Experimental and Seismic Response Study of Laminated Rubber Bearings Considering Different Friction Interfaces

Abstract

Unbonded LRBs (laminated rubber bearings) are commonly applied in small-to-medium-span bridges in China. The frictional sliding characteristics of LRBs have a vital influence on the seismic response of the bridge. Nine square LRBs were subjected to the quasi-static displacement loading test in this paper, and the differences in sliding characteristics of LRBs at the interface of steel and concrete test pad were investigated. The variation of the friction coefficient during sliding was then analyzed. Based on the experimental data, a three-fold mechanical constitutive model of LRBs that considers the breakaway-sliding friction characteristics is established. Further, the bridge seismic demands in longitudinal directions with different friction interfaces are compared by nonlinear dynamic analysis on a typical LRB-supported concrete bridge. The results show the causalities of the displacements and decreases of the friction coefficient of the LRB. The breakaway coefficient of friction of the concrete surface was generally greater than that of the steel in the pre-sliding stage, while the sliding coefficient of friction of the steel interface in the post-sliding stage was greater than that of the concrete. Moreover, the proposed three-fold constitutive model is able to simulate the frictional sliding behavior of LRBs accurately. Lastly, the seismic design of small-to-medium-span bridges should take into account the breakaway-sliding friction effect of the LRBs and the preference for steel as friction pads for LRBs is recommended.

1. Introduction

LRBs (laminated rubber bearings), as an economic and efficient seismic isolation technique, are widely adopted in earthquake regions in China for small-to-medium-span bridges. In general, unbonded LRBs are placed directly between the pier top and the beam. Sliding bearings under seismic loads can reduce the seismic forces transferred from the beam to the substructure effectively [1,2]. However, smooth-contact surfaces or excessive sliding result in uncontrolled bearing displacement (Figure 1), and the friction coefficient of different friction interfaces is ignored, which will make the bearing unable to achieve the displacement requirements. In the Wenchuan Earthquake in China, in which LRBs were placed between the pad of steel or concrete (Figure 2), a large number of uncontrolled sliding bearings and significant damages were observed in the bridge [3,4].

Studies of the sliding characteristics of LRBs have been conducted worldwide. Kelly [5] and Nagarajaiah [6] carried out experimental studies on unbonded LRBs, and the results showed that when the horizontal force was greater than the friction, the bearing will slide and then the internal force of the structure will be reduced. Large displacement loading tests have shown that LRBs in appropriate dimensions and vertical loads will change the sliding displacement and friction of the bearing effectively [7]. Since then, plenty of experimental and numerical studies of bearing sliding have been carried out. It has been found that the key factors affecting the sliding characteristics of LRBs include the geometries [8,9,10], the material properties [11,12] and the loading conditions [13,14,15] of the bearings. In particular, the roughness of the friction interface will affect the bearing sliding response as well. Steelman [16] compared the sliding characteristics of LRBs in polytetrafluoroethylene and concrete interfaces, finding that the friction coefficient and failure mode of bearings are significantly different. Consequently, the sliding characteristics of LRBs in the steel and concrete friction pads which are commonly used in bridges may vary considerably.

The development of rigorous and accurate mechanical constitutive models of LRBs is the premise for seismic response analysis of bridges. Constitutive models of LRBs have been proposed, such as a bilinear model [17], three-fold model [15], polynomial regression model [18,19], etc. Compared with these models, although some of the models considered the sliding characteristics of LRBs, the effect of steel or concrete frictional interfaces has been neglected, and the affecting factors have limitations. Therefore, the establishment of reasonable constitutive models of LRBs in different frictional surfaces remains challenging.

LRBs have a critical impact on the seismic response of a bridge, which are divided into two main aspects: the shear stiffness of pre-sliding [20] and the friction of post-sliding [16]. The shear stiffness is mainly controlled by the geometrical parameters of the bearing [20], while the contact area [18], the vertical load [13], the loading velocity [21] and the friction coefficient [16] are the main factors affecting frictional force. In particular, numerical and experimental studies have shown that post-sliding behavior of LRBs, especially the friction coefficient, is a vital factor which reduces the seismic damage of bridges and structures effectively [22,23,24,25,26] which will help to control the seismic displacement requirements of bearings and to avoid serious collapses of bridges [27]. Therefore, it is necessary to investigate the influence of the sliding characteristics of LRBs at steel or concrete interfaces on bridge seismic response.

