Machine Learning-Based Seismic Fragility Analysis of Large-Scale Steel Buckling Restrained Brace Frames

: Steel frames equipped with buckling restrained braces (BRBs) have been increasingly applied in earthquake-prone areas given their excellent capacity for resisting lateral forces. Therefore, special attention has been paid to the seismic risk assessment (SRA) of such structures, e.g., seismic fragility analysis. Conventional approaches, e.g., nonlinear finite element simulation (NFES), are computationally inefficient for SRA analysis particularly for large-scale steel BRB frame structures. In this study, a machine learning (ML)-based seismic fragility analysis framework is established to effectively assess the risk to structures under seismic loading conditions. An optimal artificial neural network model can be trained using calculated damage and intensity measures, a technique which will be used to compute the fragility curves of a steel BRB frame instead of employing NFES. Numerical results show that a highly efficient instantaneous failure probability assessment can be made with the proposed framework for realistic large-scale building structures.


Introduction
Steel frames equipped with the buckling restrained braces (BRBs) are gaining increasing popularity and have wide applications in new structures [1][2][3] or for retrofitting existing ones [4] given their excellent lateral force resistance. Fragility analysis plays an essential role for the seismic risk assessment (SRA) of the steel BRB structures in earthquake-prone areas [5]. According to the performance-based structural design method, the damage limit state, D, of a structure under seismic excitations can be divided into several levels, e.g., negligible, light, moderate, and severe damage states [5]. The probability of failure can be expressed as P f = P [DM ≥ D], where DM indicates the damage measure. Considering that the intensity measure (IM) of a given ground motion may be an uncertain value, e.g., the peak ground acceleration (PGA) or spectral acceleration (Sa), the conditional probability of failure can be modified as [6], where P [DM ≥ D | IM = x] indicates the fragility that suggests the failure probability under the ground motion IM level x, and P [IM = x] expresses the annual occurrence probabilities of the specific level x of an earthquake motion.
The Monte Carlo Simulation (MCS) based on the nonlinear finite element simulation (NFES) [7] is one of the most accurate and popular techniques. However, the MCS is computationally prohibitive because of its tremendous time cost, especially for realistic large-scale structures. Consequently, some approximate regression methods, such as the least-squares [8] and empirical regression methods [9], are commonly used to compute the fragility curve by utilizing the limited data of structural responses. Among them, the empirical regression (denoted as Reg) method is slightly more popular and expresses the fragility curve in the form of two-parameter lognormal distribution functions [10].
Given the epistemic uncertainties of structures and the aleatory uncertainties of earthquakes, numerous repeated simulations may be necessary to perform structural reliability or fragility analysis using the MCS or the Reg method. Therefore, conducting instantaneous SRA of realistic large-scale structures is problematic for researchers and engineers.
In the last few years, machine learning (ML) has become a powerful tool that supplanted conventional finite element (FE) approaches for repeated computation. The artificial neural network (ANN), an ML approach, has various applications in the reliability analysis of civil structures [11][12][13][14][15][16][17][18][19][20][21][22][23]. ML may also be extensively used in the fragility analysis of structural SRAs [24][25][26][27][28][29][30][31][32]. The basic idea of ML involves several steps. First, the actual DM of a structure under seismic loadings is calculated according to the NFES, and appropriate intensity measures (IMs) are selected to characterize the ground motion. Second, an optimal ML (e.g., the ANN) model is trained using the IMs-DM data. Finally, the fragility curve of the structure can be predicted according to the ANN model. The main concerns focus on the selection of the IMs and predicted uncertainties of the ML (or ANN) model. For the IM selection, more than one IM is needed to represent the ground motion because the random earthquake process cannot be sufficiently characterized by any single seismic IM, such as the PGA, peak ground velocity (PGV), and SA. However, the ANN model will be overfitted if given too many selected IMs and will become complex if these selected IMs have substantial correlation with one another. A parametric study was conducted to select the suitable combination of IMs that leads to close predictions of the demand [28] but ignored the correlation among these selected IMs and that between the IM and DM. A wrapper approach based on the genetic algorithm can be employed to select the optimal set that maximizes ANN representation accuracy [27], for which the correlation between the selected IMs is considered. Another filter approach based on the semi-partial correlation coefficient is proposed to consider the independent correlation between the IM and DM [32]. For the predicted uncertainties of the ML (or ANN) model, the input and output randomness, as well as the effect of data size on the accuracy of the ANN model, are studied [27]. These uncertainties are investigated in relation to the fragility analysis of nuclear power plants [32].
In this paper, an ML-based seismic fragility analysis framework is proposed to instantaneous assess the failure probability of a realistic large-scale 48-story steel BRB frame under excitations. CMES, 2020, vol.125, no.2 757 The suggested framework is introduced in Section 2. The numerical results are presented and discussed in Section 3. Conclusions are provided in Section 4.

