Direct Displacement-Based Seismic Design of Structures

Transcription

1 Drect Dsplacement-Based Sesmc Desgn of Structures M.J.N. Prestley, G.M. Calv and M.J.Kowalsky 2. European School for Advanced Studes n Reducton of Sesmc Rsk, Italy. 2.Dept. of Cvl Engneerng, North Carolna State Unversty, USA 27 NZSEE Conference ABSTRACT: The concept of desgnng structures to acheve a specfed performance lmt state defned by stran or drft lmts was frst ntroduced, n New Zealand, n 993. Over the followng years, and n partcular the past fve years, an ntense coordnated research effort has been underway n Europe and the USA to develop the concept to the stage where t s a vable and logcal alternatve to current force-based code approaches. Dfferent structural systems ncludng frames, cantlever and coupled walls, dual systems, brdges, wharves, tmber structures and sesmcally solated structures have been consdered n a seres of coordnated research programs. Aspects relatng to characterzaton of sesmc nput for dsplacement-based desgn, and to structural representaton for desgn verfcaton usng tme-hstory analyss have also receved specal attenton. Ths paper summarzes the general desgn approach, the background research, and some of the more controversal ssues dentfed n a book, currently n press, summarzng the desgn procedure. INTRODUCTION Vewed through the hstorcal prsm of the past years, sesmc structural desgn can be seen to have been n constant evoluton much more so than desgn for other load cases or actons such as gravty, wnd, traffc etc. Intally, followng structural damage n the semnal earthquakes of the early 2 th century (Kanto, Long Beach, Naper), sesmc attack was perceved n terms of smple massproportonal lateral forces, ressted by elastc structural acton. In the 94 s and 5 s the nfluence of structural perod n modfyng the ntensty of the nerta forces started to be ncorporated nto structural desgn, but structural analyss was stll based on elastc structural response. Ductlty consderatons were ntroduced n the 96 s and 7 s as a consequence of the expermental and emprcal evdence that well-detaled structures could survve levels of ground shakng capable of nducng nerta forces many tmes larger than those predcted by elastc analyss. Predcted performance came to be assessed by ultmate strength consderatons, usng force levels reduced from the elastc values by somewhat arbtrary force-reducton factors, that dffered markedly between the desgn codes of dfferent sesmcally-actve countres. Gradually ths lead to a further realzaton, n the 98 s and 9 s that strength was mportant, but only n that t helped to reduce dsplacements or strans, whch can be drectly related to damage potental, and that the proper defnton of structural vulnerablty should hence be related to deformatons, not strength. Ths realzaton has lead to the development of a large number of alternatve sesmc desgn phlosophes based more on deformaton capacty than strength. These are generally termed performance-based desgn phlosophes. The scope of these can vary from comparatvely narrow structural desgn approaches, ntended to produce safe structures wth unform rsk of damage under specfed sesmcty levels, to more ambtous approaches that seek to also combne fnancal data assocated wth loss-of-usage, repar, and a clent-based approach (rather than a code-specfed approach) to acceptable rsk. Ths paper does not attempt to provde such ambtous gudance as mpled by the latter approach. In fact, t s our vew that such a broad-based probablty approach s more approprate to assessment of desgned structures than to the desgn of new structures. The approach taken n ths paper s based on provdng the desgner wth mproved tools for selectng the best structural alternatve to satsfy Paper Number XXX

2 socetal (as dstnct from clent-based) standards for performance, as defned n what we hope wll be the next generaton of sesmc desgn codes. The bass of ths approach s the procedure termed Drect Dsplacement Based Desgn (DDBD), whch was frst ntroduced n 993 (Prestley, 993), and has been subjected to consderable research attenton, n Europe, New Zealand, and North Amerca n the ntervenng years. The fundamental phlosophy behnd DDBD s that structures should be desgned to acheve a specfed performance level, defned by stran or drft lmts, under a specfed level of sesmc ntensty. As such, we mght descrbe the desgned structures as beng unform-rsk structures, whch would be compatble wth the concept of unform-rsk spectra, to whch we currently desgn. The research effort to develop a vable and smple desgn approach satsfyng ths goal has consdered a wde range of structural types ncludng, frame buldngs, wall buldngs, dual wall/frame buldngs, brdges, sesmcally solated structures and wharves, and a range of structural materals, ncludng renforced and prestressed (precast) concrete, structural steel, masonry and tmber. The culmnaton of ths research s a book n press (Prestley et al, 27) at the tme of wrtng ths paper. The research project behnd the desgn approach, whch can now be consdered n a rather complete stage of formulaton, has had to re-examne a number of long-held basc tenets of earthquake engneerng. Ths re-examnaton was frst presented n 993, and then updated n 23. The consequences of ths re-examnaton have been: A revew of the problems assocated wth ntal-stffness characterzaton of structures expected to respond nelastcally to the desgn level of sesmcty. A revew of sesmologcal nformaton to provde more approprate nput for dsplacementbased desgn. A re-examnaton of some of the fundamentals of nelastc tme-hstory analyss, partcularly aspects related to modellng of elastc dampng. Development of equatons relatng equvalent vscous dampng to ductlty demand for dfferent structural systems. Development of alternatve methodologes for determnng desgn moments n structural members from desgn lateral forces. A re-examnaton of capacty-desgn equatons for dfferent structural systems, and development of new, ductlty-based equatons and methodologes. The focus throughout ths lengthy research project has been on the development of practcal and smple sesmc desgn methodology, sutable for ncorporaton n codes n a format reasonably smlar to that currently avalable for the Equvalent Lateral Force approach, but wth much mproved smulaton of structural response. The book summarzng the research ncludes a chapter contanng a draft buldng code, based entrely on DDBD procedures. It s hoped that ths mght become a platform for future development of sesmc codes. A bref revew of tems lsted above follows. 2 PROBLEMS WITH SEISMIC DESIGN USING INITIAL STIFFNESS AND SPECIFIED DUCTILITY CAPACITY Problems wth ntal-stffness structural characterzaton n conventonal force-based sesmc desgn, and use of a code-specfed ductlty capacty have been dentfed n several prevous publcatons (Prestley 993, Prestley 23) and wll only be brefly lsted n ths paper: Intal stffness s not known at the start of the desgn process, even f member szes have been selected. Ths s because the stffness depends on the strength. Increasng or decreasng renforcement content to satsfy results of the force-based desgn proportonally changes the member stffness. The same concluson apples to steel members: changng the flange thckness to satsfy a strength requrement changes the member stffness almost n drect proporton to the strength change. Dstrbuton of lateral forces between dfferent parallel structural elements (walls; frames, e.g.) based on elastc estmates of stffness s llogcal and tends to concentrate strength n elements wth greatest potental for brttle falure. 2

