Seismic behavior of precast segmental column bridges under near-fault forward-directivity ground motions

Xin Lifeng, Li Xiaozhen, Fu Peiyao and Mu Di

1. School of Mechanics, Civil Engineering & Architecture, Northwestern Polytechnical University, Xi′an 710072, China

2. Department of Bridge Engineering, Southwest Jiaotong University, Chengdu 610031, China

3. Department of Civil and Environmental Engineering, Hanyang University, Seoul 04763, Korea

Abstract: Precast segmental column bridges exhibit various construction advantages in comparison to traditional monolithic column bridges. However, the lack of cognitions on seismic behaviors has seriously restricted their applications and developments. In this paper, comprehensive investigations are conducted to analyze the dynamic characteristics of precast segmental column bridges under near-fault, forward-directivity ground motions. First, the finite-element models of two comparable bridges with precast segmental columns and monolithic columns are constructed by using OpenSees software, and the nonlinearities of the bridges are considered. Next, three different earthquake loadings are meticulously set up to handle engineering problems, namely recorded near- and far-field ground motions, parameterized pulses, and pulse and residual components extracted from real records. Finally, based on the models and earthquake sets, extensive explorations are carried out. The results show that near-fault forward-directivity ground motions are more threatening than far-field ones; precast segmental column bridges may suffer more pounding impacts than monolithic bridges; the “narrow band” effect caused by near-fault, forward-directivity ground motions may occur in bridges with shorter periods than pulse periods; and pulse and residual components play different roles in seismic responses.

Keywords: precast segmental column bridge; monolithic column bridge; seismic behavior; near-fault forward-directivity ground motion; pounding effect

1 Introduction

As a new technique for accelerated bridge construction, precast segmental column (PSC) bridges have gradually been attempted, accepted, and implemented in engineering practices. More than 30 PSC bridges have been built since 1961, and the number is increasingly growing. As demonstrated by these engineering facts, PSC bridges show remarkable construction superiority over the classic cast-in-place monolithic bridges as regards fabrication control, environmental impact, structural quality, etc. (Zhanget al., 2016). However, the promotion and application of PSC bridges has been severely impeded in the earthquake regions due to the serious lack of understanding about their seismic behaviors. Also, no mature specifications and codes can currently be referred to across the world. Against these backdrops, research on seismic analysis of PSC bridges is urgently sought.

To characterize the seismic effects on PSC bridges, the initial work centers on a comprehensive analysis of individual PSCs. For decades, abundant studies of the seismic performance of PSCs have been done from aspects of theory and experiment, e.g., the representative and classic work by Hewes and Priestley (2002), Ouet al.(2007, 2010), Billingtonet al.(2004), Kimet al.(2010), ElGawadyet al.(2011), etc. As pointed out by these researchers, PSCs have two unique properties when compared to conventional monolithic columns (MC): high self-centering and poor energy-dissipation capabilities. The high self-centering capacity, namely, small residual drift, fundamentally results from restoring forces exerted by prestressing steel tendons. Such properties are favorable for post-earthquake repair work. Unfortunately, poor energy-dissipation capacity, which is caused by minor plastic deformations for the concrete and prestressing tendons, will lead to large and sustained reciprocating oscillations. It therefore has a negative effect on reducing seismic damage. Yamashita and Sanders (2009) noted that severe damage may occur at the base of the first segment when an earthquake takes place. In seeking various remedies, massive investigations have been conducted with an effort to improve the energydissipation capacity of PCSs to make them adapt to moderate and strong earthquakes (Ouet al., 2010; Wanget al., 2008; Motarefet al., 2010; Roh and Reinhorn, 2010; Chou and Chen, 2006). The most commonly used techniques include applying metallic yielding components at the base of column, e.g., joint-connecting mild steel bars, steel jackets, high-performance steel bars, shape memory allow bars, exterior yielding braces, etc. Summarily, seismic explorations on individual PSCs are gradually approaching to the maturity.

