A Review of Research Progress on Friction Pendulum Isolation Bearings: Theory, Experiment, and Numerical Simulation

A Review of Research Progress on Friction Pendulum Isolation Bearings: Theory, Experiment, and Numerical Simulation

1st Chao Zhao

Faculty of Architecture and Civil Engineering Huaian University Huaian, China

2nd Peng Zhang*

Faculty of Architecture and Civil Engineering Huaian University Huaian, China

3rd Shijun Chen

Jiangsu Meicheng Architectural & Planning Design Institute Co., Ltd. Huaian, China

4th Yujie Wang

Faculty of Architecture and Civil Engineering Huaian University Huaian, China

Abstract – The friction pendulum system (FPS), as a typical recenteringenergy dissipation isolation device, has signicant application value in structural seismic engineering. This paper systematically reviews the research progress in theoretical mod- eling, experimental investigations, and numerical simulations of FPS. Starting from the fundamental mechanical mechanism of the single friction pendulum (SFP), the decoupling characteristic between isolation period and structural mass is analyzed, and the development of double friction pendulum (DFP) and triple friction pendulum (TFP) systems is outlined, highlighting their staged control capability achieved through multiple sliding in- terfaces. Based on material- and component-level experiments, the time-dependent behavior of the friction coefcient under the coupled effects of velocity, normal pressure, and tempera- ture is elucidated. Furthermore, rened nite element models and macroscopic simplied models are compared in terms of modeling strategies and applicability. The current limitations in multi-physics coupling mechanisms, three-dimensional dynamic response characterization, and life-cycle performance evaluation are identied, and future research directions are discussed. It is shown that establishing a unied analytical framework incor- porating multi-eld coupling and three-dimensional nonlinear analysis is essential for improving the accuracy and reliability of FPS-based seismic isolation systems.

Index TermsFriction Pendulum System (FPS),Multi-stage

friction pendulum,Multi-physics coupling,Numerical simula- tion,Seismic performance

1. Introduction

   As a natural disaster characterized by high randomness and destructive power, earthquakes have long posed a central challenge in the eld of civil engineering. Traditional seismic design relies on structural members entering a non-elastic phase to dissipate energy; this strategy of trading damage for safety is prone to causing signicant residual deformation and

   structural damage during strong earthquakes, resulting in high post-earthquake repair costs and, in some cases, forced demo- lition due to loss of functionality [13]. To meet the demands of resilient urban development, the philosophy of seismic engineering is shifting from ensuring life safety to achieving functional recovery [4]. Chinas 2021 Regulations on Seismic Management of Construction Projects [5] explicitly require that important public buildings in high-seismic-intensity zones prioritize the use of seismic isolation and vibration-damping technologies. Against this backdrop, the Friction Pendulum System (FPS) has become a focal point of research and application due to its unique mechanical properties [6].

   Compared to traditional lead-rubber bearings (LRB), as shown in Figure 1, the FPS offers signicant advantages: The natural period depends solely on the radius of curvature of the sliding surface, completely decoupling it from the mass of the superstructure; gravitational forces can be utilized to provide restoring forces, enabling excellent self-resetting capability and signicantly reducing residual displacement; simultaneously, energy dissipation through dry friction pro- duces a full and stable hysteresis curve, demonstrating superior energy dissipation performance [7,8]. However, as applica- tion scenarios extend into complex environments, traditional single-stage FPS struggles to meet the demands of extreme conditions. Under the action of near-fault velocity pulses, the displacement of the seismic isolation layer is prone to exceeding limits; in high-rise structures, the problem of tensile detachment caused by overturning moments is prominent; and the thermal degradation of sliding plate materials under high- speed reciprocating friction severely affects long-term service performance [9]. Therefore, the development of multi-stage friction pendulum bearings and models of their performance degradation mechanisms has become an important research

   direction in the eld of seismic isolation.

   This equation indicates that the seismic isolation period is determined solely by the radius of curvature and is in- dependent of the structural mass, exhibiting a period-mass decoupling characteristic. However, because its displacement capacity and mechanical parameters are not adjustable, SFP systems struggle to simultaneously meet the requirements for both small-to-medium and large earthquakes; therefore, multi-

   stage friction pendulum systems have emerged as a direction for further development [8, 14].

   ffi

   00ffi

   [!£IDffii00

   ffi

   Fig. 1. Comparison between traditional Lead rubber bearing and traditional friction pendulum bearing

   This paper systematically summarizes the latest advance- ments in friction pendulum bearings regarding the evolution of mechanical theories, multiscale experimental validation, and high-performance numerical simulation. It provides an in- depth analysis of the dynamic response mechanisms of multi- stage friction pendulums under complex loads, identies re- search bottlenecks and breakthrough opportunities, and offers guidance for the design and promotion of high-performance, resilient seismic isolation systems.

   Friction pendulum supports provide restoring force through their curved geometry and dissipate seismic energy via in- terfacial friction, making them a typical restoring-damping synergistic seismic isolation system [1,2,10]. Since Zayas et al. proposed the single-stage model in the 1980s, its form has evolved from single-stage to multi-stage congurations, essentially to meet performance control requirements under different seismic levels [8,11]. Based on this, we rst ana- lyze the fundamental mechanism of the single-stage friction pendulum.

