For a plane-stress state with normal stresses σx, σy, and shear stress τxy, the principal stresses are:
- σavg = (σx + σy) / 2
- R = √(((σx − σy)/2)² + τxy²)
- σ1 = σavg + R, σ2 = σavg − R
- τmax = R
- θp = 0.5 · atan2(2τxy, σx − σy) (converted to degrees)
Angle convention: θp is measured from the x-axis to the plane where σ1 acts.
- Enter σx, σy, and τxy from your stress analysis.
- Select the unit that matches your inputs (for example, MPa or psi).
- Choose your preferred decimal precision for the output.
- Click Calculate to view results above the form.
- Use Download CSV or Download PDF to save results.
| σx | σy | τxy | Unit | σ1 | σ2 | τmax | θp (deg) |
|---|---|---|---|---|---|---|---|
| 80 | 20 | 30 | MPa | 94.154 | 5.846 | 44.154 | 22.500 |
| 50 | 50 | 25 | MPa | 75.000 | 25.000 | 25.000 | 45.000 |
| -120 | 30 | 40 | MPa | 40.000 | -130.000 | 85.000 | 15.255 |
Principal Stress Calculator Guide
1) Why principal stresses matter
Many checks depend on the extreme normal stresses at a point, not the original components. Principal stresses (σ₁ and σ₂) are those extremes for a plane-stress state and are widely used in strength, crack screening, and load-case comparison.
2) Scope of the calculation
Inputs are σx, σy, and τxy for in-plane loading. This fits thin plates, surface regions, and many 2D models where out-of-plane stress is negligible. If σz is significant, use a full 3D principal stress procedure.
3) Core equations used
The method matches Mohr’s circle. The center is σavg = (σx + σy)/2 and the radius is R = √(((σx − σy)/2)² + τxy²). Then σ₁ = σavg + R, σ₂ = σavg − R, and τmax = R.
4) Principal plane angle
Orientation is reported as θp = 0.5·atan2(2τxy, σx − σy), converted to degrees. θp is measured from the x-axis to the plane where σ₁ acts, and the second principal plane is θp + 90°. Keep your shear sign convention consistent.
5) Reading signs correctly
Positive normal stress usually indicates tension and negative indicates compression. If τxy = 0, principal stresses reduce to σx and σy. When σx ≈ σy, even moderate τxy can produce a large radius R and a noticeable τmax.
6) Units and precision
Choose Pa, kPa, MPa, GPa, psi, or ksi and enter all inputs in that unit. Outputs follow the same unit. The precision option only changes formatting, which is helpful for reports and side-by-side verification.
7) Engineering use cases
Principal stresses are useful at holes, fillets, weld toes, and pressure boundaries where stress directions rotate. σ₁ helps indicate crack-opening tendency, while τmax supports ductile shear screening. They also provide a clean way to compare multiple load cases.
8) Limits and validation
Results are point-based and do not replace a full stress field analysis. Validate inputs from closed-form solutions, FEA output, or strain-gauge rosettes. If uncertainty exists, vary σx, σy, and τxy to see sensitivity in σ₁, σ₂, and θp.
Frequently Asked Questions
1) Is this calculator for 2D or 3D stress?
It is for plane stress using σx, σy, and τxy. If you have σz or additional shear components, use a 3D principal stress method.
2) What does θp represent exactly?
θp is the rotation from the x-axis to the plane where σ₁ acts. The second principal plane is perpendicular at θp + 90°.
3) Why is maximum shear stress equal to R?
In Mohr’s circle for plane stress, the radius R is the maximum deviation from the center. That deviation corresponds to the peak in-plane shear stress magnitude.
4) What if my shear sign convention is opposite?
You can still compute σ₁ and σ₂ correctly because R uses τxy². However, the reported angle θp will change sign, reflecting your chosen convention.
5) Can σ₁ be negative?
Yes. If the stress state is mostly compressive, both principal stresses may be negative. Interpret the sign based on your tension-compression convention.
6) What inputs produce θp = 45°?
A common case is σx = σy with nonzero τxy. Then atan2(2τxy, 0) gives 90°, so θp becomes 45°.
7) How should I cross-check results quickly?
Compute σavg and R by hand, then verify σ₁ = σavg + R and σ₂ = σavg − R. If τxy = 0, confirm the outputs match σx and σy.