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Mohr’s circle is a two-dimensional graphical representation of transformation equations for plane stress.
This representation has been created by Otto Mohr and based on his name, the circle is named as Mohr’s circle.
Let’s see the practical approach and correlation between the physical object and the cycle.
In this process, a small square is considered for derivations, applications, etc. Now, what is this small square?
Considerations are as follows,
To understand it in the simplest way, Principal Plane means a plane on which stress value normal to the plane that is normal stress value is maximum.
It is the stress value perpendicular to a plane.
The normal stress value in the principal plane is maximum, this stress is called principal stress.
The stress acted along the plane or perpendicular to the normal stress.
A branch of mechanics that deals with the mechanical behavior of materials considered as a continuous mass.
Cauchy stress tensor
It is used for determining stress analysis caused by small deformations in the material bodies.
According to Cauchy stress tensor, the stress at any point in an object assumed as a continuum can be completely defined by nine stress components.
The stress distribution is determined using the coordinate system represented by (x, y) we need to calculate the stress component at point O at the new position achieved by the displacement of the object due to stress represented by (x’, y’).
To find the normal stress and the shear stress acting on a particular point tensor transformation is performed.
Mohr’s circle is the graphical representation of the transformation law for Cauchy stress tensor.
In a two-dimensional state, three stress components namely – Normal stresses σx and σy and Shear stress τxy give the stress tensor at any given point O.
The Cauchy stress tensor can be written in a two-by-two symmetric matrix:
Our aim is to find the stress components involved using the Mohr’s circle on a rotated coordinate system i.e. (x’, y’). The rotated plane makes an angle θ with the original plane (x, y).
Consider a two-dimensional material element around point O with a certain unit area. Using the equilibrium forces on the given element, the magnitude of the normal stress σn and the shear stress τn can be determined by:
These equations are obtained by applying the tensor transformation law on the Cauchy stress tensor.
Let’s derive the equation and find out how we get the above equations:
Stress tensor transformation law is stated as,
σ’ = AσAT
Expanding the right-hand side,
As σx’ = σn and τx’y’ = τn,
So, we get,
Above two equations are the parametric equation of the Mohr’s circle.
2θ = Parameter
σn & τn = Coordinates
The points on the Mohr’s circle can be found by choosing the coordinates with σn and τn and giving the values to the parameter θ.
To get the non-parametric equation of Mohr’s circle, we eliminate the parameter 2θ,
Above equations can be rearranged in to one by taking σn and τn in an equation and then squaring it:-
The equation of the Mohr’s circle can be given by –
(x – a)2 + (y – b)2 = r2
r = R (radius of the circle)
(a,b) = (σavg, 0) Coordinates in the (σn, τn) coordinate system.
Now, after deriving the equation for the Mohr’s circle let’s learn how to Draw/Plot it!
To take an example we take an elastic element where the signs of the Normal stresses σx & σy are positive (Tension and stress going out of the surface) and Shear stress τxy is negative (stress coming in on the surface) as shown in the figure.
Steps needed to be followed for the construction of Mohr’s circle are given below:-
Now that you have learned how to draw Mohr’s circle and get the values from it.
Let’s take an example to understand it better.
For the below given elastic element draw Mohr circle of stress and find the maximum and minimum stress values.
From the diagram given above we get:
Now draw the Mohr’s circle based on the values given above:-
σ1 = 12 N/mm2
σ2 = -22 N/mm2
Let’s check the results using the principal stress equation:
σ1 = 12 N/mm2
σ2 = -22 N/mm2
Thus, we can see that we got the same results as we got through Mohr circle method.
Let’s see few applications,
We have learned all about the Mohr’s circle in the above article, this knowledge can be used for the construction of Mohr’s circle with the values given and finding the principal stresses acting on it.
Mohr’s circle is an easy method of finding the stress acting on a point on an elastic element with minimal calculations.
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