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Hindawi Publishing CorporationAdvances in Civil EngineeringVolume 2013 Article ID 275657 13 pageshttpdxdoiorg1011552013275657
Research ArticleAn Analytical Step-by-Step Procedure to Derive the FlexuralResponse of RC Sections in Compression
Piero Colajanni1 Marinella Fossetti2 and Maurizio Papia3
1 Dipartimento di Ingegneria Civile Universita di Messina Contrada Di Dio 98166 Messina Italy2 Facolta di Ingegneria Architettura e Scienze Motorie Universita Kore di Enna Cittadella Universitaria 94100 Enna Italy3 Dipartimento di Ingegneria Civile Ambientale Aerospaziale dei Materiali Universita di Palermo Viale delle Scienze90128 Palermo Italy
Correspondence should be addressed to Maurizio Papia mauriziopapiaunipait
Received 6 March 2013 Revised 5 August 2013 Accepted 8 August 2013
Academic Editor Andreas Kappos
Copyright copy 2013 Piero Colajanni et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper proposes an analysis procedure able to determine the flexural response of rectangular symmetrically reinforcedconcrete sections subjected to axial load and uniaxial bending With respect to the usual numerical approaches based on the fibredecompositionmethod this procedure is based on the use of analytical expressions of the contributions to the equilibrium given bythe longitudinal reinforcement and the concrete region in compression which depend on the neutral axis depth and the curvatureat each analysis step The formulation is developed in dimensionless terms after a preliminary definition of the geometrical andmechanical parameters involved so that the results are valid for classes of RC sectionsThe constitutive laws of thematerials includeconfinement effect on the concrete and postyielding behaviour of the steel reinforcement which can be assumed to be softeningbehaviour for buckled reinforcing bars The strength and curvature domains at the first yielding of the reinforcement in tensionand at the ultimate state are derived in the form of analytical curves depending on the compression level therefore the role of asingle parameter on the shape of these curves can easily be deduced The procedure is validated by comparing some results withthose numerically obtained by other authors
1 Introduction
The performance of reinforced concrete frames under severeearthquakes largely depends on the ability of the beam andcolumn sections to undergo large inelastic deformationsEspecially this ability plays a decisive role in existing build-ings the safety level of which can be estimated by employingnonlinear analysis tools (like pushover) needing a carefulinput in terms of strength domains and moment-curvaturerelationships of the critical sections
The evaluation of the moment-curvature response ofcritical sections of RC members is a complex issue mainlybecause of the interaction of various parameters constitutivelaws of materials in the elastic and plastic ranges membergeometry buckling phenomena in reinforcing steel bars andloading conditions
In order to include in a computer software the cross-sec-tion strength domain andmoment-curvature relationships ofreinforced concrete members two different approaches are
usually followed the use of strength domains [1ndash3] andmoment-curvature relationships of a usually bilinear or tri-linear prefixed shape [4ndash6] in which a degrading stiffnessmodel reproduces the effect of yielding and damage of mate-rials the layered section approach based on the fibre decom-position method [7ndash9]
The layered section approach gives a realistic and almostcomplete description of the behaviour of the critical regionsof a RC framed structure also allowing to include the bondslip effects in the moment-rotation response However theapplication of this approach to large structures with manydegrees of freedom has some limitations because numericalintegrations and a prediction of the state of materials must becarried out considering the entire layers of each region wherea plastic hinge can occur thus a high amount of informationis required to characterize each section involved and a largenumber of numerical operations is needed to reach an accept-able level of error This results in heavy computational effortsand convergence problems for nonlinear structural analysis
2 Advances in Civil Engineering
Consequently the use of moment-curvature relationships isstill the more widespread and efficient approach
The definition of moment-curvature relationships of RCsections has been a point of research interest for many years[10 11] Many analytical and numerical techniques havebeen proposed including several phenomena that affect theresponse of reinforced concrete elements
Historically moment-curvature relationships with soft-ening branch were first introduced by Wood [12] Mo [13]suggested a classical approach to reproduce the moment-curvature relationship including the softening branch basedon an FE analysis of the elastic-plastic buckling of the longi-tudinal reinforcement An alternative approachwas proposedin [14] where by using a simplified model complex nonlin-ear geometric effects were embedded in the nonlinear mate-rial behaviour of the cross-section Chandrasekaran et al [15]proposed an analytical relationship providing in explicit formthe moment-curvature response of a section by consideringnonlinear constitutive laws of materials chosen in accord-ancewith the European technical codes which can be appliedby the use of a spreadsheet
Commercial structural analysis software programs nowprovide the analyst with the option of conducting moment-curvature analyses at critical sections Although well-verifiedequations have been used in the software developments somequestions may remain (i) for a given project how can thecomputational analysis results be easily verified outside thesoftware (ii) how can general considerations on the role ofdifferent parameters characterizing the section and reinforce-ment geometry be derived
To answer to these questions a dimensionless formula-tion is required able to fully characterize the nonlinearbehaviour of an RC cross-section subjected to an assignedloading condition
In a recent paper [16] the behaviour of RC rectangularcross-sections under axial load and biaxial bending is exam-ined in dimensionless terms This paper confirms that thevalues of the ultimate bending moment and curvature in anydirection can be efficiently related to the values correspond-ing to two separate conditions of uniaxial bending
In this context the present paper proposes an analyticalmodel for the evaluation of the moment-curvature relation-ship of rectangular RC sections subjected to axial load anduniaxial bending by using an incremental strain techniquewhich could be an alternative to the classical fibre decomposi-tionmethodThemodel is based on a cross-sectional analysissatisfying strain compatibility and equilibrium conditionsand can be utilized assuming any constitutive law for theconfined concreteThemodel is able to take into account sev-eral mechanical and geometrical parameters such as sectionaspect ratio longitudinal reinforcement amount and distri-bution confinement effect on the concrete core reinforcingsteel hardening andor softening postbuckling behaviourThe formulation is carried out in dimensionless form and itis able to stress the role of the aforementioned parameters indetermining the shape of the moment-curvature relationshipand the ultimate values of the bending moment and curva-ture
The ability to analyze this role for classes of RC sectionsis the main contribution of the formulation proposed On
the other hand the following limits are related to the basicassumptions the model is applicable only to rectangularsections with symmetrical distribution of the longitudinalreinforcement no cyclic behaviour is considered Moreoverthe use of the proposedmodel to define themoment-rotationresponse of the critical region towhich the section consideredideally belongs requires a suitable modeling of the potentialldquoplastic hingesrdquo which is not discussed here
2 Analysis Model and Geometric Parameters
The analysis model adopted is an updated version of thatalready utilized in [17] where analytical expressions of theultimate moment and the corresponding compression levelwere provided for a class of sections more restricted than theone considered in the present paper
The compressed concrete rectangular section is assumedto be reinforced by longitudinal steel bars symmetricallylocated with respect to the two principal axes of inertia of thesection so that the geometrical and mechanical barycentresare coincident
The longitudinal reinforcing bars are distributed as fol-lows
(i) four bars of the same diameter are located at the cor-ners giving total sectional area 119860
119904119888
(ii) 119899119887bars of the same diameter having total area119860
119904119887 are
located on the upper and lower parts of the sectionbetween the corner bars in the direction parallel tothe width
(iii) 119899ℎbars of the same diameter having total area 119860
119904ℎ
are located on the right and on the left between thecorner bars in the direction parallel to the height ofthe section
Figure 1 shows the section geometry the reinforcing barlocation and the analysis model adopted The confined con-crete core of width 119887 and height ℎ is measured inside the stir-rup perimeter nevertheless in some applications in agree-ment with some technical codes the resisting section of theconcrete core is assumed to be defined by the axis line of theperimeter stirrups
The symbols 119888V and 119888119900in Figure 1(a) indicate the concrete
cover depths that is the distances of the horizontal and ver-tical edges of the core from the external perimeter of the sec-tion
These quantities had not been defined in the originalmodel [17] because at the ultimate state the contribution ofthe concrete cover to the bearing capacity of the section isnegligible
The mean distance of the barycentres of the sectionalarea of the upper and lower reinforcing bars from the nearesthorizontal side of the core of length 119887 is denoted as ℎ
1
By assuming that the bending moment lies in the bary-centric vertical plane of the section the upper and lower rein-forcing bars provide a primary contribution to the rotationalequilibrium of the section while the intermediate bars dis-tributedwith constant pitch119901
ℓbetween the corner bars in the
direction parallel to the height of the section give a minor
Advances in Civil Engineering 3
Gh
p1200012
b
Ash
h1
d1
As1
G
co co
Asb
c
c
p120001
Asc2
As2
d1
(a) (b)Figure 1 Section geometry and analysis model
contribution Consequently the upper and lower reinforcingbars and the intermediate reinforcing bars will be conven-tionally denoted as ldquoprincipal reinforcementrdquo and ldquosecondaryreinforcementrdquo respectively
With reference to the symbols in Figure 1(b) the follow-ing relationships must be considered
1198601199041
=119860119904119888
2+ 119860119904119887 119860
1199042= 119860119904ℎ
119901ℓ=
ℎ minus 2ℎ1
119899ℎ+ 1
1198891= 119899ℎ119901ℓ
(1)
Figure 1(b) shows that in the analysis model the secondaryreinforcement is assumed to be uniformly distributed alongthe segment 119889
1defined as shown in Figure 1(a)
Since high curvature values are involved (ultimate limitstate) unconfined concrete spalling reduces the resisting sec-tion to that of the confined concrete core all the geometricaland mechanical parameters involved in the formulation pro-posed here are referred to this reduced section
Therefore the axial load 119873 and bending moment 119872 willbe normalized to the values
119873119900= 119887ℎ119891cco 119872
119900= 119887ℎ2
119891cco (2)
where119891cco is the cylindrical strength of the confined concreteThe strains of the confined concrete and the dimension-
less curvature120593ℎwill be normalizedwith respect to the strain120576cco corresponding to119891cco and this normalization is indicatedby using the superscription () the same notation is adoptedfor the normalization of the strains of the unconfined con-crete with respect to the strain 120576co corresponding to the cyl-indrical strength 119891co The distances normalized to the heightℎ of the concrete core are denoted by using the superscription(minus
)For clarityrsquos sake the geometrical parameters thatmust be
considered assigned are denoted using Greek letters There-fore with reference to the symbols in Figure 1 the followinggeometrical parameters are defined here
120575V =119888V
ℎ 120575
119900=
119888119900
119887
120582 =ℎ1
ℎ 120572 =
1198891
ℎ=
119899ℎ
119899ℎ+ 1
(1 minus 2120582)
(3)
Stre
ss
Strain
ConfinedUnconfined
fcco
fco
120576co 120576cco 120576ccu
Figure 2 Constitutive laws of concrete
3 Constitutive Laws
31 Concrete Differently from what was necessary for theanalysis at the ultimate state [17] to derive the entiremoment-curvature curve of the class of RC sections under considera-tion the constitutive law of the concrete must be defined forboth the concrete core and the unconfined cover
Figure 2 shows typical shapes of constitutive laws of con-crete and the symbols adopted for the characteristic valuesof strains and stresses Several analytical relations expressingthese laws and confirmed by experimental investigations areavailable in the literature
The formulation proposed in the present work allows theuse of any law however the applications were carried outusing the expressions proposed in [18] which also prove tobe valid for eccentric compression and fibrous andor high-strength concrete [19ndash22]
In normalized form these expressions are written as
119904119888(120576) = (2120576 minus 120576
2
)120573
0 le 120576 le 1 (4)
119904119888(120576) = 1 + 120578 (120576 minus 1) 1 le 120576 le 120583 (5)
where 119904119888is the current stress value normalized to the cylindri-
cal strength and 120576 the strain normalized to the correspondingstrain (120576co or 120576cco for unconfined or confined concrete resp)The exponent 120573 le 1 governs the shape of the ascendingbranch of the constitutive lawThe parameter 120578 rules the neg-ative slope of the linear postpeak branch and is obtained bynormalizing the value of the softening modulus 119864
119888soft in thedimensional plane 119891
119888minus120576119888with respect to the secant modulus
expressed by the ratio between the cylindrical strength andthe corresponding strain
The symbol 120583 indicates the normalized value of the ulti-mate concrete strain and is an index of the available ldquoductilityrdquoof the material
Equations (4) and (5) are valid both for unconfined andconfined concrete upon calibration of the parameters 120573 120578and 120583
4 Advances in Civil Engineering
All quantities referring to the unconfined concrete arehere denoted by the subscript ( )
119888 the ascending branch is
modelled by assuming the value 120573119888= 1 so that for 120576co = 0002
(4) expresses the parabolic law first proposed in [23] and sub-sequently adopted by several researchers Considering thatthe softening branch is very steep for unconfined concreteone sets 120578
119888= minus08 so that (5) provides 119891
119888= 0 for 120576
119888= 120576cu =
00045 (120583119888= 225)
For confined concrete the characteristic quantities ofwhich are denoted by the subscript ( )cc the parameters 120573cc e120578cc depend on the effective confinement pressure and can becalculated by the procedure proposed in [18] The availableductility 120583cc = 120576ccu120576cco is obtained by imposing a conven-tional limit value of reduction to the postpeak strength withrespect to the peak value or by assuming the ultimate strain120576ccu in agreement with expressions validated experimentally[24 25]
In the present formulation the factors relating the cylin-drical strength and the corresponding strain of the confinedconcrete to the ones of the originally unconfined concrete aredenoted as
1198961=
119891cco119891co
1198962=
120576cco120576co
(6)
Moreover the following integral functions are defined
1198781120576
= int
120576
0
119904119888(120576) 119889120576 119878
2120576= int
120576
0
119904119888(120576) 120576 119889120576 (7)
Equations (4) and (5) show that these integrals can beexpressed in an exact analytical form only if 120573 = 1 (uncon-fined concrete) Therefore numerical analyses were carriedout in order to derive approximate analytical expressions forthe confined concrete to be utilized in practical applications
As a result it can be shown that in the field 04 le 120573 le 1which includes all real cases the following expressions implya maximum error of 8 with respect to the values calculatedby numerical integration for 120576 ge 01
1198781120576
= (1205762
minus1
31205763
)
1199031
1198782120576
= (2
31205763
minus1
41205764
)
1199032
01 le 120576 le 1
(8)
where
1199031= 076 + 048 (120573 minus 05) 119903
2= 087 + 026 (120573 minus 05)
(9)
On the other hand the analytical form expressed by (5) showsthat (7) can be exactly integrated in the field of the normalizedpostpeak strains
1198781120576
= 11987811
+ (120576 minus 1) +1
2120578(120576 minus 1)
2
1198782120576
= 11987821
+1
2(1205762
minus 1) +1
6120578(120576 minus 1)
2
(1 + 2120576)
1 le 120576 le 120583
(10)
These expressions also provide approximate values for con-fined concrete because of the approximation by which
0
Stre
ss
0
Strain
Compression
fsy Atan Esc
Atan Es02fsy
minus120576su minus120576sy
120576sy 120576lowasts
Tension
Atan Esh minusfsy
Figure 3 Constitutive law of longitudinal steel reinforcement
the addends 11987811
and 11987821
are affected while for 120573 = 1 (uncon-fined concrete) (9) give 119903
1= 1199032
= 1 and both (8) and (10)express the integral quantities (7) in the exact analytical form
32 Longitudinal Steel Reinforcement Figure 3 shows thebilinear simplified constitutive law adopted for the longitu-dinal steel bars in tension and compression and the symbolsdenoting the characteristic quantities involved
With respect to the analyses made in [17] the restrictiveassumption that the postyielding modulus in compressionmust be the same as in tension is removed
Under tension the possible hardening behaviour isdefined by themean hardeningmodulus119864
119904ℎ under compres-
sion the slope of the postyielding branch can bemodified (thisoption is shown in the figure) so that it can becomenegative ifthe transverse reinforcement is not able to prevent buckling ofthe longitudinal bars In this case the slope of the postyield-ing branch is governed by the modulus 119864
119904119888 which can be
determined by using the model proposed in [26] assumingthat the ultimate strain corresponds to a reduction of 80 ofthe yielding stress 119891
119904119910 This ultimate strain in compression is
denoted as 120576lowast119904in Figure 3
The simple equations that analytically express the lawsof the elastic and postyielding branches in Figure 3 are hereomitted for brevityrsquos sake They will be introduced in thecourse of the formulation
The dimensionless parameters characterizing the slopesof the postyielding branches and the ultimate strain in anideal normalized stress-strain diagram are denoted as
120578119904ℎ
=119864119904ℎ
119864119904
120578119904119888
=119864119904119888
119864119904
120583119903=
120576119904119906
120576119904119910
(11)
where 119864119904is the Young modulus and 120576
119904119910 120576119904119906
are the yield-ing and ultimate strains respectively For compressed steelbars subject to buckling 120576
119904119906is understood to be replaced by
120576lowast
119904in (11)A further parameter relating the characteristic strain
values of steel and concrete is
120577119904119888
=
120576119904119910
120576cco (12)
Advances in Civil Engineering 5
Finally the amount of principal and secondary longitudinalreinforcement is related to the section of the concrete core bymeans of the mechanical ratios of reinforcement
1205961
= 2
1198601199041119891119904119910
119887ℎ119891cco 120596
2= 2
1198601199042119891119904119910
119887ℎ119891cco (13)
4 Equilibrium of Section
Denoting as119873 and119872 the axial load and the bendingmomentacting on the section and as
119899 =119873
119873119900
=119873
119887ℎ119891cco 119898 =
119872
119872119900
=119872
119887ℎ2119891cco (14)
these quantities normalized with respect to the ones definedby (2) the equilibrium of the section in dimensionless formis expressed by the following equations
119899 = 1198991199041
+ 1198991199042
+ 119899119888 (15)
119898 = 1198981199041
+ 1198981199042
+ 119898119888 (16)
At the secondmember of these equations there are clearlyindicated and ordered contributions offered by the principalreinforcement secondary reinforcement and concrete
These contributions are analytically expressed in the fol-lowing sections by assuming the classical hypothesis that thesection remains plane and neglecting the tensile concretestrength
41 Contribution of Principal Reinforcement In the analyticalformulation shown in this section the following parametersare involved the geometrical parameter 120582 defined in (1) themechanical parameters characterizing the constitutive law ofthe steel reinforcement 120578
119904119888 120578119904ℎ and 120577
119904119888 defined by (11) and
(12) and the mechanical ratio of reinforcement 1205961expressed
by the first of (13)Figure 4 shows the lateral view of the RC member con-
sidered and a generic state of strain and stressBy using the symbols shown in the figure the upper and
lower reinforcement bars are subjected respectively to thestrains
Then the axial load and bending moment with respect to thebarycentre of the section that the secondary reinforcementcan bear can be calculated by the following expressions
1198731199042
=21198601199042
1198891
ℎ
2[1198911198901
(1198861+
120577119904119888
120593) + 119891
1198902(1198862+
120577119904119888
120593)
minus (Δ11989111198861+ Δ11989121198862) ]
1198721199042
=21198601199042
1198891
ℎ2
2[(1198911198901
minus 1198911198902)1205722
6minus Δ11989111198861(120572
2minus
1198861
3)
+Δ11989121198862(120572
2minus
1198862
3)]
(33)
Equations (28) (29) and (31) show that
1198911198901
(1198861+
120577119904119888
120593) + 1198911198902
(1198862+
120577119904119888
120593) = 2119891
119904119910
120593
120577119904119888
(119909119888minus
1
2) 120572
1198911198901
minus 1198911198902
= 119891119904119910
120593
120577119904119888
120572
(34)
Therefore substituting (34) into (33) in normalized form oneobtains
1198991199042
=1198731199042
119873119900
= 1205962
120593
120577119904119888
(119909119888minus
1
2) minus
1
2120572
times [(1 minus 120578119904119888) 1198862
1minus (1 minus 120578
119904ℎ) 1198862
2]
(35)
1198981199042
=1198721199042
119872119900
=1205962
2120572
120593
120577119904119888
[1205723
6minus (1 minus 120578
119904119888) (
120572
2minus
1198861
3) 1198862
1
minus (1 minus 120578119904ℎ) (
120572
2minus
1198862
3) 1198862
2]
(36)
Advances in Civil Engineering 7
h
c
c
co cob
x3
xc
Confinedconcrete
1205761
1205762
1205763
1205764
(a)
h
c
c
co cob
xc
1205761
1205762
1205763 = 1205764 = 0x3 = 0
Confinedconcrete
(b)
Figure 6 Strain state of concrete section (a) uncracked section (b) cracked section
It must be observed that if one sets 119864119904119888
= 119864119904in (30) andor
119864119904ℎ
= 119864119904in (32) (33) expresses the contributions to the axial
load and bendingmoment in the case inwhich the upper fibreandor the lower fibre of the secondary reinforcement remainin the elastic field As a consequence (35) and (36) are validfor any value of 119909
119888e 120593 under the condition that one sets
120578119904119888=1 if minus 120576
119904119910le 1205761199041
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 minus 120572
2)le1
120578119904ℎ=1 if minus 120576
119904119910le 1205761199042
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 + 120572
2)le1
(37)
43 Contribution of Concrete Figure 6 shows the strain statesof the section corresponding to the cases of uncracked andcracked sections By using the symbols in the figure the con-tributions to the equilibrium provided by the concrete coreare expressed by
119873cc = 119887int
119909119888
1199093
119891119888(119909) 119889119909 =
119887ℎ119891cco120593
int
1205762
1205763
119904119888(120576) 119889120576
119872cc = 119887int
119909119888
1199093
119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus
ℎ
2)
=119887ℎ2
119891cco (1
1205932int
1205762
1205763
119904119888(120576) 120576 119889120576minus
119909119888minus 05
120593int
1205762
1205763
119904119888(120576) 119889120576)
(38)
where
1199093=
0 for 119909119888le ℎ
119909119888minus ℎ for 119909
119888gt ℎ
(39)
and consequently
1205762= 120593119909119888
1205763=
0 for 119909119888le 1
120593 (119909119888minus 1) for 119909
119888gt 1
(40)
Considering (7) in the normalized form one obtains
119899cc =119873cc119873119900
=1
120593(11987811205762
minus 11987811205763
) (41)
119898cc =119872cc119872119900
=1
1205932(11987821205762
minus 11987821205763
) minus 119899cc (119909119888 minus1
2) (42)
The contribution of the surrounding unconfined concrete(concrete cover) can be calculated by subtracting the contri-bution of the confined region to that of the whole section
Considering that the strains involved in the functions 1198781120576
and 1198782120576
must be normalized with respect to the strain 120576co(unconfined concrete) by following the same procedure asthat leading to (41) and (42) and by introducing the con-finement efficiency factors defined by (6) for the whole sec-tion one obtains
119899un1 =1 + 2120575
119900
1198961
1
1198962120593
(11987811205761015840
1
minus 11987811205761015840
4
)
119898un1 =1 + 2120575
119900
1198961
1
(1198962120593)2(11987821205761015840
1
minus 11987821205761015840
4
) minus 119899un1 (119909119888 minus1
2)
(43)where
1205761015840
1= 1198962120593 (119909119888+ 120575V) le 120583
119888
1205761015840
4=
0 for 119909119888le (1 + 120575V)
1198962120593 [119909119888minus (1 + 120575V)] for 119909
119888gt (1 + 120575V)
(44)
The contribution to be subtracted from the above quantitiesare calculated bymodifying (41) and (42) in order to considerthe different constitutive law of the unconfined concrete andthe different base of normalization of strain and stress It caneasily be shown that one obtains
119899un2 =1
11989611198962120593
(11987811205761015840
2
minus 11987811205761015840
3
) (45)
119898un2 =1
1198961(1198962120593)2(11987821205761015840
2
minus 11987821205761015840
3
) minus 119899un2 (119909119888 minus1
2) (46)
8 Advances in Civil Engineering
where
1205761015840
2= 1198962120593119909119888le 120583119888
1205761015840
3=
0 for 119909119888le 1
1198962120593 (119909119888minus 1) for 119909
119888gt 1
(47)
On the basis of what was said above the effective contributionof the concrete section in compression in a generic loadingstate is expressed by
119899119888= 119899cc + 119899un1 minus 119899un2 (48)
119898119888= 119898cc + 119898un1 minus 119898un2 (49)
If the confinement effect is negligible (6) gives 1198961= 1198962= 1
By comparing (40) (41) and (42) with (47) (45) and (46)respectively it can easily be observed that 119899cc = 119899un2 and119898cc = 119898un2 Therefore in this case the contributions of thewhole section of reacting concrete given by (48) and (49) arereduced to the values 119899cc = 119899un1 and 119898cc = 119898un1 and areexpressed by (45) and (46) for 119896
1= 1198962= 1
5 Analysis Procedure
The flexural response of a class of RC sections characterizedby assigned values of the geometrical andmechanical param-eters defined above is obtained by imposing the equilibriumcondition expressed by (15) and (16)
For each configuration the contributions to the equi-librium provided by the principal reinforcement ((23) and(24) under the conditions expressed by (26)) the secondaryreinforcement ((28) (35) and (36) under the conditionsexpressed by (37)) and the compressed region of concrete((48) and (49) considering (41)ndash(47)) only depend on thevariables 119909
119888and 120593 which are independent of or related to one
another according to the type of analysis required (see below)In all cases it must be assumed that the proposed formulationis applied for 120593 gt 0
51 Field of Application The field of validity of the procedureproposed here is limited by the fact that the constitutive law ofthe steel reinforcement does not include unloading branchesstarting from a point of the post yielding branches in com-pression or in tension Therefore for any loading step theupper principal reinforcement and the upper fibre of the dis-tributed secondary reinforcement both potentially subject tocompression cannot be in the postyielding field in tensionthe lower principal reinforcement and the lower fibre of thedistributed secondary reinforcement both potentially subjectto tension cannot be in the postyielding field in compression
Because of these assumptions the field of values of 119909119888
consistent with a given value of 120593 is limited by the initial andend values provided by (26) or (37)
More precisely in the absence of secondary reinforce-ment (120596
2= 0) (26) gives
119909119888min = 120582 minus
120577119904119888
120593 119909
119888max = (1 minus 120582) +120577119904119888
120593 (50)
while if the secondary reinforcement is present (37) leads tomore restrictive limitations
119909119888min =
1 minus 120572
2minus
120577119904119888
120593 119909
119888max =1 + 120572
2+
120577119904119888
120593 (51)
These limitations also occur in a classical numericalapproach based on the strip decomposition of the section
However it must be observed that (50) or (51) allows oneto construct the moment-curvature curve for a large field ofvalues of assigned level of compression including the moreusual cases occurring in practical applications
52 Moment-Curvature Curves The more usual applicationof the analytical expressions derived in the previous sectionsis to derive by a step-by-step procedure the moment-curva-ture119898-120593 curve of a class of RC sections for a given value of 119899
To this purpose the procedure is applied by the followingsteps
(i) assigning the lowest value of curvature 120593 that onewants to consider increasing values of 119909
119888are consid-
ered from the minimum to the maximum expressedby (50) or (51)
(ii) for each value of the couple 120593 119909119888the corresponding
value of 1198991199041 1198991199042 and 119899
119888are calculated and their sum
increasing for each increase in 119909119888
(iii) when the assigned value of 119899 is reached with anacceptable tolerance (ie (15) is verified with verygood approximation) the assigned values of 120593 and 119899and the value of 119909
119888that was found make it possible
to calculate the corresponding values of11989811990411198981199042 and
119898119888expressed by (24) (36) and (49) and the sum
of these contributions of moment is the ordinate ofthe point having the abscissa 120593 belonging to the119898-120593curve sought
(iv) considering an increased value of 120593 the procedure isrepeated to obtain a subsequent point of themoment-curvature curve
The procedure stops when the maximum available ulti-mate confined concrete strain or the maximum available ten-sile strain of the lower reinforcement is exceeded These twooccurrences imply respectively that 120593119909
119888gt 120583cc or 120593(119909119888 minus 1 +
120582) lt minus120577119904119888120583119903 where 120583cc = 120576ccu120576cco is the available ductility of
the confined concrete defined in Section 31 and 120583119903is the steel
ductility defined in (11)If the stress-strain law of the steel reinforcement also
exhibits a softening postyielding branch corresponding to apostbuckling behaviour up to the strain value 120576
∙
119904(Figure 3)
a further limit at which the procedure must be stoppedderives from the condition that 120593(119909
119888minus 120582) gt 120577
119904119888120583119903119888 in
which 120583119903119888
= 120576lowast
119904120576119904119910 Beyond this condition the residual stress
of the compressed reinforcement should be assumed to beconstant and equal to 02119891
119904119910 but this branch of the stress-
strain law is not considered in the proposed model as statedin Section 32
Advances in Civil Engineering 9
53 Strength and Curvature Domains A typical 119898-120593 curveexhibits two characteristic points corresponding to theachievement of the ultimate state and to the first yield-ing of the principal reinforcement in tension respectivelyObviously the first point is present in all cases while thesecond can only be reached if yielding of the reinforcementin tension occurs before the brittle collapse of the concretecore due to low available concrete ductility orand a very highcompression level
The analytical expressions shown in the previous sectionsmake it possible to determine directly the coordinates of thesepoints and to derive in closed form the119898
119906-119899 120593119906-119899 and119898
119910-119899
120593119910-119899 domains which give useful information on the flexural
strength and curvature ductility of an RC section subjected toan assigned compression level
The constructions of the aforementioned curves can bemade more easily than by using a classical discretized modelof the section because (15) and (16) in a suitable specializedform provide analytical expressions where the axial com-pressive load and the corresponding bending moment onlydepend on a single variable 119909
119888and 120593 being related to one
anotherIn order to construct the 119898
119906-119899 and 120593
119906-119899 curves two dif-
ferent relationships linking the neutral axis depth to the cur-vature must be considered according to whether the collapseof the section is produced by excess of the available strain ofthe reinforcement in tension or of the available strain of theconcrete core in compression Moreover to distinguish thesetwo different events it must be observed that the maximumcurvature of the section is reachedwhen these two conditionsoccur simultaneously Considering Figure 4 it can easily bededuced that this value of curvature in normalized form is
120593119906max =
120583cc + 120577119904119888120583119903
1 minus 120582 (52)
The first branch of the domains sought is determined byimposing the condition that the collapse of the section is dueto the achievement of themaximum tensile strain of the prin-cipal reinforcement in tension (low level of compression)Therefore the following condition has to be assumed
120593119906(119909119888minus 1 + 120582) = minus 120577
119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus
120577119904119888120583119903
120593119906
(53)
Considering increasing values of the curvature from themin-imum value to themaximum expressed by (52) the second of(53) provide the corresponding value of 119909
119888 and (15) and (16)
by making explicit the contributions of the concrete and theprincipal and secondary reinforcement provide the corre-sponding values of 119899 and119898
119906
Once 120593119906max is reached the second branch of the 119898
119906-119899
120593119906-119899 curves must be determined considering that for further
decreasing values of 120593119906 the collapse is produced by the
achievement of the maximum compressive strain of the con-fined concrete Therefore the new relationship linking thecurvature to the neutral axis depth is
By using the second of these equations (15) and (16) againlink 119899 and119898
119906to the only variable 120593
119906
It must be observed that if the principal reinforcement incompression is subject to buckling after the achievement ofthe yielding stress the ultimate strain of this compressed rein-forcement is the strain denoted as 120576∙
119904in Figure 3 This strain
value can be reached before the concrete core reaches thenormalized value of strain 120583cc In this case the maximumvalue of normalized dimensionless curvature expressed by(52) must be substituted by the value 120593
119906max = 120577119904119888(120583119903119888
+
120583119903)(1minus2120582) where 120583
119903119888= 120576lowast
119904120576119904119910 and the second branch of the
ultimate strength domains must be determined by assumingthat for values of 120593
119906that decrease from this maximum the
relationship linking the neutral axis dept to the curvature is119909119888= 120582 + 120577
119904119888120583119903119888120593
The 119898119910-119899 120593119910-119899 curves can be simply determined by
assuming that whatever the value of the curvature is the prin-cipal reinforcement in tension is affected by the strain valueequal to minus120576
119904119910 This condition in dimensionless terms gives
120593119910(119909119888minus 1 + 120582) = minus 120577
119904119888997904rArr 119909119888= (1 minus 120582) minus
120577119904119888
120593119910
(55)
By introducing the second of (55) into the equilibrium Equa-tions (15) and (16) they again become two functions of thesingle variable 120593 = 120593
119910 The 119898
119910-119899 120593119910-119899 curves can be con-
structed starting froma value thatwas assumed to be themin-imum up to the maximum value corresponding to one of thepossible aforementioned occurrences the collapse of the con-crete core due to achievement of the available ductility of thematerial or the achievement of the strain 120576
∙
119904in the principal
reinforcement in compression if this reinforcement is subjectto buckling
Considering the field of validity of the procedure pro-posed the minimum value of 120593
119906that can be considered to
construct the119898119906-119899 120593119906-119899 domains is obtained by introducing
the second of (53) into the first of (50) or (51) so that
120593119906min = 120577
119904119888
120583119903minus 1
1 minus 2120582 (56)
in the absence of the secondary reinforcement and
120593119906min = 120577
119904119888
120583119903minus 1
(1 + 120572) 2 minus 120582 (57)
if the secondary reinforcement is also presentObviously the 119898
119910-119899 120593119910-119899 domains can be constructed
starting from any value 120593119910gt 0
In relation to the field of applications (RC columns) thefirst values of 120593
119906and 120593
119910that are to be actually considered are
the ones first producing 119899 ge 0
6 Applications
The reliability of the procedure proposed is shown here byusing the results derived from two RC sections considered byother researchers
Thefirst application refers to one of the sections examinedby Zahn et al [27] This was a square RC section of side119861 = 400mm and realized with concrete having cylindrical
10 Advances in Civil EngineeringN
orm
aliz
ed st
ress
1
00 1 2 3 4 5
Saatcioglu and Razvi (1992)Mander et al (1988)
Normalized strain
Figure 7 Calibration of constitutive law of confined concrete
strength 119891co = 30Nmm2 The longitudinal reinforcementconsisted of 12 steel bars of diameter 16mm having yieldingstrength 119891
119904119910= 420Nmm2 uniformly distributed along the
perimeter of the concrete core (constant pitch) The coverdepth measured from the external perimeter of the stirrupswas 13mm The transverse reinforcement consisting of steelstirrups of diameter equal to 10mm according to the calcu-lations made by the aforementioned authors provided theeffective confinement pressure 119891
ℓ119890= 011119891
1015840
co where 1198911015840
co =
085119891co is the reduced strength value prescribed by the ACI318 code [28]
The flexural response was numerically derived in [27] byusing a classical fibre model in which the stress-strain law ofthe confined concrete was assumed in agreement with Man-derrsquosmodel [25]Therefore in this application the parameters120573cc and 120578cc governing the shape of the laws expressed by (4)and (5) are calibrated so that they lead to a constitutive lawof confined concrete which is very similar to that used by theaforementioned authors
Manderrsquos model [25] for the given value of the confine-ment pressure is governed by the parameter 119903 = 13 themodel of Saatcioglu and Razvi [18] leads to a very similarshape of the stress-strain curve by assuming that 120573cc = 045
and 120578cc = minus006 Figure 7 shows the very similar shapes ofthe constitutive laws corresponding to the two models con-sidered
The procedure proposed in [25] provided the followingcharacteristic values of confined concrete strength and cor-responding strain 120576cco = 00073 119891cco = 46Nmm2 Thesevalues are also adopted for the dimensional stress-strain lawexpressed by the model used here [18]
Since for the stress-strain law of the unconfined concreteZahn et al [27] do not give any information the default valuesindicated in Section 31 are assumed here 120573
119888= 1 120578
119888= minus08
and 120576co = 0002On the basis of the above data the square cross-section of
the concrete coremeasured inside the perimeter stirrups hasa side of 119887 = ℎ = 354mm the cover depth is equal to
400
300
200
100
00 001 002 003 004 005
Mom
ent (
kNm
)
Curvature (1m)
Zahn et al (1989)Present model
Figure 8 Comparison of results119872-120593 curve for given119873
23mm the number of intermediate reinforcing bars realizingthe secondary reinforcement is 119899
119887= 119899ℎ= 2 and the distance
of the barycentre of the section of the principal reinforcementfrom the external side of the concrete core section is ℎ
1=
162 = 8mmThe principal reinforcement consists of four upper and
four lower 16mm steel bars while the secondary reinforce-ment consists of four bars (two on the right and two on theleft) of the same diameter The steel reinforcement constitu-tive law is defined by assuming 119864
119904= 210GPa 120576
119904119910= 119891119904119910119864119904=
0002) and 119864119904ℎ
= 119864119904119888
= 0Therefore the dimensionless parameters involved in the
formulation proposed take on the following values 120575V = 120575119900=
= 0Figure 8 compares the 119872-120593 curve obtained in [27] with
that derived from the procedure shown in Section 52 for acompressive constant axial load 119873 = 03119891co(119861 times 119861) corre-sponding to a compressive level 119899 = 119873(119887
2
times 119891cco) = 025 inagreement with the first of (14)The results in the figure showa very good level of agreement between the numerical pro-cedure adopted in [27] and the analytical one adopted here
Both curves in the figure clearly show the cusp producedby the yielding of the principal reinforcement in tension andthe effects of the progressive cover spalling
The second application refers to the middle-height sec-tion of an RC column of height 1640mm experimentallytested by Saatcioglu et al [19] marked as specimen C6-2 bythe authors
The loading condition was realized by imposing relativeaxial displacements so that the reactive compressive forceacted with fixed eccentricity in a plane of principal inertiaof the sections Suitable devices were applied at the columnends so that the column itself behaved like a hinged verticalRC member under an eccentric compressive load
As a consequence each point of the moment-curvaturecurve characterizing the experimental response obtained by
Advances in Civil Engineering 11
the aforementioned authors corresponds to a different valueof axial loadMoreover the actual bendingmoment acting onthe middle-height section was influenced by the 119875-Δ effect
The formulation proposed here is validated consideringsome points of the aforementioned curve for which theauthors indicate the values of the axial load119873 and curvaturethat were measured These values of 119873 are normalized withrespect to the axial load 119873
119900expressed by (2) The procedure
is applied by determining the value of 119909119888from (15) for the
assigned values of 119899 and 120593 and by calculating 119898 from (16)The results are compared with the numerical values obtainedby the strip model adopted by the authors which includedthe 119875-Δ effect and the values experimentally detected
The specimen considered had a square section of side 119861 =
210mm cover depth of 125mm and longitudinal reinforcingbars of diameter 113mm uniformly distributed along theperimeter of the concrete core having yielding strength119891
