Performance-based assessment and design of FRP ...

Engineering Structures 61 (2014) 166–183

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Performance-based assessment and design of FRP-based high damping rubber bearing incorporated with shape memory alloy wires F. Hedayati Dezfuli, M. Shahria Alam ⇑ School of Engineering, The University of British Columbia, Kelowna, BC, Canada

a r t i c l e

i n f o

Article history: Received 6 November 2012 Revised 31 December 2013 Accepted 6 January 2014 Available online 10 February 2014 Keywords: Shape memory alloy Fiber-reinforced elastomeric isolator High damping rubber Hysteretic shear response Pre-strained wires

a b s t r a c t This study deals with two new generation smart high damping rubber bearings (HDRBs) incorporated with shape memory alloy (SMA) wires. Due to the superelastic effect and the re-centering capability of SMAs, the residual deformation in SMA-based elastomeric isolators is reduced. Two different configurations of SMA wires incorporated in rubber bearings are compared by changing the aspect ratio of base isolator, the type of SMA, the thickness of wires, and the pre-strain in wires where the isolator is subjected to a vertical pressure and unidirectional cyclic lateral displacement. A performance-based design flowchart is also provided along with a design example for determining the pre-strain and the radius of cross section of wires in the SMA wire-based rubber bearings. Results demonstrated that using ferrous SMA wires (FeNCATB) with 13.5% superelastic strain in the cross configuration leads to the best performance since the strain induced in wires is significantly decreased. Seismic performance evaluation of a three-span continuous bridge isolated by smart HDRBs shows that using SMA wires with cross configuration leads to the highest energy dissipation during earthquake. However, implementing SMA wires into HDRBs has negligible effect and in some cases no effect on the pier displacement and peak deck acceleration. It was also observed that neither of the HDRBs nor the SMA-HDRBs experienced any residual deformations when they were subjected to three different ground motions scaled by Vancouver design response spectrum. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction In order to reduce the seismic demand on civil infrastructures several techniques are commonly employed in seismic prone regions. Seismic isolation, which is one of the most popular techniques, is widely used to safeguard bridges from severe damages due to strong earthquake events. A historical appraisal on seismic isolation systems is provided in the literature [1–3]. Various types of bridge isolators have been developed, tested and are in use all over the world as effective earthquake resisting systems, including lead-rubber bearing, high damping rubber bearing, friction pendulum bearing, magneto-rheological damper, and steel plate damper [4–6]. The most popular isolation devices are the lead rubber bearing (LRB) and the high damping rubber bearing (HDRB). They have been used in bridges for both new constructions and retrofit projects. The premise of using these devices is to have high flexibility, which shifts the natural period of the bridge structure to a value ⇑ Corresponding author. Address: School of Engineering, The University of British Columbia, 1137 Alumni Avenue, EME 4225, Kelowna, BC V1V1V7, Canada. Tel.: +1 250 807 9397; fax: +1 250 807 9850. E-mail address: [email protected] (M.S. Alam). http://dx.doi.org/10.1016/j.engstruct.2014.01.008 0141-0296/Ó 2014 Elsevier Ltd. All rights reserved.

beyond the critical period range of the earthquake event. In addition, they are usually endowed with damping properties that prevent the bridge piers and decks from undergoing excessive displacement. Nevertheless, most of these devices have known limitations related to ageing and durability (e.g., rubber-based dampers), maintenance (e.g., viscous fluid dampers), long-term reliability (e.g., friction dampers), temperature dependent mechanical performance (e.g., rubber-based dampers, viscoelastic dampers), and geometry restoration after a strong earthquake (for most dampers) [7–9]. In this regard, the use of superelastic shape memory alloy (SMA) can provide an effective solution to overcome several of these problems. Superelastic SMA is a unique material with the ability to undergo large deformation and potentially recover its inelastic deformation upon stress removal. Its flag-shaped hysteresis, large recoverable strain and deformation capability, excellent endurance against fatigue, and corrosion resistance make superelastic SMA an ideal material for utilizing it in seismic protection devices [10]. Several researchers have proposed different types of superelastic SMA-based smart isolation devices in the past. Choi et al. performed numerical study considering NiTi SMA wire wrapped around an elastomeric bearing to improve its re-centering

167

F. Hedayati Dezfuli, M.S. Alam / Engineering Structures 61 (2014) 166–183

capability over the lead rubber bearing [11]. However, at very large shear deformation (200% shear strain), this device will malfunction since wires experience axial strain beyond the NiTi’s superelastic strain range. Although Dolce et al. effectively implemented SMA wires in a base isolation device, the manufacturing of the device was quite complex [7]. Another SMA-based isolation device was developed by Dolce et al., which showed high sensitivity and considerable variation in forces with temperature, and inefficiency in energy dissipation capacity [7]. Liu et al. used a diagonal arrangement of large diameter SMA strands around the rubber bearing [12]. However, this arrangement did not improve the re-centering capability or the level of damping compared to the original rubber bearing. Attanasi and Aurichhio proposed an isolation device equipped with eight SMA coil springs, which is expensive due to its complex manufacturing process and the use of expensive large diameter SMA springs [13]. Attanasi et al. investigated the possibility of using shape memory alloys in base isolation systems [14]. They compared the behavior of a proposed smart isolator with that of a traditional lead rubber bearing and an equivalent linear elastic model. According to their results, the behavior of the smart isolation device with flag-shaped hysteresis loops was similar to a system with elasto-plastic hysteresis. They concluded that it is possible to replace existing LRBs with SMA-based bearing systems considering the amount of energy dissipation capacity. They suggested that SMA-based restrainers can be applied to rubber bearings or friction pendulum systems in order to provide re-centering force and control the relative horizontal displacement and upward force transmitted to the superstructure. Suduo and Xiongyan introduced three types of SMA-based dampers and one base isolator using nickel-titanium SMA wires [15]. They proposed theoretical models to estimate the behavior of the devices with high amounts of damping capacity. Suduo and Xiongyan compared the performances of lattice shell structures using SMA-based (smart) elastomeric isolator and conventional rubber bearings with those of fixed supported structures. They found out that not only the smart base isolators can efficiently mitigate the seismic response in terms of acceleration, displacement, and internal forces but also, they have superior performances relative to the existing rubber bearings. High damping rubber bearings (HDRBs) behave quite differently compared to natural rubber bearings (NRB) especially at high shear strain amplitudes. Hedayati Dezfuli and Alam explored the effect of SMAs on the performance of NRBs and concluded that using SMA wires can be beneficial to NRBs in terms of energy dissipation capacity and re-centering capability [16]. Therefore, it would be of great interest to assess the effect of SMA wires on the response of HDRBs as well and check whether it is advantageous to implement SMAs into such type of elastomeric isolators. In this study, a cross configuration of SMA wire is proposed for high damping elastomeric isolators and its performance will be compared to the straight arrangement, suggested by Choi et al. [11], using two types of SMA. Then, the most efficient SMA in terms of the superelastic strain range and compatibility with environmental thermal conditions will be determined through a performance assessment. The effective horizontal stiffness and the residual deformation of the proposed smart base isolators will be calculated from the lateral force-deflection hysteresis curves through numerical simulations. The elastomeric isolator is made of high damping rubber layers bonded to carbon fiber-reinforced polymer (CFRP) composite plates as reinforcements [17]. In this regard, a hyper-viscoelastic material model is used to describe the highly nonlinear behavior of the high damping rubber [18]. After validating the results obtained from the numerical simulation, the hysteretic shear responses of SMA-based rubber bearings (SMA-HDRBs) is evaluated using finite element (FE) analysis. The effect of different factors, including the aspect ratio of rubber

bearing (i.e. ratio of the effective height to the length), the SMA types, and the arrangement and thickness of wires, on the performance of the device are also investigated. In order to appropriately determine the pre-strain and the radius of cross section of SMA wires, a performance-based design approach is also provided along with a design example for SMA wire-based rubber bearings. In order to assess the seismic performance of the proposed SMA-HDRBs on a structure, a three-span continuous steel girder concrete-reinforced pier supported bridge is modeled and a number of dynamic time history analyses are performed under three ground motions. In this regard, the relative lateral displacement in rubber bearings, the total energy dissipated by rubber bearings, and the maximum acceleration of bridge deck for non-isolated (fixed-base) and isolated bridges are determined. 2. Smart elastomeric isolators SMA-based smart rubber bearings will have many advantages, such as stability, re-centering capability, high energy dissipation capacity and long service life. They not only will mitigate the seismic response of structures in terms of acceleration, displacement and internal forces but will also have superior performance in terms of fatigue property and energy dissipation capacity compared to those of existing rubber bearings [15]. In the present study, SMA wires are used as a supplementary element to improve the performance of a fiber-reinforced high damping rubber bearing in terms of the residual deformation. The main concern about the SMA wire is the maximum superelastic strain under large shear strains of rubber bearings. Two configurations of wires are compared by changing different factors, such as the type of SMA and the aspect ratio of rubber bearing. In this regard, seven carbon FRP-based high damping rubber bearings (CFR-HDRB) are considered with different aspect ratios (R). Their geometrical properties are listed in Table 1. The thickness of the elastomer layer, te, is kept constant (4.7 mm in all cases), while the number of high damping rubber layers, ne, increases from 6 to 18. The aspect ratio of rubber bearing is defined as a ratio of the effective height (Heff) to the length (L).

R¼

Heff L

ð1Þ

where the effective height is measured by subtracting the thickness of two supporting steel plates (ts) from the total height (H).

