Non-Linear Modeling of FRP Confined RC Columns

8/6/2019 Non-Linear Modeling of FRP Confined RC Columns

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FRPRCS- 8 University of Patras, Patras, Greece, July 16-18, 2007

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The influence of external loading conditions, namely pure compression and combined flexure andcompression has been studied in order to determine the available ductility of unstrengthened andstrengthened rectangular hollow cross sections. This evaluation consists in an experimental phaseundertaken in conjunction with analytical studies to predict and to model the results of the former tests.The development of design construction specifications and of a refined methodology to design and

assess hollow cross section members behavior under combined axial load and bending is the finaloutput of the program.

3 EXPERIMENTAL CAMPAIGN

Laboratory work has been conducted at the University of Naples “Federico II” and includes twoparts: the first series of tests focusing on the flexural-compression behavior of un-strengthened squarehollow columns in reduced scale and a second series of tests studying FRP strengthened companionhollow columns under the same loading conditions, investigating failure mode changes and ductilityand flexural strength enhancements. A total of 7 hollow square cross section concrete columns weretested. Unstrengthened and FRP jacketed square hollow piers subjected to combined axial load andbending, therefore slender specimens, whose behavior is dominated by flexure, were investigated. It isnoted that shear failure occurs only for short rectangular piers; in all other piers the collapse isgoverned by exhaustion of ductility induced by combined axial load and bending.

Test specimens reproducing in 1:5 scale typical hollow square cross section concrete bridge pierswith section dimensions of 360x360 mm 2 and walls thickness of 60 mm were tested (Figure 1).

Fig. 1 Specimen geometry and FRP wrapping layout.

The test matrix was designed in order to assess the FRP wrapping effectiveness incorrespondence of three P/M ratios. Accordingly three unstrengthened specimens (U1, U2 and U3)and other three companions strengthened with CFRP laminates (S1, S2 and S3) were tested withvarious load eccentricities kept constant during each test. The three eccentricities have been selected

in order to study the behavior of hollow members under P/M combinations carrying the ultimate neutralaxis at ultimate outside the cross section (e=50 mm, fully compressed “1” ), at mid-height (e=200 mm“2” ) and close to the compressed flange (e=300 mm “3” ). A construction issue did not allow to testboth specimen U1 and S1 with an eccentricity of 50 mm, but the actual eccentricity was respectively52 mm and 80 mm.

A total of 2 plies of CFRP wet lay-up unidirectional fabric (600 gr/m 2) have been applied in thetransverse direction in all strengthened specimens for the entire specimen height. The number of installed plies was considered an upper limit that could be derived from an economical and technicalanalysis, also accounting for the scale reduction. On this scheme the mechanical CFRP reinforcementratio is four times bigger than that of internal stirrups. Nevertheless it has been observed that theinfluence of the number of layers of FRP on the specimens under eccentric loading is not sopronounced as that of the specimen under concentric loading [2]. One unstrengthened specimen (U0)under pure axial load was also tested.

Fundamental results and global outcomes of the experimental research in terms of strength andfailure modes of the scaled hollow columns are given in [1]. In the following, a brief summary will recallthe main results in terms of strength and failure modes.




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The failure of hollow members is strongly affected by the occurrence of premature mechanisms(compressed bars buckling and unrestrained concrete cover spalling). FRP confinement does notchange actual failure mode, but it is able to delay bars buckling and to let compressive concretestrains attain higher values, thus resulting in higher load carrying capacity of the column (strengthimprovement about 7% in the case of larger eccentricity and 19% in the case of smaller eccentricity)

and significantly in ductility enhancement.The ductility increments have been estimated through the comparison of curvature ductility μχ

indexes (defined as the ratio of the curvature on the softening branch at 80% of ultimate load, χ80% ,and the yielding curvature, χy). In unstrengthened columns the curvature ductility ranged between 1(brittle failure of U1 and U2) and 1.54 (specimen U3), while in the case of strengthened columns thecurvature ductility increased significantly attaining values ranging between 3.07 and 8.27. Thisanalysis evidenced a remarkable improvement of the seismic response of the wrapped columns: after peak load they kept a good load carrying capability, that is good energy dissipation.

