Role of interfacial damage modeling in finite element analysis of intermediate crack debonding failure of FRP-plated reinforced concrete beams: A numerical investigation

Abstract

This study presents a numerical investigation into the effects of interfacial damage on the bond behaviors of the interfaces between two adjacent cracks in a fiber-reinforced polymer-plated reinforced concrete beam. The interfaces investigated herein refer to both fiber-reinforced polymer-to-concrete interface and steel-to-concrete bonded interface which were well represented in this study by a simplifying fiber-reinforced polymer-to-concrete bonded joint or steel bar-to-concrete bonded jointed joint loaded at their two ends, aiming to simulating the interaction between two adjacent flexural cracks in the fiber-reinforced polymer-plated reinforced concrete beams. Parametric studies were carried out using nonlinear finite element model built in ABAQUS to investigate the effects of a number of significant factors such as bond length, ratio of loads applied at the two ends (i.e. load ratio), and types of bond–slip models, with the main objective being to clarify the effect of interfacial damage (i.e. bondline damage) during slip reversals on the bond behaviors of the interfaces. The numerical results show that in finite element analysis of intermediate crack debonding of fiber-reinforced polymer-plated reinforced concrete beams where multiple cracks may exist, an appropriate consideration of the interfacial damage during slip reversals is necessary in order to achieve accurate predictions on the behavior of fiber-reinforced polymer-to-concrete bonded interfaces, while for the steel bar-to-concrete interface, the consideration of the interfacial damage has insignificant effect on the numerical results if the yield strength and bar diameter are in their practical range.

Keywords

bonding cracking damage debonding fiber-reinforced polymers finite element method reinforced concrete beams

Introduction

Fiber-reinforced polymer (FRP) has been increasingly used in civil engineering both for new structures (e.g. Bai et al., 2015; Feng et al., 2015; Teng et al., 2007) and for structural strengthening (e.g. Fernando et al., 2009; Teng et al., 2009; Yao et al., 2005a; Yu et al., 2012) due to the advantages of FRP, including high tensile strength, low density, excellent resistance to corrosion, high durability, and ease of installation (Bank, 2006; Hollaway and Teng, 2008; Oehlers and Seracino, 2004; Teng et al., 2002). Strengthening of reinforced concrete (RC) structures using externally bonded (EB) FRP is one of the applications of FRP in structural strengthening. FRP debonding failure, namely, FRP is detached from the concrete surface with a thin layer of concrete attached to the debonded FRP surface, is a typical failure mode for RC members strengthened with EB FRP; as a result, the FRP-to-concrete bonded interface is the weakest link among the structural system and thus plays a critical role in ensuring the effectiveness of this strengthening method (Aprile et al., 2001).

Finite element method (FEM) appears as a valuable tool for studying FRP-strengthened structures as demonstrated by a number of recent studies (Chen et al., 2011, 2012a, 2012b, 2015; Dai et al., 2015; Niu and Karbhari, 2008; Teng et al., 2015; Wu et al., 2009; Yu et al., 2010a, 2010b; Zhang and Teng, 2014; Zhao et al., 2012; Zheng et al., 2012). This is because in an FE analysis, it is easy to obtain important information, such as distributions of strain in FRP and steel bars, but in an experimental study such information can hardly be measured using a limited number of strain gauges mainly because the FRP strain in the strengthened beams is highly variable and affected by the locations of concrete cracks which are usually not known as a priori. According to the authors’ previous work on the FE analyses of debonding failures (Chen et al., 2011, 2012b, 2015), two elements were found to be critically important for the accurate simulation of debonding failures: (1) accurate modeling of the localized cracking behavior of concrete and (2) accurate modeling of the interfacial bond–slip behavior between concrete and external FRP reinforcements as well as the bond–slip behavior between concrete and internal steel reinforcement. For the second element, most of the existing FE models (except quite a few recent studies such as Chen et al., 2011, 2012b, 2015; Niu and Karbhari, 2008; Niu and Wu, 2005) did not define the unloading behavior of the interfaces (between concrete and FRP and/or between concrete and steel reinforcements) within an FRP-strengthened member; instead, they assumed a fully reversible bond–slip relationship either for FRP-to-concrete interface (e.g. Dirar et al., 2013) or for steel reinforcement-to-concrete interface (e.g. Zhao et al., 2012) or for both (e.g. Kishi et al., 2005; Smith and Gravina, 2007). The phenomenon of reversed interfacial slips has been experimentally observed and showed to have a significant detrimental effect on bond behavior of the FRP-to-concrete interface (Sato et al., 2002; Ueda et al., 2002), although there is still a lack of research on the mechanism of bond deterioration associated with slip reversals (Farah and Sato, 2011). As a result, the lack of appropriately considering the unloading bond–slip behavior of the interfaces may lead to inaccurate prediction in the FE analysis of FRP-plated RC beams. Against the above background, this article presents a numerical investigation into the role of modeling unloading bond–slip behavior of the interfaces in an FRP-plated RC beam in the FE analysis of intermediate crack (IC) debonding failure (Yao et al., 2005a). It should be noted that IC debonding is a typical failure mode for RC beams strengthened in flexure by externally bonding FRP in the forms of pultruded plate or wet-layup sheets (both are referred to as plates unless differentiation is needed) to the tensile face of the beam (which will be referred to as FRP-plated RC beams hereafter). The IC debonding usually initiates at the end of a critical flexural crack and propagates to one of the plate ends, leading to the ultimate failure of the FRP-plated beam. Although the present numerical investigation was focused on IC debonding, the conclusions achieved in this article are also valuable for understanding the FE prediction of other types of debonding failures such as FRP debonding failures in RC beams shear-strengthened with FRP (Chen and Teng, 2003) in which slip reversal also exists due to the interactions between adjacent cracks (Chen et al., 2012b).

