Component-based Probabilistic Methodology for the Vulnerability Assessment of RC Frames Retrofitted with Dissipative Braces

Transcription

1 Component-based Probablstc Methodology for the Vulnerablty Assessment of RC Frames Retroftted wth Dsspatve Braces F. Fredd & L. Ragn Marche Polytechnc Unversty, Italy E. Tubald & A. Dall Asta Unversty of Camerno, Italy SUMMARY: The paper llustrates a probablstc methodology for assessng the vulnerablty of low-ductlty renforced concrete (RC) buldngs retroftted by dsspatve braces. The am s to hghlght the most mportant parameters controllng the capacty of these systems and specfc aspects concernng the response uncertantes. The proposed methodology s based on local engneerng demand parameters (EDPs) for montorng the sesmc response and on the development of component and system fraglty curves before and after the retroft. Its capablty s tested consderng a RC frame desgned for gravty-loads only retroftted by elasto-plastc dsspatve braces. The results show the effectveness of the methodology n descrbng the changes n the response and n the falure modaltes due to the retroft. Moreover, the retroft effectveness s evaluated by ntroducng proper synthetc parameters descrbng the fraglty curves and hghlghtng the mportance of employng local rather than global EDPs n the sesmc rsk evaluaton of low-ductlty frames. Keywords: Renforced Concrete Frame, Fraglty, Vulnerablty Assessment, Sesmc retroft, Dsspatve Braces. INTRODUCTION The damage occurred durng recent earthquakes n many exstng renforced concrete (RC) buldngs desgned before the ntroducton of modern sesmc codes has shown that these structures are very vulnerable to the sesmc acton due to ther reduced ductlty capacty. Thus, there s a sgnfcant need of modern retroft technques for ncreasng ther safety and of relable tools for assessng the effectveness of the retroft. Among the varous technques currently employed for the retroft, the use of dsspatve braces appears to be very promsng (Soong and Spencer 22). These braces provde a supplemental path for the earthquake nduced horzontal actons and thus enhance the sesmc behavor of the frame by addng dsspaton capacty and, n some cases, stffness to the bare frame. It should be noted, however, that the ntroducton of a bracng system nto a low-ductlty frame often nduces remarkable changes both n the collapse modaltes and n the probablstc propertes of the sesmc response of the structure. The latter aspect assumes a consderable mportance n consequence of the hgh degree of uncertanty affectng the sesmc nput and of the dfferences n the propagaton of ths uncertanty through the two resstng systems (RC frame and dsspatve bracng). For these reasons, the evaluaton of the effectveness of ths type of retroft technque n reducng the frame vulnerablty should be performed wthn a probablstc framework. An ncreasngly popular approach for assessng n probablstc terms the sesmc vulnerablty of structural systems and the effectveness of a retroft technque nvolves the development of fraglty curves. These tools provde the probablty of exceedng a specfed lmt state (LS) or falure condton, condtonal to the strong-moton shakng severty, quantfed by means of an approprately selected ntensty measure (IM). In ths context, fraglty curves are employed by Hueste and Ba 26, Ramamoorthy et al. 26, Güneys and Altay 27, Özel and Güneys 2. Although n these

2 studes probablstc methodologes are employed for evaluatng the effectveness of dfferent retroft schemes, some modfcatons and extensons should be ntroduced n order to properly address the specfc ssues dervng from the use of dsspatve braces for the retroft of exstng low-ductlty RC frames. The frst ssue s related to the choce of approprate engneerng demand parameters (EDPs) for montorng the sesmc response and evaluatng the performance of the frame and of the retroft system. In the studes lsted above the fraglty curves are developed by usng the peak nterstory drft as unque global EDP. Ths strategy s commonly pursued snce montorng the