Let,

T = Maximum twisting torque or twisting moment

D = Diameter of the shaft

R = Radius of the shaft

J = Polar moment of Inertia

τ = Max. Permissible Shear stress (Fixed for a given material)

G = Modulus of rigidity

θ = Angle of twist (Radians) = angle D’OD

L = Length of the shaft.

Φ = Angle D’CD = Angle of Shear strain

Than Torsion equation is: T/J = τ/R = G.θ/L

Let the shaft is subjected to a torque or twisting moment ‘T’. And hence every C.S. of this shaft will be subjected to shear stress.

Now distortion at the outer surface = DD’

Shear strain at outer surface = Distortion/Unit length

tan Φ = DD’/CD

i.e. shear stress at the outer surface (tan Φ) = DD’/L

or Φ = DD’/L ...(i)

Now DD’ = R.θ or Φ = R.θ/L ...(ii)

Now G = Shar stress induced/shear strain produced

G = τ/(R.θ/L);

or ; τ/R = G.θ/L ...(A);

This equation is called Stiffness equation.

Hear G, θ, L are constant for a given torque ‘T’.

i.e., τ is proportional to R

If τ_{r} be the intensity of shear stress at any layer at a distance ‘r’ from canter of the shaft, then;

τ_{r} /r = τ/R = G.θ/L

Now Torque in terms of Polar Moment of Inertia

From fig.

Area of the ring (dA) = 2 πr⋅dr

Since, τ_{r} = (τ/R)⋅r

Turning force on Elementary Ring; = (τ/R)⋅r⋅2πrdr.

= (τ/R).2 πr^{2}.dr ...(i)

Turning moment dT = (τ/R).2 πr^{2}.dr.r

dT = (τ/R).r^{2} ⋅2π.r⋅dr = (τ/R).r^{2}⋅dA

T = (τ/R) ∫r^{2}.dA for r ∈ [0, R] ...(ii)

∫r^{2}.dA for r ∈ [0, R] = M.I. of elementary ring about an axis perpendicular to the plane passing through center of circle.

∫r^{2}.dA for r ∈ [0, R] = J Polar Moment of Inertia

Now from equation (ii) T = (τ/R).J

or τ/R = T/J; ...(B)

This equation is called strength equation

Combined equation A and B; we get

T/J = τ/R = G.θ/L

This equation is called Torsion equation.

From the relation T/J = τ/R ;; We have T = τ.J/R = τ.Z_{P }

For a given shaft I_{P} and R are constants and I_{P}/R is thus a constant and is known as POLAR MODULUS(Z_{P}). of the shaft section.

Polar modulus of the section is thus measure of strength of shaft in torsion.

TORSIONAL RIGIDITY or Torsional Stiffness (K) : = G.J/L = T/θ