All tools Print with PE stamp box Designed for sealed engineering submittals — print drops PE stamp + signature block at the end.

Euler Column Buckling

Critical buckling load and stress for slender columns. End-condition K-factor (pinned, fixed, free) selects the effective length. Reports slenderness ratio and indicates when Euler is invalid (inelastic buckling, use Johnson or AISC).

in
psi (steel: 29×10⁶; alum: 10×10⁶)
psi
in⁴
in²
—in
—in
——
——
——
—lb
—psi

Defaults: 10-ft pin-pin steel column, A36 (σ_y = 36 ksi), I = 10 in⁴, A = 4.4 in² (≈ W4×13). For real design use AISC E3 (compression chapter), not pure Euler.

Euler critical load (elastic):
$$ P_{cr} = \frac{\pi^2 E I}{(K L)^2}, \qquad \sigma_{cr} = \frac{\pi^2 E}{(K L / r)^2} $$
Transition slenderness:
$$ C_c = \pi \sqrt{\frac{2 E}{\sigma_y}} $$
For KL/r < Cc: inelastic buckling — use Johnson or AISC. Euler over-predicts.
Pcr Euler critical load · K effective length factor (end conditions) · L physical column length · E elastic modulus · I least moment of inertia (governs which axis buckles) · r = √(I/A), radius of gyration · σcr Euler critical stress · Cc slenderness above which Euler is valid.

End conditions and K-factors

The K-factor accounts for boundary conditions:

Design K-factors (AISC §C.3) are larger than theoretical because real connections aren't perfectly fixed: design K = 0.65 for theoretical 0.5, design K = 0.80 for theoretical 0.7, design K = 2.10 for theoretical 2.0. Pin-pin (K = 1.0) is unchanged.

If sidesway is permitted (no lateral bracing — moment frames), K can exceed 1.0 even for pin-pin ends. AISC alignment charts give K-factors based on column-to-girder stiffness ratios.

Why Euler isn't enough — the inelastic regime

Euler's formula assumes purely elastic buckling, which is only valid when the critical stress σcr is less than the yield stress σy. The transition point is Cc = π√(2E/σy), about 126 for A36 steel. Below Cc, the column yields before Euler's load is reached, and the actual critical load is smaller than Euler predicts.

For inelastic buckling, use:

where σe = π² E / (KL/r)² is the Euler stress.

Use the LEAST moment of inertia

A column buckles about its weak axis (the axis with smallest I). For a W-shape, that's the y-y (minor) axis. For a round tube, both axes are the same. For an unequal-leg angle, it's the diagonal axis (z-z), not x-x or y-y.

Bracing can change which axis governs. A column braced midspan in the weak direction effectively halves L for that axis, so the strong axis might govern even if its I is larger.

Real columns aren't pinned-pinned

In real construction, "pinned" connections (single-bolt clip angles) carry some moment. "Fixed" connections (welded moment frames) are flexible at design loads. AISC alignment charts and the modified-G method handle these properly. For preliminary design or homework, K = 1.0 is conservative.

Reference: AISC Specification for Structural Steel Buildings (AISC 360-22), §E (Compression). Hibbeler, R.C. (2014). Mechanics of Materials, 9th ed., Pearson, ch. 13.

Related tools

Monthly engineering case studies

One real stormwater or hydraulics design problem per month, with the math worked out and the gotchas called out. No tutorials, no fluff.

Free. Unsubscribe anytime. Privacy.

Engineer of Record — Stamp & Signature
APPLY PE STAMP HERE
Engineer Name
License No.
State
Signature
Date
Project / Sheet
By stamping and signing, the Engineer of Record certifies that the inputs, formulas, and applicability of this calculation have been reviewed for the specific design context. PE-Calc tools provide computational support only — the engineer is responsible for verifying results, applying engineering judgment, and complying with applicable codes and standards.
Calculation generated at pe-calc.com
pe-calc.com