MESOSCALE FINITE ELEMENT MODEL FOR FRP SHEETS/PLATES BONDED TO CONCRETE X.Z. Lu1,2, L.P.
Ye1, J.G.
Teng2,* and J.J.
Jiang1 Engineering Structures. 2005. 27(4) . 564 575 ABSTRACT External bonding of fiber reinforced polymer (FRP) plates or sheets has recently emerged as a popular method for the strengthening of reinforced concrete (RC) structures. The behavior of such FRPstrengthened RC structures is often controlled by the behavior of the interface between FRP and concrete, and this interfacial behavior is commonly studied through a pull test in which an FRP sheet or plate is bonded to a concrete prism and is subject to tension. In this paper, a mesoscale finite element (FE) model implemented with the MSC.MARC program is presented for the simulation of interfacial debonding failures in a pull test. In this model, very small nearly square elements (0.25 mm to 0.5mm in size) are used with the fixed angle crack model (FACM) to capture the development and propagation of cracks in the concrete layer adjacent to the adhesive layer. The effect of element size is taken into account in modeling both the tensile and shear behavior of cracked concrete. Comparisons between the predictions of this model and test results are presented to demonstrate the capability and accuracy of this FE model. The debonding mechanism is also examined using results obtained with the FE model. Finally, a method for the determination of the local bondslip curve of the FRPtoconcrete interface from the FE results is described. 
External bonding of fiber reinforced polymer (FRP) plates or sheets (referred to as plates only here after for brevity) has emerged as a popular method for the strengthening of reinforced concrete (RC) structures. In this strengthening method, the performance of the FRPtoconcrete interface in providing an effective stress transfer is of crucial importance. Indeed, a number of failure modes in FRPstrengthened RC members are directly caused by the debonding of the FRP from the concrete [1,2]. Therefore, for the safe and economic design of externally bonded FRP systems, a sound understanding of the behavior of the FRPtoconcrete interface needs to be developed. It should be noted that throughout this paper, the term “interface” is used to refer to the interfacial part of the bonded joint, including the adhesive and the adjacent concrete, responsible for the relative slip between the FRP plate and the concrete prism, instead of any physical interface in the joint.
In various debonding failure modes, the stress state of the interface is similar to that in a pull test in which a plate is bonded to a concrete prism and is subject to tension (Figure 1). As a result, a large number of studies, both experimental and theoretical, have been carried out on pull tests on FRPtoconcrete bonded joints [3, 4]. Existing studies suggest that the main failure mode of FRPtoconcrete bonded joints in pull tests is concrete failure under shear, occurring generally at a few millimeters from the adhesive layer [3]. The ultimate load (i.e. the maximum transferable load) of the joint therefore depends strongly on the strength of concrete. In addition, the platetoconcrete member width ratio also has a significant effect. A very important aspect of the behavior of these bonded joints is that there exists an effective bond length beyond which an extension of the bond length cannot increase the ultimate load. This is a fundamental difference between an externally bonded plate and an internal reinforcing bar, for which a sufficiently long anchorage length can always be found so that the full tensile strength of the reinforcement can be achieved.
Apart from experimental and analytical studies, the finite element (FE) method has also been used to study debonding in FRPtoconcrete bonded joints. Earlier FE studies of interfacial behaviour employed linear elastic analysis and were concerned with the elastic stress distribution in the interface [e.g. 5, 6]. In the latest studies, attention shifted to the nonlinear FE analysis of the concretetoFRP interface, aimed at the simulation of the entire debonding process. There are generally two approaches to the simulation of debonding in FRPstrengthened RC structures using a nonlinear FE model. One approach is to employ a layer of interface elements between the FRP and the concrete [710], in which debonding is simulated as failure of the interface elements. Obviously, the success of such an approach depends on the constitutive law (i.e. the bondslip model) specified for the interface elements. Such models are not truely predictive models, although they may be used with tests to verify/identify interfacial behaviour. In the second approach [9, 1113], the use of interface elements is avoided; instead, debonding is directly simulated by modelling the cracking and failure of concrete elements adjacent to the adhesive layer. The advantage of this approach is that the debonding behaviour can be predicted using an appropriate constitutive model for concrete, without recourse to an interfacial bondslip model. Indeed, such a model has the capability of predicting the bondslip relationship for use in a model following the first approach. The second approach also provides a useful tool for understanding the debonding failure process and mechanism as only limited experimental observations of the debonding failure process can generally be made due to the microscopic details involved in and the suddenness of a debonding failure.
