Study on the Performance of Elliptical Negative Poisson’s Ratio Structural Isolation Bearing

Abstract

The negative Poisson’s ratio structure has special deformation behavior and energy absorption characteristics and is a new structure with broad application prospects. However, most of the current research is still at the theoretical level, while research on its practical performance is sparse. Therefore, this paper proposes an elliptical negative Poisson’s ratio structural isolation bearing (NRB) for application in the field of seismic isolation engineering. The finite element simulation method is used to conduct a mechanical comparison with the traditional high damping isolation bearing (HDR), highlighting the advantages of the NRB in isolation and energy absorption. At the same time, parameter analysis is used to study the influence of the number and angle of structural holes on the stress of the NRB structure, which is 80% higher than that of the traditional isolation bearing, and incremental dynamic analysis (IDA) is also used. The overall average damage rate decreased by 70.3%, showing significant advantages in seismic energy dissipation, control of component damage, and other aspects, providing a strong data basis for the application of seismic isolation technology in practical engineering.

1. Introduction

With the rapid development of materials science and engineering technology [1], materials and structures with special mechanical properties have shown great application potential in aerospace, automotive manufacturing, building protection [2] and biomedicine. Among them, the negative Poisson’s ratio (NPR) structure, as one of the research hotspots, has opened up a new path for engineering structure design and performance optimization with mechanical properties that subvert traditional understanding—transverse expansion during tension and transverse contraction during compression [3]. Inspired by nature, researchers have observed that oval shaped objects such as olive cores and egg shells can effectively disperse stress and show excellent strength and stiffness with their unique geometric profile [4] when resisting external force extrusion and impact [5]. Based on these findings, a new elliptical negative Poisson’s ratio structure was proposed and quickly became a research focus of academia and engineering [6]. With its innovative geometry [7], the structure not only performs well in energy absorption, impact resistance and sound insulation [8], but also significantly enhances the overall stability and bearing capacity of the structure [9], which has high research value and application potential [10].

In recent years, the negative Poisson’s ratio structure has made a series of breakthroughs in the engineering practice of interdisciplinary fields with its unique mechanical properties. In the field of sports equipment, Sun et al. [11] developed a new sole with excellent shock absorption performance by introducing the negative Poisson’s ratio structure into the design of sports shoes, significantly improving the wearing comfort and sports performance. In the field of biomedical engineering, Ali et al. [12] developed a polyurethane film esophageal stent suitable for the treatment of esophageal cancer based on the deformation characteristics of negative Poisson’s ratio materials. The clinical application showed that the stent could significantly alleviate the symptoms of dysphagia and reduce the risk of postoperative complications. Although negative Poisson’s ratio structures have shown potential applications in many fields [13], their complex geometry and nonlinear mechanical response mechanism still pose challenges for structural design and engineering applications. In the field of building seismic resistance, as the core component to improve the seismic performance of building structures [14], their mechanical performance directly affects the seismic response control effect of the overall structure. This study proposes an innovative elliptical negative Poisson’s ratio isolation bearing [15] and discusses its mechanical working mechanism, seismic performance index and vulnerability characteristics, aiming to provide theoretical support and practical reference for the further research and engineering application of negative Poisson’s ratio structures in the field of building isolation.

In terms of morphology and structure, the early isolation bearings are represented by lead rubber bearings (LRBs), which have been mentioned in many studies. For example, Ren et al. [16] carried out numerical simulation analysis on the damage of lead rubber bearings and superstructure response under near-fault earthquakes. The lead rubber bearing is made of rubber layers and steel plates alternately superimposed and vulcanized, and the lead core is placed in the rubber center. This structure has both the elastic deformation capacity of rubber and the energy consumption characteristics of lead. In addition, Aiken et al. [17] conducted experimental research on the mechanical properties of three types of isolation bearings, and the structural types of traditional isolation bearings covered therein provided basic reference for subsequent research. In recent years, new seismic isolation bearings have been emerging. Wang et al. [18] proposed a new type of roller-type isolation bearing with energy dissipation capacity and conducted experimental research on its isolation system. The unique roller structure of the bearing changed the movement mode of the traditional isolation bearing, bringing new ideas to the isolation technology. Peng et al. [19] studied the sliding magnetic isolation bearing considering the site conditions, and realized the sliding isolation by using the magnetic characteristics, which provided a new direction for the shape and structural design of the isolation bearing. Wang et al. [20] experimentally studied a new type of self-centering seismic isolation bearing combined with super elastic shape memory alloy. By using the characteristics of the shape memory alloy, the bearing can be better restored to its original position after an earthquake, and the functional characteristics of the isolation bearing are improved.

