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TENSION
1. Introduction. By long tradition and common practice our subject is called "strength of materials," and, as is often the case with a traditional name, it does not strictly fit the facts. An uninitiated person would naturally think that under the heading "strength of materials" we would study the forces that various materials can withstand before they break or deform, but that subject is commonly called "properties of materials" or "materials testing." The traditional content of a course in strength of materials can more aptly be described as the "statics of deformable elastic bodies." In our subject we shall calculate the stresses and deflections or deformations in the beams, shafts, pipes, or other structures as functions of the loads imposed upon them and of the dimensions of the structure. As we shall see later, these stresses are usually independent of the material: a steel beam and a wooden beam of the same dimensions under the same loads will have the same stresses. The question of the conditions of failure of such a beam under the load, surprisingly enough, is only of secondary significance in our subject. In a typical calculation almost all of the work, say 95 per cent of it, will be "statics," independent of the material in use, and only at the very end will we substitute numerical values for an "allowable working stress" and for a "modulus of elasticity" of the material at hand.
The reason for this nomenclature is largely historical. Our subject owes much of its development to a great school of French mathematicians in the first half of the last century, of which the most outstanding names are Poisson, Lamé, Navier, Poncelet, Saint-Venant, and Boussinesq. Being mathematicians, they naturally considered their problem completely solved as soon as they had a formula relating stress to loading, and moreover they were convinced that they were working on a "practical" subject. Hence they gave to their subject the practical name résistance de matériaux, and their influence was so great that the name has persisted to this day among engineers in the English-speaking world.
Another distinction in nomenclature exists between "strength of materials" and "theory of elasticity." The general mathematical problem of finding the stresses for given applied loads on a body of arbitrary shape is extremely difficult and in fact has not been solved even by the mathematicians. On the other hand, we do possess solutions for simplified bodies, such as "beams" or "shafts," which are bodies in which the sidewise dimensions are negligibly small with respect to the length, or "plates," "slabs," and "curved shells," in which the thickness is negligibly small with respect to the other two dimensions. Although no practical structure exists in which one or more dimensions are mathematically negligible, they are negligible in practice, so that we "idealize" our given structure into equivalent beams, shafts, or plates. To this idealized structure we apply the theory. The more simple parts of that theory (bending, torsion, etc.) are called "strength of materials," while the more complicated parts are usually named "theory of elasticity." Hence the distinction between the two is not great. This book deals with the more elementary portions of strength of materials, sufficient to give a practically adequate solution for a great many of the commonly applied structures and machine elements. Therefore, to an engineer our subject is just about the most important of the subdivisions of mechanics that he must know.
Besides the questions of stress and deflection of structures, the subject also embraces the problem of stability. The main question in this category is the buckling of columns, which is the determination of the compressive load that can be placed lengthwise on a long, thin column before it buckles out sidewise, This problem is discussed in Chap. IX.
2. Hooke's Law. In the usual "tensile test" a bar of steel or other material is placed in a tensile-testing machine, and while it is slowly being pulled, readings are made of the pulling force and of the change in length (elongation) of the center portion of the bar. When these two quantities are plotted against each other, a diagram such as Fig. 1 results. In it we distinguish three stages, OA, AB, and BC. During the first stage OA, the diagram is substantially a straight line, and for ordinary steel the elongation OA is about one part in a thousand (0.001 in. elongation per 1 in. gage length), while the stress OA is about 30,000 Ib/sq in. Moreover, if the load at A is let off again, the force-elongation diagram goes back along the same line AO to O, and in particular the bar returns to point O, which means that after release of the load there is no "permanent elongation." This first stage OA is called the "elastic" stage. The next, or second, stage AB usually has some undulations in the diagram as shown, but it is substantially a horizontal piece of curve at constant force (constant stress), and the elongation will increase from 0.001 in./in. at A to about 0.020 in./in. at B. Thus the horizontal distance AB is very much greater than the horizontal distance OA, which could not be conveniently shown in the diagram. This second stage is known as the "plastic stage, in which the stress is substantially constant and equal to the "yield stress," about 30,000 Ib/sq in. for structural steel. In the third stage BC, the deformations become very large, and the test piece "necks," or shows a locally diminished diameter, and finally breaks in the center of the necking at point C in the diagram. If the load is let off at some point between A and C, before the final failure, the return curve is substantially a straight line parallel toOA, as shown by the dotted lines, and in that case the bar shows a "permanent elongation" after release of the load to zero.
Our subject of strength of materials deals only with the material in the first, or elastic, stage OA. Another branch of mechanics, named "plasticity," dealing with the second stage A to B, was started about twenty-five years ago. It is not only much younger but also much more difficult and complicated than our present subject and hence altogether outside the scope of this text.
The elastic, linear behavior of a material in the region OA of Fig. 1 is known as "Hooke's law." It states that the elongation is proportional to the force, or expressed in a formula:
s = P/A = E Δl,l
or
s/E = Δl/l = ε.
