Performance of bi-directional elliptical rolling rods for base isolation of buildings under near-fault earthquakes

Abstract

Rolling base isolation system provides effective isolation to the structures from seismic base excitations by virtue of its low frictional resistance. Herein, dynamic analysis of flexible-shear type multi-storey building mounted on orthogonally placed elliptical rolling rod base isolation systems subjected to bi-directional components of near-fault earthquake ground motions is presented. The orthogonally placed rods would make it possible to resist the earthquake forces induced in the structure in both the horizontal directions. The curved surface of these elliptical rods has a self-restoring capability due to which the magnitude of peak isolator displacement and residual displacement is reduced. The roughness of the tempered curved surface of the rollers dissipates energy in motion due to frictional damping. The seismic performance of the multi-storey building mounted on the elliptical rolling rod base isolation system is compared with that mounted on the sliding pure-friction and cylindrical rolling rod systems. Parametric studies are conducted to examine the behavior of the building for different superstructure flexibilities, eccentricities of the elliptical rod, and coefficients of friction. It is concluded that the elliptical rolling rod base isolation system is effective in mitigation of damaging effects of the near-fault earthquake ground motions in the multi-storey buildings. Even under the near-fault earthquake ground motions, the base-isolated building mounted on the elliptical rolling rods shows considerable reduction in seismic response. The isolator displacement with the elliptical rolling rod base isolation system is less in comparison to the pure-friction and cylindrical rolling rod systems.

Keywords

base isolation bi-directional building elliptical rolling rods near-fault

Introduction

The technology of seismic isolation has been used from four decades to protect engineering structures from seismic hazard. Presently, there are thousands of base-isolated structures like buildings, bridges, liquid storage tanks, and so on constructed across the world and has resisted the real-time earthquakes and hence proven the effectiveness of the seismic base isolation systems. The base isolation systems introduced in structures reduce the economic losses caused due to damage of valuable structures during seismic hazards apart from saving human lives. The main objective of isolation system is to completely decouple the structure from earthquake base excitations. For the purpose of seismic isolation, elastomeric and sliding bearings are commonly used, whereas rolling isolation systems are relatively new. In all types of sliding systems, the performance of the system depends on the value of coefficient of friction, velocity at the friction interface, and the intensity of earthquake.

Several attempts have been made by the researchers to use low value of coefficient of friction to make the isolation systems effective under all intensities of earthquake, however consequently with unduly larger isolation level displacements. Alternatively, a third type of isolation system was introduced called rolling system, based on the concept of rolling friction. When one body rolls over the surface of another body, the resistance mobilized at the interface to the motion is called rolling friction. The rolling friction is typically lower than the sliding friction. Rolling base isolation system provides effective isolation to the structures from seismic base excitations by virtue of its low frictional resistance. These isolation systems can completely decouple the structures from the seismic forces and hence reduce the seismic demand on structures. Bridge bearings and many mechanical bearings used in industries are based on this concept where balls are used as rolling device.

Lin and Hone (1993) proposed two sets of orthogonally placed circular free rolling rods under the basement and found them effective in reducing the response of the structure, but it resulted in an uncontrolled residual displacement of base mass. The use of various jacks can restore the structure back to its original position after earthquake excitation. Lin et al. (1995) further carried out experimental and analytical works on single-degree-of-freedom (SDOF) system using rolling rod system and found it effective with a re-centering force device help in reducing the residual isolator displacement. Jangid (1995) used orthogonally placed free circular rolling rods without any restoring force and found them effective in reducing the response of the structure as compared to the fixed-base structure. These rolling rods were considered without any restoring force, resulted into more peak and residual isolator displacements. Muhr et al. (1997) used isolation system consisting of steel balls rolling on rubber tracks. The isolation system was effective in reducing the seismic response of the structures and further the experimental studies were also carried out by Guerreiro et al. (2007) later on to assess the rolling-ball isolation system.

Londhe and Jangid (1998, 1999) used elliptical rolling rods (ERRs) for a multi-storey building which overcame large peak and residual isolator displacements caused due to the circular shape of the rods and found them effective in reducing seismic responses. Zhou et al. (1998) combined the advantage of free rolling rod isolators and friction pendulum system (FPS), in which the steel balls were placed in between the two concave surfaces, at the superstructure and the foundation interface. The numerical integration and harmonic balance methods were used for obtaining analytical solution in evaluating the dynamic response. Shake table test was also conducted on a three-storey masonry model for studying the effective performance of the ball system with restoring property. It was concluded that seismic response of the structure is reduced considerably as compared to the fixed-base structure. Jangid (2000) further carried out a stochastic study on the circular rolling rods placed orthogonally along with re-centering device in the form of a spring or cantilever beam, for a flexible-shear type multi-storied building subjected to non-stationary earthquake excitations. It was concluded that the rolling rods with re-centering device are quite effective in reducing seismic response of the structure. Kaplan and Seireg (2001) used steel balls in concave surfaces with a spring-cam system. It was found that response of the structure was reduced by providing the cams, which gets compressed the spring under normal conditions and released when an earthquake excitation was sensed. Calio et al. (2003) studied frequency domain behavior of the FPS with ball system having restoring property for the use in multi-storey buildings. Lee and Liang (2003) studied sloped rolling-type bearing (RTB), which utilizes the concept of a steel cylinder rolling on a V-shaped surface. Butterworth (2006) proposed a rolling isolation system consisting of rollers with non-concentric spherical upper and lower surfaces.

Tsai et al. (2006, 2007) have also used the RTB isolators in seismic isolation tests of a bridge model on shaking table, to investigate the seismic behavior of a scaled bridge model isolated by the RTBs. The test results showed the effective performance of the sloped RTB for seismic isolation of bridges. Chung et al. (2009) proposed an eccentric nonlinear rolling isolator which overcame the drawbacks of the resonance and large displacement response of the linear isolator having fixed vibration frequency when located near a fault. Kesti et al. (2010) carried out experimental work on ball-n-cone (BNC) bearing, consisting of a ball placed in between the two conical plates resulting in isolation of the structure. Ismail et al. (2010) developed a roll-n-cage (RNC) isolator; the performance analysis of this isolator was verified for wide range of structures and ground motion excitations. Ou et al. (2010) proposed roller seismic isolation bearings for highway bridges. Yang et al. (2012) proposed eccentric rolling isolation system (ERIS) considering friction damping for energy dissipation. The performance of the ERIS was found effective in reducing the response as compared to the linear isolation system. Hosseini and Soroor (2011, 2013) proposed a system called orthogonal pairs of rollers on concave beds (OPRCB) for low and medium rise buildings. Harvey and Gavin (2013) proposed the model of a rolling isolation platform generally used for protecting the objects from the hazards of horizontal excitations. Harvey et al. (2014) and Harvey and Gavin (2015) studied the performances of lightly and heavily damped rolling isolation systems (RISs) located within earthquake-excited structures analytically and experimentally. Yet, hardly any works reported in the literature wherein the seismic response of the base-isolated buildings with the rolling systems under near-fault ground motions are studied when subjected to bi-directional components of earthquake ground motions. Wei et al. (2016a, 2016b, 2017) studied the isolation performance of a spring-rolling isolation system by considering that the rolling friction force gradually and linearly increases with the relative displacement between the isolator and the ground.

