Fragility Analysis of Overturning Resistance of Hybrid Base-Isolated Structures in Diesel Engine Buildings of Nuclear Power Plants

Abstract

This paper validates the effectiveness of the modeling approach based on the finite element analysis of shaking table tests, establishing finite element models for both a base-isolated structure and a hybrid base-isolated structure designed to address overturning issues in the diesel engine building of a nuclear power plant. By using the Incremental Dynamic Analysis (IDA) method, a fragility analysis of the overturning resistance was conducted for both isolation systems. This study demonstrates that the hybrid base isolation scheme, which incorporates additional dampers, effectively enhances the structure’s overturning resistance and reduces the probability of failure. When evaluating the seismic fragility of the structure by using the TP value, which is related to the tensile stress of the isolation bearings, as a damage index, the results are more conservative compared with those obtained by using shear strain ($\gamma$). This highlights the importance of improving the tensile capacity of the isolation bearings in structural design. Furthermore, fragility assessment using $\gamma$ as a damage index can provide design references for the collision limit of the isolation moat in the base-isolated structure of the diesel engine building in nuclear power plants.

1. Introduction

The application of base isolation technology can prevent building structures from being affected by moderate-to-severe seismic vibrations, enhancing the seismic margin of structures. It has evolved into a relatively mature technology, achieving significant progress during the 1980s and 1990s [1,2,3,4,5]. The commendable performance of base-isolated buildings during the 1994 Northridge earthquake and the 1995 Kobe earthquake facilitated the widespread dissemination of base isolation technology [6].

In recent years, nuclear power safety and development have become hot topics worldwide. Nuclear power engineering construction faces increasing challenges, with seismic requirements for nuclear power structures continuously rising. The seismic uncertainty of nuclear power plant sites [7], along with the constraints imposed by internal pipelines and equipment on seismic design, presents numerous limitations. However, new development directions, such as the standardization of nuclear power plant construction [8], are also emerging. Base isolation is a crucial technology for protecting building structures against geological disasters. It is not only widely used in civil buildings [9,10] but has also been applied in operational nuclear power plants [11,12,13,14,15]. On one hand, it enables nuclear power plants to possess seismic resistance under different peak ground acceleration conditions without altering the superstructure, relying instead on modifications to the base isolation system, thereby reducing the coupling effects on equipment pipelines and components. On the other hand, it contributes to the standardization of nuclear power plant design. However, the application of base isolation technology in the nuclear field is still limited, especially in safety-related structures like diesel engine buildings. If the diesel engine building within the nuclear island is damaged during an accident, losing its role as an emergency backup power source, it could lead to the inability of reactor to continue cooling, potentially causing an explosion. Subsequent accident recovery efforts would also be hindered by the lack of power supply. Therefore, ensuring the safety of diesel engine buildings under extreme seismic conditions and enhancing the seismic margin of the structure are crucial.

Research indicates that base isolation bearings are prone to excessive horizontal displacement under high-intensity seismic motion [16], leading to tensile stress in the bearings and causing overturning effects in the structure. This has become one of the main obstacles limiting the application of isolation technology in high-rise structures. In recent years, numerous scholars have conducted extensive research on the tensile stress in bearings and structural overturning issues [17,18,19,20], proposing various bearing optimization methods. In addition to redesigning isolation bearings to address tensile stress and structural overturning, combining isolation devices with different performance characteristics can also improve the seismic performance of isolated structures. Dampers can provide excellent additional damping [21,22,23,24,25], and when combined with isolation bearings to form hybrid isolation devices, they complement the shear characteristics of the bearings, significantly enhancing the seismic performance of the isolation system. Through comparative case studies, Zhou [26] discovered that employing seismic isolators with a vertical frequency of less than 3 Hz can significantly mitigate the vertical response of nuclear power plant structures. However, when the vertical frequency decreases to 1 Hz, it becomes necessary to install anti-sway devices to control the pronounced sway effect. This finding underscores the superiority of three-dimensional seismic isolation over single-level horizontal seismic isolation, offering valuable insights for the seismic design of nuclear facilities. Chen [27] proposed an isolation device composed of friction pendulum bearings (FPBs) and viscous dampers (VDs). Finite element analysis results demonstrated that the device effectively reduces structural acceleration and deformation, significantly suppressing the dispersion caused by the randomness and type of input motion, thereby enhancing the predictability of structural performance.

Despite the continuous optimization of isolation systems, which can effectively improve the seismic capacity of structures, the uncertainty of earthquake magnitudes can still lead to structural failure. The 2011 Fukushima nuclear power plant accident, where beyond-design-basis seismic events were a major cause [28], resulted in equipment damage and severe nuclear leakage, drawing the attention of researchers and nuclear industry professionals to seismic probabilistic safety assessment. Fragility analysis is a crucial component of seismic probabilistic safety analysis, providing the conditional probability of structural failure under different earthquake magnitudes to evaluate the seismic performance of structures. The Incremental Dynamic Analysis (IDA) method, as an important dynamic elastoplastic analysis method in seismic fragility analysis, can simulate the entire process from elasticity to elastoplasticity until collapse, used for structural collapse capacity analysis and performance evaluation. Many scholars have conducted extensive research on the theory and application of the IDA method [29]. The IDA method was first proposed by Bertero [30] in 1977, and over the following 40 years, it was widely studied by scholars and applied in many practical engineering applications. In 2002, Vamvatsikos et al. [31] provided a comprehensive summary of the IDA method. The PEER Center applied IDA for the seismic performance evaluation of the University of California’s science building [32]. Azimi H. et al. [33] proposed an approximate IDA curve method based on the Pushover analysis method, termed Incremental Modified Pushover, and by comparing this method with IDA and modal dynamic analysis, they showed it to be the least time-consuming, albeit with more conservative results. Castaldo et al. [34] used the IDA method to assess the exceedance probabilities of different limit states related to reinforced concrete superstructures and isolation layers, determining seismic fragility curves.

