Evaluation of infilled frames: an updated in-plane-stiffness macromodel considering the effects of vertical loads

Transcription

1 Evaluation of infille frames: an upate in-plane-stiffness macromoel consiering the effects of vertical loas Campione, G., Cavaleri, L., Macaluso, G., Amato, G., & Di Trapani, F. (2015). Evaluation of infille frames: an upate in-plane-stiffness macro-moel consiering the effects of vertical loas. Bulletin of Earthquake Engineering, 13(8), DOI: /s x Publishe in: Bulletin of Earthquake Engineering Document Version: Peer reviewe version Queen's University Belfast - Research Portal: Link to publication recor in Queen's University Belfast Research Portal Publisher rights The final publication is available at Springer via General rights Copyright for the publications mae accessible via the Queen's University Belfast Research Portal is retaine by the author(s) an / or other copyright owners an it is a conition of accessing these publications that users recognise an abie by the legal requirements associate with these rights. Take own policy The Research Portal is Queen's institutional repository that provies access to Queen's research output. Every effort has been mae to ensure that content in the Research Portal oes not infringe any person's rights, or applicable UK laws. If you iscover content in the Research Portal that you believe breaches copyright or violates any law, please contact openaccess@qub.ac.uk. Downloa ate:18. Aug. 2018

2 Evaluation of infille frames: an upate in-plane-stiffness macro-moel consiering the effects of vertical loas L. Cavaleri*, G. Campione, G. Macaluso, Dipartimento i Ingegneria Civile, Ambientale, Aerospaziale e ei Materiali (DICAM), University of Palermo, Palermo, Italy. G. Amato School of Planning Architecture an Civil Engineering, Queen's University Belfast, UK Author s accepte manuscript. The final publication is available at Springer via Keywors: Infille frames, in-plane behaviour, equivalent iagonal pin-jointe strut moel, vertical loa influence Abstract The influence of masonry infills on the in-plane behaviour of RC frame structures is a central topic in the seismic evaluation an retrofitting of existing builings. Many moels in the literature use an equivalent strut member in orer to represent the infill but, among the parameters influencing the equivalent strut behaviour, the effect of vertical loas acting on the frames is recognize but not quantifie. Nevertheless a vertical loa causes a non negligible variation in the in-plane behaviour of infille frames by influencing the effective volume of the infill. This results in a change in the stiffness an strength of the system. This paper presents an equivalent iagonal pinjointe strut moel taking into account the stiffening effect of vertical loas on the infill in the initial state. The in-plane stiffness of a range of infille frames was evaluate using a finite element moel of the frame-infill system an the cross-section of the strut equivalent to the infill was obtaine for ifferent levels of vertical loaing by imposing the equivalence between the frame containing the infill an the frame containing the iagonal strut. In this way a law for ientifying the equivalent strut with epening on the geometrical an mechanical characteristics of the infille frame was generalize to consier the influence of vertical loas for use in the practical 1

3 applications. The strategy presente, limite to the initial stiffness of infille frames, is preparatory to the efinition of complete non-linear cyclic laws for the equivalent strut. *Corresponing author, 2

