sensitivity of seismic applications to different shape memory alloy models

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Sensitivity of Seismic Applications to Different ShapeMemory Alloy Models

Bassem Andrawes, A.M.ASCE1; and Reginald DesRoches, A.M.ASCE2

Abstract: Shape memory alloys �SMAs� are known for their superelastic properties, which have been exploited in numerous applicationsin the biomedical, aerospace, and commercial fields. More recently, these materials have been evaluated for applications in the area ofearthquake engineering. One key question that arises when using these materials is the appropriate constitutive material model to use tocapture the highly nonlinear behavior of SMAs. This paper explores the effect of using different SMA constitutive models on the resultingresponse of systems using SMAs. A sensitivity analysis is conducted by using three SMA models with various levels of complexity. Themodels are implemented in a single-degree-of-freedom system and subjected to three groups of earthquake records with various charac-teristics. Considering a more accurate trend in modeling incomplete cycles in SMAs has little impact on the structural response. Thestrength degradation and residual deformation seem to be of more importance than the sublooping behavior. The response is moresensitive to the cyclic effects in the case of records with long durations or large intensities.

DOI: 10.1061/�ASCE�0733-9399�2008�134:2�173�

CE Database subject headings: Shape memory effect; Elasticity; Earthquakes; Constitutive models; Seismic effects; Sensitivityanlysis.

Introduction

Shape memory alloys �SMAs� are metallic alloys known for theirunique thermomechanical characteristics, such as the superelasticeffect and the shape memory effect. SMAs exhibit a unique abil-ity to recover their original shape after being deformed. If thetemperature is above the austenite finish temperature, Af, theshape recovery is attained by the removal of the load �super-elasticity effect�, whereas at temperatures below Af, the shapeis recovered by heating the alloy to a temperature above Af

�shape memory effect�. These two phenomena have resulted inunique applications across many fields. The application ofSMAs in civil structures has been considered by many researcherssuch as Saadat et al. �2001�. Corbi �2003�, who proposed usingsuperelastic SMAs as who proposed using superelastic SMAs ascross bracings, and base isolators for buildings. Thomson et al.�1995�, Krumme et al. �1995�, and Yan and Nei �2003� in-vestigated using SMAs as passive control devices in buildings.Andrawes and DesRoches �2005� studied using superelasticSMAs as seismic restrainers for bridges subjected to earth-quakes. Sakai et al. �2003� focused on exploiting the re-

1Assistant Professor, Dept. of Civil and Environmental Engineering,Univ. of Illinois at Urbana-Champaign, 205 N. Mathews Ave., Urbana,IL 61801 �corresponding author�. E-mail: [email protected]

2Associate Professor, School of Civil and Environmental Engineering,Georgia Institute of Technology, Atlanta, GA 30332. E-mail: [email protected].

Note. Associate Editor: Lambros S. Katafygiotis. Discussion openuntil July 1, 2008. Separate discussions must be submitted for individualpapers. To extend the closing date by one month, a written request mustbe filed with the ASCE Managing Editor. The manuscript for this paperwas submitted for review and possible publication on January 19, 2006;approved on June 1, 2007. This paper is part of the Journal of Engineer-ing Mechanics, Vol. 134, No. 2, February 1, 2008. ©ASCE, ISSN 0733-
9399/2008/2-173–183/$25.00.
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J. Eng. Mech. 2008.1

centering capability of superelastic SMA wires in providingself-restoration for concrete beams. Previous studies haveshown that SMAs are very promising for structural applica-tions. However, the most appropriate model to use for repre-senting the behavior of SMAs in civil structures is still an openquestion.

