Random Seismic Response Analysis of Long-Span Cable-Stayed Bridges Under High-Intensity Earthquakes Based on the Improved Power Spectral Model

Abstract

To study the influence of random seismic responses on the structure of a large-span double-deck steel truss cable-stayed bridges under the effects of high-intensity rare earthquakes, a new power spectral model was proposed based on improvements to existing power spectra for fitting the improved power spectra of random seismic responses. The bridge finite element model established using ANSYS was employed as an engineering example for computational analysis to investigate whether the improved spectrum exhibited better adaptability and feasibility under high-intensity rare earthquake compared with other power spectra. The results indicated that the power spectral model, based on improvements to the original power spectra, had a more pronounced filtering effect on the low-frequency and high-frequency portions. Moreover, under the consistent three-dimensional excitation, the vertical displacement of the main beam was the greatest, indicating that the improved spectrum had better adaptability than other power spectra in studying the high-intensity rare earthquakes affecting bridges. It also reflected the feasibility of using the improved spectrum for studying the random responses to high-intensity rare earthquakes, providing a reference for bridge design concerning rare earthquakes in large-span bridges.

1. Introduction

Earthquakes are among the most destructive natural disasters affecting human society, particularly high-intensity rare earthquakes. The immense shocks released by seismic waves can cause severe damage to building clusters and infrastructure on the ground [1]. Additionally, the resulting secondary disasters such as epidemics and landslides lead to casualties and economic losses that are difficult to quantify. In recent years, with the increasing frequency of high-intensity rare earthquakes in various regions, the study of stochastic seismic responses of large-span railway bridges under rare earthquake conditions has become increasingly urgent in the field of bridge engineering. In previous seismic design concepts for bridge structures, seismic actions were often treated as deterministic events for structural analysis. Current highway and railway seismic design codes propose three methods for seismic design analysis of bridge structures: the response spectrum method, the time-history analysis method, and the power spectral density method. Traditional time-history analysis, while considering the dynamic time history of structures, treats seismic actions as deterministic events, which significantly differs from reality. The response spectrum method, on the other hand, only calculates the peak response of structures without accounting for the time-varying nature of seismic actions. As a sudden natural disaster, earthquakes exhibit strong randomness in terms of magnitude, intensity, wave propagation direction, and duration. Given these stochastic characteristics of seismic motions, employing stochastic vibration methods for seismic analysis of large-span cable-stayed bridges was considered a more scientifically sound and reasonable approach.

In 1999, Allam S.M. and Datta T.K. [2] studied the seismic response of cable-stayed bridges using a stochastic vibration power spectral model, comparing the response characteristics of cable-stayed bridges under different seismic motion parameters and structural parameters. A.A. Dumanoglu and K. Soyluk [3] investigated the random ground motion response of cable-stayed bridges considering seismic spatial effects using a stochastic vibration power spectral model, particularly focusing on structural response variations due to site effects resulting from differences in local soil conditions at various support points. K. Soyluk [4], using the Jindo Bridge as an engineering background, studied the seismic response characteristics of cable-stayed bridges considering seismic spatial coherence effects and traveling wave effects, employing the stochastic vibration power spectral method for response analysis. Comparative analyses were conducted on the seismic response patterns of cable-stayed bridges under different site conditions. The study concluded that considering spatial effects was beneficial for the seismic response analysis of cable-stayed bridges. The application of stochastic vibration methods in seismic response analysis had been relatively limited; however, the power spectral method, as a classic approach within stochastic vibration methods, could effectively consider the characteristics of different sites where structures are located and perform non-uniform seismic inputs on structures.

