1. Introduction
Using seismic isolators has been found an effective way to protect low- to middle- rise structures under earthquakes in past investigations and practices [
1]. According to the principle of structural dynamics, seismic isolators are required to have much smaller lateral stiffness than that of the protected superstructure. Although adding stiffness or damping at the isolation level is not suggested from the viewpoint of protecting the superstructure [
2], small stiffness and damping may result in excessive deformation at the isolation level toward the earthquake attack direction. In the practical applications, the allowable deformation space for isolation system is usually limited by the adjacent structures or foundations. In fact, as reported in the Northridge earthquake [
3], the limited isolation gap generated suddenly increased shear force and interstory drift in the superstructure as a result of the pounding impact. Therefore, it is advisable to achieve a balance between controlling deformation for the isolators and alleviating the counter effect of increasing seismic demand in the superstructure.
In this context, various damping devices have been proposed to assist the isolators to deform in allowable space by absorbing the input seismic energy [
4,
5]. Representatives primarily include viscous damper [
6], friction damper [
7] and lead damper [
8]. But these dampers are found tend to shift superstructures away from the initial position. Therefore, besides with controlling peak deformation demand, the residual deformation of the isolation level should be minimized as well, with the aim to restore the structure to its at-rest position. To fulfill this requirement, people made encouraging attempts in developing advanced isolators, such as the friction-pendulum base isolator [
9] and conical spring isolator [
10]. Recently, to address the issue, the research community resorted to incorporating shape memory alloys (SMAs) into the isolation systems. SMA is a class of metal materials, which are able to exhibit excellent superelasticity when the environmental temperature is above the phase transformation threshold [
11]. This feature is attributed to the solid-to-solid transformation between two crystallographic phases, namely, austenite and martensite [
12,
13]. Phase transformation can be activated by varying ambient temperature or changing stress state. The former is referred to shape memory effect and the latter is known as superelastic effect. In seismic applications, the superelastic effect is of particular interest. As can be seen in
Figure 1, the loading and unloading behaviors of superelastic SMAs are triggered by recoverable phase transformation of crystalline, forming a flag-shape hysteresis.
The studies of using SMA devices in the base isolation system of buildings are getting increasing attentions. For example, Ponzo et al. [
14] compared three different isolation systems through shaking table tests and found that the SMA damper is better than the counterparts. Gur et al. [
15] focused on near-fault seismic performance and indicated the enhancement of the isolation efficiency provided by SMA was outstanding in suppressing the acceleration and displacement demands. Ozbulut and Silwal [
16] used multi-objective genetic algorithm to optimize the design parameters for SMA damping isolator. Qiu and Tian [
17] considered the strain hardening behavior of the SMA damper in particular and highlighted the advantage of this behavior in controlling deformation demand. A proof-of-concept shake table test for shear frames isolated with SMA springs was carried out by Huang et al. [
18]. Attanasi et al. [
19] investigated the behavior of a base isolated building using conventional lead rubber bearing or SMA devices. Although the efficacy of SMA device was shown, the superstructure was not explicitly modeled in the seismic analysis, which thus ignored the seismic behavior of the superstructures. With the noticeable achievement made so far, it is still worth noting that past studies are often based on the assumption that the superstructure is always in its elastic stage. However, this assumption is actually over idealized, because the superstructure is likely to endure nonlinear deformation under moderate or large earthquakes.
In this study, SMA spring damper is proposed to be implemented in the isolation system for frame structures. Compared with prior studies [
17,
18,
19], the building nonlinearity was well modeled and discussed in the current analysis. According to the ductile design philosophy, structures are usually permitted to deform into inelastic state by moderate and severe earthquakes. Therefore, with considering the nonlinear behavior of the superstructure, the seismic performance of the structures isolated with SMA springs can be more realistically revealed. The advantages of SMA spring dampers include the concise form, stand-alone property, the extraordinary deformation capability through transforming axial deformation into shear deformation, the versatile performance by tuning the design parameters to achieve desired properties. Cyclic loading tests were carried out to assess the seismic properties of the SMA spring. Then, finite element analyses were conducted to quantify the effect of varying geometrical dimensions on the mechanical properties of SMA springs. Finally, a prototype nonlinear multi-story frame is selected in the seismic analyses, with the aim of further revealing the feasibility of using SMA springs in the isolation system of frame buildings.
3. Finite Element Simulation in ABAQUS
This section conducted numerical investigation on the cyclic behavior of SMA springs in the general nonlinear finite element (FE) analysis program ABAQUS (Dassault Systèmes SIMULIA Corp., Providence, RI, USA). The purpose is to analyze the development of stress and strain in SMA spring under seismic loading scheme. A high-fidelity three-dimensional FE model will be built to further explore the cyclic behavior of the test specimen. The FE model was firstly built and calibrated by the test results. Then parametric analyses were carried out on wire diameter and spring diameter, with the purpose of demonstrating the versatile hysteretic properties of SMA springs through varying the geometrical dimensions. The corresponding outcome helps to shed light on the seismic design of SMA springs. This section is to demonstrate the versatile hysteretic properties of SMA springs through varying the geometrical dimensions. In practice, to scale up the strength of isolators, there are many ways including using dozens of springs in parallel, utilizing SMA bars with large diameter and change the geometrical dimension of the SMA springs.
3.1. Model Calibration
As can be seen in
Figure 8, 3D FE model was built for the SMA spring. To improve the accuracy of the numerical simulation, the SMA wire was used the eight-node solid element and hourglass control (C3D8R). Considering that the meshing of component has a great influence on the efficiency and accuracy of numerical calculations, the final mesh of the spring was finely and regularly divided as shown in
Figure 8. The meshing size is 0.35 mm, making at least 5 layers of elements throughout the thickness of the spring, which is adequate for capturing the complex stress distribution within the SMA spring. The loading scheme and boundary conditions of the FE model were as the same as that adopted in the test. To simulate the loading process, the spring is fixed at one end and the other end is loaded with an axially cyclic displacement load.
