13 th World Conference on Earthquake Engneerng Vancouver, B.C., Canada August 1-6, 004 Paper o. 151 A EECTIVE ETHOD OR DISPLACEET-BASED EARTHQUAKE DESIG O BUILDIGS Jorge GUTIERREZ 1 and aurco ALPIZAR SUARY The Dsplacement-Based Plastc Desgn (DBPD) method, an effectve and ratonal desgn procedure for the desgn of buldngs, s presented. The sesmc demand s represented by constant ductlty nelastc desgn spectra rather than the elastc spectrum wth ncreased dampng to account for nelastc behavor, proposed by ATC-40 and EA 73 onlnear Statc Procedure. The constant ductlty nelastc spectra are represented n a S a -S d plot, wth S a accountng for the pseudo-acceleraton spectra and S d for the correspondng nelastc (or peak) dsplacement spectra. or the desgn process the target ductlty of the structure s selected n terms of a partcular Performance Objectve defned for the buldng, the type of structural confguraton and the local ductlty capacty of ts structural components. Concurrently, the target structural dsplacements are chosen and the correspondng structure sesmc lateral forces are determned from the constant ductlty nelastc desgn spectra correspondng to the target ductlty. Subsequently, the element strengths requred for a selected plastc mechansm are calculated usng Plastc Theory and capacty desgn prncples. The structural desgn s fnally verfed by Capacty-Demand- Dagram ethods. An llustratve example shows excellent agreement between the selected and calculated target ductlty and dsplacements. ITRODUCTIO ost current sesmc codes (IAEE [1]) use force-based desgn procedures. In the defnton of the sesmc lateral forces, these codes accept that buldngs wll deform beyond the lmt of lnear elastc behavor (nelastc response) and use elastc analyss, wth forces derved from an elastc desgn spectrum reduced by a force reducton factor, to account for nelastc behavor. The man contrbutors for the force reducton factors are the expected ductlty and overstrength of the structure. The ductlty accounts for the capacty of the buldng to deform n the nelastc range wthout sensble loss of strength, dsspatng hysteretc energy n the process. The structure overstrength represents the rato of the actual strength to lateral loads and the desgn loads defned by the code. It accounts for a number of factors ncludng nternal force 1 Professor and Char, Structural Engneerng Department, School of Cvl Engneerng, Unversty of Costa Rca, San José, Costa Rca. Emal: jorgeg@lanamme.ucr.ac.cr. Senor Structural Engneer, Research & Development Department, ESCOSA, San José, Costa Rca. ormerly, Graduate Student, Unversty of Costa Rca, San José, Costa Rca. Emal: ces@escosacr.com
redstrbutons, mnmum code requrements, actual member overstrength due to sze and materal overstrength. To estmate the nelastc dsplacements, these codes ncrease the values resultng from the elastc analyss by an amplfcaton factor that consders both ductlty and overstrength. If the nelastc dsplacements exceed specfed lmts, the structure must be modfed and recalculated. The results obtaned wth ths procedure omt mportant nformaton necessary to evaluate the sesmc performance of the structure, lke falure modes, requred global ductlty and correspondng nelastc deformatons of structural elements and components. To overcome these lmtatons t s possble to perform a Response Hstory Analyss (RHA) nvolvng a tme step soluton of the mult-degree-of freedom equatons of moton that represent the mult-storey buldng (Chopra []). Although very sophstcated, ths analyss presents some problems: a. It s cumbersome and tme consumng. b. As the drect use of desgn spectra s not possble, ndvdual analyses for a related famly of accelerograms s requred, followed by a statstcal analyss of ther response. c. It s only a verfcaton (analyss) procedure; hence, t requres a prevously desgned structure. As an ntermedate alternatve between the smple but ncomplete Response Spectrum Analyss (RSA) and the cumbersome RHA, the Capacty Spectrum ethod has been proposed (reeman [3], ATC [4], EA [5]). Ths s a non-lnear statc method that performs a pushover analyss to determne the capacty curve representng lateral base force versus a selected representatve lateral dsplacement. or each sgnfcant pont n the capacty curve, the state of absolute and relatve dsplacements and nternal deformatons, as well as ther correspondng external and nternal forces, can