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Prediction of Ultimate Strain for Rectangular Reinforced Concrete Columns Confined with Fiber Reinforced Polymers under Cyclic Axial Compression

This paper investigates the crucial design parameters for the prediction of the ultimate axial compressive deformation of reinforced concrete columns externally confined with fiber reinforced polymer (FRP) materials. Numerous test results of available columns with a square and rectangular section under cyclic axial loading were gathered in an advanced database. Herein, the database is enriched with necessary design parameters in order to address the unique tensile strain field variation of the FRP jacket. Since there is a lack of consequent recording of the FRP strain field in existing experiments, three dimensional pseudodynamic finite element analyses results from several characteristic cases of tested columns are utilized to address this gap. Therefore, a hybrid experimental–analytical database is formed, including several critical FRP strains, steel strains and deformations. A modified model is proposed to predict the ultimate axial strain for reinforced concrete columns externally confined with FRP materials. The proposed model aims to address indirectly the effects of the internal steel cage, concrete section shape and of their interaction with the external FRP jacket on the critical tensile strain of the FRP jacket at failure of the column. The predictive performance of the model over the available tests of (reinforced concrete) RC columns under cyclic compression is remarkably improved when compared against the performance of other existing models. It provides predictions with average ratio (AR) of 0.96 and average absolute error (AAE) of 36.5% and therefore may contribute to safer seismic resistant redesign.


Introduction
The use of fiber reinforced polymers (FRPs) has been widely accepted as a successful confinement technique for existing reinforced concrete (RC) columns in buildings and bridges. The benefits of this technique are the improvement of the axial strength, strain ductility and energy dissipation capacity of the confined concrete. Thus, the behavior of concrete columns confined with FRP has been extensively studied, leading to a significant number of stress-strain models, some focused on circular cross-sections [1][2][3][4][5] and others focused on square and rectangular ones [6][7][8][9][10][11][12][13][14][15][16][17]. In addition to that, some studies are concentrated on modeling the stress-strain behavior of both circular and non-circular FRP confined concrete columns [17][18][19][20][21][22]. Other studies model the overall stress-strain curves of columns under cyclic axial loading [23,24]. These models are mostly design-oriented models [6], based on empirical or semiempirical relationships. They take into account parameters that are easily understood and available to designers, such as the corner radius, the stiffness of the FRP and concrete strength. Recently, Zeng et al. [25] compared the performance of five representative stress-strain models [6,[14][15][16]21] for FRP-confined RC concrete columns against the experimental behavior of nine In order to check the accuracy of the existing strain models, an experimental database has been formed through an extensive literature research. This database was presented in the previous work by Fanaradelli and Rousakis [29] and includes only the results for FRP-confined concrete columns of square and rectangular cross-sections under cyclic axial loading. This database includes all 44 test results from specimens with internal steel reinforcement from five different studies, taking place between 2008 and 2018. The database includes all the characteristics of the specimens: naming, shape and dimension of the cross-section (h and b are the length of the longer and shorter sides of the column, respectively, H is the height of the specimens and r c is the corner radius) and the mechanical characteristics of concrete (f co is the unconfined compressive concrete strength, f cc and e cc are the peak axial compressive stress and the corresponding axial strain of FRP-confined concrete, f cu and e cu are the ultimate axial compressive stress and the corresponding strain of FRP-confined concrete) and steel materials (f y,long and f y,stirrup are the yield strength of longitudinal steel bars and stirrups, respectively; Table 1). The database also contains the material properties of the FRP strengthening (carbon FRP (CFRP) and glass FRP (GFRP)), its detailing, the number of layers, the FRP thickness (t FRP ) and its elastic modulus (E FRP ; Table 1).  The collected experiments included rectangular cross sections (19 specimens) or square sections (25 specimens). All specimens included transverse and longitudinal internal steel reinforcement. Most of the tested columns were strengthened with carbon FRP (CFRP), while only 14 specimens were confined with GFRP jackets. The layers of the FRP materials ranged from 1 to 9 and partial wrapping was not included. The corner radius (r c ) of the specimens varied from 0 to 40 mm, the unconfined compressive concrete strength (f co ) varied from 10.83 to 50 MPa and the nominal yield stress of reinforcement ranged from 200 to 500 MPa. Special care was taken to distinguish the results of 27 specimens having stress-strain curves with descending inelastic (second) branches (i.e., f cu < 0.85 × f cc < f cc ) and 17 columns having ascending second branches (i.e., f cu = f cc ). So, for columns with ascending second branches it was considered that f cu = f cc while for columns with descending second branches, if the ultimate stress f cu was lower than 15% of the maximum strength, there was a revision of this value with the 0.85 × f cc and of the corresponding value for ε cu from the experimental stress-strain curves of the test results.