Apart from the above achievements, a few studies have been conducted to analyze the seismic effects of PSCs in entire bridges. Sideriset al.(2014) employed several shake table tests on a large scale to evaluate the seismic performance of PSC bridge systems with hybrid sliding-rocking members, including slip- and rockingdominant joints. Zhang (2014) skillfully applied steel fiber-reinforced, self-consolidating concrete to unbonded PSC bridges in moderate and high seismic regions, the feasibility of which was numerically and experimentally validated. Most recently, Zhaoet al.(2017) conducted pioneering work on comparing the differences between PSC and MC bridges under the effects of three earthquake loadings, and meanwhile, the influences of the pounding, frequency ratio and gap size on seismic performances are completely clarified; later, these same researchers conducted more realistic seismic analyses by considering ground-motion spatial variations (Zhaoet al., 2018). Presently, the seismic behavior of PSC bridges has not been sufficiently understood, especially regarding near-fault earthquakes.

The site at the vicinity of the seismic source may experience near-fault effects. One of the most important characteristics for near-fault earthquakes is forwarddirectivity (FD) effects. In the strike-slip and dip-slip earthquakes, FD ground motions usually occur in a direction perpendicular to the fault plane, when the rupture propagates towards the site, with a velocity almost equal to the shear-wave velocity (Somerville, 2003). As a result, FD ground motions contain many different intrinsic features due to far-field motions, such as high-density energy and large velocity pulses at the beginning of the records (Abrahamson and Somerville, 1996; Bray and Rodriguez-Marek, 2004). Accordingly, this uniqueness will impose significant demands on elastic and inelastic structures, especially for long-period or friable structures (Mavroeidiset al., 2004; Liaoet al., 2004; Sandhya and Masato, 2020; Xuet al., 2007; Sehhatiet al., 2011; Mortezaei and Ronagh, 2013; Liet al., 2016; Parket al., 2004; Phanet al., 2007; Tonget al., 2007). In these studies, Parket al.(2004) concluded that FD pulses increased seismic demand in energy dissipating units for the Bolu Viaduct. Phanet al.(2007) carried out an experimental dynamic test on bridge tests and found that the residual deformation of the pier under the influence of near-fault FD ground motions is significantly larger than that from the impact of original earthquakes. Unfortunately, PSC bridges demonstrate a poor energy-dissipation capability and small residual displacement, as previously stated. Apparently, this necessitates investigations into the seismic behavior of PSC bridges that are subjected to near-fault FD ground motions.

In summary, this paper pays particular attention to the effects of near-fault FD ground motions on PSC bridges, as the extended studies of Zhaoet al.(2017, 2018). The paper is organized as follows: in Section 2, bridge structures with PSCs and MCs are briefly presented, including geometric configurations and modeling method; in Section 3, three types of earthquake loadings are set; in Section 4, some representative calculations are illustrated in detail to investigate the effects of nearfault FD ground motions on the two bridges.

2 Bridge description

In this paper, the five-span, two-frame highway bridge presented by Zhaoet al.(2017) is adopted as the bridge prototype. Then, two different bridges are elaborately devised and modeled using OpenSees software to facilitate the comparison, namely, between the PSC bridge and the MC bridge. As shown in Fig. 1, the two bridges are basically the same, except for the bridge piers. For reader convenience, some information about the geometric configuration, modeling method and modal shapes will be described in the following.

The bridge superstructure with an arrangement of (20+30+30+30+20) m is manufactured by a singlecell, pre-stressed concrete girder with a unified cross section, as shown in Fig. 2. The beam is modeled by using “elastic beam column” elements. The pre-stressed effect is neglected because of its minor influence on seismic responses.

The abutments on the two sides consist of the pile foundation in the vertical direction and the passive soil embankment in the longitudinal direction. The pile foundation is simulated by use of the bilinear symmetrical model, as shown in Fig. 3(a). The ultimate strength and yielding displacement are 4 MN and 25 mm, respectively. The passive soil embankment is simulated by use of the elastic-plastic spring model, with an initial stiffness of 500 MN/m and an ultimate strength of 10 MN, as shown in Fig. 3(b). These nonlinear models are all realized by employing the “zero length” element.

The pounding phenomenon between adjacent substructures is worth considering since the instantaneous and tremendous impact force caused by the pounding may produce extensive deformation, abutment tilting, concrete crushing, etc. A linear spring model presented by Raheem (2009) is employed herein to characterize the dynamic behavior by means of the “zero length” element. The displacement-force relations are drawn in Fig. 3(c). The initial interval is 10 cm, which corresponds to the initial expansion joint spacing. The spring stiffness is 6.5 GN/m, which is calculated according to the mathematical expression offered by Raheem (2009). For brevity, the complete representation is not presented herein.

