7 For seism ic design or performance assessment, r einforced concrete (RC) frame buildings with 8 masonry infills are often modeled using strut -cum-masonry constitutive models . A number of 9 models have been considered in the past, which lead to substantially different simulation results. 10 Through an analysis of 106 one -bay-one-story infilled RC frames experimentally studied in the 11 past, t his paper presents an evaluation of eight previously proposed m asonry constitutive models 12 with masonry panel represented using a set of three struts . The exercise enabled identification of 13 suitable functional forms to characterize the constitutive behavior of masonry panel . 14 Accordingly, a new constitutive model was de veloped with strength parameters defined in terms 15 of the product of masonry prism strength and cross -sectional area of struts, and stiffness 16 parameters defined in terms of the masonry elastic modulus and cross -sectional area of struts 17 divided by masonry di agonal length. Multipliers were established for ranges of masonry prism 18 strength. In comparison with existing models , the proposed model could effectively capture the 19 response of the 106 specimens despite its simplicity. Simulation results were not signifi cantly 20 correlated with relevant strength parameters of the RC frame, relative stiffness parameter, and 21 observed failure modes. The proposed model can also be used in combination with one or two 22 1 Correspond ing author. E-mail address: mkumar@iitgn.ac.in (M. Kumar). 2 struts. However, one -strut model should be avoided if capturin g shear failure in frame members 1 is important. 2 Keywords : RC frames ; Masonry infill ; Shear failure ; Flexural failure ; Strut models ; Masonry 3 constitutive models . 4 1. Introduction 5 Reinforced concrete (RC) frames with unreinforced masonry infill are commonly seen in 6 residential and commercial buildings across the world, and serve as lateral force resisting systems 7 by design or by chance. A number of these building s have performed rea sonably during 8 earthquakes . However, t he frame members are vulnerable to f lexural and shear failure s, and the 9 masonry infill walls often und ergo in-plane and out-of-plane failure s (especially in the upper 10 stories) [1]-[9]. The in -plane lateral force -displacement response of the infilled RC fram es is 11 complex due to the brittle behaviour of masonry panel, and the frame -infill interactions (e.g., 12 [10], [11]). Complexities are further enhanced when a panel has special features (e.g., openings) , 13 or when the in-plane and out -of-plane response s are linked (e.g., [12]-[19]). 14 Infilled RC frames have been modeled u sing detailed finit e element approaches 15 (e.g., [20]-[23]). Such an approach potentially allows capturing global as well local response s of 16 the frame. However, the associated computational costs for realistic building structures can be 17 prohibitive. Besides, the calculated responses may be sensitive to input parameter choices. 18 Therefore, in practical designs and research studies involving realistic or a large number of 19 structural systems , simplistic macro -modeling appro ach is often adopted (e.g., [24]-[27]). This 20 paper focuses on calculating in-plane response of infilled RC frames through the simplistic 21 models , wherein frame members are considered using line elements capable of capturing 22 flexural, shear and axial responses of the members , and the masonry panel is considered using 23 one or more struts. 24 3 Relatively well accepted approaches exist to model the line elements representing the frame 1 members. However, a number of modelling approaches for masonry panel have been used in the 2 past, and a consensus remains elusive. A macro model for masonry panel can be characterized in 3 terms of number and orientation of struts, and associated constitutive properties. Most macro 4 models consider one, two, three or more struts oriented along or around the diagonals of the 5 masonry panel. A number of constitutive models also have been proposed. These models 6 generally start with a strut model, and the parameters of the assumed constitutive model are 7 adjusted to match the experimentally observed response of one -bay-one-story [28]-[29], 8 multi -bay-one-story [28], one -bay-multi -story [10] or multi -bay-multi -story [30]-[31] infilled 9 RC frames. A constitutive model based on the masonry prism tests also has been proposed [32]. 10 The key parameters of these constitutive models typically are peak strength, initial stiffness, 11 deformation corresponding to peak strength (or associated stiffness), residual strength etc. The 12 functional forms for these parameters are chose n based on the basic principles of mechanics (e.g., 13 [10], [11], [28], [30], [32], [33]) or statistical analyses (e.g., [29]). The existing constitutive 14 models have one or more of the following limitations: the calibration of the functional form was 15 performed a gainst a small number (e.g., less than 15 ) of experimental specimens (e.g., [10], [11], 16 [28], [30], [31], [33]), shear failure in frame members was not considered during calibration (e.g., 17 [10], [29], [31]), interaction between axial force and bending moment capacities (P -M 18 interaction) was ignored (e.g., [10], [29], [31]), interaction between frame and infill was ig nored 19 (e.g., [32]), and the experimental dataset did not represent the population either because it was 20 small or because frames with a lower infill strength were more likely to be tested experimentally 21 (e.g., [34]). Further, many constitutive models identified herein consider a single strut to 22 represent the masonry panel for calibration. However, it has been reported that a single -strut 23 model may not adequately capture the lateral force -displacement behaviour of an infilled RC 24 frame, particularly in the post -peak region (e.g., [35]-[37]). 25 4 Multiple combinations of strut and constitutive models have been used i n the past. For 1 example, single -strut model has been used in combination with constitutive models considered 2 by [10] and [29], two-strut model has bee n used by [27] and [38], and three -strut mode l has been 3 used by [39] and [40]. Some studies have been devoted to understand the choice of strut and/or 4 constitutive models on the calculated response of infilled RC frames (e.g., [35], [36], [41]). 5 However, these studies are limited by number of combinations considered (e. g., less than 10), 6 number of experimental specimens (e.g., less than 15), and the parameters of response considered 7 (often limited to initial stiffness, peak strength, and/or visual comparison of the 8 force -displacement response). Expectedly, these evaluati ons led to divergent conclusions. For 9 example, Asteris et al. [35] found single -strut models inadequate and recommended that a 10 multi -strut model sho uld be used, while Noh et al. [41] based their evaluation on single -strut 11 models. On the other hand, Mohyeddin et al. [36] noted that strut models should be developed 12 specific to frame being studied, which may be impractical, but echoes the observations made by 13 Asteris et al. [35] that macro -modelling of infilled RC frames is an open -ended problem. None 14 of these studies evaluated the performance of the macro -models in cap turing the post -peak 15 response of infilled frames, or flexural and shear failures in frame members, which may be 16 important in the collapse assessment of these structures (e.g., [31], [24], [42]). 17 The choice of strut and constitutive models can impact the conclusions related to calcul ated 18 performance of a structure . Therefore, it is important to establish some general premises that can 19 guide these choices. Accordingly, the objectives of the p resent research are: (1) evaluate the 20 performance of constitutive models i n simulating the lateral force -displacement response of 21 infilled RC frames, (2) develop a new constitutive model using latest experimental data set on 22 one-bay-one-story infilled frame with functional forms of the model parameters guided by basic 23 principles of mechanics and results of the first objective, (3) validate the proposed constitutive 24 model , and (4) evaluate the efficacy of single - and two -strut models in combination with the 25 5 proposed constitutive model . Cyclic degradation parameters are beyond the scope of present 1 study, as their role is limited in realistic single -event earthquake scenarios (e.g., [43]). A 2 description of the strut and constitutive models is presented in Section 2. Details on 106 3 one-bay-one-story i nfilled RC frame specimens experimentally tested in the past are discussed 4 in Section 3. Section 4 presents the modeling approach for infilled RC frames . A numerical model 5 for each of these 106 specimens is developed considering the three -strut and eight constitutive 6 models. Pushover analy ses are carried out, and the results are characterized in terms of initial 7 stiffness of the infilled frames, peak strength, secant stiffness corresponding to peak strength, 8 residual strength, post -peak stiffness, and flexural and shear failure in frame mem bers. The 9 approach for characterization is presented in Section 5, and select r esults are presented in 10 Section 6. A new constitutive model is devel oped based on three -strut model , and its performance 11 in combination with one -, two - and three -strut models is evaluated in Section 6. Limitations of 12 the proposed model are discussed in Section 7, and the s ummary and conclusions of this study 13 are presented in Section 8. 