The strength improvement was more relevant in the case of specimens loaded with smaller eccentricity, while the ductility improvement was more relevant in the case of bigger eccentricity. Atlower levels of axial load also the brittle effect of reinforcement buckling was less noticeable.

4 THEORETICAL ANALYSIS

The results of the experimental campaign suggest that a reliable numerical procedure to predicthollow cross section behavior under combination of flexure and compression should includeappropriate models for compressed bars buckling and concrete cover spalling. Besides a reliablestress-strain behavior of concrete, and of course of other structural involved materials, is necessary.

4.1 Proposed hollow cross section confinement modelTwo approaches were proposed to simulate the effect of the FRP confinement and to assess the

behavior of the FRP confined hollow members. In the first approach (Section 4.1.1) the confinementdue to the interaction of the four walls forming the hollow member is considered. In the Section 4.1.2 asecond different approach is taken instead, which considers the confinement of the whole hollowsection forming the hollow member.

A preliminary Finite Element Method (F.E.M.) analysis has been conducted in the elastic range toevaluate the stress field generated by external wrapping on hollow concrete core.

A square solid cross section column has been wrapped with an elastic reinforcement only in thetransverse direction (no reinforcement has been applied in the longitudinal direction) and a distributedload has been applied on the top in displacement control. The boundary constraints were only verticalsupports and two pairs of supports in the plane of the load to restrain only rigid movement so that thespecimen was free to expand laterally.

To evaluate the effect of concrete confinement only, no longitudinal bars have been inserted in themodel. A cross section at mid height of the specimen has been extracted and the principal confiningstress vector field is plotted in figure 2a.

a) b)

Fig. 2 F.E.M.: Elastic Vector Plot of confining stresses.




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Depicted vector field clearly shows the well known arching effect. From the corners two struts startthat connect opposite corners and in each quarter of the section are clearly visible the arch-shapedpaths of the confining stresses.

Subsequently the inner core of concrete has been removed and the same analysis has beenconducted on a hollow square cross section column, with the same load and boundary conditions.

The principal confining stress vector field on the mid height section is plotted in figure 2b.In this second vector field plot it is still clearly noticeable an arching effect, but in the case of hollow

section, it seems that the strut between two opposite corners spreads along the walls leading to aconfining stress field compatible with four wall-like columns confinement. This fact allows consideringthe effect of confinement on the four walls interacting together. The arch-shaped paths of the confiningstresses rapidly changes in a straight distributed confinement stress field moving off the corner.

4.1.1 Walls interaction approach This approach considers the interaction of the four walls forming the hollow member. The

confinement of the walls have been analyzed according to the preliminary FEM analysis and to thebehavior observed in wall-like columns tested by Prota et al. [3] similar to the behavior of the wallsforming the hollow member.

The transverse dilation of the compressed concrete walls stretches the confining device, which

along with the other restrained walls applies an inward confining pressure. The effective pressure f l’ isreduced by a reduction factor k eff due to the so-called arching effect. The reduction factor is given bythe ratio of the effective confinement area, the core of the walls, to the total concrete area enclosed bythe FRP jacket; in the present case is k eff =0.69. For FRP-confined concrete, the ultimate compressiveaxial strain of concrete is considered to be attained when lateral strain is equal to FRP ultimate strainhowever experimental evidence shows that FRP failure did not occur in eccentrically loadedspecimens. The confining pressure is provided by an FRP jacket of the same thickness to anequivalent circular column of a diameter D equal to the average (longer) side length (in the presentcase, D ≈ 300mm).