With respect to the PSC and MC piers, the similar dimensions, reinforcement ratios, and constraint conditions are controlled to guarantee their comparability. The height of both the PSC and MC piers is 10 m. In the PSC pier, 9 segments are designed from the bottom to the top, namely, S1 to S9. The nine individual segments are connected by six pre-stressed tendons, each with seven 15 mm-diameter, seven-wire strands (7T D15). In each segment, the longitudinal mild steel bars that each has a diameter of 22 mm are used to position the transverse reinforcement. They are not extended across the segment joints. Eight energy-dissipation (ED) bars, each with a diameter of 36 mm, are extended from S1 to S5. Four D36 ED bars are extended from S1 to S3. As for the MC pier, the arrangement is basically the same as the PSC pier, except: (1) No pre-stressed tendons are placed; (2) The concrete is poured as a whole; (3) The D22 longitudinal mild steel bars are extended along with the entire height of the column. The details of the segmental and monolithic columns are displayed in Fig. 4. Conventional fiber-based simulations may display a poor performance in capturing the hysteretic characteristics of the PCS pier due to the currently unclear sliding and rocking mechanism between segments, as well as the transient separation between the segments and their re-contact (Donget al., 2017; Shresthaet al., 2017). As an alternative, the plastic hinge model can skip the details of the PSC components, thereby directly reflecting the global dynamic behaviors of the PSC pier. The drawback is that the quasi-static experimental data are necessary. Based on the hysteretic experiments by Wanget al.(2008), the PSC pier in this paper is simulated in a plastic-hinge manner by combing the rigid beam element and the “zero length” element with the “Pinching4” material, as shown in Fig. 1. The material parameters are assigned from the experiment. The modeling parameters of the “Pinching4” material are presented in the Appendix. In addition, the hysteretic comparison between the model and the experiment is also shown in Fig. A2 in the Appendix. It can be seen that the modeling is appropriate due to good agreement. As for the MC pier, each one is constructed by using the “nonlinear beam column” element, with a fiberbased cross section in which the concrete is modeled by utilizing uniaxial “Concrete 06” material, as taken from Mander and Priestley′s constitutive model, and the “‘steel 02’ material with the Giuffre-Menegotto-Pinto constitutive model is applied to the reinforcing steel bar. For either the PSC or MC bridges, the girder and the pier are linked by the pin connections, represented by the “Equal DOF” command. All piers are assumed to stand on rigid foundations.

Fig. 1 Finite element models of bridge structures with abutments: (a) PSC bridge; (b) MC bridge

Fig. 4 Details of the segmental and monolithic columns: (a) elevation view; (b) cross section of the PSC; (c) cross section of the MC (mm)

Modal damping with a ratio of 5% was applied for the entire bridge structure (Zhaoet al., 2017, 2018). Next, a modal analysis is performed to obtain the fundamental vibration characteristics of the entire bridges. It is found that the fundamental periods of one bridge frame are 1.075 and 1.323 s, respectively, for the MC and PSC bridges. The longitudinal movement of the bridge girder provides the fundamental vibration mode for both types of bridges. Also, the seismic excitations will input to the structures in the longitudinal direction.

For interested readers, more details about the modeling process and experimental validation can be consulted in the studies of Zhaoet al.(2017, 2018).

3 Input ground motions

Regarding the FD effects on PSC and MC bridges, three crucial problems faced by bridge designers are: (1) What are the differences between near- and far-fault earthquakes that influence the seismic responses of the two bridges? (2) How do the pulse details affect the dynamic behaviors of bridge systems? and (3) What roles do the pulse and residual components from the actual records play in the change of dynamic indices?

Three different earthquake loadings have been designed for the aforementioned targets. The first one is near- and far-fault seismic records, which are directly downloaded from the Pacific Earthquake Engineering Research Center (PEER), NGA database (Center, 2013). The second is idealized pulses, which are artificially generated by using an excellent wavelet expression. The third is ground motion sets including the original seismic records, the pulse, and the residual components. The pulse and the residual components are extracted from seismic records. Detailed information on the selecting, processing, and generating methods will be introduced in the following section.