The differences in the sliding characteristics of LRBs at different friction interfaces and their effect on the seismic response of bridges were explored in this paper. Nine LRBs were subjected to a quasi-static loading test, and the differences in sliding characteristics of LRBs at steel and concrete test pads were compared. Next, the variation of the breakaway coefficient of friction and sliding coefficient of friction with vertical load at the different interfaces was investigated. Based on the linear regression method, a three-fold mechanical constitutive model of LRBs that takes into account breakaway-sliding friction was developed. In addition, a typical three-span simply supported concrete bridge was subjected to non-linear dynamic analysis. The seismic response of the bridge with LRBs at different friction interfaces was compared. Finally, to ensure the pier’s safety and continuity of the bearings’ sliding force, the material of the friction pad for LRB-supported bridges was recommended.

2. Experimental Tests

2.1. Test Specimens

The test specimen contained 9 square LRBs, composed of a few layers of vulcanizate of laminated rubber and steel plates, the top and bottom of a bearing being wrapped by a rubber layer. The rubber material has a tensile strength of 18 MPa and a hardness of 60 IRHD, and the yield strength of the steel plates is 235 MPa. The specimen conformation satisfied the specification of the highway bridge elastic bearings of China [28] and shown in Figure 3, where the configuration characteristics are shown in

Table 1. Bearing specimens’ geometry characteristics.

Type	Mark	L/mm	Rubber Layer		Steel Plate		h1/mm	h2/mm	H/mm	Sum
			hr/mm	Number of Layers	hs/mm	Number of Layers
T1	GJZ400 × 400 × 99	400	11	6	4	7	2.5	5	99	3
T2	GJZ500 × 500 × 110	500	15	5	5	6	2.5	5	110	3
T3	GJZ500 × 500 × 130	500	15	6	5	7	2.5	5	130	3

.

2.2. Experimental Installation

The quasi-static shear tests were carried out on the LRBs on the FCSYJZ-3000t electro-hydraulic servo loading system in the Key Laboratory of Engineering Seismic Isolation Structures and Materials in Hebei Province, with the specimen installation as shown in Figure 4. The test pads were concrete slabs (115 cm × 70 cm × 25 cm) and steel plates (103 cm × 103 cm × 5 cm). The pull-wire displacement sensors were attached to the edge of the steel plates at each height of the LRBs. The monitoring points were placed 5 cm apart horizontally so that the sensors and the monitoring points were kept at the same altitude to reduce aberration. The shear deformation between the rubber layers was measured by the pull-wire sensors, while the residual displacement in the sliding of the LRBs was measured to prevent the bearing detachment from the test pad.

2.3. Test Program

The complete details of the test procedures are available in a previous study [18], and details of the loading conditions are given in

Table 2. Test program of LRBs.

Test Specimen	Type	Loading Displacement	Vertical Load	Interface
				Concrete Slabs	Steel Plates
J1-4-C-1	T1	400% ESS	4 MPa	X
J1-4-C-2	T1	400% ESS	4 MPa	X
J1-4-C-3	T1	400% ESS	4 MPa	X
J1-6-C-1	T1	300% ESS	6 MPa	X
J1-8-C-1	T1	300% ESS	8 MPa	X
J1-10-C-1	T1	300% ESS	10 MPa	X
J1-4-S-1	T1	400% ESS	4 MPa		X
J1-6-S-1	T1	400% ESS	6 MPa		X
J1-8-S-1	T1	400% ESS	8 MPa		X
J2-6-C-1	T2	300% ESS	6 MPa	X
J2-6-C-2	T2	300% ESS	6 MPa	X
J2-8-C-1	T2	300% ESS	8 MPa	X
J2-4-S-1	T2	300% ESS	4 MPa		X
J2-6-S-1	T2	300% ESS	6 MPa		X
J2-8-S-1	T2	300% ESS	8 MPa		X
J3-4-C-1	T3	300% ESS	4 MPa	X
J3-6-C-1	T3	300% ESS	6 MPa	X
J3-8-C-1	T3	300% ESS	8 MPa	X
J3-10-C-1	T3	300% ESS	10 MPa	X
J3-4-S-1	T3	300% ESS	4 MPa		X
J3-6-S-1	T3	300% ESS	6 MPa		X
J3-8-S-1	T3	300% ESS	8 MPa		X
J3-8-S-2	T3	300% ESS	8 MPa		X

. The range of vertical load was referenced to the vertical load capacity of LRBs and scaled down [28]. The equivalent shear strain (ESS) was defined as the measurement of loading displacement, which is the ratio of the loading displacement to the height of the bearing; for partial specimens the maximum loading displacement is 300% ESS due to the dimensional limitations of the test pad.

3. Results

3.1. Overview of the Experimental Response

The hysteresis behavior of the LRBs that dissipates seismic energy is observed in Figure 5. The force-displacement response of LRBs is divided into three stages: elastic shear deformation of the rubber layer (I), bearing warpage (II), and bearing sliding (III).

