Full Title: Quantitative Fire Risk Analysis: a Method for Defining BuildingSpecific RiskAdjusted Design Fires for Estimating Safe Egress Time
By Johannes Almås
An alternative fire safety design is usually verified by comparison with a design deemed acceptable according to the prescriptive code. A comparative analysis is, however, only a compromise for a performancebased design (PBD) and unlikely to achieve the same degree of costsafety optimization. The basic principle in PBD is calculation of available and required safe egress times. These calculation procedures have several limitations and uncertainties, mainly because of the lack of standardized inputs.
Traditionally, a fire is assumed to grow as a t^{2}fire unimpeded to reach a ventilationcontrolled fire, thereafter continuing to burn until all the fire energy within the compartment is consumed.
While several factors can limit the probability of ignition, fire development and fire spread, the only limiting factor commonly used is automatic sprinkler protection. Other factors are usually not included, perhaps because of uncertain or unknown effects on fire development.
Uncertainties are not an issue solely in the field of fire protection. Other engineering disciplines also need to deal with uncertainty. These disciplines have developed standardized inputs based on statistics and accumulated knowledge. Examples are wind loads, snow loads, earthquakes and flooding. Once these inputs are defined, the engineering strategy and documentation are based on standardized mathematical calculation methods and acceptance criteria, thus making the engineering objective. If optimization is a goal, verifiable documentation is also needed in the fire protection engineering community.
A design fire for calculating life safety (available safe egress time) differs from other types of design fires. Design fires used to evaluate fire protection measures are described with certain specific fire properties like fire area (for evaluation of smoke vent/exhaust), fire development in early stages (for evaluation of sprinklers) and fire load (for evaluation of fire wall rating). When designing for life safety, all factors that limits the probability of fire ignition, development and spread should be evaluated.
If the goal is to create a fair and optimized fire protection design for a specific building being evaluated, factors that influence consequences should also be included. Simply put, if the area and number of occupants of a building are doubled, both the probability of a fire (ignition) and its consequences are doubled. This means the risk is increased by a factor of four, and additional fire protection measures may be needed to compensate and reduce fire risk back to an acceptable level.
A riskadjusted fire is of a size appropriate for analyzing safety for the specific building or situation being evaluated. This does not mean that a fire cannot be larger. It only means that a larger fire is so rare and/or exposes so few people to risk that the larger fire does not have to be considered for the specific building or situation (low risk).
Several measures and factors can limit the probability of fire ignition, growth and spread, together with the potential life safety consequences, and should be included to define a buildingspecific and fair design fire. Only the basic principles for such a method are presented here; further research is required to develop standardized input factors.
The governing equation is that risk is the product of probability and consequence (R = P x C). Instead of calculating the number of potential deaths, this method suggest that the consequence is the number of people who could be exposed to the design fire, and that probability is a fire issue (the possible magnitude of the fire threat).
The risk [R] is a number that accounts for probable fatalities per year. The number used in the examples in Table 2 is 1x10^{7}, which is derived from the acceptable fatalities equation F(N) = 10^{7} x 1/N.
The consequence in this approach is a weighted average number of people [N] in the object/building/compartment who share the same corridor or staircase, or the maximum number of people who might need to evacuate from a fire in a large compartment (e.g., assembly area or shopping mall).
The magnitude of the risk adjusted fire is at least limited to these factors:
 The probability of ignition is dependent on occupancy and compartment area. If the probability of smoke spread to a common means of egress is to be analyzed, the number of compartments [K] with access to the same corridor or staircase should also be included. Tillander^{1} suggested both a formula and inputvalues based on accidents where fire departments were alerted (accident database Pronto). On average, the probability of fire within a compartment [P_{ign}] is set to 1x10^{5} based on values presented by Tillander^{1}.
 A probability factor that accounts for the number of ignited fires that develop to an established significant fire [P_{SF}] (e.g. fire area 1 m^{2}): This factor is set to 1x10^{2} (Fitzgerald^{2}). This factor will vary between occupancies and their likelihood to have fire alarm systems.
 The probability of no selfextinguishment. Only a fraction of established fires will continue to grow to larger fires, and only a fraction of the larger fires will continue to grow all the way up to ventilationcontrolled fires. The probability of fire growth depends on combustibility, fire load, room geometries, ventilation and insulation etc. If statistics are known, the probability can also be described as a Pareto distribution^{4}. The London fire brigade (among others) have collected a large fire database that describes the fire size upon their arrival. In these examples fire statistics from the BRE Pilot study^{3} for the effectiveness of residential sprinklers was used to estimate the ^{}factor in the Pareto distribution: P(A) = k x A^{}, where A is the reported fire damaged area, and k is a factor that describes the fire size at P_{SF }(k=1). The ^{factor is found to vary between 1 and 4 for residential buildings. It is suggested to define factor based on the rate of fire development (the factor in t2fires), and the fire load weighted to compartment interior surface area [QE]}
 Automatic sprinklers that also reduce the probability of fire growth. The fire might continue to grow if sprinklers fail. For the sake of simplicity, it is assumed that a fire is at least controlled at sprinkler actuation. Moresophisticated approaches could be used to account for the probability of fast fire extinction.

