FINITE ELEMENT METHOD

FINITE ELEMENT METHODSubjectCode :TKM. 4204The studyload :3 scsStatus:M

MEH-TM-FTUB-AA SONIEF-2013Finite Element Analysis

MEH-TM-FTUB-AA SONIEF-2013Canexplain thebasicconcept ofthe finite element method;Canformulatetechnicalproblemsin themodel;Cansolvethe problemmodelingthestatic and dynamic conditions.Canexplain the numerical and computational methodstosolve mathematical problems numerically, andits applicationin mechanical engineering.Purpose MEH-TM-FTUB-AA SONIEF-2013Introduction (1)The basics ofthe finite element method;coordinatesystem; Definitionstrain;potentialenergyfunction;Powerand employment; strain energy; (1,2)The principle ofminimumpotential energy;calculus ofvariations;formulationvaritional;balanceformulationvarisional;elasticity;Potential energyandtensileforce; (3,4)The concept ofthe finite elementmodel;MEH1-dimensional probleminmechanics;Modeluniaxialtensionelements; (5,6,7)Modelingcontinuoussystemswithmanyelements;Assembly element; theheat flow:modelelements onheat flow,heat flowto theconvention. (9,10,11)Problem2-dimensionalFEMtriangular elements,coordinate systems,coordinate transformations3- node element assembly isoparametrik;Quadrilateral4-node element isoparametrik. (12,13,14)MEHtwo-dimensionalproblemonheat transferand heat flow. (15)Axisymetricstress analysis. (16) SubjectsMEH-TM-FTUB-AA SONIEF-2013Grandin, Hartley. Jr. Fundamentals of The Finite Element Method. Mac Millan Publishing Company.Yang, T.Y. Finite Element Structural Analysis. Prentice Hall International Series.Bathe, Klaus-Jurgen. Finite Element Procedurs. Prentice Hall International Editions.Zienkiewicz, O.C. The Finite Element Method. London: Mc.Graw-Hill.Cook, Robert D. Concepts and Aplications of Finite Element Analysis. New York: John Willey & Sons Inc.Knight, Charles E. The Finite Element Method in Mechanical Design. PWS Kent Publishing Company.Liu Yijun, 1998, Introduction Finite Element Method, University of Cincinnati.Literature MEH-TM-FTUB-AA SONIEF-2013DILAKUKAN DENGAN KULIAH dan STUDENT CENTER LEARNING (SCL) :Suatu metode pembelajaran yang berintikan pada kebutuhan peserta ajar untuk menyusun bahasan dan target pembelajaran yang akan dicapai sesuai dengan kurikula yang telah ditetapkan METODE PEMBELAJARANPERIMBANGAN NILAIQuiz &UAS: 25 %DISKUSI: 25 %PR : 15 %UTS : 20 %Tugas khusus : 15 %Koefisien Presensi terhadap NILAI TOTAL :Presensi 75 - 100 %: 100 % 50 - 74% : 75 % 25 - 49 % : 50 % < 25% : 25 %Introduction and Basic ConceptsMEH-TM-FTUB-AA SONIEF-2013

MEH-TM-FTUB-AA SONIEF-2013Physical problemChange of physical problemMathematic modelGoverned by differential equationsAssumptions onGeometryKinematicsMaterial lawLoadingBoundary conditionsEtc.Improve mathematical model Finite element solution Choice ofFinite elementsMesh densitySolution parametersRepresentation ofLoadingBoundary conditionsEtc.

Refine mesh, solution parameter etc.Assessment of accuracy of finite element solution of mathematical modelFinite element solution of mathematical modelInterpretation resultRefine analysisThe process of finite element analysis

Design improvements Structural optimizationMEH-TM-FTUB-AA SONIEF-20131.0 Computational Methods

MEH-TM-FTUB-AA SONIEF-2013Mathematical ModelsA model is a symbolic device built to simulate and predict aspects of behavior of a system.Abstraction of physical realityImplicit vs. Explicit ModellingImplicit modelling consists of using existent pieces of abstraction and fitting them into the particular situation (e.g. Using general purpose FEM programs)Explicit modelling consists of building the model from scratchI.1. IdealizationMEH-TM-FTUB-AA SONIEF-2013Finite Difference DiscretizationThe solution is discretizedStability ProblemsLoss of physical meaningFinite Element DiscretizationThe problem is discretizedPhysical meaning is conserved on elementsInterpretation and Control is easierI.2. DicretizationMEH-TM-FTUB-AA SONIEF-201313 Linear System Solution AlgorithmsGaussian EliminationFast Fourier TransformRelaxation Techniques

Error Estimation and Convergence Analysis

I.3. SolutionMEH-TM-FTUB-AA SONIEF-2013Two interpretationsPhysical Interpretation:The continous physical model is divided into finite pieces called elements and laws of nature are applied on the generic element. The results are then recombined to represent the continuum.Mathematical Interpretation:The differetional equation reppresenting the system is converted into a variational form, which is approximated by the linear combination of a finite set of trial functions.II.1 Finite Element MethodMEH-TM-FTUB-AA SONIEF-2013Elements are defined by the following properties:DimensionalityNodal PointsGeometryDegrees of FreedomNodal Forces II.2 FEM NotationMEH-TM-FTUB-AA SONIEF-2013II.4 Features of Finite Element MethodsDiscretization----- use smaller elements to represent objects with more complex geometry.

