FNCATB Superelastic damper for seismic vibration mitigation

Abstract

A new shape memory alloy of Ferrous origin (Fe-Ni-Co-Al-Ta-B, referred to as FNCATB) is reported recently that shows huge Superelasticity in comparison with its earlier variants, such as Nitinol and Cu-Al-Be alloy, commonly employed in vibration damping applications. The performance of the FNCATB alloy based Superelastic damper is explored herein and compared with its conventional alternative. The optimal performance of the dampers is ensured by maximizing the equivalent damping, a closed form expression of which is derived based on the force–deformation behavior of the combined structure–damper system. The formulation closely follows that of conventional yield damper but with an additional term $(λ)$ characterizing the flag shaped hysteresis. The available experimental data from a cyclic test are fitted with the existing Auricchio model to describe the force–deformation hysteresis of FNCATB alloy. The dynamic response analysis of the structure–damper systems subjected to recorded seismic ground motions reveals the superior performance of this new damper in reducing the drift of the structure with almost identical base shear, experienced by the structure without damper. The primary constituent being ferrous, the FNCATB might be cost-effective, which is also supported by its less volume requirement, demonstrated herein.

Keywords

shape memory alloy Superelasticity damping seismic ferrous shape memory alloy

Introduction

The seismic design philosophies, adopted worldwide, assume that the structure will venture into inelastic domain to mitigate seismic hazard by allowing the structure to be partially damaged. Nonlinear behavior of the structure will allow for the reduction in design forces by using the strength reduction factor (Miranda and Bertero, 1994). The structure is detailed suitably to allow such damages to be localized at strategic locations in order to prevent global instability/collapse of the building. However, experiences in the past earthquakes (e.g. Northridge earthquake, USA, 1994; Garamendi, 2004) show that in an extreme seismic event these damages become significant and the penalties to be paid in designing suitable retrofitting and rehabilitation are extensive.

A number of alternative design approaches have been proposed, in which the structures are augmented by specially designed energy dissipating devices in the form of viscous/friction/hysteretic dampers to dissipate the input seismic energy imparted to the structure by an earthquake. A number of such dampers have been proposed, a review of which can be obtained from Symans et al. (2008).

Over almost two decades (Clark et al., 1995; Krumme et al., 1995), a new genre of damper, referred to as superelastic damper, has been proposed for seismic vibration mitigation of the structure that utilizes the superelastic force–deformation behavior of shape memory alloy (SMA). The SMA is a class of smart material with several other well-acclaimed attributes other than Superelasticity, a detailed review of which can be obtained from Ozbulut et al. (2011). By virtue of this Superelasticity, the SMA can recover their original configuration after unloading, if the strain level remains within some allowable limit, referred to as Superelastic window (SEW). For Nitinol, this SEW is as high as 8%. Other than the Superelasticity, large ductility, excellent resistance to corrosion and fatigue make SMA an attractive choice in seismic damping applications.

The potential of SMA as a material for base isolation device was first identified by Graesser and Cozzarelli (1991) and later applied as a damping material in civil structures by Clark et al. (1995), Krumme et al. (1995), and Thomson et al. (1995). A one-dimensional model for the hysteresis of SMA under cyclic loading was also proposed by them. Experimental demonstration of the SMA as damping material was done by Clark et al. (1995), Dolce et al. (2000), and Dolce and Cardone (2001). Cyclic test on most commonly known SMA (Nitinol) by Dolce et al. (2000) reveals the stable hysteresis under usual frequency range (0.2–4 Hz) of seismic excitations. Yan and Nie (2003) presented the usage of the Superelastic SMA as a damper by illustrating its superior energy dissipative capability, larger bearing strength, and much reduced residual displacement. It was demonstrated that the dissipative capability of SMA is irrespective of the exciting frequencies that are typically encountered in earthquakes. Similar study by DesRoches et al. (2004) reported the strength, equivalent viscous damping, and the re-centering capability of SMA, which shows that damping in SMA is typically less than 7% and high level of cyclic strain leads to degradation of damping and re-centering capability. A number of studies are followed thereafter. Wilde et al. (2000) performed simulation to demonstrate the effectiveness of the SMA damper in supplementing a base isolation system, isolating highway bridge. Shake table tests on structures augmented with SMA damper are also performed (Dolce et al., 2005). Casciati et al. (2009) proposed an SMA passive device to control the earthquake response behavior of bridge and demonstrated that the SMA damper can significantly reduce the peak displacement response.

Adachi and Unjoh (1999, May) developed an SMA based damping device which can absorb seismic energy and reduce the seismic force by its pseudo yield effect. The effectiveness of the SMA damper was also demonstrated on a building model by Han et al. (2003). In view of the dependence of the hysteresis of SMA on chemical composition, ambient temperature, and strain rate, a sensitivity based study of the responses with respect to these parameters has been conducted by Andrawes and DesRoches (2007). Another class of SMA, made of Cu-Al-Be alloy, shows improved performance under temperature variations and also appears to be economical (Ozbulut et al., 2011). The application of Cu-Al-Be-type SMA in damping has been presented by Zhang et al. (2009). Recently, the attenuation of seismic response of structures using SMA dampers has been demonstrated by Parulekar et al. (2012).

The reported studies in literature, as presented herein, mostly adopt the Nitinol alloy as the SMA. However, although limited, a number of studies also explored the suitability of the Cu-Al-Be alloy (Casciati and Faravelli, 2004, June; Cerda et al., 2006; Kustov and Van Humbeeck, 2008; Van Humbeeck and Kustov, 2005; Zhang et al., 2009) for similar applications, which is motivated by the fact that the Copper based SMA(s) are less expensive and easier to machine. Furthermore, Cu-Al-Be alloy has a much wider temperature range (–80°C to 100°C) in which the Superelasticity behavior is retained. This property distinguishes Cu-Al-Be from other variant of SMA, especially in outdoor environment in cold regions (Zhang et al., 2009).

