2003Lam and Teng - Design-Oriented Stress–Strain Model for FRP-confined Concrete

Construction and Building Materials 17(2003) 471–489

0950-0618/03/$ - see front matter� 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0950-0618(03)00045-X

Design-oriented stress–strain model for FRP-confined concrete

L. Lam, J.G. Teng*

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, PR China

Abstract

External confinement by the wrapping of FRP sheets(or FRP jacketing) provides a very effective method for the retrofit ofreinforced concrete(RC) columns subject to either static or seismic loads. For the reliable and cost-effective design of FRPjackets, an accurate stress–strain model is required for FRP-confined concrete. In this paper, a new design-oriented stress–strainmodel is proposed for concrete confined by FRP wraps with fibres only or predominantly in the hoop direction based on a carefulinterpretation of existing test data and observations. This model is simple, so it is suitable for direct use in design, but in themeantime, it captures all the main characteristics of the stress–strain behavior of concrete confined by different types of FRP. Inaddition, for unconfined concrete, this model reduces directly to idealized stress–strain curves in existing design codes. In thedevelopment of this model, a number of important issues including the actual hoop strains in FRP jackets at rupture, thesufficiency of FRP confinement for a significant strength enhancement, and the effect of jacket stiffness on the ultimate axialstrain, were all carefully examined and appropriately resolved. The predictions of the model are shown to agree well with testdata.� 2003 Elsevier Ltd. All rights reserved.

Keywords: Concrete; Fibre reinforced polymer; Confinement; Stress–strain model; Compressive strength; Ultimate strain; Design

1. Introduction

Fibre reinforced polymer(FRP) composites havefound increasingly wide applications in civil engineeringdue to their high strength-to-weight ratio and highcorrosion resistance. One important application of FRP-composites is as a confining material for concrete, inboth the retrofit of existing reinforced concrete(RC)columns by the provision of an FRP jacket, and inconcrete-filled FRP tubes in new construction. As aresult of FRP confinement, both the compressivestrength and ultimate strain of concrete can be greatlyenhanced. In both types of applications, an accurateaxial stress–axial strain model(referred to simply asstress–strain model hereafter) is required for FRP-confined concrete for reliable and cost-effective designs.In early studies of FRP retrofit of RC columns, the

stress–strain model of Mander et al. for steel-confinedconcretew1x was directly used in the analysis of FRP-confined concrete columnsw2,3x. Subsequent studies,however, showed that this direct use is inappropriate.This is because in Mander et al.’s modelw1x, a constant

*Corresponding author. Tel.:q852-2766-6012; fax:q852-2334-6389.

E-mail address: [email protected](J.G. Teng).

confining pressure is assumed, which is the case forsteel-confined concrete when the steel is in plastic flow,but not the case for FRP-confined concrete. As FRPcomposites remain linear elastic until final rupture, thelateral confining pressure in FRP-confined concreteincreases continuously with the applied load.Many investigations have been conducted into the

behavior of FRP-confined concrete and as a result, anumber of stress–strain models have been proposed.These models can be classified into two categories:(a)design-oriented modelsw4–11x, and(b) analysis-orient-ed modelsw12–14x. In the first category, the compressivestrength, ultimate axial strain(hereafter, often referredto as ultimate strain for brevity) and stress–strainbehavior of FRP-confined concrete are predicted usingclosed-form equations based directly on the interpreta-tion of experimental results. In the second category,stress–strain curves of FRP-confined concrete are gen-erated using an incremental numerical procedure. In thissecond approach, an active confinement model for con-crete is used to evaluate the axial stress and strain ofpassively confined concrete at a given confining pressureand the interaction between the concrete and the confin-ing material is explicitly accounted for by equilibriumand radial displacement compatibility considerations. In

472 L. Lam, J.G. Teng / Construction and Building Materials 17 (2003) 471–489

Fig. 1. Confining action of(b) FRP jacket to(a) Concrete.

the three studies cited abovew12–14x, the model ofMander et al.w1x was used as the active confinementmodel.Although analysis-oriented models have advantages

in accounting for the interaction between concrete andconfining materials including both steel and FRP com-posites, the complexity of the incremental process pre-vents analysis-oriented models from direct use in design.They are, however, suitable for incorporation in com-puter-based numerical analysis such as non-linear finiteelement analysis. Compared to analysis-oriented models,design-oriented models are particularly suitable fordirect application in design calculations. A simple andaccurate design-oriented stress–strain model offers anapproach that is familiar to engineers for determiningthe strength and ductility of FRP-confined RC structuralmembers. It may be worth noting that in Eurocode 2w15x, although a stress–strain model for concrete inuniaxial compression is provided for structural analysis,a simpler idealized model is recommended for designuse, which thus represents a similar differentiationbetween analysis-oriented and design-oriented models.This paper is concerned with the development of a

new design-oriented stress–strain model for concreteconfined by an FRP jacket in which the reinforcingfibres are only or predominantly oriented in the hoopdirection, so that the jacket has little longitudinal stiff-ness. That is, the jacket can be simplified as a unidirec-tional material providing only hoop resistance to anyexpansion of the concrete. The ultimate condition ofsuch FRP-confined concrete is reached when the FRPruptures due to hoop tensile stresses; failure of insuffi-cient vertical lap joints is excluded from considerationhere. Concrete-filled FRP tubes for new constructionalso make use of FRP confinement but they differsignificantly in behavior as a result of the substantiallongitudinal stiffness possessed by the tube(e.g. part ofthe hoop strain of the tube comes from its own Poison’seffect). Concrete confined by such FRP tubes is thusexcluded from consideration here.The paper begins with a review of the fundamental

behavior of FRP-confined concrete as established byexisting test results, followed by a discussion of thedeficiencies of existing design-oriented stress–strainmodels based on available test observations. A newdesign-oriented stress–strain model, which overcomesthese deficiencies is then presented and compared withtest data.

2. Confining action of FRP jacket

In most applications, the lateral confinement providedby an FRP jacket to concrete is passive in nature. Whenthe concrete is subject to axial compression, it expandslaterally. This expansion is confined by the FRP jacket,which is loaded in tension in the hoop direction. Differ-

ent from steel-confined concrete in which the lateralconfining pressure is constant following the yielding ofsteel, the confining pressure provided by the FRP jacketincreases with the lateral strain of concrete because FRPdoes not yield. The confining action in FRP-confinedconcrete can be schematically illustrated in Fig. 1, whereall stresses are shown in their positive directions. In theconcrete, compressive stresses and strains are defined aspositive but in the FRP, tensile stresses and strains arepositive. The lateral(radial) confining pressure actingon the concrete cores is given byr

s t 2s th hs s s (1)r R d

wheres stensile stress in the FRP jacket in the hooph

direction, tstotal thickness of the FRP jacket, andRand dsradius and diameter of the confined concretecore, respectively. If the FRP is loaded in hoop tensiononly, then the hoop stress in the FRP jackets ish

proportional to the hoop strain due to the linearity ofh

FRP and is given by

s sE ´ (2)h frp h

whereE selastic modulus of FRP.frp

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3. Experimental behavior of FRP-confined concrete

