Effect of Masonry Inﬁll Constitutive Law on the Global Response of Inﬁlled RC Buildings

: Masonry-inﬁlled reinforced concrete frames represent a very common construction typology across the Mediterranean countries. The presence of inﬁlls substantially modiﬁes the global seismic performances of buildings in terms of strength, stiffness, and energy dissipation. Although several research studies focused on the overall performances of inﬁlled reinforced concrete frames, the modeling of inﬁll panels remains an open issue due to the complex interaction between the inﬁll and the frame and the uncertainties involved in the deﬁnition of the problem. In the present paper, an existing masonry-inﬁlled RC frame designed according to obsolete seismic codes is chosen as a case study. A reﬁned three-dimensional ﬁnite element model is built for performing nonlinear static and time-history analyses in order to investigate some signiﬁcant aspects related to the modeling of inﬁlls. In particular, it is investigated the effect of different inﬁll constitutive models on the seismic performance of inﬁlled RC building expressed in terms of engineering demand parameters such as interstory drift ratios and peak ﬂoor accelerations, and on the generation of damage fragility curves.


Introduction
Reinforced concrete (RC) buildings infilled with masonry panels represent a relevant portion of the existing building stock across the Mediterranean countries [1,2]. According to the current construction practice, infill panels are encased between framing members significantly influencing the structural response under seismic loadings. Observations from past earthquakes demonstrate that masonry infill walls often behave as primary structural elements [3]. The distribution in plan and elevation of masonry panels strongly influences the behavior of the building at the global level leading to an increase of structure lateral stiffness [4][5][6], strength [7,8], and energy dissipation, and strongly affecting the seismic performances and the collapse mechanism [9] of the structures, and the building residual capacity [10][11][12][13]. Further, the presence of an irregular infill layout may lead to additional torsional effects or the development of soft-story mechanisms [14], while a regular layout contributes to the reduction of lateral deformations [6]. Even in presence of a regular layout, the abrupt failure of some panels during an earthquake may lead to modification of the geometrical configuration with a sudden manifestation of the mentioned effects. Moreover, the interaction between infills and the surrounding frame significantly influences the building behavior at the local level leading to a modification of internal force distributions that may trigger the activation of brittle collapse mechanisms at the RC member extremities [15][16][17]. The contribution of infill panels depends on a series of geometric and mechanical features of the masonry infills and the surrounding frame. Material, consistency, and opening percentage of infill panels are extremely variable across the whole Mediterranean countries depending on the material availability, construction techniques, and environmental conditions. Due to the significant contribution of masonry infills to the seismic performances of RC frames both and the local and global level, the modeling of infill panel represents a crucial aspect when the actual building response under cyclic actions needs to be reproduced. Although the actual role of infills on the lateral response of buildings is widely recognized, different modeling approaches have been proposed and developed to account for the infill contribution and interaction with the surrounding frame, and there is still a lacking of well-established and reliable analytical models for the simulation [18,19].
Most refined methods, such as micro-modeling approaches, are based on a finite element representation of frame and infills and requires an accurate definition of constitutive relationships describing the cyclic behavior of bricks and mortar joints. These modeling approaches have the advantage of allowing to capture the development of local phenomena. However, due to a large amount of information and the remarkable computational effort required, the adoption of simpler methods is generally preferred. In particular, the so-called macro-modeling approach that employs equivalent struts is one of the most adopted in the numerical analysis of infilled frames and widely used in loss estimation framework [9,20] or to generate mechanical-based fragility curves [21]. Different approaches are available in the literature depending on the number, the geometrical configuration of the equivalent struts, and the constitutive models describing the cyclic behavior of infill panels. Depending on the mechanical model and the response envelope adopted, the results in terms of building response may significantly vary [19].
The choice of the most appropriate analytical model to be used for the assessment of masonry-infilled RC buildings is of paramount importance when the global building performances have to be estimated, especially in terms of damage and repair costs. In fact, the adoption of alternative constitutive models to simulate the presence of infill panels may lead to a different distribution of deformations and accelerations along the height, possibly affecting the distribution and amount of damage and leading to different collapse modes. While several authors studied the static cyclic response of single-bays single-story frames, often focusing on the comparison between the results via analytical models and experimental tests (e.g., [9,19]), very few studies focused on the effect of different infill modeling approaches on the global seismic performances of buildings under dynamic loadings. Based on these considerations, further research is still required to provide extensive indications to account for the response of masonry infills. To this scope, this paper investigates the effect of the adopted infill constitutive model on the building response in terms of engineering demand parameters, which are directly related to damage and repair costs, for a case study building.
In the present study, an existing masonry-infilled RC frame constructed according to obsolete seismic code provisions is considered as a case study. Three different strut backbone curves are employed and the effect of the infill model on the global seismic performances of RC buildings is analyzed. Static pushover and incremental dynamic analyses are performed on a refined three-dimensional model adopting a ground motion bin consisting of 42 records in order to deduce some critical observations about the dependence of the building seismic performances on the infill model. The comparative analysis allowed us to investigate the role of the strut backbone curve of the infill on the seismic behavior of RC frames in terms of attainment of damage states, distribution of interstory drift ratios, and peak floor accelerations. Section 2 introduces the most common techniques to explicitly model the contribution of the infill panel. Focusing on the single-strut modeling approach and with reference to the different backbone curves adopted in this study, the assumptions regarding the strut hysteretic behavior and the geometrical and mechanical characterization are introduced. Section 3 presents the research methodology. Section 4 introduces the case-study building and describes the refined three-dimensional finite element model. In Section 5, the influence of the infill backbone curve is analyzed in terms of static response, while in Section 6 the influence of the infill backbone curve on the seismic performances of the building is analyzed in terms of damage intensity and distribution along the height.

Single-Strut Model
The single-strut model is widely adopted by engineers and researchers since it provides a good compromise between simplicity and accuracy in predicting analytical response. The model involves the replacement of each masonry infill through two opposite diagonal pin-joints struts. Each strut should be able to reproduce with an adequate degree of approximation the damage evolution in terms of stiffness and strength when performing nonlinear static or dynamic analyses. The identification of the equivalent diagonal strut requires the assignment of specific mechanical properties that, for existing buildings, is affected by large uncertainties related not only to masonry properties but also to manufacturing and arrangement details. The main parameters characterizing the masonry strut model are the geometry and constitutive law shape for the masonry strut.

Single-Strut Geometry
The geometry of the single-strut model varies depending on the relative position of the strut with respect to the main diagonal connecting opposite nodes. For instance, Bertoldi et al. [37], Panagiotakos and Fardis [38], Dolšek and Fajfar [7], Asteris et al. [39], among others, adopt a single compression strut for each direction in a concentric position. Conversely, Zarnic and Tomazevic [40,41] proposed an off-diagonal model in order to account for possible failure induced by infills at the top of the columns. When a concentric position is adopted, the geometry of the diagonal strut is identified by the diagonal length of the strut d w , determined by the clear length and height of the panel, l w and h w respectively, and the equivalent strut width b w (see Figure 1). stiffness shared among the struts need to be assigned to single struts, and it is not univocal (e.g., [35,36]) and affects the final response of the structure.

