Development of a bidirectional rocking isolation bearing system (Bi‐RIBS) to control excessive seismic response of bridge structures

A bidirectional rocking isolation bearing system (Bi‐RIBS) is proposed to provide seismic protection for bridge structures. By using the 3‐D rocking motion of the Bi‐RIBS, this system acts as a mechanical fuse to limit the maximum force transmitted to the bridge piers and as a restoring component to control an excessive girder response. Possible applications in bridge structures were discussed. A simple analytical model was established to characterize the dynamics of an example bridge featuring such a Bi‐RIBS. A series of dynamic analyses were performed by using the proposed model to investigate the effects of several factors on controlling the seismic responses of the bridge, for example, the design parameters of Bi‐RIBS including inclined angle and size, the damping property at the support interface, and the mass ratio. The peak ground accelerations of the bidirectional ground motion record were scaled to various levels to evaluate the maximum performance indices of the bridge structure and the response control effectiveness of the Bi‐RIBS compared to the uncontrolled counterparts. The simulation results demonstrated that the proposed Bi‐RIBS could effectively control the maximum pier displacement while keeping the bearing from overturning if suitable parameters were selected. In particular, the control effectiveness on the maximum pier response becomes more significant as the seismic intensity increases, due to its distinctive negative stiffness property.


NOVELTY
• A bi-directional rocking isolation bearing system (Bi-RIBS) is proposed to provide all-directional seismic protection for bridge structures • This system acts as a mechanical fuse through its 3-D rocking motion to limit the maximum force transmitted to the bridge piers as well as to control the girder response. • The dynamics of an example bridge with Bi-RIBS were characterized by an analytical model to offer a computationally efficient approach for seismic performance assessment. • The control effectiveness on the maximum pier response could become more significant as the seismic intensity increases due to its distinctive negative stiffness property.
concept of kinematic base isolation that allows the columns of the first story of buildings 1,7,[13][14][15] or the piers of bridges 2, [16][17][18] to rock freely under seismic excitation so as to modify the dynamic response of superstructures. The effects of the rocking mechanism on controlling the seismic response of various structures have been the subject of many discussions. Studies on slender rigid blocks undergoing rocking motion revealed some distinctive properties of this type of system. The larger block of two geometrically similar blocks was shown to have greater stability. 19 The rocking frame structures supported by rocking columns or piers were proven to become more stable as the weight of the cap beam or the superstructure increased. 5,20 Unlike ordinary fixed-base structures with positive stiffness properties to provide restoring capability, the rocking motion of structures introduces negative stiffness properties as well as restoring capability into systems. The distinctive negative stiffness dynamic features of rocking structures are advantageous in controlling an excessive seismic response as well as in avoiding the resonant response. In particular, the implementation of negative stiffness elements in civil engineering is usually limited to the vibration control of small mass systems since the preload forces are on the order of the main structure. The application of such systems in massive structures has received growing attention, such as the adaptive negative stiffness devices for buildings achieved by highly compressed springs, 21,22 the pseudo negative stiffness system for bridges achieved by semiactive control strategy, [23][24][25] or by a convex pendulum bearing. 26 Although the rocking elements in structures are not generally treated as negative stiffness devices, the rocking structures share analogous negative stiffness characteristics with these devices.
The dynamic response of even a simple rocking structure was found to be sensitive to a slight variation in its defining parameters, which to a certain degree indicates the unpredictable and chaotic dynamic characteristics of rocking structures. [27][28][29] The systematic trend of the seismic response of rocking structures was found to emerge from the perspective of probability. 28 This conclusion was validated by numerical simulation and experimental results. [30][31][32] The dynamics of rocking structures experiencing complex motions and conditions were investigated from various directions, such as multiple rocking blocks, [33][34][35][36] sliding and take-off motions, 37,38 and planar rocking frames with an eccentric top mass. 39 Studies on planar rocking frames with an asymmetric geometry 40,41 indicated that the overall stability of the structures subject to pulse-type excitations was hardly affected by their symmetry.
Although most of the studies on rocking structures focused on their 2-D dynamic response (in-plane response), the dynamics of rocking structures undergoing 3-D rocking motion have received relatively less attention. The free-standing 3-D rocking motion of rigid cylinders for ground motions was studied through simplified models. 42,43 Similar studies were performed on rocking blocks with a rectangular base. 44,45 A refined FEM model of a real bridge structure was established to examine its seismic performance under fixed-base pier, rocking pier, and rocking foundation conditions. 18 Recently, the bounded cylinders and wobbling frames, for which the 3-D rocking motions were restrained in the initial position without rolling-out motion, were numerically and experimentally studied 31,46,47 to demonstrate their expected seismic performance.
Our previous study was the first time that a unidirectional rocking isolation bearing system (Uni-RIBS) was proposed as a possible response modification technique to control the longitudinal seismic response of bridge structures. 48 Considering the 3-D nature of observed ground motions, the bidirectional rocking isolation bearing system (Bi-RIBS) is proposed in the present study to provide all-directional protection for bridge structures. Even though there are various isolation bearings capable of providing bidirectional seismic protection for bridge structures, [49][50][51][52][53] this is the first time that the use of the 3-D rocking mechanism of bearings is proposed as the isolation measure to control bridge responses. This paper is organized as follows. The motivation for developing the Bi-RIBS and its possible application in bridge structures are elaborated in Section 2. An analytical model of an example bridge featuring such a Bi-RIBS is established in Section 3. This model provides a simple and computationally efficient way to explore its dynamic characteristics and evaluate its seismic response control effectiveness. The effects of several design factors on the dynamic characteristics of the bridge, including the parameters defining the Bi-RIBS, the mass ratio of the girder to the pier, and the damping property at the support interface, are investigated through a series of dynamic analyses in Section 4. The conclusions and highlights are presented in Section 5.

