Finite Element Analysis
A Course Material onFinite Element AnalysisByMr. L.VINOTH MEASSISTANT PROFESSORDEPARTMENT OF MECHANICAL ENGINEERINGSASURIE COLLEGE OF ENGINEERINGVIJAYAMANGALAM – 638 056
QUALITY CERTIFICATEThis is to certify that the e-course materialSubject Code: ME2353Subject: Finite Element AnalysisClass: III Year Mechanicalbeing prepared by me and it meets the knowledge requirement of the university curriculum.Signature of the AuthorName:L.VINOTH M.E.Designation:ASSISTANT PROFESSORThis is to certify that the course material being prepared by Mr.L.Vinoth is of adequate quality. He hasreferred more than five books among them minimum one is from aboard author.Signature of HDName: E.R SIVAKUMAR M.E.,(PhD)SEAL
CONTENTSS.NOTOPICPAGE NOUNIT-1 FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS1.1INTRODUCTION11.1.1 A Brief History of the FEM11.1.2General Methods of the Finite Element Analysis11.1.3General Steps of the Finite Element Analysis11.1.4 Objectives of This FEM21.1.5 Applications of FEM in Engineering21.2WEIGHTED RESIDUAL METHOD21.3THE GENERAL WEIGHTED RESIDUAL STATEMENT51.4WEAK FORMULATION OF THE WEIGHTED RESIDUAL51.5PIECE WISE CONTINUOUS TRIAL FUNCTION61.6EXAMPLES OF A BAR FINITE ELEMENT81.6.1 Rigid Body13PRINCIPLE OF STATIONERY TOTAL POTENTIAL PSTP)191.7.1 Potential energy in elastic bodies191.7.2 Principle of Minimum Potential Energy191.8RAYLEIGH – RITZ METHOD (VARIATIONAL APPROACH)241.9ADVANTAGES OF FINITE ELEMENT METHOD241.10DISADVANTAGES OF FINITE ELEMENT METHOD241.7UNIT – 2 ONE DIMENSIONAL FINITE ELEMENT ANALYSIS2.1ONE DIMENSIONAL ELEMENTS252.2LINEAR STATIC ANALYSIS( BAR ELEMENT)282.3BEAM ELEMENT282.41-D 2-NODED CUBIC BEAM ELEMENT MATRICES332.5DEVELOPMENT OF ELEMENT EQUATION34
2.6BEAM ELEMENT422.6.1 ELEMENT MATRICES AND VECTORS45UNIT – 3 TWO DIMENSIONAL FINITE ELEMENT ANALYSIS3.1INTRODUCTION543.2THREE NODED LINEAR TRIANGULAR ELEMENT543.3FOUR NODED LINEAR RECTANGULAR ELEMENT55TWO-VARIABLE 3-NODED LINEAR TRIANGULAR3.43.5ELEMENT56STRAIN – STRESS RELATION603.5.1 Plane stress conditions613.5.2 Plane strain conditions61GENERALIZED COORDINATES APPROACH TO NODEL3.63.73.8APPROXIMATIONSISOPARAMETRIC ELEMENTSSTRUCTURAL MECHANICS APPLICATIONS IN 2DIMENSIONSUNIT – 4 DYNAMIC ANALYSIS USING ELEMENT METHOD656671INTRODUCTION884.1.1 Fundamentals of Vibration884.1.2 Causes of Vibrations884.1.3 Types of Vibrations884.2EQUATION OF MOTION894.3CONSISTENT MASS MATRICES944.3.1 Single DOF System944.3.2.Multiple DOF System984.4VECTOR ITERATION METHODS994.5MODELLING OF DAMPING1024.5.1 Proportional Damping (Rayleigh Damping)1024.5.2 Frequency Response Analysis1054.1
4.6TRANSIENT RESPONSE ANALYSIS1064.6.1Cautions in Dynamic Analysis107UNIT -5 APPLICATIONS IN HEAT TRANSFER &FLUID MECHANICS5.1ONE DIMENSIONAL HEAT TRANSFER ELEMENT1115.1.1Strong Form for Heat Conduction in One Dimensionwith Arbitrary Boundary Conditions1115.1.2Weak Form for Heat Conduction in One Dimensionwith Arbitrary Boundary Conditions5.2112APPLICATION TO HEAT TRANSFER 2-DIMENTIONAL1125.2.1Strong Form for Two-Point Boundary Value Problems1125.2.2Two-Point Boundary Value Problem WithGeneralized Boundary Conditions1125.2.3 Weak Form for Two-Point Boundary Value Problems1145.3SCALE VARIABLE PROBLEM IN 2 DIMENSIONS1145.42 DIMENTIONAL FLUID MECHANICS117QUESTION BANK120TEXT BOOKS:1. P.Seshu, “Text Book of Finite Element Analysis”, Prentice-Hall of India Pvt. Ltd. NewDelhi, 2007.2. J.N.Reddy, “An Introduction to the Finite Element Method”, McGraw-Hill InternationalEditions(Engineering Mechanics Series), 1993.3. Cook,Robert.D., Plesha,Michael.E & Witt,Robert.J. “Concepts and Applications ofFinite Element Analysis”,Wiley Student Edition, 2004.4. Chandrupatla & Belagundu, “Introduction to Finite Elements in Engineering”, 3rdEdition, Prentice-Hall of India, Eastern Economy Editions.
