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GATE 2020 CE (CE1) – Question 17

Solid Mechanics · Transformation of stress, Mohr circle · 1 mark · Multiple choice

A two-dimensional stress state at P is $$[\sigma]=\begin{bmatrix}\sigma_{xx}&\tau_{xy}\\\tau_{xy}&\sigma_{yy}\end{bmatrix}.$$ Find the necessary and sufficient condition for pure shear on rotated axes.

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

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Correct answer: (C) $\sigma_{xx}+\sigma_{yy}=0$

Explanation

Pure shear has zero tensor trace. Rotation preserves trace, requiring $\sigma_{xx}+\sigma_{yy}=0$. Conversely, zero trace gives equal and opposite principal stresses; rotation by 45° from principal axes gives zero normal stresses. Thus this condition is both necessary and sufficient. **C**.