THEORETICAL DEVELOPMENTS AND NUMERICAL VERIFICATION OF A DISPLACEMENT-BASED DESIGN PROCEDURE FOR STEEL BRACED STRUCTURES

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The 4 th Wod Confeene on athquake ngneeng Otobe 7, 008, Bejng, Chna THORTICAL DVLOPMNTS AND NUMRICAL VRIFICATION OF A DISPLACMNTBASD DSIGN PROCDUR FOR STL BRACD STRUCTURS G. Dea Cote, F.M. Mazzoan Assstant Pofesso, Dept.

1 The 4 th Wod Confeene on athquake ngneeng Otobe 7, 008, Bejng, Chna THORTICAL DVLOPMNTS AND NUMRICAL VRIFICATION OF A DISPLACMNTBASD DSIGN PROCDUR FOR STL BRACD STRUCTURS G. Dea Cote, F.M. Mazzoan Assstant Pofesso, Dept. of Stutua ngneeng, Unesty of Napes Fedeo II, Napes. Itay ma: gdeao@unna.t Fu Pofesso, Dept. of Stutua ngneeng, Unesty of Napes Fedeo II, Napes. Itay ma: fmm@unna.t ABSTRACT : The pape dsusses onepts and pesents poedues fo the deeopment of a dspaementbased desgn (DBD) methodoogy fo stee baed stutues. One bas onept n DBD s the ndependene of yed defomatons of stutua membes fom the stength. Ths ndependene aows the yed defomaton to be omputed befoe seetng membe oss seton popetes,.e. at the statng phase of the desgn. Usng appopate substutung tehnques, yed dspaements of the whoe stutue an aso be omputed ndependent of membe stength. In addton, postyed dspaements an be omputed befoe membe oss setons hae been ompetey detaed. Ths ony eques that the utmate mt state fo the whoe stutue (e.g. the aheement of a mtng dutty n one o moe membes) s eay dentfed. One neast taget dspaements hae been omputed, eathquake engneeng methods an be apped to ompute the stength equed n ode that dspaements ae not exeeded unde the desgnee eathquake. The ase of neted Vbang s taken as ase study, beause of spef featues emphaszng pobems that oud be enounteed n the appaton of the new methodoogy. Both stat and dynam neast esponse anayses unde a set of seeted aeeaton eods hae been aed out, wth efeene to a 0stoey fame. Resuts ae ey enouagng about the potentates of the noe methodoogy, showng that maxmum dspaements, dfts and dutty demand ae wthn the mts mposed at the desgn stage. KYWORDS: Capaty desgn, dspaementbased desgn, dynam tmehstoy anayss, neted Vbang, pushoe anayss, stee stutues. INTRODUCTION Sesm desgn of stutues s uenty odfed by stutua Codes and Standads of pate, usng a soaed foebased appoah. In ts most bas fom, ths appoah does not sgnfanty dffe fom the desgn method used fo any othe extena aton (suh as gaty oads and wnd oads). In fat, eathquake atons ae smuated by means of equaent stat foes, whose ntensty and dstbuton s fxed by the Code as a funton of () the eathquake shakng ntensty at the ste of nteest and () the stutua type (fo exampe, baed stutues ae assgned dffeent foes than moment esstng fames). In ths poedue, the stutue dspaements ae ony the fna output of the desgn poess, to be auated one a the membe oss setons hae been fxed. As t s weknown, eathquakendued foes an be appoxmated by means of quaton (Chopa 995): ( ) F = m u u V = mφ V m u u mφ N b N b ( ) () whee F s the eathquake foe at the th foo ee, m and u ae mass and dspaement at the th foo, u s a efeene dspaement (e.g. the oof dspaement) and V b s the equed base shea stength. quaton says that eathquake foes ae appoxmated by the podut of two tems: ) a heghtwse dstbuton fato and ) the base shea. The heghtwse dstbuton fato depends on the shape of atea foo dspaements ( φ = u u ). quaton eay shows that dspaement shapes ( φ = u u ) ae needed to defne foes. Besdes, basay statng

