In the field of solid mechanics, torsion is the twisting of an object brought on by a torque produced to it. Torsion is measured in newtons per square metre or Pascal (Pa). Some portions of the object are perpendicular to the torque axis, and the resulting shear stress is perpendicular to the radius. Warping occurs in non-circular cross-sections as a result of twisting. In this Physics article, we will learn the derivation of torsion equation and its assumptions.

What is Torsion?

Torsion refers to the twisting or rotation of an object around its axis. When a force is applied to one end of an object while the other end remains fixed, it creates a twisting motion. This twisting force is called torsion. It's similar to when you twist a towel to wring out water.

Torsion Units

The unit of torsion is wither newtons per square metre (\(N/m^2\)), or in pounds per square inch (psi).

What is the Torsion Equation?

The torsion equation, also referred to as the torsion constant, is a geometrical characteristic of a bar’s cross-section that involves the bar’s axis and establishes a connection between the angle of twist and the applied torque. The torsion equation is as follows:

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

Torsion Equation Derivation

When two opposing and equal torques are applied at either end of a shaft, it is said to be in torsion. Shear stress and shear strain will arise in the material of a shaft when it is subjected to a torsion or twisting moment.

Here, we’ll take a single instance of a circular shaft that will be torn, and we’ll derive the circular shaft torsion equation.

Let, T = Maximum twisting moment or torque

D = the shaft’s diameter.

R = The shaft’s radius

J = the polar moment of inertia.

τ = Maximum Permitted Shear tension

G= is the rigidity modulus.

θ = Angle of Twist

L = shaft length

θ’ = Angle of Shear Stress

Radius angle=arc/Radius

Arc AB= \(R\times\theta=L\times\gamma\)

Thus, \(\gamma=\frac{R\times\theta}{L}\)

Where,

𝛾= angle subtended by AB

Also, \(G=\frac{\tau}{\gamma}\)

Where,

G = modulus of rigidity

τ = shear stress

𝛾 = shear strain

Therefore,

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

Hence,

\(\frac{R\times\theta}{L}=\frac{\tau}{G}\) ….. (1)

Consider a tiny radius strip with \(d_r\) thickness that is exposed to shear stress.

\(\tau^{\prime}\times2\pi rd_r\)

Where,

r= radius of small strip

dr= thickness of the strip

𝛾= shear stress

The torque in the middle of the shaft is given by

\(2\pi \tau^{\prime}r^2d_r\)

Integrating both sides

\(T=\int_0^R2\pi \tau^{\prime}r^2d_r\)

Putting value of \(\tau^{\prime}\)

\(T=\int_0^R2\pi\frac{G\times\theta}{L}r^2d_r\)

\(T=2\pi\frac{G\times\theta}{L}\int_0^Rr^3d_r\)

\(T=\frac{G\times\theta}{L}\left[\frac{\pi d^4}{32}\right]\)

\(T=\frac{G\times\theta}{L}J\)

Thus,

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

Where, J is constant.

Moment of resistance

The moment of resistance is described as the couple that generates tensile stress on the elastic range and compressive force on the compressive zone to resist the bending moment caused by external loading. It is always the inverse of the bending moment. When the bending moment exceeds the moment of resistance, the member or beam fails.

When everything is in balance,

Moment of resistance = bending moment

\(\frac{M}{I}=\frac{\sigma}{y}\)

\(M=I\times\frac{\sigma}{y}\)

\(M=\sigma\times Z\) ……… (since \(Z=\frac{I}{y}\))

Assumptions of Torsion Equation

Some of the assumptions are as follows:

• The substance is uniform/ elastic.
• The entire text should adhere to Hooke’s law.
• The relationship between shear stress and strain in the material should be linear.
• The cross-sectional area should have a plane cross-sectional area.
• The circular segment must be circular to function.
• All the diameters of the material should rotate at the same angle.
• The material’s elastic limit should not be exceeded by stress.

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