Dynamic response of a high-rise chemical tower controlled by pseudo-elastic shape memory alloy cables

Abstract

The wind-induced vibration of a high-rise chemical tower controlled by pseudo-elastic shape memory alloy cables was studied in this work. Based on taking into account phase transformations of shape memory alloy, the motion equation of the shape memory alloy–controlled chemical tower was established by the finite element method, which was coupled with phase transformation equations of shape memory alloy. The incremental finite-element-based Newmark integration method and the iterative method were utilized to solve this motion equation, and the dynamic response of the shape memory alloy–controlled chemical tower was analyzed. The results were compared with those of the chemical tower without any cables and under the control of common cables. It is found that pseudo-elastic shape memory alloy cables have good vibration suppression effect on the chemical tower. In addition, the vibration suppression effect of shape memory alloy cables is affected by the operating temperature, initial pre-strain of shape memory alloy, cross-sectional size of shape memory alloy, and the angle between each shape memory alloy cable and the ground.

Keywords

Dynamic response wind-induced vibration shape memory alloy high-rise chemical tower vibration control

Introduction

High-rise chemical towers are important equipments in the petro-chemical, pharmaceutical, and civil engineering. Because of the slender shape and high flexibility, high-rise chemical towers often vibrate dramatically with larger amplitudes when they are subjected to strong winds or earthquakes (Hao, 2012). This not only can result in non-uniform exchange of the materials in chemical towers but also can result in the cracking of skirt or cylinder and even collapse of towers, which will bring serious damages and great hazards to human life. Therefore, preventing the large-amplitude vibration of high-rise chemical towers and reducing the hazards effectively become an urgent and important task in the petro-chemical field.

Currently, the main vibration control methods in practice for chemical towers consist of raising natural frequencies of towers by increasing thickness to avoid resonance, increasing damping of towers by installing rubber strips and dampers to suppress vibration and installing frames or common cables to fasten towers, and so on (Nie, 1991; Shi et al., 2013; Yuan, 2009). Although these traditional measurements are proved to have some suppression effects on resonance or large-amplitude vibration of chemical towers, most of them are at the price of sacrificing costs. In addition, the optimal vibration control effect on towers by these measurements is hard to ensure due to the randomness and unpredictability of wind and seismic loads. This hinders seriously the rapid development of petro-chemical enterprises.

As one kind of novel smart materials, shape memory alloys (SMAs) are the potential candidates for passive vibration control of structures due to their stable mechanical properties, high fatigue resistance characteristics, high pseudo-elastic hysteretic damping characteristics, and so on (Khalili et al., 2013). The mechanism of using pseudo-elastic SMA for passive vibration control is based on the stress-induced martensite transformation of this material. If the temperature is higher than the austenite finish temperature, the stress-induced martensite transformation will occur in SMA entirely in austenite phase when stress is applied to this material. While the reverse transformation to austenite will take place during unloading of the material because of the instability of martensite in the absence of stress. This recovery of high strain values upon unloading yields a characteristic hysteresis loop known as pseudo-elasticity. And the area of this hysteresis loop denotes the energy dissipated per unit volume in SMA. If pseudo-elastic SMA cables or wires are fixed on high-rise chemical towers, SMA will undergo cyclic deformation and thus absorb much energy during the vibration of towers, and then the vibration of chemical towers may be suppressed passively. Because no power input is needed and the optimal vibration control is easy to be realized, SMA can provide an effective way to develop advanced vibration control techniques for high-rise chemical towers.

Many applications of pseudo-elastic SMA have been reported in civil engineering (Song et al., 2004) such as high-rise buildings. But few were found on the applications of SMA in the vibration control of chemical towers. Based on our previous work on the application of a pseudo-elastic SMA element in a single-degree-of-freedom system (Sun and Rajapakse, 2003) and on the application of pseudo-elastic SMA braces in a frame structure (Sun et al., 2006), a high-rise chemical tower controlled by two pseudo-elastic SMA cables is considered in this article. The dynamic response of the SMA-controlled chemical tower subjected to the fluctuating wind will be studied by the finite element method. The vibration suppression effect of SMA cables on the chemical tower will be analyzed. The influence of the operating temperature, initial pre-strain of SMA, cross-sectional size of SMA, and the angle between each SMA cable and the ground on the vibration suppression effect of SMA will be discussed. This study is expected to provide some guidance for the application of SMA in passive vibration control of chemical towers and further lay a foundation for realizing intelligent vibration control of high-rise chemical towers.

Problem descriptions

The high-rise chemical tower is shown in Figure 1(a). Due to its simple internal structure and equal wall thickness, the chemical tower is simplified as a homogeneous cantilever beam with equal cross sections as shown in Figure 1(b). As shown in Figure 1(b), two pseudo-elastic SMA cables with the same material and size are connected to the top of the tower and the ground symmetrically. Assume that the fluctuating wind parallel to the ground is applied along the height of the chemical tower. To ensure both SMA cables to be always in the tension state during working, they have been pre-stretched to the same extent before being connected to the tower.

In Figure 1(b), H denotes the initial height of the chemical tower. γ represents the initial angle between each SMA cable and the ground. The chemical tower with SMA cables in dotted line represents the new configuration of the tower under the action of the fluctuating wind at some time. u_C and v_C represent the horizontal and vertical displacement components of the top of the chemical tower in the global coordinate system xOy, respectively.

Theoretical analysis

Calculation of the wind load

The fluctuating wind (Wu, 2011) shown in Figure 2(a) is employed in this article. According to the characteristics of the wind spectrum varying with the height of the chemical tower, the fluctuating wind in Figure 2(a) is simplified as piecewise uniform loads as shown in Figure 2(b), where q_i (i = 1, 2,…, n) denotes the intensity of the uniform load of the i section; Y_i denotes the top height of the i section of the chemical tower.

The intensity of the uniform load of the i section q_i can be expressed as (Wu, 2011)

q_{i} = B \times (- 0.1033 \times Y_{i}^{2} + 6.3136 \times Y_{i} + 269.5626) \times abs (\cos (ω \times t))

(1)

where B denotes the correction coefficient of wind load; ω denotes the excitation angular velocity and t denotes time.

And the total wind load of the i section W_i can be written as (Wu, 2011)

W_{i} = q_{i} \times S \times A

(2)

where S denotes the shape coefficient of the wind load; A denotes the windward area of the i section of the tower.

Characterization of the pseudo-elastic behavior of SMA

The constitutive model of SMA proposed by Brinson (1993) is adopted to characterize the pseudo-elastic behavior of SMA in this study. Brinson’s model can be described using the phase transformation diagram shown in Figure 3, where “ab” and “cd” denote the finish line and the start line of the forward phase transformation (austenite to martensite), and “ef” and “gh” denote the start line and the finish line of the reverse phase transformation (martensite to austenite), respectively. $σ_{s}^{cr}$ and $σ_{f}^{cr}$ denote the critical stresses at the start and the finish of the conversion of the martensite variants, respectively. C_M and C_A denote slopes of the forward and reverse phase transformation lines, respectively. M_s, M_f, A_s, and A_f represent four transition temperatures of martensite start, martensite finish, austenite start, and austenite finish in the stress-free state, respectively.

As shown in Figure 3, if SMA is loaded along the path $\vec{16}$ and unloaded along the path $\vec{61}$ at temperature T₁ (T₁ > A_f), a complete pseudo-elastic hysteresis loop with forward phase transformation (austenite to martensite) and reverse phase transformation (martensite to austenite) will take place in SMA (as shown in Figure 4). The area of the hysteresis loop denotes the energy dissipated per unit volume in SMA during the isothermal mechanical loading and unloading processes.

The phase transformation equations of SMA in Brinson’s model are as follows:

Conversion to martensite for T > M_s and $σ_{s}^{cr} + C_{M} (T - M_{s}) < σ < σ_{f}^{cr} + C_{M} (T - M_{s})$

ξ_{S} = \frac{(1 - ξ_{S 0})}{2} \cos {\frac{π}{(σ_{s}^{cr} - σ_{f}^{cr})} [σ - σ_{f}^{cr} - C_{M} (T - M_{s})]} + \frac{(1 + ξ_{S 0})}{2}

(3)

ξ_{T} = ξ_{T 0} - \frac{ξ_{T 0}}{(1 - ξ_{S 0})} (ξ_{S} - ξ_{S 0})

(4)

Conversion to austenite for T > A_s and $C_{A} (T - A_{f}) < σ < C_{A} (T - A_{s})$

ξ = \frac{ξ_{0}}{2} {\cos [a_{A} (T - A_{s} - \frac{σ}{C_{A}})] + 1}

(5)

ξ_{S} = ξ_{S 0} - \frac{ξ_{S 0}}{ξ_{0}} (ξ_{0} - ξ)

(6)

ξ_{T} = ξ_{T 0} - \frac{ξ_{T 0}}{ξ_{0}} (ξ_{0} - ξ)

(7)

where $σ$ denotes stress, T denotes temperature, and $ξ$ denotes martensite fraction that includes two parts, $ξ_{S}$ and $ξ_{T}$ , induced by stress and temperature, respectively. The parameter $a_{A}$ is defined as $a_{A} = \frac{π}{A_{f} - A_{s}}$ . The quantities with subscript “0” denote initial conditions.

