Engineering Structures 40 (2012) 198–204

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Engineering Structures

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Critical load of slender elastomeric seismic isolators: An experimental perspective

Donatello Cardone ⇑, Giuseppe PerroneDiSGG, University of Basilicata, Via Ateneo Lucano 10, 85100 Potenza, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 July 2011Revised 31 January 2012Accepted 5 February 2012Available online 28 March 2012

Keywords:Seismic isolationElastomeric bearingsCritical loadShear strainBuckling

0141-0296/$ - see front matter � 2012 Elsevier Ltd. Adoi:10.1016/j.engstruct.2012.02.031

⇑ Corresponding author. Tel.: +39 0971205054.E-mail addresses: donatello.cardone@unibas.it (D

alice.it (G. Perrone).

One of the most important aspects of the seismic response of elastomeric isolators is their stability underlarge shear strains. The bearing capacity of elastomeric isolators, indeed, progressively degrades whileincreasing horizontal displacement. This may greatly influence the design of elastomeric isolators, espe-cially in high seismicity regions, where slender elastomeric isolators subjected to large horizontal dis-placements are a common practice. In the current design approach the critical load is evaluated basedon the Haringx theory, modified to account for large shear strains by approximate correction factors.

In this paper the critical behavior of a pair of slender elastomeric devices is experimentally evaluated atdifferent strain amplitudes, ranging from approximately 50% to 150%. The experimental results are thencompared to the predictions of a number of semi-empirical and theoretical formulations.

The main conclusion of this study is that current design approaches are overly conservative for slenderelastomeric seismic isolators, since they underestimate their critical load capacity at moderate-to-largeshear strain amplitudes.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

An elastomeric isolation bearing consists of a number of rubberand steel layers mutually vulcanized, to provide high stiffness inthe vertical direction together with large deformability in the hor-izontal direction. The elastomeric isolators work like a filter length-ening the fundamental period of vibration of the structure, thusreducing the seismic effects (interstory drifts, floor accelerations,stresses in the structural members, etc.) generated in the super-structure. However, this reduction is accompanied by large hori-zontal displacements in the isolators, which may significantlyreduce their axial load capacity [1–4].

The earliest theoretical approach for the evaluation of the criticalaxial load of rubber bearings was introduced by Haringx [5], consid-ering the mechanical characteristics of helical steel springs and rub-ber rods. Same assumptions have been made later by Gent [6]considering multilayered rubber compression springs. Basically,the Haringx’s theory is based on a linear one-dimensional beammodel with shear deformability, within the hypothesis of smalldisplacements. The critical buckling load of elastomeric seismicisolators is expressed as:

Pcr;0 ¼2 � PE

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4p2

ðEIÞeff

ðGAsÞeff � L2

s ð1Þ

ll rights reserved.

. Cardone), giuseppe.perr@

in which: ðGAsÞeff and ðEIÞeff are the effective shear rigidity and effec-tive flexural rigidity, respectively, of the elastomeric isolators, com-puted based on the bending modulus (E) and dynamic shearmodulus (Gdyn) of rubber, moment of inertia of the bearing aboutthe axis of bending (I) and bonded rubber area (As);

PE is the Euler load for a standard elastic column:

PE ¼p2 � ðEIÞeff

L2 ð2Þ

L is the total height of rubber layers and steel plates excluding topand bottom connecting steel plates.

Various authors proposed different relations to evaluate theeffective shear and flexural rigidity of laminated rubber bearings.In this paper, reference to the formula derived by Buckle and Kelly[1], Koh and Kelly [2] has been made:

ðGAsÞeff ¼ Gdyn � As �Lte

ð3Þ

ðEIÞeff ¼ ErI �Lte

ð4Þ

where te is the total thickness of the rubber layers and Er is the elas-tic modulus of the rubber bearing evaluated based on the primaryshape factor S1 and rubber Young’s modulus E0 as:

Er ¼ E0ð1þ 0:742 � S21Þ ð5Þ

The primary shape factor S1 is defined as the ratio between theloaded area of the bearing and the area free to bulge of the singlerubber layer (S1 � D/4ti for circular bearings, where D is the diameter

u

h

P

P

F

u

Ar

D

(a)

(b)

Fig. 1. (a) Schematic deformed shape of an elastomeric bearing subjected to shearand compression; (b) effective cross section area as a function of sheardisplacement.

