plasticity-based model for circular concrete columns confined with fibre-composite sheets

Engineering Structures 29 (2007) 3301–3311www.elsevier.com/locate/engstruct

Plasticity-based model for circular concrete columns confined withfibre-composite sheets

Rami Eid, Patrick Paultre∗

Department of Civil Engineering, Faculty of Engineering, University of Sherbrooke, Sherbrooke, QC, Canada J1K 2R1

Received 4 July 2007; received in revised form 8 September 2007; accepted 11 September 2007Available online 24 October 2007

Abstract

This paper presents analytical stress–strain curves that describe the axial and the lateral behaviour of externally and internally confined concretein circular columns. The solution to the problem of an elastic cylinder confined by internal and external elastic confining materials is derived.Following the elastic solution, the analytical curves are derived based on the full elastoplastic behaviour of the confined concrete column and byusing the well-known Drucker–Prager failure criterion to represent concrete behaviour. Application of the Drucker–Prager model does not requirean iterative procedure in order to solve the problem and thus an explicit solution is obtained. The proposed model can be applied to describethe behavior of concrete confined by transverse steel reinforcement, by fibre-reinforced polymer (FRP) composites, or by both transverse steelreinforcement and FRP composites. It is shown that the analytical results are in good agreement with both experimental and the finite elementmethod results.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Concrete columns; Elasticity; Plasticity; Internal confinement; External confinement; Transverse steel reinforcement; Fibre reinforced polymercomposites

1. Introduction

In existing reinforced concrete columns that are retrofittedwith fibre-reinforced polymer (FRP) composites, the internaland external passive confinements are imposed by the actionof the transverse steel reinforcement and the action of the FRP,respectively. Most of the available models of concrete confinedby transverse reinforcement (e.g., [1–3]) cannot representthe stress–strain curves of FRP-confined concrete. Severalresearchers proposed stress–strain models, especially for FRP-confined concrete (e.g., [4–10]). A notable disadvantage ofmost available confinement models is their limitation to onespecific confining material. Moreover, most of the publishedFRP-confined concrete models are not suitable for concretecolumns confined by both transverse steel reinforcement andFRP composites.

Numerical solutions for the externally (FRP composites) andinternally (transverse steel reinforcement) confined concrete

∗ Corresponding author. Tel.: +1 819 821 7108; fax: + 1 819 821 7974.E-mail address: [email protected] (P. Paultre).

0141-0296/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2007.09.005

column were provided by few researchers. Montoya et al. [11]used a nonlinear finite element program with implementationof a compression field theory confinement model to analysefour FRP-steel confined concrete columns tested by Demersand Neale [12]. Deniaud and Neale [13] showed that theelastoplastic analysis of FRP-steel confined concrete columnswith the Drucker–Prager yield criterion gives results that agreereasonably well with experimental data. Their solution wasobtained by an incremental iterative procedure.

Using the theories of elasticity and plasticity, Eid et al. [14]proposed stress–strain curves that describe the axial and lateralbehavior of externally confined circular concrete columns. Byimplementing a similar solution strategy, this paper presentssimplified analytically derived stress–strain curves that describethe axial and lateral behavior of externally and internallyconfined concrete in circular columns. The analytical curvesare derived from the full elastoplastic behaviour of the confinedconcrete column and by using the well-known Drucker–Prageryield criterion. This model can describe the behaviour ofconcrete confined by transverse steel reinforcement, by FRPcomposites, or by both.

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3302 R. Eid, P. Paultre / Engineering Structures 29 (2007) 3301–3311

2. Analysis of externally and internally confined concretecolumn

A reinforced concrete column confined by fibre-reinforcedpolymer (FRP) composites is under dual confinement: theaction due to the internal transverse steel reinforcement and thatdue to the external FRP. The axial and lateral behaviour of sucha column is given in the following sections using the theories ofelasticity and plasticity.

2.1. Elastic analysis

The solution to the elastic concrete column confinedonly by transverse steel reinforcement was obtained by Eidand Dancygier [15]. In order to solve the problem in theelastoplastic concrete material range, however, the transversesteel reinforcement was replaced by an equivalent steeltube [16]. The normalized equivalent tube thickness, tseq1, wasderived based on the active and the passive elastic solutions ofa concrete cylinder confined by lateral steel rings [16,15]:

tseq1 =tseq

Rc=

f3

m [(ν − 1) f3 + νs1/b1](1)

where f3 is the ratio between the lateral (radial) pressure, p,which develops by the action of the ties, and the axial pressure,q . The function f3 is given by:

f3 =p

q=

ν

8(1−ν2)π

∫∞

0 Fdk1 +1

ts1m

(2)

where

F =

sin(k1b1/2)

