Design method for steel restrainer bars on railway bridges subjected to spatially varying earthquakes

Engineering Structures 159 (2018) 198–212

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Design method for steel restrainer bars on railway bridges subjected to spatially varying earthquakes

T

⁎

Cong Liu , Ri Gao School of Civil Engineering, Beijing Jiaotong University, 100044 Beijing, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Steel restrainer bar Design procedure Pounding damages Unseating damages Spatially varying earthquakes

The unseating and pounding damage at expansion joints during earthquakes emphasizes the need to restrain the relative opening and closing displacements between adjacent bridge spans. Traditional cable restrainers perform well in preventing unseating damage but may increase the pounding damage between adjacent decks sometimes. Considering the limitation of the cable restrainer, a steel restrainer bar installed between a pier and a deck is proposed to restrain both the relative opening and closing displacement to prevent unseating and pounding damage to railway bridges. In this paper, theoretical formulas for the mechanical properties of the restrainer bar, experiments and numerical analyses are conducted. A design procedure for the steel restrainer bar, which considers spatially varying earthquakes and the hysteretic behaviour of the restrainer, is developed. The procedure accounts for the dynamic out-of-phase motion characteristics between adjacent bridge spans, and the pounding is assumed to be completely avoided by the restrainer bar. A series of case studies with different pier heights and different earthquake characteristics are conducted to evaluate the effectiveness of the design method; the time history analysis results are consistent with the case study results. The restraint effect of the rails is also considered in this paper. The results of the numerical analyses reveal that both the pier height and the rails significantly influence the effectiveness of the steel restrainer bar. As the pier height increases, the effectiveness of the restrainer in reducing the relative displacement also increases, whereas the effectiveness in mitigating shear decreases with a pier height greater than 15 m. The rails reduce the effectiveness of the steel restrainer bar when subjected to spatially varying ground motions.

1. Introduction Due to the convenient construction of simply supported bridges, nearly 70% of railway bridges in China are simply supported bridges. However, during high-intensity earthquakes, adjacent bridge spans of simply supported bridges exhibit out-of-phase vibrations, which will produce a relative displacement at the expansion joint. If the relative displacement exceeds the provided clearance of the expansion joint (ranged from 40 mm to 80 mm) based on the Fundamental code for design on railway bridge and culvert [1], adjacent decks will be subjected to pounding. If the relative displacement between a deck and a pier is larger than the available seating width, unseating damage occurs [2]. To reduce unseating damage, the Chinese Department of Transportation initiated a retrofit programme that links adjacent bridge spans with cable restrainers at expansion joints. Although cable restrainers have performed well in many earthquakes, failure of the cable restrainers was observed in several bridges during the 1989 Loma Prieta earthquake [3] and the 1994 Northbridge earthquake [4]. To improve

⁎

the effectiveness of cable restrainers, many researchers have studied the influencing factors of the behaviour of the restrainers. The results of these studies have revealed that the relative displacement is sensitive to the characteristics of the ground motion, the period ratio of adjacent piers and the restrainer properties [5,6]. The results have also indicated that the yielding of the cable restrainers during strong earthquakes and inappropriate design methods are the major reasons for the failure of restrainers [7–9]. Due to the importance of restrainers, many specifications have taken the restrainer design into account. The American Association of State Highway and Transportation Official (AASHTO) [10] specification requires a positive horizontal linkage between adjacent bridge spans of the superstructure. The required linkage force is equal to the design acceleration coefficient multiplied by the weight of the lighter span of the two adjacent spans. The Japanese specification [11] is similar to the AASHTO method, in which the required restrainer force is equal to the weight of the superstructure on the pier and the effective weight of the pier multiplied by the design acceleration coefficient.

Corresponding author. E-mail address: [email protected] (C. Liu).

https://doi.org/10.1016/j.engstruct.2018.01.001 Received 27 May 2017; Received in revised form 28 December 2017; Accepted 1 January 2018 0141-0296/ © 2018 Elsevier Ltd. All rights reserved.

Engineering Structures 159 (2018) 198–212

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Nomenclature

L1 L2 l M m

Symbols list a ck Davg Deq Dr Ds Dm D1, D2 d0, d1 de dij E Fe Fy Fmax fy Ie Ix I0 Ke Kd Kr K meff Kinitial Keff Kv Kabut K1, K2 k L

coefficient of displacement damping ratio of the contact spring average independent horizontal displacement of adjacent spans relative displacement of adjacent bridge spans maximum displacement of the restrainer initial slack of the restrainer clearance of expansion joint displacement of each bridge span diameter of the section on both sides of the model diameter of the equivalent section displacement between location i and j elasticity modulus elastic limit force yield force maximum design force yield force of the material inertia moment of the equivalent section inertia moment of each section of the model inertia moment of the section with d0 diameter initial stiffness of a single restrainer post-yield stiffness of the restrainer equivalent stiffness of the steel restrainer bar effective stiffness of the bridge initial stiffness of the restrainers on the bridge effective stiffness of each bridge spans stiffness of the contact spring stiffness of the abutment elastic stiffness of the adjacent piers ratio between the diameter of both sides of the model total length of the restrainer bar

m1, m2 Pbw P1 P2 p Sa (ω) Sg (ω) T TL TS ue uy umax y δ Δ α β γ ξ ξe ξi ξω ω γij (iω) vapp ρ12 μ

width derived from the proposed relative displacement response spectra is similar to the value specified in the Japanese design specifications. However, these design procedures are determined under uniform ground excitations. Both the Trochalakis and DesRoches methods assume that the cable restrainers remain elastic during earthquakes and do not consider the pounding between adjacent decks, which will enlarge the relative displacement at expansion joints. Note that the relative displacement between adjacent bridge spans is affected not only by the different dynamic characteristics of each involved member but also by the inevitable spatially varying ground motions at multiple bridge supports. Studies [16,17] have been conducted to evaluate the influence of spatially varying earthquake on bridges; the results have indicated that spatially varying earthquakes can amplify the relative displacement of adjacent bridge spans and pounding forces. Sometimes, the pounding forces for a bridge under spatially varying ground motions can be 3–4 times larger than the pounding forces derived from uniform ground motions. Although the cable restrainer has been used widely, there are still some deficiencies. A comparison of the results from Shrestha et al. [18] reveals that the pounding force and pounding number of the bridge with cable restrainers may be greater than that of the bridge without restrainers for some earthquake cases. Due to the elastic design of the cable restrainers, the seismic energy can only be dissipated by the plastic hinges that form in piers, which are not suitable for simply supported bridges due to the possibility of producing a statically unstable structure. The elastic design method also requires the utilization of a large number of cable restrainers to limit the relative displacement

