What is the formula for cracked moment of inertia?

The formula for the cracked moment of inertia is crucial in structural engineering, particularly in the analysis of reinforced concrete sections under bending. The correct answer reflects a comprehensive understanding of how to calculate the moment of inertia of a cracked section, which consists of two components: the uncracked section and the contribution of the reinforcement.

In the correct formula, the first term, (b(kd)^3)/3, represents the moment of inertia of the concrete part of the section that is considered cracked, where 'b' is the width of the beam, 'kd' is the effective depth of the section. The use of kd instead of the total depth d acknowledges that the effective depth is reduced due to the effects of cracking.

The second term, nAs(d-kd)^2, accounts for the contribution of the reinforcing steel to the overall moment of inertia. Here, 'n' represents the number of bars, 'As' is the area of the reinforcing steel, and (d-kd) denotes the distance from the centroid of the steel to the extreme fiber — capturing its effect on the overall stiffness and the moment of inertia of the cracked section.

This formulation is comprehensive, factoring both the concrete and the reinforcing steel in the cracked state, which is