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

Authors: Marinella Fossetti; Francesco Basone; Giuseppe D’Arenzo; Giuseppe Macaluso; Alfio Francesco Siciliano
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

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. The 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.


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## Body

## 1. 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 performance-based 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. The use of FRP jackets has met with increasing popularity [1–9] 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 [10–12].This 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.The 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 FRP-confined concrete for different amounts of confinement. In [19], the combination of FRP and expansive concrete related to eighteen concrete columns was investigated. The 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 [23–29] 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. The 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, 31–34], and therefore, they can be used for each kind of FRP jacketing.The 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 R2 is evaluated in order to estimate the accuracy of the regression procedure. These values of R2 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.The 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. The 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.


## 2. 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 tests of specimens with comparable geometric and mechanical characteristics were chosen:(i)
With square or circular cross sections as shown in Figure1(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 loadsFigure 1
Details of transverse section of columns. (a) Square shape. (b) Circular shape.
(a)
(b)Therefore, 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 stress-strain curves collected in the database are defined as follows:(a)
fco and fcc are unconfined and confined maximum compressive stresses of specimens, respectively(b)
εcou and εccu are unconfined and confined ultimate axial strains of specimens, respectively(c)
Ec is elastic modulus of the unconfined specimens(d)
Eo and E are energy absorption capacities of unconfined and confined specimens, respectively(e)
μεo and με are unconfined and confined ductilities of specimens, respectivelyThe values ofμεo and με, 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)
The postyield stiffness is equal to zero(2)
The elastic stiffness of the idealized curve is calculated between 0.20 and 0.30 of the concrete strength(3)
The 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)
The areas under the original and idealized curved within the range of interest are approximately equal (Equations (1a) and (1b))Figure 2
Definition of the ductility ratio. (a) Unconfined columns. (b) Confined columns.
(a)
(b)Once the area under the original stress-strain curve is evaluated, the values offco∗/fcc∗ are obtained from the analytical expression of the area under the idealized curve:(1a)12fco∗εco∗+εcou−εco∗fco∗=Eo,(1b)12fcc∗εcc∗+εccu−εcc∗fcc∗=E,where the values of the yield strain εco∗/εcc∗ are expressed in terms of fco∗/fcc∗. From these quantities, the values of εco∗/εcc∗ are immediately evaluated, and then the values of the ductility ratios μεo=εcou/εco∗ and με=εccu/εcc∗ for unconfined and confined specimens are calculated, respectively.Descriptions of experimental tests and results are given below. The symbols have the following meanings:(a)
n is the number of fiber layers(b)
tf is the nominal thickness of the reinforcing system(c)
ρf=4ntf/l is the fiber volumetric ratio(d)
Ef is the elastic modulus of the fibersThe experimental database was assembled from the studies described below. The key information of the experimental results is given in Table1.Table 1
Details of FRP-confined column specimens and experimental key results.
Source
r (mm)
ρ
n
ρ
f (%)
f
co (MPa)
f
cc/fco
E (MJ/m3)
E
/
E
o
ε
cou (%)
μ
εo
μ
ε/μεo
Bournas et al. [29]
25
0.25
2
0.680
15.28
2.00
0.21
4.03
0.37
1.68
2.24
25
0.25
3
1.020
15.28
2.27
0.36
6.77
0.37
1.68
2.37
Karabinis and Rousakis [31]
—
1.00
1
0.234
38.50
1.12
0.24
3.06
0.41
2.08
1.06
—
1.00
1
0.234
38.50
1.08
0.22
2.51
0.41
2.08
2.63
—
1.00
1
0.234
38.50
1.19
0.22
1.31
0.41
2.08
1.69
—
1.00
2
0.468
38.50
1.34
0.38
4.01
0.41
2.08
1.34
—
1.00
2
0.468
38.50
1.30
0.22
2.36
0.41
2.08
1.37
—
1.00
2
0.468
38.50
1.43
0.37
3.90
0.41
2.08
1.32
—
1.00
1
0.234
35.70
1.19
0.32
7.00
0.20
1.51
2.62
—
1.00
1
0.234
35.70
1.18
0.29
9.93
0.20
1.51
2.36
—
1.00
1
0.234
35.70
1.15
0.08
1.80
0.20
1.51
1.99
—
1.00
2
0.468
35.70
1.40
0.23
5.07
0.20
1.51
1.08
—
1.00
2
0.468
35.70
1.36
0.37
8.13
0.20
1.51
1.15
—
1.00
2
0.468
35.70
1.40
0.40
8.76
0.20
1.51
1.12
Rousakis et al. [32]
30
0.30
1
0.234
33.04
1.16
0.14
3.39
0.17
1.49
2.88
30
0.30
1
0.234
33.04
1.17
0.11
2.70
017
1.49
2.23
30
0.30
1
0.234
34.20
1.23
0.14
3.81
0.18
2.26
2.44
30
0.30
1
0.234
34.20
1.23
0.22
6.05
0.18
2.26
2.12
30
0.30
3
0.828
33.04
1.29
0.12
2.92
0.17
1.49
6.36
30
0.30
3
0.828
33.04
1.27
0.11
2.63
0.17
1.49
3.31
30
0.30
6
1.656
37.97
1.39
0.45
5.33
0.22
2.14
2.79
30
0.30
6
1.656
37.97
1.38
0.46
5.45
0.22
2.14
2.08
Wang and Wu [7]
15
0.20
1
0.440
31.90
1.05
0.50
3.41
0.68
3.71
2.78
30
0.40
1
0.440
32.30
1.23
0.62
5.24
0.68
3.71
1.60
45
0.60
1
0.440
30.70
1.43
0.75
5.52
0.68
3.71
1.05
60
0.80
1
0.440
54.10
1.57
0.75
4.64
0.68
3.71
0.75
—
1.00
1
0.440
30.90
1.80
1.09
6.76
0.68
3.71
0.69
15
0.20
2
0.880
31.90
1.32
1.01
6.28
0.68
3.71
1.23
30
0.40
2
0.880
32.30
1.75
1.19
7.42
0.68
3.71
0.61
45
0.60
2
0.880
30.70
2.22
1.48
9.19
0.68
3.71
0.50
60
0.80
2
0.880
54.10
2.48
1.53
9.53
0.68
3.71
0.52
—
1.00
2
0.880
30.90
2.74
1.83
11.39
0.68
3.71
0.43
Wu and Wei [33]
30
0.40
1
0.445
35.30
1.17
0.28
3.84
0.30
3.35
1.64
30
0.40
2
0.891
35.30
1.71
0.77
10.57
0.30
3.35
1.48Bournas 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. The 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 (tf = 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. The specimens, wrapped by carbon FRP sheets (tf = 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. They 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 cross-sectional 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. The specimens had different strength and different thickness of the reinforcing system (tf = 0.170 mm for n = 1; tf = 0.138 mm for n=3 and n=6). The 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 (tf = 0.165 mm) in compression. The 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. The 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 (tf = 0.167 mm) wrap. The 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. The fixed parameters were the height of the specimens (300 mm) and the corner radius of the sections (30 mm). The 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. This 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.