In most of the previously mentioned studies that have exploredthe behavior of SMAs analytically in structures, SMA mechanicalbehavior was modeled by using simplified phenomenologicalmodels that do not capture the complex behavior under cyclicloading. These types of models are appropriate for the study ofSMAs under monotonic loading. However, under cyclic loading,such as in seismic events, it is well known that SMAs experiencevariation in their hysteretic shape and mechanical properties, andhence a more advanced analytical model with adaptive hystereticproperties and hysteretic shape needs to be used. This paper ex-plores the effect of using SMA analytical models with a relativelyhigh level of complexity in structural applications subjected toearthquake loading versus using simplified models. This explora-tion was done by conducting a sensitivity study that compares thebehavior of structures by using different SMA analytical modelswith various levels of complexity under seismic loading. Thelevel of complexity of the model is primarily measured by thenumber of parameters required to fully describe the model. Sincemost of the potential structural applications for SMAs are focusedon using the superelasticity behavior of SMAs, the study wasprimarily directed toward comparing the models that describe themechanically triggered forward and reverse phase transformationin SMAs. Since this paper is directed toward the seismic applica-tions of SMAs, a brief introduction to the effect of cyclic loadingon SMAs is presented in the following section. This introductionprecedes the various analytical SMA models used in the study, theparameters of the sensitivity study, and the results obtained from
the study.
L OF ENGINEERING MECHANICS © ASCE / FEBRUARY 2008 / 173

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Cyclic Loading Effects

The subject of the cyclic loading effect on the mechanical behav-ior of SMAs has been studied by a large number of researchers�Tanaka et al. 1995; Bo and Lagoudas, 1995; Lexcellent andBourbon 1996; Friend and Morgan 1999; Gall and Sehitoglu1999�, and most of them agree that changes in the mechanicalproperties that are associated with the cyclic loading in SMAs aremainly attributable to the formulation of residual martensite,which accumulates with each cycle. This residual martensite ispermanent martensite and thus never participates in subsequentphase transformation cycles.

Experiments have shown that SMAs experience change intheir mechanical properties when subjected to cyclic loading. Therepeated phase transformation that the SMA experiences undercyclic loading results in a reduction in the stress required forforward transformation, a reduction in the area of hysteresisloops, and an increase in the residual strain. Fig. 1 shows theeffect of cyclic loading with increasing the number of cycles �N�.As shown in the figure, as the number of loading cycles increases,the forward and reverse transformation stresses decrease, with theforward transformation stress decreasing more. This behaviorleads to a reduction in the hysteresis area and thus a reduction inthe damping capability. The figure also illustrates the accumula-tion of residual deformation with the number of cycles. The cyclicloading effects seem to stabilize after a certain number of cycles,and thus many researchers suggested training SMA devices be-fore using them �McCormick et al. 2005�.

SMA Analytical Models

To understand the level of complexity necessary for SMA modelsused in seismic applications to achieve a certain level of accuracy,three SMA models are investigated in this paper. The modelsinclude a simplified model, which is experimental-based; a ther-momechanical model, which takes into account the stress-strain-temperature relationship in SMAs; and a thermomechanicalmodel that also considers the cyclic loading effects in SMAs.Each of the three models is implemented and tested in a single-degree-of-freedom �SDOF� system under various seismic loadingconditions. The following two subsections describe the SMA ana-

Fig. 1. Cyclic loading effect on the stress-strain behavior ofTi-50.2 at % Ni alloy

lytical models used in the study.

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Simplified Model

A one-dimensional experimental-based superelastic phenomeno-logical model was developed to represent the simplest familyof SMA constitutive models that can be used in seismic applica-tions. The model is considered an experiment-based model,since the hysteretic properties of the model are predefined andthey are independent of the thermomechanical properties of theSMA. Fig. 2 shows a schematic of the force-deformation relation-ship resulting from the simplified model. As shown in the figure,in this model, the SMA behavior is described by a multilinearhysteresis with a backbone curve described by the followingequations:

F�x� = kix 0 � x � xy �1�

F�x� = Fy + kt�x − xy� xy � x � xt �2�

F�x� = Fh + kh�x − xt� x � xt �3�

where F�x��SMA force as a function of the deformation; x, xy

and xt�SMA deformation at the beginning and end respectively,of martensitic transformation; Fy and Fh�SMA forces at the be-ginning and end, respectively, of martensitic transformation; andki, kt, and kh�stiffness before, during, and after martensitic trans-formation. In this model, the reverse transformation of the SMAfollows a linear path, and the transformation from martensite toaustenite is assumed to be triggered at a constant flat branch,which is located at a level below the loading level. This assump-tion simplifies the sublooping �i.e., incomplete cycles� behavior inSMAs. However, experimental results have shown that the re-verse phase transformation is typically triggered at higher levelsof stress depending on the extent of phase transformation experi-enced by the alloy.

