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MINISTRY OF EDUCATION AND TRAINING DANANG UNIVERSITY NGUYEN VAN NAM MODELING OF STRUCTURES SEISMICALLY ISOLATED BY FRICTION BEARINGS MAJOR CODE : MECHANICAL ENGINEERING : 62.52.01.01 SUMMARY OF DOCTOR OF ENGINEERING DISSERTATION Danang - 2017 The work was finished at University of Science and Technology - Danang University Science Advisor: Assoc Prof Dr HOANG PHUONG HOA Assoc Prof Dr PHAM DUY HOA Reviewer 1: Prof Dr PHAN QUANG MINH Reviewer 2: Assoc Prof Dr LUONG VAN HAI Reviewer 3: Dr NGUYEN THE DUONG This Dissertation was defended at The Doctor of Engineering committee at Danang University on 17 of 4, 2017 For the detail of the dissertation, please contact: - Information and Library Center of Danang University - National Library of Vietnam INTRODUCTION Motivation of Study Earthquake is one of the serious disasters caused by natural for human life, constructions and economy in general Throughout history, the world has seen occuring of many strong earthquakes, killed a lot of human lifes, destroyed a lot of buildings and lost millions of dollars in economic each year due to the earthquakes Although Vietnam is not in the dangerous zone of the epicenter of strong earthquakes in the world, it is still a country located in areas with high seismic hazard It is reported by scientists at the International Workshop "Dangerous earthquake, tsunami and the early warning system for Asia - Pacific" hold by the Institute of Geophysics - Institute of Science and Technology Vietnam within two days (5 and 6-9.2011) Some large urban areas are currently located on the fault zone and are possible to get very strong earthquakes such as Hanoi located on the Hong river fault zone, Chay river, Ma river, Son La river is forecasted to suffer level concussions according to the Richter scale Recently, the vibration caused by earthquakes appeared in many provinces and cities, especially Hanoi, Ho Chi Minh City and Danang where have of a large number of buildings, large bridges and demands of construction of large projects are growing in number as well as in height Those are very sensitive to ground acceleration of earthquakes With these facts as above, the constructions should have seismic resistant design, particularly modern seismic resistant design standpoint associated with the term "structural vibration control" and is still new in Vietnam Therefore, the research about them is essential, has scientific meaning and high practice This is also the motivation for the author to research the title: “Modeling of structures seismically isolated by friction bearings” to provide a solution to reduce the harm caused by earthquakes for constructions Purpose of Study Research the numerical modeling for base isolated structures with the friction pendulum bearings including SFP, DFP and TFP bearings subjected to earthquake load Evaluate the seismic reduction effectiveness of the base isolation bearings for constructions From there, application research of TFP bearings for high-rise buildings built in Hanoi, Vietnam Object and Scope of Study The objects of the study in dissertation are the friction pendulum bearings including SFP bearing, DFP bearing and TFP bearing The scope of the study including: Research response of structures with single bearing (don’t consider the work together of many bearings in a construction), ignoring torsion; The superstructure behavior is linear and the bearing behavior is nonlinear Dissertation content - Identify the structure, working principle of the seismic isolation bearings: SFP, DFP and TFP - Numerical modeling for base isolated structures with the friction pendulum bearings above subjected to earthquake Evaluate the seismic reduction effectiveness of these bearings for constructions - Research to develop a new model for TFP bearing - Application research of TFP bearing for high-rise buildings in Vietnam according to ASCE 7-2010 Standard design Methodology of study Investigate a theoretical model, the research results were simulated by Matlab and compared to an experimental model results of other authors have been published by NEES (Network for Earthquake Engineering Simulation) to verify Contributions of the dissertation - Numerical modeling for base isolated structures with the friction pendulum bearings: SFP, DFP and TFP Evaluate the seismic reduction effectiveness of these bearings for constructions - Improve the TFP bearings model Via improved model, details of each slider displacement on the spherical surfaces and effects of vertical component of ground acceleration are also clearly calculated for TFP bearings - Propose the optimal parameters of TFP bearings for high-rise buildings