# Simulation Study of New Buckling Restraint Bracing Based on Bayesian Optimization Algorithm to Identify the Parameters of Modified Bouc–Wen Model

Authors: Lu Zhao; Tiangang Tu; Sunwei Ding; Xinqiao Liu
Journal: Advances in Civil Engineering
 (2023)

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## Abstract

The introduction of new materials can significantly improve the performance of existing dampers, but the constitutive model of new materials is difficult to be obtained in a short time, which makes its numerical simulation difficult. Bouc–Wen–Baber–Noori (BWBN) model can well describe the stress-strain relationship of materials, and it is helpful to introduce new materials into numerical simulation. In this paper, a new high manganese steel material is used to fabricate buckling restrained brace, and based on the improved Transition Markov Chain Monte Carlo (iTMCMC) sampling method and Bayesian reasoning, a new parameter identification of BWBN model is completed; model parameters are introduced into OpenSees to complete the response analysis of the new steel damper in the structure. It provides a new method and approach for introducing new materials into structural seismic resistance and proves its reliability.


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## Body

## 1. Introduction

The constitutive model of new materials is of great significance for engineering numerical simulation. Inaccurate constitutive parameters will lead to errors in numerical simulation [1, 2]. BWBN model is not only widely used in reinforced concrete frame structures [3], seismic residual displacement [4, 5], structural health monitoring, and other fields [6, 7], but also to simulate various dampers, such as yielding shear panel device [8], and magnetorheological dampers [9]. The application of BWBN model needs to clarify its parameters first. At present, there are many methods to identify the parameters of BWBN model based on experimental data, such as Genetic algorithm [10, 11], Cuckoo Search Algorithm [12], artificial neural network [13], and Transitional Markov Chain Monte Carlo method [14]. Because of the introduction of new materials, it may be necessary to constantly adjust the model parameters in order to obtain a better fitting effect. However, the parameter identification methods such as genetic algorithm and neural network algorithm cannot provide the possible value probability of parameters, which is not conducive to model updating. Once the BWBN model parameters are found, when the difference between the experimental data and the output of the mathematical model is small enough, the resulting model is regarded as a “good” approximation of the real experimental lag [15]. At the same time, because the improved Bouc–Wen model (BWBN, iTMCMC) can describe several modes matching the response of a series of hysteretic systems, this paper applies it to a “fuse”, which is made of elastic-plastic austenitic steel with low stacking fault energy and analyzes its reliability.Some scholars have compared several design concepts of viscous dampers in building structures [16], and it has been found that the energy-based design strategy provides the best distribution method among a wide group of analyzed design methodologies complying with an equal (target) damping ratio constraint. In addition, some scholars have summarized a variety of algorithms to invert the optimal design parameters of passive dampers in order to achieve the desired performance of buildings [17], Nowadays, an input velocity adjustment method of the critical double impulse was presented for the efficient design of viscous dampers for elastic-plastic moment frames [18]. It can be seen that there are many researches on algorithms for building structure health, but there are certain differences between the parameters of dampers deduced backwards for the purpose of building health and the real parameters obtained from the produced dampers. Such differences may lead to changes in the response of building structures, which will lead to adverse results.In order to protect buildings on soft soil foundation that are not suitable for seismic isolation technology from earthquake damages, it is necessary to install dampers or supports in the structure that can dissipate energy and have the same redundancy as the building [19–21]. Buckling restrained brace (BRB) is widely used in the field of building damping because of its economy and excellent energy dissipation capacity [22]. Traditional BRBs have poor low cycle fatigue performance, durability, ductility, and other related properties [23, 24], which may lead to serious damage to buildings due to the fact that BRBs cannot achieve the same redundancy with buildings after earthquakes. Some studies have found that, dampers effectively reduce the failure probabilities of the structural frame and the facility components and It shows that the damper with the same redundancy as the building can protect the building from earthquake damage to the greatest extent [25].


