Composite Structures 144 (2016) 177–184

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Variation of the vertical stiffness of strip-shaped fiber-reinforcedelastomeric isolators under lateral loading

http://dx.doi.org/10.1016/j.compstruct.2016.01.0890263-8223/� 2016 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Dept. of Civil Engineering, McMaster University, 1280Main St W, Hamilton, Ontario L8S 4L8, Canada.

E-mail address: konstant@mcmaster.ca (D. Konstantinidis).

Peyman M. Osgooei, Dimitrios Konstantinidis ⇑, Michael J. TaitDept. of Civil Engineering, McMaster University, Canada

a r t i c l e i n f o

Article history:Available online 4 February 2016

Keywords:Fiber-reinforced elastomeric isolatorsLaminated rubber bearingsSeismic isolationCombined lateral and vertical loadingVertical responseFinite element analysis

a b s t r a c t

A fiber-reinforced elastomeric isolator (FREI) is a relatively new type of isolator that utilizes fiber materialfor the reinforcing layers. FREIs can be installed in a bonded or unbonded application. In this study, finiteelement analysis (FEA) is carried out on bonded and unbonded strip-shaped FREIs to investigate the vari-ation in vertical stiffness as the isolator undergoes lateral displacement. The vertical stiffness of the iso-lators under pure compression obtained by FEA was in good agreement with the predictions of twoavailable closed-form solutions. As the lateral displacement increases, it was observed that for bondedFREIs the vertical stiffness decreases monotonically; whereas, for unbonded FREIs, the vertical stiffnessdecreased up to 175% shear deformation, where an increase in vertical stiffness was observed. FEA resultsconfirmed that the effective overlap area method provides reasonable estimates for the vertical stiffnessof bonded FREIs. It is observed that as the applied vertical stress increases, the vertical stiffness of bondedand unbonded FREIs increases. Finally, the paper shows that under large lateral displacements, bondedFREIs develop large tensile stresses in the regions outside the overlapping areas, while the tensile stressesthat develop in unbonded FREIs are very low and confined in a small region.

� 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Steel-reinforced elastomeric isolators (SREIs) are the most com-mon type of seismic isolators in use. This type of isolator is com-posed of layers of elastomer interleaved with steel reinforcingshims. Thick steel end plates are typically bonded to the top andbottom surfaces of the bearing for connecting purposes. The appli-cation of SREIs in North America is limited to high importancestructures (e.g., historical buildings, emergency centers), or build-ings with sensitive or valuable contents [1]. Currently it is not eco-nomically feasible to employ SREIs for residential or ordinaryimportance structures. Kelly [2,3] suggested that by reducing thecost and weight of SREIs, the application of elastomeric bearings(and in a more general case, seismic isolation) can be extendedto housing and commercial buildings.

Replacing the steel reinforcing shims with a lighter material (forexample, fiber reinforced polymer) can reduce the weight of SREIs.Unlike SREIs, which require a hot vulcanization process, a cold vul-canization process can be utilized to construct fiber-reinforcedelastomeric isolators (FREIs) [4], which could potentially reduce

the high manufacturing costs associated with SREIs. In addition,FREIs can be manufactured in large pads where individual isolatorscan be cut to the desired size, leading to further potential reduc-tions in cost and turnaround time.

FREIs can be installed in both bonded (Fig. 1a) and unbonded(Fig. 1b) applications. Unbonded FREIs do not have end plates.Due to the flexibility of the fiber reinforcing layers, unbonded FREIsundergo a unique rollover deformation (Fig. 1b). This rollover hasbeen shown to increase the seismic isolating efficiency by reducingthe effective lateral stiffness of the isolators [5]. In addition, theunbonded application results in a reduced stress demand on theelastomer and fiber reinforcement layers [5], which makes it pos-sible to use a simpler manufacturing process for unbonded FREIs.

