A METHOD FOR PUSHOVER ANALYSIS IN SEISMIC ASSESSMENT OF MASONRY BUILDINGS

Transcription

1 A METHOD FOR PUSHOVER ANALYSIS IN SEISMIC ASSESSMENT OF MASONRY BUILDINGS Gudo MAGENES 1 SUMMARY A method for the nonlnear statc analyss of masonry buldngs s presented, sutable for sesmc assessment procedures based on pushover analyses. The method s based on an equvalent frame dealzaton of the structure, and on smplfed consttutve laws for the structural elements. Applcatons on up to fve storey structures are dscussed, pontng out some ssues regardng modelng hypotheses and calculated response. A possble use of the method n sesmc assessment s presented. The procedure makes use of dsplacement response spectra and of the substtutestructure approach whch has been proposed by other authors for renforced concrete structures. A smple example of the assessment procedure on a two-storey masonry structure s presented. Open questons and future developments are ponted out. INTRODUCTION The role of non-lnear equvalent statc (pushover) analyses s beng more and more recognzed as a practcal tool for the evaluaton of the sesmc response of structures. Pushover analyses are therefore ncreasngly beng consdered wthn modern sesmc codes, both for desgn of new structures and for assessment of exstng ones. Consderng the problem of sesmc assessment of masonry buldngs, the need for non-lnear analyss had been recognzed n Italy snce the late Seventes. In 1978 and 1981, recommendatons on sesmc assessment, repar and strengthenng of masonry buldngs were ssued, suggestng the use of an equvalent statc, smplfed nonlnear assessment method whch had been proposed and developed n Slovena by Tomaževc [1978]. Such method, whch has undergone several refnements n the subsequent years [Tomaževc, 1997], s based on the so-called storey-mechansm approach, whch bascally conssts n a separate non-lnear nterstorey sheardsplacement analyss for each storey, where each masonry per s characterzed by an dealzed non-lnear sheardsplacement curve (typcally elastc-perfectly plastc wth lmted ductlty). The conceptual smplcty of the storey-mechansm method and ts adopton by the Italan recommendatons were fundamental n ts dffuson among professonals, and the method has been extensvely used n Italy snce ts frst ntroducton n code provsons. However, the smplcty of the storey mechansm approach, s pad wth a seres of lmtatons whch may restrct ts applcaton only to some classes of buldngs [Magenes and Della Fontana, 1998]. The need for more general methods of analyss has stmulated n Italy the research on the subject, and analytcal methods have made sgnfcant progress n the last decades, partcularly n the feld of fnte element analyses. However, refned nonlnear fnte element modelng does not consttute yet a sutable tool for the analyss of whole buldngs n the engneerng practce. For ths reason, several methods based on macro-element dscretzaton have been developed, requrng a low to moderate computatonal burden. Wthn ths context, t was felt by the author that several basc deas of the storey-mechansm approach could be used and extended to a broader range of valdty, mantanng concepts and dealzatons that are famlar to the engneer and obtanng results that can be compared wth those of more sophstcated analyss. Followng ths dea, a smplfed method based on an equvalent frame dealzaton of multstorey walls was developed and mplemented at the Unversty of Pava. Ths paper descrbes the model and ts possble use n assessment procedures. 1 Department of Structural Mechancs, Unversty of Pava, va Ferrata 1, 271 Pava, Italy Emal: magenes@gndt.unpv.t

2 spandrel beam λ F 2 per λ F 1 jont Fgure 1. Equvalent frame dealzaton of a masonry wall. Fgure 2. Idealzed nonlnear behavour of a per element falng n shear. ϕ γ θ = ϕ + γ j Fgure 3. Chord rotaton n a beam-column element. Fgure 4. Idealzed nonlnear behavour of a spandrel element falng n shear. A METHOD FOR THE NONLINEAR STATIC ANALYSIS OF MASONRY BUILDINGS The model heren descrbed (acronym: SAM for Smplfed Analyss of Masonry buldngs) was conceved for the global analyss of new and exstng masonry buldngs, n whch the resstng mechansm s governed by nplane response of walls. Collapse mechansms due to dynamc out-of-plane response are not consdered n the model, and should be evaluated wth separate modelng. The global sesmc analyss of an