be known for the entre structure. The sesmc demand s expressed n terms of the desgn spectra represented on a S a -S d plot wth pseudo-acceleraton S a on the vertcal axs, and dsplacement S d on the horzontal. The capacty curve s then expressed n the same S a -S d plot by smple scale factors determned from dynamcs of structures (Chopra []) and the Performance Pont s calculated. Ths pont dentfes the peak lateral dsplacements of the buldng (relatve to the ground) and the correspondng base shear assocated wth the desgn earthquake. or ths state, the correspondng nelastc deformatons of all elements and components can be evaluated and the performance objectves verfed. Ths standard procedure s summarzed n aem [6]. The method just descrbed contans two serous drawbacks: a. It uses an elastc spectrum wth an ncreased vscous dampng to account for non lnear behavour nstead of a constant-ductlty desgn spectra. Ths dea s conceptually weak and leads to wrong results as demonstrated by Chopra [7, 8]. The proposed Capacty-Demand-Dagram ethods (CDD) overcome ths lmtaton (Chopra [9]). b. It s a verfcaton procedure (analyss); therefore t requres a prevous desgn of the structure ncludng defnton of strengths and force-deformaton relatonshps of the potental plastc hnge sectons as nput data. In ths paper, an effectve method for earthquake desgn of buldngs, called Dsplacement-Based Plastc Desgn (DBPD) s presented. The method has the followng characterstcs: a. It s explctly a desgn procedure rather than an analyss of a prevously desgned structure. The requred strength of each element s the result of the desgn procedure. b. To represent the sesmc demand t uses a constant-ductlty desgn spectra convenently represented n a S a -S d plot. c. It s a dsplacement-based desgn approach. Hence, to comply wth a Performance Objectve prevously selected, the structure s desgned to obtan a specfc target dsplacement assocated to the correspondng sesmc demand. d. The desgn s based n Plastc Theory and capacty desgn concepts to nduce a prevously defned lateral collapse mechansm under the desgn load.
e. The CDD s used to verfy the desgn and the fulflment of the Performance Objectve. In DBPD, the desgn drft lmts and the global ductlty of the structure are the startng ponts, and the fnal result s the requred strength of all the elements of the structure as well as ts necessary stffness. The process begns wth the selecton of a lateral dsplacement shape n whch the maxmum relatve drft ( / h ) max equals the defned target value ( /h) tar. ext, the target nelastc dsplacement S d tar for an equvalent sngle-degree-of-freedom (SDO) system s calculated and the pseudo-acceleraton S a s obtaned from the constant-ductlty desgn spectra for the selected global ductlty, represented n a S a -S d plot. Subsequently, the desgn base shear and the lateral forces can be calculated and a desgn process usng Plastc Theory s carred out to obtan the strength of each structural element related wth a selected collapse mechansm. nally, the desgn s verfed wth CDD. To explan the DBPD method a bref summary of Plastc Theory wll be presented, followed by a step-bystep descrpton of the procedure, llustrated wth a numercal example. PLASTIC THEORY I A UTSHELL One of the major dfferences between the DBPD and other dsplacement based methods lke the Drect Dsplacement-Based Desgn ethod (Prestley [10]) s the use of Plastc Theory nstead of elastc analyss for the desgn. Plastc Theory s a conceptually sound theory (assonet [11], eal [1], Hodge [13], oy [14], Bruneau [15]) for the analyss and desgn of structures that deform n the nelastc range, as t s usually the case under severe earthquake shakng. Pushover analyses use Plastc Theory, but for a structure prevously desgned va elastc analyss. The man nnovaton n DBPD s precsely the use of Plastc Theory for the desgn of the structure, obtanng the strength of all the structural elements. Accordng to the Upper Bound Theorem of Plastc Theory: A load computed on the bass of an assumed collapse mechansm wll always be greater than or equal to the true collapse load (Bruneau [15]). Ths theorem s central to the DBPD procedure, whch ntates wth the defnton of the strength of the beams requred to sustan the gravtatonal loads. Then, t proceeds from top to bottom consderng partal lateral collapse mechansms, ncreasng the element strengths f necessary to assure