For a cast iron beam or girder--Mr. Hodgkinson found from numerous
carefully conducted experiments that, by arranging the material in the
form of an inverted T--thus creating a small top flange as well as the
larger bottom one, the resistance was increased, per unit of section,
over that of a rectangular beam, in the ratio of 40 to 23.

In this beam the areas 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 = ----