Nonlinear finite element analysis of fibre-reinforced polymer/concrete joints

Abstract

The objective of this research work is to simulate the interfacial shear response of fibre-reinforced polymer/concrete joints using a micromechanics-based concrete approach. The M4 version of the microplane concrete theory is coded in FORTRAN and implemented as a parallel user-defined subroutine into the commercial finite element software package ADINA. This article first focuses on three-dimensional nonlinear micromechanics-based finite element analyses. Then, validations are carried out using experimental results of 40 fibre-reinforced polymer/concrete joints. The objective is to assess the accuracy of the microplane approach to represent the interfacial shear behaviour of the fibre-reinforced polymer/concrete joints as an alternative to implementing interface elements. At the end of this article, numerical comparisons are presented between the predictions using a phenomenological concrete constitutive law adopted in the software package (with a smeared crack model) and the micromechanics-based analysis (microplane theory) to simulate the concrete behaviour.

Keywords

debonding fibre-reinforced polymer finite element analysis interface laminates

Introduction

The trend towards the use of sophisticated nonlinear finite element (FE) analysis is underlining the importance of adequately describing the complex behaviour of concrete. An increase in the use of these models is also being driven by the advent of high-speed and affordable computers. Although several FE simulations of concrete members strengthened using fibre-reinforced polymer (FRP) laminates are now available (Aiello and Leone, 2008; Ascione, 2009; Bilotta et al., 2011; Biscaia et al., 2013; Diab and Wu, 2008; Ferracuti et al., 2006; Liu and Wu, 2012), various challenges have been found to significantly affect the accuracy of predicted behaviour of such members. In this work, we use a micromechanics-based FE approach to overcome difficulties associated with using phenomenological or closed formula-based constitutive models. In this approach, the concrete is represented based on its microstructural nature. Our purpose is to give rational and accurate FE simulations for FRP/concrete joints.

In order to produce accurate FE and analytical simulations for concrete structures strengthened with externally bonded FRPs, an appropriate concrete constitutive law is required. Of particular interest is the fact that the experimentally determined local bond–slip curves are not identical along the bonded length, as reported in the literature and depicted in Figure 1(a) (Dai et al., 2005). The local bond strength decreases with an increase in the distance from the loaded end. This peculiarity of $τ - s$ curves has been reported in several experimental investigations in the case of direct shear tests, particularly when the length of the FRP laminates is relatively small (Dai et al., 2005; Nakaba et al., 2001). To date, no persuading explanation or FE analysis has been given for such differences in the shape of these relationships or in the value of the local bond strength (e.g. 2.6–17.3 MPa in the experimental work of Dai et al. (2005); Figure 1(b)). Several researchers have ascribed this finding either to the microstructural composition of the concrete, including the distribution of the fine and coarse aggregates along the concrete surface, or to the local bending contribution of the bonded FRP laminate (Dai et al., 2005; Lu et al., 2005a).

Figure 1.

Local stress–slip curves (Dai et al., 2005): (a) bond–slip models and (b) bond strength at various locations.

As far as the FE analysis is concerned, most studies have simulated the FRP/concrete joints using plane stress elements, thus disregarding the influence of the out-of-plane stress components on the behaviour of the bond–slip relationship. These simulations have been restricted to the plane stress assumption to avoid the higher computational demands of the three-dimensional (3D) analyses. The so-called meso-scale finite element model was introduced to describe the interfacial behaviour of only the upper layer of the FRP/concrete joints, which has a minimum thickness of 45 mm, using plane stress analysis (Lu et al., 2005a). In addition, to the authors’ best knowledge, most of the FE simulations of FRP/concrete joints have disregarded the influence of the interfacial normal stress components on the shear stress behaviour as a result of employing a smeared crack technique for the cracked concrete in plane stress simulations. The shear response in such a popular technique is simulated using an empirical shear retention factor based on the tensile principal strain value, regardless of direct interaction of the other two principal strains. In 3D simulations, the shear-normal stress coupling may be indirectly considered in calculating the tensile principle strain value but not from the constitutive law. Accordingly, the predicted bond–slip profiles obtained from these plane stress FE simulations are identical along the bonded length as in Lu et al. (2005a) model. Moreover, because of the fact that the debonding mechanism depends on the interfacial crack widths along the interface, it is not suitable to use a constant value for the shear modulus for cracked concrete as in the case of employing a smeared crack model. Besides, adopting this concept with the incremental FE analysis overestimates the predictions since the shear stress always increases with an increase in the shear strain. Generally, an accurate shear retention factor is necessary to precisely account for the shear nonlinearities of the cracked concrete and to simulate the crack width effect on the shear modulus. A similar conclusion can be found in the FE study of Lu et al. (2005a). It was observed in their study that the shear retention factor has a significant influence on the ability of the FE model to capture the debonding load of FRP/concrete joints (Lu et al., 2005a). In this research work, we aim to simulate the shear response of FRP/concrete joints using a micromechanics-based concrete theory known as microplane M4 (Bažant et al., 2000).

In the microplane theory, an alternative concept is employed to simulate the effects of shear nonlinearity, the related shear compression interaction and the crack opening. In this theory, the microscopic shear boundary is assumed to represent friction. A nonlinear frictional yield function is proposed for the dependence of the shear yield on the normal stress. Furthermore, the microscopic tensile deviatoric boundary is introduced to simulate the transverse crack opening and to control the volumetric expansion and lateral strains in unconfined compression tests. More details concerning the microplane theory are given in Appendix A.

