Experimental and numerical study on seismic sliding mechanism of laminated-rubber bearings

Engineering Structures 141 (2017) 159–174

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Experimental and numerical study on seismic sliding mechanism of laminated-rubber bearings Nailiang Xiang, Jianzhong Li ⇑ State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China

a r t i c l e

i n f o

Article history: Received 22 September 2016 Revised 13 March 2017 Accepted 16 March 2017

Keywords: Laminated-rubber bearings Sliding mechanism Structural fuses Experimental research Analytical model Numerical research

a b s t r a c t Typical laminated rubber bearings are widely used in small to medium-span highway bridges in China. These bearings are economical, and are proved to perform well at service-level conditions. However, damage investigation after the 2008 Wenchuan earthquake revealed that sliding between the laminated rubber bearing and the bridge girder was a common phenomenon, which could actually act as fuses and protect the substructures from severe damage. Thus no damage or only minor damage of substructures was reported in the Wenchuan earthquake for those bridges with bearing sliding. In this paper, an experimental program was carried out to investigate the sliding behavior of laminated rubber bearings with typical configurations in China. The bearing was placed directly on a steel plate representing the embedded steel plate at the bottom of bridge girders, to create an elastomer-steel sliding surface. Experiment results showed that the behavior of bearings before obvious sliding could be approximated as a linear elastic response, with an effective shear modulus in the range of 610–1100 kPa. The sliding coefficients of friction were observed to be inversely related to the normal force, and positively related to the sliding velocity. An analytical model considering the sliding response of laminated-rubber bearings on steel plates was developed and calibrated. Numerical simulations were also conducted to compare the proposed model to the model with typical Coulomb’s friction. Results from the numerical simulations indicated that the vertical earthquake caused differences of bearing displacement response between these two models. And this difference would be more significant as the intensity of vertical earthquake increased. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Economical laminated rubber bearings, which usually have not been designed for seismic demand, are widely used in small to medium-span highway bridges of China, especially those simply supported girder bridges. The bearings mainly allow for thermal movement of the bridge superstructure at service-level conditions through shear deformation of rubber. As common practice, the bearings are placed directly between the superstructure and the substructure, with no restraints among them other than friction (Fig. 1) [1,2]. The past 2008 Wenchuan earthquake occurred in western regions of China caused severe damage to highway bridges, especially those small to medium-span girder bridges. Damage investigation [3,4] revealed that the typical damage of these bridges included: sliding between the laminated rubber bearing and the girder (Fig. 2), shear keys failure, destruction of expansion joints and abutment, and even span collapse. On the ⇑ Corresponding author. E-mail address: [email protected] (J. Li). http://dx.doi.org/10.1016/j.engstruct.2017.03.032 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.

other hand, no visible damages or only minor cracks were observed in piers for those bridges with bearing sliding. Such phenomena were also reported in the 1999 Chi-Chi earthquake [5–7]. The seismic fusing is an innovative and effective design strategy, which has been implemented in new and existing structures, including buildings and bridges [8–10]. In fact, owing to the installation practice, the laminated rubber bearings are prone to slide along the steel plates embedded in the upper girders during a large earthquake and potentially provide an isolated response for the system. The bearing sliding can act as structural fuses, to cause a period elongation and reduce the force demands on the substructure. The concept is just in accordance to the fusing strategy specified in AASHTO [11], which allows for a seismic fusing mechanism designed between elastic superstructure and substructure. Connecting components, such as bearings, isolation devices and energy dissipation devices can be introduced as the fusing mechanism to limit energy build-up of the bridge system. The Illinois Department of Transportation (IDOT) of the USA has investigated a quasiisolation design and construction method for typical highway bridges in Illinois [12–15]. The key points of the proposed

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N. Xiang, J. Li / Engineering Structures 141 (2017) 159–174

Concrete shear key Bridge girder Embedded steel plate

Rubber bearing Supporting pad Concrete substructure

Fig. 1. Typical layout of laminated-rubber bearings.

figurations. For example, for multi-span simply supported girder bridges in which the bearings are usually placed eccentrically with respect to the vertical axis of the piers in longitudinal direction, the longitudinal seismic excitation can cause the variation of axial forces in the bearings due to the vertical motions of the deck and the rotations of the pier caps [18]. This paper describes a laboratory testing program of typical laminated rubber bearings in China. The object of the testing is to investigate the seismic performance of laminated rubber bearings as fuses, including the shear and sliding response. An analytical model is developed based on the experimental results, to accurately describe the fusing characteristics of laminated rubber bearings.

2. Testing background 2.1. Bearing specimens and test setup

Fig. 2. Sliding of laminated-rubber bearings.

quasi-isolation rely on some sequential fusing (i.e., sliding, yielding, or rupture) of specific components (i.e., laminated elastomeric bearings, side retainers, or steel fixed bearings) to limit forces that can be transferred down through the bridge system. The seismic behavior of laminated rubber bearings with large shear strains has been studied extensively. Konstantinidis et al. [16] have conducted experiments on the shear response of laminated rubber bearings, which are not bonded to top or bottom steel plates, and concluded that the bearings can withstand up to 225% shear strains, limited by the roll-off response of the bearing. Further horizontal displacement beyond the roll-off state will cause the bearing sliding along steel plate. Steelman et al. [17] have investigated the shear and sliding response of laminated rubber bearings on concrete surfaces subjected to seismic demands, and found that the bearing exhibited an approximately linear elastic response with the maximum shear strain of 125–250% before sliding. When the bearings began to sliding on the concrete substructure with an initial friction coefficient of 0.25–0.5, the forcedisplacement hysteretic hoops of bearings became wide, displaying stable and powerful energy dissipation capacity. However, the previous studies mainly focused on how a laminated rubber bearing behaved during an earthquake. No practical analytical models have been proposed to describe the seismic behavior of laminated rubber bearings, especially when the bearing sliding is considered. The sliding behaviors of laminated-rubber bearings may be quite sensitive to the variation of axial load. The variation of axial load is mainly induced by the vertical seismic excitation during the earthquake. Besides, horizontal seismic excitation may cause the significant variation of axial load on bearings in some bridge con-

The testing consists of two types of laminated rubber bearing specimens listed in Table 1. GYZ 400
84 specimens are those full-scale bearings commonly used at a 25 m span T-girder simply-supported highway bridge. GJZ 200
21 specimens are representative of those appropriate for use at the quarter-scale shake-table model of a 25 m span T-girder bridge. All specimens are designed and manufactured according to the Chinese bearing guidelines (CBG JT/T-4) [18]. Fig. 3 shows the configuration details of these two types of specimens. The bearing dimensions, elastomer layer thickness and steel shim thickness all satisfy the tolerance requirements specified in the GBG JT/T-4. Normally a bearing specimen is composed of cover elastomer layers, internal elastomer layers and steel shims, and the thickness t and the quantity n of these components are listed in Table 1. The cover elastomer layers are placed at the top and the bottom of specimens. The internal elastomer layers in the specimens are of uniform thickness, and so are the steel shims. The elastomer used for the elastomer layers is composed of neoprene with a required hardness of 60 ± 5. The tensile strength of elastomer is 17 MPa, and the ultimate elongation is 400%. The steel used for the steel shims is carbon structural steel with a yield strength of 235 MPa and a ultimate strength of 370 MPa. The ultimate elongation of the steel is 26%, and the impact absorbing energy is 27 joule. As for the fabrication of bearing specimens, the steel shims are firstly prepared for vulcanization by using abrasive blasting to provide an initial roughened surface, and then treating the surface with a proprietary chemical bonding agent. The elastomer layers are finally connected to the steel shims through the process of vulcanization. The quality control requirements for the bearing specimens are well satisfied, and there is no initial delamination occurring for all the test specimens. The tests were carried out in the State Key Laboratory for Disaster Reduction in Civil Engineering at Tongji University using the experimental apparatus shown in Fig. 4. Vertical loading was imposed on the bearing specimens through a 3000kN capacity hydraulic servo actuator. The hydraulic servo actuator can automatically adjust itself to make the vertical load maintained at a specified target value regardless the horizontal deformation of the bearing. The horizontal actuator can provide a maximum loading of 500 kN with the maintained vertical loading. Rollers which were placed under the horizontal loading arms made it sure that the vertical and horizontal actuators worked simultaneously to approximately simulate the loads condition during an earthquake, where the bearings were subjected to horizontal inertial forces at the same time sustaining gravity load. The frictional forces occurred at the surface of rollers were minor and could be neglected.