In general, the debonding of FRP from concrete occurs within a thin layer of concrete adjacent to the adhesive layer unless the adhesive is rather weak. The thickness of this concrete layer is about 2~5mm. Recent work [12] on the modelling of debonding failures using the second approach mentioned above has shown that it is difficult to simulate debonding using the concrete models available in commonly used generalpurpose FE packages such as ANSYS, MARC or ABAQUS. To simulate concrete failure within such a thin layer, with the shapes and paths of the cracks properly captured, the rotating angle crack model (RACM) [13,14] should be used if elements with a size comparable to the thickness of the concrete layer involved in the debonding failure are adopted. The RACM however has the major drawback that its constitutive parameters do not have clear physical meanings and have to be empirically derived from pull tests.
This paper presents a new FE model which can accurately simulate the entire debonding process in pull tests of FRPtoconcrete bonded joints. In this new FE approach, a fixed angle crack model (FACM) [15] is employed in conjunction with a very fine finite element mesh with element sizes being one order smaller than the thickness of the facture layer of concrete. This approach has the simplicity of the FACM, for which the relevant material parameters have clear physical meanings and can be much more easily determined than those for the RACM, but in the meantime retains the capability of tracing the paths of cracks as deformation progresses. The present model using very small elements is referred to as a mesoscale finite element model.
2.1 General
To reduce the computational effort, the threedimensional FRPtoconcrete bonded joint (Figure 1a) was modeled in the present study as a plane stress problem using 4node isoparametric elements. In real pull tests, the width of the FRP plate and that of the concrete prism may be different. In comparing FE predictions with test results, the finite element results including the applied load and axial stresses/strains in the FRP plate are adjusted based on the following width ratio factor _{ } proposed by Chen & Teng [3]:
_{ } 
(1) 
where _{ } and _{ } are the widths of the FRP plate and the concrete prism respectively. The adjustment involved the multiplication of a factor to the predicted values of the applied load and the stress and strain in the FRP plate based on the relationship of Eq. 1.
The plane stress FE model for the FRPtoconcrete bonded joint is shown in Figure 1b. It should be noted that the concrete prism included in the FE model has a height of 45 mm, which is generally much smaller than the actual concrete prism in a bond test. The exclusion of the rest of the concrete prism leads to a reduced computational effort and has little effect on the FE results. The support conditions include vertical supports to the concrete prism at the base and horizontal supports over the lower 15 mm of the loaded end of the concrete prism. All nonlinear FE analyses in the present study were carried out with the displacement control method, so that the softening branch of the loaddisplacement curve and the debonding process could be followed.
2.2 Modeling of the FRP Plate
When bonding FRP to concrete for strengthening purposes, the FRP may be in the form of a prefabricated plate, in which case the plate and the bonding adhesive layer can be clearly separated, although the exact thickness of the adhesive layer is generally not precisely controlled or defined. By contrast, if a dry fiber/fabric sheet is used in a wet layup process [1], which is a more widely used method and the method used in the preparation of all pull test specimens modeled in this study, the adhesive layer cannot be clearly separated from the fiber/fabric sheet or the FRP sheet formed from the impregnation of the fiber/fabric sheet with resin. Therefore, in the FE modeling of wet layup sheets, three options may be used: (a) a plate with a nominal thickness (generally the thickness of the fiber sheet); (b) a plate with its thickness being equal or similar to the actual thickness of the laminate including all adhesive, with the fibers assumed to be evenly distributed across the plate thickness; and (c) same as (b) but with the fibers concentrated in a thickness equal to the nominal thickness of the fiber sheet sitting in the middle of the plate. Results for the three options were compared in this study, but little differences were found between them (Figure 3) . All results presented in this paper were thus obtained using the second option mentioned above.
