Calculate Euler critical buckling load for columns with various end conditions. Determine slenderness ratio and buckling safety factor.
Enter the column length, Young's Modulus E, the area moment of inertia I, the cross-sectional area A, and the end condition. The calculator solves the Euler formula Pcr = π²EI / (KL)², where K is the effective length factor set by how the column ends are restrained.
The four end conditions use the classic K factors: 1.0 for pinned-pinned, 0.7 for fixed-pinned, 0.5 for fixed-fixed, and 2.0 for fixed-free. Because capacity varies with 1/(KL)², fixing both ends quadruples the critical load of a pinned column, while a fixed-free flagpole carries only a quarter of it.
The results also report the effective length Le = K × L, the radius of gyration r = √(I/A), the slenderness ratio Le/r, and the critical stress π²E/(Le/r)². Euler's theory assumes elastic behavior, so the prediction is only meaningful while the critical stress stays below the material's yield strength.
K rescales the column length to the distance between inflection points of its buckled shape: 1.0 pinned-pinned, 0.7 fixed-pinned, 0.5 fixed-fixed, 2.0 fixed-free. The calculator multiplies your length by K before applying Euler's formula, so end restraint alone can change capacity by a factor of 16.
Effective length divided by the radius of gyration: λ = KL / √(I/A). It condenses geometry and restraint into one number; the critical stress π²E/λ² falls with the square of slenderness, so a column twice as slender buckles at a quarter of the stress.
When the computed critical stress reaches the yield strength, the column fails by crushing or inelastic buckling before elastic buckling can occur. Compare the reported critical stress against your material's yield — 250 MPa for A36 steel in the material database — to confirm the elastic assumption holds.
With K = 2.0 its effective length is double its real length, and capacity scales with 1/(KL)². A cantilevered post therefore buckles at one quarter of the load of the same member pinned at both ends — it behaves like a pinned column twice as long.