3 Dsplacement-equvalence rules relatng dsplacement demand based on ntal elastc perods (whch are lkely, any way, to be sgnfcantly n error) to expected nelastc response are nvald, n our vew, beng based on ncorrect elastc dampng estmates used n tme-hstory analyss. Ths s dscussed n more detal subsequently. Ductlty capacty s a functon of structural geometry, not just of structural type. Hence t s napproprate to specfy a dsplacement ductlty factor for all structures of the same type (e.g. renforced concrete frames) Sesmc desgn of buldng structures wll generally be governed by drft lmts, when realstc estmates of buldng stffness are used to determne dsplacements. Current desgn approaches requre teraton to satsfy drft lmts, and codes, such as the NZ concrete code, do not recognze the stffenng effect of added strength unless member szes are also changed. 3 DIRECT DISPLACEMENT-BASED SEISMIC DESIGN The fundamentals of DDBD are very smple, and have also been presented n many earler publcatons (e.g. Prestley 2, Prestley, 23). Only a bref revew s ncluded here, wth reference to Fg. whch consders a SDOF representaton of a frame buldng (Fg.(a)), though the basc fundamentals apply to all structural types. The blnear envelope of the lateral force-dsplacement response of the SDOF representaton s shown n Fg.(b). Whle force-based sesmc desgn characterzes a structure n terms of elastc, pre-yeld, propertes (ntal stffness K, elastc dampng), DDBD characterzes the structure by secant stffness K e at maxmum dsplacement Δ d (Fg.(b)), and a level of equvalent vscous dampng ξ, representatve of the combned elastc dampng and the hysteretc energy absorbed durng nelastc response. Thus, as shown n Fg.(c), for a gven level of ductlty demand, a structural steel frame buldng wth compact members wll be assgned a hgher level of equvalent vscous dampng than a renforced concrete brdge desgned for the same level of ductlty demand, as a consequence of fatter hysteress loops. Wth the desgn dsplacement at maxmum response determned as dscussed subsequently, and the correspondng dampng estmated from the expected ductlty demand, the effectve perod T e at maxmum dsplacement response, measured at the effectve heght H e (Fg.(a)) can be read from a set of dsplacement spectra for dfferent levels of dampng, as shown n the example of Fg.(d). The effectve stffness K e of the equvalent SDOF system at maxmum dsplacement can be found by nvertng the normal equaton for the perod of a SDOF oscllator to provde K e = () 2 2 4π me / Te where m e s the effectve mass of the structure partcpatng n the fundamental mode of vbraton. From Fg.(b), the desgn lateral force, whch s also the desgn base shear force s thus F = V Base = K e Δ d (2) The desgn concept s thus very smple. Such complexty that exsts relates to determnaton of the characterstcs of the equvalent SDOF structure, the determnaton of the desgn dsplacement, and development of desgn dsplacement spectra. 3. Desgn Dsplacement The characterstc desgn dsplacement of the substtute structure depends on the lmt state dsplacement or drft of the most crtcal member of the real structure, and an assumed dsplacement shape for the structure. Ths dsplacement shape s that whch corresponds to the nelastc frst-mode at the desgn level of sesmc exctaton. Thus the changes to the elastc frst-mode shape resultng from local changes to member stffness caused by nelastc acton n plastc hnges are taken nto account at the begnnng of the desgn. Representng the dsplacement by the nelastc rather than the elastc frst- 3

4 mode shape s consstent wth characterzng the structure by ts secant stffness to maxmum response. In fact, the nelastc and elastc frst-mode shapes are often very smlar. m e F u F F n rk H e K K e Δ y Δ d Dampng Rato, ξ (a) SDOF Smulaton Elasto-Plastc Steel Frame Concrete Frame Concrete Brdge Hybrd Prestress Dsplacement (m) (b) Effectve Stffness K e Δ d ξ=.5 ξ=. ξ=.5 ξ=.2 ξ= Dsplacement Ductlty T e Perod (seconds) (c) Equvalent dampng vs. ductlty (d) Desgn Dsplacement Spectra Fg. Fundamentals of DDBD The desgn dsplacement of the equvalent SDOF structure (the generalzed dsplacement coordnate) s thus gven by Δ d = n n ( mδ ) ( mδ ) = 2 / (3) = where m and Δ are the masses and dsplacements of the n sgnfcant mass locatons respectvely. For mult-storey buldngs, these wll normally be at the n floors of the buldng. For brdges, the mass locatons wll normally be at the centre of the mass of the superstructure above each column, but the superstructure mass may be dscretzed to more than one mass per span to mprove valdty of smulaton. Wth tall columns, such as may occur n deep valley crossngs, the column may also be dscretzed nto multple elements and masses. Where stran lmts govern, the desgn dsplacement of the crtcal member can be determned by ntegraton of the curvatures correspondng to the lmt strans. Smlar conclusons apply when code drft lmts apply. For example, the desgn dsplacement for frame buldngs wll normally be governed by drft lmts n the lower storeys of the buldng. For a brdge, the desgn dsplacement wll normally be governed by the plastc rotaton capacty of the shortest column. Wth a knowledge of the dsplacement of the crtcal element and the desgn dsplacement shape, the dsplacements of the ndvdual masses are gven by 4

