Design Problem

Problem: Verify a simply supported 457UB in S355 steel spanning L = 7.5 m. The beam supports precast concrete planks that provide continuous lateral restraint to the top flange. The beam is unrestrained during construction.

Loading per EN 1991-1-1:

• Permanent action (dead): gk = 8.5 kN/m (precast slabs 5.5 + finishes 1.5 + services 0.5 + beam self-weight 1.0 kN/m)
• Variable action (imposed): qk = 15.0 kN/m (office occupancy, Category B, EN 1991-1-1 Table 6.2)

Design parameters per UK NA:

• gamma_M0 = 1.00, gamma_M1 = 1.00, gamma_M2 = 1.25
• Steel grade: S355JR to BS EN 10025-2, fy = 355 MPa (t <= 16 mm), fu = 470 MPa
• ULS combination UK NA Eq. 6.10b: 1.35 x Gk + 1.5 x Qk (where xi = 0.925 for permanent actions)

Section Properties — 457 x 191 UB 74

From SCI P363 (Blue Book) for UK Universal Beam sections:

Step 1: ULS Design Load

Per UK NA to BS EN 1990, Eq. 6.10b (the standard UK combination):

FEd = 1.35 x 8.5 + 1.5 x 15.0 = 11.48 + 22.50 = 33.98 kN/m

Design bending moment (midspan):

MEd = FEd x L2 / 8 = 33.98 x 7.52 / 8 = 239.0 kN·m

Design shear force (at support):

VEd = FEd x L / 2 = 33.98 x 7.5 / 2 = 127.4 kN

Step 2: Cross-Section Classification — EN 1993-1-1 Clause 5.5

Flange classification (outstand in compression):

c = (b - tw - 2r) / 2 = (190.4 - 9.0 - 20.4) / 2 = 80.5 mm c / tf = 80.5 / 14.5 = 5.55 epsilon = sqrt(235 / fy) = sqrt(235 / 355) = 0.814

Class 1 limit for flange: 9·epsilon = 9 x 0.814 = 7.32 5.55 < 7.32 => Flange is Class 1

Web classification (bending):

c = d = 407.6 mm c / tw = 407.6 / 9.0 = 45.3

Class 1 limit for web in bending: 72·epsilon = 72 x 0.814 = 58.6 45.3 < 58.6 => Web is Class 1

Section is Class 1 — plastic design permitted.

Step 3: Bending Moment Resistance — Clause 6.2.5

For Class 1 section:

Mc,Rd = Wpl,y x fy / gamma_M0 = 1,630 x 103 x 355 / 1.00 = 578.7 kN·m

Utilisation: MEd / Mc,Rd = 239.0 / 578.7 = 0.413 (41.3 %)

The 457UB is well within its bending capacity at 7.5 m span with this loading.

Step 4: Shear Resistance — Clause 6.2.6

Shear area for rolled I-section, load parallel to web:

Av = A - 2·b·tf + (tw + 2r) x tf (conservative) Av = 9,460 - 2 x 190.4 x 14.5 + (9.0 + 20.4) x 14.5 = 9,460 - 5,522 + 426 = 4,364 mm2

Plastic shear resistance:

Vpl,Rd = Av x (fy / sqrt(3)) / gamma_M0 Vpl,Rd = 4,364 x (355 / 1.732) / 1.00 = 894.1 kN

Shear utilisation: VEd / Vpl,Rd = 127.4 / 894.1 = 0.142 (14.2 %)

VEd < 0.5 x Vpl,Rd (127.4 < 447.1) — no reduction in bending resistance required per Clause 6.2.8.

Step 5: Lateral-Torsional Buckling — Clause 6.3.2

Construction stage (top flange unrestrained):

The precast planks provide restraint in the final condition, but during construction the top flange may be unrestrained. Check LTB for the beam self-weight plus construction load.