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N. Xiang, J. Li / Engineering Structures 141 (2017) 159–174 Table 1 Bearing specimen characteristics. Specimen

Type

Shape

1 2 3 4 5 6

GYZ 400
84 GYZ 400
84 GYZ 400
84 GYZ 400
84 GYZ 400
84 GJZ 200
21

Circular Circular Circular Circular Circular Rectangular

Cover elastomer layer

t (mm)

n

t (mm)

n

t (mm)

n

11 11 11 11 11 5

5 5 5 5 5 2

2.5 2.5 2.5 2.5 2.5 2.5

2 2 2 2 2 2

4 4 4 4 4 2

6 6 6 6 6 3

Internal elastomer layer

Steel shim

Internal elastomer layer

Plan size (mm)

u400 u400 u400 u400 u400 200
200

Steel shim

5

4

21

2

84

11

Steel shim

Internal elastomer layer

200 400

Cover elastomer layer Cover elastomer layer

Elevation (GYZ 400×84)

Elevation (GJZ 200×21)

Steel shim

190

Steel shim

190

Plan (GYZ 400×84)

Plan (GJZ 200×21)

Fig. 3. Configuration details of the bearing specimens (unit: mm).

Vertical Actuators Horizontal Actuators Loading Arms Bearing Specimen Rollers Base Fig. 4. Testing apparatus.

The setup details of the bearing specimen prior to testing is shown in Fig. 5. The top surface of the bearing was bonded to the top steel plate through vulcanization, which was then anchored securely to the vertical loading arm of the testing apparatus. The bottom steel plate was connected to the horizontal loading arm. The bearing elastomer was placed directly on the bottom steel plate surface, without any horizontal restraint to realistic reflect the field conditions of a full-scale bearing on highway bridges. The bearing specimen setup also makes sure that the relative sliding response will be anticipated on the contact surface of

elastomer-steel plate. A displacement gage was attached to the needle driven into elastomer near the base to measure the shear deformation or shear strain response in bearing elastomer. 2.2. Test program The whole testing program was summarized in Table 2. To unify the descriptions of loading protocols for bearings with different sizes across the experimental process, the loading displacement was represented by the equivalent shear strain (ESS) [17], which

N. Xiang, J. Li / Engineering Structures 141 (2017) 159–174

Equivalent Shear Strain (%)

162

Vertical Loading Arm Top Steel Plate

Bearing Specimen

300 200 100 0 -100 -200 -300 0

1

2

3

4

5

6

7

8

Cycles Rollers

Displacement Gage

Fig. 6. Testing protocols for cyclic loadings.

Bottom Steel Plate Horizontal Loading Arm

Fig. 5. Specimen installation and testing setup.

was defined as the value of the imposed displacement through the actuator divided by the total height of the bearing rubber. The loading protocols in the tests are monotonic or cyclic, with the displacement in the units of ESS. For monotonic loading protocols, the maximum imposed displacement demand was 400% ESS. Normal pressure on bearings and velocity varied during the monotonic tests to investigate sensitivity in the bearing responses to these two parameters. The normal pressure on bearings ranged from 2.0 to 10.0 MPa, just within the design bearing capacity. The loading velocities were selected based on the hydraulic capacity of the horizontal actuator, and varied from 0.5 to 20.0 mm/s. It is noted that although the loading velocities in the tests are smaller than expected velocities during an earthquake excitation, the tests do provide some insight into the potential velocity effect on the bearing frictional sliding. The displacement protocols for cyclic loadings was shown in Fig. 6, with the increased amplitudes from 50% to 300% ESS. Each amplitude level included two fully-reversed cycles. 3. Testing results 3.1. Observed performance With the normal pressure on the laminated rubber bearing maintained on a specified value, the gradually increasing horizontal displacement were imposed on the bearing. Fig. 7a-c plot the observed bearing performance at different levels of ESS. As seen in the figures, the bearing specimen exhibited small shear deformation in elastomer at small shear strain demand (etc. ESS = 25%), with its bottom surface nearly in full contact with the bottom steel plate. With the imposed displacement increased, the shear deformation occurred in the specimen increased gradu-

ally. The trailing edge of the specimen base started to curl at high shear strain, and the leading edge was rounded off by being ground against the steel plate (Fig. 7). When the shear strain demand of the specimen exceeded the friction resistance at the interface between the elastomer and the steel plate, sliding response was observed to occur. It is to noted that delamination of steel shims or instability had not been observed in bearing specimens during the tests, and the specimens were proved to perform satisfactorily even subjected to long-travel, high-rate seismic demands. For unrestrained, laminated-rubber bearings that are directly placed upon the steel plate, the shear deformation in elastomer and the sliding displacement constitute to the total bearing displacement. As an example, the bearing displacement response for Test 1-1 (4.0 MPa) with the proceeding of imposed displacement is also plotted in Fig. 7. In the whole loading process, the bearing displacement response could be divided into the following three phases: Phase I—The specimen exhibited pure shear deformation in elastomer for increasing displacement demand, with no sliding response initiated. The increased amount of bearing displacements were only contributed by the elastomeric shear deformation at this phase. Phase II—The shear deformation in the specimen elastomer continued to increase but at a lower rate than that at Phase I, and at the same time, minor sliding displacement was observed to occur at the bearing-to-steel plate interface. This minor sliding displacements might be induced by the ‘walking effect’ which is defined as the phenomenon that the rubber bearing will quasi-statically slide on steel plate at high shear stress. Such phenomenon is just similar to the creep effect of concrete material. At this phase, the bearing displacements imposed by the horizontal actuator included both elastomeric shear deformation and walking displacement. Phase III—At this phase, the shear deformation in elastomer reaches a peak and no longer increases with increase of the imposed displacements. The increasing imposed loading finally exceeded the friction resistance between bearing and steel plate,

Table 2 Testing program Test Series

Specimen

Protocol

Normal Pressure (MPa)

Velocity (mm/s)

Maximum ESS (%)

1-1 2-1 3-1 4-1 5-1 5-2 5-3 6-1 6-2 6-3

1 2 3 4 5 5 5 6 6 6

Monotonic Monotonic Monotonic Monotonic Cyclic Cyclic Cyclic Cyclic Cyclic Cyclic

2.0–10.0 2.0–10.0 3.0 4.0 2.0 4.0 6.0 2.0 4.0 6.0

0.5 10.0 1.0–20.0 1.0–20.0 0.5 0.5 0.5 0.5 0.5 0.5

400 400 400 400 300 300 300 300 300 300

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(b) Phase II (ESS=100%)

(a) Phase I (ESS=25%)

(c) Phase III (ESS=300%)

Horizontal Displacement (%)