   The single-stage friction pendulum (SFP) consists of a concave spherical slideway and a spherical slider; the contact surfaces are typically coated with self-lubricating materials such as modied PTFE [12], as shown in Figure 2. Its oper- ating mechanism is as follows: as the slider moves along the curved surface, the tangential component of gravity provides geometric self-resetting capability. Its equivalent lateral stiff- ness is determined solely by the vertical load and the radius of curvature, exhibiting geometric control characteristics that are proportional to the vertical load and inversely proportional to the radius of curvature [11,13]. Based on the motion equations of a simple pendulum, its seismic isolation period is:

   ffi

   Fig. 2. Single friction pendulum bearing

   To address the limitations of single-stage friction pendulums regarding displacement capacity and parameter exibility, Daniel M. Fenz and Michael C. Constantinou proposed the Double Friction Pendulum (DFP) [8]. This device consists of two concave sliding surfacesone upper and one lowerand a double-sided slider in between; it can essentially be viewed as a series conguratin of two SFPs. Compared to the SFP, the DFP signicantly extends the effective sliding path through coordinated sliding at both interfaces, thereby enhancing the displacement capacity of the seismic isolation layer. At the same time, by setting different friction coefcients, it achieves a staged sliding mechanism: under low shear forces, only the low-friction interface initiates sliding; as the load increases, both interfaces slide simultaneously, exhibiting a two-stage hysteresis characteristic [8,14]. This mechanism enables the structure to respond differentially to seismic motions of vary- ing intensities, thereby achieving preliminary performance- based control; however, it also imposes higher design require- ments for the stability of interface coordination.

   Building upon the DFP, Daniel M. Fenz and Michael C. Constantinou proposed the Triple Friction Pendulum (TFP) and its multi-stage extensions [8] to further meet safety requirements under extremely rare earthquakes. Compared to

   s

   T = 2 R

   g

   (1)

   the two-stage response of the DFP, the TFP achieves more rened hierarchical control through multiple slip interfaces

   TABLE I

   In-depth comparison of key mechanical properties and design characteristics of friction pendulum bearings (FPS) with different orders

   Comparison Dimension	Single Friction Pendulum (SFP)	Double Friction Pendulum (DFP)	Triple Friction Pendulum (TFP)
   Structural Composition	Single spherical slider + Sin- gle concave slider	Double-concave slider + Inter- mediate double-sided slider	Nested sliders + Four sliding interfaces + Double internal concave surfaces
   Effective Radius (Reff )	R1 p	(R1 p)+ (R2 p)	Involves complex combina- tions of 4 sets of curvature radii
   Number of Friction Interfaces ()	1 set of xed friction coef- cients	2 sets (typically 1 = 2 or asymmetric)	4 sets (1 =/2 =/3 =/4)
   Hysteretic Characteristic Curves	Bilinear: Single stiffness	Trilinear: Variable stiffness in- ection points	Multi-linear: Five-stage stiff- ness switching
   Displacement Capacity (dmax)	2(R h) sin (Limited by slider diameter)	Theoretically 2 times that of SFP with the same diameter	Extensive geometric redun- dancy, reaching 3-4 times that of SFP
   Self-Centering Capability	Provided by single curvature, direct restoring path	Collaborative resetting by double surfaces, slip sequence exists	Complex progressive reset- ting, affected by multi-level friction gradients
   Performance Adjustment Methods	Can only adjust single R and	Can adjust combinations of 2 sets of R and	Multi-degree-of-freedom ad- justment: Can precisely tune responses for different earth- quake levels
   Main Mechanical States	Sliding / Stopping	1-surface sliding / 2-surface full sliding / Stopper engage- ment	Internal sliding / Coordinated sliding / External sliding / Stiffening

   and parameter combinations. Its typical feature is a multi- stage, sequential activation mechanism: during low-intensity earthquakes, the inner interface slips rst to isolate micro- vibrations; as seismic intensity increases, the remaining in- terfaces engage sequentially, allowing the system stiffness to adjust gradually and enhancing energy dissipation capacity; under extreme conditions, the displacement-constraint mecha- nism provides additional recovery and stabilization functions [8,15]. Through the combined design of geometric parameters and friction characteristics, the TFP enables adjustable control of hysteretic behavior and mechanical response, embodying a design philosophy of multi-level energy dissipation and hierarchical protection, thereby signicantly enhancing the comprehensive seismic performance of seismic isolation struc- tures [10,15].

   Based on the aforementioned evolutionary process (see Table 1), the focus of research on friction pendulum bearings has shifted from simple displacement compensation to multi- objective performance tuning. SFP laid the foundation for pendulum-type seismic isolation, DFP addressed the geometric limitations on displacement capacity, and TFP achieved precise matching between mechanical response and seismic require- ments through a complex matrix of interface parameters. This evolutionary trajectory provides the physical basis for the subsequent chapters to delve into the theory of nonlinear dynamics and rened numerical simulations.

