TC-11 · slender witness

Wight Example 12-2, Column DE: a slender column that actually magnifies

Wight's Example 12-2 (7th ed., pp. 588–593; numerically identical in the ACI 318-19-native 8th ed.) designs the columns of a braced three-story frame. Column DE is the interior story column: genuinely slender (kl_u/r ≈ 54) with a genuine magnifier (δ_ns > 1 after the floor) — unlike the widely-cited StructurePoint non-sway example, whose δ computes below 1.0 and floors out. This page walks the full §6.6.4 chain three ways: Wight's printed solution, an independent first-principles ACI 318-19 baseline, and calcnote.

Case definition

Section	14 × 14 in tied, 4-#7 corner bars (2.5 in to bar center)
Materials	f′c = 4,000 psi, Grade 60
Effective length	l_u = 264 in, k = 0.86 (Wight's Table 12-2 value — deliberately more conservative than his nomograph's 0.63; calcnote takes k as an input)
Demand	P_u = 82.4 kip; M₂ = 68.5 kip·ft, M₁ = 51.2 kip·ft, single curvature
Sustained-load ratio	β_dns = 1.2·50 / 82.4 = 0.728

The slenderness gate

\frac{k\,l_u}{r} = \frac{0.86 \times 264}{0.30 \times 14} = 54.1 \;>\; \min\!\left(34 + 12\tfrac{M_1}{M_2},\, 40\right) = 25.0

The column is slender; §6.6.4 moment magnification applies. One convention note worth pausing on: ACI 318-14 flipped the printed sign convention — in 318-19's equations, M₁/M₂ is negative for single curvature. calcnote's input form uses the engineer-familiar convention (positive M₁ = single curvature) and applies the mapping internally, so this case is entered as M₁ = +51.2. Both forms give the same limit (25.0) and the same C_m below; the mapping is locked by regression tests in both curvature directions.

The magnification chain, three ways

C_m = 0.6 - 0.4\,\frac{M_1}{M_2}\Big|_{318\text{-}19} = 0.90 \qquad P_c = \frac{\pi^2 (EI)_{\mathrm{eff}}}{(k\,l_u)^2} \qquad \delta_{ns} = \frac{C_m}{1 - \dfrac{P_u}{0.75\,P_c}} \ge 1.0

One deliberate difference is disclosed up front. §6.6.4.4.4 offers alternative stiffness formulas: Wight's printed solution uses option (a), 0.4E_cI_g/(1+β_dns) — the natural choice at his design stage, before the reinforcement is chosen. calcnote implements option (b), (0.2E_cI_g + E_sI_se)/(1+β_dns) — the formula StructurePoint's spColumn solver uses, which credits the actual bars. Both are code-compliant; option (b) is stiffer here only if the steel contributes more than 0.2E_cI_g — with 4-#7 (I_se = 48.6 in⁴) it is less stiff, so calcnote's magnifier is slightly larger (conservative). The independent baseline computes both, bridging the printed book values and calcnote's exactly.

Quantity	Wight (printed, option (a))	Baseline, option (a)	Baseline, option (b)	calcnote (option (b))
kl_u/r	54.1	54.06	54.06	54.06
C_m	0.90	0.899	0.899	0.899
(EI)_eff (kip·in²)	2.67×10⁶	2.671×10⁶	2.151×10⁶	2.151×10⁶
P_c (kip)	511	511.5	411.9	411.9
δ_ns	1.15	1.145	1.226	1.226
M_c (kip·ft)	78.8	78.4	83.98	83.98

Reading the deltas

Option (a) column vs Wight: agreement to print precision everywhere. Wight's M_c = 78.8 comes from display-rounding δ to 1.15 before multiplying (1.15 × 68.5 = 78.8); the unrounded chain gives 78.4. Same arithmetic, different rounding point — the identical pattern to StructurePoint's display-rounding documented on the Example 11-1 page.
Option (b) vs option (a): P_c drops 19% and δ_ns rises from 1.15 to 1.23 — a real, documented consequence of the formula choice, not a disagreement. For this lightly-reinforced section, crediting the actual steel (option (b)) yields less stiffness than option (a)'s blanket 0.4E_cI_g. calcnote's larger design moment (84.0 vs 78.8 kip·ft) is on the conservative side.
End to end: with M_c = 84.0 kip·ft, calcnote's full section check returns PASS at D/C = 0.91 — consistent with Wight's conclusion that the 14×14 / 4-#7 section works (his final section is governed by the companion column CD's larger demand).

The companion column: when the magnifier floors

The same example's Column CD bends in double curvature (M₁/M₂ = +0.403 in the printed convention): Wight computes C_m = 0.438, P_c = 960 kip, δ = 0.538 — below 1.0, so §6.6.4.5.2's floor takes δ_ns = 1.0 and M_c = M₂ = 94.4 kip·ft unmagnified. (CD's figures are cited from the printed text; like every printed value on this page they were transcribed from two independently cross-checked scans of the 7th and 8th editions — the report's revalidation log tracks the physical-copy spot-check.) calcnote's floor behavior is locked by regression tests on exactly this pattern (double curvature, δ_raw < 1 → δ_ns = 1.0, M_c = M₂), alongside the two guards the floor must never mask: a hard stop at instability (P_u ≥ 0.75·P_c) and the §6.2.5.3 limit (δ_ns > 1.4 → the code requires a revised design).

Second witness: the formula chain vs StructurePoint

StructurePoint's published non-sway example (17×17 in, 10-#9, f′c = 3,000 psi — spColumn-validated) is a short column whose magnifier floors to 1.0, so it cannot witness end-to-end magnification — but its printed intermediate values witness the formula chain itself, including the exact §6.6.4.4.4(b) stiffness formula spColumn's solver uses (the reason calcnote implements option (b)). Its 10-bar two-face section is outside calcnote's input envelope, so the comparison below is at formula level (locked by a regression test), not an engine run. StructurePoint takes r = √(I_g/A_g) (§6.2.5.1(a)); calcnote uses 0.30h (§6.2.5.1(b)) — both code options, same short verdict here.

Quantity	StructurePoint (printed)	calcnote's formula chain
E_c (ksi)	3,122	3,122
I_se (in⁴)	360	360
(EI)_eff option (b) (kip·in²)	10.56×10⁶	10.56×10⁶
P_c (kip)	7,871	7,871
C_m (M₁ = 0)	0.60	0.60
δ raw → floored	0.66 → 1.00	0.66 → 1.00
M_c (kip·ft)	105	105

Sources

Wight, J. K., Reinforced Concrete: Mechanics and Design, 7th ed., Pearson, 2016 — Example 12-2, pp. 588–593 (Table 12-2, Figs. 12-22/12-23); 8th ed. (2021, ACI 318-19) carries the same example with identical values.
ACI 318-19: §6.2.5 (slenderness limits), §6.2.5.3 (1.4 second-order limit), §6.6.4.4.4 ((EI)_eff options), §6.6.4.5 (non-sway magnification: C_m, δ_ns, M_2,min).
StructurePoint, "Slenderness Effects for Concrete Columns in Non-Sway Frame — Moment Magnification Method (ACI 318-19)", v10 — the W2 chain-value source (318-14 edition numerically identical).
Independent first-principles baseline (both §6.6.4.4.4 options), no calcnote-source dependency; agreement figures above.