102. To find the scantling of a piece of timber that will sustain a given weight when supported at the ends in a horizontal position.

Case I. - When The Breadth Is Given

Rule V. - Multiply the square of the length in feet by the weight in pounds, and this product by the value of a opposite the kind of wood in the preceding Tables. (Tables VI, VII., etc.) Divide the product by the breadth in inches, and the cube root of the quotient will be the depth in inches.

Example. - A beam of Norway fir is required for a 24 feet bearing to support 900 pounds, and the breadth, to be 6 inches; what should be the depth? Here and the cube root of 827 is 9.38, the depth required in inches.

24 x 24 x 900 x .00957/6 = 827.

Case 2. - When The Depth Is Given

103. Rule VI. - Multiply the square of the length in feet by the weight in pounds, and multiply this product by the value of a opposite the name of the kind of wood in Tables VI., VII., etc. Divide the last product by the cube of the depth in inches, and the quotient will be the required breadth in inches.

Example. - The space for a beam of oak does not allow it to be deeper than 12 inches; to find the breadth, so that it may support a weight of 4000 pounds, the bearing being 16 feet.

Here 16x16 x 4000 x .0164/12x12x1 = 9 3/4 inches nearly, the breadth required.

104. But, generally, neither the breadth nor depth is given: in this case it will be best to fix on some proportion which the breadth should have to the depth; for instance, suppose it be convenient to make the breadth to the depth as 0 .6 is to 1, then the rule would become as follows:

Rule VII. - Multiply the weight in pounds by the value of a opposite the kind of wood in the foregoing Tables (Tables VI., VII., etc.); divide the product by 0.6, and extract the square root. Multiply this root by the length in feet, and extract the square root a second time, which will be the depth in inches required. The breadth is equal to the depth multiplied by the decimal 0.6. It is obvious that any other proportion of the breadth and depth may bo obtained by merely changing the decimal 0 . 6 in the rule.

Example. - A beam of Riga fir is intended to bear a ton weight in the middle of its length, the bearing is 22 feet; what should be the dimensions of the beam? A ton is

2240 lbs. Here 2240x.011/.6 = 41.066; the square root of

41 . 066 is 6 . 4 nearly. Therefore 6. 4 x 22 = 140.8; and the square root of 140.8 is 11.86 inches, the depth required. And 11.86 x . 6 = 7 .116 = the breadth.

105. When the beam is inclined the scantling will be found by the following rule:

Rule VIII. - Multiply together the weight in pounds, the cosine of the angle the beam makes with the horizon to a radius of unity, and the constant number a for the kind of wood; divide this product by 0.6, and extract the square root of the quotient. Multiply this root by the length in feet, and extract the square root again, which will give the depth in inches.

106. Otherwise, let AB (Fig. 25, page 27) be the beam, and B C a vertical line; then A C will be the horizontal distance between the points of support.

Rule IX. - Multiply together the weight in pounds, the length of the beam in feet, the horizontal distance between the supports in feet, and the constant number a for the kind of wood; divide this product by 0.6, and the fourth root of the quotient will give the depth in inches. According to either rule, the breadth is assumed to be equal to the depth multiplied by the decimal 0. 6.

Example.- - Let the length of the beam be 20 feet, and the horizontal distance between the points of support 16 feet, and the weight to be supported one ton, or 2240 pounds, by a tarn of Riga fir. Then 2240x20x16x.011/.6 = 13240, the fourth root of 13240 is 10 3/4 nearly, and 10 3/4 x .6 = 6 1/2 nearly; therefore the beam should be 10 3/4 inches by 6 1/2 inches.