# Study on the Stiffness Correction Method of Novel Antivibration Bearing for Urban Rail Transit Viaduct

Authors: Weiping Xie; Youfu Du; Liangming Sun; Agostino Marioni; Xu Liang
Journal: Shock and Vibration
 (2017)

---

## Abstract

A novel antivibration bearing is developed to reduce the train-induced vibrations for urban rail transit viaduct. It adopts four high-damping thick rubber blocks stacking slantingly to reduce the vibration and provide large lateral stiffness. But the existing stiffness calculation method of laminated rubber bearing aimed at horizontal seismic isolation is unsuitable for thick rubber bearing designed for vertical vibration reduction. First, the stiffness correction method has been proposed based on the characteristics of the novel bearing. Second, to validate the design method, mechanical property tests are performed on a specimen of the novel bearing with design frequency at 8 Hz and with 3500 kN bearing capacity. Third, damping effects of the novel bearing are investigated through impulse vibration tests on scaled models. Results show that the mechanical property of the novel bearing can satisfy the engineering demand, and the proposed method for calculating the stiffness agrees well with the test results. The overall insertion loss of the novel bearing is 13.49 dB which is 5.32 dB larger than that of steel bearing, showing that the novel bearing is very promising to be used in the field to mitigate train-induced vibrations.


---

## Body

## 1. Introduction

Using viaduct as its main structure, rail transit connects the main cities or economic regions. It plays a vital role in the public transportation. Train-induced bridge vibrations will propagate to the subsoil via the bearings, piers, and foundations and then spread to the surroundings, causing the vibration of surrounding ground, the secondary vibration of adjacent buildings, and its associated structural noise. Since most elevated rail lines close to or even run across the densely populated areas and high-tech industrial parks in cities, the vibration will not only affect the structural safety of the adjacent ancient buildings and people’s normal life, but also may have a serious impact on the normal use of that vibration-sensitive high precision instruments in the nearby hospitals, schools, and research institutions [1]. Mitigating at the vibration source and along the vibration propagation path are two major ways to circumvent such train-induced vibrations problem [2]. At the vibration source, the train-bridge system shall be treated sophisticatedly with focus on the vibration mitigation. Elastic track structure [3], tuned mass dampers [4], and fluid viscous dampers [5] are common countermeasurements applied in the field. But the design is sophisticated and costly. The other way is to control the vibration along propagation path. Open or in-filled trenches [6, 7], barriers of piles [8–10], and wave impedance blocks [11, 12] are found to be adopted to mitigate vibrations. But the construction is difficult and site conditions are restricted. Therefore, economical and effective countermeasures are being urgently sought.The bridge bearing, which connects the superstructure and piers, is on the route of the train-induced vibration propagation and plays a key role as the target of vibration control. But most existing bridge bearings are renowned for their antiseismic function, such as laminated rubber bearings [13], SMA-based rubber bearings [14], and fiber-reinforced rubber bearings [15]. Those bearings have deficient performance of vertical vibration mitigation because of the high compression stiffness. Considering the dominant direction of the train-induced vibration is vertical, retrofits have to be done on the existing rubber bearings.To mitigate the train-induced vibrations, elastic bearing pad (EBP) with the assorted shear key (SK) as the supported system of the bridge has been applied to about 5 km of Taiwan high speed railway [16]. Later, this kind of antivibration bearing was also adopted by the Yichang Yangtze River Railway Bridge in Yichang-Wanzhou Railway and the Minjiang River Railway Bridge in Fuzhou-Xiamen Railway to reduce the train-induced vibrations [17]. Sun et al. [18] testified the damping effect of EBP and pot rubber bearing when trains are running on the Yichang Yangtze River Railway Bridge. Tests result showed that the overall insertion loss between the upper and lower plates for EBP and pot rubber bearing was 13.40 dB and 9.65 dB, respectively, and EBP could reduce more vertical train-induced vibrations than pot rubber bearing in most frequency components.Li et al. [19, 20] analyzed the effect of elastic bearings on train-induced vibrations by simulating the elastic bearings as spring-damper elements. The results showed that the stiffness of elastic bearings had marginally influenced wheel unloading rate and the vertical acceleration of the vehicle, because the elastic bearings could absorb the vibrations in the medium and high frequency bands and amplify their own vibration of natural frequency band. Besides, most researches [21–25] about the elastic bearings were simulated as spring elements to explore the impact of the dynamic responses of vehicle-bridge interaction system. For example, Kawatani et al. [21] analyzed dynamic responses of vehicle-bridge interaction system of certain highway bridge with steel bearings compared to that with elastic bearings based on numerical simulation method. The results showed that elastic bearings could effectively reduce the high frequency vibration. Through the theoretical analysis, Yang et al. [23, 24] studied the impact on dynamic responses of vehicle-bridge interaction system exerted by elastic bearings under moving train loads. The results indicated that the elastic bearings could prevent the transmission or dissipation of vehicle-induced forces from the superstructure to the ground, causing the huge amount of train-induced vibrations energy accumulated and amplified on the bridge during its passage.The above works indicated that elastic bearing with certain flexibility in vertical direction can reduce the train-induced vertical vibrations transferred to the substructure. Currently, only EBP has been preliminarily applied to mitigate train-induced vibrations, but it must assort SK to resist lateral load because of its small horizontal stiffness. Therefore, based on the attenuation mechanism of antivibration bearing, a novel antivibration bearing with four high-damping thick rubber blocks stacking slantingly is proposed to mitigate vertical vibrations in this paper. Compared with EBP, the novel bearing can mitigate the train-induced vertical vibrations more effectively and provide a large lateral stiffness to limit the lateral displacement caused by train braking or turning at a high speed. The straight forward design of such rubber bearing with low manufacturing costs makes the device promising to be used in the field.Yabana and Matsuda [26] and He et al. [27] conducted mechanical property tests on thick rubber bearing. Results showed that large compression deformation of thick rubber bearing would be produced under design pressure and the calculation method of compression stiffness of laminated rubber bearing aimed at horizontal seismic isolation was not suitable for thick rubber bearing. Therefore, the calculation of stiffness of the novel bearing with thick rubber blocks must be corrected. Considering the novel antivibration bearing is modeled linearly in the train-bridge interaction system, the stiffness correction method based on multiple iterations with design compression displacement has been proposed. To validate the design method of the novel bearing, mechanical property tests are performed on a specimen. Finally, damping effects of the novel bearing are investigated through impulse vibration tests on scaled models.


## 2. Design of Novel Antivibration Bearing

### 2.1. Antivibration Mechanism

Rail transit viaduct with antivibration bearing can be treated as mass-spring-damper system. It can mitigate the sensitivity of the substructure to the train-induced vibrations over the superstructure by inserting a simple harmonic oscillator with lower natural frequency [16]. It can be described as antivibration system with single degree of freedom, as shown in Figure 1.Figure 1
Antivibration system with single degree of freedom.The mass block represents the bridge superstructure. Assume that the mass block and foundation are rigid, and the elastic spring has damping characteristic,m is the mass of the block in the antivibration system, k is spring stiffness, c is viscous damping coefficient, F(t) is external force, and x(t) is vertical vibration displacement of the mass block. When motivated by the external harmonic excitation F(t)=F0sin⁡(θt), the motion equation of the mass block is (1)mx¨t+cx˙t+kxt=F0sinθt.Based on the theory of structural dynamics [28], the transmissibility coefficient of the antivibration system is(2)T=1+2ξβ21-β22+2ξβ2,where β=θ/ω is the frequency ratio, θ is the circular frequency of the external harmonic excitation, ω is the natural circular frequency of the system, and ξ is the damping ratio.The variation curve of transmissibility coefficientT with the frequency ratio β for discrete values of the damping ratio ξ is shown in Figure 2. It is indicated that all curves of different damping ratios pass through the same point when frequency ratio β=21/2. Clearly because of this feature, the effectiveness of vibration reduction system will be enhanced by increasing the damping when β<21/2. Similarly, the effectiveness will be decreased by increasing the damping when β>21/2. Since the transmissibility coefficients for β>21/2 are generally much lower than those for β<21/2, it is more efficient to apply countermeasurements at the region with high frequency ratio. However, this is not always practical because the system must operate below β=21/2 in many cases for some intervals of time, and in many other cases it may even operate near the resonant condition β=1. Consequently, the principle of vibration mitigation is to increase the frequency ratio as much as possible by decreasing natural frequency of the vibration system, meanwhile to provide proper damping for the bearing to avoid the resonance.Figure 2
Transmissibility coefficient.