   As friction pendulum seismic isolation technology evolves from single-stage systems to multi-stage sliding structures, its mechanical behavior exhibits signicant nonlinearity and multi-physics coupling characteristics, placing higher demands on traditional design theories and analysis methods. To system- atically review research progress and key issues in this eld, this paper provides a comprehensive overview across three dimensionstheoretical models, experimental studies, and nu- merical methodsand extends to engineering applications and future trends, thereby establishing a complete analytical frame- work spanning from mechanistic understanding to engineering implementation.

   Chapter 2 focuses on the evolution of theoretical models, systematically analyzing the development of the constitutive model for the restoring force of friction pendulum bearings from an ideal bilinear model to a time-varying model that accounts for the coupling of pressure, velocity, and tem- perature, and elucidating the phased sliding mechanism and stiffness control principles of multi-stage friction pendulums (DFP/TFP). Chapter 3 centers on advances in experimental re- search, reviewing the results of friction interface material prop- erty tests and component-level dynamic loading tests. It fo- cuses on the experimental verication of friction performance evolution and multi-stage sliding behavior under the coupled effects of multiple factors, and summarizes the limitations of existing experimental systems. Chapter 4 focuses on numerical

   simulation methods, comparing the modeling strategies and applicability of rened nite element models and macroscopic equivalent models, and analyzing trends in contact algorithms, multiphysics coupling, and efcient computational methods. Building on this foundation, subsequent chapters further in- tegrate engineering applications to summarize the key issues and future directions of friction pendulum seismic isolation technology under complex operating conditions.

   Overall, through a comprehensive multi-scale and multi- method analysis, this paper aims to provide a systematic reference for the theoretical research and engineering design of high-performance friction pendulum seismic isolation systems.

2. Advances in Theoretical Research on Friction Pendulum Seismic Isolation Systems

   The mechanical constitutive model of a friction pendulum support serves as the theoretical foundation for evaluating the dynamic response, energy dissipation efciency, and post- seismic recovery capacity of seismic isolation structures. This chapter begins with the fundamental force mechanisms of a single-stage friction pendulum to analyze its restoring force characteristics and period decoupling mechanisms, and further explores the evolution of time-varying friction characteristics under multiphysics coupling conditions.

   1. Basic Mechanical Constitutive Model of a Friction Pendu- lum Bearing

      The fundamental mechanism of a single-degree-of-freedom friction pendulum is to tilize the structural self-weight to provide a restoring force and to dissipate seismic energy through the work done by dry friction at the sliding inter- face [3, 16, 17]. Based on the large-radius-of-curvature and small-deformation assumptions (sin tan u/R), the horizontal output force of the support at any given instant can be expressed as [8, 11]:

      W

      basis for the FPS to achieve cooperative seismic isolation in multi-mass structures and effectively suppress torsional effects [8].

      1. V-Dependence:Constantinou et al. proposed, based on extensive dynamic test data, that A asymptotically tran- sitions from the initial sliding friction coefcient B to the high-slip friction coefcient C as the sliding speed increases. This evolution is typically described using an exponential decay model [17]:

         = max ( max min)eav (3)

         where a represents the rate-sensitive hardening index.

      2. P-Dependence:As the axial stress p increases, the in- crease in the shear strength of composite sliding plate materials is typically smaller than the increase in normal stress, resulting in a nonlinear decrease in the coefcient of friction [3,18]. This characteristic has a signicant impact on the seismic isolation design of heavily loaded structures in high-intensity zones.

      3. T-Dependence:The heat generated by high-frequency reciprocating friction under strong vibrations causes an instantaneous temperature rise at the interface, which in turn triggers thermal softening of the sliding plate material. The thermal softening effect has been veried through numerical analysis and experimental studies of frictional heat generation [19]. Current detailed the- oretical studies have incorporated thermo-mechanical coupling algorithms to correct for energy dissipation attenuation under long-range dynamic responses.

         Although traditional single-stage constitutive mod- els offer extremely high computational efciency, their single-stage stiffness behavior cannot capture the se- quential initiation mechanism of multi-level sliding in- terfaces, making it difcult to meet the detailed require- ments of multi-level seismic design.

         F =

         Reff

         u + W sgn(u) (2)

   2. Evolution of the Series Mechanical Constitutive Model for

      Refff

      In this model, the restoring force term W u denes the systems restoring stiffness. Reff = R h represents the effective radius of curvature, and h is the perpendicular distance from the center of the slider to the sliding surface. This term reveals the unique decoupling characteristic of the FPS system with respect to its period T , namely, that the systems seismic isolation period is primarily determined by the geometric curvature R and is independent of the mass of the superstructure [8,11], which signicantly enhances the systems robustness under live load uctuations. The friction term E denes the systems characteristic strength, where W sgn(u) is the interfacial sliding friction coefcient, and sgn(u) ensures that the direction of the frictional force is always collinear with the velocity vector. In terms of macro- scopic dynamic behavior, this constitutive model exhibits typ- ical bilinear hysteresis characteristics. Its post-yield stiffness depends solely on the axial pressure and geometric radius; this proportionality between stiffness and load forms the physical

      Multi-Stage Friction Pendulum Bearings

      To address the limitations of SFP in terms of displacement capacity and performance control, multiple sliding interfaces were introduced to evolve the system from bilinear to multi- stage nonlinear behavior [20]. The key lies in the coordinated sliding and displacement distribution mechanisms among these multiple interfaces, thereby enabling hierarchical control of seismic isolation performance [15, 21].

