflecting the influence of the source intensity; τ is the Brune source coefficient, reflecting the time process characteristics of the source fracture; a, b, c and d are the empirical parameters of the equivalent frequency-wavenumber relationship; ξ g and ω g are the equivalent damping ratio and equivalent dominant circular frequency of the local site respectively. When the change of the local site is linear, ξ g and ω g It can be regarded as unchanged. Since the six two-dimensional random variables A0, τ, a, b, c, and d cannot be directly observed, they can be identified by parameters through seismic inversion (to distinguish the main shock parameters, the subscript M represents the main shock parameters, and A represents the aftershock parameters, which will not be explained below). The identification steps are as follows:

[0030] (1) Performing Fourier transform on each set of sampled measured data to obtain its Fourier amplitude spectrum and Fourier phase spectrum;

[0031] (2) Using the amplitude spectrum model of formula (1) and the least squares method, the random variables in the main shock amplitude spectrum model are identified. τ M , and The value of

[0032]

[0033] Where A(λ,ω,R) is the Fourier amplitude spectrum model of the earthquake acceleration time history; λ is the basic physical parameter vector, λ= [A0,τ,a,b,c,d,ξg ,ω g ]; ω is the circular frequency; R is the epicenter distance; K is the coefficient to measure the friction attenuation effect, where K = 10 -5 s / km.

[0034] (3) Order Repeat step (2) according to the Fourier amplitude spectrum of the aftershock to obtain and τ A .

[0035] (4) Convert the Fourier phase spectrum of the recorded seismic sequence into a phase difference spectrum by using formula (2), and calculate the distribution form of the phase difference spectrum;

[0036]

[0037] In the formula, is the frequency corresponding to the kth frequency ω k The kth phase of is the frequency corresponding to the k-1th frequency ω k-1 The k-1th phase of .

[0038] (5) When identifying the values ​​of the propagation medium coefficients a, b, c, and d, the phase difference spectrum is calculated by the Fourier phase spectrum expressed in equation (3), and step (4) is repeated until the model phase difference spectrum coincides with the actual phase difference spectrum. In order to improve efficiency, a genetic algorithm is used for identification.

[0039]

[0040] In the formula, Φ R (λ, ω, R) is the Fourier phase spectrum model of the earthquake acceleration time history, and the remaining letters have the same meanings as above.

[0041] Preferably, the specific method for performing probability statistics on the inverted model parameters in step S3 is:

[0042] According to step S2, the specific values ​​of all parameters are successfully identified. In order to obtain the optimal probability model and the probability density function of each parameter of the main shock and aftershock, it is necessary to perform statistics on these parameters based on the BIC criterion. The expression of the BIC criterion is:

[0043] BIC=klnN0-2lnL(α|x) (4)

[0044] Where k is the number of distribution parameters, N0 is the number of samples, α is the distribution parameter, and L(α|x) is the likelihood function of x, which can be expressed as:

[0045]

[0046] Where p(x|α) is the density function of the sample set x under parameter α.

[0047] Preferably, the generation of the representative point set of the main and aftershock ground motion multi-dimensional random variables in step S4 is based on the GF-deviation (generalized F-deviation) point selection technology. GF-deviation (generalized F-deviation) can be expressed as:

[0048] D GF =max 1≤i≤s {sup|F n,i (x)-F i (x)|} (6)

[0049] In the formula, F n,i (x) is the empirical marginal cumulative density function (CDF), which can be calculated by formula (7); F i (x) is the marginal cumulative density function of the ith random variable.

[0050]

[0051] In the formula, x q,i is a random variable x q The i-th component of .

[0052] The iterative screening rearrangement method can reduce the GF-deviation and effectively obtain the representative point set, which specifically includes the following steps:

[0053] (1) s-dimensional initial independent scattered point set x q of acquisition.

[0054] x q ={x q,i =(x q,1 ,x q,2 ,…,x q,s ); q=1,2,…,n; i=1,2,…,s} (8)

[0055] S n = {u q,i =(u q,1 ,u q,2 ,…,u q,s ); q=1,2,…,n; i=1,2,…,s} (9)

[0056]

[0057] In the formula, x q is the s-dimensional initial independent scattered point set; S n is the Sobol set on the unit hypercube; is the inverse cumulative density function (CDF) of the ith variable.