5 Δ Δ c = δ (4) δ c where δ s the nelastc mode shape, and Δ c s the desgn dsplacement at the crtcal mass, c, and δ c s the value of the mode shape at mass c. Specfc detals on structural mode shapes for DDBD of dfferent structural types are gven n the book (Prestley et al 27). Note that the nfluence of hgher modes on the dsplacement and drft envelopes are generally small, and are not consdered at ths stage n the desgn. However, for buldngs hgher than (say) ten storeys, dynamc amplfcaton of drft may be mportant, and the desgn drft lmt may need to be reduced to account for ths. Ths factor s consdered n detal n the relevant structural desgn chapters. 3.2 Effectve Mass From consderaton of the mass partcpatng n the frst nelastc mode of vbraton, the effectve system mass for the substtute structure s m n = e m = ( Δ )/ Δd where Δ d s the desgn dsplacement gven by Eq.(3). Typcally, the effectve mass wll range from about 7% of the total mass for mult-storey cantlever walls to more than 85% for frame buldngs of more than 2 storeys. For smple mult-span brdges the effectve mass wll often exceed 95% of the total mass. The remander of the mass partcpates n the hgher modes of vbraton. Although modal combnaton rules, such as the square-root-sum-of-squares (SRSS) or complete quadratc combnaton (CQC) rules may ndcate a sgnfcant ncrease n the elastc base shear force over that from the frst nelastc mode, there s much less nfluence on the desgn base overturnng moment. The effects of hgher modes are nadequately represented by elastc analyses and are better accommodated n the capacty desgn phase, rather than the prelmnary phase of desgn. 3.3 Structure Ductlty Demand Determnaton of the approprate level of equvalent vscous dampng requres that the structural ductlty be known. Ths s straghtforward snce the desgn dsplacement has already been determned, and the yeld dsplacement depends only on geometry, not on strength. Relatonshps for yeld curvature of structural elements, such as walls, columns, beams etc have been establshed (Prestley 23) n the general form: φ y = Cε y / h (6) where C s a constant dependent on the type of element consdered, ε y s the yeld stran of the flexural renforcement and h s the secton depth. Yeld drfts for concrete and steel frames can be expressed as Lb θ y = C2ε y (7) h b where L b and h b are the beam span and depth, respectvely, and C 2 =.5 and.65 for concrete and structural steel respectvely. For buldng structures, equatons (6) and (7) can readly be ntegrated to obtan the dsplacement at the effectve heght, H e, gven by (5) H e = n = ( m Δ H ) ( m Δ ) n / (8) = 5

6 where H are the heghts of the n storeys. The dsplacement ductlty demand for the structures s thus known at the start of the desgn, by Eq.(9), even though the strength s not yet establshed: μ = Δ d / Δ y (9) From relatonshps between structural type, ductlty demand, and equvalent vscous dampng, dscussed n the followng secton, the approprate level of elastc dampng to use n Fg.(d) can be drectly obtaned, and hence the base shear force calculated (Eq.(2)). Ths base shear force s then dstrbuted to the structural masses n accordance wth Eq.(), and the structure analysed, as also dscussed subsequently. F = V Base ( m Δ ) ( m Δ ) n / () = 4 EQUIVALENT VISCOUS DAMPING A key element of DDBD s that hysteretc dampng s modelled by equvalent vscous dampng (EVD), usng relatonshps such as those presented n Fg.(c). The total equvalent dampng s the sum of elastc, ξ el and hysteretc, ξ hyst dampng: ξ = ξ + ξ () eq el hyst Both components need some examnaton. 4. Hysteretc component The approach adopted has been to use values of EVD that have been calbrated for dfferent hysteress rules (see Fg.2, for examples) to gve the same peak dsplacements as the hysteretc response, usng nelastc tme-hstory analyss. Two ndependent studes, based on dfferent methodologes were used to derve the levels of equvalent vscous dampng. The frst nvolved the use of a large number of real earthquake accelerograms (Dwar and Kowalsky, 26), where the equvalent vscous dampng was calculated for each record, ductlty level, effectve perod and hysteress rule separately, and then averaged over the records to provde a relatonshp for a gven rule, ductlty, and perod. The second study (Grant et al, 25), usng a wder range of hysteress rules was based on a smaller number of spectrum-compatble artfcal acelerograms where the results of the elastc and nelastc analyses were separately averaged, and compared. In each case the equvalent vscous dampng was vared untl the elastc results of the equvalent substtute structure matched that of the real hysteretc model. The two studes ntally were carred out wthout addtonal elastc dampng, for reasons that wll become apparent n the followng secton. It was found that the approaches resulted n remarkably smlar relatonshps for equvalent vscous dampng for all hysteress rules except elastc-perfectly plastc (EPP), where the dscrepancy was about 2%. It s felt that the dfference for the EPP rule s a consequence of the use of real records, wth comparatvely short duratons of strong ground moton n the frst study, and artfcal records, wth longer strong ground moton duratons n the second study. EPP hysteress s known to be senstve to duraton effects. 4.2 Elastc component Elastc dampng s used n nelastc tme-hstory analyss to represent dampng not captured by the hysteretc model adopted for the analyss. Ths may be from the combnaton of a number of factors, of whch the most mportant s the typcal smplfyng assumpton n the hysteretc model of perfectly lnear response n the elastc range (whch therefore does not model dampng assocated wth the actual elastc non-lnearty and hysteress). Addtonal dampng also results from foundaton complance, foundaton non-lnearty and radaton dampng, and addtonal dampng from nteracton between structural and non-structural elements. 6

