IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 17 (2008) 035018 (10pp) doi:10.1088/0964-1726/17/3/035018

Large scale testing of nitinol shapememory alloy devices for retrofitting ofbridgesRita Johnson1, Jamie E Padgett2, M Emmanuel Maragakis1,Reginald DesRoches3 and M Saiid Saiidi1

1 Department of Civil and Environmental Engineering, University of Nevada, Reno,89557-0258, USA2 Department of Civil and Environmental Engineering, Rice University, Houston,TX 77005, USA3 School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta,GA 30332-0355, USA

E-mail: ritaj@unr.edu, Jamie.Padgett@rice.edu, maragaki@ce.unr.edu,reginald.desroches@ce.gatech.edu and saiidi@unr.edu

Received 19 December 2007, in final form 1 March 2008Published 18 April 2008Online at stacks.iop.org/SMS/17/035018

AbstractA large scale testing program was conducted to determine the effects of shape memory alloy(SMA) restrainer cables on the seismic performance of in-span hinges of a representativemultiple-frame concrete box girder bridge subjected to earthquake excitations. Anotherobjective of the study was to compare the performance of SMA restrainers to that of traditionalsteel restrainers as restraining devices for reducing hinge displacement and the likelihood ofcollapse during earthquakes. The results of the tests show that SMA restrainers performed verywell as restraining devices. The forces in the SMA and steel restrainers were comparable.However, the SMA restrainer cables had minimal residual strain after repeated loading andexhibited the ability to undergo many cycles with little strength and stiffness degradation. Inaddition, the hysteretic damping that was observed in the larger ground accelerationsdemonstrated the ability of the materials to dissipate energy. An analytical study was conductedto assess the anticipated seismic response of the test setup and evaluate the accuracy of theanalytical model. The results of the analytical simulation illustrate that the analytical model wasable to match the responses from the experimental tests, including peak stresses, strains, forces,and hinge openings.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

During earthquake events, bridges are susceptible to unseatingand collapse due to excessive longitudinal motion at in-spanhinges or supports. Such damage to bridges can causesignificant disruptions to the transportation network, posinga threat to emergency response and recovery as well asresulting in severe direct and indirect economic losses for aregion. Bridges may be retrofitted, or rehabilitated, in order toovercome their seismic vulnerabilities. This paper examines anew approach for bridge retrofit using nitinol shape memory

alloys to limit the susceptibility to bridge collapse and improvethe seismic response of bridges.

1.1. Unseating at in-span hinges

The seismic vulnerability resulting from large relativedisplacements between adjacent bridge spans has beenillustrated worldwide as in the 1971 San Fernando (US),1989 Loma Prieta (US), 1994 Northridge (US), 1999 Chi-Chi (Taiwan), and 1999 Kocaeli (Turkey) earthquakes, amongothers. Figure 1(a) exemplifies this failure mode at an

0964-1726/08/035018+10$30.00 © 2008 IOP Publishing Ltd Printed in the UK1

Smart Mater. Struct. 17 (2008) 035018 R Johnson et al

Figure 1. Unseating at in-span hinge during the 1994 Northridge earthquake for (a) an existing bridge and (b) bridge retrofit with traditionalsteel restrainer cables (NISEE Collection).

in-span hinge of the I-5/Highway 14 interchange during the1994 Northridge earthquake (NISEE 1997). Such unseatingtypically occurs when adjacent bridge frames displace out-of-phase. Hence, retrofits that target improving performancetypically include response modification devices or restrainingelements placed between adjacent frames to reduce thepotential for unseating.

