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

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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, 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: (i) a sequential maximum likelihood (ML) estimation approach, which reduces to a sequential nonlinear constrained optimization method, and (ii) a sequential maximum a posteriori (MAP) 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 (DDM). The two proposed approaches are validated using the seismic response of a five-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 three 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 for estimating 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 MAP estimation approach is shown to be significantly more efficient computationally than the sequential ML estimation approach.
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Bayesian optimal estimation for output-only nonlinear system and damage
identification of civil structures
Hamed Ebrahimian 1, Rodrigo Astroza 2, Joel P. Conte 3, Costas Papadimitriou 4
1 Postdoctoral Scholar, Department of Mechanical and Civil Engineering, California Institute of
Technology, USA
2 Assistant Professor, Faculty of Engineering and Applied Sciences, University of Los Andes,
Chile
3 Professor, Department of Structural Engineering, University of California San Diego, USA
4 Professor, Department of Mechanical Engineering, University of Thessaly, Greece
Abstract
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, 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: (i) a sequential maximum
likelihood (ML) estimation approach, which reduces to a sequential nonlinear constrained
optimization method, and (ii) a sequential maximum a posteriori (MAP) 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 (DDM). The two proposed
approaches are validated using the seismic response of a five-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 three 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 for estimating the dynamic input and system parameters. Thus, modeling
uncertainty is not considered in this paper. Although the validation study demonstrates the
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estimation accuracy of both approaches, the sequential MAP estimation approach is shown to be
significantly more efficient computationally than the sequential ML estimation approach.
Keywords: Output-only system identification; joint parameter and input estimation; nonlinear
finite element model; Bayesian method; stochastic filter; direct differentiation method; structural
health monitoring.
1. Introduction
Structural damage identification (ID) based on linear modal ID is perhaps the most popular
approach for structural health monitoring (SHM) (e.g., [1], [2], [3], [4]). In this method, the
modal parameters of an equivalent linear-elastic viscously damped model of the structure of
interest are identified before and after a potentially damage-inducing event using low-amplitude
vibration data. The structural damage is then detected as a statistically significant change in the
identified modal parameters before and after the loading event. The location and extent of
damage in the structural system is usually determined through linear finite element (FE) model
updating using modal parameters (e.g., [5]). Nevertheless, SHM based on linear modal ID has
been criticized for important shortcomings, the most important of which is the underlying
assumption of linear response behavior of civil structures [6].
Different frameworks for mechanics-based nonlinear FE model updating have been developed
recently (e.g., [7], [8], [9]). These frameworks provide an advanced approach for system and
damage ID, and SHM of civil structures, which overcome the important shortcomings of the
damage ID methods based on linear modal ID. In this approach, the measured input excitation
and output response of the structure are utilized to update, in the time domain, a state-of-the-art
mechanics-based nonlinear FE model able to capture the potential damage and failure
mechanisms of the structure of interest. The updated FE model can then be interrogated to
reconstruct the structural response during the potentially damaging event and provide detailed
information about various characteristics of damage in the structural components and system.
However, measuring the complete earthquake input excitation for real-world civil structures is
often difficult. For example, beyond the traditional case of measuring incomplete and/or
erroneous (i.e., distorted by significant measurement noise) seismic input ground accelerations
for civil structures, measuring the seismic input excitation in the case of underground structures,
multi-span bridges spanning over deep water, and buildings with multiple underground stories
can be challenging, if not impossible. Therefore, state-of-the-art input-output nonlinear FE model
updating frameworks for structural health monitoring and damage ID need to be extended to
account for the effects of unknown, and/or erroneous input earthquake excitation. This is
specifically the objective of this paper, which proposes a framework for joint structural system
parameters and input identification.
Estimating the unknown input forces in structural systems has been the subject of past studies.
Huang et al. (