Manufacturing Processes - MFRG 315 - Mohr's Circle
This transformation calculation can be carried out using equations, but it was noticed by Mohr that these transformation equations can be combined to give the equation of a circle. This enables the transformation to be carried out graphically (and the equations need not be remembered). Using the graphical construction also makes it easier to visualise alternative scenarios.
In a Mohr stress circle, direct stresses are on the horizontal axis and shear stresses are on the vertical axis. Rotations of direction in a Mohr circle are double those in the real stress distribution. Tensile stresses and a shear stress tending to cause clockwise rotation are deemed to be positive. Compressive stresses and shear stresses tending to cause counter-clockwise rotation are deemed to be negative.
Many metal forming operations involve a 3 dimensional state of stress so this is considered below.
2 Mohr's Circle for Three Dimensional States of Stress
If the direction of 1 principal stress is known, say sigma3, then we can imagine looking back along the direction of sigma3 to the plane on which it acts (as this is a principal plane there are no shear stresses acting on it) and for the stresses acting in this plane a 2 dimensional Mohr's circle can be constructed to determine the values and directions of the two principal stresses acting in it.
Given the 3 principal stresses, three circles can be plotted, one for each principal plane. We can determine the stress state within a principal plane as we rotate about the principal stress direction normal to the plane. It is not possible using this approach to consider simultaneous rotations about 2 or more principal axes.
It can be stated that:
The maximum shear stress is given by the radius of the largest circle.
Some sketches of Mohr circle for common stress states are shown below:
Determining the Stress State on Another Plane
The full procedure is:
An example calculation is shown in the diagram below:
The 3 circles are drawn for three principal stresses:
This gives a maximum shear stress of 50 MPa.
The construction shows the determination of direct and shear stresses on a plane at 45o to the sigma1 direction and 60o to the sigma2 direction (point 'P' on the diagram). From the scale diagram the direct stress value value is about 87 MPa and the shear stress value is about 39 MPa (see below).
This can be checked using the equations:
First determine the third direction cosine: n = (1 - (l2 +m2))0.5
sigma = sigmall2 + sigma2m2 + sigma3n2
sigma = 120(0.7071)2 + 80(0.5)2 + 20(0.5)2 = 85 MPa
The shear stress is given by:
shear stress = 7200 + 1600 + 100 - 7225 = 40.9 MPa.
For most purposes a reasonably careful graphical construction provides adequate accuracy.
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David J Grieve, 8th November 2002.