Enter stress inputs
Use plane stress inputs for normal stresses, shear stress, angle, safety checks, and chart range.
Example data table
Sample values below use σx = 120 MPa, σy = 40 MPa, and τxy = 25 MPa.
| Angle θ | Normal stress σn | Shear stress τn | Quick reading |
|---|---|---|---|
| 0° | 120.00 MPa | 25.00 MPa | Matches the x-face state. |
| 15° | 127.14 MPa | 1.65 MPa | Near a principal orientation. |
| 30° | 121.65 MPa | -22.14 MPa | Shear changes sign. |
| 45° | 105.00 MPa | -40.00 MPa | Large in-plane shear state. |
| 90° | 40.00 MPa | -25.00 MPa | Matches the y-face state. |
Formula used
For a plane rotated by angle θ from the x-face, this calculator uses the standard plane stress transformation equations.
σn = (σx + σy) / 2 + (σx - σy) / 2 · cos(2θ) + τxy · sin(2θ)
τn = -(σx - σy) / 2 · sin(2θ) + τxy · cos(2θ)
σ1,2 = (σx + σy) / 2 ± √[ ((σx - σy) / 2)^2 + τxy^2 ]
τmax = √[ ((σx - σy) / 2)^2 + τxy^2 ]
θp = 1/2 · atan( 2τxy / (σx - σy) )
σv = √(σx² - σxσy + σy² + 3τxy²)
Positive values usually indicate tension and the selected positive shear convention. Negative results indicate compression or opposite shear direction.
How to use this calculator
- Enter the in-plane normal stresses σx and σy.
- Enter the in-plane shear stress τxy using your sign convention.
- Type the plane angle θ where you want transformed stress.
- Optionally enter yield strength for a quick factor of safety.
- Set the chart range and step size for the angle sweep.
- Press calculate to show results above the form.
- Review the cards, graph, Mohr circle, and exported files.
Frequently asked questions
1. What does angle stress mean?
It usually means the normal and shear stresses acting on a plane rotated by a chosen angle inside a loaded material element. The values depend on both the original stress state and the plane orientation.
2. Why does the calculator use 2θ in the formulas?
Plane stress transformation follows Mohr circle geometry. A physical plane rotation of θ corresponds to a 2θ movement on the circle, which is why the equations use sine and cosine of 2θ.
3. What does a negative normal stress result mean?
A negative normal stress commonly indicates compression under the chosen sign convention. A positive value commonly indicates tension. Always keep your sign convention consistent across inputs and interpretation.
4. What does a negative shear stress result mean?
It means the transformed shear acts opposite to the positive shear convention used in the equations. The magnitude is still important, because it shows how strongly the material is being sheared on that plane.
5. How are principal stresses found?
Principal stresses occur where shear stress becomes zero on the rotated plane. The calculator computes them from the stress average and Mohr circle radius, then reports the related principal plane angle.
6. Why is the stress pattern often repeated every 180 degrees?
A plane rotated 180 degrees has the same physical orientation in a two-dimensional element. Because of that, transformed normal and shear values repeat over a 180-degree interval.
7. When should I use the yield strength field?
Use it when you want a quick screening check against yielding. The calculator compares the von Mises equivalent stress to the entered yield strength and reports a simple factor of safety.
8. Why is the Mohr circle graph useful?
It gives a fast visual summary of transformed stress, principal stresses, average stress, and maximum shear. That makes it easier to verify sign changes, extreme values, and angle-dependent behavior.