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International Symposium “Steel Structures:Culture & Sustainability 2010” 21-23 September 2010, Istanbul, Turkey

Paper No:

BENDING BEHAVIOR OF RUBBER-BASED SEISMIC ISOLATION BEARINGS

Seval PINARBASI

Kocaeli University, Department of Civil Engineering, Kocaeli, Turkey

ABSTRACT

Composed of several thin rubber layers bonded to reinforcing sheets, elastomeric bearings are now extensively used for seismic isolation of steel structures. Due to low shear stiffness of interior rubber layers, a multilayered rubber bearing is subject to a “buckling type of instability” under large vertical loads. Theoretical studies show that one of the basic parameters that control the buckling behavior of a rubber bearing is its bending stiffness. Using an advanced analytical treatment based on modified Galerkin method and removing most of the commonly used assumptions in literature, such as rigid-reinforcement, rubber-incompressibility and “pressure” assumptions, Pinarbasi and Mengi (2008) have recently derived closed-form solutions for compressive and bending behavior of long rectangular bearings. In this paper, using these advanced solutions, the significance of including reinforcement extensibility and rubber compressibility in determining bending stiffness of fiber-reinforced elastomeric bearings are discussed, first. Then, the formulation of Kelly (1997) for buckling behavior of multilayered rubber bearings is reviewed shortly. Finally, by incorporating the advanced bending modulus expression derived by Pinarbasi and Mengi (2008) into the Kelly’s formulation, which was originally derived for steel-reinforced incompressible-rubber bearings, the significance of including reinforcement extensibility and rubber compressibility in determining buckling load of fiber-reinforced elastomeric bearings are discussed. Keywords: Seismic isolation, elastomeric bearing, rubber, bending modulus, buckling load

INTRODUCTION

Composed of several thin rubber layers sandwiched between and bonded to reinforcing sheets at their top and bottom faces, elastomeric bearings are now extensively used for isolation of steel buildings from devastating effects of seismic excitations and for isolation of steel bridges from forces and rotations induced by thermal expansions, traffic and earthquake. Since the behavior of a base isolated structure is mainly controlled by the mechanical properties of the individual isolators, it is essential that the structural engineer who will design an isolated building/bridge have comprehensive knowledge on the behavior of the bearings under the combined effects of compression, bending and shear forces.

In a typical rubber-based seismic isolation bearing, the interior reinforcing sheets are much more rigid than the soft rubber layers, whose shear modulus equals approximately to 1.0MPa, which is several-order smaller than that of steel or fiber reinforcement. For this reason, in the design of a multilayered elastomeric bearing, the effect of reinforcing sheets on shear behavior of the bearing is usually ignored and horizontal stiffness of the bearing is computed as if the reinforcing sheets did not exist. Contrary to the shear behavior, the compressive or bending behavior of a rubber bearing is essentially controlled by the geometrical and material properties of a typical interior “bonded”

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rubber layer. In fact, the main purpose of the use of interior reinforcing sheets, which are typically steel or a kind of fiber, such as carbon, glass, aramid, etc., in a multilayered rubber bearing is to increase the stiffness of soft rubber layers under compression and bending so that they can support the heavy weight of the superstructure without making large deformations and rotations. It is also to be noted that the assumption that the shear behavior of the bearing is independent from its axial or flexural rigidity is realistic only when the deformations and/or loads are sufficiently small. It is now very well-known that due to its low shear stiffness, a multilayered rubber bearing is subject to a “buckling type of instability” under large vertical loads. Furthermore, when subjected to large horizontal deformations, the horizontal stiffness of a multilayered rubber bearing can decrease considerably due to the second order effects especially if its bending rigidity is not sufficiently large. (Kelly, 1997)

