Development of a novel pendulum column base isolation system with vertical negative stiffness (PC-VNS) for pile-like behavior for seismic applications

Abstract

This paper introduces a novel seismic base isolator termed the “pendulum column with vertical negative stiffness (PC-VNS),” which functions similarly to a pile foundation while incorporating quasi zero stiffness (QZS) characteristics to modify the structural foundation connection in weak soil and effectively extend the natural period. The system’s nonlinear behavior enables it to reduce both displacement and acceleration, even when employing conventional damping coefficients of 5%. The theoretical framework was validated using MATLAB analyses of multiple earthquake records and finite element modeling in ABAQUS. Results from earthquake scenarios demonstrate the system’s ability to avoid resonance phenomena while maintaining controlled responses with standard damping. Compared to a conventional fixed-base frame with a 0.4-second natural period, the PC-VNS system achieves a significant reduction in input energy in most cases. The performance of the system is strongly influenced by the maximum allowable range of motion, with improved results observed for smaller displacement limits. The pile-like behavior of the PC-VNS improves soil-structure interaction, contributing to both seismic isolation and foundation reinforcement, thereby offering a promising solution for earthquake-resistant design.

Introduction

Throughout history, earthquakes have resulted in significant financial losses and numerous fatalities¹. Various effective measures have been suggested and developed to mitigate earthquake harm². The majority of these techniques rely on the idea that the earthquake impact is transmitted from the foundation to the structure³. One way to control the response of structures is to use isolation systems⁴. The seismic isolation technique aims to avoid transmitting earthquake force directly from the foundation to the structure. Base isolation technology is effective in lowering the structure’s seismic response and preventing damage to it⁵. As the natural period of a structure lengthens due to seismic isolation, the acceleration from earth motion exerted on the structure decreases. On the other hand, weak and unsuitable soils present a significant challenge, especially when the structure requires seismic isolation. Traditional soil stabilization methods include mechanical densification through compaction, soil reinforcement using inclusions or geosynthetics, chemical modification by injecting stabilizing agents, and deep foundation approaches like driven piles. These techniques improve the soil’s bearing capacity and reduce its deformability^6,7,8. In situations requiring both ground improvement and seismic isolation, the standard approach typically involves first enhancing the soil’s geotechnical properties, followed by the installation of seismic isolation systems. However, recent developments have introduced integrated methods that simultaneously address soil stabilization and seismic isolation within a single, cohesive framework. Some of the research conducted in these two areas is briefly mentioned below:

Barghian and Shahabi⁹ introduced a new method of mushroom-shaped base isolation, which was subsequently tested in a lab by Lu and Hsu^10,11. Wei et al.^12,13 studied the use of rolling base isolation with varying friction distributions. Their findings revealed that the distribution of the friction force had an impact on the natural period of the structure. Furthermore, Wei et al.¹⁴ conducted further research on the P-F isolation system’s scalability in shaking table model tests. Karayel et al.¹⁵ suggested the use of spring tube braces for seismic isolation in construction. Chen and Xiong¹⁶ suggested an enhanced base isolation mechanism that used a combination of a conventional friction pendulum bearing (FPB) and a viscous damper (VD) for earthquake-resistant design of structures. Results indicated that the proposed FPB-VD device was effective in reducing both structural acceleration and deformation demands when compared to the prototype system without isolation and conventional FPB. The study conducted by Amir Ali et al.¹⁷ examined five different inexpensive materials that can be found locally and are commonly used for sliding base isolation systems. Results showed that the isolated model experienced a decrease in acceleration response of 40% to 53% at the top floor level in comparison to the fixed-based model. Chen et al.¹⁸ proposed the use of a base isolation system called lead rubber bearing with negative stiffness springs (LRB-NS). This involves adding pre-compressed springs to conventional lead rubber bearings installed at the base of bridge columns. Results showed that the LRB-NS system was successful in reducing seismic demands on bridge columns and limiting excessive deformation in the bearings, which is common in traditional LRB systems during strong seismic events. Dong et al.¹⁹ suggested a three-dimensional (3D) isolation device comprising a conventional horizontal bearing as well as a developed long-period vertical isolation device that had variable stiffness (LVIVS). Cimellaro et al.²⁰ presented a three-dimensional (3-D) base isolation system designed to control both the horizontal and vertical components of ground motion. The system adopted a negative stiffness device (NSD), which functioned as an adaptive passive protection system capable of altering the stiffness of the structure. Numerical analyses demonstrated that the inclusion of NSDs reduced the vertical acceleration within the structure. Wu et al.²¹ suggested eliminating unwanted stiffness in isolators by incorporating an inerter element and decreasing the stiffness itself. Zhang et al.²² presented a study on the potential of using a geotechnical seismic isolation system based on a marine sand cushion (GSI-MSC) to mitigate ground shocks caused by explosions. Dalela et al.²³ presented a novel metastructure with quasi-zero-stiffness (QZS) characteristics for low-frequency vibration control, using a high static and low dynamic stiffness mechanism. The structure, composed of four unit cells with inclined beams and semicircular arches, combines negative and positive stiffness behaviors. Static and dynamic responses were validated through finite element analysis and experiments. Results confirmed the existence of a QZS region and demonstrated effective vibration isolation, with transmissibility curves showing nonlinear jump-down behavior. Parametric studies further supported its performance under varying conditions. Deng et al.²⁴ developed a quasi-zero stiffness (QZS) isolation system integrated with magnetorheological fluids (MRFs), offering variable stiffness in both vertical and lateral directions to enhance performance and address stability limitations of conventional QZS systems. Liu et al.²⁵ proposed a new class of programmable quasi-zero-stiffness (QZS) metamaterials designed to overcome limitations in traditional QZS structures for low-frequency vibration isolation. They introduced a design mechanism based on structurally bionic representative unit cells (RUCs) with tunable QZS properties. Experimental validation confirmed the programmability and effectiveness of the QZS behavior. The results demonstrated strong potential for practical application in engineering systems. The study offered a pathway toward advanced, adaptable solutions for low-frequency vibration control. Wang et al.²⁶ conducted a study on improving the seismic performance of base-isolated structures by introducing a semi-active tuned mass damper (STMD). They proposed a new control algorithm for the STMD, which adjusted stiffness and damping in real time. Their findings showed that the STMD effectively mitigates both displacement and acceleration responses, significantly enhancing the performance of base-isolated structures under seismic loads. Zhang et al.²⁷ developed a Transformer-based semi-active tuned mass damper (STMD) to control coupled translational-torsional responses in eccentric structures. They integrated a prediction model for structural response and real-time tuning of stiffness and damping. Their results demonstrated significant reductions in both translational and torsional vibrations under bidirectional seismic excitations, proving the effectiveness of their control strategy. Qiao et al.²⁸ proposed a Hilbert-Huang transform (HHT)-based phase control algorithm for semi-active tuned mass dampers (STMD) with variable damping. They validated the effectiveness of the algorithm using a 2DOF system and a slender structure model. Their results showed that the proposed algorithm outperforms traditional ON/OFF control in reducing displacement and improving earthquake mitigation. Zhang et al.²⁹ developed a semi-active control algorithm combining long short-term memory (LSTM) and wavelet transform (WT) for seismic retrofitting of aging steel structures. They applied the algorithm using a semi-active tuned mass damper (STMD) and evaluated its performance through two case studies. Their results showed that the LSTM-WT algorithm significantly improved seismic control, especially in stiffness-degraded structures, outperforming traditional passive and STMD systems. Wang and Zhou³⁰ proposed a bi-directional adaptive eddy current pendulum tuned mass damper (AEC-PTMD) for seismic response control in nonlinear structures. They validated its performance through scaled model tests and electromagnetic finite element analysis, applying it to the Shanghai Center Tower. Their results demonstrated that the bi-directional AEC-PTMD effectively improved nonlinear seismic protection, offering substantial aseismic benefits for tall buildings. Sarlis et al.³¹ introduced a negative stiffness device (NSD) for seismic protection of structures, aimed at reducing seismic forces without causing inelastic deformations. The device emulated structural weakening by applying a force that opposes the restoring force once a prescribed displacement was reached. Their study presented the development, operation, and analytical tools for the NSD, demonstrating its potential in both conventional and seismically isolated structures. Wang et al.³² proposed a passive pseudo-negative stiffness device (PPNSD) for seismic protection in isolated buildings. They validated its feasibility with cyclic tests and evaluated its performance under different ground motions. The results indicated that the PPNSD significantly reduced both base drift and superstructure acceleration, proving to be more effective than traditional passive isolators, especially under near-fault ground motions. Wang et al.³³ developed a negative stiffness damped outrigger with damping amplification (NSDO-DA) by combining an L-shape lever, a pre-compressed disc spring brace, and a viscous damper. They conducted dynamic tests on several outrigger systems, including the NSDO-DA, and validated its performance. The results demonstrated that the NSDO-DA effectively generates negative stiffness and amplifies damping, improving seismic response and simplifying the outrigger configuration. Wang et al.³⁴ systematically compared and discussed the optimal design and seismic performance of TVMD and NSAD from the point of SSI.