However, both specifications do not consider the characteristics of the ground motion and the vibration periods of the adjacent bridge spans, which influence the relative displacement at the expansion joint greatly. Several studies have been conducted to investigate the design method of bridges with cable restrainers. Trochalakis et al. [5] proposed a modified procedure based on the equivalent static procedure provided by bridge design specifications [12]. In this procedure, the maximum relative displacement is estimated from the average displacement of adjacent spans as follows:

Deq =

Davg × TL 2 × TS

⩽ 2Davg

length of the installment segment length of the bending segment length of a quarter of the restrainer bar mass of bridge piers ratio between the length of the bending segment and the installment segment mass of adjacent girders pressure force of the abutment force at Point B and C force at Point A and D probability coefficient, usually less than 0.85 acceleration response spectrum power spectral density duration of the earthquakes longer period of the adjacent bridge spans shorter period of the adjacent bridge spans elastic limit displacement yield displacement maximum design displacement deflection equation of the model post-yield stiffness/initial stiffness ratio displacement of Point A incident angle of the earthquake coherency loss coefficient conversion ratio damping ratio effective damping ratio of the restrainer damping ratio of the pier, assumed as 0.05 effective damping ratio of bridge span frequency of adjacent bridge spans coherency loss function apparent wave velocity of the earthquake cross-correlation coefficient displacement ductility ratio

(1)

where Deq is the relative horizontal displacement between the two bridge spans and Davg is the average independent horizontal span displacement; TL and TS represent longer periods and shorter periods, respectively, of the uncoupled spans. Eq. (1) is based on a regression analysis of numerous cases. This method is based on an analysis of bridges with ductile structures; however, the level of inelastic deformation is not explicitly included. DesRoches et al. [13,14] proposed a design procedure that is based on a linear model and considers the dynamic characteristics of out-of-phase motion of adjacent bridge frames. The inelastic behaviours of bridge frames are considered using the substitute structure method [2]. Ruangrassamee et al. [15] developed relative displacement response spectra by analysing a two linear single-degree-of-freedom system and considering the influence of pounding. The researchers presented the formula for the normalized relative displacement response spectra and determined that the seat 199

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possible thermal stress, creep, and shrinkage effects. Meanwhile, the large service loads, e.g. brake load of the train, will not result in the yielding of the restrainer due to the adequate initial stiffness of the restrainer bar [21]. During the construction of a bridge, there are penetrated construction holes at the bottom of the girder for the installment of the restrainer bars. After the girder has been placed on the bearing, the restrainer bars will be installed in the steel cylinder embedded in the pier through the construction holes from the cell of the girder. Then, the steel cylinder in the girder is embedded, and the construction holes are filled with the polyethylene (PE) stuffing. During earthquakes, once the relative displacement between the deck and the pier exceeds the clearance of the steel restrainer bar, the top and the bottom of the bending segment will make contact with the cylinder walls and force the restrainer bar to deform in elastic behaviours and then deform in plastic behaviours to dissipate the earthquake energy. The steel restrainer bar exhibits a strain-hardening effect which can still restrain the relative displacement as the restrainer bar deforms in plastic. After earthquakes, the stuffing should be removed, and the destroyed restrainers can be replaced through the construction holes in the same manner as the installation.

to an acceptable range, which will induce large forces in other components of the bridge [9,19]. Considering the demand for pounding mitigation and the limitation of cable restrainers, a steel restrainer bar, which allows bending, is proposed to mitigate pounding and unseating damage in railway bridges. To understand the hysteretic behaviour of the restrainer bar, the theoretical formulas of the mechanical properties derived by the equivalent stiffness method are proposed; experiment tests are also conducted. Then, a design procedure for the steel restrainer bar is developed, which considers spatially varying earthquakes and the hysteretic behaviour of the restrainer. And then a series of case studies of a five-span railway bridge with various pier heights subjected to spatially varying earthquakes are conducted to evaluate the effectiveness of the new design procedure. Finally, the influence of the rails on the performance of the restrainer bar is also investigated in this paper. 2. Description of the steel restrainer bar Fig. 1 presents the installment diagram of the steel restrainer bar. The material of the steel restrainer bar is a type of specialized steel that is designated LY325FLL; it has a yield strength range of 315–375 MPa and an ultimate strength range of 450–600 MPa. The steel restrainer bar includes three segments: the installment segments on both sides of the steel restrainer bar and the bending segment in the middle of the steel restrainer bar. The installment segments are placed in a steel cylinder embedded in the pier and the girder for the fixation of the restrainer bar, bearing nearly no bending deformation during earthquakes. The cross-section of the restrainer bar is linear variable. It has been proved that restrainer bars with variable cross-sections can support greater plastic displacement than those with equal cross-sections [20]. There are three main components working with the steel restrainer bar: the limited ring, the retaining ring and the rubber seal. The limited rings are installed at both ends of the steel restrainer bar to prevent the steel cylinder wall from possible squeezing failure. Retaining rings at the top of the cylinders are set to ensure that the bending segment remains in bending deflection during earthquakes, and the rubber seal is placed at the substructure to prevent rainwater or dust from entering the cylinder cavity. Meanwhile, anticorrosive material should be painted on the restrainer bar before its installment to prevent corrosion. The clearance between the limited rings and the bottom of the restrainer is left on purpose to allow the installment segment to rotate during earthquakes. The clearance between the retaining rings and the restrainer bar (ranges from 20 mm to 40 mm) prevents the bar from

3. Hysteretic behaviour of the steel restrainer bar 3.1. Mechanical property To understand the hysteretic behaviour of the restrainer bar, the theoretical formulas for its mechanical properties are analysed firstly. Fig. 2 presents the hysteretic model of the restrainer bar. The customary hysteretic model can be obtained by cyclic loading tests, whereas the bilinear model is always employed in theoretical analysis. In Fig. 2, point 1, point 2 and point 3represent the elastic limit point, the yield point and the maximum design point. ue、uy and umax mean the elasticlimit displacement, the yield displacement and the maximum design displacement, whereas Fe、Fy and Fmax represent the corresponding forces of the threedisplacements. Ke is the initial stiffness of the restrainer bar, which can be expressed as K e = Fe / ue = Fy / u y ; Kd denotes the post-yield stiffness of the restrainer bar, which can be expressed as K d = δK e , and δ represents the post-yield stiffness/initial stiffness ratio. To calculate the inertia moment of the steel restrainer bar with variable cross-section, a quarter of the restrainer bar is selected as the simplified model due to symmetry. Fig. 3 presents the calculation model. In Fig. 3, the diameter of the two end sections of the model are

Fig. 1. Steel restrainer bar installment diagram: (a) complete schematic; (b) steel restrainer bar schematic.

