Equation6.27 reduces to:

/ d2 1 d / d2ф 1 d<A _

dr2 + r dr dr2 + r dr

This is an ordinary differential equation that admits the following solution:

Ф(г) — A ln r + Br2 ln r + Cr2 + D (6.28)

where A, B, C, D are constants to be determined by boundary conditions. Substituting Eq. 6.28 in Eq. 6.26 the following stress values are derived:

A

ar — — + B(1 + 2ln r) + 2C r2A

‘ ae = — A + B(3 + 2lnr) + 2C (6.29)

r 2

Tr в — 0

For r ^<x>, being the stress S a finite value, B must be zero. The other boundary conditions are:

I |

ar — S for r ar — 0 for r — a

then:

2C — S A — — Sa2

Exercise 6.3 (Stress concentration for a circular hole in a spherical thin vessel of large diameter, internally Pressurized) (a) If the hole and the thickness are small respect to the sphere diameter in order that it can be considered in membrane regime, the previous theory can be extended to the case of a spherical vessel internally pressurized with a circular hole.

(b) The concentration factor due to a hole in a plane membrane in uniform state of stress is estimated through Eq. 6.30. For r — a stress components are:

ar — 0

ae — 2S

Then the stress concentration factor, i. e. ratio between maximum circumferential stress and uniform stress, is Kt — 2.

Fig. 6.14 Stress in a plate with a circular hole loaded on its boundary |

Exercise 6.4 (Stress concentrationfor a hole loaded on its boundary, in a large thin plate) (a) This case is the over-position of the previous state plus a uniform state of stress, Fig.6.14. The circumferential stress around the hole is given by:

ae = — S + 2 S = S

(b) The circumferential stress due to a pressure S acting on the hole boundary is a tension stress with the same absolute value of the applied pressure.