Fig. 2 compares the lateral load–displacement relationship ofbonded and unbonded FREIs. While the effective (secant) stiffnessof the bonded FREI is unaffected by the change in the displacement,the effective stiffness of the isolator in an unbonded applicationvaries with lateral displacement. Note that in this discussion, geo-metric effects that can reduce the effective lateral stiffness of abonded isolator [6–8], especially for slender configurations, arenot considered. Under small displacements, the effective stiffnessof the unbonded FREI is constant. As the displacement increases,due to the rollover effect, the effective stiffness of the unbondedFREI decreases, until a point where an increase in the stiffness is

(a) (b)Fig. 1. Deformed shape of (a) a bonded FREI, and (b) an unbonded FREI.

Bonded Application

Unbonded Application

Fig. 2. Lateral load–displacement relationships of bonded and unbonded FREIs.

178 P.M. Osgooei et al. / Composite Structures 144 (2016) 177–184

observed. This increase in the effective stiffness in unbonded FREIsis due to the contact of the originally vertical faces of the isolatorwith the loading surfaces and can limit the isolation displacementunder large (i.e., maximum-considered-earthquake) groundmotions [4,9].

The vertical response of FREIs has been investigated in a num-ber of analytical [10–13] and numerical [14–18] studies. Further-more, experimental studies involving vertical compression andlateral cyclic tests have shown that FREIs achieve both adequatevertical stiffness and the required lateral flexibility to render themsuitable for seismic isolation [2–4,19–23]. However, in these stud-ies, FREIs were investigated under pure compression, and the effectof lateral displacement on the compression response characteris-tics of the isolators was neglected. As such, the variation in the ver-tical stiffness of unbonded FREIs subjected to lateral loadingrequires investigation.

In this paper, finite element analysis (FEA) is employed to studythe vertical response of FREIs, in both bonded and unbonded appli-cations, under different lateral displacement amplitudes. Twostrip-shaped isolators with the same overall dimensions but differ-ent shape factors, S (defined for a single elastomer layer as the ratioof the total loaded area to the load-free area), are modeled and ana-lyzed using MSC Marc [24], a general-purpose finite element pro-gram. First, results from the FEA under pure compression arecompared against two available closed-form solutions. Then, thevertical stiffness of FREIs is characterized at various levels of lateraldisplacement and under different values of average vertical stress.The accuracy of the effective overlap area method [25], which iswidely used for estimating the vertical stiffness of bonded SREIsunder lateral displacement, is evaluated for bonded FREIs by com-paring its predictions against FEA results. The paper examines howthe downward vertical displacement of the top support of a FREIvaries with increasing lateral displacement. Finally, the stress dis-tribution in the overlapping and non-overlapping regions ofbonded and unbonded FREIs is investigated.

Table 1Characteristics of the isolators considered in this study.

Bearing 2b (mm) S t (mm) tf (mm)

S11 300 11.2 13.4 1S22 300 22.4 6.7 1

2. Finite element analysis

Two strip-shaped isolators were considered in this study.Table 1 shows the dimensions of the isolators. Both isolators hada width of 2b ¼ 300 mm and a width-to–height aspect ratio of3.0. A shear modulus of G ¼ 0:8 MPa and a bulk modulus ofK ¼ 2000 MPa, corresponding to an elastic modulus of E ¼ 2:4 MPaand a Poisson’s ratio of m ¼ 0:4998, were used for the elastomer.The elastic modulus and the Poisson’s ratio of the fiber reinforce-ment material were selected as Kf ¼ 20 GPa and mf ¼ 0:2,respectively.

FEA of FREIs is challenging due to near-incompressibility of theelastomer material, large deformations, large strains, and complexcontact. In the literature, a number of FEA studies have been car-ried out on FREIs subjected to lateral loading [26,5,27,28,15,23,29,30]. In this study, 2D FEA was carried out using MSC Marc(2013) [24].

