Circular spiral column interaction diagram: the StructurePoint 20 in example, worked and cross-checked

StructurePoint's public design example "Interaction Diagram — Circular Spiral Reinforced Concrete Column (ACI 318-19)" develops the P-M interaction diagram for a 20 in diameter spiral column with 8-#10 bars, hand-calculated through seven control points and validated against the spColumn engineering software (with Pincheira et al., Reinforced Concrete Design 9th ed., Example 10.18.1 as its own print reference). This page compares the example's control points three ways: StructurePoint's printed values, an independent first-principles ACI 318-19 circular-segment baseline, and calcnote's engine. This is test case TC-12 of the calcnote verification report.

Case definition

Section	Circular, D = 20 in (A_g = 314.16 in²)
Reinforcement	8-#10 (A_st = 10.16 in², ρ = 3.23%), equally spaced ring, one bar on the compression axis
Transverse	#4 spiral (φ = 0.75 base per Table 21.2.2; 0.85·P_o cap per Table 22.4.2.1)
Materials	f′c = 5,000 psi (β₁ = 0.80), Grade 60 (ε_ty = 0.00207)
Cover to bar centroid (δ)	2.635 in — via 1.5 in clear cover + #4 spiral (0.500) + half a #10 bar (0.635); reproduces StructurePoint's Table 1 bar-depth ladder (2.64 / 4.79 / 10.00 / 15.21 / 17.37 in) exactly at full precision
Extreme tension-steel depth	d_t = 10 + 7.365 = 17.365 in (StructurePoint prints 17.37 in its Table 1 and 17.36 in its comparison table — both are the same exact value at table precision)

Governing relations

Pure-compression nominal capacity and the spiral-column design cap (ACI 318-19 §22.4.2, Table 22.4.2.1):

P_o = 0.85\,f'_c\,(A_g - A_{st}) + f_y\,A_{st} = 0.85(5)(304.0) + 60(10.16) = 1{,}902\ \text{kip}

\phi P_{n,\max} = 0.85\,\phi\,P_o = 0.85(0.75)(1{,}902) = 1{,}212.3\ \text{kip}

The Whitney stress block acts on the circular segment of depth a = β₁c; the segment's area and centroid follow from the block geometry:

\alpha = \cos^{-1}\!\left(\frac{D/2 - a}{D/2}\right), \qquad A_{comp} = \frac{D^2}{4}\,(\alpha - \sin\alpha\cos\alpha), \qquad \bar{y} = \frac{D^3 \sin^3\alpha}{12\,A_{comp}}

The φ transition for a conforming spiral column (Table 21.2.2 — the 0.75 base is conditioned on the §25.7.3 spiral checks, which calcnote enforces as hard errors):

\phi = 0.75 + 0.15\,\frac{\varepsilon_t - \varepsilon_{ty}}{0.003} \quad (\varepsilon_{ty} \le \varepsilon_t < \varepsilon_{ty} + 0.003)

Control-point comparison

Comparison basis, stated up front: StructurePoint publishes seven control points. Four of them are strain states calcnote also uses as diagram anchors — the maximum-compression cap, the balanced point, the tension-controlled limit, and pure bending — and those four are compared directly below. StructurePoint's f_s = 0 and f_s = 0.5f_y points are NOT calcnote anchors; they are checked in the second table as fixed-c evaluations through the same circular-section response function. StructurePoint's seventh point (maximum axial tension) is outside calcnote's positive-(P, M) design quadrant and is not compared. Values are φ-reduced, in kip and kip-ft.

Control point	Strain state	φ	SP hand / spColumn φP_n / φM_n	ACI baseline	calcnote
Cap (φP_n,max)	all compression	0.75	1212.3 / —	1212.268 / 0.000	1212.268 / 0.000
Balanced (compression-controlled limit)	ε_t = 0.00207, c = 10.28 in	0.75	389.2 / 305.80	389.200 / 305.800	389.200 / 305.800
Tension-controlled limit	ε_t = 0.00507, c = 6.46 in	0.90	26.55 / 295.31	26.553 / 295.314	26.553 / 295.314
Pure bending	P_n = 0	0.90	0.0 / 288.09	0.000 / 288.096	0.000 / 288.068

Supplementary check — StructurePoint's two remaining compression-side points, evaluated at the reference's fixed neutral-axis states through calcnote's circular-section response function (not diagram anchors; disclosed above):

Fixed-c point	Neutral axis	φ	SP hand / spColumn	ACI baseline	calcnote
f_s = 0 (zero strain, extreme bar)	c = d_t = 17.365 in	0.75	984.5 / 208.16	984.483 / 208.159	984.482 / 208.159
f_s = 0.5f_y	c = 12.912 in (SP prints 12.91)	0.75	642.2 / 281.43	642.202 / 281.429	642.202 / 281.429

Reading the deltas

Five of the six compared points agree to the printed digit; pure bending agrees within 0.008%. StructurePoint's hand values and its spColumn software values are identical on every row of its own comparison table; calcnote and the independent baseline reproduce them at table precision, with the single visible residual (288.068 vs 288.09 on pure bending) explained below.

Pure bending, 288.07 vs 288.09/288.10: StructurePoint iterates by hand to c = 6.256 in and stops when P_n rounds to zero; calcnote bisects to c = 6.2553 in with a force-balance tolerance. The 0.02 kip-ft (0.008%) difference is iteration stopping criterion, not methodology.

The bar-depth ladder is exact: calcnote places the ring from first principles (δ = 1.5 + 0.500 + 0.635 = 2.635 in, one bar on the compression axis) and reproduces StructurePoint's Table 1 depths (2.64, 4.79, 10.00, 15.21, 17.37 in) at full precision — the same convention the reference and spColumn use. calcnote pins this orientation; a rotated ring would forfeit witness comparability and is deliberately not offered.

Scope note: StructurePoint's maximum-tension point (−548.6 kip) lies in the axial-tension quadrant. calcnote constructs its design diagram in the positive-(P, M) quadrant only (the same posture disclosed on the TC-1 page), so that point is out of scope rather than disagreeing.

Sources

StructurePoint, "Interaction Diagram — Circular Spiral Reinforced Concrete Column (ACI 318-19)", May 2022 — seven hand-calculated control points, spColumn v10.00-validated (its Table 8).
Pincheira, J. et al., Reinforced Concrete Design, 9th ed., Oxford University Press, 2021 — Example 10.18.1 (StructurePoint's print reference for this section).
ACI 318-19: §22.4.2 (axial cap), §22.2.2.4.1 (equivalent stress block), Table 21.2.2 (φ, spiral column), §25.7.3 (spiral conformance conditions for φ = 0.75).