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.
