re-centring capability of flag-shaped seismic isolation systems

Bull Earthquake Eng (2012) 10:1267–1284DOI 10.1007/s10518-012-9343-1

ORIGINAL RESEARCH PAPERS

Re-centring capability of flag-shaped seismic isolationsystems

Donatello Cardone

Received: 13 August 2011 / Accepted: 16 February 2012 / Published online: 11 March 2012© Springer Science+Business Media B.V. 2012

Abstract The re-centring capability is identified by the current design codes as a funda-mental feature of seismic isolation systems. In this paper, the re-centring capability of seismicisolation systems characterised by a flag-shaped hysteretic cyclic behaviour is investigatedthrough an extensive parametric study of single-degree-of-freedom hysteretic systems sub-jected to different natural records. A remarkable example of this kind of isolation systemderives from the combination of flat steel-PTFE sliding bearings with auxiliary re-centringdevices based on the superelastic properties of Shape Memory Alloys. The results of theparametric analyses are processed statistically and regression analysis relations are derivedthat show the dependence of the residual displacement after the earthquake on the govern-ing parameters of the isolation system. Based on the analysis results, the features of theflag-shaped system that guarantee sufficient re-centring capability are identified.

Keywords Seismic isolation · Re-centering capability · Residual displacement ·Flag-shaped hysteretic model · Shape memory alloys · Sliding bearings

1 Introduction

The re-centring capability is identified by the current design codes as a fundamental require-ment for seismic isolation systems (Skinner et al. 1993). Insufficient re-centring capability ismanifested by (i) substantial residual displacements at the end of the seismic event, (ii) accu-mulation of displacements during a sequence of seismic events and (iii) greater maximum andresidual displacements for seismic ground motions with directivity effects (one-side pulse,etc.).

The re-centring capability of the isolation system is increased by the lateral force of elasticcomponents, such as the rubber stiffness force of elastomeric bearings (Tsopelas et al. 1994)and the re-centring force due to the concave sliding surface of spherical sliding bearings

D. Cardone (B)DiSGG, University of Basilicata, 85100 Potenza, Italye-mail: [email protected]

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1268 Bull Earthquake Eng (2012) 10:1267–1284

(e.g. the Friction Pendulum System (FPS)) (Al-Hussaini et al. 1994). On the contrary, there-centring capability of the isolation system is decreased by hysteretic forces (e.g. due tothe presence of hysteretic dampers, the yielding of lead core in lead rubber bearings, etc.)and/or friction forces in flat sliding bearings. The balance of these counteracting componentsdefines the re-centring capability of the isolation system.

Most of the previous researches on the re-centring capability of nonlinear systems havebeen focused on the residual displacements of low-ductility systems, that are not typicalof seismic isolation systems. Riddell and Newmark (1979) showed that the magnitude ofthe residual displacement may be strongly affected by the hysteresis loop shape of the non-linear system. Mahin and Bertero (1981) found that, for some elasto- plastic systems, theresidual displacement averaged more than 40% of the peak displacements with significantscatter. Kawashima et al. (1998) proposed a residual displacement prediction procedure forbilinear ductile systems based on the results of a parametric study of various bilinear single-degree-of-freedom (SDOF) oscillators. According to MacRae and Kawashima (1997), thebilinear oscillator property that mostly affects the magnitude of the residual displacement isthe post-yield stiffness ratio of the bilinear system.

Medeot (2004) proposed the evaluation of the re-centring capability of bilinear hystereticisolation systems based on the energy criterion ES ≥ 0.25EH, where ES is the stored elasticenergy and EH is the hysteretic dissipated energy at a given maximum deformation. Dicleliand Buddaram (2006) and Berton et al. (2006) presented the results of parametric studies onbilinear hysteretic isolation systems that show the importance of the characteristic strengthof the system on its re-centring capability.

Katsaras et al. (2008) presented the results of parametric studies on bilinear hystereticisolation systems, showing that the main parameter that affects the re-centring capability ofthe isolation system is the ratio dmax/drm, where dmax is the absolute value of the peak dis-placement over the entire time history and drm is the maximum static residual displacement,which depends on the shape of the hysteretic cycles of the isolation system. The studies byKatsaras et al. (2008), moreover, clearly prove that the re-centring capability of isolationsystems depends not only on the system properties but also on the earthquake characteristics.From the statistical analysis of more than 100 seismic records, the authors conclude thatbilinear isolation systems with dmax/drm > 0.5 exhibit small residual displacements at theend of the earthquake.

Based on the outcomes of shaking table tests, Tsopelas et al. (1994) concluded that isola-tion systems consisting of sliding bearings, rubber devices and fluid dampers exhibit sufficientrestoring force capability when the ratio of characteristic strength (at high velocity) to peakrestoring force is less or equal to 3. This requirement is equivalent to dmax/drm > 0.33, thatis in line with the numerical results by Katsaras et al. (2008). An experimental study of flag-shaped systems consisting of sliding bearings and fluid restoring force/damping devices ispresented in (Tsopelas and Constantinou 1994). In that case, permanent displacements wereavoided by a preload in the fluid device, which was selected to just exceed the minimumfriction force in the isolation system (Tsopelas and Constantinou 1994).

