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Finite Element Analysis for Engineers - A Primer


NAFEMS has produced a steady stream of publications intended for the finite element community since its inception in the mid 1980’s. The publications cover many aspects relevant to the safe and proper practice of finite element analysis including theory, practice, benchmarks, quality assurance procedures and specific project reports. In the late 1980’s, NAFEMS published a book entitled “A Finite Element Primer” and this has been a key component of the literature ever since. However, advances in finite element technology, software availability and other user requirements have indicated that a major revision of primer-level material would be desirable, particularly for the emerging generation of young engineers and scientists. The current book is a complete and independent rewrite intended to address such needs.

The book is intended to benefit graduate engineers embarking on a career in industry and scientists who require a general background knowledge of the finite element method in order to perform detailed analysis or conduct related research. It is typically aimed at the level of a graduate with one year of one-the-job experience and at an early stage of career development, who is embarking on analysis in an industrial environment, or for analysts who have attended software vendor courses and require further education in the method. For many, however, the opportunity to study the method in detail will be limited due to other professional obligations and so they will have to be limited to some optimum level of material. This book is therefore aimed at such, giving a general and embracing overview of the finite element method. Emphasis is for more description and less mathematical detail than that given in traditional finite element texts, many of which give theoretical details way beyond most users’ requirements. This book complements many existing NAFEMS publications, which are written with similar objectives in mind and to which reference can be made for more specific information.

The scope of this book covers the finite element method applied to structural mechanics in solids (as opposed to fluids), with emphasis on static linear elastic stress analysis. Most of the principles of finite element analysis, implications to software use particularly in the CAD environment, and management and other issues are well represented within this scope. More complicated issues are introduced by chapters covering non-linear behaviour.

In most of the chapters, the fundamentals behind the various finite element developments are discussed, usually involving mathematical derivations. It is possible of course to omit such sections and simply state those equations which are important in the finite element context. However, the justification for including these derivations is to show how the necessary assumptions and restrictions affect the various steps. The progression through the mathematical developments should,at least, be appreciated if not fully understood. All this helps in judging errors, realising the need to design appropriate idealisations, and in understanding the user manuals.

It is assumed that the reader has a basic understanding of applied mechanics and basic mathematical processes such as differentiation and integration. Material science considerations such as the validity of material constitutive laws are beyond the scope of this book. Specific reference to commercial finite element codes is avoided since a neutral stance has to be taken here, although examples and figures referring to such are included when necessary.

Because of the breadth of the subject that this book is required to cover, it is divided into three parts. The first part covers the basis of the finite element method, with particular emphasis on its application in stress analysis. The second part considers practical aspects of the method, such as meshing and testing. Finally, the third part covers further applications of the finite element method, extending the basic concepts described in Part 1 to thermal, dynamic and non-linear behaviour.

