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. The concept of ‘measure space’ in measure theory is used to model the whole concrete cross section along with the area measure on the cross section. We call the measure space representing the environment an RCS-space. Then the concrete and the reinforcements stresses are modelled as L1- functions. And by the additive property of L1-functions, these functions are generalised into a single L1-function coined as the general stress function. From this model, the properties of the RCS-space can be derived as theorems. Consequently, the structural moment of resistance can be formulated following from the force couple theory in mechanics as a Lebesgue integral expression of the general stress function. This measure theoretic system allows us to formulate the moment of resistance of a concrete beam with any kind of reinforcement and configuration in a unified formal mathematical theory.