Torsion Equation Derivation

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 pounds 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. the 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 the 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 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

\[2 \pi \tau' r^{2} dr\] (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)

∴ \[\frac{T}{J} = \frac{\tau }{r} = \frac{G\theta }{L}\]

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 

\[G = \frac{\tau}{r}\]

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

K = T / θ = GJ / L.

Torsion, if defined in simple terms, can be explained as the amount of twist that we give to a particular object at one end when the other end of the object is fixed and doesn’t move with the twist applied at the other end of the object. The twist here if defined more precisely and technical terms is the torque applied to one end of the object. The Torsion that the object gets due to the torque applied to one of its ends is the Torsion, it can be applied and experienced in any object. A steel bar having one of its two ends fixed and torque applied to the other end of the steel bar will get twisted/Torsion along the direction which is perpendicular to the plane in which the torque is applied. Similarly, a rope that may be fixed at one and twisted at another will get twisted and have Torsion. When we wring our clothes after washing them, to drain the water we twist them and at that time we experience a twist in our clothes, that is Torsion and the twist we apply on one end of the clothes is the torque.

There is however a limit to the extent or amount of Torsion that an object can undergo. The level to which an object can be twisted is defined by the elastic limit of that object and the brittleness of that object. The elastic limit of an object is the ability of an object to bear stress and change in its shape. The higher the elastic level more bearable the object is to the change in its form. Brittleness on the other hand is related to the object’s fragileness and inability to stand any change in shape. The elastic limit and brittleness of an object are inversely proportional. The object which has a higher elastic level and hence low brittleness will be able to bear a higher level of Torsion than the object which has a lower elastic level and thus higher brittleness. This is experienced by us at several day-to-day activities, if we apply torque at one end of the iron rod which has another end fixed, the iron rod will be able to bear a change in its shape to some level and will not break immediately. However, if in place of the iron rod we use a wooden rod, the wooden rod will not get more Torsion and will break upon application of a small torque. This is because the elastic limit of the iron is higher than the elastic limit of wood. Similarly, when we twist clothes to drain water from them after washing them, the clothes get a very high amount of Torsion, this is because the elastic limit of the cloth is very high and its brittle is very low.  

Why is Torsion in the Axis Perpendicular to the Plane of Torque applied?

Torsion is always in the axis perpendicular to the plane of torque applied because the application of force in the particular plane causes the object to get twisted on its axis where one of the ends is fixed. This is the major condition for defining the Torsion. Application of torque in any other direction will not result in Torsion in that object because the other end will not be fixed and the end will be able to move with the torque. Eg Axle in automobiles.

How is Torsion related to Brittleness of an Object?

The brittleness of an object is defined as resistance to change in its form and shape. Torsion on the other hand is the amount of change in the shape of an object upon application of torque. The two are inversely proportional to each other. The more brittle an object is the less Torsion it will be able to bear and the less brittle an object is the more Torsion it will be able to bear. Torsion and brittleness of an object are thus inversely related.  Eg wood has more brittleness and thus bears less Torsion

Do Malleable Objects have More Torsion?

Malleability of an object is the amount of change it can withstand in its shape and form. Also, Torsion is the amount of change on application of torque when one of the ends of an object is fixed and not allowed to move with the torque. Thus an object which is more Malleable will be able to bear more change in its shape upon application of torque on one of its ends when the other end is fixed. E.g a gold rod is highly Malleable and will be able to stand a great amount of Torsion as is seen in the jewelry designs which have gold twisted at very large amounts. 

What is the Difference between Torsion and Rotation?

A rotation of an object is free and the object has no ends fixed. The object can move freely in the direction of the torque applied and there is no change or twist in the object. While as in case of Torsion the object has one end fixed and the end is not able to move in the direction of torque applied. This brings a twist in the object upon the application of torque and Torsion is developed in the object.