GATE Civil Engineering
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In a two-dimensional stress analysis, the state of stress at a point $P$ is 

$$\begin{bmatrix} \sigma \end{bmatrix} = \begin{bmatrix}\sigma_{xx} &\tau_{xy} \\ \tau_{xy} &\sigma_{yy} \end{bmatrix}$$

The necessary and sufficient condition for existence of the state of pure shear at the point $P$, is

  1. $\sigma_{xx}\sigma_{yy}-\tau^2_{xy}=0$
  2. $\tau_{xy}=0$
  3. $\sigma_{xx}+\sigma_{yy}=0$
  4. $(\sigma_{xx}-\sigma_{yy})^2+4\tau^2_{xy}=0$
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