Principal Stress and Mohr's Circle Calculator (Plane Stress)
Principal stresses, maximum shear, von Mises and Tresca equivalent stress, and the stress on any rotated plane, from σx, σy and τxy.
Results
The dashed line marks the value you entered.
What this tool does
At any point of a loaded part there are two perpendicular directions where the shear stress vanishes; the normal stresses along them, σ1 and σ2, are the principal stresses. Mohr's circle has its centre at (σx+σy)/2 and a radius equal to the largest in-plane shear, and the stress on a plane turned by θ is read from the circle at 2θ — the chart shows it. For ductile metals the von Mises value is the one usually compared with the yield strength (Tresca is a little more cautious). Plane stress is assumed (no stress through the thickness), tension is positive and angles are anticlockwise.
Formula
σ1,2 = (σx+σy)/2 ± √[((σx−σy)/2)² + τxy²] · tan 2θp = 2τxy ÷ (σx−σy) · σvM = √(σx² − σxσy + σy² + 3τxy²)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
sx | Normal stress σx (tension positive) | MPa |
sy | Normal stress σy (tension positive) | MPa |
tx | Shear stress τxy | MPa |
th | Angle of the plane to look at, from the x axis | ° |
S1 | Maximum principal stress σ1 | MPa |
S2 | Minimum principal stress σ2 | MPa |
TM | Maximum in-plane shear stress (Mohr's circle radius) | MPa |
AP | Direction of σ1 from the x axis | ° |
VM | Von Mises equivalent stress (plane stress) | MPa |
TR | Tresca equivalent stress | MPa |
SN | Normal stress on that plane | MPa |
TN | Shear stress on that plane | MPa |
Worked example
- Normal stress σx (tension positive)80 MPa
- Normal stress σy (tension positive)-20 MPa
- Shear stress τxy40 MPa
- Angle of the plane to look at, from the x axis30 °
- Maximum principal stress σ194.03 MPa
- Minimum principal stress σ2-34.03 MPa
- Maximum in-plane shear stress (Mohr's circle radius)64.03 MPa
- Direction of σ1 from the x axis19.33 °
- Von Mises equivalent stress (plane stress)114.89 MPa
- Tresca equivalent stress128.06 MPa
- Normal stress on that plane89.64 MPa
- Shear stress on that plane-23.30 MPa
Limitations
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.