In terms of mechanical characteristics, many scholars have conducted in-depth research on the mechanical performance of isolation bearings under conventional conditions. Nagarajaaiah et al. [21] studied the stability of elastic isolation bearings, analyzed their deformation and instability conditions in the process of stress, and provided a theoretical basis for the design and application of elastic isolation bearings. Wei et al. [22] conducted energy response analysis and isolation strategy optimization for high-speed railway bridge track systems under earthquake, and studied the stress and energy consumption of isolation bearings in complex structural systems. Nie et al. [23] evaluated the seismic isolation performance of single-layer cylindrical latticed shells supported on four sides, and analyzed the action mechanism of isolation bearings in the seismic response of long-span spatial structures. Auad et al. [24] studied the influence of ductility on the seismic performance of cross-laminated timber structures equipped with friction isolators, and discussed the mechanical characteristics and the isolation effect of friction isolators under different ductility conditions. Xiao et al. [25] studied the stress and isolation performance of isolation bearings under the seismic conditions of special building structures through shaking table tests and finite element analysis of the isolation performance of a diesel engine room in a nuclear power plant. Gao et al. [26] considered the soil structure interaction, studied the damping effect of the double-layer isolation structure, and analyzed the mechanical characteristics of the isolation bearing in a complex soil structure system.

In the research, scholars have gradually developed isolation bearings from traditional rubber bearings in a new and multi-functional direction [27]. New structural designs continue to emerge, and new materials and structural forms are introduced to improve the performance and scope of application of the isolation bearings. As a new research direction, the elliptical negative Poisson’s ratio structural isolation bearings still have broad exploration space in many aspects [28]. In this paper, a new type of elliptical negative Poisson’s ratio structural isolation bearing is proposed, and its mechanical properties, seismic energy dissipation under seismic action and damage control of components are studied and analyzed.

2. Mechanical Comparative Verification Analysis of Elliptical and Rubber Bearings

2.1. Bearing Design

In this section, a new type of seismic isolation bearing, the rubber elliptical negative Poisson’s ratio structural isolation bearing (NRB), is proposed, which is based on the rectangular seismic isolation bearing (Figure 1) of an actual project and combined with the negative Poisson’s ratio structure, and this new type of seismic isolation bearing is compared with the traditional rubber bearing. As shown in Figure 2, the rectangular isolation bearing model is modeled using SolidWorks (2020). The size of the connecting plate is 1000 mm, the size of the rubber layer is 412 mm, the thickness of the single rubber layer is 3 mm, the thickness of the single steel plate is 2 mm, and the total height of the isolation layer is 203 mm.

$$\mathrm{S}={\displaystyle \frac{\mathrm{a}}{{\mathrm{T}}_0}}={\displaystyle \frac{411.96}{123}}=3.34\ge 3$$

(1)

where a is the coefficient of the rubber layer, and T₀ is the total thickness of the internal rubber. From Equation (1), it can be seen that the shape coefficient S is greater than the standard coefficient, which meets the specification requirements of seismic isolation bearings [29].

In order to give full play to the mechanical advantages of elliptical negative Poisson’s ratio structures, a block structure design scheme is adopted in this study. This design divides the elliptical negative Poisson’s ratio high damping rubber material into multiple independent modules, which effectively activate the lateral contraction effect of the negative Poisson’s ratio structure while ensuring that the vertical stiffness of the isolation bearing meets the requirements of the building code [30]. When the structure is subjected to external loads, the block structure can deform cooperatively, so that the characteristics of the negative Poisson’s ratio can be fully reflected, and then the energy dissipation capacity of the isolation bearings can be greatly improved.

In terms of structural connection and constraint design, targeted improvements are made with the optimization of mechanical properties as the core. The traditional flat plate upper connecting plate is replaced by a shell-like connecting plate (Figure 3). With its unique spatial surface modeling, the shell-like structure can effectively restrict the displacement of the elliptical negative Poisson’s ratio structure, guide the structure to exercise its mechanical efficiency within a reasonable deformation range, and avoid the performance degradation caused by excessive deformation. At the same time, in order to solve the friction problem that may occur when the upper and lower connecting plates bear the upper load, a 3 mm isolation joint is accurately set between them. The width of the isolation joint, which has been verified by mechanical calculation and simulation, can not only effectively eliminate the adverse effect of friction on the structural performance, but also ensure the cooperative work of the upper and lower connecting plates in an earthquake and ensure the stability of the overall mechanical performance of the isolation bearing. A structural schematic diagram of the above innovative design scheme is shown in Figure 4, showing the spatial layout and connection relationship of each component and providing a visual basis for subsequent mechanical analysis and performance verification.

2.2. Finite Element Model Design

In order to verify the rationality of the finite element analysis and the correctness of the modeling method in this paper, a horizontal displacement amplitude simulation analysis of two kinds of rubber bearings (the HDR400 is a high damping rubber bearing, and the NRB400 is a negative Poisson’s ratio structural isolation bearing) is carried out. The length to width ratio of the NRB400’s elliptical holes is 1.2, and the number is 14 × 14. The finite element simulation results are compared and analyzed. In terms of material constitution, the rubber material uses the Mooney–Rivlin model, and the constitutive parameters of steel are Young’s modulus of 210,000 Pa and Poisson’s ratio of 0.27. In the specific operation, the steel plate surface is defined as the master surface, and the rubber surface is defined as the slave surface. Through this master–slave binding mechanism, it is ensured that there will be no relative slip between the rubber and the steel plate during the stress process, which truly restores the cooperative deformation behavior in a real situation.

In order to achieve accurate data extraction and mechanical analysis, a reference point is established at the center of the connecting plate under the support in the finite element model, which is used as the observation basis of the mechanical response of the whole structure. Through kinematic coupling technology, the reference point is coupled with the upper seal plate in full degrees of freedom, so that the displacement, rotation and other mechanical behaviors of the upper seal plate can be accurately recorded and analyzed through the reference point. At the same time, a fully fixed constraint is imposed on the bottom of the support to limit its displacement in three translational degrees of freedom and three rotational degrees of freedom, which simulates the rigid connection between the support and the foundation in practical engineering.

