Finite Element Modeling of Reinforced Concrete Beam Patch Repaired and Strengthened with Fiber-Reinforced Polymers

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1 International Journal o Engineering and Technical Research (IJETR) ISSN: (O) (P), Volume-4, Issue-3, March 2016 Finite Element Modeling o Reinorced Concrete Beam Patch Repaired and Strengthened with Fiber-Reinorced Polymers Mundeli Salathiel, Mbereyaho Leopold, Pilate MOYO Abstract Repair and strengthening using Fiber Reinorced Polymers (FRP) have been one o worldwide methods used to improve the load carrying capacity and durability o reinorced concrete structural elements such as beams, in decline mainly due to corrosion o steel reinorcement resulting rom the deterioration o the concrete cover. The main objective o this study was to investigate the behavior o reinorced concrete beam patch repaired and strengthened with FRP composites. The 3D inite element analysis was conducted using Abaqus sotware, while the repair consisted o varying the length o the patch, where our patch lengths were considered. Isotropic concrete damaged plasticity model was applied to describe the behavior o both concrete and repair material. Linear elastic-perectly plastic and linear elastic isotropic model were used or steel and FRP; respectively, while cohesive bond model was used or the interace between concrete and FRP. Numerical results show good agreement with experimental indings in terms o load-delection curves, crack pattern and modes o ailure. The study also revealed that or proper structural crack distribution, it is better to use orce control while or ailure mechanism, displacement control is the best choice. Finally, once again it was proved that both patch repair and FRP strengthening increased the load carrying capacity and aected crack distribution. beam. FRP composites have been extensively used due to their advantages over steel plates such as corrosion resistance, lightweight and high strength. FRP plates are bonded to concrete substrate by means o adhesive. Such patch repaired and FRP-strengthened beams may ail in a similar way to that o unstrengthened reinorced concrete beams but quite oten they ail by debonding in two modes [25]: (1) Intermediate crack induced debonding which initiates around critical lexural cracks and propagates towards the plate end, and (2) plate end debonding initiating at the plate end due to high stress concentration and critical shear cracks (Figure 1). Thus, debonding is speeded up by crack propagation particularly in the horizontal direction [25]. Cracks may also orm in dierent locations in the interace between concrete and FRP and may cause the ailure o the interace inside the adhesive, at the interace between adhesive and concrete or at the interace between adhesive and FRP, inside the FRP or within the concrete cover [6]. Index Terms cracking, constitutive model, damage, debonding, Fiber Reinorced Polymers (FRP), inite elements, prediction, repair, strengthening. I. INTRODUCTION Reinorced concrete structural elements such as beams get deteriorated and become structurally and unctionally inadequate, due to corrosion o steel reinorcement resulting rom concrete cracking and spalling, poor design and maintenance and natural disasters such as earthquake. To upgrade the load carrying capacity and durability o such concrete elements, repair and structural strengthening have been used all over the world since last decades. While repair consists o removing damaged concrete, preparation o concrete substrate and application o a new layer o repair material, strengthening consists o bonding a steel plate or a iber reinorced polymer (FRP) plate to the tension ace o the Mr Mundeli Salathiel, Department o Civil, Environmental and Geomatic Engineering (CEGE), School o Engineering, College o Science and Technology (CST), University o Rwanda (UR), Kigali City, Rwanda. Mobile Phone No: Dr Mbereyaho Leopold, Department o Civil, Environmental and Geomatic Engineering (CEGE), School o Engineering, College o Science and Technology (CST), University o Rwanda (UR), Kigali City, Rwanda., Mobile Phone No: Pro. Pilate MOYO, University o Cape Town, Department o Civil Engineering, Rondebosch, Western Cape, South Arica. Mobile Phone No: a) Intermediate crack induced b) Plate end debonding Figure1. Debonding ailure modes Among the ailure modes o reinorced concrete strengthened beams, debonding is quite diicult to predict. In act, crushing strain o concrete is known, yielding o tension steel can be