专业英语第二单元 Unit 2 Stress and strain

Unit 2 Stress and strain

1 Introduction to mechanics of materials

Mechanics of materials is a branch of applied mechanics that deals with the behaviour of solid bodies subjected to various types of loading. It is a field of study that is known by variety of names, including “strength of materials ” and “mechanics of deformable bodies” . The solid bodies considered in this book include axially-loaded bars, shafts, beams, and columns, as well as structures that are assemblies of these components. Usually the objective of our analysis will be the determination of stresses, strains, and the deformations produced by the loads; if these quantities can be found for all values of load up to the failure load, then we will have obtained a complete picture of the mechanical behaviour of the body.

Theoretical analysis and experimental results have equally important roles in the study of mechanics of materials. On many occasions we will make logical derivations to obtain formulas and equations for predicting mechanical behaviour, but at the same time we must recognize that these formulas

can not be used in a realistic way unless certain properties of the material are known. These properties are available to us only after suitable experiments have been made in laboratory. Also, many problems of importance in engineering can not be handled efficiently by theoretical means, and experimental measurements become a practical necessity. The historical development of mechanics of material is a fascinating blend of both theory and experiment, with experiments pointing the way to useful results in some instances and with theory doing so in others. Such famous men as Leonard da vinci(1452-1519) and Galileo Galilei(1564-1642) made experiments to determine the strength of wires, bars, and beams, although they did not develop any adequate theories(by today’s standards) to explain their test results. By contrast, the famous mathematician Leonhard Euler(1707-1783) developed the mathematical theory of columns and calculated the critical load of a column in 1744, long before any experimental evidence existed to show the significance of his results. Thus, Euler’s theoretical results

remained unused for many years, although today they form the basis of column theory.

The importance of combining theoretical derivations

with

experimentally

determined

properties of materials will be evident as we proceed with our study of the subject. In this section we will begin by discussing some fundamental concepts, such as stress and strain, and then we will investigate the behaviour of simple structural elements subjected to tension, compression, and shear. 2 Stress

The concepts of stress and strain can be illustrated in an elementary way by considering the extension of prismatic bar [see Fig. 1.4(a)]. A prismatic bar is one that has constant cross section throughout its length and a straight axis. In this illustration the bar is assumed to be loaded at its ends by axial forces P that produce a uniform stretching, or tension, of the bar. By making an artificial cut(section mm) through the bar at right angles to its axis, we can isolate part of the bar as a free body[Fig. 1.4(b)]. At the right-hand end the tensile force P is

applied, and at the other end there are forces representing the action of the removed portion of the bar upon the part that remains. These forces will be continuously distributed over the cross section, analogous to the conditions distribution of hydrostatic pressure over a submerged surface. The intensity of force, that is, the per unit area, is called the stress and is commonly denoted by the Greek letter

?.

Assuming that the stress has a uniform distribution over the cross section[see Fig.1.4(b)], we can readily see that its resultant is equal to the intensity ? times the cross-sectional area A of the bar. Furthermore, from the equilibrium of the body shown in Fig1.4(b), we can also see that this resultant must be equal in magnitude and opposite in direction to the force P. Hence, we obtain

??P Aas the equation for the uniform stress in a prismatic bar. This equation shows that stress has units of force divided by area---for example, Newtons per square millimeter(N/m2) or pounds per square inch(psi). When the bar is being stretched by the force P, as

shown in the figure, the resulting stress is a tensile stress; if the forces are reversed in direction, causing the bar to be compressed, they are called compressive stresses.

A necessary for Eq.(1.3) to be valid is that the stress ? must be uniform over the cross section of the bar. This condition will be realized if the axial force P acts through the centroid of the cross section, as can be demonstrated by statics. When the load P does not act at the centroid, bending of the bar will result, and a more complicated analysis is necessary. Throughout this book, however, it is assumed that all axial forces are applied at the centroid of the cross section unless specifically stated to the contrary. Also, unless stated otherwise, it is generally assumed that the weight of the object itself is neglected, as was done when discussing the bar in Fig.1.4. 3 Strain

The total elongation of the bar carrying an axial force will be denoted by the Greek letter ?[see Fig.1.4(a)], and the elongation per unit length, or strain, is then determined by the equition

???L (1.4)

where L is the total length of the bar. Note that the strain

? is nondimensional quantity. It can be

obtained accurately from Eq.(1.4) as long as the strain is uniform throughout the length of the bar. If the bar is in tension, the strain is a tensile strain, representing an elongation or a stretching of the material; if the bar is in compression, the strain is a compressive strain, which means that adjacent cross sections of the bar move closer to one another.

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