Highlights

Abstract

Keywords

1. Introduction

2. Description of theoretical aspects

3. Results of analysis

4. Modification factors

5. Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

References

Abstract

Severe damages of adjacent structures due to structural pounding during earthquakes have emphasized the need to use some seismic retrofit strategy to enhance the structural performance. The purpose of this paper is to study the influence of using linear and nonlinear Fluid Viscous Dampers (FVDs) on the seismic collapse capacities of adjacent structures prone to pounding and proposing modification factors to modify the median collapse capacity of structures considering the effects of pounding. The factors can be used to predict the collapse capacity of structures in pounding condition. A seismic retrofit strategy employs FVDs installed in 3-, 6- and 9-story Special Moment Resisting Frames (SMRFs). The SMRFs were assumed to have different values of separation distance according to the seismic code. To model pounding phenomenon, linear viscoelastic contact elements were used in the OpenSees software. Furthermore, to determine the seismic collapse capacities of each structure, the proposed algorithm was applied to remove the collapsed structure during the incremental dynamic analysis. The results of the analyses illustrate that the existence of FVDs can substantially improve the seismic behavior of structures having a significant influence on the collapse capacities of colliding structures. Moreover, considering the adjacent SMRFs in one or two sides of the main structure can significantly affect the median collapse capacity of the main structure itself. Finally, the proposed modification factors can be successfully used to estimate the effects of pounding on the collapse capacities of adjacent structures.

1. Introduction

Different investigations show that adjacent structures with inappropriate separation distance between them may interact one with another during earthquakes, resulting in significant damage [1]. Although nowadays it is a usual procedure to consider a minimum separation distance during the design of concrete or steel structures, many existing structures were constructed with insufficient in-between gap size since it was allowed due to old seismic codes. This insufficient separation distance, which may lead to impacts between adjacent structures during earthquakes, can significantly influence the seismic structural behavior as well as the collapse capacity. Therefore, improving the seismic performance of existing structures is often challenging because such external circumstances can affect the structural behavior. In this respect, there are two strategies: first, increasing the capacity of structural members, for instance using friction devices into beam to column joints, and second, installing passive energy dissipation devices such as dampers or adding dissipative metal shear panels (see for instance [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]). The first approach is more convenient for low-rise buildings, whereas the second one can be more effective and affordable for all types of structures. Furthermore, the strengthening option for structures may interfere with the architectural plans, while the installation of seismic dampers can play a positive role. Although previous studies were mainly focused on the strengthening option, the idea of using linear and nonlinear Fluid Viscous Dampers (FVDs) for retrofitting structures is not new. Among different passive energy dissipation systems, FVDs have some performance advantages for their ability to reduce both displacements and accelerations, which can be very important for the structures with sensitive components.

1- Introduction

2- Optimal damper placement with respect to transfer function amplitude at natural frequency and optimal damper placement under critical double impulse

3- Plastic deformation characteristics of building structures with optimal damper locations for increasing level of critical double impulse

4- Conclusions

References

Abstract

The effectiveness of the optimal viscous damper placement for elastic-plastic multi-degree-of-freedom (MDOF) structures under the critical double impulse is demonstrated through the comparison with the optimal damper placement for elastic MDOF structures designed with respect to transfer function amplitudes at natural frequencies. The method for optimal viscous damper placement was proposed for elastic-plastic MDOF structures subjected to the critical double impulse as a representative of near-fault ground motions in the previous paper [1]. The double impulse is composed of two impulses with opposite directions and the critical interval is determined by using the criterion on the maximum input energy. The critical timing of the second impulse was found to be the timing which requires the vanishing of the sum of the restoring force and the damping force in the first story. Three models with different story stiffness distributions of main structures are treated to demonstrate the effectiveness of the optimal viscous damper placement under the critical double impulse. The double impulse pushover (DIP) procedure proposed in the previous paper for determining the input velocity level of the critical double impulse is examined further in this paper. It is demonstrated that the optimal viscous damper placement for elastic-plastic MDOF structures under the critical double impulse is more effective for pulse-type recorded earthquake ground motions than the optimal damper placement designed with respect to transfer function amplitudes at natural frequencies because several higher modes arising in the elastic-plastic response under pulse-type recorded earthquake ground motions can be well controlled by the design for the double impulse.