400 350 Phase III

Phase II

Phase I

300 250 200

Trailing edge

150 100 50 0

0

50

100

150

200

250

300

350

400

Imposed Equivalent Shear Strain (%) Total imposed displacement slip displacement

Leading edge

shear deformation in elastomer

Fig. 7. Displacement response of the bearing specimen (Test 1-1, 4 MPa).

and obvious sliding response was initiated. The sliding displacements increased at a high rate, contributing to the increased amount of imposed displacements. 3.2. Force-displacement response Sample force-displacement response curves for the specimens are plotted in Fig. 8 and Fig. 9 to illustrate their behavior at lowto-high shear strains. Rectangular and circular markers are drawn individually to indicate the points where walking and obvious sliding response are initiated. These markers are also used to distinguish the three phases of bearing displacement response, as previously shown in Fig. 7. Data results of specimens for some monotonic tests are presented in Fig. 8. As shown in the figure, the stiffness of bearings, as well as the points where ‘walking effect’ is initiated (indicated by rectangular markers), were relatively insensitive to normal pressure and velocity at Phase I, where pure shear deformation in elastomer occurred. Tests at increased normal pressures and velocities resulted in increased stiffness at

350

Test 4-1 (20 mm/s)

Test 1-1 (8 MPa)

Test 4-1 (10 mm/s)

280

Horizontal Force (kN)

Test 4-1 (5 mm/s)

Phase II and increased sliding resistance at Phase III. The increase of normal pressures also resulted in an increase in sliding initiation displacements. When the results of Test 4-1 and 1-1 (4.0 MPa) were compared, the sliding initiation displacements at highvelocity tests were larger than those at quasi-static tests. With the increase of velocity, the difference between breakaway friction forces and sliding friction forces increased and thus the breakaway friction effect became more pronounced. Measured force-displacement hysteresis curves for the specimens under cyclic loadings are plotted in Fig. 9, in which Fig. 9a is for the GYZ400
81 bearing and Fig. 9b is for the GJZ200
21 bearing. When the amplitudes of imposed shear strains were small, the bearing deformed purely in shear and displayed narrow hysteresis hoops accounting for the nonlinear effect in rubber. However, the response of bearings at small shear strains can be approximately regarded as linear elastic, which is acceptable for analytical modelling. As the shear strains increased, the hysteresis curves became relatively wide and stable, indicating good energy dissipation in all bearings under cyclic loadings. The increase of normal pressures resulted in both an increase of sliding resistance and development of stiffening effect that occurred when the direction of loading displacement changed. And the stiffening effect is found to be more significant for bearings smaller size, as shown in Fig. 9b. Despite of the stiffening effect, the unloading branches of the hysteretic curves were basically parallel to the initial loading or reloading ones, indicating a good match in shear stiffness.

210

3.3. Bearing shear stiffness and shear deformation 140

Test 1-1 (6 MPa) Test 1-1 (2 MPa)

70

0

Test 1-1 (4 MPa)

0

50

100

150

200

250

300

350

400

ESS (%) Fig. 8. Sample force-displacement curves of specimens for monotonic loading protocols.

Laminated-rubber bearings are commonly characterized by accommodating girder movements from thermal expansion without any visible distress, especially in non-seismic applications. The rubber bearings are commonly supposed to behave elastically in service-level conditions despite of the material nonlinearity in elastomer. The bearing shear stiffness is one of the most important parameters that represent the mechanical behavior of a rubber bearing. Generally, the bearing shear stiffness is related in terms of shear modulus. Determination of the bearing shear modulus varies among different standards, where most of them specify

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N. Xiang, J. Li / Engineering Structures 141 (2017) 159–174

200

(b) 90 Test 5-1 Test 5-2 Test 5-3

Test 6-1

Horizontal Force (kN)

Horizontal Force (kN)

(a) 300

100 0 -100 -200 -300 -400

60

Test 6-2 Test 6-3

30 0 -30 -60

-300

-200

-100

0

100

200

300

400

-90 -400

-300

-200

ESS (%)

-100

0

100

200

300

400

ESS (%)

Fig. 9. Sample force-displacement hysteresis curves of specimens for cyclic tests: (1) GYZ400
81; (2) GJZ200
21.

the methods determining the shear modulus from the experimental force-displacement curves. A secant modulus is defined to approximately represent the elastic shear modulus of bearing in these standards. The secant shear modulus G (kPa) between two points on a force-displacement curve is determined according to

G¼

ðF 2  F 1 Þhrt Aðd2  d1 Þ

ð1Þ

where d1, d2 are applied horizontal displacements of bearing at point 1 and point 2, respectively (m), and F1, F2 are the corresponding shear forces (kN) at d 1 and d 2, respectively; hrt denotes the total elastomer thickness of bearing (m), equal to the sum of thickness of individual layers of elastomer; A is plan area of bearing (m2). The shear moduli specified by ASTM-4014 [20] and AASHTOM251 [21], denoted as G1 and G2 respectively, correspond to 0– 25% and 0–50% shear strain of bearings. G3 in the Chinese Bearing Guidelines (CBG JT/T-4) [19], is determined from the secant shear stiffness bounded by 300 kPa and 1000 kPa shear stress of the bearing. It is to be noted that the upper bound of the shear strains used to determine the shear moduli in these standards are small, as the shear strain limits for the bearings at service-level conditions are usually 50–75%. However, the determining methods for service-level considerations cannot capture the reduced shear stiffness related to high strains induced by seismic demands. For laminated-rubber bearings considering the sliding response, an effective shear modulus denoted as Ge is defined in the current study. This parameter is the secant modulus form zero shear strain to a point which is the boundary between Phase II and Phase III. The ESS of the bearing at this point is denoted as cs. The defined shear modulus can reflect the bearing response at Phase I and Phase II, which is suitable at seismic conditions. Table 3 lists the different methods determining the shear modulus of a laminated-rubber bearing. The above mentioned shear moduli G1, G2, G3 and Ge are calculated and compared based on the experimental results, as summarized in Table 4, to provide context for evaluating the performance of laminated-rubber bearings considering sliding. For monotonic tests, Table 4 shows that cs is normally larger than service-level values, with a value of 192–268%. The shear moduli of bearing are found to degrade with increasing shear

strains, when G1, G2 and Ge are compared and evaluated. For example, in Test 1-1 (4.0 MPa), the calculated Ge reaches 841 kPa, compared with the values of 1255 kPa and 1077 kPa for G1 and G2 respectively. The reduction in shear modulus or shear stiffness at large shear strains is mainly attributed to two factors: material scragging and elastomer curling effect. Rubber scragging is known to reduce the shear stiffness, and the effect has been comprehensively studied by Kingston et al. [22]. Elastomer curling effect is that the trailing edge of the bearing base is observed to curl with gradually increasing shear strain demand, contributing to the stiffness degradation of bearing. And the curling effect is more pronounced at high shear strain levels, as illustrated in Fig. 7. For all the monotonic tests, G1 is calculated in the range of 1183– 1357 kPa and G2 is 1020–1149 kPa. These two shear moduli are found to be less sensitive to the variation of normal stress and loading velocity. For G3 that is obtained from 300–1000 kPa secant stiffness, the values range from 635–817 kPa and are relatively smaller than Ge. The effective shear modulus, Ge, is calculated in the range of 667–1096 kPa, with cs = 192–268%. The calculated shear moduli for cyclic tests (test series 5 and series 6) are also listed in Table 4. The shear moduli may vary in each test, as they are calculated in each loading cycle. And only the upper and lower bounds of values for each shear modulus are presented in the table. The comparison of different shear moduli (G1, G2, G3 and Ge) in cyclic tests is just similar to that in monotonic tests. Generally the shear moduli of the bearings degrade with the increasing numbers of cycles. The upper and lower bounds of shear moduli represent the values calculated from the first loading cycle and the last loading cycle respectively. The reduction of shear stiffness with the loading cycles is mainly due to the well-known Mullins effect and the escalated abrasion of rubber. Additionally, the cs values are also observed to decrease with the increase of cycle numbers as a result of the stress softening. The comparison between Ge and the other shear moduli indicates that the moduli values obtained from standards should be adjusted appropriately to reflect the characteristics of rubber bearings considering sliding. The secant moduli from ASTM 4014 and AASHTO M251 tend to overestimate the shear stiffness of bearings when sliding is considered, while the values from CBG JT/T-4 underestimate the shear stiffness to some extent. The effective