Construction load: gk,const = 1.0 kN/m (beam SW) + 0.75 kN/m (construction live) = 1.75 kN/m MEd,const = 1.35 x 1.75 x 7.52 / 8 = 1.35 x 12.3 = 16.6 kN·m

Elastic critical moment for lateral-torsional buckling:

For a simply supported beam with uniform moment, using the SCI P362 method:

Mcr = C1 x pi2 x E x Iz / L2 x sqrt(Iw / Iz + L2 x G x It / (pi2 x E x Iz))

Where C1 = 1.132 (for UDL on simply supported beam)

Mcr = 1.132 x pi2 x 210,000 x 1,670 x 104 / 7,5002 x sqrt(0.538 x 1012 / (1,670 x 104) + 7,5002 x 81,000 x 55.0 x 104 / (pi2 x 210,000 x 1,670 x 104))

After calculation: Mcr = 586 kN·m (construction stage)

Non-dimensional slenderness:

lambda_LT = sqrt(Wpl,y x fy / Mcr) = sqrt(1,630 x 103 x 355 / (586 x 106)) = sqrt(0.987) = 0.993

Buckling curve selection:

h / b = 457 / 190.4 = 2.40 > 2.0 => Buckling curve c (UK NA Table 6.5) alpha_LT = 0.49 (imperfection factor for curve c)

Reduction factor chi_LT:

Phi_LT = 0.5 x [1 + alpha_LT x (lambda_LT - 0.2) + lambda_LT2] Phi_LT = 0.5 x [1 + 0.49 x (0.993 - 0.2) + 0.9932] = 0.5 x [1 + 0.389 + 0.986] = 1.187

chi_LT = 1 / (Phi_LT + sqrt(Phi_LT2 - lambda_LT2)) chi_LT = 1 / (1.187 + sqrt(1.1872 - 0.9932)) = 1 / (1.187 + 0.651) = 0.544

LTB buckling resistance (construction stage):

Mb,Rd = chi_LT x Wpl,y x fy / gamma_M1 Mb,Rd = 0.544 x 1,630 x 103 x 355 / 1.00 = 314.7 kN·m

Utilisation (construction): MEd,const / Mb,Rd = 16.6 / 314.7 = 0.053 (5.3 %)

The beam is fully adequate for LTB during construction. For longer spans over 10 m, intermediate lateral restraint may be required at the construction stage.

Final condition (with slab restraint):

With continuous lateral restraint from precast planks, LTB does not govern. The bending check (Step 3) controls.

Step 6: Deflection — Serviceability Limit State

Loads for SLS (characteristic combination per UK NA):

Total SLS load: gk + qk = 8.5 + 15.0 = 23.5 kN/m

Deflection (total load):

delta_total = 5 x w x L4 / (384 x E x Iy) delta_total = 5 x 23.5 x 7,5004 / (384 x 210,000 x 33,300 x 104) delta_total = 5 x 23.5 x 3.164 x 1015 / (384 x 210,000 x 33,300 x 104) delta_total = 19.8 mm

Deflection limits — UK NA to EN 1993-1-1:

• Total deflection limit: L/200 = 7,500 / 200 = 37.5 mm
• Imposed load deflection: typically L/360 = 20.8 mm for office floors

delta_imposed = delta_total x (qk / (gk + qk)) = 19.8 x (15.0 / 23.5) = 12.6 mm

Both checks pass: 19.8 < 37.5 mm (total) and 12.6 < 20.8 mm (imposed)

Step 7: Web Bearing and Buckling — Clauses 6.2.6.2 and 6.2.6.3(1)

Web bearing at support (assume 75 mm stiff bearing length):

ss = 75 mm (bearing length of the support)

Effective bearing length: leff = ss + 2 x tf x (1 + sqrt(bf / tw)) (simplified) leff = 75 + 2 x 14.5 x (1 + sqrt(190.4 / 9.0)) = 75 + 29.0 x (1 + 4.60) = 237 mm

Web bearing resistance: FRd = leff x tw x fy / gamma_M0 FRd = 237 x 9.0 x 355 / 1.00 = 756.9 kN

Utilisation: VEd / FRd = 127.4 / 756.9 = 0.168 (16.8 %) — acceptable.

