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:

\(\begin{array}{l}\frac{T}{J}=\frac{\tau}{r}=\frac{G\Theta}{L}\end{array} \)

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 Hooke’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 =

\(\begin{array}{l}\frac{arc}{Radius}\end{array} \)

Arc AB = RӨ = Lγ

\(\begin{array}{l}\gamma = \frac{R\Theta }{L}\end{array} \)

Where,

A and B: two fixed points on the circular shaft

γ: angle subtended by AB

\(\begin{array}{l}G=\frac{\tau }{\gamma }\end{array} \)
(Modulus of rigidity)

Where,

𝞃: shear stress

γ: shear strain

\(\begin{array}{l}\frac{\tau }{G}=\Gamma\end{array} \)
\(\begin{array}{l}∴ \frac{R\Theta }{L}=\frac{\tau }{G}\end{array} \)

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

\(\begin{array}{l}{\Gamma }’*2\pi rdr\end{array} \)

Where,

r: radius of small strip

dr: thickness of the strip

γ: shear stress

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

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

\(\begin{array}{l}T=\frac{2\pi G\Theta }{L}\int_{0}^{R}r^{3}dr\end{array} \)
\(\begin{array}{l}=\frac{G\Theta }{L}\left [ \frac{\pi d^{4}}{32} \right ]\end{array} \)
(after integrating and substituting for R )

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

\(\begin{array}{l}∴ \frac{T}{J}=\frac{\tau }{r}=\frac{G\Theta }{L}\end{array} \)

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Frequently Asked Questions – FAQs

Q1

What is torsion?

In solid mechanics, torsion is defined as the twisting of a body due to an exerted torque. Torsion is generally expressed in the pascal unit.
Q2

What is torque?

In mechanics, torque is the force in rotational motion or movement. It is also called as turning effect, rotational force, moment or moment of force.
Q3

Define torsion constant.

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.
Q4

Write a few assumptions made for the derivation of the torsion equation.

Here are some of the assumptions:
The material should follow Hooke’s law.
The material should have shear stress proportional to shear strain.
The circular section should be circular.
The stress of the material should not exceed the elastic limit.
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