TORSION-strength of material by IIT MADRAS GATE Notes | EduRev

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 Page 1


 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
 
Torsion 
 
 
Introduction 
Click here to check the Animation 
Assumptions 
Basic Assumptions 
Torsion Formula 
Stress Formula 
Stresses on Inclined Planes 
 
Angle of twist 
Angle of Twist in Torsion 
Maximum Stress 
Torsion of Circular Elastic Bars: Formulae 
 
Table of Formulae 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 2


 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
 
Torsion 
 
 
Introduction 
Click here to check the Animation 
Assumptions 
Basic Assumptions 
Torsion Formula 
Stress Formula 
Stresses on Inclined Planes 
 
Angle of twist 
Angle of Twist in Torsion 
Maximum Stress 
Torsion of Circular Elastic Bars: Formulae 
 
Table of Formulae 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Introduction: 
Detailed methods of analysis for determining stresses and deformations in axially loaded 
bars were presented in the first two chapters. Analogous relations for members subjected 
to torque about their longitudinal axes are developed in this chapter. The constitutive 
relations for shear discussed in the preceding chapter will be employed for this purpose. 
The investigations are confined to the effect of a single type of action, i.e., of a torque 
causing a twist or torsion in a member.  
 
The major part of this chapter is devoted to the consideration of members having circular 
cross sections, either solid or tubular. Solution of such elastic and inelastic problems can 
be obtained using the procedures of engineering mechanics of solids. For the solution of 
torsion problems having noncircular cross sections, methods of the mathematical theory of 
elasticity (or finite elements) must be employed. This topic is briefly discussed in order to 
make the reader aware of the differences in such solutions from that for circular members. 
Further, to lend emphasis to the difference in the solutions discussed, this chapter is 
subdivided into four distinct parts. It should be noted, however, that in practice, members 
for transmitting torque, such as shafts for motors, torque tubes for power equipment, etc., 
are predominantly circular or tubular in cross section. Therefore, numerous applications fall 
within the scope of the formulas derived in this chapter. 
In this section, discussion is limited to torsion of circular bars only. 
 
 
Top  
 
 
 
 
Page 3


 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
 
Torsion 
 
 
Introduction 
Click here to check the Animation 
Assumptions 
Basic Assumptions 
Torsion Formula 
Stress Formula 
Stresses on Inclined Planes 
 
Angle of twist 
Angle of Twist in Torsion 
Maximum Stress 
Torsion of Circular Elastic Bars: Formulae 
 
Table of Formulae 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Introduction: 
Detailed methods of analysis for determining stresses and deformations in axially loaded 
bars were presented in the first two chapters. Analogous relations for members subjected 
to torque about their longitudinal axes are developed in this chapter. The constitutive 
relations for shear discussed in the preceding chapter will be employed for this purpose. 
The investigations are confined to the effect of a single type of action, i.e., of a torque 
causing a twist or torsion in a member.  
 
The major part of this chapter is devoted to the consideration of members having circular 
cross sections, either solid or tubular. Solution of such elastic and inelastic problems can 
be obtained using the procedures of engineering mechanics of solids. For the solution of 
torsion problems having noncircular cross sections, methods of the mathematical theory of 
elasticity (or finite elements) must be employed. This topic is briefly discussed in order to 
make the reader aware of the differences in such solutions from that for circular members. 
Further, to lend emphasis to the difference in the solutions discussed, this chapter is 
subdivided into four distinct parts. It should be noted, however, that in practice, members 
for transmitting torque, such as shafts for motors, torque tubes for power equipment, etc., 
are predominantly circular or tubular in cross section. Therefore, numerous applications fall 
within the scope of the formulas derived in this chapter. 
In this section, discussion is limited to torsion of circular bars only. 
 
 
Top  
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Basic Assumptions 
a. Plane sections perpendicular to the axis of a circular member before application 
of torque remains plane even after application of torque. 
b. Shear strains vary linearly from the central axis reaching a maximum value at the outer 
surface. 
c. For linearly elastic material, Hooke's law is valid. Hence shear stress is proportional to 
shear strain. 
 
 
 
Top 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 4


 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
 
Torsion 
 
 
Introduction 
Click here to check the Animation 
Assumptions 
Basic Assumptions 
Torsion Formula 
Stress Formula 
Stresses on Inclined Planes 
 
Angle of twist 
Angle of Twist in Torsion 
Maximum Stress 
Torsion of Circular Elastic Bars: Formulae 
 
Table of Formulae 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Introduction: 
Detailed methods of analysis for determining stresses and deformations in axially loaded 
bars were presented in the first two chapters. Analogous relations for members subjected 
to torque about their longitudinal axes are developed in this chapter. The constitutive 
relations for shear discussed in the preceding chapter will be employed for this purpose. 
The investigations are confined to the effect of a single type of action, i.e., of a torque 
causing a twist or torsion in a member.  
 
