Numerical analysis on longitudinal seismic responses of high-speed railway bridges isolated by friction pendulum bearings

2.1. Bridge and track structures

Fig. 1(a) shows 10-span simply supported bridges on a high-speed railway. Each pre-stressed box girder, with the same span of 32 m, is made of C50 concrete. Each girder end is supported by two bearings, being friction pendulum bearings (FPB) or spherical steel bearings. The structural seismic responses with those two types of bearings will be compared with each other in this paper.

The bearings are supported by piers made of C50 concrete and HRB335 steel bars. All of the piers have the same length of 13.5 m and the same rectangular section sizes of 5.434 m×2.5 m. The number of piers from left to right is 0# to 10# as shown in Fig. 1, and the 5# pier is especially emphasized in the study of this paper.

Under each pier, there are 12 circular piles with a diameter of 1.0 m. The piles are made of C30 concrete and Q235 steel bars.

The CRTS II ballastless track plate comprises base plates, track plates, rails and connecting members. The connecting members include sliding layers, shear cogging, CA layers, shear reinforcement, fasteners and lateral blocks as shown in Fig. 1(c). The base plate is made of C30 concrete, and has a width of 2.95 m and a thickness of 0.19 m. The track plate is made of C55 concrete, and has a width of 2.55 m and a thickness of 0.2 m. The rail adopts the type of CHN60. The sliding layer is set between the bridge deck and base plate, and is comprised of dual textiles and one membrane. The shear cogging is set in the sliding layer near the fixed bearings to connect the girder and base plate. The CA layer is located between the base plate and track plate. The shear reinforcement is arranged in the CA layer near the girder ends to connect the track plate and base plate to bear the deformation caused by a corner. The fasteners adopt the type of WJ-8C, and are set at intervals of 0.65 m.

Fig. 1Schematic diagram of high-speed railway bridge

a) Elevation layout of high-speed railway bridge /cm

1-1 Section

c) Schematic sketch of track structure

Fig. 2Spatial finite element model of bridge

2.2. Finite element model

The spatial finite element model of bridges as shown in Fig. 2 is built by using the OpenSEES program. The main girder, base plate, track plate and rail are all simulated by an elastic beam element in OpenSEES, and their section parameters are listed in Table 1. Each unit length for them is 0.65 m, which is the same as the interval between fasteners. Spherical steel bearings, sliding layers, CA layers and fasteners are simulated using a zero length nonlinear connection element as shown in Table 2 and Fig. 3. Friction pendulum bearing (FPB) is simulated using a special Single Friction Pendulum Bearing element as shown in Fig. 4. The pier and pile foundation are simulated by a three-dimensional elastoplastic fiber element. Considering the constraint effect of stirrups, those sections are divided into three parts, including cover concrete, core concrete and reinforcement as shown in Fig. 5. Because the fiber element accounts for material nonlinearity, geometry nonlinearity and bond slip effect of anchoring steel in joint, it is better than other simplified plastic hinge model.

Fig. 3Constitutive relation of zero length connection element

Fig. 4Friction pendulum bearing (FPB)

a) Structure of FPB

b) Isolation principle of FPB

c) Hysteresis curve of FPB

Fig. 5Models of pier and pile foundation

a) Pier model

b) Pile model

c) Stress-strain diagram of concrete

d) Stress-strain diagram of reinforcement

2.3. Parameters of friction pendulum bearing

Fig. 4 shows a type of friction pendulum bearing (FPB) studied in this paper. It can be further divided into three types, including the fixed type, unilateral sliding type and bilateral sliding type. Under common loads, the shear keys on them limit the relative motion between the upper and lower support plates of the fixed bearings and unilateral sliding bearings. However, a large earthquake will cut off those shear keys to cause those bearings to slide like the bilateral sliding bearings. This sliding motion will isolate seismic forces, and the friction force will dissipate earthquake energy. Simultaneously, the girder rises during earthquakes. However, it is almost restored to an original position due to the girder gravity after earthquakes.