The basic stress–strain relationship in Brinson’s model can be expressed as

σ - σ_{0} = D (ξ) ε - D (ξ_{0}) ε_{0} + Ω (ξ) ξ_{S} - Ω (ξ_{0}) ξ_{S 0} + Θ (T - T_{0})

(8)

where $ε$ and $Θ$ denote strain and thermoelastic modulus, respectively. $D (ξ)$ and $Ω (ξ)$ denote the elastic modulus and the transformation tensor, respectively, expressed by

D (ξ) = D_{a} + ξ (D_{m} - D_{a})

(9)

Ω (ξ) = - ε_{L} D (ξ)

(10)

where $D_{a}$ and $D_{m}$ denote the modulus of 100% austenite and 100% martensite in SMA, respectively; $ε_{L}$ is the maximum residual strain.

Deformation of SMA cables under the wind load

Let $L (0)$ represent the initial length of each SMA cable in the stress-free state. $L_{AC'} (t)$ and $L_{BC'} (t)$ represent the instantaneous length of cable 1 and cable 2 at time t, respectively. According to the geometric relationship shown in Figure 1(b), we have

L_{AC'} (t) = \sqrt{(H - v_{C})^{2} + {(\frac{H}{\tan γ} + u_{C})}^{2}}

(11)

L_{BC'} (t) = \sqrt{(H - v_{C})^{2} + {(\frac{H}{\tan γ} - u_{C})}^{2}}

(12)

Then the instantaneous strain of each SMA cable can be written as

ε_{AC'} (t) = \frac{(L_{AC'} (t) - L (0))}{L (0)}

(13)

ε_{BC'} (t) = \frac{(L_{BC'} (t) - L (0))}{L (0)}

(14)

where $ε_{AC'} (t)$ and $ε_{BC'} (t)$ denote the instantaneous strain of cable 1 and cable 2 at time t, respectively.

Motion equation of the chemical tower controlled by SMA cables

The finite element method is used to investigate the dynamic response of the chemical tower shown in Figure 1(b). The beam element with 3 degrees of freedom (u_i, v_i, θ_i) in each node is utilized to discretize the chemical tower and the link element with 2 degrees of freedom (u_i, v_i) is utilized to discretize the SMA cables in the finite element model, where u_i and v_i are the longitudinal and lateral displacement components in local coordinate systems, respectively; θ_i is the rotation angle.

The element equilibrium equations in local coordinate systems can be written as

[m_{b}] {{\overset{\cdot\cdot}{δ}}_{b}}^{e} + [c_{b}] {{\overset{\cdot}{δ}}_{b}}^{e} + [k_{b}] {δ_{b}}^{e} = {f_{b}}^{e}

(15)

[m_{l}] {{\overset{\cdot\cdot}{δ}}_{l}}^{e} + [c_{l}] {{\overset{\cdot}{δ}}_{l}}^{e} + {f_{s}} = {f_{l}}^{e}

(16)

where [m], [c], and [k] denote the mass matrix, damping matrix, and stiffness matrix of an element, respectively. ${\overset{\cdot\cdot}{δ}}^{e}$ , ${\overset{\cdot}{δ}}^{e}$ , ${δ}^{e}$ , and ${f}^{e}$ denote the nodal acceleration vector, velocity vector, displacement vector, and force vector in an element, respectively. The subscripts “b” and “l” denote the beam and the link element, respectively. And ${f_{s}}$ is the internal force vector produced by SMA. If no phase transformation occurs in SMA, then ${f_{s}}$ can be directly expressed as

{f_{s}} = [k_{l}] {δ_{l}}^{e}

(17)

where $[k_{l}]$ represents the element stiffness matrix of SMA in 100% martensite or 100% austenite state. When SMA undergoes phase transformation, ${f_{s}}$ is displacement history dependent and cannot be written as the explicit expression like equation (17).

