Question

In a two dimensional stress system, it is assumed that the principal stress ${\sigma }_{1}and{\sigma }_2$ are such that ${\sigma }_1>{\sigma }_2$; then according to the maximum shear stress theory, the failure occurs when (where ${\sigma }_y$ is the yield stress, $\mu$ is the Poisson's ratio and E is the modulus of elasticity)

• A. $\frac{1}{E}({\sigma }_1-\mu {\sigma }_2)\ge \frac{{\sigma }_y}{E}$
• B. ${\sigma }_1^2+{\sigma }_2^2+2\mu {\sigma }_1{\sigma }_2\ge {\sigma }_y^2$
• C. ${\sigma }_1-{\sigma }_2\ge {\sigma }_y$
• D. ${\sigma }_1^2+{\sigma }_2^2-{\sigma }_1{\sigma }_2\ge {\sigma }_y$

Answer 1

The correct option is C

${\sigma }_1-{\sigma }_2\ge {\sigma }_y$

$({\sigma }_1-{\sigma }_2)\ge {\sigma }_y$ : Maximum shear stress theory
$\frac{1}{E}({\sigma }_1-\mu {\sigma }_2)\ge \frac{{\sigma }_y}{E}$: Maximum principal strain theroy
$({\sigma }_1^2+{\sigma }_2^2-2\mu {\sigma }_1{\sigma }_2)\ge f_y^2$: Maximum strain energy theory
$({\sigma }_1^2+{\sigma }_2^2-{\sigma }_1{\sigma }_2)\ge f_y^2$ : Distortion energy theory