Following the 1971 San Fernando earthquake, the Cali-fornia Department of Transportation (Caltrans) implemented astate-wide bridge retrofit program to systematically address thedeficiencies of existing structures, including potential span un-seating (Caltrans 2003). Other states such as Washington, Ore-gon, and Tennessee have established similar programs to in-stall restrainer cables on bridges with short seat widths. Thereare a number of potential unseating prevention devices whichhave been examined by previous researchers or are now used inpractice to prevent unseating (Saiidi et al 2001, Hipley 1997,DesRoches and Fenves 2000, Keady et al 2000). However,some of the limitations of traditional approaches have beenhighlighted (Andrawes and DesRoches 2005). For example,traditional steel restrainer cables can transfer large forces toadjacent bridge components, or upon yielding may accumulateplastic deformations in repeated loading cycles (DesRochesand Muthukumar 2002). This can ultimately result in largehinge openings and subsequent span unseating, such as seenin the 1999 Chi-Chi and 1994 Northridge earthquakes whenbridges retrofit with restrainer cables collapsed (figure 1(b)).The challenge of reducing the potential for deck collapse dueto excessive longitudinal movement continues to be an issuefaced by designers. A new bridge retrofit device capitalizingon the unique properties of shape memory alloys (SMAs) mayovercome some of the limitations of traditional devices.

1.2. Shape memory alloys

Several past researchers have indicated the unique propertiesof shape memory alloys (SMAs) which may be beneficial forapplications in the field of earthquake engineering (Aiken et al1993, Dolce et al 2000, Dolce and Cardone 2001, Baratta andCorbi 2002, Han et al 2003, DesRoches et al 2004, Saiidiand Wang 2006, Dolce and Cardone 2006, Choi et al 2006).Characteristics of superelastic SMAs desirable for seismicapplications include their hysteretic damping, excellent low-and high-cycle fatigue properties, strain hardening at large

strains, and the formation of a stress-plateau limiting forcetransmission. One of the defining qualities of superelasticSMAs that is particularly attractive for seismic design andretrofit is the recentering capability, or the ability to returnto their original undeformed shape upon returning to a stateof zero stress after loading. In addition, hysteretic dampingis associated with the energy requirements in the austenite–martensite phase transition in stressed SMA.

Previous work has focused on the optimization of SMAproperties for use in seismic applications (DesRoches et al2004, McCormick et al 2007, Tyber et al 2007), illustratingthat with proper heat treatment, nearly ideal superelasticproperties can be obtained for both wires and bars. Andrawesand DesRoches (2005) identified key characteristics of nitinolSMAs that are conducive to restraint of bridge decks, andanalytically evaluated the use of superelastic nitinol SMArestrainer cables placed at in-span hinges of multi-framebridges. They found that the devices were highly effectivein reducing hinge opening between adjacent frames. Asubsequent analytical study compared their performance toother retrofit measures (steel restrainers or metallic dampers),concluding that the SMA restrainers had a similar impact oncolumn drifts but were superior in reducing hinge openingsfor multi-frame box girder bridges (Andrawes and DesRoches2007b).

While the use of shape memory alloy restrainer cableshas been proposed as a potential seismic retrofit approachfor bridges, their viability had not been validated throughexperimental testing. This study presents the experimentalfindings of large scale testing of shape memory alloy devicesfor seismic bridge retrofit.

A series of tests were conducted at the University ofNevada Reno’s (UNR) Large Scale Structures Laboratoryto determine the effect of SMA restrainers on the seismicperformance of in-span hinges using a representative multiple-frame concrete box girder bridge. Additionally, the results arecompared to data from a previous study conducted at UNRto assess the bridge structure response using traditional steelrestrainer cables (Sanchez-Camargo et al 2004). An analyticalevaluation of the representative frame and SMA system isconducted in order to refine and validate the model against theexperimental test data.

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Figure 2. SMA restrainer test setup.

2. Experimental test setup

This section presents the specimen and test parameters usedin this study. The specimen and parameters are the sameas those used in the steel restrainer studies previously noted(Sanchez-Camargo et al 2004), in order to provide a means ofcomparison between the steel and SMA systems. The mostcritical scenario from the previous UNR experiments, in whichthe steel restrainers either had significant displacements, orfailed, established the controlling study parameters to use in theSMA restrainer experiments. These parameters helped providethe criteria for the SMA restrainer design and test protocol.

2.1. Test specimen

The test specimen, seen in figure 2, was designed andused during the UNR steel restrainer experiments (Vlassis

et al 2000, Sanchez-Camargo et al 2004) to simulate an in-span hinge within a multi-span concrete box girder bridge.Dimensions of the specimen are based on superstructuredimensions of representative Caltrans bridges. The two boxgirder cells represent adjacent bridge spans at expansion joints.Block A, seen as the right block in figure 2(a), is the lighter ofthe concrete cells, with a weight of 92.5 kN; block B, the leftblock in figure 2(a) with additional lead added, is the heaviercell with a weight of 128.3 kN. A close-up of block B withadditional weight is shown in figure 2(b).

