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

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling Approaches to Simulate the Presence of Masonry Infills

#### 2.1. Micro- and Macro-Modeling Approaches

#### 2.2. Single-Strut Model

#### 2.3. Single-Strut Geometry

_{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).

_{h}first introduced by Stafford-Smith [29].

#### 2.3.1. Single-Strut Constitutive Law

_{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.

_{peak}is calculated considering four possible failure modes and the corresponding failure stresses: (a) Diagonal tension, σ

_{br}

_{1}; (b) Sliding shear, σ

_{br}

_{2}; (c) corner crushing, σ

_{br}

_{3}; and (d) diagonal compression, σ

_{br}

_{4}. The failure stress depends on the shear strength (τ

_{m}

_{0}), the bed joints sliding strength (τ

_{0}), the masonry compressive strength (σ

_{m}

_{0}) and the vertical stress acting on the infill (σ

_{0}):

_{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:

_{h}is defined according to Stafford-Smith [29] as follows:

_{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.

_{sec}) for the equivalent strut is calculated as:

_{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.

_{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.

_{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:

_{h}is defined in Equation (6).

_{cr}= 0.7∙F

_{peak}, where F

_{peak}is equal to the lateral cracking strength of the Panagiotakos and Fardis [38] model:

_{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}.

_{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:

_{cr}) that is assumed as a ratio of the peak strength N

_{peak}:

_{peak}and ∆

_{peak}are defined. The ∆

_{peak}can be calculated as follows:

_{deg}) up to residual strength:

_{0}, h

_{0}, l

_{w}, h

_{w}are indicated in Figure 2 and Figure 3.

#### 2.3.2. Single-Strut Cyclic Law

## 3. Research Methodology

## 4. Numerical Model for the Case-Study Building

_{w}= 30 cm (strong infill). The openings vary between each span and story. For further details refer to Gaetani d’Aragona et al. [68].

_{m}

_{0}), 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

_{m}

_{0}) and Young’s modulus (E

_{w}), while more mechanical parameters are needed for the implementation of the model proposed by Bertoldi et al. [37].

_{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

_{m}

_{0}) 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

_{m}

_{0}) is calculated as σ

_{m}

_{0}= (τ

_{m}

_{0}/0.285)

^{2}and the bed joints sliding strength τ

_{0}is obtained from the empirical relationship τ

_{0}= 2/3 τ

_{m}

_{0}[73]. The values of mechanical properties adopted in his study are reported in Table 1.

## 5. Influence of Modeling Assumptions

_{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.

_{cr}) similar considerations can be carried out to those for the initial stiffness. For the peak displacement (Δ

_{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 (Δ

_{res}) predicted by DR and HG is 2.23 and 2.31 times the value by BR, respectively.

_{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.

_{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.

## 6. Results of NRHAs

_{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:

_{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.

_{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).

_{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.

_{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.

_{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.

_{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.

_{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.