ME2353FINITE ELEMENT ANALYSISL T P C3 1 0 4INTRODUCTION (Not for examination)5Solution to engineering problems – mathematical modeling – discrete and continuum modeling – need fornumerical methods of solution – relevance and scope of finite element methods – engineering applicationsof FEAUNIT IFINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS 5 3Weighted residual methods –general weighted residual statement – weak formulation of the weightedresidual statement –comparisons – piecewise continuous trial functions- example of a bar finite element –functional and differential forms – principle of stationary total potential – Rayleigh Ritz method – piecewisecontinuous trial functions – finite element method – application to bar elementUNIT IIONE DIMENSIONAL FINITE ELEMENT ANALYSIS8 4General form of total potential for 1-D applications – generic form of finite element equations – linear barelement – quadratic element –nodal approximation – development of shape functions – element matricesand vectors – example problems – extension to plane truss– development of element equations –assembly – element connectivity – global equations – solution methods –beam element – nodalapproximation – shape functions – element matrices and vectors – assembly – solution – exampleproblemsUNIT IIITWO DIMENSIONAL FINITE ELEMENT ANALYSIS10 4Introduction – approximation of geometry and field variable – 3 noded triangular elements – fournoded rectangular elements – higher order elements – generalized coordinates approach to nodalapproximations – difficulties – natural coordinates and coordinate transformations – triangular andquadrilateral elements – iso-parametric elements – structural mechanics applications in 2-dimensions –elasticity equations – stress strain relations – plane problems of elasticity – element equations – assembly– need for quadrature formule – transformations to natural coordinates – Gaussian quadrature – exampleproblems in plane stress, plane strain and axisymmetric applicationsUNIT IVDYNAMIC ANALYSIS USING FINITE ELEMENT METHOD8 4Introduction – vibrational problems – equations of motion based on weak form –longitudinal vibration of bars – transverse vibration of beams – consistent mass matrices– element equations –solution of eigenvalue problems – vector iteration methods – normal modes –transient vibrations – modeling of damping – mode superposition technique – direct integrationmethodsUNIT VAPPLICATIONS IN HEAT TRANSFER & FLUID MECHANICS6 3One dimensional heat transfer element – application to one-dimensional heat transfer problems- scalarvariable problems in 2-Dimensions – Applications to heat transfer in 2- Dimension – Application toproblems in fluid mechanics in 2-DTEXT BOOK:1. P.Seshu, “Text Book of Finite Element Analysis”, Prentice-Hall of India Pvt. Ltd. NewDelhi, 2007. ISBN-978-203-2315-5REFERENCE BOOKS:1. J.N.Reddy, “An Introduction to the Finite Element Method”, McGraw-Hill InternationalEditions(Engineering Mechanics Series), 1993. ISBN-0-07-051355-42. Chandrupatla & Belagundu, “Introduction to Finite Elements in Engineering”, 3rdEdition, Prentice-Hall of India, Eastern Economy Editions. ISBN-978-81-203-2106-93. David V.Hutton,”Fundamentals of Finite Element Analysis”, Tata McGraw-Hill Edition2005. ISBN-0-07-239536-24. Cook,Robert.D., Plesha,Michael.E & Witt,Robert.J. “Concepts and Applications ofFinite Element Analysis”,Wiley Student Edition, 2004. ISBN-10 81-265-1336-5Note: L- no. of lectures/week, T- no. of tutorials per week1
