Design of a variable stiffness bracing system_

Soil Dynamics and Earthquake Engineering 80 (2016) 87–101

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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Design of a variable stiffness bracing system: Mathematical modeling, fabrication, and dynamic analysis Amir Fateh a, Farzad Hejazi a,n, Mohd Saleh Jaafar a, Izian Abd. Karim a, Azlan Bin Adnan b a b

Department of Civil Engineering, Faculty of Engineering, University Putra Malaysia, Serdang, Selangor, Darul Ehsan, Malaysia Faculty of Civil Engineering, University Technology Malaysia, 81310 UTM Skudai, Johor, Malaysia

art ic l e i nf o

a b s t r a c t

Article history: Received 2 November 2014 Received in revised form 7 October 2015 Accepted 8 October 2015

This paper presents a new bracing system with variable stiffness springs; this adaptive structural control system is designed to protect buildings against severe vibration and ground movement. The developed variable stiffness bracing (VSB) system comprises four nonlinear steel leaf springs that provide nonlinear and variable stiffness capacity at different frame displacements. The inelastic actions of the VSB system's nonlinear leaf springs keep the energy dissipation characteristics and ductility of moment-resisting frames. At large vibration amplitudes, the VSB device restrains unallowably story drift. Therefore, frames display ductile performance. We developed a mathematical model to simulate the mechanical behavior of the system, including the stiffness nonlinearity of the springs. Moreover, we evaluated the efficiency of the VSB implementation in a single-degree-of-freedom system by dynamically analyzing different models: a moment-resisting frame, a conventional braced frame, and a frame using the VSB system. This article discusses and proves the effectiveness of the proposed system through numerical analysis. & 2015 Elsevier Ltd. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Variable stiffness bracing system Newmark method Dynamicanalysis Mathematical model Energy dissipation Ductility

1. Introduction Active variable stiffness (AVS) system has attracted increasing attention in recent decades. Previous studies have reported the desirable effects and the enhanced seismic performance of structures equipped with AVS systems. One such system was analyzed experimentally with its implementation in a full-scale test building in Japan [1,2]. Moreover, AVS action was optimized in a multi degree structure using a computational algorithm. Previous studies [3–5] mainly focused on the optimization of structural control systems without considering damper device design and function. Two structural systems were considered: the AVS system, which uses electric power to change the device’s stiffness, and a viscous damper device, a passive control system, which lacks the ability to change the device’s stiffness during seismic excitation. The development of an adaptive earthquake energy dissipation device has become essential to optimize control of structures without depending on electric power, which is not reliable during seismic excitation. When a structural system becomes complicated and earthquake motion becomes stronger, increased energy is required to run force-type schemes. Numerous hybrid and semi-active methods have been introduced to overcome this energy problem. These n

Corresponding author. E-mail addresses: [email protected] (A. Fateh), [email protected] (F. Hejazi), [email protected] (M.S. Jaafar), [email protected] (I.Abd. Karim), [email protected] (A.B. Adnan).

methods used stiffness control devices have been used to adjust the rigidity and consequently modify the dynamic characteristics of structures. These AVS systems have been studied by several researchers [6–8]. In particular, the AVS system comprises locking and unlocking parts that are attached to the brace to withstand seismic excitation. The system’s control algorithm can lock some of its parts to increase story stiffness at a particular time to decrease the structural seismic response [9]. A new semi-active variable stiffness (SAVS) device has been developed to overcome the constraints of the conventional on-off form of variable stiffness systems. The SAVS device can smoothly and continuously change between minimum and maximum stiffness. The efficiency of a miniature scale model of the SAVS device as a variable stiffness component in a system with a single degree of freedom (SDOF), has been tested. It also verified to reach a nonresonant state by switching the stiffness continuously and consequently, this process reduces displacement and acceleration [10]. Others have proposed a control system that alters the structure’s response to a nonresonant state during earthquakes by modifying the buildings’ stiffness [11]. Stiffness is altered via locking or unlocking of certain devices located between the beams and/or the diagonal braces of a structure. However, delivering a defective signal to a variable stiffness device (VSD) system remains possible, and the activation of a VSD will modify the structural dynamic behaviors without inducing peripheral vibrations or applying unwanted forces to it. The performance of a variable stiffness mount device has been proven experimentally. The VSD was constructed by the pre-stress stiffness of a cable-based mechanism. Changing pre-stress using a piezo actuator

http://dx.doi.org/10.1016/j.soildyn.2015.10.009 0267-7261/& 2015 Elsevier Ltd. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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A. Fateh et al. / Soil Dynamics and Earthquake Engineering 80 (2016) 87–101

2

4

7 3 5

6 8

1

Fig. 1. Schematic body of the VSB System.

Fig. 2. Prototype of the VSB System (PI NO:2014701608).

and a simple on–off controller significantly changes stiffness within a short time and at low energy outlays [12]. Moreover, a cable-driven manipulator (CDM) is a unique parallel manipulator in which the mobile platform is driven by cables as an alternative to rigid links. CDMs have been comprehensively investigated in prior studies [13]. Previous researchers have investigated various adjustable vibration absorbers with adaptable stiffness in which the frequency of excitation is changed. These adjustable absorbers can be implemented in buildings, automobiles, and floating rafts, where uncontrolled vibrations induce fatigue, inconvenience, expanded maintenance expenses, and reduced performance [14–17]. Recent studies related to variable stiffness actuators have mostly concentrated on four major techniques: pneumatic actuators, electroactive polymers (EAPs), electrical motors with active compliance, and adaptable stiffness components. EAPs change shape with the application of voltage. They can tolerate large amounts of deformation but possess high nonlinearity [18,19]. Pneumatic artificial muscles are normally contractile apparatuses that use pressurized air to control muscle stiffness. They necessitate an antagonist configuration to create a force or restore movement. Most such pneumatic machines are based on the McKibben muscle [20], which shortens, bulges, and produces a contraction force upon inflation. The static and dynamic behaviors of the McKibben artificial muscle pneumatic actuator have been evaluated by experimental tests [21]. However, these devices have several main defects, such as low efficiency, poor accuracy, and the need for an air compressor. An actuator that incorporates an electrical DC motor and an elastic element with variable stiffness has been proposed [22]. The elastic part supplements the mechanical structure of the device to adapt to the dynamic effects of external forces. Another manipulator was designed with a variable stiffness system. The manipulator allowed the regulation of its stiffness and would befit various intended tasks. Device stiffness was a function of cable tension [23]. Another adjustable stiffness actuator was used as a robotic component. Modifying the shape of its springs allowed the alteration of the actuator’s global stiffness of the actuator [24]. Another study proposed the design of a diagonal braced structure equipped with AVS and a control algorithm. The control algorithm would activate the brace action in the frame based on nonresonance theory. Therefore, brace stiffness would vary with frequency. The design approach and the efficiency of the control algorithm were investigated by numerical simulation, and the results demonstrated the response diminution of