A simplifying bonded joint model

Simple pull-off tests had been adopted by the early studies (Chen and Teng, 2001; Lu et al., 2005; Teng et al., 2003; Yao et al., 2005b) in studying the FRP debonding failure. However, there are essential differences between debonding failure in a simple pull-off test and the IC debonding failure in an FRP-plated RC beam. The chief difference lies in the interaction between adjacent cracks in a plated RC beam (Figure 1) which does not exist in a simple pull-off test. As a result, a number of existing studies (Chen et al., 2007, 2012a; Teng et al., 2006) adopted the simplifying bonded joint model shown in Figure 2 to consider this interaction in the FRP-plated beam. This model is geometrically similar to a simple pull-off test except that both ends of the FRP plate are subjected to two tension loads: P₁ at the right end and P₂ at the left end. The concrete prism is assumed to be subjected to two forces, P₃ and P₄, at the right and the left ends (cracks), respectively. It should be noted that in the simplifying bonded joint model, forces P₃ and P₄ represent the axial forces created in the FRP-plated beam at the locations of the two adjacent cracks, respectively, to balance the forces P₁ and P₂ in FRP at the two cracks, the bond length L of FRP represents the spacing of the adjacent cracks, and the thickness of concrete prism t_c stands for the height of the beam. The adhesive layer is assumed to have a constant thickness and the whole model is assumed to be in a plane stress state. The deformation of the actual adhesive layer and that of a thin layer of the adjacent concrete involved in the debonding failure are lumped together and referred to as the deformation of the interface (referred to as the “bondline” hereafter), which is assumed to be in pure shear because the failure mode of the interface is predominantly mode II interfacial fracture occurring in concrete. It is assumed that all these forces (i.e. P₁, P₂, P₃, P₄) remain proportional throughout the loading process. It is further assumed that the two adherends, i.e. the plate and the concrete prism, are subjected to axial deformation only.

Figure 1.

Interfaces in an FRP-plated RC beam.

Figure 2.

FRP-to-concrete bonded joint model between two cracks.

The simplifying bonded joint model (Figure 2) captures the key features of the interaction between the adjacent cracks in an FRP-plated RC beam and was shown to provide fairly accurate prediction of the bond behaviors of the interfaces between two adjacent cracks in the plated RC beam (Chen et al., 2006). Furthermore, the simplifying model is easy/convenient to be implemented in an FE analysis, particularly when a large number of parametric studies are to be carried out. As a result, the simplifying joint model was adopted in this study for numerical investigation on the behavior (e.g. the interaction between two adjacent cracks) of the FRP-to-concrete bonded interface between two adjacent cracks in an FRP-plated beam; the same simplifying joint model was also used for the steel bar-to-concrete interface between two adjacent cracks in the FRP-plated beam. It should be noted that although simplifying computational model (Figures 2 and 4) was adopted in this study for the convenience of carrying out parametric study, the motivation of this study is to clarify the role of interfacial damage (e.g. elastic damage) modeling in predicting IC debonding failure of FRP-plated beam. As similar way of interfacial damage modeling was adopted in the advanced FE model developed by the research group of the first author (Chen et al., 2011) for accurately predicting IC debonding failure, the study presented herein can be regarded as a further investigation on the FE model presented in Chen et al. (2011).

Local bond–slip models with assumed unloading curve

As mentioned above, in most of the existing numerical/analytical studies on the FRP-plated beams (e.g. Chen et al., 2007; Kishi et al., 2005; Smith and Gravina, 2007; Teng et al., 2006), the bond–slip relationship of the FRP-to-concrete bonded interface was assumed to be fully reversible (referred to as undamaged bond–slip model hereafter) (see Figure 3(a) and (c)). Subsequently, Chen et al. (2012a) carried out a numerical investigation into the effects of bondline damage on the predicted behavior of the FRP-to-concrete interface between two cracks using the bilinear bond–slip model with linear damage as shown in Figure 3(b), and the numerical results showed that bondline damage has considerable effects on the bond behavior of the interface, in particular when the load ratio at the two plate ends is high (larger than 0.7) and the bond length (L) is reasonably large. However, the actual bond–slip relationship under cyclic loading can be very complex, with nonlinear unloading and reloading behavior; such a bond–slip model is yet to be developed (Farah and Sato, 2011). As a result, an accurate nonlinear bond–slip model based on Lu et al. (2005) with elastic damage (i.e. referred to as the damaged bond–slip model hereafter; see Figure 3(d)) was adopted in this study to further clarify the effect of bondline damage on the predicted load–displacement response of the interfaces. The elastic damage assumption is simple for implementation in FE analysis but captures the key unloading features experimentally observed (Bizindavyi et al., 2003), that is, (1) when the interface has entered the softening range, the bond stress reduces as the interfacial slip decreases due to unloading, (2) the unloading stiffness decreases rapidly with the increase in interfacial slip at unloading, and (3) in the softening range, the unloading stiffness is generally smaller than the initial loading stiffness. The elastic damage assumption thus provides sufficiently accurate results for the purpose of this study. Similar damage models have also been adopted in several previous studies capable of accurately predicting the behaviors of FRP-concrete interfaces (Chen et al., 2011, 2012a, 2012b; Niu et al., 2006; Niu and Karbhari, 2008; Niu and Wu, 2005).