[n∑

i=1cos(k1/2) (2i − 1) s1

]cos(k1s1/2)

k21 I0(k1)2/I1(k1)2 + (2ν − k2

1 − 2)k1(3)

and, b1 = φ/Rc is the normalized tie diameter, φ is the tiecross-section diameter, Rc is the concrete core radius measuredcentre-to-centre of the steel ties, s1 = s/Rc is the normalizedcentre-to-centre spacing between the ties, ν is the concretePoisson’s ratio, m is the steel-to-concrete moduli ratio, andts1m is the mixed geometrical-mechanical parameter (wherets1 = φπ/4Rc). In Eqs. (1)–(3) k1 = k Rc where k is a constantwith a dimension of length−1 (wave number), I0(k1) and I1(k1)

are order zero and one modified Bessel functions (of the firstkind) of the argument k1, and n is the number of pairs of hoopsapplied symmetrically with respect to the origin (see Fig. 1(a)).The summation in Eq. (3) represents the accumulative influenceof the confining ties, where it was shown that f3 convergesfor five pairs of hoops (i.e., for n = 5 in Eq. 2, [17]).It was also shown that the normalized equivalent steel tubethickness calculated with Eq. (1) can be assumed to be 0.47ρsfor practical cases of reinforced circular concrete columns [17],where ρs is the volumetric transverse reinforcement ratio. In thefollowing sections, the steel reinforcement will be consideredusing the equivalent thickness (see Fig. 1(b)).

The solution to the elastic cylinder confined by external andinternal lateral confining elastic tubes (Fig. 1(b)) will be derived

Fig. 1. Equivalent steel-confined concrete column concept.

in two stages. First, the solution to the linear elastic cylinderconfined by external and internal uniform lateral pressures(active solution) is derived, followed by the solution to theelastic cylinder confined by the external and internal uniformelastic tubes (passive solution). The main goals of the elasticanalysis are to evaluate the lateral strains in the confiningmaterials and the lateral pressures developed due to their action.

2.1.1. Active confinementThe first stage of the analysis is to solve the active

confinement axisymmetrical problem (Fig. 2(a)) of a longsolid elastic cylinder confined by uniformly distributed lateralpressures applied internally (diameter Dc or radius Rc) andexternally (diameter D or radius R), and subjected at its endsto axial pressure. The cylinder is allowed to deform freely inthe longitudinal direction (z axis) and, due to symmetry, everycross section remains plane, and the axial strain, εz , will beconstant (generalized plane strain problem). The solution isderived by superposing a cylinder subjected to external lateralpressure, p f (Fig. 2(b)), a cylinder subjected to the axialpressure, q (Fig. 2(d)), and a cylinder subjected to internallateral pressure, ps (Fig. 2(c)). The distributions of the stressesand strains in the first two cases are trivial. In the externallyloaded cylinder (Fig. 2(b)), the radial stress is constant andequal to the tangential stress (σr = σθ = p f ), while the axialand shear stresses are equal to zero (σz = τr z = 0). In thecylinder subjected to axial pressure (Fig. 2(d)), the stresses areequal to zero (σr = σθ = τr z = 0) except the axial stress, whichis equal to q (σz = q). The solution to the third case, namelythe internally confined cylinder (Fig. 2(c)) subjected to lateralpressure, ps , is derived by superposing a cylinder of diameterDc (denoted as cylinder number “1” in Fig. 2(e)) subjected toa uniformly distributed pressure, ps1, and a hollow cylinder

R. Eid, P. Paultre / Engineering Structures 29 (2007) 3301–3311 3303

Fig. 2. Superposition of external and internal confinements.

with an internal diameter of Dc and an external diameter ofD (denoted as cylinder number “2” in Fig. 2(f)) subjected toa uniformly distributed internal pressure, ps2. Hence, the totalactive pressure, ps , is equal to ps1 + ps2.

Three conditions are applied in order to solve the internallyconfined cylinder (Fig. 2(e) and (f)): the compatibilitycondition, which satisfies that the radial displacements areequal at the radius Rc (Eq. (4)); the condition of zerolongitudinal (axial) force (Eq. (5)); and the generalized planestrain condition for which εz is constant (Eq. (6)):

u1,r=Rc = u2,r=Rc ⇒ εθ1,r=Rc = εθ2,r=Rc (4)∑Fz = 0 ⇒ 2π

∫ D

0σzrdr = 0 ⇒

σz1

σz2= −

R2− R2

c

R2c

(5)

εz1 = εz2 (6)

where the symbols “1” and “2” in Eqs. (4)–(6) refer tothe cylinders in Fig. 2(e) and (f), respectively. SolvingEqs. (4)–(6) with ps = ps1 + ps2 gives the followingexpressions for the lateral pressures ps1 and ps2:

ps1 = ps1 + η2

− 2νη2

2(1 − ν)(7)

ps2 = ps(1 − η2)(1 − 2ν)

2(1 − ν)(8)

where, η = Rc/R. The influence of η on the lateral pressureratios ps1/ps and ps2/ps is demonstrated in Fig. 3 for ν =

0.15. It can be seen in the figure that, for the practical case of0.85 ≤ η ≤ 0.9, the ratio of the internal pressure (the pressurein the cylinder’s core) is ps1/ps ≈ 0.9. Moreover, the figureshows that the derived pressure ratios (Eqs. (7) and (8)) agreewell with the solution obtained by the finite element method(FEM) using ADINA [18] software. The FEM analysis will beexplained later in the text.