tme-hstory of the local response of all structural members may be cumbersome, especally when complex models wth a hgh number of degrees of freedoms are consdered. In the cases of exstng structures desgned before the ntroducton of modern sesmc codes, the relatonshps between local falure and global EDPs, such as the nterstory drft, may change case by case, as demonstrated by the very dfferent drft lmts present n the lterature (Hueste and Ba 26, Ramamoorthy et al. 26). Moreover, n exstng structures retroftted by means of dsspatve braces, these relatonshps could change by ncreasng the dmenson of the braces, due to the reducton of the flexural ductlty capacty of the compressed columns nvolved n the bracng system. For these reasons, the use of global EDPs wth code-specfed lmts s not recommended for the assessment of exstng RC frames. By contrast, the use of local component-specfc EDPs (Lupo et al. 22), such as the stran demand at the most crtcal element sectons or the shear demand on a beam-column jont, though more cumbersome, s not affected by the mentoned lmtatons. In addton, t permts to approprately assess the probablstc response of sngle resstng components (ncludng the braces), ther contrbuton to the system vulnerablty, and the mpact of the retroft on the local response of the ndvdual members (Padgett and Des Roches 28). A second relevant ssue n defnng a probablstc methodology of analyss concerns the evaluaton of the retroft technque effectveness, whch s accomplshed n the studes cted above by comparng the medan values of the fraglty curves of the structure before and after retroft. Ths comparson has often mpled the use of structural-ndependent IMs n past studes, such as the not very effcent peak ground acceleraton (PGA). In fact, when the natural perod of the bare frame dffers from the natural perod of the retroftted frame, the comparson between fraglty curves obtaned by usng more effcent structure-specfc IMs (Katsanos et al. 29) (e.g., the spectral acceleraton at the fundamental perod of the structure) would not drectly provde nformaton about the effectveness of the retroft (Lel et al. 2). Furthermore, a more ratonal approach to accurately compute the changes n the safety margn due to retroft should also account for the dsperson of the fraglty curve, snce ths parameter affects the estmate of the sesmc rsk. Ths paper proposes a fraglty-based methodology amng at overcomng the lmts of the studes mentoned above. The methodology s developed by combnng exstng technques already employed for dfferent structural systems and by talorng these technques to the specfc problem analyzed. Local EDPs are used to develop sngle component fraglty curves whle system fraglty curves are derved and descrbed by proper synthetc parameters sutable for use wth any IM. In the frst part of the paper, the proposed methodology s accurately llustrated hghlghtng ts advantages wth respect to exstng approaches. Then, ts capablty and effectveness s tested by consderng a realstc benchmark RC frame wth lmted ductlty capacty. The frame s retroftted by nsertng a system of BRBs wth elasto-plastc behavor desgned for several levels of the base shear capacty. The braces are desgned by applyng a wdespread method based on an equvalent sngle degree of freedom (SDOF) approxmaton (Soong and Spencer 22). The applcaton of the probablstc methodology permts to evaluate the accuracy of the smplfed desgn crteron and also to draw some mportant consderatons about the behavor of the sngle resstng components, the effectveness of the retroft technque, and the structural safety ncrement.