BIDIRECTIONAL ROCKING ISOLATION BEARING SYSTEM (BI-RIBS)
The conventional pin bearing system provides a smooth rotation capability to accommodate the repetitive deflection deformation of the girder imposed by the load from vehicles crossing the bridge. However, this type of bearing system is susceptible to strong earthquake events, and its failure may be directly associated with unrecoverable problems, for example, failure of the girder, see P212-213 and P219-220 in Ref.
[54] Two different failure modes of the conventional pin bearing system under strong earthquakes are presented in Figure 1. The pulling-out of the anchor bolts in Figure 1A is caused by the rocking motion of the bearing since the inertia force of the earthquake acting on the centroid of the girder imposes a momentum around the two corners of the bottom plate of the bearing in the longitudinal direction of the bridge. In Figure 1B, the breakage of the pin is the result of excessive seismic force acting in the transverse direction of the bridge. Motivated by the promising seismic response control effect of rocking structures as revisited above, the unidirectional rocking isolation bearing system (Uni-RIBS) was proposed in our previous study to control the excessive seismic response of bridge structures in the longitudinal direction. 48 An example bridge featuring such a Uni-RIBS is presented in Figure 2. The rocking motion of the Uni-RIBS is achieved in the longitudinal direction, whereas the transverse response of the system was limited by the restrainers. Considering the 3-D nature of earthquake ground motions and the observed transverse failure of the conventional pin bearings, as shown in Figure 1B, the bidirectional extension of the Uni-RIBS shows promise for providing all-directional protection for bridge structures.
The present study proposes the bidirectional rocking isolation bearing system (Bi-RIBS) shown in Figure 3. In the practical application for bridges, at least two Bi-RIBSs are placed on the top of each single pier to ensure a stable configuration. Once the intensity of an earthquake ground motion exceeds a certain threshold, the bidirectional rocking motion of the Bi-RIBS will be triggered to limit the force transmitted to the substructure so as to reduce the pier responses. The response of the girder supported by the Bi-RIBSs is also controlled by using the gravity of the uplifted girder as the restoring force. In terms of the energy dissipation capability, the energy can be dissipated at the contact boundaries between the bearing and the girder/pier through various manners, for example, friction force, inelastic behavior, radiation wave carrying   energy and spreading outward, and impact, as suggested in Anastasopoulos's study for rocking objects. 55 More details are discussed in Section 3.
Two possible types of Bi-RIBS were designed, namely the cylinder-shaped Bi-RIBS (Type 1) and the cone-shaped Bi-RIBS with a universal pin capped on the cone-shaped part (Type 2), as shown in Figure 4. The Type 1 system consists of the cylinder-shaped rocking bearing and two restrainers placed on the bottom and the top of the girder and pier, respectively, to constrain the possible rolling-out motion of the bearing from its initial position. The rocking motion of the bearing of the Type 1 system simultaneously occurs around its top and bottom corners, the connection of which forms the diagonal line of the cylinder-shaped bearing. On the other hand, the Type 2 system is composed of the cone-shaped bearing, the bottom restrainer, and the top cap placed on top of the universal pin of the bearing, similar to that of the pivot bearing. 56 During the rocking motion of the Type 2 system, the bottom restrainer constrains the rolling-out motion of the bearing, and the top cap allows a smooth 3-D rotation between the bearing and the girder through the universal pin (flexible joint).
Although both types of Bi-RIBS enable a bidirectional rocking motion, they have different advantages. The Type 1 system is advantageous in its ease of manufacturing, while the capacity of accommodating the rotation of the girder caused by the deflection is lower than that of the conventional pin bearings. The Type 2 system provides a 3-D articulated connection without losing the advantage of the conventional pin bearings at the cost of slightly complicated structural details. In addition, the restrainers prevent the overturning of the bearing in the case of an excessive response, and the corner of the bearing (namely the rolling edge) should be carefully designed, such as in a polygonal or round shape, to avoid stress concentration.
To satisfy varying demands for bridge performance under operational and seismic conditions, several strategies can be employed.
1. Facilitation of rocking motion and prevention of denting. The corners of the bottom plate of the bearing have been designed as a round type to facilitate the intended rocking motion and to minimize denting of the corners, as shown in Figure 5A. 2. Accommodation of thermal effect on the girder. As shown in Figure 5B, a sliding interface with a specified clearance, similar to other slide-type bearings, 52 can be created on the bottom restrainers to accommodate the thermal expansion. In this case, the rocking motion is guided by the round corners of the bearing when the bearing slides and contacts with the corner parts of the bottom restrainers. The thermal expansion can also be accommodated by other bearings (e.g., elastic or movable bearings) at other spans. Additionally, continuous girder bridges with multiple fixed supports are widely used in Japan. 57,58 In such cases, it is important to design the span length, substructures, and foundations to accommodate deformation absorption requirements due to the thermal effect of girders. 3. Accommodation of girder end rotation. As shown in Figure 5C, Type 2 bearing has a universal pin that can absorb the rotation of the superstructure naturally. Type 1 bearings can allow small rotation of the girder by placing a sealed rubber mat on the bottom surface, as in BP-type bearings. 59 However, in such case, the initiation condition of the rocking motion should be modified. On the other hand, in integral abutment structures where the girder and abutment are rigidly connected, the substructure absorbs the rotation of the girder. 4. Accommodation of different rotations of the bearings. As shown in Figure 5D, if the substructure piers have different lateral stiffness, their lateral displacement and rotation will also differ. In this case, it would be acceptable to activate the rocking motion of the bearing separately for each pier to manage the seismic load transmitted to each substructure, even if the superstructure does not start rocking simultaneously. Moreover, given that bridge girders are considered flexible due to their slender and long nature, the different elevations at the top of different piers can be accommodated by the girders. 5. Maintenance of constant compressive state. This is essential for the stable performance of the bearing during seismic events. It should be ensured during the design and validation phase of the structure. The magnitude of the load that initiates rocking and the load that induces the girder's rotation in the transverse direction can be adjusted to regulate the girder's rotation. By reducing the rocking initiation load, the girder will begin to rock before rotating, thereby allowing for better control over the girder's transverse rotation.
In addition, there are some unique features of the proposed isolation bearing that we can take advantage of: (1) The triggering mechanism. The inclined angle of the bearing enables a straightforward adjustment of the seismic intensity required to trigger the rocking motion of the bearing, after which the isolation effect emerges. This could be useful in seismic design, especially in areas with high seismic intensity. (2) The negative stiffness resulting from the rocking motion can reduce resonance during earthquakes. (3) The proposed bearing (Type 2) can easily accommodate the rotation of the end of the girders in bridges caused by the deflection of the girders during normal vehicle crossings.

