Prediction Method for Dynamic Response of Suspension Bridges under Main and Aftershocks Driven by Hybrid Physical and Data

Abstract

Method for predicting dynamic response of suspension bridge under main shock and aftershock driven by physical and data hybrid, which relates to the technical field of earthquake response prediction of suspension bridges. Determine the acceleration time history sequence of main shock and aftershock ground motions; for an anchored steel truss suspension bridge, considering the uncertainty of key parameters, establish a finite element model; collect the bridge output parameters, establish a data set and divide it into a training set and a validation set; normalize the data; establish a physical constraint equation and solve the longitudinal displacement wave at the mid-span of the main girder under physical constraints; construct a machine learning model using the training set based on ANN and LSTM; establish a loss function for the machine learning model; verify the machine learning model; use the machine learning model to predict the longitudinal displacement wave at the mid-span of the main girder. It can calculate the displacement time history considering the uncertainty of key parameters and under the action of main shock and aftershock ground motions, and make up for the adverse effects caused by the traditional prediction method due to the inability to consider the uncertainty of key parameters and the aftershock stage.

Description

Technical Field

[0001] The present invention relates to the technical field of seismic response prediction of suspension bridges, and specifically to a method for predicting the dynamic response of suspension bridges under main and aftershocks driven by a hybrid of physics and data. Background Art

[0002] In the field of bridge seismic research, the application of deep learning is still in the exploratory stage. Especially in the prediction of bridge seismic responses, there are still significant gaps in research. However, more and more scholars have recognized the importance of machine learning techniques in the field of bridge earthquake resistance. For example, after studying the seismic analysis methods of small and medium-span bridges, Wang Kehai et al. pointed out that the seismic analysis of bridges based on artificial intelligence will be the future development direction. Scholars such as Jia Junfeng and Guo Wei further believe that combining emerging technologies such as machine learning and artificial intelligence with efficient algorithms is expected to provide more efficient and scientific solutions for aspects such as reasonable seismic motion parameter identification, seismic motion input selection, bridge earthquake damage prediction, seismic damage identification, and rapid post-earthquake assessment.

[0003] As one of the most widely used and effective algorithms in the current field of machine learning, the research and application of deep learning in the field of bridge earthquake resistance are of great significance. However, current research on bridge seismic response prediction mainly focuses on beam bridges. For suspension bridges, especially the response time history prediction of anchored steel truss suspension bridges, there is still a blank. At the same time, introducing physical constraint methods in machine learning can significantly improve the accuracy of the model. In addition, during the actual bridge construction process, bridge parameters often differ from the design drawings due to construction errors, and this parameter uncertainty also has a certain impact on seismic response prediction. Therefore, it is very necessary to incorporate the uncertainty of actual parameters into the analysis when predicting bridge seismic responses. Considering the above aspects, the present invention proposes a method for predicting the dynamic response of suspension bridges under main and aftershocks driven by a hybrid of physics and data. Summary of the Invention

[0004] To solve the deficiencies in the background art, the present invention provides a method for predicting the dynamic response of suspension bridges under main and aftershocks driven by a hybrid of physics and data. Aiming at the problem of predicting the seismic response time history of anchored steel truss suspension bridges, it can calculate the displacement time history considering the uncertainty of key parameters and under the action of main and aftershock seismic motions, and make up for the adverse effects caused by the traditional prediction method's inability to consider the uncertainty of key parameters and the aftershock stage.

[0005] To achieve the above object, the present invention adopts the following technical solutions: A method for predicting the dynamic response of suspension bridges under main and aftershocks driven by a hybrid of physics and data, including the following steps:

[0006] Step 1: Determine the main and aftershock seismic motions

[0007] The main - aftershock ground motion after induction consists of three stages, including the main - shock stage, the gap stage, and the aftershock stage;

[0008] (1) Determination of the induced main - shock stage: Select n main - shock ground motions, and their peak ground accelerations should conform to a normal distribution. The peak ground acceleration of the induced main - shock stage is denoted as PGA mso ;

[0009] (2) Set the duration of the induced gap stage to 30 s;

[0010] (3) Determination of the induced aftershock stage: Calculate the peak ground accelerations of the aftershock ground motions corresponding to each main - shock ground motion, which should conform to a normal distribution. The peak ground acceleration of the induced aftershock stage is denoted as PGA zoo ;

[0011] According to the calculated PGA mso and PGA zoo Determine the acceleration time - history of the main - aftershock ground motion. Denote the acceleration time - history of the main - shock stage as Denote the gap stage as Denote the acceleration time - history of the aftershock stage as Then the analytical formula of the acceleration time - history sequence of the main - aftershock ground motion is:

[0012]

[0013] Step 2: Establish a finite - element model

[0014] For the cable - stayed suspension bridge with an earth - anchored steel truss, determine its eight key design parameters, including the main - beam span l span , the initial strain ε0 of the main cable, the diameter d of the main cable d1 , the diameter d of the suspender d2 , the height h of the bridge tower tower , the concrete strength E c , the steel strength E of the suspender sd and the steel strength E of the main beam sz ;

