Malaga Et Al. STESSA 2009

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1 INTRODUCTION

Among the different methods available for seismicanalysis, non-linear dynamic analysis (also referredto as time-history) is widely recognised as the mostreliable technique due to its ability to reflect the time

dependent nature of the seismic behaviour. Never-theless, the high level of expertise it requires as wellas the significant time and costs associated with ithampers its wider application in practical assessmentand design.

Modern earthquake resistant design philosophiesaim to achieve reliable structural performance whilstmaintaining an economic design by ensuring energydissipation capabilities under strong ground-motions. Such energy dissipation is usually attainedthrough high levels of plastic deformation in certainspecially designed elements whilst elastic behaviouris ensured in other structural members. If elastic de-formations become relatively insignificant (e.g. atlarge deformation levels), rigid-plastic models can

be adopted to estimate the structural response.The main assumption of a rigid-plastic model is

the disregard for any elastic deformation. This inturn means that the stiffness alternates between infi-nite and zero levels. This avoidance of elastic stiff-ness considerations introduces a great deal of sim-

plicity to the analysis as illustrated in subsequentsections of this paper. In general, rigid-plastic mod-

els assume that all sources of energy dissipation aredue to the plastic behaviour and hence non-hysteretic damping is ignored.

One of the first studies dealing with rigid-plasticapproximations in earthquake engineering was the

work by Newmark (1965) who devised a method toestimate earthquake induced displacements on slopestability problems where relative displacements startto accumulate whenever the acceleration exceedscertain resistance thresholds. In fact, this viscous-like idealisation has become an important field of re-search in soil mechanics (Enoki et al., 2005). A fewyears later, Augusti (1969) proposed general formu-

lae for the solution of the dynamic response offramed structures based on rigid-plastic oscillatorssubjected to simple pulse and periodic loading.Paglietti & Porcu (2001), by means of the same clas-sic rigid-plastic model, calculated the maximum dis-

placement response of several SDOF systems toseismic loading. Their study also introduced the ideaof the so called rigid-plastic spectrum defined as therelationship between a structural response parameter(such as peak structural displacement) and the plas-tic capacity of the rigid-plastic model. More re-cently, and based on such relationship, Domingues-Costa et al. (2006) proposed an earthquake resistantdesign procedure for moment-resisting frames,modifying the classical rigid-plastic model to takeinto account pinching effects typical of reinforced-concrete structures. This study highlighted the rela-tive merits of rigid-plastic approaches, which led torenewed interest in exploiting its advantages, par-ticularly in terms of efficient seismic design and as-sessment procedures.

The present study extends the advantages associ-ated with rigid-plastic models to different typologies

of steel-framed buildings under earthquake loading.To this end, new hysteretic relationships (as well asits corresponding integration algorithms) are pro-

posed. It is shown that such rigid-plastic models areable to predict global demand response histories

Seismic design and assessment of steel structures based on rigid-plasticresponse history analysis

C. Málaga-Chuquitaype & A.Y. Elghazouli Department of Civil and Environmental Engineering, Imperial College London, UK

R. Bento

Department of Civil Engineering and Architecture, ICST, IST, Technical University of Lisbon, Portugal

ABSTRACT: This paper demonstrates the ability of response history analysis based on rigid-plastic oscilla-tors to predict global seismic demands of steel framed structures with reasonable accuracy. Rigid-plastic hys-teresis models are developed for concentrically braced frames and dual structural systems consisting of mo-ment-resisting frames coupled with braced systems. It is shown that, the direct relationship existing between

the displacement response parameters and the plastic capacity of the structure can be used to determine thelevel of seismic demand given a certain performance target.



with reasonable accuracy. Finally, this paper pre-sents a brief discussion on the use of relationships

between peak displacement and the plastic capacityof rigid-plastic oscillators as a mean for obtainingthe required structural strength given a target sce-nario.

2

RIGID-PLASTIC MODELS UNDER SEISMICACTION

2.1 Moment-resisting frames (MRF)

Figure 1a presents a classic rigid-plastic oscillator ofmass m, height h and plastic moment capacity at thecolumns M 0, subjected to a ground acceleration a g (t).For such a rigid-plastic oscillator, the yield capacityof the structural system defined by F yp = − F yn can

be related to the limit moment at the plastic hinges by:

F yp =4 0

h (1)

similar conventions can be introduced for other,more complicated, rigid-plastic structures.

(a) MRF rigid-plastic oscillator. (b) Force-displacementrigid-plastic response.

(c) Rigid-plastic and elastic-plastic displacement histories ofSDOFs subjected to the JMA record from the 1995 Kobeearthquake with PGA/a y=6.

Figure 1. Rigid-plastic oscillator subjected to ground-motion.

Figure 1b presents the rigid-plastic response ofthe above described MRF oscillator as a function of

lateral force and displacement. In this figure, twomain behavioural states are evident: (i) a rigid statewhere the capacity of the member is not reached andhence no displacement takes place and, (ii) a plastic

state where deformations occur and the strength de-mand is equal to the yield strength.

Expression of dynamic equilibrium by means ofD’Alembert’s principle over the rigid-plastic oscilla-tor of Figure 1a yields the following equation for therelative acceleration ar (t) as a function of time t :

ar (t ) = F (t )

m− a g (t ) (2)

where F(t) represents the internal forces opposingthe motion. With the previous assumptions and defi-nitions, it can be immediately recognised that Equa-tion 2 can be replaced by the following set of rela-tionships, which are more practical for computer-

based numerical integration:

ar (t ) = F yp

m− a g (t ) if vr (t ) > 0

ar (t ) = F yn

m− a g (t ) if vr (t ) < 0

ar (t ) = 0 if F (t ) ∈ F yn , F yp

] [

(3)

where vr (t) is the relative velocity of the mass m, andhence a change of state occurs whenever vr (t) chan-ges from positive or negative to zero. For this pur-

pose, a numerical integration scheme assuming con-stant acceleration over a ∆t period of time wasimplemented (Málaga-Chuquitaype, 2010).

Figure 1c presents a comparison between theseismic response in terms of displacement history ofa rigid-plastic oscillator and a number of elasto-

plastic SDOF models with elastic periods varyingfrom 0.6 to 1.4 seconds, to a selected record fromthe 1995 Kobe earthquake. This record imposes a

peak demand of 6 times the yield strength of thestructure or, in other words, a ratio between peak-ground acceleration ( PGA) and ground accelerationat yield a y of 6. It should be noted, as pointed out byPaglietti & Porcu (2001) that the single most impor-tant parameter governing the accuracy of rigid-

plastic prediction is the overall contribution of elas-tic deformation, for which the ratio PGA/a y is a goodindicator.

2.2 Concentrically-braced frames (CBF)

The seismic performance of concentrically-bracedframes (CBF) depends largely on the inelastic re-sponse of their bracing members, and design ap-

proaches in current seismic codes typically aim toconcentrate inelastic deformation in these elements(Elghazouli, 2003). The main behavioural aspects ofthe cyclic response of braces are: (i) successive el-ongation that causes yielding in tension to occur atincreasing axial deformations and, (ii) gradual deg-radation of the compression resistance with increas-ing cycles. Several models have been proposed toestimate the cyclic response of bracing members(Ikeda & Mahin, 1984, Remennikov & Walpole,



1997). Nevertheless such models require consider-able information in order to define the various pa-rameters involved. In this section a simple modellingalternative is proposed through a rigid-plastic repre-sentation.

The proposed rigid-plastic relationship for CBF is based initially on the behaviour of a single-storey braced oscillator with pinned joints and rigid ele-

ments as shown in Figure 2a, where all the inelastic behaviour is concentrated in the bracing members.

(a) CBF with pinned connections subjected to ground-motion.

(b) Equivalent rigid-plastic response of CBF at storey level.

(c) Displacement histories for λ =1.00 under El Centro recordwith PGA/a y=4.

Figure 2. Rigid-plastic oscillator of a CBF subjected to ground-motion.

Figure 2b presents the corresponding rigid-plastichysteretic response in terms of inter-storey drift andforce. The tension elongation is taken into account

by allowing rigid behaviour only if the previous peak deformation is reached. Furthermore, themodel assumes that the system is not able to sustainany load reversal until the load carrying capacity isrecovered by tension action in the alternate brace,hence introducing effectively a slip-like behaviour.The adopted idealisation ignores the contribution ofthe braces in compression, which in principle is ac-ceptable for slender braces noting that the brace will

buckle at a relatively early stage in the response his-tory (Broderick et al., 2006). Where necessary, for

bracing members with relatively low slenderness, an

alternative model proposed in the next section can be used in order to take into account the contributionof the compression resistance of the bracing mem-

bers.Following the previous discussion, a new set of

equations defining the rigid-plastic model can bedevised:

ar (t ) =

F yp

m −a g (t ) if vr (t ) >0∧d r (t ) > d (t p)

ar (t ) = F yn

m−a g (t ) if vr (t ) <0∧d r (t ) < d (t n)

ar (t ) =0 if vr (t ) =0∧ F (t )∈ F yn, F yp] [ar (t ) =−a g (t ) if F (t ) =0∧d r (t )∈ d (t n),d (t p)] [

(4)

where t n and t p are the instants of time at whichthe peak negative or positive displacement isreached, respectively, within a specific interval oftime [0, t ] and, as in the previous section, the valueof acceleration is considered constant over a ∆t pe-

riod of time.The response of the rigid-plastic oscillator pre-

sented in the previous section is validated hereinagainst refined analyses where attention is placed onthe prediction of the displacement history. The sin-gle-storey two-bay SDOF braced frame shown inFigure 2a is considered with 5 metres overall widthand 2.9 metres height. A refined numerical modelwas constructed in Adaptic (Izzuddin, 1991) using

pinned connections and rigid elements for all thestructural members except the braces. The elastic

modulus of steel is assumed as 200x103

N/mm2

andits yield stress as 300 N/mm2 with 0.5% of strainhardening. The braces were modelled with eight cu-

bic elasto-plastic elements of 50 cross-sectional fi- bres. Notional loads at the midspan were used in or-der to simulate initial imperfections of the order of1/200 of the brace length.

The only parameter defining the rigid-plastic re-sponse is the storey shear capacity at yielding of the

bracing system. This was calculated considering the plastic section capacity of the brace in tension only.As an example, the rigid-plastic prediction is com-

pared with the refined fibre-element model in Figure2c for the case of a 50x2.5 SHS brace (member nor-malised slenderness λ of 1.00) to the El Centro re-cord for PGA/a y of about 4. As shown in Figure 2c,very good comparison is observed.

The same favourable correlation was confirmedthrough further analyses with other ground-motionrecords. In general, good estimates of displacementhistory can be expected for ratios of PGA/a y greaterthan three and slenderness larger than 1.5. Underthese conditions the displacement error is typically

less than 10% (Málaga-Chuquitaype, 2010).



2.3 Dual frames

The single-storey oscillator presented in Figure 3a isused here in order to derive a simple rigid-plastichysteretic approximation for dual structures. Theseismic response of such oscillator combines the dis-sipative behaviour of bracing members at the storeylevel addressed in the previous section with flexuralresistance in the columns where classic rigid-plastic

hinges are formed. The related hysteretic behaviourin terms of force and interstorey drift is presented inFigure 3b.

(a) Concentrically-braced frame with fixed connections sub-

jected to ground-motion.

(b) Equivalent rigid-plastic response of dual structure at storeylevel.

(c) Displacement histories for λ =1.30 and flexural plastichinges contributing 32% of the total resistance under El Centrorecord with PGA/a y=4.

Figure 3. Rigid-plastic oscillator of dual structure subjected toground-motion.

Typically, the lateral resistance of the bracingsystem would exceed the lateral resistance of themoment frame. Consequently, the effects of therigid-plastic hinges formed in the beams or columnswill be to eliminate the slip behaviour periods where

no resistance is present (i.e. as in the rigid-plasticmodel for CBF shown in Figure 2b). This allows thestructure to mobilise actions in reverse loading butwith a net flexural resistance ( F sn or F sp) provided bythe hinges in the MRF. In this idealisation, the pos-

sibility of internal classic rigid-plastic hysteresisloops is introduced whenever the system oscillateswithin the limits imposed by the elongated bracingmembers. The full lateral resistance is not recovereduntil the previous peak displacement is reachedagain. Hence the set of equations defining the dualrigid-plastic response are:

ar (t )=

F yp

m −a g (t ) if vr (t )>0∧d r (t )>d (t p)

ar (t )= F yn

m−a g (t ) if vr (t )<0∧d r (t )<d (t n)

ar (t )= F sp

m−a g (t ) if

F (t )= F sp∧vr (t )>0∧

d r (t )<d (t p)

ar (t )= F sn

m−a g (t ) if

F (t )= F sn∧vr (t )<0∧

d r (t )>d (t n)

ar (t )=0 if

vr (t )=0∧

F (t )∈ F sn, F sp] [∧d r (t )∈ d (t n),d (t p)] [∨

F (t )∈ F yn, F yp] [∧d r (t )∈ d (t n),d (t p)( )

⎛

⎝

⎜

⎜⎜

⎟

⎟⎟

(5)

where F sp = − F sn is the lateral resistance of theflexural plastic hinges formed in the MRF and F yp =− F yn is the overall lateral strength of the system con-sidering the strength contribution of the bracing sys-tem and other variables are as previously defined.The proposed rigid-plastic model was validatedagainst the results of detailed non-linear responsehistory analyses for single-storey frames with fixed

bases modelled in Adaptic. Cubic elasto-plastic ele-

ments with distributed plasticity were employed forthe braces and columns. The elastic modulus of steelwas taken as 200x103 N/mm2 and the yield stress as300N/mm2 with 0.5% strain hardening.

For records with very large PGA and more slen-der braces, the rigid-plastic approximation may pre-sent a slight tendency to underestimate the peak dis-

placement predictions compared to the refined non-linear model. Nevertheless, in all cases the estimatesare within the ±10% band. In general, for the fullrange of slenderness studied by Málaga-

Chuquitaype (2010) from 1.3 to 2.1, the rigid-plastic procedure provided good predictions of peak displacements when the frame was subjected to accel-eration time histories with PGA of 3 or more timesthe acceleration at yield. Figure 3c presents a typicalcomparison of displacement histories.

3 RESPONSE PREDICTION OF MULTI-STOREY BUILDINGS

In this section, the ability of the proposed rigid- plastic model is assessed against the prediction ofdisplacement histories obtained by means of a de-tailed multi-degree-of-freedom (MDOF) non-linearmodel of a four-storey building. The structure under



consideration has the elevation presented in Figure4. The dead load included the weight of partitions (1kN/m2), finishing (0.3 kN/m2) and a compositeflooring system (2.88 kN/m2 on average), while 2.5kN/m2 was considered as superimposed load. Fixed

bases were assumed at the bottom of columns andthe building was designed in accordance with Euro-code 8 (CEN, 2005).

The same modelling considerations described inSection 2 were adopted and both the detailed MDOFnon-linear model and its equivalent SDOF rigid-

plastic representation were used. The El Centro re-cord was scaled so as to induce a peak force (com-

puted from the PGA and the effective mass) of 3times the actual base shear capacity of the building.The corresponding results in terms of displacementat the effective height are presented in Figure 4b,where good agreement over the whole response his-tory can be observed.

(a) Four storey dual frame under study.

(b) Displacement histories at effective height under scaled El

Centro record with PGA/a y=3.

Figure 4. Rigid-plastic analysis of a four-storey dual frame.

4

RIGID-PLASTIC MODELS AS DESIGNTOOLS

As noted previously, the only parameter definingrigid-plastic models, such as the ones proposed ear-lier in this paper, is the plastic capacity either interms of yield force or moment. Even for dual sys-tems, the contribution of the flexural plastic hingesto the overall lateral capacity of the structure can beexpressed as a percentage of the total lateralstrength. This makes response history analysis a

relatively straightforward task once the lateralstrength of the structure has been evaluated.

From another viewpoint, the single-parametercharacteristic of rigid-plastic dynamics enables theevaluation of the required strength given a certainresponse target. In this context, plots of peak dis-

placement as a function of the lateral capacity of therigid-plastic oscillator have been proposed by Pagli-

etti & Porcu (2001) and referred to as rigid-plastic“spectra” to characterize the earthquake demand.It is worth noting that previously design proce-

dures based on rigid-plastic response history usedthe so-called “envelope” (Domigues-Costa et al.,2006) which may have little correspondence with aspecific earthquake scenario. As an alternative, andas depicted in Figure 5, the median value can beconsidered as a more meaningful representation ofsuch relationships. However, the median plus anumber of standard deviations could also have beenused but this would perhaps require a larger number

of records in order to capture with confidence thevariability associated with the process. Also, the dis-

placement-plastic resistance curve expressed interms of the median does not imply that such re-quired strength will be associated with the median

probability as the ability of the rigid-plastic oscilla-tor varies as a function of the plastic resistance itself.Further studies to define a proper consideration ofthis would be required.

Figure 5. Displacement versus base shear relationships ob-tained by means of rigid-plastic oscillators under a set of se-lected ground-motions.

In theory, a designer can follow an approachthrough which serviceability checks, associated withfrequent events, can be carried out using conven-tional elastic models. On the other hand, validationof the collapse-prevention response under extremeevents can be performed using rigid-plastic ap-

proaches. This can be implemented in a perform-

ance-based design framework of the type presentedin Figure 6. Such a design framework allows reli-ability levels to be estimated with due account of thereal earthquake variability. In addition, such frame-work facilitates the understanding and selection of



performance levels and enables different levels ofstructural optimization without the computationaldemand and inherent complexity associated withnon-linear finite element response-history analysis.

Figure 6. Seismic design framework with decoupled verifica-tion of stiffness and strength based on bi-level performance tar-gets.

5 CONCLUDING REMARKS

The applicability of rigid-plastic models in predict-ing the seismic behaviour of typical steel framed

buildings has been investigated with particular em- phasis on moment-resisting (MRF), concentrically- braced (CBF) and dual configurations. The principaladvantage of using response history analysis basedon rigid-plastic models has been highlighted as theirability to predict with good levels of accuracy the

history of structural response while keeping thecomputational costs and the detailed knowledge re-quired at a relatively low level.

This paper proposed new rigid-plastic hystereticmodels for CBF and dual systems that take into con-sideration the specific behavioural characteristics ofsuch structural configurations. It has been shownthat in general, good estimates can be achieved if theground-motion pushes the structure well into theinelastic range with the ratio PGA/a y serving as agood parameter to quantify such ability. In particu-lar, rigid-plastic models perform well for demand

levels equal or greater than 3 times the plastic capac-ity of the structural system which in turn is in linewith the collapse prevention objectives in currentseismic codes of practice. It is also observed(Málaga-Chuquitaype, 2010) that the more slender

the braces are in CBF systems, the better the re-sponse prediction obtained from the idealised rigid-

plastic model, particularly when braces with a non-dimensional slenderness greater than 1.5 are em-

ployed. Nevertheless, when the contribution of flex-ural strength is taken into account, such dependenceon the slenderness value is reduced and the responseis adequately predicted for normalized slenderness

values between 1.3 and 2.1.Finally, design methodologies based on rigid- plastic relationships between plastic capacity anddemand parameters, previously applied to reinforcedconcrete moment resisting frames, have been dis-cussed and extended for steel buildings. The medianresponse was proposed as a more meaningful valuefor which good approximation to the target demandwas achieved given a certain hazard scenario.

Overall, this paper shows that rigid-plastic modelscan be used as a simple yet reasonably accurate toolfor predicting the seismic response of typical con-

figurations of steel buildings , under severe seismicloading, for both assessment and design purposes.

REFERENCES

Augusti, G. 1969. Rigid-plastic structures subject to dynamicloads. Meccanica 5:74-84.

Broderick, B. & Elghazouli, A.Y. & Goggins, J. 2006. Earth-quake testing and response of concentrically-braced sub-frames. Journal of Constructional Steel Research 64:997-1007.

CEN 2005. EN 1998-1, Eurocode 8: Design provisions for

earthquake resistance of structures, Part 1: General rules,seismic actions and rules for buildings.

Domigues-Costa, J. & Bento, R. & Levtchich, V. & Nielsen,M. 2006. Rigid-plastic seismic design of reinforces con-crete structures. Earthquake Engineering and StructuralDynamics 36:55-76.

Elghazouli, A.Y. 2003. Seismic design procedures for concen-trically braced frames. Proceedings of the Institution o CivilEngineers, Structures and Buildings 156:381-394.

Enoki, M. & Xuan Luong, B. & Okabe, N. & Itou, K. 2005.Dynamic theory of rigid-plasticity. Soil Dynamics andEarthquake Engineering 25:635-647.

Ikeda, K. & Mahin, S. 1984. A refined physical theory model

for predicting the seismic behaviour of braces frames. Re- port UMEE 77R3, Department of Civile Engineering, Uni-versity of Michigan, USA.

Izzuddin, B. 1991. Nonlinear dynamic analysis of framedstructures. PhD Thesis, Department of Civil and Environ-mental Engineering, Imperial College London, UK.

Málaga-Chuquitaype, C. 2010. Seismic behaviour and designof steel frames incorporating tubular members. PhD Thesis,Department of Civil and Environmental Engineering, Impe-rial College London, UK.

Newmark, N. 1965. Effects of earthquakes on dams and em- bankments, Fifth Ranking Lecture. Geotechnique 2:139-160.

Paglietti, A. & Porcu, M.C. 2001. Rigid-plastic approximation

to predict plastic motion under strong earthquakes. Earth-quake Engineering and Structural Dynamics 30:115-126.

Remennikov, A.M. & Walpole, W. 1997. Analytical predictionof seismic behaviour for concentrically braced steel sys-tems. Earthquake Engineering and Structural Dynamics26:859-874.

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Malaga Et Al. STESSA 2009