When is a column slender? The kl_u/r limits in ACI 318-19

A concrete column is slender when its slenderness ratio kl_u/r is above the limit in ACI 318-19 §6.2.5. Below the limit the code lets you neglect second-order effects and design the section for the first-order moment. Above it you must account for them, usually with the moment magnification factor. The limit is not a flat 34: it depends on bracing and on how the column bends.

The limits

Slenderness effects may be neglected when:

\text{Not braced against sidesway:}\quad \frac{k\,l_u}{r} \le 22

\text{Braced against sidesway:}\quad \frac{k\,l_u}{r} \le 34 + 12\,\frac{M_1}{M_2} \quad\text{and}\quad \frac{k\,l_u}{r} \le 40

M₁ is the smaller factored end moment and M₂ the larger. In ACI 318-19 the ratio M₁/M₂ is negative for single curvature and positive for double curvature. Editions before 318-14 printed the same rule as 34 − 12(M₁/M₂) with the opposite sign convention, which is why both forms still circulate. They give identical limits as long as each is used with its own convention.

Read that way, the braced limit runs from 22 (single curvature, equal end moments, the worst case) through 34 (one end moment zero) up to the cap of 40. Do not treat 34 as a lower bound: in single curvature the limit falls to 22, so using 34 there is unconservative.

Braced limit by end-moment ratio

Bending shape	M₁/M₂ (318-19 sign)	Limit on kl_u/r
Single curvature, equal end moments	−1.0	22
Single curvature	−0.5	28
One end moment zero	0	34
Double curvature	+0.5	40
Double curvature, equal end moments	+1.0	40 (cap; formula gives 46)

Unbraced (sway) columns use 22 regardless of the moment ratio.

The three inputs: k, l_u and r

l_u is the unsupported length: the clear distance between the floor slab, beams or other members that give lateral support in the direction considered.
k is the effective length factor. For braced columns it lies between 0.5 and 1.0, and §6.6.4.4.3 permits taking k = 1.0, which is the simple conservative choice. Lower values come from the alignment charts or from frame analysis.
r is the radius of gyration of the gross section. ACI 318-19 permits the exact value or two approximations:

r = \sqrt{\frac{I_g}{A_g}} \qquad r \approx 0.30\,h \ \text{(rectangular)} \qquad r \approx 0.25\,D \ \text{(circular)}

For a rectangular column, h is the dimension in the direction stability is being considered, so a 12×24 in column has r = 3.6 in about one axis and 7.2 in about the other. The exact value for a solid rectangle is h/√12 = 0.289h and for a solid circle D/4, so the 0.30h shortcut is slightly unconservative for rectangles and exact for circles. Both approximations are code-permitted.

How long can a column be and stay short?

With k = 1.0 and the permitted r approximations, the largest unsupported length that still counts as short:

Section	r (in)	Limit 22	Limit 34	Limit 40
14×14 in	4.2	92.4 in (7.7 ft)	142.8 in (11.9 ft)	168.0 in (14.0 ft)
16×16 in	4.8	105.6 in (8.8 ft)	163.2 in (13.6 ft)	192.0 in (16.0 ft)
20 in diameter	5.0	110.0 in (9.2 ft)	170.0 in (14.2 ft)	200.0 in (16.7 ft)

Lengths are limit × r / k. Checked against calcnote's gate: a 16×16 in braced column with M₁ = 0 and k = 1.0 is short at l_u = 163 in (kl_u/r = 33.96) and slender at 164 in (34.17).

Worked checks

Short: calcnote's demo column, 16×16 in, l_u = 120 in, k = 1.0, M₁ = 0:

\frac{k\,l_u}{r} = \frac{1.0 \times 120}{0.30 \times 16} = 25.0 \;\le\; 34 \quad\Rightarrow\ \text{short}

Slender: Wight Example 12-2 Column DE, 14×14 in, l_u = 264 in, k = 0.86, M₁ = 51.2 and M₂ = 68.5 kip-ft in single curvature:

\frac{k\,l_u}{r} = \frac{0.86 \times 264}{0.30 \times 14} = 54.06 \;>\; 34 + 12\left(\frac{-51.2}{68.5}\right) = 25.03 \quad\Rightarrow\ \text{slender}

Even bent in double curvature, where the limit rises to the cap of 40, this column stays slender (54.06 > 40). Its magnified moment is worked through on the moment magnification page.

What happens above the limit

Above the limit, the design must include second-order effects (§6.2.5), by the moment magnifier method of §6.6.4 or by a second-order frame analysis. The result is capped: the moment including second-order effects may not exceed 1.4 times the first-order moment (§6.2.5.3). There is no fixed upper kl_u/r in 318-19 (the old kl_u/r > 100 rule of 318-05 is gone); in practice the 1.4 cap is the ceiling. What calcnote does: it computes kl_u/r with r = 0.30h or 0.25D on every run, applies the braced limit with your signed end moments, and for slender columns continues with §6.6.4 nonsway magnification. Its input form uses the engineer-familiar sign: enter M₁ positive for single curvature and negative for double curvature; calcnote converts to the 318-19 convention internally, so both give the limits in the table above. Sway frames are outside calcnote's scope.

Sources

ACI 318-19: §6.2.5 (slenderness limits and radius of gyration), §6.2.5.3 (1.4 limit), §6.6.4.4.3 (k for nonsway columns), §6.6.4 (moment magnification).
Wight, J. K., Reinforced Concrete: Mechanics and Design, 7th ed., Example 12-2 (Column DE).
StructurePoint, Slender Concrete Column Design in Sway Frames (ACI 318-19); spColumn Manual, v10.30.