119904119910=
517Nmm2 and average hardening modulus 119864119904ℎ
= 0013119864119904
The transverse steel reinforcement consisted of square andoctagonal 63mm stirrups with pitch 50mm
The unconfined concrete had cylindrical strength 119891co =
3440Nmm2The numerical model adopted in [19] assumedthat the perimeter of the concrete core was coincident withthe axis lines of the external stirrups
On the basis of these geometrical and mechanical dataand by evaluating the effective confinement pressure by theprocedure proposed in [18] for the confined concrete sectionone obtains 119887 = ℎ = 17870mm 119891cco = 4635 Nmm2 120576cco =
000547 120578cc = minus0052 and 120573 = 0590The further parametersinvolved in the formulation proposed take on the followingvalues 120575
= 0013Figure 9 shows that the procedure proposed provides
results that are very close to those derived by the numericalmodel adopted in [19] The experimentally detected values ofmoment prove to be underestimated by both models
It must be observed that the last point detected by usingthe present formulation corresponding to the ultimate stateof the section had been already found in [17] because thissection belongs to the more restricted class of sections con-sidered in that work
Figure 10 shows the strength and curvature domainsobtained by using the procedure described in Section 53For confined concrete and steel reinforcement the followingvalues were assumed deduced from the data shown in [19]120576ccu = 0032 (120583cc = 585) and 120576
119904119906= 0066 (120583
119903= 27) The
results confirm that a good level of confinementmakes it pos-sible to achieve acceptable ductility of curvature even underhigh levels of compression In order to show how the proce-dure proposed is easily able to evaluate the influence of all thegeometric and mechanical parameters governing the flexuralresponse of a class of RC sections Figure 10(a) also shows theinfluence of the unconfined concrete cover on the bendingmoment at the first yielding of the principal reinforcement intension As expected this influence proves to be negligible forhigh values of the compression level
The dotted curve in Figure 10(b) obviously stops at thepoint of intersection with the continuous curve Beyond this
100
80
60
40
20
00 1 2 3 4
Mom
ent (
kNm
)
ExperimentalModelPresent model
Saatcioglu et al (1995)
Curvature (1mm times 10minus4)
Figure 9 Comparison of theoretical and experimental results
point the 120593119906-119873 curve proceeds by a very brief stretch (not
very evident in the figure) that corresponds to brittle collapseof the section due to the achievement of the maximum avail-able compressive strain in the confined concrete when theprincipal reinforcement in tension is still in the elastic fieldThe end point of this curve corresponds to the achievementof the maximum neutral axis depth defined by the second of(51)
Nevertheless the axial load values that cannot be consid-ered because of these limitations are well beyond the onesthat can be assumed in the structural design of RC buildingsin seismic areas or usually found in the columns of existingbuildings
7 Conclusions
A dimensionless formulation has been proposed which pro-vides the flexural response of classes of sections of RC col-umns having the same values of the geometric and mechani-cal parameters defined in this study
At each loading stage the resisting components of the sec-tion (cover and concrete core and principal and secondarylongitudinal reinforcements) give a contribution to the equi-librium that can be expressed by analytical functions depend-ing on the normalized neutral axis depth and the curvature ofthe section
Special equilibrium conditions like the ones correspond-ing to the first yielding of the principal reinforcement intension and to the ultimate state imply an analytical link bet-ween the neutral axis depth and the curvature so that the sumof the aforementioned contributions becomes an analyticalfunction of a single variable which can express a strength orcurvature domain
For a generic equilibrium condition an iterative proce-dure to determine the neutral axis depth corresponding to
12 Advances in Civil Engineering
100
80
60
40
20
00 300 600 900 1200 1500
Mom
ent (
kNm
)
Axial load (kN)
Mu
My
My without cover
(a)
8
6
4
2
00 300 600 900 1200 1500
Axial load (kN)
120593u120593y
Curv
atur
e(1
mm
times10
minus4)
(b)
Figure 10119872-119873 and 120593-119873 domains at the ultimate state and the first yielding of reinforcement
the assigned values of curvature and compression level isrequired but each iteration step implies the simple use of thesame equilibrium equation for an updated value of the nor-malized neutral axis dept
The dimensionless form of the formulation proposedmakes also it easily possible to evaluate the influence of theparameters characterizing the class of sections examined onthe strength and curvature corresponding to a given loadingstage
Acknowledgment
Thisworkwas carried outwithin the 2010ndash2013Research Pro-ject ldquoDPC-ReLUIS (Dipartimento Protezione Civile-Rete deiLaboratori Universitari di Ingegneria Sismica)rdquo AT 1 Task112 The related financial support was greatly appreciated
References
[1] A Fafitis ldquoInteraction surfaces of reinforced-concrete sectionsin biaxial bendingrdquo Journal of Structural Engineering vol 127no 7 pp 840ndash846 2001
[2] J L Bonet P F Miguel M A Fernandez and M L RomeroldquoAnalytical approach to failure surfaces in reinforced concretesections subjected to axial loads and biaxial bendingrdquo Journalof Structural Engineering vol 130 no 12 pp 2006ndash2015 2004
[3] G Monti and S Alessandri ldquoAssessment of rc columns undercombined biaxial bending and axial loadrdquo in Proceedings of the2nd FIB Congress Naples Italy 2006
[4] RW Clough and S B Johnston ldquoEffect of stiffness degradationon earthquake ductility requirementsrdquo in Proceedings of 2ndJapan Earthquake Engineering Symposium Tokyo Japan 1966
[5] T Takeda M A Sozen and N N Nielsen ldquoReinforced con-crete response to simulated earthquakerdquo Journal of StructuralDivision vol 96 no 12 pp 2257ndash2273 1970
[6] M S L Roufaiel and C Meyer ldquoAnalytical modeling of hyster-etic behavior of reinforced concrete framerdquo Journal of StructuralEngineering vol 113 no 3 pp 429ndash444 1987
[7] A R Mari and A C Scordelis ldquoNonlinear geometric materialand time dependent analysis of three dimensional reinforcedand prestressed concrete framesrdquo USBSESM Report 8412Department of Civil Engineering University of CaliforniaBerkeley Calif USA 1973
[8] T Taucer E Spacone and F C Filippou ldquoA fiber beam-col-umn element for seismic response analysis of reinforced con-crete structuresrdquo Report EERC 91-17 Earthquake EngineeringResearch Center Berkeley Calif USA 1991
[9] Z Zhu I Ahmad and A Mirmiran ldquoFiber element modelingfor seismic performance of bridge columns made of concrete-filled FRP tubesrdquo Engineering Structures vol 28 no 14 pp2023ndash2035 2006
[10] E O Pfrang C P Siess and M A Sozen ldquoLoad-moment-cur-vature characteristics of RC cross-sectionsrdquoACI Journal vol 61no 7 pp 763ndash778 1964
[11] D J Carreira and K-H Chu ldquoThemoment-curvature relation-ship of RC membersrdquo ACI Journal vol 83 no 2 pp 191ndash1981986
[12] R H Wood ldquoSome controversial and curious developments inplastic theory of structuresrdquo inEngineering Plasticity J Heymanand F A Leckie Eds pp 665ndash691 CambridgeUniversity PressCambridge UK 1968
[13] Y LMo ldquoInvestigation of reinforced concrete frame behaviourtheory and testsrdquo Magazine of Concrete Research vol 44 no160 pp 163ndash173 1992
[14] M Jirasek and Z P Bazant Inelastic Analysis of Structures JonWiley amp Sons London UK 2002
[15] S Chandrasekaran L Nunziante G Serino and F CarannanteldquoCurvature ductility of RC sections based on Eurocode analyt-ical procedurerdquo KSCE Journal of Civil Engineering vol 15 no 1pp 131ndash144 2011
Advances in Civil Engineering 13
[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012
[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013
[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992
[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995
[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006
[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010
[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012
[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951
[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982
[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002
[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989
[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008
Consequently the use of moment-curvature relationships isstill the more widespread and efficient approach
The definition of moment-curvature relationships of RCsections has been a point of research interest for many years[10 11] Many analytical and numerical techniques havebeen proposed including several phenomena that affect theresponse of reinforced concrete elements
Historically moment-curvature relationships with soft-ening branch were first introduced by Wood [12] Mo [13]suggested a classical approach to reproduce the moment-curvature relationship including the softening branch basedon an FE analysis of the elastic-plastic buckling of the longi-tudinal reinforcement An alternative approachwas proposedin [14] where by using a simplified model complex nonlin-ear geometric effects were embedded in the nonlinear mate-rial behaviour of the cross-section Chandrasekaran et al [15]proposed an analytical relationship providing in explicit formthe moment-curvature response of a section by consideringnonlinear constitutive laws of materials chosen in accord-ancewith the European technical codes which can be appliedby the use of a spreadsheet
Commercial structural analysis software programs nowprovide the analyst with the option of conducting moment-curvature analyses at critical sections Although well-verifiedequations have been used in the software developments somequestions may remain (i) for a given project how can thecomputational analysis results be easily verified outside thesoftware (ii) how can general considerations on the role ofdifferent parameters characterizing the section and reinforce-ment geometry be derived
To answer to these questions a dimensionless formula-tion is required able to fully characterize the nonlinearbehaviour of an RC cross-section subjected to an assignedloading condition
In a recent paper [16] the behaviour of RC rectangularcross-sections under axial load and biaxial bending is exam-ined in dimensionless terms This paper confirms that thevalues of the ultimate bending moment and curvature in anydirection can be efficiently related to the values correspond-ing to two separate conditions of uniaxial bending
In this context the present paper proposes an analyticalmodel for the evaluation of the moment-curvature relation-ship of rectangular RC sections subjected to axial load anduniaxial bending by using an incremental strain techniquewhich could be an alternative to the classical fibre decomposi-tionmethodThemodel is based on a cross-sectional analysissatisfying strain compatibility and equilibrium conditionsand can be utilized assuming any constitutive law for theconfined concreteThemodel is able to take into account sev-eral mechanical and geometrical parameters such as sectionaspect ratio longitudinal reinforcement amount and distri-bution confinement effect on the concrete core reinforcingsteel hardening andor softening postbuckling behaviourThe formulation is carried out in dimensionless form and itis able to stress the role of the aforementioned parameters indetermining the shape of the moment-curvature relationshipand the ultimate values of the bending moment and curva-ture
The ability to analyze this role for classes of RC sectionsis the main contribution of the formulation proposed On
the other hand the following limits are related to the basicassumptions the model is applicable only to rectangularsections with symmetrical distribution of the longitudinalreinforcement no cyclic behaviour is considered Moreoverthe use of the proposedmodel to define themoment-rotationresponse of the critical region towhich the section consideredideally belongs requires a suitable modeling of the potentialldquoplastic hingesrdquo which is not discussed here
2 Analysis Model and Geometric Parameters
The analysis model adopted is an updated version of thatalready utilized in [17] where analytical expressions of theultimate moment and the corresponding compression levelwere provided for a class of sections more restricted than theone considered in the present paper
The compressed concrete rectangular section is assumedto be reinforced by longitudinal steel bars symmetricallylocated with respect to the two principal axes of inertia of thesection so that the geometrical and mechanical barycentresare coincident
The longitudinal reinforcing bars are distributed as fol-lows
(i) four bars of the same diameter are located at the cor-ners giving total sectional area 119860
119904119888
(ii) 119899119887bars of the same diameter having total area119860
119904119887 are
located on the upper and lower parts of the sectionbetween the corner bars in the direction parallel tothe width
(iii) 119899ℎbars of the same diameter having total area 119860
119904ℎ
are located on the right and on the left between thecorner bars in the direction parallel to the height ofthe section
Figure 1 shows the section geometry the reinforcing barlocation and the analysis model adopted The confined con-crete core of width 119887 and height ℎ is measured inside the stir-rup perimeter nevertheless in some applications in agree-ment with some technical codes the resisting section of theconcrete core is assumed to be defined by the axis line of theperimeter stirrups
The symbols 119888V and 119888119900in Figure 1(a) indicate the concrete
cover depths that is the distances of the horizontal and ver-tical edges of the core from the external perimeter of the sec-tion
These quantities had not been defined in the originalmodel [17] because at the ultimate state the contribution ofthe concrete cover to the bearing capacity of the section isnegligible
The mean distance of the barycentres of the sectionalarea of the upper and lower reinforcing bars from the nearesthorizontal side of the core of length 119887 is denoted as ℎ
1
By assuming that the bending moment lies in the bary-centric vertical plane of the section the upper and lower rein-forcing bars provide a primary contribution to the rotationalequilibrium of the section while the intermediate bars dis-tributedwith constant pitch119901
ℓbetween the corner bars in the
direction parallel to the height of the section give a minor
Advances in Civil Engineering 3
Gh
p1200012
b
Ash
h1
d1
As1
G
co co
Asb
c
c
p120001
Asc2
As2
d1
(a) (b)Figure 1 Section geometry and analysis model
contribution Consequently the upper and lower reinforcingbars and the intermediate reinforcing bars will be conven-tionally denoted as ldquoprincipal reinforcementrdquo and ldquosecondaryreinforcementrdquo respectively
With reference to the symbols in Figure 1(b) the follow-ing relationships must be considered
1198601199041
=119860119904119888
2+ 119860119904119887 119860
1199042= 119860119904ℎ
119901ℓ=
ℎ minus 2ℎ1
119899ℎ+ 1
1198891= 119899ℎ119901ℓ
(1)
Figure 1(b) shows that in the analysis model the secondaryreinforcement is assumed to be uniformly distributed alongthe segment 119889
1defined as shown in Figure 1(a)
Since high curvature values are involved (ultimate limitstate) unconfined concrete spalling reduces the resisting sec-tion to that of the confined concrete core all the geometricaland mechanical parameters involved in the formulation pro-posed here are referred to this reduced section
Therefore the axial load 119873 and bending moment 119872 willbe normalized to the values
119873119900= 119887ℎ119891cco 119872
119900= 119887ℎ2
119891cco (2)
where119891cco is the cylindrical strength of the confined concreteThe strains of the confined concrete and the dimension-
less curvature120593ℎwill be normalizedwith respect to the strain120576cco corresponding to119891cco and this normalization is indicatedby using the superscription () the same notation is adoptedfor the normalization of the strains of the unconfined con-crete with respect to the strain 120576co corresponding to the cyl-indrical strength 119891co The distances normalized to the heightℎ of the concrete core are denoted by using the superscription(minus
)For clarityrsquos sake the geometrical parameters thatmust be
considered assigned are denoted using Greek letters There-fore with reference to the symbols in Figure 1 the followinggeometrical parameters are defined here
120575V =119888V
ℎ 120575
119900=
119888119900
119887
120582 =ℎ1
ℎ 120572 =
1198891
ℎ=
119899ℎ
119899ℎ+ 1
(1 minus 2120582)
(3)
Stre
ss
Strain
ConfinedUnconfined
fcco
fco
120576co 120576cco 120576ccu
Figure 2 Constitutive laws of concrete
3 Constitutive Laws
31 Concrete Differently from what was necessary for theanalysis at the ultimate state [17] to derive the entiremoment-curvature curve of the class of RC sections under considera-tion the constitutive law of the concrete must be defined forboth the concrete core and the unconfined cover
Figure 2 shows typical shapes of constitutive laws of con-crete and the symbols adopted for the characteristic valuesof strains and stresses Several analytical relations expressingthese laws and confirmed by experimental investigations areavailable in the literature
The formulation proposed in the present work allows theuse of any law however the applications were carried outusing the expressions proposed in [18] which also prove tobe valid for eccentric compression and fibrous andor high-strength concrete [19ndash22]
In normalized form these expressions are written as
119904119888(120576) = (2120576 minus 120576
2
)120573
0 le 120576 le 1 (4)
119904119888(120576) = 1 + 120578 (120576 minus 1) 1 le 120576 le 120583 (5)
where 119904119888is the current stress value normalized to the cylindri-
cal strength and 120576 the strain normalized to the correspondingstrain (120576co or 120576cco for unconfined or confined concrete resp)The exponent 120573 le 1 governs the shape of the ascendingbranch of the constitutive lawThe parameter 120578 rules the neg-ative slope of the linear postpeak branch and is obtained bynormalizing the value of the softening modulus 119864
119888soft in thedimensional plane 119891
119888minus120576119888with respect to the secant modulus
expressed by the ratio between the cylindrical strength andthe corresponding strain
The symbol 120583 indicates the normalized value of the ulti-mate concrete strain and is an index of the available ldquoductilityrdquoof the material
Equations (4) and (5) are valid both for unconfined andconfined concrete upon calibration of the parameters 120573 120578and 120583
4 Advances in Civil Engineering
All quantities referring to the unconfined concrete arehere denoted by the subscript ( )
119888 the ascending branch is
modelled by assuming the value 120573119888= 1 so that for 120576co = 0002
(4) expresses the parabolic law first proposed in [23] and sub-sequently adopted by several researchers Considering thatthe softening branch is very steep for unconfined concreteone sets 120578
119888= minus08 so that (5) provides 119891
119888= 0 for 120576
119888= 120576cu =
00045 (120583119888= 225)
For confined concrete the characteristic quantities ofwhich are denoted by the subscript ( )cc the parameters 120573cc e120578cc depend on the effective confinement pressure and can becalculated by the procedure proposed in [18] The availableductility 120583cc = 120576ccu120576cco is obtained by imposing a conven-tional limit value of reduction to the postpeak strength withrespect to the peak value or by assuming the ultimate strain120576ccu in agreement with expressions validated experimentally[24 25]
In the present formulation the factors relating the cylin-drical strength and the corresponding strain of the confinedconcrete to the ones of the originally unconfined concrete aredenoted as
1198961=
119891cco119891co
1198962=
120576cco120576co
(6)
Moreover the following integral functions are defined
1198781120576
= int
120576
0
119904119888(120576) 119889120576 119878
2120576= int
120576
0
119904119888(120576) 120576 119889120576 (7)
Equations (4) and (5) show that these integrals can beexpressed in an exact analytical form only if 120573 = 1 (uncon-fined concrete) Therefore numerical analyses were carriedout in order to derive approximate analytical expressions forthe confined concrete to be utilized in practical applications
As a result it can be shown that in the field 04 le 120573 le 1which includes all real cases the following expressions implya maximum error of 8 with respect to the values calculatedby numerical integration for 120576 ge 01
1198781120576
= (1205762
minus1
31205763
)
1199031
1198782120576
= (2
31205763
minus1
41205764
)
1199032
01 le 120576 le 1
(8)
where
1199031= 076 + 048 (120573 minus 05) 119903
2= 087 + 026 (120573 minus 05)
(9)
On the other hand the analytical form expressed by (5) showsthat (7) can be exactly integrated in the field of the normalizedpostpeak strains
1198781120576
= 11987811
+ (120576 minus 1) +1
2120578(120576 minus 1)
2
1198782120576
= 11987821
+1
2(1205762
minus 1) +1
6120578(120576 minus 1)
2
(1 + 2120576)
1 le 120576 le 120583
(10)
These expressions also provide approximate values for con-fined concrete because of the approximation by which
0
Stre
ss
0
Strain
Compression
fsy Atan Esc
Atan Es02fsy
minus120576su minus120576sy
120576sy 120576lowasts
Tension
Atan Esh minusfsy
Figure 3 Constitutive law of longitudinal steel reinforcement
the addends 11987811
and 11987821
are affected while for 120573 = 1 (uncon-fined concrete) (9) give 119903
1= 1199032
= 1 and both (8) and (10)express the integral quantities (7) in the exact analytical form
32 Longitudinal Steel Reinforcement Figure 3 shows thebilinear simplified constitutive law adopted for the longitu-dinal steel bars in tension and compression and the symbolsdenoting the characteristic quantities involved
With respect to the analyses made in [17] the restrictiveassumption that the postyielding modulus in compressionmust be the same as in tension is removed
Under tension the possible hardening behaviour isdefined by themean hardeningmodulus119864
119904ℎ under compres-
sion the slope of the postyielding branch can bemodified (thisoption is shown in the figure) so that it can becomenegative ifthe transverse reinforcement is not able to prevent buckling ofthe longitudinal bars In this case the slope of the postyield-ing branch is governed by the modulus 119864
119904119888 which can be
determined by using the model proposed in [26] assumingthat the ultimate strain corresponds to a reduction of 80 ofthe yielding stress 119891
119904119910 This ultimate strain in compression is
denoted as 120576lowast119904in Figure 3
The simple equations that analytically express the lawsof the elastic and postyielding branches in Figure 3 are hereomitted for brevityrsquos sake They will be introduced in thecourse of the formulation
The dimensionless parameters characterizing the slopesof the postyielding branches and the ultimate strain in anideal normalized stress-strain diagram are denoted as
120578119904ℎ
=119864119904ℎ
119864119904
120578119904119888
=119864119904119888
119864119904
120583119903=
120576119904119906
120576119904119910
(11)
where 119864119904is the Young modulus and 120576
119904119910 120576119904119906
are the yield-ing and ultimate strains respectively For compressed steelbars subject to buckling 120576
119904119906is understood to be replaced by
120576lowast
119904in (11)A further parameter relating the characteristic strain
values of steel and concrete is
120577119904119888
=
120576119904119910
120576cco (12)
Advances in Civil Engineering 5
Finally the amount of principal and secondary longitudinalreinforcement is related to the section of the concrete core bymeans of the mechanical ratios of reinforcement
1205961
= 2
1198601199041119891119904119910
119887ℎ119891cco 120596
2= 2
1198601199042119891119904119910
119887ℎ119891cco (13)
4 Equilibrium of Section
Denoting as119873 and119872 the axial load and the bendingmomentacting on the section and as
119899 =119873
119873119900
=119873
119887ℎ119891cco 119898 =
119872
119872119900
=119872
119887ℎ2119891cco (14)
these quantities normalized with respect to the ones definedby (2) the equilibrium of the section in dimensionless formis expressed by the following equations
119899 = 1198991199041
+ 1198991199042
+ 119899119888 (15)
119898 = 1198981199041
+ 1198981199042
+ 119898119888 (16)
At the secondmember of these equations there are clearlyindicated and ordered contributions offered by the principalreinforcement secondary reinforcement and concrete
These contributions are analytically expressed in the fol-lowing sections by assuming the classical hypothesis that thesection remains plane and neglecting the tensile concretestrength
41 Contribution of Principal Reinforcement In the analyticalformulation shown in this section the following parametersare involved the geometrical parameter 120582 defined in (1) themechanical parameters characterizing the constitutive law ofthe steel reinforcement 120578
119904119888 120578119904ℎ and 120577
119904119888 defined by (11) and
(12) and the mechanical ratio of reinforcement 1205961expressed
by the first of (13)Figure 4 shows the lateral view of the RC member con-
sidered and a generic state of strain and stressBy using the symbols shown in the figure the upper and
lower reinforcement bars are subjected respectively to thestrains
Then the axial load and bending moment with respect to thebarycentre of the section that the secondary reinforcementcan bear can be calculated by the following expressions
1198731199042
=21198601199042
1198891
ℎ
2[1198911198901
(1198861+
120577119904119888
120593) + 119891
1198902(1198862+
120577119904119888
120593)
minus (Δ11989111198861+ Δ11989121198862) ]
1198721199042
=21198601199042
1198891
ℎ2
2[(1198911198901
minus 1198911198902)1205722
6minus Δ11989111198861(120572
2minus
1198861
3)
+Δ11989121198862(120572
2minus
1198862
3)]
(33)
Equations (28) (29) and (31) show that
1198911198901
(1198861+
120577119904119888
120593) + 1198911198902
(1198862+
120577119904119888
120593) = 2119891
119904119910
120593
120577119904119888
(119909119888minus
1
2) 120572
1198911198901
minus 1198911198902
= 119891119904119910
120593
120577119904119888
120572
(34)
Therefore substituting (34) into (33) in normalized form oneobtains
1198991199042
=1198731199042
119873119900
= 1205962
120593
120577119904119888
(119909119888minus
1
2) minus
1
2120572
times [(1 minus 120578119904119888) 1198862
1minus (1 minus 120578
119904ℎ) 1198862
2]
(35)
1198981199042
=1198721199042
119872119900
=1205962
2120572
120593
120577119904119888
[1205723
6minus (1 minus 120578
119904119888) (
120572
2minus
1198861
3) 1198862
1
minus (1 minus 120578119904ℎ) (
120572
2minus
1198862
3) 1198862
2]
(36)
Advances in Civil Engineering 7
h
c
c
co cob
x3
xc
Confinedconcrete
1205761
1205762
1205763
1205764
(a)
h
c
c
co cob
xc
1205761
1205762
1205763 = 1205764 = 0x3 = 0
Confinedconcrete
(b)
Figure 6 Strain state of concrete section (a) uncracked section (b) cracked section
It must be observed that if one sets 119864119904119888
= 119864119904in (30) andor
119864119904ℎ
= 119864119904in (32) (33) expresses the contributions to the axial
load and bendingmoment in the case inwhich the upper fibreandor the lower fibre of the secondary reinforcement remainin the elastic field As a consequence (35) and (36) are validfor any value of 119909
119888e 120593 under the condition that one sets
120578119904119888=1 if minus 120576
119904119910le 1205761199041
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 minus 120572
2)le1
120578119904ℎ=1 if minus 120576
119904119910le 1205761199042
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 + 120572
2)le1
(37)
43 Contribution of Concrete Figure 6 shows the strain statesof the section corresponding to the cases of uncracked andcracked sections By using the symbols in the figure the con-tributions to the equilibrium provided by the concrete coreare expressed by
119873cc = 119887int
119909119888
1199093
119891119888(119909) 119889119909 =
119887ℎ119891cco120593
int
1205762
1205763
119904119888(120576) 119889120576
119872cc = 119887int
119909119888
1199093
119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus
ℎ
2)
=119887ℎ2
119891cco (1
1205932int
1205762
1205763
119904119888(120576) 120576 119889120576minus
119909119888minus 05
120593int
1205762
1205763
119904119888(120576) 119889120576)
(38)
where
1199093=
0 for 119909119888le ℎ
119909119888minus ℎ for 119909
119888gt ℎ
(39)
and consequently
1205762= 120593119909119888
1205763=
0 for 119909119888le 1
120593 (119909119888minus 1) for 119909
119888gt 1
(40)
Considering (7) in the normalized form one obtains
119899cc =119873cc119873119900
=1
120593(11987811205762
minus 11987811205763
) (41)
119898cc =119872cc119872119900
=1
1205932(11987821205762
minus 11987821205763
) minus 119899cc (119909119888 minus1
2) (42)
The contribution of the surrounding unconfined concrete(concrete cover) can be calculated by subtracting the contri-bution of the confined region to that of the whole section
Considering that the strains involved in the functions 1198781120576
and 1198782120576
must be normalized with respect to the strain 120576co(unconfined concrete) by following the same procedure asthat leading to (41) and (42) and by introducing the con-finement efficiency factors defined by (6) for the whole sec-tion one obtains
119899un1 =1 + 2120575
119900
1198961
1
1198962120593
(11987811205761015840
1
minus 11987811205761015840
4
)
119898un1 =1 + 2120575
119900
1198961
1
(1198962120593)2(11987821205761015840
1
minus 11987821205761015840
4
) minus 119899un1 (119909119888 minus1
2)
(43)where
1205761015840
1= 1198962120593 (119909119888+ 120575V) le 120583
119888
1205761015840
4=
0 for 119909119888le (1 + 120575V)
1198962120593 [119909119888minus (1 + 120575V)] for 119909
119888gt (1 + 120575V)
(44)
The contribution to be subtracted from the above quantitiesare calculated bymodifying (41) and (42) in order to considerthe different constitutive law of the unconfined concrete andthe different base of normalization of strain and stress It caneasily be shown that one obtains
119899un2 =1
11989611198962120593
(11987811205761015840
2
minus 11987811205761015840
3
) (45)
119898un2 =1
1198961(1198962120593)2(11987821205761015840
2
minus 11987821205761015840
3
) minus 119899un2 (119909119888 minus1
2) (46)
8 Advances in Civil Engineering
where
1205761015840
2= 1198962120593119909119888le 120583119888
1205761015840
3=
0 for 119909119888le 1
1198962120593 (119909119888minus 1) for 119909
119888gt 1
(47)
On the basis of what was said above the effective contributionof the concrete section in compression in a generic loadingstate is expressed by
119899119888= 119899cc + 119899un1 minus 119899un2 (48)
119898119888= 119898cc + 119898un1 minus 119898un2 (49)
If the confinement effect is negligible (6) gives 1198961= 1198962= 1
By comparing (40) (41) and (42) with (47) (45) and (46)respectively it can easily be observed that 119899cc = 119899un2 and119898cc = 119898un2 Therefore in this case the contributions of thewhole section of reacting concrete given by (48) and (49) arereduced to the values 119899cc = 119899un1 and 119898cc = 119898un1 and areexpressed by (45) and (46) for 119896
1= 1198962= 1
5 Analysis Procedure
The flexural response of a class of RC sections characterizedby assigned values of the geometrical andmechanical param-eters defined above is obtained by imposing the equilibriumcondition expressed by (15) and (16)
For each configuration the contributions to the equi-librium provided by the principal reinforcement ((23) and(24) under the conditions expressed by (26)) the secondaryreinforcement ((28) (35) and (36) under the conditionsexpressed by (37)) and the compressed region of concrete((48) and (49) considering (41)ndash(47)) only depend on thevariables 119909
119888and 120593 which are independent of or related to one
another according to the type of analysis required (see below)In all cases it must be assumed that the proposed formulationis applied for 120593 gt 0
51 Field of Application The field of validity of the procedureproposed here is limited by the fact that the constitutive law ofthe steel reinforcement does not include unloading branchesstarting from a point of the post yielding branches in com-pression or in tension Therefore for any loading step theupper principal reinforcement and the upper fibre of the dis-tributed secondary reinforcement both potentially subject tocompression cannot be in the postyielding field in tensionthe lower principal reinforcement and the lower fibre of thedistributed secondary reinforcement both potentially subjectto tension cannot be in the postyielding field in compression
Because of these assumptions the field of values of 119909119888
consistent with a given value of 120593 is limited by the initial andend values provided by (26) or (37)
More precisely in the absence of secondary reinforce-ment (120596
2= 0) (26) gives
119909119888min = 120582 minus
120577119904119888
120593 119909
119888max = (1 minus 120582) +120577119904119888
120593 (50)
while if the secondary reinforcement is present (37) leads tomore restrictive limitations
119909119888min =
1 minus 120572
2minus
120577119904119888
120593 119909
119888max =1 + 120572
2+
120577119904119888
120593 (51)
These limitations also occur in a classical numericalapproach based on the strip decomposition of the section
However it must be observed that (50) or (51) allows oneto construct the moment-curvature curve for a large field ofvalues of assigned level of compression including the moreusual cases occurring in practical applications
52 Moment-Curvature Curves The more usual applicationof the analytical expressions derived in the previous sectionsis to derive by a step-by-step procedure the moment-curva-ture119898-120593 curve of a class of RC sections for a given value of 119899
To this purpose the procedure is applied by the followingsteps
(i) assigning the lowest value of curvature 120593 that onewants to consider increasing values of 119909
119888are consid-
ered from the minimum to the maximum expressedby (50) or (51)
(ii) for each value of the couple 120593 119909119888the corresponding
value of 1198991199041 1198991199042 and 119899
119888are calculated and their sum
increasing for each increase in 119909119888
(iii) when the assigned value of 119899 is reached with anacceptable tolerance (ie (15) is verified with verygood approximation) the assigned values of 120593 and 119899and the value of 119909
119888that was found make it possible
to calculate the corresponding values of11989811990411198981199042 and
119898119888expressed by (24) (36) and (49) and the sum
of these contributions of moment is the ordinate ofthe point having the abscissa 120593 belonging to the119898-120593curve sought
(iv) considering an increased value of 120593 the procedure isrepeated to obtain a subsequent point of themoment-curvature curve
The procedure stops when the maximum available ulti-mate confined concrete strain or the maximum available ten-sile strain of the lower reinforcement is exceeded These twooccurrences imply respectively that 120593119909
119888gt 120583cc or 120593(119909119888 minus 1 +
120582) lt minus120577119904119888120583119903 where 120583cc = 120576ccu120576cco is the available ductility of
the confined concrete defined in Section 31 and 120583119903is the steel
ductility defined in (11)If the stress-strain law of the steel reinforcement also
exhibits a softening postyielding branch corresponding to apostbuckling behaviour up to the strain value 120576
∙
119904(Figure 3)
a further limit at which the procedure must be stoppedderives from the condition that 120593(119909
119888minus 120582) gt 120577
119904119888120583119903119888 in
which 120583119903119888
= 120576lowast
119904120576119904119910 Beyond this condition the residual stress
of the compressed reinforcement should be assumed to beconstant and equal to 02119891
119904119910 but this branch of the stress-
strain law is not considered in the proposed model as statedin Section 32
Advances in Civil Engineering 9
53 Strength and Curvature Domains A typical 119898-120593 curveexhibits two characteristic points corresponding to theachievement of the ultimate state and to the first yield-ing of the principal reinforcement in tension respectivelyObviously the first point is present in all cases while thesecond can only be reached if yielding of the reinforcementin tension occurs before the brittle collapse of the concretecore due to low available concrete ductility orand a very highcompression level
The analytical expressions shown in the previous sectionsmake it possible to determine directly the coordinates of thesepoints and to derive in closed form the119898
119906-119899 120593119906-119899 and119898
119910-119899
120593119910-119899 domains which give useful information on the flexural
strength and curvature ductility of an RC section subjected toan assigned compression level
The constructions of the aforementioned curves can bemade more easily than by using a classical discretized modelof the section because (15) and (16) in a suitable specializedform provide analytical expressions where the axial com-pressive load and the corresponding bending moment onlydepend on a single variable 119909
119888and 120593 being related to one
anotherIn order to construct the 119898
119906-119899 and 120593
119906-119899 curves two dif-
ferent relationships linking the neutral axis depth to the cur-vature must be considered according to whether the collapseof the section is produced by excess of the available strain ofthe reinforcement in tension or of the available strain of theconcrete core in compression Moreover to distinguish thesetwo different events it must be observed that the maximumcurvature of the section is reachedwhen these two conditionsoccur simultaneously Considering Figure 4 it can easily bededuced that this value of curvature in normalized form is
120593119906max =
120583cc + 120577119904119888120583119903
1 minus 120582 (52)
The first branch of the domains sought is determined byimposing the condition that the collapse of the section is dueto the achievement of themaximum tensile strain of the prin-cipal reinforcement in tension (low level of compression)Therefore the following condition has to be assumed
120593119906(119909119888minus 1 + 120582) = minus 120577
119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus
120577119904119888120583119903
120593119906
(53)
Considering increasing values of the curvature from themin-imum value to themaximum expressed by (52) the second of(53) provide the corresponding value of 119909
119888 and (15) and (16)
by making explicit the contributions of the concrete and theprincipal and secondary reinforcement provide the corre-sponding values of 119899 and119898
119906
Once 120593119906max is reached the second branch of the 119898
119906-119899
120593119906-119899 curves must be determined considering that for further
decreasing values of 120593119906 the collapse is produced by the
achievement of the maximum compressive strain of the con-fined concrete Therefore the new relationship linking thecurvature to the neutral axis depth is
By using the second of these equations (15) and (16) againlink 119899 and119898
119906to the only variable 120593
119906
It must be observed that if the principal reinforcement incompression is subject to buckling after the achievement ofthe yielding stress the ultimate strain of this compressed rein-forcement is the strain denoted as 120576∙
119904in Figure 3 This strain
value can be reached before the concrete core reaches thenormalized value of strain 120583cc In this case the maximumvalue of normalized dimensionless curvature expressed by(52) must be substituted by the value 120593
119906max = 120577119904119888(120583119903119888
+
120583119903)(1minus2120582) where 120583
119903119888= 120576lowast
119904120576119904119910 and the second branch of the
ultimate strength domains must be determined by assumingthat for values of 120593
119906that decrease from this maximum the
relationship linking the neutral axis dept to the curvature is119909119888= 120582 + 120577
119904119888120583119903119888120593
The 119898119910-119899 120593119910-119899 curves can be simply determined by
assuming that whatever the value of the curvature is the prin-cipal reinforcement in tension is affected by the strain valueequal to minus120576
119904119910 This condition in dimensionless terms gives
120593119910(119909119888minus 1 + 120582) = minus 120577
119904119888997904rArr 119909119888= (1 minus 120582) minus
120577119904119888
120593119910
(55)
By introducing the second of (55) into the equilibrium Equa-tions (15) and (16) they again become two functions of thesingle variable 120593 = 120593
119910 The 119898
119910-119899 120593119910-119899 curves can be con-
structed starting froma value thatwas assumed to be themin-imum up to the maximum value corresponding to one of thepossible aforementioned occurrences the collapse of the con-crete core due to achievement of the available ductility of thematerial or the achievement of the strain 120576
∙
119904in the principal
reinforcement in compression if this reinforcement is subjectto buckling
Considering the field of validity of the procedure pro-posed the minimum value of 120593
119906that can be considered to
construct the119898119906-119899 120593119906-119899 domains is obtained by introducing
the second of (53) into the first of (50) or (51) so that
120593119906min = 120577
119904119888
120583119903minus 1
1 minus 2120582 (56)
in the absence of the secondary reinforcement and
120593119906min = 120577
119904119888
120583119903minus 1
(1 + 120572) 2 minus 120582 (57)
if the secondary reinforcement is also presentObviously the 119898
119910-119899 120593119910-119899 domains can be constructed
starting from any value 120593119910gt 0
In relation to the field of applications (RC columns) thefirst values of 120593
119906and 120593
119910that are to be actually considered are
the ones first producing 119899 ge 0
6 Applications
The reliability of the procedure proposed is shown here byusing the results derived from two RC sections considered byother researchers
Thefirst application refers to one of the sections examinedby Zahn et al [27] This was a square RC section of side119861 = 400mm and realized with concrete having cylindrical
10 Advances in Civil EngineeringN
orm
aliz
ed st
ress
1
00 1 2 3 4 5
Saatcioglu and Razvi (1992)Mander et al (1988)
Normalized strain
Figure 7 Calibration of constitutive law of confined concrete
strength 119891co = 30Nmm2 The longitudinal reinforcementconsisted of 12 steel bars of diameter 16mm having yieldingstrength 119891
119904119910= 420Nmm2 uniformly distributed along the
perimeter of the concrete core (constant pitch) The coverdepth measured from the external perimeter of the stirrupswas 13mm The transverse reinforcement consisting of steelstirrups of diameter equal to 10mm according to the calcu-lations made by the aforementioned authors provided theeffective confinement pressure 119891
ℓ119890= 011119891
1015840
co where 1198911015840
co =
085119891co is the reduced strength value prescribed by the ACI318 code [28]
The flexural response was numerically derived in [27] byusing a classical fibre model in which the stress-strain law ofthe confined concrete was assumed in agreement with Man-derrsquosmodel [25]Therefore in this application the parameters120573cc and 120578cc governing the shape of the laws expressed by (4)and (5) are calibrated so that they lead to a constitutive lawof confined concrete which is very similar to that used by theaforementioned authors
