Vertical Stiffness of Elastomeric and Lead–Rubber Seismic Isolation Bearings

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Vertical Stiffness of Elastomeric and Lead–RubberSeismic Isolation Bearings

Gordon P. Warn1; Andrew S. Whittaker, M.ASCE2; and Michael C. Constantinou, M.ASCE3

Abstract: An experimental study investigating the influence of lateral displacement on the vertical stiffness of elastomeric and lead–rubber seismic isolation bearings is summarized. Two identically constructed low-damping rubber and lead–rubber seismic isolationbearings were subjected to a series of tests with varying levels of combined lateral displacement and axial �compressive� loading to studythis relationship. The results of these tests showed the vertical stiffness decreases with increasing lateral displacement for each bearingtested. Additionally, the vertical stiffness data are used to evaluate four formulations for the estimation of the vertical stiffness as afunction of the lateral displacement. From this comparison, two formulations, one based on the Koh–Kelly two-spring model and the otheron a piecewise linear relationship, showed good agreement with the experimental data over the wide range of lateral displacementsconsidered in this study.

DOI: 10.1061/�ASCE�0733-9445�2007�133:9�1227�

CE Database subject headings: Elastomer; Rubber; Stiffness; Lateral displacement; Experimentation.

Introduction

Seismic isolation bearings are vertically stiff yet horizontally flex-ible elements that lengthen a structure’s period of vibrationthereby reducing inertia forces that develop in the superstructureduring earthquake ground shaking. Elastomeric bearings are onetype of isolator consisting of a number of alternating elastomeric�rubber� layers bonded to intermediate steel �shim� plates. Con-ceptually, a lead–rubber bearing differs from an elastomeric bear-ing only through the addition of a lead-core �an energy dissipationmechanism�, typically located in a central hole. The horizontalflexibility �shear stiffness� of an elastomeric bearing is dictated bythe total thickness of rubber, whereas the close spacing of theintermediate shim plates provides a large vertical �relative to theshear� stiffness for a given shear modulus and bonded rubberdiameter.

The low horizontal �shear� stiffness required to lengthen thestructure’s period of vibration is, typically, accompanied by largelateral displacements in the isolator �usually, on the order of 100–150% rubber shear strain� for design level earthquake groundshaking. These large lateral displacements might lead to sustantialreductions in the load-carrying capacity and vertical stiffness ofthe elastomeric or lead–rubber bearing. The reduction in load-

1Postdoctoral Research Associate, Dept. of Civil, Structural andEnvironmental Engineering, State Univ. of New York at Buffalo, Buffalo,NY 14260 �corresponding author�. E-mail: [email protected]

2Professor, Dept. of Civil, Structural and Environmental Engineering,State Univ. of New York at Buffalo, Buffalo, NY 14260.

3Professor, Dept. of Civil, Structural and Environmental Engineering,State Univ. of New York at Buffalo, Buffalo, NY 14260.

Note. Associate Editor: Marvin W. Halling. Discussion open untilFebruary 1, 2008. Separate discussions must be submitted for individualpapers. To extend the closing date by one month, a written request mustbe filed with the ASCE Managing Editor. The manuscript for this paperwas submitted for review and possible publication on April 12, 2006;approved on February 1, 2007. This paper is part of the Journal ofStructural Engineering, Vol. 133, No. 9, September 1, 2007. ©ASCE,
ISSN 0733-9445/2007/9-1227–1236/$25.00.
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J. Struct. Eng. 2007.1

carrying capacity has been investigated �Kelly 1997; Nararajaiahet al. 1999; Buckle et al. 2002� and is currently considered for thedesign of seismic isolation systems composed of an elastomericand/or lead–rubber seismic isolation system �Buckle and Liu1994; Naeim and Kelly 1999�. However, aside from recent workinvestigating the tension buckling of elastomeric bearings �Kelly2003� and an expression derived from the Koh–Kelly two-springmodel �Kelly 1997�, there exists little information, and particu-larly experimental data, pertaining to the reduction in verticalstiffness. In addition, the reduction in vertical stiffness is not cap-tured by the state-of-the-art mathematical models for elastomericand lead–rubber seismic isolation bearings implemented inresponse-history analysis software, which typically represent theaxial degree of freedom as a linear spring with constant stiffness.Furthermore, the impact of the reduction in vertical stiffness onthe response of isolated structures subjected to earthquake groundshaking is largely unknown, however, there are several exampleswhere this behavior might have a significant impact on the per-formance of the isolation system and surrounding components.For example, in a composite isolation system composed of flatsliding and elastomeric bearings, the elastomeric bearings willdecrease in height under lateral deformation, whereas the flat slid-ers will remain at the same height �and thus vertical stiffness�likely resulting in axial load redistribution and additional momentin the diaphragm above the plane of isolation.