## 7. Conclusions

- 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.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Polese, M.; Gaetani d’Aragona, M.; Prota, A. Simplified approach for building inventory and seismic damage assessment at the territorial scale: An application for a town in southern Italy. Soil Dyn. Earthq. Eng.
**2019**, 121, 405–420. [Google Scholar] [CrossRef] - Polese, M.; Di Ludovico, M.; Gaetani d’Aragona, M.; Prota, A.; Manfredi, G. Regional vulnerability and risk assessment accounting for local building typologies. Int. J. Disaster Risk Reduct.
**2020**, 43, 101400. [Google Scholar] [CrossRef] - Manfredi, G.; Prota, A.; Verderame, G.M.; De Luca, F.; Ricci, P. Emilia earthquake, Italy: Reinforced concrete buildings re-sponse. Bull. Earthq. Eng.
**2014**, 12, 2275–2298. [Google Scholar] [CrossRef] [Green Version] - Ricci, P.; Verderame, G.M.; Manfredi, G. Analytical investigation of elastic period of infilled RC MRF buildings. Eng. Struct.
**2011**, 33, 308–319. [Google Scholar] [CrossRef] - Gaetani d’Aragona, M.; Polese, M.; Prota, A. Influence Factors for the Assessment of Maximum Lateral Seismic Deformations in Italian Multistorey RC Buildings. In Proceedings of the 6th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Rhodes Island, Greece, 15–17 June 2017. [Google Scholar]
- Gaetani d’Aragona, M.; Polese, M.; Cosenza, E.; Prota, A. Simplified assessment of maximum interstory drift for RC buildings with irregular infills distribution along the height. Bull. Earthq. Eng.
**2019**, 17, 707–736. [Google Scholar] [CrossRef] - Dolšek, M.; Fajfar, P. The effect of masonry infills on the seismic response of a four storey reinforced concrete frame—A deterministic assessment. Eng. Struct.
**2008**, 30, 1991–2001. [Google Scholar] [CrossRef] - Penava, D.; Sarhosis, V.; Kožar, I.; Guljaš, I. Contribution of RC columns and masonry wall to the shear resistance of masonry infilled RC frames containing different in size window and door openings. Eng. Struct.
**2018**, 172, 105–130. [Google Scholar] [CrossRef] [Green Version] - Uva, G.; Porco, F.; Fiore, A. Appraisal of masonry infill walls effect in the seismic response of RC framed buildings: A case study. Eng. Struct.
**2012**, 34, 514–526. [Google Scholar] [CrossRef] - Tesfamariam, S.; Goda, K.; Mondal, G. Seismic vulnerability of reinforced concrete frame with unreinforced masonry infill due to mainshock–aftershock earthquake sequences. Earthq. Spectra
**2015**, 31, 1427–1449. [Google Scholar] [CrossRef] - Polese, M.; Gaetani d’Aragona, M.; Prota, A.; Manfredi, G. Seismic behavior of damaged buildings: A comparison of static and dynamic nonlinear approach. In Proceedings of the 4th ECCOMAS Thematic Conference on Computational Methods in Structural Dy-namics and Earthquake Engineering, Kos Island, Greece, 12–14 June 2013; pp. 608–625. [Google Scholar]
- Gaetani d’Aragona, M.; Polese, M.; Prota, A. Relationship between the variation of seismic capacity after damaging earth-quakes, collapse probability and repair costs: Detailed evaluation for a non-ductile building. In Proceedings of the 5th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Crete Island, Greece, 25–27 May 2015; pp. 1478–1495. [Google Scholar]
- Gaetani d’Aragona, M.; Polese, M.; Elwood, K.J.; Baradaran Shoraka, M.; Prota, A. Aftershock collapse fragility curves for non-ductile RC buildings: A scenario-based assessment. Earthq. Eng. Struct. Dyn.
**2017**, 46, 2083–2102. [Google Scholar] [CrossRef] - Ricci, P.; De Luca, F.; Verderame, G.M. 6th April 2009 L’Aquila earthquake, Italy: Reinforced concrete building performance. Bull. Earthq. Eng.
**2011**, 9, 285–305. [Google Scholar] [CrossRef] [Green Version] - Cavaleri, L.; Di Trapani, F. Prediction of the additional shear action on frame members due to infills. Bull. Earthq. Eng.
**2015**, 13, 1425–1454. [Google Scholar] [CrossRef] - Cavaleri, L.; Di Trapani, F.; Asteris, P.G.; Sarhosis, V. Influence of column shear failure on pushover based assessment of masonry infilled reinforced concrete framed structures: A case study. Soil Dyn. Earthq. Eng.
**2017**, 100, 98–112. [Google Scholar] [CrossRef] [Green Version] - Basha, S.H.; Kaushik, H.B. A novel macromodel for prediction of shear failure in columns of masonry infilled RC frames under earthquake loading. Bull. Earthq. Eng.