As the element size adopted in this study for the concrete prism is within the range of 0.25~0.5mm, while the actual thickness of the FRP plate per ply is about 1 mm, each ply was modeled using two layers of plane stress elements if the concrete element size was 0.5mm, or four layers of plane stress elements if the concrete element size was 0.25mm. The use of plane stress elements simplified the modeling procedure in terms of proper interaction between FRP and concrete. The FRP plate was treated as an isotropic material for simplicity with the elastic modulus being that of the longitudinal direction, as only the longitudinal behavior of the FRP plate is of interest here.
2.3 Modeling of the Concrete Prism
As the element sizes used for the concrete prism in the present FE model are much smaller than the maximum aggregate size, a true mesomechanical model requires the modeling of the mesolevel internal structure of concrete [1618], with the difference between the cement mortar and aggregate explicitly considered. The present FE model is, however, not intended as such a mesomechanical model. Instead, the present model, referred to as the mesoscale FE model, aims to use small elements to accurately follow the development and propagation of cracks; the elements are still intended to represent the average performance of concrete. Hence, the concrete in this model is still treated as a homogeneous material.
The commonly used constitutive model for concrete in MSC.MARC [19] describes concrete in compression as an elasticplastic material and adopts the smeared crack approach for the tensile behavior of cracked concrete. The constitutive laws for concrete in compression and cracked concrete in tension respectively are explained below. Such a concrete model is most suitable for finite elements of normal sizes which should be several times the maximal aggregate size [16]. However, in the present model, as the element sizes for the concrete prism are very small, damage localization is expected, which may underrate the structural strength, especially the tensile strength of concrete when the maximum stress criterion is used as the criterion of fracture. Hence, in the present study, the constitutive model for concrete in MSC.MARC was used together with additional measures aimed at the elimination of the element size effect, as detailed below.
Concrete in compression is treated as an elasticplastic material using the yield function proposed by Buyukozuturk [20] with an associated flow rule, which is already included in MSC.MARC [19]. It is also assumed that no plastic deformation appears when the equivalent stress as defined by Buyukozuturk [20] is smaller than_{ } , and before yielding concrete is assumed to be linearly elastic. According to existing studies [16,21], the compressive behavior of concrete does not depend significantly on element sizes, so the following stressstrain relationship proposed by Hognestad [22] was directly used as the equivalent stressstrain curve in the present study to simulate the compressive behavior of concrete:

_{ } if _{ } 
(2) 
_{ } if _{ } 
where _{ } , _{ } , _{ } is the compressive strength of concrete which is taken to be the cylinder compressive strength _{ } in this study, _{ } is the elastic modulus of concrete, and _{ } .
The crack band theory, which was developed by Bazant [23] and Bazant & Cedolin [24], was adopted in this study to model the behavior of cracked concrete. In finite element analysis, the incremental stressstrain relationship for cracked concrete at a given Gauss point with two orthogonal cracks in directions 1 and 2 can be written as:
_{ } 
(3) 
where _{ } and _{ } are the direct stress and strain normal to crack 1; _{ } and _{ } are the direct stress and strain normal to crack 2; _{ } and _{ } are the shear stress and shear strain; _{ } is the elastic modulus; _{ } is the elastic shear modulus of concrete; _{ } and _{ } are factors representing the tensile softening of concrete in the postcracking stage in the directions normal to cracks 1 and 2 respectively; _{ } is the shear retention factor which represents the effect of interlocking of cracked concrete, and should be found using the larger value of the two crack strains.
Based on the work of Kwak & Filippou [25], when the element size is smaller than three times the maximum aggregate size, the crack strain can be treated as uniform in the element when the smeared crack model is used. Thus, the fracture energy of _{ } can be expressed as:
_{ } 
(4) 
where _{ } is the crack band width and equals to the width of the element in the direction perpendicular to the crack. Two most commonly used tensile softening models for cracked concrete, originally proposed by Hillerborg and Peterson respectively [26] (Figure 2) were used and compared in the present study. According to the model code of CEBFIP [27], the tensile fracture energy of concrete can be expressed as
_{ } 
(5) 
where _{ } for normal concrete and _{ } is the compressive strength of concrete. Thus, when the element size _{ } , the fracture energy _{ } and tensile strength of concrete _{ } are known, the stressstrain relationship of the linear softening model and bilinear softening model can be determined.
