Performance of Innovative Controlled Buildings Under Resonant and Critical Earthquake Ground Motions - Izuru Takewaki

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Taniguchi et al. Hybrid Control Building System A B Multiple isolation & Building connection Multiple isolation building acceleration [m/s ] 0.4 2 acceleration [m/s ] 8.0 Base-isolated & Building connection Base-isolated building 2 0.2 Building connection 4.0 0.0 0.0 A 40 -4.0 -0.2 30 Story 20 -8.0 -0.4 10 0.0 1.0 2.0 3.0 4.0 0.0 20.0 40.0 60.0 80.0 C time[s] D time[s] 0 Inter-story drift of base isolation layer acceleration [m/s ] 5.0 acceleration [m/s ] 0.4 2 2 B 40 0.2 2.5 30 Story 0.0 0.0 20 10 -0.2 -2.5 0 -0.4 -5.0 0.0 20.0 40.0 60.0 80.0 0.0 10.0 20.0 30.0 C 40 time[s] time[s] 30 Story 20 acceleration [m/s ] E 0.8 2 10 0.4 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 Maximum horizontal displacement[m] -0.4 D 40 -0.8 30 Story 0.0 100.0 200.0 20 time[s] 10 0 FIGURE 6 | Input ground motions: (A) artificial pulse-type ground motion (T p = 1.0 s), (B) artificial long-period, long-duration ground E 40 30 Story motion (6.8 s), (C) artificial long-period, long-duration ground motion 20 (8.4 s), (D) JMA Kobe NS (level 2: 0.5 m/s), and (E) Tomakomai EW 10 (2003). 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Maximum horizontal displacement[m] FIGURE 8 | Maximum horizontal displacements under various ground Pulse input Long period input (8.4s) Tomakomai EW motions: (A) artificial long-period, long-duration ground motion (6.8 s), Long period input (6.8s) JMA Kobe NS (Lv.2) (B) artificial long-period, long-duration ground motion (8.4 s), A B (C) Tomakomai EW (2003), (D) artificial pulse-type ground motion, response spectrum [m/s] response spectrum [m] 0.6 1.2 and (E) JMA Kobe NS (1995). 0.5 1.0 Displacement 0.4 0.8 Velocity 0.3 0.6 and long-period, long-duration ground motions. This indicates 0.2 0.4 0.1 0.2 the high robustness of the proposed building system for various 0 0.0 kinds of ground motion. In particular, the story drifts of the base- 0 2 4 6 8 10 0 2 4 6 8 10 Natural period [s] Natural period [s] isolation story and the middle-isolation story exhibit the value of half or two-thirds of the corresponding values of the com- C D parable building systems (base-isolated and building connection response spectrum [m/s ] 2 model, multiple-isolation building model) under the long-period, Energy spectrum [m/s] 10.0 2.5 2.0 long-duration ground motions. Furthermore, the acceleration Acceleration 1.5 of the proposed building system can be reduced effectively under 1.0 1.0 the long-period, long-duration ground motions compared to 0.5 the comparable building systems (base-isolated building model, 0.1 0.0 0 2 4 6 8 10 0 2 4 6 8 10 multiple-isolation building model). Natural period [s] Natural period [s] FIGURE 7 | Various spectra of five ground motions: (A) displacement Energy Response of Proposed Building response spectra, (B) velocity response spectra, (C) acceleration response spectra, and (D) energy spectra. Model and Other Comparable Models under Several Earthquake Ground Motions In this section, the energy responses of the proposed building (2003), the artificial pulse-type ground motion, and the JMA Kobe model and other comparable models are shown for the pulse-type NS (1995) are shown in Figure 8. On the other hand, the max- ground motions and long-period, long-duration ground motions. imum accelerations under these ground motions are illustrated In particular, the effect of the energy consumption at the con- in Figure 9. Figures 8A–C and 9A–C in these figures are for the nected dampers on the response is investigated in detail. long-period, long-duration ground motions, and Figures 8D,E Figure 10 shows the time histories of energy response of and 9D,E are for the pulse-type ground motions. the proposed building model and the building connection It can be observed from Figures 8 and 9 that the proposed model (without isolation) under the artificial pulse-type ground building system is effective both for pulse-type ground motions motions. The input energy, total damping energy, kinetic energy, Frontiers in Built Environment | www.frontiersin.org 9 October 2016 | Volume 2 | Article 26 Taniguchi et al. Hybrid Control Building System Multiple isolation & Building connection Multiple isolation building Input energy Total damping energy Base-isolated & Building connection Base-isolated building Kinetic energy Damping energy (Connected damper) Building connection A 120 B 250 A 40 100 200 30 Energy [MJ] Energy[MJ] 80 Input energy Story 20 150 60 10 40 100 0 Damping energy Kinetic energy 20 (Connected damper) 50 B 40 0 0 30 0 20 40 60 80 0.0 20.0 40.0 60.0 80.0 Story 20 time[s] time[s] 10 0 FIGURE 11 | Energy response of the proposed building model and C 40 other comparable model under the artificial long-period, 30 long-duration earthquake ground motion (8.4 s): (A) proposed building Story 20 model and (B) multiple isolation building model (without connection). 10 0 0.0 0.5 1.0 2 1.5 2.0 Maximum acceleration [m/s ] The remarkable reduction of the vibration energy in the main D 40 building has also been observed also under the long-period, long- 30 Story 20 duration ground motions. 10 0 Robustness of Proposed Building Model E 40 30 and Other Comparable Models under Story 20 10 Several Earthquake Ground Motions 0 0.0 1.0 2.0 3.0 4.0 2 5.0 6.0 Figure 12 shows the response variability (inter-story drift of base- Maximum acceleration [m/s ] isolation layer, inter-story drift of middle-story isolation layer, inter-story drift of non-isolation story of the main structure, base FIGURE 9 | Maximum top-story accelerations under various ground shear, overturning moment at the base) in the proposed build- motions: (A) artificial long-period, long-duration ground motion (6.8 s), (B) artificial long-period, long-duration ground motion (8.4 s), ing model and other comparable models under various ground (C) Tomakomai EW (2003), (D) artificial pulse-type ground motion, motions. and (E) JMA Kobe NS (1995). It can be observed from Figures 12A,B that the proposed build- ing system exhibits a good performance in the inter-story drift of the base-isolation layer and the middle-story isolation layer, Input energy Total damping energy especially for long-period ground motions which are critical to Kinetic energy Damping energy (Connected damper) the base-isolation system. The good performance can be observed A 80 B 80 also in the non-isolation story drift, base shear, and overturning moment at the base (Figures 12C–E). A small response variability 60 60 Energy [MJ] Energy [MJ] in the proposed building system can also be understood from 40 40 Figures 12A–D. 20 20 It can be observed from Figures 12F,G that the base shear 0 0 and base overturning moment in the free wall of the pro- 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2.0 3.0 4.0 time[s] time[s] posed building system under the pulse-type ground motions exhibit almost equivalent or smaller values compared to the FIGURE 10 | Energy response of the proposed building model and other comparable building systems. On the other hand, while other comparable model under the artificial pulse-type earthquake ground motion: (A) proposed model (multiple isolation and building these values become slightly larger under the long-period, long- connection model) and (B) building connection model (without duration ground motions, no serious problem occurs because isolation). those response values are relatively small compared to those response values under the pulse-type ground motions. and damping energy at the connected dampers are plotted in this figure. Summary of Response and On the other hand, Figure 11 presents the time histories of Robustness Analysis energy response of the proposed building model and the multiple Table 2 shows the summary of the response characteristics of isolation building model (without connection) under the artificial the proposed building model and other comparable models long-period, long-duration ground motion (8.4 s). under representative two-type ground motions. As stated above, It can be observed that the proposed building system has a the proposed building system exhibits a good performance for larger value of the ratio of the energy consumption in the con- the pulse-type ground motion keeping the allowable response nected dampers to the overall energy consumption compared to the long-period, long-duration ground motions. For long- to other comparable building systems. This leads to the effec- period, long-duration ground motions, the largest response was tive reduction of the vibration energy in the main building. selected. Frontiers in Built Environment | www.frontiersin.org 10 October 2016 | Volume 2 | Article 26 Taniguchi et al. Hybrid Control Building System Pulse input Long period input(8.4s) Tomakomai EW Long period input(6.8s) JMA Kobe NS A E 1.00 5000 base-isolation layer [m] the main structure [MNm] inter-story drift of 0.80 4000 over-turning moment of 0.60 3000 0.40 2000 0.20 1000 0.00 0 Multiple Base-isolated Building Multiple Base-isolated Multiple Base-isolated Building Multiple Base-isolated isolation & connection isolation isolation & connection isolation & Building & Building Building connection Building connection B connection F connection middle-story isolation layer [m] 1.00 100 base shear of the 0.80 80 free wall [MN] inter-story drift of 0.60 60 0.40 40 0.20 20 0.00 0 Multiple Base-isolated Building Multiple Base-isolated Multiple Base-isolated Building Multiple Base-isolated isolation & connection isolation isolation & connection isolation & Building & Building Building connection Building connection connection connection C 0.05 G 6000 inter-story drift of non-isolation 5000 story of the main structure [m] over-turning moment of 0.04 the free wall [MNm] 4000 0.03 3000 0.02 2000 0.01 1000 0.00 0 Multiple Base-isolated Building Multiple Base-isolated Multiple Base-isolated Building Multiple Base-isolated isolation & connection isolation isolation & connection isolation & Building & Building Building connection Building connection connection connection D 100 main structure [MN] 80 base shear of the 60 40 20 0 Multiple Base-isolated Building Multiple Base-isolated isolation & connection isolation & Building Building connection connection FIGURE 12 | Response variability of proposed building model and other comparable models under various earthquake ground motions: (A) inter-story drift of base-isolation layer, (B) inter-story drift of middle-story isolation layer, (C) inter-story drift of non-isolation story of the main structure, (D) base shear of the main structure, (E) base overturning moment of the main structure, (F) base shear of the free wall, and (G) base overturning moment of the free wall. OPTIMIZATION OF CONNECTION connection dampers is adopted as the objective function. This DAMPER LOCATION quantity indicates the energy absorbed in the connection dampers under an ideal white noise-like input. A sensitivity analysis is The effective connection damper location is an interesting issue. employed as the optimization method. The initial design is the In order to find the optimal location, the maximization of the model without connection damper, and the optimization is termi- area of the energy transfer function (Takewaki, 2007) for the nated at the stage where the total quantity of damping coefficients Frontiers in Built Environment | www.frontiersin.org 11 October 2016 | Volume 2 | Article 26 Taniguchi et al. Hybrid Control Building System 26 26 25 25 24 24 23 23 22 22 21 21 20 20 19 19 18 18 17 17 16 16 15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 2x106 4x106 6x106 8x106 1x107 0 2x106 4x106 6x106 8x106 1x107 Standard model