It should be noted that, using the principle of superpositionwith Eqs. (7) and (8), the stress and strain distributions in theactively confined cylinder can be obtained.

2.1.2. Passive confinement

The solution to the passive confinement problem of a linearelastic cylinder subjected to axial pressure and confined byinternal and external stiffer elastic tubes (with zero axialstiffness) is obtained from the solution to the problem of theactive confinement, which was shown in the previous section,and from a compatibility condition of the tangential strains atthe contact between the cylinder and the tubes. The tensionstrain in the equivalent internal tube (which can represent thesteel confinement), εs , and the tension strain in the external tube(which can represent the FRP composite confinement), ε f , areobtained from the equilibrium equations on half cross-sectionsof the tubes.

εs = −ps Rc

Es tseq= −

ps

Esl(9)


Fig. 3. The lateral pressure ratios, ps1/ps and ps2/ps , versus the ratio η =

Rc/R for ν = 0.15.

ε f = −p f R

E f t f= −

p f

E f l(10)

where, tseq is the equivalent thickness of the lateral steel, t f isthe thickness of the FRP, Es is the steel modulus of elasticity,E f is the FRP modulus of elasticity, Esl = Es tseq1, E f l =

E f t f 1, tseq1 = tseq/Rc, and t f 1 = t f /R.Compatibility at the contact between the tubes and the

cylinder requires that the tube tangential (or tension) strains, εsand ε f (Eqs. (9) and (10)) be equal to the cylinder’s tangentialstrains at r = Rc and at r = R, respectively:

εs = εθ3,r=Rc + εθ4,r=Rc + εθ5,r=Rc (11)

ε f = εθ3,r=R + εθ4,r=R + εθ5,r=R (12)

where the symbols “3”, “4”, and “5” refer to the internallyconfined cylinder (Fig. 2(c)), the externally confined cylinder(Fig. 2(b)), and the cylinder subjected to axial pressure(Fig. 2(d)), respectively. The tangential strains are given asfollows:

εθ3(r) = psη2 (R/r)2 (1 + ν)(1 − 2ν) + (1 − 3ν + 4ν2)

2(1 − ν)Ecfor Rc ≤ r ≤ R (13)

εθ4 =p f

Ec(1 − ν) (14)

εθ5 = −νq

Ec. (15)

It should be noted that the strain εθ3(r) (Eq. (13)) is derivedfrom the solution to the problem of the active confinement(previous section). For r < Rc, the strain εθ3 is constant andderived from Eq. (13) with r = Rc. Moreover, note that, in thecurrent derivations, a positive sign denotes compressive stressesand strains (and vice versa).

Substituting Eqs. (11)–(15) into Eqs. (9) and (10), andsolving the latter two equations gives the expressions in Box Ifor the lateral pressures, where

C p = (1 + ν)(1 − 2ν) + η2(1 − 3ν + 4ν2) (16)

and, msl = Esl/Ec and m f l = E f l/Ec. It should be noted that,for a cylinder confined by one confining material at the cylindersurface (r = R), first equation or second equation in Box I arereduced to the expression of p =

ν1−ν+1/t1m q . Substituting the

Fig. 4. The strain ratio, εs/ε f , versus the ratio η = Rc/R for ν = 0.15,msl = 6.67, and R = 120 mm.

equations in Box I into Eqs. (11) and (12), the tangential strainratio is obtained as follows:

εs

ε f=

2(1 − ν)

msl(1 + ν)(1 − 2ν)(1 − η2) + 2(1 − ν). (17)

The influence of η on the strain ratio εs/ε f is demonstrated inFig. 4 for ν = 0.15, msl = 6.67, and R = 120 mm. It can beseen in the figure that for the practical case of 0.85 ≤ η ≤ 0.9,the strain ratio is εs/ε f ≈ 0.98. For larger cylinder diameters, itwas found that this ratio is even closer to 1.0. Fig. 4 also showsthat the derived strain ratio (Eq. (17)) agree well with the FEMsolution obtained using ADINA [18] software.

2.2. Elastoplastic analysis

The solution to an externally confined cylinder thatrepresents the confined concrete core (while neglecting theunconfined concrete cover) in the elastoplastic range of theconcrete material was obtained implicitly and explicitly byEid and Dancygier [16] and Eid et al. [14], respectively. Eidand Dancygier [16] applied the Imran and Pantazopoulou [19]concrete plasticity material model and a uniaxial multilinearsteel material model to obtain an incrementally and iterativelyderived stress–strain curve of a concrete cylinder confined by anequivalent steel tube. On the other hand, Eid et al. [14] appliedthe relatively simple Drucker–Prager [20] concrete materialmodel and an elastic or an elastic-perfectly plastic confiningmaterial to derive an explicit solution for the axial and lateralstress–strain curves of a concrete cylinder externally confinedby an equivalent tube.