As seen in Osgooei et al. [16], the number of elements along theheight of the isolator influences modeling error. In this study, atotal of 84 and 104 four-node linear plane-strain elements wereused along the height of isolators S11 and S22, respectively. Inorder to consider the interaction of the originally vertical faces ofthe unbonded FREI with the loading surfaces, a touching contactwas defined between the mid-elastomer layers and the loadingsurfaces. In this contact model, the nodal points of the elastomerlayers were allowed to detach from the loading supports whennormal stress reached zero.

Under static loading conditions, an isotropic hyperelastic mate-rial model can be used for rubbers. For an isotropic hyperelasticmodel, the strain energy function, W , is expressed in terms of thethree invariants of the Cauchy-Green deformation tensor (eitherthe left, B, or the right, C), defined by [31]

I1 ¼ k21 þ k22 þ k23 ð1Þ

I2 ¼ k21k22 þ k22k

23 þ k23k

21 ð2Þ

I3 ¼ k21k22k

23 ð3Þ

where k1, k2, and k3 are the eigenvalues of B or C, also known as theprincipal stretches. For a compressible rubber, the strain energyfunction can be expressed by [31]

W ¼ C10ðI1 � 3� 2lnJÞ þ D1ðlnJÞ2 ð4Þwhere J ¼ ffiffiffiffi

I3p

. Alternatively, the strain energy function for the com-pressible neo-Hookean material can be expressed as

W ¼ C10ð�I1 � 3Þ þ D1ðlnJÞ2 ð5Þ

n tr (mm) h (mm) Aspect ratio

7 93.8 99.8 3.013 87.1 99.1 3.0

Table 2Vertical stiffness of isolators under pure compression, K0

v (N/mm).

FEA PS Difference (%) PA Difference (%)

S11 936 822 12.1 877 6.3S22 2280 2140 6.1 2254 1.1

P.M. Osgooei et al. / Composite Structures 144 (2016) 177–184 179

where �I1 ¼ I1J�2=3. C10 and D1 are model constants which, for consis-

tency with linear elasticity, are equal to

C10 ¼ G2

and D1 ¼ K2

ð6Þ

Using the strain energy function, the stress–strain relationship foreach deformation state can be calculated. For example, assuminga simple shear deformation state with shear strain c, the first andthird invariants are

I1 ¼ 3þ c2 and I3 ¼ 1 ð7Þand the strain energy function corresponding to simple shear defor-mation becomes the familiar

W ¼ 12Gc2 ð8Þ

The fiber-reinforcement layers were modeled using a linear-elastic isotropic material model. Quadrilateral plane-strain ele-ments were used to model the elastomer and fiber-reinforcementlayers. The top and bottom loading surfaces were modeled usingrigid wire elements. Fig. 3 shows the FEA model of the isolator withS ¼ 11.

The FREI models were subjected to an average vertical stress(i.e., the vertical load divided by the loaded area) of �p ¼ 2 MPaand were subsequently loaded to lateral displacements ofu=tr ¼ 0:25, 0.50, 0.75, 1.00, 1.25, 1.50 and 1.75. At each displace-ment amplitude, the applied lateral displacement was held con-stant, while the vertical load (P) on the isolator was subjected toa �20% change, and the variation in the vertical displacement(dv ) was monitored. The vertical stiffness of the isolator at lateraldisplacement u, Ku

v , was calculated by dividing the differencebetween the maximum and minimum values of the vertical force(Pmax and Pmin, respectively) by the difference between the maxi-mum and minimum values of the vertical displacement (dv ;max

and dv ;min, respectively)

Kuv ¼ Pmax � Pmin

dv ;max � dv;minð9Þ

The notation K0v used in the following section denotes the ver-

tical stiffness of the isolator at zero lateral displacement (i.e., underpure compression). It is noted that although slipping is possible inunbonded elastomeric bearings [32–35], in the FEA the contactbetween the elastomer is modeled with a high friction coefficient,as investigating the effect of slip is beyond the scope of this study.