In the design of the loading mode, in order to reasonably simulate the actual stress of the isolation bearing in an earthquake, the horizontal reset loading scheme is adopted (as shown in Figure 5 and Figure 6). In the analysis step, firstly, the connecting plate on the support is completely fixed to limit all its degrees of freedom, and the constraint effect of the superstructure on the support is simulated. Subsequently, a horizontal reciprocating displacement with an amplitude of 60 mm is applied to the connecting plate under the bearing, and the displacement amplitude is determined according to the maximum design displacement of the isolation bearing in an earthquake. In the loading process, the method of cyclic loading for three times is adopted, and each loading cycle includes three stages: forward loading, reverse loading and unloading. Through this cyclic loading mode, the mechanical performance degradation law, energy dissipation characteristics and residual deformation of the isolation bearing under repeated loading can be effectively studied, which provides comprehensive data support for evaluating the seismic performance of the isolation bearing.

The results show that under the same loading conditions, the deformation modes of the internal isolation layer of the NRB and HDR are similar, and the damping is generated by the shear deformation of the laminated rubber steel plate, while the displacement generated by the bottom of the NRB has a squeezing effect on the surrounding elliptical structure. This shrinkage deformation not only effectively disperses the internal stress of the structure, but also promotes the formation of a more efficient stress transfer path inside the material, so as to improve the mechanical performance of the isolation bearing.

2.3. Result Analysis

Through the refined post-processing analysis of the traditional HDR high damping elliptical rubber bearing and the innovative NRB high damping elliptical rubber bearing with negative Poisson’s ratio structure, the deformation response mechanism of the two bearings under the horizontal displacement amplitude load is revealed. As shown in Figure 7, under the same load conditions, the deformation modes of the NRB internal isolation layer and the HDR (high damping rubber) isolation layer are significantly similar, and the displacement generated at the bottom of the support has a squeezing effect on the surrounding elliptical structure. This shrinkage deformation not only effectively disperses the internal stress of the structure, but also promotes the formation of a more efficient stress transfer path inside the material.

By comparing the amplitude curves of the traditional NRB high damping elliptical rubber bearings with the innovative NRB high damping elliptical rubber bearings with negative Poisson’s ratio (

Table 1. Force comparison of two structures.

	Displacement (mm)	Average Shear Force (kN)	Maximum Shear Force (kN)
NRB400	60	65.09	114.64
HDR400	60	87.56	188.78

), the research results show significant performance differences. Under the same 60 mm displacement amplitude loading condition, the isolation bearing with elliptical negative Poisson’s ratio structure shows excellent mechanical performance improvement, and the peak shear force it can withstand reaches 188.78 kN, which is 65% higher than the 114.41 kN of the traditional HDR bearing. These data show that the introduction of the negative Poisson’s ratio structure not only does not weaken the original vertical stiffness characteristics of the isolation bearing, but also significantly enhances its horizontal bearing and energy dissipation capacity by optimizing the structural deformation mode.

3. Study on Mechanical Properties of Elliptical Bearing with Different Number of Holes

3.1. Model Experiment Design

In order to better study the influence of the elliptical negative Poisson’s ratio structure on the isolation bearing, this section studies and analyzes the number of holes in the elliptical negative Poisson’s ratio structure, considers the uncertainty of an earthquake in the horizontal direction, respectively establishes the elliptical structure with the number of holes of 11 × 11, 14 × 14 and 17 × 17 as the filling structure, as shown in Figure 8, applies a displacement amplitude of 60 mm in the directions of 0°, 30° and 45°, and analyzes its mechanical properties through comparative analysis.

3.2. Result Analysis

3.2.1. Results for 0° Direction

It can be seen from Figure 9 that when the NRB structure is compressed in the 0° direction, only the elliptical negative Poisson’s ratio structural blocks in the horizontal direction are compressed, and all of them shrink inward. However, with the increase in the number of apertures, the deformation of the 17 × 17 structure is more prone to the phenomenon of hole compression and compaction. This phenomenon may be due to the fact that the distribution of apertures makes the internal force transmission path of the structure more dispersed, and the mutual constraint between adjacent elliptical structural blocks is enhanced.

According to the stress results (Figure 10 and

Table 2. Stress table of different hole structures in 0° direction.

	Angular Stress	Displacement (mm)	Average Shear Force (kN)	Maximum Shear Force (kN)
NRB400-11	0	60	77.67	160.89
NRB400-14	0	60	91.54	188.78
NRB400-17	0	60	102.83	207.37

), under the same displacement (60 mm) and angle (0°), the average shear force and maximum shear force increase with the increase in model aperture, indicating that the larger the aperture, the stronger the bearing shear capacity. But at the same time, the increase in the average shear force and the maximum shear force gradually decreases, indicating that there is a non-linear relationship between the hole diameter and the shear performance of the bearing. When the hole diameter is large, the improvement effect of continuing to increase the hole diameter will gradually weaken.