3.1 Set 1: Recorded near-fault and far-field ground motions

With an effort to prevent earthquake mechanisms and site conditions from interfering with observations and comparisons, the selected seismic records in this paper are strictly limited to the same earthquake event and the same site class (Xinet al., 2019). Next, ten near-fault FD and far-field ground motions from the 1979 Imperial Valley earthquake (strike slip fault) are collected and compiled into a sub-database. Additionally, all the records are rotated from various directions to the fault-normal and fault-parallel orientations to avoid the possible influences from propagation directions on the natural characteristics. Only fault-normal components are taken as excitation sources. In this manner, the nearfault, FD ground motions and far-field ground motions are totally comparable. The pertinent properties of the ground motion sets are listed in Tables 1 and 2, including peak ground acceleration (PGA), peak ground velocity (PGV), peak ground displacement (PGD), station, site class, hypo-central distance (Rjb) and pulse period (Tp). The elastic response spectra of two no-scaling record sets are plotted in Fig. 5.

Fig. 5 Response spectra for no-scaling near-fault and far-field motions. (a) near-fault; (b) far-field

Table 1 Recorded near-fault FD ground motions

Table 2 Recorded far-field ground motions

3.2 Set 2: Pulse model

In recent decades, there have been a substantial number of simplified pulse models that have attempted to capture the impulsive characters in FD ground motions, including sine and cosine function (Makris, 1997), square wave (Alavi and Krawinkler, 2004), wavelet form (Mavroeidis and Papageorgiou, 2003), and some other composite formulations (Fu and Menun, 2004; Menun and Fu, 2002; Kalkan and Kunnath, 2006; Moustafa and Takewaki, 2010). The wavelet form proposed by Mavroeidis and Papageorgiou (2003) (M&P′s model) is capable of comprehensively reflecting the pivotal parameters, such as pulse period, peak pulse amplitude and oscillatory characters. Meanwhile, this model can derive a closed-form solution for a single degree of freedom structure. More importantly, the model shows great superiority over other representations in regard to time-domain velocities that approximate the shape of many velocity pulses. As shown in Fig. 6, the pulse velocity derived from the wavelet form coincides well with the pulse component directly extracted through Baker′s decomposing procedure (Baker, 2007), which has been widely recognized as a good pulse form in earthquake engineering. For the reasons above, M&P′s model is adopted in this paper and given by:

Fig. 6 Comparison between the pulse from M&P′s model (Mavroeidis and Papageorgiou, 2003) and the pulse component extracted through Baker′s procedure (Baker, 2007) (Record RSN161)

Fig. 7 Decomposition of Record RSN170 (a) velocity time history; (b) response spectra

in whichAcontrols the amplitude of the signal;fpis the frequency of the prevailing frequency of the signal,pp1/fT= ;vis the phase of the harmonic;γis the parameter for defining the oscillatory character; and0tspecifies the epoch of the envelope′s peak.γvaries from a value larger than 1 to a maximum of value 3. Parametervchanges from 0 to 2π.

3.3 Set 3: Pulse and residual components

With the traveling propagation of seismic waves, the energy for each frequency band will be rapidly consumed and attenuated by soils or rocks, and this, accordingly, brings about the remarkable distinctions of wave patterns of near- and far-fault ground motions. That is, the FD ground motions are more abundant in both low- and high-frequency components than is the case for far-field components.

In this subsection, the pulse and residual components are decomposed from the records listed in Section 3.1. The pulses are obtained by matching the velocity time history using the M&P′s model, as shown in Section 3.2. The residual components are extracted as the difference between the original records and the pulses. As an illustration, Fig. 7 displays the two components for Record RSN170 and their elastic response spectra. It can be seen that there are two overlapping segments in the response spectra: one appears at the long-period band, which is mainly caused by pulse components; the other occurs at the short-period band, which is dominated by residual components.

4 Analysis and discussion

Aiming at the three problems mentioned at the outset of Section 3, this section consists of three parts, that is:

➢ Section 4.1: Comparative analysis: Compare the differences of system responses for PSC and MC bridges under the influence of near- and far-fault earthquakes.

➢ Section 4.2: Parametric analysis: Determine the most disadvantageous parameters in the pulse model.

➢ Section 4.3: Component analysis: Clarify the effects of pulse and residual components on system responses, especially the pounding effect.

4.1 Comparative analysis

In viewing Fig. 5 and Tables 1–2, it can be seen that the PGA and spectral amplitudes in near-fault FD ground motions are much higher than those that occur in far-field earthquakes. To characterize the seismic responses at the same level, the incremental dynamic analysis method is used herein. All records are varied with the PGA from 0.1 g to 1.0 g. The crucial and representative results are presented in Figs. 8–10 for the PSC and MC bridges, including deck displacement, the pounding force and pounding times in the middle gap.