4 1. Introuction Infills, although consiere non-structural members, raically moify the in-plane RC frame response uner in-plane lateral loa. The fact is that the in-plane stiffness of the frame, ue to the presence of the infill, can increase up to ten times while the frame strength can grow up to four times, as foun by many authors (e.g. Staffor Smith 1968; Cavaleri et al. 2005). Moreover, the interaction between infill an frame may or may not be beneficial to the performance of the structure uner a seismic loa: while in some cases the global performance coul improve, several ebates (e.g. NCEER 1994) an experiences in recent earthquakes have emonstrate that an otherwise well-esigne structure coul collapse ue to low seismic excitation if infills are not uniformly istribute, both horizontally an vertically. The in-plane stiffness an strength of an infille frame are affecte by several variables such as the geometrical an mechanical properties of the infill an frame members, etails of frame members, frame-infill stiffness ratio, out-of-plane infill loaing (here not consiere), workmanship an construction techniques an vertical loas transferre from the frame to the infill. One of the approaches for simulating the in-plane-influence of infill consists in replacing the infill itself with one or more equivalent struts mae of the same material as the infill (macro-moelling), see original works by Asteris (2003), Cavaleri an Papia (2003), Crisafulli an Carr (2007), Zhai et al. (2011), Chrysostomou an Asteris (2012) an review works by Moghaam an Dowling 1987, Asteris et al. (2011). Macro-moelling may be avantageous for a number of reasons an especially for esign purposes. Moreover, it avois the computational effort require by the soli/plane finite element moelling of infills (micromoelling). The micro-moelling approach has been wiely use in the literature (e.g. Koutromanos et al. 2011; Manos et al. 2011, 2012; Shing an Stavriis 2014; Asteris an Cotsovos 2012; Asteris et al. 2012, 2013). It provies etaile information on the structural response, for example on local effects on frame members; however, especially in non-linear cases, it requires long computational time an calibration of many parameters. Hence the macromoel approach may be an avantage, but at the same time the importance of the micro moel approach is not uner iscussion. 3

5 A complete efinition of the equivalent strut moel requires (a) fixing the initial axial elastic properties (Young moulus an cross-section), (b) a law for the variation of the mechanical properties uring cyclic loaing, (c) the axial strength envelope. The evaluation of the initial elastic properties has a key role for the efinition of the cyclic law. For example, Klingner an Bertero (1978), after fixing the thickness an initial elastic moulus of the equivalent strut equal to those of the infill, evaluate the with w of the equivalent strut by means of the following expressions (Mainstone 1974): w 0.4 Eit sin( 2 θ ) = 0.175( λ h') ; λ = 4 (1) 4E I h f c In Eq. 1 E i an E f are respectively the elastic iagonal moulus of the infill an the elastic moulus of the concrete, I c is the moment of inertia of the column cross-section; see Fig.1 for escription of symbols. Similarly to those of Klingner an Bertero, the cyclic laws propose by Dououmis e Mitsopoulou (1986) an by Panagiatakos an Faris (1996) epen on the efinition of the initial stiffness of the equivalent strut (elastic loaing curve of the strength envelope). The same approach was followe for the efinition of the strut cyclic laws propose by Cavaleri et al. (2005) (Fig. 2) an by Cavaleri an Di Trapani (2014) (Fig. 3). As shown in the works mentione before, ientification of the initial axial stiffness of the equivalent strut is the first step for the efinition of a complete cyclic law for the strut itself uner axial loaing. The approach for the ientification of the strut with use in Eq. 1 is not the only one. For example, Durrani an Luo (1994), on the basis of the experimental work of Mainstone (1974), propose the following analytical relation: where w 1.5 E i t h' = 0.32 sin ( 2 θ ) (2) m EcIch 4

6 6 I = + b h' m 6 1 arctg π I c l' (3) an I b is the moment of inertia of the beam cross-section (see Fig. 1 for escription of symbols). Flanagan an Bennet (1999, 2001) base on the results of a number of full-scale clay infille steel frames teste uner in-plane loaing propose to evaluate the with of the strut, w, as w = π C λ cos θ (4) C being an empirical constant varying with the in-plane rift isplacement use as an inicator of the limit state of the infill. A further moel for the ientification of the with of the equivalent strut taking into account the Poisson s ratio of the infill material was propose by Papia et al. (2003) an is explaine in the next section. Ientification of the in-plane stiffness of an infille frame is aresse by the actual coes; for example FEMA 356 (2000), in agreement with the strategy aopte by the authors mentione above, suggests the formula propose by Mainstone (Eq. 1). Eurocoe 8 (2004) confirms the nee to take the infill into account but oes not suggest a specific moel an refers the esigner to specialise literature. As for the effect of vertical loas it is recognize that it moifies the behaviour of infille frames uner in-plane lateral loaing. In micro-moels the loa influence is taken into account by moelling the frame-infill contact area: see Fig. 4. On the other han in a macro-moel the vertical loa transferre from frame to infill can only be taken into account by calibrating the strut mechanical response for ifferent levels of vertical loaing. However, few authors have quantifie this influence. In (1968) Staffor Smith investigate the influence of a uniformly istribute vertical loa impose on the upper beam of a single storey-single bay steel frame in-plane stiffness an observe a 5

