Nonlinear Modelling of Curved Masonry Structures after Seismic Retrofit through FRP Reinforcing

Bartolomeo Pantò; Francesco Cannizzaro; Salvatore Caddemi; Ivo Caliò; César Chácara; Paulo B. Lourenço

doi:10.3390/buildings7030079

Nonlinear Modelling of Curved Masonry Structures after Seismic Retrofit through FRP Reinforcing

Pantò, Bartolomeo;Cannizzaro, Francesco;Caddemi, Salvatore;Caliò, Ivo;Chácara, César;Lourenço, Paulo B. 2017-08-29 00:00:00 buildings Article Nonlinear Modelling of Curved Masonry Structures after Seismic Retroﬁt through FRP Reinforcing 1 , 1 1 1 ID Bartolomeo Pantò *, Francesco Cannizzaro , Salvatore Caddemi , Ivo Caliò , 2 2 César Chácara and Paulo B. Lourenço Department of Civil Engineering and Architecture, University of Catania, 95125 Catania, Italy; francesco.cannizzaro@dica.unict.it (F.C.); scaddemi@dica.unict.it (S.C.); icalio@dica.unict.it (I.C.) Department of Civil Engineering, Institute for Sustainability and Innovation in Structural Engineering (ISISE), University of Minho, Azurem, 4800-058 Guimarães, Portugal; c.chacara@pucp.pe (C.C.); pbl@civil.uminho.pt (P.B.L.) * Correspondence: bpanto@dica.unict.it; Tel.: +39-095-738-2275 Received: 24 April 2017; Accepted: 21 August 2017; Published: 29 August 2017 Abstract: A reliable numerical evaluation of the nonlinear behaviour of historical masonry structures, before and after a seismic retroﬁtting, is a fundamental issue in the design of the structural retroﬁtting. Many strengthening techniques have been introduced aimed at improving the structural performance of existing structures that, if properly designed and applied, provide an effective contribution to the preservation of their cultural value. Among these strategies, the use of fabric-reinforced polymeric (FRP) materials on masonry surface is being widely adopted for practical engineering purposes. The application of strips or 2D grid composite layers is a low invasive and easy to apply retroﬁtting strategy, that is able to improve both the in-plane and the out of plane behaviour of masonry elements also in the presence of complex geometries thanks to their ﬂexibility. For this reason, these techniques are frequently employed for reinforcing masonry curved elements, such as arches and vaults. In this paper, taking advantage of an existing general framework based on a discrete element approach previously introduced by the authors, a discrete element conceived for modelling the interaction between masonry and FRP reinforcement is applied to different curved masonry vaults typologies. This model, already used for evaluating the nonlinear behaviour of masonry arches, is here employed for the ﬁrst time to evaluate the effectiveness of FRP reinforcements on double curvature elements. After a theoretical description of the proposed strategy, two applications relative to an arch and a dome, subjected to seismic loads, with different reinforced conditions, are presented. The beneﬁt provided by the application of FRP strips is also compared with that associated to traditional retroﬁtting techniques. A sensitivity study is performed with respect to the structure scale factor. Keywords: macro-model approach; monumental masonry structures; masonry arches and vaults; historical structural analyses; seismic assessment; cultural heritage protection; FRP-reinforcement; HiStrA software 1. Introduction The numerical simulation of the seismic response of historical masonry structures (HMS) is a key aspect of the cultural heritage preservation, and represents a challenging issue within the structural and earthquake engineering. Historical masonry structures represent a high percentage of existing buildings in several regions of the world, and their value is relevant both from the economic and social-cultural points of view. On the contrary, they suffer from scattered structural degradation, due to static loads, chemical and physic degradation of the materials, previous earthquakes, and wrong alterations of the original structural conception. Normally, they are not able to resist to earthquakes Buildings 2017, 7, 79; doi:10.3390/buildings7030079 www.mdpi.com/journal/buildings Buildings 2017, 7, 79 2 of 17 even if characterized by a low intensity. Recent seismic events occurred in Italy, such as the Central Italy Eartkquakes (2016), the Emilia (2012), and L’Aquila (2009); they produced severe damage patterns or the complete destruction of several historical sites. Such events, well documented in terms of post earthquake technical survey, demonstrated many critical aspects, which make vulnerable the historical structures to the horizontal and vertical seismic actions. One of the most important aspects is the presence of elements with a curved geometry such as arches and vaults, which interact with the vertical elements (walls or columns) during the earthquake motion, producing a signiﬁcant effect on the seismic response of the entire structure. Curved structures have been widely adopted in the past for building purposes, since their shape allows an effective transfer of the static vertical action to the walls, and induces compression along their span; for the latter reason curved shapes are still adopted and proposed for modern structures [1,2]. When it comes to single and double curvature masonry structures, their study is still an open problem debated in the literature [3]. Aimed at the reduction of the seismic vulnerability of HMS in presence of curved masonry elements, several retroﬁtting techniques based on reinforced composite materials, applied by means of polymeric (FRP) or cementitious (FRCM) matrix, have been introduced, and widely investigated by means of experimental tests and numerical simulations during the last years. The use of these techniques is getting more and more frequent in the retroﬁtting of historical and monumental masonry buildings since they consist of reversible and low invasive interventions; with regard to the design of FRP reinforcements, several proposals have been made [4,5]. On the other hand, there is a lack of fast and reliable numerical tools to assess the effectiveness of such techniques. In fact, even considering unreinforced masonry structures, the numerical simulation of their actual behaviour is still a very complex task within the computational structural mechanics ﬁeld. The main issue is related to the difﬁculty in providing reliable simulations of the high nonlinear degrading cyclic response of masonry. To this regard, a great variability of the mechanical characteristics is encountered, thus making difﬁcult the deﬁnition of a general constitutive law that is suitable for all masonry typologies. When it comes to retroﬁtting techniques, and in particular to ﬁbre-composite strengthened structures, a crucial aspect is related to the correct simulation of the tangential stress transferred from the reinforcement to the masonry substrate, and the relative failure collapse for tensile rupture of the textile or delamination of the reinforcement from the support. On this task, several contributions have been given by different authors and now are available in the literature [6–9], also in presence of curved support [10,11]. Recently, a 3D macro-model has been proposed for the non-linear seismic simulations of masonry structures aiming at a reduced computational effort when compared to the traditional ﬁnite element approaches. The main idea of the model was to use a 2D mechanical scheme, governed by unidirectional non-linear links, which, according to different typologies, have to reproduce the main masonry failure modes [12]. Such a model has been successfully employed in the simulation of laboratory tests [13] and in the seismic assessment of ordinary and mixed masonry buildings [14–16]. Subsequently, the model has been extended to catch the out-of-plane behaviour of masonry walls [17,18] and the behaviour of structures with a curved geometry [19–21]. In this paper this discrete macro-modelling approach is used to assess the seismic capacity of some typologies of masonry vaults commonly present in historical structures, before and after a reinforcing retroﬁt through composite materials. With this aim, a new non-linear model recently proposed in the literature [22], able to simulate the presence of FRP strips or 2D-webs and to grasp the interaction with the masonry support, also in presence of a curved substrate, is here employed. The model is able to simulate the debonding of the reinforcement from the masonry support due to tangential delamination or to normal tensile detachment, which represent the most probable collapse mechanisms of FRP reinforced structures. Different strips arrangements applied either to the intrados or to the extrados surfaces are considered in the paper. The efﬁciency of the FRP retroﬁtting is compared to the traditional technique of adding steel tie-rods in order to investigate the optimal solutions with respect to the retroﬁt design and Buildings 2017, 7, 79 3 of 17 to the use of composite material. The results of the numerical simulations relative to the arches, have been compared with those obtained through limit analysis approach, and applied to simpler models. Buildings 2017, 7, 79 3 of 17 The analytical results are used to validate the numerical results and are duly discussed providing a contribution towards the understanding of reinforced curved structures subjected to seismic actions. to the arches, have been compared with those obtained through limit analysis approach, and applied The results show that the composite reinforcements can produce a signiﬁcant increment of the seismic to simpler models. The analytical results are used to validate the numerical results and are duly capacity discussed in teprov rmsiding of both a cont strength ribution and towards ductility the , wit understand hout incr ing easing of reinforce the stifd fness curvof ed the strustr ctures uctur e. subjected to seismic actions. The results show that the composite reinforcements can produce a The employed macro-model is able to effectively grasp the behaviour of unreinforced and reinforced significant increment of the seismic capacity in terms of both strength and ductility, without (both with traditional and innovative techniques) curved masonry structures, as well as providing increasing the stiffness of the structure. The employed macro‐model is able to effectively grasp the reliable results which contribute to the relevant literature towards the optimal design of historical behaviour of unreinforced and reinforced (both with traditional and innovative techniques) curved masonry structures retroﬁt. masonry structures, as well as providing reliable results which contribute to the relevant literature towards the optimal design of historical masonry structures retrofit. 2. The Modelling of Historical Masonry Structures In this work a numerical strategy based on a discrete macro-model, already available in the 2. The Modelling of Historical Masonry Structures literature, is employed for the nonlinear numerical simulations of both unreinforced and FRP reinforced In this work a numerical strategy based on a discrete macro‐model, already available in the masonry structures. According to this approach, the masonry is modelled by an equivalent mechanical literature, is employed for the nonlinear numerical simulations of both unreinforced and FRP scheme, constituted by a hinged quadrilateral endowed with one or two diagonal links to rule the reinforced masonry structures. According to this approach, the masonry is modelled by an diagonal shear cracking, and interacting with contiguous elements along its four edges by means equivalent mechanical scheme, constituted by a hinged quadrilateral endowed with one or two of nonlinear discrete interfaces which govern the ﬂexional and the sliding behaviour. Each discrete diagonal links to rule the diagonal shear cracking, and interacting with contiguous elements along interface is made of a single or multiple (according to the model) rows of transversal links for the its four edges by means of nonlinear discrete interfaces which govern the flexional and the sliding ﬂexional behaviour and single or multiple (according to the model) sliding links. The different behaviour. Each discrete interface is made of a single or multiple (according to the model) rows of stages of this discrete element are reported in Figure 1. This approach was originally introduced for transversal links for the flexional behaviour and single or multiple (according to the model) sliding modelling the in-plane behaviour of Unreinforced Masonry Structures [12], Figure 1a. This plane links. The different stages of this discrete element are reported in Figure 1. This approach was element possesses four degrees of freedom, a single row of transversal links and a single in-plane originally introduced for modelling the in‐plane behaviour of Unreinforced Masonry Structures [12], sliding Figure link, 1a.and This is pla able ne to elemen model t poss the main esses failur four degrees e mechanisms of freedom, of the a masonry single row in of its transversal own plane, links as long and a single in‐plane sliding link, and is able to model the main failure mechanisms of the masonry as a proper calibration procedure of the links is adopted. Two subsequent upgrades were achieved in its own plane, as long as a proper calibration procedure of the links is adopted. Two subsequent to expand the potentialities of the approach. First, the out of plane (spatial) behaviour, typical of upgrades were achieved to expand the potentialities of the approach. First, the out of plane (spatial) historical constructions, was added [17,18] by considering additional rows of transversal links, and behaviour, typical of historical constructions, was added [17,18] by considering additional rows of two additional out-of-plane sliding links (able to govern the out of plane shear behaviour and the transversal links, and two additional out‐of‐plane sliding links (able to govern the out of plane shear torsion), thus enabling the out of plane degrees of freedom, as shown in Figure 1b. Subsequently, behaviour and the torsion), thus enabling the out of plane degrees of freedom, as shown in Figure 1b. a further upgrade was introduced considering a shell macro-element characterized by an irregular Subsequently, a further upgrade was introduced considering a shell macro‐element characterized by geometry, variable thickness along the element, and skew interfaces [19,21] in order to deal with an irregular geometry, variable thickness along the element, and skew interfaces [19,21] in order to structures with a curved geometry, such as vaults and domes, Figure 1c. The calibration procedures, deal with structures with a curved geometry, such as vaults and domes, Figure 1c. The calibration concerning the mechanical properties of the links, were properly extended in order to account for the procedures, concerning the mechanical properties of the links, were properly extended in order to more complicated geometry of the element, but keeping the same general philosophy. account for the more complicated geometry of the element, but keeping the same general philosophy. Numerical and experimental validations of the proposed approach, with reference to full-scale Numerical and experimental validations of the proposed approach, with reference to full‐scale structures can be found in [23,24]. More recently this approach was also extended to the dynamic structures can be found in [23,24]. More recently this approach was also extended to the dynamic context [25]. context [25]. (a) (b) (c) Figure 1. Layout of the macro‐element adopted for masonry at its three stages: (a) plane element, (b) Figure 1. Layout of the macro-element adopted for masonry at its three stages: (a) plane element, spatial regular element and (c) three‐dimensional element for curved structures. (b) spatial regular element and (c) three-dimensional element for curved structures. Buildings 2017, 7, 79 4 of 17 Buildings 2017, 7, 79 4 of 17 3. The Modeling of FRP Reinforcing 3. The Modeling of FRP Reinforcing The extension of the macro-element approach to account for the presence of FRP-reinforcements was proposed The extein nsion [22 ], of and the ma is cro her‐e elem brieﬂy ent ap rp ecalled. roach to The accou pr ntesence for the of pres the ence ﬁbr of e-r FR eP infor ‐reinfo ced rce elements ments was proposed in [22], and is here briefly recalled. The presence of the fibre‐reinforced elements is is modelled by means of zero thickness rigid ﬂat elements, partially or totally lying on one of the modelled by means of zero thickness rigid flat elements, partially or totally lying on one of the surfaces of the masonry element, as shown in Figure 2. A special zero-thickness non-linear interface, surfaces of the masonry element, as shown in Figure 2. A special zero‐thickness non‐linear interface, whose kinematics is related to the relative displacements between the masonry and reinforcement whose kinematics is related to the relative displacements between the masonry and reinforcement macro-elements, was introduced to simulate a proper interaction between the FRP element and the macro‐elements, was introduced to simulate a proper interaction between the FRP element and the masonry support. In particular, the FRP-masonry interface discretization is here performed according masonry support. In particular, the FRP‐masonry interface discretization is here performed according to a discrete distribution of nonlinear links whose nonlinear laws account for the presence of the to a discrete distribution of nonlinear links whose nonlinear laws account for the presence of the adhesive, organic, or cementitious, matrix by allowing the mutual reinforcement-masonry normal adhesive, organic, or cementitious, matrix by allowing the mutual reinforcement‐masonry normal and tangential stresses. In particular, a layer of transversal links is introduced to model a ﬂexural and tangential stresses. In particular, a layer of transversal links is introduced to model a flexural detachment of the textile, while two longitudinal orthogonal links model the crucial aspect of the detachment of the textile, while two longitudinal orthogonal links model the crucial aspect of the delamination phenomenon. The calibration of the transversal links is performed according to a bilinear delamination phenomenon. The calibration of the transversal links is performed according to a constitutive law (with different compressive and tensile strengths) with a post-elastic branch calibrated bilinear constitutive law (with different compressive and tensile strengths) with a post‐elastic branch according to the relevant compressive and tensile fracture energies. On the other hand, the sliding calibrated according to the relevant compressive and tensile fracture energies. On the other hand, links are calibrated with a symmetric bilinear law whose post-elastic branch is associated to proper the sliding links are calibrated with a symmetric bilinear law whose post‐elastic branch is associated fractur to prope e ener r gy fracture , with ener a dependency gy, with a depe of the ndency current of str thength e current on the stren normal gth on action the nor on mathe l acinterface. tion on the interfac Discrete e. interfaces made of a single row of links (calibrated according to a ﬁber discretization Discrete interfaces made of a single row of links (calibrated according to a fiber discretization approach) rule the constitutive behaviour of the textile itself. This latter aspect, together with the sliding approach) rule the constitutive behaviour of the textile itself. This latter aspect, together with the links of the FRP-masonry interface, leads to a progressive transfer of the tangential forces between sliding links of the FRP‐masonry interface, leads to a progressive transfer of the tangential forces masonry and FRP reinforcement; thus, implying a numerical deﬁnition of the so called anchorage between masonry and FRP reinforcement; thus, implying a numerical definition of the so called length. In the applications reported in the following, the mechanical properties for the FRP laminates anchorage length. In the applications reported in the following, the mechanical properties for the have been assumed according to a simple elasto-fragile law attributed to a homogenous material, with FRP laminates have been assumed according to a simple elasto‐fragile law attributed to a homogenous an equivalent thickness (t ), characterized by a Young’s modulus (E ) and tensile strength (f ), incapable f f t material, with an equivalent thickness (tf), characterized by a Young’s modulus (Ef) and tensile to resist to compression loads. A schematic layout of the modelling approach of a masonry element strength (ft), incapable to resist to compression loads. A schematic layout of the modelling approach reinforced with a FRP strip is reported in Figure 2. of a masonry element reinforced with a FRP strip is reported in Figure 2. Despite its simplicity, the model is able to predict the main collapse mechanisms associated to Despite its simplicity, the model is able to predict the main collapse mechanisms associated to the reinforcement: the rupture in tensile of the ﬁber, the shear debonding, and/or the peeling of the reinforcement: the rupture in tensile of the fiber, the shear debonding, and/or the peeling of the the reinforcement. Furthermore, mixed failure mechanisms in which the masonry is involved, can reinforcement. Furthermore, mixed failure mechanisms in which the masonry is involved, can be be predicted. predicted. Figure 2. Schematic layout of the interaction between masonry elements and discrete fabric‐reinforced Figure 2. Schematic layout of the interaction between masonry elements and discrete fabric-reinforced polymeric (FRP) reinforcement elements. polymeric (FRP) reinforcement elements. 