This study first presents a persuading explanation for the variation of the bond–slip profiles along the bonded length using a rational FE analysis. This is carried out by incorporating a refined 3D nonlinear FE analysis employing a micromechanics-based (microplane) concrete theory to model the interfacial behaviour of FRP/concrete joints subjected to direct shear loading. The outcome of this FE study has been used in another contribution (Abdel Baky et al., 2012) to propose a bond–slip model for FRP/concrete joints. The model has also been implemented in various applications of FRP-strengthened reinforced concrete structures (Neale et al., 2011). Detailed description of the specimens, the structural and material modelling, computational aspects and the FE results will be given. A comparison is then made between the accuracy of using phenomenological concrete constitutive laws and that when the microplane theory is used.

Micromechanics-based nonlinear FE modelling of FRP/concrete joints

Most studies that simulated the FRP/concrete joints have adopted linear or nonlinear elasticity relations to represent the interfacial behaviour of FRP/concrete joints (Ebead and Neale, 2007; Lu et al., 2005a). The FE simulations have been restricted to the plane stress assumption to avoid the high computational demands of 3D analyses. The so-called meso-scale finite element model was introduced to describe the interfacial behaviour of FRP/concrete joints using plane stress analysis (Ebead and Neale, 2007; Lu et al., 2005a). (Note that this model is called meso-scale due to the use of small element sizes rather than representing the concrete at the microstructure level.) However, experimental tests on FRP/concrete joints subjected to direct shear loadings have shown 3D modes of failure in the concrete block (Bizindavyi and Neale, 1999). Accordingly, in this article, the complete concrete block, adhesive layer and FRP laminates are modelled. In the following sections, the predicted ultimate load carrying capacities and local bond–slip $(τ - s)$ relationships of the FRP/concrete joints are presented.

FE analysis

Description of the specimens analysed

Available experimental results of 40 specimens are used to validate the FE analyses. The database is versatile in terms of geometrical characteristics and material properties of the concrete, adhesive layer and FRP composites. These specimens are selected from those tested by Chajes et al. (1996), Bizindavyi and Neale (1999) and Dai et al. (2005). The identifications of the specimens used throughout this article correspond to the notations used in the original references, with the exception of the specimens tested by Chajes et al. (1996). In this article, we refer to these specimens as S1 through S4, as they do not have specific notations in the original reference.

The 40 specimens have been selected from the large database based on the fact that they had clear and accurate reported data for the materials used, especially in terms of the mechanical and geometrical characteristics of the adhesive layer. These studies have followed strict procedures to verify the accuracy of the reported material properties. In the Dai et al. (2005) study, the thicknesses of the adhesive layers were measured using a microscope with six-order digit accuracy. In the Chajes et al. (1996) work, biaxial tests were carried out to measure the five FRP orthotropic material parameters. All the selected specimens experienced deboning failure mode between the FRP sheet/strip and the concrete block where the failure occurred in few millimetres inside the concrete adjacent to the interface. This is a common failure mode of such application attributed to the higher mechanical characteristics of the adhesive layer compared to that of the concrete.

Four FRP/concrete joints constitute the set of specimens of Chajes et al. (1996), which have different bonded lengths of FRP laminates ranging from 50.8 to 203.2 mm. The set tested by Bizindavyi and Neale (1999) involved 10 specimens investigating the influence of thicknesses and bonded lengths of the FRPs in conjunction with two types of FRPs. These are the carbon fibre-reinforced polymer (CFRP) and glass fibre-reinforced polymer (GFRP). In this set of specimens, the thicknesses of the FRP laminates vary from 0.33 to 2.0 mm, with the bonded lengths varying from 50.0 to 180.0 mm. The last set of specimens contains 26 FRP/concrete joints with a relatively long bonded length (330.0 mm) (Dai et al., 2005). Three types of adhesive and FRP laminates (aramid fibre-reinforced polymer (AFRP), GFRP and CFRP) are considered in the Dai et al. (2005) study. Young’s moduli of the FRPs, $E_{p}$ , and thickness, $t_{p}$ , of these specimens are taken as reported in the original references. The geometrical parameters of Bizindavyi and Neale (1999), Chajes et al. (1996) and Dai et al. (2005) sets of specimens are depicted in Figure 2(a) to (c), respectively. All details concerning the geometric and material properties of the specimens can be found in the original experimental works and summarized in Abdel Baky’s (2008) thesis.

Figure 2.

Geometrical characteristics of specimens analysed: (a) Bizindavyi and Neale (1999), (b) Chajes et al. (1996) and (c) Dai et al. (2005).

Structural modelling

Although plane stress analyses have generally been employed in FE simulations of FRP/concrete joints subjected to direct shear loading, 3D analyses are more accurate, especially for considering the influence of the out-of-plane stress components and the width of the FRP laminates to that of the concrete block $(b_{p} / b_{c})$ . In fact, experiments on FRP/concrete joints invariably show 3D modes of failure in the concrete block. In addition, several studies have modelled only the upper layer of the concrete block based on the assumption that only a zone of a few millimetres inside the concrete adjacent to the interface contributes to the overall interfacial behaviour (Bizindavyi and Neale, 1999; Lu et al., 2005a). Although this assumption is acceptable for analysing the global behaviour and to predict the debonding load, it is not suitable when the local behaviour, such as the bond–slip response, is being investigated. Accordingly, in this study, the entire concrete block, adhesive layer and FRP laminates are modelled. In order to deal with the computational demands associated with 3D simulations, a parallel supercomputer is used to run the simulations. Geometrical characteristics of specimens analysed are shown in Figure 3. A typical FE model is shown in Figure 3(a) and (b). Four-node tetrahedral 3D elements with three translational degrees of freedom per node are used to represent the geometric characteristics of the concrete, FRP laminate and adhesive layer (Figure 3(c)). These elements are available in the modelling software package ADINA (Bathe, 2005). The element size of the concrete prism in the top 35.0-mm layer beneath the FRP laminate (Zone ‘I’ in Figure 3(a)) is taken as 0.5 mm × 0.5 mm × 0.5 mm. For the concrete elements adjacent to the FRP in the top 35.0 mm (Zone ‘II’), the element size is increased to 2.0 mm × 2.0 mm × 2.0 mm at a distance equal to half of the FRP laminate width. For the remainder of the concrete elements, the element size is taken as 5 mm × 5 mm × 5 mm.

