StrucA final report
- 1. Graeme Ramsey Computational MethodsforStructural Analysis: Dr.Landis Final Project The Universityof Texasat Austin Spring2015 05/08/2015 Finite Element Code, Results and Discussion
- 2. Table of Contents Course Summary 1.0 Introduction 2.0 Finite ElementsMethods andFacilitatingCode Application 3.0 ResultsandDiscussion 3.1 Problemsetup 3.2 Results 3.2.1 LightMesh 3.2.2 MediumMesh 3.2.3 HeavyMesh 3.2.4 StrategicMesh 3.3 Discussion andMeshComparison Depiction 4.0 Conclusion
- 3. List of Tables Table 1: InterpolatedStressValuesat(1,0) and (0,1) List of Figures Figure 1: ScenarioDepiction Figure 2: Mesh DensityDepiction Figure 3: LightMesh DisplacementandStressDepiction Figure 4: MediumMeshDisplacementandStressDepiction Figure 5: HeavyMesh DisplacementandStressDepiction Figure 6: StrategicMeshDisplacementandStressDepiction Figure 7: StressalongX-axisinX-direction Figure 8: StressalongX-axisinY-direction Figure 9: StressalongY-axisinX-direction Figure 10: StressalongY-axisinY-direction
- 4. Course Summary Overthe course of the semesterthiscourse,Computational MethodsforStructural Analysis,has focusedondeveloping methodsforapplyingstaticstructural elementcalculations,byhandbutprimarily viathe studentscodinglanguage of choice (MATLAB). The flow of the course focusedonbuildingupon priorknowledge to developageneral approach toanalyze manydifferentstructural applications:from trussesto framestobeamsundertorsionor bending andultimatelyanygeneralizedstructure witha finite numberof degreesof freedomunderbodyandsurface forcesthatcan be prescribedasnodal loading. The final goal of thiscourse and subjectof thisreportisto have each studentbuildhis/herown code for Finite Elementanalysisusing 3-noded elements. Initiallywe tookknowledge of equilibrium, kinematics,andmaterial lawsandappliedthemto structureswithnodal degreesof freedom. We learnedhow toassembledmatricesrelatinginternal forcesto external nodal forces(equilibrium),thennodal displacementstostrains(kinematics)andfinally mergedthose twomatricesusingmaterial lawstorelate external nodal forcestonodal displacements. Solvingthissystemof equationswasasbasicas needingasmanyboundaryconditionsondisplacements and forcesas nodal degreesof freedom. Thisallowedustosolve structural problembuildingsolelyon the knowledge we hadgoingintothe course. Learninghowto assemble the same matrix formgoverningequationsfromanalysisonan elementalbasiswasthe nextapproach. Here anelemental stiffnesswasdefinedinalocal reference frame,thenrotatedintothe global referenceframe andfinallyassembledintoaglobal stiffnessin accordance withnodal degree of freedomenumerations. Thismethodwasthenappliedtomany differentstructure typesincludingbending. Usingthe strainenergymethodwasthe nextapproach. It followedasimilarprocess,element by elementbeingrotatedintothe global frame bydegreeof freedomenumeration. The same governingequationswere calculatedby equatingpotential withstrainenergyandsettingthe spatial derivativesof thatpotential energytozero(e.g. staticequilibrium). Thisgave a goodintroductioninto the final method,whichinvolvedthe principle of virtual work. The SegwayintoFinite Elementanalysiswas anintroductionintocontinuoussystemsfollowed by virtual work whichisthe basisforfinite elementanalysis. A brief overview of continuoussystems providedknowledge of general relationshipsbetweenaxial/transverse displacementsand strain/curvature respectively. The principleof virtual workexpandeduponthese relationships,using shape functionstoapproximate nodal displacements. Byassuminganarbitrary virtual displacement effectivestiffnesscoupledwithnodal displacementsare equatedtosurface andbodyforces. Although our codingand analysisfocusedonmechanicalstructures andstaticsystems,we notedanddeveloped the principle of virtual worktoapplytodynamicsystemsandnon-mechanical structuresinvolving diffusion,electromagneticsandothergradientsthatfollow similarlyappliedkinematics,equilibriumand material designs.