3. Results and discussion

3.1. Vertical stiffness under pure compression

Table 2 shows the values of the vertical stiffness of the isolatorsunder pure compression, obtained from FEA results. The results are

Fig. 3. Finite elemen

compared against two closed-form solutions for FREIs, the PressureSolution (PS) presented by Kelly and Calabrese [15], and the Pres-sure Approach (PA) by Tsai [36]. The effective compression modu-lus (Ec) of an infinitely long strip elastomeric pad perfectly bondedto flexible reinforcement layers using the PS method is [15]:

Ec ¼ Kq

qþ c1� tanhb

b

� �ð10Þ

and q, c and b are defined in Eqs. (11), (12) and (13), respectively.

q ¼ 12GS2

Kð11Þ

c ¼ 12GtEf tf

S2 ð12Þ

b2 ¼ qþ c ð13Þwhere t is the thickness of the elastomer pad, and Ef and tf are theelastic modulus and thickness of the reinforcement layers, respec-tively. With the assumption that all elastomer layers have the samecompression modulus (neglecting the effect of boundary condi-tions), the vertical stiffness of an isolator with n layers of elastomercan be calculated by

K0v ¼ EcA

ntð14Þ

Tsai [36] derived the effective compression modulus of the ith

elastomer layer EðiÞc in a FREI bonded to rigid end plates using the

PA method:

EðiÞc ¼2lþk 1� k

kþ2lb23i�a2

0

b23i�b2

2i

!tanhb2ib

b2ibþ b2

2i�a20

b22i�b2

3i

!tanhb3ib

b3ib

" #( )

ð15Þwhere l and k are Lamé’s constants and

b22i ¼

12

a20 þ a2

2i þ a23i �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2

0 þ a22i þ a2

3iÞ2 � 4a2

0a22i

q� �ð16Þ

b23i ¼

12

a20 þ a2

2i þ a23i þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2

0 þ a22i þ a2

3iÞ2 � 4a2

0a22i

q� �ð17Þ

a0 ¼ 1t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12l

kþ 2l

sð18Þ

t model of S11.

180 P.M. Osgooei et al. / Composite Structures 144 (2016) 177–184

a1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12lð1� m2f Þ

Ef tf t

sð19Þ

a22i ¼

a21ð�f iþ1 þ 2f i � f i�1Þ

12f ið20Þ

a23i ¼

a21ðf i þ f i�1Þ

2f ið21Þ

f i ¼ 4in

1� in

� �ð22Þ

The total vertical stiffness of the isolator can be calculated using

K0m ¼

AtPn

i¼11EðiÞc

ð23Þ

It can be observed from Table 2 that the predictions of the PAassuming rigid end condition are in better agreement with theFEA results. As expected, the error between the predictions of theanalytical solutions and the FEA results reduces as the shape factorof the isolator increases. These findings are in agreement withthose in Osgooei et al. [16].

3.2. Vertical Stiffness under Lateral Displacement

Table 3 shows the FEA results of the vertical stiffness of eachisolator in both bonded and unbonded applications obtained at dif-ferent lateral displacement amplitudes. The vertical stiffness val-ues, Ku

v , are normalized with respect K0v . For both S11 and S22

and up to u=tr ¼ 1:50, the reduction in the vertical stiffness ofthe isolator is greater in the unbonded application. Atu=tr ¼ 1:75, however, an increase in the vertical stiffness ofunbonded FREI is observed, which is due to the contact of the mid-dle elastomer layers with the loading surfaces. Fig. 4 compares thedeformed shape of S11 at u=tr ¼ 1:75 in bonded (left) andunbonded (right) applications. For the unbonded isolator, three

Table 3Normalized vertical stiffness of isolators, Ku

v=K0v .

u=tr S11 S22

Bonded Unbonded Bonded Unbonded

0.25 0.98 0.90 0.97 0.940.50 0.92 0.83 0.90 0.840.75 0.82 0.69 0.80 0.771.00 0.71 0.61 0.71 0.661.25 0.59 0.54 0.61 0.571.50 0.52 0.47 0.52 0.471.75 0.42 0.56 0.44 0.69

Fig. 4. Deformed shapes of (a) bonded a

of the mid-elastomer layers are in contact with the bottom loadingsurface and thus participate in the vertical load bearing mecha-nism of the isolator. Further increase in the vertical stiffness ofthe isolator can be expected at larger lateral displacementamplitudes.