in Hanoi and evaluate the seismic reduction effectiveness of them Dissertation layout The Dissertation consists of the introduction, four chapters and conclusion, the recommendations, all content of the dissertation is contained in 133 A4 pages and is organized as follows: Introduction Chapter Overview Chapter Modeling of friction bearings Chapter Improved model of triple friction pendulum bearing Chapter Seismic reduction effectiveness of TFP bearings for high-rise buildings in Ha Noi Conclusions and recommendations Chapter OVERVIEW 1.1 Overview of earthquakes and earthquake resistant design 1.1.1 Earthquakes An earthquake is a very strong shaking phenomenon of the surface of the earth when large energy are released in a very short time due to the sudden cracked in the crust or in the upper mantle of the earth [10], [ 62] Earthquakes have the following original: Earthquake originated from plate tectonics, from the faults; Other originals: from the expansion in the hard rock crust of the earth, by explosions, volcanic activity,… The important parameters of ground motion in seismic resistant design including: the largest amplitude, duration of strong motion, frequency content, magnitude earthquake, the distance to the fault, soil conditions at site 1.1.2 Approachs of earthquake resistant design Earthquake resistant design of structures is a required duty, a major challenge for structural engineers There are two earthquake resistant design perspective: traditional earthquake resistant design and modern earthquake resistant design Earthquake resistant design with modern standpoint associated with the structural vibration control engineering with main groups as follows: passive control, active control and semi-active control 1.2 Seismic base isolation technology 1.2.1 Concept of seismic base isolation Seismic base isolation is a passive control technique for structure and very effective in earthquake resistant design The main idea of this technique is the isolation between superstructure and basement by using the soft bearings, called seismic isolation bearings 1.2.2 Types of isolation devices The types of bearing used in seismic base isolation technique usually involves two common types: rubber bearings (elastic bearing, Figure 1.6) and friction bearings made from stainless steel Friction bearings main categories: - Single friction pendulum bearings: structure shown in Fig 1.7, including spherical surface with radius R, slider slides on spherical surface with friction coefficient and ability of displacement is d R, a Cutaway view d b Cross section Figure 1.7 Single friction pendulum bearings (EPS, 2011) - Double friction pendulum bearings: structure shown in Fig 1.8, including spherical surfaces and and a slider inside a Cutaway view b Cross section Figure 1.8 Double friction pendulum bearings (Fenz, 2008e) Triple friction pendulum bearings: Structure shown Fig 1.9, including spherical surfaces with the radius R1, R2, R3 and R4 Inside, sliders slide on spherical surfaces with friction coefficient i a Cutaway view b Cross section Figure 1.9 Triple friction pendulum bearings (Fenz, 2008e) 1.2.3 Brief history of seismic base isolation technology application Seismic base isolation technology has been studied and applied in recent decades However, the idea appeared more than 100 years ago by the invention of the Touaillon In recent years, the application of this techniques for structure subjected to earthquakes becomes popular in the US, Japan, New Zealand, and some countries in Europe 1.3 Overview of research for isolated friction bearings 1.3.1 The studies outside the country - The studies of SFP bearings: the first study is published in 1987 by Zayas The outstanding studies are published by other authors: Mokha, Constantinou, Reinhorn, Nagarajaiah, Mosqueda, The studies focused on analysis structure and movement of the SFP bearing Effectiveness of reduce seismic of bearings was evaluated through theoretical and experimental models - The studies of DFP bearings: Tsai and Constantinou research groups are considered systematically In addition, many individual studies on this bearing also should concern such as Kim and Yun (2007), Malekzadeh (2010), - The studies of TFP bearings: TFP bearing with it’s advantages began to be produced around the year 2007 The outstanding studies can mention such as Constantinous and Fenz (University of Buffalo); Steve Mahin, Troy Morgan and Tracy Becker (UC Berkeley); Ryan’s research group (University of Nevada, Reno), The latest publication of TFP bearing almost belongs to this group, the publication can be listed as: Dao [36], [37, [38], Okazaki [80], Ryan [86], [87], [88] In addition, some studies of other