GA-based optimum design of a shape memory alloy device for seismic response mitigation

O E Ozbulut1, P N Roschke1, P#Y#Lin2,#C#H Loh3 1 Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 2 National Center for Research on Earthquake Engineering, Taipei, Taiwan, R.O.C. 3 National Taiwan University, Department of Civil Engineering, Taipei, 106 Taiwan, R.O.C.

Abstract: Damping systems discussed in this work are optimized so that a three-story

steel frame structure and its SMA bracing system minimize response metrics due to a custom-

tailored earthquake excitation. Multiple-objective numerical optimization that simultaneously

minimizes displacements and accelerations of the structure is carried out with a genetic algorithm

(GA) in order to optimize SMA bracing elements within the structure. After design of an optimal

SMA damping system is complete, full-scale experimental shake table tests are conducted on a

large-scale steel frame that is equipped with the optimal SMA devices. A fuzzy inference system

is developed from data collected during the testing to simulate the dynamic material response of

the SMA bracing subcomponents. Finally, nonlinear analyses of a three-story braced frame are

carried out to evaluate the performance of comparable SMA and commonly-used steel braces

under dynamic loading conditions and to assess effectiveness of GA-optimized SMA bracing

design as compared to alternative designs of SMA braces. It is shown that peak displacement of a

structure can be reduced without causing significant acceleration response amplification through a

judicious selection of physical characteristics of the SMA devices. Also, SMA devices provide a

re-centering mechanism for the structure to return to its original position after a seismic event.

1. Introduction

Over the past several decades, strong ground motions have caused significant structural and non-

structural damage to buildings that have been constructed according to conventional design concepts.

This shortfall does not correlate with the extensive efforts of researchers in the field of structural

engineering in the recent past to apply modern techniques to improve seismic response. To this end,

passive, semi-active, hybrid and active vibration control techniques have been developed to establish an

acceptable level of seismic protection [1]. Currently, a number of passive structural control techniques

are being advocated that do not require an external power source for operation of a damping device. To

date they are being widely implemented in practical civil engineering projects [2]. More recently, shape

memory alloys (SMAs) have attracted a great deal of attention as a smart material that can be used in

passive protection systems for energy dissipating and re-centering purposes [3]. The current study seeks

2

to investigate the potential of using SMAs as passive damping and re-centering devices in large-scale

civil engineering structures.

Shape memory alloys are a subclass of metals that have several unique properties such as shape

memory and superelastic effects. Both of these unusual phenomena are the result of a series of phase

transformations that the material experiences when it passes through a loading-unloading cycle. An SMA

has two stable phases: austenite and martensite, and has four characteristic temperatures that are defined

as follows: Ms and Mf are the start and finish temperatures of the transformation from austenite to

martensite, respectively, while As and Af are the temperatures at which a transformation from martensite

into austenite starts and completes. When the temperature is higher than Md, the material is stabilized in

its austenite phase.

Shape memory effect of an SMA is a temperature-induced phase transformation from martensite

to austenite. Upon experiencing a loading-unloading process, if the material is heated up to a certain

temperature (Af), it remembers its original shape and recovers all of the residual deformation that it has as

shown in figure 1(a).

Superelastic SMAs are initially in the austenite phase and transform completely to the martensite

phase when they are stressed within a certain temperature range, namely between Af and Md. A reverse

transformation to austenite that results in a full recovery of deformations together with a hysteretic loop

occurs upon unloading as illustrated in figure 1(b). The ability of an SMA to return to its original

position by shape recovery and to dissipate energy as a result of hysteretic behavior makes superelastic

SMAs an attractive material for seismic applications.

Many researchers have proposed various applications of SMAs for vibration control of structures.

Since the material needs to be heated to exhibit the shape memory effect, its use in this manner can be

classified as an active control technique. Although a few researchers have investigated the shape memory

effect for active vibration control techniques [4, 5], the SMAs considered most widely for civil

engineering applications do not involve heating and active control but, rather, exhibit the superelastic

effect.

3

ε

σ

Austenite Martensite

Austenite Martensite

Md

Af

As

Ms

Mf

Austenite stabilization – No transformation

TE

MPE

RA

TU

RE

Partial recovery of deformation

Residual strain

Austenite

Detwinned Martensite

UNLOADING

LOADING

Full Recovery

Figure 1. (a) Shape memory effect, and (b) superelastic effect

Examples of the use of superelastic SMAs are becoming more common [6-11]. For example,

Andrawes and DesRoches [12] investigated the potential application of SMA bars as restrainers in multi-

span reinforced concrete bridges. In another study [13], the same researchers attempted to determine

important temperature effects on the performance of SMA restrainers for bridges. Boroschek et al. [14]

attached CuAlBe SMA wires to a three-story steel frame as diagonal braces and performed shake table

tests to evaluate performance of the superelastic braces. Lafortune et al. [15] compared the effectiveness

of conventional steel braces and SMA braces through small-scale experimental tests and an analytical

study. They also explored effects on structural response of pre-straining the SMA braces. Dolce et al.

[16] proposed an SMA-based isolation system and evaluated the performance of different isolators. Choi

et al. [17] developed a new isolation system for seismic protection of bridges using elastomeric bearings

and SMA wires. Their analytical studies on a multi-span steel bridge illustrate that the combination of an

SMA and a rubber bearing can effectively decrease the dynamic response of a bridge.

σ

ε

T

Shape recovery upon heating

Residual strain after unloading

Detwinned Martensite

(a)

Af

As

Ms

Mf

TE

MPE

RA

TU

RE

Austenite

Detwinned Martensite Twinned Martensite

LOADING

HE

AT

ING

COOLING

(b)

4

In an effort to improve the accuracy of numerical simulations involving SMAs that are embedded

in structures, numerous researchers have reported material models for the unique and complex behavior

of superelastic SMAs. However, due to the inherent complexity of phase transformations, models that

can describe the SMA behavior at high loading rates such as are of interest for seismic applications are

rare. Recently, Motahari and Ghassemieh [18] proposed a multilinear one-dimensional material model

for civil engineering applications that is derived from thermodynamics principles. The model considers

loading rate effects and is capable of representing behavior at different temperatures. In another recent

study, Ozbulut et al. [19] used a soft computing approach, namely a neuro-fuzzy technique, to model

dynamic behavior of CuAlBe SMAs. The proposed fuzzy models account for strain rate and temperature

effects. Also, the ease of implementing these models in numerical simulations has been emphasized in a

parallel study [20].

In this paper, the performance of superelastic nickel-titanium (NiTi) SMAs for seismic

applications is investigated by extensive numerical and large-scale experimental studies. First, tensile

tests are conducted on NiTi wires that have a diameter of 1 mm. Using data from these tests, a neuro-

fuzzy model of a single NiTi wire that is rate-dependent is developed and used in optimization of SMA

damping elements. Then, a non-dominated sorting genetic algorithm with controlled elitism (NSGAII-

CE) is employed to determine the optimum number of NiTi wires in each SMA damping element that

serves as a brace for each story of a large-scale three-story steel benchmark structure. After the genetic

algorithm (GA) has identified an optimal area of NiTi for each SMA brace in the laboratory structure,

full-scale shake table tests are conducted on the frame at the National Center for Research on Earthquake

Engineering (NCREE) in Taiwan. A fuzzy model of the dynamic behavior of the bundled wires that

comprise the SMA damping elements is created using data collected from a large array of transducers that

are monitored during the shake table tests. Next, a series of nonlinear numerical simulations of the

dynamic motion of the braced building under seismic excitations are performed. Specifically, in order to

enable a comparison of results for the frames that are hypothetically upgraded from a base consideration

by either using an SMA brace or a comparable steel brace, nonlinear time history analyses of the SMA-

and steel-braced frames are conducted. Finally, in order to evaluate effectiveness of the GA optimization

of the size of each SMA element in each brace, another set of nonlinear simulations on the benchmark

structure is carried out for optimum and alternative designs of the SMA brace.

2. Tensile testing of SMA wires

In this study NiTi shape memory alloy wires that have a diameter of 1 mm are selected as the

main subcomponent of a SMA damping device. Cyclical tensile tests are conducted on a single wire in an

MTS machine to characterize dynamic response of the material prior to its use in the SMA device. Each

5

test is carried out at a different strain amplitude and loading frequency. Before formal tests, in order to

stabilize hysteretic loops [10] a training test procedure that consists of 10 load cycles with strain

amplitude of 6% at 0.04 Hz is applied to the SMA wire. Subsequently, the maximum superelastic strain

of the material in the cyclical tests is targeted to be approximately 6%. The range for the loading

frequency is selected to be between 0.2 Hz and 2 Hz in order to simulate dynamic loading conditions that

may be encountered during seismic excitation of a civil engineering structure. As an example, figure 2

shows superimposed stress-strain relationships of an SMA wire for different levels of strain at a loading

frequency of 1 Hz. Lack of stress in the initial portion of the curves is attributed to lack of complete

tautness in the SMA wire. Strain hardening is observed to occur at approximately 7% strain.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07-100

0

100

200

300

400

500

600

Strain

Stre

ss (M

Pa)

Figure 2. Stress-strain curves of NiTi wires at different strain amplitudes

In order to determine and optimize the number of SMA wires that are to be used in each SMA

device assembly, which is introduced next, a reasonably accurate model of the dynamic behavior of an

individual NiTi wire is needed. While there are a large number of studies that propose analytical models

that range from relatively simple to very complex, only a small subset of these models is considered to be

suitable and effective for application in seismic analysis of engineered structures. This is the case

because some of the SMA constitutive models are too complicated and numerically expensive to

implement into simulations, whereas simplified models proposed to date do not include dynamic effects

that considerably influence the behavior of SMAs [21]. In this study, a neuro-fuzzy technique is used to

create (i) a model that emulates complex strain-stress behavior of a single SMA wire and (ii) a model of

an SMA device described in the next section. An abbreviated description of the fuzzy representation of a

single wire is given below, since fuzzy modeling of an SMA device that is composed of multiple NiTi

wires is discussed later in detail.

6

In recent years, fuzzy inference systems (FISes) have become a popular framework that is used to

model complex and nonlinear systems by means of techniques based on fuzzy set theory and fuzzy logic.

A fuzzy inference system is a simple scheme that maps an input space to an output space using fuzzy

logic. There are four main components of a FIS. They are: (i) a fuzzifier, which transforms crisp inputs

to fuzzy variables by defining membership functions to each input; (ii) a rule base, which relates the

inputs to output by means of if-then rules; (iii) an inference engine, which evaluates the rules to produce

the system output and (iv) a defuzzifier, which transforms the output to a non-fuzzy discrete value.

