A R e-dellnition of the Stiffness of R einforcedConcrete Elem ents and its lm plications in Seism ic D esign

' : #&

/f .

f ,, ,Tom PaulayProf. EmeritusUniversity of CanterburyChristchurch. New Zealand

Tom Paulay. born in 1923 in Hungary.received his engineering education inHungary and New Zealand. -lXe majorpart of his research oq'er 4 O yzars atthe University of Canterbury addressedthe seismic design of reinforced concretestructures. He is past president of thelnternational Association of EarthquakeZngindering. jjyjyodudj*on

Summary

lt is postulated that for the purposes of seism ic design the ductile behaviour oflateral force-resisting structural components, elements and indeed the entirebuilding system . can be satisfactorily simulated by sim ple bi-linear force-dis-placement relationships. This enables the displacement l-elationships betweenthe system and its lateral force-resisting elem ents at a particular lim it state to bereadily evaluated. To this end some widely used fallacies. relevant to the transi-tion from elastic to inelastic behaviour. are exposed. A re-definition of yeld dis-placem ents and consequently stiffness. allows m uch more realistic predictions ofthe m ost important feature of seismic response. element displacements. to bem ade. The concepts introduced are rational yet very simple. Their applicationsare closely interwoven with the designer-s intensions. The strategy provides thedesigners with unexpected freedom in the assignment of strengths to lateralforce resisting elem ents. such as fram es or structura) walls. Contrary to currenldesign practice. whereby a specific global displacement ductility capacity is pre-scribed for a particular structural class- the designer can determine the accept-able displacem ent dem and to be imposed on the system. RXis should protect crit-ical elem ents against excessive displacem ent dem ands.

D esign Criteria

The primary purpose of this study wasto address means by Nvhich perfor-mance criteria. conform ing with theappropriate design limit state. may berationally executed. 'These criteria are:

- Earthquake-induced deformationsshould limit the expected displace-ment ductility dem and on any ele-ment to its ductility capacity. l-taimax.stipulated in codes.

- szlaxim um magnitudes of interstoreydisplacem ents. to be expected at lo-cations rem ote from the centre ofmass. should not exceed those con-sidered acceptable for buildings.typically 2-2.5% of the storeyheight.

- Perform ance criteria m ay requiredisplacem ents associated with a spe-cific lim it state. to be less than thoseallowed by the limits listed above.

A study of the assessm ent of the struc-tural perform ance of existing buildingswith earthquake risk triggered in-quiries adressing the likely response ofbuildings as constructed. rather thantheir compliance with a particularcode. A maior perceived need was theestimation of torsion-induced dis-placem ents of elements of ductile sys-tems g1. 2- 3j. In the process severalissues emerged with apparent conflictwith ingredients of our current designpratice. The description of progres-sively emerging fallacies. firmly en-trenched in our routine seismic designtechniques (4' 1. is the subject of thispresentation.

The m otivation for the introduction ofsom e unfamiliar. but not necessarilynew. principles. relevant to ductilestructural response. was the need toem phasize the im portance of earth-quake-induced displacelnents. ratherthan a particular m ethod of assigningstrengths to elements of a system.ldentification of structural behaviour.rationale and transparency of a viabledesign strategy. com bined with sim-plicity of application. were centralissues of this motivation.

Term inology used

- ln the study of earthquak' e-induceddisplacements of buildings. refer-ence will be made to the sntcntral5').'5'f?#??

Structural Engineering International 1/2001

Peer-reviewed by international ex-perts and accepted for publicationby IABSE Publications Committee

Paper received: August 2 1 . 2000Paper accepted: Nos ember 23. 20(-('1

36 Science and Technology

- zzN structural systelm connprises later-al force-l-esisting elements. generallyarranged in two ortllogonal direc-tions. D ue to torsional effects. ele-ments of the sl'stem nl ?-13* be subi ectto different displacements. Typicale lenAen ts are bents of ductile fral-nesor interconnected wal ls in the salne1: l ane .

- zzN lateral force-resisting e 1 ementmay conlprise several iomponents.Components will be subjected toidentical displacenAents. TypicalC O m 1) O n e n t S a r e b e 1 m S OF C ( ) 1 u m n SOr 1&a1 l S.

Th ese te rm s are i l 1 tlstrat e d in Fig. l .

Nvhereas elements with excess strengthwill yield a little later. These changesin the onset of yielding were not ex-pected to change component ductilitydem ands significantly enough to be acause of concern.