Table A1 Parameters for the Pinching4 material

As for both PSC and MC bridges, it can be seen from Fig. 8 and Fig. 9 that near-fault FD ground motions may lead to nearly twice larger deck displacement and pounding force than is the case for far-field ones, even with the same values of PGA. This implies that the potential threat of near-fault FD ground motions lies in whole waveform characteristics at the beginning of records, not solely in the PGA. Moreover, PGA cannot be a good intensity measure for predicting the structural responses and damage status in performance-based designs due to the outrageous discreteness. Furthermore, one can observe from Fig. 10 that more pounding times are produced due to near-fault FD motions with low PGA than far-field motions with the same PGA. With an increase in PGA, the differences of pounding times by near-fault FD ground motions and far-field ones are gradually smaller. The phenomena are caused by the initial pulses in near-fault FD motions. The pulse in nearfault FD ground motions makes girder displacement large enough that the girders will collide with each other even at a low PGA level.

The same structural layouts of PSC and MC bridges lead to analogous laws regarding seismic responses. However, for different dynamic indices, the influence degrees are highly different. Observe, for instance, that the maximum deck displacements for both bridges are basically the same (as shown in Fig. 8), the maximum pounding forces for the PSC bridge are slightly larger than those for the MC bridge (see Fig. 9), and more pounding times for PSC bridge are excited than are those for the MC bridge (listed in Fig. 10). As a result, for the PSC bridge the beam end should receive special attention.

Fig. 8 The maximum deck displacements of two bridges excited by two earthquakes: (a) PSC bridge; (b) MC bridge

Fig. 9 The maximum pounding forces of two bridges excited by two earthquakes: (a) PSC bridge; (b) MC bridge

Another important index is energy dissipation capacity, which can be represented by the hysteretic loop area. Figure 11 depicts the energy dissipated by one column for two bridges excited by two earthquakes. Apparently, in the entire bridge that experiences pounding effects, MC consumes more energy than PSC, especially under the influence of near-fault FD ground motions. From the perspective of ductile seismic design, the MC bridge is more suitable for near-fault FD ground motions, while there is little difference between the MC and PSC bridges in the far-field region.

4.2 Parametric analysis

As demonstrated from the above-noted analyses, the pulse components of FD ground motions may have significant effects on system responses. In this subsection, the pulse parameters will be taken as the main research subject by using the pulse model described in Section 3.2.

Before conducting the analyses, it is worth noting that there exists one drawback for the M&P′s model. Compatible ground displacement can be integrated from the velocity time history, as in Eq. (1), and given as:

in whichCis the indefinite constant.

Obviously, it can be seen from Eq. (2) that a constant displacement value may occur at the end of the time factor due to the nonzero algebraic area underneath the velocity pulse, as represented in Eq. (1). This is not consistent with the actual near-fault FD effects (Makris, 1997). It can be derived from Eq. (2) that the final displacement can be zero only ifi=..., − 1, 0, 1, ...orγ=2.

The parametersγ,vand0tare set as the constant to reduce the parameter numbers, that is,γ=2,v=2, and0=10t. Meanwhile, parametersTpandAare the variables (Sehhatiet al., 2011).

Figure 12 shows the distributions of maximum superstructure displacements for the PSC bridges excited by pulses having different parameters. Clearly, the “narrow band” effect is exhibited in Fig. 12(a), that is, when the structural fundamental period (1.32 s) is close to the pulse period (1.49 s), the seismic response is the largest. However, as shown in Fig. 12 (b), with an increase of pulse amplitudes, the pulse period that produces the maximum structural response is significantly reduced, specifically, to 1.15 s. The conventional “narrow band” effect pointed out that the elongation of structural natural periods caused by inelastic behaviors in high-intensity earthquakes would lead to the ridge lines, displayed in Fig. 12(b), moving rightward. To clarify the obscurity resulting from two different “narrow band” effects, the same PSC bridge model, minus pounding elements, is taken as the reference object. In the same manner, the seismic responses are also presented in Fig. 13. Apparently, the “narrow band” effect is in accordance with the conventional one when there is no pounding effect. In other words, the existence of poundings between adjacent beams or between beam and abutment may significantly affect the seismic responses of bridges excited by FD ground motions. Severe damage may occur in a bridge even if the pulse periods of FD motions are smaller than the structural period. Additionally, it can be seen that the poundings effectively limit girder displacements, which is beneficial to PSC piers.