      The DFP can be regarded as a series conguration of two SFPs. The forces at each interface remain consistent, and the total displacement satises the superposition principle. Its mechanical response exhibits a phased evolution. During the initiation phase, the low-friction interface slips preferentially. In the cooperative sliding phase, as the load increases, both interfaces participate simultaneously, and the system stiffness undergoes a transition. This mechanism extends the equiva- lent radius of curvature, thereby signicantly increasing the displacement capacity within a nite size and enhancing the

      TABLE II

      Performance differences of three types of friction pendulum systems

      Indicator	SFP	DFP	TFP
      Stiffness Regulation Capability	Single	Dual-stage	Multi-stage
      Energy Dissipation Capacity	Moderate	Strong	Strongest
      Parameter Complexity	Low	Medium	High
      Design Difculty	Low	Medium	High
      Engineering Applicability	Relatively Broad	Relatively Optimal	Limited

      systems resilience to strong earthquakes (particularly those near faults) [22].

      Building upon the DFP, the TFP constructs a multi-stage re- sponse system through multiple sliding interfaces, whose hys- teresis behavior can be summarized as a continuous process of stage-by-stage activationstiffness transitionand ultimate constraint. In the initial stage, the inner interface is activated to achieve micro-vibration isolation. During the transition stage, multiple interfaces slip in coordination, with stiffness evolving smoothly to reduce the acceleration response. In the ultimate stage, geometric connement triggers stiffness hardening, providing displacement restraint [6,21]. Through the combined design of friction coefcients and geometric parameters, the TFP enables precise control of hysteretic performance, demonstrating multi-level energy dissipation and hierarchical protection characteristics.

      For a generalized system comprising n sliding interfaces, its restoring forcedisplacement relationship can be uniformly described based on displacement superposition and force equi- librium, reecting the coupled control of multiple interface parameters on the overall nonlinear response [15,21]. This framework reveals a theoretical shift in multi-order friction pendulums from a single passive response to multi-parameter coordinated control. However, existing models often neglect the effects of slip eccentricity and the thermalwear coupling at interfaces. Future research should incorporate multi-eld coupling mechanisms to improve prediction accuracy under extreme seismic loads.

   3. Shortcomings and Challenges in Current Theoretical Re- search

      Although the theory of friction pendulum bearings has evolved from single-order models to multi-order series sys- tems, there are still shortcomings in its detailed description under complex operating conditions. The key issues and potential directions for future research can be summarized as follows:

      1. Inadequate Characterization of Multiphysics Coupling: Understanding Mechanisms and Approaches to Unied Modeling

         Most existing PVT models are based on empirical tting and struggle to systematically characterize the intrinsic coupling mechanisms among frictional heat generation, material thermal softening, and interfacial

         wear. This limitation stems from the lack of a uni- ed theoretical framework for energy conversion and material degradation. Future research should focus on interfacial energy conservation and dissipation mecha- nisms to develop a unied multi-eld constitutive model integrating thermal, mechanical, and wear processes, thereby enbling a time-dependent description of how the coefcient of friction and energy dissipation capacity evolve under varying operating conditions.

      2. Simplied Mechanism of Multi-Interface Cooperative Sliding: Asynchronous Evolution and Path Dependence Mechanisms

         Traditional serial models are based on the assumptions of force continuity and superposition of displacements, implicitly assuming that all slip interfaces initiate si- multaneously or in an ideal sequential order. As such, they struggle to capture the actual differences in initi- ation timing and path-dependent characteristics among interfaces. The fundamental issue lies in the lack of a description of the evolution of interface states. Future work should introduce state variables and slip criteria to develop higher-order constitutive models capable of describing asynchronous slip across multiple interfaces, energy distribution, and path-dependent evolution.

      3. Absence of Multidirectional Coupling Effects in Space: Three-Dimensional Dynamic Response and Eccentricity Mechanisms

         Existing theories are largely based on unidirectional or planar analyses, neglecting the spatial motion trajecto- ries of the sliding block under complex seismic actions and the resulting dynamic eccentricity effects. Conse- quently, it is difcult to accurately assess the spatial non- uniformity of contact pressure distribution and friction energy dissipation. In the future, it will be necessary to develop a three-dimensional nonlinear dynamic analysis framework that systematically accounts for multidirec- tional seismic inputs and geometric coupling effects.

      4. Insufcient Understanding of Response Mechanisms Un- der Extreme Conditions: Impact Behavior and Local Damage Evolution

         Existing research on the response of third-order fric- tion pendulums during the limit displacement phase has primarily focused on stiffness switching criteria, while in-depth analysis of the impact dynamics and local

         damage mechanisms caused by limit contact is lacking. The key issue is that the high-frequency response during the impact process and material failure have not been incorporated into a unied description. Future research should integrate contact impact theory with damage mechanics to rene dynamic response models under extreme seismic loads.