4. Retrofitting and Restoration of Curved Masonry Structures by FRP Materials 4. Retroﬁtting and Restoration of Curved Masonry Structures by FRP Materials In this section, the ultimate seismic strength of two typologies of curved structures is In this section, the ultimate seismic strength of two typologies of curved structures is numerically numerically simulated before and after a consolidation retrofit. Namely, a circular arch and a simulated before and after a consolidation retroﬁt. Namely, a circular arch and a spherical dome are spherical dome are considered. A standard technique, that is the application of a tie rod, and an considered. A standard technique, that is the application of a tie rod, and an innovative technique, that innovative technique, that is the application of FRP strips, are here considered and compared. is the application of FRP strips, are here considered and compared. Buildings 2017, 7, 79 5 of 17 Buildings 2017, 7, 79 5 of 17 Buildings 2017, 7, 79 5 of 17 4.1.4.Cir 1. Ci cular rcular Ar Arch ch 4.1. Circular Arch A simple circular arch with radius R is considered in this section; the other significant geometric A simple circular arch with radius R is considered in this section; the other signiﬁcant geometric A simple circular arch with radius R is considered in this section; the other significant geometric parameters are inferred as functions of the radius, that is the half bay ( ), the rise (f = R/2), parameters are inferred as functions of the radius, that is the half bay (LR L= R3/32 /2), the rise (f = R/2), parameters are inferred as functions of the radius, that is the half bay ( ), the rise (f = R/2), LR  3/ 2 the thickness (s = R/10) which is kept constant, and the width (b = s). The basic geometry of the arch is the thickness (s = R/10) which is kept constant, and the width (b = s). The basic geometry of the arch is the thickness (s = R/10) which is kept constant, and the width (b = s). The basic geometry of the arch is characterized by the value R1 = 866 mm, which corresponds to a prototype tested in the laboratory, characterized by the value R = 866 mm, which corresponds to a prototype tested in the laboratory, subj charected acteri zed to an by u the nsy va mmetri lue Rc1 al= 86 ve6r tmm ical ,static whic hlo corresponds ad [26]; then to , tw a pr o ototype addition te als tva edl uines the of lab theo rato radius ry, subjected to an unsymmetrical vertical static load [26]; then, two additional values of the radius subjected to an unsymmetrical vertical static load [26]; then, two additional values of the radius (R2 = 1500 mm and R3 = 2500 mm) are considered in order to investigate the effect of the scale factor (R = 1500 mm and R = 2500 mm) are considered in order to investigate the effect of the scale factor 2 3 (R2 = 1500 mm and R3 = 2500 mm) are considered in order to investigate the effect of the scale factor on the response of the unreinforced and reinforced systems. on the response of the unreinforced and reinforced systems. on the response of the unreinforced and reinforced systems. The arch is subjected to the self‐weight and to a horizontal mass proportional load distribution The arch is subjected to the self-weight and to a horizontal mass proportional load distribution (p0), as The represen arch ist esubject d in Fiegdu to re th 3, eincreased self‐weight until an d the to complete a horizonta collap l masss e of pro thpeortiona structure. l load Th distribut e resultsi oofn (p ), as represented in Figure 3, increased until the complete collapse of the structure. The results (p0), as represented in Figure 3, increased until the complete collapse of the structure. The results of the push‐over analyses are presented both in terms of capacity curves, and collapse mechanisms. of the push-over analyses are presented both in terms of capacity curves, and collapse mechanisms. the push‐over analyses are presented both in terms of capacity curves, and collapse mechanisms. The capacity curves report the maximum lateral displacement of the arch vs. the base shear The capacity curves report the maximum lateral displacement of the arch vs. the base shear coefﬁcient co The efficient capacity (ba scurves e shear norma report liz the ed maxi by thmu e ow mn lat weig eralh t). displac ement of the arch vs. the base shear (base shear normalized by the own weight). coefficient (base shear normalized by the own weight). Figure 3. Geometry of the arch with the indication of the seismic load condition (p0). Figure Figure 3. 3. Geometry Geometry of of the thear ar ch chwith withthe the indication indication of of the the seismic seismic load load co condition ndition (p0().p ). In order to calibrate the numerical model, an initial comparison was performed with the results of theIn experi ordermental to calibr caam tep the aign nume reporte ricald mo in de [26] l, an . In initi thea lm co entioned mpariso n pa was per pe two rfo identical rmed with arc the hes results were In order to calibrate the numerical model, an initial comparison was performed with the results of the experimental campaign reported in [26]. In the mentioned paper two identical arches were subjected to a vertical concentrated load according to the experimental layout reported in Figure 4. of the experimental campaign reported in [26]. In the mentioned paper two identical arches were subjected to a vertical concentrated load according to the experimental layout reported in Figure 4. The mechanical parameters of the masonry have been estimated by means of experimental tests [26], subjected to a vertical concentrated load according to the experimental layout reported in Figure 4. an The d m are ec h here anic reported al param ein te rsTable of the 1. mason E andr yG have represen been testimated the norm by al me andans the of ta exnpgeernitme ialnt defor al tests ma [t26] ion, The mechanical parameters of the masonry have been estimated by means of experimental tests [26], and are here reported in Table 1. E and G represent the normal and the tangential deformation moduli of masonry, σt and σc the tensile and compressive strengths, Gt and Gc the corresponding and are here reported in Table 1. E and G represent the normal and the tangential deformation moduli moduli of masonry, σt and σc the tensile and compressive strengths, Gt and Gc the corresponding values of fracture energy, c the cohesion, μ the friction factor, and w the specific self‐weight of of masonry, s and s the tensile and compressive strengths, G and G the corresponding values of t c t c values of fracture energy, c the cohesion, μ the friction factor, and w the specific self‐weight of masonry. fracture energy, c the cohesion, m the friction factor, and w the speciﬁc self-weight of masonry. masonry. The results of the macro‐element model are reported in terms of capacity curve (applied force The results of the macro-element model are reported in terms of capacity curve (applied force vs. The results of the macro‐element model are reported in terms of capacity curve (applied force vs. vertical displacement at the loaded section) with the black line, and are compared with the vertical displacement at the loaded section) with the black line, and are compared with the experimental vs. vertical displacement at the loaded section) with the black line, and are compared with the experimental capacity curves or the two specimens (grey lines). In terms of collapse mechanism, the capacity loc expe atirimen on curves and tal th capac ore the opening ittwo y curves specimens sequen or the ce two of (gr th ey spe e lines). pla cimstens ic In hin (grey terms ges lines) are of in .collapse In a gtereem rmsmechanism, ent of co with llap se the me the expe cha location nrimen ism, the tal and location and the opening sequence of the plastic hinges are in agreement with the experimental theevidence opening as sequence well. of the plastic hinges are in agreement with the experimental evidence as well. evidence as well. dimensions in mm dimensions in mm failure mechanism failure mechanism Figure 4. Validation of the numerical model. Figure 4. Validation of the numerical model. Figure 4. Validation of the numerical model. Buildings 2017, 7, 79 6 of 17 Buildings 2017, 7, 79 6 of 17 Table 1. Mechanical property of the masonry. Table 1. Mechanical property of the masonry. E (Mpa) G (Mpa) s (Mpa) s (Mpa) G (N/mm) G (N/mm) c (Mpa) m (-) W (kN/m ) t c t c Buildings 2017, 7, 79 6 of 17 E σt c μ W 2700 1080 0.30 8.53 0.01 0.30 0.26 0.6 18 G (Mpa) σc (Mpa) Gt (N/mm) Gc (N/mm) (Mpa) (Mpa) (Mpa) (‐) (kN/m ) Table 1. Mechanical property of the masonry. 2700 1080 0.30 8.53 0.01 0.30 0.26 0.6 18 Once the proposed model has been validated considering the masonry arch, as described in E σt c μ W G (Mpa) σc (Mpa) Gt (N/mm) Gc (N/mm) Figure 3, and subjected to a concentrated vertical load as reported in Figure 4, the load scenario Once the proposed model has been validated considering the masonry arch, as described in (Mpa) (Mpa) (Mpa) (‐) (kN/m ) corresponding Figure 3, an tod asubj uniform ected to horizontal a concentrat load ed verti is consider cal load ed as ireporte n the following d in Figure(Figur 4, the eload 3). In scenario particular , 2700 1080 0.30 8.53 0.01 0.30 0.26 0.6 18 corresponding to a uniform horizontal load is considered in the following (Figure 3). In particular, Figure 5 reports the capacity curves relative to the different geometries in terms of global base shear Figure 5 reports the capacity curves relative to the different geometries in terms of global base shear V (FigureOn 5a)ceand the in proposed terms of model base ha shear s been coef val ﬁcient idatedC co= nsV ider /iW ng (Figur the ma es5ob), nrybeing arch, as W described the total weight in b b b Vb (Figure 5a) and in terms of base shear coefficient Cb = Vb/W (Figure 5b), being W the total weight of Figure 3, and subjected to a concentrated vertical load as reported in Figure 4, the load scenario of the arch. It can be observed that, as the radius of the arch increases, the global resistance of the the arch. It can be observed that, as the radius of the arch increases, the global resistance of the arch corresponding to a uniform horizontal load is considered in the following (Figure 3). In particular, arch increases as well (Figure 5a). On the contrary, in terms of the base shear coefﬁcient, as the radius increases as well (Figure 5a). On the contrary, in terms of the base shear coefficient, as the radius Figure 5 reports the capacity curves relative to the different geometries in terms of global base shear increases, the peak strength reduces, and all the models tend to the same residual strength (Figure 5b). increases, the peak strength reduces, and all the models tend to the same residual strength (Figure 5b). Vb (Figure 5a) and in terms of base shear coefficient Cb = Vb/W (Figure 5b), being W the total weight of the arch. It can be observed that, as the radius of the arch increases, the global resistance of the arch R=2500mm 4 R=2500mm incr10 eases as well (Figure 5a). On the contrary, in terms of the base shear coefficient, as the radius R=1500mm R=1500mm increases, the peak strength reduces, and all the models tend to the same residual strength (Figure 5b). R=866mm 3 R=866mm R=2500mm R=2500mm 10 4 R=1500mm R=1500mm R=866mm 3 R=866mm 0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Horizontal displacement (mm) 1 Horizontal displacement (mm) (a) (b) 0 0 Figure 5. Capacity curves of the unreinforced arches, expressed in terms of (a) global base shear, and 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Figure 5. Capacity curves of the unreinforced arches, expressed in terms of (a) global base shear, and Horizontal displacement (mm) Horizontal displacement (mm) (b) base shear coefficient. (b) base shear coefﬁcient. (a) (b) With regard to the assessment of the effectiveness of the structural retrofitting of the arch, three Figure 5. Capacity curves of the unreinforced arches, expressed in terms of (a) global base shear, and different typologies of reinforcing are considered. The first one consists of the introduction of a tie With regard to the assessment of the effectiveness of the structural retroﬁtting of the arch, three (b) base shear coefficient. rod, whose diameter varies proportionally to the radius of the arch, from ф10 mm in the case of different typologies of reinforcing are considered. The ﬁrst one consists of the introduction of a tie rod, R = 866 mm, to ф30 mm in the case of R = 2500 mm, with Young’s modulus E = 200 GPa, and an With regard to the assessment of the effectiveness of the structural retrofitting of the arch, three whose diameter varies proportionally to the radius of the arch, from 10 mm in the case of R = 866 mm, ultimate tensile strength equal to fy = 200 Mpa. The diameters of the tie‐rods are empirically chosen different typologies of reinforcing are considered. The first one consists of the introduction of a tie to 30 mm in the case of R = 2500 mm, with Young’s modulus E = 200 GPa, and an ultimate tensile among commercial diameters, keeping constant the ratio between the radius of the arch and the rod, whose diameter varies proportionally to the radius of the arch, from ф10 mm in the case of strength equal to f = 200 Mpa. The diameters of the tie-rods are empirically chosen among commercial diameter of the tie‐rod. In the considered models the tie‐rods’ heights hr with respect to the base of R = 866 mm, to ф30 mm in the case of R = 2500 mm, with Young’s modulus E = 200 GPa, and an diameters, keeping constant the ratio between the radius of the arch and the diameter of the tie-rod. the arch is about R/4 (Figure 6). The yielding stress of the steel has been chosen among widely ultimate tensile strength equal to fy = 200 Mpa. The diameters of the tie‐rods are empirically chosen In the considered models the tie-rods’ heights h with respect to the base of the arch is about R/4 adopted steel typologies, and large enough to keep r the tie‐rods in the elastic field. The other two among commercial diameters, keeping constant the ratio between the radius of the arch and the (Figurstrateg e 6). The ies consist yielding ofstr the ess introduct of the steel ion of has FRP been stri chosen ps, at the among intrados widely andadopted at the extrados steel typologies, surfaces and diameter of the tie‐rod. In the considered models the tie‐rods’ heights hr with respect to the base of respectively (Figure 6). The reinforcement is constituted by strips arranged over the entire width and large enough to keep the tie-rods in the elastic ﬁeld. The other two strategies consist of the introduction the arch is about R/4 (Figure 6). The yielding stress of the steel has been chosen among widely length of the arch made of glass fiber composite material (GFRP) and organic matrix. The adopted of FRPadop strips, ted steel at the tyintrados pologies, and and larg at the e en extrados ough to keep surfaces the tie respectively ‐rods in the (Figur elastic efie 6l).d.The The rot einfor her tw cement o mechanical properties have been set according to [27], and reported in Table 2, in which Ef and ft are strategies consist of the introduction of FRP strips, at the intrados and at the extrados surfaces is constituted by strips arranged over the entire width and length of the arch made of glass ﬁber the tensile module and the ultimate tensile strength of the reinforcement, and tf is the equivalent respectively (Figure 6). The reinforcement is constituted by strips arranged over the entire width and composite material (GFRP) and organic matrix. The adopted mechanical properties have been set thickness. The bond‐slip behaviour is described by the initial shear stiffness of the matrix ks, the length of the arch made of glass fiber composite material (GFRP) and organic matrix. The adopted according to [27], and reported in Table 2, in which E and f are the tensile module and the ultimate ultimate debonding stress tf, the fracture energy Gs, and the friction factor μs. mechanical properties have been set according to [27], and reported in Table 2, in which Ef and ft are tensile strength of the reinforcement, and t is the equivalent thickness. The bond-slip behaviour is the tensile module and the ultimate tensile strength of the reinforcement, and tf is the equivalent described by the initial shear stiffness of the matrix k , the ultimate debonding stress t , the fracture thickness. The bond‐slip behaviour is described by the initial shear stiffness of the matrix ks, the hr energy G , and the friction factor m . s s ultimate debonding stress tf, the fracture energy Gs, and the friction factor μs. Tie rod 10‐20‐30 Extrados FRP strip Intrados FRP strip hr Figure 6. Different reinforcing interventions considered for the arch. Tie rod 10‐20‐30 Extrados FRP strip Intrados FRP strip Figure 6. Different reinforcing interventions considered for the arch. Figure 6. Different reinforcing interventions considered for the arch. Vb (kN) Vb (kN) Cb=Vb/W (-) Cb=Vb/W (-) Buildings 2017, 7, 79 7 of 17 Table 2. Tensile and bond-slip parameters of the FRP reinforcement. Buildings 2017, 7, 79 7 of 17 Tensile Bond-Slip Table 2. Tensile and bond‐slip parameters of the FRP reinforcement. E (GPa) t (mm) t (MPa) f (MPa) k (N/mm ) G (N/mm) m (-) f t f s f s s 450 1473 0.149 20 1.3 2.5 0.75 Tensile Bond‐Slip Ef (GPa) ft (MPa) tf (mm) ks (N/mm ) τf (MPa) Gs (N/mm) μs (‐) 450 1473 0.149 20 1.3 2.5 0.75 Figure 7 shows the failure mechanisms of the reinforced arches, respectively with R = 2500 mm and R = 866 mm. The collapse mechanism observed for the model reinforced with the tie rod is Figure 7 shows the failure mechanisms of the reinforced arches, respectively with R = 2500 mm very similar to the failure mechanism of the unreinforced arch, which is not here reported for the and R = 866 mm. The collapse mechanism observed for the model reinforced with the tie rod is very sake of conciseness. The latter aspect seems to demonstrate that the presence of the tie rod does not similar to the failure mechanism of the unreinforced arch, which is not here reported for the sake of increase the strength of the arch, at least in seismic conditions and neglecting the interaction with the conciseness. The latter aspect seems to demonstrate that the presence of the tie rod does not increase underlying walls. On the contrary, the failure mechanisms of the arches reinforced by means of FRP the strength of the arch, at least in seismic conditions and neglecting the interaction with the strips are characterized by a wide spread of the damage. It is worth to note that, due to the transfer of underlying walls. On the contrary, the failure mechanisms of the arches reinforced by means of FRP tangential stress between the FRP strip and the arch, the presence of FRP strips delays or prevents the strips are characterized by a wide spread of the damage. It is worth to note that, due to the transfer opening of plastic hinges on the surface to which the strips are applied. For all of the investigated cases, of tangential stress between the FRP strip and the arch, the presence of FRP strips delays or prevents the failure mechanism is concentrated in the masonry and in the FRP strips due to the tensile rupture; the opening of plastic hinges on the surface to which the strips are applied. For all of the investigated whereas, no shear no delamination of the reinforcement is encountered. In both cases of strips applied cases, the failure mechanism is concentrated in the masonry and in the FRP strips due to the tensile to the rupture; intrados whereas, and to no th sh e extrados, ear no del the amifailur nation e is ofassociated the reinforc toem the enactivation t is encounter of an ed.intermediate In both cases plastic of strips applied to the intrados and to the extrados, the failure is associated to the activation of an hinge. The latter is located on the extrados surface of the arch in the case of the intrados reinforcing intermediate plastic hinge. The latter is located on the extrados surface of the arch in the case of the (point A in Figure 7a and point A in Figure 7b) or at the intrados surface, closer the support of the 1 2 intrados reinforcing (point A1 in Figure 7a and point A2 in Figure 7b) or at the intrados surface, closer arch, in the case of the extrados reinforcing (point B in Figure 7a and point B in Figure 7b). It is worth 1 2 the support of the arch, in the case of the extrados reinforcing (point B1 in Figure 7a and point B2 in to note that, although the arches reported in Figure 7 are not scaled according to the relevant radius, Figure 7b). It is worth to note that, although the arches reported in Figure 7 are not scaled according they refer to different size of the arch, as better speciﬁed in the caption. to the relevant radius, they refer to different size of the arch, as better specified in the caption. A1 B1 (a) Tie rod ( 30) FRP intrados FRP extrados A2 B2 (b) Tie rod (10) FRP intrados FRP extrados Figure 7. Failure mechanisms of the reinforced (a) R = 2500 mm and (b) R = 866 mm arches. Figure 7. Failure mechanisms of the reinforced (a) R = 2500 mm and (b) R = 866 mm arches. Figure 8 shows the comparison of the considered reinforcing techniques in terms of capacity Figure 8 shows the comparison of the considered reinforcing techniques in terms of capacity curves for two of the three radiuses investigated: the smallest (866 mm) and the largest (2500 mm). curves for two of the three radiuses investigated: the smallest (866 mm) and the largest (2500 mm). The capacity curves and the failure mechanisms of the models reinforced with tie rods are very close Thetocapacity the findicurves ngs relative and the to the failur unrein e mechanisms forced modof el, the whereas models ther einfor capaci ced ty curves with tie of rthe ods arc arehes very reinforced by means of the FRP strips show significant increments, both in terms of strength and close to the ﬁndings relative to the unreinforced model, whereas the capacity curves of the arches ductility. For those models, after the achievement of the peak load a sudden drop in the global reinforced by means of the FRP strips show signiﬁcant increments, both in terms of strength and strength is encountered; such a drop is associated to the opening of a cylindrical hinge in the arch, ductility. For those models, after the achievement of the peak load a sudden drop in the global strength associated to the FRP strip tensile rupture. Then, for larger displacements, the FRP strips‐masonry is encountered; such a drop is associated to the opening of a cylindrical hinge in the arch, associated interface tends to mobilize the tangential force, progressively transferring stresses to the fibres. The to the FRP strip tensile rupture. Then, for larger displacements, the FRP strips-masonry interface latter aspect implies a higher residual force of the strengthened models with respect to the tends to mobilize the tangential force, progressively transferring stresses to the ﬁbres. The latter aspect unstrengthened model. The influence of the arrangement of the FRP reinforcement (at the extrados implies a higher residual force of the strengthened models with respect to the unstrengthened model. or at the intrados) on the global resistance is negligible in the case of R = 866 mm (Figure 8a), while The inﬂuence of the arrangement of the FRP reinforcement (at the extrados or at the intrados) on the this effect is important in the case of R = 2500 mm (Figure 8b). global resistance is negligible in the case of R = 866 mm (Figure 8a), while this effect is important