We study statics to obtain a quantitative description of forces which act on engineering structures in equilibrium. Mathematics establishes the relations between the various quantities involved and enables us to predict effects from these relations. We use a dual thought process insolving statics problems: We think about both the physical situation and the corresponding mathematical description. In the analysis of every problem, we make a transition between the physical and the mathematical. One of the most important goals for the student is to develop the ability to make this transition freely.
Making Appropriate Assumptions
We should recognize that the mathematical formulation of a physical problem represents an ideal description, or model, which approximates but never quite matches the actual physical situation. When we construct an idealized mathematical model for a given engineering problem, certain approximations will always be involved. Some of these approximations may be mathematical, whereas others will be physical. For instance, it is often necessary to neglect small distances, angles, or forces compared with large distances, angles, or forces. Suppose a force is distributed over a small area of the body on which it acts. We may consider it to be a concentrated force if the dimensions of the area involved are small compared with other pertinent dimensions. We may neglect the weight of a steel cable if the tension in the cable is many times greater than its total weight. However, if we must calculate the deﬂection or sag of a suspended cable under the action of its weight, we may not ignore the cable weight. Thus, what we may assume depends on what information is desired and on the accuracy required.
Formulating Problems and Obtaining Solutions: In statics, as in all engineering problems, we need to use a precise and logical method for formulating problems and obtaining their solutions. We formulate each problem and develop its solution through the following sequence of steps.
The Free-Body Diagram: In solving a problem, it is essential that the laws which apply be carefully ﬁxed in mind and that we apply these principles literally and exactly. In applying the principles of mechanics to analyze forces acting on a body, it is essential that we isolate the body in question from all other bodies so that a complete and accurate account of all forces acting on this body can be taken. This isolation should exist mentally and should be represented on paper. The diagram of such an isolated body with the representation of all external forces acting on it is called a freebody diagram. The free-body-diagram method is the key to the understanding of mechanics. This is so because the isolation of a body is the tool by whichcause and effect are clearly separated, and by which our attention is clearly focused on the literal application of a principle of mechanics. The technique of drawing free-body diagrams is covered in Chapter 3, where they are ﬁrst used.