Civil Engineering Reference
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torque
T
w
=
T
x
acting on the I-section is replaced by two transverse shears
V
f
=
T
x
/
d
f
,
(10.50)
one on each flange. The bending deflections
v
f
of each flange can then be deter-
minedbyelasticanalysisorbyusingavailablesolutions[4],andthetwistrotations
can be calculated from
φ
w
=
2
v
f
/
d
f
(10.51)
10.3.2.3 Non-uniform torsion
When neither the torsional rigidity nor the warping rigidity can be neglected, as
in hot-rolled steel I- and channel sections, the applied torque is resisted by a
combination of the uniform and warping torques, so that
T
x
=
T
t
+
T
w
.
(10.52)
In this case, the differential equation of non-uniform torsion is
d
x
−
EI
w
d
3
φ
T
x
=
GI
t
d
φ
d
x
3
,
(10.53)
the general solution of which can be written as
φ
=
p
(
x
)
+
A
1
e
x
/
a
+
A
2
e
−
x
/
a
+
A
3
,
(10.54)
where
a
2
=
EI
w
GI
t
=
K
2
L
2
π
2
.
(10.55)
The function
p
(
x
)
in this solution is a particular integral whose form depends
on the variation of the torque
T
x
along the beam. This can be determined by the
standard techniques used to solve ordinary linear differential equations (see [16]
forexample).Thevaluesoftheconstantsofintegration
A
1
,
A
2
, and
A
3
dependon
the form of the particular integral and on the boundary conditions.
Completesolutionsforanumberoftorquedistributionsandboundaryconditions
have been determined, and graphical solutions for the twist
φ
and its derivatives
d
φ
/d
x
,d
2
φ
/d
x
2
,andd
3
φ
/d
x
3
areavailable[5,6].Asanexample,thenon-uniform
torsion of the cantilever shown in Figure 10.28 is analysed in Section 10.8.5.
Sometimes the accuracy of the method of solution described above is not
required, in which case a much simpler method may be used. The maximum
angle of twist rotation
φ
m
may be approximated by
φ
tm
φ
wm
φ
tm
+
φ
wm
φ
m
=
(10.56)
in which
φ
tm
is the maximum uniform torsion angle of twist rotation obtained by
solving equation 10.8 with
T
t
=
T
x
, and
φ
wm
is the maximum warping torsion
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