2012Marques - Performance Analysis of Load–Strain Models

el

Analysis oriented modelCircular RC columnsAxial connement modellingDilation behaviourDesign oriented modelFRP

forte c

FRP-conned concrete based either on theoretical assumptions (analysis-oriented-models AOMs) or

ch as F

ties that characterise and quantify the performance of connedconcrete columns. This evidence will be shown ahead in thisarticle.

As in other structural systems, such as slabs and beams, thestrengthening effect can be active or passive. In the specic caseof actively-conned columns, stress state is laterally applied and

an incremental numerical process [11,15,16,26,4,27,10]. The sec-ond group includes models that are based on passive connementand in which peak/ultimate stress and strain are rst determinedbeing then the non-linear behaviour is mathematically calibratedwith experimental data [8,24,28,23,9,14,3].

The discussion around advantages or disadvantages betweenAOM and DOM in modelling the connement under axial compres-sion is still open, although some authors clearly consider AOMmore accurate and DOM easier to implement due to their directuse in design calculations [9,7].

Corresponding author at: Escola Superior de Tecnologia do Barreiro InstitutoPolitcnico de Setbal, Portugal.

E-mail addresses: [email protected] (P.F. Marques), chastre

Composite Structures 94 (2012) 31153131

Contents lists available at

Composite S

[email protected] (C. Chastre).has been the aim of several authors research in order to enhancethese elements strength and ductility. Considering the importanceofdesigncalculations fornewstructuresor the strengtheningofexist-ing ones, themodelling prediction of the performance of circular con-crete columnssubjected toaxial compression isproposedbydifferentauthors as regards stressstrain behaviour.

The existing types of conning models are based on differentpremises and quantication of the properties of the materialsand structural systems involved. Consequently, the approximationof these models data against the real behaviour of tests is differentfor each model and with different inuence on the several proper-

connement the conning stress of the strengthening is activatedby the lateral expansion of the concrete core. In practice this isthe behaviour of concrete cores conned with steel or FRP stripsor jackets, though with distinct performance between these twomaterials expressed through the dilation properties and explainedin Section 4.4 [15,16,24,26].

The existing conning models under axial compression may bedivided into two groups [9]: (i) analysis-oriented models (AOM)and (ii) design-oriented models (DOM). In the rst group most ofthese theoretical models are based on stressstrain curves of con-ned concrete obtained from active connement curves by use of1. Introduction

The study of conningmaterials su0263-8223/$ - see front matter 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compstruct.2012.04.036on mathematical calibration from testing results (design-oriented-models DOMs). This article carriesout the implementation and analysis of nine existing models for circular concrete columns in view of axi-ally tested reinforced concrete columns conned with CFRP with three different diameters: 150; 250 and400 mm. The global shape of curves, peak compressive load, stressstrain relation, axial-to-lateral rela-tion and dilation response were studied to conclude which models curves were closer to tests. Quanti-cation of errors in face of the testing results was carried out for the most important parameters ultimate load, strain and lateral stress as well as for other curve parameters. Some models are accuratein predicting the peak load, though only few can accurately predict the loadstrain and dilationbehaviour.

2012 Elsevier Ltd. All rights reserved.

RP for concrete columns

externally controlled [20], being the lateral expansion restrained,while the axial stress increases. Several of the existing conningmodels are based on this principle. On the other hand, in passiveKeywords:able for the design of this system considering, not only the peak values of load and strain, but also thecomplete stressstrain behaviour. A wide group of authors have proposed several models specic forPerformance analysis of loadstrain modwith FRP composites

Pedro Faustino Marques a,b,, Carlos Chastre ba Escola Superior de Tecnologia do Barreiro Instituto Politcnico de Setbal, PortugalbDepartment of Civil Engineering, FCT/UNIC, Universidade Nova de Lisboa, Portugal

a r t i c l e i n f o

Article history:Available online 22 May 2012

a b s t r a c t

The use of FRP compositeson strengthening of concre

journal homepage: www.elll rights reserved.s for circular columns conned

the connement of concrete has become an important aspect to considerolumns. It is important therefore that accurate modelling tools are avail-

SciVerse ScienceDirect

tructures

ier .com/locate /compstruct

Additionally, there is the fact that, in general, these models donot take into account the contribution of reinforcing steel hoopsin concrete, whose inuence varies as function of their quantityand spacing length as well as a function of quality of the connedconcrete.

The present study aims to implement and analyse nine mathe-matical models of various authors four AOM [15,16,26,4,27] andve DOM [24,28,23,14,3]. All the models were implemented con-sidering existing test results of reinforced concrete specimens150, 250 and 400 mm diameter conned with CFRP jackets.

The modelling results are compared with test results in severalparameters in order to understanding which models best t thereal behaviour of reinforced concrete (RC) columns conned withFRP jackets, including the contribution of transversal and longitu-dinal reinforcing steel.

The experimental results herein used were carried out else-where by Matthys [13] and Chastre [2] and beside different diam-eters the specimens had different CFRP Jackets and differentnumber of layers. The analysis of modelling results compared withtests results is done in varied ways loadstrain response, dilationproperties, volumetric expansion, etc. with error quantication ofseveral parameters for each model.

2. Test database

2.1. Tested columns

The experimental results, needed for the analysis of the severalconning models, were obtained from the research group of the

Universidade Nova de Lisboa (UNL) and the open literature. The150 and 250 mm columns were selected from a group of 66 col-umns tested at UNL [2,25,3]. The 150 mm diameter column repre-sents a group of three specimens with similar characteristics andthe 250 mm column represents a group of 2. The column with400 mm diameter [13] comes from a group of six columns and,although all these columns have different strengthening materialsand layers the, option was upon the specimen with a CFRPstrengthening material with characteristics closer to the ones with150 and 250 mm diameter. Additionally, this column with 400 mmdiameter was chosen due to the fact that few columns with such adimension and with steel reinforcement were tested so far.

These circular cross-section specimens were of reinforced con-crete with dimensions fairly considered as those of real existingconstruction columns. Fig. 1 shows the cross-section of the differ-ent columns whose experimental behaviour is to be comparedwith models results. Table 1 presents the detailed constitution ofthe columns. The vertical reinforcement ratio is 1% for the150 mm diameter column with 3 mm diameter stirrups spacedevery 0.10 m, 1.4% for the 250 mm diameter column with 6 mmdiameter hoops spaced every 0.15 m and 0.9% for the 400 mmdiameter columns with 8 mm diameter hoops spaced every0.14 m.

Regarding the CFRP sheets, Replark 30 bres applied with epo-therm resin-L700S (here called type A) were used for 150 mmdiameter specimen, MBrace C1-30 with MBrace Saturate resin(type B) for 250 mm diameter specimen and for 400 mm diameterC240 unidirectional sheet with Multipox T epoxy (Type C). For col-umns with 150 and 250 mm diameter, the overlap length of CFRP

vers

.10

6//

ted

3116 P.F. Marques, C. Chastre / Composite Structures 94 (2012) 31153131Table 1Constitution of available testes RC columns [13,2].

Column dimensions Steel reinforcement

/ (mm) Height (mm) Longitudinal Trans

150 750 6/6 /3//0

250 mm150 mm

3//100

663//100

6126//150

Fig. 1. Cross section of available tes250 750 6/12 /6//0.15400 2000 10/12 /8//0.14CFRP connement

e Sheet type no layers tply (mm)

A Replark 30 2 0.167

400 mm

150 6//140

10126//140

RC columns: /150; /250 and /400.B MBrace C130 2 0.176C S&P C240 5 0.117

wraps was half of the perimeter of the column and for the 400 mmdiameter the overlap length was of 200 mm.

