FRP reinforcement of stone arch bridges: Unilateral contact models and limit analysis

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1 Composites: Part B 38 (2007) FRP reinforcement of stone arch bridges: Unilateral contact models and limit analysis G.A. Drosopoulos a, G.E. Stavroulakis b,c, *, C.V. Massalas a a Department of Material Science and Technology, University of Ioannina, GR Ioannina, Greece b Department of Production Engineering and Management, Technical University of Crete, GR Chania, Greece c Department of Civil Engineering, Technical University of Braunschweig, Germany Received 23 January 2006; received in revised form 20 July 2006; accepted 9 August 2006 Available online 24 October 2006 Abstract A method for the estimation of the limit load and the failure mode of fiber-reinforced polymer (FRP) reinforced stone arch bridges is hereby presented. Unilateral contact interfaces with friction simulating potential cracks are considered in the finite element model of the bridge. FRP strips are then applied to the intrados and/or the extrados of the arch. The possible failure modes of the reinforced structure are sliding of the masonry, crushing, debonding of the reinforcement and FRP rupture. Identical failure modes arise from the computer simulation and from experiments on reinforced arches published in the literature. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: A. Layered structures; B. Delamination; B. Interface/interphase; C. Finite element analysis (FEA) 1. Introduction * Corresponding author. Address: Department of Production Engineering and Management, Technical University of Crete, GR Chania, Greece. Tel.: ; fax: addresses: me01122@cc.uoi.gr (G.A. Drosopoulos), gestavr@ dpem.tuc.gr (G.E. Stavroulakis), cmasalas@cc.uoi.gr (C.V. Massalas). FRPs have initially been proposed for the reinforcement of concrete structures. The implementation of those materials on masonry structures has been studied in the last years both experimentally and analytically, as well [1 7]. FRP is made of a polymeric matrix with different kind of fibers (glass, carbon, etc.). As a strengthening material, it presents a number of advantages such as high tensile strength, negligible self-weight and corrosion resistance. On the other hand, fibers have brittle behavior while the irregularity of the masonry surface may lead to a poor bond between the fibers and the masonry, thus to a negligible strengthening effect. In experimental research conducted in the past on FRP reinforced masonry, the reinforced arch failed due to sliding of the masonry, crushing, debonding of the reinforcement and FRP rupture [1,2]. Identical failure modes have been obtained in this study, from the finite element analysis of the reinforced arch. A stone arch bridge consists of stone blocks and mortar joints. Blocks have high strength in compression and low strength in tension while mortar has generally low strength. Thus, a safe assumption of a no tension material can be adopted at least for the purpose of limit analysis. To simulate this behavior and the ultimate load of the arch, a discrete model has been developed. In particular, the elastic model of the bridge is divided into a number of interfaces perpendicular to the center line of the arch. Those interfaces are uniformly distributed along the length of the arch. A large number of interfaces is used (e.g. forty interfaces) in order to achieve a satisfactory simulation of the behavior of the masonry structure [8]. Opening or sliding of the interfaces denotes crack initiation. Other studies which are related with the investigation of the behavior of masonry arches, as well as no-tension materials like masonry, are reported in [9 12]. Unilateral contact law governs the behavior in the normal direction of an interface, indicating that no tension /$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi: /j.compositesb

2 G.A. Drosopoulos et al. / Composites: Part B 38 (2007) forces can be transmitted in this direction and a gap may appear if the stresses in this direction become zero. For the behavior in the tangential direction it is taken into account that sliding may or may not occur, by the usage of the Coulomb friction model. The either-or decisions incorporated in the unilateral contact and the friction mechanisms make the whole mechanical model highly nonlinear. Due to the presence of non-differentiable functions within these models, they are characterized as nonsmooth mechanics models. For practical applications it is important to use carefully tuned path-following iterative techniques for the reliable numerical solution of the problem. Furthermore, the limit analysis problem is related to the solvability of the underlying mechanical problem using analogous theoretical results concerning the solvability of variational inequalities