Question $3(30$ marks$)$

Figure $3.4-$node isoparametric element subjected to force $P.$

Figure $3$ shows a $4-$node isoparametric quadrilateral element subjected to a

force $P=1\mathrm{.}2kN.$ The angle $=25.$ Calculate the following:

a$.$ The Jacobian matrix at $(,)=(0,0)$

b$.$ The stiffness matrix of the element at node $2$ using $1-$point Gaussian

quadrature

c$.$ If the nodes $1,3$ and $4$ are constrained from moving in both $x-$ and $y-$

directions, determine the displacement at node $2$

Assume that the Young's modulus $E=250$kPa. The Poisson's ratio may be

taken as zero. Assume plane stress conditions.

thllowing data depends on your student number:

$y=\frac{\text{l}\text{a}\text{s}\text{t}\text{d}\text{i}\text{g}\text{i}\text{t}of\text{y}\text{o}\text{u}\text{r}\text{s}\text{t}\text{u}\text{d}\text{e}\text{n}\text{t}\text{n}\text{u}\text{m}\text{b}\text{e}\text{r}}{10}$ unit $in$ metres $$