While there is a continuous need for development of increasingly cheaper and efficient SMA(s) (in respect to their Superelasticity and Shape Memory effects), a variant of SMA of Ferrous origin has been proposed recently by Tanaka et al. (2010) and Omori et al. (2011). This newly developed SMA is composed of elements as Fe-Ni-Co-Al-Ta-B, having percentages of 40.95, 28, 17, 11.5, 2.5, and 0.05, respectively. Following its composition, this SMA is named as FNCATB alloy. Unlike the commonly studied SMAs, such as Nickel–Titanium Alloy (Nitinol) or Copper–Aluminum–Beryllium (Cu-Al-Be) Alloy, the FNCATB has ferrous constituents in significant proportion. Thus, this alloy shows potential to be a much cheaper alternative than the conventional Nitinol-SMAs and may be amenable for cost-effective use in large structural applications. Furthermore, a comparison of the Superelastic force–deformation hysteresis of FNCATB with its older counterpart (i.e. Nitinol) reveals that the Superelasticity of the latter is of significantly higher magnitude, as obtained by comparing the respective areas of their hysteresis. These two features (remarkably higher Superelasticity and lower cost) show the potential of FNCATB as a superior material for Superelastic dampers over the existing ones.

This study proposes a Superelastic SMA damper based on FNCATB and compares its performance with the conventional Nitinol, Cu-Al-Be based Superelastic dampers. The study employs a single-bay, single-storey structural frame, in which the FNCATB Superelastic damper is employed for reducing its vibrational response under seismic excitations. A cyclic stress–strain model, proposed by Auricchio and Sacco (1997), for the Superelasticity is adopted and the available test data (obtained by request from T. Omori, School of Engineering, Tohoku University) from the cyclic test are fitted into this model. The optimal performance of the damper is ensured by choice of optimal parameters for dampers that maximizes its equivalent damping. A closed form expression of such equivalent damping is derived based on area under the force–deformation loop of the combined structure–damper system. The optimal parameter for the damper is thus obtained by maximizing this equivalent damping. By adopting such optimal parameters, a number of recorded ground motions from the real earthquakes are employed as input to the structure–damper system to carry out nonlinear dynamic analysis. Important response quantities of interest are obtained for having a comparative assessment of the performance of the FNCATB with other Superelastic dampers. Details of this are presented in subsequent sections.

SMA and Superelasticity

An SMA is a special class of smart material obtained by alloying several metallic/nonmetallic components. Such alloys recover their original shape after experiencing deformation on application of heating and the name SMA is derived from this special attribute. There are several variants of SMA(s) available for their applications to various fields. Out of these, the most common ones, tried in civil engineering passive control applications, are the Nickel–Titanium Alloy (widely referred to as Nitinol) and Copper–Aluminum–Beryllium (Cu-Al-Be) Alloy. The application of these two type SMA(s) are already tested in vibration damping of civil structures subjected to seismic excitation.

Other than the Shape Memory Effect, the SMA(s) also portray another very interesting property, referred to as Superelasticity. By virtue of this property, the SMA(s) can sustain large deformation (strain) when subjected to monotonic/cyclic loading without leaving residual deformation (strains) after unloading. Such distinct behavior is due to the changes occurring in the microstructure of the SMA under loading–unloading cycles. Two different types of crystallographic microstructure, namely the Austenite (A) and Martensite (M), can exist in SMA. The Austenite phase is more ordered and stable at high temperature and at low stress. On the other hand, the Martensite is less ordered and stable at low temperature and high stress. At ambient temperature and lower stress level, the SMA stays at Austenite phase. On application of load, the Austenite phase gradually transforms to Martensite, which is stable at higher stress. During unloading and lowering of stress, the Martensite phase back transformed to Austenite phase for being energetically favorable at lower stress. The path followed during transformation results in a loop, the area of which provides the estimate of energy dissipated during loading and unloading. No residual deformation remains in SMA after unloading. The deformation and its recovery through forward and reverse phase transition by the flag shaped hysteresis is described as Superelasticity (Graesser and Cozzarelli, 1991; Ozbulut et al., 2011; Thomson et al., 1995). This behavior is schematically shown in Figure 1(a). It is envisioned that the Superelastic force–deformation characteristic is quite different from the elasto-plastic hysteresis observed in metal that is triggered by yielding and leaves residual deformation after unloading. In contrast, no residual deformations remain in Superelasticity.

It is worth mentioning that the Superelastic behavior in SMA(s) is available only within certain strain level, specific to the SMA material. This is because beyond this SEW for strain, the SMA will eventually reach its respective yield points as do metals. Thus, effective utilization of the Superelasticity can only be realized by restricting the strain (deformation) level within this SEW for the respective SMA. The strain level characterizing the SEW varies from one to another SMA and can be tailored by varying the composition of the material and thermo-mechanical treatment.

It is already mentioned that the Superelastic behavior of SMA is dependent on the temperature and the Superelastic loop may degrade significantly at very low temperature. Such variation is schematically shown in Figure 1(b). The property of Superelasticity can only be availed at a specific range of temperature for any SMA and may widely vary from one variant of SMA to the other. The lower bound of such temperature window is known as the Austenite start temperature, denoted by $(A_{s})$ . If the temperature remains higher than the Austenite start $(A_{s})$ and lower than the Austenite finish $(A_{f})$ only partial recovery of the strain is possible. Furthermore, there exists some upper limit of critical temperature $(M_{d})$ above which Austenite phase stabilizes and Martensitic transformation cannot be triggered by applied load. For example, Nitinol starts losing Superelasticity from $0^{\circ} C$ , which is completely depleted at $- 25^{\circ} C$ . The Cu-Al-Be alloy shows Superelasticity in a much wider range starting from $- 80^{\circ} C$ to $100^{\circ} C$ (Zhang et al., 2009), implying the suitability of the latter for outdoor application. Therefore, the specific temperature range for Superelasticity must be within the limits of expected temperature variations to which the damper will be subjected. This is an important consideration for design of Superelastic SMA damper.

The SEW for the FNCATB is much wider than for its predecessor, such as Nitinol and Cu-Al-Be, which have long been tested in vibration damping application. Furthermore, the temperature range for the stability of the Superelastic force–deformation behavior of FNCATB is much wider than that for the Nitinol and Cu-Al-Be. Moreover, the most intriguing is the potential of FNCATB in providing huge Superelasticity, which is much more than that offered by the Nitinol or Cu-Al-Be.

Cyclic force–deformation behavior

Dynamic response analysis of the damper requires its force–deformation characteristics. Most of the Superelastic damper system consists of wires or similar components to utilize the Superelasticity of SMA. These devices involve axial forcing and resulting deformation. Knowing the area of such element, the force–deformation can be trivially derived from the respective stress–strain behavior of such material.

