Non-linear time-history analysis is a fundamental technique in the Performance-Based Earthquake Engineering (PBEE) theory used for probabilistic seismic hazard analysis and building fragility analysis. However, due to the strong uncertainty of the earthquake, if seismic information and structural properties are not taken into account in ground motions, it often conduce to large dispersion of the analysis results. For the seismic design of a structure, the ideal situation is to use the ground motions recorded from the site where the structure is located as the input for the time-history analysis. However, this opportunity is extremely rare, it’s extremely important to select appropriate ground motions from a large earthquake database.
Some Scholars (Trevlopoulos and Zentner 2020; Fayaz et al. 2021; Genovese et al. 2023) proposed generating synthetic earthquake waves that matched target spectrum based on the structural site conditions. However, this approach still has some limitations, and some studies had shown that the calculated results of synthetic and actual motions often show differences (Demartinos and Faccioli 2012). In addition, other scholars (Al and Abrahamson 2010; Du et al. 2019; Manfredi et al. 2022) had modified the actual ground motions by adjusting frequency such as frequency domain adjustment and time domain adjustment to achieve a better match between the response spectrum and the target spectrum over a wide range of periods. But these modifications caused changes in certain characteristics of ground motions.
At present, the primary and effective ground motions selection method is to match the target response spectrum by linear scaling. Some scholars (Kappos and Kyriakakis 2000; Shome and Cornell 1999) found that adjusting the ground motion with the target spectrum value of the structural fundamental period T1 can reduce the discreteness of the analysis results. Although this measure is simple to operate, it only emphasizes the matching of the spectral values at a certain period point and cannot reflect the matching of the response spectrum shape. On this basis, Kalkan and Chopra (2018) proposed matching the target spectrum within a certain period range of the structure and using the square difference λ of the scaled response spectrum and the target spectrum as a control parameter based on the least-squares method. However, it is difficult to match the target spectrum of a structure with dispersive main modal. Monte Carlo algorithm was used to simulate multiple response spectra that match these statistical characteristics as the target spectrum group based on the mean and variance of the target spectrum (Baker and Lee 2018). Currently, ground motions selection methods based on genetic algorithms and artificial intelligence (Mergos and Sextos 2019; Florez et al. 2022) were also being studied, but these methods are more complex and more difficult to apply in engineering.
In the structural fragility analysis, some scholars (Pan et al. 2010; Xiang et al. 2019; Chen et al. 2021) compared the accuracy of selecting intensity measures (IMs) by fitting the probabilistic seismic demand model (PSDM) with a large number of ground motions. This approach verifies the correctness of the selected index by probability statistics method, but it does not investigate the essence in the selection of ground motions, so there are large differences in the results fitted by different IMs. Other scholars (Li et al. 2017; Ren et al. 2020; Yaghmaei-SabeghS and Neekmanesh 2022) selected ground motions by using the approach of matching with the target spectrum, but the matching only considered a certain period zone without considering the impact of higher-order modes on the structure. However, for complex structures, such as cable-stayed bridges, the impact of higher-order modes must be considered, and matching the target response spectrum of a certain period is inaccurate.
Considering the aforementioned issues, this study proposes a spectral matching approach based on modal participation coefficients that is highly accurate and easy to operate and can be applied effectively in engineering. Taking a large-span cable-stayed bridge as an example, the study demonstrates how to select ground motions by this approach and analyze the impacts of frequency and amplitude on the seismic analysis of such a complex structure as a cable-stayed bridge. Based on the investigate results, suggestions are made for the selection of IMs for ground shaking in seismic susceptibility analysis.
