Bayesian Finite Element Model Updating and Assessment of Cable-Stayed Bridges Using Wireless Sensor Data
<p>Jindo Bridges. The bridge on the left is the subject of this study.</p> "> Figure 2
<p>Finite element model of the Jindo Bridge.</p> "> Figure 3
<p>(<b>a</b>) Hierarchical binary tree for mass sensitivities; (<b>b</b>) Sensitivity matrices of mass clusters. The “Mode” axis indicates the first 9 vertical, 4 lateral and 1 torsional modes.</p> "> Figure 4
<p>Cluster analysis result for four types of physical parameters for the bridge girder. Elements with the same color are from one cluster (blue box: cluster 1, green box: cluster 2, red box: cluster 3).</p> "> Figure 5
<p>Samples of the initial and final stages of TMCMC for the three cases with constant, updating and marginalizing error precisions.</p> "> Figure 6
<p>A sample measured acceleration time history from Jindo Deck on 5 June 2012.</p> "> Figure 7
<p>Identified modal frequencies of the 1st vertical mode during the monitoring period.</p> "> Figure 8
<p>Plot of samples in the structural model parameter space generated at the initial, middle and final stages of TMCMC for model class <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">M</mi> <mn>1</mn> </msub> <mo>.</mo> </mrow> </semantics></math></p> "> Figure 9
<p>Plot of samples in the structural model parameter space generated at the initial, middle and final stages of TMCMC for model class <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">M</mi> <mn>2</mn> </msub> <mo>.</mo> </mrow> </semantics></math></p> "> Figure 10
<p>Prior and posterior predictions of natural frequencies for model class <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">M</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">M</mi> <mn>2</mn> </msub> <mo>.</mo> </mrow> </semantics></math></p> ">
Abstract
:1. Introduction
2. Bayesian Framework for FE Model Updating and Assessment
2.1. Structural Model Class
2.2. Bayesian Modeling
2.3. Bayesian Updating and Model Class Assessment
2.4. Proposed Bayesian Inference Method by Using a New Treatment of Prediction-Error Precision Parameters and TMCMC Sampler
2.5. Two Other Treatments for the Uncertain Prediction-Error Precision Parameters
3. Illustrative Examples
3.1. Jindo Bridge FE Model
3.2. Selection of Uncertain Structural Model Parameters
3.3. Numerical Investigation on the Comparison of the Prediction-Error Precision Parameter Treatments
3.4. Real-World Application of Bayesian FE Model Updating on Jindo Bridge
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Perera, R.; Fang, S.; Huerta, C. Structural crack detection without updated baseline model by single and multiobjective optimization. Mech. Syst. Signal Process. 2009, 23, 752–768. [Google Scholar] [CrossRef]
- Kabban, C.; Uber, R.; Lin, K.; Lin, B.; Bhuiyan, M.Y.; Giurgiutiu, V. Uncertainty evaluation in the design of structural health monitoring systems for damage detection. Aerospace 2018, 5, 45. [Google Scholar] [CrossRef]