unrenforced masonry buldng s meanngful f proper means, such as tes and/or rng beams, prevent local and global out-of-plane collapses, whch otherwse would occur prematurely at low sesmc ntenstes. The model was developed frst for plane structures [Magenes and Della Fontana, 1998], and subsequently extended to three-dmensonal buldngs [Magenes, 1999]. Consderng a multstorey masonry wall loaded n plane by horzontal forces, f the geometry of the openngs s suffcently regular, t s possble to dealze the wall as an equvalent frame made by per elements, spandrel beam elements, and jont elements (Fgure 1). The per element and the spandrel element are modeled as beamcolumn elements wth shear deformaton, whle the jont elements are supposed nfntely resstant and stff, and are modeled by means of rgd offsets at the ends of the per and spandrel elements. The per element s supposed to have an elasto-plastc behavour wth lmted deformaton. The element dsplays a lnear elastc behavour untl one of the possble falure crtera s met. The elasto-plastc dealzaton approxmates the expermental resstance envelope under cyclc actons. The followng falure mechansms are foreseen. Flexural or rockng falure occurs when the moment M at any of the end sectons of the effectve per length attans the ultmate moment M u whch s a functon of axal force, geometry of the secton and masonry compresson strength f u. A plastc hnge s then ntroduced n the secton where M u s attaned. Dagonal shear crackng s defned by the lowest between the strength assocated to mortar jont falure and brck unt falure, accordng to what proposed n [Magenes and Calv, 1997]. When the falure crteron s met, plastc 2

3 shear deformaton occurs as n Fgure 2, where a lmt θ u to the maxmum chord rotaton s set, beyond whch the strength s zeroed. Chord rotaton s expressed as the sum of the flexural deformaton and of shear deformaton θ = ϕ + γ (Fgure 3), and s a generalzaton of the concept of drft for non-symmetrc boundary condtons of a per subjected to flexure and shear. A suggested lmt for unrenforced masonry s θ u =.5 %. Shear sldng can occur n any of the end sectons of the per, and s a functon of bedjont shear strength and of the extent of flexural crackng n the secton. Anelastc deformaton due to shear sldng s modeled smlarly to the case of dagonal shear crackng. The complete expressons for the strength crtera can be found n [Magenes and Calv, 1997] and [Magenes and Della Fontana, 1998]. The falure crtera are such that flexural strength s non-zero only n presence of axal compresson. No axal tenson s allowed,.e. the axal stffness of the per s zeroed for tensle axal deformaton. The spandrel beam element s formulated smlarly to the per element, takng nto account the dfferent orentaton of bedjonts wth respect to the axal force. The possble falure mechansms are flexure and shear. For flexural falure the formulaton s dentcal to the per element. For shear strength t s consdered that, because of the openngs above and below the spandrel element, the bedjonts have almost zero normal stress, and shear strength s therefore provded by coheson only. The nonlnear behavour of spandrels falng n shear s depcted n fgure 4, n whch strength degradaton s foreseen for ncreasng values of shear deformaton. By means of the parameters α, γ 1, γ 2 t s possble to obtan a varety of behavours, from elastc-brttle to elastcperfectly plastc. Ths more artculated consttutve hypothess allows to take nto account the tendency to a more brttle post-peak behavour of spandrels, as compared to pers, whch has some relevance on the results. To analyze three-dmensonal buldngs, the plane model was extended [Magenes, 1999] by formulatng the consttutve laws of pers and spandrels n three dmensons, assumng an ndependent behavour of the per or spandrel element n the two prncpal orthogonal planes parallel to the element axs. The out-of-plane behavour s modeled smlarly to the n-plane behavour. Composte walls (.e. flanged walls or orthogonal ntersectng walls) are decomposed n smple walls wth rectangular cross secton. If the ntersectng walls are effectvely bonded, t s possble to smulate the bond defnng approprate rgd offsets and mposng the contnuty of dsplacements at the ends of rgd offsets at the floor levels. An mportant ssue was consdered the possblty of modelng the presence of r.c. rng beams, whose