Critical Buckling Stress

What is this formula?

Critical buckling stress is the theoretical compressive stress at which an ideal slender column becomes unstable and buckles according to Euler's theory.

When to use it

Use this formula for long, slender columns where elastic buckling governs failure. It is commonly used as the basis for column stability calculations and structural design checks.

Example

A steel column has:

E = 200 GPa = 200000000000 Pa

λ = 120

Formula:

σcr = π²E / λ²

Substitution:

σcr = π² × 200000000000 / 120²

Result:

σcr = 137,078,000 Pa

Result:

σcr = 137.1 MPa

Applications

Steel columns, towers, truss members, bridge structures, industrial frames, transmission structures, and stability analysis.

Note

This equation is derived from Euler's elastic buckling theory and is valid primarily for slender columns that remain elastic before buckling. Real structures may experience imperfections, residual stresses, inelastic behavior, and boundary-condition effects that reduce the actual buckling strength. Design codes typically apply additional reduction factors and empirical adjustments.