Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

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AxIAL dEFORMATION
More powerful computational methods derived from the theory of elasticity are
available to analyze objects that involve complicated loading and geometry. Of these
methods, the finite element method is the most widely used. Although the mechanics of
materials approach presented here is somewhat less rigorous than the theory of elasticity
approach, experience indicates that the results obtained from the mechanics of materials
approach are quite satisfactory for a wide variety of important engineering problems.
One of the primary reasons for this is Saint-Venant’s principle.
a
b
c
d
P
Bar width w
a
b
c
d
Lines are
distorted near
force P.
Lines away from
the ends remain
orthogonal.
Lines near support
are distorted by
the Poisson effect.
FIGURE 5.1 Rectangular bar
subjected to compressive force.
5.2 Saint-Venant’s principle
Consider a rectangular bar subjected to a compressive axial force P (Figure 5.1). The bar is
fixed at its base, and the total force P is applied to the top of the bar in three equal portions
distributed as shown over a narrow region equal to one-fourth of the bar’s width. The
magnitude of force P is such that the material behaves elastically; therefore, Hooke’s law
applies. The deformations of the bar are indicated by the grid lines shown. In particular,
notice that the grid lines are distorted in the regions near force P and near the fixed base.
Away from these two regions, however, the grid lines are not distorted, remaining orthogonal
and uniformly compressed in the direction of the applied force P.
Since Hooke’s law applies, stress is proportional to strain (and, in turn, deformation).
Therefore, stress will become more uniformly distributed throughout the bar as
the distance from the load P increases. To illustrate the variation of stress with distance
from P, the normal stresses acting in the vertical direction on Sections a–a, b–b,
c–c, and d–d (see Figure 5.1) are shown in Figure 5.2. On Section a–a (Figure 5.2a),
normal stresses directly under P are quite large while stresses on the remainder of the
cross section are very small. On Section b–b (Figure 5.2b), stresses in the middle of
the bar are still pronounced but stresses away from the middle are significantly larger
than those on Section a–a. Stresses are more uniform on Section c–c (Figure 5.2c). On
Section d–d (Figure 5.2d), which is located below P at a distance equal to the bar
width w, stresses are essentially constant across the width of the rectangular bar. This
comparison shows that localized effects caused by a load tend to vanish as the distance
from the load increases. In general, the stress distribution becomes nearly uniform at
a distance equal to the bar width w from the end of the bar, where w is the largest
lateral dimension of the axial member (such as the bar width or the rod diameter). The
maximum stress at this distance is only a few percent larger than the average stress.
In Figure 5.1, the grid lines are also distorted near the base of the axial bar because of
the Poisson effect. The bar ordinarily would expand in width in response to the compressive
normal strain caused by P. The fixity of the base prevents this expansion, and consequently,
additional stresses are created. Using an argument similar to that just given, we
could show that this increase in stress becomes negligible at a distance w above the base.
P
P
P
P
a
a
a
a
a
a
a
a
b
c
b
c
b
c
b
c
b
c
b
c
b
c
b
w
c
d
d
d
d
d
d
d
d
w
(a) (b) (c) (d)
FIGURE 5.2 Normal stress distributions on sections.