Instead of abstract derivations of rtcle-vant relationships. some simple exam-ples. connnxonly encountered in designpractice. will be used to show the ap-plicability of certain principles. Fig. 2(a)shows four rectangular reinforced con-crete canlileq'er walls of identicalheigths and widths. The interconnec-tion of these wall components is suchthat at any stage of seisnnic response.identical displacennents will be in-l-posed on all f our el enaents. The 1 e 11 gthsof the wall col-nponents. lwi. is such thatthe relative second I'nonzents of area ofthe sections- being proportional to l).i-are 1 . 2. 4 and 8. respectively. R-lne totalrelative stiffness of the element. com-prising fottr com ponent cantileNrerwal ls. is thus 15. The relative compo-nent strengths- based on traditionallydefined component stiffness- areshown in Fig. 2 ( 1) ) . lt also shows lhat .according to traditional assun-lplions.all elements will conAmence yielding ata relatiA-e la leral displacement of .1.

Fig. 2 4c) show's the innp lications of1,.' l 5 of the total strength being redis-tributed froln component ( 4) to con'l-ponent ( 3 ) without reducing the t otalstre ngth of the four-com ponent ele-ment. The traditional definition ofcomponent stiffness. based on the se-cond mom ent of area of the section.suggests that the yield displacements

A D elinition of YieldD eform ations

of these two components will change.A departure from the strength as-signed in proportion to componentstiffness implies thus. as shown in Fig.2(c). a corresponding decrease or in-crease. respectively. of the componentyield displacem ent. The fallacies re-sulting from this traditional usage ofcom ponent stiffness are exam ined sub-sequently

Wall components

- - - jj *. * l

A wall element III I lI A frame element l W

.l .- -x .* xx .m xx l G,% , &

'

#

'

,,j ..e %'.. .>

'

'# w. .e j %oFrame components ! *

' cv $/ t c'! E l 2

. . ! l rzl# %1 - -

k > .*'j

'

IA frame element Il

! !m wall etement ll - 1

FiL.;. l ' 'h-()llell c/t-l/'lf F'ta

Traditional Concepts of theTheory of Elasticity

'T'he requiremcnts for static equilibri-um and deformation compatibility instatically indetcrminate structure. iswell established. These principles arestill widely used when strength. to lat-eral force-resisting ductile elements ofa system are assigned.

Brith the introduction of eqtliN'alentlateral static seism ic design forces andthe acceptance of ductile response. thesam e technique continued to be wide-ly used. lt implied the notilln thatstrength assigned proportionally toelement stiffness will result in the si-multaneous onset of yielding in all ele-m ents.

Subsequentll' it was realized that. be-cause. of expected significant inelas-tic earthquake-imposed deformations.som e deviation from strength distribu-tion according to elastic behaviour. isquite acceptable. Thus the practice ofstrenglh re-distribution u'as adapted.lt was assumed tllat for com pflnents areduction of flexural strength will re-sult in an earlier onset of yielding.

Structural Engineering International 1/2001

Component number: (6 Xt @)' (@)Latera/ force: 12% 15* ..JJ-.... (#.r 7 :C

V's l'=lwi : .1 2 4 e ?)2fzh.

V = 1. (K) #E

(a) Interconnected Wall Compoflenlo

15

SQ k@! 10 4i;5 K.

c.9 44D s

3- 21

(b) Slrerlgm-spfacement (c) Strength-displacementrelationships relationships with so/'ne

strength retfjstzfbfgf/bn

15

St: 4.e ? o y- Ke)*.k - - --%% 45 .. .-

3- -' g1

urhere 6y. is the l'ield strain of the steelused and (1 U is the length of the flexur-al tension zone of the elastic section.

For llne purpose of bi-linear modellingof the tlexural strength-curvature rela-tionship at the critical section of thewall component- it is convenient to in-troduce the term

(:1) Slrain Patlerns

2.0

t11

:B 1 s p=ttoz/AgU) .QC! r% M

n% I

z 1,0 -Q 1%= k.% ( My*Q: o s I

. j pm;!. twy1

i00 0.02 0.04 0.06

?w/

Jdea/fzedelasto-plastic

A =0.58 fesponse ph=4.311.0 1 5I

t/./.o111.26i? 0

.8d 111.59 s

Q2) 'Y 1/2, 0 E' 0.6 a

rwy g = 5. 0 mo 8%

..r m)$ O.4 .!:': Component c)4

.um=3.97 a:E - #A 0.2 .a Ua $52 phz- .2'2

=2. 1 . .j100 0.52 ; 1.O 1.5 2,0 2.5

c5 csRelative Lateral Displacement

Fig. 4: Bi-lill (:://,' idealisztltiol ? t? f' (.'t-.?/??/Jt???t?/? r tlll(lc/(-'/,??c/?/ forz'e-detbrnlal'ioll rt.>/t'/f/t?F?.s/?!'./p.

