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Ductility demands and residual displacements of pinching hysteretic timber structures subjected to seismic sequences
Soil Dynamics and Earthquake Engineering 114 (2018) 392–403
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Ductility demands and residual displacements of pinching hysteretic timber structures subjected to seismic sequences
T
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Wuchuan Pu , Ming Wu Department of Civil Engineering, Wuhan University of Technology, Wuhan, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Pinching hysteresis Timber structure Seismic sequence Residual displacement Ductility demand
Building structures constructed of timber components are characterized by a pinching-type hysteresis that indicates degraded stiffness and strength. Owing to the significant effect of their loading history, these types of structures may be more prone to failure when subjected to sequential seismic excitations. This study investigates the effect of seismic ground motion sequences on the ductility demands and residual displacements of building structures with pinching hysteretic models. A single-degree-of-freedom (SDOF) structure is considered, and is modeled with the hysteretic model consisting of a slip element and a bilinear element. The seismic ground motion sequences are simulated by repeating ground motion records with differing intensities. Through dynamic time history analysis, the effect of the seismic intensity, ductility level, hysteretic parameters, and site conditions are investigated. The results indicate that the seismic sequences amplify the ductility demands of pinching hysteretic structures, and this effect is more significant for short period structures. The pinching hysteretic structures have ductility amplification factors that are higher than those of bilinear hysteretic structures. The residual displacement shows a relatively strong correlation to the maximum displacement, and the ratio of residual displacement to maximum displacement approximately obeys an exponential probability distribution. Based on the numerical results, empirical formulas for estimating the ductility demand amplification and the probability density distribution of the residual displacement are proposed.
1. Introduction A large earthquake can be followed by smaller aftershocks, and may also be preceded by foreshocks. The foreshock-mainshock or mainshock-aftershock sequences can lead to ground acceleration sequences at specific sites, causing the engineering structures constructed on these sites to experience repeated seismic excitations in a short period of time. The time interval between sequential earthquakes varies widely, from several minutes to several years. During very short time intervals within which the repair of damaged structures cannot be performed, repeated seismic excitation of these damaged structures can result in an undesirable accumulation of damage, and a structure that would survive a single excitation may fail under the effect of a seismic sequence. While seismologists are trying to determine the correlation between the foreshock, mainshock, and aftershocks, at present it is still difficult to predict the occurrence of foreshocks or aftershocks [1,2]. However, for engineering design, structures are expected to suffer repeated earthquake excitations over a short period of time, particularly in areas where large numbers of sequential earthquakes have previously occurred. For example, foreshocks were observed during the 1975
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Haicheng earthquake, the 1995 Kozani-Grevena earthquake [2], and the 2016 Kumamoto earthquake, while aftershocks have occurred in many major earthquakes, including the 2008 Wenchuan earthquake, the 2010 Haiti earthquake, the 2011 East Japan earthquake, the 2011 Christchurch earthquake, the 2016 Kumamoto earthquake, and the 2016–2017 Central Italy earthquakes. The number of aftershocks can vary from a few to hundreds. The 2008 Wenchuan earthquake had a mainshock of magnitude 8.0, followed by five aftershocks with magnitudes greater than 6.0 [3]. The 2010 Haiti earthquake had a mainshock of magnitude 7, which was followed by 14 aftershocks with magnitudes of 5–6.1 [4]. The 2011 East Japan earthquake had a mainshock of magnitude 9.0, which was followed by more than 7000 aftershocks over the next year. From August 2016 to January 2017, the central Italy was hit by an earthquake sequence, within includes 9 Mw5+ earthquakes and multiple aftershocks following these major shocks [28]. In current seismic codes, however, design earthquakes are usually defined as single event, and the effect of seismic sequences is not taken into account. A number of studies have investigated the influence of ground motion sequences on various types of structures. Amadio et al.
Corresponding author. E-mail address: [email protected] (W. Pu).
https://doi.org/10.1016/j.soildyn.2018.07.037 Received 10 February 2018; Received in revised form 14 June 2018; Accepted 22 July 2018 0267-7261/ © 2018 Elsevier Ltd. All rights reserved.
Soil Dynamics and Earthquake Engineering 114 (2018) 392–403
W. Pu, M. Wu
Fig. 1. Pinching hysteretic model.
strength of stiffness, and the resulting change in structural performance may have unfavorable effects on the structural response. In the 2016 Kumamoto earthquake, a large number of timber structures, including many historical structures of great cultural value, failed as a result of the sequential excitations of foreshocks, mainshock, and aftershocks. This study focuses on timber structures, which have a very limited energy dissipation capability due to the pinching phenomenon in their hysteretic loops. These characteristics are substantially different from steel or concrete structures, which have been simulated by a bilinear model or modified Clough model in previous studies. By employing a pinching hysteretic model, this study investigates the ductility demands and residual displacements of timber structures subjected to ground motion sequences. These ground motion sequences are modeled by replicating a specific ground motion record, and the intensities of the two sequential ground motions are adjusted to simulate foreshock-mainshock and mainshock-aftershock sequences. Based on the numerical results, some insights into the effect of sequential earthquake excitations on pinching hysteretic structures are presented.
[5] simulated the effects of a seismic sequence on single-degree-offreedom (SDOF) structures using a bilinear model, a degrading stiffness hysteretic model without pinching, and Clough's model. Three ground motions were used as earthquake inputs, and the base shear, ductility, and dissipated energy were investigated with a time history analysis. A parameter, q, is defined as the ratio between the maximum accelerogram that a structure can withstand without failure and the accelerogram at which yielding first appears in the structure. The results demonstrated that multiple earthquakes can induce considerable accumulation of damage and result in a consequent reduction in the qfactor. Following this research, Fragiacomo et al. [6] conducted extensive studies considering different types of steel frame structures. Hatzigeorgiou and Beskos [7] investigated the effects of multiple earthquakes on the inelastic displacement ratio of structures. A total of 112 ground motion records obtained at different types of sites were applied to structures simulated with a bilinear model. An empirical formula for the inelastic displacement ratio was proposed, which incorporated the effects of the structural period, damping ratio, post-yield stiffness ratio, and force reduction factor. Hatzigeorgiou [8] also investigated the ductility demand and strength of an inelastic structure subjected to repeated near- and far-fault earthquakes. Di Sarno [9] employed ground motion sets obtained during the 2011 earthquake in Tohoku, Japan. The inelastic spectral responses of the ground motion sets were derived, and a two-span two-story frame structure was subjected to multiple earthquake ground motions. It was recommended that the effects of stiffness and strength degradation due to sequential earthquakes should be considered in modern codes of practice. Zhai et al. [3,10–12] carried out a series of studies investigating the ParkAng damage index, constant damage displacement ratio, and strength reduction factor of structures subjected to a mainshock-aftershock sequence. Ruiz-Garcia et al. [13] investigated the interstory drift and residual drift of reinforced concrete (RC) frame buildings under the excitation of seismic sequences, and determined that the building response depends on the ratio of the damaged period of vibration to the dominant period of the aftershock. Ruiz-Garcia and Aguilar [14] investigated the influence of modeling assumptions on the maximum and residual displacements of steel frame buildings subjected to mainshockaftershock sequences, and highlighted the importance of the building model for the assessment of structural behavior induced by aftershocks. Very recently, Hosseinpour and Abdelnaby [15] investigated the effects of irregularity, earthquake direction, aftershock polarity, and vertical component of the earthquake on structural performance. Shin and Kim [16] demonstrated that the frequency contents play an important role in the response induced by aftershocks. Nazari et al. [17] examined the collapse risk of wood frame buildings, and determined that the effect of aftershocks on the damage states are more significant than their effects on collapse for low-rise buildings. These previous studies have demonstrated that seismic sequences produce responses that differ from those that occur during a single event, and the effect depends on numerous factors, including the structural period, structural modeling approach, hysteretic model, damping ratio, and frequency contents of the ground motions. During repeated earthquakes, degradation of structures is of great concern [18], as the preceding earthquake causes reduction in the structural
2. Hysteretic model of the structure Fig. 1(a) shows the force–displacement relationship curve of the structural model considered in this study. The skeleton curve is composed of three segments: the first represents the elastic stiffness, K1; the second segment indicates the stiffness of the material after cracking, K2; and the third indicates the stiffness, K3, after reaching the yielding force, Fy. The stationary hysteresis is indicated by the thick lines. This model is characterized by a pinching phenomenon, which represents the typical hysteretic characteristics of timber systems, including X-lam buildings and light-frame walls [19]. The model can be decomposed into a slip element (Fig. 1(b)) and a bilinear element (Fig. 1(c)). The slip element has a bilinear skeleton curve, with the elastic stiffness and postyield stiffness denoted as Ks1 and Ks2, respectively. The slip element experiences slip-type hysteresis (thick line) after the virgin circle, with no energy dissipated in slip hysteresis. The yield displacement is denoted as uy. The energy dissipation of the structure is contributed only by the bilinear element. Similar to the definitions established by Matsuda and Kasai [20], Kb1 and Kb2 denote the initial elastic stiffness and the post-yield stiffness of the bilinear model, respectively, and uc is the displacement corresponding to the elastic limit of the bilinear element. Matsuda and Kasai [20] presented a set of parameters for this hysteretic model: K1 = 0.44 kN/mm, Kb1/K1 = 0.53, Kb2/Kb1 = 0.0566, uc= 4.5 mm, and uy= 18 mm. These parameters were obtained from experiments with typical polywood structures used in Japan. These parameters are adopted as benchmark values. In addition, some of these parameters are also adjusted to investigate their effect on structural response. In addition to the benchmark values, Kb2/Kb1 = 0.2 and Kb1/ K1 = 0.75 and 1.0 are also considered. Note that when Kb1/K1 is equal to 1.0, the structural model becomes a bilinear model. Fig. 2 shows the hysteretic curves with different parameter values. The structural mass is adjusted so that the elastic period, T, is 0.5 s, 0.8 s, 1.2 s, and 2.0 s. An inherent viscous damping ratio of 0.02 is assumed, and the damping coefficient is proportional to the tangent stiffness. This damping assumption is commonly used for dynamic analysis of timber structures in 393
Soil Dynamics and Earthquake Engineering 114 (2018) 392–403
W. Pu, M. Wu
Fig. 2. Hysteretic model shapes.
Japan.
ground motions. Following the analysis based on the repeated ground motion sequence, a set of real ground motion sequences will be used to verify the results. Ground motions are selected from the Pacific Earthquake Engineering Research Center (PEER) strong motion database. A total of 40 ground motions are selected, 10 in each of four soil types (types A, B, C, and D in the United States Geological Survey site classification system [24]). Table 1 lists the details of the selected ground motions. Fig. 6 shows the acceleration and displacement spectra of all of the ground motions and the average for each soil type, in which the PGA of the ground motions are normalized to 0.3g. For establishing the seismic sequences, the PGA is set as 0.1 g, 0.2 g, 0.3 g, 0.4 g, 0.5 g, and 0.6 g. For each ground motion, 36 (6 × 6) sequences with different combinations of intensities are considered. The two ground motions are combined with a time interval of 30 s, which is 15 times the maximum structural period considered in this study and thus considered long enough to allow the structure to stop vibrating after the first excitation.