and complementarity problems and applications on elastoplasticity [13]. On the top of the model described above, three types of reinforcements have been tested. FRP is placed on the whole length of the extrados, on the whole length of the intrados and both on the extrados and the intrados of the arch. FRP layers are modelled with two-dimensional elasticity elements. For the masonry-frp interface the unilateral contact-friction model is also adopted. Various models are additionally considered in the interface, including fixed or elastic bonding. The proposed model is able to predict the ultimate load and the failure mode of the unreinforced arch. This structure collapses by the development of four hinges in the arch, due to the small tensile resistant of the masonry [14,8]. According to the main idea of this study, the mechanical behavior of the structure can be improved if a thin layer of reinforcement is applied to the model, which was initially developed for the unreinforced arch. The material of the reinforcement will provide tensile resistant to the structure. As a result, the new model will be capable of predicting the alteration in the behavior of the reinforced structure. In particular, the failure mode will be changed and the reinforced arch will collapse due to the sliding or the compressive failure of the masonry, the rupture of the FRP and the debonding of the reinforcement. These failure modes are clearly depicted by the proposed model of the reinforced structure, as it will be explained in the following sections. Identical failure modes, have also been obtained by experimental research on FRP reinforced stone arches. 2. Failure of the masonry arch 2.1. Elements of the frictional-contact mechanics The behavior in the normal direction of an interface is described by the unilateral contact model. In particular, let us consider the boundary of an elastic body which comes in contact with a rigid wall. Let u be the single degree of freedom of the system, g be the initial opening and t n be the corresponding contact pressure in case contact occurs. The basic unilateral contact law is described by the set of inequalities (1), (2) and by the complementarity relation (3), [15 17] h ¼ u g 6 0 ) h 6 0 t n P 0 t n ðu gþ ¼0: Inequality (1) represents the nonpenetration relation, relation (2) implements the requirement that only compressive stresses (contact pressures) are allowed and Eq. (3) is the complementarity relation according to which either separation with zero contact stress occurs or contact is realized with possibly non-zero contact stress. The behavior in the tangential direction is defined by a static version of the Coulomb friction model. Two contacting surfaces start sliding when the shear stress in the interface reaches a maximum critical value equal to t t ¼ s cr ¼ljt n j ð4þ where t t, t n are the shear stress and the contact pressure at a given point of the contacting surfaces respectively and l is the friction coefficient. There are two possible directions of sliding along an interface, so t t can be positive or negative depending on that direction. Furthermore, there is no sliding if t t <l t n (stick conditions). The stick-slip relations of the frictional mechanism can be mathematically described with two sets of inequalities and complementarity relations, similar to (1) (3), by using appropriate slack variables [18,15] Formulation and solution of the unilateral contact-friction problem For the frictional-contact problem the virtual work equation is written in a general form s : ddv ¼ du tds þ du f dv þ du t n ds 0 V S V S 0 þ du t t ds 0 ð5þ S 0 where t n and t t are the normal and tangential traction vectors on the actual contact boundary S 0, s is the stress tensor, d is the virtual strain tensor, du is the virtual displacement vector and t, f are the surface and body force vectors, respectively. The nonlinearity in the unilateral contact problem is introduced by the variational inequality [16] du t n 6 0 ð6þ which represents the principle of virtual work in a variational form. The contact constraint is enforced with Lagrange multipliers representing the contact pressures. The nonlinearity in the frictional problem is introduced by the variational inequality du t t t 6 maxðdu t t t cr ; du t t t cr Þ ð7þ ð1þ ð2þ ð3þ

