Test and development of SMA-based negative stiffness isolation device

Abstract

Although shape memory alloy (SMA) wires/cables can improve the re-centering capacity of traditional isolation devices to prevent residual displacement, pounding, or falling of girders, they also raise the forces in the substructure since the stiffness of the device is increased. In this paper, quasi-static lateral shear tests of an SMA-based negative stiffness isolation device were conducted at first to reduce the increased forces of the substructure. The mechanical properties and the failure modes of the SMA-based negative stiffness isolation device are studied. Next, the SMA-based negative stiffness isolation device is updated based on its failure modes. Finally, the mechanical properties of the improved device were numerically studied. The results show that the new SMA-based negative stiffness isolation device can partially reduce the force responses of the substructure in addition to keeping excellent re-centering capability. Moreover, it can also limit the excessive displacement of bridges to prevent the girders from falling.

Keywords

Negative stiffness isolation device tests bridges SMA cable failure modes

Introduction

Recently, a lot of bridges were damaged by strong earthquakes, which paralyzed the transportation system and indirectly enlarged the economic losses and the number of casualties. Residual displacement, collision, and falling of girders are the most common damages of bridges. They are mainly caused by the excessive relative displacements responses between girders and piers. It shows that the re-centering capability and displacement-limiting capability of traditional isolation systems are insufficient (Guo et al., 2010; Du and Han, 2014; Wang, 2015).

Having the characteristics of deformation recovery, superelastic Shape memory alloy (SMA) wires, cables, and bars have been widely applied to the isolation devices to improve the recentering capability of bridges (Ozbulut et al., 2011; Peng et al., 2021; Zheng et al., 2019; Dezfuli et al., 2017; Shinozuka et al., 2015; Cao and Yi, 2021; Dong et al., 2019; Zhu and Qiu, 2014) SMA components have been applied to rubber bearing (Guo, 2016; Cui et al., 2019; Cao and Ozbulut, 2020) and friction bearing(Yang, 2017; Wang et al., 2020; Liang et al., 2020). Most of the study show that the SMA-based bearing has the excellent re-centering capacity and displacement-limiting capacity. However, SMA components also enlarged the stiffness of the devices which result in a larger force response of the substructure.

To reduce the enlarged forces caused by SMA components, Cao et al.(Cao et al., 2019; Chang et al., 2020) proposed a multi-level SMA/lead rubber bearing(ML-SLRB) and an SMA cable-based negative stiffness bearing. The ML-SLRB adjusts its stiffness under different ground motion levels by activating three groups of SMA cables in a sequence, which shows high isolation efficiency at small or medium earthquakes and excellent displacement-limiting capacity at strong earthquakes. SMA cable-based negative stiffness bearing generated negative stiffness by reversed friction surfaces of the bearing. The force response of bridges can be reduced consequently. Dezfuli et al. (Dezfuli et al., 2017)proposed a negative stiffness isolation system that combined an SMA damper with a track-type negative stiffness device. Variable frequency pendulum isolators were proposed to show the same multi-level performance as ML-SLRB(Han et al., 2020; Yang et al., 2021; Lu et al., 2021). All these variable stiffness isolators shed some light on the methods to simultaneously reduce the force and displacement responses of bridges.

Based on the SMA-based negative stiffness isolation device proposed by Chang et al.(Chang et al., 2020), quasi-static lateral shear tests of an SMA-based negative stiffness isolation device are carried out in this study. The mechanical properties and failure modes of the device are studied. Furthermore, an improved SMA-based negative stiffness isolation device is proposed. Last but not the least, the effectiveness of the improved SMA-based negative stiffness isolation device is verified by numerical simulation.

The SMA-based negative stiffness isolation device

In this section, the SMA-based negative stiffness isolation device is briefly introduced.

Principle

The SMA-based negative stiffness isolation device is composed of a friction pendulum bearing (FPB) with reversed friction surfaces and SMA cables. The SMA cables provide the re-centering and displacement-limiting capacities, while the friction pendulum bearing reduces the force response of bridges by the reversed surfaces. Figure 1 shows the comparison of lateral force-displacement relationships of positive stiffness FPB and negative stiffness FPB. It shows that the force of negative stiffness systems is smaller than the positive stiffness systems.

Figure 1.

Lateral force-displacement relationships of: (a). SMA system, (b). Positive and negative stiffness FPB systems, (c). SMA-based systems. The motion equation of negative stiffness device under earthquake is (Iemura et al., 2006).