in the case of R = 2500 mm (Figure 8b). Buildings 2017, 7, 79 8 of 17 Buildings 2017, 7, 79 8 of 17 12 Tie rod FRP intrados FRP extrados Tie rod FRP intrados FRP extrados Buildings 2017, 7, 79 8 of 17 12 Tie rod FRP intrados FRP extrados Tie rod FRP intrados FRP extrados 0 30 0.02.0 4.06.0 8.0 0.0 3.0 6.0 9.0 12.0 Horizontal displacement (mm) Horizontal displacement (mm) (a) (b) Figure 8. Capacity curves of the reinforced arch: (a) R = 866 mm; (b) R = 2500 mm. Figure 8. Capacity curves of the reinforced arch: (a) R = 866 mm; (b) R = 2500 mm. 0.02.0 4.06.0 8.0 0.0 3.0 6.0 9.0 12.0 Horizontal displacement (mm) Horizontal displacement (mm) The presence of the FRP composite strips produces an increment of the ultimate bending moment (a) (b) of the cross section of the arch. In order to highlight the contribution of the reinforcement on the The presence of the FRP composite strips produces an increment of the ultimate bending moment structural respo Fig nse, ure th 8. eCapaci eccent ty ri cucirty ves of of th thee re norm inforal ce dac arch: tion ( a()e R= =M 86/6N mm; ), normalize (b) R = 25d 00 wi mm th. respect to the of the cross section of the arch. In order to highlight the contribution of the reinforcement on the height H of the section, along the curvilinear abscissa (s) of the arch, normalized with respect to the structural response, the eccentricity of the normal action (e = M/N), normalized with respect to the The presence of the FRP composite strips produces an increment of the ultimate bending moment arch length Φ, is reported in Figure 9. In particular, the arches with R = 866 mm are considered, both in height H of the section, along the curvilinear abscissa (s) of the arch, normalized with respect to the of the cross section of the arch. In order to highlight the contribution of the reinforcement on the the configurations with FRP at the extrados (Figure 9a) and at the intrados (Figure 9b). The tensile axial arch length structural F, is respo reported nse, the in ec Figur centrieci9 ty . In of th particular e normal ,ac the tion ar (ches e = Mwith /N), normalize R = 866d mm with ar re esconsider pect to thed, e both force (N) is considered positive, and the bending moment (M) is considered positive if it stretches the height H of the section, along the curvilinear abscissa (s) of the arch, normalized with respect to the in the conﬁgurations with FRP at the extrados (Figure 9a) and at the intrados (Figure 9b). The tensile FRP reinforcement fibres. The zero of the abscissa is set at the left abutment, while the unit value arch length Φ, is reported in Figure 9. In particular, the arches with R = 866 mm are considered, both in corresponds to the right abutment of the arch. The reported eccentricities are associated to the axial force (N) is considered positive, and the bending moment (M) is considered positive if it stretches the configurations with FRP at the extrados (Figure 9a) and at the intrados (Figure 9b). The tensile axial peak‐load conditions, which are characterised by the opening of three hinges in both of the considered the FRP reinforcement ﬁbres. The zero of the abscissa is set at the left abutment, while the unit force (N) is considered positive, and the bending moment (M) is considered positive if it stretches the cases. In the model reinforced at the extrados (Figure 9a), two hinges are located at the intrados (at the value corresponds to the right abutment of the arch. The reported eccentricities are associated to the FRP reinforcement fibres. The zero of the abscissa is set at the left abutment, while the unit value normalized abscissa 0.29 and at the right end of the arch) and one hinge corresponding to the tensile peak-load conditions, which are characterised by the opening of three hinges in both of the considered corresponds to the right abutment of the arch. The reported eccentricities are associated to the rupture of the textile, is opened at the extrados at the left end of the arch. In the model reinforced at the cases. In pea the k‐lomodel ad conditions reinfor , wh ced ich at are the chara extrados cterised (Figu by the re op 9a), enin two g of hinges three hing areeslocated in both of at th the e co intrados nsidered (at the intrados, two hinges are located at the extrados (left end of the arch and at the normalized abscissa cases. In the model reinforced at the extrados (Figure 9a), two hinges are located at the intrados (at the normalized abscissa 0.29 and at the right end of the arch) and one hinge corresponding to the tensile 0.67), and one hinge at the intrados, located at the right end of the arch. normalized abscissa 0.29 and at the right end of the arch) and one hinge corresponding to the tensile rupture of the textile, is opened at the extrados at the left end of the arch. In the model reinforced at rupture of the textile, is opened at the extrados at the left end of the arch. In the model reinforced at the the intrados, two hinges are located at the extrados (left end of the arch and at the normalized abscissa intrados, two hinges are located at the extrados (left ende of lim (N the) arch and at the normalized abscissa elim(N) 0.67), and one hinge at the intrados, located at the right end of the arch. 0.67), and one hinge at the intrados, located at the right end of the arch. e=H/2 e=H/2 e=H/2 e=H/2 elim(N) elim(N) elim(N) elim(N) e=H/2 e=H/2 (a) (b) e=H/2 e=H/2 (a) (b)elim(N) elim(N) (a) (b) N>0 N<0 N>0 N<0 N H H <e<e e <e< R =F -N 2 lim lim 2 H c s H F s F s x Rc 2 2 (a) (b) e N G Rc=F s+N N>0 N<0 N>0 N<0 Rc Rc H G HR =F +N c s e <e<e e <e< 2 lim lim 2 H Rc=F s-N H F s F s x Rc H H 2 2 Rc F G F s 2 2 H s H Rc=F s-N <e<e e <e< e 2 lim lim 2 G Rc=F s+N R c c x (c) (d) Rc=F s+N e N H H x R c F F s s 2 2 H H Figure 9. Contribution of the FRP reinforcement at the peak load: normalized abscissa versus Rc=F s-N <e<e e <e< 2 lim lim 2 eccentricity of the acting force of the models with R = 866 reinforced at the (a) extrados and (b) (c) (d) intrados. Cross section internal equilibrium for extrados (c) and intrados (d) reinforcing. Figure 9. Contribution of the FRP reinforcement at the peak load: normalized abscissa versus Figure 9. Contribution of the FRP reinforcement at the peak load: normalized abscissa versus Iteccent is imric po ityrt aofn tthe to acting notice force that of the the di mod strieblsu ti with on R of =the 866 pla reinforced stic hinges at the in ( acor ) extr resp ados on and dence (b ) of the eccentricity of the acting force of the models with R = 866 reinforced at the (a) extrados and (b) intrados. intrados. Cross section internal equilibrium for extrados (c) and intrados (d) reinforcing. peak‐load does not correspond to the locations at collapse (Figure 7), since the rupture of the textile, Cross section internal equilibrium for extrados (c) and intrados (d) reinforcing. It is important to notice that the distribution of the plastic hinges in correspondence of the peak‐load does not correspond to the locations at collapse (Figure 7), since the rupture of the textile, It is important to notice that the distribution of the plastic hinges in correspondence of the peak-load does not correspond to the locations at collapse (Figure 7), since the rupture of the textile, Vb (kN) Vb (kN) Vb (kN) Vb (kN) Buildings 2017, 7, 79 9 of 17 Buildings 2017, 7, 79 9 of 17 corresponding to the third plastic hinge, implies an internal force redistribution which induces the opening of hinges in masonry at different locations. corresponding to the third plastic hinge, implies an internal force redistribution which induces the In Figure 9 the theoretical limit values of the eccentricity, as evaluated through the limit analysis opening of hinges in masonry at different locations. approach [26], are reported. The masonry is considered as a no-tension material and linear-elastic in In Figure 9 the theoretical limit values of the eccentricity, as evaluated through the limit analysis compression, whereas the FRP strips are not capable of transferring compressive force and elastic-brittle approach [26], are reported. The masonry is considered as a no‐tension material and linear‐elastic in in tension. The limit conditions, reported with dashed lines, are associated to the rupture in traction compression, whereas the FRP strips are not capable of transferring compressive force and of the FRP strip (curved lines), and to the tensile action on the masonry (straight lines at the elastic‐brittle in tension. The limit conditions, reported with dashed lines, are associated to the dimensionless eccentricity 0.5). The grey areas in the graphs represent the ﬁeld of the admissible rupture in traction of the FRP strip (curved lines), and to the tensile action on the masonry (straight eccentricities. It is worth noting that the left parts of the arches are in tension while the right parts lines at the dimensionless eccentricity 0.5). The grey areas in the graphs represent the field of the are in compression. This is due to the particular load scenario here considered (i.e., horizontal force admissible eccentricities. It is worth noting that the left parts of the arches are in tension while the distribution right partspr are oportional in comprte oss the ion.self-mass This is due of the to the arch). parti Incul prar oximity load sc ofenar theio abscissa here coassociated nsidered (i.to e., the change horizo of ntal sign force of dist thernormal ibution proportio force, thenal eccentricities to the self‐mas tend s ofto the diver arch ge ). In (see pro Figur ximity e 9of a,b). the In absciss order a to associated to the change of sign of the normal force, the eccentricities tend to diverge (see Figure 9a,b). clarify the equilibrium of the reinforced arch cross section, simple schemes are reported in Figure 9c,d, In order to clarify the equilibrium of the reinforced arch cross section, simple schemes are reported for the case of extrados and intrados reinforcing, respectively. In each ﬁgure the two possible scenarios in Figure 9c,d, for the case of extrados and intrados reinforcing, respectively. In each figure the two are reported: tensile (N > 0) and compressive axial force (N < 0). The grey areas (whose height is possible scenarios are reported: tensile (N > 0) and compressive axial force (N < 0). The grey areas equal to x) and the white ones (whose height is equal to H x) represent the areas in compression and (whose height is equal to x) and the white ones (whose height is equal to H − x) represent the areas in tension, respectively. The internal forces are represented by the tensile action of the FRP strip (F ) and compression and tension, respectively. The internal forces are represented by the tensile action of the the global compression on the masonry (R ). The ultimate equilibrium of the section is imposed by FRP strip (Fs) and the global compression on the masonry (Rc). The ultimate equilibrium of the considering the ultimate value of F and evaluating the corresponding value of x under the hypothesis section is imposed by considering the ultimate value of Fs and evaluating the corresponding value of of linear elastic behaviour of the masonry (conﬁrmed by the numerical simulations). Once the internal x under the hypothesis of linear elastic behaviour of the masonry (confirmed by the numerical forces are computed, the ultimate moment (M ) and the ultimate eccentricity e (N) = M /N can be u u lim simulations). Once the internal forces are computed, the ultimate moment (Mu) and the ultimate easily inferred. eccentricity can be easily inferred. eN() M /N lim u Figure 10 shows the working rates of the reinforcement at the peak load of the arches with Figure 10 shows the working rates of the reinforcement at the peak load of the arches with R = 866 mm reinforced at the extrados (Figure 10a) and at the intrados (Figure 10b). The working R = 866 mm reinforced at the extrados (Figure 10a) and at the intrados (Figure 10b). The working rates are expressed in terms of # /# , being # and # the current and the ultimate tensile strains of f fu f f rates are expressed in terms of εf/εfu, being εf and εfu the current and the ultimate tensile strains of the the textile, respectively. These rates are useful to identify the achievement of the tensile rupture of textile, respectively. These rates are useful to identify the achievement of the tensile rupture of the the reinforcement, which is here identiﬁed at the left end of the arch for the model reinforced at the reinforcement, which is here identified at the left end of the arch for the model reinforced at the extrados, and at the right end of the arch for the model reinforced at the intrados. These ruptures extrados, and at the right end of the arch for the model reinforced at the intrados. These ruptures produce the sudden drops of the global resistance, as observed in the global capacity curves. produce the sudden drops of the global resistance, as observed in the global capacity curves. (a) (b) Figure 10. Contribution of the FRP reinforcement at the peak load: normalized abscissa versus working Figure 10. Contribution of the FRP reinforcement at the peak load: normalized abscissa versus working rates of the reinforcement for the models reinforced at the (a) extrados and (b) intrados. rates of the reinforcement for the models reinforced at the (a) extrados and (b) intrados. Aiming at highlighting the shear bond behaviour of the FRP reinforcement, in Figure 11 shows Aiming at highlighting the shear bond behaviour of the FRP reinforcement, in Figure 11 shows the tangential stress τ at the interface between masonry and FRP reinforcement in correspondence of the tangential stress t at the interface between masonry and FRP reinforcement in correspondence of the peak load (continuous lines), as well as the yielding tangential stress τy(N) at the same step the(d peak ashed load lines (continuous ), dependinlines), g on the as curre wellnt as co the mp yielding ression force tangential on the str interf ess tace (N ()Nat ). Th thee same figures step ref(dashed er to the arches with R = 866 mm reinforced at the extrados (Figure 11a) and at the intrados (Figure 11b). lines), depending on the current compression force on the interface (N). The ﬁgures refer to the arches In both cases, the tangential stress is lower than the corresponding yielding value confirming that with R = 866 mm reinforced at the extrados (Figure 11a) and at the intrados (Figure 11b). In both cases, the debonding mechanism does not occur. The latter results are apparently in contrast with other the tangential stress is lower than the corresponding yielding value conﬁrming that the debonding experimental and numerical results available in the literature, obtained considering similar FRP mechanism does not occur. The latter results are apparently in contrast with other experimental and reinforced prototypes subjected to a vertical eccentric force [27]. The fact that no delamination numerical results available in the literature, obtained considering similar FRP reinforced prototypes Buildings 2017, 7, 79 10 of 17 Buildings 2017, 7, 79 10 of 17 subjected to a vertical eccentric force [27]. The fact that no delamination phenomenon occurs for the phenomenon occurs for the treated cases might be due, in part, to the geometry and in part to the treated cases might be due, in part, to the geometry and in part to the horizontal mass-proportional horizontal mass‐proportional load distribution considered. load distribution considered. (a) (b) Figure 11. Tangential stress at the interface between masonry and FRP reinforcement in correspondence Figure 11. Tangential stress at the interface between masonry and FRP reinforcement in correspondence of the peak load for the models reinforced at the (a) extrados; and (b) intrados. of the peak load for the models reinforced at the (a) extrados; and (b) intrados. A comparison among all the reinforced and unreinforced models, in terms of ultimate load caA paci comparison ty (Vb,max) anamong d increment all the of rresistance einforced (Δ and Vb), unr is re einfor porteced d in models, Table 3. Th in e terms benefi of ts ultimate in terms of load strength resistance are higher in the models with the lowest radius (R = 866 mm) and the beneficial capacity (V , ) and increment of resistance (DV ), is reported in Table 3. The beneﬁts in terms of max b b effects decrease as the radius increases. Furthermore, the comparison of the effects of the extrados strength resistance are higher in the models with the lowest radius (R = 866 mm) and the beneﬁcial and intrados arrangements of the FRP strips demonstrates that the scale effect observed in Figure 8 effects decrease as the radius increases. Furthermore, the comparison of the effects of the extrados is confirmed for all of the cases investigated: in addition, for small radius models the application of and intrados arrangements of the FRP strips demonstrates that the scale effect observed in Figure 8 FRP strips to the intrados and to the extrados provides similar effects (see the first column of Table 3), is conﬁrmed for all of the cases investigated: in addition, for small radius models the application of while in the case of large radius models the benefit associated to the extrados FRP reinforcement is FRP strips to the intrados and to the extrados provides similar effects (see the ﬁrst column of Table 3), significantly higher if compared to the intrados reinforcing. while in the case of large radius models the beneﬁt associated to the extrados FRP reinforcement is signiﬁcantly higher if compared to the intrados reinforcing. Table 3. Ultimate strength of the arches and increment of the ultimate load with respect to the unreinforced configuration. Table 3. Ultimate strength of the arches and increment of the ultimate load with respect to the R = 866 mm R = 1500 mm R = 2500 mm unreinforced conﬁguration. Model Vb,max (kN) ΔVb (%) Vb,max (kN) ΔVb (%) Vb,max (kN) ΔVb (%) Unreinforced 1.23 R ‐ = 866 mm 3.43 R = ‐ 1500 mm 9.26 R = ‐ 2500 mm Model Tie rod 1.23 0 3.43 0 9.26 0 V , (kN) DV (%) V , (kN) DV (%) V , (kN) DV (%) max max max b b b b b b Intrados FRP 10.83 780 19.14 458 34.10 268 Unreinforced 1.23 - 3.43 - 9.26 - Extrado Tie rs od FRP 1.23 10.72 0772 3.43 21.93 0539 9.26 43.00 0364 Intrados FRP 10.83 780 19.14 458 34.10 268 Extrados FRP 10.72 772 21.93 539 43.00 364 Finally, in order to investigate the influence of the fibre content on the lateral strength of the structure, a further parametric investigation on the arch with R = 866 mm reinforced at the intrados is performed. In particular, a model considering a double equivalent thickness of reinforcement Finally, in order to investigate the inﬂuence of the ﬁbre content on the lateral strength of the (tf = 0.298 mm) is investigated. structure, a further parametric investigation on the arch with R = 866 mm reinforced at the intrados In Figure 12, the corresponding capacity curve is reported (Figure 12a) together with the trend is performed. In particular, a model considering a double equivalent thickness of reinforcement of the tangential stress at the interface between masonry and FRP reinforcement (Figure 12b), and (t = 0.298 mm) is investigated. the damage pattern in correspondence of the peak load and the collapse of the arch (Figure 12c,d). In In Figure 12, the corresponding capacity curve is reported (Figure 12a) together with the trend Figure 12a,b, the pushover curve and the corresponding tangential stresses of the previously of the tangential stress at the interface between masonry and FRP reinforcement (Figure 12b), and investigated model (tf = 0.149 mm) are reported for comparison. An increment of strength and the damage pattern in correspondence of the peak load and the collapse of the arch (Figure 12c,d). ductility is associated to the model with tf = 0.298 mm if compared to the standard thickness model. In Figure 12a,b, the pushover curve and the corresponding tangential stresses of the previously However in this case the ultimate lateral capacity is limited by the activation of the delamination as investigated model (t = 0.149 mm) are reported for comparison. An increment of strength and ductility demonstrated by tangential stress distribution, which overlaps the yielding stress close to the right is associated to the model with t = 0.298 mm if compared to the standard thickness model. However in end of the arch (Figure 12b). At the peak load, the opening of the cylindrical hinges at the intrados is this case the ultimate lateral capacity is limited by the activation of the delamination as demonstrated significantly delayed by the presence of FRP reinforcement (Figure 12c), causing a significant by tangential stress distribution, which overlaps the yielding stress close to the right end of the arch delamination in the post‐peak branch (Figure 12d). (Figure 12b). At the peak load, the opening of the cylindrical hinges at the intrados is signiﬁcantly delayed by the presence of FRP reinforcement (Figure 12c), causing a signiﬁcant delamination in the post-peak branch (Figure 12d). Buildings 2017, 7, 79 11 of 17 Buildings 2017, 7, 79 11 of 17 Buildings 2017, 7, 79 11 of 17 (a) (b) (a) (b) (c) (d) (c) (d) Figure 12. Arch with R = 866 mm reinforced at the intrados with a double thickness of the textile Figure 12. Arch with R = 866 mm reinforced at the intrados with a double thickness of the textile Figure 12. Arch with R = 866 mm reinforced at the intrados with a double thickness of the textile (tf = 0.298 mm): (a) capacity curve; (b) tangential stress at the interface between masonry and FRP (t = 0.298 mm): (a) capacity curve; (b) tangential stress at the interface between masonry and FRP (tf = 0.298 mm): (a) capacity curve; (b) tangential stress at the interface between masonry and FRP reinforcement in correspondence of the peak load, damage pattern at (c) the peak load; and (d) reinforcement in correspondence of the peak load, damage pattern at (c) the peak load; and (d) collapse. reinforcement in correspondence of the peak load, damage pattern at (c) the peak load; and (d) collapse. collapse. 