Damping coefficient [Ns/m] Op mal design Damping coefficient [Ns/m] FIGURE 13 | Optimal location of connection damper. TABLE 2 | Response characteristics of various models under representative TABLE 3 | Comparison of the first three natural periods and damping ratios two-type ground motions. between the standard model and the optimal design. Pulse-type ground Long-period, long-duration 1st 2nd 3rd motion ground motion Standard model Natural period (s) 8.31 3.25 1.12 Multiple isolation Small (resonance Medium (large building Damping ratio 0.27 0.43 0.15 and building avoidance = lengthening connection effect due to large Optimal design Natural period (s) 8.03 3.26 1.09 connection model of natural period by natural period difference, large multiple isolation) resonance effect) Damping ratio 0.40 0.39 0.17 BID = 0.19, TA = 0.80 BID = 0.27, TA = 0.38 Base-isolated Medium (resonance Medium (large building from 2.12 × 107 Ns3 /m (standard model) to 2.35 × 107 Ns3 /m. and building avoidance = lengthening connection effect due to large Table 3 presents the comparison of the first three natural periods connection model of natural period by base natural period difference, large and damping ratios. It can be observed that the fundamental isolation) resonance effect) natural period of the optimal design is shorter than that of the BID = 0.19, TA = 1.52 BID = 0.33, TA = 0.43 standard model and the lowest mode damping ratio of the optimal Building Large (small building Small (resonance avoidance) design is larger than that of the standard model. connection model connection effect for pulse-type ground motion) The position of the middle-story isolation may be an interesting TA = 9.08 TA = 0.32 theme in the response reduction. This will be discussed in the Multiple isolation Small (resonance Large (large resonance effect) future. building model avoidance = lengthening of natural period by CONCLUSION multiple isolation) BID = 0.20, TA = 0.53 BID = 0.60, TA = 0.78 The following conclusions have been derived. Base-isolated Medium (resonance Large (large resonance effect) building model avoidance = lengthening (1) A new hybrid passive control building system has been pro- of natural period by posed, in which a multi-isolation (double-isolation) building base isolation) is connected to another non-isolated building (free wall) with BID = 0.20, TA = 1.86 BID = 0.77, TA = 0.89 oil dampers. BID, base-isolation layer deformation (m); TA, top acceleration (m/s2 ). (2) It was demonstrated that the transfer function of the proposed building system possesses lower values in a broader frequency range compared to other comparable building systems. In reaches the total quantity for the standard model with a uniform particular, the story drifts of the base-isolation story and the damping coefficient 2.16 × 106 Ns/m. middle-isolation story in the fundamental natural frequency Figure 13 shows the result of the optimization. It can be found have been reduced greatly together with the top-story accel- that the effective dampers are located at several stories above and eration of the main multi-isolation building at higher natural below the middle-isolation story (20th story) and larger quantities frequencies. However, the story drift of the base-isolation are allocated to the upper side. The objective function (area of the story at the second natural frequency has been increased a energy transfer function for connection dampers) was maximized little bit. Frontiers in Built Environment | www.frontiersin.org 12 October 2016 | Volume 2 | Article 26 Taniguchi et al. Hybrid Control Building System (3) It has been shown that the proposed building system is comparable building systems. While these values become a effective both for pulse-type ground motions and long- slightly larger under the long-period, long-duration ground period, long-duration ground motions. This indicates the motions, no problem occurs because those response values high robustness of the proposed building system for various are relatively small compared to those response values under kinds of ground motions. In particular, the story drifts of the the pulse-type ground motions. base-isolation story and the middle-isolation story exhibit the (7) The effective connection damper location can be found by value of half or two-thirds of the corresponding values of introducing the energy transfer function for the connection the comparable building systems (base-isolated and building damper as an objective function and using a sensitivity analy- connection model, multiple-isolation building model) under sis. The effective dampers are located at several stories above the long-period, long-duration ground motions. Further- and below the middle-isolation story (20th story), and larger more, the acceleration of the proposed building system can quantities are allocated to the upper side. be reduced effectively under the long-period, long-duration ground motions compared to the comparable building sys- In introducing the proposed system, a cost issue should be tems (base-isolated building model, multiple-isolation build- resolved. When base-isolation systems were developed and intro- ing model). duced in 1980s, the cost issue was discussed in detail. However, (4) From the viewpoint of energy response, it has been shown that the benefits obtained from such new systems resolved that issue. the proposed building system has a larger value of the ratio Furthermore, as the number of constructions of buildings using of the energy consumption in the connected dampers to the such new systems becomes large, the cost is becoming lower grad- overall energy consumption compared to other comparable ually. In addition, the property of the proposed system as a highly building systems. This leads to the effective reduction of robust system for a broad type of earthquake ground motions the vibration energy in the main building. The remarkable seems to be preferred, especially in the current situation that the reduction of the vibration energy in the main building has properties of earthquake ground motions are highly uncertain and also been observed also under the long-period, long-duration unpredictable. ground motions. (5) The response reduction in the base-isolation story of the AUTHOR CONTRIBUTIONS proposed system has been achieved by the distributed place- ment of dampers in the middle-isolation story and connection M Taniguchi formulated the problem, conducted the computa- system. The realization of the larger ratio of the fundamental tion, and wrote the paper. KF helped the computation and dis- natural periods between the main building and the free wall cussed the results. M Tsuji discussed the results. IT supervised the has made the proposed system effective. On the other hand, research and wrote the paper. the installation of dampers at the non-isolated inter-stories is not effective because of the small inter-story drift in the FUNDING base-isolated (or multiple-isolation) buildings. (6) The story shear and overturning moment in the free wall Part of the present work is supported by the Grant-in-Aid for Sci- of the proposed building system under the pulse-type entific Research (KAKENHI) of Japan Society for the Promotion ground motions exhibit smaller values compared to the other of Science (No. 15H04079). This support is greatly appreciated. REFERENCES Hashimoto, T., Fujita, K., Tsuji, M., and Takewaki, I. (2015). Innovative base- isolated building with large mass-ratio TMD at basement. Int. J. Future Cities Amadio, C., Fragiacomo, M., and Rajgelj, S. (2003). 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Frontiers in Built Environment | www.frontiersin.org 14 October 2016 | Volume 2 | Article 26 ORIGINAL RESEARCH published: 05 October 2017 doi: 10.3389/fbuil.2017.00057 Innovative Seismic Response-Controlled System with Shear Wall and Concentrated Dampers in Lower Stories Tsubasa Tani 1,2 , Ryota Maseki 1 and Izuru Takewaki 2 * 1 Technology Center, Taisei Corp., Yokohama, Japan, 2 Department of Architecture and Architectural Engineering, Graduate School of Engineering, Kyoto University, Kyoto, Japan A new structural control system using damper-installed shear walls in lower stories with reduced stiffness is proposed for vibration control of high-rise RC buildings. That system has some design variables, i.e., height of shear wall, degree of stiffness reduction at lower stories, and quantity of dampers. In this paper, some parametric studies on the shear- beam model with a stiff beam against two kinds of ground motion, a pulse-type sinusoidal wave and a resonant sinusoidal wave, are conducted to clarify the vibration characteristics of the proposed structural control system. It is shown that the optimal combination of design parameters depends on the input ground motion. It is also shown that it is possible Edited by: to prevent from increasing the response under the one-cycle sinusoidal input resonant Tomaso Trombetti, Università di Bologna, Italy to the lowest mode and reduce the steady-state response under the harmonic input Reviewed by: with the resonant fundamental period by reducing the stiffness in the lower structure and Emanuele Brunesi, increasing the damper deformation. European Centre for Training and Research in Earthquake Keywords: earthquake response, vibration control, high-rise building, soft story, shear wall, passive damper Engineering, Italy Michele Palermo, Università di Bologna, Italy INTRODUCTION *Correspondence: After the occurrence of unexpected earthquake damage, many structural engineers are striving Izuru Takewaki [email protected] for resilient building structures tough for extreme earthquake inputs and recoverable fast to an acceptable level (Bruneau and Reinhorn, 2006, Takewaki et al., 2012). It is aimed at trying to Specialty section: enhance the earthquake resilience of building structures via innovative technologies for broader This article was submitted to classes of earthquake ground motions (Amadio et al., 2003, Kobori, 2004, Takewaki et al., 2012, Earthquake Engineering, 2013, Takewaki, 2013). In developing these innovative techniques, high uncertainty in earthquake a section of the journal Frontiers in ground motions may disturb the progress (Takewaki et al., 2011a,b, 2012, 2013, Takewaki, 2013). Built Environment The variability and uncertainty in building structural properties (especially the properties of added Received: 03 July 2017 control systems) should also be taken into account appropriately (Ben-Haim, 2006, Takewaki et al., Accepted: 19 September 2017 2011b). Published: 05 October 2017 In response to these circumstances, various kinds of vibration-controlled systems have been Citation: developed in the last three decades (Housner et al., 1997, Soong and Dargush, 1997, Hanson Tani T, Maseki R and Takewaki I and Soong, 2001, Christopoulos and Filiatrault, 2006, Takewaki, 2009, Lagaros et al., 2013). Base- (2017) Innovative Seismic isolation systems, inter-story damper systems, inter-building damper systems (Fukumuto and Response-Controlled System with Shear Wall and Concentrated Takewaki, 2015), and tuned-mass damper systems are representative examples. It is well known that Dampers in Lower Stories. the introduction of large deformation in the damper location is important in the inter-story damper Front. Built Environ. 