# 11.8: Tresca Yield Condition

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The stress state in uni-axial tension of a bar depends on the orientation of the plane on which the stresses are resolved. In Chapter 2 it was shown that the shear stress $$\tau$$ on the plane inclined to the horizontal plane by the angle $$\alpha$$ is

$\tau = \frac{1}{2} \sigma_{11} \sin 2\alpha$

where $$\sigma_{11}$$ is the uniaxial tensile stress, see Figure ($$\PageIndex{1}$$).

The maximum shear occurs when $$\sin 2\alpha = 1$$ or $$\alpha = \frac{\pi}{4}$$. Thus in uniaxial tension

$\tau_{\text{max}} = \frac{\sigma_{11}}{2}$

Extending the analysis to the 3-D case (see for example Fung) the maximum shear stresses on three shear planes are

$\tau_1 = \frac{|\sigma_1 − \sigma_2|}{2}, \quad \tau_2 = \frac{|\sigma_2 − \sigma_1|}{2}, \quad \tau_3 = \frac{|\sigma_3 − \sigma_1|}{2}$

where $$\sigma_1$$, $$\sigma_2$$, $$\sigma_3$$ are principal stresses. In 1860 the French scientist and engineer Henri Tresca put up a hypothesis that yielding of the material occurs when the maximum shear stress reaches a critical value

$\tau_o = \text{ max } \left\{ \frac{|\sigma_1 − \sigma_2|}{2}, \frac{|\sigma_2 − \sigma_3|}{2}, \frac{|\sigma_3 − \sigma_1|}{2} \right\}$

The unknown constant can be calibrated from the uniaxial test for which Equation (11.4.9) holds. Therefore at yield $$\tau_o = \sigma_y/2$$ and the Tresca yield condition takes the form

$\text{ max } \{|\sigma_1 − \sigma_2|, |\sigma_2 − \sigma_3|, |\sigma_3 − \sigma_1|\} = \sigma_y$

In the space of principal stresses the Tresca yield condition is represented by a prismatic open-ended tube, whose intersection with the octahedral plane is a regular hexagon, see Figure ($$\PageIndex{2}$$).

For plane stress, the intersection of the prismatic tube with the plane $$\sigma_3 = 0$$ forms a familiar Tresca hexagon, shown in Figure ($$\PageIndex{3}$$).

The effect of the hydrostatic pressure on yielding can be easily assessed by considering $$\sigma_1 = \sigma_2 = \sigma_3 = p$$. Then

$\sigma_1 − \sigma_2 = 0$

$\sigma_2 − \sigma_3 = 0$

$\sigma_3 − \sigma_1 = 0$

Under this stress state both von Mises yield criterion (Equation (??)) and the Tresca criterion (Equation (??)) predict that there will be no yielding.

This page titled 11.8: Tresca Yield Condition is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Tomasz Wierzbicki (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.