Civil Engineering Reference
In-Depth Information
which simplifies to
d
2
4
y
2
6
S
y
bd
3
τ
=−
−
(10.7)
The distribution of vertical shear stress is therefore parabolic as shown in Fig. 10.5(b)
and varies from
τ
=
0at
y
=±
d
/
2to
τ
=
τ
max
=
3
S
y
/
2
bd
at the neutral axis (
y
=
0) of
the beam section. Note that
τ
max
=
1
.
5
τ
av
, where
τ
av
, the average vertical shear stress
over the section, is given by
τ
av
=
S
y
/
bd
.
E
XAMPLE
10.2
Determine the distribution of vertical shear stress in the I-section
beam of Fig. 10.6(a) produced by a vertical shear load,
S
y
.
It is clear fromFig. 10.6(a) that the geometry of each of the areas
A
f
and
A
w
formed by
taking a slice of the beam in the flange (at
y
y
w
), respectively,
are different and will therefore lead to different distributions of shear stress. First we
shall consider the flange. The area
A
f
is rectangular so that the distribution of vertical
shear stress,
τ
f
, in the flange is, by direct comparison with Ex. 10.1
=
y
f
) and in the web (at
y
=
D
2
−
y
f
D
y
f
S
y
BI
z
B
2
τ
f
=−
2
+
or
D
2
4
−
S
y
2
I
z
y
f
τ
f
=−
(10.8)
where
I
z
is the second moment of area of the complete section about the centroidal
axis G
z
and is obtained by the methods of Section 9.6.
A
'
f
y
τ
f
at base
of flange
y
f
A
'
w
y
w
z
d
D
τ
max
G
t
w
b
a
F
IGURE
10.6
Shear
stress distribution in
an I-section beam
B
t
f
(a)
(b)