## 2. Modified Bouc–Wen Model

Bouc–Wen model is a smooth hysteretic model represented by differential equations. On the basis of Bouc–Wen model, Wen, Baber, and Noori [26, 27] defined strength, stiffness degradation parameters, and pinching effect function, which expanded the function of Bouc–Wen model (also known as BWBN model). So far, BWBN model is widely used to describe hysteresis in various projects. The schematic diagram of Bouc–Wen single degree of freedom model is shown in Figure 1, and the equation of motion can be expressed by the following equations: (1)mx••+cx•+akx+1−akz=Ft,(2)z•=hzηtx•−vtβx•zzn−1+γx•zn.Figure 1
BWBN model diagram.In equation (1), a is the ratio of postyield stiffness to preyield elastic stiffness a=EP/E, m is the mass, c is the damping constant, k is the stiffness coefficient, and z is the hysteresis parameter. In equation (2), vt represents strength degradation, vt=1.0+δvεt, ηt represents stiffness degradation, ηt=1.0+δηεt, and hz represents pinch effect.Some scholars have introduced the derivation process of Bouc–Wen model and the detailed process of parameter identification using traditional Bayesian inference [6, 14]. In this paper, the fourth-order Runge–Kutta method is used to solve the differential equation in the BWBN elastic measurement model as follows:(3)yn+1=yn+h6K1+2K2+2K3+K4,K1=fxn,yn,K2=fxn+12h,yn+12hK1,K3=fxn+12h,yn+12hK2,K4=fxn+h,yn+hK3,where f represents the differential equation of BWBN model, xn is the displacement in the measured data, yn=Ft is the force in the measured data, h is the Runge–Kutta time step, and K1,K2,K3,K4 are the transfer parameters in Runge–Kutta.


## 3. Model Updating Techniques

To apply the BWBN model to describe the hysteretic behavior of the system, the first problem is to find better model parameters. The random sampling method can deal with more general situations than the asymptotic approximation method. The parameter identification method based on Bayesian theory can not only give the optimal values of parameters, but also obtain the covariance matrix that quantitatively describes the uncertainty of model parameters. This is of great significance and theoretical value to improve the robustness and accuracy of system identification.Worden and Hensman [28] explained how the Bayesian method provides the same and more information as the differential evolution algorithm method, and used Markov Chain Monte Carlo (MCMC) and Metropolis–Hastings (MH) techniques to estimate the parameters of the BW model of hysteresis. Cheng et al. [29] replaced Gaussian sampling with differential evolution algorithm to form an improved DE-TMCMC method to identify the parameters of sand constitutive model. Betz et al. [30] provided three ways to improve TMCMC sampling methods. This paper synthesizes the first and third improvement measures, and uses iTMCMC algorithm to identify the parameters of BWBN model. The specific implementation method is shown in the Figure 2.Figure 2
Calculation flow chart.Letθ be the set of uncertain parameters and D represents measured data. The core of parameter estimation using probability idea is to know the probability density distribution (PDF) of parameters. According to Bayesian reasoning, the posterior distribution of the undetermined parameter PθD∝PDθPθ and θ of the undetermined parameters are assumed to be Gaussian distribution with standard deviation of 1, and the mean value of the prior distribution is shown in θ∗ in Table 1. The likelihood function of the mean square error (MSE) between the calculated D and the system estimated response of the given parameter θ∗ is shown in the following equations:(4)PDθ=∏i=1Nσ22π−1exp−12σ2Jdisθ,D2,(5)Jdisθ,D=ui−u∧iθ,where N is the amount of data, σ is the variance between the test displacement and the displacement prediction error calculated by the BWBN model with the current parameter, u is the actual displacement value of the test, and u∧ represents the average value of the displacement predicted by the BWBN model with the current parameter. For the convenience of calculation, the log-likelihood function should be used in the actual implementation of iTMCMC.Table 1
Parameter sampling range.
Paramθminθ∗θmaxα0.010.200.30β1.002.004.00γ−4.00−1.004.00n1.001.203.00υ00.101.203.00δυ−2.000.404.00Α00.501.103.00δA−2.000.103.00η00.501.304.00δη−3.000.504.00After obtaining the log-likelihood function, enter iTMCMC sampling, mainly using the first and third improvements proposed by Betz et al. [30], and the main steps are as follows:(i)
Prior distribution and likelihood function of input parameters;(ii)
Calculateqj. If qj > 1, let qj = 1, set i = 0, and extract μk0,k=1,2,……,N samples from the prior distribution.(iii)
The weighting coefficient is calculated for all the samplesμk0,k=1,2,……,N, where j=0,...,m, 0=q0<q1<⋯<qm=1, and μj,k=μj,lc.(6)wj,k=Pμj,lcDqj−qj−1.(iv)
Calculate the average value of the following weighting coefficient:(7)Sj=1NS∑k=1NSwj,k.(v)
The covariance matrix is calculated, wherein the scaling factorβ=2.4/M is adaptively adjusted as follows:(8)∑j=β2•∑k=1NSwj,kSj·NS•μj−1,k−μj−•μj−1,k−μj−T.(vi)
The indexl is randomly selected from the set 1,...,NS according to the probability wj,l/∑n=1NSwj,n to obtain μc as the initial sample to start the Markov chain. The Gaussian distribution centered on the current sample in the k-th chain and the covariance matrix ∑j are taken as μc suggested distributions. Let R be a sample in a uniform distribution of 0 to 1, if r≤Pjμc/Pjμj,lc, let μj,lc=μc. Then, proceed to the following step μj,k=μj,lc and wj,l=fμj,lcDqj−qj−1;(vii)
Ifqj=1, stop the iteration; otherwise, make j=j+1 and continue the cycle from step 1.The detailed steps can be seen from the three ways of improving TMCMC proposed by Betz et al. [30], and it is proved that the average deviation of evidence estimation obtained by iTMCMC method is small, and its performance is better than that of the original TMCMC method.