In order to establish the total motion equation of the chemical tower controlled by SMA cables by the classical finite element method, equations (15) and (16) are rewritten as the following incremental form

[m_{b}] {Δ {\overset{\cdot\cdot}{δ}}_{b}}^{e} + [c_{b}] {Δ {\overset{\cdot}{δ}}_{b}}^{e} + [k_{b}] {Δ δ_{b}}^{e} = {Δ f_{b}}^{e}

(18)

[m_{l}] {Δ {\overset{\cdot\cdot}{δ}}_{l}}^{e} + [c_{l}] {Δ {\overset{\cdot}{δ}}_{l}}^{e} + {Δ f_{s}} = {Δ f_{l}}^{e}

(19)

where ${Δ f_{s}}$ can approximately be expressed as

{Δ f_{s}} = [k_{l}]_{T} {Δ δ_{l}}^{e}

(20)

where $[k_{l}]_{T}$ denotes the tangent stiffness matrix of SMA element, which can be obtained by employing the linearization method proposed by Brinson and Lammering (1993) and the iterative method proposed by Sun and Dong (2010).

Assembling all the elements in the global coordinate system, we will get the total motion equation of the SMA-controlled chemical tower as follows

[M] {Δ \overset{\cdot\cdot}{δ}} + [C] {Δ \overset{\cdot}{δ}} + [K] {Δ δ} = {Δ F}

(21)

where $[M]$ , $[C]$ , and $[K]$ are the mass matrix, damping matrix, and stiffness matrix of the SMA-controlled chemical tower in the global coordinate system, respectively. ${Δ \overset{\cdot\cdot}{δ}}$ , ${Δ \overset{\cdot}{δ}}$ , ${Δ δ}$ , and ${Δ F}$ denote the incremental nodal acceleration, velocity, displacement, and force vector in the global coordinate system, respectively. Here, the damping matrix $[C]$ is taken as Rayleigh damping

[C] = α [M] + β [K]

(22)

where coefficients α and β are determined by the following expressions (Hao, 2012)

α = \frac{2 ω_{1} ω_{2} (ζ_{1} ω_{2} - ζ_{2} ω_{1})}{ω_{2}^{2} - ω_{1}^{2}}

(23a)

β = \frac{2 (ζ_{2} ω_{2} - ζ_{1} ω_{1})}{ω_{2}^{2} - ω_{1}^{2}}

(23b)

In equations (23a) and (23b), $ω_{1}$ and $ω_{2}$ denote the first and the second natural frequencies of the SMA-controlled chemical tower. $ζ_{1}$ and $ζ_{2}$ are the corresponding damping ratios of vibration modes.

Note that the total motion equation (21) is a nonlinear differential equation coupled with phase transformation equations of SMA when phase transformation occurs. It is solved by combining the Newmark integration method (Chopra, 1995) and the iterative method in this study. The details are not given here for brevity.

Numerical simulation and discussions

In this section, the dynamic response of a high-rise medium pressure flash tower controlled by NiTi-SMA cables will be simulated by the Newmark integration method based on MATLAB programming. The tower is subjected to the fluctuating wind shown in Figure 2(a). The sizes and material parameters of this tower (Hao, 2012) are as follows: inner radius of 1400 mm, wall thickness of 14 mm, height of 30,000 mm, density of 7850 kg/m³, elastic modulus of 200 GPa, and damping ratio ζ = ζ₂ = 0.02. The material properties of NiTi-SMA cables (Sun et al., 2008) used in simulation are given in Table 1.

Table 1.

Material properties of NiTi-SMA cables.

Moduli	Transformation constants	Transition temperatures	Other constants
D_a = 41.97 GPa	C_M = 7.067 MPa/°C	M_f = −101.2°C	ε_L = 0.066
D_m = 13.99 GPa	C_A = 9.637 MPa/°C	M_s = −37.85°C	ρ = 6500 kg/m³
	$σ_{s}^{cr} = 0 MPa$	A_s = −15.12°C
	$σ_{f}^{cr} = 448 MPa$	A_f = 5.71°C

ρ denotes the density of shape memory alloy.

The parameters of the fluctuating wind are: the excitation angular velocity ω = 1.5 rad/s, the correction coefficient B = 1, the shape coefficient S = 20, the total sampling time t = 15 s, and the sampling time interval Δt = 0.1 s. Unless otherwise specified, the other parameters used in the numerical study are as follows: T = T₀ = 25°C, the initial pre-strain of SMA ε₀ = 3%, the diameter of each SMA cable of 80 mm, the angle between each SMA cable and the ground γ = 30°, and initial conditions ${δ_{0}} = {{\overset{\cdot}{δ}}_{0}} = {0}$ .

