Column Buckling Calculator (Euler)

Understanding the column buckling calculator — concept and principle

A slender column under compression can fail long before the material is crushed: at a certain load it suddenly bows sideways. This calculator finds that critical load with Euler's formula and reports the slenderness ratio KL/r, which tells you whether Euler's formula is meaningful for your column.

Halving the effective length KL quadruples the buckling load; pinned–pinned gives 311 kN.

How it is calculated

Critical load: P_cr = π²EI ÷ (KL)². E in GPa and I in cm⁴ are converted to N·m² (E × 10⁹ × I × 10⁻⁸), L is in metres, and the result is shown in kN.
Effective length factor K: 1.0 pinned–pinned, 0.5 fixed–fixed, 0.7 fixed–pinned, 2.0 fixed–free (flagpole).
Slenderness ratio: KL ÷ r, with r = √(I/A) in cm and KL converted to cm (× 100).

Use the weak-axis I, because a column buckles about the axis with the least stiffness.

The principle

A perfectly straight elastic column has two equilibrium states near the critical load: straight, or slightly bent. Bending creates a moment P × deflection, and the column's bending stiffness EI resists it. Solving the differential equation EI·y″ + P·y = 0 with the end conditions shows a non-zero bent shape is possible only when P = π²EI/L² for pinned ends. Other end conditions change the shape of the buckled curve; K expresses this as the length of an equivalent pinned column.

Worked example

Inputs: E = 200 GPa, I = 142 cm⁴ (weak axis), L = 3 m, pinned–pinned (K = 1), A = 28.5 cm².
EI = 200 × 10⁹ × 142 × 10⁻⁸ = 284,000 N·m².
P_cr = π² × 284,000 ÷ 3² = 311,441 N = 311.4409 kN.
r = √(142 ÷ 28.5) = 2.2321 cm; KL/r = 300 ÷ 2.2321 = 134.4001.

Each point is the calculator re-run with only “Length (m)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Length (m)	Critical load (kN)
0.75	4,983.0536
1.5	1,245.7634
2.25	553.6726222
3	311.44085
3.75	199.322144
4.5	138.4181556
6	77.8602125

How to read the result

Euler's formula is valid only while the buckling stress P_cr/A stays below the material's proportional limit. Here 311,441 N ÷ 2,850 mm² ≈ 109 MPa, below 235 MPa for S235 steel, so elastic buckling governs. For steel with E = 200 GPa and yield 235 MPa the transition slenderness is about π√(E/f_y) ≈ 92; shorter, stockier columns fail by yielding or inelastic buckling, for which Euler overestimates strength.

Common mistakes

Using the strong-axis I of an I-section; it buckles about the weak axis unless braced.
Treating P_cr as a safe working load. Real columns have initial crookedness and residual stresses, and design codes use buckling curves well below Euler.
Taking theoretical K values for real connections; codes often recommend 0.65, 0.8 and 2.1 instead of 0.5, 0.7 and 2.0 because real fixity is imperfect.

Keep in mind: Euler's load is an idealised upper bound for perfect, centrally loaded, elastic columns. Real compression members must be designed to your local code (for example Eurocode 3 or the AISC Specification) and checked by a qualified engineer.

Frequently asked questions

What slenderness counts as "slender"?

For mild steel, KL/r above roughly 90 to 100 is usually in the elastic (Euler) range; below that, inelastic buckling or yielding controls.

Why does a fixed–free column use K = 2?

Its buckled shape is half of a pinned column twice as long.

Does stronger steel help a slender column?

Hardly. Euler's load depends on E, not yield strength, and E is nearly the same for all steels.

Sources: S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability, McGraw-Hill · AISC 360, Specification for Structural Steel Buildings, and Commentary

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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