The shear modulus of cracked concrete depends on the crack width. As there are large changes in the crack width in the debonding process, it is not suitable to use a constant value for the shear modulus. Hence, four different shear modulus expressions for cracked concrete, derived from either numerical or experimental results and given by Rots et al. [28], ABAQUS [29], AlMahaidi [30] and Dalian University of Technology (DLUT) in China [31] were compared in this study to identify the one which leads to the most accurate predictions for the present debonding problem. Details of these four shear retention models are given in Appendix A. The commonly used constant shear retention model was also included in the comparison. The changes of shear modulus for cracked concrete with difference shear retention models are illustrated in Figure 4.
Two FE models consisting of square or nearly square elements with sizes of 0.25mm and 0.5mm (or small variations from these sizes) respectively were compared in this study to find out whether the concrete model presented above can eliminate the element size effect and whether the element is small enough to simulate the debonding process. The results are discussed in the next section of the paper.
The features outlined above for the constitutive model of concrete was implemented into the general purpose finite element package MSC.MARC [19] through a user subroutine. Before finalizing the finite element model, a number of issues were first examined for accurate predictions by comparing the FE results with the test results of two reference specimens. These issues include the shear retention model, the postpeak softening scheme and the element size. As test results inevitably show some random variations, the bond strength model proposed by Chen & Teng [3] which provides bestfit predictions of a large number of tests was also used as a reference. Two specimens, Wu2 [32], and B2 [33], were selected as the reference specimens to arrive at the best model for the shear retention factor, because the experimental bond strengths of these two specimens are close to the predictions of Chen & Teng’s bond strength model [3], and the strain distributions in the FRP plate are available. The bond length of specimen Wu2 was 250mm, the concrete had a cylinder compressive strength of 57.6MPa, and the CFRP had a nominal fiber sheet thickness of 0.501mm and an elastic modulus of 390GPa. The bond length of Ueda’s specimen B2 was 200mm, the concrete had a cylinder compressive strength of 45.9MPa, and the CFRP had a nominal fiber sheet thickness of 0.33mm and an elastic modulus of 230GPa. Other details of these two specimens can be found in Table 1.
As discussed above, different shear retention models for cracked concrete are available. In some of these models, including the constant shear retention model, the ABAQUS model and AlMahaidi’s model, some parameters are unknown. The first step in identifying the best shear retention model for the present work was to determine these parameters. To this end, a trialanderror procedure was adopted, in which different values of these parameters were tested to identify values which led to the best predictions of the bond strengths of the two reference specimens mentioned above. The bond strength was selected as the target as it is the most important parameter governing the debonding behavior. The concrete was modeled using elements of 0.5mm in size and was assumed to have linear tensilesoftening. The FRP plate was assumed to be 1 mm thick per ply, which is close to the actual ply thickness of the FRP plate. These analyses showed that the best match between the predicted and test bond strengths is achieved when _{ } for the constant shear retention model, or _{ } and _{ } for the ABAQUS model, or _{ } for AlMahaidi’s model.
Once the unknown parameters in these models were determined, the FRP strain distributions from the FE model were compared with the test distributions to identify the best shear retention model (Figure 5). It is clear that the shear retention model proposed at Dalian University of Technology [31] in China is the best performing model. Hence, this shear retention model of cracked concrete was adopted in all subsequent FE analyses.
It should be noted that for the comparison of strain distributions in Figure 5 and elsewhere in the paper, both the experimental strain distributions and the FE strain distributions are available only at discrete values of the applied load, so the two sets of results had to be matched for comparison. For the elastic stage, a FE strain distribution corresponding to a load level nearest a given experimental load level was chosen for comparison. For the stage of debonding propagation, a FE strain distribution with an effective stress transfer length nearest that of the experimental strain distribution was chosen, but this means that the applied load, for the experimental strain distribution is different from that for the chosen FE strain distribution. This approach for comparison may not be ideal, but given the inevitable, although reasonably small, difference between the FE and test ultimate loads and the large variation of slip with a small variation in load during the debonding propagation stage, this is believed to be the most rational among possible approaches. In the relevant figures, only the experimental load level P normalized with respect to the experimental ultimate load P_{u} is indicated for each set of strain distributions.