Modeling assumptions
The dynamic behavior of wobbling rigid columns standing on a shaking foundation, 42,43 see Figure 6A, and that of 3-D rocking frames, 47 see Figure 6B, have been studied by numerical simulation. Through shake table tests, recent studies have demonstrated that a 3-D rocking frame 31 and a rocking column-bridge structure, 32 which allow the bidirectional rocking motion of the columns and the piers, respectively, meet their expected seismic performance. The dynamics of an example bridge structure featuring such a Bi-RIBS, as shown in Figure 6C, mainly consist of the 3-D bounded rocking motion of the bearings, the movements of the girder in the space, the vibration of the bridge piers in the horizontal directions, and the interaction among them. The prominent differences among the three structures are (1) that the wobbling rigid columns allow seismic response reduction of itself; (2) the 3D rocking frames allow seismic response reduction of the superstructure Considering the similarity between the 3-D rocking frame 31 and the example bridge featuring the Bi-RIBS in their potential kinematic characteristics, analogous modeling assumptions are used in the present study to characterize the dynamics of the example bridge as follows.
a. The girder-type superstructure, the rocking bearing, and the contact interfaces between the rocking bearing and the structures are regarded as being rigid without undergoing any damage. b. The contact condition between the bearing and the contact surfaces is pointwise, considering the above rigid body assumptions. c. The bearing is always in contact with the interfaces, namely a constant compressive condition. The motion of the rocking bearing is constrained by the restrainers to prevent sliding and rolling-out from its initial position in the horizontal plane. d. The piers with constant and isotropic mechanical properties in the horizontal plane are considered.
Note that, from a practical standpoint, assumptions (a) and (b) may not hold due to potential denting of the corners of the bearing. If the corner experiences denting, the inclined angle of the bearing, which is a crucial design parameter as indicated in subsequent discussions, may alter. This potential deviation should be appropriately accounted for in the simulation model. Additionally, the change in the position of the contact point with the surfaces of the pier may result in a subsequent alteration of the rotation angle of the bearing, which should also be considered in the simulation model. In the present study, it is assumed that the influence of the dented corner is secondary to the dynamic response of the bridge.