2.2. Materials properties

Despite the available technical information from suppliers ofsome of the materials, tests were carried out in order to have moreaccurate results of the used samples. Table 2 summarizes the prop-erties of all involved materials: concrete, steel reinforcement andCFRP sheets. The mean compressive results of unconned concreteat 28 days age were: fc0 = 38.0 MPa (/150); fc0 = 35.2 MPa (/250);fc0 = 34.3 MPa (/400). For columns with 150 and 250 mm diameterthe yield strength of steel reinforcement is 323 MPa for /3,

Table 2Properties of tested materials [13,2].

Column Concrete Steel reinforcement CFRP sheet

/ (mm) fc0 (MPa) Diameter (mm) Yield strength (MPa) type Ef (GPa) ffu (MPa) efu (%)

150 38.0 /3 323 A 226 3339 1.44/6 391

250 35.2 /12 451 B 241 3937 1.54

400 34.3 /8 560 C 198 2356 1.19/12 620

Table 3Tests results of CFRP-conned RC columns [13,2].

Column / (mm) N0 (kN) Ncc (kN) Ncc/Nco ecc elu

150 696.3 1485.7 2.13 0.0131 0.0090250 1727.9 3741.6 2.17 0.0155 0.0093400 4310.3 7460.0 2.03 0.0119 0.0073

N0 = maximum axial load for unconned concrete with steel reinforcement con-tribution: fcoAc + fsAsNcc = maximum axial load for CFRP-conned concrete considering steel reinforce-ment contribution.ecc = axial strain at maximum axial load.elu = lateral strain at failure of CFRP in hoop direction.

(a) Axial load vs lateral and axial strains

0.00

0.50

1.00

1.50

2.00

2.50

-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 0.020Nor

mal

ized

Axi

al L

oad

-Nc

/ N0

diam 150diam 250diam 400

Axial strain - cLateral strain - l

0.50

1.00

1.50

2.00

2.50

orm

aliz

ed A

xial

Loa

d -N

c / N

0

diam 150diam 250diam 400

expansioncontraction

Fig. 2. Loadstrain relation of available tested CFRP-c

(a) Passive confinement model (dots) generated by active confinement stress-strain curves

(lines) (Mander et al. 1988) - AOM

0.00.20.40.60.81.01.21.41.61.82.0

0.0 2.0 4.0 6.0 8.0

Nor

mal

ized

axi

al s

tress

Normalized axial strain

Fig. 3. Concept bases for co

P.F. Marques, C. Chastre / Composite Structures 94 (2012) 31153131 3117(b) Axial load vs volumetric strain

0.00-0.006 -0.004 -0.002 0.000 0.002 0.004

N

Volumetric strain - v

onned RC columns: /150; /250 and /400 mm.(b) Two regions confinement model - DOM

nnement modelling.

391 MPa for /6 and 458 MPa for /12. For columns with 400 mmdiameter the yield strength is 560 MPa for /8 and 620 MPa for /12. Tests on CFRP specimens resulted in: sheet type A, Ef = 226 GPa,ffu = 3339 MPa and ecu = 1.44%; sheet type B, Ef = 241 GPa,ffu = 3937 MPa and ecu = 1.54%; sheet type C, Ef = 198 GPa,ffu = 2356 MPa and ecu = 1.19%.

2.3. Columns test results

In plain concrete columns conned with an external jacket thewhole concrete is a core. During axial compression the load in-creases until the CFRP jacket reaches failure. In RC columns thebehaviour is slightly different given the fact that the presence ofsteel hoops and longitudinal steel bars has a relevant contributionto columns compression strength.

Depending on the contribution of conning hoops, there canstill be some residual strength while buckling of longitudinal bars.As regards the available test data, the CFRP-conned specimensfailed with sudden rupture of the jacket.

The results of the main parameters are shown in Table 3 and theloadstrain curves and dilation behaviour through-out testing areshown in Fig. 2.

Due to the presence of steel reinforcement and hence diversestress development during tests until failure, instead of stress

strain relation, Fig. 2a shows the normalised loadstrain relationwhere lateral strains el are on the left side (tension negative val-ues) and axial strains ec on the right side (compressive positivevalues). Also compared to load values, Fig. 2b presents the norma-lised loadvolumetric strain (Nc/N0ev) relation, in which positivevalue of ev represent volume contraction and negative values rep-resent volume expansion. Nc is the axial compressive load and N0the maximum axial compressive load of unconned concrete.

For a fair analysis it should be borne in mind that /400 column,aside from having different CFRP sheet system, has also a differentsteel reinforcement grade.

3. Existing models for the connement of circular cross-sectionconcrete with FRP jackets

3.1. Introduction

In order to increase the compressive strength of concrete col-umns the use of connement has been proved to be effective. Sev-eral models throughout the years have been developed to reectconcrete connement and thus by this way estimating the gainin strength of a certain strengthening system. Some of the rstmodels were developed for the conning effect of steel hoops inconcrete elements [11] while others considered external steel jack-

is ar

Table 4Peak stress and strain equations for AOM models.

Analysis-oriented model Peak stress Eq. no.this article

Strain at peak stress Eq. no.this article

Mirmiran and Shahawy [15,16],Spoelstra and Monti [26], Fam and Rizkalla [4] fcc fco 2:254

1 7:94 fl

fco

s 2 fl

fco 1:254

! (5)ecc eco 1 5 fccfco 1

(6)

Based on William and Warnke [29] Based on Mander et al. [11]Teng et al. [27] fcc fco 3:5f l (7)

ecc eco 1 17:5 flfco

1:2" # (8)

based on Richart et al. [20] from Eq. (6)

3118 P.F. Marques, C. Chastre / Composite Structures 94 (2012) 31153131Table 5Peak stress and strain equations for DOM models.

Design-oriented model Peak stress Eq. no. th

Samaan et al. [24] fcc fco 6f 0:7l MPa (9)

based on Richart et al. [20]Toutanji [28]

fcc fco 1 3:5 flfco

0:85" # (11)

based on Richart et al. [20]Saa et al. [23]

fcc fco 1 2:2 flfco

0:84" # (13)

based on Richart et al. [20]Toutanji revised [14]

fcc fco 1 2:3 flfco

0:85" # (15)

Calibrated based on Toutanji [28]Chastre and Silva [3] fcc fD 5:29f l (17)

based on Richart et al. [20]

efu = failure (ultimate) strain of the CFRP material.

elu = failure (ultimate) strain of the CFRP jacket in hoop direction.f0 = intercept stress in Samaan et al.s model (Fig. 3b).fD = axial compressive strength of unconned concrete accounting for the tested core sticle Strain at peak stress Eq. no. this article

ecc fcc f0E2(10)

ecc eco 1 310:87elu 1:9 fccfco

0:85" # (12)

calibrated based on Mander et al. [11]

ecc eco 1 537elu 2:6 fccfco 1 (14)

calibrated based on Mander et al. [11]

ecc eco 1 537elu 2:6 fcc

fco 1

(16)

2nd region-elu 0:6efu

ecc 17:65eco flfD

0:7 (18)

fl obtained from elu = 0.6efu (181)lenderness fD 1:5D=H2

f co .

ets as strengthening material [20,21,1] for circular-cross sectioncolumns.

With the possibility of using FRP in construction, either forstrengthening or new construction, the previous models wereapparently an obvious way to estimate connement behaviour.However, due to the different properties of FRP, regarding steel,these models are less suitable in their present form when consid-ering these materials [12].

In view of the previous, some researchers have been developingdifferent models based on extensive experimental data[22,6,17,18,15,16,12].

As mentioned before, connement models are divided into twolarge groups analysis-oriented models (AOMs) and design-ori-ented models (DOMs) which have different principle approaches.

In AOM there is an explicit interaction between different mate-rials (conned concrete and conning material FRP, steel or oth-ers) and the calculation procedure assumes the compatibilitybetween the lateral strain el of actively-conned concrete with aconstant conning pressure fl equal to that given by the jacket.The stressstrain curve is generated by an incremental approachwhere curves with different active connement levels generate apassive connement curve (Fig. 3a). In most cases the incrementalprocedure is iterative and hence not always of simple use forengineers.

