nonlinear analysis of reinforced concrete columns with holes

INTERNATIONALJOURNALOFCIVILANDSTRUCTURALENGINEERINGVolume3,No3,2013

Copyrightbytheauthors-LicenseeIPA-UnderCreativeCommonslicense3.0ResearcharticleISSN09764399

ReceivedonFebruary2013PublishedonMarch2013655

NonlinearanalysisofreinforcedconcretecolumnswithholesEhabM.Lotfy

AssociateProfessor,CivilEngineeringDepartment,FacultyofEngineering,Ismaelia,SuezCanalUniversity,Egypt

[email protected]:10.6088/ijcser.2201203013060

ABSTRACT

Thebehaviorofreinforcedconcretecolumnswithholesunderaxial loadisnotunderstood,andresearchesinthesubjectareneededtohelpdesignersandstructuralcodeofficials.Holesdrilled out to install additional services or equipment, such as for ducts through columns,beams,orwalls,can lead to lossofstrengthandpossiblestructural failure.Untilnowlittleworkhasbeendoneonholesincolumnsand,hence,thisstudyaimstoexaminetheamountofstrengthlostduetothepresenceofholesincolumns.Nonlinearfiniteelementanalysison21-columnspecimenswasachievedbyusingANSYSsoftware.ThenonlinearfiniteelementanalysisprogramANSYSisutilizedowingtoitscapabilitiestopredicteithertheresponseofreinforced concrete columns in thepost-elastic rangeor theultimate strengthof reinforcedconcrete columns. An extensive set of parameters is investigated including differentparameters;dimensionsof theholeswithdiameter0.1,0.15,0.2and0.3ofcolumnlength,their relativeposition incolumns,and the shapeofholes;circle andsquare.Acomparisonbetween the experimental results and thosepredictedby the existingmodels arepresented.Resultsandconclusionsmaybeusefulfordesigners,havebeenraised,andrepresented.

Keyword:Inelasticfiniteelementanalysis,columns,holes,strengthandtestingofmaterials.

1.Introduction

In the construction of modern buildings, a network of pipes and ducts is necessary toaccommodate essential services like water supply, sewage, air-conditioning, electricity,telephone,andcomputernetwork.Usually, thesepipesandductsareplacedunderneath thebeam soffit and, for aesthetic reasons, are covered by a suspended ceiling, thus creating adead space. Passing these ducts through transverse openings in the columns leads to areduction in the dead space and results in a more compact design. The provision of suchopenings may result in the loss of strength, stiffness and ductility and, hence, significantstructural damage may be sustained, if the provision of the openings is not consideredadequately during the design or construction stages. This is especially true for un-bracedstructures,sincelossofstiffnessleadstoredistributionofinternalforcesandmoments.

Themechanicalbehaviorofconcretebeamsandslabswithopeningshasbeenexamined inseveralstudiesanddesignruleshavebeenrecommended(AshoufA.F.etal.,1999),(TayelM. A. et al., 2004), (Simpson D., 2003), (Jiyang Wang et al., 2008) and (Mansur, M.A.,1998). However, in the case of concrete columns and walls with transverse openings,minimal research has been carried out and, currently, there is a lack of appropriate designrules.Columnsarecriticalelements,butingeneralonlycarryafractionoftheircapacityatnormalserviceloads.

NonlinearAnalysisofReinforcedConcreteColumnswithHolesEhabM.Lotfy

InternationalJournalofCivilandStructuralEngineeringVolume3Issue32013

656

Theresearchreportedinthispaperaimstoinvestigatethecompressiveresistance-capacityofconcrete columns with transverse holes with diameters 0.1, 0.15, 0.2 and 0.3 of columnlength,theirrelativepositionincolumns;inmiddlethirdandedgethirdoftestedcolumns,and the shape of holes; circle and square. Four columns with different holes were testedexperimentally to evaluate the effect of hole geometry and location. Analysis of theexperimentalresultsisusedtoderiveappropriatedesignrecommendations.

