Mohr’s Circle

By Aina Parasher|Updated : August 26th, 2022

Mohr’s circle is one of the most important concepts in Civil Engineering which is used in the domain of stress and strain. Mohr’s circle is a graphical representation of stress and strain at different planes for a stressed body in a two-dimensional space. The concept of Mohr’s circle is used in Soil Mechanics and Strength of Materials to find the stress and strains at different planes.

Mohr’s circle is the locus of points representing the magnitude of normal and shear stress at the various planes in a given stress element. Mohr’s Circle can also be used for finding the Normal strain and shear strain for a plane strain system.

Table of Content

What is Mohr’s Circle?

In a stressed body, a combination of Shear stress and Normal stress are acting at different planes. Normal stress is represented by and shear stress is represented by τ.

mohr

According to Moment equilibrium in a stressed body τxyyx

And according to Force equilibrium, stresses on opposite planes are equal and opposite in direction. Hence, Shear stress will be equal on two mutually perpendicular planes.

normal

formula

Variation of Normal stress and Shear stress value with respect to angle Θrepresents a form of a circle and that circle is known as Mohr’s Circle.

Mohr’s circle is the circle in which each point represents a plane in a stressed body in which the x-coordinate of the point represents the Normal stress and the y-coordinate represents the Shear stress acting on the plane.

circle

Mohr’s Circle of Stress

To find out Normal stress and Shear stress on a plane with the help of Mohr’s Circle method few terms are used. These terms are as follows.

Principal stress: Principal stress is the maximum or minimum normal stress acting on the plane at which shear stress is zero. Maximum normal stress is known as the major principal stress and minimum normal stress is known as the minor principal stress and these are represented as σ1 and σ2 respectively.

Maximum Shear Stress: Maximum shear stress is the maximum shear stress acting on the plane in the overall 2-D stressed body. It is represented by τmax and it is equal to the radius of Mohr’s Circle. On the plane of maximum shear stress, normal stress is not zero

Representation of Stress in Mohr’s Circle For Plane Stress and Strain

Mohr’s circle is the representation of stress in the 2-D plane. Mohr’s circle represents normal stress and shear stress on any plane of a stressed body. In Mohr’s Circle, Normal Stress is represented on the x-axis and Shear Stress is represented on the y-axis.

representation

Some Important Facts about Mohr’s Circle

Mohr’s Circle is a very important tool for finding the stress or strain on any plane for a stressed body. Mohr’s Circle is a graphical method for finding stress. Mohr’s circle helps to visualize the normal stress and shear stress acting on the plane and helps to determine principal stress and maximum shear stresses along with their planes.

major

  • Maximum Shear stress (τmax)=(σ12)/2
  • Coordinate of Mohr’s Circle is denoted by σx, τxy
  • Center of Mohr’s Circle (C)=((σ12)/2, 0)
  • σxy12
  • The angle between the maximum shear stress plane and the principal stress plane is 450.
  • Normal stress on the maximum shear stress plane is (σ12)/2.

These points related to Mohr’s Circle can be explained with the help of a diagram as follows.

diagram

Mohr’s Circle For 3D Stress Body

Mohr’s Circle is a graphical method for finding stress on any plane of the stressed body. Mohr’s Circle can also be used for 3D representation of stress. In the case of the three-dimensional stress system, there are three principal stresses which are represented as σ1, σ2and σ3 respectively. And maximum shear stress is the maximum of τmax1, τmax2 and τmax3.

stress

Mohr’s Circle Diagram For Plane Strain System

Mohr’s Circle can also be used for finding the Normal strain and Shear strain for a plane strain system. For finding strain with the help of Mohr’s Circle, it can be used similar to that of Mohr’s Circle for stress. In the case of Mohr’s Circle, the strain maximum ordinate on the shear strain axis is γmax/2 while the maximum ordinate of Mohr’s Circle for stress is τmax.

Mohr’s Circle diagram for the strain is given below.

plane

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Mohr’s Circle FAQs

  • Mohr’s Circle is a circle which represents the relation between Normal stress and shear stress for a plane of a two-dimensional stressed body. Mohr’s Circle can be used for plane stress systems or plane strain systems.

  • Mohr’s circle for stress is the representation of Normal stress and Shear stress on any plane. In Mohr’s Circle for stress, Normal stress is represented on the x-axis and Shear stress is represented on the y- axis.

  • Mohr’s Circle for strain is the circle in which every point represents the value of Normal strain and Shear strain for a plane of a body having a two-dimensional strain. Mohr’s Circle of strain can be used for finding the normal strain and shear strain of a body on a plane.

  • Generally, we can represent stress in a theoretical way but we can also represent stress and its variables with the help of graphs. This graph is indicated as Mohr’s graphical representation of stress as in Mohr’s Circle.

  • In mohr’s Circle, the Principal axis represents the principal stress value. Maximum and minimum stress in Mohr’s circle is known as principal stress. At the principal plane value of shear stress is zero.

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