04 // Interactive Formula Compendium // ENGINEERINGP_cr = (π² · E · I) / (K · L)²
Euler Critical Column Buckling Load Formula
Euler's critical load formula P_cr = π²EI / (KL)² predicts the maximum axial force a slender column can sustain before elastic lateral instability, governed by stiffness EI and effective length KL.
Formula Reference
Main Formula
P_cr = (π² · E · I) / (K · L)²
Rearranged Forms
σ_cr = π² · E / (KL / r)²
r = √(I / A)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| Pcr | Critical Buckling Load | N |
| E | Young's Modulus | Pa |
| I | Minimum Area Moment of Inertia | m⁴ |
| K | Effective Length Factor | dimensionless |
| L | Unsupported Column Length | m |
Inputs & Parameters
[Pa]
[m⁴]
[dimensionless]
[m]
Calculated Result (Critical Buckling Load)
Pcr =1,048,645N
Formula Notes & Background
- Solves the fourth-order deflected beam differential equation EI y'' + P y = 0, yielding nontrivial sinusoidal eigenmodes sin(√(P/EI)x) = 0 at x = KL when √(P/EI)·KL = nπ.
- Standard K values: pinned-pinned K = 1.0, fixed-fixed K = 0.5, fixed-pinned K = 0.7, fixed-free flagpole K = 2.0.
Step-by-Step Examples
#1: Pinned-pinned structural steel column (E = 200 GPa, I_min = 8.5 × 10⁻⁶ m⁴, K = 1.0) of height L = 4.0 m.
E = 2.0×10¹¹ PaI = 8.5×10⁻⁶ m⁴KL = 4.0 m
P_cr = (9.8696 × 2×10¹¹ × 8.5×10⁻⁶) / 16 = 16,778,327 / 16
→ P_cr = 1.049 MN (1,049 kN)
#2: Converting the same column to fixed-free cantilever boundary conditions (K = 2.0).
KL = 2.0 × 4.0 = 8.0 m
P_cr divides by 2² = 4: 1,048.6 kN / 4
→ P_cr = 262.2 kN
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Frequently Asked Questions
Why must I_min (weak-axis moment of inertia) be used?+
An unbraced column always buckles about the principal cross-sectional axis with the lowest flexural rigidity EI_min unless laterally braced along that axis.
Does material yield strength σ_y appear in Euler's formula?+
No — elastic Euler buckling depends strictly on Young's modulus E and slenderness KL/r; however, short stocky columns yield or buckle inelastically (Johnson parabola) before reaching Euler's load.
What is the slenderness ratio?+
Slenderness ratio λ = KL / r, where r = √(I/A) is the radius of gyration of the cross-section.