14 2. Macro -models for masonry -infilled RC frames 15 As noted previously, a macro model for the masonry panel comprises of a strut model and a 16 constitutive model. In the present research, single -strut, two -strut and three -strut models are 17 considered a long with eight co mmonly used constitutive models . The strut and constitutive 18 models are denoted using capital letters S and C, respectively, followed by a serial number. 19 Recognizing that individual strut models have been used with multiple constitutive mo dels in the 20 past, all constitutive models are assumed applicable for all strut models in the present study. A 21 brief description of the three strut and eight constitutive models is provided next. 22 6 2.1. Strut models 1 Fig. 1 shows th e three strut models considered in the present study. Solid and broken lines 2 indicate that the strut members are in compression an d tension, respectively, when the fram e is 3 subjected to a rightward lateral force at the beam level. Eq uivalent strut width proposed by 4 Mainstone [44], and adopt ed in design standards and research stu dies (e.g., [25], [26], [45]), is 5 considere d herein: 6 0.40.175( )mmb h d  (1) 7 where mb is equivalent strut width, h is the height of column, md is the diagonal length of 8 masonry panel, and λh is a relative stiffness pa rameter given by Stafford Smith and Ca rter [46] 9 4sin 2 4mm c c mEthhE I h (2) 10 where mE and cE are the elastic modul i of masonry and concrete , respectively, cI is the gross 11 moment of inertia of column cross -section , and mt and mh are the thickness and height , 12 respectively, of the masonry wall . Struts are connected to the surrounding frame through a pin. 13 Strut model S1 (see Fig. 1 (a)) is a commonly used single -strut model (e.g., [10], [29], [47] , 14 [44], [46], [48]). Parameters L and  in Fig. 1(a) are length of beam and inclination of slope 15 with respect to the horizontal axis , respectively . Strut model S2 (see Fig. 1(b)) has two 16 off-diagonal parallel struts [27]. Each of the two struts account for half of the total strut width 17 (a) S1 (b) S2 (c) S3 Fig. 1. Strut models (figure partially adapted from Ghosh and Kumar [37]; Reused with permission from The Masonry Society and the 13th North American Masonry Conference) 7 specified in Eq. (1). Model S3 (see Fig. 1(c)) has one diagonal and two parallel off -diagonal 1 struts (e.g., [39]-[40]). The diagonal strut accounts for 50% of the stru t width, and the remaining 2 two struts account for 25% each. The contact length c given by Stafford Smith and Carter [46] 3 is considered in the present study. 4 2c  (3) 5 where all par ameters were defined previously . The off -diagonal struts in models S2 and S3 are 6 placed at a distance of 3c and 3tanc  from the beam -column joints along the columns and 7 beam (see Fig. 1), respectively, in line with the approach taken by Smith [49]. 8 2.2. Constitutive material models 9 Constitutive model C1 is a strut -axial stress -strain relationship p roposed by Kaushik et al. 10 [32]. Details of the model (and remaining seven models) are presente d in Table 1. The para meters 11 of the model were developed based on five -brick masonry prism tests on 84 specimens with 12 prism strength ranging between 2.9 MPa and 8.5 MPa. Constitutive model C2 was developed by 13 El-Dakhakhni et al. [28]. It was validated against five experimentally tested steel frames with 14 masonry infills. Constitutive model C3 was proposed by Klingner and Bertero [10]. Parameters 15 of the model were calibrated against experimentally observed response of three 16 three -story -one-bay masonry -infilled RC frames with masonry prism strength ranging between 17 19.0 MPa and 26.4 MPa. Constitutive model C4 is a tetra -linear axial force -displacement 18 relationship for the strut proposed by Panagiotakos and Fardis [30]. Constitutive model C5 is a 19 tri-linear lateral force -displacement relationship developed by Dolsek and Fajfar [47]. 20 Parameters of the model were established based on the tests on 13 specimens [50], wherein 21 masonry prism strength varied between 2 – 5 MPa. Constitutive model C6 was developed by 22 Decanini and Fantin [33], wherein strength correspon ding to different failure modes was 23 8 considered based on the principles of mechanics and necessary assumptions. Constitutive model 1 C7 is a tri -linear axial force -deformation relationship for the diagonal strut proposed by Burton 2 and Deierlein [31]. Peak strength  max, eF and initial stiffness 1,eK for this model was 3 originally given by Paulay and Pr iestley (1992). The expressions were calibrated against 4 experimental results for 14 infilled RC frames. Constitutive model C8 was developed by Huang 5 et al. [29] based on a dataset of 113 experiments conducted on one -bay-one-story masonry - 6 infilled RC frames. For each experimental specimen, a numerical model was developed wherein 7 the masonry panel was represented using a single strut. The parameters of the backbone and the 8 hysteretic behaviour were established through a calibration exercise. 9 Table 1. Description of constitutive models Constitutive model number and reference Constitutive behavior Equations Equation number C1 Kaushik et al. [32] ' 0.49 0.320.63m b jf f f (4) C2 El-Dakhakhni et al. [28] ' ' m mffEE (5) 4 2 2 4 0 90 0 0 901 1 2 1 1cos cos sin + sinE E E G E      (6) 0.5pEE (7) ' m p pf E (8) 1 0.001p (9) 2 .0001pεε (10) 0.01uε (11) C3 Klingner and Bertero [10] ' , ()mγν ck Aef F (12) 1,k mEAKd (13) 9 C4 Panagio takos and Fardis [30] ,cr a tp m mF f t l (14) 1,mm a mGt lKh (15) max, , 1.3a cr a FF (16) 2,m m m a mE t bKd (17) 3, 1,a 0.005 0.100aKK (18) ,0.05 0.10res x a ma ,a FF (19) C5 Dolsek and Fajfar [47] ,0.6cr h max,hFF (20) 1,mm h mGt lKh (21) 2 max 0.818 1 1,hm w tp I ILt CFf C  (22) C6 Decan ini and Fantin [33] ,0.8cr d max,dFF (23) 1, 0,34 d t s d K K K (24)     0 00 max0.12 0.88 1 6 2 60.6 0.3 (1.2sin 0.45cos ) 0.3 1.12sin cosmin ' () 1.16tan'tp m m mm ,d m m m CC m m mfσ t d θ θ τ σ t d θθF f t b Kλh K λh θf t bλh          (25) ,m m m sd mE t bKd (26) ,0.35res d max,dFF (27) C7 Burton and Deierlein [31], Paulay and Priestley [11] ,e1.3max,e crFF (28) 1,emm mE t wKd (29) ' max2sec3,e c m F α tf θ (30) ,2max,e cr edd (31) ,5collapse e max,edd (32) C8 Huang et al. [29] 0.72ycFF (33) 1.096 0.618 0.6940.0143m e m m mhK E tl   (34) '0.196 0.867 0.7920.003766c m m mF f t d (35) 0.978 0.1970.0154m cm mhd E dl   (36) 0.4res cFF (37) ' 0.357 0.5171.278pc m mK f t (38) bf : compressive strength of brick units; jf : compressive strength of mortar; ' mf : masonry prism strength; 0: sliding resistance in the joints; A : area of strut; w : equivalent strut width 4md ; 10 0E: elastic modulus parallel to bed joints of masonry wall; 90E : elastic modulus normal to bed joint;  : axial deformation in the strut;  : strength degradation parameter; tpf : cracking strength obtained from diagonal compression test; 0 : vertical stress due to gravity loads; ml: length of infill wall; md : diagonal length of strut; 0tK : elastic stiffness of frame; 1.925mImlCh ; 16CK and 26CK : constants depending on relative stiffness parameter h ; The numerical model used to develop model s C3, C5, C6 and C8 did not consider P -M 1 interaction in the frame members , which may impact the calculation of nonlinear response of 2 frame and strut representing masonry . Some other constitutive models (e.g., C1) did not consider 3 the frame -infill interaction. Many model s (e.g., C3, C5, C8) used a single strut to represent 4 masonry panel that may not adequately capture the distribution of forces along frame members 5 (e.g., [35]-[37]), which may lead to errors in the numerically calculated parameters for masonry 6 panels . Two models (C2 and C3) did not consider the stiffness associated with the stress level 7 corresponding to initial cracking in masonry, which may influence the assessment of the 8 earthquake loads on a structure. Constitutive model C8 was based on the most extensive 9 experimental dataset among all constitutive models considered here. Of the 113 specimens 10 consider ed to develop the model , 64%, 19%, 10%, 5% and 3% specimens had masonry prism 11 strength ranging between 0 – 5 MPa, 5 – 10 MPa, 10 – 15 MPa, 15 – 20 MPa and 20 – 32 MPa, 12 respectively. It should be noted that masonry prism strength significantly influences the response 13 of infilled RC frame s (e.g., [43]). This distribution of masonry prism strength in the dataset can 14 be attributed to a greater likelihood of a specimen with a lesser masonry prism strength being 15 tested in the laboratory, implying that the sample may not be representative of the population due 16 to a systematic bias . While some statistical corrections are possible in the case of “missing 17 information” (e.g., [51]), the extent of the missing data and sensitivity of the regression 18 coefficients towards the same need to be investigated. This problem can partly be addressed by 19 consider ing ranges of masonry prism strength . The constitutive models discussed in this sectio n 20 use different functional forms to characterize model parameters (e.g., peak strength). Most 21 appropriate functional forms need to be established. A model based on a simple and intuitive 22 11 functional form s is more likely to be used in further research and structural design , which 1 motivates the present study . 