Practical Analysis 4 Biaxil Column

7/30/2019 Practical Analysis 4 Biaxil Column

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2011 SLOVAK UNIVERSITY OF TECHNOLOGY24

T. BOUZID, K. DEMAGH

PRACTICAL METHOD FOR

ANALYSIS AND DESIGN OF

SLENDER REINFORCEDCONCRETE COLUMNS

SUBJECTED TO BIAXIAL

BENDING AND AXIAL LOADS

KEY WORDS

slender, Bi-axial,

bending,

RC column,

encased composite columns,

composite columns.

ABSTRACT

Reinforced and concrete-encased composite columns of arbitrarily shaped cross sectionssubjected to biaxial bending and axial loads are commonly used in many structures. For this

purpose, an iterative numerical procedure for the strength analysis and design of short and

slender reinforced concrete columns with a square cross section under biaxial bending and an

axial load by using an EC2 stress-strain model is presented in this paper. The computational

procedure takes into account the nonlinear behavior of the materials (i.e., concrete and

reinforcing bars) and includes the second - order effects due to the additional eccentricity of

the applied axial load by the Moment Magnification Method. The ability of the proposed

method and its formulation has been tested by comparing its results with the experimental ones

reported by some authors. This comparison has shown that a good degree of agreement and

accuracy between the experimental and theoretical results have been obtained. An average

ratio (proposed to test) of 1.06 with a deviation of 9% is achieved.

T. BOUZID,

K. DEMAGH

email:[email protected],[email protected]

Research field: concrete columns, axial and biaxial

loads

DepartmentofCivilEngineering,

FacultyofSciences,

UniversityofBatnavinAlgeria,

5thAvenueChahidMedBoukhlouf,

Batna05000,Algeria

2011/1 PAGES 24 32 RECEIVED 15. 2. 2010 ACCEPTED 20. 5. 2010

1. INTRODUCTION

Reinforcedandconcrete-encasedcompositecolumnsofarbitrarily

shaped cross sections subjected to biaxial bending and an axial

load are commonly used in many structures, such as buildings

and bridges. Acomposite column is acombination of concrete,

structuralsteelandreinforcedsteeltoprovideforanadequateload

carrying capacity of the member. Such composite members can

therefore provide rigidity, usable floor areasand saving for mid-

to-tall buildings. Many experimental and analytical studies have

beencarried outonreinforced andcompositemembersin thepast.

Furlong [1] has carried out analytical and experimental studies

onreinforced concretecolumnsusingthewell-known rectangular

stress block for the concrete compression zone in the analysis.

Brondum-Nielsen [2] has proposed amethod of calculating the

ultimatestrength capacity of cracked polygonal concretesections

usingarectangularstressblockintheconcretecompressionzone

of asection under biaxial bending. Hsu [3, 4] has presented

theoretical and experimental results for L-shaped and channel

- shaped reinforced concrete sections. Dundar [5] has studied

reinforcedconcreteboxsectionsunderbiaxialbendingandanaxial

load.Rangan[6]haspresentedamethodtocalculatethestrength

ofreinforcedconcreteslendercolumns,includingcreepdeflection

due to asustained load as an additional eccentricity; the method

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25PRACTICAL METHOD FOR ANALYSIS AND DESIGN OF SLENDER REINFORCED...

compares with the ACI 318-Building Code Method [7]. Dundar

and Sahin [8] have researched arbitrarily - shaped reinforced

concrete sections subjectedto biaxial bending and an axial load

using Whitneys stress block [9] in the compression zone of

aconcretesection.RodriguezandOchoa[10]andFafitis[11]have

suggested numerical methods for the computation of the failure

surfaceofreinforcedconcretesectionsofanarbitraryshape.Hong

[12]hasproposedasimpleapproachforestimatingthestrengthof

slender reinforced concrete columns withan arbitrarily - shaped

cross section using anonlinear stressstrain relationship for thematerials. SaatciogluandRazvi[13] havepresentedexperimental