Among the various fuzzy models, the Sugeno-type FIS [22] has attracted the most attention. The Sugeno-

type fuzzy models enable a systematic approach to model the dynamics of complex nonlinear systems.

Adaptive neuro-fuzzy inference system (ANFIS) is a soft computing approach that combines

fuzzy theory and neural networks [23]. Specifically, ANFIS employs neural network strategies to

develop a Sugeno-type fuzzy model whose parameters (membership functions and rules) cannot be

predetermined by the knowledge of a user. One of the main advantages of ANFIS is that it does not

require a complex mathematical model to compute the system output. ANFIS uses a hybrid algorithm to

learn from the sample data from the system and can adapt parameters inside its network. Here, ANFIS is

used to create a model of superelastic NiTi wires considering rate effects of loading.

A fuzzy inference system (FIS) that has strain and strain rate as input variables and stress of the

SMA wire as output is created to model material behavior of a single SMA wire. Each input variable has

two membership functions. The FIS maps characteristics of the inputs to sets of membership functions

and if-then rules. An adaptive neuro-fuzzy inference system (ANFIS) is used to adjust the membership

function parameters of the initial FIS. ANFIS uses a learning technique that combines a back propagation

algorithm and least squares in order to tune parameters of a given FIS. Experimental results from

aforementioned tensile tests are concatenated to compose the data used for training, checking and

validation of the FIS by ANFIS. After training with ANFIS, a fuzzy model of a single SMA wire is

obtained. Figure 3 shows the strain-stress behavior of SMA wires obtained from experimental tests and

the corresponding prediction of the developed fuzzy model. This fuzzy model of a single NiTi wire is

used to design an optimum SMA device for vibration control of a three-story structure as described in the

next section.

7

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

100

200

300

400

500

600

Strain

Stre

ss (M

Pa)

Experimental resultsFuzzy model

Figure 3. Strain-stress relationship of fuzzy model and experimental data

3. Shake table tests and optimization of SMA device

This section includes a description of the experimental setup of a large-scale steel frame at

NCREE and an array of installed SMA-based devices. It also discusses the identification of optimal

distribution of the number of SMA wires in each brace that are installed in the test frame. The steel-frame

building used for testing has three floors and four columns as shown in figure 4(a). All columns and

beams are composed of H150 × 150 × 7 × 10 rolled shapes of grade A36 steel. Overall dimensions of the

building are 3 m × 2 m × 9 m. A number of lead weights are secured to each floor to give the properties

of the building that are listed in table 1. The base of the frame is securely bolted to a large shake table.

Table 1. Characteristics of experimental building and SMA braces

Floor Mass (kg)

Stiffness (kN/m)

Damping coefficient (kN-s/m)

Number of wires per SMA

brace

Cross-sectional area of SMA wires in each brace (mm2)

1 6,500 1,595 5,388 35 27.5 2 6,500 1,038 8,055 30 23.6 3 6,500 2,488 6,041 20 15.7

Although the high cost of SMA material has decreased significantly in the past decade [24], this

has been one of the impediments to actual implementation. Nevertheless, economically feasible solutions

can be attained with NiTi-based SMAs if they are used in small devices or judiciously applied to selected

regions of a structure [25]. Figure 4(b) shows the configuration of the SMA device considered in this

study. The SMA device has a straightforward design, which avoids extra fabrication costs. In particular,

multiple SMA-wire loops are wrapped around two wheels to form a bundle for each brace. A simple steel

8

connector is used to clamp the ends of the wires together. The SMA device has a length of 31 cm in each

brace where they are installed. Each SMA device is connected to steel braces that attach to adjacent floor

levels. Four SMA braces are installed between the shake table and the first floor. Their dynamic

response is monitored by displacement and force transducers. The other two floors are similarly braced

by SMA devices but braces at these levels do not have transducers attached. Note that the SMA braces

act in tension only.

In order to determine the optimal number of SMA wires to place in each brace at each

intermediate floor level, a non-dominated multi-objective algorithm (NSGAII-CE) is employed for

optimization. This genetic algorithm takes a pool of random candidate solutions, and using a non-

dominated sorting approach, generates an optimal set of solutions. In particular, it compares each

solution with every other solution in the population to determine if it is dominated, and then evaluates the

solutions in accordance with given performance objectives. More information on the algorithm is

available in the literature [26, 27].

Here, a total of three variables, that is, the cross-sectional area of NiTi for each SMA brace in

each floor, are adjusted to find an optimal solution. Four objective functions are defined and calculated as

follows

, .1

, .

max max j cont

jj unc

uJ

u

! "# #= $ %

# #& ', , .

2, .

max max j cont

jj unc

uJ

u

! "# #= $ %

# #& '

&&&&

(1)

, .3

, .

max j cont

jj unc

uJ rms

u

! "# #= $ %

# #& ', , .

4, .

max j cont

jj unc

uJ rms

u

! "# #= $ %

# #& '

&&&&

where u and u!! denote interstory displacement and absolute story acceleration, respectively, and j

represents the story that is considered. For the controlled case SMA braces are assumed to be present in

the building. The first two indices are based on the peak relative displacement and absolute acceleration

of each floor, while the other two evaluation criteria consider the entire duration of the motion and

compute the root-mean-square (RMS) of the peak relative displacement and absolute acceleration for

each floor. The objective functions from the response of the structure to the artificial earthquake

described below are measured simultaneously and organized into a set of Pareto fronts.