Figure 6 shows a comparison of the FE predictions of strain distributions in the FRP plate obtained using the two different strain softening models for concrete in tension (Figure 2) with the test results of specimen B2. The FRP plate was assumed to be 1 mm thick per ply and the concrete was modeled using elements of 0.5mm in size. Figure 6 shows that before debonding occurs, there is almost no difference between the results from the two strain softening models. After debonding (P/P_{u}=1), a small difference exists, and this difference is mainly caused by the random appearance of local cracks in the concrete and is unimportant in terms of the overall simulation of debonding (Figure 2). Therefore, the linear softening model, which is the default strain softening model in MSC.MARC[19], was adopted in subsequent FE analyses.
In Figure 7, FE strain distributions in the FRP plate obtained with two different element sizes are compared: 0.5 mm elements versus 0.25 mm elements. It can be seen that the FRP strains from the two FE models are very close to each other. The contours of the crack strain, defined as the total tensile strain minus the elastic strain, at each Gauss point predicted by the two models are shown in Figure 8. If two orthogonal cracks exist at any Gauss point, then the larger crack strain is used in plotting the contours. As larger crack strains are represented by darker lines, these contours provide a close representation of crack patterns and are referred to as crack patterns directly in this paper. A close agreement in the crack pattern is evident between the two FE models with different element sizes, but the cracks predicted using the finer mesh are more distinct. This comparison shows that, with the present constitutive model for concrete, the effect of element size on the predicted behavior is small. Hence, the element size of 0.5mm was used in the numerical simulation of all other specimens, but the element size of 0.25mm was used to investigate the debonding failure mechanism.
All specimens listed in Table 1 were simulated with the FE model presented above. The bond strengths predicted by the FE model are compared with results from tests and from the bond strength model of Chen & Teng [3] in Table 2. It was found to be difficult to provide a precise value of the effective bond length from a test by an inspection of the strain distribution in the FRP plate, as the strain distribution may feature large local fluctuations, so a range is given for each test in Table 2. For example, the strain distribution at the debonding load shown in Figure 9(a) leads to two possibilities: (a) an effective bond length of 140mm if the local fluctuations in the strain near the loaded end are not taken to indicate local debonding; and (b) an effective bond length of 115mm if this local area of 25 mm is taken to have debonded and excluded from the effective bond zone. The FE value for the effective bond length could be more accurately determined as strain distributions at more load levels were available. The effective bond length of Chen and Teng’s model [3] is based on the variation of the ultimate load with the bond length: the effective length is defined as the bond length beyond which the ultimate load does not increase. The comparisons given in Table 2 indicate that there is a good overall agreement between the results from all three approaches. The ultimate loads form the three approaches are in close agreement: indeed, the differences are small considering that even for two nominally identical specimens of each of the three pairs tested by Tan [34] (e.g. specimens Tan1 and Tan2), significant differences exist between the two ultimate loads. The FE effective bond lengths are seen to lie within the ranges found from the tests, but those from Chen and Teng’s model may lie slightly outside these ranges. Except for specimen B2, the bond lengths from Chen and Teng’s model are a little larger than the FE predictions.
Furthermore, the strain distributions in the FRP plate from tests and FE analyses are compared for specimen B2 [33], Tan1 and Tan2 [34], Wu1 and Wu2 [32] in Figures 9(a)~(d). Because of local bending of the FRP plate caused by the interfacial cracks in the concrete, the strains in FRP plate vary slightly across its thickness, so the strains at the top surface of the FRP plate differ very slightly from those at the middle surface. This difference is shown in Figure 9(a), where “top” and “middle” represent the strains at the top and middle surfaces of the FRP plate, respectively. As this difference is small and it is difficult for strain measurements in tests to capture this local bending, in all other comparisons given in this paper, the test strains are compared with FE strains at the middle surface of the FRP plate. Figure 9 demonstrates that the strain distributions in the FRP plate from tests are closely predicted by the FE model.