7 F k F k rk Δ Δ (a) Elasto-plastc (EPP) (b) B-lnear, r =.2 (BI) F k k μ α F k k μ α Δ Δ (c) Takeda Thn (TT) (d) Takeda Fat (TF) F F y βf y k Δ k Δ (e) Ramberg-Osgood (RO) (f) Flag Shaped (FS) Fg.2 Hysteress Rules Consdered n Tme-Hstory Analyses The dampng coeffcent, and hence the dampng force depends on what value of stffness s adopted. In most nelastc analyses, ths has been taken as the ntal stffness. Ths, however, results n large and spurous dampng forces when the response s nelastc, whch, t has been argued n a prevous NZSEE conference (Prestley et al 25) s napproprate, and that tangent stffness should be used as the bass for elastc dampng calculatons. Wth tangent stffness, the dampng coeffcent s proportonately changed every tme the stffness changes, assocated wth yeld, unloadng or reloadng, etc. Ths results n a reducton n dampng force as the structural stffness softens followng yeld, and a reducton n the energy absorbed by the elastc dampng. Snce the hysteretc rules are nvarably calbrated to model the full structural energy dsspaton subsequent to onset of yeldng, ths approach to characterzaton of the elastc dampng s clearly more approprate than s ntal-stffness elastc dampng. The sgnfcance to structural response of usng tangent-stffness rather than ntal-stffness dampng has been dscussed n detal n Prestley et al (25) and Prestley et al (27). However, n DDBD, the ntal elastc dampng rato adopted n Eq.() s related to the secant stffness to maxmum dsplacement, whereas t s normal n nelastc tme-hstory analyss to relate the elastc dampng to the ntal (elastc) stffness, or more correctly, as noted above, to a stffness that vares as the structural stffness degrades wth nelastc acton (tangent stffness). Snce the response veloctes of the real and substtute structures are expected to be smlar under sesmc response, the dampng forces of the real and substtute structures, whch are proportonal to the product of the stffness and the velocty wll dffer sgnfcantly, snce the effectve stffness k e of the substtute structure s approxmately equal to k eff = k /μ (for low post-yeld stffness). Prestley and Grant (25) has determned the adjustment that would be needed to the value of the elastc dampng assumed n 7

8 DDBD (based on ether ntal-stffness or tangent-stffness proportonal dampng) to ensure compatblty between the real and substtute structures. Wthout such an adjustment, the verfcaton of DDBD by nelastc tme-hstory analyss would be based on ncompatble assumptons of elastc dampng. The adjustments depend on whether ntal-stffness dampng (conventonal practce), or tangentstffness dampng (correct procedure, we beleve) s adopted for tme-hstory analyss. If ntalstffness dampng s chosen, the elastc dampng coeffcent used n DDBD must be larger than the specfed ntal-stffness dampng coeffcent; f tangent-stffness s chosen, t must be less than the specfed tangent-stffness coeffcent. It s possble to generate analytcal relatonshps between the substtute-structure and real structure elastc dampng coeffcents that are correct for steady-state harmonc response. However, as wth the hysteretc component, these are not approprate for transent response to earthquake accelerograms, though the trends from tme-hstory response follow the form of the theoretcal predctons. Hence, to obtan the approprate correcton factors, t s agan necessary to rely on the results of nelastc tme-hstory analyses. Prestley and Grant (25) compared results of elastc substtute-structure analyses wth nelastc tme hstory results to determne the correcton factor to be appled to the elastc dampng coeffcent for the assumptons of ether ntal-stffness or tangent-stffness elastc dampng. The form of Eq.() s thus slghtly changed to: ξ = κξ + ξ (2) eq el hyst The correcton factor κ s plotted for dfferent hysteress rules, and for dfferent ntal dampng assumptons (ntal-stffness or tangent-stffness) n Fg.3. FS 2 2 TT TF.6.6 BI ξ secant /ξ elastc Dsplacement Ductlty Factor (μ) (a) Tangent Stffness Dampng. TF EPP FS TT RO BI ξ secant /ξ elastc Dsplacement Ductlty Factor (μ) (b) Intal Stffness Dampng Fg.3 Secant Stffness Equvalent Elastc Vscous Dampng Related to Intal Elastc Stffness and Elastc Dampng Model It s possble to nclude the ductlty dependency of the elastc dampng nsde the basc form of the equvalent vscous dampng equatons. Wth the usual assumpton of 5% elastc dampng, the dampng ductlty relatonshps can be expressed n the general form: μ ξeq =.5 + C3 (3) μπ where the coeffcent C 3 vares between. and.7 (for the assumpton of tangent-stffness elastc dampng) dependng on the hysteress rule approprate for the structure under desgn (Prestley et al (27)). EPP. RO 8

9 4.3 Dampng n Mxed Systems In conventonal force-based desgn there s some dffculty n assessng the approprate level of ductlty to use n determnng requred base-shear strength for mxed systems, such as wall/frames, or even wall structures wth walls of dfferent length n a gven drecton. In DDBD the procedure s straghtforward, wth the EVD of the dfferent structural lateral-force-resstng elements beng separately calculated and then combned, weghted by the proporton of base-shear force (or overturnng moment) carred. Detals are presented n the Prestley et al (27). 5 ANALYSIS UNDER LATERAL FORCES Two alternatve methods of structural analyss under the desgn vector of lateral forces represented by Eq.() are suggested as approprate for DDBD. Each s brefly descrbed here, wth reference to frame buldngs. 5. Stffness-Based Analyss Conventonal force-based desgn would analyse the structure usng estmates of elastc member stffness. As already dscussed, these stffness estmates are lkely to be sgnfcantly n error, snce the stffness depends on the strength, whch s not at ths stage known. For DDBD we examne analyss procedures wth reference to the smple four-storey frame of Fg.4. To be consstent wth the prncples of DDBD, the frame structure analysed should represent the relatve stffness of members at the peak dsplacement. Thus beams, whch are expected to sustan ductlty demands should have ther stffnesses reduced from the elastc cracked-secton stffness n proporton to the expected member dsplacement ductlty demand. For frame members of normal proportons, t wll be adequate to reduce the elastc stffness of all beam members by the system dsplacement ductlty level μ Δ. An mproved soluton wll result f the member ductltes at dfferent levels are proportonal to the drft demands (assumng that the beams at dfferent levels have constant depth). Thus the member ductlty at the frst floor beams wll be taken as.33μ Δ, and at the roof level, as.667μ Δ. F 4 V B4 V B4 Level 4 Base Overturnng Moment F 2 F 3 V B3 V B3 V B2 V B2 Level 3 Level 2 OTM OTM OTM = = T L = F H V b B + L base M + Cj M Cj V B V B F Level H 3 V C V C2 V C3 M C M C2 M C3 T L base C Level Fg.4 Sesmc Moments from DDBD Lateral Forces 9

