Simulating the lateral performance of FRP-confined RC circular columns using a new eccentric-based stress-strain model

Accepted Manuscript Simulating the Lateral Performance of FRP-Confined RC Circular Columns Using a New Eccentric-Based Stress-Strain Model Ahmed M. Ismail, Mohamed F.M. Fahmy, Zhishen Wu PII: DOI: Reference:

S0263-8223(16)33030-6 http://dx.doi.org/10.1016/j.compstruct.2017.07.075 COST 8730

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

1 January 2017 29 June 2017 21 July 2017

Please cite this article as: Ismail, A.M., Fahmy, M.F.M., Wu, Z., Simulating the Lateral Performance of FRPConfined RC Circular Columns Using a New Eccentric-Based Stress-Strain Model, Composite Structures (2017), doi: http://dx.doi.org/10.1016/j.compstruct.2017.07.075

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Simulating the Lateral Performance of FRP-Confined RC Circular Columns Using a New Eccentric-Based Stress-Strain Model Ahmed M. Ismail1, Mohamed F.M. Fahmy2 and Zhishen Wu3 Abstract In this study, a stress-strain model of fiber-reinforced polymers (FRPs)-confined concrete based on the lateral confinement stiffness was adopted to simulate the lateral response of RC columns retrofitted with external FRP jackets and tested under axial and lateral loads. The adopted model and other five-stress-strain models (established in former studies) were comparatively studied to simulate the seismic response of eight RC-circular columns retrofitted with FRP jackets and experimentally tested under both axial and lateral loads. Compared to the experimental results, the simulation results indicated that all stress–strain models could not identify properly the ultimate lateral displacements of the simulated columns. The adopted stress–strain model was revised to consider the effect of a key influential parameter (eccentricity ratio), which showed a critical impact on the simulation of the seismic response of RC-columns under combined bending and axial loadings. Finally, the proposed model was evaluated in predicting the lateral response of additional three columns and the simulation results exhibited a good agreement with the experimental results.

1

Ph.D. Candidate, International Institute for Urban Systems Engineering, Southeast Univ., Nanjing 210096, China. E-mail: [email protected] 2 Research fellow, International Institute for Urban Systems Engineering, Southeast Univ., Nanjing 210096, China. and Associate Professor, Civil Engineering Dept., Faculty of Engineering, Assiut University, Assiut, Egypt, E-mail: [email protected] 3 Professor, International Institute for Urban Systems Engineering, Southeast Univ., Nanjing 210096, China, E-mail: [email protected] (corresponding author)

Keywords: Eccentricity; design-oriented model; lateral loading; Circular column; FRP-jacket; OpenSees. 1.Introduction Applications of fiber reinforced polymer (FRP) composites in deteriorated reinforced concrete (RC) structures are gaining an increase over the last three decades. Specifically, external FRP jackets have been successfully applied as a strengthening/retrofitting technique for under-designed RC columns to increase the column strength and ductility (Xiao and Ma 1997; Breña and Schlick 2007; Paultre et al. 2016). The compressive behavior of concrete cylinders externally confined with FRP under axial loads has been extensively studied during the last three decades. Numerous stress-strain models have been established during that period. Some of these studies (Shehata et al. 2002; Vintzileou and Panagiotidou 2008; Ozbakkaloglu and Lim 2013; Keshtegar et al. 2016) focused on predicting the ultimate conditions of FRP-confined concrete, including the ultimate compressive strength (fcc) and the corresponding ultimate axial strain (εcc). Other studies tried to understand and predict the overall behavior of the stress-strain curve (Samaan et al. 1998; Lam and Teng 2003; Harajli 2006; Chastre and Silva 2010; Shirmohammadi et al. 2015). The available stressstrain models have been classified by (Lam and Teng 2003) into two categories: design-oriented models and analysis-oriented models. It would be questionable by the research community to use the design-oriented models in simulating the behavior of FRP-confined RC-columns subjected to axial and lateral loadings as shown in Fig. 1. Because they may consider this type of models is inaccurate compared to the analysis-oriented models, and it was mainly developed for the aim of design. However, a comprehensive statistical study done by Ozbakkaloglu et al. (2013) to evaluate the performance

of 68 stress-strain (design-oriented and analysis-oriented) models of FRP-confined concrete, concluded that the compressive strength, the ultimate axial strain, and the stress–strain behavior of the existing design-oriented models are more acceptable than the analysis-oriented models. In the development of the analysis-oriented models, it is assumed that the axial strain and stress of FRP-confined concrete are the same as those of actively confined concrete when the lateral pressures provided by the FRP-confinement and the active confinement are equivalent. However, Lim and Ozbakkaloglu (2015a) recognized that this assumption is inaccurate for high strength concrete. Similarly, (Li and Wu 2016) reported that the behavior of cyclically loaded FRPconfined concrete is not comparable to actively confined concrete. Recently, (Lim and Ozbakkaloglu 2015a; Lim and Ozbakkaloglu 2015b) proposed a new analysis-oriented model based on an experimental work (Lim and Ozbakkaloglu 2014b) and a comparison among the existing analysis-oriented models. The proposed model provides improved prediction of the stress-strain behavior of FRP-confined concrete compared to several examined models. Although studies established to examine the compressive behavior of FRP-confined concrete under the effect of axial and bending loadings (eccentric loadings) are limited, they concluded different views. For example, under eccentric loadings, the stress–strain model proposed by Fam et al. (2003) shows a softening trend rather than the strain hardening behavior. On the other hand, Hu et al. (2011) proposed an eccentricity-based stress-strain model for FRP-confined square columns, and they mentioned that the ultimate strain is reversely proportional to the increase in the load eccentricity meanwhile the concrete ultimate strength should be constant. Recently, the study by (Wu and Jiang 2013) indicated experimentally and analytically that the increase in the eccentricity of the axial load was associated with a considerable increase in the ultimate strength

of FRP-confined concrete. Hence, they stated that the existing stress–strain models developed for axially loaded columns are deficient for predicting the behavior of columns under eccentric loads. The analysis of structures using fiber section method has been applied in numerous studies; some of them adopted the design/analysis-oriented models to predict the lateral response of FRPconfined RC columns under the effect of lateral cyclic loadings (Gallardo-Zafra and Kawashima 2009) Liu et al. (2013). Desprez et al. (2013) adopted an analysis-oriented model to simulate four RC columns confined with FRP under cyclic loading. Recently, (Teng et al. 2016) introduced modeling of both the pseudo-static cyclic response and the dynamic response of two FRPconfined RC circular columns, and they adopted the design-oriented model by (Teng et al. 2009) to define the local behavior of concrete under compression stresses. Recently, Fahmy et al. (2017) conducted a comparative study among design-oriented stress-strain models developed for concentrically loaded FRP-confined concrete to predict the response of ten-RC circular columns confined with FRP-jackets and tested under the axial and lateral cyclic loadings. In this study, the design-oriented model developed by Fahmy and Wu (2010) based on the lateral confinement stiffness was modified to include the impact of the lateral confinement provided by the internal steel stirrups. Additional four design-oriented stress-strain models and an analysis-oriented model were studied to determine the response of eight RC circular columns confined with external FRP jackets and tested under a constant axial load and cyclic lateral loadings. A description of the behavior of the concrete core and the concrete cover of FRPjacketed RC columns after rupture of the outer fibers was defined for all examined models. In addition, a simple unloading/reloading path was introduced to predict the cyclic response of the available database. Finally, an eccentricity-based stress-strain model was proposed to accurately

predict the lateral response of FRP-Jacketed RC circular columns. Validity of the model was examined using additional three RC columns from the available database. 2.Simulation of RC columns under axial and lateral loading using Opensees There are some challenges in simulating the behavior of large-scale structures. They are represented by the time cost and the lack of accuracy. Hence, simulation of large-scale structures by fiber section method is gaining increasing interest in the recent years because it offers a lowcost time to simulate large-scale structures, and it can provide accurate results compared to the finite element method (FEM) that is used in the popular software such as ANSYS or ABAQUS. Thus, many software have been recently established based on the fiber section method, one of them is OpenSees software (OpenSees 2000), which is one of the most famous open-source software. 2.1 Database of FRP-retrofitted columns The reinforcement details of the selected columns did not conform to the requirements of the current design codes, so they were retrofitted with external CFRP jackets to enhance their performances under the effect of a constant axial load and quasi-static cyclic lateral loadings. In order to achieve objective evaluation of the nominated design/analysis-oriented models, it was required that design characteristics of the simulated columns cover an acceptable range of variations. Design details of the collected database are listed as follows: a) All columns had a single bending curvature; b) The tested columns had different diameters (D) of 270, 305, 350, and 760 mm with shear spans (H) of 2000, 2000, 800, 1750 mm, respectively; c) The unconfined concrete compressive strength ranged from 18.6 MPa to 49.0 MPa; d) The ratio (N%) between the applied constant axial load (n) and the column axial compressive strength (fcʹo*Ag) was between 5% to 52%; e) CFRP jackets were used with

different ranges of lateral confinement pressure (fl/ fcʹo) from 0.16 to 0.36; and f) These columns have different ratios of lateral steel reinforcement ratio (??st %) and different ratios of the main

steel reinforcement (??s).