Elastomeric bearing pads, which support the box girdercells and transfer loading from the shake table, simulate thesubstructure stiffness (figure 2(c)). Block A is supported bybearings with a collective stiffness of 1302 kN m−1, and blockB, the heavier of the frames, by bearings with a collectivestiffness of 683 kN m−1. This resulted in individual frameperiods of 0.53 and 0.87 s and a block period ratio of 0.6.Figure 3 shows the system design with the 25 mm gap betweenblocks, the location of the elastomeric pads, and the cablerestrainers located on the outside of the cells. The east sideof the test specimen, seen in figure 3(a), shows the heavierof the blocks on the left and the lighter on the right, leadingto anticipated out-of-phase motion when excited by the shaketable.

A close-up of the SMA restrainer cable is seen infigure 3(b). The 84-wire SMA restrainers were constructedof eighty four 0.584 mm (0.023 in) diameter wires and the130-wire SMA restrainers consisted of one hundred thirty0.584 mm (0.023 inch) diameter wires. All of the wiresnubbers were contained within thin walled amber latex tubingfor ease of handling, and the loose ends of the wire weretied together using a 180◦ bend back and twist with a heatset to hold the ends. The effective length of both the 84 and130-wire restrainers was 1.16 m. The wire restrainer cableswere all made with superelastic nitinol with a composition of

Figure 3. Schematic of the test setup and SMA restrainer cable.

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Figure 4. SMA cable restrainer used in shake table tests.

51.0 at.% nickel. Nitinol in its austenitic phase has thermalexpansion coefficient of 11 × 10−6 ◦C−1 and a Poisson’s ratioof 0.33. Its material properties include a martensite start (Ms),martensite finish (Mf), austenite start (As), and austenite finish(Af) temperatures of Ms = −50 ◦C (−58 ◦F), Mf = −70 ◦C(−94 ◦F), As = −10 ◦C (14 ◦F), and Af = 10 ◦C (50 ◦F)respectively. The approximate stress loading and unloadingplateaus are 503 and 276 MPa, and the elastic modulus is31.7 GPa. Each set of brackets had a steel strength of 248 MPaand were designed to be bolted through both sides of eachframe element. The 84-wire cable had a total cross-sectionalarea of 22.58 mm2 (0.035 in2), while the 130-wire cable had across-sectional area of 34.84 mm2 (0.054 in2). The wires of thelooped end of the SMA cables, seen in figure 4, were spreadover a 19.05 mm diameter steel pin that was part of a yokesystem welded to one side of the plates. A piece of leatherwas placed between the steel pin and cable ends to reducestress concentrations and prevent cutting action on the wiresthat could lead to early failure. The larger of the plates held theload cell to measure force in the restrainers.

Figure 5 shows the complete instrumentation plan. Theexperiment was performed on one of the 50-ton capacitybiaxial shake tables at UNR. Three linear displacementtransducers were placed on the east, west and topsideof the hinge section. These three transducers (LWG-225Novotechnik), seen in figure 5, directly measured the relativedisplacement at the hinge. Absolute displacement of theblocks was measured with two string potentiometers. Theseinstruments measured the displacement between the specimenand a fixed frame. Accelerometers centrally located onthe blocks measured impact accelerations, and a load cellmeasured the force in the restrainer cables.

2.2. Parameters of study

The critical parameters of this study as determined fromprevious steel restrainer tests are: (a) frame period ratio,(b) restrainer stiffness, (c) restrainer slack, (d) and earthquakeinput motion. Likeness of all four of these factors wasrequired for comparisons between the responses of SMA cablerestrainers and the steel cable restrainers that were previouslytested. The following section details each parameter as itis defined for the series of tests conducted on the SMArestrainers.

