桥梁结构有限元模型的仿射-区间不确定修正
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摘要
为获得桥梁结构的基准状态,考虑测试和结构参数的不确定性,将区间分析、仿射算法引入响应面有限元模型修正方法中,建立了一种新的桥梁结构有限元不确定模型修正方法。在讨论结构特点及力学行为的基础上,选择了待修正结构参数和结构响应后,采用均匀试验设计方法获得试验样本,同时结合多样本的有限元分析,采用F检验法得到结构响应的显著性参数。基于有限元模型修正的响应面方法,构建结构的响应面替代模型后,引入区间分析算法的自然拓展,将响应面模型拓展为区间响应面函数,同时采用仿射算法解决区间分析的区间扩张问题,构建桥梁结构有限元模型的仿射-区间不确定修正方法,并采用遗传算法进行区间优化求解。另外,针对区间响应面有限元模型修正的具体需求,提出了区间响应面函数的两步验证方法。用斜拉桥振动台模型桥梁在不同工况下的测试模态参数和斜拉索索力,对其进行有限元模型的不确定修正,实现了实测响应与有限元计算响应间误差的最小化。区间响应面函数的两步验证证实了参数修正范围和结构响应的有效性和正确性,修正后结构纵向、横向、竖向的一阶,二阶频率以及索力的实测响应均在计算响应范围内。验证结果表明:所提有限元不确定模型修正方法,能有效实现桥梁结构有限元模型的修正。
Interval analysis and affine arithmetic were introduced into the updating process of the structural finite element model to address uncertainties of the measurement and structural parameters, and a novel uncertainty updating method of finite element model for bridge structure was carried out to obtain the baseline of the bridge structure based on the response surface method. Once the candidate structural parameters for updating and interesting structural responses were determined in light of the structural characteristics and their mechanical behaviors, the sample-sets for structural parameters were generated using uniform test design. Then, each set of generated structural parameters was applied as the input to the finite element model to evaluate the corresponding structural responses, and the significance of the parameters for interesting structural responses were found using F-test. The surrogate models for structural responses were achieved within the response surface framework for finite element model updating, and the natural extension method was then adopted to transform the obtained surrogate models to their corresponding models, called interval response surface functions. Moreover, affine arithmetic was incorporated into the updating process to control the inevitable interval overestimation problem of interval analysis, genetic algorithm was applied to optimize the updating interval, and affine-interval uncertainty updating method of the finite element model for bridge structure was performed. Furthermore, a two-step verification method was proposed to meet the specific requirements for the interval response surface based finite element model uncertainty updating. The modal parameters and cable tensions measured from a carefully instrumented model of a cable-stayed bridge tested on a shaking table under various testing cases were utilized to update their finite element model uncertainties, and the discrepancies between the measured and calculated structural responses for the model bridge were then minimized. It is shown from the two-step verification of the interval-response surface function that the rationality and feasibility for of updated structural parameters and their corresponding structural responses are guaranteed, and all the first and second longitudinal, lateral, and vertical measured frequencies as well as measured cable-tensions fall within the variation ranges of their corresponding calculated responses. It is thus demonstrated that the proposed uncertainty updating method for the finite element model is effective and feasible for updating the finite element models of real-world bridge structures.
Interval analysis and affine arithmetic were introduced into the updating process of the structural finite element model to address uncertainties of the measurement and structural parameters, and a novel uncertainty updating method of finite element model for bridge structure was carried out to obtain the baseline of the bridge structure based on the response surface method. Once the candidate structural parameters for updating and interesting structural responses were determined in light of the structural characteristics and their mechanical behaviors, the sample-sets for structural parameters were generated using uniform test design. Then, each set of generated structural parameters was applied as the input to the finite element model to evaluate the corresponding structural responses, and the significance of the parameters for interesting structural responses were found using F-test. The surrogate models for structural responses were achieved within the response surface framework for finite element model updating, and the natural extension method was then adopted to transform the obtained surrogate models to their corresponding models, called interval response surface functions. Moreover, affine arithmetic was incorporated into the updating process to control the inevitable interval overestimation problem of interval analysis, genetic algorithm was applied to optimize the updating interval, and affine-interval uncertainty updating method of the finite element model for bridge structure was performed. Furthermore, a two-step verification method was proposed to meet the specific requirements for the interval response surface based finite element model uncertainty updating. The modal parameters and cable tensions measured from a carefully instrumented model of a cable-stayed bridge tested on a shaking table under various testing cases were utilized to update their finite element model uncertainties, and the discrepancies between the measured and calculated structural responses for the model bridge were then minimized. It is shown from the two-step verification of the interval-response surface function that the rationality and feasibility for of updated structural parameters and their corresponding structural responses are guaranteed, and all the first and second longitudinal, lateral, and vertical measured frequencies as well as measured cable-tensions fall within the variation ranges of their corresponding calculated responses. It is thus demonstrated that the proposed uncertainty updating method for the finite element model is effective and feasible for updating the finite element models of real-world bridge structures.