Figure 3.

Bond–slip models including unloading behavior: (a) fully reversible bilinear bond–slip model, (b) bilinear bond–slip model with linear damage, (c) Lu et al.’s (2005) fully reversible bond–slip model, (d) Lu et al.’s (2005) bond–slip model with elastic damage, (e) CEB-FIP’s (1993) fully reversible bond–slip model, and (f) CEB-FIP’s (1993) bond–slip model with elastic damage.

For the bond–slip behaviors between steel bars and concrete, the bond–slip models of CEB-FIP (1993) with (Figure 3(f)) and without (Figure 3(e)) elastic damage assumption were adopted in this study. Similar unloading behavior has also been adopted in several previous studies on the modeling of steel bar-to-concrete bonded interfaces (Chen et al., 2011, 2012b; Niu and Wu, 2005).

FE modeling

The FE model

As mentioned in section “Local bond–slip models with assumed unloading curve,” the simplifying FRP-to-concrete bonded joint model with the FRP plate being pulled at both ends (Figure 2) was adopted in this study to simulate the bond behavior of FRP-to-concrete bonded interfaces between two adjacent cracks in an FRP-plated beam (Figure 1). An FE model as shown in Figure 4(a) was proposed by Chen et al. (2012a) to simulate the simplifying FRP-to-concrete bonded joint shown in Figure 2; this FE model was adopted in this study. For easy understanding of the adopted FE model, the main features of the FE model are briefly described next while more detailed introduction of the model can be found in Chen et al. (2012a). The FE model was implemented in the FE software package ABAQUS 6.10 (2010). In the FE model, the FRP plate and the concrete prism were idealized as truss elements (T2D2) with their bending effects neglected, based on the assumption that the two adherends of the FRP-to-concrete bonded interface (i.e. the plate and the concrete prism) are subjected to axial deformation only. As the bondline between the FRP and the concrete prism is assumed to be under pure shear deformation which represents the lumped deformation of the actual adhesive layer and that of a thin layer of the adjacent concrete, it was modeled using an appropriate interfacial element by which the bond–slip behavior of the bondline can be defined. For the undamaged bond–slip relationships (Figure 3(c)), nonlinear spring element Spring 2 defined between a pair of nodes, acting in a prescribed direction, was used to define the fully reversible bond–slip behaviors; for damaged bond–slip relationships (Figure 3(d)), the four-node, two-dimensional interfacial cohesive element COH2D4 was used to define the elastic damage behaviors of the bondline, which is further explained below.

Figure 4.

Finite element model.

The constitutive properties of the cohesive elements include the following aspects (ABAQUS 6.10, 2010): (1) the initial elastic stiffness, (2) the damage initiation point, and (3) the damage elevation law. For the nonlinear bond–slip model with elastic damage as shown in Figure 3(d), these properties can be determined as follows.

The initial stiffness is defined using the following equation

K_{0} = \frac{τ_{0}}{s_{0}}

(1)

where s₀ and τ₀ are, respectively, the slip value and stress value corresponding to the damage initiation point. The damage initiation point is determined by choosing an initial slip value s₀ which should be small enough so that the assumed elastic segment of the bond–slip curve is not large to overshadow the nonlinear nature of the bond–slip curve. The elastic damage elevation law is then described by a scalar damage variable d according to the nonlinear bond–slip relationship using the following equation (see Figure 5)

d = 1 - \frac{τ}{K_{0} s}

(2)

where s and τ are, respectively, the slip value and stress value of a point on the bond–slip curve after the damage initiation.

Figure 5.

Definition of cohesive element according to Lu et al.’s (2005) nonlinear bond–slip models.

For bilinear bond–slip relationship, the damage elevation law is reduced to linear damage case as described in Chen et al. (2012a). In this study, the same approach as described in equation (2) was also used to model the damage of steel bar-to-concrete bonded interface between two adjacent cracks in an FRP-plated beam. The bond–slip behaviors between internal steel bar and concrete were defined according to CEB-FIP (1993) bond–slip model in which different bond–slip behaviors are provided for deformed bar and round plain bar, as shown in Figure 3(e) and (f).

It should be noted that the properties of the cohesive elements are independent of its nominal thickness if “traction-separation”-type constitutive law is used (ABAQUS 6.10, 2010). A nominal thickness of 1 mm was thus adopted in this study (Figure 4) as in Chen et al. (2012a). All the results presented in this article were obtained using the Riks method (Riks, 1979) which is capable of obtaining the full-range load–displacement response of the bonded joint as shown later.

Reference parameters

In the numerical examples presented later in this article, the following reference parameters were used unless otherwise stated.

For FRP-to-concrete bonded interface:

Geometrical and material parameters: nominal thickness of FRP sheet t_p = 0.165 mm, width of FRP sheet b_p = 50 mm, width of the concrete prism b_c = 100 mm (i.e. the FRP-to-concrete width ratio is 0.5; see Lu et al. (2005)), elastic modulus of the FRP sheet E_p = 230 GPa, concrete cylinder compressive strength $f'_{c} = 30 MPa$ , and elastic modulus of concrete E_c = 28.6 GPa ( $E_{c} = 4730 \sqrt{{f'}_{c}}$ according to ACI 318 (American Concrete Institute, 2008)).