- 5. 1.0 Introduction Thisreportis focusedondescribingthe procedure andresults regardingthe constructionof a finite elementstructural analysiscode anditsapplicationtoa triangularmeshstructure (meaning3- nodedstructural elements). A summaryof how the principal of virtual work,appliedtothree-noded elements,isusedinthe constructionof anautomatedcode helpstoprovide anoverview of the procedure thatwentintothe codingendeavor. Followingthissummaryare the resultsof thisfinal project. These resultstake the formof comparingdifferentmeshdensitiesandtheiraccuracies along witha descriptionof the codingendeavoritself. Thiscomparisonisfora particularstructural setup whichisoutlinedtherein. A discussionfollowsidentifyinghow applicable thisprocedure isforstructural analysisingeneral. The concludingsectionwill bothsummarizethisreport,itsapplicability andalso identifythe validityof asemesterworthof developingwhichresultedinthe creationof aunique finite elementanalysis code/procedure. 2.0 Finite ElementsMethods and FacilitatingCode Application Havingalreadydevelopedacode thatallowedanalysisof trussstructures,modifyingitto functionasa Finite Elementscode wasas simple asimplementingshape functionstocalculate the elementalstiffness. We,the students,were suppliedinputsof nodal positions,elementcompositions, external nodal forcesandnodal displacementconditions. If we hadn’tbeengiventhe nodal forces,the shape functionswouldhave hadtohave beenusedtoapproximate themusingthe weakformof the principal of virtual workandbody/surface forcesdescribable asfunctionsof ourdegreesof freedom (nodal positions). The onlyothermodificationtothe code alreadydevelopedwascalculationof stress usinga material propertiesmatrixforisotropicmaterials. To streamline the analysis,anadditionalscriptwascreatedasa functioncall torun overeach meshdensitydisplayitsresultsintablesand save itsoutputsformodificationandevaluationusingother code. For furtheranalysisof ourstructure,additional code wascreatedtogatherstressvaluesalongthe X and Y axisand plotthem. Alsocode wascreatedto linearlyinterpolatethe values of stressatthe boundaryof the hole. Code wasprovidedbythe TeachingAssistant,IsaacLee,whichonlyhadtobe slightlymodified to use outputsfromthe finite elementcode inorderto depictthe deformedandundeformedmeshes and alsostressconcentrationsinthe structure. All these codingfileswill be submittedelectronicallyso as to accommodate verifyingthe codes’validity. 3.0 Resultsand Discussion ThisSectionwill describethe structural setup usedincludingthe differentmeshesforaccuracy comparisons. Alsoincludedisadepictionof the results showingdeflectionandstressconcentrationsby graphicmeans. The lastportionof thissectionisan applicationof those resultstoprojectstressesat the edge of the structural flaw at Y=0 as well asat X=0 (onthe 1 unitradius hole) byinterpolatingfrom stressesalongthe x and y axisinthe structure.
- 6. 3.1 ProblemSetup The structure we are consideringinouranalysisisa 6X6 square witha hole cut init witha radius of 1, unitsdon’tmatterjustthe ratio,see figure 1. The analysiswasdone ona unitthicknessbasis,and a modulousof elasticityof 50. The structure has an applieduniaxial tensioninthe Y direciton. Inthe analysisitsself we are onlyconsernedwithone quadrantdue toasymatrical structure and loading. The top rightquadrantis our focus. σy σy Figure 1: Scenario depiction, orientation shifted from vertical: y-axis right, x-axis down. The centroidsof the triangularelementsare are depictedbelow,itservestogive anideaof elementdensitywithoutthe clutterof elementboundaries. Also,the resultsbyusingfiniteelement analysisare mostaccurate at the centroidof each element,sothisfigure servestoshow where mesh focusesitsaccuracy inthe topright graphin the figure. The use of these fourdifferentmeshesare intendtoshowhow,by strategicallyconcentratingdense mesh areas,the modelingof astructure can be just as accurate with1/4th the elements. The differentmesheswere denoted6,12, 24, and R in the code,the numberrepresentingincreasingmeshdensityandRforthe variable meshdensity.