a safety factor for lateral loads greater than unty (say 1.05) for the combned loads. The process concludes wth a desred complete lateral collapse mechansm whch should have a safety factor for lateral loads equal to 1.0 (.e. ths should be the true collapse mechansm n a pushover analyss). Other undesrable collapse mechansms, such as soft-stores, should be checked as well. When the process s completed, the strengths of all the elements of the structure have been defned and they should be able to nduce the desred collapse mechansm under the combned gravtatonal and appled lateral loads. Ths must be verfed through a pushover analyss to dscard the occurrence of any unexpected undesrable mechansm. or applcaton of the Upper Bound Theorem the prncple of vrtual work s used: W E = W I (1) where W E s the external vrtual work done by the actual external loads on the vrtual dsplacements of the assumed collapse mechansm and W I s the nternal work produced by the plastc moments on the correspondng vrtual nternal rotatons. The upper lne ndcates vrtual work. The external vrtual work s due to the vertcal gravtatonal loads and lateral sesmc forces: E E G E W = W W () where W E and WE G, represent respectvely the vrtual work done by the lateral forces and the gravty loads. In the case of lateral collapse mechansms:
E W = u W G E = = 1 = 1 w l u (3) α (4) u = h (5) Where u are the lateral vrtual dsplacements and α defnes the relatve locaton of the possble nteror beam plastc hnge (g. 1). Obvously, f there s no nteror plastc hnge n the beams (α = 0), there s no vrtual work contrbuton from the gravty loads of that partcular element. λ w u λ c tar u max λ c αl u h λ 1 w u1 λ c 1 u 1 tar max λ c 1 u 1 (a) Real Structure (b) Real Dsplacements and Plastc oments (c) Vrtual Dsplacements gure 1. Applcaton of the prncple of Vrtual Work wth nternal plastc hnges n beams The nternal vrtual work W I s the work done by the plastc moments of the assumed hnges, thus: # hnges W I = y j (6) j= 1 j where y s the plastc moment of the assumed plastc hnges n beams and columns, and j are the correspondng vrtual rotatons. The safety factor λ c s a scalar factor on the lateral loads necessary to produce the assumed mechansm. rom equaton 1: G λ W E W E = W I (7) c W I W λ c = (8) W E Obvously, a safety factor λ c should always be greater than or equal to one. If not, the strengths of beams and columns on the assumed plastc hnges should be ncreased accordngly. j G E STEP-BY-STEP DESCRIPTIO O THE ETHOD In ths secton, the DBPD method s presented n sx steps (g. ):
Step 1: Intal dmensons and dsplacement shape (g..1) Intal element strength and correspondng prelmnary dmensons are frst estmated from gravty loads followng smple Plastc Theory prncples. Accordngly, the mnmum strength of the beams should be able to wthstand the crtcal gravty load combnaton, for example 1.4D or 1.D1.6L for most Amercan codes. If equal negatve moments are assumed at both ends of the beam, the beam statc equaton s: ωul v v = (9) 8 where v, - v, are the beam postve (at the center) and negatve (at both ends) moments, ω u s the dstrbuted load correspondng to the crtcal gravty load combnaton an l s the beam length. To prevent early plastc hnges, negatve and postve moments should be properly assgned. At each structural jont, the capacty desgn prncple of strong columns-weak beams defnes the mnmum column strength: ( ) c 1. v v (10) where Σ c s the strength of the top and bottom columns at the jont. These values represent mnmum beam and column strengths necessary to support the gravty loads. In addton, mnmum code requrements must be consdered. ext, a dsplacement shape s selected that satsfes a target drft d =( /h) tar choose accordng to a partcular Performance Objectve and the structural confguraton. The target drft may be based on nonstructural consderatons or lmt deformatons of crtcal members. One possblty s to select a dsplacement shape proportonal to the frst mode of the structure φ scaled by a factor Y tar to satsfy the target desgn drft d accordng to the followng equaton: u u h h 1 1 max. = h max. = h tar = d (11) where: u= φ Y tar Alternatvely, the dsplacement shape may be obtaned from general equatons lke the ones proposed by Loedng et al. as referred by Prestley [10]: for 4: u = h (1a) d ( 4) 0.5h 4 0: u = d h 1 (1b) 16h h 0: = u d h 1 0. 5 (1c) h where s the number of stores.
