Seismic response characteristics and whiplash effect mechanism of continuous rigid-frame bridges subjected to near-fault ground motions

Continuous rigid-frame bridges (CRFBs) have been widely constructed in high seismicity areas in western China. To investigate the seismic response characteristics and whiplash effect mechanism of CRFBs under near-fault ground motions, a long-span CRFB with high piers is selected as a prototype bridge, and a nonlinear finite element model is developed based on OpenSees. Three groups of near-fault ground motions, namely having forward-directivity pulses, fling-step pulses, and non-pulse, are selected as seismic inputs. These records are intercepted using the significant duration index and scaled to 0.2, 0.4, and 0.64 g, which respectively represent frequent, basic, and rare ground motions. The seismic response characteristics of CRFBs are analyzed and the effects of bearing constraints, ground motion components, and vertical excitation on the seismic responses are discussed. The numerical results show that the longitudinal vibration, transverse whiplash effect, and vertical uplift behavior of the main girder are the main deformation characteristics of CRFBs. Compared with non-pulse earthquakes, the structural displacements, lateral drift angles, bearing deformations, internal forces, and pounding effects are all significantly increased under pulse-like earthquakes. Spatial torsional effects in the mid-span girder and main piers and pounding effects between the girder ends and transition pier top are identified. Perfectly-free and fixed bearings in the transverse direction are not recommended for the seismic design of CRFBs. An optimal stiffness ratio in friction pendulum systems that can minimize the bending degree of the main girder may exist. Furthermore, the uplift of the side-span girder under pure longitudinal excitation is closely related to the pier-girder consolidation form and the rotation of the main piers. The main piers may be tensioned under strong vertical excitation, which is an issue worthy of attention.


Introduction
Earthquakes occur frequently globally, and near-fault seismic activity, which seriously threatens human life, is characterized by a wide distribution, high intensity, high frequency, and shallow focus (Somerville and Graves 1993;Liang et al. 2020). Furthermore, large numbers of buildings and bridges are destroyed by earthquakes. Post-earthquake investigations and research show that the rupture forward-directivity effect and fling-step effect of faults are significant factors leading to structural damage (Kawashima et al. 2009;Kim et al. 2011;Xiang et al. 2019;Todorov and Billah 2021). Near-fault ground motions are characterized by strong velocity and displacement pulses and are obviously different from far-field ground motions in terms of the amplitude, frequency spectrum, and duration (Bray and Rodriguez-Marek 2004;Mavroeidis and Papageorgiou 2010;Lin et al. 2020a).
A continuous rigid-frame bridge (CRFB), which adopts the unique form of pier-girder consolidation, is composed of a T-shaped rigid frame and a continuous girder. To conform to complex mountainous terrain environments, CRFBs are constructed as irregular bridges with long spans and high piers, and the pier heights may differ by tens or even hundreds of meters (Huang and Li 2022). In the past three decades, at least 350 CRFBs have been constructed in China due to their good structural behavior and strong spanning capability ). However, as-built and in-construction CRFBs located in western China are very likely to suffer from near-fault earthquakes. For example, the Miaoziping Bridge suffered a strong near-fault ground motion during the 2008 Wenchuan earthquake (Zeng et al. 2022). The overall rehabilitation process took about one year and cost about 30,000,000 RMB (4,500,000 USD) (Guan et al. 2017).
Extensive investigations on the seismic performance of CRFBs have previously been conducted. Most experimental studies have focused on the main piers of CRFBs. The seismic behaviors of thin-walled high piers were revealed via shaking table tests (Chen et al. 2018), quasi-static cyclic tests (Sun et al. 2019), and model-updating hybrid tests (Liu et al. 2020). In a few studies, shaking table tests of a two-span RC bridge model (Saiidi et al. 2014) and a 1:10 scale three-span steel-concrete composite CRFB were performed to explore their seismic characteristics and damage modes under near-fault and far-field ground motions (Lin et al. 2020a, b, c). Extensive numerical simulations have also been carried out. The seismic behavior of CRFBs with corrugated steel webs was analyzed using the endurance time method , and the effect of spatially variable ground motions on the seismic responses of five-span CRFBs was discussed . Furthermore, the seismic performance of a horizontally curved and skewed threespan CRFB under three-dimensional near-fault earthquakes was investigated (Wilson et al. 2015). The seismic fragility and damage of as-built and retrofitted old and newly designed multi-span CRFBs (Jung and Andrawes 2018; Abbasi and Moustafa 2019) and irregular bridges with non-circular tall piers considering ground motion directionality (Shan et al. 2020) were assessed. The seismic fragility of offshore (Liang et al. 2020), sea-crossing (Zhang et al. 2021a), and deep-water CRFBs (Zhang et al. 2021b) also were evaluated. The seismic responses of deep-water CRFBs have attracted attention since the occurrence of an underwater crack in the main pier of the Miaoziping Bridge (Wei et al. 2022). Qiao et al. (2022) conducted an in-depth parametric study of the topographic effect of the seismic response and the running safety of a train-bridge coupled system on a CRFB. Park et al. (2004), Ucak et al. (2014), and Yang et al. (2021) carried out seismic response studies of ordinary and isolated bridges crossing strike-slip fault rupture zones by utilizing finite element models of the 10-span Bolu Viaduct.