The theoretical studies (e.g., Kelly, 1994) have shown that one of the basic parameters that control the buckling behavior of a multilayered rubber bearing is its flexural stiffness, which can be computed from linear analysis of its typical interior bonded rubber layer. Thus, for a rational design, it is essential that the bending stiffness of elastomeric bearings be predicted accurately. Using an advanced analytical treatment based on modified Galerkin method and removing most of the commonly used assumptions in literature, such as rigid-reinforcement, rubber-incompressibility and “pressure” assumptions, Pinarbasi and Mengi (2008) have recently derived closed-form expressions for displacement/stress distributions in long rectangular fiber-reinforced rubber bearings. They have also derived closed-form expressions for compression and bending modulus of such bearings. These expressions have clearly indicated that there are three basic parameters that control the bending behavior, thus buckling behavior, of a rubber bearing: shape factor of the bearing (i.e., the ratio of one loaded area of a single bonded rubber layer to its bulge free areas), compressibility of the rubber material and extensibility of the interior reinforcing sheets. In this paper, using the advanced solutions derived by Pinarbasi and Mengi (2008), the significance of including reinforcement extensibility and rubber compressibility in determining bending stiffness of fiber-reinforced elastomeric bearings with various shape factors are discussed, first. Then, the formulation of Kelly (1997) for buckling behavior of multilayered rubber bearings is reviewed shortly. Finally, by incorporating the advanced bending modulus expression derived by Pinarbasi and Mengi (2008) into the Kelly’s formulation, which was originally derived for “steel-reinforced” “incompressible-rubber” bearings with “high shape factors”, the significance of including reinforcement extensibility and rubber compressibility in determining buckling load of fiber-reinforced elastomeric bearings with wide-range of shape factors are discussed.

EFFECTIVE BENDING MODULUS OF RUBBER BEARINGS

It is well known that the compressive or bending behavior of a multilayered elastomeric bearing is controlled by the behavior of its individual elastomer layers. For this reason, the effective compression or bending modulus of a multilayered rubber-based isolator (Figure 1a) is determined from the analysis of its typical interior “bonded rubber layer” (Figure 1b). Since steel plates have long been used as the main reinforcing elements in conventional rubber bearings, most of the earlier analytical researches have studied rubber layers bonded to rigid surfaces. For an extensive review on these earlier studies, Pinarbasi (2007) can be referred. During his studies on influence of plate flexibility on buckling behavior of rubber bearings, Kelly (1994) realized that the thickness of the interior reinforcing sheets can be reduced without compromising much from buckling behavior of the bearings. However, he also realized that such a reduction may not always be practically possible if reinforcing plates are made of steel since very thin steel plates can bend easily while they are being sand-blasted during the manufacturing process of the bearing. Thus, in 1999, he proposed the use of fiber reinforcing sheets in place of steel reinforcing sheets (Kelly, 1999). Since the use of fiber-reinforcement in multilayered elastomeric bearings is relatively recent, there are only a few analytical studies (e.g., Tsai and Kelly, 2001; Kelly, 2002; Kelly and Takhirov 2002; Pinarbasi and Mengi, 2008) on rubber layers bonded to reinforcing sheets. It can also be seen that most of these

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studies have been concentrated on compressive behavior of bonded rubber layers. Only, for bearings in the shape of long rectangular strip, are there closed-form expressions for bending modulus which are derived including the effects of both the flexibility of the reinforcing sheets and compressibility of the rubber layers (Pinarbasi and Mengi, 2008). Since this paper concentrates on the effects of both parameters on bending modulus, the discussions in this section will be limited to “infinite-strip” bearings. However, it has to be bared in mind that similar relations can also be obtained for bearings with any other shapes.

a. Circular steel-reinforced bearing b. Typical interior “bonded” rubber layer

Figure 1. A typical circular steel-reinforced rubber bearing (taken from Kelly (2007)) with its typical bonded rubber layer (taken from Imbimbo and DeLuca (1998))