An experimental study was conducted to gain an improvedunderstanding of the influence of lateral displacement on thevertical stiffness of elastomeric and lead–rubber bearings and issummarized in this paper. To facilitate this investigation, two low-damping rubber �LDR� and two lead–rubber �LR� bearings froman earthquake simulator testing program at the University ofBuffalo �Warn and Whittaker 2006� were dedicated to this study.The LDR and LR bearings were subjected to varying levels oflateral displacement and axial �compressive� loading. The resultsof these tests were used to evaluate four formulations for thevertical stiffness of elastomeric and lead–rubber bearings sub-jected to lateral displacement. Two of these formulations were
based on physical representations of an elastomeric bearing �Koh
STRUCTURAL ENGINEERING © ASCE / SEPTEMBER 2007 / 1227

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and Kelly 1987; Kelly 2003� and two were empirically derived. Adescription of the four formulations for the vertical stiffness pre-cedes the discussion on the experimental investigation.

Formulations for the Vertical Stiffness

Two-Spring

The two-spring mechanical model introduced by Koh and Kelly�1987� is considered for the prediction of the vertical stiffness ofelastomeric bearings at a given lateral displacement. The deriva-tion of Kelly �1997� is reviewed to illustrate the theoretical basisof the formulation and to investigate the sensitivity of the formu-lation to various simplifying assumptions. An illustration of thetwo-spring model in the undeformed and deformed configurationis presented in Fig. 1. The model consists of a linear spring withstiffness, KH, a rotational spring with stiffness, K�, two friction-less rollers, and a rigid tee supported by a pin with total height, h.Also shown in Fig. 1 are the applied loads, P and F, and theresulting deformations, namely, the lateral displacement at thetop of the column, �; rotation about the pin, �; reduction inheight, �v; and deformation of the linear spring, s. Relating thespring properties of the model to the mechanical properties of anelastomeric seismic isolation bearing provides a simple physicalunderstanding of the behavior under combined lateral and verticalloading �Kelly 1997�.

An expression for the vertical stiffness is obtained by differ-entiating the total vertical displacement of a bearing subjected tocombined vertical loading and lateral displacement. Two compo-nents contribute to the total vertical displacement, first the initialvertical deformation due to an applied axial load and second thereduction in height due to the combined axial �compressive� loadand lateral deformation. An expression for the reduction in heightdue to combined loading is obtained by considering equilibriumand compatibility of the two-spring model in the deformed con-figuration; see Fig. 1. From compatibility, the global deformationquantities � and �v, are related to the local deformation quantitiess and �, through geometry according to Eqs. �1� and �2�

� = s cos�� + h sin�� 1�

�v = s sin�� + h�1 − cos�� 2�

Recognizing two components contribute to the reduction in heightpresented in Eq. �2�, the first, s sin��=component of deformationof the rotated linear spring and the second; and h�1−cos�� is

Fig. 1. Two-spring model in undeformed and deformedconfigurations

due to the rotation of the column of height, h. However, to obtain

1228 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / SEPTEMBER


a closed-form solution, the assumption of small rotations is nec-essary simplifying Eqs. �1� and �2� to

� = s + h� �3�

�v = s� +h�2

2�4�

where the second term of Eq. �4� is the result of approximatingcos�� by the first two terms of its Taylor series expansion. Fur-ther, force and moment equilibrium of the model in the deformedconfiguration is considered and again assuming small rotations,the following equilibrium equations are obtained:

�Fx�: − P� + KHs = F �5�

�Mpin: �K� − Ph�� − Ps = Fh �6�

The properties of the two-spring model are now related to themechanical properties of an elastomeric bearing to facilitate anunderstanding of the behavior of elastomeric bearings subjectedto combined lateral and vertical loading. Considering the case ofK�→�, the deformation is solely a function of KH and is equatedto the shear stiffness of an elastomeric bearing that for conve-nience is expressed as GAs /h, where As=Abh /Tr; Ab=bondedrubber area; and Tr=total rubber thickness. The stiffness of therotational spring is related to the Euler buckling load consideringthe case of KH→� �i.e., no shear deformations� and F=0. Thespring properties, GAs /h and PEh are substituted into Eqs. �5� and�6�, which are then substituted into Eq. �4� to obtained the fol-lowing equation for vertical displacement:

�v =�2

2h

�GAs + P��P + GAs + 2PE��GAs + P + PE�2 �7�

For seismic isolation bearings, it is reasonable to assume GAs

� PE and P� PE, as shown later in this paper. By neglecting theGAs and P terms where they are summed with PE, Eq. �7� reducesto Eq. �8�

�v =�GAs + P�

PE

�2

h�8�

Substituting PE=�2EIs /h2, As=Ab�h /Tr�, and Is= I�h /Tr� into Eq.�8� and assuming the rotational modulus is related to the compres-sion modulus, E�Ec /3 �which implicitly assumes incompressiblematerial behavior� results in Eq. �9�, the reduction in height due tocombined lateral and vertical loading is given by

�v =3�GAbh + PTr��2

�2EcI�9�

However, the vertical stiffness formulation is based on the totalvertical displacement, that is, the initial vertical displacement dueto the applied compressive load plus the vertical displacement dueto the lateral displacements. The total vertical displacement is,therefore