Near-fault earthquake ground motions such as Parkfield 1966, San Fernando 1971, Landers 1992, Northridge 1994, Kobe 1995, and Chi-Chi 1999 have caused huge damages to various civil structures. This has motivated the seismological and engineering communities to study the effect of the near-fault earthquakes on civil structures. The strong earthquake ground motions in the immediate vicinity of a fault are frequently characterized by an intense velocity pulse. Also, these pulses transfer large amount of energy in the structures in its near vicinity. The forward directivity and/or permanent tectonic translation are the two primary characteristics of the near-fault ground motions. These characteristics make the near-fault earthquakes distinctive as compared to the far-fault ground motions. From the previous records of the near-fault ground motions, it was demonstrated that the kinematic characteristics contain large displacement pulses, say about one or two pulses from 0.5 m to more than 1.5 m with peak velocity of 0.5 m/s or greater. Their time period is usually in the range of 1–3 s; however, it can be as long as 6 s and such variation can be observed in acceleration history as well. Such different pulses do not exist in the seismic ground motions recorded at locations away from the near-fault region. When a near-fault ground motion time history is used for the seismic analysis of a structure at a particular site, the strike normal and parallel components need to be oriented with respect to the strike of the fault that dominates the seismic hazard of that site. Long-period flexible structures such as the base-isolated structures are required to be studied under the near-fault earthquakes to assess the effectiveness of the passive control system.

There have been several studies conducted for understanding the seismic behavior of both base-isolated and fixed-base structures under near-fault motion (Hall et al., 1995; Jangid and Kelly, 2001; Makris, 1997; Malhotra, 1999; Rao and Jangid, 2001). Matsagar and Jangid (2003) studied the seismic response of base-isolated structures during impact under near-fault earthquakes. Sahasrabudhe and Nagarajaiah (2005) experimentally investigated base-isolated building with sliding bearing along with magneto-rheological dampers subjected to near-fault earthquakes. It was concluded that additional dampers help in reducing the isolator displacement; however, it increases the base shear and storey drift in the buildings. Jangid (2005) investigated the optimum parameters of FPS used in multi-storey buildings and bridges under near-fault ground motions. It was observed that for lower values of the friction coefficient, considerable sliding displacement occurs in the FPS under the near-fault motions. Jangid (2007) investigated the optimum parameters of the lead rubber bearing (LRB) used in buildings and bridges subjected to near-fault ground motions. Panchal and Jangid (2008) studied the effect of near-fault earthquakes on variable friction pendulum system (VFPS), which resulted in reduction in the base shear in the buildings without much variation in the top floor absolute acceleration response. Providakis (2009) carried out a numerical simulation for base-isolated building using the LRB and FPS in addition to the supplemented viscous dampers, in order to reduce the isolator displacement under near-fault ground motions. Saha et al. (2015) reviewed the dynamic analysis of base-isolated buildings, bridges, and cylindrical liquid storage tanks under near-fault earthquakes and concluded on retention of the effectiveness of base isolation. Fathi et al. (2015) studied the base-isolated frames along with the supplemental damping (viscous dampers) subjected to the near-fault and far-fault ground motions. It was observed that the isolator displacement decreases; however, the relative displacement and acceleration in the storeys increased for near-fault motions. Limitation on excessive isolator displacement in base-isolated building was given due to consideration by Das et al. (2015) in optimizing the isolation systems. Lately, Zelleke et al. (2015) investigated the effectiveness of the supplemental dampers in the base-isolated buildings to control large isolator displacements under earthquake excitations and observed usefulness of the hybrid control scheme for earthquake protection. Chakraborty et al. (2016) proposed in their flat sliding base isolation system effective mechanism for re-centering while limiting the peak bearing displacements. However, performance of the rolling isolation systems used in structures subjected to bi-directional components of the near-fault earthquakes requires an in-depth investigation as it has not been adequately studied thus far.

The specific objectives of the present investigation are as follows: (1) to study the performance of the flexible-shear type multi-storey building mounted on the orthogonally placed ERR base isolation systems subjected to bi-directional (normal and parallel components of) near-fault earthquake ground motions, (2) to compare the seismic response of the ERR base-isolated building with the pure-friction sliding isolation system and bi-directional cylindrical rolling rods in order to measure the effectiveness of the ERR, and (3) to investigate the influence of important parameters on the response of the buildings isolated using the ERR through a parametric study. The parameters considered in the study are the coefficient of rolling friction, eccentricity of ERR, and fundamental time period of the fixed-base building, which essentially govern the seismic response of structures.

Modeling of base-isolated building mounted on ERR

Figure 1 shows a N-storey flexible-shear type building mounted on the orthogonally placed ERR base isolation system subjected to bi-directional components (fault-normal and fault-parallel) of near-fault earthquake ground motions. The ERR base isolation system is installed between the base slab of the building (also referred to as base mass) and the foundation support of the building. A conceptual three-dimensional (3D) drawing indicating the axis of rotation for the top and bottom rollers is shown in Figure 2. The rollers have grooves to protect against sliding longitudinally. The grooves have a small size offset with same eccentricity. The major radius, coefficient of rolling friction, and eccentricity of the ERR are the three design parameters required to provide the desired isolation for the building from the near-fault earthquake ground motions. The coefficient of rolling friction is typically kept low in order to transmit minimum earthquake force into the superstructure of the building. Generally, the coefficient of rolling friction is 1/10th that of the coefficient of sliding friction for the same interacting material. The eccentricity of the ERR that provides a nonlinear restoring force defines its geometry indicating the proportion of the major and minor radii. The following assumptions are made in the dynamic analysis of the ERR base-isolated building:

The floor diaphragms of each storey of the superstructure are rigid.

Constituent material of the eccentric rollers in the ERR base isolation system is rigid, that is, rods themselves are undeformable.

Law of dry coulomb friction is valid and the coefficient of rolling friction between the ERR base mass and ERR-foundation mass is constant, that is, non-velocity dependent.

The superstructure is linear-elastic, which is typically achieved by the base isolation.

The superstructure is stable and it does not overturn during earthquake-induced shaking.

Accidental eccentricity due to mismatch in center of rigidity and center of mass at each storey level is considered zero; thus, torsional effects are neglected.

Vertical component of earthquake ground motion is ignored.

Figure 1.

Mathematical model of flexible-shear type multi-storey building mounted on elliptical rolling rods (ERRs).

Figure 2.

Conceptual 3D views of the ERR with grooves.

Kinematics of an elliptical roller in the ERR base isolation system

Figure 3(a) and (b) shows state of equilibrium and non-equilibrium position of a single roller in an ERR base isolation system assembled below the base mass as shown in Figure 1. The sum of inertial, damping, and stiffness properties of the multiple rollers along one direction is represented by an equivalent roller in that direction. Hence, in a simplified form, the ERR base isolation system can be represented by two uncoupled rollers: (1) one roller rotating about the y-axis and (2) second roller rotating on top of the first roller about the x-axis. The bottom roller, at its center of mass, has 3 degrees of freedom (horizontal, vertical, and rotation) in x-z plane as illustrated in Figure 3(a), and similarly, the top roller has 3 degrees of freedom in the y-z plane. Figure 3 shows the elliptical geometry of an equivalent roller in the ERR base isolation system with a major radius, a and a minor radius, b resting along the length of its major axis. This geometrical configuration is maintained the same for both the orthogonally placed top and bottom rollers in the ERR base isolation system. The eccentricity of the rods, e, is considered equal for both the top and bottom rollers and is given as

e = \sqrt{\frac{a^{2} - b^{2}}{a^{2}}}

(1)

Figure 3.

Relationship between displacements of the base mass and ERR in x-z plane.

