---------------- | ----------------------------------------- | | Design moment M_Ed | wL²/8 = 45 × 6.0² / 8 = 202.5 kNm | | Design shear V_Ed | wL/2 = 45 × 6.0 / 2 = 135.0 kN | | Unbraced length L_b | 6.0 m (no intermediate lateral restraint) |

Section properties (from SCI P363 Blue Book):

Material: S355JR steel to EN 10025-2

Step 1 — Section Classification (Table 5.2)

Flange Classification

The flange outstand is the compression part:

c = (b - tw - 2r) / 2 = (189.0 - 8.5 - 20.4) / 2 = 80.1 mm
c / tf = 80.1 / 12.7 = 6.31

Limiting c/tf ratios per Table 5.2 (internal compression part in bending):

Web Classification

The web depth between root radii:

cw = h - 2tf - 2r = 453.6 - 25.4 - 20.4 = 407.8 mm
cw / tw = 407.8 / 8.5 = 48.0

Limiting cw/tw ratios per Table 5.2 (web in pure bending):

Result: Section is Class 1 — plastic moment resistance may be used.

Step 2 — Moment Resistance Mc,Rd (Cl. 6.2.5)

For a Class 1 or 2 cross-section:

Mc,Rd = Wpl,y × fy / γ_M0
     = 1,470 × 10³ × 355 / 1.00
     = 521.9 × 10⁶ Nmm
     = 521.9 kNm

Utilization: M_Ed / Mc,Rd = 202.5 / 521.9 = 0.39 ✓ (39%)

The cross-section moment capacity is adequate by a wide margin. However, since the beam is laterally unrestrained, LTB will govern (Step 3).

Step 3 — Lateral-Torsional Buckling Mb,Rd (Cl. 6.3.2)

3a — Elastic Critical Moment Mcr

Using the SCI SN003 expression for doubly-symmetric I-sections:

Mcr = C₁ × (π² × E × Iz / L_c²) × √(Iw/Iz + L_c² × G × It / (π² × E × Iz))

From SCI guidance, C₁ = 1.13 for a simply supported beam with UDL and end restraints preventing warping (standard case).

π² × E × Iz / L_c² = 9.87 × 210,000 × 1,430×10⁴ / 6,000²
                    = 9.87 × 210,000 × 1,430×10⁴ / 36.0×10⁶
                    = 2.96×10¹³ / 36.0×10⁶
                    = 8.23 × 10⁵ N

Iw = 0.596 × 10¹² mm⁶
Iw / Iz = 0.596 × 10¹² / 1,430×10⁴ = 4.17 × 10⁴ = 41,700 mm²

G × It = 80,770 × 38.2×10⁴ = 3.09 × 10¹⁰ N·mm²

L_c² × G × It / (π² × E × Iz) = 36.0×10⁶ × 3.09×10¹⁰ / 2.96×10¹³
                              = 1.112×10¹⁸ / 2.96×10¹³
                              = 3.76 × 10⁴ mm²

√(Iw/Iz + correction) = √(41,700 + 37,600) = √(79,300) = 282 mm

Mcr = 1.13 × 8.23×10⁵ × 282 = 2.62 × 10⁸ Nmm = 262 kNm

3b — Non-Dimensional Slenderness

λ_LT_bar = √(Wpl,y × fy / Mcr)
         = √(1,470×10³ × 355 / 262×10⁶)
         = √(521.9×10⁶ / 262×10⁶)
         = √(1.99) = 1.41

3c — Reduction Factor χ_LT

Per UK National Annex to EN 1993-1-1 (NA 6.3.2.3), for rolled I-sections:

• α_LT = 0.21 (buckling curve a)

Φ_LT = 0.5 × [1 + α_LT × (λ_LT_bar - 0.2) + λ_LT_bar²]
     = 0.5 × [1 + 0.21 × (1.41 - 0.2) + 1.41²]
     = 0.5 × [1 + 0.21 × 1.21 + 1.99]
     = 0.5 × [1 + 0.25 + 1.99]
     = 0.5 × 3.24 = 1.62

χ_LT = 1 / [Φ_LT + √(Φ_LT² - λ_LT_bar²)]
     = 1 / [1.62 + √(1.62² - 1.41²)]
     = 1 / [1.62 + √(2.62 - 1.99)]
     = 1 / [1.62 + 0.79]
     = 1 / 2.41 = 0.415

3d — Buckling Resistance Moment

Mb,Rd = χ_LT × Wpl,y × fy / γ_M1
      = 0.415 × 1,470×10³ × 355 / 1.00
      = 216.6 × 10⁶ Nmm
      = 216.6 kNm

Utilization: M_Ed / Mb,Rd = 202.5 / 216.6 = 0.93 ✓ (93%)

The beam is adequate for LTB but the 93% utilization leaves only 7% margin. Consider increasing the section to 457×191×74 or 457×191×82 UB if future load increases are anticipated.