role can nfluence to a large extent the couplng between pers. Rng beams are modeled as elasto-plastc frame elements, whch can fal n flexure wth plastc hngng. Steel tes can be modeled as elasto-plastc truss elements. Rgd floor daphragms can be smulated mposng a knematc constrant among the nodes at the floor level. VERIFICATION OF THE METHOD The frst applcatons of the method [Magenes and Della Fontana, 1998] were made on two- and three-storey walls, comparng the results to those obtaned by refned plane-stress non-lnear fnte element analyses wth a specfc consttutve law for unrenforced brck masonry [Gambarotta and Lagomarsno, 1997]. In such analyses (an example s gven n Fgure 5) a very good agreement of the results of the two methods was found n terms of overall strength and falure mechansms, provded that n the SAM method an elastc-brttle behavour of the spandrels falng n shear was assumed. Although such assumpton s conservatve and more consstent wth the fnte element smulatons, there s lttle expermental nformaton on the post-peak behavour of unrenforced spandrel beams subjected to cyclc actons, so that the queston on what knd of modelng hypothess s more realstc stll calls for clear expermental references. Although ths modelng ssue s not crucal for one- or twostorey buldngs, t can have a strong nfluence on the results for buldngs wth more than two storeys. Further analyses on a fve-storey wall were made, to evaluate the nfluence of several modelng hypoteses concernng the strength and stffness of couplng elements (r.c. beams and masonry spandrels). The fve storey wall (Fgure 6), taken from an exstng buldng n the cty of Catana (bult crca 1952), was made of brck masonry, wth contnuous r.c. beams at each floor. Such a wall was subjected to a code pattern of sesmc forces gradually ncreasng proportonally to a scalar, usng dfferent assumptons regardng the couplng elements, as descrbed n Table 1. To handle possble softenng of the structure before global collapse was reached, the analyses were carred out controllng the dsplacement of a sngle pont of an external statcally determned system whch dstrbuted the sesmc forces to the floors keepng the desred rato among the forces. The calculated global strengths (maxmum base shear V max ) n the dfferent analyses are summarzed n Table 1, and the complete force-dsplacement curves are reported n Fgure 7. The varaton n strength s qute sgnfcant, showng that the nfluence of the couplng elements can affect the strength of a multstorey wall by as much as 5% to 1%. At the same tme, the global falure mechansm of the wall can vary from a storey mechansm (at the ground floor or at the last floor) to a global overturnng of cantlever walls (n case G, where no r.c. rng beam s present), as reflected by the dsplacement profles n Fgure 8. Such a varety of results shows how the role of the couplng elements should not be overlooked n a sesmc analyss. 3

4 Total base shear (kn) F.E.M. SAM (w. brttle spandrels) Total dsplacement at 3rd floor (m) Fgure 5. Pushover analyss of a three-storey wall wth weak spandrels. Fgure 6. Equvalent frame model of a fve storey wall. Table 1. Summary of the analyses carred out an the fve storey wall. ANALYSIS HYPOTHESES V max [kn] V max / W tot A elastc r.c. rng beam, stffness calculated accordng to the uncracked secton B elastc r.c. rng beam, cracked secton stffness (1/5 of A) C elasto-plastc rng beam; flexural strength calculated accordng to the probable exstng renforcement G only masonry spandrels wth no r.c. rng beam I couplng elements wth no flexural stffness (cantlever wall system) V max = maxmum base shear; W tot = total weght of the wall. Global angular deformaton (%) th FLOOR 12 Analyss A th FLOOR Totale base shear (kn) Analyss B Analyss C Analyss G Base shear coeffcent Heght (m) Analyss A Analyss C Analyss G 3 rd FLOOR 2 nd FLOOR 1 st FLOOR Roof dsplacement (m). Fgure 7. Results of the pushover analyses of the fve-storey wall Horzontal dsplacement (m) Fgure 8. Dsplacement profles assocated to dfferent collapse mechansms. The SAM method was also appled to perform a three-dmensonal analyss of the consdered fve-storey buldng. The model (approxmately 15 x 11 m n plan, 19 m n heght) conssted of 39 elements and 195 nodes, for a total of 432 degrees of freedom (assumng n-plane rgdty of floors). As t can be seen, such a model can easly be handeled by any modern personal computer. The calculated strength n the weakest drecton wth the most realstc hypotheses was V max / W tot =.15, whch revealed a very hgh sesmc vulnerablty. The result s presently beng compared wth the results obtaned by other researchers wth dfferent analytcal models. 4

