A polynomial analytical model of rubber bearings based on series of tests Igor Gjorgjiev ⇑ , Mihail Garevski 1 Institute for Earthquake Engineering and Engineering Seismology-IZIIS, ‘‘Ss Cyril and Methodius’’ University, Salvador Aljende 73, P.O. Box 101, 1000 Skopje, The Former Yugolav Republic of Macedonia article info Article history: Received 6 September 2012 Revised 27 February 2013 Accepted 9 April 2013 Available online 29 June 2013 Keywords: Rubber bearings Production of bearings Testing of bearings Polynomial model abstract Rubber bearings are among the most frequently applied devices in seismic isolation. Although the behav- ior of rubber bearings under strong earthquakes is nonlinear, it is often interpreted through the bilinear constitutive law. The investigations presented in this paper represent a successful attempt to simulate nonlinear force–displacement relationships. The first part of the paper covers production and testing of rubber bearings and results obtained. The second part deals with development of a simple nonlinear mathematical model of a rubber bearing involving a polynomial function and eight additional parameters obtained from biaxial tests. The polynomial model can simulate the behavior of natural rubber bearings in case of small and large deformations. The model is capable of covering the strengthening of the rubber in conditions of large deformations, including the loading history effect. Based on comparison between the analytical and experimental results, it is concluded that the proposed polynomial model is capable enough to simulate the force–displacement relationship of rubber bearings. Ó 2013 Published by Elsevier Ltd. 1. Introduction One of the most effective techniques for design of earthquake resistant structures is application of seismic isolation. It is a collec- tion of isolation units, which are to reduce the transfer of seismic energy to the upper structure. Over the last century, many isolation devices have been invented, but only a few of them have become popular. Out of these, rubber bearings are commonly used for iso- lation of buildings and bridges. The first application of a rubber iso- lation system took place in the sixties of the last century. A primary school in Macedonia was isolated by a system referred to as Swiss Full Base Isolation [22]. The basic concept of this system was pro- viding a full 3D isolation by use of non-reinforced rubber bearings. Seismic isolation has practically become a reality with the develop- ment of multi-layered elastomeric bearings produced by simulta- neous vulcanization of rubber and gluing of steel plates. The investigations done by Kelly [15,16], Naeim and Kelly [20] and Skinner et al. [21] have given a huge contribution to acquiring knowledge on the behavior of base isolated structures and differ- ent types of bearings. The application of base isolation of structures has been increased particularly after the Kobe (1995) earthquake [6], when good behavior of base isolated structures was observed [7]. Unfortunately, due to the high cost of the isolators, their appli- cation has long been limited to economically developed countries. However, today, even developing countries like Macedonia [8,9], and Armenia [18] are successfully producing and applying these devices. To promote application of base isolation in R. Macedonia and the remaining countries in the Balkan, the Institute of Earthquake Engineering and Engineering Seismology (IZIIS) proposed a project on development of low-cost rubber bearings [9], which was fi- nanced by NATO through the Science for Peace (SfP) Programme. Within this project, a number of raw rubber recipes and a technol- ogy for production of rubber bearings were developed for the first time in the region. Within this project, ample experimental tests were performed on bearings produced of 18 different rubber compounds. These tests enabled creation of a large database of force–displacement relationships for different rubber compounds that was later used for development of a new analytical model of rubber bearings. The laboratory tests on rubber bearings which have been car- ried out within these investigations and the investigations per- formed by other authors [13,17,23], have pointed out their nonlinear behavior. The stress–strain behavior of rubber bearings is manifested by high horizontal stiffness under low shear strains, low stiffness under moderate strains, and an increasing shear mod- ulus under higher strains. Usually, rubber hardening begins at 125% shear strain and continues until failure. Lateral bearing behavior is quite complex because of strain-rate dependence and the presence of the Mullins effect [19], [12]. Other factors that have an influence upon the hysteretic behavior of 0141-0296/$ - see front matter Ó 2013 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.engstruct.2013.04.006 ⇑ Corresponding author. Address: Department for Geotechnics and Special Structures, Institute for Earthquake Engineering and Engineering Seismology-IZIIS, ‘‘Ss Cyril and Methodius’’ University, Salvador Aljende 73, P.O. Box 101, 1000 Skopje, The Former Yugolav Republic of Macedonia. Tel.: +389 2 3107 701. E-mail addresses: [email protected](I. Gjorgjiev), garevski@pluti. iziis.ukim.edu.mk (M. Garevski). 1 Tel.: +389 2 3107 701. Engineering Structures 56 (2013) 600–609 Contents lists available at SciVerse ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Transcript
Engineering Structures 56 (2013) 600–609
Contents lists available at SciVerse ScienceDirect
A polynomial analytical model of rubber bearings based on series of tests
0141-0296/$ - see front matter � 2013 Published by Elsevier Ltd.http://dx.doi.org/10.1016/j.engstruct.2013.04.006
⇑ Corresponding author. Address: Department for Geotechnics and SpecialStructures, Institute for Earthquake Engineering and Engineering Seismology-IZIIS,‘‘Ss Cyril and Methodius’’ University, Salvador Aljende 73, P.O. Box 101, 1000Skopje, The Former Yugolav Republic of Macedonia. Tel.: +389 2 3107 701.