Manderrsquos model [25] for the given value of the confine-ment pressure is governed by the parameter 119903 = 13 themodel of Saatcioglu and Razvi [18] leads to a very similarshape of the stress-strain curve by assuming that 120573cc = 045
and 120578cc = minus006 Figure 7 shows the very similar shapes ofthe constitutive laws corresponding to the two models con-sidered
The procedure proposed in [25] provided the followingcharacteristic values of confined concrete strength and cor-responding strain 120576cco = 00073 119891cco = 46Nmm2 Thesevalues are also adopted for the dimensional stress-strain lawexpressed by the model used here [18]
Since for the stress-strain law of the unconfined concreteZahn et al [27] do not give any information the default valuesindicated in Section 31 are assumed here 120573
119888= 1 120578
119888= minus08
and 120576co = 0002On the basis of the above data the square cross-section of
the concrete coremeasured inside the perimeter stirrups hasa side of 119887 = ℎ = 354mm the cover depth is equal to
400
300
200
100
00 001 002 003 004 005
Mom
ent (
kNm
)
Curvature (1m)
Zahn et al (1989)Present model
Figure 8 Comparison of results119872-120593 curve for given119873
23mm the number of intermediate reinforcing bars realizingthe secondary reinforcement is 119899
119887= 119899ℎ= 2 and the distance
of the barycentre of the section of the principal reinforcementfrom the external side of the concrete core section is ℎ
1=
162 = 8mmThe principal reinforcement consists of four upper and
four lower 16mm steel bars while the secondary reinforce-ment consists of four bars (two on the right and two on theleft) of the same diameter The steel reinforcement constitu-tive law is defined by assuming 119864
119904= 210GPa 120576
119904119910= 119891119904119910119864119904=
0002) and 119864119904ℎ
= 119864119904119888
= 0Therefore the dimensionless parameters involved in the
formulation proposed take on the following values 120575V = 120575119900=
= 0Figure 8 compares the 119872-120593 curve obtained in [27] with
that derived from the procedure shown in Section 52 for acompressive constant axial load 119873 = 03119891co(119861 times 119861) corre-sponding to a compressive level 119899 = 119873(119887
2
times 119891cco) = 025 inagreement with the first of (14)The results in the figure showa very good level of agreement between the numerical pro-cedure adopted in [27] and the analytical one adopted here
Both curves in the figure clearly show the cusp producedby the yielding of the principal reinforcement in tension andthe effects of the progressive cover spalling
The second application refers to the middle-height sec-tion of an RC column of height 1640mm experimentallytested by Saatcioglu et al [19] marked as specimen C6-2 bythe authors
The loading condition was realized by imposing relativeaxial displacements so that the reactive compressive forceacted with fixed eccentricity in a plane of principal inertiaof the sections Suitable devices were applied at the columnends so that the column itself behaved like a hinged verticalRC member under an eccentric compressive load
As a consequence each point of the moment-curvaturecurve characterizing the experimental response obtained by
Advances in Civil Engineering 11
the aforementioned authors corresponds to a different valueof axial loadMoreover the actual bendingmoment acting onthe middle-height section was influenced by the 119875-Δ effect
The formulation proposed here is validated consideringsome points of the aforementioned curve for which theauthors indicate the values of the axial load119873 and curvaturethat were measured These values of 119873 are normalized withrespect to the axial load 119873
119900expressed by (2) The procedure
is applied by determining the value of 119909119888from (15) for the
assigned values of 119899 and 120593 and by calculating 119898 from (16)The results are compared with the numerical values obtainedby the strip model adopted by the authors which includedthe 119875-Δ effect and the values experimentally detected
The specimen considered had a square section of side 119861 =
210mm cover depth of 125mm and longitudinal reinforcingbars of diameter 113mm uniformly distributed along theperimeter of the concrete core having yielding strength119891
119904119910=
517Nmm2 and average hardening modulus 119864119904ℎ
= 0013119864119904
The transverse steel reinforcement consisted of square andoctagonal 63mm stirrups with pitch 50mm
The unconfined concrete had cylindrical strength 119891co =
3440Nmm2The numerical model adopted in [19] assumedthat the perimeter of the concrete core was coincident withthe axis lines of the external stirrups
On the basis of these geometrical and mechanical dataand by evaluating the effective confinement pressure by theprocedure proposed in [18] for the confined concrete sectionone obtains 119887 = ℎ = 17870mm 119891cco = 4635 Nmm2 120576cco =
000547 120578cc = minus0052 and 120573 = 0590The further parametersinvolved in the formulation proposed take on the followingvalues 120575
= 0013Figure 9 shows that the procedure proposed provides
results that are very close to those derived by the numericalmodel adopted in [19] The experimentally detected values ofmoment prove to be underestimated by both models
It must be observed that the last point detected by usingthe present formulation corresponding to the ultimate stateof the section had been already found in [17] because thissection belongs to the more restricted class of sections con-sidered in that work
Figure 10 shows the strength and curvature domainsobtained by using the procedure described in Section 53For confined concrete and steel reinforcement the followingvalues were assumed deduced from the data shown in [19]120576ccu = 0032 (120583cc = 585) and 120576
119904119906= 0066 (120583
119903= 27) The
results confirm that a good level of confinementmakes it pos-sible to achieve acceptable ductility of curvature even underhigh levels of compression In order to show how the proce-dure proposed is easily able to evaluate the influence of all thegeometric and mechanical parameters governing the flexuralresponse of a class of RC sections Figure 10(a) also shows theinfluence of the unconfined concrete cover on the bendingmoment at the first yielding of the principal reinforcement intension As expected this influence proves to be negligible forhigh values of the compression level
The dotted curve in Figure 10(b) obviously stops at thepoint of intersection with the continuous curve Beyond this
100
80
60
40
20
00 1 2 3 4
Mom
ent (
kNm
)
ExperimentalModelPresent model
Saatcioglu et al (1995)
Curvature (1mm times 10minus4)
Figure 9 Comparison of theoretical and experimental results
point the 120593119906-119873 curve proceeds by a very brief stretch (not
very evident in the figure) that corresponds to brittle collapseof the section due to the achievement of the maximum avail-able compressive strain in the confined concrete when theprincipal reinforcement in tension is still in the elastic fieldThe end point of this curve corresponds to the achievementof the maximum neutral axis depth defined by the second of(51)
Nevertheless the axial load values that cannot be consid-ered because of these limitations are well beyond the onesthat can be assumed in the structural design of RC buildingsin seismic areas or usually found in the columns of existingbuildings
7 Conclusions
A dimensionless formulation has been proposed which pro-vides the flexural response of classes of sections of RC col-umns having the same values of the geometric and mechani-cal parameters defined in this study
At each loading stage the resisting components of the sec-tion (cover and concrete core and principal and secondarylongitudinal reinforcements) give a contribution to the equi-librium that can be expressed by analytical functions depend-ing on the normalized neutral axis depth and the curvature ofthe section
Special equilibrium conditions like the ones correspond-ing to the first yielding of the principal reinforcement intension and to the ultimate state imply an analytical link bet-ween the neutral axis depth and the curvature so that the sumof the aforementioned contributions becomes an analyticalfunction of a single variable which can express a strength orcurvature domain
For a generic equilibrium condition an iterative proce-dure to determine the neutral axis depth corresponding to
12 Advances in Civil Engineering
100
80
60
40
20
00 300 600 900 1200 1500
Mom
ent (
kNm
)
Axial load (kN)
Mu
My
My without cover
(a)
8
6
4
2
00 300 600 900 1200 1500
Axial load (kN)
120593u120593y
Curv
atur
e(1
mm
times10
minus4)
(b)
Figure 10119872-119873 and 120593-119873 domains at the ultimate state and the first yielding of reinforcement
the assigned values of curvature and compression level isrequired but each iteration step implies the simple use of thesame equilibrium equation for an updated value of the nor-malized neutral axis dept
The dimensionless form of the formulation proposedmakes also it easily possible to evaluate the influence of theparameters characterizing the class of sections examined onthe strength and curvature corresponding to a given loadingstage
Acknowledgment
Thisworkwas carried outwithin the 2010ndash2013Research Pro-ject ldquoDPC-ReLUIS (Dipartimento Protezione Civile-Rete deiLaboratori Universitari di Ingegneria Sismica)rdquo AT 1 Task112 The related financial support was greatly appreciated
References
[1] A Fafitis ldquoInteraction surfaces of reinforced-concrete sectionsin biaxial bendingrdquo Journal of Structural Engineering vol 127no 7 pp 840ndash846 2001
[2] J L Bonet P F Miguel M A Fernandez and M L RomeroldquoAnalytical approach to failure surfaces in reinforced concretesections subjected to axial loads and biaxial bendingrdquo Journalof Structural Engineering vol 130 no 12 pp 2006ndash2015 2004
[3] G Monti and S Alessandri ldquoAssessment of rc columns undercombined biaxial bending and axial loadrdquo in Proceedings of the2nd FIB Congress Naples Italy 2006
[4] RW Clough and S B Johnston ldquoEffect of stiffness degradationon earthquake ductility requirementsrdquo in Proceedings of 2ndJapan Earthquake Engineering Symposium Tokyo Japan 1966
[5] T Takeda M A Sozen and N N Nielsen ldquoReinforced con-crete response to simulated earthquakerdquo Journal of StructuralDivision vol 96 no 12 pp 2257ndash2273 1970
[6] M S L Roufaiel and C Meyer ldquoAnalytical modeling of hyster-etic behavior of reinforced concrete framerdquo Journal of StructuralEngineering vol 113 no 3 pp 429ndash444 1987
[7] A R Mari and A C Scordelis ldquoNonlinear geometric materialand time dependent analysis of three dimensional reinforcedand prestressed concrete framesrdquo USBSESM Report 8412Department of Civil Engineering University of CaliforniaBerkeley Calif USA 1973
[8] T Taucer E Spacone and F C Filippou ldquoA fiber beam-col-umn element for seismic response analysis of reinforced con-crete structuresrdquo Report EERC 91-17 Earthquake EngineeringResearch Center Berkeley Calif USA 1991
[9] Z Zhu I Ahmad and A Mirmiran ldquoFiber element modelingfor seismic performance of bridge columns made of concrete-filled FRP tubesrdquo Engineering Structures vol 28 no 14 pp2023ndash2035 2006
[10] E O Pfrang C P Siess and M A Sozen ldquoLoad-moment-cur-vature characteristics of RC cross-sectionsrdquoACI Journal vol 61no 7 pp 763ndash778 1964
[11] D J Carreira and K-H Chu ldquoThemoment-curvature relation-ship of RC membersrdquo ACI Journal vol 83 no 2 pp 191ndash1981986
[12] R H Wood ldquoSome controversial and curious developments inplastic theory of structuresrdquo inEngineering Plasticity J Heymanand F A Leckie Eds pp 665ndash691 CambridgeUniversity PressCambridge UK 1968
[13] Y LMo ldquoInvestigation of reinforced concrete frame behaviourtheory and testsrdquo Magazine of Concrete Research vol 44 no160 pp 163ndash173 1992
[14] M Jirasek and Z P Bazant Inelastic Analysis of Structures JonWiley amp Sons London UK 2002
[15] S Chandrasekaran L Nunziante G Serino and F CarannanteldquoCurvature ductility of RC sections based on Eurocode analyt-ical procedurerdquo KSCE Journal of Civil Engineering vol 15 no 1pp 131ndash144 2011
Advances in Civil Engineering 13
[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012
[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013
[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992
[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995
[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006
[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010
[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012
[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951
[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982
[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002
[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989
[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008
(a) (b)Figure 1 Section geometry and analysis model
contribution Consequently the upper and lower reinforcingbars and the intermediate reinforcing bars will be conven-tionally denoted as ldquoprincipal reinforcementrdquo and ldquosecondaryreinforcementrdquo respectively
With reference to the symbols in Figure 1(b) the follow-ing relationships must be considered
1198601199041
=119860119904119888
2+ 119860119904119887 119860
1199042= 119860119904ℎ
119901ℓ=
ℎ minus 2ℎ1
119899ℎ+ 1
1198891= 119899ℎ119901ℓ
(1)
Figure 1(b) shows that in the analysis model the secondaryreinforcement is assumed to be uniformly distributed alongthe segment 119889
1defined as shown in Figure 1(a)
Since high curvature values are involved (ultimate limitstate) unconfined concrete spalling reduces the resisting sec-tion to that of the confined concrete core all the geometricaland mechanical parameters involved in the formulation pro-posed here are referred to this reduced section
Therefore the axial load 119873 and bending moment 119872 willbe normalized to the values
119873119900= 119887ℎ119891cco 119872
119900= 119887ℎ2
119891cco (2)
where119891cco is the cylindrical strength of the confined concreteThe strains of the confined concrete and the dimension-
less curvature120593ℎwill be normalizedwith respect to the strain120576cco corresponding to119891cco and this normalization is indicatedby using the superscription () the same notation is adoptedfor the normalization of the strains of the unconfined con-crete with respect to the strain 120576co corresponding to the cyl-indrical strength 119891co The distances normalized to the heightℎ of the concrete core are denoted by using the superscription(minus
)For clarityrsquos sake the geometrical parameters thatmust be
considered assigned are denoted using Greek letters There-fore with reference to the symbols in Figure 1 the followinggeometrical parameters are defined here
120575V =119888V
ℎ 120575
119900=
119888119900
119887
120582 =ℎ1
ℎ 120572 =
1198891
ℎ=
119899ℎ
119899ℎ+ 1
(1 minus 2120582)
(3)
Stre
ss
Strain
ConfinedUnconfined
fcco
fco
120576co 120576cco 120576ccu
Figure 2 Constitutive laws of concrete
3 Constitutive Laws
31 Concrete Differently from what was necessary for theanalysis at the ultimate state [17] to derive the entiremoment-curvature curve of the class of RC sections under considera-tion the constitutive law of the concrete must be defined forboth the concrete core and the unconfined cover
Figure 2 shows typical shapes of constitutive laws of con-crete and the symbols adopted for the characteristic valuesof strains and stresses Several analytical relations expressingthese laws and confirmed by experimental investigations areavailable in the literature
The formulation proposed in the present work allows theuse of any law however the applications were carried outusing the expressions proposed in [18] which also prove tobe valid for eccentric compression and fibrous andor high-strength concrete [19ndash22]
In normalized form these expressions are written as
119904119888(120576) = (2120576 minus 120576
2
)120573
0 le 120576 le 1 (4)
119904119888(120576) = 1 + 120578 (120576 minus 1) 1 le 120576 le 120583 (5)
where 119904119888is the current stress value normalized to the cylindri-
cal strength and 120576 the strain normalized to the correspondingstrain (120576co or 120576cco for unconfined or confined concrete resp)The exponent 120573 le 1 governs the shape of the ascendingbranch of the constitutive lawThe parameter 120578 rules the neg-ative slope of the linear postpeak branch and is obtained bynormalizing the value of the softening modulus 119864
119888soft in thedimensional plane 119891
119888minus120576119888with respect to the secant modulus
expressed by the ratio between the cylindrical strength andthe corresponding strain
The symbol 120583 indicates the normalized value of the ulti-mate concrete strain and is an index of the available ldquoductilityrdquoof the material
Equations (4) and (5) are valid both for unconfined andconfined concrete upon calibration of the parameters 120573 120578and 120583
4 Advances in Civil Engineering
All quantities referring to the unconfined concrete arehere denoted by the subscript ( )
119888 the ascending branch is
modelled by assuming the value 120573119888= 1 so that for 120576co = 0002
(4) expresses the parabolic law first proposed in [23] and sub-sequently adopted by several researchers Considering thatthe softening branch is very steep for unconfined concreteone sets 120578
119888= minus08 so that (5) provides 119891
119888= 0 for 120576
119888= 120576cu =
00045 (120583119888= 225)
For confined concrete the characteristic quantities ofwhich are denoted by the subscript ( )cc the parameters 120573cc e120578cc depend on the effective confinement pressure and can becalculated by the procedure proposed in [18] The availableductility 120583cc = 120576ccu120576cco is obtained by imposing a conven-tional limit value of reduction to the postpeak strength withrespect to the peak value or by assuming the ultimate strain120576ccu in agreement with expressions validated experimentally[24 25]
In the present formulation the factors relating the cylin-drical strength and the corresponding strain of the confinedconcrete to the ones of the originally unconfined concrete aredenoted as
1198961=
119891cco119891co
1198962=
120576cco120576co
(6)
Moreover the following integral functions are defined
1198781120576
= int
120576
0
119904119888(120576) 119889120576 119878
2120576= int
120576
0
119904119888(120576) 120576 119889120576 (7)
Equations (4) and (5) show that these integrals can beexpressed in an exact analytical form only if 120573 = 1 (uncon-fined concrete) Therefore numerical analyses were carriedout in order to derive approximate analytical expressions forthe confined concrete to be utilized in practical applications
As a result it can be shown that in the field 04 le 120573 le 1which includes all real cases the following expressions implya maximum error of 8 with respect to the values calculatedby numerical integration for 120576 ge 01
1198781120576
= (1205762
minus1
31205763
)
1199031
1198782120576
= (2
31205763
minus1
41205764
)
1199032
01 le 120576 le 1
(8)
where
1199031= 076 + 048 (120573 minus 05) 119903
2= 087 + 026 (120573 minus 05)
(9)
On the other hand the analytical form expressed by (5) showsthat (7) can be exactly integrated in the field of the normalizedpostpeak strains
1198781120576
= 11987811
+ (120576 minus 1) +1
2120578(120576 minus 1)
2
1198782120576
= 11987821
+1
2(1205762
minus 1) +1
6120578(120576 minus 1)
2
(1 + 2120576)
1 le 120576 le 120583
(10)
These expressions also provide approximate values for con-fined concrete because of the approximation by which
0
Stre
ss
0
Strain
Compression
fsy Atan Esc
Atan Es02fsy
minus120576su minus120576sy
120576sy 120576lowasts
Tension
Atan Esh minusfsy
Figure 3 Constitutive law of longitudinal steel reinforcement
the addends 11987811
and 11987821
are affected while for 120573 = 1 (uncon-fined concrete) (9) give 119903
1= 1199032
= 1 and both (8) and (10)express the integral quantities (7) in the exact analytical form
32 Longitudinal Steel Reinforcement Figure 3 shows thebilinear simplified constitutive law adopted for the longitu-dinal steel bars in tension and compression and the symbolsdenoting the characteristic quantities involved
With respect to the analyses made in [17] the restrictiveassumption that the postyielding modulus in compressionmust be the same as in tension is removed
Under tension the possible hardening behaviour isdefined by themean hardeningmodulus119864
119904ℎ under compres-
sion the slope of the postyielding branch can bemodified (thisoption is shown in the figure) so that it can becomenegative ifthe transverse reinforcement is not able to prevent buckling ofthe longitudinal bars In this case the slope of the postyield-ing branch is governed by the modulus 119864
119904119888 which can be
determined by using the model proposed in [26] assumingthat the ultimate strain corresponds to a reduction of 80 ofthe yielding stress 119891
119904119910 This ultimate strain in compression is
denoted as 120576lowast119904in Figure 3
The simple equations that analytically express the lawsof the elastic and postyielding branches in Figure 3 are hereomitted for brevityrsquos sake They will be introduced in thecourse of the formulation
The dimensionless parameters characterizing the slopesof the postyielding branches and the ultimate strain in anideal normalized stress-strain diagram are denoted as
120578119904ℎ
=119864119904ℎ
119864119904
120578119904119888
=119864119904119888
119864119904
120583119903=
120576119904119906
120576119904119910
(11)
where 119864119904is the Young modulus and 120576
119904119910 120576119904119906
are the yield-ing and ultimate strains respectively For compressed steelbars subject to buckling 120576
119904119906is understood to be replaced by
120576lowast
119904in (11)A further parameter relating the characteristic strain
values of steel and concrete is
120577119904119888
=
120576119904119910
120576cco (12)
Advances in Civil Engineering 5
Finally the amount of principal and secondary longitudinalreinforcement is related to the section of the concrete core bymeans of the mechanical ratios of reinforcement
1205961
= 2
1198601199041119891119904119910
119887ℎ119891cco 120596
2= 2
1198601199042119891119904119910
119887ℎ119891cco (13)
4 Equilibrium of Section
Denoting as119873 and119872 the axial load and the bendingmomentacting on the section and as
119899 =119873
119873119900
=119873
119887ℎ119891cco 119898 =
119872
119872119900
=119872
119887ℎ2119891cco (14)
these quantities normalized with respect to the ones definedby (2) the equilibrium of the section in dimensionless formis expressed by the following equations
119899 = 1198991199041
+ 1198991199042
+ 119899119888 (15)
119898 = 1198981199041
+ 1198981199042
+ 119898119888 (16)
At the secondmember of these equations there are clearlyindicated and ordered contributions offered by the principalreinforcement secondary reinforcement and concrete
These contributions are analytically expressed in the fol-lowing sections by assuming the classical hypothesis that thesection remains plane and neglecting the tensile concretestrength
41 Contribution of Principal Reinforcement In the analyticalformulation shown in this section the following parametersare involved the geometrical parameter 120582 defined in (1) themechanical parameters characterizing the constitutive law ofthe steel reinforcement 120578
119904119888 120578119904ℎ and 120577
119904119888 defined by (11) and
(12) and the mechanical ratio of reinforcement 1205961expressed
by the first of (13)Figure 4 shows the lateral view of the RC member con-
sidered and a generic state of strain and stressBy using the symbols shown in the figure the upper and
lower reinforcement bars are subjected respectively to thestrains
Then the axial load and bending moment with respect to thebarycentre of the section that the secondary reinforcementcan bear can be calculated by the following expressions
1198731199042
=21198601199042
1198891
ℎ
2[1198911198901
(1198861+
120577119904119888
120593) + 119891
1198902(1198862+
120577119904119888
120593)
minus (Δ11989111198861+ Δ11989121198862) ]
1198721199042
=21198601199042
1198891
ℎ2
2[(1198911198901
minus 1198911198902)1205722
6minus Δ11989111198861(120572
2minus
1198861
3)
+Δ11989121198862(120572
2minus
1198862
3)]
(33)
Equations (28) (29) and (31) show that
1198911198901
(1198861+
120577119904119888
120593) + 1198911198902
(1198862+
120577119904119888
120593) = 2119891
119904119910
120593
120577119904119888
(119909119888minus
1
2) 120572
1198911198901
minus 1198911198902
= 119891119904119910
120593
120577119904119888
120572
(34)
Therefore substituting (34) into (33) in normalized form oneobtains
1198991199042
=1198731199042
119873119900
= 1205962
120593
120577119904119888
(119909119888minus
1
2) minus
1
2120572
times [(1 minus 120578119904119888) 1198862
1minus (1 minus 120578
119904ℎ) 1198862
2]
(35)
1198981199042
=1198721199042
119872119900
=1205962
2120572
120593
120577119904119888
[1205723
6minus (1 minus 120578
119904119888) (
120572
2minus
1198861
3) 1198862
1
minus (1 minus 120578119904ℎ) (
120572
2minus
1198862
3) 1198862
2]
(36)
Advances in Civil Engineering 7
h
c
c
co cob
x3
xc
Confinedconcrete
1205761
1205762
1205763
1205764
(a)
h
c
c
co cob
xc
1205761
1205762
1205763 = 1205764 = 0x3 = 0
Confinedconcrete
(b)
Figure 6 Strain state of concrete section (a) uncracked section (b) cracked section
It must be observed that if one sets 119864119904119888
= 119864119904in (30) andor
119864119904ℎ
= 119864119904in (32) (33) expresses the contributions to the axial
load and bendingmoment in the case inwhich the upper fibreandor the lower fibre of the secondary reinforcement remainin the elastic field As a consequence (35) and (36) are validfor any value of 119909
119888e 120593 under the condition that one sets
120578119904119888=1 if minus 120576
119904119910le 1205761199041
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 minus 120572
2)le1
120578119904ℎ=1 if minus 120576
119904119910le 1205761199042
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 + 120572
2)le1
(37)
43 Contribution of Concrete Figure 6 shows the strain statesof the section corresponding to the cases of uncracked andcracked sections By using the symbols in the figure the con-tributions to the equilibrium provided by the concrete coreare expressed by
119873cc = 119887int
119909119888
1199093
119891119888(119909) 119889119909 =
119887ℎ119891cco120593
int
1205762
1205763
119904119888(120576) 119889120576
119872cc = 119887int
119909119888
1199093
119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus
ℎ
2)
=119887ℎ2
119891cco (1
1205932int
1205762
1205763
119904119888(120576) 120576 119889120576minus
119909119888minus 05
120593int
1205762
1205763
119904119888(120576) 119889120576)
(38)
where
1199093=
0 for 119909119888le ℎ
119909119888minus ℎ for 119909
119888gt ℎ
(39)
and consequently
1205762= 120593119909119888
1205763=
0 for 119909119888le 1
120593 (119909119888minus 1) for 119909
119888gt 1
(40)
Considering (7) in the normalized form one obtains
119899cc =119873cc119873119900
=1
120593(11987811205762
minus 11987811205763
) (41)
119898cc =119872cc119872119900
=1
1205932(11987821205762
minus 11987821205763
) minus 119899cc (119909119888 minus1
2) (42)
The contribution of the surrounding unconfined concrete(concrete cover) can be calculated by subtracting the contri-bution of the confined region to that of the whole section
Considering that the strains involved in the functions 1198781120576
and 1198782120576
must be normalized with respect to the strain 120576co(unconfined concrete) by following the same procedure asthat leading to (41) and (42) and by introducing the con-finement efficiency factors defined by (6) for the whole sec-tion one obtains
119899un1 =1 + 2120575
119900
1198961
1
1198962120593
(11987811205761015840
1
minus 11987811205761015840
4
)
119898un1 =1 + 2120575
119900
1198961
1
(1198962120593)2(11987821205761015840
1
minus 11987821205761015840
4
) minus 119899un1 (119909119888 minus1
2)
(43)where
1205761015840
1= 1198962120593 (119909119888+ 120575V) le 120583
119888
1205761015840
4=
0 for 119909119888le (1 + 120575V)
1198962120593 [119909119888minus (1 + 120575V)] for 119909
119888gt (1 + 120575V)
(44)
The contribution to be subtracted from the above quantitiesare calculated bymodifying (41) and (42) in order to considerthe different constitutive law of the unconfined concrete andthe different base of normalization of strain and stress It caneasily be shown that one obtains
119899un2 =1
11989611198962120593
(11987811205761015840
2
minus 11987811205761015840
3
) (45)
119898un2 =1
1198961(1198962120593)2(11987821205761015840
2
minus 11987821205761015840
3
) minus 119899un2 (119909119888 minus1
2) (46)
8 Advances in Civil Engineering
where
1205761015840
2= 1198962120593119909119888le 120583119888
1205761015840
3=
0 for 119909119888le 1
1198962120593 (119909119888minus 1) for 119909
119888gt 1
(47)
On the basis of what was said above the effective contributionof the concrete section in compression in a generic loadingstate is expressed by
119899119888= 119899cc + 119899un1 minus 119899un2 (48)
119898119888= 119898cc + 119898un1 minus 119898un2 (49)
If the confinement effect is negligible (6) gives 1198961= 1198962= 1
By comparing (40) (41) and (42) with (47) (45) and (46)respectively it can easily be observed that 119899cc = 119899un2 and119898cc = 119898un2 Therefore in this case the contributions of thewhole section of reacting concrete given by (48) and (49) arereduced to the values 119899cc = 119899un1 and 119898cc = 119898un1 and areexpressed by (45) and (46) for 119896
1= 1198962= 1
5 Analysis Procedure
The flexural response of a class of RC sections characterizedby assigned values of the geometrical andmechanical param-eters defined above is obtained by imposing the equilibriumcondition expressed by (15) and (16)
For each configuration the contributions to the equi-librium provided by the principal reinforcement ((23) and(24) under the conditions expressed by (26)) the secondaryreinforcement ((28) (35) and (36) under the conditionsexpressed by (37)) and the compressed region of concrete((48) and (49) considering (41)ndash(47)) only depend on thevariables 119909
119888and 120593 which are independent of or related to one
another according to the type of analysis required (see below)In all cases it must be assumed that the proposed formulationis applied for 120593 gt 0
51 Field of Application The field of validity of the procedureproposed here is limited by the fact that the constitutive law ofthe steel reinforcement does not include unloading branchesstarting from a point of the post yielding branches in com-pression or in tension Therefore for any loading step theupper principal reinforcement and the upper fibre of the dis-tributed secondary reinforcement both potentially subject tocompression cannot be in the postyielding field in tensionthe lower principal reinforcement and the lower fibre of thedistributed secondary reinforcement both potentially subjectto tension cannot be in the postyielding field in compression
Because of these assumptions the field of values of 119909119888
consistent with a given value of 120593 is limited by the initial andend values provided by (26) or (37)
More precisely in the absence of secondary reinforce-ment (120596
2= 0) (26) gives
119909119888min = 120582 minus
120577119904119888
120593 119909
119888max = (1 minus 120582) +120577119904119888
120593 (50)
while if the secondary reinforcement is present (37) leads tomore restrictive limitations
119909119888min =
1 minus 120572
2minus
120577119904119888
120593 119909
119888max =1 + 120572
2+
120577119904119888
120593 (51)
These limitations also occur in a classical numericalapproach based on the strip decomposition of the section
However it must be observed that (50) or (51) allows oneto construct the moment-curvature curve for a large field ofvalues of assigned level of compression including the moreusual cases occurring in practical applications
52 Moment-Curvature Curves The more usual applicationof the analytical expressions derived in the previous sectionsis to derive by a step-by-step procedure the moment-curva-ture119898-120593 curve of a class of RC sections for a given value of 119899
To this purpose the procedure is applied by the followingsteps
(i) assigning the lowest value of curvature 120593 that onewants to consider increasing values of 119909
119888are consid-
ered from the minimum to the maximum expressedby (50) or (51)
(ii) for each value of the couple 120593 119909119888the corresponding
value of 1198991199041 1198991199042 and 119899
119888are calculated and their sum
increasing for each increase in 119909119888
(iii) when the assigned value of 119899 is reached with anacceptable tolerance (ie (15) is verified with verygood approximation) the assigned values of 120593 and 119899and the value of 119909
119888that was found make it possible
to calculate the corresponding values of11989811990411198981199042 and
119898119888expressed by (24) (36) and (49) and the sum
of these contributions of moment is the ordinate ofthe point having the abscissa 120593 belonging to the119898-120593curve sought
(iv) considering an increased value of 120593 the procedure isrepeated to obtain a subsequent point of themoment-curvature curve
The procedure stops when the maximum available ulti-mate confined concrete strain or the maximum available ten-sile strain of the lower reinforcement is exceeded These twooccurrences imply respectively that 120593119909
119888gt 120583cc or 120593(119909119888 minus 1 +
120582) lt minus120577119904119888120583119903 where 120583cc = 120576ccu120576cco is the available ductility of
the confined concrete defined in Section 31 and 120583119903is the steel
ductility defined in (11)If the stress-strain law of the steel reinforcement also
exhibits a softening postyielding branch corresponding to apostbuckling behaviour up to the strain value 120576
∙
119904(Figure 3)
a further limit at which the procedure must be stoppedderives from the condition that 120593(119909
119888minus 120582) gt 120577
119904119888120583119903119888 in
which 120583119903119888
= 120576lowast
119904120576119904119910 Beyond this condition the residual stress
of the compressed reinforcement should be assumed to beconstant and equal to 02119891
119904119910 but this branch of the stress-
strain law is not considered in the proposed model as statedin Section 32
Advances in Civil Engineering 9
53 Strength and Curvature Domains A typical 119898-120593 curveexhibits two characteristic points corresponding to theachievement of the ultimate state and to the first yield-ing of the principal reinforcement in tension respectivelyObviously the first point is present in all cases while thesecond can only be reached if yielding of the reinforcementin tension occurs before the brittle collapse of the concretecore due to low available concrete ductility orand a very highcompression level
The analytical expressions shown in the previous sectionsmake it possible to determine directly the coordinates of thesepoints and to derive in closed form the119898
119906-119899 120593119906-119899 and119898
119910-119899
120593119910-119899 domains which give useful information on the flexural
strength and curvature ductility of an RC section subjected toan assigned compression level
The constructions of the aforementioned curves can bemade more easily than by using a classical discretized modelof the section because (15) and (16) in a suitable specializedform provide analytical expressions where the axial com-pressive load and the corresponding bending moment onlydepend on a single variable 119909
119888and 120593 being related to one
anotherIn order to construct the 119898
119906-119899 and 120593
119906-119899 curves two dif-
ferent relationships linking the neutral axis depth to the cur-vature must be considered according to whether the collapseof the section is produced by excess of the available strain ofthe reinforcement in tension or of the available strain of theconcrete core in compression Moreover to distinguish thesetwo different events it must be observed that the maximumcurvature of the section is reachedwhen these two conditionsoccur simultaneously Considering Figure 4 it can easily bededuced that this value of curvature in normalized form is
120593119906max =
120583cc + 120577119904119888120583119903
1 minus 120582 (52)
The first branch of the domains sought is determined byimposing the condition that the collapse of the section is dueto the achievement of themaximum tensile strain of the prin-cipal reinforcement in tension (low level of compression)Therefore the following condition has to be assumed
120593119906(119909119888minus 1 + 120582) = minus 120577
119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus
120577119904119888120583119903
120593119906
(53)
Considering increasing values of the curvature from themin-imum value to themaximum expressed by (52) the second of(53) provide the corresponding value of 119909
119888 and (15) and (16)
by making explicit the contributions of the concrete and theprincipal and secondary reinforcement provide the corre-sponding values of 119899 and119898
119906
Once 120593119906max is reached the second branch of the 119898
119906-119899
120593119906-119899 curves must be determined considering that for further
decreasing values of 120593119906 the collapse is produced by the
achievement of the maximum compressive strain of the con-fined concrete Therefore the new relationship linking thecurvature to the neutral axis depth is
By using the second of these equations (15) and (16) againlink 119899 and119898
119906to the only variable 120593
119906
It must be observed that if the principal reinforcement incompression is subject to buckling after the achievement ofthe yielding stress the ultimate strain of this compressed rein-forcement is the strain denoted as 120576∙
119904in Figure 3 This strain
value can be reached before the concrete core reaches thenormalized value of strain 120583cc In this case the maximumvalue of normalized dimensionless curvature expressed by(52) must be substituted by the value 120593
119906max = 120577119904119888(120583119903119888
+
120583119903)(1minus2120582) where 120583
119903119888= 120576lowast
119904120576119904119910 and the second branch of the
ultimate strength domains must be determined by assumingthat for values of 120593
119906that decrease from this maximum the
relationship linking the neutral axis dept to the curvature is119909119888= 120582 + 120577
119904119888120583119903119888120593
The 119898119910-119899 120593119910-119899 curves can be simply determined by
assuming that whatever the value of the curvature is the prin-cipal reinforcement in tension is affected by the strain valueequal to minus120576
119904119910 This condition in dimensionless terms gives
120593119910(119909119888minus 1 + 120582) = minus 120577
119904119888997904rArr 119909119888= (1 minus 120582) minus
120577119904119888
120593119910
(55)
By introducing the second of (55) into the equilibrium Equa-tions (15) and (16) they again become two functions of thesingle variable 120593 = 120593
119910 The 119898
119910-119899 120593119910-119899 curves can be con-
structed starting froma value thatwas assumed to be themin-imum up to the maximum value corresponding to one of thepossible aforementioned occurrences the collapse of the con-crete core due to achievement of the available ductility of thematerial or the achievement of the strain 120576
∙
119904in the principal
reinforcement in compression if this reinforcement is subjectto buckling
Considering the field of validity of the procedure pro-posed the minimum value of 120593
119906that can be considered to
construct the119898119906-119899 120593119906-119899 domains is obtained by introducing
the second of (53) into the first of (50) or (51) so that
120593119906min = 120577
119904119888
120583119903minus 1
1 minus 2120582 (56)
in the absence of the secondary reinforcement and
120593119906min = 120577
119904119888
120583119903minus 1
(1 + 120572) 2 minus 120582 (57)
if the secondary reinforcement is also presentObviously the 119898
119910-119899 120593119910-119899 domains can be constructed
starting from any value 120593119910gt 0
In relation to the field of applications (RC columns) thefirst values of 120593
119906and 120593
119910that are to be actually considered are
the ones first producing 119899 ge 0
6 Applications
The reliability of the procedure proposed is shown here byusing the results derived from two RC sections considered byother researchers
Thefirst application refers to one of the sections examinedby Zahn et al [27] This was a square RC section of side119861 = 400mm and realized with concrete having cylindrical
10 Advances in Civil EngineeringN
orm
aliz
ed st
ress
1
00 1 2 3 4 5
Saatcioglu and Razvi (1992)Mander et al (1988)
Normalized strain
Figure 7 Calibration of constitutive law of confined concrete
strength 119891co = 30Nmm2 The longitudinal reinforcementconsisted of 12 steel bars of diameter 16mm having yieldingstrength 119891
119904119910= 420Nmm2 uniformly distributed along the
perimeter of the concrete core (constant pitch) The coverdepth measured from the external perimeter of the stirrupswas 13mm The transverse reinforcement consisting of steelstirrups of diameter equal to 10mm according to the calcu-lations made by the aforementioned authors provided theeffective confinement pressure 119891
ℓ119890= 011119891
1015840
co where 1198911015840
co =
085119891co is the reduced strength value prescribed by the ACI318 code [28]
The flexural response was numerically derived in [27] byusing a classical fibre model in which the stress-strain law ofthe confined concrete was assumed in agreement with Man-derrsquosmodel [25]Therefore in this application the parameters120573cc and 120578cc governing the shape of the laws expressed by (4)and (5) are calibrated so that they lead to a constitutive lawof confined concrete which is very similar to that used by theaforementioned authors
Manderrsquos model [25] for the given value of the confine-ment pressure is governed by the parameter 119903 = 13 themodel of Saatcioglu and Razvi [18] leads to a very similarshape of the stress-strain curve by assuming that 120573cc = 045
and 120578cc = minus006 Figure 7 shows the very similar shapes ofthe constitutive laws corresponding to the two models con-sidered
The procedure proposed in [25] provided the followingcharacteristic values of confined concrete strength and cor-responding strain 120576cco = 00073 119891cco = 46Nmm2 Thesevalues are also adopted for the dimensional stress-strain lawexpressed by the model used here [18]
Since for the stress-strain law of the unconfined concreteZahn et al [27] do not give any information the default valuesindicated in Section 31 are assumed here 120573
119888= 1 120578
119888= minus08
and 120576co = 0002On the basis of the above data the square cross-section of
the concrete coremeasured inside the perimeter stirrups hasa side of 119887 = ℎ = 354mm the cover depth is equal to
400
300
200
100
00 001 002 003 004 005
Mom
ent (
kNm
)
Curvature (1m)
Zahn et al (1989)Present model
Figure 8 Comparison of results119872-120593 curve for given119873
23mm the number of intermediate reinforcing bars realizingthe secondary reinforcement is 119899
119887= 119899ℎ= 2 and the distance
of the barycentre of the section of the principal reinforcementfrom the external side of the concrete core section is ℎ
1=
162 = 8mmThe principal reinforcement consists of four upper and
four lower 16mm steel bars while the secondary reinforce-ment consists of four bars (two on the right and two on theleft) of the same diameter The steel reinforcement constitu-tive law is defined by assuming 119864