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Fig. 4. Force diagram of restrainer bar.

installment segments, and line BC represents the bending segment. The total length of the restrainer bar is L, and the length of the installment segments and the bending segment are assumed as L1 and L2, respectively. During earthquakes, the displacement of the restrainer bar includes the displacement of the bending segment caused by the bending moment and the displacement of the installment segments caused by rotation. Shear deformation can be neglected due to its inconspicuous influence on the deformation of the steel restrainer bar based on the studies of Meng [22]. According to the force balance, the moment about Point A is as follows:

Fig. 2. Hysteretic model of the restrainer bar.

assumed to be d0 and d1 the length of the model is l, and the diameter of an arbitrary section is represented as d(x). Because the diameter of each section varies linearly, the diameter can be obtained by Eq. (2), and k is assumed to be a coefficient calculated using the diameter of the model, as shown in Eq. (3):

d (x ) =

kd 0 d x + d 0 = 0 (kx + l) l l

d1 =k+1 d0

P1 L1−P1 (L1 + L2) + P2 (L2 + 2L1) = 0

(

∫0

l

EIx (y″)2dx =

1 2

∫0

l

EIe (y″)2dx

(

x4

y″ = 2a

Δ=

16k 4 + 105k 3 + 288k 2 + 420k + 336 I0 336

2

)

(6 + m) mP2 L13 6EIe

(7)

(8)

where Δ represents the displacement of Point A. Then, the initial stiffness of the restrainer bar can be obtained:

(4)

Ke =

6EIe (6 + m) mL13

(9)

The yield force of the material is assumed to be fy, thus,

Fe =

πde3 F f ; ue = e 32L1 y Ke

(10)

As shown in Fig. 2, the relationship between the elastic limit point and the yield point is

Fy = γFe; u y = γue

(5)

(11)

where γ represents the conversion ratio between Fy and Fe, and between uy and ue, which should be calculated from experiments and finite element analyses.

where a denotes the coefficient of displacement. By substituting Eqs.(1), (2), and (5) into Eq. (4), Ie can be obtained via the following equation, and I0 denotes the inertia moment of the section with the diameter d0.

Ie =

1

where m is the ratio between L1 and L2, which is expressed as: L2 = mL1. Based on the graphic multiplication method, the displacement of Point A can be calculated as follows:

)

l2 − x 2 l

(

(3)

where Ix represents the inertia moment of each section of the model, while Ie is the inertia moment of the equivalent section, and y denotes the deflection equation of the model. In this paper, the model is regarded as a cantilever beam due to the restraint of the limited rings and the retaining rings. The deflection equation can be expressed as follows:

y = a x 2− 6l

)

P1 = P2 1 + 2 L1 = P2 1 + 2 m

Equivalent stiffness method is applied to calculate the inertia moment of the restrainer bar, which can translate the variable section into a uniform section. For the equivalent stiffness method, the bending strain energy of the restrainer bar and the equivalent bar should be equal, and the equation can be expressed as follows:

1 2

L

(2)

3.2. Experiments and numerical analysis

(6)

Experiments and numerical analysis are conducted to calculate the post-yield stiffness/initial stiffness ratio δ and the conversion ratio γ . Experiments are conducted at the Structure Laboratory of Beijing Jiaotong University. Two specimens (S1 and S2) are tested under cyclic

According to the calculated equivalent inertia moment Ie, the diameter of the equivalent section can be obtained as de. Fig. 4 presents the force diagram of the equivalent restrainer bar with a uniform section. Line AD represents the restrainer bar, lines AB and CD represent the

Fig. 3. Calculation diagram of restrainer bar.

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the experiment results. Because the elastic limit force depends on the yield stress and the cross-section of the restrainer bar, the results of the numerical analysis and experiments are similar. From the results in Table 1, parameter γ is used to correct both the yield force and the yield displacement, which ranges between 1.3 and 2.4, and the post-yield stiffness/initial stiffness ratio δ ranges from 0.06 to 0.13.

loadings to evaluate the hysteretic behaviours of the steel restrainer bar. The maximum design displacement of the steel restrainer bar is ± 160 mm. The hysteresis tests are performed under displacement control. The loading displacement specified in Ref. [23] ranges from zero to the design displacement of the specimen, which is 160 mm for the steel restrainer bar, each with three cycles. The displacement in each loading step increases at an interval of 5 mm for values between 0 and 40 mm and at an interval of 20 mm between 40 mm and 160 mm. With a 30 mm clearance between the steel restrainer bar and the steel cylinder, the first loading step is a 40 mm displacement, and the increment is 20 mm for these experiments. Fig. 5(a) presents the dimensions of the specimens. The length of the installment segments and the length of the bending segment are 325 mm and 650 mm, respectively. The bottom foundation is manufactured by reinforced concrete to imitate the pier of a bridge. The specimens are installed in the middle cylinder, whereas cylinders on both sides are used to fix to the ground by ground anchors. Fig. 6 displays the test set-up. The finite element code ANSYS [24] was applied to model the restrainer bar. The restrainer bar and the concrete foundation were modelled by solid elements, and the contact elements were applied on both sides of the restrainer bar to model the contact between it and the concrete foundation. The bottom of the restrainer bar was restrained in the vertical direction, whereas movement towards the loading direction was allowed to ensure the deformation of the restrainer bar. Thrust from an MTS hydraulic machine was applied at the top of the restrainer. Fig. 7 compares the experimental hysteretic curves and the numerical results for 80 mm, 120 mm and 160 mm displacement amplitudes. Table 1 compares the mechanical parameters of the restrainer bar. The yield strength of the material in the theoretical analysis is 375 MPa. The forces obtained from the experiment and the numerical analyses are P1 as shown in Fig. 4, whereas the theoretical results are calculated as P2. In Table 1, all forces are translated into P1 for comparison purposes by the following equation: P1 = 2P2 . Due to the gaps between the loading device and the steel cylinder in the experiments and replacement of the elastic contacts in the experiments with rigid contact in the numerical analysis, the initial stiffness of the theoretical and numerical results are greater than the initial stiffness of the experiment results, whereas the elastic limit displacement of the theoretical and numerical results are less than the elastic limit displacement of