Stochastic vibrations could be described through power spectral density functions to characterize their probabilistic statistical features in the context of seismic engineering, where power spectral models had been commonly used to describe the frequency-domain distribution of strong ground motion energy. Scholars had generally approached the study of power spectral models from two directions: one was to propose power spectral models with clear physical meanings to simulate seismic waves, and the other was to develop power spectral models that fit a large number of measured seismic records [5,6,7,8]. Due to the relatively small statistical sample of measured strong ground motion records, most studies on power spectral models were based on the former approach. In 1947, American scholar Housner [9] proposed that seismic random motion could be simulated using a stochastic model similar to white noise; however, this kind of model has unbounded energy, which is entirely inconsistent with actual seismic waves. To address the issue of unbounded energy in white noise models, Japanese scholars Tajimi and Kanai [10] proposed the classic Kanai K. model, which considers seismic waves as ground motions emitted by bedrock and filtered through soil layers, thus having clear physical significance. However, the Kanai K. model still includes a zero-frequency component, with unbounded variance in velocity and displacement, and an energy distribution in the low-frequency region that does not match reality. Scholars Penzien, Clough [11], and Ruiz [12] introduced a series-connected filter with a single-degree-of-freedom (SDOF) linear oscillator form to modify the low-frequency part of the Kanai spectrum.

In 1960, Chinese scholar Hu Yuxian [13] proposed the Hu Yuxian model, which added a low-frequency correction term to the Kanai K. model, essentially creating a white noise filtering model that included both a low-frequency and a high-frequency filter. This model was mathematically straightforward but lacked clear physical significance. Hong Feng [14] have had a very similar idea for improvements to the Hu Yuxian model, which involved adding a third-order high-pass filter in series with the Kanai filter to improve it, making it suitable for the analysis of random seismic motions in low-, medium-, and high-frequency structures. Based on this revised filtering concept, Ou Jinping [15] assumed that the bedrock seismic motion was a colored spectrum and corrected the high-frequency filter term of the Kanai spectrum, allowing the revised model to reflect the spectral characteristics of both the bedrock and the site-filtering soil layers. Du Xiuli [16], combining ideas from seismology and earthquake engineering, proposed adding a first-order low-frequency filter and a first-order high-frequency filter in series with the Kanai filter, where the low-pass filter contained spectral parameters that reflected the spectral characteristics of the bedrock. Li Hongjing [17] improved the Kanai K. model by eliminating the zero-frequency component while simultaneously correcting the low-frequency and high-frequency parts of the Kanai filter, controlling the low-frequency and high-frequency contents of the model through two spectral parameters, thus offering better adaptability than the original Kanai K. model. Peng Lingyun [18] addressed the issue of excessive energy distribution in the low-frequency region in the Kanai K. model and the Hu Yuxian model by proposing a revised Kanai K. model and a revised Hu Yuxian model, enhancing the suppression of the original models in the low-frequency region and solving the problem of the zero-frequency component in the Kanai K. model. Li Yingmin [19] systematically compared the filtering characteristics of various power spectral models and, based on the bounded energy distribution of power spectral models and the finite values of displacement, velocity, and acceleration variances, made improvements to the original Kanai power spectrum. Compared with Du Xiuli’s spectrum, Li Yingmin’s power spectrum exhibited a narrower bandwidth, higher amplitude, and better applicability.

To address this, an improved power spectral model was proposed based on research into existing power spectral models. It suggests serially connecting the Kanai spectrum with a first-order low-pass filter and a third-order high-pass filter to achieve better filtering characteristics and enhanced adaptability. The improved power spectral model is then compared with other power spectral models using the stochastic response of a single-degree-of-freedom structure. For further validation, the study employs a finite element model of a large-span cable-stayed bridge as the research subject. This involves considering the longitudinal, transverse, vertical, and three-dimensional combined uniform excitation inputs of seismic waves to investigate the stochastic seismic response patterns of the structure. The improved power spectrum is utilized to simulate the stochastic seismic response under rare earthquake conditions, thereby verifying its effectiveness. Consequently, this provides a reference for the seismic design of large-span bridges under rare earthquake conditions.