**2019**, 17, 2219–2244. [Google Scholar] [CrossRef] - Di Trapani, F.; Macaluso, G.; Cavaleri, L.; Papia, M. Masonry infills and RC frames interaction: Literature overview and state of the art of macromodeling approach. Eur. J. Environ. Civ. Eng.
**2015**, 19, 1059–1095. [Google Scholar] [CrossRef] - Noh, N.M.; Liberatore, L.; Mollaioli, F.; Tesfamariam, S. Modelling of masonry infilled RC frames subjected to cyclic loads: State of the art review and modelling with OpenSees. Eng. Struct.
**2017**, 150, 599–621. [Google Scholar] [CrossRef] - Gaetani d’Aragona, M.; Polese, M.; Di Ludovico, M.; Prota, A. Seismic vulnerability for RC infilled frames: Simplified evaluation for as-built and retrofitted building typologies. Buildings
**2018**, 8, 137. [Google Scholar] [CrossRef] [Green Version] - Polese, M.; Gaetani d’Aragona, M.; Di Ludovico, M.; Prota, A. Sustainable selective mitigation interventions towards effective earthquake risk reduction at the community scale. Sustainability
**2018**, 10, 2894. [Google Scholar] [CrossRef] [Green Version] - Asteris, P.G. Finite Element Micro-Modeling of Infilled Frames. Elect. J. Struct. Eng.
**2008**, 8, 1–11. [Google Scholar] - Asteris, P.; Cotsovos, D.; Chrysostomou, C.; Mohebkhah, A.; Al-Chaar, G. Mathematical micromodeling of infilled frames: State of the art. Eng. Struct.
**2013**, 56, 1905–1921. [Google Scholar] [CrossRef] - Crisafulli, F. Seismic Behavior of Reinforced Concrete Structures with Masonry Infills; University of Canterbury: Canterbury, UK, 1997. [Google Scholar]
- Crisafulli, F.J.; Carr, A.J.; Park, R. Analytical modelling of infilled frame structures—A general review. Bull. N. Z. Soc. Earthq. Eng.
**2000**, 33, 30–47. [Google Scholar] - Asteris, P.G.; Antoniou, S.T.; Sophianopoulos, D.S.; Chrysostomou, C.Z. Mathematical Macromodeling of Infilled Frames: State of the Art. J. Struct. Eng.
**2011**, 137, 1508–1517. [Google Scholar] [CrossRef] - Lourenço, P.B. Computations on historic masonry structures. Prog. Struct. Eng. Mater.
**2002**, 4, 301–319. [Google Scholar] [CrossRef] - Asteris, P.; Tzamtzis, A. On the use of a regular yield surface for the analysis of unreinforced masonry walls. Electron. J. Struct. Eng.
**2003**, 3, 23–42. [Google Scholar] - Stafford Smith, B. Behaviour of square infilled frames. J. Struct. Div.
**1966**, 92, 381–403. [Google Scholar] [CrossRef] - Mainstone, R.J. On the stiffness and strength of infilled frames. In Proceedings of the Institute of Civil Engineers, London, UK, June 1971; Volume 7360S, pp. 57–89. [Google Scholar] [CrossRef]
- Mainstone, R.J. Supplementary Note on the Stiffness and Strength of Infilled Frames Current Paper CP 13/UK: Building Research Establishment; Building Research Station: Watford, UK, 1974. [Google Scholar]
- Holmes, M. Steel frames with brickwork and concrete filling. In Proceedings of the Institution of Civil Engineers, London, UK, August 1961; Volume 19, pp. 473–478. [Google Scholar] [CrossRef]
- Chrysostomou, C.Z.; Gergely, P.; Abel, J.F. A six-strut model for nonlinear dynamic analysis of steel infilled frames. Int. J. Struct. Stab. Dyn.
**2002**, 2, 335–353. [Google Scholar] [CrossRef] - El-Dakhakhni, W.W.; Elgaaly, M.; Hamid, A.A. Three-Strut Model for Concrete Masonry-Infilled Steel Frames. J. Struct. Eng.
**2003**, 129, 177–185. [Google Scholar] [CrossRef] - Combescure, D. Some contributions of physical and numerical modelling to the assessment of existing masonry infilled RC frames under extreme loading. In Proceedings of the First European Conference on Earthquake Engineering and Seismology, Geneva, Switzerland, 3–8 September 2006. [Google Scholar]
- Verderame, G.M.; De Luca, F.; Ricci, P.; Manfredi, G. Preliminary analysis of a soft-storey mechanism after the 2009 L’Aquila earthquake. Earthq. Eng. Struct. Dyn.
**2011**, 40, 925–944. [Google Scholar] [CrossRef] [Green Version] - Bertoldi, S.H.; Decanini, L.D.; Gavarini, C. Telai tamponati soggetti ad azioni sismiche, un modello semplificato: Confronto sperimentale e numerico. In Proceedings of the 6 Convegno Nazionale L’ingegneria Sismica in Italia, Perugia, Italy, 13–15 October 1993. [Google Scholar]
- Panagiotakos, T.B.; Fardis, M.N. Seismic response of infilled RC frames structures. In Proceedings of the 11th World Con-Ference on Earthquake Engineering, Acapulco, Mexico, 23–28 June 1996. [Google Scholar]