Based on the literature review, it is evident that (a) The seismic response characteristics of CRFBs under near-fault earthquakes have not been examined in detail to support actual earthquake damage, and (b) The effects of vertical seismic excitation on the vertical responses of CRFBs have not been investigated. It should be noted that the Miaoziping Bridge installed with pot rubber bearings (PRBs) at the ends of the main girder is only a special case, and whether its earthquake damage form is universal has not been verified. In addition, the peak ground acceleration (PGA) of the vertical component recorded by the station at the Zipingpu Reservoir Dam during the Wenchuan earthquake was found to reach 2.06 g (where g is the acceleration of gravity). Such a large vertical excitation may lead to very large seismic responses. Therefore, it is necessary to further investigate the actual seismic response characteristics and vertical excitation effects of CRFBs under nearfault ground motions. In this study, a long-span CRFB with high piers located in western China is selected as the prototype bridge, and its corresponding nonlinear finite element model is developed based on OpenSees. Furthermore, three groups of near-fault ground motions with forward-directivity pulses, fling-step pulses, and non-pulse are selected as seismic inputs. The seismic response characteristics of the CRFB with PRBs under nearfault ground motions are analyzed in detail. Finally, the effects of the bearing constraints, ground motion components, and vertical excitation on the seismic responses of CRFBs are discussed.

Description of prototype bridge
The main bridge of the Nan-pan River Bridge, which is located on the Jiu-xiang to Shi-lin highway, Kunming, Yunnan Province, China, is adopted as the case study for the near-fault seismic characteristics and whiplash effect mechanism of CRFBs. This main bridge is a (120 + 220 + 120)-m-long pre-stressed concrete CRFB, as shown in Fig. 1. The main girder is a single-cell variable-depth box girder with a 1.6-degree parabola at the bottom edge of the girder. The main piers 1# and 2# are double-limb thin-wall hollow piers with heights of 88 and 85 m, respectively. The transition piers are single-limb hollow piers with a height of 38 m. The main girder and all piers are made of C50 and C40 concrete, respectively. During the construction process, the main girder of each T-shaped rigid frame is divided into 32 pairs of girder segments (Fig. 1a). A total of 452 pre-stressed tendons are adopted in the main girder, and their distribution information is given in Fig. 1b. The detailed dimensions and reinforcement distribution on the key cross-sections are shown in Figs. 1c-e. Four twoway movable PRBs are set on the top of transition piers 0 and 3#, the maximum deformation capacity of which is 0.2 m. In addition, expansion joints with a width of 0.4 m are set at the girder ends. The seismic design intensity of the bridge site is 9°, the site type is Class-II, and the basic PGA is 0.4 g.

Analytical model of the bridge
The nonlinear finite element model of the CRFB is simulated in OpenSees, as shown in Fig. 2. The top and bottom elements of the main pier and the bottom elements of the transition pier are modeled using displacement-based nonlinear beam-column elements with fiber sections. For the fiber sections, concrete and reinforcing steel are modeled by adopting the uniaxial materials Concrete 02 and Steel 02, respectively, accounting for their isotropic hardening in both tension and compression. The main girder and other elements of the piers are assumed to be elastic and modeled using elastic beam-column elements. The PRBs are simulated with a bilinear ideal elastic-plastic spring element, which has the compression-only behavior in the vertical direction (z) and free-slide behavior in the horizontal direction (x, y) of the bridge. The friction force of each bearing is the product of the vertical dead load and the friction coefficient. To simulate the longitudinal pounding effects between the main girder ends and the transition pier top, the Hertz-Damp model is employed as an impact model (Jia et al. 2018;Miari et al. 2021). The main girder ends can impact the transition pier top when the longitudinal relative displacement exceeds the 0.4-m expansion joint width (the initial pounding gap). The constitutive relationships of the fiber section, bearing, and pounding models are shown in Fig. 2, and the corresponding main mechanical parameters are summarized in Table 1. The values of the pounding gap stiffnesses (initial stiffness, secondary stiffness, and effective stiffness) are calculated with reference to previous studies (Deng et al. 2015;Jia et al. 2018;Li and Xu 2023a, b).