3.1. Test database

Many tests have been conducted on FRP-confinedconcrete. In the present study, a database containing thetest results of 76 FRP-wrapped plain concrete circularspecimens was assembled from an extensive survey ofthe open literature(Table 1). These 76 specimens werereported by Picher et al.w16x, Watanable et al.w17x,Matthys et al.w18x, Purba and Muftiw19x, Kshirsagar etal. w20x, Rochette and Labossierew21x, Xiao and Wuw11x, Aire et al. w22x, Dias da Silva and Santosw23x,Micelli et al. w24x, Pessiki et al.w25x, Wang and Cheongw26x, De Lorenzis et al.w27x, and Shehata et al.w28x.The specimens included in the database have diametersd from 100 mm to 200 mm and unconfined concretestrengths from 26.2 to 55.2 MPa. Two specimensf9cowith a very small diameter(ds55 mm) tested by DeLorenzis et al.w27x have been excluded. As this studyis limited to normal strength concrete, 10 specimenstested by Aire et al.w22x with have alsof9 s69 MPaco

been excluded. It should be noted that the term ‘speci-men’ is used loosely here for convenience, as some ofthe specimens represent the average performance of upto three nominally identical physical specimens.Different types of FRP were used in the specimens

in the database, namely carbon FRP(CFRP), aramidFRP(AFRP), and glass FRP(GFRP). The carbon fibresused include high strength and high modulus carbonfibres. In the following discussion, the FRP preparedfrom high strength carbon fibres is referred to simply asCFRP, but that prepared from high modulus carbonfibres is referred to as HM CFRP. The fibres employedwere supplied in the form of unidirectional tow sheets(carbon fibres), or woven fabrics with the fibres(aramidand glass fibres) mainly in one direction. They werewrapped on concrete cylinders with the main fibresrunning in the hoop direction, so the resulting FRPjacket had an insignificant stiffness in the axial direction.A few specimens that were wrapped with FRP jacketswith a significant stiffness in the axial direction havebeen excluded from the database. These include twospecimens tested by Pessiki et al.w25x with glass fibresat 0 and"458 from the hoop direction, and threespecimens tested by Dias da Silva and Santosw23x withglass fibre woven fabrics having a fibre thickness of0.094 mm and 0.040 mm in the circumferential andaxial directions, respectively. For most specimens(Table1a), the FRP properties were determined from flatcoupon tensile testsw29x by researchers themselves. Forthe rest(Table 1b), the FRP properties were suppliedby manufacturers.Table 1 reports the compressive strength andf9cc

ultimate axial strain of confined concrete, and thecu

FRP hoop strain at rupture for all specimens. Theh,rup

axial strains are average values that were obtained eitherusing strain gauges(up to three) at the mid height ofthe specimens or from relative displacement measure-ments of the middle region or between the two endsusing linear-variable differential transformers(LVDTs)(up to two). The FRP hoop strains are also averagevalues from strain gauges(up to four), or are taken tobe the same as lateral strains deduced from measure-ments of LVDTs at the mid height of specimens exceptthose reported by Pessiki et al.w25x. In Pessiki et al.’sstudy, an array of strain gauges was used to measureFRP hoop strains, and where possible, average valuesfrom a number of gauges of the critical regions (nearlocations of rupture) were reported. It should be notedthat assuming the deformation of the confined concretecylinder is truly axisymmetric, the lateral strain and thehoop strain of the FRP jacket are always equal inmagnitude but opposite in sign according to the presentsign convention. This is taken to be true in the presentstudy, despite that a small amount of asymmetry isunavoidable in the deformation due to factors such asthe inhomogeneity of concrete and eccentricity ofloading.

3.2. Failure mode and FRP hoop rupture strain

All specimens included in Table 1 failed by therupture of the FRP jacket due to hoop tension. This isthe most common mode of failure for FRP-confinedconcrete, although premature failure due to the separa-tion of the FRP at the vertical lap joint has also beenreported for specimens with an insufficient lap lengthw19x. Specimens failing by a mode other than FRPrupture have been excluded from the present database.In existing models for FRP-confined concrete, it is

commonly assumed that the FRP ruptures when thehoop stress in the FRP jacket reaches its tensile strengthfrom either flat coupon testsw29x or ring splitting testsw30x which is herein referred to as the FRP materialtensile strength. This assumption is the basis for calcu-lating the maximum confining pressuref (the confiningl

pressure reached when the FRP ruptures) using thefollowing equation:

2f tfrpfs (3)l d

where f sFRP material tensile strength in the hoopfrp

direction. The confinement ratio of an FRP-confinedspecimen is defined as the ratio of the maximumconfining pressure to the unconfined concrete strength( ).f yf9l co

However, experimental results show that in mostcases, the FRP material tensile strength was not reachedat the rupture of FRP in FRP-confined concrete. Table2 provides the average ratios between the measured

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Table 1Test results of FRP-wrapped concrete specimens

No. Source of data d L f9co ćo Fiber type t f frp Efrp ćc ću ´h,rup f9cc f9cu fo

(mm) (mm) (MPa) (%) (mm) (MPa) (GPa) (%) (%) (%) (MPa) (MPa) (MPa)(a) FRP properties from flat coupon tests by researchers

1 Watanable et al.w17x 100 200 30.2 0.23 Carbon 0.17 2716 224.6 1.51 0.94 46.6 32.02 Watanable et al.w17x 100 200 30.2 0.23 Carbon 0.50 2873 224.6 3.11 0.82 87.2 35.03 Watanable et al.w17x 100 200 30.2 0.23 Carbon 0.67 2658 224.6 4.15 0.76 104.6 35.04 Watanable et al.w17x 100 200 30.2 0.23 HM carbon 0.14 1579 628.6 0.57 0.23 41.7 30.05 Watanable et al.w17x 100 200 30.2 0.23 HM carbon 0.28 1824 629.6 0.88 0.22 56.0 36.06 Watanable et al.w17x 100 200 30.2 0.23 HM carbon 0.42 1285 576.6 1.30 0.22 63.3 40.07 Watanable et al.w17x 100 200 30.2 0.23 Aramid 0.15 2589 97.1 1.58 2.36 39.0 30.08 Watanable et al.w17x 100 200 30.2 0.23 Aramid 0.29 2707 87.3 4.75 3.09 68.5 30.59 Watanable et al.w17x 100 200 30.2 0.23 Aramid 0.43 2667 87.3 5.55 2.65 92.1 35.010 Matthys et al.w18x 150 300 34.9 0.21 Carbon 0.12 2600 200.0 0.85 1.15 44.3 32.511 Matthys et al.w18x 150 300 34.9 0.21 Carbon 0.12 2600 200.0 0.72 1.08 42.2 32.512 Matthys et al.w18x 150 300 34.9 0.21 HM carbon 0.24 1100 420.0 0.40 0.19 41.3 31.013 Matthys et al.w18x 150 300 34.9 0.21 HM carbon 0.24 1100 420.0 0.36 0.18 40.714 Kshirsagar et al.w20x 102 204 38.0 0.22 E-glass 1.42 363 19.9 1.73 1.74 57.0 38.115 Kshirsagar et al.w20x 102 204 39.4 E-glass 1.42 363 19.9 1.60 2.07 63.1 40.016 Kshirsagar et al.w20x 102 204 39.5 E-glass 1.42 363 19.9 1.79 1.89 60.4 40.017 Rochette and Labossierew21x 100 200 42.0 Carbon 0.60 1265 82.7 1.65 0.89 73.518 Rochette and Labossierew21x 100 200 42.0 Carbon 0.60 1265 82.7 1.57 0.95 73.519 Rochette and Labossierew21x 100 200 42.0 Carbon 0.60 1265 82.7 1.35 0.80 67.620 Rochette and Labossierew21x 150 300 43.0 Aramid 1.27 230 13.6 1.11 1.53 47.321 Rochette and Labossierew21x 150 300 43.0 Aramid 2.56 230 13.6 1.47 1.39 58.922 Rochette and Labossierew21x 150 300 43.0 Aramid 3.86 230 13.6 1.69 1.33 71.023 Rochette and Labossierew21x 150 300 43.0 Aramid 5.21 230 13.6 1.74 1.18 74.424 Xiao and Wuw11x 152 305 33.7 Carbon 0.38 1577 105.0 1.20 0.84 47.9 31.225 Xiao and Wuw11x 152 305 33.7 Carbon 0.38 1577 105.0 1.40 1.15 49.7 31.226 Xiao and Wuw11x 152 305 33.7 Carbon 0.38 1577 105.0 1.24 0.87 49.4 31.227 Xiao and Wuw11x 152 305 33.7 Carbon 0.76 1577 105.0 1.65 0.91 64.6 36.028 Xiao and Wuw11x 152 305 33.7 Carbon 0.76 1577 105.0 2.25 1.00 75.2 36.029 Xiao and Wuw11x 152 305 33.7 Carbon 0.76 1577 105.0 2.16 1.00 71.8 36.030 Xiao and Wuw11x 152 305 33.7 Carbon 1.14 1577 105.0 2.45 0.82 82.9 38.431 Xiao and Wuw11x 152 305 33.7 Carbon 1.14 1577 105.0 3.03 0.90 95.4 38.432 Xiao and Wuw11x 152 305 43.8 Carbon 0.38 1577 105.0 0.60 0.98 0.81 54.8 54.4 50.433 Xiao and Wuw11x 152 305 43.8 Carbon 0.38 1577 105.0 0.39 0.47 0.76 52.1 51.4 50.434 Xiao and Wuw11x 152 305 43.8 Carbon 0.38 1577 105.0 0.28 0.37 0.28 48.7 39.2 50.435 Xiao and Wuw11x 152 305 43.8 Carbon 0.76 1577 105.0 1.57 0.92 84.0 50.436 Xiao and Wuw11x 152 305 43.8 Carbon 0.76 1577 105.0 1.37 1.00 79.2 50.437 Xiao and Wuw11x 152 305 43.8 Carbon 0.76 1577 105.0 1.66 1.01 85.0 50.438 Xiao and Wuw11x 152 305 43.8 Carbon 1.14 1577 105.0 1.74 0.79 96.5 50.439 Xiao and Wuw11x 152 305 43.8 Carbon 1.14 1577 105.0 1.68 0.71 92.6 50.440 Xiao and Wuw11x 152 305 43.8 Carbon 1.14 1577 105.0 1.75 0.84 94.0 50.441 Xiao and Wuw11x 152 305 55.2 Carbon 0.38 1577 105.0 0.28 0.69 0.70 57.9 50.342 Xiao and Wuw11x 152 305 55.2 Carbon 0.38 1577 105.0 0.41 0.48 0.62 62.9 53.143 Xiao and Wuw11x 152 305 55.2 Carbon 0.38 1577 105.0 0.31 0.49 0.19 58.1 52.0