Introduction

Viscous, visco-elastic and hysteretic dampers have been used as effective passive dampers installed at interstories. It is commonly understood that hysteretic dampers are effective in general for impulsive ground motions with short duration [2–6] and viscous dampers are useful for long-duration ground motions. Recently, it is reported after devastating earthquakes that near-fault ground motions of impulsive type with short duration are apt to cause large damage to building structures. Although viscous dampers are not necessarily effective for near-fault ground motions, they have advantages to be able to reduce both displacement and acceleration. It seems, therefore, preferable that, when designing viscous dampers, they can reduce the earthquake response to small-to-moderate level ground motions and prevent excessive deformation to large level ground motions, such as long-period ground motions. Many useful researches on optimal damper placement have been accumulated so far (see Refs. [7–9]). Takewaki [10] proposed an optimality criterion-based approach including an incremental inverse-problem formulation in which the transfer function amplitude in terms of the sum of the interstory drifts at the fundamental natural frequency is minimized under the constraint on total damper quantity. This approach has an advantage that the algorithm is simple and the obtained result is independent of input ground motions. Aydin [11] extended this approach to the transfer function amplitude in terms of the base shear at the fundamental natural frequency. Fujita et al. [12] developed an optimization method for minimizing the maximum interstory drift in the transfer function under a constant total damper quantity. Adachi et al.

An equivalent coupled-two-beam (CTB) discrete model is presented to compute the dynamic response in time domain of flexible-base buildings with linear fluid viscous dampers (FVDs). The equivalent model consists of a flexural cantilever beam and a shear cantilever beam connected in parallel by a finite number of axially rigid members that allows the consideration of intermediate modes of lateral deformation. The effects of different lateral deformations on the pre-design stage of two high-rise buildings are showed considering four soil types (hard rock, dense soil, stiff soil and soft soil) and three different distribution methods for linear FVDs: uniformdistribution (UD), story-shear-proportional-distribution (SSPD) and story-shear-strain-energy-distribution (SSSED). For UD method, the damping coefficients of FVDs increase as the flexural rigidity of the building decreases; whereas for SSPD and SSSED methods, the damping coefficients of FVDs vary along the height depending on the type of lateral deformation. For the three distribution methods and the same lateral deformation, the damping coefficients of FVDs decrease as the soil flexibility increases, which leads to a significant decrease in controlling earthquake-induced vibrations and wind-induced vibrations in the two high-rise buildings on a soft soil.

Introduction

Increasing urbanisation in recent decades has led to the construction of high-rise buildings worldwide, which are susceptible to seismic and wind loads. Currently, the structural design of tall buildings must satisfy different criteria such as strength, ductility, stability, resilience, sustainability, among others. Earthquake engineering is usually governed by the ultimate limit state, where peak displacements are of great importance to avoid damage in the structure. On the other hand, wind engineering is usually governed by the serviceability limit state, where root-mean-square (RMS) accelerations are of great importance to avoid human discomfort. The lateral deformation of buildings strongly depends on the type of lateral resisting system, which is usually depends on the height of the building. Accordingly, low-rise buildings usually deform like pure shear beams; whereas in medium-rise and high-rise buildings, the flexural deformation is as significant as the shear deformation. Simplified continuous models are one of the most used procedures for pre-design complex structures and performing parametric analyses to identify the most important variables during the design process. The continuous Timoshenko beam [1] reflects a series coupling of the beam’s bending and shear stiffnesses, while the coupled-two-beam (CTB) continuous model [2] couples the bending and shear stiffnesses in parallel.

1- Introduction

2- Problem formulation

3- Optimization procedure

4- Numerical study

5- Conclusion

References

Abstract

In this paper an efficient first-order multi-objective optimization scheme is adopted for the design of linear viscous dampers for the seismic retrofitting of frame buildings. A retrofitting cost function serves as one objective while the expected losses serve as the other objective. These two objectives are well understood by decision makers that may not be engineers. Furthermore, with the Pareto front for these two objectives at hand, the decision maker can make his decisions with the whole picture at hand. To allow achieving the Pareto front with a reasonable computational effort, a first-order multi-objective optimization approach is adopted. The gradients of the expected loss function, required for the optimization, are analytically derived using the very efficient Adjoint Variable method. This considerably improves the computational efficiency of the methodology. The efficacy of the framework is illustrated with a 2D four storey frame and an eight-storey 3D asymmetric building.

