*Article* **Vertical Seismic Isolation Device for Three-Dimensional Seismic Isolation of Nuclear Power Plant Equipment—Case Study**

**Gyeong-Hoi Koo 1, \*, Jin-Young Jung 1 , Jong-Keun Hwang 2 , Tae-Myung Shin <sup>3</sup> and Min-Seok Lee 4**


**Abstract:** The purpose of this study was to develop a vertical seismic isolation device essential for the three-dimensional seismic isolation design of nuclear power plant equipment. The vertical seismic isolation device in this study has a concept that can be integrally combined with a conventional laminated rubber bearing, a horizontal seismic isolator with a design vertical load of 10 kN. To develop the vertical seismic isolation device, the vertical spring and the seismic energy dissipation device capable of limiting the vertical displacement of the spring were designed and their performances were verified through actual tests. In this study, the target elevation of the floor is 136 ft, where safety-related nuclear equipment, such as cabinet and remote shutdown console, etc., is installed. The sensitivity studies were carried out to investigate the optimal design vertical isolation frequencies for the target building elevation. Based on the results of the sensitivity study, a disc spring and a helical coil spring were selected for the vertical stiffness design, and the steel damper was selected for the seismic energy dissipation, and their performance characteristics were tested to confirm the design performance. For the steel damper, three types were designed and their energy dissipation characteristics by hysteretic behavior were confirmed by the inelastic finite element analyses and the tests in static fully reversed cyclic conditions. Through the study of the vertical seismic isolation device, it was found that 2.5 Hz~3.0 Hz is appropriate for the optimal design vertical isolation. With results of the vertical seismic isolation performance analysis, the appropriate number of steel dampers are proposed to limit the vertical seismic displacement of the spring within the static displacement range by the design vertical load.

**Keywords:** three-dimensional (3D) seismic isolation; vertical seismic isolation device; disc spring; helical coil spring; steel damper; laminated rubber bearing; seismic energy dissipation; nuclear power plant equipment; seismic isolation frequency; hysteretic behavior

#### **1. Introduction**

After the Fukushima nuclear power plant accident, nuclear power plant safety improvement against earthquakes has emerged as a major issue. The current Safe Shutdown Earthquake (SSE) design basis required for nuclear power plant design is 0.3 g, which is 50% higher than the previous 0.2 g. In recent years, seismic design requirements are being strengthened to ensure the seismic safety of nuclear power plants in case of beyond-designbasis earthquakes, and various methods are being studied to solve this issue [1–3].

In general, it is true that the improvement of seismic capacity of nuclear power plants has been mainly focused on plant buildings and structures. To achieve this goal, the seismic base isolation design technology using laminated rubber bearings (LRBs) has been studied worldwide for a long time as one of the measures to ensure the safety of nuclear power

**Citation:** Koo, G.-H.; Jung, J.-Y.; Hwang, J.-K.; Shin, T.-M.; Lee, M.-S. Vertical Seismic Isolation Device for Three-Dimensional Seismic Isolation of Nuclear Power Plant Equipment—Case Study. *Appl. Sci.* **2022**, *12*, 320. https://doi.org/ 10.3390/app12010320

Academic Editors: Giuseppe Lacidogna and José A.F.O. Correia

Received: 24 November 2021 Accepted: 24 December 2021 Published: 29 December 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

plants against large earthquake loads. In the development of seismic isolation design technology for nuclear power plant application, the entire major nuclear power plant buildings are constructed on one common mat, and this is supported by seismic isolators. Most of the future advanced nuclear power plants currently under development adopt seismic isolation design for nuclear power plant buildings [4–6].

So far, the development of seismic isolation design technology for nuclear power plant buildings has been mainly focused on LRBs that can support the heavy building weight stably and allow horizontally flexible deformation. These LRBs have been continuously developed as a seismic isolator only for horizontal seismic isolation of nuclear power plant buildings because the horizontal design earthquake level is much larger than the vertical design level. Recently, the vertical seismic design load is required to be equal to the horizontal load level. Therefore, the unavoidable disadvantage of the LRBs amplifying the vertical earthquake response by vertical earthquakes is a big issue to overcome [7–9]. In particular, as the vertical seismic load level has recently increased and the importance of nuclear power plant design covering this has become more of an issue, horizontal seismic isolation design using only LRBs may not be able to ensure the required seismic design goal.

To overcome the disadvantage of the horizontal seismic isolation with the LRBs, much research and development on three-dimensional (3D) seismic isolators has been carried out worldwide [10–15]. However, there are still not many practical cases applied to nuclear power plants. One of the design barriers of vertical seismic isolation for whole reactor buildings may be to construct a rigid upper mat supporting the reactor buildings to prevent local uneven settlement due to the flexibility of the vertical seismic isolation device.

The goal of this study is to develop a 3D seismic isolator for individual nuclear power plant facilities that are relatively easy to install and maintain when compared with the whole building seismic isolation design. For this purpose, a study was conducted on a vertical seismic isolation device having a spring and a steel damper that can be integrally combined with lead-inserted small LRBs for horizontal seismic isolation developed for individual facilities in nuclear power plants [16,17].

There are many studies for the supplemental seismic energy dissipation using the LRBs, such as ring-type steel [18], high damping rubber [19], friction pad [20], and viscous damper [21]. In this study, various dimensions and shapes of thin plate-type steel dampers integrated with the LRBs are investigated, and the capacity of their seismic energy dissipation is verified by the tests and simulations.

A disc spring or a helical coil spring was used for the vertical stiffness design that determines the vertical seismic isolation frequency (VIF) of the vertical seismic isolation device, and a steel damper was used for the vertical seismic energy dissipation that controls the vertical relative displacement. In this study, with the goal of 3D seismic isolation design for nuclear power plant facilities installed in 136 ft of an actual nuclear power plant building, a vertical spring design for determining the optimal VIF and an optimal steel damper that can accommodate relative vertical displacement were designed and verified. The feasibility of the design concept was confirmed by performing verification tests and simulations for spring and steel damper.

#### **2. Concept of Vertical Seismic Isolation Device**

#### *2.1. Configurations and Dimensions*

The 3D seismic isolator being developed in this study is intended to be applied to individual facilities of nuclear power plants where severe vertical earthquake response amplification is expected. To this end, the horizontal seismic isolator uses a lead-inserted small LRB with proven stability and seismic isolation performance [16,17], and a springdamper-based vertical seismic isolator is mounted on the top of the LRB.

Figure 1 presents the conceptual configuration of the horizontal–vertical integrated 3D seismic isolator being developed in this study. As shown in the figure, the LRB with relatively high vertical stiffness supports vertical springs and dampers of the vertical seismic isolation device.

**Figure 1.** Overall design concept of integrated 3-dimensional seismic isolator (half-symmetric view).

The housing in Figure 1 has a function to transmit the horizontal seismic load, simultaneously guiding the vertical seismic motion of the spring due to the superstructure. Table 1 shows the specifications of the 3D seismic isolator based on the lead-inserted LRB used in this study.



As shown in Table 1, the design vertical load of the 3D seismic isolator considered in this study is 10 kN, the total height of the LRB is 34 mm, and the design horizontal seismic frequency is 2.3 Hz. The vertical seismic isolation device mounted on the top of the LRB will maintain the vertical stiffness by a disc spring or a helical spring and dissipate the vertical seismic energy by steel dampers.

In the design of the vertical seismic isolation device, the vertical spring mainly controls the VIF, and the vertical damper controls the vertical displacement of the spring. The higher the vertical stiffness, the higher the vertical seismic isolation frequency, and the lower the vertical seismic isolation performance. The lower the vertical stiffness, the better the vertical seismic isolation performance, but the displacement of the vertical spring will significantly increase so that it may not meet the design concept of integral 3D seismic isolator presented in Figure 1. Therefore, a vertical spring design having an appropriate design VIF and a damping device design capable of properly suppressing the vertical spring displacement are required in a design stage.

#### *2.2. Vertical Design Displacement Limit*

Figure 2 presents the operation concept of the 3D seismic isolator. Figure 2b shows the schematics of static vertical displacement condition due to dead weight of the super structure, and Figure 2c shows the horizontal and vertical displacement condition during earthquake events. The housing of the vertical seismic isolator transmits the horizontal load of the superstructure to the LRB and guides the vertical seismic movement. The dimension design of such a housing is determined according to the shape and dimensions of the vertical spring and steel dampers, which are determined according to the determination of the design VIF. The most important aspect in this design concept is that the housing should be designed so that interference with the LRB does not occur during the beyond-designbasis earthquakes. To do this, the vertical seismic displacement of the spring should be properly limited by the determination of a design VIF and damper design.

**Figure 2.** Schematics of 3D seismic isolator's motions: (**a**) Without loads; (**b**) with vertical static load; (**c**) with horizontal and vertical seismic loads.

In this paper, the design target is established to limit the maximum vertical seismic displacement response within the static displacement of the spring by design vertical load of 10 kN. This will prevent detachment between the housing and the vertical spring.

Table 2 presents the required vertical stiffness of the spring according to the design VIF for a design vertical load of 10 kN and their static vertical displacement values.


**Table 2.** Parameter values for simple seismic analysis model.

#### **3. Sensitivity Study on Vertical Seismic Isolation Performance with VIF**

In principle, the basic concept of seismic isolation design is to design an appropriate seismic isolation frequency to avoid resonance with input earthquakes by moving the natural frequency of the superstructure in the frequency range of a strong earthquake to a sufficiently low frequency using a seismic isolator.

In general, the vertical natural frequency of a nuclear power plant building is around 10~20 Hz, which is out of the range of peak spectral frequency band, 3 Hz to 10 Hz in the US NRC RG-1.60 design ground-response spectrum [22]. However, there is possibility of resonance seismic response if the earthquake level increases enough to invoke severe cracks in the nuclear power plant building structures. In this case, the spectral peak frequencies of the response spectrum at the floor where the nuclear power plant equipment is seismically isolated may be shifted to a lower region due to the decrease of the structural stiffness of the nuclear power plant building. This situation can cause severe vertical seismic response amplification in nuclear power plants. Therefore, in the vertical seismic isolation design of nuclear power plants against a large earthquake, it is very important to determine the design VIF that can deviate from the floor-response spectrum peak frequency band and the resonant frequency of the upper structure.

In this study, a remote shutdown console (RSC), which is installed at elevation of 136 ft of an actual nuclear power plant building, is chosen as one of the target 3D seismic isolation equipment. To investigate the sensitivity of the VIF, the vertical seismic isolation performance analyses are carried out with the preliminary specified design parameters of the vertical seismic isolation device, as shown in Table 3.

**Table 3.** Preliminary design parameter values for sensitivity analysis.


Figure 3 presents the input earthquake used in the design of the vertical seismic isolation device in this study. Figure 3a is the vertical floor-response spectrum at 136 ft of the auxiliary building, corresponding to the peak ground acceleration, PGA = 0.5 g, which is 1.67 times the Safe Shutdown Earthquake (PGA = 0.3 g). We can see that the peak spectral frequency band exists at 10 Hz to 16 Hz and the peak spectral acceleration is large at about 40 g. As shown in Figure 3b,c, the zero-period acceleration (ZPA) value is 1.2 g, and the peak/valley displacements are −211 mm/+176 mm.

**Figure 3.** Used vertical seismic input motion (PGA = 0.5 g): (**a**) Floor response spectrum; (**b**) acceleration time history; (**c**) displacement time history.

Superstructure (MASS21) Node 3 Node 3 Vertical Seismic Isolation Device **<sup>e</sup>= ( <sup>t</sup>+ <sup>1</sup> ) slide <sup>t</sup>** Restoring force Figure 4 presents the simple seismic analysis model used for the sensitivity of the vertical seismic isolation performance on the VIF. For the transient seismic time history analysis, a commercial finite element program ANSYS [23] was used. As shown in Figure 4a, the vertical spring is modeled by a simple stiffness–damping element (COMBIN14), and the vertical damper is model with a bilinear force–displacement element (COMBIN40), reflecting the hysteretic characteristics of the steel damper. It was assumed that the inertia mass of the superstructure acts as a concentrated mass (MASS21) at node 3. Figure 4b shows the concept of the bilinear model of the steel damper.

Disp.

CVS

Node 1 Node 2

K1

Fslide

CSD

Steel Damper (COMBIN40)

Kt KVS

Vertical Spring (COMBIN14)

1

10

Spectral Acceleration (g)

100

1 10 100

Input FRS (Vertical) at Aux Building EL= 136ft Damping= 5% PGA= 0.5g

Frequency (Hz)

0 3 6 9 12 15 18 21

Vertical PGA= 0.5g -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300

Displacement (mm)

0 3 6 912 15 18 21

Vertical PGA= 0.5g

Time (s)

Time (s)


Acceleration (g)

**Figure 4.** Schematics of simple seismic analysis model used for sensitivity analysis: (**a**) Finite element model of vertical seismic isolation device; (**b**) bilinear model of steel damper.

To investigate the sensitivity of the design VIF on the vertical seismic isolation performance, the design VIF was considered in the range of 1 Hz~5 Hz. The equivalent damping ratio and stiffness of the steel damper corresponding to the hysteretic bilinear model in Table 3 are 30.7% and 75,291 N/m, respectively. The equivalent stiffness of the steel damper has a corresponding natural frequency of about 1.38 HZ for a vertical design load of 10 kN. Therefore, the actual VIF will be determined by considering both spring and steel damper stiffness.

Figure 5 shows the result of the calculated vertical response spectrum for the super structure according to the design VIF determined by the stiffness value of the vertical spring. As shown in the figure, it can be seen that the lower the design VIF, the higher the seismic isolation performance is due to the frequency shift effect. When the design VIF is 1.0 Hz, the spectral acceleration response of the superstructure is significantly reduced compared with the input response spectrum throughout frequencies. When the design VIF increases to 3 Hz, the superstructure exhibits the vertical seismic isolation effect in the range of about 6 Hz to 50 Hz, and the ZPA value is almost similar to that of the input earthquake without vertical seismic isolation effect. When the design VIF exceeds 3.0 Hz, the vertical seismic isolation effect is greatly reduced in the overall frequency range, and the ZPA value of the superstructure becomes larger than that of the input earthquake.

**Figure 5.** Spectral acceleration response of superstructure for various design VIF.


Relative Disp. of Superstructure (mm)

0123456

(Up direction)

With Damper

(Down direction)

Vertical Seismic Isolation Frequency (Hz)

0123456

(Up direction) (Down direction)

w/o Damper

Vertical Seismic Isolation Frequency (Hz)


Relative Disp. of Superstructure (mm)

Figure 6 shows the maximum vertical spring displacement response according to the design VIF. As shown in Figure 6a, when only a vertical spring is used without using a steel damper, the vertical spring displacement becomes very large (e.g., 101.3 mm when VIF = 3 Hz). Therefore, it is not possible to design a vertical seismic isolation device actually accommodating the LRB dimensions in Table 1. When the steel damper with the design characteristics of Table 3 is used, the vertical spring displacement can be significantly reduced to 30 mm or less, as shown in Figure 6b.

 Input FRS VIF= 1.0Hz VIF= 2.0Hz VIF= 2.5Hz VIF= 3.0Hz VIF= 3.5Hz VIF= 4.0Hz VIF= 5.0Hz

1 10 100

Frequency (Hz)

1

10

100

Spectral Acceleration (g)

1000

In addition, it can be seen that the actual VIF is larger than the design value assumed for the vertical stiffness spring due to the stiffness effect of the steel damper, showing the hysteretic characteristic. The lower the design VIF, the larger the stiffness effect of the steel damper. When the design VIF is 1.0 Hz, the actual VIF increases by 80%.

**Figure 6.** Maximum spring displacement response for various design VIF: (**a**) Without steel damper; (**b**) with steel damper.

Table 4 summarizes the sensitivity analysis results of the vertical seismic isolation performance according to the design VIF. As shown in the table, it can be seen that as the design VIF increases, the effective frequency range in which the actual seismic isolation effect can be obtained is rapidly reduced. When the design VIF is equal to or higher than 3.0 Hz, the ZPA response becomes larger than that of the input floor response spectrum, and then the vertical seismic isolator has the opposite effect of amplifying the vertical seismic response in the high-frequency region.


<sup>1</sup> Note that ZPA of input floor response spectrum is 1.27 g.

#### **4. Vertical Spring Design for Vertical Seismic Isolation Device**

As shown in Figure 1 above, the spring for vertical stiffness is designed to be applied to a vertical seismic isolator that can be integrally mounted on a small-sized LRB with a vertical design load of 10 kN, which is a horizontal seismic isolator for individual facilities in a nuclear power plant. To substantiate the stiffness of the vertical seismic isolation device, the spring design was investigated in detail based on the sensitivity analysis results above.

− − − − − − − − − − − − − −

In this study, two types, such as disc spring and helical coil spring, were chosen for a stiffness design of the vertical seismic isolation device.

#### *4.1. Disc Spring Design*

The disc spring used in this study is a cone-shaped, thin steel structure with outer diameter (*D*), inner diameter (*d*), thickness (*t*), and height (*H*) as shown in Figure 7. When a vertical force is applied, the disc spring stably undergoes compression deformation and has a constant stiffness value within the operating displacement. Once the design stiffness value is determined, it can be designed to have the appropriate load capacity and displacement range by stacking the required number of disc springs in series or in parallel.

**Figure 7.** Schematics of the disc spring configurations and dimensional design parameters.

The relationship between the applied load (*F*) and the corresponding displacement (*y*) in the disc spring in Figure 7 can be described as the following equation [24]:

$$F = \frac{2Et}{(1 - \nu^2)ZD^2} \left[ y^3 - 3hy^2 + 2y \left( h^2 - t^2 \right) \right] \tag{1}$$

where

$$Z = \frac{1}{\pi} \left(\frac{\mathbb{C} - 1}{\mathbb{C}}\right)^2 \left[\left(\frac{\mathbb{C} + 1}{\mathbb{C}}\right) - \ln \frac{2}{\mathbb{C}}\right]^{-1} \tag{2}$$

$$\mathcal{C} = \frac{D}{d} \tag{3}$$

In the above equation, *E* and *ν* represent the elastic modulus and Poisson's ratio of the material, respectively. When the disc springs are stacked in series, the total displacement and equivalent stiffness are proportional to 1/*n* (*n* is number of disc springs). Therefore, when the design VIF is determined, it can be implemented by connecting an appropriate number of disc springs in series.

Table 5 presents the design parameters of the disc spring designed in this study. The plate spring material used in the design is JIS SUP10 spring steel.

In Table 5 above, the shut displacement (*h*) refers to the maximum displacement that the disc spring can accommodate. The disc spring is mounted on the upper flange of LRB, which functions as a horizontal seismic isolator. If the outer diameter of the disc spring is larger than the outer diameter of the LRB, bending deformation may occur in the upper flange of LRB due to the load reaction force applied to the disc spring by the seismic response of the superstructure. In this study, considering the outer diameter of 100 mm of the LRB, the outer diameter of the disc spring is determined to be 80 mm.

In general, the design-allowable displacement of the disc spring assumes 75% of the shut displacement. In this case, the design-allowable displacement for a single disc spring of Table 5 is 1.65 mm, considering the shut displacement (h = 2.2 mm). Therefore, assuming that the relationship between displacement and reaction force is linear within the design-allowable displacement of the single disc spring, the load capacity of the single disc spring becomes 22.9 kN from Equation (1) above, and, accordingly, the spring stiffness is determined to be 13,879 kN/m.

**Table 5.** Design parameter values for a single disc spring.


Figure 8 compares the stiffness test results for six single disc springs manufactured according to the design parameters in Table 5 and the stiffness values calculated from Equation (1). As shown in the figure, it was confirmed that the stiffness test results and calculation results were well matched up to the design-allowable displacement of 1.65 mm, and linearity was guaranteed.

**Figure 8.** Comparison of the linearity characteristics between tests and calculation values by formula for a single disc spring.

Table 6 presents the results of design summary for the set of disc spring that satisfies the design VIF. As shown in the table, when the design VIF is lower, the required number of disc springs greatly increases. This results in a significant increase in the total height of the spring and acts as a burden in the housing design. When the design VIF is set to 2.0 Hz or less, the required housing height of the vertical seismic isolation device is more than 500 mm. Therefore, when using a disc spring, it is desirable to determine the design VIF at least larger than 2.5 Hz with consideration of an appropriate height.


**Table 6.** Design summary of stacked disc springs for various design VIF.

1 In case of a serial stack.

#### *4.2. Helical Coil Spring Design*

In this study, a helical coil spring, which is widely used in industry, was selected as another spring type to provide vertical stiffness. The main design variables that determine the stiffness characteristics of a helical coil spring are the coil diameter, section diameter, and number of turns, as shown in Figure 9.

**Figure 9.** Configuration and dimensional parameters for helical coil spring.

=

ሺ8<sup>ଷ</sup> ሻ

In above figure, *L*<sup>f</sup> and *L*<sup>a</sup> indicate the free length and assembled length with compression force, respectively.

The relationship between the applied force (*F*) and the total stretch (*y*) of a helical coil spring in Figure 9 can be approximately described as follows [24]:

$$y = \frac{F\left(8D^3N\_a\right)}{d^4G} \tag{4}$$

where *D, d*, *Na*, and *G* represent the helical coil diameter measured from spring axis to center of section, diameter of circular section, number of active turns, and shear modulus of the material, respectively.

As the helical coil spring has relatively less rigidity than the disc spring, it is recommended to install the springs in parallel. In this study, four helical coil springs, which have a spring diameter *D* = 88 mm and coil section diameter *d* = 12 mm, are considered for a vertical stiffness design corresponding to vertical design load of 10 kN.

Table 7 presents the results of design summary for the set of helical coil springs that satisfies the design VIF.

As shown in the above table, as the design VIF decreases, the number of coil turns required for the spring stiffness increases rapidly, which leads to a great increase in the total height of the spring. In this study, the design VIF of 2.5 Hz~3.0 Hz was selected to be an appropriate total height of the helical coil spring for substantiation of the design.

Table 8 shows the detailed design parameter values for the helical coil spring that satisfies the selected design VIF of 2.5 Hz and 3.0 Hz.


**Table 7.** Design summary of helical coil spring system with design VIF.

<sup>1</sup> Parallel mounting.

**Table 8.** Summary results of final design parameter values for a single helical coil spring.


To substantiate the actual stiffness design of the helical coil springs satisfying the design VIF of 2.5 Hz and 3.0 Hz in Table 8, the springs were fabricated, and static stiffness tests were performed with 16 specimens. Figure 10 presents the comparison results between tests and the design values calculated by Equation (4). As shown in the figure, the stiffness test results reveal a deviation of less than 1%, on average, when VIF = 2.5 Hz compared to the design value, and about 2%, on average, when VIF = 3.0 Hz.

**Figure 10.** Comparison of stiffness between tests and design values of helical coil springs: (**a**) For VIF = 2.5 Hz; (**b**) for VIF = 3.0 Hz.

#### **5. Design and Verification of Vertical Steel Damper**

#### *5.1. Design Configurations and Dimensions*

The seismic energy dissipation performance of the steel damper reduces the vertical seismic displacement response of the spring with an appropriate size and configuration. As shown in the results of vertical seismic isolation performance in Table 4, the vertical damper should be designed enough to control the seismic displacement of the spring to avoid the interference with the LRB.

Figure 11 shows the configuration and dimensional parameters of the steel damper designed in this study. As shown in the figure, the steel damper is a thin beam plate having a tapered length. The steel damper is fixed by a specific jig mounted on the LRB upper flange, and its end is connected to the superstructure with pin joint.

**Figure 11.** Configurations and dimensional parameters of steel damper.

In this study, the dimensions of three shapes were considered to investigate the hysteretic damping characteristics. Table 9 presents the design dimensions considered in this study.


**Table 9.** Dimensions for steel dampers (mm).

#### *5.2. Evaluations of Energy Dissipation Performance*

To evaluate the damping performance of the considered steel dampers in Table 9, force–displacement analysis was performed for the cyclic displacement load. Figure 12 shows the detailed finite element analysis model for the steel damper. As shown in the figure, a three-dimensional solid element (SOLID181) was used for the steel damper model, and it was modeled to have a sufficient number of elements and aspect ratio to enable more accurate plastic deformation analysis. As a boundary condition, a virtual node that can transmit the load from the superstructure was set and modeled so that a cyclic displacement load could be applied to the end of the steel damper using the connecting element (MPC184). As shown in Figure 12, all displacements at two pin holes are assumed to be constrained as fixed conditions, and the displacement-controlled cyclic load is applied at the end of the pinned joint.

**Figure 12.** Finite element model of steel damper for energy dissipation analysis.

In order to accurately describe the plastic behavior of Type 316SS, the material of the steel damper used in this study, for the finite element analysis, the following Chaboche's inelastic constitutive equations [25,26] were used for the kinematic hardening model:

$$\dot{\alpha}\_{ij} = \sum\_{k=1}^{3} \left[ \frac{2}{3} \mathbb{C}\_{k} \dot{\varepsilon}\_{ij}^{p} - \gamma\_{k} \left( \alpha\_{ij} \right)\_{k} \dot{p} \right] \tag{5}$$

 *γ* where . *<sup>α</sup>ij* and . *p* indicate the revolution of back stress and an accumulated plastic strain, respectively, and *C<sup>k</sup>* and *γ<sup>k</sup>* (*k* = 1~3) are material constants to be used in the ANSYS program. For the isotropic hardening model, the inelastic Voce model [27] is used as follows:

$$
\dot{R} = b[Q - R]\dot{p} \tag{6}
$$

 where . *R* indicates the revolution of drag stress, and *b* and *Q* are material constants.

 Table 10 is the material constants for Type 316 stainless steel used in the above inelastic material constitutive equations [28].



To evaluate the hysteretic damping performance of the steel damper, the vertical cyclic displacement range was set to ±30 mm and inelastic finite element analysis was performed on the triangular waveform input.

Figure 13 shows the distribution of an equivalent plastic deformation for the steel damper of SD2 during five cycles. As shown in the figure, the maximum equivalent plastic strain occurs at the upper and lower surfaces of the middle part of the steel damper by about 2.5%, and after the final fifth cyclic load, the maximum residual plastic strain remains about 0.56% at the transition region.

**Figure 13.** Analysis results of equivalent plastic strain distributions and deformation shapes of steel damper, SD2 during ±30 mm stroke: (**a**) After 1/4 cycle; (**b**) after 1(3/4) cycles; (**c**) after 3(1/4); (**d**) after 5 cycles.

Figure 14 presents the analysis results of an equivalent plastic strain time history at the node where the maximum plastic strain occurs for SD2. As shown in the figure, the maximum equivalent plastic strain in the middle of the steel damper is about 2.6% at the time of the maximum and minimum cyclic displacement loads. Considering that Type 316 SS material used in this study has a total elongation of 40% or more, it is expected that sufficient plastic strain margin can secure the structural integrity of the steel damper without fracture during the strong vertical earthquakes.

Figure 15 presents the analysis results of hysteretic behavior for five cycles, which is the vertical displacement–reaction force relationship of the steel damper. As shown in the figure, the effective stiffness increases in order of SD1, SD2, and SD3, and energy dissipation area increases inversely, and it can be seen that all shapes of steel dampers exhibit hysteretic behavior in which all isotropic hardening properties are quickly stabilized after about three cycles.