In DOM, a specimen in concrete strengthened with FRP are con-sidered as a whole reecting the connement behaviour based(calibrated) on experimental data, implying that active or passiveconnement is already taken into account and it is represented

[27]. Those based on two regions connement model (Fig. 3b)are: Samaan et al. [24], Toutanji [28], Saa et al. [23], Toutanji re-vised (Matthys et al. [14]), Chastre and Silva [3]. All these modelsare for FRP-conned concrete columns.

3.2. Peak axial stress and corresponding strain

For unconned concrete the ascending part of stressstraincurve is adopted in a reference code as Eurocode 2 (ENV, 1992,2004) considering a parabola as described in following equation:

rc fco 2ececo ececo

2 !1

where rc the axial stress, ec the axial strain and, eco the axial strainat peak stress of concrete.

Nevertheless, this equation is not suited for representing theconnement behaviour of concrete since it cannot represent thegradual development of connement [9].

The basic concept behind the generalised modelling of peakstress of conned concrete was established by Richart et al. [20]in which the failure strength of concrete conned by a hydrostaticuid pressure (active connement) takes the following form:

fcc fco k1fl 2where fcc is the maximum strength of conned concrete, fco themaximum strength of unconned concrete, fl the lateral conningpressure and k1 the connement effectiveness coefcient.

n rel

eccrec=

0] an

E1fa

h

E1lfa

h

] baE1

E

E1lE

Abb

P.F. Marques, C. Chastre / Composite Structures 94 (2012) 31153131 3119by a two regions stressstrain relation (in some models bilinear),both axial and lateral (Fig. 3b). These models are generally of sim-pler procedure in calculating, though in some cases the proposedequations are laborious.

The group of connement models based on the theory of Fig. 3aanalysed on this article includes: Mirmiran and Shahawy [15,16],Spoelstra and Monti [26], Fam and Rizkalla [4] and Teng et al.

Table 6Stressstrain relation of each model.

Models Stressstrai

Mirmiran and Shahawy [15,16], Spoelstra and Monti [26],Fam and Rizkalla [4], Teng et al. [27] fc

ec=r 1

Popovics [3Toutanji [28], Saa et al. [23], Toutanji revised [14]

fcec 1

fcec 1

Toutanji [28Samaan et al. [24], Chastre and Silva [3]

fcec

1h

fcec 1

Richard and

fc = axial stress.fa = axial stress at intersection point between the and second zones.e1c = axial strain at intersection point between the and second zones.e1l = lateral strain at intersection point between the and second zones.E1, E1l = 1st region slope for stress vs axial and lateral strains, respectively.

E2, E2l = 2nd region slope for stress vs axial and lateral strains, respectively.n = shape factor for the axial stress axial strain relation.nl = shape factor for the axial stress lateral strain relation.Correspondingly, the axial strain ecc at which peak stress fcc isreached, depends on the previous parameters and on the axialstrain at maximum stress of unconned concrete eco and takesthe following form proposed by the same authors:

ecc eco 1 5k1 flfco

3

ation Eq. No. this article

eccrr Ec

Ec fcc=ecc (19) and (191)

d Mander et al. [11]E1ec

2e1c

E2E1e1cf 2a

iec 1e21c

E2E1f 2a

h ie2c

(20)

E1l e2e1l E2lE1le1l

f 2a

iel 1e2

1l E2l E1l

f 2a

e2l

(21)

sed on Ahmad and Shah [1] and Richart et al. [21]E2 ec

1E2 ecf0

ni1n E2 ec(22)

E2l el1lE2l el

f 0l

nl1nl E2l el(23)

ott [19]

Eqs. (2) and (3) were used by Mander et al. [11] who showed thatthe axial strain at maximum stress ecc can be expressed as a func-tion of the strength of conned concrete fcc:

ecc eco 1 5 fccfco 1

4

Another way to determine the peak axial stress is the one proposedby William and Warnke [29] which is adopted by several analysis-oriented models for concrete connement

fcc fco 2:2541 7:94 fl

fco

s 2 fl

fco 1:254

!5

Considering the purpose of studying FRP-conned concrete, the lat-eral pressure fl at which the composite jacket fails is expressed as:

fl 2tf EfD eh;rup 6

3.3. Stressstrain relation

3.3.1. AOM modelsMirmiran and Shahawy [15,16] were the rst to apply a passive

connement model based on incremental approach of actively-conned curves to FRP-conned concrete. The axial stress-axialstrain relation is based on Popovics [30] (Eqs. 19, 191) being thelateral strains obtained from their relation with axial strainsthrough dilation, since for linear elastic materials such as FRP theconning pressure rises in order to contain dilation. Therefore,the model presents equations to determine the dilation curvebased on the authors own tests.

Spoelstra and Monti [26] created a model that uses an incre-mental-iterative approach to calculate the stressstrain behaviourof the FRP-conned concrete. The stressstrain relation is alsobased on Popovics modied by Mander et al. [11] and the lateral-to-axial relation based on Pantazopoulou and Mills [31].

As the previous the model of Fam and Rizkalla [4] is based onPopovics model adapted by Mander et al. [11] but for concrete

eec

ececo

2ntaz

0:75

ial s

3120 P.F. Marques, C. Chastre / Composite Structures 94 (2012) 31153131Being tf the sheet thickness, Ef the elastic modulus of the FRP com-posite, D the diameter of the column, eh,rup the hoop strain at com-posite failure.

For the models herein studied Tables 4 and 5 show for AOM andDOM, respectively, the equations that each model uses to deter-mine peak stress and the corresponding strain.

It can be seen that all the AOM base their Eqs. (6) and (8) inRichart et al.s model modied by Mander et al. for the determi-nation of strain at peak stress (Table 4). For the peak stress Tenget al.s model uses Richart et al.s equation modied by withtheir own testing data (Eq. (7)). The remaining three AOMauthors use William and Warnkes equation (5) to determinethe peak stress.

As to DOM authors (Table 5), the peak stress was calibratedfrom experimental data using Richart et al.s Eq. (2) (Eqs. 9, 11,13, 15, 17) where Matthys et al. [14] propose a revised form (Eq.(15)) of the Eq. (13) proposed by Toutanji [28]. The strain at stresspeak in Toutanji and Saa et al.s models are based on Manderet al.s Eq. (4) and calibrated with tests results (Eq. (12) and(14)). Toutanji revised model [14] (Matthys, Toutanji, Taerwe)adopts the Eq. (14) proposed by Toutanji [28] but for the 2nd re-gion the strain values are multiplied by 0.6 (Eq. (16)). Samaanet al. have their own Eq. (10) while Chastre and Silva adopted Eq.(18) calibrated from the experimental tests [3].

Table 7Axial-to-lateral relation of each model.

Models Axial-to-axial relation

Mirmiran and Shahawy [15,16] el lec

l l0 2l0 ececo

lu lmaxl0lmaxlu

1 2 ececo

lmaxl0lmaxlu

Spoelstra and Monti [26] Implicit in the model using PaFam and Rizkalla [4] Implicit in the modelTeng et al. [27] ec

eco 0:85 1 8 rl

fco

1

(

Samaan et al. [24], Toutanji [28], Saa et al. [23]Toutanji revised [14], Chastre and Silva [3]

Implicit by the bilinear both ax

l0 = 0.2, initial dilation rate = Poissons ratio of unconned concrete.lmax = 0.7611Ln(2Ejtf/fcoD) + 4.0167, peak dilation rate.lu = 0.1375Ln(2Ejtf/fcoD) + 0.8646, asymptotic dilation rate after decrease.

el = lateral strain.ec = axial strain.eco = peak axial strain for unconned concrete.cores conned by FRP tubes. Peak stress, corresponding strainand stressstrain relation are the same of the previous models (Ta-bles 5 and 6) but with its own axial-to-lateral relationship based onGardner [32] results and thus creating an equation to quantify thevariation of Poissons ratio under constant lateral conningpressure.