3.2.2. Results for 30° Direction

It can be seen from Figure 11 that when the NRB structure is under pressure in the 30° direction, most of the stress is borne by the elliptical negative Poisson’s ratio structural blocks in the horizontal direction, which shrink inward, while the vertical and oblique structural blocks are subjected to shear and compression. With the increase in pore size, the deformation of the 17 × 17 structure enters the compaction stage earlier, and the amplitude of compression deformation changes greatly.

According to the data analysis (Figure 12 and

Table 3. Stress table of different hole structures in 30° direction.

	Angular Stress	Displacement (mm)	Average Shear Force (kN)	Maximum Shear Force (kN)
NRB400-11	30	60	80.31	172.41
NRB400-14	30	60	97.52	189.03
NRB400-17	30	60	105.53	215.07

), with the increase in the diameter of the aperture, the average shear force and the maximum shear force gradually increase, and compared with the displacement in the 0° direction, the average shear force and the maximum shear force in the 30° direction also increase, indicating that the mechanical properties of the bearing in different directions are different, and the size of the aperture and the direction of the stress will positively affect the mechanical properties of the bearing.

3.2.3. Results for 45° Direction

It can be seen from the figure that when the NRB structure is compressed in the 45° direction (Figure 13), the stress is shared by three elliptical negative Poisson’s ratio structural blocks in the horizontal, vertical and oblique directions. The horizontal and vertical negative Poisson’s ratio structures are subjected to shear and pressure, and the oblique structure is subjected to pressure. It can be observed that with the increase in pore size, the structural deformation of the 17 × 17 structure enters the compaction stage earlier, and the stress effect may be better.

As shown in the results (Figure 14 and

Table 4. Stress table of different hole structures in 45° direction.

	Angular Stress	Displacement (mm)	Average Shear Force (kN)	Maximum Shear Force (kN)
NRB400-11	45	60	90.27	181.03
NRB400-14	45	60	100.72	198.49
NRB400-17	45	60	106.54	219.5

), under the conditions of 45° direction and 60 mm displacement, the average shear force and maximum shear force of elliptical negative Poisson’s ratio structural isolation bearings NRB400-11, NRB400-14 and NRB400-17 increase with the aperture increasing from 11 mm to 17 mm, reflecting that the large-aperture bearing has stronger shear resistance and extreme bearing capacity in this direction. At the same time, compared with the 0° and 30° directions, the average shear force and the maximum shear force of each type of bearing in the 45° direction are greater, reflecting that the angle change significantly affects the bearing mechanical performance, and the effect of increasing the aperture on the improvement of bearing shear performance has a marginal decreasing trend.

4. Vulnerability Analysis of Isolated Structure

4.1. Basic Overview of the Model

In this study, ABAQUS (2020) finite element simulation software is used to model the main building, and the line element is used to construct the concrete frame structure. The material properties are shown in

Table 5. Material data sheet of frame structure.

	Model	Section Length and Width/m	Height/m
Bottom column	C30	1000 × 1000	3.6
Upper column	C30	1000 × 1000	3.6
Main beam	C30	600 × 450	6
Secondary beam	C30	500 × 300	4

. There are four floors above the ground, with a total height of 14.4 m and a standard floor height of 3.6 m. Combined with the seismic risk assessment report for the project site, the seismic fortification intensity is determined to be 9 degrees, and the design basic seismic acceleration peak is 0.4 g, which provides the basis for subsequent seismic design and analysis and ensures the reliable safety of the structure under seismic action.

The elliptical negative Poisson’s ratio structural isolation bearing is selected as the isolation system, and its mechanical model refers to the theoretical system constructed in the previous content to ensure the consistency and accuracy of mechanical parameters in the analysis process. In the actual modeling application, the seismic isolation bearings are led into the bottom of the frame column in a 5 × 6 matrix arrangement to bind, and a complete seismic isolation structure and the original structure model are constructed (Figure 15).

According to the actual situation of the structure, the minimum diameter of the bearing is obtained according to the reaction force of the bearing. In combination with the dynamic response of the structure after isolation, the wind resistance and elastic resilience of the isolation layer and other requirements, the isolation bearing is reasonably selected. The mechanical parameter amplitude curve of the isolation bearing is shown in

Table 6. Mechanical parameters of bearing.

Bearing Type	Effective Diameter (mm)	Thickness of Rubber Layer (mm)	Equivalent Water Stiffness (kN/mm)	Yield Stiffness (kN/mm)	Yield Force (kN)
NRB400	400	120	4.4	8.7	70

. A total of 24 isolation bearings are set at the bottom of the columns of the structure, as shown in Figure 16.

4.2. Damage Index Identification

In order to comprehensively analyze the seismic performance difference of elliptical negative Poisson’s ratio structural isolation bearings under a concrete-reinforced concrete isolation structure and non-isolation structure, this study uses the incremental dynamic analysis (IDA) method and the damage state division idea of “seismic code” for reference, and carries out quantitative research on the seismic performance of the two structures.

4.2.1. Reinforced Concrete Frame

Taking the isolated structure as the object, considering the above factors, the maximum story drift angle is finally selected as the damage evaluation index of reinforced concrete frame structure. According to the classification of damage states of reinforced concrete frame structures under different performance levels (LS), and in combination with the provisions of the four stage performance requirements in the seismic code on the inter story displacement index [31], the maximum inter story displacement angle limit corresponding to each damage state is scientifically defined. See

Table 7. Quantitative index of damage state of reinforced concrete frame structure.

