A high$-$pressure pipe is being fitted inside a processing unit in a chemical factory. Across section of the pipe is shown in Figure $3\mathrm{.}1$ a$.$ The pipe is much longer in the z$-$direction than its dimensions in the x$-$y plane. It has a unifom cross section along itsentire length and is symmetrical about the x$-$ axis. The pipe will be subject to a unifomintermal pressure along its entire lengthIn order to check the safety of the pipe, a finite element model of a cross$-$section hasbeen developed. The model represents half of the cross$-$section as sthown inFigure $3\mathrm{.}1$ b$.$bdThe model only represents a two$-$dimensional ross$-$section throughthe object. What is the most appropriate type of analysis to be usedplane stress, plane strain or axisymmetric stress?c$)$ The displacements at each node have been determined. For onethe compscorresponpn dienS dre shown in Figure $3\mathrm{.}1$ b anddisplacerments are given in Table $3\mathrm{.}1$State the degrees of freedom that should be constrained $($Le$.$ set tozero$)$ due to symmetry along the lines AB and CD $($Figure $3\mathrm{.}1$ b$).$b$)$Folowing the finite element process, demonstrate that the directstrains and shear strain in this element are as stated below.Show all of your working and provide bref commentary at eachstepThe pipe is made from a plastic with an elastic modulus of $0\mathrm{.}3$ GPaand a Poisson's ratio of $0\mathrm{.}3,$ and Equation $5$ gives a formula for themaximum and minimum principal stressesShow that the maximum principal stress in this element is $27$ MP$.$showing your workinge$)$ To prevent rupture of the pipe, the maximum prncipal stress mustnot exceed $35$ MPa. ir this element exhibited the largest maximumprincipal stress in the model, would you cary out further analysis?AI$(0\mathrm{.}04667)($e$)0\mathrm{.}04625$If so$,$ explain how you would improve the model. Iif not, explain whyno further analysis is necessaryacose n schenin mmlo$\mathrm{.}001417)15$Node$28$Uniform pressure acts onthe internal surface$29$CDXEquation $5(200,3\mathrm{.}0)10074$U $($mm$)0\mathrm{.}005$Figure $3\mathrm{.}1$: a$)$ A section of the pipe showing the cross$-$sectional shape. Note that the pipeecion, b$]$ ne trte eement moe ofe pipe eross section andhematic of one element showing the node$0\mathrm{.}076$Close up of one of the finite elementsAll measurements in mmTable $3\mathrm{.}1$: The nodal displacements in the x$-$direction $($U$)$ and the y direction $()$ for theelement highlighted in Figure $2$B$.$V $($mm$)[1$ mark$]0\mathrm{.}042(2$ marks$]0\mathrm{.}083[10$ marks$]0\mathrm{.}009[8$ marks$][4$ marks$]3)(21\mathrm{.}5,3\mathrm{.}8)2)(215,22)$measurements aref$)$A BFigure $3\mathrm{.}2$: The finite element model of the pipe cross$-$section and a cose up schematic ofone element which has an edge on the inner surface.YModule code: MECH$390001$A pressure of $8$ MPa is applied to the inner surtface of the pipe. Inthe model, the length of the inner surtace of the model in Figure $3\mathrm{.}2$is $50$ mm and there are $17$ evenly distributed nodes on that surface.IAJ$=$Write down the extemal force vector for the one element depictedin Figure $3\mathrm{.}2.$ Showing your working or descnbe your logic, at eachslep in the derivation of the values.CDXFor the calculation, you can consider a unit thickness $(1$ mm$)$ in theZ direction.Give two conditions which need to be met for the use of symmetryin this model to be validFor one of the two conditions you gave, describe a use of themodel which is not possible because of that condition.The following notation is used throughoutK$=$stiftfness matrxDo not remove this exam paper from the exam venue.$($u$)=$ vector of nodal displacements in localcO$-$ordinate system $($relatlve to element$)($U$)=$ vector of nodal displacementsinglobal co$-$ordinate systemMatrix Algebra$[$D$]=$ elasticity matix relating stress tostrainB$]=$ strain$-$displacement matrixKoeomer$]=$ geometric matrixM$]=$ consistent mass matrixFor a $2$x$2]$ matrixClose up of one of the elements onthe inner surfacecInnersurfacedet$[$A$]=($axd$)-($cx b$)$A$\text{'}=1$ d $-$b$]$Differentiation$\mathrm{.}60$Useful EquationsF$=$elastic modulusV$=$ Poisson's ratioodensitym$=$ mass of elementL$=$ length of element$($k$)$A$=$ cross$-$sectional area of elementFor a $3$ x$3]$ matrx:a b c$($A$]=$d e fLgh ijdet$[$A$]=$ alei $-$hf$)-$d$($bi $$ hc$)+$g$($bf$-$ec$)|=$ second moment of area of beamelementif $[$A$]=($u$)$BKu$\},$ thenA$]$o$\{$u$\}[10$ marks$]=2[$B$\backslash ($u$)[5$ marks$]$

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