Recently, fragility assessment has also been applied to nuclear power plant structures and related equipment [35]. Firoozabad et al. [36] conducted a fragility study on nuclear power plant piping systems, showing that when deriving fragility curves for pipe elbows, both open and closed modes should be considered separately, with the low-probability failure of straight pipe segments having higher confidence levels than that of elbow segments. Zhao et al. [37] performed a seismic fragility analysis on base-isolated structures of nuclear power plants, indicating that the deformation coefficient of isolation bearings can serve as a damage index to assess the seismic fragility of isolated nuclear power plant structures. Nguyen et al. [38] established the finite element model of the nuclear power plant isolation structure through SAP2000, conducted nonlinear time-history analysis via IDA, defined the damage state based on the shear deformation of the isolation bearing, and carried out the seismic vulnerability analysis of the nuclear power plant isolation structure.

Most of the aforementioned studies on nuclear power plant fragility focus on the structures and related equipment, with limited research on the fragility of isolated structures in nuclear power plants. Moreover, in beyond-design-basis seismic events, isolated structures are prone to tensile stress in the bearings, leading to structural overturning. Therefore, it is essential to conduct seismic fragility analysis on base-isolated structures in nuclear power plants, particularly addressing overturning issues.

Our research group proposed a base isolation scheme for a diesel engine building in a nuclear power plant. The effectiveness of the isolation results was verified through shaking table tests, and the rationality of the modeling method was confirmed by comparing finite element simulations with experimental results. Based on this finite element model, a hybrid isolation scheme incorporating additional dampers was proposed to address the overturning issues observed in the tests. This paper conducts a seismic fragility analysis of the diesel engine building’s isolation scheme and the hybrid isolation scheme by using the IDA method. Considering the periodic characteristics of the isolated structure, the peak acceleration response spectrum (Sa) was selected as the seismic intensity parameter for IDA in finite element analysis calculations. The shear strain and tensile stress of the isolation bearings were used to measure the structure’s overturning resistance, with both indicators defining different limit states of the structure. Finally, by comparing the fragility analysis results of the two schemes, the improvement in the overturning resistance of the isolated structure was determined.

2. Validation of Finite Element Simulation Methodology

Our research group proposed a base isolation design scheme for the diesel engine building within the nuclear island of a nuclear power plant, as shown in Figure 1. Based on the similarity ratio design theory, a scaled model with a ratio of 1:8 was designed and constructed for shaking table tests. The experimental results demonstrated that the proposed base isolation design scheme is reasonable and effective. In this study, a finite element model was established by using SAP2000v20 software, and the simulation results were compared with the experimental data to validate the effectiveness of the modeling approach.

2.1. Experimental Model

The seismic prototype for this test is the diesel engine building of a nuclear power plant. This building is a 9-story reinforced concrete shear wall structure with a raft foundation. The total weight of the structure is approximately 14,246 tons, with a plan dimension of 26.60 m × 15.00 m and a total height of 36.9 m. The base-isolated structure of the diesel engine building incorporates an isolation layer added to the seismic-resistant structure. The designed isolation layer consists of 6 natural rubber bearings (LNR800) in the center and 32 lead rubber bearings (LRB1000) on the periphery. The plan layout of the isolation bearings is shown in Figure 2.

Taking into account the overall structural testing requirements, the size limitations of the shaking table, the maximum tonnage limit of the crane, the quality requirements for the production of isolation bearings, and the quality requirements for the construction of the concrete structural model, a length similarity ratio (model/prototype) of 1/8 was selected for the construction of the scaled model. Both the isolation layer and the superstructure above the isolation layer were designed by using the same similarity ratio for scaling.

The superstructure of the isolation test model has a length of approximately 4.16 m, a width of 2.66 m, and a total height of 4.8 m, while the entire isolation structure model has a total height of 5.17 m. The layout of the isolation bearings in the scaled model was designed based on the prototype layout scheme, scaled down by a geometric similarity ratio of 1/8. The types, quantities, and arrangement of the bearings were identical to those of the prototype. The experimental model is shown in Figure 3a.

2.2. Establishment of Finite Element Model

The seismic isolation test model was simulated by utilizing SAP2000 software. The finite element model was developed by employing shell and beam elements to construct the primary structural framework. The concrete material used for beams, floor slabs, and walls was C45. The steel material used for bearings and buttresses was Q345, and the reinforcing bars were HPB300. For the simulation of the isolation bearings, Rubber Isolator elements were adopted to represent the horizontal behavior, whereas Multilinear Elastic elements were utilized to model the vertical response. The bilinear hysteresis model was employed in the horizontal direction for LRB analysis. In consideration of the disparity in stiffness exhibited by actual isolation bearings under tensile and compressive conditions, the tensile stiffness of the bearings within the finite element model was designated as one-tenth of the compressive stiffness. The average mesh size of the shear walls and slabs in the SAP2000 model was 400 mm, calibrated through a sensitivity analysis. The configuration of the finite element model is illustrated in Figure 3.

