In the pile of boards the amount of slipping is different at different places between. any two boards, being greatest near the supports and zero midway between them. Also, in any cross section the slippage is least between the upper two and lower two boards, and is greatest between the middle two. These facts indi cate that the shearing unit-stress on horizontal sections of a solid beam is greatest in the neutral surface at the support.
It can be proved that at any place in a beam the shearing unit-stresses on a horizontal and on a vertical. section are equal.
It follows that the horizontal shearing unit-stress is greatest at the neutral axis of the section for which the external shear (V) is a maximum. Wood being very weak in shear along the grain, timber beams sometimes fail under shear, the " rupture " being 87 two horizontal cracks along the neutral surface somewhat as rep resented in Fig. 42. It is therefore necessary, when dealing with timber beams, to give due attention to their strength as determined by the working strength of the material in shear along the grain.
Example. A wooden beam 3 X 10 inches in cross-section rests on end supports and sustains a uniform load of 4,000 pounds Compute the greatest horizontal unit-stress in the beam.
The maximum shear equals one-half the load (see Table B, page 55), or 2,000 pounds. Hence, by equation 7, since A 3 X 10 = 30 square inches, 2,000 2 = = M T pounds per square inch.
This is the average shearing uni -stress on the cross-sections near the supports; and the greatest value equals 2 X 667 = 100 pounds per square inch.
According to the foregoing, this is also the value of the greatest horizontal shearing unit-stress. (If of white pine, for example, the beam would not be regarded as safe, since the ulti mate shearing strength along the grain of selected pine is only about 400 pounds per square inch.) 72. Design of Timber Beams. In any case we may pro ceed as follows:—(1) Determine the dimensions of the cross section of the beam from a consideration of the fibre stresses as explained in Art. N. (2) With dimensions thus determined, com pute the value ofthe greatest shearing unit-stress from the formula, Greatest shearing unit-stress = } V ab, where V denotes the maximum external shear in the beam, and b and a the breadth and depth of the cross-section.
If the value of the greatest shearing unit-strets so computed does not exceed the Working strength in shear along the grain, then the dimensions are large enough; but if it exceeds that value, then a or b, or both, should be increased until v ab is less than the working strength. Because timber beams are very often "season checked" (cracked) along the neutral surface, it is advis 88 able to take the working strength of wooden beams, in shear along the grain, quite low. One-twentieth of the working fibre strength has been recommended* for all pine bearns.
If the working strength in shear is taken equal to one twentieth the working fibre strength, then it can be shown that, 1. For a beam on end supports loaded in the middle, the safe load de pends on the shearing or fibre strength according as the ratio of length to depth (1 ÷ a) is less or greater than 10.
2. Foi• a beam on end supports uniformly loaded, the safe load depends on the shearing or fibre strength according as / a is less or greater than 20.
Examples. 1. It is required to design a timber beam to sus , tain loads as represented in Fig. 11, the working. fibre strength being 550 pounds and the working shearing strength 50 pounds per square inch.
The maximum bending moment (see example for practice 3, Art. 43; and example for practice 2, Art. 44) equals practically 7,000 foot-pounds or, 7,000 X 12 = 84,000 inch-pounds.
Hence, according to equation 6"', I84,000 c — =- 550 = 152.7 inches'.
Since for a rectangle • I 1 — = b a 6 ' 1 ' — = 152.7, or bat = 916.2.
6 • Now, if we let b = 4, then a' = 229; or, a = 15.1 (practically 16) inches.
If, again, we let b = 6, then = 152.7; or a = 12.4 (practically 13) inches.
Either of .these sizes will answer so far as fibre stress is concerned, but there is more " timber " in the second.
The external shear in the beam equals 1,556 pounds, neglecting the weight of the beam (see example 3, Art. 37; and example 2, Art. 38). Therefore, for a 4 X 16-inch beam, 3 1,556 Greatest shearing unit-stress = 2 X 4 x 16 = 36.5 pounds per square inch; and for a 6 x 11-inch beam, it equals 3 1,556 27.7 pounds per square inch.