System identification of structures using their measured earthquake response can play a key role in structural health monitoring, structural control and improving performance-based design. Implementation using data from strong seismic shaking is complicated by the nonlinear hysteretic response of structures. Furthermore, this inverse problem is ill-conditioned; for example, even if some components in the structure show substantial yielding, others will exhibit nearly elastic response, producing no information about their yielding behavior. Classical least-squares or maximum likelihood estimation will not work with a realistic class of hysteretic models because it will be unidentifiable based on the data. It is shown here that Bayesian updating and model class selection provide a powerful and rigorous approach to tackle this problem when implemented using a recently developed stochastic simulation algorithm called Transitional Markov Chain Monte Carlo. The updating and model class selection is performed on a previously-developed class of Masing hysteretic structural models that are relatively simple yet can give realistic responses to seismic loading. The theory for the Masing hysteretic models, and the theory used to perform the updating and model class selection, are presented and discussed. An illustrative example is given that uses simulated dynamic response data and shows the ability of the algorithm to identify hysteretic systems even when the class of models is unidentifiable based on the data.
Akaike, H., 1974, “A new look at the statistical identification model ,” IEEE Transactions on Automatic Control19, 716—723.
2.
Ashrafi, S.A., Smyth, A.W., and Betti, R., 2005, “A parametric identification scheme for non—deteriorating and deteriorating non—linear hysteretic behavior.” Structural Control and Health Monitoring, 13, 108—131.
3.
Beck, J.L., 1978, “Determining models of structures from earthquake records.” Technical Report EERL 78—01, Earthquake Engineering Research Laboratory, California Institute of Technology , Pasadena, CA.
4.
Beck, J.L., 1989, “Statistical system identification of structures ,” in Proceedings of the 5th International Conference on Structural Safety and Reliability, San Francisco, CA.
5.
Beck, J.L., 1996, “System identification methods applied to measured seismic response,” in Proceedings of the 11th World Conference on Earthquake Engineering, Acapulco , Mexico.
6.
Beck, J.L. and Au, S.—K. , 2000, “Updating robust reliability using Markov Chain simulation,” in Proceedings of the International Conference on Monte Carlo Simulation, Monte Carlo, Monaco.
7.
Beck, J.L., and Au, S.—K. , 2002, “Bayesian updating of structural models and reliability using Markov Chain Monte Carlo simulation ,” Journal of Engineering Mechanics128, 380—391.
8.
Beck, J.L. and Jayakumar, P., 1996, “Class of Masing models for plastic hysteresis in structures,” in Proceedings of ASCE Structures Congress XIV, Chicago, IL.
9.
Beck, J.L. and Katafygiotis, L.S. , 1991, “Updating of a model and its uncertainties utilizing dynamic test data,” in Proceedings of the First International Conference on Computational Stochastic Mechanics, Corfu, Greece.
10.
Beck, J.L. and Katafygiotis, L.S. , 1998, “Updating models and their uncertainties — Bayesian statistical framework,” Journal of Engineering Mechanics124, 455—461.
11.
Beck, J.L. and Yuen, K.—V. , 2004, “Model selection using response measurements: Bayesian probabilistic approach,” Journal of Engineering Mechanics130, 192—203.
12.
Benedettini, F., Capecchi, D., and Vestroni, F., 1995, “Identification of hysteretic oscillators under earthquake loading by nonparametric models,” Journal of Engineering Mechanics121, 606—612.
13.
Box, G.E.P. and Jenkins, G.M., 1970, Time Series Analysis, Forecasting and Control, Holden-Day, San Francisco, CA.
14.
Box, G.E.P. and Tiao, G.C., 1992, Bayesian Inference in Statistical Analysis, Wiley, New York.
15.
Ching, J. and Chen, Y.—J. , 2007, “Transitional Markov Chain Monte Carlo method for Bayesian model updating, model class selection and model averaging,” Journal of Engineering Mechanics , in press.
16.
Ching, J., Muto, M., and Beck, J.L., 2005, “Bayesian linear structural model updating using Gibbs sampler with modal data,” in Proceedings of the 9th International Conference on Structural Safety and Reliability , Rome, Italy,
17.
Ching, J., Muto, M., and Beck, J.L., 2006, “Structural model updating and health monitoring with incomplete modal data using Gibbs sampler,” Computer-Aided Civil and Infrastructure Engineering21, 242—257.
18.
Chiang, D.-Y., 1992, “Parsimonious modeling of inelastic systems.” Technical Report EERL 92—02, Earthquake Engineering Research Laboratory , California Institute of Technology, Pasadena, CA.
19.
Cifuentes, A.O., 1984, “System identification of hysteretic structures .” Technical Report EERL 84-04, California Institute of Technology, Pasadena, CA.
20.
Cifuentes, A.O. and Iwan, W.D., “ Nonlinear system identification based on modeling of restoring force behavior ,” Soil Dynamics and Earthquake Engineering8, 2—8.