Fig. 3. VSB Installation layout in frame.

structures furnished by the developed controller device [25]. To date, little is known about techniques for retrofitting moment-resisting steel frame (MRSF) structures with smart VSB comprising a passive multivariable stiffness spring without reducing the effect of inherent ductility. The present study reports our development of a new adaptive variable stiffness device having no dependency on any sort of power or even on an active controller. The proposed device can alter stiffness, based on the load transferred to the curved steel spring, because of its geometric specifications. This property significantly distinguishes the VSB device from those reported in previous studies, which normally altered device stiffness using active controllers.

2. Development of a VSB system This study set out to design an effective method of controlling a frame’s structural response to dynamic loads and vibration. Based on the advantages of the variable stiffness concept, we developed a high-performance VSB system for framed structures. We adopted a Page: 3 design procedure that optimized different elements of the VSB system to increase its functionality and performance, thereby diminishing vibration and dynamic loading in the structure Fig. 1 shows the schematic of the VSB system, which implements four leaf springs to act in bending situations under a large displacement (Label 1). Label 2 refers to a cylindrical core that can move left and right in a longitudinal direction through a steel rail (Label 3). Rods (Label 4) pass through the side’s plates (Label 5) and are fixed to the cubic core (Label 6). When force is applied to the cable, the steel core moves and makes contact with the Cshaped member (Label 2), where the spring is clamped. The Cshaped elements help to maintain the initial spring shape and change it during system performance. The global stiffness of the nonlinear spring should be protected from curvature extension. Therefore, four quarter solid cylinders (Label 7) and two C-shaped elements act as essential supports. Furthermore, the spring protection systems (Labels 2 and 7) guarantee that the springs do not yield when they reach the highest value of curvature. This system increases the lateral stiffness of a story without any reduction to the moment frame's ductility. The VSB system does not move too much at small or medium vibration amplitudes, but does move at large amplitudes, controlling unallowably large story drift. The


1 1

1

1 1

89

1

2

Fig. 4. Action of VSB System in frame. (a) Imposed cyclic load. (b) Left to right action of VSB. (c) Right to left action of VSB.

yf

Xaf

yf(mm)

Fig. 5. Structural model of curve leaf spring. 70 60 50 40 30 20 10 0

yf(a=2) yf(a=2.2) yf(a=2.4) yf(a=2.6) yf(a=2.8) yf(a=3) yf(a=3.2) yf(a=3.4)

0

50

100

150

200

xaf(mm)

Fig. 6. Shape of curve spring for various aspect ratios.

VSB system installs easily on the bottom beam or foundation using a horizontal VSB plate (Label 8). Fig. 2 depicts the VSB prototype.

3. Mathematical modeling of proposed VSB system Identifying the curved spring dimensions is the first stage in the VSB design process. We selected a leaf spring with a width (w) of 100 10 3 m, a thickness (t) of 1 10 2 m, and a variable aspect ratio (α A ) of (2.0 and 3.0). Whenever the geometric specifications of the device are assumed, the radius of the roller, the spring’s curvature, and the C-shaped elements that prevent the spring from yielding are calculated. The radius of the mentioned elements must be selected with a higher value to ensure that the spring remains in the elastic range. The common equation for calculating the bending stress in a beam under pure bending is: M I

σ ¼ ymax 2.1. VSB installation and action in frame Fig. 3 illustrates the installation of the VSB system in a steel frame. The VSB system attaches to the frame via a wire cable. The base plate of the VSB uses bolts to connect to either the lower beam or foundation. The wire cable attaches to rods on the VSB system. Fig. 4 presents the function of the VSB device subjected to monotonic load. The lateral cyclic load is imposed at the top of the frame (node 1) from left to right and vice versa. The frame is intended to move to the right; therefore, cable 1 is operated as the compression member and buckled [Fig. 4(b)]. However, the buckling deficiency for the compression component is eliminated entirely by using a cable. By contrast, cable 2 functions as a tension member, and tensile force is transferred to the VSB. The VSB moves to the left. For the right-to-left orientation of imposed load, cables 1 and 2 operate as compression and tension elements, respectively, in the situation shown in Fig. 4(c). Here, the VSB device tends to shift to the right. Therefore, the proposed system controls the story displacement within the allowable range. Since the VSB action is designed to restrict the in-plane movement of frame, thus it must be installed on both sides of the building’s frame to capture the seismic loads transferred to the major axes of the structure, like other lateral-resisting systems.

ð1Þ

where σ is the bending stress, M is the moment at the neutral axis, and ymax is the perpendicular distance to the neutral axis (ymax ¼ t/2). The ratio between the bending moment (M) and radius of curvature (R) can be expressed as:

σ¼

EI R

ð2Þ

where EI is the flexural stiffness. The equation for the minimum limit of the spring's radius in the elastic region can be rewritten as [24]: R¼

Et 2σ y

ð3Þ

where σy is yield stress; using E ¼210 GPa, t ¼1 10 3 m, and σy ¼1700 MPa (maraging steel type 18Ni1700), the radius of the protection system (Labels 2 and 7) computes to R¼ 59.11 mm (the minimum value that can be used for spring curvature). Various mathematical models have been proposed with different aspect ratios; spring shape and stiffness values are obtained by the extension of nonlinear flexible bar theory [26]. As shown in Fig. 5, each curved spring can be considered a beamtype component fixed at one extreme and with horizontal displacement-free at the other extreme. This spring operates in a bending situation only. Thus, the normal curvature equation cannot be

90


yf(a=2) yf(a=2.5) yf(a=3) yf(a=3.5)

40 yf(mm)

30 20 10 0

0

50

100 xaf(mm)

0.4 0.2 0 -0.2 -0.4

0

10

20

30

40

50

60

Ground acceleration (g)

Ground acceleration (g)

Fig. 7. Shape of leaf spring (a) based on derived formula (b) by prior study [24].