Institute for Earthquake Engineering and Engineering Seismology-IZIIS, ‘‘Ss Cyril and Methodius’’ University, Salvador Aljende 73, P.O. Box 101, 1000 Skopje, The FormerYugolav Republic of Macedonia
a r t i c l e i n f o
Article history:Received 6 September 2012Revised 27 February 2013Accepted 9 April 2013Available online 29 June 2013
Keywords:Rubber bearingsProduction of bearingsTesting of bearingsPolynomial model
a b s t r a c t
Rubber bearings are among the most frequently applied devices in seismic isolation. Although the behav-ior of rubber bearings under strong earthquakes is nonlinear, it is often interpreted through the bilinearconstitutive law. The investigations presented in this paper represent a successful attempt to simulatenonlinear force–displacement relationships. The first part of the paper covers production and testing ofrubber bearings and results obtained. The second part deals with development of a simple nonlinearmathematical model of a rubber bearing involving a polynomial function and eight additional parametersobtained from biaxial tests. The polynomial model can simulate the behavior of natural rubber bearingsin case of small and large deformations. The model is capable of covering the strengthening of the rubberin conditions of large deformations, including the loading history effect. Based on comparison betweenthe analytical and experimental results, it is concluded that the proposed polynomial model is capableenough to simulate the force–displacement relationship of rubber bearings.
� 2013 Published by Elsevier Ltd.
1. Introduction
One of the most effective techniques for design of earthquakeresistant structures is application of seismic isolation. It is a collec-tion of isolation units, which are to reduce the transfer of seismicenergy to the upper structure. Over the last century, many isolationdevices have been invented, but only a few of them have becomepopular. Out of these, rubber bearings are commonly used for iso-lation of buildings and bridges. The first application of a rubber iso-lation system took place in the sixties of the last century. A primaryschool in Macedonia was isolated by a system referred to as SwissFull Base Isolation [22]. The basic concept of this system was pro-viding a full 3D isolation by use of non-reinforced rubber bearings.Seismic isolation has practically become a reality with the develop-ment of multi-layered elastomeric bearings produced by simulta-neous vulcanization of rubber and gluing of steel plates. Theinvestigations done by Kelly [15,16], Naeim and Kelly [20] andSkinner et al. [21] have given a huge contribution to acquiringknowledge on the behavior of base isolated structures and differ-ent types of bearings. The application of base isolation of structureshas been increased particularly after the Kobe (1995) earthquake[6], when good behavior of base isolated structures was observed
[7]. Unfortunately, due to the high cost of the isolators, their appli-cation has long been limited to economically developed countries.However, today, even developing countries like Macedonia [8,9],and Armenia [18] are successfully producing and applying thesedevices.
To promote application of base isolation in R. Macedonia andthe remaining countries in the Balkan, the Institute of EarthquakeEngineering and Engineering Seismology (IZIIS) proposed a projecton development of low-cost rubber bearings [9], which was fi-nanced by NATO through the Science for Peace (SfP) Programme.Within this project, a number of raw rubber recipes and a technol-ogy for production of rubber bearings were developed for the firsttime in the region. Within this project, ample experimental testswere performed on bearings produced of 18 different rubbercompounds. These tests enabled creation of a large database offorce–displacement relationships for different rubber compoundsthat was later used for development of a new analytical model ofrubber bearings.
The laboratory tests on rubber bearings which have been car-ried out within these investigations and the investigations per-formed by other authors [13,17,23], have pointed out theirnonlinear behavior. The stress–strain behavior of rubber bearingsis manifested by high horizontal stiffness under low shear strains,low stiffness under moderate strains, and an increasing shear mod-ulus under higher strains. Usually, rubber hardening begins at125% shear strain and continues until failure.
