Probabilistic finite element model updating method of cable-stayed bridges based on Bayesian flow

doi:10.3969/j.issn.1674-0696.2026.09.03

Abstract

Abstract: To achieve efficient and accurate probabilistic finite element model updating (PFEMU) of cable-stayed bridges, a PFEMU method of cable-stayed bridges based on Bayesian flow was proposed. Firstly, in the proposed method, Bayesian flow was adopted to establish a reversible mapping between the simulated measurement data of the FEM and the model updating parameters. Secondly, based on the actual measurement data of the bridge, the posterior distribution of the model updating parameters was inferred through Bayesian flow, thereby realizing the PFEM of the bridge. The proposed method provided an “offline training and online inference” framework for the PFEMU of a bridge. Once the training was completed, the Bayesian flow model could quickly infer the posterior distribution of the updating parameters in near real time based on new measurement data. The proposed method was applied to the FEMU of an in-service long-span cable-stayed bridge, and the results were compared with those obtained from the bridge completion load test. The results show that after model updating, the average relative errors of the calculated cable forces under dead load and modal frequencies are only 1.38% and 1.90% respectively. Compared with the results of the initial FEM, the relative error of the mid-span deflection increment of the main girder under test vehicle load calculated by the updated model is reduced from 10.37% to 1.08%, and the relative error of the maximum cable force increment is reduced from 18.98% to 5.95%. The effectiveness of the proposed method has been verified through engineering examples.

Key words: bridge engineering; cable-stayed bridges; Bayesian flow; offline training; online model correction; structural health monitoring

摘要： 为实现高效且准确的斜拉桥概率有限元模型修正，提出一种基于贝叶斯流的斜拉桥概率有限元模型修正方法：该方法先采用贝叶斯流，建立有限元模型的模拟测量数据与模型修正参数之间的可逆映射；再基于桥梁的实际测量数据，通过贝叶斯流推断模型修正参数的后验分布，从而实现桥梁概率有限元模型修正。文中方法提供了一个桥梁概率有限元模型修正的“离线训练、在线推断”框架；一旦训练完成，贝叶斯流模型可基于新的测量数据，近乎实时地快速给出修正参数的后验分布。文中方法应用于一座在役大跨斜拉桥的有限元模型修正，并与桥梁竣工验收荷载试验结果进行对比。结果表明：模型修正后计算得到的恒载索力和模态频率的平均相对误差分别仅为1.38%和1.90%；与未修正的初始有限元模型相比，由修正模型计算得到的试验车载作用下主梁跨中挠度增量的相对误差从10.37%降至1.08%，最大索力增量的相对误差从18.98%降至5.95%。工程实例应用验证了文中方法的有效性。

关键词: 桥梁工程；斜拉桥；贝叶斯流；离线训练；在线模型修正；结构健康监测

CLC Number:

U448.27

Liu Xun1,2, Lin Xuechun3, Zhuo Weidong2, Zheng Haofeng4, Gu Yin2. Probabilistic finite element model updating method of cable-stayed bridges based on Bayesian flow[J]. Journal of Chongqing Jiaotong University(Natural Science), 2026, 45(9): 24-32.

刘迅1,2，林学春3，卓卫东2，郑豪峰4，谷音2. 基于贝叶斯流的斜拉桥概率有限元模型修正方法[J]. 重庆交通大学学报（自然科学版）, 2026, 45(9): 24-32.

