FRP-Confined Concrete Columns: A New Procedure for Evaluating the Performance of Square and Circular Sections

In the last few decades, the upgrading of existing reinforced concrete columns with the use of FRP jackets has met with increasing interest for its effectiveness and ease of application. )e use of these kinds of jackets ensures an improvement of the affected column in terms of strength and ductility; however, the prediction of behavior of columns wrapped with FRP jackets is still an open question because of the many parameters that influence the effectiveness of the upgrading technique, and several semiempirical models are proposed. Because these models are often only applicable to specific cases, in this paper, a generalized criterion for the determination of the increase in strength, in ductility, and in dissipated energy for varying corner radius ratios of the cross section and fiber volumetric ratios is shown. Numerical results using a finite element analysis, calibrated on the basis of experimental data available in the literature, are carried out to calibrate the new analytical models. A comparison with some available models confirms the reliability of the proposed procedure.


Introduction
In the last few years, there has been an increasing need to upgrade existing structures; in this context, particularly challenging is the upgrading of existing reinforced concrete (RC) structures, which is often needed because of errors in design or construction, change in use and hence in design loads, damage caused by seismic events or environmental factors.In the existing RC structures, the columns always need to be strengthened in the light of the performancebased design philosophy adopted by almost all national seismic codes; the most common techniques to upgrade column performances is the use of jackets, such as RC, steel, or fiber-reinforced polymer (FRP) jackets.e use of FRP jackets has met with increasing popularity [1][2][3][4][5][6][7][8][9] for different reasons: very high strength-to-weight ratio, corrosion resistance, and ease and speed of application.For these reasons, the structural behavior and effectiveness of FRP jackets has been widely investigated, and as a result of some studies, the first guidelines have been drawn up [10][11][12].
is technique also has negative aspects, such as inapplicability on wet surfaces, high costs of epoxy resin and of specialized workers for application, and inapplicability at temperatures lower than 10 °C or higher than 30 °C.However, it remains a very good method to strengthen RC columns because it is able to generate a considerable confinement effect without adding stiffness and mass to the element, which in seismic conditions may be a disadvantageous effect.For this reason, major research has been devoted to the investigation of the parameters that influence confinement effectiveness.
Results of earlier research conducted on columns wrapped with FRP jackets demonstrate that confinement effectiveness is significantly affected by several parameters: type of fiber and resin, fiber volume and orientation, jacket thickness, concrete strength, shape and corner radius of the cross section, length to diameter (slenderness) ratio, and the bond at the interface between the concrete core and the jacket.For example, in [7], the corner radius influence was investigated, and it was found that when the corner radius is zero, no confinement effect is provided by FRP jackets.In [13], the influence of the shape of the cross section was studied, and the results showed that for a fixed number of FRP layers, the confinement effect is strongly dependent on the shape of the cross section of the column.In [14], it was shown that a small corner radius of the cross section significantly reduces the ultimate strength of the FRP jackets due to stress concentration in the corner area.In [15], a parametric study was conducted to evaluate the effects of stiffness of the FRP and presence of inner holes inside the concrete core; it was shown that the increase of the central hole size reduces the confinement effect, and the increase of the stiffness of the FRP sheets improves the confinement.In [16], an experimental program was conducted in order to evaluate the seismic behavior of concrete columns confined with steel and FRP.It was shown that section and member ductility decrease with an increase of spiral pitch and reduced amount of spiral reinforcement.Furthermore, it was observed that the column ductility decreases as the level of axial load increases.
