《纺织复合材料》课程参考文献（Mechanics of Materials，2/2）09 INTRODUCTION TO THE FINITE ELEMENT METHOD.pdf_大学文库

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

§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

§9.4 Introduction to the Finite Element Method 305 9.4.2.Identification of the type of analysis The most appropriate type(s)of analysis to be employed needs to be identified in order that the component behaviour can best be represented.The assumption of either plane stress or plane strain is a common example.The high cost of a full three-dimensional analysis can be avoided if the assumption of both geometric and load symmetry can be made.If the application calls for elastic stress analysis,then the system equations will be linear and can be solved by a variety of methods,Gaussian elimination,Choleski factorisation or Gauss-Seidel procedure.s For large displacement or post-yield material behaviour applications the system equations will be non-linear and iterative solution methods are required,such as that of Newton- Raphson.5 9.4.3.Idealisation Commercially available finite element packages usually have a number of different elements available in the element library.For example,one such package,HKS ABAQUS12 has nearly 400 different element variations.Examples of some of the commonly used elements have been given in Fig.9.1. Often the type of element to be employed will be evident from the physical application.For example,rod and beam elements can represent the behaviour of frames,whilst shell elements may be most appropriate for modelling a pressure vessel.Some regions which are actually three-dimensional can be described by only one or two independent coordinates,e.g.pistons, valves and nozzles,etc.Such regions can be idealised by using axisymmetric elements. Curved boundaries are best represented by elements having mid-side(or intermediate)nodes in addition to their corner nodes.Such elements are of higher order than linear elements (which can only represent straight boundaries)and include quadratic and cubic elements. The most popular elements belong to the so-called isoparametric family of elements,where the same parameters are used to define the geometry as define the displacement variation over the element.Therefore,those isoparametric elements of quadratic order,and above,are capable of representing curved sides and surfaces. In situations where the type of elements to be used may not be apparent,the choice could be based on such considerations as (a)number of dof., (b)accuracy required, (c)computational effort, (d)the degree to which the physical structure needs to be modelled. Use of the elements with a quadratic displacement assumption are generally recommended as the best compromise between the relatively low cost but inferior performance of linear elements and the high cost but superior performance of cubic elements. 9.4.4.Discretisation of the solution region This step is equivalent to replacing the actual structure or continuum having an infinite number of dof.by a system having a finite number of dof.This process,known as

$9.4 Introduction to the Finite Element Method 305 9.4.2. IdentiJication of the type of analysis The most appropriate type(s) of analysis to be employed needs to be identified in order that the component behaviour can best be represented. The assumption of either plane stress or plane strain is a common example. The high cost of a full three-dimensional analysis can be avoided if the assumption of both geometric and load symmetry can be made. If the application calls for elastic stress analysis, then the system equations will be linear and can be solved by a variety of methods, Gaussian elimination, Choleski factorisation or Gauss-Seidel procedure ? For large displacement or post-yield material behaviour applications the system equations will be non-linear and iterative solution methods are required, such as that of NewtonRaphson? 9.4.3. Idealisation Commercially available finite element packages usually have a number of different elements available in the element library. For example, one such package, HKS ABAQUS'* has nearly 400 different element variations. Examples of some of the commonly used elements have been given in Fig. 9.1. Often the type of element to be employed will be evident from the physical application. For example, rod and beam elements can represent the behaviour of frames, whilst shell elements may be most appropriate for modelling a pressure vessel. Some regions which are actually three-dimensional can be described by only one or two independent coordinates, e.g. pistons, valves and nozzles, etc. Such regions can be idealised by using axisymmetric elements. Curved boundaries are best represented by elements having mid-side (or intermediate) nodes in addition to their comer nodes. Such elements are of higher order than linear elements (which can only represent straight boundaries) and include quadratic and cubic elements. The most popular elements belong to the so-called isoparametric family of elements, where the same parameters are used to define the geometry as define the displacement variation over the element. Therefore, those isoparametric elements of quadratic order, and above, are capable of representing curved sides and surfaces. In situations where the type of elements to be used may not be apparent, the choice could be based on such considerations as (a) number of dof., (b) accuracy required, (c) computational effort, (d) the degree to which the physical structure needs to be modelled. Use of the elements with a quadratic displacement assumption are generally recommended as the best compromise between the relatively low cost but inferior performance of linear elements and the high cost but superior performance of cubic elements. 9.4.4. Discretisation of the solution region This step is equivalent to replacing the actual structure or continuum having an infinite number of dof. by a system having a finite number of dof. This process, known as