3. Model of ground motion sequences Previous studies have employed various methods to model seismic sequences. These methods include replicating the mainshock with or without scaling [5,7,21], randomly selected ground motions recorded in similar soil conditions [8,22], ground motion matching the spectra of the mainshock [16], and real seismic ground motion sequences [9]. Moustafa and Takewaki [23] presented a simple stochastic model for simulating strong ground sequences, in which the ground motion is represented as the product of a stationary Gaussian random process and a repeating envelope function. It has been reported that the frequency contents of aftershocks can differ significantly from those of the mainshock [22]. Therefore, it is difficult to simulate an aftershock deterministically based on the frequency contents of the mainshock. On the other hand, using a random combination of ground motions for the aftershock is also not ideal, as it is difficult to determine which ground motion is most critical for the structure under consideration. An example is presented to show the different responses that can occur for ground motions with identical peak ground acceleration (PGA). Two seismic sequences are established by combining ground motions No. 29 and No. 30 in Table 1, which were recorded in different directions at the same station during the 1994 Northridge earthquake. Fig. 3 shows the acceleration time history of the two sequences. The PGA of both ground motions are set to 0.6 g, and a structural model with a natural period of 0.5 s is subjected to the two seismic sequences. The displacement time histories and hysteretic curves obtained for these two cases are shown in Fig. 4 and Fig. 5, respectively. It is clear that the responses are governed by ground motion No. 29, regardless of whether it is modeled as the first excitation or the second excitation. This demonstrates that seismic ground motions, even those recorded during the same event and at the same station, can have completely different effects on structural response. If the structural response is completely governed by any single ground motion within a sequence, the significance of sequential excitations will be reduced. For simplicity, in this study the seismic sequences are firstly modeled by replicating and scaling the same ground motion. Even though this model is not necessarily realistic or the most critical, it is still useful for clarifying the response of pinching structures under the excitation of sequential earthquakes. By employing this model, a sequence with arbitrary increasing or decreasing intensities can be assumed, and thus the analysis can be performed over a wide range of intensity ratio. It is also noted that for a specific site, the intensity induced by an aftershock or a foreshock may be larger than that induced by the mainshock, since the intensity is affected not only by the earthquake magnitude but also by the site-source distance, source mechanism etc. In this study, it is assumed that the intensity of the mainshock is larger than that of foreshock or aftershock, as observed in most cases. Only two sequential ground motions are considered in these sequences. The results obtained from two ground motion sequences provide fundamental knowledge for characterizing the effect of ground motion sequences, as the underlying effect is the varied structural performance resulting from consecutive
4. Effect of seismic sequence on ductility demands 4.1. Effect of intensity ratio on ductility demand amplification The ductility demands of pinching structures are investigated first. The ductility factor, μ, is defined as the maximum displacement divided by uc. The effect of seismic sequences on the ductility demands can be evaluated with the maximum ductility ratio of the two sequential excitations, μ2/μ1. The ductility ratio, μ2/μ1, is clearly affected by the intensity ratio of the two ground motions. Fig. 7 shows μ2/μ1 versus PGA2/PGA1 for structures with benchmark hysteretic parameters and various structural periods and soil types. Note that only those cases in which the displacement from the first ground motion is larger than uc are shown. From these figures, two observations can be made: (1) for PGA2/PGA1 = 1, the response ratio is greater than 1, which suggests that repeating the same ground motion causes an amplification of the ductility demands of pinching hysteretic structures; and (2) in a logarithmic coordinate system, the ductility ratio varies approximately linearly with the intensity ratio. The relationship can be fitted by the following equation:
PGA2 ⎞ μ 2 / μ1 = α ⎛ ⎝ PGA1 ⎠ ⎜
β
⎟
(1)
In Eq. (1), if PGA2/PGA1 = 1, then μ2/μ1 = α, and thus α denotes the ductility demand amplification ratio for a structure subjected to a sequence with identical intensities. Also, as β increases, the ductility demand amplification effect becomes more significant. Eq. (1) can be applied to all cases, but different values for α and β are obtained depending on the structural properties and soil type. Table 2 lists the values of α and β for cases with various T, Kb2/Kb1, and soil type; Table 3 lists the values of α and β for cases with different T and Kb1/K1, in which only soil types B and D are considered. The results of Table 2 indicate that the short-period structures subjected to earthquake sequences experience significant amplification of ductility demand. As the structural period increases, the amplification of ductility demand becomes weak. For example, for T = 0.5 s and Kb2/Kb1 394
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Table 1 Selected ground motion records. No.
Date
Record name
Comp.
Station name
PGA (g)
Vs30 (m/s)
Site
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
1974/11/28 1974/11/28 1979/08/06 1979/08/06 1994/01/17 1994/01/17 1994/01/17 1994/01/17 1995/01/16 1989/10/18 1978/08/13 1971/02/09 1971/02/09 1975/06/07 1975/06/07 1978/08/13 1992/06/28 1992/06/28 1980/06/09 1980/06/09 1979/10/15 1979/10/15 1999/11/12 1999/11/12 1992/04/25 1992/04/25 1994/01/17 1994/01/17 1994/01/17 1994/01/17 1979/10/15 1979/10/15 1983/05/02 1983/05/02 2004/09/28 2004/09/28 1987/11/24 1987/11/24 1983/05/02 1987/10/01
Hollister-03 Hollister-03 Coyote Lake Coyote Lake Northridge-01 Northridge-01 Northridge-01 Northridge-01 Kobe Japan Loma Prieta Santa Barbara San Fernando San Fernando Northern Calif Northern Calif Santa Barbara Landers Landers Victoria Mexico Victoria Mexico Imperial Valley-06 Imperial Valley-06 Duzce Turkey Duzce Turkey Cape Mendocino Cape Mendocino Northridge-03 Northridge-03 Northridge Northridge Imperial Valley Imperial Valley Coalinga-01 Coalinga-01 Parkfield-02 Parkfield-02 Superstition Hills Superstition Hills Coalinga-01 Whittier Narrows
N157 N247 N230 N320 N095 N185 N175 N265 EW EW N132 N111 N201 N030 N120 N222 NS EW N315 N045 N012 N282 NS EW NS EW EW NS EW NS N230 N140 NS EW EW NS EW NS NS NS
47379Gilroy Array #1 47379Gilroy Array #1 47379Gilroy Array #1 47379Gilroy Array #1 90017LA - Wonderland Ave 90017LA - Wonderland Ave 24207Pacoima Dam (downstr) 24207Pacoima Dam (downstr) Kobe University 57180Los Gatos Lexington Dam 283Santa Barbara Courthouse 126Lake Hughes #4 126Lake Hughes #4 89005Cape Mendocino 89005Cape Mendocino 283Santa Barbara Courthouse 22170Joshua Tree 22170Joshua Tree 6604Cerro Prieto 6604Cerro Prieto 6621Chihuahua 6621Chihuahua Duzce Duzce 89509 Eureka 89509 Eureka 24279Newhall - Fire Sta 24279Newhall - Fire Sta 90021 LA-N Westmoreland 90021 LA-N Westmoreland 5057El Centro Array #3 5057El Centro Array #3 36228Parkfield - Cholame 2WA 36228Parkfield - Cholame 2WA 36228Parkfield - Cholame 2WA 36228Parkfield - Cholame 2WA 5210Imperial Valley WLA 5210Imperial Valley WLA 36407Parkfield - Fault Zone 1 90081Carson - Water St
0.100 0.141 0.094 0.117 0.103 0.159 0.416 0.434 0.312 0.411 0.101 0.198 0.156 0.117 0.207 0.202 0.274 0.284 0.633 0.645 0.270 0.254 0.404 0.515 0.154 0.178 0.107 0.205 0.333 0.432 0.097 0.138 0.110 0.113 0.624 0.373 0.131 0.133 0.143 0.110
1428.14 1428.14 1428.14 1428.14 1222.52 1222.52 2016.13 2016.13 1043.00 1070.34 514.99 600.06 600.06 567.78 567.78 514.99 379.32 379.32 471.53 471.53 242.05 242.05 281.86 281.86 337.46 337.46 269.14 269.14 315.06 315.06 162.94 162.94 173.02 173.02 173.02 173.02 179.00 179.00 178.27 160.58
A A A A A A A A A A B B B B B B B B B B C C C C C C C C C C D D D D D D D D D D
= 0.057, the value of 1.22 is obtained for α from soil type A and type C, which means that repeated excitation by the same ground motion can cause amplification of the structural displacement by as much as 22%. There are no significant differences between the results for structures with Kb2/Kb1 = 0.2 and 0.057, and only slight difference is observed in the results of different soil types. On the other hand, as Kb1/K1 increases, the effect of sequential excitations decreases (Table 3). Particularly for Kb1/K1 = 1, the structural model becomes a bilinear model, and α is approximately equal to 1. This implies that pinching hysteretic structures are more significantly affected by seismic sequence than bilinear structures do. Fig. 8 shows examples of hysteretic curves resulting from the pinching model and bilinear model, in which the intensity ratio of the sequence is 1. For the pinching model, as long as the maximum displacement is updated, the skeleton curve makes a contribution to the energy dissipation and load resistance. In terms of the
energy, the pinching model should develop larger peak displacements in the second earthquake so that the structure can dissipate approximately the same amount of energy as the energy dissipated in the first earthquake. In contrast, the bilinear model has no such loading history effect on hysteresis, and it develops almost the same maximum displacement.
4.2. Damaging effect of aftershocks The previous section demonstrates that an aftershock increases the ductility demands of pinching structures even though its intensity is smaller than the mainshock. The underlying reason is that the mainshock causes a reduction of structural seismic performance. Because PGA cannot be used to represent the damaging effects of ground motion directly (as described in Section 3), the ductility factor will be used to
Fig. 3. Acceleration time history of two seismic sequences. (a) No. 30-No. 29 ground motion sequence (b) No. 29-No. 30 ground motion sequence. 395
Soil Dynamics and Earthquake Engineering 114 (2018) 392–403
W. Pu, M. Wu
Fig. 4. Displacement time history of a structure subjected to seismic sequences. (a) No. 30-No. 29 ground motion sequence (b) No. 29-No. 30 ground motion sequence.
an aftershock should consider both the pre-damage level of the structure and the intensity of the aftershock itself. Eq. (2) can be used to predict μ2 based on μ1 and μ2,s. However, it should also be noted that a certain level of error may exist in these predictions, as the coefficients of determination (R2) obtained in Table 4 are very small.
indicate the damaging effect. To determine the damaging effect of an aftershock, the sequences with PGA1 > PGA2 are selected to represent mainshock-aftershock sequences, and the influence of the aftershock is investigated quantitatively. It is noted that the intensity of a ground motion recorded in an aftershock is not necessarily smaller than that of mainshock. “Aftershock” and “Mainshock” are defined based on the earthquake magnitude, however the intensity recorded at a station depends not only on earthquake magnitude but also on many other factors such as the location of epicenter and the frequency components of ground motions. For simplicity, in this study, the sequences with PGA1 < PGA2 are defined as foreshock-mainshock sequence, and the sequences with PGA1 > PGA2 are defined as mainshock-aftershock sequence. In the following analysis, μ2 is the ductility induced by the aftershock following the mainshock, and μ2,s is the ductility of the structure subjected to a single ground motion in an “aftershock” without prior application of a mainshock. Therefore, μ2/μ2,s is the ratio of the ductility demand of a structure pre-damaged in a mainshock to the ductility demand of an intact structure. Fig. 9 shows the distribution of μ2/μ2,s versus μ1 and μ2,s. The results obtained from the mainshockaftershock sequences are almost all located on the left side of the figure and have μ1 > μ2,s. By evaluating the relationship between μ2/μ2,s, μ1, and μ2,s with Table Curve 3D software, Eq. (2) is obtained to describe the relationship between μ2/μ2,s, μ1, and μ2,s.