In Fanaradelli and Rousakis [29] the predictive performance of several existing ultimate strain design models, originally proposed for plain rectangular concrete columns confined with FRP materials under monotonic axial loading [16] and for plain [24] and RC [23] rectangular concrete columns under cyclic axial loading, were assessed over the experimental results of RC columns under cyclic axial loading. It was concluded that for columns under cyclic axial loading with internal steel reinforcement, the prediction of ultimate axial strain ε cu presents low accuracy for all assessed design models [16,23,24].
As the original models did not address the effects of the internal steel reinforcement the AAE was higher than 50%. The model by Hany et al. [24] had AAE 62.8% and AR 0.94 and the model by Wang et al. [23] had AAE 138.9% and AR 2.23. However, it was concluded the model by Wei and Wu [16] may be used for columns with descending second branches as well and has the best overall performance among all investigated models (predicts the strain at failure with average error 56.1% and average ratio 1.14) [29].

Square and Rectangular RC FRP-Confined Concrete Columns under Monotonic Axial Loading
The experimental database by Fanaradelli et al. [28] includes the results for FRP-confined RC concrete columns of square and rectangular cross-sections under monotonic axial loading. The database includes 130 test results collected from 18 different studies, taking place between 2001 and 2018. So, the database includes 29 reinforced concrete columns with rectangular cross-section and 101 with square cross-section. All the specimens in the database are fully wrapped with FRP sheets. There are 87 columns externally reinforced with CFRP, 40 columns with GFRP and 3 with aramid FRP (AFRP). Finally, the layers of the FRP materials ranged from 1 to 9. The corner radius (r c ) of the specimens varied from 0 to 40 mm, the unconfined compressive concrete strength (f co ) varied from 13.0 to 46.3 MPa and the nominal yield stress of reinforcement ranged from 200 to 587 MPa. Finally, the database included 68 specimens having stress-strain curves with descending second branches and 62 columns having ascending second branches. All the characteristics of the specimens were available: shape and dimension of the cross-section, mechanical characteristics of concrete and steel materials, the material properties of the FRP strengthening (elastic modulus), number of layers and the FRP thickness ( Table 2). Similar with the results of the database with columns under cyclic axial loading [29], the assessment of the existing models in predicting the ultimate axial strain of concrete was proved to be insufficient [6,7,16,18,[41][42][43][44][45] with AAEs higher than 50% [28]. Despite the fact the columns under monotonic loading are not the focus of this research, the available database may be utilized for additional assessments.

Model by Wei and Wu for Plain FRP-Confined Columns
In what follows, the model by Wei and Wu [16] is used as the basis for suitable modifications and is briefly presented. The model by Wei and Wu [16] was based on a large database including data from 29 published experimental studies on plain concrete columns from the international literature. In particular, the database contained test results of 432 FRP-wrapped plain concrete short columns with concrete strength between 18 and 55 MPa and included 194 circular, 170 square and 68 rectangular columns. This database was created to determine the parameters that affect the ultimate axial strain ε cu of the FRP jacket. For a certain confinement pressure, the lateral strain of the FRP differs for FRP jackets with different rigidity and as a result it affects the axial strain of concrete in principle [59]. The grade of the concrete is also found to affect the ultimate axial strain of concrete when all other factors are fixed [15,60]. Lower-strength concrete has a larger degree of deformability than higher-strength concrete, and thus the former displays greater confinement efficiency.
