Bayesian optimal estimation for output-only nonlinear system and damage identification of civil structures

Hamed Ebrahimian, Rodrigo Astroza, Joel P. Conte*, Costas Papadimitriou

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

68 Scopus citations


This paper presents a new framework for output-only nonlinear system and damage identification of civil structures. This framework is based on nonlinear finite element (FE) model updating in the time-domain, using only the sparsely measured structural response to unmeasured or partially measured earthquake excitation. The proposed framework provides a computationally feasible approach for structural health monitoring and damage identification of civil structures when accurate measurement of the input seismic excitations is challenging (e.g., buildings with significant foundation rocking and bridges with piers in deep water) or the measured seismic excitations are erroneous and/or distorted by significant measurement error (e.g., malfunctioning sensors). Grounded on Bayesian inference, the proposed framework estimates the unknown FE model parameters and the ground acceleration time histories simultaneously, using the sparse measured dynamic response of the structure. Two approaches are presented in this study to solve the joint structural system parameter and input identification problem: (a) a sequential maximum likelihood estimation approach, which reduces to a sequential nonlinear constrained optimization method, and (b) a sequential maximum a posteriori estimation approach, which reduces to a sequential iterative extended Kalman filtering method. Both approaches require the computation of FE response sensitivities with respect to the unknown FE model parameters and the values of base acceleration at each time step. The FE response sensitivities are computed efficiently using the direct differentiation method. The two proposed approaches are validated using the seismic response of a 5-story reinforced concrete building structure, numerically simulated using a state-of-the-art mechanics-based nonlinear structural FE modeling technique. The simulated absolute acceleration response time histories of 3 floors and the relative (to the base) roof displacement response time histories of the building to a bidirectional horizontal seismic excitation are polluted with artificial measurement noise. The noisy responses of the structure are then used to estimate the unknown FE model parameters characterizing the nonlinear material constitutive laws of the concrete and reinforcing steel and the (assumed) unknown time history of the ground acceleration in the longitudinal direction of the building. The same nonlinear FE model of the structure is used to simulate the structural response and to estimate the dynamic input and system parameters. Thus, modeling uncertainty is not considered in this paper. Although the validation study demonstrates the estimation accuracy of both approaches, the sequential maximum a posteriori estimation approach is shown to be significantly more efficient computationally than the sequential maximum likelihood estimation approach.

Original languageEnglish
Article numbere2128
JournalStructural Control and Health Monitoring
Issue number4
StatePublished - Apr 2018

Bibliographical note

Funding Information:
The authors wish to thank Dr. Quan Gu at Xiamen University, China, and Dr. Frank McKenna at the Pacific Earthquake Engineering Research Center at UC Berkeley for their help with the implementation in OpenSees of the DDM for earthquake ground acceleration input as sensitivity parameters. Their assistance was most valuable and is highly appreciated. Rodrigo Astroza acknowledges the financial support from the Universidad de los Andes, Chile, through the research grant Fondo de Ayuda a la Investigación (FAI) and from the Chilean National Commission for Scientific and Technological Research (CONICYT), FONDECYT‐Iniciación research project 11160009.

Publisher Copyright:
Copyright © 2018 John Wiley & Sons, Ltd.


  • Bayesian method
  • direct differentiation method
  • joint parameter and input estimation
  • nonlinear finite element model
  • output-only system identification
  • structural health monitoring


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