Heff ¼ H 2ts

ð2Þ

2.1. Straight SMA wires In the straight configuration, two continuous SMA wires are wound in two opposite sides of the rubber bearing as shown in Fig. 1a. This type of arrangement of SMA wires was proposed by

Table 1 Geometrical properties of CFR-HDRBs. CFR-HDRB

#1

#2

#3

#4

#5

#6

#7

L (mm) W (mm) H (mm) ts (mm) te (mm) tf (mm) ne nf R

250 250 60 15 4.74 0.31 6 5 0.12

250 250 70 15 4.73 0.31 8 7 0.16

250 250 80 15 4.72 0.31 10 9 0.20

250 250 90 15 4.72 0.31 12 11 0.24

250 250 100 15 4.71 0.31 14 13 0.28

250 250 110 15 4.71 0.31 16 15 0.32

250 250 120 15 4.71 0.31 18 17 0.36

168


a

z

b

Supporting Steel Plate

y x

SMA Wire 2 SMA Wire 1 Hook Fig. 1. Smart rubber bearing; (a) straight SMA wires, (b) cross SMA wires.

Choi et al. [11]. Each wire passes through four steel hooks which are welded at each corner of steel plates. In such a configuration, the induced strain along the SMA wire due to the cyclic lateral displacement of rubber bearing noticeably decreases since one continuous wire is used instead of four wires fixed at each corner of the supporting plate. The total length of the SMA wires (LSMA) required for the straight configuration in each aspect ratio is presented in Table 2. The strain in the SMA wires (eSMA) is a function of the shear strain amplitude, c, and the aspect ratio, R. When the shear strain increases from 50% to 200%, the strain generated in SMA wires increases. The SMA strain also goes up by enhancing the aspect ratio of the base isolator. Fig. 2a shows the variation of SMA wire strain by increasing the shear strain amplitude and the aspect ratio of the rubber bearing. At 200% shear strain amplitude, the SMA wires in all of the base isolators experience a strain greater than 10%. It can be also observed that the strain induced in SMA wires will not exceed 10% when the base isolator is subjected to shear strains lower than 100%. However, at larger shear strains, the SMA wire in the base isolator with high aspect ratio can experience as high as 27% strain.

tom supporting steel plate. The SMA wires pass through these hooks (Fig. 1b). The main reason of using wires in such an arrangement is to effectively reduce the maximum strain generated along the wires compared to the straight configuration (Fig. 1a). The total length of wires needed for this arrangement is presented in Table 2. Although a larger length of SMA wire is required for this configuration relative to the straight one, the generated strain in wires due to the lateral deflection of rubber bearing is much lower. Fig. 2b shows that in the case of cross configuration there is a significant reduction in the SMA wire strain compared to the straight configuration (Fig. 2a). At 200% shear strain amplitude with an aspect ratio of 0.36, the maximum strain induced in the SMA wires does not exceed 9%. It shows that unlike straight configuration in which SMAs can operate in a limited range of c and R, different types of SMA can be used in the cross arrangement. Fig. 2b illustrates that at each shear strain level, the strain in the SMA wires reaches its maximum value when the smart elastomeric isolator has the maximum considered aspect ratio.

2.2. Cross SMA wires

Different types of SMA, such as Nickel-Titanium, Cu-based SMAs and ferrous SMAs have potential for smart structural applications. The elastic modulus in the austenite phase (EA), the austenite finish temperature (Af) and the superelastic strain (es) under the maximum applied strain (emax) for a number of SMAs are listed in Table 3.

In the cross configuration, two continuous SMA wires are wound around the rubber bearing diagonally as shown in Fig. 1b. A steel hook is mounted at each corner on the lower surface of the top supporting steel plate and on the upper surface of the bot-

2.3. Efficiency of SMA in two arrangements

Table 2 Required length of SMA wire for seven CFR-HDRBs in the straight and cross configurations. CFR-HDRB

#1

#2

#3

#4

#5

#6

#7

R LSMA (mm) (straight) LSMA (mm) (cross)

0.12 1240 2252.8

0.16 1280 2262.7

0.20 1320 2275.4

0.24 1360 2290.9

0.28 1400 2308.9

0.32 1440 2329.6

0.36 1480 2352.9

SMA Strain (%)

25 20

a

b

50% 100% 150% 200%

10

SMA Strain (%)

30

15 10 5 0 0.12

0.18

0.24

0.3

Aspect Ratio, R

0.36

8

50% 100% 150% 200%

6 4 2 0 0.12

0.18

0.24

0.3

0.36

Aspect Ratio, R

Fig. 2. Variation of strain in SMA wire as a function of shear strain amplitude and aspect ratio for (a) straight configuration and (b) cross configuration.

169


temperature in countries with cold climatic conditions may go below 20 °C, the austenite finish temperature of the SMA wire should be lower than this temperature. Therefore, in this study NiTi45 and FeNiCuAlTaB (FeNCATB) with respective Af values of 10 °C and 62 °C, are chosen for smart rubber bearing.

Table 3 Mechanical characteristics of different shape memory alloys. Alloy

emax (%)

es (%)

EA (GPa)

Af (°C)

References

Ni Ti49.5 Ni Ti45 Ti Ni40 Cu10 Ti Ni25 Cu25 CuAlBe FeNiCuAlTaB

5.7 6.8 4.1 10.0 3.0 15.0

4.6 6.0 3.4 2.5 2.4 13.5

45.3 62.5 72.0 14.3 32.0 46.9

53.0 10.0 66.6 73.0 65.0 62.0

[19] [20] [19] [21] [22] [23]

3. Finite element modeling 3.1. High damping rubber bearing

In order to investigate the efficiencies of different types of SMAs in the cross and the straight configurations, two aspect ratios, 0.12 and 0.36, and four shear strain amplitudes, 50%, 100%, 150%, and 200%, are considered. According to Table 4, when the aspect ratio is 0.12, the maximum strain in the SMA wire, eSMA, at 200% shear strain is 1% for the cross configuration and 11.1% for the straight configuration. At the same shear strain level, when the aspect ratio increases to 0.36, the SMA strain in the cross and straight configurations reaches 8.8% and 27.5%, respectively. Since most of the SMAs have superelastic strain lower than 6% (Table 3), SMA wires with the straight configuration cannot operate in a superelastic range at large shear strain amplitudes, especially in high-aspectratio elastomeric isolators. Whereas, the cross SMA wires can operate within an elastic range at large lateral cyclic displacements (150% and 200% shear strains). This fact demonstrates the effectiveness of the cross configuration over the straight one. Table 5 presents the effectiveness of various types of SMA wires in the cross configuration applied to the rubber bearing with two different aspect ratios. For a rubber bearing with a 0.12 aspect ratio, the maximum induced strain in wires with cross configuration is about 1% (see Table 4). However, in the case of straight arrangement, the strain in SMA wires exceeds the superelastic strain for most of the SMAs at shear strains higher than 150%. Another point is that, if a smart elastomeric base isolator with an aspect ratio of 0.36 is subjected to 200% shear strain, only FeNiCuAlTaB (with about 13.5% superelastic strain range) can be utilized. Based on the fact that the superelastic effect of SMA wires occurs at temperatures above the austenite finish temperature, the austenite finish temperature of the SMA wires should be lower than the ambient temperature. Since the minimum ambient

High damping rubber bearings have 10–16% equivalent viscous damping [24]. Elastomer layers in HDRBs have a much higher damping capacity compared to those of natural low damping rubbers used in standard elastomeric isolators with the critical damping ratios about 2–3% [25]. The damping property of the elastomeric isolator is improved by adding specific materials like extra-fine carbon block, oils, or resins and other proprietary fillers to the natural rubber [26]. With these additives, the elastomeric isolator will have a higher damping capacity, with a high initial shear modulus. Because of the materials added to the natural rubber during vulcanization process, it is difficult to accurately capture their mechanical properties like stress–strain relationship and fatigue properties. Many studies showed that although the hyperelastic material model can simulate the response of natural low damping rubber, this model cannot accurately capture the mechanical behavior of high damping rubber materials [6,27,28]. Finite element modeling is performed by considering appropriate element types, material properties, geometry, mesh, and boundary conditions (load and displacement). Element SOLID185 with eight nodes and three degrees of freedom (translation in x, y, and z directions) at each node is selected for both reinforcement and elastomeric layers. Hyperelasticity, stress stiffening, large deflection, and large strain can be modeled using this element. In order to simulate the large deformation of rubber layers at large shear strain amplitudes, the large-deflection effect is considered in full transient analyses. Steel shim used in HDRB is modeled as an isotropic material with a Young’s modulus of 210 GPa and a Poisson’s ratio of 0.3. Material properties of carbon fiber-reinforced sheets (carbon/epoxy) which behave as an orthotropic material are listed in Table 6. Among different types of nonlinear material models available in a finite element software like ANSYS [30], hyperviscoelastic model could correctly simulate the highly nonlinear behavior of high damping rubber under cyclic lateral displacements [16,17]. In this regard, Mooney–Rivlin hyperelastic model is combined with Prony viscoelastic model in order to precisely simulate the behavior of high damping rubber. Using curve fitting option in ANSYS, material constants of both Mooney–Rivlin and Prony models can be determined from experimental data obtained from ASTM or ISO standard tests. In this study, data are gathered from the uniaxial tension-compression tests, the biaxial tension test, and the creep test conducted on high damping rubber [31]. The uniaxial and biaxial tests are used for material characterization of hyperelastic model while, the material constants of viscoelastic

Table 4 Strain in SMA wires of SMA-HDRBs with different wire configurations and aspect ratios. Symbol

Wire configuration

R

c (%) 50

100

150

200

eSMA (%) eSMA (%) eSMA (%) eSMA (%) SMA-HDRB-C1 SMA-HDRB-S1 SMA-HDRB-C2 SMA-HDRB-S2

Cross Straight Cross Straight

0.12 0.36

0.1 1.0 0.6 2.6

0.3 3.7 2.2 9.1

0.6 7.2 5.0 17.8

1.0 11.1 8.8 27.5

Table 5 Operational range of SMAs for different shear strains and aspect ratios in cross configuration.

