Effect of hysteresis properties of shape memory alloy-tuned mass damper on seismic control of power transmission tower

Abstract

Statistics from past strong earthquakes revealed that electricity transmission towers were vulnerable to earthquake excitations. It is necessary to mitigate the seismic responses of power transmission towers to ensure the safety of such structures. In this research, a novel shape memory alloy-tuned mass damper is proposed, and seismic vibration control of power transmission tower using shape memory alloy-tuned mass damper based on three types of shape memory alloy materials (i.e. NiTi, M-CuAlBe, P-CuAlBe) is analyzed. The detailed three-dimensional finite element model of a power transmission tower incorporated with shape memory alloy-tuned mass damper is developed using numerical simulation software ANSYS. The control effects of shape memory alloy-tuned mass damper on the seismic vibration of power transmission tower are assessed using nonlinear time history analysis method. The interested seismic performance indices include displacement, acceleration, and base shear force. In addition to the shape memory alloy materials, the influence of seismic intensity and frequency ratio are conducted for the optimal design. It is shown that installing shape memory alloy-tuned mass damper well reduced the seismic responses of power transmission tower. The comparison between different shape memory alloys indicated that the damping of the shape memory alloy-tuned mass damper is beneficial to mitigate the vibrations.

Keywords

earthquake power transmission tower seismic control shape memory alloy tuned mass damper

Introduction

Power transmission tower is one of critical lifeline projects, which distributes electricity from power plants to substations (Battista et al., 2003; Chen et al., 2009; Tian et al., 2017). It is expected to remain fully functional after an earthquake, to maintain seismic resilience. During normal operations, power transmission tower mainly suffers from self-weight, wind, and ice loads. However, recent investigations have revealed that the power transmission towers destroyed as a result of seismic hazard (Chen et al., 2017; Tian et al., 2017a). For example, power supply was disrupted due to the damages of transmission tower in the 1992 Landers earthquake and 1995 Kobe earthquake (Hall et al., 1996; Shinozuka, 1995). Particularly, transmission towers were severely damaged in the 1999 Chi-Chi earthquake (National Center for Research on Earthquake Engineering (NCREE), 1999), including 15 towers collapsed and 26 transmission towers tilted. Failures of transmission towers would lead to power interruption, which delay emergency rescue and cause secondary disasters. Therefore, it is of great significance to alleviate the adverse vibrations and improve the performance of power transmission tower under seismic excitation.

Tuned mass damper (TMD) has been reported as an effective method to reduce earthquake-induced vibrations for building structures (Chen and Wu, 2001; Sadek et al., 1997). However, among the efforts in controlling excessive vibrations for transmission towers, the related studies are relatively limited. Recently, people have tried to install TMD in the transmission towers. For example, Kilroe (2000) showed that fatigue life can be well increased using TMD. Battista et al. (2003) proposed a nonlinear pendulum-like damping TMD and showed good vibration control effect. Deng et al. (2003) suggested a combination of TMD and viscous elastic damper and obtained improved control effect. Zhang et al. (2012) proposed a pounding TMD (PTMD) and found that pounding TMD generated better control effect than conventional TMD. Tian et al. (2017b) optimized the design of pounding TMD upon multi-dimension earthquake excitations.

Shape memory alloy (SMA) is a class of metallic alloy, which is able to dissipate energy and recover deformation upon cyclic loadings when the ambient temperature is above the austenite transformation limit (Fang et al., 2014, 2017; Qiu and Zhu, 2014, 2016; Sun and Huang, 2010; Zhu and Qiu, 2014). The salient properties of SMA have gained wide attention in the community of vibration control. Zhang and Zhu (2007) proposed an innovative SMA damper, which is able to exhibit desirable hysteretic behavior through proper adjustment. Andrawes and DesRoches (2010) investigated the effect of hysteretic properties of SMAs on the seismic vibrations of simple bridges and single-story braced frames. Zhu and Qiu (2014) used SMA damper to control the excessive displacement of bridges under earthquakes. Qiu and Zhu (2017) demonstrated the advantages of SMA damper within a reduced scale steel framing model by a series of shake table tests.