119904= 210GPa 120576
119904119910= 119891119904119910119864119904=
0002) and 119864119904ℎ
= 119864119904119888
= 0Therefore the dimensionless parameters involved in the
formulation proposed take on the following values 120575V = 120575119900=
= 0Figure 8 compares the 119872-120593 curve obtained in [27] with
that derived from the procedure shown in Section 52 for acompressive constant axial load 119873 = 03119891co(119861 times 119861) corre-sponding to a compressive level 119899 = 119873(119887
2
times 119891cco) = 025 inagreement with the first of (14)The results in the figure showa very good level of agreement between the numerical pro-cedure adopted in [27] and the analytical one adopted here
Both curves in the figure clearly show the cusp producedby the yielding of the principal reinforcement in tension andthe effects of the progressive cover spalling
The second application refers to the middle-height sec-tion of an RC column of height 1640mm experimentallytested by Saatcioglu et al [19] marked as specimen C6-2 bythe authors
The loading condition was realized by imposing relativeaxial displacements so that the reactive compressive forceacted with fixed eccentricity in a plane of principal inertiaof the sections Suitable devices were applied at the columnends so that the column itself behaved like a hinged verticalRC member under an eccentric compressive load
As a consequence each point of the moment-curvaturecurve characterizing the experimental response obtained by
Advances in Civil Engineering 11
the aforementioned authors corresponds to a different valueof axial loadMoreover the actual bendingmoment acting onthe middle-height section was influenced by the 119875-Δ effect
The formulation proposed here is validated consideringsome points of the aforementioned curve for which theauthors indicate the values of the axial load119873 and curvaturethat were measured These values of 119873 are normalized withrespect to the axial load 119873
119900expressed by (2) The procedure
is applied by determining the value of 119909119888from (15) for the
assigned values of 119899 and 120593 and by calculating 119898 from (16)The results are compared with the numerical values obtainedby the strip model adopted by the authors which includedthe 119875-Δ effect and the values experimentally detected
The specimen considered had a square section of side 119861 =
210mm cover depth of 125mm and longitudinal reinforcingbars of diameter 113mm uniformly distributed along theperimeter of the concrete core having yielding strength119891
119904119910=
517Nmm2 and average hardening modulus 119864119904ℎ
= 0013119864119904
The transverse steel reinforcement consisted of square andoctagonal 63mm stirrups with pitch 50mm
The unconfined concrete had cylindrical strength 119891co =
3440Nmm2The numerical model adopted in [19] assumedthat the perimeter of the concrete core was coincident withthe axis lines of the external stirrups
On the basis of these geometrical and mechanical dataand by evaluating the effective confinement pressure by theprocedure proposed in [18] for the confined concrete sectionone obtains 119887 = ℎ = 17870mm 119891cco = 4635 Nmm2 120576cco =
000547 120578cc = minus0052 and 120573 = 0590The further parametersinvolved in the formulation proposed take on the followingvalues 120575
= 0013Figure 9 shows that the procedure proposed provides
results that are very close to those derived by the numericalmodel adopted in [19] The experimentally detected values ofmoment prove to be underestimated by both models
It must be observed that the last point detected by usingthe present formulation corresponding to the ultimate stateof the section had been already found in [17] because thissection belongs to the more restricted class of sections con-sidered in that work
Figure 10 shows the strength and curvature domainsobtained by using the procedure described in Section 53For confined concrete and steel reinforcement the followingvalues were assumed deduced from the data shown in [19]120576ccu = 0032 (120583cc = 585) and 120576
119904119906= 0066 (120583
119903= 27) The
results confirm that a good level of confinementmakes it pos-sible to achieve acceptable ductility of curvature even underhigh levels of compression In order to show how the proce-dure proposed is easily able to evaluate the influence of all thegeometric and mechanical parameters governing the flexuralresponse of a class of RC sections Figure 10(a) also shows theinfluence of the unconfined concrete cover on the bendingmoment at the first yielding of the principal reinforcement intension As expected this influence proves to be negligible forhigh values of the compression level
The dotted curve in Figure 10(b) obviously stops at thepoint of intersection with the continuous curve Beyond this
100
80
60
40
20
00 1 2 3 4
Mom
ent (
kNm
)
ExperimentalModelPresent model
Saatcioglu et al (1995)
Curvature (1mm times 10minus4)
Figure 9 Comparison of theoretical and experimental results
point the 120593119906-119873 curve proceeds by a very brief stretch (not
very evident in the figure) that corresponds to brittle collapseof the section due to the achievement of the maximum avail-able compressive strain in the confined concrete when theprincipal reinforcement in tension is still in the elastic fieldThe end point of this curve corresponds to the achievementof the maximum neutral axis depth defined by the second of(51)
Nevertheless the axial load values that cannot be consid-ered because of these limitations are well beyond the onesthat can be assumed in the structural design of RC buildingsin seismic areas or usually found in the columns of existingbuildings
7 Conclusions
A dimensionless formulation has been proposed which pro-vides the flexural response of classes of sections of RC col-umns having the same values of the geometric and mechani-cal parameters defined in this study
At each loading stage the resisting components of the sec-tion (cover and concrete core and principal and secondarylongitudinal reinforcements) give a contribution to the equi-librium that can be expressed by analytical functions depend-ing on the normalized neutral axis depth and the curvature ofthe section
Special equilibrium conditions like the ones correspond-ing to the first yielding of the principal reinforcement intension and to the ultimate state imply an analytical link bet-ween the neutral axis depth and the curvature so that the sumof the aforementioned contributions becomes an analyticalfunction of a single variable which can express a strength orcurvature domain
For a generic equilibrium condition an iterative proce-dure to determine the neutral axis depth corresponding to
12 Advances in Civil Engineering
100
80
60
40
20
00 300 600 900 1200 1500
Mom
ent (
kNm
)
Axial load (kN)
Mu
My
My without cover
(a)
8
6
4
2
00 300 600 900 1200 1500
Axial load (kN)
120593u120593y
Curv
atur
e(1
mm
times10
minus4)
(b)
Figure 10119872-119873 and 120593-119873 domains at the ultimate state and the first yielding of reinforcement
the assigned values of curvature and compression level isrequired but each iteration step implies the simple use of thesame equilibrium equation for an updated value of the nor-malized neutral axis dept
The dimensionless form of the formulation proposedmakes also it easily possible to evaluate the influence of theparameters characterizing the class of sections examined onthe strength and curvature corresponding to a given loadingstage
Acknowledgment
Thisworkwas carried outwithin the 2010ndash2013Research Pro-ject ldquoDPC-ReLUIS (Dipartimento Protezione Civile-Rete deiLaboratori Universitari di Ingegneria Sismica)rdquo AT 1 Task112 The related financial support was greatly appreciated
References
[1] A Fafitis ldquoInteraction surfaces of reinforced-concrete sectionsin biaxial bendingrdquo Journal of Structural Engineering vol 127no 7 pp 840ndash846 2001
[2] J L Bonet P F Miguel M A Fernandez and M L RomeroldquoAnalytical approach to failure surfaces in reinforced concretesections subjected to axial loads and biaxial bendingrdquo Journalof Structural Engineering vol 130 no 12 pp 2006ndash2015 2004
[3] G Monti and S Alessandri ldquoAssessment of rc columns undercombined biaxial bending and axial loadrdquo in Proceedings of the2nd FIB Congress Naples Italy 2006
[4] RW Clough and S B Johnston ldquoEffect of stiffness degradationon earthquake ductility requirementsrdquo in Proceedings of 2ndJapan Earthquake Engineering Symposium Tokyo Japan 1966
[5] T Takeda M A Sozen and N N Nielsen ldquoReinforced con-crete response to simulated earthquakerdquo Journal of StructuralDivision vol 96 no 12 pp 2257ndash2273 1970
[6] M S L Roufaiel and C Meyer ldquoAnalytical modeling of hyster-etic behavior of reinforced concrete framerdquo Journal of StructuralEngineering vol 113 no 3 pp 429ndash444 1987
[7] A R Mari and A C Scordelis ldquoNonlinear geometric materialand time dependent analysis of three dimensional reinforcedand prestressed concrete framesrdquo USBSESM Report 8412Department of Civil Engineering University of CaliforniaBerkeley Calif USA 1973
[8] T Taucer E Spacone and F C Filippou ldquoA fiber beam-col-umn element for seismic response analysis of reinforced con-crete structuresrdquo Report EERC 91-17 Earthquake EngineeringResearch Center Berkeley Calif USA 1991
[9] Z Zhu I Ahmad and A Mirmiran ldquoFiber element modelingfor seismic performance of bridge columns made of concrete-filled FRP tubesrdquo Engineering Structures vol 28 no 14 pp2023ndash2035 2006
[10] E O Pfrang C P Siess and M A Sozen ldquoLoad-moment-cur-vature characteristics of RC cross-sectionsrdquoACI Journal vol 61no 7 pp 763ndash778 1964
[11] D J Carreira and K-H Chu ldquoThemoment-curvature relation-ship of RC membersrdquo ACI Journal vol 83 no 2 pp 191ndash1981986
[12] R H Wood ldquoSome controversial and curious developments inplastic theory of structuresrdquo inEngineering Plasticity J Heymanand F A Leckie Eds pp 665ndash691 CambridgeUniversity PressCambridge UK 1968
[13] Y LMo ldquoInvestigation of reinforced concrete frame behaviourtheory and testsrdquo Magazine of Concrete Research vol 44 no160 pp 163ndash173 1992
[14] M Jirasek and Z P Bazant Inelastic Analysis of Structures JonWiley amp Sons London UK 2002
[15] S Chandrasekaran L Nunziante G Serino and F CarannanteldquoCurvature ductility of RC sections based on Eurocode analyt-ical procedurerdquo KSCE Journal of Civil Engineering vol 15 no 1pp 131ndash144 2011
Advances in Civil Engineering 13
[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012
[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013
[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992
[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995
[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006
[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010
[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012
[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951
[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982
[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002
[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989
[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008
All quantities referring to the unconfined concrete arehere denoted by the subscript ( )
119888 the ascending branch is
modelled by assuming the value 120573119888= 1 so that for 120576co = 0002
(4) expresses the parabolic law first proposed in [23] and sub-sequently adopted by several researchers Considering thatthe softening branch is very steep for unconfined concreteone sets 120578
119888= minus08 so that (5) provides 119891
119888= 0 for 120576
119888= 120576cu =
00045 (120583119888= 225)
For confined concrete the characteristic quantities ofwhich are denoted by the subscript ( )cc the parameters 120573cc e120578cc depend on the effective confinement pressure and can becalculated by the procedure proposed in [18] The availableductility 120583cc = 120576ccu120576cco is obtained by imposing a conven-tional limit value of reduction to the postpeak strength withrespect to the peak value or by assuming the ultimate strain120576ccu in agreement with expressions validated experimentally[24 25]
In the present formulation the factors relating the cylin-drical strength and the corresponding strain of the confinedconcrete to the ones of the originally unconfined concrete aredenoted as
1198961=
119891cco119891co
1198962=
120576cco120576co
(6)
Moreover the following integral functions are defined
1198781120576
= int
120576
0
119904119888(120576) 119889120576 119878
2120576= int
120576
0
119904119888(120576) 120576 119889120576 (7)
Equations (4) and (5) show that these integrals can beexpressed in an exact analytical form only if 120573 = 1 (uncon-fined concrete) Therefore numerical analyses were carriedout in order to derive approximate analytical expressions forthe confined concrete to be utilized in practical applications
As a result it can be shown that in the field 04 le 120573 le 1which includes all real cases the following expressions implya maximum error of 8 with respect to the values calculatedby numerical integration for 120576 ge 01
1198781120576
= (1205762
minus1
31205763
)
1199031
1198782120576
= (2
31205763
minus1
41205764
)
1199032
01 le 120576 le 1
(8)
where
1199031= 076 + 048 (120573 minus 05) 119903
2= 087 + 026 (120573 minus 05)
(9)
On the other hand the analytical form expressed by (5) showsthat (7) can be exactly integrated in the field of the normalizedpostpeak strains
1198781120576
= 11987811
+ (120576 minus 1) +1
2120578(120576 minus 1)
2
1198782120576
= 11987821
+1
2(1205762
minus 1) +1
6120578(120576 minus 1)
2
(1 + 2120576)
1 le 120576 le 120583
(10)
These expressions also provide approximate values for con-fined concrete because of the approximation by which
0
Stre
ss
0
Strain
Compression
fsy Atan Esc
Atan Es02fsy
minus120576su minus120576sy
120576sy 120576lowasts
Tension
Atan Esh minusfsy
Figure 3 Constitutive law of longitudinal steel reinforcement
the addends 11987811
and 11987821
are affected while for 120573 = 1 (uncon-fined concrete) (9) give 119903
1= 1199032
= 1 and both (8) and (10)express the integral quantities (7) in the exact analytical form
32 Longitudinal Steel Reinforcement Figure 3 shows thebilinear simplified constitutive law adopted for the longitu-dinal steel bars in tension and compression and the symbolsdenoting the characteristic quantities involved
With respect to the analyses made in [17] the restrictiveassumption that the postyielding modulus in compressionmust be the same as in tension is removed
Under tension the possible hardening behaviour isdefined by themean hardeningmodulus119864
119904ℎ under compres-
sion the slope of the postyielding branch can bemodified (thisoption is shown in the figure) so that it can becomenegative ifthe transverse reinforcement is not able to prevent buckling ofthe longitudinal bars In this case the slope of the postyield-ing branch is governed by the modulus 119864
119904119888 which can be
determined by using the model proposed in [26] assumingthat the ultimate strain corresponds to a reduction of 80 ofthe yielding stress 119891
119904119910 This ultimate strain in compression is
denoted as 120576lowast119904in Figure 3
The simple equations that analytically express the lawsof the elastic and postyielding branches in Figure 3 are hereomitted for brevityrsquos sake They will be introduced in thecourse of the formulation
The dimensionless parameters characterizing the slopesof the postyielding branches and the ultimate strain in anideal normalized stress-strain diagram are denoted as
120578119904ℎ
=119864119904ℎ
119864119904
120578119904119888
=119864119904119888
119864119904
120583119903=
120576119904119906
120576119904119910
(11)
where 119864119904is the Young modulus and 120576
119904119910 120576119904119906
are the yield-ing and ultimate strains respectively For compressed steelbars subject to buckling 120576
119904119906is understood to be replaced by
120576lowast
119904in (11)A further parameter relating the characteristic strain
values of steel and concrete is
120577119904119888
=
120576119904119910
120576cco (12)
Advances in Civil Engineering 5
Finally the amount of principal and secondary longitudinalreinforcement is related to the section of the concrete core bymeans of the mechanical ratios of reinforcement
1205961
= 2
1198601199041119891119904119910
119887ℎ119891cco 120596
2= 2
1198601199042119891119904119910
119887ℎ119891cco (13)
4 Equilibrium of Section
Denoting as119873 and119872 the axial load and the bendingmomentacting on the section and as
119899 =119873
119873119900
=119873
119887ℎ119891cco 119898 =
119872
119872119900
=119872
119887ℎ2119891cco (14)
these quantities normalized with respect to the ones definedby (2) the equilibrium of the section in dimensionless formis expressed by the following equations
119899 = 1198991199041
+ 1198991199042
+ 119899119888 (15)
119898 = 1198981199041
+ 1198981199042
+ 119898119888 (16)
At the secondmember of these equations there are clearlyindicated and ordered contributions offered by the principalreinforcement secondary reinforcement and concrete
These contributions are analytically expressed in the fol-lowing sections by assuming the classical hypothesis that thesection remains plane and neglecting the tensile concretestrength
41 Contribution of Principal Reinforcement In the analyticalformulation shown in this section the following parametersare involved the geometrical parameter 120582 defined in (1) themechanical parameters characterizing the constitutive law ofthe steel reinforcement 120578
119904119888 120578119904ℎ and 120577
119904119888 defined by (11) and
(12) and the mechanical ratio of reinforcement 1205961expressed
by the first of (13)Figure 4 shows the lateral view of the RC member con-
sidered and a generic state of strain and stressBy using the symbols shown in the figure the upper and
lower reinforcement bars are subjected respectively to thestrains
Then the axial load and bending moment with respect to thebarycentre of the section that the secondary reinforcementcan bear can be calculated by the following expressions
1198731199042
=21198601199042
1198891
ℎ
2[1198911198901
(1198861+
120577119904119888
120593) + 119891
1198902(1198862+
120577119904119888
120593)
minus (Δ11989111198861+ Δ11989121198862) ]
1198721199042
=21198601199042
1198891
ℎ2
2[(1198911198901
minus 1198911198902)1205722
6minus Δ11989111198861(120572
2minus
1198861
3)
+Δ11989121198862(120572
2minus
1198862
3)]
(33)
Equations (28) (29) and (31) show that
1198911198901
(1198861+
120577119904119888
120593) + 1198911198902
(1198862+
120577119904119888
120593) = 2119891
119904119910
120593
120577119904119888
(119909119888minus
1
2) 120572
1198911198901
minus 1198911198902
= 119891119904119910
120593
120577119904119888
120572
(34)
Therefore substituting (34) into (33) in normalized form oneobtains
1198991199042
=1198731199042
119873119900
= 1205962
120593
120577119904119888
(119909119888minus
1
2) minus
1
2120572
times [(1 minus 120578119904119888) 1198862
1minus (1 minus 120578
119904ℎ) 1198862
2]
(35)
1198981199042
=1198721199042
119872119900
=1205962
2120572
120593
120577119904119888
[1205723
6minus (1 minus 120578
119904119888) (
120572
2minus
1198861
3) 1198862
1
minus (1 minus 120578119904ℎ) (
120572
2minus
1198862
3) 1198862
2]
(36)
Advances in Civil Engineering 7
h
c
c
co cob
x3
xc
Confinedconcrete
1205761
1205762
1205763
1205764
(a)
h
c
c
co cob
xc
1205761
1205762
1205763 = 1205764 = 0x3 = 0
Confinedconcrete
(b)
Figure 6 Strain state of concrete section (a) uncracked section (b) cracked section
It must be observed that if one sets 119864119904119888
= 119864119904in (30) andor
119864119904ℎ
= 119864119904in (32) (33) expresses the contributions to the axial
load and bendingmoment in the case inwhich the upper fibreandor the lower fibre of the secondary reinforcement remainin the elastic field As a consequence (35) and (36) are validfor any value of 119909
119888e 120593 under the condition that one sets
120578119904119888=1 if minus 120576
119904119910le 1205761199041
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 minus 120572
2)le1
120578119904ℎ=1 if minus 120576
119904119910le 1205761199042
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 + 120572
2)le1
(37)
43 Contribution of Concrete Figure 6 shows the strain statesof the section corresponding to the cases of uncracked andcracked sections By using the symbols in the figure the con-tributions to the equilibrium provided by the concrete coreare expressed by
119873cc = 119887int
119909119888
1199093
119891119888(119909) 119889119909 =
119887ℎ119891cco120593
int
1205762
1205763
119904119888(120576) 119889120576
119872cc = 119887int
119909119888
1199093
119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus
ℎ
2)
=119887ℎ2
119891cco (1
1205932int
1205762
1205763
119904119888(120576) 120576 119889120576minus
119909119888minus 05
120593int
1205762
1205763
119904119888(120576) 119889120576)
(38)
where
1199093=
0 for 119909119888le ℎ
119909119888minus ℎ for 119909
119888gt ℎ
(39)
and consequently
1205762= 120593119909119888
1205763=
0 for 119909119888le 1
120593 (119909119888minus 1) for 119909
119888gt 1
(40)
Considering (7) in the normalized form one obtains
119899cc =119873cc119873119900
=1
120593(11987811205762
minus 11987811205763
) (41)
119898cc =119872cc119872119900
=1
1205932(11987821205762
minus 11987821205763
) minus 119899cc (119909119888 minus1
2) (42)
The contribution of the surrounding unconfined concrete(concrete cover) can be calculated by subtracting the contri-bution of the confined region to that of the whole section
Considering that the strains involved in the functions 1198781120576
and 1198782120576
must be normalized with respect to the strain 120576co(unconfined concrete) by following the same procedure asthat leading to (41) and (42) and by introducing the con-finement efficiency factors defined by (6) for the whole sec-tion one obtains
119899un1 =1 + 2120575
119900
1198961
1
1198962120593
(11987811205761015840
1
minus 11987811205761015840
4
)
119898un1 =1 + 2120575
119900
1198961
1
(1198962120593)2(11987821205761015840
1
minus 11987821205761015840
4
) minus 119899un1 (119909119888 minus1
2)
(43)where
1205761015840
1= 1198962120593 (119909119888+ 120575V) le 120583
119888
1205761015840
4=
0 for 119909119888le (1 + 120575V)
1198962120593 [119909119888minus (1 + 120575V)] for 119909
119888gt (1 + 120575V)
(44)
The contribution to be subtracted from the above quantitiesare calculated bymodifying (41) and (42) in order to considerthe different constitutive law of the unconfined concrete andthe different base of normalization of strain and stress It caneasily be shown that one obtains
119899un2 =1
11989611198962120593
(11987811205761015840
2
minus 11987811205761015840
3
) (45)
119898un2 =1
1198961(1198962120593)2(11987821205761015840
2
minus 11987821205761015840
3
) minus 119899un2 (119909119888 minus1
2) (46)
8 Advances in Civil Engineering
where
1205761015840
2= 1198962120593119909119888le 120583119888
1205761015840
3=
0 for 119909119888le 1
1198962120593 (119909119888minus 1) for 119909
119888gt 1
(47)
On the basis of what was said above the effective contributionof the concrete section in compression in a generic loadingstate is expressed by
119899119888= 119899cc + 119899un1 minus 119899un2 (48)
119898119888= 119898cc + 119898un1 minus 119898un2 (49)
If the confinement effect is negligible (6) gives 1198961= 1198962= 1
By comparing (40) (41) and (42) with (47) (45) and (46)respectively it can easily be observed that 119899cc = 119899un2 and119898cc = 119898un2 Therefore in this case the contributions of thewhole section of reacting concrete given by (48) and (49) arereduced to the values 119899cc = 119899un1 and 119898cc = 119898un1 and areexpressed by (45) and (46) for 119896
1= 1198962= 1
5 Analysis Procedure
The flexural response of a class of RC sections characterizedby assigned values of the geometrical andmechanical param-eters defined above is obtained by imposing the equilibriumcondition expressed by (15) and (16)
For each configuration the contributions to the equi-librium provided by the principal reinforcement ((23) and(24) under the conditions expressed by (26)) the secondaryreinforcement ((28) (35) and (36) under the conditionsexpressed by (37)) and the compressed region of concrete((48) and (49) considering (41)ndash(47)) only depend on thevariables 119909
119888and 120593 which are independent of or related to one
another according to the type of analysis required (see below)In all cases it must be assumed that the proposed formulationis applied for 120593 gt 0
51 Field of Application The field of validity of the procedureproposed here is limited by the fact that the constitutive law ofthe steel reinforcement does not include unloading branchesstarting from a point of the post yielding branches in com-pression or in tension Therefore for any loading step theupper principal reinforcement and the upper fibre of the dis-tributed secondary reinforcement both potentially subject tocompression cannot be in the postyielding field in tensionthe lower principal reinforcement and the lower fibre of thedistributed secondary reinforcement both potentially subjectto tension cannot be in the postyielding field in compression
Because of these assumptions the field of values of 119909119888
consistent with a given value of 120593 is limited by the initial andend values provided by (26) or (37)
More precisely in the absence of secondary reinforce-ment (120596
2= 0) (26) gives
119909119888min = 120582 minus
120577119904119888
120593 119909
119888max = (1 minus 120582) +120577119904119888
120593 (50)
while if the secondary reinforcement is present (37) leads tomore restrictive limitations
119909119888min =
1 minus 120572
2minus
120577119904119888
120593 119909
119888max =1 + 120572
2+
120577119904119888
120593 (51)
These limitations also occur in a classical numericalapproach based on the strip decomposition of the section
However it must be observed that (50) or (51) allows oneto construct the moment-curvature curve for a large field ofvalues of assigned level of compression including the moreusual cases occurring in practical applications
52 Moment-Curvature Curves The more usual applicationof the analytical expressions derived in the previous sectionsis to derive by a step-by-step procedure the moment-curva-ture119898-120593 curve of a class of RC sections for a given value of 119899
To this purpose the procedure is applied by the followingsteps
(i) assigning the lowest value of curvature 120593 that onewants to consider increasing values of 119909
119888are consid-
ered from the minimum to the maximum expressedby (50) or (51)
(ii) for each value of the couple 120593 119909119888the corresponding
value of 1198991199041 1198991199042 and 119899
119888are calculated and their sum
increasing for each increase in 119909119888
(iii) when the assigned value of 119899 is reached with anacceptable tolerance (ie (15) is verified with verygood approximation) the assigned values of 120593 and 119899and the value of 119909
119888that was found make it possible
to calculate the corresponding values of11989811990411198981199042 and
119898119888expressed by (24) (36) and (49) and the sum
of these contributions of moment is the ordinate ofthe point having the abscissa 120593 belonging to the119898-120593curve sought
(iv) considering an increased value of 120593 the procedure isrepeated to obtain a subsequent point of themoment-curvature curve
The procedure stops when the maximum available ulti-mate confined concrete strain or the maximum available ten-sile strain of the lower reinforcement is exceeded These twooccurrences imply respectively that 120593119909
119888gt 120583cc or 120593(119909119888 minus 1 +
120582) lt minus120577119904119888120583119903 where 120583cc = 120576ccu120576cco is the available ductility of
the confined concrete defined in Section 31 and 120583119903is the steel
ductility defined in (11)If the stress-strain law of the steel reinforcement also
exhibits a softening postyielding branch corresponding to apostbuckling behaviour up to the strain value 120576
∙
119904(Figure 3)
a further limit at which the procedure must be stoppedderives from the condition that 120593(119909
119888minus 120582) gt 120577
119904119888120583119903119888 in
which 120583119903119888
= 120576lowast
119904120576119904119910 Beyond this condition the residual stress
of the compressed reinforcement should be assumed to beconstant and equal to 02119891
119904119910 but this branch of the stress-
strain law is not considered in the proposed model as statedin Section 32
Advances in Civil Engineering 9
53 Strength and Curvature Domains A typical 119898-120593 curveexhibits two characteristic points corresponding to theachievement of the ultimate state and to the first yield-ing of the principal reinforcement in tension respectivelyObviously the first point is present in all cases while thesecond can only be reached if yielding of the reinforcementin tension occurs before the brittle collapse of the concretecore due to low available concrete ductility orand a very highcompression level
The analytical expressions shown in the previous sectionsmake it possible to determine directly the coordinates of thesepoints and to derive in closed form the119898
119906-119899 120593119906-119899 and119898
119910-119899
120593119910-119899 domains which give useful information on the flexural
strength and curvature ductility of an RC section subjected toan assigned compression level
The constructions of the aforementioned curves can bemade more easily than by using a classical discretized modelof the section because (15) and (16) in a suitable specializedform provide analytical expressions where the axial com-pressive load and the corresponding bending moment onlydepend on a single variable 119909
119888and 120593 being related to one
anotherIn order to construct the 119898
119906-119899 and 120593
119906-119899 curves two dif-
ferent relationships linking the neutral axis depth to the cur-vature must be considered according to whether the collapseof the section is produced by excess of the available strain ofthe reinforcement in tension or of the available strain of theconcrete core in compression Moreover to distinguish thesetwo different events it must be observed that the maximumcurvature of the section is reachedwhen these two conditionsoccur simultaneously Considering Figure 4 it can easily bededuced that this value of curvature in normalized form is
120593119906max =
120583cc + 120577119904119888120583119903
1 minus 120582 (52)
The first branch of the domains sought is determined byimposing the condition that the collapse of the section is dueto the achievement of themaximum tensile strain of the prin-cipal reinforcement in tension (low level of compression)Therefore the following condition has to be assumed
120593119906(119909119888minus 1 + 120582) = minus 120577
119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus
120577119904119888120583119903
120593119906
(53)
Considering increasing values of the curvature from themin-imum value to themaximum expressed by (52) the second of(53) provide the corresponding value of 119909
119888 and (15) and (16)
by making explicit the contributions of the concrete and theprincipal and secondary reinforcement provide the corre-sponding values of 119899 and119898
119906
Once 120593119906max is reached the second branch of the 119898
119906-119899
120593119906-119899 curves must be determined considering that for further
decreasing values of 120593119906 the collapse is produced by the
achievement of the maximum compressive strain of the con-fined concrete Therefore the new relationship linking thecurvature to the neutral axis depth is
By using the second of these equations (15) and (16) againlink 119899 and119898
119906to the only variable 120593
119906
It must be observed that if the principal reinforcement incompression is subject to buckling after the achievement ofthe yielding stress the ultimate strain of this compressed rein-forcement is the strain denoted as 120576∙
119904in Figure 3 This strain
value can be reached before the concrete core reaches thenormalized value of strain 120583cc In this case the maximumvalue of normalized dimensionless curvature expressed by(52) must be substituted by the value 120593
119906max = 120577119904119888(120583119903119888
+
120583119903)(1minus2120582) where 120583
119903119888= 120576lowast
119904120576119904119910 and the second branch of the
ultimate strength domains must be determined by assumingthat for values of 120593
119906that decrease from this maximum the
relationship linking the neutral axis dept to the curvature is119909119888= 120582 + 120577
119904119888120583119903119888120593
The 119898119910-119899 120593119910-119899 curves can be simply determined by
assuming that whatever the value of the curvature is the prin-cipal reinforcement in tension is affected by the strain valueequal to minus120576
119904119910 This condition in dimensionless terms gives
120593119910(119909119888minus 1 + 120582) = minus 120577
119904119888997904rArr 119909119888= (1 minus 120582) minus
120577119904119888
120593119910
(55)
By introducing the second of (55) into the equilibrium Equa-tions (15) and (16) they again become two functions of thesingle variable 120593 = 120593
119910 The 119898
119910-119899 120593119910-119899 curves can be con-
structed starting froma value thatwas assumed to be themin-imum up to the maximum value corresponding to one of thepossible aforementioned occurrences the collapse of the con-crete core due to achievement of the available ductility of thematerial or the achievement of the strain 120576
∙
119904in the principal
reinforcement in compression if this reinforcement is subjectto buckling
Considering the field of validity of the procedure pro-posed the minimum value of 120593
119906that can be considered to
construct the119898119906-119899 120593119906-119899 domains is obtained by introducing
the second of (53) into the first of (50) or (51) so that
120593119906min = 120577
119904119888
120583119903minus 1
1 minus 2120582 (56)
in the absence of the secondary reinforcement and
120593119906min = 120577
119904119888
120583119903minus 1
(1 + 120572) 2 minus 120582 (57)
if the secondary reinforcement is also presentObviously the 119898
119910-119899 120593119910-119899 domains can be constructed
starting from any value 120593119910gt 0
In relation to the field of applications (RC columns) thefirst values of 120593
119906and 120593
119910that are to be actually considered are
the ones first producing 119899 ge 0
6 Applications
The reliability of the procedure proposed is shown here byusing the results derived from two RC sections considered byother researchers
Thefirst application refers to one of the sections examinedby Zahn et al [27] This was a square RC section of side119861 = 400mm and realized with concrete having cylindrical
10 Advances in Civil EngineeringN
orm
aliz
ed st
ress
1
00 1 2 3 4 5
Saatcioglu and Razvi (1992)Mander et al (1988)
Normalized strain
Figure 7 Calibration of constitutive law of confined concrete
strength 119891co = 30Nmm2 The longitudinal reinforcementconsisted of 12 steel bars of diameter 16mm having yieldingstrength 119891
119904119910= 420Nmm2 uniformly distributed along the
perimeter of the concrete core (constant pitch) The coverdepth measured from the external perimeter of the stirrupswas 13mm The transverse reinforcement consisting of steelstirrups of diameter equal to 10mm according to the calcu-lations made by the aforementioned authors provided theeffective confinement pressure 119891
ℓ119890= 011119891
1015840
co where 1198911015840
co =
085119891co is the reduced strength value prescribed by the ACI318 code [28]
The flexural response was numerically derived in [27] byusing a classical fibre model in which the stress-strain law ofthe confined concrete was assumed in agreement with Man-derrsquosmodel [25]Therefore in this application the parameters120573cc and 120578cc governing the shape of the laws expressed by (4)and (5) are calibrated so that they lead to a constitutive lawof confined concrete which is very similar to that used by theaforementioned authors
Manderrsquos model [25] for the given value of the confine-ment pressure is governed by the parameter 119903 = 13 themodel of Saatcioglu and Razvi [18] leads to a very similarshape of the stress-strain curve by assuming that 120573cc = 045
and 120578cc = minus006 Figure 7 shows the very similar shapes ofthe constitutive laws corresponding to the two models con-sidered
The procedure proposed in [25] provided the followingcharacteristic values of confined concrete strength and cor-responding strain 120576cco = 00073 119891cco = 46Nmm2 Thesevalues are also adopted for the dimensional stress-strain lawexpressed by the model used here [18]
Since for the stress-strain law of the unconfined concreteZahn et al [27] do not give any information the default valuesindicated in Section 31 are assumed here 120573
119888= 1 120578
119888= minus08
and 120576co = 0002On the basis of the above data the square cross-section of
the concrete coremeasured inside the perimeter stirrups hasa side of 119887 = ℎ = 354mm the cover depth is equal to
400
300
200
100
00 001 002 003 004 005
Mom
ent (
kNm
)
Curvature (1m)
Zahn et al (1989)Present model
Figure 8 Comparison of results119872-120593 curve for given119873
23mm the number of intermediate reinforcing bars realizingthe secondary reinforcement is 119899
119887= 119899ℎ= 2 and the distance
of the barycentre of the section of the principal reinforcementfrom the external side of the concrete core section is ℎ
1=
162 = 8mmThe principal reinforcement consists of four upper and
four lower 16mm steel bars while the secondary reinforce-ment consists of four bars (two on the right and two on theleft) of the same diameter The steel reinforcement constitu-tive law is defined by assuming 119864
119904= 210GPa 120576
119904119910= 119891119904119910119864119904=
0002) and 119864119904ℎ
= 119864119904119888
= 0Therefore the dimensionless parameters involved in the
formulation proposed take on the following values 120575V = 120575119900=
= 0Figure 8 compares the 119872-120593 curve obtained in [27] with
that derived from the procedure shown in Section 52 for acompressive constant axial load 119873 = 03119891co(119861 times 119861) corre-sponding to a compressive level 119899 = 119873(119887
2
times 119891cco) = 025 inagreement with the first of (14)The results in the figure showa very good level of agreement between the numerical pro-cedure adopted in [27] and the analytical one adopted here
Both curves in the figure clearly show the cusp producedby the yielding of the principal reinforcement in tension andthe effects of the progressive cover spalling
The second application refers to the middle-height sec-tion of an RC column of height 1640mm experimentallytested by Saatcioglu et al [19] marked as specimen C6-2 bythe authors
The loading condition was realized by imposing relativeaxial displacements so that the reactive compressive forceacted with fixed eccentricity in a plane of principal inertiaof the sections Suitable devices were applied at the columnends so that the column itself behaved like a hinged verticalRC member under an eccentric compressive load
As a consequence each point of the moment-curvaturecurve characterizing the experimental response obtained by
Advances in Civil Engineering 11
the aforementioned authors corresponds to a different valueof axial loadMoreover the actual bendingmoment acting onthe middle-height section was influenced by the 119875-Δ effect
The formulation proposed here is validated consideringsome points of the aforementioned curve for which theauthors indicate the values of the axial load119873 and curvaturethat were measured These values of 119873 are normalized withrespect to the axial load 119873
119900expressed by (2) The procedure
is applied by determining the value of 119909119888from (15) for the
assigned values of 119899 and 120593 and by calculating 119898 from (16)The results are compared with the numerical values obtainedby the strip model adopted by the authors which includedthe 119875-Δ effect and the values experimentally detected
The specimen considered had a square section of side 119861 =
210mm cover depth of 125mm and longitudinal reinforcingbars of diameter 113mm uniformly distributed along theperimeter of the concrete core having yielding strength119891
119904119910=
517Nmm2 and average hardening modulus 119864119904ℎ
= 0013119864119904
The transverse steel reinforcement consisted of square andoctagonal 63mm stirrups with pitch 50mm
The unconfined concrete had cylindrical strength 119891co =
3440Nmm2The numerical model adopted in [19] assumedthat the perimeter of the concrete core was coincident withthe axis lines of the external stirrups
On the basis of these geometrical and mechanical dataand by evaluating the effective confinement pressure by theprocedure proposed in [18] for the confined concrete sectionone obtains 119887 = ℎ = 17870mm 119891cco = 4635 Nmm2 120576cco =
000547 120578cc = minus0052 and 120573 = 0590The further parametersinvolved in the formulation proposed take on the followingvalues 120575
= 0013Figure 9 shows that the procedure proposed provides
results that are very close to those derived by the numericalmodel adopted in [19] The experimentally detected values ofmoment prove to be underestimated by both models
It must be observed that the last point detected by usingthe present formulation corresponding to the ultimate stateof the section had been already found in [17] because thissection belongs to the more restricted class of sections con-sidered in that work
Figure 10 shows the strength and curvature domainsobtained by using the procedure described in Section 53For confined concrete and steel reinforcement the followingvalues were assumed deduced from the data shown in [19]120576ccu = 0032 (120583cc = 585) and 120576
119904119906= 0066 (120583
119903= 27) The
results confirm that a good level of confinementmakes it pos-sible to achieve acceptable ductility of curvature even underhigh levels of compression In order to show how the proce-dure proposed is easily able to evaluate the influence of all thegeometric and mechanical parameters governing the flexuralresponse of a class of RC sections Figure 10(a) also shows theinfluence of the unconfined concrete cover on the bendingmoment at the first yielding of the principal reinforcement intension As expected this influence proves to be negligible forhigh values of the compression level
The dotted curve in Figure 10(b) obviously stops at thepoint of intersection with the continuous curve Beyond this
100
80
60
40
20
00 1 2 3 4
Mom
ent (
kNm
)
ExperimentalModelPresent model
Saatcioglu et al (1995)
Curvature (1mm times 10minus4)
Figure 9 Comparison of theoretical and experimental results
point the 120593119906-119873 curve proceeds by a very brief stretch (not
very evident in the figure) that corresponds to brittle collapseof the section due to the achievement of the maximum avail-able compressive strain in the confined concrete when theprincipal reinforcement in tension is still in the elastic fieldThe end point of this curve corresponds to the achievementof the maximum neutral axis depth defined by the second of(51)
Nevertheless the axial load values that cannot be consid-ered because of these limitations are well beyond the onesthat can be assumed in the structural design of RC buildingsin seismic areas or usually found in the columns of existingbuildings
7 Conclusions
A dimensionless formulation has been proposed which pro-vides the flexural response of classes of sections of RC col-umns having the same values of the geometric and mechani-cal parameters defined in this study
