28 CHAPTER 1. PROPERTIES OF MATTER
M N
M’ N’
M N
N’
M’
(a) (b)
Compressive
Force
Tensile
Force
F
F
neutral
surface neutral^
surface
neutral
axis
neutral
axis
Figure 1.23: Neutral filament MNNÕMÕ of a beam -(a)before bending and(b)after
bending
which opposes the bending of the beam.
Bending Moment
The moment of internal restoring couple is called the ‘moment of the resistance’ or
the ‘moment of the restoring couple’ orBending Moment(BM) of the beam.
At equilibrium, moment of the external bending couple
= moment of the internal restoring couple (BM).
The action of the bending moment can be thought of as “trying to rotate the cross-
section about the neutral axis (NNÕ) back to its original vertical orientation.”
1.9 Bending Moment of a Beam
Consider a beam whose neutral filament MN is bent into an arc of radiusRwith centre
at O as shown in Figure1.24. Then,
Stress due to bending in beams
x
The neutral axis is an axis
in the cross section of a
beam along which there are
no longitudinal stresses or
strains.
Radius of curvature of the
Neutral surface = R
Strain in layer EF = x/R
Stress due to bending at
layer EF= σ
E F
R
Young’s modulus = Y = (Stress due to bending at layer EF) / (Strain in layer EF)
Or Y = σ /(x/R) = (σR) /x
Hence, stress due to bending = σ = (Yx)/R
O
P x
P’ Q’
Q
N N’
θ
δA
M
D
Compressive
force
Tensile
force
Figure 1.24: A beam whose neutral filament is bent into an arc of radiusR
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