(a) A frame period ratio of 0.6 between the two adjacentbridge frames was determined to result in large out-of-phasemotions. This ratio between the structural periods is takenas the period of the stiffer frame over the period of themore flexible frame. During the Sanchez-Camargo restrainerstudies, a frame period ratio of 0.6 between the two adjacentbridge frames had been determined to produce significantrestrainer demands. The same physical test specimen andframe period ratio was used in the SMA restrainer tests.

(b) Comparable restrainer stiffness is essential to equatethe relative properties of two sets of restrainer systems. Eachtest set consisted of steel cable restrainers and ‘equivalent’SMA cable restrainers with the same effective stiffness as thesteel restrainers. In the previous restrainer tests, (3) and (5)cable steel restrainers were tested. The 84- and 130-wire SMArestrainers were equivalent in stiffness to the steel restrainerstested. In the first set, each cable system had stiffness equalto 0.42 kN mm−1. In the second, each system had stiffnessequal to 0.7 kN mm−1. The stiffness of the restrainers wasdetermined based on geometric properties (length and cross-sectional area), elastic modulus, and number of cables used.The 6% strain that was used for the basis of the design stiffnessof the restrainers was also used in the calculation of the chordmodulus (Johnson et al 2004).

(c) Restrainer slack is the amount of relative hingedisplacement necessary to engage the cables in tension.The testing matrix consisted of two different values ofrestrainer slack, 12.7 and 0 mm. In the steel restrainer tests,zero slack was used for the restrainer with a stiffness of0.42 kN mm−1, while a slack of 12.7 mm was used for thestiffer (0.7 kN mm−1) steel cables. These combinations ofslack and stiffness were determined to produce the maximumresponses in the case of the steel restrainers. As such, thesetesting parameters were repeated in the experiments for theSMA restrainers.

Figure 5. Instrumentation used in shake table tests.

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Figure 6. Input ground motion scaled to a peak ground accelerationof 0.25g.

Table 1. Peak displacements and maximum forces for the east siderestrainer. (Note: peak disp includes the initial slack in the cables,and is not a direct measure cable deformation.)

PGASMAcable Slack

Peakdisp

Cablestrain

Maxforce

Max cablestress

Run (g) size (mm) (mm) (%) (kN) (MPa)

1 0.05 84-wire 0 9.5 0.81 4.6 2062 0.10 84-wire 0 15.8 1.35 8.4 3733 (caseA) 0.15 84-wire 0 23.0 1.97 10.5 4654 0.20 84-wire 0 28.4 2.43 11.1 4925 0.25 84-wire 0 37.2 3.18 11.0 4876 0.05 130-wire 12.7 21.2 0.72 4.7 1347 0.10 130-wire 12.7 28.7 1.37 12.0 3458 (case B) 0.15 130-wire 12.7 32.1 1.66 17.5 5039 (case C) 0.20 130-wire 12.7 38.9 2.25 18.9 542

10 0.25 130-wire 12.7 45.2 2.78 18.8 540

(d) The synthetic earthquake input motion used for thesteel restrainer testing was developed based on the AppliedTechnology Council ATC32-E (soft soil) motion (CaliforniaDepartment of Transportation 2001) design spectrum. Thismotion, ATC32-E, is based on the expected magnitude of theearthquake (6.5), soil type of the site (E or soft soil), and peakground acceleration (PGA). Previous tests have shown that soiltype E produced the largest out of phase motion and resultedin frequent restrainer engagement during shake table testing(Sanchez-Camargo et al 2004). Five levels of shake table inputground motion with peak ground accelerations between 0.05gand 0.25g were used in the SMA cable testing. An example ofthe input ground motion scaled to a peak ground acceleration(PGA) of 0.25g is shown in figure 6.

The loading is applied to the bridge by use of a shake tablesupporting the test specimen, which simulates the earthquakeground motion. The period difference between the blocksrepresenting results in out-of-phase motion when excited bythe shake table, and produces a hinge opening that is targetedfor reduction with restrainer cables. The test parametersdescribed above, along with the experimental results, will bediscussed in section 3. In total, ten tests were conducted. Theseries of tests performed with the 84-wire cables had a slack of0 mm, while the tests with the 130-wire cables had a 12.7 mmslack. These reflect the cases from the steel restrainer testingwhich proved to have large hinge openings and indicate theneed for an improved restraining system.