e enhancement of the mechanical properties inducted by FRP wrapping has recently been investigated from different authors.In [17], a new three-dimensional unified expression to evaluate the compressive strength of circular columns confined with FRP subjected to axial load was studied for application in pile design.In [18], an experimental campaign on fifty-six columns was carried out varying the number and strength of FRP layers.A new model was studied which predicts the ultimate strength of FRPconfined concrete for different amounts of confinement.In [19], the combination of FRP and expansive concrete related to eighteen concrete columns was investigated.e experimental results showed that the expansive concrete specimens achieved significantly higher ultimate load capacities than the conventional nonexpansive concrete specimens.In [20,21], the behavior of reinforced concrete piers confined with FRP through a nonlinear pushover analysis was studied in order to define the main influential factors for the achievement of certain limit states.Furthermore, the authors focused on interaction of influential factors, and they also showed that the compressive strength does not affect the yielding and ultimate drift of the structural element.
Many mechanical models have been proposed in the literature to describe the behavior of compressed concrete columns wrapped with FRP jackets; most of these models are based on the confinement model of Mander et al. [22] originally developed for steel confinement; it correlates the increase in strength and ultimate strain with the lateral confinement pressure.Several confinement models based on the formulation in [22] have been proposed in the literature [23][24][25][26][27][28][29] for the FRP system; in all these works, the evaluation of the lateral confinement pressure is calibrated on compression tests on concrete columns wrapped by FRP fiber sheets.e accuracy of predictions provided by these models is strictly related to correct definition of the lateral confinement pressure, which is strictly related with the hoop strain in the jacket; it is difficult to evaluate the lateral confinement pressure experimentally as it depends on various parameters.Consequently, many empirical or semiempirical models given in the literature are affected by experimental calibration and then only applicable to specific cases.
In order to overcome this drawback recently, the present authors proposed a simplified model for FRP systems that relates the increase in strength, ultimate strain, and energy absorption capacity directly to a single parameter dependent on the relative stiffness of the jacket and the concrete column [30]; in this way, definition of the lateral confinement pressure is avoided.In [30], the analytical functions are obtained from the best fitting of several experimental results [7,29,[31][32][33][34], and therefore, they can be used for each kind of FRP jacketing.
e mathematical models for the strength, ductility, and dissipated energy enhancement in [30] are obtained by a best fit of experimental data with the least squares method, and the coefficient of determination R 2 is evaluated in order to estimate the accuracy of the regression procedure.ese values of R 2 are small because the data are taken from different experimental campaigns, and then very different stress-strain curves were considered for the construction of the database.
Although various experimental tests are available in literature, only some experimental results are selected to define a homogeneous database.Indeed, only some tests were chosen from whole experimental campaigns in which specimens had comparable mechanical and geometrical properties, i.e., fiber volumetric ratio, elastic modulus of the fiber, concrete strength, and cross-section dimension.
e proposed model predicts the mechanical properties of square and circular cross-section columns overcoming the effects of the proportion between the dimensions of cross sections.Moreover, the absence of internal reinforcing bars allows to capture the effective confinement contribution due to the FRP sheets on the whole cross section.In sum, the model is calibrated on a restricted range of geometrical and mechanical characteristics in order to avoid the aforementioned effects that make the interpretation of results difficult.
In order to compensate for the lack of experimental results, in this paper, a numerical investigation is carried out.Following the strategy adopted in [35,36], finite element (FE) models are calibrated on results of experimental tests available in the literature, and FE simulations of compression tests with configurations that have not yet been tested are performed.e experimental results are then integrated with the FE simulations results, and following the approach adopted in [37], the analytical models of strength, ductility, and dissipated energy variation due to the FRP system are again obtained by a new best fit.
Finally, by means of comparison with available experimental data, it is shown that the proposed approach provides predictions in good agreement with available mechanical models.

