COLUMNS.

Columns are prisms, with a length of more than ten times the least diameter. If the prism is very short it fails by simple compression as we have already seen; if it is long it fails by bending

HI

FIG. 10.

sideways. There are many kinds and shapes of columns. They are built round, square, hexagonal, etc., and are both solid and

ITI

H

F

hollow. If of wood they are usually round or square. Cast iron columns are generally round. The wrought iron column is made in a great variety of forms. They are made of I beams, or of different shaped irons riveted together as shown in Fig. 10. Columns are made symmetrical so that the liability to bending will be the same for all directions. The shape of the ends makes considerable difference as to the strength. The ends are either round or fixed. In Fig. 11, R has rounded ends, H one end rounded and the other fixed, and F has both ends fixed. Columns with both ends fixed are the most used, but in bridge construction and machines columns with rounded ends are frequently used.

R

FIG. 11.

Experiments show that columns with fixed ends are stronger than with rounded ends, so the order of strength of the above columns is F, H, R. And with the same lengths, sizes and material, the relative strengths are R, 1; II, 2 and F, 4. Other things being equal a column with both ends fixed is 4 times as strong as one with both ends rounded.

The mathematical formulas for columns are empirical owing to the imperfect tests made upon them.

A formula deduced by Euler, applicable only to columns very long in proportion to their diameter, is:

in which C = 1, 2, or 4 for the columns R, H and F respectively, E = coefficient of elasticity, I = moment of Inertia, P = weight, 7= length in inches.

1=

Find the weight on a rectangular wooden column 4 × 6 inches and 24 feet long, ends fixed.

We use the least moment of inertia since the column will bend in the direction of the least resistance; then substituting we

for a rectangular column, d, being the least side or,

for a cylindrical column of diameter d. If r be the least radius of gyration of the cross section, the formula may be written,

The value P above is not the breaking strength but the load which causes bending. This load is the one we want, for the

bending practically marks the beginning of failure. The above formulas show that the strength of a column varies directly as its cross section and directly as the square of the least diameter or least side.

The following table gives values of r2 for the most used forms.

For I beams, deck beams and other shapes, r2 is the least moment of inertia divided by the area of cross section. Both these quantities may be found from tables or calculated.

A formula for ultimate strength of wrought iron columns such as are ordinarily used in practice is :

in which S = the ultimate strength, 7 = the length in feet, r2 = the value of r2 from the table, n = 36,000 for columns with fixed ends, 24,000 for columns with one end fixed and the other round, 18,000 for both ends round. Factors of safety of 4 or 5 should be used with the above formula.

EXAMPLES FOR PRACTICE.

1. What is the stress on a wrought iron column 12 feet high, when r2 .50. Use the formula

when n = 36,000 and a factor of safety of 4 is used.

Ans. 4,647 pounds (nearly).

2. The length of a wrought iron column fixed at both ends is 12 feet. Find the diameter when a factor of safety of 4 is used and the stress is 8700 pounds.

Ans. 8 inches (approx.)

TRANSMISSION OF POWER BY SHAFTING.

Shafting is often the means by which power is transmitted from place to place. Shafts are usually made of wrought iron or steel in the form of a cylindrical bar. Cold rolled shafting is made by a special rolling process. If turned in a lathe it is called bright shafting. Black shafts are of bar iron rolled by the usual process and machined wherever bearings are placed.

It is evident that the most important resistance which a shaft must overcome in transmitting power acts to prevent the shaft from turning. Hence the strength of material with regard to the amount of torsion it can withstand must be known before we choose the material for a shaft. As has been stated, torsion occurs when applied forces tend to twist a body.

The cause of torsion in shafts is due to the stress produced by pulleys, belts, ropes, etc. The twisting of a main shaft is due to the power of the engine applied at the crank. The torsion set up in the large shafts of our ocean steamships is enormous. Hence an accurate knowledge of the value of torsion under different conditions is important. Shafts are subjected to a bending action due to their own weight and to the weight of the pulleys, belts, etc., which they support. As this bending action is difficult to determine without the aid of higher mathematics, we will consider torsion only.

Suppose one end of a horizontal shaft to be rigidly fixed, and let the free end have a lever p attached at right angles to its axis, as shown in Fig. 12. If a weight A be hung at the end of this lever,

the shaft will be twisted

so that a fibre such as is represented by the dotted line ae will

assume some such spiral form as ad. The end of the shaft will also be found to have rotated through a certain angle dce.

It can be shown that if A is not so large as to strain the material beyond its elastic limit, the angles dee and ead are proportional to the weight applied, and that when this weight is removed the lines de and da return to their former positions ec and ea. If the elastic limit is exceeded and the twisting is great enough the shaft will be ruptured.

Let us represent the perpendicular distance from the axis of the shaft to the line of action of A by p, then the moment of A, which tends to twist the shaft with regard to the axis must be Ap. This product is called the twisting moment, and whatever the number of forces acting at the end of the shaft their resultant moment may always be represented by Ap. We may now consider a simple illustrative example. A force of 80 pounds at 18 inches from the axis twists a shaft 60°. What force will produce the same result when acting 4 feet from the axis? We have seen that Ap is a constant quantity. Hence we may state the equation:

A force of 200 pounds acting at a distance of 6 inches from the axis, twists a shaft 2 feet long through an angle of 7 degrees. Through what angle will a similar shaft 4 feet long be twisted by a force of 500 pounds acting at a distance of 18 inches from the axis? Ans. 105 degrees.

When a shaft is designed to transmit power, the ratio of the greatest internal shearing unit stress S, to the power H, must be known. Otherwise the shaft might be ruptured by too great a load. Let us consider a shaft making n revolutions per minute, which transmits H horse-power. The work may be applied by a belt from the motor to a pulley on the shaft. By virtue of the