3D finite element model for hybrid FRP-confined concrete in compression using modified CDPM

The main goal of the work herein presented is to propose an accurate three-dimensional finite element model to predict the compressive behaviour of hybrid FRP-confined concrete. This was achieved through the modification of the concrete damaged plasticity model (CDPM) available in ABAQUS software, since the original CDPM has shortcomings that make it unusable to predict the compressive behaviour of confined concrete. It was demonstrated that, by turning the yield function and the flow rule dependents on the confining pressure, it is possible to use the model referred to and obtain accurate results. An analytical model was used to obtain the input parameters needed to calibrate the CDPM. A specific user subroutine was developed to modify the original CDPM. Hybrid FRP properties were calibrated as well, using an appropriate analytical model. Plasticity was assumed in the hybrid combinations for which pseudo-ductile tensile responses occurred. The performance of the proposed CDPM was validated using both experimental results and analytical predictions. It was concluded that, by using the developed 3D finite element model for hybrid FRP-confined concrete in compression, a very good agreement between experimental, analytical and numerical compressive stress-strain and lateral strain-axial strain curves is reached.

confining pressure and the axial strain) directly at the integration points of FE. This was accomplished using tabular input and a FORTRAN code to define the field variables. The proposed modifications to improve the CDPM are discussed in detail in the following sections. Comparisons with experimental and analytical results show that the predictions of the proposed model are in close agreement with measured parameters.

Theoretical background
The CDPM is a continuum plasticity-based damage model in which the concept of isotropic damaged elasticity in combination with isotropic plasticity is adopted [41]. In this way, the CDPM can be used for modelling the nonlinear deformation and irreversible damage, generally associated to cracking, of plain concrete with high accuracy [41]. As any other plasticity model, the CDPM depends on yield criterion, which is a mathematical description of the stresses under which yielding occurs (known as yield function), and the flow rule that establishes the general relations between plastic strains and the stress states.
The yield function is defined as follows: where 1 is the first effective stress invariant (see equation (5)) and 2 is the second effective deviatoric stress invariant (see equation (6)), the is the minimum principal effective stress, 〈·〉 denotes the Macauley bracket defined as 〈 〉 = (| | + )/2, fb is the concrete strength under equal biaxial compression, fc0 is the peak stress of the unconfined concrete, and are the effective compressive and tensile cohesion stresses respectively, ̃, and ̃, are the equivalent compressive and tensile plastic strains respectively, and K is the strength ratio of concrete under equal biaxial compression to triaxial compression. It should be highlighted that throughout this work the soil mechanics sign convention is adopted, whereby compressive stresses/strains are considered positive while tensile stresses/strains are considered negative.
where is the effective principal compressive stress, ,1 and ,2 are the effective principal lateral stresses. Please note that, in the present work, only the case of triaxial compression is being considered.
The term effective stress is used because all the stress quantities are understood as: where is the effective stress tensor, fij is the stress tensor (see equation (8)), and is the damage tensor which characterizes the degradation of the elastic stiffness.
where and are the strain and the plastic strain tensors and is the initial (undamaged) elasticity matrix.
For concrete with a constant confining pressure, the damage tensor is reduced to the compressive damage parameter [34]: where fc is the compressive stress of concrete, fl is the confining pressure, and * is the peak stress of actively confined concrete and can be estimated using equation (10) [32,42]: * = 0 + 5.
In the work of Hany et al. [36], it was stated that damage parameter has significant effect on the stressstrain curve when FRP-confined concrete is subjected to cyclic loading, but it is negligible for monotonic loading situations. In the present work, two scenarios were tested: (i) to adopt the damage parameter, according to equation (9), and (ii) to neglect the damage parameter. It was confirmed that, in both situations, no significant differences occurred. However, the damage parameter, computed according equation (9), was implemented in the final model.
The flow rule, , that defines the direction of plastic deformation and dictates the evolution of dilation behaviour of concrete is non-associated: where G is the Drucker-Prager hyperbolic function, defined by equation (12): tan (12) where σto is the uniaxial tensile stress at failure; ∋ is the eccentricity parameter, and is the plastic dilation angle, according to equation (13): where , is the axial plastic strain and , the lateral plastic strain.
For actively confined concrete, the referred plastic strains can be computed according to equations (14) and (15) [37]: where Ec is the elastic modulus of plain concrete and νi is the initial Poisson's ratio of concrete (εl/εc) [43]: In the present work, since the compressive behaviour of confined small-scale plain circular concrete columns is analysed, it is expected that fl = fl,1 = fl,2. However, equation (17) was implemented in the final model in order to turn the same capable of simulate the compressive behaviour of concrete under non-uniform confinement.