Lateral bearing behavior is quite complex because of strain-ratedependence and the presence of the Mullins effect [19], [12]. Otherfactors that have an influence upon the hysteretic behavior of
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rubber bearings are the axial load, ambient temperature, ageing ef-fects and compounding. All these factors have resulted in complex-ity of hysteretic behavior of rubber bearings. Because of suchcomplex behavior, the generally accepted modeling of bearingsin horizontal direction by a bilinear model [5,20] cannot com-pletely cover the actual bearing behavior. However, on the otherhand, if all these factors are included in the analytical model of arubber isolator, then this model will be ideal but also verycomplex.
To enable a more realistic dynamic analysis of structures iso-lated with rubber bearings, several advanced analytical modelshave been proposed in literature. One of the approaches is repre-senting the nature of damping as hysteretic [17]. This model in-cludes the high nonlinearity of shear strains but it does notinclude the strain rate and the effect of variation of the axial loadupon the hysteretic properties of the bearing. The other approachincludes implementation of the viscoelastic theory where theforces depend on the strain rate [13]. This proposed model is de-fined by ten parameters that are functions of the number of factorsinfluencing the behavior of the elastomer. This model includes theMullins effect, the strain rate and temperature dependence. Themodel proposed by Tsai et al. [23] also includes the effect of thestrain rate. This model is based on the Bouce-Wen’s model [2]which has been extended to seven material parameters and itcan simulate high strain rate at low strains and strengthening ofrubber at high strains. Another model where the hysteresis hasthe typical ‘‘butterfly’’ shape is proposed by Dall’ Asta and Ragni[3]. In this model, the dependence on the strain rate and the Mul-lins effect are included, as well. The rate dependence has also beeninvestigated and included in the model proposed by Jankowski[14].
The first phase of the investigations covered the development ofrubber bearings and their testing. A comprehensive study was per-formed to develop the technological process of production of rub-ber bearings reinforced by steel shims. Providing quality of the endproduct and development of an appropriate recipe were one of thekey tasks since such bearings were produced for the first time inthe Balkan region. The quality of the end products was verifiedthrough a series of dynamic tests. In this part, the results fromthe vertical and horizontal tests carried out on a few characteristicbearings are presented through force–displacement diagrams.
The analytical polynomial model proposed in the second part ofthis paper was developed based on a series of tests on rubber iso-lators. Presented further in this part are the most characteristicones. The proposed model can simulate the behavior of naturalrubber bearings in conditions of small and large deformations.The model is defined by a polynomial function and eight parame-ters and it is able to cover the strengthening of the rubber in con-ditions of large deformations. It includes the loading history effectand enables adaptation to different shapes of loading/unloading. Aleast-square regression was used to determine the ‘‘best’’ coeffi-cients in order to minimize the sum of the squares in an nth orderpolynomial model. System property modification factors for the ef-fects of aging, temperature and scragging of rubber bearing [1] canalso be included, if necessary. The consideration of the systemproperty modification factors in this analytical model enablesextension of the bilinear simplification [1] to the range of largedeformations. This model can substitute the bilinear simplificationof rubber bearings in the design procedure and was implementedin the finite element nonlinear program [11].
2. Production and testing of natural rubber bearings
One of the main goals of the NATO SfP project was developmentof a high damping rubber (HDR) compound, adoption of produc-
tion of rubber bearings and their testing and implementation.The production of rubber bearings with different rubber com-pounds was carried out in a small local workshop in Macedonia.Eighteen natural rubber compounds were developed and used forproduction of more than 100 bearings. Two types of bearings wereproduced. The first was of a square shape, side length of 200 m andtotal height of 75 mm (Fig. 1b). The second was of a circular shape,external diameter of 150 mm and total height of 100 mm (Fig. 1c).The bearings pertaining to each type were produced without inter-nal steel plates and with different number of steel plates.
Prior to the very process of vulcanization, the raw rubber wascalendared to obtain rubber sheets with the required thickness.The internal steel plates, referred to as shims, were proportioned190/190 mm and Ø140 mm. The first phase of treatment of theshims consisted of cleaning the sandblasted areas with medicalgasoline. This helped remove all the dirt occurring from the mo-ment of sandblasting to the moment of vulcanization. The secondphase involved coating of the areas with an appropriate adhesive.This coating protects the steel areas against any external effects.After 2 h, the third phase took place, i.e., treatment of the steelareas with the second coat. As in the case of the first coat, afterthe second coating, the plates were also left to dry at room temper-ature for about an hour.