4. Design procedure of the restrainer bar In previous studies, the design procedures only calculated the initial stiffness and assumed that the cable restrainer remains elastic during earthquakes. In this paper, the equivalent stiffness and the equivalent damping ratio are used to present the hysteretic behaviour of the steel restrainer bar. Once the equivalent stiffness is obtained, the initial stiffness can be calculated by the mechanical properties to determine the number of required steel restrainer bars. In most railway bridges, the distance between adjacent expansion joints is substantially smaller than the seat width of the piers; thus, the objective of the procedure proposed in this paper is to provide a sufficient restrainer stiffness to limit the relative displacement and to avoid pounding between adjacent decks. Because the pounding is assumed to be avoided completely, the simply supported bridge can be simplified as a two single-degree-of-freedom system, as presented by the model in Fig. 8. The restrainer bars are only installed with sliding bearings, and the fixed bearing is assumed to limit the relative displacement between the pier and the deck perfectly. As shown in Fig. 8, the restrainer bar can be regarded as a spring connected between adjacent bridge decks. The mass of each deck is represented as m1 and m2, while M denotes the mass of the pier. In previous studies [25,26], the seat abutment is always regarded as an important parameter in influencing the relative displacement of the bridge. However, due to the long distance of the total railway bridge in China, such as 10 km or even more, the influence of the seat abutment can be negligible in the design. In the latter numerical analyses, the influence of the seat abutment is considered as the model is a five-span railway bridge. And as the long distance of the bridge, the spatially varying ground motions should be taken as an important factor. The coherency loss function in the Sobczyk model [27] is applied to simulate the spatially varying ground motions at locations 1 and 2, which

Fig. 5. Dimensions of (a) specimens; (b) concrete foundation.

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(b)

(c)

Fig. 6. Test set-up: (a) Schematic diagram of the test set-up; (b) photo of the test set-up; and (c) photo of the steel restrainer bar installed in the test set-up.

800

can be described as follows:

Numerical results S1 S2

600

γij (iω) = exp(−βωdij2/vapp) × exp(−iωdij cosα /vapp)

Force (KN )

400

(12)

where β is the coefficient that reflects the level of coherency loss, dij is the distance between the different bridge supports i and j. vapp is the apparent wave velocity and α is the incident angle of the incoming wave to the site. Fig. 9 presents the flow chart of the entire restrainer design method, which will be described in following section.

200 0 -200 -400

4.1. Step 1: Ground motion simulation

-600 -800 -200

-150

-100

-50

0

50

100

150

Sg (ω) is the ground motion power spectral density at the origin of the earthquake. The power spectral density of locations 1 and 2 in Fig. 8 can be calculated as follows:

200

Displacement (mm)

2 Sg1 (ω) = γ10 (iω) Sg (ω) = exp(−βωd10 / vapp) × exp(−iωd10cosα / vapp) Sg (ω)

Fig. 7. Comparison of test hysteretic curves and numerical analysis results.

2 Sg 2 (ω) = γ20 (iω) Sg (ω) = exp(−βωd 20 / vapp) × exp(−iωd20cosα / vapp) Sg (ω)

(13)

Table 1 Comparison of the mechanical characteristic parameters of the restrainer bar.

Fe (KN)

ue (mm)

Fy (KN)

uy (mm)

Ke (KN/ mm)

Kd (KN/ mm)

Theoretical results

421.59

18.27

421.59 γ

18.27 γ

23.07

–

Numerical results

420.08

20

566.35

32

21.004

1.54

Experiments

419.32 421.76

29.87 30.61

521.71 537.24

43.42 44.02

14.04 13.78

1.68 1.81

S1 S2

where Sg1 (ω) and Sg2 (ω) represent the power spectral density of locations 1 and 2; d10 and d20 represent the distance between the location and the earthquake origin. The relationship between the power spectral density Sg (ω) and the acceleration response spectrum Sa (ω) can be described as follows [28]:

Sg (ω) = −

ξ 2 π Sa (ω,ξ )/ln(− lnp) πω ωT

(14)

where ξ is the damping ratio, T is the time duration, and p is the probability coefficient, usually p ⩾ 0.85 [26]. From Eq. (14), the acceleration response spectrum of location 1 and 2 can be calculated by the given power spectral density, which are denoted as Sa1 (ω1,ξ1) and Sa2 (ω2,ξ2), respectively. 4.2. Step 2: Maximum and initial relative displacements The maximum relative displacement between adjacent decks is dependent on the initial clearance of the expansion joint and the initial slack of the steel restrainer bar. The maximum displacement of the steel restrainer bar Dr is the difference value between the clearance of the expansion joint and the slack of the restrainer Ds:

Dr = Dm−Ds

(15)

where Dm is the clearance of the expansion joint. The initial relative displacement Deq0 can be calculated by CQC combination rules as follows:

Deq0 = Fig. 8. Simplified model.

D120 + D220 −2ρ12 D10 D20

(16)

where D10 and D20 are the individual span displacements, which are calculated by the following expressions:

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Fig. 9. Flow chart of the restrainer design method.

