GATE Mechanical Engineering (ME) 2020 Shift 2 Solved Paper

Question : 38 of 65

Marks: +1, -0

A thin-walled cylinder of radius r and thickness t is open at both ends, and fits snuglybetween two rigid walls under ambient conditions, as shown in the figure.

The material of the cylinder has Young's modulus E, Poisson's ratio

v

, and coefficient of thermal expansion

\alpha

. What is the minimum rise in temperature

\Delta {T}

of the cylinder (assume uniform cylinder temperature with no buckling of the cylinder) required to prevent gas leakage if the cylinder has to store the gas at an internal pressure of

p

above the atmosphere?

$∆ T=(v-\;{1}/{4}) \;{p r}/{α t E}$
$∆ {T}=\;{3 \;\text " vpr "\;}/{2 α {tE}}$
$∆ T=(v+\;{1}/{2}) \;{p r}/{α t E}$
$∆ T=\;{v p r}/{α t E}$

Validate

Solution:

Strain along the longitudinal direction due to pressure

ε_{1}=\;{σ_{1}}/{E}-v \;{σ_{C}}/{E}=-\;{V σ_{C}}/{E}

[ ∵ σ_{1}=0

(cylinder open at both ends

]

Strain in longitudinal direction due to thermal expansion

=a ∆ T

Equate the magnitude

\;{v σ_{C}}/{E}=α ∆ T

\;{v P r}/{t E}=α ∆ T

∆ T=\;{v P r}/{α t E}

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