Local nonlinear bond–slip parameters as determined from the simplified bond–slip model proposed by Lu et al. (2005): maximum shear stress τ_f = 4.317 MPa, interfacial fracture energy G_f = 0.52 N/mm², and s₁ = 0.056 mm (s₁ is the slip corresponding to the peak shear stress; see Figure 3(a) to (d)). In order to compare the numerical results between the nonlinear bond–slip model (Figure 3(c) and (d)) and the bilinear bond–slip model (Figure 3(a) and (b)), parameters of the bilinear bond–slip curve were also determined according to Lu et al. (2005) using the same geometrical and material parameters listed above, as follows: τ_f = 4.317 MPa, G_f = 0.52 N/mm², s₁ = 0.056 mm, s_f = 0.242 mm (s_f is the slip at which shear stress reduces to 0; see Figure 3(a) to (d)).

Load ratios: β = P₂/P₁ = 0.8 and µ = P₃/P₁ = 1 (it was assumed that $P_{1} > P_{2} > 0$ without loss of generality).

Bond length L = 100 mm, which represents a typical crack spacing in RC beams.

For steel bar-to-concrete interface:

Concrete has a cylinder compressive strength $f'_{c} = 30 MPa$ and a modulus of elasticity E_c = 28.6 GPa.

The steel geometrical and material parameters: the modulus of elasticity E_s = 200 GPa. The plain bars have a diameter of 10 mm and a yield strength f_y = 350 MPa. The deformed bars have a diameter D = 25 mm and a yield strength f_y = 460 MPa.

Local bond–slip parameters were determined according to CEB-FIP (1993) bond–slip model (see Figure 3(e) and (f)).

Load ratios: β = P₂/P₁ = 0.9 and µ = P₃/P₁ = 1.

A mesh with element lengths m = 1 mm was adopted in this study according to the mesh convergence study carried out in Chen et al. (2012a).

Verification of the FE model

To verify the validity of the above FE model, the load–displacement responses of FRP-to-concrete bonded joint predicted by the FE model and those from the theoretical solution of Teng et al. (2006) were compared under the assumption of the reference parameters (bilinear bond–slip relationship was assumed for both FE analysis and theoretical solution), as shown Figure 6. From Figure 6, it can be clearly seen that the FE model predicts a load–displacement response which is nearly identical to the theoretical solution, indicating the accuracy and validity of the adopted FE model.

Figure 6.

Comparison between FEM prediction and analytical solution.

Numerical results

In the parametric study presented below, when one parameter was varied, all other parameters were kept unchanged and same as the reference case.

Behaviors of FRP-to-concrete bonded interfaces between two cracks

Load–displacement responses

The full-range load–displacement responses from the undamaged bondline and the damaged bondline are shown in Figure 7 for β = 0.0, 0.5, 0.8, and 0.95, respectively. For comparison purpose, both the responses obtained from the nonlinear and the bilinear bond–slip model are shown in Figure 7 (the solid lines are for the results obtained from nonlinear bond–slip model and the dashed lines are for the results obtained from bilinear bond–slip model). For each β value, the numerical results of three different bond lengths (i.e. L = 15, 50, and 100 mm) were shown.

Figure 7.

Effect of bondline damage on full-range load–displacement responses of FRP-to-concrete bonded joints: (a) β = 0, (b) β = 0.5, (c) β = 0.8, and (d) β = 0.95.

It can be seen that for β = 0.0 and β = 0.5, the two curves of the undamaged model and the damaged model almost coincide with each other which is independent of the bond length, except that when β = 0.5 and L = 100 mm the peak load of the undamaged case is slightly higher than that of the damaged case for the nonlinear bond–slip model. It is also observed that the load–displacement curves have a longer plateau for a larger bond length, and the phenomena are similar for the cases of other β values.

For β = 0.8, the two curves fully coincide with each other only for a very small bond length L = 15, regardless of the types of adopted bond–slip models (i.e. bilinear bond–slip model or nonlinear bond–slip model). For bond length L = 50 mm, the two curves coincide with each other before P₁ reaches the peak; the responses of the damaged case diverge from the undamaged case when P₁ is around the peak value which is more pronounced for the responses obtained from nonlinear bond–slip model. For bond length L = 100 mm, the curve of the damaged case diverges from that of the undamaged case after P₁ reaches the plateau, and the undamaged bond–slip model leads to an apparently higher ultimate load (i.e. P_1,u) than the damaged bond–slip model at the same displacement. Again, it can be seen that the difference of P_1,u between the damaged case and undamaged case is more pronounced for the response obtained from nonlinear bond–slip model than that from bilinear bond–slip model.

For β = 0.95, similar observations can be achieved as for the case of β = 0.8, except that the differences of ultimate load between the damaged case and undamaged case become even more apparent, implying that larger β leads to larger loss of bond strength for the damaged interface.

Apart from the above, it is also noted that for the responses obtained from nonlinear bond–slip model, the load–displacement responses from the damaged case diverge from that of the undamaged case in the later stage of the descending branch for β > 0. 5 when the bond length is large (e.g. L = 100 mm).

