| Earthquake events over the past century have revealed that an earthquake of large magnitude often triggers numerous aftershocks.Although the magnitudes of these aftershocks are commonly less than that of the mainshock,an aftershock has considerable potential to cause structural damage,casualties,and economic losses.Due to the short interval between the mainshock and its aftershock,the buildings with the mainshock-induced damage are usually not repaired,and may suffer additional damage even collapse caused by the following aftershocks.If the aftershock is close to the city,or it is relatively shallow,the aftershock is capable of causing major damage.Therefore,the mainshock-aftershock(MS-AS)seismic sequence attracts more and more scholars' attention as a type of multi-hazard,and the effects of MSAS seismic sequence on the structures have become a hot topic.The MS-AS ground motions have been the key part of the research,as the inputs of the structures.Due to a lack of the real MS-AS ground motions,how to generate proper synthesized MSAS ground motion becomes the top priority.Because the aftershocks are triggered by the mainshock,there are connections and randomness between the mainshock and its aftershocks in the respects of magnitude and source location,etc.Due to the above reasons,the characteristics of the aftershock ground motions are correlated to those of the mainshock ground motions,for example,the strong-motion duration,the predominant period,and so on.The purpose of this study is to offer a method of selecting aftershock ground motions properly,given the mainshock ground motions.To do this,this study selectes 662 MS-AS seismic sequences from the PEER NGA-West2 ground motion database,and analyzes the correlation of the intensity measures and spectral shape(epsilon)of the ground motion between the mainshocks and the aftershocks.Firstly,the correlation coefficients of the selected parameters of the ground motions between the mainshocks and the aftershocks are analysed,then the optimum marginal distributions and copula functions are selected according to the AIC and BIC,to build the joint distribution of these parameters between the mainshocks and the aftershocks.After that,the conditional distribution,the conditional mean and the conditional standard deviation of the aftershocks' parameters are obtained given the parameters of the mainshocks.Due to the complicated computation of the copula mean,the bivariate Gaussian distribution is used to build the joint distribution between the mainshock epsilon and the aftershock epsilon,and a simplified calculation of the conditional mean and the conditional standard deviation of the aftershock epsilon given the mainshock epsilon is given.Based on the correlation of epsilon between the mainshocks and the aftershocks,the concept of the conditional mean spectrum(CMS)is extended to the aftershocks,then the study proposes the conditional mean spectrum of aftershocks(CMSA)and offers its detailed calculation procedure.Unlike the conditional mean spectrum proposed by Baker,the condition of the CMSA is the spectral acceleration of the mainshock ground motion at the same period,whereas the ‘conditional' in the CMS is the spectral acceleration of the ground motion at the interesting period.The calculation method of the CMSA is given under the following conditions:(1)all the aftershock information is known;(2)only the aftershock magnitude is known;(3)all the aftershock information is unknown.If part of or all the information is unknown,the aftershock magnitude and the location of the aftershock rupture can be determined with the Monte Carlo simulation,according to the relation of the magnitude,the location and depth of the source,the strike of the rupture and so on between the mainshocks and the aftershocks.Then the size of the aftershock rupture is obtained according to the regression model between the size of the rupture and the magnitude,the type of the rupture.Finally,all the other needed parameters of the ground motion prediction equation can be calculated according to their definitions,for example,the rupture distance,the Joyner-Boore distance,and so on.The site condition of the aftershock ground motion is normally assumed to the same as that of the mainshock ground motion,without regard to the soil liquefaction of the interesting site.Based on the CMSA,this study proposes a method of selecting aftershock ground motions given the mainshock ground motions.The aftershock ground motions are selected to matching the targeted spectrum,which is the CMSA.The aftershock ground motions are selected initially with the limitation of the magnitude and the conditional duration.When matching the targeted spectrum,the CMSA could be matched,or the CMSA and its conditional standard deviation could be matched simultaneously.Three RC frames with different floors are selected.The levels of the seismic hazard are 2%-in-50-years and 10%-in-50-years,respectively.30 mainshock ground motions are selected using CMS for every frame with every seismic hazard.Then 30 aftershock motions are selected for each group of mainshock ground motions.Meanwhile,the repeated and random aftershock ground motions are also generated for each group of mainshock ground motions.The above repeated and random aftershock ground motions are compared with the aftershock ground motions selected based on the CMSA,including the response spectra and their influence on the response of the frames.Finally,this study analyzes the influence factors on the selection of the aftershock ground motions based on the CMSA,such as the duration,the period range and polarity,spectral shape and the scaling method.Compared with the prediction spectrum calculated by the ground motion prediction equation,the CMSA proposed in this study matches the real response spectrum of the aftershock ground motion better.The method of selecting aftershock ground motion based on the CMSA can reflect the randomness of the magnitude,the location of hypocenter and so on between the mainshocks and the aftershocks,and the correlation of the intensity measures,the spectral shape,the duration,and so on,between the mainshock and the aftershock ground motions.Therefore,this method can be used to select reasonable ground motions of the aftershocks,given the mainshock ground motions. |
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