Thermomechanical Model

The second model used in this study is a thermomechanicalmodel that was developed based on the basis of the work ofTanaka et al. �1995�. This model is characterized by its abilityto capture the cyclic loading effects on SMAs in addition tothe sublooping behavior resulting from incomplete phase transfor-mation cycles. Three internal variables were presented in themodel. The local stress and strain and the residual martensiticphase accumulated irreversibly because of the cyclic forward/

Fig. 2. Force-deformation relationship of the simplified superelasticSMA model

reverse martensitic transformation. In Tanaka’s previous work

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�Tanaka and Iwasaki 1985; Tanaka, 1986�, he had shown that theuniaxial thermomechanical behavior of SMAs could be describedby the constitutive relation consisting of the following equation inrate form:

� = D� + �T + �� 4�

where �, �, and T�stress, strain and temperature, respectively;D�Young’s modulus; and −� /D and −� /D�coefficient oflinear expansion and the strain attributable to transformation,respectively. � denotes the volume fraction of the martensitephase. The transformation kinetics in martensitic transformationis described by the following relation:

�

1 − �= bMcMT − bM� 0 �5�

whereas in the case of reverse transformation, the relation is

−�

�= bAcAT − bA� 0 �6�

The terms cM and cA are material parameters that can be de-termined experimentally, whereas bM and bA are material param-eters, which will subsequently be calculated. By integratingEqs. �5� and �6�, the following expressions for the transformationkinetics in forward and reverse transformations, respectively, areobtained:

� = 1 − �A0 exp�bMcM��A0� − T� + bM�� 7�

� = �M0 exp�bAcA��M0� − T� + bA�� 8�

where �A0 and �M0�volume fraction residual austenite and mar-tensite, respectively. Both volume fractions are related to eachother with the relation, �A0+�M0=1. ��A0� and ��M0��marten-sitic and austenitic start transformation temperatures, respectively,and are assumed to be related to the residual austenite and mar-tensite. In the case of Ni-Ti alloys, researchers have found that��A0� and ��M0� can be taken as the martensite start tempera-ture Ms and the austenite start temperature As, respectively. UsingEqs. �7� and �8�, the parameters bM and bA can be determined asfollows:

bM =ln�100�A0�

cM�Mf − ��A0��9�

bA =ln�100�M0�

cA�Af − ��M0��10�

Themomechanical Model with Cyclic Effects

To take into account the effect of residual stress accumulatedduring cyclic loading, three additional internal variables are intro-duced in the thermomechanical model described in the precedingsubsection. Those variables were the residual stress, bir, corre-sponding residual strain, �ir, and the macroscopic volume fractionof the martensite phase, �ir. The martensite is understood not totake part in the subsequent transformations. The stress, �, andstrain, �, now must be understood as local values, as follows:

� = � + �ir �11�

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� = E − �ir �12�

where � and E�global stress and strain, respectively. Assumingall material parameters to be constant, the residual stress could beexpressed as

�ir = S�1 − exp�− /�� 13�

where S is a material parameter that represents the stabilizedtransformation stress; � is a parameter that governs the speed ofthe accumulation; and �intrinsic time, which flows only whenthe transformations take place. The /� ratio describes the num-ber of cycles required before the hysteretic properties of the SMAare stable. For simplicity, the residual martensite was assumed tovary linearly with the local residual stress. Fig. 3 shows a typicalstress-strain hysteresis of the thermomechanical model with cy-clic loading effects. As shown in the figure, the hysteresis loopshifts with the number of cycles and tends gradually to a limitstationary loop.