In order to obtain an explicit solution for the problem ofan internally and externally confined concrete circular columnin the elastoplastic concrete material range, the followingassumptions were applied based on the elastic analysis forpractical cases of reinforced concrete columns (previoussection): (1) the tangential (tension) strains in the internaland external confining tubes are equal εs = ε f ; and (2)the internal core pressure ratio ps1/ps in the concrete plasticrange is equal to 1 (this assumption was also confirmed bythe FEM results). Moreover, the total stress–strain curve of theinternally and externally confined column will be derived bycombining the behavior of the concrete core which is under dualconfinement, and the behaviour of the concrete cover, whichis under only external confinement. The typical behaviour of


ps =2νq(1 − ν)msl

mslC p + 2(1 − ν)[1 + m f l(1 − ν)

]+ mslm f l(1 − η2)(1 − ν2)(1 − 2ν)

p f =νqm f l

[2(1 − ν) + msl(1 + ν)(1 − 2ν)(1 − η2)

]mslC p + 2(1 − ν)

[1 + m f l(1 − ν)

]+ mslm f l(1 − η2)(1 − ν2)(1 − 2ν)

Box I.

Fig. 5. Typical proposed stress–strain curve of a concrete column confined bytransverse steel reinforcement and FRP.

the concrete core, which is under the dual confinement of anelastic-perfectly plastic transverse steel (internal confinement)and linear elastic FRP composites (external confinement), ischaracterized by four phases, as shown in Fig. 5: (1) elasticbehaviour of the concrete and confining materials; (2) concretein the plastic range, while the steel and the FRP remain elastic;(3) concrete and steel behave plastically, while the FRP remainselastic; and (4) the FRP ruptured and does not contribute to theconfinement, while the concrete and steel behave plastically.The stress-stain behaviour of the confined concrete core in thefour phases is developed in the following sections.

2.2.1. Phase IThe elastic and the first phase of the behaviour of the

confined concrete core was analysed in the previous sections.As already stated earlier, however, the derivation in this sectionis based on the assumption εs = ε f . It should be noted that,in the elastic phase of the confined concrete behaviour, the ratiops1/ps can be taken as derived in Eq. (7). Based on Hook’s law,the force equilibrium on the half cross-section of the concretecore, and strain compatibility, the following expressions for theaxial stress and the lateral stress and strain as a function of theaxial strain in the elastic range of the concrete and the confiningmaterials are derived

f Ic = AD Ecεz; σ I

`at = pI= AD BD Ecεz;

εI`at = −

AD BD Ec

Es f `

εz (18)

where

AD =1 + (1 − ν)ms f `

1 + (1 + ν)(1 − 2ν)ms f `

;

BD =ν

1 − ν + 1/ms f `

(19)

Es f ` = ξ tseq1 Es + t f 1 E f ; ms f ` = Es f `/Ec (20)

and ξ = ps1/ps as given in Eq. (7). The limit of phase I isreached when the state of stresses reaches the yield surface,which is defined in this study by the Drucker–Prager criterion:

f =

√J2 − α I1 − kD (21)

where J2 is the second deviatoric stress invariant, I1 is thefirst stress invariant and α and kD are material constantsthat can be related to the Mohr–Coulomb constants cm andΦ (the cohesion and internal friction angle of the material,respectively). Note that, in the current derivations, a positivesign denotes compressive stresses and strains (and vice versa).In the compressive meridian, α and kD are expressed in thefollowing form [21,22]:

α =2 sin Φ

√3(3 − sin Φ)

; kD =6cm cos Φ

√3(3 − sin Φ)

. (22)

For a given Φ, cm can be calculated from the uniaxial behavior(σz = f ′

c , σr = σθ = 0, where f ′c is the unconfined concrete

strength) by the following expression [21]:

cm =f ′c(1 − sin Φ)

2 cos Φ. (23)

Substituting the stress expressions (Eq. (18)) into the Drucker–Prager yield criterion gives the limit axial strain of phase I,εz,ep, which also controls the limit axial and lateral stresses andstrain, fc,ep, σlat,ep, εlat,ep, respectively (see Fig. 5):

εz,ep =−3kD

(3α + 6αBD −√

3 +√

3BD)Ec AD;

fc,ep = AD Ecεz,ep (24)

εlat,ep = −AD BD

ms f `

εz,ep; σlat,ep = AD BD Ecεz,ep. (25)

