The method of securing the ends of a column greatly influences its strength. While wooden and cast-iron columns usually occur in building construction with flat or square ends, structural-steel columns are often used, having either hinged, flat or square, and fixed ends. Where the ends are securely fixed, so that the column is likely to fail in the shaft, before the end connections are ruptured, greater strength is developed than with columns having hinged or pinned ends. Columns having flat or square ends are somewhat stronger than hinged-end columns, but not so strong as those having their ends firmly secured.

The strength of structural-steel columns may be determined by the following formulas:

For Fixed Ends.

For Square Ends.

For Pin Ends.

in which S= ultimate strength of column in pounds per square inch; U = ultimate compressive strength of the material in pounds per square inch; I = length of column in inches; *R = least radius of gyration.

The value of U for structural steel is usually taken at from 52.000 to 54,000 lb. for soft steel, and as high as 60,000 for medium steel.

Having determined S, the safe load per square inch is found by dividing S by the factor of safety. This quotient, multiplied by the area of the section, will give the safe load the column will sustain.

* See page 81.

### Example

What safe load will a square-end structural-steel column, 20 ft. long, having the section shown in Fig. 11, support, taking the ultimate compressive strength at 52,000 lb. per sq. in., and using a factor of safety of 5?

### Solution

First calculate the radius of gyration about both the axes X X and Y Y, by the method given on page 81. This will determine the least radius of gyration, which in this case is around the axes Y Y, and is approximately 2.13 in. The value of R2 will then be 2.13 X 2.13 = 4.53. The value of I, the length of the column in inches, is 20 (ft.) X 12 (in.) = 240 in.; and l2 = 240X240 = 57,600. Substituting the above values in the formula,

which is the ultimate strength of the column in pounds per square inch of section.

The total sectional area is as follows:

*Area of 4-5"x3"x5/16" angles = 2.40x4= 9.60 sq. in.

† Area of 5/16"x 10" web-plate = 10 x .3125 = 3.125 sq. in.

Total area of section = 12.725 sq. in.

Then 34,000 (ultimate strength of the column in pounds per square inch of section) X 12.72 (sq. in.) = 432,480 lb. The factor of safety required being 5, the safe load that the column will support is 432,480 / 5 = 86,496 lb.

The formulas above given being somewhat awkward to use, more convenient ones have been deduced, and while they give somewhat different values for S than those obtained by the previously stated formulas, the differences are on the side of safety. These formulas may be applied to columns whose lengths are between 50 and 150 least radii of gyration; that is, if the radius of gyration of a column is 2 in., these formulas will apply to columns whose lengths are between 100 and 300 in..

Fig. 11.

* See table, page 86.

† For table of decimal equivalents, see page 53.

Medium Steel. Soft Steel.

Fixed ends, S = 60,000 - 210(l/R) S= 54,000 - 185(l/R)

Square ends, S = 60,000 - 230(l/R) S = 54,000 - 200(l/R)

Pin ends, S = 60,000 - 260(l/R) S= 54,000 - 225(l/R)

In these formulas, S = ultimate or breaking strength of the column in pounds-per square inch of sectional area; I = length of the column in inches; R = least radius of gyration in inches. Example. - What is the ultimate strength in pounds, per square inch of section, for a fixed-end column 22 ft. long, the least radius of gyration for the section being 2.5 in. and the material being medium steel?

### Solution

The formula S = 60,000 - 210(l/R), by substituting the values for I and R, becomes

S = 60,000 - 210(264/2.5)= 37,824 lb., the ultimate strength per square inch of column section.

Structural columns are usually considered as having square ends, and a factor of safety of 4 is generally allowed. The following formulas will give the allowable stress in pounds per square inch which such columns will sustain:

Medium Steel. Soft Steel.

S = 15,000 - 57(l/R) S=13,500 -50(l/R)

The formula for medium-steel columns is for lengths over 50 times the least radius of gyration; that for soft-steel columns is for lengths over 30 times the least radius of gyration. For columns of dimensions less than these figures, a safe load of 12,000 lb. per sq. in. of section can safely be assumed.

No column should be used whose length is greater than 150 least radii of gyration, or whose length exceeds 45 times the least dimension of the column.