2 3. Experimental specimens 3 Huang et al. [29] consider ed an extensive experimental da taset on one -bay-one-story infilled 4 RC frames for the development of constitutive model C8. Of the 113 specimen s considered by 5 Huang et al. [29], seven studied by Angel et al. [52] were dropped for the present study as the 6 specimens were tested for small deformations only, two studied by Chiou and Hwang [53]) were 7 dropped because these were not strictly one -bay-one-story frames, and four studied by [54], [55], 8 [56] were dropped because the information on some geometrical or mater ial properties was 9 missing. Besides, six specimens studied by [57], [58], [59] not considered by Huang et al. [29] 10 were included in the database , making it a total of 106 specimens considered in the present study . 11 The geometric and material properties of the specimens, and key features of the experiments are 12 presented in Table 2. Scale of specimens ranged between 0.2 and 1, aspect ratio h/L ranged 13 between 0.41 and 1.36, compressive strength of concrete ranged between 9.7 MPa and 65.9 MPa, 14 and compressive strength of masonry prisms ranged between 0.8 MPa and 26.7 MPa. One 15 hundred and four specimens were tested under cyclic loading, while two were tested 16 monotonically. Vertical load s were applied on columns in 85 of 106 specimens. The 17 experimentally recorded force -displacement curves were digitized manually. It should be noted 18 that th e present study considers the curve joining the points corresponding to local maxima and 19 minima in line with prior studies (e.g., [41], [60], [61]). The experimental backbone c onsidered 20 in the present study is the average of those in the positive and negative displacement directions, 21 in case the loading was cyclic. 22 23 24 12 Table 2. List of experimental specimens Reference Scale Tag (m × m) Aspect ratio Cross section (mm × mm) Compressive strength (MPa) Elastic Modulus3 (MPa) Load1 Vertical load2 Column Beam Concrete ' cf Masonry ' mf Masonry mE Mehrabi et al. [60] 1/2 E1 1.54 × 2.31 0.66 178 × 178 152 × 229 26.8 10.6 4595.63 C Yes E2 1.54 × 2.31 0.66 178 × 178 152 × 229 20.9 13.9 8943.22 C Yes E3 1.54 × 2.34 0.66 203 × 203 152 × 229 25.8 10.1 4196.01 C Yes E4 1.54 × 2.34 0.66 203 × 203 152 × 229 33.4 13.6 9067.24 C Yes E5 1.54 × 3.12 0.49 178 × 178 152 × 229 26.9 10.6 3941.08 C Yes E6 1.54 × 3.12 0.49 178 × 178 152 × 229 25.7 11.4 9597.77 C Yes Basha and Kaushik [61] 1/2 E7 1.59 × 1.68 0.95 115 × 175 115 × 175 22.4 3.9 2700.00 C Yes E8 1.59 × 1.68 0.95 115 × 175 115 × 175 22.4 3.9 2700.00 C Yes E9 1.59 × 1.68 0.95 115 × 175 115 × 175 22.4 4.6 2800.00 C Yes E10 1.59 × 1.68 0.95 115 × 175 115 × 175 22.4 4.6 2800.00 C Yes Al-Chaar et al. [57] 1/2 E11 1.43 × 2.03 0.70 127 × 203 127 × 197 38.4 18.2 - M No E12 1.43 × 2.03 0.70 127 × 203 127 × 197 38.4 26.7 - M No Kakaletsis and Karayannis [62] 1/3 E13 0.9 × 1.35 0.67 150 × 150 100 × 200 28.5 2.6 660.66 C Yes E14 0.9 × 1.35 0.67 150 × 150 100 × 200 28.5 15.2 2837.14 C Yes Colangelo [50] 1/2 E15 1.43 × 1.90 0.75 200 × 200 200 × 250 43.7 5.1 4230.00 C Yes E16 1.43 × 1.90 0.75 200 × 200 200 × 250 46.5 5.1 4230.00 C Yes E17 1.43 × 2.50 0.57 200 × 200 200 × 250 51.2 5.1 4230.00 C Yes E18 1.43 × 2.50 0.57 200 × 200 200 × 250 48.9 5.1 4230.00 C Yes E19 1.43 × 2.50 0.57 200 × 200 200 × 250 44.5 2.7 1212.00 C Yes E20 1.43 × 2.50 0.57 200 × 200 200 × 250 54.6 2.7 1212.00 C Yes E21 1.43 × 1.90 0.75 200 × 200 200 × 250 35.6 2.2 - C Yes E22 1.43 × 1.90 0.75 200 × 200 200 × 250 41.3 2.2 - C Yes E23 1.43 × 2.50 0.57 200 × 200 200 × 250 39.6 2.2 - C Yes E24 1.43 × 2.50 0.57 200 × 200 200 × 250 42.5 2.2 - C Yes E25 1.43 × 2.50 0.57 200 × 200 200 × 250 41.7 2.2 - C Yes Combescure and Pegon [63] 1/2 E26 1.72 × 2.30 0.75 150 × 150 150 × 200 20.0 2.2 - C Yes Cai and Su [59] 1 E27 3.23 × 4.40 0.73 400 × 400 200 × 450 33.5 4.2 - C Yes E28 3.23 × 4.40 0.73 400 × 400 200 × 450 33.5 2.0 - C Yes Blackard et al. [64] 2/3 E29 2.05 × 3.66 0.56 279 × 279 279 × 368 19.1 19.1 - C Yes Alwashali et al. [58] 1/2 E30 1.60 × 2.30 0.70 200 × 200 600 × 400 24.2 17.3 - C Yes E31 1.60 × 2.30 0.70 300 × 300 600 × 400 28.3 18.6 - C Yes Dautaj et al. [65] 2/3 E32 2.10 × 2.65 0.79 150 × 150 150 × 200 24.5 3.99 2192.99 C Yes E33 2.10 × 2.65 0.79 150 × 150 150 × 200 24.5 2.03 1121.23 C Yes E34 2.10 × 2.60 0.81 150 × 200 150 × 200 24.0 3.99 2192.99 C Yes E35 2.10 × 2.60 0.81 150 × 200 150 × 200 24.5 3.99 2192.99 C Yes E36 2.10 × 2.55 0.82 150 × 250 150 × 250 25.0 3.99 2192.99 C Yes E37 2.10 × 2.65 0.79 150 × 150 150 × 200 24.0 3.69 10058.07 C Yes E38 2.10 × 2.50 0.84 150 × 300 150 × 300 24.5 5.25 2885.51 C Yes Gazic and Sigmund [66] 1/2 E39 1.41× 2.0 0.71 125 × 150 125 × 200 20.0 4.6 8182.37 C Yes E40 1.41× 2.0 0.71 125 × 150 125 × 200 20.0 4.6 8182.37 C Yes E41 1.41× 2.0 0.71 125 × 150 125 × 200 20.0 5.2 1663.85 C Yes E42 1.41× 2.0 0.71 125 × 150 125 × 200 20.0 0.8 1764.78 C Yes E43 1.41× 2.0 0.71 125 × 150 125 × 200 20.0 3.5 1651.86 C Yes E44 1.41× 2.0 0.71 125 × 150 125 × 200 20.0 5.2 1663.85 C Yes E45 1.41× 2.0 0.71 125 × 150 125 × 200 20.0 4.6 8182.37 C Yes E46 1.41× 2.0 0.71 125 × 150 125 × 200 20.0 5.2 1663.85 C Yes E47 1.41× 2.0 0.71 125 × 150 125 × 200 20.0 4.6 8182.37 C Yes E48 1.41× 2.0 0.71 125 × 150 125 × 200 20.0 5.2 1663.85 C Yes Bergami and Nuti [67] 1/2 E49 1.42× 2.5 0.57 200 × 200 200 × 250 30.0 6.2 1666.52 C Yes E50 1.42× 2.5 0.57 200 × 200 200 × 250 30.0 6.2 1666.52 C Yes Mansouri et al. [68] 1/2 E51 1.38× 2.30 0.60 200 × 200 200× 150 21.9 2.30 436.69 C Yes Misir et al. [69] 4/5 E52 2.20 × 3.60 0.61 250 × 400 250 × 400 27.1 1.5 840.92 C Yes E53 2.20 × 3.60 0.61 250 × 400 250 × 400 26.4 2.1 1158.09 C Yes E54 2.20 × 3.60 0.61 250 × 400 250 × 400 24.5 2.4 1314.82 C Yes E55 2.20 × 3.60 0.61 250 × 400 250 × 400 26.8 1.6 854.30 C Yes E56 2.20 × 3.60 0.61 250 × 400 250 × 400 25.5 1.8 1005.78 C Yes Morandi et al. [70] 1/2 E57 3.12 × 4.55 0.69 350 × 350 350 × 350 34.0 4.6 5295.35 C No2 E58 3.12 × 4.55 0.69 350 × 350 350 × 350 34.0 4.6 5295.35 C No2 E59 3.12 × 4.55 0.69 350 × 350 350 × 350 34.0 4.6 5295.35 C No2 Tizapa [71] 1/1 E60 2.10 × 2.70 0.78 125× 200 125× 200 24.0 4.5 2424.33 C No Verderame et al. [72] 1/3 E61 1.47 × 2.30 0.64 200 × 200 200 × 250 22.7 3.2 - C Yes E62 1.47 × 2.30 0.64 200 × 200 200 × 250 21.3 3.2 - C Yes hL 13 4. Modelling masonry -infilled RC frames 1 Fig. 2 presents a schematic of the model for the infilled RC frames. There are nine frame 2 elements (numbered 1 through 9) in series with nine shear hinge elements, and struts representing 3 the masonr y infill panel; solid (broken) lines indicate that the strut will be in compression 4 (tension) when a lateral load to the right is applied at the beam level. 5 Akhoundi et al. [73] 1/2 E63 1.77 × 2.86 0.62 160 × 160 160 × 270 20.0 2.0 - C Yes Chiou and Hwang [53] 1/1 E64 2.94 × 3.60 0.82 350 × 400 350 × 600 25.0 7.0 - C No E65 2.94 × 3.60 0.82 350 × 400 350 × 600 27.0 8.5 - C No Zovkic et al. [74] 1/2.5 E66 2.05 × 3.66 0.56 200 × 200 120 × 200 51.5 4.3 - C Yes E67 2.05 × 3.66 0.56 200 × 200 120 × 200 48.5 1.9 - C Yes E68 2.05 × 3.66 0.56 200 × 200 120 × 200 35.0 1.6 - C Yes Kumar [75] 1/1 E69 2.41 × 2.50 0.96 150 × 300 150 × 600 30.0 0.9 - C No Bose and Rai [76] 1/2.5 E70 1.23 × 3.00 0.41 200 × 200 200 × 200 37.6 2.4 - C Yes Leuchars and Scrivener [77] 1/1 E71 1.27× 1.96 0.65 203 × 152 203 × 152 24.8 3.8 - C No Schwarz et al. [78] 1/2 E72 1.50× 1.10 1.36 200 × 200 100 × 200 28.8 2.7 - C Yes E73 1.50× 2.24 0.67 200 × 200 100 × 200 28.8 2.7 - C Yes Zhai et al. [79] 1/1 E74 2.80 × 3.15 0.89 352× 352 350 × 400 27.7 1.9 - C Yes Stylianidis [80] 1/3 E75 0.96 × 1.60 0.60 150 × 150 100 × 200 26.5 4.2 - C No E76 0.96 × 1.01 0.95 150 × 150 100 × 200 27.9 1.9 - C No Cavaleri et al. [81] E77 1.80 × 1.80 1.00 200 × 200 200 × 400 25.0 2.7 3930.29 C Yes E78 1.80 × 1.80 1.00 200 × 200 200 × 400 25.0 2.7 3930.29 C Yes E79 1.80 × 1.80 1.00 200 × 200 200 × 400 25.0 8.7 6396.59 C Yes E80 1.80 × 1.80 1.00 200 × 200 200 × 400 25.0 8.7 6396.59 C Yes E81 1.80 × 1.90 0.95 300 × 300 300 × 400 25.0 1.7 4561.86 C Yes E82 1.80 × 1.90 0.95 300 × 300 300 × 400 25.0 1.7 4561.86 C Yes E83 1.80 × 1.90 0.95 300 × 300 300 × 400 25.0 1.7 4561.86 C Yes E84 1.80 × 1.90 0.95 300 × 300 300 × 400 25.0 1.7 4561.86 C Yes E85 1.80 × 