Torsion Equation Derivation

What is Torsion?

In the solid mechanics field, torsion is defined as the twisting of an object due to a torque applied to it. Torsion can be expressed in either pascals (Pa) or an S.I. unit Newtons per square meter, or in pound per square inch (psi). In contrast, torque is expressed in Newton-meters (Nm) or foot pound-force (ft.lbf). In the object, some sections are perpendicular to the torque axis; in these sections, the resultant shear stress is perpendicular to the radius. In non-circular cross-sections, a distortion accompanies twisting, this distortion is known as warping. In warping, transverse sections are rough.


What is Torsion Constant?

The torsion constant is considered as a  geometrical property of the cross-section of a bar that exists in the relationship between the applied torque along with the axis’s bar and the angle of twist. All this theory is applied to a homogeneous plastic bar. A bar’s torsional stiffness can be described by the torsion constant when accompanied by properties like the length.

The S.I. unit of torsion constant is m4.


Partial Derivation

The derived formula for a beam of uniform cross-section along the length:

θ = TL / GJ

Where

  • θ is the angle of twist in radians.

  • T is the torque applied to the object.

  • L is the length of the beam.

  • G is the material’s modulus of rigidity which is also known as shear modulus.

  • J is the torsional constant.

What is Torsion Equation?

A bar having uniform sections that are fixed at one end and is subject to a torque in the other extreme end which is being applied to its axis normally, will make the bar to twist to an angle which will be proportional to the torque applied. It is assumed that the bar cannot be stressed to a level that is higher than its elastic limit. This theory covers various formulas with the help of which one can calculate the angles of twist or the resulting maximum stress. The equation formed requires the following assumptions.

  • The bar should be straight and should have uniform sections.

  • The material of the bar should have uniform properties.

  • The only force that should exist is the torque which is being applied to the axis of the bar normally.

  • The bar is capable of being stressed to a level that is within its elastic limits.

Let us now learn how to derive the torsion equation and also to derive the torsion formula.


Torsion Equation Derivation

There are some assumptions made for the derivation of the torsion equation, those assumptions are as follows.

  • The material should be homogeneous and should have elastic property throughout.

  • The material should follow the theory of Hooke’s law.

  • The material should have shear stress that is proportional to the shear strain.

  • There should be a plane cross-sectional area.

  • The section should be circular.

  • Every diameter of the material must rotate at the same angle.

  • The stress of the material must not exceed the limit of its elasticity.

Consider a solid circular shaft having radius R which is exposed to a torque T at one end and the other end is also under the same torque.

Angle in radius = arc/ radius

Arc Ab = Rθ = LY

𝛾 = Rθ/L

Where, 

A and B: these are considered as the two fixed points present in the circular shaft

Y: the angle subtended by AB

G = τ/𝛾 ( modulus of rigidity)

Where,

τ : shear stress

𝛾 : shear strain

τ/G = ୮

∴ R / L = τ/G

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

୮’ * 2πr dr

Where,

r: radius of the small strip

dr: the thickness of the strip

y: shear stress

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

\[T = \int_{0}^{R} 2 \pi \tau' r^{2} dr\]

\[T = \int_{0}^{R} 2 \pi G \theta /L r^{3} dr\] (substituting for τ’)

\[T = (2 \pi G \theta/L) \int_{0}^{R} r^{3}vdr = G \theta /L [ (\pi d^{4}) /32]\] (after integrating and substituting for R)

(Gθ/L)J (substituting for the polar moment of inertia)

∴ T/J = τ/r = Gθ/L

These are the steps followed to derive the torsion equation. This process is also termed as the derivation of the torsion equation for a circular shaft.


Some Related Topics

Twisting Moment:

The twisting moment for any section along the bar or the shaft is defined to be the algebraic sum of the moments of the applied couples that lie to one side of the section without consideration.


Modulus of Elasticity in Shear:

Modulus of elasticity in shear is the ratio of the shear stress to the shear strain. It is represented by the symbol G = τ/r.


Tensional Stiffness:

The tensional stiffness K can be defined as the torque present per radius twist, i.e.,

K = T / θ = GJ / L.

FAQs (Frequently Asked Questions)

1. What is Torsional Shear Stress?

Answer: Torsional shear can be considered as a shear that is formed by the torsion exerted from a beam. Torsion takes place when two equal forces of similar values are applied in two different directions; this causes torque. For example, a traffic sign on a windy day can get twisted by the wind, and this twist causes shear stress which is exerted along the cross-section of the structure. Therefore, while making a traffic sign, the maker should estimate the value of the shear stress to design the traffic sign, which will resist stress.

2. What are Shafts?

Answer: Shafts are a kind of mechanical components that are usually of circular cross-section. The shafts are used for transmitting power or torque. The power is transmitted through their rotational motion. In this operation, they are subjected to:

  • Torsional shear stresses that are present within the cross-section of the shaft, and the maximum shear stress is present in the outer surface of the shaft.

  • Bending stresses (for example when a transmission gear shaft is supported by bearings).

  • Vibrations that are caused due to the critical speed that is being generated.