Development of rocking constraint device with vertical damping capacity for three-dimensional base-isolated frame structures

Abstract

A new rocking constraint device (RCD) is developed for three-dimensional (3D) base-isolated frame structures by connecting a custom-designed cylinder pair to provide vertical damping with replaceable damping components installed outside the cylinders when the superstructure undergoes translational motion, and rocking constraint capacity when the superstructure is susceptible to rocking. Theoretical formulas for calculating the damping and rocking constraint stiffness of the RCD are proposed. Two series of sinusoidal loading tests are conducted at different loading frequencies and amplitudes to verify the damping and rocking constraint performance of the RCD. The test results show that the cylinder without orifices on its piston can provide the desired damping with a replaceable damping component, and that the RCD can effectively suppress rocking. Although the vertical stiffness of an individual cylinder is affected by the location of the replaceable damping component and loading frequency, the average vertical stiffness of the two cylinders, which determines the rocking constraint stiffness of the RCD, is independent of the two factors. Comparisons of the test and theoretical results indicate that the errors of the proposed formulas for calculating the damping and rocking constraint stiffness of the RCD do not exceed 12.9% and 11.0%, respectively.

Graphical abstract

Keywords

three-dimensional isolation / rocking behavior / rocking constraint device / replaceable damping component / sinusoidal test

1 Introduction

As the most widely used structural response control technology, base isolation is typically adopted to extend the natural vibration period of the structure and increase the damping of the structure via the installation of isolation bearings between the foundation and superstructure, which aims to reduce the energy input to the superstructure by earthquakes. Currently, isolation bearings used in engineering practices for frame structures primarily focus on structural control in horizontal directions because the seismic failure of structures is primarily governed by the performance of structural members under the horizontal components of ground motions. However, investigations have shown that the strong vertical component of ground motions will yield significant vertical responses in base-isolated structures owing to the extremely high stiffness of typically used bearings [1]. Full-scale shaking table tests conducted in E-Defense and University of California San Diego showed that the vertical floor acceleration responses of base-isolated structures can be amplified by more than five times from the ground [1], and that the slab acceleration can reach 5g [2]. Although the strong vertical vibrations did not cause safety concerns to the structure, they resulted in significant floor responses and severe damage to nonstructural objects, such as interior equipment, furniture, partition walls, ceilings, and piping systems [1–6], which are detrimental to important structures such as hospitals, nuclear power plants, key laboratories, and data centers, owing to their adverse effect to the functionality of the structures. Therefore, the responses of vertical structures must be controlled to improve the seismic resilience of key structures and maintain their post-earthquake functionality.

The development of 3D isolation systems over the last century has provided a new approach for enhancing the seismic behavior of conventional base isolation systems [7,8]. Different materials and elements have been used to develop low-stiffness vertical isolation devices. Considering the successful application of rubber bearings for horizontal isolation, a thick-layered rubber 3D isolation system with a lower bearing shape factor was proposed and used [9]. The effectiveness of the isolation system was demonstrated during both earthquakes and traffic vibrations [8]. However, thick-layered rubber 3D isolation systems typically cannot provide effective vertical isolation because of their short isolation period (< 0.5 s) and other problems related to creep and insufficient capacity for supporting vertical loads [10]. Additionally, steel springs have been used to isolate vertical ground motions. Hüffmann [11] designed a new 3D isolation system comprising coil springs and viscous dampers. Coned disc springs were used to develop 3D isolation systems owing to their considerable vertical load-carrying capacity and vertical damping capacity afforded by the friction between adjacent coned disc springs [12,13]. Generally, steel springs present approximately linear characteristics when they operate independently, which results in large deformations and a long vertical isolation period. Typically, the isolation period afforded by steel springs is relatively short (< 0.5 s), considering the instability problem of springs with large deformations and the size of the isolators.

Conventional base isolation practices indicate that a longer isolation period typically offers better isolation. Although the frequency in the vertical component of ground motion is typically higher than that in the horizontal component, a longer vertical isolation period is preferred to improve isolation effectiveness. The feasibility of using air springs, hydraulic cylinders, and other techniques has been investigated to effectively extend the period of vertical isolation. In Japan, Kashiwazaki et al. [14] developed a 3D isolation system comprising rubber bearings for horizontal isolation, and load-carrying hydraulic cylinders filled with nitrogen gas for vertical isolation. Suhara et al. [15] proposed a 3D isolation device that serially combines a laminated rubber bearing (LRB) and an air spring to achieve horizontal and vertical isolation, respectively. Kageyama et al. [16] developed a cable-reinforced air spring that can provide both horizontal and vertical isolation in a single air-pressure-activated device. The above mentioned developments for 3D isolation systems were primarily proposed for nuclear power structures, and the achievable vertical isolation period was 2 s. Researchers at Shimizu Corporation [17,18] developed a 3D isolation system comprising rubber bearings and air springs to achieve horizontal and vertical isolation, respectively. The system achieved a vertical isolation period of 1.25 s, and it was applied to a three-story apartment building, which became the first civilian building in Japan to adopt a practical 3D isolation system. Recently, long-period vertical isolation systems have been developed for long-span structures. Chen et al. [19] proposed a vertical isolation device with variable stiffness using multiple hydraulic cylinders for long-span structures, which achieved a vertical isolation period of 1 s. Han et al. [20] developed an air-spring-FPS 3D isolation bearing for long-span spatial structures composed of a friction pendulum system for horizontal isolation and an air spring for vertical isolation.