Figure 7. Stress–strain hysteresis for the 84-wire SMA cable withincreasing earthquake motion ((a)–(c)).

3. Experimental results

3.1. SMA restrainers

Table 1 shows the peak displacement and maximum forcesthat were recorded during the shake table tests as well asthe calculated stresses and strains. The recorded hingedisplacement for the larger SMA cable includes the initial12.7 mm slack. The first column of the table shows the runnumber. Case A, B and C are SMA restrainer shake tableruns that are directly equivalent to the previous steel restrainerexperiments. These runs are used to compare the behavior ofthe SMA and steel cable restrainers. Consistent with intuition,increasing displacement was recorded as ground accelerationamplitude increased.

Figures 7(a)–(c) show the SMA response with incrementalincreases of ground acceleration of 0.05g. This illustrates thestress–strain relationships for the 84-wire cable for run 3, witha PGA of 0.15g, through run 5, with a PGA of 0.25g. Thesuperelastic effect of this material is evident through repeateddeformation cycles with minimal accumulation of residualstrain. The typical flag-shaped hysteresis characteristic of thesuperelastic SMA becomes apparent by run 5 (figure 7(c)).

The recentering ability of the SMA is most visible at thelarger accelerations. At a maximum PGA of 0.25g, seen infigure 7(c), the stress in the 84-wire cable is approximately483 MPa and the corresponding strain is approximately 3%.

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Relative Hinge Disp.(mm)

Tot

al R

estr

aine

r Fo

rce

(kN

)

Figure 8. Force–displacement relationship for 84 and 130-wire cablerestrainers under maximum earthquake motion.

The usable strain range for this material is 6–8%. Infigures 7(a) (PGA of 0.15g), a maximum stress of 465 MPa(67.4 ksi) and a corresponding strain of 1.97% were reached.Figure 7(b) (PGA of 0.2g) shows an opening of the hystereticloop that is characteristic of the superelastic effect of SMAs.The maximum stress and strain associated with a PGA of0.2g are 492 MPa and 2.43%, respectively. Due to the largedisplacement of the elastomeric bearings, and the effectivenessof the SMAs in limiting the relative hinge displacement, astrain of 6% in the SMAs was not achieved during dynamictesting. The largest strain realized for the SMA restrainersduring the experiment was 3%, as seen in figure 7(c) (PGAof 0.25g). Even at this strain, the SMA hysteresis that resultsfrom its mechanical ability to recover deformation after stressremoval is clearly evident in the typical flag shape loop that issynonymous with SMA’s superelasticity.

3.2. 84- versus 130-wires

A comparison of the relative hinge displacement betweenblocks and total restrainer force for both the 84-wire and130-wire SMA cables at a PGA of 0.25g is illustrated infigure 8. The initial slack in the larger restrainer is evidencedby the lack of force transmitted through the cable until a hingedisplacement of 12.7 mm is reached. After deducting theinitial slack from the relative hinge displacement, the restrainerelongation is 32.5 mm for the 130-wire cable versus 37.2 mmfor the smaller 84-wire cable restrainer. At a PGA of 0.25g,the calculated force in the 84-wire SMA restrainer is 11 kNwhile the calculated force in the 130-wire restrainer is almost19 kN. Similar to the stress–strain relationship seen in figure 7,the force–displacement relationship seen in figure 8 reveals therecentering capabilities of SMA to return to its point of originwith minimal residual elongation. The number of wires percable did not appear to affect the ability of the SMA restrainerto recenter.

3.3. SMA versus steel restrainer

As stated earlier, nitinol SMAs possess the desirable propertiesfor seismic resistant design and retrofit of structures. The largeelastic strain capacity, hysteretic damping, and the recentering

Figure 9. Block acceleration time histories for equivalent SMA andsteel restrainer cables (PGA = 0.20g).

capabilities are evident. In order to evaluate the nitinolrestrainers in relation to the past research performed on steelrestrainers with comparable stiffness, the test parameters wereduplicated.

