Short Answer:

Principal stresses in a 2D stress system are the maximum and minimum normal stresses that act at a point when the shear stress on that plane is zero. They are found by rotating the stress element until the shear component vanishes; the normal stresses at those orientations are the principal stresses.

In simple terms, for a plane stress state with normal stresses and shear , the principal stresses are

where is the larger (major) and the smaller (minor) principal stress.

Detailed Explanation :

Principal Stresses in 2D

We derive the principal stresses step by step for a two-dimensional stress state. Consider a small element in the plane subjected to normal stresses on the face normal to the x-axis, on the face normal to the y-axis, and shear stress (positive convention assumed). We want the normal stress acting on a plane inclined at angle to the x-axis and the shear stress on that plane. The transformation equations are:

Step 1 — Condition for principal planes.
Principal planes are those orientations where the shear stress on the plane is zero. So set :

Rearrange to obtain the angle of the principal planes:

This gives the orientation(s) of the planes where shear vanishes. There are two solutions separated by , corresponding to the two principal planes.

Step 2 — Compute principal stresses.
Substitute the angle condition into the expression for or proceed algebraically to find the extreme values of . A standard and compact route is to compute the center and radius of the stress transformation:

Define the average normal stress (centroid):

Define the radius (magnitude of deviation) as

With these, the normal stress on any rotated plane can be written as

where locates the principal direction. The extreme values of occur when the cosine equals . Therefore the two principal stresses are

Substituting and we get the familiar closed-form expressions:

Here (with plus sign) is the maximum (major) principal stress and (with minus sign) is the minimum (minor) principal stress.

Step 3 — Angle to principal plane (explicit).
From the earlier relation,

Depending on signs and quadrants, add as needed to find the second principal plane.

Relation to maximum shear.
The maximum in-plane shear stress magnitude is equal to the radius :

It acts on planes oriented at from the principal planes.

Geometric interpretation (Mohr’s Circle).
Mohr’s circle provides a graphical way to see these results: center at , radius . The intersections of the circle with the horizontal axis give and . The top and bottom points give .

Remarks and usage.

Principal stresses are invariant quantities: their values do not depend on the coordinate system orientation, only on the local stress state .
These expressions are fundamental in failure criteria (maximum principal stress, von Mises, Tresca) and in fracture or yielding checks.
For plane stress, there are only two nonzero principal stresses; in full 3D, a third principal stress exists and requires solving a cubic characteristic equation.

Conclusion

For a two-dimensional stress state with and , the principal stresses (extreme normal stresses on planes with zero shear) are given by

and the principal plane angle satisfies . These formulas give the key stresses for safe design and failure assessment in mechanical engineering.