7 consierable increase in the in-plane lateral stiffness an strength of the structure. More recently, Valiasis an Stylianies (1989), stuying RC frames infille with brick masonry walls, observe that the presence of a compressive axial loa on the columns consierably improve the in-plane lateral strength of the system. Base on these results, Staffor Smith an Valiasis consiere the vertical loa effect to be conservative an i not take it into account among the variables affecting the evaluation of the cross-section of the equivalent strut. Similar experimental an numerical results were obtaine by Manos et al. (2012), Stylianiis (2012), Valiasis an Stylianies (1989) an Valiasis et al. (1993). However, while the conclusion that the vertical loa effect is conservative can be vali for a single frame, this may not be true for multi-bay, multi-storey frames with non-uniform loa istribution since the ifferent stiffness an strength of a single frame may cause torsional an soft-storey effects. In this paper, a correlation taking into account the vertical loa influence on the initial stiffness of a strut equivalent to an infill is propose. This work integrates the research escribe in two previous papers: in Papia et al. (2003) a family of curves for estimating the with of the equivalent strut in the absence of vertical loas is provie an in Papia et al. (2004) the mechanism governing the influence of vertical loas on the infille frame response is analyse. In this paper a family of curves obtaine for rectangular infills is presente together with the curves propose by Amato et al. (2008, 2009) for square infills. In the next sections the proceure use to obtain the correlation between infill an equivalent strut is escribe. This proceure couples an analytical calculation of the frame-infill system components with a finite element micro-moelling of the infille frame system as a whole. The FE moel provies the response of a series of infille frames uner horizontal an vertical loas by using contact surface elements governe by the Coulomb friction law to moel the transmission of the compressive stresses from the frame to the infill. 6

8 2. Strategy for the equivalent strut with ientification The cross-section of the pin-jointe strut equivalent to an infill of a single storey-single bay infille frame can be ientifie by imposing the initial stiffness of the system in Fig. 1a to be equal to the initial stiffness of the equivalent brace frame in Fig. 1b. It shoul be note that these schemes o not exactly represent a generic frame of a frame structure: the lower beam is assume to be rigi an thus the bottom ens of the columns in Fig. 1b are fully restraine. This assumption is in agreement with the conclusions of many authors (Mainstone 1971, 1974; Staffor Smith an Carter 1969) showing that the contribution of infill to the in-plane lateral stiffness of a frame can be obtaine by stuying the scheme in Fig. 2b as an alternative to the scheme in Fig. 2a. Denoting as.di the stiffness of the actual system (Fig. 1a) solve by the Finite Element Metho (micro-moelling approach) an as Di the stiffness corresponing to the simplifie analytical moel (Fig. 1b), their equivalence can be written as D i = D i (5) When this equivalence is impose, assuming the thickness of the strut to be the same of the infill an the Young s moulus to be equal to the iagonal elastic moulus of the infill, the with w of the strut can be calculate. 3. Stiffness of the frame-strut equivalent system The in-plane stiffness Di of the scheme in Fig. 1-b, equivalent to the scheme in Fig. 1-a, can be evaluate with goo approximation as the sum of the horizontal forces D an D to be applie to f the schemes in Fig. 3-b an Fig. 3-c, (obtaine as the ecomposition of the scheme in Fig.1-a) to prouce a isplacement δ = 1 at the beam mile span: Di = D + Df (6) 7

