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

has a constant bending moment M, and consequently, this region is said to be in pure bending.
Pure bending occurs only in regions where the transverse shear force V is equal to zero.
Recall Equation (7.2), V = dM/dx. If the bending moment M is constant, then dM/dx = 0,
and thus, V = 0. Pure bending also implies that no axial forces act in the beam.
In contrast, nonuniform bending refers to flexure in which the shear force V is not
equal to zero. If V ≠ 0, then dM/dx ≠ 0, which means that the bending moment changes
along the span of the beam.
In the sections that follow, the strains and stresses in beams subjected to pure bending
will be investigated. Fortunately, the results obtained for pure bending can be applied to
beams with nonuniform bending if the beam is relatively long compared with its crosssectional
dimensions—in other words, if the beam is “slender.”
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FLExuRAL STRAINS
8.2 Flexural Strains
To investigate the strains produced in a beam subjected to pure bending, consider a short
segment of the beam shown in Figure 8.3. The segment, located between sections h–h and
k–k, is shown in Figure 8.4 with the deformations greatly exaggerated. The beam is assumed
to be straight before bending occurs, and the cross section of the beam is constant.
(In other words, the beam is a prismatic member.) Sections h–h and k–k, which were plane
surfaces before deformation, remain plane surfaces after deformation.
If the beam is initially straight, then all beam fibers between sections h–h and k–k are
initially the same length ∆x. After bending occurs, the fibers in the upper portions of the
cross section become shortened and the fibers in the lower portions become elongated.
However, a single surface where the fibers neither shorten nor elongate exists between the
upper and lower surfaces of the beam. This surface is called the neutral surface of the
beam, and its intersection with any cross section is called the neutral axis of the section.
All fibers on one side of the neutral surface are compressed, and those on the opposite side
are elongated.
When subjected to pure bending, the beam deforms into the shape of a circular arc.
The center O of this arc is called the center of curvature. The radial distance from the
center of curvature to the neutral surface of the beam is called the radius of curvature, and
it is designated by the Greek letter ρ (rho).
Consider a longitudinal fiber located at some distance y above the neutral surface. In
other words, the origin of the y coordinate axis will be located on the neutral surface. Before
bending, the fiber has a length ∆x. After bending, it becomes shorter, and its deformed
length will be denoted ∆x′. From the definition of normal strain given in Equation (2.1), the
normal strain of this longitudinal fiber can be expressed as
O
δ
ε x = = lim
L ∆ →
x 0
∆ x′ −∆x
∆x
ρ
Δθ
ρ – y
The beam segment subjected to pure bending deflects into the shape of a circular arc, and the
interior angle of this arc will be denoted ∆θ. According to the geometry shown in Figure 8.4,
the lengths ∆x and ∆x′ can be expressed in terms of arc lengths so that the longitudinal strain
ε x can be related to the radius of curvature ρ as
y
h
h
Δx′
Δx
k
k
ε
x
=
∆ x′ −∆x
ρ − y ∆θ − ρ∆θ
lim = lim ( ) =−
1 y
x 0 ∆x
∆θ
→0
ρ∆θ
ρ
∆ →
(8.1)
Neutral surface
FIGURE 8.4 Flexural
deformation.