9

Figure 4. (a) Structure on shake table at NCREE with twelve SMA braces, and (b) SMA device and its attachment to the frame

For evaluation of candidate solutions during NSGAII-CE optimization a seismic excitation is

required. The excitation should have amplitude and frequency characteristics that are most probable

during anticipated seismic events. Traditionally, frequency domain techniques are used to modify a seed

white noise signal to produce an artificial earthquake record. However, the resulting accelerogram

usually includes high frequency content that is not present in actual seismic records. Therefore, to obtain

a single excitation record that is representative of several anticipated excitations a response matching

algorithm, RspMatch2005 [28], is employed in this study. RspMatch2005 modifies an actual seismic

record in the time domain using wavelet operations. It makes possible the matching of an accelerogram

to pseudo acceleration or displacement spectral ordinates for a given spectrum at several damping ratios.

Since it modifies a historical record according to a given response spectrum in the time domain, a realistic

accelerogram is obtained for the desired frequency content that does not contain spurious high

(a) (b)

10

frequencies. Here, the historical Chi-Chi (1999) earthquake record is modified by RspMatch2005 to

obtain a temblor that is representative of all expected ground motions.

Using the fuzzy model developed above for an individual NiTi wire, NSGAII-CE determined that

the optimal number of wires for the 1st, 2nd, and 3rd floor SMA braces is 35, 30, and 20, respectively.

After the optimal sizes of SMA damping elements are determined by the GA for the seismic record

created by RspMatch2005, the braces are assembled and installed in the benchmark structure on the shake

table.

Numerous laboratory experiments with a variety of seismic events and intensities are performed

on the three-story frame that is equipped with the optimal number of wires for each SMA brace.

Excitation of the structure is accomplished by means of a number of records from historic earthquakes

that have different levels of peak ground acceleration (PGA). These tests include El Centro (with PGA

levels of 1.0, 1.5, 2.0, 2.5, and 3.0 m/s2), Kobe (with PGA levels of 1.0, 1.5, 2.0, and 2.5 m/s2), and Chi-

Chi (with PGA levels of 1.0, 1.5, 2.0, 2.5, 3.0, and 3.5 m/s2) temblors. The benchmark structure remains

elastic during each event. Data from the structural response are collected at increments of 0.005 sec from

a large array of displacement, velocity, and acceleration transducers that are attached to the SMA braces,

the steel frame, and the shake table.

Before testing of the SMA-braced frame is conducted, the bare frame (i.e. without the SMA

braces being installed) is also submitted to seismic excitation. For purposes of calibration of the

numerical model of the bare frame with the experimental results, figure 5 shows the maximum

displacement relative to the base and the maximum absolute acceleration response of the frame for the

Kobe excitation with a PGA of 1.0 m/s2. The results reveal that numerical simulations predict the

maximum response of the uncontrolled frame reasonably well.

Results of the experimental tests for the frame with optimal SMA bracing and numerical results

for the uncontrolled bare frame are summarized in figures 6 and 7. Maximum interstory drift for various

PGA levels of the Kobe, El Centro and Chi-Chi excitations is plotted in figure 6 for both cases. A careful

examination of the results shows that the SMA braces are predicted to decrease the maximum interstory

drift between 50% and 64% for all of the cases tested. However, it can be seen from figure 7 that

maximum floor acceleration of the three-story structure increases when SMA braces are present. Note

that both reduction in the displacement response and increase in the acceleration response of the structure

are expected due to the added stiffness of the bracing system. Therefore, further investigation is needed

to assess the performance of the SMA braces.

11

1 2 30

10

20

30

40

50

60

Floor

Peak

dis

plac

emen

t (m

m)

202847445453

1 2 30

0.5

1

1.5

2

2.5

3

Floor

Peak

acc

eler

atio

n (m

/s2 )

1.72.02.01.92.42.2

Simulation Experiment

1 2 30

10

20

30

40

50

60

Floor

Peak

dis

plac

emen

t (m

m)

20

28

4744

5453

1 2 30

0.5

1

1.5

2

2.5

3

FloorPe

ak a

ccel

erat

ion

(m/s

2 )

1.7

2.0 2.01.9

2.42.2

!

Figure 5. Numerical and experimental results for bare frame with Kobe excitation: (a) maximum floor displacement relative to the base, and (b) maximum absolute floor acceleration

2.5 2.0 1.5 1.0 3.0 2.5 2.0 1.5 1.0 3.5 3.0 2.5 2.0 1.5 1.00

10

20

30

40

50

60

70

80

Peak

inte

rsto

ry d

rift (

mm

)

Excitation PGA level (m/s2)

← Kobe Earthquake →← El Centro Earthquake → ← Chi-Chi Earthquake →

Bare frameFrame with SMA braces

Figure 6. Peak interstory drift of SMA braced frame and bare frame

(a) (b)

12

2.5 2.0 1.5 1.0 3.0 2.5 2.0 1.5 1.0 3.5 3.0 2.5 2.0 1.5 1.00

1

2

3

4

5

6

7

8

9

Excitation PGA level (m/s2)

Peak

abs

olut

e ac

cele

ratio

n (m

/s2 )

← Kobe Earthquake →← El Centro Earthquake → ← Chi-Chi Earthquake →

Bare frameFrame with SMA braces

Figure 7. Peak absolute acceleration of SMA braced frame and bare frame

In order to evaluate the overall effectiveness of the GA-optimized SMA bracing system, dynamic

analysis of the structure with linear steel bracing elements is carried out. Here, the comparable steel

braces are designed to have same initial stiffness with the SMA braces of each floor. In this way, the

initial lateral stiffness of the frame is expected to be the same as that of the frame with the SMA braces.