c (%)

6100 125 150 175 200

Ni Ti R 60.24 p p p p p

Ni Ti45 0.36 p p p p x

60.24 p p p p p

Ti Ni40 Cu10 0.36 p p p x x

60.24 p p p p p

Cu Al Be 0.36 p p x x x

60.24 p p p x x

Fe Mn Al Ni 0.36 p x x x x

60.24 p p p p p

FeNiCuAlTaB 0.36 p p p x x

60.24 p p p p p

0.36 p p p p p

170


Table 6 Material properties for FRP composite material [29]. FRP composite

Thickness (mm)

Elastic modulus (MPa)

Shear modulus (MPa)

Poisson’s ratio

Tensile strength (MPa)

Carbon/epoxy

0.31

Ex = 73300.0 Ey = 4613.8 Ez = 4613.8

Gxy = 1761.0 Gyz = 1659.5 Gzx = 1761.0

mxy = 0.310 myz = 0.390 mzx = 0.019

755.0

and Ragni [32]. Lateral force-deflection hysteresis loops for a range of shear strains are illustrated in Fig. 4. Based on the numerical results obtained from FE analyses, the effective horizontal stiffness of the HDRB is 7.76 kN/mm, 5.67 kN/mm, 4.56 kN/mm, and 4.23 kN/mm for 43%, 90%, 155%, and 200% shear strain amplitudes, respectively. The maximum difference between the numerical and experimental results in the horizontal stiffness is 5%, which happens at 200% shear strain. According to both experimental and numerical results, the maximum horizontal loads experienced at 43%, 90%, 155%, and 200% shear strains are 34 kN, 51 kN, 71 kN, and 85 kN, respectively.

Table 7 Hyper-viscoelastic material model constants. Mooney–Rivlin model

Prony model

C10 = 1.192 C01 = 0.547 C11 = -0.038 C02 = 0.047 C20 = 0.108 C12 = 0.000 C21 = 0.013 C03 = 0.000 C30 = 0.002

a1 = 0.765 t1 = 0.124 a2 = 0.061 t2 = 65.82

3.2. SMA-based HDRB 60

Force (kN)

40 20 0 -20

Experiment Numerical Model

-40 -60 -10

-8

-6

-4

-2

0

2

4

6

8

10

Ux (mm) Fig. 3. Lateral force-deflection hysteresis curve (f = 0.49 Hz, c = 90%) (experimental results are adapted from [32]).

model are determined through the creep test. The material constants of hyper-viscoelastic model are listed in Table 7 for a 9-parameter Mooney–Rivlin model and a 2-parameter Prony model. Fig. 3 depicts the hysteretic shear response of a HDRB consisting of two elastomer layers under a pure cyclic loading with 90% shear strain amplitude and a frequency of 0.49 Hz with no vertical pressure. Each elastomeric layer has a thickness of 5 mm. A good consistency is observed between the results obtained from the finite element simulation and experimental tests conducted by Dall’Asta

100

a

Lateral Force (kN)

Lateral Force (kN)

100 50 0 -50

-100 -2.5

In generating the FE model of the SMA-based high damping rubber bearing (SMA-HDRB), a method of superposition is implemented in order to simplify the system by decoupling the rubber bearing and SMA wires. Here, a smooth contact with no friction is assumed between the steel hook and the SMA wire. In the real case, the frictional force generated between the wire and the hook in the contact area may change the characteristics and accordingly, the response of SMA wires. This friction can be minimized by using steel hooks with a very smooth surface and minimizing the contact area between wire and hook. Fig. 5 shows a schematic of such a contact surface. Lubricating the contact surface can also reduce the friction generated due to back and forth movements of SMA wire inside the hook. In this study, instead of modeling steel hooks and the contact between the hooks and continuous SMA wires, exerted forces to the elastomeric isolator due to SMA wires are considered. This assumption considerably reduces the complexity of the finite element simulation. Otherwise, nonlinear full transient analyses for determining the hysteretic shear behavior of base isolators with different aspect ratios and wire configurations might not converge, especially at high shear strain levels. In order to simplify the FE model, before analyzing the system, the strain generated in SMA wires at each pre-defined time step is calculated according to the geometry of the device and the arrangement of wires. Rubber bearings have deformations along three axes: x, y, and z. Here, it is assumed that the effects of torsion, rotation about all axes, delamination due to the shear deformation in laminated pad, and vibration in the vertical direction are ignored. Based on these

-1.5

-0.5

0.5

Strain

1.5

2.5

b

50 0 -50 -100 -2.5

-1.5

-0.5

0.5

1.5

2.5

Strain

Fig. 4. Hysteretic shear behavior of high damping rubber bearing at 43%, 90%, 155%, and 200% shear strain amplitudes; (a) experimental results (adapted from [32]), (b) finite element simulations.

171


a

LSMA L0SMA ; L0SMA q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 L0SMA ¼ 2 ðL þ 2le Þ þ H2eff þ ðW þ 2we Þ2 þ H2eff qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 LSMA ¼ ðL þ 2le þ DXÞ þ H2eff þ ðL þ 2le DXÞ þ H2eff qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ2 DX 2 þ ðW þ 2we Þ2 þ H2eff

b

eSMA ¼

Steel Hook

SMA Wire

Fig. 5. Schematic of steel hook and SMA wires in contact; (a) loose, (b) tight.

simplifications, the formula of SMA strain in two cases is provided: (1) strain induced by displacements in x, y, and z directions and (2) strain induced by a displacement in the x direction. In the latter case, the effect of vertical displacement, DZ, due a constant compressive load is assumed to be negligible on the SMA strain since the vertical deflection is very small compared to the lateral displacements in the xy plane. Fig. 6 shows the rubber bearing equipped with cross SMA wires and subjected to a combination of vertical displacement in the z direction and lateral displacement in the xy plane. According to this figure, the strain in the wire is calculated using Eq. (3). In this equation, the initial length, L0, remains constant while, length L changes over the time since it is a function of displacements, DX, and DY. It should be noted that DZ has a constant value due to a constant vertical pressure applied on the isolator.

LSMA L0SMA ; L0SMA q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 L0SMA ¼ 2 ðL þ 2le Þ þ H2eff þ ðW þ 2we Þ2 þ H2eff qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 LSMA ¼ ðL þ 2le þ DXÞ þ DY 2 þ ðHeff DZÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ ðL þ 2le DXÞ þ DY 2 þ ðHeff DZÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ DX 2 þ ðW þ 2we þ DYÞ2 þ ðHeff DZÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ DX 2 þ ðW þ 2we DYÞ2 þ ðHeff DZÞ2

eSMA ¼

ð3Þ

ð4Þ

It should be noted that, for decoupling the effect of SMA wires from the rubber bearing in a general case, the effect of displacement in 3D on SMA strains can be calculated in order to find out the force in wires and added to the response of rubber bearing using the method of superposition. In this study, it is assumed that isolators are subjected to a cyclic lateral displacement only in the x direction. In the next step, the axial stress in SMA wires can be determined form the strain based on the stress–strain relationship of shape memory alloy [33]. The idealized stress–strain curve for ferrous SMA FeNiCuAlTaB (FeNCATB) and NiTi45 are plotted in Fig. 7. Then, using the value of the axial stress and the direction of wires at each time step, the force vectors (with x, y, and z components) exerted from the SMA wire to the hooks are computed. Since the SMA wires with cross arrangement (wrapped around the elastomeric bearing) has a 3D form, at each corner where the SMA wire is in contact with the steel hook, the generated force along the wires has components in all three directions of x, y, and z. In the decoupled system, all of these components are taken into account for FE simulations. Therefore, instead of modeling the SMA wires, the steel hooks, and the contact between them, the equivalent forces are applied to the rubber bearing at each time step while running the nonlinear analysis. In fact, the rubber bearing and the SMA wires are decoupled as two separate systems in FE simulations. Then, by measuring the force generated in SMA wires as a function of time, the effect of one system (SMA wires) on the other one (elastomeric isolator) is estimated. 4. Seismic response of a three-span continuous bridge

If the displacement is in one direction (x), considering aforementioned assumptions, Eq. (3) will be simplified to Eq. (4).

ΔY

The responsibility of the base isolation system is to significantly reduce the deck acceleration and as a result the force transferred to

b

ΔX

wE

a W

ΔY ΔZ ΔX

wE z

y y x

x ΔX ΔZ

lE

L

lE

ΔY y

ΔZ z Heff – ΔZ z x

Fig. 6. SMA-HDRB with cross arrangement of wires subjected to lateral displacements in the xy plane; (a) 3D view, (b) orthographic views (top, front, and side views).