2.ObjectiveofthestudyThemainobjectivesofthisstudycouldbesummarizedinthefollowingpoints

1. To investigate the reduction in load carrying capacity of the reinforced concreteshortcolumnshavingcircleandsquarecross-sectionswithholeindifferentplaces.

2. To model the RC columns using three-dimensional non-linear finite elementanalysis.

3. Providerecommendationsforthedesignengineersandthestructuralcodesforthedesignofthereinforcedconcretecolumns.

3.Experimentalprogram

Fourconcretecolumnswithdifferentholesindifferentpositionandcontrolcolumnwithoutholeswerecasttoevaluatetheeffectofsectionlossonthecompressiveresistance-capacity.Theparametersexaminedexperimentallywerethediameter,relativeposition;wherecolumnisdividedtothreepartsinthecolumnslengthandalsoinloadingdirection,middlethirdandedgethird,andtheshapeofholes;circleandsquareshape.Figure1showsthedetailsoftheholesprovidedineachcolumn.

All columns were 1600mm height, 300m length and 300mm wide and contained bothlongitudinaland transverse reinforcement.The longitudinal reinforcement rebarscomprised4#16 mm in diameter, and the transverse reinforcement consisted of shear links, 8mm indiameter@200mm.Aclearconcretecoverof25mmwasprovidedinallcolumnspecimensandastrengtheningjacketwasprovidedatbothendsofeachcolumninordertominimizetheeffectoflocalbucklingofthelongitudinalreinforcement,thetestmatrixisshownintable1.

Table1:Detailsoftestedcolumnsspecimens

No Col.No.Dimension

(mm)fcu

(N/mm2) Reinf.Shape

ofholes

Dim.Ofholes

Positionofholes Notes

1 C1

300X300 254nos

of16mm

- - - Controlspecimen

2 C2 circle D=60mm Case(a)

3 C3 circle D=60mm Case(b)

4 C4 square L=60mm

Case(c)



657

Figure1:Detailsofreinforcementoftestedcolumns

4.Numericalfiniteelement

The analysis is carried out on 21-RC columns; the parameters of study were a holesdimensionswithdiameters0.1,0.15,0.2and0.3ofcolumnlength,theirrelativepositionincolumns;inmiddlethirdandedgethirdoftestedcolumns,andtheshapeofholes;circleandsquareasshownintable2.

4.1.BasicfundamentalsoftheFEmethod.

The basic governing equations for two dimensions elastic plastic FEM have been welldocumented,andarebrieflyreviewedhere.

I.Strain-displacementofanelement

[d]=[B][dU]Where:[B]isthestrain-displacementtransformationmatrix.Thematrix[B]isafunctionofboth the location and geometry of the suggested element, it represents shape factor. Thematrix[B]foratriangleelementhavingnodalpoints1,2and3isgivenby

[ ]

=

211213313223

123123

211332

000000

21

yyxxyyxxyyxxxxxxxx

yyyyyyB

Where xi and yi represent the coordinates of the node and represents the area of thetriangularelement,i.e.



658

33

22

11

111

det2yxyxyx

=

II.Stress-strainrelationorfieldequation

[d]=[D][d]

Here,[D]isthestress-straintransformationmatrix.ForelasticelementsthematrixfromtheHooke'slawleadsto[D]=[De].Forplasticelements,thePrandtl-Reussstress-strainrelationstogetherwiththedifferentialformofthevonMisesyieldcriterionasaplasticpotentialleadsto[D]=[Dp].

Theelasticmatrix,[De],isgivenbytheelasticpropertiesofthematerialwhereastheplasticmatrix,[De],isafunctionofthematerialpropertiesintheplasticregimeandthestress-strainelevation.Obviously,fortwo-dimensionalanalysis[De]and[Dp]dependonthestress-strainstate,i.e.planestressversusplanestrain.

The plastic matrix, [Dp], depends on the elastic-plastic properties of the material and thestresselevation.Comparing[De]and[Dp],itcanbeseenthatthediagonalelementsof[Dp]are definitely less than the corresponding diagonal elements in [De]. This amounts to anapparent (crease in stiffnessor rigiditydue toplasticyielding.Therefore, theplasticactionreducesthestrengthofthematerial.