It can be observed from the values in Table 3 that for thebonded FREI, the change in the vertical stiffness at different dis-placement amplitudes is almost unaffected by the shape factor ofthe isolator. Fig. 5 plots the normalized vertical stiffness of the iso-lators at the displacement amplitudes considered in the analyses.For the bonded isolators, the reduction in the vertical stiffness var-ies almost linearly with lateral displacement. As suggested in [25]for bonded SREIs, the vertical stiffness under lateral displacementcan be estimated by considering an effective overlapping areaunder compression, as shown in Fig. 6, and neglecting the contri-bution of the two triangle-shaped areas in the vertical stiffness ofthe isolator. Extending the method to bonded FREIs, the verticalstiffness of a strip isolator under lateral displacement u can be esti-mated from

Kuv ¼ K0

v 1� u2b

� �ð24Þ

nd (b) unbonded S11 at u=tr ¼ 1:75.

Fig. 5. Normalized effective vertical stiffness of isolators at different lateraldisplacement amplitudes.

Fig. 6. The effective overlapping area of bonded isolators.

Table 4Predicted normalized values for vertical stiffness of bonded FREIs using the effectiveoverlapping area, compared with FEA results.

u=tr S11 S22

Kuv=K

0v

Difference with FEA (%) Kuv=K

0v

Difference with FEA (%)

0.25 0.92 �5.9 0.93 �4.40.50 0.84 �8.3 0.85 �5.00.75 0.77 �6.6 0.78 �2.21.00 0.69 �3.2 0.71 0.01.25 0.61 3.2 0.64 4.41.50 0.53 2.1 0.56 8.61.75 0.45 7.8 0.49 11.8

P.M. Osgooei et al. / Composite Structures 144 (2016) 177–184 181

Table 4 compares the values of the vertical stiffness of S11 andS22 in a bonded application, calculated using Eq. (24), as well asthe corresponding error with respect to the FEA results. For bothisolators, the predictions of Eq. (24) are lower than the FEA resultsfor u=tr < 1:00, and higher than the FEA results for u=tr > 1:00,although the maximum difference is less than 12%.

Eq. (24) predicts that the vertical stiffness of the isolatorbecomes zero at u ¼ 2b. For bonded isolators, designers considera limit on the minimum effective area [37]. Assuming a 40% limitfor the effective overlapping area and using Eq. (24),Kuv=K

0v ¼ 0:4 ¼ 1� u=ð2bÞ, and with the values for 2b and tr from

Table 1, the maximum allowable isolation displacement for S11and S22 would be u=tr ¼ 1:92 and u=tr ¼ 2:07, respectively. Forunbonded isolators S11 and S22, the vertical stiffness of the isola-tor reaches a minimum value of Ku

v=K0v ¼ 0:47 at u=tr ¼ 1:50, and

increases with an increase in the lateral displacements. This fea-ture of unbonded FREI allows larger lateral displacement ampli-tudes to be considered for an unbonded isolator, as they have

Fig. 7. Contours of normalized r22 (i.e., r22=�p) stres

Fig. 8. Contours of normalized r22 (i.e., r22=�p) stres

been tested up to u=tr ¼ 3:00 [38] while maintaining sufficient ver-tical load bearing capacity.