In recognition of pressing need of vibration control of power transmission tower and dynamic application potential of SMAs, in this study, we proposed a shape memory alloy-tuned mass damper (SMA-TMD) to mitigate the seismic vibration of power transmission tower. The core of SMA-TMD is superelastic helical SMA spring, which has the advantages of high flexibility and compact form. The numerical simulations of three-dimensional (3D) SMA spring were compared with the testing results. Considering the hysteretic variety of SMAs, parametric analyses on material properties were conducted by selecting a total of three types of SMAs. The adopted SMAs included Nitinol (NiTi), monocrystalline CuAlBe (M-CuAlBe), and polycrystalline CuAlBe (P-CuAlBe) (Qiu and Zhu, 2014). Incremental dynamic analysis was carried out as well, with the aim to assess the control robustness of the proposed SMA-TMD.

Hysteretic properties of SMAs

The part introduces the cyclic behavior of SMAs, as shown in Figure 1. The material testing results are extracted from a prior study (Qiu and Zhu, 2014). It is clearly that although all these SMAs show recentering capability, the material properties significantly depend on alloy compositions. To quantitatively compare the mechanical properties of these SMAs, Table 1 lists the hysteretic parameters of the adopted SMAs. It is seen that the recoverable strain of M-CuAlBe is 0.14, which is three times that of NiTi. The strength capacity of NiTi is the highest among the candidates. It is worth noting that the recentering capacity and equivalent damping of P-CuAlBe are much smaller than the other two counterparts, implying a lower vibration control capacity. Strain hardening is observed in NiTi when the forward phase transformation is completed. This behavior may lead to drastically increased strength in the damper.

Figure 1.

Cyclic behavior of SMAs (Qiu and Zhu, 2014).

Table 1.

Mechanical properties of SMAs.

	E (GPa)	C1 (MPa)	C2 (MPa)	C3 (MPa)	C4 (MPa)	C5	C6 (MPa)
NiTi	38.3	500	600	351	300	0.04	0
M-CuAlBe	17	170	271	167	56	0.14	0.03
P-CuAlBe	33.8	232	288	270	201	0.1	0.09

SMA: shape memory alloy.

Cyclic properties of SMA helical spring

The numerical simulation of SMA spring was conducted in ANSYS 14.5. The helical spring was built by solid elements, as shown in Figure 2. The SOLID185 element was used, and the spring was generated by mapped method. One end was fixed and the other end was applied with displacement-based loading scheme adopted in the experimental test. Figure 3 compares the experimental result and numerical result. Detailed fabrication process and experimental results of the selected spring can be found in the corresponding study (Qiu, 2016). The spring was made of NiTi SMA wire with a diameter of 2 mm. The outer diameter and free length of the specimen are 16 and 62 mm, respectively. The coil distance is 8 mm, and the number of coils is 6. To produce satisfactory behavior for SMAs, proper heating treatment was determined to be heating at 500°C for 30 mins and then water quenched. The testing results are shown in Figure 3, indicating the SMA spring has expected flag-shape hysteresis with zero residual deformation when the applied force is removed. It is seen that the numerical model well captures the global behavior of the SMA spring, including flag-shape hysteresis, comparable strength, and similar hysteresis width. Therefore, the numerical model is reasonably accurate and will be used in the TMD system in the following seismic control analysis of power transmission tower.

Figure 2.

Meshed 3D FE model.

Figure 3.

Comparison of testing results and numerical simulations.