Figure 8 illustrates the profiles of peak interstory drift and absolute acceleration for Kobe earthquake with

a PGA of 2.5 m/s2 and El Centro earthquake with a PGA of 3.0 m/s2. It is found that peak interstory drift

of the uncontrolled frame can be reduced 24% and 17% more for Kobe and El Centro earthquakes,

respectively, when the lateral stiffness of the frame is augmented with linear elements as compared to

SMA elements. However, it is also observed that peak absolute acceleration increases 73% and 151%

more for cited excitations when linear steel braces are used instead of SMA braces. In particular, as

compared to the uncontrolled frame, there exists 35% and 12% increase in peak absolute acceleration for

SMA braced frame when subjected to Kobe and El Centro earthquakes, respectively. On the other hand,

the same increases are 108% and 163% for the frame with steel braces. Therefore, it can be concluded

13

that the GA-optimized SMA bracing elements improve significantly the displacement response of the

structure while it does not drastically amplify the acceleration response.

0 20 40 60 800

1

2

3

Peak interstory drift (mm)

Floo

r

Kobe - PGA = 2.5 m/s2

0 5 10 150

1

2

3

Peak absolute acceleration (m/s2)Fl

oor

Bare frame Frame with SMA braces Frame with steel braces

0 20 40 60 800

1

2

3

Peak interstory drift (mm)

Floo

r

El Centro - PGA = 3.0 m/s2

0 5 10 150

1

2

3

Peak absolute acceleration (m/s2)

Floo

r

Figure 8. Profiles of peak interstory drift and absolute acceleration for (a) Kobe and (b) El Centro earthquakes

4. Fuzzy model of SMA device

In this section, a formulation of a fuzzy model of the SMA device composed of multiple

superelastic NiTi wires is described. This fuzzy model is employed to emulate response of the SMA

device during nonlinear numerical simulations of the three-story frame in a subsequent section.

Figure 9 shows a flow chart for fuzzy modeling of the SMA device. The first step is to set up

training, checking and validation data sets from experimental test results. Here, results from shake table

tests using the El Centro excitation with 1.0, 2.0, 2.5, and 3.0 m/s2 PGA and the Kobe excitation with 1.0,

1.5, 2.0, and 2.5 m/s2 PGA are employed for training and checking. Data obtained from displacement and

14

force transducers attached to the SMA braces in the first story during these excitations are concatenated to

provide a total of 56,800 data points as shown in figure 10. Odd numbered data points are used for

training and even numbered data values are employed for checking. Use of checking data prevents

overfitting of the model during training. Experimental data that result from the El Centro excitation with

a PGA of 1.5 m/s2 are reserved for validation. Note that there are two pairs of cross-braces in each floor

and only one pair is in tension at any given time. Therefore, the negative data in figure 10 basically

represent the tensile stress in SMA elements of one pair.

Train FIS with ANFIS

Validate the new model by using

the validation data set

Obtain optimized FIS from ANFIS

Set up the training, checking and

validation data sets

Experimental Data

Generate generic FIS for ANFIS

training

Figure 9. Flow chart of fuzzy modeling of SMA braces

0.5 1 1.5 2 2.5x 104

-505

Stra

in(%

)

0.5 1 1.5 2 2.5x 104

-500

50

Stra

in ra

te (%

/sec

)

0.5 1 1.5 2 2.5x 104

-500

0

500

Data point

Stre

ss (M

Pa)

Figure 10. Experimental training data: (a) strain, (b) strain rate, and (c) stress

(a)

(b)

(c)

15

After the input data have been prepared, an initial Sugeno type FIS is created. A FIS employs

membership functions and if-then rules to map the given inputs to a single-valued output. In order to

adjust the random parameters of assigned membership functions of the FIS, ANFIS is used. Figure 11

shows inputs (strain and strain rate) and output (stress) of the FIS developed for this study. After a trial

and error procedure, three and two Gaussian membership functions are selected for the strain and strain

rate inputs, respectively. The initial FIS and its membership functions have no information about the

target behavior of the bundled set of SMA wires. ANFIS modifies the initial variables of the FIS by

using the training procedure outlined earlier. Membership functions of both input variables before and

after training with ANFIS are shown in figure 12(a). In figure 12(b) the surface of the predicted stress in

the SMA brace is plotted versus the strain and strain rate of the bundled wires between the two wheels. A

portion of training data for the output variable, stress, and the corresponding fuzzy prediction are shown

in figure 13(a). Figure 13(b) presents a stress-strain curve from the training data. It is clear that the

trained FIS successfully predicts the stress on the SMA wire for given inputs. However, note that there is

slight deterioration in accuracy of the model above 6% strain as can be seen from figure 13(b). This

mismatch occurs because the training data set (see figure 10) does not contain many data points with

strain values larger than 6%.