3.2. Material Properties
To simulate the behavior of SMAs, the superelastic material model implemented in ABAQUS was used. The stress-strain relationship of the NiTi SMA at room temperature was defined according to the parameters listed in the
Table 1. The used material parameters stem from experimental data. E
A is the austenite elastic modulus; E
M is the martensitic elastic modulus; σ
Ms and σ
Mf are forward transformation stress featuring the start and complete of austenite to martensite phase transformation; σ
As and σ
Af are reverse transformation stress featuring the start and complete of martensite to austenite phase transformation; ε
t is the maximum transformation strain.
3.3. Simulation Results
Figure 9 compares the force-displacement relationships of the SMA spring between the experimental data and FE simulated results. It can be seen that the FE model successfully reproduced the cyclic behaviors of the spring in every single loading cycle. The “yield” force and elastic stiffness are measured as approximately F
y = 65.6 N and K = 2.01 N/mm, the “post-yield” stiffness coefficient α and energy dissipation coefficient β are quantified to be 0.28 and 0.5, respectively, both of which agree with the test results very well, indicating that the established FE model captures the critical hysteretic parameters. Compared with test results, the FE model generate a relatively sharp phase-transformation process, which is primarily due to the simplification of the built-in SMA material model.
Figure 10 shows the Mises stress state of the SMA spring when it was tensioned to a displacement of 140 mm. As can be seen from
Figure 10a, the spring is uniformly stretched, and the tensile stress is evenly distributed throughout the body. To observe the internal stress of the specimen,
Figure 10b presents the cross-sectional stress distribution. It is found that the maximum stress occurred at the edge of the cross section, reaching to approximately 688 MPa, which is larger than the forward phase transformation stress, indicating the local area has been partially deformed into martensite phase; while the stress level at the core of the cross section is approximately 500 MPa, which is smaller than the forward phase transformation stress, implying the core is still in austenite phase. The stress distribution is mainly caused by torsion and bending actions at the section.
3.4. Parametric Analysis
Based on the calibrated FE model established in above section, this section further conducts parametric analysis to examine the effect of varying the geometrical dimension on the cyclic behavior of SMA springs. The considered geometrical dimensions include the wire diameter and the spring diameter, as listed in
Table 2. The parametric analysis also aims to demonstrate the versatile feature of SMA springs. To quantify the effect of varying the selected parameters, the interested hysteresis indices include the peak strength, tangent stiffness, equivalent damping ratio and residual displacement corresponding to each loading cycle.
Figure 11 assembles the hysteresis loops of SMA springs with different geometrical dimensions. The hysteresis of S1 is repeated here for reference purpose. To ensure a fair and direct comparison between these springs, the applied loading history is identical to that used in the experimental tests. Overall, it can be seen that the SMA springs always attain typical flag-shape hysteresis within the applied loading amplitude, although the wire diameter and spring diameter are severely changed. Specifically, when the wire diameters of S2 and S3 were respectively increased to 3 and 4 mm and the “yield” strength of the springs remarkably reach to 337 and 822 N, which are approximately 5 times and 12 times that of the peak force of S1. Compared with S1, the spring diameter of S4 and S5 was respectively increased to 18 and 20 mm and it is seen that the peak force of the spring was reduced to 61 and 59 N, which are approximately 93% and 89% of the peak force of S1. Therefore, it shows that the strength demand of the SMA spring can be conveniently tuned by varying the wire diameter or the spring diameter.
The initial stiffness of the SMA spring is estimated as well, since this index directly determines the fundamental period of the isolated structures. According to Liang and Rogers [
26], the elastic stiffness of a spring can be expressed by the following equation:
where
G is the shear modulus of SMA material,
d is the wire diameter,
D is the spring diameter, N is the number of active coils and
C is the spring index, defined as
C =
D/
d. It is seen that the initial stiffness of the SMA spring is proportional to
d4 and inversely proportional to
D3. The calculated stiffness of S1–S5 is listed in
Table 3. Comparisons are made between the theoretical and numerical results, showing a maximum error of 6.9%. This validates the FE model again. Compared with S1, the stiffness of S2–S5 is 5.3, 16.8, 0.7 and 0.49 times that of S1, respectively, indicating the initial stiffness performs with a similar trend as the elastic strength capacity.
The peak strength, tangent stiffness, equivalent damping ratio and residual deformation as a function of the applied displacements are shown in
Figure 12.
Figure 12a is about the peak force generated in each cycle. S3 has noticeably higher strength capacity than the others, primarily because of the large wire diameter directly increases lateral stiffness, which can be estimated by Equation (1).
Figure 12b plots the tangent stiffness for each cycle. Again, S3 with a larger wire diameter exhibits higher stiffness.
Figure 12c shows the variation of the equivalent damping ratio, it can be seen that the equivalent damping ratio increases with the displacement increment. The largest equivalent damping ratio is attained by S3 with a value of 3.6%; and the smallest value is found in S5.
Figure 12d shows the residual deformation. Although the residual deformation tends to increase with the loading amplitude, the overall residual deformation is essentially very small, compared with the applied displacement. Among them, due to the strain hardening caused by the martensite phase, the largest residual deformation is generated by S3 with only 0.425 mm, which is low to 0.3% of the loading amplitude. Through the parameter analysis, it can be seen that adjusting wire diameter and spring diameter of SMA spring can effectively change the critical seismic performance indices of the damper and thus it allows the designers to properly tune the parameters to satisfy the requirements of seismic design.