Axiomatic Theory of the Reinforced Concrete Beam Undergoing a Pure Bending Moment

Abstract

This work provides an axiomatic foundational theory of the reinforced concrete (RC) beam undergoing a pure bending moment. Measure theory in mathematics is used as the underlying axiomatic theory to describe the RC beam structure. Novel concepts related to RC beams based on measure theory are developed. These concepts include RCS-space, the General Stress Function (GSF), the Moment of Resistance (MoR), and some other derivative concepts. An RCS-space is a measure space representing the cross section of a RC beam. The GSF is an L²-function describing the stresses on the RC beam cross section including the concrete and the reinforcements stresses. As a consequence, the properties of RCS-space can be stated as theorems. The MoR of a RC beam is presented as a Lebesgue integral expression involving the GSF. As a validation, we demonstrate the derivation of the nominal moment equation of the conventional RC beam following from this axiomatic theory. The result indicates that this axiomatic theory serves as the general theory for RC beams in terms of bending moment.