- Zhang, Q.; Chang, T.Y.P.; Chang, C.C. Finite-element model updating for the Kap Shui Mun cable-stayed bridge. J. Bridge Eng. 2001, 6, 285–293. [Google Scholar] [CrossRef]
- Zapico, J.; Gonzalez, M.; Friswell, M.; Taylor, C.; Crewe, A. Finite element model updating of a small scale bridge. J. Sound Vib. 2003, 268, 993–1012. [Google Scholar] [CrossRef]
- Jang, J.; Smyth, A.W. Model updating of a full-scale FE model with nonlinear constraint equations and sensitivity-based cluster analysis for updating parameters. Mech. Syst. Signal Process. 2017, 83, 337–355. [Google Scholar] [CrossRef]
- Perera, R.; Marin, R.; Ruiz, A. Static-dynamic multi-scale structural damage identification in a multi-objective framework. J. Sound Vib. 2013, 332, 1484–1500. [Google Scholar] [CrossRef]
- Beck, J.L.; Katafygiotis, L. Updating of a model and its uncertainties utilizing dynamic test data. In Computational Stochastic Mechanics; Springer: Berlin, Germany, 1991; pp. 125–136. [Google Scholar]
- Beck, J.L. Bayesian system identification based on probability logic. Struct. Control Health Monit. 2010, 17, 825–847. [Google Scholar] [CrossRef]
- Behmanesh, I.; Moaveni, B. Bayesian FE model updating in the presence of modeling errors. In Model Validation and Uncertainty Quantification; Springer: Berlin, Germany, 2014; Volume 3, pp. 119–133. [Google Scholar]
- Goller, B.; Schueller, G.I. Investigation of model uncertainties in Bayesian structural model updating. J. Sound Vib. 2011, 330, 6122–6136. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Huang, Y.; Beck, J.L. Hierarchical sparse Bayesian learning for structural health monitoring with incomplete modal data. Int. J. Uncertain. Quantif. 2015, 5, 139–169. [Google Scholar] [CrossRef]
- Behmanesh, I.; Moaveni, B. Probabilistic identification of simulated damage on the Dowling Hall footbridge through Bayesian finite element model updating. Struct. Control Health Monit. 2015, 22, 463–483. [Google Scholar] [CrossRef]
- Jang, J.; Smyth, A. Bayesian model updating of a full-scale finite element model with sensitivity-based clustering. Struct. Control Health Monit. 2017, 24, e2004. [Google Scholar] [CrossRef]
- Arangio, S.; Bontempi, F. Structural health monitoring of a cable-stayed bridge with Bayesian neural networks. Struct. Infrastruct. Eng. 2015, 11, 575–587. [Google Scholar] [CrossRef]
- Kuok, S.C.; Yuen, K.V. Investigation of modal identification and modal identifiability of a cable-stayed bridge with Bayesian framework. Smart Struct. Syst. 2016, 17, 445–470. [Google Scholar] [CrossRef]
- Ni, Y.C.; Zhang, Q.W.; Liu, J.F. Dynamic Property Evaluation of a Long-Span Cable-Stayed Bridge (Sutong Bridge) by a Bayesian Method. Int. J. Struct. Stab. Dyn. 2018. [Google Scholar] [CrossRef]
- Asadollahi, P.; Li, J.; Huang, Y. Prediction-error variance in Bayesian model updating: A comparative study. In Proceedings of the SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring, Portland, OR, USA, 25–29 March 2017. [Google Scholar]
- Simoen, E.; Papadimitriou, C.; Lombaert, G. On prediction error correlation in Bayesian model updating. J. Sound Vib. 2013, 332, 4136–4152. [Google Scholar] [CrossRef]
