# Thin Cylinder

 Question 1
The figure shows a thin-walled open-top cylindrical vessel of radius $r$ and wall thickness $t$. The vessel is held along the brim and contains a constant-density liquid to height $h$ from the base. Neglect atmospheric pressure, the weight of the vessel and bending stresses in the vessel walls.
Which one of the plots depicts qualitatively CORRECT dependence of the magnitudes of axial wall stress $(\sigma _1)$ and circumferential wall stress $(\sigma _2)$ on $y$?  A A B B C C D D
GATE ME 2023   Strength of Materials
Question 1 Explanation: Unit weight of liquid $=\gamma$
$\text { Weight of liquid }=\gamma \times \text { volume }$
$W=\gamma \times\left(\pi r^{2}\right)(h)$
$\sigma_{1}=\frac{W_{T}}{(2 \pi r \times t)}=\frac{\gamma \times \pi r^{2}(h)}{2 \pi r t}=\frac{\gamma r h}{2 t}$
$\text { Pressure at } y^{\prime}=\gamma \times y$
$\sigma_{2}(\text { circumterential stress })=\frac{P d}{2 t}=\frac{\gamma \times y \times 2 r}{2 t}=\frac{\gamma r y}{t}$ Question 2
A thin-walled cylindrical pressure vessel has mean wall thickness of $t$ and nominal radius of $r$. The Poisson's ratio of the wall material is 1/3. When it was subjected to some internal pressure, its nominal perimeter in the cylindrical portion increased by 0.1% and the corresponding wall thickness became $\bar{t}$. The corresponding change in the wall thickness of the cylindrical portion, i.e. $100\times (\bar{t}-t)/t$, is ________%(round off to 3 decimal places).
 A 0.06 B -0.06 C 0.12 D -0.12
GATE ME 2022 SET-1   Strength of Materials
Question 2 Explanation: $\pi(d+\delta d)=1.001 \pi d$
$\varepsilon _1=\frac{\delta d}{d}=0.001=\frac{Pd}{4tE}(2-v)$
$\frac{Pd}{4tE}=\frac{0.001}{(2-v)}=6 \times 10^{-4}$
$\varepsilon _2=\frac{\delta t}{t}=\frac{1}{E}\left [ \sigma _r-v(\sigma _h-\sigma _L) \right ]$
$\frac{\delta t}{t}=\frac{1}{E}\left [ -v\frac{Pd}{4t}(2+1) \right ]$
$\frac{\delta t}{t}=-\frac{Pd}{4tE}v(3)=-6 \times 10^{-4} \times \frac{1}{3}(3)=-6 \times 10^{-4}$
Therefore, The corresponding change in the wall thickness of the cylindrical portion
$=100 \times \left [ \frac{\bar{t}-t}{t} \right ]=100 \times \frac{\delta t}{t}=100 \times (-6) \times 10^{-4}=-0.06 \%$

 Question 3
Consider two concentric circular cylinders of different materials M and N in contact with each other at r=b, as shown below. The interface at r=b is frictionless. The composite cylinder system is subjected to internal pressure P. Let $(u_r^M,u_\theta ^M)$ and $(\sigma_{rr}^M, \sigma_{\theta \theta}^M)$ denote the radial and tangential displacement and stress components, respectively, in material M. Similarly, $(u_r^N,u_\theta ^N)$ and $(\sigma_{rr}^N,\sigma_{\theta \theta}^N)$ denote the radial and tangential displacement and stress components, respectively, in material N. The boundary conditions that need to be satisfied at the frictionless interface between the two cylinders are: A $u_r^M=u_r^N$ and $\sigma _{rr }^M=\sigma _{rr}^N$ B $u_r^M=u_r^N$ and $\sigma _{rr }^M=\sigma _{rr}^N$and $u_{\theta }^M=u_{\theta }^N$ and $\sigma _{\theta \theta }^M=\sigma _{\theta \theta }^N$ C $u_{\theta }^M=u_{\theta }^N$ and $\sigma _{\theta \theta }^M=\sigma _{\theta \theta }^N$ D $\sigma _{rr }^M=\sigma _{rr}^N$ and $\sigma _{\theta \theta }^M=\sigma _{\theta \theta }^N$
GATE ME 2019 SET-2   Strength of Materials
Question 3 Explanation:
As the contact is frictionless, one cylinder can rotate freely with respect other. The displacement in tangential directions need not be same at a point contact for two cylinders (i.e., $u_{\theta}^{M} \neq u_{\theta}^{N}$ ). Similarly
the Hoop stress at point of contact need not be same (i.e., $\sigma_{\theta \theta}^{M} \neq \sigma_{\theta \theta}^{N}$ ). As the interface will be always in contact the displacement in radial direction and stress in radial directions must be same for two cylinders. i.e., $u_{r}^{M}=u_{r}^{N}$ and $\sigma_{r}^{M}=\sigma_{r r}^{N}$
 Question 4
A thin cylindrical pressure vessel with closed-ends is subjected to internal pressure. The ratio of circumferential (hoop) stress to the longitudinal stress is
 A 0.25 B 0.5 C 1 D 2
GATE ME 2016 SET-2   Strength of Materials
Question 4 Explanation:
For thin cylinder
Circumferential stress
$\sigma_{h}=\frac{p d}{2 t}$
Longitudinal stress,
\begin{aligned} \sigma_{L} &=\frac{p d}{4 t} \\ \therefore \quad \frac{\sigma_{h} }{\sigma_{L}} &=2 \end{aligned}
 Question 5
A cylindrical tank with closed ends is filled with compressed air at a pressure of 500 kPa. The inner radius of the tank is 2 m, and it has wall thickness of 10 mm. The magnitude of maximum in-plane shear stress (in MPa) is ________
 A 25MPa B 34MPa C 65MPa D 45MPa
GATE ME 2015 SET-3   Strength of Materials
Question 5 Explanation:
Maximum in plane shear stress (in MPa)
\begin{aligned} \tau &=\frac{\sigma_{1}-\sigma_{2}}{2}=\frac{\frac{p d}{2 t}-\frac{p d}{4 t}}{2} \\ &=\frac{500 \times 10^{3} \times 4000}{8 \times 10}=25 \times 10^{6} \mathrm{Pa}=25 \mathrm{MPa} \end{aligned}

There are 5 questions to complete.