[0015] Considering the uncertainty of the key parameters in the actual situation, according to the requirements, add and subtract an error value to each key parameter respectively to form the corresponding allowable range of the key parameter, and divide the allowable range of each key parameter into m equal parts. Randomly select m sampling points from each partition of the allowable range of the key parameter, and then combine them to obtain the sampling matrix X as follows:

[0016]

[0017] Sampling is carried out in the sampling matrix X. One point is randomly sampled from each row, and a corresponding acceleration time history sequence of a main-aftershock ground motion is selected to form m groups of sample pairs, constructing a sample set A of the main-aftershock ground motion - earth-anchored steel truss suspension bridge structure η It is expressed as follows:

[0018]

[0019] Use finite element software to establish a finite element model, for each row of the sample pairs in the sample set A η a finite element model is established, and a total of m finite element models are established;

[0020] Step three: Collect bridge output parameters

[0021] In the finite element software, output the longitudinal displacement waves Δδ1,..., Δδ at the mid-span of the main girder corresponding to each finite element model j and establish a data set T of the corresponding input and output for each finite element model o It is expressed as follows:

[0022]

[0023] Divide the data set T o into a training set T o1 and a validation set T o2 according to 7:3, which is expressed as follows:

[0024]

[0025] Step four: Process the data set

[0026] Normalize the data in the data set T o so that it is in the range of 0 to 1, obtaining the normalized training set and validation set;

[0027] Step five: Establish physical constraint equations

[0028] Use physical constraint equations to construct the relationship between the acceleration time history sequence of the main-aftershock ground motion and the longitudinal displacement wave Δδ at the mid-span of the main girder under physical constraints p = Δδ p,1 ,..., Δδ p,j The relationship is expressed as follows:

[0029]

[0030] In the formula, M represents the mass matrix of the structure, C represents the damping matrix, and K represents the stiffness matrix;

[0031] Under the initial conditions Δδ p = 0, In the case of, the longitudinal displacement wave Δδ at the mid-span of the main girder under the physical constraints of each finite element model in the training set is solved by using the gradient recurrence formula p , which is expressed as follows:

[0032]

[0033] In the formula, Δδ p,t represents the displacement of the node at time t, t = 1,..., j, ζ represents the damping ratio, ω represents the fundamental frequency of the structure, f p represents the system mass restoring force, and Δt represents the time interval; t

[0034] Step Six: Build a machine learning model

[0035] Any row of input parameters in the training set after normalization processing is denoted as y η , and the corresponding output parameters Δδ1,..., Δδ j are denoted as c η = c1,..., c j . The transfer function s is determined by modeling through the ANN artificial neural network, which is expressed as follows:

[0036] c η = s(y η W i + b i )

[0037]

[0038] In the formula, W i and b i represent the weight and bias term of the non-linear mapping respectively;

[0039] The LSTM long short-term memory model is modeled as a machine learning model and consists of an input gate, a forget gate, and an output gate, which is expressed as:

[0040] Input gate: r η = s(W t y η + U t y η-1 + b t )

[0041] Forget gate: h η = s(W h y η + U h y η-1 + b h )

[0042] f η = tanh(W​f y η +U f y η-1 +b f )

[0043] o η =s(W o y η +U o y η-1 +b o )

[0044]

[0045] Output gate: c η =h η *c η-1 +r η *f η

[0046] h η =o η *tanh(c η )

[0047] Wherein, W t , W h , W f , W o represent the corresponding weights, U t , U h , U f , U o represent the corresponding hidden weights, b t , b h , b f , b o represent the corresponding bias terms;

[0048] Step 7: Establish the loss function of the machine learning model

[0049] Denote the longitudinal displacement wave at the mid-span of the main girder predicted by the machine learning model as χ η =χ1,...,χ j , and use the loss function to quantify the error between the predicted value and the true value. The loss function of the physical constraint is denoted as L p , and the loss function of the data constraint is denoted as L d . The mean square error is adopted as the objective function, and the total function of the constraint is denoted as L, which is expressed as follows:

[0050]

[0051] L = αL p +βL d

[0052] α + β = 1

[0053] In the formula, λ p and λ d respectively represent the overfitting adjustment values introduced by the loss functions of physical constraints and data constraints, and ω p and ω d respectively represent the optimization parameters of the loss functions of physical constraints and data constraints;

[0054] Step Eight: Verify the machine learning model

[0055] Import the normalized validation set into the machine learning model for verification, and evaluate the prediction accuracy of the machine learning model through evaluation metrics. If the result of the predicted value meets the required accuracy, it is used as the final machine learning model. If not, return to Step Six to retrain the machine learning model until the requirements are met;

[0056] Step Nine: Use the machine learning model for prediction

[0057] By inputting the acceleration time history sequence of the selected new main-aftershock ground motion into the final machine learning model, the predicted longitudinal displacement wave at the mid-span of the main girder can be output.