Deq0 ⩾ Dr , then restrainer bars must be installed and the stiffness should be calculated according to the iteration in the subsequent steps. 4.3. Step 3: Initial equivalent stiffness of the restrainer bar The initial equivalent stiffness of the restrainer can be calculated as follows:

Kr1 =

K meff =

(17)

and ρ12 is the cross-correlation coefficient proposed by Jeng and Kasai [29], which considers the time lag of the seismic travelling waves and can be expressed as follow (assumption: d20 ⩾ d10 ):

ρ12 =

4 ξ1 ξ2 ω13 ω23 (ω12−ω22)2

4ξ1 ξ2 ω1 ω2 (ω12

ω22)

+ + + 1 − ξ ω τ 1 1 (A1 cosωd,1 τ −A2 sinωd,1 τ ) e ωd,1

4(ξ12

+

ξ22) ω12 ω22

A1 = 2ωd,1 (ξ1 ω1 + ξ2 ω2)

(20)

τ = (d20−d10)/ vapp

K1 K2 K1 + K2

(24)

Fig. 10 shows the relationship between the initial stiffness and the equivalent stiffness. Based on the simulation method of equivalent stiffness proposed in AASHTO [12], the equivalent stiffness can be estimated as follows:

(18) (19)

A2 =

(23)

4.4. Step 4: Equivalent stiffness and equivalent damping ratio

·

ωd,1 = ω1 1−ξ12

(1−2ξ12) ω12−ω22−2ξ1 ξ2 ω1 ω2

Deq0

where K meff denotes the effective stiffness of the bridge structure, which is calculated by the stiffness of each pier:

Fig. 10. Relationship between initial stiffness and equivalent stiffness.

D10 = m1 × Sa1 (ω1,ξ1)/ K1; D20 = m2 × Sa2 (ω2,ξ2)/ K2

K meff (Deq0−Dr )

Kr =

μ= (21)

1 + δ (μ−1) Kinitial μ

(25)

Deqj uy

(26)

where δ represents the post-yield stiffness/initial stiffness ratio, μ is the displacement ductility ratio, Kinitial is the initial stiffness of restrainers, and Deqj is the current relative displacement. The equivalent damping ratio of the restrainer can be calculated as follows:

(22)

where ξ1 and ξ2 are the damping ratio of each pier; ω1 and ω2 are the frequencies of each pier. If Deq0 ⩽ Dr , restrainers are not required. If 204

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ξe =

2(1−δ )(1−1/ μ) π (1 + δ (μ−1))

Eq. (22). If Deqj ⩽ Dr , the equivalent stiffness of the restrainer bar is the last value of K r j . Otherwise, step 6 should be followed.

(27)

4.5. Step 5: Relative displacement 4.6. Step 6: Equivalent stiffness of the restrainer bar The relative displacement between adjacent spans can be determined from the simplified model shown in Fig. 8. To determine the relative displacement, the effective stiffness of each bridge span with current restrainers should be calculated. Then, we solve the eigenvalue problem for the system shown in Fig. 8 and calculate the effective frequency for each bridge span, the equation is presented as follow:

⎡ K1 + K r j − K r j ⎤ Φ = ω 2 ⎡ m1 0 ⎤ Φ i effi ⎢ − Krj K2 + K r j ⎥ i ⎣ 0 m2 ⎦ ⎣ ⎦

(28)

2 2 K eff1 = (m1 + M ) ωeff ; K eff2 = (m2 + M ) ωeff 1 2

(29)

When Deqj is larger than Dr, the stiffness of the restrainer bars should be modified. The modified stiffness is derived from the last value of K r j according to the following expression:

K r j + 1 = K r j + (K meff + K r j )

Once the equivalent stiffness of the restrainer bar is calculated as Kr , the initial stiffness of the restrainer Ki can also be estimated by Eq. (25) as follows:

Kinitial = (30)

Ki + K r j

Kr ·Deq (1−δ ) u y + δDeq

(33)

For a single steel restrainer bar, the initial stiffness is Ke and the total number of restrainers applied in the bridge can be calculated as Kinitial/ K e .

where D1 and D2 are the displacements of an individual bridge span with the limitation of the restrainer bars. Due to the application of restrainer bars, the damping ratio of each bridge span should be recalculated, which depends on the equivalent stiffness of the restrainer bars and the stiffness of the pier. The damping ratio of each pier can be calculated as follows:

ξωi =

(32)

4.7. Step 7: Application of the restrainer bar

D1 = m1 × Sa1 (ωeff1,ξ w1)/(K eff1 + K r j ); D2

ξe·Ki + ξi·K r j

Deqj

The increment j is increased, and we return to step 4 until Deq is smaller than Dr; and then the equivalent stiffness of the restrainer can be determined to be Kr .

where Φi denotes the mode shape (i = 1, 2); ωeff1 and ωeff2 denote the effective frequency of each bridge span; K r j denotes the current equivalent stiffness of the restrainer. Due to the energy dissipation of the steel restrainer bar, the piers are assumed to be elastic during earthquakes. With these calculated characteristics, the displacement for each individual span can be calculated as follows:

= m2 × Sa2 (ωeff2,ξ w2)/(K eff2 + K r j )

(Deqj−Dr )

5. Parameter study To understand the seismic performance of the steel restrainer bar, a series of parameter studies for typical Chinese railway bridges are conducted under spatially varying ground motions in the longitudinal direction. First, the relative displacement at each expansion joint calculated from the procedure is compared with the relative displacement derived from the time history analysis to evaluate the effectiveness of the procedure. Then, the influence of the pier height and the effect of the rails is also investigated in this paper.

i = 1,2 (31)

Substitute the damping ratio ξωi (i = 1, 2) into Eq. (18) to Eq. (22) and recalculate the cross-correlation coefficient ρ12 . The relative displacement of adjacent bridge spans limited by the current restrainer stiffness, denote Deqj , can be calculated by Eq. (16) to

Fig. 11. Schematic of the bridge model.