Experimental Database of Compression Tests Available in the Literature
In this section, a database is assembled from the experimental results obtained by several authors.In order to ensure reliable interpretation of the data, the compression 2 Advances in Civil Engineering tests of specimens with comparable geometric and mechanical characteristics were chosen: (i) With square or circular cross sections as shown in Figure 1 (ii) With cylindrical compressive strength ≤ 40 MPa (iii) Without steel reinforcement (iv) With FRP reinforcing laminates with fibers only in the direction normal to the axis (v) Subjected to monotonic loads erefore, selecting experimental results according to the criteria defined above, not many experimental results can be taken into account although several experimental investigations have been carried out by different authors in the last few years.
Hereinafter, ρ � 2r/l is the corner radius ratio of the square cross sections while the characteristic values of stressstrain curves collected in the database are defined as follows: (a) f co and f cc are unconfined and confined maximum compressive stresses of specimens, respectively (b) ε cou and ε ccu are unconfined and confined ultimate axial strains of specimens, respectively (c) E c is elastic modulus of the unconfined specimens (d) E o and E are energy absorption capacities of unconfined and confined specimens, respectively (e) μ εo and μ ε are unconfined and confined ductilities of specimens, respectively e values of μ εo and μ ε , ductility ratios of all columns, were evaluated similarly to Wang and Wu [7].In order to define the characteristic values of the stress-strain curves, a bilinear idealization was performed (Figure 2) based on the following assumptions: (1) e postyield stiffness is equal to zero (2) e elastic stiffness of the idealized curve is calculated between 0.20 and 0.30 of the concrete strength (3) e ultimate strain of the idealized curve is evaluated as the strain at 80 percent of the concrete strength when the constitutive law presents a softening branch and at the maximum strength when the column failed at the peak point (4) e areas under the original and idealized curved within the range of interest are approximately equal (Equations (1a) and (1b)) Once the area under the original stress-strain curve is evaluated, the values of f * co /f * cc are obtained from the analytical expression of the area under the idealized curve: where the values of the yield strain ε * co /ε * cc are expressed in terms of f * co /f * cc .From these quantities, the values of ε * co /ε * cc are immediately evaluated, and then the values of the ductility ratios μ εo � ε cou /ε * co and μ ε � ε ccu /ε * cc for unconfined and confined specimens are calculated, respectively.
Descriptions of experimental tests and results are given below.e symbols have the following meanings: (a) n is the number of fiber layers (b) t f is the nominal thickness of the reinforcing system (c) ρ f � 4nt f /l is the fiber volumetric ratio (d) E f is the elastic modulus of the fibers e experimental database was assembled from the studies described below.
e key information of the experimental results is given in Table 1.
Bournas et al. [29] investigated the behavior of square concrete columns with and without internal reinforcement steel bars wrapped by FRP and FRCM (textile-reinforced mortar-TRM in the original paper) jacketing systems.e specimens had a 200 × 200 mm cross section representing columns at approximately 2/3 scale.Although the paper is focused on comparison between FRP and FRCM systems, the data from FRP (t f � 0.170 mm) confined columns without steel reinforcement were very useful in the construction of the database.
Karabinis and Rousakis [31] tested concrete columns with cylindrical cross section.e specimens, wrapped by carbon FRP sheets (t f � 0.170 mm), were subjected to axial monotonic load until failure.Carbon FRP was considered at different levels of fiber reinforcement ratios, in all 22 specimens tested by the authors.ey showed that carbon sheets allow a considerable growth in strength, ductility, and energy absorption of concrete, even at low volumetric ratios.
Rousakis et al. [32] investigated 101 square crosssectional concrete columns subjected to monotonic and cyclic axial compressive loads.All specimens were externally confined using glass and carbon FRP sheets, and the dimensions of all cross-sections were 200 × 200 mm.e specimens had different strength and different thickness of the reinforcing system (t f � 0.170 mm for n � 1; t f � 0.138 mm for n � 3 and n � 6).e results highlighted that the square concrete section can reach elevated levels of strength and ductility if the specimens are properly confined.
Wang and Wu [7] carried out experimental tests on 108 short concrete columns, confined with carbon FRP (t f � 0.165 mm) in compression.
e tested columns had a 150 × 150 mm square cross section and different corner radii varying from 15 to 75 mm; moreover, the specimens were wrapped with one and two fiber layers.
e results highlighted that enhancement in confined concrete strength is directly proportional to the corner radius ratio, and a jacket with sharp corners did not provide effective enhancement of column strength, whereas it provided an enhancement of the ductility of columns.
Wu and Wei [33] studied the behavior of axially loaded short rectangular columns strengthened with CFRP (t f � 0.167 mm) wrap.e aspect ratio (longer side/shorter side) from 1 to 2 (1.0, 1.25, 1.5, 1.75, and 2.0) and number of CFRP layers were the parameters investigated.e fixed parameters were the height of the specimens (300 mm) and Advances in Civil Engineering the corner radius of the sections (30 mm).e results showed that the strength gain in the confined concrete columns decreases when the aspect ratio increases, and it becomes ineffective for an aspect ratio of 2.Moreover, it was found that for an increasing aspect ratio, the behavior of the stress-strain curve changes from monotonically increasing with strain-hardening to a strain-softening type; the aspect ratio value at which transition from strain hardening to strain softening occurs increases with an increase in FRP thickness.
is result suggests that the efficiency of FRP jackets decreases with an increase in the aspect ratio and increases with an increase in thickness of the FRP wrap.