authors on the TFP bearing also published as: Fadi [41], Ghodrati [52], Moeindarbari [67], Sarkisian [89], Tsai [103], [104] 1.3.2 The studies inside the country In Vietnam, base isolation was mentioned in Vietnam design standards 375: 2006 in 2006 The study is very limited, the outstanding studies such as: Nguyen Van Giang and Chu Quoc Thang (2006), Tran Tuan Long (2007), Le Xuan Huynh et al (2008), Do Kien Quoc (2009), Le Xuan Tung (2010, 2012) 1.4 Comments, the need of new findings - Earthquake resistant design is a necessary requirement Using seismic isolation bearings in the structural vibration control technology subjected to earthquakes is new perspective bringing high efficiency, should be studied and more widely applied - The studies of friction sliding bearing need to be implemented in this dissertation as follows: indicate the mumerical model and evaluate the sseismic reduction effectivenes of SFP, DFP and TFP bearing; research to develop a improved model of TFP bearing from the simple model of previous studies This model must have the reliability and improve than the existing models; A application research on TFP bearing for high-rise buildings built in the ground conditions in Hanoi need been implemented Chapter MODELING OF FRICTION BEARINGS 2.1 Basis of the theory 2.1.1 Basis of earthquake resistant design for buildings - Numerical model: with these assumptions in structural dynamics, numerical model of an n-storey building subjected to earthquakes will be presented as shown in Figure 2.1 Nth storey kn mn cn 2nd storey k2 1st storey k1 a ug b m2 c2 u1 m1 c1 ug u2 k1 k2 c1 m1 c2 c un kn m2 cn mn ug a N-storey frame; b Ideal numerical model; c Equivalent model Fig 2.1 Multiple DOFs structure model subject to earthquakes - Equations of motion: the differential equations of motion of the structural model are established by displacement method (stiffness matrix method) as Equation 2.1 - Determination method of structural response: this study will use the direct integration method (time-history analysis) This method is the most accurate results, reflect nature of the dynamic problem, is suitable for research problems 2.1.2 The choice of numerical method in research System of differential equations of motion of seismic isolated structures subjected to earthquakes in this research is complex We 15 From the model shown in Figure 3.2, the system of differential equations of motion of structure isolated by TFP bearing subjected to ground acceleration in each direction established on the basis of the d'Alembert principle (Equation 3.1 and 3.2) Friction coefficient of the movement in both directions is calculated by Equation 3.3, where the hysteresis variable Z is determined according to Equation 3.5 The vertical component of ground acceleration will be calculated according to Equation 3.7 and included in the model through the total weight of superstructure 3.2.2 Model verification Theoretical model simulated by Matlab is verified by experimental results performed by Ryan et al in 2013 [87] The experimental full-scale model is a 5-storey building having a total weight of over 5000 kN (Figure 3.3 to 3.6), is isolated by TFP with 1.4 m in size (Figure 3.7) subjected to 19 different ground acceleration data Extracted from some of verify results such as the following figure a Results of displacement bearing in the x and y directions 16 b Results of hysteresis loop in the x and y directions Figure 3.13 Comparing the analytical result to experimental result subjected to 90TAB motion a Results of displacement bearing in the x and y directions b Results of hysteresis loop in the x and y directions Figure 3.26 Comparing the analytical result to experimental result subjected to 115TAK motion 17 3.3 Detailed calculation slider displacements Slider displacements on surfaces are calculated as follows: Reff u1 ( F Ff Fr1 ) W R u2 ( F Ff Fr W u1 ) eff Reff W (3.9) Reff W u3 ( F Ff Fr u4 ) Reff W Reff u4 ( F Ff Fr ) W Where the force components are determined from solving the system of differential equations of motion 3.4 Effects of the vertical component of ground acceleration The model considering of the effects of vertical component of ground acceleration is verified with the model of Dao [36] Analysis of 5-storey building model with different ground acceleration data in cases with/without vertical component The results showed that displacement bearing has not changed much but absolute acceleration and shear force in the floor were increased significantly Therefore, we can not ignore this component as previous studies Figure 3.46 and 3.47 illustrate a case with 88RRS motion a Absolute acc b Shear Figure 3.46 Structural behavior with 88RRS motion 