A plane strain problem (in X-Y plane) must satisfy the condition:

Question

What is the CORRECT match between the air pollutants and treatment techniques given in the table

Air Pollutants		Treatment techniques
P	NO₂	i	Flaring
Q	SO₂	ii	Cyclonic separator
R	CO	iii	Lime scrubbing
S	Particles	iv	NH₃ injection

Answer 1

Plane strain and plane stress are the two standard 2-D simplifications of 3-D elasticity, and this question checks which strain components vanish under the plane strain assumption. Go through each option as a candidate definition.

$\sigma_{zz}=0$: this is actually the defining condition of plane STRESS (used for thin flat plates loaded in their own plane), not plane strain. Under plane strain the z-direction is restrained, not free of stress, so $\sigma_{zz}$ builds up instead of vanishing.
$\epsilon_{zz}=\epsilon_{xz}=\epsilon_{yz}=0$: plane strain applies to a long body, like a long earth dam or tunnel cross-section, where the ends are fixed and geometry/loading do not change along the length. That length direction, z, cannot elongate or shear out of plane, so every strain with a z index is exactly zero. This matches the physical setup used to derive plane strain equations.
$\sigma_{xx} \neq \sigma_{xy} \neq \sigma_{xz} \neq 0$: this only says the stress components differ from each other and are non-zero, an arbitrary inequality that could describe almost any general 3-D stress point, not a rule defining plane strain.
$\epsilon_{xx} \neq \epsilon_{yy} \neq \epsilon_{xy} \neq 0$: same problem as option 3, but for in-plane strains. It is a generic statement, not the restraint condition that produces plane strain.

So the only option stating an actual restraint condition, rather than a vague inequality, is the second one. Plane strain forces $\epsilon_{zz}=\epsilon_{xz}=\epsilon_{yz}=0$, while $\sigma_{zz}$ stays non-zero because of the trapped Poisson effect.

Let's summarize:

Plane stress: $\sigma_{zz}=0$, but $\epsilon_{zz}\neq 0$.
Plane strain: $\epsilon_{zz}=\epsilon_{xz}=\epsilon_{yz}=0$, but $\sigma_{zz}\neq 0$.

The correct option is (B).

Answer 2

Plane strain and plane stress are the two standard 2-D simplifications of 3-D elasticity, and this question checks which strain components vanish under the plane strain assumption. Go through each option as a candidate definition.

$\sigma_{zz}=0$: this is actually the defining condition of plane STRESS (used for thin flat plates loaded in their own plane), not plane strain. Under plane strain the z-direction is restrained, not free of stress, so $\sigma_{zz}$ builds up instead of vanishing.
$\epsilon_{zz}=\epsilon_{xz}=\epsilon_{yz}=0$: plane strain applies to a long body, like a long earth dam or tunnel cross-section, where the ends are fixed and geometry/loading do not change along the length. That length direction, z, cannot elongate or shear out of plane, so every strain with a z index is exactly zero. This matches the physical setup used to derive plane strain equations.
$\sigma_{xx} \neq \sigma_{xy} \neq \sigma_{xz} \neq 0$: this only says the stress components differ from each other and are non-zero, an arbitrary inequality that could describe almost any general 3-D stress point, not a rule defining plane strain.
$\epsilon_{xx} \neq \epsilon_{yy} \neq \epsilon_{xy} \neq 0$: same problem as option 3, but for in-plane strains. It is a generic statement, not the restraint condition that produces plane strain.

So the only option stating an actual restraint condition, rather than a vague inequality, is the second one. Plane strain forces $\epsilon_{zz}=\epsilon_{xz}=\epsilon_{yz}=0$, while $\sigma_{zz}$ stays non-zero because of the trapped Poisson effect.

Let's summarize:

Plane stress: $\sigma_{zz}=0$, but $\epsilon_{zz}\neq 0$.
Plane strain: $\epsilon_{zz}=\epsilon_{xz}=\epsilon_{yz}=0$, but $\sigma_{zz}\neq 0$.

The correct option is (B).

A plane strain problem (in X-Y plane) must satisfy the condition:

Show Hint

The Correct Option is B

Solution and Explanation

Questions Asked in GATE CE exam