      5. Missing Long-Term Service Performance Evolution: Time-Varying Constitutive Models and Reliability Mod- eling

         Most existing resilience models are based on initial state parameters and fail to account for the ongoing effects of material aging, corrosion, and cumulative wear on friction properties. The fundamental issue lies in the absence of a parameter evolution mechanism that incorporates the time dimension. In the future, time- dependent constitutive models covering the entire life cycle should be developed, combined with probabilistic statistical methods to describe the uncertainty in pa- rameter degradation, thereby supporting the long-term performance evaluation and reliability design of seismic isolation systems.

         Upon further examination, although the single-stage friction pendulum (SFP) model is theoretically well-established, its fundamental assumptions still have certain limitations. Tra- ditional models typically simplify the friction coefcient to a constant or consider only the velocity effect [23], making it difcult to reect the coupled effects of multiple factors such as temperature, pressure, and interface wear; simultaneously, due to its structural characteristics based on a single radius of curvature, it struggles to characterize multi-stage dynamic responses under complex seismic loads. In contrast, multi- stage friction pendulums (DFP, TFP) achieve graded control of stiffness and energy dissipation capacity by introducing multiple slip interfaces and parameter combinations, offer- ing signicant advantages in terms of displacement capacity, response adaptability, and control under extreme conditions. Therefore, in the context of high-performance seismic design, friction pendulum isolation systems are evolving from single- stage to multi-stage congurations [24]. Based on existing research ndings, the performance differences among the three types of friction pendulum systems can be summarized as follows (see Table 2).

         Therefore, developing a rened mechanical model that accounts for multi-directional spatial coupling, multi-eld evo- lution at interfaces, and long-term performance degradation remains a critical bottleneck in future theoretical research on friction pendulum seismic isolation.

3. Recent Advances in Research on Friction Pendulum Bearings

   The accuracy of theoretical constitutive models depends on high-precision experimental data. The performance of friction pendulum bearings under seismic loading is inuenced by the coupled effects of multiple factors, including material wear, heat generation at interfaces, and contact pressure distribution.

   Based on this, this paper systematically reviews recent re- search progress in China and internationally regarding material friction characteristics, mechanical properties of components, and structural shake table tests, with the aim of elucidating the relationship between theoretical predictions and actual dynamic responses.

   1. Testing of Friction Interface Characteristics and Material Properties

      The stability of the friction coefcient is a key factor determining the seismic isolation efciency of FPS systems [8, 11]. Early theoretical studies often assumed the friction coefcient to be constant; however, extensive tribological experiments at the interface have shown that its value exhibits signicant multi-parameter correlations and spatiotemporal evolution characteristics [25].

      The stability of the friction coefcient is a key factor affect- ing the seismic isolation performance of friction pendulum supports (FPS) [26]. Early studies often simplied it as a constant, but extensive tribological experiments have shown that the friction coefcient exhibits signicant multi-parameter coupling and spatiotemporal evolution characteristics [13]. First, under the coupled effects of velocity and pressure, the friction behavior at the interface exhibits typical nonlinear patterns: as sliding velocity increases, the friction coefcient rapidly transitions from static to kinetic friction and stabilizes, a process that can be described by an exponential model; conversely, as axial pressure increases, the increase in shear strength is smaller than the increase in normal stress, leading to an overall decrease in the friction coefcient [25]. This effect is particularly pronounced in heavy-load seismic isola- tion designs. Second, under long-term or large-displacement cyclic loading, frictional heat generation and material wear further alter interface properties. Experiments indicate that high-frequency reciprocating sliding can cause a signicant rise in interface temperature and lead to thermal softening of polymeric materials [27], thereby triggering hysteretic strength degradation; simultaneously, the generation and migration of wear particles alter the contact morphology, causing random uctuations in the friction coefcient of approximately 10% 20%. Furthermore, environmental factors exert a signicant in- uence on interface properties. Under low-temperature condi- tions, the viscoelasticity of materials increases, manifesting as rises in initial friction force and equivalent stiffness; corrosion and salt spray exposure, on the other hand, exacerbate sliding plate wear and roughen the sliding tracks, thereby increasing the variability of the coefcient of friction [28]. During long- term service, these factors interact to alter the microstructure of the interface, consequently affecting the stability and durability of the seismic isolation system.