10 The desgn phlosophy of weak beams/strong columns wll requre that the columns between the frst floor and the roof reman essentally elastc. Hence, the stffness of these columns should be modelled by the cracked-secton stffness, wthout any reducton for ductlty. A problem occurs wth the structural characterzaton of the columns between the ground and frst floors. The desgn phlosophy requres that plastc hnges be permtted to form at the ground floor level to complete the desred beam-sway mechansm, but that the frst-storey columns reman elastc at the frst floor level, to ensure that a soft-storey (column sway) mechansm of nelastc dsplacement cannot develop. It s thus not clear how the stffness of the ground floor columns should be represented n the structural analyss. The soluton to ths problem reles on the recognton that any structural analyss s approxmate (compare the relatve structural approxmatons nvolved n an ntal-stffness and a secant-stffness representaton, both of whch are vald at dfferent levels of sesmc response), and that the fundamental requrement s that equlbrum s mantaned between nternal and external forces. Wth ths n mnd, we realse that t s possble for the desgner to select the moment capactes of the column-base hnges, provded that the resultng moments throughout the structure are n equlbrum wth the appled forces. Snce our desgn crteron s that column hnges do not form at the undersde of the level beams, t would appear logcal to desgn n such a way that the pont of contraflexure n the columns occurs approxmately at 6% of the storey heght. The desgn moment at the column base wll thus be.6v C H where V C s the column shear and H s the frst-storey heght. Ths mples that the maxmum column moment at the base of the frst floor beam under frst-mode response s about.325v C, or about 54% of the column-base moment capacty. We note that the column shear can be drectly determned from the lateral forces, provded logcal decsons are made about the dstrbuton of storey shears between columns. Thus, wth reference to the four-storey frame of Fg.4, the total base shear of V = F + F + F + F = V + V + V (4) Base C C 2 C3 should be dstrbuted between the columns n proporton to the beam moment nput. If we desgn for equal postve and negatve moment capacty of the beams at a gven level, then the moment nput from the beams to the central column wll be twce that for the exteror columns. The correspondng dstrbuton of the base shear between the columns wll thus be V = V = 2V. 5V (5) C 2 2 C C3 = Base The desred column-base moment capactes can then be defned before the structural analyss for requred flexural strength of beam plastc hnges. The structural analyss then proceeds wth the frststorey columns modelled as havng cracked-secton stffness, and pnned-base condtons. The base moments (e.g. M =.6V C H ) are then appled as forces to the column-base hnges, n addton to the appled lateral forces F to F Equlbrum-Based Analyss A problem wth the stffness-based analyss descrbed n the prevous secton s that, as wth forcebased desgn, the member stffness values are not known untl the strength s allocated. Although the use of stffness values reduced by ductlty demand, and of defned base-moments reduces the senstvty to assumptons of relatve stffness, a more satsfyng, and smpler approach to the structural analyss can be obtaned purely from equlbrum consderatons Beam Moments We agan refer to the regular frame of Fg.4, and consder equlbrum at the foundaton level. The lateral sesmc forces nduce column-base moments, and axal forces n the columns. The total overturnng moment (OTM) nduced by the lateral forces at the base of the buldng s n m = F H = = j= OTM M + T. L cj base (6)

11 where M Cj are the column base moments (m columns) T = C are the sesmc axal forces n the exteror columns, and L base s the dstance between T and C. The tenson force T (and the compresson force C) s the sum of the beam shear forces, V B up the buldng: T = n V B = (7) Equatons (5) to (7) can be combned to fnd the requred sum of the beam shears n a bay: n = n m V B = T = F H M Cj / L = j= base (8) Any dstrbuton of total beam shear force up the buldng that satsfes Eq.(8) wll result n a statcally admssble equlbrum soluton for the DDBD. Thus as wth the choce of column-base moment capacty, t s also to some extent a desgner s choce how the total beam shear force s dstrbuted. However, t has been found, from nelastc tme-hstory analyses that drft s controlled best by allocatng the total beam shear from Eq.(8) to the beams n proporton to the storey shears n the level below the beam under consderaton. Ths can be expressed as VS, VB = T (9) n V = S, Sesmc beam moments can now be determned from the beam shears from the relatonshp M, M = V L (2) B l + B, r B B where L B s the beam span between column centrelnes, and M B,l and M B,r are the beam moments at the column centrelnes at the left and rght end of the beam, respectvely. The relatve proportons of the moments at each end can be chosen to reflect the nfluence of slab renforcement, and gravty moments, f so desred Column moments Column base moments wll be determned n the same fashon as n the prevous secton. Column desgn moments at other elevatons can then be determned from statcs, knowng the column shears n each storey, and the moment nput to the jont centrods from the beam moments. Snce the column moments wll be amplfed by capacty desgn prncples, the exact magntude s not requred. 6 CAPACITY DESIGN PROTECTION IN DDBD The concept of capacty desgn was developed n New Zealand by Dr. John Hollngs, and Profs. Tom Paulay and Bob Park. Although t s far to say that New Zealand stll leads the world n codfed capacty desgn provsons, there s room for re-examnaton, partcularly as related to DDBD. The coordnated research project outlned at the start of ths paper has had partcular emphass on examnng exstng, and developng new capacty desgn rules and procedures for a range of dfferent structural types, ncludng framed buldngs, structural wall buldngs, dual wall/frame buldngs, brdges, and margnal wharves. The general requrement for capacty protecton can be defned by Eq.(2): o φ S S = φ ωs (2) S D R E where S E s the value of the desgn acton beng capacty protected, correspondng to the desgn lateral force dstrbuton found from the DDBD process, φ o s the rato of overstrength moment capacty to