Description and simulation model of the selected cable-stayed bridge
A two-pylon cable-stayed highway bridge with a main span of 436 m and an overall length of 1266 m is taken as an example of long-span bridges and hereinafter referred to as example bridge for short, depicted in Fig. 1. The east and west approach bridges (reinforced concrete continuous box-girder bridges) are provided to consider the interaction between the main bridge and the approach bridges. The main bridge is a double-cable-plane two-pylon steel box girder cable-stayed bridge. Steel box girders of the main bridge are streamlined multi-cell flat-box girders, approximately 3 m high and 35 m wide (including bilateral wind fairings of 2.23m). The 150m-high diamond-type main bridge pylon with a crossbeam support is constructed of reinforced concrete, presented in Fig. 2. Footings for the pylon and pier are both pile-group foundations and use friction piles and bored cast-in-place piles respectively.
The simulation model of the example bridge is developed for use through spatial finite element method with SAP2000. Spatial beam elements considering geometric stiffness are used to simulate the pylons, box girders and piers. The box girders are interconnected to subsidiary and transition piers by frictional bearings in the longitudinal direction, respectively. Perfect elastic-plastic link elements are approximately used to model sliding frictional bearings. Master-slave constraints are rationally selected based on the constraint properties of bearings. The central cross-beams of two main bridge pylons and the box girders are interconnected by four viscous dampers in the longitudinal direction, respectively. The stay cables are modeled as spatial truss elements. The evaluation of an equivalent elasticity modulus for the stay cables is made using the Ernst formula to account for the cable sag effect. The connections between the stay cables and the box girders and the main pylons are modeled as rigid connections. The effect of the pile cap is simulated by the lumped-mass model at its center of mass. The piles of pylon foundations above and under the scour line are modeled as spatial beam elements and 6×6 coupling spring elements, respectively. The piles of pier foundations are modeled as 6×6 coupling spring elements at the bottom of the pile cap. Modeling of soil-structure interaction is accomplished in SAP2000 using spring elements. These elements are given elastic properties to represent the stiffness of all six degrees of freedom provided by the soil-structure system. Coupling between the degrees of freedom is considered, and the numerical model adopts a fixed damping ratio of 0.05. The nonlinear finite element model of the example bridge is displayed in Fig. 3.
Selection of intensity measures (IM) and engineering demand parameters (EDP)
The intensity measure (IM) denotes a measure of ground motion intensity corresponding to a particular level of seismic hazard evaluated using a regional hazard model. The structural analysis is performed in order to obtain a probabilistic model for structural response as a function of this intensity measure parameter in PBEE. Several choices of IMs are possible, such as peak ground acceleration (PGA), peak ground velocity (PGV), spectral acceleration (Sa), a two-parameter IM and a vector-valued IM consisting of spectral acceleration and epsilon (Cordova et al. 1999; Baker and Cornell 2008; Chaudhary 2016; Wei et al. 2020). In this paper, PGA and Sa at the foundational frequency are used as IMs respectively to show how to reasonably select IMs because they are the most common IMs for earthquake resistance of bridge structures.
Structural seismic demand needs to be characterized by a limited set of response measures that are referred to as engineering demand parameters (EDPs) which in general are functions of IMs in PBEE. In the case of bridge structures, the displacement at the top of the main pylon or the pier drift ratio is a typical response measure that can be correlated with damage and performance. Other examples of EDPs include forces, stresses, strains, and cumulative measures such as plastic deformation and dissipated energy because damage to bridges can result from movement of the foundation, substructure, or superstructure. EDPs in this paper included seismic forces and displacements. The key sections for seismic forces are Section 1 to Section 4 in Fig. 2. The seismic displacements as EDPs are the maximum longitudinal displacements at the top of thepylons and at the end of the steel box girders (i.e Dis 1and Dis 2), the maximum vertical displacement at the middle of the steel box girders (Dis 3), the maximum bearing displacement of the subsidiary piers and the transition piers (Dis 4 and Dis 5), the maximum damping-displacement (Dis 6).
Dynamic properties of the example bridge
The earthquake resistant design and analysis of bridge structures should be based on the analysis of their dynamic properties. Through the above finite element model and the modal analysis, Table 1 to Table 3 lists some important information (frequencies and periods) of several natural modes whose modal mass participation factors are more than 0.05 in the longitudinal, transverse and vertical directions of the example bridge, respectively. Bold italic implied that modal mass participation of these natural modes is significant in Table 1 to Table 3.