Because the pile foundations enter the rock layer to a certain depth and the cover layer thickness exceeds 10 m, the pile-soil-structure interaction is not considered, and the bottoms of all piers are assumed to be consolidated on the ground. The mass of the main girder and piers is assigned to nodes using the lumped mass method, and the initial internal force states of the bridge are considered. Rayleigh damping is adopted, and the damping ratio is taken as 5%. In addition, the dynamic characteristic analysis of the bridge is conducted to verify the reliability and accuracy of the model (Shi et al. 2022;Li and Xu 2023a). The first three natural vibration periods of the bridge are respectively 3.97, 3.72, and 2.60 s, and the corresponding modal shapes are the first symmetric lateral bending of the girder (modal 1), the longitudinal bending of the piers and the drift of the girder (modal 2), and the first antisymmetric lateral bending of the girder (modal 3).

Selected ground motion records
The most distinct characteristics of near-fault ground motions originate from the rupture forward-directivity and fling-step effects. Given their different characteristics, it is desirable to treat the seismic responses originating from these effects separately (Bray and Rodriguez-Marek 2004;Mavroeidis and Papageorgiou 2010). In this study, the  data of near-fault ground motion records with forward-directivity and fling-step effects are selected from the Northridge earthquake (1994, M W = 6.7) and Chi-Chi earthquake (1999, M W = 7.6). Both earthquakes are attributable to dip-slip faults and have different characteristic parameters. Velocity pulses with the fling-step effect were recorded in the Chi-Chi earthquake, but not in the Northridge earthquake; only velocity pulses with the forward-directivity effect were recorded in the latter (Yang et al. 2010). The near-fault records of the Chi-Chi earthquake were chosen from the database processed by Wang et al. (2002), which sufficiently reserved the fling-step effect. The near-fault records of the Northridge earthquake were obtained from the strong motion database of the Pacific Earthquake Engineering Research Center. Twenty-one records are divided into three groups to consider the effects of the near-fault characteristics on the seismic responses, including rupture forward-directivity pulse (GM1), fling-step pulse (GM2), and non-pulse (GM3) responses, as listed in Table 2. Because the records with fling-step pulses in the existing database are quite limited, only seven fling-step seismic records are selected for GM2. Each record has three components consisting of two horizontal components and one vertical component.
These seismic records are scaled to represent frequent, basic, and rare ground motions. The seismic design intensities and basic design accelerations correspond to those stipulated in the Chinese Code for Specifications for Seismic Design of Highway Bridges (JTG/T 2231-01-2020). In this study, because the seismic design intensity is 9° and the site type is Class-II, the PGA of the design basic ground motion is 0.4 g. Based on the basic PGA, the PGAs of frequent and rare ground motions are calculated according to the proportional relationship provided in the Code for Seismic Ground Motion Parameters Zonation Map of China (GB18306-2015). The PGAs of the frequent and rare ground motions are taken as 0.2 and 0.64 g according to 0.5 and 1.6 times the basic PGA, respectively. Three components of each scaled record are input along the three directions of the bridge based on uniform excitation . In detail, the horizontal acceleration component with a larger PGA is scaled to the target peak value (0.2, 0.4, and 0.64 g), which is input in the x-direction of the bridge. The scaled smaller horizontal acceleration component and vertical component are input in y-and z-directions, respectively. It should be noted that the scaling of ground motions only amplifies or reduces the acceleration amplitude and does not change the other characteristics of near-fault ground motions, such as the duration, frequency spectrum, and velocity pulse effect (Lin et al. 2020a). Furthermore, the 5-95% significant duration index (D s5-95% ) related to the Arias Intensity (AI) is adopted to consider the effective duration effect of all ground motions (Shi et al. 2022). In the analysis, the mean values of the peak seismic responses under seven ground motion records in each group are primarily discussed.