5 A PROCEDURE FOR SEISMIC ASSESSMENT The possble use of the proposed model wthn a smplfed sesmc assessment procedure s here outlned. It s assumed that the sesmc nput s gven by means of elastc desgn dsplacement/acceleraton spectra. The procedure proposed heren s based on the use of dsplacement spectra and on the substtute-structure concept [Shbata and Sozen, 1976], whch has been adopted n recent proposals of dsplacement-based desgn and assessment [Prestley and Calv, 1997] and whch had been outlned for masonry by Magenes and Calv [1997] n the case of sngle d.o.f. systems. Other approaches could be envsaged, based for nstance on force reducton factor and acceleraton spectra, or based on composte dsplacement-acceleraton spectra, and they wll be consdered for future developments. The goal of the procedure s to evaluate the deflected shape of the buldng at peak response. As a start, n ths context t wll be assumed that the structure s suffcently regular so that multple-mode response need not be consdered. The man steps of the procedure can be descrbed as follows. 1) Assume a deflected shape {δ () }, and defne a dstrbuton of equvalent statc nerta forces {F} as: F = F base γ, where γ = M () δ () δ M where M and δ () are respectvely the lumped mass and the horzontal dsplacement at the th degree of freedom, and F base s the total base shear. A possble frst choce for {δ () } could be obtaned by the frst mode shape assocated to the ntal elastc stffness of the buldng, or more smply by a set of dsplacements lnearly ncreasng wth heght. 2) Perform a nonlnear statc pushover analyss up to collapse of the structure under the gven dstrbuton of statc forces, mantanng the ratos determned by the coeffcents γ. Collapse may be defned as the attanment of the ultmate drft for ndvdual pers. To handle possble softenng of the structure before the attanment of the ultmate lmt state, t may be necessary to perform the statc analyss n dsplacement control, as made n the examples descrbed n the prevous secton, to assure that the desred ratos among the sesmc forces are kept. 3) Defne an equvalent s.d.o.f. system, wth the followng characterstcs: M eq = M = M tot ; δ eq = γ δ ; Feq = Fbase (2) Calculate and plot the force-dsplacement curve F eq - δ eq of the equvalent s.d.o.f. system. The evaluaton of the dynamc response of the s.d.o.f. system wll be made defnng a substtute structure whose effectve stffness s equal to the secant stffness K eq,s at a gven value of dsplacement δ eq.. 4) Defne the equvalent vscous dampng ξ eq (ncludng the effects of hystreretc energy absorpton) for the s.d.o.f. susbsttute structure, as a functon of the equvalent dsplacement δ eq, based on the evoluton of the damage mechansms obtaned n the pushover analyss, and on energy equvalence prncples. Plot the correspondng ξ eq - δ eq curve. 5) Evaluate teratvely the maxmum dsplacement of the s.d.o.f. system consstent wth the desgn elastc dsplacement spectrum δ eq,max = SD(T eq ; ξ eq ), where T eq = 2π (M eq /K eq,s ) 1/2 s the effectve perod at maxmum dsplacement response. The sequence of steps from 1 to 5 s based on the results of the pushover analyss carred out wth the set of statc forces defned at step 1 from an assumed deflected shape. However, n the pushover analyss the rato of the dsplacements at each story may vary as a consequence of the nonlnear behavour of the structure, and the dsplaced shape correspondng to the value of δ eq,max calculated at the end of step 5 wll dffer from what assumed at the begnnng of step 1. Dependng on the structure, the results of the statc analyss may be more or less senstve to the assumed pattern of statc forces, and, n general, the dsplaced shape wll vary contnuously as the analyss proceeds n the nonlnear range, dfferng from a lnear or frst-mode vbraton shape. It may be advsable therefore to repeat the procedure substtutng n equaton 1 of step 1 the dsplaced shape obtaned at the end of step 5, teratng the whole procedure untl a fnal dsplaced shape consstent wth the assumed force dstrbuton