Stage I: The deformation of LRB mainly consists of elastic shear; the upper and lower surfaces of the bearing are closely fitted to the test pad without sliding (Figure 5a). The curve recorded by the sensor intersects near the origin (Figure 6) and the bearing was able to return to its initial position after displacement loading cycles. The force-displacement relationship corresponding to this stage is basically considered elastic, and it should be noted that the peak breakaway friction force corresponds to a displacement δ₁ of approximately 150% ESS, defining the peak horizontal force at this stage as the max breakaway friction F_sf.

Stage II: It can be seen in Figure 5b that significant warpage was found at the interface between the LRB and test pad, the friction force of the bearing beginning to decline with the loading of the horizontal displacement. There was a tendency for LRBs to move from breakaway friction to sliding friction. As derived from Figure 6, the residual deformation of the bearing at this stage was approximately 47.3 mm and −61.5 mm, which corresponds to the shear deformation of 101.2 mm and 87 mm, close to the actual height of the bearing of 99 mm. This differs from the suggestions of the bearing shear displacement of the Chinese Code [17]. This stage is the transition from shear deformation to frictional sliding of the LRBs. The displacement δ₂ is defined as corresponding to stable sliding of the bearing occurring at approximately 200% ESS.

Stage III: As displacement load increased from 200% ESS to 400% ESS, the LRB performed stable frictional sliding action (Figure 5c). The horizontal force came from the sliding friction of the bearing and no longer grew, the horizontal force at this stage being defined as the sliding friction force F_df. The residual deformation of the bearing at this stage was approximately 96.1 mm and −125.7 mm, corresponding to shear deformation of 101.9 mm and 72.3 mm, as derived from Figure 6. The bearing shear deformation was confirmed to be equal to the approximate bearing height as well.

3.2. LRBs Response at the Concrete Interface

The seismic response of the LRBs at the concrete test pad was observed in the force-displacement curve shown in Figure 7. The response of the bearing meets the displacement requirements of the beam while controlling the horizontal seismic force requirements effectively, which is similar to the previous experimental study [16]. For the J1 specimen, under low vertical load (J1-4-C), the δ₁ was about 115–140 mm, while the F_sf. was approximately 245 kN. When the displacement was loaded to about 150 mm, the friction force experienced a sharp decline, which eventually remained at 150. From the hysteresis curve, it can be seen that there was a severe descent in the sliding response of the LRBs at the concrete pad, as the F_sf of the bearing includes the pure friction and the adhesion friction of the rubber to the interface [29], the value being generally greater than the F_df. The significant abrupt variation in the horizontal forces during sliding at the concrete interface was considered as an unanticipated mechanism, and the use of a bilinear model simulation would misestimate the seismic performance of the bridge.

3.3. LRBs Response at the Steel Interface

As shown in Figure 8, differing from the concrete interface, the force-displacement curve is flat and fuller during the loading process of the bearing at the steel interface, being the stable horizontal force. For the J2 specimen, the δ₁ under increasing vertical loads were 130 mm, 170 mm and 200 mm, corresponding to F_sf. of 320 kN, 400 kN and 470 kN, respectively, and F_df of 260 kN, 320 kN and 350 kN, respectively, with the difference between the breakaway and sliding frictional forces being less than 100 kN. In the J3-4-S-1 specimen, the difference between the breakaway and sliding friction was slim (35 kN approximately). The frictional response of the LRB at the steel interface was characterized by stability and continuity, which functioned as a more reliable fusing mechanism, making the seismic performance of the bridge more stable and predictable [27].

4. Analysis and Discussion

The sliding response and frictional properties of LRBs at steel and concrete test pads were compared and analyzed in the previous section. As unbonded LRBs are imposed in isolation system for small-to-medium-span bridges, mechanical features such as the coefficient of friction [15] and the horizontal shear stiffness K1 [20] will affect the bridge’s seismic performance. The key features in the force-displacement curve of the LRB shear test were counted and summarized in

Table 3. Statistics of LRB sliding response characteristics.

