UK Steel Fire Design -- BS EN 1993-1-2 Limiting Temperature Method

Fire engineering of structural steelwork is a mandatory design check under UK Building Regulations. BS EN 1993-1-2 provides three assessment methods: member analysis, analysis of parts of the structure, and global structural analysis. For the vast majority of UK building steelwork, the limiting temperature method (member analysis) is sufficient and is the standard approach in UK design offices. It determines whether a steel member, with or without applied fire protection, can maintain its load-bearing function for the required fire resistance period. This reference covers the limiting temperature method, section factor calculation, critical temperature determination, and intumescent coating thickness selection.

Fire Limit State -- Load Combination

Per EN 1990 UK NA, the accidental design situation for fire is:

E_fi,d,t = SUM(Gk,j) + psi_1,1 x Qk,1 + SUM(psi_2,i x Qk,i)

Where psi_1,1 = 0.5 for office occupancy (the leading variable action) and psi_2,i = 0.3 for other occupancies (UK NA Table A1.1). This reduced loading, typically 55-65% of the ambient-temperature ULS load, is a fundamental reason why fire engineering often succeeds: the required resistance in fire is lower than at ambient temperature.

The degree of utilisation at the fire limit state is:

mu_0 = E_fi,d / R_fi,d,0

Where R_fi,d,0 is the design resistance at time t=0 (ambient temperature, but using the fire partial factors). For steel, gamma_M,fi = 1.0 (UK NA confirms) compared to gamma_M0 = 1.00 at ambient -- so R_fi,d,0 = R_d for Class 1-3 sections.

Section Factor Am/V (Heated Perimeter / Volume)

The section factor is the single most important geometric parameter in steel fire design. It controls how rapidly the steel temperature rises during fire exposure. A high section factor means rapid heating; a low section factor means slower heating.

Am/V = heated perimeter (m) / volume per unit length (m^3/m) = units of m^-1

For UK practice, section factors are published in SCI P363 (Blue Book) and the Tata Steel sections handbook:

Key rule: Larger, more compact sections have lower Am/V and achieve higher fire resistance without protection. Hollow sections (SHS, RHS, CHS) have inherently lower Am/V than open sections (UB, UC) of similar mass because they enclose more volume per unit perimeter.

3-sided vs 4-sided exposure: Beams supporting concrete slabs have the top flange shielded (3-sided exposure). Columns, bracing members, and edge beams with edge protection omitted may have 4-sided exposure. The Am/V for 3-sided exposure is approximately 75-85% of the 4-sided value for UB sections.

Critical Temperature -- Limiting Temperature Method

The critical temperature theta_cr is the steel temperature at which the member can no longer support the applied load in the fire condition. Per EN 1993-1-2 Clause 4.2.4:

theta_cr = 39.19 x ln(1 / (0.9674 x mu_0^3.833) - 1) + 482

Where mu_0 is the degree of utilisation at t=0 (must not exceed 0.65 for the standard formula; if mu_0 > 0.65, use the simplified expression or a more detailed analysis).

Critical Temperature vs Utilisation Level

mu_0 exceeding 0.65 requires either the simplified expression or the full method accounting for non-uniform temperature distribution.

Unprotected Steel Temperature After 30 Minutes

For a standard ISO 834 fire curve, unprotected steel reaches approximately:

For 30-minute fire resistance, unprotected steel achieves the required performance when theta at 30 min < theta_cr. For a typical UK office beam with mu_0 = 0.50 (theta_cr = 611 degrees C), this requires Am/V <= 150 m^-1 -- achievable with UC sections and larger UB sections but not with lightweight beams.

Intumescent Coating Thickness -- Selection Procedure

Intumescent coatings are characterised by their dry film thickness (DFT), which determines the fire resistance period for a given section factor and critical temperature. The selection procedure follows BS EN 13381-8:

1. Determine the required fire resistance period from Approved Document B (30, 60, 90, or 120 minutes)
2. Calculate the section factor Am/V for the steel member (3-sided for beams, 4-sided for columns)
3. Determine the critical temperature theta_cr from mu_0 using the formula above
4. Select the intumescent coating DFT from the manufacturer's assessment report -- interpolating between tested configurations for the specific Am/V and theta_cr

Indicative DFT Requirements for UK Sections -- 60 Minutes FR

Based on a typical UK water-based intumescent system (Nullifire S605 or equivalent):

Important: These are indicative values for preliminary design only. The actual DFT must be taken from the manufacturer's project-specific assessment report based on fire test data to BS EN 13381-8. Interpolation between published values, or scaling from a different manufacturer's data, is not permitted under UK Building Regulations.

Worked Example -- UKB Office Floor Beam

Problem: A 457 x 191 x 67 UB in S355 supports a composite floor slab in a 5-storey London office building. The beam is simply supported at 7.5 m span, 3.0 m spacing, with an imposed load of 3.0 kN/m^2 (office occupancy). Fire resistance of 60 minutes is required per Approved Document B (building height < 18 m, multi-storey office).

Step 1 -- Degree of utilisation at fire limit state: Dead load per beam: 4.5 kN/m^2 x 3.0 m = 13.5 kN/m + self-weight 0.67 kN/m = 14.2 kN/m Imposed load (fire): 0.5 x 3.0 x 3.0 = 4.5 kN/m Total fire load: 18.7 kN/m

Ambient ULS load: 1.35 x 14.2 + 1.5 x 9.0 = 19.2 + 13.5 = 32.7 kN/m (approx)

mu_0 = 18.7 x M_ambient / 32.7 x M_ambient = 18.7 / 32.7 = 0.57

However, the composite beam gains additional resistance at ambient from the slab, so actual mu_0 for the steel section alone is typically lower. Assume from detailed design mu_0 = 0.50.

