Column Buckling Calculator

Euler's critical load for slender columns. Real‑time calculation with multiple end conditions and cross‑section shapes. Perfect for mechanical & civil engineering.

Euler's formula: \( P_{cr} = \dfrac{\pi^2 E I}{(K L)^2} \)     (valid for slender columns, σcr ≤ σyield)

Buckling Analysis Results
Critical Load Pcr
-- kN
Critical Stress σcr
-- MPa
Slenderness λ
--
I = 0 mm⁴   |   A = 0 mm²   |   r = 0 mm
Pcr = π² × 200000 × I / (K·L)²

Buckling mode shape (exaggerated) – one half sine wave

Euler buckling applies if σcr ≤ yield strength. Enter yield strength to check.
Within elastic range

Understanding Column Buckling

When a slender column is compressed, it may suddenly bend sideways — buckling. The critical load is given by Euler's formula (1757).

Euler's critical load: \( P_{cr} = \dfrac{\pi^2 E I}{(K L)^2} \)

Critical stress: \( \sigma_{cr} = \dfrac{P_{cr}}{A} = \dfrac{\pi^2 E}{(K L / r)^2} = \dfrac{\pi^2 E}{\lambda^2} \)

Effective Length Factor K

End condition K factor Mode shape
Pinned‑pinned (hinged‑hinged) 1.0 one sine wave
Fixed‑free (cantilever) 2.0 quarter sine wave
Fixed‑fixed 0.5 full sine wave (two inflection points)
Fixed‑pinned 0.7 approx. 0.7 sine wave

Slenderness Ratio λ = KL / r

Where r = √(I/A) is the radius of gyration. Euler formula is valid for λ ≥ λp (proportional limit). For stocky columns, use Johnson's parabolic formula.

Applications & Limitations

  • Structural columns in buildings, bridges
  • Mechanical components (pushrods, screws)
  • Assumptions: perfectly straight, elastic, no imperfections; failure by bending, not crushing.

Frequently Asked Questions

For low slenderness, inelastic buckling occurs. Use Johnson's formula or consult design codes (AISC, Eurocode).

E in MPa (1 GPa = 1000 MPa), dimensions in mm → I in mm⁴, A in mm², load in N (displayed in kN). 1 kN = 1000 N.

Rectangle: I = b·h³/12, A = b·h. Circle: I = π·d⁴/64, A = π·d²/4.