Figure 3.

Geometrical characteristics of specimens analysed: (a) typical finite element model, (b) detail ‘A’ and (c) 4-node 3D tetrahedral finite element.

Two elements are taken through the depth of the adhesive layer. For the FRP laminates, the element size is taken as 0.5 mm × 0.5 mm × 0.5 mm. For the cases where the thickness of the FRP is less than 0.5 mm, the element size is taken as $t_{p} \times 0.5 mm \times 0.5 mm$ . Using these small element sizes, we expect to accurately trace the propagation of the debonding cracks and to capture the variation of the bond–slip profile along the interface. In order to simulate the softening branch of the load–displacement profile and the post-debonding behaviour, a displacement-controlled technique is used in the FE analysis. The applied displacement increment is taken as 0.02 mm with a total of 100 steps. In the simulations, only one-half of each direct shear specimen is modelled due to the geometrical and loading symmetries, and suitable constraint equations are applied at the plane of symmetry. The nodes on the bottom surface of the concrete prism are prevented from vertical movement.

Material modelling

Constitutive law for the concrete

The key reason for choosing the microplane model in this work is its ability to simulate the shear nonlinearity of the concrete without recourse to the popular concept of the shear retention factor. With the microplane theory, the shear compression/tension interaction can be captured more accurately than when using the smeared crack technique. This constitutive model has been successfully utilized to simulate the complex concrete behaviour in two-way slab-to-column connections strengthened using steel and FRP plates (Ebead and Saeed, 2010), that is, based on original experimental work (Ebead and Marzouk, 2002a, 2002b, 2004, 2005).

A brief summary describing the microplane concrete theory and the parameters of the user-defined subroutine is given in Appendix A. The values of the non-dimensional material constants of the microplane model (their values are constant for all concrete batches) defining the microplane model ( $c_{1}$ to $c_{18}$ ) are taken from the original work of Caner and Bažant (2000) The non-dimensional material adjustable parameters ( $k_{1}$ to $k_{4}$ ) as well as the macroscopic Young’s modulus, $E_{c}$ , and Poisson’s ratio, $ν$ , are specified individually for each concrete batch. Since the confining pressure is not high enough in our simulations to cause pore collapse of the concrete, the values of the parameters $k_{2}$ (defining the triaxial compressive behaviour under high confining state of stress), $k_{3}$ and $k_{4}$ (describing the behaviour of the concrete under hydrostatic compression state of stress) are taken constant for all specimens as

k_{2} = 110.0, k_{3} = 12.0, k_{4} = 38.0

(1)

Poisson’s ratio $ν$ is taken as 0.18 in all simulations. The value of the parameter $k_{1}$ is computed for each individual concrete batch as

k_{1} = 2.45 \times 10^{- 4} \frac{E_{c} f'_{c}}{46 \times 25}

(2)

The values of $E_{c}$ and $f'_{c}$ in equation (2) are in GPa and MPa, respectively. In cases where the initial modulus of elasticity $E_{c}$ is not experimentally reported, they are evaluated according to the CSA-A23.3 (2004) equation as

E_{c} = 4500 \sqrt{f_{c}^{'}} (MPa)

(3)

Constitutive models for the adhesive

Two approaches have been employed to simulate the mechanical characteristics of the adhesive layer. The first approach uses an isotropic linear elastic material. The second approach considers the ultimate tensile strength of the adhesive layer as a failure criterion; in other words, the adhesive layer is no longer able to transfer stresses when the tensile stresses in the adhesive layer exceed the admissible ultimate tensile strength. No discrepancies have been observed between the two approaches in terms of the predicted ultimate load capacities. Generally, the debonding or rupture of the FRP laminate takes place at low stress levels in the adhesive layer, and this requires no further consideration of the failure criterion of the adhesive layer in our simulations. One exception is for the eight specimens tested by Dai et al. (2005). Here, the adhesive layer is modelled as an elastic-perfectly plastic material since the yield strength of the adhesive layer in these particular specimens is relatively low (11.8 and 15.9 MPa). These specimens are CR2L1 to CR2L3, CR3L1 to CR3L3, AR2L3, AR3L3, GR2L3 and GR3L3. In general, the adhesive layers simulated in this work showed a linear response up to the peak stress in the original experimental references. Therefore, both elastic and elastic–plastic models are good assumptions for the mechanical response of the adhesive layer.