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44 Xiao and Wuw11x 152 305 55.2 Carbon 0.76 1577 105.0 0.39 1.21 0.74 74.6 73.1 68.645 Xiao and Wuw11x 152 305 55.2 Carbon 0.76 1577 105.0 0.81 0.83 77.6 68.646 Xiao and Wuw11x 152 305 55.2 Carbon 1.14 1577 105.0 1.43 0.76 106.5 61.247 Xiao and Wuw11x 152 305 55.2 Carbon 1.14 1577 105.0 1.45 0.85 108.0 61.248 Xiao and Wuw11x 152 305 55.2 Carbon 1.14 1577 105.0 1.18 0.70 103.3 61.249 De Lorenzis et al.w27x 120 240 43 Carbon 0.3 1028 91.1 0.87 1.16 0.70 58.5 55.650 De Lorenzis et al.w27x 120 240 43 Carbon 0.3 1028 91.1 0.82 0.95 0.80 65.6 63.551 De Lorenzis et al.w27x 150 300 38 Carbon 0.45 1028 91.1 0.71 0.95 0.80 6252 De Lorenzis et al.w27x 150 300 38 Carbon 0.45 1028 91.1 1.24 1.35 0.80 67.3

(b) FRP properties from manufacturers53 Picher et al.w16x 152 304 39.7 Carbon 0.36 1266 83.0 1.070 0.84 56.0 42.554 Purba and Muftiw19x 191 788 27.1 Carbon 0.22 3483 230.5 0.576 0.67 53.9 43.855 Aire et al.w22x 150 300 42.0 Glass 0.149 3000 65.0 0.30 0.73 0.55 41.0 35.756 Aire et al.w22x 150 300 42.0 Glass 0.447 3000 65.0 1.74 1.30 61 35.757 Aire et al.w22x 150 300 42.0 Glass 0.894 3000 65.0 2.5 1.10 85 41.158 Aire et al.w22x 150 300 42.0 Carbon 0.117 3900 240.0 1.1 0.95 46 40.759 Aire et al.w22x 150 300 42.0 Carbon 0.351 3900 240.0 2.26 1.05 77 40.760 Aire et al.w22x 150 300 42.0 Carbon 0.702 3900 240.0 3.23 1.06 108 40.761 Dias da Silva and Santosw23x 150 600 28.2 Carbon 0.111 3700 240.0 0.39 0.26 31.4 28.262 Dias da Silva and Santosw23x 150 600 28.2 Carbon 0.222 3700 240.0 2.05 1.18 57.4 26.863 Dias da Silva and Santosw23x 150 600 28.2 Carbon 0.333 3700 240.0 2.59 1.14 69.5 29.664 Dias da Silva and Santosw23x 150 600 28.2 HM carbon 0.167 3000 390.0 0.75 0.37 41.5 2465 Dias da Silva and Santosw23x 150 600 28.2 HM carbon 0.334 3000 390.0 1.81 0.69 65.6 28.266 Dias da Silva and Santosw23x 150 600 28.2 HM carbon 0.501 3000 390.0 1.69 0.64 79.4 28.267 Micelli et al. w24x 102 204 37 Carbon 0.16 3790 227.0 1.02 1.2 60 3768 Micelli et al. w24x 102 204 32 Glass 0.35 1520 72.0 1.25 1.25 52 3369 Pessiki et al.w25x 152 610 26.2 0.22 E-glass 1.00 383 21.6 1.30 1.15 38.4 31.570 Pessiki et al.w25x 152 610 26.2 0.22 E-glass 2.00 383 21.6 1.82 1.24 52.5 31.571 Pessiki et al.w25x 152 610 26.2 0.22 Carbon 1.00 580 38.1 1.44 0.81 50.6 33.972 Pessiki et al.w25x 152 610 26.2 0.22 Carbon 2.00 580 38.1 1.65 0.72 64.0 33.973 Wang and Cheongw26x 200 600 27.9 0.16 Carbon 0.36 4400 235.0 1.52 0.85 82.8 32.574 Wang and Cheongw26x 200 600 27.9 0.16 Carbon 0.36 4400 235.0 1.43 1.07 81.2 32.575 Shehata et al.w28x 150 300 29.8 0.21 Carbon 0.165 3550 235.0 1.23 1.23 57.0 3376 Shehata et al.w28x 150 300 29.8 0.21 Carbon 0.33 3550 235.0 1.74 1.19 72.1 34


Table 2Average hoop rupture strain ratios

Type of fibre No. of FRP material Ratio of hoopspecimens ultimate rupture strain

tensile to FRPstrain material´ fromfrp ultimatecoupon tensile straintests ´ y´ (%)h,rup frp

Average S.D. Average S.D.