0.6 0.4 0.2 -1E-15 -0.2 -0.4 -0.6

0

10

Time(Sec)

y″ðxÞ 1 þ y0 ðxÞ

2

C C 32 A ¼ 0;

0 ox oxαf

ð4Þ

The boundary conditions at the left extreme (x¼ 0) indicate that the leaf spring is fastened at the mentioned points. Moreover, at the α lower extreme (roller support x¼xf ), the spring is clamped but can move longitudinally. The differential equation is exceedingly nonlinear; therefore, for this particular circumstance and after integrating two times, the equation can be reduced with discrete variables [using a new alteration in variables, such as w(x)¼y0 (x)], thereby establishing a solution in a closed form as shown in Eq. (5). The constant values for the two integrations of Eq. (4) are c and d. EI

w 0 ðx Þ ð1 þw2 ðxÞÞ3=2

¼ cx þ d

ð5Þ

By integrating both sides of the aforementioned equation, the new equation can be presented as follows: Z EI

w 0 ðx Þ ð1 þ w2 ðxÞÞ3=2

¼c

x2 þ dx þ e 2

ð6Þ

where 2 2 cosh a sinh a ¼ 1 2 w ¼ t ¼ sinh a 2 dw ¼ dt ¼ cosh a da The results of substituting the above parameters in Eq. 6 can be expressed as: Z EI

w 0 ðx Þ ð1 þ w2 ðxÞÞ3=2

Z ¼ EI

cosh a

EIt da ¼ EI tanh a ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 3 cos h a 1 þ t2

40

50

60

70

Fig. 9. Northridge Earthquake (1994) acceleration Newhall – LA County Fire Department record.

linearized and used. Using the aforementioned assumption, and considering the aspect ratio α ¼ xaf/yf, the deformed shape y(x) for the spring (as depicted in Fig. 5) can be obtained by solving the nonlinear differential equation (Eq. (4)) [26]. The horizontal and vertical distances of the leaf spring are respectively represented by xaf and yf . Therefore, as shown in Fig. 5, the boundary conditions in the structural model of the curved leaf spring are: α α y(0) ¼yf ; y0 (0) ¼ 0; y(xf )¼0; y0 (xf ) ¼0 0 1 d B BEI dx2 @

30 Time (sec)

Fig. 8. El-Centro Earthquake (1940) acceleration North–South record.

2

20

ð7Þ

Fig. 10. Single degree of freedom (SDOF) system.

By applying the boundary conditions, the solution can be presented as: Z xα φ x2 xαf :x f α α rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yðxf Þ ¼ yf þ ð8Þ 2ffi dx; for 0 o x rxf 0 α 2 2 1 φ x xf :x where φ is an unknown variable that can be obtained from numerical integration using the Gaussian points of Eq. (9). The value of y (xαf Þ is equal to zero, as shown in Fig. 5. Z xα f φ x2 xαf :x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð9Þ yðxαf Þ ¼ yf þ 2 dx ¼ 0 0 1 φ2 x2 xαf :x After using numerical Gaussian points and implementing the formula simplification based on normal mathematical sequences, φ can be calculated using Eq. (10). After calculating φ, the shape profile of the curved spring for different aspect ratios can be plotted as presented in Fig. 6. The accuracy of the obtained formula has been verified through prior study results available in the literature [24], as shown in Fig. 7. All of the parameters considered in that prior study have been implemented and applied in the derived formula, and the element shapes for different aspect ratios have been plotted. The results show a high rate of similarity with prior research.

φ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 36:01 xfα4 α2 þ 1

ð10Þ


We substitute the above variables in the stored potential energy equation and rearrange it as follows:

3.1. Stored potential energy in leaf spring Once the shape of the leaf spring has been obtained, the value of its stored elastic potential can be calculated using Eq. (11) [25]. 1 U xa f ¼ EI 2

Z 0

xα f

y″ðxÞ2 ð1 þ y'ðxÞ2 Þ5=2

ð11Þ

dx

91

The following variables can be substituted in the above equations to decrease the order of the differential equation.

1 U xa f ¼ EI 2 1 U xa f ¼ EI 2 1 U xa f ¼ EI 2

Z

xαf

2

ð1 þ sinh tÞ5=2

0

Z

xαf

xαf 0

dt 2

ðcosh tÞ

dt 2 ðcosh t sinh t þ sinh tÞ5=2 Z xα f 1 1 1 1 dt dt ¼ EI 3 2 2 0 cosh t cosh t cosh t 2

0

Z

2

ðcosh tÞ

2

ð12Þ

To solve the above integration, partial integration is applied.

y0 ¼ sinh t

u¼

y″ ¼ cosh t

Bare Frame

1 cosh t

Braced Frame

du ¼

sinh t 2

cosh t

VSB Frame

Fig. 11. Schematic forms of different case study of SDOF systems. (a) Bare Frame (b) Braced Frame (c) VSB Frame.

Fig. 12. General computation algorithm of variable stiffness system adopted in program code.