Verification of the finite element program and the numerical scheme

The high-rise medium pressure flash tower is divided into 10 beam elements and each SMA cable is divided into two link elements in the present finite element study. The fluctuating wind shown in Figure 2(a) is simplified as piecewise uniform loads shown in Figure 2(b) with 10 sections, which can be converted into equivalent nodal loads on each node of the beam elements.

In order to verify the reliability of the present finite element program and the accuracy of the numerical scheme based on the Newmark integration method, the commercial software ANSYS is also used to analyze the wind-induced vibration of the chemical tower controlled by SMA cables and that of the same tower without any cables. Both the tower and the SMA cables are meshed by the element of Solid 185 in ANSYS. It is pointed out that some material parameters of SMA used in ANSYS are based on the calculation results from MATLAB, which are given in Table 2 and obtained according to the method proposed by Wu (2012).

Table 2.

Material parameters of SMA used in ANSYS.

D_a (GPa)	D_m (GPa)	C₁ (MPa)	C₂ (MPa)	C₃ (MPa)	C₄ (MPa)	C ₅	C₆ (m/s)
41.97	13.99	444	892	386	185	0.066	0

In Table 2, C₁ and C₂ denote the start and the finish stress of the forward phase transformation (austenite to martensite), respectively; C₃ and C₄ denote the start and the finish stress of the reverse phase transformation (martensite to austenite), respectively; C₅ denotes the maximum residual strain and C₆ is the phase change rate.

Figure 5 shows the horizontal displacement response of the top node of the chemical tower with SMA cables and without any cables when the tower is subjected to the fluctuating wind. Both the results from ANSYS and those from MATLAB are given in this figure for comparison.

It can be seen from Figure 5 that the present MATLAB solutions are very close to those from ANSYS for the chemical tower whether it is controlled by SMA cables or not. This confirms the reliability of the present finite element program and the high accuracy of the present numerical scheme based on the Newmark integration method.

Dynamic response of the chemical tower under different control schemes

To observe the vibration suppression effect of SMA cables on the chemical tower, three schemes for the chemical tower with SMA cables, with common cables, and without any cables were considered to study the dynamic response of the tower. The results are given in Figures 6 to 10. Here, the material of each common cable is steel with elastic modulus of 206 GPa, Poisson’s ratio of 0.269, density of 7850 kg/m³, and tensile strength of 600 MPa. The size and the angle between each common cable and the ground are the same as those of each SMA cable.

Figure 6 shows the horizontal displacement response of the top node of the chemical tower under three schemes. We can see from Figure 6 that the displacement of the tower in every half period increases first with time gradually until it reaches the peak value and then decreases with time until it becomes about 0 for each scheme. Although the tensile stiffness of common cables is much higher than that of SMA cables, the peak value of displacement for the SMA-controlled tower is the smallest in the three schemes in every half period, which is about half of the peak value for the tower without any cables. This is mainly due to the reason that pseudo-elastic SMA cables absorbed much energy during the vibration of the tower. This can be clearly seen from the stress–strain curves of SMA cables in Figure 7. In addition, a short plateau can approximately be found in the increase or decrease in the displacement response curve for the SMA-controlled tower in every half period, which results from phase transformations of SMA. This is illustrated in Figure 8.

Figure 7 gives stress–strain curves of two SMA cables fixed on the chemical tower. When the chemical tower vibrates under the action of the fluctuating wind, two SMA cables are alternately in the state of loading and unloading. Then the pseudo-elastic hysteresis loops shown in Figure 7 for each SMA cable are formed. And the pseudo-elastic hysteresis loops of SMA become stable after about one loading–unloading cycle. In this figure, 1-2-3-4-5-6-7-8 denotes the stress–strain path of cable AC and 1-2′-3′-4′-5′-6′-7′-8′denotes the stress–strain path of cable BC.

The martensite fraction of each SMA cable varying with time is given in Figure 8 during the vibration of the chemical tower. We can see from Figure 8 that partial phase transformation occurs in each SMA cable during the whole time, but the martensite fraction of cable AC is always larger than that of cable BC. The phase transformation of each SMA cable enters into the steady state at about 3 s. The results in the above section demonstrate that pseudo-elastic SMA cables have great advantages of being used for the passive vibration control of chemical towers.