The eccentricity of the rolling rods develops an additional restoring force which brings the building back to its original position after an earthquake event. As a special case, zero eccentricity (e = 0) implies an isolation system with cylindrical rolling rod. Herein, the kinematic relationships for bottom roller rotating about y-axis are discussed. The kinematic relationships of the top roller can be derived in a similar way. The relation between the displacement of the base mass and the ERR expressed in equations (2) to (4) in the x-z plane are evaluated from Figure 3(a). If $u_{rx}$ is the horizontal displacement of the center of mass of the roller relative to ground in x-direction; $u_{rzx}$ is the vertical displacement relative to the ground in z-direction; $θ_{rx}$ is the rotation of the rolling rod about the y-axis; and $u_{bx}$ is the horizontal displacement of the base mass relative to the ground in x-direction then

u_{rx} = \frac{u_{bx}}{2}

(2)

u_{rzx} = p_{x} - b

(3)

θ_{rx} = \tan^{- 1} (\frac{b}{a} \tan θ_{x})

(4)

Furthermore, if p_x is half the vertical distance between the lower and upper pole points (A and B) as shown in Figure 3(b) of the system in the x-z plane, then

p_{x} = a \sin θ_{x} \sin θ_{rx} + b \cos θ_{x} \cos θ_{rx}

(5)

Here, $θ_{x}$ is the eccentric angle such that the coordinates of the point A are $(- a \sin θ_{x}, b \cos θ_{x})$ ; hence, the distance between center of the roller and point of contact is expressed as

OA = \sqrt{a^{2} \sin^{2} θ_{x} + b^{2} \cos^{2} θ_{x}} = a \sqrt{1 - e^{2} \cos^{2} θ_{x}}

(6)

By equating the distances $AA'$ and $AA ″$ in Figure 3(b), a relation between the angle $θ_{x}$ and the isolator displacement $u_{bx}$ can be expressed as

\frac{u_{bx}}{2} + c_{x} = \int_{0}^{θ_{x}} a \sqrt{1 - e^{2} \cos^{2} φ} d φ

(7)

where $c_{x} = a \sin θ_{x} \cos θ_{rx} - b \cos θ_{x} \sin θ_{rx}$ is half the horizontal distance between the upper and lower pole. The right-hand side of equation (7) contains a special function called as elliptical integral of second kind. For the given values of the e and $θ_{x}$ , $u_{bx}$ is determined using numerical Gauss quadrature method.

Dynamical equations of motion for the ERR

Consider the equilibrium of forces acting on a roller rotating about y-axis in a state of displaced position as shown in Figure 3(b). There are three forces acting tangentially at the two points of contact in the roller: (1) isolator inertial force, (2) isolator restoring force, and (3) isolator rolling frictional force. These three forces act for both the top and bottom rollers but in their respective orthogonal directions. However, in this study, the case is illustrated with Figure 3(b) for the bottom roller rotating about the y-axis. The inertial force of the isolator, $F_{bx}$ , acting tangentially along x-direction at the two points of contact of the bottom roller and for rotation angle $- 90 \circ \leq θ \leq 90 \circ$ (as shown in Figure 4) can be expressed as

F_{bx} = \frac{1}{2 p_{x}} J_{r} {\overset{\cdot\cdot}{θ}}_{rx} + m_{r} {\overset{\cdot\cdot}{u}}_{rzx} \frac{c_{x}}{p_{x}} + \frac{1}{2} m_{r} ({\overset{\cdot\cdot}{u}}_{rx} + {\overset{\cdot\cdot}{u}}_{gx})

(8)

Figure 4.

Free-body diagram of the base mass and rolling rods along the x-direction: (a) base mass and (b) cross-section of rolling rods.

Likewise, the isolator force, $F_{by}$ , of the top roller rotating about x-axis and acting tangentially along y-direction can be expressed as

F_{by} = \frac{1}{2 p_{y}} J_{r} {\overset{\cdot\cdot}{θ}}_{ry} + m_{r} {\overset{\cdot\cdot}{u}}_{rzy} \frac{c_{y}}{p_{y}} + \frac{1}{2} m_{r} ({\overset{\cdot\cdot}{u}}_{ry} + {\overset{\cdot\cdot}{u}}_{gy})

(9)

Here, J_r and m_r are the sum of mass moment of inertia and mass of the bottom rolling rods, respectively. Furthermore, ${\overset{\cdot\cdot}{u}}_{rzx}$ and ${\overset{\cdot\cdot}{u}}_{rzy}$ are the vertical acceleration of the base mass due to the motion of rolling rods along the x- and y-directions, respectively.

The restoring force, $F_{sx 1}$ , of the bottom roller, by virtue of its elliptical shape, acting tangentially along x-direction is given by

F_{sx 1} = (\sum_{i = 1}^{N} m_{i} + m_{b}) (g + {\overset{\cdot\cdot}{u}}_{bzx}) \frac{c_{x}}{p_{x}}

(10)

Likewise, the isolator restoring force, $F_{sy 1}$ , of the top roller rotating about x-axis acting tangentially along y-direction can be expressed as

F_{sy 1} = (\sum_{i = 1}^{N} m_{i} + m_{b}) (g + {\overset{\cdot\cdot}{u}}_{bz}) \frac{c_{y}}{p_{y}}

(11)

where $(\sum_{i = 1}^{N} m_{i} + m_{b})$ is the total mass of the building (including the base mass); ${\overset{\cdot\cdot}{u}}_{bz} = {\overset{\cdot\cdot}{u}}_{bzx} + {\overset{\cdot\cdot}{u}}_{bzy}$ is the absolute vertical acceleration of the ERR base isolation system; $c_{x}$ and $c_{y}$ are half the horizontal distance between the upper and lower pole for bottom and top roller, respectively; $p_{x}$ and $p_{y}$ are half the vertical distance between the lower and upper pole points for bottom and top roller, respectively; and g is the acceleration due to gravity (9.81 m/s²).

The rolling frictional force is directly related to the normal force acting at the contact points of the roller. This normal force is the sum of gravity force of the building mass and the vertical force due to the vertical acceleration of the building. The frictional force, thus mobilized as a result of dry coulomb law, acts tangentially at the two points of contact in the roller and opposite to the direction of motion. The isolator rolling frictional force for the bottom roller rotating about the y-axis along the x-direction is expressed as

F_{sx 2} = (\sum_{i = 1}^{N} m_{i} + m_{b}) (g + {\overset{\cdot\cdot}{u}}_{bz}) μ_{R} sgn ({\overset{\cdot}{u}}_{bx})

(12)

Likewise, the isolator rolling frictional force, $F_{sy 2}$ , of the top roller rotating about x-axis acting tangentially along y-direction can be expressed as

F_{sy 2} = (\sum_{i = 1}^{N} m_{i} + m_{b}) (g + {\overset{\cdot\cdot}{u}}_{bz}) μ_{R} sgn ({\overset{\cdot}{u}}_{by})

(13)

where μ_R is the coefficient of rolling friction between the base mass and the rolling rods, sgn(·) denotes the sign function, that is, $sgn ({\overset{\cdot}{u}}_{b}) = {\overset{\cdot}{u}}_{b} / | {\overset{\cdot}{u}}_{b} |$ .