2. Earthquake Motion Power Spectral Model

2.1. The Kanai K. Model

The Kanai K. model had clear physical significance, considering seismic excitation as being generated by bedrock movement and transmitted to the structure after being filtered through soil layers. The Kanai K. model assumed the soil layer to be a linear single-degree-of-freedom system, taking into account the filtering effect of the site’s soil layers on the propagation of seismic excitation through the use of a predominant angular frequency (${\omega }_g$) and a characteristic damping ratio (${\xi }_g$) [20]. We could consider the Kanai K. model as an absolute acceleration power spectral model of white noise after it had been filtered through a single-degree-of-freedom spring-damper system with a fixed frequency and damping ratio, which could be denoted as the filtering system:

$$S_{K-T}\left(\omega \right)={\displaystyle \frac{1+4{\xi }_g^2\frac{{\omega }^2}{{\omega }_g^2}}{{\left(1-\frac{{\omega }^2}{{\omega }_g^2}\right)}^2+4{\xi }_g^2\frac{{\omega }^2}{{\omega }_g^2}}}·S_0$$

(1)

The Kanai K. model possessed good filtering characteristics, but its failure to eliminate the zero-frequency component led to amplified energy in the low-frequency region, causing severe distortion for flexible structures with low frequencies or structures containing zero-frequency rigid-body vibration modes. The Kanai K. model had a significant singularity at zero frequency, making it impossible to compute finite variances for the displacement and velocity of seismic ground motion.

2.2. Hu Yuxian Model

Hu Yuxian modified the low-frequency region of the Kanai K. model. To eliminate the zero-frequency content, the Hu Yuxian model introduced a high-pass filter based on the Kanai filtering model.

$$S\left(\omega \right)={\displaystyle \frac{{\omega }^6}{{\omega }^6+{\omega }_c^6}}·S_{K-T}\left(\omega \right)$$

(2)

In the equation, ${\omega }_c$ represents the low-frequency cutoff frequency. The Hu Yuxian model effectively eliminated the zero-frequency content, suppressed the energy in the low-frequency portion, and ensured that both displacement and velocity are finite. The physical significance of the Hu Yuxian model was not as clear as that of the Kanai K. model, and it was more of a mathematical model.

2.3. The Du Xiuli Model

The Du Xiuli power spectrum took into account the seismic wave propagation path and local site conditions, referencing seismological methods, by connecting a first-order high-pass filter, a first-order low-pass filter, and the Kanai filter in series, achieving better filtering effects, namely:

$$S\left(\omega \right)={\displaystyle \frac{1}{1+{\left(D\omega \right)}^2}}·{\displaystyle \frac{{\omega }^2}{{\omega }^2+{\omega }_0^2}}·S_{K-T}\left(\omega \right)$$

(3)

In the equation, $D$ represents the spectral parameter reflecting the characteristics of the bedrock, and ${\omega }_0$ is the low-frequency corner frequency, which serves to suppress low-frequency energies.

2.4. The Improved Spectrum Model

Scholar Li Yingmin derived formulas based on the two fundamental requirements that random seismic motion models need to satisfy: finite energy in the seismic process and elimination of the zero-frequency component. He believed that the denominator ($\omega$) of the acceleration power spectrum ($S_x\left(\omega \right)$) should be at least four orders higher than the numerator to ensure that the variances of the first and second derivatives of the function were bounded. $S_x\left(\omega \right)$ must contain at least four quadratic terms of ω in order to have no singularities in the first and second integrals at $\omega =0$. Based on the method of cascading filters and considering the aforementioned basic requirements of the power spectral model, improvements were made to the Du Xiuli model to achieve better filtering effects and stronger adaptability to high-intensity rare earthquakes. By connecting the Kanai filter with a first-order low-pass filter and a third-order high-pass filter in series, an improved power spectral model was obtained, referred to hereafter as the improved spectrum.

$$S\left(\omega \right)={\displaystyle \frac{1}{1+{\left(D\omega \right)}^2}}·{\displaystyle \frac{{\omega }^6}{{\omega }^6+{\omega }_0^6}}·S_{K-T}\left(\omega \right)$$

