Energy-based Design Method for Passive Energy Dissipative Bracing Systems

7 In this study an energy-based method for the design of passive Energy Dissipative Bracing (EDB) 8 systems is presented, as a retrofit technique for existing reinforced concrete (RC) buildings. A 9 comprehensive literature overview concerning the design of hysteretic bracing systems based on 10 various design philosophies, such as force-, displacement- or energy-based, is provided. The efficiency 11 of the proposed method is validated by comparing the proposed methodology with two design 12 procedures selected in the literature, applied to three RC frames. The results showed that the proposed 13 method is more effective in avoiding the damage concentration at a single story and in distributing the 14 additional strength provided by the EDBs proportionally to the hysteretic energy demand along the 15 structure height. The validity of each procedure is compared based on non-linear static and non-linear 16 dynamic analyses.


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Introduction 18 Seismic design methods are either force-, displacement-or energy-based. It is widely accepted that 19 force-based approach is not a suitable tool for implementing performance-based earthquake 20 engineering concept (Bertero and Bertero 2002). Performance levels are more effectively described in 21 terms of displacements, as damage is better correlated to displacements rather than forces. Therefore, 22 efforts have been made to develop alternative methods based on displacement-and energy-based 23 concepts rather than on forces. In this respect, significant research has been carried out for developing 24 design methods of hysteretic bracing systems, based on displacement-and energy-based concepts, to 25 be used in the seismic retrofitting of existing reinforced concrete (RC) buildings. 26 The main limitations encountered in displacement-based approaches for designing energy-dissipative 27 bracing (EDB) systems are: a) conversion of a MDOF system into a SDOF system and choice of the 28 SDOF displacement as the main design parameter; b) distribution of strength and stiffness along the 29 storeys, accounting for the existing structure characteristics; c) neglecting the cyclic behaviour of 30 structural elements in design; d) not considering the effects of duration-related cumulative damage; e) 31 not accounting for near-fault effects. 32 These limitations are overcome in energy-based approaches, thus making them more attractive. Since 33 Housner (Housner 1956) proposed the energy balance concept in 1956, significant amount of research 34 has been dedicated to developing design methods based on such concept. For example, Decanini and 35 Mollaioli (Decanini and Mollaioli 2001) proposed two inelastic energy spectra, namely the input 36 energy ( ) and the hysteretic-to-input energy ratio ( / ). These spectra allow to evaluate the 37 seismic demand in terms of maximum displacement and ductility. They further studied the influence 38 of the inelastic behaviour on the input energy spectra and on the excitation characteristic, which is 39 influenced by soil type, source-to-site distance, and seismic event magnitude. Riddel and Garcia 40 (Riddell and Garcia 2001) presented a hysteretic-energy demand spectrum. Chou et al. (Chou and Uang 41 2000) proposed an attenuation relationship for establishing absorbed (strain) energy spectra, based on 42 two-stage nonlinear regression analysis on 273 ground motion records. Akiyama (Akiyama 1985), 43 using Japanese design earthquakes, introduced the input energy per unit mass of an elastic SDOF 44 structure due to a given earthquake and provided a relationship between normalized input and damage 45 (hysteretic) energy. Akbas et al. (Shen and Akbas 1999) proposed a linear relationship for the hysteretic  46 energy distribution along the building height, based on several non-linear time history analysis of 47 regular frames with a 2% damping ratio. Ye et al. (Ye et al. 2012) established a relationship between 48 the peak story responses and plastic deformation energy obtained from a pushover analysis. Uang and 49 Bertero (Uang and Bertero 1990) obtained the distribution of hysteretic energy in multi-story buildings. 50 Several energy-based procedures for the design of energy-dissipative bracing systems have been 51 proposed in the literature: Choi and Kim (Choi and Kim 2006) proposed a design method using 52 hysteretic and accumulated energy spectra and obtained the bracings cross-section by equating the 53 hysteretic energy demand to the accumulated plastic energy dissipated by the braces. Benavent-54 Climent (Benavent-Climent 2011) proposed a design procedure of EDB systems grounded on the 55 concept of energy-balance, imposing that the whole hysteretic energy demand is dissipated entirely by 56 the EDB system while the existing structure remains elastic. Dasgupta et al. (Dasgupta et al. 2004) 57 obtained the base shear of the EDB system using energy balance and compared it with the base shear 58 obtained from displacement-based design. 59 Although the efficiency of energy-based procedures is praised in the literature, there are several factors 60 that discourage the practicing engineers to use them: a) the definition of actions in terms of energy and 61 not in terms of the more familiar acceleration response spectrum; b) evaluating the energy demand 62 along the building height; c) inclusion of several parameters in energy-based procedures to account for 63 various earthquake effects, such as number of plastic cycles, earthquake duration, cyclic behaviour of 64 elements, etc. However, the recent vast literature provides enough tools to simplify the application of 65 energy-based procedures: as an example, several proposals are now available for defining both energy 66 spectra and the energy distribution along the building height (Decanini and Mollaioli 2001)(Shen and 67 Akbas 1999). Moreover, regression analyses with several earthquakes have been carried out to study 68 the duration and near-fault and far-field effects of earthquakes and simple parameters have been 69 defined to account for the site-to-fault distance (Manfredi, Polese, and Cosenza 2003). 70 The main objectives of this study are: 1) analysis of pros and cons of two existing design procedures, 71 which are the most well-founded, in the authors opinion: a) the displacement-based design procedure 72 by Ponzo-Di Cesare (Cesare and Ponzo 2017); b) the energy-based method by Benavent-Climent 73 (Benavent-Climent 2011); 2) proposal of a new energy-based methodology for the design of EDB 74 systems and comparison with the two latter procedures to evaluate its relative efficiency. 75 The comparison is performed on three 2D frames, whose EDB systems are designed, both, by the two 76 selected procedures and by the proposed method. The post-retrofit performance of the frames is 77 assessed through non-linear static and dynamic analysis. 78

