The two main requirements of a beam are stiffness and strength. The formulæ for the modulus of elasticity (E) or measure of stiffness of a rectangular prismatic simple beam loaded at the centre and resting freely on supports 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 formulæ are required. See any book on mechanics.]



P' l3
E=-----------


4 D b h3



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 formulæ 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 formulæ by substituting D = 1" and solving for P'. The formulæ then becomes:



4 E b h3
Necessary weight (P')=----------


l3

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