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

, ,

, similar to

F lE

A lEA

F l F kxl

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)

1 1 2 2 1

2 2 2 2

F K K K U

F K K U

Solution

Matrix operations used to determine unknown dof’s (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 (Newton’s 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 Analysis

See Chapter 2 of Building Better Products with FEA, Vince Adams and Abraham Askenazi, Onward Press, 1999

A Simple FEA Model

2

1

22

221

2

1

22122

221211

2122

212111

)(

)()(

0)(

0)(

U

U

KK

KKK

F

F

UKUKF

UKUKKF

KUUF

KUUKUF

KxF

Stiffness matrix

A Simple FEA Model - 2

DOF’s - 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