Three-dimensional finite element modeling of intermediate crack debonding in fiber-reinforced-polymer-strengthened reinforced concrete beams

Abstract

Intermediate crack debonding is a common failure mode of reinforced concrete beams strengthened in flexure with an externally bonded laminate (sheet or plate) of fiber-reinforced polymer. Many finite element models have been developed to predict this failure mode. Research shows that the accurate modeling of interfaces between concrete and either internal steel or external fiber-reinforced polymer reinforcements is very important. This article presents a three-dimensional finite element model based on the smeared crack approach for predicting intermediate crack debonding failure of fiber-reinforced-polymer-strengthened reinforced concrete beams (including preloaded beams). In this model steel-to-concrete and fiber-reinforced-polymer-to-concrete interfaces are more expediently modeled. The finite element results agree well with experimental results on crack patterns, fiber-reinforced polymer strain distribution, and the variation of strain and deflection with load. This model can also simulate the fiber-reinforced polymer debonding process of the beam and the response of the residual beam after the fiber-reinforced polymer reinforcement has separated from the reinforced concrete beam. In addition, parameters’ analyses are further conducted to find the differences between the two-dimensional and three-dimensional models. Simulations of fiber-reinforced polymer-plated slabs or beams with additional anchors are mostly three-dimensional problems, and their focus is also on intermediate crack debonding. This model can be used to simulate fiber-reinforced-polymer-plated slabs or beams with additional anchors such as U-jacket strips or mechanically fastened fiber-reinforced polymer strengthening systems in future research.

Keywords

bonding concrete beams fiber-reinforced polymer finite element method preloaded beams

Introduction

Reinforced concrete (RC) beams can be strengthened by bonding fiber-reinforced polymer (FRP) plates to their tension face. There are several failure modes of this strengthening method. They are summarized as (1) flexural failure by FRP rupture, (2) flexural failure by crushing of compressive concrete, (3) shear failure, and (4) failure by interfacial debonding between FRP plates and RC beams. The modes of debonding in FRP-plated beams can be classified into four types (Teng et al., 2003): (a) concrete cover separation, (b) plate end interfacial debonding, (c) intermediate flexural crack-induced interfacial debonding, and (d) intermediate flexural–shear crack-induced interfacial debonding. Many attempts have been made to explore effective solutions for debonding prevention (Kalfat et al., 2011; Wu and Huang, 2008; Zhou et al., 2017). Meanwhile, analysis of failure mechanism and prediction of ultimate load are of the same importance.

Failure modes (c) and (d) abovementioned can be collectively referred to as intermediate crack (IC) debonding (IC debonding) which is very common in FRP-strengthened RC beams. IC debonding initiates at an IC in the beam and then propagates to one of the laminate ends. Many finite element (FE) models have been made to simulate IC debonding. To accurately predict IC debonding in FRP-plated RC beams, an FE approach should consider the following three key components: (a) appropriate modeling of concrete cracking, (b) accurate modeling of the bond of FRP-to-concrete interface, and (c) accurate modeling of the bond of steel-to-concrete interface (Chen et al., 2010).

There are mainly two approaches of FE analysis to simulate the crack propagation in RC beams. One is the discrete crack approach, which is the first proposed crack model to simulate concrete cracking. The basic idea is to treat the crack as the boundary of the elements, adjust the position of the node or add new nodes once the crack appears, and remesh the elements so that the crack is between the boundaries of the elements. In this way, the discontinuity caused by cracks can be described naturally, and the location, shape, and width of cracks can be clearly expressed. Therefore, this approach can predict slip concentrations near a crack. To trace the propagation of cracking, it requires continuous remeshing in the solution process. In RC beams, many cracks need to be traced, which involves prohibitively large computational effort. Moreover, pre-definition of crack positions is required in the discrete crack approach. As a result, the discrete crack approach has only been used to investigate IC debonding resulting from one or several pre-defined cracks (Monti et al., 2003; Niu and Wu, 2005; Sun et al., 2015; Yang et al., 2003). The other is the smeared crack approach. Its essence is to “smear” actual concrete cracks into the entire element, treat concrete material as anisotropic material, and use the material constitutive model of concrete to simulate the influence of cracks. In this way, when the stress of a concrete element exceeds the cracking stress, the material constitutive matrix can be adjusted without changing the element shape or remeshing the elements, which is easy to implement using a FE program. Therefore, it has been much more popular with researchers who are concerned with IC debonding in FRP-plated RC beams (Baky et al., 2007; Chen et al., 2010; Coronado and Lopez, 2006; Fu, 2016; He et al., 2006; Kotynia et al., 2008; Li and Wu, 2018; Lu et al., 2007; Neale et al., 2006; Nour et al., 2007; Pham and Al-Mahaidi, 2005; Teng et al., 2004; Wong and Vecchio, 2003; Wu and Yin, 2003). The assumptions in the smeared crack approach lead to the mesh sensitivity problem: results of the simulation with the smeared crack approach are sensitive to the choice of size of elements. To address the mesh sensitivity problem, Bažant and Oh (1983) proposed the crack band model, which further developed the traditional smeared crack model. By introducing the concepts of fracture zone and fracture energy, the smeared crack model was combined with fracture mechanics to reduce the influence of element size.

