May 05, 2008
Volume 1: The Basis
1. Some preliminaries: the standard discrete system
2. A direct approach to problems in elasticity
The process of approximating the behaviour of a continuum by ‘finite elements’ which behave in a manner similar to the real, ‘discrete’, elements described in the previous chapter can be introduced through the medium of particular physical applications or as a general mathematical concept. We have chosen here to follow the first path, narrowing our view to a set of problems associated with structural mechanics which historically were the first to which the finite element method was applied. In Chapter 3 we shall generalize the concepts and show that the basic ideas are widely applicable.
In many phases of engineering the solution of stress and strain distributions in elastic continua is required. Special cases of such problems may range from twodimensional plane stress or strain distributions, axisymmetric solids, plate bending, and shells, to fully three-dimensional solids. In all cases the number of interconnections between any ‘finite element’ isolated by some imaginary boundaries and the neighbouring elements is infinite. It is therefore difficult to see at first glance how such problems may be discretized in the same manner as was described in the preceding chapter for simpler structures. The difficulty can be overcome (and the approximation made) in the following manner:
1. The continuum is separated by imaginary lines or surfaces into a number of ‘finite elements’.
2. The elements are assumed to be interconnected at a discrete number of nodal points situated on their boundaries and occasionally in their interior. In Chapter 6 we shall show that this limitation is not necessary. The displacements of these nodal points will be the basic unknown parameters of the problem, just as in simple, discrete, structural analysis.
3. A set of functions is chosen to define uniquely the state of displacement within each ‘finite element’ and on its boundaries in terms of its nodal displacements.
4. The displacement functions now define uniquely the state of strain within an element in terms of the nodal displacements. These strains, together with any initial strains and the constitutive properties of the material, will define the state of stress throughout the element and, hence, also on its boundaries.
5. A system of ‘forces’ concentrated at the nodes and equilibrating the boundary stresses and any distributed loads is determined, resulting in a stiffness relationship of the form of Eq. (1.3).
Once this stage has been reached the solution procedure can follow the standard discrete system pattern described earlier.
Clearly a series of approximations has been introduced. Firstly, it is not always easy to ensure that the chosen displacement functions will satisfy the requirement of displacement continuity between adjacent elements. Thus, the compatibility condition on such lines may be violated (though within each element it is obviously satisfied due to the uniqueness of displacements implied in their continuous representation).
Secondly, by concentrating the equivalent forces at the nodes, equilibrium conditions are satisfied in the overall sense only. Local violation of equilibrium conditions within each element and on its boundaries will usually arise.
The choice of element shape and of the form of the displacement function for specific cases leaves many opportunities for the ingenuity and skill of the engineer to be employed, and obviously the degree of approximation which can be achieved will strongly depend on these factors.
The approach outlined here is known as the displacement formulation.
So far, the process described is justified only intuitively, but what in fact has been suggested is equivalent to the minimization of the total potential energy of the system in terms of a prescribed displacement field. If this displacement field is defined in a suitable way, then convergence to the correct result must occur. The process is then equivalent to the well-known Rayleigh-Ritz procedure. This equivalence will be proved in a later section of this chapter where also a discussion of the necessary convergence criteria will be presented.
The recognition of the equivalence of the finite element method to a minimization process was late. However, Courant in 1943 and Prager and Synge in 1947 proposed methods that are in essence identical.
This broader basis of the finite element method allows it to be extended to other continuum problems where a variational formulation is possible. Indeed, general procedures are now available for a finite element discretization of any problem defined by a properly constituted set of differential equations. Such generalizations will be discussed in Chapter 3, and throughout the book application to non-structural problems will be made. It will be found that the processes described in this chapter are essentially an application of trial function and Galerkin-type approximations to a particular case of solid mechanics.
3. Generalization of the finite element concepts. Galerkin-weighted residual and variational approaches
We have so far dealt with one possible approach to the approximate solution of the particular problem of linear elasticity. Many other continuum problems arise in engineering and physics and usually these problems are posed by appropriate differential equations and boundary conditions to be imposed on the unknown function or functions. It is the object of this chapter to show that all such problems can be dealt with by the finite element method.
4. Plane stress and plane strain
Two-dimensional elastic problems were the first successful examples of the application of the finite element method. Indeed, we have already used this situation to illustrate the basis of the finite element formulation in Chapter 2 where the general relationships were derived. These basic relationships are given in Eqs (2.1)-(2.5) and (2.23) and (2.24), which for quick reference are summarized in Appendix C.
In this chapter the particular relationships for the plane stress and plane strain problem will be derived in more detail, and illustrated by suitable practical examples, a procedure that will be followed throughout the remainder of the book.
Only the simplest, triangular, element will be discussed in detail but the basic approach is general. More elaborate elements to be discussed in Chapters 8 and 9 could be introduced to the same problem in an identical manner.
The reader not familiar with the applicable basic definitions of elasticity is referred to elementary texts on the subject, in particular to the text by Timoshenko and Goodier, whose notation will be widely used here.
In both problems of plane stress and plane strain the displacement field is uniquely given by the u and v displacement in the directions of the Cartesian, orthogonal x and y axes.
Again, in both, the only strains and stresses that have to be considered are the three components in the xy plane. In the case of plane stress, by definition, all other components of stress are zero and therefore give no contribution to internal work. In plane strain the stress in a direction perpendicular to the xy plane is not zero. However, by definition, the strain in that direction is zero, and therefore no contribution to internal work is made by this stress, which can in fact be explicitly evaluated from the three main stress components, if desired, at the end of all computations.