Structural displacement
Figures 3, 4, and 5 compare the three-directional displacements of the main girder under near-fault ground motions with different pulse effects. It is evident that the larger the PGA, the greater the three-directional displacements. The three-directional displacements of the No. As shown in Fig. 3, the vibration form of the main girder along the longitudinal direction is translational motion. Due to the large axial stiffness, the longitudinal displacement of the main girder at any position along the girder length is almost the same. Under rare ground motions (0.64 g) with forward-directivity pulses, fling-step pulses, and non-pulse, the longitudinal displacements of the main girder are approximately 1.1, 1.7, and 0.35 m, respectively.
As shown in Fig. 4, the vibration form of the main girder along the transverse direction can be decomposed into translational motion and bending motion. At the first stage, the main girder generates translational displacement. On this basis, bending deformation obviously occurs, which is regarded as the second stage. In this stage, the bending deformation of the two side spans is defined as the transverse whiplash effect, which is much greater than that of the middle span. This is because the girder ends are only constrained by the small friction force provided by the PRBs on the transition pier top, while the girder roots are rigidly constrained by the main piers (namely pier-girder consolidation); thus, the transverse bending stiffness of the side spans is much smaller than that of the middle span. The transverse displacements of both of the side spans are asymmetric, which may be caused by the time-varying amplitudes of the ground motions. The transverse displacement of the girder ends is larger than the longitudinal displacement due to the significant transverse whiplash effect. When subjected to rare ground motions with forward-directivity pulses, fling-step pulses, and non-pulse, the transverse displacements of the girder end reach 1.8, 2.5, and 0.4 m, respectively. For example, the girder-end displacement of 2.5 m is the sum of the 1.25 m translational displacement of the girder end (equal to the displacement of the main pier top) and the 1.25 m relative bending displacement of the side-span girder. In other words, the large girder-end displacement is the result of the bending of both the main pier and the side-span girder.
As shown in Fig. 5, the vibration form of the main girder along the vertical direction is bending motion. Because the main girder roots are rigidly constrained by the main pier top and the PRBs at the girder ends provide vertical compression-only behavior, the girder ends of the side spans can be uplifted and impact the transition pier top. The uplift and pounding effect of the side spans is defined as the vertical whiplash effect. Under rare ground motions with forward-directivity pulses, fling-step pulses, and non-pulse, the vertical uplift heights of the girder end are approximately 0.12, 0.14, and 0 m, respectively. Figure 6 compares the horizontal bending displacements of the main pier (taking pier 1-2# as an example) under near-fault ground motions with different pulse effects. The bending deformation form of the main pier in the transverse direction is equivalent to that of a common single-column pier, and a bending plastic hinge occurs at only the pier bottom. In the longitudinal direction, the bending deformation of the main pier presents an "S" shape due to the pier-girder consolidation, and bending plastic hinges can occur at both the pier top and bottom. Hence, the projection of the spatial deformation of the main pier on the horizontal plane is a curve. The reverse bending point is located at about half of the pier height. Due to the pier-girder consolidation, the horizontal displacements of the main pier top are equal to those of the girder roots. This means that the horizontal displacement of the main pier top can represent the horizontal translation displacement of the main girder.
In summation, the longitudinal large-amplitude vibration, transverse whiplash effect, and vertical uplift behavior of the main girder and the S-shaped bending of the main pier are the typical deformation characteristics of CRFBs. These characteristics are consistent with the actual earthquake responses of the Miaoziping Bridge during the 2008 Wenchuan earthquake (Chen 2012;Tong et al. 2021;Zeng et al. 2022). Compared with the nonpulse near-fault ground motions, the structural displacements of the CRFB significantly increase under the pulse-like near-fault ground motions. Fling-step pulses have a larger influence on the displacement responses than rupture forward-directivity pulses due to the stronger velocity and displacement pulse effects of the former. On the other hand, the CRFB may enter the medium-damage stage under rare earthquakes with fling-step pulses, according to the damage index of the main piers provided by Li and Xu (2023a) based on the displacement ductility ratio. Moreover, some seismic records selected for this analysis, such as TCU052 and TCU068, are well-known for having high-velocity and fling-step pulses, under which the CRFB may encounter severe damage. The earthquake damage of CRFBs is not discussed in detail.   1, 2, 3, 4) , where D by,i is the transverse bending displacement of the main girder and L i is the length of the main girder cantilever (see Fig. 7). The lateral drift angles ( y ) of the main pier in the transverse direction can be expressed by the equation As shown in Fig. 8, higher the PGA, the greater the lateral drift angles of the bridge. The lateral drift angles under fling-step pulses are the largest, followed by those under forward-directivity pulses and non-pulse ground motions, respectively. Under rare ground motions with fling-step pulses, the lateral drift angles of the main pier in the longitudinal and transverse directions are about 1.7 and 1.3%, respectively. The longitudinal drift angles of the main pier are larger than the transverse drift angles. The lateral drift angles at the roots (R1 and R4) of the side-span girder are much higher than those at the roots (R2 and