Figure2a shows the undeformed configuration for a “bonded” elastomer layer in a multilayered fiber-reinforced elastomeric strip. The layer has uniform thickness t and is bonded to flexible reinforcing sheets with equivalent thickness tf at its top and bottom faces. Since the length of the layer is much larger than its width 2w and thickness t, the layer is in a state of plane strain. When the layer is uniformly compressed with axial load of P, it deforms as shown in Figure2b, where Δ is the applied vertical displacement. On the other hand, when the layer is purely bended with bending moments of M, the deformed shape takes the form shown in Figure2c, where φ is the relative rotation of the top face of the bonded elastomer layer with respect to its bottom face.

a. undeformed shape b. deformed shape under compression c. deformed shape under bending

Figure 2. Undeformed and deformed configurations for a long rectangular elastomer layer bonded to reinforcing sheets under uniform compression and pure bending

Assuming small deformations with linear behavior, the effective compression modulus (Ec) and effective bending modulus (Eb) of a bearing composed of several such bonded rubber layers can be computed from

RUBBER BEARING

Steel

SINGLE RUBBER LAYER

Loaded AreaLateral Surface Free to Bulge

RubbeRubber

h=t/2

x2

x1 h=t/2

w w

tf

tf

P

P Δ/2

Δ/2

φ/2

φ/2M

M

4

=cP AE

tΔ//

and =bM IE

tφ//

(1)

where A is the cross sectional area of the bearing and I is the moment of inertia of the bearing about the axis of bending.

Assuming strict incompressibility and rigid plates and using the pressure formulation (for details of pressure formulation, Pinarbasi (2007) can be referred), Gent and Lindley (1959) derived a simple expression for Ec of strip-shaped steel-reinforced bearings. Using the same approach, Gent and Meinecke (1970) derived a similar expression for Eb of such bearings. These well-known expressions are

( )2=4 1cE G S+ and 21=4 15bE G S⎛ ⎞+⎜ ⎟

⎝ ⎠ (2)

where G is the shear modulus of the rubber and S is the shape factor of the bearing, which is defined as the ratio of one loaded area of its typical interior rubber layer to its bulge-free areas (see Figure 1b). For a long rectangular bearing as shown in Figure 2a, S=w/t. It can be recognized that bending modulus of bearings with high shape factors are approximately equal to one-fifth of their compression modulus (Eb / Ec ≅ 1/5).

In a recent study, including the effects of both reinforcement flexibility and rubber compressibility, and using a higher order theory based on a modified version of Galerkin Method, Pinarbasi and Mengi (2008) derived advanced closed form expressions for both compression and bending moduli of long rectangular bearings. These expressions can be represented as follows:

2 2 2 210 1 112 2

1 1 1

tanh( )( )c

wEw

λ β β λ βαα β β α β

= − − and 2 2 2 2

10 1 112 2 2

1 1 1 1

3 1 ( ) 1( ) tanh( )b

wEw w

λ β β λ βαα β β β α β

⎡ ⎤= − − −⎢ ⎥

⎣ ⎦ (3)

where

210 2

12Gt

βα

= , 211

12

f

Gk t

β = , 2 2 21 10 11β β β= + ,

( )21f f

ff

E tk

ν=

− and 2Gα λ= + (4)

In above equations, λ is Lamé’s constant; kf is in-plane stiffness of the reinforcing sheets; E f and ν f are, respectively, elasticity modulus and Poisson’s ratio of the reinforcing sheets.

Since these expressions include both rubber compressibility and reinforcement flexibility, and since they are valid not only for high-shape factor bearings, but also for low-shape factor bearings, they are practical tools for investigating the effects of these important parameters on bending behavior of rubber bearings. The graphs presented in Figure 3, which are plotted using Eqs. (3), show the effects of these three basic parameters; i.e., rubber compressibility, shape factor and reinforcement flexibility, on bending modulus of a long-rectangular multilayered rubber bearing.