5. Axisymmetric stress analysis
The problem of stress distribution in bodies of revolution (axisymmetric solids) under axisymmetric loading is of considerable practical interest. The mathematical problems presented are very similar to those of plane stress and plane strain as, once again, the situation is two dimensional. By symmetry, the two components of displacements in any plane section of the body along its axis of symmetry define completely the state of strain and, therefore, the state of stress. Such a cross-section is shown in Fig. 5.1. If r and z denote respectively the radial and axial coordinates of a point, with u and u being the corresponding displacements, it can readily be seen that precisely the same displacement functions as those used in Chapter 4 can be used to define the displacements within the triangular element i, j , m shown.
The volume of material associated with an ‘element’ is now that of a body of revolution indicated in Fig. 5.1, and all integrations have to be referred to this.
The triangular element is again used mainly for illustrative purposes, the principles developed being completely general.
In plane stress or strain problems it was shown that internal work was associated with three strain components in the coordinate plane, the stress component normal to this plane not being involved due to zero values of either the stress or the strain. In the axisymmetrical situation any radial displacement automatically induces a strain in the circumferential direction, and as the stresses in this direction are certainly non-zero, this fourth component of strain and of the associated stress has to be considered. Here lies the essential difference in the treatment of the axisymmetric situation.
The reader will find the algebra involved in this chapter somewhat more tedious than that in the previous one but, essentially, identical operations are once again involved, following the general formulation of Chapter 2
6. Three-dimensional stress analysis
It will have become obvious to the reader by this stage of the book that there is but one further step to apply the general finite element procedure to fully three-dimensional problems of stress analysis. Such problems embrace clearly all the practical cases, though for some, the various two-dimensional approximations give an adequate and more economical ‘model’.
The simplest two-dimensional continuum element is a triangle. In three dimensions its equivalent is a tetrahedron, an element with four nodal corners, and this chapter will deal with the basic formulation of such an element. Immediately, a difficulty not encountered previously is presented. It is one of ordering of the nodal numbers and, in fact, of a suitable representation of a body divided into such elements.
The first suggestions for the use of the simple tetrahedral element appear to be those of Gallagher et al. and Melosh. Argyri elaborated further on the theme and Rashid and Rockenhauser were the first to apply three-dimensional analysis to realistic problems.
It is immediately obvious, however, that the number of simple tetrahedral elements which has to be used to achieve a given degree of accuracy has to be very large. This will result in very large numbers of simultaneous equations in practical problems, which may place a severe limitation on the use of the method in practice. Further, the bandwidth of the resulting equation system becomes large, leading to increased use of iterative solution methods.
To realize the order of magnitude of the problems presented let us assume that the accuracy of a triangle in two-dimensional analysis is comparable to that of a tetrahedron in three dimensions. If an adequate stress analysis of a square, two-dimensional region requires a mesh of some 20 x 20 = 400 nodes, the total number of simultaneous equations is around 800 given two displacement variables at a node. (This is a fairly realistic figure.) The bandwidth of the matrix involves 20nodes (Chapter 20), Le., some 40 variables.
An equivalent three-dimensional region is that of a cube with 20 x 20 x 20 = 8000 nodes. The total number of simultaneous equations is now some 24000 as three displacement variables have to be specified. Further, the bandwidth now involves an interconnection of some 20 x 20 = 400 nodes or 1200 variables.
Given that with direct solution techniques the computation effort is roughly proportional to the number of equations and to the square of the bandwidth, the magnitude of the problems can be appreciated. It is not surprising therefore that efforts to improve accuracy by use of complex elements with many degrees of freedom have been strongest in the area of three-dimensional analysis. The development and practical application of such elements will be described in the following chapters. However, the presentation of this chapter gives all the necessary ingredients of the formulation for three-dimensional elastic problems and so follows directly from the previous ones. Extension to more elaborate elements will be self-evident.
7. Steady-state field problems - heat conduction, electric and magnetic potential, fluid flow, etc.
While, in detail, most of the previous chapters dealt with problems of an elastic continuum the general procedures can be applied to a variety of physical problems. Indeed, some such possibilities have been indicated in Chapter 3 and here more detailed attention will be given to a particular but wide class of such situations. Primarily we shall deal with situations governed by the general ‘quasi-harmonic’ equation, the particular cases of which are the well-known Laplace and Poisson equations. The range of physical problems falling into this category is large. To list but a few frequently encountered in engineering practice we have:
Heat conduction
Seepage through porous media
Irrotational flow of ideal fluids
Distribution of electrical (or magnetic) potential
Torsion of prismatic shafts
Bending of prismatic beams,
Lubrication of pad bearings, etc.
The formulation developed in this chapter is equally applicable to all, and hence little reference will be made to the actual physical quantities. Isotropic or anisotropic regions can be treated with equal ease.
Two-dimensional problems are discussed in the first part of the chapter. A generalization to three dimensions follows. It will be observed that the same, Co, ‘shape functions’ as those used previously in two- or three-dimensional formulations of elasticity problems will again be encountered. The main difference will be that now only one unknown scalar quantity (the unknown function) is associated with each point in space. Previously, several unknown quantities, represented by the displacement vector, were sought.
In Chapter 3 we indicated both the ‘weak form’ and a variational principle applicable to the Poisson and Laplace equations (see Secs 3.2 and 3.8.1). In the following sections we shall apply these approaches to a general, quasi-harmonic equation and indicate the ranges of applicability of a single, unsed, approach by which one computer program can solve a large variety of physical problems.
To be continued…
