Seismic Retrofit of Bridge Steel Truss Piers Using a Controlled Rocking Approach

Transcription

1 Seismic Retrofit of Brige Steel Truss Piers Using a Controlle Rocking Approac Micael Pollino 1 an Micel Bruneau 2 Abstract: Tis paper investigates a seismic retrofit tecnique tat allows brige steel truss piers to uplift an rocn teir founation. Displacement-base passive energy issipation evices buckling-restraine braces, or BRBs are implemente at te uplifting location to control te rocking response wile proviing aitional energy issipation. Te ysteretic beavior of te controlle rocking system is evelope for a static cyclic loa applie to te top of a brige pier, representing te ominant moe of vibration. Some existing metos of analysis are consiere in preicting te response of te controlle rocking system in terms of maximum isplacements. A capacitybase esign proceure is establise for sizing te BRBs an a esign example provie to illustrate te key steps. Metos to preict esign response values isplacements, velocity, forces are iscusse, an a parametric stuy, base on nonlinear time istory analysis, is performe to verify te effectiveness of tese metos. Te parameters in te stuy inclue te pier aspect ratio /, te local strengt ratio L an an effective perio of vibration T eff. Results of te stuy are presente as normalize by te esign response values an are sown, in almost all cases, to be conservative. DOI: / ASCE :5 600 CE Database subject eaings: Brige piers; Briges, steel; Seismic effects; Retrofitting; Energy issipation; Trusses. Introuction 1 P.D. Caniate, Dept. of Civil, Structural, an Environmental Engineering, Univ. at Buffalo, Buffalo, NY mpollino@ eng.buffalo.eu 2 Director, MCEER, an Professor, Dept. of Civil, Structural an Environmental Engineering, Univ. at Buffalo, Buffalo, NY bruneau@buffalo.eu Note. Discussion open until February 1, Separate iscussions must be submitte for iniviual papers. To exten te closing ate by one mont, a written request must be file wit te ASCE Managing Eitor. Te manuscript for tis paper was submitte for review an possible publication on Marc 4, 2005; approve on September 18, Tis paper is part of te Journal of Brige Engineering, Vol. 12, No. 5, September 1, ASCE, ISSN /2007/ /$ Recent eartquakes, suc as te 1989 Loma Prieta, 1994 Nortrige, an 1995 Kobe eartquakes, ave emonstrate te nee for improve metos for te esign an construction of igway briges to witstan seismic force an isplacement emans. Span collapses or significant amage from tose eartquakes particularly Kobe ave left many large lifeline briges unusable until repairs were mae, contributing to substantial losses to te local economy. Higway briges eeme critical in te response an recovery efforts following a major eartquake also known as lifeline briges nee to remain operational after an eartquake, requiring te brige to respon in a mostly elastic manner wit little to no resiual isplacements. Steel truss brige piers supporting a slab-on-girer or truss brige exist in nearly every region of te Unite States. Lateral loa-resisting pier elements consisting of built-up lattice-type members wit rivete connections were prevalent at te time of construction of many of tese briges. Tese built-up lattice-type members can suffer global an local buckling, resulting in loss of pier lateral strengt an major structural amage uring an eartquake Lee an Bruneau Furtermore, existing rivete connections an eck iapragm bracing members typically possess little to no uctility Ritcie et al Anoter possible nonuctile failure location is te ancorage connection at te pierto-founation interface. Wile strengtening tese existing vulnerable elements to resist seismic emans elastically is an option, tis meto can be expensive an also gives no assurance of performance beyon te elastic limit. Terefore, it is esirable to ave structures able to eform inelastically, limiting amage to easily replaceable uctile structural fuses, able to prouce stable ysteretic beavior wile protecting existing nonuctile elements. Ieally, it woul also be esirable to prevent resiual inelastic eformations an ave structural systems tat can be self-centering following an eartquake. Releasing of te pier-to-founation ancorage connections tensile capacity or allowing tem to fail woul enable a steel truss pier to rocn its founation, effectively increasing its perio an partially isolating te pier. Aing passive energy issipation evices at te uplifting location woul restrain te uplift isplacements wile proviing aitional energy issipation. Tis retrofit strategy also is avantageous because te location of te pier ancorage tens to be easily accessible compare to oter parts of te brige. Te rocking system escribe as an inerent restoring force, capable of allowing for automatic recentering of te tower, leaving te brige wit no resiual isplacements after an eartquake. A sketc of suc a retrofitte brige pier is sown in Fig. 1. Te rocking of structures uring eartquake excitation as been observe in te past an was first investigate by Housner 1963, wo consiere te response of rigi blocks. Meek 1975 introuce aspects of structural flexibility to te seismic response of single-egree-of-freeom SDOF rocking structures. Te use of rocking structural systems for te seismic esign an retrofit of structures as been examine analytically an experimentally by Kelley an Tsztoo 1977, Priestley et al. 1978, Maner an 600 / JOURNAL OF BRIDGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 2007