3 2. PROBABILISTIC METHODOLOGY FOR VULNERABILITY ASSESSMENT The sesmc response of the frame before and after retroft s affected by uncertantes n the earthquake nput (record-to-record varablty), n the propertes of the system (model parameter uncertanty), and by lack of knowledge (epstemc uncertanty). The uncertanty affectng the earthquake nput s taken nto account by selectng a set of natural ground moton (g.m.) records that reflect the varablty n duraton, frequency content, and other characterstcs of the nput. The effects of model parameter uncertanty and epstemc uncertanty are usually less notable than the effects of record-to-record varablty and they are not consdered n ths study (Kwon and Elnasha 25). In order to generate fraglty curves, ncremental dynamc analyss (IDA) (Vamvatskos and Cornell 22) s performed by subjectng the system to a set of selected g.m. records for ncreasng values of the sesmc IM. The methodology proposed n ths study s orented to the use of structural-dependent IMs. In partcular, the spectral acceleraton S a (T) at the fundamental perod of the structure T for a dampng factor ξ=5% (Katsanos et al. 29) s employed as IM due to ts effcency. Ths choce requres scalng the g.m. records n order to obtan the same value of S a (T) for the natural perod of the structure, whch s dfferent for the bare and the retroftted frames. IDA provdes a set of samples of approprately selected EDPs montorng the system response for dscrete values of the IM. As already dscussed n the ntroducton, local EDPs, drectly related to the component falure modes, are used n order to montor the behavor of the most vulnerable system components and to capture the modfcatons to the frame response and collapse modaltes nduced by the ntroducton of the bracng system. The sesmc demand on the frame elements (beams and columns) due to flexural moments and axal forces s controlled by montorng the maxmum-over-tme values of the concrete compressve stran ε c and of the steel stran ε s at the most crtcal sectons. The non-ductle mechansms of the frames are controlled by recordng the maxmum-over-tme values of the shear force V at the crtcal sectons of each element of the frame, the dagonal tenson stress σ t, and the dagonal compresson stress σ c at each beam-column jont. Fnally, n the retroftted case, the sesmc demand mposed on the retroft system s controlled by evaluatng the maxmum-over-tme value of a damage parameter d (e.g., the maxmum-over-tme value of the ductlty demand μ d for elasto-plastc braces) for each dsspatve brace. The component fraglty curves for the bare and the retroftted frame are evaluated by consderng the followng lmt states (LSs) chosen coherently wth the montored EDPs: LS) ε c exceedng the capacty lmt ε cu at each crtcal secton, LS2) ε s exceedng the capacty lmt ε su at each crtcal secton, LS3) the shear demand V exceedng the shear resstance V u at each crtcal secton, LS4) σ c exceedng the resstance n compresson σ cu at each jont, LS5) σ t exceedng the resstance n tenson σ tu at each jont, and LS6) the damage ndex d overcomng the correspondng capacty du at each dsspatve brace, (e.g., μ d overcomng the lmt μ du for elasto-plastc braces). The system fraglty curves are then derved by assumng a seres arrangement of the components,.e., falure n one component yelds system falure. The choce of the LSs and the seres arrangement assumpton s consstent wth sesmc code prescrptons requrng that all the consdered LSs must be verfed for all the structural members. Moreover, t allows of lmtng structural damage on the exstng frame, often sought by the retroft crtera. However, t s noteworthy that the proposed methodology can be appled also wth dfferent assumptons on the system fraglty curves. Fnally, t s noted that the proposed methodology s purely numercal snce t s based on the drect comparson between the samples of the demand and the correspondng capacty. Thus, the correlaton among the varous component LSs s automatcally taken nto account. The numercal fraglty curves are approxmated by analytcal lognormal curves obtaned through least-square mnmzaton. The assumpton of lognormalty smplfes the analyss of the results and permts to synthetcally descrbe the fraglty of the systems by means of the two characterstc parameters. These are the medan fraglty capacty, IM c,5, defned as the 5 th fractle of the lognormal fraglty curve and the logarthmc standard devaton or dsperson measure, β c, gven by: IM c,84 βc = ln 2 IM c,6 (2.)