172


1000

Stress (MPa)

800 600 400

NiTi 45 FeNCATB

200 0

0

2

4

6

8

10

12

14

16

Strain (%) Fig. 7. Idealized stress–strain curve for NiTi45 and FeNCATB SMAs.

the piers. In order to investigate the seismic performance of HDRBs and SMA-based HDRBs, an scaled three-span continuous steel girder reinforced-concrete (RC) pier supported bridge is modeled and analyzed under seismic ground motion using finite element software Seismostruct [34]. In this regard, time history analyses are performed for three earthquake records: Loma Prieta, Northridge, and Kobe with the maximum peak ground acceleration of 0.46 g,

0.38 g, and 0.42 g, respectively. The magnitude of the ground motions in the Richter scale is between 6.7 and 7.0. Since it is assumed that the bridge is located in Vancouver city, all ground motion records are scaled by Vancouver design response spectrum with a damping ratio of 5% as a design ground motion level. The bridge consists of continuous RC deck-steel girder isolated by four rubber bearings installed between the steel girder and pier caps. The superstructure consists of 260 mm RC slab covered by 80 mm of asphalt layer. The height of the continuous steel girder is 600 mm. The substructures consist of RC piers and footings supported on shallow foundation. The reinforcement details of the bridge pier consist of D29 (29 mm diameter) longitudinal reinforcements in the longer direction distributed at 200 mm center to center except at the corners where the spacing is 125 mm. In the shorter direction, D29 rebars are distributed at 200 mm except at the corners where the spacing is limited to 150 mm. The hoop reinforcements in both directions are D22 (22 mm diameter) distributed at 125 mm. The isolated bridge, the bridge pier and the superstructure are shown in Fig. 8. The superstructure consisting of reinforced concrete bridge deck and steel girders is modeled using linear beam-column elements so that the superstructure remains elastic under the seismic loads applied in the longitudinal direction. It should be mentioned

Fig. 8. Geometric details of the bridge; (a) longitudinal view of the bridge, (b) transverse and longitudinal views of the bridge pier, (c) transverse section of the superstructure (all dimensions are in mm).

F. Hedayati Dezfuli, M.S. Alam / Engineering Structures 61 (2014) 166–183 Table 8 Geometry and material properties of the bridge. Properties

Specifications

Cross-section of the pier cap (mm mm) Cross-section of the pier body (mm mm) Cross-section of the girder (mm mm) Cross-section of the footing (mm mm) Height of the pier (mm) Concrete compressive strength (N/mm2) Yield strength of steel (N/mm2) Young’s modulus of steel (N/mm2)

667 4000 667 3000 667 2667 1667 3000 5000 35 485 200,000

Table 9 Material properties of bilinear kinematic hardening model used for rubber bearings. HDRB-1 HDRB-2 SMA-HDRB-S1 SMA-HDRB-C2 Initial stiffness K0 (kN/mm) 8.24 Yield force Fy (kN) 25.8 Post-yield hardening ratio, r 0.23

2.76 25.7 0.22

13.54 35.1 0.24

3.32 29.8 0.23

that the stiffness of the superstructure does not have a significant effect on the seismic response of the bridge [35]. Steel is modeled using bilinear kinematic strain hardening model. Mander et al. constitutive relationship [36] represents the behavior of concrete. Bridge piers are modeled with nonlinear beam-column elements. The pier cap and footings are modeled by using linear elastic elements. It is assumed that these components will remain elastic and sufficiently rigid during the earthquake excitations. Details of analytical model can be found in Alam et al. [37]. In order to assess the performance of HDRBs and SMA-HDRBs with two cross and straight configurations, the seismic response of the bridge equipped with these rubber bearings is evaluated and compared to that of the non-isolated bridge. The geometry and material properties of the bridge piers, bridge deck and footings are listed in Table 8. HDRBs and SMA-HDRBs are modeled as link elements using bilinear kinematic hardening model with three characteristics: the initial stiffness (K0), the yield force (FY), and the post-yield hardening ratio (r) defined in the longitudinal and transverse directions. Here, the behavior of rubber bearing is idealized using symmetric bilinear hysteresis curve as an approximation. However, it should be mentioned that an elastomeric material simulated by hyper-viscoelastic model will provide more accurate result. These characteristics are given in Table 9 for HDRB-1, HDRB-2, SMA-HDRB-S1, and SMA-HDRB-C2. It should be noted that the geometry of HDRB-1 and HDRB-2 are same as those of CFRHDRB#1 and CFR-HDRB#7, respectively, as shown in Table 1. 5. Results and discussion Based on the nonlinear material model of high damping rubber, which has been verified with the experimental tests [32], finite element analyses have been performed to simulate the behavior of the SMA-HDRB using ANSYS [30]. By calculating the hysteretic response of SMA-HDRBs subjected to a frequency of 0.2 Hz and 6 MPa vertical pressure, the influence of the shear strain amplitude, the aspect ratio of rubber bearing, the types of SMA, the wire arrangement, the thickness of SMA wires, and the pre-strain in SMA wires on the performance of base isolators are assessed. 5.1. Hysteretic behavior of SMA-HDRB: design basis By considering two wire configurations and two different aspect ratios for rubber bearings, 0.12 and 0.36, four cases are investigated (Table 4). NiTi45 wire, in the straight arrangement, undergoes a large strain which is beyond the superelastic strain range (6.8%)

173

at large shear strain amplitudes (see Table 4). Therefore, to study the effect of SMA type, ferrous (FeNCATB) and NiTi-based (NiTi45) wires are considered to be installed in low-aspect-ratio CFR-HDRBs (R = 0.12) with cross arrangement. Ferrous FeNCATB wire, as a more efficient option compared to NiTi45, is chosen to be implemented in all four cases. The hysteretic shear behavior of each SMA-HDRB is compared to that of a CFR-HDRB with the same geometrical and mechanical properties under four different shear strain levels: 50%, 100%, 150%, and 200%. Increasing the diameter of SMA wires increases the effective horizontal stiffness and the total energy dissipation capacity of the base isolation system. On the other hand, increasing the amount of pre-strain in SMA wires causes the horizontal stiffness to be decreased. Since increasing the lateral flexibility and maximizing the damping capacity of a rubber bearing are both beneficial, it is more advantageous to reach the target horizontal stiffness by changing both the diameter and pre-strain of SMA wires rather than just altering the diameter of wires. In a design procedure, in order to determine the diameter and pre-strain of wires, first, a target value for the effective horizontal stiffness (KH d) of elastomeric isolator is determined. Then, for a specific size of base isolator, the diameter of SMA wires increases from an initial value. Based on the geometry of rubber bearing and the configuration of SMA wires (cross), a minimum initial diameter r0w is obtained from a force ratio (RF), defined as a ratio of the maximum lateral force generated by SMA wires (FSMA max) to the maximum shear force of the isolator with no SMA wires (Fs max) at 100% shear strain amplitude. The force ratio is set to a value at which the effect of SMA wires can be observed on the hysteretic shear behavior of the base isolator (here considered 10%). In fact, knowing the maximum strain in SMA wire at c = 100%, from Eq. (4), and accordingly the maximum axial stress in wire, rmax, the initial radius of SMA wire, r 0w , can be obtained from FSMA max calculated using the force ratio. The stiffness margin, MK (initially considered 15%), is defined in order to determine the relative difference between the calculated effective lateral stiffness of SMA-based rubber bearing, KH, and the target value of stiffness, KH d. Under this condition, the radius of wires should increase up to a level at which KH is 15% higher than the target value. In this step, by fixing the radius, pre-strain in SMA wires goes up and the effective lateral stiffness is calculated. Considering pre-strain values lower than 5%, if the reduction in shear force due to pre-straining the SMA wires causes the lateral stiffness to be reduced and reaches the target point, with a 10% error, the values of radius and pre-strain are selected and the design procedure is finished. Otherwise, the stiffness ratio decreases by 5%, the radius of wire and the pre-strain are set to their initial values (r 0w and 0, respectively), and then, the effective horizontal stiffness is recalculated again. The flow chart of the design process is shown in Fig. 9. Based on the described design procedure, the radius of SMA wires increases from 1 mm, considered as an initial value, to the 2.5 mm. Since the stiffnesses of FeNCATB and NiTi45 at the austenite phase are close to each other (see Table 3), same thicknesses are selected for both types of SMA wires. In order to make the design procedure more clear, an example is given for SMA-HDRB-C2 in Appendix A. 5.1.1. Fe- and NiTi-based SMA wires FeNCATB and NiTi45 wires having cross configurations are used in SMA-HDRBs with aspect ratios of 0.12 (SMA-HDRB-C1). Both types of SMA wires have the same cross sectional areas (19.6 mm2). Fig. 10 shows lateral force-deflection curves of SMAHDRB-C1s consisting of different types of SMA at different shear strain amplitudes. According to Table 4, at 200% shear strain, the strain in SMA wires reaches 1.0%. As shown in Fig. 7, the strain at

174


(in austenite phase) compared to FeNCATB and as a result, a higher axial stress is generated in NiTi45 wire for the same amount of strain in wires. However, the performances of SMA-HDRB-C1s are very similar when ferrous and NiTi-based SMA wires are implemented. This fact can be understood by comparing the effective horizontal stiffnesses and residual deformations of SMA-HDRBC1s according to Table 10. This consistency in results is due to a low amount of strain in wires (1%), which is much smaller than the superelastic strain for both types of SMA wires. In other words, if the wire strain increases with increasing the aspect ratio of rubber bearing or changing the wire configuration, the behaviors of SMA-HDRBs will vary.

