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 temper ature stress produced by the change of temperature, and what is the actual unit-stress in the rod at the new temperature ? Let / 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: Now, since the rod could not shorten, it has a greater than normal length at the new temperature; that is, the fall in temperature has produced an effect equivalent to an elongation in the rod ing to D, and hence a tensile stress. This tensile stress can be computed from the elongation D by means of formula 17. Thus, S = E s; and since s, the unit-elongation, equals D0.000065 = = 0.000065, / / S = 30,000,000 X 0.000065 = 1,950 pounds per square inch. This is the value of the temperature stress; and the new unit stress equals 10,000 + 1,950 = 11,950 pounds per square inch.
Notice that the unit temperature stresses are independent of the length of the rod and the area of its cross-section.
2. Suppose that the change of temperature in the preceding example is a rise instead of a fall. What are the values of the temperature stress due to the change, and of the new unit-stress in the rod ? The temperature stress is the same as in example 1, that is, 1,950 pounds per square inch ; but the rise in temperature releases, as it were, the stress in the rod due to its being screwed up, and the final unit stress is 10,000 — 1,950 = 8,050 pounds per square inch.
1. The ends of a wrought-iron rod 1 inch in diameter are fastened to two heavy bodies which are to be drawn together, the temperature of the rod being 200 degrees when fastened to the ob jects. A fall of 120 degrees is observed not to move them. What is the temperature stress, and what is the pull exerted by the rod on each object ? Temperature stress, 22,000 pounds per square inch.
Ans.
Pull, 17,280 pounds.
n. Deflection of Beams. Sometimes it is desirable to know how much a given beam will deflect under a given load, or to design a beam which will not deflect more than a certain amount under a given load. In Table B, page 55, Part I, are given formulas for
deflection in certain cases of beams and different kinds of loading.
In those formulas, d denotes deflection; I the moment of inertia of the cross-section of the beam with respect to the neutral axis, as in equation 6; and E the coefficient of elasticity of the material of the beam (for values, see Art. 95).
In each case, the load should be expressed in pounds, the length in inches, and the moment of inertia in biquadratic inches; then the deflection will be in inches.
According to the formulas for d, the deflection of a beam varies inversely as the coefficient of its material (E) and the mo ment of inertia of its cross-section (I) ; also, in the first four and last two cases of the table, the deflection varies directly as the cube of the length (1').
Example. What deflection is caused by a uniform load of 6,400 pounds (including weight of the beam) in a wooden beam on end supports, which is 12 feet long and 6 X 12 inches in cross-section ? (This is the safe load for the beam ; see example 1, Art. 65.) The formula for this case (see Table B, page 55) is 5 d = 381 EI • Here W = 6,400 pounds ; 1 = 144 inches ; E = 1,800,000 pounds per square inch ; and I 1 = 12 6(13= 12 6 x 12'= 864 inches'. • 1. Compute the deflection of a timber built-in cantilever 8 X 8 inches which projects 8 feet from the wall and bears an end load of 900 pounds. (This is the safe load for the cantilever, see example 1, Art. 65.) Ans. 0.43 inch.
2. Compute the deflection caused by a uniform load of 40,000 pounds on a 42-pound 15-inch steel I-beam which is 16 feet long and rests on end supports.
Ans. 0.28 inch. 98. Twist of Shafts. Let Fig. 57 represent a portion of a shaft, and suppose that the part represented lies wholly between two adjacent pulleys on a shaft to which twisting forces are applied (see Fig. 54). Imagine two radii ma and nb in the ends of the portion, they being parallel as shown when the shaft is not twisted. After the shaft is twisted they will not be parallel, ma having moved to ma', and nb to nb'. The angle between the two lines in their twisted positions (ma' and nb') is called the angle of twist, or angle of torsion, for the length 1. If a'a" is parallel to ab, then the angle a"nb' equals the angle of torsion.