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Derive the Torsional equation T/J = π/R = Gθ/L.

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

τr /r = τ/R = G.θ/L 

Now Torque in terms of Polar Moment of Inertia 

From the fig 

Area of the ring (dA) = 2 πr ⋅ dr 

Since, τr = (τ/R) ⋅ r 

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

= (τ/R).2 πr2.dr ...(i) 

Turning moment dT = (τ/R).2 πr2.dr.r 

dT = (τ/R).r⋅ 2π.r⋅dr = (τ/R).r2⋅dA 

T = (τ/R)∫r2.dA for r ∈ [0, R] ...(ii) 

∫r2.dA for r ∈ [0, R] = M.I. of elementary ring about an axis perpendicular to the plane passing through center of circle. 

∫r2.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

For a given shaft IP and R are constants and IP/R is thus a constant and is known as POLAR MODULUS(ZP). 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

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