2 Fig. 2. Cyclic pusover of controlle rocking brige pier an buckling-restraine brace beavior Fig. 1. Retrofitte brige steel truss pier using controlle rocking approac Ceng 1997, Toranzo et al. 2001, an Miorikawa et al A rocking brige pier concept as been implemente or consiere for te seismic esign or retrofit of a few briges. Te Sout Rangitikei Rail Brige, locate in Mangaweka, New Zealan, was esigne an constructe in te 1970s wit pier legs allowe to uplift uner seismic loas Priestley et al Torsional steel yieling evices were ae to control te amount of uplift. Te Nort Approac of te Lions Gate Brige, locate in Vancouver, Britis Columbia, was retrofitte using a rocking brige pier approac Dowell an Hamersley Flexural yieling steel evices were place at te ancorage interface to provie ysteretic amping an limit te uplifting isplacements. Oter briges tat allow rocking or at least partial uplift of pier legs as a means of seismic resistance inclue te Carquinez Brige Jones et al an te Golen Gate Brige Ingam et al. 1995, bot of wic are locate in California. Wile many types of energy issipation evices exist, te evice consiere ere is te buckling-restraine brace unbone brace, or BRB. A BRB consists of a steel core surroune by a buckling restraining part, allowing te brace to reac full yiel in tension an compression. Te component an system beavior of BRBs as been evaluate by Watanabe et al. 1988, Waa et al. 1989, Watanabe an Nakamura 1992, Hasegawa et al. 1999, Iwata et al. 2000, an Black et al Tis paper escribes te static ysteretic cyclic response of te propose controlle rocking system for te seismic retrofit of brige steel truss piers an investigates existing simple metos of analysis for preicting te response of te system in terms of maximum evelope isplacements. A capacity-base esign proceure is evelope for sizing te BRBs suc tat response meets a set of esign constraints, an an example is presente to illustrate te esign process. A parametric stuy, base on nonlinear inelastic time istory analysis, is performe to investigate system performance an assess te simplifie metos of analysis for preicting te esign response values. Hysteretic Response of Controlle Rocking Brige Pier System Te key parameters for te static cyclic ysteretic response of te controlle rocking brige pier system consiere ere inclue te fixe-base lateral stiffness of te existing steel truss pier, te eigt-over-wit aspect ratio of te pier /, an te crosssectional area, effective lengt, an yiel stress of te BRB A ub,l ub,f yub. Ientical BRBs implemente at te base are consiere ere to beave elastoplastically an are assume to be implemente vertically suc tat tey o not transfer orizontal sear at te base of te pier. Also, te weigt excite by orizontally impose accelerations w an te vertical gravity weigt carrie by a pier w v are assume equal an expresse as w. Te moel consiers motion of te pier in a irection ortogonal to te brige eck an assumes no interaction wit oter piers or abutments troug te brige eck. Te response of an actual brige must consier beavior of te entire brige system in te longituinal an transverse irections, incluing eac pier s an abutment s properties, brige eck properties, an te connection etails between te eck an piers. However, tis will vary significantly from brige to brige, an te purpose of tis stuy was to evaluate te beavior of suc a controlle rocking pier as a component of an entire brige system. Also, te existing ancorage connection is assume to provie no resistance to vertical movement but is able to transfer te orizontal base sear. Te various steps an pysical beaviors tat evelop troug a typical alf-cycle are sown in Fig. 2. By symmetry, te process repeats itself for movement in te oter irection. Transition from first to secon-cycle response occurs wen te BRBs yiel in compression an te braces carry a portion of te weigt after te system comes to rest upon completion of te cycle a penomenon explaine later. First-Cycle Response Te orizontal loa applie at te top of te pier, P, as a function of te lateral isplacement,, before uplift begins, consists of te elastic stiffness of te pier s structural members, efine by P = Uplifting of a tower leg begins wen te restoring moment create by te tributary vertical brige weigt is overcome by te applie moment Position 2 in Fig. 2. Te orizontal force at te point of uplift uring te first cycle is efine by P up1 = 2 w 2 an te isplacement at te point of uplift in te first cycle is efine by 1 JOURNAL OF BRIDGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 2007 / 601

3 up1 = P up1 / 3 After uplift, te BRB attace to te uplifting leg is activate. Te global stiffness is reuce an becomes a function of te pier s lateral stiffness,, an te uplifting BRB s contribution to te orizontal stiffness. Te structural stiffness from uplift to te yiel point Steps 2 to 3 in Fig. 2 is efine ere as te elastic rocking stiffness an is expresse by = k 1 1 r EA ub L ub 4 Te orizontal force at te onset of brace yieling, P y, an tus te structural system yiel strengt is efine by P y = w 2 + A ubf yub = w 2 1+ L 5 were L is efine ere as te system s local strengt ratio equal to L = A ubf yub 6 w/2 Te corresponing system yiel isplacement for te first cycle, y1, is efine as y1 = w + A ubf yub 2 k r = w 1 + L 2 k r 7 Ignoring strain arening in te brace an any secon-orer effects, te system as zero postelastic stiffness an is eforme to its ultimate