9 For the scheme in Fig. 3-b the in-plane stiffness D can be calculate as follows D = k 1 + k c 2 k cos θ 1 k sen 2 θ + 4 k b cos 2 θ (7) k, k c an k b being the axial stiffness of the iagonal strut, columns an beam respectively: E t w E f Ac E f Ab k = ; kc = ; kb = (8) h' l' In Eq. 8 E an E f are the Young s moulus of the infill along the iagonal irection an the Young s moulus of the concrete use for the frame; t is the thickness of the infill; A c an A b are the column an beam cross-sectional areas; the angle θ efines the iagonal irection of the strut an h ' an l ' are the height an length of the frame, see Fig. 1. In regar to the Young s moulus of the infill along the iagonal irection it shoul be observe that since the masonry shows an orthotropic behaviour the mechanical characteristics of the equivalent strut can be estimate by combining the masonry elastic mouli along the horizontal an vertical irections as suggeste in Jones 1975, or by using the simplifie approach iscusse by Cavaleri et al on the basis of the experimental stuies reporte in Cavaleri et al The in-plane stiffness of the frame D f in Fig. 3-c, in the case of columns having the same crosssection, can be evaluate as follows E f Ic Ib h' D f = h' Ic l' 1 (9) I c an I b being the moments of inertia of the column an beam sections respectively. In the case of columns with ifferent cross-sections the mean value of the axial stiffness of the columns can be use. 8

10 4. Infille frame stiffness by the refine FE moel For the evaluation of the in-plane lateral stiffness by means of the micro-moel approach, the ADINA software was use. Both the frame an the infill were moelle using plane stress soli elements having four noes each. The noes at the base of the columns were fully restraine while three egrees of freeom were assigne to all the other ones. The infill an the frame were moelle by means of elastic homogenous an isotropic materials having elastic moulus E an E f an Poisson s ratio ν an ν f respectively. The frame-infill interaction was moelle by 2D contact surface elements (Bathe an Bouzinov 1997). Each interface element is compose of two contact surfaces, a contactor an a target surface, which may come into contact uring the loaing process. No tensile strength is associate with the joint an this makes it possible to moel the etachment between frame an infill. Because the interaction between frame an infill is strictly epenent on the length of the contact zone, which is influence by the vertical loa, this kin of finite element allows evaluation of the system in-plane lateral stiffness.di in relation to the vertical loa. With regar to the frame-infill contact surface, the value to assign to the Coulomb friction coefficient has been ebate in the literature (Manos et al. 2011, 2012; Stylianiis 2012; Valiasis an Stylianies 1989; Valiasis et al. 1993). However, in some cases it is accepte that a variation in friction coefficient oes not moify the overall response of an infille frame (e.g. Asteris 2008; Fiore et al. 2012) while in other cases it is consiere basic (e.g. Saneineja anhobbs 1995). Certainly a variation in friction between frame an infill, whose realistic characterization is not simple, though it may leave the overall response unchange, it can moify the local response, leaing to a reuction in the stresses normal to the frame-infill contact surface. This also prouces a reuction in the shear stresses on the members. 9

11 For the case analyze here, consiering that values of the Coulomb friction coefficient generally lower than 0.6 can be foun in the literature for moelling the frame-infill interface, the value 0.45 was set, being the average of the values most frequently encountere. The numerical analysis was carrie out for ifferent values of mechanical an geometrical properties of the system an for four vertical loa levels. For each analysis the in-plane lateral stiffness.di of the system was calculate as the ratio between the applie horizontal loa an the average beam isplacement. The horizontal an vertical forces acting on the frame were applie on the initial an final sections of the beam at mile epth, while the vertical loa was concentrate on the top noes of the beam-column joints, as shown in Fig Equivalent strut cross-section By substituting the value of D i obtaine from Eq. 6 in Eq. 5, one obtains D = D + D (10) i f Further, by substituting Eq.7 in Eq.10 the ratio w / can be expresse as a function of the in-plane stiffness D i of an infille frame given by the refine FE moel previously escribe an the bare frame stiffness D f given in Eq. 9: w Di D f = 1 2 E t cos θ D D i k c f h' l' k k c b 1 (11) In Eq. 10 D f is the in-plane stiffness of a bare frame uner the assumption of non-negligible axial eformation. With regar to Di ifferent experimental an numerical investigations (see Cavaleri et al 2005, Cavaleri an Di Trapani 2014, Manos et al. 2011, Manos et al. 2012, Valiasis an Stylianies 1989) have shown that the infille frame eflecte shape is flexural. In agreement with the most wiesprea tenency in the literature the ratio between the strut with an the strut length is expresse as a function of a parameter * λ which takes into account the elastic 10