Step 4 — Shear Resistance Vc,Rd (Cl. 6.2.6)

4a — Shear Area

For a rolled I-section loaded in the major axis (Cl. 6.2.6(3)):

Av = A - 2 × b × tf + (tw + 2r) × tf
   = 8,550 - 2 × 189 × 12.7 + (8.5 + 20.4) × 12.7
   = 8,550 - 4,801 + 367
   = 4,116 mm²

Alternatively by the simplified formula Av = 1.04 × hw × tw (conservative): hw = h - 2tf = 453.6 - 25.4 = 428.2 mm Av ≈ 1.04 × 428.2 × 8.5 = 3,785 mm²

Use Av = 4,116 mm² (Cl. 6.2.6(3) formula, less conservative).

4b — Shear Buckling Check

hw / tw = 428.2 / 8.5 = 50.4

Limiting value per Cl. 6.2.6(6): 72 × ε / η = 72 × 0.814 / 1.0 = 58.6 (η = 1.0 per UK NA)

50.4 < 58.6 → No shear buckling check required ✓

4c — Shear Resistance

Vc,Rd = Av × (fy / √3) / γ_M0
      = 4,116 × (355 / 1.732) / 1.00
      = 4,116 × 204.9
      = 843.7 × 10³ N
      = 843.7 kN

Utilization: V_Ed / Vc,Rd = 135.0 / 843.7 = 0.16 ✓ (16%)

Shear is not critical for this beam.

Step 5 — Bending and Shear Interaction (Cl. 6.2.8)

Bending and shear interaction must be checked when V_Ed > 50% of Vc,Rd:

0.5 × Vc,Rd = 0.5 × 843.7 = 421.9 kN
V_Ed = 135.0 kN < 421.9 kN

Interaction not required ✓ — the reduced moment resistance does not need to be calculated.

Step 6 — Serviceability Deflection (Informative)

For reference, the serviceability deflection under characteristic load:

Assuming characteristic load w_ser ≈ 33.0 kN/m (estimated as w_Ed / 1.35 for UDL):

δ = 5 × w × L⁴ / (384 × E × Iy)
  = 5 × 33.0 × 6,000⁴ / (384 × 210,000 × 29,400×10⁴)
  = 5 × 33.0 × 1.296×10¹⁵ / (384 × 210,000 × 2.94×10¹¹)
  = 2.14×10¹⁷ / 2.37×10¹⁹
  = 9.0 mm

Span/360 = 6,000 / 360 = 16.7 mm δ = 9.0 mm < 16.7 mm → OK for L/360

Additional Check — Web Bearing and Buckling (Cl. 6.2.6.2)

For beams with concentrated loads or support reactions, the web must be verified for bearing (local crushing) and buckling (web crippling) at the supports and under any point loads. While not governing for this uniformly loaded beam, these checks are essential for beams with concentrated loads.

At each support, the design reaction V_Ed = 135.0 kN must be resisted by the web. Assuming a stiff bearing length ss = 75 mm from a typical seating cleat:

ly = ss + 2 × tf × (1 + √(f_yf / f_yw))
   = 75 + 2 × 12.7 × (1 + 1) = 75 + 50.8 = 125.8 mm

F_Rd = ly × tw × f_yw / γ_M0
     = 125.8 × 8.5 × 355 / 1.00
     = 379.7 kN per support >> 135.0 kN ✓

Web bearing is not critical at this small reaction relative to the web capacity. For web buckling, a similar check using an effective column model of the web confirms adequate resistance. When point loads exceed approximately 50% of the web bearing capacity, a stiffener may be required.

Summary

Conclusion: A 457×191×67 UKB in S355 steel is adequate for the design loads governed by LTB at 93% utilisation. The bending resistance is the critical design check.

Design Sensitivity — What-If Analysis

Understanding which parameters most affect the design result helps the engineer make informed decisions. For this beam, the following sensitivity table shows how the utilisation changes under different conditions:

The most cost-effective intervention is adding intermediate lateral restraint (reducing L_b), which provides a 58% capacity increase without changing the section. Increasing the section to the next weight class (74 kg/m) provides 13% more capacity at approximately 10% more cost and weight. Upgrading the steel grade to S460 is rarely economical for LTB-governed beams because the elastic critical moment Mcr is independent of steel grade — only the non-dimensional slenderness threshold shifts slightly.

Frequently Asked Questions

How does EN 1993-1-1 beam design differ from BS 5950?

