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

Marks: 5M

Year: Dec 2016

derivetion of shape function for 1-d linear element ;

let $\phi =A(\xi+1) (\xi - 1)$

At node 1, from (\xi+1) vanishen

$\hspace{2cm}\therefore \phi = A(\xi-1)$

At node :$\xi = - 1 and \phi_1=1$

$\hspace{2cm}\therefore 1=A(-2)$

$\hspace{3cm} \therefore A=\frac{-1}{2}$

$\hspace{2cm}\therefore \phi_1=\frac{-1}{2}(\xi-1)=\frac{1}{2}(1-\xi)$

similarly At node 2, from $(\xi-1)$ vanishen

$\hspace{2cm}\phi_2=(\xi+1)$

At node 2 : $\xi=1 and \phi_2=1$

$\hspace{4cm} A=\frac{1}{2}$

$\hspace{4cm} \therefore \phi_2=\frac{1}{2}(1+\xi)$

shope function for 1-D linear element,

$\phi_1=\frac{1}{2}(1- \xi) \hspace{1cm} \& \hspace{1cm} \phi_2=\frac{1}{2}(1+ \xi)$