Civil Engineering Reference
In-Depth Information
parallel to the
y
axis but vertically downwards, i.e. in the negative
y
direction as before;
similarly we take
W
z
to act in the negative
z
direction. The distributed loads
w
y
(
x
) and
w
z
(
x
) can be functions of
x
and are also applied in the negative directions of the axes.
The bending moment
M
z
in the vertical
xy
plane is, as before, a sagging (i.e. positive)
moment and will produce compressive direct stresses in the positive
yz
quadrant of the
beam section. In the same way
M
y
is positive when it produces compressive stresses in
the positive
yz
quadrant of the beam section. The applied torque
T
is positive when
anticlockwise when viewed in the direction
x
O and the displacements,
u
,
v
and
w
are
positive in the positive directions of the
z
,
y
and
x
axes, respectively.
The positive directions and senses of the internal forces acting on the positive face
(see Section 3.2) of a beam section are shown in Fig. 9.17 and agree, as far as the
shear force and bending moment in the vertical
xy
plane are concerned, with those
in Fig. 3.7. The positive internal horizontal shear force
S
z
is in the positive direction
of the
z
axis while the internal moment
M
y
produces compression in the positive
yz
quadrant of the beam section.
S
y
M
y
M
z
P
S
z
T
F
IGURE
9.17
Internal force
system
x
DIRECT STRESS DISTRIBUTION
Figure 9.18 shows the positive face of the cross section of a beam which is subjected to
positive internal bending moments
M
z
and
M
y
. Suppose that the origin O of the
y
and
z
axes lies on the neutral axis of the beam section; as yet the position of the neutral
axis and its inclination to the
z
axis are unknown.
We have seen in Section 9.1 that a beambends about the neutral axis of its cross section
so that the radius of curvature,
R
, of the beam is perpendicular to the neutral axis.
Therefore, by direct comparison with Eq. (9.2) it can be seen that the direct stress,
σ
x
,
on the element,
δ
A
, a perpendicular distance
p
from the neutral axis, is given by
E
p
R
σ
x
=−
(9.22)
