李金朋(1994-), 男, 主要从事结构抗震不确定性研究, E-mail:
Li Jinpeng (1994-), main research interest:uncertainty analysis, E-mail:
侯和涛(通信作者), 男, 教授, 博士, 博士生导师, E-mail:
Hou Hetao(corresponding author), professor, PhD, doctorial supervisor, E-mail:
形状记忆合金(shape memory alloy,简称为SMA)具有"超弹性",即在受到应力而发生较大变形并卸载后,可以恢复原始形状,并在这个过程中耗散能量,在建筑抗震和桥梁振动控制中具有广阔的应用前景。SMA的模型参数通常由优化方法来确定,然后用于装有SMA装置的结构地震时程响应分析中。利用Metropolis-Hasting算法(简称为MH算法)中的改进算法DRAM方法(延迟拒绝及自适应采样),基于经过"预拉伸"和热处理的SMA棒材循环拉伸试验结果,对SMA改进的Graesser & Cozzarelli模型参数进行采样,从SMA的本构模型参数和耗能能力两个方面分析了SMA材料的不确定性。建立了各参数的后验分布,并得到了参数两两之间的相关性,结果可用于概率模型的建立及基础模型数学形式的研究。研究表明,在累积概率密度为15%时,材料的能量耗散能力相对误差高达20%;累积概率密度为85%时,相对误差为10%。
Shape memory alloy(SMA) has "super elasticity", that is, it can recover original shape after deformation and unloading due to stress, and dissipate energy in this process. It has broad application prospect in seismic control of buildings and bridge vibration. The model parameters of SMA are often determined through optimization and treated as deterministic for dynamic analysis of structures with SMA based devices. In this study, the modified Metropolis-Hasting algorithm-DRAM algorithm, which is a combination of delay rejection and adaptive sampling, is utilized to characterize the uncertainties in modified Graesser & Cozzarelli SMA model parameters. A series of SMA bars with the same geometric size and heat treatment were tested under cyclic loads. The Markov Chain Monte Carlo (MCMC) method is applied to analyze the uncertainties of SMA in terms of model parameters and energy dissipation capacity. The analysis provide insight into the underlying mathematical form of a model, suggest simplifications or modifications and begin to indicate the relative significance of individual parameters, based on a limited set of experimental data. Besides, research shows thatthe energy dissipation of the SMA bar could have up to a relative error of 20% and 10% corresponding to the CDF of 15% and 85%.
形状记忆合金(SMA)具备形状记忆,这使其在经历较大幅度的变形后,可通过加热或者卸载恢复原本形状[
笔者提出对模型参数进行概率建模的方法,基于SMA棒材的实验数据,采用改进的Graesser和Cozzarelli模型与MCMC算法的组合来分析模型本身固有的不确定性,将模型参数视为随机变量,采用Metropolis-Hastings算法来生成样本参数集,揭示了参数的概率特性和参数之间的潜在相关性,并从模型参数的角度研究了SMA模型中固有的不确定性及其对材料能量耗散能力预测值的影响。
基于Ozdemir的一维滞回模型[
式中:
但是,该模型无法模拟SMA由奥氏体相向马氏体相转化完成后出现的“硬化”现象,即材料的弹性模量突然增大,称为SMA的马氏体硬化特性。为了描述这一特性,Qian等[
改进后,相较于式(1),式(4)中参数
相较于式(2),式(5)添加了描述马氏体硬化的表达式。在SMA进入马氏体硬化阶段前,这部分值为0,此时式(5)与式(2)完全相同。
