The relationship between geometric and mechanical properties of shell structures

Abstract

Shell structures can be either calculated with complex mathematics and differential geometry, or with the help of finite element method (FEM) software. Neither method gives the designer sufficient insight about the behaviour of the shell in the early design stage. The first method is for most designers too complex, it requires proficiency in at higher order mathematics and it can be only used for shells the geometry of which can be described by analytical equations. The second method is more accessible for most designers, and in combination with 3D modelling software is an attractive alternative to the cumbersome mathematics. However, the disadvantage of using FEM software is that the analytical relations between the different parameters important for obtaining insight into the structural behaviour of the shell is lost.

The ideal tool for designers would take the best of both methods: the analytical insight of the mathematics and the relatively easy access of FEM and 3D modelling software. The aim of the research done presented in this thesis is to do precisely that. By extending the well-known beam-analogy and its relations for an arch to shell structures a construct of relations is available for providing the designer with insight and means to influence the geometry of the shell and the resulting stress state.

The proposed hypotheses are based on classic analytical geometry and mechanics, especially analogies and methods developed in the past to elucidate the complex mathematics and mechanics for the purpose of insight. The theory of graphic statics, reciprocal diagrams, complementary and potential energy was used to develop the method of solving the thrust network. Analogies such as the moment-hill for out-of-plane loaded slabs and the static-geometric analogy for thin shells as well as the load path theorem and stress functions were used to develop the slab – shell analogy.

Two approximate hypotheses are proposed in this thesis, the first is used to solve 3D indeterminate thrust networks by using complementary energy. The second hypothesis extends the beam – arch analogy in two directions to the slab – shell analogy; this method produces results in range of solutions found in classical shell theory.

The result of the different examples has been checked with the help of well-known solutions of classical shell mechanics, FEM calculations or graphic statics. Examples with relatively basic analytical formulas have been used to elucidate the proposed method and for other examples simple purpose made tools based on the method have been used. Some simplifications have been made to avoid unnecessary complications in the derivation of the proposed method, such as only applying a uniformly distributed load. But most of the simplifications are not technically necessary; the conclusion and recommendations section include some suggestions have been added for extending the method.

The inception of numeric methods for analyzing structures in the 1960s was the end of the development of analytical mechanics for shell structures. This thesis aims to continue this development by tying the used theories and analogies together and bridge the gap with the numeric methods and to increase the understanding of the structural performance of shell structures.