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Tresca predicts initial yielding of a ductile material when maximum shear stress reaches half the uniaxial yield strength.
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Formula
Principal stresses and yield strength in Pa; stress differences in Pa
| Symbol | Meaning | Unit |
|---|---|---|
| Three principal stresses | Pa | |
| Maximum absolute principal-stress difference | Pa | |
| Uniaxial yield strength | Pa | |
| Maximum shear stress, half the maximum difference | Pa |
\sigma_T=\max(|\sigma_1-\sigma_2|,|\sigma_2-\sigma_3|,|\sigma_3-\sigma_1|)\leq\sigma_yFor an idealized isotropic ductile material under static loading, Tresca predicts yielding at τmax=σy/2; compare all three principal-stress differences. It is a yield-onset criterion, not a fracture or fatigue check. Anisotropy, residual stresses, temperature, detailed multiaxial response and safety factors require additional assessment. Tresca is at least as conservative as von Mises for common comparisons.
For an idealized isotropic ductile material under static loading, Tresca predicts yielding at τmax=σy/2; compare all three principal-stress differences. It is a yield-onset criterion, not a fracture or fatigue check. Anisotropy, residual stresses, temperature, detailed multiaxial response and safety factors require additional assessment. Tresca is at least as conservative as von Mises for common comparisons.
Tresca predicts initial yielding of a ductile material when maximum shear stress reaches half the uniaxial yield strength.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Tresca yield criterion (maximum shear stress)” and check its unit and magnitude.
For principal stresses 120, 20 and −30 MPa, pairwise differences are 100, 50 and 150 MPa.
Thus σT=150 MPa.
With σy=200 MPa, the criterion is satisfied; equivalently τmax=75 MPa is below σy/2=100 MPa.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: Principal stresses and yield strength in Pa; stress differences in Pa.
Formula cluster
Tresca predicts initial yielding of a ductile material when maximum shear stress reaches half the uniaxial yield strength.
For principal stresses 120, 20 and −30 MPa, pairwise differences are 100, 50 and 150 MPa. Thus σT=150 MPa. With σy=200 MPa, the criterion is satisfied; equivalently τmax=75 MPa is below σy/2=100 MPa.
Assess multiaxial stress states in ductile components using a maximum-shear model.
Here is how to work through a typical Tresca yield criterion (maximum shear stress) (Maximale Hauptspannungsdifferenz ≤ Streckgrenze) task step by step:
Task
Work through the numerical example for “Tresca yield criterion (maximum shear stress)” and check its unit and magnitude.
Solution path
For principal stresses 120, 20 and −30 MPa, pairwise differences are 100, 50 and 150 MPa.
Thus σT=150 MPa.
With σy=200 MPa, the criterion is satisfied; equivalently τmax=75 MPa is below σy/2=100 MPa.
Principal stresses are normal stresses on mutually perpendicular principal planes. Maximum shear is half the largest difference between the greatest and smallest principal stresses. In three dimensions compare all pairs; for ordered values, σ1−σ3 suffices. Signs matter because tensile and compressive principal stresses can enlarge the difference. Compare all principal-stress differences; Tresca yielding is not a fatigue or fracture assessment. Application: Assess multiaxial stress states in ductile components using a maximum-shear model. The numerical example For principal stresses 120, 20 and −30 MPa, pairwise differences are 100, 50 and 150 MPa. Thus σT=150 MPa. With σy=200 MPa, the criterion is satisfied; equivalently τmax=75 MPa is below σy/2=100 MPa anchors the idea in a concrete measurement case.
Under uniaxial tension the principal stresses are σ, 0 and 0. The maximum difference is |σ|, so Tresca predicts yield at |σ|=σy, matching the uniaxial yield strength. Multiaxial loading can reach the same difference sooner. The criterion therefore transfers a measured uniaxial material property to a more complex stress state. Compare all principal-stress differences; Tresca yielding is not a fatigue or fracture assessment. The corresponding units are Principal stresses and yield strength in Pa; stress differences in Pa. For principal stresses 120, 20 and −30 MPa, pairwise differences are 100, 50 and 150 MPa. Thus σT=150 MPa. With σy=200 MPa, the criterion is satisfied; equivalently τmax=75 MPa is below σy/2=100 MPa. This checks that the reference quantities are compatible.
For 120, 20 and −30 MPa, pairwise differences are 100, 50 and 150 MPa; 150 MPa governs. With σy=200 MPa, the criterion remains below yield and τmax=75 MPa is below 100 MPa. This margin is not a component safety factor because load factors, scatter and geometry effects are absent. Compare all principal-stress differences; Tresca yielding is not a fatigue or fracture assessment. Scope: For an idealized isotropic ductile material under static loading, Tresca predicts yielding at τmax=σy/2; compare all three principal-stress differences. It is a yield-onset criterion, not a fracture or fatigue check. Anisotropy, residual stresses, temperature, detailed multiaxial response and safety factors require additional assessment. Tresca is at least as conservative as von Mises for common comparisons. A practical application is Assess multiaxial stress states in ductile components using a maximum-shear model.
Von Mises uses an energy-based equivalent stress and fits many isotropic ductile metals well. Tresca uses maximum shear and is often slightly more conservative. Both are idealized and agree in uniaxial stress, but can predict different multiaxial boundaries. Use the criterion required by the discipline or governing code. Compare all principal-stress differences; Tresca yielding is not a fatigue or fracture assessment. Check the calculation with this case: For principal stresses 120, 20 and −30 MPa, pairwise differences are 100, 50 and 150 MPa. Thus σT=150 MPa. With σy=200 MPa, the criterion is satisfied; equivalently τmax=75 MPa is below σy/2=100 MPa. Keep the model limits and units in view.
Use the three algebraic principal stresses at the same material point and load case, not arbitrary components from different axes. In plane stress the third principal stress is often zero if plane stress truly applies. In plane strain it may be nonzero. First form the appropriate stress tensor and determine its principal values. Compare all principal-stress differences; Tresca yielding is not a fatigue or fracture assessment. The relation describes Tresca predicts initial yielding of a ductile material when maximum shear stress reaches half the uniaxial yield strength. This helps distinguish its target quantity from neighbouring quantities.
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Which formula describes Tresca yield criterion (maximum shear stress)?
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