Dynamical equations of motion for the base-isolated building

For the present study, a five-storey regular and uniform building with zero eccentricity is considered mounted on an ERR base isolation system. The building and the ERR are symmetrical in the both the orthogonal directions and have uncoupled dynamic properties along both x- and y-directions. Thus, there are no coupled interactions between the responses for the structural system. The equations of motion of an N-storey shear type flexible ERR base-isolated building subjected to bi-directional earthquake ground motion can be expressed in the matrix form as

\begin{matrix} [\begin{matrix} [M_{ss}] & [M_{sb}] \\ [M_{bs}] & [M_{bb}] \end{matrix}] {\begin{matrix} {\overset{\cdot\cdot}{u}}_{s} \\ {\overset{\cdot\cdot}{u}}_{b} \end{matrix}} + [\begin{matrix} [C_{ss}] & [C_{sb}] \\ [C_{bs}] & [C_{bb}] \end{matrix}] {\begin{matrix} {\overset{\cdot}{u}}_{s} \\ {\overset{\cdot}{u}}_{b} \end{matrix}} \\ + [\begin{matrix} [K_{ss}] & [K_{sb}] \\ [K_{bs}] & [K_{bb}] \end{matrix}] {\begin{matrix} u_{s} \\ u_{b} \end{matrix}} + {\begin{matrix} O_{1} \\ F_{s 1} \end{matrix}} + {\begin{matrix} O_{1} \\ F_{s 2} \end{matrix}} \\ + {\begin{matrix} O_{1} \\ F_{b} \end{matrix}} = - [\begin{matrix} [M_{ss}] & [O_{2}] \\ {[O_{2}]}^{T} & [M_{bb}] \end{matrix}] {\begin{matrix} I_{ss} \\ I_{bb} \end{matrix}} {\begin{matrix} {\overset{\cdot\cdot}{u}}_{gx} \\ {\overset{\cdot\cdot}{u}}_{gy} \end{matrix}} \end{matrix}

(14)

or, in concise form, equation (14) can be written as

[M] {\overset{\cdot\cdot}{u}} + [C] {\overset{\cdot}{u}} + [K] {u} + {F_{s 1}} + {F_{s 2}} + {F_{b}} = - [M_{G}] {I} {{\overset{\cdot\cdot}{u}}_{g}}

(15)

where $[M]$ , $[C]$ , and $[K]$ each of order $(2 N + 2) \times (2 N + 2)$ represent the mass, damping, and stiffness matrices for the base-isolated building, respectively; $[M_{ss}]$ , $[C_{ss}]$ , and $[K_{ss}]$ each of order $2 N \times 2 N$ represent the mass, damping, and stiffness matrices for the fixed-base building, respectively; $[I_{ss}]$ of order $2 N \times 2$ and $[I_{bb}]$ of order $2 \times 2$ are the influence coefficient matrices for the superstructure and base mass, respectively; ${{\overset{\cdot\cdot}{u}}_{g}} = {{\overset{\cdot\cdot}{u}}_{gx}, {\overset{\cdot\cdot}{u}}_{gy}}^{T}$ of order $2 \times 1$ is the earthquake ground acceleration vector, where ${\overset{\cdot\cdot}{u}}_{gx} and {\overset{\cdot\cdot}{u}}_{gy}$ are the ground acceleration along x- and y-directions, respectively; ${u_{s}} = {u_{x 1}, u_{y 1}, u_{x 2}, u_{y 2}, \dots, u_{xN}, u_{yN}}^{T}$ of order $N \times 1$ is the relative floor displacement vector with respect to the base mass; ${u_{b}} = {u_{bx}, u_{by}}^{T}$ of order $2 \times 1$ is the relative base mass displacement vector with respect to the ground; ${F_{s 1}} = {F_{sx 1}, F_{sy 1}}^{T}$ of order $2 \times 1$ is the isolator restoring force vector, where $F_{sx 1}$ and $F_{sy 1}$ are the isolator restoring forces of the ERR by virtue of its geometry along the x- and y-directions, respectively; ${F_{s 2}} = {F_{sx 2}, F_{sy 2}}^{T}$ of order $2 \times 1$ is the isolator rolling frictional force vector, where $F_{sx 2}$ and $F_{sy 2}$ are the isolator rolling frictional force along the x- and y-directions, respectively.

The equations of motion for a fixed-base N-storey shear type building subjected to bi-directional earthquake ground motion is reduced from equation (15) and is expressed as

[M_{ss}] {{\overset{\cdot\cdot}{u}}_{s}} + [C_{ss}] {{\overset{\cdot}{u}}_{s}} + [K_{ss}] {u_{s}} = - [M_{ss}] {I_{ss}} {{\overset{\cdot\cdot}{u}}_{g}}

(16)

For the fixed-base building, the damping matrix $[C_{ss}]$ is obtained using Caughey’s damping by assuming all the modes to have a critical damping ratio of $ξ_{s}$ . For a five-storey building, the various matrices in equations (15) and (16) are given in Appendix 2.

Solution procedure

The governing equations of motion for the ERR base-isolated shear type building are nonlinear. Therefore, the equations of motion are solved in the incremental form by numerical time integration.

Integration of equations of motion

Equations (14) and (16) can respectively be expressed in their incremental form as

[M] {Δ \overset{\cdot\cdot}{u}} + [C] {Δ \overset{\cdot}{u}} + [K] {Δ u} = - [M_{G}] {I} {Δ {\overset{\cdot\cdot}{u}}_{g}} - {Δ F_{s 1}} - {Δ F_{s 2}} - {Δ F_{b}}

(17)

[M_{ss}] {Δ {\overset{\cdot\cdot}{u}}_{s}} + [C_{ss}] {Δ {\overset{\cdot}{u}}_{s}} + [K_{ss}] {Δ u_{s}} = - [M_{ss}] {I_{ss}} {Δ {\overset{\cdot\cdot}{u}}_{g}}

(18)

The above equations are integrated using the Newmark-β method considering linear variation of earthquake acceleration between the original data points in the discrete time interval record. The choice of equations (17) or (18) in each time step depends upon the rolling or non-rolling phase at the isolator level. If the isolator is in the rolling phase, equation (17) is integrated while if the isolator is in the non-rolling phase, equation (18) is integrated. It is to be noted that equation (17) requires the determination of ${Δ F_{s 1}}$ , ${Δ F_{s 2}}$ , and ${Δ F_{b}}$ in the current time step. This is achieved by a recursive formulation of equation (17), wherein the time step iterations of the recursive formulations are performed by initially assuming each of the incremental quantities ${Δ F_{s 1}}$ , ${Δ F_{s 2}}$ , and ${Δ F_{b}}$ as zero. In the present study, number of iterations to solve recursive formulations is taken as 10. Upon balancing the equations of motion after successive iterations, the integration is carried out to the next time step.

Conditions for rolling/non-rolling phase

The condition for rolling occurs when the velocity and acceleration of the base mass are zero (i.e. ${\overset{\cdot}{u}}_{bx} = {\overset{\cdot\cdot}{u}}_{bx} = 0$ and ${\overset{\cdot}{u}}_{by} = {\overset{\cdot\cdot}{u}}_{by} = 0$ ), and in this condition, the building does not roll on the ERR, that is, the frictional force between the rolling rods and the base mass is greater than the base shear of the building at the base mass level. The ERRs start rolling along x- or/and y-directions when the base shear exceeds the limiting value (i.e. $| F_{x} | > | F_{sx 2} |$ and $| F_{y} | > | F_{sy 2} |$ ) in the respective directions. A detailed flowchart is given in Figure 5 indicating the sequence of steps involved in checking for rolling and non-rolling conditions.

Figure 5.

Flowchart for rolling/non-rolling condition.