il-respective of the ratio and arrange-m ent of reinforcement. As an examplefor design purposes. it mal' be assumedg6j that the nominal yield curvature ofa l'ectangular wall section is:

(!) =v 2e ,/1.

ample. the reference yield displace- (2)m ent at the top of a wall component i.shown in Fig. 2(a). when subjected tolateral static forces- when based on eq.(3). can be approximated by

A.y i = Csihti = lzchszoe),vl/'lwi z'i l/lwi

As eq. (4 ) states. the yield dis-placem ent is independent of thestrength assigned to a com ponent.

(3) svhen the lateral force-resisting el-ement. shown in Fig. 2((1). is sub-jected to increasing displacements.the components will commenceyielding in a predeterm ined order.ln the example shown. component(4) will be the first and component(1) the last one to yield. At thisstage the full strength of the four-component element is developed.

(4 ) As a corollary. and contrary to tra-ditional assumptions. depicted inFig. 2(b). the simultaneous yieldingof components with differentlengths is not possible.

(5) The displacement capacity of a lat-eral force resisting element is lim it-ed by that of the component withthe sluallest yield displacem ent.i.e.. largest length. ln this exampleit has been assumed that the dis-placement ductility capacity of thewall components is l-taimas = 5.Therefore. the displacelnent of thiselement at the ultimate limit statemust be limited to the capacitl' ofcomponent (4 ) with Au4 = 5 >: 0.5 =2.5 displacement units. ln Fig. 4 theyield displacement of component( 1 ) was chosen as a displacementunit. It follows then that tahe duc-tility demand on components ( 1 ).(2) and (3) Nvill be non-critical.

(6) Because the sequence of yieldingis set and is independent of the

Structural Engineering International 1/2001

A higher degree of precision in seism icdesign is not warranted.

The important message of eq. (3) is-that the yield curvature of a reinforcedconcrete section is proportional to theyield strain of the steel used and in-versely proportional to the length ordepth of the section.

The bi-linear modelling of flexuralstrength-curvature relationship. with-out post yield stiffness. is presentedin Fig. 3(b). It is also found that m o-derate axial compression load on awall. comm only encountered in m ulti-storey buildings. P = 0.07fJAg. does notchange the yield curvature to any sig-nificance. However- the strength ofthe sections m ay be significantly in-creased. The important relationshipsare recapitulated in Fig. .:1. where fc' de-notes the compression strength of theconcrete and Ag is the gross sectionalarea of the wall.

where C is a coefficient which quanti-fies the distribution pattern of lateraldesign forces. and hwi is the height ofthe wall com ponent. Because. asshown in Fig. 2(a). the heights of thewralls. hai. and the grade of the tlexuralreinforcement used will be the samefor a1l wall components. the bracketedterm in eq.(4) Nvill be a constant.Therefore. yield displacements will beinversely proportional to wall lengths.This sim ple relationship can be conve-niently used in design B'henever rela-tive properties of components are suf-ficient to establish. for example. ductil-ity relationships. Exam ples will subse-quently show the relevance of this veryim portant relation.

Implications of the Redefined YieldD isplacem ents

The idealized bi-linear ductile behav-iour of a four-com ponent lateral force-resisting wall elem ent. shown in Fig.2((1). is presented in Fig. 4. The assign-ment of component strengths in accor-dance with traditional practice. asshown in Fig. .J(Jp )- was used. A studyof the relationships demonstrate that:

( 1) The relative yield displacement ofeach wall com ponent is inverselyproportional to its length.

Yield .ll/r/tzcczzltlzl/

Once the reference yield curvature ofa com ponent. such as a cantilever wall.is established. the corresponding dis-placement for a given set of lateralforces can be readily established at anydesil-ed level of the structure. For ex-

Science and Technologl'

strength of the com ponents. strength nents. The stiffness of each com ponentto components may be assigned in any in Fig. 4 was clearly defined by eq. (5).arbitrary manner. This provides the The superposition of the response ofdesigner wilh considerable freedom of the four components leads to a non-choice. lt can be exploited so as to im- linear transition from the elastic to theprtlve overall structural performance. post-yield behax'iour of the elem ent.as well as to arris'e at more economical For design purposes this can againsolutions. Som e examples are provided be replaced by bi-linear simulation.subsequently. shown b)' dashed lines in Ffk. 4 as total

re s p o n s e . A cc o rdi n g l y t h e s t i f f n e s s o fAn unrecognised significant advantage j

ae ojtzyuuyly jsr- tO t t his ap p roach is that i t e n abl es thedesigner to control the I'naior sourcesof t he d e t ri m e n t a l e f fects of the t 01--sional response of buildings. These are This in turn allows the nolninal yieldeccentricities with respect to the centre displacelnent of the elel-nent to be de-of the mass of the system . lf desired- fined asthe strength to elements of a systen:mal' be assigned so that the centres 01' A .c = XV i,'Y k,) nstre ngth and I-nass will coinci de g71 .

A .s. used here- is associated with theuniform translation of :111 its elel-nentsand the centre of n'lass. The definitionis thus identical to that represented byeq. ( 7 ) . l n this definition. displ ace-m C 1'1 t S d tl C t 0 t O f S i O IR a l P h C n O m e 11 a a r 0not included. ln seismic design. the in-fluence of torsion on the displacem entof the centre of mass may be consid-ered to represent unu'arranted analyti-ca1 refinem ents.

D uctility Relationships

Onc: the noluinal strengths and yielddisplacenAents. and hence the stiffness.of components or eleluents have beenset. the relationships between differentdisplacennents and displacement duc-tilities. to be considered in the design-are readily established.

Components ofLateral Force-resisting Elem ents

A Re-definition of Stiffness

ln the context t)f this studl'- stiffness re-lates the total lateral force to a corre-sponding horizontal displacelzlent.

The value of the eleluent yield dis-placenAents so derived is shown dis-tinctly in Fig. 4. Because the sinxulta-neous yielding of conlponents with dif-ferent yield displacements. -.y. is notpossible- son'le components will yieldat displacements less than that of theclement . .-y o . Larger displacenAent willbe required to develop the nolminalstrength of those connponents for wich..X) i > -X) o.. -lXe non-linear transitionfrom elastic to plast ic behaviour.shown by the full line marked rz-lbtalpi n Fig. -/, - i 11 tls t ra tes thi s fe a t ure .

The abo: e examples shou the fallacl'of the com nlon asstlmption. that theelement and system or global displace-ment ductilitq' factor is the sam e as thatspecified in codes for appropriatelydetailed components. A displacementductilitq' capacity of f of the elementdepicted in Fig. 4. im plies a ductilitydemand on component (4) of 8.t.:4 = 5 )x:0.58//0.5 = 5.8.

Bi-linear responses of 4-col-nponent e 1-elrents were presented in Fig. 4 . l t wasstated that the displacel-nent capacityof th ts elel-nent is controlled by that ofits critical component ( 4 ) with thesmall est yield displacen-lent. The sul'nof com ponent relative stiffness wastbund to be 52 k . = 1.72 . Equation ( 7 ) al-lows the yield displacelnent of the ele-ment in 1ig. 4 to be determined as .1 .CL= 1/1.72 = 0.58 displacelnent tlnits.'-rherafore. with the ultimate displace-menl lim ited to :5 )

Allocation of Strength

I t was posttllated that. irrespective ofthe way strengths were assigned tocolnponents. the sequence of yieldingwi 11 depend on ly on the yield displace-l'nents of the com ponents. A' s previous-ly stated. for given material properties( 6., ). these are controlled by the geo-nnetry rather than t he strength of

ln the first case the column shearforces and hence rele: ant strenghts.are assigned in accordance with con-N'entional l,/hSj proportionality. In theother case shear forces are madeinN'ersell' proportional to columnheights. For this case the bi-linear sim-ulation of the fivc columns is alsoshown in Fiz. 6. Svith the use of eq. (7).the bi-linear sim ulation allows the ele-ment yield displacem ents. A) u,. for eachcase. i.e.. 1 .364 and 1.667 displacementunits. respectively- to be estim ated.These are inversely proportional to el-ennent stift-ness.

- The adN'antages of arbitrary strengthallocation to components outNveighthe disadq'antage associated Nvit h t hesomewhat reduced ene rgl' dissipa-tion capacity' of the elel-ne nt . Nolethat post-l'ield stiffness. ineN'itablypresent- has not been considered inthis con-lparison.

- 'Fhe exanl ples shou'' t 1) at an al-lo'wance for a global displaceluentductility ca pacit y of 6. U hich lnaj'haq'e been adopted for ductilefral-nes. urould grossly tlnderestinlatethe ductility del-nand on the criticltlcol-nponent- i .e. col unl 1-1 ( 1 ) . ( For ex-aluple p- .. 1 ma . = 6 ) > 6 ) .

- The exanaple denxonstrates how theelennent displ acen-lent duct i I i ty de-ln an d sh 0 ul d be l ilm i t e d t o a pp- rox i -I'nately 4. if the critical ctlnlponent ist o be pro t ect e d aga i !'1 s t exces s i N'edisplacenlent dennands. As a corol-lary. the st udl' of the exan-lpl e strtlc-ture reveals that it is not ptlssible totlt ilize the d uct ility capaci t#' ol' con-l-p onents U i t 1-1 larger l'ic ld displ ace-nen ts. For exan-lple la--j = (n?''4 =1 .5%

E1 E1 L D T (D Q E1ED D E1 E1 El Q E1 ED1.0 0 ' 1.5 2.0 1'0

w- . - Vi ..*- v .

(D e Y d

A =0.5721 . 0 e - *=o q- - t . ph=5.2i

:? , i Totalso . 10

.8 , j> - y = /: 2 (arbitrarvo

. ni c jc .m 0.6= z y = j,'a (conventional) I ?Jo . ni c

# t ( ;Q' 04 I i Jta2 = 6. Oo I 2 & 4 IQ

1 /faa=4.58 o 2 3k ' I k k /ta: =3.01 d 5

J ! q.0.0 0

.5 1 1.5 2 2.5 3

Normalized Iateral displacementsof shear strength would follow theprinciples described in connectionwith the model element slzown inFig. .?( a). Details of this are not givenhere.

The relative de pths of columns f' l ) - (2 )and ( 3 ) are 1 .()- 2.() and l .5. respective-13.7. Correspondingly the relative yielddisplacen-lents. being inversely propor-t i on al to th e de p th of the sect ions ( e q .(4) )- are 1).1 = l .()00. - .l).a = ().51-.)() and-s.y .A = 0.667. respectively. The conven-tional assignment of (shear) strengths.in proportion to h?:. would result int hese 3 columns havi ng to resist 4.7 %' .3 7 . 4 8'?,' a n d l 5 . 8 %' o f t h e t o t a l b a s eshear. Vs. respectively. 'The lateralt'orce-displacement response of the 5-comp. onent elem ent so designed. issh own by the dashed line in Fig 8.

having q'ield displacements larger that0.500 displacem ent units. cannot beutilized.

R eferences

Conclusions

( 1 1 RA L- L-A'L''- 'F. .b'ei.%lllil' Desi z?? flll' 77')F'.$;/f 1lltIIJt(opt),'!.5'(7 t?./' I.)ltL'f f'/t, Bltiltlillgs. B u l l t2 t i 1'1 ()!' t heN Z Stlciet: 1''tlr E Clrtl'lq uake E ngi n eeri ng. N'ol .''' 9 . N. t3 . 3 . S k? p l c n'l la e r . p p . 1 7 S- l 9 Fl . 1 t? t?6 .

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? iL' E J'-/crA' ill B d iltl-il l t'rh . B u i -0 f I t2 t i 11 of t 1) u? N Z S ()c 1 e t ),' f or E a rt h q u a k e E n gi -

n t.l tl r i n g . N b l . 3( ) . N' k) . 3 - S e p t e n'l b e 1* . 17 17 . 2 5 2 -2 (?3 -l t) 9 7

1. The recognition of the eriteriaperformance-based seismic designnecessitates m ore attention to begiven to realistic estim ates of lateralforce-induced structural deforma-tions.

( .F j P.AL' Ln.4'N'. T. ' Jt't.'/ztl/? J'.5',?? () f .I)lf z'Iilfn B1 /jJt'J' /?7 trSb'st'Q,?l;

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s a.% ('(f'.r. L'L':,ti /') '' '.I'I.tr.%i()I: . Procvwckl n gs N ZS t 1 c 1 e t ).' fo r E a 2* t h q u a k e E n g i n e e ri n g . Tc c h n i c a l(.'t3 12 f e re nce . 5N'a i ra k e i . 1.3 p. l 1 - l 1 1hl . l tl9s .

Nlaking use of the designer's freedomin choosing any arbitrary assignm entof component strengths. it is decidedto make colum n base shear strengthsproportional to the square of the sec-tion depths. This technique leads to awell balanced design- whereby the ra-tio of the vertical reinforcement is ap-proximatell' the same in all columns.

