Pressure Vessels

When the thickness of the wall of the shell in less than $\frac{1}{10}$ to $\frac{1}{15}$ of its diameter, then shell is called thin shell.
\[t < \frac{D}{10} \quad to \quad \frac{D}{15}\]
When the thickness of the wall of the shell in greater than $\frac{1}{10}$ to $\frac{1}{15}$ of its diameter, then shell is called thick shell.
\[t > \frac{D}{10} \quad to \quad \frac{D}{15}\]

Thin Cylinders

Say L length, diameter d and t thickness of cylinder is subjected to internal pressure P. Due to this pressure, three type of stresses are developed at any point on the wall of cylinder -

1. Hoop Stress / Circumferential stress ($\sigma_h$) will be tensile in nature.
2. Longitudinal stress / Axial stress ($\sigma_L$) will be tensile in nature.
3. Radial stress ($\sigma_R$) will be compressive in nature. 

Analysis of thin cylinder

• It is assumed that stresses are uniformly distributed through the thickness of the wall.
• Radial stress varies from P at inner surface to atmospheric pressure at the outside of surface.
• If the internal pressure is very low, radial stress is negligible compared to axial and hoop stress. 

1. Hoop or circumferential stress ($\sigma_h$):

This stress is directed along to the tangent to the circumference of the cylinder. It resist the bursting effect due to internal pressure. 

At the equilibrium,

$P*(dL) = σ_h*(2tL)$

Hoop stress,                                       $\boxed{σ_h = \frac{Pd}{2t}}$

2. Longitudinal Stress ($\sigma_L$):

This stress is directed along the length of the cylinder and it tends to increase the length. 

At the equilibrium,
\(P*(\frac{\pi d^2}{4}) = σ_L*(\pi dt)\)

Longitudinal stress                  \(\boxed{σ_L = \frac{Pd}{4t}}\)

3. Longitudinal strain

\[ε_L = \frac{σ_L}{E} - μ\frac{σ_h}{E}\]

\[\boxed{ε_L = \frac{Pd}{4tE}(1 - 2μ)}\]

4. Hoop Strain

\[ε_h = \frac{σ_h}{E} - μ\frac{σ_L}{E}\]

\[\boxed{ε_h = \frac{Pd}{4tE}(2 - μ)}\]

5. Volumetric Strain

Volumetric strain = Longitudinal strain + 2*hoop strain
\[\boxed{ε_v = \frac{Pd}{4tE}(5 - 4μ)}\]

Thin Sphere

Hoop Stress/ Longitudinal Stress  

\[σ_h = σ_L = \frac{Pd}{4t}\]

Hoop Strain/ Longitudinal Strain

\[ε_h = ε_L = \frac{Pd}{4tE}(1 - μ)\]

Volumetric Strain

                                            \(ε_v = 3*ε_L = \frac{3Pd}{4tE}(1 - μ)\)

Thick Cylinders

• Radial stress in thin cylinder is neglected but it is of significant magnitude in case of thick cylinders.
• Tangential stresses assumed uniformly distributed over the wall in thin cylinder while it changes gradually from inner surface to outer surface in case of thick cylinders.  

• Axial Stress \( \sigma_z = \frac{P_i r_i^2}{r_o^2 - r_i^2}\)

• Radial Stress \(\sigma_r = A - \frac{B}{r^2}\)

• Circumferential or hoop stress   \(\sigma_h = A + \frac{B}{r^2}\)

A and B are constant which can be determine by boundary conditions:

$\sigma_r = - P_i$  at $r = r_i$

$\sigma_r = - P_o$  at $r = r_o$