Fragility Curves

Fragility curves are X-Y plots relating a hazard parameter (e.g. PGA) to the damage probability of a struc-ture for a certain damage state (U.S. FEMA, 1997). They are frequently used to estimate natural hazard, especially earthquake, damage to engineered structures such as housings, bridges, or industrial plant units. Typically, fragility curves are prepared by statistical analysis of historical natural hazard and dam-age data (O’Rourke and So, 2000). Recently, fragility curves based on computational numerical analysis and modeling studies have also been developed (Berahman and Behnamfar, 2009).

A typical fragility curve is given in Figure 1.

RAPID-N uses fragility curves in the damage estimation part of the risk assessment, details of which are given in the "Risk Assessment" section. For the definition of fragility curves, the system features a generic fragility curve framework supporting different hazard and plant unit types by utilizing damage classifications. Fragility curves can be described by various mathematical functions and their validity conditions can be indicated. The system is also capable of visualizing the fragility curves in an interactive manner. Fragility curve records are used to store fragility curve data. The data fields of the fragility records are listed in Table 1.

Table 1. Fragility Curve data fields

For each fragility curve, a descriptive name and an abbreviation should be provided. The damage classification that the fragility curve is based on should be selected. The type of plant unit, for which the fragility curve is valid (e.g. storage tank), should be specified. The list of plant unit types is linked to the damage classification and updated automatically to list plant unit types that are valid for the selected damage classification. The hazard parameter (independent variable) of the fragility curve should be selected from the list of hazard parameters. The list includes properties that are only defined for the hazard type of the selected damage classification. If the parameter is a numerical property with a base unit, the unit used for the fragility curve should be indicated explicitly (Figure 2).

The type of the fragility curve denotes the data source of the curve. Pre-defined fragility curves are based on data available in the scientific literature. Custom fragility curves are based on historical natech damage data available in the database. Support for custom fragility curve preparation is currently under development and will be available in the next version of RAPID-N. For pre-defined fragility curves, four different functional forms are supported, which are listed in Table 2. Most, if not all, fragility curves found in the literature are in one of these functional forms. Hence, there should be no difficulty in add-ing fragility curves to the RAPID-N database.

Table 2. Functional forms supported for fragility curves

For pre-defined fragility curves, curve data should be entered separately for each damage state. It is not mandatory to enter data for all damage states. Therefore, a subset of damage states can be used. Dam-age states have to be unique, i.e. it is not possible to enter different damage definitions for the same damage state. In addition to the damage state, its context either as equal (=) or equal or greater (≥) should also be indicated for each data row. The standard deviation, and mean or median should be en-tered for each data row for log-normal fragility curves. Probit curves require k1 and k2 constants. For curves defined by a dataset, semi-colon separated parameter and probability values should be entered. The number of parameter and probability values should be equal to each other. In order to calculate the damage probability for a certain value of the hazard parameter, quadratic interpolation is used for data-set-based fragility curves. Equations used for other functional forms are listed in Table 2.

Similar to the property estimators, validity conditions can be specified for fragility curves. Therefore, a fragility curve can be created specifically for a selected type of plant unit with certain kinds of proper-ties (e.g. atmospheric fixed-roof cylindrical storage tanks). See "Property Estimators" section for more details on the definition and evaluation of validity conditions. Unless "flexible fragility curve selection" is activated, only compatible fragility curves are used for damage estimation at a plant unit. If the flexi-ble selection is active, the most suitable fragility curve is used if no compatible fragility curve is found. See "Risk Assessment" section for more details on damage estimation.

For pre-defined fragility curves, bibliographic references should be indicated to provide information on the origin of the fragility curve data. At least one reference is mandatory to assure data quality. For cus-tom fragility curves, list of bibliographic references is automatically generated by the system by using the references of natech damage records used for the calculation of the fragility curve.

Fragility curve information is presented in tabular form on the view page, together with curve data, validity conditions, and bibliographic references. Interactive fragility curve plots are provided for each damage state. Fragility curves are plotted on the same graph as separate series, so that they can be compared easily. By moving the mouse pointer over the curves, numerical values of damage probabili-ties can be obtained.

A sample fragility curve information page is shown in Figure 3.