Load capacity of the interface (P_1,u)

Effect of bond length

Figure 8(a) summarizes the effect of the bond length L on the ultimate load capacity of the interface (i.e. P_1,u which is defined as the peak load of the load–displacement curves shown in Figure 7) for different values of load ratio β. It can been observed that for the numerical results obtained from nonlinear bond–slip model, P_1,u first increases with L but remains nearly constant after L reaches a certain value when the load ratio β is very small (β < 0.5); for β values larger than 0.5, P_1,u first increases with L and reaches a peak value before it decreases to a constant value as L further increases. This phenomenon is more pronounced when β is closer to 1. It should be noted that similar conclusions were drawn by Chen et al. (2012a) on the numerical results obtained from a bilinear bond–slip models (Figure 3(a) and (b)) which are also shown in Figure 8 for comparison purpose.

Figure 8.

Effect of bond length L on the ultimate load P_1,u: (a) ultimate load P_1,u versus bond length L, (b) ultimate load P_1,u versus normalized bond length L/a_u, and (c) the normalized P_1,u versus normalized bond length L/a_u.

The same results are alternatively shown in Figure 8(b) where the bond length L is normalized against the characteristic softening length a_u. Following an approach similar to that of Teng et al. (2006), the characteristic softening length a_u can be obtained by the following equation deduced based on bilinear bond–slip model

a_{u} = \frac{1}{λ_{2}} \arccos (β)

(3)

where

λ_{2} = \sqrt{\frac{τ_{f}}{δ_{f} - δ_{1}} (\frac{1}{E_{p} t_{p}} + \frac{b_{p}}{b_{c} E_{c} t_{c}})}

(4)

For nonlinear bond–slip model, the characteristic softening length a_u can be estimated by the following equation

a_{u} = (\frac{2}{π}) \arccos (β) L_{e}

(5)

in which the effective bond length L_e is determined according to Chen and Teng (2001) as $L_{e} = \sqrt{(E_{p} t_{p}) / \sqrt{{f'}_{c}}}$ where t_p and t_c are the thicknesses of the FRP plate and the concrete prism, respectively, b_f and b_c are the widths of the FRP plate and concrete prism, respectively, E_p and E_c are the elastic modulus of the FRP plate and concrete, respectively, $f'_{c}$ is the concrete cylinder compressive strength, τ_f is the maximum bond stress of the bond–slip relationship, δ₁ is the interfacial slip corresponding to τ_f, and δ_f is the interfacial slip where the interfacial bond stress decreases to 0 (in the bilinear bond–slip model only).

It should be noted that the characteristic softening length a_u calculated using equation (5) (for the nonlinear bond–slip model) is larger than a_u obtained using equation (3) for bilinear bond–slip model which reflects the feature of the shear stress distribution as explained in the section “Interfacial shear stress distributions.” It is observed that all curves in Figure 8(b) with β > 0.5 reach their peak values at about L/a_u = 1.2. For smaller β values, the peak is reached at slightly larger L/a_u values. For example, when β = 0.0, the peak loads are reached at L/a_u = 1.8 and L/a_u = 1.6 for nonlinear and bilinear bond–slip models, respectively.

From Figure 8(b), it can also be seen that for β values less than 0.5, an effective bond length exists beyond which the ultimate load does not increase any more. However, for higher values of β, an optimum bond length instead of an effective bond length exists: the ultimate load reaches a peak value at the optimum bond length; it decreases to a constant value apparently smaller than the peak value with further increase in bond length. It is worth noting that at the same bond length (larger than the optimum bond length) and same load ratio β, the ultimate value of P_1,u obtained from the nonlinear bond–slip model is smaller than that from a bilinear bond–slip model especially when load ratio β is closer to 1 and the bond length L is apparently longer than the optimum bond length, which will be further explained in the section “Interfacial shear stress distributions” by examining the interfacial shear stress distributions.

To quantitatively assess the effect of bondline damage on the ultimate load capacity of the interface (i.e. P_1,u), the value of P_1,u is normalized against the corresponding load capacity of the interfaces without considering the effects of the bondline damage and shown in Figure 8(c) versus normalized bond length L/a_u. From Figure 8(c), it can be seen that the effect of the damage is significant when L/a_u > 1.2 and β > 0.7.

Effect of load ratio β

Figure 9(a) shows the effect of load ratio β on the ultimate load P_1,u. From Figure 9(a), it can be seen that the load ratio β has a very significant effect on the ultimate load (similar conclusions can be obtained from Figure 8 as well). It can be seen that the ultimate load P_1,u increases with β when L is constant, and this trend is more pronounced for a β value larger than 0.7. To quantitatively assess the effect of bondline damage on the ultimate load capacity of the interface (i.e. P_1,u), P_1,u is normalized against the load capacity of the interfaces without considering the effects of the bondline damage and shown in Figure 9(b) versus load ratio β. From Figure 9(b), it can be seen that (1) for the same bond length L, the normalized P_1,u decreases with the increase in load ratio β, indicating that larger β leads to larger detrimental effect of bondline damage on P_1,u which is regardless of the type of bond–slip models; (2) at the same the load ratio β, the normalized P_1,u declines as bond length L increases until the bond length is about 175 mm, implying that larger bond length leads to larger bondline damage on P_1,u which is regardless of the type of bond–slip models; (3) at the same load ratio β and the same bond length L, normalized P_1,u obtained from nonlinear bond–slip model is smaller than that obtained from the bilinear bond–slip model, showing that the bondline damage leads to larger reduction in P_1,u for the nonlinear bond–slip model than for bilinear bond–slip model. The above phenomena will be further explained in the section “Interfacial shear stress distributions” by examining the interfacial shear stress distributions.

Figure 9.

Effect of load ratio β on the ultimate load P_1,u: (a) the ultimate load P_1,u versus load ratio β and (b) the normalized P_1,u versus load ratio β.

