EN 1993-1-2 Fire Design — Section Factor, Critical Temperature, Passive Protection

Quick Reference: This guide covers structural fire design of steel members to EN 1993-1-2:2005. We explain the section factor Am/V and how it controls heating rate, the critical temperature θcr as a function of load utilisation, the three simplified design approaches (load level, critical temperature, limiting temperature), and passive fire protection methods including intumescent coatings, board systems, and sprayed cementitious materials. All clauses reference EN 1993-1-2.

PRELIMINARY — NOT FOR CONSTRUCTION. All calculations are illustrative educational examples. Results must be verified by a licensed Professional Engineer before use in any design project.

1. Fire Design Philosophy — Accidental Limit State

Structural fire design differs fundamentally from ambient temperature design in two respects: the loads are reduced (accidental combination), and the material strength degrades with temperature. EN 1993-1-2 prescribes three design approaches, each appropriate for different levels of design refinement:

The fire resistance period is expressed in minutes: R30, R60, R90, R120 — these require the member to maintain its load-bearing function for 30, 60, 90, or 120 minutes under the standard ISO 834 fire curve.

2. Section Factor — Am/V (or Ap/V)

The section factor is the single most important parameter in steel fire design. It governs how quickly a bare steel section heats up when exposed to fire.

Am/V = heated perimeter / volume per unit length (m⁻¹)

Where:

• Am = exposed surface area per unit length (m²/m)
• V = volume of steel per unit length (m³/m)
• A = cross-sectional area (often reported as Ap/V = profile perimeter / volume)

For I-sections exposed on 3 sides (top flange in contact with a concrete slab — the most common floor beam case):

Am/V = (b + 2h) / A

where b = bottom flange width, h = section depth, A = cross-sectional area.

For I-sections exposed on 4 sides (columns, perimeter beams with no slab contact):

Am/V = (2b + 2h − tw) / A ≈ (2b + 2h) / A (approximately)

Example Section Factors:

Golden rule: sections with Am/V > 200 m⁻¹ heat so quickly that unprotected steel will reach critical temperature within 10–15 minutes of standard fire exposure — they require fire protection for any fire rating. Sections with Am/V < 50 m⁻¹ (heavy column sections, large hollow sections fillable with concrete) can sometimes achieve R30 without protection.

3. Unprotected Steel Temperature Rise

For unprotected steel, the temperature rise Δθa,t during each time step Δt (typically 5 seconds) is calculated per EN 1993-1-2 Cl. 4.2.5.1:

Δθa,t = ksh × (Am/V) × (1 / (ca × ρa)) × ḣnet,d × Δt

Where:

• ksh = shadow effect correction factor (ksh = 0.9 for I-sections, 1.0 for hollow sections)
• ca = specific heat of steel (J/kgK), temperature-dependent per EN 1993-1-2 §3.4.1.2 — ca peaks at 735°C due to phase change
• ρa = density of steel = 7850 kg/m³
• ḣnet,d = net heat flux = ḣnet,c + ḣnet,r (convective + radiative)

The convective component: ḣnet,c = αc × (θg − θm) where αc = 25 W/m²K (standard fire), 35 W/m²K (hydrocarbon fire).

The radiative component: ḣnet,r = Φ × εm × εf × σ × [(θr + 273)⁴ − (θm + 273)⁴] Where Φ = 1.0 (configuration factor), εm = 0.7 (steel emissivity), εf = 1.0 (fire emissivity), σ = 5.67 × 10⁻⁸ W/m²K⁴.

Worked temperature progression for an unprotected IPE 400 beam (Am/V = 116 m⁻¹):

The yield strength retention factor ky,θ drops to 0.35 at 15 minutes — meaning the beam retains only 35% of its ambient-temperature yield strength. For a beam originally utilised at 55% (μ0 = 0.55), failure occurs when ky,θ × fyd = μ0 × fyd → ky,θ = 0.55 → from the table above, approximately θcr = 580°C, reached at about 13 minutes. Unprotected R15, not R30.

4. Critical Temperature θcr

The critical temperature is the steel temperature at which the member fails under the applied fire loading. It is a function of the degree of utilisation μ0.

μ0 = Efi,d / Rfi,d,0

Where:

• Efi,d = design effect in the fire situation (reduced loads)
• Rfi,d,0 = design resistance at time t = 0 (ambient temperature), factoring in γM,fi = 1.0

Critical temperature formula (EN 1993-1-2 Cl. 4.2.4(3)):

θa,cr = 39.19 × ln(1 / (0.9674 × μ0³·⁸³³) − 1) + 482 (°C)

This formula is valid for μ0 ≥ 0.013. For μ0 < 0.013, θa,cr is taken as 1000°C (members with negligible utilisation effectively have unlimited fire resistance).

Typical μ0 values and resulting θcr:

The goal of fire engineering is to keep θa ≤ θcr for the required fire duration. This is achieved through:

1. Section choice: heavier sections (lower Am/V) heat more slowly.
2. Load reduction: accurate live load assessment to keep μ0 low.
3. Passive protection: insulation that slows the temperature rise.

5. Fire Protection Materials

When unprotected steel cannot achieve the required fire resistance, passive fire protection is needed. The three main families are:

5.1 Intumescent Coatings (EN 13381-8)

Intumescent coatings are thin-film epoxy or water-based paints that expand to 30–50 times their applied thickness when heated above 200°C. The resulting char forms an insulating foam layer.

Design procedure (EN 1993-1-2 Cl. 4.2.5.2):

The steel temperature rise is: Δθa,t = (λp / dp) × (Ap/V) × (θg,t − θa,t) × Δt / (ca × ρa) − (eφ/10 − 1) × Δθg,t

Where λp is the effective thermal conductivity of the intumescent char (temperature-dependent, provided by the manufacturer's ETA), and dp is the dry film thickness (DFT).

