About this calculator
This tool computes the axial compressive strength of a steel column using the AISC 360 Chapter E compression-member provisions, covering both the inelastic (short-to-intermediate) and elastic (slender) buckling ranges. Given the member's effective length, radius of gyration, area, and yield strength, it returns the slenderness ratio, Euler buckling stress, critical stress, and factored design strength. It is intended for structural engineers doing preliminary axial-capacity checks on braced or unbraced steel columns.
Method & formulas
Slenderness ratio: KL/r, where K is the effective length factor set by the end restraint (e.g. 1.0 for pin-pin, 0.5 for fix-fix, 2.0 for fix-free), L is the unbraced length, and r is the governing (typically minimum) radius of gyration.
Elastic (Euler) buckling stress: Fe = π²E / (KL/r)². Slenderness limit separating the inelastic and elastic regimes: (KL/r)lim = 4.71·√(E/Fy).
Fe = π²E / (KL/r)²
if KL/r ≤ 4.71√(E/Fy): Fcr = 0.658(Fy/Fe)·Fy (inelastic)
else: Fcr = 0.877·Fe (elastic)
Nominal and design strength: Pn = Fcr·A; φPn = φc·Pn, with φc = 0.90.
Worked example
Inputs: A = 6000 mm², r = 50 mm, E = 200,000 MPa, Fy = 248 MPa (A36), L = 3500 mm, K = 1.0, φc = 0.90.
KL/r = 1.0×3500/50 = 70.
(KL/r)lim = 4.71×√(200,000/248) ≈ 133.8 → since 70 ≤ 133.8, inelastic buckling governs.
Fe = π²×200,000/70² ≈ 402.8 MPa.
Fcr = 0.658(248/402.8) × 248 = 0.6580.616 × 248 ≈ 191.7 MPa.
Pn = 191.7 × 6000 / 1000 ≈ 1150 kN.
φPn = 0.90 × 1150 ≈ 1035 kN.
Assumptions & limitations
- Assumes a doubly-symmetric or otherwise torsionally-stable section failing in flexural (Euler-type) buckling; torsional and flexural-torsional buckling modes (common in singly-symmetric or thin open shapes) are not checked.
- The effective length factor K must be chosen to match the actual end restraint and frame bracing condition — an unconservative K will overstate the capacity.
- Local buckling of slender cross-section elements (width-to-thickness limits) is not checked; the tool assumes a nonslender section for compression.
- Uses a single governing r; built-up or unsymmetric sections may need separate checks about each principal axis.
- Verify results against the governing code and a licensed engineer's judgment before finalizing a design.
FAQ
- What's the physical difference between "inelastic" and "elastic" column buckling?
- Short, stocky columns partially yield before they buckle, so their strength is governed by an inelastic curve tied to Fy; long, slender columns buckle elastically at a stress well below yield, governed purely by Euler's formula. The 4.71√(E/Fy) slenderness limit marks the transition between the two behaviors.
- Why does doubling the unbraced length reduce capacity so much?
- Fe is inversely proportional to (KL/r)², so doubling KL cuts the elastic buckling stress to roughly a quarter; Fcr follows that same steep reduction (moderated somewhat by the inelastic curve at lower slenderness), which is why column capacity is very sensitive to unbraced length.
- Which radius of gyration should I use for r?
- Use the smallest (typically weak-axis) radius of gyration unless bracing effectively restrains that axis at a shorter unbraced length than the strong axis — the governing check is whichever axis produces the larger KL/r.