Damage Condition	Intact	Slight Damage	Moderate Damage	Severe Damage	Extreme Damage
Performance level	LS1	LS2~LS3	LS3~LS4	LS4~LS5	LS5~LS6
Damage factor	≤0.0018	0.0018~0.005	0.005~0.01	0.01~0.02	≥0.02

for the specific quantitative index.

4.2.2. Non-Structural Member

In view of the fact that infilled wall is the main component of this non-structural component, this study will take it as a typical research object. In terms of damage assessment, the story drift angle is a common damage index for infilled walls. Through in-plane reciprocating loading tests on six full-scale reinforced concrete frame structures, vulnerability analysis based on the inter story displacement angle shows that when the inter story displacement angle reaches 0.0018 (corresponding to the LS₁ performance level of reinforced concrete frame structure), the lower limits of the probability of obvious damage and serious damage of the infilled wall are 87.6% and 5.1%, respectively, which indicates that the quantitative standard of a reinforced concrete frame structure can not accurately reflect the actual damage of the infilled wall. Therefore, this study refers to the relevant literature [32] and redetermines the maximum inter story displacement angle limit of the infilled wall in each damage state. See

Table 8. Quantitative indicators of various damage states of non structural components.

Damage Condition	Intact	Slight Damage	Moderate Damage	Severe Damage	Damage Damage
Performance level	LS1	LS2~LS3	LS3~LS4	LS4~LS5	LS5~LS6
Damage factor	≤0.001	0.001~0.0018	0.0018~0.005	0.005~0.006	≥0.006

for specific indicators.

4.2.3. Isolation Bearing

In this study, the equivalent shear deformation is selected as the key index to evaluate the damage state. Referring to the research results from the performance analysis of isolation bearings in the relevant literature [33], the damage states of rubber isolation bearings under different performance levels (LS) are systematically divided, and the corresponding quantitative indicators of each state are scientifically defined. See

Table 9. Quantitative index of damage state of isolation bearing.

Damage Condition	Intact	Slight Damage	Moderate Damage	Severe Damage	Damage Damage
Performance level	LS1	LS2~LS3	LS3~LS4	LS4~LS5	LS5~LS6
Damage factor	≤1	1~1.75	1.75~2.5	2.5~3.5	≥3.5

for details.

4.3. Selection of Ground Motion

This study comprehensively referred to the requirements of the peer ground motion database [34] and the relevant provisions of the seismic code, and finally determined to select 24 groups of near-fault pulse-type seismic actions as the analysis input, as shown in Figure 17.

4.4. Incremental Dynamic Analysis

4.4.1. Reinforced Concrete Frame

In the IDA analysis of the isolated structure and the original structure, the maximum inter story displacement angle θ₁ of the frame structure is taken as the damage index, and PGA is selected as the strength index of ground motion input. The IDA analysis results for the frame structure are shown in the figure. It can be seen from Figure 18a that with the continuous increase in the PGA value of the input ground motion, the maximum inter story displacement angle θ₁ of the isolated structure and the original structure also increases. For isolated structures, when the PGA is less than 0.6 g, the IDA curve basically shows a linear change, which means that the frame structure is in the elastic stage. When PGA is greater than 0.6 g, the IDA curve begins to bend, showing nonlinear characteristics, indicating that the frame structure has entered the elastoplastic stage at this time. When the input PGA is constant, the maximum inter story displacement angle θ₁ of the frame structure is significantly different, because the input ground motion is uncertain, and the response of the structure will be different under different ground motions.

Comparing the average value of the maximum story displacement angle θ₁ between the isolated structure and the original structure, it is found that when the PGA is 0.6 g, the frame structure of the isolated structure is slightly damaged, while the frame structure of the original structure is nearly moderately damaged, which shows that the isolation technology can reduce the response of the frame structure. In order to compare the dispersion degree of the response of the isolated structure and the original structure, under the same PGA, take the mean value of the maximum inter story displacement angle θ₁ of the frame structure under 24 groups of ground motions as the 50% quantile value, subtract one time of the standard deviation from the mean value to obtain the 16% quantile value, add one time of the standard deviation to obtain the 84% quantile value, and draw the quantile curve. As can be seen from Figure 18b, compared with the 16% and 50% quantile lines of the original structure, the value for the isolated structure is smaller than that of the original structure; Compared with the 84% quantile line, with the increase in ground motion intensity, the story drift angles of the isolated structure and the original structure gradually increase, but the story drift angle θ₁ of the isolated structure is always smaller than that of the original structure. This shows that the isolation technology can improve the seismic performance of the structure and reduce the response of the frame structure.