### 2.2. Design Method

The novel antivibration bearing is designed for urban rail transit viaduct with simply supported box girder. The main type of train used in Chinese urban rail transit is Type-B metro vehicle, and its distance between bogies in the same carriage (Lc) or adjacent carriages (Lb) is 12.6 m or 19 m while between the axles (La) it is 2.2 m. Considering the normal operating speed as 60 km/h, the range of variation of the train excitation frequencies is from 0.88 Hz to 7.58 Hz. Therefore, it is appropriate to control the vertical natural frequency of the bridge around 8–10 Hz by proper designing the compression stiffness of the novel bearing, which can efficiently avert from the train excitation frequencies and the noise frequency of the upper structure (10–100 Hz). Generally, the antivibration bearing is modeled linearly in the train-bridge interaction system; thus its natural frequency f (Hz) can be calculated based on the deflection D (m) of bridge structure under static load if the damping is ignored (3). And from the load-deformation curve obtained through the mechanical property test, the compression stiffness of the bearing can be estimated.(3)f=g2πD≈12D,where g is the gravitational acceleration.From (3), the natural frequency of the antivibration bearing can be decreased with the increase of the deflection of the bridge, but the deflection should not exceed 3-4 mm, or it would impact the serviceability and safety of running train. Hence, the appropriate natural frequency of the novel bearing should be set around 8–10 Hz. But in some cases, the vertical natural frequency of the bridge with the novel bearing may be lower than 8 Hz but generally greater than 4 Hz (depending on the bridge type, mass, and span length). It overlaps with the axis frequency but is far greater than the bogie frequency. Actually, the axle frequency fa (v/La) exerts less influence on the vibration response of train-bridge interaction system, because the wavelength is short and the energy input from the track is small. The resonance response mainly depends on bogie frequency fb (v/Lb) or fc (v/Lc). Considering the maximum speed as 80 km/h, the resonance frequency is less than 1.76 Hz. Thus, only little effects would be conducted on the vibration response of train-bridge interaction system when the natural frequency of the bridge is lower than 8 Hz.The structure diagrams of novel antivibration bearing are shown in Figure3 where two rubber blocks are α inclined side by side on the left and another two are laid symmetrically on the right. The value range of α is 20°–70°. The lateral direction of the novel bearing is fixed and the longitudinal direction is moveable. To satisfy the different engineering demand, the novel bearing can be designed in different frequency and bearing capacity by changing the parameters of rubber block. The compression stiffness of the novel bearing can ensure the vertical capacity of urban rail transit viaduct and reduce the bridge deflection under dynamic load. Certain horizontal stiffness in lateral direction can limit the lateral displacement caused by extra horizontal load when train is braking or turning at a high speed, ensuring the safety and stability of running trains. And smaller horizontal stiffness in longitudinal direction can ensure the longitudinal displacement of the bridge caused by thermal effect of girder. The application of novel antivibration bearing can increase the frequency ratio β by decreasing the circular natural frequency ω of the bridge, and thus the frequency ratio β can be adjusted to the damping range to meet the expected damping effect.Figure 3
The structure diagram of novel antivibration bearing: (a) front view and (b) relative location arrangement diagram of rubber blocks.Notes. 1 Upper plate; 2 high-damping thick rubber block; 3 lower plate; 4 anchor stud; 5 connection plate; (6)~(9) bolt; 10 nut.
(a)
(b)


### 2.3. Stiffness Correction Method

#### 2.3.1. Stiffness Calculation for Single Rubber Block

The effective size of rubber block that constitutes the novel bearing isa×a. Based on Chinese national standard GB20688.2-2006 [29], the shape coefficient of the rubber block is(4)S1=a4trS2=antr,where tr is the thickness of single layer rubber and n is number of rubber layers.Horizontal equivalent stiffness is(5)Kh=GAntr,where G is the shear modulus of rubber and A is effective loaded area of rubber block.Compression stiffness is(6)KV=EcAntr,where Ec is the compression elasticity modulus of rubber and the empirical equation Ec=3G1+2S12 is adopted in this paper [30].


#### 2.3.2. Stiffness Calculation for Novel Antivibration Bearing

Based on the composition and separation of mechanics, the compression stiffness of the novel bearing can be calculated through the mechanical calculation model:(7)KN=KVcos2α+Khsin2α×4.Horizontal equivalent stiffness in longitudinal direction is(8)Kh1=4Kh.Horizontal equivalent stiffness in lateral direction is(9)Kh2=KVsin2α+Khcos2α×4,where α is vertical inclination angle of rubber block.


#### 2.3.3. Stiffness Correction

The stiffness calculation equations of rubber bearings are all based on design pressureP0 in the test methods of elastomeric seismic-protection isolators [31]. Different from the laminated rubber bearing aimed at horizontal seismic isolation, the thick rubber bearing has thick single layer rubber, and thus the restraining effect from the internal steel plate to the rubber layer is weaker. Weaker restraints would cause smaller compression stiffness and large compression deformation would be expected under design pressure P0. Therefore, for the thick rubber bearing, the effect of large compression deformation on the shape coefficient cannot be ignored in the stiffness calculation. Since the novel bearing is composed of thick rubber blocks, its stiffness calculation method must be corrected.Four thick rubber blocks of the novel bearing will have compression deformation and shear deformation simultaneously under design pressureP0 as shown in Figure 4. The compression deformation of each rubber block is dcos⁡α and the shear deformation is dsin⁡α when compression displacement of the novel bearing is d. The deformation of single layer rubber is shown in Figure 5, where dv=dcos⁡α/n and dh=dsin⁡α/n. The shape coefficient S1 will increase with the decrease of tr caused by compression deformation, and the effective loaded area A will decrease by shear deformation.Figure 4
Deformation of the novel bearing under compression load.Figure 5
Deformation of single layer rubber.Therefore, the horizontal equivalent stiffness of single rubber block is corrected as(10)Kh′=GAntr-dv.Shape coefficientS1 will turn into(11)S1′=a4tr-dv.Compressive elasticity modulusEc will turn into(12)Ec′=3G1+2S1′2.Compression area of single layer rubber will turn into(13)A′=a×a-dh.And compression stiffness is corrected as(14)KV′=Ec′A′ntr-dv.Putting the corrected horizontal equivalent stiffness and compression stiffness of single rubber block into calculation equations (7)–(9), the corrected stiffness of the novel bearing can be calculated.Therefore, to carry out stiffness correction of the novel bearing, the compression displacementd under design pressure P0 must be confirmed at first. Considering the novel antivibration bearing is modeled linearly in the train-bridge interaction system, d1 can be obtained through dividing the design pressure P0 by the original compression stiffness KV, and KV1 can be obtained based on d1, d2=P0/KV1,…, and then the value of d will tend toward stability after multiple iterations, and it will be utilized to correct the stiffness of the novel bearing. Besides, the displacement d can be confirmed through compression test.


## 2.1. Antivibration Mechanism

Rail transit viaduct with antivibration bearing can be treated as mass-spring-damper system. It can mitigate the sensitivity of the substructure to the train-induced vibrations over the superstructure by inserting a simple harmonic oscillator with lower natural frequency [16]. It can be described as antivibration system with single degree of freedom, as shown in Figure 1.Figure 1
Antivibration system with single degree of freedom.The mass block represents the bridge superstructure. Assume that the mass block and foundation are rigid, and the elastic spring has damping characteristic,m is the mass of the block in the antivibration system, k is spring stiffness, c is viscous damping coefficient, F(t) is external force, and x(t) is vertical vibration displacement of the mass block. When motivated by the external harmonic excitation F(t)=F0sin⁡(θt), the motion equation of the mass block is (1)mx¨t+cx˙t+kxt=F0sinθt.Based on the theory of structural dynamics [28], the transmissibility coefficient of the antivibration system is(2)T=1+2ξβ21-β22+2ξβ2,where β=θ/ω is the frequency ratio, θ is the circular frequency of the external harmonic excitation, ω is the natural circular frequency of the system, and ξ is the damping ratio.The variation curve of transmissibility coefficientT with the frequency ratio β for discrete values of the damping ratio ξ is shown in Figure 2. It is indicated that all curves of different damping ratios pass through the same point when frequency ratio β=21/2. Clearly because of this feature, the effectiveness of vibration reduction system will be enhanced by increasing the damping when β<21/2. Similarly, the effectiveness will be decreased by increasing the damping when β>21/2. Since the transmissibility coefficients for β>21/2 are generally much lower than those for β<21/2, it is more efficient to apply countermeasurements at the region with high frequency ratio. However, this is not always practical because the system must operate below β=21/2 in many cases for some intervals of time, and in many other cases it may even operate near the resonant condition β=1. Consequently, the principle of vibration mitigation is to increase the frequency ratio as much as possible by decreasing natural frequency of the vibration system, meanwhile to provide proper damping for the bearing to avoid the resonance.Figure 2
Transmissibility coefficient.