**Figure 14.** Analysis results of equivalent plastic strain time history during cyclic load for SD2.

**Figure 15.** Analysis results of hysteretic response for steel dampers: (**a**) SD1; (**b**) SD2; (**c**) SD3.

Table 11 presents the results of calculating the energy dissipation performance of the steel dampers from the following relational expression [16], assuming idealized bilinear hysteretic behavior extracted from the third cycle.

ଶ

$$\mathcal{L}\_d = \mathcal{W} / \left( 2\pi \mathcal{K}\_{eff} \mathcal{D}^2 \right) \tag{7}$$


**Table 11.** Evaluation results of energy dissipation capacity of steel dampers.

<sup>1</sup> *W* is the area of hysteresis curve indicating the energy dissipation per cycle (EDC).

In the above equation, *W*, *K*eff, and *D* represent the energy dissipation area per cyclic load, the effective stiffness, and the maximum displacement of the hysteresis curve, respectively. As shown in the table, all steel dampers designed in this study were evaluated to have damping performance with a critical damping ratio of 30% or more.

#### *5.3. Verification Tests of Steel Damper Performance*

The quasistatic test was performed to verify the hysteretic damping performance of the steel dampers in Table 11. In the verification test, three specimens were used for each steel damper *ID*. Figure 16 is a photograph of the steel damper specimen shapes and test facility with installation of the steel damper specimen. Table 12 shows the specifications of the DC motor-driven testing machine used in this test.

**Figure 16.** Photos of steel damper specimen and test facility: (**a**) Steel damper specimen; (**b**) test facility with specimen.


In order to verify the hysteretic energy dissipation performance, the test was carried out in five cycles for two ranges of maximum ±24 mm and ±30 mm, considering the target design displacement limit with quasistatic displacement control. The test speed applied to the quasistatic displacement control test is 5 mm/min. Figure 17 shows a photograph of the deformed shape under the maximum displacement load during the cyclic tests on the steel damper. Through visual inspection after the test, it was confirmed that there were no surface cracks on all steel damper specimens.

Reaction Force (N)

**Figure 17.** Deformed shapes of steel dampers after the maximum displacement loads.

Figures 18–20 present a comparison of the results of the hysteretic behavior of the steel dampers between the tests and inelastic analyses.

As shown in the figures above, we can see that the hysteretic behavior of the steel dampers by tests are in good agreement with the analysis results. In addition, the initial yield behavior and cyclic hardening characteristics are well matched, and all steel dampers are rapidly stabilized after three cycles. Therefore, it is expected that there is no significant change in the energy dissipation performance due to the increase in the yield strength according to the cyclic load. From the comparison results of these tests and analysis results, it is confirmed that the design of the steel damper presented in this study for application to the vertical seismic isolation device of the 3D seismic isolator can ensure the damping performance of 30% or more, which is the design target in terms of energy dissipation performance.

**Figure 18.** Comparison results of hysteretic behavior between tests and analyses for steel damper, SD1: (**a**) for ±24 mm stroke; (**b**) for ±30 mm stroke.

1200

**Figure 19.** Comparison results of hysteretic behavior between tests and analyses for steel damper, SD2: (**a**) for ±24 mm stroke; (**b**) for ±30 mm stroke.

1200

**Figure 20.** Comparison results of hysteretic behavior between tests and analyses for steel damper, SD3: (**a**) for ±24 mm stroke; (**b**) for ±30 mm stroke.

#### **6. Evaluations of Vertical Seismic Isolation Performance**

To investigate the seismic energy dissipation performance and find the adequate number of steel dampers required for the vertical displacement limits, the vertical seismic isolation performance evaluations are carried out for the chosen design VIF of 2.5 Hz and 3.0 Hz with three designed steel dampers of SD1, SD2, and SD3. The used finite element seismic analysis model is shown in Figure 4, and the steel damper is modeled by the bilinear stiffness, as identified in Table 11.

Figures 21–23 present the calculation results of the floor response spectrum according to the used number of steel dampers. As shown in Figure 21, in the case of the relatively flexible steel damper SD1, the variation of the actual VIF is not sensitive to the used number of steel dampers. However, in order to obtain effective vertical seismic isolation performance, it was found that two or more SD1 dampers should be used when VIF is 2.5 Hz, and three or more when design VIF is 3.0 Hz. From the results of Figure 22, it can be seen that the steel damper SD2 shows almost similar vertical seismic isolation performance characteristics to that of the steel damper SD1.

1000

for SD1, VIF= 2.5Hz

**Figure 21.** Results of vertical response spectrum for steel damper, SD1: (**a**) Design VIF = 2.5 Hz; (**b**) design VIF = 3.0 Hz.

1000

for SD1, VIF= 3.0Hz

(**b**) design VIF = 3.0 Hz.

**Figure 22.** Results of vertical response spectrum for steel damper, SD2: (**a**) Design VIF = 2.5 Hz;

In the case of the SD3 steel damper, which has relatively strong rigidity, as the used number of steel dampers increases, the actual VIF significantly increases, and the effective frequency range in which the seismic isolation effects can be obtained is greatly reduced, especially at lower frequency less than 10 Hz as shown in Figure 23. However, in case of vertical seismic isolation design for nuclear power plant equipment designed with a vertical natural frequency of 10 Hz or higher, it can be seen that an effective vertical seismic isolating effect can be obtained even if three steel dampers are used.

To check the vertical displacement limits, as discussed in Section 2.2 above, the seismic displacement responses were investigated for each case of the disc spring and helical coil spring. Table 13 presents the summary analysis results of the maximum vertical seismic displacement response of the vertical seismic isolation device according to the used number of steel dampers. In the table, the values of column 4 and column 5 represent the accommodatable vertical seismic displacement by extracting the static displacement from the shut displacement. As shown in the table, when the steel damper SD1 is used, five or more steel dampers must be used to satisfy the vertical displacement limits, and four or more steel dampers are required for SD2 and one or more steel dampers for SD3.

**Figure 23.** Results of vertical response spectrum for steel damper, SD3: (**a**) Design VIF = 2.5 Hz; (**b**) design VIF = 3.0 Hz.


**Table 13.** Results of vertical seismic displacement responses of vertical isolation device.

− − − − (1) *D*sd: Shut displacement, *D*dw: Static displacement due to design vertical load of 10 kN. (2) It will be 37.5 mm in the case of helical coil spring.

#### − − − − **7. Conclusions**

− − − − − − − − In this study, the design of a vertical seismic isolation device that can be integrally used combined with a lead-inserted small-sized laminated rubber bearing (LRB) was studied for three-dimensional seismic isolation of the nuclear power plant equipment. The overall study was based on the target equipment installed at 136 ft elevation of the typical nuclear power plant building, the input vertical seismic motions of Figure 3, and the rigid superstructure.

− − From the results of this study, some valuable conclusions are obtained, as follows:


design values obtained from the equation of a force and displacement relationship are verified by tests.


**Author Contributions:** Conceptualization, G.-H.K. and J.-Y.J.; methodology, G.-H.K., J.-Y.J. and J.-K.H.; validation, G.-H.K. and J.-Y.J.; formal analysis, G.-H.K.; investigation, T.-M.S. and M.-S.L.; writing—original draft preparation, G.-H.K.; writing—review and editing, J.-Y.J., J.-K.H., T.-M.S. and M.-S.L.; funding acquisition, G.-H.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study was funded by the Ministry of Trade, Industry and Energy through KETEP (Korea Institute of Energy Technology Evaluation Planning). (No. 20181510102380).

**Institutional Review Board Statement:** Not Applicable.

**Informed Consent Statement:** Not Applicable.

**Data Availability Statement:** Not Applicable.

**Acknowledgments:** This study was funded by the Ministry of Trade, Industry and Energy through KETEP (Korea Institute of Energy Technology Evaluation Planning). (No. 20181510102380).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Zhenyuan Gu 1 , Yahui Lei 1 , Wangping Qian 1,2, \* , Ziru Xiang 1 , Fangzheng Hao <sup>1</sup> and Yi Wang 1**


**\*** Correspondence: qianwangping@my.swjtu.edu.cn

**Abstract:** A high damping rubber bearing (HDRB) is widely utilized in base-isolation structures due to its good energy dissipation capacity and environmentally friendly properties; however, it is incapable of isolating the vertical vibration caused by earthquakes and subways effectively. Thick rubber bearings with a low shape factor have become one of the important vertical isolation forms. This paper provides an experimental comparative study on high damping rubber bearings with low shape factor (HDRB-LSF), thick lead–rubber bearings (TLRB), and lead–rubber bearings (LRB). The abilities of the bearing and energy dissipation of the above bearings are analyzed contrastively considering the influence of vertical pressure, loading frequency, shear strain, and pre-pressure. Firstly, the HDRB-LSF, TLRB, and LRB are designed according to the Chinese Code for seismic design of buildings. Secondly, cyclic vertical compression tests and horizontal shear tests, as well as their correlation tests, are conducted, respectively. The vibrational characteristics and hysteresis feature of these three bearings are critically compared. Thirdly, a corrected calculation of vertical stiffness for the thick rubber bearings is proposed based on the experimental data to provide a more accurate and realistic tool measuring the vertical mechanical properties of rubber bearings. The test results proved that the HDRB-LSF has the most advanced performance of the three bearings. For the fatigue property, the hysteresis curves of the HDRB-LSF along with TLRB are plump both horizontally and vertically, thus providing a good energy dissipation effect. Regarding vertical stiffness, results from different loading cases show that the designed HDRB-LSF possesses a better vertical isolation effect and preferable environmental protection than LRB, a larger bearing capacity, and, similarly, a more environmentally friendly property than TLRB. Hence, it can avoid the unfavorable resonance effect caused by vertical periodic coupling within the structure. All the experimental data find that the proposed corrected equation can calculate the vertical stiffness of bearings with a higher accuracy. This paper presents the results of an analytical, parametric study that aimed to further explore the low shape factor concepts of rubber bearings applied in three-dimensional isolation for building structures.

