EN 1993-1-1 Steel Beam Design — Section Classification, Moment Resistance, LTB
Quick Reference: The EN 1993-1-1 beam design verification checks: (1) cross-section resistance (Cl. 6.2.5), (2) lateral-torsional buckling (Cl. 6.3.2), and (3) shear buckling (Cl. 6.2.6) when the web is slender. The interaction of bending and shear (Cl. 6.2.8) may reduce moment capacity for high coincident shear. Use the free beam capacity calculator for instant EN 1993 checks.
Section Classification (EN 1993-1-1 Cl. 5.5)
Before calculating resistance, classify the cross-section. The class determines which moment resistance formula applies:
| Class | Behavior | Moment Resistance |
|---|---|---|
| 1 (Plastic) | Plastic hinge with rotation capacity | Mc,Rd = Wpl * fy / gamma_M0 |
| 2 (Compact) | Plastic moment, limited rotation | Mc,Rd = Wpl * fy / gamma_M0 |
| 3 (Semi-compact) | Yield in extreme fibre only | Mc,Rd = Wel * fy / gamma_M0 |
| 4 (Slender) | Local buckling before yield | Mc,Rd = Weff * fy / gamma_M0 |
Classification is based on c/t ratios (width-to-thickness) of the flange and web under the applied stress distribution. Table 5.2 gives the limiting proportions.
For hot-rolled I-sections in S355, most standard UK sections (UKB, UKC) are Class 1 or Class 2 for pure bending. Class 3 and 4 become relevant for fabricated sections, slender webs, or sections in high-strength steel (S460, S690).
Cross-Section Moment Resistance (Cl. 6.2.5)
For Class 1 and 2 cross-sections, the design moment resistance is:
Mc,Rd = Wpl * fy / gamma_M0
where Wpl is the plastic section modulus about the axis of bending, and gamma_M0 = 1.00.
For a 457x191x67 UKB in S355 (Wpl,y = 1,470 cm^3): Mc,Rd = 1,470 _ 10^3 _ 355 / 1.00 = 522 kN.m
For Class 3 cross-sections, use Wel instead of Wpl. This effectively caps the moment at first yield.
Lateral-Torsional Buckling (Cl. 6.3.2)
When the compression flange is not continuously restrained, the design buckling resistance moment is:
Mb,Rd = chi*LT * Wy _ fy / gamma_M1
where chi_LT <= 1.0 is the LTB reduction factor, Wy = Wpl,y for Class 1/2 or Wel,y for Class 3.
The reduction factor chi_LT is computed from:
chi_LT = 1 / (Phi_LT + sqrt(Phi_LT^2 - lambda_LT_bar^2)) ... but chi_LT <= 1.0
where:
- Phi_LT = 0.5 [1 + alpha_LT (lambda_LT_bar - 0.2) + lambda_LT_bar^2]
- lambda_LT_bar = sqrt(Wy * fy / Mcr)
- Mcr is the elastic critical moment for LTB
The imperfection factor alpha_LT depends on the buckling curve selected from Table 6.4. For hot-rolled I-sections, the buckling curve is typically 'a' (alpha_LT = 0.21) or 'b' (alpha_LT = 0.34) depending on h/b ratio.
Mcr — Elastic Critical Moment
Mcr can be computed using the formula in NCCI SN003 or through the general method:
Mcr = C1 _ pi^2 _ E * Iz / (kzL)^2 _ sqrt( (kz/kw)^2 _ Iw/Iz + (kzL)^2 * G _ It / (pi^2 _ E * Iz) )
For a simply supported beam under uniform moment with end warping prevented, kz = kw = 1.0 and C1 = 1.0.
For a 457x191x67 UKB with L = 6.0 m between lateral restraints:
- Iz = 1,450 cm^4, It = 37.1 cm^4, Iw = 0.594 dm^6
- Mcr ~ 840 kN.m (approximate — use software for exact value)
- lambda_LT_bar = sqrt(1,470*10^3 * 355 / 840e6) = 0.79
- chi_LT ~ 0.82
- Mb,Rd = 0.82 * 1,470*10^3 * 355 / 1.00 = 428 kN.m
The LTB check reduces capacity by ~18% for this unbraced length. For a fully restrained beam, the full Mc,Rd = 522 kN.m applies.