The adopted constitutive model of SMA is given by the software, as shown in Figure 4. There are a total of six parameters to describe the hysteretic behavior of SMAs. C1 is the onset stress of forward phase transformation, $σ_{s}^{AS}$ ; C2 is the finish stress of forward phase transformation, $σ_{f}^{AS}$ ; C3 is the onset stress of backward phase transformation, $σ_{s}^{SA}$ ; C4 is the finish stress of backward phase transformation, $σ_{f}^{SA}$ ; C5 denotes the maximum recoverable strain, ${\bar{ε}}_{L}$ ; C6 measures the stress difference between tension and compression.

Figure 4.

Constitutive model SMA in ANSYS.

Finite element models

Power transmission tower

To this study, we aim to study the vibration control problem of power transmission tower using SMA-based TMD, and current focus is particularly paid on the effect of hysteresis properties of the TMD. Therefore, a standalone transmission tower is sufficient for us to address the issue. Besides, it is an usual way to investigate the effectiveness of using different devices to control seismic induced vibrations of a single transmission tower. For example, Zhang et al. (2012) proposed the concept of using PTMD to control seismic vibration of a transmission tower.

In this study, a SZ21 type power transmission tower in the Northeast China is selected as a prototype for analysis. The height and total weight of the tower are 53.9 m and 20.23 ton, respectively. Figure 5(a) shows a graph of the transmission tower. The main member and diagonal member of the transmission tower are made of Q345 and Q235 angle steels, respectively. The elastic modulus of steel is given as 206 GPa. For the standalone tower, the mass of conductors is not considered.

Figure 5.

(a) Elevation view of the transmission tower, (b) FE model in ANSYS, and (c) first mode shape.

Detailed 3D finite element (FE) model of the transmission tower is developed using the commercial software ANSYS and Figure 5(b) shows the FE model. The transmission tower is simulated by the elastic BEAM188 element in ANSYS. The transmission tower is modeled by 1786 beam elements and 684 nodes. The base nodes of the transmission tower are fixed on the ground. The dynamic characteristics of the transmission tower can be calculated according to eigenvalue analysis, and the natural frequency is 1.797 Hz in the X direction. Figure 5(c) shows the mode shape of transmission tower in the X direction.

SMA-TMD

To reduce the structural vibration, a novel SMA-TMD is proposed. The SMA-TMD combines the mechanism of conventional TMD and the damping capacity of SMA spring. A possible configuration of the SMA-TMD is shown in Figure 6(a). The SMA-TMD device consists of a stiff box, the mass block, two SMA springs, and smooth surface. Since placed on the smooth surface, the restoring force of the mass block is entirely provided by SMA springs. The SMA-TMD is bolted at the top of the tower, and it will be activated in the seismic input direction, as shown in Figure 6(b). The proposed SMA-TMD has two characteristics: (1) when the earthquake loads are small, the SMA spring can be regarded as ordinary spring and behaves linearly; (2) when the earthquake loads are large enough, the forward phase transformation of SMA material will be activated, and the SMA spring will enter into the nonlinear stage and begin to dissipate energy. Therefore, the SMA-TMD combines the advantages of a traditional TMD and energy dissipation capacity.

Figure 6.

Proposed SMA-TMD: (a) configuration and (b) location.

The FE model of the SMA-TMD is established using ANSYS. The MASS21 element is used to simulate the mass of the SMA-TMD. SOLID185 element is used to model the SMA spring, and the mapped mesh method is adopted. The mass element is connected with the top node of the transmission tower by two SMA springs. TAGRE170 and CONTA175 contact elements are used to simulate the point-surface contact between SMA-TMD mass and SMA spring, and SMA spring and node of the transmission tower, respectively.

In the present study, the optimal frequency ratio proposed in reference (Den Hartog, 1956) is adopted, given as below:

f_{o p t} = \frac{1}{1 + μ_{m}}

(1)

where μ_m is the mass ratio of the SMA-TMD over the corresponding modal mass of the structure, taking 0.02 here. According to eigenvalue analysis, the first modal participating mass ratio of the tower is 0.48; thus, the mass of the SMA-TMD is 194 kg. Compared to the self weight of the tower, the mass of the SMA-TMD is very small; thus, the additional stress induced by the SMA-TMD can be neglected.

