When a piece of timber is supported only at the ends, and a weight or power is applied at the centre, it will, if the force is sufficient, bend or sag. If the power of resistance be great, the wood is said to be stiff; but, if it bends easily, it is said to be flexible. Should it bend much, without fracture, it is called tough.

If a beam, two feet long and an inch square, will support, at its centre, five hundred pounds, one of the same length, two inches wide and an inch deep, will support a thousand pounds. Hence we have as a rule, that beams of the same depth are to each other as their thickness. Should the beam described be turned upon its side, so as to make it an inch thick and two inches deep, it will support two thousand pounds. "We therefore have as a second rule, that beams of equal thickness are to each other as the square of their depth.

If a beam, an inch square and two feet long, will support, at its centre, five hundred pounds, one four feet long will support but two hundred and fifty pounds. A third rule, therefore, is, that beams are to each other inversely as their length.*

If a beam, sixteen feet long, supported at the ends, will support, at its centre, a weight of eight hundred pounds, it will support equally well twice that amount if eight hundred pounds be placed at points, each four feet from either side of the centre, - half-way between the centre and the points of support. Again: it will equally well support twice that amount (or 3,200 pounds), if sixteen hundred pounds be placed at points, each half-way from those last named and the points of support (two feet). A beam, therefore, that will support a thousand pounds at its centre, will support two thousand pounds if the weight be distributed equally over its entire length.

A beam, having but one end fixed in a wall, will sustain only a fourth as much weight, when applied to the end, as will one of the same dimensions with both ends in like manner supported, and the weight placed at the middle. When the weight is equally distributed over the entire length of a beam which has only one end supported, it will sustain twice the amount that would break it if applied to the middle.

* Experiments made by Buffon tend to prove that the strength of a beam does not decrease in exact geometrical progression to its length, but that it will actually bear something more than half the amount which would break one of half its length.

Should three beams be fixed at one end in a wall, and the other end left unsupported, - one of them inclined upwards, one at the same angle downwards, and the third level or at right angles with the wall, - that inclined upwards would sustain the least weight; that inclined downwards, the most; and the horizontal one, a mean between the two. In calculating the strength of an inclined beam, the distance from the end of the beam, at right angles with the wall, should be taken as the actual length of the beam; which length, as a basis, will give the strength of the beam, if, instead of being inclined, it were placed in a horizontal position.

From the foregoing data, it will be seen, that, by the aid of tables and rules, it is easy to determine the strength of inclined as well as horizontal timbers.

The following table exhibits the cross-strength of each of the several kinds of wood; the pieces being dry, an inch square, and twelve inches long between the points of support: -

Problem III. To Determine The Cross-Strength Of A Stick Of Timber

Rule. - Multiply the thickness of the stick in inches by the square of its depth in inches, and divide the product by the length of the piece in feet. With the quotient multiply the sum in the table that is set against the kind of wood; and the product will be the breaking-weight in pounds.

Example. - What weight will a spruce-beam, 18 feet long, 6 inches thick, and 8 inches deep, sustain ?

Length. Breaking-weight.

8 Depth. 18)384(21.3 590

8 36 21.3

___ ___ ___

64 Square. 24 177.0

6 Thickness. 18 590

____ ----- 1180

384 60 _______

54 12,567.0 lbs.

Problem IV. To Determine The Depth Of A Stick Of Timber That Will Sustain A Given Weight, The Thickness And Length Being Given

Rule. - Divide the weight to be sustained by the sum set against the kind of wood in the table. Multiply the quotient by the length of the stick in feet, and divide the product by the thickness of the stick in inches. The square-root of the quotient will be the depth of the stick in inches.

Example. - What depth will be required to a stick of chestnut, 19 feet long and 3 inches thick, that it may sustain 27,251 pounds?

Breaking-weight.

595)27251(45.8 45.8 290.066(5.38

23S8 19 Length. 25

______ _____ ___

3451 4122 103) 400

2975 458 309

_____ ____ ____

4760 Thickness 3) 870.2 1068) 9166

4760 ----------- 8544

290.066

Ans. 5 38/100 inches nearly.

Problem V. To Determine The Thickness Of A Stick Of Timber That Will Sustain A Given Weight, The Length And Depth Being Given

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Rule. - Divide the weight to he sustained by the sum set against the kind of wood in the table. Multiply the quotient by the length of the stick in feet, and divide the product by the square of the depth in inches. The quotient will be the thickness of the beam in inches.

Example. - What should be the thickness of a hemlock-beam, 21 feet long and 12 inches deep, that it may sustain a weight of 19,170 pounds?

Breaking-weight.

426)19170(45 45 12 Depth.

1704 21 Length. 12

___ ___ ____

2130 45 144 Square.

2130 90

___

144) 945 (6.56 864

___

810 720

___

900 864

Ans. 6 56/100 inches nearly.