As it is shown in the graphs, the bending modulus of a multilayered rubber bearing decreases with increasing rubber compressibility. The decrease in bending modulus can be significant especially if the shape factor of the bearing is large. As an example, for a bearing with shape factor of S=30, bending modulus is approximately 10 times smaller than its incompressible value when Poisson’s ratio is equal to 0.495, which seems to be almost equal to 0.5. The effect of reinforcement flexibility on Eb is also significant especially in bearings with high shape factors. The bending modulus of a bearing with flexible fiber-reinforced sheets can considerably be smaller than its equivalent steel-reinforced counterpart. For this reason, it is extremely important that the simple expressions predicted by the pressure method (such as, those given in Eq. (2)) should be used with ultimate care if the effects of rubber compressibility and/or shape factor and/or reinforcement flexibility are not negligibly small. It is also worth noting that, in such cases, Ec/Eb ratio can also significantly be

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smaller than the prediction of pressure method, i.e., 5.0, which can be seen from Figure 4 where the variation of this ratio with shape factor for bearings with different geometric and material properties are shown.

a. variation with rubber compressibility

b. variation with shape factor c. variation with reinforcement flexibility

Figure 3. Effects of rubber compressibility, shape factor and reinforcement flexibility on bending modulus of a long rectangular elastomeric bearing

Figure 4. Effects of rubber compressibility, shape factor and reinforcement flexibility on Ec/Eb ratio

1

10

100

1000

0 2 4 6log[1/(1-2ν)]

Eb/G

S=1S=5S=15S=30

ν0.495 0.49995 ≅0.50

k f /(G t)=300

1

10

100

1000

0 2 4 6log[1/(1-2ν)]

Eb/G

S=1S=5S=15S=30

ν0.495 0.49995 ≅0.50

k f /(G t) →∞

1

10

100

1000

10000

1 10 100S

Eb/G

ν≅0.5 ν=0.499ν=0.45 ν=0.3

k f /(G t) →∞

1

10

100

1000

10000

1 10 100S

Eb/G

ν≅0.5 ν=0.499ν=0.45 ν=0.3

k f /(G t)=300

0.0

0.5

1.0

1.5

1 100 10000 1000000

kf/(Gt)

Eb/

Eb,

stee

l-rei

nfor

ced

ν≅0.5 ν=0.499ν=0.45 ν=0.3

S=30

0.0

0.5

1.0

1.5

1 100 10000 1000000

kf/(Gt)

Eb/E

b,st

eel-r

einf

orce

d

ν≅0.5 ν=0.499ν=0.45 ν=0.3

S=5

0

1

2

3

4

5

1 10 100S

Ec/E

b

ν≅0.5 ν=0.499ν=0.45 ν=0.3

k f /(G t) →∞

0

1

2

3

4

5

1 10 100S

Ec/E

b

ν≅0.5 ν=0.499ν=0.45 ν=0.3

k f /(G t) = 300

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STABILITY OF MULTILAYERED ELASTOMERIC BEARINGS

Buckling load of a “steel-reinforced” “high-shape-factor” “incompressible” rubber bearings Realizing that a multilayered elastomeric bearing can be subjected to column-like buckling instability, Kelly (1997) derived simple expressions for buckling load of multilayered rubber bearings, which is mainly controlled by the flexural stiffness of the bearing. He also obtained expressions for their horizontal stiffness in the presence of large vertical loads. In his analysis, he used a theory which was essentially the extension of an earlier theory formulated by Haringx (1947) for helical steel springs and rubber vibration mountings. In this formulation, a multilayered elastomeric bearing is modeled as a continuous column. For such a simplification, it is necessary to include the effects of the interior reinforcing sheets on bearing behavior indirectly.

