Finite element study of bond-slip behaviour of CFRP and GFRP laminates on brick masonry

Chi Chiu Lam Finite Element Study of Bond- Slip Behaviour of CFRP and GFRP Laminates on Brick Masonry Finite Element Study of Bond-Slip Behaviour of CFRP and GFRP Laminates on Brick Masonry Chi Chiu Lam 2009 Italy Chi Chiu Lam Finite Element Study of Bond- Slip Behaviour of CFRP and GFRP Laminates on Brick Masonry Italy 2009 Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS i DECLARATION Name: Chi Chiu LAM Email: [email protected] Title of the Msc Dissertation: Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Supervisor(s): Prof. M. R. Valluzzi and Dr. E. Garbin Year: 2009 I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. I hereby declare that the MSc Consortium responsible for the Advanced Masters in Structural Analysis of Monuments and Historical Constructions is allowed to store and make available electronically the present MSc Dissertation. University: University of Padova Date: 21st, July, 2009 Signature: ___________________________ Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ii ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS This page is left blank on purpose. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS iii ACKNOWLEDGEMENTS The author would like to thank Prof. M. R. Valluzzi for her persistent technical guidance and encouragement throughout the period of the thesis work. The author would also like to thank Dr. E. Garbin and M. Panizza for the valuable information and suggestion to the thesis work. The author would like to express his gratitude to the European Commission for the financial support through out the master’s program. The support from the University of Macau is also much appreciated. The author would like to thank Prof. P. Roca and all the professors from Barcelona, Padova, Guimarães and Prague who have given lectures in Barcelona, Spain. The author would like to thank all of his master’s colleagues for their consistent mental support. The author would like to express his gratitude to his parents and his wife for their continuous support and encouragement. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme iv ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS This page is left blank on purpose. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS v ABSTRACT Rehabilitation and preservation of historic monuments and ancient structures is attracting more and more interest in the world. There are a certain amount of historic monuments which were built by using brick masonry. As the development of fiber reinforced polymer (FRP) is getting more popular now, applying bonded FRP to strengthen historic brick masonry monuments becomes one of the alternative strengthening method. In the past decade, many researches have been carried out to investigate the behaviour of bonded FRP to reinforced concrete structures. However, still a few contributions are available concerning debonding problem on masonry. In this report, the experimental study which was carried out by Panizza et al. (2009) about the bond-slip behaviour of CFRP and GFRP laminates on brick masonry were briefly summarized. Based on the experimental results and the proposed bond-slip equations, finite element analysis of the test specimens were carried out. Comparison of the results obtained from different finite element models were made, they are: (1) coarse mesh versus fine mesh, (2) exponential bond-slip versus bilinear bond-slip and (3) FRP with concentrated fibre property versus FRP with distributed fibre property. The finite element results were also compared with the results obtained from the analytical solution proposed by Yuan et al. (2004). With the stiffness of FRP assigned as the average value of stiffness obtained from the test, the results obtained from both finite element method and analytical solution compared well with the test results in term of both maximum load and deflection. Parametric studies were carried out based on the analytical solution for different bond length and width ratio of FRP to brick. It is found that the minimum required bond length for the current CFRP and GFRP specimens studied are about 70 mm and 59 mm, respectively. When the bond length is longer than the minimum required bond length, increasing the bond length does not increase the maximum load capacity significant, however, the maximum deflection increases with increasing bond length. For the current specimens studied, the width ratio of FRP to brick does not have significant affect to the maximum load capacity and deflection, especially for specimen with lower stiffness ratio of FRP to brick. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme vi ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS This page is left blank on purpose. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS vii ASTRATTO La riabilitazione e la conservazione dei monumenti e delle strutture storiche sta divenendo sempre più di maggiore interesse nel mondo. Molti monumenti e strutture storiche sono stati costruiti in muratura di mattoni e necessitano di adeguate tecniche d’intervento. Dato che lo sviluppo dei materiali fibro rinforzati (Fiber Reinforced Polymers FRP) sta diventando sempre più comune, la loro applicazione mediante incollaggio si configura come uno dei possibili metodi di rinforzo dei monumenti e delle strutture storiche in muratura. Nel corso dell’ultima decade, sono state effettuate molte ricerche per studiare il comportamento dell’incollaggio degli FRP a strutture in cemento armato. Tuttavia, sono ancora pochi i contributi disponibili sul comportamento di adesione alla muratura. Nel presente lavoro è brevemente riportato lo studio sperimentale di base, condotto da Panizza et al. (2009), per l’identificazione della legge di adesione locale di laminati in fibra di carbonio (CFRP) e di vetro (GFRP) applicati su muratura di mattoni. Sulla base dei risultati sperimentali e delle leggi di adesione identificate, sono state effettuate una serie di analisi di dettaglio agli elementi finiti delle prove sperimentali. I risultati numerici ottenuti da diversi approcci di modellazione sono stati tra di loro confrontati. I confronti hanno riguardato: (1) la suddivisione del dominio con elementi di media o piccola grandezza, (2) l’utilizzo di una legge locale di adesione esponenziale o bi-lineare, (3) la modellazione del materiale composito FRP secondo un approccio con fibre concentrate o distribuite. I risultati degli elementi finiti sono stati anche confrontati con i risultati ottenuti dalla soluzione analitica proposta dal Yuan et al. (2004). Utilizzando la rigidezza assiale del composito FRP pari a quella ottenuta dalle prove sperimentali, i risultati ottenuti dal metodo agli elementi finiti e dalla soluzione di analisi sono in ottimo accordo sia in termini di carico massimo che di deformazione. Attraverso l’utilizzo della soluzione analitica, sono state eseguite analisi parametriche che hanno considerato differenti lunghezze di ancoraggio e differenti rapporti di larghezza FRP/mattone. Si sono quindi determinate la minima lunghezza di ancoraggio per i compositi in CFRP e GFRP utilizzati, le quali risultano rispettivamente pari a circa 70 e 59 mm. Si è inoltre osservato che quando la lunghezza di ancoraggio è maggiore di quella minima richiesta, ogni suo eventuale incremento non aumenta il carico massimo, che rimane pressoché costante, ma aumenta solo lo spostamento ultimo. Infine, sulla base degli attuali modelli di studio, si è constatato che il rapporto di larghezza FRP/mattone non influenza significativamente il carico e lo spostamento ultimo, in particolare nel caso di prove di adesione con un basso rapporto di rigidezza tra FRP e mattone. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme xiv ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS This page is left blank on purpose. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS xv LIST OF TABLE Page Table 1 Summary of bricks mechanical properties and reinforcement components properties ..............................................................................................................................................8 Table 2 Summary of the experimental failure load, Pu (Panizza et al. 2009) ....................... 10 Table 3 Significant values for local bond of CFRP and GFRP bonded to brick specimens.. 11 Table 4. Summary of material properties of brick and FRP used in finite element analysis .14 Table 5. Summary of FE results of coarse mesh and fine mesh model of CFRP specimens19 Table 6 Summary of FE results of fine mesh model of CFRP and GFRP specimens..........20 Table 7. Summary of FE results of exponential and bilinear bond-slip curves of CFRP specimens ...........................................................................................................................26 Table 8 Specimens with exponential bond-slip behaviour (concentrated versus distributed fiber) ....................................................................................................................................33 Table 9 Specimens with bilinear bond-slip behaviour (concentrated versus distributed fiber) ............................................................................................................................................38 Table 10 Comparison of finite element resutls and analytical results...................................51 Table 11 Summary of results from finite element analysis, analytical solution and test (with equivalent stiffness of FRP equals to the average value of test results)...............................53 Table 12 Maximum deflection versus bond length (bfrp / bbrick = 0.42)..................................70 Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme xvi ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS This page is left blank on purpose. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 1 1. REHABILITATION AND PRESERVATION OF HISTORIC CLAY BRICK STRUCTURES Rehabilitation and preservation of historic monuments and ancient structures is attracting more and more interest in the world. However, the conservation of architectural heritage is a difficult task due to the complex geometry of buildings and large variability of construction materials. It is a multidisciplinary work that requires the contribution of different scientists and professionals in order to collect all the data necessary for the intervention. In the last decades, the strategic importance of historic buildings due to cultural and economical reasons caused a large increase in studies dealing with historic structures and materials. In the case of ancient clay brick masonry, studies have been focusing on the main mechanical properties, retrofitting techniques, seismic vulnerability, etc. 1.1 A brief overview of historic clay bricks Due to the aggressions of the environment (snow, rain, cold, heat, etc.) human being started to protect themselves by using the natural materials such as natural caves, tree trunks, animal’s fur, straw, clay, etc. With the appearance of the first civilization, around 9000 to 7000 BC, the construction techniques evolved stone, adobe, wood and clay brick begun to be used. The first vestiges of brick masonry buildings were found in the region of Israel (Mesopotamia) and dated from 9000 to 8000 BC (Fernandes et al 2006). Clay brick masonry is, effectively, one of the finest and most durable construction techniques ever invented by human. Masonry consist of building stable bonded stacks of small pieces by hand (Vekey, 1998). Used since the time of the first villages and cities built by human, masonry application has been growing and evolving to new uses all over the entire civilized world. It was a fundamental building material in Mesopotamian, Egyptian and Roman periods. During Roman period, the use of clay brick increased and become specialized in order to maximize its benefits. Despite several modifications of the clay brick uses, shape and manufacture along thousands of years of constant evolution, the simplicity that made its success remained. 1.2 Example of clay bricks structures in Italy There are many historical brick masonry structures in Italy. One of the examples is the Torrazzo which is a medieval brickwork tower adjacent to the Cathedral of Cremona, a town 90 km far from Milan, Italy (Binda et al. 2000). The height of the Tower is around 112 m and it is the highest medieval masonry tower in Europe. The real date of construction is not known but declared around the 13th century. It belongs to a group of monuments, the Cathedral, the Bapistery, the Town Hall Palace, the Militia Loggia and the Torrazzo itself, which form one of the most beautiful Italian squares. The Bell-Tower consists of Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 2 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS three architectural elements: the Romanesque tower, the Ghirlandina and the Bertazzola lodge at the base (Fig. 1). The Romanesque tower has a square shape 13 m for each side large with load bearing masonry walls 3.3 m thick at the ground floor. It has four vertical reinforcing built-in strips at the four corners, two semi-cylindrical ones on the facade on each side and six horizontal cornices, made of simple little arches or hanging ones. The external load-bearing walls of the tower show since years several cracks. Since the crack pattern has developed along the years a possible time dependent behaviour of the material can be supposed. This phenomenon, together with the effects of cyclic loads as wind and temperature variations can eventually cause to the structure long-term heavy damages. Binda et al. (2000) applied systematically georadar, sonic tests and flat jack tests on the walls of the Torrazzo. It is shown that an external thin leaf is partially detached in some part of the tower base. Figure 1 Torrazzo brickwork Tower adjacent to the Cathedral of Cremona Another example of brick masonry structures in Italy is the Basilica of S. Lorenzo at Cremona, Italy. The Basilica of S. Lorenzo shows all the formal contradictions deriving from centuries of continuous modifications of the load-bearing structures. The Basilica is a three naves construction built in the 12th century on the remains of a previous church destroyed by a fire. In 15th century, an important intervention introduced vaulted ceilings to the lateral naves and a monumental three-lobes chapel on the north-west corner. Several damages were reported by Anzani et al. (2007). Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 3 It is shown from the above examples that nowadays ancient buildings exhibit often serious damages, which are the result of many years of abandon, of the weathering of materials, poor or inadequate structural behaviour and damage due to earthquake. Therefore, it is necessary to carry out modification or strengthening of those structures which show potential problems. 1.3 Traditional strengthening and modern strengthening techniques Since the beginning of structural restoration, architects and engineers have envisaged and actually applied a wide variety of repair or strengthening interventions to improve the structural response of ancient masonry structures. Many historic constructions are structurally inadequate for current use. Additional material deterioration due to environmental factors and lack of maintenance has caused significant weakening of historically important structures (Drysdale et al. 1993). A few decades ago, strengthening of structures was accomplished by the materials available at that time. The criteria for choosing a particular solution must take into account not only its structural effectiveness and cost, but also the compatibility with the techniques and materials used in the construction of the monument regarding its original conception and historical value. Different techniques, such as pre-stressing cables, injected mortar, inserted steel bars, wooden planks, iron cramps and synthetic polypropylene fibers, for retrofitting historical constructions have been adopted. Some of the oldest techniques are still in use, such as dismantling and remounting with possible improved material substitution. In the process masonry element or structures contain parts that have to be removed, substituted or repaired, if a local intervention is not feasible. The main objective is to recover the functionality of a structure maintaining its historical and cultural value, modifying an erroneous design. Application of strengthening to arched structures in the last 20-25 years allowed developing several methods. Such methods include the installation of stainless steel reinforcing bars in the near surface zones of the masonry (Sumon 1997). As an example the use of near-surface mounted reinforcement (NSM) in masonry arch bridges can enhance the load carrying capacity, delaying the formation of cracks and hinges and, and at the same time, minimize any disruption to the bridge users (Garrity 1995). Traditional techniques employ the materials and building processes used originally for the construction of ancient structures. Besides the usage of traditional techniques, modern techniques such as the application of modern materials for strengthening are getting more attention nowadays. Modern approaches are based on the idea that the strengthening should be light and removable and, if possible, it should not change the structural scheme or the construction. This objective can be achieved by using advance composite materials. Strengthening of existing masonry structures can be carried out by applying bonded fiber reinforced polymer (FRP) such as glass fiber reinforced polymer (GFRP) or carbon fiber reinforced polymer (CFRP). These materials present several advantages, such as low specific weight, corrosion immunity and high tensile strength. Their flexibility and easy application allow a wide range of intervention scenarios. The FRP materials have the advantage that they are easily adapt to the surface to be strengthened, on the external face of the element locally (as a strips arrangement) or to the whole surface of the structure (as a grid reinforcement arrangement). The bond between the Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 4 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS masonry and fiber is normally obtained with the use of epoxy resins or mortar (Sanchez et al. 2007). An effective use of this technique requires certain regularity in the masonry surface. 1.4 Scope and objective It is shown previously that strengthening of brick masonry structures can be done by applying the modern materials such as fiber reinforced polymer (FRP). In the past decade, many researches have been carried out to investigate the behaviour of bonded FRP to reinforced concrete structures. However, still a few contributions are available concerning debonding problem on masonry. Recently, Panizza et al. (2009) have carried out double-lap push-pull shear tests to investigate the bond-slip behaviour of externally bonded FRP to brick. Based on the test results, bond-slip equations (one continuous and one bi-linear equation) are proposed for FRP bonded to brick element. In this thesis, finite element analysis of those test specimens is carried out in order to achieve the following goals: 1. Set up a suitable finite element model based on the bond-slip equations proposed by Panizza et al. (2009). 