Seismic protection of the horizontally curved bridge with semi-active variable stiffness damper and isolation system

Abstract

Seismic vibration due to earthquake can damage the curved bridges that are the main interconnecting component for traffic separation structures of the urban highway system. Failure of bridges during and after a seismic event is vulnerable as bridges are lifeline structures. This work investigates the application of semi-active variable stiffness damper for a seismic control of the horizontally curved bridge isolated with different passive devices. The main objectives of the study are to investigate the effectiveness of the hybrid system and to find the optimum hybrid system for the seismic control of the curved bridge with different control laws of a damper. The selected bridge is a three-span continuous concrete box girder supported on pier and rigid abutment. The bridge deck is modeled as a single spine beam and the supporting pier is modeled as linear lumped mass system. The bridge is excited with four different ground motions having different ground motion characteristics with all three-ground motion components (horizontal as well as vertical). The results of the analysis demonstrate that the use of semi-active variable stiffness damper with different isolators is very effective in controlling the response of the curved bridge. The combination of semi-active variable stiffness damper and lead rubber bearing can provide an effective way for overall seismic control of the curved bridge. The use of modified switching control law for damper is as effective as switching control law with less number of sensors.

Keywords

curved bridge elastomeric rubber bearing friction pendulum system lead rubber bearing seismic analysis seismic isolation semi-active variable stiffness damper

Introduction

Today, curved concrete bridges have become the inherent part of the traffic separation structure in the urban transportation system due to geometric constraints, pleasant aesthetic view, easy construction, and higher torsion rigidity. Bridges are highly vulnerable in the seismic event and may get failed or damaged due to shear failure of piers and excessive displacement of bridge deck and bearings; unseating, pounding, and the rotation of the superstructure; or outward displacement (Galindo et al., 2009). In the past, the damage and failure of curved bridges caused by earthquakes such as San Fernando earthquake of 9 February 1971, the Northridge earthquake of 17 January 1994, and the Wenchuan earthquake of 12 May 2008 (Fenves and Ellery, 1998; Galindo et al., 2009; Liu and Wang, 2014; Williams and Godden, 1976) were noticed. As bridges are lifeline structures, they must withstand in such catastrophic events. One of the ways to protect such structures is seismic isolation. Kunde and Jangid (2003) recognized, in their literature, the use of a variety of passive isolation devices for seismic control of bridges by a number of researchers worldwide. The use of passive isolation technique is highly effective in reducing the forces on the structure at the cost of increase in bearing and deck displacement. This increase in displacement may result into unseating at the bearing, hammering of deck, and creating uncomfortable conditions for the traffic movement on the deck. In order to protect the bridge, a hybrid system that consists of isolation device with dampers can be applied.

In the past, various researchers provided a bit of damping devices and their control law for seismic protection of the structure. It was observed that the application of these devices to control the responses of the different structures was successfully accomplished. Among the different control devices, semi-active devices are renowned due to their properties such as low-power requirements, working at varying temperature, mechanical simplicity, large force capacity, high dynamic range, and robustness. In this work, semi-active variable stiffness damper (SAVSD) is considered to be a seismic control of the curved bridge.

A number of researchers used the stiffness damper with different control algorithms and found its effectiveness. Control methods with active variable stiffness (AVS) systems for seismic control of buildings are presented by Yang et al. (1996). It was found that AVS system is effective for response control of buildings subjected to earthquake motion. Yang et al. (2000) proposed the resetting control law for semi-active stiffness damper for seismic control of structure and found control law to be effective in controlling building response depending on the types of earthquake motion. A non-resonant control algorithm for AVS was described by Nasu et al. (2001) to create a non-resonant state against seismic analysis of structure and found results to be with the minimum response. Application of SAVSD has been investigated for a benchmark cable-stayed bridge by He et al. (2001) and Agrawal et al. (2003) and found that damper is quite effective in reducing the peak response quantity of the bridge. Kori and Jangid (2007, 2008) studied the performance of SAVSD for seismic control of structures and highway bridges. It was observed that SAVSD is effective in reducing the response of the structure and is also effective in reducing the displacement of the isolated bridge. Application of SAVSD has been investigated for seismic control of the piping system by Kumar et al. (2013) and found that damper is very effective in the seismic response mitigation of piping system.

In order to protect the curved bridge against catastrophic seismic events, some researchers studied the application of different devices. Lei and Chien (2004) studied the effect of isolation on seismic performance of the curved bridge. In their study, it was found that the isolation performance on base shear reduction of the curved bridge depends on the type of earthquake and the angle of curvature of the structure. Also, it was concluded that the application of the isolator with higher flexibility in the lead rubber bearing (LRB) or with a lower frictional coefficient in the friction pendulum system (FPS) significantly reduces the base shear. Liu et al. (2011) established the three-dimensional (3D) computational model for a double-pier curved continuous girder bridge, and they added viscous dampers at the positions of sliding bearings. From the results, it was concluded that viscous dampers can reduce the difference in internal force between the inner pier and the outer pier in double-pier curved bridge, and it can effectively reduce the bending and twisting coupling effect of the curved bridge. Ates and Constantinou (2011a, 2011b) consider the isolated and non-isolated curved bridges to determine the effects of earthquake on the response of the bridge. They calculated the responses using response history analysis and response spectrum analysis, including the effect of soil–structure interaction. They found that there is a decrease in internal forces of the deck in the isolated curved bridge. Monzon et al. (2012) experimentally studied the seismic response of full and hybrid isolated curved bridges and found both the techniques to be successful in protecting the column. Although the SAVSD is studied for building’s bridges and piping system, the evaluation of optimal SAVSD for curved isolated bridge is not investigated so far. Therefore, it will be interesting to investigate the optimal device controlling parameter for the seismic control of isolated curved concrete box girder bridge subjected to different earthquake motions.

