## 1. Introduction

The use of composites, such as Carbon Fibre Reinforced Polymer (CFRP), as external strengthening material for concrete and reinforced concrete (RC) structural elements, is becoming very popular. The main reason is the improved performance of the retrofitted members, in terms of enhanced strength and ductility. CFRP is frequently used not only for confining of compressed concrete columns, but also for shear [

1,

2,

3] and bending [

4,

5] strengthening of flexural elements, e.g. beams. In contrast to flexural elements, the standard practice in the case of columns is to fully or partially wrap them with CFRP [

6]. The shear strengthening with externally bonded CFRP sheets and laminates is one common application for reinforced concrete beams. Differently from the columns, where a full wrapping of the cross-section is possible, a closed continuous CFRP loop around the whole cross-section of the beam is not always possible. Namely, the presence of the slab can prevent the full wrapping and/or the sufficient end anchorage of the externally bonded material [

1]. In such cases, the bond performance of the interface between concrete and CFRP is fundamental for preventing a premature de-bonding, i.e. brittle failure of the strengthened element.

The present study focuses on the application of CFRP as confining material for relatively small concrete columns subjected to uniaxial compressive load where the specimens are fully wrapped along the entire height.

Several experimental and numerical studies have been performed over the years to better understand the performance of RC columns retrofitted with CFRP. Differently from actively confined concrete, the behavior of CFRP-confined concrete is governed by a continuous interaction between the mechanical properties of CFRP and dilatation of concrete. Furthermore, some other factors influence the performance of passively confined concrete (see [

7]), such as: geometry of the cross-section (corner radius), type of resin and fibers used in composite, delamination between composite layers or between concrete and CFRP, and eccentricity of the load. An efficient numerical model for predicting the behavior of CFRP-confined concrete should be able to account for all of these aspects.

Mainly three categories of models are available in the literature for predicting the behavior of CFRP-confined concrete members: (1) design oriented models [

8,

9,

10], (2) analysis oriented models [

11,

12,

13], and (3) models used in the non-linear finite element analysis [

14,

15,

16,

17,

18,

19,

20,

21,

22,

23,

24,

25]. Most of the models proposed for the third category are formulated at macro-scale and they are based on plasticity-based models for concrete. The main focus of these models is to correctly reproduce the complex interaction phenomena between concrete and Fibre Reinforced Polymer (FRP), especially in non-circular sections, characterized by a non-uniform stress distribution. The shape of the cross-section (corner radius) greatly influences the behavior of concrete columns confined with CFRP, as confirmed by experimental studies [

7].

In Rousakis et al. [

14], the model formulated in [

15,

16] for steel confined concrete sections was improved to correctly capture the dilatation properties of concrete. In [

17], the plastic dilatation parameter was formulated as a function of the plain concrete strength and FRP stiffness and the proposed relation was used in [

18] for the three-dimensional (3D) FE analysis of square RC columns confined with FRP. A modified DP based model was proposed by Yu et al. [

19] to study the performance non-circular confined sections, although with some limits. Wu presented an interesting overview of the main contributions of constitutive models for FRP-confined concrete [

20], with particular attention paid to the work performed by the author’s group. Lo et al. [

21] proposed a FE model based on the 3D surface of Menétrey and Willam [

22] and the lateral strain-axial strain constitutive model for concrete reported in [

23] to study FRP-confined rectangular concrete columns under uniaxial compression. In a more recent study [

24], a modified Concrete Damage Plasticity Model (CDPM) was proposed to correctly reproduce the dilatation properties of FRP-confined columns with different cross-sections (circular, square, and rectangular).

A more sophisticated numerical approach, based on the Lattice Discrete Particle Model (LDPM), is reported in [

25] to analyze the CFRP-confined specimens that were tested by Wang and Wu [

7]. The model, which was recently developed to analyze concrete materials through the meso-scale interaction of coarse aggregate particles, provides good results with respect to the experimental data. However, as pointed out by the authors [

25], the analysis of specimens with sharpest corners requires an advanced modeling of the interaction between concrete and CFRP to capture the complex local phenomena at corner. Such phenomena cannot be easily captured by means of macroscopic modeling approach.

With these premises, the microplane model [

26] is used in this study to perform 3D FE meso-scale analysis of the available experimental tests [

7]. Some preliminary results obtained for the circular section (

R75) are published in [

27]. In Gambarelli et al. [

28], a macroscopic approach for concrete was employed to numerically analyze the CFRP-confined specimens tested in [

7]. The tests results show that the ultimate strain of CFRP (at failure) gradually reduces with the decreasing corner radius [

7]. The confinement offered by FRP decreases proportionally with the decreasing corner radius and, in the case of sharper corners (

R0,

R15), the efficiency of fibers is almost irrelevant in terms of strength gain, while it can be important in increasing the ductility of the jacketed elements, due to high stresses localized in the corner area.

This phenomenon is probably due to a combination of: (1) geometrical effect related to the shape of the cross-section and (2) concrete and fibers local damage at corners, which affects the performance of CFRP. In [

28], a reduced strength for fibers at corners was introduced to account for the second aspect. The aim of the present study is to check whether the explicit modeling of concrete heterogeneity can realistically capture the interaction between concrete and CFRP and the induced damage at corner which govern the failure behavior of the confined specimens with sharp corners.