## 2.2. Design Method

The novel antivibration bearing is designed for urban rail transit viaduct with simply supported box girder. The main type of train used in Chinese urban rail transit is Type-B metro vehicle, and its distance between bogies in the same carriage (Lc) or adjacent carriages (Lb) is 12.6 m or 19 m while between the axles (La) it is 2.2 m. Considering the normal operating speed as 60 km/h, the range of variation of the train excitation frequencies is from 0.88 Hz to 7.58 Hz. Therefore, it is appropriate to control the vertical natural frequency of the bridge around 8–10 Hz by proper designing the compression stiffness of the novel bearing, which can efficiently avert from the train excitation frequencies and the noise frequency of the upper structure (10–100 Hz). Generally, the antivibration bearing is modeled linearly in the train-bridge interaction system; thus its natural frequency f (Hz) can be calculated based on the deflection D (m) of bridge structure under static load if the damping is ignored (3). And from the load-deformation curve obtained through the mechanical property test, the compression stiffness of the bearing can be estimated.(3)f=g2πD≈12D,where g is the gravitational acceleration.From (3), the natural frequency of the antivibration bearing can be decreased with the increase of the deflection of the bridge, but the deflection should not exceed 3-4 mm, or it would impact the serviceability and safety of running train. Hence, the appropriate natural frequency of the novel bearing should be set around 8–10 Hz. But in some cases, the vertical natural frequency of the bridge with the novel bearing may be lower than 8 Hz but generally greater than 4 Hz (depending on the bridge type, mass, and span length). It overlaps with the axis frequency but is far greater than the bogie frequency. Actually, the axle frequency fa (v/La) exerts less influence on the vibration response of train-bridge interaction system, because the wavelength is short and the energy input from the track is small. The resonance response mainly depends on bogie frequency fb (v/Lb) or fc (v/Lc). Considering the maximum speed as 80 km/h, the resonance frequency is less than 1.76 Hz. Thus, only little effects would be conducted on the vibration response of train-bridge interaction system when the natural frequency of the bridge is lower than 8 Hz.The structure diagrams of novel antivibration bearing are shown in Figure3 where two rubber blocks are α inclined side by side on the left and another two are laid symmetrically on the right. The value range of α is 20°–70°. The lateral direction of the novel bearing is fixed and the longitudinal direction is moveable. To satisfy the different engineering demand, the novel bearing can be designed in different frequency and bearing capacity by changing the parameters of rubber block. The compression stiffness of the novel bearing can ensure the vertical capacity of urban rail transit viaduct and reduce the bridge deflection under dynamic load. Certain horizontal stiffness in lateral direction can limit the lateral displacement caused by extra horizontal load when train is braking or turning at a high speed, ensuring the safety and stability of running trains. And smaller horizontal stiffness in longitudinal direction can ensure the longitudinal displacement of the bridge caused by thermal effect of girder. The application of novel antivibration bearing can increase the frequency ratio β by decreasing the circular natural frequency ω of the bridge, and thus the frequency ratio β can be adjusted to the damping range to meet the expected damping effect.Figure 3
The structure diagram of novel antivibration bearing: (a) front view and (b) relative location arrangement diagram of rubber blocks.Notes. 1 Upper plate; 2 high-damping thick rubber block; 3 lower plate; 4 anchor stud; 5 connection plate; (6)~(9) bolt; 10 nut.
(a)
(b)


## 2.3. Stiffness Correction Method

### 2.3.1. Stiffness Calculation for Single Rubber Block

The effective size of rubber block that constitutes the novel bearing isa×a. Based on Chinese national standard GB20688.2-2006 [29], the shape coefficient of the rubber block is(4)S1=a4trS2=antr,where tr is the thickness of single layer rubber and n is number of rubber layers.Horizontal equivalent stiffness is(5)Kh=GAntr,where G is the shear modulus of rubber and A is effective loaded area of rubber block.Compression stiffness is(6)KV=EcAntr,where Ec is the compression elasticity modulus of rubber and the empirical equation Ec=3G1+2S12 is adopted in this paper [30].


### 2.3.2. Stiffness Calculation for Novel Antivibration Bearing

Based on the composition and separation of mechanics, the compression stiffness of the novel bearing can be calculated through the mechanical calculation model:(7)KN=KVcos2α+Khsin2α×4.Horizontal equivalent stiffness in longitudinal direction is(8)Kh1=4Kh.Horizontal equivalent stiffness in lateral direction is(9)Kh2=KVsin2α+Khcos2α×4,where α is vertical inclination angle of rubber block.


### 2.3.3. Stiffness Correction

The stiffness calculation equations of rubber bearings are all based on design pressureP0 in the test methods of elastomeric seismic-protection isolators [31]. Different from the laminated rubber bearing aimed at horizontal seismic isolation, the thick rubber bearing has thick single layer rubber, and thus the restraining effect from the internal steel plate to the rubber layer is weaker. Weaker restraints would cause smaller compression stiffness and large compression deformation would be expected under design pressure P0. Therefore, for the thick rubber bearing, the effect of large compression deformation on the shape coefficient cannot be ignored in the stiffness calculation. Since the novel bearing is composed of thick rubber blocks, its stiffness calculation method must be corrected.Four thick rubber blocks of the novel bearing will have compression deformation and shear deformation simultaneously under design pressureP0 as shown in Figure 4. The compression deformation of each rubber block is dcos⁡α and the shear deformation is dsin⁡α when compression displacement of the novel bearing is d. The deformation of single layer rubber is shown in Figure 5, where dv=dcos⁡α/n and dh=dsin⁡α/n. The shape coefficient S1 will increase with the decrease of tr caused by compression deformation, and the effective loaded area A will decrease by shear deformation.Figure 4
Deformation of the novel bearing under compression load.Figure 5
Deformation of single layer rubber.Therefore, the horizontal equivalent stiffness of single rubber block is corrected as(10)Kh′=GAntr-dv.Shape coefficientS1 will turn into(11)S1′=a4tr-dv.Compressive elasticity modulusEc will turn into(12)Ec′=3G1+2S1′2.Compression area of single layer rubber will turn into(13)A′=a×a-dh.And compression stiffness is corrected as(14)KV′=Ec′A′ntr-dv.Putting the corrected horizontal equivalent stiffness and compression stiffness of single rubber block into calculation equations (7)–(9), the corrected stiffness of the novel bearing can be calculated.Therefore, to carry out stiffness correction of the novel bearing, the compression displacementd under design pressure P0 must be confirmed at first. Considering the novel antivibration bearing is modeled linearly in the train-bridge interaction system, d1 can be obtained through dividing the design pressure P0 by the original compression stiffness KV, and KV1 can be obtained based on d1, d2=P0/KV1,…, and then the value of d will tend toward stability after multiple iterations, and it will be utilized to correct the stiffness of the novel bearing. Besides, the displacement d can be confirmed through compression test.


## 2.3.1. Stiffness Calculation for Single Rubber Block

The effective size of rubber block that constitutes the novel bearing isa×a. Based on Chinese national standard GB20688.2-2006 [29], the shape coefficient of the rubber block is(4)S1=a4trS2=antr,where tr is the thickness of single layer rubber and n is number of rubber layers.Horizontal equivalent stiffness is(5)Kh=GAntr,where G is the shear modulus of rubber and A is effective loaded area of rubber block.Compression stiffness is(6)KV=EcAntr,where Ec is the compression elasticity modulus of rubber and the empirical equation Ec=3G1+2S12 is adopted in this paper [30].


## 2.3.2. Stiffness Calculation for Novel Antivibration Bearing

Based on the composition and separation of mechanics, the compression stiffness of the novel bearing can be calculated through the mechanical calculation model:(7)KN=KVcos2α+Khsin2α×4.Horizontal equivalent stiffness in longitudinal direction is(8)Kh1=4Kh.Horizontal equivalent stiffness in lateral direction is(9)Kh2=KVsin2α+Khcos2α×4,where α is vertical inclination angle of rubber block.


## 2.3.3. Stiffness Correction

The stiffness calculation equations of rubber bearings are all based on design pressureP0 in the test methods of elastomeric seismic-protection isolators [31]. Different from the laminated rubber bearing aimed at horizontal seismic isolation, the thick rubber bearing has thick single layer rubber, and thus the restraining effect from the internal steel plate to the rubber layer is weaker. Weaker restraints would cause smaller compression stiffness and large compression deformation would be expected under design pressure P0. Therefore, for the thick rubber bearing, the effect of large compression deformation on the shape coefficient cannot be ignored in the stiffness calculation. Since the novel bearing is composed of thick rubber blocks, its stiffness calculation method must be corrected.Four thick rubber blocks of the novel bearing will have compression deformation and shear deformation simultaneously under design pressureP0 as shown in Figure 4. The compression deformation of each rubber block is dcos⁡α and the shear deformation is dsin⁡α when compression displacement of the novel bearing is d. The deformation of single layer rubber is shown in Figure 5, where dv=dcos⁡α/n and dh=dsin⁡α/n. The shape coefficient S1 will increase with the decrease of tr caused by compression deformation, and the effective loaded area A will decrease by shear deformation.Figure 4
Deformation of the novel bearing under compression load.Figure 5
Deformation of single layer rubber.Therefore, the horizontal equivalent stiffness of single rubber block is corrected as(10)Kh′=GAntr-dv.Shape coefficientS1 will turn into(11)S1′=a4tr-dv.Compressive elasticity modulusEc will turn into(12)Ec′=3G1+2S1′2.Compression area of single layer rubber will turn into(13)A′=a×a-dh.And compression stiffness is corrected as(14)KV′=Ec′A′ntr-dv.Putting the corrected horizontal equivalent stiffness and compression stiffness of single rubber block into calculation equations (7)–(9), the corrected stiffness of the novel bearing can be calculated.Therefore, to carry out stiffness correction of the novel bearing, the compression displacementd under design pressure P0 must be confirmed at first. Considering the novel antivibration bearing is modeled linearly in the train-bridge interaction system, d1 can be obtained through dividing the design pressure P0 by the original compression stiffness KV, and KV1 can be obtained based on d1, d2=P0/KV1,…, and then the value of d will tend toward stability after multiple iterations, and it will be utilized to correct the stiffness of the novel bearing. Besides, the displacement d can be confirmed through compression test.