3 146 G.A. Drosopoulos et al. / Composites: Part B 38 (2007) where t t cr is the vector of the critical shear stresses s cr, inthe tangential direction of the interfaces. Relation (7) implies that no slip occurs when j t t j < s cr = ljt n j while slip starts when t t = s cr. Lagrange multipliers are also used in the principle of the virtual work to enforce sticking conditions. The set of the nonlinear equations is solved by the Newton Raphson incremental iterative procedure. The frictionless unilateral contact problem can be written in matrix form as follows Ku þ N T r ¼ P o þ kp Nu g 6 0 r P 0 ðnu gþ T r ¼ 0: ð8aþ ð8bþ ð8cþ ð8dþ Eq. (8a) expresses the equilibrium equations of the unilateral contact problem, where for simplicity frictional terms are omitted. K is the stiffness matrix and u is the displacement vector. P o denotes the self-weight of the structure and P represents the concentrated live load. N is an appropriate geometric transformation matrix and vector g contains the initial gaps for the description of the unilateral contact joints. Relations (8b), (8c), (8d) represent the constraints of the unilateral contact problem for the whole discretized structure and are based on the local description given by relations (1) (3). The enforcement of the constraints is achieved by using Lagrange multipliers. Thus, r is the vector of Lagrange multipliers corresponding to the inequality constraints and is equal to the corresponding contact pressure ( t n ). The problem described above is a nonsmooth parametric linear complementarity problem (LCP) [19] parametrized by the one-dimensional load parameter k. All required quantities can be calculated by using finite element techniques. Using path-following the solution of the problem can be calculated in the interval 0 6 k 6 k failure, where k failure is the value of the loading factor for which the unilateral contact problem does not have a solution. This is the limit analysis load Compressive failure of the arch For the compressive failure of the arch the Drucker Prager plasticity model, in which a cap yield surface has been added, is used (Fig. 1). The addition of the cap yield surface to the Drucker Prager shear failure surface bounds the yield surface in hydrostatic compression, thus providing an inelastic hardening mechanism to represent plastic compaction. The Drucker Prager shear failure surface is written as F s ¼ t p tan b d ¼ 0 ð9þ where b and d represent the angle of internal friction and the cohesion of the material respectively, t is a deviatoric stress measure and p is the equivalent pressure stress. The cap yield surface is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 F c ¼ ðp p a Þ 2 Rt þ Rðd þ p 1 þ a a= cos b a tan bþ ¼ 0 ð10þ where R is a material parameter that controls the shape of the cap, a is a small number used to define a transition yield surface for a smooth intersection between the cap and shear failure surfaces (for the sake of simplicity the transition surface is not written explicitly here) and p a is an evolution parameter, which is obtained by substituting (p, t) with (p b,0) in Eq. (10) p a ¼ p b Rd 1 þ R tan b : ð11þ In relation (11), p b represents the hydrostatic compression yield stress and is defined by a bilinear hardening law (with an almost horizontal branch), relating p b with the plastic strain. The model described above uses associated flow rule in the cap region and nonassociated flow in the shear failure and transition regions. However, yielding in the shear (and in the transition) surface is prevented because the contact-friction model stands for the shear failure of the masonry arch. This is implemented by giving high enough values in the parameters that mainly define the shape of the shear failure surface (b and d), see Fig The reinforced arch 3.1. Failure modes of the reinforced arch Failure occurs by sliding of the masonry at the springing of the bridge and at the point of external (concentrated) Fig. 1. Drucker Prager cap model plasticity (a) in the p t plane (b) in the deviatoric plane.

4 G.A. Drosopoulos et al. / Composites: Part B 38 (2007) loading if the reinforcement is attached on the whole length of the extrados [1,2]. Debonding of the FRP at the intrados may also lead to failure. An FRP strip epoxy-bonded onto a curved surface, gives rise to stress concentrations normal to the boundary of the bonding masonry [1]. These stresses exist due to the curvature of the arch, and, in fact, are proportional to the curvature. For reinforcement placed on the intrados these stresses are tensile for the masonry-frp interface, denoting that debonding is an issue of concern. These previously mentioned normal stresses are also proportional to the tensile force of the strip [1] indicating that they mainly arise on the bricks adjacent to cracks (i.e. when a crack is about to open on a brick, the strip bonded to this brick develops tensile forces). As a result, debonding occurs close to the point of application of the load or more generally in areas of the intrados where a hinge of the collapse mechanism tends to open. In some experiments debonding of the FRP is accompanied either with ripping of a layer of a stone close to the reinforcement (local tensile failure of the masonry) or with pull-out of a whole block. In both cases debonding is related with the masonry, not the