- 7. Figure 2: Mesh nodal density depiction, element centroid locations. (clockwise, starting top-right) the number of elements are 144, 576, 2304, 576. 3.2 Results Thissectionwill outline the directresultsfrommypersonal Finite Elementcode. The primary outputsforconsiderationwere nodal displacementsandelemental stresses. The meshesoriginal placementiscoloredbrown,andthe displacedmeshisbrightgreen. The stressconcentrationsare red huedif tensile andblue huedif compressive. Tomake realisticdisplacementfieldsamodulousof elasticityof 50 wasusedfor the generationof these plots. 3.2.1 Light Mesh As showninfigure 3 below the displacementsare primarilydownwardandtothe right for the quadrant. Thisis to be expecteddue tothe simple nature of the structure andload. The nodal displacements of the course meshare the easiesttoview andare relativelyaccurate. Itis fairlyevident
- 8. that thismeshisfar too corse whenlookingatthe stressdistrobutionplots. Althoughyoucansee the general locationof stressconsentrations,the resolutionof the stressesatthese critical pointsisvery low. These critical pointsforstressinthe Y-directionare atthe edgesof the whole nearY=0 where it showsa spike upto 3.7 σy. For stress inthe X-directionthe largestcritical pointare nearthe hole (spike to near1.5 σy) at X=0 and othersmallerconsentrationsare atthe extremaof the structure at X=0 and nearthe hole atY=0. Figure 3: Light mesh original/displaced structure and stress depiction in the X and Y direction. 3.2.2 MediumMesh As showninfigure 4 belowthe displacementsare primarilydownwardandtothe rightfor the quadrant. It is fairlyevidentthatthismeshisa little toocorse whenlookingatthe stressdistrobution plots. Althoughyoucansee the general outline of stressconsentrations,the resolutionof the stresses at these critical pointsisrelativelylow. These critical pointsforstressinthe Y-directionare atthe edges of the whole nearY=0 where itshowsa spike upto 4 σy. For stressinthe X-direction the largestcritical pointare nearthe hole (spike tonear1.8 σy) at X=0 and othersmallerconsentrationsare at the extrema of the structure at X=0 and nearthe hole at Y=0.
- 9. Figure 4: Medium mesh original/displaced structure and stress depiction in the X and Y direction. 3.2.3 Heavy Mesh As showninfigure 5 belowthe displacementsare primarilydownwardandtothe rightfor the quadrant. It is fairlyevidentthatthismeshissufficientlydense whenlookingatthe stressdistrobution plots. The outline of stressconsentrationsare well defined,the resolutionof the stressesatthese critical pointsisproficient. These critical pointsforstressinthe Y-directionare at the edgesof the whole nearY=0 where itshowsa spike upto4.1 σy. For stressinthe X-directionthe largestcritical point are nearthe hole (spike tonear1.9 σy) at X=0 andother smallerconsentrationsare atthe extremaof the structure at X=0 and nearthe hole atY=0.
- 10. Figure 5: Heavy mesh original/displaced structure and stress depiction in the X and Y direction. 3.2.4 Strategic Mesh As showninfigure 6 belowthe displacementsare primarilydownwardandtothe rightfor the quadrant. It is fairlyevidentthatthismeshissufficientlydense whenlookingatthe stressdistrobution plots. The outline of stressconsentrationsare well defined,the resolutionof the stressesatthese critical pointsisproficient. Thismeshmanagedtoachieve asimilarresolutionof the stresscontourwith nearly1/4th the amountof elementsinthe mesh. These critical pointsforstressinthe Y-directionare at the edgesof the whole nearY=0 where itshowsa spike upto >4 σy. For stressinthe X-directionthe
- 11. largestcritical pointare near the hole (spike to>1.8 σy) at X=0 and othersmallerconsentrationsare at the extremaof the structure at X=0 and nearthe hole atY=0. Figure 6: Strategic mesh original/displaced structure and stress depiction in the X and Y direction. 3.3 Discussionand Mesh Comparison Depiction Figures7 and 8 belowshowstressinthe X and Y directionrespectivelyalongthe X-axis,whichis the bottomof the quadrantof the structure we considered. Itisapparentthat onthisaxis,stressinthe X directionisapproaching0whereas stressinthe Y directionisapproaching4 σy . Figures9 and 10 belowshow stressinthe Xand Y directionrespectivelyalongthe Y-axis,whichisthe leftof the quadrant of the structure we considered. Iis apparentthat onthisaxiskstressinthe X directionisapproachinga value near-2 σy whereasthe stressinthe Y directionisapproaching0. Obviouslythe densestmesh (mesh24, 2304 elements)providesthe mostdetail forthe overallstressprofile. The strategicmesh (meshR,576 elements) densitymanagestogetjustas goodof resolutionnearthe critical point(hole)as the densestmeshwith1/4th the elements. Inthe figuresthe stressinthe elementisnormalizedbythe appliedstress(σy),andthe distance isnormalizedbythe radiusof the hole (R). The wordsigmais the same as stress,the twolettersubsriptsrepresentthe directionof thatinternal stess.