Single-Degree-of-Freedom Model

An SDOF stick-mass model was developed for comparing thethree SMA models under seismic loading. Fig. 4 shows a sche-matic of the analytical model used in this study. The model con-sists of a single mass, m; a structural restoring force element withan initial stiffness, k; a dashpot that incorporates the viscousdamping; and two identical tension-only SMA links. The twoSMA links are used in this study as an alternative for using atension-compression link, since the mechanical properties ofSMAs under compression and tension are not consistent. The

Fig. 3. Stress-strain relationship for the thermomechanical SMAmodel used in the study

Fig. 4. Schematic of the SDOF model used in the study


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SMA links used in this study were assumed to be in the form ofcables consisting of bundled SMA wires. The size and number ofthese cables can vary on the basis of the type of application, aswell as the level of force at which the SMA links would serve.The ground motion was applied at the point of structural supportshown in the figure. The SDOF model was governed by the fol-lowing equation of motion:

mx + cx + fr�x, x� + fsma1�x, x� + fsma2�x, x� = − mxg�t� �14�

where m�mass of the structure; c�damping coefficient;x�structural displacement; x�velocity; x�acceleration;xg�t��input ground acceleration; fr�x , x��restoring force ofthe structure; and fsma1�x , x� and fsma2�x , x��restoring forces ofthe two SMA links. To account for the nonlinearity in the re-sponse of the structure, fr and fsma are assumed to be dependenton the structural displacement, x, and structural velocity, x. Thenonlinear behavior of the SDOF system was described by usinga Q-hyst model �Saiidi 1982�, which is typically developed todescribe the stiffness/strength degradation in the response of re-inforced concrete members under cyclic loading.

The Q-hyst model is represented by the following equations:Before yielding

fr�x, x�kx �15�

After yielding

fr�x, x� = fy + kh�x − xy� x � 0 �16�

fr�x, x� = fm − k�xy/xm�0.5�xm − x� x � 0 �17�

where k�initial stiffness of the structure; kh�stiffness duringstrain hardening; fy and fm�yield force and maximum force, re-spectively, reached by the structure; xy and xm�yield displace-ment and maximum displacement, respectively, reached by thestructure. The stiffness of the load-reversal stage is taken to beequal to the slope of the line joining the unloading branch-deformation axis intersection point and the point on the curvecorresponding to maximum displacement experienced at eitherside �Saiidi 1982�.

The equation of motion was solved numerically by usingNewmark’s method �Chopra 1995�. An average acceleration wasassumed during each time step �the parameter � was taken equalto 1 /4�. To consider the nonlinearity in the structure, the Newton-Raphson method was used to perform the iterations on the tan-gential stiffness of the structure. The analysis at a specific timestep was considered to be converged if the residual force was lessthan a tolerance value that was calculated from the followingequation:

tol. = 10−6 � �max�xg�t�� m� �18�

As illustrated by the equation, the tolerance considered in thisstudy depends on the input ground acceleration and mass.

Ground Motion Records

One of the main goals of this study was to explore the sensitivityof SMA models to ground motion characteristics. As previouslydiscussed, part of the difference between the three SMA modelsconsidered for this study is attributable to considering the incom-plete loading cycles and the strength degradation/residual defor-mation resulting from cyclic loading. As a result, the number and
intensity of the cycles that the SMA model goes through plays an


important role in defining the behavior of the model. The durationof the ground motions was considered to be one of the mainparameters controlling the number of cycles in each ground mo-tion and thus affecting the behavior of SMA models. The numberof large-intensity cycles is also included by considering the near-field ground motion records, which in this study are considered tobe less than 5 km from the epicenter. This type of ground motionnormally contains cycles with relatively large intensity.

For the previously discussed purpose, three suites of 15ground motion historical records with various parameters wereselected. Suites A and B represented far-field records with longand short durations, respectively. The record’s duration was de-fined as the time interval between the first and last exceedance ofthe ground acceleration time history to an acceleration thresholdof 0.05g. A duration greater than 14 s was assumed to define thelong and short duration. Suite C consisted of 15 records measuredat stations near the earthquake fault. The acceleration responsespectrum for each ground motion and the average response spec-trum for each group of records were calculated and are presentedin Figs. 5–7 for groups A, B, and C, respectively. The averageresponse spectra for the three ground motion groups are comparedin Fig. 8. To be consistent regarding the level of scaling of the

Fig. 5. Response spectra and average response spectrum �solid-thick�for suite A

Fig. 6. Response spectra and average response spectrum �solid-thick�for suite B

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records, the ground motion records for the three groups werescaled on the basis of the average spectral acceleration curve ofGroup A. This was conducted such that the spectral accelerationvalue of the ground motion record at the natural period of thestructure would match the average spectral acceleration value ofSuite A.