Appropriate modeling of FRP-to-concrete interfaces is essential for the accurate prediction of IC debonding. Most existing studies considered slips between the FRP plate and the concrete using a bond-slip model. The model is generally derived from pull tests in which the FRP-to-concrete interface is subject to shear by loading the FRP plate in tension. However, many researchers have found that the debonding strength of the interface is often overestimated if the bonding slip relationship obtained from pull tests is directly used to simulate the interface element (Wong and Vecchio, 2003). When the smeared crack model is employed to simulate the crack propagation, the crack is simulated evenly distributed as tensile strain over a representative zone of concrete. It does not require remeshing, but it cannot predict slip concentrations near a crack. As a result, the slip near the crack obtained from the smeared crack model is smaller than the actual slip. Therefore, if the relationship derived from pull tests is directly used to represent the shear bond stress–slip behavior of the FRP-to-concrete interface, the load capacity will be overestimated. Lu et al. (2007) proposed a dual local debonding criterion to solve this problem.

Appropriate modeling of steel-to-concrete interfaces is essential for capturing the localization of slips at cracks. Most existing studies on FE modeling of FRP-plated RC beams considered the bond behavior of steel-to-concrete interface using either perfect bonding (i.e. directly connecting the concrete elements to steel elements) or tension stiffening concrete stress–strain model (i.e. directly connecting the concrete elements to steel elements and modifying the stress–strain model for concrete elements adjacent to consider the slip between concrete and steel tension bars; Baky et al., 2007; Coronado and Lopez, 2006; Kotynia et al., 2008; Lu et al., 2007; Neale et al., 2006; Nour et al., 2007; Pham and Al-Mahaidi, 2005; Teng et al., 2004; Wong and Vecchio, 2003). Both approaches are inaccurate in predicting cracking behavior (i.e. crack patterns and widths), thus leading to inaccurate prediction of IC debonding (Chen et al., 2010). Chen et al. (2010) considered the bond behavior of both steel-to-concrete interface and FRP-to-concrete interface using interfacial elements cooperating with the appropriate bond–slip models for them to connect the concrete elements and steel elements. Numerical results presented in Chen et al. (2010) show that their approach has the ability to accurately predict the crack pattern as well as crack width, thus accurately predicting IC debonding of FRP-plated RC beams.

Based on Lu et al. (2007) and Chen et al.’s (2010) research, this article presents a three-dimensional (3D) FE model to simulate IC debonding in FRP-plated RC beams (including preloaded beams), as there is no FE model published to simulate IC debonding failure of preloaded beams. Our model is implemented in ABAQUS (Simulia, 2013). In this model, the interfaces between the concrete and both the internal steel and the external FRP reinforcements are more expediently modeled. The capability and accuracy of the proposed FE model are demonstrated through comparisons of its predictions with selected test results. In addition, parameters’ analyses that cannot be implemented in two-dimensional (2D) models are further studied to observe the differences between 2D and 3D models. The model is based on the smeared crack approach, while the crack band theory (Bažant and Oh, 1983) is adopted to overcome the mesh sensitivity problem. More information about this approach is shown in the following.