3:57. systems and inter-building damper systems (Takewaki, 2009). This is because the large deformation doi: 10.3389/fbuil.2017.00057 in the damper location makes the damper systems effective. Frontiers in Built Environment | www.frontiersin.org 15 October 2017 | Volume 3 | Article 57 Tani et al. Soft Lower-Story Vibration Control Base-isolation systems have been employed mainly in Japan, installation of large mass pendulum systems at roof and usage of New Zealand, China, and US. Various types of base-isolation upper stories as TMD masses. systems have been introduced for pulse-type ground motions Recently, large mass ratio TMDs were tackled for base-isolated (Jangid and Datta, 1994; Hall et al., 1995; Heaton et al., 1995; buildings (Villaverde, 2000; Villaverde et al., 2005; Angelis et al., Jangid, 1995; Kelly, 1999; Naeim and Kelly, 1999; Jangid and Kelly, 2012; Nishii et al., 2013; Xiang and Nishitani, 2014). While usual 2001; Morales, 2003; Takewaki, 2005, 2008; Li and Wu, 2006; Hino tall buildings exhibit large displacement near the top story because et al., 2008; Takewaki and Fujita, 2009). But their resilience for of contribution of almost uniform inter-story drifts, base-isolated earthquake has never been proved in actual situations for long- buildings show relatively large displacement at the base-isolation period ground motions with the characteristic period of 5–8 s story near ground surface. This property is very useful from the (Irikura et al., 2004; Kamae et al., 2004; Ariga et al., 2006). This view point of reduction of effect of large vertical loads due to large problem is closely associated with the resonance of the base- mass ratio TMD (Kareem, 1997; Zhang and Iwan, 2002; Mukai isolated buildings to those ground motions (Hashimoto et al., et al., 2005; Petti et al., 2010; Nishii et al., 2013; Xiang and Nishi- 2015). The long-period ground motions with 5–8 s characteris- tani, 2014). However, several issues still exist, e.g., avoidance of tic periods were greatly concerned in the structural design of excessive vertical load by large mass ratio TMD, reduction of TMD base-isolated buildings and super high-rise buildings after the stroke, and reduction of TMD support reactions (Hashimoto et al., Northridge earthquake in 1994 and the Tokachi-oki earthquake in 2015). 2003. Such input was shown as a key critical input for those kinds The use of a braced or mega-braced core is another effective of buildings during the 2011 off the Pacific coast of Tohoku earth- method for improvement of structural responses under seismic quake. It is also of great concern that, while building structures loading (Brunesi et al., 2016). Furthermore, alternative solutions with passive dampers are effective for long-duration and long- have been presented recently to obtain a kind of controlled struc- period ground motions (Takewaki, 2007; Patel and Jangid, 2011; tures by making use of dissipative devices called crescent-shaped Takewaki et al., 2011b, 2012; Kasagi et al., 2015), their effectiveness braces (Palermo et al., 2014; Kammouh et al., 2017). Finally, for pulse-type ground motions is doubtful. This is because the a bracing system (called strongback system, SB) to control the structures with viscous-type dampers can not necessarily possess deformed shape of framed structures subjected to seismic input good performance for the impulsive input like near-fault ground is currently under development (Lai and Mahin, 2014). While motions resulting from the delay of response in such viscous- these kinds of structural systems appear to be effective for long- type dampers. The overcome of these two difficulties is of great period ground motions under the condition of possessing suf- significance in the seismic resistant and control design (Koo et al., ficient damper systems, the effectiveness for near-fault ground 2009, Petti et al., 2010, Karabork, 2011). motions represented by pulse-type ground motions is not clear Base isolation has been applied even to high-rise buildings. One and further investigation may be necessary. is a base-isolated high-rise building without connection and the It is also important to develop methods for estimation of dis- other is a base-isolated building connected to another structure placement and velocity profiles for framed structures with added with dampers (Murase et al., 2013, Kasagi et al., 2016, Fukumoto dampers. The methods proposed by Palermo et al. (2015, 2016) and Takewaki, 2017). In the latter connected system, a base- are representative ones. isolated high-rise building structure is linked to another non- The present authors proposed a new vibration-controlled sys- isolated normal structure (free wall) with oil dampers. Because tem in which large deformation in lower stories is induced and of the necessity of a substructure supporting the main building, oil dampers are installed effectively (Tani et al., 2017). A sim- high-rise residential apartment houses are the main object where ilar study was conducted by Kazama and Mita (2006). In the a car parking tower is allocated as the substructure. The connected previous research (Tani et al., 2017), the large deformation of high-rise buildings without base isolation and base-isolated high- oil dampers is obtained by the rigid rotation of shear walls. To rise buildings connected to another structure have been designed ensure the safety in those lower stories, rigid shear-wall systems and constructed by Obayashi Corporation and Shimizu Corpora- are introduced in the oil-damper installation system. Figure 1A tion in Japan (Murase et al., 2013, Kasagi et al., 2016, Fukumoto shows the proposed vibration-controlled structure and Figure 1B and Takewaki, 2017). presents the modeling of the structure with the proposed system Historically tuned mass dampers (TMDs) have often been used into a multi-degree-of-freedom model and a reduced model. In for reducing building responses to wind loading and have been the MDOF model, a rigid beam element (shear walls) with a actually installed in many high-rise buildings (Soong and Dar- pin connection at the bottom is connected to the main structure gush, 1997). It should be reminded that TMD is not effective for with rigid connectors. The added damping by the oil dampers is earthquake input because of the difficulty in stroke limitation and installed in the lower stories of the main structure. On the other realization of large mass ratio TMD. hand, in the reduced model, the lower structure is modeled by Nevertheless, large mass ratio TMDs have been explored mainly a rigid bar supported by a rotational spring at the base and the for earthquake inputs (Chowdhury et al., 1987; Feng and Mita, lower structure is treated as a single-degree-of-freedom model in 1995; Arfiadi, 2000; Villaverde, 2000; Zhang and Iwan, 2002; which the mass is concentrated at the point with the equivalent Mukai et al., 2005; Villaverde et al., 2005; Tiang et al., 2008; Matta height. and De Stefano, 2009; Petti et al., 2010; Angelis et al., 2012; Nishii In this paper, parametric study on shear-beam model with et al., 2013; Xiang and Nishitani, 2014). It should be pointed stiff beam against two kinds of ground motion, pulsed sinusoidal out that several projects are being planned in Japan aiming at wave and resonant sinusoidal wave, is conducted to clarify the Frontiers in Built Environment | www.frontiersin.org 16 October 2017 | Volume 3 | Article 57 Tani et al. Soft Lower-Story Vibration Control A B C FIGURE 1 | Proposed vibration-controlled structure and its modeling into multi-degree-of-freedom model and reduced model: (A) proposed system, (B) MDOF model, and (C) reduced model. vibration characteristics. It is shown that the optimal combination story height. The number of stories is an example of a high-rise of variables depend on input ground motion, however, consid- building. The main structure is connected to a rigid bar (shear wall ering damper deformation growth by shear-wall, slight stiffness including dampers) with rigid connection. The model is assumed reduction at lower stories can achieve smaller story drift than to have a straight-line lowest mode. proportional damping. Since the analysis of the original model shown in Figure 1B seems difficult, a simpler model (equivalent to the model in NATURAL FREQUENCY AND MODE OF Figure 1B) as shown in Figure 1C is introduced. Consider the lower structure in Figure 1B without the super structure (the SYSTEM WITH SHEAR WALL IN LOWER lower part of the main structure and the rigid bar with rigid STORIES link). Let D, N, and Hw denote the top displacement of the lower structure, the number of stories of the lower structure, and the Basic Model and Reduced Model for height of the lower structure, respectively. Furthermore, let me , Natural Vibration Analysis He , and kθ denote the equivalent mass of the lower structure, Consider a 50-story shear building model, as shown in Figure 1B, the equivalent height of the lower structure, and the rotational which has an equal floor mass m at each story and an equal stiffness of the spring at the base. When the fundamental natural Frontiers in Built Environment | www.frontiersin.org 17 October 2017 | Volume 3 | Article 57 Tani et al. Soft Lower-Story Vibration Control circular frequency of the lower structure is denoted by ωl , the frequencies of the lower structure and its equivalent SDOF model, lowest-mode i-th story shear force Qi of the lower structure in free the equivalent quantities He , me and the rotational stiffness of the vibration, the lowest-mode i-th story overturning moment Mi of spring at the base can be derived as follows: the lower structure, the equivalent story shear Qe of the equivalent ∑N ∑N SDOF model, and the equivalent overturning moment Me of the He 1 i=1 n equivalent SDOF model can be expressed by = ∑N n=i (2a) Hw N n=1 n (∑N )2 ∑ N n n=1 n Qi = mω2l D (1a) me = m ∑N ∑N (2b) n=i N i=1 n=i n Hw2 ∑ He N Qe = me ω2l D (1b) kθ = ki . (2c) Hw N 2 i=1 ∑ N Hw DHw ∑ ∑ N N Mi = Qn = mω2l n (1c) As N becomes large, He /Hw converges to 2/3 and me /(Nm) n=i N N 2 j=i n=j converges to 3/4. Me = Qe He . (1d) Natural Frequency and Mode By requiring the equivalence Q1 = Qe , M 1 = M e in Eq. 1 in The model shown in Figure 1C is used for the analysis of addition to the equivalence of the fundamental natural circular natural frequencies and modes. Figure 2A shows the change A B FIGURE 2 | Change of natural period and participation vector to wall height: (A) change of natural period and (B) change of participation vector. Frontiers in Built Environment | www.frontiersin.org 18 October 2017 | Volume 3 | Article 57 Tani et al. Soft Lower-Story Vibration Control of natural periods to wall height (ratio of wall height to total deformation becomes smaller in lower parts as Hw /H becomes height). The left figure is normalized to the fundamental natural larger (see Figure 2B). period of the model without shear wall and the right one is normalized to each natural period of the model without shear RESPONSE TO RESONANT ONE-CYCLE wall. Since the lowest mode is straight, the fundamental natural period is not affected by the wall height. It can be observed that SINUSOIDAL PULSE WAVE there is no influence until 0.2 and higher natural periods are When the building with the proposed system is subjected to affected much. pulse-type ground motions, the energy dissipation by repeated Figure 2B illustrates the change of participation vector to wall vibration cannot be expected. In this case, the strict check of height. It can be seen that, as the wall height becomes larger, the strength is important in the lower parts. In this section, the lower-part higher-mode participation vectors remain straight and one-cycle sinusoidal waves resonant to the fundamental and their slopes become smaller. On the other hand, the slopes of the second natural periods are input. The maximum response is upper-part higher-mode participation vectors become larger as evaluated by the mean of “the absolute sum of the maximum the wall height becomes larger. However, its effect is small until fundamental and second vibration components” and “the SRSS Hw /H = 0.6 in the second mode and