- Huang, Y.; Beck, J.L.; Li, H. Bayesian system identification based on hierarchical sparse Bayesian learning and Gibbs sampling with application to structural damage assessment. Comput. Methods Appl. Mech. Eng. 2017, 318, 382–411. [Google Scholar] [CrossRef] [Green Version]
- Beck, J.L.; Yuen, K.-V. Model selection using response measurements: Bayesian probabilistic approach. J. Eng. Mech. ASCE 2004, 130, 192–203. [Google Scholar] [CrossRef]
- Muto, M.; Beck, J.L. Bayesian updating and model class selection for hysteretic structural models using stochastic simulation. J. Vib. Control 2008, 14, 7–34. [Google Scholar] [CrossRef]
- Jaynes, E.T. Probability Theory: The Logic of Science; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Vanik, M.W.; Beck, J.; Au, S. Bayesian probabilistic approach to structural health monitoring. J. Eng. Mech. ASCE 2000, 126, 738–745. [Google Scholar] [CrossRef]
- Beck, J.L.; Katafygiotis, L.S. Updating models and their uncertainties. I: Bayesian statistical framework. J. Eng. Mech. ASCE 1998, 124, 455–461. [Google Scholar] [CrossRef]
- Ching, J.; Chen, Y.-C. Transitional Markov chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging. J. Eng. Mech. ASCE 2007, 133, 816–832. [Google Scholar] [CrossRef]
- Ching, J.; Wang, J.-S. Application of the transitional Markov chain Monte Carlo algorithm to probabilistic site characterization. Eng. Geol. 2016, 203, 151–167. [Google Scholar] [CrossRef]
- Robert, C.P.; Casella, G. Monte Carlo Statistical Methods, 2nd ed.; Fienberg, S., Ed.; Springer: Berlin, Germany, 2004. [Google Scholar]
- Beck, J.L.; Au, S.-K. Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation. J. Eng. Mech. ASCE 2002, 128, 380–391. [Google Scholar] [CrossRef]
- Zuev, K.M.; Beck, J.L. Asymptotically independent Markov Sampling: A new MCMC scheme for Bayesian inference. Int. J. Uncertain. Quantif. 2013, 3, 445–474. [Google Scholar]
- Caicedo, J.M. Structural Health Monitoring of Flexible Civil Structures. Ph.D. Thesis, Washington University, St. Louis, MO, USA, 2003. [Google Scholar]
- Dyke, S.J.; Caicedo, J.M.; Turan, G.; Bergman, L.A.; Hague, S. Phase I Benchmark Control Problem for Seismic Response of Cable-Stayed Bridges. J. Struct. Eng. 2002, 129, 857–872. [Google Scholar] [CrossRef]
- Caicedo, J.M.; Dyke, S.J.; Moon, S.J.; Bergman, L.A.; Turan, G.; Hague, S. Phase II benchmark control problem for seismic response of cable-stayed bridges. Struct. Control. Health Monit. 2003, 10, 137–168. [Google Scholar] [CrossRef] [Green Version]
- Shahverdi, H.; Mares, C.; Wang, W.; Mottershead, J. Clustering of parameter sensitivities: Examples from a helicopter airframe model updating exercise. Shock Vib. 2009, 16, 75–87. [Google Scholar] [CrossRef]
- Everitt, B.S.; Landau, S.; Leese, M.; Stahl, D. Cluster Analysis (Wiley Series in Probability and Statistics); Wiley: Chichester, UK, 2011. [Google Scholar]
- Rice, J.A.; Spencer, B.F., Jr. Flexible Smart Sensor Framework for Autonomous Full-Scale Structural Health Monitoring. NSEL Report Series, No. 18, University of Illinois at Urbana-Champaign. 2009. Available online: http://hdl.handle.net/2142/13635 (accessed on 27 July 2018).