[0058] Furthermore, in the said Step One, the selection rule of the main shock ground motion is as follows:

[0059] ① The magnitude of the main shock ground motion is greater than 6.5;

[0060] ② The peak acceleration of each main shock ground motion record is greater than 0.2g;

[0061] ③ The peak velocity of each main shock ground motion record is greater than 15 cm / s;

[0062] ④ The focal mechanism of the earthquake is reverse fault or strike-slip fault;

[0063] ⑤ The epicentral distance of the main shock ground motion is not less than 10 km;

[0064] ⑥ The effective period of the seismic wave of the main shock ground motion is not less than 4 s.

[0065] Furthermore, in the said Step Two, the allowable range of the key parameters randomly extracts m sampling points and the error values corresponding to each key parameter are expressed as follows:

[0066] L span =(l span,1 ,...,l span,m ) T ,l span -2m ≤ l span,1 ,...,l span,m ≤ l span +2m

[0067] E0 = (ε 0,1 ,..., ε 0,m ) T , ε0 - 0.0001 ≤ ε 0,1 ,..., ε 0,m ≤ ε0 + 0.0001

[0068] D d1 = (d d1,1 ,..., d d1,m ) T , d d1 - 5mm ≤ d d1,1 ,..., d d1,m ≤ d d1 + 5mm

[0069] D d2 = (d d2,1 ,..., d d2,m ) T , d d2 - 2mm ≤ d d2,1 ,..., d d2,m ≤ d d2 + 2mm

[0070] H tower = (h tower，1 ,..., h tower，m ) T , h tower - 1m ≤ h tower，1 ,..., h tower，m ≤ h tower + 1m

[0071]

[0072] In the formula, L span , E0, D d1 , D d2 , H tower , respectively represent the vectors of the sampling point combinations extracted from the allowable ranges of the corresponding key parameters.

[0073] Furthermore, in the sixth step, the AM attention mechanism is adopted to improve the accuracy of the machine learning model, expressed as follows:

[0074] e η = vtanh(W e h η + b e )

[0075]

[0076] In the formula, eη denotes the attention scoring function, v denotes the weight vector of the attention scoring function, W e denotes the weight matrix of the attention scoring function, b e denotes the bias term of the attention scoring function, γ η attention weight, s η denotes the output layer of the attention mechanism.

[0077] Furthermore, in the eighth step, the root mean square error RMSE, mean absolute error MAE or R 2 evaluation index is adopted.

[0078] Compared with the prior art, the beneficial effects of the present invention are as follows: By introducing physical constraint equations and uncertainty modeling of key parameters, the present invention significantly improves the accuracy and reliability of seismic response prediction. At the same time, this method can consider the complex effects of key parameter uncertainty and main-aftershock ground motion on the response of the cable-stayed steel truss suspension bridge, overcoming the deficiencies of traditional prediction methods in dealing with key parameter uncertainty and the aftershock stage. It not only realizes the accurate calculation of the displacement time history of the cable-stayed steel truss suspension bridge under main-aftershock ground motion, but also effectively improves the generalization ability and applicability of the prediction results, providing more scientific and efficient technical support for the seismic performance evaluation and optimal design of suspension bridges. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] Figure 1 is the flow chart of the method of the present invention;

[0080] Figure 2 is the acceleration time history diagram of a main-aftershock ground motion listed in the embodiment. DETAILED DESCRIPTION OF THE INVENTION

[0081] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0082] As Figure 1 shown, the method for predicting the dynamic response of a suspension bridge under main-aftershock with physical and data hybrid driving includes the following steps:

[0083] Step 1: Determine the main-aftershock ground motion

[0084] The induced main-aftershock ground motion consists of three stages, including the main shock stage, the intermission stage and the aftershock stage.

[0085] (1) The method for determining the induced main shock stage is as follows:

[0086] Select n main shock ground motions that meet the site conditions, where n ≥ 200, and the selection rules are as follows:

[0087] ① The magnitude of the main shock ground motion is greater than 6.5;

[0088] ② The peak acceleration of each main shock ground motion record is greater than 0.2g;

[0089] ③ The peak velocity of each main shock ground motion record is greater than 15 cm / s;

[0090] ④ The focal mechanism of the earthquake is thrust fault or strike-slip fault;

[0091] ⑤ The epicentral distance of the main shock ground motion is not less than 10 km;

[0092] ⑥ The effective period of the seismic wave of the main shock ground motion is not less than 4 s;

[0093] The peak accelerations of the selected n main shock ground motions are denoted as PGA ms , which should conform to a normal distribution. For these n PGAs ms are summarized, and the peak acceleration in the main shock stage after summarization is denoted as PGA mso , which is determined by the following formula:

[0094]

[0095] In the formula, σ1 is the standard deviation of PGA ms , μ1 is the expected value of PGA ms , and a1 is the acceleration of the main shock ground motion;

[0096] (2) Set a duration of 30 s for the interval stage after summarization;

