At any point in a loaded part, there are two perpendicular planes with no shear stress – the principal planes. The stresses on them are the largest and smallest normal stresses, and failure criteria are built on them.
How to use the Principal Stress Calculator
- Normal stress σx
- Normal stress σy
- Shear stress τxy
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the normal stresses and shear stress with signs (tension positive).
Inputs explained
| Field | What to enter |
|---|---|
| Normal stress σx | Units: MPa, kPa, Pa, GPa, psi, ksi |
| Normal stress σy | Units: MPa, kPa, Pa, GPa, psi, ksi |
| Shear stress τxy | Units: MPa, kPa, Pa, GPa, psi, ksi |
What each result means
- Maximum principal stress σ₁ = ([Normal stress σx] + [Normal stress σy])÷2 + sqrt((([Normal stress σx] – [Normal stress σy])÷2)^2 + [Shear stress τxy]^2)
- Minimum principal stress σ₂ = ([Normal stress σx] + [Normal stress σy])÷2 – sqrt((([Normal stress σx] – [Normal stress σy])÷2)^2 + [Shear stress τxy]^2)
- Maximum in-plane shear stress = sqrt((([Normal stress σx] – [Normal stress σy])÷2)^2 + [Shear stress τxy]^2)
- Principal plane angle θp = deg(atan2(2×[Shear stress τxy], [Normal stress σx] – [Normal stress σy]))÷2
- Maximum shear plane angle θs = deg(atan2(2×[Shear stress τxy], [Normal stress σx] – [Normal stress σy]))÷2 – 45
- Von Mises stress = sqrt([Normal stress σx]^2 – [Normal stress σx]×[Normal stress σy] + [Normal stress σy]^2 + 3×[Shear stress τxy]^2)
Example values
How the main result changes when Normal stress σx changes and everything else stays at the example values above:
| Normal stress σx (MPa) | Maximum principal stress σ₁ (MPa) |
|---|---|
| 25 | 43.6008 |
| 37.5 | 52.2818 |
| 50 | 62.0156 |
| 62.5 | 72.5 |
| 75 | 83.503 |
| 100 | 106.4782 |
Formula
σ_max,min = (σx + σy)/2 ± √(((σx − σy)/2)² + τxy²), τ_max = √(((σx − σy)/2)² + τxy²)
tan 2θp = 2τxy/(σx − σy), θs = θp − 45°
Worked example
Inputs — Normal stress σx: 50 MPa; Normal stress σy: 10 MPa; Shear stress τxy: 25 MPa.
Results — Maximum principal stress σ₁ = 62.01562 MPa; Minimum principal stress σ₂ = -2.01562 MPa; Maximum in-plane shear stress = 32.01562 MPa; Principal plane angle θp = 25.6701 °; Maximum shear plane angle θs = -19.3299 °; Von Mises stress = 63.0476 MPa.
Key terms
- Principal plane
- A plane carrying no shear stress.
Frequently asked questions
Why are principal stresses important?
Failure theories compare them with the material strength.
Is there shear on principal planes?
No – by definition the shear stress is zero.
How far apart are principal and max-shear planes?
45°.
What about 3D stresses?
A third principal stress exists; for plane stress it is zero.