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Fig. 6 34.
W = 1.8643
Of the Strength of Timbers in an Inclined Position, 1622. If we suppose the vertical piece AB (fig. 613a.) to become inclined to the base. as BD, the action of a vertical force at D will tend to cause the piece to bend, and thereby to lose much of its power of resistance to a force acting in the direction of its length. Suppose the radius AB or BD to represent the vertical force acting at D on the piece BD then, by the “ resolution of forces," it can be resolved into two forces, one acting in the direction of its length, and the other acting at right angles to its length. The former will be repre. sented by the line Df, or the vertical force multiplied by the sine of the angle DBC; while the latter, at right ang'es to BD, will act at D, tending to bend the piece BD about its base B, and will be represented by the line Bf, or the vertical force inultiplied by the cosine of the angle DBC. The piece is supposed to be fixed firmly at B, and may be considered as a
s beam fixed at one end and strained at the other by the force represented by Bf tending to break it about the end B. When the piece comes into the horizontal position, as BC, the vertical force acting at C will cease to produce any strain in the direction of its length, and the transverse strain will be represented by the line BC or AB acting at C, and straining the piece about its fixed end B.
1623. Example. Let AB be a piece of Riga fir 20 feet long, the scantling being 10 inches by 6 inches. First take it in the upright position A B, then (froen par. 16316) the breaking weight W in tons of an oak pillar when the load acts vertically down AB is
18 x 63
= 11.24 tons.
146 + 200 And the strength of a pillar of Riga fir being (par. 1691c.) five-sixths that of oak, we hare W =9.37 tons for Riga fir.
1624. If we now place the piece in the horizontal position, as BC, the strain upon it from the load W, at C will be entirely transverse, and the breaking weight can be found from the formula (1629g.) for a beam supported at both ends and loaded at the middle. But as the load is at one end in the present case and the beam fixed at the other, we take one fourth of the weight given in the above. named formula, so that we have
Ā * T = 24•08 cwts. = 1.204 tons. 1625. Now let the piece be placed in the inclined position, BD, making 60° with the horizontal. A load F, acting vertically at D, will be resolved into two others at right angles to each other, one acting longitudinally and equal to F sin 60°, and the other transversely at D and equal to F x cos 60°, or 1F, which equals w, in the formula above for a beam fixed at one end and loaded at the other. Therefore the breaking weight applied vertically at D will be F=2W, = 2.408 tons.
1626. The breaking weight acting longitudinally is Fx sin 60° = W =9:37 tons, as found above. Therefore we have
= 10.82 tons.
sin 60° 806 1627. This value of F is, however, more than four times the value obtained for it when taking into consideration the transverse strain as found above, and we must take the smaller amount as the actual strength of the piece, which is therefore reduced in the proportion of 2:408 to 9.37, in consequence of its inclination from the vertical.
1628. In Practical or Constructive CARPENTRY, Chap. III. Sect. IV., Tables of scantlings for timbers are given more immediately useful to the practical architect. But in consequence of the very large amount of information obtained since the first edition of this wurk, resulting from the investigations of scientific and practical experimentalists, the following condensed summary of the new ideas on the strength of BEAMS, GIRDERS, and PILLARS, both in timber and iron, are submitted for the consideration of the student.
1628a. The term beam is applied herein to large rectangular sections; that of girder to large irregular shapes : and those of bar and irons to small rectangular and irregular forms,
16286. Beams and girders are calculated for the following classes of buildings: I. Light workshops and factories, public balls, churches, and other buildings in which
people only accumulate, with warehouses for light goods. For all these an allowance of 12 cwt., or 168 lbs., per square foot of door surface will include the weight
of the joisting, the flooring, and the load upon it. II. Storehouses for heavy goods, or factories in which heavy machinery and goods are
placed. For these an allowance of 2 cwt., or 280 lbs , per square foot of Hoor sur
face will include the same weights. III. Ordinary dwelling houses. For these an allowance of 11 cwt., or 140 lbs., per square
foot will include the same weights.
which the value was derived, since this constant expresses tue specific strength of that form of section.
1628h. Another formula for estimating the strength of beams rests on the knowledge of the resistance (or r) offered by any material to fracture by a tensile or crushing force, and the depth of the neutral axis (or n) of this area in the beam; the latter, of course,
r x breadth x depth? cannot be calculated, except from experiment. The rule is
n x length
= breaking Feight. See RESISTANCE, in Glossary.
1628i. TABLE OF THE TRANSVERSE STRENGTH OF TIMBER: 1 Inch Square, 1 Foot Long.