What this calculator does
Slenderness ratio is a structural engineering measure of how likely a column is to fail by buckling, sideways bowing under load, rather than by simply crushing under compression. It is defined as the effective length of the column divided by the radius of gyration of its cross-section, and it is one of the first checks made when sizing a compression member.
A short, thick column has a low slenderness ratio and tends to fail by the material itself crushing. A long, thin column has a high slenderness ratio and tends to fail by buckling well before the material reaches its crushing strength, which is why slender columns need extra bracing or a larger cross-section even when the load itself seems modest.
The formula
Divide the effective length, which accounts for how the column’s ends are restrained, by the radius of gyration, a property of the cross-sectional shape describing how its area is distributed around the bending axis. If you do not already know the radius of gyration, this calculator can work it out for a simple rectangular cross-section, using the smaller side, or a circular cross-section, using the diameter.
| Term | Meaning |
|---|---|
| Effective length (Le) | The length over which the column behaves as if pinned at both ends, which can be longer or shorter than its physical length depending on how the ends are actually restrained. |
| Radius of gyration (r) | A cross-section property equal to the square root of (moment of inertia divided by area); for a rectangle it is the smaller side divided by √12, and for a circle it is the diameter divided by 4. |
| Slenderness ratio (λ) | Effective length divided by radius of gyration. Higher values mean a greater relative risk of buckling. |
The inputs explained
| Field | What to enter |
|---|---|
| Known values | Choose from Effective length and radius of gyration, Effective length and a rectangular cross-section, Effective length and a circular cross-section. |
| Effective length (m) | The effective length of the column, in metres. |
| Radius of gyration (direct mode only) (m) | The radius of gyration of the cross-section, in metres, used only when entering it directly. |
| Smaller cross-section side (rectangle mode only) (m) | The smaller side of a rectangular cross-section, in metres, used only in rectangle mode. |
| Diameter (circle mode only) (m) | The diameter of a circular cross-section, in metres, used only in circle mode. |
When to use it
Sizing a compression member early in design
Before finalising a section size, calculating the slenderness ratio for a few candidate sizes flags which options risk being governed by buckling rather than material strength.
Comparing a rectangular timber post against a circular steel post
The rectangle and circle modes let two different cross-section shapes be compared on the same slenderness basis without manually working out each radius of gyration first.
Checking an existing column against a design guideline
Many design codes set slenderness limits for particular materials and applications; calculating the actual ratio is the first step before checking it against the relevant limit.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How slenderness ratio changes with column length
A 100 mm square-ish timber post at a range of effective lengths.
| Effective length | Slenderness ratio (λ) | Classification |
|---|---|---|
| 1 m | 34.64 | Intermediate column: both material strength and buckling can govern, depending on the design code used |
| 2 m | 69.28 | Intermediate column: both material strength and buckling can govern, depending on the design code used |
| 3 m | 103.92 | Long, slender column: buckling risk dominates, and Euler-type buckling analysis is normally required |
| 4 m | 138.56 | Long, slender column: buckling risk dominates, and Euler-type buckling analysis is normally required |
| 5 m | 173.21 | Long, slender column: buckling risk dominates, and Euler-type buckling analysis is normally required |
| 6 m | 207.85 | Long, slender column: buckling risk dominates, and Euler-type buckling analysis is normally required |
How slenderness ratio changes with cross-section size
A column of fixed length with a range of circular cross-section diameters.
| Diameter | Slenderness ratio (λ) | Classification |
|---|---|---|
| 80 mm | 150.00 | Long, slender column: buckling risk dominates, and Euler-type buckling analysis is normally required |
| 100 mm | 120.00 | Long, slender column: buckling risk dominates, and Euler-type buckling analysis is normally required |
| 150 mm | 80.00 | Intermediate column: both material strength and buckling can govern, depending on the design code used |
| 200 mm | 60.00 | Intermediate column: both material strength and buckling can govern, depending on the design code used |
| 300 mm | 40.00 | Intermediate column: both material strength and buckling can govern, depending on the design code used |
| 400 mm | 30.00 | Intermediate column: both material strength and buckling can govern, depending on the design code used |
Questions
What counts as a "slender" column?
There is no single universal cutoff, since it varies by material and design code, but as a general guide a slenderness ratio under about 30 is usually considered short, over about 100 is usually considered long and buckling-governed, and the range between is intermediate.
Why does effective length differ from the actual physical length?
Effective length adjusts for how the column’s ends are held. A column fixed rigidly at both ends resists buckling more than one merely pinned at both ends, so it behaves, for buckling purposes, as if it were shorter than its actual physical length.
Why use the smaller side for a rectangular section?
A rectangular column can buckle about either axis, and it will buckle about whichever axis gives the lower radius of gyration first, which is always the axis through the smaller side. Using the smaller side gives the governing, worst-case slenderness ratio.
Is a lower slenderness ratio always better?
For buckling resistance, yes, a lower ratio means less buckling risk. But a lower ratio usually comes from a larger or shorter cross-section, which has its own cost and weight trade-offs, so it is one factor in the design rather than the only one.
For how much a beam or cantilever physically bends under a load, rather than whether a column buckles, see the beam deflection calculator.