95
C H A P T E R 3
TwoDimensional
Problems in Elasticity
3.1
INTRODUCTION
As has been pointed out in Sec. 1.1, the approaches in widespread use for determin
ing the influence of applied loads on elastic bodies are the
mechanics of materials
or
elementary theory
(also known as
technical theory
) and the
theory of elasticity
.
Both must rely on the conditions of equilibrium and make use of a relationship be
tween stress and strain that is usually considered to be associated with elastic mate
rials. The essential difference between these methods lies in the extent to which the
strain is described and in the types of simplifications employed.
The mechanics of materials approach uses an assumed deformation mode or
strain distribution in the body as a whole and hence yields the
average
stress at a
section under a given loading. Moreover, it usually treats separately each simple
type of complex loading, for example, axial centric, bending, or torsion.Although of
practical importance, the formulas of the mechanics of materials are best suited for
relatively slender members and are derived on the basis of very restrictive condi
tions. On the other hand, the method of elasticity does not rely on a prescribed de
formation mode and deals with the general equations to be satisfied by a body in
equilibrium under any external force system.
The theory of elasticity is preferred when critical design constraints such as
minimum weight, minimum cost, or high reliability dictate more exact treatment or
when prior experience is limited and intuition does not serve adequately to supply
the needed simplifications with any degree of assurance. If properly applied, the
theory of elasticity should yield solutions more closely approximating the actual
distribution of strain, stress, and displacement.
Thus, elasticity theory provides a check on the limitations of the mechanics of
materials solutions.We emphasize, however, that both techniques cited are approxi
mations of nature, each of considerable value and each supplementing the other.
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96
Chapter 3
TwoDimensional Problems in Elasticity
The influences of material anisotropy, the extent to which boundary conditions de
part from reality, and numerous other factors all contribute to error.
3.2
FUNDAMENTAL PRINCIPLES OF ANALYSIS
To ascertain the distribution of stress, strain, and displacement within an elastic
body subject to a prescribed system of forces requires consideration of a number of
conditions relating to certain physical laws, material properties, and geometry.
These fundamental principles of analysis, also referred to as the
three aspects of
solid mechanics problems
, are summarized as follows:
1.
Conditions of equilibrium.
The equations of statics must be satisfied throughout
the body.
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 Spring '08
 yok
 Force, Stress, Eqs., plane stress

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