Figure 16
Three common forms of beams. 1. Simple. 2. Cantilever. 3. Continuous.
Stiffness of Beams
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]
| 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.