Proposed modifications
It has been stated that original CDPM is not successful in predicting the compressive behaviour of actively confined concrete and, consequently, the passively confined concrete [34,36,37]. This is due to the fact that a unique stress-strain curve of plain concrete cannot be used to define the compressive cohesion stress, , or, in other words, the hardening rule of concrete. As discussed in Section 2.2.1, the hardening rule has to be dependent on the confining pressure, which leads to the modification of the original yield function. Furthermore, the plastic dilation angle should vary in function of lateral strain-axial strain curve, as discussed in Section 2.2.2. In the present subsection, the modification of the hardening rule is proposed. Numerical predictions are compared with analytical results obtained through the stress-strain model for actively and passively confined concrete of Lim and Ozbakkaloglu [33]. These were computed according to both equation (18), proposed by

Hardening/softening rule
Popovics [44], and equation (19), proposed by Lim and Ozbakkaloglu [33]: where fc and εc are the compressive stress and the axial strain of confined concrete, * is the peak strain of actively confined concrete [32,42], r is the concrete brittleness [45], fc,res is the residual stress, and is the axial strain corresponding to the inflection point of the descending branch of stress-strain curve [33].
In equation (20), εc0 is the axial strain at the peak stress of the unconfined concrete ( 0 ) given by [32]: 0 = (−0.067 0 2 + 29.9 0 + 1053) × 10 −6 (24) In equation (21), Ec is defined according to the next equation [33]: The model of Lim and Ozbakkaloglu [33] was used to define the compressive stress-strain curve of plain concrete, i.e., the material input data of the numerical model that gave rise to the results of Figure 1 (a), assuming In Figure 1 (a) it is possible to observe that the shape of the numerical curves is the same for the different levels of confining pressure. According to the original CDPM, the increase of the confining pressure imposes a vertical and horizontal translation of the compressive stress-strain curve. The former is dependent on the yield function, and the latter is defined by the elastic modulus of plain concrete, i.e., since the axial stress increases with the increase of the confining pressure, the initial elastic branch is prolonged, leading to the observable lateral translation of the initial stress-strain curve.
In Figure 1 (a) it is also demonstrated that original CDPM leads to inaccurate predictions of actively confined concrete compressive behaviour, for the presented levels of confining pressure. It should be noted that a large number of actively confined and FRP-confined specimen results were used in the development of the presented stress-strain analytical model. It was proved that the analytical model provides improved predictions when compared to the previous models presented in the literature, see the details in [32,33]. For this reason, it was expected that the numerical curve matched the analytical one.
Due to the lack of the observed accuracy between the numerical and analytical curves, it is mandatory to turn the hardening rule dependent of the confining pressure in order to correct the results. In ABAQUS software [40] the introduction of the hardening rule is made in tabular format. To turn the hardening rule dependent on the confining pressure, several compressive stress-strain curves have to be defined in function of specific values confining pressure. The last is considered a field variable that should be specified through USDFLD subroutine, programmed in FORTRAN language. Furthermore, the elastic modulus has to be modified, as well, in function of the confining pressure. The following procedure was used to correct the hardening rule: i. The stress-strain analytical model, presented in equations (18) and (19), is used to define the expected compressive stress-strain curves of plain concrete subjected to different levels of constant confining pressure; ii. The yield function, presented in equation (1), is used to define the value of fc when the effective compressive cohesion stress is equal to zero; iii. An actual elastic modulus is computed dividing the fc value (obtained in step ii) by the corresponding strain.  It should be noted that the presented procedure is different from all others found in published studies. In the present work, the perfectly-plastic behaviour assumption used by Yu et al. [34] is assumed as well, i.e., after the peak stress, the stress-strain curve is truncated and the yield function remains unchanged. This option has implications in the dilation concrete behaviour predictions. If a complete stress-curve curve was considered, it would not be possible to control the concrete dilation behaviour.
In Figure 1 (b) the numerical results obtained after the correction are presented and compared against analytical ones. It is possible to observe that after an initial branch, that was intentionally assumed as elastic, there is a perfect match between numerical and analytical curves until the peak stress is reached.