      In summary, the coefcient of friction is jointly goerned by velocity, pressure, temperature, and environmental condi- tions, and its dynamic evolution determines the time-varying characteristics of the restoring force. Therefore, elucidating the mechanisms of interfacial friction is fundamental to accurately

      TABLE III

      Comparison of dominant nonlinear factors of friction pendulum bearings in different engineering scenarios

      Engineering scenario	Core mechanical char- acteristics	Dominant nonlinear factors	Requirements for constitutive model
      High-rise buildings	Large axial pressure, large height-to-width ratio causing overturning moment	Pressure dependence	Need to introduce a variable friction coefcient model that considers pressure dependence
      Large-span spatial structures	Large displacement span, long period, low- frequency response	Geometric nonlinearity	Large displacement switch must be turned on to correct restoring force characteristics
      Alpine/Severe cold regions	Extremely low ambient temperature (-40C)	Ambient temperature	Need to consider the initial sliding criterion modied by ambient temperature
      Near-fault/Strong earthquake zones	Extremely fast recipro- cating velocity, long slid- ing history	Velocity and heat gener- ation	Need to introduce a velocity- temperature coupling subrou- tine to simulate strength degra- dation of hysteresis loops
      Coastal/Heavy corrosion environments	Long-term salt spray erosion, high humidity	Surface roughness	Need to consider performance drift and parameter randomness over long service life

      characterizing the multi-stage slippage and energy dissipation behavior of multi-stage friction pendulums (DFP/TFP).

   2. Study on Mechanical Properties of Members (DFP/TFP) Under Loading Conditions

      Building on the understanding at the material level, component-scale tests further validated the overall mechanical behavior of multi-stage friction pendulum bearings [29]. Ex- tensive pseudo-dynamic and quasi-static loading tests on full- scale and large-scale models demonstrated that the synergistic interaction of multiple slip interfaces can generate a stable, multi-stage response mechanism at the macroscopic level [1, 15]. The test results show that as the external load increases, each sliding interface activates sequentially, causing a change in the stiffness of the hysteresis curve, which validates the predictive capability of the series constitutive model for multi- stage sliding sequences. At the same time, there is an uneven distribution of displacement between interfaces, and due to differences in initial friction, the actual sliding sequence may deviate slightly from the theoretical results.

      For the third-order friction pendulum [30], tests conducted during the large-displacement phase further revealed its ul- timate mechanical characteristics: when the slider contacts the geometric stop, the system stiffness increases signi- cantly, which is consistent with theoretical predictions and demonstrates an effective displacement constraint mechanism

      [8]; however, under high-speed loading conditions, the stop collision generates an instantaneous impact effect, placing higher demands on material strength and connection design [31]. Shake table tests also indicate that the friction pendulum support exhibits excellent overall dynamic performance. On the one hand, its stiffness-axial force relationship causes the stiffness center of the isolation layer to approach the mass

      center, thereby effectively suppressing torsional response; on the other hand, although friction introduces some residual dis- placement, this value is small under conditions of reasonably designed curvature radius, and the system still possesses good self-resetting capability.

      Overall, the component-level tests have veried the syn- ergistic mechanism of staged slippagestiffness switch- ingenergy dissipation in multi-stage friction pendulum systems at the macro level, providing critical experimental evidence for their engineering applications. At the same time, the results indicate that impact and multi-eld coupling effects still require further consideration under extreme operating conditions.

   3. Limitations and Challenges of Current Experimental Stud- ies

      Based on the results of comprehensive material interface tests and component-level dynamic tests, it is evident that while existing research has validated the effectiveness of friction pendulum seismic isolation systems across multiple scales, there are still shortcomings in terms of high-precision performance evaluation and adaptability to complex operating conditions. These shortcomings are primarily reected in the following aspects:

      1. Insufcient characterization of multi-eld coupling mechanisms:Although existing experiments have iden- tied the patterns by which pressure, velocity, and tem- perature inuence friction behavior, there has been a lack of systematic investigation into the coupling mechanisms underlying the evolution of frictional heating, material softening, and wear under long-term seismic loading, which has limited the ability to accurately characterize energy-dissipative degradation processes.

      2. There is insufcient data on long-term service in ex- treme environments:Most existing low-temperature and corrosion tests have been conducted under short-term conditions, and there is a lack of long-term test data that accounts for the coupled effects of aging, wear, and repeated seismic actions, making it difcult to support the evaluation of the performance of seismic isolation systems over their entire life cycle.

      3. There is a lack of research on the effects of multi- directional coupling dynamics:Existing component tests primarily involve uniaxial or biaxial loading, with in- sufcient attention paid to uctuations in vertical axial force, the spatial motion of the sliding block, and the coupled effects of triaxial seismic inputs; this limits the accuracy of mechanical model validation under complex loading conditions.

      4. Scale effects and boundary conditions have a signicant impact:Due to limitations in test equipment and cost, existing research largely relies on scaled models, which struggle to fully capture the effects of frictional heat generation, impact forces, and boundary constraints on mechanical responses in actual engineering applications. This limits the applicability of these ndings to heavy- load and large-scale structures.

      5. Existing general macro models often use a uniform friction coefcient, which can lead to signicant errors when modeling pressure uctuations in high-rise build- ings or the initial dynamic characteristics in cold regions, as shown in Table 3.

   Therefore, the development of rened numerical simula- tion methods that account for multi-eld evolution, multi- body coupling, and scale-dependent effects has become a key approach to overcoming the limitations of experiments and gaining a deeper understanding of the complex mechanical behavior of friction pendulums.

