The theory of flexure in reinforced concrete is exceptionally complicated. A multitude of simple rules, formulae, and tables for designing reinforced-concrete work have been proposed, some of which are sufficiently accurate and applicable under certain conditions. But the effect of these various conditions should be thoroughly understood. Reinforced concrete should not be designed by "rule-of-thumb" engineers. It is hardly too strong a statement to say that a man is criminally careless and negligent when he attempts to design a structure on which the safety and lives of people will depend, without thoroughly understanding the theory on which any formula he may use is based. The applicability of all formulae is so dependent on the quality of the steel and of the concrete, and on many of the details of the design, that a blind application of a formula is very unsafe. Although the greatest pains will be taken to make the following demonstration as clear and plain as possible, it will be necessary to employ symbols, and to work out several algebraic formula? on which the rules for designing will be based. The full significance of many of the terms mentioned below may not be fully understood until several subsequent paragraphs have been studied: b = Breadth of concrete beam; d = Depth from compression face to center of gravity of the steel; A = Area of the steel; p = Ratio of area of steel to area of concrete above the center of gravity of the steel, generally referred to as percentage of reinforcement, A/bd

Es = Modulus of elasticity of steel;

Ec= Initial modulus of elasticity of concrete; r = Es/Ec= Ratio of the moduli; s = Tensile stress per unit of area in steel; c = Compressive stress per unit of area in concrete at the outer fibre of the beam; this may vary from zero to c'; c' = Ultimate compressive stress per unit of area in concrete - the stress at which failure might be expected;

Es = Deformation per unit of length in the steel;

Ec = " " " " " in outer fibre of concrete;

Ec'= " " " " " in outer fibre of concrete when crushing is imminent;

Ec" = Deformation per unit of length in outer fibre of concrete under a certain condition (described later); q =Ec/Ec" = Ratio of deformations; k = Ratio of depth from compressive face to the neutral axis to the total effective depth d; x = Distance from compressive face to center of gravity of compressive stresses; ∑ X = Summation of horizontal compressive stresses; M = Resisting moment of a section.