2.3. Verification of Test Results and Finite Element Results

For isolated structures, the first three modes of vibration predominantly govern the dynamic response. The finite element analysis and experimental results exhibit a favorable agreement in the periods of the first two translational modes. A comparative analysis of modal frequencies is presented in

Table 1. Modal comparison between shaking table test model and finite element model.

Mode	Test Result		Finite Element Model of Test Structure		Error
	Frequency (Hz)	Direction	Frequency (Hz)	Direction
1	2.30	Y	2.32	Y	0.8%
2	2.33	X	2.34	X	0.4%
3	2.42	Twist	2.39	Twist	−1.3%
4	12.00	Y	16.7	Y	28.1%
5	26.00	X	25.68	X	−1.3%
6	30.50	Z	44.29	Z	31.1%

, and the mode shapes of the finite element model are shown in Figure 4.

The seismic isolation structure was subjected to tri-directional input by using the artificial ground motion acceleration of the Safe Shutdown Earthquake (SL-2) level, which represents the ultimate safety seismic motion. The peak ground acceleration (PGA) of the prototype ground motion corresponding to the SL-2 level is 0.3 g. A comparative analysis was conducted between the finite element results and the experimental data. Figure 5 presents a comparison of the hysteretic curves for the lead rubber bearing (LRB) under the same conditions, as obtained from the finite element model and the experimental model. The LRB is assigned horizontal bilinear hysteresis model through the elements, resulting in a more robust finite element hysteretic curve that exhibits a significantly more realistic response. However, the bilinear hysteresis model cannot fully capture the dynamic behavior of the horizontal stiffness of the LRB, resulting in errors between the test results of the hysteresis curve and the finite element results.

Figure 6 compares the acceleration–time-history curves of the experimental and finite element models. Subfigures (a) and (b) depict the horizontal acceleration comparisons at the first floor and the top floor, respectively, where the first-floor acceleration also represents the acceleration above the isolation layer. As illustrated in the figure, the waveforms and overall trends of the time-history curves from both models exhibit a high degree of consistency, with closely matched peak values. There are differences between the two acceleration–time-history curves around 4 s. This is attributed to the peak of ground motion acceleration occurring at approximately this time. During this period, the metal counterweights in the test model collide with each other, leading to a sudden increase in acceleration. Nevertheless, this phenomenon was not considered in the finite element model.

3. Finite Element Model of Anti-Overturning Hybrid Isolation Structure

3.1. Overturning Issue in the Test

Figure 7 presents the acceleration–time-history curves for the first, fourth, seventh, and ninth floors under tri-directional seismic input for the aforementioned isolation test scenario. It can be observed that the peak acceleration initially decreases and then increases with the rise in the floor level, indicating the presence of an overturning effect in the superstructure. This phenomenon is primarily attributed to the significant horizontal displacement at the isolation layer, which results in considerable tensile displacement of the bearings. Consequently, the superstructure exerts a substantial overturning moment on the foundation. When the isolation model is subjected to intense seismic activity, the isolation bearings are prone to tension, and since the tensile stiffness of the bearings is significantly lower than their compressive stiffness, the overturning effect becomes more pronounced.

3.2. Hybrid Seismic Isolation Scheme

To mitigate the overturning effect in the seismic isolation structure, a hybrid isolation system incorporating viscous dampers is proposed. In this scheme, 16 horizontal viscous dampers are added to the isolation layer of the experimental model to control and reduce the rocking motion. The viscous dampers have a damping coefficient of 19 kN∙s/m and a damping index of 1. The region exhibiting the maximum relative velocity is situated at the edges and corners of the isolation layer. The viscous dampers were positioned at the perimeter of the isolation layer to maximize their lever arm against overturning moments. The plan layout of the hybrid isolation system is illustrated in Figure 8a.

The equation of motion for the discretized system is expressed as

$$M\ddot{u}+C\dot{u}+Ku+F_{nl}(\dot{u},u)=-M\mathsf{\Gamma }{\ddot{u}}_g$$

(1)

where $F_{nl}$ encapsulates the nonlinear forces from isolators and dampers. $\mathsf{\Gamma }$ represent the influence vector. For clarity, the partitioned form isolating the superstructure and base is

$$\left[\begin{array}{cc}{M}_s& 0\\ 0& M_b\end{array}\right]\left\{\begin{array}{c}{\ddot{u}}_s\\ {\ddot{u}}_b\end{array}\right\}+\left[\begin{array}{cc}{C}_s& 0\\ 0& C_b\end{array}\right]\left\{\begin{array}{c}{\dot{u}}_s\\ {\dot{u}}_b\end{array}\right\}+\left[\begin{array}{cc}{K}_s& -K_s\\ -K_{\mathrm{s}}& K_s+K_b\end{array}\right]\left\{\begin{array}{c}{u}_s\\ u_b\end{array}\right\}=\left\{\begin{array}{c}0\\ -M_b{\ddot{u}}_g\end{array}\right\}+\left\{\begin{array}{c}{F}_{dampers}\\ 0\end{array}\right\}$$

(2)

where $M$, $C$, and $K$ represent the mass and the damping and stiffness matrices, respectively. $u_s$ and $u_b$ represent the displacement vectors of the superstructure and the isolation layer, respectively. ${\ddot{u}}_g$ represents the ground acceleration during an earthquake. $F_{dampers}$ represents the nonlinear force vector of the viscous damper. Here, $F_{dampers}=C\cdot sign({\dot{u}}_b)\cdot {\left|{\dot{u}}_b\right|}^n$ represents the nonlinear viscous forces, and $K_b$ includes the instantaneous stiffness of the lead rubber bearings updated at each time step.