引文
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[14] TENG J, ZHU Y, ZHOU F, et al. Finite Element Model Updating for Large Span Spatial Steel Structure Considering Uncertainties [J]. Journal of Central South University of Technology, 2010, 17 (4): 857-862.
[15] GABRIELE S, VALENTE C. An Interval-based Technique for FE Model Updating [J]. International Journal of Reliability and Safety, 2009, 3 (1/2/3):79.
[16] FANG S E, ZHANG Q H, REN W X. An Interval Model Updating Strategy Using Interval Response Surface Models [J]. Mechanical Systems and Signal Processing, 2015, 60-61: 909-927.
[17] STOLFI J, DE FIGUEIREDO L H. An Introduction to Affine Arithmetic [J]. Trends in Applied and Computational Mathematics, 2003, 4 (3): 297-312.
[18] 朱增青,陈建军,宋宗凤,等.区间参数杆系结构非概率可靠性指标的改进仿射算法[J].工程力学,2010,27(2):49-53. ZHU Zeng-qing, CHEN Jian-jun, SONG Zong-feng, et al. Non-probabilistic Reliability Index of Bar Structures with Interval Parameters Based on Modified Affine Arithmetic [J]. Engineering Mechanics, 2010, 27 (2): 49-53.
[19] 苏永华,贺启,罗正东,等.基于仿射算法的深部隧道衬砌力学状态区间分析[J].计算力学学报,2012,29(6):921-926. SU Yong-hua, HE Qi, LUO Zheng-dong, et al. Interval Analysis of Mechanical State for Deep Tunnel Lining Based on Affine Arithmetic [J]. Chinese Journal of Computational Mechanics, 2012, 29 (6): 921-926.
[20] MYERS R H, MONTGOMERY D C, ANDERSON-COOK C M. Response Surface Methodology-Process and Product Optimization Using Designed Experiments [M]. 4th ed. Hoboken: John Wiley & Sons Inc, 2016.
[21] TSHILIDZI M. Finite-element-model Updating Using Computational Intelligence Techniques [M]. Berlin: Springer, 2010
[22] 方开泰.均匀试验设计[EB/OL].(2004-10-14) [2018-11-4]. http://www.math.hkbu.edu.hk/～ktfang/. FANG Kai-tai. Uniform Design [EB/OL]. (2004-10-14) [2018-11-4]. http://www.math.hkbu.edu.hk/～ktfang/.
[23] BAKIR P G. The Combined Deterministic Stochastic Subspace Based System Identification in Buildings [J]. Structural Engineering and Mechanics, 2011, 38 (3): 315-332.
[24] 沈德建,吕西林.模型试验的微粒混凝土力学性能试验研究[J].土木工程学报,2010,43(10):14-21. SHEN De-jian, LU Xi-lin. Experimental Study on the Mechanical Property of Microconcrete in Model Test [J]. China Civil Engineering Journal, 2010, 43 (10): 14-21.
[25] HOLGER S. Cable-stayed Bridges-40 Years of Experience Worldwide [M]. Berlin: John Wiley & Sons, 2013.
[2] SIMOEN E, DE ROECK G, LOMBAERT G. Dealing with Uncertainty in Model Updating for Damage Assessment: A Review [J]. Mechanical Systems and Signal Processing, 2015, 56-57: 123-149.
[3] SHAN D, LI Q, KHAN I, et al. A Novel Finite Element Model Updating Method Based on Substructure and Response Surface Model [J]. Engineering Structures, 2015, 103: 147-156.
[4] 宗周红,高铭霖,夏樟华.基于健康监测的连续刚构桥有限元模型确认(Ⅱ)—不确定性分析与模型精度评价[J].土木工程学报,2011,44(3):85-92. ZONG Zhou-hong, GAO Ming-lin, XIA Zhang-hua. Finite Element Model Validation of the Continuous Rigid Frame Bridge Based on Structural Health Monitoring—PartⅡ:Uncertainty Analysis and Evaluation of Model Accuracy [J]. China Civil Engineering Journal, 2011, 44 (3): 85-92.
[5] 苏静波,邵国建.基于区间分析的工程结构不确定性研究现状与展望[J].力学进展,2005,35(3):338-344. SU Jing-bo, SHAO Guo-jian. Current Research and Prospects on Interval Analysis in Engineering Structure Uncertainty Analysis [J]. Advances in Mechanics, 2005, 35 (3): 338-344.
[6] 万华平,任伟新,黄天立.基于贝叶斯推理的随机模型修正方法[J].中国公路学报,2016,29(4):67-76,95. WAN Hua-ping, REN Wei-xin, HUANG Tian-li. Stochastic Model Updating Approach by Using Bayesian Inference [J]. China Journal of Highway and Transport, 2016, 29 (4): 67-76, 95.
[7] MOORE R E, KEARFOTT R B, CLOUD M J. Introduction to Interval Analysis [M].Philadelphia: Society for Industrial and Applied Mathematics, 2009.
