Ch 1 Introduction 09
-
Upload
firaz-kaotz -
Category
Documents
-
view
227 -
download
0
Transcript of Ch 1 Introduction 09
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 1/14
1
MS5019 – FEM 1
Hosted by:
I Wayan Suweca
MS5019 – FEM 2
Design
need
Design
specifications
Feasibility study with
collecting design information
Design
documentation
Design
evaluation
Design
analysis,
optimization
DesignAnalysis
model
Design
concept-
ualization
Process
planningProduction
Quality
controlPackaging Shipping
Marketing
Production
planning
Design and
procurement
of new tools
Order
materials
NC/CNC/
DNC Pro-
gramming
Design Process
Manufacturing Process
Synthesis
Analysis
CAD + CAE
CAM
Product Life Cycle
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 2/14
2
MS5019 – FEM 3
1.1. Introduction
The Finite Element Method (FEM) is an versatile
and powerful mathematical tool that has wide
applications in a multitude of physical problems
such as stress analysis, fluid flow, heat transfer,
acoustics, aero-elasticity, micro-fluidics, MEMS
(Micro-Electro-Mechanical Systems), electricaland magnetic fields, electrostatic coupling and
many others.
MS5019 – FEM 4
A. Formal Definition of FEA:
An approximate mathematical analysis tool to study
the behavior of a continua (or a system) to an external
influence such as stress or strain, heat, pressure,temperature, fluid velocity, magnetic field, etc.
This involves generating a mathematical formulation
of the physical process followed by a numerical
solution of the mathematics model.
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 3/14
3
MS5019 – FEM 5
B. History of FEA:
− Force method
− Displacement method
1955
1956
Argyrys-Denke
Argyris-Turner
Modern FEM
1953Levy & Garvey Matrix method:
Force method in aircraft industry
1940Courant Approximation by “finite elements”
1908
1915
Ritz
Galerkin
Approximation method
1864
1878
Maxwell
Castigliano
Energy theorem
1819 Navier Hyper-static structure
Figure 1-1(a) Historical background to modern FEM, after J.F. Imbert [2]
MS5019 – FEM 6
Engineers Mathematicians
Trial functions Finite differences
Variational
methods
Weighted
residualsRayleigh 1870
Ritz 1909
Gauss 1795
Galerkin 1915
Biezeno-Koch 1923
Richardson 1910
Liebman 1918
Southwell 1940
Structural analogue
substitution
Piecewise continous
trial functionCourant 1943
Prager-Synge 1947Hrenikoff 1941
McHenry 1943
Newmark 1949
Direct continuum
elements
Variational finite
differencesArgyris 1955
Turner et al1. 1956
Varga 1962
Modern FEM
Figure 1-1(b) Historical background to modern FEM, after O.C. Zienkiewics [3]
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 4/14
4
MS5019 – FEM 7
C. Basic Concept:
Division of a given domain into a set of simple sub-
domains called finite elements accompanied with
polynomial approximations of solution over each
element in terms of nodal values.
Assembly of element equation with inter-element
continuity of solution and balance of force isconsidered.
MS5019 – FEM 8
1.2. Basic Illustration
1. FE Discretization• Each line segment is an element
• Collection of these line segments is called a “mesh”
• Element are connected at nodes
2. Element equations
R
22 sin( )e θ H R=
R
θ
S eA. Circumference:
Rθ
H e
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 5/14
5
MS5019 – FEM 9
3. Assembly of equations and solution
1
2For , 2 sin( ), 2 sin( )
n
e
e
e
P H
π π π θ H R P nR
n n n
=
=
= = =
∑
0 0
As , 2
1 sin( )If 2
0
sin( ) cos( )2 2 2
1lim lim x x
n P π R
π x x P R
n x
n x
π x π x R π R π R
x→ →
• → ∞ =
• = ⇒ =
• → ∞ ⇒ →
⎡ ⎤ ⎢ ⎥⎛ ⎞ ⎛ ⎞• ∴ = =⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥
⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦
( )
Error, 2 sin
Total Error 2
e e e
e
π π E S H R
n n
nE π R P
⎡ ⎤⎛ ⎞= − = − ⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
= = −
4. Assembly of equations and solution
5. Error Estimation
1,03354E-076,2831910000
1,03354E-056,283171000
0,0010334926,28215100
0,102845426,1803410
6,2831853072,5E-161
nEePn
MS5019 – FEM 10
B. Frame Structure:
Figure 1-2 Example of discretization of a frame structure by FEM
(a) Real structure (b) Discretized structure
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 6/14
6
MS5019 – FEM 11
C. Continuous problem:
Figure 1-3 Descritization of an elasticity 2D continuous problem by FEM
(a) Continuous problem
(b) Discrete model
MS5019 – FEM 12
1.3. General Step in the FEM
Derive the element stiffness
matrix and equations
Define the strain-
displacement and stress-
strain relationship
Select a displacement
function
Discretize and Select
Element Types
Based on the concept of stiffness influence
coefficients (direct equilibrium method, work or
energy method, weighted residual method.)