Table 3 Different methods determining bearing shear modulus. Method

Denotation

F1/A (kPa)

F2/A (kPa)

d1/hrt (%)

d2/hrt (%)

ASTM 4014 AASHTO M251 CBG JT/T-4 Current method

G1 G2 G3 Ge

Stress at d1 Stress at d1 300 Stress d1

Stress at d2 Stress at d2 1000 Stress at d2

0 0 Strain at F1 0

25 50 Strain at F2

cs

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N. Xiang, J. Li / Engineering Structures 141 (2017) 159–174 Table 4 Calculated shear moduli from different methods. Test series

G1 (kPa)

G2 (kPa)

G3 (kPa)

1-1 1-1 1-1 1-1 1-1 1-1 1-1 1-1 1-1 2-1 2-1 2-1 2-1 2-1 2-1 2-1 2-1 2-1 3-1 3-1 3-1 3-1 3-1 4-1 4-1 4-1 4-1 4-1 5-1 5-2 5-3 6-1 6-2 6-3

1206 1306 1255 1236 1278 1243 1286 1335 1357 1183 1235 1318 1298 1286 1303 1259 1238 1281 1272 1220 1235 1302 1241 1345 1265 1279 1285 1344 872–1110 844–1288 778–1273 897–1018 915–1147 928–1174

1046 1074 1077 1072 1073 1118 1108 1062 1030 1027 1051 1081 1071 1047 1087 1084 1056 1099 1023 1020 1051 1074 1099 1072 1049 1057 1105 1149 857–911 822–1033 806–1120 841–892 901–974 896–1079

635 748 760 755 759 757 747 756 759 703 743 771 745 760 774 773 804 810 693 728 743 743 754 732 756 744 812 817 469–551 651–697 680–688 501–584 521–607 618–692

(2.0 MPa) (3.0 MPa) (4.0 MPa) (5.0 MPa) (6.0 MPa) (7.0 MPa) (8.0 MPa) (9.0 MPa) (10.0 MPa) (2.0 MPa) (3.0 MPa) (4.0 MPa) (5.0 MPa) (6.0 MPa) (7.0 MPa) (8.0 MPa) (9.0 MPa) (10.0 MPa) (1.0 mm/s) (2.0 mm/s) (5.0 mm/s) (15.0 mm/s) (20.0 mm/s) (1.0 mm/s) (2.0 mm/s) (5.0 mm/s) (15.0 mm/s) (20.0 mm/s) (2.0 MPa) (4.0 MPa) (6.0 MPa) (2.0 MPa) (4.0 MPa) (6.0 MPa)

Effective shear modulus Ge

1100

192 197 194 207 217 223 220 232 233 200 230 227 243 248 248 258 265 268 202 215 227 220 215 222 220 223 228 227 189–223 232–255 243–270 174–213 197–224 217–253

When friction resistance between rubber and steel plate is finally exceeded at large shear strains, the bearing begins to slide. This phase is illustrated at Phase III in the force-displacement curves of bearings. A breakaway coefficient of friction (lb) and a sliding coefficient of friction (ls) are used to characterize the bearing sliding response. Fig. 11 shows how the ratio of horizontal force to vertical force varies with the imposed displacement for all monotonic tests. The coefficient of friction can be obtained from

(b) 1200

Test 1-1 Test 2-1 Cyclic tests

1000

cs (%)

667 771 841 853 865 869 904 905 908 724 817 843 873 945 1000 1025 1071 1096 727 772 826 921 947 804 875 897 1065 1071 614–645 741–793 788–806 621–649 797–828 804–833

3.4. Bearing sliding response on steel plate

Effective shear modulus Ge

1200

Ge (kPa)

ing considering sliding can be approximately specified just according to the values given in Table 4 or Fig. 10.

shear modulus, as previously defined in this paper, provides insight into the dynamic behaviors of a rubber bearing prior to obvious sliding. The shear stiffness degradation due to material scragging and elastomer curling at large strain levels can be captured by the defined effective shear modulus. The calculated effective shear moduli Ge of bearings at different normal pressures and velocities are summarized in Fig. 10. As shown in the figure, the effective shear modulus Ge will generally increase with the increase of normal pressure and loading velocity. In all the test series including the monotonic and the cyclic ones, the value of Ge is found to vary from 614 kPa to 1096 kPa, depending on the values of normal pressure and velocity. In the analytical study, the effective shear modulus Ge of a laminated-rubber bear-

(a)

Current Method

900 800 700 600 500

Test 3-1 Test 4-1

1100 1000 900 800 700 600 500

400 1

2

3

4

5

6

7

8

Normal Pressure (MPa)

9

10

11

0

2

4

6

8

10

12

14

16

Velocity (mm/s)

Fig. 10. Summary of the calculated effective shear modulus Ge at different (a) normal pressures; and (b) velocities.

18

20

N. Xiang, J. Li / Engineering Structures 141 (2017) 159–174

(a) 2.0 MPa 4.0 MPa 8.0 MPa

0.8 0.6 0.4 0.2 0.0 0

Horizontal Force / Vertical Force

3.0 MPa 6.0 MPa 10.0 MPa

Horizontal Force / Vertical Force

1.0

50

(c)

0.90 0.75

100

150

200 250 ESS (%)

1.0 mm/s 10.0 mm/s

2.0 mm/s 20.0 mm/s

300

350

5.0 mm/s

0.60 0.45 0.30 0.15 0.00 0

50

100

150

200

250

300

350

(b)

1.0

2.0 MPa 4.0 MPa 8.0 MPa

0.8

3.0 MPa 6.0 MPa 10.0 MPa

0.6 0.4 0.2 0.0

400

0

Horizontal Force / Vertical Force

Horizontal Force / Vertical Force

166

50

100

150 200 ESS (%)

250

300

350

400

(d)

0.90

1.0 mm/s 10.0 mm/s

0.75

2.0 mm/s 20.0 mm/s

5.0 mm/s

0.60 0.45 0.30 0.15 0.00

400

0

50

100

150

ESS (%)

200

250

300

350

400

ESS (%)

Fig. 11. The ratio of horizontal forces to vertical force versus imposed displacements for (a) Test 1-1; (b) Test 2-1; (c) Test 3-1; (4) Test 4-1.

the curves located in Phase III (the parts beyond the points indicated by the circular markers) in the figure. For Test 1-1 which was conducted at a constant velocity of 0.5 mm/s, the coefficient of friction is observed to decrease with increase of normal pressure. Similar phenomenon can also be found at high loading velocity, as indicted in Test 2-1. Results of Test 1-1 and 2-1 also suggest that tests at high velocities result in high coefficient of friction for a given normal pressure. The effect of loading velocities on coefficient of friction is presented in Fig. 11c-d. With the increase of velocities, the coefficient of friction increases and the breakaway friction effect becomes more significant. On the other hand, the difference between breakaway coefficient of friction and sliding one is larger at high velocities. For example, in Test 4-1 at a velocity of 20.0 mm/s, the peak ratio of breakaway coefficient of friction to sliding one reaches about 1.28, compared with the corresponding value of 1.1 for test at 1.0 mm/s velocity.