## 3. 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]. This 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.


### 3.1. Nonlinear FE Model

The finite element discretization of columns was obtained with a mesh of eight-node solid elements. The constitutive law of the concrete is “CC3DNonLinCementitious2” in ATENA3D. The 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. The 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. The behavior of concrete used for the specimens was simulated assuming Poisson’s ratio νc = 0.2, modulus of elasticity Ec = 29450 MPa, and β = 0.5.An example in Figure3 shows the Rankine failure criterion implemented in ATENA3D and one of the FE models adopted for unconfined columns.Figure 3
Finite element model. (a) Exponential crack-opening law. (b) Deformed shape and crack pattern for unconfined columns.
(a)
(b)The finite element discretization of FRP jacketing was obtained with a mesh of shell elements. These elements were reduced from a quadratic 3D brick element with 20 nodes. The element had 9 integration points in the shell plane and layers in the direction normal to its plane. The total number of integration points was 9 × (number of layers). The contact surface between the FRP and concrete was modeled so that it 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. The 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.


### 3.2. Verification of Nonlinear FE Modeling

The 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
Details and experimental key results by Rousakis et al. [32] for FE model calibration.
r (mm)
ρ
ρ
f (%)
f
co (MPa)
f
cc (MPa)
f
cc/fco
ε
cou (%)
ε
ccu (%)
ε
ccu/εcou
30
0.30
0.234
33.04
38.44
1.16
1.71
4.50
2.63Table2 shows the average values of test results of 3 confined and 3 unconfined concrete columns. The 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. The axial strain was obtained by dividing the imposed displacement by the height of the specimens.Figure4 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.Figure 4
Comparisons between experimental results of Rousakis et al. [32] and numerical results of FE model.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.


### 3.3. 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 oftf (0.117, 0.125, and 0.187 mm) and varying r from 30 to 90 mm (30, 50, 60, 70, and 90).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.The FE numerical results are summarized in Table3. The symbols used in the table were previously described.Table 3
Details of FE models for FRP-confined columns and numerical key results.
t
f (mm)
ρ
ρ
f (%)
f
co (MPa)
f
cc/fco
E
o (MJ/m3)
E
/
E
o
ε
cou (%)
μ
εo
μ
ε/μεo
0.117
0.3
0.234
31.88
1.219
0.052
2.306
0.23
1.77
1.608
0.117
0.5
0.234
32.10
1.260
0.052
2.385
0.23
1.77
1.554
0.117
0.6
0.234
31.02
1.259
0.051
2.364
0.23
1.79
1.572
0.117
0.7
0.234
31.68
1.285
0.051
2.389
0.23
1.85
1.469
0.117
0.9
0.234
31.20
1.280
0.051
2.382
0.23
1.87
1.474
0.187
0.3
0.375
31.88
1.224
0.052
2.335
0.23
1.77
1.584
0.187
0.5
0.375
32.10
1.272
0.052
2.405
0.23
1.77
1.539
0.187
0.6
0.375
31.02
1.264
0.051
2.382
0.23
1.79
1.555
0.187
0.7
0.375
31.68
1.309
0.051
2.414
0.23
1.85
1.450
0.187
0.9
0.375
31.20
1.296
0.051
2.413
0.23
1.87
1.478
0.125
0.3
0.250
31.88
1.211
0.052
2.308
0.23
1.77
1.608
0.125
0.5
0.250
32.10
1.259
0.052
2.387
0.23
1.77
1.552
0.125
0.6
0.250
31.02
1.249
0.051
2.366
0.23
1.79
1.570
0.125
0.7
0.250
31.68
1.267
0.051
2.390
0.23
1.85
1.467
0.125
0.9
0.250
31.20
1.269
0.051
2.385
0.23
1.87
1.473Figure5 shows the stress distribution in the transverse cross sections of some specimens analyzed at the peak load. The stress distribution was consistent with the theoretical assumption for the in-plane efficiency coefficient (Figure 5(a)), commonly adopted in most analytical models [10, 23, 24, 26–28] for considering confinement effects.Figure 5
Stress distribution in MPa at peak loads in square columns with different corner radii. (a) Theoretical model. (b)r = 30 mm. (c) r = 50 mm. (d) r = 60 mm. (e) r = 70 mm. (f) r = 90 mm.
(a)
(b)
(c)
(d)
(e)
(f)In particular, an effectively confined core was highlighted and delimited by variation of concrete lateral stress in the cross section. The 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.