Coordinate systems
A simplified model of the example bridge structure can be described by four independent generalized degrees-of-freedom (DOF): the tilt angle of the rocking bearing , the rolling angle of the rocking bearing , and the displacements of the pier top in the horizontal plane u p,x and u p,y . As shown in Figure 7, the following right-handed Cartesian coordinate systems are used: XYZ is the inertial reference frame, which is fixed on the ground with origin O; x p y p z p originates at the center of the top of the bridge piers P, and it is a fixed-body coordinate system; and xyz originates at the center of the bottom of the bearing B, and it is also a fixed-body coordinate system.
At the resting state of the Bi-RIBS, the three coordinate systems have the same orientation but differ with translation, and x p y p z p and xyz coincide with each other. During the motion of the Bi-RIBS, x p y p z p translates to its new location, and xyz rotates to its new location with respect to x p y p z p . The 3-2-3 Euler angles are used to describe the 3-D rocking motion of the bearing.
a. The first rotation angle, namely the rolling angle of the bearing φ, describes the rotation around the z axis. This angle determines the location of the contact point between the bearing and the contact interfaces, and leads to a new coordinate system x 1 y 1 z 1 ; b. The second rotation angle , equivalent to the tilt angle of the bearing (with respect to the vertical axis), describes the rotation around the y 1 axis. This rotation leads to a new coordinate system x 2 y 2 z 2 ; c. The third rotation angle describes the rotation around the z 2 axis. This leads to the new coordinate system x 3 y 3 z 3 .
Since the motion of the bearing is assumed to be constrained in its initial position in the horizontal plane by the restrainers, it can be proven that ψ = −φ. 46 On the other hand, the bearing has a size of r = √ b 2 + h 2 , which is the diagonal of the Type 1 bearing (cylinder-shaped) or the slant height of the Type 2 bearing (cone-shaped). The radius of the bearing's circular bottom plate b and the height of the bearing h, which describe the inclined angle of the bearing α = arctan b∕h.
Under the rigid body assumptions, the centroid of the girder and the center of the resultant forces provided by the bearings in symmetric configuration coincide because the restoring force of every bearing is proportional to its vertical load. The resultant force acting on the centroid of the girder does not include a rotational moment, so we assume an ignorable rotation of the girder in 3-D space for the sake of simplicity. Consequently, the motions of the centroids of the system relative to the inertial reference frame XZY should be tracked to describe its dynamics. Considering that the mass of the bearing is ignorable compared to the girder, the position of the centroid of the bearing is not tracked. Only the centroids of the superstructure girder and the piers are tracked. As shown in Figure 7, the components of the position vector of the pier top mass can be directly expressed as follows: where u gx , u gy , and u gz are the three components of the ground motion in the X, Y, and Z axes, respectively; , , and are the corresponding unit vectors in the X, Y, and Z axes, respectively; and u px and u py are the horizontal components of the pier top mass in the X and Y axes, respectively. Then, the position vector of the center of the mass of the superstructure S is expressed as follows: where ′ is the vector of the ground motion translated from the origin of XYZ to the pier bottom center ′ ; ′ represents the translational motions of the pier top center in the horizontal plane with respect to the ground (it is easy to establish that = ′ + ′ ); represents the motion of the bearing bottom center with respect to the pier mass center; and ′ represents the distance from the bearing bottom center to the top center of the bearing S ′ , which is directly expressed as follows: where 3 , 3 , and 3 are the unit vectors in the x 3 , y 3 , and z 3 axes, respectively. The vector can be expressed as follows: where , , and are the unit vectors in the x, y, and z axes, respectively; d x , d y , and d z are the movement of the bottom plate center of the bearing in the x, y, and z axes, respectively. As shown in Figure 7, d x , d y , and d z are consequently expressed as follows: Combined with Equation (4), in the XYZ coordinate system can be expressed as follows: To determine ′ in the XYZ coordinate system, the coordinate systems are related through the following transformations, namely premultiplying the rotation matrices of intrinsic rotations about axes z, y 1 , and z 2 in order, where is the identity matrix; 1 , 2 , and 3 are the rotation matrices that correspond to the Euler angles, as follows: where the rotation angles are positive (counterclockwise) in the sense of the right-handed rule in the coordinate systems. Then, the reverse relationship of the vectors can be expressed as follows: where it is easy to establish the following: Combining Equations (3) and (13), the vector ′ in the XYZ coordinate system can be expressed as follows: The substitution of Equations (1), (8), and (15) into Equation (2) yields the expression of the position of the centroid of the superstructure girder in the XYZ coordinate system as follows:

Equation of motions without considering damping effect at contact interfaces
The energy dissipation between the bearing and the contact interface is first neglected to refer to the bridge model as the undamped model. The girder and the piers have the total masses of m s and m p , respectively. The stiffnesses of the piers in the X and Y directions are defined as k p,x and k p,y , respectively. The corresponding damping properties of the piers are defined as c p,x and c p,y , respectively. The Lagrangian equations of the system can be expressed as follows: where q i (i = 1, 2, 3, 4) is the ith independent DOF of the system, corresponding to , φ, u p,x , and u p,y , respectively; and L = K s − V is the Lagrangian of the system, where K s and V are the kinetic and potential energy of the system, respectively. The kinetic energy of the system consists of the translational motion related component K trans and the rotational motion related component K rot , namely K s = K trans + K rot . The girder and the pier experience only translational motions according to the above assumptions. Although the bearing experiences rotational motion in the 3-D space, the corresponding kinetic energy is ignored. As a consequence, the kinetic energy of the translational motion of the system is expressed as follows: Substituting Equations (19) and (18) into Equation (17), combined with Equations (1) and (16), gives the equation of motions of the system. Specifically, the equation of motion for the pier displacement in the X direction is as follows: The equation of motion for the pier displacement in the Y direction is as follows: The equation of motion for the tilt angle of the bearing is as follows: The equation of motion for the rolling angle of the bearing φ is as follows: (23) Finally, the stiffness and damping of the pier with constant mechanical properties can be expressed as c p,x = x c r,x , c py = y c r,y where T p,x and T p,y denote the specified natural period of the pier in the X and Y directions, respectively; x and y are the corresponding specified damping ratios; and c rx and c ry are the corresponding critical damping ratios.
In addition, the proposed analytical computational model for the example bridge structure shares some similarities in kinematic characteristics with previous studies in the literature. For example, if either the X or Y components of the motion of the model are assumed to be zero (i.e., at rest in that direction), the proposed model reduces to the model of the Uni-RIBS that we proposed in our previous study. 48 If the substructure piers are assumed to be a rigid body, the proposed model reduces to a special case of the 3-D wobbling frame structures, 46,47 where the mass of the 3-D wobbling columns is neglected.