Upon completion of the preparation works, the tailored rubberand the treated reinforcement were inserted in a mold, which waspreviously heated to 120 �C. The rubber was vulcanized for 50 minat a temperature of 150 �C (Fig. 1a). In the course of the vulcaniza-tion, the rubber was constantly exposed, from the upper and thelower side, to external pressure of 150 bars. Once the vulcanizationwas over, the element was dismantled from the mold and left tocool at room temperature for 24 h.
After the production of the bearings, two types of tests wereperformed, namely vertical and biaxial tests. All the tests on rubberbearings were carried out in the Dynamic Testing Laboratory of theInstitute of Earthquake Engineering and Engineering Seismology(IZIIS), Skopje. To perform the tests, the existing biaxial dynamicframe with a dynamic load capacity of FMAX = 100 kN and strokeof D = ±75 mm, was used. The static load capacity of the testingframe is FMAX = 200 kN and the stroke is D = ±50 mm. The set-upof the vertical and biaxial test is shown in Fig. 2a and b.
A detailed description of the selected tested specimens is givenin Table 1.
Prior to each testing of the isolators, the materials were stabi-lized by keeping them under room temperature of (22 ± 2 �C) witha duration of 24 h. First, the isolators were pre-loaded with threecycles at low frequency. Such a pre-loading is practiced in testingrubber elements for the purpose of eliminating the Mullins effect[19]. Pre-loading is carried out exclusively for elements not loadedin the course of the preceding 24 h.
The procedure for the axial tests involved loading of the ele-ment with vertical compressive force at different axial stress levels(Fig. 3). The specimens were monotonically loaded up to theachievement of the necessary stress level when harmonic excita-tion was applied. The purpose of these tests was to define the effectof the number of internal layers upon the vertical stiffness of thebearings at different load levels.
Fig. 4 shows the vertical force–displacement relationship for therubber bearings produced by different number of internal steelplates. The presented graph provides a thorough insight into thebehavior of the bearings with and without layers. As expected,the bearing with the greatest number layers (four shims) exhibitedthe highest vertical stiffness.
For this series of produced bearings, the vertical stiffness at dif-ferent axial load levels has been calculated (Table 2).
The biaxial tests consisted of application of horizontal loadwhile the element was exposed to vertical load. The bearings were
Fig. 1. (a) Vulcanization of isolator, (b) squared isolators and (c) circular isolators.
Fig. 2. (a) Set-up of axial test of squared bearing and (b) set-up of biaxial test of two circular bearings.
Fig. 4. Comparison of axial behavior of rubber bearings.
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tested under two different harmonic loads: (1) a sinusoid functionwith constant amplitude defined by two parameters: amplitudeand frequency (Fig. 5a), (2) a sinusoid function with a linearincreasing amplitude (Fig. 5b) and a linear decreasing amplitude(Fig. 5c). The biaxial tests were performed to define the behaviorof the bearings under different loading conditions. The obtaineddata on the behavior of the bearings were later used in the devel-opment of analytical lateral force – displacement relationships.
These tests provided an accurate information about the hori-zontal stiffness and the equivalent damping of the bearing for eachhysteretic cycle. Fig. 6 shows comparison of the behavior of therubber bearings made of eight different rubber compounds ex-posed to horizontal sinusoidal excitation with constant ampli-tudes. The graph provides a clear insight into the difference inthe damping abilities and the difference in shear behavior. Fromthe graphs presented in Fig. 6, it can be concluded that the shearbehavior mainly depends on the rubber compound. Analyzingthese types of behavior, it can be said that some bearings start tobehave in the nonlinear range at lower strains, while others startto behave nonlinearly at higher strains. During unloading, almost
all bearings exert high nonlinearity accompanied by differentdamping values.
A similar behavior of the specimens was observed under loadsof increasing and decreasing amplitudes (Fig. 7).
The effective shear modulus and hysteretic damping of thebearings were computed within the second hysteretic cycle, under
Table 2Vertical stiffness of rubber bearing at various vertical loads.
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an excitation with a constant amplitude and shear strain of c = 0.5(Table 1). The results given in the enclosed table show that thetested rubber bearings have different mechanical characteristics.Namely, the effective shear modulus ranges within (0.37–1.5)Mpa, while the hysteretic damping is within the range of 0.0% to12.5%. It can be said that the specimens with zero damping be-haved almost as a linear elastic material.
To enable a clear presentation of the shear behavior of the rub-ber bearings, the force–displacement curves were divided into twoparts. The first part involved the loading state (Fig. 8a) whereas thesecond involved the unloading state (Fig. 8b). From the consideredbehavior under loading, it was concluded that the bilinear approx-imation was valid up to a moderate shear strain (c � 125%). In therange of high strains, strengthening of the rubber takes place and itstarts to behave with a higher degree of nonlinearity.
