7 Axi sy mmet r ic she IIs
7.1 Introduction The problem of axisymmetric shells is of sufficient practical importance to include in this chapter special methods dealing with their solution. While the general method described in the previous chapter is obviously applicable here, it will be found that considerable simplification can be achieved if account is taken of axial symmetry of the structure. In particular, if both the shell and the loading are axisymmetric it will be found that the elements become ‘one-dimensional’. This is the simplest type of element, to which little attention was given in earlier chapters. The first approach to the finite element solution of axisymmetric shells was presented by Grafton and Strome.’ In this, the elements are simple conical frustra and a direct approach via displacement functions is used. Refinements in the derivation of the element stiffness are presented in Popov et a1.* and in Jones and S t r ~ m e .An ~ extension to the case of unsymmetrical loads, which was suggested in Grafton and Strome, is elaborated in Percy et d4and Later, much work was accomplished to extend the process to curved elements and indeed to refine the approximations involved. The literature on the subject is considerable, no doubt promoted by the interest in aerospace structures, and a complete bibliography is here impractical. References 7- 15 show how curvilinear coordinates of various kinds can be introduced to the analysis, and references 9 and 14 discuss the use of additional nodeless degrees of freedom in improving accuracy. ‘Mixed’ formulations (Chapter 11 of Volume 1) have found here some use.I6 Early work on the subject is reviewed comprehensively by Gallagher’7.18and Stricklin. I 9 In axisymmetric shells, in common with all other shells, both bending and ‘inplane’ or ‘membrane’ forces will occur. These will be specified uniquely in terms of the generalized ‘strains’, which now involve extensions and changes in curvatures of the middle surface. If the displacement of each point of the middle surface is specified, such ‘strains’ and the internal stress resultants, or simply ‘stresses’, can be determined by formulae available in standard texts dealing with shell
Straight element 245
7.2 Straight element As a simple example of an axisymmetric shell subjected to axisymmetric loading we consider the case shown in Figs 7.1 and 7.2 in which the displacement of a point on the middle surface of the meridian plane at an angle 4 measured positive from the x-axis is uniquely determined by two components ii and E in the tangential (s) and normal directions, respectively. Using the Kirchhoff-Love assumption (which excludes transverse shear deformations) and assuming that the angle 4 does not vary (i.e. elements are straight), the four strain components are given by2'-**
.=[kJ=( XS
]
dulds [ii cos 4 - E sin 4 ] / ~ -d2 ,?/ds2 -(dE/ds) COS d/Y
(7.1)
This results in the four internal stress resultants shown in Fig. 7.1 that are related to the strains by an elasticity matrix D:
[ :]
(7.21
=DE
(r=
Mo For an isotropic shell the elasticity matrix becomes
D
=1 -u2L
1'
v
'
o 0 ?/12
0
0 0 vt2/12 t2;12 0 0 vt2/12
1
(7.3)
Fig. 7.1 Axisymmetric shell, loading, displacements, and stress resultants, shell represented as a stack of conical frustra
Fig. 7.2 An element of an axisymmetric shell.
the upper part being a plane stress and the lower a bending stiffness matrix with shear terms omitted as 'thin' conditions are assumed.
7.2.1 Element characteristics - axisymmetrical loads Let the shell be divided by nodal circles into a series of conical frustra, as shown in Fig. 7.2. The nodal displacements at points 1 and 2 for a typical 1-2 element such as i a n d j will have to define uniquely the deformations of the element via prescribed shape functions. At each node the radial and axial displacements, u and w , and a rotation, ,B, will be used as parameters. From virtual work by edge forces we find that all three components are necessary as the shell can carry in-plane forces and bending moments. The displacements of a node i can thus be defined by three components, the first two being in global directions Y and z , a.I =
{;}
(7.4)
The simplest elements with two nodes, i and j , thus possess 6 degrees of freedom, determined by the element displacements a' =
{ ::}
(7.5)
The displacements within the element have to be uniquely determined by the nodal displacements a' and the position s (as shown in Fig. 7.2) and maintain slope and displacement continuity. Thus in local (s) coordinates we have
u=
{ ;,}
= N(s) a''
(7.6)
Straight element 247
Based on the strain-displacement relations (7.1) we observe that ii can be of Co type while I3 must be of type C1.The simplest approximation takes ii varying linearly with s and ii, as cubic in s. We shall then have six undetermined constants which can be determined from nodal values of u, w ,and p. At the node i,
{ (dids)i} [ :]{ i} sin$
COS$
= $: -
0
co;C
(7.7)
=Tai
Introducing the interpolations
where Ny are the usual linear interpolations in E (-1 d
< d 1)
N:=$(l-