(4)

$$S\left(\omega \right)={\displaystyle \frac{1}{1+{\left(D\omega \right)}^2}}·{\displaystyle \frac{{\omega }^6}{{\omega }^6+{\omega }_0^6}}·{\displaystyle \frac{1+4{\xi }_g^2\frac{{\omega }^2}{{\omega }_g^2}}{{\left(1-\frac{{\omega }^2}{{\omega }_g^2}\right)}^2+4{\xi }_g^2\frac{{\omega }^2}{{\omega }_g^2}}}·S_0$$

(5)

In the above equation, $D$ is the spectral parameter reflecting the characteristics of the bedrock, ${\omega }_0$ is the low-frequency cutoff frequency, ${\omega }_g$ is the dominant angular frequency, and ${\xi }_g$ is the site characteristic damping ratio, where $D$ has the same meaning as in the original Du Xiuli spectrum. When comparing the improved model with the Kanai K. model, the Hu Yuxian model, and the Du Xiuli model using the same seismic motion parameters, the spectral parameter $D$ was taken as 0.011, ${\omega }_g$ as 5$\pi$, ${\omega }_0$ as 2$\pi$, and ${\xi }_g$ as 0.72. This was shown in the following Figure 1:

It can be seen that both the improved spectrum and the Du Xiuli spectrum removed the zero-frequency content, while maintaining consistency in the high-frequency and low-frequency portions. The magnitude of the improved spectrum was higher, exhibiting a greater amplitude than the Du Xiuli spectrum and a narrower bandwidth compared with the Du Xiuli and Hu Yuxian spectrum, demonstrating a more pronounced filtering effect on the low-frequency and high-frequency portions.

The following examined the impact of the improved spectrum, the Du Xiuli spectrum, and the Kanai spectrum on the elastic response of structures. For a single-degree-of-freedom elastic structural system, the power spectral density function of the structural displacement response is given by:

$$S_u\left(\omega \right)={\left|H_{\ddot{x}u}\left(\omega \right)\right|}^2·S_{\ddot{x}}\left(\omega \right)$$

(6)

Which was the transfer function:

$${\left|H_{\ddot{x}u}\left(\omega \right)\right|}^2={\displaystyle \frac{1}{{\left({\omega }_0^2-{\omega }^2\right)}^2+4{\xi }_0^2{\omega }_0^2{\omega }^2}}$$

(7)

In the equation, ${\omega }_0$ is structural frequency, and ${\xi }_0$ is structural damping ratio. The improved spectrum was input into a single-degree-of-freedom structural system, with ${\xi }_0$ set to 0.05 and ${\omega }_0$ to 2$\pi$, 5$\pi$, 10$\pi$, 20$\pi$, and 50$\pi$, yielding the displacement response power spectra as shown in the following figure.

As shown in Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6, when the structural circular frequency (${\omega }_0$) was less than the site characteristic frequency (${\omega }_g$), the reaction peak values of the improved spectrum and the Du Xiuli spectrum were closer to each other and significantly lower than those of the Kanai spectrum. The peak values of all three spectra were at the same position, clearly much higher than in other frequency regions. When the structural circular frequency (${\omega }_0$) was equal to the site characteristic frequency (${\omega }_g$), the peak values of the reactions of all three spectra were at the same position and were close to each other. When the structural circular frequency (${\omega }_0$) was between the low-frequency cutoff frequency and the high-frequency cutoff frequency, the peak values of the reactions of all three spectra were close, and they all exhibited bimodal peaks, with the bimodal peaks of the improved spectrum being more distinct, indicating stronger adaptability. When the structural circular frequency (${\omega }_0$) was higher than the high-frequency cutoff frequency, the bandwidth of the improved spectrum was narrower than that of the Du Xiuli spectrum and the Kanai spectrum, with slightly higher amplitudes than the Du Xiuli spectrum, and the low-frequency filtering effect of the improved spectrum was evident.

In summary, the bandwidth of the improved spectrum was narrower, with better low-frequency filtering effects and stronger adaptability, and a better match with the spectral properties of high-intensity rare earthquakes.