Typical DFT requirements (Cafco or Nullifire systems, Am/V ≤ 150 m⁻¹):

Higher Am/V or higher μ0 → thicker DFT needed. Always consult the manufacturer's assessment report for the specific section factor range.

5.2 Board Protection (EN 13381-4)

Mineral fibre or calcium silicate boards are fixed mechanically around the steel section, forming a box enclosure. Board systems are common for columns and exposed beams where aesthetics are secondary (plant rooms, car parks).

Advantage: very reliable insulation, no on-site curing, visible quality assurance. Disadvantage: bulky (25–50 mm thickness), labour-intensive to install around complex connections.

The temperature of board-protected steel follows: Δθa,t = (λp / dp) × (Ap/V) × (1 / (ca × ρa)) × (θg,t − θa,t) × Δt / (1 + φ/3)

Where dp is the board thickness and φ = (cp × ρp × dp) / (ca × ρa) × (Ap/V) accounts for the thermal capacity of the board material itself.

5.3 Sprayed Cementitious / Vermiculite (EN 13381-4)

Spray-applied fire protection is applied wet and cured on-site. Used primarily for large steel-framed buildings with repetitive exposed steelwork.

• Thickness: 10–50 mm depending on rating and section factor.
• Density: 250–550 kg/m³.
• Thermal conductivity λp ≈ 0.12 W/mK — similar to board materials.

Spray systems are cost-effective for large areas but require careful on-site quality control. Thickness must be verified by gauge measurement after curing.

6. Worked Example — IPE 400 Floor Beam, R60 Rating

Given:

• Beam: IPE 400, S275, simply supported, 8.0 m span
• Am/V (3-sided) = 116 m⁻¹
• Design bending moment at ambient: MEd = 320 kN·m
• Mc,Rd (ambient) = Wpl,y × fy / γM0 = 1310 × 10³ × 275 / 1.0 = 360.3 kN·m
• Ambient utilisation: 320/360.3 = 0.888 → but load reduction in fire

Step 1 — Fire Limit State Loading (EN 1990 Accidental Combination)

Efi,d = Gk + ψ1 × Qk

For an office floor (Category B): ψ1 = 0.5

The design fire moment is approximately: Mfi,d = (MEd / (1.35 × Gk + 1.50 × Qk)) × (Gk + 0.5 × Qk) For a typical office where Gk/Qk ≈ 1.0: Mfi,d/MEd ≈ (1 + 0.5) / (1.35 + 1.50) = 1.5 / 2.85 = 0.526 Mfi,d = 0.526 × 320 = 168.3 kN·m

Step 2 — Degree of Utilisation

μ0 = Efi,d / Rfi,d,0 = Mfi,d / Mc,Rd = 168.3 / 360.3 = 0.467

Step 3 — Critical Temperature

θa,cr = 39.19 × ln(1 / (0.9674 × 0.467³·⁸³³) − 1) + 482 = 39.19 × ln(1 / (0.9674 × 0.0444) − 1) + 482 = 39.19 × ln(1 / 0.0430 − 1) + 482 = 39.19 × ln(23.26 − 1) + 482 = 39.19 × ln(22.26) + 482 = 39.19 × 3.103 + 482 = 121.6 + 482 = 603.6°C

Step 4 — Unprotected Temperature at R60

From iterative calculation (or design tables), an unprotected IPE 400 with Am/V = 116 m⁻¹ reaches:

• θa,30min ≈ 720°C
• θa,60min ≈ 870°C

Since θa,60 (870°C) > θcr (603.6°C), unprotected steel is inadequate → fire protection required.

Step 5 — Protection Selection

Required intumescent DFT for Am/V = 116 m⁻¹, R60 rating, and μ0 = 0.47 (θcr = 604°C):

From manufacturer's ETA (example — Cafco SprayFilm WB3):

• DFT = 0.95 mm for Am/V ≤ 120 m⁻¹, R60, θcr ≥ 600°C

Therefore, specify 0.95 mm DFT intumescent coating to the bottom flange and web (3-sided exposure). The top flange is shielded by the concrete slab and requires no protection.

7. Design Notes for Common Scenarios

• Composite beams with profiled decking trap heat in the deck ribs, reducing the effective exposed perimeter. Some National Annexes permit a reduced Am/V for beams partially shielded by trapezoidal decking.
• Concrete-filled hollow sections effectively reduce Am/V as the concrete core acts as a heat sink. EN 1994-1-2 provides procedures for composite columns in fire.
• Cellular beams (with web openings) have higher Am/V due to the internal perimeter of the openings. The section factor must account for the full exposed perimeter including hole surfaces. Cellular beams often require 1.5–2× the DFT of equivalent solid-web sections.
• Offshore / petrochemical applications use the hydrocarbon fire curve (EN 1993-1-2 Cl. 3.2.2), which reaches 1100°C in 5 minutes — far more severe than the ISO 834 cellulosic curve. Section factors and protection thicknesses must be recalculated for the hydrocarbon curve.
• Connection fire protection — EN 1993-1-2 Cl. 4.2.2(8) permits reduced protection (or no protection) for connections where the connection temperature is demonstrably lower than the connected member temperature. For beam-to-column joints, the column's thermal mass reduces the connection heating rate.

Related Pages

• EN 1993-1-1 Beam Design — IPE 300 Worked Example, χLT
• EN 1993-1-1 Column Design — HEA 200 Example, Buckling Curves
• EN 1993-1-1 Tension Member Design — Clause 6.2.3, Angle Example
• EN 1993-1-9 Fatigue Design — Detail Categories, S-N Curves
• EN 1994 Composite Beam Design — Shear Connectors, IPE Worked Example