Single-Strut Model
The single-strut model is widely adopted by engineers and researchers since it provides a good compromise between simplicity and accuracy in predicting analytical response. The model involves the replacement of each masonry infill through two opposite diagonal pin-joints struts. Each strut should be able to reproduce with an adequate degree of approximation the damage evolution in terms of stiffness and strength when performing nonlinear static or dynamic analyses. The identification of the equivalent diagonal strut requires the assignment of specific mechanical properties that, for existing buildings, is affected by large uncertainties related not only to masonry properties but also to manufacturing and arrangement details. The main parameters characterizing the masonry strut model are the geometry and constitutive law shape for the masonry strut.

Single-Strut Geometry
The geometry of the single-strut model varies depending on the relative position of the strut with respect to the main diagonal connecting opposite nodes. For instance, Bertoldi et al. [37], Panagiotakos and Fardis [38], Dolšek and Fajfar [7], Asteris et al. [39], among others, adopt a single compression strut for each direction in a concentric position. Conversely, Zarnic and Tomazevic [40,41] proposed an off-diagonal model in order to account for possible failure induced by infills at the top of the columns. When a concentric position is adopted, the geometry of the diagonal strut is identified by the diagonal length of the strut dw, determined by the clear length and height of the panel, lw and hw respectively, and the equivalent strut width bw (see Figure 1). The single-strut model was originally proposed by Polyakov [42] and implemented by Holmes [32], which proposed to calculate the width of the equivalent strut as 1/3 of the diagonal length. Afterwards, several authors [29][30][31]37,[43][44][45][46][47] proposed more detailed formulations mainly based on the relative infill panel to surrounding frame elastic stiffness λh first introduced by Stafford-Smith [29]. The single-strut model was originally proposed by Polyakov [42] and implemented by Holmes [32], which proposed to calculate the width of the equivalent strut as 1/3 of the diagonal length. Afterwards, several authors [29][30][31]37,[43][44][45][46][47] proposed more detailed formulations mainly based on the relative infill panel to surrounding frame elastic stiffness λ h first introduced by Stafford-Smith [29].

Single-Strut Constitutive Law
In order to perform nonlinear analyses of masonry-infilled RC frames, reliable constitutive law describing the equivalent strut in-plane behavior needs to be selected. The constitutive law for a single equivalent strut is generally described by means of a multilinear relationship calibrated on experimental data as backbone curve for simulating both Buildings 2021, 11, 57 5 of 25 the monotonic and the cyclic response. Between the constitutive model proposed in the literature, here three alternative constitutive laws are analyzed in detail: Bertoldi et al. [37], De Risi et al. [48], Huang et al. [49].
The model by Bertoldi et al. [37] adopts a four-branches backbone curve (Figure 2a) for the lateral force-displacement (F-D) relationship. The first ascending branch represents the un-cracked behavior up to the first cracking and the second branch corresponds to the post-cracking behavior up to the development of the peak strength (F peak ). The descending third branch of the backbone curve defines the post-peak strength deterioration up to the residual strength (F res ). The fourth branch is horizontal and corresponds to the residual strength of the infill panel. The main parameters required to calibrate the proposed model are the width of the equivalent strut (b w ), the secant stiffness at the complete cracking stage (K sec ), and the infill panel peak strength (F peak ). All the parameters can be defined as a function of the geometric and mechanical characteristics of the infill panel and the surrounding frame. For example, the formulation proposed by Decanini et al. [54] accounts for both the openings percentage and shape ratio: where: where l0, h0, lw, hw are indicated in Figures 2 and 3. The model proposed by Bertoldi et al. [37] is the only one that predicts and explicitly accounts for the failure mode that is likely to be exhibited by the infill panel. In fact, the maximum strength of the panel depends on the predicted failure mode defined based on the infill mechanical properties and corresponds to the minimum among four failure modes. In particular, the peak strength F peak is calculated considering four possible failure modes and the corresponding failure stresses: (a) Diagonal tension, σ br1 ; (b) Sliding shear, σ br2 ; (c) corner crushing, σ br3 ; and (d) diagonal compression, σ br4 . The failure stress depends on the shear strength (τ m0 ), the bed joints sliding strength (τ 0 ), the masonry compressive strength (σ m0 ) and the vertical stress acting on the infill (σ 0 ): σ br,3 = (1.12 sin θ cos θ) where d and θ are the equivalent strut length and inclination, respectively. The term λ h is a non-dimensional parameter that depends on relative infill panel to surrounding frame elastic stiffness, K 1 and K 2 are two constants calibrated on experimental tests that depends on λ h [44]. Finally, b w is the equivalent strut width that is calculated as follows: The parameter λ h is defined according to Stafford-Smith [29] as follows: where E w is the elastic modulus of the infill masonry, EI c is the product between the elastic modulus of the concrete and the moment of inertia of the columns of the surrounding frame, h w is the height of the masonry panel, and h is the interstory height. The secant-to-peak lateral stiffness (K sec ) for the equivalent strut is calculated as: The remaining parameters required for the definition of the backbone curve are evaluated according to De Sortis et al. [50]. In particular, the cracking-to-peak (F cr /F peak ) and the residual-to-peak (F res /F peak ) strength ratios are assumed equal to 0.8 and 0.35, respectively. And the cracking-to-peak (K 0 /K sec ) and the softening-to-peak (K deg /K sec ) stiffness ratios are defined as 4.0 and −0.02, respectively.
Another commonly adopted constitutive model is the modified version of the constitutive law originally proposed by Panagiotakos and Fardis [38]. In the original model, the backbone curve is represented by four branches. Conceptually, the four-branch backbone accounts for different stress states: (a) initial behavior of the un-cracked panel; (b) post-cracked linear response, characterized by a reduction of lateral stiffness due to the detachment of the infill from the surrounding frame; (c) post-peak softening response; (d) achievement of the residual axial strength at a given displacement value. The model by Panagiotakos and Fardis [38] is one of the most widely used in the context of singlestrut models for engineering applications. However, it was calibrated on the basis of 10 tests performed on infilled RC frames with hollow masonry bricks that mainly exhibited diagonal cracking failure. For this reason, based on the analysis of a large database of masonry infills made of hollow clay bricks collected to be representative of the Mediterranean building stock, recently De Risi et al. [48] modified the values of the lateral response curve in order to reduce the dispersion when compared to the assembled database. In particular, the cracking (F cr ) and peak (F max ) strength, the initial un-cracked stiffness (K 0 ) the cracking-to-peak (K 0 /K sec ) and the softening-to-peak (K deg /K sec ) stiffness ratios were modified (Figure 2b). The modification proposed by De Risi et al. [48] significantly reduced the CoV values for tests performed on hollow bricks with respect to the original formulation by Panagiotakos and Fardis [38]. The accuracy of the provided model is also proved by the observation that the resulting backbone curve mean relative error was lower than 3% for all required parameters.
In the De Risi et al. [48] model the first branch corresponds to the elastic behavior up to cracking, and is characterized by the initial un-cracked stiffness (K 0 ), assumed equal to 2.8 times the Mainstone's stiffness (K MS ) [31]. The K MS is obtained adopting in Equation (7) the equivalent strut width defined as follows: where λ h is defined in Equation (6). The infill cracking strength is F cr = 0.7·F peak , where F peak is equal to the lateral cracking strength of the Panagiotakos and Fardis [38] model: The second branch continues up to the peak strength (F peak ), and the secant-to-peak stiffness corresponds to 0.8·K MS . The third branch is a degrading branch up to zero residual lateral strength defined by degrading slope (K deg ) assumed equal to K deg = −0.1·K MS .
More recently, Huang et al. [49] (Figure 2c) proposed the adoption of a backbone curve calibrated on experimental results. Similar to De Risi et al. [48], the authors collected a database of 264 tests performed on masonry-infilled frames. Based on the analysis of the database, the authors developed an empirical model for estimating the backbone curve parameters for the equivalent diagonal strut. The main difference between the model by De Risi et al. [48] and Huang et al. [49] is the way the backbone curve is calibrated. In particular, De Risi et al. [48] started from the proposal by Panagiotakos and Fardis [38] and modified existing semi-empirical formulations in order to reduce the scatter with respect to their database. Instead, Huang et al. [49] developed empirical equations to relate different backbone parameters to several geometric and material properties of the infilled frame using multivariate regression analysis. In this work, only median values of the backbone parameters were adopted.
Differently from the work by Bertoldi et al. [37] and De Risi et al. [48], which provide the force-displacement curve for the infill strut, Huang et al. [49] provides formulations in terms of axial forceaxial deformation (N, ∆ a ) for the infill strut. The first branch of the lateral response backbone is defined by the elastic stiffness up to cracking (K a,0 ) which can be calculated as: (10) and the cracking strength (N cr ) that is assumed as a ratio of the peak strength N peak : where: The second branch corresponds to the post-cracking behavior, and is defined once that F peak and ∆ peak are defined. The ∆ peak can be calculated as follows: The third branch describes the post-peak behavior and is characterized by a degrading slope (K deg ) up to residual strength: Buildings 2021, 11, 57 8 of 25 Finally, the residual strength is defined as: The constitutive models reported above describe the behavior of solid infills. When openings are present to accommodate windows or doors, experimental results still highlighted that the presence of infills enhances the performance of RC frames, but with a reduced effect with respect to a solid infill wall. Generally, the presence of openings leads to a reduction in stiffness, strength, and energy dissipation and also modifies the crack pattern depending on the position and size of openings [51].
To simulate the presence of openings, the most common approach consists of indirectly reduce the strength and the stiffness of the infill panel depending on the opening size, reducing the width of the diagonal strut by adopting an approximate reduction factor [39,46,[52][53][54][55]. However, some authors evidenced that both the location [52] and the shape ratio [54] of openings influence the frame global performance and, in some cases, it can lead to the development of brittle shear failure (e.g., in the case of partiallyinfilled frame).
For example, the formulation proposed by Decanini et al. [54] accounts for both the openings percentage and shape ratio: where: where l 0 , h 0 , l w , h w are indicated in Figures 2 and 3.  Opening notation adopted in this study.