Test Specimen	δ1/mm	Fsf/kN	δ2/mm	Fdf/kN	K1/kN/m	μs	μd	γ
J1-4-C-1	114.90	231.90	140.00	154.40	2018.28	0.36	0.24	1.50
J1-4-C-2	116.90	241.70	137.80	163.70	2067.58	0.38	0.26	1.48
J1-4-C-3	137.20	249.90	193.30	145.30	1821.43	0.39	0.23	1.72
J1-6-C-1	155.40	279.90	173.60	111.50	1801.16	0.31	0.12	2.51
J1-8-C-1	159.60	323.90	182.90	100.90	2029.45	0.27	0.08	3.21
J1-10-C-1	226.80	523.38	240.66	334.09	2307.67	0.33	0.21	1.57
J1-4-S-1	133.00	250.10	183.70	204.30	1880.45	0.39	0.32	1.22
J1-6-S-1	196.14	305.17	276.10	233.60	1555.88	0.32	0.24	1.31
J1-8-S-1	221.90	329.90	284.60	235.60	1486.71	0.26	0.18	1.40
J2-6-C-1	197.10	499.90	221.30	183.50	2536.28	0.35	0.13	2.72
J2-6-C-2	101.49	347.80	113.87	173.20	3426.94	0.24	0.12	2.01
J2-8-C-1	192.10	502.20	215.30	256.80	2614.26	0.25	0.13	1.96
J2-4-S-1	129.70	319.60	209.81	259.96	2464.15	0.32	0.26	1.23
J2-6-S-1	167.50	406.50	212.90	324.10	2426.87	0.27	0.22	1.25
J2-8-S-1	202.80	474.80	265.00	351.62	2341.22	0.24	0.18	1.35
J3-4-C-1	168.80	451.00	189.60	246.80	2671.80	0.45	0.25	1.83
J3-6-C-1	170.56	361.39	178.10	182.20	2118.84	0.25	0.13	1.98
J3-8-C-1	159.20	474.30	203.10	284.10	2979.27	0.24	0.14	1.67
J3-10-C-1	249.40	581.03	321.50	333.10	2329.71	0.23	0.13	1.74
J3-4-S-1	156.80	301.20	190.50	282.70	1920.92	0.30	0.28	1.07
J3-6-S-1	242.16	447.30	314.60	316.60	1847.13	0.30	0.21	1.41
J3-8-S-1	278.10	468.10	316.70	336.90	1683.21	0.23	0.17	1.39
J3-8-S-2	229.70	465.50	288.70	356.10	2026.56	0.24	0.19	1.31

where δ₁ is displacement corresponds to the peak breakaway friction force, δ₂ is displacement corresponds to the stable sliding of the bearing, F_sf is the max breakaway friction, F_df is sliding friction force, γ is the ration of the breakaway coefficient of friction (μ_s) and sliding coefficient of friction (μ_d).

, including the δ₁, δ₂, F_sf, F_df, etc., to determine the factors affecting friction and establish the constitutive model of LRBs. The breakaway coefficient of friction (μ_s) and sliding coefficient of friction (μ_d) are calculated by Equations (1) and (2):

$${\mu }_s=\frac{F_{sf}}{\mathrm{P}}$$

(1)

$${\mu }_d=\frac{F_{df}}{\mathrm{P}}$$

(2)

where P is the vertical load.

4.1. Analysis of LRB Frictional Characteristics

Analysis of the sliding characteristics of LRBs at different interfaces is a precondition for precise constitutive model establishment. For the larger LRBs (J2 and J3), Table 3 demonstrates that the μ_s (breakaway coefficient of friction) of LRBs at the concrete surface is generally larger, due to the roughness of the concrete pad being greater than that of the steel. The μ_s of specimen J2-6-C-1 and specimen J2-6-S-1, for example, are 0.35 and 0.27, which makes the bearing more difficult to slide at the concrete interface, thus maintaining a high horizontal force. However, the μ_d (sliding coefficient of friction) of the LRBs after sliding of the steel interface is greater than that of the concrete, and the μ_d of the steel pad is 0.19 higher than that of the concrete at 0.14 with the same size and vertical load (specimen J3-8-C-1 and specimen J3-8-S-2). To further quantify the variation in friction during sliding, a friction ratio γ of the μ_s to the μ_d was introduced to represent the frictional properties of the LRBs at different interfaces. For the ratio, γ at the concrete interface is 1.5–3.2, while generally exceeding the steel interface of 1.07–1.41. For example, the ratio γ is 1.5 for specimen J1-4-C-1, while the corresponding value is 1.22 for specimen J1-4-S-1 in steel with the same size and vertical load. In the period of breakaway friction, the μ_s of the rubber and the rigid interface consists of an adhesive component and a deformation component. The concrete pad is rougher, and with vertical loading, invisible micro-cracks appeared on the concrete surface, increasing the component of adhesive friction [29,30]. When the bearing slides, a large amount of heat is generated between the bearing and the surface and the protective layer of rubber is melted. The melted rubber is more likely to adhere to the rough interface of the concrete, and the roughness change of the friction interface to reduce the μ_d. Therefore, the μ_s of the LRBs at the concrete surface is generally greater than that of the steel, and after the LRB sliding, the μ_d of the bearing at the steel surface is greater than that of the concrete. The variety of the friction coefficient of LRBs from breakaway friction to sliding friction at the concrete interface is intense and non-negligible, so it would be more reasonable to develop a three-fold mechanical constitutive model that takes into account the breakaway-sliding friction effect.