1 Intal Data (EI)Column φ u = Yφ u Target Dsplacement SDO system Sd tar. * (EI)Beam 1 h u u 1 U -U h -h 1-1 -1 = h max = max h tar * K Geometry Dsplacement Shape Lmt Dsplacement Shape Sd = tar. u n L * φ = Y tar. L * 3 Inelastc Spectrum (Sa-Sd) 4 Dstrbuton of Base Shear orce Sa µ G tar. T = el π Sd tar. Sa µ G tar. V base 1 T el Sa T sec Sd tar. Sd L V = Sa = Sa L base * * 5 Plastc Desgn λ 6 Verfcaton: Capacty-Demand-Dagram ethod Sa 1 λ λ1 Demand µ G tar. Capacty λ λ 1 1 λ 1 µ GR λ > λ > λ 1 > 1 Sd y Sd GR < Sd tar. Sd gure. Steps of Dsplacement-Based Plastc Desgn ethod (DBPD)
Step : Target desgn dsplacement for an equvalent SDO system (g..) Once the dsplacement shape s defned, the target dsplacement S d tar for the correspondng SDO system s calculated from prncples of dynamcs (Chopra []): u Ytar L Sd = = ; Γ = tar * Γφ Γ (13) or a system wth lumped-masses m at each level: where s the number of stores. L * = m = 1 = m = 1 φ ; Generalzed ass (14) φ ; Partcpaton factor (15) Step 3: Sesmc demand (g..3) The sesmc demand s obtaned from the constant-ductlty nelastc desgn spectrum correspondng to the structure global target ductlty µ G tar. Ths value s selected consderng the Performance Objectve as well as the buldng structural confguraton and the local ductlty of the structural elements and components. The constant-ductlty spectrum for an elasto-plastc system s tradtonally presented as a plot of pseudoacceleraton S a versus the ntal elastc perod T n for selected values of ductlty µ. It can be obtaned by dvdng the elastc desgn spectrum by approprate ductlty-dependent reducton factors that also depend on T n (Chopra [8]). There are numerous recommendatons for the reducton factors, from the early ones by ewmark [16, 17] to the more recent ones cted by Chopra [9]. The nelastc desgn spectrum s then represented n a S a -S d plot where S d corresponds to the nelastc peak dsplacement gven by: T = µ (16) S d S a 4π Wth the S a -S d plot, the correspondng pseudo-acceleraton S a s obtaned for the target dsplacement S d tar prevously defned. urthermore, the expected elastc perod of the structure T el can also be calculated wth the followng expresson: T π S d tar el = (17) Saµ G tar If the calculated elastc perod T n, correspondng to the stffness of the structure defned n Step 1, s greater than the calculated by equaton 17, t would not be possble to fully reach ts global target ductlty µ G tar wthout exceedng ts target dsplacements S d tar. In ths case, to satsfy both targets n the Performance Pont, the stffness of the structure should be sutably ncreased. Step 4: Dstrbuton of base shear force (g..4) Once S a s obtaned, the base shear force can be calculated from smple prncples of dynamcs (Chopra []): L = (18) V b S * a Ths force s then vertcally dstrbuted n proporton to the masses and the selected dsplacement shape, to obtan the forces at each level:
L mφ = Sa φ = Vb (19) * m φ Step 5: Plastc desgn (g..5) As already explaned n the prevous secton, a seres of partal collapse mechansms startng from the top story to the lower levels must be consdered for the desgn process. To prevent the possblty of these undesrable partal lateral collapse mechansms, they should have a safety factor λ c greater than 1.0, say 1.05. In contrast, the desred complete collapse mechansm (wth plastc hnges formng at the base columns) should be λ c = 1.0 to guarantee that t wll precede the undesred partal collapse mechansms (g.3). k 3 λ 3 3 λ 3 λ 1 3 λ λ 1 1 λ 1 1 (a) Load and Geometry (b) Partal collapse echansm 1 (c) (d) Partal collapse Complete collapse echansm echansm gure 3. Partal and complete lateral collapse mechansms for plastc desgn or load combnatons nvolvng sesmc loads the correspondng gravtatonal loads are reduced from maxmum values to expected average values, for example 1.D0.5L n some Amercan codes or 1.05D0.5L n Costa Rca (CIA [18]). The safety factor for lateral loads λ c correspondng to the top partal collapse mechansm (g. 3b) s calculated and f t s less than a mnmum value of 1.05, the strengths of beams and columns are ncreased to obtan at least that mnmum value. These ncreased strengths are consdered for the analyss of the next partal lateral mechansm (g. 3c). When calculatng the external and nternal vrtual work for a lateral collapse mechansm, each partcular beam may form ether a panel mechansm wth plastc hnges at both ends or a combned mechansm wth an nteror plastc hnge at a relatve dstance αl from the end (g. 4).