92


Stiffness(KN/mm)

120

dv ¼

a=2 a=2.5 a=3 a=3.5

100 80 60

1 EI 2

20 0

20

40

60

80

100

Z

Displacement (mm)

0

Stiffness(KN/mm)

140

a=2

120

a=2.5

100

a=3

80 40

0

20

40

60

80

100

Stifness(KN/mm)

Z

a=2 a=2.5 a=3 a=3.5

0

20

40

60

80

100

120

40 30 20 10 0 -10 -20 -30 -40

Bare Frame

Brace Frame Displacement(mm)

VSB

0

10

20

VSB

30 Time (Sec)

40

Bare Frame

50

0

10

20

30

60

Brace Frame

Time(Sec)

40

50

Z

xαf

0

xαf

1

dt 3 cosh t " # Z xα f 1 sinh t 1 dt þ ¼ 2 2 cosh t 0 cosh t " # Z xα a EI sinh t f 1 U x f ¼ dt þ 4 cosh2 t cosh t 0 2 3 Z xα f EI 6 y0 ðxÞ 1 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidx5 ¼ 4 þ 4 1 þ ðy' ðxÞÞ2 2 0 ' 1 þ ðy ðxÞÞ

Displacement(mm)

Displacement(mm)

Fig. 13. Stiffness vs. displacement for the VSB system with a spring thickness of 5 mm and different widths and curve ratios. (a) 80 mm width. (b) 100 mm width. (c) 120 mm width.

Displacement(mm)

sinh t tanh tdt 2 cosh t Z xα Z xα 2 2 f sinh t f cosh t 1 dt ¼ dt ¼ 3 3 0 0 cosh t cosh t

0

Displacement(mm)

40 30 20 10 0 -10 -20 -30 -40

0

1 dt 3 cosh t " # Z xα Z xα f f 1 sinh t 1 1 dt þ dt ¼ EI 2 3 2 cosh t 0 0 cosh t cosh t

Displacment(mm) 180 160 140 120 100 80 60 40 20 0

xαf

xαf

1 EI 2

20 0

Z

v ¼ tanh t

After substituting the following variables in Eq. (12), the closed formed of stored potential energy can be presented as:

a=3.5

60

2

1 dt 3 cosh t " # Z xα f sinh t 1 sinh t þ tanh t dt ¼ EI 2 2 2 0 cosh t cosh t

40 0

1 cosh t

60

40 30 20 10 0 -10 -20 -30 -40

40 30 20 10 0 -10 -20 -30 -40

VSB

0

10

Bare Frame

20

VSB

0

10

30 Time(Sec)

Brace Frame

40

Bare Frame

20

30 Time(Sec)

ð13Þ

50

60

Brace Frame

40

50

60

Fig. 14. Time history of displacement response in different models. (a) VSB Spring ratio of 2(t¼ 5 & w¼ 80). (b) VSB Spring ratio of 2.5(t¼ 5 & w¼ 80). (c) VSB Spring ratio of 3 (t¼ 5 & w¼80). (d) VSB Spring ratio of 3.5(t ¼ 5 & w¼80).


200

VSB

Bare Frame

200

Brace Frame Velocity(mm/sec)

Velocity(mm/sec)

150 100 50 0 -50

10

20

VSB

30 Time (Sec)

40

50

50 0 -50

60

0

10

20

30

40

50

60

Time(Sec)

Bare Frame

200

Brace Frame Velocity(mm/sec)

150 Velocity(mm/sec)

Brace Frame

100

-150 0

200 100 50 0 -50

VSB

150 100 50 0 -50 -100

-100 -150

Bare Frame

150

-100

-100 -150

VSB

93

0

10

20

30 Time(Sec)

40

50

-150

60

0

10

20

30

40

50

60

Time(Sec)

Fig. 15. Time history of velocity response in different models. (a) VSB Spring ratio of 2(t ¼5 & w¼ 80). (b) VSB Spring ratio of 2.5(t ¼5 & w¼80). (c) VSB Spring ratio of 3(t¼ 5 & w¼80). (d) VSB Spring ratio of 3.5(t ¼ 5 & w¼80).

VSB

Bare Frame

Brace Frame

1000 500 0 -500 -1000 -1500

0

10

20

30

40

50

1500 Acceleration(mm/sec2 )

Acceleration(mm/sec2)

1500

Bare Frame

0 -500 -1000 0

10

20

Time(Sec)

VSB

Bare Frame

1500

Brace Frame

1000 500 0 -500 -1000 -1500

0

10

20

30

30

40

50

60

Time(Sec)


Acce,eration(mm/sec2 )

1500

Brace Frame

500

-1500

60

VSB

1000

40

50

60

VSB

Bare Frame

Brace Frame

1000 500 0 -500 -1000 -1500

0

10

20

Time(Sec)

30

40

50

60

Time(Sec)

Fig. 16. Time history of acceleration response in different models. (a) VSB Spring ratio of 2(t ¼5 & w¼ 80). (b) VSB Spring ratio of 2.5(t¼ 5 & w¼ 80). (c) VSB Spring ratio of 3 (t¼ 5 & w¼ 80). (d) VSB Spring ratio of 3.5(t¼ 5 & w¼80).

After differentiating Eq. (8), the equation below is obtained for substitution in Eq. (13).

The closed formula for stored potential can be established as shown in Eq. (14).

φ x2 xαf :x y0 ðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 φ2 x2 xαf :x

EI U xa f ¼ 4

φ x2 –xαf :x

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 þ 1 φ2 x2 –xαf :x

xαf 0

1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffidx 1 φ2 x2 xαf :x

2

ð14Þ


40 30 20 10 0 -10 -20 -30 -40

VSB

Bare Frame


40 30 20 10 0 -10 -20 -30 -40

0

10

20

30 Time (Sec)

40

VSB

50

60

Bare Frame Displacement(mm)

Displacement(mm)

Displacement(mm)

94

0

10

20

30 Time(Sec)

40

50

60

40 30 20 10 0 -10 -20 -30 -40

40 30 20 10 0 -10 -20 -30 -40

VSB

0

10

20

VSB

0

10

Bare Frame

30 Time(Sec)

40

Bare Frame

20

30 Time(sec)

50

60

Brace Frame

40

50

60

Fig. 17. Displacement vs. time graph in different models. (a) VSB Spring ratio of 2 (t ¼5 & w¼ 100). (b) VSB Spring ratio of 2.5 (t ¼5 & w¼100). (c) VSB Spring ratio of 3 (t¼ 5 & w¼ 100). (d) VSB Spring ratio of 3.5 (t ¼5 & w¼100).