4.2. Hemisperical Dome 4.2. Hemisperical Dome 4.2. Hemisperical Dome A further A further example example relative relative to a to double a double curvatur curv eavault ture va is ult consider is cons edider in this ed in section. this sectio In particular n. In , A further example relative to a double curvature vault is considered in this section. In particular, a masonry hemi‐spherical dome, already studied in the elastic field in [28], with a a masonry hemi-spherical dome, already studied in the elastic ﬁeld in [28], with a thickness t = 20 cm, particular, a masonry hemi‐spherical dome, already studied in the elastic field in [28], with a thickness t = 20 cm, and whose geometric layout is reported in Figure 13 is here studied with and whose geometric layout is reported in Figure 13 is here studied with reference to the nonlinear thickness t = 20 cm, and whose geometric layout is reported in Figure 13 is here studied with reference to the nonlinear field. The masonry dome is initially subjected to its own self‐weight, and ﬁeld. The masonry dome is initially subjected to its own self-weight, and subsequently, a horizontal reference to the nonlinear field. The masonry dome is initially subjected to its own self‐weight, and subsequently, a horizontal force distribution proportional to the masses (p0) is applied until collapse force distribution proportional to the masses (p ) is applied until collapse in order to investigate a subsequently, a horizontal force distribution proportional to the masses (p0) is applied until collapse in order to investigate a typical load scenario in seismic conditions. typical load scenario in seismic conditions. in order to investigate a typical load scenario in seismic conditions. Figure 13. Geometry of the dome and control points. Figure 13. Geometry of the dome and control points. Figure 13. Geometry of the dome and control points. The displacements of three different nodes has been monitored according to the layout showed The displacements of three different nodes has been monitored according to the layout showed in Figure 13. The adopted mechanical properties for the numerical simulations are reported in Table 4. in The Figure displacements 13. The adopted of thr mec eehdif anic fer alent pronodes pertieshas for been the num monitor erical ed simu accor lation ding s are to re the por layout ted in Table showed 4. in The results are reported in Figure 14 in terms of collapse mechanisms, damage patterns (Figure 14a), The results are reported in Figure 14 in terms of collapse mechanisms, damage patterns (Figure 14a), Figure 13. The adopted mechanical properties for the numerical simulations are reported in Table 4. and capacity curves with respect to the three monitored nodes (Figure 14b). and capacity curves with respect to the three monitored nodes (Figure 14b). The results are reported in Figure 14 in terms of collapse mechanisms, damage patterns (Figure 14a), Table 4. Mechanical characteristics of the masonry. and capacity curves with respect to the three monitored nodes (Figure 14b). Table 4. Mechanical characteristics of the masonry. E (Mpa) G (Mpa) σt (Mpa) σc (Mpa) Gt (N/mm) Gc (N/mm) c (Mpa) μ (‐) w (kN/m ) Table 4. Mechanical characteristics of the masonry. 3 E (Mpa) G (Mpa) σt (Mpa) σc (Mpa) Gt (N/mm) Gc (N/mm) c (Mpa) μ (‐) w (kN/m ) 1200 480 0.15 2.50 0.10 0.5 0.15 0.7 25 1200 480 0.15 2.50 0.10 0.5 0.15 0.7 25 E (Mpa) G (Mpa) s (Mpa) s (Mpa) G (N/mm) G (N/mm) c (Mpa) m (-) w (kN/m ) t c t c 1200 480 0.15 2.50 0.10 0.5 0.15 0.7 25 Buildings 2017, 7, 79 12 of 17 The collapse mechanism is characterized by a large damaged area along the meridians in the positive direction of the load and two smaller damaged areas at about a latitude of 30 at the two symmetric upper and lower sides orthogonal to the direction of the load distribution. In terms of capacity curves, the structure is characterized by a signiﬁcant peak load (C = 0.6) and by a signiﬁcant residual resistance as well. It is worth to note that the horizontal displacements of the monitored points decrease as the height of the control point increases. (a) 0.8 0.6 0.4 0.2 P3 P2 P1 0 5 10 15 20 Lateral displacement (mm) (b) Figure 14. Response of the unreinforced dome in terms of (a) failure mechanism; and (b) capacity curves. Regarding the structural retroﬁtting strategies, the application of FRP strips has been adopted. The strips (which have a width equal to 120 cm) have been arranged along the parallels to prevent the occurrence of damage along the meridians. Two different levels of retroﬁtting have been considered: a soft one with two strips centred at the latitudes of 22.5 and 49.5 (Figure 15a), and a strong retroﬁtting with four strips centred, respectively, at the latitudes of 13.5 , 31.5 , 49.5 , and 67.5 (Figure 15b). The same mechanical properties of the reinforcement considered for the circular arches, are adopted (see Table 2). 1 C =V /W (-) b b Buildings 2017, 7, 79 13 of 17 Buildings 2017, 7, 79 13 of 17 Buildings 2017, 7, 79 13 of 17 (a) (b) Figure 15. Typologies of reinforcing: (a) soft and (b) strong retrofitting. Figure 15. Typologies of reinforcing: (a) soft and (b) strong retroﬁtting. Again, the results are reported in terms of collapse mechanisms (Figure 16), and pushover (a) (b) curves, considering the same three monitored displacements of the unreinforced configuration Again, the results are reported in terms of collapse mechanisms (Figure 16), and pushover curves, Figure 15. Typologies of reinforcing: (a) soft and (b) strong retrofitting. (Figure 17). As expected, increasing resistances are obtained with both the softly and strongly considering the same three monitored displacements of the unreinforced conﬁguration (Figure 17). reinforced models with respect to the unreinforced configuration. In both cases the three monitored As expected, Agai incr n, the easing resulrtesistances s are reported are obtained in terms of with collap both se mech the softly anisms and (Fig str urongly e 16), ran einfor d pushover ced mode ls nodes show closer displacements to each other. Nevertheless, only in the post peak branches the curves, considering the same three monitored displacements of the unreinforced configuration with respect to the unreinforced conﬁguration. In both cases the three monitored nodes show closer lowest of the monitored nodes have larger displacements than the other two. The latter aspect is due (Figure 17). As expected, increasing resistances are obtained with both the softly and strongly displacements to each other. Nevertheless, only in the post peak branches the lowest of the monitored to the confinement effect of the FRP strips, as demonstrated also by the damage distribution at the reinforced models with respect to the unreinforced configuration. In both cases the three monitored nodes have larger displacements than the other two. The latter aspect is due to the conﬁnement effect collapse, which show how plastic strains develop only along the unreinforced parts of the meridians. nodes show closer displacements to each other. Nevertheless, only in the post peak branches the of the FRP strips, as demonstrated also by the damage distribution at the collapse, which show how The failure mechanism of both reinforced models are characterized by a spread damage at the base lowest of the monitored nodes have larger displacements than the other two. The latter aspect is due plastic secti strains on of devel the do opmonly e, bealong low the the first unr FR einfor P strip ced. parts In the of model the meridians. reinforcedThe by failur two st erips mechanism (soft of to the confinement effect of the FRP strips, as demonstrated also by the damage distribution at the reinforcing), the damage propagates along the entire height of the dome involving a limited radial both reinforced models are characterized by a spread damage at the base section of the dome, below collapse, which show how plastic strains develop only along the unreinforced parts of the meridians. portion (Figure 16a). A different failure mode is observed for the model reinforced by four FRP the ﬁrst FRP strip. In the model reinforced by two strips (soft reinforcing), the damage propagates The failure mechanism of both reinforced models are characterized by a spread damage at the base strips (strong reinforcing): in this case the damage propagates above the lowest strip involving a along the entire height of the dome involving a limited radial portion (Figure 16a). A different failure section of the dome, below the first FRP strip. In the model reinforced by two strips (soft large portion of the dome, the damage doesn’t propagate at the top of the dome (Figure 16b). modereinforcing is observed ), thfor e damage the model propag reinfor ates alo ced ng b th ye four entire FRP height strips of the (str dome ong rin einfor volving cing): a limin ited this radial case the portion (Figure 16a). A different failure mode is observed for the model reinforced by four FRP damage propagates above the lowest strip involving a large portion of the dome, the damage doesn’t strips (strong reinforcing): in this case the damage propagates above the lowest strip involving a propagate at the top of the dome (Figure 16b). large portion of the dome, the damage doesn’t propagate at the top of the dome (Figure 16b). (a) (a) (b) Figure 16. Failure mechanisms of the reinforced models: dome with (a) soft; and (b) strong reinforcement. (b) Figure 16. Failure mechanisms of the reinforced models: dome with (a) soft; and (b) strong reinforcement. Figure 16. Failure mechanisms of the reinforced models: dome with (a) soft; and (b) strong reinforcement. Buildings 2017, 7, 79 14 of 17 Buildings 2017, 7, 79 14 of 17 1.2 1.2 Buildings 2017, 7, 79 14 of 17 1.2 1.2 0.8 0.8 1 1 0.6 0.6 0.8 0.8 0.4 0.4 0.6 0.6 0.2 0.2 P3 P2 P1 P3 P2 P1 0.4 0.4 0 5 10 15 20 0 5 10 15 20 0.2 0.2 P3 P2 P1 P3 P2 P1 Lateral displacement (mm) Lateral displacement (mm) (a) (b) 0 5 10 15 20 0 5 10 15 20 Lateral displacement (mm) Lateral displacement (mm) Figure 17. Capacity curves of the reinforced models: (a) soft and (b) strong retrofitting. Figure 17. Capacity curves of the reinforced models: (a) soft and (b) strong retroﬁtting. (a) (b) The comparison in terms of pushover curves between the unreinforced configuration and the Figure 17. Capacity curves of the reinforced models: (a) soft and (b) strong retrofitting. The comparison in terms of pushover curves between the unreinforced conﬁguration and the two retrofitted domes, as reported in Figure 18, considering as monitored displacement P1, shows two retroﬁtted domes, as reported in Figure 18, considering as monitored displacement P , shows the effectiveness of the FRP retrofitting technique, which leads to a significant 1 The comparison in terms of pushover curves between the unreinforced configuration and the the effectiveness of the FRP retroﬁtting technique, which leads to a signiﬁcant improvement in terms improvement in terms of resistance without implying any global stiffness alteration, thus two retrofitted domes, as reported in Figure 18, considering as monitored displacement P1, shows of resistance without implying any global stiffness alteration, thus guaranteeing that no signiﬁcant guaranteeing that no significant change of the seismic demand for the structure occurs. On the other the effectiveness of the FRP retrofitting technique, which leads to a significant hand, the presence of FRP strips, not only increases the peak load of the arch, but significantly delays change of the seismic demand for the structure occurs. On the other hand, the presence of FRP strips, improvement in terms of resistance without implying any global stiffness alteration, thus the loss of resistance in the post‐peak branch (from around 2.5 mm for the unreinforced dome, to not only increases the peak load of the arch, but signiﬁcantly delays the loss of resistance in the guaranteeing that no significant change of the seismic demand for the structure occurs. On the other around 10 mm for the case of the strongly retrofitted dome), thus guaranteeing to the dome a larger post-peak branch (from around 2.5 mm for the unreinforced dome, to around 10 mm for the case of hand, the presence of FRP strips, not only increases the peak load of the arch, but significantly delays ductility as well. In Table 5 the ultimate lateral resistance (Cb,max) and the percentages of strength theth str e lo ongly ss of rresist etroﬁtted ance in dome), the pos thus t‐pea guaranteeing k branch (from to aro theun dome d 2.5 amm lar ger for the ductility unreinf as orced well. do In me T,able to 5 increment (ΔCb) are reported, highlighting the enhancement associated to the FRP reinforcement around 10 mm for the case of the strongly retrofitted dome), thus guaranteeing to the dome a larger the ultimate lateral resistance (C , ) and the percentages of strength increment (DC ) are reported, max b b application. The softly retrofitted model presents a residual lateral strength close to that relative to ductility as well. In Table 5 the ultimate lateral resistance (Cb,max) and the percentages of strength highlighting the enhancement associated to the FRP reinforcement application. The softly retroﬁtted the unreinforced model, whereas the strongly retrofitted model presents a higher value of residual increment (ΔCb) are reported, highlighting the enhancement associated to the FRP reinforcement model presents a residual lateral strength close to that relative to the unreinforced model, whereas the resistance due to a larger spreading of the damage at the ultimate condition, as shown in Figure 18. application. The softly retrofitted model presents a residual lateral strength close to that relative to strongly retroﬁtted model presents a higher value of residual resistance due to a larger spreading of It is worth to point out that the numerical investigation reported in this section considers only the unreinforced model, whereas the strongly retrofitted model presents a higher value of residual the damage at the ultimate condition, as shown in Figure 18. the load scenario corresponding to a horizontal force distribution proportional to the masses, resistance due to a larger spreading of the damage at the ultimate condition, as shown in Figure 18. It is worth to point out that the numerical investigation reported in this section considers only the representative of seismic condition. Nevertheless, structures can be subjected to very different load It is worth to point out that the numerical investigation reported in this section considers only load scenario corresponding to a horizontal force distribution proportional to the masses, representative scenarios (e.g., static conditions, concentrated loads). Further investigations to assess the the load scenario corresponding to a horizontal force distribution proportional to the masses, of seismic condition. Nevertheless, structures can be subjected to very different load scenarios effectiveness of FRP strengthening technique, under different seismic loads distributions (e.g., representative of seismic condition. Nevertheless, structures can be subjected to very different load (e.g., static conditions, concentrated loads). Further investigations to assess the effectiveness of proportional to the eigenmodes) or static loads should be investigated in further studies. scenarios (e.g., static conditions, concentrated loads). Further investigations to assess the FRP strengthening technique, under different seismic loads distributions (e.g., proportional to the effectiveness of FRP strengthening technique, under different seismic loads distributions (e.g., eigenmodes) or static loads should be investigated in further studies. proportional to the eigenmodes) or static loads should be investigated in further studies. 0.8 0.6 0.8 0.4 0.6 strong reinforcement 0.2 0.4 soft reinforcement unreinforced strong reinforcement 0.2 soft reinforcement 0 5 10 15 20 unreinforced Lateral displacement (mm) 0 5 10 15 20 Figure 18. Comparison of the capacity curves of the unreinforced and reinforced models. Lateral displacement (mm) Figure 18. Comparison of the capacity curves of the unreinforced and reinforced models. Figure 18. Comparison of the capacity curves of the unreinforced and reinforced models. C =V /WC (-)=V /W (-) b b b b C =V /W C (= -)V /W (-) b b b b C =V /WC (-)=V /W (-) b b b b Buildings 2017, 7, 79 15 of 17 Table 5. Ultimate strength of the domes. Model C , (-) DC (%) b max b Unreinforced 0.60 - Softly retroﬁtted 0.75 25 Strongly retroﬁtted 1.00 67 5. Conclusions A comprehensive discrete element strategy to simulate the nonlinear behaviour of existing masonry structures is employed here. The adopted model, based on a simple but effective mechanical scheme, was initially conceived for the nonlinear simulation of the in-plane behaviour of the masonry panels, and then upgraded to account for the out-of-plane behaviour and for the presence of curved elements (such as arches and vaults). More recently, the same modelling strategy has been extended with a new discrete element to model FRP strips, able to interact with a masonry support. In this paper, the numerical results obtained with this strategy are shown, aiming at demonstrating its capability to grasp the pre- and post-retroﬁtting capacities in seismic conditions. The approach has been ﬁrst validated with a comparison with the results obtained in the nonlinear static context on a unreinforced masonry arch; then, the beneﬁts provided by traditional and innovative retroﬁtting techniques (namely insertion of tie rods and application of FRP strips) are assessed and discussed. Signiﬁcant vault typologies with a scheme of both single and double curvature masonry structures are considered. The results relative to the arches are validated by the comparison with analytical results, as obtained through the limit analysis approach in order to demonstrate the effectiveness of the proposed approach to grasp the ultimate behaviour of the reinforced masonry cross sections (activation of the plastic hinges), and the changing of the global collapse of the structure. The proposed approach, being based on a model in which masonry and FRP strips are modelled with separate elements interacting with each other by means of discrete interfaces, is able to clearly identify the actual failure mode of the structure. The seismic load scenario, which, in spite of its high risk is not very debated in the academic literature, is here investigated, and the effectiveness of widely adopted FRP reinforcement arrangements are assessed and discussed. In spite of the relevance of the achieved results, in the future further investigations will be needed to assess different retroﬁtting techniques, also considering other load scenarios and structural typologies. In addition, with regard to the application of FRP reinforcements, different disposals of the strips have to be investigated with the aim of providing useful guidelines for the optimal retroﬁtting design. Author Contributions: B.P. and F.C. conceived the investigation strategy; B.P., C.C. and F.C. developed and calibrated the numerical models, analysed the results and wrote the paper; I.C., S.C and P.B.L. supervised the research, approved the outcome of the numerical investigations and revised the paper. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Huerta, S. Structural Design in the Work of Gaudi. J. Archit. Sci. Rev. 2006, 49, 324–339. [CrossRef] 2. Foti, D.; De Tommasi, D. An Innovative Modular System for the Building of Timber Cylindrical Roofs. Int. J. Mech. 2013, 7, 226–233. 3. Fabbrocino, F.; Farina, I.; Berardi, V.P.; Ferreira, A.J.M.; Fraternali, F. On the thrust surface of unreinforced and FRP-/FRCM-reinforced masonry domes. Compos. Part B Eng. 2013, 83, 297–305. [CrossRef] 4. Carpentieri, G.; Modano, M.; Fabbrocino, F.; Feo, L.; Fraternali, F. On the minimal mass reinforcement of masonry structures with arbitrary shapes. 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Struct. 2016, 152, 499–515. [CrossRef] 10. Bertolesi, E.; Fabbrocino, F.; Formisano, A.; Grande, E.; Milani, G. FRP-Strengthening of Curved Masonry Structures: Local Bond Behavior and Global Response. In Proceedings of the Mechanics of Masonry Structures Strengthened with Composite Materials (MURICO5), Bologna, Italy, 28–30 June 2017. 11. Malena, M.; de Felice, G. Debonding of composites on a curved masonry substrate: Experimental results and analytical formulation. Compos. Struct. 2014, 112, 194–206. [CrossRef] 12. Caliò, I.; Marletta, M.; Pantò, B. A new discrete element model for the evaluation of the seismic behaviour of unreinforced masonry buildings. Eng. Struct. 2012, 40, 327–338. [CrossRef] 13. Marques, R.; Lourenço, P.B. Possibilities and comparison of structural component models for the seismic assessment of modern unreinforced masonry buildings. Comput. Struct. 2011, 89, 2079–2091. [CrossRef] 14. Caliò, I.; Pantò, B. 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Ph.D. Thesis, University of Catania, Catania, Italy, April 2007. 19. Caliò, I.; Cannizzaro, F.; Marletta, M. A discrete element for modeling masonry vaults. Adv. Mater. Res. 2010, 133–134, 447–452. [CrossRef] 20. Caddemi, S.; Caliò, I.; Cannizzaro, F.; Pantò, B. The Seismic Assessment of Historical Masonry Structures. In Proceedings of the 12th International Conference on Computational Structures Technology, Naples, Italy, 2–5 September 2014. 21. Cannizzaro, F. Un Nuovo Approccio di Modellazione Della Risposta Sismica di Ediﬁci Storici. Ph.D. Thesis, University of Catania, Catania, Italy, April 2011. 22. Caddemi, S.; Caliò, I.; Cannizzaro, F.; Lourenço, P.B.; Pantò, B. FRP-reinforced masonry structures: Numerical modelling by means of a new discrete element approach. In Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Rhodes Island, Greece, 15–17 June 2017. 23. Cannizzaro, F.; Lourenço, P.B. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Buildings Multidisciplinary Digital Publishing Institute http://www.deepdyve.com/lp/multidisciplinary-digital-publishing-institute/nonlinear-modelling-of-curved-masonry-structures-after-seismic-P6wC06Dd67

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Publisher: Multidisciplinary Digital Publishing Institute
Copyright: © 1996-2019 MDPI (Basel, Switzerland) unless otherwise stated
ISSN: 2075-5309
DOI: 10.3390/buildings7030079
Publisher site: See Article on Publisher Site

Abstract