In order to carry out the reasonable seismic design of an LRB-supported bridge, the sliding friction coefficient is specified and assumed in MOTC, AASHTO and CALTRANS. The MOTC recommends a sliding coefficient of friction of 0.2 for rubber bearings and steel plates, and 0.25 for concrete [17]. AASHTO suggests that a design sliding coefficient of friction of 0.2 may be assumed between the elastomer and clean concrete surface or steel surface [31]. CALTRANS specifies a coefficient of sliding friction of 0.4 for rubber with a concrete pad, and 0.35 with a steel pad [32]. According to the results of the test (Figure 9), it can be seen that the μ_d between the bearing and the steel surface was in the range of 0.17–0.32, which is close to the recommended value of the MOTC and AASHTO. The frictional capacity of the bearings was overestimated in CALTRANS, while the μ_d at the concrete interface was in the range of 0.08–0.26, which was below the value specified in the MOTC and unstable. By comparison, the mean values of μ_d were closer to those recommended by AASHTO.

4.2. Effect of Vertical Loads on the Friction Coefficient

In addition to the change in the roughness of the contact surfaces, the coefficient of friction also varies with vertical loads. As Table 3 displays, the increase in vertical load increases the horizontal friction force but reduces the coefficient of friction. The inverse relationship between the vertical load and the coefficient of friction is considered the overall trend, which matches the phenomena of previous tests [7]. The effect of vertical load on the μ_s and μ_d of the LRBs is shown in Figure 10. The μ_s of J1 specimens at the concrete interface were 0.38, 0.31 and 0.27 for vertical loads of 4 MPa, 6 MPa and 8 MPa, respectively, with a decrease of approximately 22.5% and 14.8% with the vertical load. Correspondingly, the μ_d were 0.26, 0.12 and 0.08, respectively, with decreases of approximately 116.6% and 50% with the vertical load. The μ_d decreased much more with an increasing vertical load than the μ_s, so it is recommended that the effect of vertical loads on the friction coefficient should be taken into consideration in the mechanical constitutive model of LRBs to make it more accurate.

4.3. Mechanical Constitutive Model of the LRBs

The frictional characteristics and hysteresis curves of the LRBs at the concrete and steel interfaces were analyzed in the previous paragraph, finding that horizontal restraint capacity will decrease in the progression from breakaway friction to sliding. This feature has an important influence on the mechanical constitutive model of LRBs [15]. Based on the experimental results, a three-fold constitutive model of LRBs with consideration of the breakaway-sliding frictional characteristics was established (Figure 11).

The mechanical constitutive model of LRBs includes four independent variables: K₁, μ_s, δ₂, and μ_d. It is clear from the previous study that the bearings will experience pure shear deformation at small displacement, and therefore K₁ was employed to express the shear stiffness. For the breakaway-sliding friction characteristics of the bearing, μ_s and μ_d were imposed to determine the variation of the coefficient of friction during sliding, respectively. The δ₂ was deployed to confirm the displacement boundary for the two frictional behaviors. The expressions for the constitutive model of LRBs are as follows.

$$\left\{\begin{array}{cc}F=K_1\cdot \delta \hfill & \hfill \delta ⩽\frac{{\mu }_s{F}_N}{K_1}\\ F={\mu }_s{F}_N+\frac{\left({\mu }_d-{\mu }_s\right)F_N}{{\delta }_2-\frac{{\mu }_s{F}_N}{K_1}}\cdot \left(\delta -\frac{{\mu }_s{F}_N}{K_1}\right)\hfill & \hfill \frac{{\mu }_s{F}_N}{K_1}<\delta ⩽{\delta }_2\\ F={\mu }_d{F}_N\hfill & \hfill \delta >{\delta }_2\end{array}\right.$$

(3)

The correlation between the vertical load and the coefficient of friction was ascertained in the previous chapter, and the correlation between the bearing area and height and the K₁ of the bearing is clear, being proportional to the area and inversely proportional to the height [8]. Based on the PYTHON programming language, the relationship between the area (A), height (H) and vertical load (P) and K₁, μ_s, δ₂ and μ_d of the constitutive model were fitted by the linear regression method, which can show the magnitude of the correlation from the polynomial. The multidimensional arrays and matrix of experimental data was normalized during the regression process. The R² of the analysis model of LRBs at concrete interface was 0.752, and the R² of the analysis model at steel interface was 0.875. The LRB constitutive models at different interfaces were as follows.