Introduction

Recent seismic events have shown that, when it comes to modern countries with modern seismic codes, loss of human lives due to collapse of buildings is considerably smaller compared to that in the past. For example, the 1994 Northridge and the 2011 Christchurch earthquakes caused a total of 57 and 185 deaths, respectively. As saving human lives has been the major task of earthquake engineering, this is indeed an achievement for the structural engineering community. However, with the relatively small number of deaths due to these events, came huge monetary losses. The 1994 Northridge and 2011 Christchurch led to losses estimated at ∼ US $44 billion and NZ $40 billion (corresponds to approximately 20% of the GDP), respectively (direct and indirect). This motivates a design considering expected losses in parallel to reducing loss of human lives [2]. Adopting such a design approach is necessary for new buildings, where new technologies to limit damage could be easily included. One such loss based design approach is the Loss Optimization Seismic Design, commonly referred as LOSD [3,4]. This may highly affect the losses to be expected in the far future. Nonetheless, in the foreseen future, most buildings to experience earthquakes are ones that have already been designed and built based on the sole goal of saving human lives. Thus, new technologies and design approaches for their retrofitting are required to reduce expected losses.

A simple formula is proposed that provides interstorey velocities of steel framed structures at specific interstorey drift levels. The proposed equation is statistically derived from the variation of floor relative velocity results along the height of the structure. These results are obtained by non-linear inelastic time history analyses of a number of plane steel moment resisting frames. The proposed equation can then be used for seismic retrofit purposes, i.e., in dimensioning the linear or non-linear viscous dampers to be inserted in a steel frame. A detailed numerical example is provided and conclusions regarding the accuracy and the potential use of interstorey velocity are drawn.

Introduction

The variation (distribution) of maximum interstorey velocities have been recognized as key parameter for the evaluation of the along-theheight effectiveness of and demand for viscous dampers [1]. This variation (distribution) seems to depend mainly on the modes (number of stories) of the structure under consideration. In building codes, e.g., ASCE 7–10 [2], peak (design) interstorey velocity (IV) is approximately calculated in the fundamental and higher modes using the design (maximum) interstorey drift (IDR). This approximation permits the calculation of the maximum damping force in terms of the design IDR. Moreover, building codes, e.g., ASCE 7–10 [2], seem to accept that the most effective way of allocation of viscous dampers is to place them where large IDRs are exhibited. Therefore, the maximum IDR is used to estimate the maximum IV, without, however, providing an insight about the real relationship of maximum IDR and IV values. In literature so far, the relationship between IDR and IV is formulated either by using simplified building models and the first mode shape [3,4] or by employing single-degree-of-freedom systems and their maximum displacement [5,6]. The importance of having reliable estimates of the true IV is initially stressed by Pekcan et al. [7] in view of the operating velocities of non-linear viscous dampers and very recently by Favvata [8] in the context of seismic pounding of adjacent structures. It is the purpose of this paper to establish a relationship between IDR and IV values for steel moment resisting frames (MRFs). More specifically, the seismic responses in terms of floor relative displacements and relative velocities are obtained by non-linear inelastic seismic analyses of 20 plane steel MRFs under 22 real and as recorded seismic motions (accelerograms). Then, for each frame-accelerogram pair, height-wise variations of IDR and IV are constructed. Considering a specific value for IDR along-height, i.e., 1.5%, the corresponding IV values are maintained. Statistical processing of these IV values permits the derivation of an equation that provides IV along-the-height of a structure for this specific IDR. The proposed equation is then used in a seismic retrofit scheme, where dimensioning of viscous dampers is based on IV and targets a specific IDR range. An example that involves the retrofit of a 12-storey with 4-bays, steel MRF is presented. This steel MRF is retrofitted either with linear viscous dampers or with non-linear ones. For each kind of dampers, two cases of target IDR are considered. The retrofitted MRFs are then subjected to non-linear inelastic time history analyses and IDR, IV and damper forces are computed. On the basis of the numerical results of the present work, it can be concluded that the proposed IV equation is quite effective in satisfying target IDR. It is also demonstrated that for a specific level of seismic demand (in terms of mean acceleration spectra) the proposed IV equation offers controlled IV and IDR values as well as damper forces under the condition that plastic hinge formations to columns due to additional axial forces induced by the damper forces are within acceptable limits.