1.80 1.00 200 × 200 200 × 400 25.0 4.6 7101.11 C Yes E86 1.80 × 1.80 1.00 200 × 200 200 × 400 25.0 4.6 7101.11 C Yes E87 1.80 × 1.80 1.00 200 × 200 200 × 400 25.0 8.7 6396.59 C Yes E88 1.80 × 1.80 1.00 200 × 200 200 × 400 25.0 8.7 6396.59 C Yes Baran and Sevil [82] 1/3 E89 0.82 × 1.40 0.58 150 × 100 150 × 150 15.6 3.2 - C Yes E90 0.82 × 1.40 0.58 150 × 100 150 × 150 10.7 3.0 - C Yes E91 0.82 × 1.40 0.58 150 × 100 150 × 150 9.7 3.0 - C Yes Crisafulli [83] 3/4 E92 2.10 × 2.67 0.79 150 × 150 150 × 200 22.5 19.3 11542.05 C Yes E93 2.10 × 2.67 0.79 150 × 150 150 × 200 31.2 19.3 11542.05 C Yes Calvi and Bolognini [84] 1/1 E94 2.90 × 4.50 0.64 300 × 300 700 × 250 29.3 1.1 1871.71 C No2 Tawfik and Essa [85] 1/2 E95 1.60× 1.98 0.80 120 × 200 120 × 200 65.0 4.0 2192.99 C No2 E96 1.60× 1.98 0.80 120 × 200 120 × 200 66.0 3.4 1896.19 C No2 E97 1.60× 1.98 0.80 120 × 200 120 × 200 66.0 1.9 1071.76 C No2 Basha and Kaushik [61] 1/2 E98 1.59 × 1.68 0.95 115 × 175 115 × 175 22.4 3.9 2698.14 C Yes E99 1.59 × 1.68 0.95 115 × 175 115 × 175 22.4 3.9 2698.14 C Yes Haider [86] 1/1 E100 2.20 × 2.26 0.97 254 × 254 254 × 254 34.3 10.8 2344.67 C No2 E101 2.20 × 2.20 1.00 305 × 305 254 × 254 35.2 12.1 2248.21 C No2 E102 2.20 × 1.80 1.22 254 × 254 254 × 254 37.5 18.1 3219.70 C No2 E103 2.20 × 1.80 1.17 305 × 305 305 × 254 38.2 17.4 3203.16 C No2 Sigmund [87] 1/2.5 E104 1.40 × 2.00 0.70 200 × 200 120 × 200 45.0 2.7 3204.16 C Yes Yuksel [88] 1/2 E105 1.36 × 1.95 0.70 200 × 250 200 × 325 18.0 25.3 7912.55 C No2 Bellington et al. [89] 1/5 E106 0.63 × 1.1 0.57 165 × 108 165 × 140 20.7 20.1 5119.30 C Yes 1M – Monotonic; C – Cyclic 2Vertical loads were applied on the beam during experiments, but the magnitude of the load was small. 3The elastic mo dulus of masonry panel was considered as  '550mmEf in case not provided. 14 Fig. 2. A schematic of a three -strut model (adapted from Ghosh and Kumar [37]); Figure reused with permission from TMS and 13NAMC) The open source softwar e program OpenSees [90] is considered for lateral pushover analyses . 1 Beams and columns of a frame are modelled using the distributed plasticity app roach, which 2 incorporates the P -M interaction in frame members. Flexural behavior of the frame members is 3 simulated using the forceBeamColumn element, which is a force -based nonlinear fiber -based 4 beam -column element . Concrete and longitudinal reinforcements are model led using fiber 5 sections. Uniaxial material Concrete01 is used to model the core (confined) and cover 6 (unconfined) concrete. Stress -strain relationships for the confined and unconfined concrete are 7 defined using the model proposed by Mander et al. [91]. Reinforcements are modelled using 8 Steel01 , which is a uniaxial bilinear steel material object with kinematic hardening. Reinforced 9 concrete members ma y be vulnerable to shear failure before or after flexural failure has taken 10 place. Several models to describe shear force -displacement behavior of RC members have been 11 proposed (e.g., [92]-[94]). In the present study, the model proposed by Elwood [93] has been 12 considered, wher ein shear strength is based on Sezen and Moehle [94]. Shear behavior of RC 13 members is modelled using zeroLength elements placed in series with frame members. 14 LimitStateMaterial material available with OpenSees is used to implement the material model. 15 Struts representing masonry panels are modelled using forceBeamColumn element . These are 16 connected to the frame through a pin, and resist forces only in compression. Uniaxial material 17 15 object Pinching4 is used to define the constitutive model for the struts. Cyclic degradation and 1 pinching parameters used for the Pinching4 material are based on Kumar et al . [43]. It should be 2 noted that the central goal of this work is to characterize the backbone curves obtained through 3 monotonic analysis. Accordingly, the degradation parameters do not influence the results 4 presented in this study . Secondly, the influence of the choice of degradation parameters on the 5 lateral force -displacement response of an infilled RC frame subjected to a realistic earthquake 6 scenario is considerably less compared to that of the b ackbone curve [43]. 7 Vertical loads were applied on most specimens. These loads are considered in the analytical 8 models as follows. Loads applied di rectly on the columns during experiments are applied directly 9 on the columns in the models. Loads applied on the beams during experiments are applied 10 directly on the top of columns considering the beam to be simply supported on the column. It is 11 recognized that a fraction of the load could have gone through the masonry panel, which is not 12 possible to simulate through the analytical models considered herein. 13 All analyses were d ispla cement -controlled with Gauss -Lobatto integration scheme. Newton 14 or ModifiedNewton algorithm was used to solve to the equilibrium equations. Command 15 NormDispIncr was invoked to establish convergence with a tolerance of 10-10 (it was increased 16 up to 10-5 in order to achieve convergence at a step). 17 5. Characterizing performance of macro models 18 5.1. Backbone parameters 19 Eight macro models (see Fig. 1(c) and Table 1) were developed for each of the 106 20 experimental specimens . Monotonic pushover analyses were carried out using each of these 21 models. Simulated results, namely, lateral force -displacement resp onse, and flexural and shear 22 failure s in frame members, were compared with corresponding experimental observations. Fig. 23 3 presents the simulated forc e-displacement response of experimental specimen s DFS of [61] 24 16 and WF of [58] using macro model S 3-C1 (combination of strut model S 3 and constitutive 1 material model C1). Masonry prism strengths for the two specimens are 4 MPa (“weak” infill) 2 and 17.3 MPa (“strong” infi ll), respectively. Also presented in the figure are the results for macro 3 models S 3-C2, S 3-C3, S 3-C4, S 3-C5, S 3-C6, S3-C7 and S 3-C8, and that observed 4 experimentally. It is clear that the choice of constitutive model significantly influences the lateral 5 force-displacement response of the infilled RC frame. In order to quantify the performance of a 6 macro model (e.g., S 3-C6), the calculated lateral force -displacement response is idealized first. 7 Fig. 4 presents a generic lateral force -displacement curve for an infilled RC frame. The response 8 up to the point corresponding to the peak strength can be characterized using the force associated 9 with first major change in stiffness 1F , initial stiffness 1K, and peak force 2F and associated 10 displacement 2D . These parameters can be compared with that for corresponding experimentally 11 obtained response. The average of the ratios of simulated and experimentally recorded forces at 12 different displacements greater than corr esponding experimental value of 2D can provide 13 information on the ability of a macro model to simulate the post -peak response. 14 (a) Specimen DFS [61] (b) Specimen WF [58] Fig. 3. Experimentally observed and simulated force -displacement response for two one -bay-one- story infilled RC specimens. 17 For an experimentally obtained force -displacement curve, 2F and 2D (see Fig. 4) were 1 identified first. The point corresponding to 1F was identified by judgement. Initial stiffness 1K 2 was determined as the slope of the first line. The parameters for simul ated force -displacement 3 curves we re determined through a similar, but automated, approach. Peak force 2F and 4 corresponding displacement 2D were identified first. Force 1F was then determined such that 5 the ratio of the slope of the line between origin and 11D ,F , and that between 11D ,F and the 6 point corresponding to 2 0.85F was the highest. This approach help ed avoid points closer to 7 22,DF , which may be associated with greater levels of damage . 8 Fig. 4. Idealized backbone curve for masonry -infilled RC frames Table 3 presents parameters 1F, 1K, 2F and 2D corresponding to experimentally recorded 9 force -displacement curve for specimen DFS of [61]. Also included in the table are parameters 10 for the simulated curves obtained using the eight macro models (also see Fig. 3). A total of 10,000 11 equispaced points were identified between zero displacement and the displacement up to which 12 the experiment was performed. Further, points beyond displacement 2D (in the post -peak 13 regime) were considered, and simulated and experimental values of forces ordinates were 14 determined. The ratios of simulated values to the corresponding experimental values were 15 calculated for all these points, and their average PPR was considered. Parameter PPR is 16 18 considered to indicate the strength of simulation in the post -peak region, namely, the simulation 1 using a macro -model is better if the parameter is closer to 1, and vice -versa. Parameter PPR is 2 also reported in Table 3. 