Concrete interface:

$$K_1=0.741A-0.453H+0.067L+0.123$$

(4)

$${\mu }_s=-0.241A+0.125H-0.549L+0.674$$

(5)

$${\mu }_d=-0.511A+0.453H-0.504L+0.768$$

(6)

$${\delta }_2=-0.126A+0.256H-0.453L+0.168$$

(7)

Steel interface:

$$K_1=1.023A-0.811H-0.181L+0.292$$

(8)

$${\mu }_s=-0.313A+0.033H-0.563L+0.875$$

(9)

$${\mu }_d=-0.224A+0.068H-0.688L+0.877$$

(10)

$${\delta }_2=-0.318A+0.411H-0.666L+0.181$$

(11)

This indicates that, for concrete interfaces, vertical loads are significantly correlated with μ_s, δ₂, and μ_d, and the area and height are correlated with μ_d. For the steel interfaces, vertical loads are strongly correlated with μ_s, δ₂, μ_d. The design parameters for the constitutive model of LRBs of the common type were extracted from the regression models in

Table 4. LRB constitutive model at concrete surface.

A/m2	Type	H/mm	K1/kN/m	δ2/mm				μs				μd
				P1	P2	P3	P4	P1	P2	P3	P4	P1	P2	P3	P4
0.16	400 × 400	99	2033.81	128.76	181.82	227.27	265.22	0.35	0.33	0.31	0.29	0.20	0.19	0.18	0.17
0.2025	450 × 450	99	2491.42	129.48	182.52	227.72	265.15	0.34	0.32	0.30	0.28	0.19	0.18	0.17	0.16
		114	2282.73	139.94	197.13	245.77	285.93	0.34	0.32	0.30	0.28	0.19	0.18	0.17	0.16
0.25	500 × 500	110	2850.27	134.44	189.04	235.17	272.92	0.33	0.31	0.29	0.27	0.17	0.16	0.15	0.14
		130	2572.15	146.96	206.45	256.56	297.38	0.32	0.30	0.28	0.26	0.17	0.16	0.15	0.14

Note: P_i represents vertical load, P₁ = 4 MPa, P₂ = 6 MPa, P₃ = 8 MPa, P₄ = 10 MPa.

and

Table 5. LRB constitutive model at steel surface.

A/m2	Type	H/mm	K1/kN/m	δ2/mm			μs			μd
				P1	P2	P3	P1	P2	P3	P1	P2	P3
0.16	400 × 400	99	1699.28	184.94	244.75	283.06	0.36	0.32	0.28	0.30	0.25	0.19
0.2025	450 × 450	99	2134.23	173.60	227.96	260.98	0.34	0.30	0.26	0.29	0.24	0.18
		114	1777.31	205.71	270.11	309.20	0.34	0.30	0.26	0.29	0.24	0.18
0.25	500 × 500	110	2359.00	176.57	229.55	259.31	0.32	0.28	0.24	0.28	0.23	0.17
		130	1883.36	217.17	282.30	318.85	0.32	0.28	0.24	0.28	0.23	0.17

Note: Pi represents vertical load, P₁ = 4 MPa, P₂ = 6 MPa, P₃ = 8 MPa.

, which can be used as a reference for bridge seismic design.

5. Bridge Seismic Response at Different Friction Interfaces

5.1. Finite Element Model and Ground Motion

A typical, simply supported concrete bridge in the Wenchuan area of China was used to compare the effects of LRB models of different surfaces on the seismic response of a highway bridge, and the middle union (3 × 30 m) was selected for analysis. Figure 12 indicates the deck of the continuous simply supported bridge, with each span consisting of five 2 m-high T-shaped precast concrete girders with a total width of 10 m. The substructure consists of 10 m-high concrete double columns with a circular cross section of 1.5 m in diameter and reinforcement ratio of 1.4%. The piers were spaced at 7 m apart and the tops of the piers were consolidated to the cape beams. The seismic design of the bearing was applied with unbonded LRBs (GJZ400 × 400 × 99), which was placed directly in the middle of the concrete cape beam and T-shaped precast concrete beam. The bridge was designed in a continuous deck so that only shear and axial forces, but not bending moments, are transferred between the single span. The joints of end piers were loaded with half the mass and load of the main beam to simulate realistic stresses. In this paper, longitudinal seismic effects were considered, with the effect of soil-structure interactions being ignored, and the piers and foundations are in the fixed restraints form.

