A Deep Learning-Based Approach for the Identification of a Multi-Parameter BWBN Model

Abstract

A restoring-force model is a versatile mathematical model that can describe the relationship between the restoring force and the deformation obtained from a large number of experiments. Over the past few decades, a large body of work on the development of restoring-force models has been reported in the literature. Under high intensity cyclic loadings or seismic excitations, reinforced concrete (RC) structures undergo a wide range of hysteretic deteriorations such as strength, stiffness and pinching degradations. These characteristic behaviors can be described by the multi-parameter Bouc-Wen-Baber-Noori (BWBN) model, which offers a wide range of applicability. This model has been applied for the response prediction and modeling restoring-force behavior in structural and mechanical engineering systems, by adjusting the distribution range of this model’s parameters. However, a major difficulty in utilizing the multi-parameter BWBN model is the parameters’ identification. In this paper, a deep neural network model is used to estimate the hysteresis parameters of the BWBN model. This model is one of the most versatile and widely used general hysteresis models that can describe the hysteretic behavior of RC columns. The experimental data of the RC columns used in this paper are collected from the database of the Pacific Earthquake Engineering Research Center (PEER). Firstly, the hysteretic loop obtained from a physical experiment is described by the BWBN model, and the parameters of the BWBN model are identified via a genetic optimization algorithm. Then a neural network is established by a backpropagation (BP) algorithm for associating the identified BWBN model parameters with physical parameters of the RC column. Finally, the regression analysis of the identified parameters is carried out to obtain the regression characteristics of the RC columns. The trained neural network model can directly identify the parameters of BWBN model based on the physical parameters of RC columns, and is effective and computationally efficient for multi-parameter BWBN model identification. The proposed approach overcomes the difficult problem of identifying the parameters of BWBN model and provides a promising approach for a wider application of this multi-parameter hysteresis model.

1. Introduction

The damage or collapse of engineering structures caused by earthquake action often causes serious casualties and property losses [1,2,3,4,5]. In order to mitigate the catastrophic impact of earthquakes, extensive research has been conducted and reported in the literature on the dynamic response of structures under earthquake action, and in particular the nonlinear dynamic analysis of concrete structures under strong earthquake action [6,7,8,9,10,11,12,13,14]. When concrete structures experience nonlinear deformation under high intensity dynamic loads, such as earthquake loadings, their strength and stiffness deteriorate through a hysteretic behavior. Quasi-static load tests are important experiments that help to understand the nature of this deterioration, especially in seismic research. A large number of quasi-static experiments have shown that the hysteresis loops of concrete members under different failure modes are significantly different. For example, the hysteresis loops of reinforced concrete (RC) columns can be divided into flexural failure (FF), flexural shear failure (FS) and shear failure (SF) modes. As part of the analysis required to study, understand and describe the characteristics of these various failure modes and the corresponding hysteresis loops, multi-parameter restoring-force models are needed. If a versatile hysteresis model can be used, by adjusting the parameter values of the model, it can accurately predict the nonlinear yield behavior of RC structures under different failure modes.

The existing models for predicting the hysteretic restoring force of concrete structures are generally divided into polygonal hysteresis models (PHM) and smooth hysteresis models (SHM) [15,16,17,18]. PHM describes the nonlinear failure behavior of RC structure by establishing the typical physical failure characteristic line segments, such as yield and cracking. This causes simplification of the nonlinear behavior and low accuracy of structural analysis. Moreover, it is difficult to describe a complex degenerative behavior of the RC structure by these models, and the calculation cost is increased by refined modeling. The SHM models use nonlinear differential equations to describe the nonlinear behavior of RC structures by adjusting the parameters of the differential equations [19,20,21]. Compared with PHM, the SHM has the advantages of high accuracy and low calculation cost. Identifying a suitable multi-parameter SHM is the focus of this paper.

Various SHM models have been proposed in the past few decades. One of the most versatile SHM models is the Bouc-Wen (BW) [22,23] model, which has been widely used in several engineering fields. Bouc developed the earliest version of this smooth and versatile hysteresis model; Baber and Wen extended Bouc’s model by considering the degradation in strength or stiffness in nonlinear hysteretic systems [23]. Baber and Noori further extended the BW model by including the complicated pinching behavior [24] and introduced a general hysteresis model known as the BWBN model [25]. The BWBN model takes into account strength and stiffness degradation as well as the shear-pinching phenomenon. Extensive research studies, utilizing the BWBN model, have been reported in the literature [26,27,28,29,30,31,32,33,34,35,36,37]. However, despite its broad application, one of the practical problems in the application of the multi-parameter smooth BWBN model is to identify its parameters. Multiple combinations of BWBN parameters can describe a broad range of structural hysteresis behavior, however, since the BWBN model is an analytical model; its parameters have no obvious physical meaning. The question is how to identify a combination of parameters for an optimal solution. The traditional approach for determining the parameters has been based on using computational schemes that rely on engineering experience and experimental studies, which are subjective and random in nature. Improving the traditional method and introducing a robust and reliable approach for parameter identification of the BWBN model is another objective of this paper.