- 12. Figure 7: Stress along the X-axis in the X direction. Figure 8: Stess along the X-axis in the Y direction.
- 13. Figure 9: Stress along the Y-axis in the X direction. Figure 10: Stress on the Y-axis in the Y direction.
- 14. Linearlyinterpolatedvaluesforthe stressatthe hole boundarywhere the axisintersectare showninTable 1 below. Asexpectedstressinthe Xdirection where the X-axisintersectsthe hole and stressinthe Y directionwhere the Y-axisintersectsthe hole are close to0. The strategicmeshseemsto be evenmore accurate thanthe dense meshforthese near0 values. The largeststressconcentrationswere predictedtobe andfoundto be inthe Y directionwhere the X-axisintersectsthe holeandstressinthe X directionwhere the Y-axisintersectsthe hole. The densestmeshseemstobe slightlymore accurate thanthe strategicmeshon these large stress consentrations. Mesh SigmaXXat (1,0) SigmaYYat (1,0) SigmaXXat(0,1) SigmaYY at (0,1) Light 0.14897 3.2205 -1.1906 -0.01527 Medium 0.093008 3.7905 -1.6496 -0.04733 Heavy 0.037149 4.0216 -1.8469 -0.02375 Strategic 0.0081625 3.9641 -1.7961 0.003498 Table 1: Interpolated Normalized Stress Values 4.0 Conclusion Overthe semesterthiscourse,Computational MethodsforStructural Analysis,hasfocusedon developingmethodsforapplyingstaticstructural elementcalculations,byhandbutprimarilyviathe studentscodinglanguage of choice (MATLAB). The flow of the course focusedonbuildinguponprior knowledge todevelopageneral approachtoanalyze manydifferentstructural applications:from trussesto framestobeamsundertorsionor bendingandultimatelyanygeneralizedstructure witha finite numberof degreesof freedomunderbodyandsurface forcesthatcan be prescribedasnodal loading. The final goal of thiscourse and subjectof thisreportwas tohave each studentbuildhis/her owncode forFinite Elementanalysisusing3-nodedelements. The largestgain fromthisproject has been the knowledge andskillacquiredfromthe coding endeavoritself. Bycreating,debuggingandevaluating aunique code thatperformsfiniteelement structural analysiswe have gainedinsightinto real worldapplicationsandthe needfor streamlining automatedproceduresforaccurate,speedyresultsthatare easyto decipheranduse. Anobviousresult fromusingdifferentmeshtypesisthatstructural flaws(holesorcracks),interfaces(boltsorfasteners) and otherlocationswhere stresstendstobe intensifiedneedtobe modeledtoafinerdegree to maintainaccurate stresscalculationsandfailure predictions. Thisapplicationtoa simple,predictablestructure wasagood meanstodebuga freshcode and gainan understandingof the general methodologyof finite elementsstructural analysis. Withthe skills gainedfromthisendeavor,we couldevaluate almostany 2Dstructure made of isotropicmaterials completelyin-house. We couldalsoapplythisprocessusingelementswithmore than3 nodes andto non-isotropicmaterialswithalittlefinesse. Thiscourse hasbeenavaluable introductionintostructural analysisandthe knowledgegainedwill be useful regardlessof whethermyoccupationinvolvesliteral structural analysis.