[0058] (2) In each dimension, for the initial s-dimensional independent scattered point set x q Perform the following rearrangement to make the probability distribution of each point more even and obtain a new s-dimensional independent scattered point set x′ m,i :

[0059]

[0060] where I(·) is an indicator function, if and only if event A: x q,i <x m,i When true, I(A) = 1, otherwise I(A) = 0; x m,i Same as x q,i The same, both belong to the random variable x q The i-th component of , in order to distinguish the order, the subscripts are changed; n is the number of points in the s-dimensional independent scattered point set.

[0061] (3) Calculate the allocation probability P according to equation (12): q , and remove all points with probability less than b / n (b represents the threshold, n represents the number of points in the set), where b = 0.7.

[0062]

[0063] Where P q is Ω q The probability of assigning the qth representative point in the region; p Θ (x) is the joint probability density function of θ; Ω q Represents the point x q The representative area can be expressed as:

[0064]

[0065] In the formula, x represents the set of s-dimensional independent points; x q 、x j Represent subdomain Ω q and Ω j Representative points in represents the set of s-dimensional real numbers.

[0066] (4) Repeat step (3) until the probability of point allocation meets the requirement or the number of points reaches the target. At this point, the number of points in the set changes from n to n'.

[0067] (5) The final representative point set M with lower GF deviation is obtained n ={x″ q =(x″ q,1 ,x″ q,2 ,…,x″ q,s ),q= 1,2,…,n′}, which can be expressed by formula (14):

[0068]

[0069] In the formula, p m is the probability of the representative point corresponding to the sequence number m.

[0070] The random representative point set is brought into the "source-path-site" model as earthquake parameters to generate N groups of random mainshock and aftershock sequences. The size of N is related to the actual project and the required accuracy, and is generally between 100 and 5000.

[0071] Preferably, the specific method of establishing the correlation between main shock and aftershock parameters in step S5 is:

[0072] When establishing the correlation between main shocks and aftershocks based on Copula functions, seven commonly used Copula functions include independent function, Gaussian function, t function, Gumbel function, Plackett function, Frank function and Clayton function. To establish the correlation between main shock and aftershock parameters, it is necessary to find the joint probability distribution of the main shock and aftershock parameters. The steps are as follows:

[0073] (1) Calculate the Kendall rank correlation coefficient τ of the main shock and aftershock parameters;

[0074] (2) Based on the marginal probability distribution f1, f2 and rank correlation coefficient τ of the main shock and aftershock parameters, the correlation parameter θ of the alternative Copula function is obtained;

[0075] (3) Based on the BIC criterion, seven commonly used Copula functions are traversed to obtain the optimal Copula function and related parameters;

[0076] (4) The joint probability distribution function of the main shock and aftershock parameters is obtained by the optimal Copula function. The specific method is as follows:

[0077] According to the Copula function theory, the joint density function of the main shock and aftershock dual variables is established:

[0078] f(x1,x2)=c 12 {F1(x1),F2(x2)}·f1(x1)·f2(x2) (15)

[0079] In the formula, c 12 (·) is the two-dimensional Copula density function

[0080] Conditional density in the case of main shock and aftershock bivariate:

[0081] f(x1|x2)=c 12 {F1(x1),F2(x2)}·f1(x1) (16)

[0082] The boundary condition distribution can be expressed as:

[0083]

[0084] From this, combined with the formula in the iterative screening-rearrangement method, the sampling method of independent variables was obtained, and the correlation between the main shock and aftershock parameters was established:

[0085]

[0086] In the formula, ω i Yes m,i The set of; u2 as the intermediate variable, ω i The set is transformed into a new set u2 with correlation; It is the inverse function of F(x1|x2) in equation (17).

[0087] Preferably, the specific method of performing probability calculation on the physical quantity extracted in step S8 in step S9 is:

[0088] The stochastic direct probability integration method (DPIM) is used to calculate the probability of the physical quantity to obtain its probability distribution (PDF / CDF) at each time.