## 3.1. Nonlinear FE Model

The finite element discretization of columns was obtained with a mesh of eight-node solid elements. The constitutive law of the concrete is “CC3DNonLinCementitious2” in ATENA3D. The 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. The 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. The behavior of concrete used for the specimens was simulated assuming Poisson’s ratio νc = 0.2, modulus of elasticity Ec = 29450 MPa, and β = 0.5.An example in Figure3 shows the Rankine failure criterion implemented in ATENA3D and one of the FE models adopted for unconfined columns.Figure 3
Finite element model. (a) Exponential crack-opening law. (b) Deformed shape and crack pattern for unconfined columns.
(a)
(b)The finite element discretization of FRP jacketing was obtained with a mesh of shell elements. These elements were reduced from a quadratic 3D brick element with 20 nodes. The element had 9 integration points in the shell plane and layers in the direction normal to its plane. The total number of integration points was 9 × (number of layers). The contact surface between the FRP and concrete was modeled so that it 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. The 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.


## 3.2. Verification of Nonlinear FE Modeling

The 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
Details and experimental key results by Rousakis et al. [32] for FE model calibration.
r (mm)
ρ
ρ
f (%)
f
co (MPa)
f
cc (MPa)
f
cc/fco
ε
cou (%)
ε
ccu (%)
ε
ccu/εcou
30
0.30
0.234
33.04
38.44
1.16
1.71
4.50
2.63Table2 shows the average values of test results of 3 confined and 3 unconfined concrete columns. The 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. The axial strain was obtained by dividing the imposed displacement by the height of the specimens.Figure4 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.Figure 4
Comparisons between experimental results of Rousakis et al. [32] and numerical results of FE model.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.


## 3.3. 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 oftf (0.117, 0.125, and 0.187 mm) and varying r from 30 to 90 mm (30, 50, 60, 70, and 90).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.The FE numerical results are summarized in Table3. The symbols used in the table were previously described.Table 3
Details of FE models for FRP-confined columns and numerical key results.
t
f (mm)
ρ
ρ
f (%)
f
co (MPa)
f
cc/fco
E
o (MJ/m3)
E
/
E
o
ε
cou (%)
μ
εo
μ
ε/μεo
0.117
0.3
0.234
31.88
1.219
0.052
2.306
0.23
1.77
1.608
0.117
0.5
0.234
32.10
1.260
0.052
2.385
0.23
1.77
1.554
0.117
0.6
0.234
31.02
1.259
0.051
2.364
0.23
1.79
1.572
0.117
0.7
0.234
31.68
1.285
0.051
2.389
0.23
1.85
1.469
0.117
0.9
0.234
31.20
1.280
0.051
2.382
0.23
1.87
1.474
0.187
0.3
0.375
31.88
1.224
0.052
2.335
0.23
1.77
1.584
0.187
0.5
0.375
32.10
1.272
0.052
2.405
0.23
1.77
1.539
0.187
0.6
0.375
31.02
1.264
0.051
2.382
0.23
1.79
1.555
0.187
0.7
0.375
31.68
1.309
0.051
2.414
0.23
1.85
1.450
0.187
0.9
0.375
31.20
1.296
0.051
2.413
0.23
1.87
1.478
0.125
0.3
0.250
31.88
1.211
0.052
2.308
0.23
1.77
1.608
0.125
0.5
0.250
32.10
1.259
0.052
2.387
0.23
1.77
1.552
0.125
0.6
0.250
31.02
1.249
0.051
2.366
0.23
1.79
1.570
0.125
0.7
0.250
31.68
1.267
0.051
2.390
0.23
1.85
1.467
0.125
0.9
0.250
31.20
1.269
0.051
2.385
0.23
1.87
1.473Figure5 shows the stress distribution in the transverse cross sections of some specimens analyzed at the peak load. The stress distribution was consistent with the theoretical assumption for the in-plane efficiency coefficient (Figure 5(a)), commonly adopted in most analytical models [10, 23, 24, 26–28] for considering confinement effects.Figure 5
Stress distribution in MPa at peak loads in square columns with different corner radii. (a) Theoretical model. (b)r = 30 mm. (c) r = 50 mm. (d) r = 60 mm. (e) r = 70 mm. (f) r = 90 mm.
(a)
(b)
(c)
(d)
(e)
(f)In particular, an effectively confined core was highlighted and delimited by variation of concrete lateral stress in the cross section. The 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.