Re-centring capability criteria have been adopted by modern seismic codes. The currentregulations, however, are not based on theoretical fundamentals but rather on semi-empir-ical approaches. The current AASHTO Guide Specifications for Seismic Isolation Design(AASHTO 2000), for instance, include the following two main requirements:

(i) The period corresponding to the tangent stiffness at any displacement up to the designdisplacement di shall be less than 6 s;

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Bull Earthquake Eng (2012) 10:1267–1284 1269

(ii) The re-centring force at the design displacement d shall be greater than the re-centringforce at 0.5d by not less than W/80, where W is the weight of the structure. Based onthe typical behaviour of isolation systems, this requirement can be expressed as:

K2 · d ≥ 0.025 · W (1)

where K2 is the post elastic stiffness of the isolation system.

The AASHTO Guise Specifications do not permit the use of isolation systems that do notsatisfy this requirement. The International Building Code (International Code Council (ICC)2006), specifies re-centring capability requirements that are largely based on the AASHTOprovisions.

The 2001 California Building Code (California Buildings Standards Commission (CBSC)2001), on the other hand, simply requires a minimum post-elastic stiffness such that the forceat the design displacement d minus the force at half the design displacement (d/2) is greaterthan 0.025W. Based on the typical cyclic behaviour of isolation systems, this requirementcan be expressed as:

K2 · d ≥ 0.05 · W (2)

Eurocode 8—Part 2 (European Committee for Standardization (CEN) 2005) introduces theconcept of static residual displacement drm , which is defined as the residual displacementthat is found when the system is unloaded under quasi-static conditions from its displace-ment capacity dmax . Eurocode 8 requires that the force at the design displacement (d) minusthe force at half the design displacement (d/2) is greater than 0.025W (drm/d). Based onthe typical cyclic behaviour of isolation systems, this requirement can be expressed in thefollowing way:

K2 · d ≥ √0.05 · q · W (3)

where q is the ratio of the characteristic strength of the isolation system to the weight ofthe structure (q = Qd/W). It should be noted that Eq. (3) collapses to Eq. (2) of the 2001CBC when q = 0.05, while it is more conservative when q > 0.05 and is less conservativeotherwise. For sliding isolation systems, for instance, the coefficient q is the friction coef-ficient at near zero velocity μfr,min (Dolce et al. 2005). Similarly, in lead-rubber bearings,the characteristic strength Qd can be identified with the yield strength of the lead core underquasi-static conditions (Skinner et al. 1993).

Equation (3) recognizes the importance of the characteristic strength of the isolation sys-tem in defining its re-centring capability. From this point of view, the Eurocode 8 follows amore rational approach for establishing sufficient re-centring capability than either the 2001CBC or the 2000 AASHTO Guide Specifications.

The design strategy of requiring strong re-centring capability is based on the experiencethat bridge failures in earthquake were primarily the result of excessive displacements. Byrequiring strong restoring force, cumulative permanent displacements are avoided and theprediction of displacement demand is accomplished with less uncertainty (Costantinou et al.2007). By contrast, seismic isolation systems with low restoring force ensure that the forcetransmitted to the structure is predicted with some certainty. However, this is accomplishedat the expense of uncertainty in the resulting maximum displacement and the possibility oflarge residual displacements (Costantinou et al. 2007).

In this paper, the re-centring capability of isolation systems with flag-shaped hystereticcyclic behavior is investigated through a parametric study of single-degree-of-freedom sys-tems subjected to different seismic records.

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Typically, a flag-shaped seismic isolation system derives from the combination of a num-ber of steel-PTFE flat sliding bearings with a number of auxiliary re-centring devices basedon the superelastic properties of Shape Memory Alloys (SMA) (Dolce and Cardone 2001;Dolce et al. 2000). The flat sliding bearings sustain the gravity loads of the superstructurewhile allowing large horizontal displacements, due to the low friction (typically 2–3% theweight of the superstructure) of lubricated steel-PTFE sliding bearings (Dolce et al. 2005).The seismic response of flag-shaped seismic isolation systems has been extensively examinedin previous studies (Cardone et al. 2006; Dolce et al. 2007). In the past, however, the attentionhas been mainly focused on the maximum seismic response of flag-shaped isolation systemswith supplemental re-centring capacity, compared to other isolation systems (elastomeric iso-lators, FPS, etc.). The re-centring capability of flag-shaped isolation systems has been neverexamined in detail, although it can significantly affect the design of the isolation systems,especially when the main scope of the design is to limit the force transmitted to the structure.

2 Theoretical considerations

The flag-shaped hysteretic model is fully defined by three independent parameters, whichare the force at zero displacement F0 (typically equal to the friction resistance μfrW of flatsliding bearings), the “post-yield” stiffness K2 (related to the strain rate effect of the SMAwires during the forward austenite transformation; Dolce and Cardone 2001) and the “yield”displacement dy (corresponding to the beginning of deformation in the superelastic SMAwires; Dolce et al. 2000). The force–displacement relationship of the flag-shaped hystereticmodel is illustrated in Fig. 1c. It is worth considering the resultant reaction force of the systemas sum of two components: a nonlinear elastic force component F1 = K1dy + K2(d − dy)

(see Fig. 1a), which is proportional to the displacement d and directed always towards theorigin, and a rigid–perfectly plastic component F2 = ±µ f r W (see Fig. 1b), which remainsapproximately constant during cyclic deformation. The rigid-plastic component is indepen-dent from the displacement d and it can be directed away from the origin, which may leadto imperfect re-centering of the isolation system.

