EulerBuckling

Euler buckling is the instability of a slender column under axial compressive load, causing a sudden lateral deflection when the load reaches a critical value. It arises from the bending stiffness being insufficient to resist sideways displacement as the column length increases. The phenomenon is named after Leonhard Euler, who analyzed the problem in the 18th century.

The critical buckling load is given by P_cr = (π^2 E I) / (K L)^2, where E is Young's modulus, I is the second moment of area of the cross-section, L is the unsupported length, and K is the effective length factor determined by the end boundary conditions. Common end conditions and their K values are: both ends pinned (K = 1), both ends fixed (K ≈ 0.5), one end fixed and the other pinned (K ≈ 0.7), and one end fixed with the other end free (K = 2). Thus, P_cr = π^2 EI/L^2 for pinned-pinned, and increases for other configurations.

Derivation typically uses the differential equation EI y'' + P y = 0 for small lateral deflections y(x), with boundary conditions set by the end supports. Solutions yield sinusoidal buckling modes and the corresponding critical loads. In practice, real columns have imperfections such as initial crookedness, residual stresses, and material inhomogeneities, which reduce the actual buckling load below the ideal P_cr.

Applications focus on safe column design in structures. Euler buckling dominates for long, slender members, while short columns tend to fail by material yielding or other modes. Design codes use effective length factors and Euler-based formulas alongside other criteria to ensure stability under axial compression.