3 Table 3. Simulated and experimental results for DFS specimen of [61] (see Fig. 2 for element numbers) Experimental specimen Macro model 1K 1F 2F 2D PPR Elements failed in shear Elements failed in flexure kN/mm kN kN mm DFS 16.34 52.15 64.09 14.11 - Column Column S3-C1 11.02 59.74 83.65 10.51 0.85 - 4,7 S3-C2 - - 75.77 8.63 0.78 - 4,7 S3-C3 - - 72.22 14.10 1.07 - 4,7 S3-C4 45.91 92.48 120.67 4.20 0.98 4 4,7 S3-C5 47.68 72.58 127.69 6.10 0.67 3,4 4,7 S3-C6 51.77 44.93 68.83 13.65 0.96 - 4,7 S3-C7 24.52 144.81 187.47 19.90 2.67 3,4,7 4,7 S3-C8 20.06 70.20 109.87 12.39 1.40 3,4 4,7 5.2. Failure in frame members 4 Specimen DFS of [61] underwent shear as well as flexural failure during the experiment. 5 Strut model S3 combined with constitutive models C4, C5, C7 and C8 could sim ulate shear 6 failure in frame members. All combinations of strut (S3) and constitutive (C1 through C8) models 7 could capture flexural failure in frame members. A similar analysis was performed for all 106 8 experimental specimens. Ratios of simulated values of 1K, 1F, 2F, and 2D to the respective 9 experimental values were computed (also see [95]). A part of these results is discussed in 10 Section 6, which guides the development of the proposed model. 11 Informat ion on the flexural and/or shear failures in frame members was reported for 74 12 experimentally studied specimens [95]. Bending moment developed in a frame member 13 exceeding the corresponding yield strength was considered as flexural failure. Similarly, shear 14 force developed in a frame member exceeding the cor responding strength ( e.g., [93]) was treated 15 as shear failure. The ability of a macro model to capture flexure (or shear) failure can be 16 quantified in terms of correct and incorrect simulations. Say , 1n is the number of specimens for 17 which a failure was observed during a simulation as well as the corresponding experiment. 18 19 Similarly, 2n is the number of specimens for which failure was not observed in either simulation 1 or experiment. These numbers correspond to correct simulations. Say, 3n 4n is the number of 2 specimens for which a failure was experimentally observed but not simulated (experimentally 3 not observed but simulated) using a numerical model. These numbers ( 3n and 4n ) correspond 4 to incorrect simulations. 5 6. Proposed macro model 6 6.1. Development 7 Strut model S3 is considered for the development of the masonry constitutive model, which 8 is defined using parameters identified in Fig. 5, namely, ,mcrF , 1,mK , 2,mK , max,mF , 3,mK , and 9 ,res mF . These parameters represent first major crack in masonry, initial stiffness, peak strength, 10 secant stiffness corresponding to peak strength, slope for the post -peak segment, and residual 11 strength of the combination of the three struts representing the masonry pan el, respe ctively 12 (also see Table 1). The development of each parameter is discussed in the sections below . 13 Fig. 5. Backbone curve for the proposed constitutive model 20 6.1.1. Peak strength  max,mF 1 An attempt was made to characterize the peak strength of masonry prism max,mF (see Fig. 5) 2 first. For the 106 one -bay-one-story infilled RC frames (see Table 2), Table 4 presents the mean 3 of the simulated -to-experimental ( S/E) ratios of peak strength of infilled RC frame 2F (see Fig. 4 4). The simulations were performed using eight macro models , namely, S3-C1 through S3 -C8 5 (see Fig. 2 for the schematic, and Table 1 for the details of masonry constitutive models C1 6 throu gh C8 developed in the prior studies ). Table 4 also presents standard deviations (STDs) and 7 coefficients of variation (COVs) associated with the S/E r atios . 8 Table 4. Mean simulated -to-experimental (S/E) ratios, corresponding standard deviations (STDs) and coefficients of variation (COVs) for peak strength of infilled RC frame '05mf MPa '5 10mf MPa '10mf MPa Parameter CM Eq. Mean STD COV Mean STD COV Mean STD COV 2F C1 (4) 0.93 0.36 0.38 0.99 0.25 0.25 1.61 0.79 0.49 C2 (5) 0.84 0.33 0.39 0.93 0.23 0.25 1.54 0.83 0.54 C3 (12) 0.82 0.29 0.36 0.89 0.23 0.26 1.46 0.83 0.57 C4 (16) 1.34 0.55 0.41 1.14 0.26 0.23 1.48 0.85 0.58 C5 (22) 1.29 0.53 0.41 1.13 0.27 0.24 1.44 0.86 0.59 C6 (25) 0.80 0.28 0.35 0.69 0.16 0.23 0.97 0.73 0.75 C7 (30) 1.65 0.72 0.44 1.59 0.38 0.24 2.13 0.74 0.35 C8 (35) 1.17 0.42 0.36 0.86 0.20 0.24 1.03 0.69 0.67 The COVs of the S/E ratios for all eight previously proposed constitutive models range 9 between 0.35 and 0.44 for infilled frames with '05mf MPa. The mean of the S/E ratios is 10 closest to 1.00 for constitutive model C1 ( mean = 0.93, COV = 0.38) , which defines max,mF as 11 the product of the stress capacity of masonry prism and cross -sectional area of strut (see Table 12 1). The functional form for model C1 is based on fundamental principles of mechanics, and the 13 model is able to simulate the peak strength of infi lled RC frames rather well. Therefore, i t is 14 proposed to define parameter max,mF as follows. 15 ' max,m max,m m m mbt Ff (39) 16 where max,m is a factor depending on ' mf , and other parameters were defined previously. 17 21 For '5 10mf MPa , model C1 led to the mean S/E rati o closest to 1.00 (mean = 0.99, 1 COV = 0.25) among all models studied . It should be noted that all eight macro models led to 2 COV between 0.23 and 0.26 for this range of ' mf . Accordingly, Eq. (39) is also considered for 3 '5 10mf MPa. 4 For '10mf MPa , the least COV (= 0.35) is obtained through constitutive model C7, which 5 defines 2F in terms of the results obtained through the diagonal compression tests on masonry 6 walls (see Table 1). It is easier to perform tests on masonry prism compared to masonry walls. 7 The next lowest COV is obtained through constitutive model C1 (mean = 1.61, COV = 0.49). 8 Given its simplicity , it is prop osed to consider the expression of Eq. (39) also for '10mf MPa. 9 It should be noted that there are only 20 samples with '10mf MPa, and more experimental 10 results may enable the selection of a better functional form. 11 Parameter max,m was set equal to 1.1, 1.0 and 0.4 for '05mf MPa , '5 10mf MPa and 12 '10mf MPa in order to achieve the mean values of S/E ratios for 2F (see Fig. 4) close to 1.0, 13 respectively. The values of the coefficient suggest that approximately 100% (40%) of masonry 14 strength is “utilized” toward s the overall strength of an infilled RC frame for ' mf smaller (greater) 15 than 10 MPa. 16 The approach described above ca n be appreciated through Fig. 6. Panel (a) of the figure 17 presents the average simulated -to-experimental ratios for parameters 1F , 1K, 2F, 2D and PPR 18 obtained using strut model S3 and the eight constitutive models described in Table 1 for 19 '05mf MPa . Panels (b) and (c) of the fig ure present results for '5 10mf MPa and 20 '10mf MPa, respectively. Panel s (d), (e) and (f) of the figure present the ratios for the three 21 ranges of ' mf , respectively, with max,mF defined using the proposed method, i.e., Eq. (39) , and 22 22 other parameters defined using the respective expressions presented in Table 1. It can be se en 1 from the figures that the average simulated -to-experimental ratio for 2F is 1.0 (indicated using a 2 red circle) . 3 Fig. 6. Simulated -to-experimental ratios for parameters 1F , 1K, 2F, 2D and PPR at different steps of the development of the proposed model 23 6.1.2. Force corresponding to first major crack in masonry ,mcrF 1 The next step was to characterize the force corresponding to the first major crack in masonry 2 ,mcrF (see Fig. 5). For this part, max,mF was given by Eq. (39) for all constitutive models, while 3 other parameters of the models were defined using the expressions given in Table 1. Strut model 4 S3 was used in combination with the variants of the eight constitutive models, and lateral 5 force -deformation response of the 106 infilled RC fra mes (see Table 2) were simulated. The ratio 6 of simulated force corresponding to the first major crack in infilled RC frame 1F (see Fig. 4) to 7 the corresponding experimental value (S/E) was determined for all 106 specimens. Notably, 8 constitutive models C1, C4, C5, C6 and C8 consider ,mcrF equal to 0.75, 0.77, 0.60 , 0.80 and 9 0.72 times max,mF , respectively. The mean S/E ratios for 1F were closest to 1.00 , when ,mcrF was 10 set equal to 0.60 max,mF (see [95]) irrespective of ' mf . Accordingly, parameter ,mcrF is proposed 11 to be defined as follows. 12 ,m ,m max,mcr cr F F (40) 13 where ,mcr is set equal to 0.60. 14 For different ranges of ' mf , panels (g) through (i) of Fig. 6 present average 15 simulated -to-experimental ratios for parameters 1F, 1K, 2F, 2D and PPR with max,mF and ,mcrF 16 defined using Eq. (39) and Eq. (40), respectively, and other parameters of the constitutive models 17 defined using the expressions presented in Table 1. The average simulated -to-experimental ratios 18 for parameters 2F and 1F are 1.0 (indicated using red circles). 