3.3. Finite Element Analysis of Hybrid Seismic Isolation Scheme

By using the modeling methodology validated by experimental results, hybrid isolation models and the original isolation model were established. Both models were subjected to tri-directional artificial ground motions corresponding to the Beyond-Design-Basis Earthquake (BDBE) level (PGA = 0.5 g), and the time-history analysis was carried out. The anti-overturning capacity of the structures was evaluated based on four key computational outcomes: tensile stress in the bearings, maximum base shear, the anti-overturning moment ratio, and the floor acceleration amplification factor.

The anti-overturning moment ratio, denoted by $\beta$, is defined as the ratio of the anti-overturning moment to the overturning moment, serving as a safety reserve coefficient to verify the structure’s resistance to overturning under rare earthquake conditions. According to Article 4.3.7 of CECS126-2001 [39], the overturning moment should be calculated based on the rare earthquake action, while the anti-overturning moment should be determined by using the representative gravity load of the superstructure. The safety factor against overturning must exceed 1.2. The damping coefficient C was calibrated through an energy-based iterative procedure. The target additional damping ratio was set to suppress overturning motion, derived from the energy dissipation deficit observed in the test, determined through iterative time-history analysis to limit the safety factor against overturning below 1.2.

The computational results are presented in

Table 2. Comparison of anti-overturning capacity parameters between two seismic isolation schemes.

Scheme	Damping Index	Maximum Tensile Stress (Mpa)	Maximum X-Direction Base Shear (kN)	Anti-Overturning Moment Ratio
Base isolation	Undamped	1.58	365.86	1.159
Hybrid seismic isolation	α = 1	0.90	242.48	2.544

. The hybrid isolation model exhibits a reduction in both the tensile stress of the bearings and the base shear in the X-direction. The tensile stress of the bearings is 0.90 MPa, meeting the design target of keeping the tensile stress below 1 MPa under the BDBE seismic level. The base shear of the hybrid isolation scheme is reduced by 33.72% compared with the non-damped isolation scheme. The anti-overturning moment ratio for the non-damped isolation scheme is 1.159, slightly below the required threshold of 1.2, confirming the presence of overturning issues in this scheme. In contrast, the hybrid isolation scheme with dampers achieves an anti-overturning moment ratio of 2.544, satisfying the requirement of being greater than 1.2 and significantly enhancing the safety reserve against overturning. The floor acceleration amplification factors are illustrated in Figure 9. For the non-damped isolation scheme, the peak floor acceleration initially decreases and then increases with floor height, while for the hybrid isolation scheme, the peak acceleration increases with floor height, indicating an improvement in mitigating the overturning effect.

Based on the four evaluation metrics—the tensile stress of the bearings, the maximum base shear, the anti-overturning moment ratio, and the floor acceleration amplification factor—it is evident that the hybrid isolation scheme with viscous dampers (damping coefficient of 19 kN∙s/m and damping exponent of 1) effectively addresses the overturning issues observed in the experiments. This scheme can be adopted to optimize the structure’s resistance to overturning.

4. Fragility Analysis Based on IDA Method

4.1. Ground Motion Records

The selection of ground motion records is a critical factor influencing the analysis results of the IDA method. To mitigate the impact of ground motion variability on structural seismic response and enhance computational efficiency, this study adopts a method that aligns with the design response spectrum. Specifically, it ensures that the response spectrum values near the fundamental period of the structure closely match those of the design response spectrum at corresponding periods. Furthermore, the US FEMA P695 [40] report outlines eight criteria for selecting ground motion records: (1) the earthquake magnitude should be no less than 6.5; (2) the fault mechanism must be either strike-slip or thrust; (3) site conditions should be hard soil or rock; (4) the epicentral distance must be at least 10 km; (5) no more than two records can come from the same earthquake; (6) the peak acceleration should not be less than 0.2 g, and the peak velocity should not be less than 15 cm/s; (7) the effective duration of the ground motion should be at least 4 s; (8) strong-motion instruments should be placed in free fields or on the ground floor of small buildings. Research indicates that using 10–20 ground motion records can reduce variability and ensure the accurate evaluation of structural seismic capacity [37,41]. FEMA P695 recommends 22 sets of ground motion records. In conjunction with GB/T50011-2010 [42], this paper selects 10 sets of ground motion records from the PEER strong-motion database for IDA model calculations. Details of the selected records are provided in

Table 3. Ten ground motion records.

No.	Earthquake Name	Year	Station Name	Magnitude
1	Imperial Valley	1979	Delta	6.5
2	Imperial Valley	1979	El Centro Array #11	6.5
3	Landers	1992	Yermo Fire Station	7.3
4	Northridge	1994	Beverly Hills-Mulhol	6.7
5	Northridge	1994	Canyon Country-W Lost Cany	6.7
6	Kobe-Japan	1995	Nishi-Akashi	6.9
7	Kocaeli, Turkey	1999	Duzce	7.5
8	Chi-Chi-Taiwan	1999	CHY101	7.6
9	Duzce, Turkey	1999	Bolu	7.1
10	Manjil, Iran	1990	Abbar	7.4

, and the comparison between the selected seismic waves and the target RG1.60 [43] response spectrum is shown in Figure 10.