**Keywords:** isolation; high damping rubber bearing; low shape factor; performance test; experiment

#### **1. Introduction**

In recent years, basic isolation technology has been extensively applied in building construction and bridge engineering in China and overseas [1–3]. By the end of 2020, China had constructed more than 8000 seismic-isolated buildings. The rubber isolation system is currently the most widely used and mature isolation technology, among which the common isolation devices include the ordinary rubber bearing, natural rubber bearing (NRB) [4], LRB [5], HDRB [6], etc. The HDRB has plentiful advantages—such as a simple structure, stable mechanical performance, strong energy dissipation capacity, large stiffness before yielding, environmental protection, etc.—that make it an excellent choice for

**Citation:** Gu, Z.; Lei, Y.; Qian, W.; Xiang, Z.; Hao, F.; Wang, Y. An Experimental Study on the Mechanical Properties of a High Damping Rubber Bearing with Low Shape Factor. *Appl. Sci.* **2021**, *11*, 10059. https://doi.org/10.3390/ app112110059

Academic Editor: Felix Weber

Received: 22 September 2021 Accepted: 20 October 2021 Published: 27 October 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

base-isolation structures. It is made of laminated rubber pads with multi-layer steel plates that are vulcanized and bonded at a high temperature, where the steel plate layers restrain the vertical deformation of the rubber layers from guaranteeing a certain vertical stiffness of the bearing. The rubber layers adapt the high damping material with graphite and other additives to be capable of simultaneously withstanding a vertical load and resist a large horizontal shear displacement. Compared with the ordinary rubber bearing, HDRB can achieve a higher equivalent damping ratio of more than 20% without matching dampers, which saves installation space. HDRB possesses greater stiffness before yielding, a better braking effect on the structure subjected to wind load, more minor damage caused by lead when processing, and is more applicable for those places with special requirements for environmental protection, especially in contrast with the lead–rubber bearing. When the structures are subjected to an earthquake, the HDRB will produce a large deformation and reduce the stiffness, thus achieving a better isolation effect. In recent years, many researchers have conducted studies on HDRB. Burtscher et al. [7] conducted experimental studies on different forms of HDRB and analyzed the influence of the shape factor and steel plate forms on the performance of the bearings. Yamamoto et al. [8] took bidirectional coupling factors into consideration when testing the horizontal shear performance of HDRB. Chen et al. [9] and Xue et al. [10] carried out vertical compression tests along with horizontal shear tests for loading on the designed HDRB. Their results proclaimed a good energy dissipation effect. Bhuiyan et al. [11] proposed a rheological model considering the nonlinear characteristics of HDRB based on the cyclic shear test, monotone relaxation test, and multi-step relaxation test. Dong et al. [12] studied the influence of shear strain and vertical compressive stress on the shear performance of HDRB through the compression–shear performance test, and made an improvement to the restoring force model of the bearing with certain accuracy. Fabio Mazza et al. [13] applied the HDRBs to two typical r.c. framed structures and conducted the structural nonlinear incremental dynamic analyses based on the refined three-spring–two-dashpot model of the bearing.

Rubber bearings, including HDRBs, are generally horizontal isolation bearings, which can reduce the horizontal seismic acceleration response of a structure by about 60%. They are incapable of decreasing or even amplifying vertical seismic effect effectively. This can be explained by the fact that ordinary rubber bearings have a vertical stiffness equivalent to thousands of times their horizontal stiffness, which means that it is likely to obtain a smaller vertical basic period of around 0.1 s and is liable to fall within the striking part of the vertical acceleration spectrum, thus resulting in poor vertical isolation. However, recent large (extra-large) earthquakes in China and abroad indicate that strong vertical ground motions that occur in high-intensity areas, especially near faults areas, may even exceed the horizontal seismic action component. After horizontal seismic action is reduced, vertical ground motion with a high peak value will become the main cause of structural destruction, particularly for medical buildings and industrial facilities. Once damaged, the function of these buildings will be disrupted and the resilience of society will be seriously affected. Hence, scholars have developed several vertical isolation devices and three-dimensional isolation devices. Jia et al. [14] designed a new three-dimensional isolation device containing lead-core rubber pads combined with dish-shaped springs and steel plate dampers. Of those devices, some have undergone complex modeling, which is not conducive to processing, while some need additional dampers to enhance their vertical damping and causing greater vertical stiffness of the bearing. Thick rubber bearings have become one of the most important vertical isolation forms. Compared with an ordinary rubber bearing, there is a conspicuous increase in the thickness of HDRB, generally 4 to 6 times higher than that of the former. The shape factor of ordinary rubber bearing typically ranges from 15 to 30 or more, while that of HDRB is designed to vary from 2 to 5. In 1989, Aiken et al. [15] set the shape factor of the individual 1/4 scale bearings as 2.4 and characterization tests were then conducted. Warn and Vu [16] found that three-dimensional isolation for low and mid-rise structures would be achieved if the shape factors of the elastomeric bearings was set to be less than 4 and significant vertical

damping was supplemented. Kanazawa et al. [17] designed a thick rubber bearing with the second shape factor of 4.1, aiming to study the damping effect on the equipment in a nuclear reactor. Zhu et al. [18] selected three thick lead-core rubber bearings with shape factors of 8.0, 12.0, and 16.7, respectively, to conduct basic mechanical properties tests on them. Wang [19] carried out studies on the basic mechanical properties of five thick rubber bearings with shape factors between 4.13 and 5.37 and with different diameters, summarized the law of vertical stiffness parameters of new thick rubber bearings, and then proposed a design method of anti-buckling low-frequency seismic isolation bearings. Liu and Huang [20] compared the vertically seismic response spectrum between prestressed and non-prestressed thick rubber bearings with the shape factor of 3.29. Li et al. [21] conducted a series of experimental studies on shear stiffness, vertical stiffness, deformation performance, fatigue performance, creep performance, and aging performance of thick rubber bearings with the second shape factor of 1.85. The research results of many scholars around the world illustrate that compared with ordinary rubber bearings, thick rubber bearings have smaller vertical stiffness and superior vertical isolation performance, which can avoid the adverse resonance effect caused by the vertical period of vibration coupling with the structure.

In this paper, three bearings, including HDRB-LSF, TLRB, and LRB, were designed. Then, vertical compression performance tests, horizontal shear performance tests, vertical compression correlation tests, and horizontal shear correlation tests were carried out. The effects of vertical compressive stress, pre-pressure, shear strain, and loading frequency on the performance of the above three bearings were studied. The performance differences of the bearings were compared. Considering that the experimental results of HDRB-LSF displayed a big discrepancy with the theoretical values, a correction method for vertical stiffness of HDRB-LSF was proposed, and the modified values were found to be in good agreement with the experimental results.

#### **2. Experimental Process**

#### *2.1. Specimen Design*

Three bearings, including HDRB-LSF, TLRB, and LRB were adapted in the test. Figure 1 only displays the configuration of HDRB-LSF. All the bearings were fabricated with a diameter of 600 mm. The thickness and the diameter of the upper and lower sealing plate were 20 mm and 600 mm, respectively. The thickness and the diameter of the upper and lower connecting plates were 25 mm and 700 mm, respectively. The three bearings were designed according to a principle that the total thickness of 156 mm for the rubber pads in the bearings equals.

The first shape factor of the rubber bearing (S1) is defined as (D − d)/(4tr), and the second shape factor (S2) is defined as D/(nrtr). D and d denote the diameter of the effective bearing surface and the diameter of the central opening for a bearing, respectively. If the central hole in the bearing is filled with rubber or lead, the opening can be ignored on the basis of the Specification GB/T20688.1-2007 [22]. n<sup>r</sup> and t<sup>r</sup> are the number and the thickness of rubber layers, respectively. S<sup>1</sup> value of LRB is 37.50, while the S<sup>1</sup> value of TLRB and HDRB-LSF is 12.50. Based on the research experience, t<sup>r</sup> and S<sup>1</sup> are important parameters determining the bearing stiffness. The S<sup>1</sup> of LRB is three times that of TLRB and HDRB-LSF, indicating that LRB possesses greater vertical stiffness and more stable bearing capacity vertically. The S<sup>2</sup> value of all the above bearings keeps unchanged at 3.85, explaining that their horizontal stiffness is consistent.

**Figure 1.** *Cont*.

**Figure 1.** The configuration of HDRB-LSF: (**a**) cross-section view; (**b**) 1/4 perspective view; (**c**) compression test view; (**d**) shear test view.