Based on these studies, the ultimate strain is affected by these parameters and can have the following mathematical form: f co , ε cu is the ultimate axial strain at the ultimate failure point, ε co is the strain of the unconfined concrete corresponding to f co , λ is a constant that relates the ultimate strain to the peak strain for unconfined concrete, f l,FRP is the lateral confining pressure (MPa), f co is the axial compressive stress of unconfined concrete (MPa), r c is the corner radius, h and b are the length of the longer and shorter sides of the column, respectively, and, f 30 is the concrete strength of unconfined grade C30 concrete. All of the variables in Equation (1) are non-dimensionalized. The constant λ = 1.75 is recommended by Eurocode 2 and Ref. [2] and relates the ultimate strain to the peak strain for unconfined concrete. For the second term in Equation (1), the ultimate axial strain of concrete increases when the confinement increases, and decreases when the concrete strength increases. Additionally, when the lateral pressure provided by the FRP f l,FRP is equal to zero, or f co is equal to infinity, the confinement effectiveness is zero. So, the effects of these two terms in Equation (1) in which, the values f 2r c b , and f h b , are two functions that have to be defined, while the value of ε co is calculated by the equation proposed by Popovics [61], unless the test value is available and α is a coefficient equal to 12.
From the assessment of the databases by Wei and Wu [16], it is observed that the value of 2r c b increases as the corner radius ratio increases from 0 to 1, as a consequence the relationship is approximately linear. Following regression analysis using all data in the database by Wei and Wu [16], the corner radius ratio factor (function f 2r c b ) is defined as: Through a similar regression analysis using the full database by Wei and Wu [16], the function for the aspect ratio is obtained as Finally, the proposed ultimate strain model for a plain square and rectangular columns is and where: E FRP is the elastic modulus of the FRP material, ε fu is the ultimate tensile strain by the manufacturers for the FRP material and, t FRP is the total thickness of the jacket.

Extended Advanced Analytical Database
The available experimental tests were carefully investigated and 11 out of them were modeled and analyzed with advanced three-dimensional finite element (3D FE) software ANSYS Explicit Dynamics software [62]. These columns were selected in order to cover the effects of a wide range of parameters, crucial for the retrofit of RC columns with FRPs for seismic resistant applications (slenderness of bars, sparse stirrups, steel quality and quantity, type and layers of FRP jacket, type of impregnation resins and reinforcing fibers, corner radius, concrete strength, predamages, etc.). The specimens were experimental tests performed by Rousakis and Karabinis [39], Ilki et al. [40] and Isleem et al. [10] and their parameters are gathered in Table 3. Then, these carefully chosen columns were modeled and analyzed for the first time pseudodynamically with three-dimensional finite elements (3D FEs). They provided numerous analytical insights into the effects of different critical parameters to allow for the enrichment of the existing databases with significant missing variables (see Table 3). More details about the analytical procedure of the 3D FE analyses are presented in Fanaradelli and Rousakis [35]. Herein, the numerous FE analytical results are further elaborated to enrich the hybrid experimental-analytical database with significant missing variables. These variables are difficult to measure in several experimental programs, as extensive local instrumentation or advanced image techniques are required, but yet are important for the prediction of ultimate strain of FRP retrofitted columns with internal steel reinforcement. The critical design parameters that were investigated through FE analyses are: (a) FRP strain at the mid-point of the side and at the level of the middle stirrup (ε FRP, mid, stirrup ); (b) FRP strain at the midpoint of the side and at the midpoint level between two stirrups (ε FRP, mid, between_stirrups ); (c) FRP strain at the corner of the section at the level of the middle stirrup (ε FRP,corner,stirrup ); (d) FRP strain at the corner of the section at the midpoint level between two stirrups (ε FRP,corner, between_stirrups ); (e) maximum strain at the stirrups (ε stirrup,max ); (f) minimum strain at the longitudinal steel bars (ε long,min ); (g) strain of the longitudinal bars at the mid-point level between two stirrups (ε long,mid ); (h) lateral deformation (Y axis) of longitudinal steel bars at the point of stirrup; (i) lateral deformation (X axis) of longitudinal steel bars at the point of stirrup; (g) lateral deformation (Y axis) of longitudinal steel bars at the midpoint between two stirrups; (k) lateral deformation (X axis) of longitudinal steel bars at the midpoint between two stirrups and (l) lateral deformation of middle stirrup (Table 3).With these analytical results the variable deformation and strain field of the column around its perimeter at different levels may be assessed throughout loading and for different characteristic columns.
The analytical results suggested that the measured FRP tensile strains are higher at the corner of the sections at the level of the stirrups for most of the specimens (see also Table 3). Closer to the ultimate strain of the whole column is the corner FRP strain at the point between two successive stirrups. The mid-distance between two stirrups (far from end boundaries) is a critical position to place strain gauges in order to measure the representative axial strain of the steel reinforcement under axial compression in experiments (probably two or four strain gauges, symmetrically placed, are the best option if the bar buckles significantly). The axial strain of the bar at the mid distance between two stirrups is expected to be close to the average global axial strain of the column. On the other hand, the axial strains of the bars at the level of the stirrups may be far higher. The variation of the FRP strains proves that there may be in some cases local fracture of the jacket at its corner, while the global behavior of the column has not reached the ultimate failure.
