Determine the enhance in volume of the vessel:

A thin spherical vessel of diameter 800 mm and thickness 6 mm is filled through a liquid at an internal pressure of 1.5 N/mm². Determine the enhance in volume of the vessel, taking E = 2 × 105 N/mm² and v = 0.3.

Solution

Diameter of the vessel, d = 800 mm.

Thickness, t = 6 mm.

Internal pressure, p = 1.5 N/mm².

 ∴ Hoop stress, σ_h  = pd /4t = 1.5 × 800 / (4 × 6 )= 50 N/mm²

∴ Hoop strain, ε  = (σ_h /E)(1 - v) =(50/2 × 10⁵) (1 - 0.3) = 1.75 × 10^{- 4}

∴ Volumetric strain, εv = 3 × εh

= 3 × (1.75 × 10- 4 ) = 5.25 × 10^{- 4}

∴ Original volume, V = π d ³ / 6

= π× 800³ /6= 2.6808 × 10⁸  mm³

∴ Increase in volume, δV = ε_v × V

= 5.25 × 10^{- 4}  × 2.6808 × 10⁸  = 140743 mm³