To find the dimensions of floor-beams for dwellings, when the rate of deflection is 0.03 inch per foot, or for ordinary stores when the load is about 150 pounds per foot, and the deflection caused by this weight is within the limits of the elasticity of the material, we have the following rule:
Rule XXXVIII. - Multiply the cube of the length by the distance apart between the beams (from centres), both in feet, and multiply the product by the value of j (Art. 152) for the material of the beam, and the product will equal the product of the breadth into the cube of the depth. Now, to find the breadth, divide this product by the cube of the depth in inches, and the quotient will be the breadth in inches. But if the depth is sought, divide the said product by the breadth in inches, and the cube root of the quotient will be the depth in inches; or if the breadth and depth are to be in proportion as r is to unity, r representing any required decimal, then divide the aforesaid product by the value of r, and extract the square root of the quotient, and the square root of this square root will be the depth required in inches, and the depth multiplied by the value of r will be the breadth in inches.
Example. - In a dwelling Or ordinary store, what must be the breadth of the beams, when placed 15 inches from centres, to support a floor covering a span of 16 feet, the depth being 11 inches, the beams of white oak? By the rule, 4096, the cube of the length, by 1 1/4, the distance from centres, and by 0.6, the value of j for white oak, equals 3072. This divided by 1331, the cube of the depth, equals 2.31 inches, or 2 5/16 inches, the required breadth. But if, instead of the breadth, the depth be required, the breadth being fixed at 3 inches, then the product, 3072, as above, divided by 3, the breadth, equals 1024; the cube root of this is 10.08, or, say, 10 inches nearly. But if the breadth and depth are to be in proportion, say, as 0.3 to 1.0, then the aforesaid product, 3072, divided by 0.3, the value of r, equals 10240, the square root of which is 101.2, and the square root of this is 10.06, the required depth. This multiplied by 0.3, the value of r, equals 3.02, the required breadth; the beam is therefore to be, say, 3 x 10 inches.
(54.) - Floor-Beams for First-Class Stores. - To find the breadth and depth of the beams for a floor of a first-class store sufficient to sustain 250 pounds per foot superficial (exclusive of the weight of the material in the floor), with a deflection of 0.04 inch per foot of the length, we have -
Rule XXXIX. - The same as XXXVIII, with the exception that the value of k (Art. 152) is to be used instead of the value of j.
Example. - The beams of the floor of a first-class store are to be of Georgia pine, with a clear bearing between the walls of 18 feet, and placed 14 inches from centres: what must be the breadth when the depth is 11 inches? By the rule, 5832, the cube of the length, and 1 1/6, the distance from centres, and 0.73, the value of k for Georgia pine, all multiplied together equal 4966.92; and this product divided by 1331, the cube of the depth, equals 3.732, the required breadth, or 3 3/4 inches.
But if, instead of the breadth, the depth be required: what must be the depth when the breadth is 3 inches?
The said product, 4966.92, divided by 3, the breadth, equals 1655.64, and the cube root of this, 11.83, or, say, 12 inches, is the depth required.
But if the breadth and depth are to be in a given proportion, say 0.35 to 1.0, the 4966.92 aforesaid divided by 0.35, the value of r, equals 14191, the square root of which is 119.13, and the square root of this square root is 10.91, or, say, 11 inches, the required depth. And 10.91 multiplied by 0.35, the value of r, equals 3.82, the required breadth, say 3 7/8 inches.
155. - Floor - Beams: Distance from Centres. - It is sometimes desirable, when the breadth and depth of the beams are fixed, or when the beams have been sawed and are now ready for use, to know the distance from centres at which such beams should be placed in order that the floor be sufficiently stiff. By a transposition of the factors in equation (44.), we obtain -
In like manner, equation (45.) produces -
These, in words at length, are as follows:
Rule XL. - Multiply the cube of the depth by the breadth, both in inches, and divide the product by the cube of the length in feet multiplied by the value of j, for dwellings and for ordinary stores, or by k, for first-class stores, and the quotient will be the distance apart from centres in feet.
Example. - A span of 17 feet, in a dwelling, is to be covered by white-pine beams 3x12 inches: at what distance apart from centres should they be placed? By the rule, 1728, the cube of the depth, multiplied by 3, the breadth, equals 5184. The cube of 17 is 4913; this by 0.65, the value of j for white pine, equals 3193.45. The aforesaid 5184 divided by this 3193.45 equals 1.6233 feet, or, say, 20 inches.