4.2. Intensity Index and Ground Motion Amplitude Modulation Rule

To obtain the complete dynamic response of the structure from elastic to elastoplastic to collapse, it is necessary to modulate the amplitude of each selected ground motion record into different levels. Various indices are used to indicate the intensity of ground motion. For instance, GB/T50011-2010 [42] employs PGA as an index, while FEMA P695 [40] uses the elastic acceleration spectrum S_a based on the fundamental period of the structure. Additionally, peak ground velocity (PGV) can serve as an amplitude modulation index. Studies have shown that S_a is more relevant and applicable as a strength index compared with PGA [44,45]. Therefore, this paper adopts S_a as the intensity measure (IM) for the amplitude modulation of ground motion records, with the fundamental period T being 2.5 s for both isolation structures.

Shinozuka et al. summarized three amplitude modulation methods suitable for IDA [46,47]. This study employs the equal-step ground motion recording method, with an intensity difference of 0.5 g and a peak ratio of 1:1:0.65 for the X, Y, and Z three-component acceleration recordings.

4.3. Damage Measure Selection

The IDA curve reflects the damage state of the structure under different ground motion intensity measures, so the horizontal and vertical axes of the IDA curve represent the structural ground motion intensity measure (IM) and the structural damage measure (DM), respectively. The structural damage index must intuitively and effectively reflect the damage and performance state of the structure, which is a crucial parameter in IDA. Common damage measures used by scholars in IDA include maximum inter-story displacement angle, structural vertex displacement, maximum floor acceleration, base shear force, and isolation bearing displacement [37,48,49].

For the overturning problem identified in shaking table tests of isolated structures, the IDA of its anti-overturning effect is conducted by using two finite element models of isolation proposed in this paper. Research shows that several factors contribute to the overturning of isolation structures: (1) excessive compressive stress in the isolation bearing; (2) excessive horizontal displacement of the isolation bearing; (3) tensile stress exceeding the limit value; (4) excessive overturning torque.

According to the results of the vibration table test of the isolated structure in this study, the compressive stress does not exceed the code limit, but the tensile stress and horizontal displacement of the bearing may exceed the limit behavior under high seismic intensity. When the bearing is under compression, tensile stress is negative. To facilitate statistical calculations, the parameters TP and $\gamma$ related to the tensile stress of the bearing are selected as the structural DMs. Tensile stress is the tension divided by the effective cross-sectional area of the bearing, and shear strain is defined as the horizontal displacement divided by the thickness of the rubber layer. These two indicators are defined in Formulas (3) and (4).

$$TP=T+5\mathrm{MPa}$$

(3)

where $T$ is the tensile stress result calculated by dynamic elastic–plastic analysis.

$$\gamma =S/h$$

(4)

where $S$ is the maximum horizontal displacement of the bearing and $h$ is the total thickness of the rubber layer of the bearing.

4.4. Definition of Limit States

The failure mechanisms of seismically isolated structures are distinct from those observed in conventional aseismic structures. Owing to the relatively low stiffness and high elasticity of isolation bearings, the isolation layer serves as the primary energy dissipation mechanism during seismic events. Consequently, the displacement of the isolation layer is markedly greater than that of the superstructure, which remains relatively rigid. Isolation bearings, acting as the critical yet vulnerable components in nuclear power plant isolation systems, can precipitate the overturning or even collapse of the superstructure in the event of their failure.

In ATC-40 [50], building damage is classified into six distinct levels: Immediate Occupancy, Damage Control, Life Safety, Limited Safety, Collapse Prevention, and Complete Collapse. GB/T 24335-2009 [51] categorizes building damage into five levels: Basically Intact, Slight Damage, Moderate Damage, Severe Damage, and Destruction. Similarly, GB/T50011-2010 [42] delineates building damage into five levels: Basically Intact, Slight Damage, Moderate Damage, Severe Damage, and Collapse.

In accordance with the Chinese codes, this study establishes five damage states for seismically isolated structures, which are further refined into four limit states. The relationship between the damage states and the corresponding limit states is detailed in

Table 4. Classification of damage levels.

Destructive State	Description of Destruction State	Value of Destruction State
Basically Intact (DS1)	The load-bearing components are intact; minor damage to individual non-load-bearing components; varying degrees of damage to accessory components.	≤LS1
Slight Damage (DS2)	Some load-bearing components have slight cracks, while some non-load-bearing components have obvious damage; the auxiliary components have varying degrees of damage.	(LS1, LS2]
Moderate Damage (DS3)	Most load-bearing components have slight cracks, with some showing obvious cracks; serious damage to individual non-load-bearing components.	(LS2, LS3]
Severe Damage (DS4)	Most load-bearing components are severely damaged or partially collapsed.	(LS3, LS4]
Destruction (DS5)	Most load-bearing components have collapsed.	≥LS4

.

According to HAD102/02-1996 [52], the diesel engine building of a nuclear power plant is classified as a Seismic Category I item related to nuclear safety, and its seismic design and analysis should be conducted based on SL-2. GB 50267-97 [53] stipulates that the peak acceleration of Operating-Basis Earthquake (SL-1) should not be less than half of the peak acceleration of the corresponding SL-2. The peak acceleration of the Safe Shutdown Earthquake is specified as 0.3 g, which corresponds to the rare earthquake level for a seismic fortification intensity of 7 degrees. GB/T50011-2010 [42] specifies that the tensile stress of rubber bearings under three-dimensional seismic action during rare earthquakes should not exceed 1 MPa. Furthermore, GB/T 51408-2021 [54] states that for structures with special seismic fortification requirements, the horizontal displacement limit of seismic isolation rubber bearings during extremely rare earthquakes can be taken as 4.0 times the total thickness of the rubber layer. Under SL-2, the maximum horizontal displacement of the isolation bearings should be less than 1.3 times the total thickness of the rubber layer and should not exceed 300 mm.