The lateral force-displacement relationship model

Figure 2 shows the original and deformed configuration of an SMA-based negative stiffness isolation system. Where x is the horizontal displacement of the upper plate (The positive x-direction was set to the right); y is the vertical displacement of the upper plate(The positive y-direction was set to down); L₀ is the length of the tiltable part of the SMA cable vertically distributed before deformation, and L is the length after deformation ; θ is the rotation angle of the center of the upper plate after deformation ; α is the included angle between the vertical distributed SMA cables and the horizontal direction after deformation. R is the curvature radius of the plates and the slider.

x = 2 R \sin θ

(1)

\frac{G}{g} \ddot{x} + F (x, \ddot{x}) = - \frac{G}{g} \ddot{z}

(2)

F (x, \ddot{x}) = - G \sin θ \cos θ + μ G sgn (\dot{x}) \cos θ

(3)

Where G is the load acting on the upper plate by the superstructure;μ is the friction coefficient between the plates and the slider.

sgn (\dot{x})

is a sign function of speed. when

\dot{x}

> 0,

sgn (\dot{x})

=1, and when

\dot{x}

< 0,

sgn (\dot{x})

=-1.

Figure 2.

The original and deformed configuration of an SMA-based negative stiffness isolation device.

From the geometric relationship:

y = R (1 - \cos θ)

(4)

The strain of SMA cable is:

ε = \frac{Δ L}{L} = \frac{\sqrt{x^{2} + {(L_{0} - y)}^{2}}}{L_{0}} - 1

(5)

Therefore, the recovery force of SMA cables is:

F_{s m a} = n A E_{s m a} ε

(6)

Where n is the number of SMA cables vertically distributed; A is the cross-sectional area of each SMA cable; E_sma is the austenite elastic modulus of SMA cable.

The SMA-based negative stiffness isolation device is composed of the SMA cables and the friction pendulum bearing (FPB) with reversed friction surfaces in parallel, so the recovery force equation of the device can be obtained by adding equation (3) and equation (6), as shown in equation (7) and equation (8). Where $F_{s m a} \cos α$ is the horizontal recovery force component of SMA cables, and $F_{s m a} \sin α$ is the vertical recovery force component of SMA cables.

F = F_{s m a} \cos α - G_{1} \sin θ \cos θ + μ G_{1} sgn (\dot{x}) \cos θ

(7)

G_{1} = G + n F_{s m a} \sin α

(8)

Since x is very small compared to R, it can be obtained according to equation (1):

θ \approx \sin θ \approx \frac{x}{2 R}

(9)

Then equation (7) can be simplified as equation (10) by substituting equation (9) into equation (7).

F - n A E_{s m a} ε \cos α - G_{1} \frac{x}{2 R} + μ G_{1} sgn (\dot{x})

(10)

In equation (10), it can be found that the recovery force of the SMA-based negative stiffness isolation device is composed of three parts: the sliding surface friction of the negative stiffness bearing, the horizontal component of the surface pressure, and the restoring force provided by the SMA cable. It can be found that the negative stiffness of the device is by taking the first derivative of equation (10).

Experimental overview

Experimental device

Negative stiffness isolation device

The negative stiffness isolation device, which is similar to a friction pendulum bearing (FPB) with reversed friction surfaces, is shown in Figure 3. The bearing is composed of three parts: upper and lower convex plates, double concave slider, all of which are made of Q345 steel. The side length of the bearing is 0.76 m and the height is 0.2 m. The curvature radius of the plates and slider is 2m, and the radius of the slider section is 0.175 m. Three grooves are set in the transverse and longitudinal directions respectively of the plates to fix the SMA cables. To avoid the intersection of cables, the depths of longitudinal and transverse grooves are different, the depth of the transverse groove is 10 mm and the depth of the longitudinal groove is 20 mm.

Figure 3.

Friction pendulum bearing: (a). 3D view, (b). section view of the bearing, (c). plan view of the bearing.

SMA cable

7*7*0.885 mm superelastic Nitinol (NiTi) SMA cable produced by Fort Wayne Metals is used in this study. Strands are formed by wrapping seven filaments of wire together to form a single strand and seven of these strands are wrapped together to form a cable. Next, the cables are heat-treated and water quenched to form the configuration. The austenite finish temperature of the SMA cable is 19.3°C. The stress-strain relation of the SMA cable is shown in Figure 4. More details about the test and simplification procedure of the stress-strain relationship of the cables can be found in (Cao et al., 2020). Six SMA cables with a length of 1980 mm were used in the test (100 mm is reserved for overlapping fixation).

Figure 4.

SMA cable: (a). section of SMA, (b). stress-strain relationship of SMA (Cao et al., 2020).