s obtaned. However, the need for teraton should not be overemphaszed. Gven the approxmaton of a pushover approach, t may be more effectve to assume two or three arbtrary dsplaced shapes consstent wth the most probable falure mechansms (e.g. storey mechansm at the frst storey, storey mechansm at the last storey, global overturnng) and then follow steps 1 to 5 once for each assumed dsplaced shape. A range of possble solutons would be obtaned, gvng a better reference for the assessment. The use of more than one load pattern would be recommended to account for possble hgher mode effects [Krawnkler and Senevratna, 1998]. Equatons (2) n step 3 are obtaned by smple dynamc and energy equvalence prncples and do not need specal dscusson. Step 4 deserves some comments wthn ths context. The evaluaton of a global equvalent vscous dampng for a masonry buldng requres expermental nformaton on the energy dsspaton propertes (1) 5

6 of sngle structural elements (e.g. pers and spandrels). Once the energy dsspaton of sngle elements s defned, t s possble to evaluate the global energy dsspaton of the whole structure, and the global equvalent dampng. Energy equvalence between the s.d.o.f. substtute structure and the buldng leads to the followng expresson for the equvalent dampng: ξeq = Ekξk Ek (3) k k where E k s the elastc stran energy assocated to the secant stffness and ξ k s the equvalent dampng of the k-th structural element. Consderng the equvalent frame dealzaton of the SAM method, the elastc energy of a sngle beam-column element can be convenently expressed n terms of moments and chord rotatons at the nodes and j as: E k = 1 ( M k, θ k, + M k, jθ k, j Ek, E 2 ) = + k, j where the work due to axal deformaton s neglected. At present, lmted expermental nformaton s readly avalable for URM structural elements n terms of equvalent dampng. Heren reference wll be made to the work of Magenes and Calv [1997] who have explctly evaluated values of equvalent dampng for brck masonry pers subjected to n-plane statc cyclc loadng. On that bass, a frst rough approxmaton can be made to quantfy the vscous dampng equvalent to hysteretc energy dsspaton of a sngle structural element, dependng on the falure mode. In the followng applcaton, t has been assumed that pers and spandrels n the lnear range are characterzed by a constant equvalent dampng equal to 5%, and that the value ncreases to 1 % when one of the shear falure crtera s met. If the element fals n flexure, the equvalent dampng assocated to hysteretc energy dsspaton remans equal to 5 %, but an addtonal 5 % due to mpact and radaton dampng s added. The equvalent dampng of structural elements wll vary therefore n a stepwse fashon. These assumptons am to gve a slghtly conservatve estmate of the equvalent dampng wth respect to expermental results. The program SAM has been therefore modfed to calculate automatcally the equvalent dampng of the buldng accordng to equatons 3 and 4 at every ncrement of the pushover analyss. To verfy the results that can be obtaned by ths crteron on a structure, the results of a full scale statc cyclc test on a two storey brck masonry buldng were processed to obtan a reference for the numercal evaluaton of the parameters of the substtute structure. The experment was carred out at the Unversty of Pava [Magenes et al., 1995], and conssted n a seres of dsplacement cycles of ncreasng ampltude, appled to the structure keepng a 1:1 rato among the forces appled at the frst and second floor. The longtudnal walls were coupled by flexble floor beams only, so that each longtudnal wall could be analyzed ndependently as a two-degreesof-freedom structure. Consderng one of the two walls (Fgure 9), the expermental response can be evaluated n terms of an equvalent s.d.o.f. structure accordng to the crtera descrbed above, obtanng the forcedsplacement dagram of Fgure 1. For each cycle t s then possble to calculate the equvalent dampng on the bass of the dsspated hysteretc energy and the secant stffness at peak dsplacement, obtanng the values reported n Fgure 11. The same wall was also analyzed wth the SAM method, carryng out a pushover analyss wth equal forces at the floor levels, and evaluatng the parameters of the s.d.o.f. substtute structure accordng to the hypotheses descrbed above. Snce the test was statc, however, mpact and radaton dampng was not taken nto account n the evaluaton of ξ eq. A lmt chord deformaton θ u =.5 % was assumed for pers falng n shear, and the numercal collapse of the structure concded wth the attanment of the lmt deformaton of the central per at the ground floor. (4) 15 W all D - Door wall 1 Base shear (kn) Fgure 9. Longtudnal wall of the masonry buldng subjected to cyclc statc testng Equvalent dsplacement δ eq (mm) Fgure 1. F base - δ eq curve calculated from the expermental response of the wall. 6

