Problem # 3.16:
Determine the shape function of a 3-D simple element:
ɸ = a1+a2x+a3y+a4z
ɳT= { 1 x y z }
[N] = [Ni Nj Nk Nl]
[NT] = { Ni Nj Nk Nl}
Ni
[N] = Nj
Nk
Nl
Ni
ʃNNtds = ʃʃʃvNNTdV = ʃʃʃv Nj [Ni Nj Nk Nl] dV
Nk
Nl
ʃʃʃv(Ni
2 Nj
2 Nk
2 Nl
2)dV
As Ni=L1
Nj = L2
Nk = L3
Nl = L4
ʃʃʃv L1
2 L2
2 L3
2 L4
2dV
According to Equation 3.81
ʃʃʃv L1
2 L2
2 L3
2 L4
2dV
= α!β!ϒ!δ!*6V/(α+β+ϒ+δ+3)!
Putting the above values we have 2!2!2!2!*6V/(2+2+2+2+3)!
Ans = 96V/11!
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