μ 2 / μ 2,s = a + blnμ1 + clnμ 2,s
4.3. Effect of a foreshock on ductility demand Similar to the previous section, the sequences with PGA1 < PGA2 are selected to represent the foreshock-mainshock sequences. It should be noted that μ2 here refers to the ductility induced by the mainshock following a foreshock and μ2,s denotes the ductility demand induced by the mainshock without a foreshock. In Fig. 9, the results on the right side with μ1 < μ2,s are obtained from the foreshock-mainshock sequences. Compared to the mainshockaftershock sequences, the foreshock-mainshock sequences have smaller values of μ2/μ2,s, most of which are near 1. This implies that foreshocks with smaller intensities have no significant effect on the ductility demand of the structure subjected to the mainshock. To some extent, the results for the foreshock-mainshock sequence can be explained by Eq. (2), which was originally obtained from the mainshock-aftershock sequences. This equation demonstrated that when the wave in the sequence has a smaller intensity (smaller μ1), the value of μ2/μ2,s is relatively small. Therefore, a structure yields a similar ductility demand under the excitation of the mainshock, regardless of whether it has experienced a foreshock. In summary, the displacement magnification effect of an aftershock is extremely dependent on the intensity ratio of the seismic sequence, and in a sequence, the aftershock is likely to lead to ductility demand that is equivalent to or greater than that induced by the mainshock (Section 4.1). The foreshock has a slight effect on structural response, while the response is still governed by the mainshock.
(2)
Based on Eq. (2), μ2/μ2,s can be expressed as linear function of ln μ1 and ln μ2,s. The identified values for the coefficients in Eq. (2), a, b, and c, are listed in Table 4. Coefficients a and b are positive and coefficient c is negative, which implies that the ratio of μ2/μ2,s is proportional to μ1 and inversely proportional to μ2,s. In other words, the more severe the pre-damage to the structures, the larger μ2/μ2,s will be. These results imply that an aftershock may lead to much larger displacement in a structure damaged by the mainshock, while this single “aftershock” causes only a small displacement in an intact structure. On the other hand, as μ2,s becomes large, the aftershock becomes more governing, and whether there is pre-damage is of less significance for the damaging effect of aftershock. Therefore, an estimation of the damaging effect of
5. Residual displacement of pinching hysteretic structures Residual displacement is an important index for quantifying postearthquake structural performance, as it provides information about the
Fig. 5. Hysteretic curves for a structure subjected to seismic sequences. (a) No. 30-No. 29 sequence (b) No. 29-No. 30 sequence. 396
Soil Dynamics and Earthquake Engineering 114 (2018) 392–403
W. Pu, M. Wu
Fig. 6. Acceleration and displacement response spectra for selected ground motions. (a) Acceleration spectra (b) Displacement spectra.
Fig. 7. Relationship between μ2/μ1 and PGA2/PGA1. 397
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that the residual displacement ratio obeys a probabilistic exponential distribution. This resulting probabilistic density function can be expressed as: f(R1/M1) = 15.24e−15.24R1/M1. This exponential distribution is then applied to fit the residual displacement ratio, R2/M2, obtained from the second ground motion, and the regression result is shown in Fig. 11(b). Strictly speaking, R2/M2 does not obey a probabilistic exponential distribution based on the results of statistical hypothesis testing, and using an exponential distribution function is an approximation to illustrate trends. The fitted exponential distribution for R2/ M2 has a smaller rate parameter, which indicates a higher probability of a large ratio of residual displacement to maximum displacement. When structures are subjected to sequential excitations, residual displacements may accumulate when they are located on the same side, thus leading to a R2/M2 ratio that is higher than R1/M1 for a specific probabilistic level. For comparison, Fig. 11(c) and (d) show the results for bilinear hysteretic structures subjected to ground motions on soil types B and D. The residual displacement ratio shows a similar distribution pattern to Fig. 11(a). Similarly, the probabilistic distribution is regressed based on an exponential distribution, and much smaller rate parameters for the exponential distribution are obtained. Compared to bilinear hysteretic structures, the residual displacement of a pinching hysteretic structure is more likely the result of a large maximum displacement. This should be carefully considered when estimating the maximum displacement using the residual displacement.
Table 2 Values of α and β obtained for Eq. (1). T
0.5 s
Kb2/Kb1
0.057
0.200
0.057
0.200
0.057
0.200
0.057
0.200
1.22 1.08 0.89 1.19 1.05 0.92 1.22 1.08 0.90 1.14 1.18 0.95
1.23 1.08 0.90 1.14 1.15 0.89 1.12 1.09 0.90 1.14 1.22 0.95
1.07 0.99 0.95 1.07 1.06 0.94 1.11 1.16 0.92 1.06 1.09 0.90
1.06 1.05 0.96 1.04 1.07 0.94 1.11 1.19 0.93 1.05 1.15 0.92
1.05 1.10 0.95 1.05 0.97 0.95 1.07 1.06 0.89 0.95 1.05 0.93
1.10 1.05 0.95 0.99 1.04 0.96 1.03 1.06 0.91 1.03 1.07 0.94
0.97 0.92 0.94 1.04 0.96 0.94 0.97 0.99 0.91 0.92 0.95 0.91
0.91 1.02 0.94 1.02 1.04 0.95 0.98 1.10 0.90 0.92 1.03 0.94
Soil Type A
Soil Type B
Soil Type C
Soil Type D
α β R2 α β R2 α β R2 α β R2
0.8 s
1.2 s
2.0 s
reparability of the structure and its structural performance during aftershocks. The previous sections have demonstrated that the ductility level of a structure induced by the mainshock substantially affects the structural performance during an aftershock. There are many ways to quantify the maximum deformation of a structure in an earthquake event, such as real-time monitoring and post-earthquake analyses. Estimating the maximum displacement based on residual displacement is another typical approach. There are a number of studies investigating the residual displacements of various types of structures, including structures modeled by bilinear models [25,26] and reinforced concrete structures [27]. The results indicate that the maximum displacement can be effectively estimated from the residual displacements. The residual displacements of pinching hysteretic structures are investigated with a statistical analysis. In Fig. 10, the residual displacement is plotted against the maximum displacement for structures with benchmark hysteretic parameters. As observed for other types of structures, a large maximum displacement generally leads to a large residual displacement. As the maximum displacement increases, the ratio of residual displacement to maximum displacement also increases. The ratio of residual displacement to maximum displacement can be modeled as a power function. The fitted power functions are shown for each subfigure in Fig. 10. The scattering in the results is relatively large, although it is similar to that obtained for RC columns [27]. This indicates that there is a certain level of uncertainty in the estimation of maximum displacement using the residual displacement. To quantify this uncertainty, the distribution of the ratio of residual displacement to maximum displacement is investigated. Fig. 11(a) shows histograms for the ratio of R1/M1 obtained from structures with benchmark parameters, where R1 and M1are the residual displacement and maximum displacement, respectively, induced by the first ground motion. The kernel density is estimated with a Gaussian distribution as the kernel function, and the kernel density curves are indicated by a blue line. The kernel estimation suggests that the residual displacement ratio might obey an exponential distribution. To verify this hypothesis, an exponential distribution is employed to fit the residual displacement ratio data. A statistical hypothesis testing is performed, and it suggests
6. The effect of seismic sequence model For the two ground motions in a real earthquake sequence, significant difference may exist in their frequency components, predominant period, and duration etc. These characteristics may cause results different with those obtained from the repeated sequences. In order to clarify this effect, a comparison study between the real sequence and repeated sequence is performed. The real ground motion sequences are selected from 1999 Chi-Chi earthquake, 2011 East Japan earthquake, 2010 Christchurch earthquake and 2016 Kumamoto earthquake. A set of 10 sequences, which were recorded in two horizontal directions at 5 stations, are selected, as given in Table 5. A subscript number is added to the component name to differentiate the two ground motion in each sequence. In order to obtain more sequence samples, the order of the two ground motions in each sequence are reversed, and thus another set of 10 seismic sequences with different intensity ratios are obtained. These reversed sequences are also considered as real sequences. Totally, 20 realistic sequences are considered. Corresponding to these real sequences, the repeated sequences are built by repeating the second wave in each real sequence, and the PGA of repeated sequence is adjusted so that it is consistent with the real sequence. Fig. 12 shows two examples of real and repeated sequences, which is built based on the sequence of No. 10 of Table 5. These real and repeated sequences are applied to the benchmark model. The index of μ2/μ1 and μ2/μ2,s are investigated again. The μ2/μ1 is shown versus the intensity ratio in Fig. 13(a) and Fig. 13(b) for real
Table 3 Values of α and β obtained for Eq. (1). T
0.5 s
Kb1/K1
0.53
0.75
1
0.53
0.75
1
0.53
0.75
1
0.53
0.75
1
1.19 1.05 0.92 1.14 1.18 0.95
1.14 1.10 0.93 1.10 1.20 0.95
1.06 1.15 0.90 1.01 1.22 0.93
1.07 1.06 0.94 1.06 1.09 0.90
1.08 1.05 0.94 1.03 1.09 0.92
1.03 1.04 0.92 1.01 1.10 0.88
1.05 0.97 0.95 0.95 1.05 0.93
1.06 0.99 0.96 0.97 1.03 0.94
1.05 0.91 0.91 1.03 0.98 0.92
1.04 0.96 0.94 0.92 0.95 0.91
1.08 0.91 0.94 0.97 0.93 0.95
1.06 0.86 0.92 1.01 0.91 0.96
Soil Type B
Soil Type D
α β R2 α β R2
0.8 s
1.2 s
398
2.0 s
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Fig. 8. Hysteretic curves of a pinching model and a bilinear model (ground motion No. 13, 0.4–0.4 g). (a) Pinching hysteretic model (b) Bilinear hysteretic model.
smaller than those of repeated sequences in case of PGA1 > PGA2, and the ductility ratios of real sequences are little larger than those of repeated sequences in case of PGA1 < PGA2. Third, larger dispersion in the results of real sequences is observed. Fig. 14(a) and Fig. 14(b) illustrate the displacement time histories obtained from sequences of PGA1 < PGA2 and PGA1 > PGA2, respectively, and Fig. 15 and Fig. 16 show the corresponding hysteretic curves. In Fig. 14(a), the first wave (EW2 with adjusted PGA1 of 597 gal) of repeated sequence causes a maximum displacement (M1 = 15.06 cm) larger than that caused by the first wave of the real sequence (EW1, PGA1 = 597 gal, M1 = 5.93 cm). The first wave in the repeated sequence is originally obtained from the mainshock, and it possesses the characteristics of a mainshock with large magnitude, such as the longer duration and the larger energy [18]. Because of these properties, the discrepancy between the maximum displacements may be resulted. Moreover, there is only slight difference between the maximum displacement M2 of the real sequence and the repeated sequence, and therefore, a larger ductility demand ratio μ2/μ1 is obtained from real sequence. Correspondingly, under the excitation of sequences of PGA1 > PGA2 (Fig. 12b), the structure develops displacement time histories as shown in Fig. 14(b) and hysteretic curves as shown in Fig. 16. In contrast to the previous observations, the first wave of repeated sequence was originally recorded in small earthquake (EW2), even though it was scaled up to the same PGA of the real wave (EW1), it caused a relatively smaller M1, and finally resulted in a larger μ2/μ1. The values of μ2/μ2,s obtained from real sequences and repeated sequences are also compared in Fig. 17. The results suggest that the μ2/ μ2,s derived from two types of seismic sequences are generally in agreement. The above analysis demonstrate that the results derived from repeated sequences can reflect the general features of structural responses induced by real seismic sequences, while a certain level of error may be caused in the estimation of the ductility demand ratio depending on the intensity ratio. This reason is that simply scaling ground motion cannot reflect the characteristics of the duration and the frequency contents, which are dependent on earthquake magnitude and play an important role in structural responses.