## 3. Mechanical Property of Novel Antivibration Bearing

### 3.1. Specimen and Testing Apparatus

To validate the design method, mechanical property tests are performed on a specimen of the novel bearing with design natural frequency at 8 Hz and with 3500 kN bearing capacity. The design parameters of the specimen are shown in Table1. The inclination angle α of rubber block is 45°.Table 1
Design parameters of the specimen of the novel bearing.
Rubber block
Rubber hardness
60
Shear modulusG (MPa)
0.8
Effective sizea×a(mm)
300
×
300
Overall thickness (mm)
99.5
Primary shape factorS1
4.55
Secondary shape factorS2
6.06
Specimen of the novel bearing
Frequency (Hz)
8
Vibration damping ratioξ
10%
Vertical capacity (kN)
3500
Lateral capacity (kN)
700
Lateral displacement (mm)
±0
Longitudinal displacement (mm)
±50The specimen and testing equipment are fixed mechanically through an external fixing plate to avoid the relative sliding between the specimen and test platform, as shown in Figure6. The mechanical property tests are performed through 75000 kN multipurpose testing machine, as shown in Figure 7. The machine can conduct dynamic control in both vertical and horizontal direction simultaneously. Performance parameters of this testing machine are as follows: vertical dynamic load 75000 kN, horizontal dynamic load 6000 kN, maximum vertical displacement 120 mm, maximum horizontal displacement ±600 mm, and maximum horizontal peak velocity 1000 mm/s.Figure 6
Test specimen of novel antivibration bearing.Figure 7
Multifunctional testing machine.


### 3.2. Testing Program

The mechanical property tests are strictly performed according to GB/T 20688.1-2007 [31]. Three-cyclic loading of 0-Pmax-0 will be conducted in vertical direction during the compressive property test, among which Pmax is maximum design pressure 3500 kN. Displacement loading in longitudinal direction and horizontal load loading in lateral direction are adopted in shear property test. Shear property correlation tests include the correlation of shearing strain, loading frequency, and loading time, which mainly study horizontal mechanical property under varying horizontal deformation. After the correlation tests, the ultimate shear property tests are performed in longitudinal positive direction and negative direction, respectively, to measure the ultimate shear displacement ability of the test specimen under the maximum design pressure. The detailed testing program is shown in Table 2.Table 2
Mechanical property test program of novel antivibration bearing.
Test types
Loading direction
Amplitude
Loading frequency/Hz
Loading waveform
Vertical loads/kN
Cycle-index
Compression property
Vertical
/
/
/
3500
3
Shear property
Longitudinal
±
50 mm
0.005
/
2600
3
Lateral
±
700 kN
0.2
/
2600
3
Shearing strain correlation
Longitudinal
25 mm
0.5
Sine
2600
3
50 mm
0.5
Sine
2600
3
75 mm
0.5
Sine
2600
3
Loading frequency correlation
Longitudinal
50 mm
0.001
Ramp
2600
1
50 mm
0.005
Ramp
2600
1
50 mm
0.01
Ramp
2600
3
50 mm
0.1
Sine
2600
3
50 mm
0.5
Sine
2600
3
50 mm
1.0
Sine
2600
3
50 mm
2.0
Sine
2600
3
Loading times correlation
Longitudinal
50 mm
0.5
Sine
2600
50
Ultimate shear property
Longitudinal
±
75 mm
/
/
3500
/


### 3.3. Definition of Mechanical Property Index

The calculation equation of compression stiffness is(15)KN=P2-P1Y2-Y1,where P1 is 0.7P0, P2 is 1.3P0, P0 is the design pressure 2600 kN in normal train operation, and Y1 and Y2 are compression displacement corresponding to the third cycle of compression loads P1 and P2, respectively.The equation of horizontal equivalent stiffness is(16)Kh=Q1-Q2X1-X2,where Q1 is the maximum shear force, Q2 is the minimum shear force, X1 is maximum displacement, and X2 is minimum displacement.The equation of horizontal equivalent damping ratio is(17)heq=2ΔWπKhX1-X22,where ΔW is envelop area of the third cycle of the hysteretic loops.


### 3.4. Results and Discussion

#### 3.4.1. Compression Property and Shear Property

Compressive load-displacement curve under three-cyclic loading is shown in Figure8. Based on (15), compression stiffness KN of the specimen is 479.8 kN/mm, which can ensure the vertical capacity of urban rail transit viaduct.Figure 8
Compressive load-displacement curve under three-cyclic loading.Horizontal load-displacement curves of the specimen under three-cyclic loading are shown in Figure9. The horizontal load shown in the figure comprises the actual shearing load and the friction of machinery equipment itself. Then, the actual shearing load is equal to horizontal load minus the friction of machinery equipment. Based on (16), horizontal equivalent stiffness of the specimen in longitudinal direction (Kh1) and lateral direction (Kh2) is 7.31 kN/mm and 861.4 kN/mm, respectively. Thus, the measured Kh1 is smaller, which can ensure the horizontal displacement of the longitudinal bridge caused by thermal effect of girder. Based on (17), the equivalent damping ratio heq in longitudinal direction is 9.6%, showing the specimen possesses great energy-dissipating capacity. Besides, Figure 9(b) indicates that the lateral displacement of the specimen is less than 1 mm when the horizontal load reaches the design lateral bearing capacity 700 kN. It shows that Kh2 of the specimen is sufficient to resist the expected horizontal load in the lateral direction of bridge and to limit the lateral displacement caused by train braking or turning at a high speed.Figure 9
Horizontal load-displacement curves under three-cyclic loading: (a) longitudinal and (b) lateral.
(a)
(b)The theoretical stiffness of the specimen can be calculated based on the stiffness calculation method mentioned in Section2.3. It should be noted that the design compression displacement d required for stiffness correction is obtained by trial. The deviation between d5=5.51 mm and d4=5.52 mm is less than 1%. Thus, the iteration number should be set to 5 and the final design compression displacement d set to 5.51 mm. Figure 8 shows that the measured compression displacement of the specimen is 5.83 mm when the compression load is 2600 kN. The error between measured compression displacement and design compression displacement is only 5.5%. It proves that the design compression displacement after multiple iterations has high precision.Comparison among the theoretical stiffness and measured stiffness of the specimen is shown in Table3. It indicates that the error between corrected value and measured value is smaller than that of standard calculated value, whether compression stiffness or horizontal equivalent stiffness. Kh2 of the specimen is much larger than theoretical value. That is because the lateral deformation is less than 1 mm (shearing strain γ<2%) in the test, and the actual shear modulus is much larger than theoretical value (γ=100%) when γ is minimal. Higher horizontal equivalent stiffness in lateral direction means stronger lateral load resistance to meet the operating requirements better. Since the damping effect of the novel bearing depends mainly on compression stiffness and the error between corrected compression stiffness and measured compression stiffness is only 1.7%, the stiffness correction method is much more consistent with real situation.Table 3
Comparison among the theoretical stiffness and measured stiffness.
Stiffness
Measured
Standard calculated
Error
Corrected value
Error
K
N (kN/mm)
479.8
372.0
22.5%
471.9
1.7%
K
h
1 (kN/mm)
7.31
5.80
20.7%
6.32
15.7%
K
h
2 (kN/mm)
861.4
372.0
56.8%
471.9
45.2%