Flag-shaped hysteretic systems can be in static equilibrium with zero resultant force undera non-zero residual displacement dres. This occurs when the force of the nonlinear elasticcomponent equilibrates the force of the rigid-plastic component, i.e. F1 + F2 = 0. This canoccur under a residual displacement dres bounded by the maximum static residual displace-ment, i.e.: −drm < dres < drm, where the limit drm is defined by the condition F1 = −F2.The value of the maximum static residual displacement drm depends on the system propertiesonly. For a flag-shaped hysteretic system it is equal to:

drm = μfr · W − (K1 − K2) · dy

K2(4)

assuming dmax > dy.The re-centring capability of the flag-shaped isolation system enhances as drm decreases,

because the residual displacements are limited by this value. For a given friction resistanceµ f r W , the re-centring capacity of a flag-shaped hysteretic model improves while increasingthe “post-yield” stiffness K2 and the elastic stiffness K1, as well as the “yield” displacementdy. A small value of drm means that the linear elastic component dominates the rigid–perfectlyplastic component. If the isolation system was based on flat sliding bearings only, the value

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Fig. 1 Flag-shaped hystereticmodel: a nonlinear elasticcomponent, b rigid-perfectlyplastic component and resultantcyclic behavior

(a)

(b)

(c)

F0

-F0

F2

d

dmax

K1

K2

drm

F0

F1+F2

d

dmax

-F0

K1

K2

dy

Fy

F1

d

dmax

Fmax

-Fj

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of drm would tend to infinity, which means that such a system would be in static equilibriumat any displacement.

A set of dimensionless parameters that are expected to govern the re-centring capabilityof flag-shaped isolation systems can be then defined:

α = K1 · dy

μfr · W(5)

r = K2

K1(6)

μ = dmax

dy(7)

The effect of the variability of the axial load on the friction resistance of sliding bearings,due to rocking movements of the building or the vertical component of the seismic excita-tion, is not examined in this study. The effect of the rate of loading on the properties of theisolation system is not examined in this work as well. The effect of the reduction of F0 withdecreasing velocity during the coda stage of the earthquake, which appears in some systemswith velocity-dependent strength (e.g. in some sliding systems and lead–rubber systems), isobviously beneficial in terms of re-centering capability of the isolation system. However, it isin part counterbalanced by the reduction of K2 with decreasing velocity during the coda stageof the earthquake, which characterizes the superelastic cyclic behavior of SMA. It is worthnoting that the energy dissipation of the isolation system is restricted to its hysteretic/frictionbehavior that is included explicitly in the nonlinear force–displacement relationship. Theeffect of possible auxiliary viscous dampers is not examined in this study.

As argued by Katsaras et al. (2008), examining the seismic response of bilinear hyster-etic isolation systems, the re-centring capacity of the isolation system depends on the entiredisplacement-time history, which is strongly affected by the characteristics of the seismicground motion. However, the re-centring capacity of the system tends to increase for seis-mic ground motions involving maximum displacements larger than drm. As a consequence,the ratio dmax/drm appears to be a suitable parameter that can be used to characterize there-centring capacity of nonlinear isolation systems.

Since the magnitude of the residual displacement depends strongly on the details of theground motion and the complex cyclic behavior of the isolation system during the earth-quake, a large and representative database of real seismic ground motions should be usedto investigate the re-centring capacity of the isolation system in a statistical way. Moreover,the selected records do not need to be compatible with any design spectrum. On the otherhand, the selected records should produce a wide variety of dmax values in order to take intoaccount the effect of periodicity of the residual displacement with respect to dmax/drm.

3 Numerical analysis

3.1 Parametric study

In this paper, the re-centring capability of flag-shaped isolation systems is investigated ina statistical way in terms of residual displacements after the earthquake. A database of 50ground motions, which corresponds to historic records from 24 different seismic eventsand includes a number of records with near-fault effects, has been used for this purpose (seeTable 1). In Fig. 2 the distribution of the characteristics of the 50 ground motions is presented.

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Table 1 Seismic ground motions considered in the nonlinear response-time history analyses

Seismic event Year Station Ms Distance ( km) Soil type PGA (g)

Tabas (Iran) 1978 9102 Dayhook 7.3 13.9 A 0.35

Imperial Valley (USA) 1979 5058 El Centro Array #11 6.5 12.6 C 0.36

5060 Brawley Airport 8.5 C 0.16

5055 Holtville Post Office 7.5 C 0.25

5028 El Centro Array #7 1.0 C 0.34

5061 Calipatria Fire Station 24.6 D 0.23

Coyote Lake (USA) 1979 57383 Gilroy Array #6 5.7 3.1 B 0.43

Morgan-Hill (USA) 1984 57382 Gilroy Array #4 6.1 11.5 D 0.27

North PalmSprings (USA)