[0097] (3) The determination method for the aftershock stage after summarization is as follows:

[0098] The peak accelerations of the aftershock ground motions corresponding to each selected main shock ground motion are denoted as PGA zo , which is determined by the following formula:

[0099]

[0100] In the formula, m s is the magnitude of the main shock ground motion, δ M is the ratio of the magnitude of the aftershock ground motion to the magnitude of the main shock ground motion, v0 is the reference shear wave velocity, with a value of 760 m / s, v s-30 is the average shear wave velocity of the selected main shock ground motion at a depth of 30 m, and d sis the distance to the main shock fault, and β1 to β4 are the regression coefficients after fitting, obtained by fitting the main - aftershock ground motion sequences from the PEER database. β1 = 0.250, β2 = 3.986, β3 = 0.206, β4 = 0.012;

[0101] The calculated PGA zo should conform to the normal distribution. For these n PGAs zo conduct induction, and the peak acceleration in the aftershock stage after induction is denoted as PGA zoo , which is determined by the following formula:

[0102]

[0103] In the formula, σ2 is the standard deviation of PGA zo , μ2 is the expected value of PGA zo , and a2 is the acceleration of the aftershock ground motion.

[0104] According to the calculated PGA mso and PGA zoo to determine the acceleration time - history of the main - aftershock ground motion, denote the acceleration time - history in the main shock stage as denote the gap stage as denote the acceleration time - history in the aftershock stage as Then the analytical formula of the acceleration time - history sequence of the main - aftershock ground motion is:

[0105]

[0106] Step 2: Establish a finite - element model

[0107] For the earth - anchored steel - truss suspension bridge, determine its eight key design parameters, including the main - girder span l span , the initial strain ε0 of the main cable, the main - cable diameter d d1 , the diameter d of the suspender d2 , the height h of the bridge tower tower , the concrete strength E c , the steel strength E of the suspender sd , the steel strength E of the main girder sz .

[0108] Considering the uncertainty of the key parameters in the actual situation, according to the requirements, add and subtract an error value to each key parameter respectively to form the allowable range of the corresponding key parameter, and divide the allowable range of each key parameter into m equal parts. Then the probability interval of each partition is 1 / m, so that the eight - dimensional space is divided into m 8 partitions. Randomly select a sampling point from each partition in the allowable range of each key parameter, that is, m sampling points are selected from the allowable range of each key parameter, and a total of 8m sampling points are selected, which are expressed as follows:

[0109] L span =(l span,1 ,...,l span,m ) T ,l span -2m ≤ l span,1 ,...,l span,m ≤ l span +2m

[0110] E0=(ε 0,1 ,...,ε 0,m ) T ,ε0 - 0.0001 ≤ ε 0,1 ,...,ε 0,m ≤ ε0 + 0.0001

[0111] D d1 =(d d1,1 ,...,d d1,m ) T ,d d1 -5mm ≤ d d1,1 ,...,d d1,m ≤ d d1 +5mm

[0112] D d2 =(d d2,1 ,...,d d2,m ) T ,d d2 -2mm ≤ d d2,1 ,...,d d2,m ≤ d d2 +2mm

[0113] H tower =(h tower，1 ,...,h tower，m ) T ,h tower -1m ≤ h tower，1 ,...,h tower，m ≤ h tower +1m

[0114]

[0115] Wherein, L span , E0, D d1 , D d2 , H tower , respectively represent the vectors of the combinations of sampling points extracted from the allowable ranges of the corresponding key parameters.

[0116] Combining the vectors of the combinations of sampling points extracted from the allowable ranges of the eight key parameters, the sampling matrix X is represented as follows:

[0117]

[0118] Sampling is carried out in the sampling matrix X. One point is randomly sampled from each row, and a corresponding acceleration time history sequence of a mainshock-aftershock ground motion is selected to form m groups of sample pairs, constructing a sample set A of the mainshock-aftershock ground motion - earth-anchored steel truss suspension bridge structure η It is expressed as follows:

[0119]

[0120] Use finite element software to establish a finite element model, for the sample set A η A finite element model is established for each row of sample pairs in it, and a total of m finite element models are established;

[0121] Step 3: Collect bridge output parameters

[0122] In the finite element software, output the longitudinal displacement waves Δδ1,..., Δδ at the mid-span of the main girder corresponding to each finite element model j and establish a data set T of the corresponding input and output for each finite element model o It is expressed as follows:

[0123]

[0124] Divide the data set T o into a training set T o1 and a validation set T o2 preferably in a 7:3 ratio, which is expressed as follows:

[0125]

[0126] Step 4: Process the data set

[0127] Normalize the data in the data set T o so that it is in the range of 0 to 1, obtaining the normalized training set and validation set. The normalization process is expressed as follows:

[0128]

[0129] In the formula, x represents the data in the data set, x min and x max represent the minimum and maximum values of each parameter respectively, and x n represents the processed data;

[0130] Step 5: Establish physical constraint equations

[0131] Use physical constraint equations to construct the acceleration time history sequence of the mainshock-aftershock ground motion The relationship between the longitudinal displacement wave Δδ at the mid-span of the main girder under physical constraints p =Δδ p,1 ,...,Δδ p,j is as follows:

[0132]

[0133] In the formula, M represents the mass matrix of the structure, C represents the damping matrix, and K represents the stiffness matrix.