The details of the selected column are given in Table 1. For more details of these columns, readers are referred to the original references (Li and Sung 2004; Ozbakkaloglu and Saatcioglu 2006; Gu et al. 2010; Paultre et al. 2016). It is worth indicating that the axial load for column FCS-1 was not declared in its reference, but this value has been confirmed by (Youssf et al. 2015). It is should be noticed that the present investigation is mainly designed to study reinforced concrete columns retrofitted with external FRP jackets: the concrete-filled FRP tube (CFFT) circular columns with high strength concrete (e.g. (Idris and Ozbakkaloglu 2013)) and the CFFT columns with embedded FRP tube in column footing as (Youm et al. 2013)) have not been discussed in the present study. Regarding the failure modes of the selected columns that are shown in Table 1. The cyclic test of the column (FCS-1) tested by Li and Sung (2004) was stopped at a drift of 6.3%, at which point FRP jacket ruptured and simultaneously buckling of the longitudinal bars was observed at the bottom of the column. For columns (CL2 and CL3), CR-3, (S150P10C1, S150P35C1, S75P10C1 and S75P35C1) which were tested by Gu et al. (2010), Ozbakkaloglu and Saatcioglu (2006) and Paultre et al. (2016), respectively, the test was stopped when FRP rupture occurred. In addition, it is worth mentioning that they did not observe any buckling or rupture of the transverse and longitudinal steel reinforcement. It is worth noting that the behavior of S150P10C1, S150P35C1, S75P10C1, and S75P35C1 columns, which were tested by Paultre et al. (2016), were not significantly affected by the PDelta effect because the test configuration had vertical actuators applying a follower force at the

tip of the specimen with a hinge located at the level of the plastic-hinge region. The P-Delta effect was also passed over for FCS-1, CR-3 CL2, CL3, and CF-R3 columns because the axial load was applied by a tension prestressed bar passing through a duct in the centerline of the column. Table.2 summarizes the peak lateral force and the ultimate lateral displacement at FRPrupture for the nominated columns.

2.2 Modeling of FRP-retrofitted RC circular columns Each of the simulated columns was divided into 3-element from the column base to the position of the applied lateral load. In addition, a zero-length element was added to consider the effect of strain penetration, as shown in Fig. 2.b. All elements out of the zero-length element were of the displacement-based beam-column element “dispBeamColumn” type with 5integration points for each element using the fiber section method, as shown in Fig. 2.b. The column cross section was discretized into numerous fibers in its circumferential and radial directions, e.g., the concrete core was divided into 30 fibers in both directions and the concrete cover was discretized into two fibers through its thickness. Fig. 2.c and d show the idealized section used for analysis. In case of the large-scale column (FCS-1), the number of division fibers of the core concrete was increased to 60 in both directions, and the concrete cover was separated into five fibers. Boundary conditions of the numerically tested columns were fixed at the column base and free to rotate and move at the top; see Fig. 2.a. 2.3 Materials modeling 2.3.1 Longitudinal steel The axial tension-compression and cyclic behavior of the longitudinal steel bars were simulated as a uniaxial material by “ReinforcingSteel” (OpenSees 2000), as shown in Fig. 2.e.

The definition of strain penetration (Bond-slip relationship) in OpenSees was incorporated in the analytical model of all columns, and it was represented at the column-footing interface by a zerolength element proposed by (Zhao and Sritharan 2007), as shown in Fig. 2.f. The key parameters of the bond-slip model are the yield and ultimate strengths of the main reinforcing bars (fy and fu, respectively) and the respective loaded-end slips (Sy and Su, respectively). " #

    ! (1) 2 + 1   = 2.54 + 0.34 8437     where db is the rebar diameter, fy is the yield strength of the reinforcement steel and α =0.4. The value of Su was calculated using the relationship Su = (30 ∼ 40) Sy. Buckling of the longitudinal steel bars was not considered in the current study because it did not affect the behavior of the tested columns as discussed previously. 2.3.2 Unconfined concrete and confined concrete with (TSR) For the concrete outside of the FRP-confined zone, the uniaxial material model "Concrete01", which is an identified model in the materials library of the OpenSees platform, was used to represent the behavior of both the concrete core (confined with TSR) and the concrete cover (unconfined concrete), as shown in Fig. 2.d. The ultimate stress-strain values of the concrete located outside of the FRP-confined zone were calculated using the model proposed by Mander et al. (1988). 2.3.3 FRP-confined concrete Lam and Teng (2003) classified the stress-strain models of FRP-confined concrete into design-oriented models (closed-form equations relied directly on the interpretation of experimental results) and analysis-oriented models (curves formed using an incremental

numerical procedure). Some of the analysis-oriented models were proposed by (Fam and Rizkalla 2001; Marques et al. 2004; Jiang and Teng 2007; Teng et al. 2014). One of the available design-oriented models is that proposed by Fahmy and Wu (2010), which was developed based on the fact that “the trend of the stress-strain relationship would be similar among concrete specimens confined with FRP jackets having the same stiffness, however, the ultimate condition of these specimens will be highly dependent on the rupture strain of the confining FRP”. Fahmy and Wu (2010) proved the significant effect for considering this fact in building the mathematical forms describing the stress-strain behavior of FRP-confined concrete, and in another study by Fahmy et al. (2017) they confirmed that design-oriented models that did not consider this fact failed to predict a similar lateral displacement at the same axial strain except for the Fahmy and Wu (2010) model. For this reason, the authors in this study selected the proposed model by Fahmy and Wu (2010), which is represented by Eq.(2):

 = & ' −

& − &) ) ' ) + 0 < Ɛ < Ɛ. 4

 =  + &) Ɛ + Ɛ. < Ɛ < Ɛ

(2)

where fo is the plastic stress at the intercept of the second slope with the stress axis ( fo= fco), fco is the unconfined concrete compressive strength, and the strain at the transition point is εct=2fo/(EcE2). The slope of the second stiffness (E2) has been defined by Eq.(3), where m1 and m2 are 0.5 and 0.83 for fco ≤ 40 MPa respectively, and they are 0.2 and 1.73 for fco > 40 MPa, respectively. In addition, the ultimate conditions of this model are defined by Eq.(4) and Eq. (5) for ultimate stress and strain, respectively, where k1 in Eq.(4) is 4.5fl,f-0.3 for fco ≤ 40 MPa and 3.75fl,f-0.3 for fco > 40 MPa. ′ &) = ) 245.61

01

+ 0.6728&2 

(3)

′ +3  ′ =  " 2,5

' =

′ −  &)

2,5 = &2 '6 =

2 75 &5 2 75 5 '6 = 8 8

(4) (5) (6)

where fl,f is the lateral pressure that provided by FRP jacket, tf is the total thickness of FRP jacket, ff is the tensile stress of FRP, D is the cross section diameter. It should be noticed that this model has been investigated for FRP confinement only and neglected the TSR effect. Therefore, the authors here suggested a modification to this model by providing the TSR effect adopted in the study of Harajli (2006). In addition, the failure behavior of an RC-column confined with FRP after FRP-rupture was not defined in any of the existing design-oriented stress-strain models. Therefore, in this study, the authors suggested defining this stage with a sudden decrease of stress to reach ultimate stress of 0.2 fco and the tangent is horizontal (i.e., tangent slope = 0), as shown in Fig. 2.c. To consider the interaction mechanism of the confinement provided by both the external FRP and the internal TSR, the following modification on Fahmy and Wu (2010) model was considered; see Fig.3. Several design-oriented models were proposed based on the average of the confinement with FRP and TSR (such as (Kawashima et al. 2000; Harajli 2006; Chastre and Silva 2010; Lee et al. 2010) ) to apply a unified model for the concrete core and cover. This assumption may be useful and easier for simulating columns with improper transverse reinforcement details, which do not satisfy the current provisions of seismic design codes. However, for the columns confined internally with higher levels of (TSR), this assumption is incorrect, as it will be shown in the numerical simulation results. As a result, it is better to simulate the concrete section using the

concept of the section division into core and cover. Where the region of concrete cover can be simulated by a stress-strain model confined by FRP only and using the stress-strain model that consider the confinement of both FRP and TSR to represent the concrete core. Fig. 3 shows the proposed modification on the stress-strain model of Fahmy and Wu (2010) for both the concrete core, where the model for the concrete cover is the original model developed by Fahmy and Wu (2010). The stress-strain model for the concrete core is improved based on the lateral confinement of TSR. So, the second stiffens (E2) branch is shifted upward by a stress amount ∆fcc,s, while it is assumed that its slope would not be affected by the existence of TSR. Hence, the overall shape of the stress-strain model has been modified. Firstly, the concrete strength at the interception point of the second branch (second slope) with the stress axis (fo) can be defined using Eq. (7). This lead to a change in the transition strain to become εts=2fo/(Ec-E2). Secondly, Eq. (8) and Eq. (9) shall be used to define the peak stress and the corresponding strain, respectively. After that, the ultimate strain can be calculated according to Eq. (10) that was adopted by Scott et al. (1982) to produce the third branch of the stress-strain relationship with a descending slope. Finally, the sustained stress which was adopted by Kent and Park (1971) is 0.2 fco value. The effect of lateral confinement provided by the TSR depends on the confinement pressure according to Eq.(11), where Kv was declared by Mander et al. (1988). ′ + 4.1 ∗  ∗ ;  =  2:

<= ? <>>

′ + 3  + 4.1 ∗  ∗ ; ′ =  " 2,5 2:

@ = &) '.: = &) '.: − '.  ′ −  ' = &) 0.9C:  'A = ' + 300 2 ∗ <:. ∗  2: = DE  ∗ >

<= ? + @ <>>

(7) (8) (9) (10) (11)

Finally, according to the previous details, the modified stress-strain model have been written in C++ language using the Dynamic-Link-Library (DLL) files and implemented into OpenSees software (Version 2.5). 2.3.4 FRP-confined concrete in cyclic compression Regarding the hysteretic rules for the selected model of FRP-confined concrete that was not defined by the original authors, a simplified linear hysteretic unloading/reloading path was suggested based on the hysteresis criteria proposed by Karsan and Jirsa (1969), as shown in Fig.3. It is worth noting that this criterion is already implemented in OpenSees platform by several concrete models. 3.The predicted cyclic response of the tested columns After implementation of the suggested models in OpenSees software and building the simulation of eight circular columns into it, using the tools command language (TCL), the simulation behavior of the selected columns under the effect of the combined actions of the cyclic lateral loads and a constant axial load will be discussed in the current section. The numerical cyclic responses of the columns (FCS-1, S150P10C1 and S75P35C1) are presented in Figs. 4.d-f compared with the experimental responses superimposed on the same figures. Figs.4.a-c show the predicted axial stress-strain curves for the confined concrete with FRP and for concrete confined with both internal steel stirrups and an external FRP jacket for these columns. As it is previously mentioned, the concrete cross-section of the examined columns has been defined using two materials one for the concrete core (confined by TSR and FRP) and the other for the concrete cover (confined by FRP only). It is clear from Figs.4.b-c that there is a considerable difference caused by the high confinement level (ρst = 1.12% and ρst = 2.24%) provided by TSR for the S150P10C1 and S75P35C1 columns, respectively. On the other

hand, in Fig.4.a this difference is slight for the FCS-1 column, in which ρst = 0.135% representing a low level of TSR confinement compared to that of the other columns. This is an indication of the importance of representing the concrete core and concrete cover with proper stress-strain relationships that consider the impact of the interaction mechanisms of the lateral reinforcements used. In Figs. 4.d-f, the predicted elastic stiffness has a good agreement with the experimental results for S150P10C1 and S75P35C1 columns. However, the simulation result of the FCS-1 column shows an overestimation in both the elastic stiffness and unloading/reloading paths, which could be attributed to the deficient definition of the strain penetration model for largescale columns (Fahmy et al. (2017)). In the inelastic stage of loading, the strain in the outer FRP increases until reaching the ultimate strain capacity, which is followed by a decrease in the column lateral load capacity. That is, the accuracy of the predicted ultimate lateral displacement of the simulated columns depends mainly on the definition of the concrete ultimate conditions (the ultimate strength of confined concrete and the corresponding ultimate strain). Fig. 4 shows that the modified stress-strain model predicts lower ultimate lateral displacements for the simulated columns with reference to the experimental values. On the other hand, the predicted lateral load capacities of the columns have a respectable matching with the experimental values except for the last column S75P35C1. It is worth noting that the definition of the FRP-failure is considered when the most outer fiber element reaches a strain value corresponding to the peak stress in the second interpolation point of the first element from the column base. Additional five columns were simulated in the same way of the previous three columns. The ratios between the predicted ultimate displacements to the experimental values are plotted in Fig. 5; results are in ascending order according to the axial load ratio. Obviously, this figure shows

that the predicted ultimate lateral displacement for all simulated columns have values lower than the experimental test results with an average of 47%. This means that the suggested model failed in predicting the ultimate lateral displacement of all examined columns. The first impression from this result is that the adopted model failed to accurately define the ultimate lateral displacement capacity of the simulated columns and in turn, the predicted failure modes of those columns are not matching well with the actual test results. These results quickly invoke our attention toward examining the accuracy of using an analysis-oriented model and/or other available design-oriented models. 4.Analysis versus design stress-strain models of FRP-confined concrete in simulating the tested columns OpenSees Software has a large material library; two of them are “FRPConfinedConcrete” and “ConfinedConcrete01”, which can be used to simulate the behavior of FRP-confined concrete sections. These models have been investigated based on the analysis-oriented model. However, “FRPConfinedConcrete” model was not stable for the analysis of RC columns until the pre-last version of OpenSees (2.4.6). The second one has no specific definition for the ultimate axial compressive strain of FRP-confined concrete at FRP-rupture. The analysisoriented “FRPConfinedConcrete” model has been adopted in this study to simulate the behavior of FRP-confined concrete; this model was originally proposed by (Megalooikonomou et al. 2012). In addition, for comparison, other four design-oriented models proposed by (Kawashima et al. 2000; Harajli 2006; Chastre and Silva 2010; Lee et al. 2010) have been nominated to simulate the behavior of the available database; all these models consider the confinement provided by both the external FRP and the internal TSR, mathematical forms of these models have been presented in Appendix (A). It is worth mentioning that the examined models have no

clear description of the stress-strain behavior after rupture of the outer fibers, so a sudden drop in the strength was assumed at the ultimate axial strain and the sustained stress was assumed to be equal to 0.2 fco. In addition, the definition of the FRP-failure is considered when the outer fiber element has a strain value corresponding to the peak stress in the second interpolation point on the first element from the column base. Using the design-oriented models of (Kawashima et al. 2000; Harajli 2006; Chastre and Silva 2010; Lee et al. 2010) and the analysis-oriented model “FRPConfinedConcrete”, the stress-strain curves identifying the behavior of the concrete core and the concrete cover of three columns (FCS-1, S150P10C1 and S75P35C1) and the predicted lateral cyclic load versus the corresponding displacement are shown in Figs. 6 and 7, respectively. From Fig.6, there is a minor difference in the stress-strain curves of the concrete core and the concrete cover for FCS-1 column using the selected models except that predicted by Lee et al. (2010) model that exhibited an unexpected increase in the strain capacity of the concrete core. In contrast, for S150P10C1 and S75P35C1 columns, the stress-strain curves of the concrete core and the concrete cover are obviously different. This confirms that it is required to accurately represent the behavior of the concrete core and the concrete cover in simulating RC columns confined with external FRP. In addition, there is a big difference between the predicted stress-strain curves using the examined models, where the lowest ultimate concrete strength values for the FCS-1, S150P10C1 and S75P35C1 columns are about 23.4, 34.8 and 34.8 MPa, respectively, which was predicted by (Kawashima et al. 2000) model; and the largest values are 37.0, 56.1 and 56.1 MPa, respectively. Moreover, the chosen models predicted ultimate strain values of (0.005 to 0.01mm/mm) for the S150P10C1 and S75P35C1 columns, and ultimate strain values of (0.007 to 0.017mm/mm) for the FCS-1 column. Although the model of Lee et al. (2010) was proposed based on a high level

of the internal confinement provided by TSR, it predicts the smallest ultimate strain values (about 0.005 and 0.007mm/mm) as shown in Figs.6.a, b and c. All of these differences are dependent on the mathematical forms of the examined models. Figs.7.a-f show the predicted lateral cyclic load-displacement relationships for the FCS-1, S150P10C1 and S75P35C1 columns using design/analysis-oriented models. In addition, the column displacement corresponding to the onset of FRP rupture according to the numerical calculation of each of the examined models is plotted on the same figure. Using any of the nominated stress-strain models, there is a slight difference in the predicted lateral load capacity for FCS-1 and S150P10C1 columns compared to the experimental value; however, there is an underestimation in the lateral load capacity of S75P35C1 column. Moreover, due to the early estimated FRP-rupture, it can be stated that the ultimate displacement of the S150P10C1 and S75P35C1 columns are inaccurately predicted. For the large-scale column FCS1, all models predict ultimate lateral displacement less than the experimental value except Harajli (2006) model that has the largest concrete compressive strength and the corresponding axial strain. All examined columns were simulated using the selected five models, and the predicted ultimate lateral displacements to the experimental values have been drawn in Fig.8; presented results are in an ascending order based on the column axial load ratio. From this figure, Lee et al. (2010) model predicted the lowest values for all columns, and Harajli (2006) model determined the highest values, where the average percentages are 26.0% and 56.0%, respectively. Generally, all models inaccurately estimate the onset of the FRP rupture: early FRP-rupture compared to the experimentally tested values where the average percentage is about 42% for the examined columns using all stress-strain models.

