g, is:--

WL squared bd cubed x 80 E
---- = bd cubed, whence W = -----------
80 E L squared

[TeX: $\frac{WL^2}{80 E} = {bd^3}$, whence $W = \frac{{bd^3} x 80 E}{L^2}$]

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

-----
/ WL squared WL squared
d = cubed/ -----, and b = ------
\/ 80 Eb 80 Ed cubed

[TeX: $d = \sqrt[3]{\frac{WL^2}{80 EB}}$, and $b = \frac{WL^2}{80 ED^3}$]

in the above formulae,

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.

Dist. of axis
Resistance Resistance from top in
to to frac's of
Material. Extension. Compression. Ratio. the depth.

Wrought Iron, 90 66 90/66 90/156, or 0.58.

Cast Iron, 20 111 20/111 20/131, or 0.15.

Wood, 2 1 2/1 2/3, or 0.66.

Thus we see that the resistance of a beam to a cross strain, as well
as to tension and compression,