## 2. Random Aggregate Structure of Concrete

A simple generation procedure that was implemented in Matlab R2013b was used in this study to generate the meso-scale structure of the C30 concrete tested in [

7]. Based on proper distance criteria, the coarse aggregate (assumed spherical), with a specific size distribution, was randomly distributed inside the concrete specimen. The coarse aggregate occupies 34% of the total specimen volume, with diameters ranging between 5 and 10 mm.

After evaluating the size distribution of the coarse aggregate (Fuller curve), the procedure reported in [

29] was adopted to generate the meso-scale structure of concrete. First, the particle centers were randomly placed inside the specimen. Subsequently, the corresponding spheres, with given diameter, were created from the previous generated centers, and the solids particles were subtracted from the external solid to obtain the two concrete phases (coarse aggregate and mortar matrix). The generated geometries were imported into the FE code MASA [

30] and meshed with 3D solid four-node finite elements. In this study, two different concrete compositions were considered in order to generate the FE meso-scale model: (a) composition I (

Figure 1a), where only the maximum diameter from the experiments (10 mm) was used to achieve a coarse aggregate volume fraction of 10% and (b) composition II (

Figure 1b), where a 30% of the coarse aggregate was obtained with three different diameters (5, 7.5, and 10 mm). The simplified meso-scale structure (

Figure 1a) was firstly used in numerical simulations.

## 3. Materials and Method

The specimens tested by Wang and Wu [

7] present a constant ratio width/height (150/300 mm) and six values of the cross-section corner radius: 0, 15, 30, 45, 60, and 75 mm (

Figure 2). Three different configurations were experimentally tested, namely: plain concrete, 1ply CFRP-confined specimens (nominal fiber thickness of 0.165 mm), and 2ply CFRP-confined specimens (nominal fiber thickness of 0.33 mm). Furthermore, two different concrete grades (C30, C50) and two different CFRP were used. In the present study, unconfined and 1ply CFRP-confined specimens that were made of C30 were numerically analyzed at the meso-scale.

The mechanical properties of CFRP (see

Table 1) were obtained from the standard coupon test concerning a single-ply specimen [

7]. The procedure for evaluating the mechanical properties of the two separate phases is reported in [

28]. The tensile strength and the elastic modulus of the CFRP were calculated based on the nominal thickness of the fibers (0.165 mm). The microplane-based non-linear constitutive law [

26] was used for the epoxy resin. A proper calibration of the microplane model parameters [

28] was carried out based on the experimental results of the coupon test. An elastic-brittle response was assumed for the fibers.

Figure 3 shows the constitutive laws that were employed for the two CFRP phases.

Regarding concrete, the constitutive law for mortar (

Figure 4) is based on the microplane model [

26], while a linear elastic behavior was assumed for the aggregate. The mechanical parameters of the mortar matrix and the aggregate were properly calibrated to correctly simulate the uniaxial compressive behavior of the normal-strength concrete (C30) used in the experiments. The ratio of the elastic moduli of the mortar matrix and the coarse aggregate was set at 1:2.4. The fracture energy of mortar, evaluated with respect to an average element size of 6mm, was assumed to be equal to 50 [J/m

^{2}]. The macroscopic elastic and fracture properties of C30 were determined in [

28].

Figure 5 shows the finite element discretization of one CFRP-confined specimen (

R60). Solid 3D four-node finite elements were used to discretize the concrete phases (

Figure 5a,b) and the epoxy resin (

Figure 5c), while one-dimensional (1D) truss elements were employed for the carbon fibers (

Figure 5d). Perfect connection was assumed between the fibers and the epoxy resin, i.e. no delamination can occur between the two phases.

The load was applied by displacement control in axial specimen direction. The loading and reaction surfaces were confined in the horizontal plane to correctly reproduce the experimental boundary conditions. The crack band method [

31] was employed as the regularization technique.

## 5. Conclusions

In the present study, the performance of plain and CFRP-confined concrete specimens loaded in uniaxial compression was numerically investigated using the meso-scale modeling approach. The concrete and the CFRP were both modeled as bi-phase composite materials; the former constituted by coarse aggregate and mortar matrix; the latter constituted by epoxy resin and carbon fibers. An interesting aspect of the study is the use of the meso-scale approach to investigate the behavior of CFRP-confined specimens with different cross-section corner radii. Two meso-scale structures for concrete were considered, namely: the concrete composition with 10% of the total coarse aggregate and the more realistic composition with higher volumetric fraction (30%). From the obtained numerical results, the following can be concluded: (1) the adopted meso-scale approach is a suitable tool for the analysis of both unconfined and CFRP-confined specimens; (2) in the case of CFRP-confined specimens, the model is able to capture the influence of the cross-section shape on the confining effect provided by CFRP, which gradually reduces with the decreasing corner radius; (3) the stress-strain curves obtained for 1ply-CFRP are in good agreement with the experimental one, even if for very small corner radius ($R0,R15$) the numerical results show the large differences with the experimental data in the post-peak region; (4) it is shown that the more realistic meso-scale model for concrete can better capture the local damage phenomena at corner, i.e. the interaction between the material heterogenity and CFRP is more realistic; and, (5) similarly to the experiments, the failure mode of the confined specimens is governed by tensile rupture of the fibers near the corner area. The interaction mechanisms between concrete and CFRP, togheter with shape of the cross-section, strongly influence the performance of the confined specimens; (6) Further studies will be carried out by the authors to investigate the effect of the following influential parameters: type of composite (fibers, matrix), jacket thickness.