Study Parameters

SDOF Model Parameters

One of the main focuses of the study was to explore the effect ofusing SMA models on the structural response at various structuralperiods. Therefore, in this study, the weight of the structure wasassumed to be 1717 KN, whereas the initial stiffness, k, of thestructure was assumed to be variable that is calculated from thenatural period of the structure. Four values were considered forthe structural natural period—0.25 s, 0.5 s, 0.75 s, and 1.0 s. Inthe initial stage of the study, the structure was assumed to belinearly elastic. The structural nonlinearity is included and is dis-cussed subsequently. The only structural damping considered in

Fig. 7. Response spectra and average response spectrum �solid-thick�for suite C

Fig. 8. Average response spectra for ground motion suites A, B,and C

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this stage was the equivalent viscous damping represented by adashpot. The equivalent viscous damping coefficient was as-sumed to be 5%.

SMA Parameters

Although the structural properties �stiffness� and the ground mo-tion scaling factors were considered as variable in the analysis,the SMA mechanical properties were assumed to be constant dur-ing the study. This assumption facilitated comparing the perfor-mance of the three SMA models at different deformation levels.As previously mentioned, instead of using a compression-tensionSMA link, two tension-only SMA links were connected to bothsides of the structure. The two links were assumed to be identicalin their mechanical properties. The parameters of the three modelswere selected so that the constitutive behavior of the three modelswould match. In other words, the three models were designed toproduce the same behavior under monotonic loading. However,under cyclic loading, each model would perform differently.

In the case of the thermomechanical model with cyclic loadingeffects, the values of the S and � parameters presented in Eq. �13�were selected so that the residual strain and strength degradationwould reach 70% of their final values after going through the firstcomplete phase transformation cycle. The model was also de-signed to reach a stable condition after six cycles. The final re-duction in the strength at the end of the sixth cycle was assumedto be approximately 18% of the original phase transformationforce; this reduction in the strength is associated with approxi-mately 0.75% residual strain. The parameter values selected todescribe the constitutive behavior of the thermomechancal modelsare presented in Table 1. These values were selected on the basisof previously conducted experimental studies �DesRoches andDelemont 2002; Dolce and Cardone 2001�.

Results

The SDOF structure was subjected to the three groups of groundmotion records. The summary of the analytical results in groupsA, B, and C is shown in Fig. 9. The results presented in thesefigures were based on the assumption that the structure is linearlyelastic. The effect of structural nonlinearity will be discussedsubsequently. The plots presented in Fig. 9 represent the relation-ship between the normalized structural lateral displacement andthe structural period. The lateral displacement is normalized byusing the response of the as-built structure, �i.e., when no SMA

Table 1. Parameters of the Thermomechanical SMA Models

Parameter Description Value

D Austenite Young’s modulus 6.894 KN /mm2

� Phase transformation modulus –3.447 KN /mm2

As Austenite start temperature 253°K

Af Austenite finish temperature 263°K

Ms Martensite start temperature 243°K

Mf Martensite finish temperature 233°K

T Environmental temperature 270°K

Ca Material parameter 0.3

Cm Material parameter 0.3

S Stabilized transformation stress 103 KN /m2

� Speed of accumulation parameter 0.85

links are used�. The figure shows a comparison among the struc-


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tural responses using the three SMA models. Each of the points inthe figures represents the average of the responses resulting fromeach group of records. Since the maximum differences observedamong the three SMA models were at a structural period equal to0.5 s, the percentage of difference between each of the two ther-momechanical models and the simplified model at this specificperiod is presented on each figure. The number presented abovethe lines is the difference resulting from using the thermome-chanical model with cyclic loading effects, whereas the numberlocated below the lines is the difference resulting from using thethermomechanical model without cyclic effects. The figures showthat the response in the case of the thermomechanical model isquite close to the response in the case of the simplified model.The average difference between the responses of the two modelswas between 4 and 5% for each of the three group of records. Themaximum difference which occurred in Group A, was 9%