A 3D finite element model was developed in OpenSEES software and non-linear dynamic analysis (NDA) was carried out to simulate the seismic response of a simply supported girder bridge. The beam and bent cap were modeled in linear elastic beam element, which are considered to be damage-free under seismic load. The fiber section was taken to emulate the plastic hinge of the pier, the section properties of which are shown in Figure 13. The concrete fibers were modeled by Concrete02, which follows the Kent–Park model [33]. The steel fibers were simulated by Steel02 material, which follows the Menegotto–Pinto model [34]. During modeling, the definition of Rayleigh damping helped to account for the energy dissipation mechanism, assuming a damping ratio of 5% [35].

In the previous sections, the sliding characteristics of LRBs with different frictional surfaces have shown that the three-fold mechanical constitutive model is considered to be a reasonable model for accurately emulating the sliding characteristics of the LRBs. Two parallel non-linear spring elements were employed to simulate the behavior of LRBs under earthquake load in OpenSEES [15,36]. One of the spring elements was modelled by Steel01 material, which mainly simulates the frictional force requirements when LRBs are under sliding friction. The other spring elements were simulated with the multi-linear constitutive laws, which are mainly used to capture the variation in horizontal forces for the max breakaway friction to stable sliding friction.

The earthquake ground motions were obtained from the PEER database as longitudinal ground motion input, and details are given in

Table 6. Information of selected ground motion.

Earthquake	Year	Station	Magnitude	PGA-Horizontal (g)	Rjb (km)	Rrup (km)
“Northridge-01”	1994	“LA-Saturn St”	6.69	0.467	21.17	27.01
“Northridge-01”	1994	“LA-Centinela St”	6.69	0.448	20.36	28.3
“Northridge-01”	1994	“Beverly Hills-14145 Mulhol”	6.69	0.443	9.44	17.15
“Loma Prieta”	1989	“Gilroy Array #4”	6.93	0.418	13.81	14.34
“Loma Prieta”	1989	“Hollister-South & Pine”	6.93	0.369	27.67	27.93
“Superstition Hills-02”	1987	“El Centro Imp. Co. Cent”	6.54	0.357	18.2	18.2
“Whittier Narrows-01”	1987	“Tarzana-Cedar Hill”	5.99	0.472	38.24	41.22

. The results of the seismic response were averaged over seven seismic responses. In this paper, the peak ground acceleration (PGA) is used as an indicator of the intensity of measurement (IM), and the ground motions are amplitude-adjusted to 0.1–1.0 g, respectively, with 0.1 g as the variation.

5.2. Comparison of Seismic Response under Different LRB Models

Four LRB analysis models were proposed to investigate the effects of different friction surfaces on the seismic response of the bridge. The mechanical model being influenced by the vertical load had been verified in the previous study. The vertical load of the LRB was determined to be 4.0 MPa by calculating the static load of the bridge. A comparison of each bearing model is shown in Figure 14 and the mechanical parameters are shown in

Table 7. Mechanical parameters of models.

Mechanical Properties of Bearing	Model 1	Model 2	Model 3	Model 4
K1	2033.81 kN/m	1699.28 kN/m	2033.81 kN/m	1699.28 kN/m
δ2	128.76 mm	184.94 mm	/	/
μs	0.35	0.36	0.25	0.2
μd	0.20	0.30	0.25	0.2
u1	62.93 mm	112.98 mm	78.67 mm	75.32 mm

where $u_1=\frac{{\mu }_d{F}_N}{K_1}$.

.

Model 1: An LRB analysis model at the concrete interface based on experimental data was established, considering the effect of breakaway-sliding frictional variation during sliding.

Model 2: An LRB analysis model at the steel interface based on experimental data was established, considering the effect of breakaway-sliding frictional variation during sliding.

Model 3: An LRB analysis model at the concrete interface assuming a constant friction coefficient of 0.25 (MOTC suggested value) was established, without considering the effect of breakaway-sliding frictional variation during sliding.

Model 4: An LRB analysis model at the steel interface assuming a constant friction coefficient of 0.2 (AASHTO suggested value) was established, without considering the effect of breakaway-sliding frictional variation during sliding.

Seismic hazards, such as girders falling and plastic deformation of piers, seriously affect the regular transportation operation in small-to-medium span bridges. The displacement of the pier top at the unite terminal and the corresponding bearing deformation were selected as the key objects to bridge seismic demands. The seismic damage criteria of small-to-medium-span bridges was defined [18], which were classified into three levels of nondamaged, slight damage and serious damage based on ductility coefficients of pier displacement 1.3, 3.04 and bearing displacements: u₁, δ₂.