isplacement u. Metos of preicting te system s ultimate isplacement will be iscusse later. As te orizontal loa is reuce, te pier first respons elastically wit stiffness k r, an te tensile force in te BRB reuces per its initial elastic properties. Te applie lateral loa at te top of te pier at te point of compressive yieling of te brace Point 5 in Fig. 2 is efine by P c = w 2 A ubf yub = w 2 1 L 8 Te corresponing isplacement at tis point is efine as c = u 2A ub F yub 2 F yubl ub E Te BRB isplaces plastically in compression an again is assume to yiel wit no significant stiffness until te uplifte pier leg returns in contact to its support Steps 5 to 6 in Fig. 2.At tis point of contact, te system stiffness is again efine by. 9 Te orizontal force at te onset of uplift can be sown equal to P c efine by Eq. 8 an is efine for te secon an subsequent cycles as P up2 = P c = 1 L w 2 P upl 10 Te yiel isplacement can be expresse as y2 = w 2 1 L + 2 L k r yl 11 Te yiel strengt of te system, P y, is uncange. Te force in te BRB canges from its compressive strengt A ub F yub to tension yieling A ub F yub for te secon an subsequent cycles tat excee eck level isplacement of y2. Note tat te controlle rocking brige pier system consiere evelops a flagsape ysteresis. Tis is ue to te combination of pure rocking response from te restoring moment, provie by te brige eck weigt, an energy issipation provie by yieling of te BRBs. Influence of Secon-Orer Effects on Hysteretic Response Te restoring moment, M r, for base rocking is provie by gravity. As te center of mass isplaces, tis restoring moment reuces as a function of te orizontal seismically inuce isplacement of te brige eck suc tat M r i = w i 2 12 Tis loss in restoring moment can be written in terms of te loss in orizontal base sear as P r i = w 2 i = w 2 w i = P up1 w i 13 Consiering tis effect along wit te strain arening of te BRB, in te form of a postyiel stiffness ratio ub, results in a global postyiel stiffness of k py = ub k ub 2 w 14 Generally, te effect of te strain arening of te BRB is greater tan te effective negative stiffness ue to te nonlinear geometric effect, tus resulting in a positive global postyiel stiffness, k py. For te pier wits an aspect ratios consiere erein representing brige piers, a moestly size BRB can result in a positive global postyiel stiffness. Te metos of analysis propose in tis paper for etermining te maximum isplacement response of te controlle rocking system o not take into account secon-orer effects, assuming tey are negligible. Secon-Cycle Response As a BRB yiels in compression an te pier settles back to its support, te BRB effectively carries a portion of te brige weigt equal to its compressive capacity assume to be A ub F yub. As a result of tis transfer of te gravity loa pat now partially troug te BRBs, a smaller orizontal force is require to initiate uplift, causing an earlier transition from stiffness to te rocking stiffness k r, tus increasing te flexibility an system yiel isplacement from te first-cycle response, as can be seen by te secon-cycle curve in Fig. 2. Metos of Analysis Consiere for Determining Maximum Pier Displacements Metos for preicting system isplacement response base on equal isplacement an equal energy teory Newmark an Hall 1982 ave been use wit reasonable confience for welletaile steel seismic lateral force resisting systems suc as moment resisting frames MRFs, eccentrically brace frames EBFs, an concentrically brace frames CBFs. However, te effectiveness of tese metos wit te flag-sape ysteretic be- 602 / JOURNAL OF BRIDGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 2007

4 avior consiere ere is unknown. Terefore, a few simplifie analysis metos are investigate analytically as part of te parametric stuy presente later for teir effectiveness in preicting te seismic response of a controlle rocking system in terms of maximum isplacements tat can be use later in te esign process of te controlle rocking approac. Te first analysis meto consiere consists of caracterizing system response in a manner similar to te nonlinear static proceure NSP escribe in FEMA 356 FEMA 2000, wile a secon is similar to te nonlinear static proceure for passive energy issipation systems foun in FEMA 274 FEMA Te first meto is typically use for MRFs, EBFs, an CBFs, wile an analysis proceure similar to te secon can be foun in te NCHRP ocument ATC/MCEER Eac of tese proceures is escribe in its respective ocuments; tus only te important parameters from tese proceures relate to te beavior of te controlle rocking system are iscusse ere. Meto 1 Te proceure escribe in te FEMA 356 ocument relies on te calculation of an effective perio of vibration, a series of factors, an a esign spectrum to calculate maximum isplacements. Assuming tat te ysteretic beavior is stable, witout strengt or stiffness egraation; tat P- effects are not significant; an tat te system s effective perio lies in te long perio range, ten eac of te factors is set to unity an te effective perio of vibration is te key parameter. Tis effective perio of vibration, T eff, is efine as T eff =2 m k eff 15 A caracterization of te effective stiffness similar to tat in FEMA 356 for systems tat experience progressive yieling an o not ave a efinite yiel point is use ere an efine as 16 k eff = up2 y2 + k r y2 up2 y2 were all terms ave been efine previously. Te effective stiffness coul also simply be taken as te rocking