Section factor

Section factor is the heated perimeter of a structural cross-section divided by its cross-sectional area. For a uniform member, it is also the heated surface area divided by the steel volume. It expresses how much boundary is available to receive heat relative to the amount of material being heated.

The quantity helps explain the thermal response of steel in fire. Under otherwise comparable conditions, a member with more heated surface relative to its steel volume warms more quickly. The inverse relationship is expressed in some sources through area-to-perimeter or mass-to-perimeter ratios. These conventions describe related geometry, but their numerical values and trends run in opposite directions.

Section factor does not by itself give a resistance time or an allowable steel temperature. Heating also depends on the exposure, thermal properties and any protection. A structural assessment then determines the significance of the resulting temperature history. The definition of heated perimeter must match the exposure and protection arrangement used by the calculation method.

Geometry, units and reciprocal conventions

For a prismatic member, the cross-section remains constant along its length. Multiplying the heated perimeter by an arbitrary length gives the corresponding heated surface area. Multiplying the steel cross-sectional area by the same length gives the steel volume. Dividing these two quantities cancels the length, leaving the perimeter-to-area ratio used in the thermal description.

The member length cancels when heated area is divided by steel volume. Based on Milke (2002), pp. 4-216–4-217 and 4-224.

S = P/A = A_heated/V

The section factor therefore has units of inverse length. When perimeter is expressed in metres and area in square metres, the result is in m⁻¹. Its reciprocal, A/P, has units of length. Confusing these conventions reverses the interpretation of a larger value and gives the wrong dimensions when the quantity enters a heat-transfer equation.

Milke also uses a mass-per-length to heated-perimeter ratio, conventionally written W/D in the cited methods. In consistent SI units, multiplying the cross-sectional area by density gives mass per unit length. Dividing that mass by heated perimeter gives density divided by section factor. The ABCB 2020 Data Sheets explicitly express this mass-to-perimeter quantity in kg/m².

m′/P = ρA/P = ρ/S

Symbol	Description	Unit
S	Heated perimeter divided by steel cross-sectional area	m⁻¹
P	Heated perimeter consistent with the chosen method	m
A	Steel cross-sectional area	m²
Aheated	Heated surface area over a selected member length	m²
V	Steel volume over the same length	m³
m′	Steel mass per unit length	kg/m
ρ	Steel density	kg/m³

The ratio does not represent the outside dimensions of an arbitrary box drawn around the section. A hollow or open section contains less steel than its bounding rectangle. The denominator is the actual steel area, while the numerator follows the method's heated-boundary convention. Both quantities are needed before a catalogue dimension can be converted into the required thermal parameter.

Unit conventions in published W/D expressions deserve separate attention. Milke presents some empirical methods using mass per foot of member length and perimeter in inches. The resulting number cannot be inserted unchanged into an SI equation using kg/m². The source also uses the traditional word weight for this parameter, while the SI thermal balance uses mass. Checking the stated units preserves the relationship between geometry, density and heat capacity.

Which perimeter receives heat

An unprotected section uses the boundary exposed to the specified heating. A member heated on all sides has a different boundary from the same member with one face shielded. A beam associated with a floor slab can therefore require a different exposure description from a column surrounded by hot gases. Milke's perimeter diagrams distinguish these arrangements rather than assigning one universal perimeter to each steel shape.

Protection changes how the relevant perimeter is defined. Contour protection follows the steel surface, including the recessed parts of an open section. Box protection encloses the member within a simpler outer arrangement. Milke's methods determine perimeter at the stated steel-protection interface or corresponding protected boundary. A number calculated for one protection profile cannot simply be reused for another.

The relevant heated perimeter depends on the protection configuration and calculation convention. Based on Milke (2002), pp. 4-216–4-217.

The same steel section can consequently have different thermal geometry parameters depending on its protection arrangement. The steel mass has not changed, but the path by which heat reaches it has changed. This explains why specifying only the section designation leaves part of the thermal problem undefined. The calculation also needs the exposed faces and the protection geometry represented by its formula.

