In thin walled pressure vessels, the longitudinal stress is _______ hoop stress. 

Concept:

Hoop stress:

\({\sigma _1} ={\sigma _h} = \frac{{pd}}{{2t}} =\sigma \)

Longitudinal stress:

\({\sigma _2} ={\sigma _L} = \frac{{pd}}{{4t}} =\frac{\sigma}{2} \)

Hence, In thin-walled cylindrical pressure vessel, hoop stress is twice the longitudinal stress.

Additional Information

As this is the case of a thin cylinder:

\({\sigma _r} =0\)

All these above stress can be considered as principal stresses as no shear stess is acting.

Maximum shear stress,

= \(\tau _{max} = max\ [\frac{{\sigma _{max}}\;-\;{\sigma _{min}}}{{2}}=\frac{{\sigma _1}\;-\;0 }{2} =\frac{{pd}}{{4t}} \)

\(\tau _{max} = max\ [| \frac{{\sigma _{1}}\;-\;{\sigma _{2}}}{{2}}|, | \frac{{\sigma _{2}}\;-\;{\sigma _{3}}}{{2}}|,| \frac{{\sigma _{3}}\;-\;{\sigma _{1}}}{{2}}|]\)

\(\Rightarrow\tau_{max} = max\ [\frac{\sigma }{4}, \ \frac{\sigma }{4}, \ \frac{\sigma }{2}] \)

.\(\Rightarrow \tau_{max} = \frac{\sigma}{2}\)

\(\Rightarrow {\tau _{max}}= \frac{{pd}}{{4t}} \)