Vibrations induce input energy to structures, and if they are not well isolated against vibrations, they may fail and cause catastrophic consequences. An example of a structure that withstood such fate includes the Onagawa nuclear power station (Ibrion et al. 2020). Vibration isolation and damping systems come in a variety of types. For instance, some use viscous damping to reduce vibrations due to friction (Graf and Lankenau 2021) and some use the concept of sloshing liquid tanks to reduce vibrations due to lateral loads such as wind and earthquake ground motions (Tsao and Huang 2021). The vibration isolation system uses both viscous damping and friction to reduce input energies originating from the vehicular and lateral loads in the bridge industry.

Most of the existing bridges use elastomeric bearings, which rely on rubber or synthetic elastomer to absorb the energy originating from vehicles, wind loads, earthquake ground motions, etc. These bearings come in two varieties: high damping rubber bearings (HDRB) and lead rubber bearings (LRBs). Other systems also exist, but they are not as commonly used.

Elastomeric bearing pads are cost-effective to reduce noise and vibration in transportation applications. They are easy to install and do not require special tools or equipment. In addition, they are reusable and can be moved from one location to another as needed.

Figure 1 shows the installation of elastomeric bearings in a steel girder bridge under construction and a completed reinforced concrete bridge box with already installed rubber bearings.

Fig. 1
figure 1

Elastomeric bearings installation in bridge structures

The LRBs (Fig. 2) are a type of energy-absorbing isolation system that has been developed to increase the energy dissipation capacity of rubber bearings. It is widely used in New Zealand and consists of a steel-reinforced elastomeric bearing with a lead plug inserted in its center. The lead plug is firmly pressed into a hole in the center, and the lead forms a positive key between the steel plates within the bearing, which helps to confine them.

Fig. 2
figure 2

Elastomeric bearing on the left and lead rubber bearing on the right (source:

The effects of the lead core temperature on the performance of lead rubber bearings have been the subject of few studies. Kalpakidis and Constantinou (2009a) developed a theory for predicting the temperature rise of the lead core and the reduction in characteristic strength and energy dissipation history of lead-rubber bearings subjected to cyclic motion. The theory includes a simplified solution in which an explicit closed-form equation for the temperature rise was derived. The theory is useful in predicting the cyclic behavior of lead-rubber bearings for simplified analysis, extrapolating experimental data from one scale to another, and developing models for dynamic response history analysis of seismically isolated structures that account for the time-dependent mechanical properties of the bearings.

The authors’ additional study (Kalpakidis and Constantinou 2009b) aimed to validate the theory presented in their previous research paper. The study used finite-element heat transfer analyses and experimental results from testing six different lead-rubber bearings. The study’s results confirmed the theory’s accuracy and showed that testing at quasi-static conditions results in a lesser increase in the lead core temperature and a lesser reduction of dissipated energy per cycle and characteristic strength of lead-rubber bearings. This difference in reduction in EDC is dependent on the geometry of the bearing, the conditions of testing, and the number of cycles imposed.

Kalpakidis et al. (2010) developed a model of the hysteretic behavior of lead rubber bearings that accounts for the temperature increase in the lead core based on first principles. The model can predict the lead core’s instantaneous temperature and its instantaneous effect on the characteristic strength of the bearing. The paper demonstrates that the model is in good agreement with experimental results. The paper further uses the model to examine the effects of lead core heating on the dynamic response of an isolated structure. The study’s results demonstrate that bounding analysis produces conservative results for predicting isolation system displacement demand, peak shear force, and peak structural responses.

The HDRB system is similar to the LRB system, but the rubber is thinner and has steel shims between the layers; this increases the vertical stiffness of the bearing while also allowing for a certain amount of lateral deformation. The LRB system uses thicker rubber sheets, which provide more lateral deformation, but also have a lower vertical stiffness.

Both types of bearings work by deflecting the energy from the earthquake away from the structure rather than absorbing it. This is done by making the structure vibrate at a frequency different from the earthquake’s frequency. The isolation system does not work as well when there is high energy in the earthquake motion at frequencies close to the isolator’s fundamental frequency. The MRPRA developed a natural rubber with enough inherent damping to overcome this issue. This damping is increased by adding extra-fine carbon block, oils or resins, and other proprietary fillers. The increased damping helps to suppress any possible resonance at the isolation frequency.

An isolator’s lateral stiffness is minimal compared to its vertical stiffness, making it almost elastic for lateral deformations within its radius. This allows the isolator to easily deflect the energy from the earthquake, which keeps the structure safe.

Yuan et al. (2020) developed a new polyurethane elastomer (PUE) material with improved shear performance. This material was used to create isolation bearings with increased vertical capacity. The mechanical properties of the PUE material were investigated, and a rheological shear constitutive model was proposed. The accuracy of the proposed constitutive model was verified by the isolation bearings experimental. Results from the experimental investigations and a basic shear mechanical study of the new PUE material suggest that the PUE bearing has a large vertical bearing capacity, promising horizontal deformability and energy dissipation capacity.