Modeling of concrete

The concrete is modeled using the 3D solid element C3D8R which is an eight-node linear brick–adopted reduced integration with hourglass control. The compress and tensile behavior is defined using concrete plasticity damage model in ABAQUS. Poisson’s ratio and the dilation angle are assumed to be 0.2 and 35°, respectively. The stress–strain relationship of concrete under uniaxial compression is defined based on the following equation proposed by Saenz (1964)

σ_{c} = \frac{α ε}{1 + (\frac{α ε_{p}}{σ_{p}} - 2) (\frac{ε}{ε_{p}}) + {(\frac{ε}{ε_{p}})}^{2}}

(1)

where σ_c and ε are the compressive stress and compressive strain, respectively; σ_p and ε_p are the maximum compressive stress and its corresponding strain, and they are set to be equal to the cylinder compressive strength $({f_{c}}^{'})$ and 0.002, respectively; and α is the initial tangent modulus which is set to be equal to the elastic modulus of concrete E_c that is estimated following the equation from the American Concrete Institute (ACI, 2008) guidelines

E_{c} = 4730 \sqrt{{f_{c}}^{'}}

(2)

The tensile behavior of concrete is defined as linear elastic up to the tensile strength (f_t), and the descending branch is determined following the stress–crack opening displacement relationship proposed by Hordijk (1991)

\frac{σ_{t}}{f_{t}} = [1 + {(c_{1} \frac{w_{t}}{w_{cr}})}^{3}] e^{(- c_{2} \frac{w_{t}}{w_{cr}})} - \frac{w_{t}}{w_{cr}} (1 + c_{1}^{3}) e^{(- c_{2})}

(3)

w_{cr} = 5.14 \frac{G_{F}}{f_{t}}

(4)

where w_t is the crack opening displacement, w_cr is the crack opening displacement at the complete loss of tensile stress, σ_t is the tensile stress normal to the crack direction, G_F is the fracture energy required to create a stress-free crack over a unit area, and c₁ = 3.0 and c₂ = 6.93 are constants determined from tensile tests of concrete. The Comite Euro-International du Beton–Federation International de la Precontrainte (CEB-FIP, 1993) equation is used in this article to estimate f_t and G_F

f_{t} = 1.4 {(\frac{{f_{c}}^{'} - 8}{10})}^{(2 / 3)} MPa

(5)

G_{F} = (0.00469 {d_{a}}^{2} - 0.5 d_{a} + 26) {(\frac{{f_{c}}^{'}}{10})}^{0.7} N / mm

(6)

where d_a is the maximum aggregate size of the concrete. In this study, it is assumed to be 20 mm if no test data are given. The stress–displacement curve determined by equations (3) to (6) can be transformed into a stress–strain curve according to the crack band model in ABAQUS.

The damage variable (d) is computed using the equations proposed by Lubliner et al. (1989), whose plastic degradation occurs only in the softening range. It can be described during both compression and tension behavior as

d = 1 - \frac{σ}{f}

(7)

where σ is the stress and f is either the compressive strength $({f_{c}}^{'})$ or tensile strength (f_t) of concrete as appropriate.

Modeling of steel and steel–concrete bond behavior

The steel is modeled using truss element T3D2 (two-node linear displacement), and the constitutive relation is shown in Figure 1, which is a bilinear relation allowing for the strain hardening slope. In Figure 1, f_su is the ultimate tensile strength, f_sy is the yield strength, and ε_su and ε_sy are the corresponding steel strains.

Figure 1.

Constitutive relation for steel.

The bond behavior between longitudinal bars and concrete is modeled using spring elements, while stirrups are modeled using the embedded element technique as the slip between stirrups and concrete is neglected. The embedded element technique is described in the following: ABAQUS will search for the geometric relationships between nodes on the embedded elements and the host elements. If a node on an embedded element lies within a host element, the degrees of freedom at the node will be eliminated by constraining them to the interpolated values of the degrees of freedom of the host element (Simulia, 2013).

The process used to create spring elements is described as follows. First, create a virtual longitudinal bar. Its property, length, position, and mesh size are the same as those of longitudinal bars, but its sectional area is very small, that is, 0.001 mm². Then, embed the virtual longitudinal bar elements into the concrete elements. Finally, connect the same position nods of virtual longitudinal bar elements and longitudinal bar elements using spring elements. In this way, the force of the longitudinal bar can be evenly transferred to the concrete elements through the virtual bar, and the modeling process is simplified at the same time. The behavior of spring elements in the direction parallel to the steel–concrete interface is defined by using the CEB-FIP (1993) bond–slip model. It can be expressed as follows for deformed bars

τ = {\begin{matrix} τ_{max} {(\frac{s}{s_{1}})}^{α} (0 \leq s \leq s_{1}) \\ τ_{max} (s_{1} \leq s \leq s_{2}) \\ τ_{max} - (τ_{max} - τ_{f}) (\frac{s - s_{2}}{s_{3} - s_{2}}) (s_{2} \leq s \leq s_{3}) \\ τ_{f} (s_{3} \leq s) \end{matrix}

(8)

where s is the slip between steel and concrete and τ is the corresponding shear stress. $τ_{max} = 2.0 \sqrt{{f_{c}}^{'}}$ , τ_f = 0.15τ_max, α = 0.4, s₁ = s₂ = 0.6 mm, and s₃ = 1.0 mm. In the direction normal to the interface, it is assumed that there is no relative displacement between the steel reinforcement and the concrete.