The optimum values of stiffness of the SMA-TMD is therefore given by

k_{opt} = f_{opt}^{2} ω^{2} m_{d}

(2)

where ω is the natural frequency of the transmission tower in the X direction, that is, the seismic input direction; m_d is the mass of the SMA-TMD.

The diameter of the SMA spring can be defined as follows (Budynas and Nisbett, 2014)

d = {(\frac{16 \cdot k \cdot n \cdot D^{3} \cdot (1 + υ)}{E})}^{\frac{1}{4}}

(3)

where k is the lateral stiffness of the SMA spring; n is the coil number of the SMA spring; D is the pitch diameter of the SMA spring; υ is Poisson’s ratio, taking 0.3; the elastic modulus E is 38.3 GPa.

To explore the effect of hysteretic properties on vibration control for transmission tower using SMA-TMD, the elastic properties of the all SMA-TMDs are tuned to be identical, including same initial elastic modulus and “yielding” strength. Whereas the nonlinear behaviors are noticeably different, which is due to the variety of material properties. Based on this, the final geometry dimensions of the SMA springs are designed, as listed in Table 2. Figure 7 shows the numerically simulated behavior of designed springs upon a loading displacement of 0.14 m. The targeted loading displacement is determined to be 0.14 m, because this is the maximum displacement applied to the spring specimen in the test. It is seen that the springs show identical initial behavior, whereas the nonlinear behaviors are significantly dependent on the material property. The cyclic behaviors of springs using NiTi and M-CuAlBe are globally comparable, whereas the spring using P-CuAlBe shows a much narrower hysteresis width than the other two springs.

Table 2.

Geometry dimensions of the designed SMA springs.

SMA	d (m)	D (m)	t (m)	n
NiTi	0.011	0.069	0.050	3
M-CuAlBe	0.018	0.102	0.070	3
P-CuAlBe	0.017	0.118	0.059	3

SMA: shape memory alloy.

Figure 7.

Cyclic behavior of designed SMA springs.

Seismic control analysis

Ground motion records

Considering the stochastic of the ground motion, three typical natural seismic waves respectively denoted as GM1 to GM3 are selected. Table 3 lists the detailed information of the selected seismic records. The seismic waves are downloaded from the Pacific Earthquake Engineering Research Center (PEER, http://peer.berkeley.edu/). Figure 8 shows the response spectra of each acceleration time histories. The peak ground acceleration (PGA) of seismic wave is adjusted to 0.4g, and the acceleration time history is applied in the X direction. The selected ground motions, including the El Centro, Taft, and Kobe ground motions, represent famous earthquakes and were widely used in many seismic analytical studies. Current purpose is to clarity the effect of using different SMA springs in the TMD system of transmission tower, instead of discussing the seismic responses due to different ground motions. Besides with the adopted records, other ground motion records can be used as well. The ground motions are input in only one direction, because this study focuses on the variance in SMA properties. The multi-direction problem is also very interesting. Actually, this problem has been well addressed by a recent study (Tian et al., 2018).

Table 3.

Selected ground motion records.

ID	Earthquake	Event date	Magnitude	Station
GM1	Imperial Valley	18 May1940	6.9	El Centro
GM2	KernCounty	21 July1952	7.4	TaftLincoln School
GM3	Kobe	16 January1995	6.9	Oka

Figure 8.

Spectral accelerations: (a) El Centro, (b) Taft, and (c) Kobe.

Vibration control effect

To investigate the control performance of the SMA-TMD, the responses of the transmission tower with the SMA-TMD subjected to different ground motions are analyzed. Furthermore, in order to show the control effect of using SMA-TMD, the responses of the transmission tower without control are presented as well. The interested seismic performance indices include displacement, acceleration, and base shear force.