Shear Resistance (Cl. 6.2.6)
The design plastic shear resistance, for unstiffened webs:
Vpl,Rd = Av * (fy / sqrt(3)) / gamma_M0
where Av is the shear area. For hot-rolled I-sections loaded parallel to the web, Av = A - 2btf + (tw + 2r)_tf (but not less than eta _ hw * tw).
A shear buckling check (Cl. 6.2.6(6)) is only required when hw/tw > 72 * epsilon / eta (for unstiffened webs).
Bending and Shear Interaction (Cl. 6.2.8)
When VEd > 0.5 * Vpl,Rd, the moment resistance is reduced:
For Class 1/2: My,V,Rd = (Wpl - rho _ Aw^2/4tw) _ fy / gamma_M0
where rho = (2*VEd/Vpl,Rd - 1)^2.
This typically only matters for short, heavily loaded beams (e.g., transfer beams) where both moment and shear are high at the same cross-section.
Cross-Section Classification — c/t Limits (Table 5.2)
The classification depends on the width-to-thickness ratio c/t of the compression elements. The limiting proportions depend on:
epsilon = sqrt(235 / fy)
For S275: epsilon = 0.92. For S355: epsilon = 0.81. For S460: epsilon = 0.71.
Internal Compression Parts (Web in Bending)
| Class | Limit (c/t) for S355 | Limit (c/t) for S275 |
|---|---|---|
| 1 | 72 * epsilon = 58.6 | 72 * epsilon = 66.5 |
| 2 | 83 * epsilon = 67.5 | 83 * epsilon = 76.7 |
| 3 | 124 * epsilon = 100.9 | 124 * epsilon = 114.6 |
Outstand Flanges (Rolled I-Sections)
| Class | Limit (c/t) for S355 | Limit (c/t) for S275 |
|---|---|---|
| 1 | 9 * epsilon = 7.3 | 9 * epsilon = 8.3 |
| 2 | 10 * epsilon = 8.1 | 10 * epsilon = 9.2 |
| 3 | 14 * epsilon = 11.4 | 14 * epsilon = 12.9 |
Classification Example: 457x191x67 UKB in S355
Section properties: b = 189.9 mm, tf = 12.7 mm, tw = 8.5 mm, r = 10.2 mm, d = 407.6 mm between fillets.
Flange: c = (b - tw - 2r) / 2 = (189.9 - 8.5 - 20.4) / 2 = 80.5 mm. c/tf = 80.5 / 12.7 = 6.34 < 7.3 → Class 1.
Web (pure bending): c = d = 407.6 mm. c/tw = 407.6 / 8.5 = 47.95 < 58.6 → Class 1.
The section is Class 1 for pure bending. Use Mc,Rd = Wpl,y * fy / gamma_M0.
Bi-Axial Bending (Cl. 6.2.9)
When a section is bent about both axes, the interaction check is:
For Class 1/2: [M_y,Ed / M_N,y,Rd]^alpha + [M_z,Ed / M_N,z,Rd]^beta <= 1.0
where alpha = 2.0 and beta = 1.0 for I-sections (beta = 2.0 for CHS/RHS).
For a 457x191x67 UKB with M_y,Ed = 300 kN.m and M_z,Ed = 15 kN.m (minor-axis bending from bracing eccentricity):
- M_N,y,Rd = 522 kN.m (Class 1, plastic)
- M_N,z,Rd = Wpl,z * fy / gamma_M0 = 188*10^3 * 355 / 1.0 = 66.7 kN.m
Check: (300/522)^2 + (15/66.7)^1 = 0.331 + 0.225 = 0.556 < 1.0 → OK.
Combined Bending and Axial Force (Cl. 6.2.9.1)
For Class 1/2 cross-sections with axial force, the reduced plastic moment resistance M_N,Rd accounts for the portion of the section carrying axial load.
When n = N_Ed / N_pl,Rd <= 0.25, no reduction is needed for bending about y-y if:
N_Ed <= 0.25 * N_pl,Rd and N*Ed <= 0.5 * hw _ tw * fy / gamma_M0
For a 457x191x67 UKB in S355 with N_Ed = 200 kN:
- A = 85.5 cm^2, N_pl,Rd = 85.5*10^2 * 355 / 1.0 = 3,035 kN
- n = 200/3035 = 0.066 < 0.25 → no reduction. Full Mc,Rd applies.