Thus, the purpose of this study is to investigate the effectiveness of the hybrid semi-active systems for seismic response control of the curved concrete box girder bridge subjected to different earthquake ground motions. The objectives of the study are (a) to investigate and compare the effectiveness of hybrid system: SAVSD+elastomeric rubber bearing (ERB), SAVSD+LRB, and SAVSD+FPS for seismic response control of the curved concrete box girder bridge; (b) to investigate the influence of variation in important parameters of the SAVSD damper on the response of the bridge; and (c) to investigate the effectiveness of different control strategies for seismic control of the curved concrete box girder bridge.

Modeling of curved bridge

The bridge model considered for this study is the design example no. 6 of “Federal Highway Administration (FHWA) Seismic Design Course,” formulated by Berger/Abam Engineers (1996), which was also studied by Ates and Constantinou (2011a, 2011b). The bridge is having three spans, cast-in-situ concrete box girder supported over the reinforced concrete column as shown in Figure 1. The span length of the bridge along the centerline is 27.25, 33.5, and 27.25 m and width of the deck is 11.8 m. The roadway alignment over the bridge is sharply horizontally curved (104°). The intermediate piers are made of rectangular cross-section having an area of 1.7 m² and 6.4 m height from the ground surface. The superstructure consists of a three-cell deck with 10% slope horizontally. The properties of the bridge system are same as considered in FHWA. It is assumed that the bridge is structurally safe (satisfying load and displacement demand) when the isolation device is introduced in between the pier and the superstructure. The deck is isolated from the supports by eight isolators (two at each pier cap and abutments), which are located at the corners of the cap. In addition, four SAVSD dampers in chord direction and four in radial direction were installed between the deck and the supporting pier and abutments. The material and geometrical properties of the bridge are given in Table 1. The following assumptions are considered for seismic analysis of the bridge:

The deck and piers of the bridge system will remain in the elastic state under the action of earthquake excitation.

The abutments of the bridge are rigid and piers are fixed at the ground level.

The soil–structure interaction is ignored for the analysis.

The deck of the bridge is modeled as a spine beam, which is made with small, straight two-node beam elements having 6 degree of freedom (DOF) at each node.

The mass of each element is distributed between the two adjacent nodes in the form of lumped mass.

The stiffness of non-structural elements (sidewalk and parapet) is neglected.

The bridge is subjected to horizontal and vertical components of earthquake ground motion.

Figure 1.

(a) Curved bridge plan and (b) developed elevation.

Table 1.

Properties of bridge deck and piers.

Properties	Deck	Piers
Cross-sectional area (m²)	6.238	1.7
Moment of inertia (m⁴)	2.7074	0.409416
Young’s modulus of elasticity (N/m²)	3.2 × 1010	3.2 × 1010
Mass density (kg/m³)	2.5 × 103	2.5 × 103
Length/height (m)	88	6.4

Figure 2 shows the idealized mathematical model of the curved bridge. The selected bridge is modeled as a multi-degrees-of-freedom (MDOF) system. Based on the detailed drawings of the curved bridge, a 3D evaluation model is made in MATLAB. The number of elements considered in the bridge deck and piers are 32 and 4, respectively.

Figure 2.

Mathematical model of the isolated bridge.

Governing equations of motion

The equations of motion for the curved bridge system subjected to seismic loads are expressed in the matrix form as follows

\begin{matrix} [M] {\overset{\cdot\cdot}{u} (t)} + [C] {\overset{\cdot}{u} (t)} + [K] {u (t)} \\ = - [M] [η] {{\overset{\cdot\cdot}{u}}_{g} (t)} + [b] {F (t)} \end{matrix}

(1)

{u (t)} = {x_{1}, y_{1}, z_{1}, x_{2}, y_{2}, z_{2}, \dots, x_{N}, y_{N}, z_{N}}^{T}

(2)

{{\overset{\cdot\cdot}{u}}_{g} (t)} = {\begin{matrix} {\overset{\cdot\cdot}{x}}_{g} (t) \\ {\overset{\cdot\cdot}{y}}_{g} (t) \\ {\overset{\cdot\cdot}{z}}_{g} (t) \end{matrix}}