[0134] Under the initial conditions Δδ p =0, , the equilibrium condition can be written in the form of a state-space vector. The longitudinal displacement wave Δδ at the mid-span of the main girder under physical constraints of each finite element model in the training set is solved using the gradient recurrence formula, which is as follows: p is as follows:

[0135]

[0136] In the formula, Δδ p,t represents the displacement of the node at time t of Δδ p , where t = 1,..., j, ζ represents the damping ratio, ω represents the fundamental frequency of the structure, f t represents the system mass restoring force, and Δt represents the time interval;

[0137] Step 6: Construct a machine learning model

[0138] Machine learning is performed using ANN-LSTM. Among them, the artificial neural network ANN is used to extract the relationship between the input and output from the data set T o , and the long short-term memory model LSTM is used to learn the long-term changes in the relationship between the input and output.

[0139] Any row of input parameters in the training set after normalization processing is denoted as y η , and the corresponding output parameters Δδ1,..., Δδ j are denoted as c η =c1,..., c j . The transfer function s is determined through modeling using the artificial neural network ANN, which is as follows:

[0140] c η =s(y η W i +b i )

[0141]

[0142] In the formula, W i and b iThey represent the weights and bias terms of the non-linear mapping respectively.

[0143] As a machine learning model, the LSTM long short-term memory model is composed of an input gate, a forget gate, and an output gate, and is expressed as:

[0144] Input gate: r η = s(W t y η + U t y η-1 + b t )

[0145] Forget gate: h η = s(W h y η + U h y η-1 + b h )

[0146] f η = tanh(W f y η + U f y η-1 + b f )

[0147] o η = s(W o y η + U o y η-1 + b o )

[0148]

[0149] Output gate: c η = h η * c η-1 + r η * f η

[0150] h η = o η * tanh(c η )

[0151] In the formula, W t , W h , W f , W o represent the corresponding weights, U t , U h , U f , U o represent the corresponding hidden weights, b t , b h , b f , b oRepresents the corresponding bias term.

[0152] The AM attention mechanism is adopted to improve the accuracy of the machine learning model, which is expressed as follows:

[0153] e η = vtanh(W e h η + b e )

[0154]

[0155] In the formula, e η represents the attention scoring function, v represents the weight vector of the attention scoring function, W e represents the weight matrix of the attention scoring function, b e represents the bias term of the attention scoring function, γ η attention weight, s η represents the output layer of the attention mechanism;

[0156] Step Seven: Establish the loss function of the machine learning model

[0157] Denote the longitudinal displacement wave at the mid-span of the main beam predicted by the machine learning model as χ η = χ1,..., χ j , and use the loss function to quantify the error between the predicted value and the true value. The loss function of the physical constraint is denoted as L p , and the loss function of the data constraint is denoted as L d . The mean square error is adopted as the objective function, and the total function of the constraint is denoted as L, which is expressed as follows:

[0158]

[0159] L = αL p + βL d

[0160] α + β = 1

[0161] In the formula, λ p and λ d respectively represent the overfitting adjustment values introduced by the loss functions of the physical constraint and the data constraint, ω p and ω d respectively represent the optimization parameters of the loss functions of the physical constraint and the data constraint;

[0162] Step Eight: Verify the machine learning model

[0163] Import the normalized validation set into the machine learning model for verification. Through the root mean square error RMSE, mean absolute error MAE or R 2Use evaluation indicators to evaluate the prediction accuracy of the machine learning model. If the result of the predicted value meets the required accuracy, it is used as the final machine learning model. If not, return to step six to retrain the machine learning model until the requirements are met;

[0164] Step Nine: Use the machine learning model for prediction

[0165] By inputting the selected new main-aftershock ground motion acceleration time history sequence into the final machine learning model, the predicted longitudinal displacement wave at the mid-span of the main girder can be output.

[0166] Embodiment

[0167] Taking the Xinghai Bay Cross-sea Bridge in Dalian as an example, predict the displacement time history data under the action of main-aftershock ground motion through the method of the present invention.

