Consider the following statements:

1. In case of a thin spherical shell of diameter d and thickness t, subjected to internal pressure p, the principal stresses at any point equal pd/4t

2. In case of thin cylinders the hoop stress is determined assuming it to be uniform across the thickness of the cylinder

3. In thick cylinders, the hoop stress is not uniform across the thickness, but it varies from a maximum value at the inner circumference to a minimum value at the outer circumference.

Which of the above statements are correct?

Concept:

In case of thin cylinders, the hoop stress is determined by assuming it to be uniform across the thickness of the cylinder but in thick cylinders, the hoop stress is not uniform across the thickness, it varies from a maximum value at the inner circumference to a minimum value at the outer circumference.

For a thick cylinder hoop stress (tangential stress) is given by,

\({σ _h} = \frac{{P\;{{\left( {{r_i}} \right)}^2}}}{{{{\left( {{r_0}} \right)}^2} - {{\left( {{r_i}} \right)}^2}}}\left[ {1 + \frac{{{{\left( {{r_0}} \right)}^2}}}{{{x^2}}}} \right]\)

For a thick cylinder Radial stress is given by,

\({σ _r} = \frac{{P\;{{\left( {{r_i}} \right)}^2}}}{{{{\left( {{r_0}} \right)}^2} - {{\left( {{r_i}} \right)}^2}}}\left[ {1 - \frac{{{{\left( {{r_0}} \right)}^2}}}{{{x^2}}}} \right]\)

r_i ≤ x ≤ r_o

Where, r_i and r_o are inner and outer radius, respectively

Thin spherical shell 

For the thin spherical shell of diameter d and thickness t, subjected to internal pressure p,

σ₁ = σ ₂ = σ_h = \(\frac{{Pd}}{{4t}}\)