4 where IM c,84 and IM c,6 are the IM values correspondng respectvely to the 84 th and the 6 th fractle of the lognormal fraglty curve,.e., the values of the IM whch yeld falure respectvely n 84 and 6 cases over. The use of lognormal fraglty curves s very common and wdely accepted n performance based earthquake engneerng, snce t permts to estmate the above defned parameters even when a lmted number of EDP samples are avalable. Moreover, t permts to easly ncorporate the effects of other sources of uncertanty n addton to the record-to-record varablty, and t smplfes the evaluaton of the sesmc rsk. As already ponted out n the ntroducton, f a structural-dependent IM such as S a (T) s employed to montor the sesmc ntensty, the comparson of the values of IM c,5 obtaned for the bare and retroftted frame would not drectly provde nformaton about the effectveness of the retroft, snce the natural perod changes due to the retroft. For ths reason, the comparson should be performed between the values of the capacty margn rato m 5 (Lel et al. 2), defned as the rato between the value of IM c,5 and the value of the IM correspondng to a reference return perod, IM TR. In the proposed methodology, IM TR s assumed as the value of S a (T) for a reference return perod such that m 5 = for the bare frame, as llustrated n Fg. 2.. By ths way, the value of m 5 obtaned for the retroftted frame drectly measures the ncrement of sesmc ntensty that can be wthstood by the retroftted structure. In a smlar way, based on the rato IM c,84 / IM TR and IM c,6 / IM TR, the factors m 84 and m 6 correspondng to the 84 th and 6 th fractles of ncrement of capacty are defned. These parameters, together wth parameter β c, may also be used to assess n probablstc terms the effectveness of the retroft, by accountng for the dsperson of the system response, whch may have a non-neglgble nfluence on the sesmc rsk assessment. IM = S a (T) Retroftted frame Bare frame IM = S a (T) m 5 IM = IM c,5 TR.5 P f T [sec] Fgure 2.. Defnton of capacty margn rato m 5 : sesmc fraglty curves before and after retroft (left), unform hazard spectrum such that m 5 = for the bare frame (rght) Fnally, t should be noted that the proposed methodology permts to draw some mportant consderatons regardng the performance of the system before and after the retroft. In fact, by drectly comparng the sngle component fraglty curves to each other and to the system fraglty curve, t s possble to evaluate the most vulnerable components and ther contrbuton to the system vulnerablty. Ths comparson permts to understand the changes n the response and n the falure modaltes of the frame due to the retroft. 3. RETROFITTING OF RC FRAME WITH ELASTO-PLASTIC BRACES A three story RC moment resstng frame buldng s consdered as case study. The buldng has been desgned for gravty loads only and wthout any sesmc detalng, applyng the desgn rules exstng before the ntroducton of modern sesmc codes. The consdered frame of the buldng s a three stores 3.66 m hgh and three bays, 5.49 m wde. Columns have a 3 3 mm 2 square secton whle beams are mm 2 at each floor. Grade 4 steel (f y = 276 MPa) and concrete wth compresson resstance f c = 24 MPa, were employed n the desgn. Fg. 3. shows the general layout of the structure and the poston of the braces. The complete detalng may be found n Bracc et al. 995.

5 B3- B3-2 B3-3 C3- C3-2 D-3 C3-3 C3-4 B2- B2-2 B2-3 C2- C2-2 D-2 C2-3 C2-4 B- B-2 B-3 C- C-2 D- C-3 C m 3.66 m 3.66 m a) 5.49 m each span Fgure 3.. General layout of the structure and braces arrangement (adapted from Bracc et al. 995) A two-dmensonal fnte element (FE) model of the structure s developed n OpenSees (McKenna et al. 26). Extended expermental results are avalable for a :3 reduced scale model of the frame and of ts subassemblages (Bracc et al. 995). The expermental nformaton nclude the results of quasstatc lateral load tests of columns and beam-column jont subassemblages and shakng table tests of the whole frame. The FE model s valdated by comparng the expermental results wth the smulated test results of the :3 scale numercal FE models showng good agreement at global and local scale. The retroft desgn method s based on the pushover analyss of the exstng frame under a dstrbuton of forces correspondng to ts