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

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

``` Page 1

Strength of Materials Prof. M. S. Sivakumar

Torsion

Introduction
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

Torsion

Introduction
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

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

Torsion

Introduction
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

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

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

Torsion

Introduction
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

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

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

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

Torsion

Introduction
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

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

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

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

max
p
TC
I
t=
and
p
TR
I
t=

Top

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