4.4.2. Non-Structural Members

Using IDA analysis for isolated structures and non-isolated structures, the infilled wall is selected as the key local component of non-structural components, with its maximum inter story displacement angle θ₂ as the damage index and PGA as the seismic input strength index. Finally, the IDA analysis results for non-structural components are obtained, as shown in Figure 19. It can be seen from Figure 19 that with the continuous increase in the input ground motion PGA, the maximum inter story displacement angle θ₂ between the isolated structure and the original structure also increases accordingly. Comparing the average value of the maximum inter story displacement angle θ₂ between the two, it is found that when the PGA reaches 0.4 g, that is, the non-structural members of the isolated structure are slightly damaged, while the non-structural members of the original structure are nearly moderately damaged. When the PGA increases to 0.6 g, the non-structural members of the isolated structure are moderately damaged, and the non-structural members of the original structure also tend to be moderately damaged, which fully shows that the isolation technology can improve the seismic performance of the structure and reduce the response of non-structural members. In addition, when the input PGA is fixed, there is a significant difference in the maximum inter story displacement angle θ₁ of non-structural members, because the input ground motion is uncertain, resulting in different responses of the structure under different ground motions.

4.4.3. Elliptical Negative Poisson’s Ratio Structural Isolation Bearing

IDA analysis is used for the isolated structure. The equivalent shear strain γ of the isolated bearing is selected as the damage index, and PGA is used as the seismic input strength index. Finally, the IDA analysis curve and quantile curve of the isolated bearing are obtained, as shown in Figure 20a. It is found that the equivalent shear strain γ of the isolation bearing increases gradually with the increase in the PGA strength of the input ground motion. For the isolated structure, under the fixed PGA input, there is a significant difference in the equivalent shear strain γ of the isolated bearing, which is due to the uncertainty of the ground motion input, resulting in different responses of the structure under different ground motion excitations.

To further explore the discrete characteristics of the isolated structure, under the same PGA conditions, take the mean value of equivalent shear strain γ of the isolated bearing under 24 groups of ground motions as the 50% quantile value, determine the 16% quantile value by subtracting one time standard deviation from the mean value, add one time standard deviation to obtain the 84% quantile value, and draw the quantile value curve accordingly. It can be seen from Figure 20b that the IDA curve of the equivalent shear strain γ of the isolation bearing changes approximately linearly, mainly because the equivalent stiffness of the isolation bearing changes slightly under this working condition, which makes the fluctuation of the equivalent shear strain relatively stable.

4.5. Vulnerability Analysis

4.5.1. Vulnerability Analysis Theory

The seismic vulnerability assessment uses the probability method, focusing on the possibility of a certain degree of damage to the structure under the action of a specific ground motion intensity. In this study, the analytical method is selected to carry out vulnerability assessment. The exceedance probability of the structure under ground motion intensity I is as shown in Equation (2).

$${\mathrm{P}}_{\mathrm{DV}}(0/\mathrm{I})=\sum {\mathrm{P}}_{\mathrm{DV}/\mathrm{I}}(0/\mathrm{C})·{\mathrm{P}}_{\mathrm{d}/\mathrm{I}}(\mathrm{D}>\mathrm{C}/\mathrm{I})$$

(2)

In the Equation, DV represents the binary indicator variable representing whether the structure exceeds a certain limit state, and the value is 0 or 1 (corresponding to the non-exceeding and exceeding states, respectively); P_DV(0/I)) refers to the probability that DV will take 0 when the ground motion intensity is I; ∑P_DV/I(0/C) is the probability that DV takes 0 when the seismic capacity of the structure reaches C, and it is generally assumed that the seismic capacity of the structure is constant (C = 1); P_d/I(D > C/I) is the probability that a seismic response index D of the structure exceeds its seismic capacity C when the ground motion intensity reaches I.

In the seismic vulnerability analysis, it is generally assumed that the ground motion intensity I and its corresponding seismic response D conform to the lognormal distribution, and the distribution relationship can be expressed by Equation (3).

$$\mathrm{D}=\mathrm{\alpha }{\mathrm{I}}^{\mathrm{\beta }}$$

(3)

In this Equation, α and β are distribution coefficients.

Assuming that the median value of structural response $\widehat{\mathrm{D}}$ is exponentially correlated with the ground motion intensity I, Equation (3) can be derived by taking logarithms on both sides of Equation (4) at the same time.

$$\ln (\widehat{\mathrm{D}})=\ln (\mathrm{\alpha })+\mathrm{\beta }\ln (\mathrm{I})=\mathrm{a}+\mathrm{bln}(\mathrm{I})$$

(4)

where a and b are the regression coefficients, which are obtained through the regression analysis of the incremental dynamic analysis results.

In this paper, the seismic peak acceleration PGA is selected as the seismic intensity index I, and the structural vulnerability is expressed by Equation (5). Let z = c − d, where C and D are independent random variables that obey normal distribution. It can be seen that Z also follows normal distribution.

$${\mathrm{p}}_{\mathrm{f}}=\mathrm{p}(\mathrm{C}/\mathrm{D}<1)$$

(5)

Set the mean and standard deviation of Z as λ_z = λ_c − λ_d, ${\mathrm{\beta }}_{\mathrm{z}}=\sqrt{{\mathrm{\beta }}_{\mathrm{C}}^2+{\mathrm{\beta }}_{\mathrm{D}}^2}$, respectively. When PGA is taken as the ground motion intensity index, β_Z is taken as 0.5. In this way, the failure probability Pf for the structure under local vibration intensity can be solved by Equation (6).

$${\mathrm{p}}_{\mathrm{f}}=\mathrm{p}(\mathrm{Z}<1)={\int }_{-\infty }^0\mathrm{f}\left(\mathrm{Z}\right){\mathrm{d}}_{\mathrm{z}}$$

(6)

In this Equation, f(Z) is the probability distribution function of Z.