The model of Teng et al. [27] is also an AOM with incremental-iterative approach but differs from the former as regards the equa-tions to determine the peak stress and its corresponding strain andthe lateral-to-axial relationship. The authors use Richart et al.sequations calibrated with their research group testing results andpropose a general equation to represent the dilation properties(Eq. (26)) that is applicable to unconned, actively conned andFRP-conned concrete.

3.3.2. DOM modelsAlthough incremental but with no need of iteration the models

of Toutanji [28] and Saa et al. [23] are DOM and based on thesame approximation based on two equations for two region behav-iour. The initial equation of Ahmad and Shah [1] is used for the 1stregion and Richart et al.s equation [21] for the 2nd region modiedby Toutanji [28] (Fig. 3b, Table 6). Toutanjis model was based ontesting of concrete cylinders conned with FRP sheets while Saaet al.s model was based on testing of FRP conning tubes. Peakstress and corresponding strain equations are therefore different

Eq. no. this article

(24)

c

o

2 (25)

opoulou and Mills [31]

ele co

0:7 exp 7 el

eco

) (26)

tress-lateral strain and axial stress-axial strain relationships

(Table 5). As for Toutanji revised model [14], this is entirely basedon the same authors model (1999) though considering for the 2ndregion the failure strain in hoop direction corresponding to 60% ofthe ultimate strain of the CFRP material.

Samaan et al. [24] proposed a design-oriented model non-incremental based on a correlation between the dilation rate ofconcrete and the hoop (lateral) stiffness of the restraining FRPsheet. The authors used a single equation by calibrating the fourparameter stressstrain relation proposed by Richard and Abbott[19] with a bilinear model conguration for the two distinct re-gions and the use shape parameter for the transition zone. This isdirectly related to the material properties of the conning FRPand the concrete core. The 2nd region is proportional to the stiff-ness of the conning jacket.

Chastre and Silva [3] proposed a model for CFRP-conned con-crete cylinders based on the same single equation of Richard andAbbott [19] and calibrated with tests (Eq. (22)) in which the bothstress-axial strain and stress-lateral-strain relationships are ofbilinear type with a shape factor. In the axial stressaxial straincurve, the slope of the 1st region is considered identical to theone of the plain concrete, as the FRP jacket has a passive behaviourand is only activated for a level of lateral deformation similar to themaximum stress of the non-conned concrete. The same type of

equation is used for the axial stress-lateral strain relationship(Eq. (23)).

3.4. Dilation properties

As mentioned before the rst connement models were devel-oped for steel-conned concrete columns. Due to distinct proper-ties between steel and FRP, and whether the nature ofconnement is active or passive, the understanding of the dilationbehaviour of concrete is essential to the accuracy of proposedmodels.

According to current knowledge and from what experimentalevidences show (Fig. 2b) when conned concrete columns are axi-ally loaded volumetric changes develop. As regards passive con-nement, at a rst stage the column shortens and a contractionof its volume takes place. This behaviour goes on until a certainpoint where the lateral pressure of the conning material is acti-vated. The subsequent development may cause contraction orexpansion (dilation), depending on the used conning material,with relevant effect on the axial stressstrain relation.

Considering actively-conned concrete, where the lateral con-ning pressure is kept constant, the volumetric response is not re-lated with the axial stressstrain behaviour and its inuence isonly negligible over axial-lateral stressstrain relation [5,15,16].

150

0

500

1000

1500

2000

TestChastre & SilvaToutanjiTeng et al.Spoelstra & MontiFam & RizkallaSamaan et al.Saafi et al.Mirmiran & ShahawyToutanji revised

Axial load -Nc (kN)

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040Axial strain

P.F. Marques, C. Chastre / Composite Structures 94 (2012) 31153131 3121(a) Axial load lateral strain (left) Axial load axial strain (right)

Axial strainLateral strain

0

200

400

600

800

1000

1200

1400

1600

1800

2000

-0.010 -0.005 0.000 0.005 0.010

Volumetric strain

Axial load -Nc(kN)

volume expansion

volume contraction(c) Axial load volumetric load relation

Fig. 4. Test results vs models results: 150 mm diam. C(b) Axial to lateral strain relation

0.000-0.020 -0.015 -0.010 -0.005 0.000

Lateral strain

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0.000 0.010 0.020 0.030 0.040

Axial strain

Volumetric strain

volume expansion

volume contraction(d) Volumetric strain Axial strain relation

FRP-conned RC column; 2 plies of tf = 0.167 mm.

In steel-(passively) conned concrete the lateral strain el acti-vates the axial stress fc which increases until steel yields. However,it is likely that yield strength is reached long before the peak axialstress fcc [5,15,16]. Consequently the dilation behaviour does notaffect fcc and have little or negligible inuence on axial strain atpeak stress ecc, resembling what happens in active connement [5].

Due to its linear elastic behaviour FRP-conned concrete withexternal passive jackets shows distinct dilation response comparedto steel-conned concrete. This property allows the lateral strainand hence conning pressure to increase until failure of the FRPjacket is reached. Accordingly, if the volumetric response is ofexpansion the passive connement will be activated. The higherthe volumetric expansion (dilation) the more actuated is thepassive connement and therefore the higher (stiffer) is the axialstressstrain relation, which can increase signicantly the axialcompressive strength.

4. Modelling results and discussion

4.1. General

The comparison of the models proposed by the authors hereinpresented was done in view of existing experimental results on

CFRP-conned RC columns with different diameters. Despite beingavailable in the open literature, with tests on specimens rangingfrom 75 to 400 mm diameter, most of the experimental programswere based on small-size cylinders under 150 mm diameter [25].

Although the mentioned practical experience regarding thequantity of tests is unbalanced favouring small-size specimens,the analytical results of models is presented and compared withtests results of axially loaded columns with 150, 250 and400 mm diameter.

It should be borne in mind that these models have unalikeassumptions also in view of the different state of the art at the timethey were developed. Therefore, the comparison does not limititself to single parameter evaluation but the whole behaviour fromunloaded sate until failure.

Regarding the basis of modelling, most models do not have intoaccount that ultimate lateral hoop strain of the column does notequal the CFRP strain of the composite sheet as reported by Lamand Teng [9] and Toutanji revised [14]: (a) nonhomogeneousdeformations due to internal concrete cracking and hence non-uni-form stress distribution in the FRP jacket; (b) additional stress con-centration on FRP originated by buckling of longitudinal steelreinforcement; (c) irregularities in the FRP composite (misalign-ment of bres); (d) multiaxial stress state due to bonding between

2505001000

1500

2000

2500

3000

3500

4000

4500


Axial load -Nc (kN)

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040Axial strain

3122 P.F. Marques, C. Chastre / Composite Structures 94 (2012) 31153131(a) Axial load lateral strain (left) Axial load axial strain (right)

0


0

500

1000

1500

2000

2500

3000

3500

4000

4500

-0.012 -0.008 -0.004 0.000 0.004 0.008

Volumetric strain

Axial load -Nc (kN)

volume expansion

volume contraction(c) Axial loadvolumetric load relation

Fig. 5. Test results vs models results: 250 mm diam. C(b) Axial-to-lateral strain relation

0.000-0.020 -0.015 -0.010 -0.005 0.000

Lateral strain

-0.012

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.000 0.010 0.020 0.030 0.040

Axial strain

Volumetric strain

volume expansion


FRP-conned RC column; 2 plies of tf = 0.176 mm.

concrete and the FRP which may introduce in the latter part of theaxial loading. In the models of the present study this aspect is onlyaccounted by Toutanji revised model [14], Teng et al. [27] andChastre and Silva [3] in which the ultimate hoop strain is takenas 60% of the composite strain failure elu = 0.6ef.