The base shears in x- and y-directions are respectively given by the following expressions

F_{x} = m_{b} {{\overset{\cdot\cdot}{u}}_{bx} + {\overset{\cdot\cdot}{u}}_{gx}} - c_{1} {\overset{\cdot}{u}}_{xN} - k_{1} u_{xN} + F_{bx} + F_{sx 1}

(19)

F_{y} = m_{b} {{\overset{\cdot\cdot}{u}}_{by} + {\overset{\cdot\cdot}{u}}_{gy}} - c_{1} {\overset{\cdot}{u}}_{yN} - k_{1} u_{yN} + F_{by} + F_{sy 1}

(20)

The ERR base isolation system starts to roll in x-direction if

| F_{x} | > | F_{sx 2} |

(21)

and the ERR base isolation system starts to roll in y-direction if

| F_{y} | > | F_{sy 2} |

(22)

The checks for the rolling condition are ${\overset{\cdot}{u}}_{bx} \neq {\overset{\cdot\cdot}{u}}_{bx} \neq 0$ and ${\overset{\cdot}{u}}_{by} \neq {\overset{\cdot\cdot}{u}}_{by} \neq 0$ . Whenever the relative velocity of the base mass becomes zero (i.e. ${\overset{\cdot}{u}}_{bx} = 0$ or ${\overset{\cdot}{u}}_{by} = 0$ ) or reverses, the phase of rolling is required to be checked to determine whether the ERR is in rolling phase or it remains fixed to the foundation (i.e. in stick condition). The check for zero relative velocity of the base mass in the rolling phase due to numerical challenges is obtained by dividing the data time interval into finer points. For the present study, each time interval in the earthquake data record is divided equally into 100 divisions and intermediate data points are obtained using linear interpolation.

Conditions for isolator stability

The isolator, if rotated beyond $π / 2$ radians in either of the horizontal direction, will result in a sudden shock and permanent displacement at the isolator level. These two circumstances are undesirable from strength and serviceability design point of view. An isolator must be structurally stable during its operation under all displacement levels and a fail-safe mechanism must be provided to mitigate the unlikely but possible occurrence of a sudden shock and permanent displacement. As a fail-safe mechanism, a maximum seismic gap (moat width) is provided along the periphery of the base slab such that the seismic gap distance is equal to the allowable maximum displacement for the isolator in the stable position. Any further movement will result in impact of the base slab with the lateral supports, thus restricting the further movement of the isolator. In the numerical study, the isolator rotation is checked for every time step. The simulation is terminated if $| θ_{rx} | \geq π / 2$ or $| θ_{ry} | \geq π / 2$ and the isolator design parameters are modified till the condition is satisfied.

Numerical study

Seismic performance of the bi-directional ERRs for base isolation of buildings subjected to near-fault earthquakes is investigated. In the present study, the superstructure is assumed symmetrical with no eccentricity. Therefore, a planar frame with equivalent dynamic properties is considered for a simplified analysis in both the orthogonal directions. For the frame, mass and stiffness of each floor are kept equal along x- and y-directions (i.e. m_i = m for i = 1, 2,…, N and k_i = k for i = 1, 2,…, N). The mass matrix of the superstructure, $[M_{ss}]$ , is diagonal. The value of k is selected such as to provide the required fundamental time period for the fixed-base superstructure (T_s). The damping matrix of the superstructure, $[C_{ss}]$ , is not known explicitly. It is developed by assuming equal modal damping ratio in all the modes of vibration. The rolling friction is typically lower than the sliding friction, but in the present study for the purpose of comparison the coefficient of rolling friction ( $μ_{R}$ ) is taken equal to the coefficient of sliding friction ( $μ_{s}$ ). Hence, in the further part of the study coefficient of sliding friction ( $μ_{s}$ ) is equal to coefficient of rolling friction ( $μ_{R}$ ).

The superstructure and the base mass of the isolated structural system under consideration can be completely characterized by the parameters, namely, the fundamental time period of the superstructure (T_s), damping ratio of the superstructure (ξ_s), number of stories in the superstructure (N), the ratio of base mass to the superstructure floor mass (m_b/m), ratio of rolling rod mass to the superstructure floor mass (m_r/m), coefficient of rolling friction ( $μ_{R}$ ), and eccentricity (e) of the ERRs. For the present study, the parameters namely, damping ratio of the superstructure, ratio of the base mass to the superstructure floor mass, and ratio of the rolling rod mass to the superstructure floor mass are kept constant: ξ_s = 0.02, m_b/m = 1, and m_r/m = 0.05. The mass of each floor is considered 1000 kg and the major radius of the ERRs (a) is considered to be 200 mm.

The seismic response of the base-isolated structure is obtained under the fault-normal and fault-parallel components of six real near-fault earthquake ground motions. The peak ground acceleration in terms of g (acceleration due to gravity) of the selected earthquake ground motions is shown in Table 1. The normal and parallel components of the near-fault motion are applied in the x- and y-directions of the base-isolated building mounted on the ERR, respectively. The building has a set of lateral displacements in each x-direction (u_xi) and y-direction (u_yi) relative to the base mass; hence, the resultant displacement, $u_{R} = \sqrt{u_{x}^{2} + u_{y}^{2}}$ . For the base-isolated building, the seismic response quantities of interest are the top floor absolute acceleration and the isolator displacement along x- and y-directions. The absolute acceleration is directly proportional to the forces exerted in the superstructure due to the near-fault earthquake ground motions. On the other hand, the relative isolator displacement is the fundamental parameter for the design of the ERR base isolation system. Among the two response quantities of interest, isolator displacement is of special interest due to the large velocity pulses prominent in the normal component of the near-fault earthquakes.

Table 1.

Near-fault earthquake ground motions.

Sr. no.	Earthquake ground motion	Recording station	Peak ground acceleration (g)
Sr. no.	Earthquake ground motion	Recording station	Normal component (along x-direction)	Parallel component (along y-direction)
1	Imperial Valley, California 15 October 1979	El Centro Array # 5	0.36	0.54
2	Imperial Valley, California15 October 1979	El Centro Array # 7	0.45	0.33
3	Landers, California 28 June 1992	Lucerne Valley	0.71	0.64
4	Northridge, California 17 January 1994	Newhall station	0.70	0.78
5	Northridge, California 17 January 1994	Rinaldi	0.87	0.38
6	Northridge, California 17 January 1994	Sylmar	0.72	0.58

Figure 6 shows the bi-directional time history response of the top floor absolute acceleration and isolator displacement of the building mounted on an ERR when subjected to the Northridge, 1994 (Sylmar station) earthquake ground motion along x- and y-directions. The fundamental time period of the superstructure in both the directions is 0.5 s; the ERR base isolation system has an eccentricity, e = 0.5, and coefficient of rolling friction, μ_r = 0.01, for both the top and bottom rollers. In Figure 7, it is compared against a pure-friction (P-F) base isolation system with coefficient of sliding friction, μ_s = 0.01, and a cylindrical rolling rod base isolation system with a coefficient of rolling friction, μ_r = 0.01, along x- and y-directions.

Figure 6.

Time history response of the top floor absolute acceleration and base displacement under Northridge 1994 earthquake (T_s = 0.5 s, e = 0.5, and µ = 0.01).

Figure 7.

Time history response of $θ, {\overset{\cdot\cdot}{u}}_{bz} / g, F_{b} / W, F_{s} / W$ under Northridge 1994 earthquake (T_s = 0.5 s, e = 0.5, and µ = 0.01).

Although it is argued that base isolation is ineffective under near-fault earthquakes due to excessive displacements, it is evident from Figure 6 that even under the near-fault earthquakes the base-isolated building mounted on the ERR base isolation system shows significant reduction in the seismic response when compared with the non-isolated building. The peak absolute acceleration responses of the fixed-base building are 16.68 m/s² along the x-direction and 31.32 m/s² along the y-direction. The peak top floor absolute acceleration for the building mounted on the ERR base isolation system along the x-direction is 1.34 m/s², while the value for the P-F and cylindrical rolling rod base isolation systems with same design parameters are 1.30 m/s² (−0.04 m/s², −3.0%) and 1.32 m/s² (−0.02 m/s², −1.5%), respectively. The values in the bracket indicates the change in value and the change in percentage with the corresponding response of the building with the ERR base isolation system. Again, the peak top floor absolute acceleration along y-direction is 1.22 m/s² for the building mounted on the ERR base isolation system while the responses obtained for P-F and cylindrical rolling rod base isolation systems are 1.45 m/s² (0.23 m/s², +18.8%) and 1.42 m/s² (0.20 m/s², +16.4%), respectively. It is noted that although the performance of the P-F and cylindrical rolling rod against the ERR base isolation systems has marginally improved in x-direction and slightly worsened in y-direction (as indicated in the magnified graph appended in Figure 6), the magnitude of change in comparison to the fixed-base response is still negligible. It is, however, worth noting that the top floor absolute acceleration has reduced by almost 92% and 96% in the ERR base-isolated building as compared to the fixed-base building in x- and y-directions, respectively. Thereby, it signifies a considerable reduction in the seismic base shear developed in the columns and the advantage of base isolation using the ERRs under near-fault earthquakes.