STRAIN

STRAIN RATE

STRESSFUZZY INFERENCE

SYSTEM (FIS)

Figure 11. Fuzzy inputs and outputs

16

-5 0 50

0.5

1

Strain (%)

Degr

ee o

f mem

bers

hip

Initial MFs for strain

-50 0 500

0.5

1

Strain rate (%/sec)

Degr

ee o

f mem

bers

hip

Initial MFs for strain rate

-5 0 50

0.5

1

Strain (%)

Degr

ee o

f mem

bers

hip

Final MFs for strain

-50 0 500

0.5

1

Strain rate (%/sec)

Degr

ee o

f mem

bers

hip

Final MFs for strain rate

-50

5-50

050

-500

0

500

Strain (%)Strain rate (%/sec)

Stre

ss (M

Pa)

Figure 12. (a) Initial and final membership functions, and (b) surface of predicted stress

1.65 1.7 1.75 1.8x 104

-500

0

500

Data point

Stre

ss (M

Pa)

Experimental resultsFIS prediction

-10 -5 0 5 10

-1000

-500

0

500

1000

Strain (%)

Stre

ss (M

Pa)

Experimental resultsFIS prediction

Figure 13. (a) Time history of stress, and (b) stress-strain curve for experimental result and fuzzy prediction

Although it is shown in figure 13 that the trained FIS is capable of reproducing experimental data,

it is important to validate the fuzzy model using a data set that has not been used during training. Here, as

discussed above, experimental results from the El Centro earthquake with a PGA of 1.5 m/s2 are used for

validation. Stress-strain curves of the validation data from both the experimental tests and the fuzzy

model prediction for the SMA wires are plotted in figure 14. As shown the fuzzy model closely predicts

the experimental test results. Note that here the results are plotted for a combination of both of the

diagonal SMA damping elements that are installed in the same floor. Since one of the cross-braces is

(a) (b)

(a) (b)

17

always assumed to have buckled in compression, the negative stress shown in the figure is simply the

tensile stress in one of the braces.

-4 -3 -2 -1 0 1 2 3-600

-400

-200

0

200

400

600

Strain (%)

Stre

ss (M

Pa)

Experimental resultsFuzzy model

Figure 14. Validation of fuzzy model: experimental results versus fuzzy prediction

5. Nonlinear analyses of three-story benchmark building

In this section, two sets of nonlinear time history analyses of the three-story benchmark building

are performed. First, in order to compare performance of the SMA-braced frame with the same structure

that has comparable steel braces installed, nonlinear simulations of the building in both configurations are

conducted. Then, in order to compare response metrics of the building for both GA-identified optimal

SMA device design and three alternative methods for the design of a set of SMA devices, additional

numerical simulations are performed.

5.1. Comparison of SMA and steel braces

Since superelastic SMAs return to their original shape upon removal of external loads, they can

provide re-centering forces to a structure at the end of a seismic excitation. Although the shake table tests

of the three-story frame that were conducted at NCREE reveal effectiveness of the SMA braces for

decreasing response of the frame to different ground motions, the ability of the SMA braces to recenter a

frame should be further explored since experimental tests were only carried out within the linear response

range of the steel material. To this end the performance of the SMA-braced frame is compared with the

simulated behavior of the same structure that has steel braces installed by means of a set of nonlinear

simulations.

The steel columns and braces are modeled as having elasto perfectly-plastic material behavior.

The fuzzy model outlined above is used to represent the material behavior of each SMA brace. Both the

18

SMA devices and the steel braces are assumed to be effective only for tensile loads. The optimum

distribution of SMA wires described above is assumed to be installed for each brace (see table 1). In

order to compare performance of the two different bracing systems, the steel braces are designed so that

they have the same initial stiffness and yield strength as those of the SMA braces.

The artificial earthquake record with a PGA of 4.0 m/s2 that is described above is used as external

excitation at the base of each configuration of the frame. A nonlinear block that uses a Bouc-Wen model

to relate the deformation history and restoring force with nonlinear characteristics is developed in

MATLAB and Simulink [29] to perform the simulations. The equation of motion for a three degree-of-

freedom system is expressed as;

13 )1()( +−=+++ iibraceigii RFRuxm δ!!!! (2)

where mi is the mass of each floor; xi and Ri are the relative displacement and restoring force of each

floor, respectively; Fbrace is the lateral force exerted by either steel or SMA braces; and δ is the Kronecker

delta which is 1 if i = 3, and 0 otherwise. Restoring force Ri is defined as follows:

iyipieiieiiii zukkxkxcR )( −++= ! ! (3)

where zi is given as:

[ ]{ }niiiiiiyi

ii zzxAux

z )sgn( !!

! γβ +−= (4)

and ci is the damping coefficient; kei and kpi are the initial elastic stiffness and post-yielding stiffness of the

steel columns, respectively; and uyi is the yielding displacement for each floor. Also, Ai , βi , γi , and n are

shape parameters for hysteresis loops, and have the values of 1, 0.5, 0.5, and 1, respectively. Note that kpi

is assumed to be 0 since each steel brace is modeled as an elasto perfectly-plastic material.

Profiles of peak relative displacement, residual displacement and peak absolute acceleration that

result from these numerical simulations for the SMA- and steel-braced frames are shown in figure 15.

Maximum relative displacement is 91 mm when the frame is braced with SMA elements, while the steel-

braced frame undergoes a maximum relative displacement of 99 mm. Especially noteworthy from the

simulation is the prediction that the SMA elements recenter the frame at the end of the motion, i.e. there is

no residual displacement at each floor. As a typical example, a time-history of the relative displacement

of the second floor is given in figure 16. From this figure the residual displacement of the steel-braced

frame is predicted to be a maximum of 54 mm. Also, as shown in figure 15, maximum absolute floor

acceleration for SMA- and steel-braced frame is 6.7 m/s2 and 7.4 m/s2, respectively. It can be concluded

from this observation that while the SMA-braced frame notably decreases the peak relative drift of the

three-story frame and recenters the frame after the motion has ceased, addition of SMA braces does not

cause a significant increase in acceleration response of the frame.