Fig. 9. Relationship among μ2/μ2,s, μ2,s, and μ1 (Kb2/Kb1 = 0.0566, 1 ≤ μ ≤ 40). Table 4 Values for a, b and c in Eq. (2).
Soil Type A
Soil Type B
Soil Type C
Soil Type D
a b c R2 a b c R2 a b c R2 a b c R2
T = 0.5 s
T = 0.8 s
T = 1.2 s
T = 2.0 s
0.45 0.91 − 0.71 0.52 0.84 0.51 − 0.42 0.45 0.35 0.62 − 0.35 0.29 0.40 0.60 − 0.40 0.32
1.00 0.21 − 0.22 0.17 0.91 0.36 − 0.37 0.37 0.81 0.30 − 0.27 0.16 0.90 0.41 − 0.45 0.22
0.98 0.12 − 0.11 0.06 0.92 0.32 − 0.33 0.32 0.26 0.56 − 0.36 0.32 1.11 0.09 − 0.21 0.24
0.91 0.38 − 0.43 0.41 1.26 0.14 − 0.25 0.24 0.69 0.39 − 0.33 0.36 1.20 0.12 − 0.22 0.15
7. Conclusions sequences and repeated sequences, respectively. In each subfigure, the regressed functions based on the form of Eq. (1) are also presented. First, the values of α and β regressed from results of repeated sequences are larger than those obtained in Section 4. The reason is probably that the number of data is less than that of Fig. 7, and thus the estimation error becomes large due to the limited number of data. Second, the regressed α of the real sequences (Fig. 13a) is smaller than that of the repeated sequences (Fig. 13b), while the regressed β of the real sequences is larger than that of the repeated sequences. Since β determines the curve shape of the regressed function, the regression results suggest that the ductility demand ratios of real sequences are little
Timber structures are modeled with pinching hysteretic models and subjected to seismic sequences composed of two ground motions with different PGA. The resulting ductility demands and residual displacements of the pinching hysteretic structures were investigated. The findings obtained from the above investigations can be concluded as follows: (1) Compared to bilinear hysteretic structures, pinching hysteretic structures are more significantly affected by seismic sequences, as they generally lead to higher ductility amplification ratios. The 399
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Fig. 10. Comparison of residual displacement and maximum displacement.
Fig. 11. Probabilistic distribution of the residual displacement ratio.
400
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Table 5 Real seismic sequences used in analysis. No.
Earthquake (year)
Date-Time
1
Chi-Chi Earthquake (1999)
2
Chi-Chi Earthquake (1999)
3
Christchurch earthquake (2011)
4
Christchurch earthquake (2011)
5
Great East Japan earthquake (2011)
6
Great East Japan earthquake (2011)
7
Kumamoto earthquake (2016)
8
Kumamoto earthquake (2016)
9
Kumamoto earthquake (2016)
10
Kumamoto earthquake (2016)
1999/09/20 1999/09/20 1999/09/20 1999/09/20 2010/09/03 2011/02/21 2010/09/03 2011/02/21 2011/03/11 2011/04/07 2011/03/11 2011/04/07 2016/04/14 2016/04/14 2016/04/14 2016/04/14 2016/04/14 2016/04/14 2016/04/14 2016/04/14
17:47 18:03 17:47 18:03 16:35 23:51 16:35 23:51 14:46 23:32 14:46 23:32 21:26 22:07 21:26 22:07 21:26 22:07 21:26 22:07
MW
Station
Comp.
PGA(m/s2)
7.6 6.2 7.6 6.2 7.0 6.2 7.0 6.2 9.0 7.1 9.0 7.1 6.5 5.8 6.5 5.8 6.5 5.8 6.5 5.8
TCU129 TCU129 TCU129 TCU129 CBGS CBGS CBGS CBGS MYG001 MYG001 MYG001 MYG001 KMM006 KMM006 KMM006 KMM006 KMMH16 KMMH16 KMMH16 KMMH16
NS1 NS2 EW1 EW2 N891 N892 S011 S012 NS1 NS2 EW1 EW2 NS1 NS2 EW1 EW2 NS1 NS2 EW1 EW2
6.12 3.90 9.85 9.28 1.52 5.42 1.86 4.43 4.01 3.75 4.13 3.79 5.99 4.51 3.59 2.21 8.02 4.59 9.51 5.87
Fig. 12. Examples of real sequences with inversed intensity ratio (based on sequence of No.10). (a) PGA1 < PGA2 (b) PGA1 > PGA2.
Fig. 13. Relationship between ductility ratio and intensity ratio. Fig. 14. Displacement time history. PGA1 < PGA2 (b) PGA1 > PGA2.
(a)
structures already damaged in the mainshock, and the damaging effect is dependent not only on the intensity of the aftershock, but also on the damage level caused by mainshock. In contrast, the foreshock has little effect on the structures subjected to a subsequent mainshock. (3) In general, a large maximum displacement causes a large residual displacement, and the residual displacement can be modeled as a
ductility amplification ratio resulting from sequential ground motion excitations can be simulated as a power function of the intensity ratio of the ground motions. When subjected to sequences composed of two same ground motions, the ductility demand amplification of pinching hysteretic structures can be as much as 22%, while almost no amplification is observed for bilinear structures. (2) An aftershock has a significant influence on pinching hysteretic 401
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Fig. 15. Hysteretic curves induced by sequences of PGA1 < PGA2.
Fig. 16. Hysteretic curves induced by sequences of PGA1 > PGA2.
aftershock sequences. It should be noted that the SDOF model employed in this study is able to reflect the global behavior of structure, but cannot reflect local damage or higher mode effect of multi-degree-of-freedom structures. In case that identification of these effects is required, further investigation should be performed based on a detailed member-to-member model. Acknowledgement The authors thank the four anonymous reviewers for their helpful comments and suggestions. References [1] Parsons T, Segou M, Marzocchi W. The global aftershock zone. Tectonophysics 2014;618(4):1–34. [2] Lin CH. Foreshock characteristics in Taiwan: potential earthquake warning. J Asian Earth Sci 2009;34(5):655–62. [3] Zhai C, Wen W, Chen Z, et al. Damage spectra for the mainshock-aftershock sequence-type ground motion. Soil Dyn Earthq Eng 2013;45(1):1–12. [4] Wang G, Wang Y, Lu W, et al. Damage demand assessment of mainshock-damaged concrete gravity dams subjected to aftershocks. Soil Dyn Earthq Eng 2017;98:141–54. [5] Amadio C, Fragiacomo M, Rajgelj S. The effects of repeated earthquake ground motions on the non-linear response of SDOF systems. Earthq Eng Struct Dyn 2003;32(2):291–308. [6] Fragiacomo M, Amadio C, Macorini L. Seismic response of steel frames under repeated earthquake ground motions. Eng Struct 2004;26(13):2021–35. [7] Hatzigeorgiou GD, Beskos DE. Inelastic displacement ratios for SDOF structures subjected to repeated earthquakes. Eng Struct 2009;31(11):2744–55. [8] Hatzigeorgiou GD. Ductility demand spectra for multiple near- and far-fault earthquakes. Soil Dyn Earthq Eng 2010;30(4):170–83. [9] Di Sarno L. Effects of multiple earthquakes on inelastic structural response. Eng Struct 2013;56(6):673–81. [10] Zhai C, Wen W, Ji D, Li S. The influences of aftershocks on the constant damage inelastic displacement ratio. Soil Dyn Earthq Eng 2015;79:186–9. [11] Zhai C, Wen W, Li S, Xie L. The ductility-based strength reduction factor for the mainshock-aftershock sequence-type ground motions. Bull Earthq Eng 2015;123(10):2893–914. [12] Zhai CH, Wen WP, Li S, et al. The damage investigation of inelastic SDOF structure
Fig. 17. Comparison of μ2/μ2,s obtained from real sequences and repeated sequences.
power function of the maximum displacement. The probabilistic density function of residual displacement ratio can be fitted by exponential distribution functions, and the rate parameter of 15.237 and 12.071 were obtained from the first excitation and the second excitation, respectively, of a sequence. Furthermore, a specific residual displacement is more likely to result from a large maximum displacement for structures with pinching hysteresis. This is critical for estimating the post-earthquake performance of timber structures. (4) The repeated sequences cannot reflect the difference of duration and frequency contents existing between ground motions in a real sequence, which may have unneglectable influence on structural response. Compared to the real sequences, the repeated sequences may slightly underestimate the ductility amplification ratios induced by foreshock-mainshock sequences, while slightly overestimate the ductility amplification ratios induced by mainshock402
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[14]
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2015;80(716):1525–35. [21] Li Q, Ellingwood BR. Performance evaluation and damage assessment of steel frame buildings under main shock-aftershock earthquake sequences. Earthq Eng Struct Dyn 2007;36(3):405–27. [22] Hatzigeorgiou GD. Behavior factors for nonlinear structures subjected to multiple near-fault earthquakes. Comput Struct 2010;88(5):309–21. [23] Moustafa A, Takewaki I. Response of nonlinear single-degree-of-freedom structures to random acceleration sequences. Eng Struct 2011;33(4):1251–8. [24] Boore DM. Some notes concerning the determination of shear-wave velocity and attenuation. In: Proceeding of geophysical techniques for site and material characterization; 1993. p. 129–34. [25] Macrae GA, Kawashima K. Post-earthquake residual displacement of bilinear oscillators. Earthq Eng Struct Dyn 1997;26(7):701–16. [26] Hatzigeorgiou GD, Papagiannopoulos GA, Beskos DE. Evaluation of maximum seismic displacements of SDOF systems from their residual deformation. Eng Struct 2011;33(12):3422–31. [27] Cheng H, Li H, Wang D, et al. Research on the influence factors for residual displacements of RC bridge columns subjected to earthquake loading. Bull Earthq Eng 2016;14(8):2229–57. [28] Di Ludovico M, Digrisolo A, Moroni C, et al. Remarks on damage and response of school buildings after the Central Italy earthquake sequence. Bull Earthq Eng 2018. https://doi.org/10.1007/s10518-018-0332-x.