1- Introduction

2- Selected IMs

3- Structural modeling and analysis

4- Investigating the efficiency of the IMs for collapse capacity prediction

5- Investigating the sufficiency of the IMs for collapse capacity prediction

6- Proposing optimal IMs

7- GMPEs for the proposed IMs

8- Conclusions

References

Abstract

Nowadays, passive energy dissipation systems are used in the seismic design of new structures and the retrofit of existing structures. Fluid Viscous Dampers (FVDs) are one of the important types of passive energy dissipation systems. Using FVDs can considerably decrease the seismic demands on structures. In this study, seismic collapse behavior of steel Special Moment Resisting Frames (SMRFs) equipped with FVDs is investigated using different scalar Intensity Measures (IMs). Incremental Dynamic Analysis (IDA) method is applied to determine the collapse capacity, IMcol, values for low- to mid-rise steel SMRFs equipped with FVDs. After determining the collapse capacity, IMcol, values by using each of the IMs, the efficiency and sufficiency of the IMs for predicting the seismic collapse capacity of the structures are investigated. Then, advanced scalar IMs, including the effects of spectral shape and ground motion duration, are proposed to reliably predict the collapse capacity of steel SMRFs equipped with FVDs. The results indicate that the proposed IMs possess high efficiency and sufficiency for collapse capacity prediction of steel SMRFs equipped with FVDs.

Introduction

Using passive energy dissipation systems, including Fluid Viscous Dampers (FVDs), hysteretic dampers, viscoelastic dampers and friction dampers, is one of the effective ways to mitigate excitations due to strong ground motions [1,2]. FVDs are a type of passive energy dissipation systems that are extensively used for the seismic design of new structures and the retrofit of existing structures [3,4]. For elastic structures, using FVDs reduces both displacements and accelerations simultaneously [5,6]. However, as pointed out by Karavasilis and Seo [7], for highly inelastic structures, FVDs may increase accelerations, as the damper forces are not out of phase with the peak drifts and internal member forces, due to the nonlinearity of the structure. FVDs provide a velocity-dependent force and can behave as linear or nonlinear elements. The force developed by a FVD is as follows: F Cv v d =⋅ ⋅sgn( ) αd (1) where C is the damper coefficient, v is the relative velocity between the two ends of the damper, αd is the velocity exponent, and sgn is the signum function. In seismic applications, the exponent αd is in the range of 0.2–1.0 [8]. When αd is equal to one, the damper is called “linear FVD”, and values of αd lower than one represent nonlinear FVDs. Several researchers have investigated the seismic response of structures equipped with FVDs (e.g., see [9,10]). Although a number of procedures have been developed for the design of these structures [11–13], the seismic collapse of these structures has not been extensively investigated. The collapse of structural systems due to strong ground motions is the primary source of casualties and loss of life during earthquakes. Seismic collapse occurs when a structural system is unable to withstand gravity loads under earthquake shaking. In recent years, due to significant advancements in the computational capability of computers and the methods of nonlinear analysis, assessing the seismic collapse of structures has become an interesting field of study for researchers. Thus, several studies have been performed to assess the seismic collapse of structures [14–16], and to develop engineering approaches for seismic collapse assessment. The ATC-63 document [17] presents a new methodology for seismic collapse assessment of structures, to assess design criteria and seismic performance factors existing in seismic codes. Recently, some studies have been performed to assess the seismic collapse of structures equipped with FVDs. For instance, Hamidia et al. [18] proposed a simplified approach to assess the seismic collapse of structures equipped with FVDs. Seo et al. [19] investigated the seismic resistance of steel Moment Resisting Frames (MRFs) with supplemental FVDs against collapse.

This study presents a methodology to evaluate the optimal parameters of fluid viscous damper for cable-stayed bridges using the system-level fragility assessment approach. Instead of investigating the impact of different isolation devices on the component’s vulnerability separately, this study focuses on evaluating the optimal parameters of fluid viscous damper to achieve the best overall performance of cable-stayed bridge as a system. Numerical model of a cable-stayed bridge with the most common configuration in China is established using OpenSEES that can account for their nonlinear response and uncertainty treatment. A joint probabilistic seismic demand model and Monte Carlo simulation are employed to obtain the system fragility of cable-stayed bridges by accounting for the contribution of multicomponents to the global damage state. The system-level fragility curves and component fragility curves are compared before and after the application of fluid viscous damper with different parameters. The results indicate that a given parameter of the fluid viscous damper may have a negative impact on some components, yet lead to a better performance of the bridge as a system. Thus, in order to obtain comprehensive knowledge of bridge performance and derive the accurate optimal parameters of fluid viscous damper, it is necessary to consider the fragility based on bridge system.