stiffness, k r Eq. 4, wic represents a lower boun of Eq. 16, tus resulting in an upper boun in te preicte isplacement using tis meto. Meto 2 Te meto propose in te FEMA 274 ocument for te esign of passive energy issipation systems uses spectral capacity pusover an eman curves. Conversion of te eman an capacity pusover curve to spectral orinates is base on moal analysis teory. Te brige piers are assume ere to beave as a single egree of freeom system representing te ominant orizontal moe of vibration. Te ae energy issipation from te BRBs is converte to equivalent viscous amping an te seismic eman curve reuce from te 2% ampe spectrum. For te flag-sape ysteretic beavior of te controlle rocking system, te equivalent viscous amping can be etermine by eff = o + ys 17 were o inerent structural amping assume to be 2% an ys ysteretic amping provie by BRBs uring rocking response. Te ysteretic amping can be approximate by moifying te equivalent amping of a bilinear system wit no strain arening by a factor q ys = q bi = L 2 1+ L 1 y2 u 18 Factors for reucing te spectrum for te ae energy issipation are given in FEMA 274. Te effectiveness of tese metos is presente following te sections on esign as part of te parametric stuy, an te steps of Meto 2 are presente in etail as part of te esign example. Design Constraints for te Controlle Rocking Approac To ensure satisfactory seismic performance of tis retrofit approac, a number of esign constraints must be ientifie an a systematic esign proceure formulate following capacity-base esign principles. Specific esign constraints inclue: 1 pier rift limits; 2 uctility emans on te steel yieling evices; 3 limits to allow for pier self-centering; an 4 maximum evelope ynamic forces witin te pier an founation capacity protection. Metos of etermining response values for esign esign response values are introuce along wit eac constraint. Pier Drift For te purpose of tis stuy an to illustrate ow suc limits woul be consiere, two isplacement limits are arbitrarily impose ere, to prevent excessive P- effects on seismic beavior, an system overturning instability. Aitional limits can be ae on a case-by-case basis, as necessary. A requirement sown to limit P- effects base on te ynamic analysis of SDOF systems wit various ysteretic relationsips is taken from te Recommene LRFD Guielines for te Seismic Design of Higway Briges ATC/MCEER Te limit is given by u P y w 19 were w orizontal reactive weigt of te brige tributary to te pier, pier eigt, an P y lateral strengt of te pier efine by Eq. 5. Te limit to ensure stability against overturning is base on preventing isplacement of te center of mass from exceeing alf of te base wit /2 wit a large factor of safety suc tat u2 20 2FS A factor of safety FS of 5 is conservatively recommene. Te metos of analysis presente previously are use to etermine maximum evelope isplacements. Bot isplacement constraints can typically be satisfie by increasing A ub an/or ecreasing L ub. Ductility Demans on Buckling-Restraine Brace Limits on te inelastic strain emans of BRBs are set to ensure tat tey beave in a stable, preictable manner. Tese limits soul be base on engineering jugment an experimental test ata on te ultimate inelastic cyclic response of suc braces. An JOURNAL OF BRIDGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 2007 / 603

5 allowable strain of 1.5% in te BRB is use ere for a seismic eman representing a maximum creible eartquake MCE wit 3% probability of exceeance in 75 years, as efine in ATC/MCEER Tis is a moest level of strain for most structural steels, an some BRBs ave been sown, troug experimental testing, to evelop twice tis strain level wit very stable ysteretic beavior Iwata et al Assuming a rigi founation, brace elongation ub is equal to te uplifting isplacement of te pier leg, an tis constraint can be expresse in terms of te uplifting isplacement as ub = uplift 0.015L ub were te uplifting isplacement is efine as 21 w 2 1+ L uplift = u 22 were all terms ave been efine previously. Tis constraint can be satisfie most effectively by increasing L ub an can be efine in terms of L ub as L ub u w + A ub F 2 yub Fig. 3. Maximum effective static forces uring rocking response Self-Centering An upper-boun on te BRB strengt is neee to ensure te self-centering ability of te system. Assuming te BRB s beavior to be elastoplastic, of strengt equal to A ub F yub, an ignoring any secon-orer effects, ten limiting te local strengt ratio, L,to less tan 1.0 will satisfy tis constraint. Tus, tis constraint can also be written in terms of BRB area as A ub3 1 2 w F yub 24 Maximum Dynamic Forces in Existing Members an Connections Capacity esign proceures an conservative assessment of maximum force emans are neee to ensure tat nonuctile elements can remain elastic an tat all inelastic action occurs in te specially etaile uctile structural elements. Strengt of existing brige elements will vary from brige to brige, an partial strengtening may be require. A meto is propose ere for creating an effective static base sear tat can be use to evaluate te aequacy of te pier s lateral bracing members, followe by a meto to etermine te ultimate emans place on te pier legs an founation. For eac case, ynamic amplification factors R v an R L are introuce as a result of te excitation of te pier s vertical moes of vibration uring impact to an uplift from te founation. Tis excitation of te mass vertically uring te orizontal