under the mainshock-aftershock sequence-type ground motions. Soil Dyn Earthq Eng 2014;59(59):30–41. Ruiz-Garcia J, Marin MV, Teran-Gilmore A. Effect of seismic sequences in reinforced concrete frame buildings located in soft-soil sites. Soil Dyn Earthq Eng 2014;63(3):56–68. Ruiz-Garcia J, Aguilar JD. Influence of modeling assumptions and aftershock hazard level in the seismic response of post-mainshock steel framed buildings. Eng Struct 2017;140:437–46. Hosseinpour F, Abdelnaby AE. Effect of different aspects of multiple earthquakes on the nonlinear behavior of RC structures. Soil Dyn Earthq Eng 2017;92:706–25. Shin M, Kim B. Effects of frequency contents of aftershock ground motions on reinforced concrete (RC) bridge columns. Soil Dyn Earthq Eng 2017;97:48–59. Nazari N, Van de Lindt JW, Li Y. Effect of mainshock-aftershock sequences on woodframe building damage fragilities. J Perform Constr Facil ASCE 2015;29(1):04014036. Takewaki I. Characteristics of earthquake ground motion of repeated sequences. In: Chapter 5: improving the earthquake resilience of buildings. London: SpringerVerlag; 2013. p. 93–113. Heiduschke A, Kasal B, Haller P. Shake table tests of small- and full-scale laminated timber frames with moment connections. Bull Earthq Eng 2009;7(1):323–39. Matsuda K, Kasai K. Passive control design method for MDOF system composed of bilinear + slip model added with visco-elastic damper. J Struct Construct Eng
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Ductility demands and residual displacements of pinching hysteretic timber structures subjected to seismic sequences
T
⁎
Wuchuan Pu , Ming Wu Department of Civil Engineering, Wuhan University of Technology, Wuhan, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Pinching hysteresis Timber structure Seismic sequence Residual displacement Ductility demand
Building structures constructed of timber components are characterized by a pinching-type hysteresis that indicates degraded stiffness and strength. Owing to the significant effect of their loading history, these types of structures may be more prone to failure when subjected to sequential seismic excitations. This study investigates the effect of seismic ground motion sequences on the ductility demands and residual displacements of building structures with pinching hysteretic models. A single-degree-of-freedom (SDOF) structure is considered, and is modeled with the hysteretic model consisting of a slip element and a bilinear element. The seismic ground motion sequences are simulated by repeating ground motion records with differing intensities. Through dynamic time history analysis, the effect of the seismic intensity, ductility level, hysteretic parameters, and site conditions are investigated. The results indicate that the seismic sequences amplify the ductility demands of pinching hysteretic structures, and this effect is more significant for short period structures. The pinching hysteretic structures have ductility amplification factors that are higher than those of bilinear hysteretic structures. The residual displacement shows a relatively strong correlation to the maximum displacement, and the ratio of residual displacement to maximum displacement approximately obeys an exponential probability distribution. Based on the numerical results, empirical formulas for estimating the ductility demand amplification and the probability density distribution of the residual displacement are proposed.
1. Introduction A large earthquake can be followed by smaller aftershocks, and may also be preceded by foreshocks. The foreshock-mainshock or mainshock-aftershock sequences can lead to ground acceleration sequences at specific sites, causing the engineering structures constructed on these sites to experience repeated seismic excitations in a short period of time. The time interval between sequential earthquakes varies widely, from several minutes to several years. During very short time intervals within which the repair of damaged structures cannot be performed, repeated seismic excitation of these damaged structures can result in an undesirable accumulation of damage, and a structure that would survive a single excitation may fail under the effect of a seismic sequence. While seismologists are trying to determine the correlation between the foreshock, mainshock, and aftershocks, at present it is still difficult to predict the occurrence of foreshocks or aftershocks [1,2]. However, for engineering design, structures are expected to suffer repeated earthquake excitations over a short period of time, particularly in areas where large numbers of sequential earthquakes have previously occurred. For example, foreshocks were observed during the 1975
⁎
Haicheng earthquake, the 1995 Kozani-Grevena earthquake [2], and the 2016 Kumamoto earthquake, while aftershocks have occurred in many major earthquakes, including the 2008 Wenchuan earthquake, the 2010 Haiti earthquake, the 2011 East Japan earthquake, the 2011 Christchurch earthquake, the 2016 Kumamoto earthquake, and the 2016–2017 Central Italy earthquakes. The number of aftershocks can vary from a few to hundreds. The 2008 Wenchuan earthquake had a mainshock of magnitude 8.0, followed by five aftershocks with magnitudes greater than 6.0 [3]. The 2010 Haiti earthquake had a mainshock of magnitude 7, which was followed by 14 aftershocks with magnitudes of 5–6.1 [4]. The 2011 East Japan earthquake had a mainshock of magnitude 9.0, which was followed by more than 7000 aftershocks over the next year. From August 2016 to January 2017, the central Italy was hit by an earthquake sequence, within includes 9 Mw5+ earthquakes and multiple aftershocks following these major shocks [28]. In current seismic codes, however, design earthquakes are usually defined as single event, and the effect of seismic sequences is not taken into account. A number of studies have investigated the influence of ground motion sequences on various types of structures. Amadio et al.
Corresponding author. E-mail address: [email protected] (W. Pu).
https://doi.org/10.1016/j.soildyn.2018.07.037 Received 10 February 2018; Received in revised form 14 June 2018; Accepted 22 July 2018 0267-7261/ © 2018 Elsevier Ltd. All rights reserved.
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Fig. 1. Pinching hysteretic model.
strength of stiffness, and the resulting change in structural performance may have unfavorable effects on the structural response. In the 2016 Kumamoto earthquake, a large number of timber structures, including many historical structures of great cultural value, failed as a result of the sequential excitations of foreshocks, mainshock, and aftershocks. This study focuses on timber structures, which have a very limited energy dissipation capability due to the pinching phenomenon in their hysteretic loops. These characteristics are substantially different from steel or concrete structures, which have been simulated by a bilinear model or modified Clough model in previous studies. By employing a pinching hysteretic model, this study investigates the ductility demands and residual displacements of timber structures subjected to ground motion sequences. These ground motion sequences are modeled by replicating a specific ground motion record, and the intensities of the two sequential ground motions are adjusted to simulate foreshock-mainshock and mainshock-aftershock sequences. Based on the numerical results, some insights into the effect of sequential earthquake excitations on pinching hysteretic structures are presented.
[5] simulated the effects of a seismic sequence on single-degree-offreedom (SDOF) structures using a bilinear model, a degrading stiffness hysteretic model without pinching, and Clough's model. Three ground motions were used as earthquake inputs, and the base shear, ductility, and dissipated energy were investigated with a time history analysis. A parameter, q, is defined as the ratio between the maximum accelerogram that a structure can withstand without failure and the accelerogram at which yielding first appears in the structure. The results demonstrated that multiple earthquakes can induce considerable accumulation of damage and result in a consequent reduction in the qfactor. Following this research, Fragiacomo et al. [6] conducted extensive studies considering different types of steel frame structures. Hatzigeorgiou and Beskos [7] investigated the effects of multiple earthquakes on the inelastic displacement ratio of structures. A total of 112 ground motion records obtained at different types of sites were applied to structures simulated with a bilinear model. An empirical formula for the inelastic displacement ratio was proposed, which incorporated the effects of the structural period, damping ratio, post-yield stiffness ratio, and force reduction factor. Hatzigeorgiou [8] also investigated the ductility demand and strength of an inelastic structure subjected to repeated near- and far-fault earthquakes. Di Sarno [9] employed ground motion sets obtained during the 2011 earthquake in Tohoku, Japan. The inelastic spectral responses of the ground motion sets were derived, and a two-span two-story frame structure was subjected to multiple earthquake ground motions. It was recommended that the effects of stiffness and strength degradation due to sequential earthquakes should be considered in modern codes of practice. Zhai et al. [3,10–12] carried out a series of studies investigating the ParkAng damage index, constant damage displacement ratio, and strength reduction factor of structures subjected to a mainshock-aftershock sequence. Ruiz-Garcia et al. [13] investigated the interstory drift and residual drift of reinforced concrete (RC) frame buildings under the excitation of seismic sequences, and determined that the building response depends on the ratio of the damaged period of vibration to the dominant period of the aftershock. Ruiz-Garcia and Aguilar [14] investigated the influence of modeling assumptions on the maximum and residual displacements of steel frame buildings subjected to mainshockaftershock sequences, and highlighted the importance of the building model for the assessment of structural behavior induced by aftershocks. Very recently, Hosseinpour and Abdelnaby [15] investigated the effects of irregularity, earthquake direction, aftershock polarity, and vertical component of the earthquake on structural performance. Shin and Kim [16] demonstrated that the frequency contents play an important role in the response induced by aftershocks. Nazari et al. [17] examined the collapse risk of wood frame buildings, and determined that the effect of aftershocks on the damage states are more significant than their effects on collapse for low-rise buildings. These previous studies have demonstrated that seismic sequences produce responses that differ from those that occur during a single event, and the effect depends on numerous factors, including the structural period, structural modeling approach, hysteretic model, damping ratio, and frequency contents of the ground motions. During repeated earthquakes, degradation of structures is of great concern [18], as the preceding earthquake causes reduction in the structural
2. Hysteretic model of the structure Fig. 1(a) shows the force–displacement relationship curve of the structural model considered in this study. The skeleton curve is composed of three segments: the first represents the elastic stiffness, K1; the second segment indicates the stiffness of the material after cracking, K2; and the third indicates the stiffness, K3, after reaching the yielding force, Fy. The stationary hysteresis is indicated by the thick lines. This model is characterized by a pinching phenomenon, which represents the typical hysteretic characteristics of timber systems, including X-lam buildings and light-frame walls [19]. The model can be decomposed into a slip element (Fig. 1(b)) and a bilinear element (Fig. 1(c)). The slip element has a bilinear skeleton curve, with the elastic stiffness and postyield stiffness denoted as Ks1 and Ks2, respectively. The slip element experiences slip-type hysteresis (thick line) after the virgin circle, with no energy dissipated in slip hysteresis. The yield displacement is denoted as uy. The energy dissipation of the structure is contributed only by the bilinear element. Similar to the definitions established by Matsuda and Kasai [20], Kb1 and Kb2 denote the initial elastic stiffness and the post-yield stiffness of the bilinear model, respectively, and uc is the displacement corresponding to the elastic limit of the bilinear element. Matsuda and Kasai [20] presented a set of parameters for this hysteretic model: K1 = 0.44 kN/mm, Kb1/K1 = 0.53, Kb2/Kb1 = 0.0566, uc= 4.5 mm, and uy= 18 mm. These parameters were obtained from experiments with typical polywood structures used in Japan. These parameters are adopted as benchmark values. In addition, some of these parameters are also adjusted to investigate their effect on structural response. In addition to the benchmark values, Kb2/Kb1 = 0.2 and Kb1/ K1 = 0.75 and 1.0 are also considered. Note that when Kb1/K1 is equal to 1.0, the structural model becomes a bilinear model. Fig. 2 shows the hysteretic curves with different parameter values. The structural mass is adjusted so that the elastic period, T, is 0.5 s, 0.8 s, 1.2 s, and 2.0 s. An inherent viscous damping ratio of 0.02 is assumed, and the damping coefficient is proportional to the tangent stiffness. This damping assumption is commonly used for dynamic analysis of timber structures in 393
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Fig. 2. Hysteretic model shapes.