2. Compare the finite element results with different assumption of FRP material properties. 3. Based on the verified finite element model, compare the finite element results with the available analytical solution in literature. 4. Carry out parametric studies of the bond-slip behaviour of FRP bonded to brick by using the verified analytical solution. 1.5 Outline of thesis The outline of thesis is as following. A brief introduction of clay brick, clay brick masonry structure and strengthening techniques is shown in Chapter 1. Discussion of bond-slip tests of external bonded FRP to concrete and masonry elements is shown in Chapter 2 together with the test results of Panizza et al. (2009). In Chapter 3, finite element analysis of the test specimens of Panizza et al. (2009) is presented. Followed to the finite element analysis of the test specimens, the finite element results were compared with the results obtained from current available analytical solution in Chapter 4. Parametric study based on the analytical solution of the FRP-to-brick bond strength with different parameters, such as bond length of FRP and width ratio of FRP to brick, is shown in Chapter 5. Summary, conclusion and suggestion for further study is shown in Chapter 6. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 5 2. STRENGTHENING OF STRUCTURES WITH FRP: BOND-SLIP BEHAVIOUR In the past decade, the application of advance materials, such as fiber reinforced polymer (FRP) for strengthening structures become more popular. One of the main reasons is due to the great effort of the searches in this area and more understanding about the behaviour of FRP to the substrate is achieved. Due to the popularity of reinforced concrete (RC) structures nowadays, most of the researches are focused on the bonded behaviour of FRP applied to RC structures. In fact, similar technique can be applied to historical brick masonry structures. However, more studies about the applied of FRP on brick masonry structures need to be carried out in order to obtain better understanding of the behaviour. 2.1 Bond-slip of FRP to RC structures A fairly large amount of bond tests for the FRP sheet-concrete interfaces under shear have been carried out in the past decades. As it is shown in Fig. 2 that test methods include single lap pullout test method (Chajes et al. 1996), double lap pullout bond tests (Sato et al. 2001) and bending tests (Lorenzis et al. 2001). (a) single lap pullout test (b) double lap pullout test (c) bending test Figure 2 Various type of bond test methods Through those experimental studies the bond mechanisms of FRP sheet-concrete interfaces, the important aspects such as the (1) bond strength, (2) interfacial fracture energy, (3) effective bond length and (4) bond stress-slip relationship have been clarified. A summary of those important aspects is reported by Dai et al. 2005 and they are listed in following. Bond strength: the FRP sheet-concrete interface fails mostly at a thin layer beneath the concrete surface. As a result, the concrete surface condition and strength are critical factors affecting the interfacial bond strength. At present, sandblasting is the most common surface treating method. Chajes et al. (1996) and Sato et al. (2000) studied the effects of concrete strength f’c and concluded that the average interface bond strengths, which are the ultimate pullout forces divided by bond areas between FRP sheets and concrete, are linearly proportional to f’c 1/2, f’c 2/3, and f’c 1/5, respectively. Besides the concrete property, the FRP and adhesive properties affect the interface bond strength as well. In general, using higher FRP stiffness (Nakaba et al. 2001) and softer adhesives (Dai et al. 2002) can increase the average bond strength. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 6 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Interfacial fracture energy: the interfacial fracture energy Gf, which is the area underneath the interfacial bond stress-slip curve, is an important parameter for the bond characteristics. Based on different types of interfacial bond stress-slip relationships, Yuan et al. (2001) proved that the maximum interfacial bond force can be expressed as a function of the Gf and FRP stiffness (Efrp tfrp). Due to the clear physical meaning of the Gf, it is very useful to apply it in numerical analysis for deriving bond strength and anchorage length models as well as for clarifying the debonding failure mechanisms of FRP sheet-concrete interfaces in more comprehensive ways (Wu and Yin 2002). Effective bond length: there exists an active bonding zone named as the effective bond length Le, along which most of the interfacial load is transferred between FRP sheets and concrete. When the bond length of FRP sheet-concrete interfaces exceeds the Le, the bond strength will not increase significantly any longer. In general, it was reported that the effective bond length increases with the stiffness of FRP sheets. However, due to the different materials used in various researches, the effective bond length was reported in a fairly big range (from 45 mm to 275 mm). Bond stress-slip relationship: the bond-slip relationship relates on the interfacial fracture energy, Gf, as discussed previously. The shape of the bond-slip model determines the predicted distribution of axial strains in the plate. Lu et al. (2005) summarized four existing bond-slip models for normal-adhesive interfaces for an FRP-to-concrete bonded joint with the following properties: f’c = 32 MPa, ft = 3.0 MPa, bf = 50 mm, bc = 100 mm, Ef tf = 16.2 GPa mm. The curves of the four bond-slip models are shown in Fig. 3 and it can be seen that the shapes of the predicted bond-slip curves differ substantially. In particular, the linear-brittle model of Neubauer and Rostasy (1999) 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. Nakaba et al. (2001) and Savioa et al. (2003) have shown that the bond-slip curve should have an ascending branch and a descending branch. A bilinear model of bond-slip curve is proposed by Monti et al. (2003) and it can be used as an approximation but the linear-brittle model proposed by Neubauer and Rostasy (1999) is unrealistic. Apart from the general shape, the slip at maximum stress and the ultimate slip at zero bond stresses, determine the accuracy of the model. It is interesting to know that the models by Nakaba et al. (2001), Monti et al. (2003) and Savioa et al. (2003) are in reasonably close agreement, and the linear-brittle model of Neubauer and Rostasy (1999) predicts a similar maximum bond stress. Lu et al. (2005) used another approach to obtain the bond-slip curve of FRP-to-concrete. Their new bond-slip models are not based on axial strain measurements on the FRP plate; instead, they are based on the predictions of a meso-scale finite element model, with appropriate adjustment to match their predictions with the experimental results for a few key parameters. The bond-slip curves proposed by Lu et al. (2005) (the precise model and the bilinear model) are shown in Fig. 3 as well. By comparing both the bond strength and strain distribution in the FRP plate, Lu et al. (2005) concluded that their new bond-slip models which is based on a combination of finite element results and the test results give better prediction than the other bond-slip models mentioned. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 7 Figure 3 Bond-slip curves from existing bond-slip models for FRP-to-concrete 2.2 Experimental study of bond-slip behaviour of FRP-to-brick by Panizza et al. (2009) As it is shown in the previous section that many researches have been focused on the bond-slip behaviour of FRP-to-concrete elements, however, only a few investigations concerning about the debonding on masonry substrate are available. Recently, some researchers such as Aiello et al. (2005) investigated the bond behaviour of FRP to natural stones and Briccoli Bati et al. (2007) investigated the bond behaviour of FRP to solid clay bricks. They adopted double-lap push-pull shear tests which consist in loading in tension two reinforcement strips, symmetrically connected to the support, in order to create shear stresses at the interface; the brittle support is subjected to compressive stresses. This set-up is based on the assumption that the applied load is equally distributed on the two strips, but it is also particularly simple and suitable for the usual common available test machine. Experimental study of the bond behaviour of CFRP and GFRP laminates on brick masonry was carried out by Panizza et al. (2009). In their study, five samples of clay brick bonded by high-strength carbon reinforcement and five samples of clay brick bonded by alkali-resistant glass reinforcement were tested to examine the bond strength of the FRP to the clay brick. Solid clay bricks (nominal dimension 250 x 120 x 55 mm) were used as substrate, and the MBrace © Wet lay-up system as reinforcement. In their study, double-lap Push-pull Shear Tests were performed to examine the bond-slip behaviour of CFRP and GFRP laminates on brick masonry. The experimental results, in terms of failure load, were compared with predictive bond-strength models proposed in literature, mainly available for concrete. Also, the interface fracture energy which based on the experimental strength was calibrated for FRP bond to brick. Bond-slip equations, an exponential equation based on simplified analytical model and a Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 14 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Table 4. Summary of material properties of brick and FRP used in finite element analysis Brick Elastic modulus Eb 16110 MPa Poisson ratio ν 0.16 Matrix Elastic modulus Ematrix 3,000 MPa Poisson ratio ν 0.33 Carbon fiber Elastic modulus Ecarb 230,000 MPa Thickness tcarb 0.165 mm CFRP composite Elastic modulus Ecfrp 21,728 MPa Thickness tcfrp 2.0 mm Glass fiber Elastic modulus Eglass 65,000 MPa Thickness tglass 0.23 mm GFRP composite Elastic modulus Egfrp 10,130 MPa Thickness tgfrp 2.0 mm All the non-linear behaviour was modelled by the interface element. The exponential bond-slip behaviour of the interface element was assigned according to the data obtained from Table 3 and Equation 5. In the finite element analysis, the continuous bond-slip equations were converted to a set of pair of data points (shear stress versus slip) as an input for the bond-slip behaviour. The corresponding bond-slip curves for CFRP and GFRP specimens are shown in Figs. 7 and 8 together with the data points used in the finite element analysis, respectively. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 15 0 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 Slip, s (mm) Tangential stress τ τ τ τ (MPa) UNIPD Data for FEA Figure 7 Expontiental bond-slip curve and data points used in finite element analysis (CFRP) 0 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 Slip, s (mm) Tangential stress τ τ τ τ (MPa) UNIPD Data for FEA Figure 8 Expontiental Bond-slip curve and data points used in finite element analysis (GFRP) Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 16 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Due to symmetry, only half of the specimen is modeled. Therefore, suitable boundary conditions were assigned to the brick and the FRP as it is shown in Fig. 9. The initial stiffnesses in the normal and tangential direction of the bond interface are assigned as 16,110,000 N/mm3 and 240.56 N/mm3. The initial stiffness in the normal direction is about 1,000 times of the elastic modulus of the brick per mm and it is believed that this high value of stiffness is good enough to avoid the FRP element to penetrate to the brick element. The initial stiffness in the tangential direction is according to the initial slope of the bond-slip curve. For the un-bonded surface between the FRP and brick, similar interface elements as used in the bonded interface are used. However, the stiffness in the tangential direction is assigned close to zero in order to simulate the free slip behaviour in this region. Figure 9 Boundary conditions of FE model Nodal forces are applied to the loading end of the FRP. In order to simulate the uniform displacement at the loading end of the FRP, suitable nodal forces should be assigned to the nodes according to their position and modeling option. Detail of the arrangement of nodal forces to achieve uniform displacement at the loading end of FRP is discussed in the later section. Non-linear analysis was performed by using an energy based method together with arc-length control method. With a monotonic incremental load assigned to the loading end of the FRP, equilibrium of the system was calculated and the corresponding reactions, internal stress/strain and deformation of the structures were obtained. 3.2 Finite element mesh study of FRP laminates on brick masonry For the finite element modeling of FRP-to-brick, one of the important issues is the size of the mesh. Therefore, mesh study was carried out in order to examine the effect of mesh size to the results of the finite element analysis. Two finite element meshes, the coarse mesh and the fine mesh, of the Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 17 FRP-to-brick model were prepared based on the dimension and material properties of the CFRP specimen. For the coarse mesh model, the size of element is about 2 x 2.5 mm. The number of elements in the coarse mesh model is about 1490 plus 115 interface elements. In order to have better resolution of the stress and strain near the interface between FRP and brick, a fine mesh model was created. The fine mesh model is a modification of the coarse mesh model by refining the element size within the interface region. It was done by introducing a transfer layer of elements near the interface region in order to reduce the size of element at that location to around 1 x 1 mm. Also, the FRP was modeled by using two layers element of around 1 x 1 mm. In this case, the number of elements in the fine mesh model is 3085 plus 230 interface elements. Therefore, the number of elements of the fine mesh model is about double of that of coarse mesh model. The corresponding coarse mesh and fine mesh finite element models are shown in Fig. 10. In the finite element mesh study, the concentrated FRP option is used for modeling the composite material. The material properties of the matrix were assigned to the element of the composite material. Then, the fiber was modeled by using the reinforcement element function given in the finite element program. The reinforcement was located at the middle height of the composite material for all cases. By introducing the reinforcement function, the stiffness of the elements representing the composite materials will be increased accordingly. 2 mm 2.5 mm FRP as reinforcement (a) coarse mesh model (Total number of elements = 1490 + 115 interface elements) (b) fine mesh model (Total number of elements = 3085 + 230 interface elements ) Figure 10 Typical (a) coarse mesh and (b) fine mesh FE models Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 18 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS As it is stated previously, nodal forces are applied to the loading end of the FRP in order to simulate the uniform displacement at the loading end of the FRP. Since the number of elements at the loading end of FRP are different for coarse mesh and fine mesh models, different percentage of nodal loading should be assigned. For the case in which fiber are considered as reinforcement and concentrated in the middle height of the composite materials, the percentage of nodal forces assigned to the loading end of the FRP for coarse mesh and fine mesh model are shown in Fig. 11. It is noticed that due to the higher stiffness of the fiber, more loading was attracted to the middle height of the composite materials. (a) coarse mesh model (b) fine mesh model Figure 11 Percentage of nodal forces assigned to the loading end of FRP (concentrated fiber model) 3.2.1 Comparison of FE load versus deflection results with test results Comparison of the results of the maximum end displacement of FRP (δ) and the corresponding load (P) obtained from the finite element analysis of the coarse mesh and fine mesh models are shown in Table 5 and the corresponding load versus displacement curves of FRP loading end is shown in Fig. 12. As it is shown in Table 5, the maximum end displacement of FRP and the maximum load for the coarse and fine mesh models are almost the same. However, it is shown in the figure that the load versus displacement curves for the fine mesh and coarse mesh models are almost the same in the ascending part but not in the region near the descending part. The results obtained from the fine mesh model are smoother compared to the results obtained from the coarse mesh model. Compare to the average test ultimate load (Pu, ave), the finite element results are about 95% of Pu, ave for both cases. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 19 Table 5. Summary of FE results of coarse mesh and fine mesh model of CFRP specimens Max. end displacement of FRP (δ) mm Max. Load (P) N P / Pu, ave Coarse mesh model 1.67 34,473 0.95 Fine mesh model 1.67 34,537 0.95 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 0 0.3 0.6 0.9 1.2 1.5 1.8 Deflection in loading direction (δ δδ δ) mm Loading (P) N Fine mesh Coarse mesh Test range of P u 0.3P 0.9P 1.0P Test Average of P u Figure 12 Comparison of the load versus deflection of CFRP specimens (coarse versus fine mesh models) Similar finite element analysis was carried out for the GFRP specimens using the fine mesh model. The load versus deflection curve of the FRP loading end is shown in Fig. 13 and the results of the maximum end displacement and maximum load of FRP is shown in Table 6 together with the results obtained from the CFRP finite element analysis. Compare to the average test result of the failure load (Pu, ave) of GFRP specimens, the finite element results is about 4% higher than the average values of test results. It is also shown from that finite element analysis that the maximum end displacement of the FRP of GFRP specimen is larger than that of the CFRP specimen due to the lower elastic modulus of the GFRP composite. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 20 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Table 6 Summary of FE results of fine mesh model of CFRP and GFRP specimens Fine mesh model Max. end displacement of FRP (δ) mm Max. Load (P) N PFE / Pu, ave CFRP 1.67 34,560 0.95 GFRP 2.77 27,843 1.04 0 5000 10000 15000 20000 25000 30000 0 0.5 1 1.5 2 2.5 3 Deflection in loading direction (δ δδ δ) mm Loading (P) N Fine mesh - GFRP Test range of P u 0.3P 0.9P 1.0P Test Average of P u Figure 13 Load versus deflection of fine mesh models of GFRP specimen 3.2.2 Comparison of FE strain results with test results The finite element strain results of the fine mesh model were compared with the test results as shown in Fig. 14. As it is shown in the figure, the strain results were compared in three different load levels (30%, 90% and 100% of maximum loading level as shown in Fig. 12). For the load levels under 90% of maximum loading level, the strain results of the fine mesh model compared well with the test results. For the maximum load levels, the strain results compared well with the test results near the free end region. However, near the loading end, the strain results over predicted the strain values compared to the test results of C1 to C4, but the finite element strain results are compared well with test results of C5. Nevertheless, it is shown that the strain results obtained from the fine mesh model compared well with the test results. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 21 0 1000 2000 3000 4000 5000 6000 7000 8000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) C1 C2 C3 C4 C5 FE (a) Load level = 30% of maximum loading 0 1000 2000 3000 4000 5000 6000 7000 8000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) C1 C2 C3 C4 C5 FE (b) Load level = 90% of maximum loading Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 22 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 0 1000 2000 3000 4000 5000 6000 7000 8000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) C1 C2 C3 C4 C5 FE (c) Load level = 100% of maximum loading Figure 14 Comparison of strain results of CFRP fine mesh models and test results Similar comparison of the finite element strain results to the test results was carried out for the GFRP specimen as well and the results are shown in Fig. 15. As it is shown in the figure, the strain results were compared in three different load levels as well (30%, 90% and 100% of maximum loading level as shown in Fig. 13). The behaviour of the strain results of GFRP specimens is similar to the CFRP specimens. It is also found that for the load levels under 90% of maximum loading level, the strain results of the fine mesh model compared well with the test results. For the maximum load levels, the strain results compared well with the test results near the free end region. However, near the loading end, the strain results over predicted the strain values compared to the test results. Due to the lower stiffness of the GFRP composite, the strain results were all higher than that of the CFRP specimens. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 23 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (mm/m) G1 G2 G3 G4 G5 FE (a) Load level = 30% of maximum loading 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (mm/m) G1 G2 G3 G4 G5 FE (b) Load level = 90% of maximum loading Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 30 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) G1 G2 G3 G4 G5 Bilinear Bond-slip Exp. Bond-slip (a) Load level = 30% of maximum loading 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) G1 G2 G3 G4 G5 Bilinear Bond-slip Exp. Bond-slip (b) Load level = 90% of maximum loading Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 31 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) G1 G2 G3 G4 G5 Bilinear Bond-slip Exp. Bond-slip (c) Load level = 100% of maximum loading Figure 21 Comparison of strain results of GFRP specimens (exponential versus bilinear bond-slip) 3.4 Finite element model of FRP (Concentrated fiber versus Distributed fiber) As it is mentioned in the beginning of this chapter, there are three options for modeling the FRP: (a) a plate with its thickness being equal or similar to the actual thickness of the laminate including all adhesive, but with the fibers concentrated in a thickness equal to the nominal thickness of the fiber sheet sitting in the middle of the plate, (b) a plate with its thickness being equal or similar to the actual thickness of the laminate including all adhesive, with the fibers assumed to be evenly distributed across the plate thickness; and (c) a plate with a nominal thickness (generally the thickness of the fiber sheet) without considering the adhesive (Lu et al. 2004). In general, option (a) and (b) are more suitable for modeling the laminated FRP since both material properties of fiber and matrix are considered in the finite element model. In the above study, option (a), in which the fiber is considered as reinforcement and is concentrated in the middle of the composite material, was applied for the modeling of FRP. In order to examine the different between option (a) and (b), finite element analysis of the test specimens were carried out by following option (b) for modeling the FRP. The fine mesh model is used and both exponential and bilinear bond-slip behaviour are considered in the following finite element study. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 32 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 3.4.1 Model with exponential bond-slip behaviour (Concentrated fiber versus Distributed fiber) In general, all the properties (such as mesh size, element type, boundary condition, bond-slip data and material properties) for the finite element models of FRP with distributed fiber are the same as previous study, except the equivalent elastic modulus of the FRP are used for the composite material instead of introducing the reinforcement element to the FRP. The equivlent elastic modulus for the CFRP and GFRP are calculated based on Equation 7 and the values are shown in Table. 4. Since the FRP is being modeled as homogenerous materials, in order to obtain a uniform displacement at the end of the composite, the following nodal forces, which are not the same as those used for model with concentrated fiber, are assigned to the end of the composite (Fig. 22). Figure 22 Percentage of nodal forces assigned to the loading end of FRP (Distributed fiber model) Comparison of the results of the maximum end displacement of FRP (δ) and the corresponding load (P) obtained from the finite element analysis of models with concentrated fiber and distributed fiber, respectively are shown in Table 8 and the corresponding load versus displacement curves of FRP loading end is shown in Figs. 23 and 24. For the CFRP specimens, as it is shown in Table 8, the maximum end displacement of FRP and maximum load for both models are almost the same. However, the model with distributed fiber property showed a slightly increase in maximum displacement for the CFRP specimen. For the GFRP specimen, it is shown that the model with distributed fiber property showed a slightly increase in maximum displacement and also for the maximum load. Nevertheless, the results obtained in both models only showed very little different in term of both maximum end displacement and maximum load. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 33 Table 8 Specimens with exponential bond-slip behaviour (concentrated versus distributed fiber) Max. end displacement of FRP (δ) mm Max. Load (P) N PFE / Pu, ave CFRP specimens Concentrated fiber 1.67 34,537 0.95 Distributed fiber 1.68 34,413 0.95 GFRP specimens Concentrated fiber 2.77 27,843 1.04 Distributed fiber 2.83 28,058 1.05 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 0 0.3 0.6 0.9 1.2 1.5 1.8 Deflection in loading direction (δ δδ δ) mm Loading (P) N Concentrated fiber Distributed fiber Test range of P u 0.3P 0.9P 1.0P Test Average of P u Figure 23 Comparison of the load versus deflection of CFRP specimens with exponential bond-slip behaviour (concentrated fiber versus distributed fiber) Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 34 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 0 5000 10000 15000 20000 25000 30000 0 0.5 1 1.5 2 2.5 3 Deflection in loading direction (δ δδ δ) mm Loading (P) N Concentrated fiber Distributed fiber Test range of P u 0.3P 0.9P 1.0P Test Average of P u Figure 24 Comparison of the load versus deflection of GFRP specimens with exponential bond-slip behaviour (concentrated fiber versus distributed fiber) Comparison of the finite element strain results of the models with exponential bond-slip behaviour (concentrated versus distributed fiber) were made with the test results as shown in Figs. 25 and 26 for CFRP and GFRP specimens, respectively. Similarly, the strain results were compared in three different load levels (30%, 90% and 100% of maximum loading level) and it is shown that for both CFRP and GFRP specimens, the finite element strain results are almost the same for models with concentrated fiber and distributed fiber, respectively. Therefore, it is concluded that the strain results obtained from both type of models are almost the same and they are both compared well with the test results especially in the lower load levels. Similar results are also observed for the GFRP specimens. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 35 0 1000 2000 3000 4000 5000 6000 7000 8000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) C1 C2 C3 C4 C5 Distributed fiber Concentrated fiber (a) Load level = 30% of maximum loading 0 1000 2000 3000 4000 5000 6000 7000 8000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) C1 C2 C3 C4 C5 Distributed fiber Concentrated fiber (b) Load level = 90% of maximum loading Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 36 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 0 1000 2000 3000 4000 5000 6000 7000 8000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) C1 C2 C3 C4 C5 Distributed fiber Concentrated fiber (c) Load level = 100% of maximum loading Figure 25 Comparison of strain results of CFRP specimens with exponential bond-slip behaviour (concentrated versus distributed fiber) 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) G1 G2 G3 G4 G5 Distributed fiber Concentrated fiber (a) Load level = 30% of maximum loading Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 37 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) G1 G2 G3 G4 G5 Distributed fiber Concentrated fiber (b) Load level = 90% of maximum loading 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) G1 G2 G3 G4 G5 Distributed fiber Concentrated fiber (c) Load level = 100% of maximum loading Figure 26 Comparison of strain results of GFRP specimens with exponential bond-slip behaviour (concentrated versus distributed fiber) Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 38 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 3.4.2 Model with bilinear bond-slip behaviour (Concentrated fiber versus Distributed fiber) Similar finite element analyses were also carried out for model with bilinear bond-slip behaviour. Comparison of the results of the maximum end displacement of FRP (δ) and the corresponding load (P) obtained from the finite element analysis of models with concentrated fiber and distributed fiber, respectively are shown in Table 8 and the corresponding load versus displacement curves of FRP loading end is shown in Figs. 27 and 28. In general, the results of model with bilinear bond-slip behaviour are in the same trend as those obtained from the model with exponential bond-slip behaviour. For the CFRP specimens, as it is shown in Table 9, the maximum end displacement of FRP and maximum load for both models are almost the same. However, the model with distributed fiber property showed a slightly increase in maximum displacement and a slightly decrease in maximum load. Nevertheless, the results obtained in both models only showed very little different in term of both maximum end displacement and maximum load. Table 9 Specimens with bilinear bond-slip behaviour (concentrated versus distributed fiber) Max. end displacement of FRP (δ) mm Max. Load (P) N P / Pu, ave CFRP specimens Concentrated fiber 1.69 34,650 0.96 Distributed fiber 1.70 34,472 0.95 GFRP specimens Concentrated fiber 2.79 28,237 1.06 Distributed fiber 2.85 28,058 1.05 Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 39 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 0 0.3 0.6 0.9 1.2 1.5 1.8 Deflection in loading direction (δ δδ δ) mm Loading (P) N Concentrated fiber Distributed fiber Test range of P u 0.3P 0.9P 1.0P Test Average of P u Figure 27 Comparison of the load versus deflection of CFRP specimens with bilinear bond-slip behaviour (concentrated fiber versus distributed fiber) 0 5000 10000 15000 20000 25000 30000 0 0.5 1 1.5 2 2.5 3 Deflection in loading direction (δ δδ δ) mm Loading (P) N Concentrated fiber Distributed fiber Test range of P u 0.3P 0.9P 1.0P Test Average of P u Figure 28 Comparison of the load versus deflection of GFRP specimens with bilinear bond-slip behaviour (concentrated fiber versus distributed fiber) Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 46 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 0 tdx d p p= τ − σ (8) 0btbt sssppp =σ+σ (9) where τ is the shear stress in the adhesive layer, σp is the axial stress in the plate and σs is the axial stress in the substrate. The constitutive equations for the adhesive layer and the two adherends can be expressed as )s(f = τ (10) dx du Ep pp =σ (11) dx du Es ss =σ (12) The interfacial slip s is defined as the relative displacement between the two adherends, that is sp uus −= (13) After substituting Eqs. 9 to 13 into Eq. 8 and introducing the parameters of local bond strength τmax and interfacial fracture energy Gf yield the following equations, 0)s(f G2 dx sd 2 2 max f 2 2 =λ τ − (14) and dx ds tG2 2 pf 2 max pλ τ =σ (15) where        + τ =λ sss p ppf 2 max 2 tEb b tE 1 G2 (16) Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 47 Equation 14 is the governing differential equation of the bonded joint and can be solved if the local bond-slip model relating the local interfacial shear stress to the local shear slip is defined. The interfacial fracture energy, which is simply the area under the local bond-slip curve, is introduced because once it is known it can be used regardless of the exact shape of the local bond-slip curve where a particular quantity depends on the interfacial fracture energy but not on the shape of the bond-slip curve. In the analytical solution, a bilinear bond-slip curve as shown in Fig. 32 which features a linear ascending branch followed by a linear descending branch is used. The bond-slip model defined above, the governing equation (Equation 14) can be solved to find the shear stress distribution along the interface and the load-displacement response of the bonded joint. The interfacial shear stress distribution and the propagation of debonding are shown in Fig. 33 and a typical load versus deflection curve is shown in Fig. 34. As it is shown in Fig. 34, the load versus deflection curve is defined in four stages: (1) elastic stage; (2) elastic-softening stage; (3) elastic-softening-debonding stage and (4) softening-debonding stage. The corresponding equations of load versus deflection are shown in following: Figure 32 Local bilinear bond-slip curve x P P max = max a = max a = ad = max a = add a = aud = max a = aud < max Figure 33 Interfacial shear stress distribution and propagation of debonding Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 48 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Figure 34 Typical full-range analytical load versus displacement curve Elastic stage (O-A) )Ltanh( s b P1 01 pmax λ ∆ λ τ = for 0 s0 ≤∆≤ (17) where        + τ = τ λ=λ sss p pp0 max max0 f 22 1tEb b tE 1 ss G2 (18) Elastic-softening stage (A-B) [ ]      λ+λ−λ λ λ λ τ =)asin()acos()aL(tanh b P221 1 2 2 pmax for d aa0 ≤≤ (19) ( ) [ ]       − +λ−λ−λ λ λ −=∆ 0f f 221 1 2 0f ss s )acos()asin()aL(tanhss for d aa0 ≤≤ (20) where Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 49        + − τ = τ− λ=λ sss p pp0f max max0f f 22 2tEb b tE 1 )ss()ss( G2 (21)         λ λ λ = 2 1 2 darctan 1 a (22) Elastic-softening-debonding stage (B-C-D) [ ]      λ+λ−−λ λ λ λ τ =)asin()acos()adL(tanh b Pd2d2d1 1 2 2 pmax for u aLd0 −≤≤ (23) )d b P 1(s pmax 2 fτ λ +=∆ for u aLd0 −≤≤ (24) where 2 u2 aλ π = (25) Softening-debonding stage (D-E) ))aL( b P 1(s u pmax 2 f− τ λ +=∆ (26) 4.2 Comparison of finite element results with analytical solution The finite element results of the load versus deflection curves were compared with the analytical solution. In the analytical solution, the FRP is considered as homogenous materials with the equivalent elastic modulus of the composite material assigned. Therefore, the results obtained from the finite element model with distributed fiber property were used for comparison. The material properties of brick Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 50 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS and FRP which were used in the finite element model with distributed fiber property are used in the analytical solution. Although the bilinear bond-slip behaviour was used in the analytical solution, the finite element results of both models with exponential and bilinear bond-slip behaviour, respectively, were compared with the analytical solution. It should be noted that in the finite element model, the deflection of the FRP is not measured directly from the end of the bonded length. It is measured from the loading end of the bond length plus a 30 mm of unbonded length of FRP. However, the result obtained from the analytical solution is the deflection at the end of the bonded length. Therefore, in order to account for the deformation due to the 30 mm unbonded length, deformation of the unbonded length is calculated based on Equation 27 and the results are added to the analytical solution. frpfrp free,frp free,frp AE PL =∆ (27) where P is the load on the FRP, Lfrp, free is the unbond length (i.e. 30 mm), Efrp and Afrp are the elastic modulus and area of the FRP. According to the equations of the analytical solution, the load (P) is the load on one FRP. Therefore, the load obtained from the equations of the analytical solution is multiplied by 2 in order to obtain the total load for the comparison to the finite element results. Also, in the analytical solution, the actual width of FRP and brick can be assigned. Therefore, two results of the analytical solution, one with the width ratio equals to 50/120 and one with the width ratio equals to 1, are included for comparison. Summary of the results obtained from the finite element analysis and the analytical solution are shown in Table. 9 and the corresponding load versus deflection curves are shown in Figs. 35 and 36. It is shown in Table. 10 that the analytical solution compared well with the finite element results. In general, for the maximum load, the results obtained from the analytical solution are slightly lower than that obtained from the finite element analysis (about 3% lower for both specimens). For the maximum deflection, the results obtained from the analytical solution are almost the same as those obtained from the finite element analysis. Meanwhile, it is observed that the maximum loads obtained from the analytical solution, which considered the width ratio of FRP and substrate equals to 50/120, are slightly larger than that obtained from the analytical solution with width ratio equals to 1 (about 3% higher for CFRP specimens and 1% higher for GFRP specimens). However, the corresponding maximum deflections obtained from the analytical solution, which considered the width ratio equals to 50/120, are slightly lower than that obtained from the analytical solution with width ratio equals to 1 (about 2% lower for CFRP specimens and 1% lower for GFRP specimens). Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 51 Table 10 Comparison of finite element resutls and analytical results Max. end disp. of FRP (δ) mm Max. Load (P) N CFRP specimens FE – Exp. bond-slip 1.68 34,413 FE – Bilinear bond-slip 1.70 34,472 Analy. solution (width ratio = 1) 1.69 33,467 Analy. solution (width ratio = 50/120) 1.66 34,374 GFRP specimens FE – Exp. bond-slip 2.83 28,058 FE – Bilinear bond-slip 2.85 28,058 Analy.solution (width ratio = 1) 2.86 27,194 Analy. solution (width ratio = 50/120) 2.83 27,547 0 5000 10000 15000 20000 25000 30000 35000 40000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Deflection (mm) Load (N) FE results - Exponential FE results - Bilinear Analytical solution (width ratio = 1) Analytical solution (width ratio = 50/120) Figure 35 Comparison of FE results and analytical solution (CFRP specimens) Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 52 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 0 5000 10000 15000 20000 25000 30000 0 0.5 1 1.5 2 2.5 3 Deflection (mm) Load (N) FE results - Exponential FE results - Bilinear Analytical solution (width ratio = 1) Analytical solution (width ratio = 50/120) Figure 36 Comparison of FE results and analytical solution (GFRP specimens) 4.3 Comparison of analytical solution results and finite element results with test results In the previous section, it is shown that the results obtained from the analytical solution compared well with the finite element results. In Chapter 3, with the equivalent elastic modulus of CFRP and GFRP predicted according to Equation 7, it is found that the finite element results under-predicted the maximum load of about 5% for the CFRP specimens and over-predicted the maximum load of about 5% for the GFRP specimens when compared to the average values of the test results. The theoretical equivalent stiffness (Efrp tfrp) of CFRP and GFRP which were used in the finite element models are 43,455 N/mm and 20,260 N/mm, respectively. Although these two values are within the range of the test results as shown in Table. 2, they are not exactly the same as the average values of the test specimens. Therefore, finite element analysis of CFRP and GFRP specimens are carried out again with the average test values of equivalent stiffness of the specimens (i.e. 46,260 N/mm and 18,673 N/mm, respectively) assigned to the FRP in the finite element models. In the finite element analysis, the models with distributed fiber property are used. Two analyzes, one with the exponential bond-slip behaviour and one with the bilinear bond-slip behaviour, are carried out for CFRP and GFRP specimens, respectively. Also, calculation was carried out again according to the analytical solution with the average test values of equivalent stiffness of FRP of the specimens. Summary of the results obtained from the finite element analysis and analytical solution for CFRP and GFRP specimens are shown in Table. 11 and the Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 53 corresponding load versus deflection curves are shown in Figs. 37 and 38. As it is shown in Table. 11 that the results obtained from both finite element analyzes and analytical solution compared very well with the test average value of maximum load. It is also found that the results obtained from the analytical solution gave almost the same value of the test average value of maximum load. Table 11 Summary of results from finite element analysis, analytical solution and test (with equivalent stiffness of FRP equals to the average value of test results) Max. end disp. of FRP (δ) mm Max. Load in (P) N ave,u P P CFRP specimens FE – Exp. bond-slip 1.63 35,372 0.98 FE – Bilinear bond-slip 1.65 35,372 0.98 Analy. solution (width ratio = 50/120) 1.60 35,422 0.99 GFRP specimens FE – Exp. bond-slip 2.96 26,379 0.99 FE – Bilinear bond-slip 2.96 26,199 0.98 Analy. solution (width ratio = 50/120) 2.95 26,466 0.99 Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 54 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Deflection (mm) Load (N) Test Average, Pu Analytical solution (width ratio = 50/120) FE Exponent BS FE Bilinear BS Test range of P u Figure 37 Load versus deflection results of CFRP specimens with test average FRP stiffness (Ecfrp tcfrp) 0 5000 10000 15000 20000 25000 30000 0 0.5 1 1.5 2 2.5 3 Deflection (mm) Load (N) Test Average, Pu Analytical solution (width ratio = 50/120) FE Exponential BS FE Bilinear BS Test range of P u Figure 38 Load versus deflection results of GFRP specimens with test average FRP stiffness (Egfrp tgfrp) Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 55 4.3.1 Comparison of FE strain results with test results The strain results obtained from the finite element analysis were compared with the test results. In order to provide a more detail comparison, the results were compared in ten different load level, (i.e. 10%, 20% up to 100% of maximum load). The corresponding results of CFRP and GFRP specimens with exponential bond-slip and bilinear bond-slip behaviour are shown in Figs. 39 and 40, respectively. It is shown from the figures that the finite element results of models with exponential bond-slip behaviour were almost the same as those of the model with bilinear bond-slip behaviour. Meanwhile, the finite element results compared well with the test results, especially when the load levels are less that 90% of the maximum load. At the maximum load level, the finite element strain results are higher than that of the test results, except for the results of specimen C5 which compared well with the finite element results. Nevertheless, it is shown that the finite element results are in the same trend as the test results for both CFRP and GFRP specimens. 0 1000 2000 3000 4000 5000 6000 7000 8000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) C1 C2 C3 C4 C5 FE Exponential BS FE Bilinear BS (a) Load level = 10% of maximum loading Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 62 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) G1 G2 G3 G4 G5 FE Exponential BS FE Bilinear BS (d) Load level = 40% of maximum loading 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) G1 G2 G3 G4 G5 FE Exponential BS FE Bilinear BS (e) Load level = 50% of maximum loading Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 63 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) G1 G2 G3 G4 G5 FE Exponential BS FE Bilinear BS (f) Load level = 60% of maximum loading 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) G1 G2 G3 G4 G5 FE Exponential BS FE Bilinear BS (g) Load level = 70% of maximum loading Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 64 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) G1 G2 G3 G4 G5 FE Exponential BS FE Bilinear BS (h) Load level = 80% of maximum loading 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) G1 G2 G3 G4 G5 FE Exponential BS FE Bilinear BS (i) Load level = 90% of maximum loading Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 65 0 2500 5000 7500 10000 12500 15000 0 50 100 150 200 250 Distance along bond length (mm) Strain of FRP (µ µ µ µm/m) G1 G2 G3 G4 G5 FE Exponential BS FE Bilinear BS (j) Load level = 100% of maximum loading Figure 40 Comparison of strain results of GFRP specimens (with mean value of Egfrp tgfrp assiged) Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 66 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 67 5. PARAMETRIC STUDY OF BOND-SLIP BEHAVIOUR OF FRP-TO-BRICK In the previous chapter, it is shown that the results obtained from the analytical solution compared very well with the finite element results in term of both maximum load and deflection. In order to examine the effect of different parameters in the maximum load and deflection of FRP-to-brick joint, the analytical solution which is proposed by Yaun et al. (2005) is applied for carrying out parametric study and the results are discussed in this chapter. 