Civil Engineering Reference
In-Depth Information
10.8.4 Warping torsion analysis of statically
indeterminate members
The warping torsion of the statically indeterminate beam shown in Figure 10.26a
may be analysed by taking the left-hand reaction torque
T
0
as the redundant.The
distribution of torque
T
x
is then given by (see Figure 10.26b)
T
x
=
T
0
−
mx
and equation 10.49 becomes
−
EI
w
φ
=
T
0
x
3
6
−
mx
4
24
+
A
1
x
2
+
A
2
x
+
A
3
.
2
Afterusingtheboundaryconditions
φ
0
=
(
d
φ/
dx
)
0
=
0, theconstants
A
2
and
A
3
are determined as
A
2
=
A
3
=
0,
while the condition
φ
L
=
0 requires that
3
+
mL
2
A
1
=−
T
0
L
12
.
The additional boundary condition is (d
2
φ
/d
x
2
)
L
=
0, and so
T
0
=
5
mL
/
8.
Thus the complete solution is
−
x
4
.
24
+
5
Lx
3
48
−
L
2
x
2
−
EI
w
φ
=
m
16
Themaximumangleoftwistoccursnear
x
=
0.58
L
,andisapproximatelyequalto
0.0054
mL
4
/
EI
w
.The warping shear stresses are greatest at the left-hand support,
where the torque
T
x
has its greatest value of 5
mL
/8. The warping normal stresses
aregreatestat
x
=
5
L
/8,where-
EI
w
d
2
φ/
d
x
2
hasitsmaximumvalueof9
mL
2
/128.
10.8.5Non-uniform torsion analysis
Thetwistedshapeofamemberinnon-uniformtorsionisgivenbyequation10.54.
FortheexampleofthecantilevershowninFigure10.28a,thetorque
T
x
isconstant
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