19 24 6.1.3. Stiffness corresponding to first major point of cracking 1,mK 1 With max,mF and ,mcrF defined using Eq. (39) and Eq. (40), respectively, the functional form 2 for the initial axial stiffness of the masonry panel 1,mK (see Fig. 5) was established next. These 3 definitions for the two parameters were used for all eight constitutive models, while the 4 remaining parameters for the models were defined using the expressions presented in Table 1. In 5 addition, a further variant of constitutive model C7 was considered, wherein the strut width 6 parameter w in Eq. (29) was defined using Eq. (1) given by Mainstone [44] instead of 4md as 7 originally considered for C7. The lateral force -displacement response for all 106 specimens was 8 computed using the nine macro models, and S/E ratios for the initial stiffness of infilled RC 9 frames 1K (see Fig. 4) were determined. Mean , STD and COV of these ratios obtained us ing the 10 nine macro model are presented in Table 5. The variant of model C7 is denoted using C7M in 11 the table. 12 Table 5. Mean simulated -to-experimental (S/E) ratios, corresponding standard deviations (STDs) and coefficient s of variation (COVs) for initial stiffness of infilled RC frame '05mf MPa '5 10mf MPa '10mf MPa Parameters CM Reference Mean STD COV Mean STD COV Mean STD COV max,mF ' max,m m m mbtf Eq. (39) cr,mF max,m 0.60 F Eq. (40) 1K C1 Table 1 0.46 0.39 0.85 0.34 0.22 0.65 0.45 0.5 1.11 C2 - - - - - - - C3 - - - - - - - C4 Eq. (15) 8.47 16.61 1.96 8.92 13.5 1.51 6.97 9.82 1.41 C5 Eq. (21) 2.65 2.03 0.77 4.57 5.41 1.18 3.74 3.08 0.82 C6 Eq. (24) 11.97 13.54 1.13 12.22 12.9 1.06 6.41 7.91 1.23 C7 Eq. (29) 1.26 1.07 0.85 0.83 0.53 0.64 1.33 0.93 0.70 C7M Eq. (29), Eq. (1) 0.64 0.48 0.75 0.39 0.21 0.54 0.64 0.37 0.58 C8 Eq. (34) 1.07 0.74 0.69 0.67 0.4 0.60 0.92 0.39 0.42 Model C8 led to the least COV for '05mf MPa among all the nine models studied 13 (mean = 1.07, COV = 0.69) . This model considers elastic modulus of masonry, and thickness, 14 height and length of the masonry panel. The expression of model C8 is somewhat intuitive. The 15 25 next lowest COV was obtained u sing C7M model (mean = 0.64, COV = 0.75) . This model 1 considers a mechanics -based definition of axial stiffness. Considering its simplicity , it is 2 proposed to define 1,mK based on the definition consid ered for constitutive model C7M. 3 1,m 1,mm m m mE b tKd (41) 4 where 1,m is a factor depending on ' mf , and other parameters were defined previously. For 5 '5 10mf MPa, the least COV was obtained for constitutive model C7M (mean = 0.39, 6 COV = 0.54). Therefore, it was decided to consider the expression of Eq. (41) for this infill 7 strength range as well. This expression was adopted also for '10mf MPa because of its 8 simplicity, despite model C8 leading to somewhat better simulation in terms of COV. Parameter 9 1,m was set equal to 1.8, 2.8 and 1.8 for '05mf MPa , '5 10mf MPa and '10mf MPa in 10 order to achieve the mean value of S/E ratio for 1K (see Fig. 4) close to 1.0, respectively. 11 For different ranges of ' mf , panels (j ) through (l) of Fig. 6 present average 12 simulated -to-experimental ratios for parameters 1F , 1K, 2F, 2D and PPR with max,mF , ,mcrF and 13 1,mK defined using Eq s. (39) through (41), respectively, and other constitutive model parameters 14 defined using the expressions presented in Table 1. The average simulated -to-experimental ratios 15 for parameters 2F , 1F and 1K are 1.0 (indicated using red circles ). 16 6.1.4. Stiffness corresponding to peak strength  2,mK 17 The next step was to establish the expression for the secant stiffness corresponding to peak 18 strength of masonry strut 2,mK (see Fig. 5). Many constitutive models define 2,mK using the 19 functional form considered for 1,mK (e.g., C2, C4, and C5). The same approach is adopted in the 20 present study as well. Accordingly, 2,mK is defined as follows. 21 26 2,m 2,mm m m mE b tKd (42) 1 where 2,m is a function of ' mf , and other parameters were defined previously. With max,mF , ,mcrF 2 and 1,mK defined using Eq. (39), Eq. (40) and Eq. (41), respectively, p arameter 2,m was set 3 equal to 0.7, 1.3 and 0.5 for '05mf MPa , '5 10mf MPa and '10mf MPa in order to 4 achieve the mean value of S/E ratio for 2D (see Fig. 4) close to 1.0, respectively. 5 For different ranges of ' mf , panels ( m) through (o) of Fig. 6 present the average 6 simulated -to-experimental ratios for parameters 1F , 1K, 2F, 2D and PPR with max,mF , ,mcrF , 7 1,mK and 2,mK defined using Eq s. (39) through (42), respectively, and the other parameters 8 defined using the respective expressions presented in Table 1. The average ratio for parameters 9 2F , 1F and 1K is 1.0 , while that for parameter 2D is less than 1.0 . The average ratio for parameter 10 2D was found sensitive to the definitions of other post -peak constitutive model parameters . 11 6.1.5. Residual strength ,mresF and slope for the post -peak segment 3,mK 12 Eqs. (39) through (42) define the proposed constitutive model for masonry strut up to peak 13 strength. The post -peak behaviour is to be characterized using the slope for the post -peak segment 14 3,mK and residual strength ,res mF (see Fig. 5). Parameter 3,mK can be defined using the functional 15 form considered for 1,mK (e.g., C4 , C6). Similarly, ,res mF can be defined using the functional 16 form for max,mF (e.g., C1, C4, C6, C8). Thirty seven out of 106 infilled RC frames were 17 considered to have been loaded till respective residual strengths ( see [95]). Expressions for these 18 two post -peak parameters are given below. 19 ,m ,m max,mres resFF (43) 20 27 3,m 3,mm m m mE b tKd (44) 1 where ,mres and 3,m are factors depending on ' mf , and other parameters were defined 2 previously. These two parameters were established together through iterations. The final value 3 of parameter 3,m ( ,mres ) was set equal to 0.09 (0.70), 0.08 (0.70) and 0.05 (0 .70) for 4 '05mf MPa , '5 10mf MPa and '10mf MPa in order to achieve the value of post -peak 5 response parameter PPR (see Section 5) close to 1.0, respectively. It should be noted that the 6 value of ,mres (= 0.70) is close to the average of the ratios of the residual to peak strength of 7 infilled RC frames for the 37 specimens considered. Most constitutive models considered in the 8 present study define the residual masonry strut strength in the range of 0 – 40% of the 9 corresponding peak strength. Kumar et al. [43] among others note that a “weak” masonry would 10 lead to a greater residual strength (e.g., 70% of the peak strength) for the infilled RC frames, and 11 vice-versa. It should be noted that 73% of the 37 specimens considered to characterize the 12 post-peak behaviour have ' mf smaller than 5 MPa. 13 For different ranges of ' mf , panels (p) through (r) of Fig. 6 present the average 14 simulated -to-experimental ratios for parameters 1F , 1K, 2F, 2D and PPR obtained using the 15 proposed constitutive model , i.e., Eqs. (39) through (44). The average simulated -to-experimental 16 ratio is 1.0 for all five backbone parameters . 17 A summary of the proposed masonry constitutive model corresponding to the S3 strut model 18 is presented in Table 6. The steps to implement the model are outlined in Fig. 7. These steps can 19 be extended to model multi -bay-multi -story infilled RC frames as well. 20 21 22 28 Table 6. Proposed masonry constitutive model (see Fig. 2 and Fig. 5) 1 Parameter Expression Multiplier '05mf MPa '5 10mf MPa '10mf MPa max, mF ' max,m m m mf b t max,m1.1 max,m1.0 max,m0.4 ,mcrF ' ,m max,mcr m m mf b t  ,m0.6cr ,m0.6cr ,m0.6cr 1, mK 1,mm m m mE b t d 1,m1.8 1,m2.8 1,m1.8 2 , mK 2,mm m m mE b t d 2,m0.7 2,m1.3 2,m0.5 ,mresF ' ,m max,mres m m mf b t  ,m0.7res ,m0.7res ,m0.7res 3,mK 3,mm m m mE b t d 3,m0.09 3,m0.08 3,m0.05 Fig. 7. Summary of procedure to use the proposed masonry constitutive model for analysis 6.2. Performance 2 This section presents a discussion on the performance of the proposed masonry constitutive 3 model in simulating the response of one -bay-one-story , and multi -bay-multi -story infilled RC 4 frames . 5 29 6.2.1. One-bay-one-story frames 1 This section presents the performance of the proposed constitutive model (along with models 2 C1 through C8 developed in the past) in simulating the response of 106 experimentally studi ed 3 infilled RC frames (see Table 2). Also discussed is the effectiveness of the proposed model in 4 combination with single - and two -strut models. Further, it is examined whether the simulation is 5 correlated with some key geometrical and material parameters of the frame and infills (e.g., 6 stiffness of infill relative to frame) . Finally, the relationship between the observed failure modes 7 in the infil led RC frame and simulated backbone parameters is studied. 