Vulnerability Models

Introduction

Vulnerability functions quantify the economic loss ratio—the repair cost of damaged structural components normalised by their replacement cost—over a range of intensity measure (IM) levels. They are generated by combining damage state probabilities from fragility functions with damage-to-loss ratios through the total probability theorem.

Expected Loss Ratio

The vulnerability function expresses the expected loss ratio as a function of an intensity measure level (IM), obtained by convolving fragility functions with damage-to-loss ratios associated with each damage state.

Let \(P(DS = ds_i | IM)\) denote the probability of the structure being in damage state \(ds_i\) at a given intensity measure level IM, and let \(\mu_{LR,i}\) be the mean loss ratio associated with that damage state. The expected loss ratio is:

\[E[LR | IM] = \sum_{i=1}^{N_{DS}} P(DS = ds_i | IM) \cdot \mu_{LR,i}\]

where:

  • \(N_{DS}\) is the total number of discrete damage states

  • \(P(DS = ds_i | IM)\) is derived from the fragility functions

  • \(\mu_{LR,i}\) is the mean loss ratio associated with damage state \(ds_i\)

Damage-to-Loss Ratios

The damage-to-loss ratios for structural components are:

Damage-to-Loss Ratios

Damage State

Mean Loss Ratio

Description

Slight (DS1)

0.05

Minor repairs, cosmetic damage

Moderate (DS2)

0.15

Moderate repairs needed

Extensive (DS3)

0.60

Major structural repairs

Complete (DS4)

1.00

Full replacement required

Damage State Probabilities

Fragility functions are expressed in terms of probabilities of exceedance. The probability of being in a specific damage state is computed as:

\[\begin{split}P(DS = ds_i | IM) = \begin{cases} P(DS \ge ds_i | IM) - P(DS \ge ds_{i+1} | IM), & i < N_{DS} \\ P(DS \ge ds_{N_{DS}} | IM), & i = N_{DS} \end{cases}\end{split}\]

Uncertainty in Loss Estimation

Method 1: Silva (2019) Semi-Empirical

When uncertainty in the damage-to-loss relationship is not explicitly modelled, the dispersion of the loss ratio can be estimated using the formulation proposed by Silva (2019):

\[\sigma_{LR|IM} = 0.5 \sqrt{\overline{LR}_{|IM} \left(-0.7 \cdot 2 \overline{LR}_{|IM} - \sqrt{6.8 \overline{LR}_{|IM} + 0.5}\right)}\]

where \(\overline{LR}_{|IM}\) is the mean loss ratio conditional on IM.

The coefficient of variation is:

\[COV(LR | IM) = \frac{\sigma_{LR|IM}}{E[LR | IM]}\]

Method 2: Explicit Statistical Propagation

Alternatively, uncertainty can be quantified by explicitly propagating uncertainty through the convolution. Using the law of total variance:

\[Var(LR | IM) = \sum_{i=1}^{N_{DS}} P(DS = ds_i | IM) \left[\sigma_{LR,i}^2 + \left(\mu_{LR,i} - E[LR | IM]\right)^2\right]\]

This formulation captures both:

  • Uncertainty within damage states due to variability in damage-to-loss ratios

  • Uncertainty due to damage-state mixing as intensity varies

Beta Distribution Interpretation

The loss ratio conditional on IM can be modelled as a Beta distribution due to its bounded support on [0, 1]:

\[LR | IM \sim Beta(\alpha(IM), \beta(IM))\]

The Beta parameters are computed from the mean and variance:

\[\kappa(IM) = \frac{\mu(IM)[1-\mu(IM)]}{\sigma^2(IM)} - 1\]
\[\alpha(IM) = \mu(IM) \cdot \kappa(IM), \quad \beta(IM) = [1-\mu(IM)] \cdot \kappa(IM)\]

Interactive Viewer

Use the interactive viewer below to explore vulnerability functions for different building classes. All decision variables are displayed in stacked plots.

Click map to select region
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Average Annual Loss Ratio

In performance-based earthquake engineering, the Average Annual Loss Ratio (AALR) quantifies the expected loss ratio due to earthquake-induced damage over one year:

\[AALR = \int_0^{\infty} LR(IM) \cdot \frac{d\lambda(IM)}{dIM} \, dIM\]

where:

  • \(LR(IM)\) is the loss ratio from the vulnerability function

  • \(\lambda(IM)\) is the annual frequency of exceedance from the hazard curve

In practice, AALR is approximated using discrete intensity levels:

\[AALR \approx \sum_{i=1}^{N} LR(IM_i) \cdot [\lambda(IM_i) - \lambda(IM_{i+1})]\]

Regional Variations

Vulnerability functions account for regional variations in:

  • Construction practices and material quality

  • Seismic design code adoption and enforcement

  • Building maintenance and age distributions

  • Economic factors affecting repair costs

References

  • Silva V. (2019). Uncertainty and Correlation in Seismic Vulnerability Functions of Building Classes. Earthquake Spectra.

  • Martins L. et al. (2016). Development and assessment of damage-to-loss models for moment-frame reinforced concrete buildings. Earthquake Engineering & Structural Dynamics.

  • Di Pasquale G. et al. (2005). New developments in seismic risk assessment in Italy. Bulletin of Earthquake Engineering.