## 4. Experiment of the New Type Buckling Restrained Brace

Buckling restrained braces (BRBs) are widely favored because they have sufficient seismic performance in compression and tension. Dizaj et al. [31] proposed a shortened BRB with low yield and proved that the shortened BRB has lower potential danger. Iwata et al. [32] said that due to the longer and longer duration of the earthquake, it was necessary to study and develop new buckling restrained braces with better fatigue performance and higher energy dissipation capacity, and put forward corresponding opinions from the structure.This paper studies the identification of mechanical properties of high ductility buckling restrained energy dissipation braces (double stage axial steel damper) made of a new type of austenite steel that Yang et al. [33] of Shanghai Research Institute of materials developed. The ductility coefficient (i.e., the ratio of the ultimate displacement to the yield displacement) of the new two-stage axial steel damper is more than 12, which can realize the energy dissipation and damping effect under different earthquake intensities and achieve the same redundancy failure as the building. The test piece and test principle of high ductility double order buckling restrained brace made of new materials are shown in Figures 3 and 4.Figure 3
Buckling restrained bracings with high ductility.Figure 4
Schematic diagram of test principle.The total length of BRB core plate is 1.5 m, the thickness is 16 mm, C45 concrete is poured in the barrel, and the test loading frequency is 0.05 Hz. Because the core plate is welded by traditional ferritin steel plate and new austenite steel plate, as shown in Figure5, under cyclic loading, the weld breaks and the strength of the component decreases, thus achieving the purpose of double-stage energy consumption.Figure 5
Schematic diagram of double order constraint support.The traditional BRB core plate material is a low yield ferritin steel, and the fatigue life of ferritin steel is low during cyclic loading deformation. The new austenite steel described in this paper changes the deformation mechanism of the traditional steel, improves the fatigue resistance and ductility of the material, and the new material can miniaturize the buckling restrained brace. In this paper, a miniaturized BRB full-scale test is carried out. The maximum tensile and compressive bearing capacity of the core element after yielding in each loading cycle is not lower than the yield load. The obtained hysteresis curves are shown in Figure6.Figure 6
Experimental hysteresis cycles.From Figure6, it shows that the new BRB has almost no stiffness degradation under 30 cycles of cyclic load, and has a certain degree of strength degradation, which meets the requirements of double-order energy consumption. The displacement is within 1/40 range, and a stable, full, repeatable, and nondecaying hysteresis curve of bearing capacity is obtained.Therefore, it can be seen that the new BRBs has the following advantages:(1)
The axial damper overcomes the shortcomings of the ductility of the traditional buckling restrained brace, maximizes the deformation capacity of the damper, achieves the same seismic redundancy with the main structure, and ensures the energy dissipation in the whole process under the earthquake.(2)
Under the same deformation design, the length of the damper is shorter and easy to install. Through bolt connection, quick replacement and repair after earthquake can be realized.(3)
Based on excellent performance, optimize the main structure, reduce the consumption of traditional concrete and steel, reduce carbon emissions, and significantly improve the seismic performance of buildings to meet the needs of resilient cities.In order to apply the new type of buckling restrained brace to engineering, it is necessary to conduct simulation analysis on it during design. However, the new product cannot form its constitutive model in a short time, which is unfavorable for engineering application. Different research institutions have different models and parameter identification methods to describe the hysteresis performance of BRB. In terms of models, commonly used models include double (multiple) linear model [34], Ramberg Osgood model [35] and Bouc–Wen model [36]. These models have the advantages of smooth curve, high simulation accuracy, and strong adaptability, but at the same time, these models do not have the characteristics of describing stiffness degradation and strength degradation.