buildings Article Nonlinear Modelling of Curved Masonry Structures after Seismic Retroﬁt through FRP Reinforcing 1 , 1 1 1 ID Bartolomeo Pantò *, Francesco Cannizzaro , Salvatore Caddemi , Ivo Caliò , 2 2 César Chácara and Paulo B. Lourenço Department of Civil Engineering and Architecture, University of Catania, 95125 Catania, Italy; francesco.cannizzaro@dica.unict.it (F.C.); scaddemi@dica.unict.it (S.C.); icalio@dica.unict.it (I.C.) Department of Civil Engineering, Institute for Sustainability and Innovation in Structural Engineering (ISISE), University of Minho, Azurem, 4800-058 Guimarães, Portugal; c.chacara@pucp.pe (C.C.); pbl@civil.uminho.pt (P.B.L.) * Correspondence: bpanto@dica.unict.it; Tel.: +39-095-738-2275 Received: 24 April 2017; Accepted: 21 August 2017; Published: 29 August 2017 Abstract: A reliable numerical evaluation of the nonlinear behaviour of historical masonry structures, before and after a seismic retroﬁtting, is a fundamental issue in the design of the structural retroﬁtting. Many strengthening techniques have been introduced aimed at improving the structural performance of existing structures that, if properly designed and applied, provide an effective contribution to the preservation of their cultural value. Among these strategies, the use of fabric-reinforced polymeric (FRP) materials on masonry surface is being widely adopted for practical engineering purposes. The application of strips or 2D grid composite layers is a low invasive and easy to apply retroﬁtting strategy, that is able to improve both the in-plane and the out of plane behaviour of masonry elements also in the presence of complex geometries thanks to their ﬂexibility. For this reason, these techniques are frequently employed for reinforcing masonry curved elements, such as arches and vaults. In this paper, taking advantage of an existing general framework based on a discrete element approach previously introduced by the authors, a discrete element conceived for modelling the interaction between masonry and FRP reinforcement is applied to different curved masonry vaults typologies. This model, already used for evaluating the nonlinear behaviour of masonry arches, is here employed for the ﬁrst time to evaluate the effectiveness of FRP reinforcements on double curvature elements. After a theoretical description of the proposed strategy, two applications relative to an arch and a dome, subjected to seismic loads, with different reinforced conditions, are presented. The beneﬁt provided by the application of FRP strips is also compared with that associated to traditional retroﬁtting techniques. A sensitivity study is performed with respect to the structure scale factor. Keywords: macro-model approach; monumental masonry structures; masonry arches and vaults; historical structural analyses; seismic assessment; cultural heritage protection; FRP-reinforcement; HiStrA software 1. Introduction The numerical simulation of the seismic response of historical masonry structures (HMS) is a key aspect of the cultural heritage preservation, and represents a challenging issue within the structural and earthquake engineering. Historical masonry structures represent a high percentage of existing buildings in several regions of the world, and their value is relevant both from the economic and social-cultural points of view. On the contrary, they suffer from scattered structural degradation, due to static loads, chemical and physic degradation of the materials, previous earthquakes, and wrong alterations of the original structural conception. Normally, they are not able to resist to earthquakes Buildings 2017, 7, 79; doi:10.3390/buildings7030079 www.mdpi.com/journal/buildings Buildings 2017, 7, 79 2 of 17 even if characterized by a low intensity. Recent seismic events occurred in Italy, such as the Central Italy Eartkquakes (2016), the Emilia (2012), and L’Aquila (2009); they produced severe damage patterns or the complete destruction of several historical sites. Such events, well documented in terms of post earthquake technical survey, demonstrated many critical aspects, which make vulnerable the historical structures to the horizontal and vertical seismic actions. One of the most important aspects is the presence of elements with a curved geometry such as arches and vaults, which interact with the vertical elements (walls or columns) during the earthquake motion, producing a signiﬁcant effect on the seismic response of the entire structure. Curved structures have been widely adopted in the past for building purposes, since their shape allows an effective transfer of the static vertical action to the walls, and induces compression along their span; for the latter reason curved shapes are still adopted and proposed for modern structures [1,2]. When it comes to single and double curvature masonry structures, their study is still an open problem debated in the literature [3]. Aimed at the reduction of the seismic vulnerability of HMS in presence of curved masonry elements, several retroﬁtting techniques based on reinforced composite materials, applied by means of polymeric (FRP) or cementitious (FRCM) matrix, have been introduced, and widely investigated by means of experimental tests and numerical simulations during the last years. The use of these techniques is getting more and more frequent in the retroﬁtting of historical and monumental masonry buildings since they consist of reversible and low invasive interventions; with regard to the design of FRP reinforcements, several proposals have been made [4,5]. On the other hand, there is a lack of fast and reliable numerical tools to assess the effectiveness of such techniques. In fact, even considering unreinforced masonry structures, the numerical simulation of their actual behaviour is still a very complex task within the computational structural mechanics ﬁeld. The main issue is related to the difﬁculty in providing reliable simulations of the high nonlinear degrading cyclic response of masonry. To this regard, a great variability of the mechanical characteristics is encountered, thus making difﬁcult the deﬁnition of a general constitutive law that is suitable for all masonry typologies. When it comes to retroﬁtting techniques, and in particular to ﬁbre-composite strengthened structures, a crucial aspect is related to the correct simulation of the tangential stress transferred from the reinforcement to the masonry substrate, and the relative failure collapse for tensile rupture of the textile or delamination of the reinforcement from the support. On this task, several contributions have been given by different authors and now are available in the literature [6–9], also in presence of curved support [10,11]. Recently, a 3D macro-model has been proposed for the non-linear seismic simulations of masonry structures aiming at a reduced computational effort when compared to the traditional ﬁnite element approaches. The main idea of the model was to use a 2D mechanical scheme, governed by unidirectional non-linear links, which, according to different typologies, have to reproduce the main masonry failure modes [12]. Such a model has been successfully employed in the simulation of laboratory tests [13] and in the seismic assessment of ordinary and mixed masonry buildings [14–16]. Subsequently, the model has been extended to catch the out-of-plane behaviour of masonry walls [17,18] and the behaviour of structures with a curved geometry [19–21]. In this paper this discrete macro-modelling approach is used to assess the seismic capacity of some typologies of masonry vaults commonly present in historical structures, before and after a reinforcing retroﬁt through composite materials. With this aim, a new non-linear model recently proposed in the literature [22], able to simulate the presence of FRP strips or 2D-webs and to grasp the interaction with the masonry support, also in presence of a curved substrate, is here employed. The model is able to simulate the debonding of the reinforcement from the masonry support due to tangential delamination or to normal tensile detachment, which represent the most probable collapse mechanisms of FRP reinforced structures. Different strips arrangements applied either to the intrados or to the extrados surfaces are considered in the paper. The efﬁciency of the FRP retroﬁtting is compared to the traditional technique of adding steel tie-rods in order to investigate the optimal solutions with respect to the retroﬁt design and Buildings 2017, 7, 79 3 of 17 to the use of composite material. The results of the numerical simulations relative to the arches, have been compared with those obtained through limit analysis approach, and applied to simpler models. Buildings 2017, 7, 79 3 of 17 The analytical results are used to validate the numerical results and are duly discussed providing a contribution towards the understanding of reinforced curved structures subjected to seismic actions. to the arches, have been compared with those obtained through limit analysis approach, and applied The results show that the composite reinforcements can produce a signiﬁcant increment of the seismic to simpler models. The analytical results are used to validate the numerical results and are duly capacity discussed in teprov rmsiding of both a cont strength ribution and towards ductility the , wit understand hout incr ing easing of reinforce the stifd fness curvof ed the strustr ctures uctur e. subjected to seismic actions. The results show that the composite reinforcements can produce a The employed macro-model is able to effectively grasp the behaviour of unreinforced and reinforced significant increment of the seismic capacity in terms of both strength and ductility, without (both with traditional and innovative techniques) curved masonry structures, as well as providing increasing the stiffness of the structure. The employed macro‐model is able to effectively grasp the reliable results which contribute to the relevant literature towards the optimal design of historical behaviour of unreinforced and reinforced (both with traditional and innovative techniques) curved masonry structures retroﬁt. masonry structures, as well as providing reliable results which contribute to the relevant literature towards the optimal design of historical masonry structures retrofit. 2. The Modelling of Historical Masonry Structures In this work a numerical strategy based on a discrete macro-model, already available in the 2. The Modelling of Historical Masonry Structures literature, is employed for the nonlinear numerical simulations of both unreinforced and FRP reinforced In this work a numerical strategy based on a discrete macro‐model, already available in the masonry structures. According to this approach, the masonry is modelled by an equivalent mechanical literature, is employed for the nonlinear numerical simulations of both unreinforced and FRP scheme, constituted by a hinged quadrilateral endowed with one or two diagonal links to rule the reinforced masonry structures. According to this approach, the masonry is modelled by an diagonal shear cracking, and interacting with contiguous elements along its four edges by means equivalent mechanical scheme, constituted by a hinged quadrilateral endowed with one or two of nonlinear discrete interfaces which govern the ﬂexional and the sliding behaviour. Each discrete diagonal links to rule the diagonal shear cracking, and interacting with contiguous elements along interface is made of a single or multiple (according to the model) rows of transversal links for the its four edges by means of nonlinear discrete interfaces which govern the flexional and the sliding ﬂexional behaviour and single or multiple (according to the model) sliding links. The different behaviour. Each discrete interface is made of a single or multiple (according to the model) rows of stages of this discrete element are reported in Figure 1. This approach was originally introduced for transversal links for the flexional behaviour and single or multiple (according to the model) sliding modelling the in-plane behaviour of Unreinforced Masonry Structures [12], Figure 1a. This plane links. The different stages of this discrete element are reported in Figure 1. This approach was element possesses four degrees of freedom, a single row of transversal links and a single in-plane originally introduced for modelling the in‐plane behaviour of Unreinforced Masonry Structures [12], sliding Figure link, 1a.and This is pla able ne to elemen model t poss the main esses failur four degrees e mechanisms of freedom, of the a masonry single row in of its transversal own plane, links as long and a single in‐plane sliding link, and is able to model the main failure mechanisms of the masonry as a proper calibration procedure of the links is adopted. Two subsequent upgrades were achieved in its own plane, as long as a proper calibration procedure of the links is adopted. Two subsequent to expand the potentialities of the approach. First, the out of plane (spatial) behaviour, typical of upgrades were achieved to expand the potentialities of the approach. First, the out of plane (spatial) historical constructions, was added [17,18] by considering additional rows of transversal links, and behaviour, typical of historical constructions, was added [17,18] by considering additional rows of two additional out-of-plane sliding links (able to govern the out of plane shear behaviour and the transversal links, and two additional out‐of‐plane sliding links (able to govern the out of plane shear torsion), thus enabling the out of plane degrees of freedom, as shown in Figure 1b. Subsequently, behaviour and the torsion), thus enabling the out of plane degrees of freedom, as shown in Figure 1b. a further upgrade was introduced considering a shell macro-element characterized by an irregular Subsequently, a further upgrade was introduced considering a shell macro‐element characterized by geometry, variable thickness along the element, and skew interfaces [19,21] in order to deal with an irregular geometry, variable thickness along the element, and skew interfaces [19,21] in order to structures with a curved geometry, such as vaults and domes, Figure 1c. The calibration procedures, deal with structures with a curved geometry, such as vaults and domes, Figure 1c. The calibration concerning the mechanical properties of the links, were properly extended in order to account for the procedures, concerning the mechanical properties of the links, were properly extended in order to more complicated geometry of the element, but keeping the same general philosophy. account for the more complicated geometry of the element, but keeping the same general philosophy. Numerical and experimental validations of the proposed approach, with reference to full-scale Numerical and experimental validations of the proposed approach, with reference to full‐scale structures can be found in [23,24]. More recently this approach was also extended to the dynamic structures can be found in [23,24]. More recently this approach was also extended to the dynamic context [25]. context [25]. (a) (b) (c) Figure 1. Layout of the macro‐element adopted for masonry at its three stages: (a) plane element, (b) Figure 1. Layout of the macro-element adopted for masonry at its three stages: (a) plane element, spatial regular element and (c) three‐dimensional element for curved structures. (b) spatial regular element and (c) three-dimensional element for curved structures. Buildings 2017, 7, 79 4 of 17 Buildings 2017, 7, 79 4 of 17 3. The Modeling of FRP Reinforcing 3. The Modeling of FRP Reinforcing The extension of the macro-element approach to account for the presence of FRP-reinforcements was proposed The extein nsion [22 ], of and the ma is cro her‐e elem brieﬂy ent ap rp ecalled. roach to The accou pr ntesence for the of pres the ence ﬁbr of e-r FR eP infor ‐reinfo ced rce elements ments was proposed in [22], and is here briefly recalled. The presence of the fibre‐reinforced elements is is modelled by means of zero thickness rigid ﬂat elements, partially or totally lying on one of the modelled by means of zero thickness rigid flat elements, partially or totally lying on one of the surfaces of the masonry element, as shown in Figure 2. A special zero-thickness non-linear interface, surfaces of the masonry element, as shown in Figure 2. A special zero‐thickness non‐linear interface, whose kinematics is related to the relative displacements between the masonry and reinforcement whose kinematics is related to the relative displacements between the masonry and reinforcement macro-elements, was introduced to simulate a proper interaction between the FRP element and the macro‐elements, was introduced to simulate a proper interaction between the FRP element and the masonry support. In particular, the FRP-masonry interface discretization is here performed according masonry support. In particular, the FRP‐masonry interface discretization is here performed according to a discrete distribution of nonlinear links whose nonlinear laws account for the presence of the to a discrete distribution of nonlinear links whose nonlinear laws account for the presence of the adhesive, organic, or cementitious, matrix by allowing the mutual reinforcement-masonry normal adhesive, organic, or cementitious, matrix by allowing the mutual reinforcement‐masonry normal and tangential stresses. In particular, a layer of transversal links is introduced to model a ﬂexural and tangential stresses. In particular, a layer of transversal links is introduced to model a flexural detachment of the textile, while two longitudinal orthogonal links model the crucial aspect of the detachment of the textile, while two longitudinal orthogonal links model the crucial aspect of the delamination phenomenon. The calibration of the transversal links is performed according to a bilinear delamination phenomenon. The calibration of the transversal links is performed according to a constitutive law (with different compressive and tensile strengths) with a post-elastic branch calibrated bilinear constitutive law (with different compressive and tensile strengths) with a post‐elastic branch according to the relevant compressive and tensile fracture energies. On the other hand, the sliding calibrated according to the relevant compressive and tensile fracture energies. On the other hand, links are calibrated with a symmetric bilinear law whose post-elastic branch is associated to proper the sliding links are calibrated with a symmetric bilinear law whose post‐elastic branch is associated fractur to prope e ener r gy fracture , with ener a dependency gy, with a depe of the ndency current of str thength e current on the stren normal gth on action the nor on mathe l acinterface. tion on the interfac Discrete e. interfaces made of a single row of links (calibrated according to a ﬁber discretization Discrete interfaces made of a single row of links (calibrated according to a fiber discretization approach) rule the constitutive behaviour of the textile itself. This latter aspect, together with the sliding approach) rule the constitutive behaviour of the textile itself. This latter aspect, together with the links of the FRP-masonry interface, leads to a progressive transfer of the tangential forces between sliding links of the FRP‐masonry interface, leads to a progressive transfer of the tangential forces masonry and FRP reinforcement; thus, implying a numerical deﬁnition of the so called anchorage between masonry and FRP reinforcement; thus, implying a numerical definition of the so called length. In the applications reported in the following, the mechanical properties for the FRP laminates anchorage length. In the applications reported in the following, the mechanical properties for the have been assumed according to a simple elasto-fragile law attributed to a homogenous material, with FRP laminates have been assumed according to a simple elasto‐fragile law attributed to a homogenous an equivalent thickness (t ), characterized by a Young’s modulus (E ) and tensile strength (f ), incapable f f t material, with an equivalent thickness (tf), characterized by a Young’s modulus (Ef) and tensile to resist to compression loads. A schematic layout of the modelling approach of a masonry element strength (ft), incapable to resist to compression loads. A schematic layout of the modelling approach reinforced with a FRP strip is reported in Figure 2. of a masonry element reinforced with a FRP strip is reported in Figure 2. Despite its simplicity, the model is able to predict the main collapse mechanisms associated to Despite its simplicity, the model is able to predict the main collapse mechanisms associated to the reinforcement: the rupture in tensile of the ﬁber, the shear debonding, and/or the peeling of the reinforcement: the rupture in tensile of the fiber, the shear debonding, and/or the peeling of the the reinforcement. Furthermore, mixed failure mechanisms in which the masonry is involved, can reinforcement. Furthermore, mixed failure mechanisms in which the masonry is involved, can be be predicted. predicted. Figure 2. Schematic layout of the interaction between masonry elements and discrete fabric‐reinforced Figure 2. Schematic layout of the interaction between masonry elements and discrete fabric-reinforced polymeric (FRP) reinforcement elements. polymeric (FRP) reinforcement elements. 4. Retrofitting and Restoration of Curved Masonry Structures by FRP Materials 4. Retroﬁtting and Restoration of Curved Masonry Structures by FRP Materials In this section, the ultimate seismic strength of two typologies of curved structures is In this section, the ultimate seismic strength of two typologies of curved structures is numerically numerically simulated before and after a consolidation retrofit. Namely, a circular arch and a simulated before and after a consolidation retroﬁt. Namely, a circular arch and a spherical dome are spherical dome are considered. A standard technique, that is the application of a tie rod, and an considered. A standard technique, that is the application of a tie rod, and an innovative technique, that innovative technique, that is the application of FRP strips, are here considered and compared. is the application of FRP strips, are here considered and compared. Buildings 2017, 7, 79 5 of 17 Buildings 2017, 7, 79 5 of 17 Buildings 2017, 7, 79 5 of 17 4.1.4.Cir 1. Ci cular rcular Ar Arch ch 4.1. Circular Arch A simple circular arch with radius R is considered in this section; the other significant geometric A simple circular arch with radius R is considered in this section; the other signiﬁcant geometric A simple circular arch with radius R is considered in this section; the other significant geometric parameters are inferred as functions of the radius, that is the half bay ( ), the rise (f = R/2), parameters are inferred as functions of the radius, that is the half bay (LR L= R3/32 /2), the rise (f = R/2), parameters are inferred as functions of the radius, that is the half bay ( ), the rise (f = R/2), LR  3/ 2 the thickness (s = R/10) which is kept constant, and the width (b = s). The basic geometry of the arch is the thickness (s = R/10) which is kept constant, and the width (b = s). The basic geometry of the arch is the thickness (s = R/10) which is kept constant, and the width (b = s). The basic geometry of the arch is characterized by the value R1 = 866 mm, which corresponds to a prototype tested in the laboratory, characterized by the value R = 866 mm, which corresponds to a prototype tested in the laboratory, subj charected acteri zed to an by u the nsy va mmetri lue Rc1 al= 86 ve6r tmm ical ,static whic hlo corresponds ad [26]; then to , tw a pr o ototype addition te als tva edl uines the of lab theo rato radius ry, subjected to an unsymmetrical vertical static load [26]; then, two additional values of the radius subjected to an unsymmetrical vertical static load [26]; then, two additional values of the radius (R2 = 1500 mm and R3 = 2500 mm) are considered in order to investigate the effect of the scale factor (R = 1500 mm and R = 2500 mm) are considered in order to investigate the effect of the scale factor 2 3 (R2 = 1500 mm and R3 = 2500 mm) are considered in order to investigate the effect of the scale factor on the response of the unreinforced and reinforced systems. on the response of the unreinforced and reinforced systems. on the response of the unreinforced and reinforced systems. The arch is subjected to the self‐weight and to a horizontal mass proportional load distribution The arch is subjected to the self-weight and to a horizontal mass proportional load distribution (p0), as The represen arch ist esubject d in Fiegdu to re th 3, eincreased self‐weight until an d the to complete a horizonta collap l masss e of pro thpeortiona structure. l load Th distribut e resultsi oofn (p ), as represented in Figure 3, increased until the complete collapse of the structure. The results (p0), as represented in Figure 3, increased until the complete collapse of the structure. The results of the push‐over analyses are presented both in terms of capacity curves, and collapse mechanisms. of the push-over analyses are presented both in terms of capacity curves, and collapse mechanisms. the push‐over analyses are presented both in terms of capacity curves, and collapse mechanisms. The capacity curves report the maximum lateral displacement of the arch vs. the base shear The capacity curves report the maximum lateral displacement of the arch vs. the base shear coefﬁcient co The efficient capacity (ba scurves e shear norma report liz the ed maxi by thmu e ow mn lat weig eralh t). displac ement of the arch vs. the base shear (base shear normalized by the own weight). coefficient (base shear normalized by the own weight). Figure 3. Geometry of the arch with the indication of the seismic load condition (p0). Figure Figure 3. 