Torsion Equation Derivation

Torsion equation or torsion constant is defined as the geometrical property of a bar’s cross-section that is involved in the axis of the bar that has a relationship between the angle of twist and applied torque whose SI unit is m4. The torsion equation is given as follows:

\(\frac{T}{J}=\frac{\tau}{r}=\frac{G\Theta}{L}\)

Torsion equation derivation

Following are the assumptions made for the derivation of torsion equation:

  • The material is homogeneous (elastic property throughout)
  • The material should follow Hook’s law
  • The material should have shear stress proportional to shear strain
  • The cross-sectional area should be plane
  • The circular section should be circular
  • Every diameter of the material should rotate through the same angle
  • The stress of the material should not exceed the elastic limit

Consider a solid circular shaft with radius R that is subjected to a torque T at one end and the other end under the same torque.

Angle in radius = \(\frac{arc}{Radius}\)

Arc AB = RӨ = Lγ

\(\gamma = \frac{R\Theta }{L}\)

Where,

A and B: two fixed points on the circular shaft

γ: angle subtended by AB

\(G=\frac{\tau }{\gamma }\) (Modulus of rigidity)

Where,

𝞃: shear stress

γ: shear strain

\(\frac{\tau }{G}=\Gamma\)

\(∴ \frac{R\Theta }{L}=\frac{\tau }{G}\)

Consider a small strip of radius with thickness dr that is subjected to shear stress.

\({\Gamma }’*2\pi rdr\)

Where,

r: radius of small strip

dr: thickness of the strip

γ: shear stress

\(2\pi {\tau }’r^{2}dr\) (torque at the center of the shaft)

\(T=\int_{0}^{R}2\pi {\tau }’r^{2}dr\)

\(T=\int_{0}^{R}2\pi \frac{G\Theta }{L}r^{3}dr\) (substituting for 𝛕’ )

\(T=\frac{2\pi G\Theta }{L}\int_{0}^{R}r^{3}dr\)

\(=\frac{G\Theta }{L}\left [ \frac{\pi d^{4}}{32} \right ]\) (after integrating and substituting for R )

\(\frac{G\Theta }{L}J\) (substituting for the polar moment of inertia)

\(∴ \frac{T}{J}=\frac{\tau }{r}=\frac{G\Theta }{L}\)

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