frst vbraton mode. The stffness of the dsspatve braces s dstrbuted at each story ensurng that the frst modal shape of the bare frame remans unvared after the retroft. The strength dstrbuton of the dsspatve braces ams at obtanng smultaneous yeldng of the devces at all the stores n order to mantan a smlar deformaton also n the post-elastc range. The nterested reader s referred to Dall Asta et al. 29 for a more detaled descrpton. Fg. 3.2a shows the pushover curve obtaned for the load dstrbuton relatve to the frst vbraton mode of the bare frame (mass partcpaton factor of 86.4%). The ultmate capacty of the frame members s evaluated by consderng the stran demand n the most crtcal concrete and steel fbres (ε c and ε s ) and the correspondng lmts ε cu =.35 and ε su =.4 accordng to Eurocode 8. The other falure modes reported n Secton 2 are not montored n the applcaton of the desgn procedure. The top story dsplacement d =.2 m denotng the falure of the most crtcal element (columns C-2) s posed n evdence n Fg. 3.2a. It corresponds to a maxmum nterstory drft of about.%, and to a base shear capacty V f = 86 kn and t s assumed as the ultmate dsplacement d u n the desgn procedure. Obvously, after ths frst falure, the bare frame stll possess a resdual capacty and can be pushed up to a top story dsplacement d =.83 m, at whch all the base story columns fal (Fg. 3.2c). a) Base shear [kn] Bare frame α =.4 α =.6 α = 3.2 d=.2m d=.83m Dsplacement [m] b) Yelded sectons Faled sectons c) Faled sectons Fgure 3.2. a) Pushover curves for bare and retroftted frame, b) mappng of plastc hnges at d=.2m, and c) mappng of plastc hnges at d=.83m The dsspatve devces adopted n ths case are BRBs typcally descrbed by an elasto-plastc behavor (Zona and Dall Asta 22). Dfferently from those commonly used n steel-structures, the dsspatve

6 devces employed here are qute short, n order to obtan low yeld dsplacements. Thus, the dsspatve dagonal brace s made by assemblng the BRB n seres wth an elastc brace characterzed by an adequate over-strength. The ductlty capacty μ u of the BRBs s assumed equal to 5, whle the ductlty capacty of the whole brace μ du s assumed equal to 2 n order to obtan adequate dmenson of the elastc braces. The bare frame s retroftted by nsertng a bracng system desgned for several retroft levels, measured by the rato α between the base shear capacty of the bracng system V d and that of the bare frame V f. Parameter α assumes dscrete values n the range from (bare frame) to 3.2. Fg. 3.2a reports the pushover curves for all the α values. In Tab. 3., the axal yeld force F d and elastc stffness K d of the dsspatve braces are gven for three retroft levels consdered. Tab. 3. also reports the fundamental vbraton perods for each retroft level consdered, calculated by consderng an effectve stffness of the RC frame elements. Table 3.. Dsspatve braces propertes at each story α=.4 (T=.67 sec) α=.6 (T=.44 sec) α=3.2 (T=.32 sec) Story F d [kn] K d [kn/m] F d [kn] K d [kn/m] F d [kn] K d [kn/m] VULNERABILITY ASSESSMENT For the purpose of developng fraglty curves, a number of 3 natural g.m. records are selected from the European database. These records are chosen n a range of magntude and source to ste dstance of and km respectvely and are compatble wth the type hazard spectrum gven n Eurocode 8, wth sol type D (S =.35) and peak ground acceleraton a g =.Sg. In order to perform IDA, the records are scaled to the same value of the spectral acceleraton at the fundamental vbraton perod of the system S a (T). It s noteworthy that the vbraton perod, and consequently the IM vary wth α and thus, the g.m. records are re-scaled for each value of α..6 C-2. C-2.4 εc [ ] εs [ ] J- J- 3 σc [MPa] 5 σt [MPa] Fgure 4.. Demand samples and correspondng capacty lmts for the case of bare frame

