Finite Element Analysis

MEEN 5330Dustin GrantKamlesh BorgaonkarVarsha MaddelaRupakkumar PatelSandeep Yarlagadda

Introduction

What is finite element analysis, FEA?

What is FEA used for?

1D Rod Elements, 2D Trusses

Basic Concepts

Loads

Equilibrium

Boundary conditions

fT

iP

0~

, ijji f

Development of Theory

Rayleigh-Ritz MethodTotal 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 vectorsElement body load vectorElement 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

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

2D Truss

2 DOF

Transformations

Modified Stiffness Matrix

Methods of Solving

2D Truss

Transformation MatrixDirection 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 ApproachEliminate Constraints

Penalty ApproachWill not discuss Today

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 TableConnectivity TableDirection Cosines Table

2D Truss

Coordinate Table

2D Truss

Connectivity Table

2D Truss

Direction Cosines Table

2

1 22

1 2y y x x le

el

x xl

1 2cos

el

y ym

1 2sin

Example 2D Truss

MATLAB Program TRUSS2D.M

3D Truss Stiffness Matrix

3D Transformation MatrixDirection 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

Conclusion

Good at Hand Calculations, Powerful when applied to computers

Only limitations are the computer limitations

References

Homework