Set a target for effective horizontal stiffness,

Calculate the initial value of radius of SMA wire, Reduce

by 5%

Calculate No Yes

Calculate Yes No

No

Yes

Fig. 9. Flow chart of design procedure to determine the diameter and pre-strain of SMA wires.

which the forward phase transformation is started in FeNCATB and NiTi45 wires is 1.6% and 0.6%, respectively. Consequently, when the SMA-HDRB-C1 equipped with NiTi45 is subjected to 200% shear strain, wires go to the phase transformation region (see Fig. 7). The maximum shear force in smart rubber bearings at c = 200% is 141 kN and 147 kN when FeNCATB and NiTI45 wires are used, respectively. The reason is that, NiTi45 has a higher elastic stiffness

200

a

Lateral Force (kN)

Lateral Force (kN)

200 100 0 -100

-200 -80

5.1.2. CFR- and SMA-HDRB (R = 0.12) Fig. 11 shows the lateral force-deflection curves of the CFRHDRBs with two aspect ratios of 0.12 and 0.36 at four different shear strain amplitudes. Fig. 12 depicts the hysteresis loops of low-aspect-ratio SMA-HDRBs (R = 0.12), which are equipped with cross wires (SMA-HDRB-C1) and straight wires (SMA-HDRB-S1) with the same cross sectional areas (19.6 mm2). By comparing the hysteretic behavior of CFR-HDRB and SMAHDRB, it is observed that the operational characteristics of smart base isolators, including the effective horizontal stiffness and the residual deformation, change due to the use of SMA wires. When the stress along the SMA wires increases, a forward phase transformation from austenite to martensite occurs in the wires. As a result, the horizontal component of the force exerted to the rubber bearing noticeably goes up in the opposite direction of the lateral cyclic displacements. This increase in the lateral force causes horizontal stiffening and re-centering in the whole system. Therefore, the effective horizontal stiffness increases and the residual deformation decreases. The strain induced in the SMA wires in the straight configuration increases with a very high rate from 1.0% at 50% shear strain to 11.1% at 200% shear strain (Table 4). As a result, the stress along the SMA wires increases rapidly. In such a situation, using FeNCATB wires instead of NiTi45 wires is advantageous since FeNCATB has a higher superelastic strain range (13.5%). Tables 11 and 12 demonstrate the changes in the effective horizontal stiffness and the residual deformation, respectively, by changing the arrangement of wires for the aspect ratio of 0.12. In the cross configuration (SMA-HDRB-C1), slight changes in the stiffness and the residual deformation are observed compared to performance characteristics of the CFR-HDRB. On the other hand, when SMA wires are installed in the straight configuration (SMAHDRB-S1), the horizontal stiffness increases and the residual deformation considerably decreases. This is because of a large amount of stress generated in SMA wires due to the higher elongation compared to the wires with cross arrangement. As can be seen in Fig. 12 and Tables 11 and 12, for 200% shear strain, although the phase transformation in the FeNCATB wires is not completed, a high amount of yield stress, defined as the

-40

0

40

Lateral Displacement (mm)

80

100

b

0 -100 -200 -80

-40

0

40

80


Fig. 10. Hysteresis curves of SMA-HDRB-C1; (a) FeNCATB SMA, (b) NiTi45 SMA (c = 50%, 100%, 150%, 200%).

175

F. Hedayati Dezfuli, M.S. Alam / Engineering Structures 61 (2014) 166–183 Table 10 Effective horizontal stiffness and residual deformation of SMA-HDRB-C1s with FeNCATB and NiTi45 wires.

c (%)

FeNCATB

50 100 150 200

NiTi45

KH (kN/mm)

DKH (%)

R.D. (mm)

DRD (%)

KH (kN/mm)

DKH (%)

R.D. (mm)

DRD (%)

3.89 2.93 2.60 2.49

0 0 0 0

4.3 10.3 18.1 29.5

11 12 14 16

3.89 2.94 2.61 2.58

0 0 0 0

4 10 18 34

10.9 11.5 13.4 14.6

Note: D is the relative difference between performance characteristics of SMA-HDRBs and CFR-HDRB.

200

a

Lateral Force (kN)

Lateral Force (kN)

200 100 0 -100

-200 -80

-40

0

40

b

100 0 -100 -200 -200

80


-100

0

100

200


Fig. 11. Hysteresis curves of CFR-HDRB; (a) R = 0.12, (b) R = 0.36 (c = 50%, 100%, 150%, 200%).

250

a

Lateral Force (kN)

Lateral Force (kN)

200 100 0 -100

-200 -80

-40

0

40

80


150

b

50 -50 -150 -250 -80

-40

0

40

80


Fig. 12. Hysteresis curves of (a) SMA-HDRB-C1, (b) SMA-HDRB-S1 (c = 50%, 100%, 150%, 200%).

observed that, for rubber bearings with low aspect ratio (0.12), the performance of SMA-HDRB-S1 is superior to that of the SMAHDRB-C1.

Table 11 Effective horizontal stiffness of CFR-HDRB and SMA-HDRBs (R = 0.12).

c (%) 50 100 150 200

CFR-HDRB

SMA-HDRB-C1

KH (kN/mm)

KH (kN/mm)

DKH (%)

KH (kN/mm)

SMA-HDRB-S1

DKH (%)

3.93 2.95 2.61 2.48

3.89 2.93 2.60 2.49

0 0 0 0

5.03 4.41 3.80 3.56

28 50 46 43

Note: D is the relative difference between effective horizontal stiffnesses of SMAHDRBs and CFR-HDRB.

Table 12 Residual deformation of CFR-HDRB and SMA-HDRBs (R = 0.12).

c (%) 50 100 150 200

CFR-HDRB

SMA-HDRB-C1

R.D. (mm)

R.D. (mm)

DRD (%)

R.D. (mm)

SMA-HDRB-S1

DRD (%)

4.8 11.6 21.1 35.2

4.3 10.3 18.1 29.5

11 11 14 16

3.1 8.5 13.0 18.3

36 27 38 48

Note: D is the relative difference between residual deformations of SMA-HDRBs and CFR-HDRB.

starting stress for the forward phase transformation, causes a significant growth of 43% in the effective horizontal stiffness and a high reduction of 48% in the residual deformation. It is also

5.1.3. CFR- and SMA-HDRB (R = 0.36) Fig. 11b depicts the lateral force-deflection curves of the CFRHDRB with an aspect ratio of 0.36 for different shear strain amplitudes. Similar to the previous cases, SMA wires used in the cross and the straight configurations have equal cross sectional area (19.6 mm2). In the cross configuration (SMA-HDRB-C2), for shear strain amplitudes up to 150%, the maximum strain induced in the SMA wires is 4.99% which is lower than the superelastic strain range of FeNCATB. In the SMA-HDRB-S2, when the aspect ratio is 0.36, the strain in the SMA wire increases from 2.6% to 27.5% with increasing the shear strain from 50% to 200%. At 150% and 200% shear strain levels, the strain generated in the straight FeNCATB wires is higher than the superelastic strain (13.5%). As a result, FeNCATB SMA wires undergo a plastic deformation. Fig. 13 shows the hysteretic behaviors of SMA-HDRB-C2 and SMA-HDRB-S2. Since the plastic deformation substantially degrades the performance of SMA wires subjected to cyclic loadings, the lateral force-deflection curves of SMA-HDRB-S2 are sketched for the first cycle at 150% and 200% shear strains. By comparing the hysteresis curves of two SMA-HDRBs (Fig. 13a and b), it is observed that the effective horizontal stiffness significantly increases when the SMA wires are mounted in the straight

176


400

a

Lateral Force (kN)

Lateral Force (kN)

400 200 0 -200

-400 -200

-100

0

100

200

200

b

0 -200 -400 -200

-100

0

100

200



Fig. 13. Hysteresis curves of (a) SMA-HDRB-C2, (b) SMA-HDRB-S2 (c = 50%, 100%, 150%, 200%).

300

Lateral Force (kN)

200 100 0 -100 -200 -300 -150

-100

-50

0

50

100

150

Lateral Displacement (mm) Fig. 14. Hysteresis curves of SMA-HDRB-S2 (c = 150%).

arrangement. In order to investigate the effect of SMA wires subjected to strains higher than superelastic strain range, the hysteretic shear behavior of SMA-HDRB-S2 is re-plotted at shear strain of 150% in Fig. 14. As can be seen in Fig. 14, the lateral force does not follow the path indicated by dashed lines. SMA-HDRB-S2 experiences a rapid change in the lateral force at about 35 mm lateral displacement. Considering a complete cycle of lateral displacement, if SMA wires undergo a plastic deformation due to exceeding the superelastic strain, the deformation will remain in the wires when the bearing returns to its initial position in the first half cycle and as a result, the wires get loose at zero lateral displacement. Thus, in the second half cycle, by increasing the lateral displacement, SMA wires will not be effective up to a level of lateral displacement (here 35 mm) at which the wires are tight in their place. In this situation, the strain in SMA wires starts to increase from the plastic strain previously generated and not from zero. When the rubber bearing returns to its initial position in the second half cycle, the stress in SMA wires reaches zero before the lateral displacement reaches zero (at about 50 mm). Therefore, compared to a case in which no plastic deformation happens in SMA wires, the shear force in the base isolator decreases. The shear behavior of SMA-HDRB-S2 at 200% shear strain can be interpreted similarly. Eq. (4) provides a relation between the lateral displacement and the strain in SMA wires. Knowing the frequency of horizontal cyclic loading, the exact time and the lateral displacement level at which the strain in SMA wires exceeds the superelastic strain limit can be specified. Subsequently, the time and the displacement at which the axial stress in wires vanishes while the rubber bearing is returning to its initial position can be determined from the stress–strain curve of SMA (Fig. 7). The maximum lateral forces in SMA-HDRB-C2 and SMA-HDRBS2 at 200% shear strain are 153 kN and 320 kN, respectively. At this shear strain, the straight wires undergo 27.5% strain, which is much higher than the superelastic strain range of FeNCATB. After

finishing the forward phase transformation in SMA wires, at strains above 15%, the stress increases rapidly with a rate equal to the martensitic modulus of elasticity. As a result, a huge amount of stress is generated in SMA wires when the strain reaches 27.5%, and consequently, the lateral stiffness of the base isolator considerably increases. Tables 13 and 14 compare the performance characteristics of CFR-HDRB, SMA-HDRB-C2, and SMA-HDRB-S2 at different shear strain levels. At 50% shear strain, the effective horizontal stiffness of CFR-HDRB increases by 4% and 46%, respectively, when cross and straight wires are used. At the maximum considered shear strain (200%), the effective horizontal stiffnesses of SMA-HDRB-C2 and SMA-HDRB-S2 are 16% and 142% greater than that of CFR-HDRB, respectively. The maximum reduction in the residual deformation of SMAHDRB-C2 is 32%, which happens in the straight configuration at 100% shear strain (Table 14). Under this condition, the effective horizontal stiffness increases about 58% (Table 13). Since SMA wires undergo plastic deformations at 150% and 200% shear strains, they cannot be fully recovered after unloading. As a result, their performance in improving the re-centering capability of rubber bearing is degraded. In the cross configuration, when the residual deformation decreases by 22% at a shear strain of 200%, the effective horizontal stiffness increases by 16%. This fact shows that although the straight configuration of SMA wires can reduce the residual deformation of rubber bearing more than the cross configuration, it has an inferior performance in terms of the lateral flexibility.

