Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 963530 13 pageshttpdxdoiorg1011552013963530
Research ArticleRecentering Shape Memory Alloy Passive Damper forStructural Vibration Control
Hui Qian1 Hongnan Li2 Gangbing Song23 and Wei Guo4
1 School of Civil Engineering Zhengzhou University Zhengzhou 45000 China2 Faculty of Infrastructure Engineering Dalian University of Technology Dalian 116024 China3Department of Mechanical Engineering University of Houston Houston TX 77204 USA4 School of Civil Engineering Central South University Changsha 410075 China
Correspondence should be addressed to Hui Qian qianhuizzueducn
Received 12 July 2013 Accepted 9 September 2013
Academic Editor Gang Li
Copyright copy 2013 Hui Qian et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper presents a preliminary study on the evaluation of an innovative energy dissipation system with shape memory alloys(SMAs) for structural seismic protection A recentering shape memory alloy damper (RSMAD) in which superelastic nitinol wiresare utilized as energy dissipation components is proposed Improved constitutive equations based on Graesser and Cozzarellimodel are proposed for superelastic nitinol wires used in the damper Cyclic tensile-compressive tests on the damper with variousprestrain under different loading frequencies and displacement amplitudes were conducted The results show that the hystereticbehaviors of the damper can be modified to best fit the needs for passive structural control applications by adjusting the pretensionof the nitinol wires and the damper performance is not sensitive to frequencies greater than 05Hz To assess the effectivenessof the dampers for structural seismic protection nonlinear time history analysis on a ten-story steel frame with and without thedampers subjected to representative earthquake ground motions was performed The simulation results indicate that superelasticSMA dampers are effective in mitigating the structural response of building structures subjected to strong earthquakes
1 Introduction
Conventional structures rely on their adequate stiffnessstrength and ductility to survive earthquakes Such a designstrategy may not be economical and may be ineffective forunexpected seismic events [1 2] In recent years the designstrategy based on the performance of civil structures hasattached increasing attentions of both practicing engineersand structure owners [3ndash5] To enhance the seismic perfor-mance of structural systems many possible strategies havebeen proposed [6 7] and one promising family of solutionsis the passive control techniques [8 9]
By and large current passive control applications arebased on the following two techniques seismic base isola-tion and energy dissipation [10] Seismic isolation systemrelies on special ductile alternate layers which are installedbetween substructure and superstructure to reduce the trans-fer of seismic energy to the superstructure thus protectingthe superstructurersquos integrity [11] Usually in order to limit
the extent of the displacement energy dissipation devicesare incorporated into the alternate layers Energy dissipa-tion system incorporates special energy dissipation devicesinto the structures to absorb or consume a portion of theseismic energy thereby reducing energy dissipation demandon primary structural members and minimizing possiblestructural damage [8] Nowadays several types of seismicenergy dissipation devices such as metallic dampers frictiondampers viscoelastic dampers and viscous fluid damperare available However current technologies present somelimitations such as problems related to aging and durabilitymaintenance reliability in the long run substitution afterstrong events and variable temperature-dependent perfor-mances among others [12] Recently the increasing researchand development of smart materials and controlling devicesopen up a new area for seismic vibration control of structuralengineering providing a basic platform for the design andexploration of new generation high-performance dampingdevices
2 Mathematical Problems in Engineering
residual strain Unloading with
Detwinning
Returning to origin upon heating
Strain 120576
Stre
ss120590
T lt Mf
(a)
Unloading without
Forward martensite transformation
Inverse martensite transformation
residual strain
A
B
C
D
O Strain 120576
Stre
ss120590
Af lt T lt Md
(b)
Figure 1 Stress-strain diagrams of NiTi shape memory alloy (a) SME 119879 lt 119872119891 (b) superelasticity 119860119891 lt 119879 lt 119872119889
0
100
200
300
400
500
600
700
800
Experimental data The improved Graesser and Cozzarelli
000 001 002 003 004 005 006 007 008 009
Stre
ss120590
(MPa
)
Strain 120576
Figure 2 Stress-strain curves of superelastic nitinol wires
Shape memory alloys [13] are a class of novel functionalmaterials that possess unique properties including shapememory effect (SME) superelasticity effect (SE) extraor-dinary fatigue resistance high corrosion resistance highdamping characteristics and temperature-dependentYoungrsquosmodulus At a low temperature SMAs exhibit the SMEmdashresidual deformations can be recovered by heating the mate-rial above the austenite finish temperature as shown inFigure 1(a) At a higher temperature SMAs exhibit the SEas shown in Figure 1(b) In the superelastic phase SMAs areinitially austenitic However upon loading stress-inducedmartensite is formed Upon unloading themartensite revertsto austenite at a lower stress level resulting in the hystereticbehavior These properties make them ideal candidates forseismic energy dissipation devices in structural engineering
A significant number of research studies have beenconducted in an effort to use SMAs for applications in seismicresistant design and structural retrofit in the past decade(eg [14ndash24]) Wilde et al [14] proposed a smart isolationsystem combining a laminated rubber bearing with a devicemade of SMA for highway bridges Their analytical resultsshow that the isolation system can limit displacement anddissipate energy for earthquake mitigation Dolce et al [15]developed two families of SMA-based energy dissipating andrecentering braces for seismic vibration control of buildingsand bridges as outcomes of the MANSIDE project (MemoryAlloys for New Seismic Isolation and Energy DissipationDevices) To assess the effectiveness of SMA braces toreduce the seismic response of reinforced concrete (RC)framed structures shaking table tests of a 133-scale three-story two-bay RC plane frame which was designed forlow seismicity and low ductility according to the Europeanseismic code were carried out by Dolce et al [16] Theirexperimental results show that the SMA braces can enhanceseismic performances at least comparable to those providedby steel braces while having an additional self-centeringfeature Indirli et al [17] retrofitted historic buildings usingsuperelastic SMA tie bars to enhance its seismic resistancecapacities DesRoches and Delemont [18] and Andrawesand Desroches [19] evaluated the efficacy of superelasticnitinol bars as restrainers to reduce the risk of collapsefrom unseating of bridge superstructures at the hinges Theyfound that the SMA bars are effective in limiting relativehinge displacements in typical multiple frame bridges Liet al [20] investigated the vibration mitigation of a staycable provided with one superelastic SMA damper Theanalytical results show that the SMA damper can reducecable vibration significantly and the control effectivenessis influenced by SMA damper parameters and locationsZhang and Zhu [21] developed reusable hysteretic damper(RHD) based on superelastic nitinol stranded wires andtheir numerical analysis indicated the effectiveness of RHD
Mathematical Problems in Engineering 3
Left pull platePrestrain adjusting plate
Connecting fitting
End cap
Adjusting bolt
Fixed bolt Retaining plate
End capGrips
Superelastic SMA wires
Inner cylinderOuter cylinder
Push-pull rodRight pull plate
Figure 3 Diagram of RSMAD
Figure 4 Photograph of experimental setup
in passive seismic response control of structures Ocel et al[22]and McCormick et al [23] investigated beam-columnconnections using SMA bars Their studies show that SMAcan enhance the seismic performance of the connectionsParulekar et al [24] proposed a damper device using austeniteNiTi wiresThe device is tested and validated using a thermo-mechanical model of SMA taking into account the residualmartensite accumulation Performance of the structure withSMA dampers is compared with that of the same structurewith yielding dampers
Previous works show that SMAs are of promise in struc-tural engineering particularly as energy dissipation compo-nents for seismic protection However significant researchis still needed In Particular designing new types of SMA-based passive seismic devices which possess not only energydissipation and recentering capabilities but also simple con-figuration for easy installation in the practical engineering isstill a challenge
This paper presents an innovative recentering shapememory alloy damper (RSMAD) for seismic structural pro-tection The damper is simple in design and easy to imple-ment Superelastic nitinol wires were utilized in the damperas kernel energy dissipating components Cyclic tensile-compressive tests on the dampermodelwith various prestrainunder different loading frequencies and displacements wereconducted To assess the effectiveness of the damper for
structural seismic protection nonlinear time history analyseson a ten-story steel frame subjected to representative earth-quake ground motions with and without the dampers wereperformed
2 Constitutive Equation ofSuperelastic SMA Wire
With the wide applications of SMAs in different fields mod-eling of the peculiar mechanical behavior of SMAs such asSME and superelasticity has been an active area of researchover the past decades So far many constitutive models forSMA have been developed [25ndash33] These models describethe thermomechanical thermoelectrical and thermochem-ical behaviors of SMAs however most of them are too com-plex to be convenient for practical application in earthquakeengineering In this paper a relatively simple model devel-oped by Graesser and Cozzarelli [34] is adopted This modelis an extension of a one-dimensional strain rate independentmodel for hysteretic behavior proposed by Ozdemir [35]Theequation is given as
= 119864 [ 120576 minus | 120576| (120590 minus 120573
119884)
119899
] (1)
where 120590 and 120576 are the one-dimensional stress and the one-dimensional strain respectively 119864 is the initial elastic modu-lus 119884 is the yield stress 119899 is a constant assumed any positiveodd real value controlling the sharpness of transition from theelastic state to the phase transformation and 120576 denote thetime derivative of the stress and strain respectively and 120573 isthe one-dimensional back stress given by
where 119891119879 119886 and 119888 are material constants controlling the typeand size of the hysteresis the amount of elastic recovery dur-ing unloading and the slope of the unloading stress plateaurespectively When 119891119879 = 0 the model is purely martensiticWhen 119891119879 gt 0 the model predicts the superelastic behavior120572 is a constant controlling the slope of the 120590 minus 120576 curve in theinelastic range given by
120572 =
119864119910
119864 minus 119864119910
(3)
where 119864119910 is the slope of the 120590minus 120576 curve in the inelastic range
4 Mathematical Problems in Engineering
0
0
200
400
600
2 4 6Displacement (mm)
Resto
ring
forc
e (N
)
0
minus600
minus400
minus200
minus2minus4minus6
(a)
1
0
0
200
400
600
2 4 6Displacement (mm)
Resto
ring
forc
e (N
)
minus600
minus400
minus200
minus2minus4minus6
(b)
2
0
0
200
400
600
2 4 6Displacement (mm)
Resto
ring
forc
e (N
)
minus600
minus400
minus200
minus2minus4minus6
(c)
4
0
0
200
400
600
2 4 6Displacement (mm)
Resto
ring
forc
e (N
)
minus600
minus400
minus200
minus2minus4minus6
(d)
Figure 5 Hysteresis loops of the RSMAD at different prestrains and displacement amplitudes (005Hz frequency of loading 20∘Ctemperature)
120576in is the inelastic strain given by
120576in = 120576 minus120590
119864 (4)
119906() is the unit step function defined as
119906 (119909) = +1 119909 ge 0
0 119909 lt 0(5)
erf() is the error function defined by
erf (119909) = 2
radic120587int
120587
0119890minus1199052119889119905 (6)
The original Graesser and Cozzarelli model has a rel-atively simple expression with the parameters that can beeasily acquired however this model excludes the martensitichardening characteristics of SMAs under large amplitudeswhich are critical for structural safety protection underextreme events
To overcome the limitation of the original model Wildeet al [14] extended the Graesser and Cozzarelli model bydividing the full loop into four parts adding two termswith six parameters into (1) The Wilde model was utilizedto simulate the cyclic behaviors of SMA devices in otherresearches [21]
In the following in order to accurately predict the cyclicbehavior of a superelastic SMA device especially capture themartensitic hardening characteristics of SMAs under largeamplitudes an improved Graesser and Cozzarelli model ispresented In the present model the backstress expression ismodified by adding a special term to capture the martensitichardening characteristic of SMA under large amplitudesThemodified model is of the form
= 119864 [ 120576 minus | 120576| (120590 minus 120573
Figure 6 Mechanical properties of RSMAD as a function of prestrain and displacement amplitude (005Hz frequency of loading 20∘Ctemperature)
+ 119891119872[120576 minus 120576Mf sgn (120576)]119898[119906 (120576 120576)]
times [119906 (|120576| minus 120576Mf) ]
(8)
The third term in (2) is used to contribute to the back-stress on the ascending branch of the hysteresis in a way thatallows for the martensitic hardening 120576Mf is the martensitefinish strain 119891119872 and119898 are material constants controlling themartensitic hardening curve sgn(119909) is the signum functiongiven by
sgn (119909) =
+1 119909 gt 0
0 119909 = 0
minus1 119909 lt 0
(9)
Figure 2 shows the stress-strain curves of superelasticnitinol wires predicted by the improved Graesser and Coz-zarelli model versus experimental data at different strain
levels The characteristic parameters used in the models are119864 = 39500MPa 119884 = 385MPa 120572 = 001 119891119879 = 114119888 = 0001 119886 = 550 119899 = 3 120576Mf = 005 119891119872 = 42500and 119898 = 3 The superelastic nitinol wires are 05mm indiameter with a composition of approximately 509 Ni and491 Ti Under zero external stress the martensite startand finish temperatures (119872119891119872119904) and the austenite startand finish temperatures (119860 119904 119860119891) measured by differentialscanning calorimeter (DSC) are minus73∘C minus55∘C minus23∘C and5∘C respectively The uniaxial tension test of the superelasticnitinol wires was carried out using an electromechanicaluniversal testing machine at room temperature of 20∘C Thenitinol wire samples with a 100mm test length between thetwo custom-made grips were subjected to triangular cyclicloading under different strain amplitudes The strains werecalculated from the elongation measured by a 50mm gagelength extensometer with the stress calculated from the axialforce which was measured by a 10KN load cell Prior to
6 Mathematical Problems in Engineering
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
001Hz
(a)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
005Hz
(b)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
01Hz
(c)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
05Hz
(d)
Figure 7 Hysteresis loops of the RSMAD at different loading frequencies and displacement amplitudes (06mm predisplacement 20∘Ctemperature)
testing the nitinol SMA specimens were cycled 20 timesat 6 strain amplitude and 12 times 10minus3 sminus1 strain rate bya ldquotrainingrdquo process to reach a steady-state condition Theexperimental data in Figure 2 are the results of cyclic tests onthe superelastic nitinol wire at 12 times 10minus3 sminus1 strain rate with1 to 8 strain levels As shown in Figure 2 the hysteresisloops based on the improved Graesser and Cozzarelli modeland experimental data match with close accuracy Moreoverthe improved model can accurately reflect the martensitichardening characteristic of SMAs under large amplitudes
3 An Innovative SMA Damper DesignExperiment and Numerical Simulation
31 Recentering SMADamper Design By utilizing the energydissipating and recentering features of superelastic nitinolSMA an innovative damper is designed As shown inFigure 3 the damper consists of outer and inner cylinders leftand right pull plates superelastic SMA wires retaining plate
prestrain adjusting plate adjusting bolt fixed bolt push-pullrod grip end caps and connecting fitting In this dampersuperelastic nitinolwires are the key components that provideboth damping and self-centering abilities
The damper will be connected to a structure via its push-pull rod and the connecting fitting The prestrain of thesuperelastic wires can be adjusted by the prestrain adjustingplate and the adjusting bolt The configuration of the dampershown in Figure 3 is in its equilibrium position The specificdesign of the damper allows the push-pull rod to move inboth left and right directions and return to its equilibriumposition when the load is removed (self-centering) Duringthis cyclic process the damper provides damping attributedto the hysteretic property of the superelastic wires
32 Experimental Tests
321 Setup and Program An SMA damper based on thedesign presented in Section 31 is fabricated To assess theperformance of the SMA damper cyclic tests were carried
Mathematical Problems in Engineering 7
001563 00625 025 1 40
300
600
900
1200
1500
1800
2100
2400
Log2 frequency (Hz)
Ener
gy d
issip
atio
n pe
r cyc
le (1
0minus3
J)
(a)
0
100
200
300
400
500
600
700
800
Resto
ring
forc
e (N
)
001563 00625 025 1 4Log2 frequency (Hz)
(b)
0
30
60
90
120
150
180
210
240
001563 00625 025 1 4Log2 frequency (Hz)
230mm345mm
460mm575mm
Seca
nt st
iffne
ss (N
mm
minus1)
(c)
000
002
004
006
008
010
012
014
016
Equi
vale
nt d
ampi
ng
001563 00625 025 1 4Log2 frequency (Hz)
230mm345mm
460mm575mm
(d)
Figure 8 Mechanical properties of RSMAD as a function of loading frequencies and displacement amplitudes (06mm predisplacement20∘C temperature)
out The SMA damper is 210mm in length with a diameterof 100mm and a stroke of plusmn92mm (the maximum allowablestrain for superelastic nitinol wires is 8) Four superelasticnitinol wires of a length of 115mm and a diameter of 05mmare used
Tests were conducted using an MTS 810 machine with a100KN load cell at room temperature of 20∘C The layout ofthe test system is shown in Figure 4 The damper was testedwith different prestrains at different loading frequenciesand at various amplitudes During the tests both force anddisplacement are recorded
Prior to the installation each nitinol wire was cycled 20times at 6 strain with 12 times 10minus3 sminus1 strain rate to minimizethe accumulation of residual strain and reach a steady-statecondition The scheme of the tests is described as follows
(1) Without prestrain the damper was subjected tocyclic loading at 005Hz frequencywith displacement
amplitudes of 23mm (2 of total length) 345mm(3 of total length) and 46mm (4 of total length)respectively
(2) Step (1) was repeated with prestrains of 1 (115mmpredisplacement) 2 (23mm predisplacement) and4 (46mm predisplacement) respectively
(3) With 05 prestrain (about 06mm predisplace-ment) the damper was subjected to cyclic loading at001Hz loading frequency with displacement ampli-tudes of 23mm 345mm 46mm and 575mmrespectively
(4) Step (3) was repeated with loading frequencies of005Hz 01 Hz 05Hz 1Hz and 2Hz respectively
To describe the performance of RSMAD as a functionof prestrains loading frequencies and displacement ampli-tudes some important mechanical properties were calcu-lated including the secant stiffness119870119904 the energy dissipation
8 Mathematical Problems in Engineering
0 2 4 6 8
0
100
200
300
400
500
Displacement (mm)
Resto
ring
forc
e (N
)
Experiment Numerical results
minus2minus4minus6minus8
minus500
minus400
minus300
minus200
minus100
Figure 9 Comparison between experimental and numerical fittingcurves of RSMAD
per cycle 119882119863 the restoring force at peak displacement percycle119865119889 and the equivalent viscous damping ratio 120585eq whichis expressed as
120585eq =119882119863
21205871198701199041205752 (10)
where 120575 is the displacement amplitude of the cycle
322 Experimental Results Figure 5 shows the hysteresisloops of RSMAD at different prestrains and displacementamplitudes The tests were carried out at 005Hz frequencyof loading and room temperature of 20∘C As seen fromFigure 5 the SMA damper shows double-flag hystereticbehavior without any residual strain
Figure 6 shows themechanical properties of RSMAD as afunction of prestrain and displacement amplitude As we cansee in Figure 6 the greater the amplitude is the greater theenergy dissipation per cycle and restoring force are Howeverthe secant stiffness reduces markedly while increasing theamplitude With the increasing of the predisplacement theenergy dissipation per cycle decreases and restoring forceand the secant stiffness increase at large displacement Theeffectiveness of a damper is generally measured by theequivalent viscous damping ratio With the increasing of thepredisplacement the equivalent damping ratio reduces Themost important finding is that when the predisplacementis less than 23mm (2 prestrain) the equivalent dampingratio increasedwith the increasing of the amplitude howeverwhen the predisplacement is 46mm (4 prestrain) themaximum of the equivalent damping ratio is at 23mm (2prestrain) This is because the maximum of the equivalentdamping ratio of theNiTi SMA specimens is at about 6 totalstrain as noted in [10]
Figure 7 shows the hysteresis loops of RSMAD with06mm predisplacement at different loading frequencies anddisplacement amplitudes at the temperature of 20∘C Figure 8
shows mechanical properties of RSMAD as a function offrequency of loading and displacement amplitude As wecan see in Figures 7 and 8 the energy dissipation percycle and the equivalent damping decrease as the loadingfrequency increases in the range of 001ndash05Hz but arenot much sensitive to frequencies greater than 05Hz Therestoring force and the secant stiffness increase slightly as theloading frequency increases in the total range of experimentalfrequency
33 Numerical Simulation Based on the improved Graesserand Cozzarelli model of SMA wire a theoretic model of theSMA damper is developed The differential equations of themodel are given as
+ 119891119872[119909 minus 119909Mf sgn (119909)]119898
times [119906 (119909)] [119906 (|119909| minus 119909Mf)]
(11)
where 119865 is restoring force 119909 is displacement 119861 is back-force1198700 is initial stiffness 119861119888 120572119891119879 119899 119886 119888 119909Mf119891119872 and 119898 and areconstants controlling the size of the hysteresis loop 119909in is theinelastic displacement and 119909in = 119909 minus 1198651198700 erf(119909) 119906(119909) andsgn(119909) are respectively the error function the step functionand the signum function which have already been listed inSection 2
Figure 9 shows the comparison of experimental resultswith numerical prediction based on the theoretical modelat different displacement amplitudes The parameters of theconstitutive equation used in this study to simulate the behav-iors of RSMAD are given as follows 1198700 = 380Nmm 119861119888 =330N 120572 = 005 119891119879 = 21 119888 = 00001 119886 = 3 119899 = 2119909Mf = 575mm 119891119872 = 42500 and 119898 = 3 To accuratelysimulate the hysteresis behavior the above parameters weredirectly obtained from the cyclic test results of the SMAdamper according to the parametersrsquo meanings and rolesThevalues also can be converted through the section area andthe length of NiTi wires However there is slight differencebetween the two parameter groups This is possible becauseof the effect of loading conditions on the cyclic behavior ofNiTi wires As can be seen in Figure 9 numerical predictionsagree well with the experimental results
Table 1 shows the comparison of the experimental dataand numerical results respectively of the energy dissipationper cycle the secant stiffness and the equivalent viscousdamping As can be seen in Table 1 the maximum differencesof energy dissipation per cycle secant stiffness and equiva-lent viscous damping are 50 12 and 51 respectivelyThese results indicate that the mechanical behavior of theSMAdamper is well predicted by the numericalmodel whichverifies its suitability for the damper
Mathematical Problems in Engineering 9
(a) Bare structure (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5
Figure 10 Different cases of the ten-story frame structure
Table 1 Comparison between the experimental and theoretical results
Peakdisplacement(mm)
Energy dissipation per cycle (Nsdotmm) Secant stiffness (Nmm) Equivalent viscous damping ()Experimental
Themain objective of including energy dissipating devices ina structure is to reduce structural response through energydissipation To protect the original structural members understrong seismic loading nonlinear deformation in energydissipating devices will be allowed In frame structures thedevices are usually incorporated in steel braces connectingtwo consecutive stories of the buildingThedynamic responseof the structure subjected to earthquake loading is governedby the following equation
where 119872119904 is the mass matrix 119862119904 is the damping coefficientmatrix 119865119904 is the vector of the frames restoring force and 119865119889is the vector of restoring force resulting from SMA dampersThe 119909 and are the structural displacement velocity andacceleration vectors respectively On the right-hand side of
the equation the vector 119868 is the influence vector and thevector 119892 is the ground motion acceleration input
With the aid of the SIMULINK module of MATLAB adynamical simulation system was developed in which theWen model [36] is utilized to simulate the restoring forcecurve of the steel frame structure and the improved Graesserand Cozzarelli model presented above is used for NiTi SMAdamper
5 Seismic Structural Control UsingSMA Dampers
In this section to assess the effectiveness of the proposedrecentering SMA dampers (RSMAD) in mitigating the seis-mic response of building structure nonlinear time historyanalysis on a multistory steel frame with and withoutthe dampers subjected to representative earthquake groundmotions was performed The improved Graesser and Coz-zarelli model for RSMAD given in Section 33 was employedin this numerical study A ten-story steel moment resisting
10 Mathematical Problems in Engineering
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0
10
20 40 60 80 100 120 140 1600123456789
Lateral displacement (mm)
Floo
r lev
elEl Centro
TaftS69E
Case 0Case 5Case 4
Case 3Case 2Case 1
PGA = 02 g
PGA = 02 g
PGA = 02 gTangshan-Beijing
(a)
0 5 10 15 20 25 30
000002004006008010
Disp
lace
men
t (m
)
Time (s)
0 5 10 15 20 25 30Time (s)
0 5 10 15 20Time (s)
Without controlWith case 5
El Centro
000002004006008010012014
Disp
lace
men
t (m
)
Taft S69E
Disp
lace
men
t (m
)
PGA = 02 g
PGA = 02 g
PGA = 02 g
minus010
minus008
minus006
minus004
minus002
minus014
minus012
minus010
minus008
minus006
minus004
minus002
000002004006008010012014016
minus014
minus016
minus012
minus010
minus008
minus006
minus004
minus002
Tangshan-Beijing
(b)
Figure 11 Lateral displacement envelopes (a) and roof displacement timehistories (b)with andwithout SMAdampers under basic (moderate)ground motions
frame structure was selected for this study The building isdesigned for a location in Beijing China The parameters ofthe structure are listed in Table 2
Three representative earthquake ground motions wereused to considering the site effect Three earthquake groundmotions namely Imperial Valley 1940 El Centro LincolnSchool 1952 Taft and Beijing Hotel 1976 Tangshan wereselected These three ground motions represent different siteconditions According to Chinese code for seismic design
of building [37] the local seismic precautionary intensity iseight degrees The peak ground accelerations (PGA) wereadjusted to 02 g and 04 g corresponding to a seismic hazardlevel of 10 and 2 probability of exceedance in a 50-yearperiod respectively
Simulation analysis is conducted on the bare structureand on the structure with five or ten SMA dampers installedas shown in Figure 10 Parameters of the SMA damper arelisted as follows 1198700 = 119870119889 = 120 kNmm 119861119888 = 360KN
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 40 80 120 160 200 240 2800123456789
10
Lateral displacement (mm)
Floo
r lev
el
El CentroPGA = 04 g
Taft S69EPGA = 04 g
PGA = 04 g
Case 0Case 5Case 4
Case 3Case 2Case 1
Tangshan-Beijing
(a)
El Centro
0 5 10 15 20
0 5 10 15 20
25 30
000
005
010
015
020
Time (s)
Disp
lace
men
t (m
)
Taft S69E
000005010015020025
Disp
lace
men
t (m
)
Time (s)
PGA = 04 g
PGA = 04 g
PGA = 04 g
minus020
minus015
minus010
minus005
minus020
minus025
minus015
minus010
minus005
0 5 10 15 20 25 30
000
008004
012016020
Time (s)
Disp
lace
men
t (m
)
minus020
minus016
minus010
minus004
minus008
Without controlWith case 5
Tangshan-Beijing
(b)
Figure 12 Lateral displacement envelopes (a) and roof displacement time histories (b) with and without SMA dampers under strong (severe)ground motions
Figures 11 and 12 show the lateral displacement envelopeswith and without SMA dampers under basic groundmotions(02 g) and strong ground motions (04 g) respectively Aswe can see for most cases the lateral displacement of thestructure decreases remarkably with the introduction of theSMA dampers However the location and the number ofthe dampers have significant effects on the control resultsCase 5 in which dampers are installed in all stories is
the best for the overall structural vibration control For other4 configurations in which five dampers are installed indifferent stores cases 3 and 4 with dampers installed inalternate stories perform better than cases 1 and 2 withdampers placed in consecutive stories Moreover case 3 isbetter than case 4 since the first story is retrofitted Case 1is slightly worse due to the whiplash effect and case 2 isthe worst since the stiffness of the lower half is significantlygreater than that of the upper half of the structure Theroof displacement time histories with (case 5) and without
12 Mathematical Problems in Engineering
SMAdampers under basic groundmotions (02 g) and strongground motions (04 g) are also respectively provided inFigures 11 and 12 which confirm that the SMA damperssignificantly reduce the structural vibration
6 Concluding Remarks
This paper presents the results of a study on evaluating theefficacy of using an innovative SMA-based damper to reducethe seismic response of structures To describe the hysteresisbehavior of the SMA damper an improved Graesser andCozzarelli model was proposed and verified by the cyclictensile test on SMA wires
Cyclic tests on the SMA damper model utilizing foursuperelastic SMA wires with 05mm diameter with variousprestrains under different loading frequencies and displace-ment amplitudes were carried outThe results show satisfyinghysteresis properties including both recentering and energydissipating features under various conditions
A comparative study on nonlinear time history analysisof the seismic response of a ten-story steel frame with theSMA dampers was performed Five cases were considered forthe location and the number of the dampers in the storiesThe numerical analysis results indicate that the proposedSMA damper is capable of significantly reducing seismicresponse of structures which verifies its effectiveness asenergy dissipating device for structures However it is alsoindicated that the location and number have significanteffects on the results of the response
In future multiobjective optimization model will beproposed to obtain the number and the location of the SMAdampers and large scale shake table tests will be performed ona steel frame buildingwith SMAdampers to prove the efficacyof these dampers in dissipating seismic energy
Acknowledgments
This work was funded by National Science Foundation ofChina (no 51108426 and no 41104106) China PostdoctoralScience Foundation (no 20100471008) and Research Fundfor the Doctoral Program of Higher Education of China (no20104101120009) These supports are greatly appreciatedTheopinions expressed in this study are those of the authors anddo not necessarily reflect the views of the sponsor
References
[1] H N Li and L S Huo ldquoAdvances in structural control in civilengineering in Chinardquo Mathematical Problems in Engineeringvol 2010 Article ID 936081 23 pages 2010
[2] G Song N Ma and H N Li ldquoApplications of shape memoryalloys in civil structuresrdquo Engineering Structures vol 28 no 9pp 1266ndash1274 2006
[3] SEAOC Vision 2000 Committee Performance-Based SeismicEngineering Structural Engineering Association of CaliforniaSacramento Calif USA 1995
[4] ATC-40 Seismic Evaluation and Retrofit of Conctete BuildingsApplied Technology Council 1996
[5] FEMA 273 NEHRP Guidelines for Seismic Rehabilitation ofBuildings Federal Emergency Management Agency 1997
[6] Y Fujino T T Soong and B F Spencer Jr ldquoStructural controlbasic concepts and applicationsrdquo in Proceedings of the ASCEStructures Congress pp 15ndash18 Chicago Ill USA April 1996