Introduction

In recent decades, cable-stayed bridges have been widely constructed around the world because of their aesthetics, efficient use of construction materials, and fast construction period. Many studies focused on the dynamic characteristics of cable-stayed bridges subjected to earthquake (Bruneau, 1992; Ren and Obata, 1999; Wilson and Gravelle, 1991). These studies showed that the seismic demand of towers would experience a significant increase in terms of bending moment and shear forces when restraining the bridge deck completely at tower locations. When there are no additional restraints in the longitudinal direction, larger deck displacement would occur during earthquake, resulting in pounding and unseating (Aiken and Kelly, 1992), which is not desirable for performancebased earthquake engineering. Therefore, there is an agreement among many researchers that mitigation devices should be equipped to allow some sort of deck movement, which would lead to the balance of the force and displacement demand of the cable-stayed bridges (Sharabash and Andrawes, 2009). Many researchers have conducted work on the isolation devices to reduce the overall seismic response of cablestayed bridges (Ali and Abdel-Ghaffar, 1994; Sharabash and Andrawes, 2009; Zhang et al., 2009; Zhu and Qiu, 2014); however, these studies are determined methods, which could not account for the earthquake uncertainties, because of the unpredictable nature of seismic events (Zhang and Huo, 2009). In order to consider the uncertainties of earthquake and structure information, it is a widely used approach to develop probabilistic model in the form of fragility curves (Basoz et al., 1999; Choi et al., 2004; Hwang et al., 2001; Nielson, 2005; Padgett and DesRoches, 2008; Pan et al., 2007; Ramanathan, 2012; Shinozuka et al., 2000a, 2000b). Fragility function is quite useful for comparing and selecting mitigation strategies. Casciati et al. (2008) evaluated the seismic vulnerability of components (bending moment, shearing force, deck displacement, cable strain, etc.) of a cable-stayed bridge with passive hysteretic devices in terms of damage exceedance probability. Barnawi and Dyke (2014) developed component (deck displacement, overturning moment, deck shear, etc.) fragility curves of a benchmark cable-stayed bridge equipped with response modification systems. These studies evaluated the effectiveness of the mitigation devices using component fragility separately, rather than the system fragility of the overall bridge performance. For example, some kind of isolation devices might decrease the fragility of bending moment, but would increase the fragility of deck displacement. Hence, a comprehensive evaluation of seismic isolation on bridges should be based on system-level fragility instead of on the component level. It is still worth noting that the above-mentioned studies did not emphasize on selecting optimum mitigation devices to achieve the best performance of cable-stayed bridges.

The use of fluid viscous dampers (FVDs) together with isolators, frequent in near-fault buildings, is effective in reducing displacements of the isolation layer. Such a hybrid system is also beneficial in the case of inter-storey isolation with the aim of limiting P-Δ effects. However, previous research on base isolation shows that this additional damping may also be detrimental, as inter-storey drifts and floor accelerations may increase. This paper analyses the effectiveness of FVDs for enhanced seismic performance of systems with inter-storey isolation. A seven-floor building, with natural and lead rubber bearings between the second and third levels, was used as a case study, and a multi-objective optimal design was performed to identify the best damper parameters. In particular, time-history analyses with various natural records were carried out and two competing objectives were examined: minimisation of the deflection of the isolation layer and minimisation of the total drift of the superstructure. The results show not only the effectiveness of optimal FVDs but also the fact that their optimal linearity degree depends to a great extent on the non-linear seismic response of the structure, i.e., on the type of earthquake. The simplest design approach, consisting of applying an optimization algorithm for each design accelerogram, did not seem, in this case, to be sufficient to identify the best overall design solution. The design consequences of these findings are discussed.

