6.2 Shells of revolution

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In an axisymmetric shell and in a stress state with rotational symmetry only

displacements normal to the meridian w, and in the tangential direction u

will develop; see Fig. 6.4. Therefore a subdivision of the meridian (= the

generator) into beam-like straight or curved elements suffices.

The relations between the arc length s on such an element and the characteristic

quantities of a shell of revolution are (see Fig. 6.4)

6.2 Shells of revolution 489

Fig. 6.4. Shell of revolution

Rϑ = r

cos ϕ

, Rs = − ds

dϕ

, sin ϕ = dr

ds

, cos ϕ = −dz

ds

. (6.14)

Here Rϑ and Rs are the principal radii of curvature. If straight elements are

used, as in a frustum or cylindrical shell the radius of curvature in the plane

of the meridian is infinite, Rs = ∞. The strains are

εs = du

d s

+ w

Rs

εϑ = u sin ϕ + w cos ϕ

r

(6.15)

κs = d

ds

& u

R

'

− d2w

d s2 κϑ =

sin ϕ

r

_

u

Rs

− dw

d s

_

, (6.16)

where εs and εϑ are the strains of the midsurface in the direction of a meridian

(arc length s) or in the tangential direction (ϑ) and κs and κϑ are the

curvatures.

The strain energy product of an element is

a(u, u) = εT

_ l

0

_

DM 0

0 DK

_

2π r dsε (6.17)

where with C = E t/(1 − ν2), D = E t3/(12(1 − ν2)),

DM = C

_

1 ν

ν 1

_

DK = D

_

1 ν

ν 1

_

ε =

⎡

⎢⎢⎣

εs

εϑ

κs

κϑ

⎤

⎥⎥⎦

, (6.18)

and the resultant stresses are

_

ns

nϑ

_

= E t

1 − ν2

_

1 ν

ν 1

__

εs

εϑ

_

(6.19)

_

ms

mϑ

_

= E t3

12 (1 − ν2)

_

1 ν

ν 1

__

κs

κϑ

_

. (6.20)

In the sense of isoparametric elements, the element is interpreted as the C1

map of a master element −1 ≤ ξ ≤ +1 on which four cubic shape functions

corresponding to the two nodes ξ1 = −1, ξ2 = +1 and ξ0 = ξi ξ, are defined:

490 6 Shells

ϕ(1)

i (ξ) =

1

4

(ξ0 ξ2 − 3 ξ0 + 2) ϕ(2)

i (ξ) =

1

4

(1 − ξ0)2 (1 + ξ0) . (6.21)

These make it possible to interpolate the shape of the element, i.e., the functions

r and z,

r(ξ) =

_2

i=1

_

ϕ(1)

i (ξ) r(ξi) + ϕ(2)

i (ξ) dr

dξ

(ξi)

_

(6.22)

(as well as z(ξ)), and the displacements u and w of the meridian in a C1

continuous fashion. The degrees of freedom at the element nodes are the displacements

and the first-order derivative with respect to the arc length s

ue = [ui, wi, u

_

i, w

_

i]T . (6.23)

The C1 continuity of the displacement u is unusual, and must be dropped at

nodes where the thickness of the shell changes, because then the strains εs

are discontinuous [258].

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