21.
Cox, R.T., 1961, The Algebra of Probable Inference, Johns Hopkins Press, Baltimore, MD.
22.
Fishman, G.S., 1996, Monte Carlo: Concepts, Algorithms, and Applications , Springer, New York.
23.
Geman, S. and Geman, D., 1984, “Stochastic relaxation, Gibbs distribution and the Bayesian restoration of images,” IEEE Transactions on Pattern Analysis and Machine Intelligence6, 721—741.
24.
Gilks, W.R., Richardson, S., and Spiegelhalter, D.J. , 1996, Markov Chain Monte Carlo in Practice, Chapman and Hall, London.
25.
Gull, S.F., 1989, “Bayesian inductive inference and maximum entropy ,” in Maximum Entropy and Bayesian Methods, J. Skilling (ed.), Kluwer, Dordrecht and Boston, MA.
26.
Hall, J.F., 2006, “Problems encountered from the use (or misuse) of Rayleigh damping,” Earthquake Engineering and Structural Dynamics35, 525—545.
27.
Hastings, W.K., 1970, “Monte Carlo sampling methods using Markov Chains and their applications,” Biometrika57,97—109.
28.
Iwan, W.D., 1966, “A distributed-element model for hysteresis and its steady-state dynamic response,” Journal of Applied Mechanics33, 893—900.
29.
Iwan, W.D., 1967, “On a class of models for the yielding behavior of continuous or complete systems,” Journal of Applied Mechanics34, 612—617.
30.
Jayakumar, P., 1987, “Modeling and identification in structural dynamics .” Technical Report EERL 87—01, California Institute of Technology, Pasadena, CA.
31.
Jayakumar, P. and Beck, J.L., 1988, “System identification using nonlinear structural models,” in Structural Safety Evaluation based on System Identification Approaches, H. G. Natke and J. T. P. Yao (eds), Vieweg, Wiesbaden.
32.
Jaynes, E.T., 2003, Probability Theory: the Logic of Science, Cambridge University Press, Cambridge and New York.
33.
Jeffreys, H., 1961, Theory of Probability, Clarendon Press , Oxford.
34.
Katafygiotis, L.S. and Lam, H.—F. , 2002, “Tangential—projection algorithm for manifold representation in unidentifiable model updating problems ,” Earthquake Engineering and Structural Dynamics31, 791—812.
35.
Kullback, S. and Leibler, R.A., 1951, “On information and sufficiency,” The Annals of Mathematical Statistics22, 79—86.
36.
Masing, G., 1926, “Eigenspannungen und verfestigung beim messing (Self—stretching and hardening for brass),” in Proceedings of the Second International Congress on Applied Mechanics , Zurich, Switzerland.
37.
MacKay, D.J.C., 1991, “Bayesian methods for adaptive models,” PhD thesis, California Institute of Technology, Pasadena, CA.
MacKay, D.J.C., 2003, Information Theory, Inference, and Learning Algorithms , Cambridge University Press, Cambridge and New York.
40.
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., and Teller, E., 1953, “Equations of state calculations by fast computing machines,” Journal of ChemicalPhysics21, 1087—92.
41.
Neal, R.M., 1993, “Probabilistic inference using Markov Chain Monte Carlo methods.” Technical Report CRG— TR—93—01, Department of Computer Science, University of Toronto, Canada.
42.
Papadimitriou, C., Beck, J.L., and Katafygiotis, L.S. , 2001, “Updating robust reliability using structural test data,” Probabilistic Engineering Mechanics16, 103—113.
43.
Papadimitriou, C., and Katafygiotis, L.S. , 2005, “Bayesian modeling and updating,” in Engineering Design Reliability Handbook , D. M. Ghiocel and N. Nikolaidis (eds), CRC Press, Boca Raton, FL.
44.
Robert, C.P. and Casella, G., 2004, Monte Carlo Statistical Methods, 2nd edn, Springer, New York.
45.
Rubinstein, R.Y., 1981, Simulation and the Monte Carlo Method, Wiley, New York.
46.
Schwartz, G., 1978, “Estimating the dimension of a model,” Annals of Statistics6, 461—464.
47.
Shannon, C.E., 1948, “A mathematical theory of communication.” The Bell System Technical Journal27, 379—423, 623—656.
48.
Thyagarajan, R.S., 1989, “Modeling and analysis of hysteretic structural behavior.” Technical Report EERL 89—03, Earthquake Engineering Research Laboratory, California Institute of Technology , Pasadena, CA.
49.
Valanis, K.C., 1971, “A theory of viscoplasticity without a yield surface ,” Architectural Mechanics23, 517—551.
50.
Wen, Y.—K., 1976, “Method for random vibration of hysteretic systems ,” Journal of Engineering Mechanics102, 249—263.