8 6.2.1.1 Simulation of backbone parameters 9 Influence of constitutive models 10 The proposed masonry constitutive model (denoted using CP in this section ) was used in 11 combination with strut model S3 to simulate the lateral force -displacement response of the 106 12 one-bay-one-story masonry -infilled RC frame specimens listed in Table 2. Fig. 8(a) presents the 13 mean and mean ± standard deviation of the S/E ratios for the peak strength 2F of infilled RC 14 frames (see Fig. 4) for '05mf MPa . Results for the proposed model are highlighted using red 15 circle s. Also presented in the panel are the results for the previously proposed constitutive models 16 C1 through C8 (see Table 1) in combination with strut model S3. Expectedly, t he proposed model 17 is able to simulate the peak streng th considerably better compared to the previously proposed 18 models . The next best simulation could be performed using model C1, which was developed 19 using the experimental results for masonry prism without considering the interaction between 20 frame and infill . This observation suggests that the influence of the frame -infill interaction on 21 the peak strength of infilled RC frames is relatively small. Similar observations can be made for 22 panel (b) of Fig. 8, which reports the results for '5 10mf MPa. It should be noted that the 23 30 mean ratios obtained using model C8 are greater than 1.0 for '05mf MPa, and smaller than 1 1.0 for '5 10mf MPa. However, performance of this model is similar to the proposed model 2 for '10mf MPa, as can be seen from Fig. 8(c). 3 Fig. 8. Ratios of simulated -to-experimental (S/E) values of lateral force -displacement parameters (see Fig. 4) for the proposed and some existing constitutive models Panels (d) through (f) of Fig. 8 present the S/E ratios for force corresponding to the first major 4 crack in frame or infill 1F (see Fig. 4) for the three ranges of infill strength, respectively. The 5 31 proposed model performs better compared to other constitutive models. Overall, a similar 1 conclusion can be draw n for other parameters, namely, 1K , 2D and PPR (see Fig. 4), as can be 2 seen from panels (g) through (o) of Fig. 8. 3 Influence of select strength and stiffness parameters 4 The proposed model was developed considering the data in three brackets based on masonry 5 prism strength. Within each bracket, the influence of strength parameters for beam and columns, 6 and ratio of masonry infill stiffness to that of frame are examined next. Accordingly, correlation 7 coefficients between S/E ratios of backbone parameters ( 2F , 1F, 1K, 2D and PPR ) and each of 8 the following parameters were calculated : (1) product of characteristic conc rete strength and 9 cross -section for column ' c c cf b d , (2) product of characteristic concrete strength and cross -section 10 for beams ' c b bf b d , and (3) a representative stiffness of infill relative to frame. The third parameter 11 was characterised as the ratio of lateral stiffness of masonry 2cosm m m mE b t d (e.g., [11]) to 12 a representative lateral stiffness of frame 3cEI h (e.g., [60]). 13 Fig. 9(a) presents the S/E ratios of 2F corresponding to ' c c cf b d . The correlation coefficients 14 between the S/E ratio s and ' c c cf b d is 0.04 for '05mf MPa, which can be considered small . 15 For this range of masonry prism strength, t he correlations are small also for parameters 1K, 2D 16 and PPR (see panels (g), (j) and (m) of Fig. 9, respectively) . However, t he correlation for 17 parameter 1F is 0.23 . For this range of ' mf , correlation coefficients between the S/E ratios and 18 ' c b bf b d are -0.08, 0.07, 0.04, -0.06 and -0.13 for parameters 2F , 1F, 1K, 2D and PPR , respectively 19 (see panels (b), (e), (h), (k) and (n) of Fig. 9). Overall, the simulations of t he backbone parameters 20 32 can be considered to be not meaningfully correlated with the strength of frame members for 1 '05mf MPa. 2 For '5 10mf MPa , correlations between S/E ratios and ' c c cf b d ( ' c b bf b d ) are 0.06, -0.09, 3 - 0.31, 0.62 and 0.18 (0.11, -0.20, -0.20, 0.60 and 0.21) for parameters 2F , 1F, 1K, 2D and PPR 4 , as can be seen from panels (a), (d), (g), (j) and (m) (panels (b), (e), (h), (k) and (n)) of Fig. 9, 5 respectively. These correlations are -0.22, -0.35, 0.70, -0.26 and -0.17 ( -0.23, -0.34, 0.04, -0.34 6 and -0.22) for '10mf MPa as can be se en from corresponding panels of Fig. 9, respectively. 7 The magnitude and signs of the correlation coefficient do not indicate a meaningful pattern of 8 the simulation results affected by the strength of frame members. It should be noted that only 9 16% and 19% of 106 one -bay-one-story specimens had '5 10mf MPa and '10mf MPa , 10 respectively. Additional data may help understand the patterns better. 11 Correlations between S/E ratios for different backbone parameters and representative 12 stiffness of infill relative to frame were studied (data presented in panels (c), (f), (i), (l) and (o) 13 of Fig. 9). The magnitudes and signs of these correlations suggest that the overall simulation of 14 the backbone parameters are not considerably dependent on the relative stiffness. 15 Influence of strut models 16 At times it is desirable to use simpler strut models (e.g., S1) for the purpose of simulating the 17 lateral force -displacement response of the infilled RC frames. Therefore, the lateral 18 force -displacement responses of the 106 specimens (see Table 2) were simulated using the 19 proposed masonry constitutive model in combination with strut models S1, S2 and S3 (see Fig. 20 1). Corresponding S/E ratios for the backbone parameters (see Fig. 4) are presented in Fig. 10. 21 33 Fig. 9. Ratios of simulated -to-experimental (S/E) values of backbone parameters (see Fig. 4) plotted against different strength and stiffness parameters of the frame 1 34 Fig. 10. Ratios of simulated -to-experimental (S/E) values of lateral force -displacement parameters (see Fig. 4) for the proposed constitutive model in combination with three strut models The choice of strut model does not influence the simulation of the lateral force -displacement 1 parameters for the m asonry -infilled RC frames materially. However, the peak strength 2F and 2 initial stiffness 1K obtained using that two -strut model are marginally smaller compared to 3 single -strut and three -strut model, which ca n be attributed to additional deformation (and the 4 35 development of force) in the frame members close to the loaded corner due to the absence of a 1 diagonal strut in the two -strut model. 2 Influence of observed failure mode in masonry panel 3 The proposed model did not consider the mode of failure in the masonry panel. However, it 4 has been noted in the past that the response of the frame may depend significantly on the failure 5 mode of infilled frames (e.g., [57], [60]). Out of 106 infilled frames considered herein, masonry 6 panels un derwent sliding shear, corner crushing and diagonal cracking failure in 42, 47 and 10 7 specimens, respectively; information on the remaining seven specimens was not available. 8 Results simulated using S3 -CP model for the 99 (= 42+47+10) specimens were consid ered 9 further. The corresponding S/E ratios for parameters 2F and PPR were relatively close to 1.0 in 10 case of sliding shear and corner crushing modes, while these were relatively close to 0.7 for 11 diagonal cracking mode. Expectedly, 1K was only marginally influenced by the mode of failure 12 in masonry. 13 Further, the S/E ratios for different backbone parameters (see Fig. 4) were computed 14 separately for specimens with reported shear (25 specimens) and flexural (49 specimens) failures 15 in RC frame members. The ratios were close to 1.0 irrespective of the failure mode. Ove rall, it 16 can be said that simulation of key response parameters for the infilled RC frames was not 17 significantly influenced by the mode of failure in masonry (except when masonry failed in 18 diagonal cracking mode, i.e., in approximately 10% specimens) or RC frame member. 19 6.2.1.2 Simulation of f lexural and shear failure in frame members 20 Fig. 11(a) presents the comparison of simulated and observed instances of flexural failure in 21 frame members for the infilled RC frames (see Section 5). The proposed model (S3 -CP; indicated 22 using S3 in the figure) was able to capture flexural failure (or its absence) correctly in 65 out of 23 36 74 infilled RC frame specimens for which flexure and/or shear failure was reported. The 1 influence of the choice of strut model was marginal. Results for shear failure are reported in Fig. 2 11(b). The proposed model could capture shear failure (or its absence) in frame memb ers for 57 3 out of the 74 specimens. Strut model S1 could lead to the correct or incorrect simulation of shear 4 failure in frame members only in five out of the 74 specimens 5 Fig. 11. Occurrence of (a) Flexure and (b) Shear failure in frame members ( 1n ≡ experimentally observed and simulated, 2n ≡ experimentally not observed and not simulated, 3n ≡ experimentally observed and not simulated, 4n ≡ experimentally not observed and simulated) 6.2.2. Multi -bay-multi -story frames 6 The proposed macro model was used to compute the lateral force -displacement response of 7 four multi -bay-multi -story infilled RC frames studied in the past. A one -bay-two-story (2S-1B) 8 and a two -bay-one-story (1S -2B) frames experimentally studied by Suzuki et al. [96], a 9 three -story -one-bay (3S -1B) frame experimentally studied by Koutas et al. [97], and a 10 three -story -three -bay (3S -3B) frame experimentally studied by Dukuze and Dawe [98] were 11 considered. Fig. 12 presents the experimentally obtained lateral force -displacement response of 12 the frames along with that computed using the proposed macro model (S3 -CP). Select details of 13 the specimens a re also specified in the figure. The proposed model could capture the response 14 for specimens 2S -1B, 1S -2B and 3S -1B. The initial stiffness of the 3S -3B specimen also could 15 37 be captured well. However, simulated peak strength was substantially lower than the 1 corresponding peak value. Accordingly, the overall simulation was inadequate for the specimen. 