The moment magnification factor δ_ns in ACI 318-19, explained

A slender column bows under load, and the axial force acting on that bow adds moment the first-order analysis never saw. ACI 318-19 §6.6.4 covers this with a single multiplier: the first-order end moment M₂ is amplified by the moment magnification factor to get the design moment M_c. This page explains each term in the formula, the choices the code leaves to you, and works one real column end to end with numbers calcnote's engine produces.

When it applies

Two conditions must both hold before the nonsway magnifier is used:

The column is slender: kl_u/r exceeds the §6.2.5 limit (34 + 12·M₁/M₂, not more than 40, for braced columns). Below the limit, slenderness may be neglected and the column is designed for M₂ directly. See when is a column slender.
The story is nonsway: §6.6.4.3 permits a nonsway analysis when the stability index Q does not exceed 0.05, or when second-order effects raise the column end moments by no more than 5 percent of the first-order end moments. Sway stories use the separate δ_s procedure of §6.6.4.6.

The formula chain

Design moment (§6.6.4.5.1) and magnifier (§6.6.4.5.2):

M_c = \delta_{ns}\,M_2 \qquad \delta_{ns} = \frac{C_m}{1 - \dfrac{P_u}{0.75\,P_c}} \ge 1.0

The 0.75 in the denominator is a stiffness reduction factor (φ_K), separate from the strength reduction factor φ used for section capacity. The floor of 1.0 means the magnifier never reduces a moment.

C_m: the moment shape factor (§6.6.4.5.3)

C_m = 0.6 - 0.4\,\frac{M_1}{M_2}

M₁ is the smaller end moment and M₂ the larger. Since 318-14 the ratio is negative for single curvature and positive for double curvature, so a column bent in single curvature with equal end moments gets C_m = 1.0 and one in double curvature gets as little as 0.2. Older references print 0.6 + 0.4·M₁/M₂ with the opposite sign convention; the numbers come out the same. If transverse load acts between the supports, C_m = 1.0 (§6.6.4.5.3(b)).

P_c: the critical buckling load (§6.6.4.4.2)

P_c = \frac{\pi^2\,(EI)_{eff}}{(k\,l_u)^2}

For nonsway columns the effective length factor k is at most 1.0, and §6.6.4.4.3 permits taking k = 1.0 without further analysis. A smaller k from the alignment charts raises P_c and lowers δ_ns.

(EI)_eff: three permitted options (§6.6.4.4.4)

(a)\ \frac{0.4\,E_c I_g}{1+\beta_{dns}} \qquad (b)\ \frac{0.2\,E_c I_g + E_s I_{se}}{1+\beta_{dns}} \qquad (c)\ \frac{E_c I}{1+\beta_{dns}}

(a) needs no reinforcement details, so it suits preliminary design.
(b) credits the actual bars through I_se, the moment of inertia of the reinforcement about the section centroid. It is the option calcnote implements, and the one StructurePoint's spColumn solver uses.
(c) uses a cracked-section moment of inertia I.

β_dns is the ratio of the maximum factored sustained axial load to the maximum factored axial load in the same load combination. It divides the stiffness to account for creep: the more of the load that is permanent, the softer the column and the larger the magnifier.

Three guards: M_2,min, the 1.4 limit and stability

§6.6.4.5.4: M₂ in the magnifier formula must be at least M_2,min = P_u(0.6 + 0.03h), h in inches, result in kip-in. When M_2,min governs, calcnote takes C_m = 1.0 (§6.6.4.5.4). See minimum eccentricity.
§6.2.5.3: the moment including second-order effects may not exceed 1.4 times the first-order moment. With the magnifier method, that means δ_ns ≤ 1.4. Past that, the code wants a stiffer or shorter column, not a bigger number.
Stability: if P_u reaches 0.75·P_c, the denominator is zero or negative and the column is at its design buckling load. calcnote stops with an explanation instead of flooring such a value to 1.0.

Worked example: Wight Example 12-2, Column DE

A braced interior column from Wight's Reinforced Concrete: Mechanics and Design: 14×14 in, 4-#7 bars, f′_c = 4,000 psi, Grade 60, l_u = 264 in, k = 0.86, P_u = 82.4 kip, M₂ = 68.5 kip-ft, M₁ = 51.2 kip-ft in single curvature, β_dns = 0.728. Every value below is what calcnote's engine returns for this input.