#### 3.4.2. Correlation of Shear Property in Longitudinal Direction

The horizontal load-displacement curves of the specimen in different shear strain are shown in Figure10. And the corresponding test results of Kh1 are shown in Table 4. It is indicated that Kh1 decreases with the increase of longitudinal displacement, which agrees with characteristics of the hysteresis loop of high-damping rubber. Based on the shearing strain 100% (50 mm), the variation range of Kh1 will be 0.93–1.08.Table 4
Test results ofKh1 in different shear strain.
Shear strain
Q
1 (kN)
Q
2 (kN)
X
1 (mm)
X
2 (mm)
K
h
1 (kN/mm)
Normalization
50%
195.0
∗
-
192.9
∗
24.59
-
24.17
7.96
1.08
100%
366.5
∗
-
359.6
49.36
-
49.13
7.37
1.00
150%
505.0
∗
-
503.1
∗
73.7
-
73.6
6.84
0.93
Notes. The value with ∗ is actual shear load which is equal to horizontal load minus the friction of machinery equipment, the same in Tables 5 and 6.Table 5
Test results ofKh1 in different loading frequency.
Frequency (Hz)
Q
1 (kN)
Q
2 (kN)
X
1 (mm)
X
2 (mm)
K
h
1 (kN/mm)
Normalization
0.001
397.3
∗
-
414.4
∗
49.14
-
50.91
8.11
1.12
0.005
398.3
∗
-
404.1
∗
49.14
-
50.93
8.07
1.12
0.01
385.2
∗
-
407.4
∗
48.97
-
50.74
7.96
1.10
0.1
367.6
∗
-
346.5
∗
49.74
-
49.30
7.21
1.00
0.5
357.9
∗
-
350.4
∗
49.29
-
49.00
7.21
1.00
1.0
333.7
∗
-
326.9
∗
48.14
-
48.27
6.85
0.95
2.0
323.3
∗
-
333.4
∗
49.04
-
49.24
6.68
0.93Table 6
Test results ofKh1 in different loading times.
Times
Q
1 (kN)
Q
2 (kN)
X
1 (mm)
X
2 (mm)
K
h
1 (kN/mm)
Normalization
1
352.4
∗
-
337.5
∗
49.23
-
49.17
7.01
1.08
3
322.7
∗
-
317.8
∗
49.21
-
49.17
6.51
1.00
5
312.5
∗
-
316.9
∗
49.19
-
49.17
6.39
0.98
10
306.1
∗
-
315.9
∗
49.17
-
49.17
6.32
0.97
30
310.9
∗
-
314.3
∗
49.96
-
49.37
6.29
0.97
50
307.2
∗
-
308.6
∗
50.20
-
49.13
6.20
0.95Figure 10
Curves of the shearing strain correlation test: (a) 50%, (b) 100%, and (c) 150%.
(a)
(b)
(c)The horizontal load-displacement curves of the specimen in different loading frequency are shown in Figure11. And the corresponding test results of Kh1 are shown in Table 5. It is indicated that Kh1 of the specimen decreases with the increase of loading frequency. Based on the loading frequency 0.5 Hz, the variation range of Kh1 will be 0.93–1.12.Figure 11
Curves of loading frequency correlation test: (a) 0.001 Hz and 0.005 Hz, (b) 0.01 Hz, (c) 0.1 Hz, (d) 0.5 Hz, (e) 1.0 Hz, and (f) 2.0 Hz.
(a)
(b)
(c)
(d)
(e)
(f)The loading time correlation test of the specimen requires repetitive loading for 50 times. Considering the capacity of energy accumulator in testing machine, the loading time correlation test will be conducted for three times with loading cycles 1–15, 16–32, and 33–50 successively. The horizontal load-displacement curves of the specimen under different loading time are shown in Figure12. And the corresponding test results of Kh1 are shown in Table 6. It is indicated that Kh1 decreases with the increase of loading time. Based on the third loading time, the variation range of Kh1 will be 0.95–1.08.Figure 12
Curves of loading time correlation test: (a) 1–15, (b) 16–32, and (c) 33–50.
(a)
(b)
(c)


#### 3.4.3. Ultimate Shear Property in Longitudinal Direction

In ultimate shear property test of the specimen, test picture when shear displacement is +75 mm (150% shear strain) is shown in Figure13. Test curves of ultimate shear property are described as Figure 14. Results show that the specimen does not have any signs of damage when shear displacement reaches its ultimate value 75 mm under the design bearing capacity, and the shear load monotonously increased with the increase of displacement. It proves that the ultimate shear property of the specimen in longitudinal direction has satisfactory performance.Figure 13
Test picture when shear displacement is +75 mm.Figure 14
Test curves of ultimate shear property: (a) 75 mm and (b) −75 mm.
(a)
(b)


## 3.1. Specimen and Testing Apparatus

To validate the design method, mechanical property tests are performed on a specimen of the novel bearing with design natural frequency at 8 Hz and with 3500 kN bearing capacity. The design parameters of the specimen are shown in Table1. The inclination angle α of rubber block is 45°.Table 1
Design parameters of the specimen of the novel bearing.
Rubber block
Rubber hardness
60
Shear modulusG (MPa)
0.8
Effective sizea×a(mm)
300
×
300
Overall thickness (mm)
99.5
Primary shape factorS1
4.55
Secondary shape factorS2
6.06
Specimen of the novel bearing
Frequency (Hz)
8
Vibration damping ratioξ
10%
Vertical capacity (kN)
3500
Lateral capacity (kN)
700
Lateral displacement (mm)
±0
Longitudinal displacement (mm)
±50The specimen and testing equipment are fixed mechanically through an external fixing plate to avoid the relative sliding between the specimen and test platform, as shown in Figure6. The mechanical property tests are performed through 75000 kN multipurpose testing machine, as shown in Figure 7. The machine can conduct dynamic control in both vertical and horizontal direction simultaneously. Performance parameters of this testing machine are as follows: vertical dynamic load 75000 kN, horizontal dynamic load 6000 kN, maximum vertical displacement 120 mm, maximum horizontal displacement ±600 mm, and maximum horizontal peak velocity 1000 mm/s.Figure 6
Test specimen of novel antivibration bearing.Figure 7
Multifunctional testing machine.


## 3.2. Testing Program

The mechanical property tests are strictly performed according to GB/T 20688.1-2007 [31]. Three-cyclic loading of 0-Pmax-0 will be conducted in vertical direction during the compressive property test, among which Pmax is maximum design pressure 3500 kN. Displacement loading in longitudinal direction and horizontal load loading in lateral direction are adopted in shear property test. Shear property correlation tests include the correlation of shearing strain, loading frequency, and loading time, which mainly study horizontal mechanical property under varying horizontal deformation. After the correlation tests, the ultimate shear property tests are performed in longitudinal positive direction and negative direction, respectively, to measure the ultimate shear displacement ability of the test specimen under the maximum design pressure. The detailed testing program is shown in Table 2.Table 2
Mechanical property test program of novel antivibration bearing.
Test types
Loading direction
Amplitude
Loading frequency/Hz
Loading waveform
Vertical loads/kN
Cycle-index
Compression property
Vertical
/
/
/
3500
3
Shear property
Longitudinal
±
50 mm
0.005
/
2600
3
Lateral
±
700 kN
0.2
/
2600
3
Shearing strain correlation
Longitudinal
25 mm
0.5
Sine
2600
3
50 mm
0.5
Sine
2600
3
75 mm
0.5
Sine
2600
3
Loading frequency correlation
Longitudinal
50 mm
0.001
Ramp
2600
1
50 mm
0.005
Ramp
2600
1
50 mm
0.01
Ramp
2600
3
50 mm
0.1
Sine
2600
3
50 mm
0.5
Sine
2600
3
50 mm
1.0
Sine
2600
3
50 mm
2.0
Sine
2600
3
Loading times correlation
Longitudinal
50 mm
0.5
Sine
2600
50
Ultimate shear property
Longitudinal
±
75 mm
/
/
3500
/


## 3.3. Definition of Mechanical Property Index

The calculation equation of compression stiffness is(15)KN=P2-P1Y2-Y1,where P1 is 0.7P0, P2 is 1.3P0, P0 is the design pressure 2600 kN in normal train operation, and Y1 and Y2 are compression displacement corresponding to the third cycle of compression loads P1 and P2, respectively.The equation of horizontal equivalent stiffness is(16)Kh=Q1-Q2X1-X2,where Q1 is the maximum shear force, Q2 is the minimum shear force, X1 is maximum displacement, and X2 is minimum displacement.The equation of horizontal equivalent damping ratio is(17)heq=2ΔWπKhX1-X22,where ΔW is envelop area of the third cycle of the hysteretic loops.


## 3.4. Results and Discussion

### 3.4.1. Compression Property and Shear Property

Compressive load-displacement curve under three-cyclic loading is shown in Figure8. Based on (15), compression stiffness KN of the specimen is 479.8 kN/mm, which can ensure the vertical capacity of urban rail transit viaduct.Figure 8
Compressive load-displacement curve under three-cyclic loading.Horizontal load-displacement curves of the specimen under three-cyclic loading are shown in Figure9. The horizontal load shown in the figure comprises the actual shearing load and the friction of machinery equipment itself. Then, the actual shearing load is equal to horizontal load minus the friction of machinery equipment. Based on (16), horizontal equivalent stiffness of the specimen in longitudinal direction (Kh1) and lateral direction (Kh2) is 7.31 kN/mm and 861.4 kN/mm, respectively. Thus, the measured Kh1 is smaller, which can ensure the horizontal displacement of the longitudinal bridge caused by thermal effect of girder. Based on (17), the equivalent damping ratio heq in longitudinal direction is 9.6%, showing the specimen possesses great energy-dissipating capacity. Besides, Figure 9(b) indicates that the lateral displacement of the specimen is less than 1 mm when the horizontal load reaches the design lateral bearing capacity 700 kN. It shows that Kh2 of the specimen is sufficient to resist the expected horizontal load in the lateral direction of bridge and to limit the lateral displacement caused by train braking or turning at a high speed.Figure 9
Horizontal load-displacement curves under three-cyclic loading: (a) longitudinal and (b) lateral.
(a)
(b)The theoretical stiffness of the specimen can be calculated based on the stiffness calculation method mentioned in Section2.3. It should be noted that the design compression displacement d required for stiffness correction is obtained by trial. The deviation between d5=5.51 mm and d4=5.52 mm is less than 1%. Thus, the iteration number should be set to 5 and the final design compression displacement d set to 5.51 mm. Figure 8 shows that the measured compression displacement of the specimen is 5.83 mm when the compression load is 2600 kN. The error between measured compression displacement and design compression displacement is only 5.5%. It proves that the design compression displacement after multiple iterations has high precision.Comparison among the theoretical stiffness and measured stiffness of the specimen is shown in Table3. It indicates that the error between corrected value and measured value is smaller than that of standard calculated value, whether compression stiffness or horizontal equivalent stiffness. Kh2 of the specimen is much larger than theoretical value. That is because the lateral deformation is less than 1 mm (shearing strain γ<2%) in the test, and the actual shear modulus is much larger than theoretical value (γ=100%) when γ is minimal. Higher horizontal equivalent stiffness in lateral direction means stronger lateral load resistance to meet the operating requirements better. Since the damping effect of the novel bearing depends mainly on compression stiffness and the error between corrected compression stiffness and measured compression stiffness is only 1.7%, the stiffness correction method is much more consistent with real situation.Table 3
Comparison among the theoretical stiffness and measured stiffness.
Stiffness
Measured
Standard calculated
Error
Corrected value
Error
K
N (kN/mm)
479.8
372.0
22.5%
471.9
1.7%
K
h
1 (kN/mm)
7.31
5.80
20.7%
6.32
15.7%
K
h
2 (kN/mm)
861.4
372.0
56.8%
471.9
45.2%