Japan.
ground motions. Following the analysis based on the repeated ground motion sequence, a set of real ground motion sequences will be used to verify the results. Ground motions are selected from the Pacific Earthquake Engineering Research Center (PEER) strong motion database. A total of 40 ground motions are selected, 10 in each of four soil types (types A, B, C, and D in the United States Geological Survey site classification system [24]). Table 1 lists the details of the selected ground motions. Fig. 6 shows the acceleration and displacement spectra of all of the ground motions and the average for each soil type, in which the PGA of the ground motions are normalized to 0.3g. For establishing the seismic sequences, the PGA is set as 0.1 g, 0.2 g, 0.3 g, 0.4 g, 0.5 g, and 0.6 g. For each ground motion, 36 (6 × 6) sequences with different combinations of intensities are considered. The two ground motions are combined with a time interval of 30 s, which is 15 times the maximum structural period considered in this study and thus considered long enough to allow the structure to stop vibrating after the first excitation.
3. Model of ground motion sequences Previous studies have employed various methods to model seismic sequences. These methods include replicating the mainshock with or without scaling [5,7,21], randomly selected ground motions recorded in similar soil conditions [8,22], ground motion matching the spectra of the mainshock [16], and real seismic ground motion sequences [9]. Moustafa and Takewaki [23] presented a simple stochastic model for simulating strong ground sequences, in which the ground motion is represented as the product of a stationary Gaussian random process and a repeating envelope function. It has been reported that the frequency contents of aftershocks can differ significantly from those of the mainshock [22]. Therefore, it is difficult to simulate an aftershock deterministically based on the frequency contents of the mainshock. On the other hand, using a random combination of ground motions for the aftershock is also not ideal, as it is difficult to determine which ground motion is most critical for the structure under consideration. An example is presented to show the different responses that can occur for ground motions with identical peak ground acceleration (PGA). Two seismic sequences are established by combining ground motions No. 29 and No. 30 in Table 1, which were recorded in different directions at the same station during the 1994 Northridge earthquake. Fig. 3 shows the acceleration time history of the two sequences. The PGA of both ground motions are set to 0.6 g, and a structural model with a natural period of 0.5 s is subjected to the two seismic sequences. The displacement time histories and hysteretic curves obtained for these two cases are shown in Fig. 4 and Fig. 5, respectively. It is clear that the responses are governed by ground motion No. 29, regardless of whether it is modeled as the first excitation or the second excitation. This demonstrates that seismic ground motions, even those recorded during the same event and at the same station, can have completely different effects on structural response. If the structural response is completely governed by any single ground motion within a sequence, the significance of sequential excitations will be reduced. For simplicity, in this study the seismic sequences are firstly modeled by replicating and scaling the same ground motion. Even though this model is not necessarily realistic or the most critical, it is still useful for clarifying the response of pinching structures under the excitation of sequential earthquakes. By employing this model, a sequence with arbitrary increasing or decreasing intensities can be assumed, and thus the analysis can be performed over a wide range of intensity ratio. It is also noted that for a specific site, the intensity induced by an aftershock or a foreshock may be larger than that induced by the mainshock, since the intensity is affected not only by the earthquake magnitude but also by the site-source distance, source mechanism etc. In this study, it is assumed that the intensity of the mainshock is larger than that of foreshock or aftershock, as observed in most cases. Only two sequential ground motions are considered in these sequences. The results obtained from two ground motion sequences provide fundamental knowledge for characterizing the effect of ground motion sequences, as the underlying effect is the varied structural performance resulting from consecutive
4. Effect of seismic sequence on ductility demands 4.1. Effect of intensity ratio on ductility demand amplification The ductility demands of pinching structures are investigated first. The ductility factor, μ, is defined as the maximum displacement divided by uc. The effect of seismic sequences on the ductility demands can be evaluated with the maximum ductility ratio of the two sequential excitations, μ2/μ1. The ductility ratio, μ2/μ1, is clearly affected by the intensity ratio of the two ground motions. Fig. 7 shows μ2/μ1 versus PGA2/PGA1 for structures with benchmark hysteretic parameters and various structural periods and soil types. Note that only those cases in which the displacement from the first ground motion is larger than uc are shown. From these figures, two observations can be made: (1) for PGA2/PGA1 = 1, the response ratio is greater than 1, which suggests that repeating the same ground motion causes an amplification of the ductility demands of pinching hysteretic structures; and (2) in a logarithmic coordinate system, the ductility ratio varies approximately linearly with the intensity ratio. The relationship can be fitted by the following equation:
PGA2 ⎞ μ 2 / μ1 = α ⎛ ⎝ PGA1 ⎠ ⎜
β
⎟
(1)
In Eq. (1), if PGA2/PGA1 = 1, then μ2/μ1 = α, and thus α denotes the ductility demand amplification ratio for a structure subjected to a sequence with identical intensities. Also, as β increases, the ductility demand amplification effect becomes more significant. Eq. (1) can be applied to all cases, but different values for α and β are obtained depending on the structural properties and soil type. Table 2 lists the values of α and β for cases with various T, Kb2/Kb1, and soil type; Table 3 lists the values of α and β for cases with different T and Kb1/K1, in which only soil types B and D are considered. The results of Table 2 indicate that the short-period structures subjected to earthquake sequences experience significant amplification of ductility demand. As the structural period increases, the amplification of ductility demand becomes weak. For example, for T = 0.5 s and Kb2/Kb1 394
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Table 1 Selected ground motion records. No.
Date
Record name
Comp.
Station name
PGA (g)
Vs30 (m/s)
Site
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
1974/11/28 1974/11/28 1979/08/06 1979/08/06 1994/01/17 1994/01/17 1994/01/17 1994/01/17 1995/01/16 1989/10/18 1978/08/13 1971/02/09 1971/02/09 1975/06/07 1975/06/07 1978/08/13 1992/06/28 1992/06/28 1980/06/09 1980/06/09 1979/10/15 1979/10/15 1999/11/12 1999/11/12 1992/04/25 1992/04/25 1994/01/17 1994/01/17 1994/01/17 1994/01/17 1979/10/15 1979/10/15 1983/05/02 1983/05/02 2004/09/28 2004/09/28 1987/11/24 1987/11/24 1983/05/02 1987/10/01
Hollister-03 Hollister-03 Coyote Lake Coyote Lake Northridge-01 Northridge-01 Northridge-01 Northridge-01 Kobe Japan Loma Prieta Santa Barbara San Fernando San Fernando Northern Calif Northern Calif Santa Barbara Landers Landers Victoria Mexico Victoria Mexico Imperial Valley-06 Imperial Valley-06 Duzce Turkey Duzce Turkey Cape Mendocino Cape Mendocino Northridge-03 Northridge-03 Northridge Northridge Imperial Valley Imperial Valley Coalinga-01 Coalinga-01 Parkfield-02 Parkfield-02 Superstition Hills Superstition Hills Coalinga-01 Whittier Narrows
N157 N247 N230 N320 N095 N185 N175 N265 EW EW N132 N111 N201 N030 N120 N222 NS EW N315 N045 N012 N282 NS EW NS EW EW NS EW NS N230 N140 NS EW EW NS EW NS NS NS
47379Gilroy Array #1 47379Gilroy Array #1 47379Gilroy Array #1 47379Gilroy Array #1 90017LA - Wonderland Ave 90017LA - Wonderland Ave 24207Pacoima Dam (downstr) 24207Pacoima Dam (downstr) Kobe University 57180Los Gatos Lexington Dam 283Santa Barbara Courthouse 126Lake Hughes #4 126Lake Hughes #4 89005Cape Mendocino 89005Cape Mendocino 283Santa Barbara Courthouse 22170Joshua Tree 22170Joshua Tree 6604Cerro Prieto 6604Cerro Prieto 6621Chihuahua 6621Chihuahua Duzce Duzce 89509 Eureka 89509 Eureka 24279Newhall - Fire Sta 24279Newhall - Fire Sta 90021 LA-N Westmoreland 90021 LA-N Westmoreland 5057El Centro Array #3 5057El Centro Array #3 36228Parkfield - Cholame 2WA 36228Parkfield - Cholame 2WA 36228Parkfield - Cholame 2WA 36228Parkfield - Cholame 2WA 5210Imperial Valley WLA 5210Imperial Valley WLA 36407Parkfield - Fault Zone 1 90081Carson - Water St
0.100 0.141 0.094 0.117 0.103 0.159 0.416 0.434 0.312 0.411 0.101 0.198 0.156 0.117 0.207 0.202 0.274 0.284 0.633 0.645 0.270 0.254 0.404 0.515 0.154 0.178 0.107 0.205 0.333 0.432 0.097 0.138 0.110 0.113 0.624 0.373 0.131 0.133 0.143 0.110
1428.14 1428.14 1428.14 1428.14 1222.52 1222.52 2016.13 2016.13 1043.00 1070.34 514.99 600.06 600.06 567.78 567.78 514.99 379.32 379.32 471.53 471.53 242.05 242.05 281.86 281.86 337.46 337.46 269.14 269.14 315.06 315.06 162.94 162.94 173.02 173.02 173.02 173.02 179.00 179.00 178.27 160.58
A A A A A A A A A A B B B B B B B B B B C C C C C C C C C C D D D D D D D D D D
= 0.057, the value of 1.22 is obtained for α from soil type A and type C, which means that repeated excitation by the same ground motion can cause amplification of the structural displacement by as much as 22%. There are no significant differences between the results for structures with Kb2/Kb1 = 0.2 and 0.057, and only slight difference is observed in the results of different soil types. On the other hand, as Kb1/K1 increases, the effect of sequential excitations decreases (Table 3). Particularly for Kb1/K1 = 1, the structural model becomes a bilinear model, and α is approximately equal to 1. This implies that pinching hysteretic structures are more significantly affected by seismic sequence than bilinear structures do. Fig. 8 shows examples of hysteretic curves resulting from the pinching model and bilinear model, in which the intensity ratio of the sequence is 1. For the pinching model, as long as the maximum displacement is updated, the skeleton curve makes a contribution to the energy dissipation and load resistance. In terms of the
energy, the pinching model should develop larger peak displacements in the second earthquake so that the structure can dissipate approximately the same amount of energy as the energy dissipated in the first earthquake. In contrast, the bilinear model has no such loading history effect on hysteresis, and it develops almost the same maximum displacement.