3.3. Stress distribution

Figs. 7 and 8 show the contours of normalized r22 (i.e., r22=�p,where r22 is the normal stress in the vertical direction, and�p ¼ 2 MPa) in the S11 and S22 isolators, respectively. The stresscontours are shown for u=tr ¼ 0 (i.e., under pure compression),1.00, and 1.75. As the isolators undergo lateral displacement, stressconcentration is observed at the corners of the isolators. For thebonded isolators, an increase in the lateral displacement resultsin a decrease in the effective overlapping area, thus increasingthe peak values of r22 in the center of the isolator. Also, for thebonded isolators under lateral displacement, the regions outsidethe effective overlapping areas experience tensile r22 stresses,which develop in order to equilibrate the unbalanced moment thatis generated when the isolator is sheared [39]. For unbonded FREIunder lateral displacement, however, the regions that experiencetensile r22 stresses are very small, and the values of these tensilestresses are significantly smaller than for the bonded case. In anunbonded FREI, a larger portion of the isolator experiences com-pressive stresses, compared to a bonded FREI. As a result, theincrease in the peak r22 stress values in the center of the unbondedFREI as the isolator is displaced is considerably lower than those inthe bonded FREI.

3.4. Vertical displacement

Figs. 9 and 10 plot the variations in the normalized downwardvertical displacement (dv=tr) of S11 and S22, respectively, underdifferent lateral displacement amplitudes u=tr . Note that the value

s distribution in S11 at u=tr ¼ 0, 1.00, and 1.75.

s distribution in S22 at u=tr ¼ 0, 1.00, and 1.75.

Fig. 9. Variation of downward vertical displacement of S11 in (a) a bonded, and (b) an unbonded application, at different lateral displacement amplitudes.

Fig. 10. Variation of downward vertical displacement of S22 in (a) a bonded, and (b) an unbonded application, at different lateral displacement amplitudes.

182 P.M. Osgooei et al. / Composite Structures 144 (2016) 177–184

of dv=tr at u=tr ¼ 0 is due to pure compression under an averagevertical stress of �p ¼ 2 MPa. For the bonded isolators, the down-ward vertical displacement increases as the applied lateral dis-placement of the isolator increases. The vertical displacementfollows a parabolic trend as a function of lateral displacement.For S11 for example, the vertical displacement at lateral displace-ments of u=tr ¼ 1:00 and u=tr ¼ 1:75 compared with the verticaldisplacement under pure compression shows 90% and 372%increase, respectively. For the unbonded FREI, the geometry changeof the isolator due to the rollover deformation tends to decreasethe downward vertical displacement as the isolator undergoes lat-eral displacement. Thus, the variation in the vertical displacementof isolators under different lateral displacement amplitudes is con-siderably lower compared to the bonded isolators. For S11 and S22,the maximum variation in the vertical displacement of the isolator

(a)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0

2 MPa4 MPa6 MPa

Fig. 11. Variation of normalized effective stiffness of S11 in (a) a bonded, and (b)

at different lateral displacement amplitudes compared to the ver-tical displacement under pure compression load is 23% and 49%,respectively. The same trend (not shown here), whereby the down-ward displacement is consistently larger for the bonded isolatorcompared to the unbonded one, was also observed for isolatorS11 under an average vertical stress of 4 and 6 MPa.

3.5. Influence of applied vertical stress

Fig. 11 compares the normalized effective stiffness of S11 calcu-lated under average vertical stress values of �p ¼ 2, 4, and 6 MPa forbonded (Fig. 11(a)) and unbonded (Fig. 11(b)) applications. Thevertical stiffness values are normalized with respect to the effec-tive stiffness value under pure compression (K0

v) predicted using

(b)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0

2 MPa4 MPa6 MPa

an unbonded application, under different average compression stress values.

