Vulnerability Functions

Linking Hazard to Damage

The vulnerability module is the bridge between physical hazard intensity and building damage. It answers a specific question for each building type at each level of hazard intensity: how badly does this type of structure get damaged? The mathematical relationships that capture this are called vulnerability functions, damage functions, or fragility curves, depending on how they are expressed.

Approaches to Vulnerability Modelling

There are two broad approaches to developing vulnerability functions:

1. Expert judgment: Noted structural engineers estimate the damage ratio that would result to a typical building of a specific type under a given intensity. Their responses are statistically combined. This approach cannot be easily updated as new data becomes available and is somewhat arbitrary by nature.
2. Engineering analysis: Building response is calculated using analytical structural models, validated against post-event damage surveys and laboratory tests. This is widely recognised as the superior approach and is the basis for modern cat model vulnerability modules.

Typical Buildings and Building Classes

Applying detailed engineering analysis to every building in a large portfolio is impractical — insurers rarely collect the level of detail needed for individual building analysis, and portfolios can contain hundreds of thousands of properties. The solution is to classify the building stock into building classes based on the most important factors affecting structural response:

• Building material (steel, reinforced concrete, masonry, wood)
• Structural system (moment frame, braced frame, shear wall)
• Height (number of storeys)
• Age (reflecting applicable building code)

A typical cat model for the United States might define 50 or more building classes. Each class is then further subdivided by secondary modifiers — details that affect vulnerability but don't define the class, such as roof pitch (for hurricane) or the presence of a cripple wall (for earthquake). For each building class, one typical building is analysed in detail, and its response is applied to all properties in the portfolio belonging to that class.

The Damage Function

A damage function relates the expected structural damage state of a building to the intensity of the event at that location. The output is typically expressed as a damage ratio — the ratio of repair cost to replacement cost, ranging from 0% (no damage) to 100% (total loss). The coefficient of variation (standard deviation divided by mean) captures the uncertainty in the damage prediction.

Earthquake Damage

Earthquake damage is typically both structural and non-structural. Engineers use fragility curves — which express the probability of reaching or exceeding a given damage state as a function of ground motion intensity. Structural damage is primarily driven by lateral building deformation (interstory drift). Non-structural damage (partitions, ceilings, mechanical systems) can dominate repair costs even when the structure itself performs well. Contents are more sensitive to peak floor acceleration than to building deformation.

Wind Damage

Wind damage is primarily non-structural, affecting the building envelope — roof coverings, windows, cladding, and garage doors. The exception is mobile homes, where roof damage can lead to partial collapse. The sequence of wind damage matters: loss of the first window or shingle allows wind to penetrate and can trigger progressive damage. Engineered reinforced concrete and steel frame buildings fare relatively well in wind but may experience major cladding and glass damage at very high wind speeds.

Separate Damage States

Modern vulnerability modules estimate separate damage states for three components of a building: the structure itself, its contents, and the time element (loss of use). These are combined to estimate overall loss. The building's construction type drives structural damage; its occupancy type (what the building is used for) drives contents damage and time element losses.