### 3.4.2. Correlation of Shear Property in Longitudinal Direction

The horizontal load-displacement curves of the specimen in different shear strain are shown in Figure10. And the corresponding test results of Kh1 are shown in Table 4. It is indicated that Kh1 decreases with the increase of longitudinal displacement, which agrees with characteristics of the hysteresis loop of high-damping rubber. Based on the shearing strain 100% (50 mm), the variation range of Kh1 will be 0.93–1.08.Table 4
Test results ofKh1 in different shear strain.
Shear strain
Q
1 (kN)
Q
2 (kN)
X
1 (mm)
X
2 (mm)
K
h
1 (kN/mm)
Normalization
50%
195.0
∗
-
192.9
∗
24.59
-
24.17
7.96
1.08
100%
366.5
∗
-
359.6
49.36
-
49.13
7.37
1.00
150%
505.0
∗
-
503.1
∗
73.7
-
73.6
6.84
0.93
Notes. The value with ∗ is actual shear load which is equal to horizontal load minus the friction of machinery equipment, the same in Tables 5 and 6.Table 5
Test results ofKh1 in different loading frequency.
Frequency (Hz)
Q
1 (kN)
Q
2 (kN)
X
1 (mm)
X
2 (mm)
K
h
1 (kN/mm)
Normalization
0.001
397.3
∗
-
414.4
∗
49.14
-
50.91
8.11
1.12
0.005
398.3
∗
-
404.1
∗
49.14
-
50.93
8.07
1.12
0.01
385.2
∗
-
407.4
∗
48.97
-
50.74
7.96
1.10
0.1
367.6
∗
-
346.5
∗
49.74
-
49.30
7.21
1.00
0.5
357.9
∗
-
350.4
∗
49.29
-
49.00
7.21
1.00
1.0
333.7
∗
-
326.9
∗
48.14
-
48.27
6.85
0.95
2.0
323.3
∗
-
333.4
∗
49.04
-
49.24
6.68
0.93Table 6
Test results ofKh1 in different loading times.
Times
Q
1 (kN)
Q
2 (kN)
X
1 (mm)
X
2 (mm)
K
h
1 (kN/mm)
Normalization
1
352.4
∗
-
337.5
∗
49.23
-
49.17
7.01
1.08
3
322.7
∗
-
317.8
∗
49.21
-
49.17
6.51
1.00
5
312.5
∗
-
316.9
∗
49.19
-
49.17
6.39
0.98
10
306.1
∗
-
315.9
∗
49.17
-
49.17
6.32
0.97
30
310.9
∗
-
314.3
∗
49.96
-
49.37
6.29
0.97
50
307.2
∗
-
308.6
∗
50.20
-
49.13
6.20
0.95Figure 10
Curves of the shearing strain correlation test: (a) 50%, (b) 100%, and (c) 150%.
(a)
(b)
(c)The horizontal load-displacement curves of the specimen in different loading frequency are shown in Figure11. And the corresponding test results of Kh1 are shown in Table 5. It is indicated that Kh1 of the specimen decreases with the increase of loading frequency. Based on the loading frequency 0.5 Hz, the variation range of Kh1 will be 0.93–1.12.Figure 11
Curves of loading frequency correlation test: (a) 0.001 Hz and 0.005 Hz, (b) 0.01 Hz, (c) 0.1 Hz, (d) 0.5 Hz, (e) 1.0 Hz, and (f) 2.0 Hz.
(a)
(b)
(c)
(d)
(e)
(f)The loading time correlation test of the specimen requires repetitive loading for 50 times. Considering the capacity of energy accumulator in testing machine, the loading time correlation test will be conducted for three times with loading cycles 1–15, 16–32, and 33–50 successively. The horizontal load-displacement curves of the specimen under different loading time are shown in Figure12. And the corresponding test results of Kh1 are shown in Table 6. It is indicated that Kh1 decreases with the increase of loading time. Based on the third loading time, the variation range of Kh1 will be 0.95–1.08.Figure 12
Curves of loading time correlation test: (a) 1–15, (b) 16–32, and (c) 33–50.
(a)
(b)
(c)


### 3.4.3. Ultimate Shear Property in Longitudinal Direction

In ultimate shear property test of the specimen, test picture when shear displacement is +75 mm (150% shear strain) is shown in Figure13. Test curves of ultimate shear property are described as Figure 14. Results show that the specimen does not have any signs of damage when shear displacement reaches its ultimate value 75 mm under the design bearing capacity, and the shear load monotonously increased with the increase of displacement. It proves that the ultimate shear property of the specimen in longitudinal direction has satisfactory performance.Figure 13
Test picture when shear displacement is +75 mm.Figure 14
Test curves of ultimate shear property: (a) 75 mm and (b) −75 mm.
(a)
(b)


## 3.4.1. Compression Property and Shear Property

Compressive load-displacement curve under three-cyclic loading is shown in Figure8. Based on (15), compression stiffness KN of the specimen is 479.8 kN/mm, which can ensure the vertical capacity of urban rail transit viaduct.Figure 8
Compressive load-displacement curve under three-cyclic loading.Horizontal load-displacement curves of the specimen under three-cyclic loading are shown in Figure9. The horizontal load shown in the figure comprises the actual shearing load and the friction of machinery equipment itself. Then, the actual shearing load is equal to horizontal load minus the friction of machinery equipment. Based on (16), horizontal equivalent stiffness of the specimen in longitudinal direction (Kh1) and lateral direction (Kh2) is 7.31 kN/mm and 861.4 kN/mm, respectively. Thus, the measured Kh1 is smaller, which can ensure the horizontal displacement of the longitudinal bridge caused by thermal effect of girder. Based on (17), the equivalent damping ratio heq in longitudinal direction is 9.6%, showing the specimen possesses great energy-dissipating capacity. Besides, Figure 9(b) indicates that the lateral displacement of the specimen is less than 1 mm when the horizontal load reaches the design lateral bearing capacity 700 kN. It shows that Kh2 of the specimen is sufficient to resist the expected horizontal load in the lateral direction of bridge and to limit the lateral displacement caused by train braking or turning at a high speed.Figure 9
Horizontal load-displacement curves under three-cyclic loading: (a) longitudinal and (b) lateral.
(a)
(b)The theoretical stiffness of the specimen can be calculated based on the stiffness calculation method mentioned in Section2.3. It should be noted that the design compression displacement d required for stiffness correction is obtained by trial. The deviation between d5=5.51 mm and d4=5.52 mm is less than 1%. Thus, the iteration number should be set to 5 and the final design compression displacement d set to 5.51 mm. Figure 8 shows that the measured compression displacement of the specimen is 5.83 mm when the compression load is 2600 kN. The error between measured compression displacement and design compression displacement is only 5.5%. It proves that the design compression displacement after multiple iterations has high precision.Comparison among the theoretical stiffness and measured stiffness of the specimen is shown in Table3. It indicates that the error between corrected value and measured value is smaller than that of standard calculated value, whether compression stiffness or horizontal equivalent stiffness. Kh2 of the specimen is much larger than theoretical value. That is because the lateral deformation is less than 1 mm (shearing strain γ<2%) in the test, and the actual shear modulus is much larger than theoretical value (γ=100%) when γ is minimal. Higher horizontal equivalent stiffness in lateral direction means stronger lateral load resistance to meet the operating requirements better. Since the damping effect of the novel bearing depends mainly on compression stiffness and the error between corrected compression stiffness and measured compression stiffness is only 1.7%, the stiffness correction method is much more consistent with real situation.Table 3
Comparison among the theoretical stiffness and measured stiffness.
Stiffness
Measured
Standard calculated
Error
Corrected value
Error
K
N (kN/mm)
479.8
372.0
22.5%
471.9
1.7%
K
h
1 (kN/mm)
7.31
5.80
20.7%
6.32
15.7%
K
h
2 (kN/mm)
861.4
372.0
56.8%
471.9
45.2%