VSB

Bare Frame

200

Brace Frame

150

Velocity(mm/sec)

Velocity (mm/sec)

200 100 50 0 -50 -150

0

200

10

20

VSB

30 Time(Sec)

40

50

Brace Frame

100 50 0 -50 -150

60

0

10

20

30

40

50

60

Time(Sec)

Bare Frame

200

Brace Frame

100 50 0 -50

VSB

Bare Frame

Brace Frame



Bare Frame

-100

-100

100 50 0 -50 -100

-100 -150

VSB

150

0

10

20

30 Time(sec)

40

50

60

-150

0

10

20

30

40

50

60

Time(sec)

Fig. 18. Velocity vs. Time graph in different models. (a) VSB Spring ratio of 2 (t¼ 5 & w¼ 100). (b) VSB Spring ratio of 2.5 (t¼ 5 & w¼ 100). (c) VSB Spring ratio of 3 (t¼ 5 & w¼ 100). (d) VSB Spring ratio of 3.5 (t ¼5 & w¼100).

Subsequently, the value of the horizontal force in each curved leaf spring can be derived as: f¼

∂U ðxa f Þ ∂xa f

ð15Þ

Hence, 8 9 > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z xα = 2 α f EI < ∂ 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx f¼ φ x –xf :x 1 φ2 x2 –xαf :x þ af > 2 4> ∂x 0 : ; 1 φ2 x2 xαf :x


VSB

0 -500 -1000 0

10

1500

20

40

50

Bare Frame

0 -500 -1000 0

10

20

30 Time(sec)

40

50

Brace Frame

0 -500 -1000 0

10

20

VSB

30 Time(Sec)

40

Bare Frame

50

60

Brace Frame

1000 500 0 -500 -1000 -1500

60

Bare Frame

500

1500

Brace Frame

500

VSB

1000

-1500

60


VSB

30 Time(Sec)

1000

-1500

1500

Brace Frame

500

-1500


Bare Frame

1000



1500

95

0

10

20

30 Time(sec)

40

50

60

40 30 20 10 0 -10 -20 -30 -40

VSB

Bare Frame


40 30 20 10 0 -10 -20 -30 -40

0

10

20

VSB

30 Time(Sec)

40

50

Bare Frame

60


Displacement(mm)

Displacement(mm)

Fig. 19. Acceleration vs. Time graph in different models. (a) VSB Spring ratio of 2 (t ¼5 & w¼ 100). (b) VSB Spring ratio of 2.5 (t¼ 5 & w¼ 100). (c) VSB Spring ratio of 3 (t¼ 5 & w¼ 100). (d) VSB Spring ratio of 3.5 (t ¼ 5 & w¼100).

0

10

20

30 Time(sec)

40

50

60

40 30 20 10 0 -10 -20 -30 -40

40 30 20 10 0 -10 -20 -30 -40

VSB

0

10

Bare Frame

20

VSB

0

10

30 Time(sec)

Brace Frame

40

Bare Frame

20

30 Time(Sec)

50

60

Brace Frame

40

50

60

Fig. 20. Record of displacement vs. time in different models. (a) VSB Spring ratio of 2 (t¼ 5 & w¼ 120). (b) VSB Spring ratio of 2.5 (t¼ 5 & w¼ 120). (c) VSB Spring ratio of 3 (t¼ 5 & w¼ 120). (d) VSB Spring ratio of 3.5 (t ¼5 & w¼ 120).

After calculating the integration and partial derivation, the equation can be written as:

After simplifying Eq. (15), Eq. (17) derives the horizontal force at each spring:

82 9 3 2 > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 2 7 EI 4 > α : ; 1 φ2 x2 xf :x

8 9 2 > > = EI 2 4 > : ; 1 φ2 x2 x xαf

ð16Þ

ð17Þ

96


VSB

Bare Frame

Brace Frame

85 35 -15 -65

Bare Frame

85 35 -15 -65

0

10

20

30

40

50

-165

60

0

10

20

Time(sec) VSB

Bare Frame

Brace Frame

85 35 -15 -65 -115 0

10

20

30

40

50

VSB


Velocity)mm/sec)

135

-165

Brace Frame

-115

-115 -165

VSB


Velocity(mm/sec)

135

40

Bare Frame

50

60

Brace Frame

85 35 -15 -65 -115 -165

60

30 Time(sec)

0

10

20

Time(Sec)

30

40

50

60

Time(Sec)

Fig. 21. Record of velocity vs. Time in different models. (a) VSB Spring ratio of 2 (t ¼5 & w¼120). (b) VSB Spring ratio of 2.5 (t¼ 5 & w¼ 120). (c) VSB Spring ratio of 3 (t ¼5 & w¼ 120). (d) VSB Spring ratio of 3.5 (t ¼5 & w¼ 120).

VSB

Bare Frame

1000 500 0 -500 -1000 -1500

1500

Brace Frame Acceleration(mm/sec2)


1500

0

10

20

30

40

50

Bare Frame

0 -500 -1000 0

10

20

Time(Sec)

VSB

Bare Frame

1000 500 0 -500 -1000 -1500

0

10

20

30

40

50

60

Time(Sec)

VSB

1500

Brace Frame Acceleration(mm/sec2)


1500

Brace Frame

500

-1500

60

VSB

1000

30 Time(Sec)

40

Bare Frame

50

60

Brace Frame

1000 500 0 -500 -1000 -1500

0

10

20

30 Time(Sec)

40

50

60

Fig. 22. Record of acceleration vs. Time in different models. (a) VSB Spring ratio of 2 (t¼ 5 & w¼120). (b) VSB Spring ratio of 2.5 (t¼ 5 & w¼120). (c) VSB Spring ratio of 3 (t¼ 5 & w¼120). (d) VSB Spring ratio of 3.5 (t¼ 5 & w¼ 120).