Constitutive models for the FRPs

FRP materials used in FRP-strengthened concrete structures are, in general, orthotropic linear elastic materials. The stress–strain relations for these materials are given by

[\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{23} \\ σ_{31} \\ σ_{12} \end{matrix}] = \frac{1}{Δ} [\begin{matrix} \frac{1 - υ_{23} υ_{32}}{E_{22} E_{33}} & \frac{υ_{21} + υ_{31} υ_{23}}{E_{22} E_{33}} & \frac{υ_{31} + υ_{21} υ_{32}}{E_{22} E_{33}} & 0 & 0 & 0 \\ \frac{1 - υ_{13} υ_{31}}{E_{33} E_{11}} & \frac{υ_{32} + υ_{31} υ_{12}}{E_{33} E_{11}} & 0 & 0 & 0 \\ \frac{1 - υ_{12} υ_{21}}{E_{11} E_{22}} & 0 & 0 & 0 \\ 2 G_{23} & 0 & 0 \\ Symmetric & 2 G_{31} & 0 \\ 2 G_{12} \end{matrix}] [\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ ε_{23} \\ ε_{31} \\ ε_{12} \end{matrix}]

(4)

Here, $Δ = (1 - υ_{12} υ_{21} - υ_{23} υ_{32} - υ_{31} υ_{13} - 2 υ_{13} υ_{23} υ_{31}) / (E_{11} E_{22} E_{33})$ and $E_{11}$ , $E_{22}$ , $E_{33}$ , $G_{12}$ , $G_{13}$ , $G_{23}$ , $υ_{12}$ , $υ_{13}$ and $υ_{23}$ are the nine independent material constants, where $E$ , $G$ and $υ$ denote Young’s modulus, the shear modulus and Poisson’s ratio, respectively. The subscripts 1, 2 and 3 refer to the three orthogonal directions (local directions of the lamina). In the case of a unidirectional-oriented FRP composite (the case of our simulations), the material becomes transversely isotropic, where five independent material constants are sufficient to construct the elastic stress–strain relationship. As far as the failure envelope is concerned, the maximum stress criterion is employed where failure occurs if any stress component reaches a value equal to its ultimate strength. More details regarding the constitutive relations for orthotropic FRPs can be found in Labossiere and Neale (1988).

For the specimens tested by Bizindavyi and Neale (1999) and those tested by Dai et al. (2005), $E_{22}$ is assumed to be one-tenth of that in the direction of the fibres, $E_{11}$ . (Throughout this article, we will refer to ‘E₁₁’ as ‘E_p’.) This assumption has been adopted since these references do not report the values of the transverse elastic moduli. For the specimens tested by Chajes et al. (1996), the value of $E_{22}$ is 11.58 GPa. Poisson’s ratio in the fibre direction, $υ_{21}$ , is 0.268 and in the other two perpendicular directions $υ_{12}$ and $υ_{13}$ are equal to 0.025. These values are reported in the experimental work of Chajes et al. (1996) and have been assumed for the other specimens. In general, since the FRP laminates used in the simulations are unidirectionally oriented composites, the numerical values of the out-of-plane properties ( $E_{22}$ , $υ_{12}$ and $υ_{13}$ ) have secondary effects on the predicted responses.

Computational aspects

The micromechanics-based FE calculations lead to significant demands in terms of both computational power and storage, due to the necessity of tracking an enormous number of microscopic variables at each integration point. The limited computational capacities of personal computers are unable to supply the computer power required for such applications. However, with supercomputers and parallelization techniques, such demands are easily met. In our simulations, the ‘Mammoth’ parallel supercomputer is used to run the FE calculations. It is a Dell Power Edge SC1425 consisting of 576 nodes with a maximum speed of 4147.2 GHz. Each node consists of two processors of 3.6 GHz with a total RAM of 8.0 GB. This supercomputer is designed to solve 650,000 equations per second when running at full capacity. In the simulations, the parallel computations are performed by concurrently executing different problems on a various number of processors, while the overall FE problem is handled by one processor. This type of parallelization is referred to as a ‘multi-input, multi-output technique’. The total calculation time for the 40 specimens was about 5804 h (around 242 days) with an average solution time for each problem ranging from 32 to 48 h. Because of the synchronous execution, the entire computer solution time was done in 336 h. A parallel FORTRAN code is implemented to organize the parallelization process of the FE runs.

FE results

The following sections include a validation of the numerical model through comparisons with available experimental results in terms of the load capacities and FRP strain profiles. Subsequently, the local bond–slip relationships are presented based on the numerical results.

Ultimate loads

The comparisons between the FE predictions and the experimental results for all the 40 specimens, in terms of the ultimate load carrying capacities, are depicted in Figure 4. As seen from the figure, there is a very good agreement between the numerically predicted load capacities and the experimental results for all the test specimens. The average numerical-to-experimental load ratio is 0.98 with a corresponding standard deviation of 0.095, which indicates an excellent predictive capability of the FE model. The scatter in the predicted results is within the admissible range for the direct shear test. The highest numerical accuracy ratio is achieved in the FE simulations of Dai et al. (2005) with an average accuracy of 99.5% and standard deviation of 9.3%.

Figure 4.

Ultimate capacity of bonded joints: (a) statistic characteristics and (b) ultimate load carrying capacity.

Strain distributions along the FRP laminates

The experimental results of Chajes et al. (1996) specimens are chosen to validate the results of the FE analysis in terms of the strain distributions along the bonded FRP laminates. The comparisons show a satisfactory agreement between the numerical predictions and the experimental data, as shown in Figure 5(a) to (d) for Specimens S1, S2, S3 and S4 (Chajes et al., 1996). In these figures, the strain in the FRP plates is measured from the loaded end at the centreline of FRP plate, where the zero point indicates a starting point of the FRP bonded to the concrete. The load levels in this figure vary up to the capacities of the joints. As can be observed in Figure 5(a) to (d), for a given load level, the strains in the FRP decrease progressively along the length of the bonded area.

Figure 5.

FRP strain distributions: numerical versus experimental (Chajes et al., 1996): (a) Specimen S1, (b) Specimen S2, (c) Specimen S3 and (d) Specimen S4.