Figure 9 shows the dynamic responses of the transmission tower with and without control under the El Centro earthquake. It can be seen from the figures that the SMA-TMD can reduce the responses of the transmission tower, irrespective of the material properties of SMAs. As shown in Figure 9(a), the reduction ratios of the peak displacement, acceleration, and base shear force under this earthquake can reach 42%, 40%, and 29%, respectively; in terms of the root-mean-square (RMS) values, the corresponding reduction ratios are 43%, 35%, and 31%, respectively. Therefore, this case study shows that the SMA-TMD can more effectively control the seismic responses of the transmission.

Figure 9.

Seismic response of power transmission tower installed with SMA-TMD under El Centro ground motion record: (a) NiTi, (b) M-CuAlBe, and (c) P-CuAlBe.

Figure 10 assembles the time history responses to compare the control effect of using different SMA-TMDs. The comparison shows that NiTi- and M-CuAlBe-based SMA-TMDs tend to better control vibrations than the P-CuAlBe-based SMA-TMD throughout the whole time history responses under the ground motion records of El Centro and Taft. This is primarily attributed to the high damping offered by SMA-TMDs using NiTi and M-CuAlBe, which absorbed more seismic energy. It is also interesting to note that the vibration magnitudes are comparable under ground motion record Kobe, which is due to the frequency characteristics of this ground motion. The cyclic behaviors of SMA springs are plotted in Figure 11. It is found that the P-CuAlBe SMA-TMD sustained larger deformation demand than the other ones. Considering the allowable movement space for TMD is usually limited, it indicates SMA-TMD using higher damping SMA is more favorable.

Figure 10.

Comparisons between SMA-TMD control effect: (a) El Centro, (b) Taft, and (c) Kobe.

Figure 11.

Cyclic behavior of SMA springs: (a) El Centro, (b) Taft, and (c) Kobe.

To quantify the control effect of using different SMA-TMDs, Table 4 lists the reduction ratios of the peak and RMS response of the transmission tower under different seismic excitations. It shows the reduction degree of displacement demand is the greatest, and followed by acceleration demand and base shear demand. The best control effect is found in the case of Kobe earthquake, and the seismic demands are well reduced by over 60%.

Table 4.

Vibration reduction ratio of using SMA-TMD.

Seismic records	Materials	Displacement		Acceleration		Base shear force
Seismic records	Materials	Peak (%)	RMS (%)	Peak (%)	RMS (%)	Peak (%)	RMS (%)
El Centro	NiTi	42	43	40	35	29	31
	M-CuAlBe	44	43	41	35	24	31
	P-CuAlBe	25	31	35	20	24	22
Taft	NiTi	41	28	27	24	26	18
	M-CuAlBe	38	28	28	24	25	18
	P-CuAlBe	40	25	15	19	17	14
Kobe	NiTi	60	56	44	42	36	44
	M-CuAlBe	61	56	44	42	37	44
	P-CuAlBe	62	55	34	40	42	42

SMA-TMD: shape memory alloy-tuned mass damper; RMS: root mean square.

Parametric analysis

To obtain a rational design of the SMA-TMD, the effects of seismic intensity and frequency ratio are discussed. Unless mentioned otherwise, the seismic intensity is 0.4g, the mass ratio is 0.02, and the frequency ratio is determined based on equation (1).

Seismic intensity

To investigate the effect of seismic intensity, nine PGAs are considered which varies from 0.05 to 0.8g. The selected ground motion record is El Centro. The mass ratio of the SMA-TMD is constantly kept to be 2%, and the stiffness of the springs is not varied. The reduction ratios of the transmission tower varied with different PGAs are plotted in Figure 12. All the SMA-TMDs show a similar trend with the variation of ground motion intensity. Because the cyclic properties of the SMA spring were tuned based on the case when the PGA of ground motion equals to 0.2g, the reduction ratio increases initially and then decreases with the increase of seismic intensity globally. Different seismic index shows different sensitivities to the variation of seismic intensity, but the maximum reduction ratio always occurs around a PGA of 0.2g. Regardless of seismic intensities, the SMA-TMDs using NiTi or M-CuAlBe exhibit comparable control effect. In terms of SMA-TMD using P-CuAlBe, the vibration control effect is smaller in almost each case.