7 Equv. vscous dampng ξ eq Wall D - Door wall 1' 2' 3' 4' 1'' 2'' 3'' 4'' Experment SAM pushover analyss 5' 5''' ' 6'' 6''' 7'' 7' Base shear (kn) ' 1'' 4' 4'' 3' 3'' 2' 2'' 5' 5''' 5'' 6' 6'' 6''' 7'' 7' Exp. 1st cycle envelope Exp. 2nd cycle envelope Exp. 3rd cycle envelope SAM pushover analyss Equvalent dsplacement δ eq (mm) Equvalent dsplacement δ eq (mm) Fgure 11. Equvalent vscous dampng assocated to hysteretc energy dsspaton. Fgure 12. Comparson between expermental envelopes and pushover analyss. In general, t can be observed from Fgures 11 and 12 that the analytcal method obtans an acceptable estmate of the nonlnear force-dsplacement curve and of the equvalent dampng. The expermental cycles beyond the dsplacement of ±15 mm (labeled 7 and 7 ) show a sgnfcant strength degradaton assocated to the collapse of a spandrel above one of the doors, whch explans the very hgh expermental value of equvalent dampng. Classfyng these cycles as beyond collapse, the comparson between experment and analyss s meanngful only up to run 6. The jumps n the ξ eq numercal curve of fg. 11 correspond to the falure of pers or spandrels n the SAM analyss, accordng to the assumptons made. A more realstc evoluton of the equvalent dampng would probably be obtaned wth a contnuous varaton of the dampng of the elements wth ncreasng angular deformaton, although ths s not expected to produce sgnfcantly dfferent results n the assessment. A Smple Example of Applcaton As an example of applcaton of the assessment procedure heren outlned, the deal two-storey structure represented by the wall of Fgure 9 was assessed, assumng a sesmc nput descrbed by the elastc response spectrum of Eurocode 8 for stff sol (sol A). The dsplacement spectrum s obtaned from the acceleraton spectrum as SD(T;ξ) = 4π 2 /T 2 SA(T;ξ). The elastc spectrum defned for a default dampng of 5% s scaled multplyng the ordnates by the factor suggested n EC8: η=[7/(2+ξ)] 1/2. A peak ground acceleraton of.25 g s assumed. The dstrbuted masses are lumped at the storey levels and summed to the masses assocated to the floors, gvng a total mass of M 1 = kn at the frst floor and M 2 = kn at the second floor. To perform the pushover analyss, a normalzed dsplaced shape correspondng to the frst mode of vbraton s assumed n step 1: {δ () } T = {.545 ; 1.} whch gves a force dstrbuton {F () } T = F base {.38 ;.62}. The pushover analyss s then carred out (step 2), and the F base - δ eq and ξ eq - δ eq curves of the substtute s.d.o.f. structure are obtaned (steps 3 and 4). Step 5 s then carred out by assumng a frst value of δ eq, equal to the ultmate dsplacement of the F base - δ eq curve. The correspondng secant stffness K eq = F eq /δ eq, perod T eq and dampng ξ eq are evaluated and the dsplacement spectrum s entered to obtan a new value of dsplacement δ eq,1. At ths frst teraton, by checkng f δ eq,1 δ eq, t s already possble to verfy f the ultmate dsplacement δ eq,ult of the structure wll not be exceeded. If ths s verfed, wth a tral- and-error procedure t s possble to converge to a fnal value of dsplacement such that δ eq,n+1 δ eq,n wthn a specfed tolerance. In the case consdered ths results n δ eq,max =12.1 mm, compared to an ultmate dsplacement δ eq,ult = 13.6 mm. At ths pont, the dsplaced shape {δ (1) } correspondng to the desgn dsplacement of 12.1 mm can be checked and compared wth the ntal assumed dsplaced shape {δ () }. In ths case, after normalzaton, the dsplaced shape {δ (1) } T = {.84 ; 1.} s obtaned, whch shows a storey mechansm at the frst storey. As t can be observed, the calculated nelastc response may lead to a dsplaced shape whch s rather dfferent from an elastc frst mode shape. If now a new dstrbuton of sesmc forces {F (1) } s calculated from {δ (1) }, obtanng {F (1) } T = F base {.475 ;.525}, steps 1 to 5 can be repeated, defnng new F base - δ eq and ξ eq - δ eq curves. It may be worth to notce that now the force dstrbuton s approachng a constant force dstrbuton, whch s consstent wth the storey mechansm obtaned. At the end of ths second global teraton, the followng results are obtaned: δ eq,max = 5.5 mm, {δ (2) } T = {.668 ; 1.}. A further teraton yelds δ eq,max = 6. mm, {δ (3) } T = {.665 ; 1.}. The smlarty of the values calculated n the last two teratons suggest that the value δ eq,max = 6. mm can be consdered an acceptable estmate of the maxmum response of the structure. 7

