How this calculator works
Principal planes carry normal stress without shear traction. For a symmetric 2D stress tensor, the two eigenvalues give the in-plane principal stresses. The larger-value direction is measured counterclockwise from x, with positive tau_xy interpreted as the tensor xy component. The smaller-value direction is perpendicular. Inverse mode rotates specified principal values back to the x-y components. The two maximum-shear orientations lie 45 degrees to either side of a principal plane. Plane stress also includes an out-of-plane principal value of zero, so the two displayed in-plane values may not be the overall maximum and minimum in 3D. Equal in-plane normal stresses with zero shear have no unique principal direction.
Formula
C = (sigma_x + sigma_y)/2; R = sqrt(((sigma_x-sigma_y)/2)^2 + tau_xy^2); sigma1,2 = C +/- R; theta = atan2(2 tau_xy, sigma_x-sigma_y)/2.
Worked example
For sigma_x = 80 MPa, sigma_y = 20 MPa and tau_xy = 40 MPa, C = 50 MPa and R = 50 MPa. Principal stresses are 100 MPa and 0 MPa, at a major direction of 26.565 degrees.
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Frequently asked questions
Why is there only one shear input?
A classical Cauchy stress tensor is symmetric: tau_xy equals tau_yx. Two independent unequal values would violate local moment equilibrium.
Why use atan2 for the direction?
The two-argument arctangent keeps the stress-difference and shear signs, placing the direction in the correct quadrant before the angle is halved.
Is the smaller in-plane stress always the smallest 3D stress?
No. Plane stress includes a third principal stress of zero, which must be included when ordering all three values.