The FPB isolation principle is shown in Fig. 4(b). When the girder relatively moves on the piers, the bearings between them vibrate like a pendulum with a certain period. The period can be changed to an expected value by adjusting the curvature radius $R$ of pendulum.

The relationship between force and displacement of FPB can be obtained from Fig. 4(b). The horizontal displacement $u$ and vertical displacement $v$ are:

The horizontal force $F$ and vertical force $W$, i.e. the gravity of superstructure, are:

In the equations, $f$ is the friction force, and $S$ is the centripetal force. A new equation can be derived from Eqs. (3) and (4):

The bearing stiffness and vibration period can be approximately obtained as follows:

As $\theta$ is usually small, $\mathrm{c}\mathrm{o}\mathrm{s}\theta \approx \text{1}$, $T=2\pi \sqrt{R/g}$, $F=W\mathrm{s}\mathrm{i}\mathrm{n}\theta +f=Wu/R+\mu W$, and the bearing hysteretic curve is shown in Fig. 4(c).

In Fig. 4(c), $D_i$ is the relatively horizontal displacement of bearing, and $\mu$ is the friction coefficient. The effective stiffness according to any $D_i$ is shown in Eq. (8):

FPB is simulated using a special Single Friction Pendulum Bearing element as shown in Fig. 3. The main parameters of bearings are shown in Table 3. The structural seismic responses, respectively adopting FPBs with the friction coefficients of 0.02, 0.03, 0.04, 0.05, 0.06, 0.07 and 0.08, will be compared with those using the fixed spherical steel bearings and the sliding spherical steel bearings with the friction coefficient of 0.02.

Fig. 6Ground motion input

4.1. Sliding layer

Fig. 7 shows the seismic relative displacement of sliding layer, located between the girder and base plate, of one span supported by the 4# and 5# piers.

Fig. 7Seismic relative displacement of sliding layer

a) Under El

b) Under taft

c) Under landers

The displacement values of sliding layer distribute among 2.0~3.0 mm near the 4# pier, and gradually decrease to 0 mm near the 5# pier, where the shear cogging is set.

At any position, the displacement values of sliding layer in the cases with FPB are much less than those in the common case with the spherical steel bearing. For example, the former is only 76 %-88 % of the latter under the El ground motion in Fig. 7(a). Moreover, the displacement values of sliding layer decrease when the friction coefficient decreases on FPB.

The displacement values of sliding layer are slightly changed under different ground motions by comparing Fig. 7(a), (b) and (c), however, the changing trends are almost the same with each other under different ground motions.

4.2. CA layer

Fig. 8 shows the seismic relative displacement of CA layer, located between the base plate and track plate, of one span supported by the 4# and 5# piers.

The displacement values of CA layer are close to 0mm at the beam ends due to the limitations of shear reinforcement, and in the midspan due to the deformation compatibility. In detail, the displacement values of CA layer at the right end of girder are always larger than those of the left one. The reason is that the shear cogging is set in the sliding layer at the right end of girder and transforms more earthquake energy to the CA layer over there.

The displacement values of CA layer near the girder ends are the largest ones due to the seismic corner responses between beams, and gradually decrease to 0mm in the midspan.

At any position, the displacement values of CA layer in the cases with FPB are much less than those in the common case with the spherical steel bearing. For example, the former is only 58 %-78 % of the latter under the El ground motion in Fig. 8(a). Furthermore, the displacement values of CA layer decrease when the friction coefficient decreases on FPB. However, the decreasing trend is feeble since all of the displacement values of CA layer are very small, being less than 0.3 mm.

The displacement values of CA layer are slightly changed under different ground motions by comparing Fig. 8(a), (b) and (c). However, the changing trends are almost the same with other under different ground motions.

Fig. 8Seismic relative displacement of CA layer

a) Under El

b) Under taft

c) Under landers

4.3. Fastener

Fig. 9 shows the seismic relative displacement of fasteners, located between the track plate and rails, of one span supported by the 4# and 5# piers. All the displacement values of fasteners are almost the same as those of CA layer as shown in Fig. 8, and the difference between them is less than 0.01 mm.