�vt =PTr

EcAb+

3�GAbh + PTr��2

�2EcI�10�

where EcAb /Tr=initial vertical stiffness, denoted Kvo; andEc=compression modulus of a given rubber layer. The verticalstiffness is obtained by inverting the result of differentiating Eq.�10� with respect to the vertical load, P. After some algebraic
manipulation the following expression is obtained:
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Kv = Kvo

1

�1 +3Ab

I

�2

�2��11�

where Kvo=initial vertical stiffness of the bearing �with no lateraldisplacement� as previously defined. If �=0, then Eq. �11� cor-rectly results in Kv=Kvo; for ��0, KvKvo. Normalizing Kv byKvo gives

Kv

Kvo=

1

�1 +3Ab

I

�2

�2��12�

Elastomeric bearings are often constructed with a central holeserving the primary purpose of allowing heat to penetrate into thecenter of the bearing during the curing process as was the case forthe LDR and LR bearings used in this study. To gain an under-standing of the sensitivity of the normalized vertical stiffness tothe geometry of the bearing cross section, a generic hollow cir-cular cross section with outer radius R and inner radius Ri isassumed. For simplicity let Ri=aR, where 0a1. SubstitutingAb=��R2−Ri

2� and I=��R2−Ri2� /4 into Eq. �12� with Ri=aR

results

Kv

Kvo=

1

�1 +12

�2� 1

1 + a2��

R2� �13�

Consider, for example, a=1/10 and �=2R; Eq. �13� givesKv /Kvo=0.172. In addition, for a=1/5, Kv /Kvo=0.176. To furtherinvestigate the influence of the central hole radius, Eq. �13� wasevaluated for several values of a �including a=0 corresponding toa solid circular cross section�; results are plotted in Fig. 2. Fromthe curves plotted in Fig. 2 and the sample calculations, the valueof Kv /Kvo is not sensitive to the value of a even for a=1/2, thatis, a hollow circular bearing with an inner radius, Ri, equal to halfthe outer radius, R. Based on the results of this analysis, it isreasonable to use the properties of a generic solid circular cross

2 4

Fig. 2. Normalized vertical stiffness expression Eq. �13� evaluatedfor various values of a

section �a=0, Ab=�R , I=�R /4�, and Eq. �13� simplifies to

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Kv

Kvo=

1

�1 +12

�2��

R2� �14�

Again, if �=0 then Eq. �14� correctly equals 1 and for ��0 thenormalized vertical stiffness, Kv /Kvo, is less than 1. The utilityof Eq. �14� is evaluated using the results of an experimental in-vestigation presented in the following section. In addition, thevalidity of the various assumptions used to derive Eq. �14� arereviewed.

Tension Buckling

Kelly �2003� presented a derivation for the vertical stiffness of anelastomeric bearing sheared through a displacement, �. This for-mulation is the result of an analytical investigation of the tensionbuckling of elastomeric bearings and is based on a rigorous sta-bility analysis of multilayer elastomeric bearings. Due to thelength of the derivation, only the resulting expression is presentedhere. For this analysis, the bearing is treated as a continuous beamwith equivalent properties and the deformed shape accounts forshear deformations, assuming plane section remain plane but notnecessarily normal to the deformed axis. Similar to the two-springderivation, an expression for the vertical stiffness is obtained bydifferentiating the expression for the vertical displacement of thetop of the bearing due to lateral displacement. Again, this deriva-tion assumes the shear rigidity �GAs� is negligible compared tothe applied load �P�. The resulting expression for the verticalstiffness is presented here in normalized form as

Kv

Kvo=

1

�1 + 3f��b��

R2� �15�

where b= P / Pcr; and Pcr=critical buckling load accounting forshear deformation calculated according to

Pcr = PEGAs �16�

In addition, f��b�, is a trigonometric function

f��b� =�b − sin��b�1 − cos��b�

�17�

The form of Eq. �15� is similar to the expression derived from thetwo-spring model �Eq. �14�� in that the vertical stiffness reducesproportional to a factor �3f��b�� times the square of the lateraldisplacement, �, normalized by the radius, R. For b= P / Pcr=0,Eq. �17� equals 0 and Eq. �15� gives Kv /Kvo=1. That is, for zeroaxial load there is no reduction in vertical stiffness although therewill be vertical displacement of the top of the bearing under lat-eral deformation. Additionally, for b= P / Pcr=1, Eq. �17� equals� /2 and 3f��b�=1.5�, which is substantially larger than 12/�2

for the two-spring formulation. Eq. �15� is evaluated using theappropriate P / Pcr for comparison with the experimental resultslater in this paper.

Overlapping Area

A concept widely accepted for the estimation of the critical loadof an elastomeric bearing subjected to combined compression andlateral displacement �Buckle and Liu 1994� is adapted here for theestimation of the reduction in vertical stiffness with lateral dis-
placement. The concept is based on a column with reduced area.