ME2353Finite Element AnalysisUNIT IFINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS1.1INTRODUCTIONThe finite element method constitutes a general tool for the numerical solution of partialdifferential equations in engineering and applied scienceThe finite element method (FEM), or finite element analysis (FEA), is based on the idea ofbuilding a complicated object with simple blocks, or, dividing a complicated object into small andmanageable pieces. Application of this simple idea can be found everywhere in everyday life as well asin engineering.Examples:Lego (kids’play) BuildingsApproximation of the area of a circle:“Element” SiiRWhy Finite Element Method? Design analysis: hand calculations, experiments, and computer simulations FEM/FEA is the most widely applied computer simulation method in engineering Closelyintegrated with CAD/CAM applications1.1.1 A Brief History of the FEM 1943 --- Courant (variational method) 1956 --- Turner, clough, martin and top(stiffness) 1960 --- Clough (finite element plan problems) 1970 --- Applications on mainframe computer 1980 --- Microcomputers, pre and post processors 1990 --- Analysis of large structural systems1.1.2 General Methods of the Finite Element Analysis1. Force Method – Internal forces are considered as the unknowns of the problem.2. Displacement or stiffness method – Displacements of the nodes are considered as theunknowns of the problem.1.1.3 General Steps of the Finite Element Analysis SCEDiscretization of structureNumbering of Nodes and ElementsSelection of Displacement function or interpolation functionDefine the material behavior by using Strain – Displacement and Stress – Strainrelationships1Department of Mechanical Engineering
ME2353Finite Element Analysis Derivation of element stiffness matrix and equationsAssemble the element equations to obtain the global or total equationsApplying boundary conditionsSolution for the unknown displacements computation of the element strains and stressesfrom the nodal displacementsInterpret the results (post processing).1.1.4 Objectives of This FEM Understand the fundamental ideas of the FEM Know the behavior and usage of each type of elements covered in this course Be able to prepare a suitable FE model for given problems Can interpret and evaluate the quality of the results (know the physics of the problems) Be aware of the limitations of the FEM (don’t misuse the FEM - a numerical tool)1.1.5 Applications of FEM in Engineering Mechanical/Aerospace/Civil/Automobile Engineering Structure analysis(static/dynamic, linear/nonlinear) Thermal/fluid flows Electromagnetics Geomechanics Biomechanics1.2 WEIGHTED RESIDUAL METHODIt is a powerful approximate procedure applicable to several problems. For non – structuralproblems, the method of weighted residuals becomes very useful. It has many types. The popularfour methods are,1. Point collocation method,Residuals are set to zero at n different locations X i, and the weighting function wiis denoted as (x - xi). (x xi) R (x; a1, a2, a3 an) dx 02. Subdomain collocation method3. Least square method, [R (x; a1, a2, a3 an)]2 dx minimum.4. Galerkin’s method. wi Ni (x) Ni (x) [R (x; a1, a2, a3 an)]2 dx 0,i 1, 2, 3, n.Problem IFind the solution for the following differential equation.SCE2Department of Mechanical Engineering
ME2353Finite Element Analysis qo 0EIThe boundary conditions areu(0) 0,(L) 0,(0) 0,(L) 0,Given: The governing differential equationEI qo 0Solution: assume a trial functionLet u(x) a0 a1x a2x2 a3x3 a4x4 .Apply 1st boundary conditionx 0, u(x) 00 a0 0a0 0Apply 2nd boundary conditionx 0, 0a1 0Apply 3rd boundary conditionx L, 0a2 -[3a3L 6a4L2]Apply 4th boundary conditionx L, 0a3 -4a4LSubstitute a0, a1, a2 and a3values in trial functionu(x) 0 0-[3a3L 6a4L2] -4a4Lu(x) a4[6 L2x2-4 Lx3 x4] a4[6 L2 (2x)-12 Lx2 4x3] 24 a4R EI qo 0a4 Substitute a4values in u(x)u(x) [x4-4Lx3 6L2x2]Result:Final solution u(x) [x4-4Lx3 6L2x2]Problem 2The differential equation of a physical phenomenon is given bySCE3Department of Mechanical Engineering