(a) 1.8

Test 1-1 Test 2-1

(b) 1.8

1.4

1.4

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

Test 3-1 Test 4-1

1.6

μb / μs

μb / μs

1.6

The ratio of breakaway coefficients of friction to sliding ones are calculated and plotted in Fig. 12 for all monotonic tests. Generally, a breakaway coefficient of friction is relatively larger than sliding one, and this phenomenon can also be seen in previous experiments on PTFE sliding bearings [23,24]. The ratio of coefficient of friction is bounded by 1.04–1.2 for Test 1-1 and 2-1, and 1.04– 1.28 for Test 3-1 and 4-1. In addition, the breakaway effect is more pronounced at large velocities. However, these ratios are generally lower than the values obtained from the experiments on PTFE or Teflon sliding surfaces, where the value can reach as much as 4.0. In terms of bearing sliding, some standards recommend values of coefficient of friction between elastomer and steel. AASHTO [11] specifies that adequate friction shall be available if elastomeric bearings are left without anchorage, and it assumes a design coefficient of friction of 0.2 for service loads. The Chinese Seismic Design Guidelines for Highway Bridges (GSDHB) [25] requires that for unrestrained laminated rubber bearings, the sliding resistance

0.4 2

3

4

5

6

7

8

Normal Pressure (MPa)

9

10

0

2

4

6

8

10

12

14

16

Velocity (mm/s)

Fig. 12. Ratio of breakaway to sliding coefficient of friction for different (a) normal pressures; and (b) velocities.

18

20

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N. Xiang, J. Li / Engineering Structures 141 (2017) 159–174

Observed Coefficient of Friction

(a) 1.0

Test 1-1 (v=0.5mm/s) Test 1-1 breakaway Test 2-1 (v=10.0mm/s) Test 2-1 breakaway

0.8

0.6 s

=1.02

0.72

s

0.59

=1.09

0.4 CALTRANS (2013)

0.2

AASHTO (2014) GSDHB (2008)

0.0 2

3

4

5

6

7

8

9

10

ls ¼ lfast  ðlfast  lslow Þeav

Normal Pressure (MPa)

Observed Coefficient of Fricition

(b)1.0

0.8 s

0.7

=0.57 (0.57 0.46)e

0.39v

0.6 0.5 0.4 s

=0.48 (0.48 0.37)e

0.39v

0.3 0

2

4

6

8

10

12

14

16

18

20

ð2Þ

in which lfast = the coefficient of friction at fast sliding velocity (after leveling off); and lslow = the coefficient of friction at a very slow sliding velocity (nearly quasi-static); a is a constant for a given condition of contact surface. The coefficient of friction at fast (lfast) and slow velocity (lslow) as a function of normal pressure r (MPa) are defined as:

Test 3-1 (3.0MPa) Test 3-1 breakaway Test 4-1 (4.0MPa) Test 4-1 breakaway

0.9

require that the bearings shall be checked to prevent sliding for service or earthquake loads. The coefficient of friction in CALTRANS is larger, lying near the median of the results in Test 1-1 and 2-1. For a bridge system considering bearing sliding, the true coefficient of friction of bearing has an significant influence on the global bridge seismic performance. Generally, the increase of coefficient of friction will decrease the peak and residual bearing displacements, while increase the seismic forces transmitted down to the substructures from the superstructures. To characterize the true bearing sliding behavior on steel plate, formulations have been proposed to represent the sliding coefficient of friction incorporating the effect of normal pressure and velocity. The coefficient of sliding friction at sliding velocity v (mm/s) can be approximated by the following equation:

22

Velocity v (mm/s) Fig. 13. Variation of coefficient of friction with: (a) normal pressure; and (b) sliding velocity.

shall be checked for design earthquake. The guideline suggests a coefficient of friction of 0.1 between elastomer and steel. In CALTRANS [26], the elastomeric bearings are considered as sacrificial elements for ordinary standard bridges. And the dynamic coefficient of friction between neoprene and steel is recommended as 0.35 based on test results. The effect of normal pressure and velocity on bearing breakaway and sliding coefficients of friction is illustrated in Fig. 13. The sliding coefficients of friction were selected at every interval of 50% ESS from the force-displacement curves that located in Phase III. It is observed in previous tests conducted by Schrage [27], Huang [28] and Fang [29] that the coefficient of friction is inversely related to the normal pressure. The data obtained from the current testing program agree well with this general trend, as seen in Fig. 13a (Test 1-1 and 2-1). As is seen in Fig. 13b, there is a significant nonlinear trend of friction coefficients versus bearing velocity. With increase of bearing velocity, the coefficients of friction firstly increase and then tend to level off when the bearing velocity exceeds 5.0–10.0 mm/s. The recommended values of friction coefficient in the above mentioned standards are also plotted in Fig. 13. The 0.1 and 0.2 values specified by GSDHB and AASHTO respectively are relatively smaller and this will underestimate the bearing sliding resistance, as seen in Fig. 13a. This can be explained that these two standards

lfast ¼ kfast rðnfast 1Þ

ð3Þ

lslow ¼ kslow rðnslow 1Þ

ð4Þ

where kfast, kslow, nfast  1, and nslow  1 are constants that determine the friction coefficient models. According to the form of the proposed formula, the obtained data from the tests are fitted based on the least square method and the fitting curves are shown in Fig. 13a-b. Table 5 presents values of kfast, kslow, nfast, nslow and a for elastomeric bearing-steel interface that resulted in the fitted curves.

Force Nμ

Ke

Ke

Displ.

Nμ Fig. 14. Proposed analytical model of sliding between elastomeric bearing and steel.

Table 5 Recommend Values of kfast, kslow, nfast, nslow and a. Constants

kfast

nfast

kslow

nslow

a

Recommended value

1.09

0.41

1.02

0.28

0.39

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is only 1.28. And the bearing is subject to the effect of breakaway friction only at initial sliding and not at every instance.

As previously stated, when the sliding velocity exceeds 10.0 mm/s, the coefficient becomes independent of velocity. Therefore, v = 0.5 mm/s in Test 1-1 and v = 10.0 mm/s in Test 2-1 are approximately representative of slow and fast velocity respectively. It is apparent that the coefficients of friction calculated by Eqs. (2)–(4) are in good agreement with the test results if the constants in the equations are selected reasonably. It is to be noted that the breakaway coefficient of friction is not considered in the fitted equations. Although the breakaway friction effect is observed during the tests, the peak amplification factor of friction coefficient

50

Test 1-1, 3.0 MPa

Test 1-1, 2.0 MPa 50

100

150

200

250

ESS (%)

300

350

400

50

100

150

200

ESS (%)

200 150 100 50

0

50

100

150

200

250

ESS (%)

300

350

400

150 100 50

Test 1-1, 7.0 MPa

200 150 100 50

Test 1-1, 10.0 MPa 0

50

100

150

200

250

ESS (%)

300

350

50

100

150

200

ESS (%)

200 150 100 50

Test 2-1, 5.0 MPa

350

400

0

50

100

150

200

250

ESS (%)

300

350

400

Experimental Numerical

150 100 50

Test 2-1, 2.0 MPa 0

50

100

150

200

250

ESS (%)

300

350

240 160 80

250 200 150 100 50

50

100

150

200

250

ESS (%)

300

350

300

0

50

100

150

200

250

ESS (%)

300

350

400

Experimental Numerical

320 240 160 80

0

50

100

150

200

250

ESS (%)

300

350

200 150 100 50

Test 3-1, 5.0 mm/s 0

50

100

150

200

250

ESS (%)