Generally, a long vertical isolation period ranging from 1 to 2 s can avoid resonance of structures with seismic waves resonance with seismic waves and result in improved control performance. However, it may result in considerable rocking of the entire structure, which causes coupled horizontal and vertical responses of the superstructure and thus degrades the isolation control efficiency. Hüffmann [11] discovered that the use of flexible springs for 3D isolation resulted in rocking of the structure, which consequently increased the horizontal floor acceleration of the structure. More importantly, the rocking behavior increased the displacement demand of vertical isolators placed on the periphery of the structure [21] and imposed large internal forces on the isolators, which might damage the components of the isolators as well as the entire structure. Previous studies have shown that rocking is affected by the height-to-width ratio of a superstructure and the isolation period in the horizontal and vertical directions [22]. As the height-to-width ratio of a superstructure increases, the ratio of the overturning movement of the superstructure to the resistance of the overturning moment, which is contributed primarily by the vertical bearings, increases; consequently, rocking becomes more intense. In addition, when the vertical isolation period increases, particularly when it approaches the horizontal isolation period, the rocking of a superstructure becomes more significant. Zhou et al. [8]. investigated the dynamics of vertical and 3D isolation systems developed for modern nuclear facilities and concluded that rocking effect was evident when the vertical isolation period approached 1 s.

To restrain rocking, different methods have been attempted in addition to controlling the height-to-width ratio of a superstructure as well as adjusting the isolation period of 3D isolation. For example, to avoid mass eccentricities in the vertical direction and control rocking, Kitamura and Morishita [23] proposed a deck isolation system featuring a reactor vessel and the primary components suspended from a large slab structure (a typical deck) while being supported by a few vertical isolation devices. Fujita et al. [24] attempted to mitigate the rocking of a structure by installing support bearings around it. Meanwhile, the typical approaches considered in the development of 3D isolators are as follows: first, a guiding device is adopted for the vertical isolator, particularly when the isolation period is long, to restrain the isolator’s lateral displacement and transfer the shear force from the superstructure to the base. For example, sliders were designed to guide air springs to propagate in the vertical direction with a larger horizontal stiffness than that of air springs [17]. Restrainers were used to allow the sliding block of a 3D oblique sliding friction seismic isolator to shift up and down as well as to transmit shear force [25]. Seitaro et al. [26] proposed a 3D isolation system comprising lead rubber bearings and metal bellows, in which inner and outer casings with metallic slide bearings were used to transmit shear forces between the LRB and base. Similar concepts have been adopted in other studies [19,27]. Although the main function of these guiding devices is to transmit shear forces, they can in fact resist the overturning moment, as the sliding surfaces of the devices can constrain the rotation of the isolators connected to the superstructure. However, the rocking constraint capacities provided by the guiding devices are relatively limited owing to the small size of the isolators, which cannot yield a significant resisting moment. The forces imposed on the guiding devices owing to rocking can eventually damage the components of the isolators under an extreme earthquake event.

A separate rocking constraint device (RCD) is required to effectively constrain the rocking behavior of a superstructure. A few new configurations have been proposed and discussed. Kageyama et al. [16] used pulleys, steel cables, and shafts to assemble an RCD for a 3D isolation system. When the superstructure rocked, one steel cable shifted downward, whereas another steel cable shifted upward. Because one side of the crossing wire cables was under tension and the movements of the two shafts were restricted by the tensile force of the wire cable, the structure was successfully prevented from rocking. Kashiwazaki et al. [14] proposed a rocking suppression cylinder system that featured vertical isolators and accumulators filled with nitrogen gas connected by horizontal rocking cylinders linked by swivel joints. The rocking movement was suppressed by the opposite flow of the fluid in the rocking cylinders. The performance of the device was verified via shaking table tests [28]. Takahashi et al. and Suhara et al. [17,18] applied air-bearing-type 3D isolations to building structures and proposed a rocking suppression device that utilizes fluid dampers. The system provided damping during the translational movement of the superstructure and suppressed rocking of the structure when rocking was generated. One disadvantage of the system, however, is the presence of orifices on the pistons, and that the fluid can flow from one chamber to another in the same cylinder. Furthermore, the rocking constraint stiffness under low-frequency excitations is low, which may cause the structure to undergo considerable rocking movements under low-frequency earthquakes or wind.

To date, several types of 3D isolation systems have been proposed. However, as one of the keys which affect the future development and practice of 3D isolation systems, the development of RCDs is insufficient. The design theory and seismic performance of these devices require further investigation. In this study, a new RCD was developed. In contrast to the devices developed in previous studies [17,18], the new RCD does not require orifices for the pistons in the cylinders. Replaceable damping components are installed outside the cylinder to allow the damping of the system to be adjusted conveniently and to allow the design requirements of both the damping and rocking constraint stiffness to be satisfied easily. More importantly, as the piston can fully separate the fluid in the two chambers of one cylinder owing to the no-orifice design, fluid cannot flow from one chamber to another of the same cylinder, thus preventing rocking even under low-frequency excitations. Theoretical formulas for calculating the damping and rocking constraint stiffness have been proposed. Sinusoidal loading tests were conducted at different loading frequencies and amplitudes to verify the damping and rocking constraint performances of the RCD under different loading conditions and the proposed theoretical formulas.