The major part of this chapter is devoted to the consideration of members having circular 
cross sections, either solid or tubular. Solution of such elastic and inelastic problems can 
be obtained using the procedures of engineering mechanics of solids. For the solution of 
torsion problems having noncircular cross sections, methods of the mathematical theory of 
elasticity (or finite elements) must be employed. This topic is briefly discussed in order to 
make the reader aware of the differences in such solutions from that for circular members. 
Further, to lend emphasis to the difference in the solutions discussed, this chapter is 
subdivided into four distinct parts. It should be noted, however, that in practice, members 
for transmitting torque, such as shafts for motors, torque tubes for power equipment, etc., 
are predominantly circular or tubular in cross section. Therefore, numerous applications fall 
within the scope of the formulas derived in this chapter. 
In this section, discussion is limited to torsion of circular bars only. 
 
 
Top  
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Basic Assumptions 
a. Plane sections perpendicular to the axis of a circular member before application 
of torque remains plane even after application of torque. 
b. Shear strains vary linearly from the central axis reaching a maximum value at the outer 
surface. 
c. For linearly elastic material, Hooke's law is valid. Hence shear stress is proportional to 
shear strain. 
 
 
 
Top 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Torsion Formula 
Since shear strains varies linearly across the section, 
max
R
C
?
?=
 
where ? is the shear strains at a point of raidus R, C is the radius of the member. 
A
Torque,  T = R dA ? t
?
 
 
A
= GR dA ?
?
 
where t = G ?, the shear stress at any point at a distance R (Refer Figure 6.1)  
 
 
Figure 6.1 
 
Hence writing in terms of shear stresses. 
 
max
A
2 max
R
T = RdA
C
  = R dA
C
t
t
?
?
 
 
2
p
RdA I =
?
 
 
 the Polar moment of Inertia of the circular section. 
  
max p
I
 T = 
C
t
? 
 
Page 5


 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
 
Torsion 
 
 
Introduction 
Click here to check the Animation 
Assumptions 
Basic Assumptions 
Torsion Formula 
Stress Formula 
Stresses on Inclined Planes 
 
Angle of twist 
Angle of Twist in Torsion 
Maximum Stress 
Torsion of Circular Elastic Bars: Formulae 
 
Table of Formulae 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Introduction: 
Detailed methods of analysis for determining stresses and deformations in axially loaded 
bars were presented in the first two chapters. Analogous relations for members subjected 
to torque about their longitudinal axes are developed in this chapter. The constitutive 
relations for shear discussed in the preceding chapter will be employed for this purpose. 
The investigations are confined to the effect of a single type of action, i.e., of a torque 
causing a twist or torsion in a member.  
 
The major part of this chapter is devoted to the consideration of members having circular 
cross sections, either solid or tubular. Solution of such elastic and inelastic problems can 
be obtained using the procedures of engineering mechanics of solids. For the solution of 
torsion problems having noncircular cross sections, methods of the mathematical theory of 
elasticity (or finite elements) must be employed. This topic is briefly discussed in order to 
make the reader aware of the differences in such solutions from that for circular members. 
Further, to lend emphasis to the difference in the solutions discussed, this chapter is 
subdivided into four distinct parts. It should be noted, however, that in practice, members 
for transmitting torque, such as shafts for motors, torque tubes for power equipment, etc., 
are predominantly circular or tubular in cross section. Therefore, numerous applications fall 
within the scope of the formulas derived in this chapter. 
In this section, discussion is limited to torsion of circular bars only. 
 
 
Top  
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Basic Assumptions 
a. Plane sections perpendicular to the axis of a circular member before application 
of torque remains plane even after application of torque. 
b. Shear strains vary linearly from the central axis reaching a maximum value at the outer 
surface. 
c. For linearly elastic material, Hooke's law is valid. Hence shear stress is proportional to 
shear strain. 
 
 
 
Top 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Torsion Formula 
Since shear strains varies linearly across the section, 
max
R
C
?
?=
 
where ? is the shear strains at a point of raidus R, C is the radius of the member. 
A
Torque,  T = R dA ? t
?
 
 
A
= GR dA ?
?
 
where t = G ?, the shear stress at any point at a distance R (Refer Figure 6.1)  
 
 
Figure 6.1 
 
Hence writing in terms of shear stresses. 
 
max
A
2 max
R
T = RdA
C
  = R dA
C
t
t
?
?
 
 
2
p
RdA I =
?
 
 
 the Polar moment of Inertia of the circular section. 
  
max p
I
 T = 
C
t
? 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
max
p
TC
I
t= 
and  
p
TR
I
t= 
 
 
 
 
 
Top 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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