- Jo, H.; Sim, S.H.; Nagayama, T.; Spencer, B.F., Jr. Development and application of high-sensitivity wireless smart sensors for decentralized stochastic modal identification. J. Eng. Mech. ASCE 2012, 138, 683–694. [Google Scholar] [CrossRef]
- Asadollahi, P.; Li, J. Statistical analysis of modal properties of a cable-stayed bridge through long-term structural health monitoring with wireless smart sensor networks. In Proceedings of the SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring, Las Vegas, NV, USA, 20–24 March 2016. [Google Scholar]
- Asadollahi, P.; Li, J. Statistical Analysis of Modal Properties of a Cable-Stayed Bridge through Long-Term Wireless Structural Health Monitoring. J. Bridge Eng. 2017, 22, 04017051. [Google Scholar] [CrossRef]
- James, G.H., III; Carrie, T.G.; Lauffer, J.P. The natural excitation technique (NExT) for modal parameter extraction from operating wind turbines. Int. J. Anal. Exp. Modal Anal. 1995, 10, 260–277. [Google Scholar]
- Juang, J.-N.; Pappa, R.S. An eigensystem realization algorithm for modal parameter identification and model reduction. J. Guid. Control Dyn. 1985, 8, 620–627. [Google Scholar] [CrossRef]
- Huang, Y.; Beck, J. Full Gibbs Sampling Procedure for Bayesian System Identification incorporating Sparse Bayesian Learning with Automatic Relevance Determination. Comput-Aided Civ. Inf. 2018, 33, 712–730. [Google Scholar] [CrossRef]
- Huang, Y.; Beck, J.; Li, H. Multi-task Sparse Bayesian Learning with Applications in Structural Health Monitoring. Comput-Aided Civ. Inf. 2018. [Google Scholar] [CrossRef]
Model Class | Bridge Girder Parameters | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Mass | Young’s Modulus | Moment of Inertia | ||||||||||
m | E | Iyy | Ixx | |||||||||
m1 | m2 | m3 | E1 | E2 | E3 | Iyy1 | Iyy2 | Iyy3 | Ixx1 | Ixx2 | Ixx3 |
Uncertain Parameters in | Coefficient | |
---|---|---|
Mass | m1 | 1.1 |
m2 | 1 | |
m3 | 0.9 | |
Young Modulus | E1 | 1.05 |
E2 | 1 | |
E3 | 0.95 | |
Moment of Inertia | Iyy1 | 0.95 |
Iyy2 | 1 | |
Iyy3 | 1.05 | |
Ixx1 | 1.05 | |
Ixx2 | 1 | |
Ixx3 | 0.95 |
Mode * | Case with Constant Error Precisions | Case with Error Precisions Updated | Case with Error Precisions Marginalized | |||
---|---|---|---|---|---|---|
Freq. Errors (%) | MAC | Freq. Errors (%) | MAC | Freq. Errors (%) | MAC | |
VM-1 | −6.91 | 0.997 | −6.70 | 0.997 | −0.30 | 1.000 |
VM-2 | −2.60 | 0.997 | −2.19 | 0.997 | 1.18 | 1.000 |
VM-3 | −2.84 | 0.990 | −2.53 | 0.992 | 0.31 | 0.999 |
VM-4 | −1.51 | 0.993 | −1.27 | 0.996 | 1.02 | 0.999 |
VM-5 | −1.33 | 0.996 | −0.85 | 0.996 | 0.87 | 1.000 |
VM-6 | −1.08 | 0.994 | −0.63 | 0.996 | −0.45 | 0.950 |
VM-7 | 0.05 | 0.997 | 0.57 | 0.997 | −0.75 | 0.994 |
VM-8 | 0.18 | 0.996 | 0.75 | 0.996 | −2.46 | 0.990 |
VM-9 | 1.60 | 0.997 | 2.25 | 0.998 | −2.00 | 0.974 |
LM-1 | −6.54 | 0.998 | −6.17 | 0.998 | −3.56 | 1.000 |
LM-2 | −3.87 | 0.998 | −3.52 | 0.997 | 0.19 | 0.999 |