researchthatinvestigatesthebehaviorofhigh-strengthconcrete

columns confinedby arectilinear reinforcement under concentric

compression. Furlong, et al. [14] have examined several design

proceduresfor anultimatestrength analysisof reinforcedconcrete

columnsandcomparedarangeofshortandslenderexperimental

columnsunderashort-termaxialloadandbiaxial bending.Mirza

[15]hasexaminedtheeffectsofvariablessuchastheconfinement

effect,theratioofstructuralsteeltoagrossarea,thecompressive

strengthofconcrete,theyieldstrengthofsteelandtheslenderness

ratio,ontheultimatestrengthofcompositecolumns.Lachance[16],

Chen,etal.[17]andSfakianakis[18]haveproposedanumerical

analysis method for short composite columns with an arbitrarily- shaped cross section.The confinement provided by lateral ties

increasestheultimatestrengthcapacityandductilityofreinforced

concrete columns under combined biaxial bending and an axial

load. Thegainsin strengthand ductility inconcreteare obtained

by many confinement parameters, e.g., the compressive strength

of the concrete, the longitudinal reinforcement, the typeand the

yieldstrengthofthelateralties,thetiespacing,etc.Duetosuch

parameters,adeterminationofthemechanicalbehaviorofconfined

concrete is not as easy as that with unconfined concrete. Some

researchers,forinstance,KentandPark[19],SheikhandUzumeri

[20],SaatciogluandRazvi[21],Chung,etal.[22]havepresented

astressstrain relationship to describe the confined concretes

behavior. Dundar, et al. [23] have carried out an experimental

investigationof thebehavior ofreinforcedconcretecolumns,and

atheoretical procedure for an analysis of both short and slender

reinforcedandcompositecolumnswithanarbitrarily-shapedcross

sectionsubjectedtobiaxialbendingandanaxialloadispresented.

In the proposed procedure, nonlinear stressstrain relations are

assumedforconcrete,reinforcedsteelandstructuralsteelmaterials.

Thecompressionzoneoftheconcretesectionandtheentiresection

of the structural steel are divided into an adequate number of

segmentsinordertousevariousstressstrainmodelsfortheanalysis.

Theslendernesseffectofthememberistakenintoaccountbyusing

theMomentMagnificationMethod(MMM).Thetestresultswere

compared with the theoretical results obtained by adeveloped

computerprogramwhichusesvariousstressstrainmodelsforthe

confined or unconfined concrete in the compression zoneof the

member. The comparison shows agood degree agreementof the

resultsobtainedbytheproposedprocedure.

Themainobjectiveofthispaperistopresentaniterativecomputing

procedure for the rapid design and ultimate strength analysis

of asquare cross-section for both short and slender reinforced

concrete elements subjected to biaxial bending and an axial

load. For this aim asimple model has been developed, which

considers various unconfined concrete stressstrain models foraconcretecompressionzoneforbothshortandslenderreinforced

concrete columns. Asimple formula to predict the resistance

capacityofbiaxiallyloadedshortreinforcedconcretecolumnswith

asquarecross-sectionisintroduced.Basedonanumericalanalysis,

acapacity factor which represents the ratio of PM interaction

diagramsinauniaxialloadingcolumntoabiaxialloadingcolumn

is proposed. The relationships between the capacity factor (K)

andallthedesignvariablesare establishedby regression,and the

required P-M interactiondiagramof the biaxial RC column can

be easily constructed without conducting refined analyses. The

slendernesseffectofthe memberis thentakenintoaccountusing

theMomentMagnificationMethod.Finally,thetheoreticalresults

obtained from using the proposed model is compared with thetheoretical andexperimentalresultsavailable in theliterature for

shortandslendercolumns.

2. ANALYTIC METHOD

2.1. Assumptions

Theproposedmethodisbasedonthefollowingassumptions:

1.Theplanesectionsremainplaneafteranydeformation(Bernoullis

assumption).

2.Arbitrary monotonic stressstrainrelationships for eachof the

threematerials(i.e.,concrete,structuralsteelandthereinforcing

bars)maybeassumed.

3. Thelongitudinalreinforcingbarsareidenticalindiameterandare

subjectedtothesameamountofstrainastheadjacentconcrete.

4.Theeffectofcreepandthetensilestrengthofconcreteandany

directtensionstressesduetoshrinkage,etc.,areignored.

5.Sheardeformationisignored.

2.2. Stressstrain models for the materials

Theanalysisutilizeswellestablishedmodelsforconcrete(confined,

and unconfined) and reinforcing steel. Figure 1 and 2 show the

stressstrainmodelsfortheconcreteandsteel.