At each loading stage the resisting components of the sec-tion (cover and concrete core and principal and secondarylongitudinal reinforcements) give a contribution to the equi-librium that can be expressed by analytical functions depend-ing on the normalized neutral axis depth and the curvature ofthe section
Special equilibrium conditions like the ones correspond-ing to the first yielding of the principal reinforcement intension and to the ultimate state imply an analytical link bet-ween the neutral axis depth and the curvature so that the sumof the aforementioned contributions becomes an analyticalfunction of a single variable which can express a strength orcurvature domain
For a generic equilibrium condition an iterative proce-dure to determine the neutral axis depth corresponding to
12 Advances in Civil Engineering
100
80
60
40
20
00 300 600 900 1200 1500
Mom
ent (
kNm
)
Axial load (kN)
Mu
My
My without cover
(a)
8
6
4
2
00 300 600 900 1200 1500
Axial load (kN)
120593u120593y
Curv
atur
e(1
mm
times10
minus4)
(b)
Figure 10119872-119873 and 120593-119873 domains at the ultimate state and the first yielding of reinforcement
the assigned values of curvature and compression level isrequired but each iteration step implies the simple use of thesame equilibrium equation for an updated value of the nor-malized neutral axis dept
The dimensionless form of the formulation proposedmakes also it easily possible to evaluate the influence of theparameters characterizing the class of sections examined onthe strength and curvature corresponding to a given loadingstage
Acknowledgment
Thisworkwas carried outwithin the 2010ndash2013Research Pro-ject ldquoDPC-ReLUIS (Dipartimento Protezione Civile-Rete deiLaboratori Universitari di Ingegneria Sismica)rdquo AT 1 Task112 The related financial support was greatly appreciated
References
[1] A Fafitis ldquoInteraction surfaces of reinforced-concrete sectionsin biaxial bendingrdquo Journal of Structural Engineering vol 127no 7 pp 840ndash846 2001
[2] J L Bonet P F Miguel M A Fernandez and M L RomeroldquoAnalytical approach to failure surfaces in reinforced concretesections subjected to axial loads and biaxial bendingrdquo Journalof Structural Engineering vol 130 no 12 pp 2006ndash2015 2004
[3] G Monti and S Alessandri ldquoAssessment of rc columns undercombined biaxial bending and axial loadrdquo in Proceedings of the2nd FIB Congress Naples Italy 2006
[4] RW Clough and S B Johnston ldquoEffect of stiffness degradationon earthquake ductility requirementsrdquo in Proceedings of 2ndJapan Earthquake Engineering Symposium Tokyo Japan 1966
[5] T Takeda M A Sozen and N N Nielsen ldquoReinforced con-crete response to simulated earthquakerdquo Journal of StructuralDivision vol 96 no 12 pp 2257ndash2273 1970
[6] M S L Roufaiel and C Meyer ldquoAnalytical modeling of hyster-etic behavior of reinforced concrete framerdquo Journal of StructuralEngineering vol 113 no 3 pp 429ndash444 1987
[7] A R Mari and A C Scordelis ldquoNonlinear geometric materialand time dependent analysis of three dimensional reinforcedand prestressed concrete framesrdquo USBSESM Report 8412Department of Civil Engineering University of CaliforniaBerkeley Calif USA 1973
[8] T Taucer E Spacone and F C Filippou ldquoA fiber beam-col-umn element for seismic response analysis of reinforced con-crete structuresrdquo Report EERC 91-17 Earthquake EngineeringResearch Center Berkeley Calif USA 1991
[9] Z Zhu I Ahmad and A Mirmiran ldquoFiber element modelingfor seismic performance of bridge columns made of concrete-filled FRP tubesrdquo Engineering Structures vol 28 no 14 pp2023ndash2035 2006
[10] E O Pfrang C P Siess and M A Sozen ldquoLoad-moment-cur-vature characteristics of RC cross-sectionsrdquoACI Journal vol 61no 7 pp 763ndash778 1964
[11] D J Carreira and K-H Chu ldquoThemoment-curvature relation-ship of RC membersrdquo ACI Journal vol 83 no 2 pp 191ndash1981986
[12] R H Wood ldquoSome controversial and curious developments inplastic theory of structuresrdquo inEngineering Plasticity J Heymanand F A Leckie Eds pp 665ndash691 CambridgeUniversity PressCambridge UK 1968
[13] Y LMo ldquoInvestigation of reinforced concrete frame behaviourtheory and testsrdquo Magazine of Concrete Research vol 44 no160 pp 163ndash173 1992
[14] M Jirasek and Z P Bazant Inelastic Analysis of Structures JonWiley amp Sons London UK 2002
[15] S Chandrasekaran L Nunziante G Serino and F CarannanteldquoCurvature ductility of RC sections based on Eurocode analyt-ical procedurerdquo KSCE Journal of Civil Engineering vol 15 no 1pp 131ndash144 2011
Advances in Civil Engineering 13
[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012
[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013
[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992
[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995
[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006
[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010
[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012
[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951
[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982
[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002
[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989
[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008
Finally the amount of principal and secondary longitudinalreinforcement is related to the section of the concrete core bymeans of the mechanical ratios of reinforcement
1205961
= 2
1198601199041119891119904119910
119887ℎ119891cco 120596
2= 2
1198601199042119891119904119910
119887ℎ119891cco (13)
4 Equilibrium of Section
Denoting as119873 and119872 the axial load and the bendingmomentacting on the section and as
119899 =119873
119873119900
=119873
119887ℎ119891cco 119898 =
119872
119872119900
=119872
119887ℎ2119891cco (14)
these quantities normalized with respect to the ones definedby (2) the equilibrium of the section in dimensionless formis expressed by the following equations
119899 = 1198991199041
+ 1198991199042
+ 119899119888 (15)
119898 = 1198981199041
+ 1198981199042
+ 119898119888 (16)
At the secondmember of these equations there are clearlyindicated and ordered contributions offered by the principalreinforcement secondary reinforcement and concrete
These contributions are analytically expressed in the fol-lowing sections by assuming the classical hypothesis that thesection remains plane and neglecting the tensile concretestrength
41 Contribution of Principal Reinforcement In the analyticalformulation shown in this section the following parametersare involved the geometrical parameter 120582 defined in (1) themechanical parameters characterizing the constitutive law ofthe steel reinforcement 120578
119904119888 120578119904ℎ and 120577
119904119888 defined by (11) and
(12) and the mechanical ratio of reinforcement 1205961expressed
by the first of (13)Figure 4 shows the lateral view of the RC member con-
sidered and a generic state of strain and stressBy using the symbols shown in the figure the upper and
lower reinforcement bars are subjected respectively to thestrains
Then the axial load and bending moment with respect to thebarycentre of the section that the secondary reinforcementcan bear can be calculated by the following expressions
1198731199042
=21198601199042
1198891
ℎ
2[1198911198901
(1198861+
120577119904119888
120593) + 119891
1198902(1198862+
120577119904119888
120593)
minus (Δ11989111198861+ Δ11989121198862) ]
1198721199042
=21198601199042
1198891
ℎ2
2[(1198911198901
minus 1198911198902)1205722
6minus Δ11989111198861(120572
2minus
1198861
3)
+Δ11989121198862(120572
2minus
1198862
3)]
(33)
Equations (28) (29) and (31) show that
1198911198901
(1198861+
120577119904119888
120593) + 1198911198902
(1198862+
120577119904119888
120593) = 2119891
119904119910
120593
120577119904119888
(119909119888minus
1
2) 120572
1198911198901
minus 1198911198902
= 119891119904119910
120593
120577119904119888
120572
(34)
Therefore substituting (34) into (33) in normalized form oneobtains
1198991199042
=1198731199042
119873119900
= 1205962
120593
120577119904119888
(119909119888minus
1
2) minus
1
2120572
times [(1 minus 120578119904119888) 1198862
1minus (1 minus 120578
119904ℎ) 1198862
2]
(35)
1198981199042
=1198721199042
119872119900
=1205962
2120572
120593
120577119904119888
[1205723
6minus (1 minus 120578
119904119888) (
120572
2minus
1198861
3) 1198862
1
minus (1 minus 120578119904ℎ) (
120572
2minus
1198862
3) 1198862
2]
(36)
Advances in Civil Engineering 7
h
c
c
co cob
x3
xc
Confinedconcrete
1205761
1205762
1205763
1205764
(a)
h
c
c
co cob
xc
1205761
1205762
1205763 = 1205764 = 0x3 = 0
Confinedconcrete
(b)
Figure 6 Strain state of concrete section (a) uncracked section (b) cracked section
It must be observed that if one sets 119864119904119888
= 119864119904in (30) andor
119864119904ℎ
= 119864119904in (32) (33) expresses the contributions to the axial
load and bendingmoment in the case inwhich the upper fibreandor the lower fibre of the secondary reinforcement remainin the elastic field As a consequence (35) and (36) are validfor any value of 119909
119888e 120593 under the condition that one sets
120578119904119888=1 if minus 120576
119904119910le 1205761199041
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 minus 120572
2)le1
120578119904ℎ=1 if minus 120576
119904119910le 1205761199042
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 + 120572
2)le1
(37)
43 Contribution of Concrete Figure 6 shows the strain statesof the section corresponding to the cases of uncracked andcracked sections By using the symbols in the figure the con-tributions to the equilibrium provided by the concrete coreare expressed by
119873cc = 119887int
119909119888
1199093
119891119888(119909) 119889119909 =
119887ℎ119891cco120593
int
1205762
1205763
119904119888(120576) 119889120576
119872cc = 119887int
119909119888
1199093
119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus
ℎ
2)
=119887ℎ2
119891cco (1
1205932int
1205762
1205763
119904119888(120576) 120576 119889120576minus
119909119888minus 05
120593int
1205762
1205763
119904119888(120576) 119889120576)
(38)
where
1199093=
0 for 119909119888le ℎ
119909119888minus ℎ for 119909
119888gt ℎ
(39)
and consequently
1205762= 120593119909119888
1205763=
0 for 119909119888le 1
120593 (119909119888minus 1) for 119909
119888gt 1
(40)
Considering (7) in the normalized form one obtains
119899cc =119873cc119873119900
=1
120593(11987811205762
minus 11987811205763
) (41)
119898cc =119872cc119872119900
=1
1205932(11987821205762
minus 11987821205763
) minus 119899cc (119909119888 minus1
2) (42)
The contribution of the surrounding unconfined concrete(concrete cover) can be calculated by subtracting the contri-bution of the confined region to that of the whole section
Considering that the strains involved in the functions 1198781120576
and 1198782120576
must be normalized with respect to the strain 120576co(unconfined concrete) by following the same procedure asthat leading to (41) and (42) and by introducing the con-finement efficiency factors defined by (6) for the whole sec-tion one obtains
119899un1 =1 + 2120575
119900
1198961
1
1198962120593
(11987811205761015840
1
minus 11987811205761015840
4
)
119898un1 =1 + 2120575
119900
1198961
1
(1198962120593)2(11987821205761015840
1
minus 11987821205761015840
4
) minus 119899un1 (119909119888 minus1
2)
(43)where
1205761015840
1= 1198962120593 (119909119888+ 120575V) le 120583
119888
1205761015840
4=
0 for 119909119888le (1 + 120575V)
1198962120593 [119909119888minus (1 + 120575V)] for 119909
119888gt (1 + 120575V)
(44)
The contribution to be subtracted from the above quantitiesare calculated bymodifying (41) and (42) in order to considerthe different constitutive law of the unconfined concrete andthe different base of normalization of strain and stress It caneasily be shown that one obtains
119899un2 =1
11989611198962120593
(11987811205761015840
2
minus 11987811205761015840
3
) (45)
119898un2 =1
1198961(1198962120593)2(11987821205761015840
2
minus 11987821205761015840
3
) minus 119899un2 (119909119888 minus1
2) (46)
8 Advances in Civil Engineering
where
1205761015840
2= 1198962120593119909119888le 120583119888
1205761015840
3=
0 for 119909119888le 1
1198962120593 (119909119888minus 1) for 119909
119888gt 1
(47)
On the basis of what was said above the effective contributionof the concrete section in compression in a generic loadingstate is expressed by
119899119888= 119899cc + 119899un1 minus 119899un2 (48)
119898119888= 119898cc + 119898un1 minus 119898un2 (49)
If the confinement effect is negligible (6) gives 1198961= 1198962= 1
By comparing (40) (41) and (42) with (47) (45) and (46)respectively it can easily be observed that 119899cc = 119899un2 and119898cc = 119898un2 Therefore in this case the contributions of thewhole section of reacting concrete given by (48) and (49) arereduced to the values 119899cc = 119899un1 and 119898cc = 119898un1 and areexpressed by (45) and (46) for 119896
1= 1198962= 1
5 Analysis Procedure
The flexural response of a class of RC sections characterizedby assigned values of the geometrical andmechanical param-eters defined above is obtained by imposing the equilibriumcondition expressed by (15) and (16)
For each configuration the contributions to the equi-librium provided by the principal reinforcement ((23) and(24) under the conditions expressed by (26)) the secondaryreinforcement ((28) (35) and (36) under the conditionsexpressed by (37)) and the compressed region of concrete((48) and (49) considering (41)ndash(47)) only depend on thevariables 119909
119888and 120593 which are independent of or related to one
another according to the type of analysis required (see below)In all cases it must be assumed that the proposed formulationis applied for 120593 gt 0
51 Field of Application The field of validity of the procedureproposed here is limited by the fact that the constitutive law ofthe steel reinforcement does not include unloading branchesstarting from a point of the post yielding branches in com-pression or in tension Therefore for any loading step theupper principal reinforcement and the upper fibre of the dis-tributed secondary reinforcement both potentially subject tocompression cannot be in the postyielding field in tensionthe lower principal reinforcement and the lower fibre of thedistributed secondary reinforcement both potentially subjectto tension cannot be in the postyielding field in compression
Because of these assumptions the field of values of 119909119888
consistent with a given value of 120593 is limited by the initial andend values provided by (26) or (37)
More precisely in the absence of secondary reinforce-ment (120596
2= 0) (26) gives
119909119888min = 120582 minus
120577119904119888
120593 119909
119888max = (1 minus 120582) +120577119904119888
120593 (50)
while if the secondary reinforcement is present (37) leads tomore restrictive limitations
119909119888min =
1 minus 120572
2minus
120577119904119888
120593 119909
119888max =1 + 120572
2+
120577119904119888
120593 (51)
These limitations also occur in a classical numericalapproach based on the strip decomposition of the section
However it must be observed that (50) or (51) allows oneto construct the moment-curvature curve for a large field ofvalues of assigned level of compression including the moreusual cases occurring in practical applications
52 Moment-Curvature Curves The more usual applicationof the analytical expressions derived in the previous sectionsis to derive by a step-by-step procedure the moment-curva-ture119898-120593 curve of a class of RC sections for a given value of 119899
To this purpose the procedure is applied by the followingsteps
(i) assigning the lowest value of curvature 120593 that onewants to consider increasing values of 119909
119888are consid-
ered from the minimum to the maximum expressedby (50) or (51)
(ii) for each value of the couple 120593 119909119888the corresponding
value of 1198991199041 1198991199042 and 119899
119888are calculated and their sum
increasing for each increase in 119909119888
(iii) when the assigned value of 119899 is reached with anacceptable tolerance (ie (15) is verified with verygood approximation) the assigned values of 120593 and 119899and the value of 119909
119888that was found make it possible
to calculate the corresponding values of11989811990411198981199042 and
119898119888expressed by (24) (36) and (49) and the sum
of these contributions of moment is the ordinate ofthe point having the abscissa 120593 belonging to the119898-120593curve sought
(iv) considering an increased value of 120593 the procedure isrepeated to obtain a subsequent point of themoment-curvature curve
The procedure stops when the maximum available ulti-mate confined concrete strain or the maximum available ten-sile strain of the lower reinforcement is exceeded These twooccurrences imply respectively that 120593119909
119888gt 120583cc or 120593(119909119888 minus 1 +
120582) lt minus120577119904119888120583119903 where 120583cc = 120576ccu120576cco is the available ductility of
the confined concrete defined in Section 31 and 120583119903is the steel
ductility defined in (11)If the stress-strain law of the steel reinforcement also
exhibits a softening postyielding branch corresponding to apostbuckling behaviour up to the strain value 120576
∙
119904(Figure 3)
a further limit at which the procedure must be stoppedderives from the condition that 120593(119909
119888minus 120582) gt 120577
119904119888120583119903119888 in
which 120583119903119888
= 120576lowast
119904120576119904119910 Beyond this condition the residual stress
of the compressed reinforcement should be assumed to beconstant and equal to 02119891
119904119910 but this branch of the stress-
strain law is not considered in the proposed model as statedin Section 32
Advances in Civil Engineering 9
53 Strength and Curvature Domains A typical 119898-120593 curveexhibits two characteristic points corresponding to theachievement of the ultimate state and to the first yield-ing of the principal reinforcement in tension respectivelyObviously the first point is present in all cases while thesecond can only be reached if yielding of the reinforcementin tension occurs before the brittle collapse of the concretecore due to low available concrete ductility orand a very highcompression level
The analytical expressions shown in the previous sectionsmake it possible to determine directly the coordinates of thesepoints and to derive in closed form the119898
119906-119899 120593119906-119899 and119898
119910-119899
120593119910-119899 domains which give useful information on the flexural
strength and curvature ductility of an RC section subjected toan assigned compression level
The constructions of the aforementioned curves can bemade more easily than by using a classical discretized modelof the section because (15) and (16) in a suitable specializedform provide analytical expressions where the axial com-pressive load and the corresponding bending moment onlydepend on a single variable 119909
119888and 120593 being related to one
anotherIn order to construct the 119898
119906-119899 and 120593
119906-119899 curves two dif-
ferent relationships linking the neutral axis depth to the cur-vature must be considered according to whether the collapseof the section is produced by excess of the available strain ofthe reinforcement in tension or of the available strain of theconcrete core in compression Moreover to distinguish thesetwo different events it must be observed that the maximumcurvature of the section is reachedwhen these two conditionsoccur simultaneously Considering Figure 4 it can easily bededuced that this value of curvature in normalized form is
120593119906max =
120583cc + 120577119904119888120583119903
1 minus 120582 (52)
The first branch of the domains sought is determined byimposing the condition that the collapse of the section is dueto the achievement of themaximum tensile strain of the prin-cipal reinforcement in tension (low level of compression)Therefore the following condition has to be assumed
120593119906(119909119888minus 1 + 120582) = minus 120577
119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus
120577119904119888120583119903
120593119906
(53)
Considering increasing values of the curvature from themin-imum value to themaximum expressed by (52) the second of(53) provide the corresponding value of 119909
119888 and (15) and (16)
by making explicit the contributions of the concrete and theprincipal and secondary reinforcement provide the corre-sponding values of 119899 and119898
119906
Once 120593119906max is reached the second branch of the 119898
119906-119899
120593119906-119899 curves must be determined considering that for further
decreasing values of 120593119906 the collapse is produced by the
achievement of the maximum compressive strain of the con-fined concrete Therefore the new relationship linking thecurvature to the neutral axis depth is
By using the second of these equations (15) and (16) againlink 119899 and119898
119906to the only variable 120593
119906
It must be observed that if the principal reinforcement incompression is subject to buckling after the achievement ofthe yielding stress the ultimate strain of this compressed rein-forcement is the strain denoted as 120576∙
119904in Figure 3 This strain
value can be reached before the concrete core reaches thenormalized value of strain 120583cc In this case the maximumvalue of normalized dimensionless curvature expressed by(52) must be substituted by the value 120593
119906max = 120577119904119888(120583119903119888
+
120583119903)(1minus2120582) where 120583
119903119888= 120576lowast
119904120576119904119910 and the second branch of the
ultimate strength domains must be determined by assumingthat for values of 120593
119906that decrease from this maximum the
relationship linking the neutral axis dept to the curvature is119909119888= 120582 + 120577
119904119888120583119903119888120593
The 119898119910-119899 120593119910-119899 curves can be simply determined by
assuming that whatever the value of the curvature is the prin-cipal reinforcement in tension is affected by the strain valueequal to minus120576
119904119910 This condition in dimensionless terms gives
120593119910(119909119888minus 1 + 120582) = minus 120577
119904119888997904rArr 119909119888= (1 minus 120582) minus
120577119904119888
120593119910
(55)
By introducing the second of (55) into the equilibrium Equa-tions (15) and (16) they again become two functions of thesingle variable 120593 = 120593
119910 The 119898
119910-119899 120593119910-119899 curves can be con-
structed starting froma value thatwas assumed to be themin-imum up to the maximum value corresponding to one of thepossible aforementioned occurrences the collapse of the con-crete core due to achievement of the available ductility of thematerial or the achievement of the strain 120576
∙
119904in the principal
reinforcement in compression if this reinforcement is subjectto buckling
Considering the field of validity of the procedure pro-posed the minimum value of 120593
119906that can be considered to
construct the119898119906-119899 120593119906-119899 domains is obtained by introducing
the second of (53) into the first of (50) or (51) so that
120593119906min = 120577
119904119888
120583119903minus 1
1 minus 2120582 (56)
in the absence of the secondary reinforcement and
120593119906min = 120577
119904119888
120583119903minus 1
(1 + 120572) 2 minus 120582 (57)
if the secondary reinforcement is also presentObviously the 119898
119910-119899 120593119910-119899 domains can be constructed
starting from any value 120593119910gt 0
In relation to the field of applications (RC columns) thefirst values of 120593
119906and 120593
119910that are to be actually considered are
the ones first producing 119899 ge 0
6 Applications
The reliability of the procedure proposed is shown here byusing the results derived from two RC sections considered byother researchers
Thefirst application refers to one of the sections examinedby Zahn et al [27] This was a square RC section of side119861 = 400mm and realized with concrete having cylindrical
10 Advances in Civil EngineeringN
orm
aliz
ed st
ress
1
00 1 2 3 4 5
Saatcioglu and Razvi (1992)Mander et al (1988)
Normalized strain
Figure 7 Calibration of constitutive law of confined concrete
strength 119891co = 30Nmm2 The longitudinal reinforcementconsisted of 12 steel bars of diameter 16mm having yieldingstrength 119891
119904119910= 420Nmm2 uniformly distributed along the
perimeter of the concrete core (constant pitch) The coverdepth measured from the external perimeter of the stirrupswas 13mm The transverse reinforcement consisting of steelstirrups of diameter equal to 10mm according to the calcu-lations made by the aforementioned authors provided theeffective confinement pressure 119891
ℓ119890= 011119891
1015840
co where 1198911015840
co =
085119891co is the reduced strength value prescribed by the ACI318 code [28]
The flexural response was numerically derived in [27] byusing a classical fibre model in which the stress-strain law ofthe confined concrete was assumed in agreement with Man-derrsquosmodel [25]Therefore in this application the parameters120573cc and 120578cc governing the shape of the laws expressed by (4)and (5) are calibrated so that they lead to a constitutive lawof confined concrete which is very similar to that used by theaforementioned authors
Manderrsquos model [25] for the given value of the confine-ment pressure is governed by the parameter 119903 = 13 themodel of Saatcioglu and Razvi [18] leads to a very similarshape of the stress-strain curve by assuming that 120573cc = 045
and 120578cc = minus006 Figure 7 shows the very similar shapes ofthe constitutive laws corresponding to the two models con-sidered
The procedure proposed in [25] provided the followingcharacteristic values of confined concrete strength and cor-responding strain 120576cco = 00073 119891cco = 46Nmm2 Thesevalues are also adopted for the dimensional stress-strain lawexpressed by the model used here [18]
Since for the stress-strain law of the unconfined concreteZahn et al [27] do not give any information the default valuesindicated in Section 31 are assumed here 120573
119888= 1 120578
119888= minus08
and 120576co = 0002On the basis of the above data the square cross-section of
the concrete coremeasured inside the perimeter stirrups hasa side of 119887 = ℎ = 354mm the cover depth is equal to
400
300
200
100
00 001 002 003 004 005
Mom
ent (
kNm
)
Curvature (1m)
Zahn et al (1989)Present model
Figure 8 Comparison of results119872-120593 curve for given119873
23mm the number of intermediate reinforcing bars realizingthe secondary reinforcement is 119899
119887= 119899ℎ= 2 and the distance
of the barycentre of the section of the principal reinforcementfrom the external side of the concrete core section is ℎ
1=
162 = 8mmThe principal reinforcement consists of four upper and
four lower 16mm steel bars while the secondary reinforce-ment consists of four bars (two on the right and two on theleft) of the same diameter The steel reinforcement constitu-tive law is defined by assuming 119864
119904= 210GPa 120576
119904119910= 119891119904119910119864119904=
0002) and 119864119904ℎ
= 119864119904119888
= 0Therefore the dimensionless parameters involved in the
formulation proposed take on the following values 120575V = 120575119900=
= 0Figure 8 compares the 119872-120593 curve obtained in [27] with
that derived from the procedure shown in Section 52 for acompressive constant axial load 119873 = 03119891co(119861 times 119861) corre-sponding to a compressive level 119899 = 119873(119887
2
times 119891cco) = 025 inagreement with the first of (14)The results in the figure showa very good level of agreement between the numerical pro-cedure adopted in [27] and the analytical one adopted here
Both curves in the figure clearly show the cusp producedby the yielding of the principal reinforcement in tension andthe effects of the progressive cover spalling
The second application refers to the middle-height sec-tion of an RC column of height 1640mm experimentallytested by Saatcioglu et al [19] marked as specimen C6-2 bythe authors
The loading condition was realized by imposing relativeaxial displacements so that the reactive compressive forceacted with fixed eccentricity in a plane of principal inertiaof the sections Suitable devices were applied at the columnends so that the column itself behaved like a hinged verticalRC member under an eccentric compressive load
As a consequence each point of the moment-curvaturecurve characterizing the experimental response obtained by
Advances in Civil Engineering 11
the aforementioned authors corresponds to a different valueof axial loadMoreover the actual bendingmoment acting onthe middle-height section was influenced by the 119875-Δ effect
The formulation proposed here is validated consideringsome points of the aforementioned curve for which theauthors indicate the values of the axial load119873 and curvaturethat were measured These values of 119873 are normalized withrespect to the axial load 119873
119900expressed by (2) The procedure
is applied by determining the value of 119909119888from (15) for the
assigned values of 119899 and 120593 and by calculating 119898 from (16)The results are compared with the numerical values obtainedby the strip model adopted by the authors which includedthe 119875-Δ effect and the values experimentally detected
The specimen considered had a square section of side 119861 =
210mm cover depth of 125mm and longitudinal reinforcingbars of diameter 113mm uniformly distributed along theperimeter of the concrete core having yielding strength119891
119904119910=
517Nmm2 and average hardening modulus 119864119904ℎ
= 0013119864119904
The transverse steel reinforcement consisted of square andoctagonal 63mm stirrups with pitch 50mm
The unconfined concrete had cylindrical strength 119891co =
3440Nmm2The numerical model adopted in [19] assumedthat the perimeter of the concrete core was coincident withthe axis lines of the external stirrups
On the basis of these geometrical and mechanical dataand by evaluating the effective confinement pressure by theprocedure proposed in [18] for the confined concrete sectionone obtains 119887 = ℎ = 17870mm 119891cco = 4635 Nmm2 120576cco =
000547 120578cc = minus0052 and 120573 = 0590The further parametersinvolved in the formulation proposed take on the followingvalues 120575
= 0013Figure 9 shows that the procedure proposed provides
results that are very close to those derived by the numericalmodel adopted in [19] The experimentally detected values ofmoment prove to be underestimated by both models
It must be observed that the last point detected by usingthe present formulation corresponding to the ultimate stateof the section had been already found in [17] because thissection belongs to the more restricted class of sections con-sidered in that work
Figure 10 shows the strength and curvature domainsobtained by using the procedure described in Section 53For confined concrete and steel reinforcement the followingvalues were assumed deduced from the data shown in [19]120576ccu = 0032 (120583cc = 585) and 120576
119904119906= 0066 (120583
119903= 27) The
results confirm that a good level of confinementmakes it pos-sible to achieve acceptable ductility of curvature even underhigh levels of compression In order to show how the proce-dure proposed is easily able to evaluate the influence of all thegeometric and mechanical parameters governing the flexuralresponse of a class of RC sections Figure 10(a) also shows theinfluence of the unconfined concrete cover on the bendingmoment at the first yielding of the principal reinforcement intension As expected this influence proves to be negligible forhigh values of the compression level
The dotted curve in Figure 10(b) obviously stops at thepoint of intersection with the continuous curve Beyond this
100
80
60
40
20
00 1 2 3 4
Mom
ent (
kNm
)
ExperimentalModelPresent model
Saatcioglu et al (1995)
Curvature (1mm times 10minus4)
Figure 9 Comparison of theoretical and experimental results
point the 120593119906-119873 curve proceeds by a very brief stretch (not
very evident in the figure) that corresponds to brittle collapseof the section due to the achievement of the maximum avail-able compressive strain in the confined concrete when theprincipal reinforcement in tension is still in the elastic fieldThe end point of this curve corresponds to the achievementof the maximum neutral axis depth defined by the second of(51)
Nevertheless the axial load values that cannot be consid-ered because of these limitations are well beyond the onesthat can be assumed in the structural design of RC buildingsin seismic areas or usually found in the columns of existingbuildings
7 Conclusions
A dimensionless formulation has been proposed which pro-vides the flexural response of classes of sections of RC col-umns having the same values of the geometric and mechani-cal parameters defined in this study
At each loading stage the resisting components of the sec-tion (cover and concrete core and principal and secondarylongitudinal reinforcements) give a contribution to the equi-librium that can be expressed by analytical functions depend-ing on the normalized neutral axis depth and the curvature ofthe section
Special equilibrium conditions like the ones correspond-ing to the first yielding of the principal reinforcement intension and to the ultimate state imply an analytical link bet-ween the neutral axis depth and the curvature so that the sumof the aforementioned contributions becomes an analyticalfunction of a single variable which can express a strength orcurvature domain
For a generic equilibrium condition an iterative proce-dure to determine the neutral axis depth corresponding to
12 Advances in Civil Engineering
100
80
60
40
20
00 300 600 900 1200 1500
Mom
ent (
kNm
)
Axial load (kN)
Mu
My
My without cover
(a)
8
6
4
2
00 300 600 900 1200 1500
Axial load (kN)
120593u120593y
Curv
atur
e(1
mm
times10
minus4)
(b)
Figure 10119872-119873 and 120593-119873 domains at the ultimate state and the first yielding of reinforcement
the assigned values of curvature and compression level isrequired but each iteration step implies the simple use of thesame equilibrium equation for an updated value of the nor-malized neutral axis dept
The dimensionless form of the formulation proposedmakes also it easily possible to evaluate the influence of theparameters characterizing the class of sections examined onthe strength and curvature corresponding to a given loadingstage
Acknowledgment
Thisworkwas carried outwithin the 2010ndash2013Research Pro-ject ldquoDPC-ReLUIS (Dipartimento Protezione Civile-Rete deiLaboratori Universitari di Ingegneria Sismica)rdquo AT 1 Task112 The related financial support was greatly appreciated
References
[1] A Fafitis ldquoInteraction surfaces of reinforced-concrete sectionsin biaxial bendingrdquo Journal of Structural Engineering vol 127no 7 pp 840ndash846 2001
[2] J L Bonet P F Miguel M A Fernandez and M L RomeroldquoAnalytical approach to failure surfaces in reinforced concretesections subjected to axial loads and biaxial bendingrdquo Journalof Structural Engineering vol 130 no 12 pp 2006ndash2015 2004
[3] G Monti and S Alessandri ldquoAssessment of rc columns undercombined biaxial bending and axial loadrdquo in Proceedings of the2nd FIB Congress Naples Italy 2006
[4] RW Clough and S B Johnston ldquoEffect of stiffness degradationon earthquake ductility requirementsrdquo in Proceedings of 2ndJapan Earthquake Engineering Symposium Tokyo Japan 1966
[5] T Takeda M A Sozen and N N Nielsen ldquoReinforced con-crete response to simulated earthquakerdquo Journal of StructuralDivision vol 96 no 12 pp 2257ndash2273 1970
[6] M S L Roufaiel and C Meyer ldquoAnalytical modeling of hyster-etic behavior of reinforced concrete framerdquo Journal of StructuralEngineering vol 113 no 3 pp 429ndash444 1987
[7] A R Mari and A C Scordelis ldquoNonlinear geometric materialand time dependent analysis of three dimensional reinforcedand prestressed concrete framesrdquo USBSESM Report 8412Department of Civil Engineering University of CaliforniaBerkeley Calif USA 1973
[8] T Taucer E Spacone and F C Filippou ldquoA fiber beam-col-umn element for seismic response analysis of reinforced con-crete structuresrdquo Report EERC 91-17 Earthquake EngineeringResearch Center Berkeley Calif USA 1991
[9] Z Zhu I Ahmad and A Mirmiran ldquoFiber element modelingfor seismic performance of bridge columns made of concrete-filled FRP tubesrdquo Engineering Structures vol 28 no 14 pp2023ndash2035 2006
[10] E O Pfrang C P Siess and M A Sozen ldquoLoad-moment-cur-vature characteristics of RC cross-sectionsrdquoACI Journal vol 61no 7 pp 763ndash778 1964
[11] D J Carreira and K-H Chu ldquoThemoment-curvature relation-ship of RC membersrdquo ACI Journal vol 83 no 2 pp 191ndash1981986
[12] R H Wood ldquoSome controversial and curious developments inplastic theory of structuresrdquo inEngineering Plasticity J Heymanand F A Leckie Eds pp 665ndash691 CambridgeUniversity PressCambridge UK 1968
[13] Y LMo ldquoInvestigation of reinforced concrete frame behaviourtheory and testsrdquo Magazine of Concrete Research vol 44 no160 pp 163ndash173 1992
[14] M Jirasek and Z P Bazant Inelastic Analysis of Structures JonWiley amp Sons London UK 2002
[15] S Chandrasekaran L Nunziante G Serino and F CarannanteldquoCurvature ductility of RC sections based on Eurocode analyt-ical procedurerdquo KSCE Journal of Civil Engineering vol 15 no 1pp 131ndash144 2011
Advances in Civil Engineering 13
[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012
[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013
[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992
[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995
[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006
[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010
[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012
[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951
[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982
[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002
[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989
[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008
Then the axial load and bending moment with respect to thebarycentre of the section that the secondary reinforcementcan bear can be calculated by the following expressions
1198731199042
=21198601199042
1198891
ℎ
2[1198911198901
(1198861+
120577119904119888
120593) + 119891
1198902(1198862+
120577119904119888
120593)
minus (Δ11989111198861+ Δ11989121198862) ]
1198721199042
=21198601199042
1198891
ℎ2
2[(1198911198901
minus 1198911198902)1205722
6minus Δ11989111198861(120572
2minus
1198861
3)
+Δ11989121198862(120572
2minus
1198862
3)]
(33)
Equations (28) (29) and (31) show that
1198911198901
(1198861+
120577119904119888
120593) + 1198911198902
(1198862+
120577119904119888
120593) = 2119891
119904119910
120593
120577119904119888
(119909119888minus
1
2) 120572
1198911198901
minus 1198911198902
= 119891119904119910
120593
120577119904119888
120572
(34)
Therefore substituting (34) into (33) in normalized form oneobtains
1198991199042
=1198731199042
119873119900
= 1205962
120593
120577119904119888
(119909119888minus
1
2) minus
1
2120572
times [(1 minus 120578119904119888) 1198862
1minus (1 minus 120578
119904ℎ) 1198862
2]
(35)
1198981199042
=1198721199042
119872119900
=1205962
2120572
120593
120577119904119888
[1205723
6minus (1 minus 120578
119904119888) (
120572
2minus
1198861
3) 1198862
1
minus (1 minus 120578119904ℎ) (
120572
2minus
1198862
3) 1198862
2]
(36)
Advances in Civil Engineering 7
h
c
c
co cob
x3
xc
Confinedconcrete
1205761
1205762
1205763
1205764
(a)
h
c
c
co cob
xc
1205761
1205762
1205763 = 1205764 = 0x3 = 0
Confinedconcrete
(b)
Figure 6 Strain state of concrete section (a) uncracked section (b) cracked section
It must be observed that if one sets 119864119904119888
= 119864119904in (30) andor
119864119904ℎ
= 119864119904in (32) (33) expresses the contributions to the axial
load and bendingmoment in the case inwhich the upper fibreandor the lower fibre of the secondary reinforcement remainin the elastic field As a consequence (35) and (36) are validfor any value of 119909
119888e 120593 under the condition that one sets
120578119904119888=1 if minus 120576
119904119910le 1205761199041
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 minus 120572
2)le1
120578119904ℎ=1 if minus 120576
119904119910le 1205761199042
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 + 120572
2)le1
(37)
43 Contribution of Concrete Figure 6 shows the strain statesof the section corresponding to the cases of uncracked andcracked sections By using the symbols in the figure the con-tributions to the equilibrium provided by the concrete coreare expressed by
119873cc = 119887int
119909119888
1199093
119891119888(119909) 119889119909 =
119887ℎ119891cco120593
int
1205762
1205763
119904119888(120576) 119889120576
119872cc = 119887int
119909119888
1199093
119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus
ℎ
2)
=119887ℎ2
119891cco (1
1205932int
1205762
1205763
119904119888(120576) 120576 119889120576minus
119909119888minus 05
120593int
1205762
1205763
119904119888(120576) 119889120576)
(38)
where
1199093=
0 for 119909119888le ℎ
119909119888minus ℎ for 119909
119888gt ℎ
(39)
and consequently
1205762= 120593119909119888
1205763=
0 for 119909119888le 1
120593 (119909119888minus 1) for 119909
119888gt 1
(40)
Considering (7) in the normalized form one obtains
119899cc =119873cc119873119900
=1
120593(11987811205762
minus 11987811205763
) (41)
119898cc =119872cc119872119900
=1
1205932(11987821205762
minus 11987821205763
) minus 119899cc (119909119888 minus1
2) (42)
The contribution of the surrounding unconfined concrete(concrete cover) can be calculated by subtracting the contri-bution of the confined region to that of the whole section
Considering that the strains involved in the functions 1198781120576
and 1198782120576
must be normalized with respect to the strain 120576co(unconfined concrete) by following the same procedure asthat leading to (41) and (42) and by introducing the con-finement efficiency factors defined by (6) for the whole sec-tion one obtains
119899un1 =1 + 2120575
119900
1198961
1
1198962120593
(11987811205761015840
1
minus 11987811205761015840
4
)
119898un1 =1 + 2120575
119900
1198961
1
(1198962120593)2(11987821205761015840
1
minus 11987821205761015840
4
) minus 119899un1 (119909119888 minus1
2)
(43)where
1205761015840
1= 1198962120593 (119909119888+ 120575V) le 120583
119888
1205761015840
4=
0 for 119909119888le (1 + 120575V)
1198962120593 [119909119888minus (1 + 120575V)] for 119909
119888gt (1 + 120575V)
(44)
The contribution to be subtracted from the above quantitiesare calculated bymodifying (41) and (42) in order to considerthe different constitutive law of the unconfined concrete andthe different base of normalization of strain and stress It caneasily be shown that one obtains
119899un2 =1
11989611198962120593
(11987811205761015840
2
minus 11987811205761015840
3
) (45)
119898un2 =1
1198961(1198962120593)2(11987821205761015840
2
minus 11987821205761015840
3
) minus 119899un2 (119909119888 minus1
2) (46)
8 Advances in Civil Engineering
where
1205761015840
2= 1198962120593119909119888le 120583119888
1205761015840
3=
0 for 119909119888le 1
1198962120593 (119909119888minus 1) for 119909
119888gt 1
(47)
On the basis of what was said above the effective contributionof the concrete section in compression in a generic loadingstate is expressed by
119899119888= 119899cc + 119899un1 minus 119899un2 (48)
119898119888= 119898cc + 119898un1 minus 119898un2 (49)
If the confinement effect is negligible (6) gives 1198961= 1198962= 1
By comparing (40) (41) and (42) with (47) (45) and (46)respectively it can easily be observed that 119899cc = 119899un2 and119898cc = 119898un2 Therefore in this case the contributions of thewhole section of reacting concrete given by (48) and (49) arereduced to the values 119899cc = 119899un1 and 119898cc = 119898un1 and areexpressed by (45) and (46) for 119896
