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72 Mechanical Engineering Design
One of the main objectives of this book is to describe how specific machine components
function and how to design or specify them so that they function safely without failing
structurally. Although earlier discussion has described structural strength in terms of
load or stress versus strength, failure of function for structural reasons may arise from
other factors such as excessive deformations or deflections.
Here it is assumed that the reader has completed basic courses in statics of rigid
bodies and mechanics of materials and is quite familiar with the analysis of loads, and
the stresses and deformations associated with the basic load states of simple prismatic
elements. In this chapter and Chap. 4 we will review and extend these topics briefly.
Complete derivations will not be presented here, and the reader is urged to return to
basic textbooks and notes on these subjects.
This chapter begins with a review of equilibrium and free-body diagrams associated
with load-carrying components. One must understand the nature of forces before
attempting to perform an extensive stress or deflection analysis of a mechanical com-
ponent. An extremely useful tool in handling discontinuous loading of structures
employs Macaulay or singularity functions. Singularity functions are described in
Sec. 3–3 as applied to the shear forces and bending moments in beams. In Chap. 4, the
use of singularity functions will be expanded to show their real power in handling
deflections of complex geometry and statically indeterminate problems.
Machine components transmit forces and motion from one point to another. The
transmission of force can be envisioned as a flow or force distribution that can be fur-
ther visualized by isolating internal surfaces within the component. Force distributed
over a surface leads to the concept of stress, stress components, and stress transforma-
tions (Mohr’s circle) for all possible surfaces at a point.
The remainder of the chapter is devoted to the stresses associated with the basic
loading of prismatic elements, such as uniform loading, bending, and torsion, and topics
with major design ramifications such as stress concentrations, thin- and thick-walled
pressurized cylinders, rotating rings, press and shrink fits, thermal stresses, curved beams,
and contact stresses.

3–1 Equilibrium and Free-Body Diagrams
Equilibrium
The word system will be used to denote any isolated part or portion of a machine or
structure—including all of it if desired—that we wish to study. A system, under this
definition, may consist of a particle, several particles, a part of a rigid body, an entire
rigid body, or even several rigid bodies.
If we assume that the system to be studied is motionless or, at most, has constant
velocity, then the system has zero acceleration. Under this condition the system is said
to be in equilibrium. The phrase static equilibrium is also used to imply that the system
is at rest. For equilibrium, the forces and moments acting on the system balance such
that

F = 0 (3–1)

M = 0 (3–2)
which states that the sum of all force and the sum of all moment vectors acting upon a
system in equilibrium is zero.