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:

\rho c_p \frac{\partial T}{\partial t} = \frac{\partial}{\partial x}\left(k \frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k \frac{\partial T}{\partial y}\right)

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):

q = h_c (T_{fire} - T_s) + \varepsilon_r \sigma (T_{fire}^4 - T_s^4)

Where:

$h_c$ = convective heat transfer coefficient (25 W/m $^2\cdot$ K for standard fire)
$\varepsilon_r$ = 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

Unexposed surfaces: adiabatic (no heat loss), which is conservative.

Fire curves

Curve	Equation	Peak temperature
ISO 834	$T = 345 \log_{10}(8t + 1) + 20$	~1050 $^\circ$ C at 120 min
ASTM E119	Tabulated time-temperature data	~1010 $^\circ$ C at 120 min
Hydrocarbon	$T = 1080(1 - 0.325e^{-0.167t} - 0.675e^{-2.5t}) + 20$	~1100 $^\circ$ C at 30 min

Thermal properties of concrete

Per EN 1992-1-2 Annex A, the thermal properties vary with temperature:

Thermal conductivity (upper bound):

k(\theta) = 2 - 0.2451 \left(\frac{\theta}{100}\right) + 0.0107 \left(\frac{\theta}{100}\right)^2 \quad \text{W/m}\cdot\text{K}

Specific heat (for siliceous aggregate, dry concrete):

c_p(\theta) \approx 900 + 80\left(\frac{\theta}{120}\right) - 4\left(\frac{\theta}{120}\right)^2 \quad \text{J/kg}\cdot\text{K}

With a moisture peak near 100 $^\circ$ C to account for evaporation.

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.

Material degradation

Concrete strength reduction

The concrete compressive strength at elevated temperature is:

f'_{c,\theta} = k_c(\theta) \cdot f'_c

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

Temperature ( $^\circ$ C)	$k_c$
20	1.00
100	1.00
200	0.95
300	0.85
400	0.75
500	0.60
600	0.45
700	0.30
800	0.15
900	0.08

Steel strength reduction

The reinforcement yield strength at elevated temperature is:

f_{y,\theta} = k_s(\theta) \cdot f_y

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

Temperature ( $^\circ$ C)	$k_s$
20	1.00
100	1.00
200	1.00
300	1.00
400	1.00
500	0.78
600	0.47
700	0.23
800	0.11

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:

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

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

Design standard treatment

Aspect	AS 3600:2018	ACI 216.1-14	EN 1992-1-2
Method	Tabulated or rational	Tabulated or rational	Tabulated, simplified, or advanced
Thermal properties	Not specified (use EN 1992-1-2)	ASTM tables	Annex A (temperature-dependent)
$k_c$ values	EN 1992-1-2 tables	Own tables	Table 3.1 (by aggregate type)
$k_s$ values	EN 1992-1-2 tables	Own tables	Table 3.2a/3.2b (by bar type)
Fire curves	ISO 834	ASTM E119	ISO 834, Hydrocarbon, parametric

ACS implements the advanced calculation method (EN 1992-1-2 Cl. 4.3) for all three codes, using 2D FE heat transfer and fibre-based capacity analysis. 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 effects (simplified treatment of evaporation)
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)