12 requred capacty of the plastc hnges, ω s the amplfcaton of the acton beng consdered, due to hgher mode effects, S D s the desgn strength of the capacty protected acton, and φ S s a strengthreducton factor relatng the dependable and desgn strengths of the acton. In DDBD the desgn strength of plastc hnges s based on conservatve but realstc estmates of materal strengths, ncludng stran hardenng and concrete confnement, where approprate, and s matched to the demand at maxmum dsplacement. No strength reducton factor s used for DDBD. As a consequence, the overstrength factor φ o to be used n desgn s sgnfcantly lower than wth current desgn. On the other hand, t has been clear for some tme that a sgnfcant omsson n NZ capacty desgn rules s the lack of a ductlty modfer. Tme-hstory analyses clearly show that hgher mode effects are closely related to ductlty demand. Modal superposton approaches, where the hgher-mode actons as well as the actons n the fundamental mode are dvded by the ductlty or force-reducton factor can be serously non-conservatve, partcularly for structures braced wth walls. The same statement can be made about exstng smplfed approaches followng the general form of Eq.(2). Comparatve results are shown n Fgs.5 and 6 for moment and shear envelopes of mult-storey walls, excted by dfferent multples (between.5 and 2) of the desgn ntensty (Prestley and Amars, 2). In these plots IR ndcates desgn ntensty, and a value of IR =.5 ndcates an exctaton of.5 tmes the desgn ntensty. Also ndcated n the plots by dashed lnes are capacty desgn envelopes correspondng to NZ code provsons (ndcated as ) and modal superposton, where the elastc modal responses have been combned by the SSRS combnaton rule and dvded by the desgn ductlty factor (ndcated as SSRS/μ). Two ponts are mmedately apparent from these plots. Frst, the current capacty desgn envelopes are generally non-conservatve at the desgn ntensty. Second, the non-conservatsm ncreases as the ntensty ncreases, clearly ndcatng a dependency on dsplacement ductlty demand, snce ths ncreases as the ntensty ncreases. The excess demand over capacty s partcularly worryng for shear strength. A basc and smple modfcaton to the modal superposton method (termed modfed modal superposton (MMS) heren) s avalable by recognzng that ductlty prmarly acts to lmt frstmode response, but has comparatvely lttle effect n modfyng the response n hgher modes. If ths were to n fact be the case, then frst-mode response would be ndependent of ntensty, provded that the ntensty was suffcent to develop the base moment capacty, whle hgher modes would be drectly proportonal to ntensty. Ths approach s very smlar to that proposed by Ebl and Kentzel (988) as a means for predctng shear demand at the base of cantlever walls. In ths approach, shear force profles were calculated based on the followng assumptons. Frst-mode shear force was equal to the shear profle correspondng to development of the base moment capacty, usng the dsplacement-based desgn force vector. However, for low sesmc ntensty, where plastc hngng was not antcpated n the wall, smple elastc frst mode response, n accordance wth the elastc response spectrum was assumed. Hgher-mode response was based on elastc response to the acceleraton spectrum approprate to the level of sesmc ntensty assumed, usng the elastc hgher-mode perods. Force-reducton factors were not appled. The basc equaton to determne the shear profle was thus: ( V + V + V + ). 5 V (22) MMS, = D, 2E, 3E,... where V MMS, s the shear at level, V D, s the lesser of elastc frst mode, or ductle (DDBD value) frst-mode response at level, and V 2E, and V 3E etc are the elastc modal shears at level for modes 2, 3 etc. Predctons based on ths approach are compared wth average tme-hstory results n Fg.7. Agreement s very close, though the MMS approach tends to become ncreasng conservatve at hgh ntensty levels (.e. hgh ductlty levels), partcularly for taller walls. A smlar approach for predctng moments n structural walls was equally successful. 2

13 It has been found that mproved predctons can be obtaned when the modal analyss s based on a structural model where the member stffness s reduced to the secant level approprate at maxmum desgn dsplacement. Although the effect s mnor for wall structures, t s a sgnfcant mprovement for frames and brdges IR= IR= Moment (knm) (a) Two-Storey Wall 2 3 Moment (knm) (b) Four-Storey Wall IR= Heght(m) 2 IR= Moment (knm) (c) Eght-Storey Wall Moment (knm) (d) Twelve-Storey Wall IR= IR= Moment (knm) Moment (knm) (e) Sxteen-Storey Wall (f) Twenty-Storey Wall Fg.5 Comparson of Capacty Desgn Moment Envelopes wth Results of Tme-Hstory Analyses for Dfferent Sesmc Intensty Ratos 3

14 6 2 IR= IR= (a) Two-Storey Wall (b) Four-Storey Wall 2 3 IR= IR= (c) Eght-Storey Wall (d) Twelve-Storey Wall IR= IR= (e) Sxteen-Storey Wall (f) Twenty-Storey Wall Fg.6 Comparson of Capacty Desgn Shear Force Envelopes wth Results of Tme-Hstory Analyses for Dfferent Sesmc Intensty Ratos 4

15 6 3 4 MMS THA MMS THA MMS THA (m) Twelve-Storey Wall, IR=.5 (q) Sxteen-Storey Wall, IR=.5 (u) Twenty-Storey Wall, IR= MMS THA 3 2 THA MMS 4 2 THA MMS (n) Twelve-Storey Wall, IR=. (r) Sxteen-Storey Wall, IR=. (v) Twenty-Storey Wall, IR= THA MMS 3 2 THA MMS 4 2 THA MMS (o) Twelve-Storey Wall, IR=.5 (s) Sxteen-Storey Wall, IR=.5 (w) Twenty-Storey Wall,IR= THA MMS THA MMS THA MMS (p) Twelve-Storey Wall, IR=2. (t) Sxteen-Storey Wall, IR=2. (x) Twenty-Storey Wall, IR=2. Fg.7 Comparson of Modfed Modal Superposton (MMS) Shear Force Envelopes wth Tme- Hstory Results, for Dfferent Sesmc Intenstes 5