2 Development of the rocking constraint device

2.1 Components and operating principle of the rocking constraint device

To prevent the rocking of a superstructure with 3D base isolation, an RCD was proposed (see Fig.1). The developed RCD comprises a hydraulic cylinder pair connected to rubber pipes and two replaceable damping components with orifices. Each cylinder featured two chambers separated by a piston without an orifice; thus, the fluid cannot flow from one chamber to another. The two cylinders of the RCD were identical in configuration and size, but their installation directions were different, as shown in Fig.1. Cylinder 1 was connected to the ground with the cap end, and Cylinder 2 was placed upside-down with the rod end connected to the ground. The two cylinders can be installed in the same direction; however, this requires the two pipes to cross each other. Hard pipes made of steel or other materials can be adopted, particularly when the pipes are long, to reduce the deformation of rubber tubes. However, to accommodate the slight lateral movement of the cylinders, short rubber tubes must be used to connect the hard pipes and cylinders in practical applications.

The RCD was developed to provide the rocking constraint stiffness of the structure by connecting the upper chamber of one cylinder to the lower chamber of another cylinder, thereby reducing the structural rocking effect. To improve the performance of the RCD under low-frequency excitations [17] and maintain the frequency-independent characteristics of the rocking constraint stiffness, no orifice was designed for the piston; thus, the fluid cannot flow from one chamber to another of the same cylinder. Meanwhile, replaceable damping components were installed on the surface of one of the cylinders, as shown in Fig.1, to provide damping to the structure when the superstructure undergoes translational vertical displacement. As replaceable damping components are outside the cylinder, they can be easily replaced and adjusted based on the requirements by changing the number and size of the orifices in the damping component.

The RCDs should be installed at the perimeter of the vertical isolation layer to increase the stiffness of the rocking constraint. They can be placed either in the longitudinal and transverse directions with the target to control the rocking in each direction separately, as shown in Fig.2(a), or in the diagonal direction with the target of each RCD to allow their operation in two directions, as shown in Fig.2(b). In addition, the RCD is more suitable for fully developing the rocking constraint stiffness of 3D isolation systems comprising vertical and horizontal isolators in different layers, as shown in Fig.2.

When the superstructure is in a static state, the RCD does not support a vertical load from the superstructure; in fact, the vertical load is supported only by the isolation bearings. Fig.3 shows the two operating states of the RCD. When the superstructure shifts downward translationally under the vertical component of earthquakes, as shown in Fig.3(a), the rod end of Cylinder 1 shifts downward, and the fluid in the lower chamber of Cylinder 1 flows into the upper chamber of Cylinder 2, which is placed upside-down through the replaceable damping component and pipe. Similarly, the cap end of Cylinder 2 shifts downward (i.e., the piston and piston rod of Cylinder 2 shift upward relative to the cap end), and the fluid in the lower chamber of cylinder 2 flows into the upper chamber of Cylinder 1 through the pipe and replaceable damping component. Notably, because the piston sizes of the two cylinders are the same, the displacements of the two cylinders are the same. When the fluid flows through the orifices in the replaceable damping components, a vertical damping force is generated to dissipate the kinetic energy of the structure owing to head loss. When the fluid flows through the orifices in the replaceable damping components, a vertical damping force is generated to dissipate the kinetic energy of the structure owing to head loss. Although pipes with a small diameter can provide a sufficient damping dissipation capacity, a replaceable damping component scheme is preferred because it allows the pipe sizes to be customized to satisfy the damping requirements of different structures. In addition, using a replaceable damping component provides more flexibility in designing the damping and stiffness of the device.

When the superstructure rocks, as shown in Fig.3(b), the rod end of Cylinder 1 needs to move downward, and the cap end of Cylinder 2 needs to shift upward (i.e., the piston and piston rod of Cylinder 2 shift downward relative to the cap end); additionally, fluid needs to flow in through both the upper chamber of Cylinder 1 and the lower chamber of Cylinder 2 connected by a pipe, whereas fluid needs to flow out from both the lower chamber of Cylinder 1 and the upper chamber of Cylinder 2 connected by another pipe. As the piston does not contain an orifice, the fluid in one chamber cannot flow into another chamber of the same cylinder. The sizes of the two cylinders are designed to be the same; thus, the pressure changes in the two chambers connected by a pipe caused by rocking are the same. Therefore, ideally, no flow should occur in the pipes from one cylinder to another, and the rocking of the superstructure under earthquakes should be suppressed. Compared with the device proposed in Refs. [17,18], the RCD proposed herein does not involve orifices in the pistons; therefore, rocking can be completely restrained by the proposed RCD if the compressibility of the fluid is disregarded because the pistons are stationary, even under low-frequency vibrations caused by earthquakes or wind. Therefore, the proposed RCD can achieve an adequate rocking constraint stiffness over the full range of excitation frequencies.

2.2 Calculation formulas for damping force and rocking constraint stiffness of the rocking constraint device

2.2.1 Damping force calculation

When the RCD is operating, the fluid flow in the tube from a larger area (the cylinder chamber or pipe) to a smaller area (the orifice of the replaceable damping component), or from a smaller area (the orifice of the replaceable damping component) to a larger area (the cylinder chamber or pipe) generates a vertical damping force. Theoretically, the damping force is proportional to the pressure difference across the replaceable damping component and can be expressed as a function of the piston velocity using a linear viscous dashpot model [29].

