Fire design theory

Heat transfer analysis and material degradation theory for fire resistance assessment of concrete cross-sections.

Introduction

Fire design of concrete members involves two coupled problems: determining the temperature distribution through the cross-section during fire exposure, and computing the reduced structural capacity at those elevated temperatures. This page covers the theory behind both stages.

Concrete performs well in fire due to its low thermal conductivity, high thermal mass, and non-combustibility. However, prolonged exposure degrades both the concrete and the embedded reinforcement, reducing the section’s load-carrying capacity.

Heat transfer

Governing equation

The temperature distribution through the cross-section is governed by the 2D heat conduction equation:

Where:

• $T$ = temperature ($^\circ$C)
• $t$ = time (s)
• $\rho$ = density (kg/m$^3$)
• $c_p$ = specific heat capacity (J/kg$\cdot$K)
• $k$ = thermal conductivity (W/m$\cdot$K)

All three thermal properties ($\rho$, $c_p$, $k$) are temperature-dependent for concrete.

Boundary conditions

Exposed surfaces (fire-side):

Where:

• $h_c$ = convective heat transfer coefficient (W/m$^2\cdot$K) — a property of the fire curve you selected, not a fixed constant; see below
• $\varepsilon_r$ = resultant emissivity (0.7 typical)
• $\sigma$ = Stefan-Boltzmann constant ($5.67 \times 10^{-8}$ W/m$^2\cdot$K$^4$)
• $T_{fire}$ = fire temperature from the fire curve
• $T_s$ = surface temperature

The convective coefficient follows the fire curve. EN 1991-1-2 fixes $h_c$ per nominal curve, so choosing a curve chooses its convective boundary condition too — selecting the hydrocarbon curve without the matching coefficient would apply half the standard’s convective flux:

ASTM E119 has no EN 1991-1-2 coefficient, and ACS does not pretend otherwise. ASTM E119 sits outside that standard’s scope, so no clause of it assigns the curve an $h_c$. ACS adopts 25 W/m$^2\cdot$K as the nearest analogue to the §3.2.1 standard curve, whose shape the implemented Lie (1992) fit closely follows (§ Fire curves below). That is a documented engineering judgement rather than a citation, and it is recorded as one — the related question of AS 1530.4 being attributed to ASTM E119 is tracked separately (#4402).

If you supply convectionCoefficientFire explicitly through the API or an MCP tool, that value is used verbatim on any curve; the per-curve value above applies only when you leave it undeclared.

:::caution[Results changed for hydrocarbon-curve sections — August 2026 (#4816)] Before this release $h_c$ was fixed at 25 W/m$^2\cdot$K for every curve. A section analysed on the hydrocarbon curve therefore received half the convective flux EN 1991-1-2 §3.2.3 specifies, which under-predicted temperatures and so over-predicted residual capacity — the unconservative direction, on the curve with the fastest early-time rise. Hydrocarbon-curve results now run hotter and capacities lower. ISO 834 and ASTM E119 results are unchanged, as is any run that declared convectionCoefficientFire by hand. :::

Unexposed outline surfaces: convection to ambient at $h_c = 9$ W/m$^2\cdot$K, 20$^\circ$C. An unexposed outer face normally has continuous construction beyond it — a slab over a beam, an adjoining member — which genuinely is a sink at roughly ambient.

Unexposed void surfaces (an interior cavity in a box or hollow-core section) take one of two boundary conditions, declared per void under Fire → Void Cavities:

• Sealed — adiabatic, no heat loss. This is the default, and what every design carries unless you change it. A sealed cavity has no ventilation and no mass to absorb heat, so its surface rises with the concrete around it; convecting it to a 20$^\circ$C reservoir would remove heat that has nowhere to go, computing a cooler wall and a larger effective section. Adiabatic is the conservative choice of the two boundary conditions — not an upper bound on the section, for the reason given under the limitations below.
• Ventilated — convection to ambient at $h_c = 9$ W/m$^2\cdot$K, 20$^\circ$C: the same boundary condition an unexposed outline surface receives. Use it for a cavity with real, sustained airflow — a ventilated service duct, or a cavity open to a ventilated space.

The two kinds are independent per void, so a member with a sealed cavity and a ventilated duct in the same cross-section is described directly rather than by picking the closer of two wrong answers.

Ventilated is the unconservative direction. It removes heat from the cavity wall, so the section computes cooler, the 500$^\circ$C isotherm sits shallower and the fire-reduced effective section is larger than under the sealed assumption. It rests entirely on your assertion that the airflow is real and sustained for the duration of the fire — a cavity that seals during the fire behaves as sealed, and geometry alone cannot tell the two apart. The editor and the report both disclose the choice wherever it reaches a result.

