Vulnerability analysis

Vulnerability curves provide the expected level of loss for a given level of ground shaking. The most frequent method to estimate vulnerability is through the convolution between fragility functions and a damage-to-loss model.

Damage to loss models

Damage-to-loss models (also know as consequence models) establish the connection between the level of damage and the expected loss (see examples in Figure 1).

Figure. 1 Example of damage-to-loss models. From left to right: Di Pasquale and Goretti (2001) (Italy); Kappos et al. (2006) (Greece); Bal et al. (2008) (Turkey) and HAZUS (FEMA 2014) (California).

Uncertainty around the mean loss can be modelled through a beta distribution (Martins et al 2016).

Computing vulnerability curves

Converting fragility into vulnerability functions is then numerically performed from the equation below using a damage-to-loss model that correlates each damage state with the respective probability distribution of loss ratio.

\[E[LR|IM]=\sum_{i=1}^{nDS}\sum_{j=1}^{m}\left(P[DS=ds_i|IM]*LR_{i,j}*P[LR_{i,j}\right)\]

Standard deviation on expected loss

When not explicitly modelled, the uncertainty in the mean loss ratio can be estimated from the equation by Silva (2019) (see below)

\[\sigma_{LR}|GMR=0.5*\sqrt{E[LR_{GMR}|IM]*\left(-0.7-2*E[LR_{GMR}|IM]+\sqrt{6.8*E[LR_{GMR}|IM]+0.5}\right)}\]

The uncertainty in the loss given an intensity measure (LR|IM) appears to have little impact on average annual losses but significantly influences return period losses. The reduction in the coefficient of variation was determined through sensitivity analysis using a range of reduction factors. These results were subsequently validated using impact assessment applications based on real-world earthquake scenarios from the Global Earthquake Impact Database.

References

Bal İE, Crowley H, Pinho R, Gülay FG. (2008) Detailed assessment of structural characteristics of Turkish RC building stock for loss assessment models. Soil Dynamics and Earthquake Engineering; 28(10–11):914–932.DOI:10.1016/j.soildyn.2007.10.005.

Di Pasquale G, Goretti A. (2001) Vulnerabilità funzionale ed economica degli edifici residenziali colpiti dai recenti eventi sismici italiani in Proceedings of the 10th National Conference ‘L’ingegneria sismica in Italia’. Potenza-Matera, Italy

FEMA. (2014) HAZUS-MH MR5, technical manual, Department of Homeland Security—Federal Emergency Management Agency

Kappos A, Panagopoulos G, Panagiotopoulos C, Penelis G. (2006) A hybrid method for the vulnerability assessment of R/C and URM buildings. Bulletin of Earthquake Engineering; 4(4):391–413. DOI:10.1007/s10518-006-9023-0.

Martins, L., Silva, V., Marques, M., Crowley, H. and Delgado, R. (2016) Development and assessment of damage-to-loss models for moment-frame reinforced concrete buildings. Earthquake Engineering & Structural Dynamics, 45(5): p. 797-817. DOI: 10.1002/eqe.2687

Silva, V. (2019) Uncertainty and correlation in seismic vulnerability functions of building classes. Earthquake Spectra DOI: 10.1193/013018eqs031m.