A procedure is offered for the analytical derivation of the seismic vulnerability of building classes, that is, probabilistic relationships between shaking and repair cost as a fraction of replacement cost new for a category of buildings. It simulates structural response, damage, and repair cost for the structural and non-structural components that contribute most to construction cost, and then scales up results to account for the components that were not simulated. It does so for a carefully selected sample of building specimens called index buildings whose designs span the domain of up to three features that are believed to most strongly influence seismic vulnerability within the building class. One uses moment matching to combine results for the index buildings to estimate behaviour and variability of the building class. One can simulate non-structural vulnerability alone by ignoring damage and repair cost for structural components. The work is written for a structural engineer with a master’s degree, skilled in structural analysis, but not necessarily experienced in loss modelling.

The procedure has five steps. In Step 1, the analyst defines the asset class with one, three, or seven specimens of the asset class; the specimens are called index buildings. The choice depends on available resources and the rigor with which the analyst wants to address variabilities within the building class and within the performance of an individual index building.  Each index building is assigned a particular structural and non-structural design, including number of stories, structural material, lateral load resisting system (LLRS), geometry, and quantities of each of the top 1 or 2 structural component categories and top 5 or 6 non-structural component categories.

Step 2 is to derive story-level vulnerability functions, without considering collapse. (Collapse is addressed in a later step.) The vulnerability functions express the repair cost of components on the story as a function of story-level excitation (drift, acceleration, or other measures of story-level structural response). Step 3 is to perform a structural analysis at each of many levels of ground motion with the objective of estimating story-level excitation and collapse probability as a function of ground motion. We offer three options for structural analysis, from a very simple approach to multiple nonlinear dynamic structural analyses; the analyst is free to choose among these, again considering available resources and desired rigor.

Step 4 is to derive a building-level vulnerability function by summing story-level losses over stories, factoring up to account for the fact that only the top 6 to 8 structural and non-structural component categories are inventoried, applying the theorem of total probability to consider the probability of collapse. By omitting the top 2 or so structural components, one can create vulnerability functions for only the non-structural components. The vulnerability function is normalized by replacement cost new to depict damage factor as a function of ground motion.

In Step 5, the mean vulnerability function and coefficient of variation of damage factor for the asset class are calculated. The mean damage factor for the asset class is calculated as a weighted average of those of the index buildings. The coefficient of variation is calculated by one of three means: using a proxy from HAZUS in the case of a single index building, as a multiple of the variability of vulnerability between index buildings in the case of three index buildings, or in the case of seven index buildings, by calculating the variance of vulnerability of the weighted sample of index-building-level vulnerability functions, including both between- and within-building variability.