(3)

where [M], [C], and [K] are the mass, damping, and stiffness matrices, respectively; ${\overset{\cdot\cdot}{u} (t)}$ , ${\overset{\cdot}{u} (t)}$ , and ${u (t)}$ are structural acceleration, structural velocity, and structural displacement vector, respectively; ${F (t)}$ is the vector of control force inputs of the devices (isolator and damper), and ${{\overset{\cdot\cdot}{u}}_{g} (t)}$ is the vector of earthquake ground acceleration acting in three directions (i.e. two horizontal and one vertical); ${\overset{\cdot\cdot}{x}}_{g} (t)$ , ${\overset{\cdot\cdot}{y}}_{g} (t)$ , and ${\overset{\cdot\cdot}{z}}_{g} (t)$ represent the earthquake ground acceleration (m/s²) in chord, vertical, and radial directions, respectively; $[η]$ is the influence coefficient matrix, relating the ground acceleration to the bridge DOF; $[b]$ is the matrix relating the forces produced by the control devices (isolator and damper) to the bridge DOF that includes 1 and 0. First, individual elemental stiffness matrix of each element with 6 DOF at each node using beam element is formed; using a transformation matrix, it is converted into the global elemental matrix. A global structural stiffness matrix is assembled using elemental global stiffness matrices. Similarly, the global mass matrix for the curved bridge is assembled. Then, structural stiffness matrix for numerical integration is obtained by dynamic condensation from the global structural stiffness matrix. The global damping matrix is a combination of the distributed 5% inherent Rayleigh damping in the first two modes. Consequently, the response of the curved bridge is evaluated step-by-step at successive increments of time using Newmark-beta method. It is assumed that the properties of the bridge system remain constant during the time increment.

Modeling of control devices

The force generated by the control device is depending on the type of isolator and damper. For this study, SAVSD and for the isolation of the curved bridge, ERB, LRB, and FPS are considered. The detail of modeling of each device is described below.

Elastomeric rubber bearing

ERB consists of thin layers of rubber and steel plates built in alternate layers. The ERB is vertically stiff and horizontally flexible. The vertical stiffness of the bearing is several hundred times the horizontal stiffness, which is due to the presence of internal steel shims, and horizontal stiffness of the bearing is controlled by the low shear modulus of elastomer. The steel shims prevent the bulging of rubber as well. The damping in the bearing can be increased by adding extra-fine carbon block, oils or resins, and other proprietary fillers (Naeim and Kelly, 1999). The dominant features of ERB system are the parallel action of linear spring and viscous damping. Figure 3(a) shows the schematic diagram of the ERB system. The restoring force developed in the bearing is given by

F_{x} = c_{b} {\overset{\cdot}{x}}_{b} + k_{b} x_{b}

(4)

F_{y} = c_{b} {\overset{\cdot}{y}}_{b} + k_{b} y_{b}

(5)

where $c_{b}$ and $k_{b}$ are the damping and stiffness of ERB system, respectively; x_b and y_b are the displacements of the bearing in chord and radial directions, respectively; and ${\overset{\cdot}{x}}_{b}$ and ${\overset{\cdot}{y}}_{b}$ are the velocity of the bearing in chord and radial directions, respectively. The stiffness and damping of ERB system are selected to provide the required values of the two parameters, namely the isolation time period (T_b) and the damping ratio (ξ_b) as

T_{b} = 2 π \sqrt{\frac{M}{k_{b}}}

(6)

ξ_{b} = \frac{c_{b}}{2 M ω_{b}}

(7)

where M is the total mass of the bridge deck and $ω_{b} = 2 π / T_{b}$ is the isolation frequency.

Figure 3.

Schematic diagram for isolation system: (a) ERB, (b) LRB, and (c) FPS.

Lead rubber bearing

LRB is similar to the elastomeric bearing, which consists of thin layers of steel and high damping rubber plates built in alternate layers with a lead plug at its center. The LRB was invented in New Zealand in 1975 (Robinson and Tucker, 1977) and has been widely used in New Zealand and other nations. This bearing provides a vertical support, horizontal flexibility, and energy absorbing capacity (Robinson, 1982). In addition, LRB can safely tolerate strain up to 200% for the occasional very large earthquakes (Tyler and Robinson, 1984). The interlocked steel plates in the bearing force the lead core to deform its entire volume in pure shear. From past test and theory analysis, it is observed that the force deformation behavior of the LRB is nonlinear in nature. For this study, Wen’s (1976) model is employed to characterize the hysteretic behavior of the LRB bearings, which is shown schematically in Figure 3(b). The restoring force developed in the isolation bearing is given by

F_{x} = c_{b} {\overset{\cdot}{x}}_{b} + α k_{b} x_{b} + (1 - α) f_{y} Z_{x}

(8)

F_{y} = c_{b} {\overset{\cdot}{y}}_{b} + α k_{b} y_{b} + (1 - α) f_{y} Z_{y}

(9)

where k_b and c_b are the initial stiffness and viscous damping of the bearing, respectively; f_y is the yield strength of the bearing; $α$ is an index which represents the ratio of post- to pre-yielding stiffness; and Z_x and Z_y are the hysteretic dimensionless displacement components governed by Wen’s model. In this study, these components are considered to satisfying the following bi-directional interaction nonlinear first-order differential equation (Park et al., 1986)

\begin{matrix} q {\begin{matrix} {\overset{\cdot}{Z}}_{x} \\ {\overset{\cdot}{Z}}_{y} \end{matrix}} \\ = [\begin{matrix} A - β sign ({\overset{\cdot}{x}}_{b}) | Z_{x} | Z_{x} - τ Z_{x}^{2} & - β sign ({\overset{\cdot}{y}}_{b}) | Z_{y} | Z_{x} - τ Z_{x} Z_{y} \\ - β sign ({\overset{\cdot}{x}}_{b}) | Z_{x} | Z_{y} - τ Z_{x} Z_{y} & A - β sign ({\overset{\cdot}{y}}_{b}) | Z_{y} | Z_{x} - τ Z_{y}^{2} \end{matrix}] \\ {\begin{matrix} {\overset{\cdot}{x}}_{b} \\ {\overset{\cdot}{y}}_{b} \end{matrix}} \end{matrix}

