COLUMN BUCKLING EXAMPLE OF A SOLID ROUND BAR

1. Cross-sectional area (solid round): A = (π d²)/4 = (π · 50²)/4 = 1963.495 mm² .
2. Second moment of area (solid round): I = (π d⁴)/64 = (π · 50⁴)/64 = 306,796.158 mm⁴ .
3. Radius of gyration: r = √(I/A) = √(306,796.158 / 1,963.495) = 12.50 mm .
4. Slenderness ratio: S = L / r = 2000 / 12.5 = 160.0 .
5. Johnson/Euler boundary: S_cr = √(2 π² E / S_y) = √(2 π² · 205,000 / 345) = 108.30 . Since S = 160 > S_cr, this is a long column → use Euler.
6. Effective length factor for pinned–pinned: K = 1.0.
7. Euler critical load:
   P_cr = (π² E I) / (K L)² = 9.8696 · 205,000 · 306,796.158 / (1 · 2000)² = 155,182.78 N = 155.18 kN.
8. Allowable load with design factor n_d=5: F_a = P_cr / n_d = 155.18 / 5 = 31.04 kN .
9. (Optional) Mass check: Volume = A · L = 1,963.495 mm² × 2000 mm = 3.92699×10⁶ mm³ = 3926.99 cm³,  M = ρ·V = 7.87 × 3926.99 ≈ 30.905 kg.

Yes. A slender, pin-ended round member in axial compression is a very common real-world case. You’ll see close analogs in practice:

• Truss/space-frame members: bars or rods connected to gusset plates with pins/bolts → behavior close to pinned–pinned (K ≈ 1).
• Construction props/shoring: temporary steel props typically checked as pin-ended columns.
• Machine struts/links: members with spherical rod ends (Heim/rose joints) approximate pinned ends.
• Architectural/bridge bracing: tension/compression bars often designed as pin-ended compression members.

Sanity Check of the Numbers

• Slenderness S = L/r = 2000/12.5 = 160 → long column regime, so Euler buckling governs (as shown).
• Computed critical load ≈ 155.18 kN and allowable ≈ 31.04 kN (with n_d=5) are plausible.
• Self-weight of the bar is small (≈ 0.31 kN) compared to the capacity → negligible influence here.

Design Caveats & Good Practice

• End restraint isn’t perfectly “pinned” in reality. Effective length factor K depends on joint/frame stiffness. Use the appropriate K from your design code or a stability analysis.
• Initial crookedness & load eccentricity always exist. Codes (e.g., AISC/EC3) handle this via column curves or secant/Perry–Robertson style formulas. Our calculator also supports eccentricity checks.
• Use your governing code’s resistance format. For final design, prefer code provisions (AISC, EC3, etc.) over a generic “design factor”.
• Section choice matters. Real columns are often tubes/rolled shapes; check local buckling limits and slender-element rules where applicable.

Bottom line: this worked example is a sound baseline and matches the calculator. For a project deliverable, apply your code’s prescribed K, imperfection model, and resistance factors/curves.