Figure 15 compares the variation of displacement demand of the pier top under ground motion of different LRB analysis models. The displacement demand of the pier top generally increases with the growth of the ground motion under different LRB analysis models, but these are considerably different under each model. When the PGA ≤ 0.2 g, this stage is usually considered to be the elastic design for seismic design of small-to-medium-span bridges in China [17]. The piers are in the nondamaged phase under all four LRB analysis models, but the displacement demand of the pier top is greater for the bearing at the concrete interface (Model 1 and Model 3) than at the steel interface (Model 2 and Model 4) due to differences in K1. Where the PGA is in the range of 0.2 g to 0.6 g, this phase includes or above the highest ground motion intensity considered for plastic seismic design of bridges in China. The pier top displacements for all models except Model 4 entered the slight damage phase, while the LRB model at the concrete interface was consistently greater than at the steel. When the PGA > 0.6 g, piers are the first to enter the severe damage stage with bearing models at concrete interfaces, while the demand of piers in the steel plate interfaces model is with the corresponding PGAs of 0.7 g (Model 2) and 0.9 g (Model 4).

Figure 16 shows the variation in bearing displacement demand with growth in ground motion under four LRB models, which is different to the law of pier response. When PGA < 0.4 g, the displacement response of the bearing increases with PGA under all models, and while PGA = 0.2 g, all bearing displacements are almost equal and enter the slight damage phase. When the ground motion increases, the peak bearing displacement of the steel interface bearings (Model 2 and Model 4) are greater than those of the concrete interface (Model 1 and Model 3), and the bearings of Model 3 and Model 4 slip first (δ > 78.67 mm). When PGA > 0.5 g, the peak bearing displacement of the steel interface model is much larger than that of the concrete, and the bearing displacement of the Model 4 model exceeds that of the other three models, mainly due to the fact that AASHTO suggests that a smaller coefficient of friction and is more prone to sliding, while the remaining three models show little change in peak bearing displacement with increasing ground motion due to the plastic hinges of the pie, which reduces its stiffness so as to impair the bearing displacement requirements.

From the bridge seismic response analysis, it can be concluded that the application of LRBs at the steel friction interface is an effective way of reducing the pier seismic response and thus protecting the bridge. For bridges supported by LRBs at steel friction interfaces (Model 2 and Model 4), failure to take into account the breakaway-sliding friction effect of the bearing will seriously underestimate the pier top displacement requirements (the difference value in pier displacement ductility coefficients between the two models is about 0.5), which makes the pier seismic design questionable. For the LRBs at concrete interfaces (Model 1 and Model 3), if the breakaway-sliding friction effect of the bearings is ignored, the displacement requirements of the piers will be overestimated and the seismic design will be more conservative, while the displacement requirements of the bearings are underestimated (the difference value is about 0.02~0.03 m). From the perspective of ensuring the safety of the pier and that the bearing sliding force is continuous, it is recommended that the steel pad be adopted as the friction interface of the LRB-supported bridge, but attention needs to be paid to the bearing displacement requirements under strong earthquakes.

6. Conclusions

• By the observation of the quasi-static cycle test of the LRBs, it was found that the friction coefficient of the bearing will experience a non-negligible decrease in the coefficient of friction with the growth of the displacement loading. The breakaway coefficient of friction of the bearing at the concrete surface is generally greater than that at the steel surface, but when the bearing slides, the sliding coefficient of friction of the bearing at the steel surface is greater than that at the concrete surface.
• The results of the force-displacement hysteresis curve of the LRBs were investigated. A three-fold mechanical constitutive model of the LRBs considering breakaway-sliding friction is established, which is determined by four parameters: the shear stiffness (K₁), the breakaway coefficient of friction (μ_s), the displacement corresponding to stable sliding (δ₂) and the sliding coefficient of friction (μ_d).
• The sliding performance of the LRBs at the steel friction interface has stable characteristics, which can effectively reduce the seismic demand of the substructure and ensure the seismic safety of the bridge. Therefore, it is recommended to choose steel friction pads for LRB-supported bridges, but attention needs to be paid to the bearing displacement requirements under strong earthquakes.
• For bridges supported by LRBs with steel friction interfaces, failure to take into account the breakaway-sliding friction effect of the bearing will undervalue the pier top displacement requirements. The piers were designed to be vulnerable. For bridges supported by LRBs at concrete friction interfaces, if the breakaway-sliding friction effect of the bearings is not taken into account, it will overestimate the pier displacement requirements, as the displacement requirements of the bearings were underestimated.

The three-fold mechanical constitutive model of LRBs considering breakaway-sliding friction proposed in this paper is based on experimental data. The analysis of the bridge seismic response is based on a typical small-to-medium-girder bridge with multiple girders. Further consideration should be given to other factors affecting the analysis model of the LRBs and additional testing should be carried out to obtain more comprehensive data and consider the effects of bridge types.