4. Recent Advances in Numerical Simulation of

   Friction Pendulum Bearings

   With the advancement of high-performance computing, nu- merical simulation has become a key method for analyzing the complex mechanical behavior of friction pendulum bearings. Compared to physical testing, it allows for the simulation of extreme operating conditions at a lower cost and reveals microscopic eld information, such as interface pressure is- tributions and heat generation dynamics. Based on this, this paper reviews recent research progress in three areas: rened nite element modeling, macroscopic constitutive algorithms, and multiphysics coupling simulations.

   1. Recent Advances in Rened Finite Element Modeling

      In general-purpose nite element platforms such as ABAQUS, the denition of contact constitutions and interface properties is critical to modeling [32]. Normal behavior is typically modeled using a hard-contact model, combined with penalty functions or the Lagrangian method to improve numer- ical stability under high-pressure contact; regarding tangential

      behavior, traditional Coulomb models struggle to capture the nonlinear evolution of friction. In recent years, coupled effects of velocity, pressure, and temperature have been introduced via user subroutines [33] to enable dynamic updates of the fric- tion coefcient, thereby enhancing the accuracy of hysteresis response simulations.

      Regarding discretization strategies, sliding blocks and tracks generally employ reduced-integral hexahedral elements to avoid numerical locking during large-deformation shear pro- cesses [34]; simultaneously, geometric nonlinearity options must be enabled to accurately describe path changes caused by spherical slippage [35]. Studies indicate that mesh density in the contact region signicantly inuences contact pressure and friction response, and local renement is an effective means of improving computational accuracy [36]. For highly nonlinear contact problems, the choice of algorithm is equally critical. Implicit methods are suitable for conventional hys- teresis analysis but may experience convergence difculties during the initial stages of multi-interface contact; explicit methods, however, offer advantages in strong seismic impact and limited-deection collision problems, and when combined with mass scaling and damping control, can effectively capture transient nonlinear responses.

      Overall, rened models can accurately characterize mesoscale mechanisms such as contact stress distribution, lo- cal stress concentration, and frictional heat generation, serving as primary tools for structural optimization and extreme con- dition analysis. However, their high computational cost limits their application in large-scale structural analysis [34,35].

   2. Recent Advances in Macro-Scale Models and Simplied Computational Units

      To improve the efciency of engineering analysis, macro- scopic models based on the principle of mechanical equiva- lence have become an important branch of numerical simula- tion [3]. These methods achieve an equivalent representation of complex contact behavior by abstracting the restoring force and energy dissipation characteristics of friction pendulum supports.

      In platforms such as ABAQUS, constructing macroscopic models using connection elements is a common strategy [34]. By combining axial constraints with slip constitutive relations, geometric restoring forces and frictional energy dissipation can be equivalently described; for multi-stage friction pendulums (DFP/TFP) [8,37], multiple elements are typically connected in series with differentiated slip thresholds to simulate multi- stage stiffness evolution. To improve numerical stability, ide- alized elastoplastic constitutions are often regularized. In plat- forms such as OpenSees [1, 38] and SAP2000, specialized friction pendulum elements further enhance analysis efciency. Based on the FenzConstantinou theoretical framework, these elements achieve multi-stage slip responses through displace- ment accumulation and state determination; simultaneously, geometric nonlinearity corrections are introduced under large- displacement conditions to account for the effects of slider lifting and axial force variations on restoring forces, thereby

      mitigating the underestimation of displacement responses in- herent in traditional linear models. From an applicability per- spective, rened models and macroscopic models each possess distinct advantages and are complementary: the former, based on solid modeling, can accurately characterize contact stresses and interface evolution mechanisms, making them suitable for mechanism studies and ultimate performance analysis; the latter signicantly reduces computational complexity through constitutive abstraction, making it suitable for dynamic time- history analysis and parameter sensitivity studies of long-span or high-rise structures.

      Therefore, current research often adopts a collaborative strategy of using detailed models to reveal mechanisms and macro models for engineering analysis to strike a balance between computational accuracy and efciency.

   3. Summary of This Chapter

      This chapter provides a systematic overview of the current state of research and technical approaches regarding friction pendulum bearings in the eld of numerical simulation. A comparative analysis reveals that existing numerical stud- ies have established a two-scale simulation framework that emphasizes both microscopic mechanisms and macroscopic responses.

      1. Rened solid nite element models, by accurately char- acterizing contact, geometric nonlinearities, and material constitutive behavior, reveal the evolution of bearing stress distribution and frictional energy dissipation, pro- viding support for rened design and ultimate limit assessment.

      2. Macroscopic equivalent models, based on the abstraction of mechanical characteristics, employ series-connected nonlinear elements to efciently simulate the variable stiffness behavior of multi-stage friction pendulums. They signicantly reduce computational costs while ensuring accuracy, making them suitable for large-scale structural analysis.

      3. Numerical methods have expanded from implicit static analysis to explicit dynamic and multi-physics coupled simulations, advancing research from single-mechanism responses to the simulation of service performance in- corporating multi-factor coupling such as heat genera- tion, wear, and corrosion.