CFRP 52 0.0148 0.0015 58.6 15.3High modulus CFRP 8 0.0045 0.0027 78.8 16.8AFRP 7 0.0223 0.0068 85.1 9.5GFRP 9 0.0280 0.0136 62.4 36.4Total 76 0.0160 0.0080 63.2 20.5

Fig. 2. Hoop strains at rupture of FRP jackets.

hoop strain at FRP rupture and the FRP materialh,rup

ultimate tensile strain for a number of categories. Itfrp

is seen that the average ratio differs for specimensconfined by a different type of FRP, and has a value of0.63 when all specimens of the present database areconsidered together. Thus, the maximum confining pres-sure given by Eq.(3) is only a nominal value. Theactual maximum confining pressure should be given by

2E t´frp h,rupf s (4)l,a d

where f is the actual maximum confining pressure.l,a

The actual confinement ratio is then given by the ratiobetweenf and .f9l,a co

Table 2 indicates that the assumption that the FRPruptures when the stress in the jacket reaches the FRPmaterial tensile strength is invalid for concrete confinedby FRP wraps. Fig. 2 shows the ratio of FRP hooprupture strain from tests to the FRP material ultimatetensile strain against the actual confinement ratio

, which illustrates the great scatter displayed byf yf9l,a co

the data. In particular, this figure indicates that inconcrete confined by a small amount of FRP, prematurerupture of the FRP at a hoop strain much lower thanthe material ultimate tensile strain is very likely. Forexample, the four specimens with very low levels ofFRP confinement failed with hoop rupture strains below20% of the material ultimate tensile strain. The maxi-mum value of of these four specimens is onlyf yf9l,a co

0.034.The difference between the FRP tensile strength or

ultimate strain from material tests and that reached intests of FRP-confined concrete specimens has beendiscussed in a number of recent papersw13,25,31–33x.Several causes have been suggested for this phenome-non. The two main causes are believed to be(a)deformation localization in cracked concrete leading toa non-uniform stress distribution in the FRP jacket andthus premature rupture of FRP, and(b) the effect of

curvature of an FRP jacket on the tensile strength ofFRP. Shahawy et al.w31x suggested that ring splittingtests could provide a closer estimation of the hooprupture strain of FRP. Thorough studies on these aspectsare not yet available. The hoop rupture strains reportedin the existing literature are in general average valuesaround the circumference, so strain distributions aroundthe circumference in FRP-confined specimens areunclear at this stage.

3.3. Stress–strain response

It has been well recognized that the stress–straincurve of FRP-confined concrete features a monotonicallyascending bi-linear shape as shown in Fig. 3a, if theamount of FRP exceeds a certain threshold. Such FRP-confined concrete is said to be sufficiently confined.This type of stress–strain curves(the increasing type)was observed in the vast majority of the tests coveredby the present database. With this type of stress–straincurves, both the compressive strength and the ultimatestrain are reached at the same point and are significantlyenhanced. However, existing tests have also shown thatin some cases such a bi-linear stress–strain behaviorcannot be expected. Instead, the stress–strain curvefeatures a post-peak descending branch and the com-pressive strength is reached before FRP rupture(thedecreasing type). This decreasing type of stress–straincurves can be further differentiated in terms of the stressin concrete at the ultimate strain (Fig. 3b,c). If thef9custress–strain curve terminates at a concrete stressf9cuabove the compressive strength of unconfined concreteas illustrated in Fig. 3b, the FRP confinement is stillf9co

sufficient to lead to strength enhancement. Such concreteis also referred to as sufficiently confined concrete inthe present study. However, if the stress–strain curveterminates at a stress as illustrated in Fig. 3c,f9 -f9cu co

the specimen is said to be insufficiently confined, where


Fig. 3. Classification of stress–strain curves of FRP-confined concrete.(a) Increasing type;(b) Decreasing type with ;(c) Decreasingf9 )f9cu co

type with .f9 -f9cu co

Fig. 4. Typical stress–strain curves of FRP-confined concrete.

little strength enhancement can be expected. The behav-ior of concrete with insufficient FRP confinement hasbeen observed in some tests conducted by Xiao and Wuw11x and Aire et al.w22x. For those specimens with thecompressive strength reached before FRP rupture, Table1 also provides the axial strain at the peak stressćc

and the stress of concrete at the ultimate strain wheref9cuavailable.A set of stress–strain curves of concrete confined by

different amounts of FRP is shown in Fig. 4, using thetest data obtained by Xiao and Wuw11x, where the axialstress is normalized by the unconfined concrete strength.In Fig. 4, specimen A had an unconfined concretestrength of 55.2 MPa and was wrapped with one layerof CFRP (specimen 41 of Table 1a). Specimens B, Cand D had an unconfined concrete strength of 43.8 MPa

and were wrapped with one, two, and three layers ofCFRP, respectively,(specimens 32, 35 and 40 in Table1a). Specimen E had an unconfined concrete strengthof 33.7 MPa and was wrapped with three layers ofCFRP (specimen 31 in Table 1a). The actual confine-ment ratios of these five specimens are 0.067, 0.097,0.221, 0.302 and 0.421, respectively. Further details ofthese specimens can be found in Table 1a. Specimen Aexhibits insignificant strength enhancement and thestress–strain curve terminates at a stress below theunconfined concrete strength(Fig. 4). Specimen B alsohas a decreasing stress–strain response after the peakstress, but the stress at the ultimate strain is higher thanthe unconfined concrete strength. The behavior of thisspecimen thus belongs to the type illustrated in Fig. 3b.Specimens C, D and E all has a stress–strain curve ofthe increasing type. For these three specimens, theenhancement in both the compressive strength and theultimate strain increases with the amount ofconfinement.The volumetric change of confined concrete under

axial compression can be represented by the volumetricstrain´ , which is defined byv

´ s´ q´ q´ s´ q2´ (5)v c r u c r

where´ scircumferential strain ands´ slateral(radi-u r

al) strain. It is commonly known that unconfined con-crete in axial compression experiences a volumetricreduction or compaction up to 90% of the peak stress,but thereafter the concrete shows volumetric expansionor dilation which becomes unstable after the peak stressw34,35x. Unstable dilation has also been observed inactively confined concrete in tri-axial compression testsw36,37x. Recently, Mirmiran and his co-authorsw6,35xcompared the volumetric responses of FRP-confinedconcrete with those of plain concrete and steel-confinedconcrete. They demonstrated that for steel-confined con-crete, unstable dilation occurs when steel yields, but for


Fig. 5. Volumetric responses of FRP-confined concrete.

FRP-confined concrete, the linearly increasing hoopstress of FRP can eventually curtail the dilation if theamount of FRP is large enough.Fig. 5 illustrates the volumetric changes of FRP-

confined concrete using the test data of Xiao and Wuw11x for the five specimens shown in Fig. 4. A positivevolumetric strain indicates compaction while a negativevalue corresponds to dilation. It can be seen that forspecimens A, B and C, the volumetric strain changesfrom compaction to dilation at an axial stress above thecompressive strength of unconfined concrete, and thisdilation continues to increase until failure. For specimenD, dilation is taken over by compaction at a normalizedaxial stress of approximately 1.75. For specimen E, nodilation is found during the entire loading history. Thedifferent volumetric responses of FRP-confined concretefrom steel-confined concrete are due to the linear elasticbehavior of FRP. This linearity of FRP provides acontinuously increasing confining pressure until rupture,and limits the lateral strain of confined concrete to thehoop rupture strain of FRP. As a result, concrete con-fined by a large amount of FRP may not show dilationat all.

