Fatigue Crack Paths 2003

Spherical Shell underInternal Pressure

In the case of an unnotched spherical shell (R1 = R 2=RS, r = 1) under an internal

pressure p, the actual hoop stress distribution 0']( PI S”) along the wall thickness is given

by [l l] :

p-(RS —t)3 IRS’ +2(R, +W—a)3

(8)

O-I(p,su) (W) =

2(RS +W—a)3 RS3 —(R, —r)3

A n approximate dimensionless SIF, K *

is deduced by using the superposition

I(p,su) ’

principle (see Eq.4, [8]):

l

6 Z Brow”)K *1(n)

K

M

(9)

*

:

1013871)

Germs”) n20

where o'reflpIsu) = p R/ (2t) is the well-known hoop stress in a pressurised thin-walled

sphere ( R =mean radius of the sphere), and the coefficients Bnumu) (see Eq. 5) are

listed in Table 2.

Table 2. Coefficients of series expansion for a spherical shell under an internal pressure.

I

_ 3

3

2

_ 3

14R. 01112-411. —atl

o-l(p,su)(wI) p ( R s t) 3 ' R s + 3 ( R s+ W 36')

B 0(1743”) : 2(RS— d ) 3 _ ( R s—l)31

2(RS +w—a) R, —(R,—t)

_

3114K. —03

R3 +2(R.. —af

_ 3pa2(RS—t)3

Rpm-.3):

B1 (or) _ (R. —a) (Rs-0.001” 20. —at

B2 (PM) - 2lR:-

(13.-at

5 W302, — [)3

R3 + 2(R, - a)3

_ 15 pa“ (R, —i)3

R3 + 2(R, - a)3

B3 (PM) : <1.—a>3lR:—(R.—0312+

(R3—a)3

B4 (PM) - (R3—a)41R3—(R.

—03) 1+ 20. as

B

21pa5(R, —t)3

[1+ Rf +2(R, —a)3

14pa6(R, —r)‘

[2 I Rf +2(RS —a)3:|

50W): (R,_a)5(R3_(R,_t)3)

2(R —a)3:|

B6 (1W)=(R.—a)61R3—(R.—ri1

(It-at

s

In the case of a notched spherical shell, the analytical solution for the hoop stress

distribution o'1(pIS)(w) is not available. Therefore, o'1(pIS)(w) must numerically be

computed in order to obtain the coefficients Bnuw). The reference stress can conveniently

be defined as o',ef(pIS) =pR/(2t'),and the dimensionless SIF, K ’ i m w ) ,can be

obtained from an expression similar to Eq. 9, Where the subscript su is replaced by s.