18 Hình 3.47 Hysteresis loop of bearing with 88RRS motion 3.5 Conclusion of Chapter Developing a new model for seismic isolated structure with TFP bearing from the one-dimensional (1D) equivalent model of Fenz et al (2008) Via this model, the displacement of the slider on surface is calculated in detail and affect of vertical component of ground acceleration is evaluated clearly Chapter SEISMIC REDUCTION EFFECTIVENESS OF TFP BEARING FOR HIGH-RISE BUILDINGS IN HA NOI 4.1 Introduction As predicted, Hanoi may occur earthquake level Earthquake resistant design of buildings for this area is required in the recent constructions However, seismic resistant design method is still traditional In the study of this chapter, seismic resistant design method with seismic base isolation technology by TFP bearing for high-rise buildings was implemented 4.2 Analysis of the effectiveness of TFP bearings for high-rise buildings in Hanoi 4.2.1 Structure parameter The 9-storey reinforced concrete building, assuming the absoluted hard slab, mass and stiffness of the floor the same 19 assumptions, including: mass mi = 100 N.s2 /mm, stiffness ki = 150 kN/mm, damping ratio fundamental period of structure T1 = s (suitable for various projects) 4.2.2 Selection of analyzed ground acceleration parameters With performance - based seismic design and time history analysis, ground acceleration are selected according to the provisions of ASCE 7-2010 including acceleration data and each motion is recorded both directions Structure is analyzed and evaluated with different levels of earthquakes including SLE (Service Level Earthquake) level, DBE (Design Base Earthquake) level and MCE (Maximum Considered Earthquake) level The magnitude of data for each level is adjusted by SF coefficient as Eq 4.3 Results acceleration data is selected as the Table 4.1 and SF coefficient is calculated as Table 4.2 Fig 4.2 and 4.3 illustrate the average of SRSS spectra acceleration and target spectra magnitude after adjustment T2 SF f ( SF ) SSRSS 1.3Sa dT T1 (4.3) T2 S SRSS dT T1 Figure 4.2 Target spectra MCE compare to the average of SRSS in isolated base structure 20 Figure 4.3 Target spectra MCE compare to the average of the SRSS in fixed base structure 4.2.3 The choice of the optimal parameters for TFP bearing with ground conditions in Hanoi As structure, there will be parameters of bearing need be chosen icluding: 2 3 ; Reff Reff ; d2 d3 ; 1; 4; Reff1 = Reff4; d1 = d4 ensuring sliding must occur on stages The criteria to select the optimal parameters are given: Adapt many earthquake levels; The structural response is the smallest Process of choosing the optimal parameters for TFP bearing is performed as schematic in Figure 4:24 The results of the optimal parameters of bearing following: R2 = R3 =500 mm; R1 = R4 = 4000 mm; 2 = 3 = 0.01 - 0.02; 1 = 0.02 - 0.06; 4 = 0.04 – 0.08; d2 = d3 = 40 mm; d1 = d4 = 170 mm 4.2.4 Seismic reduction effectiveness of the bearing for buildings We analyze the structure of the two cases: fixed base and isolated base with TFP bearing Seismic reduction effectiveness is evaluated through absolute acceleration and relative displacement of the floors With MCE level, the results shown in Figure 4.20 and 4.21 SLE and DBE levels for the same results Displacement of bearing, base shear in cases and seismic reduction effectiveness of 21 TFP bearing is calculated in detail in Table 4.4 Figure 4.20 Absolute acceleration in the floors, MCE level Figure 4.21 Relative displacement in the floors, MCE level Table 4.4 Displacement bearing and effectiveness of reducing base shear in isolated base 9-storey structures with TFP bearings Level Disp bearing, ub (mm) Base shear, Fb (kN) Fixed base Isolated base Fixed base Isolated base Effectiveness (%) SLE 65.8 2415 697 71 DBE 176 5183 1103 79 MCE 311 7791 1451 81 From the results of structural response, the horizontal force conditions in the bearing (in 17.2.4.4, ASCE 7-2010) and the initial assumption parameters are tested satisfactorily 22 - Size of structure - Construction site Design spectrum Assume TD and TM Structural parameters calculation Choice of parameters for bearing: - According to experience: Reff2 = Reff3 ; d2 = d3; 2 = 3 Choose and adjust ground accl - Preliminary: d1 = d4 - Preliminary: Varied Reff1 = Reff4 - Preliminary: Varied 1 < 4 Fixed Reff1 = Reff4, analyze with varied 1 < 4 - Determine the reasonable 1 < 4 - Analyze varied Reff1 = Reff4 with 1 < 4 above - Determine the reasonable Reff1 = Reff4 - Determine the parameters fully - - Calculate and check TD and TM Analyze structure with parameters above Check horizontal force conditions + + Optimal parameters Fig 4.24 Determining schematic the parameters for