Buckling

Buckling is a stability failure in which a slender compression member suddenly develops large lateral (or torsional) deformations at a critical load that may be well below the material yield load. The classic elastic solution is Euler’s formula for a pinned–pinned column: P_cr = π² EI / (KL)² where E is Young’s modulus, I is the minimum second-moment of area, L is the unsupported length and K is the effective-length factor that accounts for end-restraint conditions (K = 1.0 for pinned–pinned, 0.5 for fixed–fixed, 0.7 for fixed–pinned, 2.0 for free–fixed, etc.). The corresponding critical stress is σ_cr = P_cr / A = π² E / (KL/r)², where r = √(I/A) is the radius of gyration. The dimensionless ratio KL/r is the slenderness ratio; higher values produce lower buckling stresses. When σ_cr exceeds the proportional limit, inelastic buckling occurs and empirical or tangent-modulus formulas replace the pure Euler expression. Other buckling modes include local buckling of thin plate elements, torsional buckling of open sections, and lateral-torsional buckling of beams bent about their strong axis. Design codes therefore classify members by slenderness and apply reduction factors or interaction equations that incorporate residual stresses, initial imperfections and partial yielding.