measured in the laboratory and hence predicted and FRP rupture can be predicted rom rupture strain provided by the manuacturer; but debonding control is still not well understood. Dierent approaches have been used to predict the behavior o such beams and include analytical [14, 22, 24, 26], experimental [14, 15, 18, 20], numerical [4, 8, 23] and racture mechanics [3, 10, 19]. However, such approaches ace some limitations due to complex behavior o concrete and that o the interace between concrete and FRP. Such complexity or concrete is due to the nonlinear load-deormation response o concrete and diiculty in orming suitable constitutive relationships under combined stresses, progressive cracking o concrete under increasing load and the complexity in the ormulation o the ailure 47

2 Finite Element Modeling o Reinorced Concrete Beam Patch Repaired and Strengthened with Fiber-Reinorced Polymers behavior or various stress states, the consideration o steel and its interaction with concrete and time dependent eects such as creep and shrinkage o concrete [5]. Even though such approaches have been used or studying the behavior and ailure mechanisms o patch repaired and FRP-strengthened RC beams, no numerical study using inite elements method has been done so ar taking into consideration the eect o patch repair. The inite element method application requires a good understanding o the mechanical behavior o various materials involved, thus the requirements o constitutive relationships or the description o such behavior. This paper presents a inite element analysis o patch repaired and FRP-strengthened RC beams with varying patch length and loaded in our points bending. Their behavior in terms o load-displacement response, crack pattern, ailure mechanisms and strain distribution in FRP is studied and validation against experimental results obtained rom the same beams is done. II. FINITE ELEMENT MODELING The inite element analysis perormed in this study consisted o modeling the nonlinear behavior o reinorced concrete beams patch repaired and strengthened with FRP bonded to their tension ace to investigate the behavior o such RC beams under our points bending. The commercial inite element package ABAQUS sotware was used. The beam modeled in 3D is shown in Figure 2. The beam was reinorced with 2Y20 in tension and 2Y8 in compression. Shear reinorcement consisted o 8mm diameter stirrups at 80mm c/c distributed equally over the whole length o the beam as shown in Figure 2. The FRP reinorcement was placed at a distance o 50mm rom the support. While the height o the patch repair was maintained at 105mm, the length was varied as 450, 800, 1300 and 1800mm.. Figure 2. Geometry, reinorcement and FRP setup o the beam (800mm patch length) 2.1. Material properties and constitutive models Concrete There are three models available in Abaqus or the analysis o concrete at low conining pressure: Concrete smeared cracking model in Abaqus/Standard, Brittle cracking model in Abaqus/Explicit and Concrete damaged plasticity model in Abaqus/Standard and Abaqus Explicit. Concrete undergoes both inelastic deormation and stiness degradation when loaded. However, those two are not captured at the same time by concrete smeared crack model and brittle cracking model. In concrete smeared crack model, cracking is the most important aspect o the behavior and a compression yield surace is used to control plastic straining in compression whereas in brittle cracking model compressive ailure is not important [1]. Concrete damaged plasticity; available in ABAQUS; due to its capability o capturing both inelastic deormation and stiness degradation, was used in this study. The compressive strength was experimentally determined at the age o 28 days as 50MPa and 70MPa or concrete and repair material, respectively. The elastic parameters necessary or the establishment o the irst part o the model were the secant modulus o elasticity cm and mean axial tensile strength, ctm and were calculated according to [9] as: E 0.3 cm /10 MPa 2 / 3 3. Mpa Ecm ctm 0.30 ck 5 For concrete 0.3 Ecm 22[( cm )/10] 38400MPa /10 4. Mpa ctm 2.12ln 1 cm 3 (4) For repair material, the post-peak behavior in tension was represented with tension stiening to simulate the eects o concrete/steel eects such as bond slip and dowel action. Tension stiening was speciied in terms o racture energy criterion. A racture energy o 0.08N / mm and 0.15N / mm were used or concrete and repair material, respectively. The tensile stress in concrete was reduced to zero at a tensile strain o or both concrete and repair material [1]. The nonlinear uniaxial compression stress-strain curve was constructed based on the expression proposed by [9]: c cm 2 k 1 ( k 2) c Where c1 (4) k 1.05 (1) (2) c1 cm (3) and 0.31 c cm 2.8, cm is the mean value o concrete cylinder compressive strength derived rom concrete cube strength and c is the compressive stress in concrete. The curve is shown in Figure 3 or both concrete and repair material. (a) 48