Table 13 Effective horizontal stiffness of CFR-HDRB and SMA-HDRB (R = 0.36).

c (%) 50 100 150 200

CFR-HDRB

SMA-HDRB-C2

KH (kN/mm)

KH (kN/mm)

DKH (%)

KH (kN/mm)

SMA-HDRB-S2

DKH (%)

1.30 0.96 0.85 0.78

1.36 1.14 1.00 0.90

4 18 18 16

1.89 1.52 1.70 1.89

46 58 99 142

Note: D is the relative difference between effective horizontal stiffnesses of SMAHDRBs and CFR-HDRB.

Table 14 Residual deformation of CFR-HDRB and SMA-HDRBs (R = 0.36).

c (%) 50 100 150 200

CFR-HDRB

SMA-HDRB-C2

SMA-HDRB-S2

R.D. (mm)

R.D. (mm)

DRD (%)

R.D. (mm)

DRD (%)

14.9 36.4 69.2 109.4

12.4 33.0 54.9 86.6

17 9 21 22

12.7 24.7 60.8 116.3

15 32 12 6

Note: D is the relative difference between residual deformations of SMA-HDRBs and CFR-HDRB.

177


200

a

Lateral Force (kN)

Lateral Force (kN)

200 100 0 -100

-200 -80

-40

0

40

80


b

100 0 -100 -200 -80

-40

0

40

80


Fig. 15. Hysteresis curves of Ferrous SMA-HDRB-C1; (a) r = 2.5 mm, (b) r = 5 mm (c = 50%, 100%, 150%, and 200%).

The dimension of SMA wire can affect the behavior of smart rubber bearing subjected to a compressive and a cyclic shear loading. When the radius of the wire’s cross section increases, the effective force exerted to the base isolator is significantly enhanced. As a case study, the cross FeNCATB wires with two different radii: 2.5 mm and 5.0 mm are chosen to be installed in a CFR-HDRB with an aspect ratio of 0.12. The corresponding cross sectional areas of SMA wires are 19.6 mm2 and 78.5 mm2. In reality, for the second case, two 2.5 mm wires can be used rather than a thick wire with a 5 mm radius. In order to investigate the effect of SMA wires’ thickness on the performance of the rubber bearing, hysteretic behaviors of smart base isolators are compared (Fig. 15). Compared to CFR-HDRB, changes in the effective horizontal stiffness and the residual deformation of SMA-HDRBs are calculated with increasing the radius of SMA wires and shear strain amplitude (Table 15). When 2.5 mm SMA wires are used in a HDRB subjected to a 200% shear strain, the effective horizontal stiffness does not change while the residual deformation decreases about 16%. When 5.0 mm SMA wires are incorporated, the effective horizontal stiffness increases by 8% and the residual deformation decreases by 34%. This shows that using an SMA wire with a higher thickness improves the performance of the HDRB in terms of residual deformation reduction. The reason of this behavior is the increase of the force generated in the SMA wires due to an increase in the cross sectional area of wires.

efficiently use the SMA wire as a damper in the rubber bearing, the initial elastic part should be removed by pre-straining the SMA wire [11]. In the pre-strained SMA wire, the forward phase transformation occurs at a lower strain and as a result, the yield stress considerably decreases. Consequently, when the SMA wire is subjected to a tension due to the cyclic displacement in the rubber bearing, a smaller force will be transferred to the superstructure. By applying a pre-strain (e.g. 2%) to the SMA wire, the stress–strain curve shifts according to Fig. 16. In order to investigate the effect of pre-straining on the performance of smart elastomeric bearings, ferrous SMA wires with 2% pre-strain and a cross sectional area of 19.6 mm2 are mounted on a HDRB with an aspect ratio of 0.36 in cross configuration. When SMA wires are wrapped around the rubber bearing by passing through the steel hooks bolted to the steel end plates, both ends of the wire are attached to a slotted hexagonal head bolt which is mounted in a small box (housing). According to Fig. 17a, ends of the SMA wire are passed through a hole located in the middle

1000 Shift due to Pre-Straining

Stress (MPa)

5.2. Effect of SMA wire’s thickness

800 600 400 200 0 0

5.3. Effect of pre-strain in SMA wires

2

4

6

8

10

12

Strain (%) The yield strength of the ferrous FeNCATB SMA which represents the starting stress for the forward phase transformation (austenite to martensite) is about 740 MPa (Fig. 7). The increasing rate of the stress from 0 to 740 MPa, which represents the initial elastic stiffness in the austenite phase, is about 47 GPa. The large amount of yield stress, which happens at 1.7% strain, exerts a large lateral force to the rubber bearing in the opposite direction of the cyclic horizontal displacement. In such a situation, the lateral flexibility is noticeably reduced and as a result, the effective horizontal stiffness significantly increases compared to a CFR-HDRB. In order to

Table 15 Changes in the effective stiffness and the residual deformation of SMA-HDRB-C1 compared to those of the CFR-HDRB for different radii of SMA wire (R = 0.12).

c (%)

50 100 150 200

Effective horizontal stiffness (%) r (mm)

Residual deformation (%) r (mm)

2.5

5

2.5

5

0 0 0 0

0 2 5 8

11 12 14 16

12 16 24 34

Fig. 16. Stress–strain curve of FeNCATB SMA before and after pre-straining.

a

Slotted Hexagonal Head Bolt

SMA Wire

b

SMA Wire

Hole Housing

Fig. 17. Adjustable mechanism for fixing the SMA wire and applying pre-strain; (a) side view of the mechanism, (b) 3D view of slotted hexagonal head bolt with a hole in the middle.


Lateral Force (kN)

200

200

a

100 0 -100 -200 -200

HDRB SMA-RB SMA-RB (2% Pre-Strain)

-100

0

100

200


Lateral Force (kN)

178

b

100 0 -100

2% Pre-Strain 3% Pre-Strain

-200 -200

-100

0

100

200


Fig. 18. Lateral force-displacement curves of (a) CFR-HDRB, SMA-HDRB-C2, and SMA-HDRB-C2 with 2% pre-strain, (b) SMA-HDRB-C2 with 2% and 3% pre-strains (c = 200%).

of the bolt (see Fig. 17b). In order to fix the wire from sliding inside the bolt, before applying a pre-stress, wire is wound around the bolt tightened clockwise. To apply a pre-strain to SMA wires, the bolt is rotated and as a result, wires are subjected to a tensile stress. Accordingly, a specific amount of pre-strain is generated in the stretched wires (e.g. 2% pre-strain). The advantageous of the proposed mechanism is its high accuracy in adjusting the level of pre-strain by accurately tightening the bolt. In fact, the level of pre-strain can be controlled by considering the perimeter of the screw and the number of rotation applied to it. Fig. 18a depicts the hysteresis shear behavior of a CFR-HDRB, an SMA-HDRB without pre-strain (SMA-HDRB), and an SMA-HDRB with 2% pre-strain. The areas of cross section of SMA wires are the same (19.6 mm2). All rubber bearings are subjected to 200% shear strain and 6 MPa vertical pressure. When 2% pre-strained SMA wires are used, the decrease in the residual deformation is greater and the increase in the effective horizontal stiffness is lower compared to those of the non-pre-strained SMA-HDRB. These changes show that pre-strained SMA wires can improve the efficiency of the smart rubber bearing more than the unstretched wires. Changes in the effective horizontal stiffness and the residual deformation are listed in Table 16. The effective horizontal stiffness of the SMA-HDRB-C2 is 16% more than that of the HDRB reinforced by CFRP composite plates. When 2% pre-strained SMA wires are used, the effective horizontal stiffness increases to 0.87 kN/mm which is 14% higher than that of CFR-HDRB. This shows that the reduction in the lateral flexibility of SMA-HDRB can be controlled by pre-straining SMA wires. The reason for this behavior is the lower amount of stress induced in the pre-strained SMA wires after the completion of the forward phase transformation. On the other hand, compared to the CFR-HDRB, the residual deformations of the SMA-HDRB and 2% pre-strained SMA-HDRB are reduced by 22% and 8%, respectively. When the 2% pre-strained SMA wires are elongated due to the horizontal cyclic displacement, the forward phase transformation in the wires starts and finishes at lower stress and strain levels compared to a case in which unstretched SMA wires are used. Therefore, a smart HDRB with pre-strained

Table 16 Characteristics of CFR-HDRB and SMA-HDRB-C2 for different amounts of pre-strain in SMA wires (R = 0.36, c = 200%). Rubber bearing

CFR-HDRB SMA-HDRB SMA-HDRB (2% Pre-Strain) SMA-HDRB (3% Pre-Strain)

Horizontal stiffness

Residual deformation

KH (kN/mm)

DKH (%)

R.D. (mm)

DRD (%)

16 14 15

109.4 85.8 100.3 101.1

22 8 8

0.78 0.90 0.86 0.89

Note: D is the relative difference in operational characteristics of SMA-HDRB-C2 and CFR-HDRB.