(10)

where q is the yield displacement; β, τ, and A are dimensionless parameters that predict response from the model closely matches the experimental results. In this study, the values of parameters considered for LRB are as follows: q = 25 mm, A = 1, β = 0.5, and τ = 0.5. The LRB system is mainly characterized by the isolation period (T_b), damping ratio (ξ_b), and normalized yield strength (F_o). The bearing parameters T_b and ξ_b are computed by equations (6) and (7), respectively. In this study, the damping ratio that equals to 0.05 is considered. The normalized yield strength (F_o) is given by

F_{o} = \frac{\sum f_{y}}{W}

(11)

where W is the total weight of the deck.

Friction pendulum system

FPS is the most popular sliding isolation bearing. It consists of two sliding plates connected by an articulated slider. Sliding plates are made with hard, dense chrome over steel with spherical concave surface and slider wrap with a Teflon-based high bearing capacity material (Zayas et al., 1990). The FPS is working with the concept as similar to the sliding pendulum. The lateral force produced by FPS during execution is equal to the sum of the frictional and restoring forces. The frictional force is developed when the slider is in contact with the sliding plate and the restoring force is developed because of the movement of the structure upward along a circular concave surface (Jangid, 2005a, 2005b). As soon as the earthquake force exceeds the static value of friction force, the isolation system is activated. For this study, a continuous model of frictional force of a sliding system presented by Constantinou et al. (1990) is used. The isolation period (T_b) of FPS is expressed as

T_{b} = 2 π \sqrt{\frac{r}{g}}

(12)

where r is the radius of curvature of the concave surface and g is the acceleration due to gravity. Figure 3(c) shows the schematic diagram for FPS. The restoring force developed in the FPS system is considered as follows

F_{x} = k_{b} x_{b} + μ w Z_{x}

(13)

F_{y} = k_{b} y_{b} + μ w Z_{y}

(14)

where k_b is the bearing stiffness provided by the virtue of inward gravity action at the concave surface; µ and w are friction coefficient and weight shared by the individual bearing of the curved bridge; and Z_x and Z_y are Wen’s constant parameter computed using equation (10). The hysteretic displacement components, Z_x and Z_y, are bounded by peak values of ±1, to account for the conditions of sliding and non-sliding phases. The parameters of Wen’s model are selected such that the response is nearly same as the experimental results. In this study, the values of the parameters considered are q = 0.25 mm, A = 1, β = 0.9, and τ = 0.1 to provide a rigid plastic behavior (typical coulomb-friction behavior) for FPS. The isolation stiffness, k_b, is selected such that it provides the specific value of the isolation period satisfying equation (6). Thus, modeling of FPS requires the two parameters, namely the friction coefficient (µ) and the isolation period (T_b).

Semi-active variable stiffness damper

The SAVSD is a hydraulic damper. In SAVSD, the system of seismic response reduction for controlling is composed of control devices and energy dissipation devices. The control devices control the valve of the energy dissipation devices jointed to the main structure by switching off or on the valve as per the response information of the structure. When the structure reaches the maximum displacement position and starts to move in the reverse direction, the control devices open the energy dissipation devices and the stiffness of the system becomes 0. When the energy dissipation devices are closed, the system provides the certain stiffness and absorbs the seismic energy, and when it is open, the absorbed seismic energy by the system dropped into the energy dissipation device (Xinghua, 2000). Figure 4 shows the schematic model of the damper mechanism. The damper is composed of a cylinder-piston system with a valve in the bypass pipe connecting two sides of the cylinder filled with hydraulic oil (Djajakesukma et al., 2002; Yang et al., 2000). This device, by dynamically changing its damping and stiffness characteristics, can bring forth a large control action. The stiffness by the hydraulic damper (k_f) is provided by the bulk modulus of the fluid in the cylinder. The total stiffness of the damper is the combination of fluid stiffness (k_f) and device stiffness (k_c). Therefore, the equivalent effective stiffness of the entire damper device (k_hi) for the ith device unit is given by

k_{hi} = \frac{k_{f} k_{c}}{k_{f} + k_{c}}

(15)

Figure 4.

(a) Schematic diagram and (b) ideal force–displacement behavior of semi-active variable stiffness damper.

This stiffness of damper can operate in the passive or semi-active modes. When the valve is either open or closed, the damper serves as passive. In passive mode, when the valve is closed, the device adds constant stiffness (k_hi) to the structure. On the other hand, when the valve is open, the device adds a little amount of damping to the structure. When the valve of the hydraulic damper opens at a certain time interval and closes at another time interval, the damper serves as semi-active. The control force f_i(t) for SAVSD can be calculated as

f_{i} (t) = k_{hi} v_{si} u_{i}

(16)

where u_i is the relative displacement and v_si is the switching parameter that depends on the control law.

