Civil Engineering Reference
In-Depth Information
This nominal maximum shear stress occurs at the centre of the long edge of the
flange. A similar value may be calculated for the opposite edge, but the actual
stressesnearthispointareincreasedbecauseofthere-entrantcorneratthejunction
of the web and flange.An approximate estimate of the stress concentration factor
can be made by using Figure 10.7b with
r
t
=
12.7
15.6
=
0.81.
Thus the actual local maximum stress is approximately
τ
t
,
max
=
1.7
×
79.0
=
134 N/mm
2
.
10.9.4 Example 4 - elastic warping shear and normal stresses
Problem.
Forthecantileverofexample3,determinethemaximumelasticwarping
shear and normal stresses.
Solution.
The maximum warping shear stress
τ
w
occurs at the fixed end (
x
=
0)
of the cantilever (see Section 10.8.5) where the uniform torque is zero. Thus the
total torque there is resisted only by the warping torque, and so
V
f
=
T
/
d
f
,
where
V
f
is the flange shear (see Section 10.8.2).The shear stresses in the flange
vary parabolically (see Figure 10.24b), and so
τ
w
,
max
=
1.5
V
f
b
f
t
f
.
Thus
τ
w
,
max
=
1.5
×
(
5
×
10
6
)/
517.5
209.3
×
15.6
N/mm
2
=
4.4 N/mm
2
.
The maximum warping normal stress
σ
w
occurs where d
2
φ
/d
x
2
is a maximum
(seeequation10.37).Anexpressionford
2
φ
/d
x
2
canbeobtainedfromthesolution
in Section 10.8.5 for
φ
,
(
−
e
(
x
−
2
L
)/
a
+
e
−
x
/
a
)
(
1
+
e
−
2
L
/
a
)
d
2
φ
d
x
2
=
T
aGI
t
whence
.
The maximum value of this occurs at the fixed end
x
=
0 and is given by
d
2
φ
d
x
2
(
1
−
e
−
2
L
/
a
)
(
1
+
e
−
2
L
/
a
)
T
aGI
t
=
max
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