2 The 4 th Wod Confeene on athquake ngneeng Otobe 7, 008, Bejng, Chna fom the appoxmaton that one mode of baton s domnant (whh s mpt n usng quaton ), poposas hae been made fo substtutng the ea mutstoey budng wth a substtute snge degee of feedom stutue (Shbata and Sozen, 976). In ths ontext, dspaementbased desgn (DBD) methodooges hae been poposed, amng at an expt and asebyase defnton of the base shea stength equed to meet a seeted sesm pefomane objete (Pestey et a., 007). In the DBD poedues, dspaements ae used as the man nput to the desgn poess, as opposte to the foebased poedues. The dspaement shape ( φ = u u ) s usuay fxed usng some mxtue of theoy and empa knowedge of the spef stutua type unde anayss. Fequenty, the dspaement shape s deed based on the statsta haatezaton of the esponse of a age numbe of typa fames. A dspaementbased desgn method nopoatng defnton of the dspaement shape dety n the desgn poess s dsussed n ths pape. The method s based on sound mehans of east stutues, but ams at ontong both the east and neast defomaton pattens. Ths s aheed by peseetng the neast pats of the stutue and eognzng that the emanng pats of the stutue an st be deat wth as noma east stutues subjeted to mposed dspaements at the onneton ponts between east and neast egons. The ase of neted Vbang s taken as ase study, beause of spef featues emphaszng pobems that oud be enounteed n the appaton of the new methodoogy.. BASIS OF TH DSIGN MTHOD.. Genea onsdeatons Fgue a ustates a shemat epesentaton of bakbone esponse ues fo stee baes. N p s the axa foe oespondng to a fuy yeded oss seton. χ s the eduton fato fo stength n ompesson, n ase of fst oadng. Vaues of χ ae gen by stutua odes as funton of the nomazed sendeness λ. χ ges the esdua stength n ompesson, afte age ompesson defomaton and t may be assumed n the fom χ = a+ bλ t (Tembay, 00). χ s an ntemedate aue between χ and χ, and t s aso funton of the axa shotenng of the bae. h p s a oeffent aountng fo maxmum stan hadenng of the bae n tenson (Tembay, 00). Δ y ndates the axa eongaton oespondng to N p, whe Δ s the neast defomaton apaty of the bae. Tembay (00) ges nfomaton about the dutty aaabe n the ast defomaton exuson befoe faue μf = + dλ. Assumng a symmet y oadng hstoy, the monoton dutty apaty, takng nto aount y degadaton effets, may onseatey be estmated as equa to Δ Δ = μ. y F A typa pushoe esponse of a stutue wth sende baes s ustated n Fgue b. The foowng mt states may be dentfed:. Compesson bae bukng.. Tenson bae yedng. 3. Compesson bae faue (maxmum aaabe dutty). Fo stoke baes the tanston fom bukng to utmate state s not so smooth as ndated n Fgue b. The thee mt states ae aso ustated n Fgue, n tems of membe foes. The desgn (taget) dspaement, to be eahed unde the desgnee eathquake, may be oated n an ntemedate poston between bae bukng and the utmate state, aodng to the desgn hoe and/o neessty. Fgue 3 ustates a efeene state of the stutue. It s obtaned assumng that one bae at eey stoy s fuy yeded and hadened n tenson, whe negetng the othe (ompesson) bae. Ths s a tua state, whh s found onenent to be ntodued beause beam and oumn foes an be omputed ndependent of the dspaement. It s easy to see that beam bendng moments ae maxmzed n the efeene state ( apaty desgn ). It may aso be poed that ompesson oumn axa foes ae maxma at the ast two stoey ees. Ths s not tue fo owe stoes. Howee, the use of a tenson foe equa to the maxmum aue hn p p,b, at eey stoey patay ompensates fo negetng ompesson bae esdua stength. In fat, n the ea stutue, maxmum oumn axa foes woud be eahed wth ony some (o aso none) of the tenson bae axa foes eahng hn p p,b,, whe the

3 The 4 th Wod Confeene on athquake ngneeng Otobe 7, 008, Bejng, Chna emanng tenson baes beng haatezed by foes smae than hn p p,b, and the ompesson baes by some t esdua stength χ Np. Besdes, the dffeene n oumn axa foes omputed fo the assumed efeene state (Fg. 3) and the theoeta uppe bound stuaton may be sma, beause of sma esdua stength n the buked bae. Theefoe, beause of smpty and onenene n the anayta fomuaton, the sheme of membe foes shown n Fgue 3 w be used as efeene stuaton fo stength desgn of beams and oumns. Anyway, any othe pefeed efeene state (.e. any othe apaty desgn tea) oud be assumed, wthout affetng the DBD desgn poedue whh s outned n the foowng Setons. a) N b) hn N p p p V b Taget pont Utmate state χ N Δ p χ N t p χ N p Δ y Δ Δ Tenson bae yedng Compesson bae bukng u Fgue. Shemat esponse of baes (a) and baed stutues (b). Fst bae bukng Taget (desgn) state Utmate state N t p,b, χ Np,b, χb,np,b, N χ Np,b, b, Np,b, N h N b, p p,b, Fgue. Bukng, taget and utmate state n tems of bae foes. Refeene state hn p p,b, Ths s used as uppe bound to: beam bendng moments (goousy) ompesson oumn axa foes (appoxmatey) Fgue 3. Refeene state... Outne of the desgn poedue Bas onepts behnd the poposed desgn method hae been pesented n Dea Cote (006). The desgn poedue an be summazed wth the foowng st of man steps:. Compute yedng (.e. fst bae bukng) dspaements.. Compute taget neast dspaements. 3. Compute the equed base shea stength at the taget (desgn) state (e.g. by the equaent sous dampng appoah (Pestey et a., 007)). 4. Redue the base shea foe fom the taget to the bukng state.

