ts at either end is:[10]

[Footnote 10: Only this form of beam is considered since it is
the simplest. For cantilever and continuous beams, and beams
rigidly fixed at one or both ends, as well as for different
methods of loading, different forms of cross section, etc.,
other formulae are required. See any book on mechanics.]

P' l^{3}
E = -------------
4 D b h^{3}

b = breadth or width of beam, inches.
h = height or depth of beam, inches.
l = span (length between points of supports) of beam, inches.
D = deflection produced by load P', inches.
P' = load at or below elastic limit, pounds.

From this formulae it is evident that for rectangular beams of
the same material, mode of support, and loading, the deflection
is affected as follows:

(1) It is inversely proportional to the width for beams of the
same length and depth. If the width is tripled the deflection is
one-third as great.

(2) It is inversely proportional to the cube of the depth for
beams of the same length and breadth. If the depth is tripled
the deflection is one twenty-seventh as great.

(3) It is directly proportional to the cube of the span for
beams of the same breadth and depth. Tripling the span gives
twenty-seven times the deflection.

The number of pounds which concentrated at the centre will
deflect a rectangular prismatic simple beam one inch may be
found from the preceding formulae by substituting D = 1" and
solving for P'. The formulae then becomes:

4 E b h^{3}
Necessary weight (P') = -------------
l^{3}

In this case the values for E are read from tables prepared from
data obtained by experimentation on the given material.

_Strength of Beams_

The measure of the breaking strength of a beam is expressed in
terms of unit stress by a _modulus of rupture_, which is a
purely hypothetical expression for points beyond the elastic
limit. The formulae used in computing this modulus is as follows:

1.5 P l
R = ---------
b h{^2}

b, h, l = breadth, height, and span, respectively, as in
preceding formulae.
R = modulus of rupture, pounds per square inch.
P = maximum load, pounds.

In calculating the fibre stress at the elastic limit the same
formulae is used except that the load at elastic limit (P_{1}) is
substituted for the maximum load (P).

From this formulae it is evident that fo