The line AB, Fig. 50, represents Johnson's straight-line formula; and BC, Euler's formula. It will be noticed that the two lines are tangent; the point of tangency corresponds to the "limiting value" l 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 l 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 inches4, and the area of the section is 11.76 square inches. Hence r2 = 9.50/11.76 = 0.81; or r = 0.9 inch.

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Fig. 50.

Since 1 = 8 feet = 96 inches, l/r = 96/0.96=106 ⅔

This value of I 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

P/11.76 = 52,500 - 180 X 106 ⅔; or, P = 11.76 (52,500 - 180 X 106 ⅔) = 391,600 pounds.

This is the breaking load for the column according to the straight, line formula; hence the factor of safety is

391,600/100,000=3.9

2. Suppose that the length of the column described in the preceding example were 16 feet. What would its factor of safety be?

Since I = 16 feet = 192 inches; and, as before, r = 0.9 inch, I r = 2131/3. This value is greater than the limiting value (195) indicated by Table E (p. 97) for flat-ended steel columns; hence Euler's formula is to be used. Thus

P/11.76 = 666,000,000/(213⅓)2 or, P =11.76x666,000,000/(213⅓)2 =172,100 pounds.

This is the breaking load; hence the factor of safety is

172,100/100,000=1.7

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 5x8 inches and the inside 4x7 inches, with a factor of safety of 6 ?

Substituting in the formula for the radius of gyration giver in Table A, page 54, we get r =

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= 1.96 inches.

Since l = 10 feet = 120 inches, l/r=120/1.96=61.22

According to the straight-line formula for cast iron, A being equal to 12 square inches,

P/12= 34,000 - 88 X 61.22; or, P = 12 (34,000 - 88 X 61.22) = 343,360 pounds.

This being the breaking load, the safe load is

343,360/6= 57,227 pounds.

Examples For Practice

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 column 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. 2.8

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 5 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. ParaboIa=EuIer 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=S-m(l/r)2 (13)

P, A, l and r having the same meanings as in Rankine's formula, Art. 83; and S and m 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:

P/A=n/(lr)2 (14) the values of n (Johuson) being given in the following table:

Table F. Data For Mild Steel Columns

S

m

Limit (l r )

n

Hinged ends..

42,000

0.97

150

456,000,000

Flat ends..

42,000

0.62

190

712,000,000

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 l 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.

The line AB, Fig. 51, represents the parabola formula; and the line BC, Euler's formula. The two lines are tangent, and the point of tangency corresponds to the "limiting value" l r of the table.

For wooden columns square in cross-section, it is convenient to replace r by d, the latter denoting the length of the sides of the square. The formula becomes

P/A=S-m(l/d)2,

S and m for flat-ended columns of various kinds of wood having the following values according to Professor Johnson:

For White pine, S=2,500, m = 0.6;

" Short-leaf yellow pine, S = 3,300, m = 0.7;

" Long-leaf yellow pine, S = 4,000, m = 0.8;

" White oak. S = 3,500, m = 0.8.

The preceding formula applies to any wooden column whose ratio, l d, is less than 60, within which limit columns of practice are included.

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Fig. 51.