ML-Based Seismic Fragility Analysis Framework
As depicted in Fig. 1, the proposed ML-based seismic fragility analysis framework consists of three parts:  This section is organized as follows. Section 2.1 introduces the structure model and presents the input ground motions. Section 2.2 explains the selection of the IMs and the training and validation of the ANN model. Section 2.3 demonstrates the sampling methods of the IMs and the prediction of the fragility curves.

Structure Model
This study conducts fragility analysis of a 48-story steel BRB frame. The structure is designed according to the Chinese Code for Seismic Design of Buildings GB50011-2010 [33], which consists  The live load applied on each slab is 2.0 kN/m 2 , the self-weight dead loads on each slab and the roof are 3.5 kN/m 2 , and the roof has an additional dead load of 3.32 kN/m 2 . The frame is located in an earthquake-prone region with seismic fortification intensity 8 (0.2 g), site soil class II and design Group 1. The damping ratio of the structure under an 8-degree rare earthquake is taken as 0.02. The BRB frame structure is analyzed using an open source software platform, OpenSees [34,35]. During the seismic analysis of the frame under earthquake loads, each column or beam component is modeled by using an Euler-Bernoulli fiber element with five Gauss points, where the uniaxial behaviors of steel fiber is simulated by using a uniaxial Giuffre-Menegotto-Pinto (GMP) model [34,36]. Moreover, the BRB component is simulated by employing a truss element, for which the uniaxial stress-strain relationship is computed by utilizing the GMP model. The beam-column connections are assumed to be rigid and the mass is regarded as concentrated at the beam-column joints. The first five periods and frequencies of the steel BRB frame are given in Tab. 2.
In the transient analysis, a Newmark-β integration scheme is employed with the parameters γ = 0.5 and β = 0.25, and the Rayleigh damping is assumed to have a damping ratio of 0.02 that corresponds to the first and second frequencies (1.29 and 4.14 Hz) in the seismic action direction (i.e., N-S direction). A tolerance of 1.0 e-5 mm on the norm of the incremental displacement is employed as the convergence criterion in the nonlinear iterative computation. Period (s)

Input Ground Motions
To obtain the DM data used for training the ANN model, a 48-story steel BRB frame is analyzed under a suite of 60 Los Angeles ground motion records in the SAC steel research project [37]. Detailed information of all ground motions is presented in Tab. 3. The magnitude range is from 5.7 to 7.7 and the source-to-site distance of the selected record ranges from 1 km to 107 km. These 60 records are divided into three groups in terms of the probability of exceedance: the first, middle, and last 20 ground motion records for 50 years are respectively represented as 10%, 2%, and 50%.

ANN Regression Model Construction 2.2.1 IM Selection
Given that the input earthquake motion is a random process, establishing a clear and simple relationship between the input and output DMs is difficult. Consequently, some IMs that represent the earthquake motion must be selected to describe the relationship. Characterizing seismic motion generally requires more than one IM [39]. Tab. 4 lists seven commonly used IMs, i.e., the PGA, PGV, peak ground displacement (PGD), Sa, average spectral acceleration (ASA), cumulative absolute velocity (CAV ), arias intensity (Ia). Note that increasing the number of selected IMs will enhance the complexity of the ANN model and may even lead to a risk of overfitting. Therefore, selecting the most relevant IMs with the DM to train the ANN regression model is crucial.
As these IMs may be highly related to one another, conventional correlations are unsuitable for selecting the most relevant IMs with the DM. In this study, a filter approach on the basis of semi-partial correlation coefficients (SPCCs) is used to select the most relevant IMs [32], for which the independent correlation between any IM and DM can be accurately measured after controlling for (i.e., "partialling" out) the effects of the other six IMs. Tab. 5 and present the correlation coefficients and SPCCs between the logarithms of IMs and DM. IM 5 has the largest correlation coefficient (larger than 0.9) with the DM and a very small SPCC, whereas IM 3 (PGD) has the second largest correlation but the smallest SPCC (Tabs. 5 and 6). According to the correlation coefficients in Tab. 5, IM 2 , IM 3 , IM 4 , and IM 5 are the most relevant IMs with the DM, whereas IM 2 and IM 4 (Tab. 6) are the most relevant ones quantified by the SPCCs. Therefore, IM 2 and IM 4 will be adopted to train the ANN model in the following simulation.