Introduction

Inter-storey seismic isolation has attracted increasing interest in recent years, particularly in densely populated areas, as an alternative mitigation strategy to base isolation for both new and existing buildings. As the name suggests, the isolation system is incorporated between storeys rather than at the base of the structure, in view of architectural concerns, feasibility of construction, and performance benefits. Although base isolation for multi-storey buildings is a wellknown technique applied worldwide, it may sometimes clash with substantial economic and technical problems, which may limit its application. In particular, installing base isolation is straightforward for new buildings, but becomes complicated and expensive for existing ones, since excavation and temporary support works are required. Instead, the installation of inter-storey isolation is relatively simple and generally less expensive and disruption-free. It also allows extra floors to be constructed on an existing building (if its vertical capacity allows this) without increasing the total base shear demand, and thus represents an innovative and realistic retrofitting approach [1,2]. Firstly, base isolation is not as effective for medium/high-rise buildings as inter-storey isolation, because of the flexibility and bending-type behaviour of the latter [3]. Secondly, storey isolation can greatly increase design flexibility in high-rise and multipurpose buildings, by separating them into two independent structural parts which can be designed with different shapes, materials and functions, thus allowing them to become unique architectural features [4]. Examples of this application to irregular high-rise buildings are the Iidabashi First Building [4] and the Shiodome Sumitomo Building [5] in Japan, two multipurpose buildings having substructure and superstructure with different structural shape. In China, this technique was used to isolate 50 buildings (seven- or nine-storey RC frames) in Beijing, built on top of a two-storey platform covering a very large (∼3 km2 ) railway area [6]. Built relatively recently, in the National Taiwan University campus, the Civil Engineering Research Building is a nine-storey pre-cast RC structure with an inter-storey isolation system installed between the second and third floors, which also includes viscous dampers [7]. Lastly, moving the isolation layer to the upper storeys reduces the need for a seismic gap, which is necessary to accommodate the expected displacement of isolators, but also expensive and sometimes impractical in densely-built urban areas.

Energy dissipation systems are technologies able to improve the performance of structural systems subject to transient excitations. Their purpose is to dissipate or reduce part of the energy generated by the dynamic excitation and transferred to these systems. Thus, it is possible to reduce the deformation demand upon the structural systems considered. In fact, if properly designed, they can reduce specific systems’ responses of interest, such as selected displacements and accelerations. These technologies can be divided into passive systems, active and semi-active systems (Soong and Dargush 1997; Constantinou et al. 1998). In each of these categories, there are a wide variety of different technologies nowadays well developed and widely used. Examples of their applications are: The seismic retrofitting of buildings subject to seismic excitation with viscous dampers (e.g. Takewaki 1997; Lavan and Levy 2005; Kanno 2013) and tuned-mass dampers (e.g. Almazan et al. ´ 2012; Daniel and Lavan 2015); The reduction of wind and earthquake induced vibrations in high-rise buildings (e.g. Yang et al. 2004; Smith and Willford 2007; Infanti et al. 2008); The seismic protection of bridges with viscous dampers (e.g. Infanti et al. 2004; Simon-Talero et al. 2006; Infanti and Castellano 2007); The control of human-induced vibrations (Caetano et al. 2010; Casado et al. 2013); The vibration control of offshore wind turbines (e.g. Brodersen and Høgsberg 2016); The shock absorption for high-speed boat seats (e.g. Klembczyk and Mosher 2000); The structural control of dynamic blast loading (e.g. Miyamoto and Taylor 2000); The design of passive and semi-active suspension systems for cars subject to road excitation (e.g. Georgiou et al. 2007; Suciu et al. 2012). The efficacy of the energy dissipation systems is strongly related to their location in the system that needs to be controlled and to their size. For these reasons, often their design is based on optimization, to address the placement and sizing of these devices. For instance, in Lavan and Amir (2014) the authors distribute and size linear fluid viscous dampers in linear structures subject to seismic excitation with an optimization approach based on Sequential Linear Programming (SLP). This procedure is then further enhanced in Pollini et al. (2016) in order to consider a more complete and realistic objective cost function. In Kanno (2013) the author presents a mixed-integer programming approach for the optimal placement of viscous dampers in shear frames, where the damping coefficients are selected from a discrete set of values. A discrete optimization approach is presented in Dargush and Sant (2005), where different types of passive dissipation systems (i.e. metallic plate dampers, fluid viscous dampers, viscoelastic solid dampers) are sized and placed with genetic algorithms. Also in this case, the properties of the devices are selected from a predefined set of available sizes. In Georgiou et al. (2007) Georgiou, Verros, and Natsiavas present a methodology for the multi-objective optimization of the suspension damping and stiffness parameters of quarter-car models subjected to random road excitation. The authors consider both passive and semi-active suspension systems.