It worth mentioning that many nonconvergences occurred during the simulation process using the analysis-orientated models “FRPConfinedConcrete” especially after the occurrence of FRP rupture at the outer edge (it cannot complete the simulation process), see Fig7.c, which required changing the initial increment displacement and repeating the solution again for each cycle. This is due to the many necessary trails to get the stability of determining the confinement pressure between the lateral and axial strains. Moreover, it costs a lot of time compared to the designoriented model that uses a direct value for the axial strain. Furthermore, this model did not exhibit a better accuracy than the other models (design-oriented models). 5. Characteristic stress/strain values affecting the prediction of the column peak strength and ultimate displacement The studied analysis-oriented model and two of the design-oriented models (proposed by Lee et al. (2010) and Chastre and Silva (2010)) adopted a strain reduction factor (Kε) for the ultimate hoop strain of the FRP composites, where this factor affects the definition of both the ultimate axial stress and the corresponding strain of FRP-confined concrete. However, Teng et al. (2016) and Ozbakkaloglu and Saatcioglu (2006) concluded that the value of this factor should be relaxed to a higher value than that defined by Lam and Teng (2003), in addition, they endorsed that this factor needs further study for columns under cyclic lateral loading. It is worth mentioning that (Lim and Ozbakkaloglu 2014a) proposed a model that reliably predict the hoops rupture strains in FRP-confined concrete under axial compression; however, such a model is not available to predict these strains for FRP-confined concrete subjected to an eccentric loading. Here, therefore, the responses of all columns were re-simulated assuming that Kε = 1 in these models and the results (the determined ultimate displacement) are presented in Figs. 9.a, b and c. From these figures, the average percentages of the predicted ultimate lateral displacement to the

corresponding experimental value increased to become 85.0%, 36.0%, and 75.0% for these models, respectively. It is evident that the reduction factor Kε has a clear effect on the prediction of lateral displacement and the corresponding strength as result of improving the ultimate stress/strain of the examined models especially the analysis-oriented model. There is a clear improvement in the prediction of column ultimate displacement using the analysis model to reach an average close to 79% for most of the examined columns; however, it overestimates the ultimate displacement of FCS-1 column. On the other hand, the Chastre and Silva (2010) model presents an overestimation of the ultimate lateral displacement for CL3 column that refers to the largest value of the lateral confinement ratio (fl/fco) of this column. Finally, although the effect of that reduction factor was omitted from the stress-strain models, it is clear that the three models could not accurately predict the column ultimate displacement. Using the models by (Kawashima et al. 2000; Harajli 2006; Chastre and Silva 2010; Fahmy and Wu 2010) and “FRPConfinedConcrete” model), Fig. 10 shows the relationship between the ratio of the predicted ultimate displacement to the experimental value and the column axial load ratio for the eight simulated columns. From the numerical results of these columns, it can be concluded that the column axial load ratio has a significant effect on the prediction of the column ultimate displacement for most selected models. That is, the increase in the axial load ratio is associated with a decrease in the predicted column ultimate displacement. Apparently from this figure, the “FRPConfinedConcrete” model has the largest descending trend with the increase of the axial load ratio, however, the model by (Chastre and Silva 2010) has the lowest descending slope. The authors attribute this behavior to the expected difference between the stress-strain behaviors of FRP-confined concrete columns under the effect of concentric loading and that subjected to axial and lateral loadings. Wu and Jiang (2013) concluded that when circular

concrete specimens externally confined with FRP and subjected to eccentric loadings shall have a higher axial strength than that under the effect of a concentric loading. A similar conclusion was recently stated by Fahmy and Farghal (2016) for rectangular columns subjected to eccentric loadings. Both studies proved that behavior of the FRP-confined concrete under eccentric loadings is different from the plain concrete under the same loading condition. It is evident from Fig. 7 that not only the predicted column ultimate displacement is inversely proportioned to the axial load applied, but also the predicted column lateral load capacity decreases with the increase in the axial load ratio. Hence, a clear answer to the following question is required “which characteristic value of the FRP-confined concrete (ultimate stress or ultimate strain) does affect the column peak strength? In other words, the impact of the ultimate conditions of the stress-strain relationship of FRP-confined concrete (ultimate strength and strain) on the development of the theoretical moment interaction diagram of RC columns confined with FRP and subjected to different axial load ratio should be quantified. Hence, two separate cases were considered to develop the theoretical moment interaction diagram using OpenSees software for four tested columns (S150P10C1, S150P35C1, CL3, and RC-3). In the first case, it was assumed that the ultimate axial strain increases while the corresponding strength (fcc) is constant (see Fig.11.a). The second case examined the increase in the ultimate strength by an amount (∆fcc), (where ∆fcc = 0.5 fco, 1.0 fco or 1.5 fco), meanwhile the corresponding strain was constant; as shown in Fig.11.b. The interaction diagrams of the four columns are shown in Figs.11 c, e and g for case 1 and Figs.11 d, f and h for case 2. From Figs.11.c-to-h, the ultimate stress has a large effect on the predicted column strength rather than the ultimate strain, especially for the columns subjected to a high axial load level. These results agreed well with the recommendation by Wu and Jiang

(2013) that the ultimate strain of eccentrically loaded concrete could be assumed constant under the effect of any eccentric loading meanwhile the axial strength should be higher than that of a concentrically loaded concrete. The previous discussion indicates that the current understanding of using an axial stress-strain model for modeling the behavior of FRP-jacketed RC columns subjected to combined axial compression and bending loadings is inadequate, where the examined design/analysis-oriented models failed in predicting the failure modes of the studied columns. A special consideration should be made to reflect the effect of the combined axial and bending loadings on the definition of the stress-strain behavior of FRP-confined concrete. Consequently, a new model is proposed to consider the previously discussed drawbacks and to predict the load-displacement of RC columns confined with FRP and subjected to axial and lateral loadings with an acceptable level of accuracy. 6. Proposed Stress–Strain Model The authors adopted additional modification to Fahmy and Wu (2010) model by increasing the predicted ultimate strength of FRP-confined concrete when subjected to both axial and bending loadings. The evaluated ultimate strain of this model was not dependent on any reduction factor (i.e. kε), so the proposed modification has been applied to the peak strength for the concrete core section and the concrete cover externally confined with FRP at the ultimate strain (εcc); as shown Fig.12. This can be achieved by an increase in the second slope E2, and this increase shall be dependent on the eccentricity ratio (e/r, where e is equal to (M/Pn) and r is the radius of the cross section) and the confinement level fl/fco. The increase in the slope E2 is dependent on the difference between fo,and fcc and the ultimate strain. Assuming that the ultimate strain and fo, are the same for the concentrically and eccentrically loaded concrete, so the

change in the slope E2 relies on the increase in the peak strength of the confined concrete, as shown in Fig.12. In order to obtain the required increase in the ultimate strength (∆fcc), the examined columns were re-simulated assuming new values for the concrete ultimate strength of the confined concrete to simultaneously reach the experimental ultimate displacement and rupture of the outer fibers. Therefore, several trails were made to investigate the increase in ultimate stress corresponding to the experimental value at FRP-rupture for each of the examined columns. Wu and Jiang (2013) proved that for each eccentricity there is a significant difference in the second branch of the stress-strain behavior of FRP-confined concrete due to the eccentric loading (i.e., for each eccentricity a new stress-strain curve is required). For columns under the effect of a constant axial load and reversed cyclic loadings, after yielding, the rate of an increase in the lateral load capacity could be assumed negligible and in turn, the ratio between the applied bending and axial load is almost fixed. That is, one eccentricity-based stress-strain curve could be assumed to accurately simulate the column response to axial and lateral loads. Plotting a relationship between (∆fcc/ fco) and the eccentricity (e/r) in Fig. 13 for all the simulated columns exhibits a good agreement with the power equation. Whereas, the coefficient of determination (R2) was 0.983. The parameter e/r = (H.l/Pn)/r; where H is the maximum lateral load, l is the shear span, and Pn is the applied axial load. Moreover, the relationship between (∆fcc/ fco) and the percentage of axial load N= (Pn /Ag.fco) is shown in Fig. 13.b. This relationship shows a good agreement with the power trend. Furthermore, as the axial load ratio increases, the value of (∆fcc/ fco) decreases. This result supports the recommendation by Wu and Jiang (2013) regarding the impact of an increase in the axial load on the stress-strain behavior of the FRPconfined concrete.

To ensure a safety margin in the definition of the column response to the combined axial and lateral loads, the authors recommend using Eq.(12), which provides a conservative model, in which constants were defined based on the trial-and-error method. Finally, Fig.13.c shows the ratio between the predicted values of (∆fcc/fco) for all selected columns to the values defined based on the experimental results of the simulated columns. From the figure, all values are less than 1.0, and the lowest value is 0.86 for S75P35C1 column.

