Problem $2$

Create a program to study a square plate of dimension $2m2m0\mathrm{.}1m,$ simply supported on the four

edges and loaded by a uniform download distributed load $q=100000\frac{N}{m^2}$

$E=210$GPav$=0\mathrm{.}3$

Use $100$ elements and the mesh shown on the last page. The topology for this mesh as well as the vector of

active degrees of freedom for the case of simply supported on the four edges are given in the file

topoproblem$2.$m

Plot the vertical displacement along the line defined by the nodes $56$ to $66($middle of the plate$)$

Compare the results with the following theoretical results:

Maximum displacement $($at the center of the plate$)$

$d=-0\mathrm{.}00406\frac{q{L}^4}{D},D=\frac{E{t}^3}{12(1-v^2)}$

$L$ and $t$ are the length and thickness of the plate

Part $2$

$1$ Create a function $[$mp$]=$momxy$($x$,$y$,$d$)$ which calculates the moments $M_x,M_y$ and $M_{xy}$ in the element at

the coordinates $(x,y).$

dis the displacement vector for the element.

$M_y$

$M_{xy}1$

$2$ Create a function $[mx,my,mxy=$momel$($d$)$ which calculates the moments $M_x,M_y$ and $M_{xy}$ at the four

nodes of an element. $d$ is the displacement vector for the element.

at node $1$

Mxat node $2$

$M$ at node $3$

$M$ at node $4]$

at node $1$

Mxy at node $2$

Mxy at node $3$

Mxy at node $4]$

Problem $1$

Calculate the moments at the nodes for each element. Put the results on $3$ arrays $($one for each moment$)$

with dimension $9($number of element$)4($moment at node $1,$ node $2,$ node $3,$ node $4)$

The program should give the following results:

Moments at node $6($average value from the $4$ elements sharing node $6)$

$M_x=M_y=-4\mathrm{.}2805*{10}^{4}N\frac{m}{m},M_{xy}=7\mathrm{.}5615*{10}^{3}N\frac{m}{m}$

Problem $2$

Calculate the moments at the nodes for each element.

Plot the following results in $5$ figures

$Mx$ along the line defined by the nodes $56$ to $66$

Mxy along the line defined by the nodes $56$ to $66$

$Mx$ along the line defined by the nodes $23$ to $33$

Mxy along the line defined by the nodes $23$ to $33$

Compare the results with the following theoretical results:

Maximum bending moment $($at the center of the plate$)$

$M_x=-0\mathrm{.}0479q{L}^2$