FRP, due to the fact that the epoxy resin is stronger than the masonry under tension, under the assumption that the reinforcement has been properly applied. However, debonding can be related to the FRP as well, if the masonry-frp connection is not strong enough. In addition, some experiments have shown that a debonding failure mode can be accompanied with the four hinges collapse mechanism [1] introduced by Heyman [20]. A brief description of the four hinges failure mechanism is given later in this study. Simplified relations and relevant information concerning detachment of the FRP, are presented in [1,2]. In case that no debonding is permitted in an arch reinforced at the whole length of the intrados, collapse occurs by compressive failure of the masonry [1]. If this is the case, the limit load is considerably increased. Finally, if the reinforcement includes interior FRP, the structure is possible to fail by the four hinges mechanism but the positions of the hinges can be different in comparison with the collapse mechanism of the unreinforced arch [1] FRP reinforcement The failure modes of the reinforced arch obtained by experimental research, have been mentioned in the previous section. The proposed model predicts identical failure modes. In particular, if the FRP is applied to the whole length of the extrados of the arch, experiments show that the structure will collapse due to sliding of the masonry. This possibility is taken into account in the framework of the proposed model, by the usage of the Coulomb friction law in the tangential direction of the interfaces of the arch. If the reinforcement is applied to the whole length of the intrados and there is a strong connection between the masonry and the FRP, then the structure collapses due to compressive failure of the arch. This type of failure is represented in the proposed model, by the usage of the previously described Drucker Prager plasticity model, in which a cap yield surface has been added in order to put a limit on the hydrostatic pressure. If the reinforcement is applied to the intrados of the arch and the connection between the FRP and the masonry is not strong enough, experiments demonstrate that debonding of the reinforcement is possible to appear. In this case the failure is caused by the four hinges mechanism similar to the unreinforced structure. The behavior of the masonry-frp interface is simulated by the contact-friction law for all types of reinforcement. Nonlinear springs are also used in this interface, in order to depict debonding of the FRP which appears in the case of intrados reinforcement. These springs provide a finite (small) tensile resistant in the normal direction of the interface. When this limit is reached, debonding of the reinforcement takes place. If the ultimate load of the reinforced structure is considerably increased, FRP rupture may occur. Failure of the FRP is represented by the proposed model, with the von Mises yield criterion and a perfect plastic stress-plastic strain law. In all the above cases of reinforcement, the behavior of the masonry-fpr interface significantly affects the result of the reinforcement. The proposed model is able to reproduce situations in which the interface has not been adequately prepared before the application of the FRP (e.g. there is a low bond strength). As it has already been mentioned, the contact-friction model is applied in the masonry-frp interface, for all types of reinforcement. If the reinforcement is applied to the intrados of the arch and the connection between the FRP and the masonry is not strong enough, it is possible to occur debonding of the FRP. This weak connection is simulated in the proposed model with nonlinear springs, which offer a small tensile resistant in the normal direction of the masonry-frp interface. When this strength is reached, debonding of the reinforcement, together with premature failure take place. The proposed model is capable of simulating another case of premature failure of the reinforcement, which is related with the sliding of the FRP in the tangential direction of the masonry-fpr interface. Sliding of the FRP is modelled with the Coulomb friction law and a small friction coefficient. It is noted, that conditions which do not permit sliding nor debonding of the FRP from the masonry can also be represented by the proposed model, by applying sticking conditions (without a bound, contrary to the Eq. (4)). In the following sections some applications on the reinforcement of stone arches are presented. Three types of FRP reinforcement are applied on the masonry. The FRP is attached to the whole length of the extrados, to the whole length of the intrados and both to the extrados and the intrados of the arch. The mechanical properties of both the masonry and the reinforcement are given in Table 1.