Spoelstra and Monti model [4], or more widely, a general model for rectangular solid sections, canbe adapted to assess the behavior of the confined wall forming the hollow section. The behavior of thehollow member, both in terms of load deflection and of moment curvature, can be assessedconsidering a stress-strain relationship derived by solid rectangular cross section confinement modelsfor confined concrete. As clearly seen in the following numerical predictions, this approach is rather good to assess strength enhancements, while it fails to predict the post peak behavior and the hollowsection thus resulting in incorrect ductility predictions.

4.1.2 Whole section confinement approach A confinement model for circular hollow sections is proposed and extended to square hollow

sections. The model is able to estimate confinement effectiveness, which is different in the case of solid and hollow sections. More details on the proposed confinement model can be found in [5].

This approach considers the confinement of the whole hollow section forming the hollow member.Recently a model based on the assumption that the increment of stress in the concrete is achieved

without any out-of-plane strain was proposed [6]. Plain strain conditions were adopted to simulate theconfinement effect.

Fam and Rizkalla model [7] is based on equilibrium and radial displacement compatibility andthrough the equations proposed by Mander et al. [8] adopts a step-by-step strain increment techniqueto trace the lateral dilation of concrete. The failure of the confined concrete member is due to therupture of the FRP confinement (i.e. controlled by the multiaxial Tsai-Wu failure criteria).

The passive confinement on axially loaded concrete members is due to the transverse dilation of concrete and the presence of a confining device which opposes to this expansion and puts theconcrete in a triaxial state of stress. Gardener [9] tested concrete cylinders under different confiningpressures (uniform in the transverse direction) thus providing dilation ratios of concrete under differenttransverse confinement pressures.

In the hypothesis of axial symmetry the radial displacement s r is the only displacement componentand stress components σr and σθ (where r calls for the radial component and θ for the circumferentialcomponent) can be evaluated according to boundary conditions (i.e. applied external (at r=R e) inwardpressure q e and internal (at r=R i) outward q i pressure).

In the case of external inward pressure only (q i=0) the stress equations and displacement equationbecome:




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2 2e

2 2 2e

R R1

R Re i

r i

qr

σ ⎛ ⎞= −⎜ ⎟− ⎝ ⎠

(1a)

2 2e

2 2 2

e

R R1

R Re i

i

q

r

θ σ

⎛ ⎞= +⎜ ⎟

− ⎝ ⎠

(1b)

( )2 2e

2 2e

R R1( ) 1 2

R Re i

r i

qs r r

E r

ν ν

⎡ ⎤+= − +⎢ ⎥− ⎣ ⎦

(1c)

In the case of internal outward pressure only (q e=0) if the thickness t=R e-R i of the cylinder is verysmall compared to the average radius R=(R e+R i)/2 (i.e. an FRP jacket where t«R), the stressequations and displacement equation can be simplified as:

Riqt

θ σ ≈ − , ( )

2R( ) 1i

r

qs r

Et ν ≈ − − (2a,b)

A key aspects of the proposed model is that plain strain condition is considered to evaluate theradial displacement of the elements confined by FRP and in the case of hollow core sections, thedifferent contributions of radial and circumferential stresses are explicitly considered through anequivalent average confining pressure (actually the confining stress field is not equal in the twotransverse directions and the effect of confinement should be evaluated in each point of the sectionwith the effective confining pressures different in the two orthogonal direction, however this approachis currently neglected due to the massive computational efforts).

The dependence of the lateral strain with the axial strain is explicitly considered through radialequilibrium equations and displacement compatibility (the sum of the radial displacements of concrete- expansion due to axial load and contraction due to confining pressure - is equal to the confiningdevice expansion due to the reacting confining pressure if there is no detachment).

The confining device is in equilibrium with the concrete cylinder so that the inward pressure q i onconcrete cylinder is equal to the outward pressure q e=q i=q on the confining jacket.