The stress–strain behavior of SMA for Superelasticity has been modeled by different researchers in the past using different formalisms (Auricchio and Sacco, 1997; Graesser and Cozzarelli, 1991). Among the most widely employed models (Auricchio et al., 1997, 2006; Yang et al., 2010), the one suggested by Auricchio and Sacco (1997) is the simplest and is adopted presently for describing the uniaxial force–deformation hysteresis of SMA(s). This particular model considers two phases of transformations, namely, forward transformation from Austenite to Martensite and backward transformation from Martensite to Austenite. The fraction of Martensite present in the alloy at any point of microstructural phase transition in the course of deformation is denoted by $ξ_{S}$ and the fraction of Austenite is denoted as $ξ_{A}$ . Obviously, these two fractions would satisfy the following relation

ξ_{S} + ξ_{A} = 1

(1)

In the present formulation, the fraction of Martensite $(ξ_{S})$ is adopted to be an internal variable in deriving the respective stress–strain relationship. The criteria of phase transformation are described using the Drucker–Prager loading function $F (σ_{˜})$ as

F (σ_{˜}) = q + 3 α p

(2)

in which $σ_{˜}$ is the stress tensor. The parameter q is obtained from the deviatoric part of $σ_{˜}$ as

q = \sqrt{(\frac{3}{2}) σ_{˜}_{v} : σ_{˜}_{v}}

(3)

in which the deviatoric stress tensor is obtained as

σ_{˜}_{v} = σ_{˜} - {p I}_{˜}

(4)

and the hydrostatic state of stress is obtained as

p = (\frac{1}{3}) σ_{˜} : I {p I}_{˜}

(5)

where I is an identity tensor and the symbol (:) indicates the inner product between the two tensors. The evolution for the fraction of Martensite $(ξ_{S})$ during forward and backward phase transition is described using rate-type equations of the following form

\frac{\partial ξ_{S}}{\partial t} = {\begin{matrix} - H^{A S} (1 - ξ_{S}) (\frac{(\partial F (\underset{˜}{σ}) / \partial t)}{F (\underset{˜}{σ}) - σ_{f}^{A S} (1 + α)}) \begin{matrix} A \to S \end{matrix} \\ H^{S A} ξ_{S} (\frac{(\partial F (\underset{˜}{σ}) / \partial t)}{F (\underset{˜}{σ}) - σ_{f}^{S A} (1 + α)}) \begin{matrix} S \to A \end{matrix} \end{matrix}

(6)

in which t refers to pseudo-time. Symbols $H^{AS}, H^{SA}$ refer to two functions defined subsequently. The quantities $σ_{f}^{AS}, σ_{f}^{SA}$ are material parameters defining the stress level at the corresponding phase transformation point of the Superelastic force–deformation loop, shown in Figure 2. The functions $H^{AS}$ and $H^{SA}$ are defined as

\begin{matrix} H^{AS} = {\begin{matrix} 1 if {\begin{matrix} σ_{S}^{AS} (1 + α) < F (σ_{˜}) < σ_{f}^{AS} (1 + α) \\ (dF (σ_{˜}) / dF dt) > 0 \end{matrix} \\ 0 otherwise \end{matrix} \\ H^{SA} = {\begin{matrix} 1 if {\begin{matrix} σ_{f}^{SA} (1 + α) < F (σ_{˜}) < σ_{s}^{SA} (1 + α) \\ (dF (σ_{˜}) / dt) < 0 \end{matrix} \\ 0 otherwise \end{matrix} \end{matrix}

(7)

Similar to $σ_{f}^{AS}, σ_{f}^{SA}$ , the parameters $σ_{s}^{AS}, σ_{s}^{SA}$ are also intrinsic to the materials (shown in Figure 2) and are determined by conducting experiments. Another material parameter $α$ is used to characterize asymmetry (if present) among the tensile and compressive loop. For symmetric tension–compression behavior, this parameter is set to zero. However, in case of asymmetry this parameter is estimated as

α = \frac{σ_{c}^{AS} - σ_{t}^{AS}}{σ_{c}^{AS} + σ_{t}^{AS}}

(8)

in this expression, the subscripts c and t refer to the compressive and tensile stress, respectively. The stress–strain relation is expressed as

σ_{˜} = D_{˜} : (ε_{˜} - ε_{˜}_{T})

(9)

where $D_{˜}$ is the elastic stiffness tensor and $ε_{˜}_{T}$ is the transformation strain tensor. The evolution of the transformation strain tensor $(ε_{˜}_{T})$ is given by

\frac{\partial ε_{˜}_{T}}{\partial t} = \frac{\partial ξ}{\partial t} ε_{˜}_{L} \frac{\partial F (σ_{˜})}{\partial σ_{˜}}

(10)

The transformation strain tensor $(ε_{˜}_{T})$ for uniaxial case becomes a scalar quantity, which is determined from the experimental measurement.

Configurations of Superelastic dampers

A number of alternative configurations for the Superelastic dampers are proposed in the literature. These configurations along with the mechanics of such damper are briefly reviewed herein. Zhang et al. (2009) have presented a Cu-Al-Be Superelastic damper, in which the stranded Superelastic Cu-Al-Be wires are attached to two different parts using anchoring fixtures so that when the damper is subjected to either tension or compression, the Cu-Al-Be wire strands will be stretched in one direction. A hybrid device that dissipates energy and also serves as a re-centering device is proposed by Yang et al. (2010) by using energy absorbing mild steel struts and a set of SMA wires. The hybrid device also contains two high-strength steel tubes mounted in the frame bay and guides the struts and SMA wires. The SMA wires are surrounded by the steel tubes and wound by two cylindrical blocks inside the high-strength steel tubes. The length of the SMA wire is determined from the limiting strain of the SEW and also by simultaneously satisfying the required initial stiffness of the damper. Parulekar et al. (2012) proposed a Superelastic damper by making use of the energy dissipation property of Ni-Ti wires. The Superelastic damper consists of two concentric pipes mutually moving with respect to one another. Three studs are attached to the inner pipe at an angle 120° apart at two locations. Six studs are attached to the outer pipe in the center equidistant from the studs attached to the inner pipe. There are in total six wires, out of which three are in tension when loaded and three are in slack. Thus two independent groups of wire loops act as energy dissipating group. A typical Superelastic damper configuration is schematically shown in Figure 3, which consists of wires attached between two moving parts in a way that one set of wires experiences tension whereas others are in slack. The assembly gets reversed on changing the direction of forcing. In this study, the estimation of the number of required SMA wires along with their diameter (area) and length is given; those are required in order to attain the required energy dissipation as well as re-centering capability. This is also to ensure that the damper attains optimal strength ratios for a specified stiffness ratio with respect to the main structure and for a given level of ductility demand under earthquake excitations.

