02. 2001 Restrepo Floor Horizontal Accelerations

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2002; 31:693718 (DOI: 10.1002/eqe.149)

Earthquake-induced 0oor horizontal accelerations in buildings

M. E. Rodriguez1;;, J. I. Restrepo2 and A. J. Carr3

1National University of Mexico; Apartado Postal 70-290; CP 04510; Mexico City; Mexico2Department of Structural Engineering; University of California; San Diego; 9500 Gilman Drive;

La Jolla; CA 92093-0085; U.S.A.3Department of Civil Engineering; University of Canterbury; Private Bag 4800; Christchurch; New Zealand

SUMMARY

Floor horizontal accelerations are needed for obtaining forces for the design of diaphragms, for thedesign of their connections and for the design of non-structural components and equipment supportedby structures. Large 0oor horizontal accelerations have been recorded in buildings during earthquakes.Such accelerations have been responsible for inertia forces causing damage to services and are a majorreason for structural damage and even building collapse.

This paper describes an analytical investigation into earthquake-induced 0oor horizontal accelerationsthat arise in regular buildings built with rigid diaphragms. The paper also describes several methodsprescribed by design standards and proposes a new method. The method is based on modal superpositionmodiAed to account for the inelastic response of the buildings lateral force resisting system. Resultsobtained from time-history inelastic analysis are compared with the proposed method. Copyright ? 2001John Wiley & Sons, Ltd.

KEY WORDS: 0oor accelerations; diaphragms; non-structural components; non-linear analysis; seismicdemands; building codes

1. INTRODUCTION

The impact and cost of the consequences of damage caused by earthquakes worldwide duringthe past 12 years has raised the question of whether current building seismic design proceduresare satisfying the needs of modern society. Most seismic design standards are based on a life-prevention approach where structural damage is accepted providing that collapse is avoided.No other economic parameters, such as the cost of damage to equipment and stored goods andthe cost associated with loss of operation following a moderate=strong earthquake, are currently

Correspondence to: M. E. Rodriguez, National University of Mexico, Apartado Postal 70-290, CP 04510, MexicoCity, Mexico.

E-mail: [email protected]

Contract=grant sponsor: New Zealand Foundation for Research Science and Technology; contract=grant number:UOC 808.

Received 15 January 2001Revised 31 August 2001

Copyright ? 2001 John Wiley & Sons, Ltd. Accepted 31 August 2001
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694 M. E. RODRIGUEZ, J. I. RESTREPO AND A. J. CARR

Figure 1. Floor acceleration magniAcation obtained from records in instrumented buildings during theNorthridge earthquake (Adapted from Hall, 1995).

accounted for in the design process. The philosophy behind the newly proposed performance-based seismic design intends to encompass, directly or indirectly, those parameters within aset of objectives aimed at ensuring predictable behaviour of the entire building envelope [1].

Within the framework of performance-based seismic design, signiAcant eLort has been madein recent years at developing new methodologies [2; 3] and new structural systems in a waythat the design objectives can be achieved [4]. The design methodologies tend to give moreemphasis on obtaining better estimates of lateral deformations. Nevertheless, there are areasrelevant to performance-based seismic design that have been the subject of limited researchwork. For example, the determination of 0oor horizontal accelerations in buildings withemphasis on practical applications is one of them [58]. Floor accelerations are needed forobtaining in-plane forces for the design of diaphragms and their connections to the primarylateral force resisting system. Of particular importance are those diaphragms that incorporatesigniAcant openings and are built using precast concrete 0oor systems. Furthermore, 0oorhorizontal accelerations are needed for determining forces for the design of non-structuralcomponents and equipment supported on the 0oors.

It has been reported that damage to diaphragms and their connections was a major causeof poor building behaviour, and even collapse, during the 1988 Armenia [9], the 1994Northridge [10; 8] and the 1999 central Colombia [11] earthquakes. It has also been reportedthat damage to services caused business interruption in several buildings in the Northridgeearthquake [12]. In fact, records obtained during the Northridge earthquake in multistoreybuildings, other than base-isolated, showed that 0oor peak horizontal accelerations weregenerally greater than those recorded at the ground level. Figure 1 plots the maxima 0ooracceleration magniAcation versus peak ground acceleration (PGA) reported by Hall [12] for25 multistorey buildings. The maxima 0oor acceleration magniAcation was obtained as theratio of the maxima 0oor horizontal acceleration to the PGA. Floor acceleration magniAca-tions ranged between 1.1 and 4.6. These results are in agreement with those found by Soonget al. [6] for data also obtained in the 1994 Northridge earthquake. Large 0oor accelerationshave been also observed in experimental work. For example in the pseudo-dynamic test of alarge-scale Ave-storey precast concrete building reported by Priestley et al. [3] 0oor accelera-

Copyright ? 2001 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2002; 31:693718
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FLOOR HORIZONTAL ACCELERATIONS 695

tions computed from the loading algorithm led to in-plane 0oor forces that were in excess ofthose estimated with the Uniform Building Code [13]. These researchers suggested that theaugmented forces were caused by higher modes. Moreover, other researchers have reportedevidence of the importance of higher mode eLects on in-plane 0oor forces. Eberhard andSozen [14], examining the base-shear response of structures in terms of incremental stiLnessand using modal analysis, found that after a mechanism formed, further changes in forcedistribution occurred as a result of the 0uctuation of the higher modes.

The main objective of the paper is to improve the current understanding on how 0oor hori-zontal accelerations arise during earthquakes in buildings. Emphasis is given to the interactionbetween a building non-linear response and the magnitude of the 0oor accelerations. Using anapproach based on the concepts of linear structural dynamics, the authors propose a simplemethod for determining 0oor horizontal accelerations and design forces.

2. REVIEW OF CURRENT DESIGN PROVISIONS

A review of seismic design provisions revealed that several approaches are used in theevaluation of the horizontal forces needed for the design of diaphragms, non-structuralcomponents and equipment in buildings. For example, and as described later in this section, theUBC [13] requires for the design of diaphragms horizontal forces greater than those used in thedesign of the lateral force resisting system. The UBC also contains provisions for the designof non-structural components and equipment supported at the 0oors. These provisions arediLerent from those for the design of diaphragms. In Mexico the design forces for the lateralforce resisting system and for the diaphragms are generally considered equal. In New Zealand,the Loadings Standard [16], gives provisions for the design of building parts. Accordingto this standard, diaphragms can be designed as a rigid part, which is a non-structuralcomponent or equipment having a short-period of vibration such that the response ampli-Acation under earthquake loading is negligible. The approach given by the Loadings Standardaccounts for the eLects of overstrength in lateral force resisting systems and leads to forcesthat are greater than the design forces required for the primary lateral force resisting system. Itis interesting to note that in New Zealand, diaphragms can also be designed for forces derivedfrom the Concrete Structures Standard [17]. The procedure for the design of diaphragms in thisstandard is based on the horizontal forces used for the design of the primary lateral forceresisting system multiplied by an overstrength factor. Design provisions of New Zealand, UBCand Mexico are described below in more detail.

2.1. New Zealand design standards

In New Zealand, 0oor horizontal accelerations can be derived from the Loadings Standard [16]or from the Concrete Structures Standard [17]. The Loadings Standard [16] gives recommen-dations for the design of rigid parts in buildings. Its commentary implies that in-plane forcesin diaphragms can be evaluated using provisions for rigid parts. The 0oor horizontal force,Fph, inferred from the standard is,

Fph = 2WpCfiRp (1)

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where Fph is the design horizontal force, Wp is the weight of the part, Rp is the risk factorfor the part, which varies from 1 to 1.1 and Cfi is a basic horizontal seismic coePcient fora part located at level i.

At ground level, the coePcient Cfi becomes Cfo, which for the ultimate limit state isdeAned as:

Cfo = 0:25RZ (2)

where R is the building risk factor, which, for buildings other than of secondary nature, variesbetween 1 and 1.3, and Z is the seismic zone factor, which varies between 0.6 and 1.2.

The 0oor horizontal seismic coePcient for the uppermost level of a building, Cfn, is deAnedas,

Cfn =Ch(T1; o)Ch(T1; )

FnWn

(3)

where Fn is the horizontal force evaluated for the primary lateral force resisting system at theuppermost level of the building, Wn is the seismic weight at that level. Parameter Ch(T1; )is the seismic hazard acceleration coePcient obtained from the design spectrum for the fun-damental period of the building, T1, and for the design structural ductility factor ; Ch(T1; o)is the seismic coePcient obtained from the design spectrum for the period T1 and ductilityfactor o associated with overstrength. The Loadings Standard recommends the use of o = 1unless capacity design is applied to the structure to justify a larger value. The standard doesnot give explicit guidelines for calculating o. This lack of guidance has usually meant thatdesign engineers generally select the default value of =o = 1. When using unity valuesfor and o, Equation (3) simpliAes to Cfn =Fn=Wn, which is the seismic coePcient at theuppermost principal seismic weight associated with the buildings elastic response.

CoePcient Cfi for any level i below the uppermost level of a building can be foundusing two diLerent methods depending on whether the equivalent static method or the modalresponse spectrum method is being used in the structural analysis.

Where the modal response spectrum method of analysis is used, coePcient Cfi is given by,

Cfi =Ch(T1; o)Ch(T1; )

FiWi

(4)

where Fi is the horizontal force found for the primary lateral force resisting system for leveli and Wi is the seismic weight at this level.

It is interesting to note that 0oor accelerations can also be derived in New Zealand fromthe provisions for the seismic design of diaphragms given by the Concrete Structures Standard[17]. The importance of the structural role of a diaphragm has always been recognized by thestandard to the extent that design recommendations are contained in a separate chapter. Theprocedure outlined in the current standard is based on the principles of capacity design, wheredesign diaphragm forces are equal to the design lateral forces multiplied by an overstrengthfactor. The 0oor horizontal acceleration inferred from this standard can be obtained by dividingthe diaphragms design lateral force by its mass.

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2.2. Uniform Building Code [13]

The 0oor horizontal force deAned by the UBC for non-structural components and equipmentis,

Fp =apCaIpRp

(1 + 3

hxhn

)Wp (5)

where ap is the structure ampliAcation factor for the part, Rp is the response modiAcationfactor for the part, and hx and hn are the part elevation measured from the ground and theuppermost level, respectively. CoePcients ap and Rp vary from 1.0 to 2.5 and from 1.0 and4.0, respectively. When using Equation (5) Fp shall be no less than 0:7CaIpWp and need notbe more than 4:0CaIpWp. The horizontal force in elastic parts inferred from Equation (5) canbe determined using ap = 1:0 and Rp = 1:0.

The UBC requires diaphragms to be designed to resist the force Fpx given by:

Fpx =Ft +

ni=x Fin

i=x wiwpx (6)

where the horizontal force Ft is the portion of the design base shear force, V; concentrated atthe top level of the structure in addition to the horizontal force Fn. Fx is the design horizontalforce applied to level x, wi and wx are the portion of the total seismic dead weight, W; locatedor assigned to level i or x, respectively; and wpx is the weight of the diaphragm at level x.Forces Fn and Fx are derived from the equivalent static method.

Force Fpx shall not be less than 0:5CaIwpx, and nor greater than 1:0CaIwpx, where I is thebuilding importance factor. This factor varies between 1.0 and 1.5.

The design base shear force V in this code is a fraction of the force required for elasticresponse. The ratio of the force required for elastic response and the design force is deAned asthe R factor. This factor varies between 2.2 and 8.5. According to the deAnition given by theUBC, R is a numerical coePcient representing the inherent ductility capacity and overstrengthof the lateral force resisting system.

2.3. Mexico city building code [15]

As noted above, designers in Mexico use for the design of diaphragms the horizontal forcesderived for the primary lateral force resisting system. These forces are obtained by dividingthe elastic forces by a factor that recognizes the ductility and overstrength inherent in thestructural system.

2.4. Application of current design provisions

To illustrate the application of the approaches described above, forces for the design ofdiaphragms and non-structural components were obtained for a regular twelve-storey buildingwhose primary lateral force resisting system consisted on ductile reinforced concrete structuralwalls. Floor accelerations were determined by dividing the design forces by the mass of thediaphragm or partition in consideration. Floor accelerations were normalized by the peakground acceleration obtained from the design spectrum. These ratios are referred to as 0ooracceleration magniAcations, which will be used extensively in the sections below.

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Figure 2. Floor acceleration magniAcations derived for a twelve-storey wall building.

Figure 2(a) compares the 0oor acceleration magniAcations found from the two methodsgiven by the New Zealand Standards. Floor acceleration magniAcations were calculated usingthe modal response spectrum method of the Loading Standard [16] for = 5 with the defaultvalue of o = 1. The design response spectrum for intermediate soils was used in this exam-ple. Floor acceleration magniAcations were also determined from the procedure given in theConcrete Structures Standard [17] assuming an overstrength factor, = 1:8. Factor is de-Aned as the ratio of the maximum moment at the base of walls obtained from the non-linearanalysis to the design bending moment. These methods reveal contrasting diLerences. Forexample the magniAcation found from the modal response spectrum method of the LoadingsStandard gives a top 0oor magniAcation of 4.9, while a value of 0.5 is obtained from the

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Concrete Structures Standard method. In fact, the values obtained from the latter standardsuggest that there is de-ampliAcation of the ground acceleration at all levels, which is contraryto the data plotted in Figure 1.

Figure 2(b) shows the 0oor acceleration magniAcations found from the UBC [13] designrecommendations for the design of diaphragms. The base shear force for this building wasfound using the equivalent static method with R= 4:5 and I = 1:0. The peak ground accele-ration to estimate the 0oor magniAcation was assumed equal to CaI . Two observations can bemade from the results obtained: (i) the values obtained from the analyses are always within0.5 and 1.0, which indicates 0oor acceleration de-ampliAcation rather than magniAcation, and,(ii) the values depend on the seismic coePcient Ca. Also plotted in Figure 2(b) is the 0ooracceleration magniAcation derived from the recommendation for the design of parts. In thiscase the peak ground acceleration was estimated to be equal to CaIp. CoePcients ap and Rpfor this example were made equal to 1.0. As can be seen in Figure 2(b), 0oor accelerationmagniAcations using the design procedure for parts are signiAcantly greater than those foundusing the design provisions for diaphragms.

For diaphragms designed in accordance with the Mexico City Building code [15] 0ooracceleration magniAcations are expected to be lower than those derived using proceduresfrom the New Zealand Concrete Structures Standard [17] and from the UBC [13]. The reasonfor this is the later codes deAne 0oor horizontal forces that are larger than those derived forthe primary lateral force resisting system.

In summary, 0oor acceleration magniAcations derived from the diLerent approaches diLersigniAcantly in their results. In some cases the magniAcations obtained are too high whencompared to the results obtained during the 1994 Northridge earthquake. In other cases theresults actually indicate de-ampliAcation at the acceleration at the 0oors. The consequence ofsuch diLerences can be of signiAcance when performance-based is implemented in the designof diaphragms or parts.

3. ALTERNATIVE METHOD FOR EVALUATING FLOOR HORIZONTALDESIGN FORCES

3.1. E3ect of non-linear response on the magnitude of the 4oor acceleration magni5cation

Parametric non-linear time history analyses were performed on three, six and twelve-storeyhigh buildings to observe the eLects of diLerent variables on the development and magnitudeof the 0oor acceleration. In these analyses the diaphragms were considered rigid in their plane.The plan view of a typical 0oor on the three- and twelve-storey high buildings is shown inFigure 3. As can be seen there, four 4:7 m long by 250 mm thick cantilever walls providethe lateral force resistance in the short direction of the three-storey high building, whereasthe lateral force resistance in the short direction of the twelve-storey high building is pro-vided by eight 7 m long by 250 mm thick cantilever walls. The six-storey high building, notshown here, had a distribution, number and thickness of structural walls similar to those ofthe three-storey building but with lengths equal to 7 m [18]. The 0oor system was assumedto consist of one-way precast prestressed 0oor units with 80 mm of cast-in-place concretetopping. The response of the buildings to earthquake input ground motion was investigatedfor the short direction only. The seismic weight per 0oor was assumed equal to 4700 kN. It

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Figure 3. Plan view of typical 0oor of the prototype three- and twelve-storey buildings investigated.

was assumed that the walls would provide the entire primary lateral force resistance in thisdirection.

The storey heights for the twelve-storey high building were equal to 4 m for the Arst twolevels and 3 m for the remaining levels. For the three-storey high building, the height of theArst level was 4 m, while the remaining levels were 3 m high. These buildings were analyzedand designed in accordance with the New Zealand Loadings Standard [16] and the NewZealand Concrete Structures Standard [17]. The initial stiLness of these walls was determinedusing an eLective moment of inertia equal to 0:25Ig, where Ig is the moment of inertia ofgross concrete section. Designs were carried out for limited ductility response, = 3, and forfully ductile response, = 5. Further details are given in Rodriguez et al. [18].

The analyses were carried out for two earthquake records and two types of non-linearresponse. A synthetic record, named SIM1, with a PGA equal to 0:42g, was generated tomatch the design spectra for intermediate soil conditions for the highest seismic region in NewZealand. SpeciAc details of this record can be found elsewhere [18]. The other earthquakerecord was SYLM949, with a PGA equal to 0:80g, obtained during the 1994 Northridgeearthquake. The acceleration amplitude of these records was scaled to observe the gradualeLect that inelastic response has on the 0oor acceleration. The non-linear response at thebase of the walls was represented either by a Takeda or origin-centered hysteresis rules. The



Figure 4. Elastic dynamic characteristics of the three- and twelve-storey buildings investigated.

Takeda hysteresis rule is generally used to characterize behaviour of cast-in-place reinforcedconcrete walls or the behaviour of precast concrete walls designed to respond as if mono-lithic. The origin-centered hysteresis rule, termed modiAed origin-centered rule (MOC) inthe computer program Ruaumoko [19], was used to characterize the non-linear response ofself-centering precast concrete hybrid walls, which are post-tensioned with partially unbondedtendons and incorporate energy dissipation devices. The behaviour of such walls is describedelsewhere [7; 20]. The Takeda and MOC hysteresis rules used in the parametric analysis hadidentical initial and post-elastic backbone moment-rotation behaviour. The non-linear timehistory dynamic response of the building was performed using the computer program Ru-aumoko [19].

The structural model incorporated a Rayleigh damping formulation proportional to the massand initial stiLness matrices [19]. The Arst mode of vibration was given a 5 per cent dampingratio. Care was taken to ensure that damping ratios would be similar for the Arst modes ofall buildings and to ensure that the higher translational modes would not be highly damped.Figure 4 shows the mode shapes, periods of vibration, damping ratios and participation fac-tors for the three- and twelve-storey buildings. Owing to the Rayleigh damping formulation,the second and third modes were signiAcantly under-damped. While under-damping of these



Figure 5. Top 0oor acceleration magniAcation obtained for the twelve-storey wall buildings for diLerentscaled SYLM949 input ground motions.

modes could be expected to have a small eLect on the peak displacement response of thebuildings, it would be expected to have a greater eLect on the magnitude of the 0oor acceler-ations. The Newmark constant average acceleration method was used to integrate the equationof motion in the time-history analyses [19]. The integration time-step was equal to 1=1000 secto ensure satisfactory acceleration results. Data from the response was stored at 1=100 secintervals. Further details of the model are described in Reference [18].

The trends obtained for the 0oor horizontal accelerations were generally similar for theresponse of the buildings to the SYLM949 and SIM1 records. Figure 5 plots the top 0ooracceleration magniAcation against the scale factor [SF] used for the SYLM949 record for thetwo types of hysteresis rules investigated. There are three distinct regions in this Agure. Inthe Arst region, from SF0 to SF = 0:08, the magniAcation is constant because the buildingresponds elastically. In the second region, from SF = 0:08, where the building reaches theelastic limit, to about SF = 0:2, the magniAcation decreases very rapidly. In the third region,from SF = 0:2 onwards there is little dependency between the magniAcation and the SF.

Figure 6 plots envelopes for the 0oor acceleration magniAcation for the twelve-storey highbuilding, for the SYLM949 record scaled by 0.08, 0.2, 0.4 and 1, and for the Takeda hysteresisrule. Each dot corresponds to the magniAcation in a level of the building. It is clear fromFigure 6 that the maximum 0oor acceleration magniAcation occurs at the top of the building.The following additional observations can be inferred from this Agure: (i) the maximum0oor acceleration magniAcation occurs when the building responds elastically, and, (ii) the0oor acceleration magniAcation tends to diminish as the SF, and hence the ductility demand,increase.

Figure 7 is similar to Figure 6 except that the non-linear response of the building wasmodelled with the MOC hysteresis rule. In contrast with the trend observed for the responseobtained for the Takeda hysteresis rule, the maximum magniAcation does not always occurat the top of the building. This plot does not show any clear trend, which also contrastswith the response observed for the Takeda hysteresis rule. This is caused by the shock eLectresulting from the sudden change in stiLness at the origin of the non-linear response as willbe discussed later on.

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Figure 6. Floor acceleration magniAcations for each level of the twelve-storey wall building respondingto four scaled SYLM949 input ground motions with a Takeda hysteresis rule.

Figure 7. Floor acceleration magniAcation for each level of the twelve-storey wall building respondingto four scaled SYLM949 input ground motions with an origin-centered hysteresis rule.

The response plotted in Figures 57 suggests that the 0oor acceleration magniAcation, andtherefore the peak 0oor acceleration, is somewhat related to the ductility demand but is notinversely proportional to it, as is inferred from recommendations given for the design ofdiaphragms in the Mexico City Building Code [15] and in the New Zealand Standards [16; 17].In the case of the Uniform Building Code [13] the in-plane 0oor forces are often controlled bythe limits 0.5 and 1:0(CaIWpx), which means that in this case they are not related to ductilitydemand.

Floor pseudo-acceleration response spectra derived from the response of elastic one-degree-of-freedom oscillators, responding to the uppermost 0oor acceleration time-histories, willbe used here to provide an understanding of the main variables that aLect 0oor accelera-

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Figure 8. Top 0oor pseudo-acceleration response spectra (5 per cent damping ratio) normal-ized by the peak ground acceleration obtained from scaled SYLM949 input ground motions for

the twelve-storey wall building.

tions. Uppermost 0oor-response spectra for the twelve-storey building, for 5 per cent damp-ing ratio, were derived from the response of the building to the SYLM949 input groundmotion for SF equal to 0.08, 0.4 and 1.0 for the two types of hysteresis rules investi-gated. A value of SF = 0:08 demarcates the limit of the buildings elastic response while thevalue of SF = 0:4 and SF = 1:0 result in moderate and high displacement ductility demands,respectively. Figure 8 depicts the 0oor-response spectra. In this Agure the spectral ordinateshave been normalized by the peak ground acceleration obtained from the scaled input groundmotion. The solid-bold line in Figure 8 represents response spectrum corresponding to thebuildings elastic response. The three of peaks in the response are related to the translationalmodes with periods of free vibration T1, T2 and T3.

It is interesting to note that inelastic behaviour in the primary lateral force resisting system,shown by the shaded and solid-thin lines in Figure 8, results in a reduction of the responsearound the natural periods of free vibration. This reduction is greatest for the response aroundT1. The response associated with the Takeda hysteresis rule, see Figure 8(a), shows a diLerent

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Figure 9. Momentcurvature response at the base of a wall in the twelve-storey building for the Arst 7sec of the SYLM949 input ground motion for the two types of hysteresis rule.

trend to that observed for the MOC rule, see Figure 8(b). In the case of the response with aTakeda hysteresis rule, an increase in the ductility demand in the walls results in further, butnot signiAcant, reduction of the acceleration in the period band around T1 while the responsearound the higher mode periods is largely unaLected. When the response is obtained usingthe MOC hysteresis rule, see Figure 8(b), a large ductility demand also results in further,but not signiAcant reduction of the response around T1. However, the peaks in the spectralordinates associated with the higher modes periods have an amplitude much greater than thoseassociated with relative moderate ductility response, that is, with the response for SF = 0:4.

The diLerent trends observed between the response at high ductility demands for the twohysteresis rules can be explained with the aid of the momentcurvature response at the criticalregion at the base of the walls obtained from the analyses, see Figure 9. The characteristicself-centering response of the MOC hysteresis rule produces a sudden change of stiLness at theorigin, see Figure 9(a). The sudden change from low to high stiLness is felt in the structure asa shock that feeds energy into the higher modes. The shock eLect is more pronounced whenthe ratio between the reloading stiLness and the unloading stiLness is high. For example, forthe SYLM 949 record such ratio was 4.5 when SF = 0:4 and 12 when SF = 1:0 [18]. It shouldbe noted, however, in an actual building built with a self-centering system that the changein stiLness at the origin should not be as abrupt as that given by the MOC hysteresis rule.

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For this reason the shock eLect should be expected to be less pronounced than that obtainedtheoretically.

3.2. Proposed First Mode Reduced method for determining 4oor magni5cations

The 0oor acceleration Aqn corresponding to mode q at the uppermost level of the building,can be expressed by

Aqn = TqqnSa(Tq; q)Rq

(7)

where Tq is the participation factor for mode q; qn is the amplitude of mode q at level n; Sa

is the spectral acceleration, Tq and q are the period of free vibration and damping ratio,respectively, associated with mode q, and Rq is a reduction factor to account for the eLect ofductility on the primary lateral force resisting system.

Equation (7) is based on linear-elastic theory and is adapted here for evaluating the responseof non-linear systems. In linear-elastic systems, the natural modes of vibration can clearlybe deAned by ensuring orthogonality between modes with respect to the stiLness and massmatrices. As soon as the structure exceeds the elastic limit, the modes are no longer orthogonalto each other with respect to the stiLness matrix. Notwithstanding this, the modes still providea set of independent vectors that can conveniently be used in the analysis of non-linear systems[21; 14; 22]. Noteworthy is the fact that when a structure is deformed beyond the elastic limitthe modal characteristics change instantaneously every time the stiLness changes.

For hysteresis rules characterized by a relatively smooth non-linear response, such as theTakeda rule employed here, it appears from the results of the parametric analysis discussedpreviously that RqRq+1 and Rq1. For rules showing self-centering characteristics, suchas the MOC rule utilized in this study R11. However, the reduction factors for the highermodes, R2; R3 : : : Rr , seem to be larger than one for low ductility demands but could be lessthan one for high ductility demands.

Modal accelerations can be combined to obtain an approximation to the 0oor acceleration.This can be achieved using a suitable modal combination technique. The square-root-of-the-sum-of-the-squares technique [SRSS] is chosen in this investigation because of its simplicity.The limitation of the SRSS technique when combining the response of modes with closeperiods is not considered to signiAcantly aLect the results of this study. When combining theabsolute 0oor accelerations due to the translational modes in the direction begin considered,the SRSS results in the following 0oor horizontal acceleration, An, for the uppermost principalseismic mass,

An =

rq=1

[Tq

qnSa(Tq; q)Rq

]2(8)

The observations obtained from the non-linear analysis suggest that Equation (8) can besimpliAed by assuming that Arst mode is the only mode aLected by the ductility. This

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simpliAcation results in R2 =R3 = =Rr = 1. Thus, Equation (8) becomes,

An =

[T11n Sa(Tq; q)R1]2

+rq=2

[TqqnSa(Tq; q)]2 (9)

where R1 =OMe=OMo; OMe is the maximum overturning moment required for elastic responseand OMo is the maximum overturning moment found from the non-linear time-history analysisconsidering also the eLects of overstrength.

Results obtained from the non-linear time history dynamic analyses, to be presented inthe following section, indicate that the 0oor accelerations in the lower 0oors is stronglyin0uenced by the horizontal ground excitation as well as by the shape of the hysteresis rule.The following interpolation function is proposed to obtain Ai at any 0oor in the building:

Ai = UiAo (10)

where Ao is the peak ground acceleration and factor Ui is the 0oor acceleration magniAcationfactor. This factor is given by,

Ui =An=Ao for 0oors located between 0:2hi=hn61

or Ui = 5(hihn

)(AnAo

1)

+ 1 for 0oors located between 06hi=hn60:2 (11)

where hi is the height of the 0oor in consideration, and hn is the height of the uppermostlevel of the building both measured from the base.

4. COMPARISON OF FLOOR ACCELERATION MAGNIFICATIONS OBTAINEDFROM THE FIRST MODE REDUCED METHOD AND FROM

NON-LINEAR TIME-HISTORY DYNAMIC ANALYSES

In this paper results of the parametric analysis are only presented for the three- and twelve-storey buildings designed for = 5 and subjected to the full-scale SYLM949 and SIM1records. Complete results of the analyses are reported elsewhere [18]. Although only fewexamples of the First Mode Reduced method are given in this paper, the previously dis-cussed Andings of Eberhard and Sozen [14] on higher mode eLects suggest that the methodmight work for other structures with diLerent parameters.

Figures 10 and 11 plot the distribution in height of the 0oor acceleration magniAcationsfor the three- and twelve-storey buildings when subjected to the SYLM949 record. TheseAgures plot the magniAcations obtained from the Takeda and MOC hysteresis rules. Theline associated with black-circle markers in Figures 10 and 11 corresponds to those resultsobtained from the non-linear analysis. The line associated with triangular markers corre-sponds to the 0oor magniAcation predicted using the First Mode Reduced method describedabove. The reduction factors for the Arst mode, R1, for the three- and twelve-storey buildings,

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Figure 10. Floor acceleration magniAcation for all levels for the three-storey buildingsubjected to SYLM949 ground motion.

were evaluated according to the deAnition given in Section 3, and are shown in the legendbox in both Agures.

Figure 10 shows the worst case where the First Mode Reduced methods under-predictsthe 0oor acceleration magniAcations. Much better agreement was found for the 0oor acce-leration magniAcations for higher rise buildings and for all buildings when subjected to thesynthetic record [18]. For example, Figure 11 compares the magniAcation predicted by theFirst Mode Reduced method with the magniAcation derived from the non-linear analysisfor the two hysteresis rules investigated. In this case 0oor acceleration magniAcations pre-dicted with the First Mode Reduced method envelopes the magniAcations derived from thenon-linear analyses, except in the Arst 0oor of the building responding with the MOC hys-teresis rule. It is interesting to note that, as shown in Figures 10 and 11, 0oor accelerationmagniAcations associated with the origin-centered hysteresis rule were generally greater thanthose associated with the Takeda rule, particularly in the lower 0oors of the buildings. ThediLerence was particularly large in low-rise buildings responding to a near fault input groundmotion [18].



Figure 11. Floor acceleration magniAcation for all levels for the twelve-storeybuilding subjected to SYLM949 motion.

5. SIMPLIFICATION OF THE FIRST MODE REDUCED METHODFOR USE IN DESIGN

The First Mode Reduced method described in Section 3 is suitable for use in conjunctionwith a response spectrum modal analysis but is rather cumbersome for use in routine design.A simpliAcation of this method, for use in compliance with a generic design spectrum, suchas that prescribed by the New Zealand Loadings Standard [16], will be derived in this section.

5.1. Step 1

The Arst step is to assume that the damping ratio for all modes is equal to 5 per cent asis normally assumed in most seismic design standards. This assumption enables the spec-tral acceleration terms in Equations (9)(11) to be obtained from a single spectrum. Theseismic coePcient, Ch(Tq; 1), for fully elastic response corresponding to period Tq is the spec-tral acceleration for 5 per cent damping, Sa(Tq; 0:05), divided by the acceleration of gravity.



Figure 12. DeAnition of parameters for deriving a simple method compatible with a design spectrum.

That is:

Ch(T1; 1) =Sa(Tq; 0:05)

g(12)

The seismic coePcient, Cpn, for a part or diaphragm at the uppermost level is equal to,

Cpn =Ang

(13)

Substituting Equations (12) and (13) into Equation (9)

Cpn =

[T11n Ch(T1; 1)R1]2

+rq=2

[TqqnCh(Tq; 1)]2 (14)

where Ch(T1; 1) is the seismic coePcient derived for elastic response from the design spectrumfor the fundamental period in the direction being considered, see Figure 12.

In the simpliAed method described in this section 0oor acceleration magniAcations aredeAned by the ratio Cpi=Cho, where coePcients Cpi and Cho are deAned as:

Cpi =Ai=g and Cho =Ao=g (15)

For example, in the current New Zealand Loadings Standard [16] coePcient Cho is equal to0.4, 0.42 and 0.42 for hard, intermediate and soft subsoil categories, respectively.

The relationship between Cpi and Cho is obtained by substituting Equation (15) into Equa-tion (10). Thus:

Cpi = UiCho (16)

where the magniAcation factor Ui, deAned in Equation (11) in terms of acceleration, canbe written in terms of the seismic coePcients. This is achieved by substituting Equations



(13) and (15) into Equation (11),

Ui =Cpn=Cho for 0oors located between 0:2hi=hn61

or Ui = 5(hihn

)(CpnCho

1)

+ 1 for 0oors located between 06hi=hn60:2 (17)

5.2. Step 2

The second step assumes that all the natural periods of free vibration corresponding to thehigher modes in the considered direction are in the period band that coincides with the maxi-mum spectral ordinate, see Figure 12. It is also assumed that for these periods the corre-sponding elastic seismic coePcient is taken conservatively as Ch;max. Thus, this implies thatEquation (14) can be written as:

Cpn =

[$1R1Ch(T1; 1)

]2+ $2h (!Cho)2 (18)

where coePcient !=Ch;max=Cho, see Figure 12, $1 is Arst mode contribution coePcient and$h is a coePcient that accounts for the contribution of the higher modes of response. ThesecoePcients are given by,

$1 = T11n and $h =

rq=2

(Tqqn)2 (19)

Upper and lower bounds for $1 and $h can be obtained from modal analyses of 0exural andshear cantilevers representing buildings with regular mass and stiLness distribution. Flexuralbeams characterize the behaviour of slender walls where shear deformations are negligible. Incontrast, shear beams characterize the behaviour of squat walls or moment resisting frameswith stiL beams and inextensible columns. Figure 13 plots the variation of $1 and $h with thenumber of levels in a building. CoePcients $1 and $h obtained from modal analyses for thethree buildings investigated are also shown in Figure 13. It is evident in Figure 13 that, whenthe number of levels is greater than two, 0exural beams have the largest values of coePcients$1 and $h. An upper bound solution applicable to all types of buildings is obtained by deAningcoePcients $1 and $h as:

$1 = 1 for single-storey buildings and $1 = 1:5 for multi-storey buildings (20a)

and

$h = 0:53

ln(n) (20b)

where n is number of the levels in the building.

5.3. Step 3

The deAnition of Arst mode reduction factor R1 given in Section 3 is appropriate for use inconjunction with non-linear time-history dynamic analyses of multi-storey buildings. Conse-quently, a simpliAcation for use in practical design seems warranted. It should be expected



Figure 13. Variation of coePcients $1 and $ 2h with the number of levels in a building.

that the displacement ductility demand in a building be less than that chosen for design due tothe eLects of overstrength. Buildings are expected to have values of R1 ranging between =and 1, whichever is greater. CoePcient depends on the overstrength of the overall structureand on the relationship between the ordinates in the elastic and inelastic response spectra. Avalue of = 2, proposed in this paper, seems appropriate for buildings designed in accordancewith the principles of capacity design [23]. Hence:

R1 =2

or 1; whichever is greater (21)

The following expression is obtained for coePcient Cpn by substituting Equation (20b) intoEquation (18) and by making != 2:5,

Cpn =

[$1R1Ch(T1; 1)

]2+ 1:75 ln(n)C 2ho (22)

The horizontal design force, Fph, for a rigid part or diaphragm at level i between, and including,the base and the top level is,

Fph = SpRpZCpiWp (23)

where Sp is a structural performance factor, Rp is the risk factor for the part or diaphragm,Z is the seismic zone factor and Wp is the weight of the part or diaphragm inconsideration.

Figure 14 compares the 0oor acceleration magniAcations determined from the simpliAedmethod and from the non-linear analyses of the three- and twelve-storey buildings. Floor accel-



Figure 14. Floor acceleration magniAcation for the three- and twelve-storey buildings when sub-jected to the synthetic SIM1 input ground motion and the simpliAed method.

eration magniAcations determined from the simpliAed approach are in general good agreementwith the results obtained from the non-linear analysis, particularly for those cases associatedwith the Takeda hysteresis rule.

Figure 15 shows the top 0oor acceleration magniAcation against the displacement duc-tility demand for four buildings. It is evident in this Agure that 0oor acceleration mag-niAcation becomes rather insensitive to the ductility demand as the number of levels in-crease. The only case where de-ampliAcation is found, that is when the 0oor accelerationis less that the peak ground acceleration, is in the one-storey building. This is because ina single-degree-of-freedom system, the 0oor acceleration is directly controlled by the capac-ity of the critical mechanism. Such capacity is a function of ductility factor, , selectedin design. Note also that the top 0oor acceleration magniAcations found using the sim-pliAed method are in the range of values observed during the Northridge earthquake, seeFigure 1.



Figure 15. Top 0oor acceleration magniAcation obtained from the simple design approach vsdisplacement ductility and number of levels.

6. CONCLUSIONS

This paper summarized an investigation into earthquake induced 0oor horizontal accelerationsthat develop in regular buildings. The main Andings and conclusions of this investigationare:

1. A parametric non-linear time-history dynamic analysis of cantilever wall buildings wasconducted in this investigation. Three-, six-, and twelve-storey buildings, designed for limitedductility response and for fully ductile response, were subjected to two diLerentinput ground excitations. One excitation was a synthetic record designed to match one of thedesign response spectra given by the New Zealand Loadings Standard [16]. The other exci-tation was the near-fault record SYLM949 obtained during the 1994 Northridge earthquake.The response of the buildings was investigated using a Takeda hysteresis rule representingbehaviour of cast-in-place walls and an origin-centered hysteresis rule representing thebehaviour of self-centering precast concrete walls post-tensioned with partially unbonded ten-dons and incorporating mechanical energy dissipation devices.

2. The investigation found that the maxima 0oor acceleration magniAcations nearly alwaysoccur at the uppermost 0oor of a building. It was also found that the response immedi-ately beyond the elastic limit signiAcantly reduces the 0oor accelerations at the uppermost0oor.

3. Floor accelerations associated with the origin-centered hysteresis rule were generallygreater than those associated with the Takeda rule, particularly in the lower 0oors of thebuildings investigated. The diLerence was particularly large in low-rise buildings respondingto a near fault input ground motion.

4. A procedure for deriving the design horizontal forces was proposed in the paper. Theprocedure assumes that ductility only aLects 0oor accelerations associated with the Arst modeof the response. A simple version of the method, for use in routine design, was also proposedand compared, with generally good agreement, with the results from non-linear time historyanalyses.

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ACKNOWLEDGEMENTS

The Anancial support of the Public Good Science Fund administered by the New Zealand Foundationfor Research Science and Technology, Contract UOC 808 for the 1998=2000 grant cycle, is gratefullyacknowledged.

Thanks are due to Prof. J.D. Jaramillo, from University EaAt in Medellin, Colombia, to Dr B. Deam,Leicester Steven Lecturer at the University of Canterbury, and to Prof. A Rutenberg from TechnionIsrael Institute of Technology for their useful comments. Thanks are also due to Mr J.J. BlandVon, Ph.D.candidate from the National University of Mexico who helped with several analyses. Thanks are alsogiven to the anonymous reviewers for their critical reading of the manuscript and useful suggestions.

APPENDIX: NOMENCLATURE

ap = horizontal force factor according to UBC 1997Aqn = absolute acceleration of the seismic weight at level n due to mode q; mm=sec

2

Ai = absolute acceleration at 0oor i; mm=sec2

Ao = peak ground acceleration, mm=sec2

Ca = seismic coePcient (deAned for six seismic zone factors and six soils proAle types)Cfi = 0oor horizontal seismic coePcient at level i (NZS 4203:1992)Cfn = uppermost level horizontal seismic coePcient (NZS 4203:1992)Cfo = 0oor horizontal seismic coePcient applicable at or below the base of a building

(NZS 4203:1992)Ch(Tq; 1) = seismic coePcient derived for elastic response from the basic hazard acceleration

design spectra for the q translational mode of natural vibration in the direction beingconsidered

Chmax = maximum spectral acceleration coePcientCho = peak ground acceleration and acceleration of gravity ratioCpi = basic horizontal seismic coePcient for a part at level i (NZS 4203:1992)Cpn = basic horizontal coePcient for a part or diaphragmFi = horizontal force on primary lateral force resisting system at level i, kN (NZS

4203:1992)NFn = is the horizontal force found for the primary lateral force resisting system at the

uppermost level of the building, kN (NZS 4203:1992)Fph = horizontal force acting on a part according to NZS 4203:1992Fpx = design horizontal force for diaphragms, kN (UBC 1997)Ft = portion of the base shear, V , considered concentrated at the top level of the

structure in addition to Fn, kN (UBC 1997)Fx = seismic design force applied to level x, kN (UBC 1997)hi = height, measured from the principal seismic weight at level i, mhn = height, measured from the uppermost principal seismic weight, mi = index deAning a level or storey in a building. Note level i is immediately above

storey iI = building importance factor (UBC 1997)Ip = importance factor for a part (UBC 1997)



kr = reloading 0exural stiLness, kN m2

ku = unloading 0exural stiLness, kN m2

n = number of levels in buildingOMe = maximum overturning moment for elastic response, kN mOMo = overturning moment in a structure at the development of overstrength, kN mq = node numberr = highest mode of vibration considered to in0uence the absolute 0oor accelerationR = risk factor for a structure (NZS 4203:1992) also

= base shear force reduction (UBC 1997)Rp = risk factor applicable to a part (NZS 4203:1992), also

= component response modiAcation factor for the part deAned by UBC 1997Rq = reduction factor in the spectral acceleration Sa to account for the eLect of struc-

tural ductility on mode qSa = spectral acceleration, mm=sec

2

Sp = structural performance factor (NZS 4203:1992)T1 = natural period for a buildings Arst mode of translational vibration in the direction

being considered, secT2 : : : Tq = natural period of the buildings higher translational modes of vibration in the

direction being considered, secTp = natural period of vibration of the part in the direction being considered, sec

(NZS 4203:1992)Tq = natural of free vibration associated with mode of vibration q, secV = base shear force, kN (UBC 1997)W = total seismic dead load, kN (UBC 1997)wi and wx = portion of the total seismic dead load, W, located or assigned to level i or x,

kN (UBC 1997)Wi = seismic weight at level i, kNWn = seismic weight at level n, kNWp = weight of a part in NZS 4203:1992, kNwpx = weight of the diaphragm at level x according to UBC 1997, kNx = index referring to a level in a building

Greek letters

= overstrength factorTq = participation factor for mode q = damping ratio for a one-degree-of-freedom systemq = damping ratio associated with mode qqn = amplitude of mode q at level n$1 = Arst mode contribution coePcient$h = higher mode contribution coePcient = displacement ductility factor (NZS 4203:1992)o = displacement ductility factor including overstrength required by NZS 4203:1992Ui = 0oor magniAcation factor at level i! = Chmax=Cho



Abbreviations

DF = displacement ductility factorNZS = New Zealand StandardMCBC = Mexico City Building CodeMOC = modiAed origin-centered hysteresis rulePGA = peak horizontal ground accelerationPSA = pseudo-spectral accelerationSF = input ground motion scale factorSRSS = square-root-of-the-sum-of-the-squares modal combination techniqueUBC = Uniform Building Code

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