For higher axial loads (n > 0.25), use the interaction formulae from Cl. 6.2.9.1(5)-(6). The reduced plastic moment M_N,y,Rd depends on n and a, where a = (A - 2btf)/A but a <= 0.5.
Lateral-Torsional Buckling — Buckling Curves
The LTB buckling curve is selected from EN 1993-1-1 Table 6.4 based on the cross-section geometry:
| Cross-Section | Limits | Buckling Curve |
|---|---|---|
| Rolled I-sections | h/b <= 2 | b |
| Rolled I-sections | h/b > 2 | c |
| Welded I-sections | h/b <= 2 | c |
| Welded I-sections | h/b > 2 | d |
For a 457x191x67 UKB: h/b = 453.4/189.9 = 2.39 > 2 → Buckling curve c (alpha_LT = 0.49).
But wait: the UK National Annex (NA.2.18) says to use the rolled I-section buckling curves from Table 6.4, which is 'b' for h/b <= 2 and 'c' for h/b > 2. However, for cases with lateral restraint at the compression flange, the modified LTB check (6.3.2.3) with the 'a' curve is commonly applied for rolled sections subject to moment gradient.
LTB Imperfection Factors (alpha_LT)
| Buckling Curve | alpha_LT |
|---|---|
| a | 0.21 |
| b | 0.34 |
| c | 0.49 |
| d | 0.76 |
C1 Factor — Moment Gradient
The C1 factor accounts for the moment distribution between lateral restraints. From NCCI SN003:
| Loading and Support Conditions | C1 |
|---|---|
| Uniform moment (end moments M and -M) | 1.00 |
| Uniformly distributed load, simply supported | 1.13 |
| Central point load, simply supported | 1.35 |
| End moments only (psi = -0.5, double curvature) | 1.32 |
| End moments only (psi = +0.5, single curvature) | 1.07 |
| Cantilever, UDL | 2.05 |
| Cantilever, end point load | 2.30 |
psi is the ratio of end moments: M_smaller / M_larger. Negative psi = double curvature (beneficial for LTB). Positive psi = single curvature.
Modified LTB Check for Lateral Restraints (Cl. 6.3.2.3)
When the compression flange has intermediate lateral restraints at spacing L_c, the UK NA allows a modified check:
lambda_LT_bar,0 = 0.4 (plateau length, UK NA value)
For slenderness lambda_LT_bar <= 0.4, LTB does not govern — chi_LT = 1.0. For a 457x191x67 UKB under UDL with C1 = 1.13, the LTB plateau corresponds to an unrestrained length of about 2.5 m. Beyond this, chi_LT reduces below 1.0.
LTB Worked Example: 457x191x67 UKB — General Case
Same beam as before but now under a UDL (not uniform moment). Lateral restraints at 3.0 m spacing. C1 = 1.13 from table above.
- Iz = 1,450 cm^4, It = 37.1 cm^4, Iw = 0.594 dm^6, L = 3.0 m
- kz = kw = 1.0 (free to rotate on plan, end warping restrained)
- Mcr = 1.13 _ pi^2 _ 210,000 * 1,45010^4 / (3,000)^2 _ sqrt(1.0^2 _ 0.59410^12/1,45010^4 + (3,000)^2 _ 81,000 _ 37.110^4 / (pi^2 * 210,000 * 1,450*10^4))
- Mcr ~ 1,670 kN.m (CF — much higher than the 840 kN.m at 6.0 m)
With C1 = 1.13 and shorter L = 3.0 m, the Mcr increases significantly. lambda_LT_bar = sqrt(1,470*10^3 * 355 / 1,670*10^6) = 0.56.
Phi*LT = 0.5 * [1 + 0.49_(0.56 - 0.2) + 0.56^2] = 0.745
chi_LT = 1 / (0.745 + sqrt(0.745^2 - 0.56^2)) = 1 / (0.745 + 0.491) = 0.809 (but cap at 1.0)
Still less than 1.0, but chi_LT = 0.81 vs 0.82 at 6.0 m — the moment gradient benefit offsets the LTB reduction
Mb,Rd = 0.81 * 522 = 423 kN.m
Key insight: Reducing the lateral restraint spacing from 6.0 m to 3.0 m with UDL loading roughly doubles Mcr but chi_LT increases only modestly because the section enters the transition zone where LTB still controls but less severely. The real gain comes from the interaction of C1 and L — C1 > 1.0 is equivalent to a shorter effective length.