The most commonly used algorithms for parameter identification include genetic and differential algorithms. In both approaches, the hysteretic loop of the identified object is first classified and the distribution range of these hysteretic loop parameters is determined based on prior knowledge. In order to reduce the computation of multi-parameter identification, a sensitivity analysis is usually performed. Finally, the identified parameters are screened in order to determine the most suitable ones. This process of traditional methods is complicated [15,38,39,40,41,42,43,44,45]. This poses a great challenge to the application of the model. Therefore, there is a need for a more efficient approach that can quickly identify the parameters of complex hysteretic systems such as the BWBN model. In recent years, with the development of more powerful and faster computational processing, artificial intelligence (AI)-based algorithms, such as deep learning, have emerged as more promising and computationally more efficient tools for system identification, especially for treating complex problems [46,47,48,49,50,51,52,53,54,55,56,57,58,59,60]. At the same time, given the availability of more powerful data analysis tools, a wide range of quasi-static experimental data of RC structures have also been accumulated. These large banks of experimental data, combined with deep learning prediction methods, provide an opportunity for a more accurate and realistic parameter identification of multi-parameter hysteretic models, especially nonlinear hysteretic models for the response analysis of RC structures.

In this paper, a smooth multi-parameter BWBN hysteresis model is considered for the dynamic response of RC structures. This model can reproduce and predict the stiffness and strength degradation as well as the pinching characteristics of these structure. The governing equations of the BWBN model are presented and a deep learning-based system identification method is proposed to identify the parameters of the BWBN model. The input utilized for the deep neural network are the physical parameters of RC columns. To demonstrate the performance of the proposed method, the hysteresis data of RC columns in the test database of the Pacific Earthquake Engineering Research Center (PEER) [61] are used. The results show that, compared with traditional parameter identification methods, the proposed prediction method can identify the parameters of the BWBN hysteretic model of RC columns under different failure modes more efficiently.

2. The BWBN Model

2.1. Problem Definition

The BWBN model can reproduce the behavior of general nonlinear hysteretic systems. By incorporating a pinching function in the conventional Bouc-Wen model, the BWBN model captures the strength, stiffness, and pinching degradation effects. Figure 1 shows a single degree of freedom BWBN model.

Based on the D’Alembert principle, and assuming that the system is subjected to a horizontal earthquake excitation, the equation of an SDOF system with a BWBN restoring force can be described by Equation (1),

$$m\ddot{u}+c\dot{u}+f(t)=-m{\ddot{u}}_{gx}$$

(1)

where the restoring force has the form:

$$f(t)=\alpha ku+(1-\alpha )kz$$

(2)

In this equation u is the relative translational displacement, $\dot{u}$ and $\ddot{u}$ are velocity and acceleration responses, respectively, k, m and c are stiffness, mass and viscous damping coefficients, respectively, ${\ddot{u}}_{gx}$ is the ground acceleration in the x-direction, and α is the ratio of post-yield to initial stiffness. The restoring force is composed of the elastic restoring force $\alpha ku$ and the nonlinear restoring force $(1-\alpha )kz$. The hysteretic displacement z is given by [62]:

$$\dot{z}=\frac{h(z,\epsilon )}{1+{\delta }_{\eta }\epsilon }[\dot{u}-(1+{\delta }_{\nu }\epsilon )(\beta |\dot{u}||z{|}^{n-1}z+\gamma \dot{u}|z{|}^n)]$$

(3)

where β and γ are loading and unloading control parameters, n controls the yield point smoothness, and ${\delta }_{\nu }$ and ${\delta }_{\eta }$ are the strength and stiffness degradation parameters, respectively. ε is the hysteresis energy, and $h(z,\epsilon )$ is the pinching function. The hysteresis energy dissipated from $t=0$ to the current time $T$ is:

$$\epsilon =(1-\alpha )k{\displaystyle \underset{0}{\overset{T}{\int }}\dot{u}zdt}$$

(4)

The pinching function $h(z,\epsilon )$ is given by [62]:

$$h(z,\epsilon )=1-{\zeta }_s[1-\exp (-p\epsilon )]\exp [-(\frac{z\mathrm{sgn}(\dot{u})-q{[(1+{\delta }_{\upsilon }\epsilon )(\beta +\gamma )]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}}{(\psi +{\delta }_{\psi }\epsilon )(\lambda +{\zeta }_s[1-\exp (-p\epsilon )])}{)}^2]$$

