Moment under a point load in Thin Shell Structures

49 Pages·2016·1.14 MB·English

by  Karan Taneja

Moment under a point load
in Thin Shell Structures
Optimization of design equation
Karan Taneja
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M T
OMENT UNDER A POINT LOAD IN HIN
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HELL TRUCTURES
OPTIMIZATION OF DESIGN EQUATION
by
KaranTaneja
asanadditionalgraduationwork(CIE5050-09)forpartialfulfilmentof
MasterofScience
inStructuralEngineering
attheDelftUniversityofTechnology.
Studentnumber: 4505468
Supervisors: Dr.ir.P.C.J.Hoogenboom, TUDelft
Dr.ir.drs.C.R.Braam, TUDelft
P
REFACE
ThisreportwasmadeasanAdditionalthesisforthecourseCIE-5050-09. Theinspirationtoworkonthis
topiccamethroughadiscussionwithDr. PierreHoogenboom,duringthecourseShellAnalysis,Theoryand
Application(CIE-4143)whichIundertook. Iwouldliketothankhimforhisunflinchingsupportandguid-
ancethroughoutthedurationofthework.
IwouldalsoliketothankmycolleaguesatTUDelftandmyfriendsandfamilybackinIndiafortheirex-
uberantmotivationduringthecourseoftheprojectandreport.
KaranTaneja
Delft,November2016
i
S
UMMARY
• Theobjectiveofthisadditionalthesiswastoestimateandoptimizethedesignequationformoments
inx-directionunderapointloadinashellstructure. Forthat,theFiniteElementpackageANSYSMe-
chanicalwasusedtoobtainthemomentresultsandthemethodoffittingafunctiontodatawasused
toestimateaformulathatproducedresultswithinatolerableerrorpercentage.Basedonafewcritical
assumptionsabouttheeffectofthematerialproperties,thestructuralcharacteristicsandparametric
studies,theformulafoundin[1]wasthebestestimationthatcouldbeobtained.
• Tobeginwith,inchapter1,themethodusedtorunANSYSandtheimplementationofthescriptused
tocreatethefiniteelementmodelhasbeendescribed. Theimportantpartsofthescriptshavebeen
explained.
• Animportantpartofthestudywastofindouttheidealparametersofthefiniteelementmodelthat
couldbeusedtostudythevariationofmomentsduetochangeincurvatureoftheshell. Hencepara-
metricstudies(chapter3)weredoneforareferenceshape,wherethelengthsinxandy,thenumberof
elementsalongxandyaxes,theparametersusedforthedistributionofthepointloadandthethickness
ofthestructurewerevariedandstudied.
• Inthefinalandthemostcriticalpartoftheproject,itwasassumedthattheformulatocalculatethe
d
momentwasdependentonthreevariables,namelyk ·t,k ·tandthe ratio(wherek andk are
xx yy xx yy
t
thecurvaturesinxandydirectionsattheoriginoftheshellstructure,tisthethicknessoftheshelland
disthediameterofpointloaddistributionarea)andthatassumptionprovedtobetrueasequation6.1
inchapter6ofthisreportvalidatedtheclaim.
ii
C
ONTENTS
Summary ii
1 Introduction 1
2 TheANSYSScripts 2
2.1 Shell1.mac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Shell24.mac. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 ParametricStudies 7
3.1 Lengthinxandydirections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 NumberofElementsinxandydirections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 LoadPatchSize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 MomentresultsfromShell24.mac 11
5 OptimizationofConstants 14
5.1 Assumptionsforderivationoftheformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2 MethodofNon-LinearLeastSquares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.3 CurveFittingtrial1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.4 CurveFittingtrial2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6 TheeffectofPoisson’sratioontheformula 21
7 ConclusionsandDiscussions 23
Bibliography 25
A appendix-a 26
A.1 Shell1.mac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
B appendix-b 29
B.1 Shell24.mac. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
C appendix-c 34
C.1 Codeforvariationoflengthinxandy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
C.2 Codeforvariationofno.ofelementsalongxandyaxes . . . . . . . . . . . . . . . . . . . . . 37
d h
C.3 Codeforvariationof and ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
h b
D appendix-d 41
D.1 Codeforvariationofcurvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
E appendix-e 43
E.1 CodeforCurveFittinginPython . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
iii
1
I
NTRODUCTION
Shellstructuresaredesignedtocarryloadsviamembraneforcesbuteventhenmomentsoccuratedgedis-
turbanceswhichcanbeanythingfrompointloadstochangeinboundaryconditionsoranythingthatleads
toasuddenchangeintheflowofforcesthroughthestructure.Throughpreviousworks,weknowanalytically
thattheeffectsoftheseedgedisturbancesarelocalinnatureandtheydecreasetonegligibleproportionsof
theirpeakvalueswhenmeasuredatandbeyondtheinfluencelength.