Two limitations remain:

• Cavity radiation between opposing faces is not modelled. Radiation across a void would heat the shielded far face above the adiabatic result while slightly cooling the near face, so adiabatic is not a strict upper bound on the far wall of a large cavity. This is unchanged by the ventilated option, which is a boundary condition rather than an exchange between surfaces. Which way the capacity moves therefore depends on where the governing bar or tendon sits: a bottom-flange strand under a void is on the safe side of this; a bar in the shielded wall of a box column, or a compression zone above a hollow core, is not. Thin-walled box and hollow-core sections are where it bites. It is a ceiling of the model rather than a setting — there is no input that addresses it — so the editor and the report disclose it on every section that has a void.
• A partially fire-exposed void keeps adiabatic on its remaining faces. If you declare some of a void’s edges fire-exposed, the rest stay adiabatic rather than reverting to ambient convection: a cavity open to the compartment contains fire gases, so its other faces sit against something near compartment temperature, not a 20$^\circ$C reservoir. Adiabatic errs in the safe direction there. Declaring such a void ventilated is refused, not resolved — a cavity cannot be open to a ventilated space at ambient and open to the fire compartment at the same time, and any airflow through a compartment-open cavity carries fire gases rather than 20$^\circ$C air. Declare it sealed, or drop the fire-exposed edges if it really is ventilated to outside the fire.

Fire curves

$t$ is time in minutes; $t_h$ is time in hours.

EN 1992-1-2 B.1.1(1) — scope of the 500°C isotherm method. Clause B.1.1(1) qualifies the isotherm method for standard fire exposure (ISO 834). The method’s accuracy was calibrated against the ISO 834 temperature-time curve; the conservatism of the opposing-error pairing (full strength below 500°C against zero above) is not validated for non-standard fire curves such as ASTM E119 or the hydrocarbon curve. The advanced-fibre method (§4.3) integrates the actual temperature field and carries no equivalent curve restriction.

The ASTM E119 curve is a closed-form fit, not the tabulated data. ACS evaluates the Lie (1992) fit (Structural Fire Protection, ASCE Manuals and Reports on Engineering Practice No. 78) — the standard closed-form representation of ASTM E119 — rather than interpolating ASTM E119-20 Table 1. The two differ, and the difference is disclosed rather than removed because Table 1 specifies no interpolation scheme between its points:

At R240 the fit runs hotter than ASTM tabulates. Measured through the heat transfer solver on a 400 × 600 mm section exposed on four faces at R240, that bias moves the hottest bar by +3.6$^\circ$C, the 500$^\circ$C isotherm depth by +0.29 mm and the effective cool concrete area by $-0.22$% — all in the conservative direction, and none of them enough to change a design outcome. The ISO 834 and hydrocarbon curves are exact evaluations of their published equations and carry no such divergence.

Thermal properties of concrete

Per EN 1992-1-2 §3.3, the thermal properties vary with temperature:

Thermal conductivity (upper limit, EN 1992-1-2 §3.3.3 Eq. 3.8):

The ACS app always uses the upper limit. §3.3.3(2) leaves the choice to the National Annex, so an API or MCP caller must state it (useUpperConductivity). The lower limit (Eq. 3.9) is $k(\theta) = 1.36 - 0.136(\theta/100) + 0.0057(\theta/100)^2$.

Specific heat (siliceous and calcareous aggregate, EN 1992-1-2 §3.3.2), in J/kg·K:

The moisture peak represents the latent heat of evaporating free water: $c_{p,peak} = 1470$ at 1.5 % moisture by weight and 2020 at 3 %. ACS uses 3 %. At 0 % there is no peak, and the 100–200 °C branch is $900 + (\theta - 100)$.

Numerical method

ACS solves the heat equation using a 2D finite element method with triangular elements. The mesh is generated from the section outline (including voids). Time-stepping uses an implicit scheme for unconditional stability.

Moisture migration is not modelled

The heat transfer analysis is a pure-conduction model. It solves the heat equation above using the EN 1992-1-2 §3.3 temperature-dependent properties and nothing more. Free water enters the model only through the specific-heat peak, which absorbs the latent heat of evaporation where that water sits. The model does not represent moisture migration: heating drives pore water toward the cooler core, where it carries latent heat inward and releases it as it recondenses. In a real section this transport holds the interior on a plateau near 100 °C that pure conduction cannot reproduce.