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2.3. Capacity Factor

SincetheultimateresistingcapacityofanRCcolumnisgoverned

by many variables and is gradually reduced as the degree of

axial load increases (P/Agf

c), it is necessary in many cases to

conduct arefined numerical analysis that considers the material

nonlinearitiesin order to accurately predict the ultimatestrength

ofabiaxial RCcolumn.In order toanalyseand design(directly)

biaxialRCcolumns,acapacityfactorK,whichrepresentsthe ratio

of the PM interaction diagrams in an uniaxial loading column

to abiaxial loading column, is introduced. If the dimensions of

the concrete cross-section and the material properties have been

selected, the interaction diagrams for uniaxial loading are then

easilyconstructedbyintroducingthecapacityfactor(K).Onecan

easilyobtaintheinteractiondiagramsforbiaxialloadingwithany

angle ofaresultant bendingmomentMBIA

.The strengthcapacity

factor(SCF)isdefinedastheratioofthedistancefromtheorigin

(eccentricity)forauniaxialinteractiondiagramtoabiaxialloading

interactiondiagram atthesamedegreeofaxialloadlevel(P/P0),

Fig.2:

PUNI

=PBIA

(1)

(2)

(3)

Where Kis the Capacityfactor, MUNI

is the equivalent uniaxial

moment,andMBIA

istheresultantmomentforthebiaxialloading.

Fig. 1 Stress- strain relationship of confined and unconfined concrete in compression.

Fig. 2Elastic -Plastic bilinear behavior for steel.

Fig. 3Determination of the strength resistance factor K.

AxialloadP/P0

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In order to introduce aformulafor thecapacityfactor(K), some

difficulties must be overcome, because the interaction diagrams

mustbedeterminedforthebiaxialanduniaxialcross-sectionswith

thesamedesignvariablesusedbyDundaretal.[8];moreoveran

infinitenumberofpossibleRCsectionscanbeselectedforthesame

setofexternalforcesapplied.Hence,indeterminingRCinteraction

diagrams, all the variables need to be assumed on the basis of

practical limitations and the design code requirements, R.P.A.99-

03[2003].The commonly usedcompressive strengthof concrete

andstressofsteelfordesignarefc=25,30and40MPafornormalconcreteandf

y= 400MParespectively.Inaddition,thesteelratio

rangesfrom1%to4%inthecurrentzones.Duetothesymmetryof

thesectionandthereinforcement,theangleofloadingissupposed

tovaryfrom0 (uniaxial) to45 (biaxial) with anincrement of

15.Table1givestherangeofvariablesadoptedforthedesignof

experimentalplanstobeincludedintheanalysis.

Theeffectofthe differentparametersonthe interactiondiagrams

canbesummarizedas:

- the cross-sections capacity increases with the increase in the

concretesstrengthinthe sameproportionbetween0.1to0.7,but

especiallyintheregion0.2P/P00.5aroundthebalancedpoint;

-thesectionscapacityincreaseswiththeincreaseinthesteelratio

overthelengthofthecurves;

-thesectioncapacitydecreaseswhentheangleofloadingincreases

overthecurve,especiallyintheregionoftensioncontrolanduntil

P/P0=0.6,overthisvaluetheresistancecapacityisthesameforthe

differentangles.Foralltheexperimentplans,thecalculatedcapacity

factorsKforthedifferentparametersselectedandasafunctionof

theaxialloadlevelaredepictedinFigs.2-4.Ninetypicalresults

ofthestrengthcapacityfactorKcalculatedwith3 ratios,3angles

and3typesofconcretewitharangeofaxialloadsequalto0,1P0to

0,7P0

areobtained.

Figures4to6showthatthestrengthcapacityfactorK:

Decreases for large values of compressive strength, especially

intheregionoftensioncontrolP0.4P0.Inthecompression

Tab. 1Parameters variation.

Parameters Values

Cross-sectionshape Square

Biaxialbendingangle()with

respecttoastrongaxis

=0,15,30,45

Reinforcementdistribution Uniformlydistributedatfourfaces

AxialloadP/P0whereP0=fcAg 10 values from 0,1P0 to 0,7P0

Compressiveconcretestrength fc=25,30,40MPa

Steelstrength fy=400Mpa

Geometricreinforcementratio s=1%,2%,4%

K

factor

Fig. 4 Variation of K factor in accordance with concrete compression

strength.

