Experiments on the transverse strength of beams are generally made by loading in the middle beams supported at both ends. The following table, from experiments by Mr. Barlow, gives the value of c for beams supported at each end and loaded in the middle: Tons.
Cast-iron Wrought-iron 12 English oak Red pine These numbers when substituted in the formula give the breaking weight, one-third of this will he the safe load in practice. The transverse strength of. cast iron is considered so good a test of its value, that in specifications of iron work, it is generally required to be of such a quality that a bar of it, of certain dimensions, will bear a specified weight at the center; for example, "that a bar of it, 42 in. long, 2 in. deep, and 1 in. wide, set on bearings 86 in. apart, shall bear, without breaking, :h cwt, suspended in the middle." If a beam be loaded uniformly over its length, it will bear twice as much as if the load be condensed at the center. Also if the load be placed some distance from the center, the load it will bear is to the load borne at the center inversely as the rectangle of the segments into which the beam is divided by the point of application of the load are to one another, from which it follows that it will bear less weight at the center than at any other point.
Since the strength of a rectangular beam is proportional to the square of the depth, multiplied by the breadth, it is evident that by increasing the depth and diminishing the breadth we shall, up to a certain limit, increase the strength of a beam without increasing. its weight; for example, let A and B be the sections of two beams, of which A is 2 in. broad and 2 deep, and B 4 in. deep and 1 in. broad, they arc of the same sec tional area—viz., 4 sq.in., but the strength of B is to the strength of A as X 1 is to 2, or as 16 to 8, that is 2 to 1. that is to say, B is twice the strength of A. Hence arises the advantage of the double T forms so generally used in iron girders, the strength of which forms are proportional to the area of time top or bottom plates multiplied by the depth. For a beam of this form loaded at the center, the following formula will give the breaking weight: • ad • W =C Where a= the area of the top or bottom flange in sq. inches.
(„= S 4 times the destroying load per sq.in. of the material, under direct ten C-- sion or compression in tons.
d= depth of the beam in feet.
/= length between supports in feet.
W= breaking weight at the center in tons.
For cast-iron beams, when the area of the bottom flange is made 6 times that of the top, which has been found by experiment to be the best arrangement, and the strength is measured by the tensional strain, supported by the bottom flange, that is, 6 tons per sq. inch.
C=GiX4=26 tons.
For wrought-iron beams, • C=4X20=80 tons for the lower flange, and C=4X16=64 tons for the tamer flange.
Another way of throwing the great body of the material at a distance from the neutral axis is, to make it into the shape of a tube or hollow cylin der. Let B be the section of a hollow cylinder, the thick ness of whose walls is represented by the shaded ring; and A be the section of a solid cylinder of the same material. If the area of A is equal to that of the ring in B, the two cylin ders will contain the same quantity of matter, but B will be stronger than A, nearly in proportion as cg is longer than dg.
The principle of hollow structure prevails both in nature and art, wherever strength and lightness have to be combined. It is seen in the sterns of plants, especially of the grasses; the bones of animals are also hollow, and those of birds, where great lightness is required, are most so. A feather, with its hollow stem, is perhaps the best instance of the union of strength and lightness that could be given. In art, again, we have hollow metal pillars; and sheet-iron for roofing and other purposes is corrugated, or bent into ridges and furrows, to give it depth. Each ridge or furrow is, as it were, half a tube, and resists bending with twice or thrice the energy it would if flat.
The most striking application of the principle of hollow structure is seen in tubular bridges. The object being to resist a vertical strain, the form is made rectangular, and the chief mass of the material is thrown into the top and bottom. The tube may, in fact, be considered as an immense beam or girder constructed on the principle of the double T-iron girder, the top and bottom being the two flanges, and the two sides serv ing to connect them instead of the one rib in the middle. As it is constructed of plate iron, the top requires more metal than the bottom, in order to resist the compression; but instead of putting the metal into one thick plate, or into several plates, laid the one on the other,. it is made to form a set of minor tubes or cells, which give additional stiff ness and strength to the whole tube. The floor, in like manner, contains cells. Each of the tubes over the Conway bridge is 24 ft. high, 14 ft. wide (outside), and 420 ft. long, and weighs 1,300 tons; yet these enormous hollow beams sustain not only their own but the heaviest railway-trains without sensible deflection.