1= 1198962= 1
5 Analysis Procedure
The flexural response of a class of RC sections characterizedby assigned values of the geometrical andmechanical param-eters defined above is obtained by imposing the equilibriumcondition expressed by (15) and (16)
For each configuration the contributions to the equi-librium provided by the principal reinforcement ((23) and(24) under the conditions expressed by (26)) the secondaryreinforcement ((28) (35) and (36) under the conditionsexpressed by (37)) and the compressed region of concrete((48) and (49) considering (41)ndash(47)) only depend on thevariables 119909
119888and 120593 which are independent of or related to one
another according to the type of analysis required (see below)In all cases it must be assumed that the proposed formulationis applied for 120593 gt 0
51 Field of Application The field of validity of the procedureproposed here is limited by the fact that the constitutive law ofthe steel reinforcement does not include unloading branchesstarting from a point of the post yielding branches in com-pression or in tension Therefore for any loading step theupper principal reinforcement and the upper fibre of the dis-tributed secondary reinforcement both potentially subject tocompression cannot be in the postyielding field in tensionthe lower principal reinforcement and the lower fibre of thedistributed secondary reinforcement both potentially subjectto tension cannot be in the postyielding field in compression
Because of these assumptions the field of values of 119909119888
consistent with a given value of 120593 is limited by the initial andend values provided by (26) or (37)
More precisely in the absence of secondary reinforce-ment (120596
2= 0) (26) gives
119909119888min = 120582 minus
120577119904119888
120593 119909
119888max = (1 minus 120582) +120577119904119888
120593 (50)
while if the secondary reinforcement is present (37) leads tomore restrictive limitations
119909119888min =
1 minus 120572
2minus
120577119904119888
120593 119909
119888max =1 + 120572
2+
120577119904119888
120593 (51)
These limitations also occur in a classical numericalapproach based on the strip decomposition of the section
However it must be observed that (50) or (51) allows oneto construct the moment-curvature curve for a large field ofvalues of assigned level of compression including the moreusual cases occurring in practical applications
52 Moment-Curvature Curves The more usual applicationof the analytical expressions derived in the previous sectionsis to derive by a step-by-step procedure the moment-curva-ture119898-120593 curve of a class of RC sections for a given value of 119899
To this purpose the procedure is applied by the followingsteps
(i) assigning the lowest value of curvature 120593 that onewants to consider increasing values of 119909
119888are consid-
ered from the minimum to the maximum expressedby (50) or (51)
(ii) for each value of the couple 120593 119909119888the corresponding
value of 1198991199041 1198991199042 and 119899
119888are calculated and their sum
increasing for each increase in 119909119888
(iii) when the assigned value of 119899 is reached with anacceptable tolerance (ie (15) is verified with verygood approximation) the assigned values of 120593 and 119899and the value of 119909
119888that was found make it possible
to calculate the corresponding values of11989811990411198981199042 and
119898119888expressed by (24) (36) and (49) and the sum
of these contributions of moment is the ordinate ofthe point having the abscissa 120593 belonging to the119898-120593curve sought
(iv) considering an increased value of 120593 the procedure isrepeated to obtain a subsequent point of themoment-curvature curve
The procedure stops when the maximum available ulti-mate confined concrete strain or the maximum available ten-sile strain of the lower reinforcement is exceeded These twooccurrences imply respectively that 120593119909
119888gt 120583cc or 120593(119909119888 minus 1 +
120582) lt minus120577119904119888120583119903 where 120583cc = 120576ccu120576cco is the available ductility of
the confined concrete defined in Section 31 and 120583119903is the steel
ductility defined in (11)If the stress-strain law of the steel reinforcement also
exhibits a softening postyielding branch corresponding to apostbuckling behaviour up to the strain value 120576
∙
119904(Figure 3)
a further limit at which the procedure must be stoppedderives from the condition that 120593(119909
119888minus 120582) gt 120577
119904119888120583119903119888 in
which 120583119903119888
= 120576lowast
119904120576119904119910 Beyond this condition the residual stress
of the compressed reinforcement should be assumed to beconstant and equal to 02119891
119904119910 but this branch of the stress-
strain law is not considered in the proposed model as statedin Section 32
Advances in Civil Engineering 9
53 Strength and Curvature Domains A typical 119898-120593 curveexhibits two characteristic points corresponding to theachievement of the ultimate state and to the first yield-ing of the principal reinforcement in tension respectivelyObviously the first point is present in all cases while thesecond can only be reached if yielding of the reinforcementin tension occurs before the brittle collapse of the concretecore due to low available concrete ductility orand a very highcompression level
The analytical expressions shown in the previous sectionsmake it possible to determine directly the coordinates of thesepoints and to derive in closed form the119898
119906-119899 120593119906-119899 and119898
119910-119899
120593119910-119899 domains which give useful information on the flexural
strength and curvature ductility of an RC section subjected toan assigned compression level
The constructions of the aforementioned curves can bemade more easily than by using a classical discretized modelof the section because (15) and (16) in a suitable specializedform provide analytical expressions where the axial com-pressive load and the corresponding bending moment onlydepend on a single variable 119909
119888and 120593 being related to one
anotherIn order to construct the 119898
119906-119899 and 120593
119906-119899 curves two dif-
ferent relationships linking the neutral axis depth to the cur-vature must be considered according to whether the collapseof the section is produced by excess of the available strain ofthe reinforcement in tension or of the available strain of theconcrete core in compression Moreover to distinguish thesetwo different events it must be observed that the maximumcurvature of the section is reachedwhen these two conditionsoccur simultaneously Considering Figure 4 it can easily bededuced that this value of curvature in normalized form is
120593119906max =
120583cc + 120577119904119888120583119903
1 minus 120582 (52)
The first branch of the domains sought is determined byimposing the condition that the collapse of the section is dueto the achievement of themaximum tensile strain of the prin-cipal reinforcement in tension (low level of compression)Therefore the following condition has to be assumed
120593119906(119909119888minus 1 + 120582) = minus 120577
119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus
120577119904119888120583119903
120593119906
(53)
Considering increasing values of the curvature from themin-imum value to themaximum expressed by (52) the second of(53) provide the corresponding value of 119909
119888 and (15) and (16)
by making explicit the contributions of the concrete and theprincipal and secondary reinforcement provide the corre-sponding values of 119899 and119898
119906
Once 120593119906max is reached the second branch of the 119898
119906-119899
120593119906-119899 curves must be determined considering that for further
decreasing values of 120593119906 the collapse is produced by the
achievement of the maximum compressive strain of the con-fined concrete Therefore the new relationship linking thecurvature to the neutral axis depth is
By using the second of these equations (15) and (16) againlink 119899 and119898
119906to the only variable 120593
119906
It must be observed that if the principal reinforcement incompression is subject to buckling after the achievement ofthe yielding stress the ultimate strain of this compressed rein-forcement is the strain denoted as 120576∙
119904in Figure 3 This strain
value can be reached before the concrete core reaches thenormalized value of strain 120583cc In this case the maximumvalue of normalized dimensionless curvature expressed by(52) must be substituted by the value 120593
119906max = 120577119904119888(120583119903119888
+
120583119903)(1minus2120582) where 120583
119903119888= 120576lowast
119904120576119904119910 and the second branch of the
ultimate strength domains must be determined by assumingthat for values of 120593
119906that decrease from this maximum the
relationship linking the neutral axis dept to the curvature is119909119888= 120582 + 120577
119904119888120583119903119888120593
The 119898119910-119899 120593119910-119899 curves can be simply determined by
assuming that whatever the value of the curvature is the prin-cipal reinforcement in tension is affected by the strain valueequal to minus120576
119904119910 This condition in dimensionless terms gives
120593119910(119909119888minus 1 + 120582) = minus 120577
119904119888997904rArr 119909119888= (1 minus 120582) minus
120577119904119888
120593119910
(55)
By introducing the second of (55) into the equilibrium Equa-tions (15) and (16) they again become two functions of thesingle variable 120593 = 120593
119910 The 119898
119910-119899 120593119910-119899 curves can be con-
structed starting froma value thatwas assumed to be themin-imum up to the maximum value corresponding to one of thepossible aforementioned occurrences the collapse of the con-crete core due to achievement of the available ductility of thematerial or the achievement of the strain 120576
∙
119904in the principal
reinforcement in compression if this reinforcement is subjectto buckling
Considering the field of validity of the procedure pro-posed the minimum value of 120593
119906that can be considered to
construct the119898119906-119899 120593119906-119899 domains is obtained by introducing
the second of (53) into the first of (50) or (51) so that
120593119906min = 120577
119904119888
120583119903minus 1
1 minus 2120582 (56)
in the absence of the secondary reinforcement and
120593119906min = 120577
119904119888
120583119903minus 1
(1 + 120572) 2 minus 120582 (57)
if the secondary reinforcement is also presentObviously the 119898
119910-119899 120593119910-119899 domains can be constructed
starting from any value 120593119910gt 0
In relation to the field of applications (RC columns) thefirst values of 120593
119906and 120593
119910that are to be actually considered are
the ones first producing 119899 ge 0
6 Applications
The reliability of the procedure proposed is shown here byusing the results derived from two RC sections considered byother researchers
Thefirst application refers to one of the sections examinedby Zahn et al [27] This was a square RC section of side119861 = 400mm and realized with concrete having cylindrical
10 Advances in Civil EngineeringN
orm
aliz
ed st
ress
1
00 1 2 3 4 5
Saatcioglu and Razvi (1992)Mander et al (1988)
Normalized strain
Figure 7 Calibration of constitutive law of confined concrete
strength 119891co = 30Nmm2 The longitudinal reinforcementconsisted of 12 steel bars of diameter 16mm having yieldingstrength 119891
119904119910= 420Nmm2 uniformly distributed along the
perimeter of the concrete core (constant pitch) The coverdepth measured from the external perimeter of the stirrupswas 13mm The transverse reinforcement consisting of steelstirrups of diameter equal to 10mm according to the calcu-lations made by the aforementioned authors provided theeffective confinement pressure 119891
ℓ119890= 011119891
1015840
co where 1198911015840
co =
085119891co is the reduced strength value prescribed by the ACI318 code [28]
The flexural response was numerically derived in [27] byusing a classical fibre model in which the stress-strain law ofthe confined concrete was assumed in agreement with Man-derrsquosmodel [25]Therefore in this application the parameters120573cc and 120578cc governing the shape of the laws expressed by (4)and (5) are calibrated so that they lead to a constitutive lawof confined concrete which is very similar to that used by theaforementioned authors
Manderrsquos model [25] for the given value of the confine-ment pressure is governed by the parameter 119903 = 13 themodel of Saatcioglu and Razvi [18] leads to a very similarshape of the stress-strain curve by assuming that 120573cc = 045
and 120578cc = minus006 Figure 7 shows the very similar shapes ofthe constitutive laws corresponding to the two models con-sidered
The procedure proposed in [25] provided the followingcharacteristic values of confined concrete strength and cor-responding strain 120576cco = 00073 119891cco = 46Nmm2 Thesevalues are also adopted for the dimensional stress-strain lawexpressed by the model used here [18]
Since for the stress-strain law of the unconfined concreteZahn et al [27] do not give any information the default valuesindicated in Section 31 are assumed here 120573
119888= 1 120578
119888= minus08
and 120576co = 0002On the basis of the above data the square cross-section of
the concrete coremeasured inside the perimeter stirrups hasa side of 119887 = ℎ = 354mm the cover depth is equal to
400
300
200
100
00 001 002 003 004 005
Mom
ent (
kNm
)
Curvature (1m)
Zahn et al (1989)Present model
Figure 8 Comparison of results119872-120593 curve for given119873
23mm the number of intermediate reinforcing bars realizingthe secondary reinforcement is 119899
119887= 119899ℎ= 2 and the distance
of the barycentre of the section of the principal reinforcementfrom the external side of the concrete core section is ℎ
1=
162 = 8mmThe principal reinforcement consists of four upper and
four lower 16mm steel bars while the secondary reinforce-ment consists of four bars (two on the right and two on theleft) of the same diameter The steel reinforcement constitu-tive law is defined by assuming 119864
119904= 210GPa 120576
119904119910= 119891119904119910119864119904=
0002) and 119864119904ℎ
= 119864119904119888
= 0Therefore the dimensionless parameters involved in the
formulation proposed take on the following values 120575V = 120575119900=
= 0Figure 8 compares the 119872-120593 curve obtained in [27] with
that derived from the procedure shown in Section 52 for acompressive constant axial load 119873 = 03119891co(119861 times 119861) corre-sponding to a compressive level 119899 = 119873(119887
2
times 119891cco) = 025 inagreement with the first of (14)The results in the figure showa very good level of agreement between the numerical pro-cedure adopted in [27] and the analytical one adopted here
Both curves in the figure clearly show the cusp producedby the yielding of the principal reinforcement in tension andthe effects of the progressive cover spalling
The second application refers to the middle-height sec-tion of an RC column of height 1640mm experimentallytested by Saatcioglu et al [19] marked as specimen C6-2 bythe authors
The loading condition was realized by imposing relativeaxial displacements so that the reactive compressive forceacted with fixed eccentricity in a plane of principal inertiaof the sections Suitable devices were applied at the columnends so that the column itself behaved like a hinged verticalRC member under an eccentric compressive load
As a consequence each point of the moment-curvaturecurve characterizing the experimental response obtained by
Advances in Civil Engineering 11
the aforementioned authors corresponds to a different valueof axial loadMoreover the actual bendingmoment acting onthe middle-height section was influenced by the 119875-Δ effect
The formulation proposed here is validated consideringsome points of the aforementioned curve for which theauthors indicate the values of the axial load119873 and curvaturethat were measured These values of 119873 are normalized withrespect to the axial load 119873
119900expressed by (2) The procedure
is applied by determining the value of 119909119888from (15) for the
assigned values of 119899 and 120593 and by calculating 119898 from (16)The results are compared with the numerical values obtainedby the strip model adopted by the authors which includedthe 119875-Δ effect and the values experimentally detected
The specimen considered had a square section of side 119861 =
210mm cover depth of 125mm and longitudinal reinforcingbars of diameter 113mm uniformly distributed along theperimeter of the concrete core having yielding strength119891
119904119910=
517Nmm2 and average hardening modulus 119864119904ℎ
= 0013119864119904
The transverse steel reinforcement consisted of square andoctagonal 63mm stirrups with pitch 50mm
The unconfined concrete had cylindrical strength 119891co =
3440Nmm2The numerical model adopted in [19] assumedthat the perimeter of the concrete core was coincident withthe axis lines of the external stirrups
On the basis of these geometrical and mechanical dataand by evaluating the effective confinement pressure by theprocedure proposed in [18] for the confined concrete sectionone obtains 119887 = ℎ = 17870mm 119891cco = 4635 Nmm2 120576cco =
000547 120578cc = minus0052 and 120573 = 0590The further parametersinvolved in the formulation proposed take on the followingvalues 120575
= 0013Figure 9 shows that the procedure proposed provides
results that are very close to those derived by the numericalmodel adopted in [19] The experimentally detected values ofmoment prove to be underestimated by both models
It must be observed that the last point detected by usingthe present formulation corresponding to the ultimate stateof the section had been already found in [17] because thissection belongs to the more restricted class of sections con-sidered in that work
Figure 10 shows the strength and curvature domainsobtained by using the procedure described in Section 53For confined concrete and steel reinforcement the followingvalues were assumed deduced from the data shown in [19]120576ccu = 0032 (120583cc = 585) and 120576
119904119906= 0066 (120583
119903= 27) The
results confirm that a good level of confinementmakes it pos-sible to achieve acceptable ductility of curvature even underhigh levels of compression In order to show how the proce-dure proposed is easily able to evaluate the influence of all thegeometric and mechanical parameters governing the flexuralresponse of a class of RC sections Figure 10(a) also shows theinfluence of the unconfined concrete cover on the bendingmoment at the first yielding of the principal reinforcement intension As expected this influence proves to be negligible forhigh values of the compression level
The dotted curve in Figure 10(b) obviously stops at thepoint of intersection with the continuous curve Beyond this
100
80
60
40
20
00 1 2 3 4
Mom
ent (
kNm
)
ExperimentalModelPresent model
Saatcioglu et al (1995)
Curvature (1mm times 10minus4)
Figure 9 Comparison of theoretical and experimental results
point the 120593119906-119873 curve proceeds by a very brief stretch (not
very evident in the figure) that corresponds to brittle collapseof the section due to the achievement of the maximum avail-able compressive strain in the confined concrete when theprincipal reinforcement in tension is still in the elastic fieldThe end point of this curve corresponds to the achievementof the maximum neutral axis depth defined by the second of(51)
Nevertheless the axial load values that cannot be consid-ered because of these limitations are well beyond the onesthat can be assumed in the structural design of RC buildingsin seismic areas or usually found in the columns of existingbuildings
7 Conclusions
A dimensionless formulation has been proposed which pro-vides the flexural response of classes of sections of RC col-umns having the same values of the geometric and mechani-cal parameters defined in this study
At each loading stage the resisting components of the sec-tion (cover and concrete core and principal and secondarylongitudinal reinforcements) give a contribution to the equi-librium that can be expressed by analytical functions depend-ing on the normalized neutral axis depth and the curvature ofthe section
Special equilibrium conditions like the ones correspond-ing to the first yielding of the principal reinforcement intension and to the ultimate state imply an analytical link bet-ween the neutral axis depth and the curvature so that the sumof the aforementioned contributions becomes an analyticalfunction of a single variable which can express a strength orcurvature domain
For a generic equilibrium condition an iterative proce-dure to determine the neutral axis depth corresponding to
12 Advances in Civil Engineering
100
80
60
40
20
00 300 600 900 1200 1500
Mom
ent (
kNm
)
Axial load (kN)
Mu
My
My without cover
(a)
8
6
4
2
00 300 600 900 1200 1500
Axial load (kN)
120593u120593y
Curv
atur
e(1
mm
times10
minus4)
(b)
Figure 10119872-119873 and 120593-119873 domains at the ultimate state and the first yielding of reinforcement
the assigned values of curvature and compression level isrequired but each iteration step implies the simple use of thesame equilibrium equation for an updated value of the nor-malized neutral axis dept
The dimensionless form of the formulation proposedmakes also it easily possible to evaluate the influence of theparameters characterizing the class of sections examined onthe strength and curvature corresponding to a given loadingstage
Acknowledgment
Thisworkwas carried outwithin the 2010ndash2013Research Pro-ject ldquoDPC-ReLUIS (Dipartimento Protezione Civile-Rete deiLaboratori Universitari di Ingegneria Sismica)rdquo AT 1 Task112 The related financial support was greatly appreciated
References
[1] A Fafitis ldquoInteraction surfaces of reinforced-concrete sectionsin biaxial bendingrdquo Journal of Structural Engineering vol 127no 7 pp 840ndash846 2001
[2] J L Bonet P F Miguel M A Fernandez and M L RomeroldquoAnalytical approach to failure surfaces in reinforced concretesections subjected to axial loads and biaxial bendingrdquo Journalof Structural Engineering vol 130 no 12 pp 2006ndash2015 2004
[3] G Monti and S Alessandri ldquoAssessment of rc columns undercombined biaxial bending and axial loadrdquo in Proceedings of the2nd FIB Congress Naples Italy 2006
[4] RW Clough and S B Johnston ldquoEffect of stiffness degradationon earthquake ductility requirementsrdquo in Proceedings of 2ndJapan Earthquake Engineering Symposium Tokyo Japan 1966
[5] T Takeda M A Sozen and N N Nielsen ldquoReinforced con-crete response to simulated earthquakerdquo Journal of StructuralDivision vol 96 no 12 pp 2257ndash2273 1970
[6] M S L Roufaiel and C Meyer ldquoAnalytical modeling of hyster-etic behavior of reinforced concrete framerdquo Journal of StructuralEngineering vol 113 no 3 pp 429ndash444 1987
[7] A R Mari and A C Scordelis ldquoNonlinear geometric materialand time dependent analysis of three dimensional reinforcedand prestressed concrete framesrdquo USBSESM Report 8412Department of Civil Engineering University of CaliforniaBerkeley Calif USA 1973
[8] T Taucer E Spacone and F C Filippou ldquoA fiber beam-col-umn element for seismic response analysis of reinforced con-crete structuresrdquo Report EERC 91-17 Earthquake EngineeringResearch Center Berkeley Calif USA 1991
[9] Z Zhu I Ahmad and A Mirmiran ldquoFiber element modelingfor seismic performance of bridge columns made of concrete-filled FRP tubesrdquo Engineering Structures vol 28 no 14 pp2023ndash2035 2006
[10] E O Pfrang C P Siess and M A Sozen ldquoLoad-moment-cur-vature characteristics of RC cross-sectionsrdquoACI Journal vol 61no 7 pp 763ndash778 1964
[11] D J Carreira and K-H Chu ldquoThemoment-curvature relation-ship of RC membersrdquo ACI Journal vol 83 no 2 pp 191ndash1981986
[12] R H Wood ldquoSome controversial and curious developments inplastic theory of structuresrdquo inEngineering Plasticity J Heymanand F A Leckie Eds pp 665ndash691 CambridgeUniversity PressCambridge UK 1968
[13] Y LMo ldquoInvestigation of reinforced concrete frame behaviourtheory and testsrdquo Magazine of Concrete Research vol 44 no160 pp 163ndash173 1992
[14] M Jirasek and Z P Bazant Inelastic Analysis of Structures JonWiley amp Sons London UK 2002
[15] S Chandrasekaran L Nunziante G Serino and F CarannanteldquoCurvature ductility of RC sections based on Eurocode analyt-ical procedurerdquo KSCE Journal of Civil Engineering vol 15 no 1pp 131ndash144 2011
Advances in Civil Engineering 13
[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012
[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013
[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992
[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995
[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006
[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010
[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012
[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951
[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982
[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002
[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989
[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008
Figure 6 Strain state of concrete section (a) uncracked section (b) cracked section
It must be observed that if one sets 119864119904119888
= 119864119904in (30) andor
119864119904ℎ
= 119864119904in (32) (33) expresses the contributions to the axial
load and bendingmoment in the case inwhich the upper fibreandor the lower fibre of the secondary reinforcement remainin the elastic field As a consequence (35) and (36) are validfor any value of 119909
119888e 120593 under the condition that one sets
120578119904119888=1 if minus 120576
119904119910le 1205761199041
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 minus 120572
2)le1
120578119904ℎ=1 if minus 120576
119904119910le 1205761199042
le 120576119904119910
997904rArr minus1le120593
120577119904119888
(119909119888minus
1 + 120572
2)le1
(37)
43 Contribution of Concrete Figure 6 shows the strain statesof the section corresponding to the cases of uncracked andcracked sections By using the symbols in the figure the con-tributions to the equilibrium provided by the concrete coreare expressed by
119873cc = 119887int
119909119888
1199093
119891119888(119909) 119889119909 =
119887ℎ119891cco120593
int
1205762
1205763
119904119888(120576) 119889120576
119872cc = 119887int
119909119888
1199093
119891119888(119909) 119909 119889119909 minus 119873cc (119909119888 minus
ℎ
2)
=119887ℎ2
119891cco (1
1205932int
1205762
1205763
119904119888(120576) 120576 119889120576minus
119909119888minus 05
120593int
1205762
1205763
119904119888(120576) 119889120576)
(38)
where
1199093=
0 for 119909119888le ℎ
119909119888minus ℎ for 119909
119888gt ℎ
(39)
and consequently
1205762= 120593119909119888
1205763=
0 for 119909119888le 1
120593 (119909119888minus 1) for 119909
119888gt 1
(40)
Considering (7) in the normalized form one obtains
119899cc =119873cc119873119900
=1
120593(11987811205762
minus 11987811205763
) (41)
119898cc =119872cc119872119900
=1
1205932(11987821205762
minus 11987821205763
) minus 119899cc (119909119888 minus1
2) (42)
The contribution of the surrounding unconfined concrete(concrete cover) can be calculated by subtracting the contri-bution of the confined region to that of the whole section
Considering that the strains involved in the functions 1198781120576
and 1198782120576
must be normalized with respect to the strain 120576co(unconfined concrete) by following the same procedure asthat leading to (41) and (42) and by introducing the con-finement efficiency factors defined by (6) for the whole sec-tion one obtains
119899un1 =1 + 2120575
119900
1198961
1
1198962120593
(11987811205761015840
1
minus 11987811205761015840
4
)
119898un1 =1 + 2120575
119900
1198961
1
(1198962120593)2(11987821205761015840
1
minus 11987821205761015840
4
) minus 119899un1 (119909119888 minus1
2)
(43)where
1205761015840
1= 1198962120593 (119909119888+ 120575V) le 120583
119888
1205761015840
4=
0 for 119909119888le (1 + 120575V)
1198962120593 [119909119888minus (1 + 120575V)] for 119909
119888gt (1 + 120575V)
(44)
The contribution to be subtracted from the above quantitiesare calculated bymodifying (41) and (42) in order to considerthe different constitutive law of the unconfined concrete andthe different base of normalization of strain and stress It caneasily be shown that one obtains
119899un2 =1
11989611198962120593
(11987811205761015840
2
minus 11987811205761015840
3
) (45)
119898un2 =1
1198961(1198962120593)2(11987821205761015840
2
minus 11987821205761015840
3
) minus 119899un2 (119909119888 minus1
2) (46)
8 Advances in Civil Engineering
where
1205761015840
2= 1198962120593119909119888le 120583119888
1205761015840
3=
0 for 119909119888le 1
1198962120593 (119909119888minus 1) for 119909
119888gt 1
(47)
On the basis of what was said above the effective contributionof the concrete section in compression in a generic loadingstate is expressed by
119899119888= 119899cc + 119899un1 minus 119899un2 (48)
119898119888= 119898cc + 119898un1 minus 119898un2 (49)
If the confinement effect is negligible (6) gives 1198961= 1198962= 1
By comparing (40) (41) and (42) with (47) (45) and (46)respectively it can easily be observed that 119899cc = 119899un2 and119898cc = 119898un2 Therefore in this case the contributions of thewhole section of reacting concrete given by (48) and (49) arereduced to the values 119899cc = 119899un1 and 119898cc = 119898un1 and areexpressed by (45) and (46) for 119896
1= 1198962= 1
5 Analysis Procedure
The flexural response of a class of RC sections characterizedby assigned values of the geometrical andmechanical param-eters defined above is obtained by imposing the equilibriumcondition expressed by (15) and (16)
For each configuration the contributions to the equi-librium provided by the principal reinforcement ((23) and(24) under the conditions expressed by (26)) the secondaryreinforcement ((28) (35) and (36) under the conditionsexpressed by (37)) and the compressed region of concrete((48) and (49) considering (41)ndash(47)) only depend on thevariables 119909
119888and 120593 which are independent of or related to one
another according to the type of analysis required (see below)In all cases it must be assumed that the proposed formulationis applied for 120593 gt 0
51 Field of Application The field of validity of the procedureproposed here is limited by the fact that the constitutive law ofthe steel reinforcement does not include unloading branchesstarting from a point of the post yielding branches in com-pression or in tension Therefore for any loading step theupper principal reinforcement and the upper fibre of the dis-tributed secondary reinforcement both potentially subject tocompression cannot be in the postyielding field in tensionthe lower principal reinforcement and the lower fibre of thedistributed secondary reinforcement both potentially subjectto tension cannot be in the postyielding field in compression
Because of these assumptions the field of values of 119909119888
consistent with a given value of 120593 is limited by the initial andend values provided by (26) or (37)
More precisely in the absence of secondary reinforce-ment (120596
2= 0) (26) gives
119909119888min = 120582 minus
120577119904119888
120593 119909
119888max = (1 minus 120582) +120577119904119888
120593 (50)
while if the secondary reinforcement is present (37) leads tomore restrictive limitations
119909119888min =
1 minus 120572
2minus
120577119904119888
120593 119909
119888max =1 + 120572
2+
120577119904119888
120593 (51)
These limitations also occur in a classical numericalapproach based on the strip decomposition of the section
However it must be observed that (50) or (51) allows oneto construct the moment-curvature curve for a large field ofvalues of assigned level of compression including the moreusual cases occurring in practical applications
52 Moment-Curvature Curves The more usual applicationof the analytical expressions derived in the previous sectionsis to derive by a step-by-step procedure the moment-curva-ture119898-120593 curve of a class of RC sections for a given value of 119899
To this purpose the procedure is applied by the followingsteps
(i) assigning the lowest value of curvature 120593 that onewants to consider increasing values of 119909
119888are consid-
ered from the minimum to the maximum expressedby (50) or (51)
(ii) for each value of the couple 120593 119909119888the corresponding
value of 1198991199041 1198991199042 and 119899
119888are calculated and their sum
increasing for each increase in 119909119888
(iii) when the assigned value of 119899 is reached with anacceptable tolerance (ie (15) is verified with verygood approximation) the assigned values of 120593 and 119899and the value of 119909
119888that was found make it possible
to calculate the corresponding values of11989811990411198981199042 and
119898119888expressed by (24) (36) and (49) and the sum
of these contributions of moment is the ordinate ofthe point having the abscissa 120593 belonging to the119898-120593curve sought
(iv) considering an increased value of 120593 the procedure isrepeated to obtain a subsequent point of themoment-curvature curve
The procedure stops when the maximum available ulti-mate confined concrete strain or the maximum available ten-sile strain of the lower reinforcement is exceeded These twooccurrences imply respectively that 120593119909
119888gt 120583cc or 120593(119909119888 minus 1 +
120582) lt minus120577119904119888120583119903 where 120583cc = 120576ccu120576cco is the available ductility of
the confined concrete defined in Section 31 and 120583119903is the steel
ductility defined in (11)If the stress-strain law of the steel reinforcement also
exhibits a softening postyielding branch corresponding to apostbuckling behaviour up to the strain value 120576
∙
119904(Figure 3)
a further limit at which the procedure must be stoppedderives from the condition that 120593(119909
119888minus 120582) gt 120577
119904119888120583119903119888 in
which 120583119903119888
= 120576lowast
119904120576119904119910 Beyond this condition the residual stress
of the compressed reinforcement should be assumed to beconstant and equal to 02119891
119904119910 but this branch of the stress-
strain law is not considered in the proposed model as statedin Section 32
Advances in Civil Engineering 9
53 Strength and Curvature Domains A typical 119898-120593 curveexhibits two characteristic points corresponding to theachievement of the ultimate state and to the first yield-ing of the principal reinforcement in tension respectivelyObviously the first point is present in all cases while thesecond can only be reached if yielding of the reinforcementin tension occurs before the brittle collapse of the concretecore due to low available concrete ductility orand a very highcompression level
The analytical expressions shown in the previous sectionsmake it possible to determine directly the coordinates of thesepoints and to derive in closed form the119898
119906-119899 120593119906-119899 and119898
119910-119899
120593119910-119899 domains which give useful information on the flexural
strength and curvature ductility of an RC section subjected toan assigned compression level
The constructions of the aforementioned curves can bemade more easily than by using a classical discretized modelof the section because (15) and (16) in a suitable specializedform provide analytical expressions where the axial com-pressive load and the corresponding bending moment onlydepend on a single variable 119909
119888and 120593 being related to one
anotherIn order to construct the 119898
119906-119899 and 120593
119906-119899 curves two dif-
ferent relationships linking the neutral axis depth to the cur-vature must be considered according to whether the collapseof the section is produced by excess of the available strain ofthe reinforcement in tension or of the available strain of theconcrete core in compression Moreover to distinguish thesetwo different events it must be observed that the maximumcurvature of the section is reachedwhen these two conditionsoccur simultaneously Considering Figure 4 it can easily bededuced that this value of curvature in normalized form is
120593119906max =
120583cc + 120577119904119888120583119903
1 minus 120582 (52)
The first branch of the domains sought is determined byimposing the condition that the collapse of the section is dueto the achievement of themaximum tensile strain of the prin-cipal reinforcement in tension (low level of compression)Therefore the following condition has to be assumed
120593119906(119909119888minus 1 + 120582) = minus 120577
119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus
120577119904119888120583119903
120593119906
(53)
Considering increasing values of the curvature from themin-imum value to themaximum expressed by (52) the second of(53) provide the corresponding value of 119909
119888 and (15) and (16)
by making explicit the contributions of the concrete and theprincipal and secondary reinforcement provide the corre-sponding values of 119899 and119898
119906
Once 120593119906max is reached the second branch of the 119898
119906-119899
120593119906-119899 curves must be determined considering that for further
decreasing values of 120593119906 the collapse is produced by the
achievement of the maximum compressive strain of the con-fined concrete Therefore the new relationship linking thecurvature to the neutral axis depth is
By using the second of these equations (15) and (16) againlink 119899 and119898
119906to the only variable 120593
119906
It must be observed that if the principal reinforcement incompression is subject to buckling after the achievement ofthe yielding stress the ultimate strain of this compressed rein-forcement is the strain denoted as 120576∙
119904in Figure 3 This strain
value can be reached before the concrete core reaches thenormalized value of strain 120583cc In this case the maximumvalue of normalized dimensionless curvature expressed by(52) must be substituted by the value 120593
119906max = 120577119904119888(120583119903119888
+
120583119903)(1minus2120582) where 120583
119903119888= 120576lowast
119904120576119904119910 and the second branch of the
ultimate strength domains must be determined by assumingthat for values of 120593
119906that decrease from this maximum the
relationship linking the neutral axis dept to the curvature is119909119888= 120582 + 120577
119904119888120583119903119888120593
The 119898119910-119899 120593119910-119899 curves can be simply determined by
assuming that whatever the value of the curvature is the prin-cipal reinforcement in tension is affected by the strain valueequal to minus120576
119904119910 This condition in dimensionless terms gives
120593119910(119909119888minus 1 + 120582) = minus 120577
119904119888997904rArr 119909119888= (1 minus 120582) minus
120577119904119888
120593119910
(55)
By introducing the second of (55) into the equilibrium Equa-tions (15) and (16) they again become two functions of thesingle variable 120593 = 120593
119910 The 119898
119910-119899 120593119910-119899 curves can be con-
structed starting froma value thatwas assumed to be themin-imum up to the maximum value corresponding to one of thepossible aforementioned occurrences the collapse of the con-crete core due to achievement of the available ductility of thematerial or the achievement of the strain 120576
∙
119904in the principal
reinforcement in compression if this reinforcement is subjectto buckling
Considering the field of validity of the procedure pro-posed the minimum value of 120593
119906that can be considered to
construct the119898119906-119899 120593119906-119899 domains is obtained by introducing
the second of (53) into the first of (50) or (51) so that
120593119906min = 120577
119904119888
120583119903minus 1
1 minus 2120582 (56)
in the absence of the secondary reinforcement and
120593119906min = 120577
119904119888
120583119903minus 1
(1 + 120572) 2 minus 120582 (57)
if the secondary reinforcement is also presentObviously the 119898
119910-119899 120593119910-119899 domains can be constructed
starting from any value 120593119910gt 0
In relation to the field of applications (RC columns) thefirst values of 120593
119906and 120593
119910that are to be actually considered are
the ones first producing 119899 ge 0
6 Applications
The reliability of the procedure proposed is shown here byusing the results derived from two RC sections considered byother researchers
Thefirst application refers to one of the sections examinedby Zahn et al [27] This was a square RC section of side119861 = 400mm and realized with concrete having cylindrical
10 Advances in Civil EngineeringN
orm
aliz
ed st
ress
1
00 1 2 3 4 5
Saatcioglu and Razvi (1992)Mander et al (1988)
Normalized strain
Figure 7 Calibration of constitutive law of confined concrete
strength 119891co = 30Nmm2 The longitudinal reinforcementconsisted of 12 steel bars of diameter 16mm having yieldingstrength 119891
119904119910= 420Nmm2 uniformly distributed along the
perimeter of the concrete core (constant pitch) The coverdepth measured from the external perimeter of the stirrupswas 13mm The transverse reinforcement consisting of steelstirrups of diameter equal to 10mm according to the calcu-lations made by the aforementioned authors provided theeffective confinement pressure 119891
ℓ119890= 011119891
1015840
co where 1198911015840
co =
085119891co is the reduced strength value prescribed by the ACI318 code [28]
The flexural response was numerically derived in [27] byusing a classical fibre model in which the stress-strain law ofthe confined concrete was assumed in agreement with Man-derrsquosmodel [25]Therefore in this application the parameters120573cc and 120578cc governing the shape of the laws expressed by (4)and (5) are calibrated so that they lead to a constitutive lawof confined concrete which is very similar to that used by theaforementioned authors
Manderrsquos model [25] for the given value of the confine-ment pressure is governed by the parameter 119903 = 13 themodel of Saatcioglu and Razvi [18] leads to a very similarshape of the stress-strain curve by assuming that 120573cc = 045
and 120578cc = minus006 Figure 7 shows the very similar shapes ofthe constitutive laws corresponding to the two models con-sidered
The procedure proposed in [25] provided the followingcharacteristic values of confined concrete strength and cor-responding strain 120576cco = 00073 119891cco = 46Nmm2 Thesevalues are also adopted for the dimensional stress-strain lawexpressed by the model used here [18]
Since for the stress-strain law of the unconfined concreteZahn et al [27] do not give any information the default valuesindicated in Section 31 are assumed here 120573
119888= 1 120578
119888= minus08
and 120576co = 0002On the basis of the above data the square cross-section of
the concrete coremeasured inside the perimeter stirrups hasa side of 119887 = ℎ = 354mm the cover depth is equal to
400
300
200
100
00 001 002 003 004 005
Mom
ent (
kNm
)
Curvature (1m)
Zahn et al (1989)Present model
Figure 8 Comparison of results119872-120593 curve for given119873
23mm the number of intermediate reinforcing bars realizingthe secondary reinforcement is 119899
119887= 119899ℎ= 2 and the distance
of the barycentre of the section of the principal reinforcementfrom the external side of the concrete core section is ℎ
1=
162 = 8mmThe principal reinforcement consists of four upper and
four lower 16mm steel bars while the secondary reinforce-ment consists of four bars (two on the right and two on theleft) of the same diameter The steel reinforcement constitu-tive law is defined by assuming 119864