Bearing deformation
Figures 9, 10, and 11 present the horizontal hysteresis curves, motional traces, and relative deformation of the PRBs at the main girder ends (left end (LE) and right end (RE)), respectively. The bearing deformation is taken as the horizontal relative displacement (Δ x , Δ y ) between the girder end and the transition pier top. The hysteresis curves in the x-direction are asymmetrical, whereas those in the y-direction are symmetrical. This is because there is a longitudinal pounding effect between the girder ends and transition pier top when the relative displacement reaches the initial impact gap of 0.4 m. The longitudinal relative displacement along the pounding direction is within 0.4 m, which is less than that along the non-pounding direction due to the unidirectional pounding constraint. Moreover, the horizontal motional traces of the PRBs substantially exceed the deformation capacity of 0.2 m.   Figure 11 shows that the ground motion type also has an obvious influence on the bearing deformation. The relative displacements under fling-step pulses are larger than those under forward-directivity pulses. The higher the PGA, the greater the relative displacements. It should be noted that the relative displacements under non-pulse ground motions at 0.2 g are less than the 0.2-m deformation capacity, which means that the PRBs have not been destroyed. In contrast, the deformation values beyond 0.2 m under pulse-like ground motions at 0.2 g indicate that the PRBs have been completely destroyed. When the PGA are higher 0.4 g, the values far exceed the deformation limit of the bearing, regardless of the ground motion type. Under rare ground motions with forward-directivity pulses, flingstep pulses, and non-pulse, the longitudinal relative displacements are nearly 1.0, 1.5, and 0.3 m, respectively, and the transverse relative displacements reach 1.8, 2.5, and 0.4 m, respectively. Hence, the PRBs cannot meet the large horizontal displacement demand of CRFBs during near-fault earthquakes. For instance, the four PRBs on the transition piers of the Miaoziping Bridge exhibited a very large shear deformation of up to 1.03 m and completely failed (Kawashima et al. 2009;Tong et al. 2021;Zeng et al. 2022). In addition, the transverse relative displacements are significantly larger than the longitudinal relative displacements because there is no lateral constraint at the girder ends and the transverse whiplash effect of the side-span girder is very strong. In detail, the torsional moment T x about the x-axis at the roots (R1 and R4) of the side-span girder is zero, while that at the roots (R2 and R3) of the mid-span girder is relatively high, reaching 45 MN·m under fling-step pulses with a PGA of 0.64 g (Fig. 12a). Thus, the torsional effect occurs in the mid-span girder but not in the side-span girder. The torsional effect of CRFBs is a significant issue in bridge engineering. In simply supported or continuous girder bridges, the girder can only translate horizontally without relative torsion behavior. The torsional effect of the mid-span girder in CRFBs may be caused by the unequal transverse movement of the two main piers when subjected to ground motions, especially in CRFBs with a significant difference in pier height. Due to the pier-girder consolidation, the two ends of the mid-span girder are fixed by the main pier top, which can be regarded as a fixed girder. When the displacements at the tops of piers 1# and 2# are not equal at a certain instance of ground motion, the relative torsion of the cross-sections of the mid-span girder will occur. A small transverse displacement difference between the two main piers may cause torsion. However, the torque values of the main girder are very small as compared to its bending moment because there is not much difference in the height of the two main piers (88 and 85 m, respectively). The side-span girder, for which the constraining force of the PRBs on the girder ends is very small, can be regarded as a cantilever girder with no torsion. Tong et al. (2021) also demonstrated that the main girder may experience a certain torque around the longitudinal axis of the bridge, but the torsional effect of the whole structure was not deeply analyzed. The bending moments M y about the y-axis (Fig. 12b) and M z about the z-axis (Fig. 12c) of the main girder are caused by the bending movement of the main girder in the vertical and transverse directions, respectively. Due to the strong transverse whiplash effect of the side spans, the bending moment M z at the roots (R1 and R4) of the side-span girder is larger than that at the roots (R2 and R3) of the mid-span girder. Figure 13a shows that the bending moment M x about the x-axis at the pier bottom is much larger than that at the pier top, and the latter is close to 0. This is because the bending plastic hinge of the main pier along the transverse direction only occurs at the pier bottom. The bending moment M y about the y-axis at the pier top is very close to that at the pier bottom because the bending plastic hinges along the longitudinal direction can occur at both the pier top and bottom (Fig. 13b). The bending moment M x at the pier bottom is about twice M y under the same ground motions. The torsional moment T z about the z-axis at the pier top is equal to that at the pier bottom (Fig. 13c). In fact, there is only one torque value along the pier-height direction, which is consistent with the torsion of a linear bar with a uniform cross-section. It should be mentioned that the stronger spatial torsion of CRFBs may result in the cracking damage of the box girder and main piers (Chen 2012;Tong et al. 2021).

