Civil Engineering Reference
In-Depth Information
y
P
e
z
e
z
P
e
y
e
y
G
P
P
F
IGURE
9.9
Combined bending
and axial load on a beam section
z
x
Consider now a length of beam having a vertical plane of symmetry and subjected
to a tensile load,
P
, which is offset by positive distances
e
y
and
e
z
from the
z
and
y
axes, respectively (Fig. 9.9). It can be seen that
P
is equivalent to an axial load
P
plus
bending moments
Pe
y
and
Pe
z
about the
z
and
y
axes, respectively. The moment
Pe
y
is a negative or hogging bending moment while the moment
Pe
z
induces tension in
the region where
z
is positive;
Pe
z
is, therefore, also regarded as a negative moment.
Thus at any point (
y
,
z
) the direct stress,
σ
x
, due to the combined force system, using
Eqs (7.1) and (9.9), is
P
A
+
Pe
y
I
z
Pe
z
I
y
σ
x
=
y
+
z
(9.15)
Equation (9.15) gives the value of
σ
x
at any point (
y
,
z
) in the beam section for any
combination of signs of
P
,
e
z
,
e
y
.
E
XAMPLE
9.5
A beam has the cross section shown in Fig. 9.10(a). It is subjected
to a normal tensile force,
P
, whose line of action passes through the centroid of the
horizontal flange. Calculate the maximum allowable value of
P
if the maximum direct
stress is limited to
150N
/
mm
2
.
±
The first step in the solution of the problem is to determine the position of the centroid,
G, of the section. Thus, taking moments of areas about the top edge of the flange we
have
(200
×
20
+
200
×
20)
y
¯
=
200
×
20
×
10
+
200
×
20
×
120
from which
y
¯
=
65mm
