STRUCTURAL ANALYSIS
1.11 INTRODUCTION TO THE FINITE ELEMENT METHOD FOR STRUCTURAL ANALYSISSTRUCTURAL ANALYSIS
kn1u1+ kn2u2 + kn3u3 + . . . + knnun= fn
(1.50) In this way, the second equation is equivalent to
1015k22u2= 1015k22u2 or u2= u2 (1.51) which is the prescribed condition. The value of the reaction f2 is com-puted “a posteriori” by Eq.(1.49).
The issue of prescribed displacements will be treated again in Chap-ter 9.
1.11 INTRODUCTION TO THE FINITE ELEMENT METHOD FOR STRUCTURAL ANALYSIS
Most structures in practice are of continuous nature and can not be accu-rately modelled by a collection of bars. Examples of “continuous” struc-tures are standard in civil, mechanical, aeronautical and naval engineering.
Amongst the more common we can list: plates, foundations, roofs, contai-ners, bridges, dams, airplane fuselages, car bodies, ship hulls, mechanical components, etc. (Figure 1.19).
Although a continuous structure is inherently three-dimensional (3D), its behaviour can be accurately described in some cases by one- (1D) or two-dimensional (2D) structural models. This occurs, for instance, in the analysis of plates in bending, where only the deformation of the plate mid-plane is considered. Other examples are the structures modelled as 2D solids or as axisymmetric solids (i.e. dams, tunnels, water tanks, etc.)
Fig. 1.19 Continuous structures: a) Dam, b) Shell, c) Bridge, d) Plate
The analytical solution of a continuous structure is very difficult (ge-nerally impossible) due to the complexities of the geometry, the boundary conditions, the material properties, the loading, etc. This explains the need for computational models to analyse continuous structures.
The FEM is the simpler and more powerful computational procedure for the analysis of structures with arbitrary geometry and general material properties subjected to any type of loading.
The FEM allows one the behaviour of a structure with an infinite num-ber of DOFs to be modelled by that of another one with approximately the same geometrical and mechanical properties, but with a finite number of DOFs. The latter are related to the external forces by a system of algebraic equations expressing the equilibrium of the structure. We will find that the basic finite element methodology is analogous to the matrix analysis technique studied for bar structures. The analogies can be summarized by
Fig. 1.20 Analysis of a bridge by the finite element method
considering the bridge shown in Figure 1.20. Without entering into the details, the basic steps in the finite element analysis are the following:
Step 1 : Starting with the geometrical description of the bridge, its supports and the loading, the first step is to select a structural model. For example, we could use a 3D solid model (Chapter 8), a stiffened plate model (Chapter 10, Vol. 2 [On]) or a facet shell model (Chapter 7, Vol. 2 [On]). The material properties must also be defined, as well as the scope of the analysis (small or large displacements, static or dynamic analysis,
etc.). As mentioned earlier, in this book we will focus on linear static analysis only.
Step 2 : The structure is subdivided into a mesh of non-intersecting domains termed finite elements (discretization process). The problem va-riables (displacements) are interpolated within each element in terms of their values at a known set of points of the element called nodes. The number of nodes defines the approximation of the solution within each ele-ment. Some nodes are placed at the element boundaries and they can be interpreted as linking points between adjacent elements. However, nodes in the interior of the elements are needed for higher-order approximations and, hence, the nodes do not have a physical meaning as the connecting joints in bar structures. The mesh can include elements with different geometry, such as 2D plate elements coupled with 1D beam elements.
The discretization process is an essential part of the preprocessing step which includes the definition of all the analysis data. The preprocessing step typically consumes a considerable amount of human effort. The use of efficient preprocessing tools is essential for the analysis of practical structures in competitive times. More details are given in Chapter 10.
Step 3 : The stiffness matrices K(e) and the load vectors f(e) are ob-tained for each element. The computation of K(e)and f(e)is more complex than for bar structures and it usually requires the evaluation of integrals over the element domain.
Step 4 : The element stiffness and the load terms are assembled into the overall stiffness matrix K and the load vector f for the structure.
Step 5 : The global system of linear simultaneous equations Ka = f is solved for the unknown displacement variables a.
Step 6 : Once the displacements a are computed, the strains and the stresses are evaluated within each element. Reactions at the nodes restrai-ned against movement are also computed.
Step 7 : Solving steps 3-6 requires a computer implementation of the FEM by means of a standard or specially developed program.
Step 8 : After a successful computer run, the next step is the interpre-tation and preseninterpre-tation of results. Results are presented graphically to aid their interpretation and checking (postprocessing step). The use of spe-cialized graphic software is essential in practice. More details are given in Chapter 10.
Step 9 : Having assessed the finite element results, the analyst may consider several modifications which may be introduced at various stages of the analysis. For example, it may be found that the structural model selected is inappropriate and hence it should be adequately modified. Al-ternatively, the finite element mesh chosen may turn out to be too coarse to capture the expected stress distributions and must therefore be refined or a different, more accurate element used. Round-off problems arising from ill-conditioned equations, the equations solving algorithm or the com-puter word length employed in the analysis may cause difficulties and can require the use of double-precision arithmetic or some other techniques.
Input data errors which occur quite frequently must be also corrected.
All these possible modifications are indicated by the feedback loop shown in Figure 1.21 taken from [HO2].
From the structural engineer’s point of view, the FEM can be consi-dered as an extension to continuous systems of the matrix analysis pro-cedures for bar structures. The origins of the FEM go back to the early 1940’s with the first attempts to solve problems of 2D elasticity using ma-trix analysis techniques by subdividing the continuum into bar elements [Hr,Mc]. In 1946 Courant [Co] introduced for the first time the concept of
“continuum element” to solve 2D elasticity problems using a subdivision into triangular elements with an assumed displacement field. The arrival of digital computers in the 1960’s contributed to the fast development of matrix analysis based techniques, free from the limitations imposed by the need to solve large systems of equations. It was during this period that the FEM rapidly established itself as a powerful approach to solve many problems in mathematics and physics. It is interesting that the first applications of the FEM were related to structural analysis and, in par-ticular, to aeronautical engineering [AK,TCMT]. It is acknowledged that Clough first used the name “finite elements” in relation to the solution of 2D elasticity problems in 1960 [Cl]. Since then the FEM has had a tremen-dous expansion in its application to many different fields. Supported by the continuous upgrading of computers and by the increasing complexity of many areas in science and technology, today the FEM enjoys a unique position as a powerful technique for solving the most difficult problems in engineering and applied sciences.
It would be an impossible task to list here all the significant published work since the origins of the FEM. Only in 2008, the scientific publications in this field were estimated to number in excess of 25,000. The reader in-terested in bibliography on the FEM should consult the references listed
Fig. 1.21 Flow chart of the analysis of a structure by the FEM [HO2]
in [No,ZT,ZTZ] and in the Encyclopedia of Computational Mechanics [SDH,SDH2].