Stochastic seismic response of building with shape memory alloy damper

The present study focuses on the estimation of response of building structure supplemented with the superelastic shape memory alloy (SMA) damper under the stochastic seismic excitation. To this end, the stochastic response has been determined using the stochastic linearization method under random earthquakes and the control efficiency of the SMA damper is compared with the yield damper. As the response output, top floor root mean square (rms) acceleration and displacement are presented here in this study. Response analysis results show the presence of optimal values of damper strength which minimizes the responses. As observed, in this optimal condition of damper strength, compared to the yield damper, the SMA damper provided 17 % and 49 % improvements in the values of top floor rms acceleration and displacement. In the end, the parametric study has been performed under the varying strength and stiffness of the damper, time period of the structure, and earthquake excitation to establish the superior control efficiency of SMA damper over yield damper.


Introduction
The traditional way of structural response reduction was focused on improving its inherent damping capacity. However, during strong earthquakes, structural elements fail to dissipate the heavy energy through inherent damping. Thus, external response control systems are used. Various studies describe the advantages of dampers in dissipating a large amount of imparted seismic energy [1,2]. The effectiveness of metallic dampers as a vibration control system is established in past [2][3][4][5]. A previous study [2] gives a comprehensive review of various advantages of metallic dampers such as low manufacturing costs, stable hysteresis behavior, resistance to ambient temperature, reliability, and high energy dissipation capability, etc. To get the optimal utilization of damper with minimum damage of structure, optimization of damper configuration is studied in past [4]. Chan and Albermani demonstrated the energy dissipation capacity and effectiveness of using steel slit damper and vertical steel slit dampers [3]. Another study on steel plate slit dampers [5] showed the good capability of inter-story drift reduction and thereby provides satisfactory results in seismic retrofitting. However, these devices leave some permanent plastic deformation after strong earthquake motion. Thus after the disaster, they lead to replacement and retrofitting of dampers which ultimately lead to high cost of the system.
To overcome this difficulty, a novel smart material called shape memory alloy (SMA) is recommended by other literature [6,7]. Superelastic characteristics of SMA material can be of great interest for engineering applications [8]. The hysteretic behavior of NiTi SMA (a metal alloy of nickel and titanium, also known as Nitinol, which exhibits shape memory effect and superelasticity at a different temperature) has been studied to demonstrate its effectiveness in the response reduction capacity [9]. The effect of strain rate and temperature on the hysteresis behavior, energy dissipation capacity, and response control efficiency of SMA damper is also studied in past [10]. Humbeeck and Liu [11] have analyzed the active control behavior of NiTi alloy wires which control the internal energy of the structure. Kari et al [12] found that even at the very high intensity of ground motion, SMA passive damper system does not require replacement due to its high fatigue resistance characteristics. Duval et al [13] studied the feasibility of an SMA spring system in a SDOF system under random excitation. Optimal design of superelastic SMA damper had been conducted in past to minimize the seismic response of building under random earthquake [14]. Another study on SMA wire dampers is found to be very effective and potential to withstand numerous earthquake motions, without any repair [15]. Gur et al. [16] showed the superior seismic vulnerability reduction capability of SMA damper over yield damper.
Most of the above works of literature deal with specific recorded ground motion. Only a few studies considered the stochastic nature of ground motion and its effect on response statistics [14]. But, due to large variability in ground motion, it is cumbersome to compare the effectiveness of yield and SMA damper system on a certain definite range of ground excitation. Thus, it is highly important to study the performance and response control efficiency of the dampers under stochastic ground motion. In this study, the random vibration method is incorporated to account for the stochasticity of the ground motion. Effect of the various damper, structural, and ground motion parameters are adopted and the control of both yield and SMA damper devices are examined.

Numerical modeling of dampers
The force-deformation behavior of the yield damper and superelastic SMA dampers has been shown in Fig. 1 (a). Subsequently, Fig. 1 (b) shows a simple mechanical model of yield or SMA damper, which are connected between two consecutive floors via bracings. More details are provided in the subsequent section.