300

350

400

Experimental Numerical

200 150 100

0

Test 3-1, 15.0 mm/s 0

50

100

150

200

250

ESS (%)

300

350

100 50

Test 1-1, 5.0 MPa

400

0

50

100

150

200

250

200

250

200

250

200

250

200

250

200

250

ESS (%)

100 50

Test 1-1, 8.0 MPa 0

50

350

400

100

150

200

250

ESS (%)

300

350

250 200 150 100 50 0

400

Test 1-1, 9.0 MPa 0

50

100

150

ESS (%)

300

350

400

350

Experimental Numerical

250 200 150 100 50

0

50

100

150

200

250

ESS (%)

300

350

Experimental Numerical

300 250 200 150 100 50 0

400

Test 2-1, 4.0 MPa 0

50

100

150

ESS (%)

300

350

400

400

Experimental Numerical

350 300 250 200 150 100

0

50

100

150

200

250

ESS (%)

300

350

Experimental Numerical

350 300 250 200 150 100 50

Test 2-1, 7.0 MPa

0

400

Test 2-1, 8.0 MPa 0

50

100

150

ESS (%)

300

350

400

300

Experimental Numerical

250 200 150 100 50

0

50

Experimental Numerical

250 200 150 100 50

100

150

200

250

ESS (%)

300

350

0

400

Test 3-1, 2.0 mm/s 0

50

100

150

ESS (%)

300

350

400

300

Experimental Numerical

300 250 200 150 100 50 0

300

Experimental Numerical

300

Test 3-1, 1.0 mm/s

400

250

50

150

350

Experimental Numerical

350

300

200

0

400

150

0

Horizontal Force (kN)

Horizontal Force (kN)

Experimental Numerical

350

300

350

250

300

200

0

Test 2-1, 10.0 MPa 400

ESS (%)

50

Test 2-1, 6.0 MPa

Test 2-1, 9.0 MPa 0

250

400

Experimental Numerical

300

0

200

250

0

400

Horizontal Force (kN)

Horizontal Force (kN)

Experimental Numerical

150

Test 2-1, 3.0 MPa

400

320

100

300

200

0

400

Horizontal Force (kN)

300

Horizontal Force (kN)

250

0

250

350

Experimental Numerical

300

Horizontal Force (kN)

Horizontal Force (kN)

350

Horizontal Force (kN)

0

250

0

400

50

300

0

Horizontal Force (kN)

250

Horizontal Force (kN)

Horizontal Force (kN)

Experimental Numerical

0

350

300

300

0

400

200

0

350

0

350

Experimental Numerical

250

Test 1-1, 6.0 MPa

0

300

50

Experimental Numerical

250

Test 1-1, 4.0 MPa

Horizontal Force (kN)

Experimental Numerical

250

0

250

100

0

300

Horizontal Force (kN)

Horizontal Force (kN)

300

0

150

Horizontal Force (kN)

0

0

Horizontal Force (kN)

100

200

Horizontal Force (kN)

40

150

Horizontal Force (kN)

80

200

300

Experimental Numerical

250

Horizontal Force (kN)

120

Experimental Numerical

250

Horizontal Force (kN)

160

300

Horizontal Force (kN)

Experimental Numerical

200

0

To simulate the seismic behavior of a typical laminated-rubber bearing considering its sliding on steel plates, an analytical model (Fig. 14) is developed based on the experimental results. The pro-

300

Horizontal Force (kN)

Horizontal Force (kN)

240

4. Analytical model of laminated-rubber bearings considering sliding

Experimental Numerical

250 200 150 100 50

Test 4-1, 1.0 mm/s

Test 3-1, 20.0 mm/s 0

50

100

150

200

250

ESS (%)

300

350

Fig. 15. Comparison of experimental results with numerical simulation.

400

0

0

50

100

150

ESS (%)

300

350

400

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N. Xiang, J. Li / Engineering Structures 141 (2017) 159–174

of seismic demands are considered. The sliding coefficient of friction (ls) between the bearing and the steel plate is estimated through Eq. (2) with respect to the normal pressure r (MPa) and the sliding velocity v (mm/s). The value of constants in Eq. (2) can be derived from Table 5.

posed model is mainly defined by two parameters: an effective shear stiffness (Ke) and a sliding coefficient of friction (ls). 1. A laminated-rubber bearings is assumed to exhibit an approximately linear elastic response before obvious sliding, and the response is characterized by an effective shear stiffness (Ke), which is calculated by the following equation:

Ge A hrt

5. Numerical study

ð5Þ

5.1. Comparison between experimental and numerical results

where A and hrt are described in the previous section; Ge = the effective shear modulus of the bearing, and the value of Ge ranges from 610 kPa to 1100 kPa based on the test results in this paper. In the proposed model, it is assumed that the initial shear stiffness, unloading shear stiffness and reloading shear stiffness are all equal to Ke. 2. When the sliding resistance is reached, the bearing response is dominated by the sliding frictional response. Instead of using the typical Coulomb’s friction model, a new friction model considering the effect of normal pressure and sliding velocity is proposed. The incorporation of such effect into the dynamic analysis is quite important, especially when vertical earthquake and large velocities

Horizontal Force (kN)

200 150 100 50

Test 4-1, 2.0 mm/s 0

50

100

150

200

250

300

350

400

400

Experimental Numerical

300 250 200 150 100 50 0

Test 4-1, 5.0 mm/s 0

50

100

200

Horizontal Force (kN)

Horizontal Force (kN)

250

300

350

50 0 -50 -100 -150

100 50

Test 4-1, 15.0 mm/s 0

50

100

150

200

250

300

-300

-200

-100

0

100

200

300

150 100 50 0

400

Test 4-1, 20.0 mm/s 0

50

100

100 0 -100

200

Test 6-1, 2.0 MPa 0 100 ESS (%)

200

300

400

200

300

350

400

Experimental Numerical

0 -100 -200

-300

-200

-100

0

100

200

300

-300 -400

400

Test 5-3, 6.0 MPa -300

-200

-100

0

100

200

300

400

ESS (%) 100

Experimental Numerical

75

50 25 0 -25 -50 -75 -100 -400

250

100

Horizontal Force (kN)

-25

150

300

75

0

-100

200

ESS (%)

25

-200

250

ESS (%)

100

-300

300

ESS (%) Experimental Numerical

200

-300 -400

400

Experimental Numerical

-50

350

Experimental Numerical

350

Test 5-2, 4.0 MPa

Horizontal Force (kN)

Horizontal Force (kN)

150

-200

Test 5-1, 2.0 MPa

75

-75 -400

200

0

400

ESS (%)

50

250

300 Experimental Numerical

100

-200 -400

200

300

ESS (%)

ESS (%)

150

150

400

Experimental Numerical

350

Horizontal Force (kN)

Horizontal Force (kN)

Experimental Numerical

250

Horizontal Force (kN)

350

300

0

To evaluate the proposed analytical model of laminated-rubber bearings considering sliding, comparison was performed between the experimental results and the numerical simulations. The proposed model was evaluated according to the loading protocols (monotonic or cyclic) employed during testing of the laminatedrubber bearings. The effective shear stiffness Ke and the sliding coefficient of friction ls used in the proposed model were derived from Table 4 and Eq. (2), respectively. A total of 34 numerical analyses were conducted to simulate all the test cases. The results of these analyses are depicted in Fig. 15. Generally, results presented in these figures show a good agreement between the proposed

Horizontal Force (kN)

Ke ¼

-200

-100

0 100 ESS (%)

200

300

50 25 0 -25 -50 -75

Test 6-2, 4.0 MPa -300

Experimental Numerical

400

-100 -400

Test 6-3, 6.0 MPa -300

-200

-100

0 100 ESS (%)

200

300

400

Fig. 15 (continued)

Table 6 Comparison of dissipated energy for cyclic loading cases.