v ω u v ω u v 1 v l α l (1- α )l (a) Panel mechansm (b) Combned mechansm gure 4. Possble plastc hnge formaton at beams Whch mechansm wll preval depends exclusvely on the dstrbuted gravty load of the beam and ts bendng strengths. There wll be a combned mechansm f the vrtual external work due to gravty loads exceeds the ncrement on vrtual nternal work between both possble mechansms. It can be shown that the combned mechansm prevals f the followng relaton s satsfed: wul (1 α) ( v v ) (0) Once all the lateral collapse mechansms have been consdered and the requred strengths of the structural elements have been defned t s convenent to check for possble soft-story mechansms (g. 5). Indeed, accordng to the Upper Bound Theorem of Plastc Theory, the calculated safety factor may be on the unsafe sde f an unforeseen mechansm, wth a lower than unty safety factor, precedes the desred collapse mechansm. 3 3 V 1 V h V V 3 1 V 3 h 1 (a) Structure (b) Shear orces (c) Soft Story echansm 1 gure 5. Possble soft-story collapse mechansms (d) Soft Story echansm The external vrtual work for these partal collapse mechansms would be: WE = V h (1) where V s the shear force at the consdered level. In the event of any of these mechansms controllng the desgn the strength of the correspondng columns should be ncreased accordngly.
Step 6: Verfcaton procedure: Capacty-Demand-Dagram ethod (g..6) To valdate the desgn procedure the CDD must be appled. Ths wll prevent the possblty of unforeseen undesrable collapse mechansms and wll verfy the fulflment of the Performance Objectve n terms of the target dsplacements and the requred global ductlty. ILLUSTRATIVE EXAPLE The descrbed DBPD procedure wll be llustrated wth the desgn of a fve-story, two-bay specal moment frame (g. 6) consdered as typcal for a partcular buldng. The peak ground acceleraton s defned as 0.6g. or ths acceleraton the selected target drft s ( /h) tar =.04, whch may correspond to a Performance Objectve of lfe safety. The example follows the descrbed step-by-step procedure: ( EI )beam ( EI )column 5 4 3 1 5.00 m 4.00 m 4.00 m 4.00 m 4.00 m Desgn Loads (k/m) Gravty Loads DL LL 9.81 3 1...5 = 53 k s /m ( EI ) beam = 86030 k m ( EI ) = 189140 k m col. 6.00 m 6.00 m gure 6. Desgn example Step 1. Intal dmensons and dsplacement shape The ntal bendng strength of beams and columns s calculated for gravtatonal loads usng Plastc Theory ( v5 /- =37.1 k-m; c5 e =.3 k-m; c5 =44.5 k-m). The structure ntal stffness corresponds to these prelmnary dmensons of beams and column. or ths stffness a fundamental natural perod of 1.15 s and ts correspondng mode shape are calculated. The dsplacement shape s assumed as proportonal to the frst mode and the scale factor Y tar s calculated to satsfy the target desgn drft ( /h) tar =.04. T φ [. 0617.1304.1917.379.665] Y = 1. 45 = tar Step. Target desgn dsplacement for an equvalent SDO system The S d tar s found from equaton 13:
Γ = L Ytar = 4.796 S = = 0.303m * d tar Γ Step 3. Sesmc Demand or ths type of buldng, and consderng the Performance Objectve and the specal moment frame characterstcs, a structure global target ductlty µ G tar = 6 s selected. The correspondng constant ductlty nelastc desgn spectrum s calculated followng ewmark [16] and transformed to a S a -S d plot usng Equaton 16. or the selected S d tar = 0.303 m, the correspondng pseudo-acceleraton s calculated as S a =1.56 m/s, as shown n g. 7. The elastc perod of the structure s calculated from Equaton 17 as T = 1.13 s. Ths s the necessary perod for the structure to be able to develop ts global target ductlty µ G tar =6 wthout exceedng the calculated S d tar =0.303 m. As ths perod s practcally dentcal to the ntal perod (1.15s), t s not necessary to modfy the stffness of the structure. Sa (m/s ) Inelastc Desgn Spectrum ewmark & Hall rm sol - amax= 0.60 g ( ζ = 5%; µ = 6) 7 6 5 4 3 S a=1.56 1 0 S d tar=0.303 0 0. 0.4 0.6 0.8 1 1. Sd (m) gure 7. Calculaton of S a Step 4. Dstrbuton of base shear force The desgn base shear force s obtaned from Equaton 18, and the correspondng floor lateral forces are calculated from Equaton 19: V = 35.3k = [ 4.5 51.7 76.0 94.33 105. ]k b 8 Step 5. Plastc desgn In ths Step, the requred strength of beams and columns, necessary to produce the desred complete lateral collapse mechansm for the defned lateral forces s calculated usng Plastc Theory. echansm 5, correspondng to the top story partal lateral collapse mechansm (ffth story) s consdered frst (g. 8a), wth the ntal bendng moment strengths of beams and columns necessary for gravtatonal loads calculated n Step 1 ( v5 /- =37.1 k-m; c5 e =.3 k-m; c5 =44.5 k-m), as the frst tral. rom Equaton 8, the safety factor s λ 5 = 0.39 (<1.05), hence the bendng strength of columns and beams must be ncremented to the values shown of g. 8a, correspondng to a safety factor λ 5 = 1.05.
/ 105.8 5 v5 = 60.5 λ λ 105.8 4 / v5= 60.5 ( ) ( e ) C5 = 100.8 C5 = 50.4 94.3 λ 4 / v4= 161.8 ( ) ( e ) C4 C4 =87.5 =143.8 (a) (b) The calculated bendng strengths wll be used as the ntal data for the next mechansm (echansm 4, g. 8b). The safety factor λ 4 for ths partal lateral collapse mechansm s equal to 0.55. Therefore, for a λ 4 =1.05, these values are ncreased to the bendng strengths shown n g. 8b. rom equaton 0, t can be verfed that the panel mechansm prevals over the combned mechansm for the beams n ths floor. ollowng a smlar procedure, the ntal safety factor for echansm 3 s λ 3 =0.91 (<1.05) and consequently the bendng strengths are ncremented to the values shown s g. 9a. Agan, for ths case the panel mechansm prevals over the combned mechansm. The next partal lateral collapse mechansm, echansm, has a safety factor λ =1.09 (>1.05); hence, t s not necessary to ncrement the correspondng bendng strengths. nally, the safety factor for the complete lateral collapse mechansm s λ 1 =0.94, and the correspondng bendng strengths must be ncremented to the values of g. 9b for a safety factor of λ 1 = 1, correspondng to the desred mechansm of collapse under the combned loads. λ 105.8 3 gure 8. Partal lateral collapse echansms 5 (a) and 4 (b) (k-m) / v5=60.5 105.8 λ 1 / v5 = 60.5 94.3 λ 3 / v4=161.8 94.3 λ 1 / v4 = 161.8 λ 76.0 3 / v3 =83.8 76.0 λ 1 / v3 =83.8 ( ) ( e ) C3 C3 = 393.5 = 10.8 λ 51.7 1 / v =83.8 4.5 λ 1 / v =366. ( ) C =485.4 ( e ) C =4.7 (a) (b) gure 9. Partal collapse mechansm 3 (a) and complete collapse mechansm (b) (k-m)