Single-Strut Cyclic Law
When adopting the equivalent strut approach to perform nonlinear time-history analyses, it is of crucial importance to select the more appropriate cyclic law to realistically simulate the hysteretic behavior of the masonry infills. The adopted hysteretic law should be able to capture the strength and stiffness deterioration, the loading and unloading cycles, and the residual strength with sufficient accuracy. Since the cyclic behavior of masonry infills is very difficult to capture, several proposals were made to describe the actual behavior of infill panels. The first proposal of a hysteretic model was made by Klingner and Bertero [56] based on results from a single dynamic test. The model was later modified by Cavalieri et al. [57,58] to enhance the description of loading and unloading branches, and to include the effect of pinched behavior. In this case, the model only requires the definition of three parameters related to pinching and degradation behaviors.

Single-Strut Cyclic Law
When adopting the equivalent strut approach to perform nonlinear time-history analyses, it is of crucial importance to select the more appropriate cyclic law to realistically simulate the hysteretic behavior of the masonry infills. The adopted hysteretic law should be able to capture the strength and stiffness deterioration, the loading and unloading cycles, and the residual strength with sufficient accuracy. Since the cyclic behavior of masonry infills is very difficult to capture, several proposals were made to describe the actual behavior of infill panels. The first proposal of a hysteretic model was made by Klingner and Bertero [56] based on results from a single dynamic test. The model was later modified by Cavalieri et al. [57,58] to enhance the description of loading and unloading branches, and to include the effect of pinched behavior. In this case, the model only requires the definition of three parameters related to pinching and degradation behaviors. Later, Panagiotakos and Fardis [38] extended the model by Tassios [59] and proposed a three-parameters hysteretic model. Crisafulli [24,60] developed a more sophisticated cyclic model in order to simulate different failure mechanisms. However, the latter model is of difficult implementation since requiring a large number of parameters to calibrate. Lately, several authors focused on the calibration of existing hysteretic models available in different software environments to describe the cyclic behavior of infills. For instance, the Hysteretic material and Pinching4 material [61] were adopted to simulate the onlycompression strut axial behavior in OpenSees [19,62]. However, Noh et al. [19] compared two different modeling strategies to reproduce the hysteretic behavior of infill struts and found that the Pinching4 material is the most suitable to reproduce actual response under cyclic loadings of masonry infills.
The pinching4 material is often adopted due to its capability of modeling nonlinearity and degradation of stiffness and strength under cyclic loading. The material model requires the definition of 15 parameters controlling strength, unloading, and reloading stiffness degradation, as a function of ductility and/or energy demand, and 6 parameters controlling pinching behavior. However, due to the significant effort required for the calibration of the full suite of parameters, it is common procedure to relate various degradation modes only to the displacement or energy history in order to reduce the effort required during the calibration procedure. The parameters required to define this material hysteric behavior can be obtained following different authors and international code recommendations. For instance, Lima et al. [63] calibrated Pinching4 parameters by comparing numerical analyses and experimental tests on masonry walls performed by Koutromanos et al. [64]. Kumar et al. [65], based on the results of experimental tests [66], proposed two different sets of hysteretic parameters for Pinching4 material depending on the rigidity of the panel ("weak" or "strong" infills). While previous studies calibrated Pinching4 material based on a very limited number of tests, Huang et al. [49] performed a visual calibration for a large database composed of 113 experimental tests. However, the authors did not perform a regression analysis to relate pinching and cyclic degradation parameters to infill properties, only providing the appropriate range of variation for parameter values.