The measured deformations suggest there is nonuniform bulging of the retrofitted column axially and transversely. The lateral deformation of the mid-point of the longitudinal steel bars between two stirrups in most cases is lower than half the lateral deformation at the level of the stirrups, because of the bar buckling during the axial compression. For lower FRP confinement and for lower corner radius there is a high local damage development in concrete and local bulging in both concrete and FRP. In some cases, local concrete damage may trigger the failure of the specimen prior to the fracture of the FRP jacket. It seems that the amount of the FRP confinement is an important parameter for the prediction of the ultimate strain of the RC columns. Finally, it comes out that all bars under compression exhibit and maintain strains higher than the yield one, with high variation along the axis. All stirrups under tension exceed their yield strain or are very close to it (see Fanaradelli and Rousakis [35] for more details). Figure 1 shows the variation of the most critical parameter as concluded by the investigation of all additional analytical findings gathered in Table 3 (variation of strains and displacements), in order to contribute to the prediction of the ultimate strain of the RC columns. The strongest correlation arose out of the ratio of the FRP strain at the corner of the section at the level of the middle stirrup to the ultimate FRP strain according to the manufacturers (ε FRP,corner,stirrup /ε fu ) and the parameter α st × ω st + α FRP × ω FRP , which is the effective confinement mechanical volumetric ratio for the stirrups plus the one for the FRP. The ratio of strains arose as a combined parameter that could provide a more reliable estimate for the critical ultimate strain of the FRP at failure (significantly variable around and along the column), rather than a fictitious average FRP strain. The combined steel-FRP effective confinement mechanical volumetric ratio is chosen as a rough estimate of the different confining effects on the concrete core provided by both steel and FRP reinforcement. The rest of analytical FRP strain values at different characteristic positions revealed high scarcity and low correlation with the effective mechanical volumetric confinement ratio. Figure 1 suggests the FRP strain ratio increased with the increase of α st × ω st + α FRP × ω FRP . That is, for higher provided confinement or higher corner radius of the concrete section, etc., the effects of the internal steel reinforcement are lower. Further, as the number of the FRP layers is increased, the increase of the ratio of the strain is lower and even becomes equal to the case of low confinement. That is for a very high number of layers the quality of application affects the FRP strain variation. Characteristic is the specimens by Ilki et al. [40]. There, the analyzed columns have reached their ultimate FRP strain and the fibers have raptured in different places, leading to brittle local failure. The pseudodynamic FE analyses may provide the mechanical response of the columns beyond the original local failures and their successive behavior up to their ultimate global failure. At global failure state the columns have already reached in some positions local fracture of the FRP jacket while the analysis provides fictitious ultimate FRP strain that are higher than the ultimate strains provided by the manufacturers. While these exaggerated values are fictitious, they may reveal the general trend for this FRP strain variation, in case a phenomenological modeling approach is followed. correlation with the effective mechanical volumetric confinement ratio. Figure 1 suggests the FRP strain ratio increased with the increase of αst × ωst + αFRP × ωFRP. That is, for higher provided confinement or higher corner radius of the concrete section, etc., the effects of the internal steel reinforcement are lower. Further, as the number of the FRP layers is increased, the increase of the ratio of the strain is lower and even becomes equal to the case of low confinement. That is for a very high number of layers the quality of application affects the FRP strain variation. Characteristic is the specimens by Ilki et al. [40]. There, the analyzed columns have reached their ultimate FRP strain and the fibers have raptured in different places, leading to brittle local failure. The pseudodynamic FE analyses may provide the mechanical response of the columns beyond the original local failures and their successive behavior up to their ultimate global failure. At global failure state the columns have already reached in some positions local fracture of the FRP jacket while the analysis provides fictitious ultimate FRP strain that are higher than the ultimate strains provided by the manufacturers. While these exaggerated values are fictitious, they may reveal the general trend for this FRP strain variation, in case a phenomenological modeling approach is followed. Figure 1. Correlation between the ratio of the FRP strain at the corner of the stirrups and the ultimate FRP strain according to the manufacturers (εFRP,corner,stirrup/εfu) and the sum αst × ωst + αFRP × ωFRP.