Pounding effect
Under spatial earthquake excitation, the main girder exhibits a large-amplitude longitudinal vibration and a vertical uplift effect, and the girder ends may strongly impact the transition pier top. Figures 14 and 15 illustrate the pounding force histories and hysteresis curves under an earthquake in the x-and y-directions, respectively. The longitudinal pounding effect occurs when the relative displacement of the bearing reaches the initial impact gap   (Fig. 14). The vertical pounding effect will occur when the uplift of the girder end occurs, and the corresponding pounding force is much larger than the static load of 2200 kN (Fig. 15). Figure 16 compares the dynamic pounding forces and pounding times (i.e., the number of pounding) in x-and z-directions under near-fault earthquakes. The results show that the longitudinal pounding effect occurs very easily under pulse-like ground motions, even if the PGA is low (0.2 g) (Fig. 16a). Under non-pulse ground motions, the longitudinal pounding effect may occur when the PGA is 0.4 g. Under rare ground motions with forward-directivity pulses, fling-step pulses, and non-pulse, the longitudinal pounding forces reach 25, 40, and 4 MN, respectively, and the corresponding pounding times are about 11, 17, and 3, respectively. As presented in Fig. 16b, the vertical pounding effect is also obvious under pulse-like ground motions. When the PGA is 0.2 g, there is no vertical pounding effect. The vertical pounding effect occurs under pulse-like ground motions with a PGA of 0.4 g. Under rare ground motions with forward-directivity pulses, fling-step pulses, and non-pulse, the vertical pounding forces respectively reach 16, 15, and 7 MN, and the corresponding pounding times are about 3, 2.5, and 0.6, respectively. The values of the vertical pounding forces are in line with those reported by Zeng et al. (2022). It should be noted that a huge vertical impact load may result in the complete damage of the bearing and padstone. The damage of the PRBs in the Miaoziping Bridge is a typical case (Kawashima et al. 2009;Zeng et al. 2022). On the whole, the longitudinal pounding effects under flingstep pulses are larger than those under forward-directivity pulses, and those under nonpulse ground motions are very small. On the contrary, the vertical pounding effects under forward-directivity pulses are larger than those under fling-step pulses. This means that the vertical excitation effect of forward-directivity pulses is relatively stronger than that of other ground motions under the same acceleration peak. Moreover, the longitudinal pounding effect is significantly greater than the vertical pounding effect. In short, the seismic responses of CRFBs under pulse-like excitation is significantly large. The seismic damage control on CRFBs is an issue worthy of investigation. Using advanced seismic devices may be an ideal control strategy (Xu et al. 2017;Zhang and Xu 2022;Li and Xu 2023b).

