Such an analysis requires only equilibrium to be satisfied。 Referring to Fig。 2, and considering vertical forces on an axisymmetric part of the shell above that cut off by a plane  through  point P(x, y) and perpendicular  to the y axis,

2 cry N z cos & — pay。

Resolving forces perpendicular  to an element of shell at the same position   [4j

where rg, rd are, respectively, the meridional and circumferential  (principal) radii  of curvature at  P, and N , N j, respectively, are the principal forces per unit length of shell in the meridional and circumferential  directions  at  P。 The internal  pressure  is p and  the  shell  thickness  is 1。

Solving Eqns (2) and (3) gives  the  well-known  result   (5j:

N z - pr /2 cos B = pr,/2

N y = (prp/'2) [2 — n, rg)。

The geometrical expressions in Eqns (2a) and (3a) are now determined in terms of   0。

Fig。 2。 Notation for membrane analysis: (a)  geometry; (b)  forces on cap above point  P; and (c)  in-plane  forces on  element  at P。

The  gradient  of  the normal  at  P is

d 2$, d 2 (« + 2« six’ sh 6' sin 3 ()

Substituting  expressions  (6) and  (7) into  Eqn (5):

— {cos’ h ( u + 2« sin B)' + h’ sin' d}  ’, h (a + 2e sin’ B)。

The centre of curvature  relevant  to  r   is where  the  normal  at  P  meets  the J• axis,  so

ft — f p/" COS

where  r   = 。x,  and Eqn (4)  gives

Substituting in Eqn (9)

rp = b((a + 2 e sin 8)’cos’ 8 + b' sin’ 8) "  '(u = 2 e sin d)。

Using the expressions in Eqns (1) and (10) in Eqn (2a)

and using Eqns (8), (11) and (12) in Eqn (3)

Stresses (« «›) would be calculated by piding corresponding stress resultants by the thickness, i。

To determine stress distributions along the meridian, b is more tangible as a variable than 8 but it is impracticable to write Eqns (l 2) and (13) in terms of b。 For the distributions which are given later, stresses have been evaluated for different values of 8 and then converted to the corresponding values of b using the following equation which follows from Eqn (1):

For  an  unmodified  elliptical  meridian  (i。e。 c = 0), expressions (12) and  (13) become

R E S U L T S

For the dimensions quoted above, a thickness of 0。35 mm, and an internal pressure of 1 Mpa, the principal membrane stresses are plotted against the angle b in Fig。 3 (see Table 1 for corresponding

Fig。 3。 Membrane stresses for modified elliptical meridian compared with those for elliptical meridian。 ap, circumferential stress—ellipse  -— ng, meridional stress—ellipse n  circumferential stress—modified ellipse

ng meridional stress—modified ellipse -

Table l 。 Corresponding values of b and 0  for  egg- shape and ellipsoid  with a ——  30 mm, 6 = 22。5  mm

b degrees

II degrees Egg shape e — 4 mm Ellipsoid

90 90 90

75 79。91 78。63

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