Recent advancements in soil-structure interaction aim to enhance structural performance during seismic events. Traditional rigid connections between structures and the ground have evolved to include flexible and energy-absorbing systems. Techniques such as base isolation, the use of soil–rubber mixtures, sleeved piles, and geotechnical seismic isolation have shown effectiveness in reducing seismic forces by dissipating energy and minimizing stress transfer. Yaghmaei and Rahmani³⁵ evaluated the impact of seismic isolation using a rubber-soil mixture on the seismic demands of a steel moment frame located in a near-fault zone. Tsang et al.³⁶ performed a numerical study to explore the effectiveness of granulated rubber–soil mixtures (GRSM) as a seismic isolation method for low- to medium-rise buildings. By combining shredded scrap rubber tires with soil and applying the mixture around building foundations, the system acts as a cushion that absorbs seismic energy. Chew and Leong³⁷ conducted field tests and finite element modelling to evaluate the effectiveness of sand-rubber mixtures (SRM) as ground vibration barriers. Wan et al.³⁸ employed molecular dynamics simulations to explore the nanofriction behavior at the rubber–clay interface in rubber-soil mixtures (RSM) to enhance foundation stability under seismic loading. Their study revealed that denser soil structures increase frictional strength, and compacting rubber and clay enhances this effect. Yao et al.³⁹ researched a super-long pile group in stratified soils under both vertical and lateral loads. The results showed that the sleeved super-long pile group can reduce the horizontal displacement significantly and is feasible in practical applications. Muhammed et al.⁴⁰ presented laboratory measurements of mobilized local friction along piles subjected to very large numbers of axial loading cycles. A comparison was made between the maximum static shear mobilized before and after the cycles, which shows the influence of the cyclic sequence on this quantity. Xue et al.⁴¹ presented experimental tests of four RC pile specimens backfilled with different damping materials to assess their effect on the soil-structure hysteretic response. They developed a physical hysteresis model of the soil-pile interaction to evaluate the optimum length of the damping pre-hole. The research by Li et al.⁴² demonstrates the effectiveness of the unconnected piles–caisson foundation as a novel seismic isolation system. Oliaei and Tohidifar⁴³ examined the seismic stability of slopes reinforced with both sleeved and unsleeved piles through 3D finite-difference dynamic analyses. Their study focused on using compressible sleeves around concrete piles to minimize lateral load transfer to the adjacent slopes, especially at shallow depths. Zia and Komak Panah⁴⁴ conducted a 3D numerical study to investigate the seismic performance of sleeved-pile foundations incorporating soil–rubber mixtures as a damping mechanism for structures. Their results demonstrated that the sleeved-pile system significantly reduced inter-story drift and horizontal accelerations in both foundations and superstructures, effectively lowering seismic forces.