3 International Journal o Engineering and Technical Research (IJETR) ISSN: (O) (P), Volume-4, Issue-3, March 2016 Tension reinorcements were partitioned and their cross sections were reduced by 10% over a length equal to that o the repair to simulate the corrosion eects since corrosion results in steel cross section reduction and hence mass loss. (b) Figure3. Uniaxial compressive behavior: (a) Concrete, (b) Repair material The cracking strain and inelastic strain or both concrete and repair material were determined by taking the total strain minus the elastic strain corresponding to the undamaged material. On the other hand, concrete tension and compression damage were deined through tensile and compressive damage parameters deined as : Dc, t 1 c, t c, t (5) Where D is the damage parameter equal to zero or undamaged material and one or a completely damaged material, is the stress on the descending branch on the uniaxial stress strain curve, σ is the ultimate stress and the subscripts c and t stand or compression and tension, respectively. For plasticity parameters, except the dilatation angle that was taken as 370, other parameters were assigned deault values as suggested by [1]. The criterion or concrete cracking under tension was deined in terms o maximum principal stress, which was taken equal to tensile strength o concrete. Since crack propagation requires energy consumption, the energy required to open a unit area crack, G as deined by [11] was used. The same philosophy was also applied to repair material Steel reinorcement Since the properties o steel do not depend on environmental conditions or time, steel exhibits the same stress-strain curve in compression and in tension [12]. Tension, compression and shear reinorcements were assumed to behave in an elastic perectly plastic manner in both compression and tension as shown in igure 4. The modulus o elasticity o steel was taken as 200 GPa while the yield stress was taken as 460MPa or longitudinal reinorcement and 250MPa or shear reinorcement. Poisson s ratio was taken as Fiber Reinorced Polymers Material Fiber Reinorced Polymers are made o high strength ibers o glass, aramid or carbon embedded in a polymer resin called matrix. Fibers may be arranged so that they are oriented in directions o high stresses (anisotropic) or are oriented only in one direction (isotropic). FRP material behaves in a linear elastic manner up to ailure. For lexural strengthening, the elastic modulus in iber direction is o most importance. Thereore, in this study the isotropic elastic type was used or characterization o mechanical properties o Carbon Fiber Reinorced Polymer (CFRP) since they were unidirectional with high stiness. The elastic modulus in iber direction was speciied by the manuacturer as 165GPa and the Poisson s ratio was taken as 0.3. The minimum value o strain at break in iber direction was given by the manuacturer as 1.70%. The width o the CFRP used was 50mm with a thickness o 1.2mm. This thickness was reserved in order to acilitate the installation o warp abrics at the plate end to prevent early end plate debonding. Only one layer o CFRP used was bonded to concrete by means o adhesive epoxy. The adhesive layer constituted the interace between concrete and CFRP material Adhesive-Concrete/CFRP interace. The interace between concrete and FRP is critical in FRP-strengthened RC beams. It ensures the transer o both shear and normal stresses rom concrete to FRP as it resists their separation. The layer o adhesive between concrete and FRP o 1mm thickness was modeled using cohesive zone model. The model is characterized by a progressive stiness degradation driven by a damage process. Cohesive zone model used in this study is graphically represented by a simple bilinear traction separation law, i.e. linear elastic traction shown in igure 5. Figure 5. Bilinear traction separation law Figure4. Behavior o steel in tension and compression It was accepted that the interace was subjected to normal stress and shear stress in two shear directions. The initiation o damage was assumed to take place when the maximum nominal stress ratios reached a value o one [1]. 49