4.2. Damaging effect of aftershocks The previous section demonstrates that an aftershock increases the ductility demands of pinching structures even though its intensity is smaller than the mainshock. The underlying reason is that the mainshock causes a reduction of structural seismic performance. Because PGA cannot be used to represent the damaging effects of ground motion directly (as described in Section 3), the ductility factor will be used to
Fig. 3. Acceleration time history of two seismic sequences. (a) No. 30-No. 29 ground motion sequence (b) No. 29-No. 30 ground motion sequence. 395
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Fig. 4. Displacement time history of a structure subjected to seismic sequences. (a) No. 30-No. 29 ground motion sequence (b) No. 29-No. 30 ground motion sequence.
an aftershock should consider both the pre-damage level of the structure and the intensity of the aftershock itself. Eq. (2) can be used to predict μ2 based on μ1 and μ2,s. However, it should also be noted that a certain level of error may exist in these predictions, as the coefficients of determination (R2) obtained in Table 4 are very small.
indicate the damaging effect. To determine the damaging effect of an aftershock, the sequences with PGA1 > PGA2 are selected to represent mainshock-aftershock sequences, and the influence of the aftershock is investigated quantitatively. It is noted that the intensity of a ground motion recorded in an aftershock is not necessarily smaller than that of mainshock. “Aftershock” and “Mainshock” are defined based on the earthquake magnitude, however the intensity recorded at a station depends not only on earthquake magnitude but also on many other factors such as the location of epicenter and the frequency components of ground motions. For simplicity, in this study, the sequences with PGA1 < PGA2 are defined as foreshock-mainshock sequence, and the sequences with PGA1 > PGA2 are defined as mainshock-aftershock sequence. In the following analysis, μ2 is the ductility induced by the aftershock following the mainshock, and μ2,s is the ductility of the structure subjected to a single ground motion in an “aftershock” without prior application of a mainshock. Therefore, μ2/μ2,s is the ratio of the ductility demand of a structure pre-damaged in a mainshock to the ductility demand of an intact structure. Fig. 9 shows the distribution of μ2/μ2,s versus μ1 and μ2,s. The results obtained from the mainshockaftershock sequences are almost all located on the left side of the figure and have μ1 > μ2,s. By evaluating the relationship between μ2/μ2,s, μ1, and μ2,s with Table Curve 3D software, Eq. (2) is obtained to describe the relationship between μ2/μ2,s, μ1, and μ2,s.
μ 2 / μ 2,s = a + blnμ1 + clnμ 2,s
4.3. Effect of a foreshock on ductility demand Similar to the previous section, the sequences with PGA1 < PGA2 are selected to represent the foreshock-mainshock sequences. It should be noted that μ2 here refers to the ductility induced by the mainshock following a foreshock and μ2,s denotes the ductility demand induced by the mainshock without a foreshock. In Fig. 9, the results on the right side with μ1 < μ2,s are obtained from the foreshock-mainshock sequences. Compared to the mainshockaftershock sequences, the foreshock-mainshock sequences have smaller values of μ2/μ2,s, most of which are near 1. This implies that foreshocks with smaller intensities have no significant effect on the ductility demand of the structure subjected to the mainshock. To some extent, the results for the foreshock-mainshock sequence can be explained by Eq. (2), which was originally obtained from the mainshock-aftershock sequences. This equation demonstrated that when the wave in the sequence has a smaller intensity (smaller μ1), the value of μ2/μ2,s is relatively small. Therefore, a structure yields a similar ductility demand under the excitation of the mainshock, regardless of whether it has experienced a foreshock. In summary, the displacement magnification effect of an aftershock is extremely dependent on the intensity ratio of the seismic sequence, and in a sequence, the aftershock is likely to lead to ductility demand that is equivalent to or greater than that induced by the mainshock (Section 4.1). The foreshock has a slight effect on structural response, while the response is still governed by the mainshock.
(2)
Based on Eq. (2), μ2/μ2,s can be expressed as linear function of ln μ1 and ln μ2,s. The identified values for the coefficients in Eq. (2), a, b, and c, are listed in Table 4. Coefficients a and b are positive and coefficient c is negative, which implies that the ratio of μ2/μ2,s is proportional to μ1 and inversely proportional to μ2,s. In other words, the more severe the pre-damage to the structures, the larger μ2/μ2,s will be. These results imply that an aftershock may lead to much larger displacement in a structure damaged by the mainshock, while this single “aftershock” causes only a small displacement in an intact structure. On the other hand, as μ2,s becomes large, the aftershock becomes more governing, and whether there is pre-damage is of less significance for the damaging effect of aftershock. Therefore, an estimation of the damaging effect of
5. Residual displacement of pinching hysteretic structures Residual displacement is an important index for quantifying postearthquake structural performance, as it provides information about the
Fig. 5. Hysteretic curves for a structure subjected to seismic sequences. (a) No. 30-No. 29 sequence (b) No. 29-No. 30 sequence. 396
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Fig. 6. Acceleration and displacement response spectra for selected ground motions. (a) Acceleration spectra (b) Displacement spectra.
Fig. 7. Relationship between μ2/μ1 and PGA2/PGA1. 397
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that the residual displacement ratio obeys a probabilistic exponential distribution. This resulting probabilistic density function can be expressed as: f(R1/M1) = 15.24e−15.24R1/M1. This exponential distribution is then applied to fit the residual displacement ratio, R2/M2, obtained from the second ground motion, and the regression result is shown in Fig. 11(b). Strictly speaking, R2/M2 does not obey a probabilistic exponential distribution based on the results of statistical hypothesis testing, and using an exponential distribution function is an approximation to illustrate trends. The fitted exponential distribution for R2/ M2 has a smaller rate parameter, which indicates a higher probability of a large ratio of residual displacement to maximum displacement. When structures are subjected to sequential excitations, residual displacements may accumulate when they are located on the same side, thus leading to a R2/M2 ratio that is higher than R1/M1 for a specific probabilistic level. For comparison, Fig. 11(c) and (d) show the results for bilinear hysteretic structures subjected to ground motions on soil types B and D. The residual displacement ratio shows a similar distribution pattern to Fig. 11(a). Similarly, the probabilistic distribution is regressed based on an exponential distribution, and much smaller rate parameters for the exponential distribution are obtained. Compared to bilinear hysteretic structures, the residual displacement of a pinching hysteretic structure is more likely the result of a large maximum displacement. This should be carefully considered when estimating the maximum displacement using the residual displacement.
Table 2 Values of α and β obtained for Eq. (1). T
0.5 s
Kb2/Kb1
0.057
0.200
0.057
0.200
0.057
0.200
0.057
0.200
1.22 1.08 0.89 1.19 1.05 0.92 1.22 1.08 0.90 1.14 1.18 0.95
1.23 1.08 0.90 1.14 1.15 0.89 1.12 1.09 0.90 1.14 1.22 0.95
1.07 0.99 0.95 1.07 1.06 0.94 1.11 1.16 0.92 1.06 1.09 0.90
1.06 1.05 0.96 1.04 1.07 0.94 1.11 1.19 0.93 1.05 1.15 0.92
1.05 1.10 0.95 1.05 0.97 0.95 1.07 1.06 0.89 0.95 1.05 0.93
1.10 1.05 0.95 0.99 1.04 0.96 1.03 1.06 0.91 1.03 1.07 0.94
0.97 0.92 0.94 1.04 0.96 0.94 0.97 0.99 0.91 0.92 0.95 0.91
0.91 1.02 0.94 1.02 1.04 0.95 0.98 1.10 0.90 0.92 1.03 0.94
Soil Type A
Soil Type B
Soil Type C
Soil Type D
α β R2 α β R2 α β R2 α β R2
0.8 s
1.2 s
2.0 s
reparability of the structure and its structural performance during aftershocks. The previous sections have demonstrated that the ductility level of a structure induced by the mainshock substantially affects the structural performance during an aftershock. There are many ways to quantify the maximum deformation of a structure in an earthquake event, such as real-time monitoring and post-earthquake analyses. Estimating the maximum displacement based on residual displacement is another typical approach. There are a number of studies investigating the residual displacements of various types of structures, including structures modeled by bilinear models [25,26] and reinforced concrete structures [27]. The results indicate that the maximum displacement can be effectively estimated from the residual displacements. The residual displacements of pinching hysteretic structures are investigated with a statistical analysis. In Fig. 10, the residual displacement is plotted against the maximum displacement for structures with benchmark hysteretic parameters. As observed for other types of structures, a large maximum displacement generally leads to a large residual displacement. As the maximum displacement increases, the ratio of residual displacement to maximum displacement also increases. The ratio of residual displacement to maximum displacement can be modeled as a power function. The fitted power functions are shown for each subfigure in Fig. 10. The scattering in the results is relatively large, although it is similar to that obtained for RC columns [27]. This indicates that there is a certain level of uncertainty in the estimation of maximum displacement using the residual displacement. To quantify this uncertainty, the distribution of the ratio of residual displacement to maximum displacement is investigated. Fig. 11(a) shows histograms for the ratio of R1/M1 obtained from structures with benchmark parameters, where R1 and M1are the residual displacement and maximum displacement, respectively, induced by the first ground motion. The kernel density is estimated with a Gaussian distribution as the kernel function, and the kernel density curves are indicated by a blue line. The kernel estimation suggests that the residual displacement ratio might obey an exponential distribution. To verify this hypothesis, an exponential distribution is employed to fit the residual displacement ratio data. A statistical hypothesis testing is performed, and it suggests
6. The effect of seismic sequence model For the two ground motions in a real earthquake sequence, significant difference may exist in their frequency components, predominant period, and duration etc. These characteristics may cause results different with those obtained from the repeated sequences. In order to clarify this effect, a comparison study between the real sequence and repeated sequence is performed. The real ground motion sequences are selected from 1999 Chi-Chi earthquake, 2011 East Japan earthquake, 2010 Christchurch earthquake and 2016 Kumamoto earthquake. A set of 10 sequences, which were recorded in two horizontal directions at 5 stations, are selected, as given in Table 5. A subscript number is added to the component name to differentiate the two ground motion in each sequence. In order to obtain more sequence samples, the order of the two ground motions in each sequence are reversed, and thus another set of 10 seismic sequences with different intensity ratios are obtained. These reversed sequences are also considered as real sequences. Totally, 20 realistic sequences are considered. Corresponding to these real sequences, the repeated sequences are built by repeating the second wave in each real sequence, and the PGA of repeated sequence is adjusted so that it is consistent with the real sequence. Fig. 12 shows two examples of real and repeated sequences, which is built based on the sequence of No. 10 of Table 5. These real and repeated sequences are applied to the benchmark model. The index of μ2/μ1 and μ2/μ2,s are investigated again. The μ2/μ1 is shown versus the intensity ratio in Fig. 13(a) and Fig. 13(b) for real
Table 3 Values of α and β obtained for Eq. (1). T
0.5 s
Kb1/K1
0.53
0.75
1
0.53
0.75
1
0.53
0.75
1
0.53
0.75
1
1.19 1.05 0.92 1.14 1.18 0.95
1.14 1.10 0.93 1.10 1.20 0.95
1.06 1.15 0.90 1.01 1.22 0.93
1.07 1.06 0.94 1.06 1.09 0.90
1.08 1.05 0.94 1.03 1.09 0.92
1.03 1.04 0.92 1.01 1.10 0.88
1.05 0.97 0.95 0.95 1.05 0.93
1.06 0.99 0.96 0.97 1.03 0.94
1.05 0.91 0.91 1.03 0.98 0.92
1.04 0.96 0.94 0.92 0.95 0.91
1.08 0.91 0.94 0.97 0.93 0.95
1.06 0.86 0.92 1.01 0.91 0.96
Soil Type B
Soil Type D
α β R2 α β R2
0.8 s
1.2 s
398
2.0 s
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Fig. 8. Hysteretic curves of a pinching model and a bilinear model (ground motion No. 13, 0.4–0.4 g). (a) Pinching hysteretic model (b) Bilinear hysteretic model.
