Parametric fire curve

A parametric fire curve is a temperature–time curve for a post-flashover fire in a single room, calculated from a few parameters that describe the room and its contents. The parameters are usually the fire load, the size of the openings and the thermal properties of the walls, floor and ceiling. A furnace test follows the standard fire curve, which is the same for every building. A parametric curve, by contrast, covers the fully developed stage and the decay that follows, and its course depends on the fire load, the openings and the linings of the room.

A widely used version is Annex A of EN 1991-1-2, part of the Eurocodes. The Australian Building Codes Board describes it as codified, used extensively and easy to put into a spreadsheet. Other families include the Swedish design curves and the characteristic curves of T. T. Lie. Structural engineers use the curve as the thermal exposure for calculating member temperatures and the time to structural failure. The method assumes uniform burning and one gas temperature throughout the compartment, so it suits rooms of moderate size better than large open floors.

Why a room-specific curve is needed

Building elements are tested for fire resistance in furnaces that follow an agreed temperature–time curve, such as the ISO 834 standard curve. Such a curve gives every test the same starting point, but it does not reproduce a fire in a real room. The temperature in a room fire depends on the size of the room, its openings, the insulation of its boundaries and the heat release rate of the fire. The temperature is therefore a result of a fire scenario and not an input to it. For this reason the International Fire Engineering Guidelines state that a rating from the standard test should only be used together with an analysis of the expected fire severity.

Design guidance offers two broad ways of describing that severity. The first converts the expected fire into an equivalent time of exposure in the standard test. The second describes a natural or realistic fire directly as a curve of temperature against time. The Eurocodes give a similar choice between a temperature–time curve and a natural fire, whose temperature follows from the characteristics of the fire and the room. A parametric fire curve belongs to the second group, yet keeps the simplicity of a closed formula.

Lie explains why a single predicted fire is not the aim. The amount, surface area and arrangement of the combustible contents change during the life of a building and cannot be known when it is built. Surveys show that deviations of 50% or more from the most probable fire load are common. Such a spread alone can shift the temperature course by hundreds of degrees and the fire duration by 50% or more. Lie therefore proposed characteristic curves whose effect, with reasonable likelihood, will not be exceeded during the lifetime of the building.

The heat balance behind the curves

Lie divides the temperature course of a room fire into a growth period, a fully developed period and a decay period. Flashover marks the start of the fully developed period, in which gas temperatures of about 1,000 °C or more can occur. Temperatures stay relatively low during growth, so fire resistance studies can disregard that period. After flashover the room is so hot that pyrolysis gases are abundant, and ventilation usually controls the fire. The heat release rate then depends mainly on the fresh air entering through windows and doors, and part of the gases burns outside the room.

The gas temperature follows from a heat balance, first estimated by Kawagoe and Sekine and by Odeen. Combustion inside the room produces heat. Part of that heat warms the gases, part flows into the walls, floor and ceiling, and part leaves through the openings as radiation and as hot gas. Solving the balance requires the rate of heat production and the rate of heat loss at every moment of the fire. Pettersson and co-workers used this approach in 1976 to derive temperature–time curves as a function of fire load and ventilation.

Heat balance of a fully developed room fire: heat released by combustion warms the gases, flows into the walls, floor and ceiling, and leaves through the opening as radiation and hot gas. Based on Lie (2002), p. 4-202, and Australian Building Codes Board (2005), p. 2.6-3.

The air supply through an opening grows with its area multiplied by the square root of its height. This product, $A\sqrt{H}$, is commonly known as the ventilation factor. For a ventilation-controlled fire of cellulosic fuel, the International Fire Engineering Guidelines approximate the air inflow as $0.52A\sqrt{H}$ kg/s. The corresponding heat release rate is about $2.16A\sqrt{H}$ MW, with the area in square metres and the height in metres. Parametric curves usually divide the ventilation factor by the total internal surface area of the room, which gives the opening factor in m^1/2. According to Lie, the fire load determines how long the fire lasts, whereas the openings determine both its duration and its intensity.

Swedish curves and Lie's characteristic curves

The Swedish method of Magnusson and Thelandersson is based on the same mass and energy balance. It does not model the fire itself: heat release rate curves are given as input, and the energy release always stays below the stoichiometric limit. The design curves approved by the Swedish authorities assume a uniform temperature in the compartment and boundaries with normal thermal properties. They give the gas temperature against time for a range of fire load densities and for opening factors from 0.02 to 0.12. Results from the computer program SFIRE agreed well with a large number of full-scale experiments. Most of those tests used wood cribs, which burn more slowly and produce less excess fuel than furnishings.