[7] B F Spencer and S Nagarajaiah ldquoState of the art of structuralcontrolrdquo Journal of Structural Engineering vol 129 no 7 pp845ndash856 2003
[8] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering John Wiley amp Sons NewYork NY USA 1997
[9] F M Mazzolani ldquoPassive control technologies for seismic-resistant buildings in Europerdquo Progress in Structural Engineeringand Materials vol 3 no 3 pp 277ndash287 2001
[10] M Dolce and D Cardone ldquoMechanical behaviour of SMAelements for seismic applicationsmdashpart 2 austenite NiTi wiressubjected to tensionrdquo International Journal of Mechanical Sci-ences vol 43 no 11 pp 2657ndash2677 2001
[11] H N Li and X X Wu ldquoLimitations of height-to-width ratiofor base-isolated buildings under earthquakerdquo The StructuralDesign of Tall and Special Buildings vol 15 no 3 pp 277ndash2872006
[12] R Desroches J McCormick and M A Delemont ldquoCyclicproperties of superelastic shape memory alloy wires and barsrdquoJournal of Structural Engineering vol 130 no 1 pp 38ndash46 2004
[13] T W Duerig K N Melton D Stockel and C M Way-man Engineering Aspects of Shape Memory Alloys ButterworthHeinemann London UK 1990
[14] K Wilde P Gardoni and Y Fujino ldquoBase isolation systemwith shape memory alloy device for elevated highway bridgesrdquoEngineering Structures vol 22 no 3 pp 222ndash229 2000
[15] M Dolce D Cardone and R Marnetto ldquoImplementation andtesting of passive control devices based on shape memoryalloysrdquo Earthquake Engineering and Structural Dynamics vol29 no 7 pp 945ndash968 2000
[16] M Dolce D Cardone F C Ponzo and C Valente ldquoShakingtable tests on reinforced concrete frames without and withpassive control systemsrdquo Earthquake Engineering and StructuralDynamics vol 34 no 14 pp 1687ndash1717 2005
[17] M Indirli M G Castellano P Clemente and A MartellildquoDemo-application of shapememory alloy devices the rehabili-tation of the S Giorgio Church Bell-Towerrdquo in Smart Structuresand Materials 2001 Smart Systems for Bridges Structures andHighways vol 4330 of Proceedings of SPIE pp 262ndash272 New-port Beach Calif USA March 2001
[18] B Andrawes and R Desroches ldquoUnseating prevention for mul-tiple frame bridges using superelastic devicesrdquo Smart Materialsand Structures vol 14 no 3 pp S60ndashS67 2005
[19] R DesRoches and M Delemont ldquoSeismic retrofit of simplysupported bridges using shape memory alloysrdquo EngineeringStructures vol 24 no 3 pp 325ndash332 2002
[20] H Li M Liu and J Ou ldquoVibration mitigation of a stay cablewith one shape memory alloy damperrdquo Structural Control andHealth Monitoring vol 11 no 1 pp 21ndash36 2004
[21] Y Zhang and S Zhu ldquoA shape memory alloy-based reusablehysteretic damper for seismic hazard mitigationrdquo Smart Mate-rials and Structures vol 16 no 5 pp 1603ndash1613 2007
[22] J Ocel R DesRoches R T Leon et al ldquoSteel beam-columnconnections using shape memory alloysrdquo Journal of StructuralEngineering vol 130 no 5 pp 732ndash740 2004
Mathematical Problems in Engineering 13
[23] J McCormick R Desroches D Fugazza and F AuricchioldquoSeismic assessment of concentrically braced steel frames withshape memory alloy bracesrdquo Journal of Structural Engineeringvol 133 no 6 pp 862ndash870 2007
[24] YM Parulekar G R Reddy K K Vaze et al ldquoSeismic responseattenuation of structures using shape memory alloy dampersrdquoStructural Control and Health Monitoring vol 19 no 1 pp 102ndash119 2012
[25] K Tanaka ldquoA thermomechanical sketch of shape memoryeffect one-dimensional tensile behaviorrdquoResMechanica vol 18no 3 pp 251ndash263 1986
[26] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 1 no 2 pp207ndash234 1990
[27] L C Brinson ldquoOne-dimensional constitutive behavior of shapememory alloys thermomechanical derivation with non-con-stant material functions and redefined martensite internal vari-ablerdquo Journal of Intelligent Material Systems and Structures vol4 no 2 pp 229ndash242 1993
[28] F Falk ldquoModel free energy mechanics and thermodynamicsof shape memory alloysrdquo Acta Metallurgica vol 28 no 12 pp1773ndash1780 1980
[29] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashII study of the individual phenomenardquo Journal of theMechanicsand Physics of Solids vol 41 no 1 pp 19ndash33 1993
[30] J G Boyd andDC Lagoudas ldquoA thermodynamical constitutivemodel for shape memory materialsmdashpart I the monolithicshape memory alloyrdquo International Journal of Plasticity vol 12no 6 pp 805ndash842 1996
[31] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashI derivation of general relationsrdquo Journal of the Mechanics andPhysics of Solids vol 41 no 1 pp 1ndash17 1993
[32] H Qian H Li G Song and W Guo ldquoA constitutive modelfor superelastic shape memory alloys considering the influenceof strain raterdquoMathematical Problems in Engineering vol 2013Article ID 248671 8 pages 2013
[33] W J Ren H N Li and G Song ldquoA one-dimensional strain-ratedependent constitutive model for superelastic shape memoryalloysrdquo Smart Materials and Structures vol 16 no 1 pp 191ndash1972007
[34] E J Graesser and F A Cozzarelli ldquoShape-memory alloys asnew materials for aseismic isolationrdquo Journal of EngineeringMechanics vol 117 no 11 pp 2590ndash2608 1991
[35] H Ozdemir Nonlinear transient dynamic analysis of yieldingstructures [PhD thesis] University of California BerkeleyCalif USA 1976
[36] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976
[37] GB 50011-2010 Code for Seismic Design of Buildings Ministryof Housing and Urban-Rural Development of the PeoplersquosRepublic of China 2010
Figure 1 Stress-strain diagrams of NiTi shape memory alloy (a) SME 119879 lt 119872119891 (b) superelasticity 119860119891 lt 119879 lt 119872119889
0
100
200
300
400
500
600
700
800
Experimental data The improved Graesser and Cozzarelli
000 001 002 003 004 005 006 007 008 009
Stre
ss120590
(MPa
)
Strain 120576
Figure 2 Stress-strain curves of superelastic nitinol wires
Shape memory alloys [13] are a class of novel functionalmaterials that possess unique properties including shapememory effect (SME) superelasticity effect (SE) extraor-dinary fatigue resistance high corrosion resistance highdamping characteristics and temperature-dependentYoungrsquosmodulus At a low temperature SMAs exhibit the SMEmdashresidual deformations can be recovered by heating the mate-rial above the austenite finish temperature as shown inFigure 1(a) At a higher temperature SMAs exhibit the SEas shown in Figure 1(b) In the superelastic phase SMAs areinitially austenitic However upon loading stress-inducedmartensite is formed Upon unloading themartensite revertsto austenite at a lower stress level resulting in the hystereticbehavior These properties make them ideal candidates forseismic energy dissipation devices in structural engineering
A significant number of research studies have beenconducted in an effort to use SMAs for applications in seismicresistant design and structural retrofit in the past decade(eg [14ndash24]) Wilde et al [14] proposed a smart isolationsystem combining a laminated rubber bearing with a devicemade of SMA for highway bridges Their analytical resultsshow that the isolation system can limit displacement anddissipate energy for earthquake mitigation Dolce et al [15]developed two families of SMA-based energy dissipating andrecentering braces for seismic vibration control of buildingsand bridges as outcomes of the MANSIDE project (MemoryAlloys for New Seismic Isolation and Energy DissipationDevices) To assess the effectiveness of SMA braces toreduce the seismic response of reinforced concrete (RC)framed structures shaking table tests of a 133-scale three-story two-bay RC plane frame which was designed forlow seismicity and low ductility according to the Europeanseismic code were carried out by Dolce et al [16] Theirexperimental results show that the SMA braces can enhanceseismic performances at least comparable to those providedby steel braces while having an additional self-centeringfeature Indirli et al [17] retrofitted historic buildings usingsuperelastic SMA tie bars to enhance its seismic resistancecapacities DesRoches and Delemont [18] and Andrawesand Desroches [19] evaluated the efficacy of superelasticnitinol bars as restrainers to reduce the risk of collapsefrom unseating of bridge superstructures at the hinges Theyfound that the SMA bars are effective in limiting relativehinge displacements in typical multiple frame bridges Liet al [20] investigated the vibration mitigation of a staycable provided with one superelastic SMA damper Theanalytical results show that the SMA damper can reducecable vibration significantly and the control effectivenessis influenced by SMA damper parameters and locationsZhang and Zhu [21] developed reusable hysteretic damper(RHD) based on superelastic nitinol stranded wires andtheir numerical analysis indicated the effectiveness of RHD
Mathematical Problems in Engineering 3
Left pull platePrestrain adjusting plate
Connecting fitting
End cap
Adjusting bolt
Fixed bolt Retaining plate
End capGrips
Superelastic SMA wires
Inner cylinderOuter cylinder
Push-pull rodRight pull plate
Figure 3 Diagram of RSMAD
Figure 4 Photograph of experimental setup
in passive seismic response control of structures Ocel et al[22]and McCormick et al [23] investigated beam-columnconnections using SMA bars Their studies show that SMAcan enhance the seismic performance of the connectionsParulekar et al [24] proposed a damper device using austeniteNiTi wiresThe device is tested and validated using a thermo-mechanical model of SMA taking into account the residualmartensite accumulation Performance of the structure withSMA dampers is compared with that of the same structurewith yielding dampers
Previous works show that SMAs are of promise in struc-tural engineering particularly as energy dissipation compo-nents for seismic protection However significant researchis still needed In Particular designing new types of SMA-based passive seismic devices which possess not only energydissipation and recentering capabilities but also simple con-figuration for easy installation in the practical engineering isstill a challenge
This paper presents an innovative recentering shapememory alloy damper (RSMAD) for seismic structural pro-tection The damper is simple in design and easy to imple-ment Superelastic nitinol wires were utilized in the damperas kernel energy dissipating components Cyclic tensile-compressive tests on the dampermodelwith various prestrainunder different loading frequencies and displacements wereconducted To assess the effectiveness of the damper for
structural seismic protection nonlinear time history analyseson a ten-story steel frame subjected to representative earth-quake ground motions with and without the dampers wereperformed
2 Constitutive Equation ofSuperelastic SMA Wire
With the wide applications of SMAs in different fields mod-eling of the peculiar mechanical behavior of SMAs such asSME and superelasticity has been an active area of researchover the past decades So far many constitutive models forSMA have been developed [25ndash33] These models describethe thermomechanical thermoelectrical and thermochem-ical behaviors of SMAs however most of them are too com-plex to be convenient for practical application in earthquakeengineering In this paper a relatively simple model devel-oped by Graesser and Cozzarelli [34] is adopted This modelis an extension of a one-dimensional strain rate independentmodel for hysteretic behavior proposed by Ozdemir [35]Theequation is given as
= 119864 [ 120576 minus | 120576| (120590 minus 120573
119884)
119899
] (1)
where 120590 and 120576 are the one-dimensional stress and the one-dimensional strain respectively 119864 is the initial elastic modu-lus 119884 is the yield stress 119899 is a constant assumed any positiveodd real value controlling the sharpness of transition from theelastic state to the phase transformation and 120576 denote thetime derivative of the stress and strain respectively and 120573 isthe one-dimensional back stress given by
where 119891119879 119886 and 119888 are material constants controlling the typeand size of the hysteresis the amount of elastic recovery dur-ing unloading and the slope of the unloading stress plateaurespectively When 119891119879 = 0 the model is purely martensiticWhen 119891119879 gt 0 the model predicts the superelastic behavior120572 is a constant controlling the slope of the 120590 minus 120576 curve in theinelastic range given by
120572 =
119864119910
119864 minus 119864119910
(3)
where 119864119910 is the slope of the 120590minus 120576 curve in the inelastic range
4 Mathematical Problems in Engineering
0
0
200
400
600
2 4 6Displacement (mm)
Resto
ring
forc
e (N
)
0
minus600
minus400
minus200
minus2minus4minus6
(a)
1
0
0
200
400
600
2 4 6Displacement (mm)
Resto
ring
forc
e (N
)
minus600
minus400
minus200
minus2minus4minus6
(b)
2
0
0
200
400
600
2 4 6Displacement (mm)
Resto
ring
forc
e (N
)
minus600
minus400
minus200
minus2minus4minus6
(c)
4
0
0
200
400
600
2 4 6Displacement (mm)
Resto
ring
forc
e (N
)
minus600
minus400
minus200
minus2minus4minus6
(d)
Figure 5 Hysteresis loops of the RSMAD at different prestrains and displacement amplitudes (005Hz frequency of loading 20∘Ctemperature)
120576in is the inelastic strain given by
120576in = 120576 minus120590
119864 (4)
119906() is the unit step function defined as
119906 (119909) = +1 119909 ge 0
0 119909 lt 0(5)
erf() is the error function defined by
erf (119909) = 2
radic120587int
120587
0119890minus1199052119889119905 (6)
The original Graesser and Cozzarelli model has a rel-atively simple expression with the parameters that can beeasily acquired however this model excludes the martensitichardening characteristics of SMAs under large amplitudeswhich are critical for structural safety protection underextreme events
To overcome the limitation of the original model Wildeet al [14] extended the Graesser and Cozzarelli model bydividing the full loop into four parts adding two termswith six parameters into (1) The Wilde model was utilizedto simulate the cyclic behaviors of SMA devices in otherresearches [21]
In the following in order to accurately predict the cyclicbehavior of a superelastic SMA device especially capture themartensitic hardening characteristics of SMAs under largeamplitudes an improved Graesser and Cozzarelli model ispresented In the present model the backstress expression ismodified by adding a special term to capture the martensitichardening characteristic of SMA under large amplitudesThemodified model is of the form
= 119864 [ 120576 minus | 120576| (120590 minus 120573
Figure 6 Mechanical properties of RSMAD as a function of prestrain and displacement amplitude (005Hz frequency of loading 20∘Ctemperature)
+ 119891119872[120576 minus 120576Mf sgn (120576)]119898[119906 (120576 120576)]
times [119906 (|120576| minus 120576Mf) ]
(8)
The third term in (2) is used to contribute to the back-stress on the ascending branch of the hysteresis in a way thatallows for the martensitic hardening 120576Mf is the martensitefinish strain 119891119872 and119898 are material constants controlling themartensitic hardening curve sgn(119909) is the signum functiongiven by
sgn (119909) =
+1 119909 gt 0
0 119909 = 0
minus1 119909 lt 0
(9)
Figure 2 shows the stress-strain curves of superelasticnitinol wires predicted by the improved Graesser and Coz-zarelli model versus experimental data at different strain
levels The characteristic parameters used in the models are119864 = 39500MPa 119884 = 385MPa 120572 = 001 119891119879 = 114119888 = 0001 119886 = 550 119899 = 3 120576Mf = 005 119891119872 = 42500and 119898 = 3 The superelastic nitinol wires are 05mm indiameter with a composition of approximately 509 Ni and491 Ti Under zero external stress the martensite startand finish temperatures (119872119891119872119904) and the austenite startand finish temperatures (119860 119904 119860119891) measured by differentialscanning calorimeter (DSC) are minus73∘C minus55∘C minus23∘C and5∘C respectively The uniaxial tension test of the superelasticnitinol wires was carried out using an electromechanicaluniversal testing machine at room temperature of 20∘C Thenitinol wire samples with a 100mm test length between thetwo custom-made grips were subjected to triangular cyclicloading under different strain amplitudes The strains werecalculated from the elongation measured by a 50mm gagelength extensometer with the stress calculated from the axialforce which was measured by a 10KN load cell Prior to
6 Mathematical Problems in Engineering
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
001Hz
(a)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
005Hz
(b)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
01Hz
(c)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
05Hz
(d)
Figure 7 Hysteresis loops of the RSMAD at different loading frequencies and displacement amplitudes (06mm predisplacement 20∘Ctemperature)
testing the nitinol SMA specimens were cycled 20 timesat 6 strain amplitude and 12 times 10minus3 sminus1 strain rate bya ldquotrainingrdquo process to reach a steady-state condition Theexperimental data in Figure 2 are the results of cyclic tests onthe superelastic nitinol wire at 12 times 10minus3 sminus1 strain rate with1 to 8 strain levels As shown in Figure 2 the hysteresisloops based on the improved Graesser and Cozzarelli modeland experimental data match with close accuracy Moreoverthe improved model can accurately reflect the martensitichardening characteristic of SMAs under large amplitudes
3 An Innovative SMA Damper DesignExperiment and Numerical Simulation
31 Recentering SMADamper Design By utilizing the energydissipating and recentering features of superelastic nitinolSMA an innovative damper is designed As shown inFigure 3 the damper consists of outer and inner cylinders leftand right pull plates superelastic SMA wires retaining plate
prestrain adjusting plate adjusting bolt fixed bolt push-pullrod grip end caps and connecting fitting In this dampersuperelastic nitinolwires are the key components that provideboth damping and self-centering abilities
The damper will be connected to a structure via its push-pull rod and the connecting fitting The prestrain of thesuperelastic wires can be adjusted by the prestrain adjustingplate and the adjusting bolt The configuration of the dampershown in Figure 3 is in its equilibrium position The specificdesign of the damper allows the push-pull rod to move inboth left and right directions and return to its equilibriumposition when the load is removed (self-centering) Duringthis cyclic process the damper provides damping attributedto the hysteretic property of the superelastic wires
32 Experimental Tests
321 Setup and Program An SMA damper based on thedesign presented in Section 31 is fabricated To assess theperformance of the SMA damper cyclic tests were carried
Mathematical Problems in Engineering 7
001563 00625 025 1 40
300
600
900
1200
1500
1800
2100
2400
Log2 frequency (Hz)
Ener
gy d
issip
atio
n pe
r cyc
le (1
0minus3
J)
(a)
0
100
200
300
400
500
600
700
800
Resto
ring
forc
e (N
)
001563 00625 025 1 4Log2 frequency (Hz)
(b)
0
30
60
90
120
150
180
210
240
001563 00625 025 1 4Log2 frequency (Hz)
230mm345mm
460mm575mm
Seca
nt st
iffne
ss (N
mm
minus1)
(c)
000
002
004
006
008
010
012
014
016
Equi
vale
nt d
ampi
ng
001563 00625 025 1 4Log2 frequency (Hz)
230mm345mm
460mm575mm
(d)
Figure 8 Mechanical properties of RSMAD as a function of loading frequencies and displacement amplitudes (06mm predisplacement20∘C temperature)
out The SMA damper is 210mm in length with a diameterof 100mm and a stroke of plusmn92mm (the maximum allowablestrain for superelastic nitinol wires is 8) Four superelasticnitinol wires of a length of 115mm and a diameter of 05mmare used
Tests were conducted using an MTS 810 machine with a100KN load cell at room temperature of 20∘C The layout ofthe test system is shown in Figure 4 The damper was testedwith different prestrains at different loading frequenciesand at various amplitudes During the tests both force anddisplacement are recorded
Prior to the installation each nitinol wire was cycled 20times at 6 strain with 12 times 10minus3 sminus1 strain rate to minimizethe accumulation of residual strain and reach a steady-statecondition The scheme of the tests is described as follows
(1) Without prestrain the damper was subjected tocyclic loading at 005Hz frequencywith displacement
amplitudes of 23mm (2 of total length) 345mm(3 of total length) and 46mm (4 of total length)respectively
(2) Step (1) was repeated with prestrains of 1 (115mmpredisplacement) 2 (23mm predisplacement) and4 (46mm predisplacement) respectively
(3) With 05 prestrain (about 06mm predisplace-ment) the damper was subjected to cyclic loading at001Hz loading frequency with displacement ampli-tudes of 23mm 345mm 46mm and 575mmrespectively
(4) Step (3) was repeated with loading frequencies of005Hz 01 Hz 05Hz 1Hz and 2Hz respectively
To describe the performance of RSMAD as a functionof prestrains loading frequencies and displacement ampli-tudes some important mechanical properties were calcu-lated including the secant stiffness119870119904 the energy dissipation
8 Mathematical Problems in Engineering
0 2 4 6 8
0
100
200
300
400
500
Displacement (mm)
Resto
ring
forc
e (N
)
Experiment Numerical results
minus2minus4minus6minus8
minus500
minus400
minus300
minus200
minus100
Figure 9 Comparison between experimental and numerical fittingcurves of RSMAD
per cycle 119882119863 the restoring force at peak displacement percycle119865119889 and the equivalent viscous damping ratio 120585eq whichis expressed as
120585eq =119882119863
21205871198701199041205752 (10)
where 120575 is the displacement amplitude of the cycle
322 Experimental Results Figure 5 shows the hysteresisloops of RSMAD at different prestrains and displacementamplitudes The tests were carried out at 005Hz frequencyof loading and room temperature of 20∘C As seen fromFigure 5 the SMA damper shows double-flag hystereticbehavior without any residual strain
Figure 6 shows themechanical properties of RSMAD as afunction of prestrain and displacement amplitude As we cansee in Figure 6 the greater the amplitude is the greater theenergy dissipation per cycle and restoring force are Howeverthe secant stiffness reduces markedly while increasing theamplitude With the increasing of the predisplacement theenergy dissipation per cycle decreases and restoring forceand the secant stiffness increase at large displacement Theeffectiveness of a damper is generally measured by theequivalent viscous damping ratio With the increasing of thepredisplacement the equivalent damping ratio reduces Themost important finding is that when the predisplacementis less than 23mm (2 prestrain) the equivalent dampingratio increasedwith the increasing of the amplitude howeverwhen the predisplacement is 46mm (4 prestrain) themaximum of the equivalent damping ratio is at 23mm (2prestrain) This is because the maximum of the equivalentdamping ratio of theNiTi SMA specimens is at about 6 totalstrain as noted in [10]
Figure 7 shows the hysteresis loops of RSMAD with06mm predisplacement at different loading frequencies anddisplacement amplitudes at the temperature of 20∘C Figure 8
shows mechanical properties of RSMAD as a function offrequency of loading and displacement amplitude As wecan see in Figures 7 and 8 the energy dissipation percycle and the equivalent damping decrease as the loadingfrequency increases in the range of 001ndash05Hz but arenot much sensitive to frequencies greater than 05Hz Therestoring force and the secant stiffness increase slightly as theloading frequency increases in the total range of experimentalfrequency
33 Numerical Simulation Based on the improved Graesserand Cozzarelli model of SMA wire a theoretic model of theSMA damper is developed The differential equations of themodel are given as
+ 119891119872[119909 minus 119909Mf sgn (119909)]119898
times [119906 (119909)] [119906 (|119909| minus 119909Mf)]
(11)
where 119865 is restoring force 119909 is displacement 119861 is back-force1198700 is initial stiffness 119861119888 120572119891119879 119899 119886 119888 119909Mf119891119872 and 119898 and areconstants controlling the size of the hysteresis loop 119909in is theinelastic displacement and 119909in = 119909 minus 1198651198700 erf(119909) 119906(119909) andsgn(119909) are respectively the error function the step functionand the signum function which have already been listed inSection 2
Figure 9 shows the comparison of experimental resultswith numerical prediction based on the theoretical modelat different displacement amplitudes The parameters of theconstitutive equation used in this study to simulate the behav-iors of RSMAD are given as follows 1198700 = 380Nmm 119861119888 =330N 120572 = 005 119891119879 = 21 119888 = 00001 119886 = 3 119899 = 2119909Mf = 575mm 119891119872 = 42500 and 119898 = 3 To accuratelysimulate the hysteresis behavior the above parameters weredirectly obtained from the cyclic test results of the SMAdamper according to the parametersrsquo meanings and rolesThevalues also can be converted through the section area andthe length of NiTi wires However there is slight differencebetween the two parameter groups This is possible becauseof the effect of loading conditions on the cyclic behavior ofNiTi wires As can be seen in Figure 9 numerical predictionsagree well with the experimental results
Table 1 shows the comparison of the experimental dataand numerical results respectively of the energy dissipationper cycle the secant stiffness and the equivalent viscousdamping As can be seen in Table 1 the maximum differencesof energy dissipation per cycle secant stiffness and equiva-lent viscous damping are 50 12 and 51 respectivelyThese results indicate that the mechanical behavior of theSMAdamper is well predicted by the numericalmodel whichverifies its suitability for the damper
Mathematical Problems in Engineering 9
(a) Bare structure (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5
Figure 10 Different cases of the ten-story frame structure
Table 1 Comparison between the experimental and theoretical results
Peakdisplacement(mm)
Energy dissipation per cycle (Nsdotmm) Secant stiffness (Nmm) Equivalent viscous damping ()Experimental
Themain objective of including energy dissipating devices ina structure is to reduce structural response through energydissipation To protect the original structural members understrong seismic loading nonlinear deformation in energydissipating devices will be allowed In frame structures thedevices are usually incorporated in steel braces connectingtwo consecutive stories of the buildingThedynamic responseof the structure subjected to earthquake loading is governedby the following equation
where 119872119904 is the mass matrix 119862119904 is the damping coefficientmatrix 119865119904 is the vector of the frames restoring force and 119865119889is the vector of restoring force resulting from SMA dampersThe 119909 and are the structural displacement velocity andacceleration vectors respectively On the right-hand side of
the equation the vector 119868 is the influence vector and thevector 119892 is the ground motion acceleration input
With the aid of the SIMULINK module of MATLAB adynamical simulation system was developed in which theWen model [36] is utilized to simulate the restoring forcecurve of the steel frame structure and the improved Graesserand Cozzarelli model presented above is used for NiTi SMAdamper
5 Seismic Structural Control UsingSMA Dampers
In this section to assess the effectiveness of the proposedrecentering SMA dampers (RSMAD) in mitigating the seis-mic response of building structure nonlinear time historyanalysis on a multistory steel frame with and withoutthe dampers subjected to representative earthquake groundmotions was performed The improved Graesser and Coz-zarelli model for RSMAD given in Section 33 was employedin this numerical study A ten-story steel moment resisting
10 Mathematical Problems in Engineering
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0
10
20 40 60 80 100 120 140 1600123456789
Lateral displacement (mm)
Floo
r lev
elEl Centro
TaftS69E
Case 0Case 5Case 4
Case 3Case 2Case 1
PGA = 02 g
PGA = 02 g
PGA = 02 gTangshan-Beijing
(a)
0 5 10 15 20 25 30
000002004006008010
Disp
lace
men
t (m
)
Time (s)
0 5 10 15 20 25 30Time (s)
0 5 10 15 20Time (s)
Without controlWith case 5
El Centro
000002004006008010012014
Disp
lace
men
t (m
)
Taft S69E
Disp
lace
men
t (m
)
PGA = 02 g
PGA = 02 g
PGA = 02 g
minus010
minus008
minus006
minus004
minus002
minus014
minus012
minus010
minus008
minus006
minus004
minus002
000002004006008010012014016
minus014
minus016
minus012
minus010
minus008
minus006
minus004
minus002
Tangshan-Beijing
(b)
Figure 11 Lateral displacement envelopes (a) and roof displacement timehistories (b)with andwithout SMAdampers under basic (moderate)ground motions
frame structure was selected for this study The building isdesigned for a location in Beijing China The parameters ofthe structure are listed in Table 2
Three representative earthquake ground motions wereused to considering the site effect Three earthquake groundmotions namely Imperial Valley 1940 El Centro LincolnSchool 1952 Taft and Beijing Hotel 1976 Tangshan wereselected These three ground motions represent different siteconditions According to Chinese code for seismic design
of building [37] the local seismic precautionary intensity iseight degrees The peak ground accelerations (PGA) wereadjusted to 02 g and 04 g corresponding to a seismic hazardlevel of 10 and 2 probability of exceedance in a 50-yearperiod respectively
Simulation analysis is conducted on the bare structureand on the structure with five or ten SMA dampers installedas shown in Figure 10 Parameters of the SMA damper arelisted as follows 1198700 = 119870119889 = 120 kNmm 119861119888 = 360KN
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 40 80 120 160 200 240 2800123456789
10
Lateral displacement (mm)
Floo
r lev
el
El CentroPGA = 04 g
Taft S69EPGA = 04 g
PGA = 04 g
Case 0Case 5Case 4
Case 3Case 2Case 1
Tangshan-Beijing
(a)
El Centro
0 5 10 15 20
0 5 10 15 20
25 30
000
005
010
015
020
Time (s)
Disp
lace
men
t (m
)
Taft S69E
000005010015020025
Disp
lace
men
t (m
)
Time (s)
PGA = 04 g
PGA = 04 g
PGA = 04 g
minus020
minus015
minus010
minus005
minus020
minus025
minus015
minus010
minus005
0 5 10 15 20 25 30
000
008004
012016020
Time (s)
Disp
lace
men
t (m
)
minus020
minus016
minus010
minus004
minus008
Without controlWith case 5
Tangshan-Beijing
(b)
Figure 12 Lateral displacement envelopes (a) and roof displacement time histories (b) with and without SMA dampers under strong (severe)ground motions
Figures 11 and 12 show the lateral displacement envelopeswith and without SMA dampers under basic groundmotions(02 g) and strong ground motions (04 g) respectively Aswe can see for most cases the lateral displacement of thestructure decreases remarkably with the introduction of theSMA dampers However the location and the number ofthe dampers have significant effects on the control resultsCase 5 in which dampers are installed in all stories is
the best for the overall structural vibration control For other4 configurations in which five dampers are installed indifferent stores cases 3 and 4 with dampers installed inalternate stories perform better than cases 1 and 2 withdampers placed in consecutive stories Moreover case 3 isbetter than case 4 since the first story is retrofitted Case 1is slightly worse due to the whiplash effect and case 2 isthe worst since the stiffness of the lower half is significantlygreater than that of the upper half of the structure Theroof displacement time histories with (case 5) and without
12 Mathematical Problems in Engineering
SMAdampers under basic groundmotions (02 g) and strongground motions (04 g) are also respectively provided inFigures 11 and 12 which confirm that the SMA damperssignificantly reduce the structural vibration
6 Concluding Remarks
This paper presents the results of a study on evaluating theefficacy of using an innovative SMA-based damper to reducethe seismic response of structures To describe the hysteresisbehavior of the SMA damper an improved Graesser andCozzarelli model was proposed and verified by the cyclictensile test on SMA wires
Cyclic tests on the SMA damper model utilizing foursuperelastic SMA wires with 05mm diameter with variousprestrains under different loading frequencies and displace-ment amplitudes were carried outThe results show satisfyinghysteresis properties including both recentering and energydissipating features under various conditions
A comparative study on nonlinear time history analysisof the seismic response of a ten-story steel frame with theSMA dampers was performed Five cases were considered forthe location and the number of the dampers in the storiesThe numerical analysis results indicate that the proposedSMA damper is capable of significantly reducing seismicresponse of structures which verifies its effectiveness asenergy dissipating device for structures However it is alsoindicated that the location and number have significanteffects on the results of the response
In future multiobjective optimization model will beproposed to obtain the number and the location of the SMAdampers and large scale shake table tests will be performed ona steel frame buildingwith SMAdampers to prove the efficacyof these dampers in dissipating seismic energy