The spring displacement d is defined as the horizontal location of the left origin, as shown in the structural model of each spring (Fig. 5). Considering the operating VSB system and two leaf springs acting together, we can derive the global or equivalent force of the proposed system: F ðdÞ ¼ 2f ðdÞ

8 9 2 > 2 2 α = 1Þ> EI 2 > 2 : ; 1 φ2 d d xαf

ð18Þ

∂F The global stiffness can be obtained fromK ¼ ∂d . After differentiating and simplifying Eq. (18), the general stiffness formula for


the proposed system can be derived as: 2 2 5 4 α α 3 3 2 α α2 EI 6φ þ 2φ d 3d 2xf d xf 3φ d 3d 5d:xf þ 2xf K¼ 4 2 3=2 2 2 1 φ2 d d xαf ð19Þ

4. Dynamic analysis of the VSB system The dynamic performance of the developed VSB system was assessed using the Newmark technique [27], which was codified in Matlab. The results were evaluated in terms of displacement, velocity, and acceleration in three models, namely, momentresisting, brace, and VSB frames. We selected the time history of the El Centro seismic (North–South direction) and Northridge were selected as input acceleration, as shown in Figs. 8 and 9. Furthermore, evaluating the energy dissipation of the proposed system requires pseudo dynamic testing. Given the unavailability of experimental instruments in our laboratory, experimental investigations are beyond the scope of this study. 4.1. Newmark method The lateral motion of the basic SDOF model shown in Fig. 10 comprises a mass (m), supported by springs with stiffness of k, and a damper with linear viscosity, c. This SDOF system was subjected to an external disturbance characterized by f (t). The excited model responds to a lateral displacement x(t) relative to the ground, which satisfies the following equation of motion: m U x€ þ c U x_ þ k U x ¼ f ðt Þ

ð20Þ

where a superposed dot represents differentiation with respect to time. For a specified input f(t) and with known structural

VSB

Bare frame

f ðt Þ ¼ m U x€g ðt Þ

20 0 -20 -40 0

10

20

30

40

50

ð21Þ

We now consider the addition of a generic VSB system to the SDOF model, as shown in Eq. (22). This supplementary element affected the response of the system. To assess the seismic performance of the proposed system, we considered three models with the same beam and column size (Fig. 11). The stiffness of the VSB system varies in nonlinear action. After finishing each loop, the new displacement value should be obtained and substituted in the next step. Fig. 12 summarizes the computation algorithm. m U x€ þ c:x_ þ ðK VSB þ K cl þ KÞ U x ¼ f ðt Þ

4.2. Effect of cable stiffness When the energy dissipation system is installed using conventional braces between stories, the braces are assumed to operate as rigid elements, and all deformations are concentrated in the energy dissipation system for simplification. However, the VSB system installation uses a wire rope to avoid the buckling issue. The stiffness of the cable may not suffice to be considered inflexible, even though the elastic modulus of the cable is higher than that of steel braces made of mild steel [28]. Therefore, it should be considered in two stiffness stages. The lateral stiffness of cable Kcl can be obtained as follows: 1 1 2 ¼ þ K cl kch kcd cos 2 θ

ð23Þ

where Kch is the axial stiffness of the horizontal cables, and Kcd is the stiffness of the diagonal cables with slope θ from horizontal axis.

VSB

60

20 0 -20 -40 0

10

20

Bare frame

60

Brace frame

40 20 0 -20 -40 -60

10

20

30 40 Time(sec)

40

50

60

50

60

VSB

Bare frame

Brace frame

40 20 0 -20 -40 -60

0

30 Time(Sec)

Displacement(mm)

Displacement(mm)

VSB

Brace frame

40

-60

60

Bare frame

Time(Sec) 60

ð22Þ

where KVSB and Kcl presents the stiffness of VSB and attached cables, respectively.

Brace frame

40

-60

parameters, the solution of this equation can be readily obtained. In the above equation, f(t) represents an arbitrary environmental disturbance, such as wind or an earthquake (Eq. (21)).

Displacement(mm)

Displacement (mm)

60

97

0

10

20

30

40

50

60

Time(Sec)

Fig. 23. Record of Displacement vs. Time in different models subjected to Northridge earthquake. (a) VSB Spring ratio of 2 (t¼ 5 & w¼ 120). (b) VSB Spring ratio of 2.5 (t¼ 5 & w¼ 120). (c) VSB Spring ratio of3 (t ¼5 & w¼120). (d) VSB Spring ratio of 3.5 (t¼ 5 & w¼ 120).

98


300

VSB

Bareframe

200

Brace frame Velocity(mm/sec)

Velocity(mm/sec)

100 0 -100 -200 -300

VSB

150

200

Bareframe

100 50 0 -50 -100 -150

0

10

20

30

40

50

-200

60

0

10

20

Time(sec) 200

VSB

Bareframe

200

Brace frame Velocity(mm/sec)

Velocity(mm/sec)

VSB

30 40 Time(sec)

50

Bareframe

60

Brace frame

150

150 100 50 0 -50 -100

100 50 0 -50 -100 -150

-150 -200

Brace frame

0

10

20

30

40

50

60

Time(sec)

-200

0

10

20

30

40

50

60

Time(sec)

Fig. 24. Velocity time history of different models subjected to Northridge earthquake. (a) VSB Spring ratio of 2 (t ¼ 5 & w¼ 120). (b) VSB Spring ratio of 2.5 (t ¼5 & w¼120). (c) VSB Spring ratio of3 (t¼ 5 & w¼ 120). (d) VSB Spring ratio of 3.5 (t ¼5 & w¼ 120).

The dynamic evaluation of the VSB implementation in a SDOF model was conducted in the developed program using various stiffness inputs. Fig. 13(a)–(c) show the stiffness versus displacement for a specific parameter of the VSB system. Fig. 13(a) shows the graph plotted for the VSB system using an 80 cm width, 5 mm thickness, and different aspect ratios for yf ¼60 mm. Furthermore, the spring width was changed to 100 and 120 mm in Fig. 13(b) and (c), respectively, using the same assumptions for the other parameters, such as thickness and yf. After codifying the Newmark method in Matlab, we input the aforementioned stiffness versus displacement for each particular case and reported results in terms of top node displacement for three models, namely, frames with or without the VSB system and a brace system. Other essential parameters (mass, damping, frame, and cable stiffness) for dynamic analysis were considered to have the same values in all cases.