3. Engineering Example

This paper was set against the background of a double-deck, four-track, twin-tower steel truss cable-stayed bridge spanning the Yangtze River, with a total length of 877.8 m and a span layout of 75 m + 175 m + 425 m + 125 m + 62.5 m; the schematic structure is shown in Figure 7 and Figure 8. Due to requirements such as terrain, river conditions, and navigation control design, the heights of the twin towers were different, and the span layout was asymmetrical, making the bridge structure relatively complex. According to the relevant engineering context, construction design drawings, and technical specifications of the bridge, a full bridge finite element model (Figure 9) was established using the large-scale design software ANSYS 15.0. During the software simulations, different element types, boundary conditions, and coupling between elements, as well as the cable sag and initial stresses of the stay cables, were considered. The full-bridge model comprised a total of 12,932 elements and 19,540 nodes. Specific simulation details are shown in

Table 1. Unit Classification.

Element Number	Element Classification	Application Location
1	Beam188	main girder
2	Beam188	bridge tower
3	Link8	stay cables

,

Table 2. Material Properties.

Material Number	Material Type	Elastic Modulus (GPa)	Poisson’s Ratio	Density (kg/m3)	Application Location
1	C50 concrete	3.55 × 1010	0.2	2500	main girder
2	Q370	1.95 × 1011	0.3	7850	bridge tower
3	OVM250 steel strand	1.95 × 1011	0.3	7850	stay cables

,

Table 3. Bridge constraints and coupling.

Location	DX	DY	DZ	ROTX	ROTY	ROTZ
Auxiliary Pier and Main Girder		√	√
Pier 1 and Main Girder		√	√
Pier 6 and Main Girder		√
Tower 3 and Main Girder	√	√	√	√	√	√
Tower 4 and Main Girder		√	√
Bottom of Pylon	√	√	√	√	√	√

and

Table 4. Natural vibration frequency and mode characteristics of cable-stayed bridge.

Modal Number	Frequency (Hz)	Mode Characteristics
1	0.152	First-order symmetrical transverse bending of the main girder
2	0.247	Lateral displacement of the left tower
3	0.288	Transverse bending of the pylon + anti-symmetrical transverse bending of the main girder
4	0.313	First-order symmetrical vertical bending of the main girder
5	0.432	First-order anti-symmetrical transverse bending of the main girder
6	0.484	First-order symmetrical transverse bending of the main girder
7	0.790	Second-order symmetrical transverse bending of the main girder
8	0.836	Transverse bending of the pylon + symmetrical vertical bending of the main girder
9	0.976	Anti-symmetrical vertical bending of the main girder + torsion of the main girder
10	1.012	Co-directional transverse bending of both pylons + torsion of the main girder
11	1.103	Transverse bending of the left tower + torsion of the main girder
12	1.145	Co-directional longitudinal bending of the pylon + torsion of the main girder
13	1.187	Second-order anti-symmetrical transverse bending of the main girder + torsion of the main girder
14	1.308	Lateral displacement of the right tower + torsion of the main girder
15	1.342	Second-order symmetrical transverse bending of the main girder + lateral bending of the right tower + torsion of the main girder
16	1.360	Second-order symmetrical transverse bending of the main girder + torsion of the right tower
17	1.382	Second-order symmetrical transverse bending of the main girder + longitudinal drift of the right tower + torsion of the main girder
18	1.465	Torsion of the main girder + second-order symmetrical vertical bending
19	1.533	Second-order symmetrical vertical bending of the main girder + torsion of the main girder
20	1.674	Second-order symmetrical transverse bending of the main girder + second-order symmetrical vertical bending

.

In the modal extraction method, the Block Lanczos algorithm was found to be faster and more accurate compared with the traditional subspace iteration method, making it suitable for extracting multiple modes from large models. In this study, the Block Lanczos method was employed to extract the first 100 modes of the structure. The modal participation masses in the X, Y, and Z directions are shown in Figure 10. Due to space limitations, only the first 20 natural frequencies and modal shapes are listed in Table 4.