(5)

where ${\zeta }_s$ is the total slip, p is the pinch slope, q is initial slip, ψ is the pinch size, ${\delta }_{\psi }$ is the pinch rate, $\lambda$ is the pinch intensity, and $sgn\left(·\right)$ is the signum function. The BWBN model contains 12 characteristic parameters which can be divided into four categories: four shape parameters $\alpha ,\beta ,\gamma$ and n, one strength degradation parameter ${\delta }_{\nu }$, one stiffness degradation parameter ${\delta }_{\eta }$ and six pinch parameters ${\zeta }_s,p,q,\psi ,{\delta }_{\psi }$ and λ. The relative translational displacement $u$ and hysteretic displacement $z$ are normalized to dimensionless variables [7]:

$$\mu =u/u_y,{\mu }_z=z/u_y,\varphi =u_y/u_0=f_y/f_0$$

(6)

where u₀ is the peak-value displacement in yield stage, f₀ is the peak-value resisting force in yield stage. u_y is the displacement of yield point, and f_y is the force at the yield point. Φ is the normalized yield strength. The yield point is the intersection point between the elastic stage and the yield stage of backbone curve. By utilizing the normalized variables, the equation of motion of the BWBN model given by Equations (1)–(5) can be rewritten in the modal coordinate [62]:

$$\ddot{\mu }+2\zeta {\omega }_n\dot{\mu }+\alpha {\omega }_n{}^2\mu +(1-\alpha ){\omega }_n{}^2{\mu }_z=-\frac{{\ddot{\mu }}_{gx}}{\varphi {\mu }_0}$$

(7)

$${\dot{\mu }}_z=\frac{h({\mu }_z,{\epsilon }_n)}{1+{\delta }_{\eta }{\epsilon }_n}\{\dot{\mu }-(1+{\delta }_{\upsilon }{\epsilon }_n)|{\mu }_z{|}^n\dot{\mu }[sgn(\dot{\mu }{\mu }_z)\beta +\gamma ]\}$$

(8)

$${\epsilon }_n=(1-\alpha ){\displaystyle \underset{0}{\overset{T}{\int }}\dot{\mu }{\mu }_z}dt$$

(9)

$$\begin{array}{l}h({\mu }_z,{\epsilon }_n)=\\ 1-{\zeta }_s[1-\exp (-p{\epsilon }_n)]\exp [-(\frac{{\mu }_{z}sgn(\dot{\mu })-q{[(1+{\delta }_{\upsilon }{\epsilon }_n)(\beta +\gamma )]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}}{(\psi +{\delta }_{\psi }{\epsilon }_n)(\lambda +{\zeta }_s[1-\exp (-p{\epsilon }_n)])}{)}^2]\end{array}$$

(10)

where $\xi =c/2m{\omega }_n$ is the damping ratio, in which ${\omega }_n=\sqrt{k/m}$ is the natural vibration frequency, $\mu$ and ${\mu }_z$ are the displacement ductility demand and the hysteretic ductility demand, respectively, $\alpha ,\beta ,\gamma ,n,{\delta }_{\nu },{\delta }_{\eta },{\zeta }_s,p,q,\psi ,{\delta }_{\psi }$ and λ are dimensionless parameters of the BWBN model.

As can be seen from Equations (7)–(10), which describe the dynamic response of a SDOF system with a BWBN hysteretic restoring force, these equations are coupled and there is no closed-form solution. Therefore, a numerical integration algorithm is used to solve the problem through an iterative process with small time increments. $Newmark-\beta$ is a common algorithm, it is based on a linear acceleration method by introducing two parameters of $\overline{\beta }$ and $\overline{\gamma }$; the bar symbol is used to differentiate from the BWBN model parameters. A major advantage of this method is that it can find the tangent or secant stiffness in each load step. The iterative flow chart [63] is shown in Figure 2. ε and ${\epsilon }_{tal}$ denote the calculation error and allowable error respectively in Figure 2. Equations (7)–(10) of the BWBN model are discretized as:

$$\mathrm{\Delta }{\ddot{\mu }}_i+2\zeta {\omega }_n\mathrm{\Delta }{\dot{\mu }}_i+\alpha {\omega }_n{}^2\mathrm{\Delta }{\mu }_i+(1-\alpha ){\omega }_n{}^2\mathrm{\Delta }{\mu }_{zi}=-\frac{\mathrm{\Delta }{\ddot{\mu }}_{gxi}}{\varphi {\mu }_0}$$

(11)

$${({\dot{\mu }}_z)}_{i+1}=\frac{h({({\mu }_z)}_{i+1},{({\epsilon }_n)}_{i+1})}{1+{\delta }_{\eta }{({\epsilon }_n)}_{i+1}}\{{\dot{\mu }}_{i+1}-(1+{\delta }_{\upsilon }{({\epsilon }_n)}_{i+1})|{({\mu }_z)}_{i+1}{|}^n{\dot{\mu }}_{i+1}[sgn({\dot{\mu }}_{i+1}{({\mu }_z)}_{i+1})\beta +\gamma ]\}$$