Figure 12.

Control effect of the transmission tower at different seismic intensities: (a) displacement, (b) acceleration, and (c) base shear force.

Frequency ratio

The primary natural frequency of transmission tower may vary in actual engineering due to the temperature variation of transmission lines, the weight of ice and snow. The effect of frequency ratio, varying from 0.8 to 1.2 f_opt, on the vibration control of the transmission tower is studied. Seismic ground motion record El Centro is adopted in this parametric analysis. The reduction ratios vary with different frequency ratios are calculated and illustrated in Figure 13, which shows that the vibration control effect of using the SMA-TMD tends to increase when the frequency ratio is smaller than the optimized value. After f_opt, it is interesting to note that the reduction effect on displacement continues to increase until 1.2 f_opt, whereas the reduction effect on acceleration and base shear force becomes saturated. The trend is mainly due to the significant nonlinear mechanism of the SMA-TMD. Again, the vibration control effects of SMA-TMDs using NiTi and M-CuAlBe are comparable and are better than that using P-CuAlBe.

Figure 13.

Variations reduction ratio of the transmission tower with different frequency ratios: (a) displacement, (b) acceleration, and (c) base shear force.

Conclusion

This article proposed a novel SMA-TMD for the vibration control problem of power transmission tower. A total of three SMA materials were considered, including NiTi, M-CuAlBe, and P-CuAlBe. The 3D FE models of power transmission tower and SMA-TMD were built in ANSYS. Intensive nonlinear time history analyses were carried out to observe the control effect of using various SMA-TMDs. Parametric analyses on seismic intensity and frequency ratio were conducted as well. The following conclusions are obtained:

All the SMA-TMDs are able to generate noticeable vibration control effect for the power transmission tower. The most noticeable reduction is found in displacement, which can be up to over 60%. The control effect is in a positive relationship with the equivalent damping ratio of the SMA-TMD.

The vibration control effect is dependent on the characteristics of input seismic ground motion records. In this study, the maximum control effect is associated with the ground motion of Kobe earthquake.

The parametric analysis shows that the control effect is relatively sensitive to the variations of seismic intensity and frequency ratio. According to this study, the optimal control effect is achieved when the PGA is 0.2g and the f_opt is 1.0.

The SMA-TMD exhibited significant nonlinear behavior, and the control analysis is much more complicated than the linear TMD, which requires more work in future.

It is worth noting that the tower-line coupling effect is not considered in this study and will be included in future analysis.

Footnotes

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 study was financially supported by the National Natural Science Foundation of China (award nos 51578325 and 51778347), the Young Scholars Program of Shandong University, and the China Postdoctoral Science Foundation (no. 2017M622206).

References

Andrawes

DesRoches

(2010) Effect of hysteretic properties of superelastic shape memory alloys on the seismic performance of structures. Structural Control & Health Monitoring 14(2): 301–320.

Battista

Rodrigues

Pfeil

(2003) Dynamic behavior and stability of transmission line towers under wind forces. Journal of Wind Engineering and Industrial Aerodynamics 91(8): 1051–1067.

Budynas

Nisbett

(2014) Shigley’s Mechanical Engineering Design. New York: McGraw-Hill Companies.

Chen

Weng

Zhi

et al . (2017) Response control of a large transmission tower-line system under seismic excitations using friction dampers. Advances in Structural Engineering 20(8): 1155–1173.

Chen

Zheng

(2009) Control of wind-induced response of transmission tower-line system by using magnetorheological dampers. International Journal of Structural Stability and Dynamics 9(4): 661–685.