5 148 G.A. Drosopoulos et al. / Composites: Part B 38 (2007) Table 1 List of the material properties for both the masonry and the reinforcement Masonry Young s modulus 5 GPa Poisson s ratio 0.3 Density 2200 Kg/m 3 Compressive yield stress 10 MPa Width 1 m FRP Reinforcement Young s modulus 50 GPa Poisson s ratio 0.3 Thickness 1 mm Tensile yield stress 5 GPa Width 0.4 m 4. A brief description of Heyman s method The classical collapse mechanism method of Heyman [20], has been proposed for the determination of the load carrying capacity of stone arch bridges. It is based on the estimation of the thrust line (a funicular polygon which defines the resultant force in a cross-section of the arch). When the thrust line in a cross-section is adjacent to the ring of the arch, the eccentricity e of the normal force P N, from the center line of the arch at a given cross-section, becomes maximum. As a result, a bending moment M equal to P N e is developed around the center line of the arch and a hinge opens, on the assumption that the arch does not develop any tensile strength. Since a three-pin arch is a statically determinate structure, opening of a fourth hinge converts the structure into a mechanism and collapse occurs. Therefore, the four hinges collapse mechanism is the collapse mode for an unreinforced arch loaded with a vertical concentrated force in the quarter-span of the arch. The same loading position (together with the self-weight) is considered both in this study and in the experimental research on stone arches referred herein, as the quarterspan is probably the worst position of the live load according to [20]. In addition, no compressive failure for the unreinforced masonry is usually expected [20]. The geometry of the arch, in which the proposed model is applied, and the four hinges collapse mechanism if no FRP reinforcement is attached to this structure, are shown in Fig. 2. The failure load is equal to KN (no crushing of the masonry occurs). A list of the material properties for both the masonry and the FRP is represented in Table 1. It is noted that the mechanical properties of the reinforcement are similar to the ones reported for a carbon fiber-reinforced polymer (CFRP) in [21]. The finite element analysis model consists of quadrilateral plane strain elements. The arch is considered fixed to the ground and large displacement effects have been neglected FRP attached to the whole extrados Experimental work has shown that the structure collapses due to masonry s sliding at the springing and at the point of concentrated loading. The same conclusion arises from the present study (Fig. 3). While no sliding of the unreinforced arch occurs for a friction coefficient equal to 0.6 (Fig. 2(b)), sliding of the masonry is the failure mode of the reinforced structure for the same friction coefficient. The ultimate failure load is equal to KN, about 6 times greater than the one received for the arch without FRP reinforcement. After initiation of sliding in the masonry, FRP is yielding at the areas of the model where sliding of the masonry takes place, especially at the point of loading, while no compressive failure of the masonry occurs FRP attached to the whole intrados As described above, crushing of the masonry is the failure mode when no debonding of the FRP can be developed in the FRP-masonry interface. But if the FRP-masonry connection is not strong enough, detachment of the reinforcement usually together with a collapse mechanism on the body of the arch, appears to be the most possible failure mode. Both collapse modes can be reproduced by the proposed model. Crushing takes place at a considerably large ultimate load. According to the numerical results, compressive yielding of the masonry occurs in places where a hinge of 5. Results Fig. 3. Failure mode of the arch reinforced at the whole extrados. Fig. 2. (a) Geometry (m) of a forty interfaces contact-friction model and (b) four hinges collapse mechanism and failure load for the unreinforced arch.

6 G.A. Drosopoulos et al. / Composites: Part B 38 (2007) Fig. 4. (a) Compressive failure mode of the arch reinforced at the whole intrados (no debonding in the masonry-frp interface) and (b) Force displacement diagrams for FRP attached to the whole extrados and the whole intrados. the collapse mechanism of the unreinforced structure opens, in the boundary of the ring opposite to the crack. FRP yielding takes place near the point of concentrated loading as well as in the springing opposite to the loading. At those places a crack of the unreinforced arch would open at the intrados, thus the reinforcement develops tensile forces. In Fig. 4(a) crushing failure mode is shown. Grey color represents yielding of the masonry. In addition, the force displacements diagrams presented in Fig. 4(b) indicate that FRP reinforcement attached to the intrados is more effective in comparison with exterior FRP reinforcement, on the assumption that no debonding of the FRP is permitted. Sliding of the masonry, which is obtained for the case of exterior FRP placement, can be an explanation for this. In order to obtain the ultimate load of the structure when debonding of the FRP can be developed, nonlinear springs are used to model the behavior of the masonry- FRP interface. A bilinear force displacement diagram is used for those springs, with an initial stiffness equal to 10 4 KN/m. This relatively small value of the springs stiffness represents a weak FRP-masonry connection. After a very small deformation the diagram becomes horizontal indicating that the contribution of the springs in the FRP-masonry connection is minimized. In addition, FRP is considered either fixed to the springings or connected with springs to them. From the results obtained in the present study, it is clear that when the FRP is considered fixed to the springings, detachment occurs both near the point of application of the concentrated loading and at the springing opposite to the concentrated load (Fig. 5(a)). The failure load is then equal to KN, very close to the one received for the unreinforced arch (Fig. 2). If the FRP is connected with springs to the springings, detachment of the FRP takes place only near the point Fig. 6. Force displacement diagrams of the arch reinforced at the whole intrados in case debonding of the FRP is permitted. of loading while a small sliding of the FRP occurs in the springing opposite to the load (Fig. 5(b)). The failure load is equal to KN. In both cases, the four hinges collapse mechanism arise. The corresponding force displacements diagrams are shown in Fig. 6. In case that debonding of the FRP takes place, the behavior of the reinforced arch is similar to the unreinforced structure (only a small increase of the failure load occurs, see also [6]). As a result, neither FRP yielding nor crushing of the masonry arise when debonding occurs FRP attached both to the intrados and the extrados of the arch In this section the FRP is assumed to be partially attached both to the intrados and the extrados of the arch, around the positions where a hinge of the collapse mechanism would open in the unreinforced arch. Similar to the previous case, the ultimate load and the collapse mechanism are influenced by the masonry-frp connection at the intrados. If this connection is strong enough and no debonding of the FRP is permitted at the intrados, the failure load is equal to KN, a value which is about 5 times greater than the one of the Fig. 5. Failure mode of the arch reinforced at the whole intrados when debonding of the FRP is permitted (a) FRP fixed in the springings and (b) FRP connected with springs in the springings.

