How this calculator works
A symmetric plane-stress tensor has equal xy and yx shear components by moment equilibrium. Its principal values are the centre plus or minus the Mohr-circle radius. The major principal direction is measured counterclockwise from x using tau_xy as the positive tensor shear component. The circle is plotted with positive shear upward. The out-of-plane principal stress is zero and can affect the absolute 3D maximum shear. At an equal biaxial state with zero shear, principal orientation is indeterminate.
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.
Does the circle show twice the physical angle?
Yes. A physical rotation theta corresponds to an angular change of 2 theta on Mohr's circle.
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.