# 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,

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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