SMA-based negative stiffness isolation device

The SMA-based negative stiffness isolation device is composed of 6 SMA cables and friction pendulum bearing in parallel, as shown in Figure 5(a). Firstly, placed the SMA cables loop around the bearing in the grooves, and it should be noted that due to the depth of the longitudinal groove being deeper than transverse grooves, SMA cables should be placed in the longitudinal groove first. Secondly, connect both ends of each SMA cable to form a loop.

Figure 5.

SMA-based negative stiffness isolation device: (a). 3D view, (b). connection of snap, (c). prefabricated steel plate, (d). connection of bolt.

In this paper, the SMA cable was connected by snap first, as shown in Figure 5(b). After the test, it was found that due to the SMA cable being softer than steel cable, the SMA cable could not be effectively connected by snap. The SMA cables would slide relatively at the connection point. So, how to connect both ends of the SMA cable is the difficulty of device assembly.

Here, after several trials and errors, the final connection method was connecting the SMA cables with bolts. Firstly, the steel plates were fabricated as shown in Figure 5(c). There were three holes in the plate., the middle hole was used to pass through the SMA cable, the others were used to pass through the bolts. And the plates needed sufficient thickness to ensure their stability. Secondly, passed each SMA cable through two plates, and then thicken both ends of the SMA cable with aluminum buckles to fix the plates. Finally, the bolts were passed through the reserved hole of the steel plates at both ends of the cable and were fixed with the nut, as shown in Figure 5(d).

Conditions of test

Lateral cyclic loading tests were conducted by using a load-shear test machine shown in Figure 6. After the device was placed on the test machine, a vertical load of 300 kN and 400 kN was applied to the upper plate. Then, a cyclic lateral displacement-loading history was applied to the upper plate. The horizontal load adopts multi-stage displacement loading. Took 20 mm as the initial displacement and loaded every 10 mm as a stage, a lateral displacement-loading history from 0 mm to 150 mm with an incremental of 10 mm was adopted, as shown in Figure 7. The loading rate was 0.01 Hz/s. The horizontal force and horizontal displacement were recorded by the data acquisition system in real-time.

Figure 6.

Test set-up.

Figure 7.

The loading system of Horizontal displacement.

Experimental results and discussion

Cyclic lateral shear force-displacement relationships of the device

Figure 8 shows the cyclic lateral shear force-displacement curves of the device. It can be found that the force of the device is small in small displacement until a drift of 50 mm. It can be seen that the enlarged forces caused by SMA components were partially reduced by the negative stiffness device. Moreover, the force of the SMA-based negative stiffness device enlarged significantly thanks to the strain hardening of the SMA cables. As a result, the device can limit excessive displacements of bridges in a strong earthquake.

Figure 8.

The lateral shear force-displacement curves of the SMA-based negative stiffness device under a vertical load of: (a). 300 kN, (b). 400 kN.

Mechanical performance of the device

Figure 9 further illustrates the detailed properties of the device in terms of strength, secant stiffness, energy dissipation, and effective damping. The skeleton curves and stiffness curves of the device under different conditions are compared in Figure 9(a) and (b) respectively. It can be seen that the force and stiffness of the device are small when the displacement is smaller than 50 mm. In addition, the force and the stiffness of the device under a 300 kN vertical load are bigger than the device under a 400 kN vertical load at the same displacement, since a larger negative stiffness can be expected with a higher vertical load which can be expressed by equation (10).

Figure 9.

Damping performance of the device under cyclic horizontal displacement load: (a). strength, (b). stiffness, (c). energy dissipation, (d). effective damping ratio.

Figure 9(c) and (d) illustrate the energy dissipation capacity and the equivalent damping ratio of the device respectively.

In which, the equivalent damping ratio $ξ_{eq}$ and dissipated energy are defined by:

ξ_{eq} = \frac{Δ W}{4 π W}

(11)

W = \frac{1}{2} Q_{\max} X_{\max}

(12)

Where

Δ W

is the energy dissipated per cycle of loading;

Q_{\max}

is the positive force at the maximum positive displacement.

It can be seen from Figure 9(c) that the dissipated energy increased steadily with the incremental of displacement, which implicates a stable force-bearing capacity of the device. There is little difference in the energy dissipation capacity between the device under the two vertical loads. The equivalent damping ratio decreased gradually with the displacement of the device after 30 mm. This corresponds to the results from a study by Wang(Wang et al., 2020) that the equivalent damping ratio of an SMA-based device would be smaller than a common isolation device.

