Question

Yielding starts in a material when the principal stresses are 100 MPa, 100 MPa and 200 MPa. As per the von Mises criterion, yield stress (in MPa) of the material is \(\underline{\hspace{2cm}}\). [round off to nearest integer]

Hint:

In von Mises criterion, only the differences between principal stresses matter, not their absolute values.

Answer 1

Correct Answer: 101 - 99

Given principal stresses: \[ \sigma_1 = 200\ \text{MPa}, \sigma_2 = 100\ \text{MPa}, \sigma_3 = 100\ \text{MPa} \] Von Mises yield criterion: \[ \sigma_y = \sqrt{\frac{1}{2}\left[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2\right]} \] Substitute values: \[ (\sigma_1 - \sigma_2)^2 = (200 - 100)^2 = 10000 \] \[ (\sigma_2 - \sigma_3)^2 = (100 - 100)^2 = 0 \] \[ (\sigma_3 - \sigma_1)^2 = (100 - 200)^2 = 10000 \] Total: \[ \sigma_y = \sqrt{\frac{1}{2}(20000)} = \sqrt{10000} = 100\ \text{MPa} \]