Control laws

The performance, efficiency, economy, reliability, and stability of the structure with the control device are depending on the control algorithms. To execute the control algorithms, the control information should be simple and have very little computational efforts and should not depend on more complicated control theory. For this study, the following simple control laws are considered.

Switching control law

Yang et al. (1996) derived the control law which is based on the sliding mode. Using this law, SAVSD is working as the on–off device, and the valve is vibrating to open and close as per requirement. As the valve of damper closes, it adds the effective stiffness k_hi to the structure and when the valve is open, effective stiffness becomes 0. Thus, by switching the device on and off in a synchronized fashion, the structural response can be controlled. The control force f_i(t) can be calculated from equation (16) and the v_si expressed for switching control law as

v_{si} = {\begin{matrix} 1 if u_{i} {\overset{\cdot}{u}}_{i} ⩾ 0 \\ 0 otherwise \end{matrix}

(17)

where $v_{si} = 1$ and $v_{si} = 0$ if the ith semi-active stiffness damper unit is closed and open, respectively.

Modified switching control law

The control system of sliding mode or switching law is based on the information of the structural incremental displacement or velocity (Xinghua, 2000). In the switching control law, the valve opening depends upon the velocity and displacement response in the structure, whereas, in the modified control law, sign of the relative velocity is replaced by sign of the rate of change of relative displacement. Hence, the number of sensor required becomes less. Thus, in the modified switching control law, switching on and off in a control valve is based on the relative displacement and the incremental displacement. The displacement, u_i, and the incremental displacement, $Δ u_{i}$ , are obtained using only measured information from a limited number of displacement sensors is being use. Hence, the control force f_i(t) can be calculated from equation (16) and the v_si is expressed for modified switching control law as

v_{si} = {\begin{matrix} 1 if u_{i} Δ u_{i} ⩾ 0 \\ 0 otherwise \end{matrix}

(18)

where $v_{si} = 1$ if the ith semi-active stiffness damper in the ith damper unit is closed when its incremental displacement, $Δ u_{i}$ , and the displacement, u_i, are both positive and negative and on the other hand, if $Δ u_{i}$ and u_i are in the opposite direction (different signs) $v_{si} = 0$ , if it is open.

Numerical study

The structural model of the isolated curved box girder bridge with SAVSD is developed in MATLAB, and seismic time history analysis is carried out. For the analysis, four real earthquake ground motions, namely El Centro (1940), Northridge (1994), Loma prieta (1989), and Kobe (1995) earthquakes with all three components, are considered. Table 2 shows the important feature of all considered ground motions. In the analysis, all the ground motions are used with the full strength when the east-west and the north-south components are applied in chord and radial directions of the bridge, respectively.

Table 2.

Earthquake data for numerical simulation.

Earthquake	Recording station	Duration (s)	PGA (g)
Earthquake	Recording station	Duration (s)	Vertical	N-S	E-W
El Centro (1940)	117 El Centro Array #9	40	0.205	0.313	0.215
Loma Prieta (1989)	16 LGPC	24.965	0.89	0.563	0.605
Northridge (1994)	24514 Sylmar—Olive View Med FF	40	0.535	0.834	0.604
Kobe (1995)	KJMA	48	0.343	0.821	0.599

PGA: peak gravity acceleration.

In order to evaluate the performance, a set of performance evaluation criteria were considered. The evaluation criteria J₁–J₄ are described to measure the reduction in peak base shear, peak overturning moment, peak mid-span displacement, and peak mid-span acceleration of the curved box girder bridge. These criteria are calculated by normalizing the peak response quantities of the controlled bridge by the corresponding peak response quantities for the uncontrolled bridge (with rigid deck to pier connection); J₆–J₉ are described to measure the reduction in norm base shear, norm overturning moment, norm mid-span displacement, and norm mid-span acceleration of the curved box girder bridge. These criteria are calculated by normalizing the norm response quantities by the corresponding norm response of the uncontrolled bridge. J₅ and J₁₀ are associated with the peak and the norm displacement of the bearing and obtained by calculating the relative displacement of bearing end. J₁₁ is related to the peak control force generated by the device normalized by the seismic weight of the bridge. The performance of the SAVSD is mainly depending upon the effective stiffness of the damper. In order to obtain the optimum effective stiffness, a device stiffness ratio is considered. Here, the device stiffness ratio (K_r) is defined as the ratio of effective stiffness of the damper device (k_hi) to isolation stiffness (k_iso) for the bridge and is given by

K_{r} = \frac{k_{hi}}{k_{iso}}

(19)

This research work is focused on the effect of the stiffness ratio (K_r) of SAVSD on the performance of the curved bridge in order to identify the optimum value of a parameter for getting better seismic control of the curved isolated bridge. In addition, this study also aims to investigate the optimum hybrid combination when the curved bridge is controlled with SAVSD+ERB, SAVSD+LRB, and SAVSD+FPS system. The values of important parameters of isolators are as follows: for the ERB, ξ_b = 0.15; for the FPS, µ = 0.11; and for the LRB, F_o = 0.1; and the isolation period is equal to 2 s for overall better response throughout the analysis.