Table 1. Natural modes of vibration of significant modal mass participation in the longitudinal direction.
|
Mode number
|
Frequency (Hz)
|
Period (s)
|
Modal mass
participation factor
|
Cumulative modal mass
participation factor
|
|
1
|
0.128
|
7.821
|
0.235
|
0.235
|
|
2
|
0.159
|
6.294
|
0.060
|
0.295
|
|
3
|
0.198
|
5.051
|
0.059
|
0.354
|
|
27
|
1.307
|
0.765
|
0.057
|
0.468
|
|
68
|
2.877
|
0.348
|
0.129
|
0.666
|
|
70
|
2.892
|
0.346
|
0.102
|
0.768
|
|
93
|
3.967
|
0.252
|
0.075
|
0.844
|
|
94
|
3.978
|
0.251
|
0.076
|
0.920
|
Table 2. Natural modes of vibration of significant modal mass participation in the transverse direction.
|
Mode number
|
Frequency (Hz)
|
Period (s)
|
Modal mass
participation factor
|
Cumulative modal mass
participation factor
|
|
5
|
0.488
|
2.049
|
0.222
|
0.222
|
|
26
|
1.267
|
0.789
|
0.164
|
0.416
|
|
57
|
2.245
|
0.446
|
0.050
|
0.596
|
|
77
|
3.295
|
0.303
|
0.264
|
0.873
|
Table 3. Natural modes of vibration of significant modal mass participation in the vertical direction.
|
Mode number
|
Frequency (Hz)
|
Period (s)
|
Modal mass
participation factor
|
Cumulative modal mass
participation factor
|
|
123
|
4.951
|
0.202
|
0.056
|
0.213
|
|
130
|
5.320
|
0.188
|
0.243
|
0.459
|
|
212
|
9.405
|
0.106
|
0.093
|
0.702
|
|
222
|
9.984
|
0.100
|
0.126
|
0.828
|
Based on the analysis of Table 1 to Table 3, it is showed that the percentage of modal mass participation of several natural modes is considerably significant in the longitudinal, transverse and vertical directions of the example bridge, respectively. However, none of these modal mass participation factors is predominant (e.g. the biggest one is not more than 0.3 and the smallest one is not less than 0.05). For example, the factor at the fundamental frequency is biggest (i.e 0.235) and there are two factors (between 0.1 and 0.2) and five factors (between 0.05 and 0.1) in the longitudinal direction. The sum of these eight factors (i.e cumulative modal mass participation) is only 0.793. Furthermore, these eight periods are obviously different (i.e from 0.251s to 7.821s). Through the above analysis, it is concluded that the dynamic properties of the long-span bridge structures are very special and complex and the higher-mode effect on their seismic responses must be more significant than the other kinds of bridge structures.
Selection of earthquake ground motions for structural analysis
The seismic hazard analysis for the example bridge site is carried out to obtain the key characteristics of earthquake ground motions by Institute of Geology, China Engineering Administration (Institute of Geology, China Engineering Administration, 2004). Uniform hazard spectra (3% damping) are provided for 3% and 10% probability of exceedance in 50 years (Fig. 4). The uniform hazard spectra based on 10% PE in 50 years is selected as the Target Response Spectrum (TRS) in this paper.
As shown in Table 1, the cumulative modal mass participation factors in the period ranges of 0.251s≤T≤0.765s (Period Range 1) and 5.501s≤T≤7.821s (Period Range 2) are 0.452 and 0.354 respectively, and larger than other period ranges in the longitudinal direction of the example bridge. Three sets of real ground motions (i.e Bin 1 to Bin 3) are selected as longitudinal input ground motions for nonlinear dynamic time history analysis and for each bin, 10 real accelerograms recorded on the same soil are considered (PEER Ground Motion Database, 2012). The same vertical input ground motion is used for more precise analysis on the effects of longitudinal input ground motions on the longitudinal EDPs. The selection standards of longitudinal input ground motions are as follows:
(1) Bin 1: the geometric mean spectrum (GMS) of Bin 1 matched TRS over the period range of 0.25~10.0s (including the Period Range 1 and 2).