Lie fitted one analytical expression to heat-balance results for two kinds of boundary. Heavy materials, with a density of about 1,600 kg/m³ or more, include normal-weight concrete and most clay bricks. Light materials include lightweight concrete and plasterboard. His expression gives the gas temperature as a function of time and opening factor. A constant equal to 0 for heavy and 1 for light boundaries adds a higher temperature for the lighter materials. The expression is valid for opening factors between 0.01 and 0.15 m^1/2.

The duration of the fully developed period follows from the burning rate, which Lie takes as proportional to the ventilation factor. The duration τ in hours is the fire load per unit of internal surface, Q, divided by 330 times the opening factor F: $$\tau =\frac{Q}{330F}$$ After this time the temperature falls in a straight line by 600 °C in each further period τ, and it stops at 20 °C. Long fires therefore cool more slowly than short ones. Compared with measurements in an experimental fire, the characteristic curve proved somewhat conservative but satisfactory as a design curve.

The Eurocode parametric curve

Annex A of EN 1991-1-2:2002 has been codified and used extensively. The Fire Safety Verification Method Data Sheets of the Australian Building Codes Board reproduce it with guidance, and the equations below follow that presentation. In the heating phase, the gas temperature θ_g in degrees Celsius always follows the same shape. A factor Γ stretches or compresses that shape in time, so that a fictitious time t* replaces the real time t in hours:

$${\theta }_g=20+1325(1-0.324e^{-0.2t^{*}}-0.204e^{-1.7t^{*}}-0.472e^{-19t^{*}}),t^{*}=\mathrm{\Gamma }t$$

The factor Γ compares the room with a reference room. It depends on the opening factor O and on the b-factor of the boundaries. The b-factor is the square root of the product of density, specific heat and thermal conductivity:

$$\mathrm{\Gamma }=\frac{{(O/b)}^2}{{(0.04/1160)}^2},O=\frac{A_v\sqrt{h_{eq}}}{A_t},b=\sqrt{\rho c\lambda }$$

That product is the thermal inertia of the lining material. A dense, conductive lining such as concrete has a high b-factor and absorbs much heat from the gases. It gives a low Γ and a slower, cooler curve, while an insulating lining gives a high Γ and a faster, hotter fire. Lie's curves show the same tendency for light boundaries.

Evaluating both published equations minute by minute shows that, for Γ equal to 1, the heating branch stays within 2.5 °C of the ISO 834 standard curve between 30 and 120 minutes. Between 5 and 30 minutes the difference reaches about 22 °C. In the first minutes it rises more slowly, with 199 °C against 349 °C after 1 minute.

The time of the peak depends on the fire load density. The Data Sheet uses the design fire load density per unit of total internal surface, q_t,d. It is obtained from the more common design value per unit floor area, q_f,d:

$$q_{t,d}=q_{f,d}\frac{A_f}{A_t},t_{max}=\max (0.2\times 10^{-3}\frac{q_{t,d}}{O},t_{lim})$$

Here t_lim is 0.25 h for fast, 0.333 h for medium and 0.417 h for slow fire growth. When the first term is the larger one, the fire is ventilation-controlled and reaches its peak temperature θ_max at the fictitious time t*_max = Γ t_max. When t_lim governs, the fire is fuel-controlled. The Data Sheet then uses a modified factor Γ_lim, based on a limiting opening factor.

After the peak the temperature falls in a straight line. The cooling rate is expressed per hour of fictitious time and depends on t*_max. The steepest rate, 625 °C per hour, belongs to the shortest fires, and the gentlest, 250 °C per hour, applies when t*_max is 2 hours or more. Between t*_max values of 0.5 and 2 hours, the rate 250(3 − t*_max) links the two. The Data Sheet misprints the time ranges of the first two rates; only the reading used here makes the intermediate rate equal 625 at 0.5 hours and 250 at 2 hours. A factor X equals 1 for ventilation-controlled fires and takes a different value for fuel-controlled fires.

Parts of the Annex A parametric curve: the heating branch, whose speed depends on Γ, the peak at tmax and the linear cooling branch. A lower b-factor raises Γ and gives a hotter curve with the same tmax. Based on Australian Building Codes Board (2020), pp. 39–41.

The method has clear limits. The opening factor must lie between 0.02 and 0.20 m^1/2, the b-factor between 100 and 2,200 J/(m²·s^1/2·K) and q_t,d between 50 and 1,000 MJ/m². According to EN 1991-1-2:2002, the curves are valid for compartments of up to 500 m² floor area and 4 m height, without openings in the roof. The method assumes that the whole fire load burns out and should be limited to mainly cellulosic fire loads. For boundaries made of several layers, or with different materials in walls, floor and ceiling, the Data Sheet gives rules for an effective b-factor.

