Tresca Stress Criterion Calculator

The Tresca maximum shear criterion uses the largest difference between principal stresses:

• σ_T = max(|S1 − S2|, |S2 − S3|, |S3 − S1|)
• τ_max = (S1 − S3) / 2 (when S1 ≥ S2 ≥ S3)

Yield is predicted when σ_T ≥ σ_y. If σ_y is provided, the safety factor is SF = σ_y / σ_T.

1. Select an input mode: principal stresses or stress tensor.
2. Choose a consistent stress unit for your inputs.
3. Enter stress values; include yield strength for assessment.
4. Press Submit to view results above the form.
5. Download CSV or PDF for documentation if needed.

1) Why the Tresca criterion matters

The Tresca criterion predicts yielding when maximum shear stress reaches the shear yield limit. It is widely used for ductile metals where slip begins under shear-dominated loading. Compared with the von Mises criterion, Tresca is slightly more conservative for combined stresses.

2) Key outputs you should interpret

This calculator reports principal stresses (S1, S2, S3), maximum shear stress τ_max, and the Tresca equivalent stress σ_T. For ordered principal stresses, τ_max = (S1 − S3)/2. The equivalent stress is σ_T = max(|S1 − S2|, |S2 − S3|, |S3 − S1|).

3) Relationship to von Mises results

In the deviatoric plane, the Tresca yield surface circumscribes the von Mises surface. A useful comparison is the worst-case ratio between criteria: Tresca may allow about 1.1547 times the von Mises equivalent for the same stress state (2/√3). This explains why Tresca often predicts yield earlier in multi-axial loading.

4) Using stress tensor inputs correctly

When you enter tensor components (σx, σy, σz, τxy, τyz, τzx), the tool forms a symmetric matrix and computes its eigenvalues as the principal stresses. Ensure your shear sign convention is consistent across components. Mixed units across inputs can invalidate the eigenvalue calculation, so keep one unit system per case.

5) Yield strength, safety factor, and design checks

If you provide yield strength σ_y, the calculator estimates safety factor as SF = σ_y/σ_T. Typical room-temperature yield strengths range from about 200–350 MPa for many structural steels, 400–1100 MPa for heat-treated alloy steels, and 150–300 MPa for common aluminum alloys. Always confirm the exact value from your material standard and heat treatment.

6) Practical load-case comparison workflow

A robust workflow is to evaluate each load case with the same reference coordinate system, compute σ_T, then rank cases by σ_T or by minimum SF. Exporting CSV supports traceable review, while the PDF captures a snapshot for design records. Store the note field with the load case name and temperature.

7) Common pitfalls and quality checks

Watch for swapped shear components, forgotten unit conversions, and missing compressive sign. If your principal stresses appear unexpected, cross-check invariants: the mean stress should match (σx + σy + σz)/3 in tensor mode. For pure shear, you should see S1 ≈ +τ and S3 ≈ −τ.

8) What the criterion does not cover

Tresca addresses initial yielding for ductile materials, not fracture, fatigue, creep, or buckling. It also does not model post-yield hardening or anisotropy. Use this calculator as a screening tool, then apply code- or test-based rules for final acceptance, including geometry and manufacturing effects.

1) What is the Tresca equivalent stress?

It is the largest absolute difference between principal stresses: max(|S1−S2|, |S2−S3|, |S3−S1|). It maps multi-axial loading to a single value for yield screening.

2) How is maximum shear stress reported here?

When S1 ≥ S2 ≥ S3, the maximum shear stress is τmax = (S1−S3)/2. This is the shear on planes oriented 45° between the first and third principal directions.

3) Why does Tresca often look conservative?

Its yield surface is a hexagon in deviatoric space, outside the von Mises circle. For some stress states, Tresca predicts yield at about 2/√3 ≈ 1.1547 lower load than von Mises.

4) Should I use principal mode or tensor mode?

Use principal mode if you already know S1, S2, S3 from Mohr’s circle or FEA output. Use tensor mode when you have component stresses in a coordinate system and need principals computed.

5) Does compressive stress count for yielding?

Yes. Tresca uses differences between principal stresses, so compressive values can increase σT when combined with tensile stresses. Always keep the correct sign convention for compression and tension.

6) What safety factor does the calculator compute?

If you enter yield strength, it computes SF = σy/σT. SF below 1 indicates predicted yield by Tresca. Select σy from the correct standard, condition, and temperature for your part.

7) Can I use this for fatigue design?

Not directly. Fatigue depends on stress ranges, mean stress effects, surface finish, and cycles. Use σT to screen load severity, then apply an appropriate fatigue method and S–N or strain-life data.

Use results carefully and confirm with applicable material standards.