The peak isolator displacement for the building mounted on the ERR base isolation system along the x-direction is 56.4 cm while for P-F and cylindrical rolling rod base isolation systems are 91 cm (35 cm, +61%) and 87 cm (31 cm, +54%), respectively. Again, for the building mounted on the ERR base isolation system, the peak displacement is 15 cm along y-direction in comparison to 12 cm (3 cm, −20%) and 17 cm (−2 cm, −13%) for the P-F and cylindrical rolling rod base isolation systems, respectively. It is noted here that the performance of the ERR base isolation system is superior to the P-F and cylindrical rolling base isolation system with significant reduction along x-direction. In the Northridge 1994 earthquake, the prominent velocity pulse arrives in x-direction and the ERR has been able to reduce the isolator displacement with a considerable average magnitude of 33 cm.

For the Northridge 1994 earthquake, if the overall response scenario is assessed with respect to the ERR base isolation system, the top floor absolute acceleration reduced by an average 94% and the isolator displacement reduced by 58%. Also, the maximum isolator displacement developed in the ERR base isolation system based on the maximum rotational angle ( $θ = \pm 90 \circ$ ) as per equation (7) is ±0.586 m. Therefore, it shows that the peak isolator displacement response under the Northridge, 1994 earthquake ground motion is within the permissible maximum design isolator displacement, ±0.586 m. The satisfaction of this limit state indicates no occurrence of instability in the performance of the ERR base isolation system. Such reduced isolator displacement response can be attributed to the restoring capability of the ERR base isolation system. This proves the effectiveness of the ERR base isolation system against the P-F and circular rolling rod base isolation systems when subjected to a typical near-fault earthquake ground motion

Figure 7 shows the time history variations of the seismic response quantities such as ${\overset{\cdot\cdot}{u}}_{bz} / g, θ_{b}, F_{b} / W, F_{s} / W$ for the five-storey building base-isolated with the ERR base isolation system (T_s = 0.5 s, e = 0.5, and μ = 0.01) subjected to the Northridge 1994 earthquake ground motion along x- and y-directions. From Figure 7, it can be observed that the variation of θ is same as the isolator displacement in Figure 6, as per equation (7), which gives the relationship between the rotation (θ) and isolator displacement (u_b). However, it must be noted that it is not exactly a linear relationship and it is the slight nonlinearity that contributes to the restoring tendency of the ERR base isolation system. Thus, the response is very sensitive to the eccentricity of the isolator. The maximum response of rotation angle in rolling rods is 1.50 rad in x-direction and 0.43 rad in y-direction for the Northridge, 1994 earthquake ground motion, which is less than the maximum rotational angle of rolling rods, ±1.57 rad (±90°). Therefore, this indicates that rotational angle of the ERRs is within the limit and hence satisfies the isolator design criterion (i.e. maximum isolator design displacement).

The eccentricity of the ERR is responsible for the development of the undesirable vertical acceleration. Therefore, its variation with respect to time is plotted to check its magnitude and also to check its effect on the stability of the building. Figure 7 also shows the individual plot of the vertical acceleration for the bottom and top roller along x- and y-directions, respectively. The peak vertical acceleration are 0.10 g and 0.05 g for the rollers moving along x- and y-directions, respectively, which is much lower than the acceleration due to the gravity. The building frames are in general designed to take gravity loads for 1.0 g. In the worst case scenario, it may happen that reversal of member stresses may affect long-span member of the building; yet, the magnitude is quite low to inflict any significant damage for a regular well-designed building.

Again, Figure 7 shows the time history responses of isolator frictional force $F_{sx 2}$ and $F_{sy 2}$ transmitted from the rolling rods to the superstructure along the x- and y-directions, respectively. The ratio of the maximum force transmitted from the ground to the total weigh of the structure is 0.01, which is the coefficient of rolling friction. Hence at any instant of time, the force transmitted to the structure is always less than or equal to the limiting force. Thus, we can reduce further transmission of force if the design coefficient of rolling friction of the ERR base isolation system can be further reduced using any suitable material.

The time response variation of ratio of isolator inertia force (F_b) transmitted to the base mass to the total weight (W) of the building is also shown in Figure 7. The peak isolator inertial force transmitted to the foundation is around 8% of the weight of the structure, which is not very high and can be reduced using lower density material for the design of the ERR base isolation system. These plots signify the effect of the horizontal forces on the horizontal balance of the overall base-isolated structure, which is one of the main factors in the design of a stable ERR base isolation system. The time variation of F_s and F_b is less than the total weight of the building which shows the stability of building against the lateral force induced during the seismic activity.

At any time instant, the absolute vertical acceleration developed in the base slab or the building is the sum of vertical acceleration due to the motion of bottom and top rollers in x- and y-directions, respectively. The separate vertical acceleration of bottom and top roller is shown along x- and y-directions in Figure 7. The time variation response of absolute vertical acceleration is shown in Figure 8. The maximum value of absolute vertical acceleration in the base slab is 0.13 g, which is much lower than the acceleration due to gravity. To increase the margin of safety of the ERR base isolation system, the major radii or eccentricity of the isolator can be increased. However, this will be at the expense of higher top floor absolute acceleration.

Figure 8.

Time history response of sum of ${\overset{\cdot\cdot}{u}}_{bz} / g$ along x- and y- directions under the Northridge 1994 earthquake (T_s = 0.5 s, e = 0.5, and µ = 0.01).

Figure 9 shows the variation of the peak average top floor absolute acceleration and isolator displacement with respect to the eccentricity of the ERR base isolation system, which are taken as e = 0.25 and 0.5. It is compared against P-F and cylindrical rolling rod base isolation systems for various superstructure time periods of the five-storey building. The increased superstructure time period (T_s) signifies increased superstructure flexibility. A coefficient of friction µ = 0.01 is adopted for this parametric study. It is observed that the acceleration is marginally increased by increasing the eccentricity of the ERR, that is, with increased eccentricity the induced seismic forces are increased; however, the isolator displacement is reduced by increasing the eccentricity of the ERR to e = 0.5. The peak top floor absolute acceleration and bearing displacement are significantly reduced by the ERR base isolation system in comparison with the P-F and cylindrical rolling rod base isolation systems. Thus, the eccentricity (e) of the ERR is a crucial design parameter in controlling the peak top floor absolute acceleration and isolator displacement. Tradeoff exists between the seismic forces induced and isolator displacement while choosing an appropriate eccentricity of the ERR base isolation system. The optimally designed ERR would thereby be an effective isolation system for near-fault ground motions to reduce seismic acceleration induced in the buildings and to limit the isolation displacement, which otherwise is quite high that utterly demands increased moat width and specialized peripheral services to render the building in a serviceable state while accommodating the large isolator displacements in the event of a near-fault type earthquake. Furthermore, the use of the supplemental dampers in controlling the large isolator displacements in case of the base-isolated buildings constructed in the near-fault sites is not necessary if the ERR is designed accordingly.