16 6. Smplfed Capacty Desgn In many cases the addtonal analytcal effort requred to carry out modal analyss of the desgned wall structure to determne the capacty desgn dstrbuton of moments and shears wll be unwarranted, and a smpler, conservatve approach, smlar to exstng capacty desgn rules may be preferred. As part of the research effort nto DDBD, smplfed rules have been developed for frames, walls, wall/frames and brdges. As an example, we contnue wth the case of cantlever wall structures. 6.. Moment Capacty Desgn Envelope for Cantlever Walls A b-lnear envelope s defned by the overstrength base moment capacty φ o M Base, the md-heght overstrength moment demand M o.5h, and zero moment at the wall top, as llustrated n Fg.8(a) for a four-storey wall. The overstrength base moment s determned from secton and renforcement propertes usng moment-curvature analyss, or usng smplfed prescrptve overstrength factors, gudance for whch s gven n Prestley et al (27). The md-heght moment s related to the overstrength base moment by the equaton: o o M.5 Hn = C, T φ M Base (23a) where C μ φ, T = T o.4 (23b) V o n Tenson shft Requred Capacty Requred Capacty Heght M o.5hn M F φ o M F V F φ o V F φ o M Base (a) Moment Capacty Envelope V o Base= φ o ω V V Base (b) Shear Force Capacty Envelope Fg. 8 Smplfed Capacty Desgn Envelopes for Cantlever Walls In Eq.(23b), T s the ntal (elastc) cracked-secton perod of the structure. Note that μ/φ ο s the effectve dsplacement ductlty factor at overstrength, and that tenson shft effects should be consdered when termnatng flexural renforcement Shear Capacty Desgn Envelopes for Cantlever Walls The shear force capacty-desgn envelope s defned by a straght lne between the base and top of the wall, as ndcated n Fg.8(b). The desgn base shear force s defned by: o o V = φ ω V (24a) where Base V Base, 6

17 μ ω = + (24b) φ V C o 2, T and C ( T.5). 5 (24c) 2, T = The desgn shear force at the top of the wall, V o n s related to the shear at the bottom of the wall by: o Vn C3 = V (25a) o Base where C.9.3T. 3 (25b) 3 = Predctons for the rato of wall moment at md-heght to base moment, and dynamc amplfcaton factor for base shear force are compared wth values obtaned n the ITHA for dfferent elastc perods and ductlty levels n Fg.9. Moment Rato T=3.9 s T=2.7s T=.9s T=.s T=.5s Dsplacement Ductlty Demand, μ Dynamc Amplfcaton Factor, ω V T=3.9s T=2.7s T=.9s T=.s T=.5s Dsplacement Ductlty Demand, μ Fg.9 Comparson of Capacty Desgn Equatons (24) and (25) wth Tme Hstory Results for Dfferent Elastc Perods and Ductlty Levels Smlar smplfed capacty-desgn equatons are presented for dfferent structural systems. 7 SEISMIC INPUT FOR DDBD As s apparent from Fg.(d), the dsplacement response spectrum s used rather than an acceleraton spectrum to determne the requred base shear strength. It s possble to generate the dsplacement spectrum from exstng acceleraton response spectra, assumng steady-state snusodal response, but ths assumpton becomes ncreasngly naccurate at long perods. It should be noted that ths naccuracy has been cted as a crtcsm of dsplacement-based desgn, snce DDBD uses the effectve perod at maxmum dsplacement response, whch s approxmately μ tmes the elastc perod used for force-based desgn, where response s presumably better known. Ths crtcsm does not stand up to scrutny, however, snce the elastcally desgned structure wll respond nelastcally wth the same perod as used for desgn of an equvalent DDBD structure, and hence any uncertanty n the long perod response data wll be reflected n naccuraces n the dsplacement-equvalence rule (such as equal-dsplacement) used to relate elastc to nelastc dsplacements n force-based desgn. Recently consderable research attenton has been focused by sesmologsts on mprovng the accuracy of dsplacement desgn spectra. The approach tentatvely adopted n Prestley et al (27) s developed from recent work by Faccol et al (24), who analysed a large number of recent hgh-qualty dgtal accelerograms. These ndcated that desgn elastc response dsplacement spectra could be reasonably represented as lnear up to a dsplacement plateau ntatng at a corner perod T C, the value of whch 7

18 depends on the moment magntude, M w. For M w > 5.7: ( M 5.7) T sec (26) c = w wth a correspondng dsplacement ampltude for 5% dampng of ( M W 3.2) δ max = CS mm (27) r where r s the epcentral dstance (or nearest dstance to the fault plane for large earthquakes) n km, and where C S =. for frm ground. The response dsplacements resultng from Eq.(2.5) should be modfed for other than frm ground. Tentatve suggestons are as follows: Rock: C S =.7 Frm Ground: C S =. Intermedate Sol: C S =.4 Very soft Sol: C S =.8 Predctons based on Eqs.(26) and (27) wth C S =. are plotted n Fg. for dfferent M w and r. Recent unpublshed work by Faccol ndcates that t s probable that the corner perod and correspondng plateau dsplacement wll be revsed upwards by about 2%. It should be noted that there are unresolved dfferences n predctons of corner perod between US sesmologsts (based manly on sesmologcal theory) and European sesmologsts (manly emprcally based). Dsplacement-based sesmc desgn usng a secant stffness representaton of structural response requres a modfcaton to the elastc dsplacement response spectrum to account for ductle response. The nfluence of ductlty can be represented ether by equvalent vscous dampng (as shown n Fg.(d) or drectly by nelastc dsplacement spectra for dfferent ductlty levels. The use of spectra modfed by dfferent levels of dampng requres relatonshps between ductlty and dampng to be developed for dfferent structural hysteretc characterstcs as dscussed n Secton 4 above, but enables a sngle desgn spectrum to be used for all structures. The use of spectra modfed by dfferent levels of ductlty s perhaps more drect, but requres the ductlty modfers to be determned for each hysteretc rule consdered. The same nelastc tme-hstory analyses can be used to develop both approaches, whch are essentally dentcal. A commonly used expresson for relatng the dsplacement response spectrum for a dampng rato of ξ to the elastc spectrum for ξ =.5 was presented n the 998 edton of Eurocode EC8 (EC8 998), and s shown below n Eq.(28):.5.7 R ξ = (28).2 + ξ Although a more recent edton of EC8 (EC8 23) revsed the coeffcents of numerator and denomnator of Eq.(28) to. and.5 respectvely, our analyses of real and artfcal records convnce us that the form gven n Eq.(28) gves a better representaton of the damped spectra. Equaton (28) apples to stes where forward drectvty effects are not apparent. It would also be desrable to have an equvalent expresson for stes where forward drectvty velocty pulse characterstcs mght be expected. It has been suggested (Prestley 23) based on lmted data, that a modfcaton to the 998 EC8 expresson gven by.25.7 R ξ = (29).2 + ξ mght be approprate. Some qualfed support for Eq.(29) s avalable n Bommer and Mends (25) whch provdes addtonal dscusson of ths topc. 8

