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GATE 2020 AE – Question 40

Structures · Strength of materials: Strain energy, Castigliano's principles, three-dimensional Hooke's law, plane stress and strain, Euler buckling · 2 marks · Multiple choice

The isotropic strain–stress compliance matrix has A on its three normal diagonal entries, C on the other entries in its 3×3 normal block, and B on the three engineering-shear diagonal entries; other entries are zero. Thus $\boldsymbol\epsilon=\mathbf C\boldsymbol\sigma$, where $$\mathbf C=\begin{pmatrix}A&C&C&0&0&0\\C&A&C&0&0&0\\C&C&A&0&0&0\\0&0&0&B&0&0\\0&0&0&0&B&0\\0&0&0&0&0&B\end{pmatrix}.$$ The strain vector is $(\epsilon_{xx},\epsilon_{yy},\epsilon_{zz},\gamma_{yz},\gamma_{xz},\gamma_{xy})^T$ and the stress vector is $(\sigma_{xx},\sigma_{yy},\sigma_{zz},\tau_{yz},\tau_{xz},\tau_{xy})^T$. Which relation is correct?

  1. $C=A/2-B$
  2. $C=A/2+B$
  3. $C=A+B/2$
  4. $C=A-B/2$

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Correct answer: (D) $C=A-B/2$

Explanation

Hooke’s law identifies A = 1/E, C = −ν/E, and B = 1/G. With $G=E/[2(1+\nu)]$, B/2 = (1+ν)/E. Therefore $A-B/2=1/E-(1+\nu)/E=-\nu/E=C$. Select D. The engineering-shear convention is essential to the factor of two.

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