Mohr Circle-Part I

Mohr circle is a graphical method to find the principal stresses, principal planes, maximum shear stress, planes of maximum shear stress, and normal and shear stresses on any inclined plane and at any point. Mohr circle is extensively used in engineering problems because it can give result for any above mentioned quantity quite easily.

Consider the following cases:

A Cube

Image a cube of any specified material in space co-ordinates namely x, y and z respectively and the orientation of these orthogonal axis is specified. Now mark any point which is going to reside inside a cube. Now by specifying that we have consider a point means it is an infinitesimal quantity. Since it is infinitesimal small we ignored its spin along the co-ordinate axis and so the three degree of freedom due to spin is equal to zero.

From our marked point infinite number of planes can passed and we wish to find any of our desired stress component along the selected plane. So our Mohr circle is a representation of infinite number of planes passing through our point. This is what a Mohr circle physically signifies, a representation of infinite number of planes with each plane differ from each other with respect to the angle they subtend from the fixed plane of reference.

Now choosing an appropriate co-ordinate system, the following sign conventions are used:

Sign Conventions for our Mohr Circle

1. Normal stress and shear stress is taken along the x and y direction respectively because both these stresses are orthogonal(perpendicular) to each other.
2.  Tensile stress is taken along the positive x direction.
3.  Compressive stress is taken along negative x direction.
4. Positive shear stress is taken along positive y direction.

5.  Negative shear stress is taken along negative y direction

Take the following case:

Consider a triangular element ABC in which the plane AB is inclined at an angle (θ) in anti-clock wise direction.

A Triangular Element

Our Preferred Co-ordinate System

Now we do the stress analysis of the cube using the following stresses on the element:

Normal stress σ(1) is acting on the plane BC in the outward direction from the element.

Shear stress 𝛕  acting on plane BC in the negative x-direction.

Normal stress σ(2) acting on the plane AC in the outward direction.

Shear stress 𝛕 acting on the plane AC in the positive y-direction.

Construction of Mohr Circle

Procedure:

Construction of Mohr Circle

1. Draw x-y coordinate system with specified x and y co-ordinates respectively with origin O.
2.  After taking down our stresses to a suitable scale mark a point C and the line OC represents +σ(1).
3. Similarly we mark a point A and the line CA represents –𝛕.
4. Then mark point D on the positive x-axis and the line OD represents +σ(2).
5. From point D mark a point B vertically in upward direction and the line BD represents +𝛕.
6.Join point A and B and the line AB will be the diameter of your Mohr Circle.
7.Using I as the center and IA as the radius draw a circle and this complete the construction of your Mohr circle.
8. Because in our triangular element we have chosen BC as our reference plane so in your Mohr circle the line IA (radius of circle) will behave as the reference line in your Mohr circle.
9.  In our triangular element observe that the plane bearing the normal and shear stresses are normal to each other, i.e., the planes are orthogonal to each other and in our Mohr circle the co-ordinates of the end points of line IA and ED on the periphery of circle represents the normal and shear stresses respectively and they are at parallel to each other, so the angle between any two plane is taken double.
10. The line BD and CA represents ±𝛕 respectively.

The proof of principal stress will be given in the post Mohr Circle-Part II.