CHAPTER XI.

=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:

_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.