[0089] DPIM can calculate the failure probability without knowing the failure domain of the basic random variables. The PDF of the structural response is realized based on the probability density integral equation (PDIE). However, the analytical solution of PDIE is difficult to obtain. The solution of PDIE can be solved by smoothing the Dirac delta function of the Gaussian function. The instantaneous PDF of the dynamic system output can be obtained by DPIM:

[0090]

[0091] Where n is the total number of representative points; P q is the probability of assigning the qth representative point; z represents the output of the dynamic system; g(θ q ,t) is the dynamic response corresponding to the qth representative point; σ is the smoothing parameter of the Dirac delta function.

[0092] Preferably, in step S10, the specific method for calculating the dynamic reliability according to the limit state or threshold is:

[0093] The dynamic reliability can be calculated by constructing the equivalent extreme value method (EVD) or the absorption condition method (AC).

[0094] For a general random dynamic system, the dynamic reliability based on the first-pass criterion is expressed as:

[0095] R(t)=Pr{Y(Θ,t)∈Ω s ,τ∈(0,t)} (20)

[0096] Where Y(Θ,t) is the random response of the dynamic system to be studied; Θ is the random variable corresponding to the ground motion parameter; Ω s Represents a security domain.

[0097] Dynamic reliability cannot be directly obtained by integrating the PDF at instant t. According to the definition of the first-pass failure criterion in equation (20), the structure will fail when the maximum response in the time interval (0, t) exceeds the threshold. In this context, the dynamic reliability based on the first-pass criterion can be transformed into an extreme value distribution problem combined with DPIM. The specific method is:

[0098] In formula (20), the structural dynamic reliability at instant t is equivalent to

[0099] R(t)=Pr{Y ext (Θ,t)∈Ωs} (21)

[0100] Where Y ext (Θ,t) is the extreme random response of the dynamic system to be studied.

[0101] According to the method in step S10, the extreme value mapping Y of the dynamic response is obtained ext The PDF of (Θ,t) is:

[0102]

[0103] In the formula, y ext is the extreme value variable of the Dirac delta function. Finally, the dynamic reliability in the instantaneous safety domain (0, t] time period is obtained by PDF integration:

[0104]

[0105] The absorption condition method assumes that once some representative points fall into the failure domain at a certain moment, they are absorbed and cannot be transferred to the next moment. The allocation probability of the qth representative point can be divided into the failure part P q,f (t) and the effective part P q,s (t). The performance function can be expressed as:

[0106]

[0107] In the formula, is the specified threshold.

[0108] The failure event is reflected by the performance function Z≤0. The probability of the qth allocation of the failure part can be expressed as:

[0109] P q,f (t i )=0,z∈Ω Z,f ={z|g(θq ,t i )≤0} (25)

[0110] The probability is conserved during the uncertainty propagation from input to output, so the remaining distribution probability should be replaced by the effective part. Using the method in step S10 to solve the PDF of random responses in the safety domain, the final first-pass reliability can be expressed as:

[0111]

[0112] Beneficial effects of the present invention:

[0113] The present invention provides a simulation and random dynamic analysis method for a random mainshock-aftershock sequence that is more in line with actual conditions. The "source-path-site" model overcomes the limitations of empirical statistics of the phenomenological random seismic motion model, constructs the correlation between the mainshock and the aftershock based on the Copula function, and uses the DPIM method to form a mapping between the input and output ends of the random system, and introduces a delta function for smoothing. This greatly shortens the calculation time while ensuring that the equations are easy to solve, and is more suitable for solving long-duration random dynamics such as sequence earthquakes. BRIEF DESCRIPTION OF THE DRAWINGS

[0114] Figure 1 Flowchart for parameter identification;

[0115] Figure 2 These are three groups of random main shock and aftershock sequence diagrams generated by bringing random representative point sets into the “source-path-site” model as earthquake parameters; Figure 2 (a) is the first group of random main shock and aftershock sequence diagrams; Figure 2 (b) is the second set of random main shock and aftershock sequence diagram; Figure 2 (c) is the third group of random main shock and aftershock sequence diagram;

[0116] Figure 3 It is a schematic diagram of the frame structure under the action of random earthquake sequence;

[0117] Figure 4 is the skeleton curve of the IMK model, Figure 4 (a) is the skeleton curve of the IMK model, and (b) is the restoring force curve of one layer;

[0118] Figure 5 is the time history of the maximum inter-story drift (MID) of the building;