Chen

(2001) Optimal placement of multiple tune mass dampers for seismic structures. Journal of Structural Engineering 127(9): 1054–1062.

Den Hartog

(1956) Mechanical Vibrations. New York: McGraw-Hill.

Deng

Zhu

Chen

et al . (2003) Study on wind-induced vibration control of long span transmission line system. Journal of Building Structures 24(4): 60–64. (in Chinese).

Fang

Wang

et al . (2017) Self-centring behaviour of steel and steel-concrete composite connections equipped with NiTi SMA bolts. Engineering Structures 150: 390–408.

10.

Fang

Yam

MCH

Lam

ACC

et al . (2014) Cyclic performance of extended end-plate connections equipped with shape memory alloy bolts. Journal of Constructional Steel Research 94: 122–136.

11.

Hall

Holmes

Somers

(1996) Northridge Earthquake of January 17, 1994: Reconnaissance Report. Oakland, CA: Earthquake Engineering Research Institute.

12.

Kilroe

(2000) Aerial method to mitigate vibration on transmission towers. In: Proceedings of the IEEE ESMO-2000 IEEE 9th international conference on transmission and distribution construction, operation and live-line maintenance, Montreal, QC, Canada, pp. 187–194. New York: IEEE.

13.

National Center for Research on Earthquake Engineering (NCREE) (1999) Investigation Report of Chi-Chi Earthquake. Taipei, China: National Center for Research on Earthquake Engineering.

14.

Qiu

(2016) Seismic-resisting self-centering structures with superelastic shape memory alloy damping devices. PhD Thesis, The Hong Kong Polytechnic University, Kowloon, Hong Kong.

15.

Qiu

Zhu

(2014) Characterization of cyclic properties of superelastic monocrystalline Cu–Al–Be SMA wires for seismic applications. Construction and Building Materials 72: 219–230.

16.

Qiu

Zhu

(2016) High-mode effects on seismic performance of multi-story self-centering braced steel frames. Journal of Constructional Steel Research 119: 133–143.

17.

Qiu

Zhu

(2017) Shake table test and numerical study of self-centering steel frame with SMA braces. Earthquake Engineering & Structural Dynamics 46(1): 117–137.

18.

Sadek

Mohraz

Taylor

et al . (1997) A method of estimating the parameters of tuned mass dampers for seismic applications. Earthquake Engineering & Structural Dynamics 26(6): 617–636.

19.

Shinozuka

(1995) The Hanshin-Awaji Earthquake of January 17, 1995 Performance of Lifelines. Berkley, CA: US National Center for Earthquake Engineering Research.

20.

Sun

Huang

(2010) The temperature memory effect and the influence of thermo-mechanical cycling in shape memory alloys. Smart Materials and Structures 19: 1–8.

21.

Tian

Pan

et al . (2017a) Progressive collapse analysis of long-span transmission tower-line system under multi-component seismic excitations. Advances in Structural Engineering 20(12): 1920–1932.

22.

Tian

Rong

Zhang

et al . (2017b) Vibration control of a power transmission tower with pounding tuned mass damper under multi-component seismic excitations. Applied Sciences 7(5): 477.

23.

Tian

Yin

(2018) Orienting ground motion inputs to achieve maximum seismic displacement demands on electricity transmission towers in the near-fault regions. ASCE Journal of Structural Engineering 144(4): 04018017–04018011.

24.

Zhang

Song

et al . (2012) Seismic control of power transmission tower using pounding TMD. Journal of Engineering Mechanics 139(10): 1395–1406.

25.

Zhang

Zhu

(2007) A shape memory alloy-based reusable hysteretic damper for seismic hazard mitigation. Smart Material Structures 16(5): 1603–1613.

26.

Zhu

Qiu

(2014) Incremental dynamic analysis of highway bridges with novel shape memory alloy isolators. Advances in Structural Engineering 17(3): 429–438.