Torsional equation:
Derive the Torsional equation T/J = Π /R = Gθ/L
Or
Derive an expression for the shear stress in shaft subjected to a torque.
Sol.: Assume,
T = Maximum twisting torque or twisting moment
D = Diameter of shaft
R = Radius of shaft
J = Polar moment of Inertia
τ= Maximum Permissible Shear stress (Fixed for given material)
G = Modulus of rigidity
θ= Angle of twist (Radians) = angle D'OD L = Length of shaft.
?= 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? = DD'/CD
i.e. shear stress at the outer surface (tan? ) = DD'/L or = DD'/L ...(i)
Now DD' = R.θ or ?= 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'. That 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;
Now from equation (ii) T = ( τ/R) J
or τ/R = T/J; ...(B)
This equation is called as strength equation
The combined equation A and B; we get
T/J = τ/R = G. τ/L
This equation is called as Torsion equation.
From the relation T/J = τ/R ; We have T = τ.J/R = τ .ZP
For the given shaft I_{P } and R are constants and IP/R is thus constant and is called as POLAR MODULUS(Z_{P}). of the shaft section.
Polar modulus of section is thus measure of strength of shaft in the torsion.
TORSIONAL RIGIDITY or Torsional Stiffness (K): = G.J/L = T/θ