There is a good agreement between the numerical predictions and the experimental results in terms of the strain distributions at load levels less than 80% of the ultimate capacity. Since the FE analysis adopts a displacement-controlled technique, an interpolation is required to find the strain profiles at load levels that match the exact experimental load levels. This could explain the slight discrepancy between the numerical predictions and experimental results for load levels around the debonding failure load, in particular for Specimen S1 (Figure 5(a) at the load level of 8.1 kN). Moreover, it is difficult, around the debonding loads, to experimentally obtain accurate measurements of the strains in the CFRP sheets at specific load levels because of debonding. At debonding, a small variation in load causes a significant change in the corresponding slip value. As a result, it is not possible to find the same load level from the FE analysis that exactly matches the experimental one. The fluctuation in the predicted strain profiles (in particular, in Figure 5(c) at the load level of 11.9 kN) arises from the interfacial cracks along the bonded FRP as seen in previous research work (Abdel Baky et al., 2010, 2012; Ebead and Saeed, 2010).

It is worth mentioning that the predicted strain profile along the bonded length is generally not sensitive to the type of analysis, that is, linear or nonlinear simulations give almost the same prediction at load levels less than the debonding value. For example, the predictions from a linear-based analysis and nonlinear simulations (Ebead and Neale, 2007; Lu et al., 2005a) give very good agreements with experimental data in terms of the normal strain profiles along the bonded length. This is attributed to the fact that the axial strains in the FRP laminates at load levels before debonding significantly depend on the characteristics of the laminates rather than the concrete behaviour.

Local bond–slip profiles

In this section, $τ - s$ curves based on the previous FE simulations are presented. These curves for the specimen CR1L1 tested by Dai et al. (2005) are depicted in Figure 6. In Figure 6(a), $τ - s$ profiles are given at five different locations along the adhesive/concrete interface at the centreline of the block. The local bond strength in this figure has a value of 22.3, 7.4, 8.9, 6.3 and 4.4 MPa at the loaded end and, respectively, at 2.0, 4.0, 6.0 and 165.0 mm (the middle point) from the loaded end. The bond–slip relationships in Figure 6(a) are characterized by a residual stress with an average value around 10%–15% of the local bond strength. We observe that the slope of these relationships is almost independent of the distance from the loaded end. In Figure 6(b), the $τ - s$ relationships are presented for various points through the depth of the concrete at the loaded end. The maximum value of the local bond strength occurs at the adhesive/concrete interface, with a value of 22.3 MPa. From Figure 6(a) and (b), one can observe that the bond–slip curves have the same overall shapes; however, they have different values of local bond strength, $τ_{max}$ .

Figure 6.

Local bond–slip profiles: (a) along adhesive/concrete interface and (b) along the depth of concrete.

The significant difference between the predictions of this FE model and those in the literature is the shape of the output bond–slip profiles. Using the microplane theory gives a bond–slip relationship that varies along the bonded length. However, the numerical predictions in the literature, using the smeared crack approach with the associated shear retention factor, predict a constant $τ - s$ curve along the interface (Chen and Teng, 2001; Lu et al., 2005a, 2005b).

Discussion of the FE results

In Figure 6(a), $τ - s$ profiles are given at different locations along the adhesive/concrete interface. In Figure 6(b), similar curves are presented for various points through the depth of the concrete at the loaded end. In these figures, $τ$ is the shear stress in x-y direction $(τ_{xy})$ and referred to it as bond stress and $s$ is the relative displacement between the top of the FRP element node and the node inside the concrete where the bond–slip was presented. The values of $τ$ and $s$ are extracted directly from the FE models.

Although the input value of the pure shear strength of the concrete in the FE calculations has been set to 2.4 MPa as for Specimen CR1L1, the output values of the maximum shear stress $τ_{max}$ varies from 4.4 to 23.3 MPa as shown in Figure 6(a). Thus, the discrepancy between the output maximum shear strength of the interface and the input pure shear strength of the concrete indicates that FRP/concrete joints subjected to direct shear loading do not represent the case of a pure shear test. This can be explained by considering the equilibrium equations of a discrete element $dxdy$ shown in Figure 7, as follows

\frac{\partial σ_{x}}{\partial x} + \frac{\partial τ_{xy}}{\partial y} = 0

(5)

\frac{\partial σ_{y}}{\partial y} + \frac{\partial τ_{yx}}{\partial x} = 0

(6)

Figure 7.

2D equilibrium of a discrete element.

In the above equations, a two-dimensional (2D) equilibrium state is assumed since the variation in shear stresses in the z-direction, $\partial τ_{zx}$ and $\partial τ_{zy}$ is insignificant ( $τ_{zx}$ and $τ_{zy}$ are almost constant along the width of the plate). As a result of the shear stress variation ( $τ_{xy}$ in the above two equations) through the depth of the concrete or along the bonded FRP laminate, normal stress components ( $σ_{x}$ and $σ_{y}$ ) necessarily develop inside the concrete to satisfy equilibrium. These normal stresses cause the location of the local bond strength $τ_{max}$ to vary along the interface. The above results show that the interaction of the various components of stress along the bonded joint must be considered and that an adequate failure criterion for the FRP/concrete interface must be introduced before an appropriate bond–slip model can be accurately derived. The interaction between the normal and shear stress components for failure in concrete has been addressed previously in several experimental studies (Bresler and Pister, 1958; Moosavi and Bawden, 2003) and has led to the development of various failure criteria (e.g. Mohr–Coulomb and Drucker–Prager).

