The present study reveals for the first time that some real earthquake ground motions would induce the absolute maximum response (AMR) of structural systems posterior to the end of the excitation. Under these ground motions, illustrated using single-degree-of-freedom (SDOF) systems, the peak responses calculated within the during-shaking phase might be significantly lower than the actual AMR. Extending the dynamic analysis to the post-shaking phase by half of the structural period is recommended when conducting response history analyses or calculating the response spectra of these ground motions.</p

Tsang, Hing Ho

Yi, Jiang

English

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                                                                    University of DundeeSignificance of post-shaking response of SDOF systemsYi, Jiang; Tsang, Hing HoPublished in:Soil Dynamics and Earthquake EngineeringDOI:10.1016/j.soildyn.2023.108102Publication date:2023Licence:CC BY-NC-NDDocument VersionPublisher's PDF, also known as Version of recordLink to publication in Discovery Research PortalCitation for published version (APA):Yi, J., & Tsang, H. H. (2023). Significance of post-shaking response of SDOF systems. Soil Dynamics andEarthquake Engineering, 173, Article 108102. https://doi.org/10.1016/j.soildyn.2023.108102General rightsCopyright and moral rights for the publications made accessible in Discovery Research Portal are retained by the authors and/or othercopyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated withthese rights.Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 27. Oct. 2024Soil Dynamics and Earthquake Engineering 173 (2023) 108102Available online 24 June 20230267-7261/© 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Significance of post-shaking response of SDOF systems Jiang Yi a, Hing-Ho Tsang b,* a College of Civil Engineering, Guangzhou University, Guangzhou, China b School of Engineering, Swinburne University of Technology, Melbourne, Australia   A R T I C L E  I N F O   Keywords: Post-shaking Response spectrum Response history analysis Absolute maximum response (AMR) A B S T R A C T   The present study reveals for the first time that some real earthquake ground motions would induce the absolute maximum response (AMR) of structural systems posterior to the end of the excitation. Under these ground motions, illustrated using single-degree-of-freedom (SDOF) systems, the peak responses calculated within the during-shaking phase might be significantly lower than the actual AMR. Extending the dynamic analysis to the post-shaking phase by half of the structural period is recommended when conducting response history analyses or calculating the response spectra of these ground motions.   1. Introduction Earthquake response history analysis aims to obtain the absolute maximum response (AMR) for structural design. A response spectrum graphically describes the AMR of a series of single-degree-of-freedom (SDOF) systems as a function of a structure’s natural period and damping ratio. An accurate response spectrum of ground motion is vital for a wide range of applications in earthquake engineering. The theory for deriving response spectra is available in the literature [1,2]. The calculation of the AMR requires understanding of the complete time history behavior. Previous studies revealed that for a SDOF system subjected to a short-duration impulse load, such as blast and impact, the AMR would either be reached in the during-shaking phase (during the action of the impulse) or the post-shaking phase (posterior to the action of the impulse), depending on the system response at the end of the impulse [1]. To capture the AMR under a short-duration impulse load, it is generally recommended that the dynamic analysis of structures should go beyond the excitation [3,4]. However, the need of extending the dynamic analysis to the free-vibration phase after an earthquake shaking has not been investigated. This study intends to initiate this discussion. Unlike impulse loadings that vanish transiently, ground motions recorded in earthquakes usually have much longer durations. Due to the long duration of earthquake ground motions and the relatively low amplitude in the tail part of the shaking, there is an implicit assumption that the AMR develops in the during-shaking phase, and response his-tory analysis, including response spectrum calculation, ceases at the end of the shaking (the time when the recorded time series are terminated). These are the cases for some software tools (like Seismosignal) and the PEER-NGA2 website (http://peer.berkeley.edu/nga), the latter of which will be discussed further in this article. However, the present study reveals that similar to impulse loadings, some real earthquake ground motions would actually develop the AMR in the post-shaking phase. Illustrated using SDOF systems, the peak response calculated within the during-shaking phase might significantly underestimate the actual seismic demand. Therefore, it is recommended to conduct extended analysis on those ground motions when performing structural response history analysis or calculating the response spectra. The ground motions that require extended analysis (EAGMs) are iden-tified from non-pulse-like and pulse-like ground motions for different damping ratios and structural periods. 