7 150 G.A. Drosopoulos et al. / Composites: Part B 38 (2007) Fig. 7. Failure mode of the arch reinforced both at the intrados and the extrados (a) No debonding in the masonry-frp interface and (b) Debonding for FRP fixed in the springings. Fig. 8. Force displacement diagrams of the arch reinforced both at the intrados and the extrados. unreinforced structure. The arch fails by the four hinge collapse mechanism, but the position of the hinges differs from the case of the unreinforced bridge (see Figs. 2(b) and 7(a)). The hinges, both at the intrados and the extrados, open near the boundary of the FRP reinforcement strip, thus there is an offset of the hinges position. The increase of the limit load can be attributed to this offset. A significant reduction of the failure load occurs if debonding of the FRP at the intrados can be developed. In order to obtain debonding, nonlinear springs are used in the masonry-frp interface at the intrados. The same force displacement diagram described in the previous section for the springs, is used here. In addition, FRP is considered fixed to the springings. The arch fails by the four hinge collapse mechanism at KN (Fig. 7(b)) indicating premature failure of the reinforcement. The corresponding force displacement diagrams of the arch reinforced both at the intrados and the extrados, are shown in Fig Conclusions In the present study a numerical investigation of the ultimate load of an FRP reinforced masonry arch has been presented. A unilateral contact-friction finite element model is proposed for the simulation of the arch and is extended to cover the behavior of the reinforced structure. The model is able to predict the failure modes of the reinforced arch such as sliding of the masonry, debonding of the FRP and compressive failure of the masonry. Identical failure modes arise from experimental research on stone arches. Unilateral contact interfaces are used for the simulation of the arch. A contact law is applied in the normal direction of the interfaces, indicating that masonry has a zero tensile resistant in the normal direction of each interface. The Coulomb friction law (with a friction coefficient equal to 0.6), which is applied on the tangential direction of the interfaces, models the sliding failure mode for the masonry. The reinforcement is added in the model of the unreinforced arch. It has a very small thickness (equal to 1 mm) and is simulated by finite elements of two-dimensional elasticity. The mechanical properties of the reinforcement are similar to the ones reported for a carbon fiber-reinforced polymer (CFRP). The interface between the masonry and the FRP is also simulated by the contact-friction law. Conditions which do not permit sliding nor debonding of the FRP from the masonry can be included in the law of the masonry-frp interface, simulating a high bond strength between the reinforcement and the masonry. Sliding or debonding of the FRP may also be represented by this interface, indicating the existence of a weak connection between the two materials. If the reinforcement is applied on the intrados of the arch and the connection between the FRP and the masonry is not strong enough, debonding of the FRP is possible to occur. This weak connection is simulated in the proposed model with nonlinear springs, which are added in the masonry-frp interface and offer a small tensile resistant in the normal direction of this interface. When the strength of the springs is reached, debonding of the reinforcement, together with premature failure take place. Sliding of the FRP can also be represented by the usage of the Coulomb friction law in the masonry- FRP interface. Three types of reinforcement are applied to the masonry. FRP is attached to the whole length of the extrados, to the whole length of the intrados and both to the extrados and the intrados of the arch, in areas where a hinge of the unreinforced structure would be developed. Concerning interior reinforcement, debonding of the FRP