2.2.2. Phase IIIn this state, the concrete is in its plastic range while the

steel and FRP behave elastically. In the concrete plastic range,the direction of the plastic deformation is determined throughthe plastic potential function. For the nonassociated behaviourof the concrete material [19], the following potential functioncan be used:

g =

√J2 − α1 I1 − k1D (26)

where α1 and k1D are material constants (note that the casefor α1 = α is of an associated material model). Keeping inmind the two elastically-based assumptions described earlierconcerning the internal–external tangential strains (εs = ε f )and the internal core pressure ratio (ps1/ps = ξ = 1) in the


concrete plastic range, the problem of the concrete core can besolved similarly to the externally confined cylinder [14]. Thesimple boundary condition of σr = σθ = p = σlat reduces theproblem to three equations with the axial and lateral stressesfc and σlat, and the lateral strain εlat as the unknown variables.The following two equations (Eqs. (27) and (28)) are derivedfrom the incremental elastoplastic constitutive relations and oneequation Eq. (29) is derived from equilibrium at the concrete-confinement boundary:

d f IIc =

Ec

(1 + ν)dεz +

[Ecν

(1 − 2ν)(1 + ν)

]dI ε

1

−

[Ec

(1 + ν)

Sz

2√

J2−

Ecα1

(1 − 2ν)

]dλ (27)

dσ IIlat =

Ec

(1 + ν)dεlat +

[Ecν

(1 − 2ν)(1 + ν)

]dI ε

1

−

[Ec

(1 + ν)

Slat

2√

J2−

Ecα1

(1 − 2ν)

]dλ (28)

dεIIlat = dεII

s = dεIIf = −

dσ IIlat

Es f `

(29)

where I ε1 is the first strain invariant and Sz = fc − I1/3 and

Slat = σlat − I1/3 are the deviatoric stresses in the axial andlateral directions, respectively. The scalar dλ, which definesthe plastic strain magnitude, is derived using the consistencycondition (d f = 0) and is given by

dλ =f,σ Cdε

f,σ Cg,σ

=2

[(1 − 2ν)Si j dεi j − 2(1 + ν)αdI ε

1

√J2

] √J2

12(1 + ν)αα1 J2 + (1 − 2ν)Si j Si j(30)

where C is the isotropic material tensor, f,σ = ∂ f/∂σ ,g,σ = ∂g/∂σ and Si j and dεi j = dε are the deviatoricstress tensor and the incremental strain tensor, respectively.The solutions of the three equations (Eqs. (27)–(29)) give theexpressions for the incremental stresses and strains. Integratingthese expressions and adding the elastic components (phase I)yields the following axial and lateral stresses and strain in thesecond phase, respectively:

f IIc = fc,ep + Fz(εz − εz,ep) (31)

σ IIlat = σlat,ep + Flat(εz − εz,ep) (32)

εIIlat = εlat,ep −

Flat(εz − εz,ep)

Es f `

(33)

where

Fz =−(12αα1 + 2

√3α + 2α1

√3 + 1)Es f `

ms f `

[6ν − 3 − 18αα1 (1 + ν)

]+ 2

√3 (α + α1) − 2 (1 + 3αα1)

(34)

Flat =(6αα1 +

√3α − 2α1

√3 − 1)Es f `

ms f `

[6ν − 3 − 18αα1 (1 + ν)

]+ 2

√3 (α + α1) − 2 (1 + 3αα1)

.

(35)

The limit of this stage occurs when the steel material yields.This condition is reached when the lateral strain reaches theyield strain of the steel, εsy , and the lateral stress will be equal to

σlat1 = εsy Es f ` (εsy in absolute value). Substituting this stressinto Eq. (32) results in the following expression for the limitaxial strain of phase II (see Fig. 5):

εcc1 =Flatεz,ep + εsy Es f ` − σlat,ep

Flat. (36)

Furthermore, solving the yield criterion (Eq. (21)) with σlat1 =

εsy Es f ` gives the expression for the concrete confined stress ofphase II (see Fig. 5)

fcc1

f ′c

=

√3 + 6α

√3 − 3α

Ie1 + 1 = kD P Ie1 + 1 (37)

where kD P is a strength enhancement factor and Ie1 is theconfinement index (the lateral pressure normalized by theunconfined concrete strength) when the steel yields

Ie1 =εsy Es f `

f ′c

. (38)

2.2.3. Phase IIIIn this state, the concrete and the steel are in their plastic

range while the FRP behaves elastically. The solution is similarto that obtained in phase II, but the lateral stress increment isobtained from the action of the FRP composites alone. Thus,Eq. (29) will be replaced by the following:

dεIIIlat = −

dσ IIIlat

E f `

. (39)

Solving Eqs. (27), (28) and (39) gives the following axial andlateral stresses and strain in the third phase, respectively:

f IIIc = fcc1 + Fz1(εz − εcc1) (40)

σ IIIlat = σlat1 + Flat1(εz − εcc1) (41)

εIIIlat = −εsy −

Flat1(εz − εcc1)

E f `

(42)

where

Fz1 =−(12αα1 + 2

√3α + 2α1

√3 + 1)Es`

m f `

[6 ν − 3 − 18αα1 (1 + ν)

]+ 2

√3 (α + α1) − 2 (1 + 3αα1)

(43)

Flat1 =(6αα1 +

√3α − 2α1

√3 − 1)Es`

m f `

[6ν − 3 − 18αα1 (1 + ν)

]+ 2

√3 (α + α1) − 2 (1 + 3αα1)

.