Previously,solutionshavebeenexploredanalyticallyandnumericallyleadingtodesignequationsforesti-
matingmomentsunderapointloadforashellstructure.Themethodstosolvethisproblemincludedsolving
partialdifferentialequations[2]andusageofBesselfunctionstosimplifytheresults[3],[4]. Someresearch
ledtoequationswhichcouldrepresentthemomentsasaseriesthatcouldbenumericallyevaluated[5]as
wellaspresentingcurvesforthesame[6],[7].[8]foundarelativelysimplerequationformomentsbutunder
quiteafewassumptions.
Inthisreport(partsofwhichhavebeenusedin[1]),thedatageneratedviacomputationalanalysiswas
collectedandcurvefittingwastriedonit. Inotherwords, afterobservingthepatternformedbymoment
resultsobtainedfromANSYS,aformulawasestimatedandfittingtothemomentdataobtainedaspreviously
mentioned,wasconducted.
1
2
T ANSYS S
HE CRIPTS
TounderstandhowthemomentsarecalculatedinANSYSforthisstudy,itwascrucialtoknowhowtheAPDL
Scriptsusedwork.
2.1. SHELL1.MAC
ThescriptcanbefoundinAnnexA.1. Someimportantpartsofitarediscussedbelow. Thepurposeofthis
scriptwastounderstandthebasicAPDLcommands.
1. InsertionofnodesandFormationofElements.
Thishadtobeinaccordancewiththeshapewhichwasgovernedbythecurvaturevalueprovided. In
thisscript,onlythetwistcurvaturek wasprovidedandthesurfacegeneratedwouldbeahyperbolic
xy
paraboloid,theequationofwhichisprovidedbelow[9]:
z=x·y·k (2.1)
xy
Thexandyco-ordinateswereobtainedusingtheconditionssetinthenestedloopandthez-coordinate
wascalculatedusingtheaboveequation.
Next,elementswereformedbetweenthesenodesandElementtypeSHELL281fromtheANSYSEle-
mentLibrarywasused[10].
2. BoundaryConditions.
Asonlyaquarterofthemodelwascreated, additionalsupportswereaddedtoaccountforsymmetry
alongthexandyaxes.
Attheedgesofthequartermodel,Momentsinalldirectionswerereleasedandthetranslationsinthe
modelwerefixedinglobalX,YandZdirections.Thiscanbeseeninfigure2.1.
3. ApplicationofLoad.
Inthisscript,apressureloadwasconvertedtoPointLoadsinpositivez-directionwhichwereapplied
l
tothecornernodesofalltheelementsexceptnodeattheoriginandthenodesattheedgey= .This
2
canbeseeninfigure2.2.
4. MomentResults.
Asmomentresultsatthemiddleofthestructurearerequired,onlystressesatnode1wereobtained
2
2.2.SHELL24.MAC 3
Figure2.1:BoundaryConditionsforShell1.mac.TheSymmetricboundaryconditionscanbedifferentiated
fromtheModelboundaryconditionsduetothepresenceofrotationalconstraints.
Figure2.2:AppliedloadforShell1.mac.
fromthepost-processingdataofANSYS[11].UsingEuler’sbendingtheory,themomentatnode1was
derivedbyfirstcalculatingthemomentcausedbystressesinthebottomlayerandthensubtractingthe
momentcausedbystressesinthetoplayer. Thiswasdonekeepinginmindthesignconventionasz
beingpositiveinthedownwardsdirectionandthexandyaxesbeinginconjunctionwiththe‘right
handthumbrule’. Astripofhalfaunitwidth(duetosymmetry, theotherhalfwasassumedtoliein
thepartofthestructurewhichisn’tmodelled)andheightequaltothe‘thickness’usedinthemodelwas
assumedtocalculatethemomentatNode1. Whilefigure2.5describesthesituationforthesecond
script,Shell24.mac,thelogicusedwasthesame.
2.2. SHELL24.MAC
ThisscriptwasusedtoruntheANSYSsimulationsinbatchmodetocollectthedataonwhichfurtheranal-
ysiswasdone.Theimportantfunctionsofthescriptarediscussedbelow.ThisscriptispresentedinAnnexB.1.