The other dimension parameters of each bearing are listed in Table 1, where d*<sup>l</sup>* is the diameter of lead core; n<sup>s</sup> and t<sup>s</sup> are the numbers and the thickness of steel plates. In the specimens steel plates were made from Q345 material; the rubber pads of HDRB-LSF were made from high damping rubber with a shear modulus of 0.392 MPa and a rubber hardness of 56 HA (Shore hardness); the rubber pads of TLRB and LRB were made from ordinary rubber material with a shear modulus of 0.6 MPa and a rubber hardness of 42 HA. Here, it should be noted that each of the two thick rubber bearings had a small hole in the center when fabricated to facilitate vulcanization. According to the specification: after vulcanization, specimens with a total height of less than 250 mm should be rested for at least 24 h, while other specimens should be rested for more than 48 h. Hence, all the test specimens were rested for more than 48 h after vulcanization and then kept in the testing environment for another 24 h before the test.

**Table 1.** Basic parameters of the bearings.


The vertical stiffness (Kv) and the horizontal stiffness (Kh) of the ordinary rubber bearing were calculated by Equations (1) and (2) according to the specification GB/T20688.3- 2006 [23], where A is the whole effective cross-section area of the bearing; E<sup>c</sup> is the modified compressive elastic modulus of laminated rubber under vertical pressure load, which can be obtained from Equation (3); E<sup>v</sup> is the volumetric elastic modulus of rubber; Ecb is the apparent elastic modulus of rubber obtained from Equation (4); E<sup>0</sup> represents the elastic modulus of rubber; k is a correction coefficient for the elastic model of rubber material related to rubber hardness; and G is the shear modulus of laminated rubber.

$$\mathbf{K}\_{\mathbf{V}} = \frac{\mathbf{E}\_{\mathbf{C}} \mathbf{A}}{\mathbf{n}\_{\mathbf{I}} \mathbf{t}\_{\mathbf{I}}} = \frac{\pi \mathbf{D}}{4} \mathbf{E}\_{\mathbf{C}} \mathbf{S}\_{2} \tag{1}$$

$$\mathbf{K}\_{\mathbf{h}} = \frac{\mathbf{C}\mathbf{A}}{\mathbf{n} \cdot \mathbf{t} \mathbf{r}} \tag{2}$$

$$\mathbf{E\_c = \frac{E\_{c\mathbf{b}} \times E\_v}{E\_{c\mathbf{b}} + E\_v}} \tag{3}$$

$$\mathbf{E}\_{\rm cb} = \mathbf{E}\_0 \left( 1 + 2k S\_1^2 \right) \tag{4}$$

For LRB, the computational formula of vertical stiffness (Kv) is the same as an ordinary rubber bearing. However, the shear stiffness obtained from Equation (2) is merely the post-yield stiffness, and its equivalent shear stiffness can be written as

$$\mathbf{K}'\_{\mathbf{h}} = \frac{\mathbf{K}\_{\mathbf{h}}\mathbf{X} + \mathbf{Q}}{\mathbf{X}} \tag{5}$$

where X is the horizontal shear displacement of the bearing; Q is the yield force of the LRB that linearly correlated with the cross-section area of the lead core and can be acquired by the following formula [24].

$$\mathbf{Q} = \frac{8.367 \pi \mathbf{d}\_l^2}{4} + 4.682\tag{6}$$

where d*<sup>l</sup>* is the diameter of the lead core. It should be pointed out that the role of the lead core should be considered when calculating the vertical stiffness of LRB in Equation (1). The cross-section area A is not the simple addition of lead core cross-section and rubber cross-section but is the relational expression that A = A<sup>r</sup> +A<sup>l</sup> (El/E<sup>c</sup> − 1), where A<sup>r</sup> and A*<sup>l</sup>* are the cross-section area of rubber and the cross-section area of lead core, respectively; E*l* is the elastic modulus of lead core.

We calculated that the designed shear stiffness of HDRB-LSF, TLRB, and LRB are 1.09 kN/mm, 1.35 kN/mm, and 1.35 kN/mm, respectively. The vertical stiffness values of the above three bearings are 706.13 kN/mm, 581.94 kN/mm, and 2116.18 kN/mm, respectively.

#### *2.2. Test Method*

The loading device used in the test is shown in Figure 2. Vertical parameters of the compression and shear test device were: the maximum pressure was 20,000 kN, the maximum tension was 6000 kN, the maximum displacement stroke was 700 mm, and the maximum loading speed was 3 mm/s. Horizontal parameters of this device were: the maximum shear test force was ± 6000 kN, the maximum displacement stroke was ±600 mm, and the maximum loading speed was 10 mm/s. Vertical and horizontal loading adopted force control and displacement control, respectively. The acquisition of data relied on an automatic data acquisition system. The vertical pressure was assumed to be P. For the vertical compression performance test, cyclic loading was carried out within the range of P ± 0.3 P. According to the Chinese Specification [19], three loading cycles are recommended for the performance test of rubber bearings. In this test, four cycles were actually loaded to guarantee the data integrity of the third cycle, and the results of the third cycle were extracted to calculate the performance of the bearings.

#### *2.3. Test Cases*

Based on the design surface pressure of 12 MPa, according to the Code, and bearing diameter of 600 mm, the specified vertical pressure P was determined as 3400 kN. In order to analyze the influencing factors of vertical stiffness of the three specimens, loading cases with different pre-pressures, vertical pressures, and loading frequencies were successively carried out, as shown in Table 2. For the horizontal compression and shear performance test, when the specified vertical pressure P (3400 kN) was used, the corresponding shear displacements under 25%, 50%, 75%, and 100% of shear strain γ were, respectively, applied at different loading frequencies of 0.01 Hz, and 0.0082 Hz. When the loading frequency was 0.01 Hz and the shear strain γ was 100%, loading cases with different vertical pressure P of 3400 kN, 4250 kN, and 5100 kN were launched to analyze the shear strain correlation of the horizontal performance for the three specimens, which can be seen in Table 3.

**Figure 2.** The loading device in the test.



**Table 3.** The cases of the horizontal shear test.


#### **3. Test Results and Analysis**

#### *3.1. Vertical Compression Performance of Bearings*

The vertical compression performance tests with 100% shear strain were carried out on the three bearings, respectively, under the designed vertical pressure P of 3400 kN and loading frequency of 0.1 Hz (Case 8). Four loading and unloading cycles were conducted for each test case of the three bearings within the range of P ± 0.3 P, and the schematic

diagram of vertical loading mode is shown in Figure 3. Figure 4 gives the vertical load– displacement curves of the bearings. It indicates that the vertical energy dissipation effect of HDRB-LSF and TLRB is superior to LRB. With the increase of compressive stress, extrusion deformation occurred in the rubber layer causing the vertical displacement of the bearings. The slopes of hysteresis curves—i.e., the vertical stiffness of the bearings—showed an augmented tendency. Under the action of equal compressive stress, the vertical stiffness of different bearings varied, explaining that the vertical stiffness of a bearing has a close relation with the thickness and the properties of the rubber material.

**Figure 3.** The schematic diagram of vertical loading mode.

**Figure 4.** *Cont*.

**Figure 4.** Load–displacement curves of vertical compression: (**a**) HDRB-LSF; (**b**) TLRB; (**c**) LRB.

The third cyclic results of the vertical load–displacement curves of the three bearings under different loading frequencies, pre-pressures, and vertical pressures were obtained, and the effects of these parameters on the vertical stiffness of bearings were analyzed.

#### 3.1.1. Loading Frequency

Test data from cases 1–12 were extracted, and correlation tests of loading frequencies were carried out on three bearings, which aimed to study the influence on hysteresis behavior of high-damping isolation bearings. Due to space limitations, Figure 5 only gives the hysteresis curves of the three bearings at different loading frequencies when the vertical pressure is 3400 kN. It can be seen that the shape of the vertical hysteresis curves for all the bearings obtained by cyclic loading with different loading frequencies are basically the same. Loading frequency has a diverse effect degree on the vertical mechanical properties of different bearings; TLRB is most sensitive to it. With the augment of loading frequency, the vertical stiffness of each bearing presents an increasing trend. When the loading frequency was rather low, it had a slight influence on the vertical performance of the bearings, while the vertical compression displacements observably changed as the loading frequency was relatively high (0.05 Hz and 0.1 Hz).

#### 3.1.2. Vertical Pressure

Figure 6 listed the load–displacement curves of the three bearings under different vertical pressures (2900 kN, 3400 kN, and 3700 kN) at a loading frequency of 0.1 Hz. It explains that the hysteretic curve slopes of the bearings, namely the vertical compression stiffness, increases significantly with the growth of vertical pressure. Among them, the curve shape for LRB is less sensitive to vertical pressure compared to TLRB and HDRB-LSF, which have thick rubber layers. This is due to the fact that as the vertical pressure increases, the thickness of the rubber layer for TLRB and HDRB-LSF diminishes, and the constraint effect of the steel plate is strengthened, ultimately leading to the rapid enlargement of compression modulus of the rubber in the triaxial compression state.