smaller than those of repeated sequences in case of PGA1 > PGA2, and the ductility ratios of real sequences are little larger than those of repeated sequences in case of PGA1 < PGA2. Third, larger dispersion in the results of real sequences is observed. Fig. 14(a) and Fig. 14(b) illustrate the displacement time histories obtained from sequences of PGA1 < PGA2 and PGA1 > PGA2, respectively, and Fig. 15 and Fig. 16 show the corresponding hysteretic curves. In Fig. 14(a), the first wave (EW2 with adjusted PGA1 of 597 gal) of repeated sequence causes a maximum displacement (M1 = 15.06 cm) larger than that caused by the first wave of the real sequence (EW1, PGA1 = 597 gal, M1 = 5.93 cm). The first wave in the repeated sequence is originally obtained from the mainshock, and it possesses the characteristics of a mainshock with large magnitude, such as the longer duration and the larger energy [18]. Because of these properties, the discrepancy between the maximum displacements may be resulted. Moreover, there is only slight difference between the maximum displacement M2 of the real sequence and the repeated sequence, and therefore, a larger ductility demand ratio μ2/μ1 is obtained from real sequence. Correspondingly, under the excitation of sequences of PGA1 > PGA2 (Fig. 12b), the structure develops displacement time histories as shown in Fig. 14(b) and hysteretic curves as shown in Fig. 16. In contrast to the previous observations, the first wave of repeated sequence was originally recorded in small earthquake (EW2), even though it was scaled up to the same PGA of the real wave (EW1), it caused a relatively smaller M1, and finally resulted in a larger μ2/μ1. The values of μ2/μ2,s obtained from real sequences and repeated sequences are also compared in Fig. 17. The results suggest that the μ2/ μ2,s derived from two types of seismic sequences are generally in agreement. The above analysis demonstrate that the results derived from repeated sequences can reflect the general features of structural responses induced by real seismic sequences, while a certain level of error may be caused in the estimation of the ductility demand ratio depending on the intensity ratio. This reason is that simply scaling ground motion cannot reflect the characteristics of the duration and the frequency contents, which are dependent on earthquake magnitude and play an important role in structural responses.
Fig. 9. Relationship among μ2/μ2,s, μ2,s, and μ1 (Kb2/Kb1 = 0.0566, 1 ≤ μ ≤ 40). Table 4 Values for a, b and c in Eq. (2).
Soil Type A
Soil Type B
Soil Type C
Soil Type D
a b c R2 a b c R2 a b c R2 a b c R2
T = 0.5 s
T = 0.8 s
T = 1.2 s
T = 2.0 s
0.45 0.91 − 0.71 0.52 0.84 0.51 − 0.42 0.45 0.35 0.62 − 0.35 0.29 0.40 0.60 − 0.40 0.32
1.00 0.21 − 0.22 0.17 0.91 0.36 − 0.37 0.37 0.81 0.30 − 0.27 0.16 0.90 0.41 − 0.45 0.22
0.98 0.12 − 0.11 0.06 0.92 0.32 − 0.33 0.32 0.26 0.56 − 0.36 0.32 1.11 0.09 − 0.21 0.24
0.91 0.38 − 0.43 0.41 1.26 0.14 − 0.25 0.24 0.69 0.39 − 0.33 0.36 1.20 0.12 − 0.22 0.15
7. Conclusions sequences and repeated sequences, respectively. In each subfigure, the regressed functions based on the form of Eq. (1) are also presented. First, the values of α and β regressed from results of repeated sequences are larger than those obtained in Section 4. The reason is probably that the number of data is less than that of Fig. 7, and thus the estimation error becomes large due to the limited number of data. Second, the regressed α of the real sequences (Fig. 13a) is smaller than that of the repeated sequences (Fig. 13b), while the regressed β of the real sequences is larger than that of the repeated sequences. Since β determines the curve shape of the regressed function, the regression results suggest that the ductility demand ratios of real sequences are little
Timber structures are modeled with pinching hysteretic models and subjected to seismic sequences composed of two ground motions with different PGA. The resulting ductility demands and residual displacements of the pinching hysteretic structures were investigated. The findings obtained from the above investigations can be concluded as follows: (1) Compared to bilinear hysteretic structures, pinching hysteretic structures are more significantly affected by seismic sequences, as they generally lead to higher ductility amplification ratios. The 399
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Fig. 10. Comparison of residual displacement and maximum displacement.
Fig. 11. Probabilistic distribution of the residual displacement ratio.
400
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Table 5 Real seismic sequences used in analysis. No.
Earthquake (year)
Date-Time
1
Chi-Chi Earthquake (1999)
2
Chi-Chi Earthquake (1999)
3
Christchurch earthquake (2011)
4
Christchurch earthquake (2011)
5
Great East Japan earthquake (2011)
6
Great East Japan earthquake (2011)
7
Kumamoto earthquake (2016)
8
Kumamoto earthquake (2016)
9
Kumamoto earthquake (2016)
10
Kumamoto earthquake (2016)
1999/09/20 1999/09/20 1999/09/20 1999/09/20 2010/09/03 2011/02/21 2010/09/03 2011/02/21 2011/03/11 2011/04/07 2011/03/11 2011/04/07 2016/04/14 2016/04/14 2016/04/14 2016/04/14 2016/04/14 2016/04/14 2016/04/14 2016/04/14
17:47 18:03 17:47 18:03 16:35 23:51 16:35 23:51 14:46 23:32 14:46 23:32 21:26 22:07 21:26 22:07 21:26 22:07 21:26 22:07
MW
Station
Comp.
PGA(m/s2)
7.6 6.2 7.6 6.2 7.0 6.2 7.0 6.2 9.0 7.1 9.0 7.1 6.5 5.8 6.5 5.8 6.5 5.8 6.5 5.8
TCU129 TCU129 TCU129 TCU129 CBGS CBGS CBGS CBGS MYG001 MYG001 MYG001 MYG001 KMM006 KMM006 KMM006 KMM006 KMMH16 KMMH16 KMMH16 KMMH16
NS1 NS2 EW1 EW2 N891 N892 S011 S012 NS1 NS2 EW1 EW2 NS1 NS2 EW1 EW2 NS1 NS2 EW1 EW2
6.12 3.90 9.85 9.28 1.52 5.42 1.86 4.43 4.01 3.75 4.13 3.79 5.99 4.51 3.59 2.21 8.02 4.59 9.51 5.87
Fig. 12. Examples of real sequences with inversed intensity ratio (based on sequence of No.10). (a) PGA1 < PGA2 (b) PGA1 > PGA2.
Fig. 13. Relationship between ductility ratio and intensity ratio. Fig. 14. Displacement time history. PGA1 < PGA2 (b) PGA1 > PGA2.
(a)
structures already damaged in the mainshock, and the damaging effect is dependent not only on the intensity of the aftershock, but also on the damage level caused by mainshock. In contrast, the foreshock has little effect on the structures subjected to a subsequent mainshock. (3) In general, a large maximum displacement causes a large residual displacement, and the residual displacement can be modeled as a
ductility amplification ratio resulting from sequential ground motion excitations can be simulated as a power function of the intensity ratio of the ground motions. When subjected to sequences composed of two same ground motions, the ductility demand amplification of pinching hysteretic structures can be as much as 22%, while almost no amplification is observed for bilinear structures. (2) An aftershock has a significant influence on pinching hysteretic 401
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Fig. 15. Hysteretic curves induced by sequences of PGA1 < PGA2.
Fig. 16. Hysteretic curves induced by sequences of PGA1 > PGA2.
aftershock sequences. It should be noted that the SDOF model employed in this study is able to reflect the global behavior of structure, but cannot reflect local damage or higher mode effect of multi-degree-of-freedom structures. In case that identification of these effects is required, further investigation should be performed based on a detailed member-to-member model. Acknowledgement The authors thank the four anonymous reviewers for their helpful comments and suggestions. References [1] Parsons T, Segou M, Marzocchi W. The global aftershock zone. Tectonophysics 2014;618(4):1–34. [2] Lin CH. Foreshock characteristics in Taiwan: potential earthquake warning. J Asian Earth Sci 2009;34(5):655–62. [3] Zhai C, Wen W, Chen Z, et al. Damage spectra for the mainshock-aftershock sequence-type ground motion. Soil Dyn Earthq Eng 2013;45(1):1–12. [4] Wang G, Wang Y, Lu W, et al. Damage demand assessment of mainshock-damaged concrete gravity dams subjected to aftershocks. Soil Dyn Earthq Eng 2017;98:141–54. [5] Amadio C, Fragiacomo M, Rajgelj S. The effects of repeated earthquake ground motions on the non-linear response of SDOF systems. Earthq Eng Struct Dyn 2003;32(2):291–308. [6] Fragiacomo M, Amadio C, Macorini L. Seismic response of steel frames under repeated earthquake ground motions. Eng Struct 2004;26(13):2021–35. [7] Hatzigeorgiou GD, Beskos DE. Inelastic displacement ratios for SDOF structures subjected to repeated earthquakes. Eng Struct 2009;31(11):2744–55. [8] Hatzigeorgiou GD. Ductility demand spectra for multiple near- and far-fault earthquakes. Soil Dyn Earthq Eng 2010;30(4):170–83. [9] Di Sarno L. Effects of multiple earthquakes on inelastic structural response. Eng Struct 2013;56(6):673–81. [10] Zhai C, Wen W, Ji D, Li S. The influences of aftershocks on the constant damage inelastic displacement ratio. Soil Dyn Earthq Eng 2015;79:186–9. [11] Zhai C, Wen W, Li S, Xie L. The ductility-based strength reduction factor for the mainshock-aftershock sequence-type ground motions. Bull Earthq Eng 2015;123(10):2893–914. [12] Zhai CH, Wen WP, Li S, et al. The damage investigation of inelastic SDOF structure
Fig. 17. Comparison of μ2/μ2,s obtained from real sequences and repeated sequences.
power function of the maximum displacement. The probabilistic density function of residual displacement ratio can be fitted by exponential distribution functions, and the rate parameter of 15.237 and 12.071 were obtained from the first excitation and the second excitation, respectively, of a sequence. Furthermore, a specific residual displacement is more likely to result from a large maximum displacement for structures with pinching hysteresis. This is critical for estimating the post-earthquake performance of timber structures. (4) The repeated sequences cannot reflect the difference of duration and frequency contents existing between ground motions in a real sequence, which may have unneglectable influence on structural response. Compared to the real sequences, the repeated sequences may slightly underestimate the ductility amplification ratios induced by foreshock-mainshock sequences, while slightly overestimate the ductility amplification ratios induced by mainshock402
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