Column Critical Stress Calculator
This calculator computes the critical compressive stress for a structural column using both the Euler elastic buckling formula (governing for slender columns) and the AISC 360-22 inelastic column strength equation (governing for intermediate columns). Enter the material modulus and yield strength, the cross-section radius of gyration, effective column length, and cross-sectional area to determine the critical stress, design stress (with phi factor), and design compressive strength.
AISC 360-22 column strength formula
Fe = pi2 × E / (Le/r)2 (Euler elastic stress)
Limit = 4.71 × sqrt(E / Fy)
If Le/r ≤ limit: Fcr = 0.658(Fy/Fe) × Fy
If Le/r > limit: Fcr = 0.877 × Fe
phic × Pn = 0.90 × Fcr × Ag
Where: E = modulus of elasticity (MPa), Fy = yield strength (MPa), Le = effective length (mm), r = radius of gyration (mm), Ag = gross cross-sectional area (mm^2). phi_c = 0.90 is the LRFD resistance factor for compression per AISC 360-22 Section E1.
Slenderness classification
- Short column (Le/r less than 50): yielding likely governs before buckling. AISC Fcr approaches Fy.
- Intermediate column (50 to 120): inelastic buckling governs. Use AISC Chapter E inelastic formula.
- Long column (Le/r greater than 120): elastic (Euler) buckling governs. Fcr = 0.877 * Fe.
- AISC limits the maximum slenderness ratio to 200 for most compression members (Section E2).
Column critical stress calculator: frequently asked questions
What is critical stress in a column?
Critical stress (sigma_cr) is the average compressive stress at which an ideal elastic column buckles. For long (slender) columns governed by elastic buckling, sigma_cr = pi^2 * E / (Le/r)^2 (Euler's formula). For intermediate columns where the yield strength is relevant, the Johnson parabolic formula or the AISC column curve applies.
What is the slenderness ratio?
The slenderness ratio = Le / r, where Le is the effective length of the column (Le = K * L) and r = sqrt(I/A) is the radius of gyration of the cross-section. A higher slenderness ratio means the column is more slender and more susceptible to buckling at a lower stress. Short, stocky columns have low slenderness ratios and fail by yielding; long, slender columns have high slenderness ratios and fail by elastic buckling.
What is the transition between Euler and Johnson formulas?
The transition slenderness ratio Cc = sqrt(2 * pi^2 * E / Fy), where Fy is the yield strength. For slenderness ratio greater than Cc (or 4.71 * sqrt(E/Fy) per AISC 360), elastic Euler buckling governs. For slenderness ratio less than Cc, inelastic buckling governs and the Johnson formula or AISC column strength equation applies.
How does AISC 360 compute column strength?
AISC 360-22 Chapter E uses Fe = pi^2 * E / (Le/r)^2 (elastic critical stress). For Le/r <= 4.71 * sqrt(E/Fy), Fcr = Fy * 0.658^(Fy/Fe). For Le/r > 4.71 * sqrt(E/Fy), Fcr = 0.877 * Fe. The design compressive strength = phi_c * Fcr * Ag, with phi_c = 0.90 (LRFD).
What radius of gyration should I use for biaxial buckling?
Use the minimum radius of gyration r_min for the critical (weak) axis of the cross-section. For a W-shape, r about the y-y axis is much smaller than r about the x-x axis. Always check buckling about both axes with their respective effective lengths. The axis with the largest Le/r ratio controls the design.
Official sources
- AISC 360-22 Specification for Structural Steel Buildings, Chapter E: AISC 360-22.
- NIST Engineering Laboratory Structures: nist.gov/el.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.