7 .6.4 C D- εc [ ].2 µd [ ] Fgure 4.2. Demand samples and correspondng capacty lmts for α=.6 The dynamc analyses have been carred out on the numercal model developed n OpenSees and descrbed n Secton 3. For each record, for each IM value and for each element of the frame, the maxmum-over-tme values of the EDPs lsted n Secton 2 have been recorded. The maxmum-overtme values of the tenson (σ t ) and compresson (σ c ) stresses and ther capacty at each jonts of the frame have been calculated through the formulas reported n Lupo et al. 22. Coherently wth the capacty lmts assumed n the retroft desgn procedure, the lmts of the concrete and steel capacty are set equal to ε cu =.35 and ε su =.4, whle the elements shear resstance V u s evaluated accordng to the formulas reported n Lupo et al. 22. Fg. 4. and Fg. 4.2 report the results of IDA, expressed n terms of varaton wth IM of the montored EDP samples and ther capacty. In Fg. 4., the samples of the maxmum-over-tme values of the concrete compressve stran ε c and steel stran ε s at the most crtcal secton of C-2 are llustrated, for the case of the bare frame. The correspondng capacty lmts are also reported. In the same fgure, the values of σ c and σ t recorded at jont J- are also reported and compared wth the correspondng capacty lmts. Fg. 4.2 plots the values of ε c at column C-2 and the maxmum-over-tme value of the ductlty µ d experenced by dsspatve brace D- at the base story, for the case of retroftted frame wth retroft level α=.6. a).8 b).8 α=.4 α=.6 Pf.6.4 P f concrete P f steel P f jont n tenson.2 P f jont n compr. P f shear Pf.6.4 α=3.2 System.2 C-2 C-3 D Fgure 4.3. a) Lognormal fraglty curves for the dfferent falure modes and b) Fraglty curve of the system and of the most vulnerable components for three retroftted cases. The component fraglty curves are evaluated for each LS and for each frame member by comparng the demand samples wth the correspondng capacty lmts. Then, the system fraglty curves are derved by assumng a seres arrangement of the component fragltes. Fg. 4.3a reports the lognormal component fraglty curves for the case of the bare frame. It s observed that jont falure n tenson s the most crtcal LS. However, ths LS provdes only a measure of the damage of the jonts due to the concrete degradaton and t s not deemed as crtcal as the brttle falure of the jont n compresson. Hence, t s dsregarded n developng the system fraglty curve and therefore concrete crushng n compresson (LS) s the most crtcal falure modalty, whle steel rupture (LS2) s much less

8 probable and falure of jonts n compresson and shear falure have a zero probablty of occurrence. Fg. 4.3b shows the fraglty curves of the most vulnerable elements and of the system for three retrofttng levels correspondng to α=.4, α=.6 and α=3.2. The most vulnerable components of the bare frame are column C-2 and C-3, falng n concrete crushng mode (LS) and exhbtng a smlar vulnerablty. For α=.4 the vulnerabltes of the two columns reman comparable to each others, and also smlar to the vulnerablty of the most crtcal dsspatve brace (D-). Ths confrms the relablty of the smplfed desgn procedure, whch has the two man ams of avodng drastc changes to the nternal acton dstrbuton n the frame and of achevng a smultaneous falure of both the frame and the braces. Also n the case correspondng to α=.6, the fraglty curves of the most crtcal frame components and of the most crtcal dsspatve brace are very close. However, column C-2 s more vulnerable than C-3. Ths can be attrbuted to the bracng system confguraton, whch nduces a hgher axal load on column C-2 wth respect to C-3. The trend s confrmed by the results of the case correspondng to α=3.2, where the fraglty curve of column C-2 dffers sgnfcantly from the others and tends to concde wth the system fraglty curve. Ths means that system falure s manly due to C-2 column falure, as consequence of the excessve axal force transmtted by the bracng system on ths column. Pf a) b) α=. α=.4 α=.6 α= m6, m5, m m 6 m 5 m α Fgure 4.4. a) System fraglty curves for the bare frame and for the retroftted frame, and b) varaton wth α of the factors m 5, m 84 and m 6..5 a) b).4 Bare frame Retroftted frame 5 4 m 5 m 5,IDR m 5,TSD βc.3.2 m α α 3.2 Fgure 4.5. Varaton wth α of a) dsperson measure β c and b) factor m 5 correspondng to the use of dfferent local and global EDPs. Fg. 4.4a compares the system fraglty curves for all the retroft levels consdered. Parameter IM c,5 ncreases for ncreasng values of α, as expected. However, as already stressed prevously, ths parameter does not drectly provde nformaton about the effectveness of the retroft, snce the natural perods of the systems are dfferent. Fg. 4.4b reports the factors m 5, m 84, and m 6, whch have been defned n Secton 2 n order to compare the retroft effectveness when a structural dependent IM s used. It s observed that for values of α up to.6, the capacty margn rato ncreases about lnearly wth α, whle for hgher values ths relaton becomes strongly non-lnear. Ths mples that the