The dimensional parameters of the RCD are shown in Fig.4. Based on the pressure drop formula of non-Newtonian fluids in a tube [30], the total pressure drop (∆p) generated by the fluid passing through the orifice and pipe can be expressed as follows:

(1)\begin{array}{rl}\mathrm{\Delta }p=& 4k{\left(\frac{2(3m+1)\left(d_{\mathrm{p}\mathrm{t}}^2-d_{\mathrm{p}\mathrm{r}}^2\right)}{m}\right)}^m\\ & \cdot \left[l_{\mathrm{o}}{\left(\frac{1}{N{d}_{\mathrm{o}}^{\frac{3m+1}{m}}}\right)}^m+l_{\mathrm{p}}{\left(\frac{1}{d_{\mathrm{p}}^{\frac{3m+1}{m}}}\right)}^m\right]v^m,\end{array}

where v is the velocity of the piston; k is the consistency coefficient of the fluid; m is the nonlinear index; N represents the number of orifices; l_o and l_p are the length of the orifice and pipe, respectively; d_pt, d_o, d_pr, and d_p are the diameters of the piston, orifices, piston rod, and pipe, respectively.

An RCD is typically equipped with two replaceable damping components and two pipes. Therefore, the vertical damping force of the RCD is calculated as follows:

(2)\begin{array}{rl}F=& 2\text{π}k(d_{\text{pt}}^2-d_{\text{pr}}^2{)}^{m+1}{\left(\frac{2(3m+1)}{m}\right)}^m\\ & \cdot \left[l_{\text{o}}{\left(\frac{1}{N{d_{\text{o}}}^{\frac{3m+1}{m}}}\right)}^m+l_{\mathrm{p}}{\left(\frac{1}{{d_{\mathrm{p}}}^{\frac{3m+1}{m}}}\right)}^m\right]v^m.\end{array}

The damping coefficient c of the RCD is expressed as

(3)\begin{array}{rl}c=& 2\text{π}k(d_{\text{pt}}^2-d_{\text{pr}}^2{)}^{m+1}{\left(\frac{2(3m+1)}{m}\right)}^m\\ & \cdot \left[l_{\text{o}}{\left(\frac{1}{N{d_{\text{o}}}^{\frac{3m+1}{m}}}\right)}^m+l_{\mathrm{p}}{\left(\frac{1}{{d_{\mathrm{p}}}^{\frac{3m+1}{m}}}\right)}^m\right].\end{array}

2.2.2 Rocking constraint stiffness calculation

Ideally, the fluid in the different chambers of the RCD should not flow when the superstructure rocks. However, because the fluid is compressible, the pressure of the fluid caused by the rocking of the structure can significantly deform the fluid and rocking movements of the RCD. Therefore, the rocking constraint stiffness of the RCD caused by the fluid compressibility must be evaluated.

Assuming that the structure rocks under a moment force, as shown in Fig.3(b), the fluid flows under the moment force from the chambers with compressed fluid in the pipe, as schematically shown Fig.5. When no replaceable damping component is installed or when it is installed in the center of the pipe, the fluid flow is symmetrical, and the fluid at the center of the pipe remains. When the replaceable damping component is installed on Cylinder 1 to ease installation and reduce the number of connectors required, the narrow and long orifice of the replaceable damping component decelerates the transfer of high pressure from the chamber of Cylinder 1 to the center of the pipe [31].

Therefore, the compressed displacement of Cylinder 1 with a replaceable damping component installed on its surface (as shown in Fig.5(b)) is smaller than that without a replaceable damping component (as shown in Fig.5(a)). By contrast, the compressed displacement of Cylinder 2 shown in Fig.5(b), which is distant from the replaceable damping component, is larger than that shown in Fig.5(a). The differences in the compressed displacements of the two cylinders result in different vertical stiffness values of the cylinders.

The volume of the fluid (V₁) in the left section of the diagram shown in Fig.5(b), including the volumes of the chamber of Cylinder 1, the replaceable damping component, and a section of the pipe (l_p1), can be expressed as follows:

(4)V_1=A_{\text{ept}}{l}_{1\mathrm{c}}+A_{\mathrm{o}}{l}_{\mathrm{o}}+A_{\mathrm{p}} l_{\mathrm{p}1},

where A_o, A_p, and A_ept are the area of the orifices, the area of the pipes, and the effective area of the piston excluding the area of the orifices, respectively; l_lc is the height of the lower chamber of the cylinder. The volume change in V₁ (∆V₁) owing to the compressibility of the fluid is as follows:

(5)\mathrm{\Delta }{V}_1= A_{\text{ept}}\mathrm{\Delta }{L}_1,

where ΔL₁ is the displacement of the rod end of Cylinder 1 under rocking movement, as shown in Fig.3(b), caused by the compression of the fluid in the RCD. The axial force on the piston rod of Cylinder 1 caused by rocking can be obtained as follows:

(6)F=\mathrm{\Delta } p_1\times A_{\text{ept}},

where ∆p₁ is the change in pressure in the lower chamber of Cylinder 1.

The bulk modulus of the fluid (E) can be calculated as follows:

(7)E=\frac{V_1}{\mathrm{\Delta }{V}_1}\mathrm{\Delta }{p}_1.