5.1 Effect of bond length of FRP to the capacity and deflection of FRP-to-brick joint In the parametric study, the material properties for the brick and FRP are the same as those used in Chapter 4, i.e. the stiffness of brick (Ebrick tbrick) is 443,025 N/mm, the stiffness of CFRP (Ecfrp tcfrp) and GFRP (Egfrp tgfrp) is 46,260 N/mm and 18,673 N/mm, respectively. Those stiffnesses of FRP are the average values obtained from the test. The width of FRP is 50 mm while the width of the brick is 120 mm. The bilinear bond-slip curve and the interfacial fracture energy with the corresponding data assigned for CFRP and GFRP specimens according to Table 3 are used in the analytical solution. With the given material properties, geometry and bilinear bond-slip curve, the total load (which is the load applied on the FRP multiplied by 2) versus the deflection of FRP (without considering the deflection due to the unbonded length) are obtained for CFRP and GFRP specimens with four different bond length of FRP (80 mm, 120 mm, 160 mm and 200 mm), respectively. Finite element analysis was also carried out to obtain the load versus deflection curve of specimens with the mentioned four bond length. The finite element models which were used in previous study with FRP stiffness assigned as the mean value of the test results were used in the parametric studies. Bilinear bond-slip equations was assigned to the interface elements according to the given bond length. Comparison of the load versus deflection curves obtained from the finite element analysis and the analytical solution for CFRP and GFRP specimens are shown in Figs. 41 and 42, respectively. As it is shown from the figures that, in general, the results obtained from the analytical solution were stiffer than that obtained from the finite element analysis. Also, the maximum deflections predicted by the analytical solution were a bit smaller than that obtained from the finite element analysis. For the CFRP specimens, when the bond length is shorter, a higher maximum load was obtained from the finite element analysis compared to the analytical solution. As the bond length increases, the different between the maximum load obtained from the finite element analysis to the analytical solution becomes less. Nevertheless, the results obtained from the analytical solution were compared well to the finite element results in term of both maximum load and maximum deflection and the analytical solution seems to provide conservative results. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 68 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 0 5000 10000 15000 20000 25000 30000 35000 40000 0.00 0.40 0.80 1.20 1.60 Deflection (mm) Load (N) (Et)cfrp = 46260 N/mm FE results L = 80 mm L = 120 mm L = 160 mm L = 200 mm b frp / b brick = 0.42 Figure 41 Load versus deflection curve of CFRP specimens with different bond length (Analytical versus finite element results) 0 5000 10000 15000 20000 25000 30000 0.00 0.40 0.80 1.20 1.60 2.00 2.40 2.80 Deflection (mm) Load (N) (Et)gfrp = 18673 N/mm FE results L = 80 mm L = 120 mm L = 160 mm L = 200 mm b frp / b brick = 0.42 Figure 42 Load versus deflection curve of GFRP specimens with different bond length (Analytical versus finite element results) Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 69 The corresponding analytical results of CFRP and GFRP specimens with different bond length are shown in Fig. 43 (Noted that the symbol of stiffnesses of FRP is Ep tp = Efrp tfrp and brick is Es ts = Ebrick tbrick, respectively). As it is shown in the figure that increasing the bond length does not have significant effect on the maximum load. It is because that the bond length of 80 mm is already longer than the minimum bond length requirement for both specimens. The minimum required bond length can be obtained from the following equations (Yuan et al. 2004). )atan( )atan( ln 2 1 aL 221 221 1 eλλ−λ λλ+λ λ += (28) where        + τ =λ sss p pp0 max 1tEb b tE 1 s        + − τ =λ sss p pp0f max 2tEb b tE 1 )ss(        − λ = f 0f 2s ss 97.0arcsin 1 a According to Equation 28, the corresponding minimum bond length for CFRP and GFRP specimens are 70.13 mm and 59.41 mm, respectively. When the bond length is less than the minimum required bond length, the maximum load capacity of the joint can not be achieved. However, increasing the bond length has significant effect on the maximum deflection of the joint. As it is shown in Fig. 43 that for GFRP specimen, increasing the bond length from 80 mm to 200 mm (2.5 times) results in an increase of maximum deflection by about 3 times. Similar behaviour is observed for the CFRP specimen which has a higher FRP stiffness. Summary of maximum deflection versus bond length is shown in Table. 12 and the corresponding plot is shown in Fig. 44. It is shown from the figure that the maximum deflection increases almost linearly with respect to the bond length. However, the increasing rate is higher for specimen with lower FRP stiffness. Therefore, in order to achieve a certain amount of maximum of deflection, longer bond length should be applied for FRP with higher stiffness. It is also shown that higher maximum load and smaller maximum deflection are obtained for specimen with higher FRP stiffness. These results are consistent with the results obtained from test. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 70 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Table 12 Maximum deflection versus bond length (bfrp / bbrick = 0.42) bfrp / bbrick = 0.42 Maximum Deflection (mm) Bond length (mm) 80 120 160 200 CFRP specimen 0.44 0.75 1.06 1.38 GFRP specimen 0.84 1.41 1.97 2.54 0 5000 10000 15000 20000 25000 30000 35000 40000 0.00 0.40 0.80 1.20 1.60 2.00 2.40 2.80 Deflection (mm) Load (N) (Et)cfrp = 46260 N/mm (Et)gfrp = 18673 N/mm L = 80 mm L = 120 mm L = 160 mm L = 200 mm L = 80 mm L = 120 mm L = 160 mm L = 200 mm b frp / b brick = 0.42 Figure 43 Analytical load versus deflection curve of CFRP and GFRP specimens with different bond length Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 71 y = 0.008x - 0.188 R 2 = 1.000 y = 0.014x - 0.291 R 2 = 1.000 0 0.5 1 1.5 2 2.5 3 50 75 100 125 150 175 200 Bond length (mm) Max. Deflection (mm) CFRP specimens GFRP specimens Linear (CFRP specimens) Linear (GFRP specimens) b frp / b brick = 0.42 Figure 44 Maximum deflection versus bond length (bfrp / bbrick = 0.42) 5.2 Effect of width ratio of FRP-to-brick to the capacity and deflection of FRP-to-brick joint In the analytical solution, the width ratio of FRP to brick is also considered. In the experimental study and the previous analytical study, a fixed width ratio of FRP to brick of 0.42 is used. In order to examine the effect of the width ratio of FRP-to-brick to the capacity and deflection of the FRP-to-brick joint, two more width ratio of FRP-to-brick (i.e. bfrp / bbrick = 0.042 and 1.00) were studied by applying the analytical solution. The width ratios are achieved by keeping the width of brick as 120 mm while the width of FRP were changed to 5 mm and 120 mm, respectively. The results of load per unit width of FRP versus the deflection of FRP are shown in Figs. 44 and 46 for CFRP and GFRP specimens, respectively. As it is shown from the figures that when the width ratio is close to zero (i.e. the width of the FRP is very small compared to the width of the brick), the load capacity per unit width of FRP is the highest. When the width ratio is closed to unity (i.e. the width of the FRP is the same as the width of the brick), the load capacity per unit width of FRP is the lowest. Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme 78 ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS Lorenzis, L. De., Miller, B., and Nanni, A., Bond of fiber-reinforced polymer laminates to concrete, ACI Materials Journal, 2001, 98(3): 256-264 Lu, X. Z., Teng, J. G., Ye, L. P. and Jiang, J. J., Bond-slip models for FRP sheets/plates bonded to concrete, Engineering Structures, 2005, 27: 920-937 Monti M, Renzelli M, Luciani P., FRP adhesion in uncracked and cracked concrete zones. In: Proc. of 6th international symposium on FRP reinforcement for concrete structures. Singapore: World Scientific Publications; 2003, 183-92 Nakaba, K., Kanakubo, T., Furuta, T., and Yoshizawa, H., Bond behaviour between fiber-reinforced polymer laminates and concrete, ACI Structures Journal, 2001, 98(3): 359-167 Neubauer U, Rostasy FS., Bond failure of concrete fiber reinforced polymer plates at inclined cracks-experiments and fracture mechanics model. In: Proc. of 4th international symposium on fiber reinforced polymer reinforcement for reinforced concrete structures, SP-188, Farmington Hills (MI): ACI; 1999, 369-82 Panizza, M, Garbin, E., Valluzzi, M. R. and Modena, C., Bond behaviour of CFRP and GFRP Laminates on Brick Masonry, Department of Structural and Transportation Engineering, University of Padova, Italy, 2009 Sanchez, I. B., Lourenco, P., Oliverira, D. V. and Milian, A. G. A., Strengthening of arched masonry structures with composite materials, PhD thesis, University of Minho, Department of Civil Engineering, Portugal, 2007 Sato, Y., Asano, Y., and Ueda, T., Fundamental study on bond mechanism of carbon fiber sheet, Concrete Library International, JSCE, 2001, No. 37: 97-115 Savioa M, Farracuti B, Mazzotti D., Non-linear bond-slip law for FRP-concrete interface. In: Proc. of 6th international symposium on FRP reinforcement for concrete structures. Singapore: World Scientific Publications; 2003, 163-72 Sumon, S. K., Repair and strengthening of a damaged arch with built-in ring separation, Proceedings of the 7th International conference on structural faults and repair, Edinburgh: Engineering Technics Press, 1997: 69 - 75 Vekey, R. C., Brickwork and blockwork, Construction materials, Their nature and behaviour, E & FN Spon, ISBN 0-419-15470-1, 1998: 251-315 Wu, Z. S., Yuan, H., Niu, H., Stress transfer and fracture propagation in different kinds of adhesive joints. Journal of Engineering Mechanics, ASCE, 2002, 128(5): 562-273 Wu, Z., and Yin, J., Numerical analysis on interfacial fracture mechanism of externally FRP-strengthened structural members, Journal of Materials, Concrete Structures and Pavements, JSCE, 2002, 704(55): 257-270 Finite element study of bond-slip behaviour of CFRP and GFRP Laminates on Brick Masonry Erasmus Mundus Programme ADVANCED MASTERS IN STRUCTURAL ANALYSIS OF MONUMENTS AND HISTORICAL CONSTRUCTIONS 79 Yuan, H., Wu, Z., and Yoshizawa, H., Theoretical solutions on interfacial stress transfer of externally bonded steel/composite plates, Journal of Structural Mechanics and Earthquake Engineering, JSCE, 2001, 18(1): 27-39 Yuen, H., Teng, J. G., Seracino, R., Wu, Z. S. and Yao, J., Full-range behavior of FRP-to-concrete bonded joints, Engineering Structures, 2004, 26: 553-565