This book has introductory materials on FEM. The book covers the basic principles and formulations related to finite element method. The book is written with lucid and simple way keeping in mind all categories of readers. The readers interested to collect basic, starter informations on FEM would be most benefited. The full coverage of theory and problems would make the reading interesting to the readers of mechanical engineering. The attractive pedagogy with highlights of the chapters and the question banks and the short viva-voce questions would make reading interesting and preparative. The self explanatory diagrams and solution to typical problems make this book different. The Chapter-1 deals with the general introduction to FEM and attempts to give an identification to the topic. Chapter 2 covers the requirements of prerequisites in the form of numerical techniques used in FEM. Chapter-3 offers material on variational approaches that laid foundation for FEM development. The basics and introduction to theory of elasticity needed in FEM are dealt in Chapter-4. ID bar element analysis with examples is the material of Chapter-5. The FEM formulation of beam element is offered as study material in Chapter-6. The formulation related to 2D elements is the reading stuff in Chapter-7. The Chapter-8 provides detailed informations on higher order element used in FEM and convergence criteria with material on patch test. The utility of FEM in heat transfer analysis is briefed in Chapter-8. The question bank collected from the exam papers of universities is provided at the end of the book. The short viva-voce question bank that guides the preparation for FEM laboratory is also listed at the end of the book. The points that should be remembered to acquire basic knowledge in FEM are listed at the end of each Chapter. The authors are thankful to those who directly or indirectly helped and supported to bring out this book. The constructive criticism from the readers on the improvisation of the book would be well received. The authors would be delighted if the book serves the interest of the students and readers effectively.
Additional Info
  • Publisher: Laxmi Publications
  • Language: English
  • ISBN : 978-93-83828-48-7
  • Chapter 1

    Introduction to finite element method Price 2.99  |  2.99 Rewards Points

    This is a introductory chapter on Finite Element Method which gives basic familiarity of understanding the concepts. The general questions of what and why of FEM are dealt in detail to give an overview of form and structure of the method. Comparison with the other methods originated prior to FEM, highlights the supremacy of FEM. Before knowing the methodology of workouts of finite element procedures it is imperative to know the definition and objectives that lead to the development of FEM. The term Finite Element was first coined by civil engineering scientists who started the procedure for structural engineering applications. In fifty years of development FEM has stretched its arms in different interdisciplinary areas of engineering like design, heat transfer, fluid flow, dynamics and electrical engineering etc., the adaptability is enhanced by flexibility of approach, accuracy and reliability of results and speed of generation. The development of fast digital computers has supplemented the growth of FEM to a commendable extent
  • Chapter 2

    Numerical techniques in finite element method Price 2.99  |  2.99 Rewards Points

    Since finite element method is a totally computer oriented approach, the handling of equations is completely based on solving the simultaneous equations in the matrix form. Hence it is the pre- requisite to learn for the FEM engineers to learn the terminology, the algebra, solution methods and the numerical methods related to matrices. This chapter attempts to throw light on the following important issues. Terminology of different matrices definitions. The algebraic operations pertaining matrices. The determinants, inversion and calculus of matrices. The solution methods such as Gauss elimination method Cholesky's method LU decomposition method Tri-Diagonal Matrix Algorithm (TDMA) The numerical integration by Gauss Quadrature.
  • Chapter 3

    Variational principles and approximate methods Price 2.99  |  2.99 Rewards Points

    The variational calculus can be applied to continuous and discrete systems, especially to discretized systems with higher comfort. The exact solutions requires the satisfaction of compatibility and equilibrium conditions to be satisfied completely. When the exact solution does not exist, the approximate solution close to the exact is made possible by Rayleigh-Ritz method. The basis for this method is the minimum potential energy principle. More the number of Ritz constants in trial function more accurate is the solution. The Lagrange-Euler formulation of variational principle aids in establishing the governing differential equation of equilibrium describing the physical phenomenon. The physical problems like heat conduction and fluid flow cannot be addressed by Rayleigh-Ritz method as it is based on stationarity condition in potential energy functional. Such processes have the answer in weighted residual methods. Several weighted residual methods are - point collocation, subdomain collocation, least square and Galerkin's method. The Galerkin's method is popularly used in finite element method as it can be comfortably employed for elasticity and fluid/ heat related problems. The virtual displacement approach gives suitable base for FE formulation of static analysis problems. The Newton's laws of motion, the work-energy principle are the basis for the principle of virtual work which helps to derive expressions for stress, strain and displacement in an elastic structure. This chapter deals with variational approach, minimum potential energy principle, Rayleigh-Ritz method, weighted residual methods, and principle of virtual displacement and virtual work with suitable illustrative examples from elasticity and heat transfer.
  • Chapter 4