∆ H K.L = 0.44 G J  I

(12)

7.Validation of the proposed model Using the proposed model, Fig.14 presents the predicted lateral load-displacement relationships for three columns (FCS-1, S150P10C1, and S75P35C1). These columns were tested under the effect of different percentages of the axial load (5%, 10%, and 35% respectively), and they had low to high levels of TSR ratios (0.135%, 1.12% and 2.23% respectively). In addition, ratios of the external lateral FRP-confinement were 0.24%, 0.16%, and 0.16%, respectively. Figs.14.a, b, and c show that there is a good agreement between the predicted lateral response and failure mode and the corresponding experimental results of the simulated columns. Fig. 14.c shows that the decrease in the strength after rupture of the outer fibers is gradual; however, this result is not matching well with what was recorded during the experimental test, which showed several sudden drops in the lateral load capacity. This result can be attributed to the accuracy of the assumed behavior for the confined concrete with both TSR and FRP after rupture of the outer fibers: that is the slope of the third part of the stress-strain relationship still needs further study. The proposed eccentricity-based model exhibits a good ability in predicting the lateral displacement compared to the experimentally tested columns; as shown in (Figs.15.a). In which,

the average ratio between the predicted ultimate displacements to the corresponding experimental values is almost 0.94. Compared to the other studied models (design/analysis-oriented models) based on the Average Absolute Error (AAE) criterion, (see Eq.(13)), the eccentricity proposed model presents a reliable accuracy with AAE value (0.06) while the least value of AAE is 0.44 by Harajli (2006) model and the maximum value of AAE is 0.733 by Lee et al. (2010) as shown in Fig.15.b. ∑Z U[\P

QRST,U VQTWXS,U

P

(13) AAE = ] Where δPre,i is the predicted ultimate displacement value when one of the examined models was QTWXS,Y

used in the numerical simulation, δexpr,i is the experimental ultimate displacement value, and n is the number of columns of the available database. The behavior of the proposed model in predicting the value of lateral load capacity to the experimental results of all the examined columns has been drawn in Fig.16. The ratio of the predicted lateral load capacities to the experimental results shows a slight overestimation for some of the selected columns with a maximum value of 1.06 for the CR-3 column, and the least underestimated value is 0.92 for the S150P35C1 column. That is, the proposed model showed an acceptable accuracy in predicting of the lateral load capacity of the simulated columns. Further validation of the proposed model was achieved by simulating additional three columns that did not show rupture of the outer fibers during testing. The main characteristics of the columns are given in Table.3; for more details, readers are referred to the original studies (Gallardo-Zafra and Kawashima (2009) and Gallardo-Zafra and Kawashima (2009)). Regarding the analysis of the experimental data of the two columns (A3 and B3) by GallardoZafra and Kawashima (2009), they observed that the failure mode was not characterized by FRP

sheet rupture until the end of the experiment. Similarly, Saadatmanesh et al. (1996) reported that they did not observe any failure for the column C5 up to the end of loading. The predicted cyclic responses for the columns (A3, B3, and C5) are shown in Figs.17. a, b and c, respectively, using the proposed model and the model by Lee et al. (2010). In this figure, the performance of the proposed model exhibits a good ability in predicting the performance of the three columns, because until the end of loading and similar to the test results there was no indication of FRP rupture. In addition, if the original authors loaded these columns further, the ductility and the strength might be higher than the reported values as shown in the simulation results. On the other hand, compared to the experimental results, the model of Lee et al. (2010) predicts a very early rupture of the outer FRP jackets of the columns A3 and B3. In conclusion, the proposed eccentricity-based model correctly predicts the response of FRPconfined RC circular columns under a constant axial load and reversed cyclic lateral loadings. Where it presents a reliable prediction for the failure mode, ultimate displacement, and the lateral load capacity with acceptable accuracy compared to the other studied design/analysis-oriented models for all tested columns (within the selected range of concrete compressive strength (fco) of the studied columns). Furthermore, this model has additional advantages that addressed some issues as follows: (a) The proposed model fulfills the fact that ”the stress-strain curves that are provided by different FRP materials based on the same lateral stiffness are typical but they have different ultimate conditions” that was proved by Fahmy and Wu (2010) and Fahmy et al. (2017). (b) The proposed model is independent of the effect of the strain reduction factor on the ultimate stress/strain, which provides an advantage of independency of FRP type. However, validation of the model against a large database is required.

Finally, the authors acknowledge that the proposed modeling for FRP-confined RC circular columns still needs further improvement to consider the impact of the expected local buckling of the column longitudinal reinforcement (either before or after FRP rupture), failure of the internal steel stirrups, and damage accumulation effect which would require a better modeling for the unloading/reloading paths through systematic experimental studies. For sure, a serious future study should be also considered to examine the applicability of the proposed stress-strain model in this study [or rather development of new suitable models] for prediction of the seismic response of new earthquake resistant columns, such as high-strength concrete-filled FRP tube columns. 8.Conclusion The first part of this study presents a comparative study among five design-oriented models and an analysis-oriented model (nominated from literature) to predict the response of FRP-confined RC circular columns under the combined axial and lateral loadings. The main drawn conclusions from this part are (i) the existing models underestimated the lateral displacement of the simulated columns due to the early determined rupture of the outer fibers compared to the experimental results; and (ii) analysis-oriented model does not exhibit accuracy better than the design-oriented models in simulating the performance of RC columns confined with FRP and subjected to axial and lateral loadings. Careful consideration for potentially influential parameters showed that axial load ratio had a significant impact on the predicted column ultimate displacement: the higher the axial load ratio, the larger the error in the predicted column ultimate displacement. Hence, an eccentricity-based stress-strain model was developed to represent the local behavior of FRP-confined concrete in compression due to the combined effect of a constant axial load and cyclic lateral loadings. The

proposed model was originally developed by Fahmy and Wu (2010) for monotonic axial compression and has been modified in this study to consider the confinement provided by the internal steel stirrups. In addition, simplified loading-unloading paths were adopted to predict the cyclic response of FRP-confined RC circular columns. In light of the presented numerical results, the following conclusions can be outlined: 1. Accuracy in predicting the column ultimate lateral displacement and lateral load capacity is dependent on the concrete axial compressive strength rather than the ultimate strain. 2. Peak compressive strength of FRP-confined concrete under axial and bending loadings should be higher than that under axial loading and it increases with the increase in the ratio (e/r) (eccentricity to column’s radius). 3. One eccentricity-based stress-strain curve could be assumed to accurately simulate the column response to axial and lateral loads. 4. The proposed eccentricity-based modification into the adopted stress-train model provides a reliable trend with the available experimental tested FRP-RC columns. Moreover, the proposed model satisfies the fact that “different FRP materials providing the same lateral stiffness shall have the same stress-strain curve but with different ultimate strain capacity based on the type of the FRP used”. 5. Definition of the stress-strain curve of the FRP-confined concrete core after reaching the peak strength has a considerable effect on the column lateral load capacity after FRP rupture. The definition given in the current study could be safely used until a much better future definition is introduced.

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List of figures

Fig. 1. a) RC-bridge with FRP-jacket columns; b) Experimental evaluation for the behavior of FRP-confined concrete under axial loads; and c) FRP-RC circular under axial and lateral loads. Fig. 2. Numerical models for FRP-jacketed RC columns. Fig. 3. Stress-strain curves for the cover and core concrete section. Fig. 4. Behavior of the suggested model in predicting the axial stress-strain curves (a, b and c) and the predicted lateral cyclic load-displacement relationships (d, e and f) for FCS-1, S150P10C1 and S75P35C1 columns, respectively. Fig. 5. Behavior of the suggested model in predicting the lateral displacement for all columns. Fig. 6. The predicted axial stress-strain curve for FCS-1, S150P10C1 and S75P35C1 columns using the design-oriented models (a, b and c), respectively, and the analysis-oriented model (d, e and f), respectively. Fig. 7. The cyclic response of the predicted load-displacement relationships with the experimental results for FCS-1, S150P10C1 and S75P35C1 columns, using the design-oriented models (a, b and c), respectively, and the analysis-oriented model (d, e and f), respectively. Fig. 8. The predicted ultimate lateral displacements compared to the experimental values of the tested columns. Fig. 9. The predicted ultimate displacements compared to the experimental values of the tested columns after removing the strain reduction factor. Fig. 10. The ratio of the predicted column ultimate displacement to the experimental value versus the column axial load ratio. Fig. 11. The theoretical interaction diagram (M-P) and the experimental values for two different cases: (a) case1 (increasing the ultimate strain) and (b) case2 (increasing the ultimate stress) for

four columns (S150P10C1 and S150P35C1 (c and d) respectively), (CL3 (e and f), respectively) and (RC-3 (g and h) respectively). Fig. 12. The proposed stress-strain curves for concrete core and concrete cover under the action of axial and bending loadings. Fig. 13. The relationship between the increase in the axial strength to the ultimate strength (∆fcc/fcc) with the eccentricity ratio (e/r) and axial load ratio (N%) (a and b), respectively. c) The predicted (∆fcc/fcc) to the required (∆fcc/fcc) for all examined columns. Fig. 14. The cyclic responses of the simulated columns (FCS-1, S150P10C1, and S75P35C1) (a, b and c) respectively, using the proposed eccentricity-based design-oriented model. Fig. 15. a) The predicted ultimate lateral displacement compared to the experimental value for the examined columns; b) Comparison between the proposed eccentricity-based model and the others nominated models based on the AAE criterion. Fig. 16. The predicted ultimate lateral load capacity compared to the experimental value for the examined columns. Fig. 17. The performance of the proposed eccentricity-based model and the model by Lee et al. (2010) in predicting the lateral load/displacement relationship for three columns (A3, B3, and C5) (a, b and c), respectively, that did not recognize FRP-rupture up to the end of loading. List of tables Table 1 Properties of the experimentally collected columns. Table 2. Concrete dimensions and mechanical characteristics of the materials used for the selected columns. Table 3 Maximum lateral load and ultimate displacement for the selected columns.

