The preceding discussions have related to the strength of materials. We shall now consider principally the elongation of rods, deflection of beams, and twist of shafts.

FIRST FLOOR PLAN OF RESIDENCE AT DEDHAM, MASS.

FIRST-FLOOR PLAN OF RESIDENCE AT DEDHAM, MASS.

Frank Chouteau Brown, Architect, Boston, Mass. For Exteriors, See Page 10 and Page 106. Second-Floor Plan Shown on Opposite Page.

SECOND FLOOR PLAN OF RESIDENCE AT DEDHAM, MASS.

SECOND-FLOOR PLAN OF RESIDENCE AT DEDHAM, MASS.

Frank Chouteau Brown, Architect, Boston, Mass.

First-Floor Plan Shown on Opposite Page.

95. Coefficient of Elasticity. According to Hooke's Law (Art. 9, p. 7), the elongations of a rod subjected to an increasing pull are proportional to the pull, provided that the stresses due to the pull do not exceed the elastic limit of the material. Within the elastic limit, then, the ratio of the pull and the elongation is constant; hence the ratio of the unit-stress (due to the pull) to the unit-elongation is also constant. This last-named ratio is called "coefficient of elasticity." If E denotes this coefficient, S the unit-stress, and s the unit-deformation, then

E = S / s. (17)

Coefficients of elasticity are usually expressed in pounds per square inch.

The preceding remarks, definition, and formula apply also to a case of compression, provided that the material being compressed does not bend, but simply shortens in the direction of the compressing forces. The following table gives the average values of the coefficient of elasticity for various materials of construction:

Table G. Coefficients Of Elasticity

Material.

Average Coefficient of Elasticity.

Steel ....

30,000,000 pounds per square inch.

Wrougth iron ..

27.500.000 "' " " "

Cast iron...

15.000,000 " " " "

Timber .....

1,800,000

The coefficients of elasticity for steel and wrought iron, for different grades of those materials, are remarkably constant; but for different grades of cast iron the coefficients range from about 10,000,000 to 30,000,000 pounds per square inch. Naturally the coefficient has not the same value for the different kinds of wood; for the principal woods it ranges from 1,000,000 (for spruce) to 2,100,000 (for white oak).

Formula 17 can be put in a form more convenient for use, as follows :

Let P denote the force producing the deformation ; A the area of the cross-section of the piece on which P acts; b the length of the piece; and D the deformation (elongation or shortening).

Then

S = P A (see equation 1), and s = D l (see equation 2).

Hence, substituting these values in equation 17, we have

E = Pl /AD;or D = Pl /AE . (17)

The first of these two equations is used for computing the value of the coefficient of elasticity from measurements of a "test," and the second for computing the elongation or shortening of a given rod or bar for which the coefficient is known.

Examples. 1. It is required to compute the coefficient of elasticity of the material the record of a test of which is given on page 9.

Since the unit-stress S and unit-elongation s are already computed in that table, we can use equation 17 instead of the first of equations 17'. The elastic limit being between 40,000 and 45,000 pounds per square inch, we may use any value of the unit-stress less than that, and the corresponding unit-elongation.

Thus, with the first values given,

E = 5,000/0.00017=29,400,000

With the second,

E = 10,000/0.00035= 28,600,000.

This lack of constancy in the value of E as computed from different loads in a test of a given material, is in part due to errors in measuring the deformation, a measurement difficult to make. The value of the coefficient adopted from such a test, is the average of all the values of E which can be computed from the record.

2. How much will a pull of 5,000 pounds stretch a round steel rod 10 feet loner and 1 inch in diameter?

We use the second of the two formulas 17'. Since A = 0.7854 X l2 = = 0.7854 square inches, l = 120 inches, and E = 30,000,000 pounds per square inch, the stretch is:

D = [ 5,000 X 120] /[ 0.7854 X 30,000] = 0.0254 inch.

Examples For Practice

1. What is the coefficient of elasticity of a material if a pull of 20,000 pounds will stretch a rod 1 inch in diameter and 4 feet long 0.045 inch?

Ans. 27,000,000 pounds per square inch.

2. How much will a pull of 15,000 pounds elongate a round cast-iron rod 10 feet long and 1 inch in diameter?

Ans. 0.152 inch. 96. Temperature Stresses. In the case of most materials, when a bar or rod is heated, it lengthens; and when cooled, it shortens if it is free to do so. The coefficient of linear expansion of a material is the ratio which the elongation caused in a rod or bar of the material by a change of one degree in temperature bears to the length of the rod or bar. Its values for Fahrenheit degrees are about as follows:

For Steel,

0.0000065.

For Wrought iron,

.0000067.

For Cast iron,

.0000062.

Let K be used to denote this coefficient; t a change of temperature, in degrees Fahrenheit; l the length of a rod or bar; and D the change in length due to the change of temperature. Then

D=K t l. (18)

D and l are expressed in the same unit.

If a rod or bar is confined or restrained so that it cannot change its length when it is heated or cooled, then any change in its temperature produces a stress in the rod; such are called tem= perature stresses.

[Examples. 1. A steel rod connects two solid walls and is screwed up so that the unit-stress in it is 10.000 pounds per square inch. Its temperature falls 10 degrees, and it is observed that the walls have not been drawn together. What is the temperature stress produced by the change of temperature, and what is the actual unit-stress in the rod at the new temperature?

Let l denote the length of the rod. Then the change in length which would occur if the rod were free, is given by formula 18, above, thus: