## 1. Introduction

Nowadays, structural strengthening with externally bonded (EB) carbon fiber reinforced polymer (CFRP) strips has been considered as the state-of-the-art for more than three decades of application in the field [

1,

2]. Prestressing the externally bonded reinforcements represents a newer way of using the same material(s), however, with the difference of exploiting its excellent mechanical properties, such as high tensile strength, in a more efficient way [

3].

An overlaying question is the anchorage issue. Similar to conventional external prestressing with steel cables, prestressed CFRP also needs end-anchorage in order to guarantee an adequate force transfer to the structure. At the moment, most available solutions on the market are partially mechanical anchorages based on anchor bolts and plates (see

Figure 1a). In terms of structural behavior in flexion, prestressed strips usually have a direct influence on the cracking, yielding and ultimate load carrying capacity. Experimental investigations on reinforced and prestressed reinforced concrete members with mechanical anchorages are amongst other things presented by Pellegrino and Modena [

4], El-Hacha

et al. [

5,

6], França and Costa [

7] and Fernandes

et al. [

8].

**Figure 1.**
(**a**) Mechanical anchorage and (**b**) gradient anchorage for prestressed carbon fiber reinforced polymer (CFRP) strips.

**Figure 1.**
(**a**) Mechanical anchorage and (**b**) gradient anchorage for prestressed carbon fiber reinforced polymer (CFRP) strips.

An alternative anchorage technique is the “gradient anchorage” (

Figure 1b) developed at Empa [

3,

9]. This method is based on the possibility for the accelerated curing of the epoxy resin [

10,

11,

12,

13]. Each curing sequence is followed by a partial prestress force release until no pressure in the hydraulic jack remains at both strip ends. At the end, the temporary anchor bolts for the installation of the different components are cut. A pure strip-epoxy-concrete anchorage is the result. With this technique, the application time compared to conventional systems is reduced. Additionally, the end appearance is more appealing and corrosion is avoided, due to the absence of remaining metallic devices.

An early prototype of the heating device necessary for creating high temperatures in the resin has been successfully applied at the laboratory level. The system is presented in Meier

et al. [

3], Stöcklin and Meier [

9], Czaderski and Motavalli [

14], Kotynia

et al. [

15] and Motavalli

et al. [

16]. In a recently completed research project with strong industry collaboration of S&P Clever Reinforcement (Seewen, Switzerland), a newer and more suitable system for practical application on-site has been developed and extensively presented in Michels

et al. [

17,

18].

The present manuscript describes, in the first step, a short amount of information about the prestressing procedure together with the subsequent results from the static loading tests. Whereas, in an earlier publication [

18], force-deflection curves already have been presented, this paper delivers more extensive information about the failure mode, as well as the strain levels of both the CFRP strip and the concrete in compression. Furthermore, the image correlation measurement in the anchorage zone delivers information about the epoxy layer thickness in this region. In the second step, a non-commercial 1D finite element code is used for modeling the experiments, as well as for performing a parametric study. This manuscript is an extended version of a conference paper presented at the 2nd conference on Smart Monitoring, Assessment and Retrofitting of Civil Structures (SMAR) in Istanbul, 2013 [

19].

## 3. Results and Discussion

The force-deflection curve, as well as force-strain measurements are presented in

Figure 6 and

Figure 7, respectively. It is précised that the deflection value is taken into account as an average deflection of both LVDT measurements.

Table 4 summarizes the most important values in terms of ultimate load

F_{u}, midspan deflection at failure

δ_{u}, CFRP tensile strain at debonding

ε_{f,u} and localization of the last crack

l_{cr} (distance from the last crack to the support). The indicated value, Δ

ε_{f}, is the tensile strain in addition to the prestrain introduced in the strip due to static loading. Note that the force level,

F_{tot}, is the sum of the four acting forces according to the presented test setup. It is mentioned that for Beam 1, force values were measured, but could not be registered, due to technical problems with the data acquisition system. Polymers 2014, 6 121 Nevertheless, it is presented that the ultimate load carrying capacity,

F_{u}, was around an 80 kN reading on screen during the test at a midspan deflection,

δ_{u}, of 127 mm. In terms of force-deflection behavior, some of the following results have been presented earlier in Michels

et al. [

18].

For all the tested beams, failure occurred by the debonding of the prestressed strips from the concrete surface. Due to the very fast debonding process, the location of the initiation is rather difficult to determine. For Beams 1–3, though, as presented in

Figure 8, the picture of the gradient after debonding indicates a separation from the strip from the epoxy layer. For Beam 4, failure seemed to have been initiated in the free length. The highest bearing loads with approximately 80 kN could be noticed for Beams 1 and 3. This is in direct correlation to the registered maximum strip strain,

ε_{f,u}, of 1.42% and 1.38%, respectively (

Figure 7a). With an earlier presented ultimate strain of 1.6%, tensile failure is almost reached. After having tested Beam 1 and having noticed a failure by delamination at the front part of the anchorage zone (first 300 mm) between the CFRP strip and the underlying epoxy layer (see

Figure 8), the idea was to roughen the strip end with sandpaper in order to increase the bond between the two components (see

Section 2.2). However, the results from Beam 2 show that this measure was counterproductive, resulting in an even lower ultimate failure load (69.9 kN) and strip tensile strain (1.16%,

Figure 7a). It is hypothesized that the grinding procedure provoked the creation of very thin carbon dust, which subsequently could not be removed with the conventional cleaning procedure. The resulting failure mechanism is the same, but at a clearly lower load level. For Beam 3, the exact same procedure as for Beam 1 was followed; the results are almost identical. Lastly, Beam 4 was strengthened “top-down”,

i.e., the concrete substrate surface on which the CFRP strip has been applied was initially the upper surface while casting. This most likely results in lower concrete quality of the strengthened surface. For this last element, failure by debonding at midspan in the concrete substrate could be noticed at a lower force, deflection and CFRP tensile strain value than for Beams 1 and 3. Inclination changes in the force-deflection (

Figure 6), as well as in the force-strain curves (

Figure 7) indicate a cracking load,

F_{cr}, of about 30 kN and a start of steel yielding,

F_{y}, at about 50–60 kN for all beams. All four tests have in common that the compressive strain of the concrete was clearly below the failure level, as presented in

Figure 7 and

Table 4.

**Figure 6.**
Total force-midspan deflection curve of all the performed static tests.

**Figure 6.**
Total force-midspan deflection curve of all the performed static tests.

**Figure 7.**
Force-strain curves for the tested beam elements, 2–4. (**a**) Beam 2 force-strain; (**b**) Beam 3 force-strain; and (**c**) Beam 4 force-strain.

**Figure 7.**
Force-strain curves for the tested beam elements, 2–4. (**a**) Beam 2 force-strain; (**b**) Beam 3 force-strain; and (**c**) Beam 4 force-strain.

**Table 4.**
Summary of the key results after static loading (* not registered).

**Table 4.**
Summary of the key results after static loading (* not registered).
Beam | F_{cr} | δ_{cr} | F_{y} | δ_{y} | F_{u} | δ_{u} | ε_{f,u} | Ε_{c,u} | Δε_{f} [%] | l_{cr} (m) |
---|

No. | (mm) | (mm) | (kN) | (mm) | (kN) | (mm) | (%) | (%) | (%) | (m) |
---|

1 | * | * | * | * | ≈80 | 127.4 | 1.42 | 1.27 | 0.83 | 1.10 |

2 | 20 | 2.4 | 54 | 41 | 69.9 | 91.2 | 1.16 | 0.90 | 0.57 | 1.15 |

3 | 18 | 2.2 | 55 | 46 | 80.9 | 125.3 | 1.38 | 1.34 | 0.78 | 1.11 |

4 | 28 | 2.9 | 57 | 57 | 70.5 | 98.8 | 1.28 | 0.93 | 0.67 | 1.27 |

**Figure 8.**
Debonding failure between the CFRP strip and the epoxy adhesive in the gradient anchorage zone.

**Figure 8.**
Debonding failure between the CFRP strip and the epoxy adhesive in the gradient anchorage zone.

Since no reference beam is tested, cracking and ultimate moment

M_{cr} and

M_{u}, as well as the corresponding loads,

F_{cr} and

F_{u}, are estimated as follows:

where

f_{ct} = the unidirectional tensile strength of the concrete (≈4 MPa),

b,

h = the width and height of the concrete cross-section,

F_{s} = the maximum force in the steel reinforcement in tension,

z = the lever arm,

R_{m} = the ultimatestrength of the steel reinforcement (≈ 620 MPa) and

d = the static height (=190 mm). With the indicated values,

M_{cr} ≈ 32 and

M_{u} ≈ 37 kNm. By subtracting the dead load, the cracking and ultimate load (the total of the four applied forces) are 9.5 and 15 kN. In

Figure 6 and

Table 4, cracking loads are in the region of 30 kN, whilst ultimate loads are between 69.9 and 80.9 kN. Compared to the estimated reference values, the advantages for both serviceability and the ultimate state are clearly visible.

An interesting aspect of the strengthened beams is the overall structural behavior as a comparison between the deformation (deflection) at which the inner steel reinforcement starts yielding and the deflection at the ultimate load. For all the beams, yielding occurs more or less at a deflection,

δ_{y}, of about 50 mm (the second stiffness drop after cracking). When referring to Beams 1 and 3 with the higher observed deformation of about 125 mm, a ductility index for deflection

μ_{δ} =

δ_{u} /

δ_{y} = 125/50 = 2.5 is obtained. This value can be regarded as being very satisfying. In terms of forces, the ductility index

μ_{F} =

F_{u} /

F_{y} = 80/55 = 1.45 is less high. For advanced considerations on ductility issues for concrete members with an externally bonded FRP reinforcement, the reader is invited to consult the research by Oudah and El-Hacha [

24].

Crack patterns of all the tested beams are shown in

Figure 9, and the location of the last crack,

l_{cr} (measured from the support), is also given in

Table 4. It can be seen that for all elements, the last crack,

l_{cr}, is located outside of the gradient anchorage zone. This observation is confirmed by the strain evolution presented in

Figure 7. For Beams 2 and 3, the increase of the strip strain,

ε_{f,}_{6}, located in the gradient region, isalmost none (for Beam 4, no measurement in the anchorage zone has been recorded).

**Figure 9.**
Crack patterns after the test ended of all four beams tested.

**Figure 9.**
Crack patterns after the test ended of all four beams tested.

## 4. Numerical Code

Numerical simulations of the experimental results outlined in the previous section are performed by means of a 1D finite element (FE) model based on an extension of the one formulated by Faella

et al. [

25] for non-prestressed externally bonded FRP-strengthened RC beams.

Figure 10 shows the kinematics of the aforementioned finite element, whose displacement and force vectors collect the six components described in the following equations, (3) and (4):

It is worth highlighting that the interface slips considered by the model are those measured after the possible release process, which is carried out as part of the aforementioned gradient method.

**Figure 10.**
Kinematics and the degrees of freedom of the proposed finite element.

**Figure 10.**
Kinematics and the degrees of freedom of the proposed finite element.

The force and displacement vectors are related by means of the following usual matrix expression:

where

**K** is the so-called stiffness matrix and

**Q**_{0} is the vector of equivalent nodal forces. Since the model under consideration derives from the well-known Newmarks theory for two-layer beams in partial interaction, an “exact” expression is available for both

**K** and

**Q**_{0} [

26], which takes into account both the shear-flexibility of the connected layers and the partial interaction resulting from their flexible interface. As a matter of principle, such a solution completely defines the matrix,

**K**, and the contribution,

**Q**_{0,q}, of transverse distributed loads to the vector,

**Q**_{0}. Then, the complete expression of the vector

**Q**_{0} for the problem under consideration should also cover the effect of the initial strain,

, which represents the axial strain imposed on the FRP strip for applying the prestressing action to the RC beam. Although mathematical details about the whole derivation of the contribution,

**Q**_{0,p}, corresponding to the pre-stressing action cannot be reported herein due to space constraints, it is easy to recognize that the final analytical expression of

**Q**_{0} can be written as follows:

where

E · Af is the axial stiffness of the FRP strip and d the distance between the transverse section centroid of the RC beam and FRP strip. As a matter of fact, the prestressing action, as well as the applied distributed loads, is represented by a vector of equivalent nodal forces: the contribution of the former tends to compensate for the latter, as also expected from the conceptual design of the prestressing action.

The 1D FE formulated through Equations (4) and (6) can be conveniently utilized to perform linear analyses of beams strengthened in bending by externally-bonded prestressed FRP strips. However, due to the nonlinear behavior that characterizesthe experimentally observed response, the following (generalized) stress-strain relationships are considered in the present simulations:

The well-established rational law by Sargin is adopted for concrete in compression. All its relevant properties can be defined in terms of the compressive strength,

f_{c}, and an ultimate axial strain

ε_{cu} = 0.0035 is assumed as a conventional value for the concrete crushing condition ([

20]);

An elastic response of concrete in tension is assumed, up to the unidirectional tensile strength,

f_{ct}, which can be determined as a function of

f_{c} ([

20]). Then, a power law is assumed for simulating a softening branch, which corresponds to the well-known tension-stiffening effect ([

27]);

A trilinear elastic-hardening-plastic law is adopted for reinforcing steel bars. It allows the simulation of the linear behavior up to yielding stress f_{y}, a hardening branch up to the ultimate tensile strength, f_{s,u}, which is kept constant up to a conventional value of the ultimate strain of ε_{s,u} = 0.06;

A linear elastic behavior is assumed for the FRP strip in tension, depending on the elastic modulus, E_{f}, and the ultimate stress, f_{f,u}.

A bilinear elastic-softening bond-slip law is finally employed for describing the interaction between the FRP laminate and the concrete substrate ([

10],

= 8 MPa,

s_{f,el} = 0.02 mm,

s_{f,u} = 0.2 mm). The key parameters are mainly related to the fracture energy in mode II,

G_{f,II}, and can be identified via pull-out tests on FRP strips glued onto concrete blocks.

The values of the relevant parameters that define the above stress-strain laws are taken from the experiments reported in

Section 2.1.

A secant updating procedure is, then, implemented to handle the mechanical nonlinear behavior introduced by the aforementioned stress-strain relationships (as a matter of fact, this procedure works in Faella

et al. [

25]). Besides the usual failure modes of RC members, generally related to the compression failure of concrete and the tensile failure of the steel reinforcement, both the tensile failure of the CFRP strip and debonding failure at the FRP-to-concrete interface are considered in the analyses through the limit values of strains and slips that identify the adopted generalized stress-strain law. Particularly, the calibration of the bond stress-slip relationship in the gradient zone was based upon the results reported in Czaderski

et al. [

10]. The behavior observed on the RC beams reported in

Section 2 was simulated by means of the aforementioned model. Particularly, the numerical analyses focused on the beam referred to as Beam 3, which represents the regular application of the gradient method for prestressed FRP plates. To this end, the beam is discretized in a mesh of 25 finite elements: moving from the theoretical support, the first FE is a 200 mm-long regular RC beam element; the second is a non-prestressed on a 100-mm element, like the one formulated by Faella

et al. [

25]; then, three elements correspond to the three steps of the gradient method. These first five elements cover the first meter of the beam (from the theoretical support); then, another 20 (100 mm each) FE, like the ones described above, cover the distance up to the midspan; only one-half of the beam is actually schematized, as a result of the geometrical and loading system under investigation.