Integration of the Experimental Database by FE Analysis
In this section, a database is assembled from numerical results using a finite element (FE) method approach.Nonlinear FE analysis was carried out using ATENA Engineering-3D Software ( Červenka Consulting s.r.o) [37]. is software was chosen because it allows complete modeling of the member and is one of the most refined numerical methods for analyzing the constitutive behavior of concrete for tensile behavior (fracturing) and compressive behavior (plastic) at the same time.

Nonlinear FE Model.
e finite element discretization of columns was obtained with a mesh of eight-node solid elements.
e constitutive law of the concrete is "CC3DNonLinCementitious2" in ATENA3D.e fracture model is based on the classical orthotropic smeared crack formulation and the crack band model.It employs the Rankine failure criterion and exponential softening, and it can be used as a rotated or fixed crack model.e plasticity model is based on the Menétrey-Willam failure surface, and it provides for the definition of "β" parameter: if β < 0, the material is being compacted during crushing; if β � 0, the material volume is being preserved; and if β > 0, the material is dilating.e behavior of concrete used for the specimens was simulated assuming Poisson's ratio ] c � 0.2, modulus of elasticity E c � 29450 MPa, and β � 0.5.
An example in Figure 3 shows the Rankine failure criterion implemented in ATENA3D and one of the FE models adopted for unconfined columns.
e finite element discretization of FRP jacketing was obtained with a mesh of shell elements.ese elements were reduced from a quadratic 3D brick element with 20 nodes.e element had 9 integration points in the shell plane and layers in the direction normal to its plane.e total number of integration points was 9 × (number of layers).e contact surface between the FRP and concrete was modeled so that it   Advances in Civil Engineering did not allow slippage.Moreover, FRP jacketing was modeled as a smeared reinforcement layer.In particular, the matrix material constitutive law was assumed linear-elastic, and the fibers were added as linear-elastic reinforcement layers in the transverse direction considering an elasticity modulus of 240 GPa, a strain at failure of 15.5%, and a thickness of 0.117 mm.All specimens were subjected to a monotonic axial compressive load in the displacement control mode.e displacements were applied concentrically by means of a steel plate, which was also modeled with the same kind of brick element, care being taken to ensure compatibility with the topmost face of the column.

Verification of Nonlinear FE Modeling.
e FE model was calibrated on the experimental results obtained by Rousakis et al. [32].In particular, the following were taken into account: specimens with a cross-section dimension of 200 × 200 mm, a height of 320 mm, and the key information given in Table 2.
Table 2 shows the average values of test results of 3 confined and 3 unconfined concrete columns.e confined specimens were wrapped with one layer of carbon fiber perpendicularly to their axes.
In accordance with the experimental results, the numerical axial stress was calculated by dividing the axial load by the cross-section area.e axial strain was obtained by dividing the imposed displacement by the height of the specimens.
Figure 4 shows comparisons between the stress-strain curves determined with the FE model and the experimental results of Rousakis et al. [32].As can be noted, close agreement is observed between the two results, highlighting the fact that the derived FE model is verified with good accuracy.
It is observed that the numerical stress-strain curves were clipped at the axial strain of 0.4%.In fact, as indicated in [10], for higher axial strain, FRP-confined RC members lose their functionality.

Numerical Database Obtained from FE Modeling of the Compression Tests.
To build the numerical database, three series of nonlinear FE analyses were carried out, for three different values of t f (0.117, 0.125, and 0.187 mm) and varying r from 30 to 90 mm (30, 50, 60, 70, and 90).

