CHAPTER 9 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 This chapter does require some knowledge of matrix algebra,and again,students are directed to suitable texts on the subject.11 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 vibrations).Instead of approximating the entire solution region by a single assumed displace- ment 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 300
CHAPTER 9 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 diference 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 0 an insight into some of the basic concepts of the finite element method (fern.), and, hence, 0 the theoretical development associated with some relatively simple elements, enabling 0 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.'-'' This chapter does require some knowledge of matrix algebra, and again, students are directed to suitable texts on the subject.'' some basis of finite element (fe.), practice, analysis of applications which can be solved with the aid of a simple calculator, and 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 300
s9.1 Introduction to the Finite Element Method 301 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 Oll-set axis -W 1 Curved Spring Concentrated Rod Beams mass Displacement assumption Quadratic 12 Cubic Membrane and plate bending Fig.9.1(a).Examples of element types with nodal points numbered
59.1 Introduction to the Finite Element Method 30 1 an assemblage of a number of elements each with a limited, Le. 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 09.3), analysis requires the assembly and solution of a set of 1 Beams 1 1 A2 1 LIZ 3 1 -l 4 2 LT 1 3 A 7)774 4 5 2 1 Membrane end plate bending Fig. 9.l(a). Examples of element types with nodal points numbered
302 Mechanics of Materials 2 $9.2 7 Linear ■ 20 8 19 15 12 16 Quadratic 13 10 2 6 22 17 24 23 21 12 20 7 32 16 9 28 5 ◆22 24 27 252115 16 /3042 Cubic 5 26 29 4 3 18 12 201 14 17 11 19 13 10 Solid elements Fig.9.1(b).Examples of element types with nodal points numbered. simultaneous equations to provide the displacements for every node in the model.Once the displacement field is determined,the strains and hence the stresses can be derived,using strain-displacement and stress-strain relations,respectively. 9.2.Applicability of the finite element method The fem.emerged essentially from the aerospace industry where the demand for extensive structural analyses was,arguably,the greatest.The general nature of the theory makes it applicable to a wide variety of boundary value problems (i.e.those in which a solution is required in a region of a body subject to satisfying prescribed boundary conditions,as encountered in equilibrium,eigenvalue and propagation or transient applications).Beyond the basic linear elastic/static stress analysis,finite element analysis(fea.),can provide solutions
302 Mechanics of Materials 2 59.2 6 q21 Solid elements Fig. 9.l(b). Examples of element types with nodal points numbered. simultaneous equations to provide the displacements for every node in the model. Once the displacement field is determined, the strains and hence the stresses can be derived, using strain-displacement and stress-strain relations, respectively. 9.2. Applicability of the finite element method The fem. emerged essentially from the aerospace industry where the demand for extensive structural analyses was, arguably, the greatest. The general nature of the theory makes it applicable to a wide variety of boundary value problems (i.e. those in which a solution is required in a region of a body subject to satisfying prescribed boundary conditions, as encountered in equilibrium, eigenvalue and propagation or transient applications). Beyond the basic linear elastichtatic stress analysis, finite element analysis (fea.), can provide solutions
§9.3 Introduction to the Finite Element Method 303 to non-linear material and/or geometry applications,creep,fracture mechanics,free and forced vibration.Furthermore,the method is not confined to solid mechanics,but is applied successfully to other disciplines such as heat conduction,fluid dynamics,seepage flow and electric and magnetic fields.However,attention in this text will be restricted to linearly elastic static stress applications,for which the assumption is made that the displacements are sufficiently small to allow calculations to be based on the undeformed condition. 9.3.Formulation of the finite element method Even with restriction to solid mechanics applications,the fem.can be formulated in a variety of ways which broadly divides into 'differential equation',or 'variational' approaches.Of the differential equation approaches,the most important,most widely used and most extensively documented,is the displacement,or stiffness,based fem.Due to its simplicity,generality and good numerical properties,almost all major general purpose analysis programmes have been written using this formulation.Hence,only the displacement based fem.will be considered here,but it should be realised that many of the concepts are applicable to other formulations. In $9.7,9.8 and 9.9 the theory using the displacement method will be developed for a rod, simple beam and triangular membrane element,respectively.Before this,it is appropriate to consider here,a brief overview of the steps required in a fe.linearly elastic static stress analysis.Whilst it can be expected that there will be detail differences between various packages,the essential procedural steps will be common. 9.4.General procedure of the finite element method The basic steps involved in a fea.are shown in the flow diagram of Fig.9.2.Only a simple description of these steps is given below.The reader wishing for a more in-depth treatment is urged to consult some of numerous texts on the subject,referred to in the introduction. 9.4.1.Identification of the appropriateness of analysis by the finite element method Engineering components,except in the simplest of cases,frequently have non-standard features such as those associated with the geometry,material behaviour,boundary condi- tions,or excitation (e.g.loading),for which classical solutions are seldom available.The analyst must therefore seek alternative approaches to a solution.One approach which can sometimes be very effective is to simplify the application grossly by making suitable approx- imations,leading to Mechanics of Materials solutions(the basis of the majority of this text). Allowance for the effects of local disturbances,e.g.rapid changes in geometry,can be achieved through the use of design charts,which provide a means of local enhancement. In current practice,many design engineers prefer to take advantage of high speed,large capacity,digital computers and use numerical techniques,in particular the fem.The range of application of the fem.has already been noted in $9.2.The versatility of the fem.combined with the avoidance;or reduction in the need for prototype manufacture and testing offer significant benefits.However,the purchase and maintenance of suitable fe.packages,provi- sion of a computer platform with adequate performance and capacity,application of a suitably
