BONDSLIP MODELS FOR FRP SHEETS/PLATES BONDED TO CONCRETE X.Z. Lu1,2, J.G.
Teng2,*, L.P. Ye1 and J.J.
Jiang1 Engineering Structures. 2005. 27(6) . 920937 ABSTRACT An accurate local bondslip model is of fundamental importance in the modelling of FRPstrengthened RC structures. In this paper, a review of existing bond strength models and bondslip models is first presented. These models are then assessed using the results of 253 pull tests on simple FRPtoconcrete bonded joints, leading to the conclusion that a more accurate model is required. In the second half of the paper, a set of three new bondslip models of different levels of sophistication are proposed. A unique feature of the present work is that the new bondslip models are not based on axial strain measurements on the FRP plate; instead, they are based on the predictions of a mesoscale finite element model, with appropriate adjustment to match their predictions with the experimental results for a few key parameters. Through comparisons with the large test database, all three bondslip models are shown to provide accurate predictions of both the bond strength (i.e. ultimate load) and the strain distribution in the FRP plate. KEYWORDS FRP, concrete, bond, bondslip models, bond strength, pull tests, finite
element simulation, composites
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Over the past decade, external bonding of fibre 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 [1]. An important issue in the strengthening of concrete structures using FRP composites is to design against various debonding failure modes, including (a) cover separation [24] (b) plate end interfacial debonding [23]; (c) intermediate (flexural or flexuralshear) crack (IC) induced interfacial debonding [5]; and (d) critical diagonal crack (CDC) induced interfacial debonding [6]. The behaviour of the interface between the FRP and the concrete is the key factor controlling debonding failures in FRPstrengthened RC structures. Therefore, for the safe and economic design of externally bonded FRP systems, a sound understanding of the behaviour of FRPtoconcrete interfaces needs to be developed. In particular, a reliable local bondslip model for the interface is of fundamental importance to the accurate modelling and hence understanding of debonding failures in FRPstrengthened RC structures. It should be noted that throughout this paper, the term “interface” is used to refer to the interfacial part of the FRPtoconcrete bonded joint, including the adhesive and a thin layer of 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 specimen in which a plate is bonded to a concrete prism and is subject to tension (Figure 1). Such pull tests can be realized in laboratories in a number of ways with some variations [7], but the results obtained are not strongly dependent on the setup as long as the basic mechanics as illustrated in Figure 1 is closely represented [8].
The pull test not only delivers the ultimate load (referred to as the bond strength hereafter in this paper) of the FRPtoconcrete interface, but also has been used to determine the local bondslip behaviour of the interface [916]. Local bondslip curves from pull tests are commonly determined in two ways: (a) from axial strains of the FRP plate measured with closely spaced strain gauges (e.g. Nakaba et al. [12]); (b) from loaddisplacement (slip at the loaded end) curves (e.g. Ueda et al. [15]). In the first method, the shear stress of a particular location along the FRPtoconcrete interface can be found using a difference formula, while the corresponding slip can be found by a numerical integration of the measured axial strains of the plate. This method appears to be simple, but in reality cannot produce accurate local bondslip curves. This is because the axial strains measured on the thin FRP plate generally show violent variations as a result of the discrete nature of concrete cracks, the heterogeneity of concrete and the roughness of the underside of the debonded FRP plate. For example, a strain gauge located above a crack will have a much greater strain than one that sits above a large aggregate particle. The shear stress deduced from such axial strains is thus not reliable although the slip is less sensitive to such variations. Consequently, bondslip curves found from different tests may differ substantially. The second method is an indirect method and has its own problem: the local bondslip curve is determined indirectly from the loadslip curve, but it is easy to show that rather different local bondslip curves may lead to similar loaddisplacement curves.
This paper has two principal objectives: (a) to provide a critical review and assessment of existing bondslip models, and (b) to present a set of three new bondslip models. The former part aims to clarify the differences between existing bondslip models and between these models and test results, a task that does not appear to have been properly undertaken so far. The former part also sets the stage for the latter part in which three new bondslip models of different levels of sophistication are presented. A unique feature of the present work is that the new bondslip models are not based on axial strain measurements on the FRP plate, but instead they are based on the predictions of a mesoscale finite element model, with appropriate adjustment to match the experimental results of a few key parameters. As these key parameters such as the bond strength are much more reliable than local strain measurements on the FRP plate, the present approach does not suffer from the random variations associated with strain measurements nor the indirectness of the loadslip curve approach.
Before presenting a review of the existing test data and bondslip models, some fundamental aspects of the behaviour of FRPtoconcrete interfaces should be summarized to place the present work in its proper context. Existing pull tests have shown conclusively that in the vast majority of cases and except when a very weak adhesive or a high strength concrete is used, the failure of an FRPtoconcrete bonded joint is by cracking in the concrete layer adjacent to the adhesive layer. In Figure 1, the dotted lines identify a typical fracture plane in the process of debonding failure, and this plane is generally slightly wider than the width of the FRP plate (Figure 1), if the plate is narrower than the concrete prism. The fracture plane propagates from the loaded end to the free end of the FRP plate as loading/deformation increases. In Figure 1, the FRP plate is shown unbonded near the loaded end (the free zone), which has been adopted in some tests (e.g. [17]), but in some other tests, such a free zone was not included (e.g. [8], [12]). If this free zone does not exist or is small, a lump of concrete near the loaded end will generally be pulled off the concrete prism, but this variation in detail does not have a significant effect on the localbond slip behaviour elsewhere nor the general behaviour as long as the bond length is not very short. From existing theoretical and experimental studies (e.g. [7], [15], [16]), the following six parameters are known to govern the local bondslip behaviour as well as the bond strength of FRPtoconcrete bonded joints in pull tests: (a) the concrete strength, (b) the bond length L (Figure 1), (c) the FRP plate axial stiffness, (d) the FRPtoconcrete width ratio, (e) the adhesive stiffness, and (f) the adhesive strength. A very important aspect of the behaviour 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.
In this study, a database containing the results of 253 pull tests on FRPtoconcrete bonded joints was built. The database includes tests reported by Chajes et al. [18], Taljsten [19], Takeo et al. [20], Zhao et al. [21], Ueda et al. [22], Nakaba et al. [12], Wu et al. [13], Tan [17], Ren [23] and Yao et al. [8]. Both single shear tests (e.g. Yao et al. [8]) and double shear tests (e.g. Tan [17]) are included in the database. Details of these tests, except those already included in the easily accessible databases assembled by Chen and Teng [7], Nakaba et al. [12] and those from the recent study of Yao et al. [8], are given in Table A1 of Appendix A, where _{ } , _{ } ,_{ } and _{ } are the width, thickness, elastic modulus and tensile strength of the FRP plate respectively, _{ } is the width of the concrete prism, _{ } is the cube compressive strength of concrete (converted from cylinder compressive strength by a factor of 0.78 where applicable), _{ } is the tensile strength of concrete (_{ } according to the Chinese code for the design of concrete structures [24] if not available from the original source), _{ } is the total bond length, and _{ } is the bond strength. For some of these specimens, strains measured on the FRP plate are also available.
The distributions of the test data in terms of the following four key parameters are shown in Figure 2: (a) the concrete cube compressive strength (Figure 2a); (b) the axial stiffness of the plate per unit width (Figure 2b); (c) the bond length normalized by the effective bond length predicted by Chen and Teng’s model [7]; (d) the FRP platetoconcrete prism width ratio. It is clear that the test data cover a wide range of each parameter and can be expected to provide a reliable benchmark for theoretical models. It is desirable for future tests to be conducted in regions where current data are scarce.
Dai and Ueda [14] and Ueda et al. [15] recently reported that the bond strength of FRPtoconcrete interfaces can be enhanced through the use of a very soft adhesive layer with a shear stiffness _{ } (=_{ } ) being between 0.14 to 1.0 GPa/mm, where _{ } is the adhesive layer thickness and _{ } is the elastic shear modulus of the adhesive. It is clear that a small shear stiffness of the adhesive layer can be achieved by the use of a soft adhesive or a thick adhesive layer. While the properties of the adhesives used in the specimens of the present test database were not always reported, none of the relevant studies was focussed on the issue of very soft adhesive layers. At least outside Japan, the application of adhesives commonly available in the market in a procedure complying with the recommendations of the manufacturers is unlikely to lead to an adhesive layer which can be classified as being very soft (i.e. with a shear stiffness in the range studied by Dai and Ueda [14] and Ueda et al. [15]). Furthermore, relatively soft adhesives are normally used only in wet layup applications where the definition of the thickness of the adhesive layer is problematic but affects the value of the shear stiffness of the adhesive layer significantly. Indeed, since the same resin is commonly used to saturate the fibre sheet to form the FRP plate as well as to bond the FRP plate to the concrete which is often already covered with a thin layer of primer, the thickness of the adhesive layer which deforms primarily in shear cannot be easily defined and is believed to be very small by the present authors in debonding failures unless debonding occurs in the adhesive layer. Finally, in practice, the thickness of the adhesive layer cannot be precisely controlled and measured as reported in the studies of Dai and Ueda [14] and Ueda et al. [15]. Therefore, it is reasonable to assume that the bonded joints of the present database have _{ } values much greater than those studied by Dai and Ueda [14] and Ueda et al. [15] and are referred to as normaladhesive joints hereafter. A separate study by the authors to be reported in a future paper has shown that for values of _{ } ranging from 2.5 GPa/mm to 10 GPa/mm, the bondslip curve is little dependent on the shear stiffness of the adhesive layer. A shear stiffness of 5 GPa/mm for the sheardeformed adhesive layer is used in this study to represent a normaladhesive bonded joint when it is needed. As the bondslip models of Dai and Ueda [14] and Ueda et al. [15] are for very soft adhesive layers and consider the adhesive layer shear stiffness as a significant parameter, they are not included in the comparisons and discussions in this paper. The test data from their studies are also not included in the present database. The scope of the present study is therefore limited to FRPtoconcrete bonded joints whose sheardeformed adhesive layer has a shear stiffness of no less than 2.5 GPa/mm. The present work nevertheless is believed to cover at least all commercially available FRP systems for external bonding applications outside Japan.
Many theoretical models have been developed from 1996 onwards to predict the bond strengths of FRPtoconcrete bonded joints, generally on the basis of pull test results. These are commonly referred to as bond strength models. Altogether 12 bond strength models have been found in the existing literature, and 8 of them have been examined in detail by Chen and Teng [7]. These 8 models have been developed by Tanaka [25], Hiroyuki and Wu [26], van Gemert [27,28], Maeda et al. [9], Neubauer and Rostasy [29], Khalifa et al. [30], Chaallal et al. [31] and Chen and Teng [7]. The 4 models not covered by Chen and Teng [7] include three models (Izumo, Sato, and Iso) developed in Japan and described in a recent JCI report [32] and one developed by Yang et al. [33]. These 4 models are detailed in Appendix B. Table 1 provides a summary of the key parameters considered by these 12 models, while an assessment of their accuracy is given later in the paper.
Despite the difficulty in obtaining local bondslip curves from pull tests directly, local bondslip models for FRPtoconcrete interfaces have been developed, based on strain measurements or loadslip curves. Six local bondslip models available in the existing literature are summarized in Table 2, where _{ }(MPa) is the local bond (shear) stress, _{ }(mm) is the local slip, _{ } (MPa) is the local bond strength (i.e. the maximum bond/shear stress experienced by the interface), _{ } (mm) is the slip when the bond stress reaches _{ } , _{ } (mm) is the slip when the bond stress reduces to zero, _{ } is the width ratio factor, _{ } (MPa) is the cylinder compressive strength of concrete. In addition, Sato [32] proposed a model which was modified from an existing bondslip model for rebarconcrete interfaces by replacing the yield strain of steel with the ultimate tensile strain of FRP, based on strain measurements on FRPstrengthened RC tension members. As a result, the model has included the effect of tensile cracking and is not a true local bondslip model. This model is therefore not further discussed in this paper. Of the six models, the two models recently proposed by Dai and Ueda [14] and Ueda et al. [15] were based on test data for specimens with very soft adhesive layers and are not further discussed in this paper.
The predictions of all 12 bond strength models are compared with the 253 test results of the present test database in Table 3 and Figure 3. The average value and coefficient of variation of the predictedtotest bond strength ratios and the correlation coefficient of each model are given in Table 3. It can be seen that the bond strength models of Maeda et al. [9], Neubauer and Rostasy [29], Khalifa et al. [30], Iso [32], Yang et al. [33] and Chen and Teng [7] are the better models, with a reasonably small coefficient of variation and a large correlation coefficient. The test results are shown against the predictions of these betterperforming models in Figure 3. Based on Tables 1 and 3 as well as Figure 3, Chen and Teng’s model is clearly the most accurate model among the 12 existing bond strength models. If Table 3 is examined together with Table 1, it can be found that the accuracy of a model improves as more significant parameters are considered, with the effective bond length being the most influential parameter. All the 6 betterperforming models include a definition of the effective bond length. Of the other 6 models, only Sato’s model [32] takes the effective bond length into consideration.
For a bondslip model to provide accurate predictions, it needs to have an appropriate shape as well as a correct value for the interfacial fracture energy which is equal to the area under the bondslip curve. The shape of the bondslip model determines the predicted distribution of axial strains in the plate. The predictions of the four existing bondslip models for normaladhesive interfaces are shown in Figure 4 for an FRPtoconcrete bonded joint with the following properties: _{ } , _{ } ,_{ } , _{ } , _{ } . An FRPtoconcrete width ratio of 0.5 was chosen for this comparison joint as some of the bondslip models were based on test results of joints with similar width ratios and do not account for the effect of varying this ratio. It can be seen that the shapes of the predicted bondslip curves differ substantially (Figures 4). In particular, the linearbrittle model of Neubauer and Rostasy [34] is very different from the other three models. The fact that the bond stress reduces to zero at the ultimate slip dictates that there exists an effective bond length beyond which an increase in the bond length will not increase the ultimate load.
Existing studies (e.g. [12], [36]) have shown that the bondslip curve should have an ascending branch and a descending branch, similar to the curve from Nakaba et al.’s model [12] or Savioa et al.’s model [36] shown in Figure 4. The bilinear model can be used as an approximation [16], but the linearbrittle model by Neubauer and Rostasy [34] is unrealistic. Apart from the general shape, three key parameters, including the maximum bond stress, the slip at maximum stress and the ultimate slip at zero bond stress, determine the accuracy of the model. It is interesting to note that the models by Nakaba et al. [12], Monti et al. [35] and Savioa et al. [36] are in reasonably close agreement, and the linearbrittle model of Neubauer and Rostasy [34] predicts a similar maximum bond stress. It should be noted that Savioa et al.’s model [36] was obtained by some very minor modifications of Nakaba et al.’s model (Table 2).
Existing research has shown that the bond strength _{ } of an FRPtoconcrete bonded joint is directly proportional to the square root of the interfacial fracture energy_{ } regardless of the shape of the bondslip curve [16, 29, 37, 38], so a comparison of the bond strength is equivalent to a comparison of the interfacial fracture energy. As most bondslip models do not provide an explicit formula for the ultimate load, the bond strengths of bondslip models need to be obtained numerically. In the present study, they were obtained by numerical nonlinear analyses using MSC.Marc [39] with a simple model consisting of 1 mmlong truss elements representing the FRP plate connected to a series of shear springs on a rigid base representing the bondslip law of the interface. The nonlinear analyses were carried out with a tight convergence tolerance to ensure accurate predictions. The theoretical predictions of the bond strengths are compared with the 253 test results of the present test database. The average value and coefficient of variation of the predictedtotest bond strength ratios together with the correlation coefficient for each model are given in Table 4. The correlation coefficients for all four bondslip models are larger than 0.8, which demonstrates that the trends of the test data are reasonably well described by the bondslip models. The coefficients of variation of these models are nevertheless still larger than that of Chen and Teng’s model (Table 3). The test results are shown against the theoretical predictions in Figure 5, where it is clearly seen that all four bondslip models are too optimistic.
Since it is difficult to obtain accurate bondslip curves directly from strain measurements in a pull test, Lu et al. [40] recently explored a numerical approach from which the bondslip curve of any point along the interface can be obtained. The approach is based on the observation that debonding in a pull test occurs in the concrete, so if the failure of concrete can be accurately modelled, the interfacial shear stress and slip at a given location along the interface can be obtained from the finite element model. It should be noted that this numerical modelling approach relies on the accurate modelling of concrete failure near the achieve layer. Tests have shown that debonding of FRP from concrete in a pull test generally occurs within a thin layer of concrete of 2 to 5mm thick adjacent to the adhesive layer. To simulate concrete failure within such a thin layer, with the shapes and paths of the cracks properly captured, Lu et al. [40] proposed a mesoscale finite element approach in which very small elements (with element sizes being one order smaller than the thickness of the facture zone of concrete) are used in conjunction with a fixed angle crack model (FACM) [41]. The size effect of elements is duly accounted for through facture energy considerations. This approach has the simplicity of the FACM for which the relevant material parameters have clear physical meanings and can be found from well established standard tests, but in the meantime retains the capability of tracing the paths of cracks as deformations increase through the use of very small elements. To reduce the computational effort, the threedimensional FRPtoconcrete bonded joint (Figure 1) was modelled as a plane stress problem using 4node isoparametric elements, with the effect of FRPtoconcrete width ratio being separately considered using a width ratio factor devised by Chen and Teng [7].
Lu et al. [40] implemented their finite element model into the general purpose finite element package MSC.Marc [39] as a user subroutine. The finite element model was verified by detailed comparisons with the results of 10 pull tests taken from studies by Wu et al. [13], Ueda et al. [22], Tan [17], and Yuan et al. [16]. A close agreement was achieved for all 10 specimens. A Fast Fourier Transform smoothing procedure was proposed in Lu et al. [40] to process the raw finite element interfacial shear stresses before the results are used to obtain local bondslip curves. Lu et al. [40] showed that a smoothing length of 10mm is suitable and this length was used in the present study. An unbonded zone of 25 mm was included in the finite element model in all numerical simulations of the present study. Further details of the finite element model can be found in Lu et al. [40].
Using the mesoscale finite element model of Lu et al. [40], a parametric study was undertaken to study the local bondslip behaviour of the interface, considering the effects of a number of key parameters. The bonded joint modelled in this parametric study has the following properties: the axial stiffness of the FRP plate _{ } is 26 GPa.mm, which is similar to that provided by one thin layer of CFRP and is within the most popular range of FRP plate axial stiffness in pull tests (Figure 2b). For a given concrete strength, the elastic modulus of concrete was found according to the Chinese code for the design of concrete structures [24] (i.e. _{ } , in MPa) and the Poisson’s ratio was assumed to be 0.2. The shear stiffness of the adhesive layer is 5GPa/mm. The bond length of the FRP plate is 200mm, which is much longer than the effective bond length. A typical bondslip curve obtained from the finite element model is shown in Figure 6. From these finite element results, the following observations can be made:
(a) The bondslip curve is made up of an ascending branch and a descending branch, with the bond stress reducing to zero when the slip is sufficiently large.
(b) The initial stiffness of the bondslip curve is much larger than the secant stiffness at the peak stress point. This initial high stiffness, representing the stiffness of the completely linear elastic state of the interface, decreases quickly with the appearance of microcracking in the concrete as the bond stress increases.
(c) The maximum bond stress _{ } and the corresponding slip _{ } increase almost linearly with_{ } , while the interfacial fracture energy _{ } increases almost linearly with_{ } , as shown in Figure 7.
Based on the above observations, the following equations, referred to hereafter as the precise bondslip model, are proposed to describe the local bondslip relationship:
_{ } if _{ } 
(1a) 

_{ } if_{ } , 
(1b) 
where _{ } , _{ } . To closely capture the finite element bondslip curves, a variety of equation forms were tested and Eqs 1a and 1b were found to predict the finite element bondslip curves most closely without undue complexity. The maximum bond stress τ_{max} and the corresponding slip s_{0} are given by
_{ } 
(1c) 

_{ } 
(1d) 
where _{ } is the elastic component of _{ } and β_{w} is the FRPtoconcrete width ratio factor. The initial stiffness of the bondslip model is defined by:
_{ } 
(1e) 
where _{ } and _{ } . G_{c} is the elastic shear modulus of concrete and t_{c} is the effective thickness of the concrete whose deformation forms part of the interfacial slip, which can be deduced from the initial stiffness of the bondslip curve from a mesoscale FE analysis [40]. The initial part of the bondslip curve from mesoscale FE analysis given in Figure 6 is shown in Figure 8. It can be seen that t_{c} =5mm leads to a close prediction of the bondslip curve. While a precise definition of t_{c }requires more deliberation, the overall effect of such precision on the bondslip curve is very small and insignificant for practical purposes. Furthermore, it may be noted that the simplified model introduced blow does not include t_{c} as a parameter but still leads to a bondslip curve which is very closely similar to that of the precise model.
The parameter _{ } in Eq. 1b controls the shape of the descending branch and is given by
_{ } 
(1f) 
where the interfacial fracture energy can be expressed as:
_{ } 
(1g) 
while the fracture energy of the ascending branch _{ } can be calculated as:
_{ } 
(1h) 
It should be noted that Eqs 1c, 1d and 1g were found as linear bestfit lines to the finite element predictions, except for the introduction of the width effect ratio β_{w} and the elastic slip component _{ } . The width effect is introduced based on existing knowledge of how it affects the three bondslip parameters defined by Eqs 1c, 1d and 1g, while the elastic slip component is introduced to ensure that the slope of the bondslip model is equal to that given by Eq. 1e. The elastic slip component is generally very small and its inclusion into Eq. 1d has little effect on its predictions. The function f(K_{a}) is included to cater for the future extension of the model to interfaces with very soft adhesive layers but for normal adhesive layers with K_{a}≥2.5GPa/mm, f(K_{a})=1 as finite element results not presented here have shown that the effect of the adhesive layer stiffness on G_{f} is very small for such normal adhesives.
Because of some inevitable differences between the finite element predictions and the test results, the three coefficients in the proposed bondslip model _{ } , _{ } and _{ } were determined through an iterative procedure, making use of both the finite element and the test results. The planar nature of the finite element model also means that the effect of the FRPtoconcrete width ratio needs to be accounted for based on the test results. This iterative procedure is as follows:
(1) Take _{ } for a normal adhesive layer and start the process with _{ } , _{ } and _{ } , which were determined from regressions of the finite element results.
(2) Assuming that _{ } , use the precise bondslip model with the coefficients from step (1) to calculate the bond strength.
(3) Compare the predicted bond strengths with the test results to evaluate the width ratio effect and to determine a bestfit expression for the width ratio factor_{ } . Figure 9 shows the deduced values of the width ratio factor at the end of the iterative process.
(4) Using the current expression for _{ } , finetune the values for _{ } , _{ } and _{ } to reach an improved agreement between the predicted and the test bond strengths.
(5) Compare the predicted bond strengths to the test results again to refine the expression for _{ } .
(6) Repeat steps (4) and (5) until changes in_{ } , _{ } and _{ } fall below 0.1%.
The final values obtained from this process for these three coefficients are: _{ } , _{ } , and _{ } , while the width ratio factor is given by
_{ } 
(1i) 
The bondslip curve from the precise model for one of the bonded joints analysed by the finite element method is shown in Figure 6. It is clear that there is a close agreement between this precise model and the finite element curve.
In terms of the present test database, Eq. 1i represents a slight improvement to the following expression originally proposed by Chen and Teng [7]:
_{ } 
(2) 
The difference between the two expressions is however very small (Figure 9) and both equations are satisfactory for practical applications.
7.2 Simplified model
The precise model is accurate but somewhat complicated. A simplified model without a significant loss of accuracy can be easily obtained by noting that the initial stiffness of the bondslip curve is much larger than the secant stiffness at the peak point. Based on this observation, the initial stiffness can be approximated as infinity and the following simplified bondslip model can be obtained:

_{ } if _{ } 
(3a) 
_{ } if _{ } 
(3b) 
where
_{ } 
(3c) 

_{ } 
(3d) 

_{ } 
(3e) 
_{ } and _{ } can be calculated with Eqs 1c and 1i. The bondslip curve predicted by the simplified model is also shown in Figure 6, where it can be seen that there is little difference between this model and the precise model. For all practical purposes, the simplified model is sufficient for normaladhesive joints with f(K_{a})=1 but much simpler than the precise model.
7.3 Bilinear model
Further simplification can be made to the simplified model by adopting a bilinear bondslip curve which can be used to derive a simple explicit design equation for the bond strength. This bilinear model has the same local bond strength and total interfacial fracture energy, so the bond strength is unaffected by this simplification if the bond length is longer than the effective bond length. This bilinear model is described by the following equations:

_{ } if _{ } 
(4a) 
_{ } if _{ } 
(4b) 

_{ } if _{ } 
(4c) 
where
_{ } 
(4d) 
In the above equations, _{ } , _{ } and _{ } can be found using Eqs 1c, 3c and 3d, respectively. The prediction of the bilinear model is also shown in Figure 6.
Regardless of the bondslip model, the bond strength of an FRPtoconcrete bonded joint in terms of the interfacial fracture energy is given by Eq. 4e [16]
_{ } 
(4e) 
where _{ } is the bond length factor. When _{ } , _{ } , but when _{ } , _{ } is smaller than 1. The analytical solution for _{ } with a bilinear bondslip model is given by Yuan et al. [16]:
_{ } 
(4f) 
where
_{ } 
(4g) 

_{ } 
(4h) 