This study investigates a mechanism known as a pendulum column with vertical negative stiffness, designed for seismic isolation while simultaneously functioning as a pile. The mechanism operates similarly to a conventional pendulum column but incorporates a negative stiffness component. By strengthening the connection between the structure and the soil, it also provides soil reinforcement. After presenting the proposed system, the governing equations describing its behavior are derived. The system’s response under multiple earthquake events is then analyzed. Finally, to validate the accuracy of the derived equations, the analytical results are compared with numerical simulations conducted in ABAQUS. Compared to elastomeric and friction isolators, the PC-VNS system is able to implement nonlinear responses and QZS characteristics. Conventional isolators usually reduce the acceleration and increase the displacement, but the proposed system was able to control and even reduce the displacement due to the negative stiffness. At the same time, this system improves the soil behavior and reinforcement like a pile. Compared to conventional piles and sleeved soil-rubber mixture systems, the PC-VNS system is an efficient solution for special conditions such as weak soils and seismic zones due to negative stiffness.

Materials and methods

Weak soils constitute a critical challenge in the design and construction of foundations and structures due to their inherently low load-bearing capacity, high deformability, and susceptibility to failure mechanisms such as excessive settlement and soil liquefaction. The challenges become markedly more severe in seismic regions, where earthquake-induced ground shaking can amplify soil instability, triggering complex soil-structure interaction phenomena and increasing the risk of structural damage or collapse.

To address these challenges, engineers have developed a suite of methods aimed at both improving soil behavior and enhancing structural resilience. Conventional deep foundation techniques, such as pile foundations, provide a means to bypass weak near-surface soils by transferring loads to deeper, more competent soil or rock layers. Meanwhile, seismic isolation systems are implemented to decouple the structural mass from ground motion, thereby mitigating the transmission of damaging seismic forces to the superstructure and reducing acceleration demands.

Description of proposed system

This study proposes a novel adaptation of the sleeved pile concept, wherein the inner layer is suspended from the outer pile by tensile elements, creating a pendulum-like isolation system (depicted in Fig. 1a). Furthermore, the system incorporates a spring-like element designed to introduce negative stiffness, as shown in Fig. 1b. Figure 1c provides a top-down view of the proposed configuration. The suspended mass supports a portion of the structural load through cables or rods with pinned connections, while a negative stiffness mechanism supports the remaining load, enabling dynamic tuning of the system’s response. The natural frequency of this configuration is primarily dictated by the length of the tensile members, as described in Eq. 1 when excluding negative stiffness effects⁴⁵.

The pendulum isolation principle operates through gravitational restoring forces^46,47; when displaced, the suspended mass oscillates with a frequency determined by the suspension length. Horizontal movement of the support induces inertial forces on the mass, but due to the pendulum dynamics, the suspended mass experiences significantly reduced accelerations relative to the structure’s upper components. For pendulum lengths on the order of typical floor heights, this results in a notable reduction of seismic forces transmitted to the structure^48,49.

The present investigation aims to analyze the dynamic behavior and seismic performance of this suspended sleeved pile system through comprehensive analytical modeling and numerical simulation. The objectives include quantifying seismic force mitigation, assessing soil-structure interaction impacts, and evaluating the practicality of this approach for application in weak soil and seismic-prone environments. Furthermore, the incorporation of negative stiffness elements offers the potential to further optimize the system’s natural frequency and enhance energy dissipation, thus contributing to innovative, resilient foundation designs tailored for challenging geotechnical and seismic conditions.

Fig. 1

(a) A structure incorporated with the proposed system (a pile foundation and a pendulum-like isolation mechanism) (b) 3D shape of PC-VNS (c) Top view of PC-VNS.

Fig. 2

(a) Pendulum - like isolation system (b) A proposed system as negative stiffness. c) Scaled non-symmetrical geometry of combined system d) Forces on the system during deformation of angles e) Simplified form of the system by moving the supports from the bottom to the top of the tension members f) Final simplified form of the system.

Fig. 3

Simplified equivalent model of PC-VNS.

$$\:\omega\:=\sqrt{g/l}$$

(1)

Where m, g, and l are mass, acceleration due to gravity and length, respectively.

Figure 2a illustrates a schematic two-dimensional view of the proposed system configuration. Since the geometry of the proposed system was completely cylindrical and symmetric, this 2D representation that is well established and validated in the scientific literature for such geometries is considered. In this design, the inner structural element is suspended from the outer pile layer, forming a pendulum-like mechanism. This configuration allows the mass to oscillate independently, thereby reducing the transmission of seismic forces to the structure. Figure 2b shows the negative stiffness element incorporated into the system. This component consists of a vertically compressed spring connected to a rod with an attached mass. It is statically unstable due to the compressive preload in the spring. However, when integrated with the pendulum system shown in Fig. 2a, it contributes to the overall dynamic stability of the system while enhancing its energy dissipation and vibration isolation capabilities. The final configuration involves inserting the compression member into the center of the isolator to serve as negative stiffness (Fig. 2c). This member, by applying upward force, mitigates the impact of gravity and reduces the force in the tensile members, such as cables. Additionally, the angle and direction of force from the compression member produce negative stiffness in the structural system. In this system, by creating a horizontal displacement, the tensile members, because they act like a pendulum, cause centripetal behavior in the system. However, the added member to provide negative stiffness neutralizes this effect to some extent and increases the periodicity of the system. Figure 2d illustrates the forces acting during the deformation of the system in the pendulum isolator with negative stiffness. Therefore, the system can be adjusted to have an initial quasi-zero stiffness. To derive the governing equation of the system, the proposed dual-purpose system can be simplified by moving supports to the top of tensile members, as shown in Fig. 2e, while maintaining the accuracy and precision of the model. This model can be illustrated in Fig. 2f. This model shows a pendulum with a rod attached to it (the rod is located under the mass). The lower end of the rod is restrained so that it can only move up and down and does not cause lateral movement to the entire system. A spring is included under the rod to apply an upward force to the rod by means of its pre-loading.

