These two expressions must be equal to each other, or It will be seen from this that the section of the bar does not affect the length under the conditions stated. Example 7.-Two round steel bars, each of 2 inches diameter, are joined together by means of a steel pin in the manner shown in fig. 4. If the ultimate strength of the steel to resist tension be 30 tons per square inch, and its ultimate shearing strength be 24 tons per square inch, what must be the diameter of the pin so as to be equal in strength to the bars? These two expressions, according to the conditions of the question, must be equal to each other, or which is the required diameter of the pin. 11. Long Struts.-The rule embodied in the formula F = af, does not hold when applied to compressive stresses, if the diameter of the bar or column is small in proportion to its length. Long bars when subjected to a compressive stress in the direction of their length do not break entirely by crushing. They have a tendency to become deflected laterally, and to break partially by cross-fracture. This happens in bars made of iron or steel when the length of the bar is more than five times its diameter, or least thickness; and in the case of wood when the proportion exceeds 10 to 1. Members of this kind are termed "long struts," and their strengths will be investigated in Chapter XI. CHAPTER II. ELASTICITY AND FATIGUE OF MATERIALS. 12. Elasticity. There are other properties of materials which make them valuable for structures besides their tensile, compressive, and shearing strengths. One of the principal of these is elasticity. Elasticity is the term applied to that property which materials possess of returning to their original size and shape after they have been strained; and a material is said to be elastic if the strain disappears after the stress has been removed. When a gradually-increasing tensile stress is applied to a bar of iron, steel, or other material, it becomes elongated or stretched, and the amount of this elongation, or increment of length, within certain limits, is proportional to the stress applied. The same law holds good if the bar is subjected to a compressive stress. In this latter case the bar is shortened, and the amount of shortening, or decrement of length, is proportional to the stress within the limits named. This principle is known as Hooke's law of uniform elastic reaction, or the law of elasticity, to which the discoverer applied the Latin phrase " Ut tensio sic vis." The truth of this law has been proved by more than one experimenter. The late Mr. Hodgkinson instituted an elaborate series of experiments on cast and wrought iron, subjecting these materials both to compressive and tensile stresses, and the results he obtained practically prove the truth of the above law, so far as these materials are concerned. 13. Limits of Elasticity. The limits of stress, between which bodies are elastic, are termed their limits of elasticity. The range of these limits is very much greater in some materials than in others. Lead, for example, has little or no elasticity. If it be strained to any appreciable extent, it will be found that when the stress which produces the strain is removed, the strain itself does not suffer any appreciable diminution. Lead, therefore, may be called a non-elastic substance. On the other hand, glass is very elastic. If it be strained just up to its breaking-point, when the stress is removed, it regains its original dimensions. Iron and steel occupy a position intermediate between lead and glass as regards their elasticity. They are elastic, or nearly so, up to about one-half of their ultimate strength. When a body |