_{ } 
(4i) 
In Eq. 4i, a factor of 0.99 is used instead of 0.97 originally adopted in Yuan et al. [16]. The former implies that the effective bond length is one at which 99% of the bond strength of an infinitely long bonded joint is achieved while the latter requires only 97%. The former is thus a more stringent definition and leads to effective bond lengths in closer agreement with those given by Chen and Teng’s bondstrength model [7]. The effective bond length factor _{ } in Eq. 4e has been defined by Chen and Teng [7] to be
_{ } if _{ } 
(4j) 
The use of a sine function has its basis in the analytical solution [16]. The following alternative expression for _{ } proposed by Neubauer et al. [29] provides similar predictions (Figure 10):
_{ } if _{ } 
(4k) 
When compared with the present finite element results, Eq. 4k is slightly more accurate (Figure 10) but this small difference is insignificant and does mean that it provides more accurate predictions of test results. The use of either expression is thus satisfactory for design purposes, although Eq. 4k was used with Eq. 4e in the present study to obtain the results shown in Table 3.
Two of the three bondslip models proposed in this study are compared with the four existing bondslip models developed for normaladhesive bonded joints in Figure 4. It can be seen that Nakaba et al.’s model [12] and Savioa et al.’s model [36] are closer to the proposed models than the other two models. The maximum bond stress and the interfacial fracture energy of Nakaba et al.’s model and those of Savioa et al.’s model are however much larger than those of the proposed models.
In Figure 11, the bond strengths predicted using the proposed bondslip models are compared with the results of the 253 pull tests listed in Table A1. It can be found that the proposed bondslip models give results in close agreement with the test results and perform better than existing bondslip models. The results of the precise model and the simplified model are almost the same, with the precise model performing very slightly better. The average value and coefficient of variation of the predictedtotest bond strength ratios together with the correlation coefficient for the bond strength formula (Eq. 4e) are given in Table. 3. It can be seen that Eq. 4e performs significantly better than all existing bond strength models except Chen and Teng’s model [7]. The new bond strength model is only slightly better than Chen and Teng’s model [7], so Chen and Teng’s model [7] is still recommended for use in design due to its simple form.
The strain distributions in the FRP plate can be numerically calculated from the bondslip models. The comparison of strain distributions between tests and predictions for specimens PG122 and PC11C2 tested by Tan [17], specimen SCFS40025 tested by Wu et al. [13], and specimen B2, tested by Ueda et al. [22], are shown in Figure 12 (a~d). Comparisons are made for the same applied load (except for insignificant differences as the test load levels are not identical to the load levels in the numerical analysis which was conducted by displacement control) before debonding and for the same effective stress transfer length in the stage of debonding propagation. The load levels and slip values indicated here are those from numerical analysis. It can be found that both the precise model and the bilinear model are in close agreement with the test results. The precise model does provide slightly more accurate predictions, which demonstrates that the curved shape of the precise model is closer to the real situation. Additional comparisons not reported here for a number of other specimens for which strain distributions are available also showed similar agreement.
Using specimen PG122 as an example, the strain distributions predicted with different bondslip models are compared with the test results in Figure 13. Comparisons are made for the same load of P/P_{u} = 0.40 (where P_{u} is the finite element ultimate load) before debonding occurs (Figure 13a) and for the same effective stress transfer length of 125mm in the stage of debonding propagation (Figure 13b). It can be seen that at a low load in the predebonding stage, the strain distribution does not appear to be so sensitive to the bondslip model. However, in the stage of debonding propagation, the differences between the models and between the model predictions and the test results are large. Figure 13 shows that the existing models do not provide accurate predictions of test results.
This paper has provided a critical review and assessment of existing bond strength models and bondslip models, and presented a set of three new bondslip models. The assessment of theoretical models has been conducted using the test results of 253 pull specimens collected from the existing literature. The development of the new bondslip models employed a new approach in which mesoscale finite element results with appropriate numerical smoothing are exploited together with test results. Based on the results and discussions presented in this paper, the following conclusions may be drawn.
1. Among the 12 existing bond strength models, the model proposed by Chen and Teng [7] is the most accurate. The bond strength model based on the proposed bilinear bondslip model is as accurate as Chen and Teng’s model [7] but is more complicated. Chen and Teng’s model therefore remains the model of choice for use in design.
2. Typical bondslip curves should consist of an ascending branch with continuous stiffness degradation to the maximum bond stress and a curved descending branch reaching a zero bond stress at a finite value of slip.
3. While a precise bondslip model should consist of a curved ascending branch and a curved descending branch, other shapes such as a bilinear model can be used as a good approximation. An accurate bondslip model should provide close predictions of both the shape and fracture energy (area under the bondslip curve) of the bondslip curve. None of the existing bondslip models provides accurate predictions of both the shape and the interfacial fracture energy as found from tests.
4. The three new bondslip models, based on a combination of finite element results and the test results predict both the bond strength and strain distribution in the FRP plate accurately. These models are therefore recommended for future use in the numerical modelling of FRPstrengthened RC structures.
It should be noted that the scope of the present study has been limited to FRPtoconcrete bonded joints whose sheardeformed adhesive layer has a shear stiffness of no less than 2.5 GPa/mm. The studies by Dai and Ueda [14] and Ueda et al. [15] should be consulted for information on FRPtoconcrete bonded joints with a very soft adhesive layer. The present work nevertheless is believed to be applicable to at least all commercially available FRP systems for external bonding applications outside Japan.
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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[21] Zhao HD, Zhang Y, Zhao M. Research on the bond performance between CFRP plate and concrete. Proc. of 1^{st} Conference on FRPConcrete Structures of China. 2000, p. 247253.
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[23] Ren HT. Study on basic theories and long time behavior of concrete structures strengthened by fiber reinforced polymers. PhD Thesis, Dalian University of Technology, China, 2003. (in Chinese).
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[26] Hiroyuki Y, Wu Z. Analysis of debonding fracture properties of CFS strengthened member subject to tension. Proc. of 3^{rd} International Symposium on NonMetallic (FRP) Reinforcement for Concrete Structures, Sapporo, vol. 1. 1997, p. 284294.
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[40] Lu XZ, Ye LP, Teng JG, Jiang JJ. Mesoscale finite element model for FRP plates/sheets bonded to concrete. Engineering Structures, Accepted for Publication.
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(a) Range of concrete strength
(b) Range of plate stiffness
(c) Range of bond length
(d) Range of FRPtoconcrete width ratio
(a) Maximum bond stress
(b) Slip at maximum bond stress
(c) Interfacial fracture energy
Figure 8. Initial stiffness of bondslip curve
(a) Precise model
(b) Bilinear model
(a) Specimen PG122 of Tan [17]
(b) Specimen of PC11C2 of Tan [17]
(c) Specimen SCFS40025 of Wu et al. [13]
(d) Specimen B2 of Ueda et al. [22]
(a) Before debonding stage
(b) Debonding propagation stage
Bond strength model 
Concrete strength 
FRP plate stiffness 
Effective bond length 
Width ratio 