[0119] Figure 6 The probability density function of the maximum inter-layer drift obtained by different methods at typical moments; Figure 6 (a) is the probability density function of the maximum interlayer drift obtained by the DPIM method at typical time 1; Figure 6 (b) is the probability density function of the maximum inter-layer drift obtained by the MCS method at typical time 1; Figure 6 (c) is the probability density function of the maximum interlayer drift obtained by the PDEM method at typical time 1; Figure 6 (d) is the probability density function of the maximum interlayer drift obtained by the DPIM method at typical time 2; Figure 6 (e) is the probability density function of the maximum inter-layer drift obtained by the MCS method at typical time 2; Figure 6 (f) is the probability density function of the maximum interlayer drift obtained by the PDEM method at typical time 2; Figure 6 (g) is the probability density function of the maximum inter-layer drift obtained by the DPIM method at typical time 3; Figure 6 (h) is the probability density function of the maximum inter-layer drift obtained by the MCS method at typical time 3; Figure 6 (i) is the probability density function of the maximum interlayer drift obtained by the PDEM method at typical time 3;

[0120] Figure 7 CDF of inter-story displacement calculated after absorption condition method; Figure 7 (a) indicates that the CDF is calculated using the DPIM method; Figure 7 (b) indicates the calculation of CDF using the MCS method; Figure 7 (c) indicates the calculation of CDF using the PDEM method; Figure 7 (d) is the CDF when the threshold is 3 cm;

[0121] Figure 8 is the dynamic reliability under different thresholds; Figure 8 (a) Dynamic reliability under different thresholds obtained using DPIM; Figure 8 (b) Dynamic reliability under different thresholds obtained using MCS; Figure 8 (c) Dynamic reliability under different thresholds obtained using PDEM;

[0122] Fig. 9 are the PDF and CDF of the maximum absolute value of the base shear force; Fig. 9 (a) PDF of the maximum absolute value of base shear force calculated by DPIM method; Fig. 9 (b) PDF of the maximum absolute value of base shear force calculated by the MCS method; Fig. 9 (c) PDF of the maximum absolute value of base shear force calculated by PDEM method; Fig. 9 (d) CDF of the maximum absolute value of base shear force calculated by DPIM method; Fig. 9 (e) CDF of the maximum absolute value of base shear force calculated by the MCS method; Fig. 9(f) CDF of the maximum absolute value of base shear force calculated by PDEM method;

[0123] Fig.10 is the dynamic reliability under different thresholds; Fig.10 (a) Dynamic reliability under different thresholds calculated by the DPIM method; Fig.10 (b) Dynamic reliability under different thresholds calculated by the MCS method; Fig.10 (c) Dynamic reliability under different thresholds calculated by the PDEM method;

[0124] Fig.11 Attached is a flowchart of the method of the present invention. DETAILED DESCRIPTION

[0125] In order to make the objectives, technical means and advantages of the present application more clearly understood, the present application is further described in detail below in conjunction with the accompanying drawings.

[0126] The present application provides a method for simulating a random main shock and aftershock sequence and a random dynamic analysis method, which basically includes the simulation and random dynamic analysis of a random main shock and aftershock sequence. The simulation method and random dynamic analysis method of the present application are mainly used for seismic motion analysis. The present application is described below using a seismic motion as an example. The specific embodiments described are only part of the many embodiments of the present invention and do not represent all embodiments.

[0127] A method for simulating and analyzing random main shock and aftershock sequences and random dynamics comprises the following steps:

[0128] First, according to the Chinese seismic code, the site types of sequential earthquakes are divided into four categories: I, II, III, and IV, and the influence of site types on structural response and damage state is studied. A large amount of measured sequential ground motion data is screened to obtain data that meets the requirements.

[0129] See attached Figure 1 In step S2, A0, τ, a, b, c, and d are the most basic physical parameter vectors of the main and aftershocks, which cannot be obtained by direct observation and need to be identified by parameters through ground motion inversion. It can be clearly seen from the attached figure that the Fourier spectrum is identified by the least squares method, while the Fourier phase difference spectrum is identified by the genetic algorithm.

