A stick of timber, a bar of iron, etc., when used to sustain end loads which act lengthwise of the pieces, are called columns, posts, or struts if they are so long that they would bend before breaking. When they are so short that they would not bend before breaking, they are called short blocks, and their compressive strengths are computed by means of equation 1. The strengths of columns cannot, however, be so simply determined, and we now proceed to explain the method of computing them.

77. End Conditions. The strength of a column depends in part on the way in which its ends bear, or are joined to other parts of a structure, that is, on its " end conditions." There are practically but three kinds of end conditions, namely:

1. "Hinge" or "pin" ends,

2. " Flat" or " square " ends, and

3. "Fixed" ends.

(1) When a column is fastened to its support at one end by means of a pin about which the column could rotate if the other end were free, it is said to be "hinged" or "pinned" at the former end. Bridge posts or columns are often hinged at the ends.

(2) A column either end of which is flat and perpendicular to its axis and bears on other parts of the structure at that surface, is said to be "flat" or " square" at that end.

(3) Columns are sometimes riveted near their ends directly to other parts of the structure and do not bear directly on their ends; such are called " fixed ended." A column which bears on its flat ends is often fastened near the ends to other parts of the structure, and such an end is also said to be " fixed." The fixing of an end of a column stiffens and therefore strengthens it more or less, but the strength of a column with fixed ends is computed as though its ends were flat. Accordingly we have, so far as strength is concerned, the following classes of columns:

78. Classes of Columns. (1) Both ends hinged or pinned; (2) one end hinged and one flat; (3) both ends flat.

Other things being the same, columns of these three classes are unequal in strength. Columns of the first class are the weakest, and those of the third class are the strongest. Fig. 46.

70. Cross=sections of Columns. Wooden columns are usually solid, square, rectangular, or round in section; but sometimes they are "built up" hollow. Cast-iron columns are practically always made hollow, and rectangular or round in section. Steel columns are made of single rolled shapes - angles, zees, channels, etc.; but the larger ones are usually "built up" of several shapes. Fig. 46, a, for example, represents a cross-section of a "Z-bar" column; and Fig. 46, b, that of a "channel" column.

80. Radius of Gyration. There is a quantity appearing in almost all formulas for the strength of columns, which is called "radius of gyration." It depends on the form and extent of the cross-section of the column, and may be defined as follows:

The radius of gyration of any plane figure (as the section of a column) with respect to any line, is such a length that the square of this length multiplied by the area of the figure equals the moment of inertia of the figure with respect to the given line.

Thus, if A denotes the area of a figure; I, its moment of inertia with respect to some line; and r, the radius: of gyration with respect to that line; then (9)

In the column formulas, the radius of gyration always refers to an axis through the center of gravity of the cross-section, and usually to that axis with respect to which the radius of gyration (and moment of inertia) is least. (For an exception, see example 3. Art. 83.) Hence the radius of gyration in this connection is often called for brevity the "least radius of gyration," or simply the "least radius."

Examples. 1. Show that the value of the radius of gyration given for the square in Table A, page 54, is correct.

The moment of inertia of the square with respect to the axis is 1/12 a4- Since A = a2, then, by formula 9 above, 2. Prove that the value of the radius of gyration given for the hollow square in Table A, page 54, is correct.

The value of the moment of inertia of the square with respect to the axis is 1/12 (a4 - a1 4). Since A = a2 - a12, ## Example For Practice

Prove that the values of the radii of gyration of the other figures 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 Built=up 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 A A.

In example 1, Art. 54, it is shown that the moment of inertia of the section with respect to the axis AA is 429 inches4. The area of the whole section is

2 X 6.03 + 7 = 19.06; hence the radius of gyration r is ## Example For Practice

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.