ME2353Finite Element Analysis 4, 0 1The boundary conditions are: y(0) 0y(1) 1Obtain one term approximate solution by using galerkin methodSolution:Here the boundary conditions are not homogeneous so we assume a trial function as,y a1x(x-1) xfirst we have to verify whether the trial function satisfies the boundary condition or noty a1x(x-1) xwhen x 0, y 0x 1, y 1Resuldual R:Y a1x(x-1) x a1(x2-x) x a1(2x-1) 1 21Substitutevalue in given differential equation.2a1 y 4xSubstitute y vlueR 2a1 a1x(x-1) x-4xIn galerkin’s method Substitute wi and R value in equationa1 0.83So one of the approximate solution is, y 0.83x(x-1) x 0.83x2-0.83x xy 0.83 x2 0.17xProblem 3Find the deflection at the center of a simply supported beam of span length l subjected touniform distributed load throughout its length as shown using (a) point collection method (b) Subdomain method (c)least squared and (d) galerkin’n method.Solution:EI- 0, 0 The boundary condition are y 0, x 0and y EI 0 at x 0 and x Where, EILet us select the trail function for deflection as,y a sin/1.3 THE GENERAL WEIGHTED RESIDUAL STATEMENTSCE4Department of Mechanical Engineering
ME2353Finite Element AnalysisAfter understanding the basic techniques and successfully solved a few problem generalweighted residual statement can be written asR dx 0 for i 1,2, .nWhere wi NiThe better result will be obtained by considering more terms in polynomial and trigonometric series.1.4 WEAK FORMULATION OF THE WEIGHTED RESIDUAL STATEMENT.The analysis in Section as applied to the model problem provides an attractive perspective to thesolution of certain partial differential equations: the solution is identified with a “point”, whichminimizes an appropriately constructed functional over an admis- sible function space. Weak(variational) forms can be made fully equivalent to respective strong forms, as evidenced in thediscussion of the weighted residual methods, under certain smoothness assumptions. However, theequivalence between weak (variational) forms and variational principles is not guaranteed: indeed, thereexists no general method of constructing functionals I [u], whose extremization recovers a desired weak (variational) form. In thissense, only certain partial differential equations are amenable to analysis and solution byvariational methods.Vainberg’s theorem provides the necessary and sufficient condition for the equivalence of aweak (variational) form to a functional extremization problem. If such equivalence holds, the functionalis referred to as a potential.Theorem (Vainberg)Consider a weak (variational) formG(u, δu) : B(u, δu) (f, δu) (q , δu)Γq 0 ,where u U , δu U0 , and f and q are independent of u. Assume that G pos- sesses aGˆateaux derivative in a neighborhood N of u, and the Gˆateaux differen- tial Dδu1 B(u, δu2) iscontinuous in u at every point of N .Then, the necessary and sufficient condition for the above weak form to be derivable from apotential in N is thatDδu1 G(u, δu2) Dδu2 G(u, δu1) ,Namely that Dδu1 G(u, δu2) be symmetric for all δu1, δu2 U0 and all u N .Preliminary to proving the above theorem, introduce the following two lemmas:Lemma 1 Show that Dv I[u] SCElim5Department of Mechanical Engineering
ME2353Finite Element AnalysisIn the above derivation, note that operations and ω 0 are not interchangeable (as theyboth refer to the same variable ω), while lim ω 0 and ω 0 are interchangeable, conditional uponsufficient smoothness of I [u].Lemma 2 (Lagrange’s formula)Let I [u] be a functional with Gateaux derivatives everywhere, and u, u δu be any pointsof U. Then,I [u δu] I [u] Dδu I [u ǫ δu]0 ǫ 1.To prove Lemma 2, fix u and u δu in U, and define function f on R asf(ω) : I[u ω δu] .It follows thatF dfdω lim ω 0f (ω ω) f (ω)lim ω ω 0I [u ω δu ω δu] I [u ω δu] ω Dδu I [u ω δu] ,Where Lemma 1 was invoked. Then, u s i n g the standard mean-value