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5.1. Bridge model

Pbw = Ae × 239 ×

The superstructures of the selected bridge are entirely precast box girders with an equal length of 32 m. The main structural characteristics of the bridge are derived from the Chinese railway bridge standard map. A schematic of the layout of the selected bridges is shown in Fig. 11(a). The bridge decks are supported by four RC piers with coincident heights, and the seismic behaviour of the piers is assumed to be elastic-plastic. The RC piers are also derived from the Chinese railway bridge standard map. The entire bridge is modelled by ANSYS [24]. The elastic beamcolumn elements are used to model the superstructures of the bridge, which were assumed to be elastic during the ground motions. The piers are modelled by SOLID65 elements and LINK180 elements to simulate concrete and reinforcements, respectively. C35 is selected as the pier concrete, and HRB335 is selected as the steel reinforcement. Fig. 11(b) presents the reinforcements of the pier top and the reinforcements of the other part of the pier are same with the pier top. Fig. 11(c) presents the cross-section of the pier top. The pier foundations are fixed to the ground to focus on the seismic response of the bridge caused by spatially varying ground motions. The pier-deck connections are realized by sliding bearings and fixed bearings. The bearings are modelled by COMBIN14 elements with three DOFs. The stiffness of the fixed bearing in the vertical and longitudinal directions is simulated as 104KN/mm. The sliding bearings and fixed bearings are alternately placed, as shown in Fig. 11(a). The fixed bearing, which is usually brittle, are assumed to remain linear elastic until failure due to its shear failure behaviour. The sliding bearings are modelled by a rigid-perfectly plastic model to describe their frictional behaviour. In this study, a friction coefficient of 0.05 is assumed for the sliding bearings, and the maximum displacement capacity of the sliding bearings is assumed to be 100 mm in the longitudinal direction [30,31]. The steel restrainer bar is modelled using LINK180 elements with an appropriate initial strain to simulate the clearance between the steel restrainer bar and the retaining rings. CONTAC12 elements in ANSYS are employed to simulate the pounding force, and the Kelvin model is applied to calculate the mechanic parameters of the elements [32,33]. For the Kelvin model, a linear spring with a stiffness kv is employed in conjunction with a damper element (ck) that accounts for the energy dissipation during pounding. The rule of thumb is to select the stiffness of the linear contact spring (kv), which is proportional to the axis stiffness of the colliding structure (EA/L), where E and A represent the elastic modulus and the cross-section area of the superstructure, respectively, and L denotes the length of the decks, which is selected to be 32 m in this study. The stiffness of the linear spring (kv) is calculated as 1.54 × 104 KN/mm, and the damping ratio ck is 0.138. These values are calculated based on the Kelvin model. Abutments are modelled via COMBIN39 elements in the longitude direction to represent the pile stiffness and passive soil stiffness, respectively. The nonlinear abutment behaviour of each spring is derived from the design recommendation of Caltrans [34]. An effective stiffness of 7/KN/mm/pile and an ultimate strength of 119 KN/pile are applied in this study based on the Caltrans recommendations [34]. Tri-linear symmetrical models proposed by Choi [35] are used to model the pile stiffness. The first yielding of the pile occurs at the displacement of 7.5 mm, and the second yielding of the pile occurs at a displacement of 25.4 mm. A total of 24 piles are placed in each abutment. The effective width of the abutment and height of the abutment are w = 13.5 m and h = 4 m, respectively. COMBIN39 elements in the passive direction are employed to represent the passive behaviours of the abutments. The abutment stiffness and pressure force can be calculated as follows:

K abut = K ai × w ×

h 1.7

h 1.7

(35)

where Kabut is the stiffness of the abutment, and Pbw is the passive pressure force that is calculated based on an assumed elastic perfectly plastic force displacement relationship. The initial stiffness Kai = 287 KN/mm/m, and Ae is the effective area of the back wall. In this paper, only the longitudinal direction earthquakes are taken into account, and the transverse direction earthquake design is similar with the longitudinal direction. The spatially varying ground motions are modelled by the Sobczyk model[27], and the coherency loss coefficient β is selected to be 0.0025, 0.0015, and 0.0005 to describe weakly correlated motions, intermediately correlated motions, and highly correlated motions, respectively. The distance between the different sites i and j are set as follow d12 = 32 m, d13 = 64, d14 = 96 m, d15 = 128 m and d16 = 160 m. The apparent wave velocity vapp and the incident angle of the incoming wave to the site α are selected as vapp = 400 m/s and α = 60°, respectively. The approximate method for modelling spatially varying ground motions is based on the method proposed by Bi [36], which has few or even no iterations. The design spectra that consider site classification 2 and zone 7 are derived from the Guidelines for seismic design of highway bridge, China [37]. 5.2. Rail model Previous studies [38–40] have indicated that rails have a significant restraint effects on the bridge in the longitudinal direction under earthquake excitations, which can influence the dynamic characteristics of the bridge. Meanwhile, the restraint effect can also influence the effectiveness of the steel restrainer bar in reducing relative displacements, and the effect of the rails depends on the rail fastener model. In this paper, CHN60 rail is assumed to be the railway installed on the bridge, and the base metal of the rails is U71Mn(k). Four rails are installed on the bridge; the essential parameters of CHN60 rail are listed in Table 2. The resistance force of the railway is an important factor for the interaction between the superstructure and the rails. In this paper, the resistance of the railway is assumed to be the resistance of the rail fastener. The WJ-8 rail fastener is applied to the bridge, and the constitutive relation of the WJ-8 fastener is bilinear. The sliding resistance of the WJ-8 rail fastener is assumed to be 15 KN, and the initial stiffness in the longitudinal and vertical directions are assumed to be 7.5 KN/ mm and 60 KN/mm, respectively. The distance between each rail fastening is 1 m, and the boundary condition of the railway on both sides of the bridge is set to 100 m, which has been proven to achieve both accuracy and efficiency of calculations. 6. Results and discussion 6.1. Application of the design procedure The finite element code ANSYS [24] is applied to evaluate the effectiveness of the proposed procedure. A series of restrainer stiffnesses calculated by the new procedure is employed in a nonlinear time history analysis with different pier heights and different earthquake motions. Models with pier heights of 8 m, 10 m, 15 m, 20 m and 25 m are calculated; the clearance of the expansion joint is set to 50 mm for all cases. Each case is analysed under three types of spatially varying earthquakes data of highly, intermediately and weakly correlated; a Table 2 Parameters of CHN60 rail. Type

(34)

CHN60

206

Mass (kg/ m)

Elasticity modulus (MPa)

Poisson ratio

60

2.06 × 105

0.3

Cross area

Unit weight

(cm2 )

(KN/m3 )

Coefficient of expansion (°C )

77.45

78.5

1.18 × 10−5

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Table 3 Parameters of each pier. Pier height (m)

8

10

Coherency loss*

W

Stiffness (KN/mm)

1040.5

669.3

232.7

75.8

Period (s)

0.0564

0.0756

0.2060

0.3161

Equivalent stiffness (KN/mm)