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In practice the nominal value, either critical load or vertical stiff-ness, is multiplied by the ratio of the reduced area to the bondedrubber area where the reduced area is the overlapping area be-tween the top and bottom internal plates. An illustration of theoverlapping area, Ar, for a solid circular bearing subjected to alateral displacement, �, is presented in Fig. 3. The vertical stiff-ness, in normalized form, is then determined according to

Kv

Kvo= �Ar

Ab �18�

where Ab=bonded rubber area equal to �R2; and Kvo=initial ver-tical stiffness. The reduced area is calculated as

Ar =D2� − sin �

4�19�

where D=2R and is calculated according to

= 2 cos−1��

D �20�

From Eq. �20� for �=2R, =0, and Ar=0. Also, for �=0, Eq.�20� gives =�. Substituting =� into Eq. �19� yields Ar

=�D2 /4, which is equal to Ab for a solid circular cross section.Therefore, the overlapping area correctly predicts Kv /Kvo=1 at�=0 and predicts Kv /Kvo=0 at �=2R. Although the overlappingarea concept is simple and easily implemented for the design ofelastomeric isolation systems it is not based on any rigoroustheory and does not capture the physical behavior of elastomericbearings subjected to combined compressive and lateral loading.

Linear

A simple piecewise linear relationship for the reduction of vertical

Fig. 3. Illustration of overlapping area for a solid circular bearingsubjected to lateral displacement

stiffness with lateral displacement is considered. This formulation



was empirically derived based on Kv /Kvo=1 for �=0 and fromthe analysis of the two-spring model predicting Kv /Kvo�0.2 at�=2R. Assuming the vertical stiffness decreases linearly withincreasing lateral displacement up to �=2R then remains constantresults in the piecewise linear expression

Kv

Kvo= 1 − 0.4��

R for �/R � 2

Kv

Kvo= 0.2 for �/R � 2 �21�

Although not based on rigorous theory, Eq. �21� offers a simplerelationship accounting for some remaining vertical stiffness at�=2R. Eq. �21� is plotted with experimental data in the section ofthis paper on the comparison of results.

Experimental Investigation

Bearing Specimen

A total of twelve bearings were designed and fabricated for earth-quake simulator testing performed at the University at Buffalo�Warn and Whittaker 2006�. Four of these bearings, specifically,two LDR and two LR, were dedicated to characterization andlateral offset testing. For reference, these bearings were arbitrarilyassigned Numbers 5 and 6, and are referred to throughout thissection as LDR 5 and LR 6, etc. The model bearings were pro-portioned using a target model period of vibration �related to theshear stiffness of the rubber and the supported weight� of 1.25 sbased on an assumed prototype period of vibration of 2.5 s, avalue typical of elastomeric seismic isolation bearings �HITEC1998a, b�, and a time scale factor of 2. The resulting model LDRand LR bearings are identical in construction and material, withthe exception of a lead core inserted in the central hole of the LRbearings, as shown in Fig. 4. The bearings are circular in planwith an outer bonded diameter of 152 mm and an inner centralhole diameter of 30 mm consisting of 20 layers of natural rubbereach 3 mm thick bonded to 19 intermediate steel shim plates,approximately, 3 mm thick �11 gauge� with two steel internal

Fig. 4. Details for LDR and LR bearings

load plates each 25 mm thick. The shape factors for the LDR and

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LR were determined to be approximately, 10 and 12, respectively,
and calculated for an individual rubber layer according to

S =Loaded area

Area free to bulge�22�

Test Apparatus and Setup

The LDR and LR bearings were tested using the single bearingtest machine �SBTM� located in the Structural Engineering andEarthquake Simulation Laboratory �SEESL� at the University atBuffalo. The SBTM was designed for the primary purpose oftesting single seismic isolation bearing under conditions of unidi-rectional shear and combined axial load and consists of a pedestalframe, a reaction frame, a loading beam, a horizontal �dynamic�actuator, two vertical actuators and a five-channel reaction loadcell. An illustration of the SBTM is presented in Fig. 5.

A total of 12 data channels including time, recorded the per-formance of the driving actuators and the response of the seismicisolation bearing during testing. The instruments were distributedthroughout the SBTM as follows. The horizontal actuator con-tains an inline uniaxial load cell �LC� and internal linear variabledisplacement transducer �LVDT� recording actuator axial loadand relative displacement, respectively. The vertical actuatorseach contain an inline uniaxial LC and externally attached MTS±50 mm Temposonic displacement transducer recording axial

Table 1. Summary of Bearing Testing Program

Test Description Signal

5 Shear Sine

7 Axial Ramp

8 Axial Ramp

11 Axial Ramp

16 Offset Ramp

17 Offset Ramp

18 Offset Ramp

21 Offset Ramp

22 Offset Ramp

23 Offset Ramp

27 Offset Ramp

28 Offset Ramp

32 Offset Ramp

33 Offset Ramp

36 Offset Ramp

Note: NA indicates the parameter is not applicable to test type; Test 33

Fig. 5. Illustration of single bearing testing machine

0.01 Hz.

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load and relative displacement, respectively. A five-channel reac-tion LC fixed atop the pedestal frame recorded axial load �P�,shear force �F�, and bending moment �M� at the base of theisolation bearing. In addition, during lateral offset testing, two±6 mm linear potentiometers were installed between the bearingtop and bottom endplates to record relative vertical displace-ments. The average of the two linear potentiometer signals wasused for the vertical displacement measurement in lieu of a mea-surement at the center of the bearing. All signals were filtered at10 Hz via a low-pass analog filter then digitized using a PCI DAS6402 analog-to-digital �AD� card installed in a desktop PC.