(44)

The limit state of this stage occurs when the FRP ruptures.This condition is reached when the lateral strain reaches theactual rupture strain of the FRP, ε f u,a , and the lateral stress willbe equal to σlat,max = ε f u,a E f ` + εsy Es` (ε f u,a and εsy inabsolute values). The use of the actual FRP rupture strain is dueto the experimental evidence that this strain is about 58%–73%[10,23] of the FRP ultimate tensile strain, ε f u , obtained fromthe flat coupon test [24]. Substituting the stress σlat,max intoEq. (41) results in the following expression for the confinedconcrete strain (see Fig. 5):

εcc2 =Flat1εcc1 + (ε f u,a − εsy)E f `

Flat1. (45)


Solving the yield criterion (Eq. (21)) with σlat,max gives theexpression for the confined concrete strength (see Fig. 5)

fcc2

f ′c

= kD P Iemax + 1 (46)

where,

Iemax =σlat,max

f ′c

=εsy Es` + ε f u,a E f `

f ′c

. (47)

2.2.4. Phase IVIn this state, the FRP, which has ruptured, no longer

contributes to the confinement action and therefore the concretecore is confined only by the transverse reinforcement. TheDrucker–Prager model (used here) does not include softeningsurfaces. This implies that, after FRP rupture, the axial stressof the concrete core, which, in this case, is confined by only theyielded steel, will be constant and equal to the concrete strengthof the steel-confined section (for strains greater than εcc2). Analternative for this case, which was used also by Eid et al. [14],can be to incorporate a post-peak softening behaviour into thecurrent explicit solution by using the following expressions forthe residual concrete strength, fcr , and for its correspondingstrain (the ultimate concrete strain), εcu , which are based on theexpressions proposed by Eid [17]:

fcr

fcc2= (ρs f )

0.4≤ 1 (48)

εcu

ε′c

= (0.58 fyh/ f ′c + 14)ρ

(0.0035 fyh/ f ′c+0.17)

s f + 3 (49)

where ρs f is the mechanical volumetric transverse reinforce-ment ratio. This ratio is defined as ρs fyh/ f ′

c , where ρs is thevolumetric lateral reinforcement ratio.

3. Proposed stress–strain curve

Based on the above derivations, the axial and the lateralstress–strain relations of FRP-steel confined circular concretecolumns are described by quadrilinear curves, which aredefined by six points (Fig. 5):

fc =

0 for εz = 0fc,ep for εz = εz,epfcc1 for εz = εcc1fcc2 for εz = εcc2fcd for εz = εcc2fcr for εz = εcu

σlat =

0 for εlat = 0σlat,ep for εlat = εlat,epσlat1 for εlat = −εsyσlat,max for εlat = −ε f u,aεsy Es` for εlat = εlatdεsy Es` for εlat = −εs f

(50)

where εs f is the fracture strain of the transverse steelreinforcement (in absolute value), and fcd and εlatd are the

axial stress and the lateral strain after FRP rupture, respectively,derived from the steel-confined concrete curve at strain εcc2:

fcd = (εcc2 − εcc2,s)fcr − fcc2,s

εcu − εcc2,s+ fcc2,s (51)

εlatd = (εcc2 − εcc2,s)εsy − εs f

εcu − εcc2,s− εsy (52)

where fcc2,s and εcc2,s are the maximum axial stress and straindue to the action of the steel only, respectively (derived fromEqs. (45) and (46)).

4. Comparison with finite element analysis

In order to examine the analytical model, the proposedstress–strain curves (Eq. (50)) were compared against FEMsimulation. The dimensions and material properties used forthe simulation represent an FRP-steel confined concrete columntested by Eid et al. [23] (specimen C4NP4C) with D =

303 mm, Dc = 241.7 mm, and s = 100 mm. The ADINA [18]computer program was used for the FEM simulation. Thecylinder was modelled by 4-node axisymmetrical elements,while the transverse steel bars and the FRP were modelled by 1-node axisymmetrical ring elements. The axisymmetrical plane(shown in Fig. 6) was restrained in the vertical direction ofits bottom boundary. The material models implemented in theFEM analysis are similar to these used to develop the proposedmodel. The Drucker–Prager material model was used for theconcrete material and the nonlinear elastic material model wasused for the transverse steel bars and the FRP. Based on thematerial properties of specimen C4NP4C [23], the followingmaterial variables were used: Ec = 25592 MPa, ν = 0.15, α =

0.25, kD = 10.45 MPa, α1 = α/5 = 0.05, Es = 200 000 MPa,fyh = 456 MPa, E f = 78 000 MPa, and ε f u,a = 0.0119.Determination of the parameters α and α1 will be discussed inthe next section.

