COLUMN BUCKLING (EULER & JOHNSON)

Long compression members fail by buckling rather than yielding. Euler's formula P_cr = π²EI / (KL)² governs slender columns where buckling stress stays below yield. K is the end-condition factor: 0.5 (fixed-fixed), 0.7 (fixed-pinned), 1.0 (pinned-pinned), 2.0 (fixed-free). For shorter columns the actual failure stress curves below Euler's prediction; the Johnson formula handles this transition region using yield strength. The crossover slenderness is (KL/r)_t = sqrt(2π²E/Sy).

• (KL/r)_t = sqrt(2π²E / Sy)
• Euler: P_cr = π²EI / (KL)²
• Johnson: σ_cr = Sy - (Sy·KL / (2π·r))² / E
• FoS = P_cr / P_applied

References & StandardsAISC 360 Ch. E, Shigley 4.11, Norton 4.16

Apply column analysis to any compression member: pneumatic-cylinder rod extended fully (treat as fixed-pinned with K=0.7), forklift mast, screw-jack lead screw, structural strut, scaffolding leg. Compute slenderness ratio KL/r first; if above (KL/r)_t use Euler, otherwise use Johnson. Apply factor of safety 2-4 depending on dynamic loading and uncertainty in end conditions.