252
7 Torsion of Prismatic Bars
T
I p
=
τ
r
where I p = polar moment of inertia
∴
T
πa 4
32
=
τ
a
2
or
32T
πa 4 =
2τ
a
∴ τ =
16T
πa 3
(ii)
Equating (i) and (ii), we get
T
4a 2 t
=
16T
πa 3 ∴ 64a
2 t T = πa
3 T
∴ t =
πa
64
(b) The stiffness of the box section is given by
θ =
q
2G A
ds
t
Here
T = 2q A ∴ q =
T
2 A
∴ θ =
T
4G A 2
a
t
+
2a
t
+
a
t
+
2a
t
as
=
6aT
4G A 2 t
=
6aT
4G
2a 2
2 t
∴ θ =
6aT
16a 4 Gt
(iii)
The stiffness of the solid circular section is
θ =
T
G I p
=
T
G
πa 4
32
=
32T
Gπa 4
(iv)
7 Torsion of Prismatic Bars
T
I p
=
τ
r
where I p = polar moment of inertia
∴
T
πa 4
32
=
τ
a
2
or
32T
πa 4 =
2τ
a
∴ τ =
16T
πa 3
(ii)
Equating (i) and (ii), we get
T
4a 2 t
=
16T
πa 3 ∴ 64a
2 t T = πa
3 T
∴ t =
πa
64
(b) The stiffness of the box section is given by
θ =
q
2G A
ds
t
Here
T = 2q A ∴ q =
T
2 A
∴ θ =
T
4G A 2
a
t
+
2a
t
+
a
t
+
2a
t
as
=
6aT
4G A 2 t
=
6aT
4G
2a 2
2 t
∴ θ =
6aT
16a 4 Gt
(iii)
The stiffness of the solid circular section is
θ =
T
G I p
=
T
G
πa 4
32
=
32T
Gπa 4
(iv)