4 The 4 th Wod Confeene on athquake ngneeng Otobe 7, 008, Bejng, Chna 5. Desgn baes n suh a way that npent bae bukng ous unde the base shea foe omputed at step Desgn oumns and beams usng foes n the efeene state (uppe bound to beam and oumn foes, Fgue 3). Key onepts of the poedue ae befy dsussed heeafte, statng wth a smpe onestoey stutue. Bukng dspaements ( u y ) an be omputed by means of a funton hang as nput aabes the bae and oumn axa stans ( b and, subspt y ndates that the eeant quantty s omputed at the bukng state): u = f(, ) () y b,y,y Anaogousy, a funton of the bae and oumn axa stans, as we as beam defeton ( b ), an ge dspaements of the stutue n the postbukng stage (subspt d ndates that the eeant quantty s omputed at the desgn (taget) state): u = g(,, ) (3) d b,d,d b,d Deaton of quatons () and (3) an be based soey on knemats of the stutue,.e. t does not eque onsdeaton of equbum of foes. It an be shown that an addtona knemats eatonshp an be wtten, eatng the beam eta dspaement to the bae and oumn axa stans n the desgn state: = g (, ) (4) b,d b,d,d One quatons (), (3) and (4) ae known, usng the bae and oumn axa stan as desgn aabes, bukng and taget dspaements an easy be omputed, ndependent of foes. Fo exampe, wth efeene to bukng dspaements, one an fx the bae axa stan equa to the aue oespondng to npent bukng b,y = χby whe takng the oumn axa stan equa to a faton of ts bukng aue,y = ρχ y, wth ρ <. The oeffent ρ must be appopatey and aefuy seeted: n fat, afte bae bukng the base shea neases, thanks to the nease n the tenson bae foe, thus podung an nease n the ompesson oumn foe. Theefoe, the oumn axa stan at bukng must be fxed ookng at the utmate esponse of the stutue. The oeffent ρ annot be goousy defned at the statng stage of the desgn, beause t depends on the atua oss seton gen to the baes. Theefoe, t s athe to be assumed as a easonabe aue and eentuay hanged wth a few teatons on the desgn souton. In ase of mutstoy budngs, t s aso equed to fx the heghtwse dstbuton of baes and oumns axa defomatons. Futhemoe, when gong fom dea stuatons, wth ony eathquake oads, to moe genea ases, wth aso eta oads taken nto aount, one addtona dffuty s enounteed. Veta oads may podue nta stan (and stess) n the baes. Consequenty, the sequene of bae bukng may be fa away fom the assumpton made onsdeng ony eathquake effets. Ths s beause the ndene of the nta stan due to gaty oads s not unfom oe the heght of the budng, beng age at the uppe stoes whee the eathquake shea effets ae smae. Theefoe, t may be foeseen that a suffenty auate epesentaton of the fame neast esponse must neessay take nto aount the effet of gaty oads. 3. MATHMATICAL FORMULATION 3.. Pebukng dspaements The expt fom of quaton () an be obtaned usng the substutung tehnque ustated n Fgue 4. The stoey dft s obtaned as the supeposton of a gd stoy otaton, whh s due to the axa shotenng and eongaton of oumns beow the stoey unde examnaton, and the effet of bae defomatons.