Flow rule
Many authors have used the CDPM assuming a constant [46][47][48][49][50][51][52]. However, this assumption does not lead to accurate prediction of the dilation behaviour (lateral strain-axial strain curves) of actively confined concrete, as demonstrated in Figure 2 (a). In this case, a constant = 35º was assumed to obtain the lateral strainaxial strain numerical curves. As it is possible to observe, the last assumption leads to confining pressure having only a residual influence on the numerical curve developing. Therefore, all numerical curves are almost coincident and differ substantially from the analytical ones (presented as well in Figure 2 (a)). The last was obtained with the model proposed by Lim and Ozbakkaloglu [32], see equation ( where n is the curve shape parameter: The analytical model was developed based on large database of unconfined and actively confined concrete results, please see the details in [32]. It was also proved that the model is able to predict the dilation behaviour of hybrid FRP confined concrete, please see details in [3]. are not in agreement with analytical ones. For this reason, they need to be corrected. This can be done considering the evolution of , according to equation (13). In Figure 3 the referred evolution of is presented as a function 10 of axial plastic strain (see equation (14)) for the considered levels of confining pressure. It is important to note that, according to the sign convention, lateral strain was considered negative and axial strains positive in the computation of .
Similarly to what was stated for the hardening rule, in ABAQUS software [24] the introduction of is also carried out in tabular format. However, the software only admits values of between 0.1º and 56º. This limitation implies that, even with the modification of , a precise dilation behaviour of confined concrete cannot be simulated. Since is negative in some cases and higher than 56º in others, it is possible to observe in Figure 2 (b) that there are still some mismatches between numerical and analytical curves. However, the suggested modifications proved to be sufficient to reach good predictions of hybrid FRP confined concrete behaviour, as it demonstrated in Section 4.

Experimental database
In fabrics. In Table 1 the density, areal mass, fibre layer thickness (areal mass density divided by the volumetric mass density) and the basic tensile properties of the mentioned materials are presented. For each dry fabric, a large number of single fibres (see the details in Table 1) were randomly taken from the dry fabrics and tested in uniaxial tension, according to ASTM D3379-75 [53].
The tensile properties of 1 layer non-hybrid composites, determined according to ISO 527-5:2009 [54], are also presented in Table 1. All the composites referred to in the present work were laminated using an epoxy-based resin (S&P Resin Epoxy 55). According to the supplier, this epoxy resin has the following main properties [55]: In Table 1, it is possible to observe that tensile strength and, consequently, the elastic modulus of nonhybrid composites are higher than the values obtained for single fibres. This is due to the fact that, in the case of composites, the tensile properties were evaluated ignoring the contribution of the resin, according to the usual practice of the wet lay-up method and guidelines [56]. This means that tensile strength was computed considering only the dry fabric thickness, thus leading to overestimation of the tensile strength.
Relatively to hybrid combinations, 16 series were considered in the confinement application: 10 combinations with 3 reinforcing material layers and 6 combinations with 5 reinforcing material layers. Each series was composed of 3 specimens of confined concrete. Specimens with 5 layers were tested only on 2 hybrid combinations: HM carbon/glass and ST carbon/glass.
In addition to hybrid series, 4 series of 3 layer non-hybrid composites (1 for each reinforcing material) were produced. All specimens involved in the experimental campaign are listed in Table 2. In the present work, regarding composite materials nomenclature, numbers placed before letters are used for indicating the number of layers. The sequence according to which these letters appear indicate the stacking sequence of the reinforcing materials. The relative volume of LS fibres (Vol% LS) was computed and presented in the next section, according to equation (28): where tL is the half thickness of the LS layers and tH is the half thickness of the HS layers.