Figure 9.

Effect of fundamental time period of superstructure on peak average top floor absolute acceleration and bearing displacement of a five-storey base-isolated building (µ = 0.01).

Figure 10 shows the effect of coefficient of rolling friction ( $μ_{R}$ ) of the ERR base isolation system on the peak average top floor absolute acceleration and the isolator displacement of a base-isolated building. It is compared with the P-F and cylindrical rolling rod base isolation system. The eccentricities, e = 0.25 and 0.5, are considered for the ERR. With lower coefficient of friction, the top floor absolute acceleration decreases, and hence shows the effectiveness of the ERR. Nevertheless, an optimum value of the μ_R needs to be evaluated for a base-isolated building with the ERR base isolation system to achieve reduction in both acceleration and displacement responses.

Figure 10.

Effect of coefficient of friction on peak average top floor absolute acceleration and bearing displacement of a five-storey base-isolated building (T_s = 0.5 s).

From Figures 9 and 10, it is observed that peak average top floor acceleration typically increases with the increase in the superstructure flexibility whereas the isolator displacement is unaffected with the increased superstructure fundamental time period (T_s). It may be inferred from this observation that base isolation is more effective for the rigid superstructures as compared to the flexible superstructures; thereby, base isolation is most attractive choice for low- to mid-rise buildings in making them earthquake resistant.

Conclusion

Multiple dynamic analyses are conducted with varied set of superstructure and isolator parameters to study the behavior of a flexible-shear type multi-storey building mounted on ERR base isolation system subjected to bi-directional components of near-fault earthquake ground motions. The effective isolation of the building is achieved through judicious choice of the ERR base isolation system parameters: (1) maximum design isolator displacement (indirectly related to the major radius of the roller), (2) eccentricity, and (3) coefficient of rolling friction. The parametric study is conducted to investigate the effects of several important parameters such as eccentricity, coefficient of friction, superstructure time period of the building, and different near-fault earthquake ground motions. From the present study, the following conclusions have been drawn:

The ERR base isolation system is effective in mitigation of damaging effects of the near-fault earthquake ground motions in multi-storey buildings. Even under the near-fault earthquake ground motions, the base-isolated building with the ERR showed considerable response reduction. The ERRs are found effective than the pure-friction (P-F) and cylindrical rolling rod systems.

Isolator displacements for an ERR base isolation system is lower as compared to the pure-friction (P-F) and cylindrical rolling rod base isolation systems.

The top floor absolute acceleration increases with increase in superstructure flexibility, whereas the isolator displacement is unaffected with the increased time period of the building.

The peak isolator displacement is reduced with the increasing value of the eccentricity of the ERR; however, it has less effect on the absolute acceleration of the building.

The effectiveness of the ERR increases with reduced coefficient of friction, and in comparison with the pure-friction (P-F) and cylindrical rolling rod base isolation system, it is most effective when the coefficient of rolling friction is in the range of 0.05–0.10.

Application of the ERR isolators for a building is a new development, and hence, it opens possibilities for further research to enhance its viability and performance in its application. The subsequent research needs to address some of the following salient issues:

The ERR might exhibit skidding, which a finite sliding friction model can assess the residual displacement due to translational movements due to the isolator inertial force.

The vertical component of earthquake ground motion can be included to understand the positive effects of the potential vertical flexibility which the ERR offers in its non-equilibrium state.

3D finite element analysis can be carried out to understand the effect of deformations produced in the ERR on the seismic response.

Footnotes

Appendix 1

Appendix 2

If $m_{xi}$ and $m_{yi}$ are the lumped masses at the floor level i = 1, 2,…, 5 in x- and y-directions, respectively, and $k_{xi}$ and $k_{yi}$ are the stiffness in storey i = 1, 2,…, 5 in x- and y-directions, respectively; $C_{ss} (i, j)$ is a value in ith row and jth column of the superstructure damping matrix then

(23)

{O_{1}} = {\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}}^{T}

(24)

[O_{2}] = {[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]}^{T}

(25)

[M_{ss}] = [\begin{matrix} m_{x 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & m_{y 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & m_{x 2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & m_{y 2} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & m_{x 3} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & m_{y 3} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & m_{x 4} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & m_{y 4} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & m_{x 5} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & m_{y 5} \end{matrix}]

(26)

[M_{bb}] = [\begin{matrix} m_{bx} & 0 \\ 0 & m_{by} \end{matrix}]

(27)

[M_{sb}] = {[\begin{matrix} m_{x 1} & 0 & m_{x 2} & 0 & m_{x 3} & 0 & m_{x 4} & 0 & m_{x 5} & 0 \\ 0 & m_{y 1} & 0 & m_{y 2} & 0 & m_{y 3} & 0 & m_{y 4} & 0 & m_{y 5} \end{matrix}]}^{T}

(28)

[M_{bs}] = {[O_{2}]}^{T}

(29)

[C_{bb}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]

(30)

[C_{sb}] = [O_{2}]

(31)

[C_{bs}] = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - C_{x}' & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - C_{y}' \end{matrix}]

where

(32)

{\begin{matrix} C_{x}' \\ C_{y}' \end{matrix}} = {\begin{matrix} C_{ss} (2 N - 1, 2 N - 1) + C_{ss} (2 N - 3) \\ C_{ss} (2 N, 2 N) + C_{ss} (2 N, 2 N - 2) \end{matrix}}

(33)

[K_{ss}] = [\begin{matrix} k_{x 1} & 0 & - k_{x 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & k_{y 1} & 0 & - k_{y 1} & 0 & 0 & 0 & 0 & 0 & 0 \\ - k_{x 1} & 0 & k_{x 1} + k_{x 2} & 0 & - k_{x 2} & 0 & 0 & 0 & 0 & 0 \\ 0 & - k_{y 1} & 0 & k_{y 1} + k_{y 2} & 0 & - k_{y 2} & 0 & 0 & 0 & 0 \\ 0 & 0 & - k_{x 2} & 0 & k_{x 2} + k_{x 3} & 0 & - k_{x 3} & 0 & 0 & 0 \\ 0 & 0 & 0 & - k_{y 2} & 0 & k_{y 2} + k_{y 3} & 0 & - k_{y 3} & 0 & 0 \\ 0 & 0 & 0 & 0 & - k_{x 3} & 0 & k_{x 3} + k_{x 4} & 0 & - k_{x 4} & 0 \\ 0 & 0 & 0 & 0 & 0 & - k_{y 3} & 0 & k_{y 3} + k_{y 4} & 0 & - k_{y 4} \\ 0 & 0 & 0 & 0 & 0 & 0 & - k_{x 4} & 0 & k_{x 4} + k_{x 5} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & - k_{y 4} & 0 & k_{y 1} + k_{y 2} \end{matrix}]

(34)

[K_{bb}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]

(35)

[K_{sb}] = [O_{2}]

(36)

[K_{bs}] = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - k_{x 5} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - k_{y 5} \end{matrix}]

(37)

[I_{ss}] = {[\begin{matrix} 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \end{matrix}]}^{T}

(38)

[I_{sb}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was financially supported by the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Ministry of Science and Technology, Government of India to Dr Vasant Matsagar for conducting the investigations on “Innovative Bi-Directional Base Isolation System for Earthquake Protection of Structures” vide SR/FTP/ETA-0076/2011.

References

Butterworth

(2006) Seismic response of a non-concentric rolling isolator system. Advances in Structural Engineering 9(1): 39–54.

Calio

Marletta

Vinciprova

(2003) Seismic response of multi-storey buildings base-isolated by friction devices with restoring properties. Computers & Structures 81(28–29): 2589–2599.