Based on the above codes and standards, which define the limits for structural seismic response parameters under different seismic intensities, the limit values for different limit states TP and $\gamma$ defined in this paper are shown in

Table 5. Limit values for extreme states.

Limit State	LS1	LS2	LS3	LS4
TP	2.5 Mpa	5 Mpa	6 Mpa	6.5 Mpa
$\gamma$	50%	130%	300%	400%

.

4.5. Incremental Dynamic Analysis (IDA)

In this study, IDA was conducted on both the base-isolated structure and the hybrid isolation structure by subjecting them to 10 selected ground motion records. Nonlinear time-history analysis was performed, and the resulting IDA curves were compared and analyzed. The IDA curves in this study are plotted with the DMs TP and $\gamma$ as the horizontal axis and the IM S_a (T,5%) as the vertical axis.

Figure 11 presents a comparative analysis of IDA curves between the base-isolated structure and the hybrid isolation structure, using seismic record 10 as the excitation input for the computational model. The comparison is illustrated through two different DMs: shear strain $\gamma$ and TP value. As depicted in the figure, the IDA curves for both structures exhibit similar trends, with both shear strain $\gamma$ and TP value showing a positive correlation with the spectral acceleration S_a (T,5%) of the seismic record. However, the growth trends of the IDA curves differ based on the chosen DM.

In Figure 11a, the gray-shaded area indicates that the slope of the curve changes with the increase in shear strain $\gamma$. When the spectral acceleration S_a (T,5%) is relatively low, resulting in smaller shear strains, the IDA curve exhibits a steeper slope, indicating that the rubber bearings are in the elastic phase with pre-yield stiffness. As S_a (T,5%) increases, leading to larger shear strains, the slope of the IDA curve becomes gentler, reflecting the post-yield stiffness of the rubber bearings. This behavior aligns with the bilinear model of the LRB in the finite element model shown in Figure 11b, where the post-yield stiffness is less than the pre-yield stiffness. In Figure 11c, the gray-shaded area reveals an inflection point in the IDA curve around a TP value of 5 MPa, where the slope of the curve increases. This arises because, according to the definition of the TP value, when the TP value exceeds 5 MPa, the tensile stress T in the isolation bearing is positive, indicating that the bearing is in tension. Conversely, when the TP value is below 5 MPa, the tensile stress T is negative, indicating compression. The finite element model used in this study assigns different stiffness values to the isolation bearings under tension and compression, with the tensile stiffness being one-tenth of the compressive stiffness. Consequently, for the same increase in seismic intensity, the increment in TP value is greater when the structure is under compression compared with when it is under tension. For the same seismic record, the slope of the IDA curve for the hybrid isolation structure is steeper than that for the base-isolated structure. This indicates that for the same seismic intensity S_a (T,5%), the shear strain in the base-isolated structure is greater than that in the hybrid isolation structure, and the increment in TP value is also larger for the base-isolated structure. This suggests that the hybrid isolation structure provides better control over the horizontal displacement and tensile stress in the isolation bearings, reducing the likelihood of structural overturning during an earthquake and enhancing the overall seismic performance.

A single IDA curve cannot fully capture the impact of different seismic records on the seismic performance of the two isolation structures. By summarizing the IDA curves from 10 different seismic records for each structure, we obtain IDA curve clusters, as shown in Figure 12 and Figure 13. Due to the randomness and variability of seismic motions, the DM values are relatively small, and the differences between the curves are minimal when the seismic intensity S_a (T,5%) is low, resulting in a more concentrated IDA curve cluster. As the seismic intensity S_a (T,5%) increases, the variability among the IDA curves becomes more pronounced. To mitigate the influence of data variability on the final analysis, a quantile regression statistical method is applied to the IDA data, yielding three quantile curves at the 16%, 50%, and 84% levels. This approach facilitates the identification of statistical patterns in the data.

To quantitatively compare the anti-overturning capacity of the base-isolated structure and the hybrid isolation structure, the spectral acceleration Sa (T,5%) corresponding to the 50% quantile curves under different limit states was recorded, as shown in

Table 6. Comparison of Sa (T,5%) values for different limit states of 50% quantile curves.

DM	Structure	LS1	LS2	LS3	LS4
$\gamma$	Isolated structure	0.1117 g	0.2367 g	0.45 g	0.5707 g
	Hybrid isolation structure	0.1341 g	0.307 g	0.6098 g	0.7827 g
	Growth rate	16.7%	22.9%	26.2%	27.1%
TP	Isolated structure	0.109 g	0.199 g	0.391 g	0.458 g
	Hybrid isolation structure	0.124 g	0.286 g	0.473 g	0.552 g
	Growth rate	12.0%	30.4%	17.3%	17.0%

. The results indicate that both shear strain and TP value increase synchronously with the rise in seismic intensity. Furthermore, the hybrid isolation structure equipped with viscous dampers requires higher seismic intensity to reach the same limit state under different seismic excitations compared with the base-isolated structure. When shear strain $\gamma$ is used as the DM, the maximum increase is 27.1%, and when TP is used as the DM, the maximum increase is 30.4%. This is because, during seismic excitation, the dampers in the hybrid isolation structure dissipate a portion of the energy absorbed by the isolation layer, resulting in smaller horizontal displacements and tensile stresses in the isolation bearings. Consequently, the anti-overturning capacity of the structure is significantly enhanced.