Substituting Eq. (7) into Eqs. (4)–(6) yields the following equation:

(8)F=E\frac{A_{\mathrm{e}\mathrm{p}\mathrm{t}}^2\mathrm{\Delta }{L}_1}{A_{\mathrm{e}\mathrm{p}\mathrm{t}}{l}_{\mathrm{l}\mathrm{c}}+A_{\mathrm{o}}{l}_{\mathrm{o}}+A_{\mathrm{p}}{l}_{\mathrm{p}\mathrm{l}}}.

Therefore, the vertical stiffness (F/∆L₁) provided by Cylinder 1 of the RCD to the structure (k₁) can be expressed as

(9)k_1=\frac{E{A}_{\mathrm{e}\mathrm{p}\mathrm{t}}^2}{A_{\mathrm{e}\mathrm{p}\mathrm{t}}{l}_{\mathrm{l}\mathrm{c}}+A_{\mathrm{o}}{l}_{\mathrm{o}}+A_{\mathrm{p}}{l}_{\mathrm{p}\mathrm{l}}}.

Similarly, the vertical stiffness provided by Cylinder 2 (k₂) is

(10)k_2=\frac{E{A}_{\mathrm{e}\mathrm{p}\mathrm{t}}^2}{A_{\mathrm{e}\mathrm{p}\mathrm{t}}{l}_{\mathrm{u}\mathrm{c}}+A_{\mathrm{p}}{l}_{\mathrm{p}2}},

where l_uc is the height of the upper chamber of the cylinder, and l_p2 is the pipe length, as shown in Fig.5(b). When no damping component exists, the vertical stiffness of the two cylinders can be calculated by setting l_p2 in Eq. (10) to 0.5l_p, where l_p is the total length of the pipe.

The rocking constraint stiffness of the RCD (K_R) can be calculated as follows:

(11)K_{\text{R}} =\frac{Fb}{{\displaystyle \frac{\mathrm{\Delta }{L}_1+\mathrm{\Delta }{L}_2}{b}}},

where b is the distance between the two cylinders, and ΔL₂ is the displacement of the rod end of Cylinder 2 under the rocking movement, as shown in Fig.3(b). Considering the vertical stiffness of the cylinders, the rocking constraint stiffness can be expressed as

(12)K_{\text{R}} =\frac{k_1{k}_2}{k_1+k_2} b^2.

Substituting Eqs. (9) and (10) into Eq. (12) yields the following equation:

(13)K_{\text{R}} =\frac{E{A}_{\text{ept}}^2}{A_{\text{ept}}{l}_{\text{uc}}+A_{\text{o}}{l}_{\text{o}}+A_{\text{ept}}{l}_{\text{lc}}+A_{\text{p}}{l}_{\text{p}}}{b}^2.

The average vertical stiffness of the two cylinders (\overline{k}) can be expressed as

(14)\overline{k}=\frac{2F}{\mathrm{\Delta }{L}_1+\mathrm{\Delta }{L}_2},

or

(15)\overline{k}=\frac{2E{A}_{\text{ept}}^2}{A_{\text{ept}}{l}_{\text{uc}}+A_{\text{o}}{l}_{\text{o}}+A_{\text{ept}}{l}_{\text{lc}}+A_{\text{p}}{l}_{\text{p}}}.

The rocking constraint stiffness can thus be expressed as

(16)K_{\text{R}} =\frac{1}{2}\overline{k}{b}^2.

The vertical stiffness of a single cylinder is affected by the installation position of the replaceable damping component. The ideal location for installing the replaceable damping component is at the center of the pipe, at which the vertical stiffness of the two cylinders can be theoretically equal. However, the average vertical stiffness of the two cylinders and the rocking constraint stiffness are independent of install position of the replaceable damping component. When the replaceable damping component is installed on the surface of the cylinder, only three connectors are required to connect the two cylinders, one pipe, and one replaceable damping component, which implies using one less connector as compared with installing it in the pipe. Based on Eq. (15), because the diameter of the orifices is much smaller than that of the pipe, the effect of the orifice size on the rocking constraint stiffness is negligible. The dimensions b and l_p are determined by the structure size, and the height of the chambers is determined by the displacement demand of the vertical isolation. Increasing the effective area of the piston (larger-diameter cylinder) or reducing the area of the pipe (smaller-diameter pipe) can effectively increase the rocking constraint stiffness. However, a pipe with an extremely small diameter significantly increases the damping of the system, which is relatively difficult to adjust. Additionally, changing the size of the piston can affect the damping force; thus, the damping and stiffness should be designed simultaneously.

3 Experimental evaluation of damping performance of the rocking constraint device

3.1 Design of test prototype

A scaled RCD was designed and fabricated to reflect the dimensional properties listed in Tab.1. In practice, the damping force capacity and rocking constraint stiffness can be adjusted by modifying the sizes of the cylinders, damping components, and pipes. The damping and rocking constraint performances were evaluated separately using different test setups.

Each cylinder of the RCD was evaluated separately in the damping performance test. The two chambers of a single cylinder were connected using a pipe and replaceable damping component, as shown in Fig.6. The fluid flow route in the test configuration was similar to that of an individual cylinder in the RCD. The length of the pipe was the same as that of the pipe used in the rocking constraint performance tests. Two replaceable damping components with one and two orifices were evaluated.