Acknowledgments
This work was funded by National Science Foundation ofChina (no 51108426 and no 41104106) China PostdoctoralScience Foundation (no 20100471008) and Research Fundfor the Doctoral Program of Higher Education of China (no20104101120009) These supports are greatly appreciatedTheopinions expressed in this study are those of the authors anddo not necessarily reflect the views of the sponsor
References
[1] H N Li and L S Huo ldquoAdvances in structural control in civilengineering in Chinardquo Mathematical Problems in Engineeringvol 2010 Article ID 936081 23 pages 2010
[2] G Song N Ma and H N Li ldquoApplications of shape memoryalloys in civil structuresrdquo Engineering Structures vol 28 no 9pp 1266ndash1274 2006
[3] SEAOC Vision 2000 Committee Performance-Based SeismicEngineering Structural Engineering Association of CaliforniaSacramento Calif USA 1995
[4] ATC-40 Seismic Evaluation and Retrofit of Conctete BuildingsApplied Technology Council 1996
[5] FEMA 273 NEHRP Guidelines for Seismic Rehabilitation ofBuildings Federal Emergency Management Agency 1997
[6] Y Fujino T T Soong and B F Spencer Jr ldquoStructural controlbasic concepts and applicationsrdquo in Proceedings of the ASCEStructures Congress pp 15ndash18 Chicago Ill USA April 1996
[7] B F Spencer and S Nagarajaiah ldquoState of the art of structuralcontrolrdquo Journal of Structural Engineering vol 129 no 7 pp845ndash856 2003
[8] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering John Wiley amp Sons NewYork NY USA 1997
[9] F M Mazzolani ldquoPassive control technologies for seismic-resistant buildings in Europerdquo Progress in Structural Engineeringand Materials vol 3 no 3 pp 277ndash287 2001
[10] M Dolce and D Cardone ldquoMechanical behaviour of SMAelements for seismic applicationsmdashpart 2 austenite NiTi wiressubjected to tensionrdquo International Journal of Mechanical Sci-ences vol 43 no 11 pp 2657ndash2677 2001
[11] H N Li and X X Wu ldquoLimitations of height-to-width ratiofor base-isolated buildings under earthquakerdquo The StructuralDesign of Tall and Special Buildings vol 15 no 3 pp 277ndash2872006
[12] R Desroches J McCormick and M A Delemont ldquoCyclicproperties of superelastic shape memory alloy wires and barsrdquoJournal of Structural Engineering vol 130 no 1 pp 38ndash46 2004
[13] T W Duerig K N Melton D Stockel and C M Way-man Engineering Aspects of Shape Memory Alloys ButterworthHeinemann London UK 1990
[14] K Wilde P Gardoni and Y Fujino ldquoBase isolation systemwith shape memory alloy device for elevated highway bridgesrdquoEngineering Structures vol 22 no 3 pp 222ndash229 2000
[15] M Dolce D Cardone and R Marnetto ldquoImplementation andtesting of passive control devices based on shape memoryalloysrdquo Earthquake Engineering and Structural Dynamics vol29 no 7 pp 945ndash968 2000
[16] M Dolce D Cardone F C Ponzo and C Valente ldquoShakingtable tests on reinforced concrete frames without and withpassive control systemsrdquo Earthquake Engineering and StructuralDynamics vol 34 no 14 pp 1687ndash1717 2005
[17] M Indirli M G Castellano P Clemente and A MartellildquoDemo-application of shapememory alloy devices the rehabili-tation of the S Giorgio Church Bell-Towerrdquo in Smart Structuresand Materials 2001 Smart Systems for Bridges Structures andHighways vol 4330 of Proceedings of SPIE pp 262ndash272 New-port Beach Calif USA March 2001
[18] B Andrawes and R Desroches ldquoUnseating prevention for mul-tiple frame bridges using superelastic devicesrdquo Smart Materialsand Structures vol 14 no 3 pp S60ndashS67 2005
[19] R DesRoches and M Delemont ldquoSeismic retrofit of simplysupported bridges using shape memory alloysrdquo EngineeringStructures vol 24 no 3 pp 325ndash332 2002
[20] H Li M Liu and J Ou ldquoVibration mitigation of a stay cablewith one shape memory alloy damperrdquo Structural Control andHealth Monitoring vol 11 no 1 pp 21ndash36 2004
[21] Y Zhang and S Zhu ldquoA shape memory alloy-based reusablehysteretic damper for seismic hazard mitigationrdquo Smart Mate-rials and Structures vol 16 no 5 pp 1603ndash1613 2007
[22] J Ocel R DesRoches R T Leon et al ldquoSteel beam-columnconnections using shape memory alloysrdquo Journal of StructuralEngineering vol 130 no 5 pp 732ndash740 2004
Mathematical Problems in Engineering 13
[23] J McCormick R Desroches D Fugazza and F AuricchioldquoSeismic assessment of concentrically braced steel frames withshape memory alloy bracesrdquo Journal of Structural Engineeringvol 133 no 6 pp 862ndash870 2007
[24] YM Parulekar G R Reddy K K Vaze et al ldquoSeismic responseattenuation of structures using shape memory alloy dampersrdquoStructural Control and Health Monitoring vol 19 no 1 pp 102ndash119 2012
[25] K Tanaka ldquoA thermomechanical sketch of shape memoryeffect one-dimensional tensile behaviorrdquoResMechanica vol 18no 3 pp 251ndash263 1986
[26] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 1 no 2 pp207ndash234 1990
[27] L C Brinson ldquoOne-dimensional constitutive behavior of shapememory alloys thermomechanical derivation with non-con-stant material functions and redefined martensite internal vari-ablerdquo Journal of Intelligent Material Systems and Structures vol4 no 2 pp 229ndash242 1993
[28] F Falk ldquoModel free energy mechanics and thermodynamicsof shape memory alloysrdquo Acta Metallurgica vol 28 no 12 pp1773ndash1780 1980
[29] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashII study of the individual phenomenardquo Journal of theMechanicsand Physics of Solids vol 41 no 1 pp 19ndash33 1993
[30] J G Boyd andDC Lagoudas ldquoA thermodynamical constitutivemodel for shape memory materialsmdashpart I the monolithicshape memory alloyrdquo International Journal of Plasticity vol 12no 6 pp 805ndash842 1996
[31] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashI derivation of general relationsrdquo Journal of the Mechanics andPhysics of Solids vol 41 no 1 pp 1ndash17 1993
[32] H Qian H Li G Song and W Guo ldquoA constitutive modelfor superelastic shape memory alloys considering the influenceof strain raterdquoMathematical Problems in Engineering vol 2013Article ID 248671 8 pages 2013
[33] W J Ren H N Li and G Song ldquoA one-dimensional strain-ratedependent constitutive model for superelastic shape memoryalloysrdquo Smart Materials and Structures vol 16 no 1 pp 191ndash1972007
[34] E J Graesser and F A Cozzarelli ldquoShape-memory alloys asnew materials for aseismic isolationrdquo Journal of EngineeringMechanics vol 117 no 11 pp 2590ndash2608 1991
[35] H Ozdemir Nonlinear transient dynamic analysis of yieldingstructures [PhD thesis] University of California BerkeleyCalif USA 1976
[36] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976
[37] GB 50011-2010 Code for Seismic Design of Buildings Ministryof Housing and Urban-Rural Development of the PeoplersquosRepublic of China 2010
in passive seismic response control of structures Ocel et al[22]and McCormick et al [23] investigated beam-columnconnections using SMA bars Their studies show that SMAcan enhance the seismic performance of the connectionsParulekar et al [24] proposed a damper device using austeniteNiTi wiresThe device is tested and validated using a thermo-mechanical model of SMA taking into account the residualmartensite accumulation Performance of the structure withSMA dampers is compared with that of the same structurewith yielding dampers
Previous works show that SMAs are of promise in struc-tural engineering particularly as energy dissipation compo-nents for seismic protection However significant researchis still needed In Particular designing new types of SMA-based passive seismic devices which possess not only energydissipation and recentering capabilities but also simple con-figuration for easy installation in the practical engineering isstill a challenge
This paper presents an innovative recentering shapememory alloy damper (RSMAD) for seismic structural pro-tection The damper is simple in design and easy to imple-ment Superelastic nitinol wires were utilized in the damperas kernel energy dissipating components Cyclic tensile-compressive tests on the dampermodelwith various prestrainunder different loading frequencies and displacements wereconducted To assess the effectiveness of the damper for
structural seismic protection nonlinear time history analyseson a ten-story steel frame subjected to representative earth-quake ground motions with and without the dampers wereperformed
2 Constitutive Equation ofSuperelastic SMA Wire
With the wide applications of SMAs in different fields mod-eling of the peculiar mechanical behavior of SMAs such asSME and superelasticity has been an active area of researchover the past decades So far many constitutive models forSMA have been developed [25ndash33] These models describethe thermomechanical thermoelectrical and thermochem-ical behaviors of SMAs however most of them are too com-plex to be convenient for practical application in earthquakeengineering In this paper a relatively simple model devel-oped by Graesser and Cozzarelli [34] is adopted This modelis an extension of a one-dimensional strain rate independentmodel for hysteretic behavior proposed by Ozdemir [35]Theequation is given as
= 119864 [ 120576 minus | 120576| (120590 minus 120573
119884)
119899
] (1)
where 120590 and 120576 are the one-dimensional stress and the one-dimensional strain respectively 119864 is the initial elastic modu-lus 119884 is the yield stress 119899 is a constant assumed any positiveodd real value controlling the sharpness of transition from theelastic state to the phase transformation and 120576 denote thetime derivative of the stress and strain respectively and 120573 isthe one-dimensional back stress given by
where 119891119879 119886 and 119888 are material constants controlling the typeand size of the hysteresis the amount of elastic recovery dur-ing unloading and the slope of the unloading stress plateaurespectively When 119891119879 = 0 the model is purely martensiticWhen 119891119879 gt 0 the model predicts the superelastic behavior120572 is a constant controlling the slope of the 120590 minus 120576 curve in theinelastic range given by
120572 =
119864119910
119864 minus 119864119910
(3)
where 119864119910 is the slope of the 120590minus 120576 curve in the inelastic range
4 Mathematical Problems in Engineering
0
0
200
400
600
2 4 6Displacement (mm)
Resto
ring
forc
e (N
)
0
minus600
minus400
minus200
minus2minus4minus6
(a)
1
0
0
200
400
600
2 4 6Displacement (mm)
Resto
ring
forc
e (N
)
minus600
minus400
minus200
minus2minus4minus6
(b)
2
0
0
200
400
600
2 4 6Displacement (mm)
Resto
ring
forc
e (N
)
minus600
minus400
minus200
minus2minus4minus6
(c)
4
0
0
200
400
600
2 4 6Displacement (mm)
Resto
ring
forc
e (N
)
minus600
minus400
minus200
minus2minus4minus6
(d)
Figure 5 Hysteresis loops of the RSMAD at different prestrains and displacement amplitudes (005Hz frequency of loading 20∘Ctemperature)
120576in is the inelastic strain given by
120576in = 120576 minus120590
119864 (4)
119906() is the unit step function defined as
119906 (119909) = +1 119909 ge 0
0 119909 lt 0(5)
erf() is the error function defined by
erf (119909) = 2
radic120587int
120587
0119890minus1199052119889119905 (6)
The original Graesser and Cozzarelli model has a rel-atively simple expression with the parameters that can beeasily acquired however this model excludes the martensitichardening characteristics of SMAs under large amplitudeswhich are critical for structural safety protection underextreme events
To overcome the limitation of the original model Wildeet al [14] extended the Graesser and Cozzarelli model bydividing the full loop into four parts adding two termswith six parameters into (1) The Wilde model was utilizedto simulate the cyclic behaviors of SMA devices in otherresearches [21]
In the following in order to accurately predict the cyclicbehavior of a superelastic SMA device especially capture themartensitic hardening characteristics of SMAs under largeamplitudes an improved Graesser and Cozzarelli model ispresented In the present model the backstress expression ismodified by adding a special term to capture the martensitichardening characteristic of SMA under large amplitudesThemodified model is of the form
= 119864 [ 120576 minus | 120576| (120590 minus 120573
Figure 6 Mechanical properties of RSMAD as a function of prestrain and displacement amplitude (005Hz frequency of loading 20∘Ctemperature)
+ 119891119872[120576 minus 120576Mf sgn (120576)]119898[119906 (120576 120576)]
times [119906 (|120576| minus 120576Mf) ]
(8)
The third term in (2) is used to contribute to the back-stress on the ascending branch of the hysteresis in a way thatallows for the martensitic hardening 120576Mf is the martensitefinish strain 119891119872 and119898 are material constants controlling themartensitic hardening curve sgn(119909) is the signum functiongiven by
sgn (119909) =
+1 119909 gt 0
0 119909 = 0
minus1 119909 lt 0
(9)
Figure 2 shows the stress-strain curves of superelasticnitinol wires predicted by the improved Graesser and Coz-zarelli model versus experimental data at different strain
levels The characteristic parameters used in the models are119864 = 39500MPa 119884 = 385MPa 120572 = 001 119891119879 = 114119888 = 0001 119886 = 550 119899 = 3 120576Mf = 005 119891119872 = 42500and 119898 = 3 The superelastic nitinol wires are 05mm indiameter with a composition of approximately 509 Ni and491 Ti Under zero external stress the martensite startand finish temperatures (119872119891119872119904) and the austenite startand finish temperatures (119860 119904 119860119891) measured by differentialscanning calorimeter (DSC) are minus73∘C minus55∘C minus23∘C and5∘C respectively The uniaxial tension test of the superelasticnitinol wires was carried out using an electromechanicaluniversal testing machine at room temperature of 20∘C Thenitinol wire samples with a 100mm test length between thetwo custom-made grips were subjected to triangular cyclicloading under different strain amplitudes The strains werecalculated from the elongation measured by a 50mm gagelength extensometer with the stress calculated from the axialforce which was measured by a 10KN load cell Prior to
6 Mathematical Problems in Engineering
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
001Hz
(a)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
005Hz
(b)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
01Hz
(c)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
05Hz
(d)
Figure 7 Hysteresis loops of the RSMAD at different loading frequencies and displacement amplitudes (06mm predisplacement 20∘Ctemperature)
testing the nitinol SMA specimens were cycled 20 timesat 6 strain amplitude and 12 times 10minus3 sminus1 strain rate bya ldquotrainingrdquo process to reach a steady-state condition Theexperimental data in Figure 2 are the results of cyclic tests onthe superelastic nitinol wire at 12 times 10minus3 sminus1 strain rate with1 to 8 strain levels As shown in Figure 2 the hysteresisloops based on the improved Graesser and Cozzarelli modeland experimental data match with close accuracy Moreoverthe improved model can accurately reflect the martensitichardening characteristic of SMAs under large amplitudes
3 An Innovative SMA Damper DesignExperiment and Numerical Simulation
31 Recentering SMADamper Design By utilizing the energydissipating and recentering features of superelastic nitinolSMA an innovative damper is designed As shown inFigure 3 the damper consists of outer and inner cylinders leftand right pull plates superelastic SMA wires retaining plate
prestrain adjusting plate adjusting bolt fixed bolt push-pullrod grip end caps and connecting fitting In this dampersuperelastic nitinolwires are the key components that provideboth damping and self-centering abilities
The damper will be connected to a structure via its push-pull rod and the connecting fitting The prestrain of thesuperelastic wires can be adjusted by the prestrain adjustingplate and the adjusting bolt The configuration of the dampershown in Figure 3 is in its equilibrium position The specificdesign of the damper allows the push-pull rod to move inboth left and right directions and return to its equilibriumposition when the load is removed (self-centering) Duringthis cyclic process the damper provides damping attributedto the hysteretic property of the superelastic wires
32 Experimental Tests
321 Setup and Program An SMA damper based on thedesign presented in Section 31 is fabricated To assess theperformance of the SMA damper cyclic tests were carried
Mathematical Problems in Engineering 7
001563 00625 025 1 40
300
600
900
1200
1500
1800
2100
2400
Log2 frequency (Hz)
Ener
gy d
issip
atio
n pe
r cyc
le (1
0minus3
J)
(a)
0
100
200
300
400
500
600
700
800
Resto
ring
forc
e (N
)
001563 00625 025 1 4Log2 frequency (Hz)
(b)
0
30
60
90
120
150
180
210
240
001563 00625 025 1 4Log2 frequency (Hz)
230mm345mm
460mm575mm
Seca
nt st
iffne
ss (N
mm
minus1)
(c)
000
002
004
006
008
010
012
014
016
Equi
vale
nt d
ampi
ng
001563 00625 025 1 4Log2 frequency (Hz)
230mm345mm
460mm575mm
(d)
Figure 8 Mechanical properties of RSMAD as a function of loading frequencies and displacement amplitudes (06mm predisplacement20∘C temperature)
out The SMA damper is 210mm in length with a diameterof 100mm and a stroke of plusmn92mm (the maximum allowablestrain for superelastic nitinol wires is 8) Four superelasticnitinol wires of a length of 115mm and a diameter of 05mmare used
Tests were conducted using an MTS 810 machine with a100KN load cell at room temperature of 20∘C The layout ofthe test system is shown in Figure 4 The damper was testedwith different prestrains at different loading frequenciesand at various amplitudes During the tests both force anddisplacement are recorded
Prior to the installation each nitinol wire was cycled 20times at 6 strain with 12 times 10minus3 sminus1 strain rate to minimizethe accumulation of residual strain and reach a steady-statecondition The scheme of the tests is described as follows
(1) Without prestrain the damper was subjected tocyclic loading at 005Hz frequencywith displacement
amplitudes of 23mm (2 of total length) 345mm(3 of total length) and 46mm (4 of total length)respectively
(2) Step (1) was repeated with prestrains of 1 (115mmpredisplacement) 2 (23mm predisplacement) and4 (46mm predisplacement) respectively
(3) With 05 prestrain (about 06mm predisplace-ment) the damper was subjected to cyclic loading at001Hz loading frequency with displacement ampli-tudes of 23mm 345mm 46mm and 575mmrespectively
(4) Step (3) was repeated with loading frequencies of005Hz 01 Hz 05Hz 1Hz and 2Hz respectively
To describe the performance of RSMAD as a functionof prestrains loading frequencies and displacement ampli-tudes some important mechanical properties were calcu-lated including the secant stiffness119870119904 the energy dissipation
8 Mathematical Problems in Engineering
0 2 4 6 8
0
100
200
300
400
500
Displacement (mm)
Resto
ring
forc
e (N
)
Experiment Numerical results
minus2minus4minus6minus8
minus500
minus400
minus300
minus200
minus100
Figure 9 Comparison between experimental and numerical fittingcurves of RSMAD
per cycle 119882119863 the restoring force at peak displacement percycle119865119889 and the equivalent viscous damping ratio 120585eq whichis expressed as
120585eq =119882119863
21205871198701199041205752 (10)
where 120575 is the displacement amplitude of the cycle
322 Experimental Results Figure 5 shows the hysteresisloops of RSMAD at different prestrains and displacementamplitudes The tests were carried out at 005Hz frequencyof loading and room temperature of 20∘C As seen fromFigure 5 the SMA damper shows double-flag hystereticbehavior without any residual strain
Figure 6 shows themechanical properties of RSMAD as afunction of prestrain and displacement amplitude As we cansee in Figure 6 the greater the amplitude is the greater theenergy dissipation per cycle and restoring force are Howeverthe secant stiffness reduces markedly while increasing theamplitude With the increasing of the predisplacement theenergy dissipation per cycle decreases and restoring forceand the secant stiffness increase at large displacement Theeffectiveness of a damper is generally measured by theequivalent viscous damping ratio With the increasing of thepredisplacement the equivalent damping ratio reduces Themost important finding is that when the predisplacementis less than 23mm (2 prestrain) the equivalent dampingratio increasedwith the increasing of the amplitude howeverwhen the predisplacement is 46mm (4 prestrain) themaximum of the equivalent damping ratio is at 23mm (2prestrain) This is because the maximum of the equivalentdamping ratio of theNiTi SMA specimens is at about 6 totalstrain as noted in [10]
Figure 7 shows the hysteresis loops of RSMAD with06mm predisplacement at different loading frequencies anddisplacement amplitudes at the temperature of 20∘C Figure 8
shows mechanical properties of RSMAD as a function offrequency of loading and displacement amplitude As wecan see in Figures 7 and 8 the energy dissipation percycle and the equivalent damping decrease as the loadingfrequency increases in the range of 001ndash05Hz but arenot much sensitive to frequencies greater than 05Hz Therestoring force and the secant stiffness increase slightly as theloading frequency increases in the total range of experimentalfrequency
33 Numerical Simulation Based on the improved Graesserand Cozzarelli model of SMA wire a theoretic model of theSMA damper is developed The differential equations of themodel are given as
+ 119891119872[119909 minus 119909Mf sgn (119909)]119898
times [119906 (119909)] [119906 (|119909| minus 119909Mf)]
(11)
where 119865 is restoring force 119909 is displacement 119861 is back-force1198700 is initial stiffness 119861119888 120572119891119879 119899 119886 119888 119909Mf119891119872 and 119898 and areconstants controlling the size of the hysteresis loop 119909in is theinelastic displacement and 119909in = 119909 minus 1198651198700 erf(119909) 119906(119909) andsgn(119909) are respectively the error function the step functionand the signum function which have already been listed inSection 2
Figure 9 shows the comparison of experimental resultswith numerical prediction based on the theoretical modelat different displacement amplitudes The parameters of theconstitutive equation used in this study to simulate the behav-iors of RSMAD are given as follows 1198700 = 380Nmm 119861119888 =330N 120572 = 005 119891119879 = 21 119888 = 00001 119886 = 3 119899 = 2119909Mf = 575mm 119891119872 = 42500 and 119898 = 3 To accuratelysimulate the hysteresis behavior the above parameters weredirectly obtained from the cyclic test results of the SMAdamper according to the parametersrsquo meanings and rolesThevalues also can be converted through the section area andthe length of NiTi wires However there is slight differencebetween the two parameter groups This is possible becauseof the effect of loading conditions on the cyclic behavior ofNiTi wires As can be seen in Figure 9 numerical predictionsagree well with the experimental results
Table 1 shows the comparison of the experimental dataand numerical results respectively of the energy dissipationper cycle the secant stiffness and the equivalent viscousdamping As can be seen in Table 1 the maximum differencesof energy dissipation per cycle secant stiffness and equiva-lent viscous damping are 50 12 and 51 respectivelyThese results indicate that the mechanical behavior of theSMAdamper is well predicted by the numericalmodel whichverifies its suitability for the damper
Mathematical Problems in Engineering 9
(a) Bare structure (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5
Figure 10 Different cases of the ten-story frame structure
Table 1 Comparison between the experimental and theoretical results
Peakdisplacement(mm)
Energy dissipation per cycle (Nsdotmm) Secant stiffness (Nmm) Equivalent viscous damping ()Experimental
Themain objective of including energy dissipating devices ina structure is to reduce structural response through energydissipation To protect the original structural members understrong seismic loading nonlinear deformation in energydissipating devices will be allowed In frame structures thedevices are usually incorporated in steel braces connectingtwo consecutive stories of the buildingThedynamic responseof the structure subjected to earthquake loading is governedby the following equation
where 119872119904 is the mass matrix 119862119904 is the damping coefficientmatrix 119865119904 is the vector of the frames restoring force and 119865119889is the vector of restoring force resulting from SMA dampersThe 119909 and are the structural displacement velocity andacceleration vectors respectively On the right-hand side of
the equation the vector 119868 is the influence vector and thevector 119892 is the ground motion acceleration input
With the aid of the SIMULINK module of MATLAB adynamical simulation system was developed in which theWen model [36] is utilized to simulate the restoring forcecurve of the steel frame structure and the improved Graesserand Cozzarelli model presented above is used for NiTi SMAdamper
5 Seismic Structural Control UsingSMA Dampers
In this section to assess the effectiveness of the proposedrecentering SMA dampers (RSMAD) in mitigating the seis-mic response of building structure nonlinear time historyanalysis on a multistory steel frame with and withoutthe dampers subjected to representative earthquake groundmotions was performed The improved Graesser and Coz-zarelli model for RSMAD given in Section 33 was employedin this numerical study A ten-story steel moment resisting
10 Mathematical Problems in Engineering
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0
10
20 40 60 80 100 120 140 1600123456789
Lateral displacement (mm)
Floo
r lev
elEl Centro
TaftS69E
Case 0Case 5Case 4
Case 3Case 2Case 1
PGA = 02 g
PGA = 02 g
PGA = 02 gTangshan-Beijing
(a)
0 5 10 15 20 25 30
000002004006008010
Disp
lace
men
t (m
)
Time (s)
0 5 10 15 20 25 30Time (s)
0 5 10 15 20Time (s)
Without controlWith case 5
El Centro
000002004006008010012014
Disp
lace
men
t (m
)
Taft S69E
Disp
lace
men
t (m
)
PGA = 02 g
PGA = 02 g
PGA = 02 g
minus010
minus008
minus006
minus004
minus002
minus014
minus012
minus010
minus008
minus006
minus004
minus002
000002004006008010012014016
minus014
minus016
minus012
minus010
minus008
minus006
minus004
minus002
Tangshan-Beijing
(b)
Figure 11 Lateral displacement envelopes (a) and roof displacement timehistories (b)with andwithout SMAdampers under basic (moderate)ground motions
frame structure was selected for this study The building isdesigned for a location in Beijing China The parameters ofthe structure are listed in Table 2
Three representative earthquake ground motions wereused to considering the site effect Three earthquake groundmotions namely Imperial Valley 1940 El Centro LincolnSchool 1952 Taft and Beijing Hotel 1976 Tangshan wereselected These three ground motions represent different siteconditions According to Chinese code for seismic design
of building [37] the local seismic precautionary intensity iseight degrees The peak ground accelerations (PGA) wereadjusted to 02 g and 04 g corresponding to a seismic hazardlevel of 10 and 2 probability of exceedance in a 50-yearperiod respectively
Simulation analysis is conducted on the bare structureand on the structure with five or ten SMA dampers installedas shown in Figure 10 Parameters of the SMA damper arelisted as follows 1198700 = 119870119889 = 120 kNmm 119861119888 = 360KN
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 40 80 120 160 200 240 2800123456789
10
Lateral displacement (mm)
Floo
r lev
el
El CentroPGA = 04 g
Taft S69EPGA = 04 g
PGA = 04 g
Case 0Case 5Case 4
Case 3Case 2Case 1
Tangshan-Beijing
(a)
El Centro
0 5 10 15 20
0 5 10 15 20
25 30
000
005
010
015
020
Time (s)
Disp
lace
men
t (m
)
Taft S69E
000005010015020025
Disp
lace
men
t (m
)
Time (s)
PGA = 04 g
PGA = 04 g
PGA = 04 g
minus020
minus015
minus010
minus005
minus020
minus025
minus015
minus010
minus005
0 5 10 15 20 25 30
000
008004
012016020
Time (s)
Disp
lace
men
t (m
)
minus020
minus016
minus010
minus004
minus008
Without controlWith case 5
Tangshan-Beijing
(b)
Figure 12 Lateral displacement envelopes (a) and roof displacement time histories (b) with and without SMA dampers under strong (severe)ground motions
Figures 11 and 12 show the lateral displacement envelopeswith and without SMA dampers under basic groundmotions(02 g) and strong ground motions (04 g) respectively Aswe can see for most cases the lateral displacement of thestructure decreases remarkably with the introduction of theSMA dampers However the location and the number ofthe dampers have significant effects on the control resultsCase 5 in which dampers are installed in all stories is
the best for the overall structural vibration control For other4 configurations in which five dampers are installed indifferent stores cases 3 and 4 with dampers installed inalternate stories perform better than cases 1 and 2 withdampers placed in consecutive stories Moreover case 3 isbetter than case 4 since the first story is retrofitted Case 1is slightly worse due to the whiplash effect and case 2 isthe worst since the stiffness of the lower half is significantlygreater than that of the upper half of the structure Theroof displacement time histories with (case 5) and without
12 Mathematical Problems in Engineering
SMAdampers under basic groundmotions (02 g) and strongground motions (04 g) are also respectively provided inFigures 11 and 12 which confirm that the SMA damperssignificantly reduce the structural vibration
6 Concluding Remarks
This paper presents the results of a study on evaluating theefficacy of using an innovative SMA-based damper to reducethe seismic response of structures To describe the hysteresisbehavior of the SMA damper an improved Graesser andCozzarelli model was proposed and verified by the cyclictensile test on SMA wires
Cyclic tests on the SMA damper model utilizing foursuperelastic SMA wires with 05mm diameter with variousprestrains under different loading frequencies and displace-ment amplitudes were carried outThe results show satisfyinghysteresis properties including both recentering and energydissipating features under various conditions
A comparative study on nonlinear time history analysisof the seismic response of a ten-story steel frame with theSMA dampers was performed Five cases were considered forthe location and the number of the dampers in the storiesThe numerical analysis results indicate that the proposedSMA damper is capable of significantly reducing seismicresponse of structures which verifies its effectiveness asenergy dissipating device for structures However it is alsoindicated that the location and number have significanteffects on the results of the response
In future multiobjective optimization model will beproposed to obtain the number and the location of the SMAdampers and large scale shake table tests will be performed ona steel frame buildingwith SMAdampers to prove the efficacyof these dampers in dissipating seismic energy
Acknowledgments
This work was funded by National Science Foundation ofChina (no 51108426 and no 41104106) China PostdoctoralScience Foundation (no 20100471008) and Research Fundfor the Doctoral Program of Higher Education of China (no20104101120009) These supports are greatly appreciatedTheopinions expressed in this study are those of the authors anddo not necessarily reflect the views of the sponsor
References
[1] H N Li and L S Huo ldquoAdvances in structural control in civilengineering in Chinardquo Mathematical Problems in Engineeringvol 2010 Article ID 936081 23 pages 2010
[2] G Song N Ma and H N Li ldquoApplications of shape memoryalloys in civil structuresrdquo Engineering Structures vol 28 no 9pp 1266ndash1274 2006
[3] SEAOC Vision 2000 Committee Performance-Based SeismicEngineering Structural Engineering Association of CaliforniaSacramento Calif USA 1995
[4] ATC-40 Seismic Evaluation and Retrofit of Conctete BuildingsApplied Technology Council 1996
[5] FEMA 273 NEHRP Guidelines for Seismic Rehabilitation ofBuildings Federal Emergency Management Agency 1997
[6] Y Fujino T T Soong and B F Spencer Jr ldquoStructural controlbasic concepts and applicationsrdquo in Proceedings of the ASCEStructures Congress pp 15ndash18 Chicago Ill USA April 1996
[7] B F Spencer and S Nagarajaiah ldquoState of the art of structuralcontrolrdquo Journal of Structural Engineering vol 129 no 7 pp845ndash856 2003
[8] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering John Wiley amp Sons NewYork NY USA 1997
[9] F M Mazzolani ldquoPassive control technologies for seismic-resistant buildings in Europerdquo Progress in Structural Engineeringand Materials vol 3 no 3 pp 277ndash287 2001
[10] M Dolce and D Cardone ldquoMechanical behaviour of SMAelements for seismic applicationsmdashpart 2 austenite NiTi wiressubjected to tensionrdquo International Journal of Mechanical Sci-ences vol 43 no 11 pp 2657ndash2677 2001
[11] H N Li and X X Wu ldquoLimitations of height-to-width ratiofor base-isolated buildings under earthquakerdquo The StructuralDesign of Tall and Special Buildings vol 15 no 3 pp 277ndash2872006
[12] R Desroches J McCormick and M A Delemont ldquoCyclicproperties of superelastic shape memory alloy wires and barsrdquoJournal of Structural Engineering vol 130 no 1 pp 38ndash46 2004