Test Procedure

A series of tests were performed on the LDR and LR bearings todetermine the mechanical properties and to investigate the influ-ence of lateral displacement on the vertical stiffness. Table 1 pre-sents a summary of the test program and relevant test informationincluding: test number, type, input signal, axial load amplitude P,lateral offset �, displacement amplitude u, and number of cycles.For brevity, only tests relevant to the vertical stiffness investiga-tion are listed in Table 1.

Test No. 5 represents a benchmark shear test. This test wasconducted by first applying a constant compressive load of 60 kN�pressure of 3.45 MPa�, then subjecting the bearing to four fullyreversed cycles to a displacement amplitude of 60 mm �corre-sponding to a rubber shear strain, �, of 100%� using a sinusoidalinput signal with a frequency, f , of 0.01 Hz. Shear tests wereconducted under displacement control. Tests 7, 8, and 11 repre-sent axial load tests. These tests were conducted by firstcompressing the bearing to a preload of approximately 24 kN�pressure of 1.4 MPa�, then subjecting the bearings to three half-cycles to axial �compressive� load amplitudes of 60, 120, and180 kN, respectively, using a ramp signal with a frequency of0.01 Hz. This frequency corresponds to the time from preload tomaximum and then back to preload. Axial load tests were con-ducted under force control. The remaining tests represent offsettests conducted to various axial load amplitudes and at lateraloffsets of 30, 60, 90, 120, and 152 mm. Lateral offset tests wereconducted by first applying a compressive preload of 24 kN,

��

�mm�u

�mm� Cycles

NA 60 4

0 NA 3

0 NA 3

0 NA 3

30 NA 3

30 NA 3

30 NA 3

60 NA 3

60 NA 3

60 NA 3

90 NA 3

90 NA 3

120 NA 3

120 NA 3

152 NA 3

rformed on LR bearings only; and all tests were performed at a rate of

P�kN

60

60

120

180

60

120

180

60

120

180

60

120

60

120

60

was pe


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shearing the bearing to a constant lateral offset �� and then sub-
jecting the bearing to three half-cycles at a given axial load am-plitude �see Table 1� using a ramp input signal with a frequencyof 0.01 Hz. Lateral offset tests were conducted under force con-trol and the lateral offset was recorded prior to testing using theLVDT housed in the horizontal actuator. For �=152 mm, the lat-eral offset is equal to the bonded diameter corresponding to zerooverlapping area between the top and bottom internal load plates.An interface plate was designed and fabricated for offset testing,installed between the loading beam and the top endplate of thebearing, to maintain the proper boundary conditions and to allowthe vertical actuators to remain plumb for each offset �not illus-trated here�.

Test Results

Sample results from the shear, axial, and offset test performed onthe LDR and LR bearings are presented in this section. Materialproperties estimated from the third cycle results of Test No. 5�corresponding to a maximum rubber shear strain of 100% and ata frequency of 0.01 Hz� performed on the LDR and LR bearingsare summarized in Table 2. These properties include: L, the ef-fective yield stress of the lead core �LR only� calculated using therecorded zero-displacement force intercept and the cross-sectionalarea of the core �see Fig. 4�; Geff, the effective shear moduluscalculated from the effective �secant� stiffness results for the LDRand LR bearings; and �eff, the effective damping ratio. For theLDR bearings, Geff represents the effective shear modulus of therubber, whereas for the LR bearings the contribution of the leadcore is included in Geff. For LDRs 5 and 6, Geff and �eff wereestimated to be approximately equal to 0.8 MPa �116 psi� and 3%of critical, respectively. The effective yield stress for the LR bear-ings was estimated to be 8.4 MPa �1220 psi�. Additionally, Geff

and �eff for the LR bearing were estimated to be 1.3 MPa�190 psi� and 19% of critical, respectively.

Table 3 presents a summary of experimentally determined val-ues of the vertical stiffness from the axial load tests ��=0� per-

Table 2. Estimated Material Properties of the LDR and LR from TestNumber 5

Parameter

Bearing

LDR LR

5 6 5 6

L �MPa� NA NA 8.4 8.3

Geff �MPa� 0.83 0.82 1.37 1.34

�eff �%� 2.7 2.8 18.6 19.0

Note: NA indicates the parameter does not pertain to bearing type.

Table 3. Vertical Stiffness Results from Axial Tests Performed on LDR

Verticalstiffness

Testnumber

P�kN�

Kvo �kN/mm� 7 60

8 120

11 180



formed on the LDR and LR bearings. The vertical stiffness, de-noted Kvo, was calculated from the results of axial load tests �aswell as offset tests� according to

Kvo =P+ − P−

�+ − �− �23�

where P+=axial load corresponding to a pressure equal to thetarget pressure, p plus 0.35 MPa; P−=axial load corresponding toa pressure equal to p−0.35 MPa; �+=average relative vertical dis-placement coincidental with P+ and measured from the ascendingbranch of the loading curve; and �−=average relative vertical de-formation coincidental with P−. The target pressures for the axialload amplitudes of 60, 120, and 180 kN were 2.76, 5.2, and9.0 MPa, respectively.