Convert (6) to standard normal distribution to obtain (7).

$${\mathrm{p}}_{\mathrm{f}}=\mathrm{p}\left(\mathrm{t}<{\displaystyle \frac{-{\mathrm{\lambda }}_{\mathrm{Z}}}{{\mathrm{\beta }}_{\mathrm{Z}}}}\right)=\mathrm{\varphi }\left({\displaystyle \frac{\ln \widehat{\mathrm{C}}-\ln \widehat{\mathrm{D}}}{\sqrt{{\mathrm{\beta }}_{\mathrm{C}}^2+{\mathrm{\beta }}_{\mathrm{D}}^2}}}\right)$$

(7)

By calculating the transcendence probability corresponding to each performance level (LS_i), the probability value of the corresponding limit state (DS_i) can be obtained, and then the seismic vulnerability matrix can be constructed. The calculation process can be completed by Equation (8).

$$\mathrm{P}\left(\mathrm{D}{\mathrm{S}}_{\mathrm{i}}\mid {\mathrm{I}}_{\mathrm{j}}\right)=\left\{\begin{array}{l}1-{\mathrm{p}}_{\mathrm{f}}\left(\mathrm{L}{\mathrm{S}}_{\mathrm{i}}\mid {\mathrm{I}}_{\mathrm{j}}\right),\mathrm{i}=1\\ {\mathrm{p}}_{\mathrm{f}}\left(\mathrm{L}{\mathrm{S}}_{\mathrm{i}-1\mathrm{i}}\mid {\mathrm{I}}_{\mathrm{j}}\right)-{\mathrm{p}}_{\mathrm{f}}\left(\mathrm{L}{\mathrm{S}}_{\mathrm{i}}\mid {\mathrm{I}}_{\mathrm{j}}\right)\mathrm{i}=2\dots \mathrm{n}-1,\mathrm{j}=1\dots \mathrm{m}\\ {\mathrm{p}}_{\mathrm{f}}\left(\mathrm{L}{\mathrm{S}}_{\mathrm{i}}\mid {\mathrm{I}}_{\mathrm{j}}\right),\mathrm{i}=\mathrm{n}\end{array}\right.$$

(8)

where m and n are the total number of seismic levels and the total performance, respectively.

The isolation structure is composed of a reinforced concrete frame, non-structural members and elliptical negative Poisson’s ratio structural isolation bearings. Although evaluating the seismic vulnerability of key local components can facilitate understanding of the seismic capacity of a single component category, it can not fully reflect the overall seismic performance of the isolated structure. Therefore, based on the reliability theory, the isolation system is regarded as a series system to measure the overall seismic vulnerability. In the case of complete correlation and uncorrelation of key local components, the calculation can be carried out with reference to Equation (9) and Equation (10), respectively.

$${\mathrm{p}}_{\mathrm{fs}}=\stackrel{\mathrm{z}}{\underset{\mathrm{i}=1}{\max }}\left[{\mathrm{P}}_{\mathrm{i}}\right]$$

(9)

$${\mathrm{p}}_{\mathrm{fs}}=1-{\prod }_{\mathrm{i}=1}^{\mathrm{Z}}\left[1-{\mathrm{P}}_{\mathrm{i}}\right]$$

(10)

In the above Equations, P_fs represents the overall seismic vulnerability value, P_i represents the seismic vulnerability index of each key local component, and Z is the total number of key local components.

4.5.2. Establishment of Earthquake Probability Demand Model

According to Equation (4), to obtain the seismic vulnerability curve, it is necessary to perform regression analysis on the results of incremental dynamic analysis, and then determine the seismic probability demand parameters. The research takes the logarithm of the peak ground acceleration PGA as the abscissa and the logarithm of the maximum story displacement angle between the reinforced concrete frame structure and non-structural members and the equivalent shear deformation of the isolation bearing as the ordinate to carry out the linear fitting operation. The specific results are shown in Figure 21, and the corresponding seismic probability demand parameters are shown in

Table 10. Mechanical parameters of bearing.

Demand Parameter	Regression Equation	α	β
Interlayer displacement angle	ln(θ) = 1.2852ln(PGA) − 5.5063	0.0041	1.2852
Equivalent shear deformation	ln(γ) = 1.3132ln(PGA) + 0.52371	1.6883	1.3132

.

4.5.3. Local Seismic Vulnerability Analysis

In order to obtain the probability P_f of the damage state of each local component under different ground motion intensities, the damage index and demand parameters of each component are substituted into Equation (7), and the exceedance probability of component damage is calculated. With PGA as the abscissa and the exceedance probability of each damage index as the ordinate, the seismic vulnerability curve is drawn. It can be seen from Figure 22 that the exceedance probability for local components increases with the increase in ground motion intensity. Under the same PGA and performance level, the exceedance probability for non-structural components is the highest, indicating that such components are more vulnerable to damage than other structures.