All the previous equations can be explicated in the form q=q( εc), so that at each axial strain εc theconfining pressure q exerted on concrete by the FRP jacket is associated:

( ) ( )22

e2 2e e

R 1 R R1 1 2

R R R

cc

e c i f c

f c i

q

E t E

ν ε

ν ν ν

=⎡ ⎤⎛ ⎞+⎢ ⎥− + − +⎜ ⎟− ⎢ ⎥⎝ ⎠⎣ ⎦

(3)

Previous equations are based on linear elasticity theory for all the involved materials (E c and νc areconcrete elastic modulus and Poisson’s ratio respectively, while E f and νf are FRP elastic modulus andPoisson’s ratio respectively). This assumption is almost good for the elastic confining device (i.e.FRP), but it is too coarse in the case of concrete.

Before peak strength the concrete dilation in the transverse direction is very low (due to an almostconstant, elastic Poisson’s ratio). As the deformation increases after peak, confined concrete exhibitspost-peak behavior characterized by the appearance of significant cracking.

To account for the nonlinear behavior of concrete, a secant approach can be considered. Theelastic modulus and the Poisson’s ratio are function of the axial strain and of the confinement.

An iterative procedure is then performed to evaluate, at any given axial strain εc, the correspondingstress f c, pertaining to a Mander curve at a certain confining pressure f’ l.(effective lateral pressure if ashape factor k eff is adopted).

The secant elastic modulus of concrete is the slope of the line connecting the origin and thepresent stress-strain point on the stress-strain curve. To simplify and to avoid an iterative procedure todetermine the actual E c(εc), it is assumed the secant modulus of the iteration (i) as the secant modulusof the previously evaluated stress-strain point (i-1) of the confined concrete curve. The first value is theinitial tangent elastic modulus E co .

The secant Poisson’s ratio is used to obtain the lateral strain at a given axial strain in theincremental approach. The dilation of confined concrete is reduced by the confinement; therefore, thePoisson’s ratio at a given axial strain level is lower in the presence of confining pressure. Gardner [9]




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tested many concrete cylinders under different confining hydrostatic pressures. Fitting the resultscurve with a second-order polynomial a simplified linear relationship for εc under constant confiningpressure is provided, and from regression analysis [7]:

'1 0.719 1.914 '

c c l

co cc co

f f

ν ε

ν ε

⎛ ⎞

= + +⎜ ⎟⎝ ⎠(4)

Where νc is the actual Poisson’s ratio at a given axial strain εc and actual confining pressure f’ l.The actual peak compressive stress and strain (evaluated for the actual confining pressure f’ l) are f cc and εcc . The initial values are the unconfined peak concrete strength (f’ co ), and the Poisson’s ratio ( νco)usually ranging between 0.1 and 0.3.

At any given axial strain εc, the Poisson’s ratio is reduced with the increase of f’ l, because the ratioεc/ εcc decreases at growing of εc. This expression (obtained by linear regression on a definite range of confining pressures) is not suitable for concrete subjected to very high confinement pressure at lowaxial strain and for concrete subjected to very low confinement pressure at high axial strain.

Usually the Poisson’s ratio of a confining device made by uniaxial FRP fibers applied by wet lay-uptechnique, can be neglected in this procedure. But in the case of laminates with multiaxial fibers or FRP tubes, the Poisson’s ratio should be considered because generally it is greater even than theinitial concrete Poisson’s ratio.

A first trial value of the Poisson’s ratio is determined (i.e. the initial value at the beginning of theprocedure or the previously evaluated value at iteration i-1) and along with the secant elastic modulusE c, a confining pressure f’ l is evaluated as f’ l=[σr (R e)+ σθ(R e)]/2 (see Eqs. (1), (3) and (5); in the case of circular sections, f’ l=f l):

2e

2 2e

R

2 R Rr

li

q f θ

σ σ += =−

(5)

Now the peak compressive strain εcc can be evaluated and in turn an output Poisson’s ratio isdetermined. This is the new value of the Poisson’s ratio to repeat the procedure that converges whenthe output value of Poisson’s ratio is close enough to the Poisson’s ratio in input. Once the correctPoisson’s ratio is known at actual axial strain, a Mander curve at given confining pressure f’ l can bedrawn and the stress point f c can be determined corresponding to the actual strain εc. At each level of load/deformation (namely εc), the complete stress and strain regime in both the concrete cylinder andconfining device is known, i.e. the circumferential stress in the confining device is given by Eq. (2a).

