Reinforced Concrete Beam Flexure (ACI 318)
Design moment capacity φM_n for a singly-reinforced rectangular concrete beam using the Whitney equivalent rectangular stress block. Reinforcement ratio checks (ρ_min and ρ_max from tension-controlled limit), strain check, and exact moment.
Defaults: 12 × 20-in effective beam, 4-ksi concrete, Gr. 60 rebar, A_s = 2.0 in² ≈ 4-#6. ε_t output is the strain in extreme tension steel at nominal-strength condition.
Tension-controlled vs compression-controlled
ACI 318 classifies sections by the strain εt in extreme tension steel at nominal capacity:
- Tension-controlled (εt ≥ 0.005): φ = 0.90. Section yields steel before crushing concrete — ductile failure with warning. Standard design target.
- Compression-controlled (εt ≤ εy ≈ 0.002 for Gr. 60): φ = 0.65. Concrete crushes before steel yields — brittle. ACI requires φ = 0.65 (or 0.75 for spiral columns).
- Transition zone (0.002 < εt < 0.005): φ interpolates linearly from 0.65 to 0.90.
Beams should always be tension-controlled. ρmax at εt = 0.005 is the practical upper bound for primary reinforcement; many engineers cap at ρ = 0.5ρmax for ductility margin.
Minimum reinforcement (ACI 9.6.1.2)
To prevent sudden cracking failure when concrete tensile strength is exceeded, ACI requires As,min = max{3√f'c bw d / fy, 200 bw d / fy}. The ρmin for f'c = 4000 psi, fy = 60 ksi is about 0.0033. For T-beams with the flange in tension, the ratio is computed differently (use the web width).
The Whitney stress block — why it works
Concrete in compression has a parabolic stress-strain curve up to crushing strain (ε ≈ 0.003). Integrating the parabolic stress over the compression zone gives the actual compressive force. The Whitney rectangular block (uniform 0.85 f'c over depth a = β1 c) gives the same force AND moment for the typical strain profile, which is why ACI uses it for code design. It greatly simplifies hand calcs without losing accuracy.
Doubly-reinforced beams
If As exceeds ρmax, add compression steel A's. The compression steel takes the moment beyond the singly-reinforced limit. Compression steel must be tied to prevent buckling. For typical building beams, doubly-reinforced is unusual — increase d or b first.
What this calculator doesn't check
- Shear (ACI 22.5 — Vc + Vs)
- Crack width / serviceability (ACI 24.3)
- Deflection (ACI 24.2; effective Ie per Branson formula)
- Anchorage / development length (ACI 25)
- Bar spacing (clear spacing ≥ db, 1 in, or 4/3 dagg)
- T-beam flange contribution (ACI 6.3.2)
Reference: ACI 318-19, Building Code Requirements for Structural Concrete. MacGregor, J.G., Wight, J.K. (2012). Reinforced Concrete: Mechanics and Design, 6th ed., Pearson, ch. 4.