The results indicate that simplifying the sublooping behaviorin the simplified model tends to slightly reduce the structuralresponse, especially in the case of structures with high to moder-ate stiffness. However, in flexible structures, including the moreaccurate sublooping behavior in the SMA models has a minoreffect on the behavior of the model. This outcome is likely attrib-utable to the SMA’s being subjected to a higher level of load andthus being forced through more complete phase transformation

Fig. 9. Normalized response of the SDOF using different SMA modcompared with the simplified model is presented at 0.5 s intervals.�

cycles.



Including the cyclic loading effects �strength degradation andresidual strain� in the thermomechanical model with cyclic effectsseems to have more effect than the sublooping effect. The averagedifference in the response between the simplified model and thethermomechanical model, which includes the cyclic effects, wasbetween 14 and 16% for the three ground motion groups. Themaximum difference in the case of groups A, B, and C occurred atthe 0.5-s period and was equal to approximately 27, 19, and 26%,respectively. Those results showed that the structural responsewas more affected by the cyclic loading effects than by the sub-looping effect. Structures with moderate periods are more sensi-tive to the cyclic loading effects because SMA devices imple-mented in such structures would experience small to moderatedeformations and thus experience a relatively small amount ofyielding. However when the strength degradation takes place, thedevices experience more nonlinear behavior, which in turn in-creases the structural response.

The plots presented in Fig. 9 evaluate the effect of groundmotion parameters �duration and intensity� on the performance ofeach of the SMA models. The structural response in the case ofground motions with long durations �Group A� and large intensi-ties �Group C� are more sensitive to the cyclic loading effectsthan the short-duration ground motion records �Group B�. Thisresult might occur because ground motions with short durations

en subjected to record suites A, B, and C. �The percent of increase
els wh
tend to include fewer cycles than long durations. The effect of

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including the more accurate subloops in the SMA model is clearlyeffective in the case of short durations, as well as long durations.However, the near-field ground motions showed less sensitivity tothe sublooping effect because near-field records usually containrelatively large intensity cycles along with small insignificantcycles. The large ground-motion cycles would produce complete

Fig. 10. Ground acceleration time history for the 1992 Landers�Coolwater� record

Fig. 11. Force-deformation relationships for the SMA links using t

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cycles in the model, and the small cycles are not strong enough tostart the phase transformation.

To better understand the behavior of each of the three SMAmodels, a sample ground motion is selected from each group ofrecords and discussed in more detail in the following sections.The records that were selected to represent the long-duration,short-duration, and near-field records are the Coolwater record�1992 Landers earthquake�, the Gilroy Array #3 record �1989Loma Prieta earthquake�, and the Pacoima Dam record �1971 SanFernando earthquake�, respectively.

1992 Landers „Coolwater… Record

Fig. 10 shows the ground acceleration time history for the 1992Landers �Coolwater� record. In this study, this ground motion wasconsidered to be of a long duration with an effective duration�first and last exceedance of 0.05g / –0.05g� equal to 18.5 s.Fig. 11 shows the force-deformation relationship of the simpli-fied, thermomechanical, and thermomechanical with cyclic effectsSMA models under the scaled 1992 Landers �Coolwater� record.The responses of the two tension-only SMA links were superim-posed where one link is engaged when the structural displace-ment, x, takes a positive value, whereas the other link startsengaging when x is negative. The response shown is for theSDOF structure with period 0.5 s. The maximum displacementsin the case of simplified, thermomechanical, and thermomechani-

e SMA models under the scaled 1992 Landers �Coolwater� record
he thre

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cal with cyclic effect models were 15.5 mm, 17.3 mm, and38.1 mm, respectively. These results show that neglecting themore accurate sublooping behavior of SMAs in the simplifiedmodel underestimated the displacement by about 10%, whereasneglecting both the sublooping behavior and the cyclic loading

Fig. 12. Ground acceleration time history for the 1989 Loma Prieta�Gilroy Array #3� record

Fig. 13. Force-deformation relationships for SMA link using vari



effect results in underestimating the displacement by about 59%.Thus, in this case, the response of the structure is more sensitiveto the cyclic loading effects than to the sublooping effect. Theseresults agree with the results previously presented in Fig. 9�a�.

