Lecture 13 torsion in solid and hollow shafts 1

2. Unit 2- Stresses in Beams
 Lecture -1 – Review of shear force and bending
moment diagram
 Lecture -2 – Bending stresses in beams
 Lecture -3 – Shear stresses in beams
 Lecture -4- Deflection in beams
 Lecture -5 – Torsion in solid and hollow shafts.
Topics Covered

3. 3
 Torsion is a moment that twists/deforms a member
about its longitudinal axis
 By observation, if angle of rotation is small, length of
shaft and its radius remain unchanged
TORSIONAL DEFORMATION
OF A CIRCULAR SHAFT

4. 4
Torsional Deformation of
Circular Bars
 Assumptions
 Plane sections remain plane and perpendicular to the
torsional axis
 Material of the shaft is uniform
 Twist along the shaft is uniform.
 Axis remains straight and inextensible

5. 5
Torsional Deformation

B
F

F’
F’
R
€
φ is the shear strain, also remember that tanφ=φ,thus:
φ=
F'F
L
=
Rθ
L
Note that shear strain does not only change with the amount of twist, but also,
it varies along the radial direction such that it is zero at the center and increases
linearly towards the outer periphery (see next slide)
L
F
= shear strain
= angle of twist

6. Torsional Deformation
€
τ
R
=
Cθ
L
=
q
r
Shear stress at any point in the shaft is proportional to
the distance of the point
from the axis of the shaft.

7. Torque transmitted by
shaft(solid)
r
R
€
total turning moment due to turning force
=total force on the ring x Distance of the ring from the axis
=
τ
R
×2πr
3
dr
Total turning moment (or total torque) is obtained by integrating
the above equation between the limits O and R
T=dT
0
R
∫=
τ
R
×2πr
3
dr
0
R
∫
=
τ
R
×2πr
3
dr
0
R
∫ =
τ
R
×2π
r
4
4
&
'
(
)
*
+
0
R
=τ×
π
2
×R
3
=
π
16
τD
3

8. Torque transmitted by
shaft(hollow)
r
R
€
total turning moment due to turning force
=total force on the ring x Distance of the ring from the axis
=
τ
R
0
×2πr
3
dr
Total turning moment (or total torque) is obtained by integrating
the above equation between the limits O and R
T=dT
R
i
Ro
∫ =
τ
R
0
×2πr
3
dr
R
i
R
0
∫
=
τ
R
×2πr
3
dr
R
i
R
0
∫ =
τ
R
0
×2π
r
4
4
&
'
(
)
*
+
R
i
R
0
=τ×
π
2
×
R
0
4
−R
i
4
R
0
&
'
(
)
*
+
=
π
16
τ
D
0
4
−D
i
4
D
0
&
'
(
)
*
+

9. Power transmitted by
shaft
€
Power transmitted by the shafts
N = r.p.m of the shaft
T=Mean torque transmitted
ω= Angular speed of shaft
Power=
2πNT
*
60
=ω×T

10. Torque in terms of polar
moment of inertia
€
Moment dT on the circular ring
dT=
τ
R
×2πr
3
dr=
τ
R
×r
2
×2πrdr⇒(dA=2πrdr)
=
τ
R
×r
2
×dA
Total Torque =dT
0
R
∫
T=dT
0
R
∫=
τ
R
×r
2
dA
0
R
∫
=
τ
R
r
2
dA
0
R
∫
r
2
dA= moment of elemnetary ring about an axis perpendicular to the plane
and passing though the center of the circle
r
2
dA
0
R
∫ =moment of the circle about an axis perpendicular to the plane
and passing though the center of the circle
= Polar moment of inertia=
π
32
×D
4
r
R

11. Torque in terms of polar
moment of inertia
€
T=
τ
R
×J
T
J
=
τ
R
τ
R
=
Cθ
L
T
J
=
τ
R
=
Cθ
L
r
R
€
C=Modulus of rigidity
θ= Angle of twist
L= Length of the shaft

12. Polar Modulus
€
Z
p
=
J
R
For solid shaft=> J=
π
32
D
4
Z
p
=
π
32
D
4
D/2
=
π
16
D
3
For hollow shaft=> J=
π
32
D
0
4
−D
i
4
[ ]
Z
p
=
π
32
D
0
4
−D
i
4
[ ]
D
0
/2
=
π
16D
0
D
0
4
−D
i
4
[ ]
Polar modulus is defined as ration of polar moment of inertia to the radius
of the shaft.

13. Torsional rigidity
€
=C*J
Torsional rigidity is also called strength of the shaft. It is defined as product of
modulus of rigidity (C) and polar moment of inertia

14. Shaft in combined bending
and Torsion stresses
€
Shear stress at any point due to torque T
q
r
=
T
J
⇒q=
T×r
J
Shear Stress at a point on the surface of the shaft r=
D
2
τ
c
=
T×r
J
=
T
π
32
D
4
×
D
2
=
16T
πD
3
Bending stress at any point due to bending moment
M
I
=
σ
y
⇒σ=
M×y
I
Bending Stress at a point on the surface of the shaft r=
D
2
σ
b
=
M×y
I
=
M
π
64
D
4
×
D
2
=
32M
πD
3
tanθ=
2τ
c
σ
b
=
2×
16T
πD
3
32M
πD
3
=
T
M

15. Shaft in combined bending
and Torsion stresses
€
Major principal Stress
=
σ
b
2
+
σ
b
2
#
$
%
&
'
(
2
+τ
c
2
=
32M
2×πD
3
+
32M
2×πD
3
#
$
%
&
'
(
2
+
16T
πD
3
#
$
%
&
'
(
2
=
16
πD
3
M+M
2
+T
2
( )
Minor principal Stress
=
16
πD
3
M−M
2
+T
2
( )
Max shear Stress
=
Max principal Stress-Min principal Stress
2
=
16
πD
3
M
2
+T
2
( )
SOLID SHAFT

16. Shaft in combined bending
and Torsion stresses
€
Major principal Stress
=
16D
0
πD
0
4
−D
i
4
[ ]
M+M
2
+T
2
( )
Minor principal Stress
=
16D
0
πD
0
4
−D
i
4
[ ]
M−M
2
+T
2
( )
Max shear Stress
=
16D
0
πD
0
4
−D
i
4
[ ]
M
2
+T
2
( )
HOLLOW SHAFT

17. Application to a Bar
Normal Force:
F
n F
n
Shear Force:
F
t F
t
Bending Moment:
M
t M
t
Torque or Twisting Moment:
M
n
M
n