rocking motion increases te force emans witin te pier but as little influence on te pier s isplacement response. Te ynamic amplification factors epen on te impulsive nature of te transfer of loas uring pier rocking, an a proceure to calculate tese factors is presente in Pollino an Bruneau Effective Lateral Base Sear Deman Te effective base sear cause by te eartquake is etermine by te static yiel force amplifie to account for te increase eman cause by ynamic effects as a result of uplift from te founation. Tus, te ultimate base sear eman can be expresse as P u = P y R v = w 2 + A ubf yub R v P u,allow 25 were R v ynamic amplification factor for loas uring pier uplifting an P u,allow maximum allowable base sear limite by te strengt of lateral loa-carrying members or connections. Te transfer of lateral loas troug te truss pier is epicte in Fig. 3. Given a maximum allowable base sear P u,allow, tis constraint can be written in terms of BRB cross-sectional area A ub as P u,allow R v w 2 A ub4 26 F yub Limiting te BRB strengt, A ub F yub, to an acceptable level or strengtening of te weak elements along te lateral loa pat increasing P u,allow can satisfy tis constraint. Decreasing R v is also teoretically possible but may be ifficult, requiring significant moification to te existing pier. Pier Leg Demans Te emans on an impacting pier leg inclue a ynamic effect relate to te velocity upon impact followe by te transfer of gravitational an evice forces vertically troug te truss pier to te compresse leg Fig. 3. To conservatively estimate emans, te maximum response of eac action is summe, tus assuming 604 / JOURNAL OF BRIDGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 2007

6 te ynamic effects to be in-pase. Assuming elastic response of pier members consistent wit te esign objectives, te total force in te leg can be written as P u,l = P vo + P wl + P uv 27 were P vo maximum force evelope ue to te vertical velocity of te pier leg upon impact, P wl force evelope in te leg ue to its tributary gravity weigt, incluing ynamic amplification, an P uv force evelope in te leg at te first-tier level of te pier, cause by te orizontal base sear transferre troug te pier iagonals Fig. 3. Te total force evelope in a pier leg is taken ere as equal to P ul = v o m k L R L w w A ubf yub R v were v o impact velocity of te pier leg, k L axial stiffness of a pier leg, an R L amplification factor for te tributary gravity weigt of te impacte pier leg. Summation of all ynamic emans may lea to overly conservative preictions, an an SRSS or CQC combination rule applie to te ynamic effects may be more appropriate. From te perspective of seismic retrofit, te BRB strengt, A ub F yub, an te impact velocity, v o, are te primary parameters influencing emans on te pier legs. Protection of tese elements epens on bot A ub an v o. Assuming tat a value of A ub is establise uring te esign process to satisfy te effective lateral base sear limit Eq. 26 or te self-centering limit Eq. 24, te limiting impact velocity can be written as v o,allow = P ul,allow R L w w A ubf yub R v 1 2 m k L 2 29 were P ul,allow maximum allowable force, controlle by eiter te strengt of te pier leg or founation. Oter limits may nee to be efine to prevent founation settlement an/or oter serviceability requirements. Limiting v o is typically acieve by increasing A ub an ecreasing L ub ; owever, increasing te allowable impact velocity of te pier leg, v o,allow, is most effectively one by ecreasing A ub. For esign purposes, te maximum impact velocity coul simply be etermine from te inelastic pseuovelocity suc tat v o = PS vi Systematic Design Proceure 30 To acieve te esire uctile performance of a retrofitte rocking steel truss brige pier, te BRBs must be proportione to meet te relevant esign constraints. Te key steps of te esign proceure are escribe briefly below 1. Establis seismic eman parameters to construct te esign response spectrum base on site location, soil properties, etc., following ATC/MCEER Determine relevant existing pier properties. Tese values inclue te pier aspect ratio /, te fixe-base lateral stiffness of te pier, te axial stiffness of a pier leg k L, an te orizontal an vertical tributary reactive weigts for te given pier, w an w v, respectively. Te ynamic amplification factors, R v an R L, can be etermine using metos presente in Pollino an Bruneau Also, te lateral strengt of te pier P u,allow an capacity of te pier legs P ul,allow nee to be etermine. 3. Ensure tat pier uplifting an rocking motion will be initiate for te esign seismic eman. First, etermine te spectral acceleration value for te fixe-base pier. If te pier s perio of vibration, T o, is greater tan te caracteristic perio of te spectrum, T s efine in ATC/MCEER 2003, ten te elastic spectral acceleration for te fixe-base pier is given by S a,fixe = S D1 s B L T o 31 Using te spectral acceleration value given by Eq. 31, pier uplifting an rocking motion will be initiate if te following statement is true: S a,fixe g 1 w v 2 w 32 If tis statement is not true, ten uplift will not occur for te given seismic eman. In fact, te value on te left sie of te equation soul be muc greater 2 tan te rigt sie for te rocking approac to be effective. If not, te pier is likely to be relatively squat, an anoter retrofit approac woul likely be more effective Berman an Bruneau Establis limits set by te pier rift, self-centering, an effective lateral base sear constraints, Eqs. 19, 20, 24, an 26 respectively, since tese limits are inepenent of A ub or L ub. 5. Begin sizing of BRBs by assigning a yiel force to te braces A ub F yub to satisfy te effective lateral base sear constraint Eq. 26. If no BRB area can satisfy tis constraint, partial pier strengtening may be require. 