EN 1993-1-1 Column Buckling — Worked Example

This worked example demonstrates the complete verification of a pin-ended steel column under concentric axial compression, following EN 1993-1-1:2022 with UK National Annex parameters. Educational use only — always verify by a qualified engineer.

Problem Statement

Column: 203×203×60 UKC, Grade S355JR (EN 10025-2) Length: 4.0 m, pinned at both ends (L_cr,y = L_cr,z = 4.0 m) Loading: N_Ed = 1,200 kN (factored ULS axial load, including self-weight)

Section properties (from SCI P363 Blue Book — 203×203×60 UKC):

Material: S355JR steel to EN 10025-2

Step 1 — Section Classification (Table 5.2)

Flange Classification

The flange is an outstand compression part in uniform compression:

c = (b - tw - 2r) / 2 = (205.8 - 9.4 - 20.4) / 2 = 88.0 mm
c / tf = 88.0 / 14.2 = 6.20

Limiting c/tf ratios per Table 5.2 (outstand flange in compression):

Web Classification

The web is an internal compression part in uniform compression:

cw = h - 2tf - 2r = 209.6 - 28.4 - 20.4 = 160.8 mm
cw / tw = 160.8 / 9.4 = 17.1

Limiting cw/tw ratios per Table 5.2 (web in uniform compression):

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

Step 2 — Cross-Section Compression Resistance Nc,Rd (Cl. 6.2.4)

For a Class 1, 2 or 3 cross-section in uniform compression:

Nc,Rd = A × fy / γ_M0
      = 76.4 × 10² × 355 / 1.00
      = 2,712 × 10³ N
      = 2,712 kN

Utilization: N_Ed / Nc,Rd = 1,200 / 2,712 = 0.44 ✓ (44%)

The cross-section is adequate by a wide margin. However, for a slender pin-ended column, flexural buckling will govern (Step 3).

Step 3 — Flexural Buckling Resistance Nb,Rd (Cl. 6.3.1)

3a — Non-Dimensional Slenderness

λ₁ = π × √(E / fy) = π × √(210,000 / 355) = 76.4

For each axis:

λ_bar_y = (L_cr,y / iy) / λ₁ = (4,000 / 89.3) / 76.4 = 44.8 / 76.4 = 0.587
λ_bar_z = (L_cr,z / iz) / λ₁ = (4,000 / 52.0) / 76.4 = 76.9 / 76.4 = 1.007

The z-z axis governs (higher slenderness).

3b — Buckling Curve Selection (Table 6.2)

For hot-rolled I-sections under compression, the buckling curve depends on the section geometry:

h / b = 209.6 / 205.8 = 1.02 ≤ 1.2 tf = 14.2 mm ≤ 100 mm

Use curve c (governing axis with α = 0.49).

3c — Reduction Factor χ_z

Φ = 0.5 × [1 + α × (λ_bar - 0.2) + λ_bar²]
  = 0.5 × [1 + 0.49 × (1.007 - 0.2) + 1.007²]
  = 0.5 × [1 + 0.49 × 0.807 + 1.014]
  = 0.5 × [1 + 0.395 + 1.014]
  = 0.5 × 2.409 = 1.205

χ_z = 1 / [Φ + √(Φ² - λ_bar²)]
    = 1 / [1.205 + √(1.205² - 1.007²)]
    = 1 / [1.205 + √(1.452 - 1.014)]
    = 1 / [1.205 + √0.438]
    = 1 / [1.205 + 0.662]
    = 1 / 1.867 = 0.536

3d — Buckling Resistance

Nb,Rd,z = χ_z × A × fy / γ_M1
        = 0.536 × 76.4 × 10² × 355 / 1.00
        = 1,454 × 10³ N
        = 1,454 kN

Utilization: N_Ed / Nb,Rd,z = 1,200 / 1,454 = 0.83 ✓ (83%)

The column is adequate for flexural buckling. The 17% margin is typical for a moderately loaded column at this slenderness.

3e — Check y-y Axis (Informative)

For completeness, checking the y-y axis even though z-z governs:

From curve b (α = 0.34):

Φ_y = 0.5 × [1 + 0.34 × (0.587 - 0.2) + 0.587²]
    = 0.5 × [1 + 0.34 × 0.387 + 0.345]
    = 0.5 × [1 + 0.132 + 0.345]
    = 0.5 × 1.477 = 0.739

χ_y = 1 / [0.739 + √(0.739² - 0.587²)]
    = 1 / [0.739 + √(0.546 - 0.345)]
    = 1 / [0.739 + 0.448]
    = 1 / 1.187 = 0.842

Nb,Rd,y = 0.842 × 76.4 × 10² × 355 / 1.00 = 2,284 kN

Utilization: 1,200 / 2,284 = 0.53 ✓ (53%) — significantly less critical as expected.

Step 4 — Combined Buckling Curve Comparison

The difference between buckling curves under EN 1993-1-1 is significant. For this column (λ_bar_z = 1.007):

This illustrates why the correct buckling curve selection per Table 6.2 is critical — using curve d would predict failure, while curve a would be unconservative.

Step 5 — Summary

Conclusion: A 203×203×60 UKC in S355 steel is adequate for the design axial load of 1,200 kN, governed by flexural buckling about the minor axis at 83% utilisation. The section is Class 1, allowing full plastic section resistance. The correct selection of buckling curve per Table 6.2 is critical — curve c applies for UKC sections (h/b ≤ 1.2) about the minor axis.

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

See Also

• Column Capacity Calculator
• Column Buckling Guide
• Column Design Guide
• Effective Length Reference