7.2. Lognormal Fragility Functions

postprocessor.calculate_lognormal_fragility(theta, sigma_record2record, sigma_build2build=0.30, sigma_ds=0.30, intensities=np.round(np.geomspace(0.05, 10.0, 50), 3))[source]

Computes the probability of exceeding a damage state using a lognormal cumulative distribution function (CDF).

Parameters:

  • theta (float) – The median seismic intensity corresponding to an EDP-based damage threshold.

  • sigma_record2record (float) – The logarithmic standard deviation representing record-to-record variability.

  • sigma_build2build (float, optional) – The logarithmic standard deviation representing building-to-building (or model) variability. Default value is 0.30.

  • sigma_ds (float, optional) – The logarithmic standard deviation representing uncertainty in damage-state thresholds. Default value is 0.30.

  • intensities (array-like, optional) – The set of intensity measure (IM) levels for which exceedance probabilities will be computed. Default is a geometric sequence from 0.05 to 10.0 with 50 points.

Returns:

poes – An array of exceedance probabilities corresponding to each intensity measure in intensities.

Return type:

numpy.ndarray

References

1) Baker JW. Efficient Analytical Fragility Function Fitting Using Dynamic Structural Analysis. Earthquake Spectra. 2015;31(1):579-599. doi:10.1193/021113EQS025M

2) Singhal A, Kiremidjian AS. Method for probabilistic evaluation of seismic structural damage. Journal of Structural Engineering 1996; 122: 1459-1467. DOI:10.1061/(ASCE)0733-9445(1996)122:12(1459)

3) Lallemant, D., Kiremidjian, A., and Burton, H. (2015), Statistical procedures for developing earthquake damage fragility curves. Earthquake Engng Struct. Dyn., 44, 1373-1389. doi: 10.1002/eqe.2522.

4) Bird JF, Bommer JJ, Bray JD, Sancio R, Spence RJS. Comparing loss estimation with observed damage in a zone of ground failure: a study of the 1999 Kocaeli Earthquake in Turkey. Bulletin of Earthquake Engineering 2004; 2: 329-360. DOI: 10.1007/s10518-004-3804-0

Theoretical Background

The lognormal fragility function is the most widely used model in earthquake engineering. It expresses the probability of exceeding a damage state as a lognormal cumulative distribution function (CDF) of the intensity measure.

Fragility model

\[P(\text{DS} \geq ds_i \mid \text{IM}) = \Phi\!\left(\frac{\ln(\text{IM}/\theta_i)}{\beta_{\text{total}}}\right)\]

where:

  • \(\Phi(\cdot)\) is the standard normal CDF,

  • \(\theta_i\) is the median IM capacity (the IM level at which there is a 50 % probability of exceeding damage state \(i\)),

  • \(\beta_{\text{total}}\) is the total logarithmic standard deviation, combining all sources of uncertainty:

\[\beta_{\text{total}} = \sqrt{\beta_{\text{r2r}}^2 + \beta_{\text{b2b}}^2 + \beta_{\text{DS}}^2}\]

with \(\beta_{\text{r2r}}\) the record-to-record variability, \(\beta_{\text{b2b}}\) the building-to-building modelling uncertainty, and \(\beta_{\text{DS}}\) the uncertainty in the damage-state threshold.