$9.3 Introduction to the Finite Element Method 303 to non-linear material and/or geometry applications, creep, fracture mechanics, free and forced vibration. Furthermore, the method is not confined to solid mechanics, but is applied successfully to other disciplines such as heat conduction, fluid dynamics, seepage flow and electric and magnetic fields. However, attention in this text will be restricted to linearly elastic static stress applications, for which the assumption is made that the displacements are sufficiently small to allow calculations to be based on the undeformed condition. 93. Formulation of the finite element method Even with restriction to solid mechanics applications, the fem. can be formulated in a variety of ways which broadly divides into ‘differential equation’, or ‘variational’ approaches. Of the differential equation approaches, the most important, most widely used and most extensively documented, is the displacement, or stiffness, based fem. Due to its simplicity, generality and good numerical properties, almost all major general purpose analysis programmes have been written using this formulation. Hence, only the displacement based fem. will be considered here, but it should be realised that many of the concepts are applicable to other formulations. In 99.7,9.8 and 9.9 the theory using the displacement method will be developed for a rod, simple beam and triangular membrane element, respectively. Before this, it is appropriate to consider here, a brief overview of the steps required in a fe. linearly elastic static stress analysis. Whilst it can be expected that there will be detail differences between various packages, the essential procedural steps will be common. 9.4. General procedure of the finite element method The basic steps involved in a fea. are shown in the flow diagram of Fig. 9.2. Only a simple description of these steps is given below. The reader wishing for a more in-depth treatment is urged to consult some of numerous texts on the subject, referred to in the introduction. 9.4.1. Identification of the appropriateness of analysis by the finite element method Engineering components, except in the simplest of cases, frequently have non-standard features such as those associated with the geometry, material behaviour, boundary conditions, or excitation (e.g. loading), for which classical solutions are seldom available. The analyst must therefore seek alternative approaches to a solution. One approach which can sometimes be very effective is to simplify the application grossly by making suitable approximations, leading to Mechanics of Materials solutions (the basis of the majority of this text). Allowance for the effects of local disturbances, e.g. rapid changes in geometry, can be achieved through the use of design charts, which provide a means of local enhancement. In current practice, many design engineers prefer to take advantage of high speed, large capacity, digital computers and use numerical techniques, in particular the fem. The range of application of the fern. has already been noted in 09.2. The versatility of the fem. combined with the avoidance; or reduction in the need for prototype manufacture and testing offer significant benefits. However, the purchase and maintenance of suitable fe. packages, provision of a computer platform with adequate performance and capacity, application of a suitably
304 Mechanics of Materials 2 s9.4 ldentification of the appropriateness of analysis by the finite element method ldentification of the type of analysis,e.g.plane stress axisymmetric,Ninear elastic.dynamic.non linear,etc. User (pre-processing) ldealisation,i.e.choice of element type(s)e.g.beam,plate. shell.etc. Discretisation of the solution region,i.e.creation of an element mesh Creation of the material behaviour model Application of boundary conditions Creation of a data file,including specification of type of analysis,(e.g.Nnear elastic).and required output Formation of element characteristic matrices Assembly of element of matrices to produce the structure equations Computor (processing) Solution of the structure equilibrium equations to provide nodal values of field variable (displacements) Computation of element resultants (stresses) Interpretation and validation of results User (post-processing) Modification and re-run Fig.9.2.Basic steps in the finite element method. trained and experienced analyst and time for data preparation and processing should not be underestimated when selecting the most appropriate method.Experimental methods such as those described in Chapter 6 provide an effective alternative approach. It is desirable that an analyst has access to all methods,i.e.analytical,numerical and experimental,and to not place reliance upon a single approach.This will allow essential validation of one technique by another and provide a degree of confidence in the results
304 Mechanics of Materials 2 $9.4 Identification d the appmpriateness of analysis by axisymmetric. linear elastic, dynamic, non linear, etc. + Idealisation, i.a. choice of element type(s) e.g. beam, plate. shell, etc. t Discretisation of the solution region, i.e. creation of an elementmesh Creation of the matecial behaviour mcdel + t Application of boundecy conditions Creation of a data file, induding specification of tvpe of analysis, (e.g. linear elastic), and required output I + I I I Fonnatkn of ebment characteristic matrices + AgtemMy of element of matrices to produce thestruchweequations I I Sdution d the sttucture equilibrium equptions to provide t lntarpretabjon and validation of results I + Modification and re-run Fig. 9.2. Basic steps in the finite element method. trained and experienced analyst and time for data preparation and processing should not be underestimated when selecting the most appropriate method. Experimental methods such as those described in Chapter 6 provide an effective alternative approach. It is desirable that an analyst has access to all methods, i.e. analytical, numerical and experimental, and to not place reliance upon a single approach. This will allow essential validation of one technique by another and provide a degree of confidence in the results