Equation deriving

To formulate the equation governing the system’s horizontal displacement, the geometric relationships, effective forces, and member properties are presented schematically in Fig. 3. These parameters serve as the foundation for developing the dynamic model. Accordingly, the geometric constraints of the system can be expressed through the following equations: 2–4.

$$\:{l}_{2}sin{\theta\:}_{2}={l}_{1}sin{\theta\:}_{1}$$

(2)

$${l}_{1}cos{\theta\:}_{1}+{l}_{2}cos{\theta\:}_{2}=l$$

(3)

$$\delta\:={l}_{1}+{l}_{2}-l={l}_{1}(1-cos{\theta\:}_{1})+{l}_{2}(1-cos{\theta\:}_{2})$$

(4)

The angular displacement $\:\theta\:$ can be expressed based on the $\:{\theta\:}_{1},{\theta\:}_{2}$:

$$\:\theta\:=\frac{\pi\:}{2}-({\theta\:}_{1}+{\theta\:}_{2})$$

(5)

The force exerted by the vertical spring located beneath the rod can be expressed as:

$$\:{F}_{sp}=k\left(\varDelta\:-\delta\:\right)=k\left(\varDelta\:-{l}_{1}\left(1-cos{\theta\:}_{1}\right)-{l}_{2}(1-cos{\theta\:}_{2})\right)$$

(6)

Where $\:{F}_{sp}$​ is the spring force, $\:k$ ​ is the spring stiffness coefficient, $\:\delta\:$ represents the vertical displacement, and $\:\varDelta\:$ is the initial compression of the spring from its equilibrium position.

Regarding the forces acting on the system’s mass, they can be described by the following equation:

$$\:{F}_{1}\:,\:{F}_{2}=\frac{k\left(\varDelta\:-\delta\:\right)}{cos{\theta\:}_{2}}\:,\:\text{W}=mg$$

(7)

Where F₁, F₂, and W represent the tensile force, pressure force, and weight, respectively, as shown in Fig. 3.

All of the forces that affect bubbled mass can be considered below in Eq. 8:

$$\:{F}_{1\theta\:}=0 , {F}_{2\theta\:}=\frac{k\left(\varDelta\:-\delta\:\right)}{cos{\theta\:}_{2}}cos\theta\:=\frac{k\left(\varDelta\:-\delta\:\right)}{cos{\theta\:}_{2}}sin({\theta\:}_{1}+{\theta\:}_{2}), {W}_{\theta\:}=-mgsin{\theta\:}_{1}$$

(8)

For writing equilibrium equations, we need to compute the moment about $\:e$:

$$\:m{l}_{1}{\ddot{\theta\:}}_{s}={F}_{1\theta\:}+{F}_{2\theta\:}+{W}_{\theta\:}$$

(9)

Where “s” refers to mass of structure. When external excitation exists, $\:{\ddot{\theta\:}}_{s}={\ddot{\theta\:}}_{1}+{\ddot{x}}_{e}cos{\theta\:}_{1}/{l}_{1}$, otherwise $\:{\ddot{\theta\:}}_{s}={\ddot{\theta\:}}_{1}$.

By expanding Eq. 9:

$$\:m{l}_{1}{\ddot{\theta\:}}_{s}=\frac{k\left(\varDelta\:-\delta\:\right)}{cos{\theta\:}_{2}}\text{sin}\left({\theta\:}_{1}+{\theta\:}_{2}\right)-mgsin{\theta\:}_{1}, m{l}_{1}{\ddot{\theta\:}}_{s}=\frac{k\left(\varDelta\:-{l}_{1}\left(1-cos{\theta\:}_{1}\right)-{l}_{2}(1-cos{\theta\:}_{2})\right)}{cos{\theta\:}_{2}}(\text{sin}{\theta\:}_{1}\text{cos}{\theta\:}_{2}+\text{sin}{\theta\:}_{2}\text{cos}{\theta\:}_{1})-mgsin{\theta\:}_{1}$$

(10)

And by replacing sin $\:{\theta\:}_{2}$ with sin $\:{\theta\:}_{1}$ we can write:

$$\:m{l}_{1}{\ddot{\theta\:}}_{s}=\left(\frac{k\left(\varDelta\:-{l}_{1}\left(1-cos{\theta\:}_{1}\right)-{l}_{2}(1-cos{\theta\:}_{2})\right)}{cos{\theta\:}_{2}}\right(cos{\theta\:}_{2}+\frac{{l}_{1}}{{l}_{2}}\text{cos}{\theta\:}_{1})-mg)sin{\theta\:}_{1}$$

(11)

By expanding Eq. 11 and simplifying, it can be written as:

$$\begin{aligned}&\:m{l}_{1}{\ddot{\theta\:}}_{s}=\left(\frac{k\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{\text{sin}}^{2}{\theta\:}_{2}}\right)\right)}{\sqrt{1-{\text{sin}}^{2}{\theta\:}_{2}}}\left(\sqrt{1-{\text{sin}}^{2}{\theta\:}_{2}}+\frac{{l}_{1}}{{l}_{2}}\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)-mg\right)sin{\theta\:}_{1}\\&m{l}_{1}{\ddot{\theta\:}}_{s}=\left(\frac{k\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}\text{sin}}^{2}{\theta\:}_{1}}\right)\right)}{\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}}\left(\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}+\frac{{l}_{1}}{{l}_{2}}\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)-mg\right)sin{\theta\:}_{1}\\&m{l}_{1}{\ddot{\theta\:}}_{s}+\left(mg-\frac{k\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}\text{sin}}^{2}{\theta\:}_{1}}\right)\right)}{\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}}\left(\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}+\frac{{l}_{1}}{{l}_{2}}\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)\right)\text{sin}{\theta\:}_{1}=0\end{aligned}$$

(12)