Modeling of FRP and FRP–concrete bond behavior

The FRP reinforcement is modeled using 3D shell elements S4R which are four-node general-purpose shell elements adopted reduced integration with hourglass control, and it is assumed to be linear–elastic–brittle. In ABAQUS, researchers usually use cohesive elements to analyze bond–slip behavior between FRP and concrete; however, this requires a high demand of meshes and is sensitive to adhesive layer thickness. After several attempts, we put aside the idea of creating a new adhesive layer independently; instead, bond–slip behavior between FRP and concrete can be achieved by defining contact interaction analysis “Cohesive Behavior” and “Damage.” This method is similar to the zero-thickness cohesive element, which is insensitive to mesh changes and has better convergence. The “Maximum nominal stress criterion” is used as well as the “Damage stabilization” principle in which a small coefficient of viscosity is set to increase convergence.

The property of the contact interaction is defined using “Bond-Slip Model II” (Lu et al., 2007). The reason we chose the model and the details of the model are as follows. Lu et al. (2005a) developed a bond–slip model for the behavior of FRP-to-concrete bonded joints in pull tests (Lu et al., 2005b), and they examined the interfacial bond–slip curves at different locations along the interface in an FRP-strengthened plain concrete beam (Lu et al., 2007). The results showed that the bond–slip relationship is very close for the parts of the interface outside the major flexural crack zone. For the part of the interface inside the major flexural crack zone, which is the focus of IC debonding simulation, the bond–slip relationship can be closely described by the bond–slip model of Lu et al. (2005a) with a brittle post-peak branch. In this study, the property of the contact interaction in the direction parallel to the steel–concrete interface is defined by the bilinear model of Lu et al. (2005a) with a brittle post-peak branch, which is a simplified version of “Bond-Slip Model II.” It can be expressed as follows

τ = {\begin{matrix} τ_{max} \frac{s}{s_{0}} (0 \leq s \leq s_{0}) \\ 0 (s > s_{0}) \end{matrix}

(9)

where

τ_{max} = 1.5 β_{w} f_{t}

(10)

s_{0} = 0.0195 β_{w} f_{t}

(11)

β_{w} = \sqrt{\frac{2.25 - b_{f} / b_{c}}{1.25 + b_{f} / b_{c}}}

(12)

where b_f is the width of the FRP plate and b_c is the width of the concrete beam. Normal to the interface, the property of the contact interaction is assumed to behave linear elastically with a stiffness of the adhesive layer, which is also adopted by Chen et al. (2010).

Modeling techniques

To improve the convergence of the numerical solution, a dynamic approach is employed, in which an essentially static nonlinear deformation problem is treated as a dynamic problem and solved using the Hilber–Hughes–Taylor α method (Hilber, 1978; an implicit time integration method) available in ABAQUS. To employ the dynamic approach, the densities of concrete, steel, and FRP plate are set to be 2400, 7800, and 1750 kg/m³, respectively, while Rayleigh mass proportional damping and Rayleigh stiffness proportional damping for all three materials are calculated by prescribing a viscous damping ratio value of 0.0005, and the loading time is set to be 2 s. It is proved in Chen et al. (2015) that this dynamic approach with these settings can achieve a close approximation of the static solution for IC debonding in FRP-plated RC beams.

To simulate beams under sustained load, the model change technique in ABAQUS is adopted, which can be used to simulate removal and reactivation of elements or contact pairs. The progress of implementing model change technique is described as follows. First, create a virtual FRP plate whose progress is similar to the creation of virtual bar. The function of virtual FRP is to locate the FRP location at a specific loading time. Then, “Merge” the virtual FRP part and FRP part, noting that “Remove duplicate elements” in “Merge nodes” option should not be selected. In this way, a new part with the same node numbers but different element numbers has been generated. Because the virtual FRP shares nodes with the FRP, when the FRP is reactivated, it is positioned at the location of the virtual FRP. Next, create contact interaction between FRP and concrete. We set tangential and normal contact interaction behavior between virtual FRP and concrete as frictionless and “Hard” contact, respectively, and do not select “Allow separation after contact.” Finally, set the “Model change” options: deactivate FRP elements in the start and then reactivate them under a specific load and deactivate virtual FRP elements simultaneously. The corresponding contact interaction in the “Model change” is set in a similar way.