[0130] In order to obtain the optimal probability model and the probability density function of each parameter of the main shock and aftershock, all model parameters successfully identified in step S2 are statistically analyzed based on the BIC criterion in step S3. The probability distribution type and statistical information corresponding to each parameter are shown in Tables 1 and 2. When the PDF (probability density function) is a normal distribution, μ and σ are the mean and standard deviation, respectively; when the PDF is a lognormal distribution, μ and σ are the logarithmic mean and standard deviation, respectively; when the PDF is a Weibull distribution, μ and σ are the shape parameter and scale parameter, respectively.

[0131] Table 1 Distribution and statistics of earthquake characteristics

[0132]

[0133] Table 2 Distribution and statistics of site characteristics

[0134]

[0135] In step S4, based on the GF-deviation (generalized F-deviation) point selection technique, the representative points of the multidimensional random variable are obtained, and the distribution probability P is also obtained. q At the same time, the N groups of random main shock and aftershock sequence groups generated in step S4 are determined according to the actual project and the required accuracy. The method for determining N is determined according to the GF deviation and the project scale, and is generally between 100 and 5000. Figure 2 ,These 3 sets of random earthquake sequences are generated by bringing random representative point sets as earthquake parameters into the “source-path-site” model.

[0136] When establishing the main shock and aftershock correlation based on the Copula function in step S5, the optimal Copula function and related parameters θ of each parameter are first identified based on the BIC criterion, as shown in Table 3. Then, the joint probability distribution function of the main shock and aftershock parameters is obtained by the optimal Copula function.

[0137] Table 3 Optimal Copula function and related parameters

[0138]

[0139]

[0140] After the dynamic system of earthquake motion is established, engineering simulation can be carried out. The first step of simulation is to establish a finite element model, which can be established according to the structural characteristics and constraints of the physical model of the analysis object. The earthquake sequence generated in step S4 is input as an external load into the finite element model for random dynamic calculation. N groups of earthquake sequences are simulated N times by finite element simulation, and N groups of time histories of one or more physical quantities of interest are extracted according to the focus of the research, such as stress, displacement, safety factor, etc.

[0141] The following is a more detailed description of the simulation of the random main shock and aftershock sequence and the random dynamic analysis method in the present application through a specific embodiment:

[0142] In a specific embodiment, Figure 3 The frame structure under the random earthquake sequence shown in the figure is taken as an example to illustrate. Figure 3 As shown in Figure 2, the structure is a two-span, 20-story inelastic frame building. The dynamic equilibrium equation of the system can be expressed by equation (30), and the nonlinear restoring force f can be described by the Ibarra–Medina–Krawinkler model, as follows: Figure 4 The basic parameters of the system are shown in Table 4. The PGA of the earthquake sequence is 0.30g.

[0143] Table 4 Basic structural parameters in the example

[0144]

[0145]

[0146] The dynamic response of the structure can be obtained by the Newmark-β method. Taking a single sample in the random dynamic calculation as an example, the time history of the maximum inter-story drift (MID) of the building is as follows: Figure 5 As shown in Figure 2. In this case, the MID occurred on the 1st floor. After the main shock, the 1st floor drifted 0.4 cm to the right, while after the aftershock, the 1st floor drifted 0.67 cm to the left. Therefore, from the perspective of a single sample, aftershocks have a significant effect on the structural response of a structure with elastic-plastic characteristics.

[0147] To evaluate the safety level of the structure from the perspective of random dynamic analysis and reliability, it is necessary to calculate the PDF of the MID. The PDF of the MID at the typical moment is solved by using the traditional MCS method and the DPIM in the present invention. Figure 6 It is shown that the PDF obtained by DPIM is basically consistent with the PDF obtained by MCS. Although the distribution of the structural response has less variability, it does not conform to a conventional distribution, such as a Gaussian distribution, so DPIM can better describe the probabilistic characteristics of a dynamic system. In addition, although aftershocks reduce the probability density of the response in a random system, they do not significantly affect the shape of the distribution.

[0148] The CDF of MID is calculated after the absorption condition method, such as Figure 7 (d) shows that when the threshold (limit state) is 3 cm, the aftershocks reduce the structural reliability from 0.85 to 0.82. Figure 7 (a) and (b) are the results of CDF calculation using DPIM and MCS methods respectively. It can be seen that the PDF obtained by DPIM is basically consistent with the CDF obtained by MCS. Figure 8 As shown in Figure 2, it can be seen that the impact of aftershocks on structural reliability is significantly smaller than that of the main shock. Therefore, aftershocks need to be considered only when the structural performance requirements are very strict.