The procedure is repeated up to a value of axial strain that induces failure of the confining device,i.e. accounting for the triaxial state of stress in the jacket. The flow chart of the iterative procedure isdepicted in figure 3.

The bigger is the hole, the higher is the deformability of the element and the circumferentialstresses compared to the radial component: in the case of solid section the dilation of concrete isrestrained by the FRP wraps and this interaction yields a strength improvement, while in the case of thin walls, the larger deformability does not allow to gain such strength improvement, even though asignificant ductility development is achieved. In figure 4 numerical predicted stress strain relationshipsare depicted. The analysis has been conducted on a solid section with different jacket thickness andon a hollow section with different R i/R e ratios, and constant relative confinement stiffness E f t/(R e-R i).This model is not directly applicable to square hollow RC sections in which the concrete is notuniformly confined by the FRP jacket. The commonly accepted approach to deal with confinedconcrete in noncircular sections is to find an equivalent pressure in terms of the average stress,reflecting the evidence that FRP confinement has a different level of efficiency in a square sectionthan in a circular section in which all the concrete is confined to the same degree.

4.2 Refined analysis of the cross sectional behaviorThrough the use of a fiber model that meshes the concrete cross-sectional geometry into a series

of discrete elements/fibers, sections of completely arbitrary cross-sectional shape (including hollowprismatic cross sections) can be modeled. Each discrete element is assumed to have a constant

stress. For a specified neutral axis location and a specified section curvature (given a reference strainin a given point of the cross section and with neutral axis known) internal section forces are computed.




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The neutral axis position is changed until the net internal axial force in the section is in equilibriumwith the externally applied axial load acting on the section, then the internal section flexural moment iscomputed and the corresponding moment curvature diagram is plotted. Perfect bond is assumed atthe interfaces between concrete and steel reinforcing bars; plate buckling (as local buckling of a thincompression flange) is not accounted, so that the ultimate strength is generated by material failure

and/or steel reinforcement bar buckling. Tension stiffening effect, compressed bars buckling, concretecover spalling and FRP confinement of concrete are included in the model.

Fig. 3 Flow Chart of the proposed iterative confinement model procedure.

One of the major improvements in member behavior due to FRP wrapping is highlightedconsidering that in unstrengthened columns, when steel reinforcement reaches in compression thebuckling stress, as it pushes outward surrounding concrete, the concrete cover spalls out. The spallingof concrete cover and the buckling of the reinforcement can be taken into account by considering for the concrete cover fibers a brittle (step-like) concrete behavior after peak load with zero stresscorresponding to reinforcement buckling strain. The improved stress-strain relationship with stressreductions due to the buckling phenomenon has been adopted for steel in compression after [10]. Inthe case of members wrapped with FRP, the steel bars, when buckling occurs, push internal concreteunrestrained cover in the inward direction (in the hollow part) and in the numerical model concretecover spalling can be simulated adopting the linear descending branch according to size effect theory[11] after peak strain, as already done in the case of concrete in unstrengthened members.

5 EXPERIMENTAL-THEORETICAL COMPARISON

The proposed refined nonlinear methodology described in previous Section 4 has been adopted toanalyze experimental outcomes.

This algorithm allowed to draw the theoretical P-M interaction diagram as well as momentcurvature diagram. Confined concrete has been modeled according to the two approaches proposed




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in previous Section 4.1 and the stress-strain relationship of concrete under concentric loading isassumed to be representative of the behavior of concrete under eccentric loading.