3.4. Minimum amount of FRP for sufficient confinement

As explained above, FRP-confined concrete with astress–strain curve of the decreasing type and with aconcrete stress at the ultimate strain below the compres-sive strength of unconfined concrete is taken to beinsufficiently confined, as in such cases little strengthenhancement can be expected from the FRP confinementand the FRP is likely to rupture at a small hoop strain.The latter phenomenon, believed to be due to thesensitivity of a weak jacket to the non-uniform defor-mation of concrete, is particularly important as it meansthat the use of such a weak jacket leads to little strengthor strain enhancement, and any enhancement cannot bereliably predicted. It is, therefore, recommended here

that such weak confinement should not be allowed inpractical design. Consequently, the new stress–strainmodel presented in Section 5 is intended for applicationto sufficiently confined concrete only. For this limitationto be observed in practical design, the minimum amountof FRP deemed necessary to achieve sufficient confine-ment needs to be defined.The issue of insufficient confinement has been dis-

cussed in a number of previous papers. Mirmiran et al.w38x suggested that for FRP-confined rectangular con-crete specimens with rounded corners, enhancement inthe compressive strength of confined concrete shouldnot be expected if the following modified confinementratio (MCR) is less than 0.15:

B E2R fc lC FMCRs (6)D GD f9co

whereR sradius of rounded corners andDsside lengthc

of confined square section,scompressive strength off9counconfined concrete andf sequivalent maximum con-l

fining pressure. For circular specimens, this criterionreduces to -0.15. However, Spoelstra and Montif yf9l co

w13x showed that the stress of concrete at the ultimatestrain falls below if using theirf9 f9 f yf9 -0.07cu co l co

analysis-oriented model. Besides, based on experimentaldata, Xiao and Wuw11x suggested that for FRP-confinedconcrete with a post-peak2 y1E tyRf9 -0.2 (MPa ),frp co

descending branch could be expected.In Table 1, a total of five specimens have a concrete

stress at the ultimate strain below the unconfinedf9cuconcrete strength . The nominal confinement ratiof9co

is less than 0.15 for thirteen of the specimens,f yf9l co

including four with (Fig. 6a). Another spec-f9 yf9 -1cu co

imen showing has a nominal confinementf9 yf9 -1cu co

ratio of . Fig. 6b shows that in terms of thef yf9 s0.18l co

actual confinement ratio, of the seven specimens withan actual confinement ratio , five havef yf9 -0.07l,a co

(Fig. 6b). The maximum value of forf9 yf9 -1 f yf9cu co l,a co

these five specimens is 0.069. Fig. 7 shows that thecriterion of Xiao and Wuw11x for insufficient confine-ment ( ) is met by five specimens,2E tyRf9 -0.2frp co

including four with and one withf9 yf9 -1 f9 yf9 scu co cu co

. For the other specimen with ,1.01 f9 yf9 -1.01cu co

.2E tyRf9 s0.275frp co

Based on the above discussion, it can be concludedthat in judging whether sufficient confinement is avail-able, Spoelstra and Monti’sw13x criterion for insufficientconfinement can be used, with the maximum confiningpressure being the actual rather than the nominal value.That is, FRP-confined concrete with an actual confine-ment ratio can be taken to be sufficientlyf yf9 G0.07l,a co

confined.


Fig. 6. Stress of concrete at ultimate strain vs. confinement ratio.(a)Variation with nominal confinement ratio;(b) Variation with actualconfinement ratio.

Fig. 7. Stress of concrete at ultimate strain vs. Xiao and Wu’s con-finement stiffness parameter.

4. Deficiencies of existing design-oriented stress–strain models

4.1. Shape of stress–strain curve

Existing design-oriented stress–strain models forFRP-confined concretew4–11x have adopted differentapproximations to a typical bilinear stress–strain curve.In the models of Karbhari and Gaow5x and Xiao andWu w11x, the two portions of a bilinear curve areapproximated using two straight lines. This approach issimple but not realistic. Samaan et al.w6x proposed amodel for FRP-confined concrete in which the nearlylinear second portion of the stress–strain curve is char-acterized by its slopeE and its intercept with the stress2

axis. Another salient feature of this model is that thestress–strain curve is represented by a single equation,with the transition from the first portion to the secondportion being controlled by a shape parametern. Theuse of a single equation, however, necessarily leads toan equation of a more complex form. Toutanjiw9x andSaafi et al. w8x proposed an alternative form for thestress–strain curve, in which the two portions of abilinear curve are approximated using two separateequations, with both equations producing a curved

shape. A smooth transition between the two portions isalso provided. Based on the same general equations,two models were proposed separately for FRP-wrappedconcrete and concrete-filled FRP tubes by calibratingthe model parameters with corresponding test data. Themodels of Samaan et al.w6x, Toutanji w9x and Saafi etal. w8x can predict the shape of a bilinear stress–straincurve reasonably closely, provided their predictions ofthe compressive strength and ultimate strain are accurate.However, the relative complexity of these three modelsin form means inconvenience or difficulty in sectionanalysis for the determination of section capacity orductility, where integration of the stress distribution overthe section is required.Miyauchi et al. w7x used Hognestad’s parabolaw39x

followed by a straight line to describe both the increas-ing and decreasing types of stress–strain curves of FRP-confined concrete. This parabola, given by the followingequation, is commonly adopted in codes of practicesuch as BS 8110w40x and Eurocode 2w15x to describethe ascending part of the stress–strain curve of uncon-fined concrete for design use:

2w zB E2´ ć cC Fs sf9 y (7)x |c coD G´ ý ~co co

wheres and´ are the axial stress and strain, respec-c c

tively, and ´ is the axial strain at the peak stress ofco

concrete. However, the direct use of Hognestad’s parab-ola as adopted in Miyauchi et al.’s modelw7x cannotreflect the process of gradual development of confine-ment. In fact, the FRP confinement is activated oncemicro-cracks in concrete are initiated under loading.Lillistone and Jolly w10x attempted to account for thiseffect in their stress–strain model for concrete-filledFRP tubes in which the first portion of the stress–straincurve is described using Hognestad’s parabola plus anadditional term related to the hoop stiffness of the FRP


Fig. 8. Proposed stress–strain model for FRP-confined concrete.

tube (E tyR), while the second portion is a straightfrp

line. This additional term used to account for the effectof confinement(not the contribution of the longitudinalstiffness), being equal to 1.282E t´ yR, means thatfrp c

the initial slope of the predicted stress–strain curve canbe significantly greater than that of unconfined concrete,which is obviously not supported by the test results.In summary, the present authors believe the best

approach for describing a typical bilinear stress–straincurve of FRP-confined concrete is to use a modifiedparabola for the first portion and a straight line for thesecond portion, with the various parameters beingdependent on the FRP properties. The modified parabolashould be able to reflect the gradual development ofconfinement as axial stress increases. This same viewhas also recently been expressed by Montiw41x. Such aparabola leads to ease in section analysis where integra-tion of the stress–strain curve is necessary and offersan approach that is familiar to engineers. For thedefinition of the linear second portion, the use of itsslope and its intercept with the stress axis as done bySamaan et al.w6x offers a rational and simple approach.

4.2. Definition of ultimate condition

Central to any stress–strain model for FRP-confinedconcrete is the determination of the ultimate conditionof FRP-confined concrete, which is reached when theFRP ruptures. This ultimate condition is characterizedby two parameters: the ultimate axial strain and thecorresponding stress level, which is generally but notalways the compressive strength of FRP-confined con-crete. There are three major deficiencies in existingdesign-oriented stress–strain models in predicting theultimate condition of confined concrete.Firstly, it is commonly assumed that rupture of FRP

occurs when the hoop stress in the FRP jacket reachesthe tensile strength determined from material tests, withthe only exception being Xiao and Wu’s modelw11x.This assumption is, however, not valid as shown in thepreceding section, and leads to difficulty in producing aunified stress–strain model for FRP-confined concreteas the ratio of hoop rupture strain to FRP materialtensile strain varies with the type of FRP(Table 2).Existing analysis-oriented models also suffer from thisdeficiency, so they are also incapable of accurate predic-tions of the ultimate condition.Secondly, the effect of the stiffness of the FRP jacket

on the ultimate condition has not been well establishedand explicitly accounted for, although it is implied tosome degree in the ultimate strain equations of Samaanet al. w6x, Toutanji w9x and Saafi et al.w8x. The stiffnessof the FRP jacket in fact has an important effect on thestress–strain response of FRP-confined concrete, partic-ularly the ultimate axial strain as shown later in thepaper.