Dynamic response analysis of structure–damper system

A structure assisted by Superelastic damper is analyzed while subjected to seismic excitations from earthquakes. The equation of motion for a linear structure is written as

\begin{matrix} [M] {\overset{\cdot\cdot}{u}} + [C] {\overset{\cdot}{u}} + [K] {u} + {F_{d} ({u}, {\overset{\cdot}{u}})} \\ = - [M] {r} {\overset{\cdot\cdot}{u}}_{g} \end{matrix}

(11)

in which $[M], [K]$ , and $[C]$ are the mass, stiffness, and damping matrix of the structure. These matrices can be obtained by finite element numerical modeling of the structure, the details of which can be obtained from Cook (2007). The vectors ${\overset{\cdot\cdot}{u}}, {\overset{\cdot}{u}}$ , and ${u}$ are the acceleration, velocity, and displacement vectors of the structure at various degrees of freedoms given by the finite element discretization of the structure. The vector ${r}$ is the influence coefficient vector governing the distribution of inertial forces along the various degrees of freedom due to shaking of ground. The input ground motion is given in terms of the acceleration at the base, denoted by ${{\overset{\cdot\cdot}{u}}_{g}}$ . The force vector ${F_{d} ({u}, {\overset{\cdot}{u}})}$ is from the Superelastic action of the SMA(s) damper and is a nonlinear function of the relative displacement ${u}$ and velocity ${\overset{\cdot}{u}}$ of the damper. It is reminded that the damping matrix $[C]$ is constituted by assuming subcritical viscous damping in the structure and is proportional to the velocities ${\overset{\cdot}{u}}$ . The forces from the Superelastic damper ${F_{d} ({u}, {\overset{\cdot}{u}})}$ are obtained from the respective stress–strain relationship, described in equations (1)–(10) and are obtained by multiplying the stress ${σ}$ with the effective cross-sectional area $(A)$ of the dampers (wires) as

{F_{d}} = {A σ}

(12)

Although, practically most civil engineering structures have large number of degrees of freedoms, for simplicity, simplified model are often employed to understand the behavior under motion. The structure may or may not remain in its linear force–deformation regime. For linear structure, stiffness matrix $[K]$ consists of constant terms; however, for nonlinear behavior, the terms in the stiffness matrix may be the function of ${u}$ , that is, $[K]$ will be replaced by $[K (u)]$ and the equation will be modified as

\begin{matrix} [M] {\overset{\cdot\cdot}{u}} + [C] {\overset{\cdot}{u}} + [K (u)] {u} + {F_{d} ({u}, {\overset{\cdot}{u}})} \\ = - [M] {r} {\overset{\cdot\cdot}{u}}_{g} \end{matrix}

(13)

The equation of motion, shown in equations (11), (13) are nonlinear and are solved using the iterative/incremental time step integration technique assuming specified time step for integration by satisfying the stability and accuracy criteria of integration.

Presently, a single-bay, single-storey frame is considered to demonstrate the effectiveness of the FNCATB damper in controlling the seismic response of the frame subjected to ground motion. The properties of the frame are detailed later. In parallel, response analyses are also conducted by replacing the material of the Superelastic damper with conventionally employed SMA(s), such as Nitinol and Cu-Al-Be alloy. This is in order to provide a comparative assessment and to establish the superior performance of the FNCATB as a material over the others.

Estimation of equivalent damping of the damper

The behavior of structures assisted by damping largely depends on the characteristics parameter of the damper, the optimal value of which must be chosen to ensure best performance of the damper–structure system. Therefore, before studying the response behavior of damper assisted structure, an estimate of such optimal parameters is obtained through simple analysis presented subsequently.

The individual force–deformation behavior of the structure, damper, and the combined behavior under lateral loading are shown in Figure 4(a). The phase transformation of the damper must precede the yielding of the main frame. This is ensured by letting the transformation displacement (displacement corresponding to the forward transformation) of the damper lower than the yield displacement of the main frame as

u_{dy} < u_{sy}

(14)

The relative proportioning of the (pre-transformation) stiffness and the yield strength of the structure/damper are important quantities, defined as

\begin{matrix} S_{r} = \frac{K'_{d}}{K_{s}} \\ γ = \frac{F_{dy}}{F_{sy}} \end{matrix}

(15)

where $S_{r}, γ$ are the stiffness and strength ratio of the damper which are defined as the ratio of lateral stiffness of the damper $(K'_{d})$ to the main frame $(K_{s})$ and the ratio of the transformation strength of the damper $(F_{dy})$ to the yield strength of the structure $(F_{sy})$ , respectively. It is noted that the combined load–deformation diagram contains two distinct yield points; the first yield corresponds to the phase transformation of the damper, whereas the second yield corresponds to the yielding of the main frame. The strength corresponding to these points can be written as

\begin{matrix} F_{y 1} = F_{dy} + K_{s} u_{dy} \\ F_{y 2} = F_{dy} + F_{sy} + β (\frac{F_{dy}}{u_{dy}}) (u_{sy} - u_{dy}) \end{matrix}

(16)

The ultimate strength of the damper and the structure (at maximum displacement) are denoted as $F_{dm}$ and $F_{sm}$ respectively, and the respective maximum displacement for both are the same and denoted as $u_{m}$ . The ductility of the structure and damper is defined by knowing the respective yield displacement and maximum displacement as

μ_{s} = \frac{u_{m}}{u_{sy}}; μ_{d} = \frac{u_{m}}{u_{dy}}

(17)

As the analysis for the Superelastic energy dissipation is based on combined force–deformation behavior of the system, all the pertinent strengths are ultimately expressed in terms of the first and second yield strengths of the combined system. Thus, equation (16) can be used to obtain $F_{sy}$ as

F_{sy} = \frac{F_{y 2}}{1 + γ {1 + β ((u_{sy} - u_{dy}) / u_{dy})}}

(18)

Similarly, the ultimate strength can be obtained as

\begin{matrix} F_{dm} = F_{dy} (1 + β μ_{d} - β) \\ F_{sm} = F_{sy} (1 + α μ_{s} - α) \end{matrix}