119904= 210GPa 120576
119904119910= 119891119904119910119864119904=
0002) and 119864119904ℎ
= 119864119904119888
= 0Therefore the dimensionless parameters involved in the
formulation proposed take on the following values 120575V = 120575119900=
= 0Figure 8 compares the 119872-120593 curve obtained in [27] with
that derived from the procedure shown in Section 52 for acompressive constant axial load 119873 = 03119891co(119861 times 119861) corre-sponding to a compressive level 119899 = 119873(119887
2
times 119891cco) = 025 inagreement with the first of (14)The results in the figure showa very good level of agreement between the numerical pro-cedure adopted in [27] and the analytical one adopted here
Both curves in the figure clearly show the cusp producedby the yielding of the principal reinforcement in tension andthe effects of the progressive cover spalling
The second application refers to the middle-height sec-tion of an RC column of height 1640mm experimentallytested by Saatcioglu et al [19] marked as specimen C6-2 bythe authors
The loading condition was realized by imposing relativeaxial displacements so that the reactive compressive forceacted with fixed eccentricity in a plane of principal inertiaof the sections Suitable devices were applied at the columnends so that the column itself behaved like a hinged verticalRC member under an eccentric compressive load
As a consequence each point of the moment-curvaturecurve characterizing the experimental response obtained by
Advances in Civil Engineering 11
the aforementioned authors corresponds to a different valueof axial loadMoreover the actual bendingmoment acting onthe middle-height section was influenced by the 119875-Δ effect
The formulation proposed here is validated consideringsome points of the aforementioned curve for which theauthors indicate the values of the axial load119873 and curvaturethat were measured These values of 119873 are normalized withrespect to the axial load 119873
119900expressed by (2) The procedure
is applied by determining the value of 119909119888from (15) for the
assigned values of 119899 and 120593 and by calculating 119898 from (16)The results are compared with the numerical values obtainedby the strip model adopted by the authors which includedthe 119875-Δ effect and the values experimentally detected
The specimen considered had a square section of side 119861 =
210mm cover depth of 125mm and longitudinal reinforcingbars of diameter 113mm uniformly distributed along theperimeter of the concrete core having yielding strength119891
119904119910=
517Nmm2 and average hardening modulus 119864119904ℎ
= 0013119864119904
The transverse steel reinforcement consisted of square andoctagonal 63mm stirrups with pitch 50mm
The unconfined concrete had cylindrical strength 119891co =
3440Nmm2The numerical model adopted in [19] assumedthat the perimeter of the concrete core was coincident withthe axis lines of the external stirrups
On the basis of these geometrical and mechanical dataand by evaluating the effective confinement pressure by theprocedure proposed in [18] for the confined concrete sectionone obtains 119887 = ℎ = 17870mm 119891cco = 4635 Nmm2 120576cco =
000547 120578cc = minus0052 and 120573 = 0590The further parametersinvolved in the formulation proposed take on the followingvalues 120575
= 0013Figure 9 shows that the procedure proposed provides
results that are very close to those derived by the numericalmodel adopted in [19] The experimentally detected values ofmoment prove to be underestimated by both models
It must be observed that the last point detected by usingthe present formulation corresponding to the ultimate stateof the section had been already found in [17] because thissection belongs to the more restricted class of sections con-sidered in that work
Figure 10 shows the strength and curvature domainsobtained by using the procedure described in Section 53For confined concrete and steel reinforcement the followingvalues were assumed deduced from the data shown in [19]120576ccu = 0032 (120583cc = 585) and 120576
119904119906= 0066 (120583
119903= 27) The
results confirm that a good level of confinementmakes it pos-sible to achieve acceptable ductility of curvature even underhigh levels of compression In order to show how the proce-dure proposed is easily able to evaluate the influence of all thegeometric and mechanical parameters governing the flexuralresponse of a class of RC sections Figure 10(a) also shows theinfluence of the unconfined concrete cover on the bendingmoment at the first yielding of the principal reinforcement intension As expected this influence proves to be negligible forhigh values of the compression level
The dotted curve in Figure 10(b) obviously stops at thepoint of intersection with the continuous curve Beyond this
100
80
60
40
20
00 1 2 3 4
Mom
ent (
kNm
)
ExperimentalModelPresent model
Saatcioglu et al (1995)
Curvature (1mm times 10minus4)
Figure 9 Comparison of theoretical and experimental results
point the 120593119906-119873 curve proceeds by a very brief stretch (not
very evident in the figure) that corresponds to brittle collapseof the section due to the achievement of the maximum avail-able compressive strain in the confined concrete when theprincipal reinforcement in tension is still in the elastic fieldThe end point of this curve corresponds to the achievementof the maximum neutral axis depth defined by the second of(51)
Nevertheless the axial load values that cannot be consid-ered because of these limitations are well beyond the onesthat can be assumed in the structural design of RC buildingsin seismic areas or usually found in the columns of existingbuildings
7 Conclusions
A dimensionless formulation has been proposed which pro-vides the flexural response of classes of sections of RC col-umns having the same values of the geometric and mechani-cal parameters defined in this study
At each loading stage the resisting components of the sec-tion (cover and concrete core and principal and secondarylongitudinal reinforcements) give a contribution to the equi-librium that can be expressed by analytical functions depend-ing on the normalized neutral axis depth and the curvature ofthe section
Special equilibrium conditions like the ones correspond-ing to the first yielding of the principal reinforcement intension and to the ultimate state imply an analytical link bet-ween the neutral axis depth and the curvature so that the sumof the aforementioned contributions becomes an analyticalfunction of a single variable which can express a strength orcurvature domain
For a generic equilibrium condition an iterative proce-dure to determine the neutral axis depth corresponding to
12 Advances in Civil Engineering
100
80
60
40
20
00 300 600 900 1200 1500
Mom
ent (
kNm
)
Axial load (kN)
Mu
My
My without cover
(a)
8
6
4
2
00 300 600 900 1200 1500
Axial load (kN)
120593u120593y
Curv
atur
e(1
mm
times10
minus4)
(b)
Figure 10119872-119873 and 120593-119873 domains at the ultimate state and the first yielding of reinforcement
the assigned values of curvature and compression level isrequired but each iteration step implies the simple use of thesame equilibrium equation for an updated value of the nor-malized neutral axis dept
The dimensionless form of the formulation proposedmakes also it easily possible to evaluate the influence of theparameters characterizing the class of sections examined onthe strength and curvature corresponding to a given loadingstage
Acknowledgment
Thisworkwas carried outwithin the 2010ndash2013Research Pro-ject ldquoDPC-ReLUIS (Dipartimento Protezione Civile-Rete deiLaboratori Universitari di Ingegneria Sismica)rdquo AT 1 Task112 The related financial support was greatly appreciated
References
[1] A Fafitis ldquoInteraction surfaces of reinforced-concrete sectionsin biaxial bendingrdquo Journal of Structural Engineering vol 127no 7 pp 840ndash846 2001
[2] J L Bonet P F Miguel M A Fernandez and M L RomeroldquoAnalytical approach to failure surfaces in reinforced concretesections subjected to axial loads and biaxial bendingrdquo Journalof Structural Engineering vol 130 no 12 pp 2006ndash2015 2004
[3] G Monti and S Alessandri ldquoAssessment of rc columns undercombined biaxial bending and axial loadrdquo in Proceedings of the2nd FIB Congress Naples Italy 2006
[4] RW Clough and S B Johnston ldquoEffect of stiffness degradationon earthquake ductility requirementsrdquo in Proceedings of 2ndJapan Earthquake Engineering Symposium Tokyo Japan 1966
[5] T Takeda M A Sozen and N N Nielsen ldquoReinforced con-crete response to simulated earthquakerdquo Journal of StructuralDivision vol 96 no 12 pp 2257ndash2273 1970
[6] M S L Roufaiel and C Meyer ldquoAnalytical modeling of hyster-etic behavior of reinforced concrete framerdquo Journal of StructuralEngineering vol 113 no 3 pp 429ndash444 1987
[7] A R Mari and A C Scordelis ldquoNonlinear geometric materialand time dependent analysis of three dimensional reinforcedand prestressed concrete framesrdquo USBSESM Report 8412Department of Civil Engineering University of CaliforniaBerkeley Calif USA 1973
[8] T Taucer E Spacone and F C Filippou ldquoA fiber beam-col-umn element for seismic response analysis of reinforced con-crete structuresrdquo Report EERC 91-17 Earthquake EngineeringResearch Center Berkeley Calif USA 1991
[9] Z Zhu I Ahmad and A Mirmiran ldquoFiber element modelingfor seismic performance of bridge columns made of concrete-filled FRP tubesrdquo Engineering Structures vol 28 no 14 pp2023ndash2035 2006
[10] E O Pfrang C P Siess and M A Sozen ldquoLoad-moment-cur-vature characteristics of RC cross-sectionsrdquoACI Journal vol 61no 7 pp 763ndash778 1964
[11] D J Carreira and K-H Chu ldquoThemoment-curvature relation-ship of RC membersrdquo ACI Journal vol 83 no 2 pp 191ndash1981986
[12] R H Wood ldquoSome controversial and curious developments inplastic theory of structuresrdquo inEngineering Plasticity J Heymanand F A Leckie Eds pp 665ndash691 CambridgeUniversity PressCambridge UK 1968
[13] Y LMo ldquoInvestigation of reinforced concrete frame behaviourtheory and testsrdquo Magazine of Concrete Research vol 44 no160 pp 163ndash173 1992
[14] M Jirasek and Z P Bazant Inelastic Analysis of Structures JonWiley amp Sons London UK 2002
[15] S Chandrasekaran L Nunziante G Serino and F CarannanteldquoCurvature ductility of RC sections based on Eurocode analyt-ical procedurerdquo KSCE Journal of Civil Engineering vol 15 no 1pp 131ndash144 2011
Advances in Civil Engineering 13
[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012
[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013
[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992
[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995
[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006
[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010
[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012
[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951
[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982
[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002
[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989
[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008
On the basis of what was said above the effective contributionof the concrete section in compression in a generic loadingstate is expressed by
119899119888= 119899cc + 119899un1 minus 119899un2 (48)
119898119888= 119898cc + 119898un1 minus 119898un2 (49)
If the confinement effect is negligible (6) gives 1198961= 1198962= 1
By comparing (40) (41) and (42) with (47) (45) and (46)respectively it can easily be observed that 119899cc = 119899un2 and119898cc = 119898un2 Therefore in this case the contributions of thewhole section of reacting concrete given by (48) and (49) arereduced to the values 119899cc = 119899un1 and 119898cc = 119898un1 and areexpressed by (45) and (46) for 119896
1= 1198962= 1
5 Analysis Procedure
The flexural response of a class of RC sections characterizedby assigned values of the geometrical andmechanical param-eters defined above is obtained by imposing the equilibriumcondition expressed by (15) and (16)
For each configuration the contributions to the equi-librium provided by the principal reinforcement ((23) and(24) under the conditions expressed by (26)) the secondaryreinforcement ((28) (35) and (36) under the conditionsexpressed by (37)) and the compressed region of concrete((48) and (49) considering (41)ndash(47)) only depend on thevariables 119909
119888and 120593 which are independent of or related to one
another according to the type of analysis required (see below)In all cases it must be assumed that the proposed formulationis applied for 120593 gt 0
51 Field of Application The field of validity of the procedureproposed here is limited by the fact that the constitutive law ofthe steel reinforcement does not include unloading branchesstarting from a point of the post yielding branches in com-pression or in tension Therefore for any loading step theupper principal reinforcement and the upper fibre of the dis-tributed secondary reinforcement both potentially subject tocompression cannot be in the postyielding field in tensionthe lower principal reinforcement and the lower fibre of thedistributed secondary reinforcement both potentially subjectto tension cannot be in the postyielding field in compression
Because of these assumptions the field of values of 119909119888
consistent with a given value of 120593 is limited by the initial andend values provided by (26) or (37)
More precisely in the absence of secondary reinforce-ment (120596
2= 0) (26) gives
119909119888min = 120582 minus
120577119904119888
120593 119909
119888max = (1 minus 120582) +120577119904119888
120593 (50)
while if the secondary reinforcement is present (37) leads tomore restrictive limitations
119909119888min =
1 minus 120572
2minus
120577119904119888
120593 119909
119888max =1 + 120572
2+
120577119904119888
120593 (51)
These limitations also occur in a classical numericalapproach based on the strip decomposition of the section
However it must be observed that (50) or (51) allows oneto construct the moment-curvature curve for a large field ofvalues of assigned level of compression including the moreusual cases occurring in practical applications
52 Moment-Curvature Curves The more usual applicationof the analytical expressions derived in the previous sectionsis to derive by a step-by-step procedure the moment-curva-ture119898-120593 curve of a class of RC sections for a given value of 119899
To this purpose the procedure is applied by the followingsteps
(i) assigning the lowest value of curvature 120593 that onewants to consider increasing values of 119909
119888are consid-
ered from the minimum to the maximum expressedby (50) or (51)
(ii) for each value of the couple 120593 119909119888the corresponding
value of 1198991199041 1198991199042 and 119899
119888are calculated and their sum
increasing for each increase in 119909119888
(iii) when the assigned value of 119899 is reached with anacceptable tolerance (ie (15) is verified with verygood approximation) the assigned values of 120593 and 119899and the value of 119909
119888that was found make it possible
to calculate the corresponding values of11989811990411198981199042 and
119898119888expressed by (24) (36) and (49) and the sum
of these contributions of moment is the ordinate ofthe point having the abscissa 120593 belonging to the119898-120593curve sought
(iv) considering an increased value of 120593 the procedure isrepeated to obtain a subsequent point of themoment-curvature curve
The procedure stops when the maximum available ulti-mate confined concrete strain or the maximum available ten-sile strain of the lower reinforcement is exceeded These twooccurrences imply respectively that 120593119909
119888gt 120583cc or 120593(119909119888 minus 1 +
120582) lt minus120577119904119888120583119903 where 120583cc = 120576ccu120576cco is the available ductility of
the confined concrete defined in Section 31 and 120583119903is the steel
ductility defined in (11)If the stress-strain law of the steel reinforcement also
exhibits a softening postyielding branch corresponding to apostbuckling behaviour up to the strain value 120576
∙
119904(Figure 3)
a further limit at which the procedure must be stoppedderives from the condition that 120593(119909
119888minus 120582) gt 120577
119904119888120583119903119888 in
which 120583119903119888
= 120576lowast
119904120576119904119910 Beyond this condition the residual stress
of the compressed reinforcement should be assumed to beconstant and equal to 02119891
119904119910 but this branch of the stress-
strain law is not considered in the proposed model as statedin Section 32
Advances in Civil Engineering 9
53 Strength and Curvature Domains A typical 119898-120593 curveexhibits two characteristic points corresponding to theachievement of the ultimate state and to the first yield-ing of the principal reinforcement in tension respectivelyObviously the first point is present in all cases while thesecond can only be reached if yielding of the reinforcementin tension occurs before the brittle collapse of the concretecore due to low available concrete ductility orand a very highcompression level
The analytical expressions shown in the previous sectionsmake it possible to determine directly the coordinates of thesepoints and to derive in closed form the119898
119906-119899 120593119906-119899 and119898
119910-119899
120593119910-119899 domains which give useful information on the flexural
strength and curvature ductility of an RC section subjected toan assigned compression level
The constructions of the aforementioned curves can bemade more easily than by using a classical discretized modelof the section because (15) and (16) in a suitable specializedform provide analytical expressions where the axial com-pressive load and the corresponding bending moment onlydepend on a single variable 119909
119888and 120593 being related to one
anotherIn order to construct the 119898
119906-119899 and 120593
119906-119899 curves two dif-
ferent relationships linking the neutral axis depth to the cur-vature must be considered according to whether the collapseof the section is produced by excess of the available strain ofthe reinforcement in tension or of the available strain of theconcrete core in compression Moreover to distinguish thesetwo different events it must be observed that the maximumcurvature of the section is reachedwhen these two conditionsoccur simultaneously Considering Figure 4 it can easily bededuced that this value of curvature in normalized form is
120593119906max =
120583cc + 120577119904119888120583119903
1 minus 120582 (52)
The first branch of the domains sought is determined byimposing the condition that the collapse of the section is dueto the achievement of themaximum tensile strain of the prin-cipal reinforcement in tension (low level of compression)Therefore the following condition has to be assumed
120593119906(119909119888minus 1 + 120582) = minus 120577
119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus
120577119904119888120583119903
120593119906
(53)
Considering increasing values of the curvature from themin-imum value to themaximum expressed by (52) the second of(53) provide the corresponding value of 119909
119888 and (15) and (16)
by making explicit the contributions of the concrete and theprincipal and secondary reinforcement provide the corre-sponding values of 119899 and119898
119906
Once 120593119906max is reached the second branch of the 119898
119906-119899
120593119906-119899 curves must be determined considering that for further
decreasing values of 120593119906 the collapse is produced by the
achievement of the maximum compressive strain of the con-fined concrete Therefore the new relationship linking thecurvature to the neutral axis depth is
By using the second of these equations (15) and (16) againlink 119899 and119898
119906to the only variable 120593
119906
It must be observed that if the principal reinforcement incompression is subject to buckling after the achievement ofthe yielding stress the ultimate strain of this compressed rein-forcement is the strain denoted as 120576∙
119904in Figure 3 This strain
value can be reached before the concrete core reaches thenormalized value of strain 120583cc In this case the maximumvalue of normalized dimensionless curvature expressed by(52) must be substituted by the value 120593
119906max = 120577119904119888(120583119903119888
+
120583119903)(1minus2120582) where 120583
119903119888= 120576lowast
119904120576119904119910 and the second branch of the
ultimate strength domains must be determined by assumingthat for values of 120593
119906that decrease from this maximum the
relationship linking the neutral axis dept to the curvature is119909119888= 120582 + 120577
119904119888120583119903119888120593
The 119898119910-119899 120593119910-119899 curves can be simply determined by
assuming that whatever the value of the curvature is the prin-cipal reinforcement in tension is affected by the strain valueequal to minus120576
119904119910 This condition in dimensionless terms gives
120593119910(119909119888minus 1 + 120582) = minus 120577
119904119888997904rArr 119909119888= (1 minus 120582) minus
120577119904119888
120593119910
(55)
By introducing the second of (55) into the equilibrium Equa-tions (15) and (16) they again become two functions of thesingle variable 120593 = 120593
119910 The 119898
119910-119899 120593119910-119899 curves can be con-
structed starting froma value thatwas assumed to be themin-imum up to the maximum value corresponding to one of thepossible aforementioned occurrences the collapse of the con-crete core due to achievement of the available ductility of thematerial or the achievement of the strain 120576
∙
119904in the principal
reinforcement in compression if this reinforcement is subjectto buckling
Considering the field of validity of the procedure pro-posed the minimum value of 120593
119906that can be considered to
construct the119898119906-119899 120593119906-119899 domains is obtained by introducing
the second of (53) into the first of (50) or (51) so that
120593119906min = 120577
119904119888
120583119903minus 1
1 minus 2120582 (56)
in the absence of the secondary reinforcement and
120593119906min = 120577
119904119888
120583119903minus 1
(1 + 120572) 2 minus 120582 (57)
if the secondary reinforcement is also presentObviously the 119898
119910-119899 120593119910-119899 domains can be constructed
starting from any value 120593119910gt 0
In relation to the field of applications (RC columns) thefirst values of 120593
119906and 120593
119910that are to be actually considered are
the ones first producing 119899 ge 0
6 Applications
The reliability of the procedure proposed is shown here byusing the results derived from two RC sections considered byother researchers
Thefirst application refers to one of the sections examinedby Zahn et al [27] This was a square RC section of side119861 = 400mm and realized with concrete having cylindrical
10 Advances in Civil EngineeringN
orm
aliz
ed st
ress
1
00 1 2 3 4 5
Saatcioglu and Razvi (1992)Mander et al (1988)
Normalized strain
Figure 7 Calibration of constitutive law of confined concrete
strength 119891co = 30Nmm2 The longitudinal reinforcementconsisted of 12 steel bars of diameter 16mm having yieldingstrength 119891
119904119910= 420Nmm2 uniformly distributed along the
perimeter of the concrete core (constant pitch) The coverdepth measured from the external perimeter of the stirrupswas 13mm The transverse reinforcement consisting of steelstirrups of diameter equal to 10mm according to the calcu-lations made by the aforementioned authors provided theeffective confinement pressure 119891
ℓ119890= 011119891
1015840
co where 1198911015840
co =
085119891co is the reduced strength value prescribed by the ACI318 code [28]
The flexural response was numerically derived in [27] byusing a classical fibre model in which the stress-strain law ofthe confined concrete was assumed in agreement with Man-derrsquosmodel [25]Therefore in this application the parameters120573cc and 120578cc governing the shape of the laws expressed by (4)and (5) are calibrated so that they lead to a constitutive lawof confined concrete which is very similar to that used by theaforementioned authors
Manderrsquos model [25] for the given value of the confine-ment pressure is governed by the parameter 119903 = 13 themodel of Saatcioglu and Razvi [18] leads to a very similarshape of the stress-strain curve by assuming that 120573cc = 045
and 120578cc = minus006 Figure 7 shows the very similar shapes ofthe constitutive laws corresponding to the two models con-sidered
The procedure proposed in [25] provided the followingcharacteristic values of confined concrete strength and cor-responding strain 120576cco = 00073 119891cco = 46Nmm2 Thesevalues are also adopted for the dimensional stress-strain lawexpressed by the model used here [18]
Since for the stress-strain law of the unconfined concreteZahn et al [27] do not give any information the default valuesindicated in Section 31 are assumed here 120573
119888= 1 120578
119888= minus08
and 120576co = 0002On the basis of the above data the square cross-section of
the concrete coremeasured inside the perimeter stirrups hasa side of 119887 = ℎ = 354mm the cover depth is equal to
400
300
200
100
00 001 002 003 004 005
Mom
ent (
kNm
)
Curvature (1m)
Zahn et al (1989)Present model
Figure 8 Comparison of results119872-120593 curve for given119873
23mm the number of intermediate reinforcing bars realizingthe secondary reinforcement is 119899
119887= 119899ℎ= 2 and the distance
of the barycentre of the section of the principal reinforcementfrom the external side of the concrete core section is ℎ
1=
162 = 8mmThe principal reinforcement consists of four upper and
four lower 16mm steel bars while the secondary reinforce-ment consists of four bars (two on the right and two on theleft) of the same diameter The steel reinforcement constitu-tive law is defined by assuming 119864
119904= 210GPa 120576
119904119910= 119891119904119910119864119904=
0002) and 119864119904ℎ
= 119864119904119888
= 0Therefore the dimensionless parameters involved in the
formulation proposed take on the following values 120575V = 120575119900=
= 0Figure 8 compares the 119872-120593 curve obtained in [27] with
that derived from the procedure shown in Section 52 for acompressive constant axial load 119873 = 03119891co(119861 times 119861) corre-sponding to a compressive level 119899 = 119873(119887
2
times 119891cco) = 025 inagreement with the first of (14)The results in the figure showa very good level of agreement between the numerical pro-cedure adopted in [27] and the analytical one adopted here
Both curves in the figure clearly show the cusp producedby the yielding of the principal reinforcement in tension andthe effects of the progressive cover spalling
The second application refers to the middle-height sec-tion of an RC column of height 1640mm experimentallytested by Saatcioglu et al [19] marked as specimen C6-2 bythe authors
The loading condition was realized by imposing relativeaxial displacements so that the reactive compressive forceacted with fixed eccentricity in a plane of principal inertiaof the sections Suitable devices were applied at the columnends so that the column itself behaved like a hinged verticalRC member under an eccentric compressive load
As a consequence each point of the moment-curvaturecurve characterizing the experimental response obtained by
Advances in Civil Engineering 11
the aforementioned authors corresponds to a different valueof axial loadMoreover the actual bendingmoment acting onthe middle-height section was influenced by the 119875-Δ effect
The formulation proposed here is validated consideringsome points of the aforementioned curve for which theauthors indicate the values of the axial load119873 and curvaturethat were measured These values of 119873 are normalized withrespect to the axial load 119873
119900expressed by (2) The procedure
is applied by determining the value of 119909119888from (15) for the
assigned values of 119899 and 120593 and by calculating 119898 from (16)The results are compared with the numerical values obtainedby the strip model adopted by the authors which includedthe 119875-Δ effect and the values experimentally detected
The specimen considered had a square section of side 119861 =
210mm cover depth of 125mm and longitudinal reinforcingbars of diameter 113mm uniformly distributed along theperimeter of the concrete core having yielding strength119891
119904119910=
517Nmm2 and average hardening modulus 119864119904ℎ
= 0013119864119904
The transverse steel reinforcement consisted of square andoctagonal 63mm stirrups with pitch 50mm
The unconfined concrete had cylindrical strength 119891co =
3440Nmm2The numerical model adopted in [19] assumedthat the perimeter of the concrete core was coincident withthe axis lines of the external stirrups
On the basis of these geometrical and mechanical dataand by evaluating the effective confinement pressure by theprocedure proposed in [18] for the confined concrete sectionone obtains 119887 = ℎ = 17870mm 119891cco = 4635 Nmm2 120576cco =
000547 120578cc = minus0052 and 120573 = 0590The further parametersinvolved in the formulation proposed take on the followingvalues 120575
= 0013Figure 9 shows that the procedure proposed provides
results that are very close to those derived by the numericalmodel adopted in [19] The experimentally detected values ofmoment prove to be underestimated by both models
It must be observed that the last point detected by usingthe present formulation corresponding to the ultimate stateof the section had been already found in [17] because thissection belongs to the more restricted class of sections con-sidered in that work
Figure 10 shows the strength and curvature domainsobtained by using the procedure described in Section 53For confined concrete and steel reinforcement the followingvalues were assumed deduced from the data shown in [19]120576ccu = 0032 (120583cc = 585) and 120576
119904119906= 0066 (120583
119903= 27) The
results confirm that a good level of confinementmakes it pos-sible to achieve acceptable ductility of curvature even underhigh levels of compression In order to show how the proce-dure proposed is easily able to evaluate the influence of all thegeometric and mechanical parameters governing the flexuralresponse of a class of RC sections Figure 10(a) also shows theinfluence of the unconfined concrete cover on the bendingmoment at the first yielding of the principal reinforcement intension As expected this influence proves to be negligible forhigh values of the compression level
The dotted curve in Figure 10(b) obviously stops at thepoint of intersection with the continuous curve Beyond this
100
80
60
40
20
00 1 2 3 4
Mom
ent (
kNm
)
ExperimentalModelPresent model
Saatcioglu et al (1995)
Curvature (1mm times 10minus4)
Figure 9 Comparison of theoretical and experimental results
point the 120593119906-119873 curve proceeds by a very brief stretch (not
very evident in the figure) that corresponds to brittle collapseof the section due to the achievement of the maximum avail-able compressive strain in the confined concrete when theprincipal reinforcement in tension is still in the elastic fieldThe end point of this curve corresponds to the achievementof the maximum neutral axis depth defined by the second of(51)
Nevertheless the axial load values that cannot be consid-ered because of these limitations are well beyond the onesthat can be assumed in the structural design of RC buildingsin seismic areas or usually found in the columns of existingbuildings
7 Conclusions
A dimensionless formulation has been proposed which pro-vides the flexural response of classes of sections of RC col-umns having the same values of the geometric and mechani-cal parameters defined in this study
At each loading stage the resisting components of the sec-tion (cover and concrete core and principal and secondarylongitudinal reinforcements) give a contribution to the equi-librium that can be expressed by analytical functions depend-ing on the normalized neutral axis depth and the curvature ofthe section
Special equilibrium conditions like the ones correspond-ing to the first yielding of the principal reinforcement intension and to the ultimate state imply an analytical link bet-ween the neutral axis depth and the curvature so that the sumof the aforementioned contributions becomes an analyticalfunction of a single variable which can express a strength orcurvature domain
For a generic equilibrium condition an iterative proce-dure to determine the neutral axis depth corresponding to
12 Advances in Civil Engineering
100
80
60
40
20
00 300 600 900 1200 1500
Mom
ent (
kNm
)
Axial load (kN)
Mu
My
My without cover
(a)
8
6
4
2
00 300 600 900 1200 1500
Axial load (kN)
120593u120593y
Curv
atur
e(1
mm
times10
minus4)
(b)
Figure 10119872-119873 and 120593-119873 domains at the ultimate state and the first yielding of reinforcement
the assigned values of curvature and compression level isrequired but each iteration step implies the simple use of thesame equilibrium equation for an updated value of the nor-malized neutral axis dept
The dimensionless form of the formulation proposedmakes also it easily possible to evaluate the influence of theparameters characterizing the class of sections examined onthe strength and curvature corresponding to a given loadingstage
Acknowledgment
Thisworkwas carried outwithin the 2010ndash2013Research Pro-ject ldquoDPC-ReLUIS (Dipartimento Protezione Civile-Rete deiLaboratori Universitari di Ingegneria Sismica)rdquo AT 1 Task112 The related financial support was greatly appreciated
References
[1] A Fafitis ldquoInteraction surfaces of reinforced-concrete sectionsin biaxial bendingrdquo Journal of Structural Engineering vol 127no 7 pp 840ndash846 2001
[2] J L Bonet P F Miguel M A Fernandez and M L RomeroldquoAnalytical approach to failure surfaces in reinforced concretesections subjected to axial loads and biaxial bendingrdquo Journalof Structural Engineering vol 130 no 12 pp 2006ndash2015 2004
[3] G Monti and S Alessandri ldquoAssessment of rc columns undercombined biaxial bending and axial loadrdquo in Proceedings of the2nd FIB Congress Naples Italy 2006
[4] RW Clough and S B Johnston ldquoEffect of stiffness degradationon earthquake ductility requirementsrdquo in Proceedings of 2ndJapan Earthquake Engineering Symposium Tokyo Japan 1966
[5] T Takeda M A Sozen and N N Nielsen ldquoReinforced con-crete response to simulated earthquakerdquo Journal of StructuralDivision vol 96 no 12 pp 2257ndash2273 1970
[6] M S L Roufaiel and C Meyer ldquoAnalytical modeling of hyster-etic behavior of reinforced concrete framerdquo Journal of StructuralEngineering vol 113 no 3 pp 429ndash444 1987
[7] A R Mari and A C Scordelis ldquoNonlinear geometric materialand time dependent analysis of three dimensional reinforcedand prestressed concrete framesrdquo USBSESM Report 8412Department of Civil Engineering University of CaliforniaBerkeley Calif USA 1973
[8] T Taucer E Spacone and F C Filippou ldquoA fiber beam-col-umn element for seismic response analysis of reinforced con-crete structuresrdquo Report EERC 91-17 Earthquake EngineeringResearch Center Berkeley Calif USA 1991
[9] Z Zhu I Ahmad and A Mirmiran ldquoFiber element modelingfor seismic performance of bridge columns made of concrete-filled FRP tubesrdquo Engineering Structures vol 28 no 14 pp2023ndash2035 2006
[10] E O Pfrang C P Siess and M A Sozen ldquoLoad-moment-cur-vature characteristics of RC cross-sectionsrdquoACI Journal vol 61no 7 pp 763ndash778 1964
[11] D J Carreira and K-H Chu ldquoThemoment-curvature relation-ship of RC membersrdquo ACI Journal vol 83 no 2 pp 191ndash1981986
[12] R H Wood ldquoSome controversial and curious developments inplastic theory of structuresrdquo inEngineering Plasticity J Heymanand F A Leckie Eds pp 665ndash691 CambridgeUniversity PressCambridge UK 1968
[13] Y LMo ldquoInvestigation of reinforced concrete frame behaviourtheory and testsrdquo Magazine of Concrete Research vol 44 no160 pp 163ndash173 1992
[14] M Jirasek and Z P Bazant Inelastic Analysis of Structures JonWiley amp Sons London UK 2002
[15] S Chandrasekaran L Nunziante G Serino and F CarannanteldquoCurvature ductility of RC sections based on Eurocode analyt-ical procedurerdquo KSCE Journal of Civil Engineering vol 15 no 1pp 131ndash144 2011
Advances in Civil Engineering 13
[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012
[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013
[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992
[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995
[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006
[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010
[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012
[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951
[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982
[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002
[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989
[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008
53 Strength and Curvature Domains A typical 119898-120593 curveexhibits two characteristic points corresponding to theachievement of the ultimate state and to the first yield-ing of the principal reinforcement in tension respectivelyObviously the first point is present in all cases while thesecond can only be reached if yielding of the reinforcementin tension occurs before the brittle collapse of the concretecore due to low available concrete ductility orand a very highcompression level
The analytical expressions shown in the previous sectionsmake it possible to determine directly the coordinates of thesepoints and to derive in closed form the119898
119906-119899 120593119906-119899 and119898
119910-119899
120593119910-119899 domains which give useful information on the flexural
strength and curvature ductility of an RC section subjected toan assigned compression level
The constructions of the aforementioned curves can bemade more easily than by using a classical discretized modelof the section because (15) and (16) in a suitable specializedform provide analytical expressions where the axial com-pressive load and the corresponding bending moment onlydepend on a single variable 119909
119888and 120593 being related to one
anotherIn order to construct the 119898
119906-119899 and 120593
119906-119899 curves two dif-
ferent relationships linking the neutral axis depth to the cur-vature must be considered according to whether the collapseof the section is produced by excess of the available strain ofthe reinforcement in tension or of the available strain of theconcrete core in compression Moreover to distinguish thesetwo different events it must be observed that the maximumcurvature of the section is reachedwhen these two conditionsoccur simultaneously Considering Figure 4 it can easily bededuced that this value of curvature in normalized form is
120593119906max =
120583cc + 120577119904119888120583119903
1 minus 120582 (52)
The first branch of the domains sought is determined byimposing the condition that the collapse of the section is dueto the achievement of themaximum tensile strain of the prin-cipal reinforcement in tension (low level of compression)Therefore the following condition has to be assumed
120593119906(119909119888minus 1 + 120582) = minus 120577
119904119888120583119903997904rArr 119909119888= (1 minus 120582) minus
120577119904119888120583119903
120593119906
(53)
Considering increasing values of the curvature from themin-imum value to themaximum expressed by (52) the second of(53) provide the corresponding value of 119909
119888 and (15) and (16)
by making explicit the contributions of the concrete and theprincipal and secondary reinforcement provide the corre-sponding values of 119899 and119898
119906
Once 120593119906max is reached the second branch of the 119898
119906-119899
120593119906-119899 curves must be determined considering that for further
decreasing values of 120593119906 the collapse is produced by the
achievement of the maximum compressive strain of the con-fined concrete Therefore the new relationship linking thecurvature to the neutral axis depth is
By using the second of these equations (15) and (16) againlink 119899 and119898
119906to the only variable 120593
119906
It must be observed that if the principal reinforcement incompression is subject to buckling after the achievement ofthe yielding stress the ultimate strain of this compressed rein-forcement is the strain denoted as 120576∙
119904in Figure 3 This strain
value can be reached before the concrete core reaches thenormalized value of strain 120583cc In this case the maximumvalue of normalized dimensionless curvature expressed by(52) must be substituted by the value 120593
119906max = 120577119904119888(120583119903119888
+
120583119903)(1minus2120582) where 120583
119903119888= 120576lowast
119904120576119904119910 and the second branch of the
ultimate strength domains must be determined by assumingthat for values of 120593
119906that decrease from this maximum the
relationship linking the neutral axis dept to the curvature is119909119888= 120582 + 120577
119904119888120583119903119888120593
The 119898119910-119899 120593119910-119899 curves can be simply determined by
assuming that whatever the value of the curvature is the prin-cipal reinforcement in tension is affected by the strain valueequal to minus120576
119904119910 This condition in dimensionless terms gives
120593119910(119909119888minus 1 + 120582) = minus 120577
119904119888997904rArr 119909119888= (1 minus 120582) minus
120577119904119888
120593119910
(55)
By introducing the second of (55) into the equilibrium Equa-tions (15) and (16) they again become two functions of thesingle variable 120593 = 120593
119910 The 119898
119910-119899 120593119910-119899 curves can be con-
structed starting froma value thatwas assumed to be themin-imum up to the maximum value corresponding to one of thepossible aforementioned occurrences the collapse of the con-crete core due to achievement of the available ductility of thematerial or the achievement of the strain 120576
∙
119904in the principal
reinforcement in compression if this reinforcement is subjectto buckling
Considering the field of validity of the procedure pro-posed the minimum value of 120593
119906that can be considered to
construct the119898119906-119899 120593119906-119899 domains is obtained by introducing
the second of (53) into the first of (50) or (51) so that
120593119906min = 120577
119904119888
120583119903minus 1
1 minus 2120582 (56)
in the absence of the secondary reinforcement and
120593119906min = 120577
119904119888
120583119903minus 1
(1 + 120572) 2 minus 120582 (57)
if the secondary reinforcement is also presentObviously the 119898
119910-119899 120593119910-119899 domains can be constructed
starting from any value 120593119910gt 0
In relation to the field of applications (RC columns) thefirst values of 120593
119906and 120593
119910that are to be actually considered are
the ones first producing 119899 ge 0
6 Applications
The reliability of the procedure proposed is shown here byusing the results derived from two RC sections considered byother researchers
Thefirst application refers to one of the sections examinedby Zahn et al [27] This was a square RC section of side119861 = 400mm and realized with concrete having cylindrical
10 Advances in Civil EngineeringN
orm
aliz
ed st
ress
1
00 1 2 3 4 5
Saatcioglu and Razvi (1992)Mander et al (1988)
Normalized strain
Figure 7 Calibration of constitutive law of confined concrete
strength 119891co = 30Nmm2 The longitudinal reinforcementconsisted of 12 steel bars of diameter 16mm having yieldingstrength 119891
119904119910= 420Nmm2 uniformly distributed along the