4.6. Seismic Probability Demand Model of Structures

Based on the data obtained from Incremental Dynamic Analysis, a probabilistic seismic demand model is established to define the functional relationship between the IM and the DM. In this study, two DM values are adopted: shear strain $\gamma$ and TP value. The IM is defined by Sa (T,5%), and the relationship between the IM and the DM can be expressed by Formula (5) [55].

$$DM=\mathrm{a}{(IM)}^b.$$

(5)

Assuming that both the DM values and IM values follow a lognormal distribution, taking the natural logarithm of both sides of Formula (3) yields the following expression:

$$\ln (DM)=A+B\ln (IM).$$

(6)

Linear regression analysis was performed on the data points satisfying Formula (6), resulting in probabilistic demand models for DM values corresponding to both base-isolated structures and hybrid base-isolated structures. The established models, along with their statistical characteristics, are presented in

Table 7. Probabilistic seismic demand model.

Structure	Probabilistic Seismic Demand Model
Isolated structure	$\ln (\gamma )=1.971+1.208\ln (Sa(T,5\%))$
Hybrid isolation structure	$\ln (\gamma )=1.758+1.306\ln (Sa(T,5\%))$
Isolated structure	$\ln (\gamma )=2.388+0.723\ln (Sa(T,5\%))$
Hybrid isolation structure	$\ln (\gamma )=2.210+0.617\ln (Sa(T,5\%))$

.

4.7. Seismic Fragility Analysis

Seismic fragility reflects the probability of structural damage exceeding a specified limit state under varying seismic intensity. It serves as a critical metric for evaluating the safety of structures at specific seismic intensity levels and their inherent capacity to withstand seismic actions. This analysis is of significant importance for developing reliability-based seismic design codes. The probability value can be calculated by using Formula (7):

$$P_{\mathrm{f}}=\mathrm{P}\left(LS/IM\right)=\mathrm{P}(\mathrm{C}<\mathrm{D}|\mathrm{IM})$$

(7)

The seismic capacity parameter C and the seismic demand parameter D are assumed to follow a lognormal distribution. Under this assumption, the failure probability can be formulated as

$$P_{\mathrm{f}}=\varphi \left({\displaystyle \frac{\ln (-\widehat{C}/\widehat{D})}{\sqrt{{\beta }_c^2+{\beta }_{\mathrm{d}}^2}}}\right)=\varphi \left({\displaystyle \frac{\ln (a*Sa{(T,5\%)}^b/\widehat{C})}{\sqrt{{\beta }_c^2+{\beta }_{\mathrm{d}}^2}}}\right)$$

(8)

In this context, $\widehat{C}$ and $\widehat{D}$ correspond to the median values of the seismic capacity parameter and the seismic demand parameter, respectively. Similarly, ${\beta }_c$ and ${\beta }_d$ represent the logarithmic standard deviations associated with $C$ and $D$. According to HAZUS99, when defining the IM as Sa, the parameter $\sqrt{{\beta }_c^2+{\beta }_{\mathrm{d}}^2}$ is typically assigned a value of 0.4 [56]. Additionally, $\varphi \left(*\right)$ denotes the mathematical expression of the standard normal distribution function.

$$\varphi (x)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\infty }^x\exp \left(-\frac{t^2}{2}\right)}dt$$

(9)

The exceedance probabilities of structural damage measure reaching various limit states under different seismic intensities are determined by substituting the data from Table 7 into Formula (8). By connecting the data points corresponding to different limit states, the fragility curves are constructed. As demonstrated in Figure 14, this figure provides a comparative analysis of fragility curves for base-isolated structures and hybrid base-isolated structures, with shear strain and TP values acting as distinct DMs.

Compared with seismic isolation structures, hybrid seismic isolation structures have smaller values on the vertical axis of the fragility curve under different ultimate states. This indicates that under identical seismic intensities, hybrid base-isolated structures have lower failure probabilities when their seismic isolator shear strain reaches the threshold, suggesting reduced fragility and a decreased likelihood of reaching critical failure states. Additionally, under the same limit state conditions, the disparity in failure probabilities between hybrid base-isolated structures and base-isolated structures widens as seismic intensity increases. This underscores the superior seismic performance and enhanced anti-overturning capabilities of hybrid base-isolated structures during stronger earthquakes.

To quantify the fragility comparison between the two structural types by using shear strain and TP value as DMs, a fragility matrix summarizing the fragility curve data was developed, as presented in

Table 8. Fragility matrices of two structures based on shear strain $\gamma$ and TP values.