Research Methodology
Numerous modeling approaches were proposed over the last decades to simulate the lateral response of masonry-infilled RC frames. Depending on the modeling strategy and the response envelope adopted, the results in terms of building response may significantly vary. This leads to significant uncertainty as to which is the most appropriate analytical models for the assessment of seismic performances of masonry-infilled frames, with nonnegligible implications, especially when dealing with large-scale simulation purposes.
Without expecting to be exhaustive, this study analyzes the effect of alternative constitutive relationships on the global response of buildings in order to quantify the differences in terms of engineering demand parameters. Among the proposed modeling approaches, the single-strut allows accounting for the contribution of infill panels in terms of global stiffness and strength and displacement capacity still preserving a good level of accuracy [67]. As a consequence, it can be adopted to assess the influence of adopted constitutive relationship on the seismic response of building in terms of engineering demand parameters such as interstory drift ratios and peak floor accelerations for different intensities of ground motion demand. The effect of different strut constitutive models available in the literature is investigated adopting three alternative constitutive laws described in detail in Section 2. In particular, (1) the differences in terms of backbone curve for a single infill panel are evidenced both at the local and the global level adopting a pushover analysis. By performing nonlinear time history analyses employing a set of 42 far-field ground motion records, the effect of the constitutive envelope on the attainment of a given level of damage is quantified at the global level (2) by generating damage state fragility functions, and (3) by evidencing the effect of the constitutive model on the amounting and distribution of engineering demand parameters such as interstory drift ratios and peak floor accelerations.

Numerical Model for the Case-Study Building
In order to investigate the influence of the constitutive law of the equivalent strut on the seismic performance of a masonry-infilled RC frame, a four-story building located in L'Aquila designed according to obsolete seismic standards was studied. The building was constructed at the beginning of the 1980s and suffered damage due to the 2009 L'Aquila earthquake. The building structural system consists of three planar RC frames oriented in the longitudinal direction, connected each other by flat beams in the transverse direction except for the transverse perimeter frame where deep beams are adopted. The building is regular both in plan and elevation and has an in-plane shape that can be inscribed in a rectangle of 17 m × 10 m. The interstory height is equal to 3.25 m except for the roof story, for which it is equal to 2.65 m. According to the design and construction practice in force at the time of construction, the geometry and reinforcing details of each structural member are the same for all frames in the same direction and for each story. Exterior frames in both directions are infilled with hollow clay bricks with thickness t w = 30 cm (strong infill). The openings vary between each span and story. For further details refer to Gaetani d'Aragona et al. [68].
In this study, a three-dimensional finite element model is developed in OpenSees (Version 3.2.0, University of California, Berkeley, CA, USA) with the aim of evaluating the performances of RC masonry-infilled frames under earthquake loadings. Due to the reduced in plan dimensions and the presence of an inclined reinforced concrete roof, the upper story is modeled as an equivalent load mass, assuming it behaves as a rigid body. Figure 4a shows the 3D perspective view of the reference building, while Figure 4b shows a schematic representation of the three-dimensional finite element model. The openings vary between each span and story. For further details refer to Gaetani d'Aragona et al. [68].
In this study, a three-dimensional finite element model is developed in OpenSees (Version 3.2.0, University of California, Berkeley, CA, USA) with the aim of evaluating the performances of RC masonry-infilled frames under earthquake loadings. Due to the reduced in plan dimensions and the presence of an inclined reinforced concrete roof, the upper story is modeled as an equivalent load mass, assuming it behaves as a rigid body. Figure 4a shows the 3D perspective view of the reference building, while Figure 4b shows a schematic representation of the three-dimensional finite element model.
When subjected to earthquake loadings, existing infilled RC buildings designed according to obsolete seismic code provisions may exhibit brittle mechanisms due to the non-ductile behavior of RC members. For this reason, the finite element model should properly account for the effects that may develop in poorly detailed frame members (e.g., widely spaced stirrups, 90°-hooks, lacking stirrups in the panel joint region) that are characterized by non-ductile behavior. Along with mechanical and geometrical nonlinearities, the possible brittle failure of beam-column joints, of RC columns, the bar-slip at members end need to be explicitly simulated via the finite element model (Gaetani d'Aragona et al. [68,69]).  In this study, a lumped plasticity approach already proposed in Gaetani d'Aragona et al. [68,69] is adopted to develop the numerical model of the case-study building ( Figure  5). In particular, beams and columns are modeled adopting mono-dimensional beam-column elastic elements while the inelastic behavior is concentrated in zero-length rotational springs at either end. The stiffness of single in series elements composing the sub-assemblage is calibrated to reproduce the actual stiffness of sub-assemblage. The inelastic behavior of beam/column rotational springs is conveniently characterized by a multilinear moment-rotation backbone curve with cracking and yielding as initial characteristic points. Three possible failure modes (flexure, flexure-shear, or pure shear failure) are considered for RC columns accounting for possible brittle failure of non-conforming members. The hysteretic behavior of the moment-rotation springs is simulated adopting the When subjected to earthquake loadings, existing infilled RC buildings designed according to obsolete seismic code provisions may exhibit brittle mechanisms due to the non-ductile behavior of RC members. For this reason, the finite element model should properly account for the effects that may develop in poorly detailed frame members (e.g., widely spaced stirrups, 90 • -hooks, lacking stirrups in the panel joint region) that are characterized by non-ductile behavior. Along with mechanical and geometrical nonlinearities, the possi-ble brittle failure of beam-column joints, of RC columns, the bar-slip at members end need to be explicitly simulated via the finite element model (Gaetani d'Aragona et al. [68,69]).
In this study, a lumped plasticity approach already proposed in Gaetani d'Aragona et al. [68,69] is adopted to develop the numerical model of the case-study building ( Figure 5). In particular, beams and columns are modeled adopting mono-dimensional beam-column elastic elements while the inelastic behavior is concentrated in zero-length rotational springs at either end. The stiffness of single in series elements composing the sub-assemblage is calibrated to reproduce the actual stiffness of sub-assemblage. The inelastic behavior of beam/column rotational springs is conveniently characterized by a multilinear moment-rotation backbone curve with cracking and yielding as initial characteristic points. Three possible failure modes (flexure, flexure-shear, or pure shear failure) are considered for RC columns accounting for possible brittle failure of non-conforming members. The hysteretic behavior of the moment-rotation springs is simulated adopting the Pinching4 material with the hysteretic rules proposed for older reinforced concrete members. P-Delta effects are considered. The longitudinal bar-slip effect in columns, beams, and beam-column joints is modeled by adding zero-length inelastic springs at either end of the beams and columns in order to account for a possible increase in lateral deflection due to the slippage effect. No hysteretic degradation is assumed for these elements. The non-linear shear deformation of the panel joint region is simulated by adopting a "scissor" model. Rigid offset depending on the dimension of converging members is included to account for the finite dimension of the panel joint. The pinched hysteretic behavior for beam-column joints is simulated adopting the Pinching4 material. Finally, the presence of RC one-way slabs is simulated via an elastic shell element connecting the main nodes of the frame at the same height. A 5% Rayleigh damping is assigned to the first and the third modes, and damping coefficients are applied to mass and committed stiffness matrixes.  To account for the presence of openings in the infill panels, the equivalent strut width is reduced by a factor that depends on the percentage and the shape of the opening in the panel according to Decanini et al. [54]. The opening percentage varies at different stories and for each span. In particular, in the X direction, the opening percentages for different infill panels varies approximately in the range 0-34% at the different stories, while in the Y direction it varies in the range 0-12% in the first story and 0-30% for other stories.