 

Unretrofitted Column

FRP- jacketed column

(a) (c) FRP-confined concrete (b) Compression region

?

Compression region

Is it possible to use this stressstrain curve to simulate FRP-RC columns under axial and lateral loads?

 

Fig. 1. a) RC-bridge with FRP-jacket columns; b) Experimental evaluation for the behavior of FRP-confined concrete under axial loads; and c) FRP-RC circular under axial and lateral loads.                        

D

l 

Concrete core

FRP sheet

N

L-3D

Concrete core Section (B-B)

Section (A-A)

εcc) L

fc

2D

2D

L

L

B

DispBeamCo lumn element

Lateral load B

p

L-3D

Axial load

fc

For core

εcc) For core

For cover

(a) Geometric shape of the selected columns

 

fs fu Envelope curve fy Cyclic curve

D

fc = 0.2fco

Node i

Zero

Zero leng. element

D

A

0

A

Node j

For cover

Ec

Ec

Eu

εc

(c) Concrete material for

(b) Simulation of the tested columns

section (A-A) Esh fs

u

)

Es εy εsh

εu εs

)

y

K S

(e) Steel material for longitudinal steel bars

(f) Steel bars material for zerolength element

Fig. 2. Numerical models for FRP-jacketed RC columns.

Eu εc

(d) Concrete material for section (B-B)

For concrete core (FRP+TSR)

fc

For concrete cover (FRP only) Unloading/reloading path E2 (fct,s , ɛct,s)

fo fco

Δfcc,s

E2

(fct,ɛct)

(fcc , ɛcc)  

(fcc , ɛcc)

Δfcc Δfcc,s

ΔƐct,s

ΔƐct,s 0.2fco

0.2fco Eun

Ec Ɛct

Ɛct,s

ɛcu

Fig. 3. Stress-strain curves for the cover and core concrete section.

ɛc

40.0

1000

Fahmy and Wu (2010) (cover) Fahmy and Wu (2010) (core)

(a)

Lateral load (kN)

Stress (MPa)

30.0 0.013, 31.5

20.0

(d)

10.0

Exper.FCS-1 Fahmy and Wu (2010) FRP rupture (numerically)

-600

0.005

0.010 0.015 Strain (mm/mm)

0.020

0.025

0

25

50 75 100 Lateral displacement (mm)

125

80

60.0 Lateral load (kN)

0.010, 48.8

30.0 20.0 10.0 0.0 0.000

80.0 70.0 (c) 60.0 50.0 40.0 30.0 20.0 10.0 0.0 0.000

(e)

60

Fahmy and Wu (2010) (core) Fahmy and Wu (2010) (cover)

40 20 0

Exper. S150P10C1 Fahmy and Wu (2010) FRP rupture (numerically)

-20 -40

0.005

0.010 0.015 0.020 Strain (mm/mm)

0.025

Fahmy and Wu (2010) (cover) Fahmy and Wu (2010) (core)

0.010, 48.8

0

100

50

0.020 0.030 Strain (mm/mm)

0.040

0.050

300

60 40 20 0

Exper. S75P35C1 Fahmy and Wu (2010) FRP rupture (numericaly)

-20

0.010

100 150 200 250 Lateral displacement (mm)

(f)

80 Lateral load (kN)

Stress (MPa)

(b)

40.0

Stress (MPa)

200

-200

0.0 0.000

50.0

600

-40 0

50

100 150 200 250 Lateral displacement (mm)

300

Fig. 4. Behavior of the suggested model in predicting the axial stress-strain curves (a, b and c) respectively, and the predicted lateral cyclic load-displacement relationships (d, e and f) for FCS1, S150P10C1 and S75P35C1 columns, respectively.

1.2

Fahmy and Wu (2010) Disp.(Pre.)/Disp. (Expr.)

1.0 N=5%

N=10%

N=35%

N=52%

N=36%

0.8 0.6 0.4 0.2 0.0 FCS-1

S150P10C1 S75P10C1 S150P35C1 S75P35C1

CL2

CL3

CR-3

Fig. 5. Behavior of the suggested model in predicting the lateral displacement for all columns.

20

0.008, 30.6

0.012, 23.4

10

0 0.000

70 60 Stress (Mpa)

50

0.010 0.015 Strain (mm/mm)

Design-oriented models Column S150P10C1

0.020

0.009, 56.1

0.025

0.008, 43.8

40 30 20

80

0.005

0.010 0.015 0.020 Strain (mm/mm)

70 60

0.005, 44.6

(c) 0.009, 56.1

30 10 0 0.000

0.008, 43.8 0.008, 34.8 0.005

10

FRPConfinedConcrete (cover) FRPConfinedConcrete (core) 0.005

0.010 0.015 Strain (mm/mm)

0.020

0.025

0.010 0.015 Strain (mm/mm)

0.020

Analysis-oriented model Column S150P10C1

0.025

(e)

50 40 0.006, 44.6

30 20 0 0.000

0.025

40 20

0.010, 25.7

FRPConfinedConcrete (core) FRPConfinedConcrete (cover)

10

Design-oriented models Column S75P35C1

50

20

60

0.008, 34.8

0 0.000

30

70

(b)

(d)

Analysis-oriented model Column FCS-1

0 0.000

0.005, 44.6

10

Stress (MPa)

0.005

Chastre and Silva (2010) (core) Chastre and Silva (2010) (cover) Lee et al. (2010) (core) Lee et al. (2010) (cover) Harajli (2006) (core) Harajli (2006) (cover) Kawashima et al. (2000) (core) Kawashima et al. (2000) (cover)

Stress (MPa)

0.007, 27.7

0.017, 37.0

40

(a)

Stress (MPa)

Stress (Mpa)

30

Design-oriented models Column FCS-1

Stress (Mpa)

40

0.005

0.010 0.015 Strain (mm/mm)

0.020

0.025

80 Analysis-oriented model 70 Column S75P35C1 (f) 60 50 40 0.006, 44.6 30 20 FRPconfinedconcrete (cover) 10 FRPconfinedconcrete (core) 0 0.000 0.005 0.010 0.015 0.020 0.025 Strain (mm/mm)

Fig. 6. The predicted axial stress-strain curve for FCS-1, S150P10C1 and S75P35C1 columns, using the design-oriented models (a, b and c), respectively, and the analysis-oriented model (d, e and f), respectively.

1000

1000

Lateral load (kN)

600

(a)

200

Exper.FCS-1 Chastre and Silva (2010) Lee et al. (2010) Harajli (2006) Kawashima et al. (2000) FRP rupture (numerically)

-200

-600 0 80

25

50 75 100 Lateral displacement (mm)

125

Lateral load (kN)

Design-oriented models

600 200 Exper.FCS-1 FRPConfinedConcrete FRP rupture (numerically)

-600 0

80

Design-oriented models

25

50 75 100 Lateral displacement (mm)

125

Analysis-oriented model

40

(b)

20

Exper. S150P10C1 Chastre and Silva (2010) Lee et al. (2010) Harajli (2006) Kawashima et al. (2000) FRP rupture (numerically)

0

-20 -40 0

50

100.0

100 150 200 250 Lateral displacement (mm)

300

Lateral load (kN)

60 40

0

(c)

80.0

Exper. S150P10C1 FRPConfinedConcrete FRP rupture (numerically)

-20 -40 0 100

Design-oriented models

(e)

20

80 Lateral load (kN)

Lateral load (kN)

(d)

-200

60

Lateral load (kN)

Analysis-oriented model

50

100 150 200 250 Lateral displacement (mm)

Analysis-oriented model

300

(f)

60

60.0

40

40.0

20

20.0

Exper. S75P35C1 Chastre and Silva (2010) Lee et al. (2010) Harajli (2006) Kawashima et al. (2000) FRP rupture (numerically)

0.0

-20.0 -40.0 0

50

100 150 200 250 Lateral displacement (mm)

300

0

Exper. S75P35C1 FRPconfinedconcrete FRP rupture (numericaly)

-20 -40 0

50

100 150 200 250 Lateral displacement (mm)

300

Fig. 7. The cyclic response of the predicted load-displacement relationships with the experimental results for FCS-1, S150P10C1 and S75P35C1 columns, using the examined designoriented models (a, b and c), respectively, and the analysis-oriented model (d, e and f), respectively.