(19)

where the parameters $α$ and $β$ are the hardening parameters for the main frame and damper, respectively. The area of the loop formed by the hysteresis of the damper and the main frame quantifies the dissipated input energy, the estimate of which can be obtained by assessing the area of the loop. An equivalent ratio of viscous damping $(ξ_{eq})$ can be derived by taking the ratio of the area responsible for dissipation $(E_{p})$ and the total area of input energy $(E_{e})$ in a loading cycle as (Chopra, 1995)

ξ_{eq} = \frac{E_{p}}{4 π E_{e}}

(20)

The area of the loop from the Superelastic damper in one cycle, providing the estimate of Superelastic energy dissipation can be obtained as (from Figure 4(b))

\begin{matrix} E_{p d} = 2 {(F_{b y} - λ F_{b y}) (u_{m} - ε) - (u_{b y} - λ u_{b y}) F_{b y} (1 + β μ_{b} - β - λ F_{b y})} \\ = 2 (1 - λ) {F_{b y} (u_{m} - λ u_{b y}) - u_{b y} F_{b y} (1 + β μ_{b} - β - λ)} \end{matrix}

(21)

Similarly, the area of the loop in the elasto-plastic hysteresis of the structure is obtained as

E_{ps} = 4 {F_{sy} u_{m} - u_{sy} F_{sy} (1 + α μ_{s} - α)}

(22)

It may be mentioned that as the Superelastic loop is only restricted in the first and third quadrants of the force–deformation diagram, unlike the elasto-plastic hysteresis, which equally occupies areas in all quadrants, the factor 2 appears in equation (21) instead of factor 4 in equation (22). The input energy in the structure can be estimated by assuming that if the structure remains elastic, the input energy $(E_{e})$ is equal to the work done by the yield force on the ultimate deformation, as shown in the hatched triangle in Figure 5. This is expressed as

E_{e} = (\frac{1}{2}) u_{m} {F_{sy} (1 + α μ_{s} - α) + F_{by} (1 + β μ_{b} - β)}

(23)

Substituting equations (21), (22), and (23) in equation (20), the equivalent viscous damping is obtained as

2 π ξ_{eq} = \frac{4 {F_{sy} u_{m} - u_{sy} F_{sy} (1 + α μ_{s} - α)} + 2 (1 - λ) {F_{by} (u_{m} - λ u_{by}) - u_{by} F_{by} (1 + β μ_{b} - β - λ)}}{u_{m} {F_{sy} (1 + α μ_{s} - α) + F_{by} (1 + β μ_{b} - β)}}

(24)

Equation (24) can be simplified in terms of the important parameters of the system $(S_{r}, γ, μ_{s}, α, β, λ)$ as

2 π ξ_{eq} = \frac{2 γ^{2} (λ + β - λ β - 1) + 4 S_{r} (α - 1) + 2 μ_{s} S_{r} (γ λ β - γ β - γ λ - 2 α + γ + 2)}{μ_{s} S_{r} (β S_{r} μ_{s} + α μ_{s} - γ β - α + γ + 1)}

(25)

Optimal parameter for the damper

Having the closed form expression of the equivalent damping, it can be recalled that Inoue and Kuwahara (1998) have demonstrated the existence of optimal strength ratio $(γ)$ for response minimization of structure, but the stiffness ratio $(S_{r})$ does not have such value and the variation of response is almost monotonic with the varying stiffness ratios. The optimal strength ratio is expressed as a function of the ductility demand of the structure. However, Inoue and Kuwahara (1998) do not consider the strain-hardening aspect of the force–deformation behavior. The present formulation for equivalent damping, however, considers the strain-hardening aspect that closely follows the procedure by Kim and Choi (2004). Furthermore, the aspect of Superelastic force–deformation behavior is included by introducing a parameter $λ$ defining the force corresponding to Superelastic unloading, as shown in Figure 4(b). Once the damping is obtained from equation (25), the optimal value of $γ$ is found by maximizing damping in expression (25) as

\frac{\partial (ξ_{eq})}{\partial γ} = 0

(26)

Using equations (25) and (26), the optimal value of strength ratio $(γ)$ can be obtained as a function of the ductility demand and specific stiffness ratio of the structure as

\begin{matrix} γ_{opt} = \\ \frac{{S_{r} (β λ - β μ_{s}) + α (λ μ_{s} - λ - μ_{s} + α) + λ - 1 + \sqrt{χ}}}{(β λ - β - λ + 1)} \end{matrix}

(27)

where the parameters $χ$ , $ζ$ , and $ψ$ are obtained from the following relationship as

χ = (1 - λ) {S_{r} ψ + α ζ - 2 S_{r} β + 2 S_{r} - λ + 1}

(28a)

\begin{matrix} ζ = 2 α λ μ_{s} + 2 λ + 2 μ_{s} + α μ_{s}^{2} + α - α λ μ_{s}^{2} \\ - α λ - 2 μ_{s} - 2 λ μ_{s} - 1 \end{matrix}

(28b)

\begin{matrix} ψ = α β (λ μ_{s} + μ_{s}^{2} - λ μ_{s}^{2} - 3 μ_{s} + 2) \\ + μ_{s}^{2} (β S_{r} + α - β λ S_{r} - α λ) + \\ α λ μ_{s} + α μ_{s} + 3 β μ_{s} - β λ μ_{s} - λ μ_{s} - μ_{s} \end{matrix}

(28c)

Physical design of the Superelastic damper

The optimal value of the strength ratio $(γ)$ for given stiffness ratio $(S_{r})$ of the damper can be utilized to arrive at particular design details for the Superelastic damper. From the typical damper configurations discussed previously, it is clear that the number $(n)$ , diameter $(ϕ)$ , and the length $(l)$ of the SMA wires need to be found out in order to satisfy the criteria of strength and stiffness requirement. Therefore, these parameters are being correlated to the optimal strength ratio for a given stiffness ratio for the damper. The lateral stiffness of the damper is expressed as

K'_{d} = K_{d} \cos^{2} θ = (\frac{A_{d} E_{d}}{L_{d}}) \cos^{2} θ

(29)

where $θ$ is the inclination angle of the damper with the horizontal line. Furthermore, $K'_{d}$ can be expressed in terms of the stiffness ratio and the stiffness of the structure, which are known. Thus an equation in $A_{d}$ and $L_{d}$ can be obtained as