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In the FE analyses, the following parameters remain unchanged: side of the cross section, 200 mm; height of specimens, 320 mm; elastic modulus of the unconfined specimens, 29450 MPa; elastic modulus of the fibers, 240 GPa; and number of fiber layers, 1.
e FE numerical results are summarized in Table 3. e symbols used in the table were previously described.Figure 5 shows the stress distribution in the transverse cross sections of some specimens analyzed at the peak load.e stress distribution was consistent with the theoretical assumption for  6 Advances in Civil Engineering the in-plane efficiency coefficient (Figure 5(a)), commonly adopted in most analytical models [10,23,24,[26][27][28] for considering confinement effects.
In particular, an effectively confined core was highlighted and delimited by variation of concrete lateral stress in the cross section.e latter started from the edges of rounded corners and developed with a parabolic shape in the section, evidencing the effectively and ineffectively confined zones.Furthermore, it is observed that decreasing the corner radius decreases the effective confined area of the cross section, while the confining pressure at the corners increases with consequential decrease in FRP efficiency.

Definition of the Parameter η.
e experimental and FE investigation shows that strength (f cc /f co ), ductility (μ ε /μ εo ) and dissipated energy (E/E o ) are significantly enhanced using FRP jacketing in concrete columns.To define an analytical law approximating the numerical and experimental results, a parameter η was defined which depends on the mechanical and geometrical characteristics of the system consisting in the original concrete core and FRP jacketing.
e increase in mechanical properties was evaluated for different values of ρ and ρ f considering the parameter η, defined in the following equation [30]: Generally, to evaluate strength, ductility, and dissipated energy enhancement, it is necessary to define the lateral confinement pressure.By contrast, with the proposed model, it is possible to evaluate enhancement of the mechanical properties mentioned without knowing the lateral confinement pressure.Furthermore, it is particularly remarkable that this mathematical model was developed from data coming from different experimental campaigns, meaning it can be used in general applications.
It is necessary to highlight that although this model has the advantages mentioned before, it was defined by means of a best fitting of experimental data with the least squares method.e coefficients of determination R 2 , considered to evaluate the accuracy of the models, are small because of the large dispersion in the experimental data available in the literature.
In order to improve the accuracy of the analytical model proposed in [30], a new best fit of the experimental data in Section 2 and the numerical data in Section 3 was carried out with the least squares method.Performance of the new expressions is evaluated with the following indices: the coefficient of determination R 2 and the root mean square error (RMSE).
R 2 defines a relationship between predicted and experimental values as follows: where x and y are the experimental and predicted values, respectively, while  x and  y are the mean experimental and predicted values, respectively.Such index can assume value between zero, i.e., zero correlation, and one, i.e., perfect correlation.
RMSE is defined as where N is the number of data points.It is clear that R 2 is a relative measure of fit and RMSE is an absolute measure of fit without upper limit.

Strength Modeling.
e analytical expression which describes the enhanced strength as a function of the corner radius ratio and the fiber volumetric ratio provided in [30] is reported in the following equation:  Figure 6 shows the values of f cc /f co for varying ρ and η for number of layer n � 1 together with the experimental and numerical data.Good agreement between analytical results and experimental and numerical ones can be observed.is is confirmed by the value of the coefficient of determination R 2 � 0.85 and RMSE � 0.14.
Figure 7 shows the values of f cc /f co for varying ρ and for number of layers one to three.It is observed that the enhanced strength depends linearly on the corner radius ratio and increases with the corner radius ratio and with the fiber volumetric ratio.

