4. Definition of Torsion
• A moment acting about a longitudinal axis of
the member is called a torque, twisting
moment or torsional moment, T.
So, Torsion is the twisting of an object due to an applied
torque.
• Twisting of a straight bar when loaded by
moments (or torques) or torques tend to
produce rotation about the longitudinal axis
of the bar.
5. Definition of Torsional Diagram
Torsional Diagram is a two-dimensional
representation of torque. A typical
representation of Torsional Diagram is as
follows:
6. Assumptions of Torsion Theory
•
•
•
•
Material is Homogeneous;
Material is elastic;
Circular section remains circular after loading;
Plane cross-section remains plane after
loading;
• Each x-section rotates as if rigid.
7. Glossaries
• Sign Convention-------- Right Hand Rule;
• Net Torque Due to Internal Stresses
Net of the internal shearing stresses is an internal
torque, equal and opposite to the applied torque,Although the net
torque due to the shearing stresses is known, the distribution of the
stresses is not.
T = ∫ρdF = ∫ρ(τdA)
8. Torsion Loading
• Deformation of Circular Subject due to Torque T
• Deformation of Rectangular Subject due to torque T
9. Torsion Formula
Torsion for a circular shaft:
T/J = τr/r = Gθ/L
And Maximum Shear Stress, τmax = Tr/Ip
So, Generalized Torsion Formula,
τ = Tρ/Ipolar
Where, T = Torque; JT = Polar Moment of Inertia = Ip or Ipolar.
r = Radius of circular shaft, G = Shear Modulus, θ = Angle of Twist, τr =
Shearing Stress at any point, L = Length of circular member.
10. Failure Mode
The shear stress in the shaft may be resolved into principal
stresses via Mohr's circle.
If the shaft is loaded only in torsion, then one of the principal
stresses will be in tension and the other in compression.
These stresses are oriented at a 45-degree helical angle around
the shaft.
If the shaft is made of brittle material, then the shaft will fail by
a crack initiating at the surface and propagating through to the
core of the shaft, fracturing in a 45-degree angle helical
shape.
This is often demonstrated by twisting a piece of blackboard
chalk between one's fingers.
11. Summary
The shear stress at a point within a shaft is:
The highest shear stress occurs on the surface of the
shaft, where the radius is maximum.
High stresses at the surface may be compounded by stress
concentrations such as rough spots. Thus, shafts for use in high
torsion are polished to a fine surface finish to reduce the
maximum stress in the shaft and increase their service life.
The angle of twist can be found by using:
