UK Beam Design — BS EN 1993-1-1 & UK NA Worked Example

This reference covers beam design for UK steel design per BS EN 1993-1-1:2005 Clauses 6.2-6.3 and UK NA. The Eurocode 3 beam design process in the UK follows a systematic sequence: cross-section classification, moment resistance, shear resistance, lateral-torsional buckling, and serviceability deflection.

Design requirements, worked examples, and practical design guidance are provided for common design office applications.

Code Reference: BS EN 1993-1-1:2005 Clauses 6.2-6.3 and UK NA

Cross-Section Classification (BS EN 1993-1-1 Clause 5.5)

Classification depends on the flange and web slenderness relative to limits in Table 5.2:

Class Moment Capacity Basis Plastic Hinge? Rotation Capacity?
1 Wpl × fy / γM0 Yes Full
2 Wpl × fy / γM0 Yes Limited
3 Wel × fy / γM0 No None
4 Weff × fy / γM0 No None

For S275 steel (ε = √(235/275) = 0.92):

For S355 steel (ε = √(235/355) = 0.81):

Most hot-rolled UB sections in S275/S355 are Class 1 or Class 2 in bending.

Moment Resistance (BS EN 1993-1-1 Clause 6.2.5)

[ M*{c,Rd} = \frac{W*{pl} fy}{\gamma{M0}} \quad \text{(Class 1 and 2)} ]

[ M*{c,Rd} = \frac{W*{el,min} fy}{\gamma{M0}} \quad \text{(Class 3)} ]

Where γM0 = 1.00 (UK NA).

Moment Capacity Table — Selected UB Sections (S355)

Section Mass (kg/m) Wpl,y (cm³) Mc,Rd (kNm) Depth (mm) Section Class
533×210 UB 92 92 2616 928.7 533.1 1
457×191 UB 89 89 2142 760.4 463.6 1
406×178 UB 60 60 1357 481.7 406.4 1
356×171 UB 51 51 1018 361.4 355.6 1
305×165 UB 40 40 746 264.8 303.4 1
254×146 UB 37 37 551 195.6 256.0 1
203×133 UB 25 25 340 120.7 203.2 1

Mc,Rd = Wpl,y × 355 / 1.0. Compression flange restraint assumed. Reduction for coexistent shear may apply.

Shear Resistance (BS EN 1993-1-1 Clause 6.2.6)

[ V*{c,Rd} = V*{pl,Rd} = \frac{Av (f_y / \sqrt{3})}{\gamma{M0}} ]

Shear area Av for rolled sections: Av = A - 2 b tf + (tw + 2r) tf

Typical shear capacities (S355):

Section Av (mm²) Vpl,Rd (kN)
533×210 UB 92 4785 980.7
457×191 UB 89 4080 836.3
406×178 UB 60 2784 570.7

Vpl,Rd = Av × (355/√3) / 1.0. Shear buckling check required if hw/tw > 72ε/η.

Worked Example — 533×210 UB 92 in S355

Given:

Section data (533×210 UB 92):

Check 1 — Moment (Clause 6.2.5): Mc,Rd = 2616 × 355 / 1.0 × 10⁻³ = 928.7 kNm UT = 600 / 928.7 = 0.65 — Satisfactory

Check 2 — Shear (Clause 6.2.6): Vpl,Rd = 4785 × (355/√3) / 1.0 × 10⁻³ = 980.7 kN VEd / Vpl,Rd = 250 / 980.7 = 0.25 < 0.5 — No moment reduction for shear

Check 3 — Lateral-Torsional Buckling (Clause 6.3.2): Mcr = C1 × ÃƒÂÃ‚Â€²EIz / Lcr² × ÃƒÂ¢Ã‚ÂˆÃ‚Âš(Iw/Iz + Lcr²GIt/π²EIz)

Using SCI P362 design tables for Lcr = 4.0m (loaded at top flange, destabilising? No): Mb,Rd = χLT × Wpl,y × fy / γM1

For Lcr = 4.0m with curve 'c' (αLT = 0.49): χLT ≈ 0.74 (from SCI P362 design tables for rolled sections per UK NA)

Mb,Rd = 0.74 × 928.7 = 687.2 kNm UT for LTB = 600 / 687.2 = 0.87 — Satisfactory

Check 4 — Deflection (Serviceability): wmax = 5wL⁴ / (384EI) for UDL For w = 20 kN/m (unfactored live load): w = 5×20×8000⁴/(384×210000×55200×10⁴) = 18.4mm L/300 = 8000/300 = 26.7mm — 18.4 < 26.7 — Satisfactory

Web Bearing and Buckling (BS EN 1993-1-5)

At support locations, check web bearing resistance:

[ R*{w,Rd} = \frac{f*{yw} L*{eff} t_w}{\gamma*{M1}} ]

For the 533×210 UB 92 with stiff bearing length ss = 100mm: Leff = χF × ly where ly accounts for load spread through flange Typical resistance: ~400-500 kN for unstiffened web

Worked Example 2 — 406x178 UB 60 in S275

This second worked example demonstrates S275 grade steel, which is widely used in UK building construction. The lower yield strength (fy = 275 MPa vs 355 MPa) affects section classification, moment resistance, and LTB behaviour. S275 is approximately 5-8% cheaper per tonne and is the most common structural steel grade specified for UK building frames.

Given:

Section data (406x178 UB 60, S275):

Property Value Units
h 406.4 mm
b 177.9 mm
tw 7.9 mm
tf 12.8 mm
r 10.2 mm
Wpl,y 1,201 cm^3
Wel,y 1,058 cm^3
Iy 21,600 cm^4
Iz 1,203 cm^4
Av 3,211 mm^2

fy = 275 N/mm^2 (tf <= 16 mm), fu = 410 N/mm^2, E = 210,000 N/mm^2

Cross-Section Classification (Clause 5.5):

epsilon = sqrt(235/275) = sqrt(0.855) = 0.924

Flange: c = (177.9 - 7.9 - 20.4) / 2 = 74.8 mm, c/tf = 5.84. Class 1 limit: 9*epsilon = 8.32 — OK, Class 1.