proves to be very crucial for the behavior of the reinforced arch, as the ultimate load is very close to the one received for the unreinforced structure. The exterior reinforced arch collapses due to sliding of the masonry both at the point of loading and at the springing opposite loading. This can be prevented by applying a proper amount of reinforcement in those areas. Supposing that no debonding of the FRP is permitted, full interior reinforcement seems to result in a higher ultimate load in comparison with full exterior reinforcement. Moreover, compressive failure of the masonry occurs, if FRP is applied to the full length of the intrados and no debonding is permitted. For reinforcement attached

8 G.A. Drosopoulos et al. / Composites: Part B 38 (2007) both to the intrados and the extrados, failure follows the classical four hinges collapse mechanism. In comparison with the unreinforced arch an offset in the hinges position is observed. A significantly increased ultimate load is attributed to this offset. Further work is needed for the assessment of the behavior of both the unreinforced and the reinforced structure in the direction normal to the plane of the arch, especially when horizontal forces simulating seismic activity are present. The development of three-dimensional models is left for further investigation. References [1] Foraboschi P. Strengthening of masonry arches with fiber-reinforced polymer strips. ASCE J Compos Constr 2004;8(3): [2] Valluzzi MR, Valdemarca M, Modena C. Behavior of brick masonry vaults strengthened by FRP laminates. ASCE J Compos Constr 2001;5(3): [3] Triantafillou TC. Strengthening of masonry structures using epoxybonded FRP laminates. ASCE J Compos Constr 1998;2(2): [4] Albert ML, Elwi AE, Cheng JJR. Strengthening of unreinforced masonry walls using FRPs. ASCE J Compos Constr 2001;5(2): [5] Kolsch H. Carbon fiber cement matrix (CFCM) overlay system for masonry strengthening. ASCE J Compos Constr 1998;2(2): [6] Ascione L, Feo L, Fraternali F. Load carrying capacity of 2D FRP/ strengthened masonry structures. Compos Part B Eng 2005;36: [7] ElGawady AM, Lestuzzi P, Badoux M. Aseismic retrofitting of unreinforced masonry walls using FRP. Compos Part B Eng 2006;37: [8] Drosopoulos GA, Stavroulakis GE, Massalas CV. Limit analysis of a single span masonry bridge with unilateral frictional contact interfaces. Eng Struct 2006;28: [9] Lofti H, Shing P. Interface model applied to fracture of masonry structures. ASCE J Struct Eng 1994;120: [10] Del Piero G. Limit analysis and no-tension materials. Int J Plasticity 1998;14: [11] Lucchesi M, Padovani C, Pasquinelli G, ani N. On the collapse of masonry arches. Meccanica 1997;32: [12] Giacquinta M, Giusti G. Researches on the equilibrium of masonry structures. Arch Ration Mech Anal 1985;88: [13] Stavroulakis GE, Panagiotopoulos PD, Al-Fahed AM. On the rigid body displacements and rotations in unilateral contact problems and applications. Comput Struct 1991;40: [14] Heyman J. The stone skeleton. Cambridge, UK: Cambridge University Press; [15] Mistakidis ES, Stavroulakis GE. Nonconvex optimization in mechanics. Dordrecht, The Netherlands: Kluwer Academic Publishers; [16] Panagiotopoulos PD. Inequality problems in mechanics and applications: Convex and nonconvex energy functions. Boston, Basel, Stuttgart: Birkhauser; [17] Stavroulaki ME, Stavroulakis GE. Unilateral contact applications using FEM software. Int J Appl Math Comput Sci 2002;12: [18] AL-Fahed AM, Stavroulakis GE, Panagiotopoulos PD. Hard and soft fingered robot grippers. The linear complementarity approach. AMM 1991;71: [19] Stavroulakis GE. Optimization for modeling of nonlinear interactions in mechanics. In: Pardalos PM, Resende GC, editors. Handbook of applied optimization. New York: Oxford University Press; p [20] Heyman J. The masonry arch. England: Ellis Horwood Series In Engineering Science; [21] Berthelot MJ. Composite materials: mechanical behavior and structural analysis. New York: Springer-Verlag; 1999.