Section 4.6
Solid Mechanics Part I
Kelly
136
4.6 The Elementary Beam Theory
In this section, problems involving long and slender beams are addressed.
As with
pressure vessels, the geometry of the beam, and the specific type of loading which will be
considered, allows for approximations to be made to the full three-dimensional linear
elastic stress-strain relations.
4.6.1
The Beam
The term
beam
has a very specific meaning in engineering mechanics: it is a component
that is designed to support
transverse loads
, that is, loads that act perpendicular to the
longitudinal axis of the beam, Fig. 4.6.1.
The beam supports the load by
bending
only
.
Other mechanisms, for example twisting of the beam, are not allowed for in this theory.
Figure 4.6.1: A supported beam loaded by a force and a distribution of pressure
It is convenient to show a two-dimensional cross-section of the three-dimensional beam
together with the beam cross section, as in Fig. 4.6.1.
The cross section of this beam
happens to be rectangular but it can be any of many possible shapes.
It will assumed that
the beam has a
longitudinal plane of symmetry
, with the cross section symmetric about
this plane, as shown in Fig. 4.6.2.
Further, it will be assumed that the loading and
supports are also symmetric about this plane.
With these conditions, the beam has no
tendency to twist and will undergo bending only
1
.
Figure 4.6.2: The longitudinal plane of symmetry of a beam
Imagine now that the beam consists of many fibres aligned longitudinally, as in Fig.
4.6.3.
When the beam is bent by the action of downward transverse loads, the fibres near
the top of the beam contract in length whereas the fibres near the bottom of the beam
extend.
Somewhere in between, there will be a plane where the fibres do not change
length.
This is called the
neutral surface
.
The intersection of the longitudinal plane of
symmetry and the neutral surface is called the
axis of the beam
, and the deformed axis is
called the
deflection curve
.
1
certain special cases, where there is
not
a plane of symmetry for geometry and/or loading, can lead also to
bending with no twist, but these are not considered here
longitudinal plane
of symmetry
roller support
pin support
applied force
applied pressure
cross section

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