[0149] In addition, base shear during earthquakes is also an important indicator for high-rise buildings. Fig. 9 The PDF and CDF of the maximum absolute value of the base shear force are plotted. Fig.10 Figure 2 shows the dynamic reliability at different thresholds evaluated by DPIM, MCS, and PDEM. They all show that the results obtained by DPIM are in good agreement with those obtained by MCS and PDEM.

[0150] The above-described embodiments merely express the implementation methods of the present invention, but they cannot be understood as limiting the scope of the patent of the present invention. It should be pointed out that for those skilled in the art, several modifications and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention.

Claims

1. A method for simulating and analyzing random main shock and aftershock sequences, characterized in that: The following steps are involved: The first step is to simulate the random main shock and aftershock sequence. S1: Obtain a large amount of measured sequential earthquake acceleration time history data according to the type of engineering site, and screen to obtain m groups of measured data; S2: Invert m groups of measured data to obtain the corresponding number of groups of "source-path-site" model parameters; S3: Perform probability statistics on the inverted model parameters to obtain the optimal distribution and its distribution parameters; S4: Generate a set of representative points of multi-dimensional random variables of artificial main shock and aftershock earthquake motions, and generate N groups of random main shock and aftershock sequence groups; S5: Establish correlation between artificial main shock and aftershock parameters; Step 2: Calculate random dynamics S6: According to the structural characteristics of the physical model of the analysis object, constraints are imposed to establish a finite element model of the engineering structure; S7: Take each earthquake sequence as an external load and input it into the finite element model to perform a random dynamic calculation. Perform N calculations in total to traverse all samples, that is, perform N finite element simulations; S8: From the random dynamic calculation results of step S7, N groups of time histories of one or more physical quantities of interest are extracted according to the content of the study, including stress, displacement, and safety factor; The third step is to conduct random response analysis S9: using direct probability integration method to perform probability calculation on the physical quantity in step S8 to obtain its probability distribution at each time; S10: Calculate the dynamic reliability of the engineering structure according to the limit state or threshold.

2. The method for simulating and analyzing random main shock and aftershock sequences according to claim 1, characterized in that: The specific method of obtaining and screening the measured data according to the type of the engineering site in step S1 is: According to the Chinese seismic code, the site types of sequential earthquakes are divided into four categories: I, II, III, and IV. The influence of site types on structural response and damage state is studied by classification. The criteria for selecting measured data are: (1) The mainshock-aftershock combination is the "mainshock + the aftershock with the largest magnitude" in an earthquake event; (2) The selected main shock and aftershock combination should be data recorded at the same station; (3) To avoid soil-structure coupling, the station should be located in a free field or on the ground floor of a small building; (4) Earthquakes that are unlikely to have an impact on the structure should not be selected, and the magnitude of the main aftershock should not be less than 5; Finally, m groups of measured sequential earthquake acceleration time history data were screened and obtained.

3. The method for simulating and analyzing random main shock and aftershock sequences according to claim 1, characterized in that: The specific method for inverting m groups of measured data in step S2 is: Using the "source-path-site" model, the source, propagation path, and local site are modeled based on the random Fourier spectrum and random function model, and the key physical factors affecting the randomness of ground motion are abstracted as random variables; A0,τ,ξ g ,ω g , a, b, c, d are the most basic physical parameter vectors of the main and aftershocks: A0 is the source amplitude coefficient, reflecting the influence of the source intensity; τ is the Brune source coefficient, reflecting the time process characteristics of the source fracture; a, b, c and d are the empirical parameters of the equivalent frequency-wavenumber relationship; ξ g and ω g are the equivalent damping ratio and equivalent dominant circular frequency of the local site respectively. When the change of the local site is linear, ξ g and ω g It can be regarded as unchanged; the identification steps of six two-dimensional random variables A0, τ, a, b, c, d are as follows: (1) Performing Fourier transform on each set of sampled measured data to obtain its Fourier amplitude spectrum and Fourier phase spectrum; (2) Using the amplitude spectrum model of formula (1) and the least squares method, the random variables in the main shock amplitude spectrum model are identified. τ M , and The value of Among them, A(λ,ω,R) is the Fourier amplitude spectrum model of the earthquake acceleration time history; λ is the basic physical parameter vector, λ=[A0,τ,a,b,c,d,ξ g ,ω g ]; ω is the circular frequency; R is the epicenter distance; K is the coefficient to measure the friction attenuation effect, where K = 10 -5 s / km; (3) Order Repeat step (2) according to the Fourier amplitude spectrum of the aftershock to obtain and τ A ; (4) Convert the Fourier phase spectrum of the recorded seismic sequence into a phase difference spectrum by using formula (2), and calculate the distribution form of the phase difference spectrum; In the formula, is the frequency corresponding to the kth frequency ω k The kth phase of is the frequency corresponding to the k-1th frequency ω k-1 The k-1th phase of ; (5) When identifying the values ​​of the propagation medium coefficients a, b, c, and d, the phase difference spectrum is calculated by the Fourier phase spectrum expressed by equation (3), and step (4) is repeated until the model phase difference spectrum coincides with the actual phase difference spectrum; in order to improve efficiency, a genetic algorithm is used for identification; In the formula, Φ R (λ, ω, R) is the Fourier phase spectrum model of the earthquake acceleration time history, and the remaining letters have the same meanings as above.