Energy balance 79
The equation of motion for a single-degree-of-freedom (SDOF) inelastic system subjected to a ground 80 motion is given by: 81 where = mass, = damping coefficient; = restoring force; ̈= ground acceleration, and ̈, ̇, 82 are acceleration, velocity, and displacement of the system respectively. Multiplying (1) by =̇ 83 and integrating it over the entire duration of the earthquake, i.e., from = 0 to = , Eq. (1) becomes: 84 where each term can be written as: where is kinetic energy, is damping energy, is strain energy, and is input energy. The strain 85 energy = + is made of two parts: recoverable elastic strain energy , and irrecoverable 86 plastic strain energy . 87 Eq. (3) can be rewritten as: 88 is the elastic strain energy, is the plastic strain energy and = − is 89 the hysteretic energy demand. This can be found using energy spectra, e.g., those proposed in (Decanini 90 and Mollaioli 2001). 91 The elastic strain energy occurs because of the elastic deformation of the structure and becomes 92 null when vibration of the structures ends. The plastic strain energy is related to the inelastic 93 deformation that the structure undergoes during the ground motion. Unless otherwise dissipated 94 through some mechanism, generally inflicts permanent damage to the structure. The objective of 95 retrofitting gravity-load-designed structures is to dissipate through supplemental devices, the EDB 96 system, while the existing structure remains elastic. This implies that, globally: This is achieved if at each -th story the following requirement is fulfilled: 98 The above requirement is satisfied if, at each -th storey it can be ensured that: 99 where , is the ultimate displacement of the EDB system at the -th storey, and , is the yield 100 displacement of the existing frame at the -th storey. That is, the EDB system dissipates energy before 101 the frame yields at any -th storey. If this is satisfied, then the global plastic strain energy is due to 102 the dissipation capacity of the EDB system only, purposely designed to dissipate the global hysteretic 103 energy demand . 104

Shear force coefficient 106
The proposed procedure exploits the strength distribution profile of an optimally designed EDB system 107 proposed by Akiyama (Akiyama 1985), who suggested that buildings with medium height can be 108 represented by a shear strut. By solving the dynamic equation of the shear strut under ground motion, 109 Akiyama arrives at proposing a shear coefficient ratio: 110 where = , ∑ = ⁄ is the shear coefficient at the -th storey, 1 = ,1 ∑ =1 ⁄ is the shear 111 coefficient at the base, where is the shear force at the -th story taken by the EDB system, 1 is 112 the shear force at storey 1 taken by the EDB system, and ∑ = is the weight of the masses 113 above the -th story; then, = ∑ =1 ∑ = ⁄ . 114 Akiyama provides an equation to estimate ̅ , as: 115 ̅ = { 1 + 1.5927 − 11.851 2 + 42.58 3 − 59.48 4 + 30.15 5 > 0.2 , is the storey number and is the total number of storeys of the frame. 116 117 Fig. 1: Schematic of a frame equipped with an EDB system: a) gravity-load supporting frame; b) EDB system; c) frame + EDB system 118