Shear Buckling (Cl. 6.2.6(6))
Shear buckling must be checked for unstiffened webs when:
hw / tw > 72 * epsilon / eta
For a 457x191x67 UKB: hw = 407.6 mm, tw = 8.5 mm, hw/tw = 47.95. Limit = 72 * 0.81 / 1.0 = 58.6. 47.95 < 58.6 → shear buckling does not govern.
Eta = 1.0 (conservative, per UK NA). For S460 steel (epsilon = 0.71), the limit drops to 51.4 — still above 47.95. Shear buckling is relevant for fabricated plate girders with slender webs, not standard hot-rolled sections.
Vpl,Rd — Worked Shear Resistance
For a 457x191x67 UKB in S355, shear parallel to web:
- A = 85.5 cm^2, b = 189.9 mm, tf = 12.7 mm, tw = 8.5 mm, r = 10.2 mm
- Av = A - 2btf + (tw + 2r)tf = 8,550 - 2189.9*12.7 + (8.5 + 20.4)*12.7 = 8,550 - 4,823 + 367 = 4,094 mm^2
- But not less than eta _ hw _ tw = 1.0 _ 407.6 _ 8.5 = 3,465 mm^2 → OK.
- Vpl,Rd = 4,094 * (355/1.732) / 1.0 = 839 kN
For the same beam, if V_Ed = 300 kN: V_Ed / Vpl,Rd = 300/839 = 0.36 < 0.5 → no bending-shear interaction required.
If VEd = 500 kN (0.60 > 0.50): rho = (2500/839 - 1)^2 = (1.192 - 1)^2 = 0.037. The reduced moment resistance My,V,Rd = (Wpl - rho _ Aw^2 / 4tw) _ fy / gamma_M0 = (1,470*10^3 - 0.037 * 3,465^2 / 34) _ 355 / 1.0 = 517 kN.m → a 1% reduction. Bending-shear interaction is usually small for standard I-sections.
Serviceability — Deflection Limits
EN 1993-1-1 does not set specific deflection limits — these are in EN 1990 Annex A1.4 and the UK National Annex. Commonly applied limits for beams:
| Condition | Limit | Typical Value (L=8m) |
|---|---|---|
| Total deflection (dead + live) | L/200 | 40 mm |
| Live load deflection only | L/360 | 22 mm |
| Cantilever (total) | L/180 | — |
| Brittle finishes (live load) | L/500 | 16 mm |
| Pre-camber (dead load) | L/300 | 27 mm |
Deflection is checked using unfactored loads (serviceability limit state). The second moment of area Iy is used in the standard deflection formulae (e.g., delta = 5wL^4 / 384EI for a UDL on a simply supported beam). Pre-camber may be used to offset dead load deflection — specify on the fabrication drawings.
Full Design Example — EN 1993-1-1 Beam Check
Problem: Check a 457x191x67 UKB in S355, simply supported, span 8.0 m, carrying floor dead load 6.0 kN/m and imposed load 12.0 kN/m. Lateral restraint at beam ends and at 2.67 m spacing (third points). Assume Class 1 section with adequate torsional restraint at supports.
Step 1 — Design Loads
ULS combination (Eq. 6.10b, UK NA): w*Ed = 1.35 * 6.0 + 1.5 _ 12.0 = 8.1 + 18.0 = 26.1 kN/m
M*Ed = w * L^2 / 8 = 26.1 _ 8.0^2 / 8 = 208.8 kN.m V_Ed = w _ L / 2 = 26.1 _ 8.0 / 2 = 104.4 kN
Step 2 — Classification
As shown above: flange c/tf = 6.34 < 7.3, web c/tw = 47.95 < 58.6 → Class 1.
Step 3 — Cross-Section Moment Resistance
Mc,Rd = 1,470*10^3 * 355 / 1.0 = 522 kN.m Utilisation: 208.8 / 522 = 0.40 → OK.
