How to Calculate the Slenderness Ratio (KL/r) of a Column

Q: Which radius of gyration do I use for a W-shape column?

The smaller one, r y , unless the column is braced in the weak direction. Then check both axes with the unbraced length for each.

Q: How do I calculate the compressive strength of a column?

Find F e = π²E / (KL/r)², then F cr from the AISC 360 equations, then φP n = 0.90 F cr A g .

Short answer

Slenderness is KL / r, where K is the effective length factor, L the unbraced length and r the radius of gyration of the axis that buckles, usually the weak axis. For a W12x65 with K = 1.0 and L = 12 ft, KL / r_y = 144 / 3.02 = 47.7. That is well under 200, the value AISC recommends as an upper limit, and gives a design compressive strength of about 728 kips in A992 steel.

What slenderness tells you

A short, stocky column fails by crushing the steel. A long, slender column fails by buckling sideways long before the steel yields. The slenderness ratio puts a number on where a column sits between the two:

Slenderness = K × L / r

L is the unbraced length, K adjusts it for how the ends are held, and r is the radius of gyration. A bigger KL / r means a more slender column and a lower buckling strength. AISC 360 recommends, as a user note, that KL / r not exceed 200. It is not a hard requirement, but a column slenderer than that is flexible and hard to handle on site.

Effective length factor K

K is 1.0 for a column pinned at both ends and braced against sway. Fixed ends make the column stiffer and lower K. A free end raises it. AISC Commentary Table C-A-7.1 lists theoretical values and recommended design values for ideal cases:

Effective length factors for idealized end conditions
End conditions	Theoretical K	Recommended design K
Both ends fixed, no sway	0.50	0.65
One end fixed, one pinned, no sway	0.70	0.80
Both ends pinned, no sway	1.00	1.00
Both ends fixed, sway allowed	1.00	1.20
One end fixed, one free (cantilever)	2.00	2.10
One end pinned, one fixed against rotation, sway allowed	2.00	2.00

In a real frame the ends are never perfectly fixed or perfectly pinned. For frames that can sway, K comes from alignment charts or a stability analysis. For a simple pin-ended column, K = 1.0.

Radius of gyration: which axis?

The radius of gyration is r = √(I / A). A shape has one for each axis. A W-shape is much stiffer about the strong axis, so r_x is larger than r_y, and the weak axis normally governs. For W12x65, r_x = 5.28 in and r_y = 3.02 in. If the column is braced against weak-axis buckling at midheight, then K L is half as long for the weak axis, and you must check both axes.

Worked example: W12x65, 12 ft

A W12x65 column, 12 ft between pinned, braced ends, A992 steel (F_y = 50 ksi, E = 29,000 ksi). From the shape table: A = 19.1 in², r_x = 5.28 in, r_y = 3.02 in.

K L = 1.0 × 12 × 12 = 144 in
Strong axis: K L / r_x = 144 / 5.28 = 27.3
Weak axis: K L / r_y = 144 / 3.02 = 47.7, which governs.

47.7 is under 200, so the column is not too slender.

From KL/r to column strength

AISC 360 chapter E turns slenderness into a design compressive stress. First the elastic buckling stress, the Euler value:

F_e = π² E / (K L / r)² = π² × 29,000 / 47.7² = 125.9 ksi

Then the critical stress, in two ranges. The dividing slenderness is 4.71 √(E / F_y) = 113.4 for F_y = 50 ksi:

If K L / r ≤ 113.4: F_cr = [ 0.658^{(F_y / F_e)} ] × F_y
If K L / r > 113.4: F_cr = 0.877 × F_e

Here 47.7 is below 113.4, so the first line applies. F_y / F_e = 50 / 125.9 = 0.397, and:

F_cr = 0.658^0.397 × 50 = 42.3 ksi

The design compressive strength is φ_c P_n = 0.90 × F_cr × A_g = 0.90 × 42.3 × 19.1 = 728 kips.

Strength at different lengths

Here is the same W12x65 at different effective lengths about the weak axis. Notice how the strength falls as the column gets longer, and how quickly it drops once F_e becomes small.

W12x65 column, A992 steel, K L about the weak axis (r_y = 3.02 in)
K L	K L / r	F_e (ksi)	F_cr (ksi)	φP_n (kips)
6 ft	23.8	503.6	48.0	825
10 ft	39.7	181.3	44.5	766
12 ft	47.7	125.9	42.3	728
16 ft	63.6	70.8	37.2	640
20 ft	79.5	45.3	31.5	542
25 ft	99.3	29.0	24.3	418
30 ft	119.2	20.1	17.7	304

Bracing a column at midheight in the weak direction cuts K L in half. In the table, a 24 ft column braced at midheight in the weak direction acts like a 12 ft column for that axis, and its strength rises accordingly.

A check people miss: local buckling

These column formulas assume the plates of the shape do not buckle locally before the column does. In compression, the flange limit is 0.56 √(E / F_y) = 13.5 and the web limit is 1.49 √(E / F_y) = 35.9 for F_y = 50 ksi. W12x65 passes both: b_f / 2t_f = 9.92 and h / t_w = 24.9. A light shape may not. W12x26 has h / t_w = 47.1, above 35.9, so its web is slender in compression and the strength must be reduced further. Always check which of your shapes are slender before you use the plain formula.

Get r from the app

Look up the shape in Steel Table USA and tap Inspect. The Inertia, Gyration and Moduli card lists r for both axes, called rz (strong) and ry (weak) in the app. For W12x26 it shows 5.17 in and 1.51 in. Use the smaller one for the slenderness check unless the column is braced in that direction. Design capacity in the app covers tension and bending, not column buckling, so use the formulas above for compression.

Questions people ask

What is the slenderness ratio of a steel column?

It is K L / r: the effective length factor times the unbraced length, divided by the radius of gyration of the axis that buckles.

What is the maximum KL/r for a steel column?

AISC 360 recommends KL / r of no more than 200 as a user note. It is guidance for stiffness and handling rather than a strict limit.

Which radius of gyration do I use for a W-shape column?

The smaller one, r_y, unless the column is braced in the weak direction. Then check both axes with the unbraced length for each.

What is the K factor for a pinned column?

K = 1.0 for a column pinned at both ends and braced against sway.

How do I calculate the compressive strength of a column?

Find F_e = π²E / (KL/r)², then F_cr from the AISC 360 equations, then φP_n = 0.90 F_cr A_g.

Why is my column strength different from the AISC Manual table?

The Manual tabulates the same equations, but check that the axis, K L, steel grade and whether the shape has slender elements match. Rounding also matters.

Keep reading

Radius of gyration for every shape

Steel Table USA lists r, I and every section property for over 1,800 AISC shapes. Free on Google Play.

For reference, not for sign-off. Shape and bolt values quoted here are as shown in Steel Table USA, which takes rolled shape data from the AISC Shapes Database v16.0 and bolt tables from the RCSC Specification (2020), and the worked examples follow AISC 360 methods under the simplifying assumptions stated in the text. Transcription can contain errors. Check critical values against the AISC Steel Construction Manual and the RCSC Specification, and have designs checked by a licensed engineer. The AISC Specification, Manual and Shapes Database are published by the American Institute of Steel Construction, and the RCSC Specification by the Research Council on Structural Connections. Steel Table USA is an independent app and is not affiliated with or endorsed by either organization.