#### 3.1.3. Pre-Pressure

From the extracted test data from cases 1–12, the effect results of different pre-pressures (1700 kN, 2040 kN, and 2720 kN) on the three bearings were obtained, under the vertical pressure of 3400 kN and the loading frequency of 0.1 Hz. The obtained vertical compression load–displacement curves are drawn in Figure 7. As can be seen from the figure, the slopes of the hysteretic curves augment significantly with the growth of pre-pressures, indicating that the pre-pressure has a great influence on the vertical stiffness of the bearings, especially for HDRB-LSF and TLRB.

**Figure 5.** Vertical load–displacement curves under different loading frequencies: (**a**) HDRB-LSF; (**b**) TLRB; (**c**) LRB.

3.1.4. Results of Vertical Stiffness

We assumed P<sup>1</sup> and P<sup>2</sup> as the small pressure and the larger pressure in the third cycle, and gave them the values of 0.7 P and 1.3 P, respectively. We assumed Y<sup>1</sup> and Y<sup>2</sup> were the smaller displacement and the larger displacement in the third cycle, respectively. The vertical compression stiffness of all the bearings under various test cases was calculated according to Equation (7), as shown in Figures 8–10.

$$\mathbf{K}\_{\mathbf{V}} = \frac{\mathbf{P}\_2 - \mathbf{P}\_1}{\mathbf{Y}\_2 - \mathbf{Y}\_1} \tag{7}$$

**Figure 6.** Vertical load–displacement curves under different vertical pressures: (**a**) HDRB-LSF; (**b**) TLRB; (**c**) LRB.

**Figure 7.** Vertical load–displacement curves under different pre-pressures: (**a**) HDRB-LSF; (**b**) TLRB; (**c**) LRB.

**Figure 8.** Vertical stiffness of HDRB-LSF.

**Figure 9.** Vertical stiffness of TLRB.

**Figure 10.** Vertical stiffness of LRB.

The error of the vertical stiffness for the bearings is equal to the difference between the theoretical value and the test value divided by the theoretical value. Due to the limited length of the article, Table 4 only shows the comparison results of theoretical and experimental values of vertical stiffness when the vertical pressure is 3400 kN and the pre-pressure is 0 kN.

**Table 4.** Vertical stiffness error between theoretical value and test value (%).


As can be seen from Figures 8–10, under different loading cases the vertical stiffness of LRB is largest in the range between 2050 kN/m and 2450 kN/m. The vertical stiffness of HDRB-LSF takes second place within the range of 1050 kN/m and 1450 kN/m, while the vertical stiffness of TLRB is the lowest, varying from 450 kN/m and 850 kN/m. With the increase in loading frequency, the vertical stiffness of almost all bearings has an increase in different amplitude. When the loading frequency varies from 0.01 Hz to 0.02 Hz, the vertical stiffness values of the HDRB-LSF, LRB, and TLRB improve by 1.3–3.3%, about 2.7%, and 1.1–6.6%, respectively. Whereas when the loading frequency is enlarged from 0.05 Hz to 0.1 Hz, the vertical stiffness values of the HDRB-LSF, LRB, and TLRB improve by 9.1–12.1%, 3.2–4.6%, 13–19.6%. Hence, it can be concluded that the vertical stiffness of LRB is relatively stable, while that of HDRB-LSF and TLRB is sensitive to the change of loading frequency. It can be seen from the table that it is not certain whether the test value

or theoretical value is more remarkable for the vertical stiffness of TLRB, while the test values of vertical stiffness for LRB and HDRB-LSF always surpass the theoretical value. Additionally, the margin of error between the test values and the theoretical values of vertical stiffness for the LRB and TLRB are small, which is basically in line with the specific design requirements. The test values of the vertical stiffness for HDRB-LSF are larger than the theoretical value, which means that it is biased towards safety in the bearing design. However, this error has exaggerated what needs to be corrected. The vertical stiffness of LRB is larger than that of HDRB-LSF and TLRB, illustrating that: (1) in the case of equal total thickness of the rubber layer, the larger the number of rubber layers, the greater the vertical stiffness, due to the steel plate that has a strong constraint ability on the transverse deformation of rubber; (2) for the two bearings with thick rubber, the vertical stiffness of the one with the lead core is smaller than the one with high damping rubber, showing that the latter has a marginally worse isolation capability but greater bearing capacity vertically.

#### *3.2. Horizontal Shear Performance*

Horizontal shear performance tests were carried out on the three bearings, in which the pressure was slowly and continuously loaded to 3400 kN and remained constant. The shear displacement corresponding to γ = 100% was applied horizontally at a low frequency of 0.01 Hz, and the whole loading process consisted of four cycles. The horizontal stiffness K<sup>h</sup> and equivalent damping ratio heq of a bearing can be calculated according to Equations (8) and (9):

$$\mathbf{K}\_{\mathbf{h}} = \frac{\mathbf{Q}\_2 - \mathbf{Q}\_1}{\mathbf{X}\_2 - \mathbf{X}\_1} \tag{8}$$

$$\mathbf{h\_{eq}}(\gamma) = \frac{1}{\pi} \frac{\mathbf{W\_d}}{2\mathbf{K\_H}(\gamma \mathbf{T\_r}) 2} \tag{9}$$

where Q<sup>1</sup> and Q<sup>2</sup> are the maximum and the minimum shear forces of the third cycle, respectively; X<sup>1</sup> and X<sup>2</sup> denote the positive and the negative maximum displacement of the third cycle, respectively, and X<sup>1</sup> = Trγ, X<sup>2</sup> = Tr(−γ); and T<sup>r</sup> is equal to the product of n<sup>r</sup> and tr. ∆W represents the envelope area of the hysteresis curve.

The horizontal hysteresis curves of the tested bearings are shown in Figure 11. It can be seen from the figure that the curves of the three bearings are respectively in crescent shape (HDRB-LSF) and spindle shape (TLRB and LRB). Among them, the hysteresis curves of HDRB-LSF are relatively less plump, that is, the energy dissipation capacity of which is surprisingly inferior to TLRB and LRB under the same shear deformation condition. The shape of hysteretic curves of each bearing is basically consistent with the specification requirements.

Horizontal shear performances of the bearings are shown in Table 5. It shows that under the same vertical pressure and shear strain the horizontal equivalent stiffness of HDRB-LSF is smaller than those of TLRB and LRB, which indicates that under the premise of the equal total thickness of rubbers, the steel plates in HDRB-LSF have less binding force on rubbers than the bearings with lead core.

**Table 5.** Horizontal performance error between theoretical value and test value.


**Figure 11.** Horizontal shear load-displacement curves: (**a**) HDRB-LSF; (**b**) TLRB; (**c**) LRB.

The energy dissipation capacity of bearings is currently expressed by the equivalent damping ratio. The horizontal equivalent damping ratio of HDRB-LSF was approximately 9% in the third cyclic test (it was about 13% in the first cyclic test), expressing a better energy dissipation capacity than the NRB with a usual equivalent damping ratio of 3%. However, compared to TLRB and LRB with the lead core in the test, the energy dissipation capacity of HDRB-LSF designed in this paper is insufficient and needs improvement. When the vertical pressure is 3400 kN and the shear strain is 100%, the errors between the experimental values and the theoretical values of the horizontal performance of HDRB-LSF and LRB are both less than 15%. In contrast, the errors for TLRB are significant.

In order to quantitatively evaluate the horizontal shear stiffness of the three bearings and studying its relationship with shear deformation, axial pressure, and loading frequency, the third cyclic results of horizontal hysteresis curves of the bearings subjected to various test cases were acquired. Then, the influences of different parameters on the horizontal stiffness and equivalent damping ratio of the three bearings were analyzed.

#### 3.2.1. Shear Strain

The designed vertical pressure of 3400 kN was smoothly exerted in the three bearings and remained unchanged during the whole test process. Under the frequency of 0.01 Hz, horizontal sinusoidal excitation waves were loaded with the shear strains of 25%, 50%, 75%, and 100% in succession, until unloading was completed in all cycles under all shear strains. The data acquisition instrument recorded the data during the whole loading process.

The designed vertical pressure of 3400 kN was smoothly exerted to the three bearings and remained unchanged during the whole test process. Under the frequency of 0.01 Hz, horizontal sinusoidal excitation waves were loaded with the shear strains of 25%, 50%, 75% and 100% in succession, until unloading was completed in all cycles under all shear strains. The data acquisition instrument recorded the data during the whole loading process.

The horizontal shear performance of each bearing can be seen in Figure 12. As the shear displacement aggrandizes, the slopes of the curves gradually reduce, due to the reason that the compression area of the bearing core decreases, which can lessen the constraint of steel plates on the internal rubbers. The hysteresis curves of the bearings become larger as the shear strain grows, indicating that their energy dissipation capacity is strengthened. It should be noted that after multiple loads, points of contraflexure appear on the load–displacement curves, as seen in Figure 12b,c, and the degree of contraflexure on the curves gradually increases as the loading displacement grows.

**Figure 12.** *Cont*.

**Figure 12.** Horizontal shear load-displacement curves under different shear strains: (**a**) HDRB-LSF; (**b**) TLRB; (**c**) LRB.

