Estimation of the input energy of beam bridges by using near-fault input energy design spectra
Introduction
In 1956 [1], first proposed the concept of the energy-based seismic design (EBSD) and suggested that the seismic input energy (EI) can be divided into the energy stored in structures and the energy dissipated by structures. Specifically, the elastic-strain energy (ES) and kinetic energy (EK) are stored in structures and vanish when the vibration of structures ceases, while EI is dissipated by the damping energy (ED) caused by the damping of structures and the hysteretic energy (EH) caused by the inelastic deformations of structures. Therefore, the basic idea of the EBSD is that EI should be less than the sum of EH and ED to avoid the collapse of structures in earthquakes. Since then, many researchers have paid more attention to the EBSD for structures.
In the framework of the EBSD, the first task is to determine EI. In recent years, due to the convenience of design, many researchers tended to use the inelastic input energy (SEI) spectra to determine the EI of structures, respectively. Early studies [2] indicated that there is a significant influence of the period (T), mass (m), and damping ratio (ξ) of an inelastic single-degree-of-freedom (SDOF) system on the SEI spectra, while the hysteresis model and its post-yield stiffness ratio (η) have little effect on them. Other early studies [3] indicated that the displacement ductility ratio (μ) and ground motion (GM) characteristics also have a significant effect on the SEI spectra. In recent studies, many researchers [[4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]] have focused on the methods for establishing the SEI design spectra, respectively.
At present, many researchers pay attention to near-fault GMs, which exhibit the impulsive characteristics, the fling step effect ([15,16]), and the forward directivity effect [[15], [16], [17]]). Therefore, under near-fault GMs, the greater EI of structures are generated due to the interaction between the system periods and the pulses of near-fault GMs [[18], [19], [20], [21], [22], [23]]. Moreover, there is some literature focused on the near-fault acceleration and displacement spectra [[24], [25], [26], [27]], while there are still limited studies on the near-fault SEI spectra [8,9,28]).
On the other hand, in the Chinese seismic design codes [29,30], soils are divided into four sites (I ~ IV), and each site is divided into three groups (G1 ~ G3) according to the design characteristic period (Tg) again. However, the current studies [8,9,28]) only present the near-fault SEI spectra corresponding to each site, while those corresponding to the three groups of each site have not been reported, which makes the EBSD can not be added to the Chinese seismic design codes. Therefore, in this paper, according to the characteristics of near-fault GMs and the Chinese seismic design codes, 210 near-fault GM records were selected to establish the near-fault SEI design spectra corresponding to G1 ~ G3 of Sites I ~ IV by using the statistical analysis and curve fitting method.
Another work in this study was to propose an EBSD for the Chinese seismic design codes under near-fault GMs. At present, many researchers have proposed their EBSD methods which are widely used in structures with and without seismic isolation devices [8,9,[31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]]. However, these EBSD methods are not consistent with G1 ~ G3 of Sites I ~ IV in the Chinese seismic design codes, i.e. they can not be directly used in the seismic design for Chinese bridges under near-fault GMs. Therefore, according to the established near-fault SEI design spectra in this study, an energy-based procedure was proposed for estimating the near-fault input energy (EI) of beam bridges, which can also be consistent with G1 ~ G3 of Sites I ~ IV in the Chinese seismic design codes.
Section snippets
Selection of near-fault GM records
Some researchers [42,43] have compared the site classification standard of the Chinese seismic design codes [29,30] with that of the American seismic design code [44], and proposed that the sites with vs30 (shear wave velocity of the upper 30 m of soil profile) ≥ 510 m/s, 260 m/s ≤ vs30 < 510 m/s, 150 m/s ≤ vs30 < 260 m/s, and vs30 < 150 m/s correspond to Sites I, II, III, and IV of the Chinese seismic design codes, respectively.
Then, the 210 near-fault GM records were selected from the PEER [45
Basic theory of inelastic seismic energy spectra
Under the GM excitation, the motion equation of an inelastic SDOF system is written as ([8,9]):where are the relative acceleration, velocity, and displacement of the inelastic SDOF system, respectively; is the ground acceleration; m, c, and f are the mass, damping coefficient, and nonlinear restoring force. Note and integrate Eq. (1) with respect to x, it is written as ([8,9]):where EI is the
Grouping of the 210 records according to the design characteristic period
According to the Chinese seismic design codes [29,30], the 210 records of Sites I ~ IV should be grouped again by using the design characteristic period (Tg, Table 1). So, the number and the average () of Tg of the records in each group are shown in Table 2.
Near-fault elastic input energy design spectra with PGA = 0.2 g and ξ = 2%
By adjusting the PGA of the 210 records to 0.2 g and inputting them into the BISPEC software [46], the near-fault elastic input energy spectrum with each record and ξ = 2% was calculated. Then, calculate the average and standard deviation
Near-fault inelastic input energy design spectra
In Section 3.3, it is found that when establishing the near-fault SEI design spectra, the Takeda hysteretic model is recommended (Fig. 4(a)), and the influence of η is neglected (Fig. 4(b)). Therefore, in Section 5, by adjusting the PGA of the 210 records to 0.2 g and then inputting them into the BISPEC software [46], the inelastic SDOF systems with the Takeda hysteretic model, η = 0.05, and T = 0.01–10 s were adopted to calculate the near-fault SEI design spectra.
Input energy of the inelastic MDOF system
Under the GM excitation (), the motion equation of an inelastic MDOF system is written as:where f is the restoring force vector; C is the damping matrix; M is the diagonal mass matrix; I is a unit vector; v is the relative displacement vector and can be written as:where Φj is the elastic modal shape of the j-th mode; xj is the generalized displacement of the j-th mode.
Substitute Eq. (17) into Eq. (16), and then pre-multiply both sides of Eq. (16) by . Thus,
Conclusions
In this study, the near-fault inelastic input energy (SEI) spectra were established, which are corresponding to the three groups (G1 ~ G3) of Sites I ~ IV in the Chinese seismic design codes [29,30], and the main studies are as follows:
- (1)
In Sections 2 Selection of near-fault GM records, 4.1 Grouping of the 210 records according to the design characteristic period, according to the characteristics of near-fault ground motions (GMs) and the Chinese seismic design codes, the 210 near-fault GM
Data availability statement
All data, models, and code generated or used during the study appear in the submitted article.
CRediT authorship contribution statement
Yu Li: Conceptualization, Methodology, Writing – original draft, Writing – review & editing, Data curation, Funding acquisition, Supervision, Project administration. Chen Li: Data curation, Writing – original draft, Writing – review & editing. Guo-Hui Zhao: Writing – review & editing, Funding acquisition, Resources, Project administration.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The research in this paper was supported by the National Natural Science Foundation of China (No. 51408042). The authors wish to express their gratitude to the sponsors.
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