What this calculator does
This principal stress calculator finds the maximum and minimum principal stresses for a two-dimensional stress state, given the normal stresses on two perpendicular planes (σx and σy) and the shear stress between them (τxy). Principal stresses are the normal stresses on the particular plane orientation where shear stress drops to zero, which is the orientation a material is most likely to fail along.
The underlying formula comes from Mohr's circle, a graphical and algebraic method for transforming stresses from one orientation to another. Rather than checking every possible angle by hand, the principal stress formula finds the maximum and minimum normal stresses directly, along with the maximum shear stress, which sits exactly halfway between them.
The formula
The average of σx and σy gives the centre of Mohr's circle. The radius of that circle is the square root of the squared half-difference of σx and σy, plus the squared shear stress τxy. Adding and subtracting that radius from the centre gives the maximum principal stress σ₁ and minimum principal stress σ₂. The radius itself is also the maximum in-plane shear stress, and the angle from the x-axis to the principal plane comes from half the arctangent of 2τxy over (σx − σy).
| Term | Meaning |
|---|---|
| σx, σy | Normal (tension or compression) stress acting on two perpendicular planes. |
| τxy | Shear stress acting between the σx and σy planes. |
| σ₁, σ₂ | The maximum and minimum principal stresses: the normal stresses on the plane orientation where shear stress is zero. |
| Maximum shear stress | The largest shear stress at any orientation, equal to half the difference between σ₁ and σ₂. |
The inputs explained
| Field | What to enter |
|---|---|
| Normal stress σx (MPa) | Normal stress on the x-face of the stress element. Use a negative value for compression. |
| Normal stress σy (MPa) | Normal stress on the y-face of the stress element. Use a negative value for compression. |
| Shear stress τxy (MPa) | Shear stress acting on the element. Use a negative value if it acts in the opposite sense to your sign convention. |
When to use it
Checking a combined loading case
A shaft or beam under bending plus torsion produces both normal and shear stress at the same point. Converting that combined state to principal stresses shows the actual maximum normal stress the material experiences, which is what most failure criteria compare against.
Orientating a brittle material test
Brittle materials tend to fail along the plane of maximum principal stress rather than maximum shear, so finding that plane's orientation and magnitude is a standard step before predicting a crack path.
Comparing against a material's allowable stress
Design codes usually specify an allowable stress as a simple normal stress limit. Reducing a complex 2D stress state to its principal stresses puts it in the same terms as that allowable limit for a direct comparison.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How principal stresses change as shear stress increases
Fixed normal stresses of 80 MPa and 20 MPa, across a range of shear stress values.
| Shear stress τxy | Maximum principal stress (σ₁) | Minimum principal stress (σ₂) | Maximum in-plane shear stress |
|---|---|---|---|
| 0 MPa | 80.00 MPa | 20.00 MPa | 30.00 MPa |
| 10 MPa | 81.62 MPa | 18.38 MPa | 31.62 MPa |
| 20 MPa | 86.06 MPa | 13.94 MPa | 36.06 MPa |
| 30 MPa | 92.43 MPa | 7.57 MPa | 42.43 MPa |
| 50 MPa | 108.31 MPa | -8.31 MPa | 58.31 MPa |
| 80 MPa | 135.44 MPa | -35.44 MPa | 85.44 MPa |
How principal stresses change as σy varies
A fixed σx of 80 MPa and shear stress of 30 MPa, across a range of σy values.
| Normal stress σy | Maximum principal stress (σ₁) | Minimum principal stress (σ₂) | Maximum in-plane shear stress |
|---|---|---|---|
| -40 MPa | 87.08 MPa | -47.08 MPa | 67.08 MPa |
| -20 MPa | 88.31 MPa | -28.31 MPa | 58.31 MPa |
| 0 MPa | 90.00 MPa | -10.00 MPa | 50.00 MPa |
| 20 MPa | 92.43 MPa | 7.57 MPa | 42.43 MPa |
| 40 MPa | 96.06 MPa | 23.94 MPa | 36.06 MPa |
| 60 MPa | 101.62 MPa | 38.38 MPa | 31.62 MPa |
Questions
What are principal stresses, in plain terms?
They are the normal stresses on the one particular plane orientation, at a given point, where the shear stress is exactly zero. Every possible 2D stress state has such an orientation, and the principal stresses are its maximum and minimum normal stress values.
Why does the angle formula use half the arctangent?
Stress transformation follows a double-angle relationship: rotating the physical plane by an angle θ rotates the point on Mohr's circle by 2θ. Solving for the angle to the zero-shear point on the circle therefore gives 2θ, so the physical angle is half of that.
How is this different from torsional stiffness or slenderness ratio calculators?
Those calculators cover a shaft's resistance to twisting and a column's buckling tendency respectively, which are separate structural concepts. This calculator is purely about resolving a combined 2D stress state (normal plus shear) into its principal directions, a step that can feed into either of those other checks.
Can the minimum principal stress be negative?
Yes. A negative principal stress means that plane is under compression rather than tension. It is a common and expected result whenever the shear stress is large relative to the difference between σx and σy.
For loading dominated by torsion in a shaft, see the torsional stiffness calculator. For a column's buckling tendency under compressive load, see the slenderness ratio calculator.