What are the principal planes and principal stresses? Explain the Mohr’s Circle method for two perpendicular direct stresses with state of simple shear.

• A structural member may be subjected to different types of stresses (normal and shearing stresses) simultaneously. It is, therefore, necessary to find the region where the effect of these stresses will be critical from the design point of view.
• When such stresses act at a point in a stressed material, there always exist three orthogonal planes carrying entirely normal stresses i.e no shear stresses at all.
• Such planes are called principal planes and the normal stresses acting on them are called principal stresses.

Mohr’s Circle for Plane Stress: The transformation equations for plane stress can be represented in a graphical format known as Mohr’s circle. This representation is useful in visualizing the relationships between normal and shear stresses acting on various inclined planes at a point in a stressed body. Construct Mohr’s circle as follows:

1. Determine the point on the body in which the principal stresses are to be determined.
2. Treating the load cases independently and calculated the stresses for the point chosen.
3. Choose a set of x-y reference axes and draw a square element centered on the axes.
4. Identify the stresses σx, σy, and τxy = τyx and list them with the proper sign.
5. Draw a set of σ - τ coordinate axes with σ being positive to the right and τ being positive in the upward direction. Choose an appropriate scale for the each axis.
6. Using the rules on the previous page, plot the stresses on the x face of the element in this coordinate system (point V). Repeat the process for the y face (point H).
7. Draw a line between the two point V and H. The point where this line crosses the σ axis establishes the center of the circle.
8. Draw the complete circle.
9. The line from the center of the circle to point V identifies the x axis or reference axis for angle measurements (i.e. θ = 0).

Mohr Circle

Note: The angle between the reference axis and the σ axis is equal to 2θp.