Introduction To The Finite Element Method

Introduction

So far in this text we have studied the means by which components can be analysed using so-called Mechanics of Materials approaches whereby, subject to making simplifying assumptions, solutions can be obtained by hand calculation. In the analysis of complex situations such an approach may not yield appropriate or adequate results and calls for other methods. In addition to experimental methods, numerical techniques using digital computers now provide a powerful alternative. Numerical techniques for structural analysis divides into three areas; the long established but limited capability finite difference method, the finite element method (developed from earlier structural matrix methods), which gained prominence from the 1950s with the advent of digital computers and, emerging over a decade later, the boundary element method. Attention in this chapter will be confined to the most popular finite element method and the coverage is intended to provide

• an insight into some of the basic concepts of the finite element method (fem.), and, hence, some basis of finite element (fe.), practice,

• the theoretical development associated with some relatively simple elements, enabling analysis of applications which can be solved with the aid of a simple calculator, and

• a range of worked examples to show typical applications and solutions.

It is recommended that the reader wishing for further coverage should consult the many excellent specialist texts on the subject.1-10 This chapter does require some knowledge of matrix algebra, and again, students are directed to suitable texts on the subject."

9.1. Basis of the finite element method

The fem. is a numerical technique in which the governing equations are represented in matrix form and as such are well suited to solution by digital computer. The solution region is represented, (discretised), as an assemblage (mesh), of small sub-regions called finite elements. These elements are connected at discrete points (at the extremities (corners), and in some cases also at intermediate points), known as nodes. Implicit with each element is its displacement function which, in terms of parameters to be determined, defines how the displacements of the nodes are interpolated over each element. This can be considered as an extension of the Rayleigh-Ritz process (used in Mechanics of Machines for analysing beam vibrations6). Instead of approximating the entire solution region by a single assumed displacement distribution, as with the Rayleigh-Ritz process, displacement distributions are assumed for each element of the assemblage. When applied to the analysis of a continuum (a solid or fluid through which the behavioural properties vary continuously), the discretisation becomes an assemblage of a number of elements each with a limited, i.e. finite number of degrees of freedom (dof). The element is the basic "building unit", with a predetermined number of dof., and can take various forms, e.g. one-dimensional rod or beam, two-dimensional membrane or plate, shell, and solid elements, see Fig. 9.1.

In stress applications, implicit with each element type is the nodal force/displacement relationship, namely the element stiffness property. With the most popular displacement formulation (discussed in §9.3), analysis requires the assembly and solution of a set of

Concentrated

Curved

Beams

Spring

Concentrated

Curved

Beams

Displacement assumption

Linear

Quadratic

Cubic

Membrane and plate bending

Fig. 9.1(a). Examples of element types with nodal points numbered.

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