Summary of Checks

The 457 x 191 UB 74 in S355 is adequate for the 7.5 m span. Bending governs at 41.3 %. A lighter section could be considered — 457 x 191 UB 67 would increase bending utilisation to approximately 48 %, remaining within acceptable limits.

UK NA Specific Notes

• The UK National Annex adopts gamma_M0 = 1.00 and gamma_M1 = 1.00 (no change from EN 1993-1-1 recommended values)
• gamma_M2 = 1.25 for tension to fracture (same as recommended)
• For buildings, Eq. 6.10b is the standard ULS combination in UK practice
• Deflection limits: UK practice typically uses L/200 total, L/360 imposed for floors supporting brittle finishes
• SCI P362 provides comprehensive design guidance for UK steel beams, including stability checks

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Frequently Asked Questions

What is the difference between BS 5950 and EN 1993-1-1 for UK beam design?

BS 5950 (withdrawn 2010) used permissible stress and simple plastic theory. EN 1993-1-1 uses limit state design with partial safety factors. Key differences: (1) EN 1993 uses four cross-section classes (1-4) versus BS 5950's three (plastic, compact, semi-compact, slender); (2) EN 1993 LTB curves differ — more conservative for deep sections; (3) UK NA to EN 1993 retains some UK-practice preferences including the use of UB/UC section designations rather than IPE/HEA. BS 5950 is no longer maintained, and all new UK designs must use EN 1993 with the UK NA.

When does LTB govern UK beam design?

Lateral-torsional buckling governs when the compression flange is unrestrained over a significant length. For UK UB sections with h/b > 2.0 (deep beams), LTB can govern at unbraced lengths exceeding approximately 3-5 m depending on loading. The UK NA provides conservative buckling curves (curve c for h/b > 2.0) that reflect UK research on hot-rolled sections. Continuous restraint from composite slabs (Clause 6.3.2.2) or discrete secondary beam connections eliminates LTB as a design concern.

What deflection limits apply for UK steel beams?

The UK NA to EN 1993-1-1 recommends characteristic SLS combinations for deflection checks. Horizontal deflection: H/150 to H/300 depending on cladding type. Vertical deflection: L/200 for total load (typical for industrial buildings) and L/360 for imposed load only (typical for floors with brittle finishes). SCI P362 Table 1 recommends L/200 imposed for roofs without plaster ceilings and L/360 for office/residential floors.

How do I select the correct UK UB section for a given span?

Start with span-to-depth ratio: L/20 to L/25 for simply supported beams under typical floor loading (5-7.5 kN/m2 imposed). A 7.5 m span suggests 300-375 mm depth. Check bending, then shear, then deflection. If deflection governs (common for longer spans with UDL), increase depth — stiffness (I) increases with h3. The SCI Blue Book (P363) provides pre-calculated resistances for all UK UB and UC sections across multiple spans.

Related Pages

• EN 1993 Beam Design Example — European section worked example
• UK Steel Grades — S235, S275, S355, S460 UK grades
• UK Beam Sizes — Complete UB, UC, PFC tables
• UK Lateral Torsional Buckling — LTB detailed guide
• UK Deflection Limits — EN 1993-1-1 serviceability
• UK Web Bearing and Buckling — Local capacity checks
• UK Load Combinations — EN 1990 + UK NA

Educational reference only. All design values are per BS EN 1993-1-1:2005 + UK National Annex and BS EN 10025-2:2019. Verify all values against the current editions of the standards and the applicable UK National Annex for your project jurisdiction. Designs must be independently verified by a Chartered Structural Engineer registered with the Institution of Structural Engineers (IStructE) or the Institution of Civil Engineers (ICE). Results are PRELIMINARY — NOT FOR CONSTRUCTION without independent professional verification.