number, length, etc.$)$

$2.$ Modify the above version of the code to allow for arbitrary prescribed displacement. Solve in matlab with $15$ elements of equal length, but replace the force at node $3($in the figure above$)$ by a displacement value of $0\mathrm{.}1$ mm and keep $\backslash ($ F$\_\{2\}\backslash )$ unchanged. Solve the same problem by hand first as a $2-$element problem for validation purpose, then solve it using Matlab. Provide the nodal displacements and reaction forces. $```$

$\%$ This data will automatically describe a line of $2-$node elements

clear;

numele$=6$; numnod$=$numele$+1$;

$\%$ x$-$coordinates of nodes

x$=[0$:$1/$numele:$1]$;

$\%$ node stores the nodes of all the elements, i$.$e$.,$ element connectivity

node$=[1$:numele;$2$:numele$+1]$;

area $=[2\mathrm{.}0*$ones$(1,$numele$)]$;

young $=[1.$E$7*$ones$(1,$ numele$)]$;

$\%$ support conditions, ifix $($i$)=1$ if node i is fixed, else $0.$

ifix$=[1,$zeros$(1,$ numele$)]$;

ifix$($numnod$)=1$;

$\%$ externally applied loads

force$=[10$e$\backslash$theta$/$numele$*$ones $(1,$ numnod$)]$;

$\%$Initialize system matrix

bigk $=[$zeros$($numnod$,$ numnod$)]$;

$\%$

$\%$Loop over elements

$\%$

for e$=1$:numele

$\%$ compute element length

length $=$ x$($node$(2,$e$))-$ x$($node$(1,$e$))$;

c $=$ young$($e$)*$area$($e$)/$length;

$\%$ compute element stiffness matrix

$```$

$```$

bigk$($node$(1,$e$),$ node$(1,$e$))=$bigk$($node$(1,$e$),$ node$(1,$e$))+$ke$(1,1)$;

bigk$($node$(1,$e$),$ node$(2,$e$))=$bigk$($node$(1,$e$),$ node$(2,$e$))+$ke$(1,2)$;

bigk$($node$(2,$e$),$ node$(1,$e$))=$bigk$($node$(2,$e$),$ node$(1,$e$))+$ke$(2,1)$;

bigk$($node$(2,$e$),$ node$(2,$e$))=$bigk$($node$(2,$e$),$ node$(2,$e$))+$ke$(2,2)$;

end

for n$=1$:numnod

if$($ifix$($n$)==1)$

bigk$($n$,$n$)=1$E$+30$;

force$($n$)=0$;

end

end

$\%$solve stiffness equations

disp$=$force$/$bigk;

$\%$Plot displacements

subplot$(211)$; plot$($x$,$disp$)$;

$\%$title$(\text{'}10$ FEM results Elem number:$6\text{'})$;

xlabel$(\text{'}$x$\text{'})$;ylabel$(\text{'}$Displacement$\text{'})$;

$\%$ Plot stresses

for e$=1$:numele

length $=$ x$($node$(2,$e$))-$ x$($node$(1,$e$))$;

elong $=$ disp$($node$(2,$e$))-$ disp$($node$(1,$e$))$;

eps$=$elong$/$length;

sig$(2*$e$-1)=$young$($e$)*$eps;

sig$(2*$e$)=$ sig$(2*$e$-1)$;

xx$(2*$e$-1)=$ x$($node$(1,$e$))$;

xx$\}(\backslash$mp@subsup$\{2\}\{\}\{*\}$e$)=$x$(\backslash$operatorname$\{$node$(2,$e$))$;

$\%$yy$($e$)=($x$($node$(1,$e$))+$x$($node$(2,$e$))/2$

end

subplot$(212)$; plot$($xx$,$sig$)$;

xlabel$(\text{'}$x$\text{'})$;ylabel$(\text{'}$stress$\text{'})$;

$```$