The displacement values of fasteners are close to 0 mm at the beam ends and in the midspan. In detail, the displacement values of fasteners at the right end of girder are always larger than those of the left one. However, the difference between them is less than that of CA layer due to the isolation protection of the damaged CA layer there.

The displacement values of fasteners near the girder ends are the largest ones, and gradually decrease to 0mm in the midspan.

At any position, the displacement values of fasteners in the cases with FPB are much less than those in the common case with the spherical steel bearings. For example, the former is only 50 %-70 % of the latter under the El ground motion in Figure 9(a). The reduced friction coefficients on FPB further decrease the displacement values of fasteners.

In addition, different ground motions change the displacement values of fasteners by comparing Fig. 9(a), (b) and (c).

However, the influence of those parameters above on the displacement values of fasteners is very insignificant. The fasteners and rails are well protected by FPB, sliding layer and CA layer during earthquakes.

Fig. 9Seismic relative displacement of fastener

a) Under El

b) Under taft

c) Under landers

4.4. Bearings

Fig. 10 shows the seismic relative displacement of bearings, including the common spherical steel bearings and FPB with different friction coefficients.

The displacement values of sliding spherical steel bearings are between 1-2 mm, while those of the fixed ones are close to 0 mm. However, the former displacements are still less than any displacements of FPB.

In terms of FPB, the displacement values increase more significantly as the friction coefficients decrease. When the friction coefficient is 0.08 on FPB, the displacement values of FPB are only between 1.5-3 mm. And when the friction coefficient on FPB is reduced to 0.02, the displacement values of FPB reach the range of 4-5 mm.

Fig. 10Seismic relative displacement of bearings

a) Under El

b) Under taft

c) Under landers

4.5. Piers

Fig. 11 shows the seismic strain of cover concrete of the 5# pier under the common spherical steel bearings and FPB with different friction coefficients.

When using the common spherical steel bearings, the seismic strains are very large (0.55-0.7 mm/m) for the cover concrete at the bottom of the 5# pier, and decrease along with the pier height. Those strains are much larger than those of pier using FPB.

In terms of the seismic strain of cover concrete of the 5# pier under FPB, the strain values decrease as the friction coefficients on FPB decrease. When the friction coefficient on FPB is 0.08, the strain values of cover concrete at the pier bottom are among 0.05-0.1 mm/m. When the friction coefficient on FPB is reduced to 0.02, those strain values almost decrease to 0 mm/m.

Fig. 11Seismic strain of cover concrete of the 5# pier

a) Under El

b) Under taft

c) Under landers

Fig. 12Seismic strain of core concrete of the 5# pier

a) Under El

b) Under taft

c) Under landers

Fig. 12 shows the seismic strain of core concrete near cover concrete of the 5# pier under the common spherical steel bearings and FPB with different friction coefficients. The whole rules are similar to those of the seismic strain of cover concrete as shown in Fig. 11, however, the former is slightly less than the latter.

When using the common spherical steel bearings, the seismic strains are very large (0.5-0.65 mm/m) for the core concrete at the bottom of the 5# pier, and decrease along with the pier height. Those strains are much larger than those of the pier using FPB.

Fig. 13Seismic strain of longitudinal steel bar of the 5# pier

a) Under El

b) Under taft

c) Under landers

In terms of the seismic strain of core concrete of the 5# pier under FPB, the strain values decrease as the friction coefficients on FPB decrease. When the friction coefficient on FPB is 0.08, the strain values of core concrete at the pier bottom are among 0.035-0.075 mm/m. And when the friction coefficient on FPB is reduced to 0.02, those strain values almost decrease to 0 mm/m.

Fig. 13 shows the seismic strains of longitudinal steel bar in the 5# pier under the common spherical steel bearings and FPB with different friction coefficients. The whole rules are similar to those of the seismic strain of concrete as shown in Figs. 11 and 12. Meanwhile, the seismic strains of longitudinal steel bar are between those of cover concrete and core concrete, and are not further discussed here due to space limitations.