19 Spectral Dsplacement (mm) Spectral Dsplacement (mm) Spectral Dsplacement (mm) Perod (seconds) (a) r = km Perod (seconds) (b) r = 2 km Perod (seconds) (c) r = 4 km M = 7.5 M = 7. M = 6.5 M = 6. M = 7.5 M = 7. M = 6.5 M = 7.5 M = 7. M = 6.5 Fg. Influence of Magntude and Dstance on 5% Damped Dsplacement Spectra for Frm Ground Usng Eqs.(24) and (25) [after Faccol et al (25)] 8 BRIEF NOTES ON OTHER DDBD ISSUES Space lmtatons do not allow adequate coverage of the many addtonal aspects covered n Prestley et al (27). Bref notes on some of the more controversal ssues are ncluded n ths secton. 8. Torson n DDBD Paulay (2) and Castllo and Paulay (24) have shown that elastc analyss based on ntalstffness structural representaton s nadequate to predct nelastc torsonal response. Ths work has been extended n the present study to develop a desgn methodology for DDBD of structures wth 9

20 torsonal eccentrcty. Desgn dsplacement of the centre of mass s reduced by a torsonal component whch can be estmated wth some accuracy at the start of the desgn process. The approach combnes stffness and strength eccentrcty, and uses the effectve stffness of lateral force-resstng elements at maxmum dsplacement response. Thus for sesmc attack parallel to a prncpal axs of a buldng the structural elements parallel to the drecton of attack have ther elastc stffness reduced by the desgn ductlty factor, whle the orthogonal elements are assgned ther elastc (cracked-secton) stffness. The method provdes accurate estmates of both centre-of-mass dsplacements, and dsplacements of structural elements wth maxmum torsonal components of dsplacement, for both torsonally unrestraned (TU) and torsonally restraned (TR) systems, as ndcated n Fg Short Wall Dsplacement (m).3.2. Short Wall C. of M. Long Wall Dsplacement (m).3.2. C. of M. Long Wall Rato of Wall strengths (V/V2) (a)tu system, system strength ncreases (Castllo and Paulay 24 data) Wall Strength Rato (V/V2) (b) TR system, constant strength (Beyer 27 data) Fg. Comparson between Predcted Dsplacements (dashed lnes) and Average Tme-Hstory Results (sold lnes and data ponts) for TU and TR systems 8.2 P-Δ Effects n DDBD The treatment of P-Δ effects n DDBD s comparatvely straghtforward, and s llustrated n Fg. 2(c). Unlke condtons for force-based desgn, the desgn dsplacement s known at the start of the desgn process, and hence the P-Δ moment s also known before the requred strength s determned. DDBD s based on the effectve stffness at maxmum desgn dsplacement. When P-Δ moments are sgnfcant, t s the stffness correspondng to the degraded strength and the desgn dsplacement (see K e n Fg.2(c)) that must match the requred stffness. Hence, Eq.(2) defnes the requred resdual strength. The ntal strength, correspondng to zero dsplacement, s thus gven by PΔ d F = KeΔ d + C (3) H and hence the requred base-moment capacty s M B = K Δ H + C PΔ (3) e d d Note that t s more consstent to defne the P-Δ effect n terms of the base moment, than the equvalent lateral force. In Eq.(3), for consstency wth the desgn phlosophy of DDBD, we should take C=. However, examnaton of the hysteretc loops ndcates that more energy wll be absorbed, for a gven fnal desgn dsplacement and degraded strength, than for a desgn when P-Δ desgn s not requred, partcularly for concrete-lke response. It s also apparent from tme-hstory analyses that for small values of the stablty ndex, dsplacements are only slghtly ncreased when P-Δ moments are gnored, as noted above. It s also clear that steel structures are lkely to be more crtcally affected than wll concrete structures. Consderaton of these ponts leads to the specfcaton of C = for steel structures and C =.5 for concrete structures. Recent tme-hstory analyses (Pettnger and Prestley 2

21 27) have provded confrmaton of these recommendatons. F Force F Strength Enhanced for PΔ Νο PΔ Wth PΔ PΔ H 2 PΔ FH Δ y Dsplacement (a) Structure (b) Moments (c) Force-Dsplacement Response Fg.2 P-Δ Effects on Desgn Moments K e Δ u 8.3 Combnaton of Gravty and Sesmc Forces n DDBD When gravty moments are added to sesmc moments n potental plastc hnge regons to result n a total desgn moment that exceeds the sesmc moments, then the consequence s a reducton n the ductlty demand below the level selected for desgn. In DDBD the consequence would be a response dsplacement less than the selected desgn dsplacement. It s argued n Prestley et al (27) that t s llogcal to drectly add gravty and sesmc moments when dfferent effectve stffness s mpled n the analyses for the two effects. For example, n a frame buldng, at maxmum response plastc hnges wll have formed at the beam ends, greatly reducng the stffness of the end regons. An elastc analyss of the gravty load moments n the beam takng account of ths reducton n stffness of the end regons would ndcate a very substantal decrease n the fxed end moments, and an ncrease n the md-span moment from the value applyng wth elastc beam propertes. Ths s, of course, a justfcaton for moment-redstrbuton. Comparatve analyses (Pnto 997) of frames where the fxed-end moments from gravty loads were alternatvely ncluded or neglected have shown that the dsplacement response of frames was essentally unaffected. Ths result together wth the above comments lead us to recommend that plastc hnge regons of structures be desgned for the larger of moments resultng from factored gravty loads or from sesmc forces, but not combned gravty and sesmc moments. Ths smplfes desgn, and for many structures s very smlar, but more conservatve than the common gravty+sesmc+3% redstrbuton approach, as llustrated for a typcal frame desgn n Fg Draft DDBD Buldng Code In order to show how DDBD phlosophy mght be ncorporated nto a code document, the relatve provsons have been organsed n one chapter of Prestley et al (27) nto a code plus commentary type format that could act as the bass for development of an alternatve DDBD code for sesmc desgn of buldngs. Wth mnor modfcaton t could also apply to other structures such as brdges, sesmc solated structures, unrenforced masonry structures and wharves, all of whch are consdered n the book. 2