Dissipation capacity 119
In retrofitting of gravity-load-designed structures, the dissipation capacity of the EDB systems is 120 designed to balance the hysteretic energy demand , so to maintain the existing structure elastic. The 121 objective is to fulfil Eq. (5) through Eq. (6) at each storey. 122 Fig. 1 shows the schematic of a frame storey equipped with a bracing system. Bare frame and EDB 123 system are two springs in parallel, so that the coupled system response is obtained by adding the bare 124 frame and EDB responses, as shown in Fig. 2a. Considering Fig. 2b, is defined as follows: 129 where is the shear force taken by the EDB system, is the yield displacement of the frame, 130 is the ultimate displacement of the EDB system. The parameter is defined in (Manfredi,Polese,131 and Cosenza 2003) as the equivalent number of plastic cycles at the maximum value of plastic 132 excursion that the system must undergo to develop a plastic strain energy equal to . It is also used to 133 account for near-fault and far-field situations. The expression for given in (Manfredi,Polese,and 134 Cosenza 2003) is: 135 , with ̈( ) the ground motion, the peak ground acceleration, and 136 the peak ground velocity, is the initial period of the medium period region in the Newmark-Hall 137 spectrum (Newmark and Hall 1982), 1 is the fundamental period of the structure, 1 and 2 are 138 coefficients that differ for near-fault and far-field effects, and is the strength reduction factor. 139

Required strength and stiffness of the EDB system 140
The objective stated in Eq. (6) requires that, at each -th storey, the plastic strain energy , in Eq. (10) 141 be equal to the corresponding hysteretic energy demand , . From this energy balance, we obtain the 142 shear force at each -th storey in the EDB system: 143 Notice that it is implicitly assumed that is the same at all storeys. 144 In order to compute , , it is necessary to know , . That is, it is necessary to know the distribution 145 of the hysteretic energy demand among the storeys. This can be ascertained either using equations 146 proposed in the literature (Ye, Cheng, and Qu 2009) or from non-linear time history analyses. 147 An alternative approach could be to use the shear coefficient ratio ̅ in Eq. (8), which allows to express 148 the shear force at the -th story as: 149 where for ̅ the expression in Eq. (9) can be adopted and for ,1 the following expression is used: 150 In Eq.
where is the total hysteretic energy demand, computed using energy spectra, e.g., those proposed 152 in (Decanini and Mollaioli 2001), and is the number of stories. 153 6 The shear force , at each -th story is then found by replacing Eq. (15) into Eq. (14) and then, in 154 turn, replacing the latter into Eq. (13). 155 The EDB system, at each -th story, can now be designed as follows: 156 a) its shear capacity should be equal to the shear force , in Eq. (13), 157 b) its ultimate displacement should fulfil , = , , 158 c) its yield displacement , is found by dividing the ultimate displacement , by the ductility 159 capacity , of the EDB system at that storey: , = , / , . Notice that, usually, the ductility is 160 kept constant at each -th story ( , = ), where results from an iterative procedure, as explained 161 in the following section, 162 d) its stiffness is determined as: 163 This design procedure ensures that the energy dissipated by the EDB system at each story, , , is larger 164 than the corresponding hysteretic energy demand , . 165

Required stiffness, ductility, and strength of the single bracing device 166
Once the stiffness , of the EDB system is obtained from Eq. (16), the bracings can be designed. 167 Usually, they are made up from two components, as shown in Fig. 3: a) The ductility , can be found as: 178 , = , + , where is the EDB system ductility, which results from the iterative procedure shown in Fig. 4. It is 179 worth noticing that in practical applications it is usually assumed ≥ 3. Each single brace at the -th story has a shear capacity , , and a stiffness , , , whose horizontal 184 components are obtained considering, both, its angle with respect to the horizontal (see Fig. 1), and 185 the number of braces placed at the -th story, so to find: 186 where , is in Eq. (13) and , is in Eq. (16). 187 The flow chart in Fig. 4 summarizes the proposed design procedure. It is worth noticing that the first 188 step of the procedure (compute , at each storey) requires carrying out a pushover analysis for each

Seismic action 212
The seismic action is defined through the acceleration response spectrum given in (Ministero delle  213 infrastrutture e dei trasporti 2018) and the total hysteretic energy demand was estimated using the 214 energy spectra proposed in (Decanini and Mollaioli 2001). The frames are analysed under a design 215 earthquake of = 0.3 . 216