12 an geometrical features of the system w = f ( λ * ). This function must also take into account the influence of verticals loa. 6. Dimensionless infille frame parameter λ * The efinition of a parameter that efines the ratio w/ to be aopte for the ientification of a strut equivalent to an infill, can be obtaine by imposing that the ifference D D on the right i f han of Eq. 11 is the infill in-plane stiffness. Once the Poisson ratioν, the vertical loa F V an aspect ratio l / h are fixe, the in-plane stiffness of an infill can be calculate using the total stiffness from the finite element simulation D i as D = D D = ψ E t (12) i f where ψ epens on the unknown extension of the frame-infill contact regions influence by the above quantities. On the other han, setting Eq. 11 can be written in the form 2 * E t h' h' 1 A ' = c l λ + 2 E A (13) f c l' 4 Ab h' w = * cos θ ψ λ (14) Eq.14 shows that, for assigne values of l / h, ν an F v (on which ψ epens) a family of curves * w / = f ( λ ) can be efine. In orer to obtain these curves a number of infille frames characterize by ifferent values of the quantities that efine the parameter * λ were analyse using the micro-moelling proceure previously escribe. In this stuy two ifferent values of the aspect ratio l / h, namely 1 an 1.5, an two ifferent values of the Poisson s ratios, ν = 0.15 an ν = 0.25, were investigate. The analyses were repeate for four imensionless vertical loa levels: = 0, = , = , = , being efine as 11

13 F v ε v = (15) 2Ac Ec A c being the cross section area of the column, E c the Young moulus of the concrete constituting the frame an F v the total vertical loa acting on the frame. 7. Results an esign curves The analysis of the FE simulations shows that the frame-infill contact surface grows with the vertical loa magnitue an that for high level of vertical loa the mechanical behaviour of the infill switches from that of a strut element to that of a plate, see Fig. 8. As a consequence for a fixe λ * the frame in-plane stiffness an thus the strut imensionless with w/ grows as a function of In Figs the results of the numerical analyses in terms of ratio w/ versus the parameter F v. * λ are plotte for the ifferent infill Poisson s ratio an ifferent levels of vertical loa. These results confirm that the close epenence of the strut with on the parameter λ *, previously shown in Papia et al. (2003) hols in the presence of vertical loa. In orer to obtain a useful esign tool, the w/ values were fitte by the analytical expression propose in Cavaleri et al. (2005) w = c k β (16) * ( λ ) c = ν ν (17) 2 β = ν ν (18) 2 k being a coefficient that was not characterize in the above cite work an that takes the effect of vertical loa into account. For k=1 (no vertical loa acting) the function (16) assumes the form propose in (Papia et al. 2003). 12

14 The numerical investigation carrie out in this work showe quite a linear epenence of the coefficient k on vertical loa an axial strains of the columns. This relationship can be approximate by the following expression 1 λ ε l (19) h * k = 1 + ( ) v In Figs the analytical curves provie by Eqs.(16-19) have been plotte. The numerical vs moel comparison shows a goo agreement for both rectangular an square infills. For sake of completeness the comparison alreay iscusse in Papia et al (2003), between the results of the FE analyses an the analytical curves provie by Eqs. (16-19) when no vertical loa is transferre to the infills, is shown in Fig. 10. In Figs. 14 an 15 the families of analytical curves obtaine for ifferent Poisson s ratio an aspect ratio are plotte together. As it can be observe the equivalent strut with w/ increases when the level of vertical loa grows. Moreover the Poisson s ratio has a stiffening effect on the in-plane response of the infille frame. 8. Conclusions In this paper the mechanical behaviour of infille frames uner in-plane loaing has been iscusse an an analytical law taking into account the influence of vertical loa on the characteristics of the pin-jointe iagonal strut equivalent to an infill is propose. A numerical investigation on infille frames having ifferent geometric an mechanical characteristics has been carrie out an the results showe the stiffening effect of the vertical loa transferre from the frame to the infill. The analyses, carrie out using a FE moel in which frame an infill are moelle as linear shell elements an the frame-infill interface with link elements working in compression only, have shown that the etachment of the contact surface between frame 13

