Project description

Figure $3$ below shows a $4-$noded quadrilateral element with the given nodal coordinates. The element has a

uniform density $=2500k\frac{g}{m^3}$ and thickness $t=2\mathrm{.}5cm.$

$($a$)$ Use the Gaussian quadrature, with one Gauss point at the element centre

$(==0,$ weight $W=4),$ to determine the consistent mass matrix of the element.

$($b$)$ Use the Row Sum technique to compute the lumped mass matrix. Then compare it with the analytical

value $($assuming that the total element mass is evenly distributed to the $4$ nodes$).$ Also give reasons

in the report why the lumped mass matrix is the same or different from the analytical value.

$($c$)$ Then use $4$ Gauss points at $=+-0\mathrm{.}57735,=+-0\mathrm{.}57735$ and weights equal to $1$ to compute the

consistent mass matrix.

$($d$)$ If the element is translated by a vector distance of $(2,1),$ followed by a rotation of $25$

anticlockwise about the point $1,$ and then scaled by a factor of $2\mathrm{.}5,$ compute the final consistent

matrix again.

$($e$)$ Observe the two mass matrices obtained in $($c$)$ and $($d$),$ and in particular how the mass matrix is

changing when the element is being translated, rotated and scaled. Then try to explain your

observations in the report.MATLAB PROGRAM FOR THIS PROBLEM