## 3.4.2. Correlation of Shear Property in Longitudinal Direction

The horizontal load-displacement curves of the specimen in different shear strain are shown in Figure10. And the corresponding test results of Kh1 are shown in Table 4. It is indicated that Kh1 decreases with the increase of longitudinal displacement, which agrees with characteristics of the hysteresis loop of high-damping rubber. Based on the shearing strain 100% (50 mm), the variation range of Kh1 will be 0.93–1.08.Table 4
Test results ofKh1 in different shear strain.
Shear strain
Q
1 (kN)
Q
2 (kN)
X
1 (mm)
X
2 (mm)
K
h
1 (kN/mm)
Normalization
50%
195.0
∗
-
192.9
∗
24.59
-
24.17
7.96
1.08
100%
366.5
∗
-
359.6
49.36
-
49.13
7.37
1.00
150%
505.0
∗
-
503.1
∗
73.7
-
73.6
6.84
0.93
Notes. The value with ∗ is actual shear load which is equal to horizontal load minus the friction of machinery equipment, the same in Tables 5 and 6.Table 5
Test results ofKh1 in different loading frequency.
Frequency (Hz)
Q
1 (kN)
Q
2 (kN)
X
1 (mm)
X
2 (mm)
K
h
1 (kN/mm)
Normalization
0.001
397.3
∗
-
414.4
∗
49.14
-
50.91
8.11
1.12
0.005
398.3
∗
-
404.1
∗
49.14
-
50.93
8.07
1.12
0.01
385.2
∗
-
407.4
∗
48.97
-
50.74
7.96
1.10
0.1
367.6
∗
-
346.5
∗
49.74
-
49.30
7.21
1.00
0.5
357.9
∗
-
350.4
∗
49.29
-
49.00
7.21
1.00
1.0
333.7
∗
-
326.9
∗
48.14
-
48.27
6.85
0.95
2.0
323.3
∗
-
333.4
∗
49.04
-
49.24
6.68
0.93Table 6
Test results ofKh1 in different loading times.
Times
Q
1 (kN)
Q
2 (kN)
X
1 (mm)
X
2 (mm)
K
h
1 (kN/mm)
Normalization
1
352.4
∗
-
337.5
∗
49.23
-
49.17
7.01
1.08
3
322.7
∗
-
317.8
∗
49.21
-
49.17
6.51
1.00
5
312.5
∗
-
316.9
∗
49.19
-
49.17
6.39
0.98
10
306.1
∗
-
315.9
∗
49.17
-
49.17
6.32
0.97
30
310.9
∗
-
314.3
∗
49.96
-
49.37
6.29
0.97
50
307.2
∗
-
308.6
∗
50.20
-
49.13
6.20
0.95Figure 10
Curves of the shearing strain correlation test: (a) 50%, (b) 100%, and (c) 150%.
(a)
(b)
(c)The horizontal load-displacement curves of the specimen in different loading frequency are shown in Figure11. And the corresponding test results of Kh1 are shown in Table 5. It is indicated that Kh1 of the specimen decreases with the increase of loading frequency. Based on the loading frequency 0.5 Hz, the variation range of Kh1 will be 0.93–1.12.Figure 11
Curves of loading frequency correlation test: (a) 0.001 Hz and 0.005 Hz, (b) 0.01 Hz, (c) 0.1 Hz, (d) 0.5 Hz, (e) 1.0 Hz, and (f) 2.0 Hz.
(a)
(b)
(c)
(d)
(e)
(f)The loading time correlation test of the specimen requires repetitive loading for 50 times. Considering the capacity of energy accumulator in testing machine, the loading time correlation test will be conducted for three times with loading cycles 1–15, 16–32, and 33–50 successively. The horizontal load-displacement curves of the specimen under different loading time are shown in Figure12. And the corresponding test results of Kh1 are shown in Table 6. It is indicated that Kh1 decreases with the increase of loading time. Based on the third loading time, the variation range of Kh1 will be 0.95–1.08.Figure 12
Curves of loading time correlation test: (a) 1–15, (b) 16–32, and (c) 33–50.
(a)
(b)
(c)


## 3.4.3. Ultimate Shear Property in Longitudinal Direction

In ultimate shear property test of the specimen, test picture when shear displacement is +75 mm (150% shear strain) is shown in Figure13. Test curves of ultimate shear property are described as Figure 14. Results show that the specimen does not have any signs of damage when shear displacement reaches its ultimate value 75 mm under the design bearing capacity, and the shear load monotonously increased with the increase of displacement. It proves that the ultimate shear property of the specimen in longitudinal direction has satisfactory performance.Figure 13
Test picture when shear displacement is +75 mm.Figure 14
Test curves of ultimate shear property: (a) 75 mm and (b) −75 mm.
(a)
(b)


## 4. Damping Effects of Novel Antivibration Bearing

### 4.1. Test Scheme

Damping effects of the novel bearing are investigated through impulse vibration tests on scaled models. The scaled models of the novel bearing and steel bearing are shown in Figure15. The similarity relation of length ratio between the model and prototype is set as 1/5.5, and their similarity relation is shown in Table 7. It should be noted that the full-scaled specimen of the novel bearing is constituted by four rubber blocks, while scaled model is constituted by two rubber blocks. That is to say, one scaled model of the novel bearing is equivalent to half of scaled bearing specimen according to the similarity relation of length ratio 1/5.5.Table 7
Similarity relation between model and prototype.
Similarity factor
Symbol
Formula
Value (model/prototype)
Size
S
l
S
l
=
l
M
/
l
P
0.182
Elastic modulus
S
E
S
E
=
E
M
/
E
P
1.000
Acceleration
S
a
S
a
=
S
E
S
l
2
/
S
m
1.002
Mass
S
m
S
m
=
m
M
/
m
P
0.033
Time
S
t
S
t
=
(
S
l
/
S
a
)
0.5
0.426
Frequency
S
f
S
f
=
1
/
S
t
2.347
Displacement
S
u
S
u
=
S
l
0.182
Stress
S
σ
S
σ
=
S
E
1.000
Strain
S
ε
S
ε
=
1
1.000
Force
S
F
S
F
=
S
E
S
l
2
0.033Figure 15
Scaled models: (a) novel antivibration bearing and (b) steel bearing.
(a)
(b)


### 4.2. Test Process

The impulse vibration tests of scaled model are performed in Earthquake Engineering Center of Guangzhou University. Test scheme is shown in Figure16. There are four scaled bearings placed in each corner of the square ground, and every two of them are placed by T-type relations, which is equivalent to two-scaled bearing specimen. Considering design bearing capacity is 2600 kN for the normal train operation, additional weight composed of the concrete and steel slab is 171.6 kN based on the similarity ratio 0.033. Dynamic loads are excited by an electromagnetic vibrator which is placed at the center of slab. Sweep frequency is 5–230 Hz. This test applies four accelerometers to gather vertical acceleration responses as shown in Figure 16, where accelerometer 1 is located at the lower plate of the bearing, accelerometers 2 and 3 are located on the concrete slab, and accelerometer 4 is located on the vibrator.Figure 16
Test scheme of impulse vibration test on scaled models.


### 4.3. Results and Discussion

Under the impulse vibration in vertical direction, the time histories and Fourier spectra of novel antivibration bearing and steel bearing at accelerometer 2 and accelerometer 1 are shown in Figure17. It is indicated that the maximum acceleration responses of the upper and lower plates in the novel bearing are 0.11 m/s2 and 0.0048 m/s2, with main frequency components around 20 Hz or 50–60 Hz. While in steel bearing, the corresponding value of upper and lower plates are 0.073 m/s2 and 0.0021 m/s2, with main frequency components around 20–30 Hz or 65–75 Hz.Figure 17
Vertical acceleration time histories and Fourier spectra: (a) novel antivibration bearing and (b) steel bearing.
(a)
(b)In accordance with the Chinese national standards GB 1007-1988 [32], the vibration level is defined as follows:(1) Vibration acceleration level is(18)VAL=20lg⁡aRMSa0,where a0=10-6 m/s2 is a reference vibration acceleration and aRMS is the vibration acceleration RMS.(2) Z vibration level is as follows.Corrected byZ-weighted factor of the whole-body vibration, the vibration acceleration level is noted as VLZ based on ISO2631/1-1997 [33].The dominant frequency of vertical ground vibration induced by the passage of trains is 10–100 Hz, and the dominant frequency of transverse vibration is 4–100 Hz [34]. Based on GB 10071-1988 [32], only 1–80 Hz of vertical vibration needs to be assessed. Therefore, 1–80 Hz was adopted to make assessment on the damping effect of the novel bearing. The 1/3 octave band analysis is performed to discuss the vertical Z vibration level in the range of 1–80 Hz. Results show that the average VLZ of upper and lower plates for the steel bearing are 67.67 dB and 59.50 dB, respectively, and the whole insertion loss is 8.17 dB, while the average VLZ of upper and lower plates for the novel bearing are 70.29 dB and 56.80 dB, respectively, and the whole insertion loss is 13.49 dB. Therefore, the overall insertion loss of the novel bearing is 5.32 dB larger than that of steel bearing.Comparison of the insertion loss for steel bearing and the novel bearing in the frequency domain of 1–80 Hz is shown in Figure18. From the comparison, both steel bearing and the novel bearing can cause the insertion loss in most frequency components, which shows that both can reduce certain vertical vibration. However, the vibration responses of the novel bearing are amplified below 1/3 octave band center frequency 4 Hz. That is because the novel bearing as a flexible support decreases the natural frequency of the structure. The vibration responses are mitigated in the medium and high frequency bands and amplified in the low frequency bands. In this sense, the damping effects of the novel bearing is worse than that of the steel bearing below the center frequency 4 Hz. But above the center frequency 4 Hz that contains the main dominant frequency of vertical vibration induced by the passage of trains (10–80 Hz), the novel bearing can reduce more vibrations than the steel bearing. It can be concluded that the novel bearing can effectively mitigate the train-induced vertical vibrations.Figure 18
Comparison of the insertion loss for steel bearing and novel antivibration bearing.