(12)

$${({\epsilon }_n)}_{i+1}={({\epsilon }_n)}_i+\mathrm{\Delta }\mathrm{t}(1-\alpha ){\dot{\mu }}_{i+1}{({\mu }_z)}_{i+1}$$

(13)

$$\begin{array}{l}h({({\mu }_z)}_{i+1},{({\epsilon }_n)}_{i+1})=\\ 1-{\zeta }_s[1-\exp (-p{({\epsilon }_n)}_{i+1})]\exp [-(\frac{{({\mu }_z)}_{i+1}sgn({\dot{\mu }}_{i+1})-q[(1+{\delta }_{\upsilon }{({\epsilon }_n)}_{i+1})(\beta +\gamma )]\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}{(\psi +{\delta }_{\psi }{({\epsilon }_n)}_{i+1})(\lambda +{\zeta }_s[1-\exp (-p{({\epsilon }_n)}_{i+1})])}{)}^2]\end{array}$$

(14)

$$\begin{array}{ll}& \mathrm{\Delta }{\ddot{\mu }}_i=\frac{1}{\overline{\beta }\mathrm{\Delta }{t}^2}\mathrm{\Delta }{\mu }_i-\frac{1}{\overline{\beta }\mathrm{\Delta }t}{\dot{\mu }}_i-\frac{1}{2\overline{\beta }}{\ddot{\mu }}_i\\ where& \mathrm{\Delta }{\dot{\mu }}_i=\frac{\overline{\gamma }}{\overline{\beta }\mathrm{\Delta }t}\mathrm{\Delta }{\mu }_i-\frac{\overline{\gamma }}{\overline{\beta }\mathrm{\Delta }t}{\dot{\mu }}_i+(1-\frac{\overline{\gamma }}{2\overline{\beta }}){\ddot{\mu }}_i\mathrm{\Delta }t\\ & {({\mu }_z)}_{i+1}={({\mu }_z)}_i+\mathrm{\Delta }t{({\dot{\mu }}_z)}_{i+1}\\ & {\mu }_{i+1}={\mu }_i+\mathrm{\Delta }t{\dot{\mu }}_{i+1}\end{array}$$

(15)

After time-discretization of the nonlinear dynamic differential equations (DDEs), during a very small time increment, the nonlinear system can be approximately treated as a linear dynamic system. Another commonly used iteration algorithm is the Runge-Kutta method [64]. The Runge-Kutta method reduces higher-order differential equations to lower orders. DDEs can be converted into a set of first-order ordinary differential equations (ODEs). The detailed solution process is shown in [62]. To confirm the effectiveness of $Newmark-\beta$ algorithms, we consider the same input earthquake load, i.e., El Centro record. The normalized solution results of the $Newmark-\beta$ and Runge-Kutta algorithms are consistent [63]. The single degree of freedom BWBN model has 12 dimensionless characteristic parameters, which play different roles in the hysteresis model. The contributions of different characteristic parameters on hysteresis behavior can be found visually in [62].

2.2. NSGA-II Algorithm

Genetic Algorithm (GA) is a self-adaptive global search algorithm. It simulates the genetic and evolutionary process of organisms in the natural environment. A fitness function is introduced to measure the adaptability of individuals in the population. According to the fitness of the individual in the problem, the individual is selected according to the corresponding rules. With the help of the genetic operator of natural genetics, the population representing the new solution set is generated. The selection, crossover, mutation, and other operations are carried out in a probabilistic way, which increases the flexibility of the search process. This can converge to the optimal solution with a significant probability.

The GA algorithm of the multi-objective optimization problem has been improved over recent years. Srinivas and Ded (1995) proposed non-dominated sorting genetic algorithms (NSGA) [65], which are based on the Pareto optimal concept. The main difference with the simple GA is that the NSGA are stratified according to the dominant relationship between individuals before the selection operator is executed. The selection operator, crossover operator, and mutation operator are not different from a simple genetic algorithm. Before the selection operation is executed, the population is sorted according to the dominant and non-dominant relationship among individuals.

The computational complexity of the NSGA algorithm is $O\left(m{N}^3\right)$, where m is the cardinality of objective function and N is the population size. When calculating the first Pareto front line, each solution must be compared with other N-1 solutions’ m times. The total time of comparison is $O\left(m{N}^2\right)=m\left(N-1\right) N$. In the worst case, only one solution is found in the search $O\left(m{N}^2\right)$ times, and the rest N-1 times searches are needed, then the computational complexity is $O\left(m{N}^3\right)=m\left(N-1\right)N^2$. Thus, the calculation is time-consuming when the population is large. Deb (2000) further improved the algorithm by introducing the elite retention strategy [66]. A satisfactory solution is easily obtained by elitist retention strategy. The case of finding one solution in the search $O\left(m{N}^2\right)$ times is avoided. The computational complexity is reduced to $O\left(m{N}^2\right)$.