III.Elementstiffnessmatrix[Ke]

[ ] [ ][ ]dvBDBK Te =][Thetransposematrixof[B]is[B]T.Inthecaseofthewell-knowntriangularelements[k]isrepresentedby;

[ ] [ ] [ ][ ]VBDBK T=

The element volume is V and for a two-dimensional body equals the area of the element multipliedbyitsthicknesst.

IV.Theoverallstiffnessmatrix[K]

Thestiffnessmatrixes[Ke]oftheelementsareassembledtoformthematrix[K]ofthewholedomain. The overall stiffness matrix relates the nodal load increment [dP] to the nodaldisplacementincrement[du]andcanbewrittenas

[dP]=[K][du]



659

Thisstiffnessrelationformsasetofsimultaneousalgebraicequationsintermsofthenodaldisplacement,nodalforces,andthestiffnessofthewholedomain.Afterimposingappropriateboundary conditions, the nodal displacements are estimated, and consequently the stressstrainfieldforeachelementcanbecalculated.

4.2.Materialmodeling

A linear-elastic, isotropic constitutive relation is adopted to describe the behavior ofuncrackedconcreteelementsintensionorcompressionfigure2andfigure3.

Forsteelreinforcement,elasticstress-strainbehaviorwasassumedtoobeythelinearrelationofHook'slawdescribedas:

[ ] [ ] }{)1(2}{}{ ee DGDE +==

Where {} and {} are column matrices of stress ij and ij respectively, G is the shearmodulus;Eisthemodulusofelasticityand isthePoisson'sratio.

Intheplasticregimethestress-plasticstrain;-p,behaviorofsteelwasassumedtoobeyasimplepowerlawasshowninfigure4withastrainhardeningexponentof0.02.

Figure2:Stress-strainrelationforplainconcreteintension

Figure3:Stress-strainrelationforplainconcreteincompression

Figure4:Stress-Strainrelationforsteelreinforcement



660

4.3.Resumeaboutusedprogram

Theimplementationofnonlinearmaterial lawsinfiniteelementanalysiscodesisgenerallytackledby the software development industry inoneof twoways. In the first instance thematerial behaviour is programmed independently of the elements to which it may bespecified.Using thisapproach thechoiceofelement foraparticularphysicalsystemisnotlimited and best practice modelling techniques can be used in identifying an appropriateelementtypetowhichany,ofarange,ofnonlinearmaterialpropertiesareassigned.Thisisthe most versatile approach and does not limit the analyst to specific element types inconfiguring the problem of interest. Notwithstanding this however certain softwaredevelopers provide specific specialised nonlinear material capabilities only with dedicatedelementtypes.

ANSYS (ANSYS Manual Set, 1998) and (Installation Guide ANYSYS) provides adedicated three-dimensionaleightnodes solid isoparametricelement,Solid65, tomodel thenonlinearresponseofbrittlematerialsbasedonaconstitutivemodelforthetriaxialbehaviourofconcrete(William,K.J.etal.,1975).

4.4.Finiteelementmodeling

4.4.1Geometry

ThedetailsoftestedcolumnswereshowninFigure5and6.Analyseswerecarriedouton21-columns specimens, where all columns had square cross-section with a 300 mm side and1600mmheight,thelongitudinalreinforcementrebarscomprised4#16mmindiameter,andthe transverse reinforcement consisted of shear links, 8mm in diameter@200mm, a clearconcretecoverof25mmwasprovidedinallcolumnspecimens

4.4.2Elementtypes

ExtensiveinelasticfiniteelementanalysesusingtheANSYSprogramarecarriedouttostudythe behavior of the tested columns. Two types of elements are employed to model thecolumns. An eight-node solid element, solid65,was used to model the concrete. The solidelementhaseightnodeswiththreedegreesoffreedomateachnode,translationinthenodalx,y, and z directions. The used element is capable of plastic deformation, cracking in threeorthogonal directions, and crushing. A link8 element was used to model the reinforcementpolymerbar;twonodesarerequiredforthiselement.Eachnodehasthreedegreesoffreedom,translation in the nodal x, y, and z directions. The element is also capable of plasticdeformation(ANSYSUsersManual).