- Asteris, P.G.; Cavaleri, L.; Di Trapani, F.; Sarhosis, V. A macro-modelling approach for the analysis of infilled frame structures considering the effects of openings and vertical loads. Struct. Infrastruct. Eng.
**2016**, 12, 551–566. [Google Scholar] [CrossRef] - Zarnic, R.; Tomazevic, M. Study of the behaviour of masonry lnfilled reinforced concrete frames subjected to seismic loading. In Proceedings of the Seventh International Brick Masonry Conference, Melbourne, Australia, 17–20 February 1985; Volume 2, pp. 1315–1325. [Google Scholar]
- Zarnic, R.; Tomazevic, M. An experimentally obtained method for evaluation of the behaviour of masonry infilled R/C frames. In Proceedings of the Ninth World Conference on Earthquake Engineering, Tokyo, Japan, 2–6 August 1988; Volume VI, pp. 163–168. [Google Scholar]
- Polyakov, S. On the interaction between masonry filler walls and enclosing frame when loading in the plane of the wall. Transl. Earthq. Eng.
**1960**, E2, 36–42. [Google Scholar] - Liauw, T.C.; Kwan, K. Nonlinear behaviour of non-integral infilled frames. Comput. Struct.
**1984**, 18, 551–560. [Google Scholar] - Decanini, L.D.; Fantin, G.E. Modelos Simplificados de la Mamposteria Incluida en Porticos. Caractreisticas de Rigidez y Resistencia Lateral en Estrado Limite (in Spanish). J. Argent. Ing. Estruct.
**1987**, 2, 817–836. [Google Scholar] - Dawe, J.L.; Seah, C.K. Analysis of concrete masonry infilled steel frames subjected to in-plane loads. In Proceedings of the 5th Canadian Masonry Symposium, Vancouver, BC, Canada, 5–7 June 1989; pp. 329–340. [Google Scholar]
- Durrani, A.J.; Luo, Y.H. Seismic retrofit of flat-slab buildings with masonry infill. In Proceedings of the NCEER Workshop on Seismic Response of Masonry Infills, San Francisco, CA, USA, 4–5 February 1994. Report NCEER-94-0004. [Google Scholar]
- Saneinejad, A.; Hobbs, B. Inelastic Design of Infilled Frames. J. Struct. Eng.
**1995**, 121, 634–650. [Google Scholar] [CrossRef] [Green Version] - De Risi, M.T.; Del Gaudio, C.; Ricci, P.; Verderame, G.M. In-plane behaviour and damage assessment of masonry infills with hollow clay bricks in RC frames. Eng. Struct.
**2018**, 168, 257–275. [Google Scholar] [CrossRef] - Huang, H.; Burton, H.V.; Sattar, S. Development and Utilization of a Database of Infilled Frame Experiments for Numerical Modeling. J. Struct. Eng.
**2020**, 146, 04020079. [Google Scholar] [CrossRef] - De Sortis, A.; Bazzurro, P.; Mollaioli, F.; Bruno, S. Influenza delle tamponature sul rischio sismico degli edifice in calcestruzzo armato. In Proceedings of the ANIDIS Conference L’ingegneria Sismica in Italia, Pisa, Italy, 10–14 June 2007. (In Italian). [Google Scholar]
- Mosalam, K.M.; White, R.N.; Gergely, P. Static Response of Infilled Frames Using Quasi-Static Experimentation. J. Struct. Eng.
**1997**, 123, 1462–4169. [Google Scholar] [CrossRef] - Al-Chaar, G.; Lamb, G.E.; Issa, M.A. Effects of openings on structural performance of unreinforced masonry infilled frames. In Large Scale Structural Testing; Issa, M.A., Ed.; ACI Spec. Publ: Washington, DC, USA, 2003; Volume 211, pp. 247–262. [Google Scholar]
- Mohammadi, M.; Nikfar, F. Strength and Stiffness of Masonry-Infilled Frames with Central Openings Based on Experimental Results. J. Struct. Eng.
**2013**, 139, 974–984. [Google Scholar] [CrossRef] - Decanini, L.D.; Liberatore, L.; Mollaioli, F. Strength and stiffness reduction factors for infilled frames with openings. Earthq. Eng. Eng. Vib.
**2014**, 13, 437–454. [Google Scholar] [CrossRef] - Basha, S.H.; Surendran, S.; Kaushik, H.B. Empirical Models for Lateral Stiffness and Strength of Masonry-Infilled RC Frames Considering the Influence of Openings. J. Struct. Eng.
**2020**, 146, 04020021. [Google Scholar] [CrossRef] - Klingner, R.E.; Bertero, V.V. Earthquake resistance of infilled frames. J. Struct. Div.
**1978**, 104, 973–989. [Google Scholar] [CrossRef] - Cavaleri, L.; Fossetti, M.; Papia, M. Infilled frames: Developments in the evaluation of cyclic behaviour under lateral loads. Struct. Eng. Mech.
**2005**, 21, 469–494. [Google Scholar] [CrossRef] - Cavaleri, L.; Di Trapani, F. Cyclic response of masonry infilled RC frames: Experimental results and simplified modeling. Soil Dyn. Earthq. Eng.
**2014**, 65, 224–242. [Google Scholar] [CrossRef] - Tassios, T.P. Masonry Infill and RC Walls under Cyclic Actions, Proceedings of the 3rd International Symposium on Wall Structures, Warsaw, Poland, 1984; CIB: Ottawa, ON, Canada, 1984. [Google Scholar]
- Crisafulli, F.J.; Carr, A.J. Proposed macro-model for the analysis of infilled frame structures. Bull. N. Z. Soc. Earthq. Eng.