4 Finite Element Modeling o Reinorced Concrete Beam Patch Repaired and Strengthened with Fiber-Reinorced Polymers tn t s tt max,, 0 0 tn ts tt 0 t n 0 0 t s 0 t t 1 where, and represent the peak values o the nominal stress when the deormation is either purely in normal or purely in the irst or second shear directions. t 0 The values used in this study were n 4MPa and 0 0 ts tt 15MPa according to the manuacturer. Since the thickness o the adhesive layer was one mm, the ratio o elastic modulus and normal stiness, shear modulus and shear stiness were taken equal to one. Damage evolution was deined in terms o displacement at ailure. (6) III. METHODOLOGY FOR THE STUDY In this study, dierent parts making up the complete model were created in Abaqus/Standard in 3D modeling space. To ensure a perect bond between concrete and repair material; concrete and adhesive; concrete beam was portioned to the size o the adhesive and dierent length o repair material. The thickness o the adhesive was 1mm. Two-node linear 3D (T3D2) truss elements were used or longitudinal and shear reinorcements. The interace between concrete and CFRP was modeled using 8-node three-dimensional cohesive elements (COH3D8). Eight-node linear brick elements were used or concrete, repair material and CFRP. Dierent element shapes adopted in this study are shown in Figure 6. Three meshing techniques were used: structured meshing technique or concrete repair material and CFRP; swept meshing technique or the adhesive and ree meshing technique or steel. Figure 6 shows a typical mesh (made coarser or clarity) or reinorced concrete beam with 800mm patch strengthened with CFRP. increment [11]. Thus, the issue was to choose the initial increment size. In this study, the initial increment size was 10% o the total step time. IV. RESULTS AND DISCUSSIONS As stated earlier, the main objective o this study was to investigate the behavior o reinorced concrete beams patch repaired and strengthened with FRP composites. The parameter that was varied was the length o the patch by keeping constant its depth. Such behavior was studied in terms o crack initiation and propagation, load-delection relationships, damage energy release rate, ailure mechanisms and strain distribution in the FRP or strengthened beams only. For validation purpose, a comparison with experimental results obtained rom the same beams is done. This section presents the results obtained rom inite elements analysis or the above parameters Cracking initiation and evolution Cracking was initiated whenever the maximum principal stress was greater than the tensile strength o concrete which was 3.5 MPa or concrete and 4.3MPa or repair material. This may be seen rom Figure 7, which shows the distribution o tensile stresses or control beam where the maximum averaged tensile stress was 3.7MPa or concrete but not averaged value was 3.6MPa which was always greater than tensile strength. This value was ound ater the irst increment. For patch-repaired RC beams, or example 1800mm-Patch, an averaged tensile stress o 4.4MPa was ound at the third increment. At the same time, not averaged tensile stress o 4.4 was probed in the bottom center. This value is greater than the tensile strength o the repair material and this involves cracking since tensile strength was exceeded. Figure 6. Finite element model or patch-repaired and CFRP strengthened RC beam (800mm-Patch) [17] The perect bond model between concrete and steel was adopted. It was implemented by embedding steel reinorcement in concrete, i.e. incorporation o one dimensional element into a three dimensional elements allowing reinorcing bars to pass through concrete elements in an arbitrary ashion [21] thus, introducing new nodes in the concrete elements. Full Newton solution technique was used in this analysis. In this method, the solution to nonlinear problems is obtained by deining the load as a unction o time and breaking the simulation into a number o time increments and inds the approximate equilibrium at the end o each time Figure 7. Cracking initiation or control beam [17] Even i cracking or control beam took place ater the irst increment, the cracking load was ound to be higher than that o patch repaired and strengthened beams though or such beams it took place ater the third increment or 450mm, 800mm and 1800mm patched beams and ater the ourth increment or 1300mm patch repaired beam. For patch-repaired and strengthened beams cracking loads were lower as compared to control beam. Figure 8 compares cracking loads rom both experiments and inite elements analysis while Figure 9 compares structural crack pattern or control beam and 1800mm patched beam. For graphical visualization o cracks using concrete damaged plasticity 50