The optimum value of the device stiffness ratio is evaluated using the switching control law with device stiffness ratio (K_r) ranging from 0.025 to 0.25. The variation of evaluation criteria for peak base shear, peak overturning moment, peak mid-span displacement, peak mid-span acceleration, peak displacement of the bearing, and peak control force for the different K_r values of SAVSD+ERB, SAVSD+LRB, and SAVSD+FPS are shown in Figures 5 to 10. From the nature of plots, it is observed that (a) the peak base shear and overturning moment at the base of the pier increase by increasing K_r, (b) the mid-span displacement and peak bearing displacement decrease with increasing K_r, (c) the mid-span acceleration slightly increased/decreased for different earthquakes with increasing K_r, (d) the peak control force gets increased by increasing K_r, for all considered earthquakes and for all considered combinations in the chord and radial directions.

Figure 5.

Effect of stiffness ratio (K_r) of SAVSD on peak base shear (J₁) and peak overturning moment (J₂) for SAVSD + ERB (T_b = 2 s and ξ_b = 15%).

Figure 6.

Effect of stiffness ratio (K_r) of SAVSD on peak mid-span displacement (J₃), peak mid-span acceleration (J₄), peak displacement of bearing (J₅), and peak control force (J₁₁) for SAVSD + ERB (T_b = 2 s and ξ_b = 15%).

Figure 7.

Effect of stiffness ratio (K_r) of SAVSD on peak base shear (J₁) and peak overturning moment (J₂) for SAVSD + LRB (T_b = 2 s and F_o = 0.10).

Figure 8.

Figure 9.

Effect of stiffness ratio (K_r) of SAVSD on peak base shear (J₁) and peak overturning moment (J₂) for SAVSD + FPS (T_b = 2 s and µ = 0.11).

Figure 10.

It is noted that maximum reduction of the base shear and overturning moment occurs when using SAVSD+ERB in chord direction and when using SAVSD+LRB in radial direction, whereas the minimum reduction when using SAVSD+FPS combination. Lower value of K_r will be better for reducing the base shear and overturning moment, whereas the higher value prefers for lesser mid-span and bearing displacements. Furthermore, it is noted that the use of SAVSD is more effective in reducing response in radial direction as compared to chord direction. From the trend of the graph in Figures 5 to 10, it is also observed that the efficient value of K_r is depended on the earthquake ground motion. Figures show maximum reduction in the base shear, and bending moment is observed in the Kobe earthquake as compared to the other, whereas minimum displacement of bearing is observed in El-centro earthquake. For an overall better response of curved bridge and isolation devices, consider the optimum value of K_r as 0.1 for all hybrid combinations under the different earthquake motions.

Furthermore, the numerical study is performed for the two cases, modified switching control law and for the passive mode, which deal with the cases in which v_si kept changing as per control law and as a constant value 1 respectively. Tables 3 and 4 summarize the values of all the evaluation criteria for the Kobe earthquake for switching control law, modified switching control law, and passive mode for all considered combinations. From the table, it is noted that switching control law and modified switching control law provide the same control response. Switching control law and modified switching control law strategy are slightly more effective in controlling peak base shear and peak overturning moment as compared to passive case. Peak displacements of bearing in all cases in SAVSD+ERB remain same, whereas SAVSD+LRB and SAVSD+FPS combinations are quite more effective in switching and modified switching control law as compared to passive mode. It is also noted that SAVSD+ERB and SAVSD+LRB combinations are more effective in controlling the base shear and overturning moment, whereas SAVSD+FPS is more effective in controlling the mid-span and bearing displacement. From the trend of plots and result values in Tables 3 and 4, for effective reduction in base shear and overturning moment with controlling deck and bearing displacement as overall effective response control, the SAVSD+LRB combination can be more effective than other combinations in curved bridge.

Table 3.

Evaluation criteria J₁–J₄ and J₆–J₉ for Kobe (1995) earthquake.

