The Finite Element Analysis of Shells — Fundamentals

In this chapter, we briefly discuss shell structures — noting also that, actually, the analysis of shell structures gave the impetus for the development of finite element procedures — and we summarize the general approach of analysis of a shell problem. We then give the objectives of this book; namely, to present fundamentals regarding physical considerations, mathematical models and modern finite element procedures for the analysis of shells.

The geometry is essential for the definition of a shell structure. Our objective in this chapter is to survey the main geometrical concepts, to introduce the related notation and to recall some essential results that will be needed in this book.

A deeper understanding of finite element methods, and the development of improved finite element methods, can only be achieved with an appropriate mathematical and numerical assessment of the proposed techniques. The basis of such an assessment rests on identifying whether certain properties are satisfied by the finite element scheme and these properties depend on the framework within which the finite element method has been formulated.

In this chapter we describe and analyse the linear shell models that we consider in this book. We first describe the fundamental shell kinematics used. Then we discuss the “basic shell model” which is implicitly employed in general finite element solutions and from which other classical shell and plate models can be derived. We summarize the shell models that we call the “shear-membrane-bending model” and the “membrane-bending model”, and introduce the proper mathematical framework in which they define well-posed problems. As special cases of these shell models we obtain well-known plate models.

Implicit in the concept of a “shell” is the idea that the thickness is “small” compared to the other two dimensions. In practice, it is not unusual to deal with structures for which the thickness is smaller by several orders of magnitude, in which case the shell is said to be “thin” (consider, for example, the shell body of a motor car). Considering the role of the thickness parameter t in the shell models that we presented in the previous chapter (see for example Eqs. (4.36) and (4.51)), with different powers of t in the bilinear terms on the left-hand side, it is essential to determine how the properties of the models are affected when this parameter becomes small. Likewise, it is important to know whether the model converges, in some sense to be specified, towards a limit model when the thickness t “tends to zero”, so that this possibly simpler model can be used instead of the original one when t is sufficiently small, i.e. we need to study the asymptotic behavior of the shell models. Of course, our goal is also to investigate the influence of the thickness on the convergence of finite element methods, as we want to be able to identify numerical procedures for which there is no deterioration of convergence when the thickness becomes small. To that purpose, the analysis of the asymptotic behaviors of mathematical shell models clearly also represents a crucial prerequisite on which we concentrate in this chapter, whereas the issues arising in the finite element solutions themselves are addressed in the next chapters.

The influence of the thickness in the finite element analysis of thin structures is a crucial issue, as it is deeply interrelated with the motivation of modeling a 3D continuum as a shell in engineering. Why, indeed, should we use shell models and finite elements — instead of 3D models — to analyze a given structure? The answer to this question seems obvious: firstly, the use of a shell model is to reduce the analysis cost, and secondly, the use of the shell model is to reduce the complexity of the analysis including the interpretation of the results for engineering design. Clearly, the motivation to use shell models rests upon the fact that shell mathematical models and finite elements incorporate kinematical assumptions pertaining to the displacement distribution across the thickness of the structure, see previous chapters. Hence, we do not have to discretize the problem across the thickness, but only over the mid-surface (or equivalently in the 2D domain that corresponds to the in-plane coordinates). In other words, in shell analysis we “trade” the discretization in the transverse direction for a kinematical assumption in the same direction. Of course, by using a shell mathematical model instead of a 3D model we introduce a modeling error due to the difference between the exact solutions of the two mathematical models. The analysis of this modeling error goes beyond the scope of this book, but we recall that this error can be shown — under certain assumptions and using some specific convergence measures — to tend to zero when the thickness of the structure tends to zero, see Chapter 5 (and in particular Remark 5.1.8).

In Chapter 7 we discussed the difficulties encountered in the formulation of reliable and effective shell elements. The objective of the present chapter is to propose some strategies to evaluate shell finite element discretizations in the search for improved schemes. With general analytical proofs not available for the convergence behavior, the numerical assessment is a key ingredient in these strategies. As an example we present the formulation of the MITC shell elements and demonstrate how the numerical assessment of these elements can be performed.