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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.
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 1−ξ=0
ii) 3-4 i.e 1−η=0
iii) 8-5 i.e −ξ−η=1 or 1+ξ+η=0
Let, ϕ1=A(1−ξ)(1−η)(1+ξ+η)
At node 1 ϕ1=1ξ1=−1η=−1
∴1=A(1+1)(1+1)(1−1−1)
∴1=−4A
∴A=−14
∴ϕ1=−14(1−ξ)(1−η)(1+ξ+η)
Similarly,
ϕ2=−14(1+ξ)(1−η)(1−ξ+η)
ϕ3=−14(1+ξ)(1+η)(1−ξ−η)
ϕ4=−14(1−ξ)(1+η)(1+ξ−η)
To find node ϕ5,ϕ5 vanishes along
i) 2-3 i.e 1−ξ=0
ii) 3-4 i.e 1−η=0
iii) 4-1 i.e 1+ξ=0
Let, ϕ5=A(1−ξ)(1−η)(1+ξ)
At node 5 ϕ5=1ξ=0η=−1
∴1=A(1)(1+1)(1)
∴A=12
∴ϕ5=12(1−ξ2)(1−η)
Similarly,
ϕ6=12(1+ξ)(1−η2)
ϕ7=12(1−ξ2)(1+η)
ϕ8=12(1−ξ)(1−η2)