The Mander et al. model [8] in the following will be considered for unconfined concrete, becausefor the modeling of strengthened walls has been adopted for the first approach, the Spoelstra-Montiadapted model [4] and, for the second approach, the proposed hollow section confinement model, that

are both an evolution of the Mander et al. model [8] for confined concrete. The theoretical predictionsaccording to wall interaction and adapted confinement approach will be denoted by (1) mark whilepredictions according to proposed hollow section confinement proposed model with (2).

0

10

20

30

40

50

60

-0,01 -0,005 0 0,005 0,01 0,015

A x

i a l S t r e s s

Axial StrainRadial Strain

Unconfined

Fig. 4 Stress Strain relationships: Unconfined and confined solid and hollow sections.

The refined methodology developed so far with the selected material models and the assumptionsmade seem to predict well both the global behavior and the local deformability of the hollow columns.

5.1 StrengthThe comparison of theoretical P-M interaction diagrams according to the adopted concrete models

and the experimental outcomes is depicted in figure 5.All the experimental results were noted to be in good agreement with the theoretical predictions.

Compared to the experimental values, the theoretical P-M predictions are on the conservative sideand the scatter is about 10% (with the only exception of U2 test where capacity is overestimated of less than 0.5%).

U1

U2

U3

S1

S2

S3

U0

0

1000

2000

3000

4000

0 50 100 150 200 250 300M [kNm]

P [ k N ]

Mander et al.Model

ProposedModel (2)

Spoelstra-MontiModel (1) (adapted)

Fig. 5 Test results and numerical comparison on P-M diagram.




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5.2 Ductility

Figure 6 shows moment-curvature diagrams where comparison of experimental results withtheoretical previsions for the U-series and S-Series tests are reported.

Relevant (M- χ) points, corresponding to significant material strains, are indicated on the theoretical

curves. In the first linear elastic phase before concrete cracking it is possible to refer to the well knownelastic relationship: χ=M/EI; then, after cracking, the effective moment of inertia has to be consideredand tension stiffening effects take place.

It is clearly noticeable a drop in the load-carrying capacity of the section when steel incompression get close to the buckling strain and concrete cover starts spalling. The approach (1) israther good to assess strength enhancements (strength evaluated when reinforcement bucklingoccurs), while it fails to predict the post peak behavior thus resulting in incorrect ductility predictions.

0

100

200

0,0E+00 2,0E-05 4,0E-05[mm -1 ]

M [ k N

m ]

EI

Spoelstra-MontiModel (1)(adapted)

Strain at peak concrete compressive stress

Buckling strain for steel in compression

Yielding strain for steel in tension

S2

ProposedModel (2)

U2

Mander et al.Model

Fig. 6 Moment-Curvature Diagram: U2-S2 Theoretical-experimental comparison.

5.3 FRP efficiencyThe proposed model is able to trace the evolution of longitudinal (parallel to fibers) strains (and

then stresses) into FRP confining wraps. Theoretical values have been compared with experimentalstrain gauges measurements at mid-span of the compressed wall. The CFRP effectiveness can bechecked by strain rate and strain thresholds. In Figure 7, experimental FRP strain data in the middle of the compressed wall are compared with theoretical predictions throughout the entire load history for specimen S1. Good agreement is found both on the load path and the ultimate value.

0

1000

2000

0,00% 0,10% 0,20% 0,30% 0,40%

L o a d

[ k N ]

FRP Strain

S1

ProposedModel (2)

Fig. 7 Comparison: Experimental S1 FRP strains vs. Load and theoretical [2].