Geometry, element types and meshing
Taking into account the double symmetry of both the loading and the specimen, only a quarter of cylinders was modelled, as illustrated in Figure 4.
Two different FE types (three dimensional 8-node linear bricks with reduced integration (C3D8R) and 4node shell elements with reduced integration (S4R)) were used to discretize the concrete cylinders and the FRP jacket, respectively.
Care was taken to ensure that the mesh had, as much as possible, a regular geometry. Several decreasing mesh sizes were studied to evaluate the convergence of the model. It was concluded that elements with edges of about 8 mm provide accurate solutions.

Boundary conditions and interactions
Traditionally, in uniaxial compression tests of concrete specimens, the load is applied through rigid steel plates. In this way, it is inevitable frictional forces to occur at the concrete-steel interface. These forces lead to the development of a stress field which restrains the transverse expansion of concrete within the end regions of specimens [57].
In this section, it is discussed if the detail of surface-to-surface contact (steel-concrete) should or should not be considered in the numerical model. Thus, two scenarios were considered in the simulation of compressive behaviour of plain concrete:  In the first scenario, three dimensional steel plates with the same cross section as concrete and 50 mm of thickness were assembled to the ends of the concrete core. C3D8R elements were used to discretize the steel plates. It was assumed that these elements are rigid. A friction model to define the force resisting relative to tangential motion between concrete and steel plates was specified. A friction coefficient of 0.1 was assumed, according the recommendation suggested in the guideline [58]. In normal direction (i.e., in the loading direction) a hard contact was defined. It was concluded that the consideration of end restrains leads to highly non-uniform distribution of lateral displacements, as shown in Figure 5 (a). It is possible to observe that at middle height of the cylinder, the lateral deformation is maximum. The presented values of lateral displacement are relative to the peak axial stress of plain concrete;  In the second scenario, the end restrains were not detailed. This leads to uniform distribution of lateral displacement, as shown in Figure 5 (b). It is possible to observe that the values of lateral displacement are approximately 14% lower than the ones observed in the first scenario, comparing the middle sections, for the same axial displacement.
It should be noted that, in both scenarios, compatible boundary conditions with the simplifications of the geometry assumed in the model were defined, i.e., lateral displacement restraints were applied orthogonally to the sliced faces. Axial displacements were restrained in the base of the model and uniformly imposed on the top the latter. Again, the model of Lim and Ozbakkaloglu [33] was used to define the compressive stress-strain curve of plain concrete, assuming fc0 = 33.49 MPa and fl = 0.00 MPa.
Analytical axial stress-strain curves are compared with numerical ones in Figure 6. Since the analytical model of Lim and Ozbakkaloglu [33] was used to define compressive stress-strain curve of plain concrete and 13 there is no other action on the concrete than axial compression (in the case frictional forces were not contemplated), it was expected that both numerical and analytical curves were coincident.
It is possible to conclude that end restraints have no influence in the pre-ultimate compressive response of concrete and strength predictions but the post peak behaviour of concrete is dependent of friction between concrete and steel plates.
It should be highlighted that the analytical model of Lim and Ozbakkaloglu [33] was calibrated from experimental results in which the influence of end restrains were not suppressed. In this way, the obtained stressstrain curve implicitly considers the effect of end restrains. For this reason, it is not correct to consider the detail of end restrains in the numerical model, if the referred to analytical model is used to obtain the material input parameters of concrete. Otherwise, the end restraints effect would be contemplated twice.
Due to the above reasons, the influence of end restrains were not explicitly considered in the present work.