## 5. Identification

In order to further study the response of the new steel damper in the structure, it is necessary to obtain the constitutive model of the stress-strain relationship of BRB. In recent years, Bayesian model updating technology has been more and more applied to the neighborhood of hysteresis system parameter identification. Ortiz et al. [14] first used Bayesian model updating technology to identify BW model parameter that can simulate degradation effects, and found the parameter set that can accurately reproduce the system response, but did not find the accurate value of the parameters related to BWBN model. Angelikopoulos et al. [37] proposed the X-TMCMC algorithm in 2015, which combines the adaptive Kriging method with the Transitional Markov Chain Monte Carlo (TMCMC) technology to improve the parallel efficiency of the algorithm. In this paper, iTMCMC method is introduced into the parameter identification of BWBN model for the first time, which improves the calculation efficiency and achieves better model updating effect. Based on the original TMCMC method, iTMCMC algorithm mainly makes the following modifications:(i)
Adjust the sample weight after each MCMC step to reduce the average deviation of model evidence estimation(ii)
Apply aging period in MCMC step to improve posterior approximation(iii)
The target distribution scale of MCMC algorithm is adaptively selected to achieve near optimal acceptance rateiTMCMC is set to the following values: the number of samples in all stages, is set to 1000, the specified threshold of c.o.v. is set to 100%, and the aging parameter is set to 50. Each parameter is sampled independently. If the parameters are related, it is based on the edge transformation of the reconstructed joint distribution (nataf) method. The estimated value of the parameter range is shown in Figure7.Figure 7
Probability distribution of each parameter.
(a)(b)It is shown from Figure8 that the identified parameters can make the modified BW model reflect the strength degradation effect and estimate the actual second-order response of the system better. The comparison results show that the theoretical model can well simulate the mechanical performance of the composite damper under cyclic load and can replace the test results for theoretical analysis. The identification method has shown that it can find a parameter set capable of reproducing the system response. The BWBN model is a multimodal function. Although the exact values of the parameters related to the BWBN model are not estimated, the system response evaluated using the estimated parameters is sufficient for simulation purposes. The identification parameters are shown in the Table 2.Figure 8
Hysteresis cycles for large range sampling parameters.Table 2
Parameter estimation and variance.
ParameterEstimatedVarianceSignificanceα0.0511570.0012377Stiffness ratioβ4.763810.225278Hysteresis areaγ−2.599160.216215Plumpness parametern1.87050.0437203Smoothness parameter of the modelυ00.1000639.53017e − 05Strength degradationδυ0.1933830.0196162Strength degradationΑ02.961380.10239Strength and stiffness degradationδA−1.982180.0080262Strength and stiffness degradationη03.885940.0287953Stiffness degradationδη3.424360.280545Stiffness degradationSince the hysteretic system to be identified has no pinch effect, this paper does not consider the parameters controlling pinch. The BWBN model is a multimodal function [15], which allows multiple groups of parameters to achieve a good estimation of the system response. At the same time, the existing literature shows that the parameter values of the BWBN model are often in a fixed range [38, 39]. This paper finds that when parameter estimation is carried out by means of probability thinking, the parameter sampling range can be appropriately expanded and the number of sampling points can be appropriately increased to obtain a better parameter set.Comparing Figures8 and 9, it can be seen that the accuracy of the parameters taken into the model response under the sampling at a larger boundary is significantly higher than that under the sampling at a smaller boundary because of the strong correlation between the parameters [15]. Therefore, the parameter identification of the BWBN model is a global identification task, and the random sampling method may fall into a local trap. Through the iTMCMC method, adaptive selection of target distribution scale of MCMC algorithm can deal with this problem well. Using iTMCMC sampling method to estimate BWBN model parameters also has the advantage of higher efficiency. The whole calculation process is about five minutes, which is greatly improved compared with the traditional Bayesian model updating method.Figure 9
Hysteresis cycles for small range sampling parameters.