**2007**, 40, 69–77. [Google Scholar] [CrossRef] [Green Version] - Lowes, L.N.; Mitra, N.; Altoontash, A. A Beam-Column Joint Model for Simulating the Earthquake Response of Reinforced Concrete Frames; Technical Rep. No. PEER 2003/10; Pacific Earthquake Engineering Research Center: Berkeley, CA, USA, 2003. [Google Scholar]
- Jeon, J.-S.; Park, J.-H.; Desroches, R. Seismic fragility of lightly reinforced concrete frames with masonry infills. Earthq. Eng. Struct. Dyn.
**2015**, 44, 1783–1803. [Google Scholar] [CrossRef] - Lima, C.; De Stefano, G.; Martinelli, E. Seismic response of masonry infilled RC frames: Practice-oriented models and open issues. Earthq. Struct.
**2014**, 6, 409–436. [Google Scholar] [CrossRef] - Koutromanos, I.; Stavridis, A.; Shing, P.B.; Willam, K. Numerical modeling of masonry-infilled RC frames subjected to seismic loads. Comput. Struct.
**2011**, 89, 1026–1037. [Google Scholar] [CrossRef] - Kumar, M.; Rai, D.C.; Jain, S.K. Ductility Reduction Factors for Masonry-Infilled Reinforced Concrete Frames. Earthq. Spectra
**2015**, 31, 339–365. [Google Scholar] [CrossRef] - Kakaletsis, D.J.; Karayannis, C.G. Influence of Masonry Strength and Openings on Infilled R/C Frames under Cycling Loading. J. Earthq. Eng.
**2008**, 12, 197–221. [Google Scholar] [CrossRef] - Ricci, P.; De Risi, M.T.; Verderame, G.M.; Manfredi, G. Influence of infill distribution and design typology on seismic per-formance of low-and mid-rise RC buildings. Bull. Earthq. Eng.
**2013**, 11, 1585–1616. [Google Scholar] [CrossRef] - Gaetani d’Aragona, M.; Polese, M.; Di Ludovico, M.; Prota, A. The use of Stick-IT model for the prediction of direct economic losses. Earthq. Eng. Struct. Dyn.
**2020**. [Google Scholar] [CrossRef] - Gaetani d’Aragona, M.; Polese, M.; Prota, A. Stick-IT: A simplified model for rapid estimation of IDR and PFA for existing low-rise symmetric infilled RC building typologies. Eng. Struct.
**2020**, 223, 111182. [Google Scholar] [CrossRef] - Hak, S.; Morandi, P.; Magenes, G.; Sullivan, T.J. Damage Control for Clay Masonry Infills in the Design of RC Frame Structures. J. Earthq. Eng.
**2012**, 16, 1–35. [Google Scholar] [CrossRef] - FEMA 356. Prestandard and Commentary for the Seismic Rehabilitation of Buildings; Federal Emergency Management Agency: Washington, DC, USA, 2000. [Google Scholar]
- Ministero delle Infrastrutture e dei Trasporti. In Proceedings of the Circolare n.7 C.S.LL.PP 21 January 2019 Istruzioni per L’applicazione dell’Aggiornamento delle Norme Tecniche per le Costruzioni di cui al Decreto Ministeriale 17 gennaio 2018; Ministero delle Infrastrutture e dei Trasporti, C.S.LL.PP: Rome, Italy, 2019. (In Italian)
- Liberatore, L.; Noto, F.; Mollaioli, F.; Franchin, P. In-plane response of masonry infill walls: Comprehensive experimen-tally-based equivalent strut model for deterministic and probabilistic analysis. Eng. Struct.
**2018**, 167, 533–548. [Google Scholar] [CrossRef] - Applied Technology Council & United States. Quantification of Building Seismic Performance Factors—FEMA P695; US Department of Homeland Security: Washington, DC, USA, 2009. [Google Scholar]
- Baker, J.W. Efficient Analytical Fragility Function Fitting Using Dynamic Structural Analysis. Earthq. Spectra
**2015**, 31, 579–599. [Google Scholar] [CrossRef] - Grunthal, G. Cahiers du Centre Européen de Géodynamique et de Séismologie; European Macroseismic Scale: Luxembourg, 1998; pp. 1–97. [Google Scholar]
- Del Gaudio, C.; Ricci, P.; Verderame, G.M. A class-oriented mechanical approach for seismic damage assessment of RC buildings subjected to the 2009 L’Aquila earthquake. Bull. Earthq. Eng.
**2018**, 16, 4581–4605. [Google Scholar] [CrossRef] - Porter, K.A. An overview of PEER’s performance-based earthquake engineering methodology. In Proceedings of the 9th Inter-National Conference on Applications of Statistics and Probability in Civil Engineering, Francisco, CA, USA, 6–9 July 2003. [Google Scholar]