Thirdly, as a result of the above two deficiencies anddue to the use of a limited database, there is room forimprovement to the accuracy of the predictive equationsfor the compressive strength and ultimate strain of FRP-confined concrete in existing design-oriented modelsusing a larger test database.In summary, improvements to existing stress–strain

models in defining the ultimate condition should bemade in three aspects:(a) the actual hoop rupture strainshould be used instead of the ultimate material tensilestrain; (b) the effect of jacket stiffness should beexplicitly and properly reflected; and(c) the ultimatestrain and compressive strength equations should bebased on the largest test database which can be assem-bled from the open literature.

5. Assumptions and general equations of new model

5.1. Assumptions

A new stress–strain model is proposed here for FRP-confined concrete based on the various observationsdiscussed in the preceding sections. The basic assump-tions of this simple model are:(i) the stress–straincurve consists of a parabolic first portion and a straight-line second portion, as given in Fig. 8;(ii) the slope ofthe parabola at s0 (initial slope) is the same as thec

elastic modulus of unconfined concreteE ; (iii ) the non-c

linear part of the first portion is affected to some degreeby the presence of an FRP jacket;(iv) the parabolicfirst portion meets the linear second portion smoothly(i.e. there is no change in slope between the two portionswhere they meet); (v) the linear second portion ends ata point where both the compressive strength and theultimate axial strain of confined concrete are reached.These basic assumptions of the proposed model are

in accordance with the test observations of FRP-confinedconcrete with a monotonically increasing stress–strain


Fig. 9. Stress–strain curves predicted by Spoelstra and Monti’s anal-ysis-oriented model for concrete confined by different materials.

curve as illustrated in Fig. 3a. The first assumption leadsto a stress–strain curve which is similar to those adoptedby existing design codes for unconfined concrete andthus familiar to engineers. The second assumption is toaccount for the fact that the initial stiffness of FRP-confined concrete is little affected by the FRP due tothe passive nature of confinement. The third assumptionis to reflect the fact that the FRP confinement isactivated when the behavior of the concrete becomesnon-linear. This third assumption makes the new modeldifferent from Miyauchi et al.’s modelw7x in which theshape of the parabola remains the same as that forunconfined concrete and is not affected by the FRPconfinement at all. The fourth assumption ensures asmooth stress–strain curve, while the last assumption isobviously valid for FRP-confined concrete with a mon-otonically increasing stress–strain curve. For FRP-con-fined concrete whose behavior is as illustrated in Fig.3b, the last assumption is not reflective of reality, butthe present model provides a good approximation fordesign use. It may be noted that even for unconfinedconcrete, the design stress–strain curve is representedby a parabola followed by a horizontal straight line inboth BS 8110w40x and Eurocode 2w15x, despite thattest stress–strain curves display a descending post-peakbranch.Based on the assumptions listed above, the proposed

stress–strain model for FRP-confined concrete is givenby the following expressions:

2E yEŽ .c 22s sE ´ y ´ for 0F´ F´ (8a)c c c c c t4fo

and

s sf qE ´ for ´ F´ F´ (8b)c o 2 c t c cu

where f sintercept of the stress axis by the linearo

second portion. The parabolic first portion meets thelinear second portion with a smooth transition at´ ,twhich is given by

2fo´ s (9)t E yEŽ .c 2

whereE is the slope of the linear second portion, given2

by

f9 yfcc oE s (10)2ću

where scompressive strength of confined concrete.f9ccThis proposed model allows the use of test values or

values suggested by design codes for the elastic modulusof unconfined concrete. The three parameters yet to be

defined are: the ultimate strain , the compressivecu

strength , and the intercept of the stress axis by thef9ccsecond linear portionf .o

6. Ultimate strain of FRP-confined concrete

6.1. Theoretical basis

Studies on actively confined concrete and steel-con-fined concrete showed that the axial strain at thecompressive strength of confined concrete´ can becc

linearly related to the maximum confining pressurew42,43x. This approach has been adopted in some design-oriented modelsw5,7x for predicting the axial strain atthe compressive strength of FRP-confined concrete´ ,cc

which in most cases is also the ultimate strain of FRP-confined concrete . A deficiency of this approach iscu

that the stiffness of the confining jacket is not properlyaccounted for. This seems not important for steel-confined concrete because the elastic moduli of all typesof steel are similar, but is important for FRP-confinedconcrete as the elastic modulus of FRP varies over awide range. A number of studiesw6,13,44x have noticedthat similar levels of lateral pressure do not result insimilar ultimate strains of confined concrete. Althoughthe effect of jacket stiffness has not been properlyaccounted for in design-oriented models, it is alwaysaccurately represented in analysis-oriented modelsthrough equilibrium and compatibility considerations ofthe concrete and the jacket. This issue can thus beexplained by making use of predictions from an analysis-oriented model.Fig. 9 shows four stress–strain curves predicted by

the analysis-oriented model of Spoelstra and Montiw13xfor concrete cylinders confined by three different con-fining materials: steel, CFRP and GFRP with theirproperties given in Table 3. For confinement by CFRP,predictions are provided for two scenarios: the FRP


Table 3Properties of confining materials used for predictions shown in Fig. 9

Confining Elastic modulus Rupture or yield Rupture or yield Thickness of confiningmaterial (MPa) stress(MPa) strain(%) jacket(mm)

Steel 2=105 300 0.15 4CFRP 2.35=105 3530 1.5 0.34CFRP 2.35=105 2115 0.9 0.567(actual rupture strain)

GFRP 23 100 462 2.0 2.6

ruptures at its material ultimate tensile strain fromcoupon tests and the FRP ruptures at an assumed hooprupture strain of 60% of the material ultimate tensilestrain. The compressive strength of unconfined concreteis 35 MPa, while the diameter of the cylinders is 150mm. For all four cylinders, the FRP jackets are assumedto supply the same ultimate tensile capacity in the hoopdirection and thus the same maximum confining pres-sure, but they have different stiffnesses. The substantialdifferences between the predicted responses includingthe ultimate strain are due to the differences in thestiffness of the four jackets only. While one may arguethat analysis-oriented models can be inaccurate, thedifferences as shown in Fig. 9 are obviously too largeto be attributed to the inaccuracy of the active confine-ment model since the same concrete is modeled and thestress–strain curve predicted by Spoelstra and Monti’smodel w13x for a similar CFRP-wrapped specimen haspreviously been shown to match the test curve closelyw33x.The dependence of the ultimate strain of FRP-con-

fined concrete on the stiffness of the confining jacketcan also be shown by examining the constitutive modelfor concrete under a triaxial state of stress proposed byOttosenw45x. This model is based on non-linear elastic-ity, with the properties of concrete being represented bythe secant values of elastic modulus and Poisson’s ratio.This is the constitutive model recommended by theCEB-FIP Model Code 1990w46x for concrete undermulti-axial stresses. The basic equations of the modelare:

1 w zx |´ s s yn s qs (11a)Ž .1 1 sec 2 3y ~Esec

1w x´ s s yn (s qs ) (11b)2 2 sec 1 3Esec

1 w zx |´ s s yn s qs (11c)Ž .3 3 sec 1 2y ~Esec

where E ssecant modulus of elasticity andn ssec sec

secant Poisson’s ratio. For confined concrete,´ s´ s1 c

axial strain of concrete, s´ slateral (radial) strain2 r

of concretes´ s´ scircumferential strain of concrete,3 u

s ss scompressive stress of concrete, ands ss s1 c 2 3

s slateral confining pressure. The following equationr

can then be obtained from Eqs.(11a), (11b) and(11c):

21yn y2n sŽ .sec sec r´r´ sy q (12)c

n n Esec sec sec

Further, ass can be expressed as a function of ther

hoop strain in the FRP according to Eq.(1) and Eq.(2), there is

21yn y2nŽ .sec sec´ E t´r frp h´ sy q (13)c

n n E Rsec sec sec

Under the ultimate condition of FRP rupture,´ sry´ . The secant modulus of elasticity of concreteh,rup

under the ultimate conditionE is given byw45x.secu

EsecoE s (14)secu 1q4 Ay1 xŽ .

whereE ssecant modulus of elasticity at the com-seco

pressive strength of unconfined concrete ands ;f9 yćo co

As2 if the Hognestad’sw39x parabola is assumed forthe ascending part of the stress–strain curve of uncon-fined concrete; andx is given by

y yxs J yf9 y1y 3 (15)Ž .2 co f

whereJ is the second invariant of the deviatoric stresses2

ands for confined concrete. The term1 2= s ysŽ .c r3

denotes the value of the invariant under theyJ yf9Ž .2 co f

ultimate condition of FRP-confined concrete, soands sf . The compressive strength of con-s sf9c cc r l,a

fined concrete can be expressed in the following com-mon form w47x:

f9 fcc l,as1qk (16)1f9 f9co co


Fig. 10. Definition of secant modulus of elasticity.