2.3. Sensitivity Identification

The sensitivity of every parameter of BWBN model is obviously different for RC columns under different failure modes. The importance of parameters can be sorted out by sensitivity identification, and some insensitive parameters can be fixed in the calculation. This improves the efficiency of parameter identification, which can retain those parameters that are more sensitive to the structural response and lock the other parameters to reduce the calculation amount and improve the computational efficiency. The commonly used sensitivity identification method is global sensitivity analysis.

The global sensitivity analysis method can consider the influence of the interaction between multiple parameters on the output response of the model, and the representative method is the Sobol method [67,68]. The Sobol method decomposes the model into a single parameter and a function consisting of the combination of parameters. By analyzing the influence of the variance of the single parameter and the influence of the combination of the parameters on the total variance of the model, the importance of the parameters and the degree of the interaction between the parameters are evaluated. The Sobol method decomposes the model $f(X)$ into orthogonal polynomials of all subterms.

$$f(X)=f_0+{\displaystyle \sum _{i=1}^n{f}_i(x_i)}+{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=i+1}^n{f}_{i,j}}}(x_i,x_j)+\dots +f_{1,2,\dots ,n}(x_1,\dots ,x_n)$$

(16)

where f₀ is the integral of all parameters, $f_i(x_i)$ is the integral of all parameters except x_i, and $f_{i,j}(x_i,x_j)$ is the integral of all parameters except x_i and $x_j$.

$$\begin{array}{l}{f}_0={\displaystyle \int f(X)dX}=E\left(Y\right)\\ f_i(x_i)={\displaystyle {\int }_0^1\dots }{\displaystyle {\int }_0^{1}f(X)d{X}_{-i}-f_0}=E\left(Y|x_i\right)-E(Y)\\ f_{i,j}(x_i,x_j)={\displaystyle {\int }_0^1\dots }{\displaystyle {\int }_0^{1}f\left(X\right)d{X}_{-(i,j)}-f_0-f_i(x_i)-f_j(x_j)}=E\left(Y|x_i,x_j\right)-E\left(Y\right)-f_i-f_j\end{array}$$

(17)

In order to describe the influence degree of all parameters on the output response of the model, the total variance is derived by taking the variance on both sides of Equation (15) by using the orthogonal condition:

$$\begin{array}{l}V={\displaystyle \int f^2(X)dX}-f_0{}^2\\ =E(V(Y|x_i))+V\left(E\left(Y|x_i\right)\right)\end{array}$$

(18)

The partial square error of a single parameter on the output response of the model can be given by:

$$V_i={\displaystyle \int f_i{}^2(x_i)d{x}_i}$$

(19)

which can describe the influence degree of multiple parameters on the output response of the model. The partial square difference of multiple model parameters can be expressed by:

$$V_{i_1,\dots ,i_s}={\displaystyle {\int }_0^1\dots }{\displaystyle {\int }_0^1{f}^2{}_{i_1,\dots ,i_s}}(x_1,\dots ,x_s)d{x}_{i_1}\dots d{x}_{i_s}$$

(20)

The differences between conditional variance $V\left(Y|x_i\right)$ and total variance $V\left(Y\right)$ express the influence of model parameters x_i for the model $Y$. The Sobol main effect sensitivity index is defined by:

$$S_{X_i}=\frac{V\left(E\left(Y|x_i\right)\right)}{V\left(Y\right)}$$

(21)

where $S_{X_i}$ represents the influence of variables on the total variance of model. It is the main effect sensitivity index of X_i. In order to consider the interaction between the main effect index of the model parameter x_i with other remaining parameters, the total effect-sensitivity coefficient of the model parameter can be obtained by:

$$S_{X_i}^T=\frac{V\left(Y\right)-V\left(E\left(Y|x_{-i}\right)\right)}{V\left(Y\right)}$$

(22)

where $S_{X_i}^T$ includes the main effect of parameter x_i and the interaction influence of parameter x_i with other parameters.

The multi-parameter BWBN model can be used to analyze various structural hysteresis phenomena, such as mechanical engineering, civil engineering and geotechnical engineering. Although the BWBN model has strong universality, the model parameters do not have specific physical meaning. Not only the influence of the change of single parameter value, but also the combination of multiple parameters should be considered. It takes a lot of calculation to determine the parameter values of multiple model parameters, so the global sensitivity analysis can greatly reduce the calculation of parameter identification. Using the global sensitivity method, a large number of RC columns are analyzed in this paper and the results show that the sensitivity of parameters p and n is low, where n can be set to a constant value of 1, which means that the turning point of the curve is smooth.

