Short Notes: Torsion | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


Torsion
Torsion means twisting a structural Member when it is loaded by couple that 
Produces rotation about longitudinal axis.
• If
Ti
be the intensity of shear stress, on any layer at a distance r from the centre of 
shaft, then
r ~ J ~ l
Sign Convention
• Sign convention of torque can be explained by right hand thumb rule.
• A positive torque is that in which there is tightening effect of nut on the bolt. 
From either side of the cross-section. If torque is applied in the direction of 
right hand fingers than right hand thumbs direction represents movement of 
the nut.
TMD = Torsion moment diagram
Page 2


Torsion
Torsion means twisting a structural Member when it is loaded by couple that 
Produces rotation about longitudinal axis.
• If
Ti
be the intensity of shear stress, on any layer at a distance r from the centre of 
shaft, then
r ~ J ~ l
Sign Convention
• Sign convention of torque can be explained by right hand thumb rule.
• A positive torque is that in which there is tightening effect of nut on the bolt. 
From either side of the cross-section. If torque is applied in the direction of 
right hand fingers than right hand thumbs direction represents movement of 
the nut.
TMD = Torsion moment diagram
TMD
T = Torque
• Rate of tw ist:
• Total angle of tw ist:
Where, T = Torque,
• J = Polar moment of inertia
• G = Modulus of rigidity,
• 0 = Angle of twist
• L = Length of shaft,
• GJ = Torsional rigidity
GJ _
l
Torsional stiffness;
J _
GJ
Torsional flexibility
EA
I
Axial stiffness;
EA
Page 3


Torsion
Torsion means twisting a structural Member when it is loaded by couple that 
Produces rotation about longitudinal axis.
• If
Ti
be the intensity of shear stress, on any layer at a distance r from the centre of 
shaft, then
r ~ J ~ l
Sign Convention
• Sign convention of torque can be explained by right hand thumb rule.
• A positive torque is that in which there is tightening effect of nut on the bolt. 
From either side of the cross-section. If torque is applied in the direction of 
right hand fingers than right hand thumbs direction represents movement of 
the nut.
TMD = Torsion moment diagram
TMD
T = Torque
• Rate of tw ist:
• Total angle of tw ist:
Where, T = Torque,
• J = Polar moment of inertia
• G = Modulus of rigidity,
• 0 = Angle of twist
• L = Length of shaft,
• GJ = Torsional rigidity
GJ _
l
Torsional stiffness;
J _
GJ
Torsional flexibility
EA
I
Axial stiffness;
EA
Axial flexibility
Moment of Inertia About polar Axis:
• For solid circular shaft,:
16 T
k d3 32
• For hollow circular shaft:
Power Transmitted in the Shaft
Power transmitted by shaft:
— — fe n 
60000
Where, N = Rotation per minute.
Compound Shaft
An improved type of compound coupling for connecting in series and parallel are 
given below
1. Series connection: Series connection of compound shaft as shown in figure. 
Due to series connection the torque on shaft 1 will be equal to shaft 2 and the 
total angular deformation will be equal to the sum of deformation of 1st shaft 
and 2n d shaft.
Where,
01 = Angular deformation of 1st shaft
02 = Angular deformation of 2n d shaft
1. Parallel connection: Parallel connection of compound shaft as shown in 
figure. Due to parallel connection of compound shaft the total torque will be 
equal to the sum of torque of shaft 1 and torque of shaft 2 and the deflection 
will be same in both the shafts.
Swiss co n n ectio n
e = 6,+e2
T = I = T ,
Therefore,
Q_ TLX TL;
Gill G 2 J 2
Page 4


