# Analysis of Stress

The basic structure of matter is characterized by nonuniformity and discontinuity attributable to its various subdivisions: molecules, atoms, and subatomic particles. Our concern in this text is not with the particulate structure, however, and it will be assumed that the matter with which we are concerned is homogeneous and continuously distributed over its volume. There is the clear implication in such an approach that the smallest element cut from the body possesses the same properties as the body. Random fluctuations in the properties of the material are thus of no consequence. This approach is that of continuum mechanics, in which solid elastic materials are treated as though they are continuous media, rather than composed of discrete molecules. Of the states of matter, we are here concerned only with the solid, with its ability to maintain its shape without the need of a container and to resist continuous shear, tension, and compression.

In contrast with rigid-body statics and dynamics, which treat the external behavior of bodies (that is, the equilibrium and motion of bodies without regard to small deformations associated with the application of load), the mechanics of solids is concerned with the relationships of external effect (forces and moments) to internal stresses and strains. Two different approaches used in solid mechanics are the mechanics of materials or elementary theory (also called the technical theory) and the theory of elasticity. The mechanics of materials focuses mainly on the more or less approximate solutions of practical problems. On the other hand, the theory of elasticity concerns itself largely with more mathematical analysis to determine the "exact" stress and strain distributions in a loaded body. The difference between these approaches is primarily in the nature of the simplifying assumptions used, described in Sec. 3.1.

External forces acting on a body may be classified as surface forces and body forces. A surface force is of the concentrated type when it acts at a point; a surface force may also be distributed uniformly or nonuniformly over a finite area. Body forces are associated with the mass of a body, rather than its surfaces, and are distributed throughout the volume of a body. Gravitational, magnetic, and inertia forces are all body forces. They are specified in terms of force per unit volume. All forces acting on a body, including the reactive forces caused by supports and body forces, are considered to be external forces. Internal forces are the forces that hold together the particles forming the body. Unless otherwise stated, we assume in this text that body forces can be neglected and that forces are applied steadily and slowly. The latter is referred to as static loading.

In the International System of Units (SI), force is measured in newtons (N). Because the newton is a small quantity, the kilonewton (kN) is often used in practice. In the U.S. Customary System, force is expressed in pounds (lb) or kilopounds (kips). We shall define all important quantities in both systems of units. However, in numerical examples and problems, SI units will be used throughout the text consistent with international convention. (Table D.2 compares the two systems.)

The study of the behavior of members in tension, compression, and bending began with Galileo Galilei (1564–1642), although Robert Hooke (1635–1703) was the first to point out that a body is deformed subject to the action of a force. Since then many engineers, physicists, and mathematicians in the field of stress analysis have contributed to the basic knowledge on which modern methods are based. The literature dealing with various aspects of solid mechanics is voluminous. For those seeking more thorough treatment, selected references are identified in brackets and compiled at the end of the text.

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