6th International Congress on Advances in Civil Engineering, 6-8 October 2004 Bogazici University, Istanbul, Turkey

Investigation of Compressive Stiffness of Elastomeric Bearings

S. Pinarbasi, U. Akyuz Middle East Technical University, Department of Civil Engineering, Structural

Mechanics Laboratory, Ankara, Turkey

Abstract Compressive stiffness is one of the key parameters in the design of both seismic isolation bearings and bridge bearings. For this reason, the prediction of the behavior of multilayered elastomeric bearings under compressive loads is considerably important for their design. Elastomeric bearings are usually composed of thin rubber layers interleaved by and bonded to interior steel shim plates. Since the steel plates are rigid compared to the rubber layers, the behavior of an elastomeric bearing is considered to be governed by the behavior of a typical rubber layer bonded to rigid end plates. Several researchers have tried to obtain analytical solutions for compressive stiffness of constrained rubber blocks. In almost all of these analytical treatments, rubber is assumed to have linear and isotropic behavior and derivations have been made for infinitesimal strains. To introduce seismic isolation technique to Turkey, a comprehensive research program has been initiated in the Structural Mechanics Laboratory of the Middle East Technical University. One phase of this program comprises the investigation of the compressive behavior of different types of elastomeric bearings by both experimental and analytical studies. In this paper, only some part of the experimental results and their comparisons with the predictions of existing linear solutions in literature are presented.

Introduction Since the vertical frequency of a structure isolated with elastomeric bearings is mainly controlled by the stiffness of the bearings in the vertical direction; the prediction of the behavior of multilayered elastomeric bearings under compressive loads gains importance in seismic isolation design (Kelly, 1997). Similarly, compression stiffness is usually one of the key parameter in the design of bridge bearings, whose excessive deformation can lead to the serviceability failure of the bridge (Stanton, 1982). Compressive stiffness/compression modulus of an elastic layer can be simply defined as “the ratio of average compressive stress to average compressive strain” (Constantinou, 1992). It is known that the compressive stiffness of an elastic layer that

is bonded to and compressed between two rigid plates is much higher than that of the corresponding unbonded layer. Main reason of this increase is the restriction of lateral expansion of material at the bonded surfaces. It is also known that this effect on compressive stiffness of an elastic layer becomes more pronounced if the material is nearly incompressible (Tsai, 1998; Koh, 2001). Since elastomeric bearings used in seismic isolation applications are usually composed of thin rubber layers interleaved by and bonded to steel shim plates and since these interior steel plates are rigid compared to the rubber layers, the behavior of an elastomeric bearing under axial loading is considered to be governed by the behavior of a typical rubber layer bonded to rigid plates (Constantinou, 1992). Several many researchers have tried to investigate the compressive behavior of bonded rubber layers. Even though the behavior of rubber is indeed highly nonlinear and it can undergo considerable finite deformations, in most of the previous analytical treatments, rubber has been assumed to have linear and isotropic behavior and derivations have been made for infinitesimal strains because the use of finite strain analysis with nonlinear constitutive models usually lead to highly nonlinear and complex equations (Kelly, 1997). In fact, in many cases, some additional simplifying assumptions have had to be made on the deformed shape or on the state of stress to be able to obtain simple formulations for design purposes (Constantinou, 1992). In the following sections, before presenting the results of the experimental study conducted to determine the compression moduli of two different types of elastomeric bearings, reduced-scale square and full-size rectangular bridge bearings, analytical solutions available in literature -with particular emphasize on the expressions derived for square and rectangular bonded layers- are going to be summarized briefly. At the end, the predictions of existing solutions will be compared to the experimental results

Compression Stiffness of Bonded Rubber Blocks Incompressible Formulations By using an approximate method of treatment and assuming “strict incompressibility” (ν=0.5), Gent & Lindley (Gent, 1959) derived the following approximate relations for the compression moduli of bonded circular discs (Ec(circular)) and infinitely long rectangular (Ec(inf strip)) bars in terms of Young’s modulus “E” of rubber and the shape factor “S”. For a rubber layer constrained at the top and bottom faces, shape factor is defined as the ratio of one constrained area to load-free area.