4. The method for simulating and analyzing random main shock and aftershock sequences according to claim 1, characterized in that: The specific method for performing probability statistics on the inverted model parameters in step S3 is: According to step S2, the specific values ​​of all parameters are successfully identified. In order to obtain the optimal probability model and the probability density function of each parameter of the main shock and aftershock, it is necessary to perform statistics on these parameters based on the BIC criterion. The expression of the BIC criterion is: BIC=kln N0-2ln L(α|x) (4) Where k is the number of distribution parameters, N0 is the number of samples, α is the distribution parameter, and L(α|x) is the likelihood function of x, expressed as: Where p(x|α) is the density function of the sample set x under parameter α.

5. The method for simulating and stochastic dynamic analysis of a random main shock and aftershock sequence according to claim 1, characterized in that: The generation of the representative point set of the multi-dimensional random variables of the main shock and aftershock earthquake motion in step S4 is based on the GF-deviation point selection technology, which is expressed as: D GF =max 1≤i≤s {sup|F n,i (x)-F i (x)|} (6) In the formula, F n,i (x) is the empirical marginal cumulative density function CDF, calculated by formula (7); F i (x) is the marginal cumulative density function of the ith random variable; In the formula, x q,i is a random variable x q The i-th component of ; The iterative screening rearrangement method can reduce the GF-deviation and effectively obtain the representative point set, which specifically includes the following steps: (1) s-dimensional initial independent scattered point set x q the acquisition of; x q ={x q,i =(x q,1 ,x q,2 ,…,x q,s );q=1,2,…,n;i=1,2,…,s} (8) S n ={u q,i =(u q,1 ,u q,2 ,…,u q,s );q=1,2,…,n;i=1,2,…,s} (9) In the formula, x q is the s-dimensional initial independent scattered point set; S n is the Sobol set on the unit hypercube; is the inverse cumulative density function CDF of the ith variable; (2) In each dimension, for the initial s-dimensional independent scattered point set x q Perform the following rearrangement to make the probability distribution of each point more even and obtain a new s-dimensional independent scattered point set x′ m,i : where I(·) is an indicator function, if and only if event A: x q,i <x m,i When true, I(A) = 1, otherwise I(A) = 0; x m,i Same as x q,i The same, both belong to the random variable x q The i-th component of , in order to distinguish the order, the subscript is changed; n is the number of points in the s-dimensional independent scattered point set; (3) Calculate the allocation probability P according to equation (12): q , and remove all points with probability less than b / n, where b represents the threshold and n represents the number of points in the set, here b = 0.7; Where P q is Ω q The probability of assigning the qth representative point in the region; p Θ (x) is the joint probability density function of θ; Ω q Represents the point x q The representative area is expressed as: In the formula, x represents the set of s-dimensional independent points; x q 、x j Represent subdomain Ω q and Ω j Representative points in represents the s-dimensional real number set; (4) Repeat step (3) until the probability of point allocation meets the requirement or the number of points reaches the target; at this time, the number of points in the set changes from n to n'; (5) The final representative point set M with lower GF deviation is obtained n ={x″ q =(x″ q,1 ,x″ q,2 ,…,x″ q,s ),q=1,2,…,n′}, expressed by formula (14): In the formula, p m is the probability of the representative point corresponding to the serial number m; The random representative point set is brought into the "source-path-site" model as earthquake parameters to generate N groups of random mainshock and aftershock sequences. The size of N is related to the actual engineering practice and required accuracy.