Given the fact that the tested specimens were RC columns, thuswith longitudinal steel bars and transverse steel hoops, it is impor-tant to outline the fact that only the model of Chastre and Silva [3]

explicitly accounts for the presence of steel reinforcement. Never-theless, for all the models the contribution of longitudinal steelwas considered.

Figs. 46 show the behaviour of tests and of the implementedmodels with different approaches for each analysed column diam-eter: 150, 250 and 400 mm. Each gure is a set of four graphs (ad) where loadstrain and axial strainlateral strain relations aswell as dilation behaviour are presented. Tables 810 show the

Table 8Modelling results for /150 CFRP-conned column: Ncc; ecc; fl.

(a) Axial load lateral strain (left) Axial load axial strain (right)

(b) Axial-to-lateral strain relation

(d) Volumetric strain Axial strain relation(c) Axial load volumetric load relation

400

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000


Axial load -Nc (kN)


0.000

0.005

0.010

0.015

0.020

0.025

0.030

-0.014 -0.012 -0.010 -0.008 -0.006 -0.004 -0.002 0.000

Axial strain

Lateral strain

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

-0.012 -0.010 -0.008 -0.006 -0.004 -0.002 0.000 0.002 0.004

Volumetric strain

Axial load -Nc (kN)

volume expansion

volume contraction

-0.014

-0.012

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.000 0.005 0.010 0.015 0.020 0.025 0.030

Axial strain

Volumetric strain

volume expansion

volume contraction

Fig. 6. Test results vs models results: 400 mm diam. CFRP-conned RC column; 5 plies of tf = 0.117 mm.

P.F. Marques, C. Chastre / Composite Structures 94 (2012) 31153131 3123Model Ncc (kN) Err (Ncc) (%) ecc

Test /150 1486 0.0131Chastre & Silva 1493 0.5 0.0155Toutanji 1840 23.8 0.0230Teng et al. 1327 10.7 0.0135Spoelstra & Monti 1764 18.8 0.0368Fam & Rizkalla 1799 21.1 0.0253Samaan et al. 1487 0.1 0.0277Saa et al. 1459 1.8 0.0236Mirmiran & Shahawy 1778 19.7 0.0320

Toutanji revised 1482 0.2 0.0159

a The ratio is in this case fl/fD (Table 5).Err (ecc) (%) fl (MPa) fl/fco Err (fl//fco) (%)

10.7 0.272 18.5 9.1 0.270a 0.675.5 14.5 0.368 35.52.7 8.7 0.221 18.7

180.3 14.5 0.368 35.492.6 14.6 0.371 36.4

111.3 10.9 0.268 1.380.2 14.4 0.368 35.5

144.0 14.4 0.365 34.3

21.1 14.5 0.368 35.4

Err (ecc) % fl MPa fl/fco Err (fl//fco) %

ite STable 9Modelling results for / 250 CFRP-conned column: Ncc; ecc; fl.

Model Ncc kN Err (Ncc) % ecc

3124 P.F. Marques, C. Chastre / Composindividual results of axial strain at maximum load ecc, maximumload Ncc, ultimate lateral stress fl and corresponding errors as re-gards the tests results.

The option of presenting the relation between the axial load andboth axial strain and lateral strain instead of stressstrain curves isdue the presence of steel reinforcement in the concrete columns

Test / 250 3742 0.0155Chastre & Silva 3727 0.4 0.0143Toutanji 4193 12.1 0.0197Teng et al. 3085 17.5 0.0120Spoelstra & Monti 4063 8.6 0.0362Fam & Rizkalla 4128 10.3 0.0255Samaan et al. 3561 4.8 0.0270Saa et al. 3409 8.9 0.0203Mirmiran & Shahawy 4115 10.0 0.0315Toutanji revised 3454 7.7 0.0137

Table 10Modelling results for / 400 CFRP-conned column: Ncc; ecc; fl.

Model Ncc (kN) Err (Ncc) (%) ecc

Test /400 7460 0.0119Chastre & Silva 7428 0.4 0.0105Toutanji 8757 17.4 0.0121Teng et al. 6783 9.1 0.0085Spoelstra & Monti 8877 19.0 0.0249Fam & Rizkalla 8965 20.2 0.0213Samaan et al. 7493 0.4 0.0181Saa et al. 7457 0.0 0.0127Mirmiran & Shahawy 8711 16.8 0.0205Toutanji revised 7473 0.2 0.0086

0%17%

0%0%

20%19%

-9%17%

0%

-8%10%

-9%-5%

10%9%

-18%12%

0%0%

20%0%-2%

0%21%

19%-11%

24%0%

Toutanji revisedMirmiran&Shahawy

Saafi et al.Samaan et al.Fam&Rizkalla

Spoelstra&MontiTeng et al.

ToutanjiChastre&Silva

400





250





150Error (%) Ncc (model) vs. Ncc(Tests)

Fig. 7. Error models vs tests: peak load Ncc. 9.1 0.260 8.0 7.0 0.217 16.5tructures 94 (2012) 31153131and the consequent difference in the properties and behaviour ofconcrete and steel.

4.2. Loadstrain relation and axial-to-lateral strain relation

For the 150 mm diameter column Fig. 4a shows the loadstraincurves of the experimental results and the modelling results are

27.1 10.4 0.297 14.422.2 6.3 0.178 31.4133.5 10.5 0.297 14.364.6 10.4 0.294 13.374.4 9.3 0.264 1.431.2 10.5 0.297 14.3

103.4 10.5 0.297 14.411.7 10.5 0.297 14.3

Err (ecc) (%) fl (MPa) fl/fco Err (fl//fco) (%)

6.8 0.199 11.9 5.2 0.180 9.61.0 6.7 0.194 2.5

28.6 4.1 0.121 39.5109.6 6.9 0.201 0.979.0 7.0 0.204 2.552.2 7.1 0.207 4.06.8 6.8 0.199 0.1

72.3 6.3 0.183 8.027.4 6.7 0.194 2.5

7%72%

7%52%

79%110%

-29%1%

-12%

-12%103%

31%74%

65%134%

-22%27%

-8%

21%144%

80%111%

93%180%

3%76%

18%





400





250





150

Error (%) cc (model) vs. cc (Tests)

Fig. 8. Error models vs tests: axial strain at peak load ecc.

ite Sshown in Table 8 and Figs. 79. As regards the shape of the load-ax-

0%-8%

0%4%3%

1%-39%

-2%-10%

14%14%14%

1%13%14%

-31%14%

-17%

35%34%35%

-1%36%35%

-19%35%

-1%





400





250





150Error (%) fl / fco (model) vs. fl / fco (Tests)

Fig. 9. Error models vs tests: connement ratio at failure fl/fco.

P.F. Marques, C. Chastre / Composial strain curve, it appears that the models of Toutanji [28], Toutanjirevised [14] and Chaste and Silva [3] are those closer to the testcurve, although the rst is extended long beyond the test curve.Saa et al. [23] and Samaan et al. [24] clearly show themselvesmore divergent. In the load-lateral strain relation (left side of thegraph) Saa et al. [23], Samaan et al. [24] and Teng et al. [27] donot match the test curve. All the others seem aligned with the testcurve though only Toutanji revised [14] and Chastre and Silva [3]models have their curve limits close to test result. The maximumload is overestimated by Toutanji [28], Fam and Rizkalla [4], Spoesl-tra and Monti [26,15,16], while it is underestimated by Teng et al.[27]. Saa et al. [23], Samaan et al. [24] and Chastre and Silva [3]have close results. For the axial strain, Toutanji revised, Teng et al.and Chastre and Silva seem close to test values (Fig. 4b) while allothers overestimate it. As to the relation between axial and lateralstrains the closest curves are apparently those of Toutanji revised[14], Teng et al. [27] and Chastre and Silva [3], being these the onlymodels, among the ones analysed here, where the ultimate lateralstrain is close the test result of elu = 0.009m/m. The lateral strain di-rectly links the lateral failure stress and therefore consistent differ-ences between models (Table 8).