)S34

34(EE 2

)strip(infc += , )S21(EE 2)circular(c += (1)

The method of treatment developed by Gent&Lindley, which has been later called as the “pressure” method, is based on three fundamental assumptions. As kinematics assumptions, (a) horizontal plane sections parallel to the rigid plates are assumed to remain plane and parallel and (b) lateral surfaces are assumed to take a parabolic shape in the deformed configuration. As a stress assumption, it is assumed that the normal stress components are both the same and equal to the mean pressure in all three orthogonal directions. In the past fifty years, many researchers took this method of treatment as a basis in their formulations and extended the theory to the bonded blocks of different geometries and to the elastic compressible layers.

As an example, Gent&Meinecke (Gent, 1970) extended the pressure method to bonded incompressible rubber blocks for many cross sectional shape. Their formulation resulted in the following expression for a rectangular block with side dimensions of “2a” and “2b” and with a thickness of “h”.

ππ

−+

+++−= ∑

=))

a2bntanh(

n1

ba1921(

h)a2(

31

h2bahab

32

34EE

5,3,1n552

2

222

2

)rect(c (2)

The authors also presented the following formula for a square layer with a side dimension of “2a”, which is a special case of the above rectangular formulation.

+= 2

2

)square(ch

)a2(141.01EE (3)

Taking the studies of Gent&Lindley and Gent&Meinecke as basis, Kelly et. al. developed a theoretical approach to derive the compression moduli of bonded rubber blocks not only for incompressible case but also for compressible case (Kelly, 1997). The following expression derived by Koh&Kelly (Koh, 1989) corresponds to the square incompressible case.

)

)tanh(11S321(EE

n

n

1n 4n

2)square(c

αα

−∑α

+=∞

= where π−=α )

21n(n (4)

The authors also suggested the following simplified formula for the special square case:

)S25.21(EE 2)square(c += (5)

Compressible Formulations Although the assumption of incompressibility is accepted as a realistic assumption for low shape factor rubber units, the contribution of bulk compression of rubber to the total compression of the block should be considered for very thin rubber blocks. When the shape factor of a rubber layer is relatively high, incompressibility assumption leads to the overestimation of compressive stiffness of the bonded layer, which, in turn, leads to the overestimation of both buckling load and resonant frequency (Koh, 2001). Gent&Lindley (Gent, 1959) proposed the following “ad-hoc” modification for the prediction of compression modulus including the bulk compressibility (E'c) to the known incompressible Ec formulations, where “K” denotes the modulus of bulk compression.

K1

E1

E1

c'c

+= (6)

Koh&Kelly (Koh, 1989) obtained the following closed form solution for the compression stiffness of compressible square bonded rubber blocks in terms of Poisson’s ratio “ν”, Young’s Modulus “E” and shape factor “S” by including the effect of bulk compressibility to the “pressure solution.

ββ

−βαν+

+= ∑∞

=1n n

n2n

2n

2)square(c

)tanh(11S

1481EE (7)

where 2/1

22nn S

12172

ν+ν−+α=β and π−=α )

21n(n

Besides the above solution, the authors obtained two more solutions for bonded square layers. In the first solution, they eliminated the stress assumption of the pressure solution while retaining the basic kinematics assumptions.

ββ

−βαν−

ν+= ∑∞

=1n n

n2n

2n

22

2

)square(c)tanh(

11S1

961EE

where 2/1

22nn S

12124

ν−ν−+α=β and π−=α )

21n(n

(8)

In the other solution, they further eliminated the parabolic bulging assumption,

∑∞

= +π

=

5,3,1j j22

)square(c

)f1(Ej81

11E

where ∑∞

=

ββ

−βαν−

νπ=1n n

n2n

2n

22

222

j)tanh(11S

1j8f ,

2/12222

nn S1

21j2

ν−ν−π+α=β and π−=α )

21n(n

(9)

Recently, Koh&Lim (Koh, 2001) obtained analytical solutions for the compression stiffness of bonded rectangular layers in terms of “E” and “ν”, using a similar approach used in (Koh, 1989) for square pads. The following double series expressions are derived for a rectangular layer of thickness “t”, width “2a” and length “2b”.

