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

The increased normal stress magnitudes near P and near the fixed base are examples
of stress concentrations. Stress concentrations occur where loads are applied, and they
also occur in the vicinity of holes, grooves, notches, fillets, and other changes in shape that
interrupt the smooth flow of stress through a solid body. Stress concentrations associated
with axial loads will be discussed in more detail in Section 5.7, and stress concentrations
associated with other types of loading will be discussed in subsequent chapters.
The behavior of strain near points where loads are applied was discussed in 1855 by
Barré de Saint-Venant (1797–1886), a French mathematician. Saint-Venant observed that
localized effects disappeared at some distance from such points. Furthermore, he observed
that the phenomenon was independent of the distribution of the applied load as long as the
resultant forces were “equipollent” (i.e., statically equivalent). This idea is known as Saint-
Venant’s principle and is widely used in engineering design.
Saint-Venant’s principle is independent of the distribution of the applied load, provided
that the resultant forces are equivalent. To illustrate this independence, consider the
same axial bar as discussed before; however, in this instance, the force P is split into four
equal portions and applied to the upper end of the bar, as shown in Figure 5.3. As in the
previous case, the grid lines are distorted near the applied loads, but they become uniform
at a moderate distance away from the point where the load is applied. Normal stress distributions
on Sections a–a, b–b, c–c, and d–d are shown in Figure 5.4. On Section a–a (Figure
5.4a), normal stresses directly under the applied loads are quite large while stresses in the
middle of the cross section are very small. As the distance from the load increases, the peak
stresses diminish (Figure 5.4b; Figure 5.4c) until the stresses become essentially uniform at
Section d–d (Figure 5.4d), which is located below P at a distance equal to the bar width w.
To summarize, peak stresses (Figure 5.2a; Figure 5.4a) may be several times the average
stress (Figure 5.2d; Figure 5.4d); however, the maximum stress diminishes rapidly as the distance
from the point where the load is applied increases. This observation is also generally true
for most stress concentrations (such as holes, grooves, and fillets). Thus, the complex localized
stress distribution that occurs near loads, supports, or other stress concentrations will not significantly
affect stresses in a body at sections sufficiently distant from them. In other words,
localized stresses and deformations have little effect on the overall behavior of a body.
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SAINT-VENANT’S PRINCIPLE
a
b
c
d
—
P
2
—
P
2
Bar width w
a
b
c
FIGURE 5.3 Rectangular bar
with a different, but equivalent,
applied load distribution.
d
Expressions will be developed throughout the study of mechanics of materials for
stresses and deformations in various members under various types of loadings. According
to Saint-Venant’s principle, these expressions are valid for entire members, except
for those regions very near load application points, very near supports, or very near
abrupt changes in member cross section.
—
P
2
—
P
2
—
P
2
—
P
2
—
P
2
—
P
2
—
P
2
—
P
2
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
(a) (b) (c) (d)
FIGURE 5.4 Normal stress distributions on sections.
w