Form: Time History Matched to Response Spectrum
Seismic input to nonlinear dynamic analyses of structures is usually defined in terms of acceleration time series (time-history function) whose response spectra are compatible with a specified target response spectrum. Various methods have been developed to modify a reference time series so that its response spectrum is compatible with a specified target spectrum. Two of the most widely used methods, namely the Frequency Domain Method and the Time Domain Method, are available in ETABS.
A. Frequency Domain Method
This method adjusts the Fourier amplitude spectrum, based on the ratio of the target response spectrum to the time-series response spectrum while keeping the Fourier phase of the reference time history fixed. While this approach is relatively straightforward, it does not generally have good convergence properties. Also, this approach often alters the non-stationary character of the time series to such a large degree that it no longer looks like a time series from an earthquake. Matching in the frequency domain invariably tends to increase the total energy in the ground motion.
To generate a time series using spectral matching in frequency domain, do the following:
Use the command Define > Functions > Time History, select type "Matched to a Response Spectrum", and click "Add New Function".
Choose method "Spectral Matching in Frequency Domain".
Choose a target response spectrum. The default is as per ASCE-7 2010 for US Zip Code 94704, Site Class C, and long term transition period of 8 seconds.
Choose a reference time history function. The default reference time history uses data from 1940 El Centro earthquake, North-South component (Peknold Verion). The time history has 1500 data points at equal spacing of 0.02 seconds.
Specify, if desired, the frequency range for which the spectral matching to be done. If not specified, the default is to use the entire frequency range (0.01 Hz to 100.0 Hz).
Click "match Time History" to generate the new time series.
The following procedure is performed:
Generate the response spectrum for the reference time history using the damping specified for the target response spectrum.
Generate the Fourier amplitude spectrum of the reference time history through a Fast Fourier Transform (FFT).
Determine the scale factor for all frequencies in the specified range. The scale factor is computed as:
Scale Factor for a given frequency = RSATS/RSARTH
Where,
RSATS = Acceleration of the target response spectrum at the given frequency
RSARTH = Acceleration of the response spectrum for reference time history at the given frequency
Multiply the Fourier amplitudes by the computed scale factors for all frequencies in the specified frequency range.
Do an inverse FFT on the scaled Fourier amplitude spectrum to obtain the modified time history.
B. Time Domain Method
The time domain method is generally considered a better approach for spectral matching since this method adjusts the acceleration time histories in the time domain by adding wavelets. A wavelet is a mathematical function that defines a waveform of effectively limited duration which has a zero average. The wavelet amplitude typically starts out at zero, increases, then decreases back to zero. While the time domain spectral matching procedure is generally more complicated than the frequency domain approach, it has good convergence properties and in most cases preserves the non-stationary character of the reference time series.
The time domain method was first introduced Lilhanand and Tseng (1987, 1988). Lilhanand proposed an algorithm that uses reserve impulse wavelet functions to modify the initial time histories such that its response spectrum is compatible with a target spectrum. A fundamental assumption of this methodology is that the time of the peak response does not change as a result of the wavelet adjustment. This assumption is not always valid as the time of peak response may be shifted by adding the wavelet adjustments to the acceleration time history.
The algorithm for time domain matching used here is based on the program developed by N. A. Abrahamson (1993) to implement the Lilhanand and Tseng (1987, 1988) algorithm and modified in 1999 to preserve the non-stationary character of the initial ground motion at long periods.
To generate a time series using spectral matching in frequency domain, do the following:
Use the command Define > Functions > Time History, select type "Matched to a Response Spectrum", and click "Add New Function".
Choose method "Spectral Matching in Time Domain".
Choose a target response spectrum. The default is as per ASCE 7-2010 for US Zip Code 94704, Site Class C, and long-period transition period of 8 seconds.
Choose a reference time history function. The default reference time history uses data from 1940 El Centro earthquake, North-South component (Peknold Verion). The time history has 1500 data points at equal spacing of 0.02 seconds.
Specify, if desired, the frequency range for which the spectral matching to be done. If not specified, the default is to use the entire frequency range (0.01 Hz to 100.0 Hz).
Specify the number of recursions and the recursion factor for the spectral matching in time domain. NOTE: Since the short period accelerations are influenced by the long period wavelets, spectral matching is done in multiple recursions (passes). Only the short period range of the response spectrum is matched in the first recursion. Matching of long period range of the spectrum occurs in subsequent recursions. The recursion factor (>=1.0 and <=2.0) dictates the selection of time period range for recursions following the initial recursion.
Click "Match Time History" to generate the new time series.
The following procedure is performed:
Generate the response spectrum for the reference time history using the damping specified for the target response spectrum.
Compare the resulting response spectrum ordinate (peak response of the SDOF oscillator) with the target value. Determine the mismatch for each period and damping ratio, Samisfit.
Calculate the spectral sensitivity matrix C, whose elements Cij describe the amplitude of acceleration response at time peak ti of SDOF oscillator with period Ti due to wavelet adjustment with period Tj.
Calculate the set of wavelet magnitudes, b, by solving the linear set of equation {Samisfit} = [C} {b}.
Add wavelets to the acceleration time histories with the appropriate phase and amplitude with the objective of modifying the spectral ordinates. One wavelet is added for each period to be matched.
Iterate by repeating the above steps until the largest spectral mismatch is below a given tolerance.
Display
Once the spectral matching is completed, the following information is displayed:
Response Spectrum Plots
Superimposed plots of the target response spectrum, the response spectrum of reference time history, and the response spectrum of matched time history are displayed.
Time History Plots
Superimposed plot of the reference time history and spectrally matched time history are displayed.
Frequency Content Plots
The frequency content of the reference and spectrally-matched time histories can be displayed side-by-side using the Show Frequency Content button.
References
Lilhanand, K and Tseng, W. S., 1987, "Generation of Synthetic Time Histories Compatible with Multiple Damping Response Spectra," SMiRT-9, Lausanne K2/10.
Al Atik, L. and Abrahamson, N. A., 2010, "An Improved Method for Nonstationary Spectral Matching," Earthquake Spectra 26 No 3, 601-89.