Euler Buckling Formula

The Euler formula predicts the critical axial load at which a slender column will suddenly bow (buckle) laterally due to elastic instability, even though the compressive stress is well below the material's yield strength.

P_cr = pi^2 * E * I / (K*L)^2

Where: P_cr = critical buckling load, E = elastic modulus, I = minimum moment of inertia, K = effective length factor, L = actual column length.

• Critical stress form: sigma_cr = pi^2 * E / (KL/r)^2
• Assumptions: Perfectly straight column, homogeneous material, load applied at centroid, no initial stresses
• Valid for: Long slender columns (slenderness ratio > critical slenderness Cc)

Slenderness Ratio

The slenderness ratio is the primary parameter that determines whether a column fails by buckling or yielding:

lambda = K*L / r, where r = sqrt(I/A) is the radius of gyration

• Short columns (lambda < 40): Fail by material yielding (crushing), not buckling
• Intermediate columns (40 < lambda < Cc): Inelastic buckling - use Johnson parabola
• Long/slender columns (lambda > Cc): Elastic buckling - Euler formula applies

Critical slenderness: Cc = sqrt(2*pi^2*E / Sy) (transition point)

End Conditions (K Factors)

The effective length factor K accounts for how the column ends are constrained. The effective length Le = K*L represents the equivalent pinned-pinned column length.

• Fixed-Fixed (K=0.5): Both ends rigidly restrained against rotation and translation. Strongest configuration - buckles in S-shape with inflection points at L/4 from each end.
• Fixed-Pinned (K=0.7): One end fixed, other allows rotation but not translation. Common in building frames with moment connections at one end.
• Pinned-Pinned (K=1.0): Both ends free to rotate but not translate. The reference case - buckles in a single half-sine wave.
• Fixed-Free (K=2.0): Cantilever column - one end fixed, other completely free. Weakest configuration - effective length is twice the actual length.

Design values: In practice, use K = 0.65, 0.80, 1.0, and 2.1 respectively to account for imperfect end conditions.

Johnson Parabola for Short/Intermediate Columns

When lambda < Cc, the Euler formula overpredicts the buckling load because the material yields before elastic buckling occurs. The Johnson parabola provides a more accurate prediction:

sigma_cr = Sy - (Sy^2 / (4*pi^2*E)) * (KL/r)^2

• Tangent point: The Johnson curve is tangent to the Euler curve at lambda = Cc
• Physical meaning: Accounts for residual stresses and initial imperfections that cause premature yielding
• AISC approach: Modern codes use similar transition formulas with safety factors built in

Buckling Curve Explanation

The buckling curve shows how critical stress varies with slenderness ratio:

• Euler hyperbola (blue): sigma_cr = pi^2*E / lambda^2 - valid for lambda > Cc
• Johnson parabola (purple): Connects yield stress to Euler curve - valid for lambda < Cc
• Yield plateau (green): Very short columns fail by yielding at sigma = Sy
• Operating point (red dot): Your column's position on the curve