Another improvement to the elastomeric bearing has been recently proposed by Tan et al. (2022). In their publication, the authors discussed the development of a new elastomeric laminated bearing utilizing a core-and-filler system instead of a lead core to improve bearing performance. Two types of filler, namely granular and shape memory polymer, are implemented. Granular filler is prepared using silica sand, while shape memory polymer filler is prepared using epoxy resin. Also, a steel core is implemented to improve the stiffness of the filler. The performance of the proposed bearing utilizing the core-and-filler system is evaluated using finite element simulation. The numerical results revealed the efficiency of bearing with the proposed system by providing considerable damping and stiffness. The replacement of lead core with filled granular and shape memory polymer showed improvement in terms of stiffness, proving that the core-and-filler system effectively limits lateral displacement. Also, the prototype of base isolation devices with both granular and shape memory polymer fillers is fabricated and tested via cyclic shear test. The results are compared with finite element analysis results, and good agreement between experimental tests results and numerical simulation response is shown. The experimental testing results proved that implementing the core-and-filler system improves the lateral resistance of the proposed elastomeric bearing. Overall, it can be concluded that the implementation of the core-and-filler system provides a reliable improvement to the performance of conventional elastomeric bearing and can be considered as an alternative system to lead core rubber bearings.

Kumar and Whittaker (2018) discussed the importance of verifying and validating mathematical models of elastomeric seismic isolation bearings and presented a plan for doing so. The authors noted that advanced models of elastomeric seismic isolation bearings were implemented in three commercial software packages (OpenSees, ABAQUS, LS-DYNA) and that these models were verified and validated per ASME best practices and guidelines. The authors further noted that the component of the mathematical model that contributes most to the error is the heating of the lead core in the LRB device and that code-to-code verification shows good agreement between the three software packages. Verified models were first calibrated using experimental data to determine unknown model parameters and to characterize the behavior of elastomeric bearings in tension and tension/shear.

Buckling is a phenomenon that causes instability to structural elements (Schilling and Mittelstedt 2020) when not addressed during the design stage. Rahnvard et al. (2020) used finite element analysis to investigate the stability of seismic steel-rubber base isolators. They looked at the effect of single and multiple rubber cores on the stability of elastomeric bearings and found that the use of single and multiple rubber cores increases isolator stability due to the large critical vertical loads. They also found that the equations presented in the reduced area formula provide the same stability results for isolators with and without a core. By comparing all square and circular isolator models with four and nine rubber cores (with the same core area), no noticeable difference in their stability was found. Kazeminezhad et al. (2020) have also used finite element analysis. The authors discussed rotation effects on the vertical stiffness of elastomeric bearings. It is found that rotation can increase the vertical stiffness of the isolator when the critical lateral displacement is reduced to a specific value. This information could help design base isolation systems.

Earlier studies on individual elastomeric bearings (Koh and Kelly 1987; Aiken et al. 1989; Naeim and Kelly 1999) showed that these bearing devices exhibit a reduction in their lateral stiffness under increasing vertical compressive loads. Additionally, experiments conducted by Kelly et al. on LSF (low shape factor) elastomeric (Aiken et al. 1989) showed that load-history has a negligible effect on damping behavior of bearings and has some effect on reducing stiffness; however, the effect of loading rate on the response of the bearings was not significant (Aiken et al. 1989). In the same test program, Kelly et al. observed an increasing stiffness at large strains (over 125−150 %, which was demonstrated by the strain hardening characteristic of the elastomer. Furthermore, the vertical stiffness was found to be largely independent of any horizontal displacement imposed on the bearing (Aiken et al. 1989).

The author has observed some contradictions in results published by researchers in a wide range of documents; as an example, in the EERC-04-03 research report (Kelly and Takhirov 2004), Kelly et al. mentioned that:“…the theory also predicts that the vertical stiffness of the bearing is strongly dependent on the shear deformation…The theoretical vertical stiffness significantly decreases as the shear deformation of the bearing increases” this contradicts what was mentioned before in the EERC-89-13 research report (Aiken et al. 1989). However, some of the contradicted results may be due to the type of the bearing device itself and material properties from bearing to the bearing; any generalization of results should be with careful attention. In some tests, rubber bearings have been observed shown to soften in the vertical direction at large lateral deformations (Ryan and Chopra 2005). In some tests of pure tension, rubber bearings tend to cavitate or form small cavities in the rubber that blow out from negative pressure and link together to form cracks in the rubber matrix (Ryan and Chopra 2005), but other tension tests showed that no damage of this kind had been observed.

The characteristic strength of Lead Rubber Bearings (LRB), defined as the maximum force at zero horizontal bearing displacement, has been observed to increase with the increasing vertical (axial) load on the bearing but decreases from cycle to cycle; this is what was observed under a constant low-speed test (Benzoni and Casarotti 2008). The energy dissipated per cycle is related to the characteristic strength.

Several models were proposed for stability analysis of elastomeric bearings, among them, the linear models based on the Haringx theory developed by Koh and Kelly and its nonlinear version (Koh and Kelly 1987; Aiken et al. 1989), the two-spring models (Koh and Kelly 1987; Aiken et al. 1989) and the partially nonlinear extension of the two-spring model developed by Ryan (2005).

In this paper, an extension to the nonlinear extension of the two-spring model proposed by Ryan (2005) to the original model developed by Koh and Kelly (1987) is presented first. Then, two new fully nonlinear models suitable for elastomeric bearings and lead-rubber bearings are proposed to adequately account for the interaction between the horizontal and vertical loads. The accuracy of the models is verified using laboratory tests by Koh and Kelly (1988, 1987), Buckle et al. (2002), Warn et al. (2007).

Moreover, the effect of the interaction on the bearing’s response parameters (horizontal stiffness, vertical stiffness, and overall stability) is investigated. Next, based on Hill’s equation (Hill 1910), analytical models are then developed to predict the horizontal and vertical stiffnesses as a function of the critical buckling load. The two proposed mathematical models can readily be incorporated into open-source structural analysis software, such as OpenSees.

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