theorem ofcalculus,1.5 PIECE WISE CONTINUOUS TRIAL FUNCTIONIn weighted residual method the polynomial and trigonometric series are used as trial function.This trial function is a single composite function and it is valid over the entire solution domain thisassumed trial function solution should match closely to the exact solution of the differential equationand the boundary conditions, it is nothing but a process of curve fitting. This curve fitting is carriedout by piecewise method i.e., the more numbers of piece leads better curve fit. Piecewise method canbe explained by the following simple problem.We know that the straight line can be drawn through any two points.Let, ƒ(x) sin is the approximated function for straight line segments.One straight line segmentTwo straight line segmentOne Spring ElementxfiSCEijuiuj6fjDepartment of Mechanical Engineering
ME2353Finite Element AnalysisTwo nodes:Nodal displacements:Nodal forces:N/m, N/mm)i, jui, uj (in, m, mm)fi, fj (lb, Newton) Spring constant (stiffness):k (lb/in,Spring force-displacement relationship:LinearNonlinearFkDk F/( 0) is the force needed to produce a unit stretch.We only consider linear problems in this introductory course. Consider the equilibrium offorces for the spring.At node 1 we havefiFk(u jui )kuikujand at node j,fjFk(u jui )kuikujIn matrix form,kkkkuiujfifjor, where(element) stiffness matrixu (element nodal) displacement vectorf (element nodal) force vectorNote:That k is symmetric. Is k singular or non singular? That is, can we solve theequation? If not, why?SCE7Department of Mechanical Engineering
ME2353Finite Element AnalysisProblem 4To find the deformation of the shapeXK1u1F1K2u2F21u3F323For element 1,k1k 1 u2f1 2k1k1f2 2u3element 2,k2k2k2k2u2u3ff22where fI at node 2 F2M is the (internal) force acting on local node i of element Consider the quilibrium offorces at nodeChecking the ResultsDeformed shape of the structureBalance of the external forcesOrder of magnitudes of the numbersNotes about the Spring ElementsSuitable for stiffness analysisNot suitable for stress analysis of the spring itselfCan have spring elements with stiffness in the lateral direction,Spring elements for torsion, etc.1.6 EXAMPLES OF A BAR FINITE ELEMENTThe finite element method can be used to solve a variety of problem types inengineering, mathematics and science. The three main areas are mechanics of materials, heattransfer and fluid mechanics. The one-dimensional spring element belongs to the area ofmechanics of materials, since it deals with the displacements, deformations and stressesinvolved in a solid body subjected to external loading.SCE8Department of Mechanical Engineering
ME2353Finite Element AnalysisElement dimensionality:An element can be one-dimensional, two-dimensional or three-dimensional. A spring elementis classified as one-dimensional.Geometric shape of the elementThe geometric shape of element can be represented as a line, area, or volume. The onedimensional spring element is defined geometrically as:Spring lawThe spring is assumed to be linear. Force (f) is directly proportional to deformation (Δ) via thespring constant k, i.e.Types of degrees of freedom per nodeDegrees of freedom are displacements and/or rotations that are associated with a node. A onedimensional spring element has two translational degrees of freedom, which include, an axial(horizontal) displacement (u) at each node.Element formulationThere are various ways to mathematically formulate an element. The simplest and limitedapproach is the direct method. More mathematically complex and general approaches are energy(variation) and weighted residual methods.SCE9Department of Mechanical Engineering