Influence of Modeling Assumptions
In this section, a first comparison between the different constitutive models reported in Section 2.3.1 is performed at the local level, for a single infill panel, and at the global The characterization of structural members backbone curve is defined based on member geometries, reinforcement details, and material properties available in structural drawing and performed material tests. For further details about RC members' material properties, backbone curves and hysteretic parameters adopted for the finite element model refer to Gaetani d'Aragona et al. [68,69].
Regarding the presence of masonry infills, the equivalent strut modeling approach is adopted. For each panel, the pin-jointed strut is simulated by employing a concentric nonlinear truss element in which the axial stress-deformation backbone curve and the hysteretic behavior is simulated adopting the Pinching4 material [61] adopting the parameters proposed by Lima et al. [63]. Very small values are assumed for the definition of the tensile part of the backbone curve in order to obtain only-compression behavior. The three constitutive laws describing the behavior of infill panels reported in Section 2.3.1 are employed. Pinching and damage parameters defining the cycling degradation for infill masonry proposed in Lima et al. [63] are adopted.
The applicability of the adopted models requires the definition of the mechanical properties mentioned in Equations (1)- (15). In particular, shear cracking strength (τ m0 ), Young's modulus (E w ), and shear modulus (G w ) of masonry panel are required to define the backbone curve according to De Risi et al. [48]. Similarly, the model by Huang et al. [49] only requires the strength of the masonry prism (σ m0 ) and Young's modulus (E w ), while more mechanical parameters are needed for the implementation of the model proposed by Bertoldi et al. [37].
According to the original design drawings available for the case-study building, the infill walls are made of hollow clay bricks with a thickness t w = 30 cm. With reference to the work by Hak et al. [70], the mechanical properties for strong infill masonry are adopted (E w = 3240 MPa). The shear modulus of the masonry (G w ) is taken as 0.40 times the Young's modulus of the masonry (E w ) according to FEMA 356 [71]. The value of the shear strength (τ m0 ) is obtained via linear interpolation between the boundary values proposed in Circolare 7 [72] as a function of Young's modulus. The masonry compressive strength (σ m0 ) is calculated as σ m0 = (τ m0 /0.285) 2 and the bed joints sliding strength τ 0 is obtained from the empirical relationship τ 0 = 2/3 τ m0 [73]. The values of mechanical properties adopted in his study are reported in Table 1. To account for the presence of openings in the infill panels, the equivalent strut width is reduced by a factor that depends on the percentage and the shape of the opening in the panel according to Decanini et al. [54]. The opening percentage varies at different stories and for each span. In particular, in the X direction, the opening percentages for different infill panels varies approximately in the range 0-34% at the different stories, while in the Y direction it varies in the range 0-12% in the first story and 0-30% for other stories.

Influence of Modeling Assumptions
In this section, a first comparison between the different constitutive models reported in Section 2.3.1 is performed at the local level, for a single infill panel, and at the global level, to quantify the dependence of the lateral building response on the constitutive law. Firstly, the comparison is performed for a single panel in order to quantify the difference between backbone curves for given geometrical and mechanical properties.
With reference to a solid infill panel (i.e., without openings) the three formulations indicated in Section 2.3.1 lead to significantly different constitutive models for the equivalent strut both in terms of strength and stiffness. Figure 6 depicts the comparison for the panel with H = 2.80 m and L = 4.80 m in Figure 4a. In terms of stiffness, for Bertoldi et al. [37] (BR) and De Risi et al. [48] (DR) the initial un-cracked stiffness K 0 indirectly depends on the same geometric (l w , h w , I c ) and mechanical (E w , E c ) parameters of both the masonry infill and the surrounding RC frame, while for Huang et al. [49] (HG) it only depends on the masonry infill properties (l w , h w , E w ). For the selected infill panel, the initial stiffness for three backbone curves are K 0,BR = 8.01 × 10 5 kN/m, K 0,DR = 2.67 × 10 5 kN/m, K 0,HG = 1.16 × 10 5 kN/m. Taking the stiffer lateral response predicted by BR as a reference value, the initial stiffness predicted by DR is 33% and the one predicted by HG is 14% the value predicted by BR. This difference at the local level significantly influences the initial response of the building, and the elastic forces transmitted to the building at the initial uncracked stage, as can be evidenced by the fundamental period of the building. With reference to the peak force F peak (Table 2), the model by BR explicitly accounts for the different possible failure modes for the infill and leads to the lower value of F peak = 208.0 kN (corresponding to diagonal tension) with respect to other authors. In this case, DR and HG predict similar peak forces that are about F peak = 347.0 kN and F peak = 339.0 kN, respectively, which are about 166% and 162% the value predicted by BR. Similar considerations can be carried out for the value of cracking force. In terms of residual force (F res ), BR leads to F res = 72.8 kN and HG to about twice (186%) this value, while no residual force is considered by DR.
Another remarkable difference between the proposed backbone curves is in terms of lateral displacement values ( Table 2). BR is the most conservative always predicting lower values of lateral displacement, and a significant difference exists also between DR and HG. Regarding the displacement at cracking (Δcr) similar considerations can be carried out to those for the initial stiffness. For the peak displacement (Δpeak), taking as a reference the value predicted by BR, DR, and HG predict significantly larger values, equal to 4.34 and 11.44 times the value predicted by BR, respectively. Finally, the ultimate displacement (Δres) predicted by DR and HG is 2.23 and 2.31 times the value by BR, respectively.
The choice of constitutive law has a significant effect on the global response of masonry-infilled RC frames. This effect depends on the relative contribution to lateral strength of the infill panels with respect to the RC frame, and reduces as the opening percentage increases, up to possibly erase the infill contribution. The contribution of the infills on the lateral response of RC frames can be clearly quantified by performing a static pushover analysis. Adopting first mode-proportional loads, the pushover analysis is performed both along the longitudinal and transverse direction in order to visualize the lateral response under increasing loads. In particular, Figure 7a,c represents the pushover response for the actual infill opening configuration performed in the longitudinal (transversal) direction, considering the three constitutive laws reported in Section2.3.1, and the finite element model described in Section 4. For comparison purposes, the response of the bare frame is also reported.  Another remarkable difference between the proposed backbone curves is in terms of lateral displacement values ( Table 2). BR is the most conservative always predicting lower values of lateral displacement, and a significant difference exists also between DR and HG. Regarding the displacement at cracking (∆ cr ) similar considerations can be carried out to those for the initial stiffness. For the peak displacement (∆ peak ), taking as a reference the value predicted by BR, DR, and HG predict significantly larger values, equal to 4.34 and 11.44 times the value predicted by BR, respectively. Finally, the ultimate displacement (∆ res ) predicted by DR and HG is 2.23 and 2.31 times the value by BR, respectively.
The choice of constitutive law has a significant effect on the global response of masonryinfilled RC frames. This effect depends on the relative contribution to lateral strength of the infill panels with respect to the RC frame, and reduces as the opening percentage increases, up to possibly erase the infill contribution. The contribution of the infills on the lateral response of RC frames can be clearly quantified by performing a static pushover analysis. Adopting first mode-proportional loads, the pushover analysis is performed both along the longitudinal and transverse direction in order to visualize the lateral response under increasing loads. In particular, Figure 7a,c represents the pushover response for the actual infill opening configuration performed in the longitudinal (transversal) direction, considering the three constitutive laws reported in Section 2.3.1, and the finite element model described in Section 4. For comparison purposes, the response of the bare frame is also reported.

