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Finite Element AnalysisIntroduction

GE393 Computer-Aided Design, Analysis and Prototyping

FEA Introduction Numerical method used for solving problems

that cannot be solved analytically (e.g., due to complicated geometry, different materials) Well suited to computers Originally applied to problems in solid mechanics Other application areas include heat transfer, fluid flow, electromagnetism

Finite Element Method Phases Preprocessing Geometry Modeling analysis type Mesh Material properties Boundary conditions Solution Solve linear or nonlinear algebraic equations simultaneously to obtain nodal results (displacements, temperatures) Postprocessing Obtain other results (stresses, heat fluxes)

FEA Discretization Process - Meshing Continuous elastic structure

(geometric continuum) divided into small (but finite), well-defined substructures, called elements Elements are connected together at nodes; nodes have degrees of freedom Discretization process known as meshing

Spring Analogy Elements modeled as linear springs

F (l W ! , I ! , W ! EI A l EA F ! (l , similar to F ! kx l

Matrix Formulation Local elastic behavior of each element

defined in matrix form in terms of loading, displacement, and stiffness

Stiffness determined by geometry and material properties (AE/l)

Global Matrix Formulation

Elements assembled through common nodes

into a global matrix Global boundary conditions (loads and supports) applied to nodes (in practice, applied to underlying geometry) F1 K1 K 2 F ! K 2 2 K 2 U 1 K 2 U 2

Solution Matrix operations used to determine unknown

dofs (e.g., nodal displacements) Run time proportional to #nodes/elements Error messages

Bad elements Insufficient disk space, RAM Insufficiently constrained

Postprocessing Displacements used to derive strains and

stresses

FEA Prerequisites First Principles (Newtons Laws) Body under external loading Area Moments of Inertia Stress and Strain Principal stresses Stress states: bending, shear, torsion, pressure, contact, thermal expansion Stress concentration factors Material Properties Failure Modes Dynamic AnalysisSee Chapter 2 of Building Better Products with FEA, Vince Adams and Abraham Askenazi, Onward Press, 1999

A Simple FEA ModelF ! Kx F1 U1 K1 (U 2 U1 ) K 2 ! 0 F2 (U 2 U 1 ) K 2 ! 0 F1 ! ( K1 K 2 )U1 ( K 2 )U 2 F2 ! ( K 2 )U 1 K 2U 2 F1 K1 K 2 F ! K 2 2 K 2 U 1 K 2 U 2

Stiffness matrix

A Simple FEA Model - 2 DOFs - 1

Determines the # of equations needed to define the model Boundary Conditions Allows model to be solved U0 = 0 (fixed support) F1, F2 (external forces) Mesh 2 1D elements 2 nodes per element

A Simple Model - 3 Assumptions Linear spring (-> 1 DOF) Convergence Process of using smaller and smaller elements to reduce error

Finite Element Analysis

Introduction