Force-deformation modelling of yield damper
The behavior of yield damper under the cyclic loading shows a bi-linear hysteresis loop and modelled through the parametric Bouc-Wen model [17]. These hysteresis loops dissipate a significant part of input energy and successfully simulates the non-linear force-deformation characteristics of steel material. The force-deformation response in the steel member is expressed as [14] (2) Where, z F is the restoring force, z k represents initial elastic stiffness, z  is the post to pre yield stiffness ratio (i.e., rigidity ratio), yz F and z q are the yield strength and displacement, x and x  are the relative displacement and velocity, and variable Z is the non-dimensional parameter represents the hysteretic behavior of metallic material.
Parameters  ,  ,  , and  controls the shape and transition (elastic to plastic) of the hysteresis loop.
The nonlinear hysteresis behavior in Eq. 3 of the yield damper is very complex. Thus cannot readily be incorporated in the state-space formulation to determine the stochastic response. This equation can be linearized through stochastic linearization as [14]     0 are the stochastically equivalent damping and stiffness terms. They are obtained via least-square minimization of the expectation of error among the nonlinear (Eq. 3) and linear terms (Eq. 4), with the assumption of 1   , and x   , z  , and z x   resents the root mean square (rms) value of velocity, hysteresis coordinate, and both.

Force-deformation modelling of SMA damper
The SMAs are a special class of materials that can recover several percentages of strain upon heating beyond a certain temperature. This is achieved through phase transformation in the microstructure of the alloy. However, another property of SMA i.e. superelasticity is the interest for damper design in vibration control applications. SMA shows the flag shape hysteresis loop with no residual displacement after unloading. Yan and Nie [18] proposed an equivalent model to use in stochastic linearization. As per the previous studies, the 1-D force-deformation behavior of SMA is expressed as [14]  where, s F is the restoring force developed in SMA, s k is the initial stiffness of SMA in austenite phase, s  is the ratio of pre to post transformation stiffness, a and b are the maximum displacement in austenite phase and displacement triggering forward phase transition in SMA, respectively, x and x  are the relative displacement and the variable s Z is the non-dimensional parameter represents the hysteretic behavior of SMA material.
To incorporate the complex nonlinear force-deformation hysteresis loop in the state space formulation it is important to linearize them at first. Thus, the stochastic linearization method is applied to minimize the expected value of residual error between the equivalent linear and nonlinear terms in Eq. 5. As per a previous study [14], the linearized equation of the hysteresis deformation and force in SMA material is expressed as represents the stochastically equivalent damping and stiffness terms. The root mean square (rms) value of velocity and displacement are expressed as x   and x  , respectively.

Stochastic response analysis
In the present study, it has been assumed that the yield or SMA dampers will dissipate a significant part of seismic energy through the nonlinear force-deformation hysteresis loop and thus the structure will behave as the linear elastic system. Fig. 2 (a) and (b) shows the typical model of a seven-story building with the yield or SMA damper, and a typical view of the installation of damper (yield or SMA) between two consecutive floors via bracings, respectively. Where, x i , m i , k i, and c i denote displacement, mass, stiffness, and damping constant of the i th floor, respectively and i is ranging from 1 to 7.

Stochastic model of earthquake
In the present study, a stochastic model of ground motion, namely the Kanai-Tajimi model has been adopted for the simulation process [19]. This Kanai-Tajimi model is well known to simulate the stochastic stationary process of ground motion. The power spectral density function (PSDF) of the Kanai-Tajimi model is defined by [14]   (8) here, f  and f  are characteristic frequency and damping of the soil strata, and is the spectral intensity of white noise from bedrock motion 0 S . The filter frequency is normalized by the structure fundamental vibration mode frequency and expressed as normalized filter frequency i.e. s f     . Therefore, assuming the Kanai-Tajimi PSDF model for ground motion and as a stationary process, the filter equations are expressed as  (10) where, f x , f x  , and f x  are displacement, velocity, and acceleration of ground, and w   is the intensity of white noise at the bedrock with power spectral intensity 0 S .

Stochastic dynamic analysis of the structure
A multi-degree elastic shear building structure is considered in the present study as assumed in other previous studies. Thus, using the above description of stochastic ground motion (i.e. Eq. 10) and then rearranging different terms, the equation of motion of the building with yield damper and SMA damper can be written as (13) where,       The above equation of motion can be incorporated in the state space formulation and assuming the Markovian process, the covariance matrix of its response and its derivative process can be estimated as [20] (16) where   A is the augmented system matrix,   Y is the state vector define as for the and for the SMA  has all terms zero except the last diagonal as 2π 0 S . From Eq. 15 and 16, response parameters, i.e. rms value of top floor displacement and top floor acceleration, are calculated. More details about the modelling and analysis can be found in another study [14].