/ Recommend

References

［1］罗岚炘, 宋明明, 钟华强, 等. 考虑运营荷载的大跨径拱桥层次贝叶斯模型修正方法［J］. 振动与冲击, 2025, 44(1): 288-297.
Luo Lanxin, Song Mingming, Zhong Huaqiang, et al. Hierarchical Bayesian model updating method for long-span arch bridge considering operational loads［J］. Journal of Vibration and Shock, 2025, 44(1): 288-297.
［2］周红利, 周广东, 刘凯凯, 等. 基于PSO-GRNN的大跨桥梁有限元模型修正方法［J］. 东南大学学报(自然科学版), 2024, 54(6): 1489-1495.
Zhou Hongli, Zhou Guangdong, Liu Kaikai, et al. Finite element model updating method for long span bridges based on PSO-GRNN［J］. Journal of Southeast University (Natural Science Edition), 2024, 54(6): 1489-1495.
［3］ Goller B, Schuller G I. Investigation of model uncertainties in Bayesian structural model updating［J］. Journal of Sound and Vibration, 2011, 330(25): 6122-6136.
［4］ Wan Huaping, Ren Weixin. Stochastic model updating utilizing Bayesian approach and Gaussian process model［J］. Mechanical Systems and Signal Processing, 2016, 70-71: 245-268.
［5］ Asadollahi P, Huang Yong, Li Jian. Bayesian finite element model updating and assessment of cable-stayed bridges using wireless sensor data［J］. Sensors, 2018, 18(9): 3057.
［6］ Mao Jianxiao, Wang Hao, Li Jian. Bayesian finite element model updating of a long-span suspension bridge utilizing hybrid Monte Carlo simulation and Kriging predictor［J］. KSCE Journal of Civil Engineering, 2020, 24(2): 569-579.
［7］ Fang Shengen, Chen Shan, Lin Youqin, et al. Probabilistic damage identification incorporating approximate Bayesian computation with stochastic response surface［J］. Mechanical Systems and Signal Processing, 2019, 128: 229-243.
［8］ Kitahara M, Bi Sifeng, Broggi M, et al. Nonparametric Bayesian stochastic model updating with hybrid uncertainties［J］. Mechanical Systems and Signal Processing, 2022, 163: 108195.
［9］ Yin Tao, Zhu Hongping. An efficient algorithm for architecture design of Bayesian neural network in structural model updating［J］. Computer-Aided Civil and Infrastructure Engineering, 2020, 35(4): 354-372.
［10］ Radev S T, Mertens U K, Voss A, et al. BayesFlow: learning complex stochastic models with invertible neural networks［J］. IEEE Transactions on Neural Networks and Learning Systems, 2022, 33(4): 1452-1466.
［11］ Warey A, Raul V, Kaushik S, et al. Generative inverse design of aerodynamic shapes using conditional invertible neural networks［J］. Journal of Computing and Information Science in Engineering, 2023, 23(3): 031006.
［12］ Bister T, Erdmann M, Kthe U, et al. Inference of cosmic-ray source properties by conditional invertible neural networks［J］. The European Physical Journal C, 2022, 82(2): 171.
［13］ Kobyzev I, Prince S J D, Brubaker M A. Normalizing flows: an introduction and review of current methods［J］. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021, 43(11): 3964-3979.
［14］ Helton J C, Davis F J. Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems［J］. Reliability Engineering & System Safety, 2003, 81(1): 23-69.
［15］ Zhang Yong, Hou Zhichao. A model updating method based on response surface models of reserved singular values［J］. Mechanical Systems and Signal Processing, 2018, 111: 119-134.
［16］ Shi Zhou, Hong Yu, Yang Shili. Updating boundary conditions for bridge structures using modal parameters［J］. Engineering Structures, 2019, 196: 109346.
［17］ Boscato G, Russo S, Ceravolo R, et al. Global sensitivity-based model updating for heritage structures［J］. Computer-Aided Civil and Infrastructure Engineering, 2015, 30(8): 620-635.
［18］张立业, 汪波, 王兵见, 等. 基于响应面法的斜拉桥动力模型修正方法［J］. 公路交通科技, 2022, 39(8): 46-52.
Zhang Liye, Wang Bo, Wang Bingjian, et al. A method for modifying dynamic model of cable-stayed bridge based on response surface method［J］. Journal of Highway and Transportation Research and Development, 2022, 39(8): 46-52.
［19］齐东春, 王宇飞, 李琦, 等. 基于联合静动力修正模型的斜拉桥换索方案［J］. 重庆交通大学学报(自然科学版), 2023, 42(11): 1-8.
Qi Dongchun, Wang Yufei, Li Qi, et al. Cable replacement scheme of cable-stayed bridge based on joint static and dynamic modification model［J］. Journal of Chongqing Jiaotong University (Natural Science), 2023, 42(11): 1-8.
［20］秦世强, 袁永刚, 韩硕, 等. 基于试验数据和位移置信准则的铁路斜拉桥多目标模型修正［J］. 铁道学报, 2023, 45(6): 151-160.
Qin Shiqiang, Yuan Yonggang, Han Shuo, et al. Multi-objective model updating of cable-stayed bridge based on experimental data and displacement assurance criterion［J］. Journal of the China Railway Society, 2023, 45(6): 151-160.

