How To Draw Mohr’s Circle Efficiently?

Mohr’s circle is a very useful method to define the unknowns of specific problems of classical mechanics. In a typical stress analysis, there are tensile and compression stresses and torsional stresses appear. The strength state of systems can be assessed in mechanics by using these values in general. 

Strength of Materials, Third Edition

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Principal stresses and principal planes can be obtained from these stress values. There is a numeric method to do it. But also you can use Mohr’s circle method to define principal stresses and principal planes of a system. 

How To Draw Mohr’s Circle?

Representation of Mohr’s circle(Image Souce: D.K. Singh: Strength Of Materials, Springer).

As we stated above, you need to start with the definition of stress situations in a stress element. You need to define;

  • Tensile and compression stresses on that stress element,
  • Torsional stresses on that stress element.

Draw X And Y Axes To Place Mohr’s Circle On It

You need to select a proper set of X and Y axis values to represent stresses on it. Y-axis will be the axis of torsional stresses and the X-axis will be de axial stresses. 

Place The Stress Values That You Have On X-Axis

As we stated above, you need to have two planar stresses both can be compression or tensile stresses. Place these axial stress values on the X-axis of the drawn coordinate system. 

Define The Torsional Stress Places Ib XY Plane

We stated that the Y-axis is for torsional stresses. And torsional stresses are represented both negatively and positively. 

  • Define the point on the XY plane which equals to (τxy, σx). τxy is the positive value of shear stress and σx is the smaller axial stress value on the stress element. 
  • Define the point on the XY plane which equals to (-τxy, σy). Here, (-τxy) is the negative value of shear stress and σy is the bigger axial stress value acting on the stress element. 

Connect The Points That Found

Connect these two points with a straight line, then draw the circle that this line is the diameter. You obtained Mohr’s circle. 

Find The Angle Of Principal Stress Element

You can find the angle of the principal stress element on the Mohr’s circle as stated in the illustration above. The slope of the diameter line that is drawn is two-fold of the angle of the principal stress element. 

Special Conditions On Mohr’s Circle

There can be some special conditions that may occur as stress elements. 

  • Pure Shear: If there is a pure shear on the stress element in which there are no axial stresses, mohr’s circle will be located on the center of the XY coordinate system. 
  • Pure Biaxial Tension: If there are two axial tensions on the stress element, the diameter line will be on the X-axis which means shear stress is zero. 
  • Pure Uniaxial Tension: If there is pure uni-axial tension from one side of the stress element, the diameter line will be on the X-axis, and Y-axis will be tangent on the Mohr’s circle at the positive side of the X-axis. 
  • Pure Uniaxial Compression: If there is a uniaxial compression instead of tension, the tangency will be on the negative side of the X-axis. 


This is the most basic explanation of drawing Mohr’s circle to find principal stresses and angles from stresses acting on stress elements. 

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