perimeter of the concrete core (constant pitch) The coverdepth measured from the external perimeter of the stirrupswas 13mm The transverse reinforcement consisting of steelstirrups of diameter equal to 10mm according to the calcu-lations made by the aforementioned authors provided theeffective confinement pressure 119891
ℓ119890= 011119891
1015840
co where 1198911015840
co =
085119891co is the reduced strength value prescribed by the ACI318 code [28]
The flexural response was numerically derived in [27] byusing a classical fibre model in which the stress-strain law ofthe confined concrete was assumed in agreement with Man-derrsquosmodel [25]Therefore in this application the parameters120573cc and 120578cc governing the shape of the laws expressed by (4)and (5) are calibrated so that they lead to a constitutive lawof confined concrete which is very similar to that used by theaforementioned authors
Manderrsquos model [25] for the given value of the confine-ment pressure is governed by the parameter 119903 = 13 themodel of Saatcioglu and Razvi [18] leads to a very similarshape of the stress-strain curve by assuming that 120573cc = 045
and 120578cc = minus006 Figure 7 shows the very similar shapes ofthe constitutive laws corresponding to the two models con-sidered
The procedure proposed in [25] provided the followingcharacteristic values of confined concrete strength and cor-responding strain 120576cco = 00073 119891cco = 46Nmm2 Thesevalues are also adopted for the dimensional stress-strain lawexpressed by the model used here [18]
Since for the stress-strain law of the unconfined concreteZahn et al [27] do not give any information the default valuesindicated in Section 31 are assumed here 120573
119888= 1 120578
119888= minus08
and 120576co = 0002On the basis of the above data the square cross-section of
the concrete coremeasured inside the perimeter stirrups hasa side of 119887 = ℎ = 354mm the cover depth is equal to
400
300
200
100
00 001 002 003 004 005
Mom
ent (
kNm
)
Curvature (1m)
Zahn et al (1989)Present model
Figure 8 Comparison of results119872-120593 curve for given119873
23mm the number of intermediate reinforcing bars realizingthe secondary reinforcement is 119899
119887= 119899ℎ= 2 and the distance
of the barycentre of the section of the principal reinforcementfrom the external side of the concrete core section is ℎ
1=
162 = 8mmThe principal reinforcement consists of four upper and
four lower 16mm steel bars while the secondary reinforce-ment consists of four bars (two on the right and two on theleft) of the same diameter The steel reinforcement constitu-tive law is defined by assuming 119864
119904= 210GPa 120576
119904119910= 119891119904119910119864119904=
0002) and 119864119904ℎ
= 119864119904119888
= 0Therefore the dimensionless parameters involved in the
formulation proposed take on the following values 120575V = 120575119900=
= 0Figure 8 compares the 119872-120593 curve obtained in [27] with
that derived from the procedure shown in Section 52 for acompressive constant axial load 119873 = 03119891co(119861 times 119861) corre-sponding to a compressive level 119899 = 119873(119887
2
times 119891cco) = 025 inagreement with the first of (14)The results in the figure showa very good level of agreement between the numerical pro-cedure adopted in [27] and the analytical one adopted here
Both curves in the figure clearly show the cusp producedby the yielding of the principal reinforcement in tension andthe effects of the progressive cover spalling
The second application refers to the middle-height sec-tion of an RC column of height 1640mm experimentallytested by Saatcioglu et al [19] marked as specimen C6-2 bythe authors
The loading condition was realized by imposing relativeaxial displacements so that the reactive compressive forceacted with fixed eccentricity in a plane of principal inertiaof the sections Suitable devices were applied at the columnends so that the column itself behaved like a hinged verticalRC member under an eccentric compressive load
As a consequence each point of the moment-curvaturecurve characterizing the experimental response obtained by
Advances in Civil Engineering 11
the aforementioned authors corresponds to a different valueof axial loadMoreover the actual bendingmoment acting onthe middle-height section was influenced by the 119875-Δ effect
The formulation proposed here is validated consideringsome points of the aforementioned curve for which theauthors indicate the values of the axial load119873 and curvaturethat were measured These values of 119873 are normalized withrespect to the axial load 119873
119900expressed by (2) The procedure
is applied by determining the value of 119909119888from (15) for the
assigned values of 119899 and 120593 and by calculating 119898 from (16)The results are compared with the numerical values obtainedby the strip model adopted by the authors which includedthe 119875-Δ effect and the values experimentally detected
The specimen considered had a square section of side 119861 =
210mm cover depth of 125mm and longitudinal reinforcingbars of diameter 113mm uniformly distributed along theperimeter of the concrete core having yielding strength119891
119904119910=
517Nmm2 and average hardening modulus 119864119904ℎ
= 0013119864119904
The transverse steel reinforcement consisted of square andoctagonal 63mm stirrups with pitch 50mm
The unconfined concrete had cylindrical strength 119891co =
3440Nmm2The numerical model adopted in [19] assumedthat the perimeter of the concrete core was coincident withthe axis lines of the external stirrups
On the basis of these geometrical and mechanical dataand by evaluating the effective confinement pressure by theprocedure proposed in [18] for the confined concrete sectionone obtains 119887 = ℎ = 17870mm 119891cco = 4635 Nmm2 120576cco =
000547 120578cc = minus0052 and 120573 = 0590The further parametersinvolved in the formulation proposed take on the followingvalues 120575
= 0013Figure 9 shows that the procedure proposed provides
results that are very close to those derived by the numericalmodel adopted in [19] The experimentally detected values ofmoment prove to be underestimated by both models
It must be observed that the last point detected by usingthe present formulation corresponding to the ultimate stateof the section had been already found in [17] because thissection belongs to the more restricted class of sections con-sidered in that work
Figure 10 shows the strength and curvature domainsobtained by using the procedure described in Section 53For confined concrete and steel reinforcement the followingvalues were assumed deduced from the data shown in [19]120576ccu = 0032 (120583cc = 585) and 120576
119904119906= 0066 (120583
119903= 27) The
results confirm that a good level of confinementmakes it pos-sible to achieve acceptable ductility of curvature even underhigh levels of compression In order to show how the proce-dure proposed is easily able to evaluate the influence of all thegeometric and mechanical parameters governing the flexuralresponse of a class of RC sections Figure 10(a) also shows theinfluence of the unconfined concrete cover on the bendingmoment at the first yielding of the principal reinforcement intension As expected this influence proves to be negligible forhigh values of the compression level
The dotted curve in Figure 10(b) obviously stops at thepoint of intersection with the continuous curve Beyond this
100
80
60
40
20
00 1 2 3 4
Mom
ent (
kNm
)
ExperimentalModelPresent model
Saatcioglu et al (1995)
Curvature (1mm times 10minus4)
Figure 9 Comparison of theoretical and experimental results
point the 120593119906-119873 curve proceeds by a very brief stretch (not
very evident in the figure) that corresponds to brittle collapseof the section due to the achievement of the maximum avail-able compressive strain in the confined concrete when theprincipal reinforcement in tension is still in the elastic fieldThe end point of this curve corresponds to the achievementof the maximum neutral axis depth defined by the second of(51)
Nevertheless the axial load values that cannot be consid-ered because of these limitations are well beyond the onesthat can be assumed in the structural design of RC buildingsin seismic areas or usually found in the columns of existingbuildings
7 Conclusions
A dimensionless formulation has been proposed which pro-vides the flexural response of classes of sections of RC col-umns having the same values of the geometric and mechani-cal parameters defined in this study
At each loading stage the resisting components of the sec-tion (cover and concrete core and principal and secondarylongitudinal reinforcements) give a contribution to the equi-librium that can be expressed by analytical functions depend-ing on the normalized neutral axis depth and the curvature ofthe section
Special equilibrium conditions like the ones correspond-ing to the first yielding of the principal reinforcement intension and to the ultimate state imply an analytical link bet-ween the neutral axis depth and the curvature so that the sumof the aforementioned contributions becomes an analyticalfunction of a single variable which can express a strength orcurvature domain
For a generic equilibrium condition an iterative proce-dure to determine the neutral axis depth corresponding to
12 Advances in Civil Engineering
100
80
60
40
20
00 300 600 900 1200 1500
Mom
ent (
kNm
)
Axial load (kN)
Mu
My
My without cover
(a)
8
6
4
2
00 300 600 900 1200 1500
Axial load (kN)
120593u120593y
Curv
atur
e(1
mm
times10
minus4)
(b)
Figure 10119872-119873 and 120593-119873 domains at the ultimate state and the first yielding of reinforcement
the assigned values of curvature and compression level isrequired but each iteration step implies the simple use of thesame equilibrium equation for an updated value of the nor-malized neutral axis dept
The dimensionless form of the formulation proposedmakes also it easily possible to evaluate the influence of theparameters characterizing the class of sections examined onthe strength and curvature corresponding to a given loadingstage
Acknowledgment
Thisworkwas carried outwithin the 2010ndash2013Research Pro-ject ldquoDPC-ReLUIS (Dipartimento Protezione Civile-Rete deiLaboratori Universitari di Ingegneria Sismica)rdquo AT 1 Task112 The related financial support was greatly appreciated
References
[1] A Fafitis ldquoInteraction surfaces of reinforced-concrete sectionsin biaxial bendingrdquo Journal of Structural Engineering vol 127no 7 pp 840ndash846 2001
[2] J L Bonet P F Miguel M A Fernandez and M L RomeroldquoAnalytical approach to failure surfaces in reinforced concretesections subjected to axial loads and biaxial bendingrdquo Journalof Structural Engineering vol 130 no 12 pp 2006ndash2015 2004
[3] G Monti and S Alessandri ldquoAssessment of rc columns undercombined biaxial bending and axial loadrdquo in Proceedings of the2nd FIB Congress Naples Italy 2006
[4] RW Clough and S B Johnston ldquoEffect of stiffness degradationon earthquake ductility requirementsrdquo in Proceedings of 2ndJapan Earthquake Engineering Symposium Tokyo Japan 1966
[5] T Takeda M A Sozen and N N Nielsen ldquoReinforced con-crete response to simulated earthquakerdquo Journal of StructuralDivision vol 96 no 12 pp 2257ndash2273 1970
[6] M S L Roufaiel and C Meyer ldquoAnalytical modeling of hyster-etic behavior of reinforced concrete framerdquo Journal of StructuralEngineering vol 113 no 3 pp 429ndash444 1987
[7] A R Mari and A C Scordelis ldquoNonlinear geometric materialand time dependent analysis of three dimensional reinforcedand prestressed concrete framesrdquo USBSESM Report 8412Department of Civil Engineering University of CaliforniaBerkeley Calif USA 1973
[8] T Taucer E Spacone and F C Filippou ldquoA fiber beam-col-umn element for seismic response analysis of reinforced con-crete structuresrdquo Report EERC 91-17 Earthquake EngineeringResearch Center Berkeley Calif USA 1991
[9] Z Zhu I Ahmad and A Mirmiran ldquoFiber element modelingfor seismic performance of bridge columns made of concrete-filled FRP tubesrdquo Engineering Structures vol 28 no 14 pp2023ndash2035 2006
[10] E O Pfrang C P Siess and M A Sozen ldquoLoad-moment-cur-vature characteristics of RC cross-sectionsrdquoACI Journal vol 61no 7 pp 763ndash778 1964
[11] D J Carreira and K-H Chu ldquoThemoment-curvature relation-ship of RC membersrdquo ACI Journal vol 83 no 2 pp 191ndash1981986
[12] R H Wood ldquoSome controversial and curious developments inplastic theory of structuresrdquo inEngineering Plasticity J Heymanand F A Leckie Eds pp 665ndash691 CambridgeUniversity PressCambridge UK 1968
[13] Y LMo ldquoInvestigation of reinforced concrete frame behaviourtheory and testsrdquo Magazine of Concrete Research vol 44 no160 pp 163ndash173 1992
[14] M Jirasek and Z P Bazant Inelastic Analysis of Structures JonWiley amp Sons London UK 2002
[15] S Chandrasekaran L Nunziante G Serino and F CarannanteldquoCurvature ductility of RC sections based on Eurocode analyt-ical procedurerdquo KSCE Journal of Civil Engineering vol 15 no 1pp 131ndash144 2011
Advances in Civil Engineering 13
[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012
[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013
[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992
[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995
[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006
[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010
[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012
[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951
[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982
[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002
[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989
[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008
Figure 7 Calibration of constitutive law of confined concrete
strength 119891co = 30Nmm2 The longitudinal reinforcementconsisted of 12 steel bars of diameter 16mm having yieldingstrength 119891
119904119910= 420Nmm2 uniformly distributed along the
perimeter of the concrete core (constant pitch) The coverdepth measured from the external perimeter of the stirrupswas 13mm The transverse reinforcement consisting of steelstirrups of diameter equal to 10mm according to the calcu-lations made by the aforementioned authors provided theeffective confinement pressure 119891
ℓ119890= 011119891
1015840
co where 1198911015840
co =
085119891co is the reduced strength value prescribed by the ACI318 code [28]
The flexural response was numerically derived in [27] byusing a classical fibre model in which the stress-strain law ofthe confined concrete was assumed in agreement with Man-derrsquosmodel [25]Therefore in this application the parameters120573cc and 120578cc governing the shape of the laws expressed by (4)and (5) are calibrated so that they lead to a constitutive lawof confined concrete which is very similar to that used by theaforementioned authors
Manderrsquos model [25] for the given value of the confine-ment pressure is governed by the parameter 119903 = 13 themodel of Saatcioglu and Razvi [18] leads to a very similarshape of the stress-strain curve by assuming that 120573cc = 045
and 120578cc = minus006 Figure 7 shows the very similar shapes ofthe constitutive laws corresponding to the two models con-sidered
The procedure proposed in [25] provided the followingcharacteristic values of confined concrete strength and cor-responding strain 120576cco = 00073 119891cco = 46Nmm2 Thesevalues are also adopted for the dimensional stress-strain lawexpressed by the model used here [18]
Since for the stress-strain law of the unconfined concreteZahn et al [27] do not give any information the default valuesindicated in Section 31 are assumed here 120573
119888= 1 120578
119888= minus08
and 120576co = 0002On the basis of the above data the square cross-section of
the concrete coremeasured inside the perimeter stirrups hasa side of 119887 = ℎ = 354mm the cover depth is equal to
400
300
200
100
00 001 002 003 004 005
Mom
ent (
kNm
)
Curvature (1m)
Zahn et al (1989)Present model
Figure 8 Comparison of results119872-120593 curve for given119873
23mm the number of intermediate reinforcing bars realizingthe secondary reinforcement is 119899
119887= 119899ℎ= 2 and the distance
of the barycentre of the section of the principal reinforcementfrom the external side of the concrete core section is ℎ
1=
162 = 8mmThe principal reinforcement consists of four upper and
four lower 16mm steel bars while the secondary reinforce-ment consists of four bars (two on the right and two on theleft) of the same diameter The steel reinforcement constitu-tive law is defined by assuming 119864
119904= 210GPa 120576
119904119910= 119891119904119910119864119904=
0002) and 119864119904ℎ
= 119864119904119888
= 0Therefore the dimensionless parameters involved in the
formulation proposed take on the following values 120575V = 120575119900=
= 0Figure 8 compares the 119872-120593 curve obtained in [27] with
that derived from the procedure shown in Section 52 for acompressive constant axial load 119873 = 03119891co(119861 times 119861) corre-sponding to a compressive level 119899 = 119873(119887
2
times 119891cco) = 025 inagreement with the first of (14)The results in the figure showa very good level of agreement between the numerical pro-cedure adopted in [27] and the analytical one adopted here
Both curves in the figure clearly show the cusp producedby the yielding of the principal reinforcement in tension andthe effects of the progressive cover spalling
The second application refers to the middle-height sec-tion of an RC column of height 1640mm experimentallytested by Saatcioglu et al [19] marked as specimen C6-2 bythe authors
The loading condition was realized by imposing relativeaxial displacements so that the reactive compressive forceacted with fixed eccentricity in a plane of principal inertiaof the sections Suitable devices were applied at the columnends so that the column itself behaved like a hinged verticalRC member under an eccentric compressive load
As a consequence each point of the moment-curvaturecurve characterizing the experimental response obtained by
Advances in Civil Engineering 11
the aforementioned authors corresponds to a different valueof axial loadMoreover the actual bendingmoment acting onthe middle-height section was influenced by the 119875-Δ effect
The formulation proposed here is validated consideringsome points of the aforementioned curve for which theauthors indicate the values of the axial load119873 and curvaturethat were measured These values of 119873 are normalized withrespect to the axial load 119873
119900expressed by (2) The procedure
is applied by determining the value of 119909119888from (15) for the
assigned values of 119899 and 120593 and by calculating 119898 from (16)The results are compared with the numerical values obtainedby the strip model adopted by the authors which includedthe 119875-Δ effect and the values experimentally detected
The specimen considered had a square section of side 119861 =
210mm cover depth of 125mm and longitudinal reinforcingbars of diameter 113mm uniformly distributed along theperimeter of the concrete core having yielding strength119891
119904119910=
517Nmm2 and average hardening modulus 119864119904ℎ
= 0013119864119904
The transverse steel reinforcement consisted of square andoctagonal 63mm stirrups with pitch 50mm
The unconfined concrete had cylindrical strength 119891co =
3440Nmm2The numerical model adopted in [19] assumedthat the perimeter of the concrete core was coincident withthe axis lines of the external stirrups
On the basis of these geometrical and mechanical dataand by evaluating the effective confinement pressure by theprocedure proposed in [18] for the confined concrete sectionone obtains 119887 = ℎ = 17870mm 119891cco = 4635 Nmm2 120576cco =
000547 120578cc = minus0052 and 120573 = 0590The further parametersinvolved in the formulation proposed take on the followingvalues 120575
= 0013Figure 9 shows that the procedure proposed provides
results that are very close to those derived by the numericalmodel adopted in [19] The experimentally detected values ofmoment prove to be underestimated by both models
It must be observed that the last point detected by usingthe present formulation corresponding to the ultimate stateof the section had been already found in [17] because thissection belongs to the more restricted class of sections con-sidered in that work
Figure 10 shows the strength and curvature domainsobtained by using the procedure described in Section 53For confined concrete and steel reinforcement the followingvalues were assumed deduced from the data shown in [19]120576ccu = 0032 (120583cc = 585) and 120576
119904119906= 0066 (120583
119903= 27) The
results confirm that a good level of confinementmakes it pos-sible to achieve acceptable ductility of curvature even underhigh levels of compression In order to show how the proce-dure proposed is easily able to evaluate the influence of all thegeometric and mechanical parameters governing the flexuralresponse of a class of RC sections Figure 10(a) also shows theinfluence of the unconfined concrete cover on the bendingmoment at the first yielding of the principal reinforcement intension As expected this influence proves to be negligible forhigh values of the compression level
The dotted curve in Figure 10(b) obviously stops at thepoint of intersection with the continuous curve Beyond this
100
80
60
40
20
00 1 2 3 4
Mom
ent (
kNm
)
ExperimentalModelPresent model
Saatcioglu et al (1995)
Curvature (1mm times 10minus4)
Figure 9 Comparison of theoretical and experimental results
point the 120593119906-119873 curve proceeds by a very brief stretch (not
very evident in the figure) that corresponds to brittle collapseof the section due to the achievement of the maximum avail-able compressive strain in the confined concrete when theprincipal reinforcement in tension is still in the elastic fieldThe end point of this curve corresponds to the achievementof the maximum neutral axis depth defined by the second of(51)
Nevertheless the axial load values that cannot be consid-ered because of these limitations are well beyond the onesthat can be assumed in the structural design of RC buildingsin seismic areas or usually found in the columns of existingbuildings
7 Conclusions
A dimensionless formulation has been proposed which pro-vides the flexural response of classes of sections of RC col-umns having the same values of the geometric and mechani-cal parameters defined in this study
At each loading stage the resisting components of the sec-tion (cover and concrete core and principal and secondarylongitudinal reinforcements) give a contribution to the equi-librium that can be expressed by analytical functions depend-ing on the normalized neutral axis depth and the curvature ofthe section
Special equilibrium conditions like the ones correspond-ing to the first yielding of the principal reinforcement intension and to the ultimate state imply an analytical link bet-ween the neutral axis depth and the curvature so that the sumof the aforementioned contributions becomes an analyticalfunction of a single variable which can express a strength orcurvature domain
For a generic equilibrium condition an iterative proce-dure to determine the neutral axis depth corresponding to
12 Advances in Civil Engineering
100
80
60
40
20
00 300 600 900 1200 1500
Mom
ent (
kNm
)
Axial load (kN)
Mu
My
My without cover
(a)
8
6
4
2
00 300 600 900 1200 1500
Axial load (kN)
120593u120593y
Curv
atur
e(1
mm
times10
minus4)
(b)
Figure 10119872-119873 and 120593-119873 domains at the ultimate state and the first yielding of reinforcement
the assigned values of curvature and compression level isrequired but each iteration step implies the simple use of thesame equilibrium equation for an updated value of the nor-malized neutral axis dept
The dimensionless form of the formulation proposedmakes also it easily possible to evaluate the influence of theparameters characterizing the class of sections examined onthe strength and curvature corresponding to a given loadingstage
Acknowledgment
Thisworkwas carried outwithin the 2010ndash2013Research Pro-ject ldquoDPC-ReLUIS (Dipartimento Protezione Civile-Rete deiLaboratori Universitari di Ingegneria Sismica)rdquo AT 1 Task112 The related financial support was greatly appreciated
References
[1] A Fafitis ldquoInteraction surfaces of reinforced-concrete sectionsin biaxial bendingrdquo Journal of Structural Engineering vol 127no 7 pp 840ndash846 2001
[2] J L Bonet P F Miguel M A Fernandez and M L RomeroldquoAnalytical approach to failure surfaces in reinforced concretesections subjected to axial loads and biaxial bendingrdquo Journalof Structural Engineering vol 130 no 12 pp 2006ndash2015 2004
[3] G Monti and S Alessandri ldquoAssessment of rc columns undercombined biaxial bending and axial loadrdquo in Proceedings of the2nd FIB Congress Naples Italy 2006
[4] RW Clough and S B Johnston ldquoEffect of stiffness degradationon earthquake ductility requirementsrdquo in Proceedings of 2ndJapan Earthquake Engineering Symposium Tokyo Japan 1966
[5] T Takeda M A Sozen and N N Nielsen ldquoReinforced con-crete response to simulated earthquakerdquo Journal of StructuralDivision vol 96 no 12 pp 2257ndash2273 1970
[6] M S L Roufaiel and C Meyer ldquoAnalytical modeling of hyster-etic behavior of reinforced concrete framerdquo Journal of StructuralEngineering vol 113 no 3 pp 429ndash444 1987
[7] A R Mari and A C Scordelis ldquoNonlinear geometric materialand time dependent analysis of three dimensional reinforcedand prestressed concrete framesrdquo USBSESM Report 8412Department of Civil Engineering University of CaliforniaBerkeley Calif USA 1973
[8] T Taucer E Spacone and F C Filippou ldquoA fiber beam-col-umn element for seismic response analysis of reinforced con-crete structuresrdquo Report EERC 91-17 Earthquake EngineeringResearch Center Berkeley Calif USA 1991
[9] Z Zhu I Ahmad and A Mirmiran ldquoFiber element modelingfor seismic performance of bridge columns made of concrete-filled FRP tubesrdquo Engineering Structures vol 28 no 14 pp2023ndash2035 2006
[10] E O Pfrang C P Siess and M A Sozen ldquoLoad-moment-cur-vature characteristics of RC cross-sectionsrdquoACI Journal vol 61no 7 pp 763ndash778 1964
[11] D J Carreira and K-H Chu ldquoThemoment-curvature relation-ship of RC membersrdquo ACI Journal vol 83 no 2 pp 191ndash1981986
[12] R H Wood ldquoSome controversial and curious developments inplastic theory of structuresrdquo inEngineering Plasticity J Heymanand F A Leckie Eds pp 665ndash691 CambridgeUniversity PressCambridge UK 1968
[13] Y LMo ldquoInvestigation of reinforced concrete frame behaviourtheory and testsrdquo Magazine of Concrete Research vol 44 no160 pp 163ndash173 1992
[14] M Jirasek and Z P Bazant Inelastic Analysis of Structures JonWiley amp Sons London UK 2002
[15] S Chandrasekaran L Nunziante G Serino and F CarannanteldquoCurvature ductility of RC sections based on Eurocode analyt-ical procedurerdquo KSCE Journal of Civil Engineering vol 15 no 1pp 131ndash144 2011
Advances in Civil Engineering 13
[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012
[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013
[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992
[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995
[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006
[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010
[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012
[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951
[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982
[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002
[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989
[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008
the aforementioned authors corresponds to a different valueof axial loadMoreover the actual bendingmoment acting onthe middle-height section was influenced by the 119875-Δ effect
The formulation proposed here is validated consideringsome points of the aforementioned curve for which theauthors indicate the values of the axial load119873 and curvaturethat were measured These values of 119873 are normalized withrespect to the axial load 119873
119900expressed by (2) The procedure
is applied by determining the value of 119909119888from (15) for the
assigned values of 119899 and 120593 and by calculating 119898 from (16)The results are compared with the numerical values obtainedby the strip model adopted by the authors which includedthe 119875-Δ effect and the values experimentally detected
The specimen considered had a square section of side 119861 =
210mm cover depth of 125mm and longitudinal reinforcingbars of diameter 113mm uniformly distributed along theperimeter of the concrete core having yielding strength119891
119904119910=
517Nmm2 and average hardening modulus 119864119904ℎ
= 0013119864119904
The transverse steel reinforcement consisted of square andoctagonal 63mm stirrups with pitch 50mm
The unconfined concrete had cylindrical strength 119891co =
3440Nmm2The numerical model adopted in [19] assumedthat the perimeter of the concrete core was coincident withthe axis lines of the external stirrups
On the basis of these geometrical and mechanical dataand by evaluating the effective confinement pressure by theprocedure proposed in [18] for the confined concrete sectionone obtains 119887 = ℎ = 17870mm 119891cco = 4635 Nmm2 120576cco =
000547 120578cc = minus0052 and 120573 = 0590The further parametersinvolved in the formulation proposed take on the followingvalues 120575
= 0013Figure 9 shows that the procedure proposed provides
results that are very close to those derived by the numericalmodel adopted in [19] The experimentally detected values ofmoment prove to be underestimated by both models
It must be observed that the last point detected by usingthe present formulation corresponding to the ultimate stateof the section had been already found in [17] because thissection belongs to the more restricted class of sections con-sidered in that work
Figure 10 shows the strength and curvature domainsobtained by using the procedure described in Section 53For confined concrete and steel reinforcement the followingvalues were assumed deduced from the data shown in [19]120576ccu = 0032 (120583cc = 585) and 120576
119904119906= 0066 (120583
119903= 27) The
results confirm that a good level of confinementmakes it pos-sible to achieve acceptable ductility of curvature even underhigh levels of compression In order to show how the proce-dure proposed is easily able to evaluate the influence of all thegeometric and mechanical parameters governing the flexuralresponse of a class of RC sections Figure 10(a) also shows theinfluence of the unconfined concrete cover on the bendingmoment at the first yielding of the principal reinforcement intension As expected this influence proves to be negligible forhigh values of the compression level
The dotted curve in Figure 10(b) obviously stops at thepoint of intersection with the continuous curve Beyond this
100
80
60
40
20
00 1 2 3 4
Mom
ent (
kNm
)
ExperimentalModelPresent model
Saatcioglu et al (1995)
Curvature (1mm times 10minus4)
Figure 9 Comparison of theoretical and experimental results
point the 120593119906-119873 curve proceeds by a very brief stretch (not
very evident in the figure) that corresponds to brittle collapseof the section due to the achievement of the maximum avail-able compressive strain in the confined concrete when theprincipal reinforcement in tension is still in the elastic fieldThe end point of this curve corresponds to the achievementof the maximum neutral axis depth defined by the second of(51)
Nevertheless the axial load values that cannot be consid-ered because of these limitations are well beyond the onesthat can be assumed in the structural design of RC buildingsin seismic areas or usually found in the columns of existingbuildings
7 Conclusions
A dimensionless formulation has been proposed which pro-vides the flexural response of classes of sections of RC col-umns having the same values of the geometric and mechani-cal parameters defined in this study
At each loading stage the resisting components of the sec-tion (cover and concrete core and principal and secondarylongitudinal reinforcements) give a contribution to the equi-librium that can be expressed by analytical functions depend-ing on the normalized neutral axis depth and the curvature ofthe section
Special equilibrium conditions like the ones correspond-ing to the first yielding of the principal reinforcement intension and to the ultimate state imply an analytical link bet-ween the neutral axis depth and the curvature so that the sumof the aforementioned contributions becomes an analyticalfunction of a single variable which can express a strength orcurvature domain
For a generic equilibrium condition an iterative proce-dure to determine the neutral axis depth corresponding to
12 Advances in Civil Engineering
100
80
60
40
20
00 300 600 900 1200 1500
Mom
ent (
kNm
)
Axial load (kN)
Mu
My
My without cover
(a)
8
6
4
2
00 300 600 900 1200 1500
Axial load (kN)
120593u120593y
Curv
atur
e(1
mm
times10
minus4)
(b)
Figure 10119872-119873 and 120593-119873 domains at the ultimate state and the first yielding of reinforcement
the assigned values of curvature and compression level isrequired but each iteration step implies the simple use of thesame equilibrium equation for an updated value of the nor-malized neutral axis dept
The dimensionless form of the formulation proposedmakes also it easily possible to evaluate the influence of theparameters characterizing the class of sections examined onthe strength and curvature corresponding to a given loadingstage
Acknowledgment
Thisworkwas carried outwithin the 2010ndash2013Research Pro-ject ldquoDPC-ReLUIS (Dipartimento Protezione Civile-Rete deiLaboratori Universitari di Ingegneria Sismica)rdquo AT 1 Task112 The related financial support was greatly appreciated
References
[1] A Fafitis ldquoInteraction surfaces of reinforced-concrete sectionsin biaxial bendingrdquo Journal of Structural Engineering vol 127no 7 pp 840ndash846 2001
[2] J L Bonet P F Miguel M A Fernandez and M L RomeroldquoAnalytical approach to failure surfaces in reinforced concretesections subjected to axial loads and biaxial bendingrdquo Journalof Structural Engineering vol 130 no 12 pp 2006ndash2015 2004
[3] G Monti and S Alessandri ldquoAssessment of rc columns undercombined biaxial bending and axial loadrdquo in Proceedings of the2nd FIB Congress Naples Italy 2006
[4] RW Clough and S B Johnston ldquoEffect of stiffness degradationon earthquake ductility requirementsrdquo in Proceedings of 2ndJapan Earthquake Engineering Symposium Tokyo Japan 1966
[5] T Takeda M A Sozen and N N Nielsen ldquoReinforced con-crete response to simulated earthquakerdquo Journal of StructuralDivision vol 96 no 12 pp 2257ndash2273 1970
[6] M S L Roufaiel and C Meyer ldquoAnalytical modeling of hyster-etic behavior of reinforced concrete framerdquo Journal of StructuralEngineering vol 113 no 3 pp 429ndash444 1987
[7] A R Mari and A C Scordelis ldquoNonlinear geometric materialand time dependent analysis of three dimensional reinforcedand prestressed concrete framesrdquo USBSESM Report 8412Department of Civil Engineering University of CaliforniaBerkeley Calif USA 1973
[8] T Taucer E Spacone and F C Filippou ldquoA fiber beam-col-umn element for seismic response analysis of reinforced con-crete structuresrdquo Report EERC 91-17 Earthquake EngineeringResearch Center Berkeley Calif USA 1991
[9] Z Zhu I Ahmad and A Mirmiran ldquoFiber element modelingfor seismic performance of bridge columns made of concrete-filled FRP tubesrdquo Engineering Structures vol 28 no 14 pp2023ndash2035 2006
[10] E O Pfrang C P Siess and M A Sozen ldquoLoad-moment-cur-vature characteristics of RC cross-sectionsrdquoACI Journal vol 61no 7 pp 763ndash778 1964
[11] D J Carreira and K-H Chu ldquoThemoment-curvature relation-ship of RC membersrdquo ACI Journal vol 83 no 2 pp 191ndash1981986
[12] R H Wood ldquoSome controversial and curious developments inplastic theory of structuresrdquo inEngineering Plasticity J Heymanand F A Leckie Eds pp 665ndash691 CambridgeUniversity PressCambridge UK 1968
[13] Y LMo ldquoInvestigation of reinforced concrete frame behaviourtheory and testsrdquo Magazine of Concrete Research vol 44 no160 pp 163ndash173 1992
[14] M Jirasek and Z P Bazant Inelastic Analysis of Structures JonWiley amp Sons London UK 2002
[15] S Chandrasekaran L Nunziante G Serino and F CarannanteldquoCurvature ductility of RC sections based on Eurocode analyt-ical procedurerdquo KSCE Journal of Civil Engineering vol 15 no 1pp 131ndash144 2011
Advances in Civil Engineering 13
[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012
[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013
[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992
[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995
[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006
[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010
[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012
[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951
[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982
[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002
[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989
[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008
Figure 10119872-119873 and 120593-119873 domains at the ultimate state and the first yielding of reinforcement
the assigned values of curvature and compression level isrequired but each iteration step implies the simple use of thesame equilibrium equation for an updated value of the nor-malized neutral axis dept
The dimensionless form of the formulation proposedmakes also it easily possible to evaluate the influence of theparameters characterizing the class of sections examined onthe strength and curvature corresponding to a given loadingstage
Acknowledgment
Thisworkwas carried outwithin the 2010ndash2013Research Pro-ject ldquoDPC-ReLUIS (Dipartimento Protezione Civile-Rete deiLaboratori Universitari di Ingegneria Sismica)rdquo AT 1 Task112 The related financial support was greatly appreciated
References
[1] A Fafitis ldquoInteraction surfaces of reinforced-concrete sectionsin biaxial bendingrdquo Journal of Structural Engineering vol 127no 7 pp 840ndash846 2001
[2] J L Bonet P F Miguel M A Fernandez and M L RomeroldquoAnalytical approach to failure surfaces in reinforced concretesections subjected to axial loads and biaxial bendingrdquo Journalof Structural Engineering vol 130 no 12 pp 2006ndash2015 2004
[3] G Monti and S Alessandri ldquoAssessment of rc columns undercombined biaxial bending and axial loadrdquo in Proceedings of the2nd FIB Congress Naples Italy 2006
[4] RW Clough and S B Johnston ldquoEffect of stiffness degradationon earthquake ductility requirementsrdquo in Proceedings of 2ndJapan Earthquake Engineering Symposium Tokyo Japan 1966
[5] T Takeda M A Sozen and N N Nielsen ldquoReinforced con-crete response to simulated earthquakerdquo Journal of StructuralDivision vol 96 no 12 pp 2257ndash2273 1970
[6] M S L Roufaiel and C Meyer ldquoAnalytical modeling of hyster-etic behavior of reinforced concrete framerdquo Journal of StructuralEngineering vol 113 no 3 pp 429ndash444 1987
[7] A R Mari and A C Scordelis ldquoNonlinear geometric materialand time dependent analysis of three dimensional reinforcedand prestressed concrete framesrdquo USBSESM Report 8412Department of Civil Engineering University of CaliforniaBerkeley Calif USA 1973
[8] T Taucer E Spacone and F C Filippou ldquoA fiber beam-col-umn element for seismic response analysis of reinforced con-crete structuresrdquo Report EERC 91-17 Earthquake EngineeringResearch Center Berkeley Calif USA 1991
[9] Z Zhu I Ahmad and A Mirmiran ldquoFiber element modelingfor seismic performance of bridge columns made of concrete-filled FRP tubesrdquo Engineering Structures vol 28 no 14 pp2023ndash2035 2006
[10] E O Pfrang C P Siess and M A Sozen ldquoLoad-moment-cur-vature characteristics of RC cross-sectionsrdquoACI Journal vol 61no 7 pp 763ndash778 1964
[11] D J Carreira and K-H Chu ldquoThemoment-curvature relation-ship of RC membersrdquo ACI Journal vol 83 no 2 pp 191ndash1981986
[12] R H Wood ldquoSome controversial and curious developments inplastic theory of structuresrdquo inEngineering Plasticity J Heymanand F A Leckie Eds pp 665ndash691 CambridgeUniversity PressCambridge UK 1968
[13] Y LMo ldquoInvestigation of reinforced concrete frame behaviourtheory and testsrdquo Magazine of Concrete Research vol 44 no160 pp 163ndash173 1992
[14] M Jirasek and Z P Bazant Inelastic Analysis of Structures JonWiley amp Sons London UK 2002
[15] S Chandrasekaran L Nunziante G Serino and F CarannanteldquoCurvature ductility of RC sections based on Eurocode analyt-ical procedurerdquo KSCE Journal of Civil Engineering vol 15 no 1pp 131ndash144 2011
Advances in Civil Engineering 13
[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012
[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013
[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992
[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995
[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006
[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010
[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012
[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951
[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982
[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002
[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989
[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008
[16] M Fossetti andM Papia ldquoDimensionless analysis of RC rectan-gular sections under axial load and biaxial bendingrdquo Engineer-ing Structures vol 44 pp 34ndash45 2012
[17] P ColajanniM Fossetti andM Papia ldquoAnalytical prediction ofultimate moment and curvature of RC rectangular sections incompressionrdquo Bulletin of Earthquake Engineering 2013
[18] M Saatcioglu and S R Razvi ldquoStrength and ductility of con-fined concreterdquo Journal of Structural Engineering vol 118 no 6pp 1590ndash1607 1992
[19] M Saatcioglu A H Salamat and S R Razvi ldquoConfined col-umns under eccentric loadingrdquo Journal of Structural Engineer-ing vol 121 no 11 pp 1547ndash1556 1995
[20] G Campione M Fossetti and M Papia ldquoSimplified analyticalmodel for compressed high-strength columns confined bytransverse steel and longitudinal barsrdquo in Proceedings of the 2ndFIB Congress Naples Italy 2006
[21] GCampioneM Fossetti andMPapia ldquoBehavior of fiber-rein-forced concrete columns under axially and eccentrically com-pressive loadsrdquo ACI Structural Journal vol 107 no 3 pp 272ndash281 2010
[22] G Campione M Fossetti G Minafo and M Papia ldquoInfluenceof steel reinforcements on the behavior of compressed highstrength RC circular columnsrdquo Engineering Structures vol 34pp 371ndash382 2012
[23] E Hognestad A Study of Combined Bending and Axial Load inReinforcedConcreteMembers Bulletin SeriesNo 399 Engineer-ing Experiment Station University of Illinois Urbana Ill USA1951
[24] B D Scott R Park and M J N Priestley ldquoStress-strain behav-iour of concrete confined by overlapping hoops at low and highstrain raterdquo ACI Journal vol 79 no 2 pp 13ndash27 1982
[25] J B Mander M J N Priestley and R Park ldquoTheoretical Stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[26] R P Dhakal and KMaekawa ldquoModeling for postyield bucklingof reinforcementrdquo Journal of Structural Engineering vol 128 no9 pp 1139ndash1147 2002
[27] FA Zahn R Park andM JN Priestley ldquoStrength andductilityof square reinforced concrete column sections subjected tobiaxial bendingrdquo ACI Structural Journal vol 86 no 2 pp 123ndash131 1989
[28] Building Code Requirements for Structural Concrete and Com-mentary ACI 318 American Concrete Institute (ACI) 2008