(2) Bin 2: GMS of Bin 2 matched TRS over the period range of 0.25~2.0s (only including the Period Range 1).
(3) Bin 3: GMS of Bin 3 matched TRS over the period range of 2.0~10.0s (only including the Period Range 2).
The matching of GMS of three bins to TRS and the dispersions are shown in Fig. 5 to Fig. 7, respectively.
Fig. 5 to Fig. 7 show that when Sa(T2＝0.35s), Sa(T1=7.82s) and PGA are selected as IMs and firstly scaled to target 10% PE in 50 years IM level, namely 0.55g, 0.07g and 0.19g respectively, the differences in geometric means and dispersions of response spectra of individual records from three bins in the Period Range 1 are significantly different from those in the Period Range 2. It can be used for analyzing the effects of the amplitude and frequency content of earthquake ground motions on the seismic response of long-span bridge structures.
Probabilistic seismic demand analysis
In this paper, nonlinear dynamic time history analysis was carried out using spectral acceleration and PGA as IMs to study the effects of amplitudes and frequency content of earthquake ground motions on probabilistic seismic demand assessment of long-span bridge structures. There are three cases as follows:
(1) Case 1: Take Sa(T2) as an IM and all records are scaled to target 10% PE in 50 years IM level (i.e Sa(T2)=0.55g).
(2) Case 2: Take Sa(T1) as an IM and all records are scaled to target 10% PE in 50 years IM level (i.e Sa(T1)=0.07g).
(3) Case 3: Take PGA as an IM and all records are scaled to target 10% PE in 50 years IM level (i.e PGA= 0.19g).
In three cases, all records scaled to target IM levels are served as longitudinal input ground motions and the same vertical input ground motion was used for nonlinear dynamic time history analysis. In this paper, the geometric mean is used as the logical estimator of the median value to evaluate the seismic response of structures, and the standard deviation of the natural logarithms of the data is used as the “dispersion”. The smaller dispersion, the probabilistic assessment of seismic response is more precise and efficient (Benjamin and Cornell, 1970).
Seismic displacement analysis
For Case 1, GMS of three bins are well matched to TRS in the Period Range 1. GMS of Bin 1 match TRS better than those of Bin 2 and Bin 3 in the Period Range 2, that is Bin 1>Bin 2>Bin 3 for geometric means of spectral accelerations at the most of periods in the Period Range 2. Results show that longitudinal seismic displacements are Bin 1>Bin 2>Bin 3 because of the effects of GMS in the Period Range 1 and 2 on seismic displacements (Fig. 5(a) and Fig. 8).
For longitudinal seismic displacements, it can be concluded form Fig. 5(b) that the dispersions of response spectra of individual records of three bins are Bin 2<Bin 1<Bin 3 in the Period Range 1 and Bin 1<Bin 2<Bin 3 in the Period Range 2, respectively. The dispersions of longitudinal seismic displacements are Bin 1≈Bin 2<Bin 3 because of the effects of dispersions of response spectra of selected records in the Period Range 1 and 2 on longitudinal seismic displacements.
For Case 2, longitudinal seismic displacements are Bin 1<Bin 2<Bin 3 because geometric means of response spectral values of ground motions from three bins at the same period are Bin 1<Bin 2<Bin 3 in the Period Range 1 and Bin 2<Bin 1<Bin 3 in the Period Range 2 where the differences in geometric mean spectral values are significantly smaller than those in the Period Range 1. GMS of Bin 2 and Bin 3 in the Period Range 1 are dramatically bigger than TRS so that seismic responses calculated by ground motions from Bin 2 and Bin 3 can severely deviate from real seismic disasters and mislead structural designers to make excessively conservative seismic design (Fig. 6(a), 6(c) and Fig. 9).
