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

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 tangential stress in the pipe wall in MPa is:

A. 13

B. 100

C. 130

D. 1305

\[\mathrm{Pressure\ head} = \dfrac{P_i}{\rho g} = 10000 \times 10^{-2} = 100\ \mathrm{m}\] \begin{align*} P_i &= h \rho g\\ P_i &= 100 \times 1000 \times 9.81\\ \sigma_t &= \dfrac{P_i \cdot r}{t}\\ \sigma_t &= \dfrac{100 \times 1000 \times 9.81 \times 1}{(0.75 \times 10^{-2})}\\ \sigma_t &= 130\ \mathrm{MPa} \end{align*}

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 $\mu$, t, E are Poisson ratio, thickness and modulus of elasticity respectively]

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:

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

where,
\(r_{i}\) = inside radius
\(r_{o}\) = outside radius
For vessels with end caps:

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