Civil Engineering Reference
In-Depth Information
be determined as the torque resultant of these shear flows, and the torsion section
constant
I
t
can then be found from
T
t
G
d
φ/
d
x
.
I
t
=
10.7.2 Analysis of statically indeterminate members
The uniform torsion of the statically indeterminate beam shown in Figure 10.17a
maybeanalysedbytakingtheleft-handreactiontorque
T
0
astheredundantquan-
tity. The distribution of torque
T
x
is therefore as shown in Figure 10.17b, and
equation 10.8 becomes
GI
t
d
φ
d
x
=
T
0
−
T
x
−
a
0
,
where the value of the second term is taken as zero when the value inside the
Macaulay brackets
is negative.
Thesolutionofthisequationwhichsatisfiestheconditionthattheangleoftwist
at the left-hand support is zero is
GI
t
φ
=
T
0
x
−
T
x
−
a
.
(10.87)
The other condition which must be satisfied is that the angle of twist at the right-
hand support must be zero. Using this in equation 10.87,
0
=
T
0
L
−
T
(
L
−
a
)
,
and so
T
0
=
T
(
L
−
a
)
L
.
Thusthemaximumtorqueis
T
0
when
a
islessthan
L
/
2,whilethemaximumangle
of twist rotation occurs at the loaded point, and is equal to
φ
a
=
Ta
(
L
−
a
)
GI
t
L
.
10.8 Appendix - non-uniform torsion
10.8.1 Warping deflections
Whenamembertwists,linesoriginallyparalleltotheaxisoftwistbecomehelical,
and any element
δ
x
×
δ
s
×
t
of the section wall rotates
a
0
d
φ/
d
x
about the line
a
0
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