1 
Tanaka [25] 
No 
No 
No 
No 
2 
Hiroyuki and Wu [26] 
No 
No 
No 
No 
3 
van Gemert [27] 
Yes 
No 
No 
No 
4 
Maeda et al. [9] 
Yes 
Yes 
Yes 
No 
5 
Neubauer and Rostasy [29] 
Yes 
Yes 
Yes 
Yes 
6 
Khalifa et al. [30] 
Yes 
Yes 
Yes 
No 
7 
Chaallal et al. [31] 
No 
Yes 
No 
No 
8 
Chen and Teng [7] 
Yes 
Yes 
Yes 
Yes 
9 
Izumo [32] 
Yes 
Yes 
No 
No 
10 
Sato [32] 
Yes 
Yes 
Yes 
No 
11 
Iso [32] 
Yes 
Yes 
Yes 
No 
12 
Yang et al. [33] 
Yes 
Yes 
Yes 
No 
Table 2. Existing bondslip models
Bondslip model 
Ascending branch _{ } 
Descending branch _{ } 
_{ } 
_{ } 
_{ } 
_{ } 
Remarks 
Neubauer and Rostasy [34] 
_{ } 
_{0} 
_{ } 
_{ } 
_{ } 
A linear ascending branch and a sudden drop 

Nakaba et al. [12] 
_{ } 
_{ } 
_{ } 
A single curve 

Monti et al. [35] 
_{ } 
_{ } 
_{ } 
_{ } 
_{ } 
_{ } 

Savioa et al. [36] 
_{ } 
_{ } 
_{ } 
A single curve 

Dai and Ueda [14]* 
_{ } 
_{ } 
_{ } 
_{ } 
_{ } ,_{ } _{ } , _{ } 

Ueda et al. [15]* 
[_{ } , _{ } ] 
A single curve 
*: regressed from specimens with very soft adhesive layers.
Bond strength model 
Average Predictedtotest bond strength ratio 
Coefficient of variation 
Correlation coefficient 

1 
Tanaka [25] 
4.470 
0.975 
0.481 
2 
Hiroyuki and Wu [26] 
4.290 
0.611 
0.028 
3 
Sato [32] 
1.954 
0.788 
0.494 
4 
Chaallal et al. [31] 
1.683 
0.749 
0.240 
5 
Khalifa et al. [30] 
0.680 
0.293 
0.794 
6 
Neubauer and Rostasy [29] 
1.316 
0.168 
0.848 
7 
Izumo [32] 
1.266 
0.506 
0.656 
8 
van Gemert [27] 
1.224 
0.863 
0.328 
9 
Maeda et al. [9] 
1.094 
0.202 
0.773 
10 
Iso [32] 
1.087 
0.282 
0.830 
11 
Yang et al. [33] 
0.996 
0.263 
0.766 
12 
Chen and Teng [7] 
1.001 
0.163 
0.903 
13 
Proposed strength formula (Eq. 4e) 
1.001 
0.156 
0.908 
Bondslip model 
Average predictedtotest bond strength ratio 
Coefficient of variation 
Correlation coefficient 

1 
Neubauer and Rostasy [34] 
1.330 
0.209 
0.873 
2 
Nakaba et al. [12] 
1.326 
0.231 
0.846 
3 
Savioa et al. [36] 
1.209 
0.199 
0.847 
4 
Monti et al. [35] 
1.575 
0.164 
0.888 
5 
Proposed, precise model 
1.001 
0.155 
0.910 
6 
Proposed, simplified model 
1.001 
0.155 
0.910 
7 
Proposed, bilinear model 
1.001 
0.156 
0.908 
The following symbols are used in this paper:
_{ } ,_{ } =parameters in the proposed precise model;
_{ } = width of concrete prism;
_{ } = width of FRP plate;
_{ } = elastic modulus of adhesive;
_{ } = elastic modulus of FRP;
_{ } = concrete cylinder compressive strength;
_{ } = concrete tensile strength;
_{ } = shear modulus of adhesive;
_{ } = elastic shear modulus of concrete;
_{ } = interfacial fracture energy;
_{ } = interfacial fracture energy for the ascending branch;
_{ } = _{ } , shear stiffness of adhesive layer;
_{ } = _{ } , shear stiffness of concrete;
_{ } = bond length;
_{ } = effective bond length;
_{ } = ultimate load or bond strength;
_{ } = local slip;
_{ } = elastic component of local slip;
_{ } = local slip when bond stress _{ } reduces to zero;
_{ } = local slip at _{ } ;
_{ } = thickness of adhesive layer;
_{ } = effective thickness of concrete contributing to shear deformation;
_{ } = thickness of FRP plate;
_{ } ,_{ } ,_{ } =coefficients in proposed bondslip models;
_{ } = bond length factor;
_{ } = width ratio factor;
_{ } = local bond stress;
_{ } = maximal local bond stress;
_{ } = average bond stress;
Table A1 Database of pull tests
Source 
Specimen 
FRP plate 
Concrete prism 
Ultimate load P_{u }(kN) 

Thickness t_{f} (mm) 
Width b_{f} (mm) 
Bond length L(mm) 
Elastic modulus E_{f} (GPa) 
Tensile strength f_{f} (MPa) 
Cube strength f_{cu} (MPa) 
Tensile strength f_{t} (MPa) 
Width b_{c} (mm) 

Tan [17] 
PG111 
0.169 
50 
130 
97 
2777 
37.60 
2.90 
100 
7.78* 
PG112 
0.169 
50 
130 
97 
2777 
37.60 
2.90 
100 
9.19* 