1.2

Disp.(Pre.)/Disp. (Expr.)

1.0

Lee et al. (2010) Fahmy and Wu (2010) N=5%

N=10%

Chastre and Silva (2010) Harajli (2006) N=35%

Kawashima et al. (2000) FRPConfinedConcrete N=52%

N=36%

0.8 0.6 0.4 0.2 0.0 FCS-1

S150P10C1 S75P10C1 S150P35C1 S75P35C1

CL2

CL3

CR-3

Fig. 8. The predicted ultimate lateral displacement compared to the experimental values of tested columns.

1.4 Chastre and Silva (2010)

Chastre and Silva (2010) (kε=1)

(a)

Disp.(Pre.)/Disp. (Expr.)

1.2 1.0 0.8

N=5%

N=10%

N=35%

N=36%

N=52%

0.6 0.4 0.2 0.0 FCS-1

S150P10C1 S75P10C1 S150P35C1 S75P35C1

CL2

CL3

CR-3

1.4 Lee et al. (2010)

Disp.(Pre.)/Disp. (Expr.)

1.2

(b)

Lee et al. (2010) (kε=1)

1.0 0.8 0.6

N=5%

N=10%

N=35%

N=36%

N=52%

0.4 0.2 0.0 FCS-1

S150P10C1 S75P10C1 S150P35C1 S75P35C1

CL2

CL3

CR-3

1.4 FRPConfinedConcrete

Disp.(Pre.)/Disp. (Expr.)

1.2 1.0 0.8

N=5%

N=10%

(c)

FRPConfinedConcrete (kε=1)

N=35%

N=36%

N=52%

0.6 0.4 0.2 0.0 FCS-1

S150P10C1 S75P10C1 S150P35C1 S75P35C1

CL2

CL3

CR-3

Fig. 9. The predicted ultimate displacement compared to the experimental values of the tested columns after removing the strain reduction factor.

1.0

FRPConfinedConcrete Fahmy and Wu (2010) Chastre and Silva (2010) Harajli (2006) Kawashima et al. (2000)

disp.(pre.)/disp. (expr.)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0

10

20

30 40 Axial Load (N%)

50

60

Fig. 10. The ratio of the predicted column ultimate displacement to the experimental value versus the column axial load ratio.

(b) Case 2

(a) Case 1

The varying of the stress-strain curve for concrete cover

8000

6000 5000 4000 3000

(c)

2000

6000 5000 4000 3000

(d)

2000 1000

1000

0

0 50

14000 12000 10000

100 150 200 Moment (kN m) Expr. CL3 Fahmy and Wu (2010) Fahmy and Wu (2010) (Δεcc=1 eco) Fahmy and Wu (2010) (Δεcc=3 eco) Fahmy and Wu (2010) ( Δεcc=5 eco)

8000 6000

(e)

4000

0

50

100 150 Moment (kN m)

200

250

Expr. CL3 Fahmy and Wu (2010) Fahmy and Wu (2010) (Δfcc=0.5 fco) Fahmy and Wu (2010) (Δfcc=1.0 fco) Fahmy and Wu (2010) (Δfcc=1.5 fco)

14000 12000 Axial load (kN)

0

Axial load (kN)

Expr.( S150P10C1 &S150P35C1 ) Fahmy and Wu (2010) Fahmy and Wu (2010) (Δfcc=0.5 fco) Fahmy and Wu (2010) (Δfcc=1.0 fco) Fahmy and Wu (2010) (Δfcc=1.5 fco)

7000 Axial load (kN)

7000 Axial load (kN)

8000

Expr.( S150P10C1 &S150P35C1 ) Fahmy and Wu (2010) Fahmy and Wu (2010) (Δεcc=1 eco) Fahmy and Wu (2010) (Δεcc=3 eco) Fahmy and Wu (2010) ( Δεcc=5 eco)

10000 8000

(f)

6000 4000

(f)

2000

2000

0

0 0 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0

100

200 300 400 Moment (kN m)

500

10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0

(g) 50

100 150 Moment (kN m)

200

100

250

200 300 400 Moment (kN m)

500

600

Expr. RC-3 Fahmy and Wu (2010) Fahmy and Wu (2010) (Δfcc=0.5 fco) Fahmy and Wu (2010) (Δfcc=1.0 fco) Fahmy and Wu (2010) (Δfcc=1.5 fco)

Axial load (kN)

Axial load (kN)

Expr. RC-3 Fahmy and Wu (2010) Fahmy and Wu (2010) (Δεcc=1 eco) Fahmy and Wu (2010) (Δεcc=3 eco) Fahmy and Wu (2010) ( Δεcc=5 eco)

0

0

600

(h) 0

50

100 150 Moment (kN m)

200

250

Fig. 11. The theoretical interaction diagram (M-P) and the experimental values for two different cases: (a) case1 (increasing the ultimate strain) and (b) case2 (increasing the ultimate stress) for four columns (S150P10C1 and S150P35C1 (c and d) respectively), (CL3 (e and f), respectively) and (RC-3 (g and h) respectively).

fc

Concrete Cover (fct,ɛct)

Δfcc

fo fo,0

fc (fcc,ɛcc) Δfcc (fcc,a,ɛcc)

(fct,ɛct) Δfcc

(fct,ɛct) 0.2fco

For axial and bending loads For axial load

Concrete Core

fo fo,0

(fcc,ɛcc) Δfcc (fcc,a,ɛcc)

(fct,ɛct) 0.2fco

For axial and bending loads For axial load

ɛcc

ɛc

ɛcc

ɛc

Fig. 12. The proposed stress-strain curves for concrete core and concrete cover under the action of axial and bending loadings.

2

Δfcc/fco (req.) Δfcc/fco(pre.) Power (Δfcc/fco (req.)) R² = 0.9833

1.2

Δfcc/fco

Δfcc/fco

1.6

(a)

0.8 0.4 0 1

2

Δfcc/fco(pre. )/ Δfcc/fco (req.)

0

3

4

5

e/r

1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

6

7

8

9 10 11 12

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

R² = 0.8728

(b)

0

10

20

N%

30

40

50

60

(c)

FCS1

S150P10C1 S75P10C1 S150P35C1 S75P35C1

CL2

CL3

RC-3

Fig. 13. The relationship between the increase in the axial strength to the ultimate strength (Δfcc/fcc) with the eccentricity ratio (e/r) and axial load ratio (N%) (a and b), respectively. c) The predicted (Δfcc/fcc) to the required (Δfcc/fcc) for all examined columns.

1000

80 (b) 60 40 20 0 -20 -40 Exper. S150P10C1 Proposed model -60 FRP rupture -80 -330-270-210-150 -90 -30 30 90 150 210 270 330 Lateral displacement (mm)

(a)

200

-200 Exper.FCS-1 Proposed model FRP rupture

-600

-1000 -130

-80

-30 20 70 Lateral displacement (mm)

Lateral load (kN)

Lateral load (kN)

600

120

Lateral load (kN)

100 (c) 80 60 40 20 0 -20 -40 -60 -80 -100 -300 -250 -200 -150 -100 -50

Exper. S75P35C1 Proposed model FRP rupture 0

50 100 150 200 250 300

Lateral displacement (mm)

Fig. 14. The cyclic responses of the simulated columns (FCS-1, S150P10C1, and S75P35C1) (a, b and c) respectively, using the proposed eccentricity-based design-oriented model.

Disp.(Pre.)/Disp. (Expr.)

1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

proposed model

(a)

FCS-1

S150P10C1 S75P10C1 S150P35C1 S75P35C1

CL2

CL3

CR-3

1.0 AAE

0.8

(b)

0.6

0.440

0.616 0.628 0.526 0.565

0.733

0.4 0.2

0.060

0.0

Fig. 15. a) The predicted ultimate lateral displacement compared to the experimental value for the examined columns; b) Comparison between the proposed eccentricity-based model and the others nominated models based on the AAE criterion.

Lateral load.(pre.)/Lateral load(expr.)

1.2 1.1

N=5%

N=10%

N=36%

N=35%

N=52%

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 FCS-1

S150P10C1 S75P10C1 S150P35C1 S75P35C1

CL2

CL3

CR-3

Fig. 16. The predicted ultimate lateral load capacity compared to the experimental value for the examined columns.

140

140 (b) 100 60 60 20 20 -20 -20 Exper. B3 -60 Exper. A3 -60 Lee et al.(2010) Lee et al.(2010) -100 Proposed model Proposed model -100 FRP rupture (numericaly) FRP rupture (numericaly) -140 -140 -100 -80 -60 -40 -20 0 20 40 60 80 100 -100 -80 -60 -40 -20 0 20 40 60 80 100 Lateral displacement (mm) (a)

Lateral load (kN)

Lateral load (kN)

100

Lateral displacement (mm)

Lateral Load (kN)

100 (c) 80 60 40 20 0 -20 Exper. C5 -40 Lee et al. (2010) -60 Proposed model -80 FRP rupture (numerically) -100 -225 -175 -125 -75 -25 25 75 125 175 225 Lateral Displecment (mm)

Fig.17. The performance of the proposed eccentricity-based model and the model by Lee et al. (2010) in predicting the lateral load/displacement relationship for three columns (A3, B3, and C5) (a, b and c), respectively, that did not recognize FRP-rupture up to the end of loading.