4.4.3Materialproperties

Normalweightconcretewasusedinthefabricatedtestedcolumns.Thestress-straincurveislinearlyelasticuptoabout30%ofthemaximumcompressivestrength.Abovethispoint,thestress increases gradually up to the maximum compressive strength fc\, after that the curvedescendsintosofteningregion,andeventuallycrushingfailureoccursatanultimatestrain.

4.4.4Loadingandnonlinearsolution



661

Theanalyticalinvestigationcarriedouthereisconductedon21-RCcolumns;allcolumnsareraisedinverticalpositionwithbyverticalloadontopsurface.Ataplaneofsupportlocation,the degrees of freedom for all the nodes of the solid65 elements were held at zero. Innonlinearanalysis,theloadappliedtoafiniteelementmodelisdividedintoaseriesofloadincrementscalledloadstep.Atthecompletionofeachloadincrement,thestiffnessmatrixofthe model is adjusted to reflect the nonlinear changes in the structural stiffness beforeproceeding to the next load increment. The ANSYS program uses Newton-Raphsonequilibriumiterationsforupdatingthemodelstiffness.Forthenonlinearanalysis,automaticstepping in ANSYS program predicts and controls load step size. The maximum andminimumloadstepsizesarerequiredfortheautomatictimestepping.

Thesimplifiedstress-straincurveforcolumnmodelisconstructedfromsixpointsconnectedbystraightlines.Thecurvestartsatzerostressandstrain.PointNo.1,at0.3fc\iscalculatedforthestress-strainrelationshipoftheconcreteinthelinearrange.PointNos.2,3and4areobtainedfromEquation(1),inwhich iscalculatedfromEquation(2).PointNo.5isatandfc.Inthisstudy,anassumptionwasmadeofperfectlyplasticbehaviorafterPointNo.5asshowninfigure7,whichshowsthesimplifiedcompressiveaxialstress-strainrelationshipthatwasusedinthisstudy

..(1)

..(2)

..(3)

Figure5:Finiteelementmeshforatypicalcolumnmodel



662

Case(a) Case(b) Case(c) Case(d) Case(e)

Figure6:Detailsoftestedcolumnsspecimens

Figure7:Simplifiedcompressiveaxialstress-strainrelationship



663

Table2:Detailsoftestedcolumnsspecimens

No Col.No.Dim.(mm)

fcu(N/mm2) Reinf.

Shapeofholes

DimofholesNotes

1 C1

300*300 25 4#16mm

- - Controlspecimen

2 C2

Case(a)0.1L

Circleholes

3 C3 0.15L4 C4 0.2L5 C5 0.3L6 C6

300*300 25 4#16mm Case(b)0.1L

7 C7 0.15L8 C8 0.2L9 C9 0.3L

10 C10

300*300 25 4#16mm Case(c)0.1L

11 C11 0.15L12 C12 0.2L13 C13 0.3L14 C14

300*300 25 4#16mm Case(d)0.1L

15 C15 0.15L16 C16 0.2L17 C17 0.3L18 C18

300*300 25 4#16mm Case(e)0.1L

Squareholes

19 C19 0.15L20 C20 0.2L21 C21 0.3L

5.Inelasticanalysisresultsanddiscussion

Theparametricstudiesincludedinthisinvestigationareholesdimensionswithdiameters0.1,0.15,0.2and0.3ofcolumnlength,theirrelativepositionincolumns;case(a),(b),(c)and(d),andtheshapeofholes;case(a)and(d).Table3showstheanalyticallyresultsoftheultimateloads,deformationsandcompressivestressofconcrete,respectively.

Table3:Theoreticalresultsoftestedcolumnsspecimens

No Col.No.

