The concept of equivalent fire severity is used to relate the severity of an expected real fire to the standard test fire. This is important when designers want to compare published fire‐
resistance ratings from standard tests with estimates of the severity of a real fire. The behav- iour of post‐flashover fires has been described in Chapter 3. This section describes methods of comparing real fires to the standard test fire.
4.3.1 Equal Area Concept
Early attempts at time equivalence compared the area under time–temperature curves.
Figure 4.3 illustrates the concept, first proposed by Ingberg (1928), by which two fires are considered to have equivalent severity if the areas under each curve are equal, above a certain reference temperature. This has little theoretical significance because the units of area are not meaningful. Even though Ingberg was aware of its technical inadequacy he used the equal area concept as a crude but useful method of comparing fires. After carrying out furnace tests, he developed a relationship between fire load in a room and the required fire resistance of the surrounding elements. This approach, subsequently used by US code writers to specify fire resistance ratings, has been useful, but ignores the effects of ventilation and fuel geometry on fire severity.
A2
Termperature
A1
te Time
Standard fire
Real fire
Figure 4.3 Equivalent fire severity on equal area basis
Real fire Gas temperature
Steel temperature
Temperature
Time te
Standard fire
Figure 4.4 Equivalent fire severity on temperature basis would be much less affected in the cooler fire.
4.3.2 Maximum Temperature Concept
A more realistic concept, developed by Law (1971), Pettersson et al. (1976) and others, is to define the equivalent fire severity as the time of exposure to the standard fire that would result in the same maximum temperature in a protected steel member as would occur in a complete burnout of the fire compartment. This concept is shown in Figure 4.4, which compares the temperatures in a protected steel beam exposed to the standard fire with those when the same protected beam is exposed to a particular real fire.
In principle, this concept is applicable to insulating elements if the temperature on the unex- posed face is used instead of the steel temperature, and is also applicable to materials which have a limiting temperature, such as the 300 °C temperature at which charring of wood gener- ally begins. The maximum temperature concept is widely used, but it can give misleading results if the maximum temperatures used in the derivation of a time equivalent formula are much greater or lower than those which would cause failure in a particular building.
4.3.3 Minimum Load Capacity Concept
In a similar concept based on load capacity, the equivalent fire severity is the time of exposure to the standard fire that would result in the same load‐bearing capacity as the minimum which would occur in a complete burnout of the firecell. This concept is shown in Figure 4.5 where the load‐bearing capacity of a structural member exposed to the standard fire decreases con- tinuously, but the strength of the same member exposed to a real fire increases after the fire enters the decay period and the steel temperatures decrease. This approach is the most realistic time equivalent concept for design of load‐bearing members. The minimum load concept is difficult to implement for a material which does not have a clearly defined minimum load capacity, for example with wood members where charring can continue after the fire temper- atures start to decrease.
4.3.4 Time Equivalent Formulae
A number of time equivalent formulae have been developed by fitting empirical curves to the results of many calculations of the type shown conceptually in Figure 4.4. The resulting formulae are based on the maximum temperature of protected steel members exposed to real- istic fires.
4.3.4.1 CIB Formula
The most widely used time equivalent formula is that published by the CIB W14 group (CIB, 1986), derived by Pettersson (1973) based on the ventilation parameters of the compartment and the fuel load. The equivalent time of exposure to an ISO 834 test te (in minutes) is given by:
te k w ec f (4.5)
1.0
Time te
Standar
d fire Real fire
Load capacity
Figure 4.5 Equivalent fire severity on load‐bearing capacity basis
4.3.4.2 Law Formula
A similar formula was developed by Margaret Law on the basis of tests in small‐scale com- partments (Thomas and Heselden, 1972) and larger scale compartments (Law, 1973). The formula is given by:
t A e
H A A Av
e
f f
c v t
(4.7)
where ΔHc is the calorific value of the fuel (MJ/kg).
The CIB formula and the Law formula are only valid for compartments with vertical openings in the walls. They cannot be used for rooms with openings in the roof. The Law formula gives similar results to the CIB formula, generally with slightly larger values of equivalent time.
4.3.4.3 Eurocode Formula
The above formulae were later modified and incorporated into the Eurocode 1 Part 1.2 (CEN, 2002b), referred to often as the ‘Eurocode formula’ giving te (in minutes) as:
te k w eb f (4.8)
where kb replaces kc and the ventilation factor w is altered to allow for horizontal roof open- ings. The ventilation factor is given by:
w Hr b
v
v h
6 0 0 62 90 0 4
1 0 5
0 3 4
. . .
.
.
(4.9)
where Hr is the compartment height (m) and
v A Av/ f 0 025. v 0 25. (4.10)
h A Ah/ (4.11)f
bv 12 5 1 10. v v2 10 0. (4.12)
Af is the floor area of the compartment (m2), Av is the area of vertical openings in the walls (m2) and Ah is the area of horizontal openings in the roof (m2).
The derivation of the Eurocode formula is based on work by Schneider et al. (1990). It is understood to have come from an empirical analysis of calculated steel temperatures in a large number of fires simulated by a German computer program called Multi‐Room‐Fire‐Code.
An important difference from the CIB formula is that the Eurocode equivalent time is independent of opening height, but depends on the ceiling height of the compartment, so the two formulae can give different results for the same room geometry. The results are similar for small compartments with tall windows, but the Eurocode formula gives much lower fire sever- ities for large compartments with tall ceilings and low window heights.
Values of the terms kc and kb are given in Table 4.3, where they are shown to depend on the compartment lining materials (roughly inversely proportional to the thermal inertia). The
‘general’ case is that recommended for compartments with unknown materials. Note that kc and kb have slightly different numerical values and units, because of the different ventilation factors in the respective formulae. The bottom line marked ‘large compartments’ is a modifi- cation to the Eurocode formula recommended by Kirby et al. (1999) for large spaces, after several experimental fires in a large compartment measuring 23 × 5.5 m by 2.7 m high. Using typical thermal properties of materials from Table 4.3, a building constructed with steel walls is in the ‘high’ category, normal and lightweight concrete are ‘medium’, and gypsum plaster and any materials with better insulating properties are in the ‘low’ category.
4.3.4.4 Validity
Time equivalent formulae are empirical. They have generally been derived by calculation, for a particular set of design fires for small rooms, using the maximum temperature concept for certain protected steel members with various thicknesses of insulation. As such the formulae may not be applicable to other shapes of time–temperature curve, to larger rooms, to other types of protection, or to other structural materials. None of the formulae described above have well documented derivations which describe their limitations. It is generally accepted
Table 4.3 Values of kc or kb in the time equivalent formulae
Formula Term Units b = √(kρcp) General
High
>2500
Medium 720–2500
Low
<720
CIB W14 kc min m2.25/MJ 0.05 0.07 0.09 0.10
Eurocode kb min m2/MJ 0.04 0.055 0.07 0.07
Large compartments kb min m2/MJ 0.05 0.07 0.09 0.09
k, thermal conductivity (W/mK); ρ, density (kg/m3); cp, specific heat (J/kgK).