Earthquake Level	Structure	Extreme State (DM: $\mathit{\gamma }$)				Extreme State (DM:TP)
		LS1	LS2	LS3	LS4	LS1	LS2	LS3	LS4
SL-1	Isolated structure	26.8%	1.6%	0	0	22.0%	1.6%	0.7%	0.5%
	Hybrid isolation structure	8.8%	0.3%	0	0	25.5%	1.9%	0.6%	0.5%
SL-2	Isolated structure	87.3%	30.8%	1.5%	0.3%	73.3%	18.3%	9.5%	6.7%
	Hybrid isolation structure	71.6%	11.3%	0.2%	0	73.2%	17.2%	8.7%	6.1%
BDBE	Isolated structure	99.7%	78.0%	14.2%	4.4%	94.1%	46.8%	30.4%	23.8%
	Hybrid isolation structure	98.1%	52.7%	3.9%	0.8%	91.8%	41.1%	25.6%	19.6%
Extreme rare earthquake	Isolated structure	100%	99.1%	65.4%	37.9%	99.5%	80.9%	66.5%	58.7%
	Hybrid isolation structure	100%	95.8%	40.1%	17.5%	99.1%	72.3%	55.2%	46.2%

. Under limit states LS1 to LS4, the maximum fragility differences between hybrid base-isolated and base-isolated structures are 19.5%, 25.3%, 25.3%, and 28.1% for the $\gamma$-based DM, and 3.5%, 8.6%, 11.3%, and 12.5% for the TP-based DM. This variation is attributed to the increase in PGV with higher seismic intensities. Since the viscous dampers in hybrid base-isolated structures are velocity-dependent, their damping force escalates with PGV, enhancing energy dissipation and reducing failure probabilities under stronger seismic events. Consequently, hybrid base-isolated structures demonstrate improved anti-overturning performance and reduced overturning effects at higher seismic intensity.

When the DM surpasses the threshold associated with LS4, the structure transitions to the DS5 collapse state. Consequently, the fragility curve for LS4 holds significant importance. As illustrated in Figure 15, a comparative analysis is conducted between the fragility curves representing LS4 for two damage metrics: the TP value and shear strain. The results reveal that the fragility curve for hybrid base-isolated structures demonstrates a smoother trend compared with conventional base-isolated structures, indicating reduced structural fragility and enhanced anti-overturning performance.

To better evaluate the improvement in anti-overturning capability of hybrid base-isolated structures and perform a quantitative comparison of the collapse resistance between the two types of base-isolated structures, this study adopts the Collapse Margin Ratio (CMR) method for assessing structural collapse resistance, as recommended by FEMA P58 [57].

The CMR is defined as the ratio of the seismic intensity corresponding to a 50% probability of collapse to the seismic intensity corresponding to the design earthquake. In this study, considering the adopted seismic intensity levels, the BDBE acceleration peak corresponds to the design earthquake acceleration peak. Therefore, the expression for the CMR is

$$CMR=S\mathrm{a}{(T,5\%)}_{50\%}/S\mathrm{a}{(T,5\%)}_{BDBE}$$

(10)

The CMR for base-isolated structures and hybrid base-isolated structures with two different damage measures are summarized in

Table 9. Anti collapse reserve coefficient based on shear strain $\gamma$ and TP values as damage measures.

Structure	Shear Strain ($\mathit{\gamma }$)	TP
Isolated structure	1.88	1.52
Hybrid isolation structure	2.30	1.79

. The results indicate that hybrid base-isolated structures exhibit consistently higher CMR values compared with conventional base-isolated structures, reflecting a significant enhancement in their collapse resistance. Furthermore, the viscous dampers demonstrate effective energy dissipation under BDBE, contributing to improved structural performance.

Additionally, the CMR values based on the TP value as the damage metric are consistently lower than those based on shear strain, indicating a more conservative assessment of collapse resistance when using the TP value. This finding is consistent with the increased sensitivity of the TP value in capturing tensile-related damage, as previously discussed.

5. Conclusions

This study established a finite element model for the base-isolated structure of a nuclear power plant’s diesel building and validated the modeling methodology’s effectiveness through shake table test results. To address the overturning issue observed in the tests, a hybrid base-isolation scheme was proposed, and a corresponding finite element model was developed. IDA was employed to conduct seismic fragility analyses on both the base-isolated and hybrid base-isolated structural models. The following conclusions were drawn:

• The IDA curves using shear strain ($\gamma$) as the damage measure exhibit a reduction in slope near the seismic intensity corresponding to isolator yielding. In contrast, IDA curves using the TP value as the damage metric show an inflection point near the seismic intensity corresponding to isolator tension and reduced vertical stiffness. Under identical earthquake records, the slope of the IDA curves for hybrid base-isolated structures is steeper than that for conventional base-isolated structures, indicating that the isolators are less prone to damage. Additionally, at the same seismic intensity, the damage measure values for hybrid base-isolated structures are smaller, reducing the likelihood of overturning under seismic loads and improving overall seismic performance.
• Under the same limit state, the fragility curves for hybrid base-isolated structures are flatter compared with those for conventional base-isolated structures. This indicates that hybrid base-isolated structures have a lower probability of failure at a given limit state, demonstrating superior anti-overturning performance.
• Based on a statistical analysis of fragility curve data, hybrid base-isolated structures exhibit lower fragility and greater collapse resistance reserves compared with conventional base-isolated structures. This is reflected in both the fragility matrices and the CMR.
• Using the TP value as the damage metric for seismic fragility assessment yields more conservative results. Therefore, in structural design, particular attention should be paid to enhancing the tensile capacity of isolators to mitigate potential damage.
• The fragility assessment based on shear strain ($\gamma$) provides valuable references for determining collision trench limits in seismic isolation systems for nuclear power plant diesel buildings.
• The collaborative design of isolation for non-structural components will be regarded as the next research direction in the future. By using dynamic characteristics such as floor response spectra obtained from the shaking table test, the dynamic coupling effect between the isolation layer and non-structural components such as building equipment (diesel generators) and pipelines will be optimized to avoid chain failure caused by local damage. Moreover, uncertainty analysis and seismic fragility assessment will be conducted for the failure modes of non-structural components.