=123. Beams of Combined Steel and Concrete.=[2]—A reference has already been made to a class of beams and arches recently come into use and now quite widely employed, composed of steel and concrete, the former being completely surrounded by and imbedded in the latter. These composite beams are very extensively used in the floors of fire-proof buildings as well as for other purposes. Arches of combined concrete and steel were probably first built in Germany and but a comparatively few years ago. During the past ten years they have been largely introduced into this country, and many such structures have not only been designed but built. The most prominent design of arches of combined concrete and steel are those of the proposed memorial bridge across the Potomac River at Washington, for which a first prize was awarded as the result of a national competition in the early part of 1900. So far as the bending or flexure of these composite beams and arches is concerned, the theory is identically the same for both, the formulæ for each of which are given below. In order to express these formulæ the following notation will be needed: [2] For a complete and detailed statement of this whole subject, including design work, reference should be made to the author’s “Elasticity and Resistance of Materials.” [Illustration: MEMORIAL BRIDGE ACROSS THE POTOMAC AT WASHINGTON D.C. WM. H. BURR, Civil Engineer. E. P. CASEY, Associated Architect. PLAN NO. 2. Plan Awarded First Prize in National Competition. River spans 192 feet clear. Total length of structure 3615 feet.] [Illustration: WM. H. BURR, Civil Engineer. E. P. CASEY, Associated Architect. PLAN NO. 1. The Towers of this Plan were Recommended by Board of Award to be Substituted for Those in Plan No. 2. River spans 283 feet clear. Total length of structure 3437 feet.] _P_ is the thrust along the arch determined by the methods explained in the consideration of arched ribs. _l_ is the distance of the line of the thrust _P_ from the axis of the arched rib. _E₁_ and _E₂_ are coefficients of elasticity for the two materials. _A₁_ and _A₂_ are areas of normal section of the two materials. _I₁_ and _I₂_ are moments of inertia of _A₁_ and _A₂_ about the neutral axes of the composite beam or arch sections. _k₁_ and _k₂_ are intensities of bending stress in the extreme fibres of the two materials. _h₁_ and _h₂_ are total depths of the two materials. _d₁_ and _d₂_ are distances from the neutral axes to farthest fibres of the two materials; distances to other extreme fibres would be (_h₁-d₁_) and (_h₂-d₂_). _W₁_ and _W₂_ are loads, either distributed or concentrated, carried by the two portions. _W = W₁ + W₂_ is total load on the beam or arch. _q₁ = W₁/W_ and _q₂ = W₂/W_; ∴ _q₁ + q₂_ = 1; _e = E₂/E₁_. The application of the theory of flexure to the case of a beam or arch of two different materials, steel and concrete in this case, will give the following results: