Manufacturing Processes - MFRG 315 - Mohr's Circle

1 Introduction
In two dimensional stress analysis (plane stress) there is often information available about direct (sigma_x and sigma_y) and shear stresses (tau_xy) acting on a particular plane and either the values of the direct and shear stresses acting on a plane inclined at some angle to the first plane are required, or alternatively the values of the 2 principal stresses (sigma₁ and sigma₂) and the inclination of the plane on which they act are required.

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
A general approach to Mohr's circle in 3 dimensions is complicated, however this is hardly ever required as in virtually all metal forming problems we know the directions of at least 1 of the 3 principal stresses.

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 stresses on any plane at any rotation, when plotted in the 3 dimensional Mohr's circle diagram will be represented by a point either on one of the 3 circles, or within the (shaded) area between the largest and 2 smaller circles.

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
A further graphical construction enables the state of stress on any plane whose normal is inclined at angles 'alpha', 'beta' and 'gamma' to the sigma₁, sigma₂ and sigma₃ directions respectively to be determined. (With direction cosines to the principal axes directions: l, m and n, respectively - For information about direction cosines).

The full procedure is:

• 1. Draw the direct stress (horizontal) and the shear stress (vertical) axes.
• 2. Mark the three principal stresses on the horizontal axis, sigma₁ greater than sigma₂ greater than sigma₃.
• 3. Draw the 3 circles
• 4. Draw the line sigma1 - A - B inclined at angle alpha from the vertical through sigma1.
• 5. Draw the lines sigma₂ - C and sigma₂ - D with inclinations of beta, either side of the vertical through sigma₂.
• 6. With centres C₁₃ and C₂₃ draw arcs DC and AB
• 7. The direct and shear stress at P are given by its coordinates.

An example calculation is shown in the diagram below:

The 3 circles are drawn for three principal stresses:

• sigma₁ = 120 MPa
• sigma₂ = 80 MPa
• sigma₃ = 20 MPa.

The construction shows the determination of direct and shear stresses on a plane at 45^o to the sigma₁ direction and 60^o 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

The shear stress is given by:

For most purposes a reasonably careful graphical construction provides adequate accuracy.