21-11-2012, 11:54 AM
Mechanics of Solids
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Introduction
Engineers use the concepts and methods of mechanics of solids in designing and evaluating tools,
machines, and structures, ranging from wrenches to cars to spacecraft. The required educational background
for these includes courses in statics, dynamics, mechanics of materials, and related subjects. For
example, dynamics of rigid bodies is needed in generalizing the spectrum of service loads on a car,
which is essential in defining the vehicle’s deformations and long-term durability. In regard to structural integrity and durability, the designer should think not only about preventing the catastrophic failures of
products, but also of customer satisfaction. For example, a car with gradually loosening bolts (which is
difficult to prevent in a corrosive and thermal and mechanical cyclic loading environment) is a poor
product because of safety, vibration, and noise problems. There are sophisticated methods to assure a
product’s performance and reliability, as exemplified in Figure 1.1.1. A similar but even more realistic
test setup is shown in Color Plate 1.*
It is common experience among engineers that they have to review some old knowledge or learn
something new, but what is needed at the moment is not at their fingertips. This chapter may help the
reader in such a situation. Within the constraints of a single book on mechanical engineering, it provides
overviews of topics with modern perspectives, illustrations of typical applications, modeling to solve
problems quantitatively with realistic simplifications, equations and procedures, useful hints and reminders
of common errors, trends of relevant material and mechanical system behaviors, and references to
additional information.
Vectors. Equilibrium of Particles. Free-Body Diagrams
Two kinds of quantities are used in engineering mechanics. A scalar quantity has only magnitude (mass,
time, temperature, …). A vector quantity has magnitude and direction (force, velocity, ...). Vectors are
represented here by arrows and bold-face symbols, and are used in analysis according to universally
applicable rules that facilitate calculations in a variety of problems. The vector methods are indispensable
in three-dimensional mechanics analyses, but in simple cases equivalent scalar calculations are sufficient.
Vector Components and Resultants. Parallelogram Law
A given vector F may be replaced by two or three other vectors that have the same net effect and
representation. This is illustrated for the chosen directions m and n for the components of F in two
dimensions (Figure 1.2.1). Conversely, two concurrent vectors F and P of the same units may be
combined to get a resultant R (Figure 1.2.2).
Forces on Rigid Bodies
All solid materials deform when forces are applied to them, but often it is reasonable to model components
and structures as rigid bodies, at least in the early part of the analysis. The forces on a rigid body are
generally not concurrent at the center of mass of the body, which cannot be modeled as a particle if the
force system tends to cause a rotation of the body.
Equilibrium of Rigid Bodies
The concept of equilibrium is used for determining unknown forces and moments of forces that act on
or within a rigid body or system of rigid bodies. The equations of equilibrium are the most useful
equations in the area of statics, and they are also important in dynamics and mechanics of materials.
The drawing of appropriate free-body diagrams is essential for the application of these equations.
Calculation of Unknown Forces and Moments
In solving for unknown forces and moments, always draw the free-body diagram first. Unknown external
forces and moments must be shown at the appropriate places of action on the diagram. The directions
of unknowns may be assumed arbitrarily, but should be done consistently for systems of rigid bodies.
A negative answer indicates that the initial assumption of the direction was opposite to the actual
direction. Modeling for problem solving is illustrated in Figures 1.2.11 and 1.2.12.
Related Free-Body Diagrams
When two or more bodies are in contact, separate free-body diagrams may be drawn for each body. The
mutual forces and moments between the bodies are related according to Newton’s third law (action and
reaction). The directions of unknown forces and moments may be arbitrarily assumed in one diagram,
but these initial choices affect the directions of unknowns in all other related diagrams. The number of
unknowns and of usable equilibrium equations both increase with the number of related free-body
diagrams.