until Hw /H = 0.4 in the third value.” The accuracy of this evaluation method will be investigated mode. later. Transfer Function The model shown in Figure 1C is also used for the analysis of Original Model and Comparison Model transfer functions. Figure 3 shows the transfer functions (top In this section, a continuum shear-beam model with uniform acceleration/base acceleration) for various wall heights. Figure 3A mass density is dealt with in order to enable the closed-form indicates those of the model with proportional damping (damping mathematical treatment. A massless rigid bar with pin at the base ratio: 0.03) and Figure 3B presents those of the model with con- is connected to this shear-beam model in the lower part. The centrated damping in lower parts (total damping quantity is the lowest mode of the upper part is assumed based on data on the same as the left one: damping ratio of 0.01 is distributed uniformly realistic high-rise building models. and the remaining damping ratio of 0.02 is concentrated in the lower parts). The damping ratio 0.03 is usually used in the design Investigation Model of reinforced concrete buildings and its comparable quantity of In most high-rise buildings, it is often the case that the inter-story damping is given to the lower part of the present controlled build- drifts are almost uniform in the middle stories and they decrease ings. The stiffness-proportional damping (proportional to overall toward the top and the bottom. Furthermore, in the proposed structure or partial lower portion) is employed to represent the model, a rigid bar is installed in the lower part. Based on these damping concentration appropriately. It can be seen in the model information, the fundamental mode is assumed to be expressed with proportional damping that, as Hw /H becomes larger, the by Eq. 3: amplitude at the second mode becomes smaller. This is because the higher-mode damping ratio becomes larger in the model  with proportional damping. On the other hand, in the model  Ax + Bx + C (Ht ≥ x > Hl ) 2 with concentrated damping in lower parts, the amplitude at the 1 φ(x) = Dx + E (Hl ≥ x > Hw ) (3)   second mode becomes larger. This is because the higher-mode Fx (x ≤ Hw ) A B 30 30 Hw/H=0.0 Hw/H=0.0 25 25 Hw/H=0.2 Hw/H=0.2 Hw/H=0.4 Hw/H=0.4 20 20 H /H=0.6 H /H=0.6 Amplitude Amplitude w w Hw/H=0.8 Hw/H=0.8 15 15 10 10 5 5 0 0 0 1 2 3 4 0 1 2 3 4 Frequency(Hz) Frequency(Hz) FIGURE 3 | Transfer function: (A) proportional damping and (B) concentrated damping. Frontiers in Built Environment | www.frontiersin.org 19 October 2017 | Volume 3 | Article 57 Tani et al. Soft Lower-Story Vibration Control where Damping Φ′ t − 1 Since the structural damping is the stiffness-proportional one, the A= φ {2Hw (Φ w − 1) + Ht + Hl + Ht Φ′ t − Hl Φ′ t } (Ht − Hl ) t ′ sum of damping coefficients of the model becomes smaller for the (4a) model with a smaller sum of the story stiffnesses in the lower part ( ′ ) and longer fundamental natural period. The additional damping 2 Ht − Φ t Hl B= φ coefficients are assumed not to change. {2Hw (Φ′ w − 1) + Ht + Hl + Ht Φ′ t − Hl Φ′ t } (Ht − Hl ) t The relations of mass, stiffness, and damping between the (4b) ( ′ ) ( ′ ) investigation model and the comparison model are expressed as 2Hw (Ht − Hl ) Φ w − 1 + Hl Φ t − 1 2 follows: C= φ {2Hw (Φ′ w − 1) + Ht + Hl + Ht Φ′ t − Hl Φ′ t } (Ht − Hl ) t M (4c) m(x) = m ¯ (x) = (5a) Ht 2 ∫Ht ∫Ht D= φ (4d) 2Hw (Φ′ w − 1) + Ht + Hl + Ht Φ′ t − Hl Φ′ t t k (x) dx = ¯k (x) dx (5b) ( ) 2Hw Φ′ w − 1 Hw Hw E= φ (4e) 2Hw (Φ′ w − 1) + Ht + Hl + Ht Φ′ t − Hl Φ′ t t c (x) ∝ k (x) , ¯c (x) ∝ ¯k (x) (5c) ′ 2Φ w ∫ Hw ∫Ht F= φ (4f) 2Hw (Φ′ w − 1) + Ht + Hl + Ht Φ′ t − Hl Φ′ t t Δc (x) dx = Δ¯c (x) dx (5d) ′ φw 0 0 Φ′ w = (4g) φ′ l where m(x), k(x), c(x), Δc(x), and m(x), ¯ ¯k(x), ¯c(x), Δ¯c(x) are the ′ mass, stiffness, structural damping coefficient, and additional φt Φ′ t = (4h) φ′ l damping coefficient distributions of the investigation model and the comparison model, respectively. In addition, M is the total φ′ l (Hl + 0) = φ′ l (Hl − 0). (4i) mass. Figure 4 shows an example of the fundamental mode and its In Eq. 3, Hw , Hl , and Ht (=H) denote the height of the lower story drift angle. part, the height of the part with straight fundamental mode, and the building total height and φw , φl , φt , φ′ w , φ′ l , φ′ t indicate the Model to Be Considered (Investigated displacements of the fundamental mode at Hw , Hl , Ht , and the slopes of the fundamental mode at Hw , Hl , Ht . Φ′ w and Φ′ t Region of Fundamental Mode Shape) indicate the ratio of the slope at x = Hw (top of the lower structure) Realistic models are considered here. Figure 5 shows the investi- of the lowest mode to that at x = Hl (the top of straight-line gated region of the fundamental mode shape with Hw and Φ′ w lowest-mode shape) and the ratio of the slope at x = Ht (top of as parameters. 0.2Ht ≤ Hw ≤ 0.6Ht and 0.5 ≤ Φ′ w ≤ 2.0 are the upper structure) of the lowest mode to that at x = Hl . In this considered here. investigation, it is assumed that Hl /Ht = 0.6 and Φ′ t = 0.6. The The additional damping quantity D1 Δ¯h is defined as the sum of structural damping is assumed to be stiffness-proportional over the damping coefficients in the comparison model with a stiffness- the whole height and the additional damping is added only to the proportional damping of the lowest-mode damping ratio 1 Δ¯ h. In lower part as stiffness-proportional one. the investigation model, the additional damping coefficients with the same value of D1 Δ¯h are allocated to the lower part. Comparison Model The fundamental mode has the property of Eq. 3 with Hl /Ht = 0.6, A B Φ′ t = 0.6, and Φ′ w = 1.0. The comparison model does not have the rigid bar in the lower part. The structural damping and the additional damping are the same as the investigation model. However, the additional damping is distributed over the total height. The quantities of the comparison model are designated by the over-bar. Stiffness The sum of the story stiffnesses in the upper part is the same between the investigation model and the comparison model. The sums of the story stiffnesses in the lower part and their distri- butions are different depending on the adopted mode. In case of Φ′ w > 1.0, the sum of the story stiffnesses of the whole part in the FIGURE 4 | Fundamental mode and its story drift angle: (A) fundamental investigation model becomes larger than that of the comparison mode shape and (B) story drift angle. model. Then, the fundamental natural period becomes longer. Frontiers in Built Environment | www.frontiersin.org 20 October 2017 | Volume 3 | Article 57 Tani et al. Soft Lower-Story Vibration Control ¯= Φ′ t − 1 A φ (10a) (Ht + Hl + Ht Φ′ t − Hl Φ′ t ) (Ht − Hl ) t ( ) 2 Ht − Φ ′ t Hl ¯= B φ (10b) (Ht + Hl + Ht Φ′ t − Hl Φ′ t ) (Ht − Hl ) t ( ) Hl 2 Φ′ t − 1 ¯ C= φ (10c) (Ht + Hl + Ht Φ′ t − Hl Φ′ t ) (Ht − Hl ) t ¯= 2 D φ (10d) Ht + Hl + Ht Φ′ t − Hl Φ′ t t 2¯ h ¯c(x) = 1 ¯k(x). (10e) 1 ω ¯ Similarly, the additional damping coefficient of the comparison model can be expressed by 21 Δ¯h¯ FIGURE 5 | Investigated region of fundamental-mode shape. Δ¯c(x) = k(x). (11) 1ω¯ Damping Coefficient of Added Damper and The sum of the additional damping coefficients of the compar- ison model can then be obtained as Second Natural Frequency and Mode Fundamental Mode ∫Ht ∫Ht 2 Δ¯ h When the fundamental mode is expressed by D1 Δ¯h = Δ¯c(x)dx = 1 ¯k(x)dx. (12) 1 ω ¯ t) = 1 φ(x)exp(i1 ωt), 0 0 1 u(x, (6a) the dynamic equilibrium of the part from x to Ht can be described Since the additional damping coefficient of the investigation by model is given only at the lower parts as one proportional to the ∫Ht corresponding stiffness, the additional damping coefficient of the M ∂21u ∂ u dx + k (x) 1 = 0. (6b) investigation model is expressed as follows: Ht ∂t2 ∂x x { 0 (x > Hw ) Substitution of Eqs. 3 and 6a into Eq. 6b leads to the following Δc(x) = k(x) . (13) form of stiffness. ∫ Hw D ¯ (x ≤ Hw ) k(x)dx 1 Δ h 0 k(x) = { } Equations 7 and 12 should be substituted into Eq. 13. The    M1 ω2 A (H3t − x3 )+ B2 (H2t − x2 ) fundamental natural circular frequency of the investigation model   3 (Ht ≥ x > Hl )   Ht (2Ax+B) + C(Ht − x) can be determined from the condition that the sum of the stiff-     nesses in the upper part is constant in the investigation model.   { ( ) ( ) }  A H3t − H3l + B2 H2t − H2l + C(Ht − Hl ) The sums of the stiffnesses in the upper parts of the investigation M1 ω 2 3 ( ) (Hl ≥ x > Hw ) .   t HD + D2 H2l − x2 + E(Hl − x) model and the comparison model are expressed by     { ( }   ) ( )   M1 ω2 A H3t − H3l + B2 H2t − H2l + C(Ht − Hl ) ∫Ht   3 ( ) (x ≤ Hw ) ( )  Ht F + D2 H2l − H2w + E(Hl − Hw )+ F2 (H2w − x2 ) k(x)dx = f Hw , Φ′ w , Φ′ t 1 ω2 (14) (7) Hw The structural damping coefficient of the investigation model ∫Ht ( ) 2 can then be expressed by ¯k(x)dx = g Φ′ t 1 ω ¯ (15) 21 h Hw c(x) = k(x). (8) 1ω where f(Hw , Φ′ w , Φ′ t ) is the function derived by substituting Eq. 7 Consider the additional damping coefficient of the investiga- into the left-hand side of Eq. 14 and g(Φ′ t ) is the function derived tion model. First of all, the stiffness and structural damping coef- by substituting Eq. 9 into the left-hand side of Eq. 15. By equating ficient of the comparison model can be obtained by substituting Eqs 14 and 15, the fundamental natural circular frequency of the Φ′ w = 1.0 into Eqs 7 and 8 (then D = F, E = 0): investigation model can be obtained as follows:  ¯ ¯ √   ¯ 2 A3 (H3t −x3 )+ B2 (H2t −x2 )+C(H M1 ω ¯ t −x) (Ht ≥ x > Hl )   H t ¯ ¯ 2Ax+ B g (Φ′ t ) { ( ) ( )} 1ω = 1ω ¯. (16) ¯k(x) = ¯ A ¯ H3t − H3l + B2 H2t − H2l f (Hw , Φ′ w , Φ′ t )   ¯M ¯2 ω 3  1  Ht D ( ) (x ≤ Hl ) ¯ t − Hl )+ D¯ H2l − x2 +C(H The lowest-mode additional damping ratio 1 Δh of the 2 (9) investigation model is predicted approximately by using the Frontiers in Built Environment | www.frontiersin.org 21 October 2017 | Volume 3 | Article 57 Tani et al. Soft Lower-Story Vibration Control undamped lowest mode. Using Eq. 3, 1 Δh can then be evaluated a0 , a1 , · · · , aL−1 are obtained to be the L-th order polynomials of 2 by 2ω . By using Eqs 7 and 19, the continuity condition of the story ∫ Hw ( )2 shear force at x = Hw leads to Δc(x) dφ dx ω 0 dx ∑L 1 Δh = 1 ( ) . ∑ L n 2 ∫ Ht dφ 2 k (Hw + 0) nan xn−1 = k (Hw − 0) n=0 an Hw (21) 0 k(x) dx dx Hw ∫ Hw n=1 ω Δc(x)F2 dx = 1 ∫ Ht 0 ∫H (17) 2ω 2 satisfying Eq. 21 is regarded as an approximate value of the 2 k(x)(2Ax + B)2 dx + Hlw k(x)D2 dx Hl ∫H second natural frequency. + 0 w k(x)F2 dx The second damping ratio of the investigation model is evalu- Second Mode ated by using an undamped second natural mode. The additional It is known that the second mode of a shear-beam with uni- damping ratios (the additional damping quantity D2% ) of the form mass density and a straight-line fundamental mode can be investigation model for various shear wall heights are shown in expressed by a cubic function in the comparison model. However, Figure 6A with respect to Φ′ w for the fundamental mode and some modification is necessary in the investigation model. For in Figure 6B for the second mode. It can be understood that the this reason, the second mode above the point x = Hw is modeled fundamental additional damping ratio becomes larger than that by an L-th order function (L = 5 is employed here). Then, the for the proportional damping case as the stiffness in the lower part second mode can be expressed by decreases. On the other hand, the second additional damping ratio  L depends greatly on Hw . While it increases as the stiffness in the  ∑ lower part decreases in the case of small values of Hw , it is almost   an x n (x > Hw ) constant irrespective of the stiffness in the lower part in the case 2 φ(x) ≈ n=0 . (18)   ∑ of large values of Hw .  