For a doubly reinforced concrete beam, under what specific condition will the compression steel yield before the concrete reaches its ultimate strain?

For a doubly reinforced concrete beam, the compression steel will yield before the concrete reaches its ultimate strain under the specific condition that the strain in the compression steel (εs') is equal to or greater than the yield strain of the steel (εy) at the very moment the extreme compression fiber of the concrete reaches its ultimate compressive strain (εcu). The ultimate compressive strain of concrete, εcu, is typically taken as 0.003 in design codes like ACI 318, representing the point where the concrete is considered to fail in compression. The yield strain of the steel, εy, is calculated as the steel's yield strength (fy) divided by its modulus of elasticity (Es), for example, fy/Es. This condition arises from the principle of strain compatibility, which assumes that plane sections remain plane and perpendicular to the neutral axis before and after bending. Therefore, the strains in the concrete and steel are linearly proportional to their distance from the neutral axis. The strain in the compression steel, εs', can be expressed as εs' = εcu (c - d') / c, where 'c' is the depth of the neutral axis from the extreme compression fiber and 'd'' is the distance from the extreme compression fiber to the centroid of the compression steel. Consequently, the compression steel yields before the concrete reaches εcu if εcu (c - d') / c ≥ εy. This inequality implies that the depth of the neutral axis (c) must be sufficiently shallow relative to the depth of the compression steel (d'). A shallower neutral axis means that the compression steel is located further from the neutral axis (relative to 'c'), thus experiencing a higher compressive strain that reaches or exceeds its yield strain before the extreme concrete fiber reaches its ultimate capacity.