The horizontal component of Eq. 12 is:

$$\begin{aligned}&\:m({\ddot{x}}_{s}-{\ddot{x}}_{e}{sin}^{2}\theta\:)\\&+\Bigg(mg-\frac{k\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}\text{sin}}^{2}{\theta\:}_{1}}\right)\right)} {\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}}\\& \times \left(\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}+\frac{{l}_{1}}{{l}_{2}}\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)\Bigg)\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\text{sin}{\theta\:}_{1}=0\end{aligned}$$

(13)

By adding an external damper and simplifying Eq. 13, the following equation can be derived:

$$\begin{aligned}&\:m\left({\ddot{x}}_{s}-{\ddot{x}}_{e}{sin}^{2}\left(\frac{{x}_{s}-{x}_{e}}{{l}_{1}}\right)\right)+c\left({\dot{x}}_{s}-{\dot{x}}_{e}\right)\\&+\Bigg(mg-\frac{k\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{\left(\frac{{x}_{s}-{x}_{e}}{{l}_{1}}\right)}^{2}}\right)-{l}_{2}\left(1-\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\left(\frac{{x}_{s}-{x}_{e}}{{l}_{1}}\right)}^{2}}\right)\right)}{\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\left(\frac{{x}_{s}-{x}_{e}}{{l}_{1}}\right)}^{2}}}\\&\times\left(\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\left(\frac{{x}_{s}-{x}_{e}}{{l}_{1}}\right)}^{2}}+\frac{{l}_{1}}{{l}_{2}}\sqrt{1-{\left(\frac{{x}_{s}-{x}_{e}}{{l}_{1}}\right)}^{2}}\right)\Bigg)\sqrt{1-{\left(\frac{{x}_{s}-{x}_{e}}{{l}_{1}}\right)}^{2}}\left(\frac{{x}_{s}-{x}_{e}}{{l}_{1}}\right)=0\end{aligned}$$

(14)

Where $\:{\theta\:}_{1}=\left(\frac{{x}_{s}-{x}_{e}}{{l}_{1}}\right)\:$has been considered in Eq. 14.

Stability condition

The stability of the system is crucial and one of the most important aspects of the system under study⁵⁰. If the force from the member providing negative stiffness exceeds a certain limit, it leads to instability. According to Eq. 13, reducing the effect of negative stiffness allows us to adjust design parameters to maintain stability and optimize the system.

The stability condition is expressed as follows:

$$\:mg-\frac{k\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}\text{sin}}^{2}{\theta\:}_{1}}\right)\right)}{\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}}\left(\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}+\frac{{l}_{1}}{{l}_{2}}\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)\ge\:0$$

(15)

Requiring for any $\:{\theta\:}_{1}$:

$$\:mg\ge\:\frac{k\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}\text{sin}}^{2}{\theta\:}_{1}}\right)\right)}{\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}}\left(\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}+\frac{{l}_{1}}{{l}_{2}}\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)$$

(16)

And for a special case of $\:{\theta\:}_{1}=0$.

$$\:mg\ge\:k\varDelta\:\left(1+\frac{{l}_{1}}{{l}_{2}}\right)$$

(17)

Therefore, always it is.

$$\:mg\ge\:\frac{k\varDelta\:}{{l}_{2}}\left({l}_{1}+{l}_{2}\right)\ge\:\frac{k\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}\text{sin}}^{2}{\theta\:}_{1}}\right)\right)}{\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}}\left(\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}+\frac{{l}_{1}}{{l}_{2}}\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)$$

(18)

So only by calculating the initial critical point for stability, the optimal state can be obtained. Therefore, the maximum efficiency of the proposed seismic isolator is derived from Eq. 19:

$$\:k\varDelta\:=\frac{mg{l}_{2}}{\left({l}_{1}+{l}_{2}\right)}$$

(19)

By rewriting Eq. 19 as below, the relation between the effectiveness of parameters in the stability of the system can be demonstrated in Fig. 4.

$$\it \:\frac{k\varDelta\:}{\text{m}\text{g}}=\frac{{l}_{2}}{\left({l}_{1}+{l}_{2}\right)}, {l}_{2}=\frac{\frac{k\varDelta\:}{mg}{l}_{1}}{1-\frac{k\varDelta\:}{mg}}$$

(20)

In this figure, for the ratio of the initial force from the negative stiffness member $\:k\varDelta\:$ to the weight $\:mg$, the relationship between l₁ and l₂ - in the optimal case where the best effectiveness of negative stiffness exists - is shown. This figure shows that by reducing this ratio, the length of l₂ should be reduced so that its angular changes in displacements increase. This increase in angle causes the image of the force required for negative stiffness to have a greater effect on the displacement axis. Also, by increasing the aforementioned ratio, the length of l₂ increases so that the horizontal effect of the force causing negative stiffness is reduced. The diagram shows that for a force $\:k\varDelta\:$ greater than the weight $\:mg$ per $\:{l}_{2}$, the system becomes unstable. So the maximum value that can be considered for $\:k\varDelta\:$ is $\:mg$.

Fig. 4

Effectiveness of parameters in the stability of the system.

Discussion on negative stiffness performance

The force produced by the compression component is nonlinear and relies on the extension of the compressed member (illustrated in Fig. 3) as well as the angle it forms. This force is maximum when the compressed member is in a vertical position; meanwhile, it is zero in the direction of horizontal movement in the structure. Then, by creating an angle in the member, the negative stiffness-generated force, F_2x, can contribute to the horizontal effect. However, due to the increased length of the compressed member, as depicted in Fig. 3, the source of the F_2x, the F_sp force, decreases, and the dominance of positive stiffness occurs following a substantial displacement.

Fig. 5

The horizontal reaction force comparison of the pendulum column and PC-VNS.

Figure 5 compares the horizontal reaction force of the proposed system (PC-VNS) with and without negative stiffness (PC). It shows that in the displacement range of 0–20 cm, typical for earthquake-induced motion, the horizontal reaction force generated in the PC-VNS due to QZS is nearly zero. This results in much less force being applied to the structure during an earthquake. When the compressed length (∆-δ) in Eq. 6 is reduced to zero, the horizontal reaction force of the pendulum column (PC-VNS without negative stiffness) and PC-VNS is equal.