Interfacial shear stress distributions

To further explain the effect of bondline damage on load–displacement response, especially on the ultimate load-carrying capacity P_1,u, the interfacial shear stress distributions at a number of load levels were plotted for three bond lengths L = 15, 50, and 150 mm, respectively; for each bond length L, the results of two typical load ratios, namely, β = 0.5 and 0.95, are shown, respectively. As a result, a total of six cases will be discussed as detailed next. Figures 10(a) to (f) show comparison of interfacial stress distributions of the above six cases for damaged and undamaged interfaces, respectively. Only the results of nonlinear bond–slip model are shown in Figure 10 for the damaged and undamaged cases while comparisons of the results from nonlinear and bilinear bond–slip models are shown in Figure 11 for the damaged cases.

Figure 10.

Effect of bondline damage on interfacial shear stress distribution: (a) L = 15 mm, β = 0.5; (b) L = 15 mm, β = 0.95; (c) L = 50 mm, β = 0.5; (d) L = 50 mm, β = 0.95; (e) L = 150 mm, β = 0.5; and (f) L = 150 mm, β = 0.95.

Figure 11.

Effect of type of bond–slip models on interfacial shear stress distribution: (a) L = 15 mm, β = 0.5; (b) L = 15 mm, β = 0.95; (c) L = 50 mm, β = 0.5; (d) L = 50 mm, β = 0.95; (e) L = 150 mm, β = 0.5; and (f) L = 150 mm, β = 0.95.

Figure 10(a) shows the case of L = 15 mm and β = 0.5: for all the load levels shown, the shear stress distribution of the undamaged model and the damaged model nearly coincides with each other. This is because the local slip reversal as a result from the interaction between the loads applied at the two ends does not appear in this case. The two curves at the load levels of 2/4/6 kN illustrate this situation. Figure 10(b) shows for the case of L = 15 mm and β = 0.95: for all the load levels shown, the shear stress distributions of the undamaged model and the damaged model nearly coincide with each other, although slight difference appears. More detailed analysis showed that this is because the bondline near the left side with a lower load (i.e. $P_{2}$ ) does not enter the softening range before the local slip reversal caused by the interaction between the two loads, even though the load ratio is large; the two curves at the load level of 20 kN illustrate this situation. As a result, the damage has nearly no effect on the interfacial shear distribution, and thus, the full-range load–displacement responses from the damaged model and the undamaged model are nearly the same as shown in Figure 7(d). Figures 10(c) and (e) show the cases of $β = 0.5$ for $L = 50 mm$ and $L = 150 mm$ , respectively. Similar observations as those for $L = 15 mm$ and $β = 0.95$ (Figure 10(b)) can be made. Figures 10(d) and (f) show the cases of $β = 0.95$ for $L = 50 mm$ and $L = 150 mm$ , respectively. It can be seen that before $P_{1}$ reaches the plateau (cf. Figure 7(d)), the bondline damage does not produce any significant difference in the shear stress distribution as shown by the two curves at $P_{1} = 13 kN$ for $L = 50 mm$ (Figure 10(d)) and those at $P_{1} = 10 kN$ for $L = 150 mm$ (Figure 10(f)). The shear stress peaks on the right and the left are far apart and no local slip reversal has occurred at this stage. Note that the shear stress peaks represent the fronts of the softening zones on the right and the left, respectively (which are also the boundaries between elastic zone and softening zone) and they are referred to as the softening fronts in the remainder of this article. When $P_{1}$ is further increased (e.g. 21 and 18 kN for L = 50 mm and L = 150 mm, respectively, as shown in Figures 10(d) and (f)), the right softening front moves forward to the left zone where the bondline has already been damaged (or softened) under $P_{2}$ . The damaged and undamaged bond–slip models lead to different interfacial stress distributions in this case because part of the left softening zone has experienced slip reversals. The damaged model also leads to a slightly smaller debonded zone on the right (where the interfacial shear stress is 0). This is because the bondline on the left is weaker (partially damaged) in this case as indicated by the lower shear stress of the damaged model compared to that of the undamaged model (Figures 10(d) and (f)); a longer bond length is thus mobilized to resist the load difference between $P_{2}$ and $P_{1}$ (Figures 10(d) and (f)). This explains why the damaged model leads to a smaller displacement (compared to that of the undamaged model) at the right end at the same load level above the plateau (Figure 7(d)) given that under the same load level, less debonded length means less deformation. When $P_{1}$ increases to the ultimate load $P_{1, u}$ (P_1,u = 31 kN for the undamaged model and P_1,u = 23 kN for the damaged model when L = 150 mm), the apparent differences in shear stress distributions also exist. In particular, it is observed that the direction of shear stress near the left end remains the same as that at lower load levels (e.g. $P_{1} = 13 kN$ for $L = 50 mm$ and $P_{1} = 10 kN$ for $L = 150 mm$ ) for the damaged bondline while changes for the undamaged bondline; furthermore, the resultant force (equal to $P_{1} - P_{2}$ ) carried by the mobilized part (i.e. with nonzero stress value) of the damaged bondline is smaller than that of the undamaged bondline. This explains why the bondline damage leads to smaller load capacity of the interface as discussed in the section “Load capacity of the interface (P_1,u).”

