Chapter 2: Updating Equivalent Linear Models Using Response Time Histories for Tracing
2.2 Example Applications
2.2.6 Case 3: Six-story Building, Earthquake Excitation
approximately 40% higher than the nominal value. Similar discrepancies between the estimated and nominal values for the linear case (scale x0.2) are observed for the modal damping ratios and modal frequencies.
We thus identify the transition into nonlinear behavior by comparing the higher scale curves (x1.0 and x2.0) to the x0.2 scale curve, rather than comparing to the nominal values. Keeping this in mind, one can identify the expected behaviors following a transition to the non-linear regime. More specifically, in the time interval from 5 to 20 s, as the excitation level increases (scales x1.0 and 2.0), the values of the two stiffness related parameters decrease (Figure 2.9A-B), the values of the modal damping ratios increase (Figure 2.9C-E), the measure-of-fit values increase (Figure 2.10A) and the values of the modal frequencies decrease (Figure 2.10C-E) in relation to the corresponding values identified for the linear case (scale x0.2). The higher the excitation, the higher the change of these properties. The changes observed as a function of time are similar to those observed for the one-story building, although due to model error the expected decrease or increase in these model properties is not as evident as in the one-story building. Similar behaviors of the properties of the equivalent linear system are also observed in the time interval from approximately 25 to 35 seconds where, as can be seen in Figure 2.8, the acceleration response time histories show an intensity that increases as the scale of the earthquake increases from x0.2 to x1.0 and eventually to x2.0.
For the time interval from approximately 20 s to 25 s, an unexpected behavior of the properties of the equivalent linear system is observed which can be attributed to the fact that the response time histories in Figure 2.8 are relatively small and the nonlinearities may not have been activated.
Specifically, for the scale of x1.0 and x2.0 earthquake input the responses in Figure 2.8 barely reach the value of 1 to 2 m/s2 which is of the same order of magnitude as the strong motion part of the scale of x0.2 earthquake input for which the structure behaves linearly. In this time interval (from 20 to 25 s), the nonlinearities are probably not activated and thus the observed time variation of the properties of the equivalent linear system is reasonable. The observed variability of the modal properties for the three different scales (x0.2, x1.0 and x2.0) is expected and is likely due to model error.
The condition number indicates that the estimation is well-posed for most window segments with centers ranging from 5 to approximately 35 s. For window segments centered after 35 s, the condition number increases dramatically which may indicate an ill-posed estimation problem. This may be due to the fact that less than three modes participate in the response at later times (after 35 seconds). Thus a number of model parameters, such as the percent of critical damping for the third and possibly also the second mode cannot be identified from the measured response time histories, giving rise to the ill-conditioning. This ill-conditioning is also confirmed by the unreasonably high damping ratio values estimated from approximately 40 to 50 s for the third mode in Figure 2.9E.
The results in Figures 2.9 and 2.10 suggest that the estimates, especially for the modal damping ratio for the third mode, should be ignored due to ill-conditioning.
For the latest time interval from 40 s to 50 s, the measure-of-fit value is very close to zero for all excitation levels, indicating that the equivalent linear system can adequately fit the response time histories and that nonlinearities are not activated due to low amplitude response levels. The two stiffness parameters of the linear model and the modal frequencies of the three modes approach the corresponding stiffness values and the modal frequencies of the nominal model used to generate the simulated data. Differences from the nominal values and for the values obtained for the three different excitation levels are due to the model error. This is expected since the response time histories are different, and the ill-conditioning arising from the fact that one or more modes might not contribute to the response. Model error due to modal truncation is not an issue here since less than three modes are contributing to the response. In addition, the fact that less than three modes contribute to the response provides higher flexibility for the finite element model predictions to fit the data by changing the values of the two stiffness parameters and the damping ratios of the contributing modes, reducing the size of model error. This reduced model error may also explain the fact that the values of the model properties (two stiffness parameters and modal frequencies) are closer to the properties corresponding to the nominal values used to generate the simulated data.
Figure 2.8. The acceleration time history data used in case 3, simulated from the nonlinear 6- story building model using the 1995 Kobe Japan earthquake (Nishi-Akashi station) ground
excitation.
Figure 2.9. Estimated parameters of the equivalent linear model for the six-story building, using acceleration measurements (case 3).
Figure 2.10. Estimated properties of the equivalent linear model for the six-story building, using acceleration measurements (case 3).