Ductility Modeling.
e analytical expression which describes the enhanced ductility as a function of the corner radius ratio and the fiber volumetric ratio provided in [30] is reported in the following equation: In the present paper, the new best fitting leads to the following values of the coefficient: a � 0.020; b � 1.11; c � −14.07; d � −0.95.
Figure 8 shows the values of μ ε /μ εo for varying ρ and η for number of layers n � 1 together with experimental and numerical data.A fine agreement between analytical result and experimental and numerical one can be observed.is sharing of the results is confirmed by the value of the coefficient of determination R 2 � 0.91.Nevertheless, the RMSE � 1.19 shows a higher dispersion of ductility predictions than the strength ones due to the difficulty to capture the postelastic behavior of the stress-strain curves with simple bilinear curves.
Figure 9 shows the values of μ ε /μ εo for varying ρ and for number of layers one to three.It is observed that the enhanced ductility is related to the corner radius ratio with a power law and decreases with the corner radius ratio and with the fiber volumetric ratio.

Dissipated Energy Modeling.
e analytical expression which describes the enhanced dissipated energy as a function of the corner radius ratio and the fiber volumetric ratio provided in [30] is reported in the following equation: In the present paper, the new best fitting leads to the following values of the coefficient: a � 161.03; b � −1.20; c � 25.20; d � −0.59.
Figure 10 shows the values of E/E o for varying ρ and η for number of layers n � 1 together with the experimental and numerical data.In this case, R 2 is equal to 0.47 and the RMSE is 1.82.Such values confirm that the energy capacity model does not show a significant increase in accuracy with respect to the one reported in [30]. is is due to the fact that E is strongly dependent on the postelastic branch regarding strength and ductility, the latter only being influenced by elastic and ultimate strain values.In the FE analyses, the slope of the postelastic branch is not easy to reproduce accurately and consequently, there is only a small improvement in the coefficient of determination.
Figure 11 shows the values of E/E o for varying ρ and for number of layers one to three.It is observed that the enhanced dissipated energy is related to the corner radius ratio with a power law d and increases with the corner radius ratio and with the fiber volumetric ratio.5) in terms of f cc /f co with the experimental data (blue points) and the numerically simulated data (red points).0 0.8 0.1 0.2 0.3 0.4 0.5 0.6 ρ 0.7 0.8 0.9 1 Figure 7: cc /f co -ρ curves of columns confined by the FRP system for n � 1, n � 2, and n � 3.

Comparison between the Proposed Model and Existing
Models for FRP. is section shows comparisons between the values of enhanced strength (f cc /f co ) calculated by Equation ( 5) and those obtained from models available in the literature reported in Table 4.
Usually, in this model, the value of f cc /f co is evaluated as a function of the confinement pressure (f l,e ) by means of the following expression: where Equation ( 8) involves the nondimensional parameters α, k 1 , and m, all experimentally assessable, while k 1 � αk 1,R take into account the reduced effectiveness provided by jackets and resins (α is a coefficient related to a specified jacketing system, and αk 1,R is a nondimensional parameter experimentally calibrated).10

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In Equation ( 9), E f is the elastic modulus in the lateral direction, ε f is the strain of the jackets in the same direction, and k e is a coefficient that counts the variation in the confinement pressure in square and rectangular crosssection specimens compared to the circular ones. is coefficient is calibrated by the authors of each model.
Table 4 shows the parameters and coefficients of the various models considered for evaluating the theoretical values of f cc /f co .
e comparison was carried out for concrete columns with square and circular sections with l � 300 mm and different values of corner radius ratio, fiber volumetric ratio, and unconfined maximum compressive strength and considering two kinds of fibers.
e first kind of carbon fiber used in the confined specimens (fiber#1) has the following mechanical properties: ultimate tensile strain 1.54%; ultimate tensile strength 3700 MPa; elastic modulus 240 GPa; nominal thickness of the sheets 0.117 mm. e second kind of carbon fiber (fiber#2) has the following mechanical properties: ultimate tensile strain 2.00%; ultimate tensile strength 4364 MPa; elastic modulus 219 GPa; nominal thickness of the sheets 0.165 mm.Both kinds of fiber have the primary fiber direction unidirectional.It is possible to observe that fiber#1 presents lower tensile strengths and ultimate deformation capacity than fiber#2.
Table 5 summarizes a comparison between results obtained with models available in the literature and results of the proposed model in terms of strength enhancement f cc /f co .
e strength predictions of the proposed model are in good agreement with the models of Campione and Miraglia [24], Pellegrino and Modena [23], and Mirmiran et al. [27], while they are more conservative than the models of CNR-DT200 [10], Triantafillou et al. [26], and Di Ludovico et al. [28].Furthermore, the model of Youssef et al. [38] underestimates the strength predictions compared to ones of the other authors.By and large, the proposed model gives suitable results for evaluating the performance of square and circular columns wrapped with FRP jackets.
Finally, it is pointed out again that in the confinement models available in literature, the accuracy of the predictions is strongly dependent on the definition of the effective confinement pressure and effectiveness coefficients.e proposed model overcomes this drawback and provides a simplified model for a very simple evaluation of the strength and ductility columns.