19

0 25 50 75 100

1

2

3

Peak interstory drift (mm)

Floo

r

0 25 50 75 100

1

2

3

Residual displacement (mm)0 2.5 5 7.5 10

1

2

3

Peak absolute acceleration (m/s2)

Frame with steel braces Frame with SMA braces

0 25 50 75 100

1

2

3

Peak interstory drift (mm)

Floo

r

0 25 50 75 100

1

2

3

Residual displacement (mm)0 2.5 5 7.5 10

1

2

3

Peak absolute acceleration (m/s2)

Figure 15. Profiles of peak story drift; residual displacement; and peak absolute acceleration for steel-braced and SMA-braced frames

0 10 20 30 40 50 60 70-100

-50

0

50

100

Time (sec)

Dis

plac

emen

t (m

m)

Frame with steel bracesFrame with SMA braces

Figure 16. Relative displacement time history of the second floor for steel-braced and SMA-braced frames

5.2. Comparison of optimum and alternative designs of SMA device

In order to compare performance of the GA-identified optimal design of SMA wires for each

floor with alternative designs of SMA wires, additional nonlinear numerical simulations are conducted in

this section. First, optimum design of SMA braces for each floor is pursued by considering nonlinear

behavior of columns. To this end, the seismic record created by RspMatch2005 as discussed above is

scaled to have a PGA of 6.0 m/s2 and as before, an NSGAII-CE optimization is employed to determine

20

the optimal design. In comparison with the case in which the area of the SMA wire is optimized for

linear behavior of the frame, here both the length and number of SMA wires for each floor are defined as

optimization parameters. The objective functions given in equation (1) are minimized simultaneously. A

large fixed penalty is added to all objective functions if the maximum strain of SMA wires at a given

floor exceeds 7%. This is done to ensure that an optimal length for the SMA wires at each floor is

selected by the GA so that the superelastic effect of the SMA braces is exploited. Note that a long wire

length results in small strains in the SMA wires, i.e. small energy dissipation during seismic excitation; on

the other hand keeping SMA wires short can lead to large strains in the wire and yields residual

deformations at the end of the seismic excitation. After running the GA with a population size of 50 for

100 generations, the optimum length and number of SMA wires for each floor is obtained as given in

table 2.

In order to provide a comparison with simulated performance results of this optimal design, three

additional design configurations of SMA wires are considered. The first configuration (Custom I) is a

uniform distribution in which the number and length of wires for each floor is required to be the same.

This design configuration has the advantage of requiring only one type of SMA device for each floor.

After several trials, it is found that 50 SMA wires with a length of 1.5 m for each device yield the

optimum performance for this configuration. In the second and third configurations, a proportional

stiffness criterion [30] which is also used for designing steel braces is employed to design SMA braces at

each floor. Here, it is assumed that the elastic lateral story-stiffness due to the braces proportional to that

of the unbraced frame. In order to vary stiffness of SMA braces at each floor, a constant wire length is

chosen for the second configuration (Custom II) and the number of SMA wires is changed for each floor

accordingly. On the other hand, for the third configuration (Custom III) a constant area of SMA wires for

each floor is required while the length of the SMA wires is suitably adjusted at each floor. Also, peak

SMA wire strain for each floor is required to be less than 7% for the chosen configuration. After a trial

and error procedure, it is found that the quantities given in table 2 for custom configuration II and III

result in the best performance in terms of reducing interstory drifts while simultaneously controlling floor

accelerations. The total weight of the SMA wire for each configuration is normalized by the total weight

for the optimum design and is also given in the table. It is noted in passing that the optimal configuration

of the SMA device requires an average of 23% less material compared to the custom design

configurations. Since the cost of NiTi SMAs is often cited as one of the barriers to actual implementation

[31], this reduction in the required material can be considered as important advantage.

21

Table 2. Characteristics of SMA device for each design case

Case 1st Floor 2nd Floor 3rd Floor Normalized

weight ratio Number Length (m) Number Length (m) Number Length (m)

Optimum 55 1.45 55 1.35 30 1.00 1.00 Custom I 50 1.50 50 1.50 50 1.50 1.21 Custom II 50 1.50 75 1.50 30 1.50 1.25 Custom III 50 1.50 50 2.20 50 0.80 1.23

Near-fault ground motions, which are often characterized by intense velocity, high amplitude and

short duration impulses, are accorded special consideration in seismic engineering. Substantial increases

in story drift and base shear of structures have been observed for this kind of earthquake [32, 33].

Therefore, a total of six historical near-fault ground motions, namely the 1986 N Palm Springs, 1992

Erzincan, 1994 Northridge, 1995 Kobe, 1999 Chi-Chi, and 1999 Duzce temblors are used for the

simulations to evaluate re-centering ability of the various SMA devices against near-fault earthquakes.

Nonlinear time history analyses of the three-story building are performed in order to compare

performance of the various SMA configurations.