Dimethyl silicone oil, with a viscosity of 1000 cSt, was used as the fluid in the device. Detailed properties, such as the bulk modulus of the fluid, were affected by the detailed operations in the device assembly, which might introduce air bubbles. Based on experience, the actual values of the properties of a fluid may vary significantly from the test values under ideal conditions [32]. Therefore, the values for the properties of the fluid were determined from the test data.

3.2 Experimental setup and loading program

Fig.7 shows the test setup used for conducting sinusoidal excitation tests to verify the damping performance of a single cylinder based on the RCD prototype. During the test, a cylinder was installed in the horizontal direction; it was pinned to a reaction frame at the cap end and at the loading head at the rod end using spherical bearings. The loading head was installed on two horizontal guide rails fixed to the ground beam to ensure the loading direction. When the end of the piston rod shifts back and forth, the piston rod may shift laterally because the piston rod and loading direction are not set ideally in the same direction. To limit the lateral movement of the piston rod, two segments of short rubber tubes were placed on the pins of the spherical bearings. A laser displacement sensor and a load cell were used to measure the displacement of the piston rod and the load applied to it. Sinusoidal motions were applied to the piston rod of the cylinder using an MTS-series 244 hydraulic actuator. Owing to the limited flow-rate capacity of the actuator, loading frequencies greater than 3 Hz were not considered. Tab.2 lists the loading programs for different frequencies and displacement amplitudes.

3.3 Experimental results and analysis

Fig.8 shows the displacement hysteresis curves for different loading frequencies and displacements using the replaceable damping component with one orifice. For low-velocity loadings with low frequencies or amplitudes (see Fig.8(a) and Fig.8(b)), the hysteresis curve was approximately rectangular. The friction force between the seal on the piston and the inner wall of the cylinder, which was approximately 600 N, constituted a significant proportion of the total reaction force. When the loading velocity increased, the hysteresis curve shape gradually changed from rectangular to elliptical, which indicates that the damping force was dominated by viscous damping. As the fluid is compressible, the change in fluid volume may be accompanied by the development of a spring-like restoring force. Generally, the effect of the restoring force in the test is negligible, and the force–displacement relationships of the cylinder can be described using a viscous dashpot model, as described in Subsubsection 2.2.1.

Fig.9 shows the relationship between the peak damping forces and peak velocities. At a low loading velocity, the force was dominated by friction in the cylinder. As the velocity increased, the peak damping forces showed a linear relationship with the peak velocities of the cylinder. Thus, the nonlinear index m in Eq. (3) was determined to be 1. Linear curve fitting was performed to calculate the damping coefficient c for Case A with two orifices in the replaceable damping component using the test data, and the calculated value was used to identify the consistency coefficient k of the fluid using Eq. (3). Subsequently, the identified consistency coefficient k and nonlinear index m were used to calculate the theoretical coefficient c for Case B with one orifice in the replaceable damping component based on Eq. (3). As shown in Tab.3, the calculated coefficient c is 12.9% lower than the test value, which indicates that the damping calculation equation of Eq. (3) is sufficiently accurate for a cylinder with a replaceable damping component.

4 Experimental evaluation of rocking constraint performance of the rocking constraint device

4.1 Experimental setup and loading program

To evaluate the rocking constraint stiffness of the RCD, the two cylinders of the device should be loaded antisymmetrically, as shown in Fig.3(b), which requires two actuators to achieve a synchronous loading. A new testing configuration was proposed, as shown in Fig.10, by adjusting the direction of one of the cylinders and the pipe connections. Two upper and two lower chambers were connected to each other. When the two cylinders were subjected to a compression force as shown in the figure, both the upper chambers of the two cylinders tended to expand their volumes, and both the lower chambers of the two cylinders tended to compress the fluid. The movement of the piston rods depends on the compressive displacement of the fluid in the cylinders and pipes. Because the volumes of the upper and lower chambers are identical, the compressive displacement owing to the test configuration, as shown in Fig.10, is expected to be identical to that shown in Fig.3(b) when rocking occurs. Using the new configuration, only one actuator is required to rock the RCD and to evaluate the stiffness of the rocking constraint. As the two cylinders may not exhibit the same performance owing to manufacturing errors, they may not move synchronously. Connecting the two cylinders directly to the actuator through a rigid beam results in unfavorable internal forces on the cylinders. Therefore, two flexible springs were installed at the ends of the two cylinders to accommodate the slight difference in their displacement. The onsite loading setup is shown in Fig.11.

Two test cases were devised, i.e., one involving the RCD installed with replaceable damping components, and the other without. The dimensions of the cylinders and replaceable damping components were the same as those listed in Tab.1. For each cylinder, one displacement sensor was used to monitor the deformation of the spring, and another was used to measure the displacement of the rod end relative to the cylinder. Two load cells were installed between the springs and loading beam, as shown in Fig.11. The recorded force was regarded as the resistance force of the piston rod because the contribution of the inertial force of the spring and that of the connection plate between the spring and load cell to the recorded force were limited owing to the light weights of the components. A displacement-controlled loading mode was adopted in the test; additionally, different frequencies and displacement amplitudes were specified for the sinusoidal motion to examine the performance of the RCD under different conditions, as shown in Tab.4. A maximum force of approximately 10 kN was successfully loaded under a maximum displacement of 100 mm based on the designed stiffness of the springs, i.e., 9.91 kN/m.