    Introduction to theory of elasticity Price 2.99  |  2.99 Rewards Points

    The theory of elastic structures is the basis for the development of finite element method which is a computational mechanics procedure. This chapter gives an introduction to basic relations like strain-displacement relation, stress-strain equation and the equilibrium equation. The applied forces on a continuous system develops a resistive force internal to the system. For the system to be in equilibrium there has to be balance between these forces which leads to the establishment of equilibrium equations. The external forces applied also results in strains in the system which are related to the deflections of the system by the strain-displacement equation. The resistive force per unit area is the stress induced. By knowing the property of the material used for the system, the relations between the stress-strain are established. If the Newton's Law helps in arriving at the force equilibrium, the Hooke's law is required to derive the stress-strain relation. The above stated three relations provide the strong foundational basis for understanding the concepts of FEM. The components of stress, and the components of strain are given brief thought in this chapter. The plane stress and plane strain approximations based on stress-strain relations with the illustrative examples are also covered. In total this chapter spans the key issues in theory of elasticity useful for the basic knowledge for the concepts in finite element method (FEM).
  • Chapter 5

    Fem formulations of bar element Price 2.99  |  2.99 Rewards Points

    Many problems in engineering and applied science are governed by differential or integral equations. The solution to these equations would provide exact closed-form solution to the particular problem being studied. However the complexity in the geometry, properties and in the boundary conditions that are seen in the most of the real world problems usually means that exact solution cannot be obtained or obtained in a reasonable amount of time. The current product design cycle times imply that the engineers must obtain design solutions in a short period of time. They have content to get approximate solutions that are easily obtained in a reasonable time frame and with reasonable effort. The FEM is one such approximate technique.
  • Chapter 6

    Finite element formulation of beam element Price 2.99  |  2.99 Rewards Points

    The finite element procedure applied to beam element is presented in this chapter. A two noded beam element with two degrees of freedom is considered for the analysis. To establish the force- displacement and moment-rotation relationship the basics from the elementary mechanics of materials has been used. The flexure equation, the deflection relations and the shear force and bending moment formulations have been used in the analysis. The shape functions that represent the deformation pattern are derived by displacement transformation method. The derivation of shape functions in local Cartesian as well as natural co-ordinate system have been demonstrated. The easier method to derive shape function using the Hermitian polynomial and the respective boundary condition is also presented. The consistent load vectors are obtained for different load cases established in the mechanics. The different loads like body force, centrifugal force and distributed and point load cases are highlighted.
  • Chapter 7

    Finite element formulation of 2-dimensional elements Price 2.99  |  2.99 Rewards Points

    To represent the displacement variation at the element level 2-D element level and to develop the shape functions or the linear interpolation functions consider a triangular element with three nodes. Let the vertices or the nodes at the corner be denoted by the lebels '1', '2' and '3'. Let the X-Y co-ordinates of these nodes be (xi, yi) for i = 1, 2 and 3 as shown in the Fig. 7.1. At this instant let us consider one degree of freedom at each node of the element. Hence total degrees of freedom works out to be three.
  • Chapter 8

    Higher order elements and convergence criteria Price 2.99  |  2.99 Rewards Points

    In order to derive the shape functions it was assumed that the displacement field is a polynomial of any degree for all cases considered. It has been previously shown that for the two noded bar element with one degree of freedom at each node it is convenient to assume a linear polynomial with two generalized co-ordinates in the derivation of the shape function. But this has the limitation on the accuracy of result obtained when a problem of real world physics is considered. Hence an attempt was made to formulate higher order elements in one-dimension and two dimension, also three dimension as well.
  • Chapter 9

    Heat transfer and finite element method Price 2.99  |  2.99 Rewards Points

    Heat transfer is one of the important processes studied by mechanical engineers. The principles of finite element method can also be applied to heat transfer and fluid flow problems effectively. Since the governing equation describing the heat flow is differential equation the variational principles and weighted residual methods can be applied to obtain the temperature distribution. FEM can be successfully employed to obtain the temperature distribution in composite wall, extended surfaces like fins. Heat transfer through composite furnace wall, reactor wall, steam pipes are some of the examples which can be analysed using the procedures of FEM.

About the Author