108. When a solid cylinder is supported at both ends. Let D be the diameter of the cylinder; then

WxL2x constant quantity = D4. [9]

Now it is shown by Dr. Young that the stiffness of a cylinder is to that of its circumscribing rectangular prism, as three times the bulk of the cylinder is to four times the bulk of the prism;* and as the stiffness of the prism is = D4/axL2;; therefore 3x.7854xD4/4axL= W, or 1.7 axL3 x W = D4. [10]

From want of proper experiments this method of obtaining the stiffness of cylinders has been resorted to as the only experiments on cylinders, where the first deflections are given, are those of Duhamel, which were made on very small specimens.†

109. To find the diameter of a solid cylinder, so that it may be capable of supporting a given weight, without deflecting more than 1/40 of an inch for each foot in length.

Rule X. - Multiply the value of a for the kind of wood from the Tables (Arts. 93, 94, &c), by 1.7, and multiply this product by the weight in pounds. Then multiply the square root of the last product by the length in feet, and the square root of the quotient will be the diameter of the cylinder in inches.

Example. - A solid cylinder of elm is intended to support 10 hundredweight (or 1120 pounds), the length of bearing 10 feet; required the diameter? The constant number for elm being .0212, by one of the experiments in Table VIII., Art. 95.

In this case we have 1.7 X .0212 x 1120 = 40.3648, the square root of which is 6.35, therefore 10 x 6.35 = 63.5; and the square root 7.97, or nearly 8 inches, which is the diameter required.

The Stiffness of Beams Supported at Both Ends, and the Weight uniformly distributed over the Length.

110. Where the load is uniformly distributed over the length of the beam, the deflection does not increase in the same proportion as when it acts at one point. For where the weight is uniformly diffused it increases as the length, and the deflection will be as the fourth power of the length; consequently, according to the definition given in Art. 85, the stiffness will be as the cube of the length.

The stiffness of a beam uniformly loaded may be derived from the general proportion B D3 is as L3 W.

This proportion applies to the case of the rafters, and purlins of a roof, to ceiling joists, and binding joists that support ceilings only, but it does not apply to flooring joists, because their stiffness is measured by the resistance offered to a strain at one point; a floor might seem stiff enough to support a uniform load, and yet shake very much by the weight of a single person moving over it.

111. The above method may be applied in cases where beams are similarly loaded, as in rafters, ceiling joists, etc, but another manner of determining the stiffness may be used in other cases.

For let W be the weight that is uniformly distributed over a beam supported at both ends, then the deflection produced by this weight would be to the deflection produced by the same weight collected in the middle of the length as 5: 8, or as 0.625: 1.* Therefore, in the rules in Arts. 102, 103, 104, and 105, it is only necessary to employ the weight in pounds multiplied by 0.625 instead of the whole weight, and the rest of the operation is the same as in those rules; therefore it will not be necessary to repeat them.