Fig.9 and Fig. 6(b) show that the dispersions of longitudinal seismic displacements are Bin 1<Bin 2≤Bin 3 because the dispersions of response spectra of individual records from three bins are both Bin 1 < Bin 2≤Bin 3 in the Period Range 1 and 2.
Fig.10 and Fig. 7 show that longitudinal seismic displacements are Bin 1>Bin 2>Bin 3 and the dispersions are Bin 1<Bin 2<Bin 3. The relation between the calculation results and the geometric means and dispersions of response spectra of individual records from three bins in Case 3 is similar to that in Case 1 and 2. The tight link between seismic displacements and frequency content of earthquake ground motions is clear.
Seismic force analysis
Table 4. Statistical analysis of seismic forces from three Bins for scaled records (Case 1).
|
Bin
|
Bin1
|
Bin2
|
Bin3
|
|
Statistical analysis of
EDPs
|
Geometric
mean
|
Dispersion
|
Geometric
mean
|
Dispersion
|
Geometric
mean
|
Dispersion
|
|
Damping force（kN）
|
8.21E+03
|
0.152
|
8.32E+03
|
0.091
|
7.94E+03
|
0.246
|
|
Section 1
|
Axial force（kN）
|
1.60E+04
|
0.015
|
1.64E+04
|
0.065
|
1.67E+04
|
0.056
|
|
Shear force（kN）
|
6.27E+03
|
0.350
|
6.11E+03
|
0.144
|
5.45E+03
|
0.692
|
|
Bending moment（kN•m）
|
1.94E+05
|
0.362
|
1.98E+05
|
0.227
|
1.83E+05
|
0.756
|
|
Section 2
|
Axial force（kN）
|
2.02E+04
|
0.010
|
2.07E+04
|
0.054
|
2.05E+04
|
0.042
|
|
Shear force（kN）
|
1.25E+04
|
0.323
|
1.30E+04
|
0.204
|
1.21E+04
|
0.678
|
|
Bending moment（kN•m）
|
2.03E+05
|
0.328
|
2.04E+05
|
0.192
|
1.94E+05
|
0.704
|
|
Section 3
|
Axial force（kN）
|
2.35E+04
|
0.008
|
2.39E+04
|
0.043
|
2.36E+04
|
0.034
|
|
Shear force（kN）
|
1.81E+04
|
0.310
|
2.06E+04
|
0.185
|
1.83E+04
|
0.701
|
|
Bending moment（kN•m）
|
5.53E+05
|
0.357
|
5.24E+05
|
0.201
|
4.97E+05
|
0.682
|
|
Section 4
|
Axial force（kN）
|
8.73E+04
|
0.009
|
8.70E+04
|
0.030
|
8.67E+04
|
0.021
|
|
Shear force（kN）
|
1.34E+05
|
0.162
|
1.43E+05
|
0.154
|
1.33E+05
|
0.672
|
|
Bending moment（kN•m）
|
1.54E+06
|
0.337
|
1.60E+06
|
0.193
|
1.43E+06
|
0.680
|
It should be noted from Table 4 that for Case 1, the geometric means of maximum seismic forces for different cross-sections are approximately Bin 1≈Bin 2≥Bin 3 but the geometric means and dispersions of axial forces are similar to each other for three bins because vertical input ground motions have significant effects on axial forces rather than longitudinal input ground motions. The dispersions of shear force and bending moment are approximately Bin 3>Bin 1>Bin 2. Through the analysis of Fig. 5, it is found that GMS of three bins is well matched to TRS and the dispersions of response spectra of individual records from three bins are Bin 3>Bin 1>Bin 2 in the Period Range 1. GMS of three bins is Bin 1>Bin 2>Bin 3 and the dispersions of response spectra are Bin 3>Bin 2>Bin 1 in the Period Range 2.
Compared seismic displacements with seismic forces, it can be concluded that the effects of response spectra in the short period range on seismic responses of long-span bridge structures are more significant than those in the long period range. These effects on seismic forces are more remarkable than those on seismic displacements.
