Solve full tensor stresses using accurate invariant relationships. Compare principal results, von Mises, and Tresca. Export findings, inspect plots, and validate structural loading confidently.
Enter direct stresses and shear stresses for the full symmetric 3D stress tensor.
The calculator solves the symmetric stress tensor below.
[σ] = [[σx, τxy, τzx], [τxy, σy, τyz], [τzx, τyz, σz]]
Stress invariants are computed using these relations.
I1 = σx + σy + σz
I2 = σxσy + σyσz + σzσx - τxy² - τyz² - τzx²
I3 = det([σ])
Principal stresses are the tensor eigenvalues.
det([σ] - λ[I]) = 0
Equivalent and shear metrics use standard engineering expressions.
σv = sqrt(((σx-σy)² + (σy-σz)² + (σz-σx)² + 6(τxy² + τyz² + τzx²))/2)
τmax = (σ1 - σ3) / 2
σm = I1 / 3
τoct = sqrt(((σ1-σ2)² + (σ2-σ3)² + (σ3-σ1)²)/9)
Enter the three normal stress components.
Enter the three independent shear stress components.
Type the preferred stress unit label.
Click the calculate button once.
Read the principal stresses and invariants first.
Check von Mises for ductile yielding studies.
Review maximum shear for Tresca comparisons.
Use the chart to compare tensor components visually.
Export the results as CSV or PDF anytime.
| σx | σy | σz | τxy | τyz | τzx | σ1 | σ2 | σ3 | Von Mises | Max Shear | Unit |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 120.0000 | 80.0000 | 40.0000 | 20.0000 | 10.0000 | 15.0000 | 131.7025 | 72.0412 | 36.2563 | 83.5165 | 47.7231 | MPa |
It evaluates a complete three-dimensional stress tensor. It returns principal stresses, invariants, mean stress, maximum shear, von Mises stress, octahedral stresses, and a comparison chart.
For a symmetric Cauchy stress tensor, τxy equals τyx, τyz equals τzy, and τzx equals τxz. That leaves three independent shear inputs.
Principal stresses are the normal stresses acting on planes where shear stress becomes zero. They are the eigenvalues of the stress tensor and are usually ordered from largest to smallest.
Use von Mises stress for ductile material yielding checks. It combines the multiaxial stress state into one equivalent scalar value for comparison with yield strength.
Maximum shear stress shows the largest shear action possible inside the element. It is useful for Tresca-style checks and understanding potential slip or distortion tendency.
Yes. Enter compressive stresses as negative values if that matches your sign convention. The calculator preserves the entered sign pattern during all derived computations.
No. The numeric relationships stay the same. The unit field only labels the results, chart, and exports. Use one consistent unit system throughout the calculation.
No. It is a fast engineering aid for tensor interpretation. Final design checks should still consider boundary conditions, stress gradients, code requirements, and material behavior.
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.