3.3.1. Block acceleration. The acceleration history of blockB (the soft block) from the SMA experiment was comparedto the block B acceleration histories from the previous steelrestrainer tests for cases A, B and C. During the seismicloading tests, all three cases produced lower acceleration inthe blocks with SMA restrainers compared to those beingrestrained by steel. In case A, the maximum block Baccelerations for the SMA versus steel restrainer shake tabletests were 2.7g versus 6.3g, respectively. There were similarresults for case B and C. In Case C, the acceleration of BlockB was more than 3.5 times larger (11.6g versus 3.2g) whenrestrained by the steel restrainers than the block accelerationwith SMA restraining devices, as shown in figure 9. Thedecreased block accelerations seen in the SMA versus steelrestrainer experiments is most likely the result of the reductionof displacement and velocity due to the unique recenteringproperties of the shape memory alloys.

3.3.2. Hinge opening. Figure 10 shows the force–displacement relationships for the SMA restrainer cases A, B,and C and their equivalent cases from the past steel restrainertests. The restrainer force–displacement relationships seen infigure 10 reveal fairly equivalent steel and SMA restrainerforce but a larger relative hinge displacement for the steelrestrainers. The data collected during these experimentsmeasuring maximum restrainer force and maximum relativehinge displacement for these three cases is summarized intable 2.

The maximum hinge displacements for steel in case A andB are nearly double that of the SMA restrainers. In case A,the 3-cable steel restrainer has an elongation of 43 mm whilethe equivalent 84-wire SMA restrainer has an elongation of23 mm. The maximum hinge displacement for the larger 5-cable steel restrainer and 130-wire SMA restrainer in case B is61 and 32 mm, respectively. As listed in table 1, case A and Bare tested at a peak ground acceleration of 0.15g. Figure 10(c)reveals an extremely large relative hinge displacement in the5-cable steel restrainer at a PGA of 0.20g. As shown in this

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Relative Hinge Disp (mm) Relative Hinge Disp (mm) Relative Hinge Disp (mm)

Tot

al F

orce

(kN

)

Tot

al F

orce

(kN

)

Tot

al F

orce

(kN

)

Figure 10. Total restrainer force and relative hinge displacement for cases A, B, and C.

Table 2. Max force and displacement and total energy dissipationfor cases A, B, and C.

Total force Max disp Energy dissipation(kN) (mm) (kN mm)

Case A

Steel 27 43 307SMA 21 23 249

Case B

Steel 30 61 112SMA 31 32 263

Case C

Steel 36 120 2111SMA 35 39 448

figure for case C, there was a restrainer failure in two of the fivecables in the steel restrainer resulting in a maximum restrainerdisplacement three times greater than in the steel cable. Thedisplacement of the steel cable restrainer in case C is 120 mmwhile that of the SMA cable restrainer is 39 mm. Figure 10(c)also reveals that while the steel restrainer has failed, the SMArestrainer has only reached yield, beyond which the SMA canundergo large elastic deformation with reversibility.

3.3.3. Energy dissipation. Table 2 shows a comparisonof the energy dissipation between the SMA restrainers andsteel restrainer cables. It is interesting to note that in mostcases, the steel restrainer cables dissipate more energy thanthe SMA restrainers. In fact, in case C, the steel restrainersdissipate approximately five times as much energy as theSMA restrainers. The energy dissipation in steel cables isa result of yielding in the cable, which also results in largemaximum hinge displacements and large permanent (residual)displacements in the cables. The results from this studyare consistent with previous studies of SMAs which haveshown that their ability to limit hinge opening is due to theirrecentering capability, rather than their ability to dissipateenergy (Andrawes and DesRoches 2007a, 2007b). In general,SMAs have equivalent viscous damping values which are much

Figure 11. Uniaxial constitutive model used for nitinol shapememory alloys.

lower than traditional civil engineering dampers (DesRocheset al 2004).

4. Analytical evaluation

An analytical study conducted to assess the anticipated seismicresponse of the test setup and evaluate the analytical models’capability to duplicate the response of the shape memory alloyrestrainer cables. The results of the analytical simulation arecompared to the experimental results for the in-span hinge ofadjacent frames retrofit with large and small SMA cables.