As already mentioned, the main effect of the interior reinforcing sheets on behavior of a multilayered elastomeric bearing is to increase its compressive and bending stiffnesses. The horizontal stiffness of a multilayered elastomeric bearing can realistically be accepted to be unaffected from the presence of interior reinforcing sheets unless the horizontal displacement of the bearing or the vertical load on the bearing is considerably large. The flexural stiffness (EbI) of a multilayered rubber bearing can be computed from bending modulus of its typical interior bonded rubber layer. While the reinforcing layers do not influence the horizontal behavior of the bearing directly, they do reduce the effective height of the “equivalent” column since they do not contribute to the total deformation of the “equivalent” column due to their large axial stiffness. Thus, it is necessary to increase both shear and flexural stiffness of the bearing in “equivalent column” model by multiplying them with h/tr where h is the total height of the bearing and tr is the total rubber thickness in the bearing. Then, the equivalent column stiffnesses of a bearing can be written in the following form:

=Sr

hGA GAt

and =S br

hEI E It

(5)

After defining the shear and flexural rigidities of the equivalent column system, elastic Euler buckling load (PE) for a bearing can be defined as if it were a column:

2

2S

EEIPh

π= (6)

For a bearing with the boundary conditions shown in Figure 5, Kelly (1997) showed that the critical buckling load (Pcr) can be obtained from the solution of the following differential equation:

2 0S S EP PP P P+ − = (7)

where PS = GAS. The solution of Eq. (7) is

2 42

S S S Ecr

P P P PP

− + += (8)

Since for a typical seismic isolation bearing, PS is much smaller than PE, this equation can further be simplified, as shown by Kelly (1997), to

cr S EP P P= (9)

Substituting the expressions for PS and PE into the above formula, one can obtain Pcr as follows:

bcr

r

EGArPt G

π= (10)

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where r is the radius of gyration of the bearing about bending axis, i.e., r I A= / . Similarly, critical pressure (pr) can be obtained by dividing Pcr to A.

bcr

r

EGrpt G

π= (11)

Figure 5. Boundary conditions for buckling analysis of a seismic isolation bearing (Kelly, 1997)

Using Eq. (11), Kelly (1997) derived a much simpler equation for circular steel-reinforced bearings. Since the bending modulus of a “thin” circular rubber layer, with radius R and thickness t, bonded to “steel” plates at its top and bottom faces, is one-third of its compression modulus Ec if rubber is assumed to be strictly incompressible (Gent and Meinecke, 1970) and since Ec of such a bonded rubber layer (Gent and Lindley, 1959) equals to

26cE GS= (12)

where S=R/2t for such a circular bearing, bending modulus of the bearing simply equals to 22bE GS= (13)

If the expressions given in Eqs. (12) and (13), which are derived for circular bearings, are compared with those in Eq. (2), which are derived for long rectangular bearings, one can see that the former does not include the shape factor independent terms in the expressions. Assuming that for typical seismic isolation bearings the effects of these constant terms become negligibly small, Kelly (1997) made such a simplification in his theory. Comparison of the circular and strip-shaped expressions also verifies the earlier statement that “similar relations can also be obtained for bearings with any other shapes”.

Since the radius of gyration of a circular bearing simply equals to its half radius, the critical pressure for a circular steel-reinforced bearing can be obtained as

28crp GSSπ=* (14)

where S2 is called aspect ratio or second shape factor of the bearing and equals, for a circular bearing, to 2R/tr.

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Buckling load of a “fiber-reinforced” “compressible” rubber bearings with various shape factors The effects of the reinforcement flexibility, rubber compressibility and shape factor on buckling load of the bearings can be investigated by incorporating the advanced analytical expression derived by Pinarbasi and Mengi (2008) for bending modulus of long rectangular bearings, given in Eq. (3), into the general buckling pressure expression presented in Eq. (11), which is valid for any shape of bearing. Realizing that the radius of gyration of a long rectangular bearing equals to 3r w= / , where w is the half width of the bearing, as shown in Figure 2a and the second shape factor of the bearing is 2 2 rS w t= / , the critical pressure for long rectangular fiber-reinforced bearing can be expressed as

22 3b

crEp GSG

π= (15)