Failure modes

The failure modes of the device are shown in Figure 10. It is found that the SMA cables fractured at the corner of the bearing. The premature fracturing of the SMA cable may be caused by the local high concentrated stress developed by squeezing and pressing at the corners. As a result, the SMA cables cannot slide freely at the corners and the vertical part of the SMA cable can not transform to the horizontal part. As a result, the SMA cable fractured at a drift far away from the design drift.

Figure 10.

Failure modes of the SMA-based negative stiffness device.

Comparative analysis of experimental and simulation results

Finite element model

A finite element model of the SMA-based negative stiffness isolation device is established by Abaqus 2021 software package (Smith, 2020), as shown in Figure 11. The upper and lower plates and the slider are simulated by an eight-node linear hexahedron element. The friction coefficient between the contact surfaces is 0.04. The SMA cables are simulated by two-node linear three-dimensional truss elements. The steel plates and SMA cables are connected by MPC constraint. All freedoms of the lower plate are fixed, while u_z, r_x, r_y, and r_z of the upper plated are fixed. The cyclic lateral displacement-loading history is applied at the u_x of the nodes on the upper plate.

Figure 11.

Finite element model: (a). 3D view, (b). section view.

It’s complex to account for the friction between steel plates and SMA cables. In the paper, the friction behavior is ignored and the SMA cables are simplified as straight cables between the upper and lower steel plates, as shown in Figure 11(a).

The condition of SMA cables in the failure modes

To analyze the slipping states of SMA cables at the corners of the device in the failure modes of the test, two finite element models are built. One is built with an assumption that the SMA cables don’t slip over the corners, while the other one is built with an assumption that the SMA cables slip freely around the corners. In the non-slipping model, the horizontal part of SMA cables is neglected and each vertical of SMA cables is simulated as an independent cable. In the freely-slipping model, the strain of the SMA material is enlarged by a scalar which equals the ratio of the length of half loop of SMA cables to the length of its vertical part, while its elastic modulus is scaled down by the same scalar to ensure similar mechanical properties of the SMA cables in simulation as its actual behaviors in the test.

Figure 12 compares the experimental and simulated results of the hysteretic curves of the SMA-based negative stiffness device. The horizontal restoring force of the device obtained from the test is between the forces obtained from the two numerical models with non-sliding and freely sliding assumptions. As a result, a partial sliding, which is between non-sliding and freely sliding, of the SMA cables maybe happen.

Figure 12.

Comparison between test and numerical simulation.

To scrutinize the stress state of the SMA cable, the von mises stress distribution of the two models at a drift of 100 mm is shown in Figure 13. They correspond to points A and B in Figure 12 respectively. The stress of the SMA cables in the freely sliding model is even smaller than the martensite transformation stress, 370Mpa, at a drift of 100 mm which implies that the SMA cables will not fracture at this drift if they slip freely, while the SMA cables of the non-slipping model finish the austenite transformation range which indicates that they may be fractured. All these confirm the partial sliding state of the SMA cables at the experimental device.

Figure 13.

Von mises stress distribution of the models at a drift of 100 mm: (a). Non-sliding model, (b). Freely sliding model.

Comparison of experimental and simulated results

Figure 14 compares the experimental and simulated results of the cyclic lateral force-displacement relationships of the device. Where the strain of the SMA material is enlarged by a scalar of 1.5 and its elastic modulus are scaled down by the same scalar in the finite element model of the SMA-based negative stiffness isolation device. The scalar, which corresponds to the partial sliding behavior, is obtained by parametric analysis to fit the experimental results with simulated results. The numerical results are bigger than the experimental results when the drift is smaller than 50 mm. The reason may be that the SMA cables were relaxed at the beginning. As a result, the SMA cables should be prestressed in the future to improve the performance of the device. However, the numerical results match the experimental results well when the drift is bigger than 50 mm. Generally, the simulation approach can well predict the mechanical behaviors of the isolation device.

Figure 14.

Comparison of the cyclic lateral force-displacement relationships of the experimental and numerical results under a vertical load of (a). 300 kN, (b). 400 kN.

Development of the SMA-based negative stiffness isolation device

A new SMA-based negative stiffness isolation device with straight SMA cables only is proposed in this section to prevent the fracture of SMA cables at the corners and to raise the stiffness of the device at small drifts. The finite element model of the new SMA-based negative stiffness isolation device is shown in Figure 15. Two steel plates are welded on each side of the upper and lower plates of the device. As shown in Figure 15 (b), the overall dimension of the device is 1874 mm*760 mm*200 mm. Section area of an SMA cable in the improved model is enlarged by a scalar of 3, which denotes 3 SMA cables. All the other dimensions and properties are the same as the experimental device in section 3.1.