Training and Validation of the ANN Model
Subsequently, the ANN model can be constructed by using the IMs-DM data which consists of an input layer, hidden layer, and output layer (Fig. 3).
The hidden layer contains n independent neurons (or nodes), and each node receives all input IMs from the input layer and computes the output, thereby yielding where IM i is the ith input variable, m is the number of IMs, y j is the output of the jth neuron in the hidden layer, w ij is the weight coefficient contributed by the IM i to the output y j , b j is the bias of the jth neuron in the hidden layer, and f is the activation function. The choice of the activation function plays an important role in the construction of the ANN. Activation functions occur as linear and nonlinear types. A nonlinear activation function with the sigmoid function as an example is expressed as T 0 |a (t)| 2 dt g: gravity acceleration and can be used to model the nonlinear behavior of the ANN, in which α is the sigmoid slope which is taken as 1 herein.
Finally, the DM calculated in the output layer can be expressed as where v j is the weight coefficient between the output and the jth neuron in the hidden layer and d is the bias in the output layer.
During the training of the ANN model, the discrepancy between the predicted DM and actual DM is used to measure the performance of the ANN. A mean squared error (MSE) can typically be adopted to quantify the difference and is defined as in which N is the number of testing data, DM i is the ith predicted value and DM i is the ith actual value.
Once the MSE is less than the pre-defined target, the training of the ANN model is terminated and an ANN regression model is built. Otherwise, the training continues and the control parameters, e.g., the weight coefficient and bias, are updated according to a training algorithm (such as the Levenberg-Marquardt training algorithm). Subsequently, a new output is computed to examine the convergence of the system.
To improve the training efficiency of the ANN model, all input and output variables must be normalized beforehand by mapping the original values to the dimensionless range [0,1] or [−1,−1]. As the fragility analysis is conducted in the logarithmic space, the following scaled equation should be used: where p and p indicate the scaled and unscaled values, respectively; andp and σ respectively denote the mean and logarithmic standard deviation.

ML-Based Seismic Fragility Predictions
In this section, seismic fragility analysis is performed according to ML. First, in view of the prediction uncertainty of the ANN model, an optimal regression model is chosen from many trained ones (Figs. 1e-1h). Subsequently, the input IMs should be determined, including the sample range and size. Finally, the DMs can be predicted using the optimal ANN regression model and will be used to calculate the fragility curve. The following sections discuss the sampling method of the selected IMs and the computation of the fragility curve.

Sampling Methods of Selected IMs
In ML-based seismic fragility analysis, the size of the IMs-DM sample data considerably affects the accuracy of the prediction. If the sample data is sufficiently large, then the failure probability of the structure under an earthquake can be accurately computed. However, the values of the IMs cannot be in an unlimited range [0, inf). To reasonably take these values, the statistical distributions of the selected IMs according to 60 earthquake waves can be used (Figs. 1d-1g). This paper adopted two sampling methods to ascertain the IMs using their statistical distributions: (1) truncated random distribution (TRD) and (2) truncated uniform distribution (TUD). In the former, the IMs are randomly taken in a specific range; whereas in the latter, the IMs are uniformly distributed in a given interval.

Computation of Fragility Curves
In this section, the MCS and Reg method are adopted to compute the fragility curves of the structure under an earthquake. In the ML-based MCS, all selected IMs for a given PGA level x can be determined using the above sampling methods and statistical distributions. Then, the DMs can be predicted using the ANN model, (Figs. 1g-1i). Finally, the probability of the structure reaching or exceeding the damage state D i can be computed as [15] where n is the number of ground motions; DM j is the DM value (i.e., maximum IDR) of the structure subjected to jth seismic motion; and 1 DM j ≥ D i is 1 when DM j ≥ D i and 0 otherwise.
In the Reg method, the fragility of the structure or component can be expressed in a form of lognormal cumulative distributions, and the conditional probability of failure for the specific damage state D i can be written as [10].
in which Φ is the standard normal cumulative distribution function (CDF), α i is the median value of the PGA at which the DM reaches or exceeds the threshold D i , and β i is the corresponding logarithmic standard deviation.
Given the limited sample data, the structure may not reach the damage state D i when the PGA takes the maximum value in some samples. Consequently, the conventional method for computing the mean and logarithmic standard deviation is inapplicable here. In this study, a multiple stripes analysis (MSA) method is adopted, in which two parameters α i and β i are estimated by using the maximum likelihood method [10]. The likelihood function corresponding to the damage state D i can be written as (9) in which m is the number of PGA levels, z i j is the number of ground motions at which the structure reaches or exceeds the damage state D i at PGA = x j , and is a product over all PGA levels. α i and β i in Eq. (8) are computed asα i andβ i by maximizing the logarithm of the likelihood function L i [10]. Thus,