It is interesting to note the deviation in the value of Kvo for agiven type of isolator and axial load as presented in Table 3.Unlike the mechanical properties estimated from the shear testthat compare well for a given type of bearings, the vertical stiff-ness results vary by 7–15% for the LR and 12–19% for the LDRfor a given axial load amplitude. This variation is due to differ-ences in the thickness of the individual rubber layers observedafter cutting the bearings in half upon completion of the testingprogram.

Fig. 6 presents photographs of the LR bearings taken duringaxial �Fig. 6�a�� and offset tests �Figs. 6�b–d��. In addition, thesephotographs show the placement of the west ±6 mm linear poten-

Bearings

Bearing

LDR LR

5 6 5 6

1.4 93.8 138.7 116.0

9.7 86.8 154.1 131.8

4.3 90.4 163.3 145.1

Fig. 6. Photographs taken during testing: �a� �=0 mm; �b��=30 mm; �c� �=90 mm; and �d� �=152 mm

and LR

8

7

8

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tiometer. Fig. 7 presents axial load versus vertical displacementloops from LDR 5 for �=0, 30, 90, and 152 mm, respectively.The loops presented in Fig. 7 clearly show the increase in maxi-mum vertical displacement and corresponding reduction in verti-cal stiffness with increasing lateral offset.

The estimated material properties from Test 5 ��=100% andf =0.01 Hz� presented in Table 2 were utilized to calculate variousbearing quantities to evaluate the assumptions made in the deri-vation of the simplified expression of Eq. �14�. A summary ofthese quantities is presented in Table 4, including: the shear rigid-ity, GAs �approximated using GeffAs�; the Euler buckling load, PE;and critical load Pcr. The Euler buckling load was calculatedusing PE=�2EIs /h2 and the critical load �Kelly 1997� calculatedaccording to Eq. �16�. In addition, Table 4 includes the dimen-sionless quantity h /Tr which is equal to 1.95 for both the LDRand LR bearings. It is worth noting the model bearings wereconstructed with shim plate thickness typical of prototype bear-ings resulting in an unusually large h /Tr value for an elastomericbearing. For a typical prototype bearing with the same number ofrubber layers each 10 mm thick, the h /Tr ratio would equal 1.3.

The primary assumptions for the derivation of Eq. �14� wereGAs� PE, P� PE, and E�Ec /3 �incompressible material�. Me-chanical properties determined from characterization tests wereused to calculate various quantities to evaluate these assumptions.The Euler buckling load, PE, for the LDR and LR bearings �6,157and 15,030 kN, respectively� are orders of magnitude larger that

Table 4. Various Calculated Properties of the LDR and LR Bearings

Quantity Unit

Bearing

LDR LR

h /Tr — 1.95 1.95

GeffAs kN 27 47

PE kN 6,157 15,030

Pcr kN 408 841

Kvo kN/mm 89 174

Fig. 7. Axial load versus vertical displacement loops from LDR 5for: �a� �=0 mm; �b� �=30 mm; �c� �=90 mm; and �d� �=152 mm

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the shear rigidity, GeffAs �27 and 47 kN, respectively� supportingthe assumption GAs� PE. Similarly, the critical load, Pcr, of theLDR and LR bearings �408 and 841 kN, respectively� is an orderof magnitude less than PE, indicating that P� PE is a reasonableassumption. Table 4 presents calculated values of the initial ver-tical stiffness �Kvo=EcAb /Tr� for the LDR and LR bearings, de-termined to be 89 and 174 kN/mm, respectively. For the verticalstiffness calculation, the compression modulus Ec was estimatedaccording to

1

Ec=

1

6GS2F+

4

3K�24�

where K=bulk modulus assumed to be equal to 2,000 MPa�a typical value for lightly filled natural rubber �Constantinouet al. 1992��; F=function of the bearing geometry �Constantinouet al. 1992� equal to 0.69 for the bearings used in this study;and G=shear modulus assumed for this calculation to be theeffective shear modulus Geff calculated at a shear strain corre-sponding to the average shear strain due to the compressive load��avg=6S�c� and the compressive strain ��c=� /Tr� from the re-sults of Test 11. For the compression modulus calculation, Geff,for the LDR and LR bearings was determined to be 0.88 MPa��avg=75% � and 1.6 MPa ��avg=50% �, respectively. For theLDR and LR bearings the values of Kvo computed using an as-sumed bulk modulus, K, of 2,000 MPa compare well with theexperimentally determined values from Test 11 presented in Table3; the maximum differences are 6 and 20%, respectively. Thevertical stiffness was also calculated �Warn and Whittaker 2006�assuming the material to be incompressible �infinitely large K�.The computed values substantially overestimated the experimen-tally determined vertical stiffness for the LDR and LR bearings,suggesting that material compressibility should be considered forthe calculation of Kvo=EcAb /Tr.

