Short Answer:

Principal stresses are the normal stresses acting on special planes (principal planes) where shear stress is zero. In a 2D stress element with normal stresses  and shear , the two principal stresses are the extreme values of normal stress and are given by a compact formula that uses the average stress and a square-root term involving the stress differences and shear.

These principal stresses are

and the principal plane orientation satisfies . These equations let you find magnitudes and directions where only normal stresses act.

Detailed Explanation :

Principal Stresses

Derivation

Start with a small 2-D stress element subjected to normal stresses  and  on the x and y faces, and shear stress . Consider a plane through the element inclined at angle  measured from the x-axis. The normal stress  and shear stress  on that inclined plane (using standard transformation formulas) are:

 

(Signs depend on the sign convention; these forms are the commonly used ones.)

Principal planes are those orientations where the shear stress on the plane is zero, i.e. . Set the expression for  to zero:

Rearrange to obtain the angle of principal planes:

This gives the orientations  of the principal planes (two solutions differing by ). Once  is known, substitute into the expression for  to get the principal stresses. However, a cleaner algebraic route avoids computing  explicitly.

We want the extreme (maximum and minimum) values of  with respect to . An extremum of  occurs when the derivative . Differentiating the expression for :

Setting this to zero,

which is algebraically identical to the condition for  and gives the same . Now evaluate  at those . A standard manipulation yields the principal stresses without substituting  explicitly:

Define the average normal stress  and the radius term :

Then the two principal stresses (the extreme values of ) are

Writing this explicitly:

This is the standard formula for principal stresses in a 2-D stress system.

Principal plane angle

To find the angle  of the principal plane (where  acts), use

From  you get . Note that  gives the orientation of the plane on which shear vanishes and the normal stress equals  or . The other principal plane is .

Alternate viewpoint: Mohr’s Circle

Graphically, these results are represented by Mohr’s circle. The circle center is at  on the -axis and radius . The intersection points of the circle with the -axis are  and . The top/bottom points of the circle give the maximum shear . Mohr’s circle offers a quick visual way to get principal stresses and orientations.

Remarks and uses

•  is the major principal stress (largest),  the minor (smallest) in plane stress.
• These formulas assume plane stress (no  or ); for full 3-D stress there are three principal stresses from the characteristic equation of the stress tensor.
• Principal stresses are used in failure criteria (e.g., maximum principal stress, Tresca, von Mises), and in locating critical planes for yielding or fracture.

Conclusion

The principal stresses in a 2-D stress element are given by a simple closed form: the average normal stress plus or minus a radius term that combines half the normal stress difference and the shear. The principal directions come from . These results let engineers find the planes of pure normal stress and the extreme normal stresses—key information for safe and efficient design.