Column Buckling Load Calculator

A slender column may fail by instability before the material reaches its yield stress. Euler's formula dominates for long columns, while Johnson's parabola gives a practical estimate for intermediate columns.

The stress curve shows where the current column falls on the slenderness axis. The shape plot shows the assumed buckled mode for the selected end condition, making the role of the effective length factor easier to see.

A slender column under axial compression can suddenly deflect sideways — buckling — before reaching yield. The ideal elastic (Euler) buckling load and stress are:

$P_{cr} = \dfrac{\pi^2 E I}{(K L)^2}, \qquad \sigma_{cr}=\dfrac{P_{cr}}{A}=\dfrac{\pi^2 E}{\lambda^2}, \qquad \lambda=\dfrac{KL}{i},\ i=\sqrt{\dfrac{I}{A}}$

$L_{cr}=KL$ is the effective length, $\lambda$ the slenderness ratio, and $i$ the radius of gyration. Stronger end restraint gives a smaller $K$ and greater buckling resistance.

Buckling occurs about the weakest axis (smallest $I$). Second moments of area: solid circle $I=\pi d^4/64$, hollow circle $I=\pi(D^4-d^4)/64$, rectangle $I=bh^3/12$.

The Euler formula is valid only for long (slender) columns. For short, stocky columns the buckling stress exceeds the yield stress $\sigma_y$, so compressive yielding occurs instead of buckling. The boundary is the critical slenderness:

$\lambda_1 = \pi\sqrt{\dfrac{E}{\sigma_y}}$

Because buckling is sudden and catastrophic, real designs use large safety factors and account for initial imperfections and eccentricity. Reducing slenderness (thicker, shorter, better restrained) is the basic countermeasure.

Use the simulator for early sizing of columns, braces, struts, and machine frames. Real designs should also check imperfections, eccentric loading, connection stiffness, local buckling, and the governing building or machine-design code.