Step 2 -- Critical temperature: theta_cr = 39.19 x ln(1 / (0.9674 x 0.50^3.833) - 1) + 482 theta_cr = 39.19 x ln(1 / (0.9674 x 0.0699) - 1) + 482 theta_cr = 39.19 x ln(1 / 0.0676 - 1) + 482 theta_cr = 39.19 x ln(14.79 - 1) + 482 theta_cr = 39.19 x ln(13.79) + 482 = 39.19 x 2.624 + 482 = 585 + 482 approximately = 585 degrees C (recalculate)

Recalculation: mu_0^3.833 = 0.50^3.833 = 0.0699 0.9674 x 0.0699 = 0.0676 1 / 0.0676 - 1 = 14.79 - ... Actually: 1 / 0.0676 = 14.79 14.79 - 1 = 13.79 ln(13.79) = 2.624 39.19 x 2.624 = 102.8 102.8 + 482 = 584.8 approximately 585 degrees C

Wait -- that seems high. Let me recalculate more carefully. EN 1993-1-2 Eq 4.22:

theta_cr = 39.19 x ln(1 / (0.9674 x mu_0^(3.833)) - 1) + 482

For mu_0 = 0.50: mu_0^3.833 = exp(3.833 x ln(0.50)) = exp(3.833 x -0.6931) = exp(-2.657) = 0.0701

0.9674 x 0.0701 = 0.0678

1/0.0678 - 1 = 14.75 - 1 = 13.75

ln(13.75) = 2.621

39.19 x 2.621 = 102.7

theta_cr = 102.7 + 482 = 584.7 approximately 585 degrees C

This seems reasonable. The critical temperature is approximately 585 degrees C for mu_0 = 0.50.

Step 3 -- Temperature at 60 minutes: Am/V for 457 x 191 UB 67 (3-sided) = 200 m^-1 approximately. From ISO 834 with Am/V = 200 m^-1 and Am/V correction, unprotected temperature at 60 min exceeds 900 degrees C >> theta_cr = 585 degrees C. Protection required.

Step 4 -- Intumescent DFT: From manufacturer's assessment for Am/V = 200 m^-1, theta_cr = 585 degrees C, 60 min FR: DFT approximately 2000 microns (2.0 mm) for a typical water-based intumescent.

Verify that the manufacturer's assessment covers this exact combination of Am/V and theta_cr, and that the primer and top-coat specification matches the tested system.

Design Resources

• UK Fire Protection Guide -- Intumescent, board, and spray systems
• UK Fire Rating Guide -- EN 1993-1-2 design overview and Approved Document B
• UK Steel Properties -- Elevated Temperature -- fy and E reduction factors with temperature
• UK Beam Sizes and Section Factors -- Am/V values for UKB/UKC sections
• UK Composite Beam Design -- Fire design of composite floors
• UK Steel Design EN 1993 Guide -- Complete Eurocode 3 overview

Frequently Asked Questions

What is the difference between the limiting temperature method and the load-bearing capacity method?

The limiting temperature method (Clause 4.2.4) works backwards: given the applied load at the fire limit state, calculate the temperature at which the member would fail, then verify that the steel temperature after the required fire duration stays below this critical temperature. The load-bearing capacity method works forwards: calculate the member's resistance at the design steel temperature (after fire exposure) and verify it exceeds the applied load. Both are valid; the limiting temperature method is faster for preliminary design, while the load-bearing method provides a direct capacity utilisation for the fire condition.

Can I use the critical temperature formula for any degree of utilisation?

The standard formula (Eq 4.22) is valid for mu_0 up to 0.65. Above 0.65, the formula becomes unconservative because it assumes uniform temperature distribution and neglects the effect of thermal expansion restraint. For mu_0 > 0.65, either use the simplified expression (which gives a lower, more conservative critical temperature) or adopt the full member analysis method accounting for non-uniform temperature distribution and thermal expansion effects.

How much does fire protection add to UK steel frame costs?

Fire protection is a significant cost component, typically representing 10-15% of the structural steel package cost for a UK multi-storey office building requiring 60-minute fire resistance. Intumescent coating costs approximately 20-35 GBP per square metre of protected steel surface (including application and site touch-up), varying with the required DFT and location (shop-applied is cheaper than site-applied). Board protection costs 25-45 GBP per square metre. For a typical 5,000 m^2 office floor plate, fire protection costs are in the range of 15,000-30,000 GBP per floor.

When is the section factor not the governing parameter for steel fire design?

Section factor governs for members heated uniformly (tension members, unrestrained beams). However, for members with non-uniform temperature distribution (columns in walls, beams with partial protection, members where the fire attack is localised), the thermal gradient through the section and the restraint to thermal expansion may govern. In these cases, the simple limiting temperature method is not directly applicable and either the advanced calculation method (Clause 4.3) or a global structural fire analysis is required.

Educational reference only. All design values are per BS EN 1993-1-2:2005 + UK National Annex and UK Building Regulations Approved Document B (2019 edition, as amended). Intumescent coating DFT values are indicative only and must be confirmed by the manufacturer's project-specific assessment report to BS EN 13381-8. Fire engineering designs must be independently verified by a Chartered Structural Engineer. Results are PRELIMINARY -- NOT FOR CONSTRUCTION without independent professional verification.