Table 1Elastomeric bearings details.

Outer diameter De (mm) 200Inner diameter D (mm) 180Rubber layer thickness ti (mm) 8Number of rubber layers nti 10Steel shim thickness ts (mm) 2Number of steel shims nti 9Total height of rubber te (mm) 80Primary shape factor S1 5.63Secondary shape factor S2 2.25

D. Cardone, G. Perrone / Engineering Structures 40 (2012) 198–204 199

of the isolator and ti the thickness of a single rubber layer). The rub-ber Young’s modulus E0 is usually taken equal to 3.3 Gdyn to 4 Gdyn.

The Haringx’s theory has been later applied by Naeim and Kelly[7], with a series of simplified assumptions, for commercial elasto-meric seismic isolators. According to Naeim and Kelly [7], the criticalbuckling load of elastomeric seismic isolators can be expressed interms of the primary and secondary shape factors S1 and S2, the latterbeing defined as the ratio between the maximum dimension of thecross section of the isolator and the total height of rubber. For circu-lar elastomeric isolator, for instance, Naeim and Kelly [7] provides:

Pcr;0 ¼p

2ffiffiffi2p � ðGAsÞeff � S1 � S2 ð6Þ

Subsequently, Kelly [8] derived a more refined formulation of thebuckling load of elastomeric isolator:

Pcr;0 ¼p

2ffiffiffi3p � ðGAsÞeff �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:742 � E0

G

r� S1 � S2 ð7Þ

The secondary shape factor S2 is defined as the ratio between thebearing maximum dimension and the total thickness of all the rub-ber layers (S2 � D/te for circular bearings, where te is the total thick-ness of all the rubber layers). It is interesting to note that the criticalbuckling load capacity evaluated considering the expressions (6)and (7), differs by 10–20%, depending on the value (between 3.3Gand 4G) assumed for the rubber Young’s modulus.

Lanzo [9] modified the Haringx’s expression by taking into ac-count the axial stiffness of the rubber bearing (EA)eff:

Pcr;0 ¼2 � PE

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4p2 � EIð Þeff

EAð Þeff �L2 þEIð Þeff

GAsð Þeff �L2

� �s ð8Þ

where ðEAÞeff is the effective axial stiffness of the rubber bearing,evaluated as:

ðEAÞeff ¼ Er � A �Lte

ð9Þ

In Italy, the current design approach [10] refers to a formulation ofthe critical load similar to (but more conservative than) that ini-tially proposed by Naeim and Kelly [7] (see Eq. (6)):

Pcr;0 ¼ Gdyn � As � S1 � S2 ð10Þ

where Gdyn is the dynamic shear modulus derived from the qualifi-cation tests of the elastomeric isolator.

More recently, a less conservative variant of the Naeim andKelly formulation has been adopted in the new European StandardEN11529 [11]:

Pcr;0 ¼ 1:3 Gdyn � As � S1 � S2 ð11Þ

For all the above mentioned formulations, the buckling load at thetarget shear displacement (u) is evaluated as a function of the ratiobetween the effective area of the inner shim plate (A) and the over-lap area of the displaced bearing (Ar) (see Fig. 1):

Pcr ¼ Pcr;0 �Ar

Að12Þ

For circular bearings, for instance, the overlap area at the targetdisplacement is given by:

Ar ¼ ðu� sin uÞ � D2

4ð13Þ

with

u ¼ 2 arccosuD

� �ð14Þ

Several experimental studies of the buckling behavior of elasto-meric seismic isolators have been carried out in the past

[12,1,3,4,13]. In this paper, the critical load of a couple of slender(low shape factors) elastomeric bearings is experimentally evalu-ated. Test set-up and experimental program are presented first indetail. Then, the experimental results are compared with the pre-dictions of the theoretical formulations presented in the previousparagraph.

2. Experimental tests

2.1. Test specimens

Test specimens are a couple of 1:2 scaled circular elastomericbearings with 200 mm diameter and 10 rubber layers with 8 mmthickness. Bearing geometrical properties are summarized inTable 1.

