Consider a long thin structure under an axial load. A single beam finite element isused to represent this structure and to predict the critical buckling load $($Figure $2\mathrm{.}1)$The model in Figure $2\mathrm{.}1$ is constralned at node$1$ in all degrees of freedom and iscomplete$$y unconstrained at node $2.$a$)$b$)$c$)$e$)$Give a definition of buckling and describe a real world example of aan $0$cCur.Scenario when buckling canIn the derivation of the buckling formulation, Equation $6$ is used toexpress the total potential energy. where U$,$ stands for potentialenergy due to bending and V$,$ the potential due to a distributed loadperpendicular to the long axis of the structure. What does U$,$ standfor?$($u$)$d$)$ Select the parts of the stiffness and geometric matrices whichcorrespond to the unknown displacements and write down thefomula which will be used to find the critical buckling load, leavingE, I, L and P as unknowns.Considering the definition that you have gven in pat a$),$ sta$),$ state whichten in Equation $6$ can be set to zero and give a reason.$($Where E is the Young's modulus, l is the second moment of area,Lis the length of the element andP is the critical buckling force.$)$Simpity the expression in part d$)$ using the substitutions a$($Equation $7)$ and B $($Equation $8).$ Then reduce the expression tothat given in Equation $9,$ showing all of your working.TK$]=$ stiffness matrixThe following notation is used throughoutI$=$U$,+$U$,-$VL$(12$a$\backslash$deg $+156$a$+1358)=0($U$=$ vector of nodal displacements inglobal co$-$ordinate systemMatrix Algebra$[$D$]=$ elasticity matrix relating stress tostrain$[$B$]=$ strain$-$displacement matrix$[$Kgeometrc$]=$ geometric matrix$[$M$]=$ consistent mass matrixFor a $[2$ x$2]$ matrix:$|$A$-$fa b$=$ vector of nodal displacements in localv$=$ Poisson's ratioO$-$ordinate system $($relative to element$)$c dB$=3071$Pdet$[$A$]=($a xd$)-($cx b$)$Differentiation:dAT"detA$-$cUseful Equations$-$baEquation $6$E$=$elastic modulus$($Equation $7)($Equation $8)$p$=$ density$($Equation $9)$m$=$mass of element$($A$]=$d e fL$=$ length of elementFor a $(3$ x $3]$ matrix:a b$-$d$($bi $$ hc$)+$g$($bf $-$ ec$)$A $=$ cross$-$sectional area of elemif $($AJ$=\{$u$)$BXu$),$ theng h i$||=$ second moment of area of be$.$elementdet$[$A$]=$ alei $-$ hf$)$c$[5$ marks$]$AJ$,[1$ mark$]-2[$B$\backslash ($u$)[3$ marks$][5$ marks$][6$ marks$]$Question $2$ continued$9)$Find the roots of Equation $9$ for either a or B$.$ then use Equations$7$ and $8$ to detemine an expression for the twO roots in terms of E$,,$L and P$.$Provide full working and briet commentary at each step.State which root you would choose to represent the critical buckingload and explain whywhyYL$($s$)->$Xaz Xi$-$Ykb$=$yj$-1$Figure $2\mathrm{.}1$: a single beam finite element representing a long thin structure under axialload$2$D Constant strain triangle element: For a $2$D constant strain triangular elementassuming a linear shape function.and A is the area of the elementnode t$[111]$Shape function corresponding toLN$=$a$,$thr$+$ cy$)$For plane stress$[$D$]=$E$1-$v$1$ VV $1$For plane strainM$]=21-$v$20$V $1-$vFor plane stress or plane strainb$10$ b$20$ b$30$B$]=0$ c $0$ c$20$ c$32$A$00$LCt b$$ c$2$ b$2$ C$3$ b$3[201010][7$ marks$]020101$m$10201012010201101020$Lo $10102$J$[3$ marks$]$P

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