Learning Objectives:

• Classify a cylindrical pressure vessel as thin or thick based on its thickness and other dimensions.

• Calculate tangential (hoop) stress and radial stress for thin cylinders.

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If the thickness of the cylinder is 10% or less of inside radius, the cylinder can be considered as a thin cylinder.

In such case, the stress calculations can be simplified as:

For longitudinal (axial) stress determination, \[\pi r^2\times P_i = \sigma_a \times \pi 2r \times t\]

\[\sigma_a = \dfrac{P_{i}r}{2t}\]

For hoop stress determination, \[P_i (2r) l = 2\sigma_t \times t \times l\]

\[\sigma _{t} = \dfrac{P_{i}r}{t}\]

Comparing, \(\sigma_t\) = 2\(\sigma_a\)
where t = wall thickness
\(P _{i}\) = internal pressure
\(\sigma _{t}\)= tangential (hoop) stress
\(\sigma _{a}\)= axial stress

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Solved Example: 93-1-12

Water is flowing in a pipe of 200 cm diameter under a pressure head of 10000 cm. The thickness of the pipe wall is 0.75 cm. The tensile stress in the pipe wall in MPa is: (SSC JE ME Dec 2020 Morning Session)

A. 13

B. 100

C. 130.5

D. 1305

Correct Answer: C

Solved Example: 93-1-13

The volumetric strain of fluid-filled inside the thin cylinder (diameter = D) under the pressure (P) is given by [where μ, t, E are Poisson ratio, thickness and modulus of elasticity respectively] (RPSC ME Lecturer 2014)

A. $\dfrac{{PD\left( {1 - 4\mu} \right)}}{{4tE}}$

B. $\dfrac{{PD\left( {5 - \mu} \right)}}{{4tE}}$

C. $\dfrac{{PD\left( {5 - 4\mu} \right)}}{{4tE}}$

D. $\dfrac{{PD\left( {1 -\mu} \right)}}{{4tE}}$

Correct Answer: C

Learning Objectives:

• Calculate tangential (hoop) stress and radial stress for thick cylinders.

• Calculate axial stress for vessels with end caps.

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For internal pressure only, the stresses at the inside wall are:

\[\sigma_{t}=P_{i}\left(\frac{r_{o}^{2}+r_{i}^{2}}{r_{o}^{2}-r_{i}^{2}}\right), \ \ \sigma_r = - P_i\]

For external pressure only, the stresses at the outside wall are:

\[\sigma_{t}= - P_{o}\left(\dfrac{r_{o}^{2}+r_{i}^{2}}{r_{o}^{2}-r_{i}^{2}}\right),\ \ \sigma_r = - P_o\] where,
\(r_{i}\) = inside radius
\(r_{o}\) = outside radius
For vessels with end caps:

\[\sigma_{a} = P_{i}\left(\frac{r_{i}^{2}}{r_{o}^{2}-r_{i}^{2}}\right)\]

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