In summary, the above micromechanics-based FE simulations show that the bond–slip profiles have essentially the same overall shape and the associated local bond strength values vary along the bonded length and through the concrete depth. This conclusion agrees with several experimental observations (Dai et al., 2005; Nakaba et al., 2001). Moreover, an FRP/concrete joint subjected to direct shear loading is not identical to a state of pure shear in the concrete test. Consequently, it is not possible to simply select one $τ - s$ profile along the interface to represent the global behaviour of the FRP/concrete joint. To determine a refined bond–slip model, a relation between $τ_{max}$ and the normal stresses at failure along the bonded length is necessary. Accordingly, in another work of the authors, they will consider a relation of the form $τ_{max} = g (σ_{ij}, f_{t})$ in terms of the state of stress $σ_{ij}$ and concrete tensile strength $f_{t}$ . This relation is developed based on a suitable failure criterion for the concrete that considers the interaction between the various stress components. Subsequently, in our analysis, a unified function is proposed to define the bond–slip model in terms of an expression for $τ_{max}$ . This function comes from a best fit of the micromechanics-based FE results.

Interfacial stress distribution along the bonded joint

This analysis is used to evaluate the main interfacial normal stresses along the bonded length that have significant values compared to the shear stress. This FE analysis is carried out for the specimen BN5 tested by Bizindavyi and Neale (1999) to examine the distribution of various stress components along the interface. The element size is taken as 0.25 mm × 0.25 mm × 0.25 mm in the top 30-mm layer and 1.0 mm × 1.0 mm × 1.0 mm for the rest of the concrete prism. The average number of elements is 2.08 × 10⁶. Four-node 3D tetrahedral elements are employed to represent the adhesive, FRP laminate and concrete. An iterative solver with a free-form meshing format is used instead of the direct solver to solve the huge number of equations and to avoid exceeding the memory capacity (around one million equations).

As a result of the fact that the debonding failure normally takes place a few millimetres inside the concrete, close attention is paid to the profiles of the normal and shear stress distributions at a distance 1.0 mm away from the adhesive/concrete interface line inside the concrete. These profiles are presented in Figure 8(a) to (c). In these figures, it is referred to the normal stresses in the direction perpendicular to the interface (y-direction in Figure 8(a) to (c)) as ‘peeling-off’ stresses $σ_{y}$ and in the horizontal direction (x-direction in Figure 8(a) to (c)) as ‘normal stresses’ $σ_{x}$ . It is observed that the peeling-off stress $σ_{y}$ and normal stress in the z-direction have negligible values along the bonded length except at the loaded end (Point 1). By contrast, the normal stresses $σ_{x}$ have significant values compared to the shear stress values.

Figure 8.

Stress distributions along the interface (1.0 mm away from the interface inside the concrete): (a) normal stress $σ_{x}$ , (b) peeling-off stresses $σ_{y}$ and (c) normal stress $σ_{z}$ .

In order to confirm the aforementioned conclusion regarding the influence of the normal stresses on the predicted value of $τ_{max}$ of the bond–slip relationship, the Mohr–Coulomb criterion for concrete is used to study the states of stress at the loaded end as an example. The values of the normal and shear stresses at this location are summarized in Table 1. Figure 9 shows the application of the Mohr–Coulomb criterion to the state of stress at Point ‘1’ giving a concrete tensile strength of 3.5 MPa and pure shear strength of 2.4 MPa (assuming the friction angle $φ$ equals 35°). In Figure 9(a), the solid Mohr’s circle represents the elastic state of stress at this point (values in Table 1).

Table 1.

Components of the state of stress at Point ‘1’.

	$σ_{x} (MPa)$ Normal stress	$σ_{y} (MPa)$ Peeling-off stress	$σ_{z} (MPa)$	$τ_{yx} (MPa)$
Point ‘1’	−4.06	−3.16	−1.47	1.77

Figure 9.

Mohr’s circles for the 3D state of stress at Point ‘1’: (a) at early elastic state of stress and (b) at state of stress corresponding to failure.

The predicted Mohr’s circle corresponding to the state of stress of Point ‘1’ at failure is the circle that is tangent to the failure envelope (solid line in Figure 9(b)) and intersects the line of maximum shear stress of Point 1 (dashed line in Figure 9(b)). This line corresponds to that which connects between the origin (Point ‘o’) and the point of the maximum shear stress at the current state of stress (Point ‘a’). In both Figure 9(a) and (b), it is assumed that the out-of-plane stresses $σ_{z} = 0$ . Consequently, the 3D state of stresses is represented in Figure 9(a) and (b). More details regarding the Mohr–Coulomb criterion and its applications can be found in Chen (2007).

Mohr’s circle at failure for the state of stress at Point ‘1’ is represented schematically by the dashed circle in Figure 9(b). Accordingly, the predicted $τ_{max}$ at Point ‘1’ is 12.51 MPa. This value is significantly higher than the pure shear strength of concrete (2.4 MPa). This confirms the conclusion that was drawn in the previous section. It is of interest to mention that the predicted local bond strength value at Point ‘1’, for example, using the Mohr–Coulomb criterion (12.51 MPa) is less than that resulted from the FE analysis (22.2 MPa). This is ascribed to the fact that the prediction using Mohr’s circle depends on the friction angle $φ$ (slope of the failure envelop in Figure 9(b)). Assuming different values of the $φ$ angle (associated with various assumptions for the pure shear strength of concrete) leads to different predictions of $τ_{max}$ . For example, taking the pure shear strength equal to 3.0 MPa with the tensile strength of 3.5 MPa (corresponding to a friction angle of 40°) leads to the prediction of $τ_{max}$ of 18.2 MPa. Some recent studies such as modified compression field theory (Vecchio and Collins, 1986) or the five-parameter failure envelope of Willam and Wranke (Willam, 1975) assumes that the concrete friction angle $φ$ is not constant but is a function of the principal normal compressive stress value (ranges from 18° to 40°). In general, the analysis in this section using the Mohr–Coulomb criterion aims to introduce a persuading interpretation of the variation of the bond–slip profiles along the bonded length.

