Chapter 2: Updating Equivalent Linear Models Using Response Time Histories for Tracing
2.3 Conclusions
Figure 2.13. Estimated properties of the equivalent linear model for the six-story building, using acceleration measurements (case 4).
regime as a function of time during a strong intensity earthquake event and under various sources of model error manifested in the analysis was demonstrated using simulated data.
Overall, the methodology does a reasonable job of identifying the transition in the plastic regime and tracing the nonlinear behavior with time. The results for the single-story building display clearly the expected behavior as the ground acceleration intensity increases and the nonlinearities get stronger. The results for the six-story building demonstrate that the proposed methodology can identify reasonably well the nonlinearities and the extent of nonlinearities activated during an earthquake, although the expected behavior of the identified structural and modal properties is not as clear as in the single-story building due to the various sources of model error manifested in the analysis performed. More specifically:
❑ As the equivalent linear model tries to fit strongly non-linear behavior, the result for the fit between the simulated response time histories and the model predicted response time histories worsen. Thus, higher misfit values are identified, as expected, for larger scale levels of the excitation that progressively activate nonlinearities.
❑ Because of softening nonlinearities, the equivalent stiffness parameter values drop and the equivalent modal frequencies decrease as compared to the stiffness parameter values and the modal frequency values corresponding to excitations that do not activate nonlinear phenomena.
❑ The plastic (hysteresis loop) energy dissipation raises, as expected, the values of the percent critical damping identified using the equivalent linear viscous model.
Although results are shown for four representative cases, a number of additional cases have been considered and demonstrate that the framework is promising for identifying the nonlinearities with time during an earthquake by fitting an equivalent linear finite element model to different time windows using a moving time window approach. One can use the framework with different inputs, stiffness parameter selections, number of modes, time windows, as well as measures-of-fit functions to perform the structural identification.
The method presented in this chapter is a single-stage approach for directly estimating stiffness- related model parameters and modal damping ratios of an equivalent linear finite element model of a structure using measured response time histories. Using the estimated model parameters, the equivalent linear finite element model can be used to compute the modal frequencies and the modeshapes of the structure. Using a moving time window approach, the proposed method traces the model properties along each window.
The proposed method should be contrasted to the previously published two-stage linear finite element model updating methods. In the first stage the modal properties (modal frequencies, modal damping ratios and modeshape components at the measured DOF) are identified using input- output response time histories (e.g. accelerations) (Beck, 1979; Beck & Jennings, 1980). Using a moving window approach, the first stage is useful to detect damage or even the nonlinearities activated in each time window by tracing the change in global properties of the structure, such as modal frequencies, modal damping ratios and modeshapes. However, the first stage is not sufficient to map the changes in the modal properties to changes in the stiffness characteristics of the structure. This is achieved in the second stage where the stiffness-related properties of the finite element model are estimated using the modal frequencies and modeshapes identified in the first stage. Finite element model updating techniques based on modal properties has been developed and widely used since the early work by Friswell and Mottershead (Mottershead & Friswell, 1993;
Friswell & Mottershead, 1995). A review can also be found in Moaveni (2007) for deterministic methods and Huang et al. (2019b) for probabilistic methods. In particular, Moaveni et al. (2010;
2011) have applied linear finite element model updating based on modal properties on a reduced- scale 7-story concrete building to trace the identified stiffness changes of the equivalent linear model from the damage distribution in building floors during progressively stronger excitation levels.
The two-stage method is applicable to linear systems, since it involves modal properties defined only for linear models. In the present approach a single-stage model updating method is used based on response time histories. The single-stage approach presented here is based directly on response time histories and not the modal features; it is thus more general since it can be extended to update
nonlinear finite element models as well. The two approaches (single stage and two-stage) are not expected to give exactly the same results but they are expected to give similar results for the identified stiffness properties as well as modal properties. The first step in the two-stage approach has certain advantages since the change in the modal properties can be sufficient for providing the necessary information about the system nonlinearities and the strength of nonlinearities or for tracing these nonlinearities over the different time windows. However, in order to associate the modal property changes to changes in the equivalent stiffness properties, one needs to use the second stage as well.