3.11. Filtered Incremental Velocity

imcalculator.get_FIV3(period, alpha, beta)[source]

Computes the filtered incremental velocity (FIV3) intensity measure for a given ground-motion record.

FIV3 is computed following Dávalos and Miranda (2019). A second-order low-pass Butterworth filter is applied to the acceleration record; the filtered incremental velocity (FIV) is then obtained by integrating successive alpha*T windows. FIV3 is the maximum of the sum of the three largest peaks and the absolute sum of the three deepest troughs.

The FIV computation is fully vectorised using a cumulative-sum approach for the sliding-window trapezoidal integrals, avoiding the per-window Python loop.

Parameters:

  • period (float) – The period (s) used to define the filter cut-off frequency and integration window length.

  • alpha (float) – Period factor defining the integration window length (window duration = alpha * period).

  • beta (float) – Cut-off frequency factor for the low-pass Butterworth filter (f_c = beta / period).

Returns:

  • FIV3 (float) – FIV3 intensity measure (Eq. 3 of Dávalos & Miranda 2019).

  • FIV (numpy.ndarray) – Filtered incremental velocity time series (Eq. 2).

  • t (numpy.ndarray) – Time instants corresponding to each FIV value (s).

  • ugf (numpy.ndarray) – Low-pass-filtered acceleration time history (g).

  • pks (numpy.ndarray) – Up to three largest peaks of the FIV series.

  • trs (numpy.ndarray) – Up to three deepest troughs of the FIV series.

References

Dávalos H, Miranda E. “Filtered incremental velocity: A novel approach in intensity measures for seismic collapse estimation.” Earthquake Engineering & Structural Dynamics, 2019, 48(12), 1384-1405. DOI: 10.1002/eqe.3205.

Theoretical Background

The Filtered Incremental Velocity (FIV3) is a duration- and pulse-sensitive intensity measure designed to improve collapse prediction for structures with period elongation (Dávalos & Miranda, 2019).

Velocity time history

The ground velocity is obtained by integrating the acceleration record:

\[v(t) = \int_0^t \ddot{u}_g(\tau)\, d\tau\]

Band-pass filtering

The velocity is band-pass filtered to retain energy in the period range \([\alpha T,\, \beta T]\), where \(T\) is the structural period of interest and \(\alpha\), \(\beta\) are user-defined scale factors (typically \(\alpha = 1.0\), \(\beta = 0.7\) relative to \(T\)). The filter isolates the frequency content most damaging to the structure.

Incremental velocity pulses

The filtered velocity \(v_f(t)\) is scanned for pulses — intervals between consecutive zero-crossings. The incremental velocity of each pulse is its peak-to-trough amplitude:

\[\Delta v_k = \max_{t \in \text{pulse}_k} v_f(t) - \min_{t \in \text{pulse}_k} v_f(t)\]

FIV3

FIV3 is the sum of the three largest incremental velocity pulses:

\[\text{FIV3} = \Delta v_{(1)} + \Delta v_{(2)} + \Delta v_{(3)}\]

where \(\Delta v_{(1)} \geq \Delta v_{(2)} \geq \Delta v_{(3)}\) are the three largest values in descending order.