Fig. 7 shows the experimental axial and lateral stress–straincurves of specimen C4NP4C’s concrete core with the curvesobtained from the proposed model and FEM analysis. Thefigure shows excellent agreement between the proposed modeland FEM results. It is also shown that, when the steel ismodelled similar to the proposed model, i.e., as an equivalenttube, the FEM and model yield nearly identical results. The factthat the internal pressure ratio ps1/ps is less than one at thebeginning of the plastic concrete range accounts primarily forthe small difference between the results. As already stated, theFEM analysis has shown that the ratio ps1/ps at the full plasticconcrete range is about one.

The comparison between the model and the FEM resultsremains valid up to FRP rupture. Then, the proposed modeluses a softening behaviour, while the FEM axial stress remainsconstant and equal to the maximum stress of the concretecolumn confined only by the transverse steel. It should bealso noted that the FEM analysis was solved using incrementaliterative procedure, while the proposed curve (Eq. (50)) wasderived using basic algebra calculations.


Fig. 6. Axisymmetrical finite element model [18] for column C4NP4C [23].

5. Comparison with test results

The proposed model is based mainly on two concretevariables: the friction angle, Φ, and the dilatancy angle, Ψ .The friction angle determines the slope of the yield function,α (see Eq. (22)); the dilatancy angle determines the slopeof the potential function, α1 (see Eq. (26)), according to thesame relation (that in Eq. (22)). Previous studies used differentvalues for these variables (Φ = 28◦–56.6◦, Ψ = 0◦–Φ,[21,25–27]). Based on the explicit solution that was obtainedfor circular confined concrete columns, a sensitivity analysiswas performed to obtain the best values of the friction anddilatancy angles to fit the experimental data (114 steel-confinedand 93 FRP-confined concrete specimens - see [14]). Thirty-sixFRP-confined and 15 FRP-steel confined concrete specimens(from Eid et al. [23]) were added to this test data, for atotal of 258 experimental results. According to the proposedmodel, the confined concrete strength is influenced only bythe friction angle, while its corresponding strain is mainlyinfluenced by the dilatancy angle. It is well-known that thefriction angle, and therefore the parameter α, decreases withincreasing confining pressure [27]. This behaviour was borneout by the experimental results. These tests lead to a frictionangle that is larger than 50◦ for a very small lateral pressure(caused by a small amount of lateral confinement or Iemax = 0).It decreases to a constant average value of 32◦ for a relativelyhigh lateral pressure (caused by a relatively high amount oflateral confinement Iemax ≥ 0.2). In order to account for theconcrete’s behaviour according to the test results, the followingexpression is suggested for the frictional angle as a function ofthe normalized maximum lateral pressure (which depends onlyon the column’s properties):

Φ = 50◦− 90Iemax ≥ 32◦. (53)

Fig. 7. Experimental, FEM, and proposed stress–strain behaviour of FRP-steelconfined concrete core (specimen C4NP4C) [23].

Moreover, based on the confined strain of the proposed model(Eq. (45)), the dilatancy angle, Ψ , or the potential surface slope,α1, are mainly functions of the lateral stiffness ratio, ms f `.By introducing the experimental axial strains, into Eq. (45), itwas found that increasing the lateral stiffness ratio results in adecrease in the dilatancy angle. Based on these test results, thefollowing expression of the dilatancy angle is suggested:

Ψ = 30◦− 500ms f ` ≥ −20◦. (54)

Note that for the determination of Fz1 and Flat1 in the thirdphase (steel in yielding), Ψ (and α1) will be derived from Eq.(54) with ms f ` = m f `. It should be noted that Eqs. (53) and(54) were estimated from the test results which were foundto have a large scatter. Instead of using constant values forthe frictional and the dilatancy angles, these bilinear equationsgive better estimation for these parameters as a function of thenormalized confining pressure and stiffness.

Table 1 and Fig. 8 compare the proposed model withexperimental results of FRP-steel confined concrete columnstested by Eid et al. [23] and by Demers and Neale [12]. Theexperimental stress–strain curves of the FRP-steel confined


Tabl

e1

Com

pari

son

betw

een

expe

rim

enta

lres

ults

and

the

prop

osed

mod

el

Ref

eren

ceSp

ecim

enN

o.D (m

m)

ca (mm

)f′ c (M

Pa)

ε′ c

FRP

com

posi

teT

rans

vers

est

eel

f cc2

ε cc2

t (mm

)E

f(M

Pa)

εfu

εfu

,aε

fuTy

pef y

h(M

Pa)

s (mm

)φ (m

m)

Test

(MPa

)M

odel

(MPa

)E

rror

(%)

Test

Mod

elE

rror

(%)

Eid

etal

.[23

]C

4NP4

C30

325

31.7

0.00

21.