3. Geometry Geometry of of the thear ar ch chwith withthe the indication indication of of the the seismic seismic load load co condition ndition (p0().p ). In order to calibrate the numerical model, an initial comparison was performed with the results of theIn experi ordermental to calibr caam tep the aign nume reporte ricald mo in de [26] l, an . In initi thea lm co entioned mpariso n pa was per pe two rfo identical rmed with arc the hes results were In order to calibrate the numerical model, an initial comparison was performed with the results of the experimental campaign reported in [26]. In the mentioned paper two identical arches were subjected to a vertical concentrated load according to the experimental layout reported in Figure 4. of the experimental campaign reported in [26]. In the mentioned paper two identical arches were subjected to a vertical concentrated load according to the experimental layout reported in Figure 4. The mechanical parameters of the masonry have been estimated by means of experimental tests [26], subjected to a vertical concentrated load according to the experimental layout reported in Figure 4. an The d m are ec h here anic reported al param ein te rsTable of the 1. mason E andr yG have represen been testimated the norm by al me andans the of ta exnpgeernitme ialnt defor al tests ma [t26] ion, The mechanical parameters of the masonry have been estimated by means of experimental tests [26], and are here reported in Table 1. E and G represent the normal and the tangential deformation moduli of masonry, σt and σc the tensile and compressive strengths, Gt and Gc the corresponding and are here reported in Table 1. E and G represent the normal and the tangential deformation moduli moduli of masonry, σt and σc the tensile and compressive strengths, Gt and Gc the corresponding values of fracture energy, c the cohesion, μ the friction factor, and w the specific self‐weight of of masonry, s and s the tensile and compressive strengths, G and G the corresponding values of t c t c values of fracture energy, c the cohesion, μ the friction factor, and w the specific self‐weight of masonry. fracture energy, c the cohesion, m the friction factor, and w the speciﬁc self-weight of masonry. masonry. The results of the macro‐element model are reported in terms of capacity curve (applied force The results of the macro-element model are reported in terms of capacity curve (applied force vs. The results of the macro‐element model are reported in terms of capacity curve (applied force vs. vertical displacement at the loaded section) with the black line, and are compared with the vertical displacement at the loaded section) with the black line, and are compared with the experimental vs. vertical displacement at the loaded section) with the black line, and are compared with the experimental capacity curves or the two specimens (grey lines). In terms of collapse mechanism, the capacity loc expe atirimen on curves and tal th capac ore the opening ittwo y curves specimens sequen or the ce two of (gr th ey spe e lines). pla cimstens ic In hin (grey terms ges lines) are of in .collapse In a gtereem rmsmechanism, ent of co with llap se the me the expe cha location nrimen ism, the tal and location and the opening sequence of the plastic hinges are in agreement with the experimental theevidence opening as sequence well. of the plastic hinges are in agreement with the experimental evidence as well. evidence as well. dimensions in mm dimensions in mm failure mechanism failure mechanism Figure 4. Validation of the numerical model. Figure 4. Validation of the numerical model. Figure 4. Validation of the numerical model. Buildings 2017, 7, 79 6 of 17 Buildings 2017, 7, 79 6 of 17 Table 1. Mechanical property of the masonry. Table 1. Mechanical property of the masonry. E (Mpa) G (Mpa) s (Mpa) s (Mpa) G (N/mm) G (N/mm) c (Mpa) m (-) W (kN/m ) t c t c Buildings 2017, 7, 79 6 of 17 E σt c μ W 2700 1080 0.30 8.53 0.01 0.30 0.26 0.6 18 G (Mpa) σc (Mpa) Gt (N/mm) Gc (N/mm) (Mpa) (Mpa) (Mpa) (‐) (kN/m ) Table 1. Mechanical property of the masonry. 2700 1080 0.30 8.53 0.01 0.30 0.26 0.6 18 Once the proposed model has been validated considering the masonry arch, as described in E σt c μ W G (Mpa) σc (Mpa) Gt (N/mm) Gc (N/mm) Figure 3, and subjected to a concentrated vertical load as reported in Figure 4, the load scenario Once the proposed model has been validated considering the masonry arch, as described in (Mpa) (Mpa) (Mpa) (‐) (kN/m ) corresponding Figure 3, an tod asubj uniform ected to horizontal a concentrat load ed verti is consider cal load ed as ireporte n the following d in Figure(Figur 4, the eload 3). In scenario particular , 2700 1080 0.30 8.53 0.01 0.30 0.26 0.6 18 corresponding to a uniform horizontal load is considered in the following (Figure 3). In particular, Figure 5 reports the capacity curves relative to the different geometries in terms of global base shear Figure 5 reports the capacity curves relative to the different geometries in terms of global base shear V (FigureOn 5a)ceand the in proposed terms of model base ha shear s been coef val ﬁcient idatedC co= nsV ider /iW ng (Figur the ma es5ob), nrybeing arch, as W described the total weight in b b b Vb (Figure 5a) and in terms of base shear coefficient Cb = Vb/W (Figure 5b), being W the total weight of Figure 3, and subjected to a concentrated vertical load as reported in Figure 4, the load scenario of the arch. It can be observed that, as the radius of the arch increases, the global resistance of the the arch. It can be observed that, as the radius of the arch increases, the global resistance of the arch corresponding to a uniform horizontal load is considered in the following (Figure 3). In particular, arch increases as well (Figure 5a). On the contrary, in terms of the base shear coefﬁcient, as the radius increases as well (Figure 5a). On the contrary, in terms of the base shear coefficient, as the radius Figure 5 reports the capacity curves relative to the different geometries in terms of global base shear increases, the peak strength reduces, and all the models tend to the same residual strength (Figure 5b). increases, the peak strength reduces, and all the models tend to the same residual strength (Figure 5b). Vb (Figure 5a) and in terms of base shear coefficient Cb = Vb/W (Figure 5b), being W the total weight of the arch. It can be observed that, as the radius of the arch increases, the global resistance of the arch R=2500mm 4 R=2500mm incr10 eases as well (Figure 5a). On the contrary, in terms of the base shear coefficient, as the radius R=1500mm R=1500mm increases, the peak strength reduces, and all the models tend to the same residual strength (Figure 5b). R=866mm 3 R=866mm R=2500mm R=2500mm 10 4 R=1500mm R=1500mm R=866mm 3 R=866mm 0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Horizontal displacement (mm) 1 Horizontal displacement (mm) (a) (b) 0 0 Figure 5. Capacity curves of the unreinforced arches, expressed in terms of (a) global base shear, and 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Figure 5. Capacity curves of the unreinforced arches, expressed in terms of (a) global base shear, and Horizontal displacement (mm) Horizontal displacement (mm) (b) base shear coefficient. (b) base shear coefﬁcient. (a) (b) With regard to the assessment of the effectiveness of the structural retrofitting of the arch, three Figure 5. Capacity curves of the unreinforced arches, expressed in terms of (a) global base shear, and different typologies of reinforcing are considered. The first one consists of the introduction of a tie With regard to the assessment of the effectiveness of the structural retroﬁtting of the arch, three (b) base shear coefficient. rod, whose diameter varies proportionally to the radius of the arch, from ф10 mm in the case of different typologies of reinforcing are considered. The ﬁrst one consists of the introduction of a tie rod, R = 866 mm, to ф30 mm in the case of R = 2500 mm, with Young’s modulus E = 200 GPa, and an With regard to the assessment of the effectiveness of the structural retrofitting of the arch, three whose diameter varies proportionally to the radius of the arch, from 10 mm in the case of R = 866 mm, ultimate tensile strength equal to fy = 200 Mpa. The diameters of the tie‐rods are empirically chosen different typologies of reinforcing are considered. The first one consists of the introduction of a tie to 30 mm in the case of R = 2500 mm, with Young’s modulus E = 200 GPa, and an ultimate tensile among commercial diameters, keeping constant the ratio between the radius of the arch and the rod, whose diameter varies proportionally to the radius of the arch, from ф10 mm in the case of strength equal to f = 200 Mpa. The diameters of the tie-rods are empirically chosen among commercial diameter of the tie‐rod. In the considered models the tie‐rods’ heights hr with respect to the base of R = 866 mm, to ф30 mm in the case of R = 2500 mm, with Young’s modulus E = 200 GPa, and an diameters, keeping constant the ratio between the radius of the arch and the diameter of the tie-rod. the arch is about R/4 (Figure 6). The yielding stress of the steel has been chosen among widely ultimate tensile strength equal to fy = 200 Mpa. The diameters of the tie‐rods are empirically chosen In the considered models the tie-rods’ heights h with respect to the base of the arch is about R/4 adopted steel typologies, and large enough to keep r the tie‐rods in the elastic field. The other two among commercial diameters, keeping constant the ratio between the radius of the arch and the (Figurstrateg e 6). The ies consist yielding ofstr the ess introduct of the steel ion of has FRP been stri chosen ps, at the among intrados widely andadopted at the extrados steel typologies, surfaces and diameter of the tie‐rod. In the considered models the tie‐rods’ heights hr with respect to the base of respectively (Figure 6). The reinforcement is constituted by strips arranged over the entire width and large enough to keep the tie-rods in the elastic ﬁeld. The other two strategies consist of the introduction the arch is about R/4 (Figure 6). The yielding stress of the steel has been chosen among widely length of the arch made of glass fiber composite material (GFRP) and organic matrix. The adopted of FRPadop strips, ted steel at the tyintrados pologies, and and larg at the e en extrados ough to keep surfaces the tie respectively ‐rods in the (Figur elastic efie 6l).d.The The rot einfor her tw cement o mechanical properties have been set according to [27], and reported in Table 2, in which Ef and ft are strategies consist of the introduction of FRP strips, at the intrados and at the extrados surfaces is constituted by strips arranged over the entire width and length of the arch made of glass ﬁber the tensile module and the ultimate tensile strength of the reinforcement, and tf is the equivalent respectively (Figure 6). The reinforcement is constituted by strips arranged over the entire width and composite material (GFRP) and organic matrix. The adopted mechanical properties have been set thickness. The bond‐slip behaviour is described by the initial shear stiffness of the matrix ks, the length of the arch made of glass fiber composite material (GFRP) and organic matrix. The adopted according to [27], and reported in Table 2, in which E and f are the tensile module and the ultimate ultimate debonding stress tf, the fracture energy Gs, and the friction factor μs. mechanical properties have been set according to [27], and reported in Table 2, in which Ef and ft are tensile strength of the reinforcement, and t is the equivalent thickness. The bond-slip behaviour is the tensile module and the ultimate tensile strength of the reinforcement, and tf is the equivalent described by the initial shear stiffness of the matrix k , the ultimate debonding stress t , the fracture thickness. The bond‐slip behaviour is described by the initial shear stiffness of the matrix ks, the hr energy G , and the friction factor m . s s ultimate debonding stress tf, the fracture energy Gs, and the friction factor μs. Tie rod 10‐20‐30 Extrados FRP strip Intrados FRP strip hr Figure 6. Different reinforcing interventions considered for the arch. Tie rod 10‐20‐30 Extrados FRP strip Intrados FRP strip Figure 6. Different reinforcing interventions considered for the arch. Figure 6. Different reinforcing interventions considered for the arch. Vb (kN) Vb (kN) Cb=Vb/W (-) Cb=Vb/W (-) Buildings 2017, 7, 79 7 of 17 Table 2. Tensile and bond-slip parameters of the FRP reinforcement. Buildings 2017, 7, 79 7 of 17 Tensile Bond-Slip Table 2. Tensile and bond‐slip parameters of the FRP reinforcement. E (GPa) t (mm) t (MPa) f (MPa) k (N/mm ) G (N/mm) m (-) f t f s f s s 450 1473 0.149 20 1.3 2.5 0.75 Tensile Bond‐Slip Ef (GPa) ft (MPa) tf (mm) ks (N/mm ) τf (MPa) Gs (N/mm) μs (‐) 450 1473 0.149 20 1.3 2.5 0.75 Figure 7 shows the failure mechanisms of the reinforced arches, respectively with R = 2500 mm and R = 866 mm. The collapse mechanism observed for the model reinforced with the tie rod is Figure 7 shows the failure mechanisms of the reinforced arches, respectively with R = 2500 mm very similar to the failure mechanism of the unreinforced arch, which is not here reported for the and R = 866 mm. The collapse mechanism observed for the model reinforced with the tie rod is very sake of conciseness. The latter aspect seems to demonstrate that the presence of the tie rod does not similar to the failure mechanism of the unreinforced arch, which is not here reported for the sake of increase the strength of the arch, at least in seismic conditions and neglecting the interaction with the conciseness. The latter aspect seems to demonstrate that the presence of the tie rod does not increase underlying walls. On the contrary, the failure mechanisms of the arches reinforced by means of FRP the strength of the arch, at least in seismic conditions and neglecting the interaction with the strips are characterized by a wide spread of the damage. It is worth to note that, due to the transfer of underlying walls. On the contrary, the failure mechanisms of the arches reinforced by means of FRP tangential stress between the FRP strip and the arch, the presence of FRP strips delays or prevents the strips are characterized by a wide spread of the damage. It is worth to note that, due to the transfer opening of plastic hinges on the surface to which the strips are applied. For all of the investigated cases, of tangential stress between the FRP strip and the arch, the presence of FRP strips delays or prevents the failure mechanism is concentrated in the masonry and in the FRP strips due to the tensile rupture; the opening of plastic hinges on the surface to which the strips are applied. For all of the investigated whereas, no shear no delamination of the reinforcement is encountered. In both cases of strips applied cases, the failure mechanism is concentrated in the masonry and in the FRP strips due to the tensile to the rupture; intrados whereas, and to no th sh e extrados, ear no del the amifailur nation e is ofassociated the reinforc toem the enactivation t is encounter of an ed.intermediate In both cases plastic of strips applied to the intrados and to the extrados, the failure is associated to the activation of an hinge. The latter is located on the extrados surface of the arch in the case of the intrados reinforcing intermediate plastic hinge. The latter is located on the extrados surface of the arch in the case of the (point A in Figure 7a and point A in Figure 7b) or at the intrados surface, closer the support of the 1 2 intrados reinforcing (point A1 in Figure 7a and point A2 in Figure 7b) or at the intrados surface, closer arch, in the case of the extrados reinforcing (point B in Figure 7a and point B in Figure 7b). It is worth 1 2 the support of the arch, in the case of the extrados reinforcing (point B1 in Figure 7a and point B2 in to note that, although the arches reported in Figure 7 are not scaled according to the relevant radius, Figure 7b). It is worth to note that, although the arches reported in Figure 7 are not scaled according they refer to different size of the arch, as better speciﬁed in the caption. to the relevant radius, they refer to different size of the arch, as better specified in the caption. A1 B1 (a) Tie rod ( 30) FRP intrados FRP extrados A2 B2 (b) Tie rod (10) FRP intrados FRP extrados Figure 7. Failure mechanisms of the reinforced (a) R = 2500 mm and (b) R = 866 mm arches. Figure 7. Failure mechanisms of the reinforced (a) R = 2500 mm and (b) R = 866 mm arches. Figure 8 shows the comparison of the considered reinforcing techniques in terms of capacity Figure 8 shows the comparison of the considered reinforcing techniques in terms of capacity curves for two of the three radiuses investigated: the smallest (866 mm) and the largest (2500 mm). curves for two of the three radiuses investigated: the smallest (866 mm) and the largest (2500 mm). The capacity curves and the failure mechanisms of the models reinforced with tie rods are very close Thetocapacity the findicurves ngs relative and the to the failur unrein e mechanisms forced modof el, the whereas models ther einfor capaci ced ty curves with tie of rthe ods arc arehes very reinforced by means of the FRP strips show significant increments, both in terms of strength and close to the ﬁndings relative to the unreinforced model, whereas the capacity curves of the arches ductility. For those models, after the achievement of the peak load a sudden drop in the global reinforced by means of the FRP strips show signiﬁcant increments, both in terms of strength and strength is encountered; such a drop is associated to the opening of a cylindrical hinge in the arch, ductility. For those models, after the achievement of the peak load a sudden drop in the global strength associated to the FRP strip tensile rupture. Then, for larger displacements, the FRP strips‐masonry is encountered; such a drop is associated to the opening of a cylindrical hinge in the arch, associated interface tends to mobilize the tangential force, progressively transferring stresses to the fibres. The to the FRP strip tensile rupture. Then, for larger displacements, the FRP strips-masonry interface latter aspect implies a higher residual force of the strengthened models with respect to the tends to mobilize the tangential force, progressively transferring stresses to the ﬁbres. The latter aspect unstrengthened model. The influence of the arrangement of the FRP reinforcement (at the extrados implies a higher residual force of the strengthened models with respect to the unstrengthened model. or at the intrados) on the global resistance is negligible in the case of R = 866 mm (Figure 8a), while The inﬂuence of the arrangement of the FRP reinforcement (at the extrados or at the intrados) on the this effect is important in the case of R = 2500 mm (Figure 8b). global resistance is negligible in the case of R = 866 mm (Figure 8a), while this effect is important in the case of R = 2500 mm (Figure 8b). Buildings 2017, 7, 79 8 of 17 Buildings 2017, 7, 79 8 of 17 12 Tie rod FRP intrados FRP extrados Tie rod FRP intrados FRP extrados Buildings 2017, 7, 79 8 of 17 12 Tie rod FRP intrados FRP extrados Tie rod FRP intrados FRP extrados 0 30 0.02.0 4.06.0 8.0 0.0 3.0 6.0 9.0 12.0 Horizontal displacement (mm) Horizontal displacement (mm) (a) (b) Figure 8. Capacity curves of the reinforced arch: (a) R = 866 mm; (b) R = 2500 mm. Figure 8. Capacity curves of the reinforced arch: (a) R = 866 mm; (b) R = 2500 mm. 0.02.0 4.06.0 8.0 0.0 3.0 6.0 9.0 12.0 Horizontal displacement (mm) Horizontal displacement (mm) The presence of the FRP composite strips produces an increment of the ultimate bending moment (a) (b) of the cross section of the arch. In order to highlight the contribution of the reinforcement on the The presence of the FRP composite strips produces an increment of the ultimate bending moment structural respo Fig nse, ure th 8. eCapaci eccent ty ri cucirty ves of of th thee re norm inforal ce dac arch: tion ( a()e R= =M 86/6N mm; ), normalize (b) R = 25d 00 wi mm th. respect to the of the cross section of the arch. In order to highlight the contribution of the reinforcement on the height H of the section, along the curvilinear abscissa (s) of the arch, normalized with respect to the structural response, the eccentricity of the normal action (e = M/N), normalized with respect