NFES Results
In this section, the 60 ground motion records in Section 2.1.2 are selected as the seismic inputs. To increase the sample size, the PGA of each earthquake wave is scaled from 0.1 g to 1.5 g in increments of 0.1 g. Then, the IM 2 and IM 4 (shown in Tab. 4) of each record can be computed. A total of 900 IMs-DM sample sets are thus generated. The damage states in terms of the DM are defined in four levels in this research: negligible (D 1 ), light (D 2 ), moderate (D 3 ) and severe (D 4 ). The corresponding threshold values are 0.5%, 1.5%, 2.5%, and 3.8%. The PGA-DM cloud data calculated according to the NFES are depicted in Fig. 4. The statistical numbers of the ground motions in each PGA level when the steel BRB frame reaches or exceeds the four damage states are summarized in Tab. 7.
The two parameters in Eq. (8) at the four damage states are given in Tab. 8. In addition, the fragility curves using the NFES are plotted in Fig. 5, for which the discrete data points were computed using the MCS and the counterparts for the solid lines were obtained with the Reg method.

Training and Validation of the ANN
Subsequently, the feedforward ANN model is trained using the input variables IM 2 and IM 4 (i.e., PGV and Sa) and actual output DM (i.e., maximum IDR). Given the limited sample data, a k-fold cross validation method is adopted to improve the generality of the ANN model (k = 15 herein), in which the 900 IMs-DM samples are randomly and equally divided into 15 subsets. Note that the ANN model is trained 15 times, and one subset is used for validation while the other 14 ones are utilized for training in each training process. Moreover, the validation subsets in any two processes vary. Herein, a three-layer ANN model with six neurons is used, and the MSE is adopted to measure the performance of the model. The minimum MSE (i.e., goal of performance) is 0.001, the learning rate is 0.001, the momentum is 0.9, the epoch is 100, and the maximum validation check is 6. To overcome the overfitting, the training stops when the validation error reaches its minimum value [40] (Fig. 6).  Fig. 7. In line with Fig. 7, the prediction uncertainty of the ANN regression model cannot be ignored. For accurate simulation, the optimal ANN regression model with a minimum error is selected, saved, and used in the following computation. The predicted IM 2 -DM and IM 4 -DM data points according to the optimal model are compared with the NFES results in Fig. 8. An acceptable discrepancy occurs between the simulations using the ANN and NFES (Fig. 8). Therefore, the accuracy and reliability of the ANN are verified.

ANN-Based Fragility Prediction
In Section 3.3, the fragility curves are calculated using the trained ANN model. The statistical logarithmic means of the IM 2 (PGV ) and IM 4 (Sa) obtained from the 60 earthquake waves are listed in Tab. 9. The logarithmic standard deviation of the PGV and Sa (denoted as σ 2 and σ 4 ) for each PGA level are approximately 0.4881 and 0.9713, respectively. The means show strong linear relationships with the PGA in the log-log space (Fig. 9). The statistical distributions of the normalized logarithmic PGV and Sa at each PGA level are depicted in Fig. 10. Clearly, the PGV follows a good logarithmic normal distribution whereas the Sa does not. Subsequently, the PGA is increased from 0.1 g to 2.0 g in increments of 0.1 g (for a total 20 PGA levels) and transformed from the logarithmic space. Then, the logarithmic means ln μ 2 and ln μ 4 are calculated according to the linear regression relationships shown in Fig. 9. The logarithmic IM 2 and IM 4 at each PGA level and in accordance with the statistical distributions are shown in Fig. 10. Finally, the normalized ln IM 2 and ln IM 4 obtained from Eq. (6) are used to predict the normalized ln DM , and the fragility curve can thus be computed on the basis of the ln DM. In the following sections, the effects of the sampling methods and sample size on the fragility curve, as well as on the prediction uncertainty of the ANN model, are respectively studied.