Based on the structural characteristics of the bridge, a structural finite element model was established using the finite element analysis method. Modal response analysis was performed on the model to closely observe the dynamic characteristics of the bridge structure. The dynamic characteristics of the steel truss cable-stayed bridge structure included longer natural periods, uniformly distributed and dense natural frequencies, significant overall stiffness of the main beam but lesser transverse stiffness, strong torsional resistance, and smaller transverse stiffness of the main piers. The three-dimensional coupled nature of the vibration modes of the main beam and the piers was evident, leading to complex dynamic characteristics. Further research into the seismic response of long-span cable-stayed bridges was deemed necessary.

According to China’s Seismic Zonation Map (GB 18306-2015) [21], the basic seismic intensity corresponding to this case was 8 degrees. This bridge, being a long-span cable-stayed bridge, was designed for seismic protection class A, with a basic peak ground acceleration of 0.40 g, a structural damping ratio of 0.05, and a natural period of 6.58 s. It was located in a Class II site in the western region.

4. Instance Analysis

Based on the aforementioned long-span steel truss cable-stayed bridge, a study was conducted on the random seismic response under consistent and inconsistent seismic excitations. The research considered the effects of longitudinal, transverse, vertical, and combined three-dimensional consistent excitation inputs on the random seismic response of the structure. Utilizing the previously proposed improved power spectral model, the Kanai K. model, the Hu Yuxian model, the Du Xiuli model, and their associated seismic motion parameter calculations, the study accounted for the effects of rare earthquakes, with the site category being Class II and the site design group being Group I. The finite element analysis software ANSYS APDL was used to simulate the rare earthquake random responses of the long-span cable-stayed bridge model, primarily focusing on the consistent excitation inputs in the longitudinal, transverse, and vertical directions. The aim was to explore the spatial coupling characteristics of longitudinal, transverse, and vertical seismic waves and identify the structurally vulnerable areas for seismic resistance, thereby further validating the rationality of the improved power spectral model for studying the random responses to rare earthquakes.

As shown in Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15, under the consistent longitudinal, transverse, and vertical excitations of a rare earthquake, the deformation that occurred in the bridge was mainly lateral bending of the midspan of the main beam, indicating that the transverse stiffness of the steel truss girder was lower. During a rare earthquake, due to the midspan being larger than the side spans, a greater lateral deformation occurred. The pylon experienced lateral bending, and the girder underwent vertical bending, which was consistent with the deformation pattern of a double-tower cable-stayed bridge under longitudinal seismic loads. However, the deformations of the main beam and the pylon under the consistent three-dimensional excitation of a rare earthquake were not significant, which was sufficient to demonstrate that the flexural, shear, and seismic strength of the main beam and the pylon were within reasonable limits.

By comparing the results of the random seismic response analysis of the bridge’s finite element model using the improved spectrum and other power spectral models, it was observed that the improved power spectral model, based on modifications to the original power spectrum, resulted in the greatest vertical displacement of the main beam and significant lateral bending of the pylon under three-dimensional consistent excitation. This led to the most unfavorable seismic response scenarios under high-intensity rare earthquakes, indicating that the improved spectrum could provide a reference for seismic design aspects of long-span steel truss cable-stayed bridges. Additionally, this demonstrated that the improved spectrum had good adaptability for studying the high-intensity rare earthquakes affecting bridges and also reflected the feasibility of using the improved spectrum for the study of high-intensity rare random seismic responses.

The comparison results of bridge displacement under different power spectra are shown in

Table 5. Comparison of bridge displacements with different power spectra.

Power Spectrum Type	Location
	Pier 2 (Auxiliary Pier)	Tower 3 (Saddle Tower)	Mid-Span	Tower 4 (Bridge Tower)
The Kanai K. Model	3.95%	0.63%	20.14%	16.47%
Hu Yuxian Model	155.04%	44.47%	57.33%	4.86%
Improved Spectrum Model	100%	100%	100%	100%
Du Xiuli Model	59.34%	39.96%	46.70%	42.21%

.

