OpenSpecEngineering

Finds the tension steel a singly reinforced rectangular concrete beam needs to carry a factored bending moment, by ACI CODE-318-25 [1]. It also gives the code minimum steel and the most steel the section can take before it stops being tension-controlled. Designers use it to size the bottom bars of a beam once its width and depth are set.

The flexure equations

R_n = M_u / (φ·b·d²)
ρ = (0.85f′_c/f_y)·[1 − √(1 − 2R_n/(0.85f′_c))]
A_s = ρ·b·d

M_u is the factored moment, b the beam width, d the effective depth (top of the beam to the center of the tension bars), f′_c the concrete strength and f_y the steel yield strength. φ = 0.90 is the strength reduction factor for a tension-controlled section [1]. These come from the equivalent rectangular stress block: 0.85f′_c over a depth a = β₁c, with a concrete crushing strain of 0.003 [1], [2]. β₁ is 0.85 up to 4000 psi and falls to 0.65 at 8000 psi.

ρ_tc = 0.85β₁(f′_c/f_y)·0.003/(0.006 + f_y/E_s)
A_s,min = max(3√f′_c, 200)·b_w·d/f_y

ρ_tc is the steel ratio at which the tension strain reaches ε_ty + 0.003, the edge of the tension-controlled zone, with E_s = 29,000,000 psi. A_s,min is in psi units, with f_y taken at most 80,000 psi [1]. The governing area is the larger of the required and minimum areas. Where 4/3 of the required area is smaller than A_s,min, the page shows it too, because the Code accepts that amount instead.

Limits and common mistakes

• d is the effective depth, not the overall depth. Subtract the cover, the stirrup and half a bar diameter.
• The page assumes the steel yields and φ = 0.90. If ρ exceeds ρ_tc, it warns you: make the beam deeper or wider, use stronger concrete, or add compression steel, which this page does not design.
• It covers rectangular sections only. For a T-beam whose stress block stays in the flange, enter the flange width as b.
• The rebar grades use nominal metric yield strengths (Grade 60 = 420 MPa) [3], [4].
• The result is an area. Choose bars that give at least that area and fit the width with the required spacing.

Worked example

A 300 mm wide beam with a 500 mm effective depth carries a factored moment of 250 kN·m. It is cast in 4000 psi concrete with Grade 60 bars. Find the tension steel.

1. Enter Factored Moment (M_u) 250000 N·m.
2. Enter Beam Width (b) 300 mm and Effective Depth (d) 500 mm.
3. Choose Concrete Strength 4000 psi (27.6 MPa).
4. Choose Rebar Grade ASTM A615 Grade 60 — fy = 420 MPa.
5. Press Solve. The Solution shows: β1: 0.85
   Coefficient of Resistance (R_n): 3.7037 MPa
   Required Reinforcement Ratio (ρ): 0.00965309
   Tension-controlled limit ρ_tc: 0.0175701
   Required As: 1,447.96 mm²
   Minimum A_s,min: 492.483 mm²
   Governing As = max(As, As,min): 1,447.96 mm²

Check: the stress block depth is a = 1,447.96 × 420 / (0.85 × 27.579 × 300) = 86.47 mm. Then φM_n = 0.9 × 1,447.96 × 420 × (500 − 86.47/2) = 250.0 kN·m, the moment entered. Three 25M bars (1,500 mm²) would do.