3. Deep Learning Identification Model

3.1. The Network Architecture of the Deep Learning Model

The artificial neural network (ANN) is an information-processing mathematical model established by imitating the brain’s neural network structure and function. This method involves a repeated learning and training of the known information, and incudes a scheme for gradually adjusting the learning weights of the neurons to mimic the relationship between the input and the output data. Compared with the traditional GA method, the neural network technique has obvious advantages in dealing with fuzzy data, random data, and nonlinear data, and is especially suitable for large-scale, complex structures and unclear information systems.

Deep learning is an artificial neural network structure with multi-hidden layers. Depth usually refers to the number of hidden layers in the neural network structure. Learning is a cognitive process from unknown to known, corresponding to the input and output process in the neural network structure. Deep learning strengthens the ability of feature learning and makes the prediction more accurate by the hierarchical expression of input information and layer-by-layer feature transformation. A multi-layer neural network has excellent feature-learning ability and can learn more essential data features. For the multi-parameter identification problem of the BWBN model, traditional identification methods have better identification performance, but the identification process is complex and needs a large number of calculations. The deep learning model utilized in this work contains multi-hidden layers. By incorporating different activation functions for hidden layers, this deep learning model can handle more complex problems such as identification, segmentation and regression analysis. Obviously, the deep learning model can estimate the model parameters of the BWBN model by making regression analysis on a large number of existing recognition data. The optimal value of parameters can be identified in a simple and direct manner. The comparison with the traditional method is shown in Figure 3.

The problem of multi-parameter identification of the BWBN model belongs to the problem of multiple vector regression in supervised learning. Since the parameters of the BWBN model are dimensionless, the input parameters of the input layer are also dimensionless. Herein, the dimensionless parameters provided by the database of PEER include the aspect ratio (i.e., the ratio of shear span to column depth), the axial load ratio, the longitudinal reinforcement ratio, and the transverse reinforcement ratio. The schematic diagram of the neural network structure is shown in Figure 4, the network structure includes one input layer, i hidden layers, i greater than 2, and one output layer.

The BWBN model has a wide range of applications. Because this model is a general mathematical model, when it is used in experimental analysis of structural systems, its parameters cannot strictly correspond to the physical parameters of the physical experimental model.

Obviously, there is more than one mathematical solution of the BWBN model corresponding to the physical experimental model. Therefore, a double objective optimization function can be constructed to satisfy the following requirements: (a) the parameters of the network model, i.e., weights and hyperparameters of the network model; (b) the corresponding solution of the physical model. The objective function of the network model is the regression evaluation index, which are R² and adjusted R² (adjR²). The evaluation index R² can be obtained by:

$${\mathrm{R}}^2=1-\frac{{\displaystyle \sum _i{({\widehat{y}}_i-y_i)}^2}}{{\displaystyle \sum _i{({\overline{y}}_i-y_i)}^2}}$$

(23)

where ${\widehat{y}}_i,{\overline{y}}_i$ and y_i are the predicted value of the test set, the mean value of the test set, and the actual value of the test set. The evaluation index adjR² offsets the effect of sample size in the evaluation index R².

$${\mathrm{adjR}}^2=1-\frac{\left(1-R^2\right)\left(N-1\right)}{{\rm N}-P-1}$$

(24)

where N is the number of samples and P is the number of input features. The objective function is the restoring force error function of the physical model,

$$\begin{array}{l}{f}_1(P)=\frac{1}{T}{\displaystyle \sum _{i=1}^T\{[R(t_i)-\widehat{R}(t_i|\mathrm{P})]}{\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ f_2(P)=\underset{1\le i\le T}{\max }\left|[R(t_i)-\widehat{R}(t_i|\mathrm{P})\right|\\ f_3(P)=\frac{1}{T}{\displaystyle \sum _{i=1}^T\{[\frac{d(R(t_i))}{d(\mu (t_i))}-\frac{d(\widehat{R}(t_i|\mathrm{P}))}{d(\mu (t_i|{\mathrm{P}}_i))}]}{\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ f_4(P)=\left|{\epsilon }_{tot}(t_i)-{\widehat{\epsilon }}_{tot}(t_i|\mathrm{P})\right|\end{array}$$

(25)

where R is the normalized restoring force at the loading step t_i, and fi is the objective function. P is the parameter set of BWBN model, and the superscript indicates the prediction result of BWBN model.

