Composite Steel Beam Neutral Axis — Transformed Section Worked Example
A W18×35 floor beam acting compositely with a 4.5-in normal-weight slab. We find the effective width, transform the concrete to steel, locate the elastic neutral axis, and use the composite moment of inertia for unshored construction stresses and live-load deflection. The punchline: the slab more than triples the beam's stiffness, and the neutral axis ends up inside the slab.
The floor beam
| Beam | W18×35, A992, simple span 30 ft, beams at 8 ft on center, unshored |
| Steel (plate model) | d = 17.70 in, flanges 6.00 × 0.425 in, web 0.300 × 16.85 in (fillets ignored) |
| Slab | 4.5 in normal-weight concrete on top of the flange, f′c = 4,000 psi, full composite action |
| Loads | Construction: slab 56 psf + beam 35 plf. After cure: superimposed dead 15 psf, live 100 psf |
Neutral-axis tool: load the "Composite" preset, then set the slab to 90 × 4.5 at y = 17.70 with n = 0.1243.
Open the neutral axis calculator →How much slab works with the beam
Ec = 57,000√4,000 = 3,605 ksi → n = Es/Ec = 29,000/3,605 = 8.04
Transformed slab width = 90/8.04 = 11.19 in (enter n = Ec/Es = 0.1243 in the tool)
The section that carries the wet concrete
ȳ = 8.85 in (symmetric) · I = 500.2 in⁴ (AISC 510) · S = 500.2/8.85 = 56.5 in³
The plate model loses about 2% of I to the missing fillets. That is acceptable for a hand check; use the tabulated values for a final design.
Locate the elastic neutral axis
| Piece | nA (in²) | ȳi (in) | nAȳ (in³) |
|---|---|---|---|
| Steel W18×35 | 10.15 | 8.85 | 89.8 |
| Slab 0.1243 × 90 × 4.5 | 50.35 | 19.95 | 1,004.4 |
| Total | 60.50 | 18.09 | 1,094.2 |
Itr = [500.2 + 10.15(18.09 − 8.85)²] + [0.1243(90)(4.5)³/12 + 50.35(19.95 − 18.09)²] = 1,626 in⁴
Sbottom = 1,626 / 18.09 = 89.9 in³ · ctop of slab = 22.20 − 18.09 = 4.11 in
Two stages, two sections
fbot = ftop = 54.6 × 12 / 56.5 = 11.6 ksi
Stage 2 (composite): w = 8 × (15 + 100) = 920 plf → M2 = 103.5 kip-ft
fbot = 103.5 × 12 / 89.9 = 13.8 ksi · concrete top = 0.1243 × 1,242 × 4.11 / 1,626 = 0.39 ksi
Total bottom flange = 11.6 + 13.8 = 25.4 ksi at service
The top flange carries 11.6 ksi compression during the pour, with only the deck to brace it, so check lateral-torsional buckling for that stage. After the slab cures, the top flange sits just below the neutral axis and picks up a small tension of 0.3 ksi.
What composite action buys
| Section | I (in⁴) | Sbot (in³) | Live-load deflection |
|---|---|---|---|
| Steel alone | 500 | 56.5 | 1.01 in (L/358) |
| Long-term, 3n = 24.1 | 1,308 | 83.0 | — |
| Short-term, n = 8.04 | 1,626 | 89.9 | 0.31 in (L/1,165) |
Tools used in this example
Neutral axis calculator for the transformed section, beam deflection for Δ, and AISC LRFD for the steel strength checks. Guide: How to find the neutral axis.
Frequently asked questions
Where is the neutral axis of a composite beam?
Usually in or just below the concrete slab. The transformed slab is much larger than the steel section, so the centroid moves up. In this example it sits 0.39 in above the top of the steel, inside the slab. When it falls inside the slab, the concrete below it is in tension and is normally ignored for a cracked analysis; here the effect on I is under 1%.
What modular ratio should I use for a composite beam?
n = Es/Ec, with Ec = 57,000√f′c psi for normal-weight concrete (ACI 318-19 §19.2.2). For f′c = 4,000 psi, Ec = 3,605 ksi and n = 8.04, so the transformed slab width is b/n. For sustained loads, many designers use 2n or 3n to account for creep.
Why is construction stress computed on the steel alone?
In unshored construction, the steel beam carries the wet concrete and its own weight before the concrete hardens. Those stresses are locked in. Only loads applied after the slab cures (superimposed dead and live) act on the composite section. The final stress is the sum of the two stages.
How is the effective slab width chosen?
AISC 360-22 §I3.1a: on each side of the beam centerline, the least of one-eighth of the span, half the distance to the adjacent beam, and the distance to the slab edge. For a 30 ft span and 8 ft spacing that is min(45 in, 48 in) each side, so 90 in total.