Fig. 11. Dependence of ultimate secant Poisson’s ratio on confinementstiffness ratio.

where k sconfinement effectiveness coefficient. Eq.1

(15) then becomes

k y1 k y1B E B EB EŽ . Ž .1 1f E t ´l,a frp h,rupC F C FC Fxs s (17)D G D GD Gf9 E R ý yco seco co3 3

Substituting Eqs.(14) and (17) into Eq. (13), thefollowing equation can be obtained for the normalizedultimate strain of confined concrete:

21yn y2n B EŽ .secu secu´ ´ E tcu h,rup frpC Fs qD G´ n ´ n E Rco secu co secu seco

24 k y1 1yn y2nB E Ž . Ž .1 secu secu´h,rupC F= qD G´ nyco secu3

2 2B E B EE t ´frp h,rupC F C F= (18)D G D GE R ´seco co

where n ssecant Poisson’s ratio of the confinedsecu

concrete under the ultimate condition and the termE tyfrp

E R is the confinement stiffness ratio, representingseco

the stiffness ratio between the FRP jacket and theconcrete core.The definitions of the secant moduliE and Esec secu

are given in Fig. 10, which can be found by subtractingEq. (11b) from Eq. (11a). The secant Poisson’s ratiocan be found from the constitutive equations (Eqs.(11a),(11b) and(11c)) and related to the lateral-to-axial strainratio ´ y´ through the following equation:r c

s ´r rys ć c

n s (19)sec´ s sr r r1y2 q´ s sc c c

6.2. Proposed equation

Eq. (18) shows clearly the dependence of the ultimatestrain of FRP-confined concrete on the confinementcu

stiffness ratio and the strain capacity of FRP at hooprupture. Existing test data on FRP-confined concretealso show that the secant Poisson’s ratio of FRP-confined concrete under the ultimate conditionnsecudepends strongly and predominantly on the confinementstiffness ratio(Fig. 11). Thus, the ultimate strain ofFRP-confined concrete can be taken to be a functioncu

only of the confinement stiffness ratio andE ty E RŽ .frp seco

the strain ratio y´ . The following general equationh,rup co

is, therefore, proposed to predict the ultimate strain:

a bB E B EE t ´frp h,rupC F C F´ y´ scqk (20)cu co 2D G D GE R ´seco co

wherecsnormalized ultimate strain of unconfined con-crete,k sstrain enhancement coefficient, anda andb2

are exponents to be determined. The effect of the secantPoisson’s ration is reflected by the choice of appro-secu

priate values fora, b and .k2Eq. (20) explicitly accounts for the stiffness and the

actual ultimate condition of the jacket. If botha andbare taken as unity, Eq.(20) reduces to

fl,a´ y´ scqk (21)cu co 2 f9co

which then relates the normalized ultimate strain to theactual confinement ratio only.

6.3. Determination of a and b

The values of the two exponents,a and b, aredetermined here using the test data of the present


Fig. 12. Strain enhancement ratio vs. actual confinement ratio.(a)CFRP wraps;(b) AFRP wraps.

database. In determining these exponents, the strain atthe compressive strength of unconfined concrete´ wasco

taken as 0.002, a value, which was assumed in all datainterpretation in the present study.Fig. 12 shows two plots of the strain enhancement

ratio against the actual confinement ratio from the testsfor CFRP wraps and AFRP wraps, respectively. A linearrelationship clearly exists in both cases but the twotrend lines are very different. These diagrams indicatethat the ultimate strain of FRP-confined concrete can berelated linearly to the actual confinement ratio for agiven type of FRP, but separate expressions are neededfor different types of FRP due to differences in stiffness.To achieve a unified expression for the ultimate strainof FRP-confined concrete, the confinement stiffness rationeeds to be included, which means that the exponents,a and b, cannot both be unity. Thus, the followingexpression is suggested for FRP-wrapped concrete basedthe trends of the test data:

1.45B EB EE t ´frp h,rupC FC F´ y´ s1.75q12cu coD GD GE R ´seco co

0.45B EB Ef ´l,a h,rupC FC Fs1.75q12 (22)D GD Gf9 ćo co

Fig. 13a,b show that the trends of the test data aresimilar for both CFRP and AFRP wraps. For HM CFRP-wrapped specimens(Fig. 13c), if one of the threespecimens tested by Dias da Silva and Santos(2001) isexcluded(shaded) as a statistical outlier, the remainingspecimens also show a trend similar to those observedfor CFRP or AFRP wraps. A large scatter is observedfor GFRP-wrapped specimens(Fig. 13d), the cause ofwhich is difficult to pinpoint at the present, but thepredictions of Eq.(22) are well covered by the scatterof the test data. When all test data are plotted together(Fig. 13e), a close overall agreement between the testdata and Eq.(22) is observed. Eq.(22), therefore,provides a unified expression for the ultimate strain ofFRP-confined concrete that is applicable to differenttypes of FRP. Obviously, more test data for GFRP wrapsshould be obtained in the future for further verificationof Eq. (22).

7. Compressive strength of confined concrete

The compressive strength of FRP-confined concretehas been discussed in detail in Lam and Teng(2002b).In that paper, the compressive strength of FRP-confinedconcrete is related to the nominal confinement ratiof9ccthrough

f9 fcc ls1q2 (23)f9 f9co co

Fig. 14 is a plot of the strengthening ratiof9 yf9cc co

against the actual confinement ratio of thef yf9l,a co

present test data. The trend line of these test data canbe closely approximated using the following equation:

f9 fcc l,as1q3.3 (24)f9 f9co co

This equation implies that in terms of the actualconfinement ratio, the confinement effectiveness coeffi-cient k becomes 3.3 instead of 2. It should be noted1

that the stiffness of the confining jacket also has aneffect on the compressive strength of concrete with thesame confinement ratio, as demonstrated in Fig. 9. Asthis effect on the compressive strength is far less thanthat on the ultimate strain, it is ignored in Eq.(24).It should be reminded that a significant strength

enhancement can only be expected with an actualconfinement ratio . The use of Eq.(24) isf yf9 G0.07l,a co

recommended to be subjected to this condition. For thecase of FRP-confined concrete with , nof yf9 -0.07l,a co

strength enhancement is assumed.