Equation of motions with considering damping effect at contact interfaces
The damping effect at contact boundaries of rocking objects is affected by several factors, such as materials and contact conditions. Anastasopoulos's study 55 suggested that the inelastic behavior of the support surface, the wave radiation carrying energy and spreading outward to the support surface, and the impact at the interface due to uplifting motion jointly contribute to the damping effect. It may be more reasonable in practice to estimate the energy dissipation by a vibration test rather than derive it theoretically. The energy dissipation due to the inelastic behavior of the support surface is neglected due to the rigid body assumption. The energy dissipation due to impact and wave radiation can be considered as the inherent damping mechanism of the support surface. The former can be theoretically derived in an extreme case where the object undergoes an almost instantaneous change of the contact point (spike-likė, similar to the discontinuous state switch during the sign-reverse of 2-D rocking motion), and then the energy will be dissipated by the intense variation of the contact force, viz. the impact force. The impact-dissipated energy of rocking structures can be evaluated by the coefficient of restitution (COR) and undergo extensive analytical studies 5,14,19,28,48 or empirical research. 10,60,61 However, in the present study the Bi-RIBS is assumed to experience a continuous 3-D rocking motion that implies the abrupt sign-reverse of the contact point does not always occur. It will be shown in the following numerical simulation results that only at the last stage of the 3-D rocking motion the spike-likėappears. On the other hand, the energy dissipated by a 3-D rocking object under rigid body assumption was found to be much smaller than the amount of kinetic and potential energy in the system. 46 To capture the energy dissipation behavior of a 3D rocking object at contact boundaries, the Winkler foundation model with distributed springs across the support interface offers an option, for example, in Kawashima's study, 62 whereas the implementation of the contact point search algorithm based on local indentation of the body is cumbersome. Alternatively, Vassiliou's study 46 presented a simple and computationally efficient model for the problem of 3-D rocking cylinders freely standing on a rigid support surface. The inherent damping mechanism at the contact boundaries was considered as the variation of the contact force, which generates vibrations at the support surface. This damping was modeled through a fictitious linear spring element at the contact point, and the simulation results show a good agreement with Housner's COR model. 46 Following a similar idea, a fictitious element with a linear spring and a linear viscous damper is set at the contact point of the Bi-RIBS, see Figure 8. The fictitious element dissipates energy as the contact point vibrates around its equilibrium position in the global Z direction.
The equation of motion of the example bridge can be obtained by slightly modifying the above equations for undamped Bi-RIBS. Specifically, the Z component in the vector , Equation (16), is modified as u gz + b sin + h cos + (w − m s g∕k im ), where w and k im are the deformation and stiffness of the fictitious spring, respectively. Position w = 0 F I G U R E 8 Fictitious spring element to model the damping effect at contact in.
corresponds to the equilibrium position of the system at rest. The corresponding potential energy of the system, Equation (19), is also modified. As a result, the equation of motion for the tilt angle of the bearing becomes as follows: The equation of motion for the deformation of the fictitious spring w is as follows: where c im = im (2m s im ) and im = √ k im ∕m s . The stiffness should be set as a sufficiently large value to meet rigid body assumption of the support interface.

Initiation of bidirectional rocking motion and negative stiffness property of Bi-RIBS
The uplifting of the centroid of the girder following the 3-D rocking motion of the bearing occurs when the absolute acceleration of the sum of the piers and the ground motion in the horizontal plane satisfy the following relationship: √ (ü g,x +ü p,x ) 2 + (ü g,y +ü p,y The direction of the uplifting in the horizontal plane, namely the initial angle of the rolling motion of the bearing φ 0 , is the direction of the d'Alembert inertia force vector at the instance of the uplifting. This direction φ 0 can be determined as follows: Or sin φ 0 = −ü g,y +ü p,y √ (üg,x+üp,x) 2 +(ü g,y +ü p,y ) 2 (30) Note that the maximum force transmitted to the pier from the superstructure is limited by the initiation acceleration, namely . According to d'Alembert's principle, the resultant force horizontally acting on the girder can be directly computed from the resultant inertial acceleration acting on it. The interaction force between the bearing and the pier (the force transmitted to the pier from the girder) in a static equilibrium condition (see Figure 9) tends to decrease as the tilt angle increases, representing the inherent negative stiffness property of rocking structures. That is to say, the maximum force transmitted to the substructure can be easily adjusted by the designed inclined angle . On the other hand, the system behaves in a fixed bearing condition before the 3-D rocking motion of the bearing is triggered. The corresponding motion in the horizontal plane is described by the motion equation of two independent F I G U R E 9 Static equilibrium condition of the bearing.