Control type			Passive			Switching semi-active			M. S. semi-active
Peak value	Location		ERB	LRB	FPS	ERB	LRB	FPS	ERB	LRB	FPS
J ₁	Pier 1	Chord	0.1603	0.1544	0.1618	0.1374	0.1385	0.1527	0.1374	0.1386	0.1527
	Pier 1	Radial	0.0323	0.0156	0.0555	0.0312	0.0136	0.0552	0.0312	0.0136	0.0552
	Pier 2	Chord	0.1081	0.1063	0.1017	0.0921	0.0951	0.0910	0.0921	0.0951	0.0910
	Pier 2	Radial	0.0325	0.0184	0.0514	0.0312	0.0147	0.0510	0.0312	0.0147	0.0510
J ₂	Pier 1	Chord	0.1725	0.1645	0.1746	0.1482	0.1481	0.1647	0.1482	0.1481	0.1647
	Pier 1	Radial	0.0346	0.0179	0.0580	0.0335	0.0166	0.0576	0.0335	0.0166	0.0576
	Pier 2	Chord	0.1139	0.1129	0.1075	0.0966	0.1008	0.0957	0.0966	0.1008	0.0957
	Pier 2	Radial	0.0336	0.0220	0.0512	0.0323	0.0176	0.0509	0.0323	0.0176	0.0509
J ₃	Deck	Chord	1.2322	0.9806	0.9020	1.0421	0.8270	0.8014	1.0421	0.8270	0.8014
		Radial	1.3678	0.0559	1.3746	1.3177	0.0517	1.3522	1.3177	0.0517	1.3522
		Vertical	2.0776	2.0746	1.9597	2.1119	2.0728	2.0013	2.1119	2.0728	2.0013
J ₄	Deck	Chord	0.5857	0.6411	0.6046	0.5512	0.5992	0.5637	0.5512	0.5992	0.5637
		Radial	0.3627	0.1354	0.3532	0.3692	0.1280	0.3447	0.3692	0.1280	0.3447
		Vertical	1.1569	1.1925	1.1956	1.1740	1.1895	1.2017	1.1741	1.1895	1.2017
J ₆	Pier 1	Chord	0.1723	0.1937	0.2188	0.1361	0.1671	0.2081	0.1361	0.1671	0.2081
	Pier 1	Radial	0.0573	0.0209	0.0937	0.0484	0.0178	0.0930	0.0484	0.0178	0.0930
	Pier 2	Chord	0.1471	0.1710	0.1834	0.1152	0.1477	0.1722	0.1152	0.1477	0.1722
	Pier 2	Radial	0.0404	0.0176	0.0825	0.0357	0.0148	0.0828	0.0357	0.0148	0.0828
J ₇	Pier 1	Chord	0.1851	0.2066	0.2333	0.1462	0.1781	0.2214	0.1462	0.1781	0.2214
	Pier 1	Radial	0.0608	0.0240	0.0955	0.0512	0.0205	0.0944	0.0512	0.0205	0.0944
	Pier 2	Chord	0.1561	0.1830	0.1937	0.1219	0.1580	0.1817	0.1219	0.1580	0.1817
	Pier 2	Radial	0.0406	0.0202	0.0823	0.0359	0.0171	0.0828	0.0359	0.0171	0.0828
J ₈	Deck	Chord	1.5488	1.3690	1.1888	1.3253	1.2081	1.0576	1.3253	1.2081	1.0576
		Radial	1.7787	0.0415	1.3267	1.6972	0.0406	1.2183	1.6972	0.0406	1.2183
		Vertical	2.5271	2.4020	2.2768	2.5325	2.3991	2.2814	2.5326	2.3992	2.2814
J ₉	Deck	Chord	0.5884	0.6220	0.5917	0.5540	0.5921	0.5685	0.5540	0.5921	0.5685
		Radial	0.3401	0.0997	0.3336	0.3256	0.0959	0.3196	0.3256	0.0959	0.3196
		Vertical	1.3536	1.3221	1.2648	1.3570	1.3208	1.2690	1.3571	1.3208	1.2690
J ₁₁	Bridge	Chord	0.0926	0.0750	0.0749	0.0801	0.0638	0.0666	0.0801	0.0638	0.0666
J ₁₁	Bridge	Radial	0.0610	0.0151	0.0579	0.0554	0.0119	0.0548	0.0554	0.0119	0.0548

ERB: elastomeric rubber bearing; LRB: lead rubber bearing; FBS: friction pendulum system; M. S.: modified switching control law.

Table 4.

Evaluation criteria J₅ and J₁₀ for Kobe (1995) earthquake.