Torsion
Torsion means twisting a structural Member when it is loaded by couple that 
Produces rotation about longitudinal axis.
• If
Ti
be the intensity of shear stress, on any layer at a distance r from the centre of 
shaft, then
r ~ J ~ l
Sign Convention
• Sign convention of torque can be explained by right hand thumb rule.
• A positive torque is that in which there is tightening effect of nut on the bolt. 
From either side of the cross-section. If torque is applied in the direction of 
right hand fingers than right hand thumbs direction represents movement of 
the nut.
TMD = Torsion moment diagram
TMD
T = Torque
• Rate of tw ist:
• Total angle of tw ist:
Where, T = Torque,
• J = Polar moment of inertia
• G = Modulus of rigidity,
• 0 = Angle of twist
• L = Length of shaft,
• GJ = Torsional rigidity
GJ _
l
Torsional stiffness;
J _
GJ
Torsional flexibility
EA
I
Axial stiffness;
EA
Axial flexibility
Moment of Inertia About polar Axis:
• For solid circular shaft,:
16 T
k d3 32
• For hollow circular shaft:
Power Transmitted in the Shaft
Power transmitted by shaft:
— — fe n 
60000
Where, N = Rotation per minute.
Compound Shaft
An improved type of compound coupling for connecting in series and parallel are 
given below
1. Series connection: Series connection of compound shaft as shown in figure. 
Due to series connection the torque on shaft 1 will be equal to shaft 2 and the 
total angular deformation will be equal to the sum of deformation of 1st shaft 
and 2n d shaft.
Where,
01 = Angular deformation of 1st shaft
02 = Angular deformation of 2n d shaft
1. Parallel connection: Parallel connection of compound shaft as shown in 
figure. Due to parallel connection of compound shaft the total torque will be 
equal to the sum of torque of shaft 1 and torque of shaft 2 and the deflection 
will be same in both the shafts.
Swiss co n n ectio n
e = 6,+e2
T = I = T ,
Therefore,
Q_ TLX TL;
Gill G 2 J 2
Parallel connection
0 , = 0,
T = Tj + T 2
Therefore,
Til _ T 2L 
Gi Ji G2J2
Strain energy (U) stored in shaft due to torsion:
U = - T.Q = - .Volume o f sh a ft
2 2 G .] 4G 1 3
• G = Shear modulus
• T = Torque
• J = Moment of inertia about polar axis
Effect of Pure Bending on Shaft
The effect of pure bending on shaft can be defined by the relation for the shaft,
tJ
Pure bending on shaft
Where, a = Principal stress
• D = Diameter of shaft
• M = Bending moment
3 2 M 
a = -----—
7 7 D'
Effect of Pure Torsion on Shaft
• It can be calculated by the formula, which are given below
Pure torsion on shaft
167
Page 5


Torsion
Torsion means twisting a structural Member when it is loaded by couple that 
Produces rotation about longitudinal axis.
• If
Ti
be the intensity of shear stress, on any layer at a distance r from the centre of 
shaft, then
r ~ J ~ l
Sign Convention
• Sign convention of torque can be explained by right hand thumb rule.
• A positive torque is that in which there is tightening effect of nut on the bolt. 
From either side of the cross-section. If torque is applied in the direction of 
right hand fingers than right hand thumbs direction represents movement of 
the nut.
TMD = Torsion moment diagram
TMD
T = Torque
• Rate of tw ist:
• Total angle of tw ist:
Where, T = Torque,
• J = Polar moment of inertia
• G = Modulus of rigidity,
• 0 = Angle of twist
• L = Length of shaft,
• GJ = Torsional rigidity
GJ _
l
Torsional stiffness;
J _
GJ
Torsional flexibility
EA
I
Axial stiffness;
EA
Axial flexibility
Moment of Inertia About polar Axis:
• For solid circular shaft,:
16 T
k d3 32
• For hollow circular shaft:
Power Transmitted in the Shaft
Power transmitted by shaft:
— — fe n 
60000
Where, N = Rotation per minute.
Compound Shaft
An improved type of compound coupling for connecting in series and parallel are 
given below
1. Series connection: Series connection of compound shaft as shown in figure. 
Due to series connection the torque on shaft 1 will be equal to shaft 2 and the 
total angular deformation will be equal to the sum of deformation of 1st shaft 
and 2n d shaft.
Where,
01 = Angular deformation of 1st shaft
02 = Angular deformation of 2n d shaft
1. Parallel connection: Parallel connection of compound shaft as shown in 
figure. Due to parallel connection of compound shaft the total torque will be 
equal to the sum of torque of shaft 1 and torque of shaft 2 and the deflection 
will be same in both the shafts.
Swiss co n n ectio n
e = 6,+e2
T = I = T ,
Therefore,
Q_ TLX TL;
Gill G 2 J 2
Parallel connection
0 , = 0,
T = Tj + T 2
Therefore,
Til _ T 2L 
Gi Ji G2J2
Strain energy (U) stored in shaft due to torsion:
U = - T.Q = - .Volume o f sh a ft
2 2 G .] 4G 1 3
• G = Shear modulus
• T = Torque
• J = Moment of inertia about polar axis
Effect of Pure Bending on Shaft
The effect of pure bending on shaft can be defined by the relation for the shaft,
tJ
Pure bending on shaft
Where, a = Principal stress
• D = Diameter of shaft
• M = Bending moment
3 2 M 
a = -----—
7 7 D'
Effect of Pure Torsion on Shaft
• It can be calculated by the formula, which are given below
Pure torsion on shaft
167
• Where, t = Torsion
• D = Diameter of shaft
Combined effect of Moment & Torque
Bending and torsfon effect
• The equivalent bending moment
may be defined as the bending moment which will produce the same direct 
stress as produced by the bending moment and torque acting separately.
• Similarly, the equivalent torque
K
, may be defined as the torque which will produce the same maximum shear 
stress as produced by the bending moment and torque acting separately.
For a circular shaft of diameter, d = 2r
4 P P
jr d 2 jr r 2
32 M 4 M
Tid1
16 T 2 T
7i d 3
3
7i r
Principal stress
Maximum Shear Stress
If P = 0