Web: cw = 406.4 - 25.6 - 20.4 = 360.4 mm, cw/tw = 45.6. Class 1 limit: 72*epsilon = 66.5 — OK, Class 1.

Check 1 — Moment (Clause 6.2.5): Mc,Rd = 1,201 x 10^3 x 275 / 1.00 = 330.3 kN.m UT = 275.6 / 330.3 = 0.834 — OK.

Check 2 — Shear (Clause 6.2.6): Vpl,Rd = 3,211 x (275/1.732) / 1.00 = 509.8 kN VEd/Vpl,Rd = 0.309 < 0.50 — No shear-moment interaction.

Check 3 — Lateral-Torsional Buckling (Clause 6.3.2):

L_cr = 3.5 m, h/b = 2.28 > 2 — UK NA curve 'b' (alpha_LT = 0.34).

Using SCI P362 tables for L_cr = 3.5 m: chi_LT approx 0.85.

Mb,Rd = 0.85 x 330.3 = 280.8 kN.m LTB utilisation: 275.6 / 280.8 = 0.981 — OK but tight.

S275 vs S355 comparison for 406x178 UB 60 at L_cr=3.5m:

Parameter S275 S355 Difference
Mc,Rd 330.3 kN.m 426.4 kN.m +29%
Mb,Rd (Lcr=3.5m) 280.8 kN.m 362.4 kN.m +29%
LTB utilisation 0.981 0.760 -23%
Moment utilis. 0.834 0.646 -23%

The LTB utilisation at 98% is tight. If restraint spacing increases to 4.0 m, chi_LT drops to approx 0.78, Mb,Rd = 257.6 kN.m — FAIL. For S355 at Lcr=4.0m, utilisation would be 0.829 — still OK. The decision between S275 and S355 involves balancing material cost against robustness to restraint assumptions.

Check 4 — Deflection (Serviceability):

Imposed load qk approx 18.0 kN/m:

delta = 5 x 18.0 x 7000^4 / (384 x 210,000 x 21,600 x 10^4) = 12.4 mm

Limit L/300 (floors, UK NA) = 7000/300 = 23.3 mm — OK (53% utilised).

Summary — 406x178 UB 60 in S275:

Check Clause Resistance Demand Ratio Status
Section class 5.5 Class 1 — — OK
Moment Mc,Rd 6.2.5 330.3 kN.m 275.6 kN.m 0.834 OK
Shear Vpl,Rd 6.2.6 509.8 kN 157.5 kN 0.309 OK
LTB Mb,Rd 6.3.2 280.8 kN.m 275.6 kN.m 0.981 OK
Deflection EN 1990 23.3 mm 12.4 mm 0.532 OK

The 406x178 UB 60 in S275 is adequate for the 7.0 m office floor beam. LTB governs at 98% utilisation. For typical UK office construction with secondary beams at 3.0-3.5 m centres, this beam and grade combination is economical and widely used.

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Frequently Asked Questions

How is beam flexure checked per BS EN 1993-1-1 with UK NA?

Flexural capacity Mc,Rd = Wpl fy / γM0 per BS EN 1993-1-1 Clause 6.2.5. UK NA specifies γM0 = 1.00. For a 533×210 UB92 (S355): Mc,Rd = 2616 x 355 x 10⁻³ / 1.0 = 928.7 kNm. The classification of the cross-section (Class 1, 2, 3, or 4) determines whether plastic (Wpl) or elastic (Wel) section modulus is used.

What are the UK NA lateral-torsional buckling modifications?

UK NA modifies the LTB curve selection per BS EN 1993-1-1 Table 6.4: for rolled UB sections, buckling curve 'b' for h/b ≤ 2 (αLT = 0.34) and curve 'c' for h/b > 2 (αLT = 0.49). The UK NA adopts the rolled-section specific method of Clause 6.3.2.3 with λLT,0 = 0.4 and β = 0.75 for χLT determination.

When is shear buckling a concern for UK beams?

Shear buckling requires checking when hw/tw > 72ε/η per BS EN 1993-1-1 Clause 6.2.6(6). For S355: 72 × 0.81 / 1.0 = 58.5. Most standard UB sections have web slenderness below this limit at ambient temperature. For example, 533×210 UB92 has hw/tw = 476.5/8.8 = 54.1 < 58.5, so no shear buckling check needed. Heavier sections and fabricated plate girders may exceed this limit.

What deflection limits apply to UK steel beams?

UK NA to BS EN 1993-1-1 and EN 1990 NA recommends: roof beams L/200 (vertical under variable loads), floor beams L/300, plastered ceilings L/360. Cantilevers: L/150 for floors, L/100 for roofs. Refer to the UK deflection limits guide for comprehensive coverage of serviceability criteria including dynamic and horizontal drift limits.

How does coexistent shear affect moment capacity?

Per BS EN 1993-1-1 Clause 6.2.8, when VEd > 0.5 Vpl,Rd, the yield strength must be reduced for the moment resistance calculation: fy,red = (1 - ρ) fy where ρ = (2VEd/Vpl,Rd - 1)². This is a ductile shear-moment interaction. In most UK beam designs, shear utilisation is well below 50%, so no reduction is required.

Reference only. Verify all values against the current edition of BS EN 1993-1-1:2005 Clauses 6.2-6.3 and UK NA. This information does not constitute professional engineering advice.