Case I: Comparison of TRD and TUD
According to the statistical distributions of the PGV and Sa, the TRD and TUD introduced in Section 2.3.1 are adopted to take the values of PGV and Sa. For TRD, 10000 ln IM 2 and 10000 ln IM 4 are randomly taken in the range of ln μ i −2 ln σ i and ln μ i +2 ln σ i , whereas for TUD, they are uniformly distributed in [ln μ i − 2 ln σ i , lnμ i + 2 ln σ i ] with an increment of 0.01σ i . A total of 160801 data were involved. Fig. 11 shows the fragility curves using TRD and TUD. The fragility curves for the two sampling methods are almost the same (Fig. 11), but the probability of exceedance using TRD is obviously higher than that using TUD for the moderate and severe damage states. Therefore, simulation using TUD is more conservative than that using TRD. TUD is consequently adopted in the following analyses.

Case II: Effect of sample size
Herein, the effect of sample size on the fragility curve is studied. Fig. 12 presents the results of TUD using four sample sizes of 1681, 6561, 40401, and 160801. The maximum relative errors for those values are listed in Tab. 10. Fig. 12 and Tab. 10 indicate that the results of the MCS and Reg method using various sample sizes are very close to each other, and the relative maximum error decreases when the sample size increases. Furthermore, sample size has a slight influence on the fragility. In this study, TUD with a sample size of 1681 can provide a reasonable and acceptable prediction.  In this section, the prediction uncertainties of the ANN model are discussed, for which the variability of the output is illustrated in Fig. 7. Herein, the effects of the input IMs range, as well as the training and validation ratios, on the fragility curve are investigated. Fig. 13 depicts the fragility curves using three various IMs ranges, i.e., [ln μ i − 1.5σ i , ln μ i + 1.5σ i ], [ln μ i − 2.0σ i , ln μ i + 2.0σ i ], and [ln μ i − 2.5σ i , ln μ i + 2.5σ i ]. As evident from Fig. 7, as the value range of the IMs increases, the probabilities of exceedance at the four damage states increase at the low PGA levels, and decrease at the high PGA levels. Further, prediction uncertainties due to the training and validation ratios are studied. The outcomes of three simulations using the 10-, 15-, and 20-fold cross validation methods are given in Tab. 11 and their influences on the fragility curves are depicted in Fig. 14. The above analyses indicate that the prediction uncertainties of the ANN model cannot be ignored. In particular, the value range of the IMs exerts a considerable effect on the fragility analysis. The training and validation ratios have slight influences on the fragility curves of the structure at the negligible and severe damage states, whereas they have noticeable effects given light and moderate damage states.

Computational Cost
Finally, the time costs of the fragility analyses using the NFES and ANN are summarized in Tab. 12. In this study, all computations are conducted on a HP Workstation with an Inter Xeon W-2155 CPU with 10 physical cores at 3.3 GHz, and 64 GB of RAM. As the FE model is large scale, complex, and entails a substantial matrix size, the NFES uses a sixprocessor parallel method and requires approximately 2380 h. By contrast, the trainings of the ANN model based on the 10, 15, and 20-fold cross validation methods merely require approximately 8, 13, and 15 min, respectively. Furthermore, the time cost of he prediction using the ANN model in Section 3.3 is only 2 min. Therefore, the ANN model exhibits extremely high computational efficiency.

Conclusions
In this paper, a machine learning (ML)-based seismic fragility analysis framework is presented to examine the failure probability of a realistic large-scale steel BRB frame under earthquakes. The framework consists of a few sequential steps: building the model for nonlinear finite element simulation (NFES), training and validating an artificial neural network (ANN) model, and performing the fragility analysis using the ANN model. In the NFES, 60 earthquake records are selected and a total of 900 sample data can be generated by scaling the peak ground acceleration (PGA) of each record from 0.1 g to 1.5 g in increments of 0.1 g. The damage measures (DMs) of the structure, such as the maximum inter-story drift ratios, under these ground motions, are computed based on the NFES. Subsequently, the two most relevant IMs (i.e., PGV and Sa) with the DM are filtered by using the semi-partial correlation coefficient and an optimal ANN model is obtained using the IMs-DM data. The extended IMs from using the two sampling methods are inputted to the optimal ANN model to compute the DMs. Then, the fragility curves are calculated using the Monte Carlo Simulation and empirical regression method. The following conclusions can be obtained through the simulations: (1) The ML-based seismic fragility analysis framework is of significantly high computational efficiency and can instantaneously assess the failure risk of a large-scale structure under an earthquake. The trained ANN regression model based on the two IMs (i.e., PGV and Sa) and DM is reliable and accurate. (2) The truncated uniform sampling method outperforms the truncated random one. The two methods have almost similar fragility curves of the steel BRB frame at the negligible and light damage states. However, the result of truncated uniform sampling is slightly more reliable than that of its random counterpart at the moderate and severe damage states. In addition, the former is not sensitive to sample size.