Plate Critical Buckling Stress
Elastic critical buckling stress of a thin rectangular plate — pick the k for your edge conditions and load case, get σ_cr and load per unit width.
Your recent runs (stored only in your browser)
No calculations yet — results land here so you can compare runs.
The engineering
A thin plate loaded in its own plane buckles at a stress set by its width-to-thickness ratio, not its length — unlike a column, a longer plate just forms more buckle half-waves at essentially the same stress. All the boundary-condition and load-case detail collapses into the coefficient k: 4.0 for a plate simply supported on all four edges in uniaxial compression, 0.425 with one unloaded edge free (the classic outstanding flange), 6.97 for all edges clamped, and 5.34 + 4(b/a)² for shear on a simply supported panel. This card is the sizing check for aircraft skin panels, webs, and any cold-formed or built-up section where local buckling governs before yield.
Two sanity checks: σ_cr goes with (t/b)², so doubling thickness quadruples the buckling stress — if a panel is marginal, thickness is the strong lever. And if the computed σ_cr comes out above roughly half the yield stress, you are leaving the elastic range and this number is optimistic; switch to an inelastic or effective-width method. Remember also that a plate supported on both unloaded edges keeps carrying load well past σ_cr (postbuckling reserve), while an outstanding flange essentially does not.
Where this math comes from
G. H. Bryan solved the simply supported compressed plate in 1891 using an energy method, decades before anyone had a pressing use for it. Stephen Timoshenko, working first in Kyiv and then at Westinghouse and Stanford after emigrating in 1922, systematically tabulated k for the edge conditions and load cases engineers actually face; his Theory of Elastic Stability — refined with James Gere into the 1961 second edition — is still the reference this card leans on.
The aircraft industry made plate buckling a daily calculation. Stressed-skin construction in the 1930s meant every panel between stringers was a buckling problem, and Theodore von Kármán's 1932 effective-width concept showed the skin keeps working after it buckles. NACA spent the 1940s and 50s testing and cataloging the coefficients, culminating in Gerard and Becker's Handbook of Structural Stability — the charts still living in every airframe stress group's methods manual.
- 1891G. H. BryanFirst energy solution for buckling of a simply supported plate in compression.
- 1910Stephen TimoshenkoTabulates buckling coefficients k for practical edge conditions and load cases.
- 1932Theodore von KármánEffective-width concept — plates carry load beyond σ_cr.
- 1957Gerard & Becker (NACA)Handbook of Structural Stability compiles the coefficient charts airframers still use.
- 1961Timoshenko & GereTheory of Elastic Stability, 2nd edition — the canonical reference for this formula.
See the full timeline of the math behind every calculator →
Runs entirely in your browser — nothing you enter leaves this page. Your recent runs are stored only on your device.