Torsion of Circular Shafts

In this article, I will describe the torsion of solid circular shafts and hollow circular shafts.

If equal and opposite couples are applied at the ends of a circular shaft, they will either equilibrate or rotate at the same speed. In either case, the stress that causes torsion and occurs in each section is the stress. At any point in the shaft cross-section there is a simple shear stresses, the two shear planes, i.e. the plane in which the stress is completely tangent, are the cross-section itself and the plane through the point, the axis of the shaft.

There are some assumptions to be followed to drive the formulas of Torsion of Circular Shafts.

Assumptions in Torsion of Circular Shafts :

Circular sections remain circular.
A flat area remains flat and does not deform.
Straight Radial Lines in Section The projection of a section remains straight.
Shafts are loaded in pairs of rotations in a plane perpendicular to the axis of the shaft.
The Stresses do not exceed the proportional limit.
The rod material is homogeneous, perfectly elastic, and obeys Hooke’s law.

The formula of Torsion:

r = radius at a point

= maximum shear stress at the surface of a shaft

R = radius of the shaft

G = shear modulus of the material

= angle of twist

l = length of the shaft

T = torque

J = polar moment of inertia

is also termed as twist per unit length.

The J/R formula can be said to be the torsional modulus of elasticity of a section and is similar to the section modulus given in the bending formula. The term GJ refers to torsional stiffness. The angle of twist of the shaft during torsion corresponds to the deflection of the beam under transverse load and is an indicator of the stiffness of the shaft.

To ensure sufficient stiffness of the shaft, the torsion angle is usually limited to 1⁰ for a length of 20 diameters.

Polar Moment of Inertia:

J = 2I

For a solid circular section of diameter D,