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find the shape function for two dimensional eight noded element.

Mumbai University > Mechanical Engineering > Sem 6 > Finite Element Analysis

Marks: 8M

Year: Dec 2015

1 Answer
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Consider quadrilateral element with eight nodes,

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i)23:ξ1ξ=0ii)34:η=11η=0iii)85:y=mx+c

η=(0(1)10)ξ+(1)η=ξ1η+ξ+1=0

To find ϕ1, ϕ1 vanishes along lines,

Let, ϕ1=A(1ξ)(1η)(1+ξ+η)

At node 1: ϕ1=1,ξ=1 and η=1

1=A(2)(2)(111)A=14ϕ1=14(1ξ)(1η)(1+η+ξ)

Similarly,

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

To find ϕ5,ϕ5 vanishes along,

i)23:ξ=11ξ=0ii)34:η=11η=0iii)41:ξ=11+ξ=0

ϕ5=A(1ξ)(1η)(1+ξ)

At node 5, ϕ5=1,ξ=0 and η=1

1 = A(1)(2)(1) A=12ϕ5=12(1ξ2)(1η)

Similarly,

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

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