2. Ground motions requiring extended analysis (EAGMs) The governing equation of motions of a single-degree-of-freedom (SDOF) system under earthquake ground motions can be expressed as: ü(t)+ 2ξωu̇(t) + ω2u(t) = − üg(t) (1)  where ω and ξ are the circular frequency and the damping ratio of the SDOF system respectively; and üg is the input ground acceleration. The maximum duration (denoted by tmax hereafter) is an important param-eter of earthquake ground motions. The response of a SDOF system under a ground motion can be divided into two phases: the during- shaking phase (0 ≤ t ≤ tmax), when the SDOF system experiences forced vibration; and the post-shaking phase (t > tmax), when the SDOF system vibrates freely based on the motions of the structure at the end of * Corresponding author. E-mail address: htsang@swin.edu.au (H.-H. Tsang).  Contents lists available at ScienceDirect Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn https://doi.org/10.1016/j.soildyn.2023.108102 Received 12 June 2023; Accepted 21 June 2023   Soil Dynamics and Earthquake Engineering 173 (2023) 1081022the shaking (t = tmax). A ground motion requires an extended analysis (denoted by EAGM hereafter) for calculating the response spectra when the time of occur-rence of AMR, tAMR, is greater than tmax for any structural periods be-tween 0 and 20 s. Otherwise, it is denoted as a non-EAGM. Fig. 1 shows the variation of tAMR against the structural periods under two sample ground motions. It is shown that the H2 component of RSN94 is an EAGM, whereas the H1 component of RSN101 is a non-EAGM (both records can be found in the PEER-NGA2 database). If the response spectrum of an EAGM is calculated only within the duration of the shaking, the response spectral values would be underestimated. Fig. 1 also shows that tAMR is smaller than tmax + 0.5T for all structural periods. In the present study, AMRdur and AMRpost are used for depicting the absolute maximum response in the during-shaking phase and the post- shaking phase, respectively, and the real AMR is the larger value of AMRdur and AMRpost. Structural response history analysis conducted only in the during-shaking phase might underestimate the real AMR of the SDOF system. To quantify the underestimation, a ratio between AMRdur and AMR is defined as: β=AMRdurAMR≤ 1 (2) β = 1 indicates that the AMR occurs within the during-shaking phase, whereas β < 1 indicates that the AMR develops during the post-shaking Fig. 1. Time of occurrence of the absolute maximum response (AMR), tAMR, for acceleration response spectra under (a) H2 component of RSN94: Wrightwood - 6074 Park Dr during 1971 San Fernando earthquake (a ground motion record requiring extended analysis, EAGM) and (b) H1 component of RSN101: Cape Mendocino during 1975 Northern California earthquake (non-EAGM). Fig. 2. Percentage of ground motions that require extended analysis (EAGMs) to the total number of ground motions: (a) non-pulse-like ground motions and (b) pulse-like ground motions. Fig. 3. (a) Acceleration, (b) velocity and (c) displacement response spectra of H2 component of RSN94 with a 5% damping ratio with (AMR) and without (AMRdur) considering post-shaking response and underestimation ratios for (d) acceleration, (e) velocity and (f) displacement response spectra. J. Yi and H.-H. Tsang                                                                                                                                                                                                                         Soil Dynamics and Earthquake Engineering 173 (2023) 1081023phase. The smaller the value of β, the larger the extent of the underes-timation. Structural response history analysis should be extended to the post-shaking phase if β < 1. To investigate the percentage of ground motions belonging to EAGMs, 255 non-pulse-like three-component records and 139 pulse-like ones are selected from the PEER-NGA2 database. The selected records represent a wide range of intensities, from relatively weak motions to very strong shakings, regardless of geographical origins and whether the earthquakes were of crustal origin or associated with subduction zones. Three components of a record, namely two horizontal ground motions and a vertical one, are considered separately to calculate the response spectra, which results in 765 non-pulse-like ground motions and 417 pulse-like ones to be investigated. The response spectra of selected ground motions were calculated for structural periods from 0.05 s to 20 s, at intervals of 0.05 s. Two ap-proaches