The process of crack propagation is difficult to observe in laboratory tests but can be explored in detail by examining the FE results from the present FE model. It should however be noted that the actual process of debonding is more complicated than is predicted by the present model, as real concrete is an inhomogeneous material. Nevertheless, the present finite results provide useful insight into the failure mechanism.
The loadslip curve at the loaded end for specimen Wu1 from the FE model is shown in Figure 10. For the illustration of the debonding process, some representative points on this loadslip curve are marked as points A to F. The crack strain contours in the concrete corresponding to the representative points of Figure 10 are shown in Figure 11.
Based on the axial strains/stresses in the FRP plate obtained from a FE simulation, the local bond stress distribution can be calculated from Eq. (6):
_{ } 
(6) 
where _{ } is the axial stress in the plate and _{ } is the thickness of the FRP plate. The local bond stress distributions found using Eq. (6) for specimen Wu1 for different points on the loadslip curve are shown in Figure 12 as the “original” distributions. Large fluctuations are seen in these curves, as a result of interfacial cracks in the concrete. Another set of curves labeled “smoothed” is also shown in Figure 12. These curves were obtained by numerical smoothing with a 5mm Fast Fourier Transform (FFT) to remove the local jumps. The smoothed curves show clearly the effective bond zone at each level of loading (Figure 12). The effective bond zone is defined herein as the length of the interface with significant bond stresses.
Figure 11a shows that even at a load of 19% of the ultimate load, the concrete under the FRP plate near the loaded end has experienced extensive cracking. The cracks observed in the concrete can be classified into 3 types: (a) interfacial shallow cracks; (b) interfacial deep cracks; and (c) microcracks within a cracked zone under the FRP plate. The same three types of cracks also exist at later stages of deformation as shown in Figures 11b~f. The depths of the interfacial shallow cracks are about 0.5~1mm, whose widths are small and relatively uniform. This thin layer is subject to high tensile stresses as well as high shear stresses as it is directly under the FRP plate. The depths of the interfacial deep cracks are 2~5mm. The widths of these cracks are relatively large. They are caused by the interfacial shear stresses and will control the final debonding strength and the slip of the interface. The overall depth of the cracked zone of concrete is about 5~15mm. They are also caused by the interfacial shear stresses and some will develop into interfacial deep cracks as deformation further increases.
Based on the FE results, the failure process of the interface may be explained as follows. At a low level of loading, the formation of interfacial shallow cracks at angles of 45~60°to the interface leads to the appearance of small cantilever columns (i.e. mesocantilevers). With further increases in loading, these cantilevers may grow longer as the shallow cracks grow into deep cracks or may fail if the shear force acting on the cantilever reaches a critical value. The shear force F on the cantilever leads to axial compressive stresses which can cause crushing failure, or flexural tensile stresses at the root which are responsible for flexural failure at the root (Figure 13). An inspection of the finite element results revealed that debonding as observed in laboratory tests corresponds to the progressive flexural failure of the mesocantilevers, except near the loaded end where debonding is mainly due to the crushing of the mesocantilevers.
Figure 12(a) shows that in the early stage of loading, significant bond stresses are developed near the loaded end. As the load increases, the bond stresses increase and the effective bond zone expands (Figure 12(b)). When the maximum bond stress reaches the local bond strength, softening (microcracking) of the interface commences. The bond stresses near the loaded end then decrease (Figures 12(c) and 12(d)), and the effective bond zone continues to expand towards the free end. Finally, when full debonding has been reached (Figure 13(e)), the bond stresses close to the loaded end are very small, indicating the formation of a macrocrack. This macrocrack corresponds to the surface of the debonded concrete layer observed in a laboratory test. At this stage, the loadslip curve is already highly nonlinear. Afterwards, the effective bond zone moves away from the loaded end towards the free end as the macrocrack propagates towards the free end. During this stage, the load increases much more slowly than the slip. After Point F, the load remains almost constant until the FRP plate is entirely debonded from the concrete prism. The same phenomenon has been explained by Yuan et al. [4].