Taking a closer look at the force-deformation relationships andcomparing the behaviors in Figs. 11�a and b� shows that in bothcases the SMA went through the same number of cycles. How-ever, the unloading path was the main difference between the twobehaviors. In the thermomechanical model shown in Fig. 11�b�,the more realistic unloading path of SMAs was considered. It ledto a reduction in the level of force during unloading comparedwith the force level in the simplified model and hence, caused aslight increase in the maximum deformation. Including the cyclicloading effects in the thermomechanical model resulted in a sig-nificant difference in the performance of the SMA. As noticed inFig. 11�c�, a 6.9-mm residual deformation and a 1326-KNstrength reduction were observed. The relatively large deforma-tion occurred because the structure experienced the cycle withpeak intensity after the cyclic loading effects already took place.The presence of the record’s peak cycle towards the second halfof the record �see Fig. 10� played an important role in increasingthe sensitivity of the structural response to the cyclic loadingeffects.

A models under the 1989 Loma Prieta �Gilroy Array #3� record
ous SM
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1989 Loma Prieta „Gilroy Array #3… Record

The 1989 Loma Prieta �Gilroy Array #3� record presented inFig. 12 represents records with short duration in which peakcycles occur early in the record. The effective duration of the

Fig. 14. Ground acceleration time history for the 1971 San Fernando�Pacoima Dam� record

Fig. 15. Force-deformation relationships of the SMA link using difrecord

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record was found to be 10 s. The force-deformation relationshipsfor the SMA links using the three SMA models under the scaledrecord are presented in Fig. 13. The maximum displacements inthe simplified, thermomechanical, and thermomechanical with cy-clic effect models were 23.1 mm, 25.4 mm, and 25.4 mm, respec-tively. This outcome shows that considering the more accuratebehavior during the unloading in SMAs resulted in a difference ofabout 9%, which is close to the previously observed difference inthe 1992 Landers �Coolwater� record. This result implies that inshort-duration records, the SMA behavior is not sensitive to theapproach used in modeling incomplete cycles in SMAs.

Considering the effects of cyclic loading in the SMA constitu-tive model changed the behavior of the link and forced it toexperience a larger number of cycles; however the maximum de-formation was not affected. This behavior is understood throughreviewing the time history of the record in Fig. 12 and noticingthe relatively small cycles that exist during the early time of therecord before the peak cycle, which occurred at a time of about5 s. The small cycles were not enough to develop a significantamount of residual martensite. When the peak cycle struck earlyin the record, the strength degradation and residual deformationaccumulation were not entirely complete, and thus the structuredid not reach the maximum flexibility that was expected as aresult of the cyclic loading effects. This outcome indicates that

SMA models under the scaled 1971 San Fernando �Pacoima Dam�
ferent

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the structural response is less sensitive to the type of SMA modelwhen subjected to short durations, which usually contains fewercycles.

1971 San Fernando „Pacoima Dam… Record

The Pacoima Dam record, shown in Fig. 14, from the 1971 SanFernando earthquake was selected from Group C, which repre-sents near-field records. The station where it was recorded wasapproximately 2.8 km from the earthquake’s epicenter. As shownin Fig. 14, the record is characterized by a large number of cycleswith an intensity that exceeded 0.5g. The peak ground accelera-tion �PGA� for this record was 1.16g. The force-deformation re-lationships of the SMA link using the three types of SMA modelsunder the record are presented in Fig. 15. The maximum displace-ments in the simplified model, the thermomechanical model, andthe thermomechanical model with cyclic effect were 9.9 mm,11.7 mm, and 27.4 mm, respectively. A difference of approxi-mately 15% was observed in the maximum displacement whenthe accurate behavior of SMAs during unloading was neglected.This difference is considered to be minor and in the same range as

Fig. 16. Nonlinear SDOF normalized response using different SMAincrease compared with the simplified model is presented at 0.5 s int

the differences observed in far-field records. When the cyclic ef-



fects of SMAs were considered, the maximum response of thestructure was reduced by approximately 64% compared with thesimplified model response. This difference was attributable tothe existence of a relatively large number of cycles with largeintensity in the record. During the first few seconds of the record,a number of large-intensity cycles weakened the structure becauseof the cyclic effects of the SMA; and thus, once the peak cyclehit, it produced a large displacement.

