STIFFNESS OF RODS, BEAMS, AND SHAFTS.
The preceding discussions have related to the strength of materials. We shall now consider principally the elongation, oy rods, deflection, of beams, and twist of shafts.
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 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 com pressing forces. The following table gives the average values of the coefficient of elasticity for various materials of construction: 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 1,600,000 (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 ; / the length of the piece ; and D the deformation (elongation or shortening).
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 O.
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, 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 long and 1 inch in diameter We use the second of the two formulas 17'. Since A = 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 eooled, 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: Let K be used to denote this coefficient; t a change of tem perature, in degrees Fahrenheit; l the length of a rod or bar; and I) the change in length due to the change of temperature. Then D = K t /. (i8) D and / are expressed in the same unit.