SMA wires cannot reduce the residual deformation of CFR-HDRB as much as SMA-HDRB with unstretched wires. In order to evaluate how the amount of pre-strain in SMA wires affect the performance of the smart rubber bearing, considering same size of cross section, SMA wires with 3% pre-strain are used in the SMA-HDRB. The hysteretic behavior and performance characteristics of the SMA-HDRB with 3% pre-strained wires are compared to those of the SMA-HDRB with 2% pre-strained SMA wires. Fig. 18b demonstrates that changes in the hysteretic response of SMA-HDRBs are negligible when the magnitude of pre-strain increases in the SMA wire. According to Table 16, the increase in the effective horizontal stiffness of the SMA-HDRB with 3% prestrained wires is a little higher than that of the 2% pre-strained SMA-HDRB. Under the maximum lateral displacement, a higher strain is generated in the 3% pre-strained SMA wires compared to the 2% pre-strained wires. As a result, the maximum stress at which SMA wires are fully martensite will be larger in the SMA wires with 3% pre-strain. Therefore, a higher lateral force is applied to the rubber bearing from the 3% pre-strained wires and consequently, the horizontal stiffness of the 3% pre-strained SMA-HDRB increases by 15%. Since the stresses of starting and finishing phase transformation in the 2% pre-strained SMA wire are close to those of the 3% pre-strained wire, the performances of SMA-HDRBs with 2% and 3% pre-strains in reducing the residual deformation of the CFR-HDRB are the same (see Table 16).

5.4. Dynamic time history analysis The acceleration of the bridge deck (Ad), the displacement of pier and deck, the relative lateral displacement of rubber bearings, and the total energy dissipated by all elastomeric isolators installed in the bridge are calculated for four cases in which HDRB1, HDRB-2, SMA-HDRB-S1, and SMA-HDRB-C2 are implemented. Since the elastomeric rubber bearings isolate the bridge deck from piers, the acceleration of the deck and consequently, the forces trasnferred to piers are noticeably reduced. In this regard, it is of great interest to study the effect of elastomeric isolators on the deck acceleration. Here, three earthquake records: Loma Prieta, Northridge, and Kobe scaled by Vancouver design response spectrum (with a 5% damping ratio) are considered for time history analyses. Time variation of the lateral displacement of bridge deck and pier cap, as well as the relative displacement of rubber bearing during Loma Prieta earthquake is plotted in Fig. 19 while isolating the bridge with HDRBs and SMA-HDRBs. It can be seen that in all four cases, the peak displacement of the pier is considerably lower than that of the bridge deck because of the presence of elastomeric bearings. In fact, rubber bearings laterally isolate the deck from the pier and as a result, when the deck undergoes a large deflection, the displacement is not transferred to the pier and accordingly, the base shear force noticeably decreases. When the bridge is isolated by

179

Dispalcement (mm)


150 100 50 0 -50 -100 -150

Pier Deck Rubber Bearing

HDRB-1

0

5

10

15

20

25

Dispalcement (mm)

Time (s) 150 100 50 0 -50 -100 -150


HDRB-2

0

5

10

15

20

25

Dispalcement (mm)

Time (s) 150 100 50 0 -50 -100 -150


SMA-HDRB-S1

0

5

10

15

20

25

Dispalcement (mm)

Time (s) 150 100 50 0 -50 -100 -150


SMA-HDRB-C2

0

5

10

15

20

25

Time (s) Fig. 19. Time variation of lateral displacement of pier cap, deck, and rubber bearing for bridges isolated by different elastomeric isolators and subjected to Loma Prieta earthquake scaled by Vancouver design response spectrum.

HDRBs or SMA-HDRBs with high or low aspect ratios, no residual deformation is observed in piers after the earthquake. However, by increasing the total thickness of rubber layers (increasing the aspect ratio), the elastomeric isolator can tolerate higher shear strain amplitudes and accordingly its limit to prevent the bridge piers from being plastically deformed enhances. In Table 17, the peak relative displacement of four rubber bearings installed in

the three-span continuous bridge is given for each case (e.g. HDRB-1, HDRB-2, SMA-HDRB-S1, and SMA-HDRB-C2) under three scaled ground motions. Results show that for Loma Prieta earthquake, when HDRB-1 and SMA-HDRB-S1 are used, the maximum shear strain in rubber bearings reaches 274% and 180%, respectively. While, by using HDRB-2 or SMA-HDRB-C2, shear strain does not exceed 120% (see Table 17). Under Northridge and Kobe

Table 17 Maximum lateral displacements and corresponding shear strains in four rubber bearings along the bridge subjected to earthquakes scaled by Vancouver design response spectrum. RB#1

RB#2

RB#3

Ux (mm)

c (%)

Ux (mm)

c (%)

Loma prieta HDRB-1 HDRB-2 SMA-HDRB-S1 SMA-HDRB-C2

78.1 101.9 51.1 97.5

274 120 180 115

76.1 101.1 49.6 96.7

267 119 174 114

Northridge HDRB-1 HDRB-2 SMA-HDRB-S1 SMA-HDRB-C2

79.0 100.5 60.4 95.5

278 119 212 113

76.9 98.2 59.1 94.2

Kobe HDRB-1 HDRB-2 SMA-HDRB-S1 SMA-HDRB-C2

79.3 105.1 56.9 99.9

279 124 200 118

77.0 104.3 55.1 99.1

RB#4

c (%)

Ux (mm)

c (%)

73.7 98.7 47.2 93.5

259 117 166 110

69.1 91.5 44.0 87.1

243 108 155 103

270 116 208 111

74.1 97.0 57.2 92.4

261 114 201 109

69.2 89.4 53.5 84.8

243 105 188 100

271 123 194 117

73.7 101.1 53.0 95.6

259 119 186 113

68.4 94.0 48.9 88.6

240 111 172 105

Ux (mm)

180


Acceleration (g)

0.50

HDRB-1 HDRB-2 SMA-HDRB-S1 SMA-HDRB-C2 Fixed-Base

Loma Prieta

0.25 0.00 -0.25 -0.50

0

5

10

15

20

25

Time (s) Acceleration (g)

0.50


Northridge

0.25 0.00 -0.25 -0.50

0

3

6

9

12

15

Time (s) Acceleration (g)

0.50


Kobe

0.25 0.00 -0.25 -0.50

0

5

10

15

20

25

30

35

40

Time (s) Fig. 20. Time variation of deck acceleration for non-isolated and isolated bridges under Loma Prieta, Northridge, and Kobe earthquakes scaled by Vancouver design response spectrum.

earthquake records, the maximum shear strain of rubber bearings exceeds 270% for HDRB-1 and 200% for SMA-HDRB-S1. Under same ground motions, the maximum lateral deflection of HDRB-2 and SMA-HDRB-C2 does not go beyond the 125% and 120% of the total thickness of rubber layers (84.7 mm), respectively. Such behaviors prove that (1) using SMA wires in the configuration of either cross or straight restricts the lateral flexibility of the rubber bearings and (2) increasing the total thickness of elastomeric layers makes the base isolation system safer in terms of the horizontal deflection range. Fig. 20 shows variations of the deck acceleration for nonisolated and isolated bridges subjected to three earthquakes. It is observed that implementing rubber bearings into the three-span bridge causes a significant reduction in the acceleration of the deck when HDRBs and SMA-HDRBs with high aspect ratio are used. Table 18 demonstrates the peak deck accelerations of the nonisolated and isolated bridges under three considered ground motions. When the bridge isolated by HDRB-1 and SMA-HDRB-S1 is subjected to the scaled Loma Prieta earthquake with PGAmax = 0.36 g, the maximum peak deck acceleration decreases

Table 18 Peak deck acceleration in the non-isolated and isolated bridges under different earthquake records scaled by Vancouver design response spectrum. Loma prieta Amax (g) Non-isolated bridge HDRB-1 HDRB-2 SMA-HDRB-S1 SMA-HDRB-C2

0.357 0.219 0.197 0.186 0.201

Northridge

D (%)

Amax (g)

39 45 48 44

0.423 0.220 0.191 0.233 0.194

Kobe

D⁄ (%)

Amax (g)

D (%)

48 55 45 54

0.366 0.217 0.202 0.216 0.204

41 45 41 44

39% and 48%, respectively. This fact indicates that SMA-HDRB-S1 can decrease the acceleration more efficiently. Under the same ground motion, HDRB-2 and SMA-HDRB-C2 decrease the maximum deck acceleration to 0.20 g (with a negligible difference) which is about 45% lower than the maximum peak deck acceleration in the non-isolated bridge. When the bridge is subjected to Northridge ground motion, HDRB-1, HDRB-2, SMA-HDRB-S1, and SMA-HDRB-C2 diminish the maximum peak deck acceleration by 48%, 55%, 45% and 54%, respectively. The higher amount of reduction in the deck acceleration, by installing HDRB-2 and SMA-HDRBC2, is due to their higher effective thickness of rubber layers compared to HDRB-1 and SMA-HDRB-S1. In Kobe earthquake, similar behavior is observed. Using HDRB-2 and SMA-HDRB-C2 leads to about 45% and 44% decrease in the maximum peak deck acceleration, respectively. Implementing SMA wires with cross configuration into HDRBs has negligible effect on the peak acceleration of bridge deck. The total energy dissipated by HDRBs and SMA-based HDBRs during Loma Prieta, Northridge, and Kobe ground motions is listed in Table 19. Under scaled Loma Prieta ground motion, SMA-HDRBS1 and SMA-HDRB-C2 respectively dissipate 146 kJ and 95 kJ energy while HDRB-1 and HDRB-2 have an energy dissipation capacity of 106 kJ and 89 kJ, respectively. When the isolated bridge

