Understanding Stress in Thin Cylinder Pressure Vessels

Thin cylinder pressure vessels, like pipes and tanks, are subjected to internal pressure. This pressure creates stresses within the walls of the cylinder. The main stresses considered in thin cylinders are the hoop stress (circumferential stress) and the longitudinal stress (axial stress).

Calculating Hoop Stress and Longitudinal Stress

When a thin cylinder pressure vessel with internal pressure 'p', diameter 'd', and wall thickness 't' is considered, the stresses developed are:

• Hoop Stress (\(\sigma_h\)): This stress acts along the circumference of the cylinder, resisting the pressure that tries to burst it open lengthwise. It is given by the formula:
  \(\sigma_h = \frac{pd}{2t}\)
• Longitudinal Stress (\(\sigma_l\)): This stress acts along the length of the cylinder, resisting the pressure that tries to pull it apart axially. It is given by the formula:
  \(\sigma_l = \frac{pd}{4t}\)

In thin cylinders, the radial stress (\(\sigma_r\)) acting perpendicular to the wall is often considered negligible on the outer surface (\(\sigma_r \approx 0\)) compared to the hoop and longitudinal stresses.

Determining Maximum Shear Stress

The maximum shear stress (\(\tau_{max}\)) in a material is determined by the principal stresses acting at a point. For a state of stress with three principal stresses \(\sigma_1\), \(\sigma_2\), and \(\sigma_3\), the maximum shear stress is half the difference between the maximum and minimum principal stresses.

\(\tau_{max} = \text{max}\left(\frac{|\sigma_1 - \sigma_2|}{2}, \frac{|\sigma_2 - \sigma_3|}{2}, \frac{|\sigma_3 - \sigma_1|}{2}\right)\)

In a thin cylinder, the principal stresses on the outer surface are approximately the hoop stress (\(\sigma_h\)), the longitudinal stress (\(\sigma_l\)), and the radial stress (\(\sigma_r \approx 0\)).

So, the principal stresses are:

• \(\sigma_1 = \sigma_h = \frac{pd}{2t}\)
• \(\sigma_2 = \sigma_l = \frac{pd}{4t}\)
• \(\sigma_3 = \sigma_r \approx 0\)

Now, we calculate the shear stress values between each pair of principal stresses:

• Shear stress between \(\sigma_h\) and \(\sigma_l\):
  \(\frac{|\sigma_h - \sigma_l|}{2} = \frac{|\frac{pd}{2t} - \frac{pd}{4t}|}{2} = \frac{|\frac{2pd - pd}{4t}|}{2} = \frac{|\frac{pd}{4t}|}{2} = \frac{pd}{8t}\)
• Shear stress between \(\sigma_h\) and \(\sigma_r\):
  \(\frac{|\sigma_h - \sigma_r|}{2} = \frac{|\frac{pd}{2t} - 0|}{2} = \frac{pd}{4t}\)
• Shear stress between \(\sigma_l\) and \(\sigma_r\):
  \(\frac{|\sigma_l - \sigma_r|}{2} = \frac{|\frac{pd}{4t} - 0|}{2} = \frac{pd}{8t}\)

Comparing these values (\(\frac{pd}{8t}\), \(\frac{pd}{4t}\), \(\frac{pd}{8t}\)), the maximum shear stress is the largest value.

\(\tau_{max} = \text{max}\left(\frac{pd}{8t}, \frac{pd}{4t}, \frac{pd}{8t}\right) = \frac{pd}{4t}\)

Summary of Stresses

The maximum shear stress in a thin cylinder pressure vessel is found to be \(\frac{pd}{4t}\).