[13] T W Duerig K N Melton D Stockel and C M Way-man Engineering Aspects of Shape Memory Alloys ButterworthHeinemann London UK 1990
[14] K Wilde P Gardoni and Y Fujino ldquoBase isolation systemwith shape memory alloy device for elevated highway bridgesrdquoEngineering Structures vol 22 no 3 pp 222ndash229 2000
[15] M Dolce D Cardone and R Marnetto ldquoImplementation andtesting of passive control devices based on shape memoryalloysrdquo Earthquake Engineering and Structural Dynamics vol29 no 7 pp 945ndash968 2000
[16] M Dolce D Cardone F C Ponzo and C Valente ldquoShakingtable tests on reinforced concrete frames without and withpassive control systemsrdquo Earthquake Engineering and StructuralDynamics vol 34 no 14 pp 1687ndash1717 2005
[17] M Indirli M G Castellano P Clemente and A MartellildquoDemo-application of shapememory alloy devices the rehabili-tation of the S Giorgio Church Bell-Towerrdquo in Smart Structuresand Materials 2001 Smart Systems for Bridges Structures andHighways vol 4330 of Proceedings of SPIE pp 262ndash272 New-port Beach Calif USA March 2001
[18] B Andrawes and R Desroches ldquoUnseating prevention for mul-tiple frame bridges using superelastic devicesrdquo Smart Materialsand Structures vol 14 no 3 pp S60ndashS67 2005
[19] R DesRoches and M Delemont ldquoSeismic retrofit of simplysupported bridges using shape memory alloysrdquo EngineeringStructures vol 24 no 3 pp 325ndash332 2002
[20] H Li M Liu and J Ou ldquoVibration mitigation of a stay cablewith one shape memory alloy damperrdquo Structural Control andHealth Monitoring vol 11 no 1 pp 21ndash36 2004
[21] Y Zhang and S Zhu ldquoA shape memory alloy-based reusablehysteretic damper for seismic hazard mitigationrdquo Smart Mate-rials and Structures vol 16 no 5 pp 1603ndash1613 2007
[22] J Ocel R DesRoches R T Leon et al ldquoSteel beam-columnconnections using shape memory alloysrdquo Journal of StructuralEngineering vol 130 no 5 pp 732ndash740 2004
Mathematical Problems in Engineering 13
[23] J McCormick R Desroches D Fugazza and F AuricchioldquoSeismic assessment of concentrically braced steel frames withshape memory alloy bracesrdquo Journal of Structural Engineeringvol 133 no 6 pp 862ndash870 2007
[24] YM Parulekar G R Reddy K K Vaze et al ldquoSeismic responseattenuation of structures using shape memory alloy dampersrdquoStructural Control and Health Monitoring vol 19 no 1 pp 102ndash119 2012
[25] K Tanaka ldquoA thermomechanical sketch of shape memoryeffect one-dimensional tensile behaviorrdquoResMechanica vol 18no 3 pp 251ndash263 1986
[26] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 1 no 2 pp207ndash234 1990
[27] L C Brinson ldquoOne-dimensional constitutive behavior of shapememory alloys thermomechanical derivation with non-con-stant material functions and redefined martensite internal vari-ablerdquo Journal of Intelligent Material Systems and Structures vol4 no 2 pp 229ndash242 1993
[28] F Falk ldquoModel free energy mechanics and thermodynamicsof shape memory alloysrdquo Acta Metallurgica vol 28 no 12 pp1773ndash1780 1980
[29] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashII study of the individual phenomenardquo Journal of theMechanicsand Physics of Solids vol 41 no 1 pp 19ndash33 1993
[30] J G Boyd andDC Lagoudas ldquoA thermodynamical constitutivemodel for shape memory materialsmdashpart I the monolithicshape memory alloyrdquo International Journal of Plasticity vol 12no 6 pp 805ndash842 1996
[31] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashI derivation of general relationsrdquo Journal of the Mechanics andPhysics of Solids vol 41 no 1 pp 1ndash17 1993
[32] H Qian H Li G Song and W Guo ldquoA constitutive modelfor superelastic shape memory alloys considering the influenceof strain raterdquoMathematical Problems in Engineering vol 2013Article ID 248671 8 pages 2013
[33] W J Ren H N Li and G Song ldquoA one-dimensional strain-ratedependent constitutive model for superelastic shape memoryalloysrdquo Smart Materials and Structures vol 16 no 1 pp 191ndash1972007
[34] E J Graesser and F A Cozzarelli ldquoShape-memory alloys asnew materials for aseismic isolationrdquo Journal of EngineeringMechanics vol 117 no 11 pp 2590ndash2608 1991
[35] H Ozdemir Nonlinear transient dynamic analysis of yieldingstructures [PhD thesis] University of California BerkeleyCalif USA 1976
[36] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976
[37] GB 50011-2010 Code for Seismic Design of Buildings Ministryof Housing and Urban-Rural Development of the PeoplersquosRepublic of China 2010
Figure 5 Hysteresis loops of the RSMAD at different prestrains and displacement amplitudes (005Hz frequency of loading 20∘Ctemperature)
120576in is the inelastic strain given by
120576in = 120576 minus120590
119864 (4)
119906() is the unit step function defined as
119906 (119909) = +1 119909 ge 0
0 119909 lt 0(5)
erf() is the error function defined by
erf (119909) = 2
radic120587int
120587
0119890minus1199052119889119905 (6)
The original Graesser and Cozzarelli model has a rel-atively simple expression with the parameters that can beeasily acquired however this model excludes the martensitichardening characteristics of SMAs under large amplitudeswhich are critical for structural safety protection underextreme events
To overcome the limitation of the original model Wildeet al [14] extended the Graesser and Cozzarelli model bydividing the full loop into four parts adding two termswith six parameters into (1) The Wilde model was utilizedto simulate the cyclic behaviors of SMA devices in otherresearches [21]
In the following in order to accurately predict the cyclicbehavior of a superelastic SMA device especially capture themartensitic hardening characteristics of SMAs under largeamplitudes an improved Graesser and Cozzarelli model ispresented In the present model the backstress expression ismodified by adding a special term to capture the martensitichardening characteristic of SMA under large amplitudesThemodified model is of the form
= 119864 [ 120576 minus | 120576| (120590 minus 120573
Figure 6 Mechanical properties of RSMAD as a function of prestrain and displacement amplitude (005Hz frequency of loading 20∘Ctemperature)
+ 119891119872[120576 minus 120576Mf sgn (120576)]119898[119906 (120576 120576)]
times [119906 (|120576| minus 120576Mf) ]
(8)
The third term in (2) is used to contribute to the back-stress on the ascending branch of the hysteresis in a way thatallows for the martensitic hardening 120576Mf is the martensitefinish strain 119891119872 and119898 are material constants controlling themartensitic hardening curve sgn(119909) is the signum functiongiven by
sgn (119909) =
+1 119909 gt 0
0 119909 = 0
minus1 119909 lt 0
(9)
Figure 2 shows the stress-strain curves of superelasticnitinol wires predicted by the improved Graesser and Coz-zarelli model versus experimental data at different strain
levels The characteristic parameters used in the models are119864 = 39500MPa 119884 = 385MPa 120572 = 001 119891119879 = 114119888 = 0001 119886 = 550 119899 = 3 120576Mf = 005 119891119872 = 42500and 119898 = 3 The superelastic nitinol wires are 05mm indiameter with a composition of approximately 509 Ni and491 Ti Under zero external stress the martensite startand finish temperatures (119872119891119872119904) and the austenite startand finish temperatures (119860 119904 119860119891) measured by differentialscanning calorimeter (DSC) are minus73∘C minus55∘C minus23∘C and5∘C respectively The uniaxial tension test of the superelasticnitinol wires was carried out using an electromechanicaluniversal testing machine at room temperature of 20∘C Thenitinol wire samples with a 100mm test length between thetwo custom-made grips were subjected to triangular cyclicloading under different strain amplitudes The strains werecalculated from the elongation measured by a 50mm gagelength extensometer with the stress calculated from the axialforce which was measured by a 10KN load cell Prior to
6 Mathematical Problems in Engineering
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
001Hz
(a)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
005Hz
(b)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
01Hz
(c)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
05Hz
(d)
Figure 7 Hysteresis loops of the RSMAD at different loading frequencies and displacement amplitudes (06mm predisplacement 20∘Ctemperature)
testing the nitinol SMA specimens were cycled 20 timesat 6 strain amplitude and 12 times 10minus3 sminus1 strain rate bya ldquotrainingrdquo process to reach a steady-state condition Theexperimental data in Figure 2 are the results of cyclic tests onthe superelastic nitinol wire at 12 times 10minus3 sminus1 strain rate with1 to 8 strain levels As shown in Figure 2 the hysteresisloops based on the improved Graesser and Cozzarelli modeland experimental data match with close accuracy Moreoverthe improved model can accurately reflect the martensitichardening characteristic of SMAs under large amplitudes
3 An Innovative SMA Damper DesignExperiment and Numerical Simulation
31 Recentering SMADamper Design By utilizing the energydissipating and recentering features of superelastic nitinolSMA an innovative damper is designed As shown inFigure 3 the damper consists of outer and inner cylinders leftand right pull plates superelastic SMA wires retaining plate
prestrain adjusting plate adjusting bolt fixed bolt push-pullrod grip end caps and connecting fitting In this dampersuperelastic nitinolwires are the key components that provideboth damping and self-centering abilities
The damper will be connected to a structure via its push-pull rod and the connecting fitting The prestrain of thesuperelastic wires can be adjusted by the prestrain adjustingplate and the adjusting bolt The configuration of the dampershown in Figure 3 is in its equilibrium position The specificdesign of the damper allows the push-pull rod to move inboth left and right directions and return to its equilibriumposition when the load is removed (self-centering) Duringthis cyclic process the damper provides damping attributedto the hysteretic property of the superelastic wires
32 Experimental Tests
321 Setup and Program An SMA damper based on thedesign presented in Section 31 is fabricated To assess theperformance of the SMA damper cyclic tests were carried
Mathematical Problems in Engineering 7
001563 00625 025 1 40
300
600
900
1200
1500
1800
2100
2400
Log2 frequency (Hz)
Ener
gy d
issip
atio
n pe
r cyc
le (1
0minus3
J)
(a)
0
100
200
300
400
500
600
700
800
Resto
ring
forc
e (N
)
001563 00625 025 1 4Log2 frequency (Hz)
(b)
0
30
60
90
120
150
180
210
240
001563 00625 025 1 4Log2 frequency (Hz)
230mm345mm
460mm575mm
Seca
nt st
iffne
ss (N
mm
minus1)
(c)
000
002
004
006
008
010
012
014
016
Equi
vale
nt d
ampi
ng
001563 00625 025 1 4Log2 frequency (Hz)
230mm345mm
460mm575mm
(d)
Figure 8 Mechanical properties of RSMAD as a function of loading frequencies and displacement amplitudes (06mm predisplacement20∘C temperature)
out The SMA damper is 210mm in length with a diameterof 100mm and a stroke of plusmn92mm (the maximum allowablestrain for superelastic nitinol wires is 8) Four superelasticnitinol wires of a length of 115mm and a diameter of 05mmare used
Tests were conducted using an MTS 810 machine with a100KN load cell at room temperature of 20∘C The layout ofthe test system is shown in Figure 4 The damper was testedwith different prestrains at different loading frequenciesand at various amplitudes During the tests both force anddisplacement are recorded
Prior to the installation each nitinol wire was cycled 20times at 6 strain with 12 times 10minus3 sminus1 strain rate to minimizethe accumulation of residual strain and reach a steady-statecondition The scheme of the tests is described as follows
(1) Without prestrain the damper was subjected tocyclic loading at 005Hz frequencywith displacement
amplitudes of 23mm (2 of total length) 345mm(3 of total length) and 46mm (4 of total length)respectively
(2) Step (1) was repeated with prestrains of 1 (115mmpredisplacement) 2 (23mm predisplacement) and4 (46mm predisplacement) respectively
(3) With 05 prestrain (about 06mm predisplace-ment) the damper was subjected to cyclic loading at001Hz loading frequency with displacement ampli-tudes of 23mm 345mm 46mm and 575mmrespectively
(4) Step (3) was repeated with loading frequencies of005Hz 01 Hz 05Hz 1Hz and 2Hz respectively
To describe the performance of RSMAD as a functionof prestrains loading frequencies and displacement ampli-tudes some important mechanical properties were calcu-lated including the secant stiffness119870119904 the energy dissipation
8 Mathematical Problems in Engineering
0 2 4 6 8
0
100
200
300
400
500
Displacement (mm)
Resto
ring
forc
e (N
)
Experiment Numerical results
minus2minus4minus6minus8
minus500
minus400
minus300
minus200
minus100
Figure 9 Comparison between experimental and numerical fittingcurves of RSMAD
per cycle 119882119863 the restoring force at peak displacement percycle119865119889 and the equivalent viscous damping ratio 120585eq whichis expressed as
120585eq =119882119863
21205871198701199041205752 (10)
where 120575 is the displacement amplitude of the cycle
322 Experimental Results Figure 5 shows the hysteresisloops of RSMAD at different prestrains and displacementamplitudes The tests were carried out at 005Hz frequencyof loading and room temperature of 20∘C As seen fromFigure 5 the SMA damper shows double-flag hystereticbehavior without any residual strain
Figure 6 shows themechanical properties of RSMAD as afunction of prestrain and displacement amplitude As we cansee in Figure 6 the greater the amplitude is the greater theenergy dissipation per cycle and restoring force are Howeverthe secant stiffness reduces markedly while increasing theamplitude With the increasing of the predisplacement theenergy dissipation per cycle decreases and restoring forceand the secant stiffness increase at large displacement Theeffectiveness of a damper is generally measured by theequivalent viscous damping ratio With the increasing of thepredisplacement the equivalent damping ratio reduces Themost important finding is that when the predisplacementis less than 23mm (2 prestrain) the equivalent dampingratio increasedwith the increasing of the amplitude howeverwhen the predisplacement is 46mm (4 prestrain) themaximum of the equivalent damping ratio is at 23mm (2prestrain) This is because the maximum of the equivalentdamping ratio of theNiTi SMA specimens is at about 6 totalstrain as noted in [10]
Figure 7 shows the hysteresis loops of RSMAD with06mm predisplacement at different loading frequencies anddisplacement amplitudes at the temperature of 20∘C Figure 8
shows mechanical properties of RSMAD as a function offrequency of loading and displacement amplitude As wecan see in Figures 7 and 8 the energy dissipation percycle and the equivalent damping decrease as the loadingfrequency increases in the range of 001ndash05Hz but arenot much sensitive to frequencies greater than 05Hz Therestoring force and the secant stiffness increase slightly as theloading frequency increases in the total range of experimentalfrequency
33 Numerical Simulation Based on the improved Graesserand Cozzarelli model of SMA wire a theoretic model of theSMA damper is developed The differential equations of themodel are given as
+ 119891119872[119909 minus 119909Mf sgn (119909)]119898
times [119906 (119909)] [119906 (|119909| minus 119909Mf)]
(11)
where 119865 is restoring force 119909 is displacement 119861 is back-force1198700 is initial stiffness 119861119888 120572119891119879 119899 119886 119888 119909Mf119891119872 and 119898 and areconstants controlling the size of the hysteresis loop 119909in is theinelastic displacement and 119909in = 119909 minus 1198651198700 erf(119909) 119906(119909) andsgn(119909) are respectively the error function the step functionand the signum function which have already been listed inSection 2
Figure 9 shows the comparison of experimental resultswith numerical prediction based on the theoretical modelat different displacement amplitudes The parameters of theconstitutive equation used in this study to simulate the behav-iors of RSMAD are given as follows 1198700 = 380Nmm 119861119888 =330N 120572 = 005 119891119879 = 21 119888 = 00001 119886 = 3 119899 = 2119909Mf = 575mm 119891119872 = 42500 and 119898 = 3 To accuratelysimulate the hysteresis behavior the above parameters weredirectly obtained from the cyclic test results of the SMAdamper according to the parametersrsquo meanings and rolesThevalues also can be converted through the section area andthe length of NiTi wires However there is slight differencebetween the two parameter groups This is possible becauseof the effect of loading conditions on the cyclic behavior ofNiTi wires As can be seen in Figure 9 numerical predictionsagree well with the experimental results
Table 1 shows the comparison of the experimental dataand numerical results respectively of the energy dissipationper cycle the secant stiffness and the equivalent viscousdamping As can be seen in Table 1 the maximum differencesof energy dissipation per cycle secant stiffness and equiva-lent viscous damping are 50 12 and 51 respectivelyThese results indicate that the mechanical behavior of theSMAdamper is well predicted by the numericalmodel whichverifies its suitability for the damper
Mathematical Problems in Engineering 9
(a) Bare structure (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5
Figure 10 Different cases of the ten-story frame structure
Table 1 Comparison between the experimental and theoretical results
Peakdisplacement(mm)
Energy dissipation per cycle (Nsdotmm) Secant stiffness (Nmm) Equivalent viscous damping ()Experimental
Themain objective of including energy dissipating devices ina structure is to reduce structural response through energydissipation To protect the original structural members understrong seismic loading nonlinear deformation in energydissipating devices will be allowed In frame structures thedevices are usually incorporated in steel braces connectingtwo consecutive stories of the buildingThedynamic responseof the structure subjected to earthquake loading is governedby the following equation
where 119872119904 is the mass matrix 119862119904 is the damping coefficientmatrix 119865119904 is the vector of the frames restoring force and 119865119889is the vector of restoring force resulting from SMA dampersThe 119909 and are the structural displacement velocity andacceleration vectors respectively On the right-hand side of
the equation the vector 119868 is the influence vector and thevector 119892 is the ground motion acceleration input
With the aid of the SIMULINK module of MATLAB adynamical simulation system was developed in which theWen model [36] is utilized to simulate the restoring forcecurve of the steel frame structure and the improved Graesserand Cozzarelli model presented above is used for NiTi SMAdamper
5 Seismic Structural Control UsingSMA Dampers
In this section to assess the effectiveness of the proposedrecentering SMA dampers (RSMAD) in mitigating the seis-mic response of building structure nonlinear time historyanalysis on a multistory steel frame with and withoutthe dampers subjected to representative earthquake groundmotions was performed The improved Graesser and Coz-zarelli model for RSMAD given in Section 33 was employedin this numerical study A ten-story steel moment resisting
10 Mathematical Problems in Engineering
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0
10
20 40 60 80 100 120 140 1600123456789
Lateral displacement (mm)
Floo
r lev
elEl Centro
TaftS69E
Case 0Case 5Case 4
Case 3Case 2Case 1
PGA = 02 g
PGA = 02 g
PGA = 02 gTangshan-Beijing
(a)
0 5 10 15 20 25 30
000002004006008010
Disp
lace
men
t (m
)
Time (s)
0 5 10 15 20 25 30Time (s)
0 5 10 15 20Time (s)
Without controlWith case 5
El Centro
000002004006008010012014
Disp
lace
men
t (m
)
Taft S69E
Disp
lace
men
t (m
)
PGA = 02 g
PGA = 02 g
PGA = 02 g
minus010
minus008
minus006
minus004
minus002
minus014
minus012
minus010
minus008
minus006
minus004
minus002
000002004006008010012014016
minus014
minus016
minus012
minus010
minus008
minus006
minus004
minus002
Tangshan-Beijing
(b)
Figure 11 Lateral displacement envelopes (a) and roof displacement timehistories (b)with andwithout SMAdampers under basic (moderate)ground motions
frame structure was selected for this study The building isdesigned for a location in Beijing China The parameters ofthe structure are listed in Table 2
Three representative earthquake ground motions wereused to considering the site effect Three earthquake groundmotions namely Imperial Valley 1940 El Centro LincolnSchool 1952 Taft and Beijing Hotel 1976 Tangshan wereselected These three ground motions represent different siteconditions According to Chinese code for seismic design
of building [37] the local seismic precautionary intensity iseight degrees The peak ground accelerations (PGA) wereadjusted to 02 g and 04 g corresponding to a seismic hazardlevel of 10 and 2 probability of exceedance in a 50-yearperiod respectively
Simulation analysis is conducted on the bare structureand on the structure with five or ten SMA dampers installedas shown in Figure 10 Parameters of the SMA damper arelisted as follows 1198700 = 119870119889 = 120 kNmm 119861119888 = 360KN
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 40 80 120 160 200 240 2800123456789
10
Lateral displacement (mm)
Floo
r lev
el
El CentroPGA = 04 g
Taft S69EPGA = 04 g
PGA = 04 g
Case 0Case 5Case 4
Case 3Case 2Case 1
Tangshan-Beijing
(a)
El Centro
0 5 10 15 20
0 5 10 15 20
25 30
000
005
010
015
020
Time (s)
Disp
lace
men
t (m
)
Taft S69E
000005010015020025
Disp
lace
men
t (m
)
Time (s)
PGA = 04 g
PGA = 04 g
PGA = 04 g
minus020
minus015
minus010
minus005
minus020
minus025
minus015
minus010
minus005
0 5 10 15 20 25 30
000
008004
012016020
Time (s)
Disp
lace
men
t (m
)
minus020
minus016
minus010
minus004
minus008
Without controlWith case 5
Tangshan-Beijing
(b)
Figure 12 Lateral displacement envelopes (a) and roof displacement time histories (b) with and without SMA dampers under strong (severe)ground motions
Figures 11 and 12 show the lateral displacement envelopeswith and without SMA dampers under basic groundmotions(02 g) and strong ground motions (04 g) respectively Aswe can see for most cases the lateral displacement of thestructure decreases remarkably with the introduction of theSMA dampers However the location and the number ofthe dampers have significant effects on the control resultsCase 5 in which dampers are installed in all stories is
the best for the overall structural vibration control For other4 configurations in which five dampers are installed indifferent stores cases 3 and 4 with dampers installed inalternate stories perform better than cases 1 and 2 withdampers placed in consecutive stories Moreover case 3 isbetter than case 4 since the first story is retrofitted Case 1is slightly worse due to the whiplash effect and case 2 isthe worst since the stiffness of the lower half is significantlygreater than that of the upper half of the structure Theroof displacement time histories with (case 5) and without
12 Mathematical Problems in Engineering
SMAdampers under basic groundmotions (02 g) and strongground motions (04 g) are also respectively provided inFigures 11 and 12 which confirm that the SMA damperssignificantly reduce the structural vibration
6 Concluding Remarks
This paper presents the results of a study on evaluating theefficacy of using an innovative SMA-based damper to reducethe seismic response of structures To describe the hysteresisbehavior of the SMA damper an improved Graesser andCozzarelli model was proposed and verified by the cyclictensile test on SMA wires
Cyclic tests on the SMA damper model utilizing foursuperelastic SMA wires with 05mm diameter with variousprestrains under different loading frequencies and displace-ment amplitudes were carried outThe results show satisfyinghysteresis properties including both recentering and energydissipating features under various conditions
A comparative study on nonlinear time history analysisof the seismic response of a ten-story steel frame with theSMA dampers was performed Five cases were considered forthe location and the number of the dampers in the storiesThe numerical analysis results indicate that the proposedSMA damper is capable of significantly reducing seismicresponse of structures which verifies its effectiveness asenergy dissipating device for structures However it is alsoindicated that the location and number have significanteffects on the results of the response
In future multiobjective optimization model will beproposed to obtain the number and the location of the SMAdampers and large scale shake table tests will be performed ona steel frame buildingwith SMAdampers to prove the efficacyof these dampers in dissipating seismic energy
Acknowledgments
This work was funded by National Science Foundation ofChina (no 51108426 and no 41104106) China PostdoctoralScience Foundation (no 20100471008) and Research Fundfor the Doctoral Program of Higher Education of China (no20104101120009) These supports are greatly appreciatedTheopinions expressed in this study are those of the authors anddo not necessarily reflect the views of the sponsor
References
[1] H N Li and L S Huo ldquoAdvances in structural control in civilengineering in Chinardquo Mathematical Problems in Engineeringvol 2010 Article ID 936081 23 pages 2010
[2] G Song N Ma and H N Li ldquoApplications of shape memoryalloys in civil structuresrdquo Engineering Structures vol 28 no 9pp 1266ndash1274 2006
[3] SEAOC Vision 2000 Committee Performance-Based SeismicEngineering Structural Engineering Association of CaliforniaSacramento Calif USA 1995
[4] ATC-40 Seismic Evaluation and Retrofit of Conctete BuildingsApplied Technology Council 1996
[5] FEMA 273 NEHRP Guidelines for Seismic Rehabilitation ofBuildings Federal Emergency Management Agency 1997
[6] Y Fujino T T Soong and B F Spencer Jr ldquoStructural controlbasic concepts and applicationsrdquo in Proceedings of the ASCEStructures Congress pp 15ndash18 Chicago Ill USA April 1996
[7] B F Spencer and S Nagarajaiah ldquoState of the art of structuralcontrolrdquo Journal of Structural Engineering vol 129 no 7 pp845ndash856 2003
[8] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering John Wiley amp Sons NewYork NY USA 1997
[9] F M Mazzolani ldquoPassive control technologies for seismic-resistant buildings in Europerdquo Progress in Structural Engineeringand Materials vol 3 no 3 pp 277ndash287 2001
[10] M Dolce and D Cardone ldquoMechanical behaviour of SMAelements for seismic applicationsmdashpart 2 austenite NiTi wiressubjected to tensionrdquo International Journal of Mechanical Sci-ences vol 43 no 11 pp 2657ndash2677 2001
[11] H N Li and X X Wu ldquoLimitations of height-to-width ratiofor base-isolated buildings under earthquakerdquo The StructuralDesign of Tall and Special Buildings vol 15 no 3 pp 277ndash2872006
[12] R Desroches J McCormick and M A Delemont ldquoCyclicproperties of superelastic shape memory alloy wires and barsrdquoJournal of Structural Engineering vol 130 no 1 pp 38ndash46 2004
[13] T W Duerig K N Melton D Stockel and C M Way-man Engineering Aspects of Shape Memory Alloys ButterworthHeinemann London UK 1990
[14] K Wilde P Gardoni and Y Fujino ldquoBase isolation systemwith shape memory alloy device for elevated highway bridgesrdquoEngineering Structures vol 22 no 3 pp 222ndash229 2000
[15] M Dolce D Cardone and R Marnetto ldquoImplementation andtesting of passive control devices based on shape memoryalloysrdquo Earthquake Engineering and Structural Dynamics vol29 no 7 pp 945ndash968 2000
[16] M Dolce D Cardone F C Ponzo and C Valente ldquoShakingtable tests on reinforced concrete frames without and withpassive control systemsrdquo Earthquake Engineering and StructuralDynamics vol 34 no 14 pp 1687ndash1717 2005
[17] M Indirli M G Castellano P Clemente and A MartellildquoDemo-application of shapememory alloy devices the rehabili-tation of the S Giorgio Church Bell-Towerrdquo in Smart Structuresand Materials 2001 Smart Systems for Bridges Structures andHighways vol 4330 of Proceedings of SPIE pp 262ndash272 New-port Beach Calif USA March 2001
[18] B Andrawes and R Desroches ldquoUnseating prevention for mul-tiple frame bridges using superelastic devicesrdquo Smart Materialsand Structures vol 14 no 3 pp S60ndashS67 2005
[19] R DesRoches and M Delemont ldquoSeismic retrofit of simplysupported bridges using shape memory alloysrdquo EngineeringStructures vol 24 no 3 pp 325ndash332 2002
[20] H Li M Liu and J Ou ldquoVibration mitigation of a stay cablewith one shape memory alloy damperrdquo Structural Control andHealth Monitoring vol 11 no 1 pp 21ndash36 2004
[21] Y Zhang and S Zhu ldquoA shape memory alloy-based reusablehysteretic damper for seismic hazard mitigationrdquo Smart Mate-rials and Structures vol 16 no 5 pp 1603ndash1613 2007
[22] J Ocel R DesRoches R T Leon et al ldquoSteel beam-columnconnections using shape memory alloysrdquo Journal of StructuralEngineering vol 130 no 5 pp 732ndash740 2004
Mathematical Problems in Engineering 13
[23] J McCormick R Desroches D Fugazza and F AuricchioldquoSeismic assessment of concentrically braced steel frames withshape memory alloy bracesrdquo Journal of Structural Engineeringvol 133 no 6 pp 862ndash870 2007
[24] YM Parulekar G R Reddy K K Vaze et al ldquoSeismic responseattenuation of structures using shape memory alloy dampersrdquoStructural Control and Health Monitoring vol 19 no 1 pp 102ndash119 2012
[25] K Tanaka ldquoA thermomechanical sketch of shape memoryeffect one-dimensional tensile behaviorrdquoResMechanica vol 18no 3 pp 251ndash263 1986
[26] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 1 no 2 pp207ndash234 1990
[27] L C Brinson ldquoOne-dimensional constitutive behavior of shapememory alloys thermomechanical derivation with non-con-stant material functions and redefined martensite internal vari-ablerdquo Journal of Intelligent Material Systems and Structures vol4 no 2 pp 229ndash242 1993
[28] F Falk ldquoModel free energy mechanics and thermodynamicsof shape memory alloysrdquo Acta Metallurgica vol 28 no 12 pp1773ndash1780 1980
[29] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashII study of the individual phenomenardquo Journal of theMechanicsand Physics of Solids vol 41 no 1 pp 19ndash33 1993
[30] J G Boyd andDC Lagoudas ldquoA thermodynamical constitutivemodel for shape memory materialsmdashpart I the monolithicshape memory alloyrdquo International Journal of Plasticity vol 12no 6 pp 805ndash842 1996
[31] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashI derivation of general relationsrdquo Journal of the Mechanics andPhysics of Solids vol 41 no 1 pp 1ndash17 1993
[32] H Qian H Li G Song and W Guo ldquoA constitutive modelfor superelastic shape memory alloys considering the influenceof strain raterdquoMathematical Problems in Engineering vol 2013Article ID 248671 8 pages 2013
[33] W J Ren H N Li and G Song ldquoA one-dimensional strain-ratedependent constitutive model for superelastic shape memoryalloysrdquo Smart Materials and Structures vol 16 no 1 pp 191ndash1972007
[34] E J Graesser and F A Cozzarelli ldquoShape-memory alloys asnew materials for aseismic isolationrdquo Journal of EngineeringMechanics vol 117 no 11 pp 2590ndash2608 1991
[35] H Ozdemir Nonlinear transient dynamic analysis of yieldingstructures [PhD thesis] University of California BerkeleyCalif USA 1976
[36] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976
[37] GB 50011-2010 Code for Seismic Design of Buildings Ministryof Housing and Urban-Rural Development of the PeoplersquosRepublic of China 2010
Figure 6 Mechanical properties of RSMAD as a function of prestrain and displacement amplitude (005Hz frequency of loading 20∘Ctemperature)
+ 119891119872[120576 minus 120576Mf sgn (120576)]119898[119906 (120576 120576)]
times [119906 (|120576| minus 120576Mf) ]
(8)
The third term in (2) is used to contribute to the back-stress on the ascending branch of the hysteresis in a way thatallows for the martensitic hardening 120576Mf is the martensitefinish strain 119891119872 and119898 are material constants controlling themartensitic hardening curve sgn(119909) is the signum functiongiven by
sgn (119909) =
+1 119909 gt 0
0 119909 = 0
minus1 119909 lt 0
(9)
Figure 2 shows the stress-strain curves of superelasticnitinol wires predicted by the improved Graesser and Coz-zarelli model versus experimental data at different strain
levels The characteristic parameters used in the models are119864 = 39500MPa 119884 = 385MPa 120572 = 001 119891119879 = 114119888 = 0001 119886 = 550 119899 = 3 120576Mf = 005 119891119872 = 42500and 119898 = 3 The superelastic nitinol wires are 05mm indiameter with a composition of approximately 509 Ni and491 Ti Under zero external stress the martensite startand finish temperatures (119872119891119872119904) and the austenite startand finish temperatures (119860 119904 119860119891) measured by differentialscanning calorimeter (DSC) are minus73∘C minus55∘C minus23∘C and5∘C respectively The uniaxial tension test of the superelasticnitinol wires was carried out using an electromechanicaluniversal testing machine at room temperature of 20∘C Thenitinol wire samples with a 100mm test length between thetwo custom-made grips were subjected to triangular cyclicloading under different strain amplitudes The strains werecalculated from the elongation measured by a 50mm gagelength extensometer with the stress calculated from the axialforce which was measured by a 10KN load cell Prior to