\frac{A_{d}}{L_{d}} = \frac{(S_{r} K_{s})}{(E_{d} \cos^{2} θ)}

(30)

Another equation is obtained from the optimal strength ratio as

γ = \frac{F_{dy}}{F_{sy}} = \frac{A_{d} σ_{dy}}{F_{sy}}

(31)

From these, the required area of the damper $A_{d}$ can be obtained. Once $A_{d}$ is known, the length of the SMA wires is obtained from equation (28). The required number of wires for available diameter can be decided as

A_{d} = n (\frac{π}{4}) ϕ^{2}

(32)

Numerical illustration on parametric variations and response of the damper assisted structures

The response behavior of the structure supplemented with Superelastic damper is demonstrated through numerical simulation. For this, parametric studies are presented first to demonstrate the variation of the equivalent viscous damping with respect to varying parameters for the structure–damper system. This is to show the existence of the optimal parameters for the damper and also to check the variation of the damping capability with respect to the varying range of important parameters of the dampers. As the aim of this study is to demonstrate the efficiency of FNCATB damper, a simple model of structure, consisting of a single-storey, single-bay frame is employed for numerical elucidation of the proposed optimization as well as response evaluation. The single-bay, single-storey frame is shown in Figure 6. The damper is attached diagonally between the bay and base of the structure. A schematic of the typical configuration of the damper has already been shown in Figure 3. The structure is assumed to be made of structural steel having bilinear elasto-plastic hysteresis with strain-hardening behavior as indicated in preceding sections. The property of the frame and damper material along with the physical parameters of the dampers are tabulated in Tables 1 to 3, respectively. All the parameters mentioned in Table 1 are defined except $T_{s}$ , which refers to the fundamental time period of the structure.

Table 1.

Properties of the structure.

Parameters	$K_{s}$	$T_{s}$	$F_{sy}$	$α$
Values	7069 kN/m	0.5 S	264.6 kN	0.0208

Table 2.

Properties of the SMAs.

Parameters	Nitinol	Cu-Al-Be	FNCATB
$E_{AS} (N / m^{2})$	2.068 × 10¹⁰	3.2 × 10¹⁰	5.0 × 10¹⁰
$σ_{s}^{AS} (N / m^{2})$	3.585 × 10⁸	1.5 × 10⁸	7.0 × 10⁸
$σ_{f}^{AS} (N / m^{2})$	4.137 × 10⁸	2.35 × 10⁸	10.0 × 10⁸
$σ_{s}^{SA} (N / m^{2})$	2.068 × 10⁸	1.3 × 10⁸	3.5 × 10⁸
$σ_{s}^{SA} (N / m^{2})$	1.379 × 10⁸	1.2 × 10⁸	1.0 × 10⁸
$ε_{L}$	0.07	0.05	0.12

SMAs: shape memory alloys.

Table 3.

Properties of the dampers.

Material	$λ$	$β$	$S_{r}$	$γ$	Diameter of wires (m)	Length of wires (m)	Volume of materials (cm³)
FNCATB	0.1428	0.05	3.5	0.5	0.0156	0.3819	0.7295
Nitinol	0.384	0.038	3.5	0.5	0.0216	0.3085	1.1298
Cu-Al-Be	0.8	0.053	3.5	0.5	0.0336	1.1408	10.1101

With these values as the default values of the parameters, the optimal strength ratio value of the damper $(γ)$ is obtained through parameter optimization. Furthermore, range of variations in the values of the parameters is also considered to study the effect of parameter variations. It is important to note that the FNCATB damper requires much lesser material compared with the Nitinol and Cu-Al-Be. This is obtained by multiplying the area and length of the required number of wires in the damper. Along with the potentially low cost of FNCATB due to its Ferrous component, lesser amount of the required material (in volume) will also contribute to its cost-effectiveness, as indicated in Table 3.

The variations of the equivalent damping under varying stiffness ratios of the dampers are shown in Figure 7(a) to (c) for the Nitinol, Cu-Al-Be, and FNCATB dampers respectively for a specified level of ductility (varying from 1 to 3) of the structure. It is observed that the equivalent damping initially increases with increasing stiffness ratio, but after a certain value of the stiffness ratio, damping reduces. With increasing level of ductility, the damping increases as increasing ductility results in enhanced displacement of the damper to adequately mobilize the dissipation capability. Comparing all three Superelastic dampers, the FNCATB offers highest amount of equivalent damping, which is then followed by the Nitinol, whereas the Cu-Al-Be damper has the lowest damping. It is interesting to note that the damping capability of the FNCATB remains largely insensitive to the variation in stiffness ratio, especially under higher ductility demand, indicating the higher degree of robustness of the FNCATB over the others.

The variations of the optimal value of the strength ratios with varying stiffness ratios are shown in Figure 8(a) to (c) for FNCATB, Nitinol, and Cu-Al-Be dampers respectively for different levels of structural ductility. With the properties of the Cu-Al-Be damper, it is not able to provide an optimal strength ratio for higher level of structural ductility, implying less robustness of such damper for controlling responses under high intensity motions. However, other Superelastic dampers provide specific value of the optimal strength ratios as shown in Figure 8(a) and (b). The optimal strength ratio for the FNCATB is observed to be higher than that for the respective Nitinol, especially for higher value of stiffness ratio and ductility.

With the adopted value of optimum strength ratio, the variations in equivalent damping are shown for different values of ductility demand and the stiffness ratios in Figure 9(a) to (c) for the FNCATB, Nitinol, and Cu-Al-Be dampers, respectively. Much superior damping capability of the FNCATB over the Nitinol and Cu-Al-Be is obvious from these plots. It is observed that the damping reduces significantly for combination of higher ductility and lower stiffness ratios for all the Superelastic dampers. However, for increasingly higher ductility and larger stiffness ratios, the damping in FNCATB is increasingly higher than the Nitinol, which shows the potential of the superior performance of the FNCATB over the others in extreme events (e.g. high intensity earthquake) imposing large ductility demand on the structures. It is observed that the damping capability of Cu-Al-Be is much inferior than that of the others, which deteriorates further for high ductility and hence this variant is not particularly suitable for damping application, although it is known to provide better re-centering property.

With the aforementioned studies on the variations of equivalent damping capability under parametric variations of the structure/damper properties, in next, the performances of the dampers are tested by subjecting them to real seismic excitations. The optimal performances of the structure–damper system are ensured through selection of optimal parameters presented earlier. Two recorded earthquake ground motions are selected for this demonstration, the details of which are provided in Table 4.

