Rebar Area Concrete Beam

Reinforcement steel area in concrete beam

The longitudinal tensile steel area comes from A_s = M_d / (f_yd · z), where the internal lever arm is taken as z ≈ 0.85·d and d is the effective depth, measured from the compressed fibre down to the centroid of the steel. With CA-50 rebar, f_yk = 500 MPa and f_yd = 500/1.15 = 434.8 MPa. Take a rectangular beam of 20×40 cm with d = 37 cm and M_d = 80 kN·m: z = 0.85·37 = 31.45 cm, so A_s = 80·10² / (43.48·31.45) ≈ 5.85 cm², which works out to roughly 3 ⌀ 16 mm bars.

Always satisfy the minimum reinforcement ratio ρ_min = 0.15% · A_c that NBR 6118 requires (for fck ≤ 30 MPa); without it the section can fail brittlely the moment it cracks. And verify both states: ULS, the ultimate limit state that governs flexural capacity, and SLS, the serviceability side that covers crack width and deflection.

Applications

Sizing the longitudinal tensile reinforcement in reinforced-concrete beams, checking minimum and maximum steel ratios against NBR 6118 §17, pre-designing slabs and ribbed slabs, estimating bar count and gauge quickly for a budget, and sanity-checking what structural software spits out (TQS, Eberick, CYPECAD).

FAQ

Why z ≈ 0.85·d? The lever arm tracks the neutral-axis depth x. In under-reinforced sections (domain 2/3) z lands somewhere between 0.80·d and 0.90·d, so 0.85 is a safe middle value to lean on for preliminary sizing.

What is M_d? It is the design moment, M_d = γ_f · M_k, where γ_f = 1.4 for permanent and accidental loads under NBR 6118. Feed M_d into this formula, not the characteristic M_k.

Does this cover shear? No. It only sizes the longitudinal flexural steel. Stirrups, the transverse reinforcement, go through their own check using V_Rd2 and V_Rd3 in NBR 6118 §17.4.