In this example, the cable length is considered 3 m, the compressed length is 0.2 m, and the compressed member’s force is half its weight.

Then, the equivalent stiffness can be considered for the member providing negative stiffness. According to Eqs. 7 and 8, the force defining negative stiffness can be rewritten as:

$$\begin{aligned}&\:{F}_{2}=k\frac{\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}\right)\right)}{\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}}\\&{F}_{2\theta\:}=k\frac{\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}\right)\right)}{\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}}sin({\theta\:}_{1}+{\theta\:}_{2})\\&\:{F}_{2\theta\:}=k\frac{\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}\right)\right)}{\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}}(\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}+\frac{{l}_{1}}{{l}_{2}}\sqrt{1-{{sin}}^{2}{\theta\:}_{1}})\:sin{\theta\:}_{1}\end{aligned}$$

If $\:{F}_{2\theta\:}$ is defined as:

$$\:{F}_{2\theta\:}={k}_{N\theta\:}{\theta\:}_{1}$$

(21)

Then $\:{k}_{N\theta\:}$, as the nonlinear stiffness of the member that provides negative stiffness, in the $\:{\theta\:}_{1}$ direction, can be defined as:

$$\:{k}_{N{\theta\:}_{1}}=k\left(\frac{\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}\text{sin}}^{2}{\theta\:}_{1}}\right)\right)}{{\theta\:}_{1}\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}}\left(\sqrt{1-{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{\text{sin}}^{2}{\theta\:}_{1}}+\frac{{l}_{1}}{{l}_{2}}\sqrt{1-{\text{sin}}^{2}{\theta\:}_{1}}\right)\right)sin{\theta\:}_{1}$$

(22)

If F₂ force x direction component is defined as F_2x, then it can be written:

$$\begin{aligned}&\:{F}_{2x}=k\frac{\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}\right)\right)}{\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}}sin{\theta\:}_{2}\\&\:{F}_{2x}=k\frac{{l}_{1}}{{l}_{2}}\frac{\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}\right)\right)}{\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}}sin{\theta\:}_{1}\end{aligned}$$

(23)

Then $\:{k}_{Nx}$, as the nonlinear stiffness of the member that provides negative stiffness, in the $\:x$ direction, can be defined as:

$$\:{k}_{Nx}=k\frac{\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}\right)\right)}{{l}_{2}\:\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}}\:$$

(24)

Finally for F₂ component in y direction we can write:

$$\begin{aligned}&\:{F}_{2y}=k\frac{\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}\right)\right)}{\:\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}}\:cos{\theta\:}_{2}\\&{F}_{2y}=\:k\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}\right)\right)\end{aligned}$$

(25)

By considering $\:\varDelta\:y={l}_{1}(1-{cos}{\theta\:}_{1})$, the $\:{k}_{Ny}$, as the nonlinear stiffness of the member that provides negative stiffness, in the $\:y$ direction, can be defined as:

$$\:{k}_{Ny}=k\frac{\left(\varDelta\:-{l}_{1}\left(1-\sqrt{1-{{sin}}^{2}{\theta\:}_{1}}\right)-{l}_{2}\left(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}}\right)\right)}{{l}_{1}(1-\sqrt{1-{{\left(\frac{{l}_{1}}{{l}_{2}}\right)}^{2}{sin}}^{2}{\theta\:}_{1}})}$$

(26)

Damping coefficient calculation for design

As the PC-VNS seismic isolator’s horizontal stiffness is variable due to non-linearity, the damping coefficient cannot be determined conventionally as in linear systems. However, because the pendular part of the PC-VNS seismic isolator has a linear behavior, its damping coefficient is introduced as equivalent to the whole system’s damping coefficient. Assuming the seismic isolator is without negative stiffness, the angular frequency of the system will be equal to Eq. 1. Therefore, for this system without negative stiffness, the damping coefficient will be equal to Eq. 27.

$$\:{c}_{cr}=2m\omega\:=2m\sqrt{\frac{g}{l}}, \xi\:=\frac{c}{{c}_{cr}}$$

(27)

Assuming the same premise, Eq. 27 defines the equivalent damping coefficient for the damper in the PC-VNS system’s design. Since the negative stiffness system has a lower angular frequency, this damping coefficient will be the highest possible limit for achieving the desired amount of damping.

The reason for this is that in the case of quasi-zero stiffness, the periodicity of the system becomes theoretically infinite, and it will not be possible to calculate the damping coefficient. Therefore, the damping coefficient in the case of no negative stiffness is considered as an upper bound for the damping coefficient.

Effect of proposed system on soil

As previously noted, this integrated dual-purpose system provides not only seismic isolation but also enhances structural performance through soil reinforcement. From a static standpoint, even without the influence of negative stiffness, the comparative behavior of the proposed system versus a conventional pile under identical displacement conditions is clearly depicted in Fig. 6. In this analysis, a horizontal displacement of 10 cm and a vertical displacement of 1 cm are applied at the pile head. The results indicate that when the proposed isolator is employed, these displacements produce minimal disturbance to the surrounding soil, demonstrating the system’s capability to effectively isolate structural motion while maintaining soil integrity.

Fig. 6

Proposed model effect in soil compared to a common pile.

To evaluate the impact of replacing conventional piles with the proposed hybrid pile on soil–structure interaction and the dynamic stress–strain response of the surrounding soil, two models were developed in ABAQUS. One model included conventional piles, while the other used the proposed hybrid pile configuration. As depicted in Fig. 7, the soil domain was idealized as a two-dimensional rectangular continuum with dimensions of 16 × 4 m. The soil was modeled as an elasto-plastic material, following the Mohr–Coulomb constitutive law. Elastic properties were defined as E = 25e6 N/m², ν = 0.3, and ρ = 2000 kg/m3. Plastic behavior was characterized by a cohesion c = 75e3 N/m², friction angle ϕ = 30°, and dilation angle ψ = 5°. A surcharge mass of 275 tons was applied at the pile head. The soil-pile contact is modeled as “tie constraint” to ensure complete interaction between the soil and the pile. To eliminate the influence of initial in-situ stresses and ensure that the results only reflect stresses and strains induced by external loading, gravitational acceleration is applied exclusively to the surcharge, while the soil body is assumed to be stress-free at the start of the analysis.

