13 Tavio&Hemawan Sdh Okey
18
Chapter 26 Page 1 G. Subbarayan ELEMENT SHAPE FUNCTIONS Quadratic elements Shape functions of one and two-dimensional elements
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Transcript of 13 Tavio&Hemawan Sdh Okey
Chapter 26Page 1
G. Subbarayan
ELEMENT SHAPE FUNCTIONS
Quadratic elementsShape functions of one and two-dimensional elements
Chapter 26Page 2
G. Subbarayan
Local Node Numbers
When number of nodes in element increases, need alternative notation for nodes– I,j… no longer convenient
Element node numbers are called local node numbers
Chapter 26Page 3
G. Subbarayan
Chapter 26Page 4
G. Subbarayan
Global Node Numbers
Local numbers and global numbers are related
Global #e
2365245873
12541
Chapter 26Page 5
G. Subbarayan
The equation for variable in each element is:
Superscript = element #Subscript = node #
( )1 (1) (1) (1) (1)1 4 2 5 3 2 4 1N N N Nφ = Φ + Φ + Φ + Φ
Chapter 26Page 6
G. Subbarayan
Evaluating Shape Functions
Earlier we started with:
To obtain:
Difficult to apply when number of nodes increases!
1 2a a xφ = +
ii j
x XXj xL L
φ −− ⎛ ⎞⎛ ⎞= Φ + Φ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
Chapter 26Page 7
G. Subbarayan
Alternative procedure is to assume:
Fβ is zero at specific nodes/sidesGβ contains unknown coefficients evaluated by requiring shape function to be one at its own node
N F Gβ β β=
Chapter 26Page 8
G. Subbarayan
One-Dimensional Quadratic Element
2ξ( )
11ξ + ( )
31ξ −
( )
( )( )
( )
1
1
2
3
111 12
1 1
12
at
Similarly,
N C
N C
N
N
ξ ξ
ξ
ξ ξ
ξ ξ
= −
= = − ⇒ =
= − + −
= +
Chapter 26Page 9
G. Subbarayan
Quadratic Triangular Elements
The interpolation is:
2 21 2 3 4 5 6
( )( )
( )( )
this is of the form
Therefore,
a a x a y a x a xy a y
C L d L d
N C L d L d
α α δ δ
β α α δ δ
φ
φ
= + + + + +
= − −
= − −
Chapter 26Page 10
G. Subbarayan
Chapter 26Page 11
G. Subbarayan
11 1 1 2
2 1 2
1 2
( 0)( )2
( 0)( 0)4
Similarly,
N C L LC
N C L LL L
= − −
⇒ =
= − −=
Chapter 26Page 12
G. Subbarayan
Chapter 26Page 13
G. Subbarayan
Quadrilateral Elements
Linear quadrilateral element
Chapter 26Page 14
G. Subbarayan
11 4
12 4
13 4
14 4
(1 )(1 )(1 )(1 )(1 )(1 )(1 )(1 )
NNNN
ξ ηξ ηξ ηξ η
= − −
= + −
= + +
= − +
Chapter 26Page 15
G. Subbarayan
Lagrangian Element
9-noded element
Chapter 26Page 16
G. Subbarayan
Chapter 26Page 17
G. Subbarayan
Eight-Node Quadratic Element
Chapter 26Page 18
G. Subbarayan
