Slenderness ratio = effective length divided by radius of gyration, λ = KL / r. Work it out about both axes and use the larger value. For a 3 m MB 300 column pinned at both ends, λ = 3000 / 28.7 = 104.5 about the minor axis, and only 24.4 about the major axis, so the minor axis governs.
What slenderness ratio tells you
A short, stocky column fails by crushing the steel. A long, thin column fails earlier, by bending sideways, which is called buckling. Slenderness ratio is the number that says which kind of column you have. The higher it is, the sooner the column buckles and the lower the load it can carry.
λ = KL / rK is the effective length factor, L is the actual length between restraints and r is the radius of gyration. KL together is the effective length.
Radius of gyration
The radius of gyration packs the shape of the cross section into one length:
r = √( I / A )A section has one for each axis. For MB 300, the table gives rz = 12.3 cm about the major axis and ry = 2.87 cm about the minor axis. The minor axis value is more than four times smaller, so a column of this section buckles about the minor axis long before the major one. The section table guide shows where these values sit in the row.
Effective length and the K factor
The effective length is the length of an equivalent pinned-end column. It depends on how the ends are held. The classic theoretical values are:
| End conditions | K |
|---|---|
| Both ends pinned | 1.0 |
| Both ends fixed | 0.5 |
| One end fixed, one pinned | 0.7 |
| One end fixed, one free (cantilever) | 2.0 |
IS 800 gives recommended design values in its table of effective lengths. For the restrained cases they are a little higher than the ideal ones, because a real fixed end is never perfectly fixed, for example 0.65 instead of 0.5 and 0.8 instead of 0.7. Use the code table for design, and use the values above only to build intuition.
Bracing changes L. If a column is braced at mid-height against movement about its minor axis, the effective length about that axis is half.
Worked example: MB 300 column
A 3.0 m column of MB 300, pinned at both ends, so K = 1.0. From the table, rz = 123 mm and ry = 28.7 mm.
- About the major axis: λz = 3000 / 123 = 24.4
- About the minor axis: λy = 3000 / 28.7 = 104.5
The larger value, 104.5, governs. Now brace the column at mid-height against minor axis movement. The effective length about that axis drops to 1500 mm and λy = 1500 / 28.7 = 52.3. One brace halves the slenderness, and that is often the cheapest way to make a column work.
Compare a smaller section. MB 150 has ry = 1.56 cm. A 3 m pinned column of it has λy = 3000 / 15.6 = 192.3, which is above the usual limit below.
| Section | ry | λy, no brace | λy, braced at mid-height |
|---|---|---|---|
| MB 300 | 2.87 cm | 104.5 | 52.3 |
| MB 150 | 1.56 cm | 192.3 | 96.2 |
The slenderness limit
IS 800 sets a maximum slenderness for compression members so that they are not too flexible to handle and do not vibrate or bow badly. For a member that carries compression from dead and imposed loads the usual limit is 180, and the code table gives other limits for other cases. The unbraced MB 150 column at 192.3 fails that limit before any strength check is made. The unbraced MB 300 at 104.5 and the braced MB 150 at 96.2 pass it.
Why slenderness matters: the Euler stress
The elastic buckling stress of a slender column, from Euler, is:
fcc = π² × E / λ²With E = 200,000 MPa and λ = 104.5, fcc = 9.87 × 200,000 / 10,920 = 180.8 MPa. That is already below the 250 MPa yield strength of E250 steel, so this column buckles before it yields. At λ = 52.3 the Euler stress is 721 MPa, far above yield, and the column behaves much more like a short one.
Euler stress is only an indicator. IS 800 turns slenderness into a design compressive stress using buckling classes that allow for imperfections and residual stresses, and the result is lower than Euler for stocky columns. That full check is what the column check in a design calculator does.
Check it in the app
In Steel Table India, every section shows rz and ry in the Inspect panel, so the slenderness arithmetic takes a calculator and a few seconds. The design calculator checks a column against IS 800, with capacity, interaction and a pass or fail with the numbers shown. For the strength side of the same member see the capacity guide, and for how I and r are worked out see the moment of inertia guide.
Questions people ask
What is the slenderness ratio of a column?
It is the effective length divided by the radius of gyration, λ = KL / r. A higher value means the column buckles more easily.
Which axis do I use for slenderness?
Calculate it about both axes and use the larger value. For a rolled I-section the minor axis usually governs, because its radius of gyration is much smaller.
What is the effective length factor K for a pinned column?
K is 1.0 for a column pinned at both ends, so the effective length equals the actual length.
What is the maximum slenderness ratio for a steel column?
For a compression member carrying dead and imposed loads, IS 800 limits slenderness to 180. Other cases have other limits in the code table.
How do I reduce the slenderness of a column?
Brace it at intermediate points, which shortens the effective length, or choose a section with a larger radius of gyration about the weak axis.
Is the Euler stress the design strength?
No. Euler stress is the elastic buckling stress of a perfect column. IS 800 uses buckling classes to give a lower design stress that allows for imperfections.
Keep reading
Radius of gyration on every section
Steel Table India shows rz and ry for every IS 808:2021 section and checks columns to IS 800. Free on Google Play.
For reference, not for sign-off. Section and bolt values quoted here are as shown in Steel Table India, transcribed from the published IS 808:2021 tables and worked to IS 800:2007 methods. Transcription can contain errors, and these worked examples make simplifying assumptions stated in the text. Check critical values against the official Bureau of Indian Standards publications and have designs checked by a qualified engineer. IS 808 and IS 800 are published by the Bureau of Indian Standards.