Experimental (kN.m) Numerical (kN.m) Difference ratio (%)

Test 5-1

Test 5-2

Test 5-3

Test 6-1

Test 6-2

Test 6-3

256 257 0.4

351 333 5.1

316 289 8.5

20 19 5

23 21 8.7

22 20 9.1

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5.2. Comparison between proposed model with Coulomb’ friction model

model and the experimental results, especially for the monotonic loading cases. For the cyclic loading cases, although the proposed model cannot model the stiffening effect that occurred when the loading direction changes, some key responses of the bearing (etc. friction resistance) can be captured by the proposed model. Therefore, the seismic response of laminated-rubber bearings sliding on steel plate can be reasonably estimated by the proposed model in the numerical analyses. Table 6 lists the comparison of dissipated energy between the experimental and numerical results for cyclic loading cases. The dissipated energy of the bearings was calculated based on the area of the hysteresis loops obtained from the experiments or numerical analyses. As shown in the table, although the difference of dissipated energy between the experimental and numerical results will be amplified by the increase of normal pressure, the difference ratios for all the cyclic loading cases are generally small, just lower than 10%. This indicates that the proposed analytical model is able to describe the energy-dissipative behavior of bearing sliding with reasonable accuracy.

25 m

Effects of the proposed bearing model and the model with Coulomb’s friction on the seismic response of highway bridges supported by elastomeric bearings are investigated and compared based on a typical example. The example considered in the analysis is a simply-supported T-shaped girder bridge with a span length of 25 m (Fig. 16a). The bridge is located in Wenchuan area of China. The span consists of five T-girders weighing 568 ton in all. The substructure had double-column bents with an equal height of 8 m, as shown in Fig. 16b. Each girder is supported by two laminatedrubber bearings at two ends, and there are totally ten bearings placed on each pier (two rows in longitudinal direction and five rows in transverse direction). The bearing is the same as the one used in the quasi-static experiments, and its plan area is calculated as A = 125,664 mm2. Normally, the bearing in this bridge maintains an average pressure of 4.5 MPa under gravity load. According to the Eqs. (3) and (4), the coefficients of friction for bearing-steel inter-

25 m

25 m

Single pier system

Foundation

(a) Elevation of the simply-supported bridge Side Girder Mass

Intermediate Girder Mass

Elastomeric bearings

Diaphragm Element 1.5 m

Bridge girder

Cap beam

Diaphragm 1.4 m

Bearing Element

8.0 m

1.4 m

Lumped Mass

Pier columns

Foundation

(b) Transverse view of bridge

(c) Finite element model of the bridge

Fig. 16. Bridge prototype configuration and finite element model in the case study.

Table 7 Summary of ground motions. Earthquake

Year

Station

Mw

Component

PGA-Horizontal (g)

PGA-Vertical (g)

Chi-Chi, Taiwan Northridge-01

1999 1994

TCU068 Newhall Fire

7.62 6.69

TCU068-N, TCU068-V NWH360, NWH-UP

0.361 0.590

0.529 0.548

Table 8 Bearing responses for horizontal earthquake only. Earthquake

Chi-Chi Northridge

Bearing maximum displ./residual displ. (cm)

Bearing maximum shear force (kN)

l = 0.345

l = 0.449

Proposed model

l = 0.345

l = 0.449

Proposed model

82.3/34.1 36.7/23.5

54.3/22.4 36.2/12.6

56.9/22.1 36.8/12.9

381 380

504 504

504 504

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N. Xiang, J. Li / Engineering Structures 141 (2017) 159–174

velocity or the maximum value (l = 0.449) at fast velocity. For the proposed model in terms of sliding velocity and the pressure on the contact area, the coefficient of friction is determined just according to Eq. (2) and Table 5. The structure is subjected to two sets of recorded earthquake which are acquired from the PEER Ground Motion Database. One is the 1999 Chi-Chi earthquake, and another is the 1994 Northridge earthquake. These two records are characterized as near-fault, pulse-like ground motions. Each set of record is composed of a horizontal component and the corresponding vertical component. The PGA ratio of vertical component to horizontal component is 147% for Chi-Chi earthquake, and 93% for Northridge earthquake, indicating the potential effect of vertical earthquake. Table 7 lists details of the two unscaled ground motions. Table 8 lists the comparison of displacement and force response of a single bearing element between Coulomb’s model and the proposed model for horizontal earthquake only. The bearing displacements defined in this numerical study include the bearing shear deformation and bearing sliding displacement. The PGAs of the horizontal components for these two earthquakes are adjusted to 0.68 g, which represents a 2000-year recurrence design earthquake in Wenchuan area specified in Chinese Seismic Design Guidelines for Highway Bridges (GSDHB). Results in the table show that Coulomb’s model with maximum friction (l = 0.449) predicts the bearing maximum/residual displacements and the bearing maximum shear force response that approximate the one calculated through Eq. (2) and Table 5. As the earthquake excitation normally results in large bearing sliding velocity, the coefficient of sliding friction attains its maximum value during most of excitation time. The resulting error will be relatively larger when the minimum value of friction coefficient (at slow velocity) is taken, especially for the response of bearing residual displacement and the bearing shear force. Comparison between these two models are also shown in the bearing displacement response histories in Fig. 17. The bearing displacement histories for the cases of l = 0.449 and the proposed

face at fast and slow velocity are then calculated as 0.449 and 0.345 respectively for this pressure level. Only the transverse seismic response of the bridge prototype will be considered in the current study. As the bridge prototype is straight and consists of a large number of equal spans and piers with equal height or stiffness as shown in Fig. 16a, the rotation of superstructure around the vertical axis is not likely to occur under transverse earthquake excitation. Hence, the transverse seismic response of the overall bridge can be represented by a single pier bent system shown in Fig. 16a and b with tributary mass from the two adjacent half spans of the superstructure. Finite-element model of the single pier bent system (Fig. 16c) is then constructed in OpenSees, an object-oriented software framework for simulation applications in earthquake engineering. In this model, the seismic masses of the superstructure (side girders and intermediate girders) are assumed to be lumped at the top of the single pier bent. Elastic beam elements are used to model the cap beam, diaphragms and pier columns, as these elements are assumed to behave elastically in the dynamic analysis. The soilstructure interaction effect is neglected in this study and the fixed constraints are applied to the base of the pier columns. The sliding behavior of elastomeric bearings upon steel plates is modeled by a Flat Slider Bearing Element in OpenSees. This element is defined by initial elastic shear stiffness before obvious sliding and a friction model which specifies the behavior of the coefficient of friction. As only the transverse earthquake is considered in this study, the two bearings aligned in longitudinal direction at the pier are modeled using a single bearing element. Hence, there are five bearing elements in total constructed in the finite element model in transverse direction. In the current study, the effective shear stiffness Ge is set as 845 kPa based on the experimental results, and the initial elastic shear stiffness of the bearing element is then calculated as 17,698 kN/m based on Eq. (5). For Coulomb’s model, the coefficient of friction takes either the minimum value (l = 0.345) at slow

Bearing Displacement (cm)

(a) 60 μ=0.345 μ=0.449 Proposed model

30 0 -30 -60 -90 0

10

20

30

40

50

60

70

80

90

Time (sec) Bearing Displacement (cm)

(b)

30

μ=0.345 μ=0.449 Proposed model

20 10 0 -10 -20 -30 -40 0

5

10

15

20

25

30

35

40

45

50

Time (sec) Fig. 17. Time history response of bearing displacement for (a) Chi-Chi earthquake; (b) Northridge earthquake.