Influence of some parameters on the dynamic response of the SMA-controlled chemical tower

The dynamic response of the chemical towers controlled by SMA cables is further investigated in four aspects in this study, which includes the influence of operating temperature, initial pre-strain of SMA cables, cross-sectional size of SMA cables, and the angle between each SMA cable and the ground. For each aspect, only the horizontal displacement response of the top node of the chemical tower is given in the following for brevity.

Figure 9 shows the displacement response of the SMA-controlled chemical tower at three operating temperatures of 10°C, 25°C, and 40°C. It is found from Figure 9 that the trend of each curve varying with time is very similar. But the peak values of displacement in every half period are different at the three temperatures. The lower the temperature, the smaller the peak value of displacement. For instance, the peak value of displacement at 40°C is about 0.22 m when time reaches about 13.5 s, while it is about 0.2 m at 10°C at this time, which is nearly 10% lower than that at 40°C. This demonstrates that pseudo-elastic SMA cables show better vibration suppression effect on the chemical tower at lower operating temperatures. This is probably because that pseudo-elastic SMA cables can absorb much more energy at lower temperatures, which can be verified by comparing the area of hysteresis loops of SMA at the three operating temperatures.

The influence of initial pre-strains of SMA on the displacement response of the tower is given in Figure 10, where three initial pre-strains of SMA with 2%, 3%, and 4% are considered. From Figure 10, we can see that the influence of initial pre-strains of SMA on the displacement response of the tower is not obvious before the vibration reaches the steady state. But the peak value of displacement of the SMA-controlled tower decreases slightly with the decrease in pre-strains of SMA in the steady state. This is probably because that SMA cables with smaller pre-strains not only can dissipate energy but also can keep higher stiffness. Figure 11 shows the displacement response of the SMA-controlled chemical tower when the radius R of each SMA cable is 40, 45, and 50 mm.

From Figure 11, we can see that the influence of cross-sectional size of SMA cables on the displacement response of the tower is dramatic. The peak value of displacement decreases with increase in the radius of SMA cables. For instance, the peak value of displacement is about 0.2 m when R = 40 mm at the time range between 12 and 15 s, while it is about 0.17 m when R = 50 mm at this time range, which is 15% lower than that when R = 40 mm. This results from an increase in stiffness with an increase in the radius of SMA cables. The effect of the angle between each SMA cable and the ground on the displacement response of the SMA-controlled chemical tower is shown in Figure 12.

We can see from Figure 12 that the peak value of displacement in every half period decreases with the decrease in the angle. For instance, the last peak value of displacement when γ = 30° is about 10% lower than that when γ = 60° during the total sampling time. When the angle between each SMA cable and the ground becomes smaller, the horizontal component of the resultant tensile force produced by two SMA cables will become larger. This horizontal component of the resultant tensile force will reduce the maximum displacement because of its opposite direction to the fluctuating wind.

Conclusion

The dynamic response of an SMA-controlled chemical tower subjected to the fluctuating wind was studied by the finite element method in this article. The phase transformations of SMA were taken into account. The motion equation of the SMA-controlled tower that is coupled with phase transformation equations of SMA was solved by combining the incremental finite-element-based Newmark integration method and the iterative method. The results were compared with those of the chemical tower without any cables and under the control of common cables. The effects of the operating temperature, initial pre-strain of SMA, cross-sectional size of SMA, and the angle between each SMA cable and the ground on the displacement response of the chemical tower were discussed. The following conclusions can be drawn:

Compared with common cables, pseudo-elastic SMA cables have good vibration suppression effect on chemical towers because of the pseudo-elastic hysteresis damping characteristics.

Pseudo-elastic SMA cables show better vibration suppression effect on chemical towers at lower operating temperatures.

Pseudo-elastic SMA cables with lower initial pre-strains can reduce the maximum vibration displacement of chemical towers more obviously.

Increasing the cross-sectional sizes of SMA cables and decreasing the angle between each SMA cable and the ground can improve the vibration suppression effect of SMA on chemical towers.

This research shows that pseudo-elastic SMA cables are very suitable for passive vibration control of chemical towers. This study can provide the theoretical basis for the application of SMA in passive vibration control of chemical towers and will further lay a foundation for realizing the intelligent vibration control of chemical towers.

Footnotes

Declaration of Conflicting Interests

The authors 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 was supported by the Natural Science Foundation of Shandong Province (No. ZR2014AL011), the Project of Shandong Province Higher Educational Science and Technology Program (No. J13LB08), and the Project of Science and Technology Development Plan of Qingdao (No.13-1-4-150-jch).

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