Course
Year
: S0912 - Introduction to Finite Element Method
: 2010
2D OF FINITE ELEMENT METHOD
Session 7 – 9
COURSE 5
Content:
• 2D Element Types
• 2D Element Modelling
• 2D Solution
• Example/Case Study
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2D ELEMENT TYPES
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2D ELEMENT MODELLING
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2D SOLUTION
Triangle Linear Plane Element
General Equation
Detail Equation
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2D SOLUTION
Solution
Element Strain
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2D SOLUTION
Strain – Displacement Relation
N : Shape Function
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2D ELEMENT EXAMPLE
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2D ELEMENT EXAMPLE
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2D ELEMENT EXAMPLE
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2D ELEMENT EXAMPLE
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2D ELEMENT EXAMPLE
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2D Truss
• 2 DOF
• Transformations
• Modified Stiffness Matrix
• Methods of Solving
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2D Truss
• Transformation Matrix
– Direction Cosines
le 
l m 0 0 
[ L]  

0
0
l
m


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x2  x1 2   y2  y1 2
x2  x1
l  cos  
le
y 2  y1
m  sin  
le
2D Truss
• Element Stiffness Matrix
 l2
lm

m2
E e Ae  lm
[k e ] 
l e   l 2  lm

2
 lm  m
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 lm 
2
 lm  m 
l2
lm 
2 
lm
m 
l2
Methods of Solving
• Elimination Approach
– Eliminate Constraints
• Penalty Approach
– Will not discuss
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Elimination Method
• Set defection at the constraint to equal zero
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Elimination Method
• Modified Equation
• DOF’s 1,2,4,7,8 equal to zero
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2D Truss
• Element Stresses
Ee

   l  m l mq
le
• Element Reaction Forces
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

R  K Q
2D Truss
• Development of Tables
– Coordinate Table
– Connectivity Table
– Direction Cosines Table
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2D Truss
• Coordinate Table
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2D Truss
• Connectivity Table
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2D Truss
• Direction Cosines Table
le 
x2  x1 2   y2  y1 2
le
l  cos   2
x  x1
le
m  sin   2
y  y1
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Example 2D Truss
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MATLAB Program TRUSS2D.M
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