to the The presence of the FRP composite strips produces an increment of the ultimate bending moment arch length Φ, is reported in Figure 9. In particular, the arches with R = 866 mm are considered, both in height H of the section, along the curvilinear abscissa (s) of the arch, normalized with respect to the of the cross section of the arch. In order to highlight the contribution of the reinforcement on the the configurations with FRP at the extrados (Figure 9a) and at the intrados (Figure 9b). The tensile axial arch length structural F, is respo reported nse, the in ec Figur centrieci9 ty . In of th particular e normal ,ac the tion ar (ches e = Mwith /N), normalize R = 866d mm with ar re esconsider pect to thed, e both force (N) is considered positive, and the bending moment (M) is considered positive if it stretches the height H of the section, along the curvilinear abscissa (s) of the arch, normalized with respect to the in the conﬁgurations with FRP at the extrados (Figure 9a) and at the intrados (Figure 9b). The tensile FRP reinforcement fibres. The zero of the abscissa is set at the left abutment, while the unit value arch length Φ, is reported in Figure 9. In particular, the arches with R = 866 mm are considered, both in corresponds to the right abutment of the arch. The reported eccentricities are associated to the axial force (N) is considered positive, and the bending moment (M) is considered positive if it stretches the configurations with FRP at the extrados (Figure 9a) and at the intrados (Figure 9b). The tensile axial peak‐load conditions, which are characterised by the opening of three hinges in both of the considered the FRP reinforcement ﬁbres. The zero of the abscissa is set at the left abutment, while the unit force (N) is considered positive, and the bending moment (M) is considered positive if it stretches the cases. In the model reinforced at the extrados (Figure 9a), two hinges are located at the intrados (at the value corresponds to the right abutment of the arch. The reported eccentricities are associated to the FRP reinforcement fibres. The zero of the abscissa is set at the left abutment, while the unit value normalized abscissa 0.29 and at the right end of the arch) and one hinge corresponding to the tensile peak-load conditions, which are characterised by the opening of three hinges in both of the considered corresponds to the right abutment of the arch. The reported eccentricities are associated to the rupture of the textile, is opened at the extrados at the left end of the arch. In the model reinforced at the cases. In pea the k‐lomodel ad conditions reinfor , wh ced ich at are the chara extrados cterised (Figu by the re op 9a), enin two g of hinges three hing areeslocated in both of at th the e co intrados nsidered (at the intrados, two hinges are located at the extrados (left end of the arch and at the normalized abscissa cases. In the model reinforced at the extrados (Figure 9a), two hinges are located at the intrados (at the normalized abscissa 0.29 and at the right end of the arch) and one hinge corresponding to the tensile 0.67), and one hinge at the intrados, located at the right end of the arch. normalized abscissa 0.29 and at the right end of the arch) and one hinge corresponding to the tensile rupture of the textile, is opened at the extrados at the left end of the arch. In the model reinforced at rupture of the textile, is opened at the extrados at the left end of the arch. In the model reinforced at the the intrados, two hinges are located at the extrados (left end of the arch and at the normalized abscissa intrados, two hinges are located at the extrados (left ende of lim (N the) arch and at the normalized abscissa elim(N) 0.67), and one hinge at the intrados, located at the right end of the arch. 0.67), and one hinge at the intrados, located at the right end of the arch. e=H/2 e=H/2 e=H/2 e=H/2 elim(N) elim(N) elim(N) elim(N) e=H/2 e=H/2 (a) (b) e=H/2 e=H/2 (a) (b)elim(N) elim(N) (a) (b) N>0 N<0 N>0 N<0 N H H <e<e e <e< R =F -N 2 lim lim 2 H c s H F s F s x Rc 2 2 (a) (b) e N G Rc=F s+N N>0 N<0 N>0 N<0 Rc Rc H G HR =F +N c s e <e<e e <e< 2 lim lim 2 H Rc=F s-N H F s F s x Rc H H 2 2 Rc F G F s 2 2 H s H Rc=F s-N <e<e e <e< e 2 lim lim 2 G Rc=F s+N R c c x (c) (d) Rc=F s+N e N H H x R c F F s s 2 2 H H Figure 9. Contribution of the FRP reinforcement at the peak load: normalized abscissa versus Rc=F s-N <e<e e <e< 2 lim lim 2 eccentricity of the acting force of the models with R = 866 reinforced at the (a) extrados and (b) (c) (d) intrados. Cross section internal equilibrium for extrados (c) and intrados (d) reinforcing. Figure 9. Contribution of the FRP reinforcement at the peak load: normalized abscissa versus Figure 9. Contribution of the FRP reinforcement at the peak load: normalized abscissa versus Iteccent is imric po ityrt aofn tthe to acting notice force that of the the di mod strieblsu ti with on R of =the 866 pla reinforced stic hinges at the in ( acor ) extr resp ados on and dence (b ) of the eccentricity of the acting force of the models with R = 866 reinforced at the (a) extrados and (b) intrados. intrados. Cross section internal equilibrium for extrados (c) and intrados (d) reinforcing. peak‐load does not correspond to the locations at collapse (Figure 7), since the rupture of the textile, Cross section internal equilibrium for extrados (c) and intrados (d) reinforcing. It is important to notice that the distribution of the plastic hinges in correspondence of the peak‐load does not correspond to the locations at collapse (Figure 7), since the rupture of the textile, It is important to notice that the distribution of the plastic hinges in correspondence of the peak-load does not correspond to the locations at collapse (Figure 7), since the rupture of the textile, Vb (kN) Vb (kN) Vb (kN) Vb (kN) Buildings 2017, 7, 79 9 of 17 Buildings 2017, 7, 79 9 of 17 corresponding to the third plastic hinge, implies an internal force redistribution which induces the opening of hinges in masonry at different locations. corresponding to the third plastic hinge, implies an internal force redistribution which induces the In Figure 9 the theoretical limit values of the eccentricity, as evaluated through the limit analysis opening of hinges in masonry at different locations. approach [26], are reported. The masonry is considered as a no-tension material and linear-elastic in In Figure 9 the theoretical limit values of the eccentricity, as evaluated through the limit analysis compression, whereas the FRP strips are not capable of transferring compressive force and elastic-brittle approach [26], are reported. The masonry is considered as a no‐tension material and linear‐elastic in in tension. The limit conditions, reported with dashed lines, are associated to the rupture in traction compression, whereas the FRP strips are not capable of transferring compressive force and of the FRP strip (curved lines), and to the tensile action on the masonry (straight lines at the elastic‐brittle in tension. The limit conditions, reported with dashed lines, are associated to the dimensionless eccentricity 0.5). The grey areas in the graphs represent the ﬁeld of the admissible rupture in traction of the FRP strip (curved lines), and to the tensile action on the masonry (straight eccentricities. It is worth noting that the left parts of the arches are in tension while the right parts lines at the dimensionless eccentricity 0.5). The grey areas in the graphs represent the field of the are in compression. This is due to the particular load scenario here considered (i.e., horizontal force admissible eccentricities. It is worth noting that the left parts of the arches are in tension while the distribution right partspr are oportional in comprte oss the ion.self-mass This is due of the to the arch). parti Incul prar oximity load sc ofenar theio abscissa here coassociated nsidered (i.to e., the change horizo of ntal sign force of dist thernormal ibution proportio force, thenal eccentricities to the self‐mas tend s ofto the diver arch ge ). In (see pro Figur ximity e 9of a,b). the In absciss order a to associated to the change of sign of the normal force, the eccentricities tend to diverge (see Figure 9a,b). clarify the equilibrium of the reinforced arch cross section, simple schemes are reported in Figure 9c,d, In order to clarify the equilibrium of the reinforced arch cross section, simple schemes are reported for the case of extrados and intrados reinforcing, respectively. In each ﬁgure the two possible scenarios in Figure 9c,d, for the case of extrados and intrados reinforcing, respectively. In each figure the two are reported: tensile (N > 0) and compressive axial force (N < 0). The grey areas (whose height is possible scenarios are reported: tensile (N > 0) and compressive axial force (N < 0). The grey areas equal to x) and the white ones (whose height is equal to H x) represent the areas in compression and (whose height is equal to x) and the white ones (whose height is equal to H − x) represent the areas in tension, respectively. The internal forces are represented by the tensile action of the FRP strip (F ) and compression and tension, respectively. The internal forces are represented by the tensile action of the the global compression on the masonry (R ). The ultimate equilibrium of the section is imposed by FRP strip (Fs) and the global compression on the masonry (Rc). The ultimate equilibrium of the considering the ultimate value of F and evaluating the corresponding value of x under the hypothesis section is imposed by considering the ultimate value of Fs and evaluating the corresponding value of of linear elastic behaviour of the masonry (conﬁrmed by the numerical simulations). Once the internal x under the hypothesis of linear elastic behaviour of the masonry (confirmed by the numerical forces are computed, the ultimate moment (M ) and the ultimate eccentricity e (N) = M /N can be u u lim simulations). Once the internal forces are computed, the ultimate moment (Mu) and the ultimate easily inferred. eccentricity can be easily inferred. eN() M /N lim u Figure 10 shows the working rates of the reinforcement at the peak load of the arches with Figure 10 shows the working rates of the reinforcement at the peak load of the arches with R = 866 mm reinforced at the extrados (Figure 10a) and at the intrados (Figure 10b). The working R = 866 mm reinforced at the extrados (Figure 10a) and at the intrados (Figure 10b). The working rates are expressed in terms of # /# , being # and # the current and the ultimate tensile strains of f fu f f rates are expressed in terms of εf/εfu, being εf and εfu the current and the ultimate tensile strains of the the textile, respectively. These rates are useful to identify the achievement of the tensile rupture of textile, respectively. These rates are useful to identify the achievement of the tensile rupture of the the reinforcement, which is here identiﬁed at the left end of the arch for the model reinforced at the reinforcement, which is here identified at the left end of the arch for the model reinforced at the extrados, and at the right end of the arch for the model reinforced at the intrados. These ruptures extrados, and at the right end of the arch for the model reinforced at the intrados. These ruptures produce the sudden drops of the global resistance, as observed in the global capacity curves. produce the sudden drops of the global resistance, as observed in the global capacity curves. (a) (b) Figure 10. Contribution of the FRP reinforcement at the peak load: normalized abscissa versus working Figure 10. Contribution of the FRP reinforcement at the peak load: normalized abscissa versus working rates of the reinforcement for the models reinforced at the (a) extrados and (b) intrados. rates of the reinforcement for the models reinforced at the (a) extrados and (b) intrados. Aiming at highlighting the shear bond behaviour of the FRP reinforcement, in Figure 11 shows Aiming at highlighting the shear bond behaviour of the FRP reinforcement, in Figure 11 shows the tangential stress τ at the interface between masonry and FRP reinforcement in correspondence of the tangential stress t at the interface between masonry and FRP reinforcement in correspondence of the peak load (continuous lines), as well as the yielding tangential stress τy(N) at the same step the(d peak ashed load lines (continuous ), dependinlines), g on the as curre wellnt as co the mp yielding ression force tangential on the str interf ess tace (N ()Nat ). Th thee same figures step ref(dashed er to the arches with R = 866 mm reinforced at the extrados (Figure 11a) and at the intrados (Figure 11b). lines), depending on the current compression force on the interface (N). The ﬁgures refer to the arches In both cases, the tangential stress is lower than the corresponding yielding value confirming that with R = 866 mm reinforced at the extrados (Figure 11a) and at the intrados (Figure 11b). In both cases, the debonding mechanism does not occur. The latter results are apparently in contrast with other the tangential stress is lower than the corresponding yielding value conﬁrming that the debonding experimental and numerical results available in the literature, obtained considering similar FRP mechanism does not occur. The latter results are apparently in contrast with other experimental and reinforced prototypes subjected to a vertical eccentric force [27]. The fact that no delamination numerical results available in the literature, obtained considering similar FRP reinforced prototypes Buildings 2017, 7, 79 10 of 17 Buildings 2017, 7, 79 10 of 17 subjected to a vertical eccentric force [27]. The fact that no delamination phenomenon occurs for the phenomenon occurs for the treated cases might be due, in part, to the geometry and in part to the treated cases might be due, in part, to the geometry and in part to the horizontal mass-proportional horizontal mass‐proportional load distribution considered. load distribution considered. (a) (b) Figure 11. Tangential stress at the interface between masonry and FRP reinforcement in correspondence Figure 11. Tangential stress at the interface between masonry and FRP reinforcement in correspondence of the peak load for the models reinforced at the (a) extrados; and (b) intrados. of the peak load for the models reinforced at the (a) extrados; and (b) intrados. A comparison among all the reinforced and unreinforced models, in terms of ultimate load caA paci comparison ty (Vb,max) anamong d increment all the of rresistance einforced (Δ and Vb), unr is re einfor porteced d in models, Table 3. Th in e terms benefi of ts ultimate in terms of load strength resistance are higher in the models with the lowest radius (R = 866 mm) and the beneficial capacity (V , ) and increment of resistance (DV ), is reported in Table 3. The beneﬁts in terms of max b b effects decrease as the radius increases. Furthermore, the comparison of the effects of the extrados strength resistance are higher in the models with the lowest radius (R = 866 mm) and the beneﬁcial and intrados arrangements of the FRP strips demonstrates that the scale effect observed in Figure 8 effects decrease as the radius increases. Furthermore, the comparison of the effects of the extrados is confirmed for all of the cases investigated: in addition, for small radius models the application of and intrados arrangements of the FRP strips demonstrates that the scale effect observed in Figure 8 FRP strips to the intrados and to the extrados provides similar effects (see the first column of Table 3), is conﬁrmed for all of the cases investigated: in addition, for small radius models the application of while in the case of large radius models the benefit associated to the extrados FRP reinforcement is FRP strips to the intrados and to the extrados provides similar effects (see the ﬁrst column of Table 3), significantly higher if compared to the intrados reinforcing. while in the case of large radius models the beneﬁt associated to the extrados FRP reinforcement is signiﬁcantly higher if compared to the intrados reinforcing. Table 3. Ultimate strength of the arches and increment of the ultimate load with respect to the unreinforced configuration. Table 3. Ultimate strength of the arches and increment of the ultimate load with respect to the R = 866 mm R = 1500 mm R = 2500 mm unreinforced conﬁguration. Model Vb,max (kN) ΔVb (%) Vb,max (kN) ΔVb (%) Vb,max (kN) ΔVb (%) Unreinforced 1.23 R ‐ = 866 mm 3.43 R = ‐ 1500 mm 9.26 R = ‐ 2500 mm Model Tie rod 1.23 0 3.43 0 9.26 0 V , (kN) DV (%) V , (kN) DV (%) V , (kN) DV (%) max max max b b b b b b Intrados FRP 10.83 780 19.14 458 34.10 268 Unreinforced 1.23 - 3.43 - 9.26 - Extrado Tie rs od FRP 1.23 10.72 0772 3.43 21.93 0539 9.26 43.00 0364 Intrados FRP 10.83 780 19.14 458 34.10 268 Extrados FRP 10.72 772 21.93 539 43.00 364 Finally, in order to investigate the influence of the fibre content on the lateral strength of the structure, a further parametric investigation on the arch with R = 866 mm reinforced at the intrados is performed. In particular, a model considering a double equivalent thickness of reinforcement Finally, in order to investigate the inﬂuence of the ﬁbre content on the lateral strength of the (tf = 0.298 mm) is investigated. structure, a further parametric investigation on the arch with R = 866 mm reinforced at the intrados In Figure 12, the corresponding capacity curve is reported (Figure 12a) together with the trend is performed. In particular, a model considering a double equivalent thickness of reinforcement of the tangential stress at the interface between masonry and FRP reinforcement (Figure 12b), and (t = 0.298 mm) is investigated. the damage pattern in correspondence of the peak load and the collapse of the arch (Figure 12c,d). In In Figure 12, the corresponding capacity curve is reported (Figure 12a) together with the trend Figure 12a,b, the pushover curve and the corresponding tangential stresses of the previously of the tangential stress at the interface between masonry and FRP reinforcement (Figure 12b), and investigated model (tf = 0.149 mm) are reported for comparison. An increment of strength and the damage pattern in correspondence of the peak load and the collapse of the arch (Figure 12c,d). ductility is associated to the model with tf = 0.298 mm if compared to the standard thickness model. In Figure 12a,b, the pushover curve and the corresponding tangential stresses of the previously However in this case the ultimate lateral capacity is limited by the activation of the delamination as investigated model (t = 0.149 mm) are reported for comparison. An increment of strength and ductility demonstrated by tangential stress distribution, which overlaps the yielding stress close to the right is associated to the model with t = 0.298 mm if compared to the standard thickness model. However in end of the arch (Figure 12b). At the peak load, the opening of the cylindrical hinges at the intrados is this case the ultimate lateral capacity is limited by the activation of the delamination as demonstrated significantly delayed by the presence of FRP reinforcement (Figure 12c), causing a significant by tangential stress distribution, which overlaps the yielding stress close to the right end of the arch delamination in the post‐peak branch (Figure 12d). (Figure 12b). At the peak load, the opening of the cylindrical hinges at the intrados is signiﬁcantly delayed by the presence of FRP reinforcement (Figure 12c), causing a signiﬁcant delamination in the post-peak branch (Figure 12d). Buildings 2017, 7, 79 11 of 17 Buildings 2017, 7, 79 11 of 17 Buildings 2017, 7, 79 11 of 17 (a) (b) (a) (b) (c) (d) (c) (d) Figure 12. Arch with R = 866 mm reinforced at the intrados with a double thickness of the textile Figure 12. Arch with R = 866 mm reinforced at the intrados with a double thickness of the textile Figure 12. Arch with R = 866 mm reinforced at the intrados with a double thickness of the textile (tf = 0.298 mm): (a) capacity curve; (b) tangential stress at the interface between masonry and FRP (t = 0.298 mm): (a) capacity curve; (b) tangential stress at the interface between masonry and FRP (tf = 0.298 mm): (a) capacity curve; (b) tangential stress at the interface between masonry and FRP reinforcement in correspondence of the peak load, damage pattern at (c) the peak load; and (d) reinforcement in correspondence of the peak load, damage pattern at (c) the peak load; and (d) collapse. reinforcement in correspondence of the peak load, damage pattern at (c) the peak load; and (d) collapse. collapse. 4.2. Hemisperical Dome 4.2. Hemisperical Dome 4.2. Hemisperical Dome A further A further example example relative relative to a to double a double curvatur curv eavault ture va is ult consider is cons edider in this ed in section. this sectio In particular n. In , A further example relative to a double curvature vault is considered in this section. In particular, a masonry hemi‐spherical dome, already studied in the elastic field in [28], with a a masonry hemi-spherical dome, already studied in the elastic ﬁeld in [28], with a thickness t = 20 cm, particular, a masonry hemi‐spherical dome, already studied in the elastic field in [28], with a thickness t = 20 cm, and whose geometric layout is reported in Figure 13 is here studied with and whose geometric layout is reported in Figure 13 is here studied with reference to the nonlinear thickness t = 20 cm, and whose geometric layout is reported in Figure 13 is here studied with reference to the nonlinear field. The masonry dome is initially subjected to its own self‐weight, and ﬁeld. The masonry dome is initially subjected to its own self-weight, and subsequently, a horizontal reference to the nonlinear field. The masonry dome is initially subjected to its own self‐weight, and subsequently, a horizontal force distribution proportional to the masses (p0) is applied until collapse force distribution proportional to the masses (p ) is applied until collapse in order to investigate a subsequently, a horizontal force distribution proportional to the masses (p0) is applied until collapse in order to investigate a typical load scenario in seismic conditions. typical load scenario in seismic conditions. in order to investigate a typical load scenario in seismic conditions. Figure 13. Geometry of the dome and control points. Figure 13. Geometry of the dome and control points. Figure 13. Geometry of the dome and control points. The displacements of three different nodes has been monitored according to the layout showed The displacements of three different nodes has been monitored according to the layout showed in Figure 13. The adopted mechanical properties for the numerical simulations are reported in Table 4. in The Figure displacements 13. The adopted of thr mec eehdif anic fer alent pronodes pertieshas for been the num monitor erical ed simu accor lation ding s are to re the por layout ted in Table showed 4. in The results are reported in Figure 14 in terms of collapse mechanisms, damage patterns (Figure 14a), The results are reported in Figure 14 in terms of collapse mechanisms, damage patterns (Figure 14a), Figure 13. The adopted mechanical properties for the numerical simulations are reported in Table 4. and capacity curves with respect to the three monitored nodes (Figure 14b). and capacity curves with respect to the three monitored nodes (Figure 14b). The results are reported in Figure 14 in terms of collapse mechanisms, damage patterns (Figure 14a), Table 4. Mechanical characteristics of the masonry. and capacity curves with respect to the three monitored nodes (Figure 14b). Table 4. Mechanical characteristics of the masonry. E (Mpa) G (Mpa) σt (Mpa) σc (Mpa) Gt (N/mm) Gc (N/mm) c (Mpa) μ (‐) w (kN/m ) Table 4. Mechanical characteristics of the masonry. 