( ) )F

1ab12

t1tab

1961(EE

mn2n

2m1n

12n

2m

2

1m22

2

)rect(cβα

Ωβ+Ωα+

ν−ν+= ∑∑

∞

=

−∞

=

where ( )[ ]

( )

Ωβ+Ωαν−ν−+

ν−ν−+

Ωβ+Ωαν−+βα=

−

−

12n

2m2

24n

24m

2n

2m

2

mn

143

121

tab12

)1(2ab6t

F ,

π−=α )21m(m , π−=β )

21n(n , and a/b=Ω

(10)

Most recently, using the Gent’s pressure method and including the rubber compressibility, Banks et. al. (Banks, 2002) obtained the following relation for Ec of bonded rectangular blocks, with side dimensions of “l=2a” and “w=2b” and thickness of “h”, in terms of the material moduli “G”, “E” and “K”.

λλπ

−λπ

+

+++−

=

∑∞

= 5,3,1n

n3n

222n

222

222

2

)rect(c

2ltanh

nw8

nwl4

wltG24

h2bahab

32

34E

E where )KtG12

wn( 22

222n +π=λ (11)

Experimental Program at METU A comprehensive research program has been initiated in the Structural Mechanics Laboratory of the Middle East Technical University (METU), Ankara, Turkey to investigate the behavior of different types of elastomeric bearings under different loading conditions. One phase of this program, which is still continuing, comprises the investigation of the compressive behavior of elastomeric bearings. Test Bearings In the initial stage of the program, two different types of elastomeric bearings were tested to failure under compression. Having an overall side dimension of 150 mm, reduced-scale square bridge bearings (Figure 1.a) are composed of a single 2 mm thick steel shim plate (shim side dimension = 130 mm) between 20 mm thick rubber layers, which leads to a considerably low shape factor, S=1.63. It should be noted that rubber cover at the lateral sides of these bearings was removed prior to testing. Besides the square bearings, real-size rectangular bridge bearings with overall dimensions of 200 mm × 400 mm × 52 mm were also tested. Composed of relatively thin rubber layers, 8 mm, sandwiched between 2 mm thick steel plates, as shown in Figure 1.b, these multi-layer bearings have a moderate shape factor, S=8.33. While both types of bearings are composed of natural rubber, square bearings with a shear modulus of G=0.65 are softer than the rectangular bearings of shear modulus of G=1.00. The above-mentioned values of shear modulus for each type of bearings have been determined in the shear tests of the bearings (Pinarbasi, 2003; Pınarbaşı 2004).

a. Square bearings b. Rectangular bearings

Figure 1. Test bearings Test Setup Test Procedure After placing between two rigid loading plates, the bearings were concentrically compressed at the test setup (Figure 2.a), which was developed within the research program. The test machine has a compressive capacity of 3000 kN. In the tests, the amount of compressive deformation applied to the test bearings are measured by means of the four LVDT’s (Linear Variable Displacement Transducers) symmetrically placed under the bottom rigid plates (Figure 2.b). The corresponding load values are measured by the load cell incorporated into the test machine. In the compression failure tests, the test bearings were loaded monotonically up to failure, which is defined by either shim plate rupture or debonding of elastomer from the shim plates. Failure could reach in reduce-scale square bearings, which failed due to

2 mm steel shim

42 mm

5x2 mm steel shims

top & bottom rubber layers: 2x5 mm interior rubber layers: 4x8 mm

52 mm

shim rupture. For the real size rectangular bearings, the tests had to be terminated when the capacity of the loading system was reached

a. Test setup b. Loading and measurement systems

Figure 2. Test setup, loading and measurement systems Test Results Loading parts of the typical stress-strain curves for square and rectangular bearings in compression failure tests are presented in Figure 3a and 3b, respectively.