The mechanical properties of the specimens have been derivedfrom a number of standard cyclic tests, performed in accordancewith the test procedure prescribed in the Italian seismic code[10] for the qualification of elastomeric bearings. The static shearmodulus (Gstat), in particular, has been derived from a quasi-statictest consisting of five cycles at 0.1 Hz frequency of loading and100% shear strain amplitude. According to the NTC 2008 [10], Gstat

(a)

(b)

Fig. 2. Hysteresis cycles of the test specimens at 100% shear strain amplitudeduring (a) quasi-static test for the evaluation of Gstat and (b) dynamic test for theevaluation of Gdyn and neq.

Fig. 3. Test set-up.

200 D. Cardone, G. Perrone / Engineering Structures 40 (2012) 198–204

is defined as the slope of the shear–strain curve between 27% and58% shear deformation with reference to the third cycle of the test(see Fig. 2a). In the case under consideration, a value of Gstat =0.113 MPa has been obtained. The dynamic shear modulus (Gdyn)and the equivalent damping (neq) have been derived from adynamic test consisting of five cycles at 0.5 Hz frequency of loadingand 100% shear strain amplitude with reference to the third cycleof the test (see Fig. 2b). According to the NTC2008 [10], Gdyn isdefined as the secant stiffness to the origin of the axis of the cycleof maximum strain amplitude, neq is evaluated as the ratio betweenthe energy dissipated in one complete cycle (equal to the areaenclosed in the cycle) and the strain energy stored at the maximumstrain amplitude (see Fig. 2b). In the case under consideration, avalue of Gdyn = 0.37 MPa and neq = 16% have been obtained.

2.2. Test set-up

The specimens have been tested using the uniaxial bearing testfacility available at the Laboratory of Structures at the University ofBasilicata (see Fig. 3). The test apparatus constitutes a self-balanced system designed for cyclic testing of a pair of identicalcounteracting specimens. The apparatus subjects the test speci-mens to axial loads in the horizontal direction and shear forcesand displacements in the vertical direction. The test rig consistsof two steel stiff beams connected by three steel columns. Two col-umns are linked to the base beam through a double pendulumhinge while the third column (on the left in Fig. 3) by a simplehinge. This has been purposely done to preserve the load directions

and the co-planarity of the bases of the specimens during the test.The hinge connections accommodate the axial displacementresulting from the axial load being applied to the specimens.

The shear displacement is vertically applied by a Schenkhydraulic actuator, connected to the specimens through a T-shaped steel plate. The actuator is served by two pumps, each with80 l/min maximum capacity, and equipped with two servo-valveswith maximum capacity of 63 l/min each. The Shenck actuatorcan apply 250 kN maximum force and ±125 mm maximum dis-placement. The force of the actuator is measured by a load cellmounted on the top of the cylinder. The displacement of the actu-ator is measured and controlled by an internal transducer.

The compression load is horizontally applied and kept constantduring the tests by an Enerpac double-effect hydraulic jack (seeFig. 3). The hydraulic jack is served by a pump, with 5 l/min capac-ity, and equipped with an electro-valve with 5 l/min maximumcapacity. The hydraulic jack can apply 300 kN maximum force.The force of the hydraulic jack is monitored and measured by anumber of load cells of different capacity mounted on the top ofthe cylinder.

The shear displacements of the specimens have been measuredby two linear transducers connected to the T-shaped steel plate.The axial displacements of the specimens have been measuredby a couple of linear transducers, positioned between the cornerof the outer plates of the two specimens, in order to evaluate pos-sible rotations of the specimens around the out-of-plane axis.

0

45

90

0

u (m

m)

0

150

300

0

P (K

N)

Fig. 4. Steps of a typical stability test (for more details see Table 2).

D. Cardone, G. Perrone / Engineering Structures 40 (2012) 198–204 201

The acquisition frequency has been set equal to 10 Hz. The sig-nals recorded during the tests have been filtered, amplified andconverted from analog signals to 16 bit digital signals in real timeby a proper signal processing software.

2.3. Test procedure

The testing procedure followed in this study is similar, to someextent, to that adopted by Buckle et al. [13]. All the stability tests,in particular, were performed on the same couple of specimens, atdifferent levels of shear deformations.

Basically, each test consisted in the application of two consecu-tive ramps of axial load with a ramp of shear displacementbetween them (see Fig. 4). More precisely, each test can be dividedin the following six steps:

1. Applying an initial axial load of 150 kN, corresponding to adesign compression stress of 6 MPa.

2. Applying a given target displacement (hence shear deformation).3. Increasing the axial load until the horizontal force became

negative.4. Decreasing the axial load to the initial value of 150 kN.5. Decreasing the initial displacement to 0.6. Decreasing the axial load to 0.