Ln=0 an (Hw )n x (x ≤ H ) Hw w The slope of the second mode can be obtained as Maximum Response Displacement Consider the maximum story drift angle of the model subjected to  L  ∑ the resonant (first mode and second mode) one-cycle sinusoidal   nan xn−1 (x > Hw ) d2 φ n=1 wave with constant velocity amplitude of 1 m/s. The total response = . (19) dx   ∑ is evaluated approximately by superposing the first and second  Ln=0 an (Hw )n (x ≤ H ) mode vibrations. The maximum response is evaluated by the Hw w mean of the absolute sum of the first and second mode vibrations The dynamic equilibrium of the part from x = Hw to Ht can be and the SRSS value. described by The displacement response of the SDOF model under a one- cycle sinusoidal acceleration wave α sin ωt is well known (Clough ∑ M2 ω2 ∑ an ( n+1 ) L L k(x) nan xn−1 = Ht − xn+1 (x > Hw ). and Penzien, 1975, Yasui et al., 2010) and can be expressed by n=1 Ht n=0 n + 1 (20) u(t) = e−h¯ωt (X cos ω ¯ d t + Y sin ω ¯ d t) for x = x1 , x2 , · · · , xL > Hw . L points have been used to ¯ 2 − ω2 ) sin ωt − 2hω α{(ω ¯ ω cos ωt} − 2 (22) determine L coefficients. When aL is treated as a given value, (ω − ω ) + (2hω ω)2 ¯ 2 2 ¯ A B FIGURE 6 | Damping ratio with respect to Φ ′ w : (A) first mode and (B) second mode. Frontiers in Built Environment | www.frontiersin.org 22 October 2017 | Volume 3 | Article 57 Tani et al. Soft Lower-Story Vibration Control where ω¯, ω ¯ d , h are the undamped natural circular frequency, the shows a constant distribution. In case of Φ′ t = 0.6, the lowest damped natural circular frequency, and the damping ratio and mode exhibits the maximum story drift angle in the upper struc- X, Y are given by ture in Hw < x ≤ Hl and the second mode exhibits the maximum story drift angle in the upper structure at x = Ht . Let i u′ j (x) and −2hω ¯ ωα ¯′ i u j (x) denote the j-th mode maximum story drift angle at x X= (23a) ¯ 2 − ω2 )2 + (2hω (ω ¯ ω)2 of the investigation model subjected to the one-cycle sinusoidal ω ¯ 2 − ω2 − 2h2 ω ω ¯2 input resonant to the mode i and that of the comparison model. Y=α 2 . (23b) Furthermore, let i u′ max and i u¯′ max denote the maximum story ω ¯ 2 − ω2 ) + (2hω ¯ d (ω ¯ ω)2 drift angle of the investigation model subjected to the one-cycle Equation 22 is the solution during the input and the response sinusoidal input resonant to the mode i and that of the comparison after the input can be expressed in terms of free vibration. The time model. at which the maximum displacement occurs in the undamped model is expressed by the following one (Kamei et al., 2010): Resonant Wave to First Mode Figures 7A–C show 1 u′ j (Ht ), 1 u′ j (Hl ), and 1 u′ j (0) and Figure 7D n presents 1 u′ max for the structural damping ratio 0.01 and the tdmax = Tp before the input termination (24a) (Tp /T + 1) additional damping quantity D2% . It should be noted that the story kT + Tp drift angles are expressed under the condition of Ht = 1. (If the tdmax = after the input termination (24b) fundamental natural period = 1.0 s and the total height = 40 m, 2 the number 60 in the vertical axis means 0.015 rad.) It can where T is the natural period of the SDOF model, n is a positive be observed that the influence of the second mode is small integer satisfying tdmax ≤ Tp , and k is a positive integer satisfying and its small effect exists only around the top. As the stiff- tdmax ≥ Tp . Therefore, this expression is used approximately for ness of the lower structure decreases, the story drift angle in damped vibration. the upper structure becomes small and the story drift angle in Since the lower structure exhibits a straight-line displacement the lower structure increases. This phenomenon is prominent due to the existence of the rigid shear wall, the story drift angle for the larger value of Hw in the upper structure and for the A B C D FIGURE 7 | Maximum story drift angle for lowest-mode resonance pulse: (A) 1 u′j (Ht ), (B) 1 u′j (Hl ), (C) 1 u′j (0), and (D) 1 u′max . Frontiers in Built Environment | www.frontiersin.org 23 October 2017 | Volume 3 | Article 57 Tani et al. Soft Lower-Story Vibration Control smaller value of Hw in the lower structure. The model with the Accuracy Check by Time–History Response Analysis original stiffness distribution exhibits the smallest value irrespec- for MDOF Model tive of Hw and this value is almost the same as the compari- Figure 9 shows the comparison of the maximum story drift angles son model. It can also be seen that the increase of additional by the proposed method with those by the MDOF model as shown damping in the lower structure cannot prevent the response in Figure 1B. It can be observed that, although the proposed esti- amplification due to the reduction of the stiffness in the lower mation gives a slightly larger response under the input resonant to structure. the lowest mode, the accuracy is almost satisfactory. On the other hand, the proposed estimation provides a slightly smaller response Resonant Wave to Second Mode under the input resonant to the second mode in Φ′ w ≤ 1. This is Figures 8A–C show 2 u′ j (Ht ), 2 u′ j (Hl ), and 2 u′ j (0) and Figure 8D because the maximum story drift angle becomes the largest at the presents 2 u′ max for the structural damping ratio 0.01 and the top and the third-mode effect at the top becomes large compared additional damping quantity D2% . It can be observed that, to the response in the lower structure. In addition, the third-mode although the response by the second mode becomes larger than vibration is easy to be induced in the response resonant to the that by the fundamental mode around the top, the response by second mode than that resonant to the lowest mode. the fundamental mode becomes dominant except around the top. Since the difference of the fundamental and second natural Effect of Damping periods becomes large as Hw becomes large, the response exhibits Figure 10 shows the maximum story drift angle with respect to a( different) property even if the lowest mode shape is the same Φ′ w under various damping quantities for Hw /H = 0.4. It can Φ′ w = 1 . Since the effect of the second mode becomes small be observed that, although the response reduction is possible by as the stiffness in the lower structure becomes small, the response the introduction of the additional damping, its effect is small. in the lower structure due to the second mode does not become When the stiffness in the lower structure is reduced, the response larger. The region exists where the response of the investigation reduction effect becomes larger slightly. Furthermore, when the model becomes smaller than that of the comparison model. This stiffness in the lower structure is reduced, the response under the region becomes wider as Hw becomes larger. one-cycle sinusoidal wave resonant to the lowest mode becomes A B C D FIGURE 8 | Maximum story drift angle for second-mode resonance pulse: (A) 2 u′j (Ht ), (B) 2 u′j (Hl ), (C) 2 u′j (0), and (D) 2 u′max . Frontiers in Built Environment | www.frontiersin.org 24 October 2017 | Volume 3 | Article 57 Tani et al. Soft Lower-Story Vibration Control A B FIGURE 9 | Comparison of approximation with time-history response analysis: (A) 1 u′max and (B) 2 u′max . FIGURE 10 | Maximum story drift angle under various damping quantities FIGURE 11 | Necessary damper deformation growth rate. (Hw /H:0.4). larger than that under the one-cycle sinusoidal wave resonant to the second mode. When the stiffness in the lower structure is reduced, the response of the investigation model under the one-cycle sinu- soidal wave resonant to the lowest mode becomes larger than that of the comparison model. The proposed model has an advan- tage that the reduction of the stiffness in the lower structure enhances the performance of the dampers. Figure 11 presents the necessary damper deformation growth rate with respect to φw under various shear-wall heights for D2% and D5% . As the damper quantity becomes larger, the necessary damper growth rate becomes smaller. The influence of Hw is very small. FIGURE 12 | Correspondence of participation vectors and story shear forces between shear-beam model and 2DOF model. STEADY-STATE RESPONSE TO Reduction of Original Model into 2DOF RESONANT HARMONIC BASE INPUT Model Since the long-period, long-duration ground motions can be rep- It is well known that, since the steady-state response to the reso- resented approximately by long-duration sinusoidal waves and nant harmonic input can be influenced greatly by the magnitude steady-state vibrations are predominant in such vibration, the and distribution of damping in the structure, it may be difficult resonant input to the fundamental natural period is considered to use the undamped vibration mode for response evaluation. here. For this reason, the shear beam treated in the previous section Frontiers in Built Environment | www.frontiersin.org 25 October 2017 | Volume 3 | Article 57 Tani et al. Soft Lower-Story Vibration Control is reduced to the 2DOF model in which one mass is located mˆ 2 ˆk2 1 ω2 at x = Hw = H 1 and the other mass is located at x = H 2 . The ˆk1 = ˆ 1 1 ω2 +m (26b) ˆk2 − m ˆ 2 1 ω2 accuracy of this model will be investigated later. The height H 2 is determined from the equivalence of the participation vectors of 21 h ˆ ˆc2 = k2 (27a) these two models. In addition, the fundamental natural frequen- 1ω cies, the participation vectors at x = Hw , the lowest-mode story 2h shears at x = 0, Hw are made equal (see Figure 12: equivalence ˆc1 = 1 ˆk1 . (27b) 1ω conditions). The damping coefficients are determined so that the The quantities with that indicate the quantities of the 2DOF energy dissipations by the damping are equal. The parameters are model. The additional damping coefficient in the lower part can Hw , Φ′ w , and D1 Δ¯h . be obtained as ( )2 ˆ ˆ ˆ ˆ2 21 Δh k2 φ2 − φ1 + k1 φ1 ˆ Parameter of Reduced 2DOF Model Let 1 β denote the lowest-mode participation factor of the shear- Δˆc1 = . (28) 1ω ˆ2 φ 1 beam model. The masses, stiffnesses, and damping coefficients of the reduced 2DOF model can be expressed as follows in terms The story shear 1 Q (Hw ) at x = Hw in the lowest mode of the of the parameters Hw and Φ′ w from the equivalence conditions shear-beam model may be expressed by introduced just above: d1 φ 1 Q (Hw ) = k (Hw ) (Hw ) . (29) dx 2 1 β 1 Q(Hw ) The height H 2 at which the equivalence of the participation m ˆ2 = (25a) ˆφ {(1 − 1 βφ w ) φ w m1 1 ω + 1 Q (Hw )} 1 ω 2 2 vectors of the two models, i.e., AH2 2 +BH2 +C = 1 β ˆ , is satisfied 2 MHw can be derived as m ˆ1 = (25b) √ ( ) 2Ht ˆφ −B + B2 − 4A C − 1 β ˆ 2 ˆ 21 ω 2 H2 = − (30) ˆk2 = 1 Q (Hw ) m 2A (26a) 1 Q (Hw ) −m ˆ 21 ω2 φ w where A, B, C have been defined in Eq. 3. A B C D FIGURE 13 | Maximum story drift angle ratio (variable: Hw /H): (A) upper structure (D2% ), (B) lower structure (D2% ), (C) upper structure (D5% ), and (D) lower structure (D5% ). Frontiers in Built Environment | www.frontiersin.org 26 October 2017 | Volume 3 | Article 57 Tani et al. Soft Lower-Story Vibration Control Maximum Response Displacement model under various shear-wall heights for the additional damp- The amplitude of the steady-state displacement of the 2DOF ing level of D2% and Figure 13B presents the corresponding one model under the harmonic input with constant velocity amplitude in the lower structure. Figures 13C,D illustrate the corresponding is computed for various shear wall heights. These values are drawn ones for the additional damping level of D5% . with respect to the parameter Φ′ w . Figure 14A shows the ratio of the story drift angle in the Figure 13A shows the ratio of the story drift angle in the upper upper structure of the investigation model to that