A cylindrical steel pressure vessel has mean wall radius and wall thickness…

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

Bending, Ringing, Breaking and Flowing, Ch Fracture: stress concentration, cracks and toughness

Problem

A cylindrical steel pressure vessel has mean wall radius

r = 0.50\,\mathrm{m}

and wall thickness

t = 25\,\mathrm{mm}

(thin-walled,

t \ll r

), and carries internal pressure

p = 8.0\,\mathrm{MPa}

with closed ends. For a thin-walled vessel the hoop stress in the wall is

\sigma_\theta = pr/t

and the axial stress is

\sigma_z = pr/2t

, each uniform through the wall thickness. A small hole is drilled radially through the wall to fit a pressure-tap fitting; its diameter is much smaller than

t

and

r

, so the wall around it may be treated as a flat plate carrying the remote biaxial stresses

\sigma_\theta

(circumferential direction) and

\sigma_z

(axial direction) in its plane. ((a)) Find

\sigma_\theta

and

\sigma_z

in the wall away from the hole. ((b)) For a small circular hole in a plate under remote biaxial tension

\sigma_1

(larger) and

\sigma_2

(smaller), superposing two Kirsch solutions along the two directions gives a peak edge stress

\sigma_{\max} = 3\sigma_1 - \sigma_2

, at the two edge points on the diameter perpendicular to

\sigma_1

, where the hoop stress runs parallel to

\sigma_1

. Find the peak stress at the edge of the tap hole.

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