$9.4 Introduction to the Finite Element Method 307 equal to the larger lateral dimension),from the position of the disturbance.Advantage can therefore be taken of this principle by reducing the necessary extent of a finite element model.A rule-of-thumb is that a model need only extend to one-and-a-half times the larger lateral dimension from a disturbance,see Fig.9.4. (a)Actual component (b)Boundary for finite element idealisation 3b Fig.9.4.Idealisation of a shouldered tension strip. Element size and grading The relative size of elements directly affects the quality of the solution.As the element size is reduced so the accuracy of solution can be expected to increase since there is better representation of the field variable,e.g.displacement,and/or better representation of the geometry.However,as the element size is reduced,so the number of elements increases with the accompanying penalty of increased computational effort.Needlessly small elements in regions with little variation in field variable or geometry will be wasteful.Equally,in regions where the stress variation is not of primary interest then a locally coarse mesh can be employed providing it is sufficiently far away from the region of interest and that it still provides an accurate stiffness representation.Therefore,element sizes should be graded in order to take account of anticipated stress/strain variations and geometry,and the results required.The example of stress analysis of a semi-infinite tension plate with a central circular hole,Fig.9.3,serves to illustrate how the size of the elements can be graded from small-size elements surrounding the hole (where both the stress/strain and geometry are varying the most),to become coarser with increasing distance from the hole. Number of elements The number of elements is related to the previous matter of element size and,for a given element type,the number of elements will determine the total number of dof.of the model, and combined with the relative size determines the mesh density.An increase in the number of elements can result in an improvement in the accuracy of the solution,but a limit will be reached beyond which any further increase in the number of elements will not significantly improve the accuracy.This matter of convergence of solution is clearly important,and with experience a near optimal mesh may be achievable.As an alternative to increasing the number of elements,improvements in the model can be obtained by increasing the element order. This alternative form of enrichment can be performed manually (by substituting elements)

99.4 Introduction to the Finite Element Method 307 equal to the larger lateral dimension), from the position of the disturbance. Advantage can therefore be taken of this principle by reducing the necessary extent of a finite element model. A rule-of-thumb is that a model need only extend to one-and-a-half times the larger lateral dimension from a disturbance, see Fig. 9.4. (a) Actual component P +q7;TEy- (b) Boundary for finite element idealition _p 2 ._.___.__.____ Fig. 9.4. ldealisation of a shouldered tension strip. Element size and grading The relative size of elements directly affects the quality of the solution. As the element size is reduced so the accuracy of solution can be expected to increase since there is better representation of the field variable, e.g. displacement, and/or better representation of the geometry. However, as the element size is reduced, so the number of elements increases with the accompanying penalty of increased computational effort. Needlessly small elements in regions with little variation in field variable or geometry will be wasteful. Equally, in regions where the stress variation is not of primary interest then a locally coarse mesh can be employed providing it is sufficiently far away from the region of interest and that it still provides an accurate stiffness representation. Therefore, element sizes should be graded in order to take account of anticipated stresshtrain variations and geometry, and the results required. The example of stress analysis of a semi-infinite tension plate with a central circular hole, Fig. 9.3, serves to illustrate how the size of the elements can be graded from small-size elements surrounding the hole (where both the stresdstrain and geometry are varying the most), to become coarser with increasing distance from the hole. Number of elements The number of elements is related to the previous matter of element size and, for a given element type, the number of elements will determine the total number of dof. of the model, and combined with the relative size determines the mesh density. An increase in the number of elements can result in an improvement in the accuracy of the solution, but a limit will be reached beyond which any further increase in the number of elements will not significantly improve the accuracy. This matter of convergence of solution is clearly important, and with experience a near optimal mesh may be achievable. As an alternative to increasing the number of elements, improvements in the model can be obtained by increasing the element order. This alternative form of enrichment can be performed manually (by substituting elements)