The value of the moment of inertia of the square with respect to the axis is (a' — a,'). Since A = a' — Prove that the values of the radii of gyration of the other fig ures given in Table A, page 54, are correct. The axis in each case is indicated by the line through the center of gravity.
81. Radius of Gyration of Sections. The radius of gyration of a built-up section is computed similarly to that of any other figure. First, we have to compute the moment of inertia of the section, as explained in Art. 54; and then we use formula 9,as in the examples of the preceding article.
Example. It is required to compute the radius of gyration of the section represented in Fig. 30 (page 52) with respect to the axis AA.
In example 1, Art. 54, it is shown that the moment of inertia of the section with respect to the axis AA is 429 inches'. The area of the whole section is hence the radius of gyration r is Compute the radii of gyration of the section represented in Fig. 31, a, with respect to the axes AA and BB. (See examples for practice 1 and 2, Art. 54.) Ans. 2.87 inches.
2.09 " 82. Kinds of Column Loads. When the loads applied to a
column are such that their resultant acts through the center of gravity of the top section and along the axis of the column, the column is said to be centrally loaded. When the resultant of the loads does not act through the center of gravity of the top section, the column is said to be eccentrically loaded. All the following formulas refer to columns centrally loaded.
83. Rankine's Column Formula. When a perfectly straight column is centrally loaded, then, if the column does not bend and if it is homogeneous, the stress on every cross-section is a uniform compression. If P denotes the load and A the area of the cross section, the value of the unit-compression is P A.
On account of lack of straightness or lack of uniformity in material, or failure to secure exact central application of the load, the load P has what is known as an "arm " or " leverage " and bends the column more or less. There is therefore in such a column a bending or flexural stress in addition to the direct com pressive stress above mentioned; this bending stress is compressive