PG11W1 
0.169 
75 
130 
97 
2777 
37.60 
2.90 
100 
10.11* 

PG11W2 
0.169 
75 
130 
97 
2777 
37.60 
2.90 
100 
13.95* 

PG11L11 
0.169 
50 
100 
97 
2777 
37.60 
2.90 
100 
6.87* 

PG11L12 
0.169 
50 
100 
97 
2777 
37.60 
2.90 
100 
9.20* 

PG11L21 
0.169 
50 
70 
97 
2777 
37.60 
2.90 
100 
6.46* 

PG11L22 
0.169 
50 
70 
97 
2777 
37.60 
2.90 
100 
6.66* 

PG121 
0.338 
50 
130 
97 
2777 
37.60 
2.90 
100 
10.49* 

PG122 
0.338 
50 
130 
97 
2777 
37.60 
2.90 
100 
11.43* 

PC11C1 
0.111 
50 
130 
235 
3500 
37.60 
2.90 
100 
7.97* 

PC11C2 
0.111 
50 
130 
235 
3500 
37.60 
2.90 
100 
9.19* 

Zhao et al. [21] 
NJ2 
0.083 
100 
100 
240 
3550 
20.50 
2.08 
150 
11.00 
NJ3 
0.083 
100 
150 
240 
3550 
20.50 
2.08 
150 
11.25 

NJ4 
0.083 
100 
100 
240 
3550 
36.70 
2.87 
150 
12.50 

NJ5 
0.083 
100 
150 
240 
3550 
36.70 
2.87 
150 
12.25 

NJ6 
0.083 
100 
150 
240 
3550 
36.70 
2.87 
150 
12.75 

Takeo et al. [20] 
111 
0.167 
40 
100 
230 
3481 
36.56 
2.86 
100 
8.75 
112 
0.167 
40 
100 
230 
3481 
33.75 
2.74 
100 
8.85 

121 
0.167 
40 
200 
230 
3481 
36.56 
2.86 
100 
9.30 

122 
0.167 
40 
200 
230 
3481 
33.75 
2.74 
100 
8.50 

131 
0.167 
40 
300 
230 
3481 
36.56 
2.86 
100 
9.30 

132 
0.167 
40 
300 
230 
3481 
33.75 
2.74 
100 
8.30 

141 
0.167 
40 
500 
230 
3481 
36.56 
2.86 
100 
8.05 

142 
0.167 
40 
500 
230 
3481 
36.56 
2.86 
100 
8.05 

151 
0.167 
40 
500 
230 
3481 
33.50 
2.73 
100 
8.45 

152 
0.167 
40 
500 
230 
3481 
33.50 
2.73 
100 
7.30 

211 
0.167 
40 
100 
230 
3481 
31.63 
2.64 
100 
8.75 

212 
0.167 
40 
100 
230 
3481 
31.63 
2.64 
100 
8.85 

213 
0.167 
40 
100 
230 
3481 
33.13 
2.71 
100 
7.75 

214 
0.167 
40 
100 
230 
3481 
33.13 
2.71 
100 
7.65 

215 
0.167 
40 
100 
230 
3481 
30.88 
2.61 
100 
9.00 

221 
0.334 
40 
100 
230 
3481 
31.63 
2.64 
100 
12.00 

222 
0.334 
40 
100 
230 
3481 
31.63 
2.64 
100 
10.80 

231 
0.501 
40 
100 
230 
3481 
31.63 
2.64 
100 
12.65 

232 
0.501 
40 
100 
230 
3481 
31.63 
2.64 
100 
14.35 

241 
0.165 
40 
100 
373 
2942 
30.88 
2.61 
100 
11.55 

242 
0.165 
40 
100 
373 
2942 
30.88 
2.61 
100 
11.00 

251 
0.167 
40 
100 
230 
3481 
33.13 
2.71 
100 
9.85 

252 
0.167 
40 
100 
230 
3481 
33.13 
2.71 
100 
9.50 

261 
0.167 
40 
100 
230 
3481 
33.13 
2.71 
100 
8.80 

262 
0.167 
40 
100 
230 
3481 
33.13 
2.71 
100 
9.25 

271 
0.167 
40 
100 
230 
3481 
33.13 
2.71 
100 
7.65 

272 
0.167 
40 
100 
230 
3481 
33.13 
2.71 
100 
6.80 

281 
0.167 
40 
100 
230 
3481 
63.25 
3.87 
100 
7.75 

282 
0.167 
40 
100 
230 
3481 
63.25 
3.87 
100 
8.05 

291 
0.167 
40 
100 
230 
3481 
30.88 
2.61 
100 
6.75 

292 
0.167 
40 
100 
230 
3481 
30.88 
2.61 
100 
6.80 

2101 
0.111 
40 
100 
230 
3481 
31.63 
2.64 
100 
7.70 

2102 
0.111 
40 
100 
230 
3481 
33.13 
2.71 
100 
6.95 

Ren [23] 
DLUT152G 
0.507 
20 
150 
83.03 
3271 
28.70 
2.50 
150 
5.81 
DLUT155G 
0.507 
50 
150 
83.03 
3271 
28.70 
2.50 
150 
10.60 