Modern technologies for seismic hazard mitigation in building structures, such as passive dampers, have been extensively studied over the past 20 years and are now considered ready for frequent use in seismic-resistant design practice (Christopoulos and Filiatrault 2006). These technologies make it possible to design economically viable buildings that (a) experience significantly less damage than conventional buildings designed according to seismic codes; and (b) return to service within an acceptable short, if not immediate, time after a strong earthquake. The latter is of significant importance as recent strong earthquakes resulted in high losses due to long disruption of the use or occupation of a large number of buildings (New Zealand Treasury Budget Speech 2013). Fluid viscous dampers are a particular type of passive dampers with well-known advantages including stable seismic energy dissipation capacity and force output that is velocity dependent, and thus, typically out-of-phase with the peak internal forces in the structural members of a building (Symans et al. 2008). As their name implies, fluid viscous dampers consist of a cylinder that encloses fluid and a piston. The dynamic motion of the latter results in force output, FD, given by (Seleemah and Constantinou 1997): FD ¼ C j j m a sgnðmÞ ð1Þ where v is the velocity across the damper, a is the velocity exponent, C is the damping coefficient, and sgn stands for the signum function. Many research efforts showed that viscous dampers significantly improve the seismic performance of new or existing buildings by reducing story drifts and plastic deformations in main structural members (Symans et al. 2008). Moreover, research efforts evaluated the effectiveness of using viscous dampers to reduce residual displacements as well as damage in velocity-sensitive and acceleration-sensitive non-structural components of buildings (Karavasilis and Seo 2011; Pavlou and Constantinou 2006; Wanitkorkul and Filiatrault 2008). A design procedure for buildings with passive dampers (e.g. yielding, viscous, viscoelastic), which uses a highly damped elastic single-degree-of-freedom system as substitute of the real inelastic multi-degree-of-freedom building, is included in ASCE 7-10 (2010). The total damping of the substitute elastic system is the sum of the inherent, supplemental viscous, and hysteretic (due to yielding) damping. According to ASCE 7-10, the frame of the building that includes the viscous dampers (i.e. the damping system) is designed with the aid of the substitute highly damped system for three different seismic loading situations, i.e. those associated with the peak displacement, acceleration, and velocity. The ASCE 7-10 procedure has been validated with numerical simulations of the seismic response of highly damped steel MRFs of up to six stories under the design basis earthquake (DBE) and the maximum considered earthquake (MCE) intensities (Ramirez et al. 2002a, b). An alternative design procedure adopts a graphic tool to estimate the peak response of yielding structures with passive dampers with the goal of satisfying multiple target performance objectives (Guo and Christopoulos 2013).

For many years, since the early developments of modern earthquake engineering in the ‘60s, the seismic analysis and design of buildings have been carried out using methods essentially based on the concept of equivalent lateral forces. Nowadays, much more sophisticated analysis tools, such as nonlinear dynamic analyses, are available also in commercial software, and often professional engineers analyze and design building structures making use of such numerical tools. Nonetheless, actual seismic codes still admit the use of equivalent static analysis for the design of relatively regular and simple structures [10,22]. Furthermore, equivalent static analyses are particularly useful for preliminary design, since they permit a fast evaluation of the magnitude of the internal actions in the structural members and their pre-sizing necessary for the development of finite element models and for the computation of the detailed structural response with non-linear dynamic analyses. The use of dynamic analyses for major structures was first introduced in 1974 by the SEAOC Code [33] which recommended its use for “structures with highly irregular shapes, large differences in lateral resistance or stiffness between adjacent storeys” [15]. Later on, with the rapid development of computer programs, the use of dynamic analyses has been established as standard practice for the seismic design of building structures. As such, when in the ‘80s novel technologies for the seismic protection of buildings, such as seismic isolation and dissipative devices, were first adopted in the practice, the use of computer based simulations for the design of structures incorporating such new technologies appeared an obvious choice. Fundamental research works devoted to the development and evaluation of procedures for analysis and design of buildings with passive energy dissipation systems were carried out from the 1990s at the University at Buffalo [13,14,30,30,44]. The works are summarized in MCEER-00-0010 report [32] and ASCE 7-16, Chapter 18 [2] procedures, which are nowadays used in USA by professional engineers. Later on, most of the research works on viscous dampers [36–38,34,21,36,20] basically proposed sophisticated numerical algorithms for dampers optimization (i.e. dampers size and location), sometimes leading to complex design procedures. Alternative approaches leading to design procedures for the sizing of viscous dampers have been proposed in the last years: (i) LopezGarcia in 2001 [21] developed a simple algorithm for optimal damper configuration (placement and properties) in MDOF structures, assuming a constant inter-storey height and a straight-line first modal shape; (ii) Christopoulos and Filiatrault in 2006 [12] suggested a design approach for estimating the damping coefficients of added viscous dampers consisting in a trial and error procedure; (iii) Silvestri et al. in 2010 [35] proposed a direct design approach, referred to as the “five-step procedure”. In the present work, a simplified procedure for the sizing of viscous dampers and structural elements of framed structures equipped with inter-storey viscous dampers is presented. The procedure is grounded on some previous works by the authors [40,40,23,24,26] since it collects and exploits all fundamental results in the framework of the original “five-step procedure” [35]. The procedure, originally developed for elastic frames equipped with viscous dampers, is extended to the case of yielding structures making use of an overall reduction factor [23].