3.2. The Network Parameters of the Deep Learning Model

The back-propagation (BP) neural network is a multi-layer feed-forward neural network (FNN), which is trained according to the error backpropagation algorithm. Feed forward refers to the data flow direction and reverse refers to the error flow direction. The BP neural network is composed of two modules: a forward-propagation network of information and a back-propagation network of error. The signal propagates forward and the error propagates backward. The process of the BP neural network is mainly divided into two stages. The first stage is the forward propagation of the signal, from the input layer to the hidden layer and finally to the output layer. The second stage is the error back-propagation, from the output layer to the hidden layer, and finally to the input layer, adjusting the weight and bias from the hidden layer to the output layer, the weight and bias from the input layer to the hidden layer in turn. Commonly used activation functions that are utilized include linear rectification function (ReLU), Sigmoid function and tanh function. Among them, the nonlinear activation functions are sigmoid, ReLU, and tanh. In this paper, we choose the ReLU function for the nonlinear identification problem. The BP model reversely adjusts the value of neuron weight by the loss function. The commonly used loss functions are mean square error (MSE), the sum of the square error (SSE), and mean absolute error (MAE). Among them, MSE is selected, which is more suitable for continuous value regression. The process of numerical iteration to approach the optimal solution needs an optimization algorithm. The common optimization algorithms include the gradient descent method, the Newton method, momentum, the Nesterov Momentum, the Adagrad, the Adam, and the Rmsprop [69,70]. The Rmsprop is used to solve the problem of excessive swing in the optimization process and is suitable for parameter identification of the BWBN model. The learning rate of this paper is 0.9.

The learning data come from the experimental data of the PEER database. In total there are 416 sets of data which are divided into FS, FF and SF failure modes. In order to study the generality of the present method, only some failure data are deducted, the Failure data are some test distortion data. The remaining 386 groups of data are used as learning data. The training process is that the supervised label of each group of experimental data, namely BWBN model parameters, is established by using the traditional NSGA-II algorithm. Its evaluation index is given by Equation (25). Then, the regression characteristic solution of the feasible solution region is found. The regression analysis adopts the BP deep learning network model. Its evaluation index is provided by Equations (23) and (24). The appropriate parameter solution can be found through the training of neural networks. The optimal hyperparameters of the network model can be found. The number of hidden layers and neurons of the deep learning model are shown in the following

Table 1. The number of neurons of deep learning model.

BWBN Parameters	α	β	γ	n	δv	δη	ζs	p	q	δψ	λ
Hidden layer 1	11
Hidden layer 2	17
Hidden layer 3	15
Hidden layer 4	17
Hidden layer 5	11

. It can be seen that the more layers and neurons there are, the better. The prediction accuracy is not positively correlated with the model complexity. Simply increasing the model complexity will produce over-fitting phenomenon, which can add a dropout layer to the model. Therefore, for different engineering problems, the complexity of the model needs to be determined separately, for example, the stress problems of RC beam and RC column need to be determined separately.

4. Results and Discussion

4.1. Results

Based on the predicted parameters of the deep learning model, the predicted hysteretic loop can be obtained by substituting them into the BWBN model then comparing them with the hysteretic loops of the experimental data. There are 18 groups of data randomly extracted from the 3 failure modes of the database, with 6 failure modes for each failure mode. The selected samples are shown in

Table 2. Test samples of RC Columns with different mode.

No.	Authors/Specimen	Failure Mode	Span-to-Depth Ratio	Axial Load Ratio	Longitudinal Reinforcement Ratio	Transverse Reinforcement Ratio
S1	Atalay and Penzien, 1975, No. 5S1	FF	5.50	0.20	0.02	0.02
S2	Azizinamini et al., 1988, NC-2		3.00	0.21	0.02	0.02
S3	Sugano 1996, UC15L		2.00	0.35	0.02	0.00
S4	Matamoros et al., 1999,C10-05S		3.00	0.05	0.02	0.01
S5	Arakawa et al., 1988, No. 21		2.18	0.11	0.04	0.63
S6	Hamilton, 2002, UCI2		4.56	0.00	0.01	0.53
S7	Nagasaka, 1982, HPRC19-32	FS	1.5	0.35	0.0127	0.014
S8	Wight and Sozen, 1973, No. 40.033 (West)		2.87	0.114	0.0245	0.003
S9	Lynn et al., 1998, 2CMH18		3.22	0.284	0.0194	0
S10	Ang et al., 1985, No. 1		2	0	0.032	0.51
S11	Hamilton, 2002, UCI5		2.58	0.00	0.01	0.26
S12	Sezen and Moehle No. 1		3.22	0.15	0.02	0.00
S13	Lynn et al., 1998, 3CMH18	SF	3.22	0.262	0.0303	0
S14	Ang et al., 1985, No. 18		1.5	0.1	0.032	0.51
S15	McDaniel, 1997, S2		2	0.002	0.0136	0.13
S16	Arakawa et al., 1987, No. 1		1.09	0	0.0385	0.47
S17	Arakawa et al., 1987, No. 8		1.09	0.109	0.0385	1.34
S18	Arakawa et al., 1988, No. 28		1.64	0.166	0.0385	0.63

. The comparison results are shown in Figure 5.