Plasticity parameters
Beyond hardening and flow rules, there are several parameters that have to be specified in order to use the CDPM, namely ∋, fb/fc0, viscosity, and K.
The definition of ∋ as null implies that the potential function tends to a straight line. In this way, the strategy exposed in section 2.2.2 to compute and to predict the dilation behaviour of concrete is valid. According to the work of Kupfer et al. [60], fb/fc0 assumes a constant value of 1.16. However, in the work of Papanikolaou and Kappos [61] it was suggested that fb/fc0 should vary according to equation (29): According to this equation, if fc0 varies between 19 and 30 MPa, fb/fc0 will vary approximately between 1.16 and 1.20. Teng et al. [59] found that, within this range, the variation of fb/fc0 has no significant effect on the numerical predictions. Due to the lack of information about the subject, and taking into account that 1.16 is the most used value in published studies, it was decided to maintain this value in the present work.
Few studies [36,62] on the viscosity parameter state that it has residual influence on the prediction's accuracy. For this reason, this parameter was ignored in the present work. 14 Although the influence of K on the compressive behaviour predictions of confined concrete was studied in the present work, since only triaxial compression situations are analysed, a simplification of equation (1) was adopted [34]: This equation is known as Drucker-Prager yield function and it defines a conic yield surface in the principal stress space, as presented in Figure 7 (a). It is possible to observe that changing the value of K leads to substantial changes in the diameter of the yield surface, as seen for instance in the deviatoric plane defined as 1 3 ⁄ = 100 (see Figure 7 (b)).
In Figure 8, it is possible to observe that, for a constant level of confining pressure (fl = fl,1 = fl,2), the peak stress of confined concrete ( * ) decreases with the increase of K. A sensitivity analysis was carried out to find out the value of K leading to lowest initial prediction errors of the peak stress of actively confined concrete. From this analysis, it was concluded that K = 0.68 leads to the lowest differences between unmodified CDPM and analytical peak stress predictions. For this reason, this value (0.68) was assumed in the present work.

FRP
In a unidirectional (UD) hybrid FRP submitted to uniaxial tension, the first damage mode is always the failure of the LS fibres. However, other damage modes depend on the properties and configuration of the composite reinforcing materials [28]. The analytical approach proposed by Jalalvand et al. [4], validated in the scope of the present work in [63], considers that four different damage modes may occur after LS fibres failure: (i) premature  where SL is the reference strength of the LS material, α and β are respectively the elastic modulus and thickness ratios of the LS to HS fibre, GIIC is the mode II interlaminar fracture toughness of the interface between LS layers and HS layers of the hybrid composite, EH is the elastic modulus of the HS fibres, tH is the half thickness of the HS fibre, mH is the Weibull strength distribution modulus of the HS fibre, SH is the reference strength of the HS material, Kt is the stress concentration factor in the high strain material, and V is the volume of the specimen (free length × width × total fibre layer thickness). The details of the adopted parameters are fully discussed in [63].
Knowing the magnitude of all three possible stresses allows assessing their order of occurrence and, consecutively, the identification of the damage modes, according to Table 3. In the present case, the studied combinations of materials (presented in Table 4) lead to the appearance of 3 damage modes, namely premature failure, catastrophic delamination and combination of fragmentation of LS material and dispersed delamination.
When the last damage mode occurs, the behaviour of hybrid composite is pseudo-ductile because a flat-topped stress-strain curve is achieved.
After the determination of the damage modes, it is possible to plot the tensile stress-strain curve of hybrid FRP. Five stress-strain coordinates are sufficient to define all possible tensile stress-strain curves of hybrid FRP, see Figure 9. The sets of coordinates that define the stress-strain curves for all the studied combinations are presented in Table 4. Please note that, when premature failure occurs, only the linear elastic branch is defined with two sets of stress-strains coordinates.
The FRP properties were specified using "Lamina" material type [40] in which the elastic modulus in the fibre direction is defined in accordance with the value provided in Table 1, in case of non-hybrid confinement, and For all non-hybrid as well as for hybrid composites in which premature failure occurred, only elastic properties were defined. For the remaining cases, a plastic behaviour was defined using as input the coordinates exposed in Table 4. In these cases, strains have to be converted first in plastic strains, according to equation (34): where ,ℎ is the tensile plastic strain, ,ℎ is the total tensile stain, ℎ is the tensile stress, and ℎ is the elastic modulus of hybrid composite.
All simulations carried out showed that the tensile behaviour of both 1G/1CHM/1G/1CHM/1G and 1G/3CHM/1G series has to be defined as linear elastic, otherwise the numerical analysis will abort. This is due to the fact that the stress drop that occurs in the stress-strain curve of these series leads to convergence errors.