6 Mathematical Problems in Engineering
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
001Hz
(a)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
005Hz
(b)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
01Hz
(c)
Displacement (mm)
Resto
ring
forc
e (N
)
0
0
200
400
600
2 4 6 8minus600
minus400
minus200
minus2minus4minus6minus8
05Hz
(d)
Figure 7 Hysteresis loops of the RSMAD at different loading frequencies and displacement amplitudes (06mm predisplacement 20∘Ctemperature)
testing the nitinol SMA specimens were cycled 20 timesat 6 strain amplitude and 12 times 10minus3 sminus1 strain rate bya ldquotrainingrdquo process to reach a steady-state condition Theexperimental data in Figure 2 are the results of cyclic tests onthe superelastic nitinol wire at 12 times 10minus3 sminus1 strain rate with1 to 8 strain levels As shown in Figure 2 the hysteresisloops based on the improved Graesser and Cozzarelli modeland experimental data match with close accuracy Moreoverthe improved model can accurately reflect the martensitichardening characteristic of SMAs under large amplitudes
3 An Innovative SMA Damper DesignExperiment and Numerical Simulation
31 Recentering SMADamper Design By utilizing the energydissipating and recentering features of superelastic nitinolSMA an innovative damper is designed As shown inFigure 3 the damper consists of outer and inner cylinders leftand right pull plates superelastic SMA wires retaining plate
prestrain adjusting plate adjusting bolt fixed bolt push-pullrod grip end caps and connecting fitting In this dampersuperelastic nitinolwires are the key components that provideboth damping and self-centering abilities
The damper will be connected to a structure via its push-pull rod and the connecting fitting The prestrain of thesuperelastic wires can be adjusted by the prestrain adjustingplate and the adjusting bolt The configuration of the dampershown in Figure 3 is in its equilibrium position The specificdesign of the damper allows the push-pull rod to move inboth left and right directions and return to its equilibriumposition when the load is removed (self-centering) Duringthis cyclic process the damper provides damping attributedto the hysteretic property of the superelastic wires
32 Experimental Tests
321 Setup and Program An SMA damper based on thedesign presented in Section 31 is fabricated To assess theperformance of the SMA damper cyclic tests were carried
Mathematical Problems in Engineering 7
001563 00625 025 1 40
300
600
900
1200
1500
1800
2100
2400
Log2 frequency (Hz)
Ener
gy d
issip
atio
n pe
r cyc
le (1
0minus3
J)
(a)
0
100
200
300
400
500
600
700
800
Resto
ring
forc
e (N
)
001563 00625 025 1 4Log2 frequency (Hz)
(b)
0
30
60
90
120
150
180
210
240
001563 00625 025 1 4Log2 frequency (Hz)
230mm345mm
460mm575mm
Seca
nt st
iffne
ss (N
mm
minus1)
(c)
000
002
004
006
008
010
012
014
016
Equi
vale
nt d
ampi
ng
001563 00625 025 1 4Log2 frequency (Hz)
230mm345mm
460mm575mm
(d)
Figure 8 Mechanical properties of RSMAD as a function of loading frequencies and displacement amplitudes (06mm predisplacement20∘C temperature)
out The SMA damper is 210mm in length with a diameterof 100mm and a stroke of plusmn92mm (the maximum allowablestrain for superelastic nitinol wires is 8) Four superelasticnitinol wires of a length of 115mm and a diameter of 05mmare used
Tests were conducted using an MTS 810 machine with a100KN load cell at room temperature of 20∘C The layout ofthe test system is shown in Figure 4 The damper was testedwith different prestrains at different loading frequenciesand at various amplitudes During the tests both force anddisplacement are recorded
Prior to the installation each nitinol wire was cycled 20times at 6 strain with 12 times 10minus3 sminus1 strain rate to minimizethe accumulation of residual strain and reach a steady-statecondition The scheme of the tests is described as follows
(1) Without prestrain the damper was subjected tocyclic loading at 005Hz frequencywith displacement
amplitudes of 23mm (2 of total length) 345mm(3 of total length) and 46mm (4 of total length)respectively
(2) Step (1) was repeated with prestrains of 1 (115mmpredisplacement) 2 (23mm predisplacement) and4 (46mm predisplacement) respectively
(3) With 05 prestrain (about 06mm predisplace-ment) the damper was subjected to cyclic loading at001Hz loading frequency with displacement ampli-tudes of 23mm 345mm 46mm and 575mmrespectively
(4) Step (3) was repeated with loading frequencies of005Hz 01 Hz 05Hz 1Hz and 2Hz respectively
To describe the performance of RSMAD as a functionof prestrains loading frequencies and displacement ampli-tudes some important mechanical properties were calcu-lated including the secant stiffness119870119904 the energy dissipation
8 Mathematical Problems in Engineering
0 2 4 6 8
0
100
200
300
400
500
Displacement (mm)
Resto
ring
forc
e (N
)
Experiment Numerical results
minus2minus4minus6minus8
minus500
minus400
minus300
minus200
minus100
Figure 9 Comparison between experimental and numerical fittingcurves of RSMAD
per cycle 119882119863 the restoring force at peak displacement percycle119865119889 and the equivalent viscous damping ratio 120585eq whichis expressed as
120585eq =119882119863
21205871198701199041205752 (10)
where 120575 is the displacement amplitude of the cycle
322 Experimental Results Figure 5 shows the hysteresisloops of RSMAD at different prestrains and displacementamplitudes The tests were carried out at 005Hz frequencyof loading and room temperature of 20∘C As seen fromFigure 5 the SMA damper shows double-flag hystereticbehavior without any residual strain
Figure 6 shows themechanical properties of RSMAD as afunction of prestrain and displacement amplitude As we cansee in Figure 6 the greater the amplitude is the greater theenergy dissipation per cycle and restoring force are Howeverthe secant stiffness reduces markedly while increasing theamplitude With the increasing of the predisplacement theenergy dissipation per cycle decreases and restoring forceand the secant stiffness increase at large displacement Theeffectiveness of a damper is generally measured by theequivalent viscous damping ratio With the increasing of thepredisplacement the equivalent damping ratio reduces Themost important finding is that when the predisplacementis less than 23mm (2 prestrain) the equivalent dampingratio increasedwith the increasing of the amplitude howeverwhen the predisplacement is 46mm (4 prestrain) themaximum of the equivalent damping ratio is at 23mm (2prestrain) This is because the maximum of the equivalentdamping ratio of theNiTi SMA specimens is at about 6 totalstrain as noted in [10]
Figure 7 shows the hysteresis loops of RSMAD with06mm predisplacement at different loading frequencies anddisplacement amplitudes at the temperature of 20∘C Figure 8
shows mechanical properties of RSMAD as a function offrequency of loading and displacement amplitude As wecan see in Figures 7 and 8 the energy dissipation percycle and the equivalent damping decrease as the loadingfrequency increases in the range of 001ndash05Hz but arenot much sensitive to frequencies greater than 05Hz Therestoring force and the secant stiffness increase slightly as theloading frequency increases in the total range of experimentalfrequency
33 Numerical Simulation Based on the improved Graesserand Cozzarelli model of SMA wire a theoretic model of theSMA damper is developed The differential equations of themodel are given as
+ 119891119872[119909 minus 119909Mf sgn (119909)]119898
times [119906 (119909)] [119906 (|119909| minus 119909Mf)]
(11)
where 119865 is restoring force 119909 is displacement 119861 is back-force1198700 is initial stiffness 119861119888 120572119891119879 119899 119886 119888 119909Mf119891119872 and 119898 and areconstants controlling the size of the hysteresis loop 119909in is theinelastic displacement and 119909in = 119909 minus 1198651198700 erf(119909) 119906(119909) andsgn(119909) are respectively the error function the step functionand the signum function which have already been listed inSection 2
Figure 9 shows the comparison of experimental resultswith numerical prediction based on the theoretical modelat different displacement amplitudes The parameters of theconstitutive equation used in this study to simulate the behav-iors of RSMAD are given as follows 1198700 = 380Nmm 119861119888 =330N 120572 = 005 119891119879 = 21 119888 = 00001 119886 = 3 119899 = 2119909Mf = 575mm 119891119872 = 42500 and 119898 = 3 To accuratelysimulate the hysteresis behavior the above parameters weredirectly obtained from the cyclic test results of the SMAdamper according to the parametersrsquo meanings and rolesThevalues also can be converted through the section area andthe length of NiTi wires However there is slight differencebetween the two parameter groups This is possible becauseof the effect of loading conditions on the cyclic behavior ofNiTi wires As can be seen in Figure 9 numerical predictionsagree well with the experimental results
Table 1 shows the comparison of the experimental dataand numerical results respectively of the energy dissipationper cycle the secant stiffness and the equivalent viscousdamping As can be seen in Table 1 the maximum differencesof energy dissipation per cycle secant stiffness and equiva-lent viscous damping are 50 12 and 51 respectivelyThese results indicate that the mechanical behavior of theSMAdamper is well predicted by the numericalmodel whichverifies its suitability for the damper
Mathematical Problems in Engineering 9
(a) Bare structure (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5
Figure 10 Different cases of the ten-story frame structure
Table 1 Comparison between the experimental and theoretical results
Peakdisplacement(mm)
Energy dissipation per cycle (Nsdotmm) Secant stiffness (Nmm) Equivalent viscous damping ()Experimental
Themain objective of including energy dissipating devices ina structure is to reduce structural response through energydissipation To protect the original structural members understrong seismic loading nonlinear deformation in energydissipating devices will be allowed In frame structures thedevices are usually incorporated in steel braces connectingtwo consecutive stories of the buildingThedynamic responseof the structure subjected to earthquake loading is governedby the following equation
where 119872119904 is the mass matrix 119862119904 is the damping coefficientmatrix 119865119904 is the vector of the frames restoring force and 119865119889is the vector of restoring force resulting from SMA dampersThe 119909 and are the structural displacement velocity andacceleration vectors respectively On the right-hand side of
the equation the vector 119868 is the influence vector and thevector 119892 is the ground motion acceleration input
With the aid of the SIMULINK module of MATLAB adynamical simulation system was developed in which theWen model [36] is utilized to simulate the restoring forcecurve of the steel frame structure and the improved Graesserand Cozzarelli model presented above is used for NiTi SMAdamper
5 Seismic Structural Control UsingSMA Dampers
In this section to assess the effectiveness of the proposedrecentering SMA dampers (RSMAD) in mitigating the seis-mic response of building structure nonlinear time historyanalysis on a multistory steel frame with and withoutthe dampers subjected to representative earthquake groundmotions was performed The improved Graesser and Coz-zarelli model for RSMAD given in Section 33 was employedin this numerical study A ten-story steel moment resisting
10 Mathematical Problems in Engineering
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0
10
20 40 60 80 100 120 140 1600123456789
Lateral displacement (mm)
Floo
r lev
elEl Centro
TaftS69E
Case 0Case 5Case 4
Case 3Case 2Case 1
PGA = 02 g
PGA = 02 g
PGA = 02 gTangshan-Beijing
(a)
0 5 10 15 20 25 30
000002004006008010
Disp
lace
men
t (m
)
Time (s)
0 5 10 15 20 25 30Time (s)
0 5 10 15 20Time (s)
Without controlWith case 5
El Centro
000002004006008010012014
Disp
lace
men
t (m
)
Taft S69E
Disp
lace
men
t (m
)
PGA = 02 g
PGA = 02 g
PGA = 02 g
minus010
minus008
minus006
minus004
minus002
minus014
minus012
minus010
minus008
minus006
minus004
minus002
000002004006008010012014016
minus014
minus016
minus012
minus010
minus008
minus006
minus004
minus002
Tangshan-Beijing
(b)
Figure 11 Lateral displacement envelopes (a) and roof displacement timehistories (b)with andwithout SMAdampers under basic (moderate)ground motions
frame structure was selected for this study The building isdesigned for a location in Beijing China The parameters ofthe structure are listed in Table 2
Three representative earthquake ground motions wereused to considering the site effect Three earthquake groundmotions namely Imperial Valley 1940 El Centro LincolnSchool 1952 Taft and Beijing Hotel 1976 Tangshan wereselected These three ground motions represent different siteconditions According to Chinese code for seismic design
of building [37] the local seismic precautionary intensity iseight degrees The peak ground accelerations (PGA) wereadjusted to 02 g and 04 g corresponding to a seismic hazardlevel of 10 and 2 probability of exceedance in a 50-yearperiod respectively
Simulation analysis is conducted on the bare structureand on the structure with five or ten SMA dampers installedas shown in Figure 10 Parameters of the SMA damper arelisted as follows 1198700 = 119870119889 = 120 kNmm 119861119888 = 360KN
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 40 80 120 160 200 240 2800123456789
10
Lateral displacement (mm)
Floo
r lev
el
El CentroPGA = 04 g
Taft S69EPGA = 04 g
PGA = 04 g
Case 0Case 5Case 4
Case 3Case 2Case 1
Tangshan-Beijing
(a)
El Centro
0 5 10 15 20
0 5 10 15 20
25 30
000
005
010
015
020
Time (s)
Disp
lace
men
t (m
)
Taft S69E
000005010015020025
Disp
lace
men
t (m
)
Time (s)
PGA = 04 g
PGA = 04 g
PGA = 04 g
minus020
minus015
minus010
minus005
minus020
minus025
minus015
minus010
minus005
0 5 10 15 20 25 30
000
008004
012016020
Time (s)
Disp
lace
men
t (m
)
minus020
minus016
minus010
minus004
minus008
Without controlWith case 5
Tangshan-Beijing
(b)
Figure 12 Lateral displacement envelopes (a) and roof displacement time histories (b) with and without SMA dampers under strong (severe)ground motions
Figures 11 and 12 show the lateral displacement envelopeswith and without SMA dampers under basic groundmotions(02 g) and strong ground motions (04 g) respectively Aswe can see for most cases the lateral displacement of thestructure decreases remarkably with the introduction of theSMA dampers However the location and the number ofthe dampers have significant effects on the control resultsCase 5 in which dampers are installed in all stories is
the best for the overall structural vibration control For other4 configurations in which five dampers are installed indifferent stores cases 3 and 4 with dampers installed inalternate stories perform better than cases 1 and 2 withdampers placed in consecutive stories Moreover case 3 isbetter than case 4 since the first story is retrofitted Case 1is slightly worse due to the whiplash effect and case 2 isthe worst since the stiffness of the lower half is significantlygreater than that of the upper half of the structure Theroof displacement time histories with (case 5) and without
12 Mathematical Problems in Engineering
SMAdampers under basic groundmotions (02 g) and strongground motions (04 g) are also respectively provided inFigures 11 and 12 which confirm that the SMA damperssignificantly reduce the structural vibration
6 Concluding Remarks
This paper presents the results of a study on evaluating theefficacy of using an innovative SMA-based damper to reducethe seismic response of structures To describe the hysteresisbehavior of the SMA damper an improved Graesser andCozzarelli model was proposed and verified by the cyclictensile test on SMA wires
Cyclic tests on the SMA damper model utilizing foursuperelastic SMA wires with 05mm diameter with variousprestrains under different loading frequencies and displace-ment amplitudes were carried outThe results show satisfyinghysteresis properties including both recentering and energydissipating features under various conditions
A comparative study on nonlinear time history analysisof the seismic response of a ten-story steel frame with theSMA dampers was performed Five cases were considered forthe location and the number of the dampers in the storiesThe numerical analysis results indicate that the proposedSMA damper is capable of significantly reducing seismicresponse of structures which verifies its effectiveness asenergy dissipating device for structures However it is alsoindicated that the location and number have significanteffects on the results of the response
In future multiobjective optimization model will beproposed to obtain the number and the location of the SMAdampers and large scale shake table tests will be performed ona steel frame buildingwith SMAdampers to prove the efficacyof these dampers in dissipating seismic energy
Acknowledgments
This work was funded by National Science Foundation ofChina (no 51108426 and no 41104106) China PostdoctoralScience Foundation (no 20100471008) and Research Fundfor the Doctoral Program of Higher Education of China (no20104101120009) These supports are greatly appreciatedTheopinions expressed in this study are those of the authors anddo not necessarily reflect the views of the sponsor
References
[1] H N Li and L S Huo ldquoAdvances in structural control in civilengineering in Chinardquo Mathematical Problems in Engineeringvol 2010 Article ID 936081 23 pages 2010
[2] G Song N Ma and H N Li ldquoApplications of shape memoryalloys in civil structuresrdquo Engineering Structures vol 28 no 9pp 1266ndash1274 2006
[3] SEAOC Vision 2000 Committee Performance-Based SeismicEngineering Structural Engineering Association of CaliforniaSacramento Calif USA 1995
[4] ATC-40 Seismic Evaluation and Retrofit of Conctete BuildingsApplied Technology Council 1996
[5] FEMA 273 NEHRP Guidelines for Seismic Rehabilitation ofBuildings Federal Emergency Management Agency 1997
[6] Y Fujino T T Soong and B F Spencer Jr ldquoStructural controlbasic concepts and applicationsrdquo in Proceedings of the ASCEStructures Congress pp 15ndash18 Chicago Ill USA April 1996
[7] B F Spencer and S Nagarajaiah ldquoState of the art of structuralcontrolrdquo Journal of Structural Engineering vol 129 no 7 pp845ndash856 2003
[8] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering John Wiley amp Sons NewYork NY USA 1997
[9] F M Mazzolani ldquoPassive control technologies for seismic-resistant buildings in Europerdquo Progress in Structural Engineeringand Materials vol 3 no 3 pp 277ndash287 2001
[10] M Dolce and D Cardone ldquoMechanical behaviour of SMAelements for seismic applicationsmdashpart 2 austenite NiTi wiressubjected to tensionrdquo International Journal of Mechanical Sci-ences vol 43 no 11 pp 2657ndash2677 2001
[11] H N Li and X X Wu ldquoLimitations of height-to-width ratiofor base-isolated buildings under earthquakerdquo The StructuralDesign of Tall and Special Buildings vol 15 no 3 pp 277ndash2872006
[12] R Desroches J McCormick and M A Delemont ldquoCyclicproperties of superelastic shape memory alloy wires and barsrdquoJournal of Structural Engineering vol 130 no 1 pp 38ndash46 2004
[13] T W Duerig K N Melton D Stockel and C M Way-man Engineering Aspects of Shape Memory Alloys ButterworthHeinemann London UK 1990
[14] K Wilde P Gardoni and Y Fujino ldquoBase isolation systemwith shape memory alloy device for elevated highway bridgesrdquoEngineering Structures vol 22 no 3 pp 222ndash229 2000
[15] M Dolce D Cardone and R Marnetto ldquoImplementation andtesting of passive control devices based on shape memoryalloysrdquo Earthquake Engineering and Structural Dynamics vol29 no 7 pp 945ndash968 2000
[16] M Dolce D Cardone F C Ponzo and C Valente ldquoShakingtable tests on reinforced concrete frames without and withpassive control systemsrdquo Earthquake Engineering and StructuralDynamics vol 34 no 14 pp 1687ndash1717 2005
[17] M Indirli M G Castellano P Clemente and A MartellildquoDemo-application of shapememory alloy devices the rehabili-tation of the S Giorgio Church Bell-Towerrdquo in Smart Structuresand Materials 2001 Smart Systems for Bridges Structures andHighways vol 4330 of Proceedings of SPIE pp 262ndash272 New-port Beach Calif USA March 2001
[18] B Andrawes and R Desroches ldquoUnseating prevention for mul-tiple frame bridges using superelastic devicesrdquo Smart Materialsand Structures vol 14 no 3 pp S60ndashS67 2005
[19] R DesRoches and M Delemont ldquoSeismic retrofit of simplysupported bridges using shape memory alloysrdquo EngineeringStructures vol 24 no 3 pp 325ndash332 2002
[20] H Li M Liu and J Ou ldquoVibration mitigation of a stay cablewith one shape memory alloy damperrdquo Structural Control andHealth Monitoring vol 11 no 1 pp 21ndash36 2004
[21] Y Zhang and S Zhu ldquoA shape memory alloy-based reusablehysteretic damper for seismic hazard mitigationrdquo Smart Mate-rials and Structures vol 16 no 5 pp 1603ndash1613 2007
[22] J Ocel R DesRoches R T Leon et al ldquoSteel beam-columnconnections using shape memory alloysrdquo Journal of StructuralEngineering vol 130 no 5 pp 732ndash740 2004
Mathematical Problems in Engineering 13
[23] J McCormick R Desroches D Fugazza and F AuricchioldquoSeismic assessment of concentrically braced steel frames withshape memory alloy bracesrdquo Journal of Structural Engineeringvol 133 no 6 pp 862ndash870 2007
[24] YM Parulekar G R Reddy K K Vaze et al ldquoSeismic responseattenuation of structures using shape memory alloy dampersrdquoStructural Control and Health Monitoring vol 19 no 1 pp 102ndash119 2012
[25] K Tanaka ldquoA thermomechanical sketch of shape memoryeffect one-dimensional tensile behaviorrdquoResMechanica vol 18no 3 pp 251ndash263 1986
[26] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 1 no 2 pp207ndash234 1990
[27] L C Brinson ldquoOne-dimensional constitutive behavior of shapememory alloys thermomechanical derivation with non-con-stant material functions and redefined martensite internal vari-ablerdquo Journal of Intelligent Material Systems and Structures vol4 no 2 pp 229ndash242 1993
[28] F Falk ldquoModel free energy mechanics and thermodynamicsof shape memory alloysrdquo Acta Metallurgica vol 28 no 12 pp1773ndash1780 1980
[29] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashII study of the individual phenomenardquo Journal of theMechanicsand Physics of Solids vol 41 no 1 pp 19ndash33 1993
[30] J G Boyd andDC Lagoudas ldquoA thermodynamical constitutivemodel for shape memory materialsmdashpart I the monolithicshape memory alloyrdquo International Journal of Plasticity vol 12no 6 pp 805ndash842 1996
[31] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashI derivation of general relationsrdquo Journal of the Mechanics andPhysics of Solids vol 41 no 1 pp 1ndash17 1993
[32] H Qian H Li G Song and W Guo ldquoA constitutive modelfor superelastic shape memory alloys considering the influenceof strain raterdquoMathematical Problems in Engineering vol 2013Article ID 248671 8 pages 2013
[33] W J Ren H N Li and G Song ldquoA one-dimensional strain-ratedependent constitutive model for superelastic shape memoryalloysrdquo Smart Materials and Structures vol 16 no 1 pp 191ndash1972007
[34] E J Graesser and F A Cozzarelli ldquoShape-memory alloys asnew materials for aseismic isolationrdquo Journal of EngineeringMechanics vol 117 no 11 pp 2590ndash2608 1991
[35] H Ozdemir Nonlinear transient dynamic analysis of yieldingstructures [PhD thesis] University of California BerkeleyCalif USA 1976
[36] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976
[37] GB 50011-2010 Code for Seismic Design of Buildings Ministryof Housing and Urban-Rural Development of the PeoplersquosRepublic of China 2010
Figure 7 Hysteresis loops of the RSMAD at different loading frequencies and displacement amplitudes (06mm predisplacement 20∘Ctemperature)
testing the nitinol SMA specimens were cycled 20 timesat 6 strain amplitude and 12 times 10minus3 sminus1 strain rate bya ldquotrainingrdquo process to reach a steady-state condition Theexperimental data in Figure 2 are the results of cyclic tests onthe superelastic nitinol wire at 12 times 10minus3 sminus1 strain rate with1 to 8 strain levels As shown in Figure 2 the hysteresisloops based on the improved Graesser and Cozzarelli modeland experimental data match with close accuracy Moreoverthe improved model can accurately reflect the martensitichardening characteristic of SMAs under large amplitudes
3 An Innovative SMA Damper DesignExperiment and Numerical Simulation
31 Recentering SMADamper Design By utilizing the energydissipating and recentering features of superelastic nitinolSMA an innovative damper is designed As shown inFigure 3 the damper consists of outer and inner cylinders leftand right pull plates superelastic SMA wires retaining plate
prestrain adjusting plate adjusting bolt fixed bolt push-pullrod grip end caps and connecting fitting In this dampersuperelastic nitinolwires are the key components that provideboth damping and self-centering abilities
The damper will be connected to a structure via its push-pull rod and the connecting fitting The prestrain of thesuperelastic wires can be adjusted by the prestrain adjustingplate and the adjusting bolt The configuration of the dampershown in Figure 3 is in its equilibrium position The specificdesign of the damper allows the push-pull rod to move inboth left and right directions and return to its equilibriumposition when the load is removed (self-centering) Duringthis cyclic process the damper provides damping attributedto the hysteretic property of the superelastic wires
32 Experimental Tests
321 Setup and Program An SMA damper based on thedesign presented in Section 31 is fabricated To assess theperformance of the SMA damper cyclic tests were carried
Mathematical Problems in Engineering 7
001563 00625 025 1 40
300
600
900
1200
1500
1800
2100
2400
Log2 frequency (Hz)
Ener
gy d
issip
atio
n pe
r cyc
le (1
0minus3
J)
(a)
0
100
200
300
400
500
600
700
800
Resto
ring
forc
e (N
)
001563 00625 025 1 4Log2 frequency (Hz)
(b)
0
30
60
90
120
150
180
210
240
001563 00625 025 1 4Log2 frequency (Hz)
230mm345mm
460mm575mm
Seca
nt st
iffne
ss (N
mm
minus1)
(c)
000
002
004
006
008
010
012
014
016
Equi
vale
nt d
ampi
ng
001563 00625 025 1 4Log2 frequency (Hz)
230mm345mm
460mm575mm
(d)
Figure 8 Mechanical properties of RSMAD as a function of loading frequencies and displacement amplitudes (06mm predisplacement20∘C temperature)
out The SMA damper is 210mm in length with a diameterof 100mm and a stroke of plusmn92mm (the maximum allowablestrain for superelastic nitinol wires is 8) Four superelasticnitinol wires of a length of 115mm and a diameter of 05mmare used
Tests were conducted using an MTS 810 machine with a100KN load cell at room temperature of 20∘C The layout ofthe test system is shown in Figure 4 The damper was testedwith different prestrains at different loading frequenciesand at various amplitudes During the tests both force anddisplacement are recorded
Prior to the installation each nitinol wire was cycled 20times at 6 strain with 12 times 10minus3 sminus1 strain rate to minimizethe accumulation of residual strain and reach a steady-statecondition The scheme of the tests is described as follows
(1) Without prestrain the damper was subjected tocyclic loading at 005Hz frequencywith displacement
amplitudes of 23mm (2 of total length) 345mm(3 of total length) and 46mm (4 of total length)respectively
(2) Step (1) was repeated with prestrains of 1 (115mmpredisplacement) 2 (23mm predisplacement) and4 (46mm predisplacement) respectively
(3) With 05 prestrain (about 06mm predisplace-ment) the damper was subjected to cyclic loading at001Hz loading frequency with displacement ampli-tudes of 23mm 345mm 46mm and 575mmrespectively
(4) Step (3) was repeated with loading frequencies of005Hz 01 Hz 05Hz 1Hz and 2Hz respectively
To describe the performance of RSMAD as a functionof prestrains loading frequencies and displacement ampli-tudes some important mechanical properties were calcu-lated including the secant stiffness119870119904 the energy dissipation
8 Mathematical Problems in Engineering
0 2 4 6 8
0
100
200
300
400
500
Displacement (mm)
Resto
ring
forc
e (N
)
Experiment Numerical results
minus2minus4minus6minus8
minus500
minus400
minus300
minus200
minus100
Figure 9 Comparison between experimental and numerical fittingcurves of RSMAD
per cycle 119882119863 the restoring force at peak displacement percycle119865119889 and the equivalent viscous damping ratio 120585eq whichis expressed as
120585eq =119882119863
21205871198701199041205752 (10)
where 120575 is the displacement amplitude of the cycle
322 Experimental Results Figure 5 shows the hysteresisloops of RSMAD at different prestrains and displacementamplitudes The tests were carried out at 005Hz frequencyof loading and room temperature of 20∘C As seen fromFigure 5 the SMA damper shows double-flag hystereticbehavior without any residual strain
Figure 6 shows themechanical properties of RSMAD as afunction of prestrain and displacement amplitude As we cansee in Figure 6 the greater the amplitude is the greater theenergy dissipation per cycle and restoring force are Howeverthe secant stiffness reduces markedly while increasing theamplitude With the increasing of the predisplacement theenergy dissipation per cycle decreases and restoring forceand the secant stiffness increase at large displacement Theeffectiveness of a damper is generally measured by theequivalent viscous damping ratio With the increasing of thepredisplacement the equivalent damping ratio reduces Themost important finding is that when the predisplacementis less than 23mm (2 prestrain) the equivalent dampingratio increasedwith the increasing of the amplitude howeverwhen the predisplacement is 46mm (4 prestrain) themaximum of the equivalent damping ratio is at 23mm (2prestrain) This is because the maximum of the equivalentdamping ratio of theNiTi SMA specimens is at about 6 totalstrain as noted in [10]
Figure 7 shows the hysteresis loops of RSMAD with06mm predisplacement at different loading frequencies anddisplacement amplitudes at the temperature of 20∘C Figure 8
shows mechanical properties of RSMAD as a function offrequency of loading and displacement amplitude As wecan see in Figures 7 and 8 the energy dissipation percycle and the equivalent damping decrease as the loadingfrequency increases in the range of 001ndash05Hz but arenot much sensitive to frequencies greater than 05Hz Therestoring force and the secant stiffness increase slightly as theloading frequency increases in the total range of experimentalfrequency
33 Numerical Simulation Based on the improved Graesserand Cozzarelli model of SMA wire a theoretic model of theSMA damper is developed The differential equations of themodel are given as
+ 119891119872[119909 minus 119909Mf sgn (119909)]119898
times [119906 (119909)] [119906 (|119909| minus 119909Mf)]
(11)
where 119865 is restoring force 119909 is displacement 119861 is back-force1198700 is initial stiffness 119861119888 120572119891119879 119899 119886 119888 119909Mf119891119872 and 119898 and areconstants controlling the size of the hysteresis loop 119909in is theinelastic displacement and 119909in = 119909 minus 1198651198700 erf(119909) 119906(119909) andsgn(119909) are respectively the error function the step functionand the signum function which have already been listed inSection 2
Figure 9 shows the comparison of experimental resultswith numerical prediction based on the theoretical modelat different displacement amplitudes The parameters of theconstitutive equation used in this study to simulate the behav-iors of RSMAD are given as follows 1198700 = 380Nmm 119861119888 =330N 120572 = 005 119891119879 = 21 119888 = 00001 119886 = 3 119899 = 2119909Mf = 575mm 119891119872 = 42500 and 119898 = 3 To accuratelysimulate the hysteresis behavior the above parameters weredirectly obtained from the cyclic test results of the SMAdamper according to the parametersrsquo meanings and rolesThevalues also can be converted through the section area andthe length of NiTi wires However there is slight differencebetween the two parameter groups This is possible becauseof the effect of loading conditions on the cyclic behavior ofNiTi wires As can be seen in Figure 9 numerical predictionsagree well with the experimental results
Table 1 shows the comparison of the experimental dataand numerical results respectively of the energy dissipationper cycle the secant stiffness and the equivalent viscousdamping As can be seen in Table 1 the maximum differencesof energy dissipation per cycle secant stiffness and equiva-lent viscous damping are 50 12 and 51 respectivelyThese results indicate that the mechanical behavior of theSMAdamper is well predicted by the numericalmodel whichverifies its suitability for the damper
Mathematical Problems in Engineering 9
(a) Bare structure (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5
Figure 10 Different cases of the ten-story frame structure
Table 1 Comparison between the experimental and theoretical results
Peakdisplacement(mm)
Energy dissipation per cycle (Nsdotmm) Secant stiffness (Nmm) Equivalent viscous damping ()Experimental
