1.8. BENDING OF BEAMS
axis in the plane of symmetry is an axis of symmetry of the cross sections. In addition, all
loads are assumed to act in this plane. As a consequence, the bending deflections occur
in this same plane, which, therefore, is also known as the plane of bending.
A beam can be considered to be made up of a large number of thin plane layers called
filaments placed one above another.
Consider a beam bent by the application of an external bending couple as shown in
Figure1.23. Layers in the upper half are elongated while layers in the lower half are
compressed. However, at the middle there is a layer (MNNÕMÕ) which is neither elongated
nor compressed. This layer is called the ‘neutral surface’ of the beam. The intersection
of the neutral layer with any cross-sectional plane is called the neutral axis of the cross
section (NNÕin Figure1.23). The neutral axis is an axis in the cross section of a beam
M NM’ N’M NN’M’(a) (b)Compressive
ForceTensile
ForceF
F
neutral
surface neutral^
surfaceneutral
axisneutral
axisFigure 1.23: Neutral filament MNNÕMÕof a beam -(a)before bending and(b)after
bending
along which there are no longitudinal stresses or strains. All fibres on one side of the
neutral axis are in a state of tension, while those on the opposite side are in compression.
The extended filaments above the neutral axis (the layer MNNÕMÕin Figure1.23) which
are in state of tension exert an inward pull on the filaments above them. The shortened
filaments below the neutral filament MNNÕMÕwhich are in a state of compression exert
an outward push on the filaments below to them. As a result, tensile and compressive
stresses respectively develop in the upper and the lower halves of the beam (the stresses
due to restoring forces of equal magnitudeF and opposite directions, as indicated by
oppositely directed arrows above and below the neutral axisNNÕ) and form a couple
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”