The result is a known, measured bias that runs in opposite directions at different depths:

• Deep interior, during the evaporation plateau: under-predicted. Zehfuß et al. (2020) report measured interior temperatures 13–33 K hotter than pure conduction predicts, and find that the EN specific-heat peak underestimates the effect. ACS was compared with the measured furnace test of a 305 × 305 mm siliceous RC column under ASTM E119 (NRC Canada DBR Internal Report 484, Lie & Lin 1983, specimen 7). At 60 min, ACS reads 92 °C at 101.6 mm depth against a measured 117 °C (−21 %). At the section centre it reads 66 °C against a measured 101 °C (−35 %). A higher moisture content does not close the gap: a larger peak absorbs more heat locally and still carries none of it inward.
• Cover zone: over-predicted, which is conservative. In the same test, at 38.1 mm and 63.5 mm depth (the band around the roughly 48 mm cover to the main bars), ACS reads 5–14 % hotter than measured at 60, 120 and 180 min. It reads hotter than measured at every instrumented depth from the surface to 63.5 mm, and that band contains the measured 500 °C isotherm at all three durations.
• After the plateau: over-predicted. By 120 and 180 min the interior has passed the plateau, and ACS reads 4–12 % hotter than measured at every instrumented depth.

The limitation is accepted because the model errs hot where fire capacity is normally governed: the reinforcement in the cover zone, and the concrete between the exposed faces and the 500 °C isotherm. The interior it under-predicts sits at or near 100 °C, where EN 1992-1-2 Table 3.1 keeps concrete at essentially full strength: $k_c(\theta) = 1.00$ up to 100 °C, falling to 0.95 (siliceous) or 0.97 (calcareous) at 200 °C. The effect on the reduced capacity is small, but it is not zero. A bar or tendon placed deep in a large section will read cooler during the plateau than a furnace test would measure.

These comparisons use the upper-limit conductivity, which is what the ACS app uses. Zehfuß et al. found that the upper limit matched their measured interior temperatures to within about 3.9 K, while the lower limit under-predicted them by about 32 K. An API or MCP caller that selects the lower limit should expect a larger interior under-prediction, and the cover-zone conservatism described above has not been validated for it.

Material degradation

Concrete strength reduction

The concrete compressive strength at elevated temperature is:

Representative values of $k_c$ (siliceous aggregate, per EN 1992-1-2 Table 3.1):

Steel strength reduction

The reinforcement yield strength at elevated temperature is:

Representative values of $k_s$ (hot-rolled bars, per EN 1992-1-2 Table 3.2a):

Steel retains full strength up to approximately 400$^\circ$C, then degrades rapidly. This is why cover is critical — it delays the time for the reinforcement temperature to reach the critical threshold (typically 500$^\circ$C for conventional reinforcement).

Elastic modulus reduction

Both concrete and steel elastic moduli also reduce with temperature, affecting stiffness and deflection but not directly used in the simplified capacity calculation.

Fire capacity calculation

The reduced capacity is computed by:

1. Extracting the temperature at each concrete fibre from the heat transfer solution
2. Applying $k_c(\theta)$ to get the reduced concrete strength at each fibre
3. Extracting the temperature at each reinforcement bar from the heat transfer solution
4. Applying $k_s(\theta)$ to get the reduced steel yield strength at each bar
5. Running the standard flexural (or interaction) analysis with these reduced properties

This produces a fire-rated interaction diagram that sits inside the ambient diagram.

500°C isotherm applicability (EN 1992-1-2 Annex B.1)

The 500°C isotherm method (EN 1992-1-2 Annex B.1) is a simplified alternative to the full heat-transfer + fibre analysis described above. It replaces the temperature field computation with a geometric rule: concrete above 500°C is discarded entirely and the remaining cool zone is analysed at its ambient strength. The method’s accuracy rests on two opposing errors cancelling — full strength below 500°C (an overestimate) against zero strength above it (an underestimate) — and EN 1992-1-2’s calibration confirmed the pairing for sections of adequate minimum width.

Minimum section width (EN 1992-1-2 Table B1)

Clause B.1.1(2) states the method “is valid for minimum width of cross-section given in table B1”:

Below these widths the 500°C isotherm can extend across the full section, the two errors no longer cancel, and the method loses its accuracy basis. This is a hard applicability limit on the method’s scope, not a conservative recommendation.

The relevant width is the thinnest limb attacked from two opposing exposed faces. Exposure is part of the measurement: a face marked unexposed is not a fire front, and a member exposed on only one face has no opposing-face pair and is not subject to the table at all. A hollow section’s wall thickness is what the table bounds, not the section’s envelope dimension.