**Figure 6.**Lateral force-displacement equivalent strut backbone curves for a solid infill panel adopting DR, HG, and BR.

**Figure 7.**Pushover curves in the longitudinal (

**a**,

**b**) and the transverse (

**c**,

**d**) direction for (

**a**,

**c**) the actual opening configuration and (

**b**,

**d**) simulating no openings, obtained adopting DR, HG, and BR constitutive laws.

**Figure 8.**Fragility curves corresponding to the attainment of EMS-98 damage states in the (

**a**) longitudinal and (

**b**) transverse direction adopting DR, HG, and BR constitutive laws.

**Figure 9.**Interstory drift profiles in the longitudinal direction for PGA corresponding to (

**a**) 0.1 g, (

**b**) 0.4 g, (

**c**) 0.6 g, (

**d**) 1.2 g.

**Figure 10.**PFA profiles in the longitudinal direction for PGA corresponding to (

**a**) 0.1 g, (

**b**) 0.4 g, (

**c**) 0.6 g, (

**d**) 1.2 g.

**Figure 11.**Maximum interstory drift ratios for (

**a**) longitudinal and (

**b**) transverse direction and maximum peak floor accelerations for (

**c**) longitudinal and (

**d**) transverse direction.

t_{w}(mm) | E_{w}(MPa) | G_{w}(MPa) | τ_{m}_{0}(MPa) | τ_{0}(MPa) | σ_{m}_{0}(MPa) |
---|---|---|---|---|---|