Results of NRHAs
This section presents the results obtained via a nonlinear response history analysis (NRHA) performed on the three-dimensional finite element model (FEM) described in Section 4. The NRHA is performed adopting 42 earthquake records from the far field set reported in FEMA-P695 [74]. Analyses are performed for different values of the seismic action, by scaling the earthquake intensity, expressed in terms of peak ground acceleration (PGA), between 0.05 g and 1.5 g with a step of 0.05 g. The building response is analyzed separately in the longitudinal and the transverse direction.
The first comparison is performed in terms of damage state fragility function, which defines the probability of exceeding a damage state as a function of a ground motion intensity (IM): Note that for the case-study building, no brittle failure of RC members is detected when analyzing the building response in the longitudinal direction, while shear failure after rebar yielding was detected in the transverse direction. For this reason, in the longitudinal direction, the pushover curve for bare frame shows a horizontal branch after peak base shear, while in the transverse direction a degrading slope is displayed. In order to quantify the maximum contribution to lateral strength of infill as a function of the selected constitutive law, the pushover response simulating that no openings are present in the infill panels is also reported for the longitudinal direction in Figure 7b and the transverse direction in Figure 7d. If no openings are present in the masonry panels, infills provide the maximum contribution to lateral strength. By comparing the different infill backbone curves in terms of maximum strength: BR, DR, and HG show a maximum base shear (V b ) with respect to the bare frame that is 227%, 368% and 365% greater, respectively. If the actual infill configuration is considered, and the equivalent strut width is modified according to Equations (16) and (17), the contribution of infills to lateral strength reduces to 152%, 223%, and 243% for BR, DR, and HG, respectively. Due to the presence of opening, the maximum base shear reduces by about 33% for BR and HG, while of 39% for DR. The relative difference, in terms of V b , between the different backbone curves remains almost unaltered between the no-opening and actual opening configurations, suggesting that the distribution of openings is sufficiently regular to avoid the development of soft-story mechanism or changing the collapse mechanism of the building with respect to uniform distribution of openings (i.e., no openings). In the transverse direction, the maximum contribution to base shear (i.e., no openings) is about the same that was evidenced in the longitudinal direction, being equal to 229%, 366%, and 375% for BR, DR, and HG, respectively. When the actual opening configuration is considered, this contribution to lateral resistance drops to 165%, 259%, and 261%, which corresponds to a reduction of maximum base shear of 27%, 29%, and 30% for BR, DR, and HG, respectively.
The adoption of different constitutive law significantly influences the initial elastic stiffness of the building. While for the bare frame analyzed in the longitudinal direction, the fundamental vibration period is T 1X,bare = 0.68 s, for the different constitutive laws, and actual opening configuration, the vibration periods are T 1X,BR = 0.11 s, T 1X,DR = 0.17 s and T 1X,HG = 0.23 s. The differences in terms of vibration period may significantly affect the seismic demand at the uncracked stage of the structure, and the evolution of damage also in the nonlinear stage. In the transverse direction, due to the reduced in plan extension with respect to the longitudinal one and to the lower number of spans and infill panels, the structure results more deformable, with T 1Y,bare = 0.83 s, for the different constitutive laws, and actual opening configuration, the vibration periods are T 1Y,BR = 0.13 s, T 1Y,DR = 0.20 s and T 1Y,HG = 0.24 s.