Influence of bearing constraints on the transverse displacement of the main girder
It can be seen from Sect. 5.1 that the transverse whiplash effect of the side spans is significantly obvious and intense in a CRFB with two-way movable PRBs. To study the influence of bearing constraints on the transverse displacement of the main girder, the bearing model in the transverse direction is set as an elastic-perfectly-plastic model, a bilinear elastic-plastic model, and a linear elastic model in turn to represent the PRB, friction pendulum bearing and transversely-fixed bearing, respectively. The bearing model in the longitudinal direction is kept as an elastic perfectly-plastic model. Figure 17a shows the constitutive models of the bearing in the transverse direction. Herein, the initial elastic tangent E 0 is taken as 2.2 × 10 4 kN/m, and b is the tangent ratio between the post-yield tangent and the initial elastic tangent, the values of which are different. Figure 17b compares the transverse displacements of the main girder with different bearing constraints under forward-directivity pulses with a PGA of 0.4 g. When the bearing is a two-way free-sliding bearing (b = 0), the transverse bending displacements of the side-span girder can reach the maximum (case 1). When the bearing is a transversely-fixed bearing (b = 1), the displacements of the girder ends are limited to nearly zero, while the bending displacements of the mid-span girder are obviously large (case 2). Meanwhile, great torsion will occur in the main piers due to the large horizontal rotation of the main girder. Hence, both cases are disadvantageous to the structural safety of CRFBs and should be avoided during seismic design. With the increase of the tangent ratio from 0 to 1, the transverse displacement state gradually varies from case 1 to case 2. At the tangent ratio of around 0.04, the transverse deformation of the main girder tends to be a straight line. Thus, there is an optimal tangent ratio in the bearing that can minimize the bending degree of the main girder. However, it is difficult to determine this ratio because it varies with different ground motion parameters and structural systems. Moreover, the transverse displacements at the main pier top (at the pier-girder consolidation) basically remain constant (around 0.6 m) under all different bearing constraints. Therefore, the bearing constraints can significantly change the transverse bending deformation of the side-span and mid-span girders but have little influence on the transverse displacements of the main piers.

Vertical excitation and girder end uplift effect
To investigate the influence of vertical earthquake excitation on the seismic responses of the CRFB, the PGA of the horizontal acceleration components is scaled to 0.4 g, and the ratios V/H of the vertical PGA to the horizontal PGA are taken as 0.5, 1, 2, 3, 4, and 5, respectively. Figure 19 compares the vertical displacements of the main girder under the near-fault ground motions with different V/H values. The vertical whiplash effect of the main girder is obvious under the strong vertical excitation. The whole uplift process can be divided into three stages, namely initial compression, girder end uplifting, and fall-back pounding. At the V/H ratio of 1 (a vertical PGA of 0.4 g), the girder ends of the side spans and the midspan girder begin to uplift under pulse-like ground motions. The uplift height gradually increases with the increase of the V/H ratio. At the V/H ratio of 5 (a vertical PGA of 2.0 g), the uplift height of the girder ends can reach 0.5 m under pulse-like ground motions. The reason for the uplift of the girder ends is that the main girder roots are constrained by the main piers, while the PRBs have no anti-uplift capacity. Under non-pulse ground motions, the uplift of the girder ends is difficult. Figure 20 compares the vertical normalized pounding forces and pounding times at the ends of the main girder under near-fault ground motions with different V/H ratios. The normalized pounding force is defined as the ratio of the dynamic pounding force to the static compression force (4.4 MN). As the V/H ratio increases, the normalized pounding forces and pounding times gradually increase. When the V/H ratio is 0.5 or 1, the vertical pounding effects are very small and negligible. When the V/H ratio reaches 2, the pounding effects begin to increase significantly with the uplifting of the girder end. At the same V/H ratio, the vertical pounding forces and pounding times are the largest under forwarddirectivity pulses, the second-largest under fling-step pulses, and the smallest under nonpulse ground motions. This indicates that the vertical excitation effect of forward-directivity pulses is stronger than that of fling-step pulses and non-pulse ground motions under the same acceleration amplitude, thus causing the greater uplift of the girder end (Fig. 19) and the stronger vertical pounding effect (Fig. 20). Figure 21 exhibits the axial reaction forces at the pier bottom under near-fault ground motions with different V/H ratios. The normalized compression force at the pier bottom is defined as the ratio of the dynamic compression force to the static compression force. The static compression forces at the bottoms of the main piers 1-1#, 1-2#, 2-1#, 2-2# and the transition piers 0 and 3#, are 108. 49, 112.46, 111.04, 106.80, 16.99, and 16.99 MN, respectively. As shown in Fig. 21a, the axial compression forces at the bottoms of all piers increase linearly with the increase of the V/H ratio. Under the same V/H ratio, the compression forces under pulse-like ground motions are larger than those under nonpulse ground motions. Figure 21b presents the axial tension forces occurring in the piers. At the V/H ratio of 2, i.e., the vertical PGA is 0.8 g, small axial tension forces begin to appear in parts of the main piers under pulse-like ground motions. When the V/H ratio reaches 5, i.e., the vertical PGA is 2.0 g, relatively larger axial tension forces occur in all main piers. The transition piers are always in a compression state. In theory, therefore, the main piers of CRFBs may be axially tensioned under strong three-directional ground motions with very large vertical excitation. Despite poor experimental tests, the findings of a few studies support this theory. For instance, Stathas et al. (2017) conducted a hybrid simulation of bridge pier uplifting via substructure pseudo-dynamic testing. Two types of the tests were conducted, the first of which considered a two-span deck connected monolithically to a central pier and supported at each end on the abutment with a pair of only-compression elastomeric bearings; the second one was a test of a 1:2 scale model of a bridge pier. In the first type of test, at a PGA of 0.15 g, the footing was uplifted, and the bearings developed tension, but the pier stayed elastic. In the second type of test, at a PGA of 0.4 g, the uplifting of the footing did not prevent plastic hinging at the pier base, but significantly reduced the pier damage.
