Two-Way Slab Design per ACI 318-19: Step-by-Step Worked Example

This article walks through a complete two-way flat plate design for a square interior panel using the ACI 318-19 Direct Design Method (DDM), as defined in §8.10. Every number is calculated from scratch.

Problem Statement

Parameter	Value
Panel size (ln × ln)	6.0 m × 6.0 m clear span
Concrete strength f’c	25 MPa (3,625 psi)
Steel yield strength fy	420 MPa (60,900 psi)
Slab system	Flat plate (no drop panels, no column capitals)
Panel type	Interior (all four edges continuous)
Loads	Dead: 7.0 kPa (superimposed) + self-weight; Live: 3.0 kPa

Step 1 — Minimum Thickness (Table 8.3.1.1)

For an interior flat plate panel, the minimum thickness from ACI 318-19 Table 8.3.1.1 is:

h_min = ln × (0.8 + fy/200,000) / 33

Converting fy to psi: 420 MPa × 145.038 = 60,916 psi
ln in inches: 6.0 m / 0.0254 = 236.2 in (using clear span)

h_min = 236.2 × (0.8 + 60,916/200,000) / 33
      = 236.2 × (0.8 + 0.305) / 33
      = 236.2 × 1.105 / 33
      = 261.0 / 33
      = 7.91 in  →  round up to 8 in (203 mm)

Minimum slab thickness = 203 mm. Use h = 210 mm (round to nearest 10 mm construction increment).

Step 2 — Self-Weight and Factored Load

Self-weight: 0.210 m × 24 kN/m³ = 5.04 kN/m²
Superimposed dead: 7.0 kN/m²
Total dead load wD = 5.04 + 7.0 = 12.04 kN/m²
Live load wL = 3.0 kN/m²

Factored load per ACI 318-19 §5.3.1 (Comb. 1.2D + 1.6L):

wu = 1.2 × 12.04 + 1.6 × 3.0
   = 14.45 + 4.80
   = 19.25 kN/m²

Step 3 — Total Static Moment M₀ (§8.10.3.2)

The total static moment for one panel strip:

M₀ = wu × l₂ × ln² / 8

Where:

l₂ = span in the perpendicular direction = 6.0 m (centre-to-centre for interior panels = clear span + column width ≈ 6.0 m assuming 300 mm square columns)
ln = clear span in design direction = 6.0 m

M₀ = 19.25 × 6.0 × 6.0² / 8
   = 19.25 × 6.0 × 36.0 / 8
   = 4,158 / 8
   = 519.8 kN·m

Total static moment M₀ = 519.8 kN·m per panel strip.

Step 4 — Distribute M₀ to Column and Middle Strips (§8.10.4–8.10.5)

For an interior span of a flat plate (no beams, αf = 0):

Strip	Negative moment	Positive moment
Column strip	0.75 × 0.65 M₀	0.60 × 0.35 M₀
Middle strip	0.25 × 0.65 M₀	0.40 × 0.35 M₀

The distribution at interior supports (§8.10.4.2): 65% negative, 35% positive.

Negative moments:

Column strip: 0.75 × (0.65 × 519.8) = 0.75 × 337.9 = 253.4 kN·m
Middle strip: 0.25 × 337.9 = 84.5 kN·m

Positive moments:

Column strip: 0.60 × (0.35 × 519.8) = 0.60 × 181.9 = 109.2 kN·m
Middle strip: 0.40 × 181.9 = 72.7 kN·m

Step 5 — Effective Depth and Reinforcement

Assume #13 bars (13 mm dia), clear cover = 20 mm:

d = h - cover - bar_dia/2
  = 210 - 20 - 13/2
  = 210 - 20 - 6.5
  = 183.5 mm  ≈ use d = 183 mm

Column strip width = l₂/2 = 6.0/2 = 3.0 m each side, total column strip width = 3.0 m.
Middle strip width = 6.0 - 3.0 = 3.0 m.

Column strip negative moment — As calculation:

Using the approximate method (low ρ, a ≈ As·fy / 0.85·f’c·b):

Mu_neg,col = 253.4 kN·m per 3.0 m strip

ρ_required = [0.85 f'c/fy] × [1 - √(1 - 2Mu/(φ·0.85·f'c·b·d²))]

b = 3,000 mm, d = 183 mm, f'c = 25 MPa, fy = 420 MPa, φ = 0.90

Rn = Mu / (φ·b·d²)
   = (253.4 × 10⁶) / (0.90 × 3,000 × 183²)
   = 253.4 × 10⁶ / (0.90 × 3,000 × 33,489)
   = 253.4 × 10⁶ / 90,420,300
   = 2.80 MPa

ρ = (0.85 × 25/420) × [1 - √(1 - 2×2.80/(0.85×25))]
  = 0.05060 × [1 - √(1 - 0.2635)]
  = 0.05060 × [1 - √0.7365]
  = 0.05060 × [1 - 0.8582]
  = 0.05060 × 0.1418
  = 0.00717

As = ρ × b × d = 0.00717 × 3,000 × 183 = 3,939 mm²

Minimum As (§8.6.1.1): ρ_min = 0.0018 (for fy = 420 MPa, Grade 60), As_min = 0.0018 × 3,000 × 210 = 1,134 mm² — flexural demand governs.

Bar selection: Use #13 @ 150 mm → As = (3,000/150) × 132 = 2,640 mm²/strip.

Recalculate with #13: As per m = 132 mm²/bar. Bars needed = 3,939/132 = 29.8 → 30 bars over 3.0 m = 1 bar per 100 mm.
→ #13 @ 100 mm o.c. for column strip negative steel (As = 3,960 mm²/strip).

Summary Table

Location	Mu (kN·m)	As_req (mm²/strip)	Bar selection
Col. strip negative	253.4	3,939	#13 @ 100 mm
Middle strip negative	84.5	1,177	#13 @ 250 mm
Col. strip positive	109.2	1,476	#13 @ 200 mm
Middle strip positive	72.7	972	#13 @ 250 mm

All bars are Grade 420 (fy = 420 MPa), slab h = 210 mm.

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