Appendix A Stress-strain equations for the nominated models. Reference

Chastre and Silva (2010)

Kawashima et al. (2000)

 =

^_`^1 a_

+ &) ' , k = 2, & = 3950lm

First /Second branch/Ultimate stress/ultimate strain h

d Vd e b"cG _ 1 _J i fg\

′ ; m = 

".Lcmjn

′ ; m = 

".Lcmju

)

\j h

?, &) = 0.8

5__

a__



5o,f c5o,p 5q

,  = m + 1.282,5 + 2,: 

 = m r1 + 5.29 G5 o Js,----- ' = 17.65' G5 o J )

5

q

5

q

K.t

?,------- 2 = 2,v55 + 2,: ,---- 3a = 0.6 wI CFRP

-----------------------------------------------------------------------------------Hardening Behavior:1 &) ' 0`" y z 0 < Ɛ < Ɛ.  = & ' x1 − ;1 − ? ; ?  & '.  = . + &) Ɛ − '.  z Ɛ. < Ɛ < Ɛ & − &) Ɛ. = & Ɛ. − . Softening Behavior:1 ' 0`" y z 0 < Ɛ < Ɛ.  = & ' x1 − ; ?  '.  = . + &) Ɛ − '.  z Ɛ. < Ɛ < Ɛ & Ɛ. = , & Ɛ. − . 2&2,5 Ɛ5 2,: ? + 0.0107 ; ′ ? Ɛ. = 0.003 + 0.00939 ; ′  2 2,: ′ . =  + 1.93Ɛ5 2&2,5 + 2.2 , Ɛ5 = 0.0015 2 ′ ) 2&2,5  &) = −0.658 + 0.078{ & 2&2,5 Ɛ5 + 0.09822,:  &5 5

 = . + &) ' − '. , ' = 0.00383 + 0.1014 G )

|f5}~ K.tL ′ 5_g

J

------------------------------------------------------------------------------------

 = . b

)Ɛ_ Ɛ

− G _J i Ɛ

Ɛ

 = lD) − D − D . = Harajli (2006)

′



if 0 < Ɛ < Ɛ.

if Ɛ. < Ɛ < Ɛ

+ 4.1Ɛ2 b&2,5 + &2,: ;

‚__ ‚ƒ

?i,

. ′ − 1?i  < ′ − 0.5D & Ɛ „  † D = 0.0031D" &2,5 −  " 2,: 2 <… < Ɛ )  ′ + D ′ & Ɛ ′ & ;  + 0.9? D =  − 0.0032D"  "  2,: 2 2,5 <… Ɛ. Ɛ2 = 0.002 or Ɛ‰Š Ɛ. = Ɛ b1 + 310.57Ɛ2 + 1.9 ;

′ r1 + 4.1 o s , ---- Ɛ = Ɛ r1 + 310.57Ɛ + 1.9€ G __ − 1Js  =    6 5′ 5′ 5

_g

)

5

_g

------------------------------------------------------------------------------------

Lee et al. (2010)

′ − & ' G  = & ' +   

a_

a_g

J z 0 < Ɛ < Ɛ

Ɛ6 €

K.L

′ +  −  ′ € G  =  : 

 = : +  − :  G

a_` a_g

a_p ` a_g

a_ ` a_p

a__` a_p

J

J

K.t

K.t

z

Ɛ < Ɛ < Ɛ:

z Ɛ: < Ɛ < Ɛ

of z 2,5 ≥ 2,: , ': = ' r0.85 + 0.03 G 5 Js , : = 0.95

5

op a_p K.Œ

z 2,5 < 2,: , ': = 0.7' , : =   ′ r1 + 2 G  = 

f 5o,Žff cp5o,p ′ 5_g

a__

Js, --- ' = ' [1.75 + 5.25 G

If 2,v55 ≥ 2,: → 3: = 1, 35 = 1

&2,: =

‚:∗^:

)∗m∗:

=

If 2,v55 < 2,: → 3: = 2 − 

|p ^p )

, &2,5 =

)”.f ^f m

=

|f ^ f )



5o,Žff m

′ 5_g

JG

Ɛ,‘~’ K.ŒL a_g

J

, 35 = 1, DƐ as (Lam and Teng, 2003)

5o,p Œ”.f

, C5 =

f 5o,Žff cp5o,p

, C: = m∗: Œ‚:

Note: fc is stress in confined concrete, εc is strain in confined concrete; fcʹo is the unconfined concrete compressive strength, fcc is ultimate strength of confined concrete, εcc is ultimate strain of confined concrete corresponding to fcc,,ft is transition strength, εt is transition strength corresponding to ft; fl,f, is ultimate FRP confinement pressure, fl,eff, is effective confinement pressure, El,f is lateral modulus of elasticity of FRP, ρf is FRP confinement ration; fju is ultimate tensile strength of FRP jacket, Ef is the tensile modulus of FRP, εj is ultimate tensile strain of FRP jacket, εh,rup is hoop rupture strain of FRP jacket, kε is FRP hoop strain reduction factor, n is number of FRP layers, tf is the thickness of one layer of FRP; fl,s is confinement pressure due to transverse steel reinforcement (steel hoops), El,s is lateral modulus of elasticity of lateral steel, ρs is lateral steel ratio, Es is the tensile modulus of the transverse lateral steel, εys is the yield strain of the transverse lateral steel, As is the cross section of the transverse lateral steel, Dc is diameter of concrete core to centerline of steel hoops, s is the distance between the transverse lateral steel; D is the column diameter, L is the column height, Ag is gross area of section, Acc is area of concrete core.

.

Table 4 Properties of the experimentally collected columns. Ref. Li and Sung (2004) Ozbakkaloglu and Saatcioglu (2006) (Gu et al. 2010)

Paultre et al. (2016)

FCS1

Axial load f cʹ o D l Ej fl/ fcʹo fy Per. ?? ??st (mm) (mm) (GPa) (MPa) (MPa) s % 18.6 760 1750 232 0.243 426 1.88 0.135 5.0

RC-3

49.7 270

2000 227

0.186 500

2.81 -

52.0

CL2 CL3 S150P10C1 S150P35C1 S75P10C1 S75P35C1

34.9 34.9 33.6 34.6 36.0 33.6

800 800 2000 2000 2000 2000

0.262 0.367 0.169 0.164 0.157 0.169

5.79 5.79 2.48 2.48 2.48 2.48

33.8 33.8 10.1 31.9 8.9 35.3

Column name

Dimension

360 360 305 305 305 305

FRP

260 260 78.12 78.12 78.12 78.12

Main steel

382 382 415 415 415 415

TSR

0.269 0.269 1.12 1.12 2.23 2.23

Note: fcʹo is the unconfined concrete compressive strength, D is the diameter of column and l is the shear span. The tensile stress of FRP, modulus of FRP and ultimate strain are fj, Ej, εj respectively, Tj is the thickness of FRP jacket. fy is the yield stress of the longitudinal bars. fys is the yield stress of the transverse lateral steel.

Table 2 Maximum lateral load and ultimate displacement for the selected columns. Ref.

Column name

Li and Sung (2004) Ozbakkaloglu and Saatcioglu (2006)

FCS-1 RC-3 CL2 CL3 S150P10C1 S150P35C1 S75P10C1 S75P35C1

Gu et al. (2010)

Paultre et al. (2016)

At FRP-rupture Pmax δp,max (kN) (mm) 981.8 122.5 81.0 160 634.4 48 664.8 48 64.0 266.4 83.3 155.8 64.2 288.1 91.7 217.5

Note: Pmax is peak lateral load, δp,max is the corresponding displacement at peak lateral load.

Table 3 Concrete dimensions and mechanical characteristics of the materials used for the selected columns. Ref.

Gallardo-Zafra and Kawashima (2009) Saadatmanesh et al. (1996)

Dimension

FRP

Main steel

27.5

D (mm) 400

l (mm) 1350

Ej fl/ fcʹo (GPa) 266 0.18

fy (MPa) 374

??s (%) 1.92

??st (%) 0.128

Axial load Per. % 5.4

A3

27.5

400

1350

266

0.18

374

1.92

0.256

5.4

C5

36.5

305

1801

18.6

0.47

358

2.43

0.149

16.7

Column name

fcʹo

B3

TSR

Note: fcʹo is the unconfined concrete compressive strength, D is the column diameter and l is the shear span. The tensile modulus of FRP and the confinement ratio of FRP are fj and fl/ fcʹo, respectively. fy is the yield stress of the longitudinal bars, and fys is the yield stress of the transverse lateral steel.