ME 4120/6120:

Finite Element Analysis

Dr. Ha-Rok Bae

Up-front virtual design validation in VPD

Industrial Experience: Computational Mechanics

Champaign

Simulation Center

Production Build Test Virtual Validation Design

Machine performance Casting simulation Stress analysis CFD analysis Forming Analysis Concept Design

Field Test

Picture from

http://machinedesi

gn.com Virtual Vibration

w/ FE model

Life Prediction Lab Vibration Test (Fatigue Test)

Excitation input

Virtual Field Test

Structural Design Validation Road Map

Concept

Designs

Generic process for

Automotive and aerospace

industries

Production Build Test Design Simulation based

Design Exploration

Advanced VPD

Using design optimization and uncertainty

quantification technologies

System Model

Subsystem simulations

Multi-physics Multi-body

dynamic simulation

Simulink

Stateflow

ADAMS

Nastran

Abaqus

Fe-safe

iSIGHT

Dakota

Nessus

Python

Matlab-OPT

Hyperworks

Fluent

Excel+VBA

Etc.

Virtual model Verification &

Validation (V&V) study

Aerospace

Automotive

Heavy machinery

Civil infra-structures

Marine

Medical

Energy (offshore Petroleum, wind

energy, life-line)

FEA Applications

Course Introduction

Instructor: Dr. Ha-Rok Bae, Office: 227 Russ Center,

Email: ha-rok.bae@wright.edu

Phone: 775-5089

TA: Mr. Ahmed, Sazzad hossain

Email: ahmed.65@wright.edu

Lecture: MWF 2:30pm - 3:25pm, 146 Russ Center

Office Hour: MW 1:30pm – 2:30pm other times by appointment

Course Evaluations:

10% Homework

20% ANSYS Labs + extra ANSYS problems

10% Final Project

30% Midterm Exam (Tentatively on Oct. 19th Monday)

30% Final Exam (Dec. 18th Friday, 2:45~4:45pm)

(5+)% Bonus Quizzes

Communication method:

(Mainly) Pilot News and Emails

Syllabus in Pilot!

Tentative Course Outline Segerlind, L.J., Applied Finite Element Analysis, 2nd ed., John Wiley & Sons, 1984

• Introduction and 1-D Model problem

• Variational Statement of the Problem

• Approximate Variational Methods

• Galerkin Finite Element Method; 1-D Element Shape Functions

• Element Matrices and Assembly

• Axial Force Member; Truss Element

• Beam Element

• Beam Element Continued; Plane Frame Element

• 2-D Field Problems; Linear Triangular and Rectangular Elements

• 2-D Linear Elements Continued; Application to Torsion of Noncircular

Sections

• Generalized Formulation for Theory of Elasticity

• Application to Two-Dimensional Elasticity

• Higher Order and Mapped Elements

• Numerically Integrated Elements

Course Introduction cont’d

Formulation of Boundary Value Problems

Mathematical models of physical situations

Differential equations with boundary and

initial conditions

These differential equations are derived by

applying the fundamental laws and principles of

mass, momentum, and energy balance and

conservation.

Boundary Value Problems in Engineering

Mathematical Solutions

Analytical Solution:

Direct integration of differential equations

(Example: Beam equation, 1-D Heat

equation, 1-D Navier-stokes equation for a

round pipe etc.)

Numerical Solution: Finite Element Method

(FEM) and Finite Difference Method (FDM),

Boundary Element Method etc.

Analytical Solution Requires

Simpler Geometries

Neglect Many Field Variables Components

Linear Relations

Constant material properties

Simpler Applied Loads

Simpler Boundary Conditions

Exact Solutions

What is FEA?

FEA is a computational technique used to obtain approximate solutions to complex engineering problems.

With FEA, engineers seek to calculate field quantities: displacement, stress or strain, temperature or heat flux, fluid velocity, mass concentration or flux.

This course is structured with a mix of theoretical information and hands-on practices.

Why Use FEA?

Provides a non-destructive

means of testing products.

Faster prototyping for

“what if” scenarios.

Design optimization.

Speed up time to market by

shortening the design

cycle.

