Fire design of steel beams: lateral-torsional buckling

What is checked on a beam

For a member in bending, the critical temperature is the lowest of the temperatures at which it fails by:

cross-section bending resistance (cl. 4.2.3.3), for beams restrained against lateral buckling;
lateral-torsional buckling (cl. 4.2.3.3(4) and (5)), for beams with an unrestrained compression flange;
shear resistance (cl. 4.2.3.3(6)), followed by the interaction between shear and bending.

Cross-section class in bending

ε is reduced by the factor 0.85. Details are in the guide on cross-section classes in fire.

Moment resistance of the cross-section

For Class 1 or 2 cross-sections with a uniform temperature θ_a:

M_fi,θ,Rd = k_y,θ · W_pl,y · f_y / γ_M,fi eq. (4.8)

For Class 3 cross-sections with a uniform temperature θ_a:

M_fi,θ,Rd = k_y,θ · W_el,y · f_y / γ_M,fi cl. 4.2.3.4; for a non-uniform temperature, k_y,θ is taken at the maximum temperature in the section

When the temperature is not uniform over the section or along the beam, the moment resistance is increased through the adaptation factors κ₁ and κ₂:

M_fi,t,Rd = M_fi,θ,Rd / (κ₁ · κ₂) eq. (4.10)

Situation	Factor
Beam exposed on all four sides	κ₁ = 1.00
Unprotected beam exposed on three sides, with a concrete slab on the fourth	κ₁ = 0.70
Protected beam exposed on three sides, with a concrete slab on the fourth	κ₁ = 0.85
At the supports of a statically indeterminate beam	κ₂ = 0.85
In all other cases	κ₂ = 1.00

For beams that cannot buckle laterally, the critical temperature follows directly from eq. (4.22), with the degree of utilisation μ₀ = M_fi,Ed / M_fi,0,Rd, where M_fi,0,Rd = W · f_y / (κ₁ · κ₂ · γ_M,fi).

Lateral-torsional buckling

A beam whose compression flange is not laterally restrained can fail by lateral-torsional buckling. In fire, the buckling resistance moment becomes (Class 1 or 2):

M_b,fi,t,Rd = χ_LT,fi · W_pl,y · k_y,θ,com · f_y / γ_M,fi eq. (4.12); k_y,θ,com at the maximum temperature of the compression flange

χ_LT,fi = 1 / [ φ_LT,θ + √(φ_LT,θ² − λ̄_LT,θ²) ] φ_LT,θ = ½ · [ 1 + α · λ̄_LT,θ + λ̄_LT,θ² ] α = 0.65 · √(235 / f_y)

λ̄_LT,θ = λ̄_LT · √(k_y,θ / k_E,θ) λ̄_LT = √(W_y · f_y / M_cr)

The factors κ₁ and κ₂ do not apply to lateral-torsional buckling. As for columns, a single buckling curve is used in fire, through the factor α.

Elastic critical moment M_cr

EN 1993-1-1 gives no formula for M_cr. The general expression for doubly symmetric sections (ENV 1993-1-1, Annex F) is commonly used:

M_cr = C₁ · π² · E · I_z / (k · L)² · { √[ (k/k_w)² · I_w/I_z + (k · L)² · G · I_t / (π² · E · I_z) + (C₂ · z_g)² ] − C₂ · z_g }

C₁ depends on the shape of the bending moment diagram: 1.00 for a uniform moment, 1.132 for a uniformly distributed load on a simply supported beam, 1.365 for a point load at mid-span.
C₂ · z_g accounts for where the load is applied: a load on the top flange (z_g = +h/2) lowers M_cr, a load on the bottom flange raises it.
k and k_w are effective length factors for lateral rotation and for warping (1.0 for ends free to rotate, 0.5 for fixed ends).

Shear and its interaction with bending

The shear resistance is reduced by k_y,θ of the web (cl. 4.2.3.3(6)):

V_fi,t,Rd = k_y,θ,web · A_v · f_y / (√3 · γ_M,fi)

If the shear force exceeds half the shear resistance, the moment resistance is reduced as at normal temperature (EN 1993-1-1, cl. 6.2.8), with ρ = (2 · V_fi,Ed / V_fi,t,Rd − 1)².

Minor-axis bending and hollow sections

An I/H section bent about its minor axis cannot buckle laterally; only the cross-section resistance and shear matter.
Square and circular hollow sections (SHS, CHS) do not suffer lateral-torsional buckling. Rectangular hollow sections (RHS) bent about the major axis can, but it rarely governs.

Informative material summarising the provisions of EN 1993-1-2. It does not replace the text of the standard, the National Annex or verification by a qualified engineer.