Euler Buckling — Elastic Instability

Elastic instability: when compressive load P exceeds the critical Euler load P_cr, a slender column buckles into characteristic mode shapes.

Boundary Conditions

Pinned-Pinned (K=1) Clamped-Clamped (K=0.5) Cantilever (K=2) Clamped-Pinned (K≈0.7)

Parameters

Applied Load P/P_cr 0.80

Slenderness L/r 100

Mode n 1

P_cr = π²EI/(KL)²

K factor: 1.00

P/P_cr: 0.80

STABLE (P < P_cr)

Euler (1744): critical load is independent of material strength — only stiffness matters.

P_cr = n²π²EI / (KL)²

Above P_cr, column buckles into mode n. Higher modes require greater load.