By comparing the damage exceedance probability for the isolated structure with the original structure, it is found that when the PGA is fixed, the damage exceedance probability for the isolated structure is lower than that for the original structure, which indicates that the structural response is significantly reduced after using the isolation technology. Both the frame structure and non-structural members take the inter story displacement angle as the damage index, but due to the difference in the definition of performance level, combined with practical engineering experience, non-structural members are more prone to damage than the frame structure, resulting in a higher damage probability, so the seismic performance of non-structural members should be paid more attention in practical engineering.

In order to analyze the probability of damage for each local component under different fortification levels, the vulnerability data of the component are obtained through the ground motion strength PGA for different fortification levels. For the frame structure, the severe damage probability for the isolated structure in a 9-degree fortification earthquake (0.4 g), rare earthquake (0.62 g) and extremely rare earthquake (1 g) are 0.05%, 7.49% and 31.58%, respectively. The corresponding probabilities for the original structure are 11.56%, 34.45% and 24.06%, and the probabilities of damage under an extremely rare earthquake are 18.15% and 67.3%, indicating that the isolation technology can effectively reduce the seismic damage risk for the frame structure, improve the safety performance, and meet the requirements of the “isolation standard” [35] and the seismic fortification requirements of “moderate earthquake is not bad, and large earthquake is repairable”.

In terms of non-structural components, according to the data, the probability of serious damage to the structure using isolation technology in the fortification earthquake, rare earthquake and extremely rare earthquake is 0.75%, 3.27% and 20.76%, respectively; Under the corresponding seismic conditions, the probability of serious damage of the original structure is 21.44%, 43.46% and 19.65%, respectively. Especially under the action of extremely rare earthquake, the damage probability of isolated structure and original structure is 20.56% and 82.16% respectively. It can be seen that the isolation technology significantly reduces the damage degree of non structural members in rare earthquakes, and meets the seismic fortification requirements of “medium earthquake is not bad, and large earthquake is repairable” in the “isolation standard”.

The analysis of the isolation bearing shows that the probability of serious damage is 1.52%, 10.12% and 26.23% in the fortified earthquake, rare earthquake and extremely rare earthquake, respectively, and the probability of damage in the extremely rare earthquake is 29.47%. The results show that the performance of the isolation bearing meets the seismic fortification requirements.

Based on the vulnerability analysis results for local components, the seismic performance of the isolated structure is significantly improved compared with the original structure. The structural members not only meet the fortification standard, but also have high safety redundancy. At the same time, it is worth noting that non-structural members are more likely to be damaged in the earthquake than structural members. Therefore, in the subsequent design and construction process, it is necessary to focus on strengthening the seismic performance optimization of non-structural members in isolated structures.

4.5.4. Overall Seismic Vulnerability

Figure 23 shows a comparison of the overall seismic vulnerability of isolated structures and non-isolated structures. It can be clearly seen from the data in the figure that under the conditions of different performance levels, the overall seismic vulnerability value for the isolated structure is higher than that for the non-isolated structure. Further, for the 9 degree (0.4 g) frequent earthquake, fortification earthquake and rare earthquake conditions, the overall seismic vulnerability of the two structures is quantitatively compared.

When the seismic fortification intensity is 9 degrees (0.4 g), the damage probability intervals corresponding to the overall seismic vulnerability of the isolated structure under the conditions of fortification earthquake, rare earthquake and extremely rare earthquake are 0.65%, 1.98%, 30.45% and 0.23%, 13.87%, 20.52%, respectively. In contrast to the original structure, under the above earthquake conditions, the upper bound of the damage probability corresponding to the overall seismic vulnerability is 65.54%, 85.78%, 98.13%, and the lower bound is 56.75%, 86.45%, 97.32%, and the overall average damage rate is reduced by 70.3%. The data comparison shows that the seismic isolation technology significantly reduces the damage risk and effectively improves the seismic performance of the structure.

5. Conclusions

This paper systematically studies the elliptical negative Poisson structure isolation bearing (NRB) and analyzes its stress and deformation.

(1) In this study, two kinds of rubber bearings, HDR and NRB, are used for structural design. The stress and deformation of the two kinds of bearings in horizontal shear are analyzed by finite element analysis software ABAQUS, and the results are compared. The innovative NRB high damping elliptical rubber negative Poisson’s ratio structural isolation bearing has a peak shear force of 88.78 kN under 60 mm displacement amplitude loading, which is 65% higher than that of the traditional HDR bearing, and does not weaken the vertical stiffness, effectively enhancing the horizontal bearing and energy dissipation performance.

(2) The number of holes in the elliptical negative Poisson structure isolation bearing has a significant impact on the mechanical performance. With the number of holes increasing from 11 to 17, the peak shear force increases to 207.37 kN under horizontal unidirectional loading, and the bearing capacity increases by 28% compared with the control group and 80% compared with the traditional bearing. Among the other arrangements provided in this study, the 17 × 17 hole array achieves the best mechanical performance.

(3) Under the 9 degree (0.4 g) seismic fortification, the damage probability for local components and overall seismic vulnerability of the isolated structure under the fortification earthquake, rare earthquake and extremely rare earthquake are less than the damage probability and the overall seismic vulnerability of the original structure under the fortification earthquake, rare earthquake and extremely rare earthquake. The results show that the overall average damage probability for the structure is reduced by 70.3%, which effectively improves the overall seismic performance and verifies its rationality, and provides a strong data basis for the application of elliptical negative Poisson’s ratio structural isolation technology in practical engineering.