It should be noted that only when the relative vertical vibrations of the main girder and the main piers of CRFBs are large enough can the vertical inertia force completely overcome the dead weight of the whole bridge, which can cause the main piers to have a tensile stress effect. However, it is difficult for the tensile stress effect of the piers to occur in bridge engineering; usually, during the three-directional ground excitation, bending-shear and bending damage may occur in the main piers before being tensioned. Hence, the axial tension effect of the main piers of CRFBs in rare earthquakes with strong vertical excitation is a practical issue worthy of attention.

Conclusions
This study numerically investigates the seismic response characteristics of CRFBs under near-fault ground motions with forward-directivity pulses, fling-step pulses, and nonpulse. The results indicate that the longitudinal vibration and transverse and vertical whiplash effects of the main girder, as well as the S-shaped bending of the main pier, are typical deformation characteristics of CRFBs under three-directional seismic excitation and are consistent with actual earthquake responses. With the increase of the PGA, the structural displacements, lateral drift angles, bearing deformations, internal forces, and pounding effects of the CRFB all significantly increase. These responses are the largest under fling-step pulses, the second-largest under forward-directivity pulses, and the smallest under non-pulse ground motions; however, the vertical pounding effect is an exception. There are spatial torsional effects in the mid-span girder and the main pier, which are very uncommon in bridges. The former may be caused by the asynchronous and unequal transverse motion of two main piers, while the latter is the result of the horizontal rotation of the main girder. Moreover, the pounding effects between the girder ends and transition pier top will occur in the longitudinal and vertical directions when the relative displacement reaches the initial impact gap and during the occurrence of girder end uplifting occurs, respectively. The longitudinal pounding effect is much greater than the vertical pounding effect.
A comparative study revealed that the PRBs cannot meet the large displacement demand of CRFBs in near-fault earthquakes. Different bearing constraints can significantly change the transverse bending deformation of the main girder but have little effect on that of the main piers. The use of perfectly free and fixed bearings in the transverse direction is not recommended for the seismic design of CRFBs. There is an optimal stiffness ratio in friction pendulum systems that can minimize the bending degree of the main girder (around 0.04 in this study), which is determined by the ground motion parameters and structural systems. The uplift of the side-span girder under pure longitudinal excitation is an interesting phenomenon that is closely related to the piergirder consolidation form and the rotation of the main piers. Furthermore, the girder end uplift and vertical pounding effects under three-directional forward-directivity pulses are greater than those under fling-step pulses and non-pulse ground motions at the same PGA. Under pulse-like ground motions, the girder ends begin to uplift when the vertical PGA is 0.4 g and reach 0.5 m under 2.0 g. The main piers may be tensioned under strong vertical excitation, which is a practical issue. In this study, axial tension forces are found to occur in the main piers when the vertical PGA reaches 0.8 g.
In summary, the seismic characteristics and whiplash effect mechanism of CRFBs in near-fault earthquakes are presented in detail. In particular, the spatial torsional effect of the main girder, the vertical uplift and pounding effects of the girder ends, and the axial tension effect of the main piers under strong pulse-like excitations, which have attracted little attention in previous studies, are presented. However, there are some limitations of the analysis that require discussion. For instance, several findings on the relative contributions of forward directivity and fling step to various response quantities may depend on the selected ground motion dataset (which includes records with unique characteristics from the 1999 Chi-Chi earthquake). More near-fault earthquake records must be input into the dynamic models of CRFBs for performance analysis. Although the purely numerical results are consistent with the response characteristics during the actual earthquake, there is a lack of a comparison of the numerical and experimental investigations. In addition, the analysis results did not indicate the detailed damage condition of CRFBs during the shaking of pulse-like ground motions. These issues are worthy of in-depth investigation in future work.
Funding The authors gratefully acknowledge the partial support of this research by the National Natural Science Foundation of China under Grant No. 52125804 and 52078036.