Coordinates of the nodes of finite element are given by P (4.0) and Q (5.0) Find the expression of x in terms of $\xi$ when :

1) Third node R is taken at (6.0)

2) Third node R is taken at (5.0)

Comment on the result.

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

Marks: 10M

Year: Dec 2016

Quadratic element with nodes 3.

$\hspace{1cm}x=\phi, x_1+\phi_2x_2+ \phi_3 x_3 $

where, $\phi,=\frac{1}{2}\xi(\xi-1), \phi_2=\frac{1}{2}\xi (\xi+1),\phi=(1-\xi)(1+\xi)$

$x=\frac{1}{2}\xi(\xi-1)x_1+\frac{1}{2}\xi(\xi+1)x_2+(1-\xi)(1+\xi)x_3$

$\hspace{0.4cm}=\frac{1}{2}\xi(\xi-1)4+\frac{1}{2}\xi(\xi+1)8+(1-\xi^2)6$

$\hspace{0.4cm}= 2 \xi^2-2\xi +4 \xi^2 +4\xi +6-6\xi^2$

$\hspace{0.4cm}=2\xi+6$

(ii)

$x=\frac{1}{2}\xi(\xi-1)4+\frac{1}{2}\xi(\xi+1)8+(1-\xi^2)5$

$\hspace{0.4cm}= 2 \xi^2-2\xi +4 \xi^2 +4\xi +5-5\xi^2$

$\hspace{0.4cm}=\xi^2+2\xi+5$

when c taken at midpoint of the element then transformation becomes linear but when c is taken away from midpoint the transformation becomes non linear.