Tresca Yield Criterion
Tresca Yield Criterion: Theoretical Foundations
Tresca Yield Condition
Professor, how is the Tresca yield condition different from von Mises?
The Tresca criterion states that yielding occurs when the maximum shear stress reaches a critical value:
It forms a regular hexagon in deviatoric stress space. It is inscribed within the von Mises circle.
Is Tresca more conservative than von Mises?
The Tresca yield surface lies inside the von Mises surface (inscribed hexagon). For the same stress state, Tresca predicts yielding first. Therefore, Tresca is more conservative (safer side). The maximum difference is 15%.
Use in FEM
The Tresca criterion involves complex numerical handling at the corners of the yield surface. In practice, von Mises is overwhelmingly more common. Design codes like ASME BPVC sometimes evaluate using Tresca stress (stress intensity = $\sigma_1 - \sigma_3$).
Summary
Historical Background of the Tresca Criterion
In 1864, Henri Tresca reported to the Paris Academy of Sciences, based on extrusion experiments with lead, iron, and copper, that yielding occurs when the maximum shear stress reaches a material-specific critical value. The criterion formula is (σ₁-σ₃)/2=k (k=τy), forming a hexagonal prism in principal stress space. Saint-Venant (1870) provided the mathematical formulation, laying the foundation for mechanical engineering design in the 19th century.
Computational Methods for the Tresca Yield Criterion
Tresca in FEM
The Tresca criterion has complex Return Mapping at corners. Support in commercial solvers:
- Abaqus: No direct Tresca support (uses von Mises)
- Nastran: No direct Tresca support
- Ansys: von Mises or DP criterion
Is there no dedicated implementation for Tresca?
The difference between von Mises and Tresca is at most 15%. von Mises is sufficient for most problems. If Tresca is needed, implement via user subroutines (UMAT).
Summary
Corner Treatment of Hexagonal Yield Surface
The Tresca yield surface is a hexagonal prism with corners in principal stress space, meaning the normal vector is not uniquely defined when the stress state is near a corner. This is handled by applying Koiter's (1953) corner rule, combining the normals of two adjacent faces. In implementation, approximations like switching Tresca to Drucker-Prager near σ₁≈σ₂ are also used.
Tresca Yield Criterion in Practice
Tresca in Practice
ASME BPVC stress classification evaluates using Stress Intensity ($S_I = \sigma_1 - \sigma_3$). This corresponds to the Tresca criterion. Calculate with von Mises in FEM, and also output Stress Intensity in post-processing.
Practical Checklist
Adoption in Pressure Vessel Design Codes
The ASME Boiler and Pressure Vessel Code (Section VIII) uses the Tresca criterion as the basis for design, defining allowable stress as the lesser of 1/3 of tensile strength or 2/3 of yield strength. It has been continuously adopted since the first edition in 1914 and still functions as a standard for regulatory design of oil refinery plants and nuclear pressure vessels.
Tresca Yield Criterion: Software & Solver Comparison
Tools
All solvers use von Mises as standard. Tresca stress (Stress Intensity) can be output in post-processing.
Origin of Tresca Yield Rule: 19th Century Metalworking Research
The Tresca yield rule is a maximum shear stress criterion derived by Henri Tresca in 1864 from lead extrusion experiments for the Paris Exposition. It is 7% more conservative than the Mises rule, so ASME Section VIII and EN 13445 (pressure vessel codes) require the safer Tresca rule. In Nastran, stress output options MISES/TRESCA can be switched for comparison, with documented cases where yield pressure for pipe elbow design was evaluated 11% lower.
Advanced Technology
Advanced
Yield Prediction Difference from von Mises
The Tresca criterion gives τy=σy/2 in pure shear, which is about 15.5% smaller than von Mises' τy=σy/√3. They coincide under equibiaxial tension (σ₁=σ₂). In pure shear tests, von Mises is often closer to experimental values, while Tresca gives conservative (safer side) predictions.
Tresca Yield Criterion: Common Issues & Debugging
Troubles
Dealing with Convergence Issues at Corners
When solving the Tresca model with FEM, convergence can fail when principal stresses are nearly equal (Lode
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