## 4. Proposed Simplified Procedure and Analytical Models

### 4.1. Definition of the Parameterη

The experimental and FE investigation shows that strength (fcc/fco), ductility (με/μεo) and dissipated energy (E/Eo) 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. The increase in mechanical properties was evaluated for different values of ρ and ρf considering the parameter η, defined in the following equation [30]:(2)η=l4ntf⋅EcEf=EcρfEf.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. The coefficients of determinationR2, 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 R2 and the root mean square error (RMSE).R
2 defines a relationship between predicted and experimental values as follows:(3)R2=∑x−x˜y−y˜∑x−x˜2y−y˜22,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(4)RMSE=∑x−y2N,where N is the number of data points. It is clear that R2 is a relative measure of fit and RMSE is an absolute measure of fit without upper limit.


### 4.2. Strength Modeling

The 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:(5)fccfco=aηbρ+cηd.In the present paper, the new best fit leads to the following values of the coefficient:a = 1629.4; b = −2.55; c = 2.21; d = −0.17.Figure6 shows the values of fcc/fco 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. This is confirmed by the value of the coefficient of determination R2 = 0.85 and RMSE = 0.14.Figure 6
3D figure of Equation (5) in terms of fcc/fco with the experimental data (blue points) and the numerically simulated data (red points).Figure7 shows the values of fcc/fco 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.Figure 7
f
cc
/
f
co-ρ curves of columns confined by the FRP system for n=1, n=2, and n=3.


### 4.3. Ductility Modeling

The 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:(6)μεμεο=aηbρcηd.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.Figure8 shows the values of με/μεo 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. This sharing of the results is confirmed by the value of the coefficient of determination R2 = 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 8
3D figure of Equation (6) in terms of με/μεo with the experimental data (blue points) and the numerically simulated data (red points).Figure9 shows the values of με/μεo 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.Figure 9
μ
ε
/
μ
ε
o-ρ curves of columns confined by the FRP system for n = 1, n = 2, and n = 3.


### 4.4. Dissipated Energy Modeling

The 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:(7)EEo=aηbρ+cηd.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.Figure10 shows the values of E/Eo for varying ρ and η for number of layers n=1 together with the experimental and numerical data. In this case, R2 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]. This 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 10
3D figure of Equation (7) in terms of E/Eo with the experimental data (blue points) and the numerically simulated data (red points).Figure11 shows the values of E/Eo 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.Figure 11
E
/
E
o-ρ curves of columns confined by the FRP system for n=1, n=2, and n=3.


## 4.1. Definition of the Parameterη

The experimental and FE investigation shows that strength (fcc/fco), ductility (με/μεo) and dissipated energy (E/Eo) 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. The increase in mechanical properties was evaluated for different values of ρ and ρf considering the parameter η, defined in the following equation [30]:(2)η=l4ntf⋅EcEf=EcρfEf.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. The coefficients of determinationR2, 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 R2 and the root mean square error (RMSE).R
2 defines a relationship between predicted and experimental values as follows:(3)R2=∑x−x˜y−y˜∑x−x˜2y−y˜22,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(4)RMSE=∑x−y2N,where N is the number of data points. It is clear that R2 is a relative measure of fit and RMSE is an absolute measure of fit without upper limit.


## 4.2. Strength Modeling

The 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:(5)fccfco=aηbρ+cηd.In the present paper, the new best fit leads to the following values of the coefficient:a = 1629.4; b = −2.55; c = 2.21; d = −0.17.Figure6 shows the values of fcc/fco 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. This is confirmed by the value of the coefficient of determination R2 = 0.85 and RMSE = 0.14.Figure 6
3D figure of Equation (5) in terms of fcc/fco with the experimental data (blue points) and the numerically simulated data (red points).Figure7 shows the values of fcc/fco 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.Figure 7
f
cc
/
f
co-ρ curves of columns confined by the FRP system for n=1, n=2, and n=3.


## 4.3. Ductility Modeling

The 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:(6)μεμεο=aηbρcηd.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.Figure8 shows the values of με/μεo 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. This sharing of the results is confirmed by the value of the coefficient of determination R2 = 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 8
3D figure of Equation (6) in terms of με/μεo with the experimental data (blue points) and the numerically simulated data (red points).Figure9 shows the values of με/μεo 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.Figure 9
μ
ε
/
μ
ε
o-ρ curves of columns confined by the FRP system for n = 1, n = 2, and n = 3.


## 4.4. Dissipated Energy Modeling

The 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:(7)EEo=aηbρ+cηd.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.Figure10 shows the values of E/Eo for varying ρ and η for number of layers n=1 together with the experimental and numerical data. In this case, R2 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]. This 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 10
3D figure of Equation (7) in terms of E/Eo with the experimental data (blue points) and the numerically simulated data (red points).Figure11 shows the values of E/Eo 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.Figure 11
E
/
E
o-ρ curves of columns confined by the FRP system for n=1, n=2, and n=3.