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6 CONCLUSIONS

The present work is included into a wider activity that aims to improve the knowledge and developa cost and time effective design method for fast FRP strengthening of hollow bridge columns so thatbridge function can be quickly restored. The strengthening intends to upgrade seismic capacity in

terms of strength and ductility.The failure of hollow members is strongly affected by the occurrence of premature mechanisms

(compressed bars buckling and unrestrained concrete cover spalling). Confinement did not change theactual failure mode (steel reinforcement compressive bars buckling and concrete cover spalling), but itwas able to delay bars buckling and to let compressive concrete strains attain higher values, thusresulting in higher load carrying capacity of the column (strength improvement is about 15%) andsignificantly in ductility enhancement. The increase in confined concrete strength turns into loadcarrying capacity increase mainly in the columns loaded with small eccentricity (see figure 5 where it isclear that close to pure bending load the effect of concrete strength enhancement - i.e. due toconfinement - is insignificant because failure swaps to tension side). However in small loadingeccentricity cases, concrete ductility and strains are increased significantly thus resulting in ductilityimprovements. At lower levels of axial load also the brittle effect of reinforcement buckling is lessnoticeable.

Results of experimental tests and theoretical analyses show that a good agreement was achieved(the same strength increments and curvature ductility increments as experimentally found inperformed tests are predicted). The proposed confinement model, coupled with the proposedcomputation algorithm is able to predict the fundamentals of the behavior of the hollow squaremembers confined with FRP both in terms of strength and ductility giving a clear picture of themechanisms affecting the response of this kind of elements. The model is able to trace the occurrenceof the brittle mechanisms, namely concrete cover spalling and reinforcement buckling (generally thesetwo mechanisms are liable of the failure mode of hollow members), the evolution of stress and strainsin the concrete and in the confinement jacket, allowing to evaluate at each load step the multiaxialstate of stress and eventually the failure of concrete or the failure of external reinforcement. The mainoutput of the proposed model is also the assessment of the member deformability in terms of bothcurvature and displacement ductility.

REFERENCES

[1] Lignola GP, Prota A, Manfredi G and Cosenza E. “Experimental performance of RC hollowcolumns confined with CFRP”. ASCE Journal of Composites for Construction , 11, 1, 2007.

[2] Li J, Hadi MNS. “Behavior of externally confined high-strength concrete columns under eccentric loading”. Journal of Composite Structures . 62, 2, 2003, pp 145-153.

[3] Prota A., Manfredi G., Cosenza E.” Ultimate behavior of axially loaded RC wall-like columnsconfined with GFRP”. Composites: part B, ELSEVIER , 37, pp 670-678.

[4] Spoelstra MR, Monti G. “FRP-confined concrete model”. ASCE Journal of Composites for Construction , 3, 3, 1999, pp 143-150.

[5] Lignola GP. “RC hollow members confined with FRP: Experimental behavior and numerical modeling“ Ph.D. Thesis, University of Naples, Italy, 2006

[6] Braga, F., and Laterza, M. “A new approach to the confinement of RC columns.” Proc., 11 th European Conf. on Earthquake Engineering , Balkema, Rotterdam, The Netherlands, 1998

[7] Fam, Amir Z. and Rizkalla, Sami H. “Confinement Model for Axially Loaded Concrete Confinedby FRP Tubes,” ACI Structural Journal , 98, 4, 2001, pp 251-461.

[8] Mander, J.B., Priestley, M.J.N., Park, R., (1988a). “Theoretical stress–strain model for confined concrete”. ASCE Journal of the Structural Division 114, 1988, pp 1804–1826.

[9] Gardner, N. J., (1969), “Triaxial Behavior of Concrete,” ACI JOURNAL Proceedings , 66, 2,1969, pp. 136-146.

[10] Cosenza E and Prota A “Experimental Behavior and Numerical Modeling of Smooth SteelBars under Compression", Journal of Earthquake Engineering . 10, 3, 2006, pp 313-329.

[11] Hillerborg A. “The compression stress-strain curve for design of reinforced concrete beams”.Fracture Mechanics: Application to Concrete, ACI SP-118:1989, pp 281-294.

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Non-Linear Modeling of FRP Confined RC Columns