FINITE ELEMENT METHOD

INTRODUCTION

What is finite element analysis, FEM?

A Brief history of FEM

What is FEM used for?

1D Rod Elements, 2D Trusses

FINITE ELEMENT METHOD – WHAT IS IT?

The Finite Element Method (FEM) is a numerical method of solving systems of partial differential equations (PDEs)

It reduces a PDE system to a system of algebraic equations that can be solved using traditional linear algebra techniques.

 In simple terms, FEM is a method for dividing up a very complicated problem into small elements that can be solved in relation to each other.

MGWS

Overview of the Finite Element Method

Strong

form

Weak

form

Galerkin

approx.

Matrix

form

1. Lord John William Strutt Rayleigh (late 1800s), developed a

method for predicting the first natural frequency of simple

structures. It assumed a deformed shape for a structure and

then quantified this shape by minimizing the distributed energy

in the structure.

2. Ritz then exp Walter ended this into a method, now known as

the Rayleigh-Ritz method, for predicting the stress and

displacement behavior of structures.

6

3. Dr. Ray Clough coined the term “finite element” in 1960. The 1960s saw

the true beginning of commercial FEA as digital computers replaced analog

ones with the capability of thousands of operations per second.

4. In the 1950s, a team form Boeing demonstrated that complex surfaces

could be analyzed with a matrix of triangular shapes.

5. In 1943, Richard Courant proposed breaking a continuous system into

triangular segments. (The unveiling of ENIAC at the University of

Pennsylvania.)6. In the early 1960s, the MacNeal-Schwendle Corporation (MSC) develop a

general purpose FEA code. This original code had a limit of 68,000

degrees of freedom. When the NASA contract was complete, MSC

continued development of its own version called MSC/NASTRAN, while the

original NASTRAN become available to the public and formed the basis of

dozens of the FEA packages available today. Around the time

MSC/NASTRAN was released, ANSYS, MARC, and SAP were introduced.

7

8. standards such as IGES and DXF. Permitted limited geometry transfer

between the systems.

9. In the 1980s,CAD progressed from a 2D drafting tool to a 3D surfacing tool,

and then to a 3D sIn the 1980s, the use of FEA and CAD on the same

workstation with developing geometry olid modeling system. Design

engineers began to seriously consider incorporating FEA into the general

product design process.

10. As the 1990s draw to a place, the PC platform has become a major force in

high end analysis. The technology has become to accessible that it is

actually being “hidden” inside CAD packages.

7. By the 1970s, Computer-aided design, or CAD, was introduced later in the decade.

BASIC CONCEPTS

Loads

Equilibrium

Boundary conditions

fT

iP

0~

, ijji f

DEVELOPMENT OF THEORY

Rayleigh-Ritz Method Total potential energy equation

Galerkin’s Method

1D ROD ELEMENTS

To understand and solve 2D and 3D problems we must understand basic of 1D problems.

Analysis of 1D rod elements can be done using Rayleigh-Ritz and Galerkin’s method.

To solve FEA problems same are modified in the

Potential-Energy approach and Galerkin’s approach

1D ROD ELEMENTS

Loading consists of three types : body force f , traction force T, point load Pi

Body force: distributed force , acting on every elemental volume of body i.e. self weight of body.

Traction force: distributed force , acting on surface of body i.e. frictional resistance, viscous drag and surface shear

Point load: a force acting on any single point of element

1D ROD ELEMENTS

Element strain energy

Element stiffness matrix

Load vectors Element body load vector Element traction-force vector

qkqU eTe

][

2

1

11

11][

e

eee

l

AEk

1

1

2

flAf eee

1

1

2ee Tl

T

Element -1 Element-2

2D TRUSS

2 DOF

Transformations

Modified Stiffness Matrix

Methods of Solving

2D TRUSS

Transformation Matrix Direction Cosines

ml

mlL

00

00][

212

212 yyxxle

el

xxl 12cos

el

yym 12sin

2D TRUSS

Element Stiffness Matrix

22

22

22

22

][

mlmmlm

lmllml

mlmmlm

lmllml

l

AEk

e

eee

METHODS OF SOLVING

Elimination Approach Eliminate Constraints

Penalty Approach

ELIMINATION METHOD

Set defection at the constraint to equal zero

ELIMINATION METHOD

Modified Equation DOF’s 1,2,4,7,8 equal to zero

2D TRUSS

Element Stresses

Element Reaction Forces

qmlmll

E

e

e

QKR

2D TRUSS

Development of Tables

Coordinate Table Connectivity Table Direction Cosines Table

2D TRUSS

Coordinate Table E.g;

2D TRUSS

Connectivity Table E.g;

2D TRUSS

212

212 yyxxle

el

xxl 12cos

el

yym 12sin

3D TRUSS STIFFNESS MATRIX

3D Transformation Matrix Direction Cosines

nml

nmlL

000

000][

212

212

212 zzyyxxle

el

xxl 12cos

el

yym 12cos

el

zzn 12cos

3D TRUSS STIFFNESS MATRIX

3D Stiffness Matrix

22

22

22

22

22

22

][

nmnlnnmnln

mnmlmmnmlm

lnlmllnlml

nmnlnnmnln

mnmlmmnmlm

lnlmllnlml

l

AEk

e

eee

EXAMPLE 1D ROD ELEMENTSExample 1Problem statement: (Problem 3.1 from Chandrupatla and Belegunda’s book)Consider the bar in Fig.1, determine the following by hand calculation: 1) Displacement at point P 2) Strain and stress

3) Element stiffness matrix 4) strain energy in element

21.2eA in

630 10E psi 1 0.02q in

2 0.025q in

Given:

Solution:

1) Displacement (q) at point P

We have

12 1

2( ) 1

( )

2(20 15) 1 0.25

(23 15)

x xx x

Now linear shape functions N1( ) and N2( ) are given by

1

1( ) 0.375

2N

And 2

1( ) 0.625

2N

EXAMPLE 2D TRUSS

CONCLUSION

Good at Hand Calculations, Powerful when applied to computers

Only limitations are the computer limitations

MATLAB PROGRAM TRUSS2D.M

REFERENCES