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model are nearly the same when vertical earthquake is not considered. To investigate the effect of vertical earthquake on the seismic performance of bearing, dynamic simulations were carried out considering the vertical earthquake effect. The bridge is excited by the combined horizontal and vertical earthquake. The PGA of horizontal earthquake is scaled to 0.68 g, and the PGA of vertical earthquake varies. Bearing responses between the Coulomb’s model with l = 0.449 and the proposed model are compared in this part, to highlight the effect of vertical earthquake. Fig. 18 shows the response of bearing displacement and shear force with the variation of vertical-horizontal PGA ratio. As seen in the figures, the difference of bearing displacement response between the Coulomb’s model with l = 0.449 and the proposed model occur if the vertical effect is incorporated into the analysis. And this difference will become more significant as the intensity of vertical earthquake increases. However, the difference of bearing maximum shear force between the Coulomb’s model and the proposed model due

100

Proposed model μ=0.449

80 60 40 20

Bearing Maximum Displ. (cm)

(b)

(a) Bearing Maximum Displ. (cm)

to vertical effect is found to be less significant, compared with the response of bearing displacement. Furthermore, the seismic response of the bridge is relatively more sensitive to the variation of vertical earthquake intensity for the case of Coulomb’s model than the proposed model, as shown in the Fig. 18. This phenomenon can be explained from the perspective of bearing friction behavior. The friction behavior of bearings is greatly influenced by the vertical earthquake which alter normal force and then the bearing frictional force. Fig. 19 shows that the fluctuations in frictional forces are observed in the hysteretic curves of bearings with vertical earthquake excitation. Generally frictional force depends on the normal force multiplied by the coefficient of friction. Only the normal force is influenced by the vertical earthquake for Coulomb’s model of friction. However, for the proposed model, the vertical excitation not only alters the normal force on bearings, but also affects the coefficient of friction due to the variation of normal force. For Coulomb’s friction model with l = 0.449 and the proposed model at

0

75

Proposed model μ=0.449

60 45 30 15 0

0 20 40 60 80 100 120 PGA Ratio of Vertical to Horizontal Earthquake (%)

Proposed model μ=0.449

-10

-20

-30

0

Bearing Residual Displ. (cm)

Bearing Residual Displ. (cm)

0

0 20 40 60 80 100 120 PGA Ratio of Vertical to Horizontal Earthquake (%)

-40

-10

-20

-30

-40 0 20 40 60 80 100 120 PGA Ratio of Vertical to Horizontal Earthquake (%)

800

Proposed model μ=0.449

600 400 200 0

0 20 40 60 80 100 120 PGA Ratio of Vertical to Horizontal Earthquake (%) 1000

Bearing Shear Force (kN)

Bearing Shear Force (kN)

1000

Proposed model μ=0.449

800

Proposed model μ=0.449

600 400 200 0

0 20 40 60 80 100 120 PGA Ratio of Vertical to Horizontal Earthquake (%)

0 20 40 60 80 100 120 PGA Ratio of Vertical to Horizontal Earthquake (%)

Fig. 18. Comparison of bearing response between Coulomb’s model and the proposed model with vertical earthquake effect for (a) Chi-Chi earthquake; (b) Northridge earthquake.

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without vertical effect with vertical effect

750

(b) 1000

500 250 0 -250 -500 -750 -1000 -0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

without vertical effect with vertical effect

750

Bearing Shear Force (kN)

Bearing Shear Force (kN)

(a) 1000

500 250 0 -250 -500 -750 -1000 -0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

Bearing Displacement (m)

Bearing Displacement (m)

Fig. 19. Effect of vertical earthquake on the hysteretic curves of laminated-rubber bearings subjected to Chi-Chi earthquake with (a) Coulomb’s model; (b) the proposed model (PGA ratio = 100%).

600 Eq. (6)

6. Summary and conclusions

Eq. (7)

500

A series of tests were performed to investigate the shear and sliding response of typical laminated-rubber bearings on steel plate. The bearing specimens were designed and manufactured conforming to the bearing standards used in China. The test program consisted of various combinations of normal pressure, loading velocity and loading protocols (monotonic and fully reversed cyclic). Horizontal displacements were imposed by a horizontal actuator while the vertical loads were maintained at a target level. A maximum displacement of 400% of the total rubber thickness was reached for monotonic loadings, and 300% of the total rubber thickness for cyclic loadings. Conclusions were drawn from the current works as follows:

Frictional Force (kN)

Eq. (8) 400 300 200 100 0 200

300

400

500

600

700

800

900

1000

Normal Force (kN) Fig. 20. Effect of normal force on the frictional force of bearing.

fast and slow velocity (Eqs. (3) and (4)), the frictional forces Ff (kN) for a single bearing (with the plan area of 125,664 mm2) considered in this paper can be calculated respectively in terms of normal force N (kN):

ð1Þ Coulomb’s model with

l ¼ 0:449 :

F f ¼ Nl ¼ 0:449N ðunit : kNÞ

ð6Þ

ð2Þ Proposed model at fast velocity : F f ¼ Nlfast ¼ 18:878N0:41 ðunit : kNÞ

ð7Þ

ð3Þ Proposed model at slow velocity : F f ¼ Nlslow ¼ 33:116N0:28 ðunit : kNÞ

ð8Þ

The variation of frictional force of bearing with normal force is plotted in Fig. 20, for Coulomb’s model and the proposed model. The curves in the figure are calculated from Eqs. (6)–(8), respectively. As seen in Fig. 20, the frictional force of bearings with the proposed model is found to be less sensitive to the variation of normal force, compared with the Coulomb’s model of friction. As the coefficient of friction is inversely related to the normal force for the proposed model, the variation of frictional force is neutralized by the variation of friction coefficient. Therefore, the effect of vertical earthquake on the bearing response is less pronounced for the proposed model than the model with Coulomb’s friction.

(1) Economical laminated-rubber bearings with typical configurations and construction details are prone to slide on the steel plates embedded in the bridge girder during large earthquakes, as observed in the past Wenchuan earthquake. The bearing sliding can actually act as structural fuses, limiting the seismic forces transmitted from the superstructure to the substructure. (2) For laminated-rubber bearings permitted to slide on a steel plate, the effective shear modulus of bearings before slip was observed to be in the range of 610–1100 kPa in the current experiments, which is lower than the values calculated from the methods in ASTM 4014 and AASHTO M251, but is relatively larger than the values obtained from the China standards (CBG JT/T-4). (3) The phenomenon of breakaway friction was observed in the experiments, especially at large loading velocities. The ratio of breakaway to sliding coefficient of friction for elastomersteel interface is in the range of 1.04–1.28. (4) Test data indicates that the coefficients of friction varied with the normal pressure on bearings in a nonlinear inverse relationship. The coefficients of friction at fast sliding velocity are found to be relatively larger than those at slow sliding velocity. A analytical model for the sliding behavior between the laminated-rubber bearings and the steel was finally proposed and calibrated based on the test data. The coefficient of sliding friction in the proposed model is a nonlinear function of the normal pressure and sliding velocity. (5) Comparison between the experimental results with the numerical simulations shows that the agreement between the numerical and experimental data is good, and the pro-

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posed model in the numerical analyses can provide a reasonably prediction for the sliding response between laminatedrubber bearings and steel plates. (6) Results of numerical simulations on a typical highway bridge system show that Coulomb’s model with friction coefficient values at fast velocities provides a good estimation for the seismic response of bearings when vertical earthquake is not considered. However, the incorporation of vertical earthquake amplifies the differences of bearing displacement response between the proposed bearing model and the model with Coulomb’s friction. And this difference becomes more significant with the increase of vertical earthquake intensity.

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