4.2 Experimental results and analysis

4.2.1 Effect of replaceable damping component on vertical stiffness of cylinder

During the test, the RCD performed stably under different loading conditions. No fluid leakage occurred during the test, even during cases involving maximum loading forces of approximately 10 kN (loading displacement of 100 mm). The pipes vibrated slightly during the test because the pressure changes in the pipes caused the fluid to deform and flow, which indicates that the compressibility of the fluid is not negligible when calculating the rocking constraint stiffness.

Fig.12 shows the partial displacement time histories of the spring, as well as the displacement and force time histories of the piston rod end for Cylinder 1 (see Fig.10) of the specimen with replaceable damping components. Based on the figure, the motion of the piston rod end is primarily affected by the loading displacement amplitude, which is correlated with the loading force. When the amplitude of the loading displacement is small, the amplitude of the applied force is at the same level as that of the friction force; hence, the displacement of the piston rod is negligible, as shown in Fig.12(d) and Fig.12(e). When the amplitude of the loading displacement is large, the amplitude of the applied force is much larger than the friction force and the piston rod shifts owing to the compressed deformation of the fluid in the pipe, as shown in Fig.12(a)–Fig.12(c). Based on the test, the maximum displacement of the piston rod end was approximately 2.5 mm when the loading amplitude was 100 mm.

Fig.13(a) shows a comparison of the force–displacement relationships for Cylinders 1 and 2 (see Fig.10) with replaceable damping components installed under loading at a frequency of 0.5 Hz and a displacement amplitude of 60 mm. Fig.13(b) shows a comparison of the force–displacement relationships for Cylinder 1 with and without replaceable damping components installed under the same dynamic load. As shown in Fig.13(a), the displacement of the piston rod end for Cylinder 1 is smaller than that for Cylinder 2, which indicates that the stiffness of Cylinder 1 is greater. As discussed in Subsection 2.2, this is caused by the installation of the damping components, which changed the fluid flow significantly (see Fig.5). The narrow and long orifice of the replaceable damping component decelerated the transfer of high pressure from the chamber of Cylinder 1 to the center of the pipe, as shown in Fig.5(b). By contrast, because the pressure in the pipe near the right end of the damping component was relatively low, the compressed fluid in the right section of the pipe could flow beyond the center of the pipe. The difference in the compressed fluid volume in the left and right sections, shown in Fig.5(b), resulted in different magnitudes of vertical stiffness in the cylinders. In summary, the cylinder with replaceable damping components installed in close proximity to each other can provide a higher vertical stiffness than the cylinder with the other components. This finding is generally valid, even under different frequency excitations.

Based on Fig.13(b), the displacement of Cylinder 1 without the replaceable damping component indicated a larger displacement than Cylinder 1 with the replaceable damping component, the reason of which is similar to that presented above. However, under extremely low-frequency (e.g., 0.05 Hz) excitations, the effect of the replaceable damping component on the transfer of pressure was weak, and the displacement of the cylinders without the replaceable damping component might be larger than that of the cylinders with the replaceable damping component. This is attributable to the length of the replaceable damping component and the possible slight differences between the two specimens, such as the amount of air bubbles mixed in the fluid.

4.2.2 Effect of loading frequencies on vertical stiffness of cylinder

Fig.14(a) shows the force–displacement relationship of Cylinder 1 with replaceable damping components under sinusoidal motions of different frequencies and the same displacement amplitude of 80 mm. Based on the figure, although the applied maximum force (under the same displacement amplitude of 80 mm) was identical under different frequencies, the displacement of the cylinder differed for the specimen with replaceable damping components. When the frequency increased, the displacement decreased. This is primarily caused by the short displacement accumulation time in the loading cycle of the high-frequency loading, as well as because the high pressure could not propagate far through the narrow and long orifices. Consequently, the compressed fluid volume of the cylinder with the replaceable damping component installed in close proximity to each other decreased at higher frequency loadings, and the displacement of the piston rod end decreased. By contrast, the displacement of Cylinder 2 with replaceable damping components installed far from it under high-frequency loading was greater than that under low-frequency loading.

Fig.14(b) shows the average force–displacement relationships based on Eq. (14) for the three loadings shown in Fig.14(a). The variation in the displacements based on the three different loading frequencies was less significant than that of Cylinder 1. Therefore, although the vertical stiffness of a single cylinder was dependent on the loading frequency when replaceable damping components were installed, the average vertical stiffness of the two cylinders, which determines the rocking constraint stiffness, was independent of it. Notably, the variation in the displacements of Cylinder 1 with the three different loading frequencies for the specimen without the replaceable damping component was insignificant, as shown in Fig.14(c), which indicates that the effect of the loading frequency on the cylinder stiffness was due to the installation of the damping component.

Meanwhile, the average maximum displacement of the two cylinders under 0.05 Hz of loading was similar to that under 0.5 Hz of loading. Therefore, the proposed RCD is expected to be operable under different frequency conditions, including long-period motions and pulse-like motions involving low-frequency components [6,33].