2. lm proN'ed techniques for the esti-mation of yield displacements arepostulated. The latter depend onlyon matcrial properties. such as lim-iting strains. and the geometry ofcomponents of elements of thestructure. For design purposes gen-erally yield displacements may beconsidered to be independent ofthe strength assigned to compo-nents or elements.

l 41 PRIES-I'L E N-. 5,1 . J. N. ..f j'lll5 tl?lt/ I.'tlIILl(.i(z.% fFJIL' ('?rr/?(jI/(f/..'t' ILtl v//3f.,t.z?'j

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? ft - (.-t??3 ''-liL'l ff ll(.,l'b.b'ecllflcif/tv? (l!(I /?c/?D'r'k' . .Anlerican Ciln crtt tc I n>t i -tute ( SP- l 577 ). Rtlcent Developmcn t in LatklrlForce Tran s fe r i 11 B u i 1 J 1' ngs. pp. 2-7 1 ---' 5. 1 t) 95 .

( 51 PA RK. R. alpd PAL ' LAS-. T. Relbl /'f ?7'('('f/ Cf 1ll -('r(,:(, .h'rF-l/t'!I/?'t4.y . Jtlh n 'sN'ile).' and Sons. Neu' 'York .pp. 7 6 C.) . 1 97 5 .

(6 ! PR IESTLE 5'. 5l. .l. N.. and KIIB'.ALS KA'. $. 1.J. z.l %j3 6 Q-1.% f) 4* /) l'iti /F2/ dqllf L'ililb' U-tlJ?fJt'/J )' () /' Jt (>c'-.?(.111 trd Ift'f ?' ..$'i'?'/d(..i'lf 2'tlI '$1 -t111.%. (B u 11 r.'.L t i .1'1 (Jf t llt2 'Z'N;IZSocie t)' for Earthq uakrz En gineerin g. N '01 . .3 1 .N o. 2 . pp. 7 3-85 . l gt? 7 .

7 1 P/%.t,-L-A5'. T. .,4 .'.f?)z?/J/t4' f-'.:'?'.'/?.//k!T't,???t>??/ f 3..3114.3:-.f?/')j/Ff '

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-l')tlscd JltzA'jtrp? .$'lrt/

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itzfk''$' tk)?' J?gJ'?? t't l?'c'tat'/

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C-ol.rkol'e f'l,?j,J/i'F2 zs. Proceedi n gs t ) 1' t l)e 1 2th5A' 't) rl d Co n fe re 1: ce o 1) L( a rl h qua k k2 E n gi n ,J c ri ng.P a p e r N 0 . ( ) ( ) (n 2 . 2 t ) ( ) ( ) .

As the full lines in Fig. 8 shoNv. thestrengths of the three types of colum nswo u l d t h e n c o 1' re s p o n d t o 8 . 2 % . 32 . 6 9, ()and 1 8 .4 g'o respecti vely. of the totalbase shear. V n. The bi-linear idealiza-tion for each co1 umn componen t so de-signed is also sh own in Fig. 8. lt is seenthat the diffttrence between the oqeerallresponses. based on the conventionaland arbitrary assignment of compo-nent strengths. is negligible. lt shouldbtt noted that the yield displacem ents01' the columns are the sam e in b0thcases. Assuming again that the dis-placement capacity of the columns is 6.t 11 e e 1 em en t di sp. l acem ent cap aci ty iscontrolled b),' that of columns (2 ) and(4 ) - i .e.. p-ai aly i = 6 X 0.5 = 3.0 displace-l'nent units. Therefore. the displace-ment ductility dem and on the 15 ve-col-umn element shotlld be lim ited to p-ac:G 6.0/0.572 = 5.2. 'The full displace-n'lent ductilitl' capacity of the t'olumns

structural Engineering lnternational 122001

3. Beca use the sequcnce of the onsetof yielding of components of a plas-tic m echanism is independent oftheir strength. wilhn rational limits.strengths mal' be assigned to com-ponents in an),' way that suits thedesigner's intentitlns.

4. Re-defined stiffness. relating freell'chosk?n strengths to strength-inde-pendent yield displacements. en-ables 24 more real istie assessment ofthe stiffness of elem ents or of a sys-t e m t o be m ade .

5. Clearly defined yield displacennentsof conzponents en able displ acen-lentand tlisplacel-nent ductility denAandson the systelm to related to the dis-placement ductilitl' capacity of thecrilical conlponents.

S ci e n ce a T'l d '-IQ c'h u ( ) ? ( 'g r