DLUT157G 
0.507 
80 
150 
83.03 
3271 
28.70 
2.50 
150 
18.23 

DLUT301G 
0.507 
20 
100 
83.03 
3271 
45.30 
3.22 
150 
4.63 

DLUT302G 
0.507 
20 
150 
83.03 
3271 
45.30 
3.22 
150 
5.77 

DLUT303G 
0.507 
50 
60 
83.03 
3271 
45.30 
3.22 
150 
9.42 

DLUT304G 
0.507 
50 
100 
83.03 
3271 
45.30 
3.22 
150 
11.03 

DLUT306G 
0.507 
50 
150 
83.03 
3271 
45.30 
3.22 
150 
11.80 

DLUT307G 
0.507 
80 
100 
83.03 
3271 
45.30 
3.22 
150 
14.65 

DLUT308G 
0.507 
80 
150 
83.03 
3271 
45.30 
3.22 
150 
16.44 

DLUT501G 
0.507 
20 
100 
83.03 
3271 
55.50 
3.60 
150 
5.99 

DLUT502G 
0.507 
20 
150 
83.03 
3271 
55.50 
3.60 
150 
5.90 

DLUT504G 
0.507 
50 
100 
83.03 
3271 
55.50 
3.60 
150 
9.84 

DLUT505G 
0.507 
50 
150 
83.03 
3271 
55.50 
3.60 
150 
12.28 

DLUT506G 
0.507 
80 
100 
83.03 
3271 
55.50 
3.60 
150 
14.02 

DLUT507G 
0.507 
80 
150 
83.03 
3271 
55.50 
3.60 
150 
16.71 

DLUT152C 
0.33 
20 
150 
207 
3890 
28.70 
2.50 
150 
5.48 

DLUT155C 
0.33 
50 
150 
207 
3890 
28.70 
2.50 
150 
10.02 

DLUT157C 
0.33 
80 
150 
207 
3890 
28.70 
2.50 
150 
19.27 

DLUT301C 
0.33 
20 
100 
207 
3890 
45.30 
3.22 
150 
5.54 

DLUT302C 
0.33 
20 
150 
207 
3890 
45.30 
3.22 
150 
4.61 

DLUT304C 
0.33 
50 
100 
207 
3890 
45.30 
3.22 
150 
11.08 

DLUT305C 
0.33 
50 
100 
207 
3890 
45.30 
3.22 
150 
16.10 

DLUT306C 
0.33 
50 
150 
207 
3890 
45.30 
3.22 
150 
21.71 

DLUT307C 
0.33 
80 
100 
207 
3890 
45.30 
3.22 
150 
22.64 

DLUT501C 
0.33 
20 
100 
207 
3890 
55.50 
3.60 
150 
5.78 

DLUT504C 
0.33 
50 
100 
207 
3890 
55.50 
3.60 
150 
12.95 

DLUT505C 
0.33 
50 
150 
207 
3890 
55.50 
3.60 
150 
16.72 

DLUT506C 
0.33 
80 
100 
207 
3890 
55.50 
3.60 
150 
16.24 

DLUT507C 
0.33 
80 
150 
207 
3890 
55.50 
3.60 
150 
22.80 

Ueda et al. [22] 
Ueda_A1 
0.11 
50 
75 
230 
3479 
29.74 
2.55 
100 
6.25* 
Ueda_A2 
0.11 
50 
150 
230 
3479 
52.31 
3.48 
100 
9.2* 

Ueda_A3 
0.11 
50 
300 
230 
3479 
52.31 
3.48 
100 
11.95* 

Ueda_A4 
0.22 
50 
75 
230 
3479 
55.51 
3.60 
100 
10.00* 

Ueda_A5 
0.11 
50 
150 
230 
3479 
54.36 
3.56 
100 
7.30* 

Ueda_A6 
0.165 
50 
65 
372 
2940 
54.36 
3.56 
100 
9.55* 

Ueda_A7 
0.22 
50 
150 
230 
3479 
54.75 
3.57 
100 
16.25* 

Ueda_A8 
0.11 
50 
700 
230 
3479 
54.75 
3.57 
100 
11.00* 

Ueda_A9 
0.11 
50 
150 
230 
3479 
51.03 
3.43 
100 
10.00* 

Ueda_A10 
0.11 
10 
150 
230 
3479 
30.51 
2.59 
100 
2.40* 

Ueda_A11 
0.11 
20 
150 
230 
3479 
30.51 
2.59 
100 
5.35* 

Ueda_A12 
0.33 
20 
150 
230 
3479 
30.51 
2.59 
100 
9.25* 

Ueda_A13 
0.55 
20 
150 
230 
3479 
31.67 
2.64 
100 
11.75* 

Ueda_B1 
0.11 
100 
200 
230 
3479 
31.67 
2.64 
500 
20.60 

Ueda_B2 
0.33 
100 
200 
230 
3479 
52.44 
3.49 
500 
38.00 

Ueda_B3 
0.33 
100 
200 
230 
3479 
58.85 
3.71 
500 
34.10 

Wu et al. [13] 
DCFS15030a 
0.083 
100 
300 
230 
4200 
58.85 
3.71 
100 
12.20* 
DCFS15030b 
0.083 
100 
300 
230 
4200 
73.85 
4.21 
100 
11.80* 

DCFS15030c 
0.083 
100 
300 
230 
4200 
73.85 
4.21 
100 
12.25* 

DCFS30030a 
0.167 
100 
300 
230 
4200 
73.85 
4.21 
100 
18.90* 

DCFS30030b 
0.167 
100 
300 
230 
4200 
73.85 
4.21 
100 
16.95* 

DCFS30030c 
0.167 
100 
300 
230 
4200 
73.85 
4.21 
100 
16.65* 

DCFS60030a 
0.333 
100 
300 
230 
4200 
73.85 
4.21 
100 
25.65* 

DCFS60030b 
0.333 
100 
300 
230 
4200 
73.85 
4.21 
100 
25.35* 

DCFS60030c 
0.333 
100 
300 
230 
4200 
73.85 
4.21 
100 
27.25* 

DCFM30030a 
0.167 
100 
300 
390 
4400 
73.85 
4.21 
100 
19.50* 

DCFM30030b 
0.167 
100 
300 
390 
4400 
73.85 
4.21 
100 
19.50* 

DAR28030a 
1 
100 
300 
23.9 
4400 
73.85 
4.21 
100 
12.75* 

DAR28030b 
1 
100 
300 
23.9 
4400 
73.85 
4.21 
100 
12.85* 

DAR28030c 
1 
100 
300 
23.9 
4400 
73.85 
4.21 
100 
11.90* 

SCFS40025a 
0.222 
40 
250 
230 
4200 
73.85 
4.21 
100 
15.40 

SCFS40025b 
0.222 
40 
250 
230 
4200 
73.85 
4.21 
100 
13.90 

SCFS40025c 
0.222 
40 
250 
230 
4200 
73.85 
4.21 
100 
13.00 

SCFM30025a 
0.167 
40 
250 
390 
4400 
73.85 
4.21 
100 
12.00 

SCFM30025b 
0.167 
40 
250 
390 
4400 
73.85 
4.21 
100 
11.90 

0.5 
40 
250 
390 
4400 
73.85 
4.21 
100 
25.90 

SCFM90025b 
0.5 
40 
250 
390 
4400 
73.85 
4.21 
100 
23.40 

SCFM90025c 
0.5 
40 
250 
390 
4400 
73.85 
4.21 
100 
23.70 
* doubleshear test; P_{u} is equal to half of the total applied load at failure. If the literature provides only the cylinder strength, then _{ } .
This appendix provides a summary of four bond strength models which are believed to be not widely accessible for the convenience of readers. Three of them are described in a recent JCI report [32] while the fourth one was developed in China. The following units are used: N for forces, MPa for stresses and elastic moduli, and mm for lengths.
B.1 Izumo’s model
The bond strength model proposed by Izumo [32] is given by
_{ } for carbon fibre sheets
and
_{ } for aramid fibre sheets.
B.2 Sato’s model
The bond strength model given by Sato [32] is described by the following equations:
_{ }
_{ }
if _{ } , then _{ }
_{ }
_{ } mm is the working width of concrete.
B.3 Iso’s model
The bond strength model proposed by M. Iso [32] is given by
_{ }
_{ }
_{ }
where if _{ } if _{ } .
B.4 Yang’s model
The bond strength model proposed by Yang et al. [33] is
_{ }
where
_{ }
_{ } mm