By taking advantage of symmetry, only a quarter of the specimen was modeled for all the RC beams. Since the FE model is a quarter-symmetric model, the corresponding symmetric constraints should be set at the midspan section along the length direction and the mid-section along the width direction, and vertical constraints should be set at the centerline of the base of the support.

Comparison between FE and experimental results

Matthys (2000) reported the results of experimental study of RC beams strengthened in flexure with CFRP reinforcement, which included load–deflection relationship, cracking behavior, moment–strain behavior, and FRP strain distribution. Comparisons have been made between experimental data and FE results in such behaviors. Matthys (2000) tested nine RC beams, two of which were reference beams (BF1 and BF7) and others were FRP-plated beams (BF2–BF6, BF8, and BF9). It should be noted that all of the FRP-plated beams were failed by IC debonding. Beam BF5 was preloaded with 110 kN. Beams BF4 and BF6 were precracked and anchored at the plate ends, respectively, which are beyond the scope of this study. The test behavior of beam BF3 was almost the same as that of BF2. Therefore, six beams (BF1, BF2, BF5, and BF7–BF9) were simulated to compare with the test results in predicting IC debonding failure. Geometrical and material properties of those beams are given in Figure 2 and Table 1.

Figure 2.

Simple 3D model with geometrical and loading data.

Table 1.

Geometrical and material properties of RC beam specimens.

Specimen		BF1	BF2	BF5	BF7	BF8	BF9
Beam	Type of strengthening (U = unstrengthened; S = strengthened; P = preloaded)	U	S	P	U	S	S
	Span/shear span (mm)	3800/1250
	Width (mm)	200
	Height/effective depth (mm)	450/410
Concrete	Cylinder compressive strength (MPa)	33.7	36.5	37.4	38.5	39.4	33.7
	Modulus of elasticity (MPa)	27,460	28,580	28,927	29,350	29,690	27,460
	Tensile strength (MPa)	2.627	2.814	2.873	2.944	3.002	2.627
Tension bars	Nominal diameter (mm)	4Y16 (deformed)			2Y16 (deformed)
	Yield strength/tensile strength (MPa)	590/690
	Ultimate strain (%)	12.4
	Elastic modulus (GPa)	200
Stirrups	Nominal diameter (mm)	Y8 at 100 (deformed, double legs)
	Yield strength/tensile strength (MPa)	560/620
	Ultimate strain (%)	12.4
	Elastic modulus (GPa)	200
FRP plates	Nominal thickness (mm)	None	1.2	1.2	None	1.2	2 × 0.111
	Strip width (mm)		100	100		100	100
	Strip length (mm)		3660	3660		3660	3660
	Tensile strength (MPa)		3200	3200		3200	3500
	Elastic modulus (GPa)		159	159		159	233

FRP: fiber-reinforced polymer; RC: reinforced concrete.

Mesh convergence study

The result of mesh convergence study of specimen BF2 is shown in Figure 3. To reduce the computing time, meshes in width direction were controlled to be a single element as forces in that direction are very small. Figure 3 shows the influence of different concrete element sizes and the effect of meshing one element in width direction, where “20 mm in all directions” was meshed in three coordinate system directions of 20 mm and others are meshed into one element in width direction. The predicted ultimate loads of beams of element size of 10, 20, 40, and 20 mm in all directions are 176.9, 177.4, 175.8, and 175.1 kN, respectively, with differences of only 0.34%, 0.62%, 0.28%, and 0.68%, respectively, as the mean of these predicted ultimate loads is 176.3 kN. The corresponding deflections are 32.2, 33.7, 31.8, and 33.3 mm, respectively, with differences of only 1.68%, 2.90%, 2.90%, and 1.68%, respectively, as the mean of these corresponding deflections is 32.75 mm. Nevertheless, the load–deflection curve of the 40 mm mesh has some defects, perhaps because the mesh is too large. The result shows that different concrete element sizes have some influence on IC debonding in simulation, but the influence is very small compared to the variance of the test results. Meanwhile, reducing the element number in width direction also has a very small influence. Owing to the influence of element size on the clarity of crack pattern, we chose the smallest element size as far as possible within the acceptable computing time. In this study, these six beams were modeled using a concrete element size of 10 mm while meshing only one element in the width direction. Take the beam BF2 for instance; Figure 4 shows the meshed 3D model after debonding, in which CSLIP1 SPOS is the interfacial slip distance between FRP and concrete.