5 The 4 th Wod Confeene on athquake ngneeng Otobe 7, 008, Bejng, Chna Stoy dft Rgd stoy otaton Bae defomatons α L h Fgue 4. Fame substutung and shematzaton fo auaton of bukng dspaements. Consequenty, the dft oespondng to the fst sgnfant nonnea eent n the baed stutue, ndated heeafte as yed dft, an be expessed as gen by quaton (5): ν,y b,,y θ,y = + (5) L senα whee: b,,y s the bae axa stan at the th stoey, due to eathquake oads; L, and α ae the baed bay ength and the ange of the bae on the hozonta axs, espetey (Fg. 6).,y,j,yhj j= = s the eta dspaement at the base of the th stoey oumn, due to the axa shotenng/eongaton of oumns (,,y s the oumn axa stan at the th stoy, due to eathquake oads, h j s the ntestoey heght, Fg. 4). 3.. Postbukng dspaements In the postbukng ange, the dft neases beause of the nease of oumn and bae axa defomatons, but aso beause of beam fexua defomatons due to the unbaaned eta foe tansmtted by the tenson and ompesson baes (Fg. 5). Stoy dft Rgd stoy otaton Membe defomatons Fgue 5. Fame substutung and shematzaton fo auaton of desgn (taget) dspaements. Usng the sheme of Fgue 5, quaton (6) oud be deed fo the th stoey dft n the postbukng ange: + θ = + + α + α (6) tenson,d,d b,,d b,,d,,d,,d,d tg tg L senα h whee: tenson b,,d s the tenson bae axa stan, at the th stoey;

6 The 4 th Wod Confeene on athquake ngneeng Otobe 7, 008, Bejng, Chna b,,d s the beam mdspan eta defeton, at the th stoey;,,d and,,d ae the ght and eft oumn axa stans, at the th stoey (taken poste f they ae ompesson stans); and,d ae eta dspaements, at the base of the th stoey ght and eft oumn, due to,d eongaton/shotenng of oumns fom the fst to the ()th stoey. and othe symbos hae the meanng aeady deaed. Usng the sheme of Fgue 5, an addtona quaton an be deed, eatng the beam mdspan eta defeton to the baes axa defomatons, as seen nto quaton (7) ( μ s the desgn dutty fo the th ompesson bae):,d tenson b,,d,,d +,,d b,,d + + sen α = μ,dy h (7) 3.3. Reatonshp between desgn and yedng mt state stength At step 4 of the desgn poedue peousy outned, the base shea foe demand auated fo the desgn state must be saed to the fstbukng stuaton n ode to desgn baes and, subsequenty, oumns and beams. The eatonshp between the base shea foe at the desgn stuaton and the same quantty eauated at bukng s gen by quaton (8): V V b,d b,y ( + ) χ Np,b,osα = = ( + χ ) Ω Np,b, osα Ω (8) whee s a desgn paamete goenng the dstane of the taget pont fom the utmate state (Fg. ) and Ω = Np,b, y N = = χ (N b,,y s the st stoey bae axa foe at bukng). b,,y b,,y b, 3.4. Desgn tea The man desgn nput to the DBD poedue s the nomazed sendeness of baes and oumns. Fo baes, a nea dstbuton wth deeasng aues fom the top downwads has been found effete, suh as fo exampe λ b, = a b( N ) whee N s the numbe of stoeys, a and b ae two desgn paametes. The sendeness dstbuton hosen fo the oumns s muh ess ta and a fstta unfom dstbuton may be found appopate. Appaton of the DBD poedue aso eques seeton of desgn aues fo the bae and oumn axa defomatons at the eeant mt states. The assumed stan dstbutons hae sgnfant onsequenes on the fna dspaed shape (q. 5) and onsequent stength assgnments (q. ). One possbe dstbuton, whh has been found effete to get eey bae bukng n ompesson, s gen by quaton (9): = ρ χ = χ (9) b,,y b, b, y b,n y quaton (9), jonty wth the assumed bae sendeness dstbuton, mpes that the fst bae to buke unde the aton of atea foes s that one at the top stoey = χ and subsequenty the othe baes gong fom the b,n,y b,n y top downwads. quaton (9) assumes that the nta bae stan due to gaty oads s a sma faton of the tota bae stan. The oumn defomaton at the bae bukng mt state must be seeted n suh a way to aod oumn bukng at the utmate state. Ths mpes the need to seet aues appopatey sma at the bae bukng state, n ode to pemt the equed stength nease to be deeoped at the utmate state (Fg. b). quaton (0) was found effete.

