written 8.1 years ago by | • modified 3.2 years ago |
Mumbai University > Mechanical Engineering > Sem 6 > Finite Element Analysis
Marks: 8M
Year: Dec 2015
written 8.1 years ago by | • modified 3.2 years ago |
Mumbai University > Mechanical Engineering > Sem 6 > Finite Element Analysis
Marks: 8M
Year: Dec 2015
written 8.1 years ago by |
Consider quadrilateral element with eight nodes,
i)2−3:ξ∴1−ξ=0ii)3−4:η=1∴1−η=0iii)8−5:y=mx+c
η=(0−(−1)−1−0)ξ+(−1)η=−ξ−1∴η+ξ+1=0
To find ϕ1, ϕ1 vanishes along lines,
Let, ϕ1=A(1−ξ)(1−η)(1+ξ+η)
At node 1: ϕ1=1,ξ=−1 and η=−1
∴1=A(2)(2)(1−1−1)A=−14∴ϕ1=−14(1−ξ)(1−η)(1+η+ξ)
Similarly,
ϕ2=−14(1+ξ)(1−η)(1+η−ξ)ϕ3=−14(1+ξ)(1+η)(1−η−ξ)ϕ4=−14(1−ξ)(1+η)(1+ξ−η)
To find ϕ5,ϕ5 vanishes along,
i)2−3:ξ=1∴1−ξ=0ii)3−4:η=1∴1−η=0iii)4−1:ξ=−1∴1+ξ=0
∴ϕ5=A(1−ξ)(1−η)(1+ξ)
At node 5, ϕ5=1,ξ=0 and η=−1
1 = A(1)(2)(1) A=12ϕ5=12(1−ξ2)(1−η)
Similarly,
ϕ6=12(1+ξ)(1−η2)ϕ7=12(1−ξ2)(1+η)ϕ8=12(1−ξ)(1−η2)