of the top and bottom flanges are inversely
proportional to the power of the iron to resist compression and
extension. Mr. Hodgkinson's formula for the dimensions of his girder,
is

26 ad
W = ------
L

[TeX: $W = \frac{26 ad}{L}$]

The factor of safety being 6 for cast iron beams--the formula for the
working load will be,

26 ad
W = ------
6 L

[TeX: $W = \frac{26 ad}{6 L}$]

and, to find area of lower flange, we shall have

6 WL
a = ----
26 d

[TeX: $a = \frac{6 WL}{26 d}$]

The general proportions of his girders are as follows:

Length, 16
Height, 1
Area Top Flange, 1.0
Area Bottom Flange, 6.1

In the above formula for cast iron beams,

W = weight in tons.
a = area in square inches of bottom flange.
d = depth in inches.
h = length in inches.

The web uniting the two flanges must be made solid--as any opening, by
causing irregularity in cooling, would seriously affect the strength
of the beam.

_Example._--Required the dimensions of a Hodgkinson girder--for a span
of 60 feet--with a load of 10 tons in the centre.

6 x 10 x 60 x 12
a = ---------------- = 37 inches nearly.
60 x 12
26 x -------
16

[TeX: $a = \frac{6 \times 10 \times 60 \times 12}{26 \times \frac{60
\times 12}{16}} = 37$]

and the area of the top flange will be,

37
-- = 6.16 inches--
6

[TeX: $\frac{37}{6} = 6.16$]

so that our dimensions will be as follows:

Length, 30 feet.
Depth, 45 inches.
Area Top Flange, 6.16 inches.
Area Bottom Flange, 37 inches.

[Illustration: Pl. 1.]

The thickness of web is usually a little greater at the bottom than
at the top, and varies from 1/14 to 1/24 of the depth of the girder.
The bottom rib is usually made from six to eight times as wide as it
is thick, and the top rib from three to six times as wide as thick, so
that, in the example above given, we could have as dimensions for the
parts

Top Flange, 4 1/4 x 1 1/2 inches nearly.
Bottom Flange, 6 x 2 1/2 inches nearly.
Web, 1 1/2 inches thick.

The simplest bridge, consisting of a single stick, to span openings of
20 feet and under, is calculated according to the formula

------
/ 4WL
d = / ------ --
\/ 5000 b

[TeX: $d = \sqrt{\frac{4 WL}{5000 b}}$]

_Example._--The depth of