Using Eq. (3) for Eb term appearing in Eq. (15), one can easily plot the variation of buckling pressure for strip-shaped rubber bearings with shape factor, rubber compressibility and reinforcement flexibility. However, it can be recognized that pcr also depends on the aspect ratio of the entire bearing, S2. Contrary to the other parameters, the dependence of pcr on S2 is apparent; thus, not requires further detailed study. For this reason, it can be practical to plot the critical pressure variations in a normalized form so that the dependence of pcr on S2 can be eliminated. A limiting value of buckling pressure for “steel-reinforced” bearing with “high shape factors” with the assumption that rubber is “strictly incompressible”, denoted as crp* , can be obtained by letting ν → 0.5 and kf → ∞. In fact, crp* can also be obtained by using 24 5bE GS= / , which is predicted by pressure method, in Eq. (11) since as shown in Figure 3, Eq. (3) converges to 24 5bE GS= / as ν → 0.5 and kf → ∞. Thus, crp* is

215crp GSSπ=* (16)

The graphs presented in Figure 6 plot normalized buckling pressure ( cr crp p*/ ) for long rectangular fiber-reinforced bearings as a function of reinforcement flexibility for different levels of rubber compressibility and for various shape factors. From the plots, it can easily be seen that cr crp p= * only if the reinforcement stiffness is large, rubber is “strictly” incompressible (i.e., ν=0.5) and shape factor of the bearing is sufficiently large. The effect of rubber compressibility on buckling load of high shape factor bearings is considerably large. As an example, a steel-reinforced natural rubber (ν≅0.499) bearing with S=30 can only carry 60% of the buckling load computed on the basis of incompressible rubber behavior. Thus, it is extremely important that the effect of rubber compressibility be included in the calculation of buckling load of a high shape factor bearing. The effect of slight rubber compressibility decreases as shape factor of the bearing decreases. For bearings with S ≤ 5, the effect of slight rubber compressibility can realistically be ignored.

From the graph plotted for bearings with S=2.5 in Figure 6, one can also recognize that buckling loads of such low shape factor bearings can be larger than crp* if the reinforcement flexibility and rubber compressibility are considerably low. This is simply because that the buckling formulation of Kelly (1997) ignores, as mentioned previously, the shape factor dependent term in bending modulus of multilayered rubber bearings, which gains importance as shape factor of the bearing decreases. The graphs also reveal that as reinforcement flexibility increases, buckling load of a rubber bearing decreases considerably. For this reason, unless the axial stiffness of the reinforcing sheets are considerably large (e.g., as large as steel sheets in steel-reinforced bearings), their effects on buckling behavior should be included in fiber-reinforced rubber bearings.

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Figure 6. Effects of rubber compressibility, shape factor and reinforcement flexibility on “normalized” buckling pressure of a long rectangular elastomeric bearing

CONCLUSION Composed of several thin rubber layers sandwiched between and bonded to reinforcing sheets at their top and bottom faces, multilayered rubber bearings are now extensively used for isolation of steel structures from devastating effects of seismic excitations and for isolation of steel bridges from forces and rotations induced by thermal expansions, traffic and earthquake.

Due to the low shear stiffness of interior rubber layers, a multilayered rubber bearing is subject to a “buckling type of instability” under large vertical loads. Theoretical studies show that one of the basic parameters that control the buckling behavior of a rubber bearing is its bending stiffness and that there are three basic parameters that control the bending behavior of a rubber bearing: shape factor of the bearing (i.e., the ratio of one loaded area of a single bonded rubber layer to its bulge free areas), compressibility of the rubber material and extensibility of the interior reinforcing sheets.