5 model or concrete and repair material, it was assumed that cracking initiated at points where the tensile equivalent plastic strain was greater than zero and the maximum principal plastic strain was positive [13]. It is important to note that in this study no precracking was considered. Force control was used or structural crack distribution. International Journal o Engineering and Technical Research (IJETR) ISSN: (O) (P), Volume-4, Issue-3, March 2016 not anchored as in experiment. In addition, as reported by [7], concrete damaged plasticity overestimates the stresses in concrete and this could also be a reason o discrepancies. It is observed that or control RC beam ater yielding o steel no urther increment in load carrying capacity was experienced until ailure. However, or patch repaired and FRP strengthened beams, this was not the case. This shows that repair and strengthening increase the load carrying capacity o reinorced concrete beams in addition to increasing their service lie. Figure 8: Cracking loads [17] (a) Control beam (b) 1800mm patched beam Figure 9: Structural crack pattern 4.2. Load delection relationships. Generally, the behavior o reinorced concrete beams subjected to bending is analyzed through load-delection curves. Mid-span load-delection relationships represent ailure stages as the applied load increases. Those stages include cracking initiation, yielding o steel reinorcement, debonding loads or strengthened beams and ultimate load. Load delection curves obtained rom control beam and our patch repaired and strengthened beams are shown in Figure 10 in comparison to experimental results. As it can be seen, there is a close agreement between numerical results and experimental indings. This clearly, shows the ability o the proposed model or analyzing the overall behavior o patch-repaired beams and strengthened with FRP composites. There are however, some dierences between numerical and experimental results, which were due to the assumptions made in the inite elements ormulation and the act that beams were 51

6 Finite Element Modeling o Reinorced Concrete Beam Patch Repaired and Strengthened with Fiber-Reinorced Polymers The igure 11 compares the load delection curves or all the beams numerically analyzed in this study. It can be seen that cracking or all beams initiated at nearly the same load as discussed in section 4.1 but at dierent load increments Figure 11: Load delection curves or all beams analyzed [26] This means that the eect o patch repair was to delay crack initiation. For beams repaired over a length o 450mm, 800mm and1300mm the behavior was more or less the same except the dierences in yielding and debonding which occurred at dierent loads and displacement. Beam with 1800mm length patch repair showed a slightly dierent load displacement relationship. In comparison with the behavior o other beams, the behavior is relatively brittle since the debonding load was high but at small delection as compared to other beams. The reason may probably be attributed to the reduction in tension steel cross section and large size o repair with high tensile strength Damage energy release rate. Damage energy release rate is another parameter that can be used to study the overall behavior o reinorced concrete beams under bending. In act, the our stages o ailure, i.e. cracking initiation, yielding o steel, debonding load or strengthened beams and crushing load may be readily identiied on the load-energy relationship, as it is seen in Figure 12. This is the energy dissipated when concrete is undergoing damage ailure. From the igure, it can be observed that ater yielding, the energy release rate is highly nonlinear indicating extensive deormation (cracking) o dierent reinorced concrete beams. The output in terms o load versus energy rom Abaqus clearly shows that concrete starts to release energy ater a number o load increments, and that beore cracking the energy release rate is zero and then starts to increase linearly up to yielding. Figure 12. Damage energy release rates 4.4. Strain distribution in FRP Debonding o FRP plate rom concrete is a special ailure mode associated with strengthened RC beams. It is caused by high interacial stress