7 The 4 th Wod Confeene on athquake ngneeng Otobe 7, 008, Bejng, Chna χ = ρ χ = (0) Ω, y,,y,,y, y G+ hp N,,max N,,max ( ) The oumn axa foe due to gaty oads s not known at the statng of the desgn poess. Theefoe, a ta aue must be assumed and then oeted f age aues ae auated. Ths mpes a few teatons n the desgn poess. The taget dspaements an be auated usng quatons (6) and (7), wth the foowng appoxmate and smpfed desgn assumptons fo membe defomatons ( γ bk s a pata safety fato aganst oumn bukng): tenson b,,d = ; y = μ ; omp b,,d d, y χ, y = ; γ (,,max,,max ),,d G+ hp N N bk,,d,,d = 0 fo =, n = fo = n,,d () The most appopate aue to be assgned to the paamete may depend on the spef desgn ase. A fstta auaton wth = s suggested, but thee oud be ases whee a aue smae than s moe appopate. 4. RSULTS FROM AN XAMPL OF APPLICATION 4.. Genea data A tenstoey onebay fame has been desgned. The ength of the bay s equa to 6 m, whe the ntestoey heght s 3.5 m exepton made fo the fst stoy whee t s equa to 4 m. The foo mass s equa to about 55 kns /m at the oof ee, 5 kns /m at othe foos. The desgn spetum s taken fom C8: type, gound type C, PGA = 0.35g. The uopean S 75 stee has been used (expeted aeage yed stess equa to 36 MPa). uopean Ishaped wdefange oss setons hae been used fo beams and oumns, whe ua hoow setons hae been adopted fo baes. The oss seton sze obtaned at eey foo ee fo the fame membes s gen n Tabe. Subsequenty to the desgn, the stutue has been anayzed usng both stat and dynam anayss. In patua, dynam tmehstoes anayss has been aed out usng a set of 7 aeeaton eods, seeted and saed n ode to get an aeage dspaement esponse spetum as ose as possbe to the desgn spetum. Tabe. Coss setons of membes obtaned fo the anayzed ase. Stoey # Coumns Beams Baes Wdefange setons Wdefange setons Cua hoow setons (damete x thkness) 0 H B 80 H B x 4 9 H B 80 H B x 6 8 H B 60 H B x 7 7 H B 60 H B x 9 6 H B 360 H M x 0 5 H B 360 H M x 0 4 H B 550 H M x 0 3 H B 550 H M x 0 H M 550 H M x 0 H M 550 H M x Numea esuts Fgue 6 summazes the esuts of the stat and dynam anayss of the desgned Vbang. Namey, Fgue 6a shows the fst baton mode of the stutue and ompaes t wth the pedton made at the desgn stage (nomazed pebukng dspaements). Fgue 6b shows a sma ompason, but wth efeene to the desgn (postbukng) dspaements. Fgue 6 summazes the peak dspaement demand fom tmehstoy esponse anayss. The aeage dspaement demand s smae than the taget dspaements, beause of two man easons: () the atua stffness and stength of the stutue ae age than the desgn aues beause ommea pofes of membes do not exaty math the desgn output; () the Rayegh modeng of sous dampng, whh has been based on the

8 The 4 th Wod Confeene on athquake ngneeng Otobe 7, 008, Bejng, Chna nta stffness athe than the tangent stffness. The satteng n dspaement esponse s many attbuted to eodtoeod aabty of dspaement speta. Fnay, Fgue 6d ustates the aeage dutty demand to baes togethe wth taget and apaty aues. It s nteestng to note that the stutue oestength has a benefa effet on the mtaton of oa dutty demand due to hghe mode effets. Foo Foo T desgn =.3 s T anayss =.5 s Desgn Anayss a) b) φ Taget 6 RHA Aeage s_r 5 s_r 4 s_r3 3 s_r4 s_r5 s_r6 s_r u (m) ) d) Foo Desgn Anayss u taget (m) 0 Compesson Tenson Taget 3 Capaty RHA μ f Fgue 6. Resuts fom numea anayss of an neted Vbang. Foo 5. CONCLUSIONS A noe methodoogy fo dspaement based desgn of stee baed stutues has been pesented. The man dstnte featue of the poedue s the det auaton of the stutue dspaements oespondng to seeted mt states. Ths s made on the bass of smpe anayta eatonshps between stoey dft and membe defomatons, based on the seeton of the yedng zones. The poposed appoah pemt to onto both the east and neast dspaements. Resuts fom one appaton to a ase study hae aso been shown n the pape. The ompason of the anayta desgnstage pedtons and the numea fnte eement modeng esuts shows that the method may wok popey. Futhe eseah s needed n ode to assess the effet of hghe modes of baton. RFRNCS Chopa, A.K. (995), Dynams of stutues, Pente Ha, Uppe Sadde Re, New Jesey, USA. Dea Cote, G. (006), Vbaton mode s. oapse mehansm onto fo stee fames, Po. of the Fouth Intenatona Confeene on Behaou of Stee Stutues n Sesm Aea (STSSA 006), Yokohama, Japan, 47 August, pp Pestey, M.J.N., Ca, G.M., Kowask, M.J. (007), Dspaement based sesm desgn of stutues, IUSS Pess, Paa. Tembay, R. (00), Ineast sesm esponse of stee bang membes, Jouna of Constutona Stee Reseah, 58,

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