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Ebook The finite element method (Vol 2: Solid Mechanics - 5th edition): Part 2

(BQ) Written by the pre-eminent professors in their fields, this new edition of the Finite Element Method maintains the comprehensive style of the earlier editions and authoritatively incorporates the latest developments of this dynamic field. Volume Two: Solid and Structural Mechanics is intended for readers studying structural mechanics at a higher level. | 7 Axisymmetric shells 7.1 Introduction The problem of axisymmetric shells is of su cient 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 simpli®cation 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 ®rst approach to the ®nite element solution of axisymmetric shells was presented by Grafton and Strome.1 In this, the elements are simple conical frustra and a direct approach via displacement functions is used. Re®nements in the derivation of the element sti ness are presented in Popov et al.2 and in Jones and Strome.3 An extension to the case of unsymmetrical loads, which was suggested in Grafton and Strome, is elaborated in Percy et al.4 and others.5;6 Later, much work was accomplished to extend the process to curved elements and indeed to re®ne 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.16 Early work on the subject is reviewed comprehensively by Gallagher17;18 and Stricklin.19 In axisymmetric shells, in common with all other shells, both bending and `inplane' or `membrane' forces will occur. These will be speci®ed uniquely in terms of the generalized `strains', which now involve extensions and changes in curvatures of the middle surface. If the