Buckling Stability (Euler)

Critical force Fcr, slenderness λ and stability check — Euler cases I–IV

← Back to Engineering

Boundary Conditions Case I

mm
Geometric length between supports
kN
Design compressive load
mm
mm
I = b·h³/12 (about weak axis)
mm
I = π·d⁴/64
mm
mm
I = π·(D⁴−d⁴)/64
mm⁴
mm²

Results

—
Enter values
Effective length Leff = β·L
—
Moment of inertia I
—
Cross-section area A
—
Radius of gyration i
—
Slenderness λ = Leff / i
—
Critical force Fcr
—
Applied force F
—
Utilisation η = F / Fcr
—
Reserve capacity
—
Elastic buckling (Euler) governs — λ above limiting slenderness λlim.
λ below λlim — inelastic / material failure may govern before elastic buckling. Verify per EC3 §6.3.
Utilisation > 80% — consider larger cross-section or shorter effective length.
Inner diameter ≥ outer diameter — invalid hollow section.

Diagram

Method — Euler Buckling Theory

Critical force

Euler formula for elastic buckling:

Fcr = π² × E × I / Leff²

where Leff = β × L is the effective (buckling) length.

Slenderness

Radius of gyration: i = √(I / A)

Slenderness ratio: λ = Leff / i

Limiting slenderness: λlim = π × √(E / fy)

Euler formula valid for λ > λlim

Boundary conditions β

  • Case I — pin–pin: β = 1.0
  • Case II — fixed–free: β = 2.0
  • Case III — fixed–pin: β = 0.7
  • Case IV — fixed–fixed: β = 0.5

Notes

  • Calculator uses pure Euler elastic buckling — conservative for stocky columns
  • For full EC3 buckling check use reduction factor χ per §6.3.1
  • Imperfections and residual stresses are not included
  • Verify the buckling plane — use minimum I