采用粒子群优化(PSO)方法[
SMA试件原材料的化学成分
Chemical composition of the testing SMA bar
化学成分 | 含量下限/% | 含量上限/% | |
Ti | 43.96 | 44.75 | |
Ni | 55.21 | 56.01 | |
C | 00.00 | 0.009 | |
N | 0.000 | 0.003 | |
H | 0.000 | 0.006 | |
O | 0.024 | 0.027 |
SMA循环拉伸试验
Cyclic tests of SMA bar
将热处理后的试件进行循环拉伸试验,其加载制度如
加载制度和试验及优化模型模拟结果示意图
Loading protocol and comparison between experiment results and PSO methods
基于循环拉伸试验数据的PSO优化参数值
Model parameter values from cyclic tests of SMA bars using PSO method
34 641.69 | 662.83 | 57.73 | 0.003 6 | 1.233 | 0.023 | 23 609.67 | 0.017 7 | 3.649 | 1.216 |
马尔可夫链蒙特卡洛(MCMC)是一种通过建立一条按照提议分布π(
采用MCMC方法中两个重要方法的结合,即延迟拒绝法(DR法,Delaying Rejection)和自适应采样(AM法,Adaptive Metropolis Samplers),简称为DRAM方法[
似然函数定义为给定的一组参数值下,模型模拟结果与实验数据一致的概率,也可以将其视为模型预测和实验测量之间的误差概率。
式中:
DRAM模拟方法的具体流程如
DRAM模拟方法的流程示意图
Flow chart of simulation process
在模拟开始之前,首先要建立参数的先验概率,包括参数的范围及其在该范围内的分布。研究表明,先验分布并不是MCMC模拟得到参数所收敛的后验分布的决定性因素,而是影响收敛速度的关键因素[
参数的先验概率设置
Prior of model parameters
参数 | 初始值 | 下限值 | 上限值 | 先验均值 | 先验方差 |
注:NaN为MATLAB中“非数值”的IEEE®算术表示; | |||||
34 641.69 | 1.5×104 | 5.0×104 | NaN | Inf | |
662.83 | 400 | 1 200 | |||
57.73 | 0 | 500 | |||
0.003 6 | 0 | 0.04 | |||
1.233 | 0 | 5 | |||
0.023 | 0 | 0.1 | |||
23 609.67 | 0 | 1.0×105 | |||
0.017 7 | 0 | 0.1 | |||
3.649 | 0 | 100 | |||
1.216 | 0 | 4 |
基于峰值应变为8%的加载循环试验数据得到的参数不确定性分析结果如
参数分布的均值、标准差和偏度
Moments of model parameters
参数 | 均值 | 标准差 | 偏度 |
3.570×104 | 6 229.640 7 | -0.291 1 | |
611.118 | 50.072 4 | -0.235 9 | |
277.056 | 128.087 7 | -0.497 2 | |
0.163 | 0.068 8 | -0.540 1 | |
1.348 | 0.882 7 | -1.208 4 | |
0.047 | 0.024 6 | -0.533 2 | |
5.414×104 | 2.803 4×104 | -0.093 2 | |
0.019 | 0.0126 | -0.695 8 | |
4.110 | 0.496 2 | -0.568 4 | |
2.197 | 0.833 0 | -0.492 4 |
参数频率分布直方图
Histogram of model parameters
参数相关性示意图
Correlation between the simulated parameters
为了更好地说明不确定性研究的必要性,研究通过模型参数的概率分布建立材料耗能能力的概率特征,其概率密度示意图如
基于能量耗散能力预测的不确定性分析结果示意图
The results based on the energy dissipation analysis
能量耗散通常通过等效粘性阻尼(EVD)
式中:
基于不同应变峰值的EVD不确定性分析结果示意图
Uncertainty analysis of EVD for different peak-strain
1) 基于形状记忆合金棒材循环拉伸试验数据的DRAM算法采样得到的马尔可夫链体现出模型参数的概率特性。样本的分布特征(均值、方差等)体现出优化方法可能存在偏差,部分参数之间存在线性相关性,在进行数值模型研究时应予以重视。
2) 数值模型的不确定性也体现在模型的耗能预测上,在累积概率密度为15%时,材料的能量耗散能力相对误差高达20%;累积概率密度为85%时,相对误差为10%。加载应变峰值对材料的耗能性能有明显影响,等效粘滞阻尼分布显示,加载峰值应变为6%时,材料耗能性能较其他对比组更好。
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