LM-3 | −5.37 | 0.988 | −4.99 | 0.988 | 0.27 | 0.999 |
LM-4 | −4.75 | 0.985 | −4.43 | 0.987 | 1.92 | 0.878 |
TM-1 | −6.27 | 0.992 | −5.93 | 0.992 | 1.42 | 0.987 |
Uncertain Parameters | Lower Bound | Upper Bound | ||
---|---|---|---|---|
Mass | m | m1 | 8 | 18 |
m2 | ||||
m3 | ||||
Young Modulus | E | E1 | 9.5 | 10.5 |
E2 | ||||
E3 | ||||
Moment of Inertia | Iyy | Iyy1 | 9.5 | 10.5 |
Iyy2 | ||||
Iyy3 | ||||
Ixx | Ixx1 | 9.5 | 10.5 | |
Ixx2 | ||||
Ixx3 |
Model Class | Model Class | ||||||
---|---|---|---|---|---|---|---|
Parameters | Mean | C.O.V. | Changes (%) | Parameters | Mean | C.O.V. | Changes (%) |
m | 17.55 | 2.51 × 10−3 | 75.53 | m1 | 12.20 | 8.42 × 10−4 | 21.97 |
m2 | 15.55 | 4.51 × 10−4 | 55.55 | ||||
m3 | 17.95 | 3.97 × 10−4 | 79.54 | ||||
E | 10.06 | 1.92 × 10−3 | 0.65 | E1 | 10.21 | 2.89 × 10−4 | 2.13 |
E2 | 9.67 | 4.78 × 10−5 | −3.32 | ||||
E3 | 9.98 | 5.01 × 10−4 | −0.22 | ||||
Iyy | 9.62 | 8.03 × 10−4 | −3.81 | Iyy1 | 9.50 | 3.51 × 10−4 | −4.96 |
Iyy2 | 10.19 | 1.91 × 10−4 | 1.92 | ||||
Iyy3 | 9.67 | 2.56 × 10−3 | −3.29 | ||||
Ixx | 9.74 | 1.46 × 10−3 | −2.65 | Ixx1 | 9.79 | 4.56 × 10−4 | −2.14 |
Ixx2 | 9.95 | 8.63 × 10−4 | −0.46 | ||||
Ixx3 | 9.95 | 2.31 × 10−4 | −0.46 |
Mode | Identified Natural Freq. (Hz) | Prediction Means of Natural Frequencies (Hz) (Error %) | ||
---|---|---|---|---|
Prior Predictions | Posterior Predictions ( | Posterior Predictions ( | ||
VM-1 | 0.44 | 0.61 (−39.12%) | 0.47 (−7.76%) | 0.48 (−9.87%) |
VM-2 | 0.65 | 0.88 (−36.08%) | 0.68 (−5.23%) | 0.71 (−8.58%) |
VM-3 | 1.03 | 1.35 (−31.10%) | 1.03 (−0.18%) | 1.06 (−2.63%) |
VM-4 | 1.34 | 1.53 (−13.85%) | 1.16 (13.71%) | 1.20 (10.26%) |
VM-5 | 1.57 | 1.67 (−6.45%) | 1.26 (19.70%) | 1.32 (15.75%) |
VM-6 | 1.65 | 2.06 (−24.88%) | 1.59 (3.46%) | 1.65 (0.25%) |
VM-7 | 1.88 | 2.59 (−37.77%) | 1.93 (−2.88%) | 2.00 (−6.15%) |
VM-8 | 2.27 | 3.22 (−41.76%) | 2.42 (−6.44%) | 2.50 (−10.07%) |
VM-9 | 2.82 | 3.72 (−31.88%) | 2.78 (−1.27%) | 2.81 (−0.22%) |
LM-1 | 0.33 | 0.46 (−39.51%) | 0.35 (−6.32%) | 0.36 (−8.34%) |
LM-2 | 0.82 | 1.17 (−42.81%) | 0.89 (−8.70%) | 0.88 (−7.18%) |
LM-3 | 1.81 | 2.07 (−13.34%) | 1.60 (−12.50%) | 1.66 (−9.25%) |
LM-4 | 3.36 | 2.95 (−12.33%) | 2.24 (−33.34%) | 2.32 (−31.09%) |
TM-1 | 1.83 | 1.92 (−6.02%) | 1.47 (−18.52%) | 1.53 (−15.45%) |
Model Class | ||||
---|---|---|---|---|
0.00 | −394,469 | −394,591 | 121 | |
1.00 | −357,125 | −357,278 | 153 |
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Asadollahi, P.; Huang, Y.; Li, J. Bayesian Finite Element Model Updating and Assessment of Cable-Stayed Bridges Using Wireless Sensor Data. Sensors 2018, 18, 3057. https://doi.org/10.3390/s18093057
Asadollahi P, Huang Y, Li J. Bayesian Finite Element Model Updating and Assessment of Cable-Stayed Bridges Using Wireless Sensor Data. Sensors. 2018; 18(9):3057. https://doi.org/10.3390/s18093057
Chicago/Turabian StyleAsadollahi, Parisa, Yong Huang, and Jian Li. 2018. "Bayesian Finite Element Model Updating and Assessment of Cable-Stayed Bridges Using Wireless Sensor Data" Sensors 18, no. 9: 3057. https://doi.org/10.3390/s18093057