Themain objective of including energy dissipating devices ina structure is to reduce structural response through energydissipation To protect the original structural members understrong seismic loading nonlinear deformation in energydissipating devices will be allowed In frame structures thedevices are usually incorporated in steel braces connectingtwo consecutive stories of the buildingThedynamic responseof the structure subjected to earthquake loading is governedby the following equation
where 119872119904 is the mass matrix 119862119904 is the damping coefficientmatrix 119865119904 is the vector of the frames restoring force and 119865119889is the vector of restoring force resulting from SMA dampersThe 119909 and are the structural displacement velocity andacceleration vectors respectively On the right-hand side of
the equation the vector 119868 is the influence vector and thevector 119892 is the ground motion acceleration input
With the aid of the SIMULINK module of MATLAB adynamical simulation system was developed in which theWen model [36] is utilized to simulate the restoring forcecurve of the steel frame structure and the improved Graesserand Cozzarelli model presented above is used for NiTi SMAdamper
5 Seismic Structural Control UsingSMA Dampers
In this section to assess the effectiveness of the proposedrecentering SMA dampers (RSMAD) in mitigating the seis-mic response of building structure nonlinear time historyanalysis on a multistory steel frame with and withoutthe dampers subjected to representative earthquake groundmotions was performed The improved Graesser and Coz-zarelli model for RSMAD given in Section 33 was employedin this numerical study A ten-story steel moment resisting
10 Mathematical Problems in Engineering
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0
10
20 40 60 80 100 120 140 1600123456789
Lateral displacement (mm)
Floo
r lev
elEl Centro
TaftS69E
Case 0Case 5Case 4
Case 3Case 2Case 1
PGA = 02 g
PGA = 02 g
PGA = 02 gTangshan-Beijing
(a)
0 5 10 15 20 25 30
000002004006008010
Disp
lace
men
t (m
)
Time (s)
0 5 10 15 20 25 30Time (s)
0 5 10 15 20Time (s)
Without controlWith case 5
El Centro
000002004006008010012014
Disp
lace
men
t (m
)
Taft S69E
Disp
lace
men
t (m
)
PGA = 02 g
PGA = 02 g
PGA = 02 g
minus010
minus008
minus006
minus004
minus002
minus014
minus012
minus010
minus008
minus006
minus004
minus002
000002004006008010012014016
minus014
minus016
minus012
minus010
minus008
minus006
minus004
minus002
Tangshan-Beijing
(b)
Figure 11 Lateral displacement envelopes (a) and roof displacement timehistories (b)with andwithout SMAdampers under basic (moderate)ground motions
frame structure was selected for this study The building isdesigned for a location in Beijing China The parameters ofthe structure are listed in Table 2
Three representative earthquake ground motions wereused to considering the site effect Three earthquake groundmotions namely Imperial Valley 1940 El Centro LincolnSchool 1952 Taft and Beijing Hotel 1976 Tangshan wereselected These three ground motions represent different siteconditions According to Chinese code for seismic design
of building [37] the local seismic precautionary intensity iseight degrees The peak ground accelerations (PGA) wereadjusted to 02 g and 04 g corresponding to a seismic hazardlevel of 10 and 2 probability of exceedance in a 50-yearperiod respectively
Simulation analysis is conducted on the bare structureand on the structure with five or ten SMA dampers installedas shown in Figure 10 Parameters of the SMA damper arelisted as follows 1198700 = 119870119889 = 120 kNmm 119861119888 = 360KN
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 40 80 120 160 200 240 2800123456789
10
Lateral displacement (mm)
Floo
r lev
el
El CentroPGA = 04 g
Taft S69EPGA = 04 g
PGA = 04 g
Case 0Case 5Case 4
Case 3Case 2Case 1
Tangshan-Beijing
(a)
El Centro
0 5 10 15 20
0 5 10 15 20
25 30
000
005
010
015
020
Time (s)
Disp
lace
men
t (m
)
Taft S69E
000005010015020025
Disp
lace
men
t (m
)
Time (s)
PGA = 04 g
PGA = 04 g
PGA = 04 g
minus020
minus015
minus010
minus005
minus020
minus025
minus015
minus010
minus005
0 5 10 15 20 25 30
000
008004
012016020
Time (s)
Disp
lace
men
t (m
)
minus020
minus016
minus010
minus004
minus008
Without controlWith case 5
Tangshan-Beijing
(b)
Figure 12 Lateral displacement envelopes (a) and roof displacement time histories (b) with and without SMA dampers under strong (severe)ground motions
Figures 11 and 12 show the lateral displacement envelopeswith and without SMA dampers under basic groundmotions(02 g) and strong ground motions (04 g) respectively Aswe can see for most cases the lateral displacement of thestructure decreases remarkably with the introduction of theSMA dampers However the location and the number ofthe dampers have significant effects on the control resultsCase 5 in which dampers are installed in all stories is
the best for the overall structural vibration control For other4 configurations in which five dampers are installed indifferent stores cases 3 and 4 with dampers installed inalternate stories perform better than cases 1 and 2 withdampers placed in consecutive stories Moreover case 3 isbetter than case 4 since the first story is retrofitted Case 1is slightly worse due to the whiplash effect and case 2 isthe worst since the stiffness of the lower half is significantlygreater than that of the upper half of the structure Theroof displacement time histories with (case 5) and without
12 Mathematical Problems in Engineering
SMAdampers under basic groundmotions (02 g) and strongground motions (04 g) are also respectively provided inFigures 11 and 12 which confirm that the SMA damperssignificantly reduce the structural vibration
6 Concluding Remarks
This paper presents the results of a study on evaluating theefficacy of using an innovative SMA-based damper to reducethe seismic response of structures To describe the hysteresisbehavior of the SMA damper an improved Graesser andCozzarelli model was proposed and verified by the cyclictensile test on SMA wires
Cyclic tests on the SMA damper model utilizing foursuperelastic SMA wires with 05mm diameter with variousprestrains under different loading frequencies and displace-ment amplitudes were carried outThe results show satisfyinghysteresis properties including both recentering and energydissipating features under various conditions
A comparative study on nonlinear time history analysisof the seismic response of a ten-story steel frame with theSMA dampers was performed Five cases were considered forthe location and the number of the dampers in the storiesThe numerical analysis results indicate that the proposedSMA damper is capable of significantly reducing seismicresponse of structures which verifies its effectiveness asenergy dissipating device for structures However it is alsoindicated that the location and number have significanteffects on the results of the response
In future multiobjective optimization model will beproposed to obtain the number and the location of the SMAdampers and large scale shake table tests will be performed ona steel frame buildingwith SMAdampers to prove the efficacyof these dampers in dissipating seismic energy
Acknowledgments
This work was funded by National Science Foundation ofChina (no 51108426 and no 41104106) China PostdoctoralScience Foundation (no 20100471008) and Research Fundfor the Doctoral Program of Higher Education of China (no20104101120009) These supports are greatly appreciatedTheopinions expressed in this study are those of the authors anddo not necessarily reflect the views of the sponsor
References
[1] H N Li and L S Huo ldquoAdvances in structural control in civilengineering in Chinardquo Mathematical Problems in Engineeringvol 2010 Article ID 936081 23 pages 2010
[2] G Song N Ma and H N Li ldquoApplications of shape memoryalloys in civil structuresrdquo Engineering Structures vol 28 no 9pp 1266ndash1274 2006
[3] SEAOC Vision 2000 Committee Performance-Based SeismicEngineering Structural Engineering Association of CaliforniaSacramento Calif USA 1995
[4] ATC-40 Seismic Evaluation and Retrofit of Conctete BuildingsApplied Technology Council 1996
[5] FEMA 273 NEHRP Guidelines for Seismic Rehabilitation ofBuildings Federal Emergency Management Agency 1997
[6] Y Fujino T T Soong and B F Spencer Jr ldquoStructural controlbasic concepts and applicationsrdquo in Proceedings of the ASCEStructures Congress pp 15ndash18 Chicago Ill USA April 1996
[7] B F Spencer and S Nagarajaiah ldquoState of the art of structuralcontrolrdquo Journal of Structural Engineering vol 129 no 7 pp845ndash856 2003
[8] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering John Wiley amp Sons NewYork NY USA 1997
[9] F M Mazzolani ldquoPassive control technologies for seismic-resistant buildings in Europerdquo Progress in Structural Engineeringand Materials vol 3 no 3 pp 277ndash287 2001
[10] M Dolce and D Cardone ldquoMechanical behaviour of SMAelements for seismic applicationsmdashpart 2 austenite NiTi wiressubjected to tensionrdquo International Journal of Mechanical Sci-ences vol 43 no 11 pp 2657ndash2677 2001
[11] H N Li and X X Wu ldquoLimitations of height-to-width ratiofor base-isolated buildings under earthquakerdquo The StructuralDesign of Tall and Special Buildings vol 15 no 3 pp 277ndash2872006
[12] R Desroches J McCormick and M A Delemont ldquoCyclicproperties of superelastic shape memory alloy wires and barsrdquoJournal of Structural Engineering vol 130 no 1 pp 38ndash46 2004
[13] T W Duerig K N Melton D Stockel and C M Way-man Engineering Aspects of Shape Memory Alloys ButterworthHeinemann London UK 1990
[14] K Wilde P Gardoni and Y Fujino ldquoBase isolation systemwith shape memory alloy device for elevated highway bridgesrdquoEngineering Structures vol 22 no 3 pp 222ndash229 2000
[15] M Dolce D Cardone and R Marnetto ldquoImplementation andtesting of passive control devices based on shape memoryalloysrdquo Earthquake Engineering and Structural Dynamics vol29 no 7 pp 945ndash968 2000
[16] M Dolce D Cardone F C Ponzo and C Valente ldquoShakingtable tests on reinforced concrete frames without and withpassive control systemsrdquo Earthquake Engineering and StructuralDynamics vol 34 no 14 pp 1687ndash1717 2005
[17] M Indirli M G Castellano P Clemente and A MartellildquoDemo-application of shapememory alloy devices the rehabili-tation of the S Giorgio Church Bell-Towerrdquo in Smart Structuresand Materials 2001 Smart Systems for Bridges Structures andHighways vol 4330 of Proceedings of SPIE pp 262ndash272 New-port Beach Calif USA March 2001
[18] B Andrawes and R Desroches ldquoUnseating prevention for mul-tiple frame bridges using superelastic devicesrdquo Smart Materialsand Structures vol 14 no 3 pp S60ndashS67 2005
[19] R DesRoches and M Delemont ldquoSeismic retrofit of simplysupported bridges using shape memory alloysrdquo EngineeringStructures vol 24 no 3 pp 325ndash332 2002
[20] H Li M Liu and J Ou ldquoVibration mitigation of a stay cablewith one shape memory alloy damperrdquo Structural Control andHealth Monitoring vol 11 no 1 pp 21ndash36 2004
[21] Y Zhang and S Zhu ldquoA shape memory alloy-based reusablehysteretic damper for seismic hazard mitigationrdquo Smart Mate-rials and Structures vol 16 no 5 pp 1603ndash1613 2007
[22] J Ocel R DesRoches R T Leon et al ldquoSteel beam-columnconnections using shape memory alloysrdquo Journal of StructuralEngineering vol 130 no 5 pp 732ndash740 2004
Mathematical Problems in Engineering 13
[23] J McCormick R Desroches D Fugazza and F AuricchioldquoSeismic assessment of concentrically braced steel frames withshape memory alloy bracesrdquo Journal of Structural Engineeringvol 133 no 6 pp 862ndash870 2007
[24] YM Parulekar G R Reddy K K Vaze et al ldquoSeismic responseattenuation of structures using shape memory alloy dampersrdquoStructural Control and Health Monitoring vol 19 no 1 pp 102ndash119 2012
[25] K Tanaka ldquoA thermomechanical sketch of shape memoryeffect one-dimensional tensile behaviorrdquoResMechanica vol 18no 3 pp 251ndash263 1986
[26] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 1 no 2 pp207ndash234 1990
[27] L C Brinson ldquoOne-dimensional constitutive behavior of shapememory alloys thermomechanical derivation with non-con-stant material functions and redefined martensite internal vari-ablerdquo Journal of Intelligent Material Systems and Structures vol4 no 2 pp 229ndash242 1993
[28] F Falk ldquoModel free energy mechanics and thermodynamicsof shape memory alloysrdquo Acta Metallurgica vol 28 no 12 pp1773ndash1780 1980
[29] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashII study of the individual phenomenardquo Journal of theMechanicsand Physics of Solids vol 41 no 1 pp 19ndash33 1993
[30] J G Boyd andDC Lagoudas ldquoA thermodynamical constitutivemodel for shape memory materialsmdashpart I the monolithicshape memory alloyrdquo International Journal of Plasticity vol 12no 6 pp 805ndash842 1996
[31] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashI derivation of general relationsrdquo Journal of the Mechanics andPhysics of Solids vol 41 no 1 pp 1ndash17 1993
[32] H Qian H Li G Song and W Guo ldquoA constitutive modelfor superelastic shape memory alloys considering the influenceof strain raterdquoMathematical Problems in Engineering vol 2013Article ID 248671 8 pages 2013
[33] W J Ren H N Li and G Song ldquoA one-dimensional strain-ratedependent constitutive model for superelastic shape memoryalloysrdquo Smart Materials and Structures vol 16 no 1 pp 191ndash1972007
[34] E J Graesser and F A Cozzarelli ldquoShape-memory alloys asnew materials for aseismic isolationrdquo Journal of EngineeringMechanics vol 117 no 11 pp 2590ndash2608 1991
[35] H Ozdemir Nonlinear transient dynamic analysis of yieldingstructures [PhD thesis] University of California BerkeleyCalif USA 1976
[36] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976
[37] GB 50011-2010 Code for Seismic Design of Buildings Ministryof Housing and Urban-Rural Development of the PeoplersquosRepublic of China 2010
Figure 8 Mechanical properties of RSMAD as a function of loading frequencies and displacement amplitudes (06mm predisplacement20∘C temperature)
out The SMA damper is 210mm in length with a diameterof 100mm and a stroke of plusmn92mm (the maximum allowablestrain for superelastic nitinol wires is 8) Four superelasticnitinol wires of a length of 115mm and a diameter of 05mmare used
Tests were conducted using an MTS 810 machine with a100KN load cell at room temperature of 20∘C The layout ofthe test system is shown in Figure 4 The damper was testedwith different prestrains at different loading frequenciesand at various amplitudes During the tests both force anddisplacement are recorded
Prior to the installation each nitinol wire was cycled 20times at 6 strain with 12 times 10minus3 sminus1 strain rate to minimizethe accumulation of residual strain and reach a steady-statecondition The scheme of the tests is described as follows
(1) Without prestrain the damper was subjected tocyclic loading at 005Hz frequencywith displacement
amplitudes of 23mm (2 of total length) 345mm(3 of total length) and 46mm (4 of total length)respectively
(2) Step (1) was repeated with prestrains of 1 (115mmpredisplacement) 2 (23mm predisplacement) and4 (46mm predisplacement) respectively
(3) With 05 prestrain (about 06mm predisplace-ment) the damper was subjected to cyclic loading at001Hz loading frequency with displacement ampli-tudes of 23mm 345mm 46mm and 575mmrespectively
(4) Step (3) was repeated with loading frequencies of005Hz 01 Hz 05Hz 1Hz and 2Hz respectively
To describe the performance of RSMAD as a functionof prestrains loading frequencies and displacement ampli-tudes some important mechanical properties were calcu-lated including the secant stiffness119870119904 the energy dissipation
8 Mathematical Problems in Engineering
0 2 4 6 8
0
100
200
300
400
500
Displacement (mm)
Resto
ring
forc
e (N
)
Experiment Numerical results
minus2minus4minus6minus8
minus500
minus400
minus300
minus200
minus100
Figure 9 Comparison between experimental and numerical fittingcurves of RSMAD
per cycle 119882119863 the restoring force at peak displacement percycle119865119889 and the equivalent viscous damping ratio 120585eq whichis expressed as
120585eq =119882119863
21205871198701199041205752 (10)
where 120575 is the displacement amplitude of the cycle
322 Experimental Results Figure 5 shows the hysteresisloops of RSMAD at different prestrains and displacementamplitudes The tests were carried out at 005Hz frequencyof loading and room temperature of 20∘C As seen fromFigure 5 the SMA damper shows double-flag hystereticbehavior without any residual strain
Figure 6 shows themechanical properties of RSMAD as afunction of prestrain and displacement amplitude As we cansee in Figure 6 the greater the amplitude is the greater theenergy dissipation per cycle and restoring force are Howeverthe secant stiffness reduces markedly while increasing theamplitude With the increasing of the predisplacement theenergy dissipation per cycle decreases and restoring forceand the secant stiffness increase at large displacement Theeffectiveness of a damper is generally measured by theequivalent viscous damping ratio With the increasing of thepredisplacement the equivalent damping ratio reduces Themost important finding is that when the predisplacementis less than 23mm (2 prestrain) the equivalent dampingratio increasedwith the increasing of the amplitude howeverwhen the predisplacement is 46mm (4 prestrain) themaximum of the equivalent damping ratio is at 23mm (2prestrain) This is because the maximum of the equivalentdamping ratio of theNiTi SMA specimens is at about 6 totalstrain as noted in [10]
Figure 7 shows the hysteresis loops of RSMAD with06mm predisplacement at different loading frequencies anddisplacement amplitudes at the temperature of 20∘C Figure 8
shows mechanical properties of RSMAD as a function offrequency of loading and displacement amplitude As wecan see in Figures 7 and 8 the energy dissipation percycle and the equivalent damping decrease as the loadingfrequency increases in the range of 001ndash05Hz but arenot much sensitive to frequencies greater than 05Hz Therestoring force and the secant stiffness increase slightly as theloading frequency increases in the total range of experimentalfrequency
33 Numerical Simulation Based on the improved Graesserand Cozzarelli model of SMA wire a theoretic model of theSMA damper is developed The differential equations of themodel are given as
+ 119891119872[119909 minus 119909Mf sgn (119909)]119898
times [119906 (119909)] [119906 (|119909| minus 119909Mf)]
(11)
where 119865 is restoring force 119909 is displacement 119861 is back-force1198700 is initial stiffness 119861119888 120572119891119879 119899 119886 119888 119909Mf119891119872 and 119898 and areconstants controlling the size of the hysteresis loop 119909in is theinelastic displacement and 119909in = 119909 minus 1198651198700 erf(119909) 119906(119909) andsgn(119909) are respectively the error function the step functionand the signum function which have already been listed inSection 2
Figure 9 shows the comparison of experimental resultswith numerical prediction based on the theoretical modelat different displacement amplitudes The parameters of theconstitutive equation used in this study to simulate the behav-iors of RSMAD are given as follows 1198700 = 380Nmm 119861119888 =330N 120572 = 005 119891119879 = 21 119888 = 00001 119886 = 3 119899 = 2119909Mf = 575mm 119891119872 = 42500 and 119898 = 3 To accuratelysimulate the hysteresis behavior the above parameters weredirectly obtained from the cyclic test results of the SMAdamper according to the parametersrsquo meanings and rolesThevalues also can be converted through the section area andthe length of NiTi wires However there is slight differencebetween the two parameter groups This is possible becauseof the effect of loading conditions on the cyclic behavior ofNiTi wires As can be seen in Figure 9 numerical predictionsagree well with the experimental results
Table 1 shows the comparison of the experimental dataand numerical results respectively of the energy dissipationper cycle the secant stiffness and the equivalent viscousdamping As can be seen in Table 1 the maximum differencesof energy dissipation per cycle secant stiffness and equiva-lent viscous damping are 50 12 and 51 respectivelyThese results indicate that the mechanical behavior of theSMAdamper is well predicted by the numericalmodel whichverifies its suitability for the damper
Mathematical Problems in Engineering 9
(a) Bare structure (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5
Figure 10 Different cases of the ten-story frame structure
Table 1 Comparison between the experimental and theoretical results
Peakdisplacement(mm)
Energy dissipation per cycle (Nsdotmm) Secant stiffness (Nmm) Equivalent viscous damping ()Experimental
Themain objective of including energy dissipating devices ina structure is to reduce structural response through energydissipation To protect the original structural members understrong seismic loading nonlinear deformation in energydissipating devices will be allowed In frame structures thedevices are usually incorporated in steel braces connectingtwo consecutive stories of the buildingThedynamic responseof the structure subjected to earthquake loading is governedby the following equation
where 119872119904 is the mass matrix 119862119904 is the damping coefficientmatrix 119865119904 is the vector of the frames restoring force and 119865119889is the vector of restoring force resulting from SMA dampersThe 119909 and are the structural displacement velocity andacceleration vectors respectively On the right-hand side of
the equation the vector 119868 is the influence vector and thevector 119892 is the ground motion acceleration input
With the aid of the SIMULINK module of MATLAB adynamical simulation system was developed in which theWen model [36] is utilized to simulate the restoring forcecurve of the steel frame structure and the improved Graesserand Cozzarelli model presented above is used for NiTi SMAdamper
5 Seismic Structural Control UsingSMA Dampers
In this section to assess the effectiveness of the proposedrecentering SMA dampers (RSMAD) in mitigating the seis-mic response of building structure nonlinear time historyanalysis on a multistory steel frame with and withoutthe dampers subjected to representative earthquake groundmotions was performed The improved Graesser and Coz-zarelli model for RSMAD given in Section 33 was employedin this numerical study A ten-story steel moment resisting
10 Mathematical Problems in Engineering
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0
10
20 40 60 80 100 120 140 1600123456789
Lateral displacement (mm)
Floo
r lev
elEl Centro
TaftS69E
Case 0Case 5Case 4
Case 3Case 2Case 1
PGA = 02 g
PGA = 02 g
PGA = 02 gTangshan-Beijing
(a)
0 5 10 15 20 25 30
000002004006008010
Disp
lace
men
t (m
)
Time (s)
0 5 10 15 20 25 30Time (s)
0 5 10 15 20Time (s)
Without controlWith case 5
El Centro
000002004006008010012014
Disp
lace
men
t (m
)
Taft S69E
Disp
lace
men
t (m
)
PGA = 02 g
PGA = 02 g
PGA = 02 g
minus010
minus008
minus006
minus004
minus002
minus014
minus012
minus010
minus008
minus006
minus004
minus002
000002004006008010012014016
minus014
minus016
minus012
minus010
minus008
minus006
minus004
minus002
Tangshan-Beijing
(b)
Figure 11 Lateral displacement envelopes (a) and roof displacement timehistories (b)with andwithout SMAdampers under basic (moderate)ground motions
frame structure was selected for this study The building isdesigned for a location in Beijing China The parameters ofthe structure are listed in Table 2
Three representative earthquake ground motions wereused to considering the site effect Three earthquake groundmotions namely Imperial Valley 1940 El Centro LincolnSchool 1952 Taft and Beijing Hotel 1976 Tangshan wereselected These three ground motions represent different siteconditions According to Chinese code for seismic design
of building [37] the local seismic precautionary intensity iseight degrees The peak ground accelerations (PGA) wereadjusted to 02 g and 04 g corresponding to a seismic hazardlevel of 10 and 2 probability of exceedance in a 50-yearperiod respectively
Simulation analysis is conducted on the bare structureand on the structure with five or ten SMA dampers installedas shown in Figure 10 Parameters of the SMA damper arelisted as follows 1198700 = 119870119889 = 120 kNmm 119861119888 = 360KN
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 40 80 120 160 200 240 2800123456789
10
Lateral displacement (mm)
Floo
r lev
el
El CentroPGA = 04 g
Taft S69EPGA = 04 g
PGA = 04 g
Case 0Case 5Case 4
Case 3Case 2Case 1
Tangshan-Beijing
(a)
El Centro
0 5 10 15 20
0 5 10 15 20
25 30
000
005
010
015
020
Time (s)
Disp
lace
men
t (m
)
Taft S69E
000005010015020025
Disp
lace
men
t (m
)
Time (s)
PGA = 04 g
PGA = 04 g
PGA = 04 g
minus020
minus015
minus010
minus005
minus020
minus025
minus015
minus010
minus005
0 5 10 15 20 25 30
000
008004
012016020
Time (s)
Disp
lace
men
t (m
)
minus020
minus016
minus010
minus004
minus008
Without controlWith case 5
Tangshan-Beijing
(b)
Figure 12 Lateral displacement envelopes (a) and roof displacement time histories (b) with and without SMA dampers under strong (severe)ground motions
Figures 11 and 12 show the lateral displacement envelopeswith and without SMA dampers under basic groundmotions(02 g) and strong ground motions (04 g) respectively Aswe can see for most cases the lateral displacement of thestructure decreases remarkably with the introduction of theSMA dampers However the location and the number ofthe dampers have significant effects on the control resultsCase 5 in which dampers are installed in all stories is
the best for the overall structural vibration control For other4 configurations in which five dampers are installed indifferent stores cases 3 and 4 with dampers installed inalternate stories perform better than cases 1 and 2 withdampers placed in consecutive stories Moreover case 3 isbetter than case 4 since the first story is retrofitted Case 1is slightly worse due to the whiplash effect and case 2 isthe worst since the stiffness of the lower half is significantlygreater than that of the upper half of the structure Theroof displacement time histories with (case 5) and without
12 Mathematical Problems in Engineering
SMAdampers under basic groundmotions (02 g) and strongground motions (04 g) are also respectively provided inFigures 11 and 12 which confirm that the SMA damperssignificantly reduce the structural vibration
6 Concluding Remarks
This paper presents the results of a study on evaluating theefficacy of using an innovative SMA-based damper to reducethe seismic response of structures To describe the hysteresisbehavior of the SMA damper an improved Graesser andCozzarelli model was proposed and verified by the cyclictensile test on SMA wires
Cyclic tests on the SMA damper model utilizing foursuperelastic SMA wires with 05mm diameter with variousprestrains under different loading frequencies and displace-ment amplitudes were carried outThe results show satisfyinghysteresis properties including both recentering and energydissipating features under various conditions
A comparative study on nonlinear time history analysisof the seismic response of a ten-story steel frame with theSMA dampers was performed Five cases were considered forthe location and the number of the dampers in the storiesThe numerical analysis results indicate that the proposedSMA damper is capable of significantly reducing seismicresponse of structures which verifies its effectiveness asenergy dissipating device for structures However it is alsoindicated that the location and number have significanteffects on the results of the response
In future multiobjective optimization model will beproposed to obtain the number and the location of the SMAdampers and large scale shake table tests will be performed ona steel frame buildingwith SMAdampers to prove the efficacyof these dampers in dissipating seismic energy
Acknowledgments
This work was funded by National Science Foundation ofChina (no 51108426 and no 41104106) China PostdoctoralScience Foundation (no 20100471008) and Research Fundfor the Doctoral Program of Higher Education of China (no20104101120009) These supports are greatly appreciatedTheopinions expressed in this study are those of the authors anddo not necessarily reflect the views of the sponsor
References
[1] H N Li and L S Huo ldquoAdvances in structural control in civilengineering in Chinardquo Mathematical Problems in Engineeringvol 2010 Article ID 936081 23 pages 2010
[2] G Song N Ma and H N Li ldquoApplications of shape memoryalloys in civil structuresrdquo Engineering Structures vol 28 no 9pp 1266ndash1274 2006
[3] SEAOC Vision 2000 Committee Performance-Based SeismicEngineering Structural Engineering Association of CaliforniaSacramento Calif USA 1995
[4] ATC-40 Seismic Evaluation and Retrofit of Conctete BuildingsApplied Technology Council 1996
[5] FEMA 273 NEHRP Guidelines for Seismic Rehabilitation ofBuildings Federal Emergency Management Agency 1997
[6] Y Fujino T T Soong and B F Spencer Jr ldquoStructural controlbasic concepts and applicationsrdquo in Proceedings of the ASCEStructures Congress pp 15ndash18 Chicago Ill USA April 1996
[7] B F Spencer and S Nagarajaiah ldquoState of the art of structuralcontrolrdquo Journal of Structural Engineering vol 129 no 7 pp845ndash856 2003
[8] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering John Wiley amp Sons NewYork NY USA 1997
[9] F M Mazzolani ldquoPassive control technologies for seismic-resistant buildings in Europerdquo Progress in Structural Engineeringand Materials vol 3 no 3 pp 277ndash287 2001
[10] M Dolce and D Cardone ldquoMechanical behaviour of SMAelements for seismic applicationsmdashpart 2 austenite NiTi wiressubjected to tensionrdquo International Journal of Mechanical Sci-ences vol 43 no 11 pp 2657ndash2677 2001
[11] H N Li and X X Wu ldquoLimitations of height-to-width ratiofor base-isolated buildings under earthquakerdquo The StructuralDesign of Tall and Special Buildings vol 15 no 3 pp 277ndash2872006
[12] R Desroches J McCormick and M A Delemont ldquoCyclicproperties of superelastic shape memory alloy wires and barsrdquoJournal of Structural Engineering vol 130 no 1 pp 38ndash46 2004
[13] T W Duerig K N Melton D Stockel and C M Way-man Engineering Aspects of Shape Memory Alloys ButterworthHeinemann London UK 1990
[14] K Wilde P Gardoni and Y Fujino ldquoBase isolation systemwith shape memory alloy device for elevated highway bridgesrdquoEngineering Structures vol 22 no 3 pp 222ndash229 2000
[15] M Dolce D Cardone and R Marnetto ldquoImplementation andtesting of passive control devices based on shape memoryalloysrdquo Earthquake Engineering and Structural Dynamics vol29 no 7 pp 945ndash968 2000
[16] M Dolce D Cardone F C Ponzo and C Valente ldquoShakingtable tests on reinforced concrete frames without and withpassive control systemsrdquo Earthquake Engineering and StructuralDynamics vol 34 no 14 pp 1687ndash1717 2005
[17] M Indirli M G Castellano P Clemente and A MartellildquoDemo-application of shapememory alloy devices the rehabili-tation of the S Giorgio Church Bell-Towerrdquo in Smart Structuresand Materials 2001 Smart Systems for Bridges Structures andHighways vol 4330 of Proceedings of SPIE pp 262ndash272 New-port Beach Calif USA March 2001
[18] B Andrawes and R Desroches ldquoUnseating prevention for mul-tiple frame bridges using superelastic devicesrdquo Smart Materialsand Structures vol 14 no 3 pp S60ndashS67 2005
[19] R DesRoches and M Delemont ldquoSeismic retrofit of simplysupported bridges using shape memory alloysrdquo EngineeringStructures vol 24 no 3 pp 325ndash332 2002
[20] H Li M Liu and J Ou ldquoVibration mitigation of a stay cablewith one shape memory alloy damperrdquo Structural Control andHealth Monitoring vol 11 no 1 pp 21ndash36 2004
[21] Y Zhang and S Zhu ldquoA shape memory alloy-based reusablehysteretic damper for seismic hazard mitigationrdquo Smart Mate-rials and Structures vol 16 no 5 pp 1603ndash1613 2007
[22] J Ocel R DesRoches R T Leon et al ldquoSteel beam-columnconnections using shape memory alloysrdquo Journal of StructuralEngineering vol 130 no 5 pp 732ndash740 2004
Mathematical Problems in Engineering 13
[23] J McCormick R Desroches D Fugazza and F AuricchioldquoSeismic assessment of concentrically braced steel frames withshape memory alloy bracesrdquo Journal of Structural Engineeringvol 133 no 6 pp 862ndash870 2007
[24] YM Parulekar G R Reddy K K Vaze et al ldquoSeismic responseattenuation of structures using shape memory alloy dampersrdquoStructural Control and Health Monitoring vol 19 no 1 pp 102ndash119 2012
[25] K Tanaka ldquoA thermomechanical sketch of shape memoryeffect one-dimensional tensile behaviorrdquoResMechanica vol 18no 3 pp 251ndash263 1986
[26] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 1 no 2 pp207ndash234 1990
[27] L C Brinson ldquoOne-dimensional constitutive behavior of shapememory alloys thermomechanical derivation with non-con-stant material functions and redefined martensite internal vari-ablerdquo Journal of Intelligent Material Systems and Structures vol4 no 2 pp 229ndash242 1993
[28] F Falk ldquoModel free energy mechanics and thermodynamicsof shape memory alloysrdquo Acta Metallurgica vol 28 no 12 pp1773ndash1780 1980