The results of the 250 mm diameter column are shown in Figs. 5and 7, 8, 9 and Table 9. The shape of loadstrain curves show thatChastre and Silva [3] model appears perfectly superposed to thetest curve in both axial and lateral cases. For the load-axial strainrelation, Toutanji revised curve [14] show slightly underestimatedvalues compared to the test curve, while Toutanji [28] curve seemsalso close, though slightly overestimating the test values in the 2ndregion of the curve and with load, axial and lateral strain values be-yond the limits of test. All remaining curves present a lower shapedevelopment in terms of axial load until the ultimate axial strain ofthe test, though these models extend their curves outside the re-ferred limit and some of them long beyond this limit [15,16,26].For the load lateral strain curve Saa et al. [23], Samaan et al.[24] and Teng et al. [27] show themselves more distant comparingwith the others. As to the axial-to-lateral relation (Fig. 5b) Chastreand Silva [3] curve is visibly the closest to the test curve and withits end coincident with the test ultimate lateral strainelu = 0.0093 m/m.

Following the same criteria analysis for the column with400 mm diameter, the results are presented in Fig. 6 and Table10. Several models have a load-axial strain curve progression closethe test [26,23,27,3]. Fam and Rizkalla [4] is slightly under the testcurve, while Mirmiran and Shahawy [15,16] and Toutanji revised[14] are slightly over it. Samaan et al. [24] (under) and Toutanji[28] (over) are considerably more distant. In the case of load-lateralstrain curve, Samaan et al. [24], Saa et al. [23] and Teng et al. [27]present their curves progress below the test curve while the otherare fairly superposed to this. Yet, again Mirmiran and Shahawy[15,16], Spoelstra and Monti [26], Toutanji [28] and Fam and Rizka-lla [4] go beyond the failure lateral strain, while, except for the tran-sition zone between the curves 1st and 2nd regions, Toutanjirevised [14] and Chastre and Silva curves [3] match the test curve,including the ultimate lateral strain. As to the axial-to-lateral rela-tion Spoelstra and Montis model [26] appears to have the closesttrend to test curve at the beginning, even though it moves awayalong the ending part of this one and it goes on past the limit of lat-eral strain of elu = 0.007 m/m. Chastre and Silvas model is the onethat most ts the second half of the test curve.

It is interesting to verify that although Toutanji [28] and Saa etal.s [23] models have the same basis the differences regard thecalibration of FRP sheets and FRP tubes, respectively they havedistinct curves both load-axial strain and load-lateral strain (Figs.46a). However, concerning the axial-to-lateral relation theircurves match perfectly between themselves.

Having an overall observation of the three diameter results Figs. 46a it is possible to realise that the load lateral strainrelation (left side) has more modelling curves matching the testcurves than the load axial strain relation (right side). Moreover,between the three sets of results, for each diameter, it is not possi-ble to see or conclude any consistent evolution of the models inview of the diameter increase or decrease.

4.3. Error (Deviation) of Ncc, ecc and fl/fco for each model

It is possible from Figs. 79 to have a global overview of themain parameters of each model for the three columns diameters:150, 250 and 400 mm. The error in face of test results is quantiedanalysing the peak load Ncc, the axial strain at peak load ecc, and theconnement ratio fl/fco. The individual values of each model arepresented in Tables 810.

Fig. 7 shows the deviation (in %) of the axial peak load. Chastreand Silva [3] have a match in all three diameters followed bySamaan et al. [24] with 0%, 5% and 0% and Toutanji revised [14]with 0%, 8% and 0% for 150, 250 and 400 mm diameters, respec-tively. The remaining models present errors between 9% and 24%.However, although Samaan et al.s model seems to be among thosewith less deviation as per Ncc results, Figs. 46a clearly show thatthese authors curves are the farthest from the tests curves. Thisindicates that the analysis of this sole parameter does not accu-rately represent the structural behaviour of axially connedcolumns.

As to the axial strain at peak load, the deviation is shown in thegraph of Fig. 8. The models with least deviation are Chastre and Sil-va [3]: 18%, 8%,12%, Toutanji revised [14]: 21%, 12%, 7% andTeng et al. [27]: 3%, 22%, 29%. Toutanji [28] and Saa et al.[23] give both a good approximation for 250 and 400 mm diameter

tructures 94 (2012) 31153131 3125columns (27%, 1%). Spoelstra and Monti [26]: 180%, 134%, 110% andMirmiran and Shahawy [15,16]: 144%, 103%, 72% present valuesfarther from tests results.

The results of the error of the connement ratio are shown inFig. 9 and this factor is dependent of the failure lateral strain elu.For the 150 mm diameter column Chastre and Silvas [3] modelhas a little deviation of 1%, Teng et al. [27] has 19%, while theremaining models, except for Samaan et al. [24], have errors ofapproximately 35%. For the 250 and 400 mm diameter Tenget al.s model presents deviations of 31% and 39%, respectively.Once more excluding Samaan et al. [24], the results for the 250 mmdiameter are between 13% and 17%. In the 400 mm diameter col-umn, except for Teng et al.s results, errors are between 10%and 4%. In this particular parameter Samaan et al. [24] show thebest t for the three diameters with errors between 1% and 4%,even though the loadstrain relation shows these models curvesas those farther from tests curves. This means that the connementratio alone has no sensitivity as regards the connement modelperformance.

As it is, it appears that comparing modelling curves with testscurves, together with the previous parameters, is the most suitedway of evaluating the performance of a connement model.

4.4. Error (Deviation) of Wc, Wl and Wv for each model

Even considering the previous analysis, it is possible that thequantication of the observed curves is required for a complete ap-

proach to the analysis of all the models comparing with the testsresults.

In view of this, additional parameters were created in order toobserve the deviation between models and tests. Each of theseparameters consists of the area underneath the curves to be ana-lysed: Wc is the area of the axial loadaxial strain curve; Wl isthe area of the axial loadlateral strain curve; Wv is the area ofthe axial loadvolumetric strain. In case of any physical meaningthese parameters units would be kN.m/m, the purpose is howeverto have an additional measure of the deviation of each curve.Hence, these parameters are calculated according to the followingexpression:

W Xni1

12yi yi1xi1 xi

27

where y is the ordinate of the graphic, x is the abscissa of the gra-phic, i is the index of summation and n is the upper bound ofsummation.

Herewith, Tables 1113 present the results of the mentionedparameters and Figs. 1012 show the error percentage of eachmodel for all three tested columns: 150, 250 and 400 mm,respectively.

Table 11Parameters Wc, Wl, Wv results for /150 CFRP-conned column.

Model Wc (kNm/m) Err (Wc) (%) Wl (kNm/m) Err (Wl) (%) Wv (kNm/m) Err (Wv) (%)

Test /150 13 11 8Chastre & Silva 17 26.4 10 4.8 3 58.2Toutanji 13 4.6 10 7.8 7 13.3Teng et al. 14 2.3 9 14.3 4 157.2Spoelstra & Monti 51 285.3 20 85.0 12 258.0Fam & Rizkalla 32 139.2 20 87.6 8 0.6Samaan et al. 31 132.4 11 6.7 8 208.4Saa et al. 11 15.6 9 18.1 6 22.5Mirmiran & Shahawy 44 233.1 19 76.5 7 191.4Toutanji revised 17 27.4 9 14.8 2 74.3

Wl (

28282523494933424826

Wl (

4445444185866073

3126 P.F. Marques, C. Chastre / Composite Structures 94 (2012) 31153131Table 12Parameters Wc, Wl, Wv results for /250 CFRP-conned column.