The possblty of any undesrable soft-story mechansm must be checked now (g. 10). 105.8 λ 1 3 4 94.3 λ 76.0 λ 51.7 λ 4.5λ Soft Story 1 Soft Story Soft Story 3 Soft Story 4 gure 10. Possble soft-story mechansms or ths partcular example the correspondng safety factors λ are greater than one ( 1.10, 1.0, 1.43 and 1.44 respectvely). Hence, t s not necessary to ncrement the strength of the columns. Step 6. Verfcaton procedure g. 11a presents all the computed beam and column bendng strengths. These desgn values must be verfed usng CDD. In ths example, the pushover analyss was carred out wth the on Lnear SAP000 software program and g. 11b presents the capacty curve for base lateral force V base vs top dsplacement δ. The calculated lateral strength s 365.9 k, and the collapse mechansm was the expected mechansm (g. 1). However, ths s not an essental requrement, as some unforeseen but acceptable collapse mechansms, wth smlar lateral strengths, may result from the analyss. / v5=60.5 ( e ) C5 =50.4 ( ) C5 =100.8 v4= 161.8 V base (k) 400 350 365.88 k ( e ) C4 =143.8 ( ) C4 =87.5 / v3=83.8 300 50 ( e ) C3 =10.8 ( ) C3 =393.5 / v=83.8 00 150 100 ( e ) C =10.8 ( ) C =393.5 / v1=366. 50 0 0.00 0.0 0.40 0.60 0.80 1.00 ( e ) C1 =4.7 ( ) C1 =485.4 δ (m) (a) (b) gure 11. Desgn strengths (a) and pushover Capacty Dagram (b) To calculate the structure lateral dsplacements, the requred SDO dsplacement at the Performance Pont S d GR s determned usng CDD. or ths purpose the capacty curve s frst dealzed as blnear and the correspondng yeld dsplacement S dy =0.053 s obtaned (g. 1).
Sa (m/s ) 7.00 6.00 µ =4 5.00 µ =6 4.00 3.00.00 Complete Collapse echansm µ GR =5.80 ( /H) GR =0.0 1.00 Sd y =0.053 Sd GR =0.306 0.00 0.00 0.0 0.40 0.60 0.80 1.00 1.0 Sd (m) gure 1 Capacty-Demand-Dagram ethod for Desgn Example The Performance Pont s graphcally determned as the pont of the capacty curve whose global ductlty µ G = S dgr / S dy ntersects a constant ductlty desgn spectra wth the same ductlty. In our example, a value of S dgr =0.306 m, correspondng to a requred global ductlty of µ G = 0.306/0.053 = 5.8, s nterpolated from the graph (g. 1); remarkably close to the ntally selected value µ G tar = 6. Once the Performance Pont s known, all the subsequent nformaton necessary to verfy the performance of the buldng, such as lateral dsplacements and drfts, element nternal deformatons and forces, deformaton of non-structural components, floor shears and over-turnng moments, may be nterpolated from the pushover analyss. In ths partcular example, the maxmum drft at the Performance Pont was 0.00, close to the target drft of 0.04. IAL REARKS The Dsplacement-Based Plastc Desgn (DBPD) method s a very effectve tool for Performance-Based desgn as t allows for the selecton of a target global ductlty and drft lmts assocated wth the desred performance of the buldng. In ths method Plastc Theory s used to determne the requred strength of each structural element, necessary to produce a desred collapse mechansm and to reach a selected Performance Pont under a desgn earthquake defned by constant ductlty desgn spectra. Hence, the DBPD method s an explct desgn procedure and Capacty-Demand-Dagram ethods (CDD) are used as analytcal tools to verfy the desgn. Wth small modfcatons the DBPD method can be ncorporated nto exstng sesmc codes as a tool for Performance-Based Desgn. or nstance, t can be appled wth the new Costa Rcan Sesmc Code (CIA [18]), as ths code ncludes constant ductlty desgn spectra for the defnton of the sesmc demand and offers CDD as an alternatve analytcal tool for the verfcaton process. The DBPD method has been extended to consder addtonal features as P- effects, overstrength of structural elements and rgd-fnte-jont dmensons (Alpízar [19]), whch have been omtted from ths paper due to space lmtatons.
