GATE 2022 AE – Question 29

Structures · Special topics: Equilibrium and compatibility equations in two-dimensional elasticity · 1 mark · Multiple select

An isotropic linear elastic material point under plane strain condition in the x-y plane always obeys:

$\epsilon_{zz}=0$
$\sigma_{zz}=0$
$\tau_{xz}=0$
$\gamma_{xz}=0$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $\epsilon_{zz}=0$; (C) $\tau_{xz}=0$; (D) $\gamma_{xz}=0$

Explanation

**Plane strain in the $x$-$y$ plane** is defined by zero strain in the out-of-plane direction:
$$\epsilon_{zz}=0,\qquad\gamma_{xz}=0,\qquad\gamma_{yz}=0 .$$

**A. $\epsilon_{zz}=0$.** ✓ (by definition)
**D. $\gamma_{xz}=0$.** ✓ (by definition)
**C. $\tau_{xz}=0$.** For an isotropic linear elastic material $\tau_{xz}=G\gamma_{xz}=0$. ✓
**B. $\sigma_{zz}=0$.** ✗ In plane strain the out-of-plane normal stress is generally not zero: $\sigma_{zz}=\nu(\sigma_{xx}+\sigma_{yy})$. (It is zero in plane **stress**.)

Answer **A, C and D**.