Concrete
A ready-mix concrete was used in the experimental campaign [3]. The experimental campaign of the confined concrete specimens herein described was conducted in 15 consecutives working days. During this time, the concrete age varied between 294 and 315 days. Until the testing date, all specimens were kept in standard laboratory conditions (temperature of circa 20 ºC, and relative humidity of approximately 50%). In the end, three identical plain cylindrical concrete specimens were submitted to compressive tests. The mean values of the elastic modulus, according to [64], and compressive strength, according to [65], were 30.29 GPa (CoV = 6.57%) and  (25) and (16) were used to define these parameters, in order to maintain coherence with the analytical model that was used to calibrate the numerical parameters.
Although the tensile behaviour of concrete is not important for the simulations conducted in the present study, it was defined aiming at widening the field of application of the model. The parameters required to define the tensile behaviour (the tensile strength, , and the fracture energy, ) were obtained according to equations where da is the maximum aggregate size (assumed to be 12.5 mm in the present work). In the last equation, fc0 is in MPa and da is in millimetres. In this way, , obtained from equation (35), is in MPa and , obtained from equation (36), is in N/m.

COMPARISON OF FE MODEL PREDICTIONS WITH EXPERIMENTAL RESULTS
The performance of the proposed CDPM is validated with the experimental results described in Section 3.1. In the work of Ribeiro et al. [3], it has already been proven that the analytical model here presented allows to accurately simulate both the dilation behaviour and the compressive stress-strain behaviour of all confined concrete specimens analysed. Since the material parameters for the modified CDPM were obtained from the referred to analytical model, predictions obtained with both models (analytical and numerical) are expected to have similar accuracy.

Non-hybrid FRP-confined concrete
In Figure 10 the evolution of for the studied non-hybrid FRP-confined concrete combinations is presented. This parameter was computed according to equations (13) and (26) and it was introduced in CDPM in order to define the flow rule, taking into account the software limitations exposed in Section 2.2.2, i.e., negative values and values higher than 56º were not considered. However, in the computation of , an infinite axial tensile strain was assumed to generate the input parameters, and then the lateral strain at failure of the different material combinations was assumed as the average of the corresponding experimental values.
In Figure 11 and Figure 12 the lateral strain-axial strain and the compressive stress-strain curves of nonhybrid FRP-confined concrete are presented. In the specimen designation, the last number (i.e., 1, 2 or 3) was used to make the distinction between the three identical specimens. Analytical and numerical curves are also plotted to allow the comparison with the corresponding experimental curves. A good agreement between theoretical and experimental results is observed.
As expected, all the numerical predictions obtained using the proposed modified CDPM are in close agreement with test results. Nevertheless, some differences are observed between analytical and numerical values. This is essentially due to two factors: (i) the allowed range of Ψ have influence in the prediction of lateral strainaxial strain curves and (ii) the fact that stress-strain curves of FRP-confined concrete are interpolated from a reduced and fixed number of stress-strain curves for actively confined concrete (see Figure 1 (b)) leads to slope variations after the compressive stress-strain peak of plain concrete (fc0, εc0), represented in simulations with a red circle. As expected, the development of the compressive stress-strain curves follows approximately a bilinear law, where the slope of the first branch depends primarily on the properties of plain concrete. For this reason, the initial phase of the curves is similar for all combinations. The second branch depends on the confining pressure applied by the confining materials. The higher the elastic modulus of the jacket, the higher the slope of the second branch.

