Strength of materials - Boyd

This book is intended to give the student a grasp of the physical and mathematical ideas underlaying the Mechanics of Materials, together with enough of the experimental facts and simple applications to sustain his interest, fix his theory, and prepare him for the technical subjects as given in works on Machine Design, Reinforced Concrete, or Stresses in Structures.

It is assumed that the reader has completed the Integral Calculus, and has taken a course in Theoretical Mechanics which includes statics and the moment of inertia of plane areas. Chapters XVI and XVII give a brief discussion of center of gravity and moment of inertia. Students who have not mastered these subjects should study these chapters before taking up Chapter V (preferably before beginning Chapter I).

The problems, which are given with nearly every article, form an essential part of the development of the subject. They were prepared with the twofold object of fixing the theory and enabling the student to discover for himself important facts and applications. The first problems of each set usually require the use of but one new principle, - the one given in the text which immediately precedes; the later problems aim to combine this principle with others previously studied and with the fundamental operations of Mathematics and Mechanics. The constants given in the data or derived from the results of the problems fall within the range of the figures obtained from actual tests of materials. Many of the problems are taken directly from such measurements. Some of them are from tests made by the author or his colleagues at the Ohio State University; others are from bulletins of the University of Illinois Engineering Experiment Station, from "Tests of Metals" at the Watertown Arsenal, and from the Transactions of the American Society of Civil Engineers.

This book is designed for use with "Cambria Steel," to which references are made by title instead of by page, so that they are adapted to any edition of the handbook.

1. Strength of Materials. That branch of Mechanics which treats of the changes in form and dimensions of elastic solids and the forces which cause these changes is called The Mechanics of Materials. When
the physical constants and the results of experimental tests upon the materials of construction are included with the theoretical discussion of the ideal elastic solid, the entire subject is called The Strength of Materials or The Resistance of Materials.

2. Tension. Support one end of a band of soft rubber, and attach a small hook to the other end, as shown in Fig. 1. Now apply a small weight to the hook. The rubber band is stretched; its length is increased by an amount a, while its cross section is diminished. Add a second weight. If the second weight is equal to the first one, the elongation b, which it causes, is equal to that caused by the first weight. Remove the weights, and the rubber band returns to its original length and cross section.

If steel, iron, wood, concrete, stone, or other solid material is used instead of rubber, the results are similar. There is this apparent difference: while the rubber may be stretched to twice or three times its original length and still return to its original size and shape after the load is removed, one of the other materials may be stretched only a very small amount (usually less than 0.002 of its length), without receiving a permanent change in its dimensions. Again, the force required to produce a relatively small increase in the length of a rod of wood or steel, for instance, is many times greater than that necessary to double the length of a soft rubber band of equal cross section. These differences between the behavior of soft rubber and other solid materials are differences of degree and not of kind. Essentially they are alike.

The rubber bands shown in Fig. 1 are subjected to the action of two forces: the force of the weights pulling downward, and the reaction of the support pulling upward. The bands are in tension. A body is said to be in tension when it is subjected to two sets of forces whose resultants are in the same straight line, opposite in direction, and directed away from each other.

3. Compression. When a body is subjected to two sets of forces whose resultants are in the same straight line, opposite in direction, and directed toward each other, it is said to be in compression. In Fig. 2, the block B is in compression under the action of the 50 pounds pushing down and the reaction of the support pushing up. The effect of compression upon a body is to shorten it in the line of the forces and increase its dimensions in the plane perpendicular to this line.

Tension and compression may be represented as in Fig. 3, where the arrows represent the forces, and the small rectangles represent the bodies, or portions of a body, upon which the forces act. The rectangles are often omitted; a pair of arrows with Fi 3 their heads together indicate compression, and a pair with their heads in the opposite sense indicate tension.

4. Stress; Total Stress. The force exerted by one body upon another at their surface of contact is called the stress between the bodies or the total stress between the bodies. If a single body be regarded as cut by an imaginary surface, the force exerted across this surface by either portion of the body upon the other portion is the total internal stress in the body at the section.

In the case of an internal stress, if the forces are such that the portions of the body are pushed together at the imaginary surface, the stress is compressive. If the forces tend to pull the portions apart, the stress is tensile.

Compressive stress at the surface of contact of two separate bodies is called bearing stress. All parts of the bar AB, Fig. 4, are under tensile stress. The total tensile stress at any section CD is the load L and the weight of the hook and of that portion of the bar below the section.

All parts of the block in Fig. 5 are in compression. The total compressive stress at any section JK is 10 pounds plus the weight of the portion of the block above the section; or, since action and reaction are equal, it is the upward reaction at the base minus the weight of the portion below JK.

5. Unit Stress; Intensity of Stress. The unit stress at any surface is the total stress at the surface divided by its area. Unit stress is frequently called intensity of stress. In American engineering practice, unit stresses are usually expressed in pounds per square inch. Compressive stresses in masonry are sometimes given in tons per square foot; the bearing pressure of masonry upon soils is always so expressed. English engineers frequently use long tons per square inch to express the intensity of stress in steel and similar solids. Continental engineers, of course, use kilograms per square centimeter. Physicists, the world over, prefer dynes per square centimeter. In the case of tensile or compressive stresses, the surface considered is a plane section perpendicular to the direction of the forces, unless otherwise stated.

6. Working Stress; Allowable Unit Stress. Working stresses are the unit stresses to which the materials of a structure or machine are subjected. The allowable unit stress for a given material is the maximum working stress which, in the judgment of some engineer or other authority, should be applied to that material.

7. Deformations; Unit Deformation. The changes in dimensions which occur when forces are applied to a body are called deformations. In Fig. 1, the increase of length, a, which takes place when the first load is applied is the deformation due to that load, the increase b is the deformation due to the second load, and a + b is the deformation due to the two loads. The deformation produced by a tensile force or pull is an elongation; that caused by a compressive force or push is a compression. A deformation which remains after the force is removed is called a set.

8. Elastic Limit. When a stress is applied to a solid body and then removed, the body returns to its original size and shape, provided the stress has not exceeded a certain limit. If the stress has gone beyond this limit, the body does not return entirely to its original dimensions, but retains some permanent deformation or set. This limit is called the elastic limit of the material. A wrought-iron rod in tension is stretched about 0.006 inch in a length of 8 inches by a load of 20,000 pounds per square inch. When the load is removed, it returns to its original length. A stress of 20,000 pounds per square inch, or the corresponding unit elongation of 0.00075 inch, is below the elastic limit of wrought iron. If the load is increased to 30,000 pounds per square inch, the elongation in 8 inches becomes, perhaps, 0.075 inch. Upon the removal of the load the rod shortens only 0.009 inch and the residual 0.066 inch remains as a permanent set. The elastic limit is below 30,000 pounds per square inch.

9. Modulus of Elasticity. For all stresses below the elastic limit the unit stress bears a constant ratio to the unit deformation. The quotient obtained by dividing unit stress by the accompanying unit deformation is called the modulus of elasticity of the material, or Young's modulus.