Horizontal tests with variable amplitude were conducted toanalyze the effect of the loading history (Fig. 9a). The graph showsthat, while horizontal deformation decreases linearly, horizontalforce at zero horizontal deformation (Dh = 0) deteriorates nonlin-early. In addition to the variation of horizontal force at Dh = 0, thereis also a variation in horizontal stiffness (Fig. 9b).
To get an insight into the effect of displacement upon horizontal(tangent) stiffness, a horizontal stiffness-displacement graph wasestablished (Fig. 10a). The horizontal tangent stiffness was calcu-lated according to expression Kh = DF/DD. The force incrementDF and the displacement increment DD were obtained from the ac-tual hysteretic curve. This figure shows the horizontal stiffness-dis-placement relationship during the second and the last loading/unloading cycle. In the case of loading in the second cycle (Dh = 70-vv), the stiffness was almost unchanged until displacement of 40vv(c � 60%). After this point, the stiffness continuously increased.During the last cycle (Dh = 18 vv), the stiffness continuously deteri-orated. This happened since, during the test, the bearing wasloaded starting from a larger to a lower amplitude of excitation.In the tests performed with an opposite order of loading (fromsmaller to larger amplitudes), the stiffness tended to increase withthe increase of the horizontal displacement (Fig. 10b).
In the case of unloading in the second cycle (Dh = 70 vv), thestiffness considerably deteriorated from the beginning of unload-ing until displacement of 40 vv (c � 60%). After this point, the
stiffness remained constant. During the last cycle (Dh = 18vv), thestiffness continuously deteriorated. In both loading cycles, the hor-izontal stiffness was almost equal at the beginning of the unload-ing, whereas in the case of Dh = 0, the stiffness was higher duringthe loop with smaller displacements.
The effect of the previously experienced deformation upon thehorizontal stiffness of the bearings was investigated through shak-ing table testing of a base isolated liquid storage tank [10]. In theseinvestigations, the base isolated structures were exposed to excita-tion of different intensity. Also, in this case, it was concluded thatthe bearings behaved in almost the same nonlinear manner. Start-ing from these two investigations, additional parameters control-ling the stiffness of the bearings depending on the previouslyexperienced deformation were implemented in the proposed ana-lytical model.
The results from the performed experimental investigationswere also additionaly used for analysis of the Mullins effect(Fig. 11). The Mullins effect was not only observed in a non-previ-ously loaded material but also in a previously loaded material thatwas not exposed to any load for a day. This phenomenon was alsopreviously investigated by Dall’ Asta and Ragni [3], who concludedthat the behavior of the bearings during the first loading cycle de-pended on the time elapsed between the application of the loads.
The influence of Mullins (scragging) effect on the response of abase isolated structure was investigated by Grant et al. [4]. In theirwork, two finite element models were analized. In the first model,the rubber bearings were modeled as scragged isolators, while inthe second one, the scragging effect was included. From the pre-sented results, it can be seen that the inclusion of scragging inthe model clearly results in much larger peak displacements. How-ever, the peak shear forces are not significantly different. If the pro-posed model is applied for design purposes, the neglecting of thescragging effect is usually on the conservative side and is thereforeacceptable. On the other hand, in the case of seismic analysis per-formed by consideration of near fault conditions, taking the Mul-lins effect into account could enable a more accurate analysis. Ifsystem property modification factors [1] are included in the pro-posed model, it can also be applicable for running near fault earth-quake analysis.
3. Mathematical formulation of the proposed model
Based on the performed tests on the bearings made of 18 differ-ent natural rubber compounds, an analytical model was proposedto mathematically describe the behavior of the natural rubberbearings. The model includes the linear-elastic behavior, thepost-elastic behavior at loading and the post-elastic behavior atunloading (Fig. 12).
The linear elastic state includes elastic behavior of the bearingat low strains and it is presented by the following expression:
F ¼ K1 � D ð1Þ
(c)
increasing amplitudes and (c) decreasing amplitudes.
(a) (b) (c)
(d) (e) (f)
(g) (h)
Fig. 6. Comparison of horizontal behavior of bearings made of eight different rubber compounds.