15 an infill, prouce by in-plane loaing, ecreases when vertical loa are transferre to the infill. This prouces a stiffening of the structural system. The stuy focuses on the initial unamage stiffness of a frame-infill system, which is of key importance for the correct efinition of the cyclic response uner seismic loaing. This slope influences the strength envelope, in particular the slope of the linear branches of the envelope. For given infill thickness an material, the characterization of the initial stiffness of the equivalent strut epens only on the ientification of the strut with. The imensionless with w/, in this work is given as a function of a synthetic parameters (λ*), epening on the mechanical parameters of frame an infill. A family of w/ vs λ* curves, each one epening on the infill Poisson s ratio an level of mean axial strain prouce by the vertical loa on the frame columns, has been obtaine. It has also been shown that as the parameter λ* increases the corresponing values of w/ can be escribe by a power law epening on the level of vertical loa transferre from the frame to the infill. These curves provie an effective tool for taking into account the contribution of infills to the global structural stiffness. They allow a quick evaluation of the contribution of the infill to the in-plane stiffness of the generic frame of a frame structure an can be use in any commercial structural analysis software when a simplifie approach for the preiction of the response is preferre to a etaile micro-moelling. Obviously the ientification of the initial characteristics of the equivalent strut also nees the ientification of the further parameters able to efine a complete non-linear law for the strut. In the case of the Pivot moel escribe by Cavaleri an Di Trapani (2014) three further parameters - one for the egraation of stiffness an two for the strength envelope, are sufficient to characterize completely the equivalent strut for seismic analysis. 14

16 References ADINA Theory an Moeling Guie - Volume I: Solis & Structures (2012) ADINA R & D, Inc. USA Amato G, Cavaleri L, Fossetti M, Papia M (2008) Infille frames: influence of vertical loa on the equivalent iagonal strut moel. Proc. of 14th WCEE, Beijing, China, CD-ROM, Paper Amato G, Fossetti M, Cavaleri L, Papia M (2009) An upate moel of equivalent iagonal strut for infill panels. Proc. of Eurocoe 8, Perspectives from the Italian stanpoint Workshop. Doppiavoce Eitor. Napoli, Italy, Asteris PG (2003) Lateral Stiffness of Brick Masonry Infille Plane Frames. J Struct Eng ASCE 129(8): Asteris PG (2008). Finite Element Micro-Moeling of Infille Frames. Electronic J. of Structural Engineering 8: Asteris PG, Antoniou ST, Sophianopoulos DS, Chrysostomou CZ (2011) Mathematical Macromoeling of Infille Frames: State of the Art. J Struct Eng ASCE 137(12): Asteris PG, Cotsovos DM (2012) Numerical investigation of the effect of infill walls on the structural response of rc frames. Open Construction an Builing Technology Journal, 6 (1): Asteris PG, Giannopoulos IP, Chrysostomou CZ (2012) Moeling of infille frames with openings. Open Construction an Builing Technology Journal, 6 (1): Asteris PG, Cotsovos DM. Chrysostomou CZ, Mohebkhah A, Al-Chaar GK (2013) Mathematical micromoeling of infille frames: State of the art. Eng Struct 56: Baran M, Sevil T (2010) Analytical an experimental stuies on infille RC frames. Int J Phys Sciences 5(13): Bathe KJ, Bouzinov PA (1997) On the constraint function metho for contact problems. Computer an Structures 64(5/6): Cavaleri L, Di Trapani F (2014) Cyclic response of Masonry infille RC frames: Experimental results an simplifie moeling. Soil Dynamics an Earthquake Engineering 65: Cavaleri L, Di Trapani F, Macaluso G, Papia M (2012) Reliability of coe-propose moels for assessment of masonry elastic mouli. Ingegneria Sismica 1: Cavaleri L, Fossetti M, Papia M (2005) Infille frames: evelopments in the evaluation of cyclic behaviour uner lateral loas. Struct Eng Mech 21(4): Cavaleri L, Papia M (2003) A new ynamic ientification technique: application to the evaluation of the equivalent strut for infille frames. Eng Struct 25: Cavaleri L, Papia M, Macaluso G, Di Trapani F, Colajanni P. (2014) Definition of iagonal Poisson s ratio an elastic moulus for infill masonry walls. Materials an Structures 47(1-2): Chrysostomou CZ, Asteris PG (2012) On the in-plane properties an capacities of infille frames. Eng Struct 41:

17 Crisafulli FJ, Carr AJ (2007) Propose macro-moel for the analysis of infille frame structures. Bull New Zealan Soc Earthquake Eng 40(2): Durrani AJ, Luo YH (1994) Seismic retrofit of flat-slab builings with masonry infill. Proc. of the NCEER Workshop on Seismic Response of Masonry Infills, Technical report NCEER , D.P. Abrams, New York. Eurocoe 8 (2004). Design of structures for earthquake resistance - Part 1: General rules, seismic actions an rules for builings. FEMA 356 (2000). Prestanar an commentary for the seismic rehabilitation of builings. Fiore A, Netti A, Monaco P (2012). The influence of masonry infill on the seismic behavior of RC frame builings. Eng. Struct. 44: Flanagan RD, Bennett RM (1999) In-plane analysis of masonry infill materials. Pract. Perio. Struct. Des. Constr. 6(4): Flanagan RD, Bennett RM (1999) In-plane behaviour of structural clay tile infille frames. J. Struct. Eng. 125(6): Jones RM (1975) Mechanics of composite materials. McGraw-Hill, Tokio. Koutromanos I, Stavriis A, Shing PB, Willam K (2011) Numerical moeling of masonry-infille RC frames subjecte to seismic loas. Computers an Structures 89: Mainstone RJ (1971) On stiffness an strength of infille frames. Proceeings of Institution of Civil Engineers 7360s: Mainstone RJ (1974) Supplementary note on the stiffness an strength of infille frames. Builing Research Station, U.K, CP 13/74. Manos GC, Soulis VJ, Thauampteh J (2011) The behaviour of masonry assemblages an masonry-infille R/C frames subjecte to combine vertical an cyclic horizontal seismic-type loaing. J Av Eng Softw 45: Manos GC, Soulis VJ, Thauampteh J (2012) A nonlinear numerical moel an its utilization in simulating the in-plane behaviour of multi-story R/C frames with masonry infills. Open Constr Buil Technol J 6: Moghaam HA, Dowling PJ (1987) The state-of-art in infille frames. ESEE research report Imperial College of Science an Technology, Lonon. NCEER (1994), Workshop on "Seismic response of masonry infills", Technical report NCEER , San Francisco. Papia M, Cavaleri L, Fossetti M (2003) Infille frames: evelopments in the evaluation of the stiffening effect of infills. Structural Engineering an Mechanics 16(6): Papia M, Cavaleri L, Fossetti M (2004) Effect of vertical loas on lateral response of infille frames. Proceeings 13th Worl Conference on Earthquake Engineering, Vancouver, Canaa, No

