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now, having the weight given, and assuming the dimensions of the cross-section—we shall have

and

in the above formulæ,

W = weight in pounds.
L = length in feet.
E = a constant.
b = breadth in inches.
d = depth in inches.
 

Transverse Strains. The strain caused by any weight, applied transversely, to a beam supported at both ends, is directly as the breadth, and square of the depth, and inversely as the length. It causes the beam to be depressed towards the middle of its length, forming a curve, concave to the horizontal and below it. In assuming this form—the fibres of the upper part of the beam are compressed, and those of the lower part are extended—consequently there must be some line situated between the upper and lower surfaces of the beam where the fibers are subjected to neither of these two forces, this line is called the neutral axis.

These two strains of compression and extension must be equal in amount—and upon the relative strength of the material to resist these strains, as well as its form and position, the situation of this axis depends. If wood resists a compression of 1000 lbs. per square inch of section, and a tension of 2000 lbs. the axis will be twice as far from the top as from the bottom in a rectangular beam.

The following table by Mr. G.L. Vose gives, with sufficient accuracy for practice, the relative resisting powers of wood, wrought, and cast iron, with the corresponding positions of the axis.

Thus we see that the resistance of a beam to a cross strain, as well as to tension and compression, is affected by the incompressibility and inextensibility of the material.

The formula for the dimensions of any beam to support a strain transversely is

S = the ultimate strength in lbs.
b = the breadth in inches.
d = the depth in inches.
l = the length in inches.

Detrusion. Detrusion is the crushing against some fixed point, such as obtains where a brace abuts against a chord, or where a bridge rests on a bolster; and the shearing of pins, bolts and rivets, also comes under this head.
 

General Abstract. The resistance to the above mentioned strains varies as the the area of the cross section; so that by doubling the area we double the strength. Any material will bear a much greater strain for a short time than for a long one. The working strength of materials, or the weight which does not injure them enough to render them unsafe, is a mooted point, and varies, according to the authority, from 1-3 to 1-10 of the ultimate strength. The ratio of the ultimate strength to the working strength is called the factor of safety.

The following is a table of ultimate and working strengths of materials, and factors of safety:

Lateral Adhesion. Lateral adhesion is the resistance offered by the fibres to sliding past each other in the direction of the grain, as when a brace is notched into a chord, or tie beam, at its foot, it is prevented by the lateral adhesion of the fibres from crowding off the piece, to the depth of the notch, against which it toes. Barlow's experiments give the lateral adhesion of fir as 600 lbs. per square inch, and the factor of safety employed varies in practice from 4 to 6, giving a working strength of from 150 to 100 lbs. per square inch.

TABLE OF COMPRESSIVE RESISTANCE OF TIMBER.