Table 5. Statistical analysis of seismic forces from three Bins for scaled records (Case 2).
|
Bin
|
Bin1
|
Bin2
|
Bin3
|
|
Statistical analysis of
EDPs
|
Geometric
mean
|
Dispersion
|
Geometric
mean
|
Dispersion
|
Geometric
mean
|
Dispersion
|
|
Damping force（kN）
|
8.74E+03
|
0.114
|
1.28E+04
|
0.287
|
1.44E+04
|
0.440
|
|
Section 1
|
Axial force（kN）
|
1.60E+04
|
0.016
|
2.48E+04
|
0.622
|
2.86E+04
|
0.645
|
|
Shear force（kN）
|
7.70E+03
|
0.246
|
2.89E+04
|
1.011
|
3.94E+04
|
1.344
|
|
Bending moment（kN•m）
|
2.40E+05
|
0.273
|
8.25E+05
|
0.943
|
1.34E+06
|
1.457
|
|
Section 2
|
Axial force（kN）
|
2.02E+04
|
0.009
|
2.88E+04
|
0.548
|
3.25E+04
|
0.569
|
|
Shear force（kN）
|
1.46E+04
|
0.234
|
5.25E+04
|
1.036
|
8.06E+04
|
1.434
|
|
Bending moment（kN•m）
|
2.45E+05
|
0.244
|
8.26E+05
|
0.929
|
1.34E+06
|
1.445
|
|
Section 3
|
Axial force（kN）
|
2.35E+04
|
0.008
|
3.16E+04
|
0.497
|
3.54E+04
|
0.510
|
|
Shear force（kN）
|
2.16E+04
|
0.220
|
8.40E+04
|
1.010
|
1.27E+05
|
1.452
|
|
Bending moment（kN•m）
|
6.53E+05
|
0.255
|
2.07E+06
|
0.975
|
3.30E+06
|
1.379
|
|
Section 4
|
Axial force（kN）
|
8.74E+04
|
0.009
|
9.71E+04
|
0.290
|
1.04E+05
|
0.330
|
|
Shear force（kN）
|
1.57E+05
|
0.152
|
5.40E+05
|
0.987
|
8.91E+05
|
1.399
|
|
Bending moment（kN•m）
|
1.85E+06
|
0.230
|
6.34E+06
|
0.932
|
9.82E+06
|
1.428
|
Table 5 and Fig. 6 clearly indicate that for Case 2, the geometric means and dispersions of maximum seismic forces for different cross-sections are approximately Bin 1<Bin 2<Bin 3 because GMS of three bins are Bin 1 <Bin 2<Bin 3 in the Period Range 1 and Bin 2<Bin 1<Bin 3 in the Period Range 2 respectively and dispersions of response spectra are both Bin 1 <Bin 2<Bin 3 in the Period Range 1 and 2. Furthermore, the differences in geometric means and dispersions of response spectra of three bins in the short period range (Period Range 1) are significantly larger than those in the long period range (Period Range 2).
Moreover, it is different from Case 1 that the geometric means and dispersions of axial forces are Bin 1<Bin 2<Bin 3 in Case 2. It can be included that if the differences of response spectra are very significant in the period ranges where response spectra of longitudinal input ground motions have great effects on longitudinal seismic response, longitudinal input ground motions can have important influence on axial forces.
Table 6. Statistical analysis of seismic forces from three Bins for scaled records (Case 3).
