Reinforced Concrete Nominal Moment Capacity

Concrete's stress distribution in compression is a curve nobody wants to integrate, so Charles Whitney proposed in 1937 that it be replaced by an equivalent rectangle: uniform stress $0.85f'_c$ over a depth $a$, chosen so the block has the same total force and the same centroid as the real curve. Everything after that is a couple. The steel pulls with $T = A_sf_y$, the block pushes back with an equal force, and the moment is the force times the lever arm between them: $M_n = A_sf_y(d - a/2)$. With 2000 mm² of 400 MPa rebar, $d = 500$ mm and $a = 100$ mm, the tension is 800 kN, the lever arm 450 mm, and $M_n = 360$ kN·m.

The block depth is not independent — it follows from horizontal equilibrium, $a = A_sf_y/(0.85f'_cb)$ — which means the concrete strength enters the capacity only through the lever arm, and weakly at that. Double $f'_c$ and $a$ halves while $d - a/2$ barely moves. That is the single most useful thing this formula teaches a beginner: in a normally reinforced beam, flexural capacity is governed by the steel, not the concrete. Specifying 40 MPa concrete instead of 25 buys almost no bending strength. It buys shear capacity, stiffness, durability and column capacity, all of which are real, but not this.

Two conditions carry the result. The formula assumes the steel has actually yielded, which is only true if the section is under-reinforced — an over-reinforced beam crushes its concrete first, fails suddenly with no warning, and this equation overstates it. ACI 318 enforces the good behaviour through a minimum tensile strain and a $\phi$ factor that drops from 0.90 as a section moves toward compression control. And $M_n$ is nominal: the design strength is $\phi M_n$, 324 kN·m in our example, which is what actually goes up against the factored demand. This page gives the general form of the ACI 318 expression so the algebra can be worked and rearranged; the $\beta_1$ factor, the strain limits and the $\phi$ values all belong to the adopted edition of the code.