MECH593 Finite Element Methods

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MECH593 Introduction to Finite Element Methods
FEM of 1-D Problems Applications Dr. Wenjing Ye
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Bending Beam
y
Review
M
M
x
Pure bending problems
Normal strain
Normal stress
Normal stress with bending moment
Moment-curvature relationship
Flexure formula
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Bending Beam
y
Review
q(x)
x
Relationship between shear force, bending moment
and transverse load
Deflection
Sign convention
-

M
M
M
-

V
V
V
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Governing Equation and Boundary Condition

• Governing Equation

0ltxltL

• Boundary Conditions -----

Essential BCs if v or is specified at the
boundary. Natural BCs if or
is specified at the boundary.
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Weak Formulation for Beam Element

• Governing Equation

• Weighted-Integral Formulation for one element

• Weak Form from Integration-by-Parts ----- (1st
  time)

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Weak Formulation

• Weak Form from Integration-by-Parts ----- (2nd
  time)

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Weak Formulation

• Weak Form

q(x)
Q1
y(v)
Q3
Q2
Q4
x
x x2
x x1
L x2-x1
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Ritz Method for Approximation
q(x)
Q1
y(v)
Q3
Q2
Q4
x
x x2
x x1
L x2-x1
where
Let w(x) fi (x), i 1, 2, 3, 4
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Ritz Method for Approximation
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Ritz Method for Approximation
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Selection of Shape Function
The best situation is -----
Interpolation Properties
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Derivation of Shape Function for Beam Element
Local Coordinates
How to select fi???
and
where
Let
Find coefficients to satisfy the interpolation
properties.
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Derivation of Shape Function for Beam Element
How to select fi???
e.g.
Let
Similarly
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Derivation of Shape Function for Beam Element
In the global coordinates
In the local coordinates
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Shape Function and Interpolation
C0 Interpolation f is continuous at the
interface between elements. f,x is not
continuous at the interface between elements.
C1 Interpolation f and f,x are
continuous at the interface between elements.
f,xx is not continuous at the interface
between elements.
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Element Equations of 4th Order 1-D Model
q(x)
u1
y(v)
u3
u2
u4
x
x x1
L x2-x1
x x2
f4
f1
1
1
f3
f2
xx2
xx1
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Element Equations of 4th Order 1-D Model
q(x)
u1
y(v)
u3
u2
u4
x
x x2
x x1
L x2-x1
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Finite Element Analysis of 1-D Problems -
Applications
Example 1.
Governing equation
Weak form for one element
where
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Finite Element Analysis of 1-D Problems
Example 1.
Approximation function
f4
f1
f3
xx1
f2
xx2
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Finite Element Analysis of 1-D Problems
Example 1.
Finite element model
Discretization
P2 , v2
P3 , v3
P4 , v4
P1 , v1
II
III
I
M1 , q1
M3 , q3
M4 , q4
M2 , q2
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Matrix Assembly of Multiple Beam Elements
Element I
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Matrix Assembly of Multiple Beam Elements
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Solution Procedures
Apply known boundary conditions
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Solution Procedures
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Shear Resultant Bending Moment Diagram
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Plane Flame
Frame combination of bar and beam
Q1 , v1
E, A, I, L
Q3 , v2
P1 , u1
P2 , u2
Q4 , q2
Q2 , q1
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Finite Element Model of an Arbitrarily Oriented
Frame
y
q
x
y
q
x
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Finite Element Model of an Arbitrarily Oriented
Frame
local
global
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Plane Frame Analysis - Example
Beam II
Rigid Joint
F
F
Beam I
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Plane Frame Analysis
Q3 , v2
Q4 , q2
P2 , u2
P1 , u1
Q2 , q1
Q1 , v1
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Plane Frame Analysis
Q3 , v3
Q1 , v2
P1 , u2
P2 , u3
Q4 , q3
Q2 , q2
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Plane Frame Analysis
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Plane Frame Analysis
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Heat Transfer Mechanisms

• Conduction heat transfer by molecular
  agitation within a material without any motion of
  the material as a whole.
• Convection heat transfer by motion of a fluid.
• Radiation the exchange of thermal radiation
  between two or more bodies. Thermal radiation is
  the energy emitted from hot surfaces as
  electromagnetic waves.

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Heat Conduction in 1-D
Heat flux q heat transferred per unit area per
unit time (W/m2)
Governing equation
Q heat generated per unit volume per unit time
C mass heat capacity
k thermal conductivity
Steady state equation
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Thermal Convection
Newtons Law of Cooling
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Thermal Conduction in 1-D
Boundary conditions
Dirichlet BC
Natural BC
Mixed BC
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Weak Formulation of 1-D Heat Conduction(Steady
State Analysis)

• Governing Equation of 1-D Heat Conduction -----

0ltxltL

• Weighted Integral Formulation -----

• Weak Form from Integration-by-Parts -----

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Formulation for 1-D Linear Element
Let
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Formulation for 1-D Linear Element
Let w(x) fi (x), i 1, 2
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Element Equations of 1-D Linear Element
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1-D Heat Conduction - Example
A composite wall consists of three materials, as
shown in the figure below. The inside wall
temperature is 200oC and the outside air
temperature is 50oC with a convection coefficient
of h 10 W(m2.K). Find the temperature along the
composite wall.
t3
t2
t1
x
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Thermal Conduction and Convection- Fin
Objective to enhance heat transfer
Governing equation for 1-D heat transfer in thin
fin
w
t
x
dx
where
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Fin - Weak Formulation(Steady State Analysis)

• Governing Equation of 1-D Heat Conduction -----

0ltxltL

• Weighted Integral Formulation -----

• Weak Form from Integration-by-Parts -----

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Formulation for 1-D Linear Element
Let w(x) fi (x), i 1, 2
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Element Equations of 1-D Linear Element