TFP bearing 4.3 Conclusion of Chapter TFP bearing device is very effective in performance-based seismic design Application TFP bearing for high-rise buildings subjected to earthquake in Hanoi according to ASCE 7-2010 bring high efficiency (about 70% - 80%) A determining method of the optimum parameters for TFP bearing is presented Accordingly, the optimal parameters used for high-rise building in Hanoi were found 23 CONCLUSIONS AND RECOMMENDATIONS Conclusions Within the scope of the dissertation, conclusions are drawn as follows: Based on the the structure and movement principles of the friction bearings including: SFP, DFP and TFP bearings from previous studies, dissertation established the model of isolated base structures with the bearings subjected to earthquakes Contents including: indicate numerical model, establish differential equations of motion and suggest methods to solve them to find out the response of structures Dissertation performed a simulation example of the structure subjected to different ground acceleration data to evaluate the effectiveness of the devices The results show that the bearings are very effective in seismic reduction From the results of simulation examples, the advantages of TFP bearing is considered the best in comparison to SFP and DFP bearing With the advantages of TFP bearings evaluated, the dissertation established an improved model for TFP bearings The reliability of the model was verified with experimental results From improved model, the calculation of each slider displacement inside the bearing is performed The results are evaluted detailly of each location of slider at each time of the movement The significance of this work is to determine precisely the surphus displacement of each slider at the end of the earthquake Effects of vertical component of ground acceleration on response of the structures isolated by TFP bearings is carefully analyzed in the dissertation This result shows that, with some earthquakes having large peak value or large vertical acceleration 24 will cause the response of structures increased significantly, can not be ignored in the design, which is almost neglected in previous studies, especially the absolute acceleration of the floor Application research of TFP bearings for high-rise buildings with seismic conditions in Hanoi is performed in the dissertation Content of this study is to find an optimal parameters for TFP bearing with seismic conditions in Hanoi, evaluate the seismic reduction effectiveness of the TFP bearing for buildings Recommendations In terms of research and the scope of the dissertation, a number of issues have not yet been mentioned and clarified, need to have the following research, specifically: It’s necessary to study a reasonable impact model to simulate structural behavior when the slider reaches the displacement limit The impact occurs between the slider and the hard boundary will effect on the structural response When the slider slides on the spherical surface, the temperature at the contact surface between the slider and the surface will increase, take effect on the friction coefficient of the bearing This problem also need to be studied and clarified Seismic base isolation system usually has a large horizontal displacement It is nessesary to study structural control combining other devices with those bearings to limit horizontal displacement of large superstructure It’s nessesary to be further study of the problem of multiobjective optimization to find the optimal parameters for TFP bearings with all buildings and different ground conditions in Vietnam LIST OF PUBLISHED WORKS Published scientific papers 1.1 Domestic Publishing Nguyen Van Nam, Hoang Phuong Hoa, Pham Duy Hoa (2013), Seismic reduction effectiveness of TFP friction isolation bearings, compare to SFP bearings, Proceedings of the 9th Vietnam National Congress on Mechanics, Hanoi, December 08-09, 2012, ISBN: 978-604-911-435-9, pp 397- 405 Pham Dinh Trung, Nguyen Van Nam, Nguyen Trong Phuoc (2014), Analyzing the seismic reduction effectiveness of friction bearings combines with Magneto-Rheological damper connecting two structures subjected to earthquakes”, Journal of Science Ho Chi Minh City Open University, Vol (34) 2014, ISSN 1859-3453, pp 102-115 Tran Quoc Khanh, Hoang Phuong Hoa, Nguyen Van Nam (2014), Seismic reduction effectiveness of friction pendulum bearing for continuous beam bridge subjected to earthquake load, Proceedings of the Vietnam National Congress on Mechanical engineering for the 35th Anniversary of the Foundation of the Institute of Mechanics, Hanoi, April 09, 2014, ISBN: 978-604-913-233-9, pp 81-86 Nguyen Van Nam, Hoang Phuong Hoa, Pham Duy Hoa (2014), Seismic