Chakraborty

Roy

Ray-Chaudhuri

(2016) Design of re-centering spring for flat sliding base isolation system: theory and a numerical study. Engineering Structures 126: 66–77.

Chung

Yang

Chen

et al . (2009) Dynamic behavior of nonlinear rolling isolation system. Structural Control & Health Monitoring 16(1): 32–54.

Das

Gur

Mishra

et al . (2015) Optimal performance of base isolated building considering limitation on excessive isolator displacement. Structure and Infrastructure Engineering 11(7): 904–917.

Fathi

Makhdoumi

Parvizi

(2015) Effect of supplemental damping on seismic response of base-isolated frames under near and far field accelerations. KSCE Journal of Civil Engineering 19(5): 1359–1365.

Guerreiro

Azevedo

Muhr

(2007) Seismic tests and numerical modeling of a rolling-ball isolation system. Journal of Earthquake Engineering 11(1): 49–66.

Hall

Heaton

Halling

et al . (1995) Near-source ground motion and its effects on flexible buildings. Earthquake Spectra 11(4): 569–605.

Harvey

Jr Gavin

(2013) The nonholonomic and chaotic nature of a rolling isolation system. Journal of Sound and Vibration 332(14): 3535–3551.

10.

Harvey

Jr Gavin

(2015) Assessment of rolling isolation system using reduced order structural models. Engineering Structures 99: 708–725.

11.

Harvey

Jr Zehil

Gavin

(2014) Experimental validation of a simplified model for rolling isolation system. Earthquake Engineering & Structural Dynamics 43(7): 1067–1088.

12.

Hosseini

Soroor

(2011) Using orthogonal pairs of rollers on concave beds (OPRCB a base isolation system—part I: analytical, experimental and numerical studies of OPRCB isolators. Structural Design of Tall and Special Buildings 20(8): 928–950.

13.

Hosseini

Soroor

(2013) Using orthogonal pairs of rollers on concave beds (OPRCB) as a base isolation system—part II: application to multi-storey and tall buildings. Structural Design of Tall and Special Buildings 22(2): 192–216.

14.

Ismail

Rodellar

Ikhouane

(2010) An innovative isolation device for aseismic design. Engineering Structures 32(4): 1168–1183.

15.

Jangid

(1995) Seismic response of structures isolated by free rolling rods. European Earthquake Engineering 9: 3–11.

16.

Jangid

(2000) Stochastic seismic response of structures isolated by rolling rods. Engineering Structures 22(8): 937–946.

17.

Jangid

(2005) Optimum friction pendulum system for near-fault motions. Engineering Structures 27(3): 349–359.

18.

Jangid

(2007) Optimum lead-rubber isolation bearings for near-fault motions. Engineering Structures 29(10): 2503–2513.

19.

Jangid

Kelly

(2001) Base isolation for near-fault motions. Earthquake Engineering & Structural Dynamics 30(5): 691–707.

20.

Kaplan

Seireg

(2001) Optimal design of a base isolated system for high-rise steel structure. Earthquake Engineering & Structural Dynamics 30(2): 287–302.

21.

Kesti

Mowrtage

Erdik

(2010) Earthquake risk reduction of structures by a low-cost base isolation device: experimental study on BNC bearings. In: Proceeding of the 14th European conference on earthquake engineering (ECEE), Ohrid, Macedonia, 30 August - 3 September.

22.

Lee

Liang

(2003) A sloping surface roller bearing and its lateral stiffness measurement. In: Proceeding of the 19th US-Japan bridge engineering workshop, Tsukuba, Japan, 27–29 October.

23.

Lin

Hone

(1993) Base isolation by free rolling rods under basement. Earthquake Engineering & Structural Dynamics 22(3): 261–273.

24.

Lin

Chern

Hone

(1995) Experimental study of base isolation by free rolling rods. Earthquake Engineering & Structural Dynamics 24(12): 1645–1650.

25.

Londhe

Jangid

(1998) Effectiveness of elliptical rolling rods for base isolation. Journal of Structural Engineering 124(4): 469–472.

26.

Londhe

Jangid

(1999) Dynamic response of structures supported on elliptical rolling rods. Structural Mechanics & Earthquake Engineering 16(1): 11–20.

27.

Makris

(1997) Rigidity-plasticity-viscosity: can electro-rheological dampers protect base-isolated structures from near source ground motions? Earthquake Engineering & Structural Dynamics 26(5): 571–591.

28.

Malhotra

(1999) Response of buildings to near-field pulse like ground motions. Earthquake Engineering & Structural Dynamics 28(11): 1309–1326.

29.

Matsagar

Jangid

(2003) Response of base-isolated structures during impact to near-fault motions. In: Proceeding of 4th international conference of earthquake engineering and seismology, Tehran, Iran, 12–14 May, pp. 1071–1078.

30.

Muhr

Sulong

Thomas

(1997) Rolling-ball rubber layer isolators. Journal of Natural Rubber Research 12(4): 199–214.

31.

Song

Lee

(2010) A parametric study of seismic behavior of roller seismic isolation bearings for highway bridges. Earthquake Engineering & Structural Dynamics 39(5): 541–559.

32.

Panchal

Jangid

(2008) Variable friction pendulum system for near-fault ground motions. Structural Control & Health Monitoring 15(4): 568–584.

33.

Providakis

(2009) Effect of supplemental damping on LRB and FPS seismic isolators under near-fault ground motions. Soil Dynamics and Earthquake Engineering 29(1): 80–90.

34.

Rao

Jangid

(2001) Performance of sliding systems under near-fault motions. Nuclear Engineering and Design 203(2–3): 259–272.

35.

Saha

Matsagar

Jain

(2015) Reviewing dynamic analysis of base-isolated cylindrical liquid storage tanks under near-fault earthquakes. IES Journal Part A: Civil & Structural Engineering 8(1): 41–61.

36.

Sahasrabudhe

Nagarajaiah

(2005) Experimental study of sliding base-isolated buildings with magneto-rheological dampers in near-fault earthquakes. Journal of Structural Engineering 131(7): 1025–1034.

37.

Tsai

Chang

(2006) Seismic isolation of a scaled bridge model using rolling-type bearings. In: Proceeding of 4th international conference on earthquake engineering, Taipei, Taiwan, 12–13 October, Paper no. 181.

38.

Tsai

Chang

et al . (2007) Shaking table tests of a scaled bridge model with rolling-type seismic isolation bearings. Engineering Structures 29(5): 694–702.

39.

Wei

Wang

et al . (2016a) Seismic response of spring-damper-rolling systems with concave friction distribution. Earthquakes and Structures 11(1): 25–43.

40.

Wei

Wang

Weian

et al . (2016b) Impact of the concave distribution of rolling friction coefficient on the seismic isolation performance of a spring-rolling system. International Journal of Non-Linear Mechanics 83(1): 65–77.

41.

Wei

Wang

Yang

et al . (2017) Seismic response of rolling isolation systems with concave friction distribution. Journal of Earthquake Engineering 21(2): 325–342.

42.

Yang

Hsieh

Chung

et al . (2012) Effectiveness of an eccentric rolling isolation system with friction damping. Journal of Vibration and Control 18(14): 2149–2163.

43.

Zelleke

Elias

Matsagar

et al . (2015) Supplemental dampers in base-isolated buildings to mitigate large isolator displacement under earthquake excitations. Bulletin of the New Zealand Society for Earthquake Engineering 48(2): 100–117.

44.

Zhou

Wang

et al . (1998) Dynamic analysis of structures base-isolated by a ball system with restoring property. Earthquake Engineering & Structural Dynamics 27(8): 773–791.