Proposed Strain Model for RC Columns Confined with FRPs
The axial strain for concrete confined with internal steel stirrups is estimated with the relationship by Mander et al. [63] , = 1 + 5 , − 1 where fcc,st is the triaxial compressive strength of concrete because of the stirrup confinement, as calculated according to the Model Code 90. Further, the model takes into account the effects of the FRP confinement based on the Wei and Wu [16] model. The axial strain εcc,st replaces the constant parameter λ in the Equation (1). For the prediction of axial strain at ultimate conditions for RC columns, the combined parameter αst,FRP is imported and multiplied to Equation (7) and its new form is: In Equation (10), fl,FRP is calculated according to Wei and Wu model and thus εfu is the ultimate tensile strain by the manufacturer as already mentioned. Several studies [64][65][66] have shown that the ultimate FRP confining stress could be calculated using the ultimate hoop rupture strain rather than the ultimate strain of FRP coupons under direct tension. Hoop rupture strains, obtained from FRP split-disk tests and the models [25,67] based on this method are generally more accurate than those

Proposed Strain Model for RC Columns Confined with FRPs
The axial strain for concrete confined with internal steel stirrups is estimated with the relationship by Mander et al. [63] ε cc,st = ε co 1 + 5 f cc,st f co − 1 (9) where f cc,st is the triaxial compressive strength of concrete because of the stirrup confinement, as calculated according to the Model Code 90. Further, the model takes into account the effects of the FRP confinement based on the Wei and Wu [16] model. The axial strain ε cc,st replaces the constant parameter λ in the Equation (1). For the prediction of axial strain at ultimate conditions for RC columns, the combined parameter α st,FRP is imported and multiplied to Equation (7) and its new form is: In Equation (10), f l,FRP is calculated according to Wei and Wu model and thus ε fu is the ultimate tensile strain by the manufacturer as already mentioned. Several studies [64][65][66] have shown that the ultimate FRP confining stress could be calculated using the ultimate hoop rupture strain rather than the ultimate strain of FRP coupons under direct tension. Hoop rupture strains, obtained from FRP split-disk tests and the models [25,67] based on this method are generally more accurate than those based on the FRP deformability provided by the manufacturer. However, this method is mainly used in circular specimens [64][65][66]. For rectangular cross sections, a strain efficiency factor was proposed by [7,16,19,39,68] among others, to account for the reduced average deformability of the FRP jacket. Herein, the parameter α st,FRP is proposed according to the findings discussed in Figure 1. It takes into account the number of the layers of the FRP jacket and the combined effects of steel and FRP confinement through α st × ω st + α FRP × ω FRP. This parameter affects the efficiency of the FRP confinement by changing indirectly the ultimate FRP strain of FRP confinement of the model by Wei and Wu [16] (ε fu ). This modified ultimate FRP strain is indirectly related to the one measured at the corner of the section at the level of the stirrup (critical parameter ε FRP,corner,stirrup ) of the RC columns, as resulted by the FE analyses. Therefore, in order to address the ultimate axial strain of RC concrete columns confined with FRP under cyclic loading, which is the aim of this study, α st,FRP is expressed as: where: α st is the confinement effectiveness factor for the stirrups, α FRP is the confinement effectiveness factor for the FRP jacket, ω st is the mechanical volumetric ratio of the stirrups, ω FRP is the mechanical volumetric ratio of the FRP jacket, n FRP is the number of the layers of the FRP material. The parameter α st,FRP ranged from 0.29 to 1.55 for the 11 specimens. Herein, it should be reminded that in some cases of the 3D FE analyses [35], the local failure of the concrete core may trigger the global failure of the column or several prior local failures of the FRP jacket were revealed. Such issues need further investigation.
The confinement effectiveness factor a st for the stirrups is calculated according to the EN 1998-1:2004 [69] a st = a n * a s = where: b i is the distance between consecutive bars engaged by a corner of a tie or by a cross-tie in a column with b i < 200 mm (see Figure 2a), b o is the width of confined core (to the centerline of the hoops), h o is the depth of confined core (to the centerline of the hoops), n is the total number of tied longitudinal bars and, s is the distance between two successive stirrups (see Figure 2b) with s < bi is the distance between consecutive bars engaged by a corner of a tie or by a cross-tie in a column with bi < 200 mm (see Figure 2a), bo is the width of confined core (to the centerline of the hoops), ho is the depth of confined core (to the centerline of the hoops), n is the total number of tied longitudinal bars and, s is the distance between two successive stirrups (see Figure 2b) with .