Approximation----- may not satisfy all boundary conditions exactly, but may satisfy in an integral sense.MEH-TM-FTUB-AA SONIEF-2013Discretization1-D

2-D3-D

?-D

HybridMEH-TM-FTUB-AA SONIEF-2013

ApproximationNumerical InterpolationNon-exact Boundary ConditionsMEH-TM-FTUB-AA SONIEF-2013Applications of Finite Element Methods Structural & Stress Analysis Thermal Analysis Dynamic Analysis Acoustic Analysis Electro-Magnetic Analysis Manufacturing Processes Fluid Dynamics Etc.MEH-TM-FTUB-AA SONIEF-2013Finite Difference vs. Finite Element Turn differential equations into algebraic equations!!!

MEH-TM-FTUB-AA SONIEF-2013

III.1 Element TypesMEH-TM-FTUB-AA SONIEF-2013Simple mechanical problemIntroduction of basic mechanical conceptsIntroduction of governing equationsMechanical concepts used in mathematical derivationIII. 2 Mechanical ApproachMEH-TM-FTUB-AA SONIEF-2013

III. 3 The Problem Setup

MEH-TM-FTUB-AA SONIEF-2013Hookes Law:

where

Strain Energy Density:3.1 Strain Energy

MEH-TM-FTUB-AA SONIEF-2012Integrating over the Volume of the Bar:

All quantities may depend on x.

3.2 Strain Energy (contd)MEH-TM-FTUB-AA SONIEF-20133.3 External EnergyDue to applied external loadsThe distributed load q(x)The point end load P. This can be included in q.

External Energy:

3.4 The Total Potential Energy FunctionalThe unknown strain Function u is found by minimizing the TPE functional described below:

MEH-TM-FTUB-AA SONIEF-2012A FEM model in solid mechanics can be thought of as a system of assembled springs. When a load is applied, all elements deform until all forces balance.F = KdK is dependant upon Youngs modulus and Poissons ratio, as well as the geometry.Equations from discrete elements are assembled together to form the global stiffness matrix.Deflections are obtained by solving the assembled set of linear equations.Stresses and strains are calculated from the deflections.FEM Applied to Solid Mechanics Problems

Create elementsof the beam

dxi 1dxi 2dyi 1dyi 21243Nodal displacement and forcesMEH-TM-FTUB-AA SONIEF-2013Classification of Solid-Mechanics ProblemsAnalysis of solidsStaticDynamicsBehavior of SolidsLinearNonlinearMaterialFractureGeometricLarge DisplacementInstabilityPlasticityViscoplasticityGeometricClassification of solidsSkeletal Systems1D ElementsPlates and Shells2D ElementsSolid Blocks3D ElementsTrussesCablesPipes

Plane StressPlane StrainAxisymmetricPlate BendingShells with flat elementsShells with curved elementsBrick ElementsTetrahedral ElementsGeneral Elements

ElementaryAdvancedStress StiffeningMEH-TM-FTUB-AA SONIEF-2013[K] {u} = {Fapp} + {Fth} + {Fpr} + {Fma} + {Fpl} + {Fcr} + {Fsw} + {Fld}[K]=total stiffness matrix{u}=nodal displacement{Fapp} = applied nodal force load vector{Fth}=applied element thermal load vector{Fpr}=applied element pressure load vector{Fma}=applied element body force vector{Fpl}=element plastic strain load vector{Fcr}=element creep strain load vector{Fsw}=element swelling strain load vector{Fld}=element large deflection load vectorGoverning Equation for Solid Mechanics ProblemsBasic equation for a static analysis is as follows:MEH-TM-FTUB-AA SONIEF-2013Step 1 - Discretization: The problem domain is discretized into a collection of simple shapes, or elements.Step 2 - Develop Element Equations: Developed using the physics of the problem, and typically Galerkins Method or variational principles.Step 3 - Assembly: The element equations for each element in the FEM mesh are assembled into a set of global equations that model the properties of the entire system.Step 4 - Application of Boundary Conditions: Solution cannot be obtained unless boundary conditions are applied. They reflect the known values for certain primary unknowns. Imposing the boundary conditions modifies the global equations.Step 5 - Solve for Primary Unknowns: The modified global equations are solved for the primary unknowns at the nodes.Step 6 - Calculate Derived Variables: Calculated using the nodal values of the primary variables.Six Steps in the Finite Element MethodMEH-TM-FTUB-AA SONIEF-2013Displacements DOF constraints usually specified at model boundaries to define rigid supports.Forces and Moments Concentrated loads on nodes usually specified on the model exterior.Pressures Surface loads usually specified on the model exterior. Temperatures Input at nodes to study the effect of thermal expansion or contraction.Inertia Loads Loads that affect the entire structure (ex: acceleration, rotation).Step 4: Boundary Conditions for a Solid Mechanics ProblemMEH-TM-FTUB-AA SONIEF-2013Process Flow in a Typical FEM AnalysisStartProblemDefinitionPre-processor

Reads or generates nodes and elements (ex: ANSYS)Reads or generates material property data.Reads or generates boundary conditions (loads and constraints.)Processor

Generates element shape functionsCalculates master element equations Calculates transformation matricesMaps element equations into global systemAssembles element equationsIntroduces boundary conditionsPerforms solution proceduresPost-processor

Prints or plots contours of stress components.Prints or plots contours of displacements.Evaluates and prints error bounds.Analysis anddesign decisionsStopStep 1, Step 4Step 6Steps 2, 3, 5MEH-TM-FTUB-AA SONIEF-2013