Fig. 5 Variation of K factor in accordance with the steel ratio.

K

factor

Fig. 6 Variation of K factor in accordance with the loading angle.

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controlregion,thefactordecreasesproportionallytolargeaxial

loadvalues;

Increases withlargevaluesofthesteel ratio,butthevaluesof

Karelowerintheregionofcompressioncontrol;

Increaseswithlargevaluesofaloadingangle,especiallyinthe

regionoftensioncontrol.

Inordertodetermineareasonableregressionformula,theeffectofeachdesignvariablewas studied,Figs.4-6.Sincethe coefficients

aregraduallyincreasedordecreasedaccordingtochangesineach

design variable and represent anonlinear characteristic, asecond

orderpolynomialisassumedintermsofthedesignvariables.The

regressionformularepresentedinEq.4isfinallyselectedforthe

capacityfactorK:

K=a0+a

1f

c+a

2

s+a

3 ( ) + a

4+a

5f

c+a

6

s+a

7( )2+

a82+a

9fc

s+a

10f

c ( ) + a

11fc+a

12

s( ) + a

13

s+

a14

( ) (4)

Wherefcisthe compressivestrengthofconcrete(MPa),P/P

0the

loadingleveloftheaxialforce,sthesteelratio(100.A

st/A

c)and

theloadingangle(),thevariablestakethevalues:

3. SHORT COLUMNS

An experimental analysisis carried out inorder tocompare the

numerical results obtained by the proposed formula with the

experimentsresults.Forthispurpose,acomparisonofthecapacity

factor (K) iscalculated with the proposed formula and those of

tests for the specimens selected. Table 3 shows the comparison

withthe experiment resultsin Hsu[1988]; itappearsclearlythat

theproposedformulationgivesgoodresults:anerrorof7%forthe

specimens governedbytensioncontroland6% forthespecimens

governedbycompressioncontrol,withadeviationof3%.

4. SLENDER COLUMNS

AccordingtotheACI318-99provisionsforthedesignofslender

columns,strength isdefinedas thecross-section strength; on the

otherhand,theappliedexternalmomentismagnifiedduetosecond

ordereffects,i.e.,themomentmagnificationmethodisused.

Tab. 2 Values of coefficients.

a1

a2

a3

a4

a5

a6

a7

0,002271 0,009508 0,796264 0,006587 0,000005 -0,010319 -1,177950

a8

a9

a10

a11

a12

a13

a14

-0,000022 0,000268 -0,010901 -0,000054 0,126399 0,000725 -0,007453

Tab. 3 Comparison with experimental results Hsu [23].

Experimental

investigation

Column

specimen

Values

from

Tests

Ktest

Values

from

Formula

KK

Ratio

Ktest

/ KK

Error

%

Tension Control

Hsu

U-1 1,025 1,102 0,930 7

U-2 1,024 1,122 0,913 9

U-3 1,015 1,138 0,892 11

U-6 0,859 0,912 0,942 6

Ramamurthy

B-3 1,061 1,081 0,98 2

B-4 1,086 1,095 0,992 0

B-7 1,032 1,123 0,919 8

B-8 1,03 1,151 0,895 10

average 7%

Compression Control

Hsu

S-1 1,198 1,092 1,097 10

S-2 1,118 1,084 1,031 3

U-4 1,047 1,148 0,912 9

RamamurthyB-1 0,97 0,995 0,975 2

B-6 1,063 1,151 0,923 8

Heimdahland

Bianchini

BR-3 1,108 1,059 1,046 5

BR-4 1,173 1,059 1,107 11

CR-3 1,164 1,131 1,026 3

CR-4 1,184 1,134 1,044 4

ER-1 0,946 1,042 0,907 9ER-2 1,09 1,055 1,033 3

Fr-1 1,054 1,124 0,938 6

Fr-2 1,13 1,154 0,979 2

average 6%

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4.1 Cross-section strength

Fordesignpurposes,whenamemberissubjectedtoanaxialload

PandmomentM,itisusuallyconvenienttoreplacetheaxialload

andmomentwithanequalloadPappliedateccentricitye=M/P.The

computationofthedesignstrengthisthenobtainedthroughastrain

compatibilityanalysis.Theactualcompressivestressdistributionis

replacedbyanequivalentrectangulardistribution.Incomputingthe

valueofPandM,whichproducethestateofincipientfailure,the

widthofthestressblockistakenas0,85f candthedepthisa=1.c(cisthedepthoftheneutralaxis).Thefactor