Performance assessment of the bare frames 217
The bare frames performance is assessed through pushover analysis. The lateral load follows the first 218 mode shape. From the deformed shapes in Fig. 6 and the interstorey drift profiles in Fig. 7, it is observed 219 that soft-story mechanisms are formed in Case 1, at the base, and in Case 3, at mid height due to the 220 irregularity; in Case 2 damage is distributed along the height, with a higher concentration at storey 1. 221 The interstorey drift profiles highlight the need for an intervention aiming at obtaining a more uniform 222 drift distribution, as presented in the following section. In Fig. 9 the stiffness profiles of the designed EDB systems are shown. It can be observed that: 243 a) the methodology by Benavent-Climent (BC in the figures) results in an irregular stiffness distribution 244 along the frame height. This irregularity causes damage concentration at the relatively weaker storeys. 245 In all three cases, the first storey remains significantly less stiff, while the stories above are provided 246 with much higher stiffness, causing the damage to occur at the first storey with the consequent soft 247 story mechanism. Moreover, in all three cases, the overall stiffness provided to the frames is 248 comparatively much higher than for the other design methodologies, which implies that the EDB 249 system will be more costly and less efficient, 250 b) Ponzo-Di Cesare (Ponzo in the figures) distribute the EDB system strength proportional to the story 251 strength of the existing structure. The underlying concept is not to alter the strength profile along the 252 structure height. However, in most existing frames, the strength profile is not regular, so that, after the 253 application of the bracing system, such irregularity will remain, thus resulting in damage concentration 254 at the weaker storeys, as shown in Fig. 11,  255 c) the proposed design methodology follows the optimum strength distribution concept, which makes 256 sure that all stories contribute evenly to the demand energy dissipation, and as a result prevents damage 257 concentration at weaker storeys. In a sense, this approach aims at regularizing the frame response. The outcomes of the three procedures (shortly, BC, Ponzo, and proposed) are compared in Fig. 10,  270 which shows the pushover curves of the braced frames along with the relevant performance points, and 271 in Fig. 11, which shows the interstorey drift profiles. By comparing the outcomes in both figures, the 272 different strategies of the three methods become apparent: BC aims mainly at stiffening the frame, to 273 reduce the damage in the existing structure, while Ponzo aims mainly at energy dissipation by attaining 274 larger displacements. This results in higher drifts in Ponzo's case and lower drifts in BC's case. 275 However, both these drift profiles are irregular, with heavier damage in the existing structure localized 276 at the weakest storeys. The proposed method is a compromise between the previous two strategies, 277 mainly aiming at obtaining a higher global strength and a more regular drift profile. As a consequence, 278 the performance point lays closer to (as defined in Eq. (22)), which ensures an optimal even 279 distribution of energy dissipation in the EDB system. 280 281

Non-linear dynamic analysis of the braced frames 282
Non-linear dynamic analysis is carried out using real accelerogram recorded in Kobe, Japan scaled to 283 match the design response spectra in the given site (Fig. 12). consistency of the design method. In all three cases, the proposed method shows ratios that are closer 301 to the optimal ratio, while BC and Ponzo show disperse and more scattered ratios, thus confirming a 302 higher discrepancy between the pushover-based target drifts used in design and those attained in the 303 actual dynamic response. 304 Furthermore, Fig. 15 shows the comparison of the storey-wise overall dissipated energy , for the 305 three cases designed with the proposed and the selected literature design procedures. The EDB systems 306 designed according to BC dissipate almost the entire energy in the first storey, while the upper stories 307 remain elastic with a few exceptions. Overall, the method results in insufficient energy dissipation. The 308 Ponzo procedure provides higher energy dissipation for Cases 1 and 2. However, this occurs at the cost 309 of damaging the existing structure, due to the attainment of larger drifts, as shown in Fig. 13, thus  310 violating the pre-fixed threshold. In Case 3, the method results in insufficient energy dissipation. 311 The proposed procedure ensures an efficient energy dissipation throughout all storeys, keeping the 312 storey drifts within the predefined threshold given in Eq. (22), thus maximizing the efficiency of the 313 EDB systems. Furthermore, the shape of the dissipated energy , is linear, with the slight exception 314 for Case 3, which indicates that the designed strength provided by EDBs enhance both the strength and 315 regularity of the frames. 316 Conclusions 326 The study has been focused, first, on reviewing two design methodologies available in the literature 327 for the design of passive EDB (energy-dissipative bracing) systems and, subsequently, on introducing 328 a new energy-based method. The efficiency of the three methods has been validated on three reinforced 329 concrete frames, retrofitted using such systems, whose performance has been compared through 330 pushover and non-linear dynamic analyses. Based on the results obtained, the following conclusions 331 can be drawn: 332 333 Story-wise stiffness and strength distribution 334 The procedures selected from literature are not particularly efficient in regularizing the response of the 335 frames through the insertion of EDB systems, since the assigned additional stiffness and strength cause 336 vertical irregularity of the frames. On the other hand, the proposed procedure provides additional 337 strength to the storeys as a function of the demand, thus resulting in an optimal strength distribution. 338 339 Interstorey drifts 340 The drift profile obtained from pushover and non-linear dynamic analysis showed that the selected 341 procedures from the literature fail to some extent, both, in distributing damage, and in achieving a 342 uniform drift profile. The proposed method results in more uniform drift profiles and prevents damage 343 concentration at a single story, thanks to the optimal distribution along the frame height of the EDB 344 properties. 345 346 Story-wise energy distribution 347 Non-linear dynamic analyses showed that the selected literature procedures result either in energy 348 dissipation being mainly concentrated at a single story or in lower global dissipation, while the 349 proposed procedure ensures that energy dissipation occurs at all stories, thus maximizing the 350 effectiveness of the EDB systems. 351 352