Results and discussion
This section provides the simulation results. Different parameters are given a wide range of variations for parametric study. At any instant, only a single parameter has been varied, but other parameters are kept constant to their default values. These default values of the parameters are taken from a previous study [14]. Detail discussion is provided subsequently. Fig. 3 (a1) shows that, with the increase in normalized strength, top floor acceleration first decreases to an optimum point and then increases, for both of the dampers. This is because above optimum point, increase in damper's strength increase the rigidity of the structure, which in turn leads to stiffening of the building, resulting in increased acceleration. As can be observed, the requirement of optimal strength for SMA damper is slightly more than yield damper, but provide much better control efficiency. With increasing stiffness ratio, the structure gets stiffened resulting in increased acceleration, as shown in Fig. 3 (b1), top floor rms acceleration shows a monotonic increasing trend with the increase in the stiffness ratio, for both yield and SMA the dampers. Next, with the increasing time period of building, variation of top floor rms acceleration of yield and SMA damper connected building is shown in Fig. 3 (c1). They both show a similar trend of decreasing acceleration with the increase in the time period of the structure, as the structure is getting more flexible with the increasing time period. The trend of top floor rms displacement variation with the normalized strength and stiffness ratio of both the dampers (yield and SMA) are shown in Fig. 3 (a2) and (b2), respectively. It can be observed, with increasing damper strength, initially rms displacement decreases at a faster rate, and then it reaches some constant values. Similarly, with increasing damper stiffness ratio, rms displacement decreases at a faster rate initially and then reaches some constant values. This can be explained by the fact that the increase in both the normalized strength and the stiffness ratio make the structure stiffer, resulting in a decrease in displacement. Next, the effect of the building time period on top floor rms displacement is shown in Fig. 3 (c2). With the increase in the time period of the building structure, the structure becomes more flexible, resulting in increasing top floor displacement. It can be seen that, for all these six results, acceleration and displacement in the case of SMA damper are lesser than the yield damper, which is due to the better seismic energy dissipation ability of SMA damper. Compared to yield damper, SMA damper reduces top floor rms acceleration and displacement by a maximum of 17 % and 45 %.
Next, the effect of different ground motion parameters on the responses is discussed. Top floor rms acceleration, as well as displacement, increases with the increase in seismic intensity, for both the dampers, as shown in Fig. 4 (a1) and (a2), respectively. With the increase in normalized ground frequency, top floor rms acceleration increases, and after a certain point, the rate of increase decreases for both SMA and yield dampers, as depicted in Fig. 3 (b1). With the increase in normalized ground frequency, top floor rms displacement initially increases for both the dampers and then for the yield damper it decreases till λ = 7.00 and becomes almost constant thereafter, whereas, for SMA damper it becomes almost constant after λ = 6.00. From these four results, it is observed that compared to yield damper, SMA damper provides much more reduction in structure's displacement. Also, the SMA damper provides superior acceleration control efficiency than the yield damper. Compared to the yield damper the SMA damper reduces the top floor rms acceleration and displacement by a maximum of 16 % and 49 % respectively. Fig. 4. Variation of top floor rms acceleration ((a1)-(b1)) and displacement ((a2)-(b2)) with respect to seismic intensity and ground (or filter) frequency.

Conclusion
The present study focuses on the stochastic seismic response analysis of building installed with SMA damper and compares the control efficiency of SMA damper with yield damper. As the response output rms values of top floor acceleration and displacement are estimated. To compare the control efficiency of yield and SMA damper, a wide range of parameters such as damper strength, stiffness ratio of damper, time period of building structure, ground motion intensity, and filter frequency are considered. Both the damper shows the presence of optimal damper strength which minimizes the rms top floor acceleration and displacement of the building. Parametric study results show that the SMA damper slightly improves the acceleration control efficiency, but largely enhances the displacement control efficiency than the yield damper. In contrast to the yield damper, the SMA damper provided 17 % and 49 % more reduction in the top floor rms acceleration and rms displacement, respectively. Thus, taken together all the aforementioned results it can be stated that the superelastic SMA damper provides superior control efficiency than the yield damper, and therefore behaves much better seismic control device for building structure.