24

9

0

75

64

57

148

131

116

179

175

168

182

175

169

Initial stiffness (KN/mm)

40

15

0

123

105

93

236

209

185

261

258

248

267

258

248

Number of restrainers

2

2

2

6

6

6

10

10

10

10

10

10

10

10

10

I

H

W

15 I

H

W

20 I

H

W

25 I

H

W

I

H

43.9 0.4495

* W: weakly correlated ground motions; I: intermediately correlated ground motions; H: highly correlated ground motions.

the dimensions of the pier, the maximum number of restrainer bars is ten, although additional restrainer bars are needed for the bridge with 20 m and 25 m pier heights. As shown in Table 3, the restrainer stiffness increases with an increase in pier height. However, the increase in the restrainer stiffness decelerates when the pier height exceeds 20 m. For a flexible pier, a minimal increase in restrainer stiffness can significantly mitigate the relative displacement. Fig. 12 presents the displacement at expansion joints calculated by the time history analysis of the bridge with the restrainer stiffnesses listed in Table 3 for different pier heights and different ground motions. From Fig. 12, the relative displacements of most cases are smaller than the expansion joint clearance, which indicates that the pounding force

total of 15 cases are investigated. The damping ratio of each pier is assumed to be 0.05, whereas the damping ratios of the restrainer bars are calculated via the proposed method. The stiffness and period of each pier height are listed in Table 3; the equivalent and initial stiffness of the restrainer bars calculated by the proposed procedure are also listed. In this study, the yield displacement of the restrainer bars is 23.17 mm, and the postyield stiffness/initial stiffness ratio δ is assumed to be 0.07. The initial stiffness of a single restrainer bar is 25 mm, and the number of restrainer bars on each type of pier is calculated and shown in Table 3. For convenient construction, at a given pier height, the same number of restrainer bars is for all ground motions, as shown in Table 3. Regarding

Fig. 12. Relative displacement of each expansion joint comparison between the proposed procedure and numerical analysis: (a) Pier 1; (b) Pier 2; (c) Pier 3; and (4) Pier 4.

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pier, which contribute to unseating damage, are presented in Fig. 13. The bridges with steel restrainer bars and the bridges as built are compared. It can be observed that the relative displacements of the bridge with restrainer bars are significantly smaller than that of the bridges as built, which indicates that the restrainer bars can significantly reduce the relative displacement between the deck and the pier to mitigate unseating damage. As the pier height increases, the relative displacement substantially increases for the bridge as built, and the sliding bearings are destroyed for all cases with the pier height greater than 10 m and some cases with the 10 m pier height. However, for the bridge with restrainer bars, the relative displacement increases slowly and rarely exceeds 100 mm, which indicates that minor destruction would be expected in the sliding bearings. A comparison of Figs. 12 and 13 reveals that the relative displacements for the bridge with restrainer bars are similar in both figures for the pier height less than 15 m. For the pier height greater than 20 m, the relative displacements in Fig. 13 are significantly larger than the displacements in Fig. 12, which indicates that pounding damage continues to occur in the bridges with restrainer bars for higher piers and enlarges the relative displacements between the pier and the deck. The results of the comparison conclude that pounding will significantly increase the relative displacement between the pier and the deck even though restrainer bars are installed in the bridge. For the pier height less than 15 m, the restrainer stiffness calculated by the proposed procedure can totally avoid pounding damage and substantially mitigates the relative displacement to avoid unseating damage. For the pier height greater than 15 m, considerable mitigation in relative displacement is also observed, although the pounding damage cannot be completely avoided

will not occur at these expansion joint during earthquakes. For the cases with a 25 m pier height, the relative displacements under intermediately and weakly correlated earthquakes exceed 50 mm slightly, which indicates that pounding may occur. In some cases with a 20 m pier height, pounding damage can also be observed when the bridge is subjected to intermediately and weakly correlated earthquakes. This result is expected because the stiffness of the restrainer bar cannot satisfy the requirement for the bridge with 20 and 25 m pier height. For the bridge with the pier height of no more than 15 m, the restrainer stiffness calculated by the design procedure is found to be practical and instructive, which enables carefully designed structures that totally avoid pounding damage.

6.2. Effect of pier height In the majority of previous studies, the ratio of the period for adjacent piers has been regarded as an important factor influencing the relative displacement between the pier and the deck. Most of these parametric studies have been conducted under uniform ground motions, and have regarded the different periods of adjacent piers as the major reason of the relative displacement between adjacent bridge spans [6,41]. However, the period substantially influences the displacement of a bridge greatly, and the spatially varying ground motion can increase the relative displacement. Thus, the period of each pier is selected as an important factor in this paper. The expansion joint clearance is assumed to be 50 mm, and the stiffnesses of the restrainer bars listed in Table 3 are applied. The maximum relative displacements between the deck and the

Fig. 13. Relative displacement of each pier: (a) Pier 1; (b) Pier 2; (c) Pier 3; and (4) Pier 4.

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Table 4 Average reduction ratio of relative displacement and shear force. Pier height (m)

8

Coherency loss*

W

10 I

H

W

15 I

H

W

20 I

H

W

25 I

H

W

I

H

Reduction ratio of displacement (%)

42.5

47.7

46.6

49.9

47.7

47.2

54.8

65.9

64

67.5

68.8

68.8

65.4

68.9

67.9

Reduction ratio of shear force (%)

63.8

63.4

64.3

59

59.6

59.1

57.9

61

60.9

50.9

52.7

52.2

36.1

35.2

39.6

* W: weakly correlated ground motions; I: intermediately correlated ground motions; H: highly correlated ground motions.