[29] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashII study of the individual phenomenardquo Journal of theMechanicsand Physics of Solids vol 41 no 1 pp 19ndash33 1993
[30] J G Boyd andDC Lagoudas ldquoA thermodynamical constitutivemodel for shape memory materialsmdashpart I the monolithicshape memory alloyrdquo International Journal of Plasticity vol 12no 6 pp 805ndash842 1996
[31] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashI derivation of general relationsrdquo Journal of the Mechanics andPhysics of Solids vol 41 no 1 pp 1ndash17 1993
[32] H Qian H Li G Song and W Guo ldquoA constitutive modelfor superelastic shape memory alloys considering the influenceof strain raterdquoMathematical Problems in Engineering vol 2013Article ID 248671 8 pages 2013
[33] W J Ren H N Li and G Song ldquoA one-dimensional strain-ratedependent constitutive model for superelastic shape memoryalloysrdquo Smart Materials and Structures vol 16 no 1 pp 191ndash1972007
[34] E J Graesser and F A Cozzarelli ldquoShape-memory alloys asnew materials for aseismic isolationrdquo Journal of EngineeringMechanics vol 117 no 11 pp 2590ndash2608 1991
[35] H Ozdemir Nonlinear transient dynamic analysis of yieldingstructures [PhD thesis] University of California BerkeleyCalif USA 1976
[36] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976
[37] GB 50011-2010 Code for Seismic Design of Buildings Ministryof Housing and Urban-Rural Development of the PeoplersquosRepublic of China 2010
Figure 9 Comparison between experimental and numerical fittingcurves of RSMAD
per cycle 119882119863 the restoring force at peak displacement percycle119865119889 and the equivalent viscous damping ratio 120585eq whichis expressed as
120585eq =119882119863
21205871198701199041205752 (10)
where 120575 is the displacement amplitude of the cycle
322 Experimental Results Figure 5 shows the hysteresisloops of RSMAD at different prestrains and displacementamplitudes The tests were carried out at 005Hz frequencyof loading and room temperature of 20∘C As seen fromFigure 5 the SMA damper shows double-flag hystereticbehavior without any residual strain
Figure 6 shows themechanical properties of RSMAD as afunction of prestrain and displacement amplitude As we cansee in Figure 6 the greater the amplitude is the greater theenergy dissipation per cycle and restoring force are Howeverthe secant stiffness reduces markedly while increasing theamplitude With the increasing of the predisplacement theenergy dissipation per cycle decreases and restoring forceand the secant stiffness increase at large displacement Theeffectiveness of a damper is generally measured by theequivalent viscous damping ratio With the increasing of thepredisplacement the equivalent damping ratio reduces Themost important finding is that when the predisplacementis less than 23mm (2 prestrain) the equivalent dampingratio increasedwith the increasing of the amplitude howeverwhen the predisplacement is 46mm (4 prestrain) themaximum of the equivalent damping ratio is at 23mm (2prestrain) This is because the maximum of the equivalentdamping ratio of theNiTi SMA specimens is at about 6 totalstrain as noted in [10]
Figure 7 shows the hysteresis loops of RSMAD with06mm predisplacement at different loading frequencies anddisplacement amplitudes at the temperature of 20∘C Figure 8
shows mechanical properties of RSMAD as a function offrequency of loading and displacement amplitude As wecan see in Figures 7 and 8 the energy dissipation percycle and the equivalent damping decrease as the loadingfrequency increases in the range of 001ndash05Hz but arenot much sensitive to frequencies greater than 05Hz Therestoring force and the secant stiffness increase slightly as theloading frequency increases in the total range of experimentalfrequency
33 Numerical Simulation Based on the improved Graesserand Cozzarelli model of SMA wire a theoretic model of theSMA damper is developed The differential equations of themodel are given as
+ 119891119872[119909 minus 119909Mf sgn (119909)]119898
times [119906 (119909)] [119906 (|119909| minus 119909Mf)]
(11)
where 119865 is restoring force 119909 is displacement 119861 is back-force1198700 is initial stiffness 119861119888 120572119891119879 119899 119886 119888 119909Mf119891119872 and 119898 and areconstants controlling the size of the hysteresis loop 119909in is theinelastic displacement and 119909in = 119909 minus 1198651198700 erf(119909) 119906(119909) andsgn(119909) are respectively the error function the step functionand the signum function which have already been listed inSection 2
Figure 9 shows the comparison of experimental resultswith numerical prediction based on the theoretical modelat different displacement amplitudes The parameters of theconstitutive equation used in this study to simulate the behav-iors of RSMAD are given as follows 1198700 = 380Nmm 119861119888 =330N 120572 = 005 119891119879 = 21 119888 = 00001 119886 = 3 119899 = 2119909Mf = 575mm 119891119872 = 42500 and 119898 = 3 To accuratelysimulate the hysteresis behavior the above parameters weredirectly obtained from the cyclic test results of the SMAdamper according to the parametersrsquo meanings and rolesThevalues also can be converted through the section area andthe length of NiTi wires However there is slight differencebetween the two parameter groups This is possible becauseof the effect of loading conditions on the cyclic behavior ofNiTi wires As can be seen in Figure 9 numerical predictionsagree well with the experimental results
Table 1 shows the comparison of the experimental dataand numerical results respectively of the energy dissipationper cycle the secant stiffness and the equivalent viscousdamping As can be seen in Table 1 the maximum differencesof energy dissipation per cycle secant stiffness and equiva-lent viscous damping are 50 12 and 51 respectivelyThese results indicate that the mechanical behavior of theSMAdamper is well predicted by the numericalmodel whichverifies its suitability for the damper
Mathematical Problems in Engineering 9
(a) Bare structure (b) Case 1 (c) Case 2 (d) Case 3 (e) Case 4 (f) Case 5
Figure 10 Different cases of the ten-story frame structure
Table 1 Comparison between the experimental and theoretical results
Peakdisplacement(mm)
Energy dissipation per cycle (Nsdotmm) Secant stiffness (Nmm) Equivalent viscous damping ()Experimental
Themain objective of including energy dissipating devices ina structure is to reduce structural response through energydissipation To protect the original structural members understrong seismic loading nonlinear deformation in energydissipating devices will be allowed In frame structures thedevices are usually incorporated in steel braces connectingtwo consecutive stories of the buildingThedynamic responseof the structure subjected to earthquake loading is governedby the following equation
where 119872119904 is the mass matrix 119862119904 is the damping coefficientmatrix 119865119904 is the vector of the frames restoring force and 119865119889is the vector of restoring force resulting from SMA dampersThe 119909 and are the structural displacement velocity andacceleration vectors respectively On the right-hand side of
the equation the vector 119868 is the influence vector and thevector 119892 is the ground motion acceleration input
With the aid of the SIMULINK module of MATLAB adynamical simulation system was developed in which theWen model [36] is utilized to simulate the restoring forcecurve of the steel frame structure and the improved Graesserand Cozzarelli model presented above is used for NiTi SMAdamper
5 Seismic Structural Control UsingSMA Dampers
In this section to assess the effectiveness of the proposedrecentering SMA dampers (RSMAD) in mitigating the seis-mic response of building structure nonlinear time historyanalysis on a multistory steel frame with and withoutthe dampers subjected to representative earthquake groundmotions was performed The improved Graesser and Coz-zarelli model for RSMAD given in Section 33 was employedin this numerical study A ten-story steel moment resisting
10 Mathematical Problems in Engineering
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0
10
20 40 60 80 100 120 140 1600123456789
Lateral displacement (mm)
Floo
r lev
elEl Centro
TaftS69E
Case 0Case 5Case 4
Case 3Case 2Case 1
PGA = 02 g
PGA = 02 g
PGA = 02 gTangshan-Beijing
(a)
0 5 10 15 20 25 30
000002004006008010
Disp
lace
men
t (m
)
Time (s)
0 5 10 15 20 25 30Time (s)
0 5 10 15 20Time (s)
Without controlWith case 5
El Centro
000002004006008010012014
Disp
lace
men
t (m
)
Taft S69E
Disp
lace
men
t (m
)
PGA = 02 g
PGA = 02 g
PGA = 02 g
minus010
minus008
minus006
minus004
minus002
minus014
minus012
minus010
minus008
minus006
minus004
minus002
000002004006008010012014016
minus014
minus016
minus012
minus010
minus008
minus006
minus004
minus002
Tangshan-Beijing
(b)
Figure 11 Lateral displacement envelopes (a) and roof displacement timehistories (b)with andwithout SMAdampers under basic (moderate)ground motions
frame structure was selected for this study The building isdesigned for a location in Beijing China The parameters ofthe structure are listed in Table 2
Three representative earthquake ground motions wereused to considering the site effect Three earthquake groundmotions namely Imperial Valley 1940 El Centro LincolnSchool 1952 Taft and Beijing Hotel 1976 Tangshan wereselected These three ground motions represent different siteconditions According to Chinese code for seismic design
of building [37] the local seismic precautionary intensity iseight degrees The peak ground accelerations (PGA) wereadjusted to 02 g and 04 g corresponding to a seismic hazardlevel of 10 and 2 probability of exceedance in a 50-yearperiod respectively
Simulation analysis is conducted on the bare structureand on the structure with five or ten SMA dampers installedas shown in Figure 10 Parameters of the SMA damper arelisted as follows 1198700 = 119870119889 = 120 kNmm 119861119888 = 360KN
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 40 80 120 160 200 240 2800123456789
10
Lateral displacement (mm)
Floo
r lev
el
El CentroPGA = 04 g
Taft S69EPGA = 04 g
PGA = 04 g
Case 0Case 5Case 4
Case 3Case 2Case 1
Tangshan-Beijing
(a)
El Centro
0 5 10 15 20
0 5 10 15 20
25 30
000
005
010
015
020
Time (s)
Disp
lace
men
t (m
)
Taft S69E
000005010015020025
Disp
lace
men
t (m
)
Time (s)
PGA = 04 g
PGA = 04 g
PGA = 04 g
minus020
minus015
minus010
minus005
minus020
minus025
minus015
minus010
minus005
0 5 10 15 20 25 30
000
008004
012016020
Time (s)
Disp
lace
men
t (m
)
minus020
minus016
minus010
minus004
minus008
Without controlWith case 5
Tangshan-Beijing
(b)
Figure 12 Lateral displacement envelopes (a) and roof displacement time histories (b) with and without SMA dampers under strong (severe)ground motions
Figures 11 and 12 show the lateral displacement envelopeswith and without SMA dampers under basic groundmotions(02 g) and strong ground motions (04 g) respectively Aswe can see for most cases the lateral displacement of thestructure decreases remarkably with the introduction of theSMA dampers However the location and the number ofthe dampers have significant effects on the control resultsCase 5 in which dampers are installed in all stories is
the best for the overall structural vibration control For other4 configurations in which five dampers are installed indifferent stores cases 3 and 4 with dampers installed inalternate stories perform better than cases 1 and 2 withdampers placed in consecutive stories Moreover case 3 isbetter than case 4 since the first story is retrofitted Case 1is slightly worse due to the whiplash effect and case 2 isthe worst since the stiffness of the lower half is significantlygreater than that of the upper half of the structure Theroof displacement time histories with (case 5) and without
12 Mathematical Problems in Engineering
SMAdampers under basic groundmotions (02 g) and strongground motions (04 g) are also respectively provided inFigures 11 and 12 which confirm that the SMA damperssignificantly reduce the structural vibration
6 Concluding Remarks
This paper presents the results of a study on evaluating theefficacy of using an innovative SMA-based damper to reducethe seismic response of structures To describe the hysteresisbehavior of the SMA damper an improved Graesser andCozzarelli model was proposed and verified by the cyclictensile test on SMA wires
Cyclic tests on the SMA damper model utilizing foursuperelastic SMA wires with 05mm diameter with variousprestrains under different loading frequencies and displace-ment amplitudes were carried outThe results show satisfyinghysteresis properties including both recentering and energydissipating features under various conditions
A comparative study on nonlinear time history analysisof the seismic response of a ten-story steel frame with theSMA dampers was performed Five cases were considered forthe location and the number of the dampers in the storiesThe numerical analysis results indicate that the proposedSMA damper is capable of significantly reducing seismicresponse of structures which verifies its effectiveness asenergy dissipating device for structures However it is alsoindicated that the location and number have significanteffects on the results of the response
In future multiobjective optimization model will beproposed to obtain the number and the location of the SMAdampers and large scale shake table tests will be performed ona steel frame buildingwith SMAdampers to prove the efficacyof these dampers in dissipating seismic energy
Acknowledgments
This work was funded by National Science Foundation ofChina (no 51108426 and no 41104106) China PostdoctoralScience Foundation (no 20100471008) and Research Fundfor the Doctoral Program of Higher Education of China (no20104101120009) These supports are greatly appreciatedTheopinions expressed in this study are those of the authors anddo not necessarily reflect the views of the sponsor
References
[1] H N Li and L S Huo ldquoAdvances in structural control in civilengineering in Chinardquo Mathematical Problems in Engineeringvol 2010 Article ID 936081 23 pages 2010
[2] G Song N Ma and H N Li ldquoApplications of shape memoryalloys in civil structuresrdquo Engineering Structures vol 28 no 9pp 1266ndash1274 2006
[3] SEAOC Vision 2000 Committee Performance-Based SeismicEngineering Structural Engineering Association of CaliforniaSacramento Calif USA 1995
[4] ATC-40 Seismic Evaluation and Retrofit of Conctete BuildingsApplied Technology Council 1996
[5] FEMA 273 NEHRP Guidelines for Seismic Rehabilitation ofBuildings Federal Emergency Management Agency 1997
[6] Y Fujino T T Soong and B F Spencer Jr ldquoStructural controlbasic concepts and applicationsrdquo in Proceedings of the ASCEStructures Congress pp 15ndash18 Chicago Ill USA April 1996
[7] B F Spencer and S Nagarajaiah ldquoState of the art of structuralcontrolrdquo Journal of Structural Engineering vol 129 no 7 pp845ndash856 2003
[8] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering John Wiley amp Sons NewYork NY USA 1997
[9] F M Mazzolani ldquoPassive control technologies for seismic-resistant buildings in Europerdquo Progress in Structural Engineeringand Materials vol 3 no 3 pp 277ndash287 2001
[10] M Dolce and D Cardone ldquoMechanical behaviour of SMAelements for seismic applicationsmdashpart 2 austenite NiTi wiressubjected to tensionrdquo International Journal of Mechanical Sci-ences vol 43 no 11 pp 2657ndash2677 2001
[11] H N Li and X X Wu ldquoLimitations of height-to-width ratiofor base-isolated buildings under earthquakerdquo The StructuralDesign of Tall and Special Buildings vol 15 no 3 pp 277ndash2872006
[12] R Desroches J McCormick and M A Delemont ldquoCyclicproperties of superelastic shape memory alloy wires and barsrdquoJournal of Structural Engineering vol 130 no 1 pp 38ndash46 2004
[13] T W Duerig K N Melton D Stockel and C M Way-man Engineering Aspects of Shape Memory Alloys ButterworthHeinemann London UK 1990
[14] K Wilde P Gardoni and Y Fujino ldquoBase isolation systemwith shape memory alloy device for elevated highway bridgesrdquoEngineering Structures vol 22 no 3 pp 222ndash229 2000
[15] M Dolce D Cardone and R Marnetto ldquoImplementation andtesting of passive control devices based on shape memoryalloysrdquo Earthquake Engineering and Structural Dynamics vol29 no 7 pp 945ndash968 2000
[16] M Dolce D Cardone F C Ponzo and C Valente ldquoShakingtable tests on reinforced concrete frames without and withpassive control systemsrdquo Earthquake Engineering and StructuralDynamics vol 34 no 14 pp 1687ndash1717 2005
[17] M Indirli M G Castellano P Clemente and A MartellildquoDemo-application of shapememory alloy devices the rehabili-tation of the S Giorgio Church Bell-Towerrdquo in Smart Structuresand Materials 2001 Smart Systems for Bridges Structures andHighways vol 4330 of Proceedings of SPIE pp 262ndash272 New-port Beach Calif USA March 2001
[18] B Andrawes and R Desroches ldquoUnseating prevention for mul-tiple frame bridges using superelastic devicesrdquo Smart Materialsand Structures vol 14 no 3 pp S60ndashS67 2005
[19] R DesRoches and M Delemont ldquoSeismic retrofit of simplysupported bridges using shape memory alloysrdquo EngineeringStructures vol 24 no 3 pp 325ndash332 2002
[20] H Li M Liu and J Ou ldquoVibration mitigation of a stay cablewith one shape memory alloy damperrdquo Structural Control andHealth Monitoring vol 11 no 1 pp 21ndash36 2004
[21] Y Zhang and S Zhu ldquoA shape memory alloy-based reusablehysteretic damper for seismic hazard mitigationrdquo Smart Mate-rials and Structures vol 16 no 5 pp 1603ndash1613 2007
[22] J Ocel R DesRoches R T Leon et al ldquoSteel beam-columnconnections using shape memory alloysrdquo Journal of StructuralEngineering vol 130 no 5 pp 732ndash740 2004
Mathematical Problems in Engineering 13
[23] J McCormick R Desroches D Fugazza and F AuricchioldquoSeismic assessment of concentrically braced steel frames withshape memory alloy bracesrdquo Journal of Structural Engineeringvol 133 no 6 pp 862ndash870 2007
[24] YM Parulekar G R Reddy K K Vaze et al ldquoSeismic responseattenuation of structures using shape memory alloy dampersrdquoStructural Control and Health Monitoring vol 19 no 1 pp 102ndash119 2012
[25] K Tanaka ldquoA thermomechanical sketch of shape memoryeffect one-dimensional tensile behaviorrdquoResMechanica vol 18no 3 pp 251ndash263 1986
[26] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 1 no 2 pp207ndash234 1990
[27] L C Brinson ldquoOne-dimensional constitutive behavior of shapememory alloys thermomechanical derivation with non-con-stant material functions and redefined martensite internal vari-ablerdquo Journal of Intelligent Material Systems and Structures vol4 no 2 pp 229ndash242 1993
[28] F Falk ldquoModel free energy mechanics and thermodynamicsof shape memory alloysrdquo Acta Metallurgica vol 28 no 12 pp1773ndash1780 1980
[29] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashII study of the individual phenomenardquo Journal of theMechanicsand Physics of Solids vol 41 no 1 pp 19ndash33 1993
[30] J G Boyd andDC Lagoudas ldquoA thermodynamical constitutivemodel for shape memory materialsmdashpart I the monolithicshape memory alloyrdquo International Journal of Plasticity vol 12no 6 pp 805ndash842 1996
[31] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashI derivation of general relationsrdquo Journal of the Mechanics andPhysics of Solids vol 41 no 1 pp 1ndash17 1993
[32] H Qian H Li G Song and W Guo ldquoA constitutive modelfor superelastic shape memory alloys considering the influenceof strain raterdquoMathematical Problems in Engineering vol 2013Article ID 248671 8 pages 2013
[33] W J Ren H N Li and G Song ldquoA one-dimensional strain-ratedependent constitutive model for superelastic shape memoryalloysrdquo Smart Materials and Structures vol 16 no 1 pp 191ndash1972007
[34] E J Graesser and F A Cozzarelli ldquoShape-memory alloys asnew materials for aseismic isolationrdquo Journal of EngineeringMechanics vol 117 no 11 pp 2590ndash2608 1991
[35] H Ozdemir Nonlinear transient dynamic analysis of yieldingstructures [PhD thesis] University of California BerkeleyCalif USA 1976
[36] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976
[37] GB 50011-2010 Code for Seismic Design of Buildings Ministryof Housing and Urban-Rural Development of the PeoplersquosRepublic of China 2010
Themain objective of including energy dissipating devices ina structure is to reduce structural response through energydissipation To protect the original structural members understrong seismic loading nonlinear deformation in energydissipating devices will be allowed In frame structures thedevices are usually incorporated in steel braces connectingtwo consecutive stories of the buildingThedynamic responseof the structure subjected to earthquake loading is governedby the following equation
where 119872119904 is the mass matrix 119862119904 is the damping coefficientmatrix 119865119904 is the vector of the frames restoring force and 119865119889is the vector of restoring force resulting from SMA dampersThe 119909 and are the structural displacement velocity andacceleration vectors respectively On the right-hand side of
the equation the vector 119868 is the influence vector and thevector 119892 is the ground motion acceleration input
With the aid of the SIMULINK module of MATLAB adynamical simulation system was developed in which theWen model [36] is utilized to simulate the restoring forcecurve of the steel frame structure and the improved Graesserand Cozzarelli model presented above is used for NiTi SMAdamper
5 Seismic Structural Control UsingSMA Dampers
In this section to assess the effectiveness of the proposedrecentering SMA dampers (RSMAD) in mitigating the seis-mic response of building structure nonlinear time historyanalysis on a multistory steel frame with and withoutthe dampers subjected to representative earthquake groundmotions was performed The improved Graesser and Coz-zarelli model for RSMAD given in Section 33 was employedin this numerical study A ten-story steel moment resisting
10 Mathematical Problems in Engineering
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0 10
10
20 30 40 50 60 70 80 90 100 110 1200123456789
Lateral displacement (mm)
Floo
r lev
el
0
10
20 40 60 80 100 120 140 1600123456789
Lateral displacement (mm)
Floo
r lev
elEl Centro
TaftS69E
Case 0Case 5Case 4
Case 3Case 2Case 1
PGA = 02 g
PGA = 02 g
PGA = 02 gTangshan-Beijing
(a)
0 5 10 15 20 25 30
000002004006008010
Disp
lace
men
t (m
)
Time (s)
0 5 10 15 20 25 30Time (s)
0 5 10 15 20Time (s)
Without controlWith case 5
El Centro
000002004006008010012014
Disp
lace
men
t (m
)
Taft S69E
Disp
lace
men
t (m
)
PGA = 02 g
PGA = 02 g
PGA = 02 g
minus010
minus008
minus006
minus004
minus002
minus014
minus012
minus010
minus008
minus006
minus004
minus002
000002004006008010012014016
minus014
minus016
minus012
minus010
minus008
minus006
minus004
minus002
Tangshan-Beijing
(b)
Figure 11 Lateral displacement envelopes (a) and roof displacement timehistories (b)with andwithout SMAdampers under basic (moderate)ground motions
frame structure was selected for this study The building isdesigned for a location in Beijing China The parameters ofthe structure are listed in Table 2
Three representative earthquake ground motions wereused to considering the site effect Three earthquake groundmotions namely Imperial Valley 1940 El Centro LincolnSchool 1952 Taft and Beijing Hotel 1976 Tangshan wereselected These three ground motions represent different siteconditions According to Chinese code for seismic design
of building [37] the local seismic precautionary intensity iseight degrees The peak ground accelerations (PGA) wereadjusted to 02 g and 04 g corresponding to a seismic hazardlevel of 10 and 2 probability of exceedance in a 50-yearperiod respectively
Simulation analysis is conducted on the bare structureand on the structure with five or ten SMA dampers installedas shown in Figure 10 Parameters of the SMA damper arelisted as follows 1198700 = 119870119889 = 120 kNmm 119861119888 = 360KN
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 40 80 120 160 200 240 2800123456789
10
Lateral displacement (mm)
Floo
r lev
el
El CentroPGA = 04 g
Taft S69EPGA = 04 g
PGA = 04 g
Case 0Case 5Case 4
Case 3Case 2Case 1
Tangshan-Beijing
(a)
El Centro
0 5 10 15 20
0 5 10 15 20
25 30
000
005
010
015
020
Time (s)
Disp
lace
men
t (m
)
Taft S69E
000005010015020025
Disp
lace
men
t (m
)
Time (s)
PGA = 04 g
PGA = 04 g
PGA = 04 g
minus020
minus015
minus010
minus005
minus020
minus025
minus015
minus010
minus005
0 5 10 15 20 25 30
000
008004
012016020
Time (s)
Disp
lace
men
t (m
)
minus020
minus016
minus010
minus004
minus008
Without controlWith case 5
Tangshan-Beijing
(b)
Figure 12 Lateral displacement envelopes (a) and roof displacement time histories (b) with and without SMA dampers under strong (severe)ground motions
Figures 11 and 12 show the lateral displacement envelopeswith and without SMA dampers under basic groundmotions(02 g) and strong ground motions (04 g) respectively Aswe can see for most cases the lateral displacement of thestructure decreases remarkably with the introduction of theSMA dampers However the location and the number ofthe dampers have significant effects on the control resultsCase 5 in which dampers are installed in all stories is
the best for the overall structural vibration control For other4 configurations in which five dampers are installed indifferent stores cases 3 and 4 with dampers installed inalternate stories perform better than cases 1 and 2 withdampers placed in consecutive stories Moreover case 3 isbetter than case 4 since the first story is retrofitted Case 1is slightly worse due to the whiplash effect and case 2 isthe worst since the stiffness of the lower half is significantlygreater than that of the upper half of the structure Theroof displacement time histories with (case 5) and without
12 Mathematical Problems in Engineering
SMAdampers under basic groundmotions (02 g) and strongground motions (04 g) are also respectively provided inFigures 11 and 12 which confirm that the SMA damperssignificantly reduce the structural vibration
6 Concluding Remarks
This paper presents the results of a study on evaluating theefficacy of using an innovative SMA-based damper to reducethe seismic response of structures To describe the hysteresisbehavior of the SMA damper an improved Graesser andCozzarelli model was proposed and verified by the cyclictensile test on SMA wires
Cyclic tests on the SMA damper model utilizing foursuperelastic SMA wires with 05mm diameter with variousprestrains under different loading frequencies and displace-ment amplitudes were carried outThe results show satisfyinghysteresis properties including both recentering and energydissipating features under various conditions
A comparative study on nonlinear time history analysisof the seismic response of a ten-story steel frame with theSMA dampers was performed Five cases were considered forthe location and the number of the dampers in the storiesThe numerical analysis results indicate that the proposedSMA damper is capable of significantly reducing seismicresponse of structures which verifies its effectiveness asenergy dissipating device for structures However it is alsoindicated that the location and number have significanteffects on the results of the response
In future multiobjective optimization model will beproposed to obtain the number and the location of the SMAdampers and large scale shake table tests will be performed ona steel frame buildingwith SMAdampers to prove the efficacyof these dampers in dissipating seismic energy
Acknowledgments
This work was funded by National Science Foundation ofChina (no 51108426 and no 41104106) China PostdoctoralScience Foundation (no 20100471008) and Research Fundfor the Doctoral Program of Higher Education of China (no20104101120009) These supports are greatly appreciatedTheopinions expressed in this study are those of the authors anddo not necessarily reflect the views of the sponsor
References
[1] H N Li and L S Huo ldquoAdvances in structural control in civilengineering in Chinardquo Mathematical Problems in Engineeringvol 2010 Article ID 936081 23 pages 2010
[2] G Song N Ma and H N Li ldquoApplications of shape memoryalloys in civil structuresrdquo Engineering Structures vol 28 no 9pp 1266ndash1274 2006
[3] SEAOC Vision 2000 Committee Performance-Based SeismicEngineering Structural Engineering Association of CaliforniaSacramento Calif USA 1995
[4] ATC-40 Seismic Evaluation and Retrofit of Conctete BuildingsApplied Technology Council 1996
[5] FEMA 273 NEHRP Guidelines for Seismic Rehabilitation ofBuildings Federal Emergency Management Agency 1997
[6] Y Fujino T T Soong and B F Spencer Jr ldquoStructural controlbasic concepts and applicationsrdquo in Proceedings of the ASCEStructures Congress pp 15ndash18 Chicago Ill USA April 1996
[7] B F Spencer and S Nagarajaiah ldquoState of the art of structuralcontrolrdquo Journal of Structural Engineering vol 129 no 7 pp845ndash856 2003
[8] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering John Wiley amp Sons NewYork NY USA 1997
[9] F M Mazzolani ldquoPassive control technologies for seismic-resistant buildings in Europerdquo Progress in Structural Engineeringand Materials vol 3 no 3 pp 277ndash287 2001
[10] M Dolce and D Cardone ldquoMechanical behaviour of SMAelements for seismic applicationsmdashpart 2 austenite NiTi wiressubjected to tensionrdquo International Journal of Mechanical Sci-ences vol 43 no 11 pp 2657ndash2677 2001
[11] H N Li and X X Wu ldquoLimitations of height-to-width ratiofor base-isolated buildings under earthquakerdquo The StructuralDesign of Tall and Special Buildings vol 15 no 3 pp 277ndash2872006
[12] R Desroches J McCormick and M A Delemont ldquoCyclicproperties of superelastic shape memory alloy wires and barsrdquoJournal of Structural Engineering vol 130 no 1 pp 38ndash46 2004
[13] T W Duerig K N Melton D Stockel and C M Way-man Engineering Aspects of Shape Memory Alloys ButterworthHeinemann London UK 1990
[14] K Wilde P Gardoni and Y Fujino ldquoBase isolation systemwith shape memory alloy device for elevated highway bridgesrdquoEngineering Structures vol 22 no 3 pp 222ndash229 2000
[15] M Dolce D Cardone and R Marnetto ldquoImplementation andtesting of passive control devices based on shape memoryalloysrdquo Earthquake Engineering and Structural Dynamics vol29 no 7 pp 945ndash968 2000
[16] M Dolce D Cardone F C Ponzo and C Valente ldquoShakingtable tests on reinforced concrete frames without and withpassive control systemsrdquo Earthquake Engineering and StructuralDynamics vol 34 no 14 pp 1687ndash1717 2005
[17] M Indirli M G Castellano P Clemente and A MartellildquoDemo-application of shapememory alloy devices the rehabili-tation of the S Giorgio Church Bell-Towerrdquo in Smart Structuresand Materials 2001 Smart Systems for Bridges Structures andHighways vol 4330 of Proceedings of SPIE pp 262ndash272 New-port Beach Calif USA March 2001
[18] B Andrawes and R Desroches ldquoUnseating prevention for mul-tiple frame bridges using superelastic devicesrdquo Smart Materialsand Structures vol 14 no 3 pp S60ndashS67 2005
[19] R DesRoches and M Delemont ldquoSeismic retrofit of simplysupported bridges using shape memory alloysrdquo EngineeringStructures vol 24 no 3 pp 325ndash332 2002
[20] H Li M Liu and J Ou ldquoVibration mitigation of a stay cablewith one shape memory alloy damperrdquo Structural Control andHealth Monitoring vol 11 no 1 pp 21ndash36 2004
[21] Y Zhang and S Zhu ldquoA shape memory alloy-based reusablehysteretic damper for seismic hazard mitigationrdquo Smart Mate-rials and Structures vol 16 no 5 pp 1603ndash1613 2007
[22] J Ocel R DesRoches R T Leon et al ldquoSteel beam-columnconnections using shape memory alloysrdquo Journal of StructuralEngineering vol 130 no 5 pp 732ndash740 2004
Mathematical Problems in Engineering 13
[23] J McCormick R Desroches D Fugazza and F AuricchioldquoSeismic assessment of concentrically braced steel frames withshape memory alloy bracesrdquo Journal of Structural Engineeringvol 133 no 6 pp 862ndash870 2007
[24] YM Parulekar G R Reddy K K Vaze et al ldquoSeismic responseattenuation of structures using shape memory alloy dampersrdquoStructural Control and Health Monitoring vol 19 no 1 pp 102ndash119 2012
[25] K Tanaka ldquoA thermomechanical sketch of shape memoryeffect one-dimensional tensile behaviorrdquoResMechanica vol 18no 3 pp 251ndash263 1986
[26] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 1 no 2 pp207ndash234 1990
[27] L C Brinson ldquoOne-dimensional constitutive behavior of shapememory alloys thermomechanical derivation with non-con-stant material functions and redefined martensite internal vari-ablerdquo Journal of Intelligent Material Systems and Structures vol4 no 2 pp 229ndash242 1993
[28] F Falk ldquoModel free energy mechanics and thermodynamicsof shape memory alloysrdquo Acta Metallurgica vol 28 no 12 pp1773ndash1780 1980
[29] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashII study of the individual phenomenardquo Journal of theMechanicsand Physics of Solids vol 41 no 1 pp 19ndash33 1993
[30] J G Boyd andDC Lagoudas ldquoA thermodynamical constitutivemodel for shape memory materialsmdashpart I the monolithicshape memory alloyrdquo International Journal of Plasticity vol 12no 6 pp 805ndash842 1996
[31] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashI derivation of general relationsrdquo Journal of the Mechanics andPhysics of Solids vol 41 no 1 pp 1ndash17 1993