(a) (b) The parameter ω w is the actual mechanical volumetric ratio of confining reinforcement (EN 1998(EN -1:2004) volume o f con f ining hoops volume o f concrete core · f y,trans f co (13) where f y,trans is the stress of transverse reinforcement (yield stress of steel stirrup or stress of FRP, E FRP × ε fu ). The parameter α FRP is the confinement effectiveness coefficient according to fib Bulletin [70] where b' and h' are the clear distances without the rounded corners (b' = b -2 × r c and h' = h − 2 × r c ), A g is the gross cross-sectional area and, ρ sg is the reinforcement ratio of the longitudinal steel reinforcement with respect to the gross cross-sectional area (ρ sg = A s A g ).

Performance of the Proposed Model
The ultimate axial strain of the specimens under cyclic axial loading is predicted using the proposed model. The performance of the model is assessed based the average absolute error (AAE) calculated with the following expression: where ε cu,TH is the theoretical value of concrete ultimate strain, ε cu,EXP is the corresponding experimental value and, N = total number of data. The second parameter for the average ratio (AR) is defined as: Through the AAE, the model absolute error with respect to the experimental data was evaluated. The average ratio provides significant additional information; in fact, it shows if the predicted quantity overestimates (AR > 1) or underestimates (AR < 1) the experimental quantity. The combination of AR close to unity and AAE close to zero suggests the model is accurate enough.
The proposed model provides the ultimate axial strain of the 11 columns being analyzed with 3D FE, with an average absolute error of 23.9% and average ratio of 1.09 (Figure 3a). The performance of the Wei and Wu [16] model prior to the proposed modifications was AAE of 61.8% and AR of 1.36, for the same specimens. In addition to that, for the total of the 44 specimens under cyclic axial loading, the AAE of the proposed model was 36.5% and the AR was 0.96 (Figure 3b), while before the incorporation of the parameter a st,FRP the AAE of the most accurate existing model (Wei and Wu [16]) was 53.4% and AR was 1.18. The regression coefficient for the 11 FE analyzed columns was 0.91 and for the 44 columns under cyclic axial loading was 0.85.  The proposed model improved the performance over the 130 RC square and rectangular concrete columns under monotonic axial compression gathered in the second database. The average values provided by the Wei and Wu [16] model before the modification were 69.7% and 1.23 for the AAE and AR respectively. The AAE of the modified model was 48.1% and the AR was 0.86 ( Figure  3c). The modified model, while not oriented for columns under monotonic loading, provided better predictive performance than the models by Shehata et al. [41] (AAE = 62.1% and AR = 1.09), Chaallal et al. [42] (AAE = 69.5% and AR = 1.21), Lam and Teng [6] (AAE = 75.5% and AR=1.37), Vintzileou and Panagiotidou [18] (AAE = 76.2% and AR = 1.36), Cao et al. [47] (AAE = 68.7% and AR = 1.33) and Lim and Ozbakkaloglu [7] (AAE = 57.1% and AR = 1.07) that were used through the assessment of the database for the prediction of the ultimate strain for columns under monotonic axial loading [28]. The regression coefficient for the 130 specimens under monotonic axial compression was 0.68.
Further, the proposed ultimate strain model provided satisfactory predictions for 27 of the RC columns of the database with descending second branches under cyclic axial loading, revealing an AAE equal to 31.0% and AR 0.93. This performance was very important as these columns constitute the case commonly met in real practice of FRP confined RC columns of a non-circular section with a low achievable corner radius of the section and inadequate steel reinforcement detailing in seismic resistant redesign. For the 97 of columns with descending second branches under monotonic axial loading the AAE was 54.1% and AR was 1.06. It seems that the proposed model provides the best  [16] model before the modification were 69.7% and 1.23 for the AAE and AR respectively. The AAE of the modified model was 48.1% and the AR was 0.86 (Figure 3c). The modified model, while not oriented for columns under monotonic loading, provided better predictive performance than the models by Shehata et al. [41] (AAE = 62.1% and AR = 1.09), Chaallal et al. [42] (AAE = 69.5% and AR = 1.21), Lam and Teng [6] (AAE = 75.5% and AR=1.37), Vintzileou and Panagiotidou [18] (AAE = 76.2% and AR = 1.36), Cao et al. [47] (AAE = 68.7% and AR = 1.33) and Lim and Ozbakkaloglu [7] (AAE = 57.1% and AR = 1.07) that were used through the assessment of the database for the prediction of the ultimate strain for columns under monotonic axial loading [28]. The regression coefficient for the 130 specimens under monotonic axial compression was 0.68.