(a)
-1.5 -1 -0.5 0 0.5 1 1.5
Shear strain [m/m]
-1500
-1000
-500
0
500
1000
1500
Shea
r stre
ss [k
Pa]
spec-4f=0.3Hzexcitation-3
-1.5 -1 -0.5 0 0.5 1 1.5
Shear strain [m/m]
-1500
-1000
-500
0
500
1000
1500 spec-4f=0.3Hzexcitation-4
(b)
Shea
r stre
ss [k
Pa]
Fig. 7. Comparison of horizontal behavior of bearings under excitation with (a) increasing amplitudes (b) decreasing amplitudes.
604 I. Gjorgjiev, M. Garevski / Engineering Structures 56 (2013) 600–609
where F is the horizontal force, D is the horizontal deformation andK1 is the elastic (initial) stiffness of the bearing. The linear elasticstate of this model takes place at the beginning of loading of thebearing and at the transition from unloading into loading conditions(Fig. 13). The input parameters for this state are: elastic stiffness ofthe bearing K1 and yielding deformation.
The post-elastic state at loading involves the behavior of thebearing after the yielding point. Once the yielding point is
exceeded, the force–displacement relationship is defined by thepolynomial function given in the following equation:
Fi ¼ Fiom þ a1Di þ a2D2
i þ a3D3i þ ::: ð2Þ
where Fi is the horizontal force, Di – the horizontal deformation andFi
om, a1, a2, a3, . . . are the coefficients of the polynomial functionwhich are calculated for best fitting the experimental curve.
(a) (b)
D [mm]
0
10
20
30
40
spec-8spec-9spec-10
loading part
0 15 30 45 60 75 0 15 30 45 60 75
D [mm]
-10
0
10
20
30
40
spec-8spec-9spec-10
unloading part
F [k
N]
F [k
N]
Fig. 8. Comparison of horizontal behavior of bearings under (a) loading and (b) unloading.
Fig. 9. (a) Effect of previously achieved deformation upon horizontal force at zero deformation and (b) effect of previously achieved deformation upon horizontal force at zerodeformation during the first and the last loading cycles.
0 25 50 75
D [mm]
0
500
1000
K [k
N/m
]
second cyclelast (8) cycle
spec-12excitation-3
unloadingpart unloading
part
loadingpart
loadingpart
-20 -10 0 10 20D [mm]
-10
-5
0
5
10
F [k
N]
excitation-4excitation-3
spec-9(a) (b)
Fig. 10. (a) Horizontal (tangent) stiffness-displacement relationship during the second and the last cycle and (b) bearing behavior under increasing and decreasing amplitude.
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The polynomial coefficient FiomBFi
o is defined based on the cur-rent and previously experienced horizontal deformation. Thedependence of Fi
o on the previously achieved deformation is de-fined as linear and it is given by expression (3).
Fio ¼
Fmino when DKP
i < Dmino
Fmino þ ðFmax
o � Fmino Þ
DKPi �Dmin
o
Dmaxo �Dmin
owhen Dmax
o < DKPi < Dmin
o
Fmaxo when DKP
i > Dmaxo
8>>><>>>:
ð3Þ
where
DKPi ¼maxðDKP;DiÞ ð4Þ
Forces Fmino and Fmax
o and deformations Dmino and Dmax
o are parametersdefined through tests on bearings.
The post-elastic state at unloading includes the behavior of thebearing when |Di�1| > |Di|, where |Di�1| is the previous horizontaldeformation, |Di| is the current horizontal deformation of the bear-ing. The mathematical force–displacement relationship for post-elastic behavior at unloading is formulated by the same polyno-mial function for post-elastic behavior at loading which is givenin Eq. (2). To include different types of behavior of the bearingsat unloading, modification is made in the computation of coeffi-cient Fi
om given by:
Fiom ¼ Fi
oð2 � kFio � 1Þ ð5Þ
-40 -20 0 20 40D [mm]
-40
-20
0
20
40
first testtime span 20'
spec-7F
[kN
]
Fig. 11. Bearing behavior in a time span of 20 min.
D
F
FoMIND3
F3FoMAX
F2
Foi
D2D1 Dy1
1
2
3
F1 FyFi=K Di1
polynomialfunction
modifiedpolynomialfunctionKP1 KP3
KP2
Fig. 12. Polynomial analytical model of bearings made of rubber.
D
F
1
1
Fi=K1 DiYieldPoint
Fi=K1Di
Fig. 13. Linear-elastic state at beginning of loading and at transition fromunloading to loading.
(a)
(b)
Fig. 15. Post-elastic state at unloading for different exponents.
-0.06 -0.04 -0.02 0 0.02 0.04 0.06D [m]
-30
-20
-10
0
10
20
30F
[kN
]analytical
biaxial test
spec-9
Fig. 16. Force–deformation relationship obtained from test and the analyticalsolution.