When no width is measured — one-face exposure, and why that is not a clean bill of health

One case yields no opposing-face thickness for the table to bind on: a member with no pair of opposing exposed faces, of which a slab exposed on its soffit alone is the ordinary example. There is no confronted limb, so Table B1 is not applied — at any thickness, and regardless of the member’s overall size. A 300 × 200, a 500 × 200 and a 2000 × 200 soffit-exposed member all reach the same outcome at R120, because they share an exposure and a depth and differ only in a length that has no bearing on whether two fronts meet.

Where it happens, read it as “not evaluated”, never as “passed”. Table B1’s limit bounds the error where two thermal fronts meet and burn a limb through, and that failure mode is absent here — but the 500°C isotherm method still takes full $f_{ck}$ over the whole cool zone, and concrete a little below 500°C retains nearer 60% of it. The check is inapplicable, not satisfied, and ACS states that on the result rather than letting the absent check read as a passed one. The backstop is physical rather than tabulated: a section the 500°C carve leaves with no cool concrete at all is refused outright, which is exactly what happens to a soffit-exposed slab shallower than the 500°C depth for its period.

A tapering limb is measured, not exempted. Its fronts converge towards a corner rather than meeting head-on, so what has to survive heating from two sides at once is the thickness of the largest circle that fits inside it, not its overall width — and those differ. For a triangle of sides $a \ge b \ge c$ the inradius over the minimum width is $r/w = a/(a+b+c)$, confined to $[1/3, 1/2)$ and equal to $1/3$ only for an equilateral section. The confronted thickness is the diameter $2r$, so $2r/w$ lies in $[2/3, 1)$: a tapered member’s confronted thickness can be as little as two thirds of the width you would read off the drawing, and is exactly two thirds for an equilateral section. ACS scores that inscribed thickness against the tabulated row, so a wedge whose overall width clears the row can still be refused — and should be. An equilateral pier of minimum width 165 mm at R120 is scored at 110 mm against the 160 mm row; at that lower bound the member is entirely above 500°C while its width still clears the table, at every period the table covers.

Void and hollow section handling

EN 1992-1-2 Annex B.1 describes no procedure for internal cavities. For the Table B1 width measurement only, an internal void counts as an additional fire boundary — the wall between the void and the section’s outer face is the limb that gets measured, rather than the section’s envelope dimension. This can cause a hollow section to be refused where a solid one of the same wall thickness would pass.

This is a measurement rule, not a thermal one: it does not make the void a heat source. The heat-transfer model treats an unexposed void surface as adiabatic, per Boundary conditions above. The two are decided independently because Table B1’s applicability rests on calibration coverage — it derives from a D-series of solid beams containing no cavity section at all — rather than on how the cavity conducts heat.

After the 500°C carve, an authored void is treated as a hole in one region, not a separate cool zone. If the carve happens to divide the remaining concrete into disconnected fragments — uncommon, because such a limb is usually already below Table B1’s minimum — the method cannot proceed and ACS refuses on that basis.

In either case, the full heat-transfer + fibre analysis described in the previous sections carries no width limit and handles arbitrary section shapes including hollow and voided sections.

Design standard treatment

ACS implements the advanced calculation method (EN 1992-1-2 Cl. 4.3) for every design code, using 2D FE heat transfer and fibre-based capacity analysis. Today that means AS 3600 designs; the method carries across to ACI 318-19 and EN 1992-1-1 when those codes become selectable. This is more accurate than the tabulated method, especially for non-standard section shapes.

Limitations and assumptions

• Uniform fire exposure along the member length (no thermal gradients in the longitudinal direction)
• No spalling modelling (explosive spalling of concrete cover can occur in high-strength concrete or rapid heating; not captured by the analysis)
• No moisture migration. Evaporation is represented only by the EN 1992-1-2 specific-heat peak, so the deep interior reads cooler than measured during the ~100 °C plateau, while the cover zone reads hotter (conservative). See Moisture migration is not modelled
• Thermal properties are for normal-weight siliceous aggregate concrete; calcareous and lightweight aggregate have different properties
• No mechanical strain effects on thermal analysis (weak coupling; thermal drives mechanical, but not vice versa)

Further reading

• Buchanan, A.H. and Abu, A.K., Structural Design for Fire Safety, 2nd ed., John Wiley & Sons, 2017.
• EN 1992-1-2:2004, Eurocode 2: Design of Concrete Structures — Part 1-2: Structural Fire Design.
• Purkiss, J.A. and Li, L.Y., Fire Safety Engineering Design of Structures, 3rd ed., CRC Press, 2013.
• Lie and Lin, Fire Tests on Reinforced Concrete Columns, Specimen No. 7, NRC Canada Division of Building Research Internal Report 484, 1983.
• Zehfuß et al., Civil Engineering Design, article cend.202000001, 2020.