6. Design L ub to satisfy te uctility eman constraint Eq. 23. An initial effective lengt can be etermine by estimating te uplifting isplacement from Eq. 22 suc tat w SD1Teffo s 2 L ubo = uplift,o = L 33 wit te initial effective perio of vibration, T effo, set equal to 1.2T o. 7. Establis limits set by te BRB uctility eman an pier leg eman constraints, Eqs. 23 an 29, using te previous iteration s brace imensions. 8. Analyze response of te retrofitte system wit te initially size BRB using a simplifie analysis meto to etermine te esign isplacement response. 9. Evaluate te esign velocity response an etermine if all constraints efine in Steps 4 an 7 are satisfie. 10. Moify te BRB imensions, as necessary, to satisfy constraints. Guiance on varying brace imensions to satisfy eac constraint was provie in te constraints section. JOURNAL OF BRIDGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 2007 / 605

7 Table 1. Controlle Rocking System Example Design iteration Units A ub 2,000 2,000 1,500 mm 2 L ub 1,900 1,900 2,750 mm Constraints u1 Eq mm u2 Eq mm uplift Eq mm A ub4 Eq. 26 2,926 2,926 2,926 mm 2 v o,allow Eq mm/s A ub3 Eq. 24 3,681 3,681 3,681 mm 2 Design response values TH analysis u Fig mm uplit Eq mm v o Eq mm/s P u Eq kn P u,l Eq. 28 3,920 2,262 kn Grapical Design Proceure As as been sown, te esign of retrofitte brige piers using te controlle rocking approac requires tat a number of esign constraints be met to acieve te esire performance. To assist in te esign process, a visual ai is propose to illustrate te ranges of compliance an noncompliance of te esign constraints as a function of two key esign parameters. Te constraints are expresse as bounaries enclosing a solution space were all constraints are satisfie, an tus a range of esign solutions can be foun. A similar approac as been propose by Sarraf an Bruneau Te two key esign parameters use ere are A ub an L ub. Eac bounary line is efine wit A ub as a function of L ub.in some cases te constraint bounary lines can be simply efine algebraically, suc as te effective lateral base sear Eq. 26 an self-centering limit Eq. 24 constraints. However, all oter constraints are epenent on te ultimate eck-level isplacement, u, wic itself is epenent on A ub an L ub an is etermine using analysis Meto 2; u cannot be efine algebraically in terms of A ub an L ub, but it can be etermine iteratively for eac pair of esign parameters. Te grapical proceure will be presente, along wit te step-by-step approac presente previously, in te esign example. Fig. 4. Example grapical esign proceure 3,980 kn, respectively. Te ynamic amplification factors, R v an R L, are taken as 1.56 an 1.87, respectively, an ave been calculate using concepts presente in Pollino an Bruneau Te BRBs are assume to be implemente vertically an ave a steel core wit a yiel stress of 235 MPa LYP 235, Nakasima Since te fixe-base perio of vibration, T o, is greater tan T s, te retrofitte brige will ave an effective perio of vibration in te long perio range T s. Terefore, te elastic spectral acceleration for te fixe-base system can be etermine by Eq. 31. Wit te orizontal an vertical reactive weigts assume equal, Eq. 32 is evaluate as g s s g 1 1,730 kn 2 1,730 kn 1 = tus inicating tat uplift an rocking motion will occur. Te BRB area is initially size to satisfy te effective lateral base sear limit Eq. 26, resulting in a value of 2,000 mm 2. Te BRB effective lengt L ub is initially etermine from Eq. 33, resulting in a lengt of 1,900 mm. Initial esign constraints are ten calculate esign iteration 0 ; results are sown in Table 1. Also, Fig. 4 sows te grapical esign meto for te example consiere. Eac constraint bounary line 1 4 is sown an te esign solution space is sae. Te initially selecte brace imensions are marke an sown to lie outsie te solution space. Example To illustrate te above-propose esign proceure, a brige is assume to be locate on site class B an te site coefficients, F a an F v are equal to 1. Te seismic eman is obtaine from a esign spectrum specifie in ATC/MCEER 2003 wit onesecon S 1 an sort-perio S s spectral acceleration values of 0.5 an 1.25g, respectively, corresponing to 5% amping. Tis leas to a caracteristic spectral perio, T s, of 0.4 s, typical of a rock site. Te example uses a pier wit an aspect ratio of 4 =29.26 m, =7.32 m wit orizontal an vertical tributary weigts w an w v equal to 1730 kn. Te pier lateral stiffness is taken as 12.6 kn/mm, an tus te fixe-base perio of vibration T o is equal to 0.74 s. Also, te axial stiffness of a pier leg is taken to be 212 kn/mm, an te lateral strengt of te pier P u,allow an capacity of a pier leg P ul,allow is taken as 605 an Fig. 5. Example capacity spectrum analysis iterations 606 / JOURNAL OF BRIDGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 2007

