Slenderness decides whether a column fails by crushing or by buckling. Short stocky columns crush; long slender ones buckle at loads far below their crushing strength. Building codes limit the slenderness of steel and timber members.
How to use the Slenderness Ratio Calculator
- Column length L
- End conditions
- Section
- Width b (smaller side / flange)
- Height h
- Wall / web thickness t
- Flange thickness t_f
- Outer diameter D
- Inner diameter d
- Minimum moment of inertia I
- Area A
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the column length, choose the end conditions and section shape and enter the dimensions. The weaker axis is used.
Inputs explained
| Field | What to enter |
|---|---|
| Column length L | Units: m, cm, mm, km, in, ft … |
| End conditions | Choose one: Fixed–fixed (theory 0.5), Fixed–pinned (theory 0.7), Pinned–pinned (theory 1.0), Fixed–guided (theory 1.0) … |
| Section | Choose one: Rectangle, Hollow rectangle, Solid circle, Hollow circle … |
| Width b (smaller side / flange) | Units: mm, cm, m, in, µm |
| Height h | Units: mm, cm, m, in, µm |
| Wall / web thickness t | Units: mm, cm, m, in, µm |
| Flange thickness t_f | Units: mm, cm, m, in, µm |
| Outer diameter D | Units: mm, cm, m, in, µm |
| Inner diameter d | Units: mm, cm, m, in, µm |
| Minimum moment of inertia I | Units: mm⁴, cm⁴, m⁴, in⁴ |
| Area A | Units: mm², cm², m², in² |
What each result means
- Radius of gyration r = [r]
- Slenderness ratio K·L/r = [End conditions]×[Column length L]÷[r]
Example values
How the main result changes when Column length L changes and everything else stays at the example values above:
| Column length L (m) | Slenderness ratio K·L/r |
|---|---|
| 1.5 | 51.9615 |
| 2.25 | 77.9423 |
| 3 | 103.923 |
| 3.75 | 129.9038 |
| 4.5 | 155.8846 |
| 6 | 207.8461 |
Formula
λ = K·L / r, r = √(I_min / A)
Worked example
Inputs — Column length L: 3 m; End conditions: Pinned–pinned (theory 1.0); Section: Rectangle; Width b (smaller side / flange): 100 mm; Height h: 150 mm.
Results — Radius of gyration r = 28.8675 mm; Slenderness ratio K·L/r = 103.923; Column type (structural steel) = Intermediate (inelastic buckling).
Key terms
- Radius of gyration
- √(I/A), how spread out the area is.
Frequently asked questions
What is a good slenderness ratio?
Steel codes typically limit compression members to KL/r ≤ 200.
Why use the minimum radius of gyration?
The column buckles about its weakest axis.
How do end conditions change it?
Fixed ends reduce the effective length K·L.
What are the class boundaries?
They depend on the material – about 40 and 120 are common for structural steel.