6. The method for simulating and analyzing random main shock and aftershock sequences according to claim 1, characterized in that: The specific method of establishing the correlation between main shock and aftershock parameters in step S5 is: When establishing the correlation between main shocks and aftershocks based on the Copula function, the joint probability distribution of the main shock and aftershock parameters needs to be obtained. The steps are as follows: (1) Calculate the Kendall rank correlation coefficient τ of the main shock and aftershock parameters; (2) Based on the marginal probability distribution f1, f2 and rank correlation coefficient τ of the main shock and aftershock parameters, the correlation parameter θ of the alternative Copula function is obtained; (3) Based on the BIC criterion, seven commonly used Copula functions are traversed to obtain the optimal Copula function and related parameters; (4) The joint probability distribution function of the main shock and aftershock parameters is obtained by the optimal Copula function. The specific method is as follows: According to the Copula function theory, the joint density function of the main shock and aftershock dual variables is established: f(x1,x2)=c 12 {F1(x1),F2(x2)}·f1(x1)·f2(x2) (15) In the formula, c 12 (·) is the two-dimensional Copula density function Conditional density in the case of main shock and aftershock bivariate: f(x1|x2)=c 12 {F1(x1),F2(x2)}·f1(x1) (16) The boundary condition distribution is expressed as: From this, combined with the formula in the iterative screening-rearrangement method, the sampling method of independent variables is obtained, and the correlation between mainshock and aftershock parameters is established: In the formula, ω i Yes m,i The set of; u2 as the intermediate variable, ω i The set is transformed into a new set u2 with correlation; It is the inverse function of F(x1|x2) in equation (17).

7. A method for simulating and stochastic dynamic analysis of a random main shock and aftershock sequence according to claim 6, characterized in that: The Copula functions include independent functions, Gaussian functions, t functions, Gumbel functions, Plackett functions, Frank functions and Clayton functions.

8. The method for simulating and analyzing random main shock and aftershock sequences according to claim 1, characterized in that: In step S10, the dynamic reliability is calculated by constructing the equivalent extreme value method EVD or the absorption condition method AC according to the limit state or threshold, and the specific method is: For a general random dynamic system, the dynamic reliability based on the first-pass criterion is expressed as: R(t)=Pr{Y(Θ,t)∈Ω s ,τ∈(0,t)}(20) Where Y(Θ,t) is the random response of the dynamic system to be studied; Θ is the random variable corresponding to the ground motion parameter; Ω s Represents the security domain; The dynamic reliability based on the first-pass criterion is transformed into an extreme value distribution problem combined with DPIM. The specific method is as follows: In formula (20), the structural dynamic reliability at instant t is equivalent to: R(t)=Pr{Y ext (Θ,t)∈Ωs}(21) Where Y ext (Θ,t) is the extreme random response of the dynamic system to be studied; According to the method in step S10, the extreme value mapping Y of the dynamic response is obtained ext The PDF of (Θ,t) is: In the formula, y ext is the extreme value variable of the Dirac delta function; finally, the dynamic reliability in the instantaneous safety domain (0, t] time period is obtained by PDF integration: The absorption condition method assumes that once some representative points fall into the failure domain at a certain moment, they are absorbed and cannot be transferred to the next moment; the distribution probability of the qth representative point is divided into the failure part P q,f (t) and the effective part P q,s (t); the functional function is expressed as: In the formula, is the prescribed threshold; The failure event is reflected by the performance function Z≤0; the probability of the qth allocation of the failure part is expressed as: P q,f (t i )=0,z∈Ω Z,f ={z|g)θ q ,t i )≤0}(25) The probability is conserved during the uncertainty propagation from input to output, so the remaining distribution probability should be replaced by the effective part; the PDF of the random response in the safety domain is solved by the method in step S10, and the final first-pass reliability is expressed as:

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