a. Square bearings b. Rectangular bearings

Figure 3. Typical loading curves for test bearings in compression failure tests

As stated by Musceralla&Yura (Musceralla, 1995), ASTM does not include a standard test for the determination of compression moduli of elastomeric bearings. On the other hand, it is fairly common to determine the compression stiffness of bearings from the regression analysis of test results in the stress range of approximately 3-10 MPa because this range includes the most common design limits for typical bridge

0

20

40

60

80

100

120

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Vertical Strain

Ver

tical

Str

ess (

MPa

)

0

5

10

15

20

25

30

0.00 0.02 0.04 0.06 0.08 0.10

Vertical Strain

Ver

tical

Str

ess (

MPa

)

bearings. In this stress range, the behavior of the rectangular test bearings is almost linear and the corresponding strain range (εc:0.028-0.049) is reasonably small. The regression analysis in this stress range leads to a compression modulus of Ec=335.22 MPa for the rectangular bearings. In contrast to the rectangular bearings, for the square bearings whose shape factor is considerably small, stress-strain curve is highly nonlinear in this range. Moreover, the corresponding strain range (εc:0.23-0.42) is significantly large. For this reason, the compression stiffness of the square bearings was determined from the regression analysis of the data at a lower strain range. The behavior of bearings is observed to be somewhat linear in the strain range 0.05-0.20, which corresponded to a stress range of approximately 0.5-3 MPa. The result of the regression analysis in this range gave a linear, small deformation compression modulus of Ec=12.93 MPa for the square bearings. Predictions of Existing Solutions For compression moduli of the square and rectangular test bearings, the predictions of existing solutions, which were computed from the expressions given in the previous section, are presented in Table 1. While for rectangular bearings, only the rectangular formulations could be used, it was possible to use the rectangular formulations besides the square formulations for square bearings. For compressible formulations to be used, the bulk moduli of the elastomers used in the bearings should be known. However, the measurement of the bulk modulus of an elastomer is considerably difficult. Roeder et al (Roeder, 1987) stated a value of 0.001 for E/K ratio of elastomers used for typical bridge bearings. This value of E/K ratio leads to the values of Ks=1950 and Kr1=3000 for bulk moduli of the square and rectangular test bearings, respectively. Although the square bearings may not be considered as a typical bridge bearing due to their considerably low shape factors, the effect of bulk compression is indeed so small for these bearings that the approximation added to the compressible formulations by the selection of K can easily be accepted. On the other hand, the results of the compressible formulations for the rectangular bearings are highly sensitive to the selection of the K value. For this reason, in addition to Kr1, compressible formulations were also computed for another K value, Kr2=2070. This value of K was previously reported by Chalhoub&Kelly (Chalhoub, 1990) for a natural rubber compound used in the half-scale seismic isolation bearings tested at the University of California at Berkeley. This value of K, indeed, appears to be a better selection for bulk modulus of the tested rectangular bearings since it corresponds to a natural rubber compound of shear modulus of 0.952 MPa, which seems to be very close to the compound of the tested rectangular bearings composed of natural rubber with shear modulus of 1.0 MPa. Comparison of Experimental and Analytical Results The predictions of the available incompressible (Ec,i) and compressible (Ec,c) solutions for the compressive moduli of the test bearings are compared with the experimentally obtained (Ec,e) results and listed in Table 1. In this table in addition to the ratios of incompressible and compressible predictions to the experimentally obtained results; i.e., Ec,i/ Ec,e and Ec,c/ Ec,e, respectively, the ratios of the compressible predictions to the incompressible predictions; i.e., Ec,c/ Ec,i are also provided in order to be able to evaluate the effect of compressibility in the analytical predictions.