Table 19 Total energy dissipated by different types of rubber bearings for different earthquake records scaled by Vancouver design response spectrum. Rubber bearings

D: Relative difference between the peak deck acceleration in the isolated bridge and that in the non-isolated bridge

HDRB-1 HDRB-2 SMA-HDRB-S1 SMA-HDRB-C2

Dissipated energy (kJ) Loma prieta

Northridge

Kobe

106.3 88.9 145.6 95.3

118.0 81.9 163.0 95.7

191.0 161.0 217.8 178.5


is excited by scaled Northridge earthquake, the total energy dissipated by HDRB-1 and SMA-HDRB-S1 is 118 kJ and 163 kJ, respectively, while HDRB-2 and SMA-HDRB-C2 dissipate 82 kJ and 96 kJ, respectively. Here, three points should be noted: (1) SMA-HDRBs have higher capability in dissipating the earthquake’s energy compared to HDRBs due to the presence of SMA wires acting as supplementary dampers with a large flag-shaped hysteresis, (2) Since SMA wires in the straight configuration are subjected to a higher strain level compared to cross wires, they generate a larger force-deflection hysteresis curve and subsequently, they increase the total dissipated energy more than cross SMA wires, and (3) although HDRB-2 and SMA-HDRB-C2 with a high aspect ratio (R = 0.36) have a greater energy dissipation capability (because of the higher thickness of elastomer), HDRB-1 and SMA-HDRB-S1 dissipate a higher amount of earthquake’s energy, since they reach a higher level of shear strain and accordingly produce a larger amount of shear force under same excitation. Overall, by considering three factors for high damping rubber bearings: the energy dissipation capacity, the maximum lateral displacement, and the capability of reducing the acceleration of bridge deck, it can be concluded that implementing SMA wires with cross configuration into the high-aspect-ratio HDRBs (R = 0.36) leads to the best result in controlling and regulating the seismic response of the three-span continuous bridge. However, in terms of reducing the deck acceleration, performances of HDRB-2 and SMA-HDRB-C2 are almost the same. The cost of SMA is an important factor in deciding about the best configuration of wire, since a higher amount of wire should be used in the cross configuration compared to the straight one. In this regard, factors such as material, fabrication, and installation costs play important roles. Since both Nickel and Titanium are expensive, Ni–Ti based SMAs become expensive (e.g. Ni–Ti is about $80 per lb.). Ferrous SMA is a less expensive source which can be used instead of Ni–Ti. Another important point is that increasing use of SMAs in civil infrastructures will potentially reduce the cost of SMAs as has been observed for fiber reinforced polymers in the construction industry. Therefore, in order to come up with an optimized case, both cost and efficiency of SMA wires should be taken into account by defining a multi-criteria decision making problem in which the cost is minimized and the strain of SMA wires is maximized. 6. Summary and conclusions In this research, two different configurations of SMA wires are considered for CFR-HDRB. The effect of the aspect ratio of rubber bearing, the thickness of SMA wires, the type of shape memory alloy, and the amount of pre-strain in SMA wires are investigated on the performance of the devices when they are subjected to a combination of vertical pressure and a cyclic lateral displacement in one direction (x). The results are categorized as follows: In both straight and cross configurations, SMA wires enhance the lateral stiffness and residual deformation of the elastomeric isolator with increasing aspect ratio. In a high-aspect-ratio (e.g. R = 0.36) SMA-HDRB, since the amplitude of cyclic lateral displacements increases, the amount of strain in SMA wires goes up and subsequently, the changes in the lateral flexibility and the residual deformation is augmented. When an SMA-HDRB is subjected to 200% shear strain, the maximum strain generated in the cross SMA wires does not exceed 9% while, the strain in the straight SMA wire goes beyond 25%, which is much higher than the superelastic strain limit of FeNACTB. It means that an unrecoverable plastic deformation occurs in the SMA wire of SMA-HDRB-S, and as a result, the performance of SMA wire is degraded in terms of re-centering capability.

181

The hysteretic shear behavior of SMA-HDRBs with different aspect ratios, different SMAs, and different configurations of wire indicate that both the lateral flexibility and the residual deformation reduce compared to HDRBs. However, using SMA wire in CFR-HDRBs is not very effective in terms of residual deformation reduction. Knowing this fact that a lower horizontal stiffness (higher flexibility) leads to a more efficient behavior, in the best case, the residual deformation decreases about 16% while the horizontal stiffness does not alter. In order to prevent SMA-based rubber bearings from being subjected to excessive shear force and avoid reducing the lateral flexibility, SMA wires are pre-strained. An important point is that stresses of the starting and finishing phase transformation in SMA wires do not considerably change by increasing the amount of pre-strain above the 2%. The reason is that, when the magnitude of pre-strain is higher than the strain at which the forward phase transformation starts, an increase in the prestrain level will have a negligible effect on the performance of the smart elastomeric isolator. To apply a pre-strain to an SMA wire, after installation, it is subjected to a pre-tension load using an adjustable rotational mechanism equipped with a slotted hexagonal head bolt. Based on the proposed performance-based design approach (flowchart), an SMA wire with 2% pre-strain and 2.5 mm radius is chosen for high-aspect-ratio SMA-HDRB-C. Since the majority of the pier base shear is due to the bridge deck, it is of great interest to isolate the deck from the pier. Results demonstrate that implementing rubber bearings into a three-span continuous steel girder concrete-reinforced pier supported bridge causes a significant reduction in the acceleration of the deck when HDRBs and SMA-HDRBs with high aspect ratio are used. As an example, under Northridge earthquake scaled by Vancouver design response spectrum, installing HDRB-1, SMA-HDRB-S1, HDRB-2, and SMA-HDRB-C2 causes the deck acceleration to be reduced by 48%, 45%, 55%, and 54%, respectively. Considering the energy dissipation capacity, the maximum lateral displacement, and the capability of reducing the acceleration of bridge deck, it is observed that for lowaspect-ratio HDRBs (R = 0.12), implementing SMA wires with straight configuration leads to the best results in regulating the seismic response of the bridge and for high-aspect-ratio HDRBs (R = 0.36), using SMA wires in the cross arrangement is the most efficient way. However, implementing SMA wires into HDRBs has negligible effect and in some cases no effect on the pier displacement and the peak deck acceleration. Using SMA wires, as supplementary dampers, with a large flagshaped hysteresis, causes SMA-HDRBs to have a higher capability in dissipating the earthquake’s energy compared to HDRBs. considering the wires arrangement, the straight configuration leads to a larger force-deflection hysteresis curve and subsequently a higher total dissipated energy. Another finding is that, although HDRB-2 and SMA-HDRB-C2 with high aspect ratio have a greater energy dissipation capacity, HDRB-1 and SMAHDRB-S1 can dissipate a higher amount of energy, since they reach a higher level of shear strain and accordingly produce a larger amount of shear force under same excitation. The main purposes of using SMA wires in rubber bearings are to improve the re-centering capability and the energy dissipation capacity of the base isolator. SMA wires can effectively improve these two characteristics for natural rubber bearings [16]. However, results show that implementing SMA wires into HDRBs is not efficient in terms of reducing the residual deformation. This fact implies that SMA wires do not considerably affect the recentering capability of the HDRBs. Though, the influence of SMA wires on the energy dissipation capacity of HDRBs is significant. As a proof, at 100% shear strain, the energy dissipated per cycle for high-aspect-ratio SMA-HDRBs with straight and

182


cross ferrous SMA wires respectively increases by 38% and 15%, compared to HDRB. Results obtained from dynamic time history analyses performed on the three-span continuous bridge isolated by HDRBs and SMA-HDRBs also demonstrate that SMA wires can noticeably improve the energy dissipation capacity of HDRBs.

Acknowledgements The financial contribution of Natural Sciences and Engineering Research Council of Canada (NSERC) through Discovery Grant was critical to conduct this research and has been gratefully acknowledged. Appendix A Design procedure for determining the radius and pre-strain of SMA wires in SMA-HDRB-C2 1. Set a target value for the effective horizontal stiffness, KH d:

KH

d

¼ 1:00 kN=mm

This value corresponds to the maximum shear force Fs max = 81.5 kN at c = 100%. 2. Considering a force ratio of 10%, the maximum lateral force generated by SMA wire, FSMA max can be calculated.

F SMA max ¼ RF F s max ¼ 8:2 kN 3. Knowing the maximum strain in SMA wires at c = 100%, according to Eq. (4), and accordingly the maximum axial stress in wires, rmax, the initial radius of SMA wires, r 0w , can be obtained from FSMA max.

r0w ¼ ðF SMA

max =ð

prmax ÞÞ0:5 1:75 mm

4. Choosing a radius for SMA wire, calculating the effective horizontal stiffness, KH, and comparing it with KH d. KH should be greater than KH d, with a minimum relative difference of 15% (MK)

r1w ¼ 1:2r 0w 2:10 mm K H ¼ 1:11 kN=mm K H < 1:15K H

d

Since KH is lower than the desired value, radius of wire increases and step 4 is repeated.

r2w ¼ 1:2r 1w 2:50 mm K H ¼ 1:14 kN=mm K H < 1:15K H

d

5. Considering a zero initial pre-strain, for the first try, the prestrain in SMA wire, e10 , is 1%.

e10 ¼ 0 þ 0:01 ¼ 0:01 K H ¼ 1:12 kN=mm jK H K H d j=K H

d

> 10%

6. Considering the pre-strain value lower than 5%, since the relative difference between the new and the target values of the effective horizontal stiffness is higher than 10%, step 5 should be repeated.

e20 ¼ e10 þ 0:01 ¼ 0:02 K H ¼ 1:07 kN=mm jK H K H d j=K H

d

< 10%

7. Final result: rw = 2.5 mm and e0 = 2%.

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