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

The most common stress–strain relationship encountered in engineering is the equation
for a linear elastic material, defined by Hooke’s law: σ = Eε. If the strain relationship
defined in Equation (8.1) is combined with Hooke’s law, then the variation of normal stress
with distance y from the neutral surface can be expressed as
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NORMAL STRESSES IN bEAMS
E
σx = Eεx = − y = − E κ y
(8.3)
ρ
Equation (8.3) shows that the normal stress σ x on the transverse section of the beam varies
linearly with distance y from the neutral surface. This type of stress distribution is shown
in Figure 8.5a for the case of a bending moment M that produces compressive stresses
above the neutral surface and tensile stresses below the neutral surface.
While Equation (8.3) describes the variation of normal stress over the depth of a beam,
its usefulness depends upon knowing the location of the neutral surface. Moreover, the
radius of curvature ρ is generally not known, whereas the internal bending moment M is
readily available from shear-force and bending-moment diagrams. A more useful relationship
than Equation (8.3) would be one that related the normal stresses produced in the
beam to the internal bending moment M. Both of these objectives can be accomplished by
determining the resultant of the normal stress σ x acting over the depth of the cross section.
In general, the resultant of the normal stresses in a beam consists of two components:
(a) a resultant force acting in the x direction (i.e., the longitudinal direction) and
(b) a resultant moment acting about the z axis.
If the beam is subjected to pure bending, the resultant force in the longitudinal direction
must be zero and the resultant moment must equal the internal bending moment M in
the beam. Then, on the basis of the stress distribution shown in Figure 8.5a, two equilibrium
equations can be written: ΣF x = 0 and ΣM z = 0. From these two equations,
(a) the location of the neutral surface can be determined and
(b) the relationship between bending moment and normal stress can be established.
Since plane cross sections
remain plane, the normal stress
σ x caused by bending is also
uniformly distributed in the
z direction.
Location of the Neutral Surface
The cross section of the beam is shown in Figure 8.5b. We will consider a small element dA
of the cross-sectional area A. The beam is assumed to be homogeneous, and the bending
stresses are produced at an arbitrary radius of curvature ρ. The distance from the area
dA to the neutral axis is measured by the coordinate y. The normal stresses acting on area dA
The intersection of the neutral
surface (which is a plane) and
any cross section of the beam (also
a plane surface) is a line, which is
termed the neutral axis.
y
y
M
Neutral surface
(a) Side view of beam, showing
normal stress distribution
O
FIGURE 8.5 Normal stresses in a beam of linearly elastic material.
σ x
x
z
c top
c bot
O
Neutral axis
(b) Beam cross section
dA
y
In Figure 8.5a, compressive
stresses are indicated by arrows
pointing toward the cross section
and tensile stresses are indicated
by arrows pointing away from
the cross section.