Comparison of Results

The vertical stiffness results obtained from the offset tests, Kv,were normalized by the initial vertical stiffness determined fromthe axial load tests, Kvo, for the corresponding axial load ampli-tude. Normalizing the data in this manner facilitates a directgraphical comparison of the experimental data to the various for-mulations for the reduction in vertical stiffness presented earlier.In addition, this normalization allowed a residual analysis used asa quantitative measure of the fit of a particular formulation to theexperimental data.

Vertical stiffness data from the LDR and LR bearings plottedagainst � /R are presented in Figs. 8�a and b�, respectively. Alsoplotted in Figs. 8�a and b� is the predicted reduction in verticalstiffness from Eq. �14� derived from the two-spring model andshown by a solid line. In addition, a reference �dashed� line cor-responding to Kv /Kvo=1 is also plotted. From Fig. 8�a� the reduc-tion in vertical stiffness with lateral displacement is evident fromthe experimental data with one exception where Kv /Kvo is ob-served to be slightly greater than 1. The experimental data forLDRs 5 and 6 show reasonably good agreement with Eq. �14� forP=60 kN over the range of � /R considered. For LDR 5 withP=60 kN, and � /R�2.0 �a lateral displacement equal to thebonded diameter corresponding to zero overlapping area�, the pre-dicted and experimental values show good agreement withKv /Kvo�0.2. In Fig. 8�b� �LR�, the reduction in vertical stiffness
with lateral displacement is again apparent. The experimental data

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for both LRs 5 and 6 again show good agreement with Eq. �14�for each axial load amplitude over the range of � /R consideredeven for the case of � /R�2.0.

The predicted vertical stiffness from the tension buckling for-mulation �Eq. �15�� is plotted in Figs. 9�a and b� again with nor-malized experimental data from the LDR and LR bearings, re-spectively. Fig. 9�a� presents normalized vertical stiffness data forthe LDR bearings plotted with Eq. �15� evaluated for P / Pcr

=0.15, 0.29, and 0.44, corresponding to P=60, 120, and 180,respectively, and Pcr=408 kN �see Table 4�. Fig. 9�b� presentsnormalized vertical stiffness data for the LR bearings plotted withEq. �15� evaluated for P / Pcr=0.07, 0.14, and 0.22, correspondingto P=60, 120, and 180, respectively, and Pcr=841 kN �see Table4�. From these plots, Eq. �15� does not compare particularly wellwith the experimental data from the bearings tested in this study.

Figs. 10�a and b� present plots of the normalized experimentaldata from the LDR and LR bearings, respectively, with the pre-diction of the overlapping area formulation �Eq. �18��. From Figs.10�a and b�, Eq. �18� is observed to overpredict the reduction invertical stiffness for the range of � /R considered. In addition, Eq.�18� predicts Kv /Kvo=0 at � /R=2.0, whereas the experimentaldata suggest approximately 20% of the initial vertical stiffness atthis displacement.

Figs. 11�a and b� presents the predicted reduction in verticalstiffness from the piecewise linear formulation with normalizedexperimental data from the LDR and LR bearings, respectively. In

Fig. 8. Comparison with two-spring predictions: �a� LDR bearings;�b� LR bearings

Fig. 11�a� �LDR�, the experimental data are observed to agree



reasonably well with Eq. �21�, which conservatively predicts thereduction in vertical stiffness, with the exception of LDR 6 andP=60 kN. In Fig. 11�b� �LR�, again the experimental data showreasonably good agreement with Eq. �21�, although there aremore instances where Eq. �21� underpredicts the reduction in ver-tical stiffness.

To quantify the fit between the experimental data and eachformulation, the sum of the residual between the experimentaldata and the predicted value from a particular formulation wascalculated for each lateral offset and axial load amplitude accord-ing to

R = �� Kv

Kvo

e

− � Kv

Kvo

t� �25�

where the subscript e represents the experimentally determinedvalue; and the subscript t represents the predicted value from aparticular formulation. Cumulative residual values calculated foreach bearing, axial load amplitude, and formulation are presentedin Table 5. Also included are residual subtotals �that is, for eachaxial load amplitude� and total for each formulation. From thetotal residual values presented in Table 5, the linear formulationprovides the lowest residual value �2.6� followed by the two-spring model �4.7�. The total residual for the tension bucklingformulation �5.2� is slightly larger than for the two-spring formu-lation, whereas the overlapping area formulation results in thehighest residual value �8.0�, approximately two times the two-

Fig. 9. Comparison with tension buckling predictions: �a� LDRbearings; �b� LR bearings

spring and three times the linear values.

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Summary and Conclusions

An experimental investigation of the influence of lateral displace-ment on the vertical stiffness of elastomeric and lead–rubberseismic isolation bearings was presented. The results of this ex-perimental investigation were used to evaluate four formulationsfor the reduction in vertical stiffness.

The conclusions based on the results of the experimental in-vestigation and comparison with the four formulations for thevertical stiffness are as follows:1. The results of the experimental investigation showed the ap-

parent vertical stiffness of the LDR and LR bearings de-creased with increasing lateral displacement.

2. For a given type of bearing, the experimentally determinedvalues of the initial vertical stiffness varied considerably al-though the horizontal properties were nearly identical and isattributed to variations in the thickness of the individual rub-ber layers.