Over the past fifty years, a large part of research has been dedicated to earthquake-resistant systems developed to raise seismic performance levels while keeping construction costs within reasonable levels. This aspect is particularly evident in the case of existing buildings that are unable to satisfy the seismic requirements provided by current codes. The retrofit objective of satisfying the seismic requirements of new structures is often economically prohibitive and very difficult to achieve. In these cases an innovative technique as the dissipation of energy by added damping devices may be very promising in improving the seismic performance [1–12]. In rehabilitation interventions, fluidviscous dampers offer some advantages [3,4], as their behaviour is independent of frequency and their dissipative capacity is very high. Performance-Based Earthquake Engineering (PBEE) is a framework used to identify and define desired structural performances for specified seismic intensity levels. Currently, the most advanced PBEE methodologies, such as the PEER PBEE procedure [13,14], have been developed to evaluate seismic performance in terms of expected economic losses, a parameter of particular interest to decision makers. The PEER PBEE procedure consists of four steps of analysis: hazard analysis for the specific site of the building, structural analysis, damage analysis and loss analysis. In all the steps, the related uncertainties are explicitly considered in a probabilistic manner. The second step, in particular, involves a series of nonlinear time-history analyses executed to obtain a probabilistic description of the seismic demand for the building at increasing seismic intensity levels. In this framework, the widespread probabilistic SAC-FEMA approach [15] can be useful for developing simplified procedures [16,17]. This method provides a closed form expression for evaluating the annual probability of exceeding a given limit state, while accounting for record-to-record variability and modelling uncertainties. The method is based on the first two steps in the PBEE process, hazard analysis and structural analysis, and, in general, it requires a series of nonlinear time-history analyses to be performed. This probabilistic approach has been followed in the present study. The purpose of this paper is to study the correlation between the results of the probabilistic assessment method for structures with and without dampers, with particular emphasis on the results in terms of dispersion caused by ground motion variability. This paper has studied, in particular, the variability and influence of the parameters in the closed form expression, such as the dispersion of seismic demand. This research takes into account the near collapse limit state, a wide set of ground motions and different methods to approximate the hazard curves. The analyses were performed without applying scaling factors to the earthquake records, and different records for increasing values of seismic intensity were instead used. Considering the current trend of developing simplified procedures to assess performance while avoiding expensive nonlinear time-history analyses, a second purpose of the paper is to apply a direct procedure for assessing the response of nonlinear structures with nonlinear viscous dampers. This procedure was recently proposed by some of the authors and was applied within the considered probabilistic approach.

Purely viscous dampers are devices that can be used in bracing systems to dissipate large part of input seismic energy [1,2]. They consist of a cylinder, filled with a silicone fluid inside which a piston slides, allowing the device to produce a force F c sign u u =⋅ ⋅ ( ̇) ̇ α, where c is the damping constant characteristic of the device, u̇ is the velocity between the two device ends and α is an exponent typical of the device type. Optimisation of linear viscous damper systems (i.e. α = 1) has been widely studied in the literature (see for instance [3–5]). In comparison, nonlinear devices have been investigated in less works, due to inherent difficulties in the analytical formulation. In [6], a simplified procedure for damper design aimed at a given target in terms of displacement reduction factor was developed. Reference [7] proposes a practical method for optimum design of non-linear oil dampers with relief mechanism. A comprehensive probabilistic design methodology considering life-cycle cost criteria together with uncertainty in structural response and earthquake loading is proposed in [8]. Most of these methods rely on simplified assumptions that may limit their use, i.e. very specific objectives, reduction of the structure to simpler SDOF systems, linearization of damper behaviour, limitation of design variables. However, any designer knows that objectives and constraints are often given by the specific problem at hand, and thus a flexible procedure which can be adapted to the case under study may be more desirable than more efficient yet problem-dependent methodologies. In this work, an optimisation-based approach to the design of nonlinear viscous dampers for seismic retrofitting of frame structures is investigated. Different objectives and constraints are proposed to show the flexibility of the methodology, and comparisons with state-of-theart methods of the literature are provided.