Dynamic excitation was introduced in the form of a lateral cyclic displacement load with an amplitude of 0.01 m and an angular frequency of ω = 10 rad/s, corresponding to typical earthquake-induced vibration characteristics. In this modeling, first the soil model alone was subjected to frequency analysis, and then, after obtaining the first and second mode frequencies, the Rayleigh coefficients, i.e., the α = 1.64 and β = 0.00151 coefficients were calculated. Then the main model was designed with the CIN element, considering two absorbent boundaries on the right and left sides of the model.

The simulation results show that the proposed hybrid pile significantly reduces the stresses and strains in the surrounding soil compared to conventional piles. A representative set of these results is presented in Figs. 8 and 9. The red point marked in Fig. 7 was specifically examined.

Fig. 7

(a) Common pile embedded in soil (b) Proposed hybrid pile in soil.

The proposed hybrid pile led to a significant change in the response of the pile-soil system. The results indicated that the stress transmitted to the surrounding soil was considerably reduced compared to the conventional pile. Additionally, the strains observed around the pile equipped with the isolator were much lower than those recorded in the conventional case. According to the cyclic loading results, the soil behavior around the proposed hybrid pile became more stable and uniform, with a significant reduction in the intensity of cumulative deformations and phenomena caused by successive loading.

It should be noted that the primary focus of this research is on the structural performance of the proposed system; its behavior from a geotechnical aspect has been assessed to illustrate its effectiveness in mitigating responses in soil.

Fig. 8

a) Displacement that applied to soil b) stress comparison c) strain comparison.

Fig. 9

Stress-strain hysteresis loop (a) Comparison between rigid pile and PC-VNS (b) PC-VNS.

Assessment of the 3D model for PC-VNS

In order to evaluate the performance of the proposed PC-VNS and the stress distribution in the system components, a 3D model of the system was subjected to a vertical axial load of 100 tons. In this modeling, the system height and cable length were considered to be 8 and 4 m respectively, the wall thickness was 1 cm, and the radius of the outer cylinder was 40 cm. The material was considered to be steel. To define the mechanical properties of the material, the modulus of elasticity, Poisson’s ratio and the yield stress of steel were considered as 2E11 N/m², 0.3 and 2E8 N/m², respectively. For the member providing negative stiffness, the modulus of elasticity was 5E8 N/m² and the initial pre-compressed value was 20 cm.

Fig. 10

Stress distribution in the PC-VNS under vertical load.

To represent the stress distribution, the Mises criterion was used. As can be seen in Fig. 10, the stress contours show an acceptable distribution for the stress.

Results and discussions

In this section of the research, the dynamic equations derived from the system’s mathematical model are analyzed. Initially, records from six earthquakes are selected as input data. Subsequently, the system’s dynamic response under these seismic loads is determined in MATLAB.

Earthquakes record information

To analyze how the PC-VNS performs during earthquakes, 6 earthquake records with varying features were chosen. Details about these recordings are listed in Table 1, and visual representations can be found in Fig. 11.

Table 1 Earthquake information.

Fig. 11

Earthquakes response spectra, (a) displacement (b) pseudo-velocity (c) pseudo-acceleration damping ratio = 2%.

Responses of the system under excitation

To investigate and assess the performance of the PC-VNS from the kinematic perspective, the system equation was examined for a single degree of freedom model that represents the derived equation, using the MATLAB program. The PC-VNS was configured with a height of 3 m for cables and members providing negative stiffness length. Due to negative stiffness, the compression amount “Δ” in the compressed spring was considered 0.2 m, and the initial upper-lifting force in it was supposed to be half the weight of the mass on the isolator. Damping coefficients equal to 0, 0.05, 0.1, 0.15, and 0.2 were considered.

FFT analysis was employed to explore the proposed seismic isolator’s behavior in the frequency domain. Fast Fourier Transform (FFT) can show the frequency range of the earthquake components and help identify the dominant frequencies in the ground motion or structural response. In the FFT spectra of this study, the frequency content of the input signals to the system (time history) is compared with the output (system response). These spectra help evaluate the performance of the system in filtering the input vibrations and the resonances created in the responses.

To calculate the energy of a system there are two approaches. In the first approach, the energy is calculated in real and absolute terms. In this case, the movement of the ground is included in the energy calculation. However, in the second approach, the energy is calculated in relative terms and the ground is assumed to be stationary. Since the strain energy stored in the structural components is the criterion for their failure, and calculating the energy in relative terms, provides values ​​that are more proportional to the strain energy, the second approach is more common for calculating the input energy and calculating the energy in relative terms. Therefore, the isolated system with negative stiffness is evaluated through the input energy diagrams under various damping coefficients. The results are compared to those of a conventional frame with a period of 0.4 s by analyzing the normalized form of input energy for a 1 kg mass.

The findings of this study are illustrated in Figs. 12, 13, 14, 15, 16 and 17. In energy diagrams, for input energy, the system with negative stiffness indicated by (Ei-n) and non-isolated frames by (Ei-f) in the input energy diagrams has several damping coefficients.

Fig. 12

The isolator response to the Friuli (Italy) earthquake for different damping ratios b) FFT analyses of acceleration c) Input energy comparison.

Fig. 13

(a) The isolator response to the Hollister (USA) earthquake for different damping ratios (b) FFT analyses of acceleration (c) Input energy comparison.

Fig. 14

(a) The isolator response to the Kocaeli (Turkey) earthquake for different damping ratios (b) FFT analyses of acceleration (c) Input energy comparison.