Figure 11 shows that stress distributions of nonlinear bond–slip model are different from those of the bilinear bond–slip model especially for large bond length ( $L \geq 50$ ) and high load ratio (i.e. $β \geq 0.8$ based on further analyses not presented here). For example, when $L = 150 mm$ and $β = 0.95$ , the nonlinear bond–slip model leads to shorter debonded length on the right end at a load level higher than the plateau and lower resultant force of the mobilized interface (with nonzero shear stress). As a result, compared to the bilinear bond–slip model, the nonlinear bond–slip produces a slightly smaller displacement of interface at the same load level (higher than the plateau) and lower ultimate load capacity (cf. Figure 7(d)).

The above results (shown in Figures 7 to 11) suggest that in the FE analysis of IC debonding failure of FRP-plated RC beam where interaction between the two adjacent cracks exists, an accurate nonlinear bond–slip model (e.g. the simplified and accurate bond–slip models of Lu et al. (2005)) should be used, and the effects of bondline damage should be duly considered for the FRP-to-concrete interfaces.

Behaviors of steel bar-to-concrete bonded interfaces between two cracks

Figures 12 and 13 show the full-range load–displacement responses for the deformed bars and the plain bars, respectively. In each of Figures 12 and 13, the numerical results obtained from the undamaged bond–slip model and the damaged bond–slip model are shown for comparison. It should be noted that in the remainder of this section, parameters of the reference case (see section “Reference parameters” for reference) were used to obtain the numerical results if not otherwise specified.

Figure 12.

Effect of bondline damage on full-range load–displacement responses for deformed bars: (a) varied bar diameter D, (b) varied concrete compression strength $f'_{c}$ , (c) varied bond length L, and (d) varied reinforcement yield strength f_y.

Figure 13.

Effect of bondline damage on full-range load–displacement responses for plain bars: (a) varied bar diameter D, (b) varied concrete compression strength $f'_{c}$ , (c) varied bond length L, and (d) varied reinforcement yield strength f_y.

From Figures 12(a) to (d), it can be seen that the bondline damage has no effect on the load–displacement response of the steel bar-to-concrete bonded interfaces (referred to as “the interfaces” for simplicity in the remainder of this section if not otherwise stated) which is regardless of the bar diameter (Figure 12(a)), concrete strength (Figure 12(b)), bond length (Figure 12(c)), and yielding strength of steel bars (Figure 12(d)). A more detailed analyses on the numerical results shown in Figures 12(a) to (d) revealed that yield strength in the practical range (e.g. 460–660 MPa) has led to the yielding of the steel bars; as a result, the ultimate load capacities of the interfaces are governed by the yielding of the steel bars. Similar conclusions can be drawn for plain bars from Figure 13(a) to (d), namely, the bondline damage has no effect on the load–displacement response.

To further investigate the effect of bondline damage on the bond behavior of the interface for both deformed bars and plain bars, the “cut-off” effect of yield strength on the bond behavior of the interface should be removed. For this purpose, the yield strength of deformed bar and plain bar was artificially increased to 4600 MPa (i.e. 10 times yield strengths of deformed bar) and their bar diameters were equally set as 25 mm. Figures 14 and 15 show the results of the deformed bars and plain bars under the above assumptions. From Figure 14(a), it can be seen that for the deformed bars, when $L = 200 mm$ , the bondline damage has significant effect on the ultimate load of the interface only when $β \geq 0.95$ ; from Figure 14(b), it can be seen that when $β = 0.95$ , bondline damage has a significant effect on the bond behavior of the interface (in particular, on the ultimate load of the interface) only when $L \geq 200 mm$ . However, from Figure 15(a) and (b), it can be seen that the bondline damage has nearly no effect on the bond behavior of the interface for plain bars.

Figure 14.

Effect of bondline damage on full-range load–displacement responses for deformed bar when D = 25, f_y = 4600: (a) various β for L = 200 mm and (b) various L for β = 0.95.

Figure 15.

Effect of bondline damage on full-range load–displacement responses for plain bar when D = 25, f_y = 4600: (a) various β for L = 200 mm and (b) various L for β = 0.95.

To further investigate the mechanism of the above phenomena, the interfacial shear stress distributions were obtained and are shown in Figures 16 and 17 for deformed bar and plain bar, respectively, for the case of $β = 0.95$ . From Figures 16(a) to (c), it can be seen that when bond length is small (e.g. $L = 50 mm$ ), the shear stress distributions are nearly the same for the damaged and undamaged bondline. When bond is large enough (e.g. $L = 400 mm$ ), the bondline damage has a significant effect on the shear stress distribution especially near the left end; as a result, the resultant force (equal to $P_{1} - P_{2}$ ) carried by the mobilized part (i.e. with nonzero stress value) of the damaged bondline is smaller than that of the undamaged bondline, leading to smaller load capacity of the interface as shown in Figure 14. From Figure 16(b) and (c), it can also be found that the damaged model also leads to a slightly smaller constant stress zone on the right end (where the interfacial shear stress is reduced a small constant value resulting from friction); this explains why the damaged model leads to a slightly smaller displacement (compared to that of the undamaged model) at the right end at the same load level within the second segment of the load–displacement curves (Figure 14(a) and (b)). Similar observations can be obtained from Figure 17(a) and (b) for plain bars except that (1) the difference of shear stress distribution exists only in the small region close to the left end of the interface and (2) the shear stress eventually becomes constant along the length of the interface due to the feature of bond–slip model of plain bars as shown in Figure 3(e) and (f). This explains why the load capacities of the interfaces eventually become the same for both damaged and undamaged interfacial, although they slightly diverge from each other during the loading process when $L \geq 200 mm$ and $β \geq 0.95$ , as shown in Figure 15(a) and (b).

