Civil Engineering Reference
In-Depth Information
Thus the direct stress varies through the depth of the beam from compression in the
upper fibres to tension in the lower. Clearly the direct stress is zero for the fibres that
do not change in length; we have called the plane containing these fibres the
neutral
plane
. The line of intersection of the neutral plane and any cross section of the beam
is termed the
neutral axis
(Fig. 9.3(b)).
The problem, therefore, is to determine the variation of direct stress through the depth
of the beam, the values of the stresses and subsequently to find the corresponding beam
deflection.
ASSUMPTIONS
The primary assumption made in determining the direct stress distribution produced
by pure bending is that plane cross sections of the beam remain plane and normal to
the longitudinal fibres of the beam after bending. Again, we saw this from the lines on
the side of the eraser. We shall also assume that the material of the beam is linearly
elastic, i.e. it obeys Hooke's law, and that the material of the beam is homogeneous.
Cases of composite beams are considered in Chapter 12.
DIRECT STRESS DISTRIBUTION
Consider a length of beam (Fig. 9.4(a)) that is subjected to a pure, sagging bending
moment,
M
, applied in a vertical plane; the beam cross section has a vertical axis of
symmetry as shown in Fig. 9.3(b). The bending moment will cause the length of beam
to bend in a similar manner to that shown in Fig. 9.3(a) so that a neutral plane will
M
M
pl
Neutral
axis
F
IGURE
9.3
Beam
subjected to a pure
sagging bending
moment
(a)
(b)
y
y
M
J
P
M
M
A
y
1
S
T
y
z
y
Neutral
axis
O
x
I
O
K
G
Q
y
2
N
F
IGURE
9.4
Bending of a
symmetrical
section beam
x
(a)
(b)
