Unit-6

Torsion of Circular Shaft

6.1 Analysis of circular shaft subjected to torsion

“A member which is subjected to the couple which produce rotation about longitudinal axis is known as Twisting Moment or Torsion.”

• Due to torque applied on shaft, the shaft produced shear stress and shear strain in the materials.
• In practice, torque is produced in shaft of motors or power equipment. e.g. Fans, axles of vehicles, pumps, etc.

When a force is applied tangentially to a cylindrical rod, at a certain radius, in the plane of its transverse cross-section, the rod is said to be subjected to twisting moment or torque. Torsion is also defined as the product of tangential force and the radius.

Figure below shows a circular shaft of length l and diameter d acted upon by a torque T.

Fig.1 Circular shaft subjected to torque

The shear strain of elements at radial distance r form the axis is constant for a particular value of T. The line OA thus twists to OB.

Let be the angle of twist of a cross-section relative to another cross-section at a distance l apart. Then,

But is the polar moment of inertia of the shaft,

Torsion equation shows that for a given shaft, shear stress is proportional to the radius of shaft. Figure below shows a shear stress distribution in solid shaft of Radius R.

Fig.2 Shear stress distribution

Assumptions in Torsion Theory

• The bar is acted upon by a pure torque.
• The section under consideration is remote from the point of application of the load and from a change in diameter.
• Adjacent cross sections originally plane and parallel remain plane and parallel after twisting, and any radial line remains straight.
• The material obeys Hooke’s law
• Cross-sections rotate as if rigid, i.e. every diameter rotates through the same angle

Basic Torsion Formulae:

The ability of shaft to resist the motion of twisting moment is called as strength of a shaft.

Torsion in stepped and composite shaft:

Stepped or composite shaft are either in series or parallel

6.2 Power transmitted to circular shaft

Torque on strength basis

When the shaft is subjected to pure torsional moment, the torsional shear stress is given,

Where, is in degree.

Torque found by strength basis and by rigidity basis is maximum torque.

Power transmission by shaft

It is the product of average torque and corresponding angle turned per unit duration of time.

6.3 Shafts subjected to combined bending, torsion and axial trust

The condition of combined bending, torsion and axial thrust is frequently encountered. Consider a shaft subjected to combined bending and twisting moment.

In such cases, stresses are set up due to bending moment, torque and shear force. However, shear stress due to shear force is usually unimportant as its maximum value occurs at the neutral axis where the bending stress is zero.

For vertical loading on shaft, the maximum values of bending stress occurs at the ends of vertical diameter whereas maximum value of shear stress at outer surface. As there is no normal stress on the longitudinal planes of the shaft,

Maximum principal stresses are,

Example 1: An 800 mm long shaft with a diameter of 80 mm carries a flywheel weighing 4 kN at its midway. The shaft transmits 24 kW at a speed of 240 rpm. Determine the principle stresses and the maximum shear stress at the end of a vertical and horizontal diameter in a plane near the flywheel.

Solution:

Sol:

Combined bending, twisting and axial thrust

Now, consider a shaft subjected to combined bending, twisting and direct axial thrust.

Let P be the axial thrust acting on shaft with diameter D and area of cross section A.

The resultant maximum principal stresses are given by

References

1. Strength of Material by S. S. Ratan