Model Wc (kNm/m) Err (Wc) (%)

Test / 250 43Chastre & Silva 39 9.2Toutanji 26 39.6Teng et al. 29 32.9Spoelstra & Monti 119 174.6Fam & Rizkalla 76 76.9Samaan et al. 74 71.6Saa et al. 55 28.6Mirmiran & Shahawy 103 139.2Toutanji revised 36 15.6

Table 13Parameters Wc, Wl, Wv results for /400 CFRP-conned column.

Model Wc (kNm/m) Err (Wc) (%)

Test /400 70Chastre & Silva 58 17.7Toutanji 38 45.3Teng et al. 45 36.3Spoelstra & Monti 179 154.4Fam & Rizkalla 143 102.9Samaan et al. 106 51.1Saa et al. 75 6.8

Mirmiran & Shahawy 145 106.5 71Toutanji revised 49 30.7 42kNm/m) Err (Wl) (%) Wv (kNm/m) Err (Wv) (%)

131.3 16 25.2

10.4 24 87.216.8 18 37.0

75.8 20 254.273.6 21 62.618.4 8 159.150.9 29 125.670.0 8 161.1

7.3 15 16.0

kNm/m) Err (Wl) (%) Wv (kNm/m) Err (Wv) (%)

180.4 31 70.01.7 49 165.97.6 37 102.8

90.7 10 154.592.9 28 54.134.5 13 29.165.3 71 290.2

59.1 4 123.2

5.4 35 91.8

Concerning the load-axial strain relation and the associatedparameter Wc (Fig. 10), for the three diameters, Chastre and Silva[3] (26%, 9%, 18%), Toutanji [28] (5%, 40%, 45%), Teng et al. [27](2%, 33%, 36%), Saa et al. [23] (16%, 29%, 7%) and Toutanjirevised [14] (27%, 15%, 74%) show closer values to the test curvesthan Spoelstra and Monti [26], Fam and Rizkalla [4], Samaan et al.[24] and Mirmiran and Shahawy [15,16] which range from51% to285%.

For the loadlateral strain relation and the parameter Wl(Fig. 11) the closest models to the tests results are Chastre andSilva [3], Toutanji [28] and Teng et al. [27] ranging from 0% to17%. The remaining models show results between 18% and 93%.

The parameter Wv is the one with less direct approach whenloadvolumetric strain curves are observed once these representdifferent behaviour stages that may vary from volume contractionto volume expansion. In this case the signal is important and whennegative it means that throughout the loading the volume expan-sion is prevailing in face of volume contraction. Table 13 andFig. 12 show that the results are highly scatter and that the modelwith less deviation for all diameters is Fam and Rizkallas [4] (2%,54%, 63%) followed by Fam and Rizkallas [28] (13%) for150 mm diameter, Toutanji revised [14] (16%) and Chastre andSilva [3] (25%) for 250 mm diameter and Samaan et al. [24](29%) for 400 diameter. Yet, Figs. 46c show that the model ofFam and Rizkalla [4] seems less close to the tests curve when com-pared to Toutanji revised [14] and Chastre and Silva [3].

As it can be understood, despite the analysis of these parame-ters, similar values between two models do not mean that theircurves progression is close in terms of load or strain values or even

4.5. Dilation behaviour

The linear elastic behaviour of FRPs has a relevant effect on thedevelopment of deformations on concrete columns conned withthis material. In view of what was described in Section 3.4, FRPsprovide a passive connement when applied on concrete cores,which means that the composite material is only activated sub-

5%-59%

-65%-35%

-93%-91%

8%2%0%

7%-70%

-51%-18%

-74%-76%

17%10%

1%

15%-77%

18%-7%

-88%-85%

14%8%

5%





400





250





150

Error (%) Wl(model) vs. Wl (Tests)

Fig. 11. Error of models vs tests: Wl area of load vs lateral strain relation (Ncxel).

-16%161%

-126%159%

-63%254%

-37%-87%

-25%

74%191%

22%208%

1%258%

157%13%

58%





250





150Error (%) Wv(model) vs. Wv (Tests)

P.F. Marques, C. Chastre / Composite Structures 94 (2012) 31153131 3127that their shape is similar. The assessment should always cross theinformation of all parameters and the visual observation of graphs.

It is nevertheless important to quantify these parameters Wc,Wl andWv to observe and explain the consistency of the columnsbehaviour with regard to different modelling approaches.Fig. 10. Error of models vs tests: Wc area of load vs axial strain relation (Ncxec).-92%

123%-290% 29%

-54%154%

-103%-166%

-70%





400Fig. 12. Error of models vs tests: Wv area of load vs volumetric strain relation(Ncxev).

jected to increasing concrete expansion, which means that lateralexpansion FRP stress increases until failure. As a result, FRP perfor-mance distinguishes from active connement, where externalpressure is constant, and from steel connement, where the lateralstress remains constant after steel yielding. Expansion has, thus,negligible inuence stressstrain relation in both these cases.

Accordingly, the comprehension of the stressstrain (or loadstrain) behaviour of FRP-conned concrete columns is evidentlylinked to its dilation behaviour. Progression of contraction andexpansion in the course of load increase are governing as regardsthe activation of the connement FRP sheet. Figs. 4c and d, 5cand d, 6c and d present the results that express the dilation behav-iour of tests and implemented models.

In the present study, with regard to AOM authors, Teng et al.model has a loadvolumetric strain curve with shape similar totest, though with lesser contraction, with start of expansion at low-er load and lesser expansion. Fam and Rizkalla [4] have also a sim-ilar shape (contraction and then expansion) but at signicanthigher load. Mirmiran and Shahawy [15,16] and Spoelstra andMonti [26] present their models with contractionexpansion-contraction which differs from the test curve. In the volumetricstrainaxial strain (Figs. 46d) relation the model of Mirmiranand Shahawy [15,16] has a very different development comparingto the test curve and Fam an Rizkalla [4] prolongs its contraction

beyond the failure point. The main differences respecting thisbehaviour come from the axial-to-lateral relationship (Table 7).

For DOM it should be taken into consideration that the 2nd re-gion slope depends on the FRP jacket stiffness which highly inu-ences the axial-to-lateral relationship. Certainly, calibrationinuences the behaviour. Toutanji [28] and Saa et al. [23] arebased on the same model, with slight calibration differences, andhave divergent development as regards loadvolumetric strainbut close curves in volumetric-axial strains relation, once the maincalibration differences are related to peak stress and correspondingstrain equations (Table 5) and not the axial-to-lateral relation.

The curve of Chastre and Silva [3] maintains a shape with sim-ilar progress to the test curve for the three specimens, both forloadvolumetric strain (Figs. 46c) and volumetric strainaxialstrain (Figs. 46d) and it is one of the models closer to the testcurve. Moreover, for 250 mm diameter column these authorscurve fairly matches the test curves, which is probably explainedby the fact that these authors have calibrated their model with alarge sample of 250 mm diameter specimens.

The models of Toutanji revised [14] and Teng et al. [27] seem tohave the relation loadvolumetric strain (Fig. 6c) near to the testcurve for 400 mm diameter, while for this relation Toutanjis model[28] has a development close to the test curve but thismodels curveis largely prolonged presenting considerably higher expansion.

150500

1000

1500

2000

TestChastre & Silva

Axial load -Nc (kN)

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018Axial strain

3128 P.F. Marques, C. Chastre / Composite Structures 94 (2012) 31153131(a) Axial load lateral strain (left) Axial load axial strain (right)

0-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 0.020

Teng et al.Toutanji revised


0

200

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1000

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Axial load -Nc(kN)

volume expansion


Fig. 13. Test results vs Chastre and Silva [3], Toutanji revised [14] and Teng et al. [27] m(b) Axial-to-lateral strain relation

0.000

0.002

-0.010 -0.005 0.000

Lateral strain

-0.006

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-0.002

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volume expansion


odelling results: 150 mm diam. CFRP-conned RC column; 2 plies of tf = 0.167 mm.