Figure 3.

Load versus deflection curve of specimen BF2.

Figure 4.

Finite element model of specimen BF2 after debonding.

Load–deflection relationship

Figure 5 shows the comparison of load–deflection behavior between simulation and test results. The predicted ultimate loads of Beams BF1, BF2, BF5, BF7, BF8, and BF9 are 143.9, 176.9, 169.3, 78.6, 115.5, and 95.2 kN, respectively, which are close to the corresponding test values of 144.2, 185.0, 177.0, 80.7, 111.3, and 95.8 kN (Matthys, 2000), with differences of only 0.21%, 4.4%, 4.4%, 2.6%, 3.8%, and 0.63%, respectively. The post-debonding behavior is also obtained in this simulation, and the results show that the capacities of strengthened beams after debonding are very close to the reference beams. The load–deflection behavior of the preloaded beam BF5 is almost the same as the reference beam BF1 before the load was applied to 110 kN, after which the load–deflection diverges as the FRP reinforcement was strengthened to the beam. It follows that the behavior of beams strengthened under sustained load is accurately simulated.

Figure 5.

Comparison of load–deflection behavior between simulation and test results (Matthys, 2000).

Crack patterns

Figure 6 shows the comparison of crack patterns at the ultimate load between simulation and test results (Matthys, 2000), which match closely. The numbers of cracks of beams BF1, BF2, and BF5 are obviously more than those of beams BF7, BF8, and BF9, which is also shown via simulation. On account of the dynamic effect and quarter-modeling of the specimen, the FRP plate would entirely separate from the concrete beam after totally debonding in simulation as shown in Figure 6. The debonding behavior in simulation will be explained in detail later in this article, which has good agreement with the test results shown in Figure 6.

Figure 6.

Comparisons of crack patterns at the ultimate load between simulation and test results (Matthys, 2000).

Strain of steel reinforcement and concrete

In Figures 7 and 8, the steel reinforcement tensile strain and concrete compressive strain are given. The strains of simulation in the following figures are obtained by calculating the average strain of the elements within the length of mechanical deformeters which has a gauge length of 200 mm in this study (Matthys, 2000). The strain of steel reinforcement and concrete shows good agreement between simulation and test results.

Figure 7.

Comparison of strain of steel reinforcement between simulation and test results (Matthys, 2000).

Figure 8.

Comparison of concrete strain between simulation and test results (Matthys, 2000).

FRP strain and its distribution

Figure 9 shows the comparison of load–FRP strain behavior between the simulation and test results. The predicted maximum FRP strain values of beams BF2, BF5, BF8, and BF9 are 5948 με, 5030 με, 6211 με, and 10630 με, respectively, which are close to the test maximum strain values of 6700 με, 5700 με, 5800 με, and 10,000 με, respectively (Matthys, 2000). The load–FRP strain curve of beam BF5 is quite different from the others, because of the preload. When the load reaches 110 kN and the FRP plate is attached, the FRP strain begins to increase. When FRP debonds, strain value will fluctuate sharply. Therefore to make Figure 9 appear clear, the simulated curve is truncated after debonding.

Figure 9.

Comparison of FRP strain behavior between the simulation and test results (Matthys, 2000).

The FRP strain distribution along its length is shown in Figure 10, where Figure 10(a) shows the strain distribution of beams BF2 and BF5 under the load of 150 kN, while Figure 10(b) shows the strain distribution of beams BF2, BF8, and BF9 under the load of 70 kN. Owing to cracks, the simulated strain distribution is not very smooth. The strains for beam BF5 are considerably lower than BF2, as the beam was already loaded before strengthening. For a given load level, higher strains are measured for beams BF8 and BF9 than for beam BF2 (Figure 10(b)). Indeed, the beams BF8 and BF9 have a lower amount of steel reinforcement, so that the FRP has to act to a larger extent to build up the same total (steel and FRP) tensile force in the section as for beam BF2. The strains increase rapidly as the load increases after the tensile steel yields, which can be seen in Figure 9. For load levels above the yielding of the internal steel reinforcement, the tensile force in the section can only further increase, thanks to the FRP (the tensile force in the steel remains constant after yielding). As a result, FRP strains shall increase to a larger extent. At that stage, once the simulation and test results have a little difference, the strains under the same load would differ considerably. The strain distribution for beam BF5 under the load of 150 kN does not match very well; nevertheless, the test results under the load of 150 kN and the simulation results under the load of 153 kN show good agreement.

