Calculator · Structural Mechanics
Slenderness Ratio Calculator
How slender a column is, expressed as the ratio of its effective length to the radius of gyration of its cross-section. A high slenderness means the column will lose stability long before the material reaches its strength; a low one means it is a stub that will squash rather than buckle.
Inputs
About the axis the column is being checked for.
Effective-length factor for these end conditions: K = 1. These are the theoretical values for ideal restraints; a real connection rarely achieves them, and design practice often uses a larger K.
Results
| Effective-length factor K | 1 |
|---|---|
| Unbraced length L | 3 m |
| Effective length KL | 3 m |
| Radius of gyration r | 25 mm |
| Slenderness ratio λ = KL/r | 120 |
Assumptions
- · λ = K L / r is dimensionless and describes the column, not its material strength.
- · The effective-length factors are idealised theoretical values; a real frame or connection rarely realises them exactly, and a larger K is often appropriate.
- · I and r must refer to the axis about which the column is being checked for buckling.
Technical reference
r = √(I / A)
Le = K L
λ = K L / r
- L
- unbraced length of the column[m]
- K
- effective-length factor for the end restraints[—]
- I
- second moment of area about the axis being checked[m⁴]
- A
- cross-sectional area[m²]
- r
- radius of gyration about the same axis[m]
- λ
- slenderness ratio[—]
The theoretical factors are K = 1.0 pinned–pinned, K = 2.0 fixed–free, K = 0.5 fixed–fixed and K = 0.699 fixed–pinned. They assume perfect restraints; real frames and connections seldom deliver them, so design codes recommend larger values.
I and r must be taken about the axis the column is free to buckle about. A column braced about one axis buckles about the other, and that is often not the axis with the smaller I.
λ is a geometric property of the column. On its own it says nothing about capacity: whether the column buckles elastically, inelastically or simply yields depends on the material as well.
A column with different unbraced lengths or restraints about its two axes has a slenderness ratio for each; the larger one usually governs.