Design Indications for Practical Applications.
e proposed procedure can easily be extended for evaluating the performance of FRP-confined concrete columns in practical design applications.
for continuous jackets with fibers ⊥ to the member axis, A g � cross-section area of the specimen Triantafillou et al. [26] for continuous jackets with fibers ⊥ to the member axis Campione and Miraglia [24] α � 1 )] f fu � ultimate strength of the fiber, k i � 0.2121 Pellegrino and Modena [23] for continuous jackets with fibers ⊥ to the member axis Youssef et al. [38] For circular sections For circular sections Di Ludovico et al. [28] Youssef et al.Generally, in the design phase, the known parameters are the cross-section dimension l, the elastic modulus of unconfined specimens E c , the nominal thickness of the reinforcing system t f , and the elastic modulus of the fibers E f .Two procedures can be used to design the reinforcement, as indicated in [30]: the first where the designer chooses the increased ductility and evaluate the increased strength and dissipated energy; the second where the designer chooses the Once the increased ductility or strength has been chosen, the fiber volumetric ratio ρ f and the parameter η, for different values of the number of layers n, have to be evaluated.Known ρ f and η, in both design case, the ρ has to be evaluated and radius can be evaluated by means of Equation ( 6) (if μ ε /μ εo is fixed) or by means of Equation ( 5) (if f cc /f co is fixed).For 0 < ρ < 1 and r min ≤ r ≤ r max (r min � 20 mm and r max � (2 + � 2 √ )c, see detail in [30]), the value of f cc /f co , μ ε /μ εo , and E/E o are evaluated by means of Equations ( 5)- (7), respectively.

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Figures 12 and 13 display the design procedure for fixed increased ductility and fixed increased strength, respectively.

Conclusions
In this paper, an empirical model for concrete columns wrapped with FRP systems is proposed.
e model was obtained by fitting the experimental data of several authors and numerical data derived from FE analyses.e proposed model is capable of taking into account the mechanical and geometrical properties of the reinforcement and of the corner radius ratio of the cross section.It is shown that with this model, the definition of the strength increase does not require definition of the lateral confinement pressure.Comparison with results of some available confinement models shows that the proposed model is capable of capturing experimental results coming from different experimental campaigns, while most of the existing models often are only able to reproduce experimental results from the experimental campaign they were calibrated with.Finally, a very simple design procedure for practical applications is provided.

Figure 1 :
Figure 1: Details of transverse section of columns.(a) Square shape.(b) Circular shape.

Figure 3 :
Figure 3: Finite element model.(a) Exponential crack-opening law.(b) Deformed shape and crack pattern for unconfined columns.

R 2 Figure 6 :
Figure 6: 3D figure of Equation (5) in terms of f cc /f co with the experimental data (blue points) and the numerically simulated data (red points).

Figure 10 :Figure 11 : 1 Figure 9 :Figure 8 :
Figure 10: 3D figure of Equation (7) in terms of E/E o with the experimental data (blue points) and the numerically simulated data (red points).

Figure 12 :Figure 13 :
Figure 12: Flowchart for design procedure in the case of fixed increased ductility.

Table 1 :
Details of FRP-confined column specimens and experimental key results.

Table 3 :
Details of FE models for FRP-confined columns and numerical key results.

Table 4 :
Parameters and coefficients for evaluating the strength enhancement from existing analytical expressions.