[0168] S1. Determine the main-aftershock ground motion

[0169] (1) Determination of the main shock stage after induction

[0170] Select 200 main shock ground motions that meet the site conditions. Taking one of the selected main shock ground motions (Duzce, Turkey) as an example:

[0171] ① Magnitude m of the main shock ground motion s = 7.14 magnitude;

[0172] ② Peak ground acceleration PGA of each main shock ground motion record = 0.35g;

[0173] ③ Peak ground velocity PGV of each main shock ground motion record = 59.99 cm / s;

[0174] ④ The focal mechanism of the earthquake is reverse fault or strike-slip fault;

[0175] ⑤ Epicentral distance x of the main shock ground motion ee = 20 km;

[0176] ⑥ Effective period T of the seismic wave of the main shock ground motion e = 4.86 s;

[0177] For the peak ground acceleration PGA of these 200 main shock ground motions ms Carry out induction, and the peak ground acceleration PGA in the main shock stage after induction mso is as follows:

[0178]

[0179] (2) Set the duration of the gap stage after induction to 30 s;

[0180] (3) Determination of the aftershock stage after induction

[0181] The peak ground acceleration PGA of the aftershock ground motion corresponding to each selected main shock ground motion zo is as follows:

[0182]

[0183] The calculated PGA zo should conform to the normal distribution. For these 200 PGAs zo perform induction. The peak ground acceleration PGA in the aftershock stage after induction zoo is as follows:

[0184]

[0185] Based on the calculated PGA mso and PGA zoo determine the acceleration time history sequence of the main - aftershock ground motion. Taking one of them as an example, the acceleration time history sequence of this main - aftershock ground motion is combined with Figure 2 as shown.

[0186] S2. Establish a finite - element model

[0187] Considering the uncertainty of key parameters in the actual situation, for each key parameter, add and subtract an error value according to the requirements to form the corresponding allowable range of the key parameter, and divide the allowable range of each key parameter into 10 equal parts. Then the probability interval of each partition is 1 / 10, and the eight - dimensional space is divided into 10 8 partitions. Randomly select a sampling point from each partition in the allowable range of each key parameter, that is, 10 sampling points are selected from the allowable range of each key parameter, and a total of 80 sampling points are selected. The designed key parameters and the parameters extracted considering the uncertainty of key parameters in the actual situation are shown in Table 1:

[0188] Table 1 Designed key parameters and parameters extracted considering the uncertainty of key parameters in the actual situation

[0189]

[0190]

[0191] Combine the vectors of the sampling points extracted from the allowable ranges of the eight key parameters to obtain the sampling matrix X, which is shown as follows:

[0192]

[0193] Perform sampling in the sampling matrix X. Randomly sample one point from each row, and correspondingly select an acceleration time history sequence of a main - aftershock ground motion to form 10 groups of sample pairs, and construct the sample set A of the main - aftershock ground motion - cable - stayed steel truss suspension bridge structureη It is shown as follows:

[0194]

[0195] Use finite element software to establish a finite element model, and for each sample pair in sample set A η establish a finite element model, and a total of 10 finite element models are established.

[0196] S3. Collect bridge output parameters

[0197] Output the longitudinal displacement waves Δδ1,..., Δδ at the mid-span of the main girder corresponding to each finite element model in the finite element software j and establish a data set T for the input and output corresponding to each finite element model o The data set T o Some of the data in it are shown in Table 2:

[0198] Table 2 Collected output parameters (partial)

[0199]

[0200]

[0201] S4. Process the data set

[0202] Normalize the data in the data set T o so that it is in the range of 0 to 1, and obtain the normalized training set and validation set. Some of the data are shown in Table 3:

[0203] Table 3 Normalized data set (partial)

[0204]

[0205]

[0206] S5. Establish physical constraint equations

[0207] Call the TensorFlow open source library to build a spatio-temporal correlation mechanism model, and select an optimizer and a loss function. The establishment of the physical constraint equation is completed in the compute_physics_constraint function, and the longitudinal displacement wave eta_p at the mid-span of the main girder under physical constraints is calculated.

[0208] S6. Build a machine learning model

[0209] Create an ANN layer: This part constructs an ANN network through the ANN_model method to extract features from the input data.

[0210] Create the input layer: The input layer receives data through a Conv1D layer, where the input shape is (4838, 1), meaning the data dimension for each time step is 4838 and there is only 1 feature for each time step.

[0211] Create the LSTM layer: It is implemented through the LSTM layer in TensorFlow. Since features have been extracted through the ANN layer previously, matrix operations (tf.matmul) can be used in the LSTM layer to calculate eta_t and eta_tt.

[0212] Create the output layer: The role of the output layer is to predict the target variables eta_pred and eta_tt_pred based on the calculation results of the model. Through the self.net_structure() function, the model can calculate these outputs and finally obtain the predicted values through the network structure.

[0213] Specify the number of training epochs as 10,000 iterations. Train the model through the train method, set the batch size to 32 for batch gradient update. Use the Adam optimizer with a learning rate set to 1e-3 and train with the loss function combined with physical constraints.

[0214] Implement the AM attention mechanism through the Multi Head Attention layer in TensorFlow.

[0215] S7. Establish the loss function of the machine learning model

[0216] The loss function is established through the calculate_loss function. The data loss calculates the mean square error between eta_tt and eta_tt_pred, and the physical constraint loss calls the compute_physics_constraint function to establish the mean square error between eta_p and eta_tt.