ACKOWLEDGEETS The authors wsh to thank the aterals and Structural odels atonal Laboratory (LAAE) of the School of Cvl Engneerng, Unversty of Costa Rca, whch provded the necessary condtons for the research reported n ths paper. The frst author thanks hs colleagues n the Costa Rcan Sesmc Code Commttee and hs students n the Graduate Program on Cvl Engneerng of the Unversty of Costa Rca for the frutful dscussons, feedback and numercal results necessary to test ths method. The second author thanks ESCOSA for ther confdence and support durng hs graduate studes. REERECES 1. IAEE. Regulatons for Sesmc Desgn - A World Lst 1996 and Supplement 000, Internatonal Assocaton for Earthquake Engneerng, 1996, 000.. Chopra AK. Dynamcs of Structures: Theory and Applcatons to Earthquake Engneerng. nd Edton, ew Jersey: Prentce Hall, 001. 3. reeman SA. Development and Use of Capacty Spectrum ethod. Proc. 6 th US atonal Conference on Earthquake Engneerng, Seattle, 1998: 1 pp. 4. ATC. ATC-40: The Sesmc Evaluaton and Retroft of Concrete Buldngs. volumes, Appled Technology Councl, Redwood Cty, CA, 1996. 5. EA. EHRP Gudelnes for the Sesmc Rehabltaton of Buldngs. Developed by the Buldngs Sesmc Safety Councl for the ederal Emergency anagement Agency, Report º EA 73, Washngton, D.C., 1997. 6. aem. Edtor. The Sesmc Desgn Handbook. nd Edton, Kluwer Academc Publshers, 001. 7. Chopra AK, Goel RK. Evaluaton of SP to Estmate Sesmc Deformaton: SD Systems, Journal of Structural Engneerng, ASCE 000; 16(4): 48-90. 8. Chopra AK, Goel RK. Drect Dsplacement-Based Desgn: Use of Inelastc vs. Elastc Desgn Spectra. Earthquake Spectra 001; 17(1): 47-64. 9. Chopra AK, Goel RK. Capacty-Demand-Dagram ethods Based on Inelastc Desgn Spectrum. Earthquake Spectra 1999; 15(4): 637-56. 10. Prestley J, Kowalsky J. Drect Dsplacement-Based Sesmc Desgn of concrete buldngs. Bulletn of The ew Zealand Socety for Earthquake Engneerng 000; 33(4): 41-44. 11. assonnet CE, Save A. Plastc Analyss and Desgn-Volume 1 Beams and rames. Blasdell Publshng Company, 1965. 1. eal BG. The Plastc ethods of Structural Analyss. 3 rd Edton, Scence Paperback, 1985. 13. Hodge PG. Plastc Analyss of Structures. cgraw-hll. 1981. 14. oy SSJ. Plastc ethods for Steel and Concrete Structures. nd Edton, Great Brtan: acmllan Press Ltd., 1996. 15. Bruneau, Uang C, Whttaker A. Ductle Desgn of Steel Structures. cgraw Hll, 1998. 16. ewmark, Hall WJ. Earthquake Spectra and Desgn. Earthquake Engneerng Research Insttute, Calforna, USA, 1987. 17. ewmark, Rddell R. Inelastc Spectra for Sesmc Desgn, Proc. 7 th WCEE, Istanbul, Turkey, 1980; (4): 19-36. 18. CIA. Códgo Sísmco de Costa Rca 00. Colego ederado de Ingeneros y de Arqutectos de Costa Rca, Edtoral Tecnológca de Costa Rca, 003. (In Spansh) 19. Alpízar. Idonedad sísmca de Edfcos Prefabrcados con Juntas Secas Tpo Híbrda..Sc. Thess, Cvl Engneerng Graduate Program, Unversty of Costa Rca, San José, Costa Rca, 00. (In Spansh)