ME2353Finite Element AnalysisThe direct method uses the fundamentals of equilibrium, compatibility and spring law from asophomore level mechanics of material course. We will use the direct method to formulate the onedimensional spring element because it is simple and based on a physical approach.The direct method is an excellent setting for becoming familiar with such basis concepts oflinear algebra, stiffness, degrees of freedom, etc., before using the mathematical formulationapproaches as energy or weighted residuals.AssumptionsSpring deformationThe spring law is a linear force-deformation as follows:f kΔf - Spring Force (units: force)k - Spring Constant (units: force/length)Δ - Spring Deformation (units: length)Spring Behaviour:A spring behaves the same in tension and compression.Spring Stiffness:Spring stiffness k is always positive, i.e., k 0, for a physical linear system.Nodal Force Direction:Loading is unia
UNIT-1 FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS 1.1 INTRODUCTION 1 1.1.1 A Brief History of the FEM 1 1.1.2General Methods of the Finite Element Analysis 1 1.1.3General Steps of the Finite Element Analysis 1 1.1.4 Objectives of This FEM 2 1.1.5 Applications of FEM in Engineering 2 1.2 WEIGHTED RESIDUAL METHOD 2
Finite element analysis DNV GL AS 1.7 Finite element types All calculation methods described in this class guideline are based on linear finite element analysis of three dimensional structural models. The general types of finite elements to be used in the finite element analysis are given in Table 2. Table 2 Types of finite element Type of .
Figure 3.5. Baseline finite element mesh for C-141 analysis 3-8 Figure 3.6. Baseline finite element mesh for B-727 analysis 3-9 Figure 3.7. Baseline finite element mesh for F-15 analysis 3-9 Figure 3.8. Uniform bias finite element mesh for C-141 analysis 3-14 Figure 3.9. Uniform bias finite element mesh for B-727 analysis 3-15 Figure 3.10.
1 Overview of Finite Element Method 3 1.1 Basic Concept 3 1.2 Historical Background 3 1.3 General Applicability of the Method 7 1.4 Engineering Applications of the Finite Element Method 10 1.5 General Description of the Finite Element Method 10 1.6 Comparison of Finite Element Method with Other Methods of Analysis
2.7 The solution of the finite element equation 35 2.8 Time for solution 37 2.9 The finite element software systems 37 2.9.1 Selection of the finite element softwaresystem 38 2.9.2 Training 38 2.9.3 LUSAS finite element system 39 CHAPTER 3: THEORETICAL PREDICTION OF THE DESIGN ANALYSIS OF THE HYDRAULIC PRESS MACHINE 3.1 Introduction 52
Finite Element Method Partial Differential Equations arise in the mathematical modelling of many engineering problems Analytical solution or exact solution is very complicated Alternative: Numerical Solution – Finite element method, finite difference method, finite volume method, boundary element method, discrete element method, etc. 9
3.2 Finite Element Equations 23 3.3 Stiffness Matrix of a Triangular Element 26 3.4 Assembly of the Global Equation System 27 3.5 Example of the Global Matrix Assembly 29 Problems 30 4 Finite Element Program 33 4.1 Object-oriented Approach to Finite Element Programming 33 4.2 Requirements for the Finite Element Application 34 4.2.1 Overall .
The Finite Element Method: Linear Static and Dynamic Finite Element Analysis by T. J. R. Hughes, Dover Publications, 2000 The Finite Element Method Vol. 2 Solid Mechanics by O.C. Zienkiewicz and R.L. Taylor, Oxford : Butterworth Heinemann, 2000 Institute of Structural Engineering Method of Finite Elements II 2
Nonlinear Finite Element Method Lecture Schedule 1. 10/ 4 Finite element analysis in boundary value problems and the differential equations 2. 10/18 Finite element analysis in linear elastic body 3. 10/25 Isoparametric solid element (program) 4. 11/ 1 Numerical solution and boundary condition processing for system of linear
