Eccentric Column Buckling (2D)

A slender pinned column with a bow and an off-centre load: watch deflection blow up as P approaches the Euler load.

Live readout

Numerical checks

The science

A pinned column has the Euler load Pcr = pi^2 E I / L^2. With an offset load e and an initial bow e0 the beam equation EI y'' + P (y + y0 + e) = 0 gives y(x) = e [cos(k (x - L/2)) / cos(k L / 2) - 1] + e0 (P/Pcr) / (1 - P/Pcr) sin(pi x / L), k = sqrt(P / EI) (the secant formula plus Southwell amplification). The deflection diverges as P approaches Pcr; maximum stress is P/A + P (y + e) c / I.

Simplifications: linear elasticity and small rotations, so beyond P = Pcr the model has no equilibrium (real columns then yield or collapse); pinned ends; the animation is a damped first-mode response toward the static shape.
Checks solve the beam equation by finite differences, independent of the closed forms.

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