9 effectveness of the retroft ncreases weakly for values of α larger than.6, n consequence of the premature falure of column C-2 manly due to the hgh axal forces nduced by the braces. Fg. 4.5a plots the dsperson measure β c evaluated accordng to Eqn. 2. for ncreasng values of α and shows that a sgnfcant ncrease of the dsperson occurs when elasto-plastc braces are ntroduced nto the bare frame. Ths s consequence of the ncrease of the number of the vulnerable components (frame members and dsspatve braces) and of the more pronounced nonlnear behavor nduced by the ntroducton of BRBs. Accountng for ths ncrease of dsperson s mportant due to ts nfluence on the estmate of the sesmc rsk. Fnally, n order to quantfy the dfferences n the retroft effectveness evaluaton when local and global EDPs are used, system fraglty curves are evaluated also by consderng global EDPs, such as the maxmum nterstory drft (IDR) and the top story drft (TSD). Fg. 4.5b reports the comparson between the values of prevously defned parameter m 5 and the values of parameters m 5,IDR and m 5,TSD evaluated on the bass of the fraglty curves developed by consderng the IDR and the TSD respectvely. In order to make ths comparson, the global EDPs lmts IDR u and TSD u are chosen so that IM c,5 = IM c,5;idr (m 5,IDR =) and IM c,5 = IM c,5;tsd (m 5,TSD =) for the case of bare frame. The lmts obtaned are IDR u =.32% and TSD u =.29%. It s evdent from Fg. 4.8b that the use of global EDPs nstead of more accurate local EDPs results n a sgnfcant overestmaton of the sesmc ncrement capacty of the retroftted frames, especally for large α values. In fact, as already dscussed n the ntroducton, local phenomena such as the ncrement of axal force n the columns adjacent to the dsspatve braces are not accounted for by these global EDPs. Ths confrms that local EDPs must be adopted to accurately estmate the effectveness of the retroft based on dsspatve braces. Otherwse, f global EDPs are consdered, proper lmts need to be estmated for each retroft level. 4. CONCLUSIONS The paper llustrates a probablstc methodology for assessng the vulnerablty of RC buldngs wth lmted ductlty capacty and the effectveness of the retroft by means of dsspatve braces. The methodology s based on the development of fraglty curves of the bare and the retroftted frames. It employs an effcent structure-dependent ntensty measure (IM) and nvolves performng non lnear IDA to account for the randomness of the earthquake exctaton. Local EDPs are used to capture the modfcatons of the frame response nduced by the ntroducton of the bracng system. Numercal fraglty curves are derved by comparng the samples of the demand wth the correspondng capacty lmts. The component fraglty curves are bult for each sngle structural component and for each sngle LS consdered. The system fraglty curves are derved by assumng a seres arrangement of the component LSs. Fnally, proper synthetc parameters descrbng the system fraglty curves are ntroduced n order to accurately compute the ncrement n the safety acheved by the retroft whle employng a structure dependent IM, such as the spectral ntensty at the natural perod of the structure. The capablty and effectveness of the proposed methodology s tested by consderng a realstc benchmark RC frame wth lmted ductlty capacty retroftted by elasto-plastc braces. The braces are desgned by applyng a wdespread method based on an equvalent nonlnear SDOF approxmaton and by consderng dfferent values of the shear capacty of the bracng system. On the bass of the analyss of the results, the followng conclusons can be drawn. The comparson of the sngle components fraglty curves permts to ndvduate the most vulnerable elements of the frame that may change by ncreasng the retroft level. In the case study consdered, these elements concde wth the two columns nvolved n the bracng system, falng n concrete crushng mode. However, for low retroft levels, the fraglty curves of these columns are very smlar to each other and they are also smlar to the fraglty curve of the most crtcal dsspatve brace, whereas for large retroft levels the fraglty curve of the most compressed column sgnfcantly dffers from the other fraglty curves and tends to concde wth the system fraglty curves. Ths s a consequence of the very dfferent axal

