This may be definitely ascertained when the proportion which the thickness shall bear to the breadth shall have been previously determined. For example, when the proportion is as 6 to 8, then 1 1/3 times 6 equals 8, and the proportion is as 1 to 1 1/3; again, when the proportion is as 8 to 10, then 1 1/4 times 8 equals 10, and in this case the proportion is as 1 to 1 1/4. Let the latter figure of the ratio 1 to 1 1/4, 1 to 1 1/3, etc., be called n, or so that the proportion shall be as 1 to n, then -
To ascertain the thickness of a post to sustain safely a given weight, when the height is at least ten times the thickness; the action of the weight coinciding with the axis; we have -
Rule X. - Multiply the given weight by the factor of safety, and divide the product by twice the value of C for the material of the post, found in Table I., multiplied by n, as above explained; reserve the quotient, calling it G. Now multiply 432 times the value of e for the material of the post,. found in Table III., by the square of the height in feet, and by the above quotient G; to the product add the square of G; extract the square root of the sum, and to it add the value of G; then the square root of this sum will be the required thickness of the post; or -
G= Wa/2 C n. (12.)
Example. - What should be the thickness of a white-pine rectangular post 20 feet high to sustain safely 30,000 pounds, the pressure coinciding with the axis, and the proportion between the thickness and breadth to be as 10 to 12, or as 1 to 1.2? Proceeding according to the rule, we have the product of 30000, the given weight, by 6, as a factor of safety, equals 180000; this divided by twice Cx n, or 2 x 6650 x 1.2, (=15960) gives a quotient of 11.278, the value of G. Then, we have e - .0014, the square of the height equals 400; therefore,432 x .0014x400x11.278 = 2728.43.. Tothisadd-ing 127.2, the square of G, we have 2855 63, the square root of which is 53.438; and this added to G gives 64.716, the square root of which is 8.045, the required thickness of the post. Now, since the thickness is in proportion to the breadth as 1 to 1.2, therefore 8.045 x 1.2 = 9.654, the required width. The post, therefore, may be made 8x9 3/4 inches.