1986 5071 Morengo Valley 6.0 10.1 B 0.22

12149 Desert Hot Springs 8 B 0.27

Chalfant Valley (USA) 1986 54171 Bishop-LADWP South St. 6.2 17.1 D 0.21

54428 Zack Brothers Ranch 6.2 7.6 D 0.42

Superstition Hills (USA) 1987 Imp. Co. Center 6.7 37 C 0.34

Spitak (Armenia) 1988 Gukasian 6.7 36 C 0.18

Loma Prieta (USA) 1989 47379 Gilroy Array #1 6.9 11.2 A 0.41

47380 Gilroy Array #2 12.7 C 0.37

1028 Hollistar City Hall 28.2 C 0.25

47006 Gilroy Gavilan Coll. 11.6 B 0.36

CapeMendocino/Petrolia(USA)

1992 89005-Cape Mendocino 7.1 8.5 A 1.49

89156 Petrolia 9.5 C 0.59

89324 Rio Dell’Overpass-FF 14.3 C 0.42

Landers (USA) 1992 22170 Joshua Tree 7.3 11.6 B 0.27

12149 Desert Hot Springs 23.3 B 0.17

23 Coolwater 21.2 B 0.28

Northrigde (USA) 1994 24207 Pocoima Dam 6.7 8 A 1.58

Rinaldi 10 D 0.83

24389 LA-Century City North 25.7 B 0.26

90056 Newhall-Pico Canyon Rd. 7.1 B 0.45

90057 Canyon Country-W Last 12.4 C 0.43

Kozani (Greece) 1994 Kozani Prefecture 6.5 17 A 0.21

Dinar (Turkey) 1995 Meteoroloji Mudurlugu 6.4 8 A 0.32

Kobe (Japan) 1995 Takatori 6.9 2 C 0.62

Nishi-Akashi 11.1 D 0.51

Tadoka 32 D 0.29

Kalamata (Greece) 1997 Koroni 6.4 43 A 0.12

Adana (Turkey) 1998 Ceyhan-Tarim Ilce Mudurlugu 6.3 30 B 0.27

Duzce,Turckey 1999 ERD 99999 Mudurnu 7.14 34,3 A 0.09

Kocaeli (Turckey) 1999 Gebze 7.4 17 A 0.24

Izmit 4.8 A 0.22

Izmit-Kocaeli (Turkey) 1999 Izmit-Karayollari 7.6 39 C 0.13

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Table 1 Continued

Seismic event Year Station Ms Distance ( km) Soil type PGA (g)

Yarimka-Petkim 20 C 0.30

Gebze-Arcelik 55 B 0.21

Gebze-Tubitak 48 B 0.24

Sakarya 34 B 0.36

Chi-Chi (Taiwan) 1999 Taichung TCU068 7.3 12 C 0.50

Chiayi CHY024 10 A 0.28

Miyagi-Oki (Japan) 2003 Fujisawa IWTH05 7.0 29 B 0.58

Parkfield (USA) 2004 Vineyard Canyon 6.0 13 D 0.56

Coalinga 32 D 0.35

Niigata-ken Chuetsu (Japan) 2004 Kamo-NIG06 6.5 44 C 0.42

0

5

10

15

20

5.0-6.5 6.5-7.0 > 7.00

5

10

15

20

0-5 Km 5-10 Km 10-20 Km 20-50 Km

0

5

10

15

20

25

30

0-0.2 g 0.2-0.4g 0.4-0.6g > 0.6g0

5

10

15

20

A B C D

Magnitude Ms

Num

ber

of r

ecor

ds

Num

ber

of r

ecor

ds

Fault Distance

PGA Soil conditions (USGS classification)

Num

ber

of r

ecor

ds

Num

ber

of r

ecor

ds

Fig. 2 Distribution of the ground motion characteristics for the 50 seismic records considered in the analysis

Two of the four parameters governing the cyclic behaviour of flag-shaped systems (r andμfr , precisely), have been varied during the parametric analysis to cover the range of thetypical values of the friction coefficient of PTFE-steel interfaces, i.e. 3% ≤ μfr ≤ 6% (Dolceet al. 2005), and post-elastic stiffness ratio of superelastic SMA wires, i.e. 0% ≤ r ≤ 15%(Dolce et al. 2000). A different approach has been followed in selecting the values of theother two parameters of the analysis (dy and α, precisely). As far as the “yield” displacementis concerned, it has been varied between 1 mm and 35 mm, thus resulting one or two orderof magnitude lower than the typical design displacement (100–350 mm) of seismic isolationsystems. As far as the strength ratio a is concerned, it has been varied between 0.25 and 0.75in order to examine the re-centring capability of strongly nonlinear isolation systems, whosecyclic behaviour is dominated by the friction of sliding bearings, with the restoring force ofthe auxiliary re-centring devices maintained as low as possible.

For each case of study, a nonlinear time-history analysis of the corresponding SDOF flagshaped system and ground motion record has been performed. The maximum displacement

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Table 2 Examined cases in theparametric study

Parameter Values considered in the analysis

α 0.25, 0.50, 0.75

μfr (%) 3, 6

r (%) 0.1, 0.5, 1, 3, 5, 10, 15

dy ( mm) 1, 5, 10, 15, 20, 25, 30, 35

α

-240

0

240

-200 0 200

Displacement (mm)

Forc

e (K

N)

-240

0

240

-200 0 200

Displacement (mm)

Forc

e (K

N)

-240

0

240

-200 0 200

Displacement (mm)

Forc

e (K

N)

-240

0

240

-200 0 200

Displacement (mm)

Forc

e (K

N)

-240

0

240

-200 0 200

Displacement (mm)

Forc

e (K

N)

-240

0

240

-200 0 200

Displacement (mm)

Forc

e (K

N)

-240

0

240

-200 0 200

Displacement (mm)

Forc

e (K

N)

-240

0

240

-200 0 200

Displacement (mm)

Forc

e (K

N)

-240

0

240

-200 0 200

Displacement (mm)

Forc

e (K

N)

Fig. 3 Changes in the cyclic response of the isolation system (μfr = 3%, dy = 10 mm) with increasingstrength ratio (α = 0.25, 0.5, 0.75) and post-yield stiffness ratio (r = 0.1, 5, 15%) (Seismic event: CapeMendocino/Petrolia, year: 1992, M = 7.1, station: 89005-Cape Mendocino, fault distance: 8.5 km, PGA =1.49 g)

dmax and the residual displacement at the end of the earthquake dres have been recordedand then statistically processed. The cases of study examined in the parametric analysis arepresented in Table 2. A total of 336 different SDOF systems have been examined and 16,800nonlinear time-history analyses have been performed.

3.2 Estimation of the residual displacement

Figures 3 and 4 show the changes in the cyclic response of the isolation system, registeredduring a given seismic event (Earthquake: Cape Mendocino/Petrolia, year: 1992, M = 7.1,station: 89005-Cape Mendocino, fault distance: 8.5 km, PGA = 1.49 g), due to differentstrength ratios (α=0.25, 0.5, 0.75), post-yield stiffness ratios (r = 0.1, 5, 15%) and yielddisplacements (dy = 1, 20, 35 mm), respectively.

Figure 5 compares the displacement response-time histories of a number of selectedcases, characterized by different strength ratios (Fig. 5a), different post-yield stiffness ratios(Fig. 5b) and different yield displacements (Fig. 5c), respectively. As can be seen, whilethe maximum displacement recorded during the seismic event under consideration is littleaffected by strength ratio and post-yield stiffness ratio, the residual displacement changessignificantly, passing from approximately 3 to 30 mm while decreasing the strength ratio

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α

-150

0

150

Displacement (mm)

Forc

e (K

N)

-150

0

150

-200 0 200

Displacement (mm)

Forc

e (K

N)

-150

0

150

-200 0 200 -200 0 200

Displacement (mm)

Forc

e (K

N)

-150

0

150

-200 0 200

Displacement (mm)

Forc

e (K

N)

-150

0

150

-200 0 200

Displacement (mm)

Forc

e (K

N)

-150

0

150

-200 0 200

Displacement (mm)Fo

rce

(KN

)

-150

0

150

-200 0 200

Displacement (mm)

Forc

e (K

N)

-150

0

150

-200 0 200

Displacement (mm)

Forc

e (K

N)

-150

0

150

-200 0 200

Displacement (mm)

Forc

e (K

N)

Fig. 4 Changes in the cyclic response of the isolation system (μfr = 3%, r = 0.5%) with increasing strengthratio (α = 0.25, 0.5, 0.75) and “yield” displacement (dy = 1, 20, 35 mm). (Seismic event: Cape Mendocin-o/Petrolia, year: 1992, M = 7.1, station: 89005-Cape Mendocino, fault distance: 8.5 km, PGA = 1.49 g)

from 0.75 to 0.25 (see Fig. 5a) and from approximately 10 to 35 mm while decreasing thepost-yield stiffness ratio from 15 to 0.1% (see Fig. 5b). Conversely, the residual displacementis little affected by the yield displacement (see Fig. 5c), while the maximum displacementchanges considerably.

In Fig. 6 the residual displacements obtained from nonlinear response-time history anal-yses are reported as a function of the corresponding maximum displacements. The attentionis focused on the range 50–500 mm of the maximum displacements, where the design dis-placement of currently used seismic isolation systems usually falls (Cardone et al. 2010).This reduced the number of analysis cases considered from 16,800 to approximately 13,000.As can be seen in Fig. 6, the residual displacements recorded at the end of the selected earth-quakes range from approximately 1 mm to some 230 mm. The residual displacement tendsto increase while increasing the maximum displacement experienced by the isolation system(low-strength isolation systems under strong earthquakes).

In Fig. 7 the distribution of the residual displacement dres normalized with respect to themaximum displacement dmax is presented in the form of histograms as a function of the ratiodmax/drm. For this purpose the observed data are grouped into data bins defined by intervalsof dres/dmax with width equal to 0.05. Four different ranges of the ratio dmax/drm are con-sidered, namely: 0 < dmax/drm ≤ 0.1; 0.1 < dmax/drm ≤ 0.5; 0.5 < dmax/drm ≤ 1 and1 < dmax/drm ≤ 10. It is observed that when dmax/drm tends to zero (i.e. isolation systemswith low strength ratio and low post-yield stiffness ratio) the probability to find a signif-icant residual displacement is greater. This indicates that the re-centring capability of theisolation system increases significantly as the ratio dmax/drm increases. For all the casesthere is a significant variance in the observed data which reflects the strong dependence onthe details of the seismic motion history.

In Fig. 8 the observed values of the normalized residual displacement dres/dmax are pre-sented as a function of the ratios dmax/drm, separately for α = 0.75, α = 0.5 and α = 0.25.As expected, the lower is the strength ratio the worse is the re-centring capacity of the

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-200

-100

0

100

200

0 10 20 30 40 50

Time (sec)

Dis

plac

emen

t (m

m)

r = 0.1%

r = 5%

r = 15%

-200

-100

0

100

200

(a)

(b)

(c)

0 10 20 30 40 50

Time (sec)

Dis

plac

emen

t (m

m)

α = 0.25

α = 0.5

α = 0.75

-300

-200

-100

0

100

200

300

0 10 20 30 40 50

Time (sec)

Dis

plac

emen

t (m

m)

dy = 1 mm

dy = 20mm

dy = 35mm

Fig. 5 Comparison between the displacement response-time histories of a number of selected cases, charac-terized by a different strength ratios (r = 0.5%, μfr = 3%, dy = 35 mm), b different post-yield stiffness ratios (α= 0.25, μfr = 3%, dy = 5 mm) and c different yield displacements (α = 0.25,μfr = 3%, r = 5%), respectively

0

50

100

150

200

250

50 100 150 200 250 300 350 400 450 500

Maximum displacement (mm)

Res

idua

l dis

plac

emen

t (m

m)

Fig. 6 Residual displacements (dres) vs. maximum displacement (dmax) obtained from non linear response-time history analyses

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0

300

600

900

1200

1500

1800

0.0-0

.05

0.05-

0.1

0.1-0

.15

0.15-

0.2

0.2-0

.25

0.25-

0.3

0.3-0

.35

0.35-

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0.4-0

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0.45-

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dres/dmax

Freq

uenc

y

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Freq

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dres/dmax

Freq

uenc

y

0

300

600

900

1200

1500

1800

0.0-0

.05

0.05-

0.1

0.1-0

.15

0.15-

0.2

0.2-0

.25

0.25-

0.3

0.3-0

.35

0.35-

0.4

0.4-0

.45

0.45-

0.5

0.5-0

.55

0.5-0

.6

dres/dmax

Freq

uenc

y

dmax/drm = 0.0-0.1 dmax/drm = 0.1-0.5

dmax/drm = 0.5-1.0 dmax/drm = 1.0-10

Fig. 7 Distribution of the normalized residual displacement (dres/dmax) as a function of the ratio dmax/drm

isolation system. Moreover, the residual displacement tends to increase as the ratio dmax/drm

reduces. It is also observed that when dmax/drm is greater than 5, the normalized residualdisplacement dres/dmax is practically independent of α and dmax/drm. As a consequence,it can be concluded that the design value of the residual displacement dres is the same forall the earthquakes that induce maximum displacement dmax greater than 5drm. For thesecases the residual displacement dres is small compared with dmax and not sensitive withrespect to the particular strength ratio of the isolation system.

Because of the large scatter in the observed data, the 90th percentile (i.e. 90% of theobserved values do not exceed this value) is proposed as a possible reference value for designconsiderations. In Fig. 9, the 90th percentile of the observed values of normalized residualdisplacement dres/dmax is reported as a function of the ratio dmax/drm, for strength ratiosequal to (a) 0.25, (b) 0.5 and (c) 0.75, respectively. It should be noted that the 90th percentileof the observed data has been evaluated considering intervals of dmax/drmwith width equal to0.1 for 0 < dmax/drm ≤ 3 and width equal to 0.5 for 3 < dmax/drm ≤ 10. The 90th percentilevalues point out that there is a strong dependence of the normalized residual displacementon the strength ratio α and the ratio dmax/drm, as it was already observed in the previousdiagrams; however, there is no remarkable trend with respect to the post yield stiffness ratior and the yield displacement dy. This indicates that the effect of the post-yield stiffness ratioand yield displacement dy is adequately included in the maximum residual displacementlimit drm. Values of (dres/dmax)90th less than 0.1 are found for dmax/drm greater than 3, evenfor the lowest values of α(= 0.25) considered in this study.

123


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5 6 7 8 9 10

α = 0.75

α = 0.50

α = 0.25

dmax/drm

dres

/dm

ax

Fig. 8 Observed values of the normalized residual displacement dres/dmax as a function of the ratio dmax/drmfor different values of the strength ratio α

In the analyses that follow, the only considered independent variables are the strengthratio α and ratio dmax/drm. The latter can be alternatively expressed as the product betweenthe ductility ratio dmax/dy and the ratio dy/drm.

The 90th percentile of the observed normalized residual displacements is described well bythe following relation which has been derived from regression analysis (R2 ≈ 0.75 − 0.85):

dres(90th) =[

2.2 · (1.35 − α) − (1 − α) · ln

(dmax

drm

)]· 0.058 · dmax (8)

By substituting dmax/drm with (dmax/dy) · (dy/drm) the following relation is obtained:

dres(90th)

dmax= 0.058 ·

[2.2 · (1.35 − α) − (1 − α) · ln

(dmax

dy

dy

drm

)](9)

In Fig. 10, the proposed design value of the residual displacement dres(90th), normalized withthe maximum displacement dmax, is presented as a function of the strength ratio α and theductility ratio μ = dmax/dy for different values of dy/drm = 0.05, 0.1, 0.15, 0.2, 0.25, 0.3. Asimilar representation is reported in Fig. 11, in which μ = dmax/dy(instead of α) is assumedas main independent variable. The plots of Figs. 10 and 11 are useful tools for design purposesbecause the residual displacement is compared with the maximum displacement dmax whichdefines the required minimum displacement capacity of the isolation system. As expected,the re-centring capability of flag-shaped isolation systems increases when the strength ratio α

and the ductility ratio μ = dmax/dy increase. Assuming a given strength ratio α and ductilityratio μ, the re-centring capability of flag-shaped isolation systems tends to improve whileincreasing the ratio dy/drm.

Assuming dy/drm = 0.05, for instance, the residual displacement of flag-shaped isolationsystems becomes negligible (say less than 10% dmax) when the strength ratio α is greaterthan 0.8, regardless the ductility ratio μ (>5). Conversely, a sufficient re-centring capabilityis ensured when the ductility ratio μ is larger than 60, regardless the strength ratio α.

Assuming dy/drm = 0.3, for instance the residual displacement of flag-shaped isolationsystems becomes negligible (say less than 10% dmax) when the strength ratio α is greaterthan 0.5, regardless the ductility ratio μ (>5). Conversely, a sufficient re-centring capabilityis ensured when the ductility ratio μ is larger than 10, regardless the strength ratio α.

123


y = -0.0469Ln(x) + 0.1459

R2 = 0.8329

0.00

0.05

0.10

0.15

0.20

0.25

0.30 (a)

(b)

(c)

dmax/drm

(dre

s/d m

ax)

90th

per

cent

ile

y = -0.0254Ln(x) + 0.1026

R2 = 0.7721

0.00

0.05

0.10

0.15

0.20

0.25

0.30

dmax/drm

(dre

s/d m

ax)

90th

per

cent

ile

y = -0.0196Ln(x) + 0.0818

R2 = 0.7483

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

dmax/drm

(dre

s/d m

ax)

90th

per

cent

ile

Fig. 9 Regression analysis (90th percentile) of the observed values of normalized residual displacementdres/dmax as a function of the ratio dmax/drm, for strength ratios equal to a 0.25, b 0.5 and c 0.75, respectively

Many other combinations of α,μ and dy/drm can be found by examining the plots ofFigs. 10 and 11. The aforesaid combinations establish the features of the isolation system thatguarantee small residual displacements compared with its maximum displacement capacity,and therefore sufficient re-centring capability.

123


dy/drm = 0.05

0.00

0.05

0.10

0.15

0.20

0.25

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1

α

(dre

s/dm

ax)

90th

per

cent

ile5

dmax/dy

10

20

30405060

100

200

dy/drm = 0.10

0.00

0.05

0.10

0.15

0.20

0.25

α

(dre

s/dm

ax)

90th

per

cent

ile

dmax/dy

5

10

20

3040

60

100

50

dy/drm = 0.15

0.00

0.05

0.10

0.15

0.20

0.25

α

(dre

s/dm

ax)

90th

per

cent

ile

dmax/dy

5

10

20

30

5040

60

dy/drm = 0.20

0.00

0.05

0.10

0.15

0.20

0.25

α

(dre

s/dm

ax)

90th

per

cent

ile

dmax/dy

5

10

20

30

4050

dy/drm = 0.25

0.00

0.05

0.10

0.15

0.20

0.25

α

(dre

s/dm

ax)

90th

per

cent

ile

dmax/dy

5

10

20

30

40

dy/drm = 0.30

0.00

0.05

0.10

0.15

0.20

0.25

α

(dre

s/dm

ax)

90th

per

cent

ile

dmax/dy

5

10

20

30

Fig. 10 Proposed design values of the residual displacement dres,90th as a function of the strength ratio α,for different values of the ratios dmax/dy and dy/drm

4 Evaluation of current code requirements

Based on the results of this study, the main parameter that affects the re-centring capabil-ity of flag-shaped seismic isolation systems is the ratio dres/dmax (see Fig. 8). It has beenconcluded that isolation systems with dmax/drm greater than 3 exhibit good re-centring capa-bility (dres/dmax < 0.1), regardless the strength ratio α and regardless the characteristics ofthe earthquake (see Fig. 8).

For a flag-shaped isolation system, the ratio dmax/drm (see Eq. 4) can alternatively beexpressed also as (see Fig. 1): (Fmax − Fj )/(F0 − Fj ) ≈ (Fmax − Fj )/(F0(1−α)), i.e. as theratio between the increment of the restoring force between displacements 0 and dmax to thecyclic force near zero displacement F0(1−α). This is a rational comparison because the forceincrement of the linear elastic component with stiffness K2, which increases the re-centeringcapability, is compared with the unbalanced nonlinear force in proximity of zero displacementF0(1 − α) of the nonlinear component, which induces the imperfect re-centering.