Numerical simulation
The simulation method for rocking structures can be found in previous studies. 5,28,46 As shown in Figure 10, the numerical integration was carried out for different motion equations, depending on the activation of the initiation and switch conditions of the 3D rocking motion. In particular, the switch condition is to stop the 3D rocking motion and return to the fixed bearing condition so that the Bi-RIBS remains at rest at subsequent steps until the initiation condition is met again. The 3D rocking motion was assumed to enter its final stage if both the tilt angle and its velocity became very small ( < 10 −5 rad,θ < 10 −5 rad/s in this study), and then the simulation was reset to the fixed bearing condition. This process can be regarded as a perfectly inelastic final collision (discontinuous state switch) between the bearing and the interfaces to stop the 3-D rocking motion in practice. On the other hand, if the simulation is not reset to the fixed bearing condition, numerical divergence issues may appear since the motion of the rolling angle would last for a long time, and its velocity would become extremely large, which is unrealistic.

DYNAMIC CHARACTERISTICS OF THE BRIDGE WITH BI-RIBS
This section presents the results of limited case studies that were conducted to investigate the dynamic characteristics of the example bridge structure featuring the Bi-RIBS under bidirectional ground motions. The EW and NS components of the observed ground motion record of the 1971 San Fernando earthquake, Pacoima Dam Station (see Figure 11) were used as inputs aligned along the X and Y directions of the bridge, respectively. This bidirectional accelerogram was selected for its high PGA value, up to nearly 1200 gal, which is directly related to the initiation of the Bi-RIBS rocking motion. The seismic response of the Bi-RIBS was compared to that of the conventional pin bearing system subjected to the same bidirectional ground motion to investigate its response control effectiveness. A general and simplified example of bridges with continuous girders and an array of single piers in Japan was considered, see Figure 12. The fixed-bearing counterpart was selected from existing bridges in Japan with old bearing systems, where the conditions of the middle spans are fixed using pin bearings, while the other spans are movable using pin-roller bearings. These movable bearings can accommodate thermal effects. The bridge piers were assumed to have the same mechanical characteristics, and the conventional pin bearing system was assumed to perfectly restrain the relative movement between the girder and the pier in the horizontal plane, which led to a fixed bearing condition (or referred to as an uncontrolled condition). According to seismic design specifications for bridges in Japan, 63 a single vibration unit, consisting of the girder and piers in the fixed bearing condition and both ends in the movable bearing condition, can be used for seismic performance assessment. As a result, the single vibration unit for the fixed-bearing counterpart can be described by Equation (31), while that of the Bi-RIBS can be represented by the proposed analytical model. The parameters of the Bi-RIBS and those of the example bridge are presented in Table 1.

Free vibration and seismic responses of undamped and damped models
The dynamic analysis for the example bridge with the undamped model of the Bi-RIBS (α = 50 • , r = 0.9 m) was carried out numerically using MATLAB. The free vibration responses of the bridge structure were firstly investigated (see Figure 13) with the initial conditions ( 0 = 0.8 ,̇0 = 1, others remained at zero), which provided the necessary conditions for the onset of the 3-D rocking motion. The time-history responses of various parts of the bridge structure, that is, girder displacements, bearing rotations, and pier displacements, are presented in Figures  rolling angleφ shows a spike-like pattern, implying an unstable state of the 3-D rocking motion. The switch condition is then activated to stop the unstable state so that the system enters the fixed bearing condition under which the pier keeps vibration.
The same free vibration simulation was carried out for the damped model with low and high damping ratios, im = 0.05 and 50, and a natural angular frequency of the linear spring of 200 rad/s (the same parameters were used in Vassiliou's study for 3D rigid rocking columns standing on a rigid support surface 46 ). A comparison between the undamped and damped models is presented in Figure 14. It is seen that most quantities of interest get little affected by the damping properties at the support surface when the vibration magnitude is prominent for the first 4 s. After 4 s, while the rolling angle φ shows significant change, only a minor difference among the three models emerges in pier response displacements in the form of phase lag, and the tilt angle of the bearing at this moment is very small, indicating that the 3D rocking motion will soon switch to the fixed bearing condition. As the damping value increases, the 3D rocking motion stops faster, see the rolling angle.
The time-history responses of the bridge structure with undamped and damped models of the Bi-RIBS subject to the seismic excitation are presented in Figure 15. It is seen that the rocking motion of the Bi-RIBS is triggered after 6 s, indicating that the resultant acceleration of the ground motion and the pier exceeded the initiation value (g tan ≈ 1.2 g). Before that, the bridge structure behaved at the fixed bearing condition, in which the girder relative displacement and the quantities associated with the rocking motion of the bearing remained at rest as only the pier entered the F I G U R E 1 4 Comparison between undamped and damped models. vibration mode. During the rocking motion of the Bi-RIBS, the overturning of the bearing, defined as 100% of the tilt angle , did not occur, and the maximum tilt angle was less than 10 • , indicating a sufficient safety margin (α = 50 • ). The tilt angle returned to nearly zero after 10 s, and the system returned to the fixed bearing condition at nearly 13 s.
Regarding the effect of damping properties at the support interface, the simulation results between undamped and damped models almost coincide for the first 8 s, whereas the solutions show divergence after 8 s. More importantly, the influence on maximum responses of different quantities becomes different. The maximum tilt angle is reduced by nearly 15% from 8.04 rad (undamped model) to 6.90 rad (two damped models). The maximum girder relative displacement in the X direction reduces from 66 mm (undamped model) to 58 mm (two damped models), whereas that in the Y direction increases from 58 to 73 mm. The maximum pier displacements in the X and Y directions get little affected. Note that, as the assumed damping of the interface increases, the maximum responses of the three fundamental variables ( , u p,x , u p,y ) become smaller. The girder responses, on the other hand, are combination of the fundamental variables ( , φ) and the geometric features of the bearing, see Equation (16). The different amplitude and phase characteristics of the fundamental variables contribute to the girder responses, meaning that the maximum value of the girder response does not always occur at the same time as the maximum values of the fundamental variables.
Overall, the influence of the damping property at the support interface is considered secondary from the perspective of seismic performance assessment of the bridge with Bi-RIBS. The undamped model tends to give a slightly conservative assessment for the quantities directly related to safety, that is, the maximum tile angle of the bearing and the maximum pier displacement. Although a more comprehensive investigation is required in the future study, the present study focuses on exploring the possibility of the proposed Bi-RIBS being a seismic isolation device for protecting the bridge pier. For the sake of simplicity, the undamped model was used in the following numerical simulation.