concentrations around lexural or lexural-shear cracks or at the plate end. A common measure used to control debonding is to limit the strain in the FRP material to a usable or debonding strain which, when exceeded at some locations, FRP will separates rom the beam. Strain in FRP is mostly a governing actor in design o strengthened RC beams. The igure 13 shows the strain distribution or analyzed beams rom where it is observed that as the length o the patch is becoming larger, strain distribution was becomes uniorm. This shows that the discontinuity in the material to which FRP is bonded aects the strain distribution in FRP which in turns aect the debonding ailure. It is clear that there is a peak strain in FRP at which debonding is initiated. [2] recommends that in order to prevent debonding o the FRP laminate, a limitation should be placed on the strain level developed in the laminate. ACI proposed the ollowing expression or debonding or usable strain: d where m u u (7) is the design rupture strain given by the manuacturer as 1.7% and coeicient given by m (8) where t m is the bond dependent E is the tensile modulus o elasticity o FRP (MPa), is the nominal thickness o one ply o FRP reinorcement u in mm, is the design rupture strain o FRP in mm/ mm and n is the number o plies used. 52

7 International Journal o Engineering and Technical Research (IJETR) ISSN: (O) (P), Volume-4, Issue-3, March 2016 Figure 13: Strain distribution in FRP The table 1 compares the peak strain rom our inite elements analysis with ACI prediction. It is clear that there is a good agreement between the two, hence the integrity o our model. From our inite elements analysis, the peak strain in FRP was in the constant bending moment or just under loading points. That is where debonding initiated. Figure 13. Intermediate crack induced debonding (a) Failure in adhesive showing debonding The Table about Comparison o debonding strain: FEM vs ACI 440.2R-02is presented bellow. Table 1: Comparison o debonding strain: FEM vs ACI 440.2R-02. Beam type FEM* ACI 440.2R-02** % dierence 450mm-Patched beam mm-Patched beam 1300mm-Patched beam 1800mm-Patched beam *Finite Elements Method, the numerical analysis method used here **American Concrete Institute recommendations, This is also in accordance with indings o [23] which stipulates that ACI accurately predicts strain in FRP or relatively long laminates and or steel reinorcement ratios that exceeds 0.35% Failure mechanism As discussed in section 4.4 maximum debonding strain in FRP was in the constant bending moment region or just under loading points and debonding initiated at those locations. Thus, the ailure mode was intermediate crack induced debonding ollowed by concrete crushing. This is shown by Figure 13. Figure 13. Intermediate crack induced debonding (b) V. CONCLUSION In this study, a inite element model was developed or analysis o reinorced concrete beams patch repaired and strengthened with FRP composites by varying the length o the patch. Concrete damaged plasticity model was used or the behavior o both concrete and repair material; linear elastic-perectly plastic model or steel; cohesive zone model or the interace between concrete and FRP and elastic isotropic model or CFRP. Results rom inite elements analysis were in good agreement with experimental indings in terms o load delection relationships, ailure mechanism and structural crack distribution. Cracking loads were accurately ound rom damage energy release rate. Also strain distribution in FRP rom the present analysis was in good agreement with ACI recommendation about the limiting strain or debonding strain in FRP. The study conirmed the possibility and importance o structural strengthening o deicient structures, as it increases the load carrying capacity as long as the overall level o perormance o the concerned structure is still acceptable. The results rom this study clearly show the ability o the proposed model or analyzing the overall behavior o patch-repaired beams and strengthened with FRP composites. Despite these good results, uture researches should continue putting much eort in the energy approaches to study the behavior o patch repaired and strengthened reinorced concrete beams particularly the debonding ailure. Acknowledgment 53

