Shear Strain: Definition, Formula, Practice Problems and FAQs

Let’s consider a rod whose one end is fixed on the wall and the other end is free. Now if you start applying force on its free end in the direction tangent to the cross-sectional area of the rod, it will start bending. If the rod is heavy even if we don’t apply the force rod will start bending slowly because of its own weight. Let’s study about shear strain in detail.

Table of Contents

• Shear Strain
• Torsion of cylinder
• Practice problems
• FAQs

Shear Strain

This type of strain is produced when the deforming force causes change in the shape of the body. It is defined as the ratio of the displacement of a layer to its distance from the fixed layer. Let the height of the body be and the displacement of the layer is x.

Shear strain,

is the angle through which a face originally perpendicular to the fixed face is turned as shown in the figure.

For very small angle we can write,

Note:

Shear strain is unitless and dimensionless. 

A shearing strain is never accompanied by a change in volume. That means in shear strain the body gets distorted resulting only in an alternation in its shape but not in size. 

Change in shape or size need not necessarily imply strain. For example, if a body is heated to expand, its volume changes. It acquired new size due to expansion. However, the strain remains zero. Unless and until internal elastic forces operate to bring the body to the original state, no strain exists. 

Torsion of Cylinder

In the figure,

or

, where

Work done in twisting through is

For a hollow cylinder of inner and outer radii and , the restoring couple per unit twist is 

Therefore,

If the mass of a solid cylinder is equal to mass of hollow cylinder

Hence a hollow cylinder is stronger than a solid cylinder of same mass and length in case of twisting. 

Practice Problems

Question.1 Find the shear strain in the object shown below having angle .

Answer. 

Shear strain,

Question.2 A rubber cube of side has one side fixed while a tangential force equal to is applied to the opposite face. Find the shearing strain? [The modulus of rigidity of rubber is ]

Answer. , ,

We know that shearing stress and shearing strain are given as follows.

Question.3 A rubber cube of side has one side fixed while a tangential force equal to is applied to the opposite face. Find the lateral displacement of the strained face? [The modulus of rigidity of rubber is ]

Answer. , ,

We know that shearing stress and shearing strain are given as follows.

Question.4 The shear modulus for the metal is . Suppose that shear force of is applied to the upper surface of a cube of this metal having side of . Find the shear strain?

Answer.

Question.5 Two metallic cylinders made of the same material and radii r and 2r are joined as shown in the figure. If the top end is fixed and the lower end of cylinder B is twisted by angle θ, then the angle of twist for cylinder A is (both cylinders have the same length)?

Answer. The force is applied to produce the torque on B and a restoring torque is produced in A to counter the torque of B. This continued until both the rods did not achieve the conditions of equilibrium. 

We know,

Therefore,

By solving two equations we get,

FAQs

Question.1 Does the volume of the object change during shear strain?

Answer. No, shear strain does not change the volume of the object. It only changes the shape of the object.

Question.2 What is the direction of applied force in case of shear strain?

Answer. Shear strain is caused due to the application of tangential force.

Question.3 Does change in shape or size of an object guarantee that the object is under strain?

Answer. No. Unless and until internal elastic forces operate to bring the body to the original state, no strain exists. 

Question.4 Can an object undergo shear strain and normal strain simultaneously?

Answer. Yes. If we apply tangential force and normal force on the area of a cross section of an object. It will undergo both shear strain and normal strain.