[1]	ZHAO Hua, LU Keyu, SUN Tao, LIU Rong, LU Lizhi. Mechanical Performance and Parametric Analysis of Novel Steel-UHPC Joints without Cell [J]. Journal of Chongqing Jiaotong University(Natural Science), 2023, 42(1): 27-35.
[2]	WANG Zhen1,2, ZHOU Jianting1,3, ZHANG Jinquan1,4, WU Haijun1, WU Yuexing1. Influence Analysis of Cast-in-Situ Segment Length on Mechanical Behavior of Ten-Thousand-Ton Horizontal Swivel Cable-Stayed Bridge [J]. Journal of Chongqing Jiaotong University(Natural Science), 2020, 39(12): 44-52.
[3]	LI Yang. Risk Assessment of Dismantling Project of Long-Span Cable Stayed Bridges [J]. Journal of Chongqing Jiaotong University(Natural Science), 2019, 38(07): 34-39.
[4]	WANG Gaosi. Temperature Effect of Cable-Stayed Bridge with Single Inclined Pylon with Different Side-Anchor Styles [J]. Journal of Chongqing Jiaotong University(Natural Science), 2017, 36(8): 6-11.
[5]	WANG Gaosi. Mechanical Characteristics of EarthAnchored CableStayed Bridge with Single Inclined Pylon and Single Span [J]. Journal of Chongqing Jiaotong University(Natural Science), 2017, 36(5): 1-6.
[6]	WANG Feng, Chen Fuqing, WEN Xiaoxu,PENG Zhang. Analysis of Cable Parametric Vibration Model and Response with Consideration of Temperature Effect [J]. Journal of Chongqing Jiaotong University(Natural Science), 2016, 35(2): 1-5.
[7]	Zou Li, Peng Xumin, Wei Dongsheng. Accurate Calculation Method of Equal Tensile Force of Parallel Strand Stayed-Cable [J]. Journal of Chongqing Jiaotong University(Natural Science), 2015, 34(2): 7-9.
[8]	Zhao Lei, Jia Shaomin, Gu Xiang. Simplified Calculation Method of Unstressed Length of Cable-Stayed Bridge with Separate Twin-Decks [J]. Journal of Chongqing Jiaotong University(Natural Science), 2014, 33(6): 5-8.
[9]	PU Qianhui,Zhao Hu. Influence of Ratio of Side-Span to Mid-Span and Length of Unsupported Deck on Static and Dynamic Performance of Single Pylon Cable-Stayed Bridges [J]. Journal of Chongqing Jiaotong University(Natural Science), 2013, 32(6): 1101-1105.
[10]	Liu Yiming , Liu Dayang , Liu Shanhong. Optimization of Cable Forces of Cable-Stayed Bridges Based on MATLAB and ANSYS Software under Dead Loads [J]. Journal of Chongqing Jiaotong University(Natural Science), 2013, 32(6): 1111-1114.
[11]	PU Qianhui,Zhao Hu. Influence of Ratio of Side-Span to Mid-Span and Length of Unsupported Deck on Static and Dynamic Performance of Single Pylon Cable-Stayed Bridges [J]. Journal of Chongqing Jiaotong University(Natural Science), 2013, 32(6): 1101-1105.
[12]	Liu Yiming , Liu Dayang , Liu Shanhong. Optimization of Cable Forces of Cable-Stayed Bridges Based on MATLAB and ANSYS Software under Dead Loads [J]. Journal of Chongqing Jiaotong University(Natural Science), 2013, 32(6): 1111-1114.
[13]	Shan Deshan，Ding Dehao，Li Qiao，Huang Zhen. Finite Element Model Updating of Single Pylon Cable-Stayed Bridges Based on ＲBF-ANN [J]. Journal of Chongqing Jiaotong University(Natural Science), 2013, 32(4): 555-559.
[14]	Zhang Yangyong. Limitation Span of Self-Anchored Cable-Stayed Bridge ( II) [J]. Journal of Chongqing Jiaotong University(Natural Science), 2013, 32(3): 369-373.
[15]	XieMingzhi ，BuYizhi ，Wei Ran 3，Li Shaopeng. UnstressedCableLengthandGeometricLinearControl of HybridGirder Cable-StayedBridgewith1000mSaleSpan [J]. Journal of Chongqing Jiaotong University(Natural Science), 2013, 32(3): 374-378.