Using an advanced analytical treatment based on modified Galerkin method and removing most of the commonly used assumptions in literature, such as rigid-reinforcement, rubber-incompressibility and “pressure” assumptions, Pinarbasi and Mengi (2008) have recently derived closed-form solutions for compressive and bending behavior of long rectangular bearings. Since these solutions include both rubber compressibility and reinforcement flexibility, and since they are valid not only for high-shape factor bearings, but also for low-shape factor bearings, they are practical tools for investigating the effects of these important parameters on bending stiffness, in turn, on buckling load, of rubber bearings. In this paper, using these advanced solutions, the significance of including reinforcement extensibility and rubber compressibility in determining bending stiffness, which directly related to the buckling load, of fiber-reinforced elastomeric bearings are examined for bearings of various shape factors and concluded that it is extremely important to include the effect

0.0

0.5

1.0

1.5

1 100 10000 1000000kf/(Gt)

p cr/p

* cr

ν=0.3 ν=0.45ν=0.499 ν≅0.5

S=30

0.0

0.5

1.0

1.5

1 100 10000 1000000kf/(Gt)

p cr/p

* cr

ν=0.3 ν=0.45ν=0.499 ν≅0.5

S=15

0.0

0.5

1.0

1.5

1 100 10000 1000000kf/(Gt)

p cr/p

* cr

ν=0.3 ν=0.45ν=0.499 ν≅0.5

S=5

0.0

0.5

1.0

1.5

1 100 10000 1000000kf/(Gt)

p cr/p

* cr

ν=0.3 ν=0.45ν=0.499 ν≅0.5

S=2.5

10

of both reinforcement flexibility and rubber compressibility in bending and buckling analysis of multilayered fiber-reinforced bearings especially if the shape factor of the bearing is high. Use of simple expressions derived based on rigid reinforcement and incompressible rubber behavior assumptions can overestimate the actual bending stiffness and buckling load of the bearing considerably.

REFERENCES Gent A.N., Lindley P.B. (1959), The Compression of Bonded Rubber Blocks, Proceedings of the Institution of Mechanical Engineers, Vol. 173:3, 111-122.

Gent A.N., Meinecke E.A. (1970), Compression, Bending and Shear of Bonded Rubber Blocks, Polymer Engineering and Science, Vol. 10: 2, 48-53.

Haringx J.A. (1947), On Highly Compressible Hellical Springs and Rubber Rods and Their Application for Vibration-Free Mountings – Part III, Philips Research Reports, Vol.4: 206-220.

Imbimbo M., De Luca A. (1998), F.E. Stress Analysis of Rubber Bearings under Axial Loads, Computers and Structures, Vol. 68: 1, 31-39.

Kelly J.M. (1994), “The Influence of Plate Flexibility on the Buckling Load of Elastomeric Isolators”, Report UCB/EERC-94/03, Earthquake Engineering Research Center, University of California, Berkeley, California, USA.

Kelly J.M. (1997), Earthquake Resistant Design with Rubber, Springer-Verlag, London.

Kelly J.M. (1999), Analysis of Fiber-reinforced Elastomeric Isolators, Journal of Seismology and Earthquake Engineering, Vol. 2: 1, 19-34.

Kelly J.M. (2002), Seismic Isolation Systems for Developing Countries. Earthquake Spectra, Vol.18: 3, 385-406.

Kelly J.M., Takhirov S.M. (2002), Analytical and Experimental Study of Fiber-reinforced Strip Isolators, PEER Report 2002/11, Pacific Earthquake Engineering Research Center, University of California, Berkeley, California, USA.

Kelly J.M. (2007), PowerPoint Presentation, A Short Course on the Theory and Implementation of Rubber-Based Isolation Systems, June 2007, Ankara, Turkey.

Naeim F., Kelly J.M. (1999), Design of Seismic Isolated Structures, John Wiley & Sons, Inc, New York.

Pinarbasi S. (2007), A New Formulation for the Analysis of Bonded Elastic Layers, Ph. D. Dissertation, Middle East Technical University, Ankara, Turkey.

Pinarbasi, S., Mengi, Y. (2008), Elastic Layers Bonded to Flexible Reinforcements, International Journal of Solids and Structures, Vol. 45: 794-820.

Tsai H.-C., Kelly J.M. (2001), Stiffness Analysis of Fiber-Reinforced Elastomeric Isolators, PEER Report 2001/05, Pacific Earthquake Engineering Research Center, University of California, Berkeley, California, USA.