8 Fig. 6. Deck-level an uplifting isplacement results of example TH analysis Meto 2 is now use to more accurately preict te maximum system isplacement an te uplifting isplacement. As iscusse previously, tis meto uses spectral capacity pusover an eman curves to preict system response. System response for eac esign iteration is sown grapically in Fig. 5. Te initial estimate of maximum system isplacement implicit in Step 6 is sown in te figure as o. Te first iteration using analysis Meto 2, base on te initial brace imensions from te initial analysis, results in a maximum system isplacement of 158 mm an an uplifting isplacement Eq. 22 of 32.9 mm. Te impact velocity is preicte using te system s pseuovelocity. As seen in Table 1, te values of uplifting isplacement an impact velocity o not satisfy teir respective constraints. Following te guiance provie in te constraints section of tis paper to satisfy tese two constraints, L ub is increase to satisfy te brace uctility eman limit, an A ub is ecrease in tis case to increase te allowable impact velocity v o,allow to satisfy te pier leg eman limit. Te grapical esign meto Fig. 4 sows ow te brace imensions nee to be moifie to satisfy te constraints. For te secon iteration, te BRB cross-sectional area A ub is taken as 1,500 mm 2 an effective lengt L ub as 2,750 mm. Te two constraints from Step 7 are recalculate along wit eac response preiction an given in Table 1. Finally, after reanalyzing using Meto 2 an te above properties, all esign Fig. 8. Pier ysteretic beavior uring example TH analysis values are foun to satisfy te specifie esign constraints, an te esign point in Fig. 4 is sown to lie witin te esign solution space. To illustrate te effectiveness of te propose esign proceure, a nonlinear inelastic time istory analysis is performe to calculate te actual response of te retrofitte system. Details of te analytical moel an eartquake loaing use are ientical to tose use for te parametric stuy an are iscusse in te following section. Results are sown in Table 1, confirming tat all esign constraints are satisfie. Te isplacement results u, uplift are sown to be preicte accurately; but te force response values, especially te pier leg axial force P ul, may possibly be preicte as overly conservative an a combination rule may nee to be applie, as note previously. Results for selecte response quantities are given in Figs. 6 an 7 an te resulting pier ysteretic beavior is given in Fig. 8. Fluctuation of te base sear sown in Fig. 8, compare to te iealize flag-sape ysteretic beavior of te controlle rocking system, is ue to te excitation of vertical moes of vibration of te truss uring pier uplift an is accounte for by te ynamic amplification factor, R v, as iscusse previously. Te eck-level isplacement results Fig. 6 sow te system returning to its original uneforme position at te en of te excitation ue to te self-centering ability of te system. Parametric Stuy to Assess Design Response Values Te esign process escribe earlier relies on te preiction of response values neee for te esign of te controlle rocking approac isplacement, velocity, base sear, an pier leg forces of te system to ensure tat tey meet te specifie esign constraints. A parametric stuy using nonlinear inelastic time istory analysis serves ere to assess te accuracy of te preicte esign response values. Te analyses were performe wit te structural Fig. 7. Base sear an leg force results of example TH analysis Table 2. Representative Pier Properties / m m w kn kn/mm T o s k L kn/mm R v R L JOURNAL OF BRIDGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 2007 / 607

9 Fig. 9. Normalize eck-level isplacement results Meto 1 Fig. 10. Normalize eck-level isplacement results Meto 2 analysis program SAP2000 version 7.40 Wilson 2000 wit an equivalent viscous amping of 2% assigne to eac moe. Te parameters ere inclue a set of pier properties, given in Table 2; local strengt ratios, L, varie from zero elastic rocking to te point at wic te self-centering ability is lost L =1.0 ; an an effective perio of vibration caracterize by Eq. 15 wit k eff efine by Eq. 16. Results are presente for aspect ratios of 4, 3, an 2; local strengt ratios L of 0, 0.25, 0.5, 0.75, an 1.0; an tree effective perios of vibration 1.0T o,1.25t o, an 1.5T o, were T o is te pier s fixe base perio of vibration, an are given in Table 2. In te case of L =0, only te initial perio of vibration of te pier is relevant since no BRB is use. Eartquake Loaing Spectra-compatible groun acceleration time istories use for te ynamic analyses are generate using Target Acceleration Spectra Compatible Time Histories TARSCTHS software, wic was evelope by te Engineering Seismology Laboratory ESL at te University at Buffalo an is te implementation of te meto escribe in Deoatis Syntetic groun motions were generate by TARSCTHS matcing an elastic response spectral sape efine by NCHRP ATC/MCEER 2003.A esign 1 s spectral acceleration values S D1 of 0.5g is consiere ere, wit te esign sort perio spectral acceleration value S DS assume equal to 2.5 times S D1. Tis results in a caracteristic perio, T s, of 0.4 s, typical of a rock site. Seven ranomly generate syntetic motions were prouce wit a uration of 15 s Restraints are provie at te ancorage level tat prevent movement in te orizontal irection tus assuming tat sliing is prevente, but provie no resistance to vertical movements. Results an Discussion Te mean result of te seven syntetic groun motions is sown for eac case consiere. A total of 39 cases an 273 analysis were performe. Results are presente for eck-level isplacements u, maximum base sear P u,th, an maximum pier leg axial force P ul,th normalize by teir respective esign response value, for eac system parameter