|
Bin
|
Bin1
|
Bin2
|
Bin3
|
|
Statistical analysis of
EDPs
|
Geometric
mean
|
Dispersion
|
Geometric
mean
|
Dispersion
|
Geometric
mean
|
Dispersion
|
|
Damping force（kN）
|
7.99E+03
|
0.097
|
8.07E+03
|
0.070
|
7.84E+03
|
0.228
|
|
Section 1
|
Axial force（kN）
|
1.60E+04
|
0.015
|
1.64E+04
|
0.061
|
1.67E+04
|
0.057
|
|
Shear force（kN）
|
5.75E+03
|
0.281
|
5.49E+03
|
0.110
|
5.23E+03
|
0.617
|
|
Bending moment（kN•m）
|
1.81E+05
|
0.108
|
1.78E+05
|
0.136
|
1.75E+05
|
0.688
|
|
Section 2
|
Axial force（kN）
|
2.01E+04
|
0.010
|
2.06E+04
|
0.052
|
2.05E+04
|
0.042
|
|
Shear force（kN）
|
1.13E+04
|
0.231
|
1.17E+04
|
0.193
|
1.15E+04
|
0.616
|
|
Bending moment（kN•m）
|
1.84E+05
|
0.089
|
1.85E+05
|
0.126
|
1.85E+05
|
0.642
|
|
Section 3
|
Axial force（kN）
|
2.35E+04
|
0.006
|
2.39E+04
|
0.041
|
2.36E+04
|
0.032
|
|
Shear force（kN）
|
1.67E+04
|
0.173
|
1.86E+04
|
0.140
|
1.74E+04
|
0.656
|
|
Bending moment（kN•m）
|
4.91E+05
|
0.217
|
4.72E+05
|
0.150
|
4.72E+05
|
0.604
|
|
Section 4
|
Axial force（kN）
|
8.74E+04
|
0.008
|
8.70E+04
|
0.029
|
8.66E+04
|
0.022
|
|
Shear force（kN）
|
1.24E+05
|
0.166
|
1.31E+05
|
0.136
|
1.27E+05
|
0.623
|
|
Bending moment（kN•m）
|
1.41E+06
|
0.178
|
1.45E+06
|
0.122
|
1.36E+06
|
0.617
|
From Table 6 and Fig. 7, longitudinal seismic forces are similar to each other and the dispersions are Bin 3>Bin 1>Bin 2 in Case 3. Seismic forces are closely related to response spectra of earthquake ground motions.
Based on the above analysis, the geometric means and dispersions of response spectra of earthquake ground motions selected for dynamic analysis have significant effects on probabilistic seismic demand assessment of long-span bridge structures and the effects are closely related to the higher mode effects. In general, the natural modes of vibration that have significant effects on seismic response are more than one, even in some period ranges, and none of all modes has the predominant effects on seismic responses of long-span bridge structures. If response spectra of earthquake ground motions are well match to the target response spectra, probabilistic seismic demand assessment can be consistent with real structural earthquake hazards and more precise than those calculated by ground motions selected on the basis of magnitude and distance. The precision and efficiency of probabilistic seismic demand assessment can be remarkably improved through decreasing the dispersions of response spectra of earthquake ground motions. If the differences in cumulative modal mass participation factors in different period ranges are small, the effects of response spectra in the short period range on seismic demands are more significant than those in the long period range because response spectral values in the short period range are larger than those in the long period range. If the cumulative modal mass participation factors in the short period ranges are less than these in the long period ranges, the above effects can be decreased. Consequently, the methods of matching to the target response spectra are based on the differences in cumulative modal mass participation factors and response spectra in different period ranges.
In general, response spectral values as IMs, especially those at the fundamental periods of bridge structures or at the periods with the largest modal participation mass ratios in the different directions of bridge structures, can more significantly improve the precision and efficiency of probabilistic seismic demand assessment of girder bridges than PGA as an IM (Hu et al. 2022). However, the periods with the largest modal participation mass ratios in the different directions are so long for long-span bridges, even longer than 10s, that response spectral values can be questionable for correctness in earthquake engineering and unavailable for the IMs (Baker 2005). If input ground motions for dynamic analysis are rationally selected on the basis of the response spectral shape, PGA and response spectral values at the other periods with comparatively large modal participation mass ratios can be chosen as IMs. It can improve the precision and efficiency of probabilistic seismic demand assessment of long-span bridges, even better than response spectral values at the fundamental periods as IMs (Chen and Li 2011).