are applied to calculate the response spectra: the first adopts AMRdur, where the maximum response is calculated within the duration of the shaking (t ≤ tmax); and the other adopts AMR, where the calcu-lation is extended by half of the structural period after the excitation has ended (t ≤ tmax+ 0.5T). The percentage of EAGMs in the total number of ground motions is determined under different damping ratios for non-pulse-like and pulse- like ground motions [5], as shown in Fig. 2. For small damping ratios (ξ = 1–2%), there is a significant chance for a non-pulse-like ground mo-tion to be an EAGM (exceeding 10%), while that for a pulse-like EAGM is much smaller (less than 10%). With the increase of damping ratios, the percentages of non-pulse-like and pulse-like EAGMs are gradually reduced. As the damping ratio increases to 20%, all ground motions belong to non-EAGMs. Under the same damping ratio, non-pulse-like ground motions show a much greater chance of being EAGMs than pulse-like ones. Meanwhile, the ratios for acceleration and displacement spectra are generally comparable, while that of the velocity spectra is generally less than the other two. The above results indicate that a ground motion is more likely to be an EAGM for low damping ratios, non-pulse-like ground motions and when calculating acceleration and displacement spectra. 3. Structural period windows for EAGMs Fig. 3(a)–(c) shows the response spectra of a sample ground motion – the H2 component of RSN94, in which AMRdur represents the resulting spectra adopting the calculated values of AMRdur, AMR represents the accurate spectra. The spectral acceleration and displacement can be underestimated by over 20%, and the spectral velocity can be under-estimated by over 10%, which indicates that the difference between the response spectra based on AMR and AMRdur, respectively, can be very significant at certain structural period windows. Fig. 3(d)–(f) shows the underestimation ratios versus periods for the acceleration, velocity and displacement spectrum, respectively. Taking 0.95 as the threshold value for significant underestimation, the period window is 2.7–3.9 s and 4.8–5.9 s for acceleration spectra, 6.15–6.65 s for velocity spectra, and 2.6–3.85 s and 4.8–5.8 s for displacement spectra, respectively. The PEER-NGA2 website (http://peer.berkeley.edu/nga) is the most commonly adopted tool for selecting spectrum-compatible earthquake records [6]. Fig. 3(a) also presents the spectral data extracted from the PEER-NGA2 website (denoted by “PEER”). It illustrates that the PEER-NGA2 spectrum is consistent with the AMRdur, which implies that only the during-shaking phase is taken into consideration. This, on one hand, indicates a general unawareness of the fact that the AMR would occur in the post-shaking phase and, on the other hand, raises the caution of considering the EAGMs when using PEER-NGA2 for selecting spectrum-compatible ground motions. Fig. 4 shows the percentage of EAGMs for a wide range of structural periods for non-pulse-like and pulse-like ground motions. At small damping (ξ = 1%), structural periods that require extended analysis for the calculation of acceleration and displacement spectra cover a wide range, namely 0.9–20 s for non-pulse-like ground motions and 2.5–20 s for pulse-like ground motions. With the increase of damping ratios, the period window is narrowed gradually. For instance, it is reduced to 2.3–10 s and 5.2–5.55 s for non-pulse-like ground motions and pulse-like ones, respectively, under a 5% damping ratio. Compared to acceleration and displacement spectra, velocity spectrum has a narrower period window for structural periods requiring extended analysis, namely 0.9–11.3 s for non-pulse-like ground motions and 2.5–7.8 s for pulse-like ones under a 1% damping ratio. The results above indicate that struc-tural systems of longer natural periods typically require extended analysis, which raises the cautions in the seismic analysis of long-period structures, like base-isolated structures, high-rise buildings and long- span bridges [7–10]. Fig. 4 also shows that the percentage of EAGMs for each structural Fig. 4. Percentages of EAGMs (tAMR > tmax) for different structural periods for non-pulse-like and pulse-like ground motions.  J. Yi and H.-H. Tsang                                                                                                                                                                                                                         Soil Dynamics and Earthquake Engineering 173 (2023) 1081024period are alike for acceleration and displacement spectra, but different for velocity spectrum. Since acceleration spectra are used more frequently than the other two, only acceleration spectra are discussed hereafter. When the damping ratio is small (ξ = 1%), a large number of EAGMs might occur at a given structural period, with the highest per-centage more than 2.5% under non-pulse-like ground motions and 1% under pulse-like ones. With the increase of the damping ratio, the number of EAGMs decreases significantly for the same period. For instance, under the default damping ratio of 5%, the highest percentage of EAGMs is reduced to 1.2% for non-pulse-like ground motions and 0.2% for pulse-like ones, respectively. When the damping ratio becomes larger than 10%, only a small fraction of non-pulse-like ground motions (less than 0.5%) belong to EAGM for some specific periods. 