5. Results and discussion We used dynamic analysis to evaluate the efficiency of different VSB layouts in a structure subjected to the El Centro earthquake time history Fig. 14 shows the displacement response versus time of different models. The displacement time history of the SDOF model equipped by the VSB system with spring curve ratios of 2 and 2.5 is presented in Fig. 14(a) and (b), respectively. The frame equipped with VSB had a smaller peak displacement response than the momentresisting frame. This result indicates that this particular VSB system can satisfy the design criteria for applying the VSB system as a supplementary energy dissipation system – VSB having spring geometry aspect ratios equal to 2 and 2.5 can reduce the absolute displacement response by approximately 53.5 and 50 percent, respectively. Furthermore, implementing the brace component can reduce the displacement response by about 51 percent. As shown in Fig. 14 (c) and (d), adding a VSB device to the frame decreases the maximum

absolute displacement value by roughly 21 and 8 percent for a VSB device with spring aspect ratios of 3 and 3.5, respectively. Fig. 15(a) and (b) illustrate the velocity response of VSB implementation with spring ratios of 2 and 2.5 compared with moment-resisting and braced frame models. The absolute maximum value of the velocity decreases by 41 and 32 percent for VSB springs ratios of 2 and 2.5, respectively. By contrast, the velocity response of the braced frame model shows a higher value about 1.1 times than that of the moment-resisting frame. Moreover, the aforementioned value is reduced by about 21 and 44 percent for VSB spring ratios of 3 and 3.5, respectively (Fig. 15(c) and 15(d)). Fig. 16 reports the acceleration response versus time for VSB springs with different geometric ratios. The absolute maximum acceleration for a ratio of 2 increases by 1.15 times and declines by 23, 22, and 25 percent for spring ratios of 2.5, 3, and 3.5, respectively, compared with the moment-resisting frame; this acceleration is also raised by 2.24 times for the braced frame model. Fig. 17 illustrates the displacement response of the considered models with the new geometry of VSB spring. In this case, the width of the spring increases from 80 mm to 100 mm (25 percent increase in spring width), whereas the other parameters (thickness and material properties) remain the same. As shown in Fig. 17 (a) and (b), the application of the VSB system with geometry aspect ratios of 2 and 2.5 in the moment-resisting frame causes about 50 and 28 percent reduction in maximum displacement, respectively. Furthermore, the results for the other considered aspect ratios show a decreased absolute peak displacement value by about 27 and 9 percent for a ¼3 and a ¼3.5, respectively. Figs. 18 and 19 present the velocity and acceleration changes versus time for VSB system with various spring aspect ratios. Generally, the VSB element was implemented in a momentresisting frame to diminish the absolute maximum velocity and acceleration values. For instance, in the VSB model with spring


VSB

1500

Bareframe

2000

Brace frame Acceleration(mm/sec2)


2000 1000 500 0 -500 -1000 -1500

VSB

1500

Bareframe

Brace frame

1000 500 0 -500 -1000 -1500 -2000

-2000 0

10

20

30 40 Time(Sec)

50

0

60

VSB

Bareframe

20

30

40

50

60

2000

Brace frame Acceleration(mm/sec2)

1500

10

Time(Sec)

2000 Acceleration(mm/sec2)

99

1000 500 0 -500 -1000 -1500 -2000

VSB

1500 1000 500 0 -500 -1000 -1500 -2000

0

10

20

30 40 Time(Sec)

50

60

0

10

20

30 40 Time(Sec)

50

60

Fig. 25. Acceleration vs. time records of different models subjected to Northridge earthquake. (a) VSB Spring ratio of 2 (t¼ 5 & w ¼120). (b) VSB Spring ratio of 2.5 (t¼ 5 & w¼ 120). (c) VSB Spring ratio of3 (t ¼5 & w¼120). (d) VSB Spring ratio of 3.5 (t¼ 5 & w¼ 120).

aspect ratios of 2, 2.5, 3, and 3.5, the maximum value of velocity was reduced by around 48, 33, 31, and 24 percent, respectively. Acceleration records also show the reduction trend once the VSB component was applied in a moment-resisting frame. The absolute maximum values decrease by 28, 15, 20, and 29 percent for VSB furnished with spring ratios of 2, 2.5, 3, and 3.5, respectively, as presented in Fig. 19. Fig. 20 shows the displacement time history of the SDOF system equipped with a VSB device with a 120 mm-wide spring. All the structural materials and dimensions are assumed to have the same values, which are considered in the aforementioned models. As shown in Fig. 20(a), the VSB system with a spring geometry aspect ratio of 2 shows an approximately 60 percent reduction in maximum displacement. As illustrated in Fig. 20(b)–(d), dissimilar VSB configurations with aspect ratios of 2.5, 3, and 3.5 cause the maximum displacement to decrease by 9, 26, and 29 percent, respectively. Fig. 21 demonstrates the velocity record for VSB systems comparing four different geometries to moment-resisting and braced frames. The absolute maximum velocity value decreases by 0.5, 0.45, 0.35, and 0.21 times whenever the VSB system is furnished by spring geometry ratios of 2, 2.5, 3, and 3.5, respectively. Fig. 22 reports the acceleration time history for the abovementioned models. Generally, VSB implementation in momentresisting frame causes the decline in the maximum absolute value of acceleration by 21, 34, 26, and 23 percent for VSB devices equipped with spring geometry aspect ratios of 2, 2.5, 3, and 3.5, respectively. Additionally, the addition of brace elements increased the acceleration 2.24-fold. Fig. 23 shows the displacement response of the VSB device (with a thickness and width of 5 and 120 mm, respectively) subjected to the Northridge earthquake. In the case of the VSB curve aspect ratio of 2, the maximum displacement of VSB with frame is reduced by around 15 percent. Moreover, the application of brace element in frame produces a reduction effect and causes a decrease in

