Civil Engineering Reference
In-Depth Information
(+)
T
Twist rotation prevented
0
= 0
(b) Torque
T
x
T
TL
T
x
(+ )
GI
t
L
(c) Twist rotation
(a) Cantilever
Figure 10.15
Uniform torsion of a statically determinate cantilever.
Clockwise +ve (RH system
about S axis parallel to
x
)
δ
s
O
a
0
C
y
E
0
(+ve)
C
S(
y
0
,
z
0
)
z
y
x, u
(b) Sign convention for
0
S(
y
0
,
z
0
)
B
+ve
' ve
'
+ve
T
t
-
-
T
x
,T
t
,T
w
ve
T
t
z
”
= 0
T
x
,T
t
,T
w
x
+ve
” -ve
”
ve
”', +ve
T
w
B
-
(a) Positive torsion actions
(c) Conventions for
',
”,
”',
T
t
,
T
w
Figure 10.16
Torsion sign conventions.
Asanexampleofthisprocedure,theindeterminatebeamshowninFigure10.17
isanalysedinSection10.7.2.Thetheoreticaltwistedshapeofthisbeamisshown
inFigure10.17c,anditcanbeseenthatthereisajumpdiscontinuityinthetwistper
unitlengthd
φ/
d
x
attheloadedpoint.Thisdiscontinuityisaconsequenceoftheuse
of the uniform torsion theory to analyse the behaviour of the member. In practice
such a discontinuity does not occur, because an additional set of local warping
stresses is induced. These additional stresses, which may have high values at the
loaded point, have been investigated in [9, 10]. High local stresses can usually
be tolerated in static loading situations when the material is ductile, whether the
stresses are induced by concentrated torques or by other concentrated loads. In
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