Figure 16.

Effect of bondline damage on interfacial shear stress distribution for deformed bar when β = 0.95, D = 25, f_y = 4600: (a) L = 50 mm, (b) L = 200 mm, and (c) L = 400 mm.

Figure 17.

Effect of bondline damage on interfacial shear stress distribution for plain bar when β = 0.95, D = 25, f_y = 4600: (a) L = 50 mm, (b) L = 200 mm, and (c) L = 400 mm.

Effects of damage localization

Under the same assumption of elastic damage (see equation (2) and Figures 3(b), (d), and (f)), the different effects of bondline damage on the load–displacement responses and the ultimate load capacity of the interfaces (FRP-to-concrete bonded interfaces and steel bar-to-concrete bonded interfaces) can be further explained in terms of the effects of damage localization. A comparison of Figures 10(f), 16(c), and 17(c) unveils that during the later stage of loading, the mobilized interfaces capable of transferring shear stress (i.e. nonzero shear stress) have different localization features corresponding to the different softening properties of the bond–slip models shown in Figures 3(a) to (f): for FRP-to-concrete bonded interface, the interfacial shear stresses are fully localized into a region which can be further characterized by the characteristic softening length a_u as explained previously; for deformed bar-to-concrete bonded interface, the interfacial shear stresses are partially localized into a region significantly larger than that of FRP-to-concrete interface, with a small constant shear stress distribution for the rest region; for the plain bar-to-concrete bonded interface, there is no such shear stress localization phenomenon. As a result, the maximum effects of bondline damage on ultimate load capacity are most pronounced for the FRP-to-concrete bonded interface (Figure 7), followed by the deformed bar-to-concrete interface (Figure 14), and there is nearly no damage effect for the plain bar-to-concrete interface (Figure 15).

Conclusions

This study has presented a numerical investigation into the role of modeling unloading bond–slip behavior of the interfaces in FE analyses of the IC debonding failure of FRP-plated RC beams. The numerical investigation had been based on a simplifying bonded joint representing the bonded interface between two adjacent cracks in FRP-plated beams. Both the interfaces between FRP and concrete and those between steel bars and concrete were investigated. Elastic damage was assumed for the unloading behavior of the damaged interfaces while fully reversible bond–slip behavior was assumed for the undamaged interfaces. From the numerical results and discussions presented in this article, the following conclusions can be drawn:

Interfacial slip reversals usually appear in the interfaces due to the interaction of two adjacent cracks in an FRP-strengthened beam; because of the interfacial slip reversal, the unloading bond–slip behavior of the interfaces (FRP-to-concrete interfaces and steel bar-concrete interfaces) should be considered in the FE analyses of FRP-strengthened beams.

For the FRP-to-concrete bonded interfaces, the damage of the interfaces has significant effect on the load–displacement response of the interface especially when the bond length is larger (i.e. L/a_u > 1.2) and the load ratio is larger than 0.7 (i.e. $β \geq 0.7$ ). Furthermore, the shape of the bond–slip model affects the shear stress distributions and thus has considerable effect on the bond behavior of the interfaces (e.g. ultimate load capacity) under a certain condition (i.e. $β \geq 0.7$ and L/a_u > 1.2), so a more accurate nonlinear bond–slip model should be used instead of a bilinear one for the accurate prediction on the bond behavior of FRP-to-concrete bonded interface.

For the steel bar-to-concrete bonded interfaces, the damage of the interfaces has nearly no effect on the load–displacement response of the interfaces when the yield strength of the bar is in the practical ranges (e.g. 250–450 MPa for plain bars and 460–660 MPa for deformed bars) mainly because the failures of the interfaces are governed by the yielding of the steel bars. For deformed bars, the bondline damage may have a considerable effect on the bond behavior of the interface only when the yield strength is very high (e.g. 10 times the yield strength of its normal range) so that the failure of the interface is not governed by the yielding of the steel bars; for plain bars, even when the yield strength is very high, the effect of the bondline damage is negligible as the damage only affects a very small part of the interfacial shear stress distribution due to the feature of the bond–slip model.

Footnotes

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: The authors acknowledge the financial support received from the National Natural Science Foundation of China (Project Nos 51108097, 51378130, and 51308271) and Guangdong Natural Science Foundation (Project No. S2013010013293). The authors are also grateful for the financial support received from the Department of Education of Guangdong Province for Excellent Young College Teacher of Guangdong Province (Project No. Yq2013056). G.C. would like to thank the China Scholarship Council (CSC) for the scholarship awarded to him as a visiting scholar (File No. 201408440320) in the Department of Civil and Environmental Engineering, University of California, Berkeley, USA.

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Role of interfacial damage modeling in finite element analysis of intermediate crack debonding failure of FRP-plated reinforced concrete beams: A numerical investigation

Abstract

Keywords

Introduction

A simplifying bonded joint model

Local bond–slip models with assumed unloading curve

FE modeling

The FE model

Reference parameters

Verification of the FE model

Numerical results

Behaviors of FRP-to-concrete bonded interfaces between two cracks

Load–displacement responses

Load capacity of the interface (P1,u)

Effect of bond length

Effect of load ratio β

Interfacial shear stress distributions

Behaviors of steel bar-to-concrete bonded interfaces between two cracks

Effects of damage localization

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References

Load capacity of the interface (P_1,u)