From the relative error calculations presented above, it can be seen that, compared with the improved spectrum model, the other three models exhibited varying degrees of error across all positions. Specifically: The Kanai K. model showed predictions very close to those of the improved spectrum model at the positions of pier 2 and tower 3, with relatively small errors (3.95% and 0.63%, respectively). However, its errors were significantly larger at other positions, especially at mid-span and pier 5. The Hu Yuxian model demonstrated large relative errors at all positions, particularly showing significant discrepancies at pier 2 and pier 5 (155.04% and 135.46%, respectively). This might have reflected its inadequate adaptability within specific frequency ranges. The Du Xiuli model’s errors fell between those of the Kanai K and Hu Yuxian models at all positions, indicating that its predictive capability was intermediate. Notably, the error at mid-span was relatively high (46.70%), but it was comparatively lower at tower 4 (42.21%).

In summary, the improved spectrum model exhibited higher numerical accuracy at most positions. This could be attributed to its integration of various types of filters to optimize performance, thereby enhancing its adaptability and filtering effects under high-intensity rare earthquakes. Therefore, the improved spectrum model is more suitable for structural design and analysis under high-intensity rare stochastic seismic responses.

5. Conclusions

Based on the research into existing power spectral models, this paper proposed a revised power spectral model by connecting the Kanai spectrum with a first-order low-pass filter and a third-order high-pass filter, resulting in an improved power spectral model with better filtering characteristics and stronger adaptability. The improved power spectrum was compared with other power spectral models based on the random response of a single-degree-of-freedom structure. Using a large-span double-deck four-track steel truss cable-stayed bridge as the engineering background, a finite-element numerical model was established. The improved power spectrum was utilized to simulate the random seismic response under the influence of a rare earthquake, verifying its feasibility and providing a reference for the seismic design of large-span bridges under rare earthquakes. The research findings were as follows.

(1)

Based on the basic requirements of power spectral models and the characteristics of various filters, an improved power spectral model was proposed. Comparisons between the filtering characteristics and the displacement response spectra of single-degree-of-freedom systems for the improved power spectrum and other power spectral models showed that the improved power spectrum did not contain a zero-frequency component. Its filtering curve amplitude was greater than that of the Du Xiuli spectrum, and it had a narrower bandwidth compared with the Du Xiuli and Hu Yuxian spectra, demonstrating a more pronounced filtering effect on the low-frequency and high-frequency portions. When calculating the displacement response spectra of single-degree-of-freedom systems, when the structural circular frequency was between the low-frequency and high-frequency cutoff frequencies, the peak reactions of the improved spectrum were close to those of the Du Xiuli and Kanai spectra, all showing bimodal peaks, with the bimodal peaks of the improved spectrum being more distinct, thus highlighting the improved spectrum’s better filtering characteristics and stronger adaptability.

(2)

Through the calculation of the displacement and velocity variances of multi-degree-of-freedom systems under high-intensity rare earthquake, the results indicated that the displacement and velocity variances of the improved power spectrum under high-intensity rare earthquake was smaller than those of other power spectra, showing stronger adaptability to high-intensity rare earthquakes.

(3)

The results of the random seismic response analysis conducted using a bridge model instance more intuitively demonstrated the adaptability of the improved spectrum model compared with other power spectral models, as well as the feasibility of using the improved spectrum for the study of random seismic responses under high-intensity rare earthquake. At the same time, it had provided a reference for the seismic design of large-span bridges with regard to rare earthquakes.

(3)

This paper proposed an improved power spectral model that is better adapted for high-intensity rare earthquakes, providing some reference value for seismic research on similar types of bridges under rare earthquake actions. However, large-span double-deck steel truss cable-stayed bridges exhibit significant structural nonlinearity characteristics. The computational requirements for studying seismic response using nonlinear stochastic vibration methods were substantial, and the computational process was complex and cumbersome, warranting further research.