Table 1. Comparison of analytical and experimental results for the test bearings

(a) SQUARE BEARINGS (Ec,e=12.93 MPa)

Ec,i (MPa)

Ec,c (MPa)

Proposed By Eq. #

K→∞ K=1950(MPa)

Ec,c/Ec,i Ec,i/Ec,e Ec,c/Ec,e

(Gent, 1970) 2 13.532 NA** NA 1.05 NA (Gent, 1970) 3 13.531 NA NA 1.05 NA (Koh, 1989) 4 13.532 NA NA 1.05 NA (Koh, 1989) 5 13.536 NA NA 1.05 NA (Gent, 1959) 6* 13.532 13.439 0.99 1.05 1.04 (Koh, 1989) 7 13.532 13.254 0.98 1.05 1.03 (Koh, 1989) 8 13.532 13.426 0.99 1.05 1.04 (Koh, 1989) 9 13.916 13.805 0.99 1.08 1.07 (Koh, 2001) 10 13.699 13.591 0.99 1.06 1.05 (Banks, 2002) 11 11.289 11.289 1.00 0.87 0.87

(b) RECTANGULAR BEARINGS (Ec,e=335.22 MPa)

Ec,i (MPa)

Ec,c1 (MPa)

Ec,c2 (MPa)

Proposed By Eq #

K→∞ K1=3000 (MPa)

K2=2070 (MPa)

Ec,c1/Ec,i Ec,c2/Ec,i Ec,i/Ec,e Ec,c1/Ec,e Ec,c2/Ec,e

(Gent, 1970) 2 431.978 NA** NA NA NA 1.29 NA NA (Gent, 1959) 6* 431.978 377.605 357.395 0.87 0.95 1.29 1.13 1.07 (Koh, 2001) 10 432.121 362.877 338.747 0.84 0.93 1.29 1.08 1.01 (Banks, 2002) 11 429.681 362.277 338.562 0.84 0.93 1.28 1.08 1.01

SYMBOLS AND NOTES: Ec,e : Experimentally obtained compression modulus Ec,I : Analytically predicted compression modulus for incompressible case Ec,c : Analytically predicted compression modulus for square bearings for compressible case (K=1950 MPa) Ec,c1 : Analytically predicted compression modulus for rectangular bearings for compressible case (K1=3000 MPa) Ec,c2 : Analytically predicted compression modulus for rectangular bearings for compressible case (K2=2070 MPa) * : In Eq 6, as Ec the incompressible result obtained from Eq. 2 is used. ** : Not Applicable (NA) The results obtained for the compressive stiffness of the square test bearings are summarized in Table 1(a). All incompressible solutions result in an approximate value of 13.53 MPa, which is approximately 5% greater than the experimental result. From the compressible solutions, the initial two solutions derived by (Koh, 1989); i.e., equations 7 and 8, successfully converge to the above-mentioned value for incompressible case as K approaches to infinity. Except the equation proposed by (Banks, 2002), all compressible formulations yielded very close results (changing from

13.254 to 13.805 MPa) for the compressible compressive stiffness of the bearings. The prediction of the last expression derived by (Koh, 1989) and the prediction of the formula derived by (Koh, 2001), corresponding to the equations (9) and (10) respectively, are a bit higher than the predictions of the other formulations but even the ratios of these values to the experimentally measured result are not more than 1.07. It should also be noted that these equations yield slightly higher values for incompressible case, as well. The comparison of the analytical results with each other clearly indicates that inclusion of bulk compressibility in the derivations affects the results almost insignificantly (less than 3%). Furthermore, the derivation given in eqn (11) seems not to be applicable to the low shape factor bearings since not only its compressible prediction is much lower than the predictions of the other methods but also it cannot converge to the incompressible solution as K goes to infinity. Specific to rectangular bearings, there are only one incompressible (Gent, 1970) and two compressible (Koh, 2001 and Banks, 2002) solutions in literature. Surely, Gent&Lindley’s ad-hoc modification (Gent, 1959) can also be used provided that an incompressible prediction is available. The predictions of these four expressions for the rectangular test bearings are summarized in Table 1(b). Since compressible formulations were solved for two different values of K values, two different Ec,c values were obtained. The predictions obtained for K=3000 MPa are denoted as Ec,c1 while the predictions obtained for K=2070 MPa are denoted as Ec,c2. As it can easily be understood from the table, for the rectangular test bearings, the available compressible formulations successfully converges to the incompressible value, which is approximately equal to 432 MPa, as K approaches to infinity. It is seen that the incompressible predictions overestimate the experimentally measured compression stiffness by about 30%. Unlike the case of the square test bearings, the compressible formulations used for the prediction of Ec for the rectangular test bearings are considerably sensitive to the value of K. The compressible predictions are significantly smaller than the incompressible values, and as the value of K decreases, this difference increases. The predictions based on the ad-hoc modification proposed by Gent&Lindley (Gent, 1959) are somewhat higher than the predictions of the other rectangular compressible formulations, which yield almost the same result even for different K values. Although the compressible predictions of (Koh, 2001) and (Banks, 2002) for K=3000 MPa still overestimates the experimentally obtained result (about 335 MPa) by about 8%, their predictions for K=2070 are only 1% greater than the experimental result. Conclusions Consequently, based on the limited test results, the following conclusions can be made. 1. In the linear range, even the existing incompressible formulations satisfactorily

