397. We shall be able by a little consideration to understand the reason why a bar is stronger edgewise than flatwise. Suppose I try to break a bar across my knee by pulling the ends held one in each hand, what is it that resists the breaking? It is chiefly the tenacity of the fibres on the convex surface of the bar. If the bar be edgewise, these fibres are further away from my knee and therefore resist with a greater moment than when the bar is flatwise: nor is the case different when the bar is supported at each end, and the load placed in the centre; for then the reactions of the supports correspond to the forces with which I pulled the ends of the bar.
398. We can now calculate the strength of any rectangular beam of pine:
Let us suppose it to be 12' long, 5" broad, and 7" deep. This is five times as strong as a beam 1" broad and 7" deep for we may conceive the original beam to consist of 5 of these beams placed side by side (Art 391); the beam 1" broad and 7" deep, is 7 times as strong as a beam 7" broad, 1" deep ([Art. 393]). Hence the original beam must be 35 times as strong as a beam 7" broad, 1" deep; but the beam 7" broad and 1" deep is seven times stronger than a beam the section of which is 1" × 1", hence the original beam is 245 times as strong as a beam 12' long and 1" × 1" in section; of which we can calculate the strength, by [Art. 388], from the proportion—
144" : 40" :: 152 : Answer.
The answer is 42·2 lbs., and thus the breaking load of the original beam is about 10,300 lbs.
399. It will be useful to deduce the general expression for the breaking load of a beam l" span, b" broad, and d" deep, supported freely at the ends and laden in the centre.
Let us suppose a bar l" long, and 1" × 1" in section. The breaking load is found by the proportion—
l : 40 :: 152 : Answer;
and the result obtained is 6080/l. A beam which is d" broad, l" span, and 1" deep, would be just as strong as d of the beams l" × 1" × 1" placed side by side; of which the collective strength would be—
| 6080 | |
| ————— | × d. |
| l |