For solid or hollow shafts of uniform circular cross-section and constant wall thickness, the torsion relations are:
- R is the outer radius of the shaft.
- τ is the maximum shear stress at the outer surface.
- φ is the angle of twist in radians.
- T is the torque (N·m ).
- ℓ is the length of the object the torque is being applied to or over.
- G (OR) C is the shear modulus or more commonly the modulus of rigidity and is usually given in gigapascals (GPa),or N/mm2
- J is the polar moment of inertia for a round shaft or concentric tube only.
- the product GJ is called the torsional rigidity.
The shear stress at a point within a shaft is:
- r is the distance from the center of rotation
Note that the highest shear stress is at the point where the radius is maximum, the surface of the shaft. 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 its service life.
The angle of twist can be found by using:
Polar moment of inertia : J = Ixx + Iyy
The polar moment of inertia for a solid shaft is:
where r is the radius of the object.
The polar moment of inertia for a pipe is:
where the o and i subscripts stand for the outer and inner radius of the pipe.
For a thin cylinder
- J = 2π R3 t
where R is the average of the outer and inner radius and t is the wall thickness.
POWER TRANSMITTED BY THE SHAFT:
A shaft rotating with a constant revolutions for minute (r.p.m) N is being acted by a twisting moment T. The power (P) transmitted by the shaft is