It can be seen that the predicted hysteretic loop is basically consistent with the experimental hysteretic loop, which indicates that the deep learning prediction model has reached the expected setting. As shown in Figure 3, the current research method can easily and directly identify the parameters of the multi-parameter model without establishing a priori estimation or classifying the failure types. This proposed method is simple, thus, it solves the difficult problems encountered in the application of multi-parameter model. For further verifications of the correctness of the parameter identification method, the BWBN material constitutive model is embedded in OPENSEES software [71].

The parameters of the BWBN material constitutive are identified by the deep learning method. Finally, the assigned BWBN material constitutive model is added to the beam column joint element. The Eom’s cast-in-situ cross joint experiment was selected as the experimental data [72]. Cyclic loading is carried out by the displacement loading method. The geometric dimensions and reinforcement of joints can be found in [72,73]. Joint2d-1 joint is used as the joint numerical model, and the hysteretic material in the core area of the joint is simulated by the BWBN model. The simulation results of hysteretic loop is shown in Figure 6. It can be seen that the BWBN model can be well used for nonlinear simulation of stiffness, strength and pinching of RC structures.

4.2. Discussion

Since the training dataset uses estimated parameters with NSGA-II, the effect of errors introduced by this genetic algorithm will be present in the neural network predictions in addition to the network’s own errors. To eliminate the superposition effect of the two errors, the method of parallel connection of multiple evaluation functions is also used in the construction of a neural network in this paper. It can be seen from the identification example results in Figure 5 that for the specimens with ideal results, such as S1, S5 and S11, good identification accuracy is achieved, whereas for the specimens with unsatisfactory test results, these cause a large noise due to improper test process or operation, such as S6, S12 and S13, and the predicted results are within the acceptable error range. For reducing error propagation, the R² and adjR² values of 12 parameters of the BWBN model are given in Figure 7. The abscissa of Figure 7 is the prediction value, and the ordinate is the test value. The closer the prediction point is to the blue line, the better the prediction effect will be. The x-axis is the predicted value of the present model, and the y-axis is the predicted value of the traditional NSGA-Ⅱ algorithm. It can be seen that the correlation coefficient R² and adjR² are higher than 0.8, which indicates that the deep learning model in this paper is feasible. The corresponding training loss and verification loss are shown in the Figure 8. The abscissa of Figure 8 is the epochs of iterations, and the ordinate is the loss function value. It can be seen that the decline of the loss function tends to be gentle after about 20 generations, which means that the recognition results basically meet the error conditions.

5. Conclusions

As described in this paper, the multi-parameter BWBN model is a versatile model that can predict the complex hysteretic behaviors observed in a wide range of engineering applications. However, the difficulty in utilizing the multi-parameter BWBN model in practical applications is parameter identification. The traditional parameter identification methods use prior experience, sensitivity analysis, and optimization algorithms. Although those method have demonstrated good results, the cumbersome process used in those approaches is not suitable for practical application. The most noteworthy advantage of the proposed deep learning-based approach is the high efficiency and straightforward implementation and accuracy in identifying nonlinear characteristics of input parameters. Overall, the trained neural network model can easily obtain the parameters of the BWBN model. The key advantages of the proposed method can be summarized as follows:

(1)

The multi-parameter BWBN model has a wide range of applicabilities. It can be applied in different fields by adjusting the parameter distribution range, such as mechanical engineering, civil engineering, and structural health monitoring and analysis of RC columns. Based on the findings of this study, the appropriate parameter range can be given by:

$$\begin{array}{l}0<\alpha <0.2,0<\beta <1.5,0<\gamma <0.5,n=1, 0<{\delta }_{\nu }<1, 0<{\delta }_{\eta }<0.1, 0.25<{\zeta }_s<2,\\ 0<p<0.3,0<q<1.15,0<\psi <0.2,0<{\delta }_{\psi }<0.1,0<\lambda <0.1\end{array}$$

(2)

The parameter identification is a bottleneck problem in the application of the BWBN model. The traditional optimization-based identification methods focus on finding a group of suitable parameter values. However, those approaches result in more than one solution in the feasible solution region. The proposed research focuses on regression analysis. A regression analysis through deep learning leads to ideal results. Once the regression analysis model is trained well, a group of feasible solutions can be easily obtained for similar problems, which significantly reduces the workload of traditional methods.

(3)

The parameter identification method proposed herein offers a more efficient and concise technique for the application of BWBN model in engineering fields. For example, based on the efficient parameter identification introduced in this work, in the open-source software OPENSEES the macroseismic model of beams and columns can be established. The BWBN macro model is different from the finite element method, which needs refined modeling and multiple material models to simulate the complex response of the structure. The multi-parameter BWBN macro model can easily capture the complex response only by changing the parameter values.