Symbol	Description	Unit
${\theta }_g$	gas temperature in the compartment	°C
${\theta }_{max}$	maximum gas temperature	°C
$t$	time	h
$t^{*}$	fictitious time	h
$t_{max}$, $t_{max}^{*}$	real and fictitious time of the peak	h
$t_{lim}$	limiting time for fuel-controlled fires	h
$\mathrm{\Gamma }$	time scaling factor	dimensionless
$O$	opening factor	m1/2
$A_v$	total area of vertical openings in all walls	m2
$h_{eq}$	weighted average height of the openings	m
$A_t$	total area of walls, floor and ceiling, including openings	m2
$A_f$	floor area	m2
$b$	b-factor of the boundaries	J/(m2·s1/2·K)
$\rho$	density of the boundary material	kg/m3
$c$	specific heat of the boundary material	J/(kg·K)
$\lambda$	thermal conductivity of the boundary material	W/(m·K)
$q_{f,d}$	design fire load density per unit floor area	MJ/m2
$q_{t,d}$	design fire load density per unit of total internal surface	MJ/m2
$\tau$	duration of the fully developed period in Lie's method	h
$Q$	fire load per unit of internal surface, as wood, in Lie's method	kg/m2
$F$	opening factor in Lie's method	m1/2

Worked example

Consider an office of 8 m by 6 m with a ceiling height of 3 m, concrete walls, floor and ceiling, and one window of 4.0 m by 1.5 m. The floor area is 48 m², the total internal area A_t is 180 m² and the window area is 6.0 m². The opening factor is O = 6.0 × √1.5 / 180 = 0.0408 m^1/2, inside the valid range. For concrete, Introduction to Fire Safety Engineering lists a thermal conductivity of 2.0 W/(m·K), a specific heat of 880 J/(kg·K) and a density of 2,300 kg/m³. These values give a b-factor of 2,012 J/(m²·s^1/2·K).

For illustration, the fire load is taken from New Zealand practice. In New Zealand, C/VM2 (second edition, 2025) gives a design fire load energy density of 800 MJ/m² for offices. For the fire resistance of structural elements in an unsprinklered firecell, the modification factor is 1.0, so the value stays at 800 MJ/m². The Annex A equations, as presented in the Australian Data Sheet, then give the following results.

The fire load density per unit of total area becomes q_t,d = 800 × 48 / 180 = 213 MJ/m². The ventilation term gives t_max = 0.2 × 10⁻³ × 213 / 0.0408 = 1.05 h, which exceeds every value of t_lim, so ventilation controls the fire. The factor Γ = (0.0408/2,012)² / (0.04/1,160)² = 0.346, so fictitious time runs at about a third of real time. At t*_max = 0.362 h the heating equation gives θ_max = 799 °C, about 63 minutes after the start of the curve. Because t*_max is below 0.5 h, the gas cools at 625 °C per fictitious hour, or about 216 °C per real hour, and returns to 20 °C after about 279 minutes.

Lining the same room with gypsum plaster changes the result sharply. With 0.48 W/(m·K), 840 J/(kg·K) and 1,440 kg/m³, the b-factor falls to 762 J/(m²·s^1/2·K). The factor Γ then rises to 2.41 and t*_max to 2.52 h. The peak still occurs after about 63 minutes, because t_max depends only on the fire load and the opening factor. It now reaches 1,082 °C, and cooling follows the rate of 250 °C per fictitious hour, about 10 °C per real minute. The gas is back at 20 °C after about 168 minutes.

Parametric fire curves for the office in the worked example with a concrete and a gypsum plaster lining, compared with the ISO 834 standard curve. Based on Australian Building Codes Board (2020), pp. 39–41, and Lie (2002), p. 4-206.

Quantity	Concrete lining	Gypsum plaster lining
b-factor (J/(m2·s1/2·K))	2,012	762
Γ	0.346	2.41
t*max (h)	0.362	2.52
θmax (°C), rounded up	800	1,090
Time of the peak (min)	63	63
Cooling rate (°C per real hour)	216	604
Back at 20 °C (min), rounded up	280	170

The same fire load and window therefore produce peaks that differ by about 280 °C, only because of the lining. The concrete room gives a long, moderate exposure, while the plaster-lined room gives a short, intense one. Elements of construction can be sensitive to heating rates and maximum temperatures, so the more severe exposure depends on the element and cannot be judged from the peak alone. For that reason the curve normally serves as input to a thermal analysis of the element.

Use in design and regulation

A parametric curve normally supplies the first of three calculation stages: the fire exposure, the thermal response of the element and its ability to keep performing its function. From temperatures or heat fluxes over time, the temperatures inside a cross-section can be calculated. With known material properties at elevated temperatures, the structural performance and the time to failure follow. Analysis methods range from simple lumped-mass calculations to detailed finite element analysis. Because a full time history is available, the failure of an element can be placed on the timeline of a fire scenario, including interventions.