## 5. Comparison with Some Available Models and Design Indications

### 5.1. Comparison between the Proposed Model and Existing Models for FRP

This section shows comparisons between the values of enhanced strength (fcc/fco) calculated by Equation (5) and those obtained from models available in the literature reported in Table 4.Table 4
Parameters and coefficients for evaluating the strength enhancement from existing analytical expressions.
Model
Coefficients
Other parameters and coefficients
CNR-DT200 [10]
α = 1k1 = 2.6m = 2/3
k
e
=
k
H
k
V
k
α
,
k
H
=
1
−
2
l
−
r
2
/
3
A
g
k
v = kα = 1 for continuous jackets with fibers ⊥ to the member axis, Ag = cross-section area of the specimen
Triantafillou et al. [26]
α = 1k1 = 2.79m = 1
k
e
=
k
H
k
V
k
α
,
k
H
=
1
−
2
l
−
r
2
/
3
A
g
k
v = kα = 1 for continuous jackets with fibers ⊥ to the member axis
Campione and Miraglia [24]
α = 1k1 = 2m = 1
f
ie
=
2
n
t
f
f
r
/
l
,
f
r
=
f
fu
1
−
2
/
2
k
i
2
r
/
l
+
k
i
2
/
2
f
fu = ultimate strength of the fiber, ki = 0.2121
Pellegrino and Modena [23]
α = 1k1=kAkRm = 1
k
A
=
A
f
l
,
e
/
f
co
−
β
,
A
=
2.25
and
β
=
−
0.25
k
R
=
1
−
2.5
0.3
−
2
r
/
l
,
if
2
r
/
l
<
0.3
,
1
,
if
2
r
/
l
≥
0.3
,
Mirmiran et al. [27]
α = 1k1=6.0fl,e−0.30m = 1
k
e
=
2
r
/
l
Di Ludovico et al. [28]
α = 1k1 = 2.94m = 1
k
e
=
k
H
k
V
k
α
,
k
H
=
1
−
2
l
−
r
2
/
3
A
g
,
k
v = kα = 1 for continuous jackets with fibers ⊥ to the member axis
Youssef et al. [38]
For circular sections
For circular sections
α = 1.0k1 = 2.25m = 1.25
k
e = 1.0ρ1 = longitudinal reinforcement ratio
For square sections
For square sections
α = 0.50k1 = 1.225m = 0.6
k
e
=
1
−
2
ℓ
−
2
r
2
/
3
ℓ
2
−
ρ
l
/
1
−
ρ
lUsually, in this model, the value offcc/fco is evaluated as a function of the confinement pressure fl,e by means of the following expression:(8)fccfco=α+k1f1,efcom,where(9)fi,e=12ρfEfεfke.Equation (8) involves the nondimensional parameters α, k1, and m, all experimentally assessable, while k1=αk1,R take into account the reduced effectiveness provided by jackets and resins (α is a coefficient related to a specified jacketing system, and αk1,R is a nondimensional parameter experimentally calibrated).In Equation (9), Ef is the elastic modulus in the lateral direction, εf is the strain of the jackets in the same direction, and ke is a coefficient that counts the variation in the confinement pressure in square and rectangular cross-section specimens compared to the circular ones. This coefficient is calibrated by the authors of each model.Table4 shows the parameters and coefficients of the various models considered for evaluating the theoretical values of fcc/fco.The comparison was carried out for concrete columns with square and circular sections withl = 300 mm and different values of corner radius ratio, fiber volumetric ratio, and unconfined maximum compressive strength and considering two kinds of fibers.The 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. The 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.Table5 summarizes a comparison between results obtained with models available in the literature and results of the proposed model in terms of strength enhancement fcc/fco.Table 5
Comparison (fcc/fco) between the proposed model and existing models for FRP.
ρ
f (%)
ρ
f
co (MPa)
Proposed model
CNR-DT200 [10]
Triantafillou et al. [26]
Campione et al. [24]
Pellegrino et al. [23]
Mirmiran et al. [27]
Di Ludovico et al. [28]
Youssef et al. [38]
#1
#2
#1
#2
#1
#2
#1
#2
#1
#2
#1
#2
#1
#2
#1
#2
0.156
0.13
20
1.08
1.13
1.45
1.63
1.20
1.34
1.03
1.04
1.05
1.09
1.08
1.12
1.21
1.35
0.75
0.84
0.312
0.13
20
1.22
1.29
1.71
2.01
1.40
1.67
1.05
1.08
1.12
1.22
1.14
1.20
1.42
1.71
0.88
1.02
0.468
0.13
20
1.34
1.44
1.94
2.32
1.60
2.01
1.08
1.13
1.19
1.37
1.18
1.26
1.63
2.06
0.99
1.16
0.156
0.17
20
1.08
1.13
1.47
1.67
1.22
1.36
1.03
1.05
1.06
1.12
1.10
1.15
1.23
1.38
0.76
0.86
0.312
0.17
20
1.23
1.30
1.75
2.06
1.43
1.72
1.06
1.09
1.15
1.28
1.17
1.24
1.46
1.76
0.90
1.04
0.468
0.17
20
1.35
1.46
1.98
2.38
1.65
2.08
1.08
1.14
1.24
1.46
1.23
1.32
1.68
2.14
1.01
1.19
0.156
0.13
25
1.07
1.12
1.39
1.55
1.16
1.27
1.02
1.03
1.04
1.07
1.07
1.10
1.17
1.28
0.72
0.80
0.312
0.13
25
1.21
1.28
1.62
1.87
1.32
1.54
1.04
1.07
1.09
1.17
1.11
1.16
1.34
1.57
0.83
0.96
0.468
0.13
25
1.32
1.42
1.81
2.14
1.48
1.81
1.06
1.10
1.15
1.28
1.14
1.21
1.51
1.85
0.93
1.08
0.156
0.17
25
1.07
1.12
1.41
1.57
1.17
1.29
1.02
1.04
1.05
1.09
1.08
1.12
1.18
1.30
0.73
0.81
0.312
0.17
25
1.22
1.28
1.65
1.91
1.35
1.58
1.04
1.07
1.11
1.21
1.14
1.20
1.36
1.61
0.85
0.98
0.468
0.17
25
1.33
1.44
1.85
2.19
1.52
1.87
1.07
1.11
1.18
1.35
1.18
1.26
1.55
1.91
0.95
1.11
0.156
0.13
30
1.06
1.11
1.34
1.48
1.13
1.22
1.02
1.03
1.03
1.06
1.06
1.08
1.14
1.24
0.70
0.77
0.312
0.13
30
1.20
1.26
1.54
1.77
1.27
1.45
1.03
1.06
1.07
1.13
1.09
1.13
1.28
1.47
0.80
0.91
0.468
0.13
30
1.31
1.40
1.71
2.01
1.40
1.67
1.05
1.08
1.12
1.22
1.12
1.17
1.42
1.71
0.88
1.02
0.156
0.17
30
1.06
1.11
1.36
1.51
1.14
1.24
1.02
1.03
1.04
1.07
1.07
1.10
1.15
1.25
0.71
0.78
0.312
0.17
30
1.20
1.27
1.57
1.81
1.29
1.48
1.04
1.06
1.09
1.17
1.11
1.16
1.30
1.51
0.81
0.93
0.468
0.17
30
1.32
1.42
1.75
2.06
1.43
1.72
1.06
1.09
1.15
1.28
1.15
1.22
1.46
1.76
0.90
1.04
0.156
1.00
20
1.10
1.16
1.71
2.01
1.40
1.67
1.29
1.48
1.20
1.38
1.63
1.90
1.42
1.71
1.20
1.38
0.312
1.00
20
1.34
1.51
2.13
2.60
1.80
2.34
1.58
1.96
1.48
1.90
2.02
2.47
1.85
2.42
1.48
1.90
0.468
1.00
20
1.67
2.07
2.49
3.09
2.21
3.02
1.86
2.45
1.79
2.50
2.36
2.95
2.27
3.12
1.79
2.50
0.156
1.00
25
1.09
1.15
1.62
1.87
1.32
1.54
1.23
1.39
1.15
1.29
1.50
1.72
1.34
1.57
1.15
1.29
0.312
1.00
25
1.32
1.47
1.98
2.38
1.64
2.08
1.46
1.77
1.36
1.68
1.82
2.17
1.68
2.13
1.36
1.68
0.468
1.00
25
1.62
1.97
2.28
2.80
1.97
2.61
1.69
2.16
1.60
2.13
2.09
2.56
2.02
2.70
1.60
2.13
0.156
1.00
30
1.08
1.14
1.55
1.77
1.27
1.45
1.19
1.32
1.12
1.23
1.42
1.60
1.28
1.47
1.12
1.23
0.312
1.00
30
1.29
1.44
1.87
2.22
1.54
1.90
1.38
1.64
1.29
1.54
1.68
1.98
1.57
1.94
1.29
1.54
0.468
1.00
30
1.57
1.90
2.13
2.60
1.80
2.34
1.58
1.96
1.48
1.90
1.91
2.30
1.85
2.42
1.48
1.90The 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. The proposed model overcomes this drawback and provides a simplified model for a very simple evaluation of the strength and ductility columns.