P.M. Osgooei et al. / Composite Structures 144 (2016) 177–184 183

the PA method (Table 2). It can be observed that as the applied ver-tical stress increases, the vertical stiffness of the bearings increases.This stiffening behavior is induced by the geometric nonlinearitycaused by larger bulging of elastomer layers as well as the shorten-ing of the elastomer layers as the vertical load increases. Under dif-ferent vertical compression loads, the same overall trend in thevertical stiffness (reduction in bonded isolators and reductionand then increase in unbonded isolators) is observed as the isolatorundergoes lateral displacement. As the lateral displacementincreases, the overall reduction in the vertical stiffness of the isola-tor decreases with an increase in the vertical compression load. Atu=tr ¼ 1:00, for example, the vertical stiffness of the bonded isola-tor shows 29%, 24% and 16% reduction compared to the verticalstiffness under pure compression load, for �p ¼ 2, 4, and 6 MPa,respectively. For unbonded application, the reduction in the verti-cal stiffness is 39%, 35%, and 31%, for �p ¼ 2, 4, and 6 MPa,respectively.

4. Conclusions

In this paper, FEA is carried out on two strip-shaped FREIs toinvestigate the variation in the vertical stiffness of the isolatorsunder different lateral displacement amplitudes. Two shape factorvalues of S ¼ 11 (S11) and S ¼ 22 (S22) were considered for the iso-lators and their vertical compression response was studied in bothbonded and unbonded applications. Under pure compression load,the FEA results were compared with the values from two analyticalsolutions: the pressure solution, presented by Kelly and Calabrese[15], and the pressure approach, presented by Tsai [36]. It wasfound that the predictions of the PA method are in better agree-ment with the FEA results. The error between the predictions ofthe analytical solutions and the FEA results decreases as the shapefactor of the isolator increases. In general, the FEA results were ingood agreement with the results from analytical solutions, with amaximum discrepancy of 12.1% for S11 and 6.1% for S22.

FEA results showed that for bonded isolators, the vertical stiff-ness decreases as the lateral displacement increases. This reduc-tion in the vertical stiffness of bonded FREIs can be predictedusing the effective overlapping area method. A similar trend isobserved for FREIs in unbonded application up to 150% sheardeformation. For u=tr > 1:50, an increase is observed in the verticalstiffness of unbonded isolators. This increase is due to the contactof mid-elastomer layers with the loading support surfaces due tothe rollover, and is beneficial as it allows considering larger isola-tion displacement values for design.

For the bonded isolators, the increase in the lateral displace-ment results in an increase the peak values of r22 in the centerof the isolators, and the regions outside the effective overlappingareas experience appreciable tensile r22 stresses. However, for anunbonded FREI under lateral displacement, the tensile r22 stressesthat develop are very low and confined in a small region. Theincrease in peak compressive r22 stress in the central region ofthe unbonded isolator due to increasing lateral displacement isconsiderably lower than in the bonded isolators.

Under a constant vertical load, as the applied lateral displace-ment increases, the vertical displacement of the bonded isolatorincreases with a parabolic trend. The variation in the vertical dis-placement under different lateral displacement amplitudes is con-siderably lower in the unbonded application relative to the bondedapplication. This is attributed to the fact that under rollover defor-mation, the isolator tends to push the loading surfaces against thedirection of the applied vertical load.

The response of the isolator with the shape factor of S ¼ 11 wasinvestigated under three vertical load values, corresponding toaverage vertical pressure values of �p ¼ 2, 4, and 6 MPa. It was

observed that as the applied vertical compression load increases,the vertical stiffness of the isolator in both bonded and unbondedapplications increase. As the lateral displacement increases, theoverall reduction in the vertical stiffness of the isolator decreaseswith an increase in the vertical compression load.

The present study used 2D FEA to investigate the variation ofthe vertical properties of FREIs subjected to lateral loading. Thecharacterization of the vertical-lateral response interaction inunbonded and bonded FREIs of various geometries requires furtherinvestigation using 3D FEA and experimental testing.

Acknowledgments

This research was carried out as part of the mandate of the Cen-tre for Effective Design of Structures (CEDS) at McMaster Univer-sity and is partially funded by the Ontario Ministry of EconomicDevelopment and Innovation and by the Natural Sciences andEngineering Research Council of Canada (NSERC).

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