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Derive the shape function in natural co-ordinate system for eight nodded quadrilateral element.
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Serendipity elements:
They are the rectangulars element which have no interior nodes i.e all nodes lie on the boundary of element only.

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Shape Function for eight noded element.
Considder a quadrilateral element with eight node as shown in Fig.

Now to determine ϕ1 we note that ϕ1 vanishes along lines.
i) 2-3 i.e 1ξ=0
ii) 3-4 i.e 1η=0
iii) 8-5 i.e ξη=1 or 1+ξ+η=0

Let, ϕ1=A(1ξ)(1η)(1+ξ+η)
At node 1 ϕ1=1ξ1=1η=1

1=A(1+1)(1+1)(111)
1=4A
A=14

ϕ1=14(1ξ)(1η)(1+ξ+η)

Similarly,
ϕ2=14(1+ξ)(1η)(1ξ+η)
ϕ3=14(1+ξ)(1+η)(1ξη)
ϕ4=14(1ξ)(1+η)(1+ξη)

To find node ϕ5,ϕ5 vanishes along
i) 2-3 i.e 1ξ=0
ii) 3-4 i.e 1η=0
iii) 4-1 i.e 1+ξ=0

Let, ϕ5=A(1ξ)(1η)(1+ξ)

At node 5 ϕ5=1ξ=0η=1
1=A(1)(1+1)(1)
A=12

ϕ5=12(1ξ2)(1η)

Similarly,
ϕ6=12(1+ξ)(1η2)
ϕ7=12(1ξ2)(1+η)
ϕ8=12(1ξ)(1η2)

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