1isgivenby(with

theconcretecompressivestrengthfcgiveninMPa):

1=1.09-0.008f

c (5)

With:0.651

0.85

Thesolutionoftheequationsstatingtheequilibriumbetweenthe

external and internal forces as well as the external and internal

moments,withintheconstraintsofstraincompatibility,determines

thenominalaxialloadPn,whichcanbeappliedataneccentricitye

foranyeccentricallyloadedcolumn.

4.2 Magnified moments

Forashort column,the momentmagnificationdue toslenderness

effects is negligibly small; on the other hand, if the column is

sufficientlyslender, themaximummomentactingon thecolumn

increases nonlinearly as P increases. For the same externally

appliedmomentM,thestrengthoftheslendercolumnisreducedas

comparedtothestockycolumn.

The ACI 318-99 specifies that axial loads and end moments in

columns may be determined by aconventional elastic frame

analysis. The member isthen tobe designed for that axial load

and asimultaneous magnified column moment. TheACI 318-99

equation for amagnified moment Mc for columns in non-sway

framesis:

Mc=

ns.M

2 (6)

thenon-swaymomentmagnificationfactorns

isgivenby:

ns

=Cm

/(1-Pu/P

c)1 (7)

where M2 = the larger factored end moment; C

m = equivalent

uniformmomentdiagramfactor(Cm

=1forthecaseofsupportswith

equalbendingatbothends:purecurvature);

Pu

=factoredaxialloadactingonthecolumn;=capacityreduction

factor(=1toperformthiscomparativeanalysis,whichisdesigned

toconsidertheinevitablerandomvariabilityofthematerials);Pc=

thecriticalbucklingloadgivenby:

Pc=EI/(kl

u) (8)

Where:EI=effectiverigidity;

4.3 Magnified moments

Forashort column,the momentmagnificationdue toslenderness

effects is negligibly small; on the other hand, if the column is

sufficientlyslender, themaximummomentactingon thecolumn

increases nonlinearly as P increases. For the same externally

appliedmomentM,thestrengthoftheslendercolumnisreduced

ascomparedto thestockycolumn.TheACI318-99specifiesthat

axialloads and end moments incolumns may bedeterminedby

aconventional elastic frame analysis.The member is then to be

designedforthataxialloadandasimultaneousmagnifiedcolumn

moment.TheACI318-99equationforamagnifiedmomentMcfor

columnsinnon-swayframesis:

Mc=ns.M2 (9)

Thenon-swaymomentmagnificationfactorns

isgivenby:

ns

=Cm

/(1-Pu/P

c)1 (10)

WhereM2 isthelarger factoredend moment,C

m the equivalent

uniformmomentdiagramfactor(Cm

=1forthecaseofsupportswith

equalbendingatbothends:purecurvature),Pu

isthefactoredaxial

loadactingonthecolumn,thecapacityreductionfactor(=1to

performthiscomparativeanalysis,whichisdesignedtoconsiderthe

inevitablerandomvariabilityofthematerials)andPcisthecritical

bucklingloadgivenby:

Pc=EI/(klu) (11)

Where:EI=effectiverigidity;

4.4 Computer analysis of reinforced concrete

columns tested by Dundar et al.

Dundar, et al. have tested [23] fifteen (15) reinforced concrete

columns, twelve specimens of square tied columns and three

L-shaped sections. The cross section details and dimensions of

eachspecimenareshownintheTable3.Thereinforcedconcrete

column specimens were cast horizontally inside aformwork in

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Structural Laboratory at Cukurova University, Adana, Algeria.

The longitudinal reinforcement consisted of 6mm and 8mm

diameterdeformedbarswithayieldstrengthof630and550MPa,

respectively.Lateralreinforcementswerearrangedusing6mmand

6.5mmdiameterdeformedreinforcingbarswithayieldstrengthof

630MPaforthespecimens.Theparametersofthespecimensandthe

resultsarepresentedinTable4.Thestress-strainmodel CEC [26]isthesameoneisusedforthedeterminationofthecapacityfactorK.

