Reinforced Concrete Beam Moment Capacity Calculator (ACI 318)
Find the design flexural strength φMn of a singly reinforced rectangular concrete beam to ACI 318 — stress block depth, neutral axis, steel strain, φ factor, minimum steel and a check against your factored moment.
How it works
As = n × bar area · a = As·fy / (0.85·f′c·b) · c = a/β₁ (β₁ = 0.85 up to 28 MPa, −0.05 per 7 MPa, ≥ 0.65) · εt = 0.003(d − c)/c · Mn = As·fy·(d − a/2) · φ = 0.90 when εt ≥ εty + 0.003, 0.65 when εt ≤ εty, linear between (ACI 318-19 Table 21.2.2) · As,min = max(0.25√f′c/fy, 1.4/fy)·b·d.
ACI 318 replaces the curved concrete stress distribution with an equivalent rectangular (Whitney) block of 0.85f′c over a depth a = β₁c. Force equilibrium (0.85f′c·a·b = As·fy, assuming the steel yields) gives a, and the moment of the two forces about each other gives Mn. The strain in the extreme tension steel decides how ductile the section is and the strength reduction factor φ; ACI 318-19 §9.3.3.1 also requires εt ≥ 0.004 for beams, and §9.6.1.2 a minimum area of steel so the beam does not fail suddenly when the concrete cracks. Metric bar sizes and kg/cm² strengths are listed for Latin American practice, which follows ACI 318 (e.g. NSR-10, E.060, NTC).
Worked example
A 300 mm wide beam with d = 500 mm, four 20 mm bars (As = 4 × 314.16 = 1,257 mm²), f′c = 28 MPa and fy = 420 MPa: a = 1,256.6 × 420 / (0.85 × 28 × 300) = 73.9 mm, c = 73.9/0.85 = 87.0 mm and εt = 0.003 × (500 − 87.0)/87.0 = 0.01425, so the section is tension-controlled and φ = 0.90. Mn = 1,256.6 × 420 × (500 − 37.0) = 244.4 kN·m and φMn = 219.95 kN·m. With Mu = 150 kN·m the beam works at 68.2% of its capacity, and As exceeds As,min = 1.4/420 × 300 × 500 = 500 mm².
Sources: ACI 318-19 Building Code Requirements for Structural Concrete — §22.2 (flexural strength), §21.2 (φ), §9.6.1 (minimum reinforcement) · Reinforced concrete — Whitney stress block
Frequently asked questions
What is the difference between Mn and φMn?
Mn is the nominal (theoretical) moment strength of the section. φMn is the design strength: ACI multiplies Mn by φ (0.90 for ductile, tension-controlled beams, down to 0.65) to cover material and construction variability. Design requires φMn ≥ Mu, the factored moment.
What does tension-controlled mean?
The tension steel strains well past yield (εt ≥ εty + 0.003, about 0.005 for Grade 60) before the concrete crushes at 0.003, so the beam cracks and deflects visibly before failing. Over-reinforced beams fail by sudden concrete crushing, which codes avoid.
How do I get the effective depth d?
Overall depth minus clear cover, stirrup diameter and half the main bar diameter. For a 550 mm beam with 40 mm cover, 10 mm stirrups and 20 mm bars: d = 550 − 40 − 10 − 10 = 490 mm.
Does this work for T-beams or doubly reinforced beams?
No — it is for rectangular sections with tension steel only. For a T-beam whose stress block stays in the flange (a ≤ flange thickness), enter the effective flange width as b and the result still applies.
What is β₁?
The ratio of the stress-block depth to the neutral-axis depth: 0.85 for f′c up to 28 MPa (4,000 psi), reduced by 0.05 for each 7 MPa (1,000 psi) above that, with a minimum of 0.65.
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