300 | 3240 | 1296 | 0.34 | 0.23 | 1.53 |

_{w}: thickness; E

_{w}: Young’s modulus; G

_{w}: shear modulus; τ

_{m}

_{0}: shear strength; τ

_{0}: shear sliding strength of bed joints; σ

_{m}

_{0}: masonry compressive strength.

Backbone Points | BR | DR | HG |
---|---|---|---|

Δ_{cr} (m) | 2.07 × 10^{−4} | 9.04 × 10^{−4} | 2.10 × 10^{−3} |

Δ_{peak} (m) | 1.04 × 10^{−3} | 4.52 × 10^{−3} | 1.19 × 10^{−2} |

Δ_{ult} (m) | 1.80 × 10^{−2} | 4.02 × 10^{−2} | 4.16 × 10^{−2} |

F_{cr} (kN) | 166.4 | 242.7 | 244.2 |

F_{peak} (kN) | 208.0 | 347.0 | 339.0 |

F_{res} (kN) | 72.8 | 0.0 | 135.6 |

**Table 3.**Damage state thresholds in terms of interstory drift ratios defined according to EMS-98 scale.

DS_{1} | DS_{2} | DS_{3} | DS_{4} | DS_{5} |
---|---|---|---|---|

0.03% | 0.32% | 1.03% | 3.30% | 3.70% |

Concrete cracking/Onset infill cracking | Rebar yielding/ Moderate infill cracking | Rebar buckling/ cover spalling/First column shear failure/Extensive infill cracking | First column axial failure/ ultimate capacity | All story columns exhibit axial failure/ultimate capacity |

Longitudinal Direction | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

DS_{1} | DS_{2} | DS_{3} | DS_{4} | DS_{5} | ||||||

Backbone model | θ | β | θ | β | θ | β | θ | β | θ | β |

BR | 0.16 | 0.21 | 0.46 | 0.36 | 0.95 | 0.35 | 1.91 | 0.24 | 2.15 | 0.25 |

DR | 0.08 | 0.21 | 0.47 | 0.34 | 0.77 | 0.30 | 1.66 | 0.40 | 1.85 | 0.42 |

HG | 0.06 | 0.24 | 0.41 | 0.34 | 0.75 | 0.31 | 1.66 | 0.36 | 1.83 | 0.38 |

Transverse Direction | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

DS_{1} | DS_{2} | DS_{3} | DS_{4} | DS_{5} | ||||||

Backbone model | θ | β | θ | β | θ | β | θ | β | θ | β |

BR | 0.10 | 0.31 | 0.26 | 0.48 | 0.56 | 0.42 | 1.56 | 0.47 | 1.66 | 0.43 |

DR | 0.05 | 0.21 | 0.29 | 0.41 | 0.50 | 0.41 | 1.05 | 0.47 | 1.19 | 0.51 |

HG | 0.04 | 0.28 | 0.26 | 0.45 | 0.48 | 0.44 | 1.14 | 0.43 | 1.28 | 0.48 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gaetani d’Aragona, M.; Polese, M.; Prota, A.
Effect of Masonry Infill Constitutive Law on the Global Response of Infilled RC Buildings. *Buildings* **2021**, *11*, 57.
https://doi.org/10.3390/buildings11020057

**AMA Style**

Gaetani d’Aragona M, Polese M, Prota A.
Effect of Masonry Infill Constitutive Law on the Global Response of Infilled RC Buildings. *Buildings*. 2021; 11(2):57.
https://doi.org/10.3390/buildings11020057

**Chicago/Turabian Style**

Gaetani d’Aragona, Marco, Maria Polese, and Andrea Prota.
2021. "Effect of Masonry Infill Constitutive Law on the Global Response of Infilled RC Buildings" *Buildings* 11, no. 2: 57.
https://doi.org/10.3390/buildings11020057