3 E (Mpa) G (Mpa) σt (Mpa) σc (Mpa) Gt (N/mm) Gc (N/mm) c (Mpa) μ (‐) w (kN/m ) 1200 480 0.15 2.50 0.10 0.5 0.15 0.7 25 1200 480 0.15 2.50 0.10 0.5 0.15 0.7 25 E (Mpa) G (Mpa) s (Mpa) s (Mpa) G (N/mm) G (N/mm) c (Mpa) m (-) w (kN/m ) t c t c 1200 480 0.15 2.50 0.10 0.5 0.15 0.7 25 Buildings 2017, 7, 79 12 of 17 The collapse mechanism is characterized by a large damaged area along the meridians in the positive direction of the load and two smaller damaged areas at about a latitude of 30 at the two symmetric upper and lower sides orthogonal to the direction of the load distribution. In terms of capacity curves, the structure is characterized by a signiﬁcant peak load (C = 0.6) and by a signiﬁcant residual resistance as well. It is worth to note that the horizontal displacements of the monitored points decrease as the height of the control point increases. (a) 0.8 0.6 0.4 0.2 P3 P2 P1 0 5 10 15 20 Lateral displacement (mm) (b) Figure 14. Response of the unreinforced dome in terms of (a) failure mechanism; and (b) capacity curves. Regarding the structural retroﬁtting strategies, the application of FRP strips has been adopted. The strips (which have a width equal to 120 cm) have been arranged along the parallels to prevent the occurrence of damage along the meridians. Two different levels of retroﬁtting have been considered: a soft one with two strips centred at the latitudes of 22.5 and 49.5 (Figure 15a), and a strong retroﬁtting with four strips centred, respectively, at the latitudes of 13.5 , 31.5 , 49.5 , and 67.5 (Figure 15b). The same mechanical properties of the reinforcement considered for the circular arches, are adopted (see Table 2). 1 C =V /W (-) b b Buildings 2017, 7, 79 13 of 17 Buildings 2017, 7, 79 13 of 17 Buildings 2017, 7, 79 13 of 17 (a) (b) Figure 15. Typologies of reinforcing: (a) soft and (b) strong retrofitting. Figure 15. Typologies of reinforcing: (a) soft and (b) strong retroﬁtting. Again, the results are reported in terms of collapse mechanisms (Figure 16), and pushover (a) (b) curves, considering the same three monitored displacements of the unreinforced configuration Again, the results are reported in terms of collapse mechanisms (Figure 16), and pushover curves, Figure 15. Typologies of reinforcing: (a) soft and (b) strong retrofitting. (Figure 17). As expected, increasing resistances are obtained with both the softly and strongly considering the same three monitored displacements of the unreinforced conﬁguration (Figure 17). reinforced models with respect to the unreinforced configuration. In both cases the three monitored As expected, Agai incr n, the easing resulrtesistances s are reported are obtained in terms of with collap both se mech the softly anisms and (Fig str urongly e 16), ran einfor d pushover ced mode ls nodes show closer displacements to each other. Nevertheless, only in the post peak branches the curves, considering the same three monitored displacements of the unreinforced configuration with respect to the unreinforced conﬁguration. In both cases the three monitored nodes show closer lowest of the monitored nodes have larger displacements than the other two. The latter aspect is due (Figure 17). As expected, increasing resistances are obtained with both the softly and strongly displacements to each other. Nevertheless, only in the post peak branches the lowest of the monitored to the confinement effect of the FRP strips, as demonstrated also by the damage distribution at the reinforced models with respect to the unreinforced configuration. In both cases the three monitored nodes have larger displacements than the other two. The latter aspect is due to the conﬁnement effect collapse, which show how plastic strains develop only along the unreinforced parts of the meridians. nodes show closer displacements to each other. Nevertheless, only in the post peak branches the of the FRP strips, as demonstrated also by the damage distribution at the collapse, which show how The failure mechanism of both reinforced models are characterized by a spread damage at the base lowest of the monitored nodes have larger displacements than the other two. The latter aspect is due plastic secti strains on of devel the do opmonly e, bealong low the the first unr FR einfor P strip ced. parts In the of model the meridians. reinforcedThe by failur two st erips mechanism (soft of to the confinement effect of the FRP strips, as demonstrated also by the damage distribution at the reinforcing), the damage propagates along the entire height of the dome involving a limited radial both reinforced models are characterized by a spread damage at the base section of the dome, below collapse, which show how plastic strains develop only along the unreinforced parts of the meridians. portion (Figure 16a). A different failure mode is observed for the model reinforced by four FRP the ﬁrst FRP strip. In the model reinforced by two strips (soft reinforcing), the damage propagates The failure mechanism of both reinforced models are characterized by a spread damage at the base strips (strong reinforcing): in this case the damage propagates above the lowest strip involving a along the entire height of the dome involving a limited radial portion (Figure 16a). A different failure section of the dome, below the first FRP strip. In the model reinforced by two strips (soft large portion of the dome, the damage doesn’t propagate at the top of the dome (Figure 16b). modereinforcing is observed ), thfor e damage the model propag reinfor ates alo ced ng b th ye four entire FRP height strips of the (str dome ong rin einfor volving cing): a limin ited this radial case the portion (Figure 16a). A different failure mode is observed for the model reinforced by four FRP damage propagates above the lowest strip involving a large portion of the dome, the damage doesn’t strips (strong reinforcing): in this case the damage propagates above the lowest strip involving a propagate at the top of the dome (Figure 16b). large portion of the dome, the damage doesn’t propagate at the top of the dome (Figure 16b). (a) (a) (b) Figure 16. Failure mechanisms of the reinforced models: dome with (a) soft; and (b) strong reinforcement. (b) Figure 16. Failure mechanisms of the reinforced models: dome with (a) soft; and (b) strong reinforcement. Figure 16. Failure mechanisms of the reinforced models: dome with (a) soft; and (b) strong reinforcement. Buildings 2017, 7, 79 14 of 17 Buildings 2017, 7, 79 14 of 17 1.2 1.2 Buildings 2017, 7, 79 14 of 17 1.2 1.2 0.8 0.8 1 1 0.6 0.6 0.8 0.8 0.4 0.4 0.6 0.6 0.2 0.2 P3 P2 P1 P3 P2 P1 0.4 0.4 0 5 10 15 20 0 5 10 15 20 0.2 0.2 P3 P2 P1 P3 P2 P1 Lateral displacement (mm) Lateral displacement (mm) (a) (b) 0 5 10 15 20 0 5 10 15 20 Lateral displacement (mm) Lateral displacement (mm) Figure 17. Capacity curves of the reinforced models: (a) soft and (b) strong retrofitting. Figure 17. Capacity curves of the reinforced models: (a) soft and (b) strong retroﬁtting. (a) (b) The comparison in terms of pushover curves between the unreinforced configuration and the Figure 17. Capacity curves of the reinforced models: (a) soft and (b) strong retrofitting. The comparison in terms of pushover curves between the unreinforced conﬁguration and the two retrofitted domes, as reported in Figure 18, considering as monitored displacement P1, shows two retroﬁtted domes, as reported in Figure 18, considering as monitored displacement P , shows the effectiveness of the FRP retrofitting technique, which leads to a significant 1 The comparison in terms of pushover curves between the unreinforced configuration and the the effectiveness of the FRP retroﬁtting technique, which leads to a signiﬁcant improvement in terms improvement in terms of resistance without implying any global stiffness alteration, thus two retrofitted domes, as reported in Figure 18, considering as monitored displacement P1, shows of resistance without implying any global stiffness alteration, thus guaranteeing that no signiﬁcant guaranteeing that no significant change of the seismic demand for the structure occurs. On the other the effectiveness of the FRP retrofitting technique, which leads to a significant hand, the presence of FRP strips, not only increases the peak load of the arch, but significantly delays change of the seismic demand for the structure occurs. On the other hand, the presence of FRP strips, improvement in terms of resistance without implying any global stiffness alteration, thus the loss of resistance in the post‐peak branch (from around 2.5 mm for the unreinforced dome, to not only increases the peak load of the arch, but signiﬁcantly delays the loss of resistance in the guaranteeing that no significant change of the seismic demand for the structure occurs. On the other around 10 mm for the case of the strongly retrofitted dome), thus guaranteeing to the dome a larger post-peak branch (from around 2.5 mm for the unreinforced dome, to around 10 mm for the case of hand, the presence of FRP strips, not only increases the peak load of the arch, but significantly delays ductility as well. In Table 5 the ultimate lateral resistance (Cb,max) and the percentages of strength theth str e lo ongly ss of rresist etroﬁtted ance in dome), the pos thus t‐pea guaranteeing k branch (from to aro theun dome d 2.5 amm lar ger for the ductility unreinf as orced well. do In me T,able to 5 increment (ΔCb) are reported, highlighting the enhancement associated to the FRP reinforcement around 10 mm for the case of the strongly retrofitted dome), thus guaranteeing to the dome a larger the ultimate lateral resistance (C , ) and the percentages of strength increment (DC ) are reported, max b b application. The softly retrofitted model presents a residual lateral strength close to that relative to ductility as well. In Table 5 the ultimate lateral resistance (Cb,max) and the percentages of strength highlighting the enhancement associated to the FRP reinforcement application. The softly retroﬁtted the unreinforced model, whereas the strongly retrofitted model presents a higher value of residual increment (ΔCb) are reported, highlighting the enhancement associated to the FRP reinforcement model presents a residual lateral strength close to that relative to the unreinforced model, whereas the resistance due to a larger spreading of the damage at the ultimate condition, as shown in Figure 18. application. The softly retrofitted model presents a residual lateral strength close to that relative to strongly retroﬁtted model presents a higher value of residual resistance due to a larger spreading of It is worth to point out that the numerical investigation reported in this section considers only the unreinforced model, whereas the strongly retrofitted model presents a higher value of residual the damage at the ultimate condition, as shown in Figure 18. the load scenario corresponding to a horizontal force distribution proportional to the masses, resistance due to a larger spreading of the damage at the ultimate condition, as shown in Figure 18. It is worth to point out that the numerical investigation reported in this section considers only the representative of seismic condition. Nevertheless, structures can be subjected to very different load It is worth to point out that the numerical investigation reported in this section considers only load scenario corresponding to a horizontal force distribution proportional to the masses, representative scenarios (e.g., static conditions, concentrated loads). Further investigations to assess the the load scenario corresponding to a horizontal force distribution proportional to the masses, of seismic condition. Nevertheless, structures can be subjected to very different load scenarios effectiveness of FRP strengthening technique, under different seismic loads distributions (e.g., representative of seismic condition. Nevertheless, structures can be subjected to very different load (e.g., static conditions, concentrated loads). Further investigations to assess the effectiveness of proportional to the eigenmodes) or static loads should be investigated in further studies. scenarios (e.g., static conditions, concentrated loads). Further investigations to assess the FRP strengthening technique, under different seismic loads distributions (e.g., proportional to the effectiveness of FRP strengthening technique, under different seismic loads distributions (e.g., eigenmodes) or static loads should be investigated in further studies. proportional to the eigenmodes) or static loads should be investigated in further studies. 0.8 0.6 0.8 0.4 0.6 strong reinforcement 0.2 0.4 soft reinforcement unreinforced strong reinforcement 0.2 soft reinforcement 0 5 10 15 20 unreinforced Lateral displacement (mm) 0 5 10 15 20 Figure 18. Comparison of the capacity curves of the unreinforced and reinforced models. Lateral displacement (mm) Figure 18. Comparison of the capacity curves of the unreinforced and reinforced models. Figure 18. Comparison of the capacity curves of the unreinforced and reinforced models. C =V /WC (-)=V /W (-) b b b b C =V /W C (= -)V /W (-) b b b b C =V /WC (-)=V /W (-) b b b b Buildings 2017, 7, 79 15 of 17 Table 5. Ultimate strength of the domes. Model C , (-) DC (%) b max b Unreinforced 0.60 - Softly retroﬁtted 0.75 25 Strongly retroﬁtted 1.00 67 5. Conclusions A comprehensive discrete element strategy to simulate the nonlinear behaviour of existing masonry structures is employed here. The adopted model, based on a simple but effective mechanical scheme, was initially conceived for the nonlinear simulation of the in-plane behaviour of the masonry panels, and then upgraded to account for the out-of-plane behaviour and for the presence of curved elements (such as arches and vaults). More recently, the same modelling strategy has been extended with a new discrete element to model FRP strips, able to interact with a masonry support. In this paper, the numerical results obtained with this strategy are shown, aiming at demonstrating its capability to grasp the pre- and post-retroﬁtting capacities in seismic conditions. The approach has been ﬁrst validated with a comparison with the results obtained in the nonlinear static context on a unreinforced masonry arch; then, the beneﬁts provided by traditional and innovative retroﬁtting techniques (namely insertion of tie rods and application of FRP strips) are assessed and discussed. Signiﬁcant vault typologies with a scheme of both single and double curvature masonry structures are considered. The results relative to the arches are validated by the comparison with analytical results, as obtained through the limit analysis approach in order to demonstrate the effectiveness of the proposed approach to grasp the ultimate behaviour of the reinforced masonry cross sections (activation of the plastic hinges), and the changing of the global collapse of the structure. The proposed approach, being based on a model in which masonry and FRP strips are modelled with separate elements interacting with each other by means of discrete interfaces, is able to clearly identify the actual failure mode of the structure. The seismic load scenario, which, in spite of its high risk is not very debated in the academic literature, is here investigated, and the effectiveness of widely adopted FRP reinforcement arrangements are assessed and discussed. In spite of the relevance of the achieved results, in the future further investigations will be needed to assess different retroﬁtting techniques, also considering other load scenarios and structural typologies. In addition, with regard to the application of FRP reinforcements, different disposals of the strips have to be investigated with the aim of providing useful guidelines for the optimal retroﬁtting design. Author Contributions: B.P. and F.C. conceived the investigation strategy; B.P., C.C. and F.C. developed and calibrated the numerical models, analysed the results and wrote the paper; I.C., S.C and P.B.L. supervised the research, approved the outcome of the numerical investigations and revised the paper. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Huerta, S. Structural Design in the Work of Gaudi. J. Archit. Sci. Rev. 2006, 49, 324–339. [CrossRef] 2. Foti, D.; De Tommasi, D. An Innovative Modular System for the Building of Timber Cylindrical Roofs. Int. J. Mech. 2013, 7, 226–233. 3. Fabbrocino, F.; Farina, I.; Berardi, V.P.; Ferreira, A.J.M.; Fraternali, F. On the thrust surface of unreinforced and FRP-/FRCM-reinforced masonry domes. Compos. Part B Eng. 2013, 83, 297–305. [CrossRef] 4. Carpentieri, G.; Modano, M.; Fabbrocino, F.; Feo, L.; Fraternali, F. On the minimal mass reinforcement of masonry structures with arbitrary shapes. Meccanica 2017, 1, 1561–1576. [CrossRef] 5. Fraternali, F.; Carpentieri, G.; Modano, M.; Fabbrocino, F.; Skelton, R.E. A tensegrity approach to the optimal reinforcement of masonry domes and vaults through ﬁber-reinforced composite materials. Compos. Struct. 2015, 134, 247–254. [CrossRef] Buildings 2017, 7, 79 16 of 17 6. Foti, D. On the numerical and experimental strengthening assessment of tufa masonry with FRP. Mech. Adv. Mater. Struct. 2013, 20, 163–175. [CrossRef] 7. Capozucca, R. Experimental FRP/SRP-historic masonry delamination. Compos. Struct. 2010, 92, 891–903. [CrossRef] 8. Valluzzi, M.R.; Oliveira, D.V.; Caratelli, A.; Castori, G.; Corradi, M.; De Felice, G.; Garbin, E.; Garcia, D.; Garmendia, L.; Grande, E.; et al. Round Robin Test for composite-to-brick shear bond characterization. Mater. Struct. 2012, 45, 1761–1791. [CrossRef] 9. Napoli, A.; de Felice, G.; De Santis, S.; Realfonzo, R. Bond behaviour of Steel Reinforced Polymer strengthening systems. Compos. Struct. 2016, 152, 499–515. [CrossRef] 10. Bertolesi, E.; Fabbrocino, F.; Formisano, A.; Grande, E.; Milani, G. FRP-Strengthening of Curved Masonry Structures: Local Bond Behavior and Global Response. In Proceedings of the Mechanics of Masonry Structures Strengthened with Composite Materials (MURICO5), Bologna, Italy, 28–30 June 2017. 11. Malena, M.; de Felice, G. Debonding of composites on a curved masonry substrate: Experimental results and analytical formulation. Compos. Struct. 2014, 112, 194–206. [CrossRef] 12. Caliò, I.; Marletta, M.; Pantò, B. A new discrete element model for the evaluation of the seismic behaviour of unreinforced masonry buildings. Eng. Struct. 2012, 40, 327–338. [CrossRef] 13. Marques, R.; Lourenço, P.B. Possibilities and comparison of structural component models for the seismic assessment of modern unreinforced masonry buildings. Comput. Struct. 2011, 89, 2079–2091. [CrossRef] 14. Caliò, I.; Pantò, B. A macro-element modelling approach of Inﬁlled Frame Structures. Comput. Struct. 2014, 143, 91–107. [CrossRef] 15. Caliò, I.; Cannizzaro, F.; D’Amore, E.; Marletta, M.; Pantò, B. A new discrete-element approach for the assessment of the seismic resistance of composite reinforced concrete-masonry buildings. Proc. AIP Am. Inst. Phys. 2008, 1020, 832–839. 16. Caddemi, S.; Caliò, I.; Cannizzaro, F.; Pantò, B. A new computational strategy for the seismic assessment of inﬁlled frame structures. In Proceedings of the 2013 Civil-Comp Proceedings, Sardinia, Italy, 3–6 September 2013. 17. Pantò, B.; Cannizzaro, F.; Caliò, I.; Lourenço, P.B. Numerical and experimental validation of a 3D macro-model for the in-plane and out-of-plane behaviour of unreinforced masonry walls. Int. J. Archit. Herit. 2017, 1–19. [CrossRef] 18. Pantò, B. La Modellazione Sismica Degli Ediﬁci in Muratura. Un Approccio Innovativo Basato su un Macro-Elemento spaziale. Ph.D. Thesis, University of Catania, Catania, Italy, April 2007. 19. Caliò, I.; Cannizzaro, F.; Marletta, M. A discrete element for modeling masonry vaults. Adv. Mater. Res. 2010, 133–134, 447–452. [CrossRef] 20. Caddemi, S.; Caliò, I.; Cannizzaro, F.; Pantò, B. The Seismic Assessment of Historical Masonry Structures. In Proceedings of the 12th International Conference on Computational Structures Technology, Naples, Italy, 2–5 September 2014. 21. Cannizzaro, F. Un Nuovo Approccio di Modellazione Della Risposta Sismica di Ediﬁci Storici. Ph.D. Thesis, University of Catania, Catania, Italy, April 2011. 22. Caddemi, S.; Caliò, I.; Cannizzaro, F.; Lourenço, P.B.; Pantò, B. FRP-reinforced masonry structures: Numerical modelling by means of a new discrete element approach. In Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Rhodes Island, Greece, 15–17 June 2017. 23. Cannizzaro, F.; Lourenço, P.B. 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Extrados strengthening of brick masonry arches with PBO–FRCM composites: Experimental and analytical investigations. Compos. Struct. 2016, 149, 184–196. [CrossRef] 27. Basilio, I. Strenghtening of Arched Masonry Structures with Composite Materials. Ph.D. Thesis, University of Minho, Guimares, Portugal, July 2007. 28. Caddemi, S.; Caliò, I.; Cannizzaro, F.; Occhipinti, G.; Pantò, B. A parsimonious discrete model for the seismic assessment of monumental structures. In 2015 Civil-Comp Proceedings; Civil-Comp Press: Stirlingshire, UK, 2015. © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Nonlinear Modelling of Curved Masonry Structures after Seismic Retrofit through FRP Reinforcing

Nonlinear Modelling of Curved Masonry Structures after Seismic Retrofit through FRP Reinforcing

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Nonlinear Modelling of Curved Masonry Structures after Seismic Retrofit through FRP Reinforcing

Nonlinear Modelling of Curved Masonry Structures after Seismic Retrofit through FRP Reinforcing

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