Effect of the material constitutive law

Two distinct concrete models have been employed for the analyses in this section to represent the concrete characteristics: one is based on the macroscopic concrete behaviour (phenomenological model in the ADINA software (Bathe, 2005), Appendix B), while the other model is the micromechanics-based law (microplane approach, Appendix B). The objective of this investigation is to assess the feasibility of using the phenomenological and microplane concrete constitutive relations to simulate FRP-strengthened concrete structures. More details regarding numerical simulations of FRP-strengthened concrete structures using phenomenological models are introduced in the works of Abdel Baky et al. (2007, 2010) and Kotynia et al. (2008).

In the FE simulations of FRP/concrete joints, two approaches have been investigated: one considers the adhesive layer between the FRP and concrete layers, while the second analysis ignores the adhesive layer and simulates the FRP/concrete joint assuming full bond between the concrete and FRP nodes. We aim to assess the influence of the adhesive layers in transferring shear stresses from the FRP to the concrete and to understand the role of the adhesive layer on the debonding phenomena.

Figure 10(a) and (b) shows the load–displacement relationships using the phenomenological and microplane constitutive laws, respectively, in the numerical simulations of FRP/concrete joints. Obviously, the FE model employing the phenomenological concrete approach adopted in ADINA (Bathe, 2005) fails to capture the debonding load and the analysis continues until concrete crushing or rupture of the FRP occurs. By contrast, the microplane concrete model accurately captures the debonding of the FRP laminate off the concrete block (Figure 10(a)). In the phenomenological concrete approach, because of the use of the smeared crack model with such a concrete constitutive law, the shear response is simulated using an empirical shear retention factor based on the tensile principal strain value regardless of the interaction of the other two principal strains. Because of the fact that the debonding mechanism depends on interfacial crack widths along the interface, it is not suitable to use a constant value for the shear modulus for cracked concrete. Generally, an accurate shear retention factor is necessary to accurately account for the shear nonlinearities of the cracked concrete and to simulate the crack width effect on the shear modulus of the concrete. The FE study of Lu et al. (2005a) focused on predicting the appropriate shear retention factor to accurately capture the debonding process of FRP/concrete joints subjected to direct shear loading. In their study, the effects of four different shear retention expressions for cracked concrete were compared to identify the expression that led to the most accurate predictions. Lu et al. (2005a) observed that the shear retention factor has a significant influence on the ability of the FE model to capture the debonding load. Another significant discrepancy between the predictions of the FE model using the two concrete formulations is the shape of output bond–slip profiles. Using the microplane theory gives a bond–slip relationship that varies along the bonded length. However, the numerical predictions using the phenomenological formulations and smeared crack model predict a constant $τ - s$ curve along the interface. It is concluded from Figure 10(a) and (b) that the adhesive has a significant effect on the predicted ultimate load carrying capacity.

Figure 10.

Effect of the concrete constitutive law on the predictions for the direct shear test: (a) hypoelastic concrete model and (b) microplane concrete model.

It was noticed that using a softer adhesive layer increases the bond strength of the FRP/concrete joint. This is the same conclusion reported experimentally (Dai et al., 2005). This is attributed to the fact that the debonding occurs inside the concrete block rather than in the adhesive layer. With that, the role of the adhesive layer in the FRP/concrete joints is different from that in FRP-strengthened concrete beams. In the earlier application, the adhesive layer transfers shear stresses from the FRP laminates to the concrete block. However, in the latter case, the flexural behaviour and the associated widening of the flexural and shear cracks govern the force transfer from the concrete beam to the FRP laminate.

Conclusion

In this article, nonlinear micromechanics-based FE models were developed using the microplane concrete theory to investigate the interfacial behaviour of FRP/concrete joints subjected to direct shear loadings. First the M4 version of the microplane concrete approach is coded in FORTRAN environment and implemented as a user-defined subroutine into the commercial FE software package ADINA (Bathe, 2005). Then, it was used to run the simulations. The comparisons between the FE predictions and the experimental results showed a very good agreement. It is clearly observed that, when using a refined FE mesh, the FE model with the microplane constitutive law successfully captures the debonding load. The use of a small element size allows capturing the debonding load as strain discontinuity is accounted for by considering various elements in the vicinity of the cracks.

The microplane model is proven adequate in providing accurate analyses for FRP/concrete joints. The numerical comparisons between a phenomenological concrete constitutive law and micromechanics-based model (microplane theory) to simulate the concrete behaviour in the application of FRP/concrete joints subjected to direct shear loadings are presented. It is concluded that the microplane approach rather than the phenomenological relation successfully simulates the nonlinearities of the interfacial shear behaviour in the application of FRP/concrete joints subjected to direct shear loadings. The FE results indicate that the local bond–slip models are not the same along the bonded length and the associated maximum bond strength depends on the location from the loaded end. This implies that an FRP/concrete joint subjected to direct shear loadings is not the case of a pure shear test.

Footnotes

Appendix 1

Appendix 2 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC).

References

Abdel Baky

(2008) Nonlinear micromechanics–based finite element analysis of the interfacial behaviour of FRP–strengthened reinforced concrete beams. PhD Thesis, Department of Civil Engineering, University of Sherbrooke, Sherbrooke, QC, Canada.

