Rotate a plane and see stress components clearly. Supports steel, aluminum, and custom materials too. Export results to PDF or CSV for reports fast.
This tool uses the 2D plane-stress transformation equations for a plane rotated by θ. Stresses are assumed uniform at a point, and sign conventions follow standard mechanics texts.
| Normal stress | σn = (σx+σy)/2 + (σx−σy)/2 · cos(2θ) + τxy · sin(2θ) |
|---|---|
| Shear stress | τnt = −(σx−σy)/2 · sin(2θ) + τxy · cos(2θ) |
| Principal stresses | σ1,2 = (σx+σy)/2 ± √(((σx−σy)/2)² + τxy²) |
| Maximum shear | τmax = √(((σx−σy)/2)² + τxy²) |
| Von Mises (plane) | σvm = √(σx² − σxσy + σy² + 3τxy²) |
Notes: θp = 0.5 · atan2(2τxy, σx−σy) gives a principal-plane angle. The reported σT is based on |σ1−σ2| for a simple Tresca check.
Example values below use the same formulas as this calculator. Inputs and outputs are shown in MPa and degrees for clarity.
| # | σx (MPa) | σy (MPa) | τxy (MPa) | θ (°) | σn (MPa) | τnt (MPa) | σ1 (MPa) | σ2 (MPa) |
|---|---|---|---|---|---|---|---|---|
| 1 | 120 | 40 | 25 | 30 | 113.301 | -12.859 | 129.046 | 30.954 |
| 2 | 80 | 20 | 15 | 45 | 50.000 | -30.000 | 85.000 | 15.000 |
| 3 | 50 | -10 | 20 | 15 | 45.981 | 5.490 | 62.361 | -22.361 |
| 4 | 0 | 0 | 35 | 10 | 23.939 | 32.889 | 35.000 | -35.000 |
| 5 | 200 | 120 | 0 | 60 | 140.000 | -34.641 | 200.000 | 120.000 |
A stress state is defined on selected faces. When the plane is rotated by θ, the components on that plane change to σn (normal) and τnt (shear). Engineers check multiple angles to find the most critical orientation at the same point.
The inputs σx, σy, and τxy are the in-plane stresses. They commonly come from strain gauges, FEA output, or combined loading cases. Use consistent signs: tensile normal stress is positive, and keep a single shear convention throughout your work.
Mohr’s circle shows that a physical rotation of θ maps to a movement of 2θ on the circle. That is why the transformation uses sin(2θ) and cos(2θ), and why principal direction uses atan2(2τxy, σx−σy).
σ1 and σ2 are the extreme normal stresses that occur when the shear on the plane is zero. The maximum in-plane shear is τmax, the Mohr’s circle radius. These results help locate planes where yielding or fatigue damage is most likely.
For ductile metals, Von Mises is widely used: σvm = √(σx² − σxσy + σy² + 3τxy²). This tool also reports a simple Tresca measure based on |σ1−σ2|. Add a yield strength or allowable limit to compute safety factors.
Rough room-temperature starting points are often: mild steel ~250 MPa yield, 6061‑T6 aluminum ~276 MPa yield, and annealed copper ~70 MPa yield. Stainless 304 often starts near ~215 MPa yield, depending on condition. Always replace these with approved datasheet values, temperature knockdowns, and code-required safety factors for your application.
A practical method is scanning θ from 0° to 180° in 5°–10° steps and recording peak σn and |τnt|. Normal stress patterns repeat every 180°. At 10° increments you evaluate 19 angles, while 5° increments give 37 angles, which is often enough for quick screening. The principal angle output gives a fast estimate of where shear becomes zero.
Keep units consistent: if stresses are MPa, limits must be MPa. For brittle materials, principal stress limits may govern; for ductile metals, σvm is often preferred. If notches or weld toes exist, apply stress concentration factors or validated FEA because local peaks can dominate. Record assumptions in your report.
σx is the normal stress on the x-face. σn is the normal stress on a plane rotated by θ. They match only when θ is 0°.
This calculator is for plane stress. Plane strain requires accounting for out-of-plane stress (often using Poisson’s ratio and E). Use a dedicated plane-strain formulation when thickness constraints prevent out-of-plane deformation.
Shear sign depends on rotation direction and your coordinate convention. A negative value usually means the shear acts opposite to your positive sign definition. Magnitude is what many design checks use, but keep signs for consistency.
Testing 0° to 180° is sufficient for normal stress because the transformation repeats every 180°. If you are checking shear direction, also observe where the sign flips; the pattern repeats every 90° for shear orientation effects.
For ductile metals, Von Mises is commonly used. Tresca is more conservative for some cases. For brittle materials, principal stress criteria may be better. Follow your design code, material model, and verification plan.
No. You can compute transformed and principal stresses without it. Yield strength or a custom limit is only needed to calculate factors of safety and show OK/Exceeds indicators for Von Mises and Tresca measures.
Differences usually come from sign conventions, whether θ is measured to the plane or to the plane normal, and whether shear is defined on the x-face or y-face. Ensure your inputs match the same convention as your reference.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.