[32] H Qian H Li G Song and W Guo ldquoA constitutive modelfor superelastic shape memory alloys considering the influenceof strain raterdquoMathematical Problems in Engineering vol 2013Article ID 248671 8 pages 2013
[33] W J Ren H N Li and G Song ldquoA one-dimensional strain-ratedependent constitutive model for superelastic shape memoryalloysrdquo Smart Materials and Structures vol 16 no 1 pp 191ndash1972007
[34] E J Graesser and F A Cozzarelli ldquoShape-memory alloys asnew materials for aseismic isolationrdquo Journal of EngineeringMechanics vol 117 no 11 pp 2590ndash2608 1991
[35] H Ozdemir Nonlinear transient dynamic analysis of yieldingstructures [PhD thesis] University of California BerkeleyCalif USA 1976
[36] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976
[37] GB 50011-2010 Code for Seismic Design of Buildings Ministryof Housing and Urban-Rural Development of the PeoplersquosRepublic of China 2010
Figure 11 Lateral displacement envelopes (a) and roof displacement timehistories (b)with andwithout SMAdampers under basic (moderate)ground motions
frame structure was selected for this study The building isdesigned for a location in Beijing China The parameters ofthe structure are listed in Table 2
Three representative earthquake ground motions wereused to considering the site effect Three earthquake groundmotions namely Imperial Valley 1940 El Centro LincolnSchool 1952 Taft and Beijing Hotel 1976 Tangshan wereselected These three ground motions represent different siteconditions According to Chinese code for seismic design
of building [37] the local seismic precautionary intensity iseight degrees The peak ground accelerations (PGA) wereadjusted to 02 g and 04 g corresponding to a seismic hazardlevel of 10 and 2 probability of exceedance in a 50-yearperiod respectively
Simulation analysis is conducted on the bare structureand on the structure with five or ten SMA dampers installedas shown in Figure 10 Parameters of the SMA damper arelisted as follows 1198700 = 119870119889 = 120 kNmm 119861119888 = 360KN
Mathematical Problems in Engineering 11
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 20 40 60 80 100 120 140 160 1800123456789
10
Lateral displacement (mm)
Floo
r lev
el
0 40 80 120 160 200 240 2800123456789
10
Lateral displacement (mm)
Floo
r lev
el
El CentroPGA = 04 g
Taft S69EPGA = 04 g
PGA = 04 g
Case 0Case 5Case 4
Case 3Case 2Case 1
Tangshan-Beijing
(a)
El Centro
0 5 10 15 20
0 5 10 15 20
25 30
000
005
010
015
020
Time (s)
Disp
lace
men
t (m
)
Taft S69E
000005010015020025
Disp
lace
men
t (m
)
Time (s)
PGA = 04 g
PGA = 04 g
PGA = 04 g
minus020
minus015
minus010
minus005
minus020
minus025
minus015
minus010
minus005
0 5 10 15 20 25 30
000
008004
012016020
Time (s)
Disp
lace
men
t (m
)
minus020
minus016
minus010
minus004
minus008
Without controlWith case 5
Tangshan-Beijing
(b)
Figure 12 Lateral displacement envelopes (a) and roof displacement time histories (b) with and without SMA dampers under strong (severe)ground motions
Figures 11 and 12 show the lateral displacement envelopeswith and without SMA dampers under basic groundmotions(02 g) and strong ground motions (04 g) respectively Aswe can see for most cases the lateral displacement of thestructure decreases remarkably with the introduction of theSMA dampers However the location and the number ofthe dampers have significant effects on the control resultsCase 5 in which dampers are installed in all stories is
the best for the overall structural vibration control For other4 configurations in which five dampers are installed indifferent stores cases 3 and 4 with dampers installed inalternate stories perform better than cases 1 and 2 withdampers placed in consecutive stories Moreover case 3 isbetter than case 4 since the first story is retrofitted Case 1is slightly worse due to the whiplash effect and case 2 isthe worst since the stiffness of the lower half is significantlygreater than that of the upper half of the structure Theroof displacement time histories with (case 5) and without
12 Mathematical Problems in Engineering
SMAdampers under basic groundmotions (02 g) and strongground motions (04 g) are also respectively provided inFigures 11 and 12 which confirm that the SMA damperssignificantly reduce the structural vibration
6 Concluding Remarks
This paper presents the results of a study on evaluating theefficacy of using an innovative SMA-based damper to reducethe seismic response of structures To describe the hysteresisbehavior of the SMA damper an improved Graesser andCozzarelli model was proposed and verified by the cyclictensile test on SMA wires
Cyclic tests on the SMA damper model utilizing foursuperelastic SMA wires with 05mm diameter with variousprestrains under different loading frequencies and displace-ment amplitudes were carried outThe results show satisfyinghysteresis properties including both recentering and energydissipating features under various conditions
A comparative study on nonlinear time history analysisof the seismic response of a ten-story steel frame with theSMA dampers was performed Five cases were considered forthe location and the number of the dampers in the storiesThe numerical analysis results indicate that the proposedSMA damper is capable of significantly reducing seismicresponse of structures which verifies its effectiveness asenergy dissipating device for structures However it is alsoindicated that the location and number have significanteffects on the results of the response
In future multiobjective optimization model will beproposed to obtain the number and the location of the SMAdampers and large scale shake table tests will be performed ona steel frame buildingwith SMAdampers to prove the efficacyof these dampers in dissipating seismic energy
Acknowledgments
This work was funded by National Science Foundation ofChina (no 51108426 and no 41104106) China PostdoctoralScience Foundation (no 20100471008) and Research Fundfor the Doctoral Program of Higher Education of China (no20104101120009) These supports are greatly appreciatedTheopinions expressed in this study are those of the authors anddo not necessarily reflect the views of the sponsor
References
[1] H N Li and L S Huo ldquoAdvances in structural control in civilengineering in Chinardquo Mathematical Problems in Engineeringvol 2010 Article ID 936081 23 pages 2010
[2] G Song N Ma and H N Li ldquoApplications of shape memoryalloys in civil structuresrdquo Engineering Structures vol 28 no 9pp 1266ndash1274 2006
[3] SEAOC Vision 2000 Committee Performance-Based SeismicEngineering Structural Engineering Association of CaliforniaSacramento Calif USA 1995
[4] ATC-40 Seismic Evaluation and Retrofit of Conctete BuildingsApplied Technology Council 1996
[5] FEMA 273 NEHRP Guidelines for Seismic Rehabilitation ofBuildings Federal Emergency Management Agency 1997
[6] Y Fujino T T Soong and B F Spencer Jr ldquoStructural controlbasic concepts and applicationsrdquo in Proceedings of the ASCEStructures Congress pp 15ndash18 Chicago Ill USA April 1996
[7] B F Spencer and S Nagarajaiah ldquoState of the art of structuralcontrolrdquo Journal of Structural Engineering vol 129 no 7 pp845ndash856 2003
[8] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering John Wiley amp Sons NewYork NY USA 1997
[9] F M Mazzolani ldquoPassive control technologies for seismic-resistant buildings in Europerdquo Progress in Structural Engineeringand Materials vol 3 no 3 pp 277ndash287 2001
[10] M Dolce and D Cardone ldquoMechanical behaviour of SMAelements for seismic applicationsmdashpart 2 austenite NiTi wiressubjected to tensionrdquo International Journal of Mechanical Sci-ences vol 43 no 11 pp 2657ndash2677 2001
[11] H N Li and X X Wu ldquoLimitations of height-to-width ratiofor base-isolated buildings under earthquakerdquo The StructuralDesign of Tall and Special Buildings vol 15 no 3 pp 277ndash2872006
[12] R Desroches J McCormick and M A Delemont ldquoCyclicproperties of superelastic shape memory alloy wires and barsrdquoJournal of Structural Engineering vol 130 no 1 pp 38ndash46 2004
[13] T W Duerig K N Melton D Stockel and C M Way-man Engineering Aspects of Shape Memory Alloys ButterworthHeinemann London UK 1990
[14] K Wilde P Gardoni and Y Fujino ldquoBase isolation systemwith shape memory alloy device for elevated highway bridgesrdquoEngineering Structures vol 22 no 3 pp 222ndash229 2000
[15] M Dolce D Cardone and R Marnetto ldquoImplementation andtesting of passive control devices based on shape memoryalloysrdquo Earthquake Engineering and Structural Dynamics vol29 no 7 pp 945ndash968 2000
[16] M Dolce D Cardone F C Ponzo and C Valente ldquoShakingtable tests on reinforced concrete frames without and withpassive control systemsrdquo Earthquake Engineering and StructuralDynamics vol 34 no 14 pp 1687ndash1717 2005
[17] M Indirli M G Castellano P Clemente and A MartellildquoDemo-application of shapememory alloy devices the rehabili-tation of the S Giorgio Church Bell-Towerrdquo in Smart Structuresand Materials 2001 Smart Systems for Bridges Structures andHighways vol 4330 of Proceedings of SPIE pp 262ndash272 New-port Beach Calif USA March 2001
[18] B Andrawes and R Desroches ldquoUnseating prevention for mul-tiple frame bridges using superelastic devicesrdquo Smart Materialsand Structures vol 14 no 3 pp S60ndashS67 2005
[19] R DesRoches and M Delemont ldquoSeismic retrofit of simplysupported bridges using shape memory alloysrdquo EngineeringStructures vol 24 no 3 pp 325ndash332 2002
[20] H Li M Liu and J Ou ldquoVibration mitigation of a stay cablewith one shape memory alloy damperrdquo Structural Control andHealth Monitoring vol 11 no 1 pp 21ndash36 2004
[21] Y Zhang and S Zhu ldquoA shape memory alloy-based reusablehysteretic damper for seismic hazard mitigationrdquo Smart Mate-rials and Structures vol 16 no 5 pp 1603ndash1613 2007
[22] J Ocel R DesRoches R T Leon et al ldquoSteel beam-columnconnections using shape memory alloysrdquo Journal of StructuralEngineering vol 130 no 5 pp 732ndash740 2004
Mathematical Problems in Engineering 13
[23] J McCormick R Desroches D Fugazza and F AuricchioldquoSeismic assessment of concentrically braced steel frames withshape memory alloy bracesrdquo Journal of Structural Engineeringvol 133 no 6 pp 862ndash870 2007
[24] YM Parulekar G R Reddy K K Vaze et al ldquoSeismic responseattenuation of structures using shape memory alloy dampersrdquoStructural Control and Health Monitoring vol 19 no 1 pp 102ndash119 2012
[25] K Tanaka ldquoA thermomechanical sketch of shape memoryeffect one-dimensional tensile behaviorrdquoResMechanica vol 18no 3 pp 251ndash263 1986
[26] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 1 no 2 pp207ndash234 1990
[27] L C Brinson ldquoOne-dimensional constitutive behavior of shapememory alloys thermomechanical derivation with non-con-stant material functions and redefined martensite internal vari-ablerdquo Journal of Intelligent Material Systems and Structures vol4 no 2 pp 229ndash242 1993
[28] F Falk ldquoModel free energy mechanics and thermodynamicsof shape memory alloysrdquo Acta Metallurgica vol 28 no 12 pp1773ndash1780 1980
[29] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashII study of the individual phenomenardquo Journal of theMechanicsand Physics of Solids vol 41 no 1 pp 19ndash33 1993
[30] J G Boyd andDC Lagoudas ldquoA thermodynamical constitutivemodel for shape memory materialsmdashpart I the monolithicshape memory alloyrdquo International Journal of Plasticity vol 12no 6 pp 805ndash842 1996
[31] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashI derivation of general relationsrdquo Journal of the Mechanics andPhysics of Solids vol 41 no 1 pp 1ndash17 1993
[32] H Qian H Li G Song and W Guo ldquoA constitutive modelfor superelastic shape memory alloys considering the influenceof strain raterdquoMathematical Problems in Engineering vol 2013Article ID 248671 8 pages 2013
[33] W J Ren H N Li and G Song ldquoA one-dimensional strain-ratedependent constitutive model for superelastic shape memoryalloysrdquo Smart Materials and Structures vol 16 no 1 pp 191ndash1972007
[34] E J Graesser and F A Cozzarelli ldquoShape-memory alloys asnew materials for aseismic isolationrdquo Journal of EngineeringMechanics vol 117 no 11 pp 2590ndash2608 1991
[35] H Ozdemir Nonlinear transient dynamic analysis of yieldingstructures [PhD thesis] University of California BerkeleyCalif USA 1976
[36] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976
[37] GB 50011-2010 Code for Seismic Design of Buildings Ministryof Housing and Urban-Rural Development of the PeoplersquosRepublic of China 2010
Figures 11 and 12 show the lateral displacement envelopeswith and without SMA dampers under basic groundmotions(02 g) and strong ground motions (04 g) respectively Aswe can see for most cases the lateral displacement of thestructure decreases remarkably with the introduction of theSMA dampers However the location and the number ofthe dampers have significant effects on the control resultsCase 5 in which dampers are installed in all stories is
the best for the overall structural vibration control For other4 configurations in which five dampers are installed indifferent stores cases 3 and 4 with dampers installed inalternate stories perform better than cases 1 and 2 withdampers placed in consecutive stories Moreover case 3 isbetter than case 4 since the first story is retrofitted Case 1is slightly worse due to the whiplash effect and case 2 isthe worst since the stiffness of the lower half is significantlygreater than that of the upper half of the structure Theroof displacement time histories with (case 5) and without
12 Mathematical Problems in Engineering
SMAdampers under basic groundmotions (02 g) and strongground motions (04 g) are also respectively provided inFigures 11 and 12 which confirm that the SMA damperssignificantly reduce the structural vibration
6 Concluding Remarks
This paper presents the results of a study on evaluating theefficacy of using an innovative SMA-based damper to reducethe seismic response of structures To describe the hysteresisbehavior of the SMA damper an improved Graesser andCozzarelli model was proposed and verified by the cyclictensile test on SMA wires
Cyclic tests on the SMA damper model utilizing foursuperelastic SMA wires with 05mm diameter with variousprestrains under different loading frequencies and displace-ment amplitudes were carried outThe results show satisfyinghysteresis properties including both recentering and energydissipating features under various conditions
A comparative study on nonlinear time history analysisof the seismic response of a ten-story steel frame with theSMA dampers was performed Five cases were considered forthe location and the number of the dampers in the storiesThe numerical analysis results indicate that the proposedSMA damper is capable of significantly reducing seismicresponse of structures which verifies its effectiveness asenergy dissipating device for structures However it is alsoindicated that the location and number have significanteffects on the results of the response
In future multiobjective optimization model will beproposed to obtain the number and the location of the SMAdampers and large scale shake table tests will be performed ona steel frame buildingwith SMAdampers to prove the efficacyof these dampers in dissipating seismic energy
Acknowledgments
This work was funded by National Science Foundation ofChina (no 51108426 and no 41104106) China PostdoctoralScience Foundation (no 20100471008) and Research Fundfor the Doctoral Program of Higher Education of China (no20104101120009) These supports are greatly appreciatedTheopinions expressed in this study are those of the authors anddo not necessarily reflect the views of the sponsor
References
[1] H N Li and L S Huo ldquoAdvances in structural control in civilengineering in Chinardquo Mathematical Problems in Engineeringvol 2010 Article ID 936081 23 pages 2010
[2] G Song N Ma and H N Li ldquoApplications of shape memoryalloys in civil structuresrdquo Engineering Structures vol 28 no 9pp 1266ndash1274 2006
[3] SEAOC Vision 2000 Committee Performance-Based SeismicEngineering Structural Engineering Association of CaliforniaSacramento Calif USA 1995
[4] ATC-40 Seismic Evaluation and Retrofit of Conctete BuildingsApplied Technology Council 1996
[5] FEMA 273 NEHRP Guidelines for Seismic Rehabilitation ofBuildings Federal Emergency Management Agency 1997
[6] Y Fujino T T Soong and B F Spencer Jr ldquoStructural controlbasic concepts and applicationsrdquo in Proceedings of the ASCEStructures Congress pp 15ndash18 Chicago Ill USA April 1996
[7] B F Spencer and S Nagarajaiah ldquoState of the art of structuralcontrolrdquo Journal of Structural Engineering vol 129 no 7 pp845ndash856 2003
[8] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering John Wiley amp Sons NewYork NY USA 1997
[9] F M Mazzolani ldquoPassive control technologies for seismic-resistant buildings in Europerdquo Progress in Structural Engineeringand Materials vol 3 no 3 pp 277ndash287 2001
[10] M Dolce and D Cardone ldquoMechanical behaviour of SMAelements for seismic applicationsmdashpart 2 austenite NiTi wiressubjected to tensionrdquo International Journal of Mechanical Sci-ences vol 43 no 11 pp 2657ndash2677 2001
[11] H N Li and X X Wu ldquoLimitations of height-to-width ratiofor base-isolated buildings under earthquakerdquo The StructuralDesign of Tall and Special Buildings vol 15 no 3 pp 277ndash2872006
[12] R Desroches J McCormick and M A Delemont ldquoCyclicproperties of superelastic shape memory alloy wires and barsrdquoJournal of Structural Engineering vol 130 no 1 pp 38ndash46 2004
[13] T W Duerig K N Melton D Stockel and C M Way-man Engineering Aspects of Shape Memory Alloys ButterworthHeinemann London UK 1990
[14] K Wilde P Gardoni and Y Fujino ldquoBase isolation systemwith shape memory alloy device for elevated highway bridgesrdquoEngineering Structures vol 22 no 3 pp 222ndash229 2000
[15] M Dolce D Cardone and R Marnetto ldquoImplementation andtesting of passive control devices based on shape memoryalloysrdquo Earthquake Engineering and Structural Dynamics vol29 no 7 pp 945ndash968 2000
[16] M Dolce D Cardone F C Ponzo and C Valente ldquoShakingtable tests on reinforced concrete frames without and withpassive control systemsrdquo Earthquake Engineering and StructuralDynamics vol 34 no 14 pp 1687ndash1717 2005
[17] M Indirli M G Castellano P Clemente and A MartellildquoDemo-application of shapememory alloy devices the rehabili-tation of the S Giorgio Church Bell-Towerrdquo in Smart Structuresand Materials 2001 Smart Systems for Bridges Structures andHighways vol 4330 of Proceedings of SPIE pp 262ndash272 New-port Beach Calif USA March 2001
[18] B Andrawes and R Desroches ldquoUnseating prevention for mul-tiple frame bridges using superelastic devicesrdquo Smart Materialsand Structures vol 14 no 3 pp S60ndashS67 2005
[19] R DesRoches and M Delemont ldquoSeismic retrofit of simplysupported bridges using shape memory alloysrdquo EngineeringStructures vol 24 no 3 pp 325ndash332 2002
[20] H Li M Liu and J Ou ldquoVibration mitigation of a stay cablewith one shape memory alloy damperrdquo Structural Control andHealth Monitoring vol 11 no 1 pp 21ndash36 2004
[21] Y Zhang and S Zhu ldquoA shape memory alloy-based reusablehysteretic damper for seismic hazard mitigationrdquo Smart Mate-rials and Structures vol 16 no 5 pp 1603ndash1613 2007
[22] J Ocel R DesRoches R T Leon et al ldquoSteel beam-columnconnections using shape memory alloysrdquo Journal of StructuralEngineering vol 130 no 5 pp 732ndash740 2004
Mathematical Problems in Engineering 13
[23] J McCormick R Desroches D Fugazza and F AuricchioldquoSeismic assessment of concentrically braced steel frames withshape memory alloy bracesrdquo Journal of Structural Engineeringvol 133 no 6 pp 862ndash870 2007
[24] YM Parulekar G R Reddy K K Vaze et al ldquoSeismic responseattenuation of structures using shape memory alloy dampersrdquoStructural Control and Health Monitoring vol 19 no 1 pp 102ndash119 2012
[25] K Tanaka ldquoA thermomechanical sketch of shape memoryeffect one-dimensional tensile behaviorrdquoResMechanica vol 18no 3 pp 251ndash263 1986
[26] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 1 no 2 pp207ndash234 1990
[27] L C Brinson ldquoOne-dimensional constitutive behavior of shapememory alloys thermomechanical derivation with non-con-stant material functions and redefined martensite internal vari-ablerdquo Journal of Intelligent Material Systems and Structures vol4 no 2 pp 229ndash242 1993
[28] F Falk ldquoModel free energy mechanics and thermodynamicsof shape memory alloysrdquo Acta Metallurgica vol 28 no 12 pp1773ndash1780 1980
[29] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashII study of the individual phenomenardquo Journal of theMechanicsand Physics of Solids vol 41 no 1 pp 19ndash33 1993
[30] J G Boyd andDC Lagoudas ldquoA thermodynamical constitutivemodel for shape memory materialsmdashpart I the monolithicshape memory alloyrdquo International Journal of Plasticity vol 12no 6 pp 805ndash842 1996
[31] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashI derivation of general relationsrdquo Journal of the Mechanics andPhysics of Solids vol 41 no 1 pp 1ndash17 1993
[32] H Qian H Li G Song and W Guo ldquoA constitutive modelfor superelastic shape memory alloys considering the influenceof strain raterdquoMathematical Problems in Engineering vol 2013Article ID 248671 8 pages 2013
[33] W J Ren H N Li and G Song ldquoA one-dimensional strain-ratedependent constitutive model for superelastic shape memoryalloysrdquo Smart Materials and Structures vol 16 no 1 pp 191ndash1972007
[34] E J Graesser and F A Cozzarelli ldquoShape-memory alloys asnew materials for aseismic isolationrdquo Journal of EngineeringMechanics vol 117 no 11 pp 2590ndash2608 1991
[35] H Ozdemir Nonlinear transient dynamic analysis of yieldingstructures [PhD thesis] University of California BerkeleyCalif USA 1976
[36] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976
[37] GB 50011-2010 Code for Seismic Design of Buildings Ministryof Housing and Urban-Rural Development of the PeoplersquosRepublic of China 2010
SMAdampers under basic groundmotions (02 g) and strongground motions (04 g) are also respectively provided inFigures 11 and 12 which confirm that the SMA damperssignificantly reduce the structural vibration
6 Concluding Remarks
This paper presents the results of a study on evaluating theefficacy of using an innovative SMA-based damper to reducethe seismic response of structures To describe the hysteresisbehavior of the SMA damper an improved Graesser andCozzarelli model was proposed and verified by the cyclictensile test on SMA wires
Cyclic tests on the SMA damper model utilizing foursuperelastic SMA wires with 05mm diameter with variousprestrains under different loading frequencies and displace-ment amplitudes were carried outThe results show satisfyinghysteresis properties including both recentering and energydissipating features under various conditions
A comparative study on nonlinear time history analysisof the seismic response of a ten-story steel frame with theSMA dampers was performed Five cases were considered forthe location and the number of the dampers in the storiesThe numerical analysis results indicate that the proposedSMA damper is capable of significantly reducing seismicresponse of structures which verifies its effectiveness asenergy dissipating device for structures However it is alsoindicated that the location and number have significanteffects on the results of the response
In future multiobjective optimization model will beproposed to obtain the number and the location of the SMAdampers and large scale shake table tests will be performed ona steel frame buildingwith SMAdampers to prove the efficacyof these dampers in dissipating seismic energy
Acknowledgments
This work was funded by National Science Foundation ofChina (no 51108426 and no 41104106) China PostdoctoralScience Foundation (no 20100471008) and Research Fundfor the Doctoral Program of Higher Education of China (no20104101120009) These supports are greatly appreciatedTheopinions expressed in this study are those of the authors anddo not necessarily reflect the views of the sponsor
References
[1] H N Li and L S Huo ldquoAdvances in structural control in civilengineering in Chinardquo Mathematical Problems in Engineeringvol 2010 Article ID 936081 23 pages 2010
[2] G Song N Ma and H N Li ldquoApplications of shape memoryalloys in civil structuresrdquo Engineering Structures vol 28 no 9pp 1266ndash1274 2006
[3] SEAOC Vision 2000 Committee Performance-Based SeismicEngineering Structural Engineering Association of CaliforniaSacramento Calif USA 1995
[4] ATC-40 Seismic Evaluation and Retrofit of Conctete BuildingsApplied Technology Council 1996
[5] FEMA 273 NEHRP Guidelines for Seismic Rehabilitation ofBuildings Federal Emergency Management Agency 1997
[6] Y Fujino T T Soong and B F Spencer Jr ldquoStructural controlbasic concepts and applicationsrdquo in Proceedings of the ASCEStructures Congress pp 15ndash18 Chicago Ill USA April 1996
[7] B F Spencer and S Nagarajaiah ldquoState of the art of structuralcontrolrdquo Journal of Structural Engineering vol 129 no 7 pp845ndash856 2003
[8] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering John Wiley amp Sons NewYork NY USA 1997
[9] F M Mazzolani ldquoPassive control technologies for seismic-resistant buildings in Europerdquo Progress in Structural Engineeringand Materials vol 3 no 3 pp 277ndash287 2001
[10] M Dolce and D Cardone ldquoMechanical behaviour of SMAelements for seismic applicationsmdashpart 2 austenite NiTi wiressubjected to tensionrdquo International Journal of Mechanical Sci-ences vol 43 no 11 pp 2657ndash2677 2001
[11] H N Li and X X Wu ldquoLimitations of height-to-width ratiofor base-isolated buildings under earthquakerdquo The StructuralDesign of Tall and Special Buildings vol 15 no 3 pp 277ndash2872006
[12] R Desroches J McCormick and M A Delemont ldquoCyclicproperties of superelastic shape memory alloy wires and barsrdquoJournal of Structural Engineering vol 130 no 1 pp 38ndash46 2004
[13] T W Duerig K N Melton D Stockel and C M Way-man Engineering Aspects of Shape Memory Alloys ButterworthHeinemann London UK 1990
[14] K Wilde P Gardoni and Y Fujino ldquoBase isolation systemwith shape memory alloy device for elevated highway bridgesrdquoEngineering Structures vol 22 no 3 pp 222ndash229 2000
[15] M Dolce D Cardone and R Marnetto ldquoImplementation andtesting of passive control devices based on shape memoryalloysrdquo Earthquake Engineering and Structural Dynamics vol29 no 7 pp 945ndash968 2000
[16] M Dolce D Cardone F C Ponzo and C Valente ldquoShakingtable tests on reinforced concrete frames without and withpassive control systemsrdquo Earthquake Engineering and StructuralDynamics vol 34 no 14 pp 1687ndash1717 2005
[17] M Indirli M G Castellano P Clemente and A MartellildquoDemo-application of shapememory alloy devices the rehabili-tation of the S Giorgio Church Bell-Towerrdquo in Smart Structuresand Materials 2001 Smart Systems for Bridges Structures andHighways vol 4330 of Proceedings of SPIE pp 262ndash272 New-port Beach Calif USA March 2001
[18] B Andrawes and R Desroches ldquoUnseating prevention for mul-tiple frame bridges using superelastic devicesrdquo Smart Materialsand Structures vol 14 no 3 pp S60ndashS67 2005
[19] R DesRoches and M Delemont ldquoSeismic retrofit of simplysupported bridges using shape memory alloysrdquo EngineeringStructures vol 24 no 3 pp 325ndash332 2002
[20] H Li M Liu and J Ou ldquoVibration mitigation of a stay cablewith one shape memory alloy damperrdquo Structural Control andHealth Monitoring vol 11 no 1 pp 21ndash36 2004
[21] Y Zhang and S Zhu ldquoA shape memory alloy-based reusablehysteretic damper for seismic hazard mitigationrdquo Smart Mate-rials and Structures vol 16 no 5 pp 1603ndash1613 2007
[22] J Ocel R DesRoches R T Leon et al ldquoSteel beam-columnconnections using shape memory alloysrdquo Journal of StructuralEngineering vol 130 no 5 pp 732ndash740 2004
Mathematical Problems in Engineering 13
[23] J McCormick R Desroches D Fugazza and F AuricchioldquoSeismic assessment of concentrically braced steel frames withshape memory alloy bracesrdquo Journal of Structural Engineeringvol 133 no 6 pp 862ndash870 2007
[24] YM Parulekar G R Reddy K K Vaze et al ldquoSeismic responseattenuation of structures using shape memory alloy dampersrdquoStructural Control and Health Monitoring vol 19 no 1 pp 102ndash119 2012
[25] K Tanaka ldquoA thermomechanical sketch of shape memoryeffect one-dimensional tensile behaviorrdquoResMechanica vol 18no 3 pp 251ndash263 1986
[26] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 1 no 2 pp207ndash234 1990
[27] L C Brinson ldquoOne-dimensional constitutive behavior of shapememory alloys thermomechanical derivation with non-con-stant material functions and redefined martensite internal vari-ablerdquo Journal of Intelligent Material Systems and Structures vol4 no 2 pp 229ndash242 1993
[28] F Falk ldquoModel free energy mechanics and thermodynamicsof shape memory alloysrdquo Acta Metallurgica vol 28 no 12 pp1773ndash1780 1980
[29] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashII study of the individual phenomenardquo Journal of theMechanicsand Physics of Solids vol 41 no 1 pp 19ndash33 1993
[30] J G Boyd andDC Lagoudas ldquoA thermodynamical constitutivemodel for shape memory materialsmdashpart I the monolithicshape memory alloyrdquo International Journal of Plasticity vol 12no 6 pp 805ndash842 1996
[31] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashI derivation of general relationsrdquo Journal of the Mechanics andPhysics of Solids vol 41 no 1 pp 1ndash17 1993
[32] H Qian H Li G Song and W Guo ldquoA constitutive modelfor superelastic shape memory alloys considering the influenceof strain raterdquoMathematical Problems in Engineering vol 2013Article ID 248671 8 pages 2013
[33] W J Ren H N Li and G Song ldquoA one-dimensional strain-ratedependent constitutive model for superelastic shape memoryalloysrdquo Smart Materials and Structures vol 16 no 1 pp 191ndash1972007
[34] E J Graesser and F A Cozzarelli ldquoShape-memory alloys asnew materials for aseismic isolationrdquo Journal of EngineeringMechanics vol 117 no 11 pp 2590ndash2608 1991
[35] H Ozdemir Nonlinear transient dynamic analysis of yieldingstructures [PhD thesis] University of California BerkeleyCalif USA 1976
[36] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976
[37] GB 50011-2010 Code for Seismic Design of Buildings Ministryof Housing and Urban-Rural Development of the PeoplersquosRepublic of China 2010
[23] J McCormick R Desroches D Fugazza and F AuricchioldquoSeismic assessment of concentrically braced steel frames withshape memory alloy bracesrdquo Journal of Structural Engineeringvol 133 no 6 pp 862ndash870 2007
[24] YM Parulekar G R Reddy K K Vaze et al ldquoSeismic responseattenuation of structures using shape memory alloy dampersrdquoStructural Control and Health Monitoring vol 19 no 1 pp 102ndash119 2012
[25] K Tanaka ldquoA thermomechanical sketch of shape memoryeffect one-dimensional tensile behaviorrdquoResMechanica vol 18no 3 pp 251ndash263 1986
[26] C Liang and C A Rogers ldquoOne-dimensional thermomechan-ical constitutive relations for shape memory materialsrdquo Journalof Intelligent Material Systems and Structures vol 1 no 2 pp207ndash234 1990
[27] L C Brinson ldquoOne-dimensional constitutive behavior of shapememory alloys thermomechanical derivation with non-con-stant material functions and redefined martensite internal vari-ablerdquo Journal of Intelligent Material Systems and Structures vol4 no 2 pp 229ndash242 1993
[28] F Falk ldquoModel free energy mechanics and thermodynamicsof shape memory alloysrdquo Acta Metallurgica vol 28 no 12 pp1773ndash1780 1980
[29] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashII study of the individual phenomenardquo Journal of theMechanicsand Physics of Solids vol 41 no 1 pp 19ndash33 1993
[30] J G Boyd andDC Lagoudas ldquoA thermodynamical constitutivemodel for shape memory materialsmdashpart I the monolithicshape memory alloyrdquo International Journal of Plasticity vol 12no 6 pp 805ndash842 1996
[31] Q P Sun and K C Hwang ldquoMicromechanics modelling for theconstitutive behavior of polycrystalline shape memory alloysmdashI derivation of general relationsrdquo Journal of the Mechanics andPhysics of Solids vol 41 no 1 pp 1ndash17 1993
[32] H Qian H Li G Song and W Guo ldquoA constitutive modelfor superelastic shape memory alloys considering the influenceof strain raterdquoMathematical Problems in Engineering vol 2013Article ID 248671 8 pages 2013
[33] W J Ren H N Li and G Song ldquoA one-dimensional strain-ratedependent constitutive model for superelastic shape memoryalloysrdquo Smart Materials and Structures vol 16 no 1 pp 191ndash1972007
[34] E J Graesser and F A Cozzarelli ldquoShape-memory alloys asnew materials for aseismic isolationrdquo Journal of EngineeringMechanics vol 117 no 11 pp 2590ndash2608 1991
[35] H Ozdemir Nonlinear transient dynamic analysis of yieldingstructures [PhD thesis] University of California BerkeleyCalif USA 1976
[36] Y KWen ldquoMethod for random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976
[37] GB 50011-2010 Code for Seismic Design of Buildings Ministryof Housing and Urban-Rural Development of the PeoplersquosRepublic of China 2010