10 load nduced n the columns by the braces acton n the case of hgh retroft level. In order to compute accurately the ncrement of capacty due to retroft for ncreasng values of α (retroft levels) whle usng a structure-dependent IM, parameter m 5 s ntroduced. The values assumed by m 5 for all the retroft cases consdered demonstrate that the effectveness of the retroft system ncreases only weakly for hgh retroft levels n consequence of the prevously descrbed varatons n the component vulnerablty. Moreover, the response dsperson, evaluated by means of parameter β c, sgnfcantly ncreases when elasto-plastc braces are ntroduced nto the bare frame. Ths result s mportant snce the sesmc rsk estmate s strongly affected by the dsperson of the system response. Fnally, the capacty margn rato evaluated by usng local EDPs for montorng the system response s compared wth the capacty margn ratos evaluated by montorng the response through global EDPs such as the maxmum nterstory drft and the top story drft. The results obtaned show that the use of these global EDPs results n a sgnfcant overestmaton of the sesmc ncrement capacty due to dsspatve braces acton, especally for large retroft levels. Thus, t s concluded that the accurate estmaton of the effectveness of the retroft by means of dsspatve braces should be carred out by employng local EDPs capable of accountng for local phenomena. REFERENCES Bracc, J.M., Renhorn, A.M., Mander, J.B. (995). Sesmc resstance of renforced concrete frame structures desgned for gravty loads: performance of structural system. ACI Structural Journal. 92:5, Dall'Asta, A., Ragn, L., Tubald, E., Fredd, F. (29). Desgn methods for exstng RC frames equpped wth elasto-plastc or vscoelastc dsspatve braces. Proceedngs of XIII Natonal Conference ANIDIS. Fredd, F. (22). Local engneerng demand parameters for sesmc rsk evaluaton of low ductlty renforced concrete buldngs. Ph.D. Thess, Polytechnc Unversty of Marche. Güneys, E.M., Altay, G. (28). Sesmc fraglty assessment of effectveness of vscous dampers n R/C buldngs under scenaro earthquakes. Structural Safety. 3:5, Hueste, M.D., Ba, J.W. (26). Sesmc Retroft of a Renforced Concrete Flat-Slab Structure: Part II - Sesmc Fraglty Analyss. Engneerng Structures. 29:6, Katsanos, E.I., Sextos, A.G., Manols, G.D. (29). Selecton of earthquake ground moton records: A state-ofthe-art revew from a structural engneerng perspectve. Sol Dynamcs and Earthquake Engneerng. 3:4, Kwon, O.S., Elnasha, A. (26). The effect of materal and ground moton uncertanty on the sesmc vulnerablty curves of RC structure. Engneerng Structures. 28:2, Lel, A.B., Haselton, C.B., Deerlen, G.G. (2). Sesmc Collapse Safety of Renforced Concrete Buldngs: II. Comparatve Assessment of Non-Ductle and Ductle Moment Frames, Journal of Structural Engneerng. 37:4, Lupo, G., Lupo, A., Pnto, P.E. (22). Sesmc rsk assessment of rc structures wth the 2 SAC/FEMA" method. Journal of Earthquake Engneerng. 6:4, McKenna, F., Fenves, G.L., Scott, M.H. (26). OpenSees: Open system for earthquake engneerng smulaton. Pacfc Earthquake Engneerng Center, Unversty of Calforna, Berkeley, CA. Özel, A.E., Güneys, E.M. (2). Effects of eccentrc steel bracng systems on sesmc fraglty curves of mdrse r/c buldngs: a case study. Structural Safety. 33:, Padgett, J.E., Des Roches, R. (28). Methodology for the development of analytcal fraglty curves for retroftted brdges. Earthquake Engneerng and Structural Dynamcs. 37:8, Ramamoorthy, S., Gardon, P., Bracc, J. (26). Probablstc Demand Models and Fraglty Curves for Renforced Concrete Frames. Journal of Structural Engneerng. 32:, Soong, T.T., Spencer, B.F. (22). Supplemental energy dsspaton: state-of-the-art and state-of-the-practce. Engneerng Structures. 24:3, Vamvatskos, D., Cornell, C.A. (22). Incremental dynamc analyss. Earthquake Engneerng and Structural Dynamcs. 3:3, Zona, A., Dall Asta, A. (22). Elastoplastc model for steel bucklng restraned braces. Journal of Constructonal Steel Research. 68:,8-25