Fig. 14. Shear force of each pier: (a) Pier 1; (b) Pier 2; (c) Pier 3; and (d) Pier 4.

heights, which may be attributed to the fact that the restrainer bars also mitigate the pounding forces, and the displacement amplification effects of pounding damage are reduced. Table 4 also lists the average reduction percentages of the shear force of each pier with different pier heights and different correlated earthquakes. As shown in Table 4, the reduction percentages of the shear force are similar for circumstances with pier heights ranging from 8 m to 15 m, and only the reduction percentage in the 8 m case is slightly larger than the other values. When the pier height exceeds 15 m, the reduction percentage decreases rapidly, which indicates that the restrainer bars become less effective at reducing the shear force because the flexible pier of the bridge can dissipate a considerable amount of the earthquake energy input; in this case, the shear force is predominantly mitigated by the pier rather than the restrainer bars. Fig. 14 presents the shear force comparison between the bridge with restrainer bars and the bridge as built. From the figure, it can be observed that the shear force significantly decreases from an 8 m pier

for some cases due to the limitation of the number of restrainer bars, and the relative displacements at the expansion joints are significantly greater than the relative displacements for the piers with a height of less than 15 m. The average reduction percentages of the relative displacement for each pier height under different ground motions are listed in Table 4. The reduction percentages of the relative displacement increase with increasing pier height. For the cases of weakly correlated ground motions, the reductions are less than those of the other two ground motions (except for a 10 m pier height). Between pier heights of 8 m and 20 m, the reduction percentages rapidly increase, whereas the values tend to be similar when the pier height is greater than 20 m. The results demonstrate that the restrainer bars become more effective at reducing the relative displacement between the pier and the deck as the pier becomes more flexible. The results also reveal that pounding does not influence the effectiveness of the steel restrainer bars in reducing displacement, as demonstrated by the cases with 20 m and 25 m pier 209

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Fig. 15. Ratio between the bridge with rail and without rail: (a) Relative displacement and (b) shear force.

Fig. 16. Comparison of reduction ratio between the bridge with rail and without rail for each pier: (a) Relative displacement reduction ratio and (b) shear force reduction ratio.

earthquakes are less than one, which indicate that the rails may restrain the relative displacement between the pier and the deck of these piers. Previous studies have shown that rails can restrain the displacement of the superstructure and reduce the relative displacement between the pier and the deck [38–40]. The difference between this paper and previous studies is attributed to spatially varying earthquakes. Previous studies have been performed under uniform earthquakes, whereas the bridge in this paper is subjected to spatially varying earthquakes, and out-of-phase vibrations of the rails are allowed, which may cause a larger relative displacement. However, the restrainer bars mitigate the influence of the out-of-phase movement, which explains why the ratio of the bridge with restrainer bars is less than one in some cases. Based on the ratio of the shear force in Fig. 15(b), the rails enlarge the shear force for the bridge with restrainer bars, whereas the shear force of the as built bridge with the rails is similar to the shear force of the bridge without the rails for Pier 1, Pier 2 and Pier 3. The shear force in Pier 4 is significantly larger than the shear force of the bridge without the rails. Fig. 16 presents the average reduction ratio of the relative displacement and shear force for the restrainer bars in each pier. It can be seen that the reduction ratio of both the relative displacement and the shear force for the bridge with the rail are smaller than the bridge without the rails, which indicates that the effectiveness of the restrainer bars is limited by the rails. Fig. 17 shows the cumulative energy dissipation of the restrainer bars on each pier under weakly correlated earthquakes. The energy dissipation of the restrainer bars with the rails

height to a 20 m pier height for the bridge with restrainer bars and the bridge as built, whereas the shear force tends to be similar for a pier height above 20 m. However, a slight increase in the shear force is observed for a 25 m pier height with restrainer bars, which may be caused by pounding. The decreased reduction percentages of the shear force are also confirmed by Fig. 14; the shear force of the bridge without restrainer bars is similar to that of the bridge with restrainer bars for the 20 m and 25 m cases.

6.3. Effect of the rails The restraint effect of the rails can significantly influence the relative displacement and the shear force of the bridge. In this paper, the effect of the rails is also investigated for three types of earthquakes, and the pier height and the expansion joint clearance are assumed to be 10 m and 50 mm, respectively. Thus, the number of restrainer bars is six, and the total initial stiffness of the restrainer bars is 150KN/mm. Fig. 15(a) presents the ratio of the relative displacement for the bridge with rails and without rails for each pier, and Fig. 15(b) shows the ratio of the shear force. As shown in Fig. 15(a), the relative displacements of most cases with rails are substantially larger than the cases without rails, which indicate that the presence of the rails enlarges the displacement under spatially varying earthquakes, especially for the bridge as built. For the bridge with restrainer bars, the ratios of Pier 1, Pier 2 and Pier 3 under highly and intermediately correlated 210

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Fig. 17. Cumulative energy dissipation of restrainers on each pier: (a) Pier 1; (b) Pier 2; (c) Pier 3; and (4) Pier 4.

pier height, both the requirement of the restrainer bars and the effectiveness of the restrainer bars in restraining the relative displacement increase, whereas the effectiveness in mitigating the shear force decreases when the pier height exceeds 15 m. The effect of the rails in this paper differs from previous studies due to the inclusion of spatially varying earthquakes. The restraint effect of the rails is not distinct in the cases in this paper. For most cases, the relative displacements are substantially larger than the cases without rails. The reduction ratio of relative displacement to the shear force is also reduced for the bridge with rails, and the rails limit the effectiveness of the restrainer bars by reducing the energy dissipation of the restrainer bars.

is significantly smaller than the energy dissipation without the rails, which explains why the reduction ratio of the relative displacement and shear force are reduced compared to the bridge without the rails. 7. Conclusions In high-intensity earthquake areas, bridge structures may be exposed to unseating and pounding damage due to different vibrations between adjacent bridge structures and spatially varying ground motions. Steel restrainer bars are proposed to restrain the relative displacement between the pier and the deck and between adjacent decks to mitigate unseating and pounding damage. In this paper, the theoretical formulas for the mechanical properties of the restrainer bars are derived using the equivalent stiffness method. Numerical analysis and experiments are conducted to calculate the conversion ratio of both the yield force and the yield displacement γ , which ranges between 1.3 and 2.4, and the post-yield stiffness/initial stiffness ratio δ is calculated from 0.06 to 0.13. A design procedure based on a linear model is developed. The design procedure considers spatially varying earthquakes and the hysteretic behaviour of the restrainer bars. Fifteen cases with various pier heights and different types of earthquakes are conducted to evaluate the effectiveness of the design procedure. The relative displacements between adjacent decks derived from a time history analysis are compared with the values calculated via the new procedure. The consistency between the design procedure and the time history analysis is confirmed by case studies. The results indicate that the pier height has a significant influence on the effectiveness of the restrainer bars. With an increase in

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