4.6. Models with performance closer to tests behaviour

Among the several models analysed, those proposed by Chastreand Silva [3], Toutanji revised [14] and Teng et al. [27] seem to becloser to the tests results for the three studied diameters in all rela-tions: loadaxial and lateral strain; axial strainlateral strain; loadvolumetric strain and volumetric strainaxialstrain.

Chastre and Silvas model have an almost perfect match for thethree diameters for both loadstrain relations (Figs. 1315a). Tout-anji revised model presents curves close to test results thoughslightly underestimated for 150 and 400 mm diameter and withslight overestimate for 250 mm diameter. Teng et al.s model iscloser to the test results of the 400 mm diameter column as re-gards the shape of the curves. However, this model underestimatesthe axial load and even the values of axial strain for 150 and400 mm diameter.

In the case of the lateral-to-axial relationship, Teng et al.s mod-el [27] underestimates in general the axial strains as does Toutanjirevised [14] for 250 and 400 mm diameter. For 250 mm diameterChastre and Silvas model [3] has almost a perfect match, thoughwith not such good performance for the columns with 150 mmand 400 mm diameter.

The dilation properties expressed through the relation betweenaxial load and the volumetric strain (Figs. 1315c) as well as therelation between the volumetric and the axial strain show the dif-culty of models in expressing this property which means that, ex-cept for Chastre and Silva [3] with the 250 mm diameter column,none of the models herein presented could even be close to testsresults. The main reason for this difference is the fact the volumet-ric strain depends on both axial and lateral strain and (ev = ec + 2el)and in their relationship, which seems to be quite difcult to modelin view of the experimental results.

5. Conclusions

This article has presented the analysis of 9 connement modelsfor FRP-conned concrete, in view of tests results of conned con-crete columns with diameters of 150, 250 and 400 mm. Four ofthese models are based on an analysis oriented stressstrain rela-tion while the remaining 5 have a design oriented approach.

From the several compared parameters, the study of the loadstrain relations (both axial and lateral) and the dilation behaviour,Chastre and Silvas model [3] appears to be the most accurate pre-dictive model among those herein studied.

250500

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Axial load -Nc (kN)

0.002

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P.F. Marques, C. Chastre / Composite Structures 94 (2012) 31153131 3129(a) Axial load lateral strain (left) Axial load axial strain (right)

0-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 0.020

Toutanji revised


0

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-0.008 -0.004 0.000 0.004

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Axial load -Nc (kN)

volume expansion


Fig. 14. Test results vs Chastre and Silva [3], Toutanji revised [14] and Teng et al. [27] m.008

.006

.004

.002

.000

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volume expansion(b) Axial-to-lateral strain relation

0.000000.0500.0-010.0-

Lateral strain

.004

.006Volumetric strain



00

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ite S(a) Axial load lateral strain (left) Axial load axial strain (right)

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Axial load -Nc (kN)


Axial load -Nc (kN)

3130 P.F. Marques, C. Chastre / ComposThe discussion of all modelling results in face of the tests resultsherein presented lead to the following conclusions:

Some of the models predict accurately (say error < 10%) thepeak load for the three tested diameters Chastre and Silva[3], Toutanji revised [14], Samaan et al. [24] and Saa et al.[23]. Yet, only Chastre and Silvas model has in all threecases a loadaxial strain curve shape close to tests curvesin all three cases.

For the axial strain at peak load most models give poor pre-dictions (say error: 50144%) except for: Chastre and Silva[3] (18%, 8%, 12%), Toutanji revised [14] (21%, 12%, 7%)and Teng et al. [27] (3%, 22%, 29%), for the three diame-ters; Toutanji [28] (27%, 1%) and Saa et al. [23] (31%, 7%)for 250 and 400 mm diameter, respectively.

As to the axial-to-lateral relationship, Figs. 46b show thatChastre and Silva [3] have the most accurate results for thethree diameters while Toutanji revised [14] present a closecurve for the 250 mm diameter test curve, Teng et al. [27],Toutanji [28] and Saa et al. [23] present also closer curvesfor the 150 mm diameter test curve, although the two lastauthors extend their both lateral and axial strains farbeyond tests limits. From this general analysis it is fair toconclude that the models that best represent the axial-to-

(c) Axial load-volumetric load relation

0

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volume expansion

volume contraction

-0

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-0

-0

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0

0

Fig. 15. Test results vs Chastre and Silva [3], Toutanji revised [14] and Teng et al. [27] m(b) Axial-to-lateral strain relation

.000

.002

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.008

.010

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.014

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Axial strain

Lateral strain

Volumetric strain

tructures 94 (2012) 31153131lateral relation of connement are those that best suit allparameters;

The importance of the dilation behaviour on the stressstrain (loadstrain) response of an FRP-conned concrete(passive connement and linear elastic behaviour) hasbeen proved consistent with the accuracy of results. In fact,the model that best captured the dilation behaviour Chas-tre and Silva [3] (Figs. 4c and d, 5c and d, 6c and d) cor-responds to the one of most accurate load-axial straincurve. As exposed in Section 3.4 the dilation has negligibleinuence on the load-lateral strain response;

The parameters Wc, Wl and Wv, used to quantify the shape(area underneath each curve) of the several curves thatexpress the conning behaviour of columns, are an addi-tional useful way of assessing the performance of existingmodels in view of the experimental results;

Taking into account circular cross-section RC columns from150 to 400 mm diameter conned with CFRP sheets, theseveral curves of results and all the presented parameters,the model with best performance among the nine modelsherein studied is Chastre and Silvas [3], although in someof the analysed parameters followed by the modellingresults of Toutanji revised [14]. Among all models, Chastreand Silvas [3] is the only that explicitly accounts for the

(d) Volumetric strain Axial strain relation

.008

.006

.004

.002

.000

.002

.004

.006

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014

Axial strain

volume expansion

volume contraction


connement contribution given of steel hoops in the con-crete columns.

The purpose of the present study is to provide a contribution tothe investigation on the behaviour of RC circular columns connedwith CFRP since the start of compressive loading until failure. Eventhough several parameters such as peak load, corresponding strainand lateral failure stress were outlined and analysed, the analysisof the shape of each curve was considered of major relevance forthe comparison among all models. This is why the additionalparameters Wc, Wl and Wv were created: the shape of each model-ling curve can thus be assessed in view of the experimental curves.However, as the ultimate strength and strain in experimental testspresented some variability, it is advisable to carry out and continuethis comparative analysis with more experimental tests.

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P.F. Marques, C. Chastre / Composite Structures 94 (2012) 31153131 3131by circular ber-reinforced polymer tubes. ACI Struct J 2001;98(4):45161.[5] Grassl P. Modelling of dilation of concrete and its effect in triaxial compression.

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strengthening of concrete due to connement.1. Experimental studies. J ReinfPlast Compos 1995;14(9):100830.

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Performance analysis of loadstrain models for circular columns confined with FRP composites1 Introduction2 Test database2.1 Tested columns2.2 Materials properties2.3 Columns test results

3 Existing models for the confinement of circular cross-section concrete with FRP jackets3.1 Introduction3.2 Peak axial stress and corresponding strain3.3 Stressstrain relation3.3.1 AOM models3.3.2 DOM models

3.4 Dilation properties

4 Modelling results and discussion4.1 General4.2 Loadstrain relation and axial-to-lateral strain relation4.3 Error (Deviation) of Ncc, cc and fl/fco for4.4 Error (Deviation) of Wc, Wl and W? for each 4.5 Dilation behaviour4.6 Models with performance closer to tests behaviour

5 ConclusionsReferences

Documents

2012Marques - Performance Analysis of Load–Strain Models