1.8 Bending of Beams................................
Self Learning Activity : Torsion PendulumUsing the torsion pendulum simulator, study the behaviour of
di erent torsion pendula and determine moments of inertia of
the given discs and rigidity moduli of the available materials.
URL:http://vlab.amrita.edu/?sub=1&brch=2801.8 Bending of Beams
Beam theory began with Galileo Galilei (1564-1642), who investigated the behaviour of
various types of beams. His work in mechanics of materials is described in his famous
bookTwo New Sciences, first published in 1638.
A structural member that is designed to resist forces acting laterally to its axis is
called a beam. Beams di er from bars in tension and torsion, primarily because of the
directions of the loads that act upon them. A bar in tension is subjected to loads directed
along the axis, and a bar in torsion is subjected to torques having their vectors along the
axis. By contrast, the loads on a beam are directed normal to the axis.
Before a load is applied, the longitudinal axis of the beam is a straight line. Lateral
loads acting on a beam cause the beam to bend, or flex, thereby deforming the axis of
the beam into a curve, called the deflection curve of the beam.
We consider beams which are symmetric about a plane (usually taken as the plane of
the paper). This plane is called a plane of symmetry of the beam. Therefore, the vertical
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
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