1. Insertionofnodesandformationofelements.
Inthescript,parametersgx,gy,mx,my,qxandqywerecreatedspecificallywiththeaimtogeneratethe
xandyco-ordinatesofthenodes.Thezco-ordinatewascalculatedusingtheequationofaparaboloid
[9],presentedbelow.
2.2.SHELL24.MAC 4
1 1
z= k ·x2+ k ·y2 (2.2)
xx yy
2 2
Inthisscript,thecurvaturesattheorigin,k andk ,wereprovidedandcurvaturek wasnotcon-
xx yy xy
sidered.Theadvantageofusingcurvaturesattheoriginwasthatthesecanbeorientedinthedirection
oftheprincipalcurvaturesandhencek couldbeignoredintheformationofthesurfacewithoutlos-
xy
ingoutonanything.Hadcurvaturesbeenassociatedwithanyotherpointonthesurface,theuseofk
xy
wasnecessary.
Aftercreationofthenodes,elementsoftypeSHELL281[10]wereinsertedbetweenthem.
2. BoundaryConditions.
Asonlyaquarterofthestructurewasmodelledbasedontheprincipleofsymmetry,appropriatebound-
aryconditionswereassignedtotheaxesofsymmetry(2.3).
Theboundaryconditionsattheedgeswereassignedto"carrythenormalforcesandthein-planeshear
forces"[1]. HencenodalaxesofthenodesattheedgeswererotatedtocarryforcesintheGlobalXand
Ydirectionssoastoavoidsurplusreactionswhichwouldmaketheschematizedboundaryconditions
invalid.
Figure2.3:BoundaryConditionsforShell24.mac.TheSymmetricboundaryconditionscanbe
differentiatedfromtheModelboundaryconditionsduetothepresenceofrotationalconstraints.
3. Applicationofload.
Thepointloadwasnotsimplyputatthemiddlenodeofthestructurebecausesingularitiesorvery
highresultsunderthepointloadneedtobeavoided.Itwasbrokendownintomanynodalloadswhich
were distributed within circle whose radius was assumed to be a linear function of t. This was the
reasonthatthemeshwasfinerattheoriginandtobecomputationallyefficient,becamecoarseraway
fromtheload. Also,becauseitwasknownviapreviousstudies[12]thattheeffectofsuchloadswas
local,itmadesensetomakeafinermeshneartheloadandcoarserawayfromit.
Thefirstpartofthecodecalculatesthenumberofelementsalongxandywhichliewithintheradiusof
influence.
Theloadwasdistributedinthoseelementsasfollows:
(a) Withintheradiusofinfluence,theareawasdividedinto‘n’ringswherenwascalculatedbasedon
adesignatedloadpatchsizewhichwassuccessivelysubtractedfromtheradiusofinfluenceuntil
itbecamezero.
(b) Theringwidthwascalculatedbydividingtheradiusby‘n’.
2.2.SHELL24.MAC 5
Figure2.4:AppliedLoadforShell24.mac.Node1canalsobedistinguishedasthenodeattheorigin.
(c) Thenringsweretheniteratedoverandforeachringthefollowingstepswereundertaken:
• Numberofsectorswithinaringwerecalculatedbysubtractingminimumsectorsizewhich
istheringwidthfromthecircumferenceofthering.
• LoadwithinasectordP,wascalculatedbasedontheproportionofthesectorareafromthe
wholecircleandappliedontheCenterofGravity.
• Thisloadwasthenshiftedtothenearestnodebyfindingoutthenodewiththeleastdistance
fromtheCenterofGravityofthesector.
• Momentswerealsoaddedtothenodesbasedontheleverarmsinxandydirections.
(d) Thisprocesswasthenrepeateduntilallthesectorsinalltheringshadbeeniteratedthrough.
4. MomentResults.
Theyareobtainedthesamewayasinthepreviousscript,Shell1.mac. Figure2.5offerstheanalytical
modelusedincalculatingtheresults.
Figure2.5:Analyticalmodelusedforcalculationofmoment.Here,theviewfromthexzplaneisgiven.The
’strip’ofhalfaunitwidthshouldbeimaginedtobealongthey-axis.
5. InfluenceLength.
Theinfluencelengthmathematicallymaybedefinedasthedistancefromonerootofthefunctionof
anedgedisturbancetoanotherorsimplythedistancefromadisturbanceatwhichtheinfluenceofthe
disturbancecanbeneglected.Basedonthislogic,theinfluencelengthinadirectionwascalculatedas
follows: