Modeling strength degradation in lead-rubber bearings under earthquake shaking

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2010; 39:1533–1549Published online 8 September 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.1039

Modeling strength degradation in lead–rubber bearings underearthquake shaking

Ioannis V. Kalpakidis∗,†, Michael C. Constantinou and Andrew S. Whittaker

Department of Civil, Structural and Environmental Engineering, 212 Ketter Hall, University at Buffalo,State University of New York, Buffalo, NY 14260, U.S.A.

SUMMARY

Lead–rubber bearings are seismic isolators that have been used extensively to protect buildings, bridges andmission-critical infrastructure from the damaging effects of earthquake shaking. Under large-displacementcyclic motion, the strength of a lead–rubber bearing reduces due to energy dissipation and the resultantheating of the lead core. This paper proposes a method to incorporate strength degradation due to lead coreheating in modeling the hysteretic behavior of lead–rubber bearings. The validity of the proposed modelis investigated through comparing numerical and experimental results. The model is used to examine theeffects of lead core heating on the dynamic response of an isolated structure. Copyright q 2010 JohnWiley & Sons, Ltd.

Received 30 December 2009; Revised 16 April 2010; Accepted 7 June 2010

KEY WORDS: lead–rubber bearing; energy dissipation; lead core heating; hysteretic modeling; boundinganalysis; seismic isolation

1. INTRODUCTION

Numerous mathematical models of behavior of elastomeric bearings have been proposed. Thesemodels may be classified into two broad categories:

(1) Finite element formulations using models of material behavior for rubber-like materials andplasticity [1, 2].

(2) Hysteretic models of various complexities that include (a) bilinear hysteretic models thathave been widely used in dynamic analysis programs such as the 3D-BASIS class [3, 4] andSAP2000 [5], and (b) improved formulations of hysteretic models based on phenomenolog-ical constructions that require calibration on the basis of experimental data [6–8].

∗Correspondence to: Ioannis V. Kalpakidis, Department of Civil, Structural and Environmental Engineering, 212 KetterHall, University at Buffalo, State University of New York, Buffalo, NY 14260, U.S.A.

†E-mail: [email protected]

Copyright q 2010 John Wiley & Sons, Ltd.

1534 I. V. KALPAKIDIS, M. C. CONSTANTINOU AND A. S. WHITTAKER

While these models may account for complex displacement-dependent (such as elastomer hard-ening) and rate-dependent behavior, none account for lead core heating effects on the characteristicstrength of lead–rubber bearings. Moreover, models based on phenomenological constructionsrequire calibration using experimental data obtained in the testing of bearings. Accordingly, thevalidity of these models is limited to bearing configurations that are similar to the tested bear-ings. Statements on similarity or proportionality (geometric similarity) are often included indescriptions of phenomenological models [8]. It is known from the work presented by Kalpakidisand Constantinou [9] that geometric similarity cannot be used to extrapolate experimental data onthe temperature increase in the lead core of one bearing to predict the response of lead–rubberbearings of different geometries.

This paper presents a mathematical model of the mechanical behavior of lead–rubber bearingsthat accounts for lead core heating effects. The model is based on the theory of Kalpakidis andConstantinou [9–11] and does not require calibration on the basis of experimental data from testingof bearings. It simply predicts the instantaneous value of the strength of a lead–rubber bearing basedon the calculation of the instantaneous temperature of the lead core. The model is formulated as thebi-directional smooth bilinear hysteretic model that is currently employed in computer programs3D-BASIS and SAP2000. Moreover, a discussion is presented on the application of the model infinite element formulations of the behavior of lead–rubber bearings based on thermo-mechanicalanalysis with temperature-dependent lead properties.

This paper also presents comparisons of experimental force–displacement loops to predictions ofthe presented model in harmonic motion of lead–rubber bearings. The comparisons provide valida-tion of the model and also further demonstrate the validity of the theory on lead core heating [9–11].

Finally, the validated model for lead–rubber bearing behavior is utilized to study the responseof a seismically isolated structure and compare with results obtained using the standard bilinearhysteretic model for lead–rubber bearings within the context of bounding analysis [12–14], which isthe current state-of-practice in the analysis of seismically isolated structures. In bounding analysis,variations in material properties at the time of bearing construction, and changes in the mechanicalproperties of the isolators over their installed lifetime due to the effects of history of loading,aging and environmental conditions, are accounted for by conducting two analyses, one usinglower bound and one using upper bound properties. No exceedance probabilities are assigned tothese bounds although Huang et al. [15] have proposed that such bounds be at the two-sigmalevel. Comparison of analysis results obtained with this model against those of upper and lowerbound analyses serves toward evaluating the significance of accounting for lead core heating inthe prediction of the dynamic response of seismically isolated structures.

2. MODELING

2.1. Proposed model

The proposed model of lead–rubber bearing behavior is a smooth bilinear hysteretic model withcharacteristic strength that is dependent on the instantaneous lead core temperature. Figure 1illustrates the model. The parameters of the model are: characteristic strength Qd , yield force FY ,yield displacement Y , elastic stiffness Kel and post-elastic stiffness Kd .

The bi-directional smooth bilinear hysteretic model in program 3D-BASIS (also SAP2000) wasoriginally proposed by Park et al. [16] and, after some modification, has been shown to produce

Copyright q 2010 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2010; 39:1533–1549DOI: 10.1002/eqe

MODELING STRENGTH DEGRADATION IN LEAD–RUBBER BEARINGS 1535

LATERAL FORCE

LATERAL DISPLACEMENT

dK

YFdQ

Y

elK

Figure 1. Bilinear hysteretic model for lead–rubber bearings.

acceptable results on the behavior of isolators in bi-directional motion [3, 17]. In its isotropicformulation, which is appropriate for describing the behavior of lead–rubber bearings, the modelin terms of forces Fx and Fy and displacements Ux and Uy along orthogonal directions x and y,respectively, is:

{Fx

Fy

}= cd ·

{Ux

Uy

}+Kd ·

{Ux

Uy

}+(�YL(TL)AL) ·

{Zx

Zy

}(1)

Y ·{Zx

Z y

}= (A ·[I ]−B ·[�]) ·

{Ux

Uy

}(2)

[�] ={

Z2x ·[sgn(Ux Zx )+1] Zx Zy ·[sgn(Uy Zy)+1]

Zx Zy ·[sgn(Ux Zx )+1] Z2y ·[sgn(Uy Zy)+1]

}(3)

In the above equations, an overdot denotes differentiation with respect to time, [I ] is an identitymatrix and dimensionless parameters Zx and Zy , governed by the system of differential equa-tions (2) and (3), assume values in the range of −1 to 1. Quantities A and B should be related(A=2B) for proper behavior [17, 18]. Herein, the values A=1, B=0.5 are used.

Equations (1)–(3) describe the bi-directional model for lead–rubber bearings. In these equations,�YL is the effective yield stress of lead, which depends on the increase in the temperature ofthe lead core, TL, AL is the cross-sectional area of the lead core, Y is the yield displacement ofthe bearing and cd is a parameter corresponding to a viscous element that accounts for energydissipation in the rubber.

The sum of the first and second terms in (1) accounts for the resisting force in the elastomer,whereas the third term in (1) accounts for the resisting force in the lead core. The rise, TL, in leadcore temperature from its starting value varies with time in accordance with the following set ofequations [9, 10]:

TL =�YL(TL) ·

√Z2x +Z2

y

√U 2x +U 2

y

�LcLhL− kS ·TLa ·�LcLhL

·(1

F+1.274 ·

(tsa

)·(�)−1/3

)(4)



F =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

2 ·( �

�

)1/2− �

�·[2−

( �

4

)−

( �

4

)2− 15

4

( �

4

)3], �<0.6

8

3�− 1

2(� ·�)1/2 ·[1− 1

3 ·(4�) + 1

6 ·(4�)2 − 1

12 ·(4�)3], ��0.6

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(5)

� = �St

a2(6)

�YL(TL) = �YL0 ·exp(−E2TL) (7)

The following parameters appear in (4)–(7) (refer to Figure 2 for the geometric parameters hL, a andts): �L is the density of lead, cL is the specific heat of lead, hL is the height of the lead core, a is theradius of the lead core, �S is the thermal diffusivity of steel, kS is the thermal conductivity of steel,ts is the total shim plate thickness, �YL0 is the effective yield stress of lead at the reference (starting)temperature, � is the dimensionless time and t is the time since the beginning of motion. Notethat Equation (7) describes the dependency of the lead core strength on its increase in temperatureand the value of the lead core strength at the starting temperature through parameter E2 [9–11].The material parameter values are �L=11200kg/m3, cL=130J/(kg◦C), kS=50W/(m◦C), �S=1.41×10−5m2/s, and E2=0.0069/◦C. These values are used in the analyses presented herein.Parameter cd that describes the small ability of the rubber portion of the bearing to dissipate energymay be estimated based on an assumed value of effective damping in rubber as described in [9, 10].

Equation (4) includes an expression under a square root to represent the absolute value ofthe instantaneous resultant velocity of the top of the bearing with respect to its bottom (see

[9, 10] for details). Note that the term√Z2x +Z2

y

√U 2x +U 2

y equals the resultant velocity√U 2x +U 2

y

when the bearing undergoes inelastic action (resultant displacement exceeding yield displace-ment Y ) and otherwise is less than the resultant velocity. This modification has no effect when

the bearings undergo large deformations (for which√Z2x +Z2

y =1), but is important in cases of

low-displacement amplitude because it properly accounts for energy dissipation in the lead core in

proportion with parameter√Z2x +Z2

y , which then has a value less than unity. When this parameter

is replaced by unity, the energy dissipation in the lead core is overestimated when the amplitudeof displacement is less than the yield displacement.

STEEL SHIMS

RUBBER LAYERS

LEAD CORE

a

stLh

Figure 2. Schematic of a lead–rubber bearing.



2.2. Modifications to account for other behaviors

The model described above only accounts for the effect of heating of the lead core on the char-acteristic strength of the bearing. It does not account for strain rate effects on the strength andpost-elastic stiffness and it does not account for strain effects on stiffness. Although some infor-mation on the strain rate effects on strength may be found in Constantinou et al. [13], it appearsthat there is insufficient data to allow for the development of a model based on first principles.

The model can be modified to account for displacement-dependent post-elastic stiffness and forrate-dependent stiffness and strength on the basis of experimentally determined rules. Althoughsuch modifications are entirely phenomenological, they can provide for complex behavior asdemonstrated by Demetriades et al. [19], Kikuchi and Aiken [6] and Abe et al. [8]. The interestedreader is referred to those studies for further details.

2.3. Thermo-mechanical finite element analysis

Detailed finite element modeling of lead–rubber bearings is useful primarily in analysis of singlebearings in order to predict behavior in lieu of testing or prior to testing. For example, Ali andAbdel-Ghaffar [2] and Doudoumis et al. [20] reported elaborate finite element formulations forrubber, steel and lead in lead–rubber bearings, but these were restricted to mechanical behavioronly. Coupled thermo-mechanical analysis is required to capture the changing characteristics ofthe bearings due to heating of the lead. An important aspect of this analysis is the specification ofthe temperature-dependent mechanical properties for lead. The data presented by Kalpakidis andConstantinou [9] on the behavior of lead indicate that a simple model for lead would be that ofelastoplastic behavior with temperature-independent elastic modulus and temperature-dependentyield stress described by (7) or alternative forms described in [9], including some based on thetemperature rather than temperature rise.

3. COMPARISON OF ANALYTICAL AND EXPERIMENTAL DATA

The large-size lead–rubber bearing of Figure 3 was tested at the University at Buffalo under anormal load of 1441 kN. It was subjected to 25 cycles of harmonic motion at 114mm amplitude(corresponding to shear strain of 75%) and a peak velocity of 250mm/s. Figure 4 shows therecorded force–displacement loops together with loops obtained by the model of Equations (1)–(7)with the parameters listed in Table I.

The force–displacement loops displayed in Figure 4 demonstrate that the model is capable ofpredicting well the changes in strength of the tested lead–rubber bearing, but it does not capturewell the ascending branch of the loop on first loading nor does it capture well the portion of theloop at each reversal of motion. The shape of the ascending branch of the loop on initial loadingis likely affected by unrealistic rate effects (and likely measurement errors) as the test started at restwith the test machine attempting to impose an instantaneous velocity of 250mm/s. These effectsare much less pronounced in low-speed testing [9, 11]. Nevertheless, part of the difference betweenthe experimental and analytical results on the ascending initial part of the loop is due to the bilinearhysteretic model employed, which is based on a fixed value of yield displacement. As also seen in theshapes of the experimental loops on each reversal of motion, the actual behavior justifies the use of avariableyielddisplacement.Suchabehaviorcanonlybecapturedwithphenomenological adjustmentsof the presented model as, for example, described by Kikuchi and Aiken [6] and Abe et al. [8].



Figure 3. Tested large size lead–rubber bearing (1′′ =25.4mm).

-150

Lateral Displacement (mm)

-400

-200

0

200

400

Late

ral F

orce

(kN

)

AnalysisExperiment

-100 -50 0 50 100 150

Figure 4. Analytically predicted and experimentally obtainedforce–displacement loops for the large-size bearing.

A second example involves the small-size bearing shown in Figure 5 subjected to non-harmonichorizontal motion of large amplitude and a constant axial compressive load of 89 kN. Figure 6presents the history of the imposed motion of which the peak displacement corresponds to ashear strain (equal to the lateral displacement divided by the total thickness of rubber) of about200% in the rubber and an average shear strain in the lead of about 125%. Figure 7 enables a



Table I. Parameters used in models of tested lead–rubber bearings.

Bearing cd (Ns/mm) �YL0 (MPa) Y (mm) Kd (N/mm) AL(mm2) hL (mm) a (mm) ts (mm)

Large size 128 13.0 7 1080 15394 224 70 71Small size 3.6 16.2 5 190 491 89 12.5 32

3.2mm Cover

25mm Dia. Lead Core

184mm

127mm

19mm End Plate

18 Rubber Layers @3.2mm=57mm 17 Steel Shims 14 gauge (t=1.9mm)

Figure 5. Tested small-size lead–rubber bearing.

0Time (s)

-150

-100

-50

0

50

100

150

Dis

plac

emen

t (m

m)

40 80 120

Figure 6. Imposed history of motion in testing of small-size lead–rubber bearing.

comparison of the force–displacement-loop predicted by the model of Equations (1)–(7) to thatrecorded in the experiment. The model parameters are presented in Table I. Note that the startingvalue of the effective yield strength of lead is larger in the small bearing than in the large bearing.In general, this quantity depends on the size and conditions of confinement of the lead core [13].Overall, the prediction by the model of the recorded force–displacement loops of the bearing isgood. Nevertheless, we still observe differences in the loop shape on reversal of motion at smallamplitudes of motion. This is due to inability of the model to describe the hysteretic behavior



-150Lateral Displacement (mm)

-30

-15

0

15

30

Late

ral F

orce

(kN

)

AnalysisExperiment

-100 -50 50 100 1500

Figure 7. Comparison of analytically predicted and experimentally obtainedforce–displacement loops for small-size bearing.

of the bearing at all amplitudes of motion, a problem that could be corrected by incorporatingdisplacement-dependent post-elastic stiffness.

4. EFFECTS ON THE RESPONSE OF ISOLATED STRUCTURES

4.1. Introduction

Analysis of seismically isolated structures is currently carried out using suites of ground motionsand two sets of isolator properties representing the likely upper bound and the likely lower boundmechanical properties of the isolators. These two bounds are determined on the basis of test data onthe actual or similar isolators with due consideration given to uncertainty in material properties atthe time of isolator fabrication, effects of environmental conditions, aging, contamination, historyof loading and the effects of heating during the seismic event. Typically, upper and lower boundvalues of characteristic strength and post-elastic stiffness of each isolator are determined and usedin analysis.

For lead–rubber bearings, a significant portion of the difference between the upper and lowerbound values of the characteristic strength results from heating effects, which are most oftenconservatively estimated on the basis of experimental data. For example, Constantinou et al. [12]presented analysis and design examples in which the lower bound value of the characteristicstrength of large size bearings is based on the average value of the effective yield stress of lead inthree cycles of harmonic motion at large amplitude consistent with expected seismic demand, �L3 ,whereas the upper bound value is based on the yield stress in the first cycle, �L1 . Proposed valuesto use were �L3 =10–12MPa (range to account for uncertainties) and �L1 =1.35�L3 . This leadsto lower and upper bound values of strength, excluding any effects of low temperature and aging,that are based on lead yield stress values of 10MPa and 1.35×12=16.2MPa, respectively. Thissignificant range of values is based on experimental data for bearings with large lead core areaand undergoing large shear strains in the lead core at high speeds for a number of cycles. While



ISOLATION BASEMAT WEIGHT bW

SUPERSTRUCTURE WEIGHT sW

STIFFNESS sk ,

DAMPING s

ISOLATION SYSTEM (100 BEARINGS)

DRIFT

ISOLATOR DISPLACEMENT

Figure 8. Analyzed seismically isolated structural system.

these conditions may be appropriate for applications in areas of high seismicity with at least threecycles of large amplitude motion, they likely lead, when used for applications in areas of lowerseismicity and for motions that result in a smaller number of cycles (such as for near-fault highvelocity pulses), to conservative estimation of displacement demands, isolation shear forces andstructural responses.

To demonstrate the likely conservatism of bounding analysis, a seismically isolated structureis analyzed by first utilizing the lead–rubber bearing model described earlier with temperature-dependent characteristic strength, and then again by formal application of bounding analysisutilizing a bilinear hysteretic model with upper and lower bound values of characteristic strength.Comparisons of responses calculated for a number of earthquake motions reveal the significanceof accounting for the lead core heating effects.

4.2. Description of analyzed structure and earthquake ground motions

The analyzed structure is represented as a two-degree-of-freedom system in each principal directionwith one degree describing the structural drift and another describing the isolation system displace-ment. Figure 8 illustrates the system. Torsional coupling of the degrees-of-freedom is neglectedby assigning zero eccentricities between the centers of mass of each rigid floor and the centers ofresistance of the structural elements. The structure total weight is W =1026600kN, comprised ofsuperstructure weight Ws=0.8W and basemat weight Wb=0.2W. The structural stiffness ks wasselected such that the fundamental period of the non-isolated structure (when fixed at the basematlevel=2�

√Ws/(gks)) is 0.5 s and the corresponding structural damping ratio is �s=0.05. These

properties approximately represent the properties of the seismically isolated Erzurum Hospital inTurkey [11].

The isolation system consists of 100 lead–rubber bearings of the geometry shown in Figure 9and with force–displacement loops (for a vertical load of 10,266 kN per bearing) presented inFigure 10 in high-speed testing. The mathematical model of the lead–rubber bearing is as describedby Equations (1)–(7) with parameters listed in Table II. Note that in this model the effective yieldstress of lead at the reference temperature (start of motion) is �YL0=16.9MPa. This value isbased on analysis of the test data for the bearing shown in Figure 10 and slightly higher than thesuggested upper bound value [12].

Upper and lower bound analyses are performed using a temperature-independent bilinearhysteretic model (described by (1)–(3) but with constant �YL) with the parameters as follows.



Figure 9. Lead–rubber isolator in isolated structure analysis example.

Figure 10. Force–displacement loops of bearing of Figure 9. Load=10266kN, displacementamplitude=483mm and frequency=0.333Hz (peak velocity=1000mm/s) (Constantinou et al. [13]).

Table II. Parameters used in the model of lead–rubber bearing of Figure 9..

cd (Ns/mm) �YL0 (MPa) Y (mm) Kd (N/mm) AL (mm2) hL (mm) a (mm) ts (mm)

89 16.9 30 2000 73 542 333 153 125

For the upper bound condition Qd =1243kN, Kd =2.0kN/mm and Y =30mm based on a yieldstrength of 16.9MPa. For the lower bound condition, Qd =735kN, Kd =2.0kN/mm and Y =30mm based on total yield strength of 10MPa.

The effective properties per the definition of ASCE/SEI 7-05 [21] of the isolated structure ata representative isolator displacement of 500mm for the characteristic properties of the isolatedstructure are presented in Table III.



Table III. Characteristic and effective properties of isolated structure.

Condition Qd/W

Period based onpost-elasticstiffness (s)

Effective periodat displacementof 500mm (s)

Effective damping atdisplacement of 500mm(% critical damping)

Upper Bound∗ 0.12 4.54 3.03 33Lower Bound 0.07 4.54 3.45 25

∗Also initial condition in temperature-dependent model.

Table IV. Ground motions used in analyses∗.

Analysis Record (X) Record (Y) Event Year Bin from [22]1 NF17 — Kobe 1995 Near-field2 BOL000 BOL090 Duzce, Turkey 1999 Large-magnitude small-distance3 NF02 — Tabas, Iran 1978 Near-field4 NF13 — Northridge 1994 Near-field5 TCU065-N TCU065-W Chi Chi, Taiwan 1999 Near-field

∗See [9, 22] for more details.

-800


-0.3

-0.2

-0.1

0

0.1

0.2

0.3

She

ar F

orce

/ W

eigh

t

Upper BoundLower BoundProposed

-400 0 400 800

Figure 11. Isolation system force–displacement loops (Analysis #1).

The ground motions used for analysis were selected from a study of Warn andWhittaker [22] thatpresented response history analysis results using a large number of ground motions organized intoeight bins. Motions with large peak ground acceleration from two out of these bins are used for thisstudy, namely: near-field and large-magnitude small-distance (bins 1 and 2M in [22], respectively).Table IV provides information on the ground motions used in the analyses. It is noted that theserecords were modified by ignoring the first few seconds (where the acceleration is nearly zeroand no motion and heating occurs). This, although of minor importance, is theoretically necessarybecause the model is based on the dimensionless time �+ defined to start at initiation of heating.



Lateral Displacement X (mm)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

She

ar F

orce

X /

Wei

ght


-200

Lateral Displacement Y (mm)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

She

ar F

orce

Y /

Wei

ght


-100 0 100 200

-200 -100 0 100 200


4.3. Results

The seismically isolated structure was analyzed for each of the motions of Table IV using thethree isolator models: (1) upper bound bilinear hysteretic, (2) lower bound bilinear hysteretic and(3) the lead–rubber bearing model introduced in Section 2 that explicitly accounts for heating.These motions were applied to the model of the isolated structure either as one-directional (cases1, 3 and 4) or as bi-directional excitation (cases 2 and 5). The results in the form of (a) calculatedforce–displacement loops in the isolation system are presented in Figures 11–15, (b) calculatedlead core temperature rise histories are presented in Figure 16 and (c) calculated peak responsequantities of structural drift and structural acceleration in each principal direction and resultantisolation system displacement, isolation system shear force and resultant structural shear force arepresented in Table V.

The results clearly demonstrate that the use of bounding analysis typically results in conservativeestimation of isolator displacements, isolation shear force, structural shear, structural drift and



-500


-0.3

-0.2

-0.1

0

0.1

0.2

0.3

She

ar F

orce

/ W

eig

ht


-250 0 250 500


-500


-0.3

-0.2

-0.1

0

0.1

0.2

0.3

She

ar F

orce

/ W

eig

ht


-250 0 250 500


structural acceleration. Particularly important is the case of earthquake motions with dominantnear-fault characteristics, such as the NF02 motion (analysis #3). In this case, the lower boundmodel significantly overpredicts the displacement demand because it is based on assumptions forlead core heating (several cycles of large amplitude motion) that result in low value for the lowerbound characteristic strength. The actual conditions primarily consist of a single large amplitudecycle without significant lead core heating effects so that the characteristic strength of the isolationsystem remains large and marginally affected by heating. It should be noted that the significanceof the strength of isolation systems on the response of seismically isolated structures has beenknown for near-fault ground motions [23].



-1200

Lateral Displacement X (mm)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

She

ar F

orce

X /

Wei

ght


Lateral Displacement Y (mm)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

She

ar F

orce

Y /

Wei

ght


-800 -400 0 400 800 1200

-1200 -800 -400 0 400 800 1200


Of interest is to mention analysis case 5 for the Chi-Chi earthquake, where the predictedtemperature increase in the lead core is so high that the strength drops much below the assumedlower bound value. For an increase in the temperature of the lead core of 200◦C, as is the casefor this analysis, the effective yield stress of lead drops to 25% its initial value or 4.2MPa, whichis much less than the assumed lower bound value of 10MPa. For such extreme scenarios, themodel described herein can provide accurate predictions of the effects of strength degradation oflead–rubber bearings that cannot be a priori estimated as required in bounding analysis. In suchconditions is likely that bounding analysis will not provide a conservative estimate of the responseand importantly, the isolator displacement demands may be underestimated (due to the use ofan incorrectly high value for the characteristic strength). Interestingly, this does not happen inanalysis case 5 because the maximum displacement demand on the isolation system occurs earlyin the history of excitation when the bearings still have high characteristic strength. This maybe detected in the sharp increase of lead core temperature at time of about 15 s in the graph ofFigure 16.



0

Time (s)

0

50

100

150

200

250

Tem

pera

ture

(o C

)

1

23

4

5

10 20 30 40 50 60 70 80

Figure 16. Predicted lead core temperature rise histories (Analyses #1–5).

Table V. Peak response of analyzed isolated structure.

Resultant Resultant ResultantDrift Accel

Analysis isolator isolation structural(mm) (g)

Lead coreMotion model displ. (mm) shear/W shear/Ws X Y X Y temp. increase (◦C)

1 Upper 466 0.21 0.26 16 — 0.27 — —Lower 553 0.18 0.20 12 — 0.20 — —Proposed 539 0.19 0.23 15 — 0.24 — 87

2 Upper 177 0.15 0.19 10 12 0.16 0.19 —Lower 210 0.10 0.12 6 7 0.10 0.11 —Proposed 180 0.14 0.16 9 10 0.15 0.16 59



5 Upper 707 0.24 0.25 10 15 0.17 0.23 —Lower 1048 0.25 0.25 9 15 0.15 0.24 —Proposed 1018 0.24 0.25 9 14 0.14 0.23 207

5. CONCLUSIONS

This paper has presented a model of the hysteretic behavior of lead–rubber bearings that accountsfor the temperature increase in the lead core on the basis of first principles. The model is capableof predicting the instantaneous temperature of the lead core and its instantaneous effect on thecharacteristic strength of the bearing. Predictions of force–displacement relations for lead–rubberbearings subjected to a large number of cycles of high-speed harmonic motion and to randommotions were shown to be in good agreement with experimental results. Further improvements



of the model would require consideration of (a) the effects of strain and of strain rate on thepost-elastic stiffness and (b) the effects of strain rate on the characteristic strength. Understandingof these phenomena requires experimental investigations but there is no theoretical complexity inaccounting for these effects on the basis of phenomenological models.

A study of the dynamic response of an idealized structure—approximately representative of aseismically isolated hospital in Turkey—isolated with lead–rubber bearings has been presented.The analysis was based on the proposed first-principles analytical model. The results of the analysiswere compared with results of analysis based on currently available models of hysteretic behavior oflead–rubber bearings that can consider the effects of lead core heating through the use of boundinganalysis. Bounding values of characteristic strength are established on the basis of experimentalresults may be associated with very small exceedance probabilities. Herein, the bounds addressedonly the effects of heating and did not consider variations in material properties at the time oftesting, aging, contamination, etc. The results of the study demonstrated that bounding analysisproduces conservative results for the prediction of isolation system displacement demand, isolationsystem peak shear force and peak structural responses. The conservatism is particularly pronouncedfor ground motions with strong near-fault pulses. Under these conditions, the temperature increasein the lead core is not substantial and the bearings maintain their characteristic strength. Thisobservation will not hold for cases of strong motion that result in three or more cycles of highamplitude response of the isolation system. In such cases bounding analysis is not necessarilyconservative.

The model presented herein cannot entirely replace bounding analysis methods because of theunavoidable uncertainty in the initial strength of the bearing and in the mechanical properties ofrubber. However, the model, when used in combination with bounding analysis, provides morerealistic response results with less conservatism. This may be particularly useful in areas ofmoderate seismicity as bounds on the mechanical properties of isolators are currently based ontests results that are representative of behavior in areas of high seismicity. The model may alsobe useful in establishing (through a parametric study, for instance) response-dependent strengthbounds for use within the concept of bounding analysis instead of relying on limited test data.

ACKNOWLEDGEMENTS

Financial support for the studies described herein was provided by MCEER (www. mceer.buffalo.edu)and the New York State. This support is gratefully acknowledged. The lead–rubber bearings for which testdata are presented in this paper have been manufactured by Dynamic Isolation Systems, Inc. of McCarran,Nevada.

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Modeling strength degradation in lead-rubber bearings under earthquake shaking