Reinforced Concrete Section Bending Calculator

How is the moment resistance of a concrete section found?

In a reinforced concrete beam in bending the top fibres are in compression and the bottom fibres in tension. The tensile strength of concrete is ignored: the steel carries the tension and the concrete the compression. For the ultimate limit state the curved compressive stress distribution is replaced by an equivalent rectangular stress block.

EN 1992-1-1 uses a stress of η × f_cd over a depth of λ × x, with λ = 0.8 and η = 1.0 for concrete up to C50/60. Force equilibrium gives the block depth a = A_s × f_yd / (η × f_cd × b) and moment equilibrium gives M_Rd = A_s × f_yd × (d − a / 2). The Turkish code TS 500 follows the same logic with a stress of 0.85 × f_cd and a depth of k1 × c.

Required reinforcement

In design the moment is known and the steel is wanted. The block depth is a = d − √(d² − 2 × M_Ed / (η × f_cd × b)) and the steel area is A_s = M_Ed / (f_yd × (d − a / 2)).

Example

Take a section 300 mm wide with an effective depth of 520 mm, C25/30 concrete, B500 steel and 1470 mm² of tension steel, with α_cc = 0.85. Then f_cd = 14.17 MPa and f_yd = 434.8 MPa. The block depth is 1470 × 434.8 / (14.17 × 300) = 150 mm and the moment resistance is 1470 × 434.8 × (520 − 75) = 284 kN·m.

Why balanced reinforcement matters

With a moderate amount of steel, the steel yields first, the beam cracks and deflects, and failure gives warning. With too much steel the concrete crushes before the steel yields and failure is sudden. The boundary between the two is balanced reinforcement. The calculator reports the ratio to balanced and the minimum reinforcement ratio, and it flags an over-reinforced section.

Limits

The calculation covers rectangular, singly reinforced sections in pure bending only. Flanged beams, compression steel, axial load, shear and torsion need separate calculations. Seismic ductility rules, cover, bar spacing and anchorage length are also outside this tool. Use the result for preliminary sizing and for checking hand calculations.