When the thickness of a post is fixed, and the breadth required; then, to ascertain the breadth of a rectangular post to sustain safely a given weight, the direction of the pressure of which coincides with the axis of the post, we have -
Rule XI - Divide the height in inches by the given thick-ness, and multiply the quotient by itself, or take its square; multiply this square by the value of e for the material of the post, found in Table 111.; to the product add its half, and to the sum add unity; multiply this sum by the given weight, and by the factor of safety; divide the product by the product of the given thickness multiplied by the value of C for the material of the post, found in Table I., and the quotient will be the required breadth; or -
What b = Wa(I+3/2er2) (14.)
Example. - What should be the breadth of a spruce post 18 feet high and 6 inches thick to sustain safely 25,000 pounds, the pressure coinciding with the axis of the post? According to the rule, 216 (= 12 x 18), the height in inches, divided by 6, the given thickness, gives a quotient of 36, the square of which is 1296; the value of e for spruce is .00098; this multiplied by 1296, the above square, equals 1 .7; which increased by .635, its half, amounts to 1.905; this increased by unity, the sum is 2.905; which multiplied by the given weight, and by the factor of safety, gives a product of 435749; and this divided by 6 (the given thickness) times 7850 (the value of C for spruce) == 47100, gives a quotient of 9 . 2516, the required breadth of the post. The post, therefore, requires to be 6 x 91/4 inches.
Observe that when the breadth obtained by the rule is less than the given thickness, the result shows that the conditions of the case are incompatible with the rule, and that a new computation must be made; taking now for the breadth what was before understood to be the thickness, and proceeding in this case, by Rule X., to find the thickness.