Critical loads for circular ring, circular arch and narrow rectangular beam — 5 classic buckling cases
This calculator finds the critical load at which a structure loses its stable equilibrium — a threshold that must not be exceeded in design. It covers five classic elastic stability cases from thin rings and arches to narrow rectangular beams, all using exact closed-form classical stability equations.
Ring & arch cases deal with in-plane buckling under radial pressure:
• Case 8 — Circular ring, full uniform radial pressure: p' = 3EI/r^3. The entire ring buckles when the external pressure exceeds this value.
• Case 9 — Hinged circular arch, half-angle \alpha: p' = \tfrac{EI}{r^3}\!\left(\tfrac{\pi^2}{\alpha^2}-1\right). When \alpha = 90° this reduces to the ring formula.
Narrow rectangular beam cases deal with lateral-torsional buckling (LTB) — the cross-section twists sideways before the material yields. The Saint-Venant torsion correction factor C_b = 1 - 0.63b/d adjusts for the beam not being perfectly narrow. All three cases require b < d:
• Case 11 — Pure bending: critical moment M' = \tfrac{\pi b^3 d \sqrt{E G C_b}}{6l} for lateral-only restraint, or twice that if the beam can also twist.
• Case 12 — Cantilever, end load at tip: P' = 0.669 \tfrac{b^3 d \sqrt{C_b E G}}{l^2}\!\left[1 - \tfrac{a}{2l}\sqrt{E/(G C_b)}\right], where a is the height offset of the load attachment point above the neutral axis.
• Case 13 — Simply supported, center load: P' = 2.82\tfrac{b^3 d \sqrt{C_b E G}}{l^2}\!\left[1 - 1.74\tfrac{a}{l}\sqrt{E/(G C_b)}\right]; the equivalent uniformly distributed critical load is W' \approx 1.67 P'.
Model limits: All cases assume linear elastic material, small displacements and perfect geometry (no initial imperfections). Results are classical theoretical critical loads; real structures buckle at lower loads due to imperfections, residual stresses and inelasticity. Apply appropriate buckling reduction factors per your design standard.
Covered cases of this table:
Loading the interactive calculator…