8. Intercept of the stress axis by the linear secondportion

The intercept of the stress axis by the linear secondportion, f , is an important parameter in the proposedo


Fig. 13. Performance of proposed equation for the ultimate strain of FRP-confined concrete.(a) CFRP wraps;(b) AFRP wraps;(c) HM CFRPwraps;(d) GFRP wraps;(e) all specimens.

stress–strain model as it, together with the ultimatepoint, determines the slope of the second portion.Samaan et al.w6x proposed the following expression off based on experimental data available to him:o

f s0.872f9 q0.371fq6.258 (MPa) (25)o co l

In the present database, values off obtained by theo

present authors from test stress–strain curves have beenincluded. These values are normalized by the compres-sive strength of unconfined concrete and are plottedf9coagainst the actual confinement ratio as shown inf yf9l,a co

Fig. 15. The values of are seen to fall betweenf yf9o co

1.0 and 1.2 in most cases, which appear to be independ-ent of the confinement ratio. For the 63 specimens, forwhich the values of the intercept are available, theaverage ratio of is 1.09 with a standard deviationf yf9o co

of 0.13. It is, therefore, suggested for simplicity that inthe proposed model

f sf9 (26)o co

It should be noted that with , the second linearf sf9o co

portion of the proposed stress–strain model reduces toa horizontal straight line(i.e.E s0) as assumed in both2

BS 8110 w40x and Eurocode 2w15x, and at the same


Fig. 14. Strengthening ratio vs. actual confinement ratio.

Fig. 15. Intercept of stress axis by the second linear portion vs. actualconfinement ratio.

time, Eq. (8a) reduces to Hognestad’s parabola forunconfined concrete. That is, the present model for FRP-confined concrete reduces directly to the design stress–strain models in BS 8110w40x and Eurocode 2w15x forunconfined concrete, provided the same initial elasticmodulus is used.

9. FRP efficiency factor

In the preceding sections, the definitions of the ulti-mate strain, compression strength and minimum amountof FRP for sufficient confinement for the proposedstress–strain model are all in terms of the actual con-finement ratio, so the actual hoop rupture strain of theFRP is required. To facilitate the application of theproposed stress–strain model, an FRP efficiency factoris, therefore, proposed here, which is defined as theratio of the actual FRP hoop rupture strain(´ ) inh,rup

FRP-confined concrete to the FRP rupture strain fromflat coupon tests(´ ). For the 52 CFRP-wrappedfrp

specimens, out of the total of 76 specimens in thepresent database(Table 1), this efficiency factor is 0.586on average. Making use of this efficiency factor, theultimate strain of CFRP-confined concrete can beexpressed as

0.45B EB Ef ´l,a frpC FC F´ y´ s1.75q5.53 (27)cu coD GD Gf9 ćo co

With this equation, the user only needs to know thetensile strain´ from flat coupon tests. This averagefrp

efficiency factor of 0.586 for CFRP-wrapped concretealso unifies Eq.(23) and Eq.(24).For other types of FRP, insufficient information exists

to define this efficiency factor with confidence. Evenfor CFRP-confined concrete, there is a considerablescatter in the efficiency factor deduced from test results.The present authors, therefore, recommend that for cost-effective and safe applications, a small number(say 3)of FRP-confined concrete cylinder tests, analogous to

plain cylinder tests, be conducted to determine theefficiency factor of a given FRP product in confinementapplications. For this purpose, a standard confined cyl-inder test method should be formulated in the future.FRP manufacturers can then supply the results of suchtests as part of their product information. If this infor-mation is not available from the manufacturer, the usershould conduct these tests instead.

10. Comparison with test data

The proposed stress–strain model for FRP-confinedconcrete is compared with the test data obtained byXiao and Wuw11x on CFRP-wrapped concrete cylinders,as shown in Fig. 16. Details of these specimens can befound in Table 1a. Fig. 16a and b are for specimenshaving an unconfined concrete strength of 55.2 MPaf9coand wrapped with one(specimens 41–43) and two(specimens 44 and 45) layers of CFRP, respectively.Fig. 16c is for specimens having andf9 s43.8 MPaco

wrapped with three layers of CFRP(specimens 38–40),and Fig. 16d is for specimens havingf9 s33.7 MPaco

and wrapped with three layers of CFRP(specimens 30and 31). Each figure shows test stress–strain curvesfrom nominally identical specimens and predictions ofthe present model. For each case, two predicted curvesare shown: one with the FRP hoop rupture strain´h,rup

estimated on the basis of the average efficiency factorfor CFRP of 0.586 wrapped specimens(Table 2) andthe other with the hoop rupture strain being the averageof the actual values recorded during the tests. The elasticmodulus of unconfined concrete was taken asE sc

w48x and the strain at the compressive strengthy4730 f9coof unconfined concrete was taken as´ s0.002 forco

calculating the ultimate strain from Eq.(22). Thespecimens used in Fig. 16a had an average actualconfinement ratio of , so these specimensf yf9 s0.048l,a co


Fig. 16. Comparison between proposed model and test stress–strain curves.(a) ; (b) ; (c) ; (d)f yf9 s0.048 f yf9 s0.150 f yf9 s0.281l,a co l,a co l,a co

.f f9 s0.403l,a co

are insufficiently confined ones. Consequently,k s01

was used in predicting the compressive strength ofconfined concrete. The average actual confinement ratiosfor the specimens included in Fig. 16b,c and d are0.150, 0.281 and 0.403, respectively. For these cases,k s3.3 was used. It can be seen from Fig. 16 that the1

predictions compare well with the test results. It shouldbe noted that the ultimate strains of specimens 41–43are all considerably overestimated using the estimated´ (Fig. 16a), which further justifies the exclusion ofh,rup

such insufficiently confined concrete from practical con-siderations and from the range of applicability of theproposed stress–strain model.

11. Conclusions

This paper has been concerned with the developmentof a stress–strain model for concrete confined bywrapped FRP with fibres only or predominantly in thehoop direction. Existing experimental data have beenthoroughly reviewed and discussed, and the deficienciesof existing stress–strain models are highlighted. A sim-

ple and accurate stress–strain model for FRP-confinedconcrete has been presented for design use. The resultsand discussions presented in this paper also allow thefollowing conclusions to be drawn:

1. The average hoop strain in FRP at rupture in FRP-wrapped concrete can be much lower than the FRPmaterial ultimate tensile strain from flat coupon tests,indicating the assumption that FRP ruptures when theFRP material tensile strength reached is not valid inthe case of concrete confined by wrapped FRP. Basedon this observation, a unified stress–strain model forconcrete confined by different types of FRP must bebased on the actual hoop rupture strain of FRP ratherthan the ultimate material tensile strain.

2. The stress–strain curve of FRP-confined concrete canbe in one of several forms, but in the vast majorityof cases, this curve is or can be approximated as amonotonically ascending bi-linear curve. Such FRP-confined concrete is said to be sufficiently confined.Any FRP-confined concrete with an actual confine-ment ratio less than 0.07 is said to be insufficiently-


confined. Such concrete is not expected to possess acompressive strength significantly above that ofunconfined concrete and the FRP may rupture at alow hoop strain. Such insufficiently-confined concreteshould not be allowed in design.

3. The new design-oriented stress–strain model pro-posed in this paper is in a form that is familiar toengineers. This model is simple, so it is suitable fordirect use in design, but in the meantime, it capturesall the main characteristics of the stress–strain behav-ior of FRP-confined concrete. The model reducesdirectly to idealized stress–strain curves adopted byexisting design codes for unconfined concrete.

4. The stiffness of the FRP jacket and the actual ultimatecondition of the jacket are explicitly accounted for inthe proposed model. As a result, the proposed modelis applicable to concrete confined by different typesof FRP. The predictions of the model have beenshown to agree well with a set of test data.

5. For the application of the proposed model in design,the FRP efficiency factor(ratio between the actualhoop rupture strain of FRP in FRP-confined concreteand the ultimate tensile strain from material tests)needs to be established. For this purpose, confinedcylinder tests have been suggested as a standard typeof test to supply this information.

Acknowledgments

The work presented in this paper forms part of aresearch project(Project No: PolyU 5064y01E) fundedby the Research Grants Council of Hong Kong SAR.The first author has been financially supported by TheHong Kong Polytechnic University through a postdoc-toral fellowship and through the Area of StrategicDevelopment(ASD) Scheme. The authors are gratefulto both organizations for their financial support.

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2003Lam and Teng - Design-Oriented Stress–Strain Model for FRP-confined Concrete