## 4.1. Test Scheme

Damping effects of the novel bearing are investigated through impulse vibration tests on scaled models. The scaled models of the novel bearing and steel bearing are shown in Figure15. The similarity relation of length ratio between the model and prototype is set as 1/5.5, and their similarity relation is shown in Table 7. It should be noted that the full-scaled specimen of the novel bearing is constituted by four rubber blocks, while scaled model is constituted by two rubber blocks. That is to say, one scaled model of the novel bearing is equivalent to half of scaled bearing specimen according to the similarity relation of length ratio 1/5.5.Table 7
Similarity relation between model and prototype.
Similarity factor
Symbol
Formula
Value (model/prototype)
Size
S
l
S
l
=
l
M
/
l
P
0.182
Elastic modulus
S
E
S
E
=
E
M
/
E
P
1.000
Acceleration
S
a
S
a
=
S
E
S
l
2
/
S
m
1.002
Mass
S
m
S
m
=
m
M
/
m
P
0.033
Time
S
t
S
t
=
(
S
l
/
S
a
)
0.5
0.426
Frequency
S
f
S
f
=
1
/
S
t
2.347
Displacement
S
u
S
u
=
S
l
0.182
Stress
S
σ
S
σ
=
S
E
1.000
Strain
S
ε
S
ε
=
1
1.000
Force
S
F
S
F
=
S
E
S
l
2
0.033Figure 15
Scaled models: (a) novel antivibration bearing and (b) steel bearing.
(a)
(b)


## 4.2. Test Process

The impulse vibration tests of scaled model are performed in Earthquake Engineering Center of Guangzhou University. Test scheme is shown in Figure16. There are four scaled bearings placed in each corner of the square ground, and every two of them are placed by T-type relations, which is equivalent to two-scaled bearing specimen. Considering design bearing capacity is 2600 kN for the normal train operation, additional weight composed of the concrete and steel slab is 171.6 kN based on the similarity ratio 0.033. Dynamic loads are excited by an electromagnetic vibrator which is placed at the center of slab. Sweep frequency is 5–230 Hz. This test applies four accelerometers to gather vertical acceleration responses as shown in Figure 16, where accelerometer 1 is located at the lower plate of the bearing, accelerometers 2 and 3 are located on the concrete slab, and accelerometer 4 is located on the vibrator.Figure 16
Test scheme of impulse vibration test on scaled models.


## 4.3. Results and Discussion

Under the impulse vibration in vertical direction, the time histories and Fourier spectra of novel antivibration bearing and steel bearing at accelerometer 2 and accelerometer 1 are shown in Figure17. It is indicated that the maximum acceleration responses of the upper and lower plates in the novel bearing are 0.11 m/s2 and 0.0048 m/s2, with main frequency components around 20 Hz or 50–60 Hz. While in steel bearing, the corresponding value of upper and lower plates are 0.073 m/s2 and 0.0021 m/s2, with main frequency components around 20–30 Hz or 65–75 Hz.Figure 17
Vertical acceleration time histories and Fourier spectra: (a) novel antivibration bearing and (b) steel bearing.
(a)
(b)In accordance with the Chinese national standards GB 1007-1988 [32], the vibration level is defined as follows:(1) Vibration acceleration level is(18)VAL=20lg⁡aRMSa0,where a0=10-6 m/s2 is a reference vibration acceleration and aRMS is the vibration acceleration RMS.(2) Z vibration level is as follows.Corrected byZ-weighted factor of the whole-body vibration, the vibration acceleration level is noted as VLZ based on ISO2631/1-1997 [33].The dominant frequency of vertical ground vibration induced by the passage of trains is 10–100 Hz, and the dominant frequency of transverse vibration is 4–100 Hz [34]. Based on GB 10071-1988 [32], only 1–80 Hz of vertical vibration needs to be assessed. Therefore, 1–80 Hz was adopted to make assessment on the damping effect of the novel bearing. The 1/3 octave band analysis is performed to discuss the vertical Z vibration level in the range of 1–80 Hz. Results show that the average VLZ of upper and lower plates for the steel bearing are 67.67 dB and 59.50 dB, respectively, and the whole insertion loss is 8.17 dB, while the average VLZ of upper and lower plates for the novel bearing are 70.29 dB and 56.80 dB, respectively, and the whole insertion loss is 13.49 dB. Therefore, the overall insertion loss of the novel bearing is 5.32 dB larger than that of steel bearing.Comparison of the insertion loss for steel bearing and the novel bearing in the frequency domain of 1–80 Hz is shown in Figure18. From the comparison, both steel bearing and the novel bearing can cause the insertion loss in most frequency components, which shows that both can reduce certain vertical vibration. However, the vibration responses of the novel bearing are amplified below 1/3 octave band center frequency 4 Hz. That is because the novel bearing as a flexible support decreases the natural frequency of the structure. The vibration responses are mitigated in the medium and high frequency bands and amplified in the low frequency bands. In this sense, the damping effects of the novel bearing is worse than that of the steel bearing below the center frequency 4 Hz. But above the center frequency 4 Hz that contains the main dominant frequency of vertical vibration induced by the passage of trains (10–80 Hz), the novel bearing can reduce more vibrations than the steel bearing. It can be concluded that the novel bearing can effectively mitigate the train-induced vertical vibrations.Figure 18
Comparison of the insertion loss for steel bearing and novel antivibration bearing.


## 5. Conclusion

For the bridge bearing, the principle to mitigate vertical vibration is to increase the frequency ratio by appropriately decreasing natural frequency of the vibration system and also provide proper damping to avoid the resonance. In this paper, a novel antivibration bearing that adopts four high-damping thick rubber blocks stacking slantingly is proposed to reduce vertical vibrations and provide a large lateral stiffness simultaneously. Considering the novel bearing is modeled linearly in the train-bridge interaction system, the associated stiffness correction method is proposed based on multiple iterations with design compression displacement.To validate the design method, mechanical property tests are performed on a specimen of the bearing device with design natural frequency at 8 Hz and with 3500 kN bearing capacity. Test results show that the measuredKN of the specimen is 479.8 kN/mm, which can ensure the vertical capacity of urban rail transit viaduct; the measured Kh1 is 7.31 kN/mm, which can ensure the horizontal displacement of the longitudinal bridge caused by thermal effect of girder; the measured Kh2 is 861.4 kN/mm, which can resist the expected lateral load of bridge. The equivalent damping ratio heq in longitudinal direction is 9.6%, showing the specimen possesses great energy-dissipating capacity. The tests also prove that the specimen has stable performance in longitudinal direction. Therefore, the mechanical property of the novel bearing can satisfy the engineering demand. The error between corrected value and measured value is smaller than that of standard calculated value, whether compression stiffness or horizontal equivalent stiffness. So, the test results agree well with the prediction of the proposed stiffness correction method.Damping effects of the novel bearing are investigated through impulse vibration tests on scaled models. When electromagnetic vibrator is operating on test model, it shows that vibration amplitudes of the novel bearing at the upper and lower plates are smaller than those of steel bearing. The whole insertion loss of the novel bearing is 13.49 dB which is 5.32 dB larger than that of steel bearing in the range of 1–80 Hz. The damping effect of the novel bearing is worse than that of the steel bearing below the center frequency 4 Hz because of the vibration amplification effect of flexure supporting structure in the low frequency bands. But above the center frequency 4 Hz that contains the main dominant frequency of vertical vibration induced by the passage of trains (10–80 Hz), the novel bearing can reduce more vibrations than the steel bearing. Therefore, the novel bearing can effectively mitigate the train-induced vertical vibrations and it will benefit the circumvention of the vibration problem along the elevated rail line.



---

*Source: 1402301-2017-10-18.xml*