Fig. 10 The pounding times of two bridges excited by two earthquakes: (a) PSC bridge; (b) MC bridge

Fig. 11 The energy dissipated by one column excited by two earthquakes: (a) FD motions; (b) Far-field motions

Fig. 12 Maximum girder displacements of the PSC bridge with pounding elements, under the effect of pulse models having different amplitudes: (a) A = 15 cm/s; (b) A = 82.5 cm/s

Fig. 13 Maximum girder displacements of the PSC bridge without pounding elements, under the effect of pulse models with different amplitudes: (a) A=15 cm/s; (b) A=82.5 cm/s

Fig. 14 Maximum girder displacements of the MC bridge with pounding elements, under the effects of pulse models with different amplitudes: (a) A=15 cm/s; (b) A=82.5 cm/s

Fig. 15 Maximum girder displacements of the MC bridge without pounding elements, under the effect of pulse models with different amplitudes: (a) A=15 cm/s; (b) A=82.5 cm/s

Concerning the MC bridge, similar patterns can be observed in Figs. 14 and 15, that is, the pounding effect significantly hinders the “narrow band” effect moving to a larger period, and moreover reduces the bridge motion amplitudes. Therefore, the pounding effects are a doubleedged sword. A crash-resistant device or material then can be used at the end of the beam to protect the beam when the crash takes place.

4.3 Component analysis

In this subsection, ground motion set 3, described in Section 3.3 is set as the excitation source. The maximum girder displacement and pounding force in the middle gap are depicted in Figs. 16 and 17, respectively. It can be seen that the effects of single pulse components on the structural responses are far less powerful than those of residual components. This is because the pulse periods are significantly larger than the structural periods; accordingly, the “narrow band” effect is not markedly exhibited. Nevertheless, the residual components cannot replace the original ground motions as the inputs for predicting responses, especially for the MC bridge. Logically, the natural period of the MC bridge is far away from the pulse periods of the selected motions, when compared to the PSC bridge. This means that the residual components may make better predictions for the seismic responses of the MC bridge than for the PSC bridge (Mavroeidiset al., 2004). This seeming paradox can be explained by considering the hysteretic characteristics, as shown in Fig. 18. It can be seen that the PSC bridge has an approximate linear hysteretic curve, while the MC′s hysteretic curve is quite “plump”. The nonlinearity induced by the pulse components at the beginning of the records affects the follow-up seismic behaviors for MC bridges.

Fig. A1 Constitutive model of Pinching4 material

Fig. A2 Segmental column calibrations: comparison between the FE simulation and experimental data (Zhao et al., 2017)

Fig. 16 The maximum deck displacements of the two bridges excited by three components: (a) PSC bridge; (b) MC bridge

Fig. 17 The maximum pounding forces of the two bridges excited by three components: (a) PSC bridge; (b) MC bridge

Fig. 18 Comparison of the hysteretic behaviors of MC and PSC (Zhao et al., 2017)

5 Conclusions

In this paper, attempts were made to investigate the effects of near-fault FD ground motions on PSC and MC bridges, in which the two bridge models with material and geometric nonlinearities are simulated by using the finite-element software OpenSees, and three earthquake inputs were set regarding three crucial problems, including near- and far-field records, parameterized pulses, and pulse and residual components extracted from real records. Based on the numerical illustrations, several meaningful conclusions can be drawn, as follows:

(1) Near-fault FD ground motions cause more violent vibrations than do far-field for either PSC bridges or MC bridges. A PSC bridge may suffer more poundings and consume less energy, which renders it unsuitable for seismic design in a near-fault region.

(2) The pounding effect reduces the bridge vibrations since the pounding effect which invisibly provides structural stiffness. The “narrow band” effect may occur in a bridge with shorter periods than pulse periods because of the pounding effect. Therefore, it should be noted in the bridge design that the “narrow band” effect may need to be considered for the bridge with a short fundamental period.

(3) Residual components cannot replace the original FD ground motions, as structural inputs, even if the pulse period is far from the structural period, especially for the MC bridge.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant Nos. U1434205 and 51678490, the Major Research Plan of China National Railway Ministry of China under Grant Nos. 2015G002-B and P2018G007, and the National Key R&D Program of China under Grant No. 2017YFC1500803.

Appendix