Civil Engineering Reference
In-Depth Information
10.8.3 Warping torsion analysis of statically
determinate members
The twisted shape of a statically determinate member in warping torsion is given
byequation10.49.Forexample,thecantilevershowninFigure10.25a,thetorque
T
x
isconstantandequaltotheappliedtorque
T
, andsothetwistedshapeisgiven
by
−
EI
w
φ
=
Tx
3
6
+
A
1
x
2
+
A
2
x
+
A
3
.
(10.88)
2
Theconstantsofintegration
A
1
,
A
2
,and
A
3
dependontheboundaryconditions.
At the support, it is assumed that twisting is prevented, so that
φ
0
=
0
and that warping is also prevented, so that (see equation 10.34)
d
φ
d
x
=
0.
0
Attheloadedend,thememberisfreetowarp(i.e.thewarpingnormalstresses
σ
w
are zero), so that (see equation 10.37)
d
2
φ
d
x
2
=
0.
L
By substituting these conditions into equation 10.88, the constants of integration
can be determined as
A
1
=−
TL
,
A
2
=
0,
A
3
=
0.
Ifthesearesubstitutedintoequation10.88,thecompletesolutionforthetwisted
shape is obtained as
−
EI
w
φ
=
Tx
3
6
−
TLx
2
.
2
The maximum angle of twist occurs at the loaded end and is equal to
φ
L
=
TL
3
3
EI
w
.
Thewarpingshearstressdistribution(seeequation10.42)isconstantalongthe
member, but the maximum warping normal stress (see equation 10.37) occurs at
the support where d
2
φ
/d
x
2
reaches its highest value.
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