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EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2002; 31:11311150 (DOI: 10.1002/eqe.152)

Seismic response of self-centring hysteretic

SDOF systems

Constantin Christopoulos, Andre Filiatrault; and Bryan Folz

Department of Structural Engineering; University of California at San Diego; 9500 Gilman Drive, Mail Code

0085; La Jolla; CA 92093; U.S.A.

SUMMARY

The seismic response of single-degree-of-freedom (SDOF) systems incorporating ag-shaped hystereticstructural behaviour, with self-centring capability, is investigated numerically. For a SDOF system witha given initial period and strength level, the ag-shaped hysteretic behaviour is fully dened by a post-yielding stiness parameter and an energy-dissipation parameter. A comprehensive parametric studywas conducted to determine the inuence of these parameters on SDOF structural response, in terms ofdisplacement ductility, absolute acceleration and absorbed energy. This parametric study was conductedusing an ensemble of 20 historical earthquake records corresponding to ordinary ground motions havinga probability of exceedence of 10% in 50 years, in California. The responses of the ag-shaped hystereticSDOF systems are compared against the responses of similar bilinear elasto-plastic hysteretic SDOFsystems. In this study the elasto-plastic hysteretic SDOF systems are assigned parameters representativeof steel moment resisting frames (MRFs) with post-Northridge welded beam-to-column connections. Inturn, the ag-shaped hysteretic SDOF systems are representative of steel MRFs with newly proposedpost-tensioned energy-dissipating connections. Building structures with initial periods ranging from 0.1to 2:0sand having various strength levels are considered. It is shown that a ag-shaped hysteretic SDOF

system of equal or lesser strength can always be found to match or better the response of an elasto-plastic hysteretic SDOF system in terms of displacement ductility and without incurring any residualdrift from the seismic event. Copyright ? 2002 John Wiley & Sons, Ltd.

KEY WORDS: hysteretic models; SDOF systems; non-linear analysis; self-centring systems; seismicresponse

1. INTRODUCTION

Following the unexpected failures of beam-to-column connections in more than one hundredsteel moment-resisting frames (MRFs) during the 1994 Northridge, California earthquake, a

comprehensive research programthe SAC Joint Venturewas initiated in the United Statesto investigate and remediate the causes of these failures [1]. It was concluded from the

Correspondence to: Andre Filiatrault, Department of Structural Engineering, University of California at San Diego,9500 Gilman Drive, Mail Code 0085, La Jolla, CA 92093, U.S.A.

E-mail: [email protected]

Received 23 April 2001Revised 8 September 2001

Copyright ? 2002 John Wiley & Sons, Ltd. Accepted 10 September 2001

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1132 C. CHRISTOPOULOS, A. FILIATRAULT AND B. FOLZ

Figure 1. Concept of PTED steel connection: (a) steel frame with PTED connections; (b) deformedconguration of exterior PTED connection.

investigation phase of this project that the major cause of these failures was the unexpectedlylow rotational capacity of beam-to-column welded connections. In the remediation phase ofthe SAC project, studies on new construction led to a better understanding of the cyclic

behaviour of welded and bolted steel moment-resisting connections and to the developmentof more stringent welding practices [2]. However, even with these enhanced requirements,inelastic deformations as well as residual drifts are expected to occur in steel MRFs underseismic loading.

In parallel with the post-Northridge steel research, moment-resisting connections using post-tensioning concepts were developed for precast concrete construction [3]. A series of innova-

tive beam-to-column connections [4], combining self-centring characteristics as well as energydissipation, were proposed. It was demonstrated that the performance of these connections wasexcellent under simulated seismic loading. The most signicant characteristic of these connec-tions was their capacity to ensure small residual drifts, through self-centring capabilities, evenwhen signicant inelastic transient deformations were mobilized during the seismic response.

Recently, this post-tensioning technology has been extended to steel MRFs [5; 6]. Exper-imental and numerical results obtained from these studies show that these post-tensionedconnections are capable of achieving stiness and strength characteristics comparable to tradi-tional welded moment-resisting connections. In addition, these connections can be tailored to

provide a specied amount of energy dissipation. This structural behaviour can be achievedwithout introducing inelastic deformations in the beam or column and without residual drift.The concept for the particular post-tensioned energy dissipating (PTED) connection devel-

oped by the authors [6] is illustrated in Figure 1. This PTED connection incorporates highstrength steel post-tensioned (PT) bars designed to remain elastic during the seismic response,and conned energy-dissipating (ED) bars designed to yield both in tension and compression.Figure 2 shows a moment-rotation relationship obtained experimentally from a large scalePTED connection [6]. The self-centring capacity and energy dissipation characteristics of theconnection are evident. Figure 3 shows an idealization of the ag-shaped hysteretic behaviourof a PTED connection, which is easily amenable to numerical modelling. The overall response

Copyright ? 2002 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn.2002; 31:11311150

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SELF-CENTRING HYSTERETIC SDOF SYSTEMS 1133

Figure 2. Experimental momentrotation curve of PTED connection (after Reference [6]).

Figure 3. Idealized hysteretic behaviour of the PTED beam-to-column connection: (a) contributionof PT Bars; (b) contribution of ED bars; and (c) momentrotation relationship of PTED connection.

of the connection can be decomposed into the non-linear elastic contribution from the PT-bars and the bilinear elasto-plastic hysteretic contribution from the ED-bars. It is of interest tonote that this ag-shaped hysteretic response has also been achieved using specialized energydissipating dampers or materials [7; 8].

To date, there is limited information on the non-linear dynamic response of hysteretic self-centring systems under seismic loading. The main objective of this paper is to shed some lighton this issue by investigating numerically the inelastic response of single degree-of-freedom(SDOF) ag-shaped hysteretic systems, under code prescribed levels of seismic input. Forthis purpose, an ensemble of 20 historical records representative of ordinary ground motionshaving a probability of exceedence of 10% in 50 years in California was considered. Although

the parametric study that is conducted focuses on structural systems that display a ag-shapedhysteretic response similar in form to that developed by a PTED framing system, the resultsobtained can also be applicable in part to the other self-centring hysteretic systems cited above.Finally, to assess the advantages and disadvantages of using this new type of framing system,the dynamic responses of SDOF systems exhibiting ag-shaped hysteresis are compared tothe responses of SDOF systems exhibiting bilinear elasto-plastic hysteresis, typical of post-

Northridge steel MRFs incorporating welded fully restrained moment-resisting connections.


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2. ANALYSIS PROCEDURE

2.1. Hysteretic models

Two hysteretic models are considered in this study: a bilinear elasto-plastic model and a ag-shaped model. The bilinear elasto-plastic hysteretic model is representative of the behaviourof steel MRFs incorporating post-Northridge welded beam-to-column fully restrained moment-resisting connections. It is assumed that following the new recommendations on welded beam-to-column connections [2; 9], a large number of plastic rotation cycles can be achieved withoutany fracture of the welds. A post-yielding stiness of 0.02 of the initial stiness is alsoassumed. The idealized hysteretic forcedisplacement relationship of a system incorporatingthese types of welded connections is shown in Figure 4(a). Note that this idealization is anupper bound of the actual response of steel MRFs considering that strength degradation isexpected under cyclic loading.

The ag-shaped hysteretic model considered is representative of the behaviour of steel MRFsincorporating PTED connections both at all beam-to-column connections and at the base ofeach column. Figure 4(b) shows the idealized hysteretic forcedisplacement relationship of asystem incorporating these PTED connections. Associated with this hysteretic model are twoindependent response parameters and . In this study the post-yielding stiness coecient, expressed as a fraction of the initial stiness, ranges in value from 0.02 to 0.35. Thecoecient reects the energy dissipation capacity of the system. A lower bound of = 0:0

produces a piecewise non-linear elastic system. An upper bound of = 1:0 is required toensure the self-centring capability of the hysteretic model.

2.2. Normalized equations of motion

The equation of motion of a SDOF system under seismic input is given by

mx+c x+F(x) = mxg (1)

Figure 4. Idealized pseudo forcedisplacement relationships: (a) system incorporating weldedconnections; and (b) system incorporating PTED connections.


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where m is the mass of the system, c is the viscous damping coecient and F(x) is thenon-linear restoring force dened by the hysteretic model of the system. The displacement,velocity and acceleration of the system, relative to the ground are denoted by x, x and x,respectively. The ground acceleration is designated by xg.

Two key parameters that can be used in dening the dynamic response of a non-linearSDOF system are the initial period T0 and the strength ratio :

T0= 2

m=k0 (2)

= Fy

mg (3)

where k0 is the initial stiness of the system, Fy is the yield force and g is the accelerationof gravity.

Using Equation (2), Equation (1) can be rewritten as

x+ 20

2

T0

x+

2

T0

2f(x) = xg (4)

with 0 denoting the initial fraction of critical damping of the system,

0 = c

2

k0m(5)

and f(x) representing the non-linear pseudo-restoring of the system:

f(x) = F(x)

k0(6)

The yield displacement xy of the system is given by

xy = Fy

k0= fy (7)

as shown in Figure 4. Using Equations (2) and (3), the yield displacement xy can be expressedin terms of the key parameters of the system T0 and :

xy=T20g

42 (8)

With this formulation, for a specied level of critical damping 0, initial period T0 and strength

level the SDOF is completely dened for the case when the restoring forcedisplacementrelationship is bilinear elasto-plastic (Figure 4(a)) and requires only the additional parameters and to be assigned if the restoring forcedisplacement relationship exhibits a ag-shapedhysteresis (Figure 4(b)).

For the time-history dynamic analyses performed in this study, the normalized non-linearequation of motion given by Equation (4) is integrated using the Newmark constant averageacceleration scheme. The analyses were continued for 10 s of zero ground acceleration at the


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end of each record to allow the system to oscillate under viscously damped free vibrations inorder to return to rest.

2.3. Energy balance

The energy balance at time tfor the normalized equation of motion can be written as

Ek(t) +Ed(t) +Es(t) =Ein (9)

where Ek(t); Ed(t), Es(t) and Ein are the kinetic energy at time t, the energy dissipated byviscous damping up to time t, the strain energy at time t (recoverable elastic and dissipatedhysteretic) and the relative seismic input energy, respectively. These energy quantities can bedened as follows:

Ek(t) = 12x(t)2

Ed(t) = 20

2

T0

x(t)0

x(t) dx (10)

Es(t) =

2

T0

2 x(t)0

f(t) dx

Ein= x(t)

0

xg(t) dx

Equations (9) and (10) determine how the seismic input energy is distributed in the systemover time, as well as allowing for a check on the accuracy of the time integration scheme.

2.4. System response indices

The inelastic response of SDOF systems under seismic input can be characterized in part bythe following normalized non-dimensional response indices:

(i) The maximum displacement ductility :

=max06t6tD |x(t)|

xy(11)

where tD is the total duration of the seismic input.

In performance-based earthquake engineering, the maximum inelastic displacement isone of the primary response indices to determine both the structural and non-structuraldamage to buildings under seismic loading [10; 11].

(ii) The normalized absolute maximum acceleration amax:

amax=max06t6tD |x(t) + xg(t)|

g (12)


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This index is a measure of the damage potential to acceleration-sensitive non-structuralelements, as well as an indicator of potential injury to occupants during an earthquakeevent. In addition, this response index is a direct indicator of the force level inducedinto the system by the seismic input.

(iii) The normalized maximum absorbed energy Eabs:

Eabs=max06t6tD |Es(t)|

xymg (13)

This index is a measure of potential structural damage including duration eects.(iv) The normalized residual displacement xres:

xres=|x(tD)|

xy(14)

This index is an indicator of the structural damage sustained after an earthquake and ofthe extent of repair costs. Residual displacements are only computed for the bilinear

elasto-plastic hysteretic model (see Figure 4(a)). The ag-shaped hysteretic model(see Figure 4(b)), by virtue of its self-centring capabilities, does not have residualdisplacements.

2.5. Ground motions considered in parametric study

An ensemble of 20 historical strong ground motion records from California representative ofordinary earthquakes having a probability of exceedence of 10% in 50 years are used in thisstudy [12]. These records are free of any forward directivity eects (near-fault eects). Allrecords were recorded on soil types C or D, and were generated by earthquakes of momentmagnitude M

w ranging from 6:7 t o 7:3. The hypocentral distance for these records range

between 13 and 25 km. Table I gives further details on the characteristics of these earthquakerecords.

Following the method proposed in NEHERP Provisions for the seismic rehabilitationof buildings [13], a 5% damped design elastic acceleration response spectrum for a seis-mic zone 4 and a soil type C or D was constructed and used as the target spectrum.Each of the 20 earthquake records was then scaled to minimize the square of the error

between its 5% damped response spectrum and the target NEHERP spectrum at ve pe-riod values: T= 0:1; 0:25; 0:5; 1:0 and 2:0 s. The resulting scaling factors are listed inTable I.

The mean and the envelopes of the maximum and minimum spectral values of the 20 scaledrecords along with the NEHERP target spectrum are shown in Figure 5. A good match is

obtained between the mean spectral values and the target spectrum in the range of periodsof interest (0:12:0s). However, the envelopes of maximum and minimum spectral valuesindicate the large variability that exists between the records. Table I also lists the scaled peakground accelerations (PGA), and scaled peak ground velocities (PGV). The mean value ofthe PGA of the 20 scaled records is 0:43g, which is very close to the eective peak accel-eration Ca= 0:40g specied in the NEHERP provisions for a seismic zone 4 and soil typesC and D.


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TableI.Characteristic

sofgroundmotionsconsidered(afterReference[12]).

Earthquakeevent

Year

Mw

Station

Rclosest

Soiltype

Duration

Scaling

Scaled

Scaled

(km)

(NEHRP)

(s)

factor

PGA(g)PG

V(cm=s)

SuperstitionHills1987

6.7

Brawley

18.2

D

22.0

2.7

0.313

46.44


6.7

ElCentroImp.

Co.Cent.

13.9

D

40.0

1.9

0.490

77.71


6.7

PlasterC

ity

21.0

D

22.2

2.2

0.409

45.32

Northridge

1994

6.7

BeverlyHills14

145Mulhol

19.6

C

30.0

0.9

0.374

53.10

Northridge

1994

6.7

CanogaParkT

opangaCan

15.8

D

25.0

1.2

0.427

38.52

Northridge

1994

6.7

GlendaleLa

sPalmas

25.4

D

30.0

1.1

0.393

13.53

Northridge

1994

6.7

LAHollywoo

dStorFF

25.5

D

40.0

1.9

0.439

34.77

Northridge

1994

6.7

LANFaringRd

23.9

D

30.0

2.2

0.601

34.76

Northridge

1994

6.7

N.HollywoodC

oldwaterCan

14.6

C

21.9

1.7

0.461

37.74

Northridge

1994

6.7

SunlandMtGleasonAve

17.7

C

30.0

2.2

0.345

31.90

LomaPrieta

1989

6.9

Capitola

14.5

D

40.0

0.9

0.476

32.85

LomaPrieta

1989

6.9

GilroyArray#3

14.4

D

39.9

0.7

0.386

24.99

LomaPrieta

1989

6.9

GilroyArray#4

16.1

D

40.0

1.3

0.542

50.44

LomaPrieta

1989

6.9

GilroyArray#7

24.2

D

40.0

2.0

0.452

32.80

LomaPrieta

1989

6.9

HollisterDi

.Array

25.8

D

39.6

1.3

0.363

46.28

LomaPrieta

1989

6.9

SaratogaWV

alleyColl.

13.7

C

40.0

1.4

0.465

86.10

CapeMendocino

1992

7.1

FortunaFortu

naBlvd

23.6

C

44.0

3.8

0.441

114.00

CapeMendocino

1992

7.1

RioDellOverpassFF

18.5

C

36.0

1.2

0.462

52.68

Landers

1992

7.3

DesertHotSprings

23.3

C

50.0

2.7

0.416

56.43

Landers

1992

7.3

YermoFire

Station

24.9

D

44.0

2.2

0.334

65.34


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Figure 5. Elastic response spectra of 20 scaled accelerograms.

3. PARAMETRIC STUDY

3.1. Range of key system and hysteretic parameters

The parametric study presented herein focuses on the seismic response of steel MRFs rangingin number of stories from one to twenty. From the seismic provisions of the 1997 edition ofthe uniform building code (UBC) [14], the natural period range of these structures can beestimated by the equation:

T0= Cth3=4n (15)

where Ct= 0:0853 for steel MRFs, and where hn is the height of the building in meters.Using Equation (15) and assuming a storey height of 3:4m, the range of periods T0 for a

single storey and for a 20 storey building, respectively, is:

0:2s6T062:0s (16)

The strength factor which corresponds to the ratio Vy=W in the 1997 UBC [14] is denedas

=VyW

=CvI

RT0(17)

where Vy is the design base shear, W is the weight of the structure, I, taken as 1, is the

importance factor, Cv is computed as 0:64 for a Zone 4 with soil type D and R is the forcereduction factor ranging from 4.5 to 8.5 for an ordinary steel MRF and a special steel MRF,respectively. A lower bound for Equation (17) for seismic Zone 4 is

=Vy

W0:11CaI (18)

where Ca is computed as 0.44 for a seismic Zone 4 and soil type D.


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Table II. SDOF system values used in parametric study.

T0 (s)

1.00 0.05 0.02 0.00

0.25 0.10 0.10 0.300.50 0.20 0.20 0.601.00 0.30 0.35 1.001.50 0.502.00 1.00

For ag-shaped hysteretic model only.

Substituting the lower and upper period bounds into Equation (17) and verifying Equation(18), the range of strength factors is found to be

0:05660:71 (19)

In addition, the ag-shaped hysteretic model requires the specication of the parameters and to completely dene the system. Table II lists the complete set of parameters T0, , and considered in this parametric study. These values result in 576 dierent ag-shapedhysteretic systems. The resulting forcedeection relationships of the ag-shaped hystereticsystems are illustrated qualitatively in Table III for the specied range of values of and. Throughout this study the fraction of critical damping 0 is taken as 0.05 for all SDOFsystems.

3.2. Non-linear dynamic response of ag-shaped hysteretic systems

Mean values over the ensemble of earthquakes of the displacement ductility are shownin Figure 6 for all ag-shaped hysteretic systems considered. For all values of and , themean displacement ductility generally increases for decreasing values of initial period T0 anddecreasing values of strength ratio . The mean displacement ductility is reduced in all casesfor increasing values of and . This reduction of with increasing values of and ismore signicant for low period structures (T061:0s) and for structures with lower strengthratios (60:3) where the values of are also the largest.

Mean values over the ensemble of earthquakes of the maximum absolute acceleration amaxare shown in Figure 7. Note that when = 1:0 the SDOF oscillator responds in the elas-tic range for most of the earthquake records considered and for all values of and .For this case, the plot of the mean maximum absolute acceleration versus period tends

towards the elastic response spectrum. The mean maximum absolute acceleration is insen-sitive to the value of as seen in Figure 7. When is increased, the accelerations ofsystems with lower values of are increased. For increasing values of ; amax for allvalues of tends towards the elastic response spectrum. For small values of initial pe-riod (T060:5s); amax remains high even when the strength ratio is reduced. This is dueto the combination of non-zero post-yielding stiness with large values of displacementductility.


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Table III. Qualitative forcedeection relationships of ag-shaped hysteretic systems for all valuesof and considered in parametric study.

Energy-dissipation coecient,

Post-yielding stiness, 0.0 0.30 0.60 1.0

0:02

0:10

0:20

0:35

Mean values over the ensemble of earthquakes of the absorbed energy Eabs are shownin Figure 8. In general, the energy absorbed increases for decreasing initial period and fordecreasing strength ratios. This trend is similar to that observed for the displacement ductility(see Figure 6). However for smaller values of , the absorbed energy (Figure 8) does notincrease as much as the displacement ductility (Figure 6).

The mean absorbed energy Eabs is insensitive to increasing values of, but highly dependenton the value of . In general, the mean absorbed energy Eabs doubles when the value of changes from 0 to 1. This increased absorbed energy indicates a higher amount of hysteretic

damping but also larger cumulative inelastic excursions in the system.

3.3. Non-linear dynamic response of bilinear elasto-plastic hysteretic systems

Figure 9 collectively shows mean values, over the ensemble of earthquakes, of displacementductility , normalized maximum absolute acceleration amax, normalized absorbed energy Eabsand normalized residual displacement xres for the bilinear elasto-plastic hysteretic systems.


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Figure 6. Mean displacement ductility for ag-shaped hysteretic systems.

Similar to the ag-shaped hysteretic systems, the mean displacement ductility increasesfor decreasing values of initial period T0 and decreasing values of the strength ratio , asshown in Figure 9(a).

Mean maximum absolute accelerations amax for the case where the strength ratio is takenas 1 as shown in Figure 9(b), are similar to those of the elastic response spectrum. Themaximum accelerations decrease for decreasing values of the strength ratio. Similar to the ag-

shaped hysteretic systems, the accelerations for systems with short initial periods (T0= 0:1s) donot decrease when the strength ratio is decreased (60:3). This is also due to the combinationof the post-yielding stiness of the bilinear elasto-plastic hysteretic systems, = 0:02, and largedisplacement ductility values for these systems.

The mean absorbed energy Eabs, as shown in Figure 9(c), increases for decreasing valuesof initial period T0 and decreasing values of strength ratio similarly to the displacementductility. As noted for the ag-shaped hysteretic systems, for lower strength systems, lowering


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Figure 7. Mean normalized maximum accelerations for ag-shaped hysteretic systems.

the strength ratio causes a larger increase in the displacement ductility than in the absorbedenergy.

The residual displacements shown in Figure 9(d) increase for decreasing values of initialperiod T0 and decreasing values of strength ratio . For the highest strength ratio ( = 1:0),there are no residual displacements at the end of the earthquake. For lower strength values(60:3), residual displacements are more pronounced and more dependent upon the initial

period.

3.4. Comparative response of the ag-shaped and elasto-plastic hysteretic systems

The response of ag-shaped hysteretic and bilinear elasto-plastic hysteretic systems are qual-itatively very similar as seen by comparing Figures 68 with Figure 9. The following threeobservations can be made on the comparative response of these two types of hysteretic


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Figure 8. Mean normalized absorbed energy for ag-shaped hysteretic systems.

systems:

(i) For each bilinear elasto-plastic system, there is at least one ag-shaped hystereticsystem of similar initial period and strength ratio that can achieve equal or smallerdisplacement ductility. In general, the intermediate values of and are sucient to

achieve this.(ii) The maximum absolute accelerations are similar between these two hysteretic models

for low values of . For larger values of , maximum accelerations are larger for theag-shaped hysteretic systems, especially for systems with lower strength ratios.

(iii) The energy absorbed is in general signicantly larger for the bilinear elasto-plastichysteretic systems than the ag-shaped hysteretic systems, especially for low valuesof .


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Figure 9. Response of bilinear elasto-plastic hysteretic systems: (a) displacement ductility; (b) normal-ized maximum accelerations; (c) normalized absorbed energy; and (d) normalized residual displacements.

To further compare the response of these two hysteretic models, three particular systemsare considered. The rst system is characterized by an initial natural period T0= 0:25s anda strength ratio = 0:5, representing a one-storey steel MRF structure. The second systemis characterized by an initial natural period T0= 1:0s and a strength ratio = 0:1, represent-ing a 7-storey steel MRF structure. The last system considered is characterized by an initialnatural period T0= 2:0s and a strength ratio = 0:05, representing a 20-storey steel MRFstructure.

For each of these structural congurations, an elasto-plastic hysteretic system (EP) andthree ag-shaped hysteretic systems (FS) are dened and considered for comparative pur-

poses. The FS systems have the same initial period T0 and strength ratio as the corre-sponding EP system. As listed in Table IV, each FS system has a dierent combination

of post-yielding stiness coecient and energy dissipating coecient . The mean re-sponse values over the ensemble of earthquakes for these systems are presented in TableIV. The maximum absolute accelerations are increased for increasing values of . The en-ergy absorbed, also as discussed earlier, is larger for the elasto-plastic hysteretic systems andis also increased for the ag-shaped hysteretic systems for larger values of . Finally, un-like the bilinear elasto-plastic systems, the ag-shaped hysteretic systems sustain no residualdisplacements.


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Table IV. Comparative response of three systems exhibiting bilinear elasto-plastic hysteresis (EP)and ag-shaped hysteresis (FS).

amax Eabs xres

System 1: T0 = 0:25s; = 0:50EP 2.28 0.55 2.39 0.61FS ( = 0:10; = 1:0) 2.10 0.56 2.46 0.00FS ( = 0:20; = 0:6) 2.26 0.64 1.92 0.00FS ( = 0:35; = 0:3) 2.27 0.73 1.48 0.00

System 2: T0 = 1:0s; = 0:10EP 6.19 0.13 2.71 1.10FS ( = 0:02; = 1:0) 6.07 0.13 2.27 0.00FS ( = 0:20; = 0:6) 6.45 0.22 2.00 0.00FS ( = 0:35; = 0:6) 6.19 0.29 1.98 0.00

System 3: T0 = 2:0s; = 0:05EP 4.41 0.06 0.83 1.25

FS ( = 0:02; = 1:0) 4.78 0.06 0.70 0.00FS ( = 0:20; = 0:6) 4.50 0.09 0.55 0.00FS ( = 0:35; = 1:0) 4.21 0.11 0.69 0.00

Figure 10. Loma Prieta record (Hollister dierential array) scaled at 130%: (a) accelerogram; and(b) elastic response spectrum for 5% damping for accelerogram and for ensemble of earthquake records.

3.5. Examples of time-history analyses

To this point, results have been given in terms of non-dimensional mean value responsequantities over the ensemble of 20 earthquake records. To further compare the two hysteretic

models, and to provide some insight in their response over time, a series of ve structuralsystems were subjected to the 1989 Loma Prieta earthquake recorded at the Hollister Dier-ential Array and scaled at 130% of its amplitude (see Table I). The scaled accelerogram forthis record is presented in Figure 10(a). As shown in Figure 10(b), the 5% damped elasticresponse spectrum of the scaled record is in good agreement with the mean spectrum forthe ensemble of 20 records used in the parametric study. All systems considered have aninitial period of T0= 1:0s and a mass of 4:0kN s

2=mm. The resulting initial stiness k0 is


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Table V. Response of SDOF systems with T0 = 1:0s, m= 4kN s2=mm and k0= 157:9kN=mm under

130% of Loma Prieta record.

System y Fy (max) A (max) Eabs (res)(mm) (kN) (mm) (g) (kN mm) (mm)

EP 0.02 0.10 24.85 3924 124.70 0.13 2:0 106 22.95FS1 0.25 0.30 0.10 24.85 3924 110.72 0.20 7:4 105 0.00FS2 0.25 0.30 0.07 17.40 2747 117.88 0.17 8:3 105 0.00

FS3 0.15 0.50 0.10 24.85 3924 110.00 0.16 1:1 106 0.00

FS4 0.15 0.50 0.07 17.40 2747 115.22 0.14 1:1 106 0.00

157:9kN=mm. The structural system so dened is a representation of a 7-storey steel MRF.As shown in Figure 10, the peak ground acceleration is 0 :36 g while the spectral accelerationat a period of 1:0s is 0:72g.

The rst system designated by EP incorporates the elasto-plastic hysteretic model. The fourother systems designated by FS1 through FS4 utilize the ag-shaped hysteretic model. Theyield force for systems EP, FS1 and FS3 were set equal to 3924 kN which corresponds toa strength ratio = 0:1. The yield force of systems FS2 and FS4 were set equal to 2747 kNwhich is equal to 70% of the yield force of systems EP, FS1 and FS3 and correspondsto a strength ratio = 0:07. As noted earlier, intermediate values of and result in dis-

placement ductility values for the ag-shaped hysteretic model that are similar to systemswith large values of combined with low values of and vice-versa. For systems FS1and FS2, and were set to 0.25 and 0.30, respectively. For systems FS3 and FS4, and were set to 0.15 and 0.50, respectively. The dening parameters for these ve

systems are summarized in Table V. Response values of maximum relative displacementmax, maximum absolute acceleration Amax, absorbed energy Eabs as well as residual dis-placement res obtained from dynamic time-history analyses are presented in Table V. It isnoted that these response values obtained for the 130% Loma Prieta earthquake record arein close agreement with the mean values obtained over the ensemble of earthquakes andfollow similar trends as discussed earlier. In all cases, all four ag-shaped hysteretic sys-tems achieve smaller maximum displacements than the elasto-plastic system. Systems FS1and FS2 have greater maximum accelerations and lower absorbed energy than systems FS3and FS4.

Figure 11 shows the time-histories of displacement, acceleration, absorbed energy for theEP and FS4 systems along with their forcedisplacement responses. Note that the elasto-plasticsystem deforms inelastically primarily in one direction, while the FS4 system has a similar

amount of inelastic excursions in both directions. For the elasto-plastic system, the one-sidedinelastic deformations will in fact be accentuated by P-delta eects [15] that have not beentaken into account in this study. The FS4 system, with a strength ratio equal to 0.70 of the EPsystem achieves a smaller maximum displacement, while the maximum absolute accelerationsare similar. The energy absorbed is considerably smaller for the FS4 system. Finally, unlikethe EP system that sustains a residual displacement of 23 mm, the FS4 system returns to itsinitial zero position after the end of the earthquake.


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Figure 11. Comparative response of EP and FS4 systems under 130% of Loma Prieta record (Hollisterdierential array): (a) displacement time-history; (b) acceleration time-history; (c) absorbed energytime-history; and (d) forcedisplacement response.