Structural Nonlinearity Effect

The effect of structural nonlinearity on the sensitivity of the struc-tural response to the SMA models is discussed in this section. Aductility factor equal to 4.0 was considered in this study. In thiscontext, the ductility factor is defined as the ratio of the maximumstructural response to the structural response at yield. The param-eters used in this study for the SDOF model and the SMA modelswere identical to the parameters used for the elastic case.

Fig. 16 shows a summary of the results where the SDOFmodel is subjected to ground motion groups A, B, and C. Thepercentage of difference in the structural response resulting from

els and subjected to record suites A, B, and C. �The percentage of.�

modervals

using the three SMA models at a period equal to 0.5 s is presented

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in each figure. As shown in the figure, a relatively minor effect inthe range of 4 to 6% was observed when the more accuratesublooping behavior was considered in the SMA model. Themaximum differences in any of the three ground motion groupcases did not exceed 8%. The small influence on the structuralresponse associated with considering the accurate sublooping be-havior of SMAs during unloading was also observed in the elasticstructure.

Considering the effects of cyclic loading in the SMA, thethermomechanical constitutive model had more impact on thestructural response than when only sublooping behavior was con-sidered. The average response differences between the simplifiedmodel and the thermomechanical model with cyclic effects inground motion groups A, B, and C were 16, 8, and 18%, respec-tively. A maximum difference of 23, 15, and 39% was also ob-served in ground motion groups A, B, and C, respectively. Thoseresults showed good agreement with previously presented resultsfor the elastic case. The structural response seems to be moresensitive to cyclic loading effects, especially in ground motionrecords with long duration or large intensity. The differences inthe responses of the three SMA models decrease at a high level ofstructural flexibility because of the large displacements of thestructure, which force the SMA model to exceed the elasticrange—where the phase transformation takes place—and act inthe martensitic elastic range. In the martensitic elastic rangeminor differences exist between the three SMA models, resultingin minor differences in the structural response.

Conclusions

A sensitivity analysis was conducted to investigate the effect ofusing SMA models with various levels of complexity on the per-formance of an SDOF system under seismic loading. The modelsvaried in their ability to capture the hysteretic shape of the super-elastic SMAs. One of the models was a simplified multilinearphenomenological model, whereas the other two were thermo-mechanical models that were capable of capturing the effect ofincomplete phase transformation cycles �sublooping� in SMAs.One of these two models was also capable of capturing thestrength degradation and residual deformation associated withcyclic loading. The SDOF system was subjected to three groups�A, B, and C� of ground motion records with different character-istics. Group A represents fars field records with long duration,Group B represents far-field records with short duration, andGroup C represents near-field records. The fundamental period ofthe SDOF system was varied between 0.25 and 1.0 s. The effectof structural nonlinearity was also examined by imposing a struc-tural ductility demand equal to =4.0 on the system.

The results showed that the simplified model was in a goodagreement with the more complex thermomechanical model withno cyclic effects. Considering a more accurate trend in modelingthe incomplete phase transformation cycles in SMAs resulted in adifference in the maximum response that was less than 9% onaverage compared with the more simple approach. This differenceis considered to be insignificant, and hence accounting for theexact location of the reverse transformation trigger line in theSMA models has a small influence on the structural response.However, the cyclic loading effects seem to have more impact onthe structural response, which could result in a difference of ap-proximately 39% on average compared with the situation with no

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cyclic effects included. The results also showed that the structuralresponse is more sensitive to the cyclic loading effects of theSMAs in ground motion records with long durations or large in-tensities. The same trend of results was observed SDOFs withnonlinear behavior.

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