maximum displacement value of 57 percent. When the curve aspect ratio increases to 2.5, 3, and 3.5, the maximum displacement of frame furnished by VSB decreases by 16, 25, and 17 percent, respectively. Fig. 24 shows the velocity time history of different VSB models subjected to the Northridge earthquake. Whenever the VSB curve aspect ratio was set to 2, the maximum velocity increased by 3 percent compared with a frame without the VSB system. Furthermore, aspect ratios of 2.5, 3, and 3.5 reduced the maximum velocity by 14, 21, and 18 percent, respectively. By contrast, adding brace elements to frames increased this value by about 6 percent. Fig. 25 illustrates the acceleration versus time of various VSB models. The application of a VSB device with a spring curve ratio of 2 amplified the maximum acceleration by approximately 18 percent, whereas this value declined by about 2, 6, and 21 percent in models furnished with VSB having spring aspect ratios of 2.5, 3, and 3.5, respectively. By contrast, this value in the braced frame increases by 32 percent. Such structural responses as displacement, velocity, and acceleration are affected by several parameters specified in Eq. (22). VSB with a curve ratio of 2 has greater stiffness than other types, intensifying the acceleration response of the model equipped with this specific type of VSB device. Generally, the implementation of brace element in frames increases the maximum absolute values of velocity and acceleration. Acceleration plays a crucial role in the force induced by vibration, so larger structural sections must be selected to sustain the force. Hence, the effects of using a brace element system to retrofit an existing structure must be considered meticulously to avoid any earlier failure in the main structural components. Nevertheless, in frames with VSB systems that have a spring ratio of 2.5 or above, the maximum acceleration is diminished and the lateral force subsequently decreases. Consequently, a VSB device can be used to retrofit frames without any additional side effects in structural components.

100


Displacement(mm)

Acceleration(g)

0.4 0.2 0 -0.2 -0.4 0

5

10

15

20

25

30

15 10 5 0 -5 -10 -15

VSB

0

5

10

Time(sec)

0.2 0 -0.2 10

20

30

40

50

30 20 10 0 -10 -20 -30

VSB

0

60

10

20

Time(Sec)

Fig. 27. Time history of El-Centro seismic record with time interval of 0.02 s.

30

30 Time(Sec)

40

Bare Frame

50

60

Fig. 30. Displacement response of models under El-Centro excitation with 0.02 s time interval.

0.4 0.2 0 -0.2 -0.4

0

10

20

30

40

50

60

70

80

Time(Sec)

Displacement(mm)

Acceleration(g)

25

Fig. 29. Displacement response of models under El-Centro excitation with 0.01 s time interval. Displacement(mm)

Acceleration(g)

0.4

0

20

Time(Sec)


-0.4

15

Bare frame

60 40 20 0 -20 -40 -60 -80

VSB

0

10


6. Frequency evaluation of VSB system We evaluated frames with and without VSB devices using the El Centro excitation records with different frequencies. The time intervals in the seismic records are 0.01, 0.02, and 0.03 seconds, with the same acceleration values shown in Figs. 26–28. We selected a VSB device with the same geometric specifications, including a spring curve aspect ratio of 2.5. The spring's width and thickness were 120 and 8 mm, respectively. The sensitivity of the VSB application at different frequencies was evaluated and compared to moment-resisting frames in terms of displacement response. The stiffness of the VSB system depends on the geometry of the leaf spring and the displacement applied to the device; therefore, altering the frequency does not affect its performance. Fig. 29–31 show the displacement responses of frames with and without VSB devices at the aforementioned frequencies. The results show a reduction in absolute displacement of about 51, 46, and 33 percent for models subjected to El Centro excitation records with time intervals of 0.01, 0.02, and 0.03 seconds, respectively, compared with a moment-resisting frame (Fig. 29–31).

7. Conclusions This study proposes a novel design for a variable stiffness spring as an adaptive stiffness component for use in framed structures. The proposed device employs four curved springs with nonlinear elastic deformations, and system stiffness can be changed by altering displacement, velocity, and acceleration. The most important characteristic of the developed VSB system is its large nonlinear stiffness range. This distinction allows the spring to be used for different kinds of structures. This study also establishes a mathematical model that can define the spring’s dimensions for any particular function. The developed mathematical model for the VSB system uses the Newmark method, and we evaluated the dynamic behavior of the VSB system, comparing it with conventional braced and moment-resisting frames. The developed software applies the time history of the El Centro seismic record as the external vibration

20

30

40 Time(Sec)

50

60

70

Bare Frame

80

Fig. 31. Displacement response of models under El-Centro excitation with 0.03 s time interval.

acceleration. The dynamic results of different VSB configurations are reported in terms of displacement. The results reveal that the VSB spring geometry, particularly the horizontal-to-vertical aspect ratio, plays an important role in the dynamic structural response. The structural response can be dramatically reduced by implementing a specific layout of the VSB system in structures, depending on structural geometry, mass, and inherent damping. The VSB system can adjust the stiffness of the frame structure based on the input displacement transferred to the VSB device through wire cable action. This technique is proposed for implementation in new and, as a retrofitting system, in existing steel MRSFs. Generally, the ductility approach implemented in MSRFs is designed to dissipate earthquake energy through plastic hinge formation. MRSFs become damaged because of unacceptable story drift. When retrofitting MRSF structures, new brace components cannot be simply applied without meticulous consideration of the excessive force that might be transferred by attaching braces to the frame. Moreover, the braces will reduce the ductility of the frame. Therefore, the VSB technique is proposed to allow the MRSFs to dissipate energy via ductile action in the initial stage. However, whenever the frame begins to pass the allowable story drift, the VSB prevents it. The response of a structure equipped with VSB will be affected by the device’s geometry, specifically the VSB curve’s aspect ratio, and the spring material. Thus, the selection of the curve aspect ratio must take into consideration the desired performance level, the type of structural system, the lateral stiffness of the structural system, and any architectural limitations. However, the initial design assumption for the curve aspect ratio can be limited within the range of 2.5 to 3.5.

Acknowledgments This work received financial support from the Ministry of Science, Technology, and Innovation of Malaysia under Research Project no. 5524254. This support is gratefully acknowledged. A


patent grant has been filed for the VSB system at Malaysia Intellectual Property Organizations (MyIPO) under Application no. PI 2014701608.

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