(1) For bending in the first plane, the strength of the column is to be computed from the formula for a pin-ended column. Hence, for this case, r2 = 386 -4- 23.5 = 16; and the breaking load is

P= -(50,000 X 23.5)/ == 1,041,600 pounds.


The safe load for this case equals 1,041,600/4 = 260,400 pounds.

(2) If the supports of the pins are rigid, then the pins stiffen the column as to bending in the plane of their axes, and the strength of the column for bending in that plane should be computed from the formula for the strength of columns with flat ends. Hence, r2 = 214 23.5 = 9.11, and the breaking load is

(50,000 X 23.5)/(16X122)2 = 1,056,000 pounds P = 1+(16x12)2/(36,000X9.11).

The safe load for this case equals 1,056,000/4 = 264,000 pounds.

Examples For Practice

1. A 40-pound 12-inch steel I-beam 10 feet long is used as a column with flat ends sustaining a load of 100,000 pounds.

What is its factor of safety?

Ans. 4.1

2. A cast-iron column 15 feet long sustains a load of 150,000 pounds. Its section being a hollow circle, 9 inches outside and 7 inches inside diameter, what is the factor of safety?

Ans. 8.9

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. What is the safe load for the column with a factor of safety of 4 ?

Ans. 247,000 pounds.

4. A cast-iron column 13 feet loner has a hollow circular cross-section 7 inches outside and 5 inches inside diameter. What is its safe load with a factor of safety of 6 ?

Ans. 121,142 pounds.

5. Compute the safe load for a 40-pound 12-inch steel I-beam used as a column with flat ends, its length being 17 feet. Use a factor of safety of 5.

Ans. 52,470 pounds. 84. Graphical Representation of Column Formulas. Column (and most other engineering) formulas can be represented graphically. To represent Rankine's formula for flat-ended mild-steel columns,

P/A=50,000/ 1+/(lr)2/36,00 we first substitute different values of l r in the formula, and solve for P A. Thus we find, when l r = 40, P A = 47,900 ; l r = 80, P-A = 42,500; l r = 120, P + A = 35,750 ; etc., etc.

Now, if these values of I r be laid off by some scale on a line from O, Fig. 48, and the corresponding values of P A be laid off vertically from the points on the line, we get a series of points as a, b, c, etc.; and a smooth curve through the points a, b, c, etc., represents the formula. Such a curve, besides representing the formula to one's eye, can be used for finding the value of P A for any value of I r ; or the value of l r for any value of P A. The use herein made is in explaining other column formulas in succeeding articles.

Strength Of Columns Part 3 030076

Fig. 48.

85. Combination Column Formulas. Many columns have been tested to destruction in order to discover in a practical way the laws relating to the strength of columns of different kinds. The results of such tests can be most satisfactorily represented graphically by plotting a point in a diagram for each test. Thus, suppose that a column whose I r was 80 failed under a load of 276,000 pounds, and that the area of its cross-section was 7.12 square inches. This test would be represented by laying off Oa, Fig. 49, equal to 80, according to some scale; and then ah equal to 276,000 7.12 (P A), according to some other convenient scale. The point b would then represent the result of this particular test. All the dots in the figure represent the way in which the results of a series of tests appear when plotted.

It will be observed at once that the dots do not fall upon any one curve, as the curve of Rankine's formula. Straight lines and curves simpler than the curve of Rankine's formula have been fitted to represent the average positions of the dots as determined by actual tests, and the formulas corresponding to such lines have been deduced as column formulas. These are explained in the following articles.

Strength Of Columns Part 3 030077

Fig. 49.

86. Straight-Line and Euler Formulas. It occurred to Mr. T. H. Johnson that most of the dots corresponding to ordinary lengths of columns agree with a straight line just as well as with a curve. He therefore, in 1886, made a number of such plats or diagrams as Fig. 49, fitted straight lines to them, and deduced the formula corresponding to each line. These have become known as "straight-line formulas," and their general form is as follows:



Frank Chouteau Brown, Architect, Boston, Mass. For Garden Front, See Page 10; for Plans, See Page 122.

P/A=S-m l/r, (11)

P, A, I, and r having meanings as in Rankine's formula (Art. 83), and S and m being; constants whose values according to Johnson are given in Table E below.

For the slender columns, another formula (Euler's, long since deduced) was used by Johnson. Its general form is-

P/A = n/(lr)2, (12) n being a constant whose values, according to Johnson, are given in the following table:

Table E. Data For Mild-Steel Columns



Limit (l r)


Hinged ends..





Flat ends ..





The numbers in the fourth column of the table mark the point of division 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 I r for a steel column with hinged 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 is greater 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.