2 It should be noted that ' mf for specimen 3S -3B was 39.6 MPa, which is beyond the range of ' mf 3 considered to develop the constitutive model, i.e., 0.8 MPa – 26.7 MPa . 4 Fig. 12. Comparison of the simulated and experimentally obtained force -displacement responses using the proposed constitutive model 5 38 7. Limitations of the proposed model 1 The first limitation of the model is that it captures only the in -plane response of infilled 2 frames. As noted previously, the masonry infill panels are vulnerable also to out -of-plane 3 toppling (e.g., [19], [99]-[102] ), particularly in the upper stories of a building. As an example, 4 Fig. 13 shows in -plane failure of masonry panels in the lower as well as upper stories, and 5 out-of-plane failure of the panels primarily in the upper stories. Masonry is expected to resist 6 lateral loads through in -plane actions, particularly in the lower stories of the buildings. 7 Consequently, the proposed mod el is expected to be useful in calculating global responses, for 8 example, maximum forces generated in a building during an earthquake, inter -story drift ratio in 9 the ground story, and risk of collapse. The model is also expected to be useful if masonry wal ls 10 are restrained against out -of-plane failure. However, the model will need to be updated if 11 combined effect of out -of-plane and in -plane response of masonry panel is to be simulated (e.g., 12 [103] - [105] ). 13 (a) In-plane damage [102] (b) In-plane and out -of-plane damages [9] Fig. 13. Damages to masonry infill walls observed during past earthquakes 39 A yet another limitation of the proposed model is that it does not consider the effect of 1 openings in the masonry infill wall. The openings are commonly provided to enable the passage 2 of ducts, windows and doors. Presence of the openings is known to influe nce the response of 3 infilled frames during earthquakes depending on position, size and type of openings (e.g., [12], 4 [14], [15], [18]). The proposed model will need to be updated to account for these effects. 5 The proposed model has been developed considering the experimental results for 106 6 one-bay-one-story infilled RC frame specimens, and these very results have also been used for 7 validation. Ideally, a fraction of these results should have been used for development, and 8 remaining for validation. However, the size of the available database is s mall, particularly if 9 individual ranges of masonry prism strength (e.g., '5 10mf MPa) are considered. Therefore, 10 it was decided to use all 106 specimens for development as well as validation. This approach is 11 in line with that taken for all constitutive models considered herein, namely, C1 through C8, 12 which were developed as well as validated based on the entire dataset available with the team. 13 Further, the proposed model was validated also against four multi -bay-multi -story infilled RC 14 frames . 15 Finally, the model was developed using the available experimental data. The ranges of 16 different parameters (e.g., height of specimens, masonry prism strength) are specified in 17 Section 3. The coefficients in the proposed model may need to be updated as more data becomes 18 available, particularly for specimens beyond the range considered herein. 19 8. Summary and c onclusions 20 This paper presents an evalu ation of the eight macro models in simulating the lateral 21 force -displacement response of 106 one -bay-one-story reinforced concrete frames with masonry 22 infill. Frame members were modelled using line elements capable of capturing flexural and shear 23 failures, and interaction between axial loads and bending moment. Masonry was represented 24 40 using three parallel struts. The constitutive behavior of the struts was defined using eight existing 1 models that were based on mechanics, semi -empirical, or empirical. The si mulated response of 2 the frames up to peak strength was compared with experimental results in terms of peak strength 3 and corresponding secant stiffness, and strength corresponding to the first major crack in frame 4 or infill and corresponding stiffness. Furt her, the average of ratios of simulated to experimental 5 values of forces at different displacements was considered to characterize the simulation in the 6 post-peak range. A new constitutive model was developed, and its performance was evaluated. 7 Major obser vations and conclusions of the study are as follows. 8 (1) The dataset of 106 experimentally studied specimens had 69, 17 and 20 specimens with 9 masonry prism strength ranging between 0 – 5 MPa, 5 – 10 MPa, and 10 – 27 MPa. 10 Recognizing that the available data may not be representative of the population and that 11 the means to account for the “missing” information may be limited, it was decided to 12 perform the evaluation separately for the three ranges of masonry prism strength. 13 (2) Eight masonry constitutive models pr oposed in the past were evaluated. These models 14 considered a range of expressions to characterize the model parameters. Expressions 15 based on fundamental principles of mechanics could lead to minimum (or close to 16 minimum) coefficient of variation for the si mulated -to-experimental ratios of relevant 17 backbone parameters of masonry -infilled RC frames. 18 (3) A new masonry constitutive model was proposed wherein all strength parameters were 19 defined as a factor multiplied by the product of masonry prism strength and th e area of 20 diagonal strut representing masonry. Further, all stiffness parameters of the model were 21 defined as a factor multiplied by the product of elastic modulus of masonry multiplied 22 by the area of strut divided by the diagonal length of masonry panel. The factors were 23 calibrated up to one significant figure for the three ranges of masonry prism strength. 24 41 (4) The p eak lateral strength of infilled frames could be captured effectively using the 1 proposed constitutive model, which explicitly consider ed the inte raction between frame 2 and infill . This model is very similar to one proposed in the past, which was developed 3 based on the experimental results for 84 masonry prisms (with strength less than 10 4 MPa) without considering the frame -infill interaction. Accordi ngly, it can be said that 5 the consideration of frame -infill interaction does not materially influence the lateral 6 force -displacement response of infilled RC frames for masonry prism strength less than 7 10 MPa. 8 (5) In comparison with previously proposed masonry constitutive models, t he proposed 9 model could simulate the response of 106 one -bay-one-story infilled RC frames with 10 effectively , despite its simplicity . The model was also shown to simulate the lateral 11 response of three multi -bay-multi -story infilled RC frames. However, the model could 12 not perform well for a multi -bay-multi -story frame with a rather high masonry prism 13 strength. The model is expected to be useful in practical designs, and seismic 14 performance assessment of a class of infilled RC frame buildings. 15 (6) Correlation coefficients between simulated -to-experimental ratios of backbone 16 parameters of infilled frames and strength parameters for RC frame members were 17 either small or had different signs for the three ranges of masonry prism strength , 18 implying that the strength of response calculation was not materially affected by the 19 strength parameters for frame members . A similar observation was made for stiffness 20 of masonry panel relative to RC frame. 21 (7) The average simulated -to-experimental ratio of pe ak strength of infilled RC frame was 22 close to 1.0 irrespective of whether shear or flexural failure was observed in the frame 23 members. A similar observation was made regarding the shear sliding and corner 24 crushing model of failure in masonry panel. 25 42 (8) Influe nce of the choice of the strut model was small in simulating the lateral 1 force -displacement response, which further suggests that consideration of frame -infill 2 interaction does not substantially affect the lateral force -displacement response 3 calculation for infilled RC frame s. Accordingly, the proposed constitutive models can 4 be used in combination with single - or two -strut models. The single -strut model should 5 not be used if capturing the shear failure in frame members is important . 6 Acknowledgements 7 Financ ial support was provided by Science and Engineering Research Board (File No. 8 YSS/2015/1514) of Department of Science and Technology, Government of India. 9