524

7800

00.

0134

0.89

Spir

al45

610

011

.369

.38

68.7

1−

0.97

0.02

080.

0212

1.92

Eid

etal

.[23

]C

2NP2

C30

325

31.7

0.00

20.

762

7800

00.

0134

0.44

Spir

al45

665

11.3

49.6

049

.82

0.44

0.01

320.

0144

9.09

Eid

etal

.[23

]C

4NP2

C30

325

31.7

0.00

20.

762

7800

00.

0134

0.46

Spir

al45

610

011

.345

.73

46.6

31.

970.

0077

0.01

0738

.96

Eid

etal

.[23

]C

2N1P

2N25

30

360.

002

0.76

278

000

0.01

340.

63Sp

iral

456

6511

.366

.61

64.8

4−

2.66

0.01

550.

0176

13.5

5E

idet

al.[

23]

A3N

P2C

303

2531

.70.

002

0.76

278

000

0.01

340.

67H

oops

602

709.

550

.60

51.5

81.

940.

0124

0.01

4113

.71

Eid

etal

.[23

]C

2MP4

C30

325

50.8

0.00

241.

524

7800

00.

0134

0.80

Spir

al45

665

11.3

92.0

888

.17

−4.

250.

0164

0.02

0424

.39

Eid

etal

.[23

]C

2MP2

C30

325

50.8

0.00

240.

762

7800

00.

0134

0.64

Spir

al45

665

11.3

73.0

373

.90

1.19

0.01

040.

0146

40.3

8D

emer

s&

Nea

le[1

2]U

25-2

300

2023

.90.

002

0.9

8400

00.

015

0.38

Hoo

ps40

015

011

.336

.60

37.3

52.

050.

0104

0.00

94−

9.62

Dem

ers

&N

eale

[12]

U40

-330

020

43.7

0.00

270.

984

000

0.01

50.

42H

oops

400

300

6.4

54.8

055

.63

1.51

0.00

680.

0078

14.7

1

aC

oncr

ete

cove

r.


Fig. 8. Experimental and analytical stress–strain behaviour of FRP-steel confined concrete columns tested by Eid et al. [14] and Demers and Neale [12].

concrete columns were derived by dividing the concrete load,Pc, by the confined concrete cross-sectional area. This areais equal to the concrete total cross-sectional area, Ac, andto the core cross-sectional area, Acc, before spalling of theconcrete cover (which corresponds to FRP rupture in FRP-confined concrete) and after cover spalling, respectively. Thepredicted stress–strain curves were derived by combining thebehaviour of the concrete core (confined by the dual action ofFRP and steel) and the concrete cover (confined by only FRP)and by using the actual FRP rupture strain. At this point oftime, and due to the difficulty of estimating the actual FRPrupture strain in confined concrete columns, this strain shouldbe determined by performing tests on FRP-confined standardconcrete cylinders [10]. However, for design purposes of largescale FRP-confined concrete columns this strain can be fairlyestimated as 50% of the FRP rupture strain obtained from theflat coupon test [24]. Moreover, it should be noted that the

normalized equivalent steel tube thickness was calculated as0.47ρs . Fig. 8 and Table 1 show that the overall predictedstress–strain behaviour and the predicted confined concretestrength are in good agreement with the experimental results,although the degree of agreement drops when the confinedstrain is considered. The idealization of the FRP and thelateral steel as elastic and elastic-perfectly plastic materials,respectively, and the use of the Drucker–Prager yield criterionwithout hardening surfaces can influence the prediction of theconfined strain due to the path dependency of the concretedeformations. Moreover, it should be noted that the rupture ofFRP applied to a concrete column is sudden and explosive. Asa result, after this rupture the measurement devices attachedto the FRP are not able to record more data. Therefore, thelateral stress–strain behavior in Figs. 7 and 8 do not show thedescending part of the curves.


6. Conclusions

This paper uses the theories of elasticity and plasticityto present simplified analytically derived stress–strain curvesthat describe the axial and lateral behaviour of externallyand internally confined concrete in circular columns. Thesolution to the problem of an elastic cylinder confined byinternal and external elastic confining materials is derived.Following the elastic solution, the analytical curves are derivedfrom the full elastoplastic behaviour of the confined concretecolumn and by using the well-known Drucker–Prager failurecriterion to represent concrete behaviour. Application of theDrucker–Prager model does not require an iterative procedurein order to solve the problem, so an explicit solution is obtained.The proposed model can be applied to describe the behaviourof concrete confined by transverse steel reinforcement, by FRPcomposites, or by both. It is shown that the analytical resultsare in good agreement with both experimental and the finiteelement method results.

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