Load-slip relationship of tension reinforcement in reinforced concrete members

An accurate evaluation of the moment-rotation relationship of reinforced concrete members at both serviceability and ultimate limit states is a very important aspect as rotation has a significant contribution to the deflection of the member and also has a direct impact on the magnified moment, the ability to absorb energy and the redistribution of moments. The rotation in the un-cracked or homogenous parts of a reinforced concrete member can be determined by integrating the curvature using standard procedures. However, in the cracked or non-homogenous regions, rotations are found to have sudden or discrete changes at each crack between their crack faces. This can be quantified by the crack opening produced by the slip Δ between the reinforcement and the concrete at the crack face induced by the force in the reinforcing bar P. In this paper, closed form solutions are derived for the P-Δ relationships which are applicable to any type and shape of reinforcement. It is then shown how these closed form solutions can be conveniently used to derive the moment-rotation relationship at a crack.