Both forces and displacements have been applied very slowlyduring the tests, in order to avoid dynamic effects. Moreover, an

Table 2Stability test details (see Fig. 4).

Test no. u1 (mm) c (%) P1 (kN) P2 (kN) Step 1 Step 2

t1 (s) t2 (s) t3 (s)

1 48 60 150 280 15 25 352 64 80 150 280 15 25 383 80 100 150 280 15 25 424 96 120 150 280 15 25 455 112 140 150 200 15 25 486 125 160 150 200 15 25 51

interval of 10 s was allowed to elapse between one step and thenext. The test control has been carried out manually in order toprevent, in real time, possible damages to either specimens or testapparatus.

A total number of 6 stability test have been performed (seeTable 2), while increasing the target shear displacement from 48to 125 mm, corresponding to six different shear strain amplitude,equal to 60%, 80%, 100%, 120%, 140% and 160%, respectively.

2.4. Test results

The first direct result of the critical load tests is the time historyof the shear force F resulting from the application of an increasingaxial load P under a given shear displacement u. As an example,Fig. 5 shows the shear force–time history derived from Test 3(see Table 2), featuring an imposed shear deformation of 100%.From Fig. 5 it is evident that as the axial force P is increased, theshear force F decreases until it becomes negative, while the sheardisplacement remains constant.

The experimental outcomes have been employed to evaluatethe critical buckling load for each imposed shear deformation. Itis worth observing that the critical load cannot be simply esti-mated as the value of the axial load at which the horizontal shearforce is zero because this would give a ‘‘constrained’’ evaluation ofthe critical load, since the critical load is approached while thebearing is locked against further horizontal displacements. In thecommon practice (e.g. during a seismic event), instead, the hori-zontal displacement is unconstrained and free to increase, as nec-essary, when buckling takes place. Reference to the procedureproposed by Nagarajaiah and Ferrell [4], based on equilibriumpaths, has been made to determine the unconstrained critical load.

Fig. 6 shows the variation of the shear force F with increasingaxial load P, for each test performed at a given shear deformationc = 60%, 80%, 100%, 120%, 140% and 160%.

The diagram of Fig. 6 has been entered alternatively with anumber of F-lines and P-lines to determine a series of equilibriumpaths (i.e. a number of smooth curves passing through discretepoints) corresponding to given levels of shear force (Fig. 7) andaxial load (Fig. 9), respectively. The equilibrium paths are unstablepast a limit point, which is the critical load Pcr. This observation canbe exploited to estimate the critical load of the rubber bearings un-der consideration, as explained below.

Fig. 7 shows the axial load vs. shear strain curves, correspondingto different levels of shear force in the specimens. The critical loadcan be identified with the value of the axial load that must be re-duced, to maintain constant the shear force in the specimens byfurther increments of shear displacement. As expected, the criticalload decreases while increasing the horizontal displacement, henceshear strain, applied (see Fig. 8).

Fig. 9 shows the shear force vs. shear strain curves, correspond-ing to different values of axial load applied to the specimens. Ascan be seen, each curve passes through a maximum as the horizon-tal displacement increases, under constant axial load. The shear

Step 3 Step 4 Step 5 Step 6

t4 (s) t5 (s) t6 (s) t7 (s) t8 (s) t9 (s) t10 (s) t11 (s)

45 58 68 81 91 101 111 12648 61 71 84 94 108 118 13352 65 75 88 98 114 124 13955 68 78 91 101 121 131 14658 63 73 78 88 112 122 13761 66 76 81 91 117 127 142

0

25

50

75

100

125

0 50 100 150 200 250 300 350u

(m

m)

t (s)

0

100

200

300

0 50 100 150 200 250 300 350

P (

KN

)

t (s)

-20

-10

0

10

20

0 50 100 150 200 250 300 350

F (

KN

)

t (s)

(a)

(b)

(c)

Fig. 5. Typical results of stability test: shear displacement, axial load and shear force time histories.

135

155

175

195

215

235

0 1 2 3 4 5 6 7

P (

kN)

F (kN)

F =

2.2

kN

P = 162 kN

Fig. 6. Axial load vs. shear force variations as a function of the imposed shear strainamplitude.