Table 4.

Characteristics of ground motions.

Ground motions	Earthquake	Year	Station	Magnitude	PGA (g)
GM1	Northridge	1994	CASTAIC OLD RIDGE RT	6.69	0.568
GM2	Loma Prieta	1989	GILROY #3 SEWAGE PLANT	6.93	0.367

PGA: peak ground acceleration.

The time history of displacement at the top of the frame under two ground motions is shown in Figure 10(a) and (b) for GM1 and GM2, respectively. The augmentation by damper is shown to reduce the peak as well as the root mean square (RMS) displacement substantially. More precise estimate of these reductions can be obtained from Tables 5 and 6. Table 5 shows the response quantities of interest, whereas Table 6 compares the efficiencies with respect to the reduction of various responses. It is observed that the reduction in the peak displacement achieved by the FNCATB damper from the unassisted frame is around 56% which is 53% for the Nitinol and 44% for the Al-Cu-Be. For the RMS, these values are 76%, 74%, and 50% respectively for FNCATB, Nitinol, and Cu-Al-Be. These figures clearly indicate the improved performance of the FNCATB damper while compared with its alternatives.

Table 5.

Comparison among the peak, RMS displacement, and base shear.

Responses	Input motions	Alternative structure/Superelastic damper
		Responses
		Without damper	FNCATB	Nitinol	Cu-Al-Be
Peak displacement (m)	GM1	0.078	0.0364	0.0374	0.0548
	GM2	0.0621	0.0268	0.0290	0.0346
RMS disp. (m)	GM1	0.0159	0.0045	0.0052	0.0081
	GM2	0.0127	0.0031	0.0032	0.0042
Peak base shear (kN)	GM1	270.94	314.76	323.42	393.43
	GM2	247.25	264.88	270.85	304.37
RMS base shear (kN)	GM1	62.17	63.21	67.58	88.24
	GM2	56.45	54.86	54.76	64.16

RMS: root mean square.

Table 6.

Comparison of efficiencies with respect to the peak and RMS displacement and base shear.

Responses	Input motions	Alternative structure/Superelastic damper
		Efficiency
		FNCATB	Nitinol	Cu-Al-Be
Peak displacement (m)	GM1	53.3	52.0	29.7
	GM2	56.8	53.3	44.28
RMS displacement (m)	GM1	71.7	67.3	49.05
	GM2	75.6	74.8	66.9
Peak base shear (kN)	GM1	−16.2	−19.4	−45.2
	GM2	−7.13	−9.54	−23.1
RMS base shear (kN)	GM1	−1.66	−8.7	−41.9
	GM2	2.82	3.00	−13.65

RMS: root mean square.

An important point to note is that whereas the structure without damper suffers residual displacement while subjected to GM1 (Northridge) motion, this is completely eliminated with the addition of damper. This is owing to the Superelastic property of the damper which helps in re-centering.

When assisted by damper, a structure is expected to be stiffer due to added stiffness. This added stiffness might result into attraction of more inertial forces to the structure and can potentially reduce the efficiency of the damper. Thus, it is imperative to check for this aspect to ensure that added damper stiffness should not adversely affect the structural performance. This is ensured by checking the base shear, as a measure of inertial force transmission to the structure. If the base shear from the damper assisted and original structure is of comparable magnitude, then it is ensured that the stiffness of the damper does not significantly affect the force transmission to the structure. Keeping this in view, a comparison among the base shear time histories is presented for the damper augmented structure and the original structure in Figure 11(a) and (b) for GM1 and GM2, respectively. It can be observed that the base shear is not significantly modified by the addition of damper and the values from both systems are of comparable magnitude. Thus, the addition of Superelastic damper does not compromise with the stiffness of the structure. It is also noted that the base shear experienced by the FNCATB damper is lesser than the other two (Nitinol and Cu-Al-Be), further adding to its enhanced efficiency. More precise estimates of the peak and RMS values of base shear are shown in Tables 5 and 6. Comparison of the numerical figures confirms the negligible increase (around 3%−7%) of base shear due to the addition of damper. Furthermore, comparison among the Superelastic dampers reveals the superior performance of the FNCATB over the others, in terms of reduction of peak as well as RMS values of the base shear.

In next, comparison among the force–deformation hysteresis are presented for the damper assisted structure and the unassisted structure in Figure 12(a) and (b) for the two different ground motions GM1 and GM2, respectively. This is because the area of the force–deformation loop provides an idea of the amount of energy dissipation by the dampers and thus provides an estimate of the relative efficiency of one damper over the other in terms of channelizing the input energy to the damper rather than the structure. The general trends observed in both Figure 12(a) and (b) show that the addition of the damper reduces the peak displacement response of the structure–damper system significantly from that of unassisted structure. It can also be observed that the reduction in the drift of the structure by damper is such that the structure may even remain in linear regime of its behavior and the whole inelastic dissipation can be provided by the damper itself. Furthermore, comparison among the dampers reveals the lowest drift of the structure when assisted by the FNCATB damper. The slight increase in base shear in the damper assisted structure due to additional damper stiffness is also observed from these plots. Also important to note is the elimination of any residual displacements in the damper augmented structure, which remain otherwise in the unassisted structure, which can be attributed to the Superelastic force–deformation behavior of the dampers. Comparison among the area of the loops shows that the dissipation capability of the FNCATB and Nitinol are somehow comparable, whereas that of the Cu-Al-Be is much lower.

Conclusion

The superior performance of the Ferrous based SMA (FNCATB) is presented over the alternative Superelastic dampers, such as Nitinol and Cu-Al-Be SMA, in mitigating the seismic vibration of structure. The superior performance of the FNCATB damper is demonstrated in terms of the enhanced reduction in storey drift accompanied by almost identical level of base shear experienced by the structure. The dynamic response analysis of the structure–damper system, presented herein, is based on the Superelastic force–deformation hysteresis of the damper, which is obtained by fitting the available experimental data in the literature with the well known Auricchio model for SMA. A simplified closed form expression for the optimal design parameter for the damper is presented and employed to ensure optimal performance of the dampers to facilitate comparison. The optimization is based on maximization of the equivalent damping, a closed form expression of which is derived by considering the force–deformation behavior of the structure–damper system. The formulation closely follows the procedure applied to the conventional yield damper, except that a new parameter $(λ)$ is introduced to take care of the Superelastic nature. The expression of equivalent damping and respective optimal parameter for the damper can be useful to establish an initial design of such system. The FNCATB damper also has potential for being a much economical alternative (in addition to its superior performance as a damper) because of its Ferrous constituent and much lesser volume requirement, as demonstrated in this study.