18 Saneineja A, Hobbs B (1995) Inelastic esign of infille frames. J. Struct. Eng. ASCE, 121(4): Shing PB, Stavriis A (2014) Analysis of Seismic Response of Masonry Infille RC Frames through Collapse. ACI Structural Journal, Special Paper Staffor Smith B (1968) Moel tests results of vertical an horizontal loaing of infille frames. Proceeings of American Concrete Institute 65(8): Staffor Smith B, Carter C (1969) A metho for analysis for infille frames. Proceeings of Institution of Civil Engineers 7218: Stylianiis KC (2012) Experimental investigation of masonry infille RC frames. Open Constr Buil Technol J 6: Valiasis T, Stylianies K (1989) Masonry Infille R/C Frames uner Horizontal Loaing. Experimental Results. European Earthquake Engineering 3: Valiasis T, Stylianies K, Penelis G (1993) Hysteresis Moel for Weak Brick Masonry Infills in R/C Frames uner Loa Reversals. European Earthquake Engineering 1: 3-9. Valiasis TN, Stylianiis KC (1989) Masonry infills in R/C frames uner horizontal loaing, Experimental results. European Earthquake Engineering 3: Wang M, He M, Lu J, Huang W (2011) Stuy on finite element moel of infille walls of steel frames. Avance Materials Research : Willam KJ, Citto C, Shing PB (2010) Recent results on masonry infill walls. Avance Materials Research : Yang W, Ou JP (2010) Finite-element moeling of nonlinear behavior of masonry-infille RC frames. Journal of Structural Engineering 136(3): Zhai C, Wang X, Li S, Xie L (2011) An improve three-strut moel for masonry infills in RC frame structures. Avance Materials Research :

19 l F/2 F/2 h' h h' θ l' l' a) b) Fig.1 Infille frame sketch uner horizontal loa: (a) actual system; (b) macro-moel. 18

20 S (strut axial force - tension) (0) (1) (2) 2 tension curve (no strength) 2 G2 G1 G0 M 0 M 1 M 2 ωs2 S2 τo D2 F2 O2 F1 F0 τ1 τ2 βτo O1 O0 τo A0 αsc D1 D0 B2 B Sc 1 B0 M 0 M 1 M 2 (strut axial eformation - tension) elastic loaing curve strength envelope curve δc Fig.2 Strut cyclic law propose in Cavaleri et al

21 Fig.3 Strut cyclic law propose in Cavaleri an Di Trapani

22 Fig.4. Variation of frame-infill interaction ue to vertical loa 21

23 a) b) Fig.5 Infille frame uner horizontal loa: (a) real scheme, (b) simplifie scheme. 22

24 δ P δ P δ P F/2 P F/2 F/2 P F/2 F/2 P F/2 h' θ θ l' l' l' a) b) c) Fig. 6 Decomposition of the macro-moel in two schemes 23

25 Fig. 7 Finite element moel of infille frame 24

26 a) b) Fig. 8 Variation of the infill-frame contact area for two ifferent vertical loa levels: (a) = ; (b) =

27 w/ 0.50 ν =0.15 FEM analyses Eqs = λ w/ w/ = λ = λ Fig. 9 Results of the FEM analyses an comparison with the propose analytical curves: square infills; ν =

28 w/ 0.50 ν =0.25 FEM analyses Eqs = λ w/ = λ w/ = λ Fig. 10 Results of the FEM analyses an comparison with the propose analytical curves: square infills; ν =

29 w/ 0.50 ν =0.15 FEM analyses Eqs = λ w/ = λ w/ = λ Fig. 11 Results of the FEM analyses an comparison with the propose analytical curves: l rectangular infills 1.5 h = ; ν =

30 w/ 0.50 ν = 0.25 FEM analyses Eqs = λ * 0.50 w/ = λ* 0.50 w/ 0.10 = λ* Fig. 12 Results of the FEM analyses an comparison with the propose analytical curves: l rectangular infills 1.5 h = ; ν =

31 0.50 ν = ν =0.25 w/ w/ = 0 l / h= = 0 l / h= λ λ w/ 0.50 ν =0.15 FEM analyses Eqs w/ 0.50 ν = = 0 l / h=1.5 λ = 0 l / h=1.5 λ* Fig. 13 Results of the FEM analyses an comparison with the propose analytical curves: no vertical loa on infills 30

32 w/ =0 = = = w/ =0 = = = ν =0.15 ν =0.25 λ 0.15 a) b) λ Fig. 14 Propose analytical curves for ifferent vertical loa levels - square infills - (a) ν =0.15; (b) ν =

33 w/ =0 = = = w/ =0 = = = ν = 0.15 λ 0.15 ν = 0.25 λ a) b) Fig. 15 Propose analytical curves for ifferent vertical loa levels - rectangular infills - (a) ν =0.15; (b) ν =