### 5.2. Design Indications for Practical Applications

The proposed procedure can easily be extended for evaluating the performance of FRP-confined concrete columns in practical design applications.Generally, in the design phase, the known parameters are the cross-section dimensionl, the elastic modulus of unconfined specimens Ec, the nominal thickness of the reinforcing system tf, and the elastic modulus of the fibers Ef. 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 increased strength and evaluate the increased ductility and dissipated energy.Once the increased ductility or strength has been chosen, the fiber volumetric ratioρf and the parameter η, for different values of the number of layers n, have to be evaluated. Known ρf and η, in both design case, the ρ has to be evaluated and radius can be evaluated by means of Equation (6) (if με/μεo is fixed) or by means of Equation (5) (if fcc/fco is fixed). For 0 < ρ < 1 and rmin ≤ r ≤ rmax (rmin = 20 mm and rmax=2+2c, see detail in [30]), the value of fcc/fco , με/μεo, and E/Eo are evaluated by means of Equations (5)–(7), respectively.Figures12 and 13 display the design procedure for fixed increased ductility and fixed increased strength, respectively.Figure 12
Flowchart for design procedure in the case of fixed increased ductility.Figure 13
Flowchart for design procedure in the case of fixed increased strength.


## 5.1. Comparison between the Proposed Model and Existing Models for FRP

This section shows comparisons between the values of enhanced strength (fcc/fco) calculated by Equation (5) and those obtained from models available in the literature reported in Table 4.Table 4
Parameters and coefficients for evaluating the strength enhancement from existing analytical expressions.
Model
Coefficients
Other parameters and coefficients
CNR-DT200 [10]
α = 1k1 = 2.6m = 2/3
k
e
=
k
H
k
V
k
α
,
k
H
=
1
−
2
l
−
r
2
/
3
A
g
k
v = kα = 1 for continuous jackets with fibers ⊥ to the member axis, Ag = cross-section area of the specimen
Triantafillou et al. [26]
α = 1k1 = 2.79m = 1
k
e
=
k
H
k
V
k
α
,
k
H
=
1
−
2
l
−
r
2
/
3
A
g
k
v = kα = 1 for continuous jackets with fibers ⊥ to the member axis
Campione and Miraglia [24]
α = 1k1 = 2m = 1
f
ie
=
2
n
t
f
f
r
/
l
,
f
r
=
f
fu
1
−
2
/
2
k
i
2
r
/
l
+
k
i
2
/
2
f
fu = ultimate strength of the fiber, ki = 0.2121
Pellegrino and Modena [23]
α = 1k1=kAkRm = 1
k
A
=
A
f
l
,
e
/
f
co
−
β
,
A
=
2.25
and
β
=
−
0.25
k
R
=
1
−
2.5
0.3
−
2
r
/
l
,
if
2
r
/
l
<
0.3
,
1
,
if
2
r
/
l
≥
0.3
,
Mirmiran et al. [27]
α = 1k1=6.0fl,e−0.30m = 1
k
e
=
2
r
/
l
Di Ludovico et al. [28]
α = 1k1 = 2.94m = 1
k
e
=
k
H
k
V
k
α
,
k
H
=
1
−
2
l
−
r
2
/
3
A
g
,
k
v = kα = 1 for continuous jackets with fibers ⊥ to the member axis
Youssef et al. [38]
For circular sections
For circular sections
α = 1.0k1 = 2.25m = 1.25
k
e = 1.0ρ1 = longitudinal reinforcement ratio
For square sections
For square sections
α = 0.50k1 = 1.225m = 0.6
k
e
=
1
−
2
ℓ
−
2
r
2
/
3
ℓ
2
−
ρ
l
/
1
−
ρ
lUsually, in this model, the value offcc/fco is evaluated as a function of the confinement pressure fl,e by means of the following expression:(8)fccfco=α+k1f1,efcom,where(9)fi,e=12ρfEfεfke.Equation (8) involves the nondimensional parameters α, k1, and m, all experimentally assessable, while k1=αk1,R take into account the reduced effectiveness provided by jackets and resins (α is a coefficient related to a specified jacketing system, and αk1,R is a nondimensional parameter experimentally calibrated).In Equation (9), Ef is the elastic modulus in the lateral direction, εf is the strain of the jackets in the same direction, and ke is a coefficient that counts the variation in the confinement pressure in square and rectangular cross-section specimens compared to the circular ones. This