Step 4
Both relationships are necessary for deriving the
equations for each element.
Step 3
Choosing a displacement function within each
element
Step 2
Dividing the body into an equivalent system of
finite elements with associated nodes and
choosing the most appropriate element type.
Step 1
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 7/14
7
MS5019 – FEM 13
Interpret the results
Solve for the element strains
and stresses
Solve for the unknown
degrees of freedom (or
generalized displacements)
Assemble the elementequations to obtain the
global equations and
introduce boundary
conditions
The final goal is to interpret and analyse the
results for use in the design/analysis process.
Step 8
For the structural stress-analysis problem, strains
and stress (or moment and force) can be
obtained.
Step 7
Global equations obtained from step 5 is a set of
simultaneous algebric equations. These
equations can be solved by using an elimination
method (Gauss’s method) or an iterative method
(Gauss-Seidel, etc.)
Step 6
Individual element equations generated in step 4is added together using a method of
superposition (called the direct stiffness
method ).
Step 5
MS5019 – FEM 14
n : total number of nodes
Element
Global
Level of
FormulationNodal
Displ.
Defor.
Energy
Work of
Ext. forcesStiffness
Matrix
Nodal
Forces
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
=
k
j
i
d
d
d
d
⎪⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪
⎨
⎧
⋅
⋅
=
n
k
j
i
d
d
d
d
d
d
M
M
1
eeeeU dKd
T
21= eee Fd
T
=ℑ
Virtual Work Principle
FddKd
U
d
TT
0
δ δ
δ δ
δ
=
ℑ=
≠∀
eK eF
AssemblageK F
Linear Equation
System
FdK =
dSolution
ei
k
j
ei
k
j u j
v j
⎭⎬⎫
⎩⎨⎧
= j
j
j v
ud
KddT U 21= FdT =ℑ
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 8/14
8
MS5019 – FEM 15
Pointelement
0D
Frameelement
Trusselement1D
(LineEle-
ment)
GeometryNameClass
Figure 1-4 (a) Different type of elements
MS5019 – FEM 16
Coque
Bendingplate
Elasticity 2D(tin) shell
2D(PlanEle-
ment)
GeometryNameClass
Figure 1-4 (b) Different type of elements
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 9/14
9
MS5019 – FEM 17
Coqueaxisymetric
Torusaxisymetric
Axi-symet
ric
GeometryNameClass
Figure 1-4 (c) Different type of elements
MS5019 – FEM 18
Gap element that have stiffnessonly for compression direction.
Special Element
Thick Coque
Volume3D
(Volu
meEle-ment)
GeometryNameClass
Figure 1-4 (d) Different type of elements
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 10/14
10
MS5019 – FEM 19
1.4. Analysis Type
Non-linear dynamic
Direct integration step by step
Modal
Dynamic response
− Modal superposition
− Direct integration step by step
Dynamic
Static Non-linear
Non-linear stability
Linear static
Initial stabilityStatic
Non-linearLinear Analysis
FKq =
[ ]G λ+ =K K X F
[ ]
2, λ λ ω− = =K M X 0
( )t + + =Mq Cq Kq F&& &
MS5019 – FEM 20
1.5. Computer Code
Start
Input Data
FE modeling
Element
Characteristics
Ke, Fe
• Assemblage
• Restraints
K, F
Solution LES
q
Element’s stress
calculation
End
Sub program for
matrix calculation
Element’s
Library
Print Result
Figure 1-7
Simplified flowchart
for static analysis
(displacement method)
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 11/14
11
MS5019 – FEM 21
1.6. Application Structural areas:
Stress analysis, including truss and frame analysis both for
structural and non-structural concentration problems typically
associated with holes, fillets, or other changes in geometry in
a body.
Buckling problem
Vibration analysis
Non-structural problems: Heat transfer
Fluid flow, including seepage thtough porous media
Distribution of electric or magnetic potemtial
MS5019 – FEM 22
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 12/14
12
MS5019 – FEM 23
MS5019 – FEM 24
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 13/14
13
MS5019 – FEM 25
MS5019 – FEM 26
7/23/2019 Ch 1 Introduction 09
http://slidepdf.com/reader/full/ch-1-introduction-09 14/14
14
MS5019 – FEM 27
MS5019 – FEM 28
References:
1. Logan, D.L., 1992, A First Course in the Finite ElementMethod, PWS-KENT Publishing Co., Boston.
2. Imbert, J.F.,1984, Analyse des Structures par
Elements Finis , 2nd Ed., Cepadues.3. Zienkiewics, O.C., 1977, The Finite Eelement Method,
3rd ed., McGraw-Hill, London.