Footnotes

Acknowledgements

The authors would like to sincerely thank and highly acknowledge the anonymous reviewer for his or her meticulous effort that has greatly helped to improve the quality of the article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Adachi

Unjoh

(1999) Development of shape memory alloy damper for intelligent bridge systems. Proceedings of SPIE: Smart Structures and Materials 3671: 31–42.

Andrawes

DesRoches

(2007) Effect of hysteretic properties of superelastic shape memory alloys on the seismic performance of structures. Structural Control & Health Monitoring 14(2): 301–320.

Auricchio

Sacco

(1997) A one-dimensional model for superelastic shape-memory alloys with different elastic properties between austenite and martensite. International Journal of Non-Linear Mechanics 32(6): 1101–1114.

Auricchio

Fugazza

DesRoches

(2006) Numerical and experimental evaluation of the damping properties of shape-memory alloys. Journal of Engineering Materials and Technology: Transactions of the ASME 128(3): 312–319.

Auricchio

Taylor

Lubliner

(1997) Shape-memory alloys: macromodelling and numerical simulations of the superelastic behavior. Computer Methods in Applied Mechanics and Engineering 146(3): 281–312.

Casciati

Faravelli

(2004) Experimental characterisation of a Cu-based shape memory alloy toward its exploitation in passive control devices. Journal de Physique IV 115: 299–306.

Casciati

Faravelli

Al Saleh

(2009) An SMA passive device proposed within the highway bridge benchmark. Structural Control & Health Monitoring 16(6): 657–667.

Cerda

Boroschek

Farias

. (2006) Shaking table test of a reduced-scale structure with copper-based SMA energy dissipation devices. In: Proceedings of 8th US national conference on earthquake engineering, San Francisco, CA, 18–22 April.

Chopra

(1995) Dynamics of Structures. Upper Saddle River, NJ: Prentice Hall.

10.

Clark

Aiken

Kelly

. (1995) Experimental and analytical studies of shape-memory alloy dampers for structural control. Smart Structures and Materials 95: 241–251.

11.

Cook

(2007) Concepts and Applications of Finite Element Analysis. New York: John Wiley & Sons.

12.

DesRoches

McCormick

Delemont

(2004) Cyclic properties of superelastic shape memory alloy wires and bars. Journal of Structural Engineering: ASCE 130(1): 38–46.

13.

Dolce

Cardone

(2001) Mechanical behavior of shape memory alloys for seismic applications 2. Austenite NiTi wires subjected to tension. International Journal of Mechanical Sciences 43(11): 2657–2677.

14.

Dolce

Cardone

Marnetto

(2000) Implementation and testing of passive control devices based on shape memory alloys. Earthquake Engineering & Structural Dynamics 29(7): 945–968.

15.

Dolce

Cardone

Ponzo

. (2005) Shaking table tests on reinforced concrete frames without and with passive control systems. Earthquake Engineering & Structural Dynamics 34(14): 1687–1717.

16.

Garamendi

(2004) Press release on eve of Northridge earthquake’s 10th anniversary, cautions of huge risk as more. In: Proceedings of the Californians decline earthquake insurance, California Department of Insurance, Sacramento, CA, 16 January.

17.

Graesser

Cozzarelli

(1991) Shape-memory alloys as new materials for aseismic isolation. Journal of Engineering Mechanics: ASCE 117(11): 2590–2608.

18.

Han

. (2003) Structural vibration control by shape memory alloy damper. Earthquake Engineering & Structural Dynamics 32(3): 483–494.

19.

Inoue

Kuwahara

(1998) Optimum strength ratio of hysteretic damper. Earthquake Engineering & Structural Dynamics 27(6): 577–588.

20.

Kim

Choi

(2004) Behavior and design of structures with buckling-restrained braces. Engineering Structures 26(6): 693–706.

21.

Krumme

Hayes

Sweeney

(1995) Structural damping with shape-memory alloys: one class of devices. Proceedings of the SPIE: Smart Structures and Materials 2445: 225–240.

22.

Kustov

Van Humbeeck

(2008) Damping properties of SMA. Materials Science Forum 583: 85–109.

23.

Miranda

Bertero

(1994) Evaluation of strength reduction factors for earthquake-resistant design. Earthquake Spectra 10(2): 357–379.

24.

Omori

Ando

Okano

. (2011) Superelastic effect in polycrystalline ferrous alloys. Science 333(6038): 68–71.

25.

Ozbulut

Hurlebaus

DesRoches

(2011) Seismic response control using shape memory alloys: a review. Journal of Intelligent Material Systems and Structures 22: 1531–1549.

26.

Parulekar

Reddy

Vaze

. (2012) Seismic response attenuation of structures using shape memory alloy dampers. Structural Control & Health Monitoring 19(1): 102–119.

27.

Symans

Charney

Whittaker

. (2008) Energy dissipation systems for seismic applications: current practice and recent developments. Journal of Structural Engineering 134(1): 3–21.

28.

Tanaka

Himuro

Kainuma

. (2010) Ferrous polycrystalline shape-memory alloy showing huge superelasticity. Science 327(5972): 1488–1490.

29.

Thomson

Balas

Leo

(1995) The use of shape memory alloys for passive structural damping. Smart Materials and Structures 4(1): 36–41.

30.

Van Humbeeck

Kustov

(2005) Active and passive damping of noise and vibrations through shape memory alloys: applications and mechanisms. Smart Materials and Structures 14(5): 171.

31.

Wilde

Gardoni

Fujino

(2000) Base isolation system with shape memory alloy device for elevated highway bridges. Engineering Structures 22(3): 222–229.

32.

Yan

Nie

(2003) Study of a new application form of shape memory alloy superelasticity. Smart Materials and Structures 12(6): N14.

33.

Yang

CSW

DesRoches

Leon

(2010) Design and analysis of braced frames with shape memory alloy and energy-absorbing hybrid devices. Engineering Structures 32(2): 498–507.

34.

Zhang

Zhu

(2009) Seismic performance of benchmark base-isolated bridges with superelastic Cu–Al–Be restraining damping device. Structural Control & Health Monitoring 16(6): 668–685.