coefficient is calibrated by the authors of each model.Table4 shows the parameters and coefficients of the various models considered for evaluating the theoretical values of fcc/fco.The comparison was carried out for concrete columns with square and circular sections withl = 300 mm and different values of corner radius ratio, fiber volumetric ratio, and unconfined maximum compressive strength and considering two kinds of fibers.The 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. The 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.Table5 summarizes a comparison between results obtained with models available in the literature and results of the proposed model in terms of strength enhancement fcc/fco.Table 5
Comparison (fcc/fco) between the proposed model and existing models for FRP.
ρ
f (%)
ρ
f
co (MPa)
Proposed model
CNR-DT200 [10]
Triantafillou et al. [26]
Campione et al. [24]
Pellegrino et al. [23]
Mirmiran et al. [27]
Di Ludovico et al. [28]
Youssef et al. [38]
#1
#2
#1
#2
#1
#2
#1
#2
#1
#2
#1
#2
#1
#2
#1
#2
0.156
0.13
20
1.08
1.13
1.45
1.63
1.20
1.34
1.03
1.04
1.05
1.09
1.08
1.12
1.21
1.35
0.75
0.84
0.312
0.13
20
1.22
1.29
1.71
2.01
1.40
1.67
1.05
1.08
1.12
1.22
1.14
1.20
1.42
1.71
0.88
1.02
0.468
0.13
20
1.34
1.44
1.94
2.32
1.60
2.01
1.08
1.13
1.19
1.37
1.18
1.26
1.63
2.06
0.99
1.16
0.156
0.17
20
1.08
1.13
1.47
1.67
1.22
1.36
1.03
1.05
1.06
1.12
1.10
1.15
1.23
1.38
0.76
0.86
0.312
0.17
20
1.23
1.30
1.75
2.06
1.43
1.72
1.06
1.09
1.15
1.28
1.17
1.24
1.46
1.76
0.90
1.04
0.468
0.17
20
1.35
1.46
1.98
2.38
1.65
2.08
1.08
1.14
1.24
1.46
1.23
1.32
1.68
2.14
1.01
1.19
0.156
0.13
25
1.07
1.12
1.39
1.55
1.16
1.27
1.02
1.03
1.04
1.07
1.07
1.10
1.17
1.28
0.72
0.80
0.312
0.13
25
1.21
1.28
1.62
1.87
1.32
1.54
1.04
1.07
1.09
1.17
1.11
1.16
1.34
1.57
0.83
0.96
0.468
0.13
25
1.32
1.42
1.81
2.14
1.48
1.81
1.06
1.10
1.15
1.28
1.14
1.21
1.51
1.85
0.93
1.08
0.156
0.17
25
1.07
1.12
1.41
1.57
1.17
1.29
1.02
1.04
1.05
1.09
1.08
1.12
1.18
1.30
0.73
0.81
0.312
0.17
25
1.22
1.28
1.65
1.91
1.35
1.58
1.04
1.07
1.11
1.21
1.14
1.20
1.36
1.61
0.85
0.98
0.468
0.17
25
1.33
1.44
1.85
2.19
1.52
1.87
1.07
1.11
1.18
1.35
1.18
1.26
1.55
1.91
0.95
1.11
0.156
0.13
30
1.06
1.11
1.34
1.48
1.13
1.22
1.02
1.03
1.03
1.06
1.06
1.08
1.14
1.24
0.70
0.77
0.312
0.13
30
1.20
1.26
1.54
1.77
1.27
1.45
1.03
1.06
1.07
1.13
1.09
1.13
1.28
1.47
0.80
0.91
0.468
0.13
30
1.31
1.40
1.71
2.01
1.40
1.67
1.05
1.08
1.12
1.22
1.12
1.17
1.42
1.71
0.88
1.02
0.156
0.17
30
1.06
1.11
1.36
1.51
1.14
1.24
1.02
1.03
1.04
1.07
1.07
1.10
1.15
1.25
0.71
0.78
0.312
0.17
30
1.20
1.27
1.57
1.81
1.29
1.48
1.04
1.06
1.09
1.17
1.11
1.16
1.30
1.51
0.81
0.93
0.468
0.17
30
1.32
1.42
1.75
2.06
1.43
1.72
1.06
1.09
1.15
1.28
1.15
1.22
1.46
1.76
0.90
1.04
0.156
1.00
20
1.10
1.16
1.71
2.01
1.40
1.67
1.29
1.48
1.20
1.38
1.63
1.90
1.42
1.71
1.20
1.38
0.312
1.00
20
1.34
1.51
2.13
2.60
1.80
2.34
1.58
1.96
1.48
1.90
2.02
2.47
1.85
2.42
1.48
1.90
0.468
1.00
20
1.67
2.07
2.49
3.09
2.21
3.02
1.86
2.45
1.79
2.50
2.36
2.95
2.27
3.12
1.79
2.50
0.156
1.00
25
1.09
1.15
1.62
1.87
1.32
1.54
1.23
1.39
1.15
1.29
1.50
1.72
1.34
1.57
1.15
1.29
0.312
1.00
25
1.32
1.47
1.98
2.38
1.64
2.08
1.46
1.77
1.36
1.68
1.82
2.17
1.68
2.13
1.36
1.68
0.468
1.00
25
1.62
1.97
2.28
2.80
1.97
2.61
1.69
2.16
1.60
2.13
2.09
2.56
2.02
2.70
1.60
2.13
0.156
1.00
30
1.08
1.14
1.55
1.77
1.27
1.45
1.19
1.32
1.12
1.23
1.42
1.60
1.28
1.47
1.12
1.23
0.312
1.00
30
1.29
1.44
1.87
2.22
1.54
1.90
1.38
1.64
1.29
1.54
1.68
1.98
1.57
1.94
1.29
1.54
0.468
1.00
30
1.57
1.90
2.13
2.60
1.80
2.34
1.58
1.96
1.48
1.90
1.91
2.30
1.85
2.42
1.48
1.90The 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. The proposed model overcomes this drawback and provides a simplified model for a very simple evaluation of the strength and ductility columns.


