A hollow steel sphere with bore radius and outer radius holds an internal pressure

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Hooke's law for an isotropic solid

Problem

A hollow steel sphere with bore radius

a = 50\,\mathrm{mm}

and outer radius

b = 100\,\mathrm{mm}

holds an internal pressure

p = 100\,\mathrm{MPa}

, with its outer surface free. With no body force, the radial solution of the Navier equation in a sphere is

u(r) = Ar + B/r^2

. ((a)) In spherical coordinates the radial strain is

\varepsilon_{rr} = \mathrm du/\mathrm dr

, and the two angular strains are both

u/r

. Use the isotropic law to show that

\sigma_{rr} = 3KA - 4GB/r^3

and

\sigma_{\theta\theta} = 3KA + 2GB/r^3

. ((b)) Fit

A

and

B

to the two boundary conditions and show that

\sigma_{\theta\theta} = [pa^3/(b^3-a^3)](1 + b^3/2r^3)

, the spherical result quoted in the chapter. ((c)) Evaluate the hoop stress at the bore, and compare it with the bore hoop stress of a long tube with the same radii and pressure.

Browse the problem library