of the com- structure of the investigation model to that of the comparison parison model under various additional damping levels for the A B C D FIGURE 14 | Maximum story drift angle ratio (variable: D h ): Δ¯ (A) upper structure (Hw /H:0.2), (B) lower structure (Hw /H:0.2), (C) upper structure (Hw /H:0.5), and 1 (D) lower structure (Hw /H:0.5). A B FIGURE 15 | Comparison of time–history responses between 2DOF and MDOF: (A) Hw /H:0.2, D5% and (B) Hw /H:0.6, D5% . Frontiers in Built Environment | www.frontiersin.org 27 October 2017 | Volume 3 | Article 57 Tani et al. Soft Lower-Story Vibration Control shear-wall height of Hw /Ht = 0.2 and Figure 14B presents the Since the difference of the fundamental and second natural corresponding one in the lower structure. Figures 14C,D illustrate periods becomes large as Hw becomes large, the lowest mode the corresponding ones for the shear-wall height of Hw /Ht = 0.5. is hard to be induced and the response in the lower struc- It can be observed that, as the stiffness in the lower struc- ture becomes smaller (see Figure 8). It may be possible to ture decreases, the story drift angle in the upper structure also reduce the response by decreasing the stiffness in the lower decreases. While the story drift angle in the lower structure also structure. decreases according to the decrease of the stiffness in the lower (3) The steady-state vibration resonant to the lowest mode structure in the case of large additional damping in the lower becomes smaller by reducing the stiffness in the lower struc- structure, it increases in the case of small additional damping ture (see Figure 13). However, the region exists that the in the lower structure. This means that the effect of response response becomes larger partially when the additional damp- reduction due to the damping in the steady-state vibration is ing quantity is extremely small. larger than the effect of response amplification due to the stiffness (4) Since the rigid shear wall makes the story drift angle dis- reduction in the case where a certain level of additional damping tribution uniform and reduces the maximum response, the is introduced. Furthermore, the influence of Hw on the response response amplification by the second mode in the lower is rather small. structure is prevented as Hw becomes large. For this reason, Figure 15 presents the comparison of the time–history analysis when the additional dampers are concentrated in the lower result by the 2DOF model with that by the MDOF model. “2DOF part, the second damping ratio becomes smaller compared upper” indicates the maximum story drift angle in the upper to the stiffness-proportional damping model and it does not structure by the 2DOF model and “MDOF upper” means that increase even if the stiffness in the lower structure is reduced by the MDOF model. Furthermore, “2DOF lower” indicates the (see Figures 2B and 6B). maximum story drift angle in the lower structure by the 2DOF (5) It may be possible to prevent from increasing the response model and “MDOF lower” means that by the MDOF model. It under the one-cycle sinusoidal input resonant to the low- can be seen that the accuracy of the 2DOF model is satisfactory. est mode and reduce the steady-state response under the harmonic input with the resonant fundamental period by CONCLUSION reducing the stiffness in the lower structure and increasing the damper deformation (see Figure 11). A new structural control system using damper-installed shear (6) The maximum response evaluation under the resonant one- walls with a pin connection at the bottom has been proposed for cycle sinusoidal input using the undamped natural modes vibration control of high-rise RC buildings. The response analysis is accurate and reliable. However, as the higher-mode effect for one-cycle sinusoidal ground motions resonant to the lowest becomes larger, the accuracy may deteriorate slightly (see and second natural modes and for the harmonic ground motion Figure 9). has been performed. The obtained results are summarized as (7) The response amplitude evaluation under the harmonic input follows. can be made within a reliable accuracy by using the 2DOF (1) The lowest-mode component is predominant in the response model which is reduced from the shear-beam model (see resonant to the lowest mode and the maximum response Figure 15). becomes the smallest in the model with a straight-line lowest mode. It is more effective to control the stiffness and the AUTHOR CONTRIBUTIONS mode shape than concentrating the additional dampers into the location with larger story drift angles (see Figure 7). TT carried out the theoretical and numerical analysis. RM carried (2) The lowest-mode component is predominant in the response out the theoretical investigation. IT supervised the theoretical of the lower structure resonant even to the second mode. analysis. REFERENCES Bruneau, M., and Reinhorn, A. 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Aseismic roof isolation system The reviewer MP and handling editor declared their shared affiliation. built with steel oval elements. Earthq. Spec. 21, 225–241. doi:10.1193/1.1850528 Xiang, P., and Nishitani, A. (2014). Optimum design for more effective tuned mass damper system and its application to base-isolated buildings. Struct. Control Copyright © 2017 Tani, Maseki and Takewaki. This is an open-access article dis- Health Monitor. 21, 98–114. doi:10.1002/stc.1556 tributed under the terms of the Creative Commons Attribution License (CC BY). Yasui, M., Nishikage, T., Mikami, T., Kamei, I., Suzuki, K., and Hayashi, Y. (2010). The use, distribution or reproduction in other forums is permitted, provided the Theoretical solutions and response properties of maximum response of a single- original author(s) or licensor are credited and that the original publication in this degree-of-freedom system for pulse-wave ground motions. J. Struct. Construct. journal is cited, in accordance with accepted academic practice. No use, distribution Eng. 650, 731–739. doi:10.3130/aijs.75.731 or reproduction is permitted which does not comply with these terms. Frontiers in Built Environment | www.frontiersin.org 30 October 2017 | Volume 3 | Article 57 ORIGINAL RESEARCH published: 06 February 2018 doi: 10.3389/fbuil.2018.00002 A Simple Response Evaluation Method for Base-Isolation Building-Connection Hybrid Structural System under Long-Period and Long-Duration Ground Motion Kohei Hayashi, Kohei Fujita, Masaaki Tsuji and Izuru Takewaki * Department of Architecture and Architectural Engineering, Graduate School of Engineering, Kyoto University, Edited by: Kyotodaigaku-Katsura, Nishikyo, Kyoto, Japan Fabio Mazza, University of Calabria, Italy An innovative hybrid control building system of base-isolation and building-connection Reviewed by: has been proposed in the previous study. This system has two advantages, (i) to resist an Christian Málaga-Chuquitaype, Imperial College London, impulsive earthquake input through the base-isolation system and (ii) to withstand a long- United Kingdom duration earthquake input through the building-connection system. A simple response Ivo Caliò, Università degli Studi di Catania, Italy evaluation method without the need of non-linear time–history response analysis is *Correspondence: proposed here for this hybrid building system under a long-period and long-duration Izuru Takewaki ground motion. An analytical expression is derived in the plastic deformation of an [email protected] elastic–perfectly plastic single-degree-of-freedom (SDOF) model with viscous damping under the multi-impulse, which is the representative of long-period and long-duration Specialty section: This article was submitted to ground motions. A transformation procedure of a base-isolation building-connection Earthquake Engineering, hybrid structural system into an SDOF model is proposed by introducing two steps, a section of the journal Frontiers in Built Environment one is the reduction of the main base-isolated building to an SDOF system, and the Received: 12 October 2018 other is the reduction of the connecting oil dampers supported on a free-wall to an oil Accepted: 08 January 2018 damper with a newly introduced compensation factor on a rigid wall. Application of the Published: 06 February 2018 analytical expression of the plastic deformation to the reduced SDOF model including Citation: Hayashi K, Fujita K, Tsuji M and the compensation factor on the connecting oil dampers enables the development of a Takewaki I (2018) A Simple Response simplified, but rather accurate response evaluation method. The time–history response Evaluation Method for Base-Isolation analysis of the multi-degree-of-freedom model and the comparison with the proposed Building-Connection Hybrid Structural System under Long-Period and simplified formula make clear the accuracy and reliability of the proposed simplified Long-Duration Ground Motion. response evaluation method. Front. Built Environ. 4:2. doi: 10.3389/fbuil.2018.00002 Keywords: base-isolation, building-connection, hybrid control, passive control, long-period long-duration motion Frontiers in Built Environment | www.frontiersin.org 31 February 2018 | Volume 4 | Article 2 Hayashi et al. Base-Isolation Building-Connection Hybrid System INTRODUCTION withstand impulsive loading effectively. The overcome of these two difficult issues is of great concern in the field of earthquake- Resilience of infrastructures against natural disasters is becom- resistant and control design (Koo et al., 2009; Petti et al., 2010; ing a key theme recently, and many earthquakes in the last Karabork, 2011). few decades raised some issues which should be overcome for In this article, an innovative hybrid passive control building the continuing use of infrastructures in the field of earthquake system is treated in which a base-isolated building model is sup- structural engineering (Bruneau and Reinhorn, 2006; Takewaki ported by (or connected to) another earthquake-resistant, non- et al., 2012). Bruneau and Reinhorn (2006) proposed four fac- isolated building (called free-wall) with oil dampers (Murase et al., tors (robustness, redundancy, resourcefulness, and rapidity) as 2013; Kasagi et al., 2016; Fukumoto and Takewaki, 2017). This the principal elements of resilience. To take into account the innovative system has been developed by Obayashi Corporation earthquake resilience of building structures in the design stage, and Shimizu Corporation in Japan as an apartment house with a it is inevitable to make scenarios for building structures to resist car parking tower and has been actually constructed (Murase et al., devastating earthquakes without severe damage, which disturbs 2013; Kasagi et al., 2016). It has been demonstrated that this hybrid their continuing use (Amadio et al., 2003; Kobori, 2004; Takewaki passive building control system is effective and robust for differ- et al., 2012, 2013; Takewaki, 2013). Since properties of earthquake ent types of earthquake ground motions, i.e., pulse-type ground ground motions are intrinsically uncertain, it seems difficult to motions and long-period, long-duration ground motions. It has predict the future events within an allowable accuracy in time, also been demonstrated using the energy analysis that, although space, and character (Takewaki et al., 2011, 2012, 2013; Takewaki, the connecting oil dampers in the proposed hybrid system do not 2013). Because the structural properties of buildings, especially work effectively for pulse-type ground motions, those function in advanced buildings systems such as base-isolation systems and effectively for long-period and long-duration ground motions. At passive control systems are not certain (Ben-Haim, 2006) and the same time, it has