Circular Spiral Column — ACI 318-19 Interaction Diagram Worked Example
A 20-in round column in a braced frame carries Pu = 450 kip and Mu = 140 kip-ft. We check slenderness, build the interaction diagram, read the moment capacity at the factored axial load, size the spiral pitch, and price what the spiral buys over ordinary ties: 23% more axial capacity and 17% more moment at this load.
The column
| Section | D = 20 in round, 8-#8 bars equally spaced, #3 spiral, 1.5 in clear cover |
| Materials | f′c = 5,000 psi (β1 = 0.80), Grade 60 bars and spiral |
| Frame | Braced (nonsway), unsupported length 14 ft, k = 1.0, double curvature with M1/M2 = +0.3 |
| Loads | Pu = 450 kip, Mu = 140 kip-ft |
Column tool: Circular, Spiral, D 20, 8 bars #8, tie #3, f′c 5000, Pu 450, Mu 140, ℓu 14, M1/M2 0.3.
Open the column calculator →Can second-order effects be ignored?
Nonsway limit = 34 + 12(M1/M2) = 34 + 12(0.3) = 37.6 (≤ 40) → short column, use Mu directly
P0 and the spiral cap
P0 = 0.85 × 5(314.2 − 6.32) + 60 × 6.32 = 1,308.3 + 379.2 = 1,688 kip
φPn,max = 0.75 × 0.85 × 1,688 = 1,076 kip (ties: 0.65 × 0.80 × 1,688 = 878 kip)
Key points and the capacity at Pu
Bars sit on a 15.25-in circle (D − 2 × 1.5 − 2 × 0.375 − 1.0), one at the compression face. For each neutral-axis depth the concrete block is a circular segment of depth a = 0.80c; its area and centroid come from the segment formulas, and each bar's stress from its own strain. Selected points of the design curve:
| Pu (kip) | φMn (kip-ft) | Note |
|---|---|---|
| −341 | 0 | pure tension, 0.9fyAst |
| 0 | 199 | pure bending |
| 200 | 257 | transition zone |
| 300 | 266 | near the peak |
| 401 | 259 | balanced: Pb = 535, Mb = 345 nominal, × 0.75 |
| 450 | 257 | this column |
| 700 | 232 | compression-controlled |
| 900 | 181 | |
| 1,076 | — | φPn,max plateau |
The same bars with ties instead of a spiral give φMn = 219 kip-ft at 450 kip (0.64). The spiral's higher φ is worth 17% more moment here and 23% more axial cap.
Pitch from the volumetric ratio
ρs,min = 0.45(314.2/227.0 − 1)(5/60) = 0.0144
s = 4Asp/(Dcρs) = 4(0.11)/(17.0 × 0.0144) = 1.80 in → use #3 spiral at 1¾ in pitch
Clear pitch = 1.75 − 0.375 = 1.375 in, within 1–3 in ✓
Clear distance between longitudinal bars
Tools used in this example
RC column interaction diagram. Guide: How to build a column interaction diagram.
Frequently asked questions
Why do spiral columns get a higher phi?
Spirals confine the core so the column keeps carrying load after the shell spalls, giving a more ductile failure. ACI 318-19 rewards that with φ = 0.75 for compression-controlled sections (0.65 for tied) and Pn,max = 0.85P0 (0.80P0 for tied).
How do you calculate spiral pitch?
From the minimum volumetric ratio ρs = 0.45(Ag/Ach − 1)f′c/fyt (ACI 318-19 §25.7.3.3) and the relation ρs = 4Asp/(Dc·s), where Dc is the core diameter to the outside of the spiral. Solve s = 4Asp/(Dc·ρs) and keep the clear pitch between 1 and 3 in.
Why does moment capacity rise and then fall as axial load increases?
Below the balanced point, axial compression offsets tension in the steel, so the section can take more moment before the steel yields. Above it, the concrete is already highly stressed by axial load and has less capacity left for bending. In this column φMn peaks at about 266 kip-ft near Pu = 300 kip.
What is the radius of gyration of a round column?
For slenderness, ACI 318-19 §6.2.5.2 allows r = 0.25D for circular columns (0.30h for rectangular). A 20-in column has r = 5 in.