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Nonlinear behaviour of RC frames under repeated strong ground motionsGeorge D. Hatzigeorgiou a, , Asterios A. Liolios ba Department of Environmental Engineering, Democritus University of Thrace, GR-67100 Xanthi, Greeceb Department of Civil Engineering, Democritus University of Thrace, GR-67100 Xanthi, Greece

a r t i c l e i n f o

Article history:Received 19 December 2009Received in revised form

6 April 2010Accepted 12 April 2010

a b s t r a c t

This paper presents an extensive parametric study on the inelastic response of eight reinforced concrete(RC) planar frames which are subjected to forty ve sequential ground motions. Two families of regularand vertically irregular (with setbacks) frames are examined. The rst family has been designed forseismic and vertical loads according to European codes while the second one only for vertical loads, tostudy structures which have been constructed before the introduction of adequate seismic design codeprovisions. The whole range of frames is subjected to ve real seismic sequences which are recorded bythe same station, in the same direction and in a short period of time, up to three days. In such cases,there is a signicant damage accumulation as a result of multiplicity of earthquakes, and due to lack of time, any rehabilitation action is impractical. Furthermore, the examined frames are also subjected toforty articial seismic sequences. Comprehensive analysis of the created response databank isemployed in order to derive important conclusions. It is found that the sequences of ground motionshave a signicant effect on the response and, hence, on the design of reinforced concrete frames.Furthermore, it is concluded that the ductility demands of the sequential ground motions can beaccurately estimated using appropriate combinations of the corresponding demands of single groundmotions.

& 2010 Published by Elsevier Ltd.

1. Introduction

The past four decades have seen a rapid development of knowledge in seismic analysis and design of reinforced concrete(RC) framed structures. Various rational approaches have beenproposed, which are mainly based on the inelastic materialsbehaviour. The majority of modern seismic design codes, as forexample the European code EC8 [1], permit the structuralnonlinearity. However, even these codes have their drawbackswhere two of them are important. The rst is that the structuralbehaviour is usually estimated by a rather irrational manner.More specically, the design procedure is divided into two stages

[1,2] : evaluation of member internal forces by linear analysis of the whole structure followed by the ultimate limit state design of individual cross-sections assuming the ideal nonlinear propertiesof the materials. This design procedure provides no verication of the compatibility between the isolated member and the memberas part of the whole structure. Nowadays, where the capabilitiesof the structural modelling are much larger, an engineer canperform analyses which give considerably better predictionsof stresses, displacements, limit loads and mechanisms of the

damage. The second limitation of these codes is the exclusiveadoption of the isolated and rare design earthquake while theinuence of repeated earthquake phenomena is ignored. Despitethe fact that the problem has been qualitatively acknowledged [3] ,very few studies have been reported in the literature regardingthe multiple earthquake phenomena. To be sure, Amadio et al. [4]examined the effect of repeated earthquake ground motions onthe nonlinear response of single degree of freedom (SDOF)systems. However, as the authors themselves recognized, theirwork cannot be considered exhaustive since they examined onlyone natural and two articial ground motions. Recently, Hatzi-georgiou and Beskos [5] and Hatzigeorgiou [6,7] examined the

inuence of multiple earthquakes in numerous SDOF systems andfound that seismic sequences lead to increased displacementdemands in comparison with the design earthquake. However,these works are concerned with SDOF and not with RC framedstructures. To the best of the authors knowledge, there is onlyone work that has evaluated the effect of repeated earthquakes onconcrete structures where an RC bridge is examined under onereal and one articial seismic sequence [8] . Thus, the need for thedevelopment of an efcient methodology for the inelastic ana-lysis of RC framed structures, as multi-storey buildings, undersequential ground motions is apparent.

This paper presents an extensive parametric study on theinelastic response of eight reinforced concrete planar frames

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Soil Dynamics and Earthquake Engineering

0267-7261/$ - see front matter & 2010 Published by Elsevier Ltd.

doi: 10.1016/j.soildyn.2010.04.013

Corresponding author. Tel./fax: +30 2541079373.E-mail address: [email protected] (G.D. Hatzigeorgiou).

Please cite this article as: Hatzigeorgiou GD, Liolios AA. Nonlinear behaviour of RC frames under repeated strong ground motions. SoilDyn Earthquake Eng (2010), doi: 10.1016/j.soildyn.2010.04.013

Soil Dynamics and Earthquake Engineering ] (]]]] ) ] ]] ]]]
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under forty ve seismic sequences. This study employs, for therst time, as-recorded seismic sequences to determine thenonlinear behaviour of RC framed structures. More specically,the rst ve multiple earthquakes have been recorded by thesame station, in the same direction and in a short period of time,up to three days. In such cases, there is a signicant damageaccumulation as a result of multiplicity of earthquakes, and due tolack of time, any rehabilitation action is impractical. Furthermore,

the examined RC frames are also subjected to forty articialseismic sequences. Two families of regular and vertically irregular(with setbacks) frames are examined. The rst family of frameshas been designed for seismic and vertical loads according toEuropean codes while the second one for vertical loads only, tostudy structures which have been constructed before theintroduction of adequate seismic design code provisions. Thetime-history responses of these concrete frames are evaluated by

means of the structural analysis software RUAUMOKO [9].Comprehensive analysis of the created response databank isemployed in order to derive signicant conclusions. Morespecically, this study focuses on the results which are relatedto the following critical parameters: local or global structuraldamage, maximum displacements, interstorey drift ratios, devel-opment of plastic hinges and response using the incrementaldynamic analysis ( IDA) method [10] . The last one is also known as

dynamic pushover analysis [10] and can be characterized as one of the most accepted methods for determining seismic response.It has been used in many applications as for evaluation of the seismic performance of structures [11] , for studies related todamage measure [12] and for the validation of simplied proce-dures for the prediction of approximate IDA curves [13,14] .However, to the best of the authors knowledge, the IDA techniquehas not yet been applied to examine the structural behaviour of

Fig. 1. Frames A1 and B1 properties.

G.D. Hatzigeorgiou, A.A. Liolios / Soil Dynamics and Earthquake Engineering ] ( ]]]] ) ] ]] ]]]2

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RC structures under multiple or sequential ground motions.Examining the results of this study, it is found that the sequencesof ground motions have a signicant effect on the response and,hence, on the design of reinforced concrete frames. Additionally,the accumulation of permanent displacements due to multipleearthquakes is also examined. Finally, a simple and effectivecombination of ductility demands of single events is proposed toestimate the corresponding demand of the sequential ground

motions.

2. Description of structures and modelling

Four structures (Family A Frames: A1, A2, A3 and A4) areconsidered to represent low-rise (three-storey) and medium-rise(eight-storey) RC buildings for study. They consist of four typicalbeamcolumn RC frame buildings without shear walls, located in

a high-seismicity region of Europe considering both gravity andseismic loads where a design/peak ground acceleration ( PGA) of 0.2g and soil class B according to EC8 [1] are assumed. Thesestructures have been designed for the following loading combina-tions [1,15] :

a) 1.35 G+1.50 Q b) 1.00 G+c Q +1.00 E

c) 1.00 G+c Q 1.00 E where G, Q and E correspond to dead, live and earthquakeloads, respectively, and c is the combination coefcient for liveload, assumed to be c 1.00 in this study.

Most of the existing reinforced concrete buildings weredesigned according to early seismic provisions or, sometimes,without applying any seismic provision. In order to examine suchbuildings designed for gravity only, another family of structures

Fig. 2. Frames A2 and B2 properties.

G.D. Hatzigeorgiou, A.A. Liolios / Soil Dynamics and Earthquake Engineering ] ( ]]]] ) ] ]] ]]] 3



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(Family B Frames: B1, B2, B3 and B4) is also considered. Thus,this group has been designed only for the aforementioned rstloading combination.

Families A and B have the same geometry and loads but theyhave different reinforcement. This paper, examining thesefamilies of structures, does not focus into a rigorous comparisonbetween them but mainly into the effect of multiple earthquakeson both of them. The case of shear failure is not examined here

assuming adequate transverse reinforcement. However, thetransverse reinforcement of older RC frames appears to bevery light for columns and for this reason the connement of concrete is not taken into account for Family B. It is well-knownthat the cross-sectional dimensions of frames designed forseismic loading tend to be larger than those that are not designedto resist seismic forces. However, the identical geometryand loads allow the comparison of structural responses betweenthem since their elastic dynamic properties (periods of vibration,etc.) are also identical. The dead loads (excluding self-weight) andlive loads are equal to 20 and 10 kN/m, respectively, and aredirectly applied on the beams. All oors are assumed to be rigid inplan to account for the diaphragm action of concrete slabs.Material properties are assumed to be 20 MPa for the concretecompressive strength (concrete grade C20) and 500 MPa for the

yield strength of both longitudinal and transverse reinforcements(steel grade S500s). Both the examined 3- and 8-storey buildingshave 3 equal bays with total length equal to 15m. Typicaloor-to-oor height is equal to 3.0 m, while for the rst oor of the 8-storey buildings the height is equal to 4.0 m. Thecharacteristic interior frames of these structures, as shown inFigs. 16 , represent 2-D models of these buildings. Modalproperties of the rst three modes are given in Table 1 . The

column and beam dimensions used in this study are typicalframe element proportions in the existing buildings. The3-storey buildings are 9.0 m in elevation. All their beams are30cm 40 cm, all their columns are square with side 30cm andthe selected longitudinal reinforcement amount (number of barsand diameters in mm) and arrangement, is shown in Figs. 1 and 2 .The 8-storey buildings are 25.0 m in elevation. All beams havewidth equal to 30 cm and height equal to 4060cm, all columnsare square with side 3040 cm and the selected longitudinalreinforcement, i.e., amount (number of bars and diameters inmm) and arrangement, is shown in Figs. 36 . It is evident that thereinforcement amount for columns of Family A buildings is higherthan that of Family B buildings. This difference has to do with theconcept of capacity design, i.e., strong columnsweak beams.More specically, the design of columns of Family A structures

Fig. 3. Frame A3 properties.




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satises the following condition of y4.4.2.3 of EC8 [1] , at all jointsof beams with columns:

XM Rc Z 1:3 XM Rb 1where PM Rc and PM Rb are the sums of the design values of themoments of resistance of the columns and beams framingthe joint, respectively. This provision is waived at the top levelof the examined buildings of Family A.

The behaviour factors, q, for the seismic design of Family Abuildings are compatible with the provisions of y5.2.2.2 of EC8 [1]and satisfy the ductility capacity medium (DCM) criteria. Morespecically, regular buildings A1 and A3 have been designed forq 3.9, while buildings A2 and A4 for q 0.8 3.9 3.12, sinceaccording to the abovementioned provisions, for buildings whichare not regular in elevation, the value of behaviour factor shouldbe reduced by 20%.

Reduced values of member moments of inertia, I ef , wereconsidered in the design to account for the cracking; for beamsI ef 0.5 I g and for the columns I ef 0.9 I g , where I g is the moment of inertia of the corresponding gross section [16] .

An inelastic structural multi-degree of freedom (MDOF)

system with viscously damped force-deformation relationship

is used to investigate the structural response for actual records.The dynamic equilibrium equation of these systems is given inincremental form [17]

M

u C _

u K T u Ma g 2

where M is the mass matrix, u the relative displacement vector,C the viscous damping matrix, K T the tangent (inelastic) stiffnessmatrix, a g the acceleration vector of the ground motion and theupper dots stand for time derivatives. The solution of the equationof motion has been performed using the RUAUMOKO program [9] ,which is an advanced program for seismic analysis of framedstructures. A brief description of the modelling details is providedin the following.

A two-dimensional model of each structure is created inRUAUMOKO [9] to carry out nonlinear dynamic analysis. The soil-structure interaction phenomenon is not taken into account,considering xed base conditions. Second-order effects ( P Deffects) are taken into account. Beam and column elements aremodelled as nonlinear frame elements with lumped plasticity bydening plastic hinges at both ends of the beams and columns. Onthe beams, axial forces were assumed to be zero since all oorsare assumed to be rigid in plan to account for the diaphragm

action of concrete slabs. Characteristic input data for strength that

Fig. 4. Frame B3 properties.




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are required by RUAUMOKO [9] are the bending momentaxialforce interaction diagrams for columns and bending strengthvalues for beams. In this work and for each column and beam, theprogram RESPONSE-2000 [18] is used for the section modelling. Itshould be noted that for the entire group of analyses andmembers, the modied Takeda [19,20] hysteretic model isadopted. The parameters of this model are affected by the endresistances of beams and columns, which are obviously differentfor each family of structures resulting in different adoptedhysteresis models. Thus, stiffness and strength degradation aretaken into account in family B buildings, which exhibit descend-ing stiffness after yielding. An explanation of these parametersand the shape of the hysteresis model are analytically presentedin the RUAUMOKO user manual by Carr [9]. Another criticalparameter which is required has to do with the plastic hingelength to relate the rotation values with the curvatures. Severalplastic hinge lengths have been proposed in the literature. In thiswork and without loss of generality, the simplest approach, whichhas been proposed by Park and Paulay [21] , is adopted where theplastic hinge length, l ph for each member is equal to half of itssections height, H , i.e.

l ph 0:5H 3

3. Seismic input

3.1. Real seismic sequences

The rst strong ground motion database that has been usedhere consists of ve real seismic sequences, which have beenrecorded during a short period of time (up to three days), by thesame station, in the same direction, and almost at the same fault

distance. These seismic sequences are namely: Mammoth Lakes(May 19805 events), Chalfant Valley (July 19862 events),Coalinga (July 19832 events), Imperial Valley (October 19792events) and Whittier Narrows (October 19872 events) earth-quakes. The complete list of these earthquakes, which weredownloaded from the strong motion database of the PacicEarthquake Engineering Research (PEER) Center [22] , appears inTable 2 . These records are compatible with the soil class B, andtherefore compatible with the design process as mentioned in theprevious section. Every sequential ground motion records fromthe PEER database becomes a single ground motion record (serialarray) where between two consecutive seismic events a time gapis applied, which is equal to 100 s. This gap has zero accelerationordinates and is absolutely enough to cease the moving of any

structure due to damping. Thus, Fig. 7 shows the time histories of

Fig. 5. Frame A4 properties.




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the examined seismic sequences. The elastic spectra for this data-base and for viscous damping ratio x 5%, an appropriate valuefor RC structures [1], are presented in Fig. 8. For compatibilityreasons with the design process (Section 2), the seismic sequencesare normalized to have maximum PGA equal to 0.2g ( Table 2 ,right column). Thus, the aforementioned sequential groundmotions are multiplied by: 0.4525 (Mammoth Lakes), 0.4474(Chalfant Valley), 0.2729 (Coalinga), 0.9050 (Imperial Valley) and

0.9434 (Whittier Narrows).

3.2. Articial seismic sequences

The second strong ground motion database that has been usedhere consists of forty articial seismic sequences. More speci-cally, 10 articial accelerograms are considered to generate 20synthetic sequences of two events and 20 synthetic sequences of three events. The single ground motions are compatible with thedesign process, i.e., with Type 1 spectrum of EC8 [1] , Soil B localconditions and PGA 0.2g, as shown in Fig. 9, and are generatedby the specialized software SRP [23] . Between two consecutiveseismic events a time gap is also applied in this case to cease themoving of any structure due to damping.

4. Results

The inelastic behaviour of the examined RC framed structures,which are subjected to the abovementioned ve real and fortyarticial seismic sequences, is investigated in this section. Thisstudy focuses on the following basic design parameters: local orglobal damage index according to Park and Ang approach [24] ,maximum horizontal oor displacements and interstorey driftratios. Furthermore, the development of plastic hinges and the

structural response according to advanced methods as the

Fig. 6. Frame B4 properties.

Table 1Dynamic characteristics of structures: periods and mass participation factor (MPF).

Structure First mode Second mode Third mode Sum of (MPFs)

T 1 ( s) (MPF) T 2 (s) (MPF) T 3 ( s) (MPF)

Frame A1 0.6382 (0.891) 0.2048 (0.092) 0.1215 (0.017) (1.000)Frame A2 0.5001 (0.853) 0.2037 (0.127) 0.1308 (0.020) (1.000)Frame A3 1.2330 (0.800) 0.4718 (0.143) 0.2651 (0.042) (0.985)Frame A4 0.9673 (0.731) 0.4469 (0.194) 0.2746 (0.055) (0.980)Frame B1 0.6382 (0.891) 0.2048 (0.092) 0.1215 (0.017) (1.000)Frame B2 0.5001 (0.853) 0.2037 (0.127) 0.1308 (0.020) (1.000)Frame B3 1.2330 (0.800) 0.4718 (0.143) 0.2651 (0.042) (0.985)Frame B4 0.9673 (0.731) 0.4469 (0.194) 0.2746 (0.055) (0.980)




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Table 2Seismic input data.

No. Seismic sequence Station Comp. Date (time) Magnitude ( M L ) Recorded PGA (g) Normalized PGA (g)

1 Mammoth Lakes 54099 Convict Creek NS 1980/05/25 (16:34) 6.1 0.442 0.2001980/05/25 (16:49) 6.0 0.178 0.0811980/05/25 (19:44) 6.1 0.208 0.0941980/05/25 (20:35) 5.7 0.432 0.1951980/05/27 (14:51) 6.2 0.316 0.143

2 Chalfant Valley 54428 Zack Brothers Ranch EW 1986/07/20 (14:29) 5.9 0.285 0.1281986/07/21 (14:42) 6.3 0.447 0.200

3 Coalinga 46T04 CHP NS 1983/07/22 (02:39) 6.0 0.605 0.1651983/07/25 (22:31) 5.3 0.733 0.200

4 Imperial Valley 5055 Holtville P.O. HPV315 1979/10/15 (23:16) 6.6 0.221 0.2001979/10/15 (23:19) 5.2 0.211 0.191

5 Whittier Narrows 24401 San Marino NS 1987/10/01 (14:42) 5.9 0.204 0.1921987/10/04 (10:59) 5.3 0.212 0.200

Fig. 7. Ground acceleration records of the examined seismic sequences.




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incremental dynamic analysis ( IDA) technique are also investi-gated. Finally, the development of permanent displacements isalso investigated.

4.1. Determination of cumulative damage

The ParkAng model [24] is the best known and most widelyused damage index ( DI ), which is dened as a combination of maximum deformation and hysteretic energy

DI dm

du

b

duP y Z dE h 4

where dm is the maximum deformation of the element, du theultimate deformation, b a model constant parameter (usually,b 0.050.20) to control strength deterioration, R dE h the hystereticenergy absorbed by the element during the earthquake and P y theyield strength of the element. In this work, parameter b is set equalto 0.20, as suggested by Bertero and Bertero [25] . This damagemodel can also be extended to the storey and overall scales (globaldamage index), by summation of damage indices using appropriatemultiplication weights. ParkAng indices for various damage statesare shown for completeness reasons in Table 3 , which are adoptedfrom Ref. [26] . Fig. 10a depicts the local DI for the base joint of the

left base column of Frame A2, for the Mammoth Lakes seismic

Fig. 8. Response spectra of the examined seismic sequences.




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sequence. Furthermore, Fig. 10b shows the global DI , for the wholeFrame B2, under the Whittier Narrows earthquakes. It is evidentthat, in any case, seismic sequences lead to increased damage, bothin local and global level. Therefore and according to Table 3 , theseismic sequences can qualitatively and quantitatively upgrade thestructural damage. However, the majority of the existinginvestigations examine these parameters only for the idealizedcase of isolated earthquakes.

4.2. Maximum displacements and ductility demands

The maximum horizontal displacement proles, both for singleand sequential ground motions appear in Fig. 11 . More specically,the Frames A2, A3, A4 and B1, under the Mammoth Lakes, Coalingaand Chalfant Valley seismic sequences are examined. It is found that

due to the multiplicity of earthquakes, increased displacementdemands are required, in any case under consideration. It is well-known that inelastic exible systems present permanentdisplacements for single strong earthquakes. For any otheroncoming ground motion, permanent displacements are obviouslycumulated [57] and therefore the maximum displacements appearto be increased. Furthermore, the maximum displacements aredirectly related to the ductility demands. The cumulative ductility,

due to sequential ground motions, is analytically examined inSection 5, where an empirical expression is proposed to determineit by the corresponding demands of single earthquakes.

4.3. Interstorey drift ratio (IDR)

The interstorey drift ratio ( IDR) is the maximum relativedisplacement between two stories normalized to the storeyheight. It should be mentioned that from the analysis of test dataon components and small-scale structures, it was found that anIDR value smaller than 1% corresponds to damage of non-structural components, while values of IDR larger than 4% mayresult in irreparable structural damage or collapse [27] . Generally,the IDR does not account for effects of cumulative damage due torepeated inelastic deformation [28] . However, the examined IDRvalues include the multiplicity earthquakes effect. Examples of IDR values appear in Fig. 12, both for single and sequential groundmotions. It is evident that seismic sequences lead to larger IDR incomparison with the corresponding single events. Additionally,despite the limitation of damage into non-structural elements inthe case of single/isolated earthquakes ( IDRo 1%), the sequentialground motions lead to structural damage, i.e. IDR4 1%.

4.4. Development of plastic hinges

The nal development/distribution of plastic hinges of FramesA1 and B2, under the Imperial Valley and Mammoth Lakes seismicsequences appears in Figs. 13 and 14 , respectively. The formationof plastic hinges in columns of the non-ductile Frame B2 isFig. 9. Response spectra of the examined seismic sequences.

Table 3The relation between damage index and damage state [26] .

Degree of damage Physical appearance Damage index State of building

Slight Sporadic occurrence of cracking o 0.10 No damageMinor Minor cracks; partial crushing of concrete in columns 0.100.25 Minor damageModerate Extensive large cracks; spalling of concrete in weaker elements 0.250.40 RepairableSevere Extensive crashing of concrete; disclosure of buckled reinforcement 0.401.00 Beyond repairCollapse Partial or total collapse of building 4 1.00 Loss of building

Fig. 10. Local and global damage index according to the Park and Ang model [24] .




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obvious. It should be noted that due to the multiplicity of earth-quakes, the distribution of plastic hinges seems to be differentthan the corresponding one for single/isolated seismic events,both for Frames A1 and B2.

4.5. Incremental dynamic analysis (IDA) technique for sequential ground motions

All the examined structures have been analyzed using the IDA

technique. Fig. 15 shows selected results for Family-A buildings

under the real and articial, single and multiple strong groundmotions. It is evident that the seismic sequences lead tonoticeably different response in comparison with the corres-ponding single seismic events and require increased displacementdemands, in any case under consideration. As it is expected, theincreased displacement demands lead to higher values of drift anddamage. The primary goal of IDA is to quantify the reservecapacity of a structure against collapse. Since the analyses takeinto account the collapse state considering stiffness and strength

degradation, the IDA seems to be very useful for this study.

Fig. 11. Maximum horizontal displacement proles under single and sequential ground motions.

Fig. 12. Maximum interstorey drift ratios (IDR) under single and sequential ground motions.




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Imperial Valley 5055 Holtville P.O. (HPV315)

1979/10/15 (23:16) _ PGA = 0.200g

Seismic sequence _ PGA=0.200g

1979/10/15 (23:19) _ PGA = 0.191g

Frame A1

Fig. 13. Plastic hinges distribution for single and sequential ground motions Frame A1.

80/05/25 (16:34) _ PGA = 0.200g

80/05/25 (19:44) _ PGA = 0.094g

80/05/27 (14:51) _ PGA = 0.143g Seismic sequence _ PGA = 0.200g

80/05/25 (16:49) _ PGA = 0.081g

80/05/25 (20:35) _ PGA = 0.195g

Frame B2Mammoth Lakes - 54099 Convict Creek (N-S)

Fig. 14. Plastic hinges distribution for single and sequential ground motions Frame B2.



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Assuming that the collapse state is represented by extremelylarge horizontal displacements, Fig. 15 shows, for the examinedstructures and PGAs, that the collapse appears only for the casesof seismic sequences and not for the isolated single groundmotions.

4.6. Development of permanent displacements for sequential groundmotions

In order to satisfy the targeted performance levels under pre-dened seismic hazard levels, the permanent displacementshould be accurately estimated. In this work, it is found that themultiplicity of earthquakes strongly inuences the permanentdisplacements and therefore multiple earthquakes phenomenashould be taken into account to achieve dependable estimation of

permanent displacements. Fig. 16 shows selected results forstructures subjected to real and articial seismic sequences wherethe time history of horizontal displacement for the top of thesestructures is presented. The accumulation of permanentdisplacement is obvious, in any case under consideration.

5. Estimation of ductility demands for multiple earthquakes

This section examines the estimation of ductility demandsfor sequential strong ground motions. As shown in Section 4,multiple earthquakes require increased displacement and ducti-lity demands in comparison with the corresponding single events.The global displacement ductility factor, m, can be dened in

terms of the maximum displacement u max at the top level of the

examined buildings and the corresponding yield displacement u y,as [16,17]

m u maxu y

5

The denition of yield displacement is that according to Paulay[29] . In order to estimate the cumulative ductility for a sequenceof strong ground motions, various empirical expressions can bedeveloped. This work proposes the following simple and rationalrelation:

mseq 1 Xn

i 1/ mi 1S p 1= p 6

where the cumulative ductility, mseq , for a sequence of strongground motions consists of n-seismic events, results from thecorresponding ductility demands, mi, for each one of them.Furthermore, p is a parameter controlling the combination of single ductilities and / S symbolizes the Macauley brackets usedhere in order to eliminate the inuence of weak ground motions,i.e., those for mi o 1. For example, for a triple seismic sequencewith m1 4 1.0, m2 o 1.0 and m3 o 1.0, Eq. (6) provides with theexpected ductility demand, mseq m1 . Parameter p can be equal to1.0, for a simple and direct summation of ductility demands, or2.0, which corresponds to the square root of the sum of thesquares SRSS combination rule, a well-known procedure inearthquake engineering to obtain seismic design response. Thesetwo simple cases are portrayed in Fig. 17a and b where the exactdynamic inelastic analyses results, for the aforementioned 20cases of real records ( 4 frames 5 real seismic sequences), arecompared with the empirical ones of Eq. (6). It is evident that the

abovementioned values lead to overestimated (for p 1.0) or

Fig. 15. Application of IDA for sequential ground motions Family A buildings.




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underestimated (for p 2.0) cumulative ductility demands. Inorder to achieve the best t for parameter p and for the examinedstructures, this work uses the nonlinear solver of the MS-EXCEL program, which gives the optimum value of parameter p 1.3048 ffi 1.3. Therefore, Eq. (6) becomes

mseq 1 Xn

i 1/ mi 1S

1:3" #1=1:3

7

Table 4 presents the ductility demands for the well-designedRC frames (Family-A), both for the single events and the seismicsequences under consideration. Furthermore, the estimatedcumulative ductility demands using Eq. (7) are also shown inTable 4 (in parentheses). Moreover, Fig. 17c illustrates the relationbetween the exact cumulative ductility demands and those of the model of Eq. (7). It is clear that the proposed combination of ductility demands of single events is in good agreement with theresults obtained from the dynamic inelastic analyses consideringdirectly the sequential ground motions. The aforementionedprocess is also applied for the cases of articial seismic sequences.Thus, Fig. 17d shows the exact and the estimated cumulativeductility demands for 40 cases ( 4 frames 10 articial seismicsequences), also for p 1.3. It is evident that empirical Eq. (7) canalso be adopted for articial seismic sequences giving reliableestimation of cumulative ductility demands.

Building structures are typically governed by drift limits[10,12,25,28] and therefore, according to the principles of displacement-based design method, a high accuracy estimationof displacements is required. Adopting the proposed method,

erroneous response characteristics for the structures are avoided

since all the signicant seismic events are taken into account. Itshould be noted that Eq. (7) can be applied to the well-designedframes according to Eurocodes [1,2] . On the other hand, a similarexpression for buildings designed only for gravity seems to beproblematic since these structures in reality exhibit a greatvariability in their reinforcement details and in the amount of transverse reinforcement and thus in many cases, the case of shear failure is unavoidable.

In order to control the ductility demands m (and therefore thestructural damage), one can adopt appropriate/reduced behaviourfactors, q, in the case of sequential ground motions. This reductionin the behaviour factor should be considered in earthquake-proneregions, where the reappearance of seismic events may have ahigh probability of occurrence. For example, according to thewell-known NewmarkHall [30] method, behaviour factors canbe related with ductility demands as

q m for T 4 0:5sec

ffiffiffiffiffiffiffiffiffiffiffi2m 1p for 0 :1 o T o 0:5sec1 for T o 0:03sec8>:8

The fundamental periods of vibration of the examinedstructures (see Table 1 ) are greater than 0.5 s. Thus, examiningthe case of medium- to long-periods of vibration ( T 4 0.5s), thebehaviour factor is assumed to be equal to the ductility demand, astate which is also known as equal displacement rule . Therefore, inorder to achieve the target/design ductility, m, which is smallerthan the cumulative ductility mseq , one should apply a reducedbehaviour factor, qred where qred o q. For a given sequential

ground motion and target ductility demand, the qred should be

Fig. 16. Permanent displacements.




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Fig. 17. Comparison of various models with exact results for cumulative ductility demands.

Table 4Single, mi, and cumulative ductility demands, mseq , of sequential ground motions.

Event Frame A1 Frame A2 Frame A3 Frame A4

Mammoth Lakes 54099 Convict Creek (NS)1980/05/25 (16:34) m1 3.47 5.91 6.97 8.511980/05/25 (16:49) m2 1.42 3.26 3.66 2.531980/05/25 (19:44) m3 1.75 3.14 2.78 2.571980/05/25 (20:35) m4 1.67 3.78 2.89 2.781980/05/27 (14:51) m5 1.43 2.91 3.91 3.36

Exact mseq (Model Eq. (7)) 4.79 (4.54) 11.06 (10.85) 11.79 (11.85) 11.32 (11.94)

Chalfant Valley 54428 Zack Brothers Ranch (EW)1986/07/20 (14:29) m1 1.42 1.60 1.48 1.461986/07/21 (14:42) m2 4.40 2.36 5.21 3.17


Coalinga 46T04 CHP (NS)1983/07/22 (02:39) m1 1.81 2.62 3.03 1.761983/07/25 (22:31) m2 4.33 6.94 4.85 2.89


Imperial Valley 5055 Holtville P.O. (HPV315)1979/10/15 (23:16) m1 2.70 3.77 4.15 7.641979/10/15 (23:19) m2 2.89 3.68 1.33 1.99


Whittier Narrows 24401 San Marino (NS)1987/10/01 (14:42) m1 5.00 4.95 4.27 3.381987/10/04 (10:59) m2 2.38 2.70 1.40 1.67





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ARTICLE IN PRESS

obtained through an iterative and time-consuming process, whichrequires re-design of RC structures in every step. As a practicalalternative, the approximation

qred ffi mmseq !qdes 9

can sufciently used. Finally, the designer can also use empiricalexpressions to estimate behaviour factors proposed for SDOF

systems under multiple earthquakes [7] .

6. Conclusions

This paper examines the inelastic behaviour of planar RCframes under sequential strong ground motions. Two families of regular and vertically irregular (with setbacks) frames, whichhave been designed either for vertical or both for vertical andseismic loads, are examined. A detailed study of the problem leadsto the following conclusions:

1. Multiple earthquakes require increased displacement de-mands in comparison with single seismic events. Furthermore,

the seismic damage for multiple earthquakes is higher thanthat for single ground motions. These characteristics are veryimportant and should be taken into account for the seismicdesign of structures either by the conventional force-based orespecially by the more recent displacement-based designmethod, which requires a high accuracy estimation of displacements. Therefore, the traditional seismic design pro-cedures, which are essentially based on the isolated designearthquake, should be reconsidered since the multiple earth-quakes phenomenon cannot be ignored.

2. Sequential ground motions strongly affect the development/distribution of plastic hinges, which can be different than thatfor the case of single/isolated seismic events.

3. The incremental dynamic analysis ( IDA) technique undersequential ground motions has been investigated for the rsttime. It is found that the seismic sequences lead to quitedifferent responses than the corresponding ones for singleseismic events. Furthermore, for the examined structures andPGAs, the collapse appears only for the cases of seismicsequences and not for the isolated single ground motions.

4. A simple and effective empirical expression, which combinesthe ductility demands of single ground motions, can be used toestimate cumulative ductility demands due to sequentialground motions. There is good correlation between thecumulative ductility evaluation procedure and the exactductility demands from dynamic inelastic analyses. This wasdemonstrated for the cases of three-storey and eight-storey,regular and vertically irregular well-designed reinforcedconcrete frames. In order to provide a general-acceptableempirical expression for RC framed structures, additionalinvestigation is required.

5. The ductility demands of structures appear to be increasedunder sequential ground motions. These demands can becontrolled using appropriately reduced behaviour factors,taking into account the multiplicity effect of earthquakes.

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