140

160

180

200

220

40 80 120 160

Axi

al lo

ad (

kN)

Shear strain (%)

F=0.2KN F=1.3KN

F=4.5KN

F=1.5KN F=2.2KN

Fig. 7. Axial load vs. shear strain curves as a function of shear force.

202 D. Cardone, G. Perrone / Engineering Structures 40 (2012) 198–204

force and shear strain at which the maximum occurs decrease withincreasing axial load. Finally, the horizontal tangent stiffness tendsto zero for shear strains lower than the maximum test amplitude;moreover it decreases while increasing axial load and horizontaldisplacement.

2.5. Comparison between analytical and experimental results

Fig. 10 compares the experimental values of Pcr derived follow-ing the ‘‘equilibrium path’’ procedure described in the last section,to the numerical results provided by the formula proposed byHaringx [5] (Eq. (1)), Naeim and Kelly [7] (Eq. (6)), Kelly [8] (Eq.(7)), Lanzo [9] (Eq. (8)), and those adopted in the Italian seismiccode [10] (Eq. (10)) and in the EN15129 European standard [11]

0

100

200

300

60 80 100 120 140 160

Pcr

(kN

)

Shear strain (%)

experimental values

Fig. 8. Critical load as a function of shear strain amplitude.

0

1

2

3

4

5

0 40 80 120 160

She

ar fo

rce(

kN)

Shear strain(%)

Fig. 9. Shear force–strain curves as a function of axial load.

0

100

200

300

60 80 100 120 140

Pcr

(kN

)

Shear strain(%)

Fig. 10. Comparison between experimental results and critical loads expectedbased on different theoretical and semi-empirical formulations.

D. Cardone, G. Perrone / Engineering Structures 40 (2012) 198–204 203

(Eq. (11)). Among the theoretical formulations considered, theformula proposed by Haringx leads to the most conservative

results, while the formula proposed by Kelly provides the highestvalues of critical load. As can be seen, for the slender elastomericbearings under consideration (with secondary shape factor lessthan 3 and rubber thickness of 8 mm), the experimental valuesof Pcr result significantly greater than those predicted by the theo-retical formulas, especially at large shear amplitudes (100–150%)where values approximately 2.5–3.5 times greater than expectedare found. The relationship adopted in the current Italian seismiccode appears to be too conservative for low shape factors elasto-meric bearings (S1 < 12, S2 < 3), such as those considered in thisstudy. The formulation of Pcr proposed in the new EN11529 Euro-pean Standard, being less conservative and fully compatible withmany theoretical formulations, seems to be more suitable for slen-der elastomeric isolators.

3. Conclusions

In this paper the critical behavior of a couple of slender elasto-meric seismic isolators has been experimentally evaluated at dif-ferent strain amplitudes, ranging approximately from 50% to160%. The tests were performed with the specific objectives offinding the effect of shear strain on the critical load of slender(low shape factors) elastomeric seismic isolators and to evaluatethe existing design approaches. During the tests, the shear dis-placement of the specimens was held at a specified value whilethe axial load was progressively increased until critical load condi-tions occurred. For the purpose of this experiment, the isolator wasconsidered to be in critical state when the horizontal force becamezero or negative. The critical load has been evaluated, for eachstrain amplitude, based on a series of equilibrium paths derivedfrom the experimental results.

Based on the results of this study the following conclusions canbe drawn:

(i) The critical load decreases with increasing shear strain.(ii) The horizontal stiffness decreases with increasing axial load

and horizontal displacement.(iii) Current design procedures seem to be overly conservative

for slender (low shape factor) elastomeric isolators becausethey underestimate the experimentally determined criticalload by a factor of approximately 1.5 at low shear strainamplitudes and approximately 3.5 at large shear strainamplitudes.

In the near future, it is desirable to extend this experimentalinvestigation considering more specimens, characterized by differ-ent shape factors and/or different rubber thickness, in order toevaluate the influence of such parameters on the critical behaviorof slender elastomeric seismic isolators.

Acknowledgements

The authors are grateful to Domenico Nigro (University of Basil-icata) for his valuable help in setting up the testing apparatus andexecuting the experimental tests. This work has been partiallyfunded by MIUR, COFIN 2007.

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