(1)
The quantity ε = Δl/l is the "unit elongation," or "strain," expressed in inches per inch and hence dimensionless. The quantity s is the stress or the force per unit area, expressed in pounds per square inch. Finally, E is a proportionality constant, which also must be expressed in pounds per square inch. It is known as the "modulus of elasticity," or also as "Young's modulus," after its inventor Thomas Young (1773–1829). The formula states that E has the dimension of a stress. In fact, it is the stress required to make Δl/l = 1, or to elongate the test piece to double its original length. Now this is an artificial statement, because steel obeys the law (1) only up to elongations of about one part in a thousand (Fig. 1), and the test piece will have broken long before it has doubled its length, but still the statement is useful for the visualization it conveys of the size of E. For steel, E = 30,000,000 lb/sq in.; the yield stress sv is about 30,000 lb/sq in.; and hence the yield unit elongation, or yield strain, is 0.001.
Within the limit of elasticity, then, the elongation is very small compared with the original length, and it will be assumed infinitesimally small in the sense of the calculus. This assumption leads to conclusions that are in good agreement with experimental facts, except for rubber or similar materials, which can double their lengths and in which the strain consequently can be large. The behavior of rubber under stress is not treated in strength of materials. It forms a subject in itself called "elasticity of large deformations," which is extremely complicated and still in the beginning of its development.
Problems 1 to 21.
3. Simple and Compound Bars. If a bar of a cross section A, constant along its length, is subjected to a tensile or compressive force P, that is, to a pair of forces directed along the length, then the elongation of that bar is determined by Eq. (1):
Δl = Pl/ΔE.
(1a)
If the cross section varies along the length x, that is, if A is a function of x, then the same relation still holds for a small element of length dx (Fig. 2). The elongation Δ dx of an element originally of length dx is:
Δ dx = P dx/ΔE,
and the total elongation of the entire bar is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(2)
As we shall see later, on page 204, this relation is strictly valid only if the bar changes its cross section gradually, i.e., if the curve of A plotted against x does not show steep slopes. In particular, it does not apply strictly to the practical case of Fig. 3a, where two pieces, each of constant cross section, are attached to each other, forming an abrupt change in that section. For that case Eq. (2) becomes:
Δl = P/E(l1/A1 + l2/A2),
and it is only approximately true. However, for the truncated cone of Fig. 3b the cross section changes sufficiently gradually so that the expression is perfectly good. In this case, the diameter of the bar changes linearly with the length, so that we can write:
rx = r1 + x/l (r2 - r1),
Ax = πr2x,
By Eq. (2) :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
This is a result of the same dimension as Eq. (1a), and it reduces to (1a) for the special case of a constant cross section rl = r2. We also see that the elongation Δl becomes infinitely large for a full cone rl = 0. This result does not appear strange any more when we reason that the stress at the apex of the cone is infinite, so that the problem then reduces to a practically impossible mathematical abstraction.
Next we consider the case of Fig. 4, a steel bolt with a bronze bushing shrunk hard around it, so that the two pieces form a solid unit. The modulus of steel Ea is different from that of bronze Eb; also the cross sections As and Ab are different. The length l and a possible elongation Δl must be the same for the steel and the bronze, but the total force P = Ps + Pb divides itself into unequal parts between the two. Thus we can write:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
From these two equations we solve for Ps and Pb with the result:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The reader should now verify that, for the case that Ab = As/ 2 and for Es = 30 × 106lb/sq in. and Eb = 12 × 106 lb/sq in., one-sixth of the load is carried by the bronze and five-sixths by the steel. How does the stress in the bronze then compare with the stress in the steel? Could the fact that these stresses come out in ratio of the two E's have been seen immediately without calculation?
Another problem is when the compound bolt of Fig. 4 is not subjected to a tensile load P as shown but instead is heated to a temperature T° above room temperature. We continue to assume that the shrink fit is so strong that no slippage occurs. The bronze will tend to expand longitudinally more than the steel but cannot do it. As a result, the bronze will not quite expand to its full length freely and hence will find itself in compression, while the steel is pulled by the bronze to expand more than it likes to and hence is in tension. Before bringing this story into formulas, we remark that the free thermal expansion of bronze is abTl, where αb is the expansion coefficient expressed in inches expansion per inch length per degree temperature rise. If the bronze so expands, there is no stress in it; but if it is elongated by a different amount, that difference must be caused by a pull which is a stress. Hence:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Excerpted from STRENGTH OF MATERIALS by J. P. Den Hartog. Copyright © 1977 J. P. Den Hartog. Excerpted by permission of Dover Publications, Inc..
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Overview
Developed at MIT, this distinguished introductory text is popular at engineering schools around the world. It also serves as a refresher and reference for professionals. In addition to coverage of customary elementary subjects (tension, torsion, bending, etc.), it features advanced material on engineering methods and applications, plus 350 problems and answers. 1949 edition.