606 I. Gjorgjiev, M. Garevski / Engineering Structures 56 (2013) 600–609
where kFio is the decay coefficient and it is computed according to
the following equation:
kFio ¼
eeDKP
i �Di
DKPi
� �� 1
eðeDKPi Þ � 1
ð6Þ
Fig. 14. (a) Dependence of the exponent on the horizontal d
where e is the natural logarithmic base (e = 2.71828. . .), DKPi is the
horizontal deformation of the isolator at the moment of beginningof unloading and eDKP
i is the exponent at the moment of beginningof unloading.
The value of eDKPi is computed according to Eq. (7) where eDKP
i islinearly dependent on DKP
i . The remaining variables are inputparameters and they are defined on the basis of horizontal tests.
eDKPi ¼ eDmin þ ðeDmax � eDminÞ
DKPi � Dmin
e
Dmaxe � Dmin
e
ð7Þ
eformation of the bearing and (b) exponential function.
(a) (b)
-0.02 -0.01 0 0.01 0.02
D [m]
-10
-5
0
5
10
F [k
N]
analyticalexperiment
K2 K2(D=8)
spec-9
-0.06 -0.04 -0.02 0 0.02 0.04 0.06
D [m]
-30
-20
-10
0
10
20
30
F [k
N]
analyticalexperiment
K2 K2D=65
spec-9
Fig. 17. Force–displacement relationship in conditions of (a) low deformations (b) large deformation.
Fig. 18. Verification of the analytical model with the most characteristic force–deformation relationship obtained by experimental testing.
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608 I. Gjorgjiev, M. Garevski / Engineering Structures 56 (2013) 600–609
where eDmin is exponent at deformation Dmine and eDmax is exponent
at deformation Dmaxe .
This expression is valid only when deformation DKPi is within the
range of [Dmine ;Dmax
e ]. In the case the deformation at the character-istic point is beyond the above domain, then it is assigned the limitvalues (Dmin
e or Dmaxe ) of the interval (Fig. 14a). The graphical inter-
pretation of the dependence of eDKPi on DKP
i is shown in Fig. 14a.The computation of the decay coefficient kFi
o includes fourparameters (Di, DKP
i , eDKPi and Fi
o) that directly depend on the cur-rent deformation of the bearing and the history of deformations.The current deformation is included in parameter Di and its valueat unloading ranges between DKP
i and zero. The history of deforma-tion of the bearing is included in parameters DKP
i and eDKPi . Param-
eter DKPi is the maximum achieved deformation at loading, while
parameter eDKPi directly depends on DKP
i and it is calculated accord-ing Eq. (7). Parameter Fi
o is defined according to expression (3) anddepends on the current deformation and the history ofdeformation.
From Eq. (6), it can be seen that the value of the decay coeffi-cient kFi
o ranges within the following limits:
forDi ¼ 0 ) kFi
o ¼ 0
Di ¼ DKPi ) kFi
o ¼ 1
(
Accordingly:
whenDi ¼ 0 then Fi
om ¼ �Fio
Di ¼ DKPi then Fi
om ¼ Fio
(
With this, it is proved that force Fiom satisfies both ultimate states:
– When there is no deformation of the bearing, the force is equalto the value of coefficient Fi
o.– At the beginning of unloading, the force at loading and that at
unloading are equal.
To get an insight into the effect of eDKPi upon the horizontal
force in the bearing, three curves with different values ofDKP
i ¼ 2;5 and 10 were derived (Fig. 15). Fig. 15 shows that, inthe case of curves with higher value of eDKP
i , horizontal force F dur-ing unloading, decreases faster and more intensively. In all threecurves, the force has an identical value at the beginning and atthe end of unloading which points to the fact that the boundaryconditions are satisfied.
The subsequently presented graph in Fig. 15 shows the depen-dence of kFi
o on the (Di=DKPi ) ratio. From the enclosed curves
(eDKPi ¼ 2;5; and 10), it can be concluded that, at higher values of
eDKPi , there is a considerable decay of the value of kFi
o when the cur-rent displacement is within the limits of Di = [0.80 � 1.0] DKP
i . Thischaracteristic of kFi
o enables control over the total restoring force ateach deformed position.