      4. To intuitively compare the applicability and perfor- mance differences of various modeling methods, Table 4 presents a systematic comparison between macroscopic models and rened nite element models.

         Overall, numerical simulation studies not only serve as an effective complement to theoretical models and experimental research, but also play a crucial role in advancing friction pendulum technology from component development to per- formance design across the entire engineering lifecycle.

   4. Shortcomings and Challenges in Current Numerical Sim- ulation Studies

      Although numerical simulations have made signicant progress in elucidating the multiscale mechanical behavior of friction pendulum bearings, there remain critical shortcomings in terms of high-precision prediction and engineering applica- bility, primarily in the following areas:

      1. Inadequate characterization of multiphysics coupling: Most existing models treat heat generation, wear, and corrosion effects separately, lacking a unied thermal- mechanical-wear coupling framework, which makes it difcult to accurately describe the synergistic evolution of interfacial temperature rise, material softening, and performance degradation.

      2. Limited descriptive capability of macro-scale models: Macro-scale equivalent models are typically based on low-dimensional assumptions, making it difcult to cap- ture the three-dimensional motion of the sliding block under multi-directional seismic loading, uctuations in vertical axial force, and strong geometric nonlinear effects, which limits the accuracy of response predictions under complex conditions.

      3. Insufcient synergy between computational accuracy and efciency: Altough rened nite element models can capture the microscopic mechanisms of contact, they are computationally expensive; existing methods still lack effective approaches to reduce computational complexity and improve efciency while maintaining delity in key physical processes.

      4. Inadequate full-lifecycle performance simulation: Cur- rent simulations are primarily based on initial parameters and do not fully account for the time-varying character- istics of parameters caused by material aging, corrosion, and cumulative damage, making it difcult to reliably assess the long-term performance of seismic isolation systems.

   In summary, the development of a unied numerical frame- work that combines multi-eld coupling capabilities, three- dimensional nonlinear response analysis, and high computa- tional efciency is a key direction for advancing friction pen- dulum bearings from mechanistic simulation to full-lifecycle performance prediction.

5. Summary and Outlook

1. Summary of the Entire Text

   This paper systematically reviews research progress on friction pendulum bearings from three perspectives: theory, experimentation, and numerical simulation. Theoretically, a framework was established for the evolution from a rst- order bilinear model to a time-varying constitutive model that accounts for the coupling of velocity, pressure, and temperature, and mechanisms for staged slippage and variable stiffness in multi-order friction pendulums were developed. Experimentally, the study reveals the nonlinear evolution of the friction coefcient under the coupled effects of multiple

   TABLE IV

   Comparison of applicability between macro-models and refined models for friction pendulum bearings

   Comparison Dimension	Macro-model	Rened model
   Modeling Method	Constitutive model	Solid elements + contact
   Computational Efciency	High	Low
   Computational Accuracy	Global response	Local behavior
   Simulatable Content	Displacement, acceleration, hysteresis	Stress distribution, wear, heat generation
   Geometric Nonlinearity	Considered equivalently	Considered explicitly
   Applicable Scale	Global structural analysis	Component-interface mechanisms
   Typical Applications	Time-history analysis, parameter analysis	Structural optimization, limit states

   factors and veries the synergistic mechanism of multi-stage slippage and energy dissipation. In terms of numerical simu- lation, a multi-scale analysis system has been established that integrates complementary ne-scale and macro-scale models. Overall, research on friction pendulums is shifting from single- mechanical analysis toward multi-eld coupling and full- lifecycle performance evaluation.

2. Current Research Gaps and Future Directions

Although systematic progress has been made in friction pendulum seismic isolation technology at the theoretical, experimental, and numerical simulation levels, several key challenges remain to be addressed in order to meet complex engineering requirements and achieve high-performance seis- mic resistance:

First, the mechanisms of multi-eld coupling need to be rened. Existing studies often treat thermal effects, wear, and environmental inuences separately, lacking a unied frame- work for thermalmechanicalwear coupling. Future research should focus on interfacial energy conversion and material degradation to develop constitutive models that reect the time-dependent evolution of friction properties. Second, the characterization of spatial multi-directional dynamic responses remains inadequate. Existing models are often based on simplied assumptions and provide limited consideration of triaxial seismic inputs and the spatial motion of sliding blocks. Subsequent research should develop three-dimensional non- linear analysis methods to improve prediction accuracy under complex operating conditions. Third, performance evaluation over the entire life cycle remains weak. Current research is largely based on initial parameters and does not fully account for the effects of aging, corrosion, and cumulative damage. Future efforts should combine long-term monitoring with accelerated testing to establish time-dependent models and conduct reliability analyses. Finally, the level of intelligence in seismic isolation systems needs to be improved. Traditional passive friction pendulums struggle to meet the demands of multi-level seismic events; future research could integrate smart materials to develop active-passive cooperative control technologies.

In summary, establishing an integrated theoretical and methodological framework that combines multi-eld coupling

capabilities, three-dimensional dynamic response analysis, and long-term performance prediction is a key direction for ad- vancing friction pendulum seismic isolation technology toward higher precision and reliability.

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