4.1. Analytical model of the shape memory alloy restrainercable

The superelastic behavior of the SMA restrainers is developedusing a uniaxial constitutive model, as shown in figure 11.The model, which is a modification of the model proposedby Auricchio and Saco (1997), is capable of capturing thematerial behavior under non-uniform loading typically foundin earthquake excitations, where the response is primarilycomposed of sub-hysteresis loops internal to the main loopassociated with the phase transformation. The modelformulation relies on the assumption that the relationshipbetween stresses and strains is represented by a seriesof straight lines whose form is determined by the extentof the transformation experienced. Further assumptionsinclude no strength degradation during cycling (Bernardini

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and Brancaleoni 1999), and that the austenite and martensitebranches have the same modulus of elasticity. Previous studieshave shown that these parameters have little effect on theglobal response of structural systems, such as the bridge systemstudied in this paper (Andrawes and DesRoches 2008, 2007b).

The model works with one scalar internal variable, ξs,representing the martensite fraction, and with two processeswhich may produce its variation: the conversion of austeniteto martensite and the conversion of martensite to austenite.For both processes, linear kinetic rules are chosen to describethe evolution in time of the martensite fraction. Limiting thediscussion to the small deformation regime, we assume thefollowing additive decomposition of the total strain ε:

ε = εe + εLξS sgn(σ ),

where εe is the elastic strain, εL is the maximum residual strain,and sgn(·) is the sgn function. Finally, the elastic strain isassumed to be linearly related to stress as follows:

σ = Eεe,

where E is the elastic modulus of both the austenite andmartensite branch.

The modified constitutive model used in this study wasimplemented in OpenSees by Auricchio et al (2006). Theprimary advantages of the model adopted for this studyare the robustness and simplicity of implementation, easein obtaining material parameters from typical uniaxial testsconducted on wires or bars, and the ability to reproducepartial (i.e. sub-loops) and complete transformation patterns(i.e. from fully austenite to fully martensite) in both tensionand compression. The model does not account for rate andtemperature dependency. However, for the moderate loadingrates typically observed during earthquakes, this does notappear to be an important parameter.

4.2. Analytical model of the bridge system

A two degrees-of-freedom model is developed in OpenSees(McKenna and Fenves 2005) to capture the out-of-phasemotion of the blocks representing the multi-frame concretebridge. The substructure and frame stiffness for blocks Aand B were modeled with elastic springs having a stiffness of1303 and 683 kN m−1, respectively, based on the measuredcollective bearing stiffness. The nodal block masses are0.787 and 1.09 kg. In addition to 1.5% damping, energyis dissipated during vibration through friction. A hystereticfriction element was placed between the blocks with a yieldforce of 2.9 kN based on Fy = μN , where the coefficient offriction was assumed to be 0.3 and the normal force at overlapwas approximated as 7.5% of the block B weight. Impact wasmodeled between the decks using a bi-linear contact elementbased on the recommendations by Muthukumar (2003). Theproposed hysteretic model reflects the energy dissipated duringpounding. However, a gap of 10 mm was provided betweenthe blocks before the impact element engages based on the testsetup and experimental data.

Figure 12. Experimental and analytical stress–strain response of the84-wire shape memory alloy restrainer cable under 0.25g loading.

The shape memory alloy devices were modeled asdescribed in the section above. Gaps elements represented theinitial slack in the cables. The series of tests with 130-wirecables were specified to have 12.7 mm slack and zero slackfor the 84-wire runs. A slight additional slack of 2.54 mmwas defined in the analytical model to capture the inability toprecisely obtain zero gap in the test specimen.

4.3. Comparison with experimental results

Run 5 is used to compare the analytical and experimentalresults for the test setup with 84-wire SMA cables subjectedto the ground motion scaled to a peak ground acceleration of0.25g. The loading is simulated by applying the ground motionas an input acceleration to the base of the modeled structure.Figure 12 shows the stress–strain plot for the dynamic responseof the SMA restrainer. The results indicate that the analyticalmodel is able to capture the sub-looping, and the loading andunloading plateau. The peak strain from the analysis is 2.89%,and stress is 471 MPa, which is within 1% of the experimentalresults. The modeling assumption of zero strain accumulationduring cycles reasonably predicts the hysteretic response of theSMA cable recorded in the experiment.