predicts the compressive stiffness of the low shape factor square test bearings since the inclusion of compressibility to the solutions changes the results only slightly. On the other hand, for the prediction of Ec at higher strains, nonlinear formulations must be used, since the behavior is highly nonlinear.

2. For the case of the moderate shape factor rectangular bearings tested in the study, incompressible formulations highly overestimate the compression stiffness of the bearings. On the other hand, the compressible formulations successfully predict the experimentally obtained value in the linear range for a value of bulk modulus of 2070 MPa, which was reported by Chalhoub&Kelly (Chalhoub, 1990).

References Banks H.T., Pinter G.A., Yeoh O.H. (2002). Analysis of Bonded Elastic Blocks. Mathematical and Computer Modelling, 36, 875-888. Chahoub M.S., Kelly J.M. (1990). Effect of Bulk Compressibility on the Stiffness of Cylindrical Base Isolation Bearings. International Journal of Solids and Structures, Vol. 26, No. 7, 743-760. Constantinou, M.S., Kartoum, A., Kelly J.M. (1992). Analysis of Compression of Hollow Circular Elastomeric Bearings. Engineering Structures, Vol. 14, No 2, 103-111. Gent, A.N., Lindley, P.B. (1959). The Compression of Bonded Rubber Blocks. Proceedings of the Institution of Mechanical Engineers, Vol. 173, No 3, 111-122. Gent A.N., Meinecke, E.A. (1970). Compression, Bending and Shear of Bonded Rubber Blocks. Polymer Engineering and Science, 10(1), 48-53. Kelly, J.M. (1997). Earthquake Resistant Design with Rubber, Springer-Verlag London Limited. Koh C.G, Kelly J.M. (1989). Compression Stiffness of Bonded Square Layers of Nearly Incompressible Material. Engineering Structures, Vol 11, 9-15. Koh C.G, Lim H.L. (2001). Analytical Solution for Compression Stiffness of Bonded Rectangular Layers. International Journal of Solids and Structures, Vol. 38, 445-455. Muscarella J.V., Yura J.A. (1995). An Experimental Study of Elastomeric Bridge Bearings with Design Recommendations. Research Report 1304-3, Center for Transportation Research, Bureau of Engineering Research, the University of Texas at Austin. Pinarbasi, S., Akyuz, U. (2003). Seismic base isolation: philosophy, development and applications. International Conference on Earthquake Engineering, 40 SWEE, Ohrid, Macedonia. Pınarbaşı, S., Akyüz, U. (2004). Çelik Plakalı Elastomerik Köprü Yastıklarının Düşük Sıcaklıkta Kayma Deneyleri. Turkish Civil Engineering 17th Technical Congress and Exhibition, Turkish Chamber of Civil Engineers, Istanbul, Turkey. Roeder C.W., Stanton C.F., Taylor A.W. (1987). Performance of Elastomeric Bearings. National Cooperative Highway Research Program Report 298, Transportation Research Board, National Research Council, Washington D.C. Stanton, J.F., Roeder, C.W. (1982). Elastomeric Bearings, Design, Construction and Materials. National Cooperative Highway Research Program Report 248, Transportation Research Board, National Research Council, Washington D.C. Tsai H.C., Lee C.C. (1998). Compressive Stiffness of Elastic Layers Bonded Between Rigid Plates. International Journal of Solids and Structures, Vol. 35, No 23, 3053-3069.