Further, the proposed ultimate strain model provided satisfactory predictions for 27 of the RC columns of the database with descending second branches under cyclic axial loading, revealing an AAE equal to 31.0% and AR 0.93. This performance was very important as these columns constitute the case commonly met in real practice of FRP confined RC columns of a non-circular section with a low achievable corner radius of the section and inadequate steel reinforcement detailing in seismic resistant redesign. For the 97 of columns with descending second branches under monotonic axial loading the AAE was 54.1% and AR was 1.06. It seems that the proposed model provides the best predictive performance for columns with descending second branches under cyclic axial load.

Conclusions
The present study investigated the effects of the internal steel reinforcement and of the external FRP confinement on the ultimate axial strain of RC columns of the square and rectangular section, subjected to cyclic axial loading. The assessed strain models in the previous work by Fanaradelli et al. [28,29] are proved to be inefficient and presented AAEs higher than 50%. The model by Wei and Wu [16], which can be used for columns with ascending and descending second stress-strain branches, had the best overall predictive performance among all investigated models and herein was used as the basis for suitable modifications. Fanaradelli and Rousakis [35] conducted advanced 3D FE analyses of 11 characteristic RC columns confined with FRP and investigated the critical parameters important for the improvement of strain predictions. These columns cover a wide range of different characteristics, which are important for the retrofit of RC columns with FRPs for seismic resistant applications. These parameters are the internal steel rebar of different slenderness, volumetric ratio (and quality) under compression, the stirrups placed at different spacing, the external FRP jacket applied at different layers and the concrete core with different side ratios, corner radii and concrete strength on the ultimate axial strain of the columns.
The results of the 3D FE analyses allowed the enrichment of the existing database (forming a hybrid experimental-analytical one) for columns under cyclic compression with significant additional parameters. These parameters are the strain of the FRP materials at characteristic levels and positions around the specimens (related to the position of stirrups), and the lateral deformations and strains that are developed on the steel bars and stirrups at different characteristic positions. The strongest correlation comes from the ratio of the FRP strain at the corner of the section at the level of the middle stirrup to the ultimate FRP strain according to the manufacturers (ε FRP,corner,stirrup /ε fu ) and the parameter α st × ω st + α FRP × ω FRP , which is the effective confinement mechanical volumetric ratio for the stirrups plus the one for the FRP. Based on this, the parameter α st,FRP was proposed, which also includes the number of the layers of the FRP material.
The proposed model herein is a modification of the model by Wei and Wu [16] (originally proposed for plain concrete columns). The new model utilizes the Mander et al. [63] model for the effects of the steel stirrup confinement and includes a new parameter α st,FRP , which affects the efficiency of the FRP confinement by changing indirectly the critical FRP strain of FRP confinement at ultimate of the model by Wei and Wu [16], based on the FE analytical results, in order to address the ultimate axial strain of RC concrete columns under cyclic loading. The achieved enhancement of the performance of the proposed model may contribute to safe seismic resistant redesign as these columns constitute the case commonly met in the real practice of FRP confined RC columns of the non-circular section with a low achievable corner radius of the section and inadequate steel reinforcement detailing in seismic resistant redesign. The modified model predicts the ultimate strain values for the 11 columns of the 3D FE investigation with an average absolute error of 23.9% and average ratio of 1.09. For all the columns of the database under cyclic axial loading the corresponding predictions provided AAE of 36.5% and AR of 0.96. Finally, for columns with descending second branches the model seemed to be more accurate with AAE being 31.0% and AR being 0.93. This performance is very important as these columns constitute the case commonly met in real practice of FRP confined RC columns of the non-circular section with a low achievable corner radius of the section and inadequate steel reinforcement detailing in seismic resistant redesign. The proposed model provides improved predictions for 130 RC square and rectangular concrete columns under monotonic axial compression but the AAE remained as high as around 50%. Future research should identify additional critical parameters and suitably address them experimentally and analytically to improve the ultimate strain predictions.