4. Underestimation ratios for EAGMs For different damping ratios, the values of β for acceleration spectra are calculated and shown in Fig. 5 for a wide range of structural periods. For non-pulse-like ground motions, the values of β can be as small as 0.7 at a small damping ratio (ξ = 1%), which indicates a great extent of underestimation in spectral acceleration based on AMRdur instead of AMR. The underestimation is slightly alleviated along with the increase of the damping ratio, but very low values, like β = 0.75, can still be observed at the default damping ratio ξ of 5%, and the underestimation is still significant for higher damping ratios, e.g., ξ = 10%. Taking β of 0.95 as the threshold value for significant underestimation, there is a great chance for an EAGM to result in significant underestimation for small damping ratios. For ξ = 1%, the percentage of EAGMs with sig-nificant underestimation is 9.3% (71 out of 765) within the period window of 1.4–18.1 s, while it becomes 1.8% (12 out of 765) within 2.75–8.85 s, as the damping ratio increases to 5%. For pulse-like ground motions, the underestimation is not as significant as non-pulse-like ones. The minimum value of β is 0.87 for ξ = 1%, whereas the percentage of EAGMs with significant underestimation are 2.6% (11 out of 417) at period ranges of 4.45–10.7 s and 15.3–17.25 s. As the damping ratio increases to ξ = 5%, all the values of β become greater than 0.95. 5. Conclusions This study reveals for the first time that some real ground motions might induce AMR posterior to the shaking, and the peak response calculated in the during-shaking phase might be much smaller than the AMR. In such cases, it is recommended that extended analysis should be conducted on these ground motions (EAGMs) by half of the structural period after the excitation has ended. Post-shaking analysis is more important when calculating accelera-tion and displacement spectra under low damping ratios. With the in-crease of damping ratios, the number of EAGMs and the extent of underestimation are significantly reduced, and the structural period window that requires extended analysis is narrower. The present study has important implications especially for long- period structures, including base-isolated structures, high-rise build-ings and long-span bridges, and low-damping systems, such as masonry structures and non-structural components. 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. Data availability Data will be made available on request. References [1] Clough RW, Penzien J. Dynamics of structures. Berkeley: Computers & Structures, Inc; 2003. [2] Chopra A. Dynamics of structures – theory and applications to earthquake engineering. One Lake Street, Upper Saddle River: Prentice Hall; 2007. [3] Botta F, Cerri G. Shock response spectrum in plates under impulse loads - ScienceDirect. J Sound Vib 2007;308(3–5):563–78. [4] Soize C. Information theory for generation of accelerograms associated with shock response spectra. Comput Aided Civ Infrastruct Eng 2010;25(5):334–47. [5] Shahi S, Baker J. NGA-West2 models for ground motion directionality. Earthq Spectra 2013;30:1285–300. [6] Bozorgnia Y, Abrahamson NA, Atik LA, et al. NGA-West2 research project. Earthq Spectra 2014;30(3):973–87. [7] Yi J, Zhou J, Ye X. Seismic control of cable-stayed bridge using negative stiffness device and fluid viscous damper under near-field ground motions. J Earthq Eng 2020;26(5):2642–59. [8] Yi J, Yu DE, Zhou J. Seismic control of cable-stayed bridges with bearing uplift using tie-down rope-spring device. J Earthq Eng 2021;26(13):6969–92. [9] Hu RP, Xu YL, Zhao X. Optimal multi-type sensor placement for monitoring high- rise buildings under bidirectional long-period ground motions. Struct Control Health Monit 2020;27(6):e2541. [10] Tsang HH. Analytical design models for geotechnical seismic isolation systems. Bull Earthq Eng 2023;21(8):3881–904. Fig. 5. Underestimation ratio (β) for acceleration spectra of EAGMs (one color represents one EAGM).  J. Yi and H.-H. Tsang                                                                                                                                                                                                                         

Significance of post-shaking response of SDOF systems

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Significance of post-shaking response of SDOF systems

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