The test values of horizontal equivalent stiffness and the equivalent damping ratio of the bearings are displayed in Figure 13. It can be seen that under the same shear strain, the horizontal equivalent stiffness of LRB and TLRB are adjacent throughout the test, and the stiffness of the former is slightly larger than that of the latter, while that of HDRB-LSF is the smallest. TLRB has most significant horizontal equivalent damping ratio, followed by LRB and HDRB-LSF, according to priority. The shear strain has an unnoticeable effect on HDRB-LSF and TLRB, but has an obvious impact on LRB whose growth of shear displacement is faster than the increase of the hysteretic curve area [14].

#### 3.2.2. Loading Frequency

Similarly, the remaining vertical pressure of 3400 kN was exerted to the three bearings. Under a shear strain of 100%, horizontal reciprocating loads were applied to all the bearings at successive frequencies of 0.01 Hz and 0.0082 Hz, respectively. The horizontal hysteresis curves, horizontal equivalent stiffness, and equivalent damping ratio of the bearings obtained by the test are shown in Figure 14 and Table 6, respectively. It can be seen that the horizontal equivalent stiffness and equivalent damping ratio of the three bearings all decline with the diminution of loading frequency within a narrow varying range.

**Figure 13.** *Cont*.

**Figure 13.** Horizontal performance of the bearings under different shear strains: (**a**) equivalent stiffness; (**b**) equivalent damping ratio.

**Figure 14.** *Cont*.

**Figure 14.** Horizontal shear load-displacement curves under different loading frequency: (**a**) HDRB-LSF; (**b**) TLRB; (**c**) LRB.

**Table 6.** Horizontal performance of the bearings under different loading frequencies.


#### 3.2.3. Vertical Pressure

Due to the effect of the horizontal overturning moment and vertical seismic action, the vertical pressure on the rubber bearing will change considerably, which will have a certain influence on the shear performance of the bearing. Therefore, it is important to study the pressure correlation of the shear performance for the bearings. The three bearings were slowly and continuously loaded vertically with the vertical pressure loaded to 3400 kN, 4250 kN (1.25 P), and 5100 kN (1.5 P), respectively, and the pressure remained unchanged during the subsequent test. The shear strain was 100% and the horizontal reciprocating load was successively applied to each bearing at the loading frequency of 0.01 Hz. The hysteretic curves of the bearings obtained by the test are shown in Figure 15. The horizontal equivalent stiffness and equivalent damping of the bearings were calculated and are listed in Table 7.

**Table 7.** Horizontal performance under different vertical pressures.


**Figure 15.** Horizontal shear load-displacement curves under different vertical pressures: (**a**) HDRB-LSF; (**b**) TLRB; (**c**) LRB.

As can be seen from the figure and table, HDRB-LSF and LRB are more sensitive to the change of vertical pressure. The horizontal equivalent stiffness of HDRB-LSF increases and then reduces with the growth of vertical pressure, while that of TLRB and LRB gradually decreases with it, which is due to the axial compression and 100% shear deformation caused the out-of-plane distortion of the steel plates, thus reducing the horizontal constraint of the steel plates on the rubber pads. Therefore, the designed surface pressure of the bearing must be strictly limited when it is employed. The equivalent damping ratios of all the three bearings enlarged as the vertical pressure grew, owing to the fact that the higher vertical pressure boosted the triaxial stress of the rubber layer, making the rubber material inside the bearing denser, and magnifying the intermolecular friction of the rubber material, leading to the increase of the hysteretic curve area and equivalent damping ratio. Although the HDRB-LSF has a certain horizontal energy dissipation capacity, the values of the equivalent damping ratio for HDRB-LSF are rather low, owing to the fact that the equivalent damping ratio of high damping rubber material was set to be around 10% when

fabricated and considering the constant cost as compared to TLRB and LRB. This paper emphasizes studying the isolation performance law of all the rubber bearings vertically and horizontally used in the base-isolation structures. Hence, the low equivalent damping ratio for HDRB-LSF is acceptable. Of course, a follow-up study of HDRB-LSF will enhance its equivalent damping ratio and reach the corresponding level of the bearings with a lead core.

#### **4. Formula Modification**

Therefore, the vertical stiffness of HDRB-LSF and TLRB with thick rubber layer deviates significantly from the theoretical results calculated according to the Lindley Formula and has exceeded the allowable range of engineering design. Hence, the vertical stiffness formulas for the bearing with a thick rubber layer will be revised in this section.

#### *4.1. Theoretical Derivation*

As can be seen from Table 4, the test values of vertical stiffness for HDRB-LSF are larger than the theoretical values. When the total thickness of the rubber is constant, the rubber pads of the ordinary thin rubber bearing under pressure are constrained by the steel plates, causing a sizeable vertical stiffness value. Whereas for the thick rubber bearing, the rubber thickness of a single layer is great, leading to the limited constraint effect of the steel plates on the rubber pads and large transverse deformation of the rubber layers, so that its vertical stiffness is more minor. The error of the vertical stiffness between the test value and the theoretical value calculated according to the code is too large to ignore, which cannot adapt to the requirements of engineering design. Hence, the theoretical formula of vertical stiffness must be modified.

It is known from the study that the error of the vertical stiffness for the bearing between the test value and the theoretical value is related to the compressive stress σ and the loading frequency f of the bearing. The theoretical formula can be modified with the following formulas:

$$\mathbf{K}\_{\rm V} = \boldsymbol{\varrho}\_{\rm V} \frac{\mathbf{E}\_{\rm c} \mathbf{A}}{\mathbf{n}\_{\rm r} \mathbf{t}\_{\rm r}} \tag{10}$$

$$
\zeta\_V = \zeta\_1 \frac{\sigma}{\sigma\_0} + \zeta\_2 \mathbf{f} + \zeta\_3 \tag{11}
$$

where *ς*<sup>V</sup> is the correction coefficient; *ς*<sup>1</sup> , *ς*<sup>2</sup> , and *ς*<sup>3</sup> are the correction coefficients, which all can be obtained by the fitting of a polynomial based on the experimental data, and their values are 0.762, 3.599, and 0.8631, respectively; σ<sup>0</sup> is the design compressive stress, i.e., 12 Mpa.

#### *4.2. Modification Results*

Figure 16 exhibits the theoretical values, the test values, and the modified theoretical values of the vertical stiffness for HDRB-LSF.

It explains that the modified theoretical values of the vertical stiffness are closer to the test values than the original theoretical values. All the errors between them are less than 1.5%, indicating that the modified method is basically feasible. Moreover, the smaller the vertical pressure, the smaller the error. However, the accuracy of the fitting formula depends on the test samples of the fitting analysis. The test data of the bearings used are few, thus the coverage range of the sample parameters is limited. The applicability of the fitting Equation (9) needs to be further verified in future research.

**Figure 16.** Comparison between theoretical value and test value of HDRB-LSF.

#### **5. Conclusions**

In this paper, three rubber bearings (HDRB-LSF, TLRB, and LRB) were analyzed contrastively via comprehensive experimental studies. The properties of vertical stiffness, horizontal equivalent stiffness, horizontal equivalent damping ratio, and hysteretic curves were tested under different vertical pressures, loading frequencies, shear strain values, and pre-pressures. A corrected calculation for the vertical stiffness of the bearing is proposed based on the results of the study. The results show that the HDRB-LSF possesses stable and reliable performance in the rubber isolation system among all the bearing types. Its vertical stiffness could be accurately calculated by our proposed modified equation. The main conclusions are as follows:


(4) In summary, unlike the LRB, which is usually used for horizontal isolation, the application of thick rubber bearings including HDRB-LSF and TLRB can simultaneously help important buildings avoid suffering the adverse effects of horizontal and vertical earthquakes on the structure itself and indoor equipment, as well as to improve the post-earthquake recovery of building structures. Compared with TLRB, HDRB-LSF is better suited for the structures when considering their larger bearing capacity and more environmentally friendly characteristics. It should be noted that the surface pressure on the thick rubber bearings should be strictly limited in engineering applications. This paper presents the results of an analytical, parametric study that aimed to further explore the low shape factor concept applied in three-dimensional isolation for building structures.

**Author Contributions:** Conceptualization, Z.G. and W.Q.; Data curation, Z.G. and Y.W.; Investigation, Y.L., W.Q. and F.H.; Methodology, Z.G., Y.L. and W.Q.; Project administration, W.Q.; Resources, Z.G. and W.Q.; Software, Z.X. and Y.W.; Visualization, Z.X. and F.H.; Writing—original draft, Z.G., Y.L., F.H. and Y.W.; Writing—review & editing, Z.X. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by The National Natural Science Foundation of China (Grant No. 51808298), the Jiangsu Construction System Science and Technology Project (Grant No. 2019ZD013 and Grant No. 2019ZD017), Nantong Science and Technology Plan Project (Grant No. JC2020124 and Grant No. JC2021169), Natural Sciences Fund for Colleges and Universities in Jiangsu Province (Grant No. 20KJD560002), and Nantong University College Students' innovation and entrepreneurship training program (Grant No. 2021247). The authors appreciatively acknowledge the financial support of the abovementioned agencies.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data used to support the findings of this research are available from the corresponding author upon request.

**Acknowledgments:** The authors would like to equally thank Chenhui Zhu from the Nantong University, China, for linguistic assistance during the preparation of this manuscript. The insightful comments and significant suggestions from the anonymous reviewers of Applied Sciences are sincerely appreciated.

**Conflicts of Interest:** The authors declare that there are no conflict of interest.

#### **References**


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