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Steel Beam Design: worked example
A simply supported steel I-beam checked for section class, bending, shear, lateral-torsional buckling and deflection.
EN 1993-1-1:2005 — Eurocode 3: Design of steel structures — Part 1-1: General rules and rules for buildings · SI units · Calculator v1.0
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Inputs
| Section | IPE400 |
| Grade | S355 |
| Span | 6.00 m |
| Support condition | simply supported |
| Unbraced length Lb | = span (not overridden) |
| C1 | 1.14 (auto) |
| Design moment MEd/Mu | 162.0 kN·m |
| Design shear VEd/Vu | 108.0 kN |
| Self-weight included | No |
Design checks All checks pass
| Check | Required | Provided | Utilisation | Result |
|---|---|---|---|---|
| Classification | Class 1–3 | Class 1 | — | PASS |
| Bending | 162.0 kN·m | 464.0 kN·m | 34.9% | PASS |
| Shear | 108.0 kN | 875.0 kN | 12.3% | PASS |
| LTB | 162.0 kN·m | 212.9 kN·m | 76.1% | PASS |
| Deflection, live (L/360) | 5.21 mm | 16.67 mm | 31.3% | PASS |
| Deflection, total (L/250) | 8.69 mm | 24.00 mm | 36.2% | PASS |
Calculation steps
Classification — EN 1993-1-1 Table 5.2 (4 steps)
Table 5.2
Material coefficient
ε = √(235 / fy)
ε = √(235 / 355)
ε = 0.814
Table 5.2
Flange (outstand in compression)
c = (b − tw − 2r)/2; limits 9ε | 10ε | 14ε
c/tf = 64.7/13.5 = 4.79 vs 7.32 | 8.14 | 11.39
Flange: Class 1
Table 5.2
Web (internal, in bending)
c = h − 2tf − 2r; limits 72ε | 83ε | 124ε
c/tw = 331.0/8.6 = 38.49 vs 58.6 | 67.5 | 100.9
Web: Class 1
§5.5.2(6)
Section class = least favourable element class
max(flange, web)
max(1, 1)
Section: Class 1
Bending — §6.2.5 (3 steps)
§6.2.5(2)
Section modulus for Class 1
Class 1/2 → plastic modulus
Wpl,y = 1307.0 × 10³ mm³
Wpl,y = 1307000 mm³
§6.2.5(2)
Design resistance for bending
Mc,Rd = Wpl,y · fy / γM0
Mc,Rd = 1307000 × 355 / 1
Mc,Rd = 464.0 kN·m
§6.2.5(1)
Cross-section bending check
MEd / Mc,Rd ≤ 1.0
162.0 / 464.0
Ratio = 0.349 → OK
Shear — §6.2.6 (4 steps)
§6.2.6(3)a
Shear area, rolled I-section
Av = A − 2b·tf + (tw + 2r)·tf ≥ η·hw·tw
Av = 8446 − 2×180×13.5 + (8.6 + 2×21)×13.5 = 4269; min 3849
Av = 4269 mm²
§6.2.6(2)
Plastic shear resistance
Vpl,Rd = Av · (fy/√3) / γM0
Vpl,Rd = 4269 × (355/√3) / 1
Vpl,Rd = 875.0 kN
§6.2.6(6)
Web shear buckling check
hw/tw ≤ 72ε/η, else EN 1993-1-5 §5 governs
43.4 vs 48.8
OK — no shear buckling check needed
§6.2.6(1)
Shear check
VEd / Vpl,Rd ≤ 1.0
108.0 / 875.0
Ratio = 0.123 → OK
Lateral-torsional buckling — §6.3.2 (7 steps)
§6.3.2.2
Moment-distribution factor (k = kw = 1, load at shear centre)
C1 taken as the AISC F1-1 Cb of the factored diagram (or manual)
Lcr = 6000 mm
C1 = 1.136
§6.3.2.2
Elastic critical moment
Mcr = C1·(π²EIz/Lcr²)·√(Iw/Iz + Lcr²·G·It/(π²EIz))
Iz = 13.18×10⁶ mm⁴, It = 51.1×10⁴ mm⁴, Iw = 0.4900×10¹² mm⁶
Mcr = 260.9 kN·m
§6.3.2.2
Non-dimensional slenderness
λ̄LT = √(Wy·fy / Mcr)
λ̄LT = √(1307000 × 355 / 260900852)
λ̄LT = 1.334
Table 6.5
Buckling curve (rolled section)
h/b ≤ 2 → curve b, else c
h/b = 2.22
Curve c → αLT = 0.49
§6.3.2.3(1)
Reduction factor (f-factor refinement omitted — conservative)
ΦLT = 0.5·[1 + αLT(λ̄LT − 0.4) + 0.75·λ̄LT²]; χLT = 1/(ΦLT + √(ΦLT² − 0.75·λ̄LT²)) ≤ min(1, 1/λ̄LT²)
ΦLT = 1.3956
χLT = 0.4589
§6.3.2.1(3)
Design buckling resistance moment
Mb,Rd = χLT · Wy · fy / γM1
Mb,Rd = 0.4589 × 1307000 × 355 / 1
Mb,Rd = 212.9 kN·m
§6.3.2.1(1)
LTB check
MEd / Mb,Rd ≤ 1.0
162.0 / 212.9
Ratio = 0.761 → OK
Bending + shear — §6.2.8 (2 steps)
§6.2.8(2)
Is the shear high enough to reduce the moment resistance?
VEd > 0.5·Vpl,Rd ?
108.0 vs 0.5 × 875.0 = 437.5
NO — no reduction required
§6.2.8(2)
Interaction conclusion
MEd / Mc,Rd ≤ 1.0 (unreduced)
162.0 / 464.0
Ratio = 0.349 → OK
Deflection (SLS) (3 steps)
Elastic deflections from the beam analysis (unfactored SLS loads)
δ from double integration of M(x)/EI
EI = 48573 kN·m²
δ_dead = 3.47 mm, δ_live = 5.21 mm, δ_total = 8.69 mm
EN 1990 A1.4.3
Deflection limits (editable — verify against project spec)
Live: L/360; Total: L/250
L = 6000 mm
Limits: 16.67 mm (live), 24.00 mm (total)
EN 1990 A1.4.3
Serviceability check
δ_live ≤ L/360 and δ_total ≤ L/250
5.21 vs 16.67; 8.69 vs 24.00
Live: OK; Total: OK
Design basis and assumptions
- National Annex: none applied as a whole — the EN recommended values are used, except where noted below. There is no National Annex selector; check the values against the annex for your project.
- γM0 = 1.0 and γM1 = 1.0 (EN 1993-1-1 §6.1 recommended values). Both are editable.
- Section capacities assume a doubly-symmetric I-section in major-axis bending with loads applied through the shear centre.
- Lateral bracing defaults to the two end supports only (Lb = full span) unless intermediate bracing is explicitly checked.
- The moment-gradient factor (C1/Cb) is calculated automatically only when the unbraced length equals the full span; overriding Lb switches to a manual, conservative default (1.0).
- Deflection limits (L/360 live, L/250 or L/240 total by default) are common defaults, not a single code-mandated value — actual project/client requirements vary and are user-editable.
- Checks a single prismatic member between its two supports — doesn’t model continuous-beam or multi-span behavior (use Frame Analysis for that).
- Verify all factors against the governing code edition and applicable National Annex before use.
Example values for illustration only. Results are checked against a published validation suite; for real projects, have a licensed engineer review the results.