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Torsion of a Cylinder

Let a cylinder of length l and radius R be clamped at one end and twisted at the other end by applying a torque τ about the axis of the cylinder. Each cross-section of the cylinder is rotated about the axis by an amount proportional to its distance from fixed end; hence the twist is maximum at the free end.

Consider the fig. let the cylinder be divided into a large number of thin co-axial cylindrical shells. Consider a shell of radius r and thickness dr. When the cylinder is twisted, a generating line PQ of this shell is displaced to position PQ’. The angle θ represents angle of shear, while angle Ø (Q O’ Q’) is called angle of twist. From geometry, we have

QQ’ = l θ = r Ø

Or, θ = rθ/l

The angle of shear θ is same on the entire surface of the shell. It increases linearly with r and hence the shearing strain is maximum at the outermost surface of the cylinder.

Let F be the tangential force acting on the lower face of the given shell; then the shearing stress is given by

Stress = F/(2 π r dr)

where, 2 π r dr is the face area of the shell. Now modulus of rigidity η, by definition is

η = (F/(2 π r dr))/θ

or, F = 2 π η θ rdr

= (2 π η Ø )/l r² dr

The torque applied by this force about the axis OO’ of the cylinder is,

d τ = Fr = (2 π η Ø )/l r³ dr

The above is the torque required to twist a thin cylindrical shell of radius r by an angle Ø. The torque required to twist the entire cylinder is therefore,

The quantity τ/Ø is called torsional rigidity of the cylinder. It is equal to torque required to twist the cylinder by one radian (Ø = 1).