This exercise shows how the safety against flow is calculated according to the TRESCA shear stress hypothesis. The equivalent stress is calculated from the principal stresses and then compared with the yield stress of the material.

## Task

A load case with the stress tensor is given

\[ \newcommand{\myvec}[1]{{\begin{bmatrix}#1\end{bmatrix}}} \]\[ S = \myvec{-160 & 120 & 0\\120 & 80 & 0\\0 & 0 & 0} \, MPa \]

The yield point for the material is σ_{F} = 650 MPa. Is there a double safety against flowing according to TRESCA?

## Solution

The yield condition according to TRESCA (shear stress hypothesis) is

\[ \tag{1} |\tau_{max}| \overset{!}{=} \frac{\sigma_0}{2} \]

with

\[ \tag{2} \frac{1}{2}\,Max \left[ |\sigma_1 -- \sigma_2|,|\sigma_2 -- \sigma_3|,|\sigma_1 -- \sigma_3| \right] = \frac{\sigma_0}{2} \]

Since it is a plane stress state, only σ_{1} and σ_{2} have to be considered.

The dimensions of the tensions are not noted in the following equations to improve readability. The angle of the principal stresses is

\[ \tag{3} \phi_0 = \frac{arctan\left( \frac{2\tau_{xy}}{\sigma_x -\sigma_y} \right)}{2}\]

\[ \tag{4} \phi_0 = \frac{arctan\left( \frac{2 \cdot 120}{-160 -80} \right)}{2}\]

\[ \tag{5} \phi_0 = -22.5° \]

The principal stresses are

\[ \tag{6} \sigma_1 = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x-\sigma_y}{2} \cdot cos 2 \phi_0 + \tau_{xy} \cdot sin 2 \phi_0 \]

\[ \tag{7} \sigma_1 = \frac{-160 + 80}{2} + \frac{-160-80}{2} \cdot cos (-45°) + 120 \cdot sin (-45°) \]

\[ \tag{8} \sigma_1 = -210 \, MPa \]

\[ \tag{9} \sigma_2 = \frac{\sigma_x + \sigma_y}{2} -- \frac{\sigma_x-\sigma_y}{2} \cdot cos 2 \phi_0 -- \tau_{xy} \cdot sin 2 \phi_0 \]

\[ \tag{10} \sigma_2 = \frac{-160 + 80}{2} -- \frac{-160-80}{2} \cdot cos (-45°) -- 120 \cdot sin (-45°) \]

\[ \tag{11} \sigma_2 = 130 \, MPa \]

This results in the decisive tension

\[ \tag{12} \sigma_0 = |-210 -- 130| \, MPa = 340 \, MPa \]

The safety against flow S_{F} is

\[ \tag{13} S_F = \frac{\sigma_F}{\sigma_0} = \frac{650\,MPa}{340\,MPa} = 1.91 \]

This shows that there is no double safety against flow according to the shear stress hypothesis according to TRESCA.