also been clarified that this hybrid control the direct treatment of their variation is inevitable in the reliable system has a high degree of redundancy and robustness for a broad seismic resistant design of building structures, the concepts of class of earthquake ground motions and an effective connecting robustness and redundancy are becoming also very important. In damper location can be investigated using a sensitivity-type opti- fact, it is absolutely required in Japan to consider the uncertainties mization approach (Taniguchi et al., 2016b; Tamura et al., 2017). of structural properties of isolators and dampers in the design of However, only the time–history response analysis has been used base-isolated buildings and passively controlled buildings. In such for response evaluation and, if the non-linear response in the design procedure, the worst combination of structural properties base-isolation story is taken into account, this response evaluation of isolators and dampers plays a key role for reliable design (Ben- method requires heavy computational load. Haim, 2006; Elishakoff and Ohsaki, 2010; Takewaki et al., 2012; Fujita et al. (2017) developed a new method of robustness Fujita et al., 2017; Kanno et al., 2017). evaluation for an elastoplastic base-isolated high-rise build- It appears that, if it is aimed at designing building struc- ing considering simultaneous uncertainties of structural param- tures with high resilience, base-isolation or structural control is eters. It has been shown that, by using the derived upper inevitable. It is well recognized that, while base-isolated buildings bound of the critical response to a double impulse, the robust- are effective for pulse-type ground motions with predominant ness function (Ben-Haim, 2006), a measure of the robustness, periods shorter than about 2 s or random earthquake ground of elastoplastic structures can be evaluated efficiently. How- motions without clear predominant period (Jangid and Datta, ever, it is difficult to derive a simple response evaluation 1994; Hall et al., 1995; Heaton et al., 1995; Jangid, 1995; Jangid method for a base-isolation building-connection hybrid structural and Banerji, 1998; Kelly, 1999; Naeim and Kelly, 1999; Jangid system. and Kelly, 2001; Morales, 2003; Takewaki, 2005, 2008; Li and In this article, a simple response evaluation method using a Wu, 2006; Hino et al., 2008; Takewaki and Fujita, 2009), their single-degree-of-freedom (SDOF) model is proposed for a base- earthquake resilience is not clear for long-period ground motions isolation building-connection hybrid structural system under with the characteristic period of 5–8 s (Irikura et al., 2004; Kamae a long-period and long-duration ground motion. An analyti- et al., 2004; Ariga et al., 2006). The long-period ground motions cal expression is derived in the plastic deformation of an elas- with the characteristic period of 5–8 s have been argued in the tic–perfectly plastic SDOF model with viscous damping under structural design of base-isolated and super high-rise build- the multi-impulse, which is the representative of long-period and ings since the Tokachi-oki earthquake in 2003 and have been long-duration ground motions. A transformation procedure of a treated as one of the most critical inputs for such buildings after base-isolation building-connection hybrid structural system into the 2011 Tohoku earthquake. The long-period pulse with the an SDOF model is proposed by introducing two steps, one is the clear period of 3 s and the large amplitude of velocity is under reduction of the main base-isolated building to an SDOF system, critical discussion in Japan after the Kumamoto earthquake in and the other is the reduction of the connecting oil dampers 2016. On the other hand, while building structures using passive supported on a free-wall to an oil damper with a compensation energy dissipating systems are effective for long-duration and factor on a rigid wall. Application of the analytical expression of long-period ground motions (Takewaki, 2007; Patel and Jangid, the plastic deformation to the reduced SDOF model including 2011; Takewaki et al., 2011, 2012; Kasagi et al., 2015), they are the compensation factor on the connecting oil dampers enables not necessarily effective for pulse-type ground motions. This the development of a simplified, but rather accurate response is because passive dampers requiring energy dissipation cannot evaluation method. Frontiers in Built Environment | www.frontiersin.org 32 February 2018 | Volume 4 | Article 2 Hayashi et al. Base-Isolation Building-Connection Hybrid System FIGURE 1 | Multi-impulse input, (A) basic model and (B) realistic model with half amplitude in the first impulse and without gradual drift (Kojima and Takewaki, 2015b). ANALYTICAL EXPRESSION OF MAXIMUM Elastic–Perfectly Plastic SDOF Model with RESPONSE OF ELASTIC–PERFECTLY Viscous Damping PLASTIC SDOF MODEL WITH VISCOUS Consider a viscously damped elastic–perfectly plastic SDOF DAMPING UNDER CRITICAL model of mass m and stiffness k. The yield deformation √ and the yield force are denoted by dy and fy . Let ω1 = k/m, h, MULTI-IMPULSE u, and f denote the undamped natural circular frequency, the damping ratio, the displacement of the mass relative to the ground Transformation of Long-Period and (deformation of the system) and the restoring force of the model, Long-Duration Ground Motion into respectively. Vy (≡ω1 dy ) denotes the input level of velocity of one Multi-Impulse Input impulse at which the undamped SDOF system at rest just attains Kojima and Takewaki (2015b, 2017) showed that a long-period the yield deformation after one impulse of such velocity and is and long-duration ground motion can be well represented used for normalizing the input velocity level. The time derivative by a multi-impulse with an equal time interval as shown in is denoted by an over-dot. Figures 1A,B (see also Application to Recorded Ground Motion). Figure 1A is a basic model with a common velocity amplitude, and Maximum Response of Elastic–Perfectly Figure 1B is a realistic model with the half amplitude in the first impulse. The red arrow indicates the Dirac delta function. V is Plastic SDOF Model with Viscous Damping the given velocity (the input velocity level), and t 0 is the equal to Critical Multi-Impulse time interval between two consecutive impulses. In terms of the An analytical expression of the plastic deformation is derived Dirac delta function δ(t), the multi-impulse in Figure 1A can be in this section for an elastic–perfectly plastic SDOF model with expressed by the following equation: viscous damping to the critical multi-impulse. It should be emphasized that only the critical multi-impulse (resonant to the ¨g (t) = Vδ(t)−Vδ(t−t0 )+Vδ(t−2t0 )−Vδ(t−3t0 )+· · · . (1) u fundamental natural mode) is treated here which maximizes On the other hand, the multi-impulse in Figure 1B can be the plastic deformation for varied impulse timing. The critical- described by the following equation: ity was demonstrated in the references (Kojima and Takewaki, 2015b; Kojima et al., 2017), and its detailed explanation will ¨g (t) = 0.5Vδ(t)−Vδ(t−t0 )+Vδ(t−2t0 )−Vδ(t−3t0 )+· · · . (2) u appear later in this section. Under such critical multi-impulse, Frontiers in Built Environment | www.frontiersin.org 33 February 2018 | Volume 4 | Article 2 Hayashi et al. Base-Isolation Building-Connection Hybrid System FIGURE 2 | Spring and dashpot force–deformation relations of elastic–perfectly plastic single-degree-of-freedom (SDOF) model with viscous damping to critical multi-impulse, (A) restoring force–deformation relation and (B) damping force–deformation relation. the fundamental natural vibration mode governs most of the As in Kojima et al. (2017) for the damped SDOF model under vibration component. This fact supports the validity of the the critical double impulse, the damping force–deformation rela- modeling of a multi-degree-of-freedom (MDOF) model into tion after one impulse is approximated by a quadratic func- an SDOF model. Since a residual deformation could exist in tion with the vertex (u, fD ) = (u1 , 0) and passing the point the elastic–perfectly plastic model and it is sensitive to the (u, fD ) = (u1 − (up + dy ), c(vc + V)) as shown in Figure 2B input motion, the plastic deformation is the focus of this √ article. fD = c(vc + V) (u1 − u)/(up + dy ). (4) Figure 2 shows the spring and dashpot force–deformation rela- tions of the elastic–perfectly plastic SDOF model with viscous The work done by the damping force can be obtained by damping to the critical multi-impulse. (a) Presents the restor- integrating Eq. 4 from u = u1 − (up + dy ) to u = u1 ing force–deformation relation and (b) indicates the damping force–deformation relation. The impulses in Figure 2 are two ∫ u1 consecutive elements of the multi-impulse. fD du = (2/3)c(vc + V)(dy + up ). (5) u1 −(up +dy ) Kojima and Takewaki (2015a) showed that the critical timing of the second impulse corresponds to the zero restoring force The energy balance law between the point of one impulse and in the first unloading stage in the case where an undamped the point attaining the maximum deformation can be expressed elastic–perfectly plastic SDOF system is subjected to the dou- as follows by using Eq. 5 ble impulse. It was also demonstrated by Kojima and Takewaki (2015b) that this fact can be extended to the undamped elas- m(vc + V)2 /2 = kdy 2 /2 + fy up + (2/3)c(vc + V)(dy + up ). (6) tic–perfectly plastic SDOF model subjected to the multi-impulse. Furthermore, this critical timing was confirmed by Kojima et al. The left-hand side indicates the kinetic energy input at the (2017) for the damped elastic–perfectly plastic SDOF model sub- timing of one impulse and the right-hand side presents the sum jected to the double impulse. Therefore, it is assumed here again of the elastic strain energy, the dissipation energy by plastic that this critical timing is valid for the damped elastic–perfectly deformation and the dissipation energy by viscous damping. From plastic SDOF model subjected to the multi-impulse. Under this Eqs 3 and 6, the plastic deformation up can be expressed by the assumption, an analytical expression of the plastic deformation is following equation: derived here. {( )2 ( )}/ It was shown by Kojima et al. (2017) that the maximum vc V 8 vc V elastic–plastic responses of the damped SDOF model under the u p = dy + −1− h + Vy Vy 3 Vy Vy critical double impulse can be derived by an energy approach { ( )} without solving directly the equation of motion. This approach is 8 vc V 2+ h + . (7) applied here to the damped elastic–perfectly plastic SDOF model 3 Vy Vy subjected to the multi-impulse. Let vc denote the velocity at the zero restoring-force timing, and let up denote the steady-state plastic deformation after one MDOF HYBRID MODEL OF impulse. Since the response process in the unloading stage of BUILDING-ISOLATION the damped SDOF model under the critical multi-impulse is (ELASTIC–PERFECTLY PLASTIC essentially the same as that of the damped SDOF model under the critical double impulse, vc derived in Kojima et al. (2017) can be BASE-ISOLATION STORY) AND used and expressed by the following equation: BUILDING-CONNECTION [ √ { √ }] Consider a 40-story base-isolated building connected to a vc = Vy exp (−h/ 1 − h2 ) 0.5π + arctan(h/ 1 − h2 ) . 