Some elements, such as fire doors and penetration seals, are too complex for such calculations. For these, the Data Sheets suggest comparing the time to reach a critical temperature under the standard heating regime and under the design fire. In one worked case, an element reached 454 °C after 63 minutes in the standard test but after only 45 minutes of exposure to the parametric design fire. Its failure time in that scenario was therefore taken as 45 minutes. This conversion considers thermal performance only, not deflections, thermal stresses or degradation such as spalling.

In Australia, the Fire Safety Verification Method Data Sheets present Annex A as a typical way to derive a time-dependent fire exposure, which many applications of that method require. The Data Sheets also exclude the fire load adjustments of Annex E of EN 1991-1-2 when compliance with the National Construction Code is shown. Sprinklers, fire brigade intervention and similar factors should instead be addressed explicitly in the analysis. In New Zealand, C/VM2 (second edition, 2025) requires a full burnout design fire for structural design, based on complete burnout of the firecell without intervention. One of its three permitted approaches is a parametric time versus gas temperature formula, which supplies the thermal boundary conditions for a structural response model.

Computer programs that solve the full heat balance for post-flashover temperatures offer an alternative to closed-form curves. Examples named in the International Fire Engineering Guidelines are COMPF2, OZone, FASTLite and Branzfire. OZone, for example, calculates natural fires according to EN 1991-1-2 and then the temperature and fire resistance of steel members according to EN 1993-1-2.

Limits and uncertainty

The method rests on uniform burning, a single gas temperature throughout the compartment and burnout of the whole fire load inside the room. These assumptions become doubtful in large or deep compartments. A travelling fire spreads progressively through a large enclosure instead of involving all combustible surfaces at the same time. Experiments in deep enclosures with a single opening showed fires that first burned near the opening and only later moved to the back. Elements close to the opening received a significantly higher exposure than those at the rear. Such behaviour lies outside the 500 m² scope of Annex A.

Methods for calculating fire temperature curves are invariably empirical, and many rest on limited and specific test conditions. The Swedish curves were checked mainly against wood crib fires, and Annex A should be limited to mainly cellulosic fire loads. Compartments dominated by other fuels therefore fall outside the basis of the method. The result is also only as reliable as the chosen fire load, whose variability alone can change the fire duration by 50% or more.

The decay phase is the least certain part of any curve. Lie notes that decay rates can only be set empirically or with conservative or idealised assumptions, and that published studies disagree widely. Kawagoe measured about 10 °C per minute for fires with a fully developed period shorter than 1 hour and 7 °C per minute for longer fires. Harmathy found 15 to 20 °C per minute for fires of relatively short duration.

Lie adds that variations in the decay phase have relatively little influence on the maximum temperatures reached in building components. The International Fire Engineering Guidelines allow decay to start once 80% of the fuel is consumed, at a linear, measured or otherwise justified rate. Structural capacity can still keep falling during cooling, because heat continues to move into a section and thermal stresses change.

Common misconceptions

A parametric curve is sometimes read as a prediction of one real fire. Fire curves are agreed temperature courses and do not, in principle, match the heat development of an actual fire. Lie intended his characteristic curves as curves whose effect will not be exceeded with reasonable likelihood, not as forecasts.

A second misconception is that more ventilation always makes a fire worse. Larger openings raise the peak temperature but shorten the fire, while smaller openings give a cooler but longer fire. The worked example shows a comparable trade-off for the lining. A dense concrete lining lowers the peak but stretches the cooling phase over several hours.

Parametric curves for the concrete office of the worked example with a small and a large window: larger openings give a hotter but shorter fire. Based on Australian Building Codes Board (2020), pp. 39–41, and Lie (2002), p. 4-205.

A third misconception treats the parametric curve as a scaled copy of the standard curve. For Γ equal to 1 its heating branch is close to the ISO 834 curve after the first few minutes. Unlike the standard curve, however, it reaches a maximum and then cools down. The standard curve keeps rising for as long as the test lasts.

See also

• Fire load
• Structural fire engineering
• Time equivalence

References

• Australian Building Codes Board. (2020a). Data sheets: Fire Safety Verification Method (Handbook annex, Print version 1.1).
• Lie, T. T. (2002). Fire temperature-time relations. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 4-201–4-208). National Fire Protection Association; Society of Fire Protection Engineers.
• Ministry of Business, Innovation and Employment. (2025b). Verification Method C/VM2: Framework for fire safety design (2nd ed.).
• National Research Council of Canada, International Code Council, Department of Building and Housing, & Australian Building Codes Board. (2005). International fire engineering guidelines (2005 ed.). Australian Building Codes Board.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.
• Walton, W. D., & Thomas, P. H. (2002). Estimating temperatures in compartment fires. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 3-171–3-188). National Fire Protection Association; Society of Fire Protection Engineers.