Strength of Materials 91

For the slender columns, another formula (Euler's, long since deduced) was used by Johnson. Its general form is—, n being a constant whose values, according to Johnson, are given in the following table: The numbers in the fourth column of the table mark the point of divi sion between columns of ordinary length and slender columns. For the former kind, the straight-line formula applies; and for the second, Euler's. That is, if the ratio 1 r for a steel column withlinged end, for example, is less than 160, we must use the straight-line formula to compute its safe load, factor of safety, etc.; but if the ratio than 160, we must use Euler's formula.

For cast-iron columns with flat ends, S = 34,000, and m = 88; and since they should never be used "slender," there is no use of Euler's formula for cast-iron columns.

The line AB, Fig. 50, represents Johnson's straight-line for mula; and BC, Euler's formula. It will be noticed that the two lines are tangent; the point of tangency corresponds to the "lim iting value" / r, as indicated in the table.

Examples. 1. A 40-pound 10-inch steel I-beam column 8 feet long sustains a load of 100,000 pounds, and the ends are flat. Compute its factor of safety according to the methods of this article.

The first thing to do is to compute the ratio / r for the column, to ascertain whether the straight-line formula or Euler's formula should be used. From Table C, on page 72, we find that the moment of inertia of the column about the neutral axis of its cross-section is 9.50 inches', and the area of the section is 11.76 square inches. Hence This value of 1 r is less than the limiting value (195) indicated by the table for steel columns with flat ends (Table E, p. 97), and we should therefore use the straight-line formula; hence This is the breaking load for the column according to the straight line formula; hence the factor of safety is 2. Suppose that the length of the column described in the preceding example were 16 feet. What would its factor of safety be? Since l = 16 feet = 192 inches; and, as before, 9° = 0.9 inch, / r = 214. This value is greater than the limiting value (195) indicated by Table E (p. 97) for flat-ended steel col. tonne; hence Euler's formula is to be used. Thus 11.76 = ; 11.76 x 666,000,000 ____ 172,100 pounds.

or, P = This is the breaking load; hence the factor of safety is 3. What is the safe load for a cast-iron column 10 feet long with square ends and hollow rectangular section, the outside dimensions being 5 X 8 inches and the inside 4 x 7 inches, with a factor of safety of 6 I Substituting in the formula for the radius of gyration given in Table A, page 54, we get 8 x 5' – 7 x r 12 (8 x 5 – 7 X 4) — 1.96 inches.

Since / = 10 feet = 120 inches, 1. A 40-pound 12-inch steel I-beam 10 feet long is used as a flat-ended column. Its load being 100,000 pounds, compute the factor of safety by the formulas of this article.

Ans. 3.5 2. A cast-iron columa 15 feet long sustains a load of 150,000 pounds. Its section being a hollow circle of 9 inches ,outside and 7 inches inside diameter, compute the factor of safety by the straight-line formula.

Ans. 4.8 3. A steel Z-bar column (see Fig. 46, a) is 24 feet long and has square ends; the least radius of gyration of its cross section is 3.1 inches; and the area of the cross-section is 24.5 square inches. Compute the safe load for the column by the formulas of this article, using a factor of safety of 4.

Ans. 219,000 pounds.

4. A hollow cast-iron column 13 feet long has a circular cross-section, and is 7 inches outside and 51 inches inside in diameter. Compute its safe load by the formulas of this article, using a factor of safety of 6.

Ans. 68,500 pounds 5. Compute by the methods of this article the safe load for a 40-pound 12-inch steel I-beam used as a column with flat ends, if .the length is 17 feet and the factor of safety 5.

Ans. 35,100 pounds.

87.

uler Formulas. As better fitting the results of tests of the strength of columns of " ordinary lengths," Prof. J. B. Johnson proposed (1892) to use parabolas instead of straight lines. The general form of the "parabola formula" is P, A, 1 and r having the same meanings as in Rankine's formula, Art. 83; and S and vs denoting constants whose values, according to Professor Johnson, are given in Table F below.

Like the straight-line formula, the parabola formula should not be used for slender columns, but the following (Euler's) is applicable: The point of division between columns of ordinary length and slender columns is given in the fourth column of the table. That is, if the ratio 1±r for a column with hinged ends, for example, is less than 150, the parabola formula should be used to compute the safe load, factor of safety, etc.; but if the ratio is greater than 150, then Euler's formula should be used.