Slenderness check (r = 0.30h = 4.2 in):

\frac{k\,l_u}{r} = \frac{0.86 \times 264}{4.2} = 54.06 \;>\; 34 + 12\left(\frac{-51.2}{68.5}\right) = 25.03 \quad\Rightarrow\ \text{slender}

Stiffness, option (b), with E_c = 57√4000 = 3,605 ksi, I_g = 3,201 in⁴ and I_se = 48.6 in⁴:

(EI)_{eff} = \frac{0.2(3605)(3201) + 29000(48.6)}{1 + 0.728} = \frac{2.308\times10^6 + 1.409\times10^6}{1.728} = 2.151\times10^6\ \text{kip-in}^2

P_c = \frac{\pi^2 (2.151\times10^6)}{(0.86 \times 264)^2} = 411.9\ \text{kip}

C_m = 0.6 - 0.4\left(\frac{-51.2}{68.5}\right) = 0.899

\delta_{ns} = \frac{0.899}{1 - \dfrac{82.4}{0.75(411.9)}} = \frac{0.899}{0.733} = 1.226

M_c = 1.226 \times 68.5 = 84.0\ \text{kip-ft}

M_2,min = 82.4(0.6 + 0.03·14) = 84.0 kip-in = 7.0 kip-ft, far below M₂, so it does not govern, and δ_ns = 1.226 is inside the 1.4 limit. The section check at M_c = 84.0 kip-ft returns PASS at D/C = 0.91. Wight's printed solution uses option (a), which gives P_c = 511.5 kip and δ_ns = 1.145 for the same column; the difference is the stiffness option, not an error. The Wight Example 12-2 verification page puts both options side by side against the printed book values.

How much the end moments matter

Same column, same P_u and M₂, only M₁ changed. P_c stays at 411.9 kip throughout; C_m alone moves the result.

End moments	M₁/M₂ (318-19 sign)	C_m	δ_ns	M_c (kip-ft)
Equal, single curvature	−1.000	1.000	1.364	93.4
Wight's case, single curvature	−0.747	0.899	1.226	84.0
One end pinned (M₁ = 0)	0	0.600	1.000*	68.5
Double curvature	+0.747	0.301	1.000*	68.5

* The formula gives less than 1.0 (0.818 and 0.411), so the §6.6.4.5.2 floor sets δ_ns = 1.0. The column is still slender in every row (54.06 exceeds even the 40 cap), yet in two of them it needs no extra moment: a column bent in double curvature barely bows at midheight. The single-curvature, equal-moment row sits close to the 1.4 limit. To reproduce a row in calcnote, note its input convention: M₁ is entered positive for single curvature and negative for double curvature (M₁ = 68.5, 51.2, 0 and −51.2 for the four rows), and calcnote converts to the 318-19 sign internally.

What calcnote does

When a column fails the §6.2.5 short-column check, calcnote asks for β_dns and runs the chain above with option (b) and the r = 0.30h (0.25D for circular) approximation, then checks the magnified moment against the interaction diagram. It stops with an explanation at instability or when δ_ns exceeds 1.4. Scope note: sway frames (§6.6.4.6) and columns with transverse load between supports are outside calcnote's scope; k is an input, so alignment-chart work stays with the designer.

Sources

ACI 318-19: §6.2.5 (slenderness limits), §6.2.5.3 (1.4 limit), §6.6.4.3 (nonsway designation), §6.6.4.4.2 (P_c), §6.6.4.4.3 (k), §6.6.4.4.4 ((EI)_eff options), §6.6.4.5.1 to §6.6.4.5.4 (nonsway magnification, C_m, M_2,min), §6.6.4.6 (sway frames).
Wight, J. K., Reinforced Concrete: Mechanics and Design, 7th ed., Example 12-2 (Column DE).
StructurePoint, Slenderness Effects for Columns in Non-Sway Frame, Moment Magnification Method (ACI 318-19) and Slender Concrete Column Design in Sway Frames (ACI 318-19); spColumn Manual, v10.30.