4. Verification of the model
The aim of the verification was to prove the stability of themodel in different strain conditions. Three states were verified:elastic, post-elastic at loading and post-elastic at unloading. To val-idate the proposed model, the values of forces and displacementsobtained from the polynomial model were compared to the exper-imental ones. The presented experimental results were obtainedfrom biaxial tests carried out on square and circular bearings pro-duced by use of different rubber compounds. In horizontal direc-tion, the bearings were loaded with harmonic excitation of bothconstant and increasing amplitude (Fig. 5a and b).
The graph in Fig. 16 shows the force–deformation relationshipobtained from the rubber bearing test ‘‘spec-9’’ and that obtainedin the analytical solution. The dotted curve represents the data
from the test, whereas the solid curves represent the analyticalsolution.
In order to investigate the behavior of the polynomial model atlow and large strains, the first and the last two cycles for specimen‘‘spec-9’’ were separated (Fig. 17a and b). From the presented com-parison of the results, it could be concluded that the polynomialmodel was not able to completely follow the behavior of the bear-ing at low deformations. This was due to the parameters of thepolynomial model which were fitted to large strains and the modelwas not able to completely follow the behavior of the bearing atlow strains. In the case of large deformations (Fig. 17b), the polyno-mial model completely followed the behavior of the bearing. Theother loading cycles (>2) were simulated by the analytical modelwith an acceptable accuracy.
The applicability of the proposed analytical model for the bear-ings produced of different rubber is presented through comparisonof four force–displacement relationships (Fig. 18a throughFig. 18e). Fig. 18a shows the nonlinear behavior of the rubber with-out damping, while Fig. 18b–e show different forms of behavior inthe case when the compound possesses a certain internal damping.
From the presented graphs, it can be concluded, that the poly-nomial model is sufficiently flexible to be applied for modeling ofbearings made of different rubber compounds. The input parame-ters of the proposed polynomial model cover sufficiently well, themain characteristics of behavior of rubber bearings. The modelshows satisfactory accuracy not only in the case of high dampingrubber bearings but also in the case of low damping rubberbearings.
The stability and accuracy of the polynomial model was alsoverified by nonlinear dynamic analysis of a seismically isolatedsteel reservoir [11]. The time histories of acceleration analyticallyobtained from several earthquake excitations [11] were comparedwith the experimental ones, whereat the analytical model provedto be sufficiently precise.
5. Conclusions
This paper deals with production of rubber bearings, their test-ing and development of an analytical model of a rubber bearing.The process of production involved development of high dampingrubber compound and establishment of a stable bond between therubber and the steel surfaces. Vertical and biaxial tests were car-ried out for square and circular bearings produced of 18 differentrubber compounds. Based on the performed horizontal tests, adatabase was created and used to develop a polynomial analyticalmodel of a rubber bearing. From the tests on the bearings, it wasconcluded that the behavior of the bearings was bilinear untilmoderate shear strain. At higher strains, the rubber bearings be-haved in the nonlinear range and were characterized by an in-creased stiffness. It was also concluded that the horizontalstiffness at zero horizontal deformation of the bearing dependedon the loading history. In the case of change of horizontal deforma-tion at the beginning of the unloading, the force at zero deforma-tion changes, as well.
The proposed analytical model of a rubber bearing is based on apolynomial function. The behavior of the rubber bearing is pre-sented through eight parameters plus the polynomial coefficients.The analytical model and the methodology presented in this paperare based on establishment of empirical model parameters bymatching the actual bearing test results. This analytical modelwas developed on the basis of the results of experimental testson bearings made of different rubber compounds. The model in-cluded large strain behavior and loading history effect. The Mullinseffect, the strain rate and the dependence on the axial load werenot taken into account in these investigations.
I. Gjorgjiev, M. Garevski / Engineering Structures 56 (2013) 600–609 609
The parameters of the material for the proposed model were de-fined through a simple biaxial test on a rubber isolator. Using theleast square method, the coefficients of the polynomial functionwere defined. From the performed comparison between the ana-lytical and the experimental results, it can be concluded that thepolynomial model is flexible to be applied for modeling of isolatorsmade of different rubber compounds.
The proposed analytical model can be used instead of bilinearsimplification of rubber bearing for the purpose of better simula-tion of nonlinear behavior of rubber. The system property modifi-cation factors for the effects of aging, temperature and scraggingdefined in [1] code provisions can be included in this model bychanging the polynomial coefficients.
Acknowledgement
The authors are indebted to Prof. James Kelly from the Univer-sity at Berkeley, Civil Engineering Department, California, for theunselfish professional assistance provided during the realizationof the NATO SfP project 978028. Gratitude is also accorded forthe financial assistance obtained through the NATO Science forPeace Programme which enabled the performance of the presentedample investigations.
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