Effects of varying the bearing parameters
The seismic response control effectiveness of the Bi-RIBS is mainly characterized by r and , analogous to the effects of the size and slenderness of other rocking structures. Several values of the Bi-RIBS size (r =0.9, 1.5, and 2.1 m) were first considered in comparison with the conventional pin bearing system (referred to as fixed bearing). The simulation results of the four independent generalized variables of the system under seismic excitation are presented in Figure 16. It is seen that the maximum X and Y pier displacements in these cases are nearly 45% smaller than those in the fixed bearing condition (denoted by black lines), indicating the expected response control effectiveness of the Bi-RIBS. As the size of the bearing r increased, the maximum pier displacements tended to decrease (from 84 to 66 and 59 mm in the X direction, and from 76 to 74 and 62 mm in the Y direction), but a larger r did not necessarily relate to a smaller bearing response-although the maximum tilt angle was smaller in the 2.1 m case than in the 0.9 m case, it was smallest in the r = 1.5 m case. This may be attributed to the complex nonlinear behaviors of rocking structures; previous studies have pointed out that the response of rocking structures is more unpredictable than that of the conventional fixed base structures. 28,31 After 15 s, the vibration of the Bi-RIBS attenuated to zero. The simulation results of the example bridge with various inclined angles (α = 20, 30, and 40 • , and r = 0.9 m) are presented in Figure 17. It is clearly seen that moment of the onset of the 3-D rocking motion became later as increased, since a higher initiation acceleration is required by a larger . The maximum tilt angle decreased as increased; in particular, overturning occurred in the case of being 20 • . The maximum pier displacement tended to increase with a larger (from 42 to 51 and 62 mm in the X direction, and from 30 to 55 and 80 mm in the Y direction), but was still nearly 50% smaller than that in the fixed bearing condition. These observations indicate that the dynamic characteristics of the bridge structure can also be effectively modified by the inclined angle , which is consistent with the findings for the Uni-RIBS. 48 This is important for the application of the Bi-RIBS as an isolation bearing since the adjustment of a Bi-RIBS with various inclined angles is easier than the adjustment of its size due to the limited space around the bearing part. In special situations, a large bearing size can be expected.

Maximum response to various intensities of the ground motion
To further explore the effects of various parameters on the maximum seismic response, the resultant PGA of the bidirectional accelerogram was scaled to various levels, ranging from 400 to 4000 gal. Two controlled performance indices, corresponding to the maximum tilt angle of the bearing and the maximum pier displacement, were investigated in their normalized form, as follows: where u p,abs = √ u 2 p,x (t) + u 2 p,y (t) denotes the absolute displacement of the pier top in the horizontal plane, and "uncontrolled" denotes the maximum absolute pier displacement in the fixed bearing condition. Specifically, J I =100% is defined as an overturning situation for the bearing, and J II <100% corresponds to a response reduction situation for the pier.
The parameters of the example bridge model are as follows: the mass ratio of the girder to the pier is selected as 3, 5, and 10, which are the usual values for bridge structures; the inclined angle ranges from 10 • to 60 • ; and the size r is selected The normalized maximum performance indices against various mass ratios are first presented in Figures 18A-F, where the size r is 0.9 m. While the PGA required to overturn the bearing tends to slightly drop as the mass ratio increases, the two seismic performance assessment indices (J I and J II ) show similar distributions and tendencies in the function space, indicating an insensitivity to the variation in the mass ratio.
It is interesting to see that for a given the pier performance index J II decreases as the PGA increases (see Figures 18E,F). This reveals an attractive effect of the rocking structure on controlling the pier response, as mentioned earlier: once the Bi-RIBS rocking motion is triggered, the maximum force transmitted to the pier is limited by the initiation acceleration (namely determined by ), regardless of the earthquake intensity. By contrast, in the conventional fixed bearing condition, the maximum force transmitted to the pier is always proportional to the earthquake intensity. As a result, the control effectiveness of the Bi-RIBS compared to the fixed bearing condition becomes more significant as the PGA increases prior to the overturning of the bearing.
Another finding is that for a given PGA the increase in the inclined angle tends to result in a smaller bearing tilt angle J I , but a larger pier displacement J II . This trade-off relationship exists in the seismic design for most isolation systems in bridges. The satisfied design parameter of the Bi-RIBS can be obtained by minimizing the bearing response at an allowable pier displacement level, or by minimizing the pier response at an allowable bearing rotation level.
The same simulation results in the cases with various sizes (r =0.9, 2.1, and 3.0 m) are shown in Figures 19A-F, where the mass ratio is 5. It is seen that as r increases, the PGA required to overturn the bearing for a given becomes larger, indicating greater stability of the rocking motion; meanwhile, the pier response reduction effectiveness is also improved for a given PGA level. This indicates that a greater Bi-RIBS size improves the control effect on both the maximum bearing and pier responses.
A comparison example for the performance indices of the bridge under various PGAs (1200, 2400, 3600 gal) is presented in Figure 20. As the absolute PGA was amplified to three times, the maximum pier displacements under the fixed bearing condition (referred to as uncontrolled) increased proportionally (e.g., from 82 to 164 and 246 mm in the X direction), whereas the increases in the Bi-RIBS cases were less evident (e.g., from 67 to 95 and 127 mm in the X direction). The peculiar seismic isolation effect of the Bi-RIBS on bridges was confirmed. On the other hand, the growth in the maximum  tilt angle was nonlinear, from 5.2 • to 14.6 • and 20 • . The vibration pattern of the tilt angle was considerably altered: three close-value peaks, which were not observed in the 1200 and 2400 gal cases, appeared in the 3600 gal case.