3. The two-spring formulation �Eq. �14�� conservatively pre-dicts the experimentally observed reduction in vertical stiff-ness for each lateral offset with two exceptions from testspreformed on the LDR and LR.

4. Although based on the most rigorous analysis, the tensionbuckling formulation did not provide a more accurate predictthe reduction in vertical stiffness when compared to the two-spring formulation, based on the residual values.

Fig. 10. Comparison with overlapping area predictions: �a� LDRbearings; �b� LR bearings

5. The overlapping area formulation �Eq. �18�� substantial over

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predicts the reduction in vertical stiffness and predicts zerovertical stiffness at a lateral displacement equal to two timesthe radius contrary to the experimental data.

6. The piecewise linear formulation �Eq. �21�� agrees well withthe experimental data and resulted in the lowest cumulativeresidual. However, the formulation was empirical derivedwith no theoretical basis and might not predict well for bear-ings with different proportions.

7. Both the experimental results and the two-spring formulationsuggest the vertical stiffness of the bearing at a lateral dis-placement equal to the bonded rubber diameter is approxi-mately 20% of the initial vertical stiffness.

Acknowledgments

The writers gratefully acknowledge the financial support of theMultidisciplinary Center for Earthquake Engineering Researchand the Federal Highway Administration through Task D1.5 ofFederal Highway Administration Contract DTFH 61-98-C-0094.The writers also wish to thank Dr. Amarnath Kasalanati of DISInc. for generously providing the model bearings used in thisstudy. The opinions expressed in this paper are those of the writ-ers and do not reflect the opinions of the Multidisciplinary Centerfor Earthquake Engineering Research or the Federal Highway Ad-

Fig. 11. Comparison with linear prediction: �a� LDR bearings; �b�LR bearings

ministration. No guarantee regarding the results, findings, and


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recommendations are offered by either the Multidisciplinary Cen-ter for Earthquake Engineering Research or the Federal HighwayAdministration.

References

Buckle, I., Nagarajaiah, S., and Ferrell, K. �2002�. “Stability of elasto-meric isolation bearings: Experimental study.” J. Struct. Eng., 128�1�,3–11.

Buckle, I. G., and Liu, H. �1994�. “Experimental determination of criticalloads of elastomeric isolators at high shear strain.” NCEER Bull.,8�3�, 1–5.

Constantinou, M. C., Kartoum, A., and Kelly, J. M. �1992�. “Analysis ofcompression of hollow circular elastomeric bearings.” Eng. Struct.,14�2�, 103–111.

HITEC. �1998a�. Evaluation findings for Skellerup base isolation elasto-meric bearings, Civil Engineering Research Foundation, Washington,D.C.

Table 5. Residual, R, from Analysis

FormulationP

�kN�

LDR

5 6

Two spring 60 0.57 0.30

120 0.56 0.40

180 0.48 0.37

Tension buckling 60 0.33 0.52

120 0.05 0.13

180 0.19 0.09

Overlapping area 60 1.11 0.57

120 0.80 0.62

180 0.63 0.52

Linear 60 0.15 0.18

120 0.29 0.12

180 0.36 0.25



HITEC. �1998b�. Evaluation findings for dynamic isolation systems, in-cluding elastomeric bearings, Civil Engineering Research Founda-tion, Washington, D.C.

Kelly, J. M. �1997�. Earthquake-resistant design with rubber, Springer,London.

Kelly, J. M. �2003�. “Tension buckling in multilayer elastomeric bear-ings.” J. Eng. Mech., 129�12�, 1363–1368.

Koh, C. G., and Kelly, J. M. �1987�. Effects of axial load on elastomericisolation bearings, Springfield, Va., Earthquake Engineering ResearchCenter, College of Engineering, Univ. of California at Berkeley, Ber-keley, Calif.

Naeim, F., and Kelly, J. M. �1999�. Design of seismic isolated structures:From theory to practice, Wiley, New York.

Nagarajaiah, S., and Ferrell, K. �1999�. “Stability of elastomeric seismicisolation bearings.” J. Struct. Eng., 125�9�, 946–954.

Warn, G., and Whittaker, A. S. �2006�. “A study of the coupledhorizontal–vertical behavior of elastomeric and lead–rubber seismicisolation bearings.” Technical Rep. MCEER-06-0010, Multidisci-plinary Center for Earthquake Engineering Research, State Univ. of

ring

Subtotal Total

LR

5 6

0.48 0.42 1.77

0.22 0.59 1.77 4.7

0.13 0.22 1.20

1.12 0.86 2.83

0.78 0.55 1.51 5.2

0.32 0.21 0.81

1.01 0.79 3.48

0.29 1.04 2.75 8.0

0.27 0.37 1.79

0.16 0.26 0.75

0.22 0.46 1.09 2.6

0.04 0.11 0.76

Bea

New York at Buffalo, Buffalo, N.Y.

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Vertical Stiffness of Elastomeric and Lead–Rubber Seismic Isolation Bearings

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Transcript of Vertical Stiffness of Elastomeric and Lead–Rubber Seismic Isolation Bearings