The choice of perimeter is method-specific rather than a licence to select whichever value produces a favourable answer. The empirical correlations reviewed by Milke were developed for particular section and protection configurations. Their dimensional definitions form part of those correlations. Substituting an outer insulation perimeter without checking the method can therefore alter the physical meaning of the input.

Why section factor changes heating rate

A thermal balance compares heat entering through the exposed boundary with energy stored in the steel. Increasing heated perimeter increases the area available for transfer over the same member length. Increasing steel area increases the mass that must be heated. The section factor combines these competing geometric effects when the member can be represented by one temperature.

Milke gives a lumped heat-capacity expression for unprotected steel using the mass-to-perimeter convention. Rewriting its geometry in terms of S gives the expression below. The effective heat-transfer coefficient represents the surface-transfer treatment selected by the model. For the later illustration, only convection is included so that the effect of changing exposed area can be isolated.

dT_s/dt = [Sh/(ρc_s)](T_f − T_s)

Symbol	Description	Unit
Ts	Uniform steel temperature	°C
Tf	Surrounding fluid temperature in the convection illustration	°C
t	Time	s
h	Surface heat-transfer coefficient used in the model	W/(m²·K)
cs	Steel specific heat capacity	J/(kg·K)

The temperature difference may be expressed in kelvins or degrees Celsius because the size of a temperature interval is the same. The equation predicts a changing rate as the steel approaches its surroundings. It does not imply a constant temperature increase merely because S remains constant. The exposure history and the current steel temperature remain part of every evaluation.

Radiation can also contribute to heating in a fire. Milke includes radiative and convective contributions in the surface-transfer treatment of the general steel model. A calculation omitting radiation therefore represents a restricted heating problem. It can illustrate the role of geometry, but cannot automatically represent the total heat transfer experienced by an exposed member in a compartment fire.

Worked example for a thin steel plate

Consider the 4 mm steel thickness used in an example in Introduction to Fire Safety Engineering. For this separate geometric illustration, the plate is wide enough that edge heating is neglected. One case heats only one broad face, while the other heats both broad faces equally. These are selected boundary conditions for the comparison, rather than the book's steady roof heat-loss calculation.

For a plate of thickness L with one heated broad face, heated surface divided by steel volume is 1/L. With two equally heated faces, it becomes 2/L. Taking L as 0.004 m therefore gives section factors of 250 m⁻¹ and 500 m⁻¹ respectively. The change comes entirely from the number of heated faces; the plate thickness and steel volume are unchanged.

The exposed faces change section factor and the initial rate in the stated convection-only example. Based on Van Rossum (2027), §§5.2 and 5.5, and Milke (2002), p. 4-222.

The book lists steel density as 7,850 kg/m³ and specific heat as 460 J/(kg·K). Using that density, the one-face mass-to-heated-area ratio is 31.4 kg/m². For two-face exposure, it is 15.7 kg/m². These calculated values decrease as section factor increases, illustrating the reciprocal trend of the two conventions.

For an initial-rate illustration, take the book's temperatures of 634 °C and 20 °C as the gas and initially uniform steel temperatures. Select a convective coefficient of 25 W/(m²·K), within the range discussed by Milke. Assume fixed thermal properties and neglect radiation. The initial temperature difference is then 614 K for each heated face.

The calculated initial heating rate is approximately 1.06 K/s for one-face exposure. Heating both broad faces at the same coefficient and temperature difference gives approximately 2.13 K/s. Doubling the section factor doubles this initial rate within the selected model. The comparison does not establish the later temperature history under a real fire, because its radiation, changing exposure and material behaviour were deliberately excluded.

The figures also separate thermal input from structural performance. They describe how rapidly the assumed uniform temperature initially rises. They do not specify the load carried by the plate or the temperature at which its intended function is lost. A mechanical calculation and an appropriate exposure history would still be needed for a structural resistance assessment.