Control type			Passive			Switching semi-active			M. S. semi-active
Peak value	Location		ERB	LRB	FPS	ERB	LRB	FPS	ERB	LRB	FPS
J ₅	B-1	Chord	0.2243	0.1879	0.1926	0.1962	0.1613	0.1710	0.1962	0.1613	0.1710
	B-1	Radial	0.1773	0.0595	0.1447	0.1542	0.0455	0.1430	0.1542	0.0455	0.1430
	B-2	Chord	0.2143	0.1798	0.1921	0.1890	0.1567	0.1702	0.1890	0.1567	0.1702
	B-2	Radial	0.1593	0.0487	0.1431	0.1414	0.0393	0.1413	0.1414	0.0393	0.1413
	B-3	Chord	0.2273	0.1770	0.1693	0.1927	0.1463	0.1526	0.1927	0.1463	0.1526
	B-3	Radial	0.1396	0.0245	0.1295	0.1289	0.0218	0.1277	0.1289	0.0218	0.1277
	B-4	Chord	0.2201	0.1761	0.1774	0.1899	0.1497	0.1587	0.1899	0.1497	0.1587
	B-4	Radial	0.1397	0.0217	0.1316	0.1304	0.0184	0.1297	0.1304	0.0184	0.1297
	B-5	Chord	0.2359	0.1755	0.1698	0.2016	0.1482	0.1496	0.2016	0.1482	0.1496
	B-5	Radial	0.1403	0.0219	0.1456	0.1349	0.0177	0.1355	0.1349	0.0177	0.1355
	B-6	Chord	0.2259	0.1744	0.1731	0.1944	0.1463	0.1532	0.1944	0.1463	0.1532
	B-6	Radial	0.1395	0.0173	0.1416	0.1343	0.0144	0.1347	0.1343	0.0144	0.1347
	B-7	Chord	0.2277	0.1863	0.1902	0.1985	0.1587	0.1699	0.1985	0.1587	0.1699
	B-7	Radial	0.1524	0.0542	0.1550	0.1304	0.0413	0.1327	0.1304	0.0413	0.1327
	B-8	Chord	0.2146	0.1815	0.1910	0.1899	0.1564	0.1712	0.1899	0.1564	0.1712
	B-8	Radial	0.1463	0.0461	0.1464	0.1314	0.0332	0.1287	0.1314	0.0332	0.1287
J ₁₀	B-1	Chord	0.0404	0.0360	0.0331	0.0355	0.0321	0.0297	0.0355	0.0321	0.0297
	B-1	Radial	0.0357	0.0110	0.0223	0.0313	0.0092	0.0192	0.0313	0.0092	0.0192
	B-2	Chord	0.0391	0.0348	0.0328	0.0346	0.0312	0.0295	0.0346	0.0312	0.0295
	B-2	Radial	0.0337	0.0090	0.0216	0.0300	0.0075	0.0189	0.0300	0.0075	0.0189
	B-3	Chord	0.0394	0.0340	0.0296	0.0343	0.0302	0.0266	0.0343	0.0302	0.0266
	B-3	Radial	0.0286	0.0039	0.0192	0.0267	0.0033	0.0175	0.0267	0.0033	0.0175
	B-4	Chord	0.0388	0.0338	0.0302	0.0340	0.0302	0.0272	0.0340	0.0302	0.0272
	B-4	Radial	0.0285	0.0034	0.0196	0.0268	0.0029	0.0178	0.0268	0.0029	0.0178
	B-5	Chord	0.0406	0.0339	0.0302	0.0352	0.0301	0.0271	0.0352	0.0301	0.0271
	B-5	Radial	0.0268	0.0036	0.0219	0.0258	0.0031	0.0202	0.0258	0.0031	0.0202
	B-6	Chord	0.0395	0.0338	0.0306	0.0346	0.0301	0.0276	0.0346	0.0301	0.0276
	B-6	Radial	0.0267	0.0031	0.0215	0.0258	0.0027	0.0198	0.0258	0.0027	0.0198
	B-7	Chord	0.0405	0.0360	0.0331	0.0356	0.0321	0.0298	0.0356	0.0321	0.0298
	B-7	Radial	0.0273	0.0107	0.0223	0.0252	0.0090	0.0201	0.0252	0.0090	0.0201
	B-8	Chord	0.0390	0.0349	0.0326	0.0347	0.0313	0.0295	0.0347	0.0313	0.0295
	B-8	Radial	0.0267	0.0087	0.0219	0.0251	0.0073	0.0199	0.0251	0.0073	0.0199

ERB: elastomeric rubber bearing; LRB: lead rubber bearing; FBS: friction pendulum system; M. S.: modified switching control law

Figure 11 shows the comparison of time variation of base shear, for the uncontrolled and controlled (switching control and passive mode) curved bridge in chord and radial directions for SAVSD+LRB control combination of Kobe (1995) earthquake motions. These results show that there is a considerable reduction in the base shear switching control and passive mode strategy for curved isolated bridge system with SAVSD damper as compared to uncontrolled cases. The energy dissipation characteristic of a damper is usually understood by its hysteresis loop. Figure 12 shows the force–displacement variation loops at D1 (damper 1 in chord direction at left abutment) for the curved bridge system under the different time histories for switching control law. It is observed from the hysteresis loops that good amount of energy is absorbed by the SAVSD under all the time histories.

Figure 11.

Time variation of base shear of pier 1 along the chord and radial directions of bridge with SAVSD + LRB combination under Kobe (1995) earthquake.

Figure 12.

Force–displacement behavior of SAVSD (D1) with switching control law under different earthquake motions.

Conclusion

The seismic response of the horizontally curved concrete box girder bridge with semi-active hybrid control systems consisting of different passive isolators and semi-active stiffness damper for different control strategies is investigated. The effectiveness of SAVSD is studied by varying important parameters for assessment of its performance for different combinations. Based on the investigation carried out, the following conclusions have been drawn:

Significant seismic response reduction of the curved bridge can be achieved by applying a hybrid semi-active control system consisting of ERB, LRB, and FPS, with semi-active stiffness damper.

The reduction of the seismic responses depends on the types of hybrid semi-active system as well as the types of earthquake ground motions.

The lower value of K_r will be better for reducing the base shear and overturning moment, whereas the higher value prefers for lesser mid-span and bearing displacements.

Modified switching control law can be used as it provides the same control response as the switching control law with a lesser number of sensors.

The SAVSD+ERB and SAVSD+LRB combinations are more effective in controlling base shear and overturning moment than SAVSD+FPS.

For a maximum reduction in base shear and overturning moment with minimum mid-span and bearing displacement as an overall efficient response control, the SAVSD+LRB combination can be more efficient than other combinations.

Footnotes

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) received no financial support for the research, authorship, and/or publication of this article.

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