Figure 10.

Comparison of FRP strain distribution between the simulation and test results: (a) Beam BF2 and BF5 under the load of 150 kN and (b) Beam BF2, BF8 and BF9 under the load of 70 kN(Matthys, 2000).

Debonding process analysis

The debonding process in the simulation is described as follows: This study takes the beam BF8 for example and selects three representative states during the debonding process: state 1 when FRP plate starts debonding; state 2 when FRP plate is in the debonding process; and state 3 when complete debonding of the FRP plate occurs. Because the debonding process happened abruptly, the midspan deflections of the beam at these three states are almost the same and they are 28.0854, 28.1046, and 28.1058 mm, respectively. The distributions of interfacial shear stress, interfacial slip, and FRP strain at these three states are shown in Figure 11. It can be seen from Figure 11(a) that interfacial debonding started at the intermediate section of the FRP plate and then propagated toward the nearer plate end as the area where the interfacial shear stress is zero has already debonded. At state 3, interfacial debonding had propagated to the plate end and the FRP reinforcement had separated from the RC beam, which can be seen in Figure 11(b). It can be seen from Figure 11(c) that FRP strain varies very little at the debonding area. In addition, FRP strain reduces considerably at state 3 and has not reduced to zero yet on account of the dynamic effect.

Figure 11.

Debonding process of the beam BF8: (a) interfacial shear stress distribution, (b) interfacial slip distribution and (c) FRP strain distribution.

Parameter analysis

The effect of FRP placement on IC debonding is discussed as follows:Figure 12 shows a schematic diagram of the FRP plate location. We choose four configurations. Configuration 1 is the same as beam BF2, and other configurations divide an FRP plate into the same two pieces and place them in different positions. These beams were modeled using a concrete element size of 20 mm in three coordinate system directions. The results of these different configurations are shown in Figure 13. The predicted ultimate loads of beams of these configurations are 175.4, 175.0, 175.9, and 175.0, respectively. The predicted ultimate loads before completely debonding are 34.8, 37.3, 34.1, and 32.4 mm respectively. It can be seen that the ultimate loads of these beams are almost the same, but the deflections are slightly different. We can obtain an approximation indicating that the more dispersed the FRP plates are, the more likely the beam is to debond. In addition, if the location and total amount of the plates are the same, it is more difficult to debond for a few plates than a single one. However, the difference in the results between these beams is not significant.

Figure 12.

Schematic diagram of the FRP plate location: (a) configuration 1, (b) configuration 2, (c) configuration 3, and (d) configuration 4.

Figure 13.

Load–deflection curves of different configurations.

Conclusion

This article has presented a 3D FE model on the basis of the smeared crack approach for predicting IC debonding failure of FRP-strengthened RC beams. Compared with the experimental results, the FE model not only shows good prediction of IC debonding failure but also shows good agreement on details such as load–deflection relationship, crack patterns, load–strain relationship, and FRP strain distribution. The model also has the following characteristics:

This model can simulate preloaded beams.

Compared with other models on the basis of the smeared crack approach, the interfaces between the concrete and both the internal steel and the external FRP reinforcements are more expediently modeled.

This model can simulate the FRP debonding process of the beam and the response of the residual beam after the FRP reinforcement has separated from the RC beam.

The parametric analysis shows that the 3D model is more advanced than the 2D model, although it is not obvious in simulating IC debonding of RC beams with purely bonded FRP laminate. Nevertheless, this model can be used to simulate FRP-plated slabs or beams with additional anchors such as U-jacket strips or mechanically fastened FRP strengthening system in future research.

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 are grateful for the financial support received from the National Natural Science Foundation of China (Grant No. 51878664) and the National Key R&D Program of China (National Key Project No. 2017YFC0703506). The authors are also grateful to the anonymous reviewers for helping to significantly improve the quality of the manuscript.

ORCID iD

En-Li Xie

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