Civil Engineering Reference
In-Depth Information
Now rotate the eraser so that its shorter sides are vertical and apply the same pressure
with your fingers. The eraser again bends but now requires much less effort. It follows
that the geometry and orientation of a beam section must affect its
bending stiffness
.
This is more readily demonstrated with a plastic ruler. When flat it requires hardly any
effort to bend it but when held with its width vertical it becomes almost impossible to
bend. What does happen is that the lower edge tends to move sideways (for a hogging
moment) but this is due to a type of instability which we shall investigate later.
We have seen in Chapter 3 that bending moments in beams are produced by the action
of either pure bending moments or shear loads. Reference to problem P.3.4 also shows
that two symmetrically placed concentrated shear loads on a simply supported beam
induce a state of pure bending, i.e. bending without shear, in the central portion of the
beam. It is also possible, as we shall see in Section 9.2, to produce bending moments
by applying loads parallel to but offset from the centroidal axis of a beam. Initially,
however, we shall concentrate on beams subjected to pure bending moments and
consider the corresponding internal stress distributions.
9.1 S
YMMETRICAL
B
ENDING
Although symmetrical bending is a special case of the bending of beams of arbitrary
cross section, we shall investigate the former first, so that the more complex general
case may be more easily understood.
Symmetrical bending arises in beams which have either singly or doubly symmetrical
cross sections; examples of both types are shown in Fig. 9.2.
Suppose that a length of beam, of rectangular cross section, say, is subjected to a pure,
sagging bending moment,
M
, applied in a vertical plane. The length of beam will bend
into the shape shown in Fig. 9.3(a) in which the upper surface is concave and the lower
convex. It can be seen that the upper longitudinal fibres of the beam are compressed
while the lower fibres are stretched. It follows that, as in the case of the eraser, between
these two extremes there are fibres that remain unchanged in length.
Axis of symmetry
F
IGURE
9.2
Symmetrical
section beams
Double
(rectangular)
Double
(I-section)
Single
(channel section)
Single
(T-section)