4. CONCLUSIONS

The seismic response of SDOF systems incorporating either a bilinear elasto-plastic hystereticmodel or a ag-shaped hysteretic model have been investigated and compared through time-history dynamic analyses. All systems were subjected to an ensemble of 20 historical recordsrepresentative of ordinary ground motions having a probability of exceedence of 10% in 50years in California. For a SDOF system with a given initial period and strength level, the

incorporated ag-shaped hysteretic model is dened through a post-yielding parameter andenergy-dissipation parameter . These two independent parameters allow for exibility intailoring the response of this type of SDOF system. A comprehensive parametric study wasconducted to determine the inuence of these parameters on SDOF structural response, interms of displacement ductility, absolute acceleration and absorbed energy. It was found thatreduced displacement ductility in systems with short initial periods and low strength levelswas most eectively achieved by increasing the value of as opposed to increasing the value


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of. The reverse is true for long period, high strength systems: increasing is more eectivethan increasing .

It was also shown that the seismic response of ag-shaped hysteretic systems was qual-itatively similar to the elasto-plastic hysteretic systems. In addition, by adjusting the values

of and , a ag-shaped hysteretic system could be made to quantitatively match or betterthe response of an elasto-plastic system in terms of displacement ductility. Values of and to achieve this were not unique. In general, this match in performance can be realizedusing intermediate values of and . Such values are physically achievable using the newly

proposed post-tensioned energy-dissipating (PTED) connections.With respect to absolute acceleration, the ag-shaped hysteretic system tends to produce

higher values than the comparable elasto-plastic system. The greatest dierence is seen withhigher values of. Also, the absorbed energy by the ag-shape hysteretic system is always lessthan the comparable elasto-plastic system. However, the importance of this response index forsteel structures incorporating PTED connections is minimal since cumulative damage is limitedto the replaceable energy-dissipating bars within the beam-to-column and base connections.

Finally, residual drifts occurred in all of the elasto-plastic hysteretic SDOF systemsconsidered. Residual drifts were largest in systems with low strength and short periods. In allof the ag-shaped hysteretic SDOF systems there was no residual drift due to the self-centringcapability of the forcedisplacement model.

ACKNOWLEDGEMENT

The nancial assistance of the Deans oce of the Irwin and Joan Jacobs School of Engineering atthe University of California, San Diego in support of this study is greatly appreciated.

REFERENCES

1. SAC. Proceedings of the invitational workshop on steel seismic issues. SAC Report No. 94-01, Sacramento,CA, 1994.

2. FEMA. Recommended seismic design criteria for new steel moment-frame buildings. FEMA No. 350. FederalEmergency Management Agency, Washington, DC, 2000.

3. Priestley MJN. Seismic design philosophy for precast concrete frames. Structural Engineering International1996; 6(1):2531.

4. Stanton JF, Stone WC, Cheok GS. A hybrid reinforced frame for seismic regions. PCI Journal 1997; 42(2):2032.

5. Ricles MJ, Sause R, Garlock MM, Zhao C. Postensioned seismic-resistant connections for steel frames. Journalof Structural Engineering, ASCE 2001; 127(2):113121.

6. Christopoulos C, Filiatrault A, Uang C-M, Folz B. Post-tensioned energy dissipating connections for moment-resisting steel frames. Journal of Structural Engineering, ASCE 2002, accepted.

7. Witting PR, Cozzarelli FA. Shape memory structural dampers: material properties, design and seismic testing.NCEER Report No. 92-0013. National Center for Earthquake Engineering Research, Bualo, NY, 1992.

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8(3):403428.11. Priestley MJN. Displacement-based seismic assessment of existing reinforced concrete buildings. Bulletin of the

New Zealand National Society for Earthquake Engineering 1996; 29(4):256272.12. Krawinkler H, Parisi F, Ibarra L, Ayoub A, Medina AR. Development of a testing protocol for wood frame

structures. CUREE Report No. W-02, Richmond, CA, 2000.


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13. Federal Emergency Management Agency. NEHERP provisions for the rehabilitation of buildings. FEMA No.273 (Guidelines) and 274 (Commentary), Washington, DC, l997.

14. International Conference of Building Ocials. Uniform Building Code, vol. 2, Whittier, CA, 1997.15. MacRae GA, Kawashima K. Post-earthquake residual displacements of bilinear oscillators. Earthquake

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