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

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bENdINg
Equation (8.1) indicates that the normal strain developed in any fiber is directly proportional
to the distance of the fiber from the neutral surface. Equation (8.1) is valid
for beams of any material, whether the material is elastic or inelastic, linear or nonlinear.
Notice that the strain determined here occurs in the x direction, even though the
loads applied to the beam act in the y direction and the beam bends about the z axis.
For a positive value of ρ (as defined shortly), the negative sign in Equation (8.1) indicates
that compressive strain will be developed in the fibers above the neutral surface
(i.e., where y values are positive) while tensile strain will occur below the neutral
surface (where y values are negative). Note that the sign convention for ε x is the same
as that defined for normal strains in Chapter 2; specifically, elongation is positive and
shortening is negative.
Curvature κ (Greek letter kappa) is a measure of how sharply a beam is bent, and it
is related to the radius of curvature ρ by
κ = 1 (8.2)
ρ
If the load on a beam is small, then the beam deflection will be small, the radius of curvature
ρ will be very large, and the curvature κ will be very small. Conversely, a beam with
a large deflection will have a small radius of curvature ρ and a large curvature κ. For the
x–y–z coordinate axes used here, the sign convention for κ is defined such that κ is positive
if the center of curvature is located above a beam. The center of curvature O for the beam
segment shown in Figure 8.4 is located above the beam; therefore, this beam has a positive
curvature κ, and in accordance with Equation (8.2), the radius of curvature ρ must be
positive, too. To summarize, κ and ρ always have the same sign. They are both positive if
the center of curvature is located above the beam, and they are both negative if the center
of curvature is located below the beam.
Transverse Deformations
Longitudinal strains ε x in the beam are accompanied by deformations in the plane of the
cross section (i.e., strains in the y and z directions) because of the Poisson effect. Since
most beams are slender, the deformations in the y–z plane due to Poisson effects are very
small. If the beam is free to deform laterally (as is usually the case), normal strains in the
y and z directions do not cause transverse stresses. This situation is comparable to that of
a prismatic bar in tension or compression, and therefore, the longitudinal fibers in a beam
subjected to pure bending are in a state of uniaxial stress.
8.3 Normal Stresses in Beams
For pure bending, the longitudinal strain ε x that occurs in the beam varies in proportion to
the fiber’s distance from the neutral surface of the beam. The variation of normal stress σ x
acting on a transverse cross section can be determined from a stress–strain curve for the
specific material used to fabricate the beam. For most engineering materials, the stress–
strain diagrams for both tension and compression are identical in the elastic range.
Although the diagrams may differ somewhat in the inelastic range, the differences can be
neglected in many instances. For the beam problems considered in this book, the tension and
compression stress–strain diagrams will be assumed identical.