Abdel Baky

Ebead

Neale

(2007) Flexural and interfacial behavior of FRP–strengthened reinforced concrete beams. Journal of Composites for Construction 11(6): 629–639.

Abdel Baky

Ebead

Neale

(2010) Statistical analyses and parametric study for reinforced concrete beams strengthened in flexure with FRPs. Advances in Structural Engineering 13: 805–822.

Abdel Baky

Ebead

Neale

(2012) Nonlinear micromechanics–based bond–slip model for FRP/concrete interfaces. Engineering Structures 39: 11–23.

Aiello

Leone

(2008) Interface analysis between FRP EBR system and concrete. Composites Part B: Engineering 39: 618–626.

Ascione

(2009) Mechanical behaviour of FRP adhesive joints: a theoretical model. Composites Part B: Engineering 40: 116–124.

Bathe

(2005) ADINA 8.5 ed. Watertown, MA: ADINA R & D, Inc.

Bažant

Caner

Carol

. (2000) Microplane model M4 for concrete. I: formulation with work–conjugate deviatoric stress. Journal of Engineering Mechanics 126: 944–953.

Bilotta

Ludovico

Nigro

(2011) FRP-to-concrete interface debonding: experimental calibration of a capacity model. Composites Part B: Engineering 42P: 1539–1553.

10.

Biscaia

Chastre

Silva

MAG

(2013) Linear and nonlinear analysis of bond–slip models for interfaces between FRP composites and concrete. Composites Part B: Engineering 45: 1554–1568.

11.

Bizindavyi

Neale

(1999) Transfer lengths and bond strengths for composites bonded to concrete. Journal of Composites for Construction 3: 153–160.

12.

Bresler

Pister

(1958) Strength of concrete under combined stresses. ACI Journal Proceedings 55(3): 137–147.

13.

Caner

Bažant

(2000) Microplane model M4 for concrete. II: algorithm and calibration. Journal of Engineering Mechanics 126: 954–961.

14.

CSA-A23.3 (2004) Design of Concrete Structures. Clause 86. Rexdale, ON, Canada: Canadian Standard Association, 358 pp.

15.

Chajes

Finch

Jr Januszka

. (1996) Bond and force transfer of composite–material plates bonded to concrete. ACI Structural Journal 93: 295–303.

16.

Chen

Teng

(2001) Anchorage strength models for FRP and steel plates bonded to concrete. Journal of Structural Engineering 127: 784–791.

17.

Chen

(2007) Plasticity in Reinforced Concrete. J. Ross Publishing. Plantation, FL, USA.

18.

Dai

Ueda

Sato

(2005) Development of the nonlinear bond stress–slip model of fiber reinforced plastics sheet–concrete interfaces with a simple method. Journal of Composites for Construction 9: 52–62.

19.

Diab

(2008) A linear viscoelastic model for interfacial long-term behavior of FRP–concrete interface. Composites Part B: Engineering 39: 722–730.

20.

Ebead

Marzouk

(2002a) Strengthening of two-way slabs subjected to moment and cyclic loading. ACI Structural Journal 99: 435–444.

21.

Ebead

Marzouk

(2002b) Strengthening of two-way slabs using steel plates. ACI Structural Journal 99: 23–32.

22.

Ebead

Marzouk

(2004) Fiber-reinforced polymer strengthening of two-way slabs. ACI Structural Journal 101: 650–659.

23.

Ebead

Marzouk

(2005) Tension–stiffening model for FRP–strengthened RC concrete two-way slabs. Materials and Structures 38: 193–200.

24.

Ebead

Neale

(2007) Mechanics of fibre–reinforced polymer–concrete interfaces. Canadian Journal of Civil Engineering 34: 367–377.

25.

Ebead

Saeed

(2010) Modeling of reinforced concrete slabs strengthened with fiber-reinforced polymer or steel plates. ACI Structural Journal 107: 218–227.

26.

Ferracuti

Savoia

Mazzotti

(2006) A numerical model for FRP–concrete delamination. Composites Part B: Engineering 37: 356–364.

27.

Kotynia

Abdel Baky

Neale

Ebead

(2008) Flexural strengthening of RC beams with externally bonded CFRP systems: test results and 3D nonlinear FE analysis. Journal of Composites for Construction 12(2): 190–201.

28.

Labossiere

Neale

(1988) A general strength theory for orthotropic fiber-reinforced composite laminae. Polymer Composites 9: 306–317.

29.

Liu

Y–F

(2012) Analytical identification of bond–slip relationship of EB–FRP joints. Composites Part B: Engineering 43: 1955–1963.

30.

Teng

. (2005a) Bond–slip models for FRP sheets/plates bonded to concrete. Engineering Structures 27: 920–937.

31.

Teng

. (2005b) Meso–scale finite element model for FRP sheets/plates bonded to concrete. Engineering Structures 27: 564–575.

32.

Moosavi

Bawden

(2003) Shear strength of Portland cement grout. Cement and Concrete Composites 25: 729–735.

33.

Nakaba

Kanakubo

Furuta

. (2001) Bond behavior between fiber–reinforced polymer laminates and concrete. ACI Structural Journal 98: 359–367.

34.

Neale

Godat

Abdel Baky

Ebead

(2011) Approaches for finite element simulations of FRP–strengthened concrete beams and slabs. Architecture Civil Engineering Environment 4: 59–72.

35.

Vecchio

Collins

(1986) The modified compression–field theory for reinforced concrete elements subjected to shear. ACI Journal 83: 219–231.

36.

Willam

KWEP

(1975) Colloquium on Concrete Structures Subjected to Triaxial Stresses. Zurich: ISMES, p. 30.