consiere. Displacement esign response values are etermine using analysis metos 1 an 2. Te base sear P u an pier leg axial force P ul esign response values are given by Eqs. 25 an 28, respectively. Deck-Level Displacements Results of te normalize eck-level isplacements are sown in Figs. 9 an 10 for metos 1 an 2, respectively. As can be seen in te figures, Meto 2 is able to more accurately preict isplacements for all ranges of parameters consiere ere. Meto 1 works well for systems wit L 0.5, but for smaller values of L te meto tens to unerpreict te maximum isplacements. Te primary ifference between te two metos is seen at small values of te local strengt ratio, L, were large increases Analytical Moel For te propose controlle rocking brige pier system consiere ere, te piers are excite solely in te orizontal irection. Eac pier is assume to carry an equal inertia mass bot vertically an orizontally. A 2D moel of eac representative truss pier is use wit alf of te mass applie to eac of te top two noes of te truss. Te pier itself is moele wit its elastic properties, an all nonlinear action is moele to occur at te founation interface. A compression-only gap element an a isplacementbase ysteretic element are place in parallel across te ancorage interface, at te base of eac pier leg, to moel te rocking beavior. Te gap elements represent te founation wit no tensile capacity an a large linear-elastic stiffness in compression. Te ysteretic element is base on te moel propose by Wen Fig. 11. Normalize base sear results 608 / JOURNAL OF BRIDGE ENGINEERING ASCE / SEPTEMBER/OCTOBER 2007

10 in isplacements are preicte by Meto 2 ue to te ecrease in system strengt an energy issipation wit small BRB areas. Meto 1 unerpreicts isplacements for smaller values of L, as seen in Fig. 9, especially for te case of te bilinear elastic rocking system L =0. Wile Meto 2 oes not necessarily provie te exact solution, it captures trens in flag-sape beavior tat ave significant influence on response. Tese trens inclue system strengt, initial stiffness, postelastic stiffness, an energy issipating ability, wile Meto 1 is completely epenent on an effective initial elastic stiffness. Base Sear Results Te time istory analysis base sear results, P u,th, normalize by its esign response value, P u Fig. 11, sow tat te preiction of te ultimate base sear, incluing ynamic effects, is conservative for L 0.25, except for a pier aspect ratio of 2 wit L =0.25. It is also unconservative for te case of bilinear, elastic rocking i.e., L =0. Pier Leg Force Results Te time istory results of forces evelope in te pier legs, P ul,th, normalize to its esign response value, P ul efine by Eq. 28, are sown in Fig. 12. Te conservative assumptions mae in te erivation of Eq. 28 namely te in-pase response of all ynamic effects uring impact an uplift, resulte in conservative estimates of te pier leg emans in all cases consiere ere. Conservative preictions of te emans to te pier legs are key, given tat te pier legs resist te pier s gravity loa. Conclusions Fig. 12. Normalize pier leg axial force results Tis paper investigate a seismic retrofit tecnique tat allows brige steel truss piers to uplift an rocn teir founation, wit passive energy issipation evices BRBs use to control te rocking response wile proviing aitional energy issipation. Tis controlle rocking system as an inerent restoring force tat can be esigne to provie pier self-centering an leave te brige wit no resiual isplacements following an eartquake. Te ysteretic beavior of te controlle rocking system wit isplacement-base, steel yieling evices BRBs implemente at te ancorage location an te cange in te beavior uring cyclic loaing secon-cycle response ave been presente. A capacity-base esign proceure for te controlle rocking approac is propose. A set of esign constraints is establise to acieve uctile seismic performance, wic inclues pier rift, uctility emans on te BRB, self-centering, an maximum evelope force limits. Metos to etermine esign response values isplacements, velocity, forces are also presente. A esign example is provie to sow te key steps in te esign proceure. A series of iterations is performe an a set of BRB properties selecte to satisfy all esign constraints. Time istory analysis is performe, using te example s pier properties an te final selecte brace imensions, an te response is sown to satisfy all esign constraints. Te propose metos for etermining esign response values are evaluate troug a parametric stuy base on nonlinear inelastic time istory analysis. Te analysis results are presente as normalize to te esign response values. Results relevant for capacity esign base sear an pier leg forces were foun to be conservative in almost all cases consiere. Acknowlegments Tis researc was supporte in part by te Feeral Higway Aministration uner Contract No. DTFH61-98-C to te Multiisciplinary Center for Eartquake Engineering Researc. However, any opinions, finings, conclusions, an recommenations presente in tis paper are tose of te writers an o not necessarily reflect te views of te sponsors. References ATC/MCEER NCHRP recommene LRFD guielines for te seismic esign of igway briges. I: Specification. ATC/ MCEER Joint Venture, Buffalo, N.Y. Berman, J. W., an Bruneau, M Approaces for te seismic retrofit of brace steel brige piers an proof-of-concept testing of an eccentrically brace frame wit tubular link. 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