8 Finite Element Modeling o Reinorced Concrete Beam Patch Repaired and Strengthened with Fiber-Reinorced Polymers Authors would like to acknowledge the support rom University o Cape Town / Department o Civil Engineering, and University o Rwanda/College o Science and Technology/Department o Civil, Environmental and Geomatics Engineering, where this study started and completed respectively. We are also grateul to Mr. Thabiso Dladla who provided experimental results or validation o models and Mr.Kabani Matongo or his contribution during results analysis. Finally, our gratitude goes to all people who in one way or another, contributed during the research process. REFERENCES [1] Abaqus/CAE User's Manual, [2] American Concrete Institute. Guide or the design and construction o externally bonded FRP systems or strengthening concrete structures. ACI R, Detroit. [3] Au, C. & Buyukozturk, O Debonding o FRP plated concrete: A tri-layer racture treatment. Engineering Fracture Mechanics. 73(3): [4] Baky, H.A., Ebead, U.A. & Neale, K.W Flexural and interacial behavior o FRP-strengthened reinorced concrete beams. Journal o Composites or Construction. 11(6): [5] Buyukozturk, O Nonlinear analysis o reinorced concrete structures. Journal o Composites and Structures. 7: [ 6] Camata, G., Spacone,E. & Zarnic, R Experimental and nonlinear inite element studies o RC beams strengthened with FRP plates. Journal o Composites: Part B. 38: [7] Chaudhari, S.V & Chakrabarti, M.A Modeling o concrete or nonlinear analysis using inite element code ABAQUS. International Journal o Computer Applications. 44 (7): [8] Chen, G.M., Teng, J.G., Chen, J.F & Rosenboom, O.A Finite element model or intermediate crack debonding in RC beams strengthened with externally bonded FRP reinorcement. Proceedings o the ourth International Conerence on FRP Composites in Civil Engineering July 2008, Zurich, Switzerland. [9] Eurocode Design o concrete structures. Part 1-1: General rules and rules or buildings. [10] Hearing, B.P Delamination in Reinorced Concrete Retroitted with Fiber Reinorced Plastics. Ph.D disseltation, Department o Civil and Environmental Engineering, Massachusetts Institute o Technology, Cambridge, Massachusetts. [11] Hillerborg, A., Modéer, M. & Petersson, P-E Analysis o crack ormation and crack growth in concrete by means o racture mechanics and inite elements. Journal o Cement and Concrete Research. 6: [12] Kwak, H-G & Filippou, F.C Finite element analysis o reinorced concrete structures under monotonic loads. (Report N0 UCB/SEMM-90/14). University o Caliornia: Structural Engineering, Mechanics and Materials. [13] Lubliner, J., Oliver, J., Oller, S. & Onate, E A plastic damage model or concrete. International Journal o Solids and Structures. 25(3): [14] Malek, A.M., Saadatmanesh, H. & Mohammad, R.E Prediction o Failure Load o R/C Beams Strengthened with FRP Plate Due to Stress Concentration at the Plate End. ACI Structural Journal. 95(1): [15] Malumbela, G Measurable parameters or perormance o corroded and repaired RC beams under load. PhD Thesis. University o Cape Town. [16] Mattys, S Structural behavior and design o concrete members strengthened with externally bonded FRP reinorcement. PhD Thesis. Ghent University. [17] MUNDELI, S Behavior o RC Beams Patch Repaired and Strengthened with FRP Composites. Msc Thesis. University o Cape Town. [18] Obaidat, Y.T., Heyden, S. & Dahlblom, O The eect o CFRP and CFRP/concrete interace models when modeling retroitted RC beams with FEM. Journal o Composites Structures. 92: [19] Rabinovitch, O. & Frostig, Y Delamination Failure o RC Beams Strengthened with FRP Strips-A Closed -Form -High Order and Fracture Mechanics Approach. Journal o Engineering Mechanics. 127(8): [20] Rio, O., Andrade, C., Izquierdo, D. & Alonso, C Behavior o Patch-repaired concrete structural elements under increasing static loads to lexural ailure. Journal o Materials in Civil Engineering. 17(2): [21] Simonelli, G Finite element analysis o RC beams retroitted with ibre reinorced polymers. PhD Thesis. Università degli Studi di Napoli Federico II, London. [22] Smith, S.T. & Teng, J.G Interacial stresses in plated beams. [Engineering Structures]. 23(7): [23] Supaviriyakit, T. Pornpongsaro, P. & Pimanmas, A Finite element analysis o FRP-strengthened RC beams. Songklanakarin J. Sci. Technol. 26 (4): [24] Täljsten, B Strengthening o concrete prisms using the plate-bonding technique. International Journal o Fracture. 82: [25] Teng, J.G. & Chen, J.F Mechanics o debonding in FRP-plated RC beams. Proceedings o the Institution o Civil Engineers, Structures and Buildings. 162(SB5): [26] Tounsi, A. and others Interacial stresses in FRP-plated RC beams: eect o adhered shear deormations. International Journal o Adhesion and Adhesives. 29(4):