reduction effectiveness of TFP friction bearings for high-rise buildings, Proceedings of the Vietnam National Congress on Mechanical engineering for the 35th Anniversary of the Foundation of the Institute of Mechanics, Hanoi, April 09, 2014, ISBN: 978-604-913-233-9, pp 155-150 Nguyen Van Nam, Hoang Phuong Hoa, Pham Duy Hoa (2015), Modeling of friction bearings in structures subjected to earthquakes: SFP bearing, Proceedings of the Vietnam National Congress on Mechanical engineering, Danang University, ISBN 978-604-84-1273-9, pp 479 - 486 Nguyen Van Nam, Hoang Phuong Hoa, Nguyen Hoang Vinh (2015), Modeling of friction bearings in structures subjected to earthquakes: DFP and TFP bearings, Proceedings of the Vietnam National Congress on Mechanical engineering, Danang University, ISBN 978-604-84-1273-9, pp 487 - 494 Hoang Phuong Hoa, Nguyen Van Nam, Pham Duy Hoa (2016), Study the effects of dimension parameter on behavior of double friction bearings subjected to earthquakes, Journal of Construction (Vietnam Ministry of Construction), ISSN 08660762, Vol 01(2016), pp 87-90 Nguyen Van Nam, Hoang Phuong Hoa, Pham Duy Hoa (2016), Effects of vertical components of strong earthquakes on the response of base isolated structures with TFP bearings, Journal of Science and Technology - Danang University, ISSN 18591531, Vol 1(98), pp 46-49 Hoang Phuong Hoa, Nguyen Van Nam, Pham Duy Hoa (2016), Designing the optimal dimensions single friction bearings for high-rise buildings subjected to earthquake, Journal of Construction (Vietnam Ministry of Construction), ISSN 08660762, Vol 3(2016), pp 106-109 10 Nguyen Van Nam, Hoang Phuong Hoa, Pham Duy Hoa (2016), The effectiveness of SFP seismic isolation bearings for high-rise buildings subjected to earthquakes considering of vertical excitation component, Journal of Construction (Vietnam Ministry of Construction), ISSN 0866-0762, Vol 3(2016), pp 34-36 11 Nguyen Hoang Vinh, Nguyen Van Nam, Hoang Phuong Hoa (2016), General modeling of seismic isolated structures with SFP bearings, Journal of Construction (Vietnam Ministry of Construction), ISSN 0866-0762, Vol 3(2016), pp 102-105 12 Hoang Phuong Hoa, Nguyen Van Nam, Ngo Thanh Nhan (2016), Effect of SFP bearing parameters on response of seismic isolated structures, Journal of Transportation (Vietnam Ministry of Transport), ISSN 2354-0818, Vol 6(2016), pp 52-54 13 Thai Van Ngai, Nguyen Ba Ngo, Hoang Phuong Hoa, Phan Hoang Nam, Nguyen Van Nam (2016), Research the methods of seismic reduction for cable-stayed bridge at Nga Ba Hue intersection - Danang City subjected to earthquakes, Proceedings of National Conference on Transport infrastructure with sustainable development, Danang, September 17-18, 2016, ISBN 978-604-82-1809-6, pp 603-608 14 Dang Xuan Binh, Nguyen Van Duan, Hoang Phuong Hoa, Phan Hoang Nam, Nguyen Van Nam (2016), Research the methods of seismic reduction for roundabout structure at Nga Ba Hue intersection - Danang City subjected to earthquakes, Proceedings of National Conference on Transport infrastructure with sustainable development, Danang, September 17-18, 2016, ISBN 978-604-82-1809-6, pp 597-602 1.2 International Publishing Nam V.Nguyen, Hoa P.Hoang and Hoa D.Pham (2016), Predicting the responses of triple friction pendulum bearings using an improved model with variant friction coefficient, Proceedings of The Fourteenth East Asia-Pacific Conference on Structural Engineering and Construction (EASEC14), HochiMinh City, 6-8 Jan, 2016, ISBN: 978-604-82-1684-9, pp 1578-1585 V Nam Nguyen, P Hoa Hoang, H Nam Phan and Fabrizio Paolacci (2016), A Modeling Approach of Base Isolated HighRise Building with Double Friction Pendulum Bearings, International Conference on Advanced Technology Sustainable Development ICATSD2016 Ho Chi Minh City, 22-23, August 2016, ISBN 978-604-920-040-3, pp 235-240 Science research work Ministry level research: Application Research of seismic isolation solutions for constructions subjected to earthquakes Code: B2016.DNA.03 (In progress, expected to report in 2017, Responsibility: members) ... title: Modeling of structures seismically isolated by friction bearings to provide a solution to reduce the harm caused by earthquakes for constructions Purpose of Study Research the numerical modeling. .. 2.11 Hysteresis loop of TFP bearings ( : stage I to IV) 2.4.2 Modeling of seismic isolated structure with TFP bearings The model of seismic isolated structures with TFP bearings subjected to... =d 1+d Figure 2.8 Hysteresis loop of DFP bearings ( : stage I, II) 2.3.2 Modeling of seismic isolated structure with DFP bearings The model of seismic isolated structures subjected to earthquake
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