The key differences: (1) EN 1993 uses partial factors (gamma_M0, gamma_M1) applied to resistance rather than a single global factor of safety; (2) lateral-torsional buckling uses the chi_LT reduction factor approach based on the elastic critical moment Mcr rather than BS 5950's Pb method; (3) section classification follows Table 5.2 with different c/t limits; (4) deflection limits are specified in the National Annex rather than in the base code. UK engineers transitioning from BS 5950 should consult the SCI P362 publication for detailed guidance.

When should I use the simplified LTB check vs the full Mcr method?

The simplified method (Cl. 6.3.2.3 for rolled sections) can be used for common UB and UC sections under uniform moment with standard restraint conditions. The full Mcr method (demonstrated here) is required when the loading is not uniform, the member has non-standard restraint conditions, the section is not a standard I-section, or the designer wants the most economical result. The full method typically yields a 10–20% more favourable result than the simplified method for beams under UDL.

What deflection limits apply under the UK National Annex?

The UK NA specifies L/200 for vertical deflection from variable actions for general floors and roofs, L/360 for floors supporting brittle finishes, and L/180 for cantilevers. These limits apply to the variable action component only (not total deflection). The total deflection (permanent plus variable) should also be checked if it could damage finishes. SCI P362 Table 3.1 provides recommended limits for common applications including L/500 for masonry walls on beams and L/300 for glazing in facades.

How do I design a beam for the construction stage?

The construction stage check verifies the bare steel beam under the weight of wet concrete, deck, and construction live load. The slab does not provide lateral restraint until it hardens, so the full unbraced length must be considered. Construction stage typically uses reduced load factors and a reduced deflection limit (L/180 is common). If the beam fails the construction check, either provide temporary props at midspan or increase the section. The Beam Capacity Calculator can check both stages.

This worked example is for educational purposes. All designs must be verified by a qualified engineer. Use the EN 1993 beam design calculator to check other sections or load cases.

Material Selection for UKB Sections

The choice of steel grade has a measurable impact on beam design economy. For UB sections governed by lateral-torsional buckling (as in this worked example), the benefit of upgrading from S355 to S460 is modest because Mcr is elastic, depending only on E and G — which are constant for all structural steels. The higher Fy only shifts the non-dimensional slenderness threshold, not the elastic critical moment itself. For this 457x191x67 UKB, upgrading to S460 reduces the utilisation from 0.93 to approximately 0.86, a 7% capacity gain for a material cost premium of roughly 15%.

For beams that are laterally restrained (composite beams, floor beams with continuous slab restraint), the cross-section resistance Mc,Rd scales directly with Fy. In these cases, upgrading from S355 to S460 can increase moment capacity by approximately 30%, potentially saving one section size. The cost-efficiency of higher-grade steel is best for: composite beams where continuous lateral restraint eliminates LTB, short-span beams where section depth is constrained, and heavily loaded transfer beams. For deflection-critical members (where deflection depends on E and I rather than Fy), choose a deeper section rather than a higher grade — stiffness is governed by geometry, not yield strength.

For UK construction, S355J2 is the standard specification that balances cost, availability, and performance for the vast majority of building applications. S275J0 may be specified for lightly loaded secondary members where neither strength nor impact toughness are critical. S460M becomes economical for long-span restrained beams and for columns in tall buildings where axial capacity governs. When specifying steel for fracture-critical applications (tie members, exposed bridge steel), S355K2 or S460K2 with guaranteed Charpy values at minus 20 degrees Celsius should be specified per BS EN 1993-1-10.

Design Checklist for UKB Beams

Before finalising a UB beam design, verify the following items:

1. Section classification confirmed — the beam is Class 1 or 2 for the design stress state and steel grade
2. Construction stage checked — the bare steel beam carries wet concrete and construction live load without lateral restraint from the slab
3. LTB check completed — the unbraced length is correctly identified (before and after slab hardening)
4. Shear interaction verified — if V_Ed exceeds 50% of Vc,Rd at any point, reduced moment resistance is checked
5. Deflection limits met — L/200 general or L/360 brittle finishes per UK NA, checked under variable action only
6. Vibration acceptable — for long-span (over 10 m) office floors, check SCI P354 criteria
7. Web bearing/buckling OK — at supports and any point loads, particularly for unstiffened webs
8. Connections detailed — end plate or fin plate capacity is consistent with the beam utilisation
9. Fire resistance addressed — beam protection (SFRM, intumescent, or fire engineering) specified per the required fire rating period

For automated verification with full calculation trace, use the EN 1993 Beam Capacity Calculator.

See Also

• Beam Capacity Calculator
• Beam Displacement and Sag Tool
• Steel Beam Sizes Reference
• Beam Design Guide
• Beam Span Reference