PSE competencies addressed by this publication

CodeCompetence StatementReference
FEAkn2Define the meaning of degree of freedom.Section 3.2
FEAkn3List the nodal degrees of freedom and the associated force actions for common beam, 2D solid, 2D axisymmetric, 3D solid and shell elements, for the Displacement FEM.Chapters 5, 6, 7, 8 & 9
FEAkn4Define the meaning of adaptive mesh refinementSection 12.8
FEAkn5State the variational principle involved in the formulation of the Displacement Finite Element Method and identify the solution quantity assumed within each element.Sections 4.2 & 4.3
FEAkn7Name other finite element methods.Chapter 4
FEAkn8List the requirements for an axisymmetric analysis to be valid.Section 6.2.3
FEAkn9List the degrees of freedom to be constrained on a symmetric boundary.Section 10.4
FEAkn10List the degrees of freedom to be constrained on a asymmetric boundary.Sections 10.4 & 10.5
FEAkn11Sketch problems showing the various form of symmetry.Sections 10.2, 10.4-10.6
FEAkn12List the advantages of using symmetry.Sections 10.1 & 10.2
FEAkn13State the word length or arithmetic precision of calculations for any system used.Section 11.7
FEAkn15List 2 common solvers for large sets of simultaneous equations.Chapter 11
FEAkn16List the various forms of element distortion.Sections 12.11 & 14.5
FEAco1Describe the sources of error inherent in finite element analysis, in general terms.Sections 11.7 & 14.2
FEAco2Discuss checks that may be used post-solution to check for the presence of inaccuracy.Sections 11.7, 14.3 & 14.4
FEAco4Explain the meaning of convergence, including h and p types.Sections 4.5, 12.6 & 14.4
FEAco5Discuss the difficulties that can arise in using a CAD model as the basis for carrying out analysis and simulation.Sections 12.7 & 13.6
FEAco6Discuss the need for a consistent set of units in any analysis and illustrate possible pitfalls.Sections 2.10 & 13.2
FEAco7Explain why strains and stresses are generally less accurate than displacements for any given mesh of elements, using the Displacement FEM.Sections 7.9 & 14.9
FEAco8Discuss the validity of using symmetry techniques to model non-symmetric problems.Sections 10.4-10.7
FEAco9Explain the meaning of the term ill-conditioned when used in the context of a set of solution equations and illustrate physical situations where this might reflect reality.Section 11.7
FEAco11Discuss the finite element / spring analogy.Section 3.5
FEAco12Outline a common method employed to solve the large sets of sparse symmetric common in FEA.Chapter 11
FEAco13Explain how the structural stiffness matrix is assembled from the individual element matrices.Section 3.7
FEAco14Discuss the nature of the structural stiffness matrix.Section 3.7
FEAco15Discuss the integral equation for element stiffness, highlighting the variables which it is dependent upon.Section 3.6
FEAco16Discuss the salient features of the integral equation for Consistent Nodal Loading.Sections 10.10 & 10.12
FEAco17Explain the process of Gaussian Quadrature and the terms Reduced Integration, Shear Locking and Mechanisms.Sections 7.4-7.7 & 14.5
FEAco18Explain the term Isoparametric Element.Chapter 7
FEAco19Discuss the general requirements for suitable Displacement Functions.Chapter 3
FEAco21Describe the Patch Test and explain its significance.Section 14.6
FEAco22Explain why displacement elements may not always lead to a behaviour that is too stiff in practice.Section 7.5
FEAco23Explain the Equilibrium and Compatibility conditions, normally found within and between displacement elements.Sections 2.6 & 2.4
FEAco24Discuss the relationship between shape function and strain/stress prediction for simple 2D linear and parabolic elements.Section 7.3
FEAco26Discuss the significance of computer memory to solution elapse time for large models.Section 11.1
FEAco27Explain how unwanted cracks can be produced in 2D and 3D solid meshes and describe which plot type is useful in detecting these.Section 14.4
FEAco28Explain why element distortion generally results in poorer results.Subsection 12.9.4
FEAco29Discuss the term Flying Structure or Insufficiently Constrained Structure.Sections 3.8 & 10.2
FEAco30Explain why stress averaging is not appropriate at junctions between elements of different thickness.Section 7.9
FEAco31Explain why most finite elements do not represent a circular boundary exactly and highlight how this approximation manifests itself.Chapter 7
FEAco32Explain the concept of substructuring, where applicable and highlight common limitations of use.Section 12.3
FEAco33Describe the process of nested or submodelling.Section 12.3
FEAco34Explain the term hybrid model and highlightSubsection 20.5.4
FEAco38Outline how Initial Strains and Initial Stresses are commonly handled in the finite element solution.Section 10.11
FEAco39Discuss the Geometric Stiffness Matrix and highlight  situations where it becomes important.Section 18.6
FEAco40Explain the rationale behind the use of 1-D, 2-D and 3-D elements used in the analysis of components within your organisation.Chapter 12
FEAap2Demonstrate effective use of available results presentation facilities.Chapter 15
FEAap4Illustrate the various steps in the Displacement Finite Element Method from assumed displacement polynomial to determination of stresses.Section 3.6
FEAap5Illustrate possible applications of 0D, 1D, 2D and 3D  elements in your industry sector.Chapter 15
FEAap7Employ symmetric boundary conditions effectively.Section 10.4
FEAap8Employ asymmetric boundary conditions effectively.Sections 10.5 & 10.6
FEAap9Employ cyclic symmetric boundary conditions effectively, where appropriate.Section 10.6
FEAap10Illustrate various physical situations which will result in a Stress Singularity and explain why it is not appropriate to use finite element results at such locations directly.Section 20.2
FEAap11Illustrate consistent nodal loadings for uniform loading on a range of common linear and quadratic shell, 2D and 3D solid elements and note any unusual features.Section 10.12
FEAev1Select appropriate idealisation(s) for components / structures, which are consistent with the objectives of the analyses.Chapter 12

Document Details

AuthorsHellen. T Becker. A
Date 1st December 2013
OrganisationNAFEMS Education and Training Working Group


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