A quadratic trial solution within the range $x=[0,L]$ may be expressed as

widehat$(u)(x)=N_1^q(x)hat(u{)}_1+N_2^q(x)hat(u{)}_2+N_3^q(x)hat(u{)}_3$

where $N_1^q,N_2^q$ and $N_3^q$ are quadratic shape functions defined as

$N_1^q(x)=1-\frac{3}{L}x+\frac{2}{L^2}{x}^2$

$N_2^q(x)=\frac{4}{L}x-\frac{4}{L^2}{x}^2$

$N_3^q(x)=-\frac{1}{L}x+\frac{2}{L^2}{x}^2$

and parameters hat$(u{)}_1,$hat$(u{)}_2$ and hat$(u{)}_3$ are the values of the trial solution hat$(u)(x)$ at $x=0,x=\frac{L}{2}$

and $x=L,$ respectively.

Derive the quadratic shape functions above. The process of derivation is similar to the $2^{nd}$

derivation of the linear shape functions shown in class.