Results of NRHAs
This section presents the results obtained via a nonlinear response history analysis (NRHA) performed on the three-dimensional finite element model (FEM) described in Section 4. The NRHA is performed adopting 42 earthquake records from the far field set reported in FEMA-P695 [74]. Analyses are performed for different values of the seismic action, by scaling the earthquake intensity, expressed in terms of peak ground acceleration (PGA), between 0.05 g and 1.5 g with a step of 0.05 g. The building response is analyzed separately in the longitudinal and the transverse direction.
The first comparison is performed in terms of damage state fragility function, which defines the probability of exceeding a damage state as a function of a ground motion intensity (IM): The PGA was selected as IM since it is independent of the elastic characteristics of the buildings and allows a comparison between results obtained from structures with different initial elastic characteristics. Under the hypothesis that the IM values of ground motions causing the attainment of a DS are lognormally distributed, the functional form for the cumulative distribution function is: where Φ() is the standard normal cumulative distribution function, θ is the median of the fragility function and β is the logarithmic standard deviation. The parameters θ and β are estimated by adopting the maximum likelihood estimation (MLE) technique [75] to fit the results, which searches for the parameter values that are most likely to have produced the data. By assuming a binomial distribution to describe the probability that z j ground motion records with a given PGA, over n j total ground motions, caused the attainment of damage state ds i , the likelihood function considering m levels of the seismic action is expressed as the product of binomial probabilities at each PGA level: where ∏ denotes the product over all ground motion levels. The MLE technique allows the calculation of the fragility function parameters by maximizing the Likelihood function (Equation (20)). Further, this procedure provides reliable results also in case of truncated IDA (i.e., when the analysis is performed up to a given value of the IM) [75]. Note that neither variability in material properties nor in the geometrical features are considered in this study, and the dispersion in results for a structural model with a given constitutive law only depends on the record-to-record variability. Five damage states (DSs) are defined compatibly with the EMS-98 scale [76] and expressed in terms of interstory drift ratio (Table 3). Starting from the simplified procedure proposed in Gaetani d'Aragona et al. [20], which allows to perform a simplified Pushover analysis by adopting a closed-form procedure, the attainment of EMS98-like DSs [77] are identified during the generation of the pushover curve and transformed in the corresponding interstory drift ratios. The damage states are defined depending on the attainment of local phenomena exhibiting by infill panels (i.e., extension of cracking) or the RC frame (i.e., rebar yielding, buckling, concrete spalling, shear, axial failure, or attainment of maximum chord rotation for ductile members). The displacement corresponding to different DSs slightly varies depending on the constitutive law adopted for infills and on the analysis direction. For simplicity purposes, a mean value is assumed for each DS between different constitutive laws and directions.
The adoption of different constitutive laws for the equivalent strut may significantly affect the PGA value at which the damage state DS i is attained. To quantify this effect, the Damage State fragility functions expressed in terms of PGA are reported in Figure 8 for the three constitutive laws adopted in this study (BR, DR, HG) by separating the results in the longitudinal (Figure 8a) and the transverse (Figure 8b) directions. The medians (θ) and the logarithmic standard deviations (β) are also reported for the longitudinal (Table 4) and the transverse (Table 5) direction.
Damage State fragility functions expressed in terms of PGA are reported in Figure 8 for the three constitutive laws adopted in this study (BR, DR, HG) by separating the results in the longitudinal (Figure 8a) and the transverse (Figure 8b) directions. The medians (θ) and the logarithmic standard deviations (β) are also reported for the longitudinal (Table  4) and the transverse (Table 5)  Observing damage state fragility curves in Figure 8, it can be noted that they reflect the variation in lateral stiffness of the system, which is related to the adopted infill backbone curve. In fact, according to observations in Section 5, the BR backbone curve leads to the most rigid behavior, while for DR and HG the initial stiffness reduces. This observation is valid for both directions. For DS1, where first nonlinearities occur in the structure, the seismic response is expected to be close to that in the linear range. In fact, DS1 firstly occurs for HG, which corresponds to the more deformable system, followed by DR and BR. This trend is confirmed also for higher damage states despite the scatter between fragility curves for the three backbone curves is not constant. With reference to the longitudinal direction, by adopting BR as a reference response, the difference in terms of median PGA that leads to DSi for DR backbone curve is −50% for DS1, while drops to −1% and −20% for DS2 and DS3, respectively. For both DS4 and DS5 the difference again grows to about −13%. For the HG backbone curve, the difference with respect to BR is about −60% for DS1, while for DS2 drops to about −10% and from DS3 to DS5 is about −15%. Similar trends are shown in the transverse direction, where the scatter with respect to BR backbone curve is comparable for both DR and HG, except that for DS4 and DS5 where the scatter is between 1.5 to 2.5 times that in the longitudinal direction. It is also worth noting that in both the directions, for DS1 to DS3, the DSi is attained for lower PGA values for HG with respect to DR, while for DS4 and DS5, the trend is inverted, since the DSi first occurs for DR. This effect is more clear when observing the transverse direction (Figure 8b).  Observing damage state fragility curves in Figure 8, it can be noted that they reflect the variation in lateral stiffness of the system, which is related to the adopted infill backbone curve. In fact, according to observations in Section 5, the BR backbone curve leads to the most rigid behavior, while for DR and HG the initial stiffness reduces. This observation is valid for both directions. For DS 1 , where first nonlinearities occur in the structure, the seismic response is expected to be close to that in the linear range. In fact, DS 1 firstly occurs for HG, which corresponds to the more deformable system, followed by DR and BR. This trend is confirmed also for higher damage states despite the scatter between fragility curves for the three backbone curves is not constant. With reference to the longitudinal direction, by adopting BR as a reference response, the difference in terms of median PGA that leads to DS i for DR backbone curve is −50% for DS 1 , while drops to −1% and −20% for DS 2 and DS 3 , respectively. For both DS 4 and DS 5 the difference again grows to about −13%. For the HG backbone curve, the difference with respect to BR is about −60% for DS 1 , while for DS 2 drops to about −10% and from DS 3 to DS 5 is about −15%. Similar trends are shown in the transverse direction, where the scatter with respect to BR backbone curve is comparable for both DR and HG, except that for DS 4 and DS 5 where the scatter is between 1.5 to 2.5 times that in the longitudinal direction. It is also worth noting that in both the directions, for DS 1 to DS 3 , the DS i is attained for lower PGA values for HG with respect to DR, while for DS 4 and DS 5 , the trend is inverted, since the DS i first occurs for DR. This effect is more clear when observing the transverse direction (Figure 8b).
In the transverse direction, the attainment of different DSs generally corresponds to lower PGA values with respect to the longitudinal direction. This is due to the reduced lateral stiffness of the system in the transverse direction when compared to the longitudinal one, confirmed by higher values of the fundamental period of vibration (see Section 5), which leads to higher lateral deformations for the same PGA value. In particular, the scatter between the two orthogonal directions is around 40% for DS 1 to DS 3 , and reduces as the damage increases with a minimum value for DS 4 and DS 5 (18-36%). Finally, it can be noted that higher dispersion occurs in the transverse direction for all considered DSs. The influence of the constitutive law on the structural behavior can be also evidenced by means of other response quantities. One way to express the seismic performances of a building is the use of engineering demand parameters (EDPs) such as interstory drift ratios (IDRs) and peak floor accelerations (PFAs). The EDPs are response quantities of particular interest when the damage to both structural and nonstructural components need to be estimated and are at the base of the PBEE framework [78] procedures that allow the estimation of repair costs. Figure 9 shows the IDR profiles in the longitudinal direction for PGA values corresponding to median PGAs for damage states from DS 1 to DS 4 , that are 0.1 g-0.4 g-0.6 g-1.2 g (Table 4), respectively. The IDR profiles are obtained as median value between the 42 ground motion responses. For the lower value of the seismic intensity (Figure 9a), for which it is expected that the structure almost behaves elastically, a high scatter between the IDRs between the three models is attained. DR and HG predict IDRs equal to 2.7 and 5.9 times that predicted by BR, respectively. Further, the IDR is very similar for the first and the second story, but for DR and HG the maximum IDR occurs at the second story and in the first story for DR. As the intensity increases, the difference in prediction between different models reduces and the maximum IDR concentrates in the first story. For PGA = 0.4 g (Figure 9b) HG still predicts larger IDRs with respect to BR, while DR leads to lower IDRs. In particular, DR leads to IDR max which is 4% lower with respect to BR, and HG to 32% higher. For PGA = 0.6 g (Figure 9c), the scatter with respect to BR is 16% and 28% for DR and HG, respectively. For PGA = 1.2 g (Figure 9d) and higher earthquake intensities, the IDRs for DR and HG are close to each other and predict 56% and 50% larger IDRs with respect to BR, respectively. Note that for PGA varying from 1.0 g to 1.5 g, DR always predicts larger IDRs with respect to HG, while the trend is inverted for intensities lower than 1.0 g.  In the transverse direction (figure not shown here for brevity reasons), the analyses led to similar results to the longitudinal direction. However, some major difference arises. In particular, for PGA = 0.1 g the different backbone curves still lead to a high scatter between the three results, but in this case, the maximum IDR always occurs in the first story. While in the longitudinal direction, where the drift demand is well distributed along the height, in the transverse direction, as the intensity of ground motion increases, the damage concentrates in the first story, while upper stories are relatively less damaged. For instance, for PGA = 0.4 g the IDR at the second story is 20% to that in the first one. This trend is confirmed for higher intensities, where the IDR at the second story is generally comprised between 20% and 30%. This may be ascribable to the higher deformability of the structure in the transverse direction with respect to the longitudinal one, which leads to IDRs that almost twice those in the transverse direction for the same PGA value.
In terms of peak floor accelerations, the median profiles along the height are reported in Figure 10 for PGA equal to 0.1 g-0.4 g-0.6 g-1.2 g. It is interesting to note that for PGA = 0.1 (Figure 10a), when the structure is almost elastic, the PFA increases almost linearly In the transverse direction (figure not shown here for brevity reasons), the analyses led to similar results to the longitudinal direction. However, some major difference arises. In particular, for PGA = 0.1 g the different backbone curves still lead to a high scatter between the three results, but in this case, the maximum IDR always occurs in the first story. While in the longitudinal direction, where the drift demand is well distributed along the height, in the transverse direction, as the intensity of ground motion increases, the damage concentrates in the first story, while upper stories are relatively less damaged. For instance, for PGA = 0.4 g the IDR at the second story is 20% to that in the first one. This trend is confirmed for higher intensities, where the IDR at the second story is generally comprised between 20% and 30%. This may be ascribable to the higher deformability of the structure in the transverse direction with respect to the longitudinal one, which leads to IDRs that almost twice those in the transverse direction for the same PGA value.
In terms of peak floor accelerations, the median profiles along the height are reported in Figure 10 for PGA equal to 0.1 g-0.4 g-0.6 g-1.2 g. It is interesting to note that for PGA = 0.1 (Figure 10a), when the structure is almost elastic, the PFA increases almost linearly with the height. For PGA = 0.4 g, which about corresponds to DS 2 , lower values along the height with respect to those expected during elastic behavior are predicted. In particular, for BR the reduction of PFAs along the height indicates a higher level of damage with respect to DR and HG. As the damage spreads through the structure, the ratio PFA/PGA decreases in uppers stories. This effect is particularly evident for PGA = 1.2 g where the PFA decreases along the height.  In the transverse direction, the PFA distributions, not reported here for brevity reasons, show a similar trend. However, since the same level of IDR (i.e., damage) is attained in the transverse direction for lower PGAs with respect to the longitudinal one, the effect of nonlinearities on the PFA distribution (reduction of ratio PFA/PGA) is more evident starting from lower intensities. Figure 11 shows the trend of maximum IDRs (a, b) and PFAs (c, d), respectively, for increasing seismic intensities (PGA). For each intensity, the value is obtained as maximum median value occurred at the different stories for both IDRs and PFAs. Observing Figure  11a, it is interesting to note that while for PGA < 0.5 g the backbone curve by DR and BR In the transverse direction, the PFA distributions, not reported here for brevity reasons, show a similar trend. However, since the same level of IDR (i.e., damage) is attained in the transverse direction for lower PGAs with respect to the longitudinal one, the effect of nonlinearities on the PFA distribution (reduction of ratio PFA/PGA) is more evident starting from lower intensities. Figure 11 shows the trend of maximum IDRs (a, b) and PFAs (c, d), respectively, for increasing seismic intensities (PGA). For each intensity, the value is obtained as maximum median value occurred at the different stories for both IDRs and PFAs. Observing Figure 11a, it is interesting to note that while for PGA < 0.5 g the backbone curve by DR and BR predicts very similar IDRs, for PGA ≥ 0.5 g this phenomenon occurs for DR and HG. Further, DR and HG distances from BR prediction as PGA increases up to PGA = 1.5 g where IDR max,DR /IDR max,BR is almost 1.5. Similarly, this phenomenon can be also observed in the transverse direction (Figure 11b), despite PGA ≥ 0.57 g predictions by DR and HG tend to diverge. predicts very similar IDRs, for PGA ≥ 0.5 g this phenomenon occurs for DR and HG. Further, DR and HG distances from BR prediction as PGA increases up to PGA = 1.5 g where IDRmax,DR/IDRmax,BR is almost 1.5. Similarly, this phenomenon can be also observed in the transverse direction (Figure 11b), despite PGA ≥ 0.57 g predictions by DR and HG tend to diverge.
(a) (b) (c) (d) In the longitudinal direction (Figure 11c), three backbone curves predict similar PFAmax up to PGA = 0.15 g. While DR and HG produce PFAmax very close to each other for every intensity, for PGA ≥ 0.15 g they distance from BR for joining up again for PGA ≥ 1.2 g. This trend occurs also in the transverse direction (Figure 11c), but DR and HG predictions tend to diverge for PGA ≥ 0.5 g.