Uniform temperature and internal gradients

The single-temperature balance assumes that heat spreads through the steel quickly compared with the rate at which its surface supplies energy. This is related to the distinction between thermally thin and thick materials. A section can then be treated as storing heat at one representative temperature. If internal differences become significant, a distributed temperature calculation is needed instead.

The Biot number compares internal conduction with surface heat transfer. For the one-face heated plate with an insulated rear face, Rockett and Milke express it as hL/k. Here k is the thermal conductivity and L the plate thickness in their formulation. Small values support the assumption that conduction within the plate is fast relative to surface heating.

Using the illustrative h of 25 W/(m²·K), thickness of 0.004 m and the book's steel conductivity of 45.8 W/(m·K) gives Bi approximately 0.0022. This calculation supports the selected one-face convection example's uniform-temperature approximation. It does not establish that every member with the same section factor has negligible gradients. Geometry, boundary conditions and thermal conductivity remain relevant.

For steel beams associated with concrete slabs, Milke discusses differences between average temperature and the temperature distribution through the section. Heat transfer to the slab can modify the thermal response. Where that distribution affects the assessment, the handbook calls for another analytical treatment. Section factor preserves an area-to-volume relationship but cannot represent every spatial detail of heating.

Protected members and comparison methods

A protective layer introduces thermal resistance between the fire and the steel. It can also store energy, so its heat capacity may influence the temperature rise. The protected-member equation reproduced by the ABCB includes insulation conductivity, thickness, density and specific heat alongside the steel mass-to-perimeter ratio. Geometry remains relevant, but it is only one group of inputs.

An intumescent material is one of the protection types listed in Milke's review of steel protection. The thermal treatment must represent the properties and configuration relevant to the chosen system. A section factor does not supply those material properties or prove the continuing attachment of a coating. The geometry calculation consequently cannot replace evidence for the protective assembly.

The ABCB 2020 Data Sheets also use a protected steel member as a possible generic target for comparing different heating regimes. Its steel mass and insulation properties can be selected to represent features relevant to the comparison. The method then calculates the target's thermal response under each exposure. This is a defined modelling construction, rather than a claim that every protected element responds like steel.

Changing section factor in such a comparison changes the selected target's thermal response. It can therefore influence the times associated with reaching a chosen temperature. The target geometry, protection and temperature criterion need to remain consistent between the compared cases. Otherwise, a difference attributed to the fire exposure may instead arise from changing the modelled element.

Structural meaning and practical limits

The critical steel temperature is established from a structural condition or a specified endpoint. It is distinct from the section factor governing part of the heating response. Two members with similar thermal geometry can have different loads or support conditions. Conversely, two members with the same limiting temperature can reach it at different times because their thermal geometry differs.

Empirical fire-resistance expressions often contain a mass-to-perimeter term because it represents this geometric influence. Milke explains that these expressions were fitted to tests on particular assemblies. Their permitted section and protection ranges form part of their validity. A physically meaningful ratio does not remove the need to remain within the conditions supporting the overall correlation.

Local protection damage presents a further limitation of a single geometry value. Milke describes numerical analyses in which missing protection changed the heating and resistance of a steel column. The nominal steel section factor remained a geometric property of the member, but it did not describe the altered thermal boundary. Spatial modelling or appropriately supported assumptions are needed when a uniform protection condition no longer represents the assembly.

The ABCB guidance separates thermal response from structural capability and requires the method to suit the assessed application. It also identifies variability in material properties and installation quality. Section factor can describe a well-defined geometric input within that process. It cannot compensate for an incorrect exposure, missing protection or a structural model that omits relevant interaction between members.

References

• Australian Building Codes Board. (2020). Fire safety verification method data sheets: Handbook annex (Version 1.1).
• Milke, J. A. (2002). Analytical methods for determining fire resistance of steel members. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., Chapter 4-9). National Fire Protection Association.
• Rockett, J. A., & Milke, J. A. (2002). Conduction of heat in solids. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., Chapter 1-2). National Fire Protection Association.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.