## 5.2. Design Indications for Practical Applications

The proposed procedure can easily be extended for evaluating the performance of FRP-confined concrete columns in practical design applications.Generally, in the design phase, the known parameters are the cross-section dimensionl, the elastic modulus of unconfined specimens Ec, the nominal thickness of the reinforcing system tf, and the elastic modulus of the fibers Ef. 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 increased strength and evaluate the increased ductility and dissipated energy.Once the increased ductility or strength has been chosen, the fiber volumetric ratioρf and the parameter η, for different values of the number of layers n, have to be evaluated. Known ρf and η, in both design case, the ρ has to be evaluated and radius can be evaluated by means of Equation (6) (if με/μεo is fixed) or by means of Equation (5) (if fcc/fco is fixed). For 0 < ρ < 1 and rmin ≤ r ≤ rmax (rmin = 20 mm and rmax=2+2c, see detail in [30]), the value of fcc/fco , με/μεo, and E/Eo are evaluated by means of Equations (5)–(7), respectively.Figures12 and 13 display the design procedure for fixed increased ductility and fixed increased strength, respectively.Figure 12
Flowchart for design procedure in the case of fixed increased ductility.Figure 13
Flowchart for design procedure in the case of fixed increased strength.


## 6. Conclusions

In this paper, an empirical model for concrete columns wrapped with FRP systems is proposed. The model was obtained by fitting the experimental data of several authors and numerical data derived from FE analyses. The 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.



---

*Source: 2543850-2018-10-11.xml*