123


dy/drm=0.05

0.000.020.040.060.080.100.120.140.160.180.20

dmax/dy

(dre

s/dm

ax)

90th

per

cent

ile0.2

α

1.0

0.8

0.4

0.6

dy/drm=0.10

0.000.020.040.060.080.100.120.140.160.180.20

dmax/dy

(dre

s/dm

ax)

90th

per

cent

ile

α

dy/drm=0.15

0.000.020.040.060.080.100.120.140.160.180.20

dmax/dy

(dre

s/dm

ax)

90th

per

cent

ile

α

dy/drm=0.20

0.000.020.040.060.080.100.120.140.160.180.20

dmax/dy

(dre

s/dm

ax)

90th

per

cent

ileα

dy/drm=0.25

0.000.020.040.060.080.100.120.140.160.180.20

0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100

0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50

0 3 6 9 12 15 18 21 24 27 30 0 3 6 9 12 15 18 21 24 27 30

dmax/dy

(dre

s/dm

ax)

90th

per

cent

ile

α

dy/drm=0.30

0.000.020.040.060.080.100.120.140.160.180.20

dmax/dy

(dre

s/dm

ax)

80th

per

cent

ile

α

Fig. 11 Proposed design values of the residual displacement dres,90th as a function of the ductility ratiodmax/dy, for different values of the ratios α and dy/drm

A thorough evaluation of current re-centring capability code provisions has been carriedout by Katsaras et al. (2006). Both the AASHTO Guide Specifications (2000) and the Califor-nia Building Code (California Buildings Standards Commission (CBSC) 2001), essentiallycompare the increment of the post-elastic force to a certain fraction of the weight of the struc-ture. They do not take specifically into account the magnitude of the unbalanced nonlinearforce near zero displacement (equal to F0(1 − α) for flag-shaped isolation systems), whichis the main reason for insufficient re-centring capability of the isolation system.

The relation proposed in this study for guaranteeing an adequate re-centring capability offlag-shaped isolation systems (dmax/drm ≥ 3), can be expressed in the following alternativeway:

K2 · d ≥ 3 · μfr · (1 − α) · W (10)

which is directly comparable with the relationships presented in the current seismic codes(see Eqs. 1–3).

Assuming μfr = 0.06 and α = 0.25 (i.e. the worst conditions for re-centring consideredin this study), for instance, the term 3 ·μfr · (1 − α) results equal to 0.135, that is greater thanthe corresponding value prescribed by the 2000 AASHTO (0.025), the 2001 CBC (0.05) andthe 2005 EC8 (

√0.05 · μfr = 0.054). Under the aforesaid hypothesis, therefore, the require-

123


ment that flag-shaped isolation systems must comply with for ensuring negligible residualdisplacements at the end of the earthquake (quite surprisingly) results more conservativethan those prescribed by current seismic codes for bilinear hysteretic systems. Probably,this can be ascribed to different tolerance levels in the definition of the acceptable residualdisplacement.

Assuming μfr = 0.03 and α = 0.75 (i.e. the most favourable conditions for re-centringconsidered in this study), on the contrary, the term 3 ·μfr ·(1 − α) results equal to 0.0225, thatis lower than the corresponding values prescribed by the 2001 CBC (0.05) and the 2005 EC8(√

0.05 · μfr = 0.0387) but fully compatible with the values imposed by the 2000 AASHTO(0.025).

The International Building Code (International Code Council (ICC) 2006) building codeallows isolation systems that do not fulfill the re-centring capability criteria to be used ifthey remain stable under the full load and up to 3.0 times the design displacement (e.g.due to accumulation of residual displacement during some aftershocks). This requirementappears excessive for flag-shaped isolation systems for which the residual displacement undera generic earthquake (single ground motion) never exceed 0.6 dmax, based on the results ofthis study (see Fig. 8).

5 Conclusions

According to the analysis presented in this paper, the main parameter that affects the re-cen-tring capability of flag-shaped seismic isolation systems is the ratio dmax/drm, where dmax

is the maximum seismic displacement and drm is the maximum residual displacement underwhich the system can be in static equilibrium, given by relation (4) for flag-shaped systems.The maximum earthquake displacement dmax includes the effect of the excitation, whereasdrm is a parameter of the isolation system, independent from the excitation. In consequence tothis fact, the re-centring capability depends not only on the system properties but also on theseismic input. More specifically, for the same system, the re-centring capability is expectedto be, on average, better for seismic ground motions determining larger displacements thanfor seismic ground motions determining smaller displacements.

The residual displacement strongly depends on the ground motion characteristics. A reli-able estimation of the re-centring capability of the isolation system can be achieved onlystatistically, using a relatively large database of real seismic ground motions. From the sta-tistical analysis of 50 seismic ground motions presented in this paper, it is concluded thatflag-shaped isolation systems with dmax/drm > 3 experience negligible residual displace-ments compared with their maximum displacements. Therefore, such systems can be deemedto have good re-centring capability and, as a consequence, are not subjected to accumulationof residual displacements during a sequence of earthquakes.

Comparisons between the proposed re-centring capability criterion (dmax/drm > 3) andthe re-centring capability provisions included in the current seismic codes show that theselatter are suitable (and a little conservative) for flag-shaped isolation systems with relativelyhigh strength ratios (e.g. α = 0.75) while they are inadequate (and basically not conservative)for flag-shaped isolation systems with relatively low strength ratios (e.g. α = 0.25).

As a general rule of thumb, the re-centring capability provisions included in the cur-rent seismic codes may be extended to flag-shaped models by considering a fraction of theweight of the structure (see Eqs. 1–3) directly proportional to the friction coefficient μfr andinversely proportional to the strength ratio α of the flag-shaped isolation system (e.g., in firstapproximation: (μfr/2

√α)W in place of 0.025W or

√0.05 · q · W).

123


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