Concretestress(N/mm2)

UltimateDef.(mm)

UltimateLoad(KN)

Notes

1 C1 25 1.30 139.00 Controlspecimen

2 C2 25 1.24 138.503 C3 25 1.24 137.404 C4 24.8 1.24 127.445 C5 24.6 1.19 112.326 C6 25 1.22 135.507 C7 25 1.15 131.768 C8 24.8 1.10 123.129 C9 24.6 0.90 103.68

10 C10 25 1.05 121.4011 C11 25 0.89 107.6012 C12 24.7 0.79 93.4013 C13 24.6 0.58 65.70



664

14 C14 25 0.96 116.5015 C15 25 0.89 106.0016 C16 24.7 0.86 102.1017 C17 24.6 0.82 98.0018 C18 25 1.12 130.5019 C19 25 1.06 119.5020 C20 24.8 1.03 111.2021 C21 24.6 0.98 101.52

5.1.Experimentalvalidation

The validity of the proposed analytical model is checked through extensive comparisonsbetweenanalyticalandexperimentalresultsofRCcolumnsundercompressionload.Figure8showsthetheoreticalandexperimentalload-deformationcurveoffromC1toC4andcontrolcolumn.

Thetheoreticalresultsfromfiniteelementanalysisshowedingeneralagoodagreementwiththeexperimentalvalues.

Figure8:Thetheoreticalandexperimentalload-deformationcurveoftestedcolumnsfromC1toC4andcontrolcolumn.

5.2.Holesdimensions

Figures9,10,and11showthetheoreticalload-deformationofcolumns(C1,C2,C3,C4andC5), (C1, C6, C7, C8 and C9) and (C1, C18, C19, C20, and C21); which have hole



665

dimensions 0.00, 0.10, 0.15, 0.2, and 0.3 of columns length respectively; increasing holedimensionsdecreasethetoughnessandductilityoftestedcolumns.

FromTable3,itcanbeseenthat,ultimateloads,andultimatestrainofC2,C3,C4andC5toC1are(99.6,98.8,91.6and80.5%),and(95.3,95.3,95.3and91.5%)respectively.

Ultimate loads, and ultimate strain of C6, C7, C8 and C9 to C1 are (97.4, 94.7, 88.5 and74.5%),and(93.4,88.4,84.6and69.2%)respectively.

Ultimateloads,andultimatestrainofC18,C19,C20andC21toC1are(93.8,91.9,80.2and72.8%), and (86.2, 81.5, 79.2 and 75.4%) respectively. Figure 12 shows the effect of theincreasingholedimensionsontheultimateloadofcolumnsresists,wheretheincreasingofholedimensionsmorethan0.15oftestedcolumnslengthleadstoreductioninultimateloadsoftestedcolumnsto80%.Theincreasingofholedimensionmorethan0.15ofcolumnlengthdecreasethetoughnessandductilityofcrosssection,whereitisincreasethebucklingeffectoftestedcolumn,soithasasignificanteffectonultimatestrain,andultimateloadsthatthecolumnsresist.

Figure9:Thetheoreticalload-deformationofcolumnsC1,C2,C3,C3andC5


Figure11:Thetheoreticalload-deformationofcolumnsC1,C18,C19,C20,andC21

Figure12:Ultimateloadoftestedcolumnstocontrolandholedimensions/col.Lengthratio

5.3.Positionofholesincolumns

Figures13and14showthetheoreticalload-deformationofcolumns(C3,C7,C11,C15andC1)and(C4,C8,C12,C16andC1)respectively;whichhavepositionofholescase(a),case



666

(b),case(c),case(d),andcontrolspecimen;holesintheedgethirdhassignificanteffectontheultimate loadsanddeformationsof testedcolumns,henceaffect the toughnessof testedspecimens,butholesinmiddlethirdhaslimitedeffectontheultimateloadsanddeformationsoftestedcolumns

FromTable3,itcanbeseenthat,ultimateloads,andultimatestrainofC3,C7,C11,C15andC1are(98.8,94.8,77.4and76.2%),and(95.3,88.4,68.4and68.9%)respectively.Ultimateloads,andultimatestrainofC4,C8,C12,C16andC1are(91.6,88.5,77.4and73.4%),and(95.3,84.6,68.4and66.15%)respectively

Figure15showsthat;holewithcase(c)and(d)hasasignificanteffectontheultimateloadoftestedcolumnswithholedimensions0.15and0.2ofcolumnlength