[0217] S8. Validate the machine learning model

[0218] The partial validation results are shown in Table 4:

[0219] Table 4 Prediction Results of Validation Set Data (Partial)

[0220]

[0221]

[0222] The calculated root mean square error RMSE = 0.017, mean absolute error MAE = 0.014, R 2 = 0.989. S9. Make predictions using the machine learning model

[0223] Some of the predicted results are shown in Table 5:

[0224] Table 5 Results Predicted by the Model (Partial)

[0225]

[0226]

[0227] For those skilled in the art, it is obvious that the present invention is not limited to the details of the above-described exemplary embodiments, and the present invention can be implemented in other forms without departing from the spirit or basic characteristics of the present invention. Therefore, from any point of view, the embodiments should be regarded as exemplary and non-limiting. The scope of the present invention is defined by the appended claims rather than the above description. Therefore, all changes falling within the meaning and scope of the equivalent conditions of the claims are intended to be encompassed within the present invention. Any reference signs in the claims should not be construed as limiting the claims involved.

[0228] In addition, it should be understood that although this specification is described in terms of embodiments, not every embodiment only contains an independent technical solution. This narrative manner of the specification is only for clarity. Those skilled in the art should regard the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A method for predicting the dynamic response of a suspension bridge under main and aftershocks driven by a hybrid of physics and data, characterized in that: It includes the following steps: Step 1: Determine the mainshock-aftershock ground motion The summarized mainshock-aftershock ground motion consists of three stages, including the main shock stage, the intermission stage, and the aftershock stage; (1) Determination of the main shock stage after induction: Select n main shock ground motions, the peak acceleration of which should conform to a normal distribution. The peak acceleration of the main shock stage after induction is denoted as PGA mso ; (2) The duration of the summarized intermission stage is set to 30 s; (3) Determination of the aftershock stage after induction: Calculate the peak acceleration of the aftershock ground motion corresponding to each main shock ground motion, which should conform to a normal distribution. The peak acceleration in the aftershock stage after induction is denoted as PGA zoo ; According to the calculated PGA mso and PGA zoo Determine the acceleration time history of the mainshock-aftershock ground motion. Denote the acceleration time history of the main shock stage as Denote the gap stage as Denote the acceleration time history of the aftershock stage as Then the analytical formula of the acceleration time history sequence of the mainshock-aftershock ground motion is: Step 2: Establish a finite element model For an earth-anchored steel truss suspension bridge, eight key parameters for its design are determined, including the main girder span l span , the initial strain ε0 of the main cable, the diameter d of the main cable d1 , the diameter d of the suspender d2 , the height h of the bridge tower tower , the concrete strength E c , the steel strength E of the suspender sd and the steel strength E of the main girder sz ; Considering the uncertainty of key parameters in the actual situation, an error value is added to and subtracted from each key parameter according to the requirements to form the corresponding allowable range of key parameters. The allowable range of each key parameter is equally divided into m parts, and m sampling points are randomly selected from each partition of the allowable range of key parameters. Then, a sampling matrix X is combined and expressed as follows: Sampling is carried out in the sampling matrix X. One point is randomly sampled from each row, and a corresponding acceleration time history sequence of a main-aftershock ground motion is selected, forming m groups of sample pairs to construct a sample set A of the main-aftershock ground motion - earth-anchored steel truss suspension bridge structure η It is expressed as follows: Use finite element software to establish a finite element model, sample set A η Establish a finite element model for each sample pair in each row, and a total of m finite element models are established; Step 3: Collect bridge output parameters Output the longitudinal displacement waves Δδ1,..., Δδ at the mid-span of the main girder corresponding to each finite element model in the finite element software j and establish the input-output data set T corresponding to each finite element model o which is expressed as follows: Divide the dataset T o into a training set T o1 and a validation set T o2 , as shown below: Step 4: Process the data set Normalize the data in dataset T o so that it is within the range of 0 to 1, obtaining the normalized training set and validation set; Step 5: Establish a physical constraint equation Constructing the acceleration time history sequence of main shock and aftershock ground motions using physical constraint equations and the longitudinal displacement wave Δδ at the mid-span of the main girder under physical constraints p = Δδ p,1 ,..., Δδ p,j The relationship is expressed as follows: In the formula, [M] represents the mass matrix of the structure, [C] represents the damping matrix, and [K] represents the stiffness matrix; Under the initial condition Δδ p = 0, the longitudinal displacement wave Δδ p at the mid-span of the main girder under the physical constraints of each finite element model in the training set is solved by using the gradient recurrence formula, which is expressed as follows: where, Δδ p,t represents the displacement of the node at time t, t = 1, ..., j, ζ represents the damping ratio, ω represents the fundamental frequency of the structure, f p represents the restoring force of the system mass, and Δt represents the time interval; t ​ Step 6: Construct a machine learning model Any row of input parameters in the normalized training set is denoted as y η , and the corresponding output parameters Δδ1,..., Δδ j are denoted as c η = c1,..., c j , and the transfer function s is determined by modeling through the ANN artificial neural network, which is expressed as follows: c η = s(y η W i + b i ) Wherein, W i and b i respectively represent the weight value and the bias term of the non-linear mapping; The LSTM long short-term memory model is modeled as a machine learning model, which consists of an input gate, a forget gate, and an output gate, and is expressed as: Input gate: r η = s(W t y η + U t y η-1 + b t ) Forgotten gate: h η = s(W h y η + U h y η-1 + b h ) f η = tanh(W f y η + U f y η-1 + b f ) o η = s(W o y η + U o y η-1 + b o ) Output gate: c η = h η * c η-1 + r η * f η h η = o η *tanh(c η ) Where, W t , W h , W f , W o represent the corresponding weights, U t , U h , U f , U o represent the corresponding hidden weights, b t , b h , b f , b o represent the corresponding bias terms; Step 7: Establish a loss function of the machine learning model Denote the longitudinal displacement wave at the mid-span of the main girder predicted by the machine learning model as χ η = χ1,..., χ j , use the loss function to quantify the error between the predicted value and the true value. The loss function of physical constraints is denoted as L p , and the loss function of data constraints is denoted as L d , adopt the mean square error as the objective function, and the total function of constraints is denoted as L, which is expressed as follows: L = αL p + βL d α+β＝1 where λ p and λ d respectively represent the overfitting adjustment values introduced by the loss functions of physical constraints and data constraints, and ω p and ω d respectively represent the optimization parameters of the loss functions of physical constraints and data constraints; Step 8: Verify the machine learning model The normalized validation set is imported into the machine learning model for verification, and the prediction accuracy of the machine learning model is evaluated through evaluation indicators. If the result of the predicted value meets the required accuracy, it is used as the final machine learning model. If not, return to Step 6 to retrain the machine learning model until the requirements are met; Step 9: Use the machine learning model for prediction By inputting the acceleration time history sequence of the selected new mainshock-aftershock ground motion into the final machine learning model, the predicted longitudinal displacement wave at the mid-span of the main girder can be output.

