This exercise compares the shear stress hypothesis and the maximum shear strain energy criterion for a plane stress state.
Task
A thin-walled pipe is is under load of tension and torsion at the same time. The corresponding stress tensor is
\[ \newcommand{\myvec}[1]{{\begin{bmatrix}#1\end{bmatrix}}} \]\[ S = \myvec{\sigma^* & \tau & 0\\ \tau & 0 & 0\\0 & 0 & 0} \]
Set up the flow conditions according to TRESCA and HUBER-v.MISES. We are looking for the respective value for σ * that flow begins.
The yield point for the material is σF = 300 MPa, the shear stress τ = 145 MPa.
Solution
Yield condition according to TRESCA, shear stress hypothesis
In the following equations, the unit MPa is omitted for better readability. The equivalent stress according to TRESCA is for the plane stress state
\[ \newcommand{\myvec}[1]{{\begin{bmatrix}#1\end{bmatrix}}} \]\[ \tag{1} \sigma_v = \sqrt{\left( \sigma_x - \sigma_y \right)^2 + 4 \tau^2} \]
The equivalent stress is set equal to the yield point, so that the following results:
\[ \tag{2} 300 = \sqrt{\left( \sigma^* \right)^2 + 4 \left( 145 \right)^2} \]
\[ \tag{3} \sigma^* = \sqrt(5900) = 76.8 \, MPa \]
The stress state at the moment of the start of flow is according to TRESCA
\[ \tag{4} S = \myvec{76.8 & 145 & 0\\ 145 & 0 & 0\\0 & 0 & 0} \,MPa \]
Yield condition according to HUBER-v.MISES, maximum shear strain energy criterion
The equivalent stress according to HUBER-v.MISES is for the plane stress state
\[ \newcommand{\myvec}[1]{{\begin{bmatrix}#1\end{bmatrix}}} \]\[ \tag{5} \sigma_v = \sqrt{\sigma_x^2+\sigma_y^2-\sigma_x \sigma_y + 3 \tau^2} \]
The equivalent stress is set equal to the yield point, so that the following results:
\[ \tag{6} 300 = \sqrt{(\sigma^*)^2-\sigma^* \cdot 0 + 63075} \]
\[ \tag{7} \sigma^* = \sqrt{26925} = 164.1 \,MPa \]
According to HUBER-v.MISES, the state of stress at the moment the flow begins
\[ \tag{8} S = \myvec{164.1 & 145 & 0\\ 145 & 0 & 0\\0 & 0 & 0} \,MPa \]
The shear stress hypothesis calculates a significantly lower failure stress and is therefore the conservative approach.
