Calculator
Provide plane stress components. Use sign convention consistently.
Formula used
For plane stress components σx, σy, and τxy, the Mohr circle center and radius are:
- C = (σx + σy) / 2
- R = √(((σx − σy)/2)² + τxy²)
Principal stresses and maximum shear are:
- σ1 = C + R
- σ2 = C − R
- τmax = R
Principal plane orientation is computed with:
- tan(2θp) = 2τxy / (σx − σy)
For a plane rotated by angle θ, the transformed stresses are:
- σn = C + ((σx − σy)/2)cos(2θ) + τxy sin(2θ)
- τnt = −((σx − σy)/2)sin(2θ) + τxy cos(2θ)
How to use this calculator
- Enter σx, σy, and τxy using a consistent sign convention.
- Select your preferred unit so inputs and outputs match.
- Optionally enter θ to evaluate a rotated plane.
- Press Calculate to view results above the form.
- Use Download CSV or Download PDF for reports.
Example data table
Example values are illustrative for quick verification.
| σx | σy | τxy | θ (deg) | Units | σ1 | σ2 | τmax |
|---|---|---|---|---|---|---|---|
| 80 | 20 | 30 | 25 | MPa | 92.426407 | 7.573593 | 42.426407 |
| -60 | 15 | -25 | 10 | MPa | 22.569000 | -67.569000 | 45.069000 |
Technical notes
1) Plane stress and Mohr’s circle overview
Mohr’s circle is a compact way to map how normal and shear stresses change with orientation in a two‑dimensional stress state. It converts transformation equations into geometry, so principal stresses, maximum shear, and key angles can be read from the circle parameters quickly and consistently.
2) Inputs and sign convention
This calculator assumes a plane stress set: σx, σy, and τxy. Positive normal stress usually represents tension, while negative represents compression. For shear, keep a single convention across your problem statement, free‑body diagram, and input values to avoid flipped angles.
3) Circle center and radius meaning
The center C=(σx+σy)/2 is the average normal stress and does not change with rotation. The radius R captures the magnitude of stress variation with angle. Together, C and R define the full circle and the reachable stress states.
4) Principal stresses and their angle
Principal stresses occur when shear on the plane becomes zero. On the circle, these are the extreme points on the σ‑axis: σ1=C+R and σ2=C−R. The principal angle θp is computed using tan(2θp)=2τxy/(σx−σy) and is reported in degrees.
5) Maximum shear stress interpretation
Maximum in‑plane shear is equal to the circle radius, τmax=R. It occurs on planes oriented 45° from principal planes, so the calculator also reports θs=θp+45°. In design checks, this value is commonly compared against allowable shear or a yield criterion.
6) Stresses on a rotated plane
If you enter an angle θ, the calculator returns the transformed normal stress σn and shear stress τnt for that plane. These come directly from the standard transformation equations using cos(2θ) and sin(2θ), matching the geometric rotation on the circle.
7) Where these results are used
Mohr’s circle supports quick analysis in shafts under combined loading, thin‑walled pressure vessels, welded joints, fastener groups, and plate bending regions where plane stress is a good approximation. It is also useful for verifying finite‑element output and for turning measured strain data into stresses. In lab work, it helps interpret rosette strain readings before deeper modeling.
8) Practical accuracy tips
Use consistent units and rounding only at the end of a workflow. When σx≈σy, the principal angle becomes sensitive to small shear changes, so double‑check signs. If you need out‑of‑plane effects, switch to a full 3D stress analysis rather than forcing a plane result. Document assumptions beside results for traceable reviews and repeatable approvals.
FAQs
1) What does this calculator compute?
It computes Mohr’s circle center and radius, principal stresses, maximum in‑plane shear, principal and shear plane angles, plus the transformed σn and τnt for a chosen rotation angle.
2) Can it be used for three‑dimensional stress states?
This tool is for plane stress only. For 3D loading you need three principal stresses from the full stress tensor and a 3D yield check. Use plane stress only when out‑of‑plane components are negligible.
3) Why is maximum shear equal to the radius?
On Mohr’s circle, shear is the vertical coordinate. The largest magnitude of that coordinate occurs at the top and bottom of the circle, which are exactly one radius away from the center, so τmax=R.
4) What does the principal angle θp mean?
It is the rotation from the x‑plane to a plane where shear becomes zero. At that orientation the normal stresses are principal stresses. Because the formula uses tan(2θp), multiple equivalent angles exist.
5) Do negative stresses cause issues?
No. Compression is typically entered as negative normal stress. The circle and formulas work for any sign combination, as long as your sign convention is consistent across σx, σy, and τxy.
6) What happens if σx equals σy?
Then the average stress is C=σx and tan(2θp) depends mainly on shear. If τxy=0 too, every plane is principal and the circle collapses to a point.
7) How should I pick the unit option?
Select the unit that matches your inputs. The calculator converts internally for consistency and then returns outputs in your selected unit. Mixing units across inputs will produce incorrect results even if the math is correct.