Figure 15.

Finite element model of the improved device: (a). 3D view, (b). sectional view.

Comparative analysis of the original and improved SMA-based negative stiffness isolation device

Figure 16 compares the simulated results of the hysteretic curves of the original and improved device. It shows that the SMA cables of the original device fracture at a displacement of 100 mm (Point A), while they fracture at a displacement of 150 mm (Point C) in the improved device. The ultimate lateral displacement and the lateral forces of the device are greatly improved in the new configuration.

Figure 16.

Hysteresis curve of the device before and after optimization.

The von mises stress distribution of the original and improved device corresponding to points A, B, and C in Figure 16 are shown in Figure 17(a)–(c) respectively. The displacement at point B is 100 mm. It can be seen in Figure 17(a) that the stress of the SMA cables is bigger than its fracture stress, which is 660 MPa as shown in Figure 4(b). However, Figure 17(b) shows that the forces of the SMA cables of the improved device are about 400 MPa, which means the SMA cables are still in the martensite transformation range at a drift of 100 mm where the SMA cables fractured in the original device. In addition, Figure 17(c) shows that the SMA cables attain their fracture stress at a drift of 150 mm. It implicates that the improved device shows a better performance compared to the original device.

Figure 17.

Contours of the device: (a). point A, (b). point B, (c). point C.

Analysis of the mechanical performance of the improved device

Figure 18 compares the mechanical properties of the experimental device and the improved device in terms of force, secant stiffness, energy dissipation, and effective damping respectively. There is some discrepancy between the simulated results and experimental results of the original isolator. As mentioned before, it may be caused by the relaxation of SMA cables in the experimental sample, while we didn’t consider this issue in the simulation. However, when the drift is larger than 50 mm, the simulated results of the original isolator fit the experimental results well.

Figure 18.

Comparison of mechanical performance of the experimental device and the improved device: (a). force, (b). stiffness, (c). energy dissipation, (d). effective damping ratio.

The force and the stiffness of the improved device are bigger at small drifts and smaller at large drifts than the experimental device, as shown in Figure 18(a) and (b). Consequently, the improved device shows a better re-centering capability at small drift and smaller forces at large drift. In addition, the force and the stiffness of the improved device increase greatly when the SMA cables finish the martensite transformation at a drift of 120 mm. It indicates that the improved device also has excellent displacement-limiting capacity.

Figures 18(c) and (d) compare the energy dissipation capacity and the equivalent damping ratio of the two devices respectively. There is a little discrepancy in the energy dissipation capacity between the original and improved device at the same displacement. In addition, the dissipated energy of the improved device increases smoothly with the incremental displacement. Moreover, it can be seen from Figure 18(d) that the improved device has a smaller equivalent damping ratio compared to the original device since the SMA cables were activated in a longer journey.

Conclusions and recommendations

(1) In the SMA-based isolation device, the SMA cables are usually wrapped around the device with an assumption of free sliding between the cables and the device to improve the lateral displacement capacity. However, it was found that the SMA cables can not slide freely at the corner in this layout. The SMA cables fractured at the corner far away from the design displacement capacity since the stress is concentrated there because of squeezing and stretching. As a result, it is suggested that SMA cable should be laid in a straight line to improve the performance of SMA-based isolation devices.

(2) The reversed curved surface effectively generated negative stiffness in the new isolation device, which partially reduced the forces in the substructure. The novel SMA-based negative stiffness isolation device can not only reduce displacements of the SMA isolation device but also partially reduce the force responses of bridges.

(3) When the drift of the device is so big that pounding and falling of girders may happen, the SMA cables enter the strain hardening stage and limit the displacement of bridges which will prevent these damages.https://fanyi.baidu.com/-##

(4) Compared with the SMA-based negative stiffness device with a loop layout of SMA cables, the improved device with a straight-line layout of SMA cables showed a larger ultimate-displacement capacity, a smaller stiffness, and no local damage.

Footnotes

Acknowledgements

The support is gratefully acknowledged - zh/en/javascript:void(0)

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the National Natural Science Foundation of China (Grants 52178124, 51608136, 51278134), Natural Science Foundation of Guangdong Province (Grant 2020A1515010231), Science and Technology Planning Project of Guangdong Province (Grant 2020A1414010271), and Scientific Research Fund of Institute of Engineering Mechanics, China Earthquake Administration (Grant 2019D19).

ORCID iD

Sasa Cao

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