## 6. Example Analysis of Seismic Response

In order to carry out finite element simulation and shaking table test of earthquake simulation, a four story and four span space reinforced concrete frame model is designed in this paper. All beams and columns adopt the size of 18@100, Steel 02 constitutive model is selected for reinforcement, and the cross-section of the beam is 300 mm × 600 mm, the cross-section of the column is 500 mm × 500 mm, the concrete constitutive model is concrete01 constitutive model, and the restraint effect of transverse stirrup is considered for the core concrete. In order to consider the crushing phenomenon of the concrete in the protective layer, the ultimate strength of the concrete in the protective layer is taken as 30% of the peak strength of the concrete, and the ultimate strain is taken as 0.006. A uniformly distributed load of 10 kN/m is applied to each beam, and the mass of each floor is simplified to the node mass of 50 t. The BRB members are simulated by the BWBN material constitution in OpenSees, and the seismic effect is simulated by applying the inertial force on the base, and a four-story reinforced concrete frame structure is established, STKO model is shown in Figures10 and 11.Figure 10
Model diagram.Figure 11
Model diagram.According to the seismic influence coefficient curve of building structure, Tabas wave and El Centro wave are respectively adopted in this paper, and the peak acceleration of seismic wave is taken as 0.1 g, 0.2 g, and 0.25 g through amplitude modulation, and only the seismic wavex is input in the form of bottom inertial force. The ideal elastic-plastic model is adopted to consider the effect of rigid floor. Specific seismic waves are shown in Figures 12 and 13.Figure 12
Tabas, Iran 09/16/78 seismic wave.Figure 13
EI centro 270 seismic wave.Under the action of earthquake, the BRB member makes the main structure basically in the elastic range by yielding energy dissipation, to avoid the damage to the main member. Because there are many dampers arranged in this paper, the force in the damper is small under small acceleration, and a relatively complete hysteresis loop cannot be formed. Figure14 show the hysteresis curves of the dampers of the first to fourth layers under 0.25 g seismic wave.Figure 14
Hysteresis cycles of dampers under TABAS seismic waves. (a) First floor. (b) Second floor. (c) Third floor. (d) Fourth floor.
(a)(b)(c)(d)It can be seen from the figure that the unloading stiffness and the initial stiffness of the damper are almost the same, which is also in line with the experimental results. At the same time, the yield layer of BRB can be obtained through analysis, and the effect of each layer of dampers under moderate earthquakes can be seen. This can effectively guide the design and arrangement of dampers, and has strong engineering practical significance. The horizontal stiffness and bearing capacity of the empty frame are low, and the energy dissipation capacity is poor. However, the BRB reinforced frame has better seismic resistance. The horizontal force displacement hysteresis curve of the new double order buckling restrained brace in the structure is full and stable.


## 7. Conclusion

In order to determine the relationship between the force and deformation of the buckling restrained brace, a new iTMCMC method is adopted to identify the parameters of the improved BW model, and then the parameters are brought into the BW material constitutive model of OpenSees to obtain the response of the new double-order buckling restrained brace under the actual earthquake, to realize the simulation of the application of new materials in the structure. This paper mainly obtains the following conclusions:(i)
The iTMCMC method is successfully applied to the parameter identification of the improved BW model lag model. The identification method has shown its ability to find a parameter set that can accurately reproduce the system response.(ii)
In the analysis model of BRB reinforced concrete frame, the improved BW model constitutive element is adopted for the BRB frame, and the inertial force at the bottom is considered for the seismic action of the structure. The numerical analysis results of reinforced concrete frame show that the simulation results are in good agreement with the test results. The reinforced concrete frame analysis model can accurately predict the seismic performance of empty frame and BRB reinforced frame.(iii)
The improved BW model can better simulate the hysteresis characteristics of BRB. Compared with the classical BW model, it can better reflect the characteristics of plastic hardening of materials and asymmetry of mechanical parameters, to guide the design and engineering arrangement of dampers.(iv)
In view of the results of numerical experiments, the proposed iTMCMC method should be applied to other hysteresis models, such as biaxial models or asymmetric hysteresis models, to evaluate the efficiency of the algorithm in dealing with other nonlinear hysteresis models.



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