Fig. 15

(a) The isolator response to the Landers (USA) earthquake for different damping ratios (b) FFT analyses of acceleration (c) Input energy comparison.

Fig. 16

(a) The isolator response to the Loma Prieta (USA) earthquake for different damping ratios (b) FFT analyses of acceleration (c) Input energy comparison.

Fig. 17

(a) The isolator response to the Trinidad (USA) earthquake for different damping ratios (b) FFT analyses of acceleration (c) Input energy comparison.

The results indicate that the proposed system has a very low displacement and velocity response to ground vibrations. This leads to a considerable reduction in the acceleration of the structure, making it nearly undetectable. This decrease in force on the structure enhances the safety of its occupants by minimizing movements. Unlike a conventional isolation system, which not only reduces the force applied to the structure but also increases its displacement, the results from the response analysis indicate that the displacement of the system has decreased as well. The reason for this is the quasi-zero stiffness in the neighborhood of the initial equilibrium point. The results also reveal that as the amplitude of the base motion decreases, the negative stiffness effect becomes more pronounced, leading to a greater reduction in system responses. Additionally, the impact of damping coefficients on the system response, when compared to the undamped state, demonstrates that a minimum damping rate improves system performance, stability, and the convergence of the response to ground displacement.

The FFT analysis indicates the contributions of different frequencies to the seismic response, breaking down the graph into frequency components. The results reveal that the majority of amplitudes during earthquakes have been significantly reduced.In most seismic isolation systems, frequency components are filtered, but near the natural frequency of the system, the amplitudes of these frequencies are often significantly amplified. A key and notable feature of this system, observed in most of the responses, is the reduction across the entire frequency range. This reduction is primarily due to the quasi-zero stiffness (QZS) and the nonlinear behavior of the system. This highlights the effectiveness of QZS and nonlinear characteristics in enhancing the performance of seismic isolation systems.

It is important to examine the energy input to the structure, as the damage to the structure is directly related to the strain potential energy stored in its elastic components. Unlike elastic systems, the proposed system stores energy in the form of gravitational potential energy. Therefore, increasing this energy does not cause the system to collapse. The results confirm that the seismic isolator absorbs less energy than comparable models, rendering the structure healthier during an earthquake. Therefore, however, during the Kocaeli earthquake, the input energy of the system was higher than that of a common frame. Nevertheless, parameters such as acceleration and FFT analysis demonstrate that the proposed system performs remarkably well during an earthquake. Because the acceleration and, as a result, the force applied to the structural components can be acceptable.

A notable point is the vibration amplitude of this earthquake, which is about 0.5 m. Such events are very rare, while the system also for this record reduces the response acceleration well.

Pseudo-zero stiffness and damping interaction showed that in all earthquake records except the Kocaeli earthquake, increasing damping increased the acceleration applied to the structure. This means that the low amplitude of the ground vibration in this record increases the effectiveness of the quasi-zero-stiffness and adding damping, such as additional forces to the system, causes more force to be transferred from the ground to the structure through the damper. However, in the Kocaeli earthquake, due to the high amplitude of the ground motion, the effect of quasi-zero-stiffness was reduced. This caused the earthquake force to be applied through the elastic property, which increased damping caused no energy accumulation and resonance in this sample, which caused the response acceleration.

Validation of derived equations

To validate the results obtained from the analytical equations, a finite element model with equivalent specifications was developed using ABAQUS (Fig. 18). The tensile member above the mass and the compression member below the mass - which transfers the force resulting from the negative stiffness to the mass - were considered to be 3 m long. The compression member providing the negative stiffness was considered to be 3 m long and pre-compressed to be 20 cm. The up-side force of the member providing the negative stiffness -in the optimum state- was considered to be equal to half the weight of the middle mass. Damping was considered to be 0.1 and 0.2 critical damping. The comparison between the analytical and numerical results is presented in Fig. 19. In this figure, two results relate to the Hollister and Trinidad earthquakes for base excitation. As illustrated, there is a strong correlation between the two sets of results, demonstrating good agreement and confirming the accuracy and reliability of the proposed analytical approach.

Fig. 18

ABAQUS model for verification.

Fig. 19

Validation of displacement (a) Hollister earthquake (b) Trinidad earthquake.

Conclusions

The technique of extending the natural period is utilized to increase the resilience of buildings against earthquakes, usually by seismic isolators. Meanwhile, on the other hand, the unsuitable soil in earthquake high-risk situations adds other challenges to this issue. For this purpose, in this study, the method of connecting the foundation was altered by a pile-shape hybrid seismic isolation system to raise the period of the structure, and meanwhile, the pile-like shape reinforces the weak soil. The quasi-zero stiffness part that exists in the system has improved its performance. After deriving the system dynamic equation, six different earthquake records were chosen, and then four different damping ratios were used to evaluate system responses in earthquake situations. Then the soil-structure interaction was evaluated. A 3D model was modeled in ABAQUS under a load of 100 tons and the stress distribution was obtained. The study showed that the proposed PC-VNS hybrid pile-seismic isolator with a height of 3 m and the same length for cables and members providing negative stiffness was effective in reducing the acceleration. The system was modeled in MATLAB based on derived equations and compared with the equivalent instance that was modeled in ABAQUS software to assess its kinematic performance. Results showed that:

• The proposed system reduced displacement, unlike common isolators, and also acceleration, even with low damping.

• FFT analysis found that most acceleration responses were significantly reduced.

• The input energy to the system was significantly decreased.

• It was concluded that the performance of the system is determined by the maximum range of motion, and the lower the range, the better the results.

• The soil-structure interaction showed that when using the proposed system, very low stresses and strains are generated in the soil compared to a conventional pile.

• 3D modeling of the system showed that the stress distribution in the components was much lower than the allowable stress.

• Validation showed complete agreement with the results obtained from the extracted equations.

Finally, it should be said that the proposed system, with simultaneous improvement of soil and dynamic responses, can be considered as an effective component in structural engineering.