Civil Engineering Reference
In-Depth Information
Torque
T
F
IGURE
11.8
Torque-angle of twist
relationship for a gradually applied torque
Angle of twist
u
11.2 S
TRAIN
E
NERGY
D
UE TO
T
ORSION
It can be seen from Eq. (11.3) that for a bar of a given material, a given length,
L
, and
radius,
R
, the angle of twist is directly proportional to the applied torque. Therefore a
torque-angle of twist graph is linear and for a gradually applied torque takes the form
shown in Fig. 11.8. The work done by a gradually applied torque,
T
, is equal to the
area under the torque-angle of twist curve and is given by
1
2
T
θ
The corresponding strain energy stored,
U
, is therefore also given by
Work done
=
1
2
T
θ
Substituting for
T
and
θ
from Eq. (11.4) in terms of the maximum shear stress,
τ
max
,
on the surface of the bar we have
U
=
1
2
τ
max
J
R
×
τ
max
L
GR
U
=
or
τ
max
π
R
4
2
1
4
G
π
R
2
L
since
U
=
J
=
Hence
τ
max
U
=
4
G
×
volume of bar
(11.9)
Alternatively, in terms of the applied torque
T
we have
T
2
L
2
GJ
1
2
T
θ
U
=
=
(11.10)
11.3 P
LASTIC
T
ORSION OF
C
IRCULAR
S
ECTION
B
ARS
Equation (11.4) apply only if the shear stress-shear strain curve for the material of
the bar in torsion is linear. Stresses greater than the yield shear stress,
τ
Y
, induce