From the FE results, the localbond
slip curve of a point along the interface can be obtained by plotting the
smoothed bond stress value at that point versus the slip of that point. To
obtain more reliable bondslip curves this way, the chosen point should be
located in a relatively stable zone of the original bond stress distribution
in which local fluctuations are comparatively small. A set of three bondslip
curves for a point located in the zone of 1530 mm 19.5mm from the
loaded end in specimen Wu1 is shown in Figure 14 for three different FFT
smoothing lengths.
As mentioned in the preceding section, numerical smoothing is needed to obtain usable local bond stresses. By an analysis of the frequency content of the original bond stress distribution obtained from Eq. (6), a FFT smoothing length of 5mm to 20mm was found to be suitable to filter out local “noises” caused by interfacial cracks in the bond stress distribution, without a loss of the characteristics of the resulting bondslip curve. Figure 14 demonstrates that a smoothing length of 10mm is the most suitable for specimen Wu1 in terms of the accurate prediction of the local bondslip curve.
From the smoothed bondslip curves shown in Figure 14, it may be concluded that the bondslip curve is made up of two branches: an ascending branch and a descending branch. With the propagation of interfacial cracks, the stiffness of concrete under the interface decreases with the local slip, which leads to a curved ascending branch. When the peak bond stress is reached, the bond stress drops quickly in the early stage of the descending branch, followed by a more gradual decrease to zero, which reflects the interlocking effect of the cracked concrete.
A mesoscale finite element model implemented with the generalpurpose program MSC.MARC has been presented for the simulation of debonding behavior in FRPtoconcrete bonded joints. In this model, the concrete in compression is treated as an elasticplastic material, while the behavior of cracked concrete is represented using the smeared crack approach. Very small elements (0.25 mm to 0.5mm in size) are used with the crack band model to capture the development and propagation of cracks in the concrete layer adjacent to the adhesive layer. The effect of element size is taken into account in modeling both the tensile and shear behavior of cracked concrete. Comparisons between the predictions of this model and selected test results have shown that the ultimate load, effective bond length and strain distributions in the FRP plate at different load levels can all be closely predicted. The predicted crack patterns are realistic. By an examination of the crack patterns and the interfacial shear stresses from a finite element analysis, it may be concluded that debonding in such bonded joints is due to the progressive formation and failure of inclined mesocantilever columns in the concrete. Finally, a method for the determination of the local bondslip curve of the FRPtoconcrete interface from the FE results was illustrated.
The authors are grateful for the financial support received from the Research Grants Council of the Hong Kong SAR (Project No: PolyU 5151/03E), the Natural Science Foundation of China (National Key Project No. 50238030) and The Hong Kong Polytechnic University provided through its Area of Strategic Development (ASD) Scheme for the ASD in Urban Hazard Mitigation.
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[24] Bazant ZP, Cedolin L. Blunt crack band propagation in finite element analysis. Journal of Engineering Mechanics Division, ASCE. 1979; 105(2): 297315.
[25] Kwak HG, Filippou FC. Finite element analysis of reinforced concrete structures under monotonic load. Research report of Department of Civil Engineering, U.C. Berkeley, No. UCB/SEMM90/14, 3339, 1990.
[26] Rots JG., Nauta P, Kusters MA, Blaauwendraad J. Smeared crack approach and fracture localization in concrete. HERON. 1985;30(1).
[27] CEBFIP. Model Code 90. Lausanne; 1993.
[28] Rots JG, Kusters GMA, Nauta P. Variable reduction factor for the shear resistance of cracked concrete. TNOreport BI8433/68.8.2001, Rijswijk, the Netherlands, 1984.
[29] ABAQUS. User’s manual. HKS Corporation; 2001.
[30] Walraven JB, Reinhardt HW. Theory and experiments on the mechanical behavior of cracks in plain and reinforced concrete subject to shear load. HERON. 1981;26(1A).
[31] Kang QL. Finite element analysis for reinforced concrete, Beijing: China Water Power Press; 1996, p. 120126.
[32] Wu ZS, Yuan H, Yoshizawa H, Kanakubo T. Experimental/analytical study on interfacial fracture energy and fracture propagation along FRPconcrete interface. ACI International SP2018. 2001, p. 133152.