8 CONCLUSIONS From what presented n ths paper, t appears that the recent trends whch are beng followed n sesmc desgn and assessment of structural types such as renforced concrete and steel structures can be pursued also for masonry buldngs. On one hand, the proposed model for nonlnear statc analyss has so far produced satsfactory results. Stll, further comparsons wth other methods of analyss on dfferent structural confguratons are needed and are presently carred out as a part of the ongong research. The features that make the SAM method attractve for the applcatons are manly the low computatonal burden and a good versatlty. Ths second feature allows the engneer to select among a range of possble solutons and hypotheses, to compare the most realstc wth the most conservatve, allowng to draw sounder conclusons for the assessment, especally when the knowledge of the exstng structural system s ncomplete, as can be the case for hstorcal buldngs. On the other hand, t s clear that a satsfactory model for monotonc analyss s not suffcent for a relable predcton the dynamc response under sesmc exctaton. The proposed assessment procedure based on the substtute structure concept appears to be a step forward wth respect to current codfed practces, however ts effectve capablty of predctng correctly the maxmum dynamc response needs further verfcaton by comparson wth dynamc analyses and wth experments. In fact, unrenforced masonry structures presents specfc features (hstory-dependent degradaton of stffness and strength under cyclc actons, senstvty to the duraton, frequency and energy content of the sesmc nput) whch must be carefully consdered for the defnton of a relable assessment procedure. The future research wll therefore be dedcated to the study of such aspects. ACKNOWLEDGEMENTS The research heren descrbed was funded by Gruppo Nazonale per la Dfesa da Terremot (GNDT). The precous collaboraton of the former students Claudo Braggo and Davde Bolognn s acknowledged. REFERENCES CEN (1995), Eurocode 8 Desgn provsons for earthquake resstance of structures. Part 1-1: general rules and rules for buldngs ENV , Brussels. Gambarotta, L., and Lagomarsno, S. (1997), Damage models for the sesmc response of brck masonry shear walls. Part II: the contnuum model and ts applcaton, Earthq. Engn. and Struct. Dyn., Vol. 26, pp Magenes, G., (1999). Smplfed models for the the sesmc analyss of masonry buldngs, to be publshed as G.N.D.T. Report (n Italan). Krawnkler, H., and Senevratna, G.D.P.K., (1998), Pros and cons of a pushover analyss for sesmc perfomance, Engneerng Structures, Vol. 2, pp Magenes, G. and Calv, G.M., (1997). In-plane sesmc response of brck masonry walls, Earthq. Engn. and Struct. Dyn., Vol. 26, pp Magenes, G., Calv, G.M. and Kngsley, G.R. (1995). Sesmc testng of a full-scale, two-story masonry buldng: test procedure and measured expermental response, n Expermental and Numercal Investgaton on a Brck Masonry Buldng Prototype Numercal Predcton of the Experment, GNDT Report 3., Pava, Italy. Magenes, G., and Della Fontana, A. (1998), Smplfed non-lnear sesmc analyss of masonry buldngs, Proc. of the Brtsh Masonry Socety, Vol. 8, pp Prestley, M.J.N., and Calv, G.M. (1997), Concepts and procedures for drect dsplacement-based desgn and assessment, n Sesmc Desgn Methodologes for the Next Generaton of Codes, Fajfar and Krawnkler (eds.),, Balkema, Rotterdam, pp Shbata, A., and Sozen, M.A. (1976), Substtute-structure method for sesmc desgn n R/C, Journ. of Struct. Dv., ASCE, Vol. 12, no. ST1, January, pp Tomaževc, M. (1978), The computer program POR, Report ZRMK, (n Slovene). Tomaževc, M. (1997), Sesmc resstance verfcaton of buldngs: followng the new trends, n Sesmc Desgn Methodologes for the Next Generaton of Codes, Fajfar and Krawnkler (eds.), Balkema, Rotterdam, pp