The reinforced concrete column specimens were tested with

pinned conditions at both ends under ashort-term axial load and

biaxial bending. These specimens were also analyzed for their

ultimate strength capacities using acomputer program. In the

ultimate strength analysis, various stressstrain models and the

experimentalstressstrainrelationshipsobtainedfromthecylinder

specimensofthecolumnsbytheauthorswereusedfortheconcrete

compression zone in order to compute the theoretical ultimate

strengthcapacityandtocompareitwiththeexperimentalresultsof

thecolumn specimens.Agooddegreeof agreementwas obtained

between the theoretical results according to each of the concrete

stressstrainmodelsandtheexperimentalresults.Themeanratios

ofthe comparative resultsindicatethatthe shape ofthe concretestressstrainrelationshiphas little effect on the ultimatestrength

capacityofthecolumnmembers;ontheotherhand,themaximum

permissible strain plays the most important role on the ultimate

strengthcapacity.

These columns are then solved by the proposed method for the

ultimatestrengthanalysisusingthe parabola-rectangledefinedby

theEC2,whichisappliedtoobtainthecapacityfactorofthesection

Tab. 4 Specimen details of RC columns.

SpecimenL

(mm)

fc

(MPa)

ex

(mm)

ey

(mm)

/s

(mm/cm)

lateral

Ratio

= As/A

g

%

Ntest

(kN)CEC

C1 870 19.18 25 25 6/12.5 1.13 89 88.95

C2 870 31.54 25 25 6/15 1.13 121 126.78C3 870 28.13 25 25 6/10 1.13 125 116.51

C4 870 26.92 30 30 6/8 1.13 99 93.57

C5 870 25.02 30 30 6/10 1.13 94 88.83

C11 1300 32.27 35 35 6.5/10.5 2.0 104 88.81

C12 1300 47.86 40 40 6.5/10.5 2.0 95 91.39

C13 1300 33.10 35 35 6.5/10.5 2.0 98 90.00

C14 1300 29.87 45 45 6.5/12.5 2.0 58 63.02

C21 1300 31.7 40 40 6.5/10.5 0.89 238 233.41

C22 1300 40.76 50 50 6.5/10.5 0.89 199 208.83

C23 1300 34.32 50 50 6.5/10.5 0.89 192 189.38

Fig. 7 Variation of factor in accordance with slenderness. Fig. 8 Variation of factor in accordance with the value of d.

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K;thentheultimatebendingmomentofthesectionMUT

isobtained.

TheultimatemomentoftheslendermemberMUMS

iscomputedby

usingthemagnificationfactor.Thetheoreticalresultsobtainedfor

themaximumresistingmomentcapacityaswellasthetestresults

arepresentedinTable6forcomparison.

Agooddegreeofaccuracyhasbeenobtainedbetweenthetheoretical

andtheexperimentalresults;anaverageratioof1.06withavariation

coefficientof9%wasachieved.

CONCLUSIONS

An iterative numerical procedure for the strength analysis and

design of short and slender reinforced concrete columns which

square cross sectionsunder biaxial bending andan axial load by

using theEC2 stress-strainmodelis presentedin this paper. The

computationalproceduretakesintoaccountthenonlinearbehavior

ofthematerials(i.e.,concreteandreinforcingbars)andincludesthe

secondordereffectsduetotheadditionaleccentricityoftheapplied

axialloadbytheMomentMagnificationMethod.

Thecapabilityoftheproposedmethodanditsformulationhasbeen

testedbymeansofcomparisonswiththeexperimentalresultsreported

by some authors. The theoretical and experimental results show

thatagooddegree of accuracy hasbeen obtained;an averageratio

(proposedtotest)of1.06withadeviationof9%hasbeenachieved.On the other hand, the compressive strength of concrete and its

correspondingcompressivestrainarethemosteffectiveparametersof

theultimatestrengthcapacityofcolumnmembers.Consequently,the

proposedformulationcansimulatethebehaviorofslendermembers

underbiaxialloadingwithagooddegreeofaccuracy.

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