Hybrid FRP-confined concrete
All the -axial plastic strain curves used as input in the simulation of compressive behaviour of hybrid FRP-confined concrete are presented in Figure 13. It is possible to observe that, in hybrid jackets with non-linear tensile behaviour (see Table 4) there is an increase of comparatively to cases in which premature failure of hybrid composites take place. This is due to the fact that tensile non-linear behaviour of jackets allows the rapid increase of the lateral strain of concrete, and, consequently the increase of . Again, it should be noted that in the computation of , an infinite tensile strain of hybrid FRP was assumed. However, the numerical simulations were interrupted when the mean lateral strain at the failure was reached.
Experimental lateral strain-axial strain and compressive stress-strain curves of hybrid FRP-confined are compared with corresponding analytical and numerical curves in Figure 14 to Figure 23. It is possible to observe that predictions are generally in reasonable agreement with test results.
It is also possible to observe that pseudo-ductile branches were predicted in 3 combinations, namely in 2G/1CHM/G, 1G/1CHM/1G, and 1B/1CHM/1B. This was expected since pseudo-ductile tensile responses (i.e., simultaneous multiple fractures of LS fibres and dispersed delamination) occurred in tensile tests of these combinations [63]. Although pseudo-ductile tensile response has occurred as well in 1C/1CHM/1C combination [63], this behaviour did not have a significant influence in the compressive results. This is due to the fact that (i) pseudo-ductile strain (defined as the extra strain between the final failure strain and the strain on the extrapolated initial slope line at the failure stress of the stress-strain diagram, as it is shown in Figure 9) of this combination is very low, and (ii) the hybrid FRP failure of the jacket occured sooner that it was expected [3]. In this way, the application in practice of this combination in confinement would lead to fragile failures. In all cases, analytical and numerical curves are almost coincident. This validates the strategy of CDPM modification that is proposed in the present work.

CONCLUSIONS
In the present study, the performance of a modified concrete damage plasticity model (CDPM) is validated against both experimental and analytical results. The modifications were implemented in both hardening and flow rules, turning them confining dependent. An analysis-oriented confinement model, based on the approach of Lim and Ozbakkaloglu [32,33], was adopted to define the input parameters of the CDPM. Major conclusions drawn from this study are presented in the following paragraphs.
The need to contemplate friction forces in surface-to-surface contact between rigid steel plates and concrete was analysed. It was concluded that, if an analytical model that implicitly considers the frictions forces (such as the one considered in the present work) is used, to compute the input parameters of the numerical model, the end restrain does not need to be detailed.
It was observed that K parameter (strength ratio of concrete under equal biaxial compression to triaxial compression) has a significant influence in the prediction of the peak stress of confined concrete. Then, a sensitivity analysis was carried out and it was concluded that assuming K = 0.68 leads to residual errors in initial numerical predictions. For this reason, this value was assumed in the remaining part of the work.
The analytical model of Jalalvand et al. [4] was used to predict the input tensile parameters of hybrid FRP.
In the 2G/1CHM/G, 1G/1CHM/1G, 1B/1CHM/1B, and 1C/1CHM/1C combinations, plasticity was defined in order to take into account the pseudo-ductile tensile responses of the latter. In the remaining cases, a linear elastic behaviour was assumed. This approach has shown to provide accurate predictions for all FRP-confined circular concrete columns. As expected, pseudo-ductility was predicted in cases in which it was experimentally observed.
Moreover, it should be highlighted that analytical and numerical curves are almost coincident.
As a final conclusion, taking into account what has been stated previously, it can be concluded that the developed 3D finite element model, using modified CDPM, can be adopted to accurately predict the compressive behaviour of hybrid FRP-confined concrete.   Table 3 -Summary of different damage modes in function of stress level (adapted from [28]).