Figure 13 shows an experimental versus analyticalcomparison of the time history response of the hinge openingbetween the blocks. The experimental plot is the average of themeasured opening from the displacement transducers on eitherside of the in-span hinge. The analytical simulation representsthe dynamic response of the in-span hinge reasonably well.The peak hinge openings are 33.8 and 33.9 mm for theexperimental and analytical studies, respectively. Table 3presents a comparison of the peak responses for low (0.15g)and high (0.25g) PGAs, and the small (84-wire) and large(130-wire) cables. The calculated peak strains and hingeopenings differ by 1%–22% relative to the experimentalresults, and the stresses differ by 1%–13%. In general, themodel more accurately captures the response at the largeramplitude motions. Inconsistencies between the analyticaland experimental response are attributed to some sticking

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Figure 13. Time history of hinge opening from analysis (peak of 33.5 mm) plotted relative to the experimental test data.

Table 3. Comparison of peak responses from the analytical and experimental studies.

Peak from experimental Peak from analytical Per cent difference

PGA Opening Strain Force Stress Opening Strain Force Stress Opening Strain Force StressRun (g) (mm) (mm mm−1) (kN) (MPa) (mm) (mm mm−1) (kN) (MPa) (%) (%) (%) (%)

3 0.15 21.5 0.0186 10.27 450.3 16.8 0.0145 8.78 389.6 −22 −22 −15 −135 0.25 33.8 0.0292 10.90 477.9 33.5 0.0289 10.62 471.2 −1 −1 −3 −18 0.15 31.8 0.0275 15.72 450.6 28.8 0.0249 13.60 389.9 −9 −9 −13 −13

10 0.25 40.0 0.0346 17.34 497.0 36.8 0.0319 17.08 489.5 −8 −8 −2 −2

between the blocks as well as slight rotation of the blockswhich are not captured by the analytical model. However,the proposed model adequately captures the seismic responseof the simplified bridge system retrofit with SMA restrainercables.

The analytical model was used to determine the relativeperformance of the system before and after retrofit. Usingrun 5 as an example, the as-built response of the dual-framesystem was evaluated at a PGA of 0.25g, in order to estimatethe improvement realized by using the shape memory alloydevices. The SMA retrofit was found to reduce the peak hingeopenings by 48%. This reveals the considerable benefit ofretrofitting the bridge in order to improve the seismic responseand limit the potential for unseating at the in-span hinge.

5. Conclusions

The potential advantages of using nitinol shape memory alloysin the seismic restraint of bridges have been realized andillustrated as a part of this large scale testing and analyticalstudy. The results of the experimental testing have revealedthat the SMA restrainers not only served as effective bridgeretrofits, but also result in superior performance relative toequivalent traditional steel restrainer systems. Key conclusionsfrom the study are summarized below.

(1) The SMA restrainers were effective in limiting relativehinge displacements compared to steel restrainers. This wouldreduce the possibility of unseating of frames at the in-spanhinge of bridges during a seismic event.

(2) SMA restrainers produced lower block accelerationsduring earthquake excitation compared to equivalent steelrestrainers.

(3) The forces in the SMA and steel restrainers werecomparable. However, the SMA cable restrainers had minimalresidual strain after repeated loading cycles and exhibited theability to undergo many cycles of loading with little strengthand stiffness degradation. This would negate the need toreplace them after a major seismic event unlike traditional steelrestrainers.

(4) The proposed analytical model was capable of reason-ably predicting and reproducing the dynamic characteristicsof the representative multi-frame bridge retrofit with SMA re-strainer cables. Peak stresses, strains, forces, and hinge open-ings were well matched.

(5) Utilizing the analytical model, comparisons betweenthe as-built and retrofitted system could be made, and theresults revealed that using SMA restrainer cables reduced thepeak hinge openings by nearly 50% for some cases.

Acknowledgments

This work was funded by the California Departmentof Transportation. This financial support is gratefullyacknowledged. Special thanks to Patrick Laplace and PaulLucas for all of the effort required to put the test specimen inplace, for instrumentation of the experiments and for operatingthe experiment with successful acquisition of the data reportedin this study.

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