Step 4 — Shear Resistance
Vpl,Rd = 839 kN (calculated above). Utilisation: 104.4 / 839 = 0.124 < 0.5 → no bending-shear interaction.
Step 5 — Lateral-Torsional Buckling
L_c = 2.67 m between lateral restraints. Conservatively assume C1 = 1.0 (uniform moment between restraints is conservative for a continuous beam with lateral restraints at third points).
- Mcr ~ 1,910 kN.m at L = 2.67 m, C1 = 1.0
- lambda_LT_bar = sqrt(1,470*10^3 * 355 / 1,910*10^6) = 0.52
- Buckling curve c (h/b = 2.39 > 2): alpha_LT = 0.49
- Phi*LT = 0.5 * [1 + 0.49_(0.52 - 0.2) + 0.52^2] = 0.714
- chi_LT = 1 / (0.714 + sqrt(0.714^2 - 0.52^2)) = 1 / (0.714 + 0.490) = 0.830
- Mb,Rd = 0.830 * 522 = 433 kN.m
- Utilisation: 208.8 / 433 = 0.482 → OK.
Step 6 — Serviceability Deflection
SLS load (unfactored imposed only): w = 12.0 kN/m delta = 5 _ 12.0 _ 8,000^4 / (384 _ 210,000 _ 29,40010^4) = 5 * 12.0 * 4.09610^15 / (384 _ 210,000 _ 2.9410^8) = 2.45810^17 / 2.373*10^16 = 10.4 mm
Limit: L/360 = 8,000/360 = 22.2 mm → 10.4 mm < 22.2 mm → OK.
Summary — 457x191x67 UKB passes all checks. Moment utilisation 40%, LTB utilisation 48%, shear 12%, deflection 47%. The section is lightly loaded for this span — a lighter 406x178x54 UKB could also work but would need full re-check.
EN 1993-1-1 vs Other Steel Codes
| Check | EN 1993-1-1 | AISC 360-22 | AS 4100-2020 |
|---|---|---|---|
| Cross-section resistance | Cl. 6.2.5 | Chapter F | Clause 5 |
| Lateral-torsional buckling | Cl. 6.3.2 (chi_LT) | Chapter F (Cb factor) | Clause 5.6 |
| Shear resistance | Cl. 6.2.6 | Chapter G | Clause 5.11 |
| Classification | Class 1-4 (Cl. 5.5) | Compact/noncompact/slender | Section slenderness (Cl. 5.2) |
| Partial factors | gamma_M0 = 1.00 | phi_b = 0.90 | phi = 0.90 |
| Steel grades | EN 10025-2 (S235-S460) | ASTM A6/A992 (A36, A992) | AS/NZS 3679.1 |
The EN 1993 approach uses a more nuanced classification (4 classes vs 3 in AISC/AS 4100) and a theoretically more rigorous LTB treatment through the chi_LT / lambda_LT_bar framework. However, the AISC Cb factor approach and the AS 4100 alpha_m approach are functionally very similar to the EN 1993 C1 factor.
For designers working across codes, the critical differences are: (1) partial factors — 1.00 in EN 1993 vs 0.90 in AISC/AS 4100, (2) the UK NA's gamma_M1 = 1.00 vs the main text's 1.00 (both equal), and (3) the LTB buckling curve selection (curve 'c' for rolled I-sections with h/b > 2 vs 'b' for h/b <= 2).
Related Pages
- UK Steel Beam Sizes — UB, UC, PFC Chart — Complete UK section tables per SCI Blue Book
- EN 1993 Column Buckling — Column buckling curves a0-d with Perry-Robertson
- EN 1993 Connection Design — Bolted and welded connections per EN 1993-1-8
- EN 1993 Steel Grades — S235, S275, S355, S460 grade properties
- Steel Beam Capacity Calculator — Free EN 1993-1-1 beam capacity calculator
- Understanding Eurocode 3 — EN 1993 Design Guide — EN 1993 overview for non-UK designers
This page is for educational reference. All resistance formulae are per EN 1993-1-1:2005 + A1:2014 with the UK National Annex. Verify the applicable National Annex for your project jurisdiction. Section properties from SCI P363 Blue Book. Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent PE/SE verification.