Response Spectrum Interpolation Best Practice

Why interpolation matters

Response spectra are typically provided as tabulated values at discrete frequencies. The modal frequencies of the structure rarely coincide exactly with the tabulated frequencies, so interpolation is needed to obtain the spectral acceleration at each modal frequency. The interpolation method affects the accuracy of the spectral acceleration and hence the seismic demand.

Log-log interpolation

Response spectra are plotted on logarithmic axes because the spectral acceleration varies over several orders of magnitude across the frequency range. Log-log interpolation is the standard method: the logarithm of the spectral acceleration is linearly interpolated between the logarithms of the tabulated frequencies. This captures the curvature of the spectrum better than linear interpolation.

Handling spectrum peaks

The spectrum peak is the most sensitive region for interpolation. If the modal frequency falls between two tabulated points that straddle the peak, linear or log-log interpolation may underestimate the peak value. The spectrum should be tabulated with sufficient resolution near the peak — typically at least every 0.5 Hz or finer in the amplified region. If the spectrum is coarse near the peak, the peak value should be used directly for any modal frequency near the peak.

Interpolation in the rigid region

In the rigid region (above the amplified region), the spectral acceleration is constant at the ZPA. Interpolation in this region is trivial — the value is the ZPA. However, if the spectrum is tabulated with decreasing values above the peak (not yet at the ZPA plateau), interpolation between the peak and the ZPA plateau should be performed carefully to avoid overestimating the spectral acceleration.

Verification

• Check interpolated values at several modal frequencies against hand interpolation
• If the spectrum is provided as a curve, confirm that the solver reads the curve correctly
• If the spectrum is provided as a table, confirm the frequency points and values are entered correctly
• For modal frequencies near the spectrum peak, compare the interpolated value with the peak value to confirm the interpolation is not underestimating
• If the solver uses a different interpolation method (e.g. linear), assess whether the difference is significant

Common errors

• Using linear interpolation instead of log-log, underestimating the spectral acceleration near the peak
• Tabulating the spectrum with insufficient resolution near the peak
• Not confirming the units of the tabulated spectrum (g vs m/s^2)
• Extrapolating beyond the tabulated frequency range instead of using the ZPA or the peak value