Conclusions
This paper investigates the influence of the models adopted for masonry infill panels on the global performance of RC frames. To this end, nonlinear static and time-history In the longitudinal direction (Figure 11c), three backbone curves predict similar PFA max up to PGA = 0.15 g. While DR and HG produce PFA max very close to each other for every intensity, for PGA ≥ 0.15 g they distance from BR for joining up again for PGA ≥ 1.2 g. This trend occurs also in the transverse direction (Figure 11c), but DR and HG predictions tend to diverge for PGA ≥ 0.5 g.

Conclusions
This paper investigates the influence of the models adopted for masonry infill panels on the global performance of RC frames. To this end, nonlinear static and time-history analyses are performed on a refined three-dimensional model where the infills are modeled alternatively adopting truss elements with backbone curves proposed by Bertoldi et al. [37] (BR), De Risi et al. [48] (DR) and Huang et al. [49] (HG).
The main findings can be resumed in the following points: • The presence of infills has a significant effect on the global response of RC frames both in terms of lateral stiffness and strength. This effect depends on the relative contribution to the lateral strength of the infill panels with respect to that of the RC frame and reduces as the opening percentage increases. For fully filled frames, the base shear strength increases between 227% and 365% with respect to that obtained for the corresponding bare frame configuration. The infill contribution reduces to a value comprised between 152% and 261% considering the actual opening configuration for the case-study building. • The adopted constitutive model significantly influences the probability of attainment of a given damage state. The scatter in prediction between different constitutive models in terms of median PGAs is comprised between 2% and 60% depending on the adopted constitutive model and the selected damage state. The dispersion of results, which is related only to record-to-record variability, is slightly influenced by the constitutive law adopted while it mainly depends on the selected damage state and is comprised between 0.21 and 0.51 in terms of logarithmic dispersion.

•
In terms of interstory drift ratios, the presence of infills lead to a more uniform distribution along the height with respect to the corresponding bare frame. For lower seismic intensities, a uniform distribution of lateral deformations along the height occurs, and the scatter of IDR due to different infill constitutive models may be very high and comprised between 270% and 590%. For increasing intensities, the scatter due to different infill constitutive models significantly lowers to values comprised between 4% and 56%. However, the distribution of lateral deformation shape along the height is not influenced by the employed constitutive model.

•
In terms of peak floor accelerations, the distribution linearly increases along the height for lower seismic intensities. As the seismic intensity increases and the damage spreads throughout the structure, the acceleration demand reduces in upper stories with respect to the base acceleration. For higher seismic intensities, as the damage attained is very high, the acceleration demand at upper stories is lower than the base acceleration.
The results obtained in this study emphasize the effect of the selected infill constitutive model on the seismic performances of RC buildings. Although the study refers to a specific case-study building, the results may be useful to highlight relevant trends on the response and effects of relative infills model stiffness and strength on the evaluation of building damage fragility curves and performance evaluation. Future studies should analyze the dependence of the seismic performances of infilled RC frames on the other modeling assumptions, such as the adopted cyclic law and the dependence on the formulation to account for the presence of openings.
Author Contributions: M.G.d. and M.P. collected scientific articles for state of the art. M.G.d. performed the analyses, analyzed the results and contributed to write the paper. M.P. organized the work and contributed to the analyses of the results. A.P. revised the paper for overall consistency and scopes. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy reasons.