FEA Solution Types

Linear Static

Elastic Buckling

Geometric Non-linearity

Material Non-linearity

Modal Analysis (Free Vibration)

Linear Dynamic

Non-linear Dynamic

Contact problems

Explicit transient (impact) load

Thermal Load (Heat Transfer) Problems

Fluid Flow Problems

FEA Can Handle Complex Problems

Multi Components Structure

Arbitrary Geometry Shapes and Sizes

Multi Axial and Large Dynamic Loading

Large Strains and Deformations

Complex Material Behavior

Complex Boundary Conditions

Best Practices

FEA requires engineering judgment. In

the best case, you should know the

approximate answer before you begin

Proper selection of elements, materials,

loads, constraints and analysis

parameters comes from experience.

Best Practices

Understand that the computer

model never matches reality (it’s

only an approximation).

The surest route to failure in FEA is

to underestimate the complexity of

the technology.

BEGIN

CHAPTER 1 – SEGERLIND

Till

Potential Energy Formulation

1.4 Finite Element Method

FEA is a mathematical solution

to engineering problems where

a physical model is divided into

discrete components.

FEA models are defined by

nodes and elements

(commonly called a mesh).

1.4 Finite Element Method

Utilizes an integral formulation to

generate a system of algebraic

equations

Uses some form of Galerkin, TPE,

Variational method etc (Chapter 1)

Uses continuous piecewise smooth

functions to approximate unknown

variables of the physical problem.

Steps Of FEA Method

1. Divide the physical domain into finite elements

(See Figure 1.3 for Beam) – Create mesh with

elements and nodes

2. Assume approximate equation for the nodal DOF

within each element

3. Use Variational, Galerkin, TPE methods etc

(Chapter 1) to develop Element stiffness matrices

and them assemble them into Global stiffness

matrix. This results in a system of algebraic

equations.

4. Solve the System of linear simultaneous equations

for nodal degrees of freedom.

5. Calculate primary and secondary field variables.

Elements/Nodes

In FEA, finite sized sub-domains are used to

resemble fragments of a structure.

These sub-domains are often called elements.

Elements are bound by their edges and vertices.

These vertices are also called nodes.

Nodes appear on element boundaries and they

connect elements together.

Node

A node is a point in space, where the

DOF is defined.

An element is a finite small domain

bound by nodes and lines connecting

them.

Elements can be lines (beams), areas

(2-D plates) or solids (bricks).

What is a DOF?

The unknowns in a finite element problem

are referred to as degrees of freedom (DOF).

Displacement, temperature, etc.

DOF is pre-defined at nodes.

DOF varies with the types of elements and

analyses.

Node

Uy

Rot x

Rot y

Uz Rot z

Ux

Nodes and Elements

For structural analyses, DOF includes translations and rotations in 3D.

The type of element being used will characterize which type of DOF a node will have.

Some analysis types have only one DOF at a node. Examples of these analysis types are temperature in a heat transfer analysis and velocity in a fluid flow analysis in a pipe.

Element Connectivity

Elements can only transfer loads to

one another via common nodes.

No Communication Between the Elements

Communication Between the Elements

Types of Finite Elements Dr. Yijun Liu’s notes

General Case

The DOF components of each element

combine to form a matrix equation:

[K(e)] {u(e)} = {A(e)}

[K(e)] = element stiffness components

{u(e)} = DOF results (unknown)

{A(e)} = action value (e.g., force,

temperature)

Structural FEA Equation

To determine the displacement of

a simple linear spring under load,

the relevant equation is:

[K] {u} = {f} Known

Unknown

where [K] = (Global) stiffness matrix

{u} = displacement vector

{f} = force vector

FEA Equation Solution

This can be solved with

matrix algebra by

rearranging the equation as

follows:

{u} = [K] {f} -1

Dynamic Equation

For a more complex analysis, more terms are needed. This is true in a dynamic analysis, which is defined by the following equation:

{f} = [K] {u} + [c] {v} + [m] {a} where {f} = force vector

[K] = stiffness matrix

{u} = displacement vector

[c] = damping matrix

{v} = velocity vector

[m] = mass matrix

{a} = acceleration vector

Spring Element Dr. Yijun Liu’s notes

Spring Element Dr. Yijun Liu’s notes

Spring Element Dr. Yijun Liu’s notes