=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.