_M = Pl_; hence _M₁ = q₁Pl_ and _M₂ = q₂Pl_. (59) _W₁ E₁I₁_ _q₁_ = --- = ----------- (60) _W E₁I₁ + E₂I₂_ _W₂ E₂I₂_ _q₂_ = --- = ------------ (61) _W E₁I₁ + E₂I₂_ _p Md_ _k₁_ = ---------- + --------- (62) _A₁ + eA₂ I₁ + eI₂_ ( _P Md_ ) _k₂_ = _e_(---------- + ------------) (63) ( _A₁ + eA₂ I₁ + eI₂_ ) These formulæ exhibit some of the main features of the analysis which must be used in designing either beams or arches of combined steel and concrete. In the use of these equations care must be taken to give the proper sign to the bending moment _M_. They obviously apply to the combination of any two materials, although at the present time the only two used in such composite structures are steel and concrete. If the subscript 1 belongs to the concrete portion, and the subscript 2 to the steel portion, there may be taken _E₁_ = 1,500,000 to 3,000,000 and _E₂_ = 30,000,000. Hence _e_ = 20 to 10. The purpose of introducing the steel into the concrete is to make available in the composite structure the high tensile resistance of that metal. A very small steel cross-section is sufficient to satisfactorily accomplish that purpose. The percentage of the total composite section represented by the steel will vary somewhat with the dimensions of the structure and the mode of using the material; it will usually range from 0.75 per cent to 1.5 per cent of the total section. The large mass of concrete in which the steel should be completely imbedded serves not only to afford a large portion of the compressive resistance required in both arches and beams, but also to preserve the steel effectively from corrosion. Many experiments have shown that it requires but a small per cent of steel section to give great tensile resistance to the composite mass. CHAPTER XII.