2. The method for predicting the dynamic response of a main and aftershock suspension bridge driven by a physical and data hybrid according to claim 1, wherein: In the above Step 1, the selection rules for the main shock ground motion are as follows: ① The magnitude of the main shock ground motion is greater than 6.5; ② The peak acceleration of each main shock ground motion record is greater than 0.2g; ③ The peak velocity of each main shock ground motion record is greater than 15 cm / s; ④ The focal mechanism of the earthquake is reverse fault or strike-slip fault; ⑤ The epicentral distance of the main shock ground motion is not less than 10 km; ⑥ The effective period of the seismic wave of the main shock ground motion is not less than 4 s.

3. The method for predicting the dynamic response of a main and aftershock suspension bridge driven by a physical and data hybrid according to claim 1, wherein: In the above Step 2, m sampling points randomly selected from the allowable range of key parameters and the corresponding error values of each key parameter are expressed as follows: L span = (l span,1 ,..., l span,m ) T , l span -2m ≤ l span,1 ,..., l span,m ≤ l span +2m E0 = (ε 0,1 ,..., ε 0,m ) T , ε0 - 0.0001 ≤ ε 0,1 ,..., ε 0,m ≤ ε0 + 0.0001 D d1 = (d d1,1 ,..., d d1,m ) T , d d1 -5 mm ≤ d d1,1 ,..., d d1,m ≤ d d1 +5 mm D d2 =(d d2,1 ,...,d d2,m ) T ,d d2 -2mm ≤ d d2,1 ,...,d d2,m ≤ d d2 +2mm H tower =(h tower，1 ,...,h tower，m ) T ,h tower -1m ≤ h tower，1 ,...,h tower，m ≤ h tower +1m E c -1 GPa ≤ E c,1 ,..., E c,m ≤ E c +1 GPa E sd -2 GPa ≤ E sd,1 ,..., E sd,m ≤ E sd +2 GPa E sz -2 GPa ≤ E sz,1 ,..., E sz,m ≤ E sz +2 GPa where L span , E0, D d1 , D d2 , H tower , respectively represent vectors of combinations of sampling points extracted from the allowable ranges of corresponding key parameters.

4. The method for predicting the dynamic response of a main and aftershock suspension bridge driven by a physical and data hybrid according to claim 1, wherein: In the above Step 6, the AM attention mechanism is used to improve the accuracy of the machine learning model, and it is expressed as follows: e η = vtanh(W e h η + b e ) where, e η represents the attention scoring function, v represents the weight vector of the attention scoring function, W e represents the weight matrix of the attention scoring function, b e represents the bias term of the attention scoring function, γ η is the attention weight, s η represents the output layer of the attention mechanism.

5. The method for predicting the dynamic response of a main and aftershock suspension bridge driven by a physical and data hybrid according to claim 1, wherein: In the eighth step, the root mean square error RMSE, the mean absolute error MAE or R 2 is used as the evaluation index.

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