CROSS STRAIN.
145. The amount of strain caused by any weight applied in a transverse direction, to a beam supported at both ends, is as the breadth, as the length inversely, and as the square of the depth. Whatever depression takes place, tends to shorten the upper, and to extend the under-side; whence the fibres of the top part suffer compression, and those of the bottom extension. The amounts of compression and extension must of course be equal, and therefore if any material resists these two strains in a different degree, the number of fibres opposing each will also be different.
The top being compressed, while the bottom is extended, of course at some point within the beam there exists a line which suffers neither compression nor extension. The position of this line (the neutral axis) depends upon the relative power of the material to oppose the strains, upon its form and upon its position. Thus if wood resists two thousand pounds per square inch of extension, and one thousand pounds of compression, the axis will be twice as far from the top as from the bottom.
In some materials the neutral axis changes its place while the bar is at work; thus wrought iron, after being a little compressed, will bear a great deal more compression than when in its original state; also the lower fibres, after being extended, will resist less than at first; the effect of which two actions is to move the neutral axis up.
146. The following table shows the relative resisting powers of wood, wrought and cast-iron; with the corresponding positions of the axis, with sufficient accuracy for practice.
| Material. | Resistance to extension. | Resistance to compression. | Ratio. | Distance of axis from top, in fractions of the depth. | |
|---|---|---|---|---|---|
| Wrought iron, | 90 | 66 | 90 66 | 90 156 | or 0.58 |
| Cast-iron, | 20 | 111 | 20 111 | 20 131 | or 0.15 |
| Wood, | 2 | 1 | 2 1 | ⅔ | or 0.66 |
Thus in beams subjected to a cross strain, as well as to a direct extensile or compressive one, the resistance is effected by the incompressibility and inextensibility of the material.
147. The formula for dimensioning any beam to support a given weight transversely is
S = 4bd2
e,
Where S represents the ultimate strength in lbs.
b represents the breadth in inches,
d represents the depth in inches,
e represents the length in inches,