1 Putting d=3, S=4x3=12, P=60, ƒ=36, a= 500' we get 1=167.1 inches = 13.9 feet. TABLE XXVIII.-BREAKING WEIGHT IN TONS PER SQUARE INCH OF SOLID OR HOLLOW CAST-IRON COLUMNS, THE ENDS BEING SECURELY Example 14.-What must be the section of a square column, 15 feet long, so as just to fail with a load of 100 tons; its ends being fixed? If d = side of the square and 7= length, from equation (13) we get P= fd4 d2 + a 12' By substituting for f, P, a, and 7, their proper values, we get 36 d1- 100 d2 - 6,480 = 0, d2 = 14.88, 160. Rankine's Rules for the Strength of Columns.-The late Professor Rankine has given rules for calculating the strengths of columns and struts which are expressed in terms of the least radius of gyration of the section; these rules are of the greatest importance, and the following is a summary taken from his Useful Rules and Tables: Let P breaking load of the column, = g = least radius of gyration of its cross-section, ƒ and c = coefficients whose value depends on the nature of the material. 2nd. For a column with both ends rounded or jointed (15). 3rd. For a column with one end fixed and the other rounded The following are the values of the constants ƒ and c :— Definition. The square of the radius of gyration of a surface about a given axis is equal to the moment of inertia of the surface about the axis divided by its area. Let r = radius of gyration of the surface, I = A Then moment of inertia = area of the surface. TABLE XXIX.-VALUES OF 2 FOR DIFFERENT FORMS OF Rectangular cell breadths b and b1, depths d and dı, . Square cell sides b and b1, . 222 12 a2 12 16 II. WROUGHT-IRON COLUMNS AND STRUTS. Columns made of wrought iron are much more reliable than those made of cast iron. All risk of flaws, blow-holes, shifting of cores, and irregularity in section are avoided; and they possess the further advantage (which is a very important one when used in buildings) of being better able to resist the attacks of fire. The rules which regulate their strength are very similar to those which apply to cast-iron columns. 161. Wrought-iron Columns of Solid Section. The strength of solid rectangular columns may be found from Gordon's formulæ by giving to the constants ƒ and a the following values : Example 15.-What is the breaking weight of a solid pillar of wrought iron 20 feet long and 6 inches square? which is about one-half of the former. Example 16.-What must be the section of a solid square pillar of wrought iron, 15 feet long, whose breaking weight is 50 tons; the ends being fixed? 8 d1 - 25 d2 = 270; ď2 = 7·6, or d=2.75 inches. == Example 17.-If the breaking load of a solid rectangular wrought-iron pillar, 10 feet long and 2 inches in thickness, be 30 tons-What will be the width of the pillar if its ends be imperfectly fixed? 162. Rankine's Rules for Wrought-iron Columns.-Rankine's rules, given in equations (15), (16), and (17), may be applied for determining the strengths of wrought-iron columns and struts of any section, by giving to the constants the following values : These rules might be applied to solve examples 15, 16, and 17. For example, in 15 we get when both ends are fixed, which agrees very nearly with the result as found by Gordon's formula. When both ends are rounded This latter is rather more than a mean between the first two. 163. Solid Round Wrought-iron Columns.-The strengths of solid round wrought-iron columns per square inch of section is less than that of similar solid square columns. Rankine's formula may be applied for determining these strengths. |