Figure 16 shows that; hole with case (b), case (c ) and (d) has a significant effect on thedeformationoftestedcolumnswithholedimensions0.15and0.2ofcolumnlength



Figure15:PositionofholesandPu/PcontrolforholeDim.(0.15Land0.2L)

Figure16:PositionofholesandDef./Def.controlforholeDim.(0.15Land0.2L)

5.4.Shapeofholes

Figures17and18showthetheoretical load-deformationof testedcolumns(C3andC19toC1)and(C4andC20 toC1);whichconfirmthatusingsquarehole in testedcolumnhasasignificant effect on theultimate loads anddeformation so it decreased the toughness andductilityoftestedcolumns.



667

FromTable3,itcanbeseenthat,ultimateloads,andultimatestrainofC3andC19toC1are(98.8,and85.9%),and(95.3and81.53%)respectively,ultimateloads,andultimatestrainofC4andC20toC1are(91.6,and80%),and(97.6and79.2%)respectively.

Usingsquareholeintestedcolumnhasasignificanteffectonthebehavioroftestedcolumns;whereitreducedtheductility,toughness,ultimateloadandincreaseddeformation

Figure17:Thetheoreticalload-deformationofcolumnsC3,C19,andC1

Figure18:Thetheoreticalload-deformationofcolumnsC4,C20,andC1

5.5Conclusion

Theinelasticbehaviorof21columnsareinvestigatedinthecurrentstudyundertheeffectofincreasingloadingemployingtheinelasticFEanalysisprogramANSYS.Severalparametersare investigated including theparametersof study were aholesdimensionswithdiameters0.1,0.15,0.2and0.3ofcolumnlength,theirrelativepositionincolumns;inmiddlethirdandedgethird,andtheshapeofholes;circleandsquare.Thestudyfocusesontheconsequencesof the investigated parameters on the deformation and ultimate resisting load. Theconclusionsmadefromthisinvestigationare:

1. The theoretical results from Finite Element Analysis showed in general a goodagreementwiththeexperimentalvalues.

2. Theholewithdiameter more than0.15of columns lengthhas significant effect of thecolumnbehavior;reducingtheductilityandtoughnessoftestedcolumns.

3. The increasing of hole dimensions to more than 0.15 of columns length leads toreductioninultimateloadsoftestedcolumnsto80%.

4. Using square hole in tested column has a significant effect on the behavior of testedcolumns

5. Holescanbemadeinmiddlethirdofcolumnswithdiameterupto0.15columnlength.

6.References

1. Ashouf A.F. and Rishi G., (1999), Tests of reinforced concrete continuous deepbeamswithwebopenings,ACIstructuraljournal,97(3),pp418-426.

2. TayelM.A.,SolimanM.H.andIbrahimK.A.,(2004),Experimentalbehaviorofflatslabs with openings under the effect of concentrated loads, Alexandria engineeringjournal,43(2),pp203-214.



668

3. SimpsonD., (2003),Theprovisionofholes in reinforced concretebeams,Concrete(London),37(3),pp24-25.

4. Jiyang Wang, Masanobu SAKASHITA, Susumu Kono, Hitoshi Tanaka, MakotoWarashina., (2008), A macro model for reinforced concrete structural walls havingvariousopeningratios,14thworldconferenceonearthquakeengineering,October12-17,Beijing,China

5. Mansur,M.A.,(1998),EffectofopeningsonthebehaviorandstrengthofR/Cbeamsinshear,Cementandconcretecomposites,ElsevierscienceLtd.,20(6),pp477-486.

6. ANSYS Manual Set, (1998), ANSYS Inc., Southpoint, 275 Technology Drive,Canonsburg,PA15317,USA.

7. Installation Guide (2010), ANYSYS VERSION 10, Computer software forstructuralengineering.

8. William,K.J.andWarnke,E.D.,(1975),ConstitutivemodelfortheTriaxialbehaviorof concrete, Proceedings of the international association for bridge and structuralengineering,19,p174,ISMES,Bergamo,Italy

nonlinear analysis of reinforced concrete columns with holes

Documents