[8] DENG Z, GUO Z, ZHANG X. Interval Model Updating Using Perturbation Method and Radial Basis Function Neural Networks [J]. Mechanical Systems and Signal Processing, 2017, 84: 699-716.
[9] 方圣恩,张秋虎,林友勤,等.不确定性参数识别的区间响应面模型修正方法[J].振动工程学报,2015,28(1):73-82. FANG Sheng-en, ZHANG Qiu-hu, LIN You-qin, et al. Uncertain Parameter Identification Using Interval Response Surface Model Updating [J]. Journal of Vibration Engineering, 2015, 28 (1): 73-82.
[10] 姜东,费庆国,吴邵庆.基于区间分析的不确定性结构动力学模型修正方法[J].振动工程学报,2015,28(3):352-358. JIANG Dong, FEI Qing-guo, WU Shao-qing. Updating of Structural Dynamics Model with Uncertainty Based on Interval Analysis [J]. Journal of Vibration Engineering, 2015, 28 (3): 352-358.
[11] 何成,陈国平,何欢.径向基模型的不确定性模型区间修正与确认[J].机械工程学报,2013,49(11):128-134. HE Cheng, CHEN Guo-ping, HE Huan. Interval Model Updating and Validation with Uncertainty Based on the Radial Basis Function [J]. Journal of Mechanical Engineering, 2013, 49 (11): 128-134.
[12] 王登刚,秦仙蓉.结构计算模型修正的区间反演方法[J].振动工程学报,2004,17(2):89-93. WANG Deng-gang, QIN Xian-rong. Interval Method for Computational Model Updating of Dynamic Structures [J]. Journal of Vibration Engineering, 2004, 17 (2): 89-93.
[13] KHODAPARAST H H, MOTTERSHEAD J E, BADCOCK K J. Interval Model Updating with Irreducible Uncertainty Using the Kriging Predictor [J]. Mechanical Systems and Signal Processing, 2011, 25 (4): 1204-1226.
[14] TENG J, ZHU Y, ZHOU F, et al. Finite Element Model Updating for Large Span Spatial Steel Structure Considering Uncertainties [J]. Journal of Central South University of Technology, 2010, 17 (4): 857-862.
[15] GABRIELE S, VALENTE C. An Interval-based Technique for FE Model Updating [J]. International Journal of Reliability and Safety, 2009, 3 (1/2/3):79.
[16] FANG S E, ZHANG Q H, REN W X. An Interval Model Updating Strategy Using Interval Response Surface Models [J]. Mechanical Systems and Signal Processing, 2015, 60-61: 909-927.
[17] STOLFI J, DE FIGUEIREDO L H. An Introduction to Affine Arithmetic [J]. Trends in Applied and Computational Mathematics, 2003, 4 (3): 297-312.
[18] 朱增青,陈建军,宋宗凤,等.区间参数杆系结构非概率可靠性指标的改进仿射算法[J].工程力学,2010,27(2):49-53. ZHU Zeng-qing, CHEN Jian-jun, SONG Zong-feng, et al. Non-probabilistic Reliability Index of Bar Structures with Interval Parameters Based on Modified Affine Arithmetic [J]. Engineering Mechanics, 2010, 27 (2): 49-53.
[19] 苏永华,贺启,罗正东,等.基于仿射算法的深部隧道衬砌力学状态区间分析[J].计算力学学报,2012,29(6):921-926. SU Yong-hua, HE Qi, LUO Zheng-dong, et al. Interval Analysis of Mechanical State for Deep Tunnel Lining Based on Affine Arithmetic [J]. Chinese Journal of Computational Mechanics, 2012, 29 (6): 921-926.
[20] MYERS R H, MONTGOMERY D C, ANDERSON-COOK C M. Response Surface Methodology-Process and Product Optimization Using Designed Experiments [M]. 4th ed. Hoboken: John Wiley & Sons Inc, 2016.
[21] TSHILIDZI M. Finite-element-model Updating Using Computational Intelligence Techniques [M]. Berlin: Springer, 2010
[22] 方开泰.均匀试验设计[EB/OL].(2004-10-14) [2018-11-4]. http://www.math.hkbu.edu.hk/～ktfang/. FANG Kai-tai. Uniform Design [EB/OL]. (2004-10-14) [2018-11-4]. http://www.math.hkbu.edu.hk/～ktfang/.
[23] BAKIR P G. The Combined Deterministic Stochastic Subspace Based System Identification in Buildings [J]. Structural Engineering and Mechanics, 2011, 38 (3): 315-332.
[24] 沈德建,吕西林.模型试验的微粒混凝土力学性能试验研究[J].土木工程学报,2010,43(10):14-21. SHEN De-jian, LU Xi-lin. Experimental Study on the Mechanical Property of Microconcrete in Model Test [J]. China Civil Engineering Journal, 2010, 43 (10): 14-21.
[25] HOLGER S. Cable-stayed Bridges-40 Years of Experience Worldwide [M]. Berlin: John Wiley & Sons, 2013.