In order to quantitatively evaluate results of numerical simulations for each excitation, four

performance indices (peak and RMS drift as well as peak and RMS acceleration) as defined in equation

(1) are computed for each case (table 3). Also, peak residual story drifts for the uncontrolled (bare) frame

and the frame with SMA devices are given in the table. In comparison with the bare frame response, the

peak relative displacement, J1, is decreased by a range of 11% to 60% when SMA devices are present in

the optimal configuration. Note that this reduction in peak story drift is achieved without a significant

increase in peak acceleration response (J2) for all considered seismic loadings except the Erzincan

earthquake in which the peak story drift is decreased by 52% at the expense of a 57% increase in peak

acceleration. It can be seen that in terms of reducing peak and RMS story drift the optimal design

produces slightly better results for all excitation cases except the Kobe earthquake. Also, note that among

the custom configurations, the best result for each excitation varies. For example, among all alternative

designs Custom II gives the best results for the N Palm Spring excitation. In particular, when compared

to the optimal design, Custom II has only 2% performance degradation for J1 while Custom III has 12%

performance degradation. On the other hand for the Erzincan earthquake Custom II performs the worst

and has an 11% increase for J1 compared to the optimal design whereas the same increase is only 3% for

Custom III. Also, it can be seen that SMA devices at both optimal and alternative configurations

effectively reduce residual drifts for all excitation cases, while selected near-fault earthquakes cause large

residual column drifts on the bare frame.

22

Table 3. Characteristics of SMA device for each design case

Earthquake Case J1 J2 J3 J4 Residual Drift (mm)

Bare Frame SMA Frame

N Palm Spr Optimum 0.57 1.02 0.34 0.82 23 3 Custom I 0.60 1.02 0.37 0.80 23 2 Custom II 0.59 0.97 0.35 0.83 23 3 Custom III 0.69 1.03 0.40 0.85 23 3

Erzincan Optimum 0.48 1.57 0.13 1.27 140 6 Custom I 0.49 1.54 0.15 1.18 140 10 Custom II 0.59 1.68 0.15 1.22 140 6 Custom III 0.51 1.45 0.16 1.14 140 12

Northridge Optimum 0.75 0.97 0.47 1.31 25 2 Custom I 0.78 0.94 0.48 1.24 25 2 Custom II 0.80 0.91 0.49 1.29 25 2 Custom III 0.80 0.93 0.48 1.16 25 1

Kobe Optimum 0.79 1.00 0.29 1.41 57 2 Custom I 0.73 0.93 0.30 1.35 57 2 Custom II 0.90 0.95 0.34 1.39 57 3 Custom III 0.88 0.81 0.35 1.25 57 2

Chi-Chi Optimum 0.40 1.16 0.09 1.22 169 4 Custom I 0.41 1.13 0.09 1.18 169 3 Custom II 0.47 1.23 0.10 1.19 169 4 Custom III 0.40 1.10 0.10 1.14 169 2

Duzce Optimum 0.89 0.98 0.69 1.23 10 2 Custom I 0.99 0.95 0.75 1.20 10 2 Custom II 0.90 0.97 0.72 1.23 10 2 Custom III 1.01 0.98 0.83 1.15 10 1

6. Conclusion

The goal of this study is to investigate advantageous use of superelastic shape memory alloy

(SMA) damping devices for amelioration of earthquake response in a three-story steel frame structure.

To this end, a novel lateral bracing system is investigated for its ability to reduce undesirable responses

during a strong motion seismic event as well as to minimize residual deflection after the excitation ceases.

23

First, tensile testing is carried out on a single NiTi wire to characterize and model its behavior. Then, in

order to design an optimum device that is made of SMA wires and is to be installed in a three-story

building, a multi-objective genetic algorithm is used to determine the number of SMA wires that are

bundled together in the form of a cable for each floor brace. After design and installation of a large

number of SMA braces in a large-scale experiment, testing on a shake table at NCREE is conducted for a

number of ground motions with various levels of peak ground acceleration in which material in the steel

columns remains within the linear elastic range. As expected, results show that the SMA braces

effectively decrease the drift of each floor without increasing lateral floor accelerations. Data collected

from shake table tests are also used to develop a fuzzy model of the dynamic behavior of the braces that

are comprised of bundled SMA wires. In turn, this model is used to conduct nonlinear time history

analyses of the three-story building that has various arrays of SMA devices installed.

Nonlinear numerical simulations of a three-story building that have either a steel brace or an

SMA brace installed between each floor are conducted to evaluate performance of both types of bracing

systems. Special efforts are made to proportion member sizes of both types of braces (steel and SMA) so

that a valid comparison can be made with respect to the response of each braced structure to an identical

seismic motion. It is demonstrated that the SMA-braced frame not only significantly reduces

displacement and maintains approximately the same peak acceleration as the steel-braced frame, but also

that the SMA bracing system markedly reduces residual displacements by restoring the frame to its

original undeformed geometry after the excitation terminates. Finally, in order to show the relative

effectiveness of SMA braces where the individual distribution of wires between the floors is identified by

a GA in comparison with those braces where the distribution of the number of wires between the floor

levels is determined by alternative design methods, another set of nonlinear simulations are performed.

Results from these numerical simulations show that the GA-optimized design of the SMA devices can

effectively improve the seismic response of the three-story building against a suite of historical near-fault

earthquakes. This approach to design also provides a reduction of approximately 23% in the required

SMA material as compared to alternative design methods. The results of this study show that application

of optimized SMA braces appears to offer a promising substitute to traditional approaches of designing

lateral structural members in frames.

Acknowledgements

The authors gratefully acknowledge support of the National Center for Research on Earthquake

Engineering, Taipei, Taiwan.

24

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