26-story free-wall for car parking by nc -story oil dampers (allo- (3) cated to 4, 8, 12, 16, 18, 20, 22, 24, and 26th stories) as shown in Frontiers in Built Environment | www.frontiersin.org 34 February 2018 | Volume 4 | Article 2 Hayashi et al. Base-Isolation Building-Connection Hybrid System FIGURE 3 | Base-isolation and building-connection hybrid system with elastic–perfectly plastic restoring force–deformation relation of base-isolation story and its modeling into multi-degree-of-freedom model. TABLE 1 | Parameters of hybrid system. of the base-isolation story has a positive post-yield stiffness, the positive post-yield stiffness is neglected for simplicity. Therefore, Main frame mU (kg) 1.70 × 10 6 Fundamental natural period T up (s) 3.0 the base-isolation story has an elastic–perfectly plastic restoring- Damping ratio (lowest mode) 0.03 force characteristic with viscous damping as shown in Figure 3 Free-wall mF (kg) 2.20 × 105 (Qi : story shear in the base-isolation story, ui : story deformation Fundamental natural period T sub (s) 0.63 in the base-isolation story). kI , fy , and dyI indicate the initial Damping ratio (lowest mode) 0.03 Connecting oil damper c = 5 × 106 Ns/m (per story) allocated elastic stiffness, the yield force and the yield deformation of the to 4, 8, 12, 16, 18, 20, 22, base-isolation story, respectively. Let T up , T sub , and cI denote the 24, and 26 stories fundamental natural period of the main structure with fixed base- Number of stories including isolation story, the fundamental natural period of the free wall, connecting oil dampers nc 9 and the damping coefficient of oil dampers in the base-isolation Base-isolation story mI (kg) 5.10 × 106 kI (N/m) 2.61 × 106 story. It is assumed that cI is given so that the damping ratio of Yield deformation dyI (m) 0.01 oil dampers attains 0.15 for the equivalent stiffness kIeq of the base-isolation story at the base-isolation deformation uI = 0.4 m. The parameters of the base-isolation building-connection hybrid system are shown in Table 1. Figure 3. The base-isolation story consists of natural rubber isola- tors, steel dampers, and oil dampers. This hybrid building system is modeled into the MDOF mass-spring-dashpot model as shown REDUCTION OF MDOF MODEL TO SDOF in Figure 3. For simple presentation, the super-structure has a MODEL common floor mass mU , and the free-wall has a common floor mass mF . The mass of the base-isolation story is denoted by mI , To use the analytical expression of the plastic deformation of and the common damping coefficient of connecting oil dampers the base-isolation story shown in Section “Analytical Expression is denoted by c. Although the total restoring-force characteristic of Maximum Response of Elastic–Perfectly Plastic SDOF Model Frontiers in Built Environment | www.frontiersin.org 35 February 2018 | Volume 4 | Article 2 Hayashi et al. Base-Isolation Building-Connection Hybrid System FIGURE 4 | Modeling of multi-degree-of-freedom (MDOF) base-isolation and building-connection hybrid system into single-degree-of-freedom (SDOF) model. Frontiers in Built Environment | www.frontiersin.org 36 February 2018 | Volume 4 | Article 2 Hayashi et al. Base-Isolation Building-Connection Hybrid System FIGURE 5 | Modeling of 2DOF base-isolation system into single-degree-of-freedom model. with Viscous Damping under Critical Multi-Impulse,” the MDOF most of the vibration component in the present hybrid model. model shown in Figure 3 is reduced to an SDOF model. This This fact supports the validity of the modeling of an MDOF model model reduction consists of two parts. One is the reduction of into an SDOF model. the base-isolated building, and the other is the reduction of the connecting oil dampers. Reduction of Base-Isolated Building to In the base-isolated building, the upper structure is reduced SDOF Model to an SDOF model, and the reduced system of an SDOF super- The reduced 2DOF base-isolated building model is further structure and the base-isolation story is further reduced to the reduced to the SDOF model in this section as shown in Figure 5. final SDOF model by neglecting the base-isolation mass, i.e., The super-structure mass MU of the 2DOF model is the summa- reduction using the series model. tion of the super-structure masses. The super-structure stiffness On the other hand, in the reduction of the connecting oil and damping coefficient kU and cU of the 2DOF model are deter- dampers, it is necessary to take into account the effects of the free- mined by the equivalence of the fundamental natural period and wall height, the connecting oil damper location and the free-wall the lowest-mode damping ratio between the SDOF model with the stiffness on the damping coefficient in the SDOF model. Since the fixed base-isolation story and the MDOF model. number of stories with the connecting oil dampers (the common Let Me , ke , dye , and cmain denote the mass, the initial stiffness, damping coefficient per floor is c) is nc , the total damping coeffi- the yield deformation, and the damping coefficient of the reduced cient of the connecting oil dampers is nc c. When the modification SDOF model of the base-isolated building. factor is denoted by βd (see Figure 4), the compensated damping In the case where the base-isolation story mass is negligible coefficient of the total connecting dampers can be expressed by compared with the super-structure mass MU , Me can be regarded the following equation: as follows: Me = MU . (9) C = βd nc c. (8) In addition, if the base-isolation story mass is negligible com- The factor βd may include some effect of damper location. pared with the super-structure mass MU , ke can be expressed in However it seems to reflect mainly the effect of flexibility of the following form by using a series spring modeling the free-wall. To determine βd , consider the virtual intermedi- ate model, called the RMDOF model as shown in Figure 4, in ke = 1/(1/kI + 1/kU ). (10) which the connecting oil dampers with the damping coefficient Since the yield story shear forces are the same in the SDOF βd c are concentrated to upper consecutive nc floors in the base- model and the 2DOF model with zero base-isolation story mass, isolated building which is supported on a rigid wall by nc -floor oil the equivalent yield deformation of the SDOF model can be dampers. The modification factor βd is determined by equating described by the following equation: the lowest-mode damping ratios, by the complex modal analysis, between the RMDOF model and the MDOF model. In both dye = kI dyI /ke . (11) RMDOF model and MDOF model, the equivalent stiffness kIeq of the base-isolation story is determined by conducting the repetitive Let uI , uU , ue , and fy denote the base-isolation story displace- computation of uI for convergence which will be explained later. ment, the relative super-structure displacement, the displacement It should also be remarked that the structural damping of the of the SDOF model, and the yield force in the base-isolation story. super-structure and the damping in the base-isolation story are Figure 6 shows the relation of the restoring force–deformation neglected only in evaluating the damping ratios of the RMDOF characteristic between the base-isolation story of the 2DOF model model and the MDOF model for determination of βd . and the total SDOF model. It should be remarked that the plastic It should be emphasized again that only the critical multi- deformation dpe of the SDOF model is equal to the plastic defor- impulse (resonant to the fundamental natural mode) is treated mation dpI of the base-isolation story in the 2DOF model due to here. In this case, the fundamental natural vibration mode governs the series modeling. Frontiers in Built Environment | www.frontiersin.org 37 February 2018 | Volume 4 | Article 2 Hayashi et al. Base-Isolation Building-Connection Hybrid System FIGURE 6 | Relation of restoring force–deformation characteristic between base-isolation story of 2DOF model and total single-degree-of-freedom (SDOF) model. FIGURE 7 | Procedure for determining β d without iteration using multi-degree-of-freedom (MDOF) and RMDOF models. FIGURE 9 | Flowchart for repetitive evaluation of uI . complex spring modeling 1 1 1 = + , (12) ke + iωe cmain kI + iωe cI kU + iωe cU where i is the imaginary unit and the natural circular frequency ωe of the SDOF model is defined by the following equation: FIGURE 8 | Steady-state restoring force–deformation relation in √ base-isolation story. ωe = Me /ke . (13) From Eqs 10, 12, and 13, cmain can be expressed by the following equation: Since the super-structure and the base-isolation story have (kI cU + kU cI )(kI + kU ) − (kI kU − ωe 2 cI cU )(cI + cU ) cmain = . different damping coefficients, the equivalent damping coefficient (kI + kU )2 + ωe 2 (cI + cU )2 cmain of the SDOF model can be obtained by using the series (14) Frontiers in Built Environment | www.frontiersin.org 38 February 2018 | Volume 4 | Article 2 Hayashi et al. Base-Isolation Building-Connection Hybrid System FIGURE 10 | Comparison of plastic deformation in base-isolation story between multi-degree-of-freedom (MDOF) model (time–history response analysis) and single-degree-of-freedom (SDOF) model (simple evaluation method) for four levels of connecting dampers (free-wall: 26 stories), (A) c = 1 × 106 Ns/m, (B) c = 3 × 106 Ns/m, (C) c = 5 × 106 Ns/m, and (D) c = 7 × 106 Ns/m. Reduction of Connecting Damper place of βd nc c. Then, if we assume the linearity of the lowest-mode As stated earlier, in the reduction of the connecting oil dampers, it damping ratio hR for the RMDOF model with respect to the total is necessary to take into account the effects of the free-wall height, damping coefficient, we can obtain directly βd as βd = hM /h∗R the connecting oil damper location and the free-wall stiffness on from Figure 7. the damping coefficient in the SDOF model. It was confirmed that In both the RMDOF model and the MDOF model, the equiv- these effects can be taken into account properly by introducing alent stiffness kIeq of the base-isolation story at the base-isolation the RMDOF model in which the connecting oil dampers with deformation uI is adopted as the base-isolation story stiffness for the compensated damping coefficient βd c are concentrated to the the complex eigenvalue analysis as shown in Figure 8 upper consecutive nc stories in the base-isolated building and the base-isolated building is supported on a rigid wall by nc -story oil kIeq = fy /uI . (16) dampers. The modification factor βd is determined by equating the lowest-mode damping ratios, by the complex modal analysis, In the evaluation of uI , a repetitive procedure is required as between the RMDOF model (hR ) and the MDOF model (hM ) shown in Figure 9. hR = hM . (15) NUMERICAL INVESTIGATION ON As stated earlier, since only the critical multi-impulse (resonant ACCURACY OF PROPOSED SIMPLE to the fundamental natural mode) is treated here, the fundamental RESPONSE EVALUATION METHOD USING natural vibration mode governs most of the vibration component SDOF MODEL in the present hybrid model. For this reason, the equivalence of the lowest-mode damping ratio seems to provide a good corre- To investigate the accuracy of the proposed simple response eval- spondence of both models (RMDOF and MDOF). It should be uation method using the simplified SDOF model, three models remarked that the determination of the modification factor βd via with different numbers of stories of free-wall (26, 13, and 40) are Eq. 15 is difficult because some iterations are required. To avoid considered. The fundamental natural period of the 13-story free- this iteration, we employ another procedure as shown in Figure 7. wall is 0.32 s and that of the 40-story free wall is 0.95 s in addition First of all, we compute the lowest-mode damping ratio hM for to 0.63 s of the 26-story free wall (Table 1). The base-isolated the MDOF model and also h∗R for the RMDOF model with nc c in building is a 40-story model and four levels of the connecting Frontiers in Built Environment | www.frontiersin.org 39 February 2018 | Volume 4 | Article 2