4.2.3 Rocking constraint stiffness of rocking constraint device

Fig.15 shows the relationship between the amplitude of the piston rod displacements and forces. Linear curve fitting was performed to calculate the vertical stiffness of each cylinder. Based on Fig.15(a), the test data values of Cylinder 1 were scattered widely when the replaceable damping components were installed because of the frequency-dependence issue discussed in Subsubsection 4.2.2. The curve-fitted vertical stiffness of Cylinder 1 was 3507 kN/m, which is 25% higher than that of Cylinder 2 (2806 kN/m), as shown in Fig.15(b). The installation of a replaceable damping component can considerably affect the individual vertical stiffness of the cylinder. The average vertical stiffness of the two cylinders calculated using the test data based on Eq. (14), as shown in Fig.15(c), was 3321 kN/m, which is between the values of the two cylinders. In addition, Fig.15(c) shows that the correlation between the force and displacement is stronger for the average vertical stiffness of the two cylinders than that for an individual cylinder. When no replaceable damping component was installed, the average vertical stiffness of the two cylinders was 3793 kN/m, which is 14% higher than the average value of the two cylinders with installed replaceable damping components. Considering that the correlation between the force and displacement is strong for the case without a replaceable damping component, as shown in Fig.15(d), the average vertical stiffness of 3793 kN/m was used to calibrate the bulk modulus of the fluid E. Using the obtained value, the theoretical values of the vertical stiffness of the cylinder and the rocking constraint stiffness of the RCD were calculated based on the test data and theory provided in Subsubsection 2.2.2. A comparison between the test and theoretically calculated results is presented in Tab.5.

Because Case A was used to calibrate the bulk modulus, the prediction error percentage of the rocking constraint stiffness, which is calculated as the difference between the test and theoretical values relative to the test value, was 0. The calculated rocking constraint stiffness for Case B with replaceable damping components installed was 11527 kN·m/rad, based on a distance of 2.5 m between the two cylinders (b in Eq. (16)), which included the lengths of the pipe, damping components, connection thread, pipe connector, and outside diameter of the cylinder; in this case, the error percentage was 11.0%. The difference was primarily caused by the disturbance to the fluid flow under different frequency loadings due to the installation of the replaceable damping components. By comparing the test data and the theoretically calculated vertical stiffness shown in Eqs. (9) and (10), the pipe length values of l_p1 and l_p2 (see Fig.5) were determined to be 41% and 59% of l_p, respectively. The test values of the vertical stiffness of the two cylinders were 18% lower than their theoretical values. A greater difference indicates that the frequency-dependence issue significantly affects the individual cylinder when the replaceable damping component is installed asymmetrically.

Nonetheless, because the rocking constraint stiffness is determined by the average vertical stiffness, the proposed formula for calculating the rocking constraint stiffness presented in Subsubsection 2.2.2 is accurate, and the RCD can operate as intended to provide adequate rocking constraint stiffness. In practical applications, the RCD may be larger because the required rocking constraint stiffness may be higher depending on the specific building information. Because the damping component is replaceable, it can be readily designed based on the procedures described in Section 2. Additionally, multiple RCDs can be designed to increase the total capacity of the devices. Notably, an appropriate design of the RCD should account for the performance of the superstructure and 3D isolators under a considered earthquake level [22,34].

The calibrated bulk modulus E (390.5 MPa) was relatively low, primarily because the pressure in the pipe was low (up to 1.7 MPa), and air bubbles were mixed in the fluid during manual fluid filling operations [35].

In a linear isolation system, the structure is expected to return to its original position after an earthquake, including in the rocking directions, owing to the high rocking constraint stiffness and low friction of the RCDs.

5 Conclusions

To improve the rocking behavior of isolated 3D structures, an RCD comprising a pair of cylinders connected by pipes and replaceable damping components installed outside the cylinders was proposed. Its operating principle was explained, and calculation formulas for the damping force and rocking constraint stiffness were proposed. Sinusoidal excitation tests were performed to verify the mechanical performance of the device and the effectiveness of the proposed calculation formulas. The following conclusions were inferred.

1) The rocking constraint capacity of the proposed RCD was realized by connecting the upper chamber of one cylinder to the lower chamber of another cylinder. Replaceable damping components were designed to allow damping to be adjusted easily. As the piston can fully separate the fluid in the two chambers of one cylinder owing to the no-orifice design, the fluid cannot flow from one chamber to another of the same cylinder; hence, rocking is prevented even under low-frequency excitations. The theoretical formula for calculating the rocking constraint stiffness shows that the latter is independent of the input frequency and is determined by the average vertical stiffness of the two cylinders. Increasing the effective area of the piston and reducing the area of the pipe can effectively increase the stiffness of the rocking constraint.

2) Sinusoidal loading test results for each cylinder at different loading frequencies and amplitudes showed that the cylinders were able to provide damping, as intended. The force–displacement relationship of the cylinder can be described using a viscous dashpot model, and the error between the damping coefficient predicted using the theoretical formula and that obtained experimentally did not exceed 12.9%.

3) Sinusoidal loading test results for the RCD at different loading frequencies and amplitudes indicated that the asymmetric installation of the replaceable damping components led to the different vertical stiffness behaviors of the two cylinders. The cylinder with replaceable damping components installed on its surface showed a greater vertical stiffness than the other cylinders. In addition, the vertical stiffness of an individual cylinder of the RCD with replaceable damping components varied with the loading frequency. A higher frequency generally results in a greater vertical stiffness. However, the effect of the excitation frequency on the average vertical stiffness of the two cylinders, which determines the rocking constraint stiffness, was limited. The test results showed that the RCD can provide the designed rocking constraint stiffness, and that the error between the rocking constraint stiffness calculated using the test data and the proposed formula did not exceed 11%.

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