CONCLUSIONS
Considering the 3-D nature of observed earthquake ground motions, the bidirectional rocking isolation bearing system (Bi-RIBS) was proposed to provide seismic protection for bridge structures. The Bi-RIBS makes use of the 3-D rocking motion of the bearing as an isolation component to reduce the pier response and the gravity of the uplifted girder as the restoring force to control the girder displacement. The proposed Bi-RIBS can be regarded as a negative stiffness device, considering the inherent negative stiffness properties of rocking structures. As a result, the maximum force transmitted to the pier is limited by the initiation acceleration of the system and can be easily adjusted by the designed inclined angle . The possible application and structural details of the Bi-RIBS in bridge structures were discussed. A simplified model was developed to characterize the dynamics of an example bridge structure featuring such Bi-RIBS and to offer a computationally efficient approach for the seismic performance assessment. An extension model considering the damping property at the contact interface was also established for comparison purposes. A series of dynamic analyses were performed for the proposed models to investigate the dynamic characteristics of the bridge. The PGA of the ground motion was scaled to various levels to investigate the maximum seismic response control effectiveness of the Bi-RIBS in terms of the normalized maximum bearing tilt angle and the normalized maximum pier displacement. The following conclusions can be drawn.
1. The two design parameters of the Bi-RIBS, namely the size r and the inclined angle , are important for its seismic response control effectiveness. As increases, the maximum bearing tilt angle (coupled with the girder displacement) is effectively reduced while the maximum pier displacement tends to increase, resulting in a trade-off relationship between the minimization of the girder and pier responses. As r increases, both the maximum bearing tilt angle and the maximum pier displacement tend to decrease. However, a greater r requires a greater space in the bearing part of the bridge. 2. The maximum pier displacements are effectively reduced from that in the conventional pin bearing system. This effect becomes more significant as the PGA level rises since the maximum force transmitted to the piers is limited by the specified of the Bi-RIBS regardless of the earthquake intensity. 3. The influence of the damping property at the support interface under rigid body assumption is considered secondary from the perspective of seismic performance assessment of the bridge with Bi-RIBS. The undamped model tends to give a slightly conservative assessment for the quantities directly related to safety, that is, the maximum tile angle of the bearing and the maximum girder displacement. Nevertheless, this damping effect in practice is the combination of the material properties and contact conditions such that further investigation is required.
It is suggested that further investigation should be conducted in the following aspects: 1. Effect of vertical ground motions. Although the factor of vertical ground motion was included in the modeling procedure, we did not perform a specific analysis for simplicity. Previous research has indicated that the vertical component of ground motions has different dominant frequency and amplitude compared to its two horizontal components. Considering its inherent characteristics, such as relatively smaller PGAs and/or short-period predominant motions, its influence may not be as noticeable. 2. Comparative analysis with previous seismic isolation bearings. A comprehensive comparison with previous seismic isolation bearings has been omitted. However, it is recommended that a future study should perform a probabilistic assessment method in combination with an extensive parametric study for a selected bridge with different isolation bearings to provide a more comprehensive understanding of the performance of the proposed bearing. 3. Experimental verification and refined FEM verification. These are critical in developing a new isolation device.

A C K N O W L E D G M E N T S
The research described in this paper was partially supported by the Japan Society for the Promotion of Science (JSPS), KAKENHI, Grant-in-Aid for Scientific Research (A) (Grant No. 20H00255) and Japan Society for the Promotion of Science (JSPS), KAKENHI, Grant-in-Aid for Young Scientists (Grant No. 22K14313).

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.