Fire barriers that reduce the probability of fire spread from a fire compartment to a common egress corridor or staircase. If surrounding walls and decks are firerated, the weakest spot is usually fire doors. Because fire doors and their appliances can have defects, be blocked open or be left open during evacuations, they are assumed to either fail (left open) or be a closed, tight barrier. A moresophisticated approach than discussed here will lead to probabilities relating to partially open fire doors (or other small openings). The probability may also vary between occupancies and their fire prevention measures, the location of fire doors, technical features, and other factors.
Based on the probability of fire, potential consequences and risk, a fire can be assumed to grow up to the estimated riskadjusted maximum fire size (Q_{max}). (The method is only valid for fire sizes less than those that can cause flashover.) Fire brigade arrival time should be less than 10 minutes, and fire barriers, materials and fire protection systems should comply with building codes.
Where;
 P_{ign} is the probability of ignition per compartment.
 I_{a} is a factor that adjusts the probability of ignition by including the number of fire compartments [K], compartment area and businesses
 The factor [F] is used to simplify the equation and accounts for risk [R], probability of ignition [P_{ign}] and probability of fire development to 1 m^{2} fire area [P_{SF}]; these numbers equalize each other.
 A is firedamaged area; damaged area is divided by 2 because the burning area is assumed to be not more than 50% of the firedamaged area.
 ^{} is the fire development factor
(a suggested formula derived from London data; more research is needed), where [t_{a}] is the fire development time up to 1 MW and Q_{E} is the total heat load in the fire compartment divided by the area of surrounding surfaces (see Tables 1 and 2).
 is the probability of failure of fire protection barriers and other fire protection systems
 The consequence [C] is the weighted number of occupants [N] divided by the number of independent means of egress [E]: C = N / E. E should be set to 1 when evaluating evacuation from fire in a compartment.
 RHR_{f} is the Rate of Heat Release per square meter (see Table 3).
When Q_{max} is known, fire simulation models can be used to evaluate fire development in the compartment and fire spread through barrier openings. Simulation of fire spread from a fire compartment to a target of interest (e.g., the staircase) should be done with open fire doors, because the probability of a closed fire door already accounts for reduction of the fire size.
Tables 1 and 2. Table values to differentiate the probability of fire ignition, NSEN 199112, Appendix E.
Figure 1. Estimated Paretocurves for residential occupancies (London data)
It should be noted that the fire development probabilitycurve can be depend greatly on defined single burning item and the probability of fire spread to another item (e.g., spread between cars in a parking garage). The distance between the items and their ignitability are, therefore, barriers that causes significant vertical steps in the probability curve. This effect is not included here.
Table 5. Examples of risk adjusted max design fire sizes for residents (Q_{max})
The numbers used in quantitative analysis are usually derived from historical data, but a weakness is that historical data may not be appropriate for anticipating future reliability. However, historical data is also a fundamental principle behind the prescriptive codes. Prescriptive measures are, therefore, also encumbered with uncertainty and perhaps unsuitability as new building design and products develop.
The main difference between prescriptive and quantitative approaches is that quantitative data are more complicated to collect and to apply into the design. Even though it is easier to learn from historical fires and develop practical prescriptive methods, the disadvantage with prescriptive methods is that they may not produce a costeffective design.
Maximum heat release rate may be the mostimportant factor in the description of a design fire. This article presents and suggests a systematic method to calculate this parameter. The main intention of this article is to show that the description of a design fire should be based on building or situation specific fire risk, and that such an approach is possible. Further research is needed to verify and standardize a method and input values.
Johannes Almås, is with Norconsult AS, Hamar, Norway
References
^{1}Tillander, Kati. Utilization of statistics to assess fire risks in buildings, ESPOO 2004, VTT Publications 537, Finland
^{2}Fitzgerald, Richards, Beyler. 1991. Fire safety analysis of the Polar Icebreaker Replacement Design. Journal of Fire Protection Engineering, vol. 3, issue 4, pp. 137–150.
^{3}Williams, FraserMitchell, Campbell. Effectiveness of sprinklers in residential premises – Section 3: Pilot study. BRE 2004, England.
^{4}Ramachandran. Charters. 2011. Quantitative Risk Assessment in Fire Safety.
^{5}Eurocode 1: Actions on structures – Part 1–2: General action – Actions on structures exposed to fire. NSEN 199112:2002+NA2006.