[33] Ueda T, Sato Y, Asano Y. Experimental study on bond strength of continuous carbon fiber sheet. Proceedings of 4^{th} Int. Sym. on Fiber Reinforced Polymer Reinforcement for Reinforced Concrete Structure. 1999, p. 407416.
[34] Tan Z. Experimental research for RC beam strengthened with GFRP. Master Thesis, Tsinghua University, 2002.
The shear modulus for cracked concrete proposed by Rots et al. [28] is:
_{ } 
(A1) 
where _{ } is the crack strain normal to the crack.
The shear modulus for cracked concrete adopted by ABAQUS [29] is:
_{ } 
(A2) 
where _{ } is a parameter which represents the shear stiffness of a closed crack, and _{ } is a parameter which represents the crack strain when the shear stiffness of the cracked concrete is completely lost.
The shear modulus for cracked concrete proposed by AlMahaidi [30] is:
_{ } 
(A3) 
where _{ } is a parameter to be defined.
In the shear retention model proposed by DLUT (Dalian University of Technology) [31], the shear stress _{ } on the crack surfaces is given by:
_{ } 
(A4) 
where _{ } (mm) and _{ } (mm) are the width and slip of the crack, respectively. Assuming that the crack strain is uniformly distributed in the element, Eq. A(4) can be expressed as:
_{ } 
(A5) 
As a result, the tangent shear modulus of the cracked concrete is found to be:
_{ } 

_{ } 
(A6) 
No. 
Specimen name 
FRP plate 
Concrete prism 

Thickness (mm) 
Width (mm) 
Bond length (mm) 
Elastic modulus (GPa) 
Cube strength (MPa) 
Width (mm) 
P_{u} (kN) 
Width ratio factor _{ } from Eq. 1 

1 
Wu1 
0.22 
40 
250 
230 
72.0* 
100 
14.1 
1.069 
2 
Wu2 
0.501 
40 
250 
390 
72.0* 
100 
23.5 
1.069 
3 
B2 
0.33 
100 
200 
230 
57.4* 
500 
38 
1.225 
4 
Tan1 
0.167 
50 
130 
97 
37.6 
100 
7.775 
1 
5 
Tan2 
0.167 
50 
130 
97 
37.6 
100 
9.185 
1 
6 
Tan3 
0.33 
50 
130 
97 
37.6 
100 
10.49 
1 
7 
Tan4 
0.33 
50 
130 
97 
37.6 
100 
11.425 
1 
8 
Tan5 
0.111 
50 
130 
235 
37.6 
100 
7.965 
1 
9 
Tan6 
0.111 
50 
130 
235 
37.6 
100 
9.185 
1 
10 
Yuan 
0.165 
25 
190 
265 
29.74 
150 
5.74 
1.254 
Data sources: 1~2: Wu et al. (2001); 3: Ueda et al. (1999); 4~9: Tan (2002); 10: Yuan et al. (2004).
*Computed from cylinder compressive strength using _{ } and the tensile strength of concrete is computed using _{ } (MPa).
Specimen name 
Test 
FEA 
Chen & Teng (2001) 

Ultimate Load (kN) 
Effective bond length (mm) 
Ultimate Load (kN)^{*} 
Effective bond length (mm) 
Ultimate Load (kN) 
Effective bond length (mm) 

Wu1 
14.1 
61~90 
13.8 
76.8 
11.44 
81.5 
Wu2 
23.5 
120~165 
24.7 
146.0 
22.35 
159.0 
B2 
38.0 
115~140 
38.3 
125.0 
37.74 
105.2 
Tan1 
7.78 
45~65 
6.49 
45.5 
6.40 
54.6 
Tan2 
9.19 
45~65 

Tan3 
10.5 
50~70 
9.37 
63.6 
9.05 
77.3 
Tan4 
11.4 
60~80 

Tan5 
7.97 
45~65 
8.44 
46.3 
8.08 
68.9 
Tan6 
9.19 
50~70 

Yuan 
5.74 
70~85 
5.82 
85.0 
6.07 
93.1 
* The effect of width ratio has been included according to the Chen & Teng formula [3].
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
(c)
(d)
(a) (b)
(c) (d)
(e) (f)