Figure 11a presents the comparison between the experimental results in terms of applied force

F_{tot} and midspan displacement

δ measured after having applied the prestressed CFRP strip. A very good agreement can be observed between the results and the numerical simulation, which stops when the maximum axial strain

ε_{f,u} =

ε_{f,p} + Δ

ε_{f,calc} = 0.006 + 0.010 = 0.016 is reached in the CFRP strip. This indicates that a tensile failure of the CFRP strip was found to be the dominant failure mode. Compared to the experimental results, the numerical model slightly overestimates the final bearing capacity of the beam. This is probably due to the fact that the model cannot simulate local effects, such as discrete crack formation and local irregularities of the concrete surface and epoxy layer, which are often responsible for the debonding phenomenon. Moreover, the same figure depicts the simulation of a similar beam, possibly tested without prestressing action (not tested); the significant difference in terms of cracking and yielding load (with respect to the prestressed beam) points out the importance of prestressing in recovering a significant part of the stresses induced by the dead loads (just the self-weights in this experimental case). For the initially unstressed strip configuration, the cracking load is more or less identical to the unstrengthened configuration (see

Section 3). This represents an additional argument towards prestressing; as for serviceability purposes, unstressed externally bonded strips do not induce a significant improvement of the structural behavior. In the end, tensile failure in the composite reinforcement is, as for the prestressed case, found to be the dominant case from a numerical point of view. Eventually, the differences in structural stiffness between a loaded element with a prestressed or an initially unstressed strip are evident.

Figure 11b shows the same comparison between the prestressed and unstressed case compared to the experimental results in terms of the CFRP tensile strain at the midspan. Again, a good concordance between the measured and calculated values can be observed. The same comment as stated earlier,

i.e., an overestimation of the total load carrying capacity is visible with higher strip strains.

**Figure 11.**
Experimental and numerical (**a**) force-deflection, total force F_{tot}-midspan deflection δ and (**b**) force-strain at the midspan curves, total force F_{tot}-CFRP strip strain ε_{f} at the midspan.

**Figure 11.**
Experimental and numerical (**a**) force-deflection, total force F_{tot}-midspan deflection δ and (**b**) force-strain at the midspan curves, total force F_{tot}-CFRP strip strain ε_{f} at the midspan.

The numerical values of the increasing axial strain,

ε_{f}, along the strip axis with a growing load are shown in

Figure 12. It can be seen that no cracks and, therefore, no significant strain increase can be noticed in the gradient anchorage area. This was confirmed by the location of the last cracks shown in

Table 4 by visual inspection after the test ended and the strain measurement as discussed above.

**Figure 12.**
CFRP strain ε_{f}-distribution over the beam length as a function of the applied force, F_{tot}.

**Figure 12.**
CFRP strain ε_{f}-distribution over the beam length as a function of the applied force, F_{tot}.

Finally, the model outlined in this section can be regarded as a design-oriented tool for quantifying the effect of external prestressing and the effectiveness of the gradient anchorage in preventing debonding right after the application of the prestressed FRP plate. Conversely, determining the actual failure load and, particularly, detecting the occurrence of debonding possibly induced by the applied loads should be considered as open issues: their complete understanding deserves further studies under both the experimental and theoretical standpoints. As presented in the previous section about the experimental results, failure occurred due to a strip debonding between the CFRP strip and the adhesive layer in the gradient anchorage zone. Additionally, experimental and theoretical works in the field of gradient anchorage design presented in Czaderski [

11] have to be pointed out. The first implementations in a cross-section analysis tool in the framework of a master thesis at ETHZürich (Harmanci [

28]) were considered very successful and deserve further attention with more sophisticated calculation tools.